Emtp Theory Book

19/6/2020 EMTP THEORY BOOK
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CONTENTS
(Click topic to jump to its location)

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CHAPTER PAGE
1. INTRODUCTION TO THE SOLUTION METHOD USED IN THE EMTP 6
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2. LINEAR, UNCOUPLED LUMPED ELEMENTS 14
3. LINEAR, COUPLED LUMPED ELEMENTS 45
4. OVERHEAD TRANSMISSION LINES 64
5. UNDERGROUND CABLES 150
6. TRANSFORMERS 192
7. SIMPLE VOLTAGE AND CURRENT SOURCES 238
8. THREE-PHASE SYNCHRONOUS MACHINE 251
9. UNIVERSAL MACHINE 311
10. SWITCHES 339
11. SURGE ARRESTERS AND PROTECTIVE GAPS 354
12. SOLUTION METHODS IN THE EMTP 360
13. TRANSIENT ANALYSIS OF CONTROL SYSTEMS (TACS) 395
APPENDIX
I – NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS 413
II – RE-INITIALIZATION AT INSTANTS OF DISCONTINUITIES 431
III – SOLUTION OF LINEAR EQUATIONS, MATRIX REDUCTION AND
INVERSION, SPARCITY 434
IV – ACTUAL VALUES VERSUS PERUNIT QUANTITIES 452
V – RECURSIVE CONVOLUTION 459
VI – TRANSIENT AND SUBTRANSIENT PARAMETERS OF SYNCHRONOUS
MACHINES 460
VII – INTERNAL IMPEDANCE OF STRANDED CONDUCTORS 470
REFERENCES 472
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Since the early fall of 1986 when BPA received the manuscript from the contractor,
there has been an effort to obtain permission for BPA to publish all portions of the book that
were copyrighted by others. This has been completed to the satisfaction of the BPA contracting
officer, who Lust recently gave his approval for BPA to print this work, and to distribute copies
to others.
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This manual discusses by and large only those solution methods which are used in the EMTP. It is therefore
not a book on the complete theory of solution methods for the digital simulation of electromagnetic transient
phenomena. The developers of the EMTP chose methods which they felt are best suited for a general purpose
program, such as the EMTP, and it is these methods which are discussed here. For analyzing specific problems,
other methods may well be competitive, or even better. For example, Fourier transformation methods may be
preferable for studying wave distortion and attenuation along a line in cases where the time span of the study is so
short that reflected waves have not yet come back from the remote end.
The EMTP has been specifically developed for power system problems, but some of the methods have
applications in electronic circuit analysis as well. While the developers of the EMTP have to some extent been aware
of the methods used in electronic circuit analysis programs, such as T4#C or EC#P, the reverse may not be true.
# survey of electronic analysis programs published as recently as 197 [22] does not mention the EMTP even once.
Computer technology is changing very fast, and new advances may well make this manual obsolete by the
time it is finished. #lso, better numerical solution methods may appear as well, and replace those presently used
in the EMTP. Both prospects have been discouraging for the writer of this manual what has kept him going is the
hope that those who will be developing better programs and who will use improved computer hardware will find
some useful information in the description of what exists today.
Digital computers cannot simulate transient phenomena continuously, but only at discrete intervals of time
(step size )t). This leads to truncation errors which may accumulate from step to step and cause divergence from
the true solution. Most methods used in the EMTP are numerically stable and avoid this type of error build up.
The EMTP can solve any network which consists of interconnections of resistances, inductances,
capacitances, single and multiphase B circuits, distributed parameter lines, and certain other elements. To keep the
explanations in this introduction sample, only single phase network elements will be considered and the more
complex multiphase network elements as well as other complications will be discussed later. Fig. 1.1 shows the
details of a larger network just for the region around node 1. Suppose that voltages and currents have already been
computed at time instants 0, )t, 2)t, etc., up to t )t, and that the solution must now be found at instant t. #t any
instant of time, the sum of the currents flowing away from node 1 through the branches must be
eSual to the injected current i :
K (V) % K (V) % K (V) % K (V) ' K (V) (1.1)
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(KI Details of a larger network around node no. 1
Node voltages are used as state variables in the EMTP. It is therefore necessary to express the branch
currents, i , etc., as functions of the node voltages. For the resistance,
1
K (V) ' {X (V)& X (V)} (1.2)
4 Traduciendo...
For the inductance, a simple relationship is obtained by replacing the differential eSuation
FK
X'.
FV
with the central difference eSuation
X(V) % X(V&)V) K(V) & K(V&)V)

'.
2 )V
This can be rewritten, for the case of Fig. 1.1, as
)V
K (V) ' 6X (V) & X (V)> % JKUV (V&)V) (1.3a)
2.
with hist known from the values of the preceding time step,
)V
JKUV (V&)V) ' K (V&)V) % 6X (V&)V) & X (V&)V)> (1.3b)
2.
The derivation for the branch eSuation of the capacitance is analogous, and leads to
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2%
K (V) ' 6X (V) & X (V)> % JKUV (V&)V) (1. a)
)V
with hist again known from the values of the preceding time step,
2%
JKUV (V&)V) ' &K (V&)V) & 6X (V&)V) & X (V&)V)> (1. b)
)V
4eaders fresh out of University, or engineers who have read one or one too many textbooks on electric
circuits and networks, may have been misled to believe that Laplace transform techniSues are only useful for hand
solutions or rather small networks, and more or less useless for computer solutions of problems of the size typically
analyzed with the EMTP. Since even new textbooks perpetuate the myth of the usefulness of Laplace transforms,
#ppendix I has been added for the mathematically minded reader to summarize numerical solution methods for
linear, ordinary differential eSuations.
For the transmission line between nodes 1 and 5, losses shall be ignored in this introduction. Then the wave
eSuations
MX
& ' .) MK
MZ MV
MK
& ' %) MX
MZ MV
where
L , C inductance and capacitance per unit length ,
x distance from sending end,
have the well known solution due to d #lembert:
K ' ((Z & EV) & H(Z % EV)
X ' <((Z & EV) % <H(Z % EV) (1.5a)
with
F(x ct)
f(x ct) ‡ functions of the composite expressions x ct and x ct,
Traduciendo...
< surge impedance < %L /C (constant),
c velocity of wave propagation (constant).
If the current in ES. (1.5a) is multiplied by < and added to the voltage, then
The prime is used on L , C to distinguish these distributed parameters from lumped parameters L, C.
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X % <K ' 2<((Z & EV) (1.5b)
Note that the composite expression v <i does not change if x ct does not change. Imagine a fictitious observer
travelling on the line with wave velocity c. The distance travelled by this observer is x x ct (x location
of starting point), or x ct is constant. If x ct is indeed constant, then the value of v <i seen by the observer
must also remain constant. With travel time
J line length / c ,
an observer leaving node 5 at time t J will see the value of v (t J) <i (t J), and upon arrival at node 1 (after
the elapse of travel time J), will see the value of v (t) <i (t) (negative sign because i has opposite direction of i ).
But since this value seen by the observer must remain constant, both of these values must be eSual, giving, after
rewriting,
1
K (V) ' X (V) % JKUV (V & J) (1. a)
<
where the term hist is again known from previously computed values,
1
JKUV (V&J) ' & X (V&J) & K (V&J) (1. b)
<
Example: Let )t 100 zs and J 1 ms. From eSuations (1. ) it can be seen that the known history
of the line must be stored over a time span eSual to J, since the values needed in ES. (1. b) are those computed 10
time steps earlier. ES. (1. ) is an exact solution for the lossless line if )t is an integer multiple of J if not, linear
interpolation is used and interpolation errors are incurred. Losses can often be represented with sufficient accuracy
by inserting lumped resistances in a few places along the line, as described later in Section .2.2.5. # more
sophisticated treatment of losses, especially with freSuency dependent parameters, is discussed in Section .2.2. .
If ES. (1.2), (1.3a), (1. a) and (1. a) are inserted into ES. (1.1), then the node eSuation for node 1 becomes
1 )V 2% 1 1 )V 2%
% % % X (V) & X (V) & X (V) & X (V) '
4 2. )V < 4 2. )V
K (V) & JKUV (V&)V) & JKUV (V&)V) & JKUV (V&J) (1.7)
which is simply a linear, algebraic eSuation in unknown voltages, with the right hand side known from values of
preceding time steps.
For any type of network with n nodes, a system of n such eSuations can be formed ,
[)] [X(V)] ' [K(V)] & [JKUV] (1. a)
Brackets are used to indicate matrix and vector Suantities.
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with [G] n x n symmetric nodal conductance matrix,
[v(t)] vector of n node voltages, Traduciendo...
[i(t)] vector of n current sources, and
[hist] vector of n known history terms.
Normally, some nodes have known voltages either because voltage sources are connected to them, or because the
node is grounded. In this case ES. (1. a) is partitioned into a set # of nodes with unknown voltages, and a set B
of nodes with known voltages. The unknown voltages are then found by solving
[)##][X#(V)] ' [K#(V)] & [JKUV#] & [)#$][X$(V)] (1. b)
for [v (t)].
#
The actual computation in the EMTP proceeds as follows: Matrices [G ] and [G ] are##built, and [G#$] ##
is triangularized with ordered elimination and exploitation of sparsity. In each time step, the vector on the right hand
side of ES. (1. b) is assembled from known history terms, and known current and voltage sources. Then the
system of linear eSuations is solved for [v (t)], using

# the information contained in the triangularized conductance
matrix. In this repeat solution process, the symmetry of the matrix is exploited in the sense that the same
triangularized matrix used for downward operations is also used in the backsubstitution. Before proceeding to the
next time step, the history terms hist of ES. (1.3b), (1. b) and (1. b) are then updated for use in future time steps.
Originally, the EMTP was written for cases starting from zero initial conditions. In such cases, the history
terms hist , hist and hist in ES. (1.7) are simply preset to zero. But soon cases arose where the transient
simulation had to be started from power freSuency (50 or 0 Hz) ac steady state initial conditions. Originally, such
ac steady state initial conditions were read in , but this put a heavy burden on the program user, who had to use
another steady state solution program to obtain them. Not only was the data transfer bothersome, but the separate
steady state solution program might also contain network models which could differ more or less from those used
in the EMTP. It was therefore decided to incorporate an ac steady state solution routine directly into the EMTP,
which was written by J.W. Walker.
The ac steady state solution shall again be explained for the case of Fig. 1.1. Using node eSuations again,
ES. (1.1) now becomes
+% +% +% +'+ (1.9)
where the currents I are complex phasor Suantities *I* e now. ForL"the lumped elements, the branch eSuations are
obvious. For the resistance,
1
+' (8 & 8 ) (1.10)
4
This option is still available in the EMTP, but it has become somewhat of a historic relic and has seldom
been used after the addition of a steady state solution routine. For some types of branches, it may not even work
([1], p. 37c).
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for the inductance,
1
+' (8 & 8 ) (1.11)
LT.
and for the capacitance,
+ ' LT%(8 & 8 ) (1.12)
For a line with distributed parameters 4 , L , G , C , the exact steady state solution is
1
;UGTKGU % ;UJWPV &;UGTKGU
+ 2 8
' (1.13)
+ 1 8
&;UGTKGU ;UGTKGU % ;UJWPV
2
if the eSuivalent B circuit representation of Fig. 1.2 is used, with
Traduciendo...
(KI ESuivalent B circuit for ac steady state

solution of transmission line
1 sinh((ý)
;UGTKGU ' , YKVJ <UGTKGU ' ý(4) % LT.))
<UGTKGU (ý
(ý
tanh
1 ý 2
;UJWPV ' ()) % LT%)) (1.1 )
2 2 (ý
2
and sometimes eSually useful,
1
;UGTKGU % ;UJWPV ' cosh((ý) @ ;UGTKGU
2
where ( is the propagation constant,
( ' (4) % LT.)) ()) % LT%)) (1.15)
Page 12
For the lossless case with 4 0, and G 0, ES. (1.1 ) simplifies to
sin(Tý .)%))
<UGTKGU ' ý@LT.) @
Tý .)%)
Tý
tan .)%)
1 ý 2
;UJWPV ' @LT%) @ (1.1 )
2 2 Tý
.)%)
2
1
;UGTKGU % ;UJWPV ' cos(Tý .)%)) @ ;UGTKGU
2
If the value of Tý is small, typically ý # 100 km at 0 Hz for overhead lines, then the ratios sinh(x) / x and
tanh(x/2) / x/2 in ES. (1.1 ), as well as sin(x) / x and tan(x/2) / x/2 in ES. (1.1 ) all become 1.0. This simplified
B circuit is usually called the nominal B circuit,
<UGTKGU ' ý@(4) % LT.))
1 ý
;UJWPV ' ()) % LT%)) KH Tý KU UOCNN. (1.17)
2 2
With the eSuivalent B circuit of Fig. 1.2, the branch eSuation for the lossless line finally becomes
1
+ ' (;UGTKGU % ;UJWPV)8 & ;UGTKGU8 (1.1 )
2
Now, we can again write the node eSuation for node 1, by inserting ES. (1.10), (1.11), (1.12) and (1.1 )
into ES. (1.9),
1 1 1 1 1
% %LT%%;UGTKGU% ;UJWPV 8 & 8& 8 &LT%8 &;UGTKGU8 ' +
4 LT. 2 4 LT.
Traduciendo...
(1.19)
For any type of network with n nodes, a system of n such eSuations can be formed,
[;] [8] ' [+] (1.20)
with [;] symmetric nodal admittance matrix, with complex elements,
[V] vector of n node voltages (complex phasor values),
[I] vector of n current sources (complex phasor values).
#gain, ES. (1.20) is partitioned into a set # of nodes with unknown voltages, and a set B of nodes with known
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voltages. The unknown voltages are then found by solving the system of linear, algebraic eSuations
[;##][8#] ' [+#] & [;#$][8$] (1.21)
Bringing the term [; ][V ] from

#$ the
$ left hand side in ES. (1.20) to the right hand side in ES. (1.21) is the
generalization of converting Thevenin eSuivalent circuits (voltage vector [V ] behind admittance

$ matrix [; ]) into #$
Norton eSuivalent circuits (current vector [; ][V ] in parallel

#$ $ with admittance matrix [; ]). #$
Norton eSuivalent circuits (current vector [; ][V ] in parallel

#$ $ with admittance matrix [; ]). #$
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Traduciendo...
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Linear, uncoupled lumped elements are resistances 4, self inductances L, and capacitances C. They usually
appear as parts of eSuivalent circuits, which may represent generators, transformers, short sections of transmission
lines, or other components of an electric power system, or they may represent a component by itself.
4GUKUVCPEG 4
4esistance elements are used to represent, among other things,
(a) closing and opening resistors in circuit breakers,
(b) tower footing resistance (as a crude approximation [ ] of a complicated, freSuency dependent
grounding impedance),
(c) resistance grounding of transformer and generator neutrals,
(d) metering resistance in places where currents of branch voltages cannot be obtained in other
ways by the EMTP,
(e) as parts of eSuivalent networks, e.g., in parallel with inductances to produce proper freSuency
dependent damping (see Section 2.2.2).
(f) for the representation of long lines in lightning surge studies if no reflection comes back from the
remote end during the duration t of theOCZ

study.
Example (f) is easily derived from ES. (1. b) if it is assumed that the initial conditions on the line are zero. In that
case, hist (t J) 0 for t J since it takes time J for any nonzero condition occurring in node 5 after t $ 0 to
show up in node 1. If nothing is connected to node 5 ( open ended line ), then I would remain zero for t 2 J.
The EMTP recognizes this simplification if
(1) J t , andOCZ
(2) zero initial conditions .
If both conditions (1) and (2) are met, then the EMTP represents the line simply as two shunt resistances (Fig. 2.1).
This simplification saves not only
computer time but storage space as
well, because no history terms have
to be stored for the line. This long
line model is mostly used in lightning
surge studies and

(KI ESuivalent circuit of
long line if no reflections come
back from other end
It is possible to modify this simplification for cases starting from linear ac steady state conditions as well in
that case, nodes 5 and 1 in Fig. 2.1 would have ac steady state current sources connected to them.
Unfortunately, the EMTP does not yet contain this modification.
21
Page 15
in transient recovery voltage studies, where the unfaulted lines leaving the substation under study are preferably
modelled this way.
The eSuation of a lumped resistance 4 between nodes k and m,
1
iMO(t) ' vM(t) & vO(t) (2.1)
4
is solved accurately by the EMTP, as long as the value of 4 is not unreasonably small.
'TTQT #PCN[UKU
Very large values of 4 are acceptable and do not degrade the solution of the complete network. In the
limiting case, 4 4, its reciprocal value 1/4 simply gets lost in [G] of ES. (1. ), that is, it will not have any
influence on the network solution, as it should be. # practical limit for very large resistances is the approximate
sSuare root of the largest real number which the computer can handle (e.g.,Traduciendo...
4 10 if the computer accepts
numbers up to 10 ). This is because intermediate expressions of the form 4 X are computed in the steady state
solution in the conversion from impedances to admittances. Extremely large values of 4 have been used in the past
to obtain voltage differences between nodes with such metering resistance branches in newer EMTP versions,
voltage differences can be obtained directly.
Very large resistances can be used to replace the series 4 L elements in symmetric multiphase B circuits,
if one is only interested in the capacitive coupling among the phases, as explained in Fig. 2.2. This trick reduces
the number of nodes, but more importantly, it avoids accuracy problems which may occasionally show up if the B
circuit represents a very short line section . In the steady state solution of ES. (1.20) the value of the series
It may be worth adding a diagnostic printout in the EMTP if the admittances of the series and shunt elements
are too far apart in orders of magnitude. This would reSuire a comparison of 1/TL and TC in the steady state
solution, and of )t/2L and 2C/)t in the transient solution.
22
Page 16
(KI Conversion of nominal B circuit for short line into B circuit with capacitive coupling only
element is entered as 1/ý[< ] into [;], where [< ] is the impedance per unit length and ý the length of the short
section. For a short length, ý is small and 1/ý[< ] accordingly relatively large. #t the same time, the shunt
susceptances 1/2ý jT[C ] entered into [;] become relatively small. #s ý is decreased, the capacitive coupling effect
will eventually get lost in the solution. In a practical case of capacitive coupling between 500 kV circuits at 0 Hz,
this accuracy problem showed up with the shortest line section being 1. km it was discovered accidentally because
the single precision solution (accuracy approx. 7 decimal digits) on an IBM 370 differed unexpectedly by 10 from
the double precision solution (accuracy
approx. 1 decimal digits). For reasonable step sizes of )t, the problem is less severe in the transient calculation,
as can easily be seen if jT in ES. (1.19) is replaced by 2/)t in ES. (1.7). In this example, with )t 100 zs, the
value of the series element would be smaller by a factor of 53, while the value of the shunt element would be larger
Traduciendo...
by a factor of 53. Or in other words, a similar accuracy problem would appear during the transient simulation if
the line were shortened by a factor of 53 (ý 30 m instead of 1. km).
Very small values of 4 do create accuracy problems, for the same reason as discussed in the preceding
paragraph: Very small values of 4 create very large conductance values 1/4 in the matrix [;] of steady state
solutions and in the matrix [G] of transient solutions, which can swamp out the effects of other elements connected
to that resistance. Very small values of 4 have been used in the past primarily to separate switches, since earlier
EMTP versions allowed only one switch to be connected to a node with unknown voltage. In newer versions, this
limitation on the location of switches no longer exists, and the need for using very small values of 4 should therefore
no longer exist in these later versions.
Hints about the use of small resistances are given in [1], pp. b c.
23
Page 17
'ZCORNG HQT 0GVYQTM YKVJ 4GUKUVCPEGU
Practical examples for purely resistive networks are rather limited. # simple case is shown in Fig. 2.3.
#ssume a dc voltage source with negligible source impedance is connected to a line through a circuit breaker with
a closing resistor of value 4 . If we are interested in what happens after closing of contact I in the first short time
ENQUG
period during which reflections have not yet come back from the remote end, then this case can be studied with the
circuit of Fig. 2.3(b). If we choose 4 <, we seeENQUG

that the voltage at the sending end will be 0.5 p.u., which
will double to 1.0 p.u. at the open receiving end. Therefore, no overvoltage will appear as long as contact II is still
open. # real case is obviously more complicated because
(a) network (b) eSuivalent circuit
(KI Energization of a very long line (< surge

impedance, J travel time)
! contact II will close (typically to 10 ms later) as well,
! the system is three phase,
! the line is not lossless,
! multiple reflections will occur as we study a longer time period,
! closing of contact I does not necessarily occur at maximum voltage (approximated as a dc source in Fig.
2.3), but may occur anywhere on the sine wave,
! the source impedance is not zero,
! the three poles in a three phase system do not close simultaneously, and because of many other factors.
In a typical system, maximum switching surge overvoltages may be 2. to 2. p.u. without closing resistors
(versus 2.0 p.u. in Fig. 2.3), which would typically be reduced to 1.5 to 2.2 p.u. with closing resistors (versus 1.0
p.u. in Fig. 2.3) [ 9].
5GNH +PFWEVCPEG .
Magnetically coupled circuits are so prevalent in power systems, starting from the generator, through the
transformer, to the magnetically coupled phase conductors of a three phase line, that inductances usually appear as
coupled inductances. There are cases, however, of uncoupled self inductances. #mong other things, self
Traduciendo...
Page 18
inductances are used to represent
(a) normal three (b) four reactor

phase connection compensation
scheme
(KI Shunt reactor connections
(a) single phase shunt reactors and neutral reactors in shunt compensation schemes (Fig. 2. ),
(b) part of discharge circuits in series capacitor stations,
(c) eSuipment in HVDC converter stations, such as smoothing reactors, anode reactors, parts of filters
on the ac and dc side,
(d) inductive part of source impedances in Thevenin eSuivalent circuits for the rest of the system
when positive and zero seSuence parameters are identical (Fig. 2.5),
(e) inductive part of single phase nominal B circuits in the single phase representation of balanced
(positive or negative seSuence) operation or of zero seSuence operation (Fig. 2. ),
(f) part of eSuivalent circuit for loads (Fig. 2.7), even though load modelling at higher freSuencies is
a very complicated topic [9], and loads are therefore, or for other reasons, often ignored,
(KI Thevenin eSuivalent circuit

with < <RQU
< PGI \GTQ
25
Page 19
(KI Typical positive seSuence network representation
Traduciendo...
(KI Load model for harmonics studies [9]
(g) part of surge arrester models to simulate the dynamic characteristics of the arrester [10],
(h) parts of electronic circuits.
Choke coils used for power line carrier communications are normally ignored in switching surge studies,
but may have to be modelled in studies involving higher freSuencies. Current transformers are usually ignored,
unless the current transformer itself is part of the investigation (e.g., in studying the distortion of the secondary
current through saturation effects).
The eSuation of a self inductance L between nodes k and m is solved accurately in the ac steady state
solution with ES. (1.11). The only precaution to observe is that TL should not beMO
extremely small, for the same
reasons as explained in Section 2.1.1 for the case of small resistance values.
For the transient simulation, the exact differential eSuation
diMO
vM & vO ' L (2.2)
dt
is replaced by the approximate central difference eSuation
vM(t) & vO(t) % vM(t&)t) & vO(t&)t) iMO(t) & iMO(t&)t)

'L (2.3)
2 )t
The same difference eSuation is obtained if the trapezoidal rule of integration is applied to the integral in
1 V
iMO(t) ' iMO(t&)t) % vM(u) & vO(u) du (2. )
LmV&)V
giving
Page 20
)t
iMO(t) ' iMO(t&)t)% {vM(t)&vO(t)%vM(t&)t)&vO(t&)t)} (2.5)
2L
ES. (2.3) and (2.5) can be rewritten into the desired branch eSuation
)t
iMO(t) ' vM(t)&vO(t) % histMO(t&)t) (2. )
2L
with the history term hist (t )t) known

MO from the solution at the preceding time step,
)t
histMO(t&)t) ' iMO(t&)t) % vM(t&)t) & vO(t&)t) (2.7)
2L
ES. (2. ) can conveniently be represented as an eSuivalent resistance 4 2L/)t,

GSWKX in parallel with a known current
source hist (tMO

)t), as shown in Fig. 2. . Once all the node voltages have been found at a particular time step at
instant t, the history term of ES. (2.7) must be updated for
(KI ESuivalent resistive circuit

for transient solution of lumped induc
tance
Traduciendo...
each inductive branch for use in the next time step at t )t. To do this, the branch current must first be found from
ES. (2. ), or alternatively, if both eSuations are combined, the recursive updating formula
)t
histMO(t) ' vMO(t) & vO(t) % histMO(t&)t) (2. )
L
can be used. If branch current output is reSuested, then ES. (2. ) is used.
'TTQT #PCN[UKU
Since the differential eSuation (2.2) is solved approximately, it is important to have some understanding
about the errors caused by the application of the trapezoidal rule of integration. #s explained in Section I. of
#ppendix I, the trapezoidal rule is numerically stable, and the solution does therefore not run away (see Fig. I.
in #ppendix I). Fortunately, there is also a physical interpretation of the error, because ES. (2.5) resulting from the
27
Page 21
trapezoidal rule is identical with the exact solution of the short circuited lossless line in the arrangement of Fig. 2.9
This was first pointed out to the writer by H. Maier, Technical University Stuttgart, Germany, in a personal
communication in 19 , for the case of a shunt inductance. From a paper by P.B. Johns [12], it became obvious
that this identity is valid for any connection of the inductance. To derive the parameters of such a stub line
representation, it is reasonable to start with the reSuirement that the distributed inductance L , multiplied by the stub
line length ý, should be eSual to the value of the lumped inductance
L)ý ' L (2.9)
(KI Lumped inductance replaced by short circuited stub line with <
2L/)t and J )t/2
With Lý known, the next parameter to be determined is the travel time J of the stub line. Since
J ' (L)ý) (C)ý) (2.10)
the shorter the travel time, the smaller will be the value of the parasitic but unavoidable capacitance Cý. The
shortest possible travel time for a transient simulation with step size )t is
)t
J' (2.11)
2
With this value, conditions at terminal 1 at t )t arrive at the shorted end at t )t/2 and get reflected back to
terminal 1 at t. Therefore, the best possible stub line representation has
2L )t
<' , J' (2.12)
)t 2
#ssume that the smoothing reactor on a dc line has L 0.5H, and that the step size is 100 zs. Then < 10,000
S, and the unavoidable total capacitance Cý becomes 5 nF, which appears to be negligible, at least if the reactor
has a shunt capacitor of 1.2 zF connected to it anyhow, as in the case of the HVDC Pacific Intertie [11]. Now it
remains to be shown that the exact solution for the lossless stub line with parameters from ES. (2.12) is identical with
ES. (2.5). #s explained in Section 1, the expression (v <i) along a lossless line for a fictitious observer riding
Traduciendo...
2
Page 22
on the line with wave speed remains constant, or going from 1 to 2 in Fig. 2.9,
)V
X (V & )V) % <K (V & )V) ' <K V &
2
and travelling back again from 2 to 1,
)V
&<K V & ' X (V) & <K (V)
2
which, combined, yields
1 1
K (V) ' X (V) % X (V&)V) % K (V&)V) (2.13)
< <
which is indeed identical with ES. (2.5). This identity explains the numerical stability of the solution process: The
chosen step size may be too large, and thereby create a fairly inaccurate stub line with too much parasitic capacitance
()V)
%)ý ' (2.1 )
.
from ES. (2.10), but since the wave eSuation is still solved accurately , the solution will not run away. The
mathematical oscillations sometimes seen on voltages across inductances, and further explained in Section 2.2.2,
are undamped wave oscillations travelling back and forth between terminals 1 and 2 (Fig. 2.9).
The identity of the trapezoidal rule solution with the exact stub line solution makes it easy to assess the error
as a function of freSuency [13]. #ssume that an inductance L is connected to a voltage source of angular freSuency
T, through some resistance 4 for damping purposes. The transient simulation of this case will eventually lead to
the correct steady state solution of the stub line (or not drift away from the steady state answer if the simulation starts
from correct steady state initial conditions). This steady state solution at any angular freSuency T is known from
the exact eSuivalent B circuit of Fig. 1.2. By short circuiting terminal 5, the input impedance becomes
1
<KPRWV '
1 (2.15)
;UGTKGU % ;UJWPV
2
or with ES. (1.1 ),
Except for round off errors caused by the finiteness of the word length in digital computers, which are
normally negligible. There is no interpolation error, which occurs in the simulation of real transmission lines
whenever J is not an integer multiple of )t (see Section .2.2.2).
29
Page 23
)V
tan T
2
<KPRWV ' LT. @ (2.1 )
)V
T
2
Therefore, the ratio between the apparent inductance resulting from the stub line representation or from the
trapezoidal rule solution, and the exact inductance becomes

Traduciendo...
)V
tan T
.VTCRG\QKFCN 2
' (2.17)
. )V
T
2
The phase error is zero over the entire freSuency range. Since power systems are basically operated as constant
voltage networks, it makes sense in many cases to assume that the voltage V (jT) across the
. inductance is more or
less fixed, and that the current I (jT) follows

. from it. If we compare this current of the stub line representation or
trapezoidal rule solution with the current of the exact solution for the lumped inductance, then we obtain the
freSuency response of Fig. 2.10, where the ratio (the reciprocal of ES. (2.17)) is shown as a function of the NySuist
freSuency
1
H0[SWKUV ' (2.1 )
2)V
(KI #mplitude ratio I /I through

VTCRG\QKFCN an
GZCEV inductance as a
function of freSuency
2 10
Page 24
This freSuency is the theoretically highest freSuency of interest for a step size )t, amounting to 2 samples/cycle.
From a practical standpoint, at least to samples/cycle are needed to reproduce a particular freSuency even
crudely. From Fig. 2.10 or from ES. (2.17) it can be seen that the error in the current will be 5.2 at a crude
sampling rate of samples/cycle, or 0. at a more reasonable sampling rate of 20 samples/cycle. Furthermore,
Fig. 2.10 also shows that the trapezoidal rule filters out the higher freSuency currents, since the curve goes down
as the freSuency increases, as pointed out by 4.W. Hamming [1 ].
Because of the error in ES. (2.17), there is a small discrepancy between the initial conditions found with
ES. (1.11), and the response to power freSuency in the time step loop. For 0 Hz, this error would be 0.012 with
)t 100 zs, or 1.2 with )t 1 ms. It is debatable whether ES. (2.17) should be used for the steady state
solution, instead of ES. (1.11), to match both solutions perfectly. This issue appears with other network elements
as well. If a perfect match is desired, then it may be best to have two options for steady state solutions, one intended
for initialization (using ES. (2.17) in this example), and the other one intended for steady state answers at one or
more freSuencies (using ES. (1.11) in this example).
Very large values of L are acceptable as long as (TL) or 2L/)t is not larger than the largest floating point
number which the computer can handle. To obtain flux I vdt across a branch, a large inductance can be added
in parallel and current output be reSuested. The need for this may arise if a flux current plot is reSuired for a
nonlinear inductance. With L 10 H, 10 times the current would be the flux.
Very small values of TL or of 2L/)t do create accuracy problems the same way as small resistances (see
Section 2.1.1).
&CORKPI QH 0WOGTKECN 1UEKNNCVKQPU YKVJ 2CTCNNGN 4GUKUVCPEG

Traduciendo...
While the trapezoidal rule filters out high freSuency currents in inductances connected to voltage sources,
it unfortunately also amplifies high freSuency voltages across inductances in situations where currents are forced into
them. In the first case, the trapezoidal rule works as an integrator, for which is performs well, whereas in the second
case it works as a differentiator for which is performs badly. The problem shows up as numerical oscillations in
cases where the derivative of the current changes abruptly, e.g., when a current is interrupted in a circuit breaker
(Fig. 2.11). The exact solution for v is shown. as a solid line, with a sudden jump to zero at the
2 11
Page 25
(KI Voltage after current interruption
instant of current interruption, whereas the EMTP solution is shown as a dotted line. Since
2.
X.(V) ' K(V) & K(V&)V) & X.(V&)V) (2.19)
)V
and assuming that the voltage solution was correct prior to current interruption, it follows that v (t) v (t )t) in . .
points 2, 3, ,... as soon as the currents at t )t and t both become zero therefore, the solution for v will oscillate .
around zero with the amplitude of the pre interruption value.
There are cases where the sudden jump would be an unacceptable answer anyhow, and would indicate
improper modelling of the real system. #n example would be the calculation of transient recovery voltages, since
any circuit breaker would reignite if the voltage were to rise with an infinite rate of rise immediately after current
interruption. For a transient recovery voltage calculation, the cure would be to include the proper stray capacitance
from node 1 to ground (and possibly also from 1 to 2 and from 2 to ground).
On the other hand, there are cases where the user is not interested in the details of the rapid voltage change,
and would be happy to accept answers with a sudden jump. # typical example would be sudden voltage changes
caused by transformer saturation with two slope inductance models for the nonlinearity, as indicated in
Traduciendo...
2 12
Page 26
(KI Voltage jumps caused by transformer saturation
Fig. 2.12. It should be pointed out that these numerical oscillations always oscillate around the correct answer
(around zero in Fig. 2.11) and plots produced with a smoothing option would produce the correct curves.
Nonetheless, it would be nice to get rid of them, especially since they can cause numerical problems in other parts
of the network, as has happened occasionally in turbine generator models [2 , p. 5].
The textbook answer would be re initialization of variables at the instant of the jump. This would be
fairly easy if the eSuations were written in state variable form [dx/dt] [#][x]. With nodal eSuations as used in
the EMTP, re initialization was thought to be very tricky, until B. Kulicke showed how to do it [15]. His method
is summarized in #ppendix II. Whether re initialization should be implemented is debatable, since the damping
method described next seems to cure this problem, and also seems to have a physical basis as shown in Section 2.2.3.
V. Brandwajn [1 ] and F. #lvarado [17] both describe a method for damping these numerical oscillations
with parallel damping resistances (Fig. 2.13). For a given current injection, the trapezoidal rule solution of the
parallel circuit of Fig. 2.13 becomes
2.
4R&
1 )V
X(V) ' 6K(V)&K(V&)V)> & X(V&)V) (2.20)
)V 1 2.
% 4R%
2. 4R )V
If a current impulse is injected into this circuit (in a form which the EMTP can handle, e.g., as
(KI Parallel damping
2 13
Page 27
a hat, with i rising linearly to I between 0OCZ

and )t, dropping linearly back to zero between )t and 2)t, and
staying at zero thereafter), then, after the impulse has dropped back to zero, the first term in ES. (2.20) will
disappear, and we are left with the second term which causes the numerical oscillations,
X(V) ' &"@X(V&)V)
Traduciendo...
with
2.
4R &
)V
"' (2.21)
2.
4R %
)V
being the reciprocal of the damping factor. This oscillating term will be damped if " 1 it is shown in Fig. 2.1
for 4 10
R 2L/)t or " 9/11, and for 4 2 2L/)t or " 1/3. The
R oscillation would disappear in one time
step for 4 2L/)t

R or " 0 (critically damped case) . If 4 is too large, then the
R damping effect is too small. On
the other hand, if 4 is reduced

R until it approaches the value 2L/)t (ideal value for damping), then too much of an
error is introduced into the inductance representation. Fig. 2.15 shows the magnitude and phase error of the
impedance for 4 2L/)t

R and 4 2L/)t, as well
R as the magnitude error from ES. (2.17) which already exists
for the inductance alone with the trapezoidal rule. It is interesting that the magnitude error with a parallel resistance
is actually slightly smaller than the error which already exists for the inductance alone because of the trapezoidal rule.
Therefore, the parallel resistance has no detrimental effect on the magnitude freSuency response. It does introduce
losses, however, as expressed by the phase error. #s shown in the next section, these losses are often not far off
from those which actually occur in eSuipment modelled with inductances. From a purely numerical standpoint, a
good compromise between reasonable damping
(KI Oscillating term [17]. 4eprinted by permission of F. #lvarado
The critically damped trapezoidal rule with 4 2L/)t is identical

R with the backward Euler method, as
explained in #ppendix I.9.
21
Page 28
Traduciendo...
(KI Phase and magnitude error with parallel resistance [1 ]
(4 as
R low as possible) and acceptable phase error (4 as high as possible)
R leads to values of
2. 2.
5. # 4R # 9. CEEQTFKPI VQ $TCPFYCIP [1 ] (2.22)
)V )V
or
2 15
Page 29
20 2.
4R ' @ CEEQTFKPI VQ #NXCTCFQ [17] (2.23)
3 )V
with Brandwajn s lower limit determined by specified acceptable phase error at power freSuency.
The errors introduced into the parallel connection of Fig. 2.13 through the trapezoidal rule are seen in Fig.
2.1 , in which 4 jX
UGTKGU 4 (jX)/(4
UGTKGU R jX) is shown
R for the exact solution with X TL, and for the
trapezoidal rule solution with TL from ES.

VTCRG\QKFCN (2.17).
Traduciendo...
21
Page 30
(KI #pparent series resistance and series inductance for the parallel connection of Fig.
2.13, with 4 (20/3)

R ! (2L/)t). The region to the right of f/f 0.5
0[SWKUV is of little practical
interest because the sampling rate would be too low to show these freSuencies adeSuately
Whether the EMTP will be changed to include parallel resistances automatically remains to be seen. It is
interesting to note that the electronic analysis program S;SC#P of 4ockwell International Corp., which seems to
use techniSues very similar to the EMTP, has 4 and 4 of Fig.

R 2.1 built
U into the inductor model, with default
values of 4 0.1
U S and 4 10 S [22,Rp. 715]. The possibility of numerical oscillations is mentioned as well,
in cases where the time constants of the inductor model of Fig. 2.1 are small compared with )t [22, p. 773].
(KI L/4 ratio of the short circuit impedance of typical

transformers [1 ]. 4eprinted by permission of CIG4E
2 17
Page 31
Traduciendo...
(KI ESuivalent circuit for the short

circuit impedance of a transformer
2J[UKECN 4GCUQPU HQT 2CTCNNGN 4GUKUVCPEG
There are many situations in which inductances should have parallel resistances for physical reasons. In
some cases, the values of these resistances will be lower than those of ES. (2.22) or (2.23), which will make the
damping of the numerical noise even better. Typical applications of damping resistances are described next. These
examples may not cover all applications, but should at least be representative.
(a) Short circuit impedance of transformers
The short circuit impedance of transformers does not have a constant L/4 ratio instead, the L/4 ratio
decreases with an increase in freSuency, as shown in Fig. 2.17 taken from [1 ]. If we use the curve for
the 100 MV# transformer, and assume L 1H (or mH, or p.u.) as well as )t 100 zs, then a value of
4 1R 3,000 S (or mS, or p.u.) will produce the proper L/4 ratio at 1 kHz. This value lies nicely in
between the limits of 10 ,000 S and 1 ,000 S recommended in ES. (2.22). # series resistance of 4 U
9. S (or mS, or. p.u.) can then be added to obtain the correct L/4 ratio at 50 Hz, which leads to the
eSuivalent circuit of Fig. 2.1 for the short circuit impedance of the transformer. With < from ES. KPRWV
(2.1 ), the L /4 ratio

VTCRG\QKFCN of this eSuivalent circuit is shown as a dotted line in Fig. 2.19, which is a
reasonably good match for the experimental curve (solid line), and much better than a constant L/4 ratio
without 4 . RIt is interesting that a CIG4E Working Group on Interference Problems recommends the same
eSuivalent circuit of Fig. 2.1 for the analysis of harmonics [9], with
504R
13 30 (2.2 )
80
and
80
90 110 (2.25)
504U
In the EMTP, the L/4 ratio without 4 would actually

R increase with freSuency, since L of ES.
VTCRG\QKFCN (2.17)
increases with freSuency.
21
Page 32
Traduciendo...
(KI L/4 ratio of the short circuit impedance of a 100 MV#

transformer (dotted line from eSuivalent circuit of Fig. 2.1 with L being
solved by trapezoidal rule, solid line from [1 ]). 4eprinted by
permission of CIG4E
where S is0 the rated power and V the rated

0 voltage of the transformer. If we assume X 0.05 to
UJQTV EKTEWKV 0.10 p.u.
at 50 Hz ([9] talks about 50 Hz), then ES. (2.2 ) becomes with TL (0.05 to 0.10) V /S , 0 0
( 0, 1 to 1, 1) L 4 (9 ,2 to 1 , 9 ) LR (2.2 )
or with a typical step size of )t 100 zs,
(2.0 to .0 ) 2L/)t 4 ( .07 to 9. ) 2L/)t,

R
with the higher numbers for the lower short circuit reactance of 0.05 p.u. #gain, the value of 4 lies in the R
same range as ES. (2.22). ES. (2.25) implies an L/4 ratio at 50 Hz of 0.01 to 0.01 for a 0.05 p.u. short
circuit reactance, or of 0.02 to 0.035 for a 0.10 p.u. short circuit reactance, which is lower than the values
at 50 Hz in Fig. 2.17.
(b) Magnetizing impedance of transformers and iron core reactors
#s discussed in more detail in Section . , parallel resistances are added to the magnetizing inductance of
transformers for a crude approximation of the hysteresis and eddy current losses. Similarly, the eSuivalent
circuit of Fig. 2.1 is recommended for iron core reactors [20], with 4 representing I 4 losses
U in the
winding, and 4 representing

R iron core losses.
(c) Synchronous generators
2 19
Page 33
# CIG4E Working Group on Interference Problems recommends a resistance in parallel with the negative
seSuence inductance (which is practically identical to (L L )/2), but feels

F that isSit premature to
propose a probable order of magnitude [9]. # typical curve for L /4 ratios of generators,
F similar to Fig.
2.19 has been published in [1 ], and could be used to find reasonable values of 4 .R
(d) Nominal B circuits
Cascade connections of nominal B circuits are used to represent transmission lines on transient network
analyzers. To suppress the spurious oscillations which are caused by the lumpy approximation of distributed
parameters, it is customary to add parallel resistances (Fig. 2.20). Typical values appear to be 4 5 R
< UWTIG
, which would lead to a value of 4 5 2L/)t inRthe stub line representation of the inductance in Fig.
2.9. 4easonable values of 4 for cascade

R connections are discussed in [1 ].
(KI Damping resistance in

nominal B circuit
(e) Source impedances
The Thevenin eSuivalent circuit of Fig. 2.5 is obviously a crude approximation for the rest of the system
at freSuencies different from the power freSuency. To make the freSuency response of this circuit more
realistic, damping resistances are often connected in parallel with the 4 L branch.
'ZCORNG HQT 0GVYQTM YKVJ +PFWEVCPEGU
# simple yet realistic example of an 4 L circuit arises from short circuit calculations. #ssume that a three
phase system has been reduced to a steady state Thevenin eSuivalent circuit seen from the fault location, similar to
that of Fig. 2.5, with the seSuence impedances < < 4 jX andRQU
< 4 jX then
PGI known.
RQU RQU \GTQ \GTQ \GTQ
#s shown by ES. (3. ), these seSuence parameters can be converted to self and mutual impedances. # single phase
to ground fault can then be simulated with a switch closure in the circuit of Fig. 2.21(a), where it is assumed that
Traduciendo...
the self impedance < consists
U of a resistance 4 in seriesU with an inductance L . Fig. 2.21(b)Ushows the fault current
This is one of the few examples with per unit Suantities, simply to show that they can be used. The writer
prefers actual values, for reasons explained in #ppendix IV.
This assumption is obviously only correct at power freSuency, but seems to be reasonable over a wider
freSuency range in many cases. It would give wrong answers if the system were to consist of a power
plant/transmission line/series capacitor connection, since this reSuires an 4 L C representation, possibly with a
capacitor protection circuit similar to Fig. 2.2 (see Section 2.3. as well). # detailed fault calculation with the
2 20
Page 34
obtained with the EMTP for 4 0.1 p.u.

U and L 0.712 p.u. If 4 X
U at power freSuency, which
U is notU
Suite true here, then there is a minimum offset ( symmetrical fault current ) if the fault occurs when the voltage is
at its peak value, and a maximum offset ( asymmetrical fault current ) if the fault occurs at zero crossing of the
voltage. The influence of )t on the results, as well as the exact solution
8OCZ &V4U .U
K(V) ' 6sin(TV%¾&n) & sin(¾&n)G >
4
U %(T.U)
with
n ' tan& (T.U/4U)
are shown in Fig. 2.21(c). The EMTP results with )t # 500 zs are indistinguishable from the exact solution.
EMTP, which shows travelling wave effects and compares results with field tests in the Hydro 3uebec System,
is described in [19].
2 21
Page 35
Traduciendo...
(KI Single phase to ground fault

(a) ESuivalent circuit,
(b) Fault currents for 4 /X 0.252
U U , f 0 Hz, )t # 500 zs, for different closing times,
(c) Influence of )t
2 22
Page 36
%CRCEKVCPEG %
Capacitance elements are used to represent, among other things,
(a) series and shunt capacitors,
(b) shunt capacitances in nominal B circuit representations of transmission lines,
(c) eSuipment in HVDC converter stations, such as parts of snubber circuits and filters, and surge capacitors,
(d) stray capacitances of transformers, generators, etc., especially in transient recovery voltage and lightning
surge studies, where impedances TL become so high at higher freSuencies that the parallel impedances of
stray capacitances 1/TC become dominant,
(e) capacitive potential transformers and capacitive voltage dividers,
(f) parts of surge generators.
The eSuation of a lumped capacitance C between nodes k and m is solved accurately in the ac steady state
solution with ES. (1.12). The only precaution to observe is that TC should not be extremely large, which is unlikely
to occur in practice anyhow, for the same reasons as explained for small resistances in Section 2.1.1.
Traduciendo...
For the transient simulation, the exact differential eSuation
F(XM&XO)
KMO ' % (2.27)
FV
is replaced by the approximate central difference eSuation
KMO(V)%KMO(V&)V) 6XM(V)&XO(V)> & 6XM(V&)V)&XO(V&)V)>

'% (2.2 )
2 )V
which gives the desired branch eSuation
2%
KMO(V) ' 6XM(V)&XO(V)> % JKUVMO(V&)V) (2.29)
)V
with the history term hist (t )t) known

MO from the solution at preceding time step,
2%
JKUVMO(V&)V) ' &KMO(V&)V)& 6XM(V&)V) & XO(V&)V)> (2.30)
)V
#gain, analogous to inductance, identical results would be obtained from an integration of ES. (2.27) with the
trapezoidal rule. ES. (2.29) can be represented as an eSuivalent resistance 4 )t/2C,

GSWKX in parallel with a known
current source hist (t )t),

MOas shown in Fig. 2.22. Once all
2 23
Page 37
(KI ESuivalent resistive circuit

for transient solution of lumped
capacitance
the node voltages have been found at a particular time step at instant t, the history term of ES. (2.30) must be updated
for each capacitive branch for use in the next time step at t )t. To do this, one must first find the current from
ES. (2.29). #lternatively, the recursive updating formula
%
JKUVMO(V) ' & 6XM(V)&XO(V)> & JKUVMO(V&)V) (2.31)
)V
can be used, which is the same as ES. (2. ) for the inductance if followed by a sign reversal.
(KI Lumped capacitance replaced by stub line with <

)t/2C and J )t/2
'TTQT #PCN[UKU
Not surprisingly, the error analysis is analogous to that of the inductance. For a physical interpretation of
Traduciendo...
the errors, the stub line representation of Fig. 2.23 is used, in which the lumped capacitance is replaced by an open
ended lossless line. To obtain the parameters, it is reasonable to make the total distributed capacitance eSual to the
lumped capacitance,
%)ý ' % (2.32)
With Cýknown, the next parameter to be determined is travel time J. ES. (2.10) shows that the shorter the travel
time, the smaller will be the value of the parasitic but unavoidable inductance Lý. For a step size )t, the shortest
possible travel time is
22
Page 38
)V
J' (2.33)
2
With ES. (2.32) and (2.33) the surge impedance becomes < )t/2C.
Without going through the details, let it simply let it be said that the exact solution for the stub line of Fig.
2.23 is identical with the trapezoidal rule solution of ES. (2.29) and (2.30). This identity will again be used to assess
the error as a function of freSuency. #ssume that a capacitance C is connected to a source with angular freSuency
T, through some network with damping. The transient simulation will then settle down to the correct steady state
solution of the stub line of Fig. 2.23, or not drift away from it if the simulation was started from correct steady state
initial conditions. This steady state solution is known from the exact eSuivalent B circuit of Fig. 1.2, with terminal
5 being open ended,
1 ; UGTKGU
;KPRWV ' (;UGTKGU % ;UJWPV) &
2 1 (2.3 )
;UGTKGU % ;UJWPV
2
or after some manipulations with ES. (1.1 ),
)V
tan T
2
;KPRWV ' LT% @ (2.35)
)V
T
2
This is analogous with ES. (2.1 ) for the inductance, except that the analogous error now applies to the capacitance
C rather than to the inductance L, or
)V
tan T
%VTCRG\QKFCN 2
' (2.3 )
% )V
T
2
#gain, the phase error is zero over the entire freSuency range. If we force a current I (jT) into the capacitance,
E then
the voltage across the stub line, compared with the exact solution, will have the freSuency response of Fig. 2.2 ,
which is identical with Fig. 2.10 if the current ratio is replaced by the voltage ratio. The trapezoidal rule filters out
the higher freSuency voltages.
2 25
Page 39 Traduciendo...
(KI #mplitude ratio V /V

VTCRG\QKFCN of a capacitance
GZCEV as a
function of freSuency
#gain, there is a small discrepancy between the initial conditions found with ES. (1.12), and the response to power
freSuency from ES. (2.35) in the time step loop (at 0 Hz, 0.012 error with )t 100 zs, or 1.2 with )t 1
ms). Whether it should be eliminated has already been discussed in the second last paragraph of Section 2.2.1.
Very small values of C are acceptable as long as (1/TC) or )t/2C is not larger than the largest floating point
number which the computer can handle. Very large values of C do create accuracy problems the same way as small
resistances (see Section 2.1.1), but they are unlikely to occur in practice.
&CORKPI QH 0WOGTKECN 1UEKNNCVKQPU YKVJ 5GTKGU 4GUKUVCPEG
While the trapezoidal rule filters out high freSuency voltages across capacitances for given current
injections, it also amplifies high freSuency currents for given voltages across C. The numerical oscillations discussed
for the inductance in Section 2.2.2 would appear in capacitance currents if there is an abrupt change in dv /dt. For E
some reason, numerical oscillations have seldom been a problem in capacitances, either because there are very few
situations where they would appear, or simply because currents through capacitances are seldom included in the
output. #nalogous to the inductance, these numerical oscillations could be damped with series resistances 4 (Fig. U
2.25). Using #lvarado s arguments [17], the trapezoidal rule solution for a voltage impulse applied to the circuit
of Fig. 2.25 would be
)V
&4U
1 2%
K(V) ' 6X(V)&X(V&)V)> & K(V&)V) (2.37)
)V )V
%4U %4U
2% 2%
22
Page 40
(KI Series damping resistance
#fter the voltage impulse v has dropped back to zero, we are left with the second term, which causes the numerical
oscillations,
K(V) ' &"@K(V&)V)
)V
& 4U
2%
YKVJ "' (2.3 )
)V
% 4U
2% Traduciendo...
In analogy to ES. (2.23), a reasonable value for the damping resistance would be
)V
4U ' 0.15 (2.39)
2%
2J[UKECN 4GCUQPU HQT 5GTKGU 4GUKUVCPEG
None are known to the writer at this time which would justify a series resistance as high as that of ES.
(2.39). G.W.#. Dummer [23] suggests the eSuivalent circuit of Fig. 2.2 , and says that 4 is dominant at very
U high
freSuencies, while 4 is dominant

R at very low freSuencies, but
(KI ESuivalent circuit for

capacitor with losses
his comments refer to capacitors used in electronics. The typical textbook circuit has no series resistance, which
would imply that the loss factor decreases inversely proportional with freSuency. This contradicts the curve in Fig.
2.27 given by #. 4oth for high voltage capacitors [2 ]. #ssuming C 1 zF and )t 100 zs and using CVTCRG\QKFCN
instead of C to duplicate the EMTP behavior, a value of 4 0.3 S (ignoring

U 4 ) would more or lessRmatch tan*
at 2 kHz, as shown in Fig. 2.27. Note that this value of 4 is one orderU of magnitude lower than the recommended
damping resistance of ES. (2.39). S;SC#P, an electronic analysis program with solution techniSues similar to the
EMTP, has 4 andR 4 of Fig.

U 2.2 built into the capacitor model, with default values of 4 0.1 S and 4 10 U R
S [22, p. 715].
2 27
Page 41
(KI Loss factor [2 ]. 4eprinted by permission of Springer Verlag and #.W.

4oth
Note that capacitors which may be subjected to short circuits often have series resistors built in. Similarly, the
overvoltage protection of series capacitors with spark gaps (Fig. 2.2 ) includes current limiting 4 L elements in the
discharge circuit, with a typical ringing freSuency of 00 Hz during discharge.
(KI Spark gap protection of

series capacitor
'ZCORNG HQT 0GVYQTM YKVJ %CRCEKVCPEGU
Let us modify the fault current study of Section 2.2. for a case in which the transmission line is series
Traduciendo...
compensated with capacitors (Fig. 2.29). Let us further assume that L in Section 2.2.
U represented the net reactance
X PGV
TL 1/TC at 0 Hz, to make both results directly comparable. With 4 0.1 p.u., X 1.0 33 p.u.,U TC U
2. 95 p.u. and )t 100 zs, the fault current of Fig. 2.30 is obtained (data taken from [7 ], with connection from
fault location to infinite bus left off). For comparison purposes, the fault current with the net reactance represented
by L ,Uas done in Section 2.2. , is shown as well it differs appreciably from the more accurate solution with the
circuit model of Fig. 2.29. This difference has conseSuences for the accuracy of stability simulations, since net
reactances are practically always used in stability studies. Fig. 2.31 compares the swing curves obtained with a net
reactance and an L C representation for a case similar to the IEEE benchmark model for subsynchronous resonance
studies [21].
22
Page 42
(KI Single phase to ground fault in a system

with a series capacitor
(KI Fault current in series compensated network of Fig. 2.29 (line without symbols). For comparison,
results from Fig. 2.21 with net reactance are shown as well (line with symbols)
(KI Swing curves with 4 L and 4 L C representations
2 29
Traduciendo...
Page 43
5GTKGU %QPPGEVKQP QH 4 . %
If lumped elements 4, L, C freSuently occur in pairs as series connections of 4 L, 4 C, or L C, or as a
series connection of all three elements 4 L C, then it becomes more efficient to treat the series connection as a single
branch, thereby reducing the number of nodes and nodal eSuations. This has been implemented in the EMTP for
the series connection of 4 L C (Fig. 2.32). For the steady state solution, the branch eSuation is simply
1
+MO ' (8M&8O)
4%L(T.&1/T%)
(KI Series connection of 4, L, C
To derive the branch eSuation for the transient simulation, add the three voltage drops across 4, L, and C
XM & XO ' X4 % X. % XE
with the voltage drops expressed as a function of the current with ES. (2.1), (2. ) and (2.29),
2. )V 2. )V (2. 0)
XM(V)&XO(V) ' 4% % KMO(V) & JKUV.(V&)V) & JKUV%(V&)V)
)V 2% )V 2%
#fter replacing the history terms hist and hist

. with the%expressions of ES. (2.7) and (2.30), this leads to the branch
eSuation
KMO(V) ' )UGTKGU6XM(V)&XO(V)> % JKUVUGTKGU(V&)V) (2. 1a)
with
1
)UGTKGU '
2. )V (2. 1b)
4% %
)V 2%
and the combined history term
2. )V (2. 2)
JKUVUGTKGU(V&)V)')UGTKGU&4& K(V&)V)%XM(V&)V)&XO(V&)V)&2X%(V&)V) .
)V 2%
For updating this history term, the new current is first calculated from ES. (2. 1a), and the new capacitor voltage
v%
from
)V
X%(V) ' X%(V&)V) % 6K(V) % K(V&)V)>
2%
2 30
Page 44
ES. (2. 2) is not the only way of expressing the combined history term, but it is the one being used in the EMTP.
5KPING 2JCUG 0QOKPCNB %KTEWKV
This is a special case of the M phase nominal B circuit discussed in Section 3. . Earlier EMTP versions
recognize the special case of M 1, and use scalar eSuations in place of matrix eSuations, whereas newer EMTP
versions go through the matrix manipulations with M 1. Since single phase B circuits are seldom used, it is
reasonable to eliminate the special code for the scalar case.
Traduciendo...
2 31
Page 45
.+0'#4 %172.'& .7/2'& '.'/'065
Coupled lumped elements appear primarily in the M phase B circuit representation of transmission lines,
in the representation of transformers as coupled impedances, and as source impedances in cases where positive and
zero seSuence parameters are not eSual.
%QWRNGF 4GUKUVCPEGU =4?
Coupled resistances, in the form of branch resistance matrices [4], appear primarily
(a) as part of the series impedance matrix in M phase nominal B circuits,
(b) as long line representations in lightning surge studies if no reflections come back from the remote
end during the duration t of the study.

OCZ
The diagonal elements of [4] are the self resistances, and the off diagonal elements are the mutual
resistances. The off diagonal terms in the series resistance matrix of an M phase line are caused by the presence
of the earth as a potential current return path. The earth is not modelled as a conductor as such instead, it is used
as a reference point for measuring voltages. If it were explicitly modelled as a conductor, its eSuation for a three
phase line could have the form
F8'
& ' <)
FZ '#+# % <) '$+$ % <) '%+% % <) ''+'
Since the voltages are measured with respect to earth, V 0, and therefore,
'
<) '# <) '$

<) '%
+' ' & +# & +Traduciendo...
$& +%
<) '' <) ''
<) ''
which, when inserted into the voltage drop eSuations for the phases #, B, C, produces
F8# <) '# <) <)

& ' <) #'<) +# % <) #'<) '$ +$ % <) #'<) '%
+%
##& #$& #%&
FZ <) '' <) <) ''
''
and similar for B, C. This is the form used in M phase B circuits, with earth being an implicit, rather than explicit,
current conductor. #ssuming purely inductive coupling < jX , the termsKM

< < /< etc.KM
will obviously #' '$ ''
contain real parts since the self impedance of the earth < contains a'' real part. Whether the real part thus produced
can strictly be treated as a resistance for all freSuencies is open to debate, as explained in Section .1.2. .
The EMTP automatically converts a long line with distributed parameters into a shunt resistance matrix [4]
if
(1) J t for all

OCZ M modes of the M phase line,
Modes are explained in Section .1.5.
31
Page 46
(2) zero initial conditions.
This representation is simply an M phase generalization of the single phase case discussed in Section 2.1. For the
high freSuency lossless line model, which is often used in lightning surge studies and described in more detail in
Section .1.5.2, this shunt resistance matrix has the elements
2JK &KM
4KK ' 0ýP , 4KM ' 0ýP (3.1)
TK FKM
with h average
K height above ground, r conductor radius,
K D distance from conductor
KMi to image of
conductor k, d direct
KM distance between conductors i and k. These are the well known self and mutual surge
impedances of an M phase line [ ].
The eSuations for coupled resistances
[KMO(V)] ' [4]& [XM(V)] & [XO(V)] (3.2)
are solved accurately by the EMTP, as long as [4] is non singular and not extremely ill conditioned. In all cases
known so far, [4] is symmetric, and the EMTP has therefore been written in such a way that it only accepts
symmetric matrices [4].
The EMTP does not have an input option for coupled resistances by themselves instead, they must be
specified as part of the M phase nominal B circuit of Section 3. , with L and C left zero. For long lines with J
t OCZ
and zero initial conditions, the EMTP converts the distributed parameter model internally to the form of ES.
(3.2). Since [4] is symmetric, the EMTP stores and processes the elements of these and all other coupled branch
matrices as one dimensional arrays in and above the diagonal (e.g. 4 stored in X(1), 4 in X(2), 4 in X(3), 4
in X( ), etc.).
'TTQT #PCN[UKU
#s already mentioned, [4] must be non singular if a resistance matrix is read in. If its inverse [4] is read
in, then this reSuirement can be dropped, since [4] is allowed to be singular without causing any problems. #lso,
the resistances shouldn t be so small that [4] becomes so large that it swamps out the effect of other connected
elements, as mentioned in Section 2.1.1. On the other hand, very small values of [4] are acceptable (see very
large resistances in Section 2.1.1).
+PUGTVKQP QH %QWRNGF $TCPEJGU KPVQ 0QFCN 'SWCVKQPU
Since coupled branches have not been discussed in the introduction to the solution methods, their inclusion
into the system of nodal eSuations shall briefly be explained. #ssume that three branches ka ma, kb mb, kc mc are
coupled (Fig. 3.1). In forming the nodal eSuation for node ka, the current i is needed, MC OC
Traduciendo...
32
Page 47
(KI Three coupled

resistances
DTCPEJ DTCPEJ DTCPEJ

KMC OC ' ) CC XMC& XOC % ) CD XMD&XOD % ) CE XME&XOE
DTCPEJ
with G KMbeing elements of the branch conductance matrix [4] . This means that in the formation of the nodal
DTCPEJ DTCPEJ
eSuation for node ka, G CCenters into element G of
MC MC the nodal conductance matrix in ES. (1. a), G CCinto
DTCPEJ DTCPEJ
G MC OC
,G CDinto G ,G
MC MD CDinto G , etc.
MC OD If this is done systematically, the matrix [4] will be added to
two diagonal blocks, and subtracted from two off diagonal blocks of the nodal conductance matrix [G], as indicated
in Fig. 3.2. Unfortunately, rows and columns ka, kb, kc and ma, mb, mc do not follow each other that neatly, and
the entries in [G] will therefore be all over the place, but this is simply a programming task. It is worth pointing out
that the entry of coupled branches into the nodal conductance matrix can always be explained with an eSuivalent
network of uncoupled elements. For three coupled resistances, the eSuivalent network with uncoupled elements
would contain 15 uncoupled resistances (see Fig. 7 in Chapter II of [2 ]). Such eSuivalent networks with
(KI Contributions of three coupled branches to the

nodal conductance matrix
uncoupled elements are useful for assessing the sparsity of a matrix, but they can be misleading by seemingly
indicating galvanic connections where none exist. For example, the steady state branch eSuations for two winding
transformers, which are well known from power flow and short circuit analysis,
33
Page 48
+MC OC ; &V; 8MC &8OC

' (3.3)
+MD OD &V; V ; 8MD &8OD
simply imply the connection of Fig. 3.3(a), and nothing more. The eSuivalent network with uncoupled elements is
shown in Fig. 3.3(b), which produces the well known transformer model of Fig. 3.3(c) if
nodes ma and mb are grounded.

Traduciendo...
(a) Coupled elements (b) Uncoupled elements (c) Uncoupled elements with
nodes ma and mb
grounded
(KI Two winding transformer as two coupled branches
'ZCORNG HQT %QWRNGF 4GUKUVCPEG
#ssume that a lightning stroke, represented by a current source i(t), hits phase # of a three phase line (Fig.
3. ). Let us then find the voltage build up in all 3 phases over a time span
(KI Lightning stroke to phase # of a three phase

line
during which reflections have not yet come back from the remote ends of the line, using the high freSuency lossless
line model of ES. (3.1). #ssume a flat tower configuration typical of 220 kV lines, with an average height above
ground 12.5 m, spacing between conductors 7.5 m, and conductor radius 1 .29 mm. Then from ES. (3.1),
.02 7 .2 39. 1
[4] ' 7 .2 .02 7 .2 S
39. 1 7 .2 .02
The left as well as the right part of the line is then represented by [4] connected from #, B, C to ground, and the
Page 49
voltages become
v #(t) 22 .01 i(t)
v $(t) 37.12 i(t)
v%
(t) 19.71 i(t)
or 1 . of v appears
# in phase B, and . in phase C. #n interesting variation of this case is the calculation of
the effect which this lightning stroke has on the eSuipment in a substation. #ssume that the travel time J between
the stroke location and the substation is such that no reflection comes back from the stroke location during the time
tOCZ
of the study, with the time count starting when the waves hit the substation (Fig. 3.5). In such cases, the waves
coming into the substation can be represented as a three phase voltage source with amplitudes eSual to twice the value
of the voltages at the stroke location, behind the resistance matrix [4]. This, in turn, can be converted to a current
source in parallel with a shunt resistance matrix [4]. Since
K(V)
1
[XUQWTEG] ' 2 @ [4] 0
2
0
it follows that the eSuivalent current source injected into the substation simply becomes eSual to the lightning current
at the stroke location [i(t), 0, 0] which together with the shunt resistance matrix [4], represents the waves coming
into the substation as long as no reflections have come back yet from the stroke location.
Traduciendo...
35
Page 50
(KI Waves coming into substation
%QWRNGF +PFWEVCPEGU =.?
Coupled inductances, in the form of branch inductance matrices, are used to represent magnetically coupled
circuits, such as
(a) inductive part of transformers,
(b) inductive part of source impedances in three phase Thevenin eSuivalent circuits for the rest of the
system when positive and zero seSuence parameters differ,
(c) inductive part of M phase nominal B circuits.
The diagonal elements of [L] are the self inductances, and the off diagonal elements are the mutual
inductances. In all cases known so far, [L] is symmetric, and the EMTP only accepts symmetric matrices, with the
storage scheme described in the last paragraph before Section 3.1.1.
The source impedances mentioned earlier under (b) above are often specified as positive and zero seSuence
parameters < , <RQU

which can be converted to self and mutual impedances
\GTQ Traduciendo...
1 1
<U ' 2<RQU % <\GTQ , <O ' <\GTQ & <RQU (3. )
3 3
Page 51
of the coupled impedance matrix
<U <O <O
[<] ' <O <U <O (3.5)
<O <O <U
Of course, these self and mutual impedances can in turn be converted back to seSuence parameters,
<RQU ' <U & <O , <\GTQ ' <U % 2<O (3. )
For a generalization of this data conversion to any number of phases M, see ES. ( . 0) in Section .1.3.2.
The eSuations for coupled inductances between a set of nodes ka, kb,... and a set of nodes ma, mb,... (Fig.
3. ) are solved accurately in the ac steady state solution with
1
[+MO] ' [.]& [8M] & [8O] (3.7)
LT
The only precaution to observe is that [L] should not be extremely large, for reasons explained in Section 2.1.1.
(KI Four coupled

inductances
For the transient simulation, ES. (2. ) and (2.7) for the scalar case are simply generalized for the matrix
case, which produces the desired branch eSuations
)V
[KMO(V)] ' [.]& 6[XM(V)]&[XO(V)]> % [JKUVMO(V&)V)] (3. )
2
with the history term [hist (t )t)]MO

known from the solution at the preceding time step,
)V
[JKUVMO(V&)V)] ' [KMO(V&)V)] % [.]& 6[XM(V&)V)] & [XO(V&)V)]> (3.9)
2
Just as in the uncoupled case, ES. (3. ) can be represented as an eSuivalent resistance matrix [4 ] (2/)t)[L],
GSWKX
in parallel with a vector [hist (t )t)] ofMO

known current sources. The matrix [4 ] enters into the nodal
GSWKX
37
Page 52
conductance matrix of the transient solution in the same way as described in Section 3.1.2 (for the steady state
solution, simply replace [4 ] byGSWKX
(1/jT)[L] ). While the current source hist of an uncoupled inductance
MO enters
only into two components k and m of the right hand side in ES. (1. b), the vector [hist ] must now be subtracted
MO
from components ka, kb, kc,..., and added to components ma, mb, mc,.... Traduciendo...
Once all the node voltages have been found at a particular time step at instant t, the history term of ES. (3.9)
must be updated for each group of coupled inductances. This could be done recursively with the matrix eSuivalent
of the scalar eSuation (2. ). The EMTP does not have an input option for coupled inductances alone instead, they
must be specified as part of the M phase nominal B circuit of Section 3. , where the updating formulas used by the
EMTP are discussed in more detail.
There are situations where [L] may not exist, but where [L] can be specified as a singular matrix. Such
an example is the transformer model of ES. (3.3). If resistances are ignored, ES. (3.3) can be used for transient
studies with
; &V;
[.]& ' LT (3.10)
&V; V ;
where ; 1/(jX), with X being the short circuit input reactance of the transformer measured from winding ka ma.
It is therefore advisable to have input options for [L] as well as for [L], as further discussed in Section 3. .2.
'TTQT #PCN[UKU
The errors are the same as for the uncoupled inductance, that is, the ratio tan(T )t/2)/(T )t/2) of ES. (2.17)
applies to every element in the matrix [L], or its reciprocal to every element in [L] . The stub line representation
of Fig. 2.9 becomes an M phase stub line, if M is the size of the matrix [L]. There is no need to use modal analysis
for this stub line because all travel times are eSual, as mentioned in Section .1.5.2. In that case, the single phase
line eSuations can be generalized to M phase line eSuations by simply replacing scalars with matrix Suantities. ES.
(2.9), (2.12) and (2.1 ) therefore become
2 )V
ý[.)] ' [.], [<] ' [.], CPF ý[%)] ' ( ) [.]&
)V 2
&CORKPI QH 0WOGTKECN 1UEKNNCVKQPU YKVJ %QWRNGF 2CTCNNGN 4GUKUVCPEGU
#gain, the explanations of Section 2.2.2 for the uncoupled inductance are easily generalized to the matrix
case if all elements of [L] are to have the same ratio 4 /L. SinceR[L] is used in ES. (3. ), it is preferable to express
the parallel resistances in the form of a conductance matrix, e.g., with #lvarado s recipe of ES. (2.23),
)V
[)R] ' 0.15 [.]& (3.11)
2
If [L] is singular, [G ] would

R be singular as well, but the singularity would not cause any problems. If the coupled
Page 53
inductances go from nodes ka, kb,...to nodes ma, mb,... (Fig. 3. ), then [G ] would be connected
R in the same way
from nodes ka, kb,... to nodes ma, mb,...
2J[UKECN 4GCUQPU HQT %QWRNGF 2CTCNNGN 4GUKUVCPEGU
The reasons are the same as those listed in Section 2.2.3 in those situations in which the single phase case
can be generalized to the M phase case.
'ZCORNG HQT 0GVYQTM YKVJ %QWRNGF +PFWEVCPEGU
Let us go back to the example of the single phase to ground fault described in Section 2.2. , but treat it as
a three phase Thevenin eSuivalent circuit now, with coupled resistances and inductances (Fig. 3.7). #ssume that
< RQU
0.02 j0. 0 p.u. and < 0.5 j1.329 p.u., or with ES. (3. ) < 0.1 j0.712 p.u., < 0.1
\GTQ U O
j0.30 p.u. There are three voltage sources now,
X#&5174%' ' 8OCZsin(TV)
X$&5174%' ' 8OCZsin(TV&120E)
X%&5174%' ' 8OCZsin(TV%120E)

Traduciendo...
(KI Single phase to ground fault with three phase

Thevenin eSuivalent circuit
With the same values of 4 and L Uas in Section

U 2.2. , the fault current will be identical with the curves of Fig.
2.21(b) and (c). In addition, we can now obtain the overvoltages in the unfaulted phases B and C, which are shown
in Fig. 3. .
39
Page 54
(KI Overvoltages in unfaulted phases B and C
The steady state solution can of course be easily obtained from the phasor eSuations
8# 8#&5174%' <U <O <O +#

8$ ' 8$&5174%' & <O <U <O 0 (3.12a)
8% 8%&5174%' <O <O <U 0
The first row produces, with V 0, #
8#&5174%'
+# ' (3.12b)
<U
and the second the third rows produce the voltage changes in the unfaulted phases
<O
)8$ ' )8% ' & 8#&5174%' (3.12c)
<U
If these voltage changes are shown in a phasor diagram (Fig. 3.9), then it becomes obvious why the overvoltages
in phases B and C are uneSual, unless the ratio < /< is a real
O U (rather than complex) number. In the latter case the
Traduciendo...
dotted changes become vertical in B and C in Fig. 3.9, and the overvoltages become eSual.
3 10
Page 55
(KI Phasor diagram of voltage changes caused by single phase to

ground fault
%QWRNGF %CRCEKVCPEGU =%?
Coupled capacitances, in the form of branch capacitance matrices, appear as the shunt elements of M phase
nominal B circuits (Fig. 3.10). One could argue that the capacitances are not really coupled, since they appear as
uncoupled capacitances in Fig. 3.10. However, the same argument can be made for coupled resistances and
inductances, as explained in Fig. 3.3 of Section 3.1.2, and the fact remains that the shunt capacitances of M phase
lines appear as matrix Suantities in the derivation of the eSuations.
(KI Three phase nominal B circuit
Since the only known application of coupled capacitances is as shunt elements of M phase nominal B
circuits, the EMTP accepts them only in that form, that is, as eSual branch capacitance matrices 1/2 [C] at each end
of the B circuit, from nodes ka, kb,... to ground, and from nodes ma, mb,... to ground. In all cases, [C] is
symmetric, and this symmetry is exploited with the storage scheme described in the last paragraph before Section
3 11
Page 56
3.1.1. The diagonal element C of [C]MCMC

is the sum of all capacitances between phase a and the other phases b, c,...
Traduciendo...
as well as between phase a and ground, whereas the off diagonal element C is the negative valueMCMD
of the
capacitance between phases a and b.
Sometimes, shunt capacitances of three phase lines are specified as positive and zero seSuence parameters
C RQU
, C , which
\GTQ can be converted to the diagonal and off diagonal elements
1 1
%U ' (2%RQU % %\GTQ) , %O ' (%\GTQ & %RQU) (3.13)
3 3
of the coupled capacitance matrix
%U %O %O
[%] ' %O %U %O (3.1 )
%O %O %U
C Omust be negative because the off diagonal element is the negative value of the coupling capacitance between two
phases, therefore, C C . For

\GTQa generalization
RQU of this data conversion to any number of phases M, see ES.
( . 1) in Section .1.3.2.
The steady state eSuations for coupled capacitances in the shunt connection of Fig. 3.10, and with the factor
1/2, are
1 1
[+M ] ' LT[%][8M], [+O ] ' LT[%][8O] (3.15)
2 2
with subscripts k0 and m0 indicating that the currents flow from nodes ka, kb,... to ground ( 0 ), and from
nodes ma, mb,... to ground. ES. (3.15) is solved accurately in the steady state solution. The only precaution to
observe is that T[C] should not be extremely large, for reasons explained in Section 2.1.1, but this is very unlikely
to occur in practice anyhow.
For the transient simulation, ES. (2.29) and (2.30) are again generalized for the matrix case, which produces
the desired branch eSuations (taking care of the factor 1/2!),
1
[KM (V)] ' [%][XM(V)] % [JKUVM (V&)V)] (3.1 )
)V
with the history term [hist (t )t)]Mknown from the solution at the preceding time step,
It might be worthwhile to have the EMTP check for the negative sign, and automatically make it negative,
with an appropriate warning message, in cases where the negative sign was forgotten. The writer is not aware of
any situation in which the off diagonal element would not be negative.
3 12
Page 57
1
[JKUVM (V&)V)] ' & [%][XM(V&)V)] & [KM (V&)V)] (3.17)
)V
The eSuations for the shunt capacitance 1/2 [C] at the other end (nodes ma, mb,...) are the same if subscript k is
replaced by m. #s in the uncoupled case, ES. (3.1 ) can be represented as an eSuivalent resistance matrix )t[C] ,
in parallel with a vector [hist (t )t)] ofMO

known current sources. The matrix 1/)t [C] enters into the nodal
conductance matrix of the transient solution only in the diagonal block of rows and columns ka, kb,... and in the
diagonal block of rows and columns ma, mb,... (Fig. 3.2), because of the shunt connection, while the vector [hist ]M
must be subtracted from components ka, kb,... (analogous for [hist ]). O
Once all the node voltages have been found at a particular time step at instant t, the history term of ES.
(3.17) is updated recursively,
[JKUVM (V)] ' & 2 [%][XM(V)] & [JKUVM (V&)V)] (3.1 )
)V
and analogous for [hist ]. 4ecursive

O updating is efficient here, in contrast toTraduciendo...
coupled inductances, because the
branches consist only of capacitances here, unless currents are to be computed as well. In the latter case, [i (t)] is M
first found from ES. (3.1 ), and then inserted into ES. (3.17) to obtain the updated history term, with both formulas
using the same matrix 1/)t [C].
'TTQT #PCN[UKU
The errors are the same as for the uncoupled capacitance, that is, the ratio tan(T )t/2) / (T )t/2) of ES.
(2.35) applies to every element in the matrix 1/2 [C]. The stub line representation of Fig. 2.23 becomes an M phase
stub line, with the second set of nodes being ground in this case. There is no need to use modal analysis, as
explained in Section 3.2.1.
&CORKPI QH 0WOGTKECN 1UEKNNCVKQPU YKVJ 5GTKGU 4GUKUVCPEGU
#gain, the explanations of Section 2.3.2 for the uncoupled capacitance are easily generalized to the matrix
case if all elements of 1/2 [C] are to have the same time constant 4 C. ES. (2.39)Uwould then become
[4U] ' 0.15 )V[%]& (3.19)
(factor 1/2 of ES. (2.39) disappeared because the eSuations have been written for 1/2 [C] here). #s mentioned in
Section 2.3.2, numerical oscillations in capacitive currents have seldom been a problem.
2J[UKECN 4GCUQPU HQT %QWRNGF 5GTKGU 4GUKUVCPEGU
None are known to the writer at this time. The discussions of Section 2.3.3 do not apply to shunt
capacitances of overhead lines, but they may be relevant to the capacitances of underground or submarine cables.
3 13
Page 58
'ZCORNG HQT 0GVYQTM YKVJ %QWRNGF %CRCEKVCPEGU
#ssume that a power plant with a number of generator transformer units in parallel is connected into the
230 kV switchyard through a number of parallel underground cables. The circuit breakers at the end of the cables
are open, when a single phase to ground fault occurs on the power plant side of the breakers (Fig. 3.11).
(KI Cable circuit with single phase to ground fault. Fault occurs in phase # when source voltage in # is at
its peak. Generator transformers represented as three phase voltage sources of 230 kV (4MS, line to line) behind
coupled reactances with X S, XRQU
2. S (referred to\GTQ
230 kV side). Cables represented as three phase
nominal B circuit with < < 0.015
RQU3 S, \GTQ
TC TC 97. zS, 4 RQU \GTQ (#7.6 1 S
The data resembles the situation at Ground Coulee before the Third Powerhouse was built, except that < < \GTQ RQU
for the cables is an unrealistic assumption. #lso note that the shunt capacitances of the nominal B circuit are actually
uncoupled in this case because C 0, which Ois always true in high voltage cables where each phase is
electrostatically shielded. Nonetheless, this cable circuit was chosen because it illustrates the effects of shunt
capacitances better than a case with overhead lines where C û 0. O
Fig. 3.12(a) shows the voltages in the two unfaulted phases at the fault location, with oscillations
superimposed on the 0 Hz so typical of cable circuits. Fig. 3.12(b) shows the fault current the high freSuency
oscillations at the beginning are caused by discharging the shunt capacitance through the fault resistance of 1 S.
With zero fault resistance, this discharge would theoretically consist of an infinite current spike at t 0, which leads
to the undamped numerical oscillations across the correct 0 Hz values discussed in Section 2.3.2 (Fig. 3.12(c)).
These numerical oscillations would not appear if the cables were modelled as lines with distributed parameters
Traduciendo...
instead, physically based travelling wave oscillations would appear which would still look similar to those of Fig.
3.12(b).
31
Page 59
(a) Overvoltages
(b) Fault current for 4 (#7.6 1 S (negative value shown)
3 15
Page 60
Traduciendo...
(c) Fault current for 4 (#7.6 0 (different scale than in (b), but value again
negative)
(KI
Overvoltages and fault current for a single phase to ground fault in the cable circuit of Fig. 3.11 ()t 10zs small
step size chosen to allow comparisons with distributed parameter model of cable with < .2S, J 10zs)
UWTIG
/ 2JCUG 0QOKPCNB %KTEWKV
Series connections of coupled resistances and coupled inductances first appeared as part of M phase nominal
B circuits (Fig. 3.10) when the EMTP was developed. It was therefore decided to handle such series connections
as part of an M phase nominal B circuit input option. By allowing the shunt capacitance 1/2 [C] to be zero, this B
circuit input option can then be used for series connections of [4] and [L] as well.
The eSuations for the shunt capacitance matrices 1/2 [C] at both ends are solved as discussed in Section 3.3.
[C] 0 is not recognized by the EMTP as a special case instead, the calculations are done as if [C[ were nonzero.
What remains to be shown is the series connection of [4] and [L] as one single set of M coupled branches.
The derivation of the coupled branch eSuations is similar to that of the scalar case discussed in Section 2. , if scalar
Suantities are replaced by matrices. When the series [4] [L] connection was first implemented in the EMTP, it
was not recognized that [L] may not always exist. With the appearance of singular [L] matrices, e.g., in the
transformer model of ES. (3.3), an alternative formulation was developed. Both formulations have been
implemented, as discussed in the next two sections.
31
Page 61
5GTKGU %QPPGEVKQP QH =4? CPF =.?
For the steady state solution, the branch eSuations are
+MO ' 6[4] % LT[.]>& 6[8M] & [8O]> (3.20)
They are solved accurately. For the transient simulation, the branch eSuations are derived by adding the voltage
drops across [4] and [L]. From ES. (3.2) and (3. ),
[KMO(V)] ' [)UGTKGU]6[XM(V)] & [XO(V)]> % [JKUVUGTKGU(V&)V)] (3.21a)
with
2
[4UGTKGU] ' [4] % [.] , CPF [)UGTKGU] ' [4UGTKGU]& (3.21b)
)V
Traduciendo...
and the history term
[JKUVUGTKGU(V&)V)] ' [)UGTKGU]6[XM(V&)V)] & [XO(V&)V)] %
2
[.] & [4] [KMO(V&)V)]> (3.22)
)V
Direct updating of the history term with ES. (3.22) involves three matrix multiplications because [i ] must first be MO
found from ES. (3.21a). Unless currents must be computed anyhow, as part of the output Suantities, updating with
the following recursive formula is more efficient,
[JKUVUGTKGU(V) ' [*]6[XM(V)] & [XO(V)] % [4UGTKGU][JKUVUGTKGU(V&)V)]> &
[JKUVUGTKGU(V&)V)] (3.23a)
since it involves only two matrix multiplications. Matrix [H] is
[*] ' 26[)UGTKGU] & [)UGTKGU][4][)UGTKGU]> (3.23b)
#ll matrices [4 ], [G UGTKGU

UGTKGU ] and [H] are still symmetric, which is exploited by the EMTP with the storage scheme
discussed in the last paragraph before Section 3.1.1. Symmetry is not automatically assured. For instance, the
alternative updating formula
[JKUVUGTKGU(V)] ' [(][KMO(V)] & [JKUVUGTKGU(V&)V)]
which, in combination with ES. (3.21a), would be preferable in situations where current output is reSuested, has an
unsymmetric matrix [F],
[(] ' [*][4UGTKGU]
3 17
Page 62
#ll eSuations in this section can handle the special case of either [4] 0 or [L] 0 as long as [4 ] of ES. (3.21b) UGTKGU
can be inverted.
5GTKGU %QPPGEVKQPU QH =4? CPF =.?
Singular matrices [L] appear in transformer representations if exciting currents are ignored. By itself, [L]
is easily handled with ES. (3. ) and (3.9). In series connections with [4], however, the eSuations of the preceding
section cannot be used directly because [L] does not exist.
For the steady state solution, the matrix [4] jT [L] is rewritten as
[4] % LT[.] ' [LT.]6[LT.]& [4] % [7]>
with [U] being the identity matrix, which upon inversion, produces the inverse reSuired in ES. (3.20),
6[4] % LT[.]>& ' 6[7] % [LT.]& [4]>& [LT.]& (3.2 )
ES. (3.2 ) produces a symmetric matrix, even though the matrix [U] [jTL] [4] needed as an intermediate step
is unsymmetric. The symmetry of the result from ES. (3.2 ) can be shown by rewriting the matrix [4] jT[L] as
[4] % LT[.] ' [LT.]6[LT.]& [4][LT.]& % [LT.]& > [LT.]
from which the inverse is obtained as
6[4]%LT[.]>& ' [LT.]& 6[LT.]& [4][LT.]& % [LT.]& >& [LT.]& (3.25)
Each of the three factors of the product is a symmetric matrix, which is obvious for the two outer factors and which
can easily be proved for the inner factor by showing that its transpose is eSual to the original. With all three factors
being symmetric, the triple product [#][B][#] is symmetric, too. The EMTP uses ES. (3.2 ) rather than (3.25),
because the latter would fail if [4] 0 and [L] singular. The EMTP does not use complex matrix inversion,
followed by matrix multiplication with an imaginary matrix, however. Instead, ES. (3.2 ) is reformulated as the
solution of a system of N linear eSuations with N right hand sides,

Traduciendo...
6L[7] % [T.]& [4]> [;] ' [T.]& (3.2 )
where the inverse [;] is now directly obtained as the N solution vectors. To avoid complex matrix coefficients, ES.
(3.2 ) is further rewritten as 2N real eSuations,
[T.]& [4][;T] & [;K] ' [T.]& (3.27)
[;T] % [T.]& [4][;K] ' 0 (3.2 )
By replacing [; ] inTES. (3.27) with the expression from ES. (3.2 ), the imaginary part of [;] is found by solving
the N real eSuations
31
Page 63
[T.]& [4] % [7] [;K] ' &[T.]& (3.29)
and the real part is then calculated from ES. (3.2 ).
For the transient simulation, [4 ] of ES.

UGTKGU (3.21b) is rewritten as
2 2 )V
[4] % [.] ' [.] [.]& [4] % [7]
)V )V 2
which, upon inversion, produces the matrix [G ] reSuired

UGTKGU in ES. (3.21a),
)V
[)UGTKGU] ' 6[7] % [.]& [4]>& )V [.]& (3.30)
2 2
#gain, the matrix [U] )t/2 [L] [4] needed as an intermediate step is unsymmetric, while the final result [G ]
UGTKGU
becomes symmetric. Symmetry is proved with ES. (3.25) by simply replacing jT by 2/)t. #s in the steady state
case, the inverse of ES. (3.30) is found by solving N linear eSuations
)V )V
6[7] % [.]& [4]>[)UGTKGU] ' [.]& (3.31)
2 2
To initialize the history term [hist ], ES.UGTKGU

(3.21a) can be used directly. To update it, neither ES. (3.22)
nor ES. (3.23a) can be used because [L] and [4 ] do not

UGTKGU exist. Instead, ES. (3.23a) is rewritten as
[JKUVUGTKGU(V)] ' [*]6[XM(V)] & [XO(V)]> % [JKUVUGTKGU(V&)V)] %
[)UGTKGU][&24][JKUVUGTKGU(V&)V)] (3.32)
By storing the symmetric matrices [H], [G ] and

UGTKGU 2[4], the updating with ES. (3.32) can be done with
three matrix multiplications, starting with the product 2[4][hist (t )t)]. #n alternative
UGTKGU updating formula, which
reSuires the storage of only two symmetric matrices [G ] and

UGTKGU 2[4], and produces the currents [i ] as a by MO
product, is
[JKUVUGTKGU(V)] ' [)UGTKGU]6[XM(V)] & [XO(V)] % [&24][KMO(V)]> % [KMO(V)] (3.33)
if the current is first found from ES. (3.21a), followed by the multiplication 2[4][i (t)], etc. ES. (3.33)
MO is derived
from ES. (3.22) by rewriting 2/)t [L] [4] as [4 ] 2[4].

UGTKGU
#ll eSuations in this section have symmetric matrices, and can handle the special case of either [4] 0 or
[L] 0 as long as [U] )t/2 [L] [4] in ES. (3.30) can be inverted. Note, however, that [L] 0 implies infinite
inductances, that is, the M coupled branches are really M open switches.
3 19
Traduciendo...
Page 64
18'4*'#& 64#05/+55+10 .+0'5
.KPG 2CTCOGVGTU
The parameters 4 , L , and C of overhead transmission lines are evenly distributed along the line , and can,
in general, not be treated as lumped elements. Some of them are also functions of freSuency therefore, the term
line constants is avoided in favor of line parameters. For short circuit and power flow studies, only positive
and zero seSuence parameters at power freSuency are needed, which are readily available from tables in handbooks,
or can easily be calculated from simple formulas. For the line models typically needed in EMTP studies, however,
these simple formulas are not adeSuate enough. Usually, the line parameters must therefore be computed, with either
one of the two supporting routines LINE CONST#NTS or C#BLE CONST#NTS.
These supporting routines produce detailed line parameters for the following types of applications:
(a) Steady state problems at power freSuency with complicated coupling effects. #n example is the calculation
of induced voltages and currents in a de energized three phase line which runs parallel with an energized
three phase line. Both lines would be represented as six coupled phases in this case.
(b) Steady state problems at higher freSuencies. Examples are the analysis of harmonics, or the analysis of
power line carrier communication, on untransposed lines.
(c) Transients problems. Typical examples are switching and lightning surge studies.
Line parameters could be measured after the line has been built this is not easy, however, and has been
done only occasionally. #lso, lines must often be analyzed in the design stage, and calculations are the only means
available for obtaining line parameters in that case.
The following explanations describe primarily the theory used in the supporting routines LINE
CONST#NTS and C#BLE CONST#NTS, though other methods are occasionally mentioned, especially if it appears
that they might be used in EMTP studies some day. The supporting routine LINE CONST#NTS is heavily based
on the work done by M.H. Hesse [27], though some extensions to it were added.
.KPG 2CTCOGVGTU (QT +PFKXKFWCN %QPFWEVQTU
The solution method is easier to understand for a specific example. Therefore, a double circuit three phase
line with twin bundle conductors and one ground wire will be used for the explanations (Fig. .1). There are 13
conductors in this configuration. They will be called
The prime in 4 , L and C is used to indicate distributed parameters in S/km, H/km and F/km.
Page 65
Traduciendo...
(KI Line parameters
individual conductors , to distinguish them from the eSuivalent phase conductors which are obtained after pairs
have been bundled into phase conductors and after the ground wire has been eliminated.
5GTKGU +ORGFCPEG /CVTKZ
It is customary to describe the voltage drop along a transmission line in the form of partial differential
eSuations, e.g., for a single phase line as
MX )
& '4 K % . ) MK ( .1)
MZ MV
The parameters 4 and L of overhead lines are not constant, however, but functions of freSuency. In that case it
is improper to use ES. ( .1) instead, the voltage drops must be expressed in the form of phasor eSuations for ac
steady state conditions at a specific freSuency. For the case of Fig. .1,
F8
FZ
+
F8 ) ) )
< < ... < +
FZ
<) <) ... <) .
& . ' ( .2a)
.......... .
.
<) )
... <) .
. <
+
F8
FZ
In the output of the supporting routine LINE CONST#NTS, they are called physical conductors.
Page 66
with V uvoltage phasor, measured from conductor i to ground,
I ucurrent phasor in conductor i,
or in general
F8 )
& ' [< ][+] ( .2b)
FZ
with [V] vector of phasor voltages (measured from conductor to ground), and
[I] vector of phasor currents in the conductors.
Implied in ES. ( .2) is the existence of ground as a return path, to which all voltages are referenced. The matrix
[< ] [4 (T)] jT [L (T)] is called the series impedance matrix it is complex and symmetric. The diagonal
element < 4 jTL

uu is the
uu series self
uu impedance per unit length of the loop formed by conductor i and ground
return. The off diagonal element < < 4 jTLuwis the series
wu mutual
wu impedance
wu per unit length between
conductors i and k, and determines the longitudinally induced voltage in conductor k if a current flows in conductor
i, or vice versa. The resistive terms in the mutual coupling are introduced by the presence of ground, as briefly
explained in Section 3.1.
Formulas for calculating < and <uuwere developed

uw by Carson and Pollaczek in the 1920 s for telephone
circuits [2 , 29]. These formulas can also be used for power lines. Both seem to give identical results for overhead
lines, but Pollaczek s formula is more general inasmuch as it can also be used for buried (underground) conductors
or pipes. Carson s formula is easier to program than Pollaczek s and is therefore used in both supporting routines
LINE CONST#NTS and C#BLE CONST#NTS, except that the latter includes an extension of Carson s formula
for the case of multilayer stratified earth [30] as well. Carson s, Pollaczek s and other earth return formulas are
compared in [31].
Two recent new approaches to the calculation of earth return impedances are those of Hartenstein, Koglin
and 4ees [32], and of Gary, Deri, Tevan, Semlyen and Castanheira [33, 3 ].Traduciendo...
Hartenstein, Koglin and 4ees treat
the ground as a system of conducting layers 1, 2, 3...n, with uniform current distribution in each layer (Fig. .2(a)).
Their results come close to those obtained with Carson s formula. One advantage of their method is the fact that
it is very easy to assume difference earth resistivities for each of the layers. Gary, Deri, et al. calculate self and
mutual impedances with the simple formulas originally proposed by Dubanton,
) ) z 2(JK%R) )
< ýP %: ( .3)
KK ' 4 K&KPVGTPCN % 2B
LT K&KPVGTPCN
TK
and
) z (JK%JM%2R) % Z KM
< ýP (.)
KM ' LT 2B FKM
in which p represents a complex depth,
Page 67
D
R' ( .5)
LTz
#ll other parameters are explained after ES. ( . ), except for x horizontal distance
uw between conductors i and
k (Fig. . ), and D earth resistivity. The results agree very closely with those obtained from Carson s formula,
with the differences peaking at 9 in the freSuency range between 100 Hz and 10 kHz and being lower elsewhere.
This is a very good agreement, indeed, and ES. ( .3) and ( . ) may therefore supplant Carson s formula some day.
Fig. .2(b) shows a comparison of positive and zero seSuence parameters for a typical 500 kV line.
(KI C #lternative to Carson s

formula: Ground represented as layers 1,
2,...n
%CTUQP U HQTOWNC
Carson s formula for homogeneous earth is normally accurate enough for power system studies, especially
since the data for a more detailed multilayer earth return is seldom available. The supporting routine C#BLE
CONST#NTS does have an option for multilayer or stratified earth, however. Carson s formula is based on the
following assumptions:
(a) The conductors are perfectly horizontal above ground, and are long enough so that three dimensional end
effects can be neglected (this makes the field problem two dimensional). The sag is taken into account
indirectly by using an average height above ground (Fig. .3).
Traduciendo...
Page 68
(KI D #lternative to Carson s formula: formula by Gary, Deri et al. (comparison with Carson s
formula for a typical 500 kV line with bundle conductors skin effect in conductors ignored)
(b) The aerial space is homogeneous without loss, with permeability z and permittivity g .
(c) The earth is homogeneous with uniform resistivity D, permeability z and permittivity g , and is bounded
by a flat plane with infinite extent, to which the conductors are parallel. The earth behaves as a conductor,
i.e., 1/D Tg , and hence the displacement currents may be neglected. #bove the critical freSuency
f i„u†uigx1/(2Bg D), other formulas [35, 3 ] must be used (for D 10,000 Sm in rocky ground, f i„u†uigx
1. MHz, which is still on the high side for most EMTP line models).
(d) The spacing between conductors is at least one order of magnitude larger than the radius of the
conductors, so that proximity effects (current distribution within one conductor influenced by current in an
adjacent conductor) can be ignored.
The conductor profile between towers (Fig. .3) can be described
(a) as a parabola for spans # 500 m,
(b) as a catenary for 500 # spans # 2000 m, and
(c) as an elastic line for spans 2000 m.
Page 69
(KI Conductor profile between towers
If the parabola is accurate enough, then the average height above ground is
Traduciendo...
1 (.)
J'JGKIJV CV OKFURCP% UCI,
3
(.)
which is the formula used by both supporting routines LINE CONST#NTS and C#BLE CONST#NTS. The
elements of the series impedance matrix can then be calculated from the geometry of the tower configuration (Fig.
. ) and from characteristics of the conductors. For the self impedance,
z 2JK
<) ) )
ln %: ) )
( .7)
KK ' (4 K&KPVGTPCN%)4
KK) % L(T
2B TK K&KPVGTPCN%):
KK)
(KI Tower geometry
Page 70
and for the mutual impedance
z &KM
<) ) ) ln % ): )
(.)
KM ' < MK ' )4 KM % L(T2B FKM KM)
with z permeability of free space. Using
z /2B ' 2@ 10 & */MO

( .9)
produces impedances in S/km. The parameters in ES. ( .7) and ( . ) are
4 ac
u u€†q„€gx resistance of conductor i in S/unit length,
hu average height above ground of conductor i,
D uw distance between conductor i and image of conductor k,
d uw distance between conductors i and k,
ru radius of conductor i,
X u u€†q„€gxinternal reactance of conductor i,
T 2Bf with f freSuency in Hz,
)4 , )X Carson s correction terms for earth return effects.
Carson s correction terms )4 and )X in ES. ( .7) and ( . ) account for the earth return effect, and are
functions of the angle N (N 0 for self impedance, N N in Fig. . for mutual

uw impedance), and of the parameter
a:
Traduciendo...
&@ & @ H
C ' B 5 @ 10 ( .10)
D
with D 2h in m ufor self impedance,
D uw
in m for mutual impedance,
D earth resistivity in Sm.
)4 and )X become zero for a 6 4 (case of very low earth resistivity). Carson gives an infinite integral for )4
and )X , which he developed into the sum of four infinite series for a # 5. 4earranged for easier programming,
it can be written as one series, and for impedances in S/km, becomes
)4 T!10 {B/" )X T!{1/2(0. 159315 1na)
b a!cosN b a!cosN
g
b [(c 1na)a cos2N a sin2N] d a cos2N

! !
b a cos3N
! b a cos3N
!
" " "
d a" cos N b [(c
" " 1na)a cos N Na sing N]
# #
b a# cos5N b a cos5N
#
Page 71
b [(c$ 1na)a
$ cos N$ Na sin N $ $
d a$ cos N
% %
b a cos7N
% b a cos7N
%
d &
a&cos N b [(c & &
& &1na)a cos N Na sin N]
...} ...}
in S/km ( .11)
Each successive terms for a repetitive pattern. The coefficients b, c and d are constants,
u uwhich can ube
precalculated and stored in lists. They are obtained from the recursive formulas:
2
D' HQT QFF UWDUETKRVU,
UKIP
DK'D & YKVJ VJG UVCTVKPI XCNWG
K(K%2)
1 ( .12)
D' HQT GXGP UWDUETKRVU,
1
1 1
EK'E & % % YKVJ VJG UVCTVKPI XCNWG E '1.3 59315,
K K%2
B
FK' @DK,
with sign v1 changing after each successive terms (sign v1 for i 1, 2, 3, sign 1 for i 5, , 7,
etc.).
The trigonometric functions are calculated directly from the geometry,
JK%JM ZKM
cosNKM ' CPF sinNKM '
&KM &KM
and for higher order terms in the series from the recursive formulas
C KEQUKN ' [CK& cos(K&1)N@cosN&CK& sin(K&1)N@sinN]@C
C KUKPKN ' [CK& cos(K&1)N@sinN%CK& sin(K&1)N@cosN]@C ( .13)
For power circuits at power freSuency only few terms are needed in the infinite series of ES. .11.
However, at freSuencies and for wider spacings (e.g., in interference calculations) more and more terms must be
taken into account as the parameter a becomes larger and larger [37, discussion by Dommel]. Once Carson s series
starts to converge, it does so fairly rapidly. How misleading the results can be with too few terms in the series of
ES. .11 is illustrated for the case of a and N 0: If the series were truncated after the 1st, 2nd,..., 15th term,
the percent error in 4e{< } would
uu be
312, 7 , 1 , 79 , 1 , 3 5, 121, 93, 2 , 15, 5.2,

Traduciendo...
1.7, 0.35, 0.1 , 0.0
For a 5 the following finite series [3 ] is best used:
Page 72
&
) cosN 2cos2N cos3N 3cos5N 5cos7N T@10
)4 ' & % % & @
C C C C C 2
&
) cosN cos3N 3cos5N 5cos7N T@10
): ' & % % @ KP S/MO ( .1 )
C C C C 2
+PVGTPCN KORGFCPEG CPF UMKP GHHGEV
In the old days of slide rule calculations, the internal reactance X „qgi†g€iqand external reactance T z /2B ýn
2h/r for lossless earth were often combined into one expression, by replacing radius r with the smaller geometric
mean radius GM4 to account for the internal magnetic field,
z 2J z 2J
T ýP %: ) ýP ( .15)
2B T KPVGTPCN ' T2B )/4
GM4 was often included in conductor tables. Instead of or in addition to GM4, North #merican handbooks have
also freSuently given the reactance at 1 foot spacing X , which is! related
5 to GM4,
z 1(HQQV)
: ) ýP ( .1 )
#' T 2B )/4(HGGV)
with GM4 in feet (or in m if X is to be5the reactance at 1 m spacing).
The concept of geometric mean radius was originally developed for nonmagnetic conductors at power
freSuency where uneven current distribution (skin effect) can be ignored. In that case, its meaning is indeed purely
geometric, with GM4 being eSual to the geometric mean distance among all elements on the conductor cross section
area if this area were divided into an infinite number of eSual, infinitesimally small elements. For a solid, round,
nonmagnetic conductor at low freSuency,
&
)/4/T ' G
This formula changes to
&zT
)/4/T ' G
if the conductor is made of magnetic material with relative permeability z its geometric meaning
„ is then lost. If
skin effect is taken into account, its geometric meaning is lost as well. The name geometric mean radius is therefore
!
The name comes form the positive seSuence reactance formula X T z /2B ýn GMD/GM4
‚… discussed in
ES. ( .5 ), for the case where the spacing among the three phases (expressed as geometric mean distance GMD)
is 1 foot, with GM4 given in feet as well.
Page 73
misleading, and it is Suestionable whether it should be retained.
ES. ( .15) gives the conversion formula between GM4 and internal reactance,
The internal reactance can be calculated for certain types
z of conducts [39, 0] as part of the internal impedance
:)KPVGTPCN T ( .17)
4 jX
u€†q„€gx . Since
u€†q„€gx ' G is only a very Bsmall part of the total reactance for nonmagnetic conductors, its
X)/4/Tu€†q„€gx
Traduciendo...
accurate determination is somewhat academic. More important is the calculation of 4 , because the increase
u€†q„€gx
of resistance with freSuency due to skin effect can be considerable.
The internal impedance of solid, round wires can be calculated with well known skin effect formulas, with
4 being of more practical interest than X

u€†q„€gx . Stranded
u€†q„€gx conductors can usually be approximated as solid
"
conductors of the same cross sectional area [ 1]. It has been claimed that steel reinforced aluminum cables (#CS4)
can usually be approximated as tubular conductors when the influence of the steel core is negligible, which is more
likely to be the case with an even number of layers of aluminum strands, since the magnetization of the steel core
caused by one layer spiralled in one direction is more or less cancelled by the next layer spiralled in the opposite
direction. The supporting routine LINE CONST#NTS uses this approximation of an #CS4 as a tubular conductor.
If the magnetic material of the steel core is of influence, then calculations probably become unreliable, and current
dependent, measured values should be used instead. Since the solid conductor is a special case of the tubular
conductor, the supporting routine LINE CONST#NTS uses only the formula for the latter, which is described as
ES. (5.7b) in Section 5.1.
Table .1 shows the increase in resistance and the decrease in internal inductance due to skin effect for a
tubular conductor with 4 0.039pi S/mile, ratio inside radius/outside radius S/r 0.225 (Fig. .5), and z „
1.0. The internal inductance of a tube at dc is [ , p. ]
) & S T 3S &T
. ýP & */MO
FE ' 2 @ 10 S
(T &S ) (T &S )
"
or 0. 5 10 H/km in this case. #t high freSuencies, 4 X
u€†q„€gx , with
u€†q„€gx both components being
proportional to %T. This is the region of pronounced skin effect. From Table .1 it can be seen that 4 and
u€†q„€gx
X are
u€†q„€gx almost eSual at 10 kHz (difference 2.2 ), with the difference decreasing to 0.7 at 100 kHz, or 0.2
at 1 MHz.
"
There are cases, however, where this approximation is not good enough. More accurate formulas are
needed, for instance, for calculating the attenuation in power line carrier problems [39], as explained in
#ppendix VII.
10
Page 74
6CDNG Skin effect in a tubular conductor
f(Hz) 4 /4gi pi L u€†q„€gx/L

gi u€†q„€gx pi
2 1.0002 0.99992
1.0007 0.99970
1.0015 0.99932
1.002 0.99 79
10 1.00 1 0.99 12
20 1.01 0.9925
0 1.0 32 0.97125
0 1.13 7 0.93 9
0 1.2233 0. 99
100 1.3213 0. 5 39
200 1.79 3 0. 232
00 2. 55 0. 700
00 2.9 21 0.3 503
00 3.3559 0.33 1
1000 3.7213 0.2992
2000 5.15 1 0.2120
000 7.1 7 0.1500
000 .7 71 0.1225
000 10.0 22 0.10 17
10000 11.2209 0.09 97
20000 15.7 7 0.0 717
0000 22.19 0.0 750
0000 27.1337 0.03 79
0000 31.29 2 0.03359
Traduciendo...
100000 3 .9597 0.0300
200000 9.3 13 0.0212
00000 9. 02 0.01502
00000 5.2 70 0.01227
00000 9.1 0.010 2
1000000 110.0357 0.00950
2000000 155.515 0.00 72
000000 219. 33 0.00 75
(KI Tubular conductor
11
Page 75
(KI Current distribution within an conductor bundle [ 2]. l

197 IEEE
'ZCORNG HQT WUKPI UGTKGU KORGFCPEG OCVTKZ QH KPFKXKFWCN EQPFWEVQTU
The matrix of ES. ( .2) can be used to study the uneven current distribution within a bundle conductor.
Fig. . shows measured and calculated values for the uneSual current distribution in the subconductors of an
asymmetrical bundle for various degrees of asymmetry [ 2]. #symmetrical bundling was proposed to reduce audible
noise, but this advantage is offset by the uneSual current distribution. The currents in this case were found from ES.
( .2) with an x matrix, assuming eSual voltage drops in the conductors,
) & [F8/FZ]
[+] ' &[< ] ( .1 )
Traduciendo...
5JWPV %CRCEKVCPEG /CVTKZ
The voltages from the 13 conductors in Fig. .1 to ground are a function of the line charges:
12
Page 76
X 2) 2) )
S
... 2
) ) )
X 2 2 S
... 2
. ... .
' ( .19a)
. ... .
. ... .
X ) ) ) S
2
2 ... 2
with S charge
u per unit length on conductor i, or in the general case
)
[X] ' [2 ] [S] ( .19b)
Maxwell s potential coefficient matrix [P ] is real and symmetric. Its elements are easy to compute from the
geometry of the tower configuration and from the conductor radii if the following two assumptions are made: (a) the
air is lossless and the earth is uniformly at zero potential, (b) the radii are at least an order of magnitude smaller than
the distances among the conductors. Both assumptions are reasonable for overhead lines. Then the diagonal element
becomes
1 2JK
2) ln ( .20)
KK ' 2B, TK
and the off diagonal element
1 &KM
2 ) )
ln ( .21)
KM ' 2 MK ' 2B, FKM
with g permittivity of free space. The factor 1/(2Bg ) in these eSuations is c z /2B, where c is the speed of
"
light. With c 299,792.5 km/s and z /(2B) 2 10 H/km, it follows that
1/(2B, ) ' 17.975109 @ 10 MO/( ( .22)
The inverse relationship of ES. ( .19) yields the shunt capacitance matrix [C ],
) ) )&
[S] ' [% ] [X], YKVJ [% ] ' [2 ] ( .23)
The supporting routine LINE CONST#NTS uses a version of the Gauss Jordan process for this matrix inversion
which takes advantage of symmetry [ 3]. This process was chosen because it can easily be modified to handle matrix
reductions as well, which are needed for eliminating ground wires and for bundling conductors. #ppendix III
explains this Gauss Jordan process in more detail.
The capacitance matrix [C ] is in nodal form. This means that the diagonal element C is the sum of the uu
shunt capacitances per unit length between conductor i and all other conductors as well as ground, and the off
diagonal element C C is uw
the negative
wu shunt capacitance per unit length between conductors i and k. #n
example for a three phase circuit from [ , p. 57] is shown in Fig. .7, with
13
Page 77
12.1 1 &2. 25 &2. 25

)
[% ] ' &2. 25 11.729 &1.3 9 Traduciendo...
P(/OKNG
&2. 25 &1.3 9 11.729
or C y‡†‡gx 2. 25, C ! y‡†‡gx 2. 25, C ! y‡†‡gx 1.3 9, C s„ ‡€p 7.755 nF/mile, etc.
(KI Mutual and shunt capacitances
For ac steady state conditions, the vector of charges (as phasor values) is related to the vector of leakage
currents [ dI/dx] by
1 F+
[3] ' & ( .2 )
LT FZ
Therefore, the second system of differential eSuations is
F+ )
& ' LT [% ] [8] ( .25)
FZ
which, together with ES. ( .2), completely describes the ac steady state behavior of the multi conductor line. Shunt
conductances G have been ignored in ES. ( .25), because their influence is negligible on overhead lines, except at
very low freSuencies approaching dc, where the line behavior is determined by 4 and G , with TL and TC
becoming negligibly small. With G , the complete eSuation is
F+ )
& ' [; ] [8] ( .2 a)
FZ
where
) ) )
[; ] ' [) ] % LT [% ] ( .2 b)
#t very high freSuencies, the shunt capacitances are also influenced by earth conduction effects, and
correction terms must then be added to ES. ( .20) and ( .21). However, the earth conduction effect is normally
Page 78
negligible below 100 kHz to 1 MHz [ 5]. In that case, the capacitances are constant, in contrast to series resistances
and series inductances which are functions of freSuency.
.KPG 2CTCOGVGTU HQT 'SWKXCNGPV 2JCUG %QPFWEVQTU
ESuations ( .2) and ( .19) for all individual conductors contain more information than is usually needed.
Generally, only the phase Suantities are of interest. For the case of Fig. .1, the reduction from 13 eSuations to
eSuations for the phases 4, S, T, I, V, W is accomplished by introducing the following conditions,
for grounding conductor 13: dV /dx 0 !in ( .2), v 0 in ( .19), !
for bundling conductors 1 and 2 into phase 4:
I I I , dV /dx dV
R /dx dV /dx in ( .2), R
and
S S S , v v v inR( .19) R
and analogous for bundling the other phases. With these conditions, the matrices can be reduced to x , as
explained next. These reduced matrices will be called matrices for the eSuivalent phase conductors.
'NKOKPCVKQP QH )TQWPF 9KTGU Traduciendo...

#
Normally, ground wires are continuous and grounded at every tower , which are typically 250 to 350 m
apart. In that case it is permissible for freSuencies up to approximately 250 kHz to assume that the ground wire
potential is continuously zero [ ]. This allows a reduction in the order of the [< ] and [P ] matrices, with the
reduction procedure being the same for both. Let the matrices and vectors in ES. ( .2) be partitioned for the set u
of ungrounded conductors, and for the set g of ground wires,
) )
[F8W/FZ] [< [+W]
WW] [< WI]
& ' ( .27)
[F8I/FZ] [< ) ) [+I]
IW] [< I I]
Since [V ] sand [dV /dx]s are zero, ES. ( .27) can be reduced by eliminating [I ], s
F8W )
& ' [< ( .2 a)
FZ TGFWEGF] [+W]
where
) ) ) ) & [< )
[< ( .2 b)
TGFWEGF] ' [< WW] & [< WI] [< I I] IW]
4ather than using straightforward matrix inversion and matrix multiplications in ES. ( .2 b), the more efficient
Gauss Jordan reduction process of #ppendix III is used in the supporting routine LINE CONST#NTS. [P ] is
reduced in the same way, and [C „qp‡iqp] is found by inverting [P „qp‡iqp]. #t first sight it may appear as if less work
#
Non continuous segmented ground wires are discussed in Section .1.2.5.
15
Page 79
were involved in reducing [C ], where the reduction simply consists of scratching out the rows and columns for
ground wires g. However, [C ] must first be found from the inversion of [P ], and it is faster to reduce a matrix
than to invert it.
$WPFNKPI QH %QPFWEVQTU
On high voltage power lines, bundle conductors are freSuently used, where each phase or bundle
conductor consists of two or more subconductors held together by spacers (typically 100 m apart). The bundle is
usually symmetrical (S 1.0 in Fig. . ), but unsymmetrical bundles have been proposed as well. Two methods
can be used for calculating the line parameters of bundle conductors. With the first method, the parameters are
originally calculated with each subconductor being represented as an individual conductor. Since the voltages are
eSual for the subconductors within a bundle, this voltage eSuality is then used to reduce the order of the matrices
to the number of eSuivalent phase conductors. With the second method, the concept of geometric mean distances
is used to replace the bundle of subconductors by a single eSuivalent conductor. Both methods can be used with the
supporting routine LINE CONST#NTS. The supporting routine C#BLE CONST#NTS is limited to the second
method.
Method 1 Bundling of subconductors by matrix reduction
#s in the elimination of ground wires, the matrix reduction process is the same for [< ] and [P ], and will
therefore only be explained for [< ]. Let us assume that the individual conductors i, k, l, m are to be bundled to
make up phase 4. Then the conditions
+K % +M % +
ý % +O ' +4
and
F8K F8M F8 ý F8O F84

' ' ' '
FZ FZ FZ FZ FZ
R
must be introduced into ES. ( .2). The first step is to get I into the eSuations. This is done by writing I in place R
of I .u By doing this, an error is of course made, which amounts to the addition of terms
)
< ý % +O)
zK(+M % +
Traduciendo...
in all rows z they must obviously be subtracted again to keep the eSuations correct. In effect, this means subtraction
of column i from columns k, ý, m. These changes are shaded in Fig. . .
Page 80
(KI First step in bundling procedure
Columns k, ý, m are assumed to be the last ones in the matrix to make the explanation easier. The currents I , I , wý
I yare still in the eSuations after execution of the first step of Fig. . . To be able to eliminate them, there should
be zeros in the left hand side of the respective rows. This is easily accomplished by subtracting row i from rows
k, ý, m, which produces zeros because dV/dx dV /dx

u etc. These
w changes are shaded in Fig. .9. The eSuations
are now in a form which permits elimination of I , I , I , in the

w ý ysame way as elimination of ground wires in ES.
( .2 ). The four rows and columns for subconductors i, k, ý, m are thereby reduced to a single row and column
for bundle conductor 4.
Method 1 is more general than method 2 discussed next. For instance, it can easily handle the uneSual
current distribution in asymmetrical bundles described in Fig. . .
(KI Second step in bundling procedure
Method 2 4eplacing bundled subconductors with eSuivalent single conductor
This method was developed for hand calculations [ 7], and while theoretically not limited to symmetrical
bundles, formulas have usually only been derived for the more important case of symmetrical bundles. The
following formulas are based on the assumption that
(a) the bundle is symmetrical (S 1.0 in Fig. . ), and
(b) the current distribution among the individual subconductors within a bundle is uniform.
17
Page 81
Traduciendo...
(KI Symmetrical bundle with N individual

subconductors
With these assumptions, the bundle can be treated as a single eSuivalent conductor in ES. ( .15) by replacing
GM4 with the eSuivalent geometric mean radius of the bundle,
0
)/4GSWKX ' 0 @ )/4 @ #0& ( .29)
where
GM4 geometric mean radius of individual subconductor in bundle,
# radius of bundle (Fig. .10).
Similarly, the radius r in ES. ( .20) must be replaced with the eSuivalent radius
0
TGSWKX ' 0 @ T @ #0& ( .30)
Comparison between methods 1 and 2
Both methods for bundling conductors give practically identical answers, at least in the example chosen for
this comparison. The example was a 500 kV three phase line with horizontal tower configuration, with phases 0
feet apart at an average height above ground of 50 feet. The symmetrical bundle consisted of subconductors spaced
1 inches apart. Conductor diameter 0.9 inches, dc resistance 0.1 S/mile, GM4 0.3 72 inches, rqƒ‡uˆ
7. 052 inches from ES. ( .30), and GM4 qƒ‡uˆ7. 1 3 inches from ES. ( .29). Table .2 compares the results
in the form of positive and zero seSuence parameters at 0 Hz. Obviously, the results are practically identical.
6CDNG Comparison between methods 1 and 2 for bundling
Positive and zero seSuence Method 1 (Bundling by matrix Method 2 (ESuivalent

parameters at 0 Hz reduction) conductors)
Page 82
4 (S/mile)
‚… 0.0 2223 0.0 2205
X (S/mile)
‚… 0.5339 0.53399
C (zF/mile)
‚… 0.021399 0.021397
4 (S/mile)
’q„ 0.317 0 0.3173
X (S/mile)
’q„ 2.00 5 2.00 5
C (zF/mile)
’q„ 0.013 5 0.013 55
4GFWEGF /CVTKEGU HQT 'SWKXCNGPV 2JCUG %QPFWEVQTU
For the case of Fig. .1, elimination of ground wires and bundling of subconductors reduces the 13 x 13
matrices for the individual conductors to x matrices for the phases, e.g., for the series impedances,
84 < ) 44 < )
45 <
)
46 <
)
47 <
)
48 <
) 49 +4
) ) ) ) ) )
85 < 59 +5
54 < 55 < 56 < 57 < 58 <
) ) ) ) ) )
86 < 69 +6
F 64 < 65 < 66 < 67 < 68 <
Traduciendo...
& '
FZ 87 ) ) ) ) ) ) +7
< 79
74 < 75 < 76 < 77 < 78 <
88 <) ) ) ) ) )
89
+8
84 < 85 < 86 < 87 < 88 <
89 ) ) ) ) ) ) +9
< 99
94 < 95 < 96 < 97 < 98 <
or in general,
F8RJCUG )
& ' [< ( .31)
FZ RJCUG] [+RJCUG]
and
F+RJCUG )
& ' LT[% ( .32)
FZ RJCUG] [8RJCUG]
For a three phase single circuit with phases #, B, C, ES. ( .31) would have the form
F8#
FZ <) ) )
#% +#
## < #$ <
F8$ ) ) )
& ' < $%
+$ ( .33)
FZ $# < $$ <
) ) ) +%
<
F8% %# < %$ < %%
FZ
The diagonal element < in ES.

ww( .33) is the series self impedance of phase k for the loop formed by phase k with
return through ground and ground wires, and the off diagonal element < is the series mutual
uw impedance between
phases i and k. The self impedance of phase k is not the positive seSuence impedance. To obtain impedances which
19
Page 83
come close to the positive seSuence values, we would have to assume symmetrical currents in ES. ( .33),
+$ ' C +# CPF +% ' C +# , YKVJ C ' GL E
and then express the voltage drop in phase # as a function of I only, 5
F8#
& '< ) ) ) ) )
( .3 a)
FZ #&U[OO+#, YKVJ < #&U[OO ' (< ## % C< #$ % C< #%)
and similarly for phases B and C,
F8$
& '< ) ) ) ) )
( .3 b)
FZ $&U[OO+$, YKVJ < $&U[OO ' (< $$%C< #$%C < $%)
F8% ) ) ) ) )
& '< ( .3 c)
FZ %&U[OO+%, YKVJ < %&U[OO ' (< %%%C < #%%C< $%)
The values of the three impedances < 5 …‘yy , < 6 …‘yy, < 7 …‘yy in ES. ( .3 ) are not exactly eSual, but their
average value is the positive seSuence impedance. Because of slight differences in the three values, the voltage drops
are slightly unsymmetrical (or the currents become slightly unsymmetrical for given symmetrical voltage drops).
#s discussed in Section .1.3, transposing a line eliminates or reduces these unsymmetries at power freSuency,
though not necessarily at higher freSuencies.
In the capacitance matrix of a three phase line, C would be

55 the sum of the coupling capacitances to phases
B and C and of the capacitance to ground, and C would be

56 the negative value of the coupling capacitance between
phases # and B. #ssuming symmetrical voltages, ES. ( .32) would show slight unsymmetry in [dI /dx], ‚tg…q
analogous to that of ES. ( .3 ).
0QOKPCNB %KTEWKV HQT 'SWKXCNGPV 2JCUG %QPFWEVQTU
The matrices in ES. ( .31) and ( .32) are the basis for practically all EMTP line models. Even in studies
where ground wires must be retained, it is still these matrices which are used, with phase numbers assigned to the
Traduciendo...
ground wires as well. # three phase line with one ground wire is conceptually a four phase line, with phase no. 1,
2, 3 for phase conductors #, B, C and phase no. for the ground wire.
One type of line representation uses cascade connections of nominal B circuits, as discussed in Sections
.2.1.1 and .2.2.1. This polyphase nominal B circuit with a series impedance matrix and eSual shunt capacitance
matrices at both ends, as shown in Fig. 3.10, is directly obtained from the matrices in ES. ( .31) and ( .32),
)
[4] % LT[.] ' ý @ [< ( .35)
RJCUG]
and
20
Page 84
1 1 )
LT[%] ' LTý [% ( .3 )
2 2 RJCUG]
where ý is the length of the line.
The cascade connection of nominal B circuits approximates the even distribution of the line parameters
reasonably well up to a certain freSuency. It does ignore the freSuency dependence of the resistances and inductances
per unit length, however, and is therefore reasonably accurate only within a certain freSuency range.
Strictly speaking, it may not be Suite correct to treat the real part of [< ] as a resistance, and the
‚tg…q
imaginary part as a reactance, as done in ES. ( .35), especially for lines with ground wires. For a three phase line
with phases #, B, C and ground wire g, the original x matrix is reduced to a 3 x 3 matrix with elements
) )
) ) <
< MI<
KI&QTKIKPCN &QTKIKPCN
KM&TGFWEGF ' <
KM&QTKIKPCN & ( .37)
<)
I I&QTKIKPCN
Even if < could

uw „usu€gx be separated into resistance and reactance without any doubt, the real part of the second term
in ES. ( .37) depends on the imaginary parts of the three impedances as well, unless the 4/X ratios of all three
impedances were eSual. There is also some doubt about separating < into
uw „usu€gx resistance and reactance because
of the earth as an implied return conductor, as mentioned in Section 3.1. Nonetheless, experience has shown that
nominal B circuits do give reasonable answers in many cases, and they are at least correct at the freSuency at which
the matrices were calculated (and probably reasonably accurate in a freSuency range around that specific freSuency).
Example for using nominal B circuits
Electrostatic and magnetic coupling effects from energized power lines to parallel objects, such as fences
or de energized power lines, are important safety issues, and have been well described in two IEEE Committee
4eports [37, 9]. # case of a fence running parallel to a power line (Fig. .11) is discussed here, as an application
$
example for nominal B circuits. By simply treating the fence as a fourth phase conductor, the following series
impedance and shunt capacitance matrices are obtained:
0. 05 %L .9 59 U[OOGVTKE!
) 0.057 %L . 2 5 0. 05 %L .9 59
[< S/MO
RJCUG] ' 0.057 %L . 2 5 0.057 %L .37 2 0. 05 %L .9 59
0.05 1%L .31 0.05 1%L .3291 0.05 1%L .30 1. 07%L .9953
and
$
For electrically short lines, as in this example, electrostatic coupling effects can be solved by themselves
with [C ], ‚tg…q
and magnetic coupling effects by themselves with [< ]. For solving such cases with the EMTP,
‚tg…q
it is usually easier to use nominal B circuits which combine both effects. With that approach, electrically long
lines can be studied as well, provided an appropriate number of B circuits are connected in cascade.
21
Traduciendo...
Page 85
7.5709 U[OOGVTKE
)
&1. 2 7.30
[% P(/MO
RJCUG] ' &1. 30 &0. 3 9 7.2999
&0.1 &0.275 &0.11 9 .9727
From these matrices, the nominal B circuit matrices are calculated with ES. ( .35) and ( .3 ).
Power line conductors:

4 u€†q„€gx
0.3 S/km
X 0.
5 755 S/km ( 0 Hz)
(reactance at 1 m spacing)
diameter 12.7 mm
freSuency 0 Hz
Fence:
4 u€†q„€gx
1. 02 S/km
solid conductor (nonmagnetic)
diameter .0 mm
length 2 km
(KI Fence running parallel with power line phase conductors 1, 2, 3
#ssume that the fence is insulated from the posts and nowhere grounded. To find the voltage on the fence
due to capacitive coupling, simply connect voltage sources to phases 1, 2, 3 at the sending end, and leave 1, 2, 3
at the receiving end as well as at both ends open ended. #ssuming V 3 5 kV 4MS, line to line, the fence
voltage becomes V 3.97

" kV. If phase 1 were at zero potential because of a phase to ground fault, with phases
2 and 3 still at rated voltage 3 5/%3 kV, then the fence voltage would increase to V . kV. These answers
" are
practically independent of fence length.
Now assume that the 2 km long fence is grounded at the sending end and open ended at the receiving end.
To find the voltage in the fence for a load current of 1 k# 4MS, simply add current sources to phases 1, 2, 3 at the
receiving end, with symmetrical voltage sources at the sending end. Phase is connected to ground at the sending
end and open ended at the receiving end. The answer will be V 0.0
" „qiquˆu€sÂq€p 3 kV, which increases dramatically
to . 2 kV if the currents are changed to I 10 k#, I I 0 to simulate a phase! to ground fault. For this last
case, the fence current would be 1.52 k# if the fence were grounded at both ends. These answers are practically
independent of the voltage on phases 1, 2, 3, which can easily be verified by setting them zero.
%QPVKPWQWU CPF 5GIOGPVGF )TQWPF 9KTGU
(a) Circulating Currents in Continuous Ground Wires
#ssume that ground wire no. 13 of Fig. .1 is grounded at each tower. If the ground wire is not eliminated,
22
Page 86
then the series impedance matrix for eSuivalent phase conductors will be a 7 x 7 matrix. Its elements can then be
used to calculate the longitudinally induced voltage in the ground wire,
F8I
& '< ) ) ) )
( .3 )
FZ I4 +4 % < I5 +U % ... < I9 +9 % < II +I
If tower and tower footing resistances are ignored, then V 1 at all towers
s as long as span wavelength, or
) ) )
<
+I ' & I4 +4 % < I5 +5 % ... < I9 +9
< )II ( .39)
Since the mutual impedances from the phase conductors to the ground wire are never exactly eSual, the numerator
in ES. ( .39) does not add up to zero even if the phase currents are symmetrical. Therefore, there is a nonzero
Traduciendo...
ground wire current I , produced

s by positive seSuence currents, which circulates through ground wire, towers and
ground (Fig. .12). This circulating current produces additional losses, which show up as an increase in the value
of the positive seSuence resistance, compared with the resistance of the phase conductors. Handbook formulas would
not contain this increase, but the elimination of the ground wires discussed in Section .1.2.1 will produce it
automatically. In one particular case of a single circuit 500 kV line, this increase was .5 .
(KI Circulating current in ground wire
The inclusion of tower and tower footing resistances may change the results of ES. ( .39) somewhat. If
we assume eSual resistance at all towers, then it appears that the voltage drop produced by the current in the left loop
(Fig. .13) is canceled by the voltage drop produced by the current in the middle loop, and ES. ( .39) should
therefore still be correct, except in the very first and very last span of the line. This assumes that the phase currents
do not change from one span to the next, which is reasonable up to a certain freSuency.
(KI Cascade connection of

loops
23
Page 87
(b) Segmented Ground Wires
To avoid the losses associated with these circulating currents, some utility companies use segmented ground
wires which are grounded at one tower, and insulated at adjacent towers to both ends of the segmentation interval,
where they are interrupted as well (Fig. .1 ).
T configuration
in segmentation interval
) insulator
(KI Segmented ground wires
They still act as electrostatic shields for lightning protection, but when struck by lightning, the segmentation gaps
and the small insulators will flash over, thereby making the ground wire continuous again. The supporting routine
%
LINE CONST#NTS has an option for segmented ground wires, which ignores them in the calculation of the series
impedance matrix since they have no influence on the voltage drops in the phase conductors, but takes them into
account in the calculation of the capacitance matrix because the electrostatic field is not influenced by segmentation.
(c) 4eduction Effect of Continuous Ground Wires on Interference
Interference from power lines in parallel telephone lines becomes a problem if there are high zero seSuence
currents in the power line, e.g., in case of a single phase to ground fault. #ssume a three phase line with one
ground wire g and a parallel telephone line P as shown in Fig. .15. For zero seSuence currents, which implies
eSual currents in phases #, B, C, the voltages in P induced by currents in #, B, C will add up in the same direction
Traduciendo...
(Fig. .1 ). The voltage induced by the ground wire current I will have opposite polarity, however, since this
s
%
#n exception are studies where it can be assumed that the gaps and insulators have flashed over. For such
studies, ground wires must be treated as continuous, as suggested by W.#. Lewis. Switching and lightning
surge studies may fall into this category.
Page 88
(KI Parallel telephone line P close to a

power line with phases #, B, C and ground
wire g
current flows in opposite direction, thereby reducing the total induced voltage dV /dx. Part of this beneficial
‚
reduction may be offset by an increase in the zero seSuence currents because ground wires also reduce the zero
seSuence impedance of lines (typically by 5 to 15 with one
dV /dx
P < IPs s
6< =
6<<<
< IP5 5 < IP6< 6I P7 7
(KI Induced voltage caused by currents I I I and by I 5 6 7 s
steel ground wire, or 15 to 30 with one #CS4 ground wire). The reduction effect of the ground wire on
interference can be included in the calculations in two different ways:
(a) Obtain the mutual impedances from matrices in which ground wires have been eliminated and in which the
parallel telephone lines has been retained as an additional conductor. Then the reduction effect of the
ground wires is automatically contained in calculating the magnetically induced voltage from
F82
& '< ) ) )
( . 0a)
FZ 2#&TGFWEGF+# % <
2$&TGFWEGF+$ % <
2%&TGFWEGF+%
and, if needed, the electrostatically induced voltage for an insulated parallel telephone line from
) ) ) )
0'% ( . 0b)
2#&TGFWEGF8#%%2$&TGFWEGF8$%%2%&TGFWEGF8%%%
22&TGFWEGF82
(b) Calculate the mutual impedances from P to the phases as well as to the ground wires (or obtain them from
matrices in which the ground wires were retained), and recover the value of the ground wire currents with
a screening matrix from the phase currents. By setting V 0 in ES. ( .27),s the ground wire currents
are obtained as
) & [<IW][+W]
[+I] ' &[< ( . 1)
I I]
[(UETGGP]
25
Traduciendo...
Page 89
with u indicating ungrounded phase currents here. The screening matrix [F ] is

…i„qq€ the transpose of the
distribution factor matrix [D ] of ES. (III.1 ) in #ppendix III, and as indicated there, can easily be obtained
as a by product of the matrix reduction process. #s an example, Fig. .17 shows the standing waves of
the phase currents of the sixth harmonic of 0 Hz in the two poles #, B of the Pacific Intertie HVDC line,
as well as the currents in the two ground wires recovered with ES. ( . 1) [11].
(KI Currents of sixth harmonic in HVDC line [11]. l

19 9 IEEE
2QUKVKXG CPF <GTQ 5GSWGPEG 2CTCOGVGTU QH $CNCPEGF .KPGU
# balanced transmission line shall be defined as a line where all diagonal elements of [< ] and [C ] ‚tg…q ‚tg…q
are eSual among themselves, and all off diagonal elements are eSual among themselves,
<) ) )
O %) ) )
O
U< O ..... < U% O ..... %
) ) ) ) ) )
< O % O
O< U ..... < O% U ..... %
. . . . . . . .
( . 2)
. . . . . . . .
. . . . . . . .
<) ) )
U %) ) )
U
O< O ..... < O% O ..... %
&
#lso called continuously transposed in the EMTP 4ule Book.
Page 90
(KI Bipolar dc line
Traduciendo...
The only line which is truly balanced is the symmetric bipolar dc line (Fig. .1 ), where < < < and < …
< . Single
y circuit three phase lines become more or less balanced if the line is transposed, as shown in Fig.
.19, provided the length of the barrel ( 3 sections, or one cycle of the transposition scheme) is much less than
the wavelength of the freSuencies involved in the particular study. While the Westinghouse 4eference Book [51,
p. 777] mentions that a barrel may be 0 to 1 0 km in length on long lines, a German handbook [52, p. 555]
recommends that one barrel be no longer than 0 km (at 50 Hz, or 7 km at 0 Hz) for lines with triangular
conductor configuration, or 0 km (at 50 Hz, or 33 km at 0 Hz) for other conductor configurations. Whatever the
length of the barrel, it is important to realize that while
(KI Transposition scheme for single three phase

circuit
'
a line may be reasonably balanced at power freSuency, there may be enough unbalance at higher freSuencies . If
the barrel length is much shorter than the wavelength, then series impedances can be averaged by themselves through
the three sections, and shunt capacitances can be averaged by themselves, e.g., for the impedances of the line in Fig.
.19,
) ) ) ) ) ) ) ) ) ) ) )
< KO < MK < OM < O
KK < KM < MM < MO < OO < OK < U< O<
1 ) ) ) ) ) ) ) ) ) ) ) )
< % < OK % < ' <
3 MK < MM < MO OM < OO < KO < KK < KM O< U< O
<) ) ) <) ) ) <) ) ) <) ) )

OK < OM < OO KM < KO < KK MO < MK < MM O< O< U
with
'
#t the time of writing, studies at B.C. Hydro seem to indicate that transposed single circuit lines with
horizontal conductor configuration cannot be treated as balanced lines in switching surge studies.
27
Page 91
) 1 ) ) )
< (<
U' KK % <MM % <OO)
3
) 1 ) ) )
< (< ( . 3)
O' 3 KM % <MO % < OK)
The averaging process for the shunt capacitances is analogous.
2QUKVKXG CPF <GTQ 5GSWGPEG 2CTCOGVGTU QH 5KPING %KTEWKV 6JTGG 2JCUG .KPGU
Balanced single circuit three phase lines can be studied much easier with symmetrical or ", $, 0 components
because the three coupled eSuations in the phase domain,
) ) )
< O
U< O<
F8RJCUG ) ) )
& ' < O +RJCUG (.)
FZ O< U<
) ) )
< U
O< O<
become three decoupled eSuations with symmetrical components,
)
&F8\GTQ/FZ ' <
\GTQ+\GTQ
)
&F8PGI/FZ ' <
) RQU+PGI
&F8RQU/FZ ' < ( . 5)
RQU+RQU
or with ", $, 0 components,

Traduciendo...
)
&F8\GTQ/FZ ' <
\GTQ+\GTQ
)
&F8 " /FZ ' < (.)
RQU"+
)
&F8 $/FZ ' <
RQU$
+
Since transformation to symmetrical components involves complex coefficients, symmetrical components
are not well suited for transient analysis where all variables are real, and are therefore only briefly discussed in
Section .1. . The impedances needed in both systems ( . 5) and ( . ) are the same, however, namely < and ’q„
< . ‚The
… balanced distributed parameter line models in the EMTP use transformations to ", $, 0 components, due
to Edith Clarke [ ],
& [XRJCUG]
[XRJCUG] ' [6][X "$ ] [X "$ ] ' [6]
and
Page 92
[KRJCUG] ' [6][K "$ ] [K "$ ] ' [6] & [KRJCUG]

( . 7)
where
[X "$ ] ' X"
X$
with
1 2 0
1 3
1 1&
[6] ' 2 2
3
1 3
1& &
2 2
and
1 1 1
1 1
2& &
&' 1 2 2
[6] (.)
3
3 3
0 &
2 2
The columns in [T] and [T] are normalized in that case [T] is orthogonal,
[6] & ' [6]V ( . 9)
#pplying this transformation to ES. ( . ) produces
) )
F8 /FZ < 0 0 +
U%2< O
) )
& F8 " /FZ ' 0 < 0 +"
U&< O
F8 $/FZ 0 0 <) ) +$
U&< O
which is identical with ES. ( . ), with
) ) )
< ( .50a)
\GTQ ' < U% 2< O
Traduciendo...
29
Page 93
ES. ( .50) and its inverse relationship is the same as discussed previously in ES. (3. ) and (3. ). Going from the
) ) )
< O ( .50b)
RQU ' < U& <
three coupled eSuations in ( . ) to the three decoupled eSuations in ( . ) allows us to solve the line as if it
consisted of three single phase lines, which is much simpler than trying to solve the eSuations of a three phase line.
The positive seSuence inductance of overhead lines is practically constant, while the positive seSuence
resistance remains more or less constant until skin effect in the conductors becomes noticeable, as shown in Fig.
.20. <ero seSuence inductance and resistance are very much freSuency dependent, due to skin effects in the earth
return.
(KI Positive and zero seSuence resistance and inductance of a three phase line
The shunt capacitance matrix of a balanced three phase line becomes diagonal in ", $, 0 components as
well, with
) ) )
% O ( .51a)
\GTQ ' % U% 2%
) ) )
% O ( .51b)
RQU ' % U& %
30
Page 94
which is the inverse relationship of ES. (3.13). The capacitances are constant over the freSuency range of interest
to power engineers.
Comparison with results from handbook formulas
The positive and zero seSuence parameters obtained from the supporting routines LINE CONST#NTS and
Traduciendo...
C#BLE CONST#NTS may differ from those obtained with handbook formulas. Since some EMTP users may make
comparisons, it may be worthwhile to explain the major differences for a specific example. #ssume a typical 500
kV line with horizontal phase configuration, with phases 0 feet apart at an average height above ground of 50 feet.
Each phase consists of a symmetrical bundle with subconductors spaced 1 inches apart. Subconductor diameter
0.9 inches, dc resistance 0.1 S/mile, GM4 0.3 72 inches. Throughout this comparison, the bundle
conductors are represented as eSuivalent conductors with r 7. 052 inches

qƒ‡uˆfrom ES. ( .30) and GM4 qƒ‡uˆ
7. 1 3 inches from ES. ( .29).
For positive seSuence capacitance, most handbooks give the formula
) 2Bg
%
RQU ' FO ( .52)
ln
T/
!
with d %d
y d d (geometric
56 57 67 mean distance among the three phases).
This produces a value approx. lower than the more accurate value from ES. ( .51) for the 500 kV line described
above. The formula for zero seSuence capacitance in [52] and [53],
) 2Bg
% (5KGOGPU)
\GTQ '
2JO&O ( .53)
ln
TGSWKXFO
with
h y%h h!h 567 (geometric mean height),
D y%D D! D 56 57 67 (geometric mean distance between one phase and image of another phase),
can be derived by averaging the diagonal and off diagonal elements in the [P ] matrix among themselves
‚tg…q to
account for transposition. ES. ( .51) has this averaging process implied in the [C ] matrix. Both ‚tg…q
give practically
the same answer, with results from ES. ( .53) 0.23 lower than those from ES. ( .51). In [51], ES. ( .53) is
further simplified by assuming D . 2h , y y
) 2Bg
% (9GUVKPIJQWUG)
\GTQ '
(2JO) ( .5 )
ln
TGSWKXFO
which produces a value higher than the value from ES. ( .51). While ES. ( .5 ) is theoretically less accurate,
the value may actually be closer to measured values because the influence of towers, which is neglected in all
31
Page 95
formulas, typically increases the calculated zero seSuence capacitance by about to 9 on 110 kV lines, about
on 220 and 3 0 kV lines, and about on 700 kV lines [5 , p. 21 ].
The formulas for zero and positive seSuence impedances in most handbooks are based on the assumption
that parameter a in ES. ( .10) is so small that only the first term in the series of ES. ( .11) must be retained. For
normal phase spacings this is probably a reasonable assumption at power freSuency 50 or 0 Hz. Then, after all
diagonal and off diagonal elements have been averaged out among themselves through transposition,
&
TB@10
)4 ) ) KP S/MO
U' )4 O. 2
and
) & [0. 159315 & ln(2JOM H

): )] KP S/MO ( .55)
U. 2T@10
D
) & [0. 159315 & ln(&OM H

): )] KP S/MO
O. 2T@10
D
with
&
M ' B @ 5 @ 10
Traduciendo...
This leads to the expression
) ) & ln
FO
< KP S/MO ( .5 )
RQU ' 4 CE % L2T @ 10 )/4GSWKX
with 4 acgiresistance of eSuivalent phase conductor. It is interesting that the influence of ground resistivity and
of conductor height, which is present in < and < , completely

… ydisappears here in taking the difference, < ‚…
< <…. ES. y( .5 ) is the formula found in most handbooks. Table .3 compares results from ES. ( .50) with
results from ES. ( .5 ) for the 500 kV line described above with the following additional assumptions: Earth
resistivity 100 Sm skin effect within conductors ignored to limit differences to influence of earth return (that is,
4 4giand GM4
pi qƒ‡uˆ constant).
6CDNG #ccurate and approximate positive seSuence resistance and inductance
#CCU4#TE #PP4OXIM#TE
4 and
‚ … L from ES.
‚ … ( .50) 4 and
‚ … L from ES.
‚ … ( .5 )
f 4 L 4 L
(Hz) (S/mile) (mH/mile) (S/mile) (mH/mile)
32
Page 96
10 $ 0.0 215 1. 17 0.0 215 1. 17

10 0.0 215 1. 1 0.0 215 1. 17
100 0.0 229 1. 1 0.0 215 1. 17
1 000 0.05003 1. 1 0.0 215 1. 17
10 000 0.352 1. 13 0.0 215 1. 17
100 000 .229 1. 01 0.0 215 1. 17
Table .3 shows that L from ‚ES.

… ( .5 ) is Suite accurate over a wide freSuency range, whereas 4 becomes less ‚…
accurate as the freSuency increases (0.33 error at 100 Hz, but wrong by orders of magnitude at 100 kHz). The
increase in 4 in the
‚ … higher freSuency range is caused by eddy currents in the earth, as indicated in Fig. .21 for
a bipolar dc line. Ground wires also influence the positive seSuence impedance, as mentioned in Section .1.2.5
(a). Both influences are ignored in ES. ( .5 ), but automatically included in the method described here.
(KI Eddy currents in earth
The zero seSuence impedance obtained from ES. ( .55) is
D
5.7
) ) 3TB@10& H ( .57)
< ) % L T@10 & ln KP S/MO
\GTQ ' (4 CE% 2
3 )/4GSWKX@F O
with f in Hz, D in Sm, and GM4 qƒ‡uˆ and d in

y m. ES. ( .57) is the same eSuation as in [51, 52, 53]. Table .
compares the approximate results from ES. ( .57) with the accurate results from ES. ( .50). The inductance L’q„
is reasonably accurate over a wide freSuency range ( 0.75 error at 100 Hz, 33 error at 100 kHz), but the
resistance 4 is less
’q„ accurate ( . error at 100 Hz, 159 error at 100 kHz).
6CDNG #ccurate and approximate zero seSuence resistance and inductance
#CCU4#TE #PP4OXIM#TE
4 and
’q„ L from ES.
’q„ ( .50) Traduciendo...4 and
’q„ L from ES.
’q„ ( .57)
f 4 L 4 L
(Hz) (S/mile) (mH/mile) (S/mile) (mH/mile)
33
Page 97
10 $ 0.0 215 13.9 0.0 215 13.9

10 0.0 905 .170 0.0 9 0 .15
100 0. 9 0 5.0 0.051 7 5.0
1 000 .1 9 .052 . 07 3.93
10 000 32.12 3.1 7. 9 2. 23
100 000 1 .0 2.5 7. 1.711
2QUKVKXG CPF <GTQ 5GSWGPEG 2CTCOGVGTU QH $CNCPEGF / 2JCUG .KPGU
The EMTP can handle balanced distributed parameter lines not only for the case of a three phase line, but
for any number of phases M. For this general case, the ", $, 0 transformation of ES. ( . 7) has been generalized
to M phases, with the transformation matrix [55]
1 1 1 1 1
.... ...
/ 2 ,(,&1) /(/&1)
1 1 1 1 1
& .... ...
/ 2 ,(,&1) /(/&1)
1 2
0 & . . . .
/
[6] ' ( .5 )
(,&1)
. . . .& . .
,(,&1)
. . . . 0 . .
. . . . . . .
1 (/&1)
0 0 . 0 .&
/ /(/&1)
where again
[6] & ' [6]V ( .59)
[T] of ES. ( . ) is a special case of ES. ( .5 ) for M 3 if we assume that the phases are numbered 2, 3, 1 in ES.
( . 7) and if the ", $, 0 Suantities are ordered 0, $, " (sign reversal on ").
#pplying this M phase ", $, 0 transformation to the matrices of M phase balanced lines produces diagonal matrices
of the form
In the UBC EMTP, and in older versions of the BP# EMTP, Karrenbauer s transformation [57] is used
instead, which produces the same diagonal matrices, but does not have the property of ES. ( .59). This property
is important because it makes the balanced line just a special case of the untransposed line discussed in Section
.1.5.
Page 98
Z’zero
Z’pos
Z’pos
Traduciendo...
.
.
.
Z’ pos
with the first diagonal element being the zero sequence (ground mode) impedance, and the next M-1 diagonal elements
being the positive sequence (aerial mode) impedance,
) ) )
< O ( . 0a)
\GTQ ' < U% (/ & 1)<
) ) )
< O ( . 0b)
RQU ' < U& <
and similarly for the capacitances,
) ) )
% O ( . 1a)
\GTQ ' % U% (/ & 1)%
) ) )
% O ( . 1b)
RQU ' % U& %
To refer to the two distinct diagonal elements as zero and positive sequence may be confusing, because the
concept of sequence values has primarily been used for three-phase lines. "Ground mode" and "aerial mode" may be
more appropriate. Confusion is most likely to arise for double-circuit three-phase lines, where each three-phase line
has its own zero and positive sequence values defined by Eq. (4.50) and (4.51) with symmetrical components used for
each three-phase circuit, while in the context of this section the double-circuit line is treated as a six-phase line with
different zero and positive sequence values defined by Eq. (4.60) and (4.61). The fact that the terms zero and positive
sequence are used for M û 3 as well comes from the generalization of symmetrical components of Section 4.1.4 to M
phases with the transformation matrix [56, p. 155]
U U ... U /
U U ... U /
[5/&RJCUG] ' ( . 2a)
. .. .
U/ U/ ... U//
35
Page 99
with
1 2B
UKM ' exp6&L (K&1)(M&1)> ( . 2b)
/ /
11
A special case of interest for symmetric bipolar dc lines is M = 2. In this case [T] of Eq. (4.58) and [S] of
Eq. (4.62a) are identical,
1 11
[6 &RJCUG] ' ( . 3)
2 1 &1
4.1.3.3 Two Identical Three-Phase Lines with Zero Sequence Coupling Only
Just as a transposed single-circuit three-phase line can usually be approximated as a balanced line, so two
identical and parallel three-phase lines can often be approximated as "almost balanced" lines with an impedance matrix
of the form
<) ) ) ) ) )
R
U< O< O< R< R<
) ) ) ) ) )
< R
O< U< O< R< R<
Traduciendo...
<) ) ) ) ) )
R
O< O< U< R< R<
(.)
) ) ) ) ) )
< O
R< R< R< U< O<
) ) ) ) ) )
< O
R< R< R< O< U<
<) ) ) ) ) )
U
R< R< R< O< O<
The transposition scheme of Fig. 4.22 would produce such a matrix form, which implies that the two circuits are only
coupled in zero sequence, but not in positive or negative sequence. Such a complicated transposition scheme is seldom,
if ever, used, but the writer suspects that positive and negative sequence couplings in the more common double-circuit
transposition scheme of Fig. 4.23 is often so weak that the model discussed here may be a useful approximation for the
case of Fig. 4.23 as well.
To be consistent, lines with M 1 and M 2 are called single phase and two phase lines,
respectively, in this manual. This differs from the IEEE Standards [7 , p. 7], in which circuits with one phase
conductor and one neutral conductor (which could be replaced by ground return), as well as circuits with two
phase conductors and one neutral conductor (or ground return) are both called single phase circuits for historical
reasons. For M $ 3, the definition in the IEEE Standards is the same as in this manual.
Page 100
(KI Double circuit transposition scheme with zero seSuence coupling

only
The
matrix of Eq. (4.64) is diagonalized by modifying the transformation matrix of Eq. (4.58) to
11 31 0 0
11&31 0 0
1 11 0 &2 0 0
[6] ' ( . 5)
1 &1 0 0 31
1 &1 0 0&31
1 &1 0 0 0 &2
-1 t
with [T] = [T] again, which produces the diagonal matrix
Z’ G
Z’IL
Z’L
Z’ L (4.66)
Z’L
Z’L
If each circuit has three-phase sequence parameters Z’ , Z’ , and

zero if the
pos three-phase zero sequence coupling between
the two circuits is Z’ zero-coupling , then the ground mode G, inter-line mode IL and line mode L values required by the EMTP
are found from
<) ) )
)' < \GTQ % < \GTQ&EQWRNKPI
) ) )
< ( . 7)
+. ' < \GTQ & < \GTQ&EQWRNKPI
<) )
RQU
.' < Traduciendo...
with identical equations for the capacitances.
If the two three-phase circuits are not identical, then the transformation matrix of Eq. (4.65) can no longer be
used; instead, [T] depends on the particular tower configuration.
37
Page 101
4.1.4 Symmetrical Components
Symmetrical components are not used as such in the EMTP, except that the parameters of balanced lines after
transformation to M-phase , , 0-components are the same as the parameters of symmetrical components, namely zero
and positive sequence values. The supporting routine LINE CONSTANTS does have output options for more detailed
symmetrical component information, however, which may warrant some explanations.
In addition to zero and positive sequence values, LINE CONSTANTS also prints full symmetrical component
matrices. Its diagonal elements are the familiar zero and positive sequence values of the line; they are correct for the
untransposed line as well as for a line which has been balanced through proper transpositions. The off-diagonal
elements are only meaningful for the untransposed case, because they would become zero for the balanced line. For
the untransposed case, these off-diagonal elements are used to define unbalance factors [47, p. 93]. The full symmetrical
component matrices are no longer symmetric, unless the columns for positive and negative sequence are exchanged [27].
This exchange is made in the output of the supporting routine LINE CONSTANTS with rows listed in order "zero, pos,
neg,..." and columns in order "zero, neg, pos,...". With this trick, matrices can be printed in triangular form (elements
in and below the diagonal), as is done with the matrices for individual and equivalent phase conductors.
Symmetrical components for two-phase lines are calculated with the transformation matrix of Eq. (4.63), while
those of three-phase lines are calculated with
& [XRJCUG]
[XRJCUG] ' [5][XU[OO] CPF [XU[OO] ' [5] ( . a)
identical for currents,
X\GTQ
YJGTG [XU[OO] ' XRQU
XPGI
111
1
[5] ' 1 CC
3
1CC
111
1
[5] & ' 1CC ( . b)
3
1 CC
and a = e . j120E
12
The columns in these matrices are normalized ; in that form, [S] is unitary,
[5] & ' [5 Â ]V ( . 9)
The electric utility industry usually uses unnormalized transformation, in which the factor for the [S] matrix
is 1 instead of 1 / %3, and for the [S] matrix 1/3 instead of 1 / %3. The symmetrical component impedances are
identical in both cases, but the seSuence currents and voltages differ by a factor of %3.
Page 102
Traduciendo...
where "*" indicates conjugate complex and "t" transposition.
For M $ 3, the supporting routine LINE CONSTANTS assumes three-phase lines in parallel. Examples:
M = 6: Two three-phase lines in parallel
M = 9: Three-phase lines in parallel
M = 8: Two three-phase lines in parallel, with equivalent phase conductors no. 7 and 8 ignored in the
transformation to symmetrical components.

The matrices are then transformed to three-phase symmetrical components and not to M-phase symmetrical components
of Eq. (4.62). For example for M = 6 (double-circuit three-phase line),
&
) [5] 0 ) [5] 0
[< < RJCUG
( .70)
U[OO] ' 0 [5] & 0 [5]
with [S] defined by Eq. (4.68), Eq. (4.70) produces the three-phase symmetrical component values required in Eq.
(4.67).
Balancing of double-circuit three-phase lines through transpositions never completely diagonalizes the
respective symmetrical component matrices. The best that can be achieved is with the seldom-used transposition scheme
of Fig. 4.22, which leads to
)
< 0 0 <\GTQ&EQWRNKPI0 0
\GTQ&+
)
0 < 0 0 0 0
RQU&+
)
0 0 < RQU&+
0 0 0
[< )
U[OO] ' )
<\GTQ&EQWRNKPI0 0 < 0 0
\GTQ&++
)
0 0 0 0 < 0
RQU&++
)
0 0 0 0 0 <
RQU&++
(4.71)
If both circuits are identical, then Z’ = Z’ = Z’ , and

zero-I zero-IIZ’ = Z’
zero= Z’ ; in that
pos-I case,pos-II
the transformation
pos matrix
of Eq. (4.65) can be used for diagonalization. The more common transposition scheme of Fig. 4.23 produces positive
and zero sequence coupling between the two
39
Page 103
(a) barrels rolled in (b) barrels rolled in

opposite direction same direction
Fig. 4.23 - Double-circuit transposition scheme
circuits as well, with the nonzero pattern of the matrix in Eq. (4.71) changing to
: 0 0*: 0 0
0 : 0*0 : 0
0 0 :*0 0 :
: 0 0*: 0 0
0 : 0*0 : 0
0 0 :*0 0 :
where "X" indicates nonzero terms. Re-assigning the phases in Fig. 4.23(b) to CI, BI, AI, AII, BII, CII from top to
bottom would change the matrix further to cross-couplings between positive sequence of one circuit and negative
Traduciendo...
sequence of the other circuit, and vice versa,
: 0 0*: 0 0
0 : 0*0 0 :
0 0 :*0 : 0
.
: 0 0*: 0 0
0 0 :*0 : 0
0 : 0*0 0 :
4.1.5 Modal Parameters
From the discussions of Section 4.1.3 it should have become obvious that the solution of M-phase transmission
line equations becomes simpler if the M coupled equations can be transformed to M decoupled equations. These
decoupled equations can then be solved as if they were single-phase equations. For balanced lines, this transformation
is achieved with Eq. (4.58).
Many lines are untransposed, however, or each section of a transposition barrel may no longer be short
compared with the wave length of the highest frequencies occurring in a particular study, in which case each section
must be represented as an untransposed line. Fortunately, the matrices of untransposed lines can be diagonalized as
well, with transformations to "modal" parameters derived from eigenvalue/eigenvector theory. The transformation
matrices for untransposed lines are no longer known a priori, however, and must be calculated for each particular pair
of parameter matrices [Z’ ] and

phase[Y’ ]. phase
To explain the theory, let us start again from the two systems of equations (4.31) and (4.32),
Page 104
F8RJCUG
& ' [< ) ( .72a)
FZ RJCUG] [+RJCUG]
and
F+RJCUG )
& ' [; ( .72b)
FZ RJCUG][8RJCUG]
with [Y’ phase

] = j [C’ ] if shunt
phase conductances are ignored, as is customarily done. By differentiating the first equation
with respect to x, and replacing the current derivative with the second equation, a second-order differential equation for
voltages only is obtained,
F 8RJCUG ) )
' [< ( .73a)
RJCUG] [; RJCUG] [8RJCUG]
FZ
Similarly, a second-order differential equation for currents only can be obtained,
F +RJCUG ) )
' [; ( .73b)
RJCUG] [<RJCUG] [+RJCUG]
FZ
where the matrix products are now in reverse order from that in Eq. (4.73a), and therefore different. Only for balanced
matrices, and for the lossless high-frequency approximations discussed in Section 4.1.5.2, would the matrix products
in Eq. (4.73a) and (4.73b) be identical.
With eigenvalue theory, it becomes possible to transform the two coupled equations (4.73) from phase
quantities to "modal" quantities in such a way that the equations become decoupled, or in terms of matrix algebra, that
the associated matrices become diagonal, e.g., for the voltages,
F 8OQFG
' [7] [8OQFG] ( .7 )
FZ
with [ ] being a diagonal matrix. To get from Eq. (4.73a) to (4.74), the phase voltages must be transformed to mode
voltages, with
Traduciendo...
[8RJCUG] ' [6X] [8OQFG] ( .75a)
and
[8OQFG] ' [6X] & [8RJCUG]

( .75b)
Then Eq. (4.73a) becomes
F 8OQFG ) )
' [6X] & [< ( .7 a)
RJCUG] [; RJCUG] [6X] [8OQFG]
FZ
Page 105
which, when compared with Eq. (4.74), shows us that
[7] ' [6X] & [< ) )

( .7 b)
RJCUG] [; RJCUG] [6X]
To find the matrix [T ] which

v diagonalizes [Z’ ][Y’ phase
] is the eigenvalue/eigenvector
phase problem. The diagonal elements
of [ ] are the eigenvalues of the matrix product [Z’ ][Y’ ], and

phase [T ] phase
is the matrix of
v eigenvectors or modal matrix
of that matrix product. There are many methods for finding eigenvalues and eigenvectors. The most reliable method
for finding the eigenvalues seems to be the QR-transformation due to Francis [3], while the most efficient method for
the eigenvector calculation seems to be the inverse iteration scheme due to Wilkinson [4, 5]. In the supporting routines
LINE CONSTANTS and CABLE CONSTANTS, the "EISPACK"-subroutines [67] are used, in which the eigenvalues
and eigenvectors of a complex upper Hessenberg matrix are found by the modified LR-method due to Rutishauser. This
method is a predecessor of the QR-method, and where applicable, as in the case of positive definite matrices, is more
efficient than the QR-method [68]. To transform the original complex matrix to upper Hessenberg form, stabilized
elementary similarity transformations are used. For a given eigenvalue , the corresponding
k eigenvector [t ] (= k-th vk
column of [T ])vis found by solving the system of linear equations
6[< ) )
( .77)
RJCUG] [; RJCUG] & 8M[7]> [VXM] ' 0
with [U] = unit or identity matrix. Eq. (4.77) shows that the eigenvectors are not uniquely defined in the sense that they
13
can be multiplied with any nonzero (complex) constant and still remain proper eigenvectors , in contrast to the
eigenvalues which are always uniquely defined.
Floating-point overflow may occur in eigenvalue/eigenvector subroutines if the matrix is not properly scaled.
Unless the subroutine does the scaling automatically, [Z’ ][Y’ ] should
phase be scaled
phase before the subroutine call, by
2
dividing each element by -( g µ ), as suggested
00 by Galloway, Shorrows and Wedepohl [39]. This division brings
2
the matrix product close to unit matrix, because [Z' ][Y' ]phase
is a diagonal
phase matrix with elements - g µ if resistances, 00
internal reactances and Carson's correction terms are ignored in Eq. (4.7) and (4.8), as explained in Section 4.1.5.2. The
2
eigenvalues from this scaled matrix must of course be multiplied with - g µ to obtain 0the
0 eigenvalues of the original
matrix. In [39] it is also suggested to subtract 1.0 from the diagonal elements after the division; the eigenvalues of this
modified matrix would then be the p.u. deviations from the eigenvalues of the lossless high-frequency approximation
of Section 4.1.5.2, and would be much more separated from each other than the unmodified eigenvalues which lie close
together. Using subroutines based on [67] gave identical results with and without this subtraction of 1.0, however.
In general, a different transformation must be used for the currents,
[+RJCUG] ' [6K] [+OQFG] ( .7 a)
!
This is important if matrices [T ] obtained
ˆ from different programs are compared. The ambiguity can be
removed in a number of ways, e.g., by agreeing that the elements in the first row should always be 1.0, or by
normalizing the columns to a Euclidean vector length of 1.0, that is, by reSuiring t t t t ... 1.0, ˆˆ ˆˆ
with t conjugate complex of t. In the latter case, there is still ambiguity in the sense that each column could
v" have vector length 1.0.
be multiplied with a rotation constant e and still
Page 106
Traduciendo...
and
[+OQFG] ' [6K] & [+RJCUG] ( .7 b)
because the matrix products in Eq. (4.73a) and (4.73b) have different eigenvectors, though their eigenvalues are
identical. Therefore, Eq. (4.73b) is transformed to
F +OQFG
' [7] [+OQFG] ( .79)
FZ
with the same diagonal matrix as in Eq. (4.74). While [T ] is different

i from [T ], both are fortunately
v related to each
other [58],
V&
[6K] ' [6 ( . 0)
X]
where "t" indicates transposition. It is therefore sufficient to calculate only one of them.
Modal analysis is a powerful tool for studying power line carrier problems [59-61] and radio noise interference
[62, 63]. Its use in the EMTP is discussed in Section 4.1.5.3. It is interesting to note that the modes in single-circuit
three-phase lines are almost identical with the , , 0-components of Section 4.1.3.1 [58]. Whether the matrix products
in Eq. (4.73) can always be diagonalized was first questioned by Pelissier in 1969 [64]. Brandao Faria and Borges da
Silva have shown in 1985 [65] that cases can indeed be constructed where the matrix product cannot be diagonalized.
It is unlikely that such situations will often occur in practice, because extremely small changes in the parameters (e.g.,
in the 8th significant digit) seem to be enough to make it diagonalizable again. Paul [66] has shown that diagonalization
can be guaranteed under simplifying assumptions, e.g., by neglecting conductor resistances.
The physical meaning of modes can be deduced from the transformation matrices [T ] and [T ]. Assume,
v fori
example, that column 2 of [T ] has

i entries of (-0.6, 1.0, -0.4). From Eq. (4.78a) we would then know that mode-2 current
flows into phase B in one way, with 60% returning in phase A and 40% returning in phase C.
4.1.5.1 Line Equations in Modal Domain
With the decoupled equations of (4.74) and (4.79) in modal quantities, each mode can be analyzed as if it were
a single-phase line. Comparing the modal equation
F 8OQFG&M
' 8M 8OQFG&M
FZ
with the well-known equation of a single-phase line,
F8
' (8
FZ
with the propagation constant defined in Eq. (1.15), shows that the modal propagation constant mode-k is the square
Page 107
root of the eigenvalue,
(OQFG&M ' "M % L$M ' (M ( . 1)
with
k = attenuation constant of mode k (e.g., in Np/km),
k = phase constant of mode k (e.g., in rad/km).
The phase velocity of mode k is
RJCUG XGNQEKV[ 'T ( . 2a)
$M
and the wavelength is Traduciendo...
2B
YCXG NGPIVJ ' ( . 2b)
$M
While the modal propagation constant is always uniquely defined, the modal series impedance and shunt
admittance as well as the modal characteristic impedance are not, because of the ambiguity in the eigenvectors.
Therefore, modal impedances and admittances only make sense if they are specified together with the eigenvectors used
in their calculation. To find them, transform Eq. (4.72a) to modal quantities
F8OQFG
& ' [6X] & [< ) ( . 3)
FZ RJCUG] [6K] [+OQFG]
The triple matrix product in Eq. (4.83) is diagonal, and the modal series impedances are the diagonal elements of this
matrix
[< ) & [< )

( . a)
OQFG] ' [6X] RJCUG] [6K]
or with Eq. (4.80),
) V] [<)
[< ( . b)
OQFG] ' [6K RJCUG] [6K]
Similarly, Eq. (4.72b) can be transformed to modal quantities, and the modal shunt admittances are then the diagonal
elements of the matrix
) & [; )
[; ( . 5a)
OQFG] ' [6K] RJCUG] [6X]
or with Eq. (4.80),
) V] [; )
[; ( . 5b)
OQFG] ' [6X RJCUG] [6X]
Page 108
The proof that both [Z’ ] and [Y’ ] are diagonal

mode mode is given by Wedepohl [58]. Finally, the modal characteristic
impedance can be found from the scalar equation
<)
<EJCT&OQFG&M ' OQFG&M
( . a)
; )
OQFG&M
or from the simpler equation
(OQFG&M
<EJCT&OQFG&M ' ( . b)
; )
OQFG&M
A good way to obtain the modal parameters may be as follows: First, obtain the eigenvalues and the k
eigenvector matrix [T ] of
v the matrix product [Z’ ][Y’ phase
]. Then find
phase [Y’ ] from Eq. (4.85b),
mode and the modal series
impedance from the scalar equation
) 8M
< ( . c)
OQFG&M ' )
;
OQFG&M
The modal characteristic impedance can then be calculated from Eq. (4.86a), or from Eq. (4.86b) if the propagation
constant from Eq. (4.81) is needed as well. If [T ] is needed,

i too, it can be found efficiently from Eq. (4.85a)
[6K] ' [; ) ) &

RJCUG] [6X] [; OQFG]
( . 5c)
because the product of the first two matrices is available anyhow when [Y’ ] is found,
mode and the post-multiplication with
-1
[Y’ mode
] is simply a multiplication of each column with a constant (suggested by Luis Marti). Eq. (4.85c) also
i
establishes the link to an alternative formula for [T ] mentioned in [57],
)
[6K] ' [;
RJCUG] [6X] [&]
-1 t -1
with [D] being an arbitrary diagonal matrix. Setting [D] = [Y’ ] leads usmode
to Traduciendo...
the desirable condition [T] = [T ] of i v
Eq. (4.80). If the unit matrix were used for [D], all modal matrices in Eq. (4.84) and (4.85) would still be diagonal, but
with the strange-looking result that all modal shunt admittances become 1.0 and that the modal series impedances
become . Eq.
k (4.80) would, of course, no longer be fulfilled. For a lossless line, the modal series impedance would
then become a negative resistance, and the modal shunt admittance would become a shunt conductance with a value of
1.0 S. As long as the case is solved in the frequency domain, the answers would still be correct, but it would obviously
be wrong to associate such modal parameters with
MX ) MK )
& '4 K CPF & ') X
MZ MZ
(with R’ negative and G’ = 1.0) in the time domain.
Page 109
4.1.5.2 Lossless High-Frequency Approximation
In lightning surge studies, many simplifying assumptions are made. For example, the waveshape and amplitude
of the current source representing the lightning stroke is obviously not well known. Similarly, flashover criteria in the
form of volt-time characteristics or integral formulas [8] are only approximate. In view of all these uncertainties, the
use of highly sophisticated line models is not always justified. Experts in the field of lightning surge studies normally
use a simple line model in which all wave speeds are equal to the speed of light, with a surge impedance matrix [Z ] surge
in phase quantities, where
<KK&UWTIG ' 0 ln(2JK/TK) ( . 7a)
<KM&UWTIG ' 0 ln(&KM/FKM) ( . 7b)
CNN YCXG URGGFU ' URGGF QH NKIJV ( . 7c)
with r being
i the radius of the conductor, or the radius of the equivalent conductor from Eq. (4.30) in case of a bundle
conductor. 14
Typically, each span between towers is represented separately as a line, and only a few spans are normally
modelled (3 for shielded lines, or 18 for unshielded lines in [8]). For such short distances, losses in series resistances
and differences in modal travel times are negligible. The effect of corona is sometimes included, however, by modifying
the simple model of Eq. (4.87) [8].
It is possible to develop a special line model based on Eq. (4.87) for the EMTP, in which all calculations are
done in phase quantities. But as shown here, the simple model of Eq. (4.87) can also be solved with modal parameters
as a special case of the untransposed line. The simple model follows from Eq. (4.72) by making two assumptions for
a "lossless high-frequency approximation":
1. Conductor resistances and ground return resistances are ignored.
2. The frequencies contained in the lightning surges are so high that all currents flow on the surface of
the conductors, and on the surface of the earth.
Then the elements of [Z’ ] become

phase
z z
<) ln(2JK/TK) , <) ln(&KM/FKM) (.)
KK ' LT 2B KM ' LT 2B
while [Y’] is obtained by inverting the potential coefficient matrix,
) ) & ( . 9)
[; ] ' LT[2 ]
with the elements of [P’] being the same as in Eq. (4.88) if the factor j µ /(2 ) is replaced
0 by 1/(2 g ). Then both 0matrix
"
Ground wires are usually retained as phase conductors in such studies. If they are eliminated, the method
of Section .1.2.1 must be used on [< ].
…‡„sq
Traduciendo...
Page 110
products in Eq. (4.73) become diagonal matrices with all elements being
8M ' &T g z , M'1,.../ ( .90)
These values are automatically obtained from the supporting routines LINE CONSTANTS and CABLE
CONSTANTS as the eigenvalues of the matrix products in Eq. (4.73), by simply using the above two assumptions in
the input data (all conductor resistances = 0, GMR/r = 1.0, no Carson correction terms). The calculation of the
eigenvector matrix [T ] or
v [T ] needed
i for the untransposed line model of Section 4.2 breaks down, however, because
the matrix products in Eq. (4.73) are already diagonal. To obtain [T ], let us first
v assume equal, but nonzero conductor
resistances R’. Then the eigenvectors [t ] are defined

vk by
) )
(&T g z [7] % LT4 [2 ] & ) [VXM] ' 8M[VXM] ( .91)
with the expression in parentheses being the matrix product [Z’ ][Y’ ], and [U] = unit
phase phase matrix. Eq. (4.91) can be
rewritten as
[2 )] & [VXM] ' 8M&OQFKHKGF [VXM] ( .92)
with modified eigenvalues
)
LT4 8M&OQFKHKGF ' 8M % T g z ( .93)
Eq. (4.92) is valid for any value of R’, including zero. It therefore follows that [T ] is obtainedvas the eigenvectors of
-1
[P’] , or alternatively as the eigenvectors of [P’] since the eigenvectors of a matrix are equal to the eigenvectors of its
-1
inverse. The eigenvalues of [P] are not needed because they are already known from Eq. (4.90), but they could also
be obtained from Eq. (4.93) by setting R’ = 0.
For this simple mode, [T ] is va real, orthogonal matrix,
[6X] [6X]V ' [7] ( .9 )
and therefore,
[6K] ' [6X] ( .95)
D.E. Hedman has solved this case of the lossless high-frequency approximation more than 15 years ago [45]. He
recommended that the eigenvectors be calculated from the surge impedance matrix of Eq. (4.87), which is the same as
calculating them from [P’] since both matrices differ only by a constant factor.
One can either modify the line constants supporting routines to find the eigenvectors from [P’] for the lossless
high-frequency approximation, as was done in UBC’s version, or use the same trick employed in Eq. (4.91) in an
unmodified program: Set all conductor resistances equal to some nonzero value R’, set GMR/r = 1, and ask for zero
Carson correction terms. If the eigenvectors are found from [P’], then it is advisable to scale this matrix first by
multiplying all elements with 2 g .0
The lossless high-frequency approximation produces eigenvectors which differ from those of the lossy case
Page 111
at very high frequencies [61]. This is unimportant for lightning surge studies, but important for power line carrier
problems.
Example: For a distribution line with one ground wire (Fig. 4.24) the lossless high-frequency approximation produces
the following modal surge impedances and transformation matrix,
mode Z surge-mode ( )
1 993.44
2 209.67
3 360.70
4 310.62Traduciendo...
(KI Position of phase conductors #, B,

C and ground wire D (average height, all
dimensions in m). Conductor diameter
10.1092 mm
0.5299 0. 2 0 &0.1 0
0. 90 0 &0.21322 0. 2
0. 90 0 &0.21322 0. 2
0. 721 &0. 7170 &0.73
t
Converted to phase quantities, the surge impedance matrix becomes [T ][Z v surge-mode ][T ]v , or
90.33
17 .95 . 9 U[OOGVTKE
[<UWTIG&RJCUG] ' S
17 .95 17 .27 . 9
190.7 1 .2 1 .2 95.31
Page 112
The elements from Eq. (4.87) are slightly larger, by a factor of 300,000/299,792, because the supporting routine LINE
CONSTANTS uses 299,792 km/s for the speed of light, versus 300,000 km/s implied in Eq. (4.87).
Representation in EMTP then would be by means of a 4-phase, constant-parameter, lossless line. The following
branch cards are for the first of 4 such cascaded sections:
-11A 2A 993.44 .3E-6 2 4

-11B 2B 993.44 .3E-6 2 4
-11C 2C 993.44 .3E-6 2 4
-11D 2D 993.44 .3E-6 2 4
The modelling of long lines as coupled shunt resistances [R] = [Z surge-phase ] has already been discussed in Section
3.1.3. In the example above, such a shunt resistance matrix could be used to represent the rest of the line after the 4
spans from the substation. Simply using the 4 x 4 matrix would be unrealistic with respect to the ground wire, however,
because it would imply that the ground wire is ungrounded on the rest of the line. More realistic, though not totally
accurate, would be a 3 x 3 matrix obtained from [Z’ ] and phase

[Y’ ] in which the ground wire has been eliminated. This
phase
model implies zero potential everywhere on the ground wire, in contrast to the four spans where the potential will more
or less oscillate around zero because of reflections up and down the towers.
Comparison with More Accurate Models: For EMTP users who are reluctant to use the simple model described in this
section, a few comments are in order. First, let us compare exact values with the approximate values. If we use constant
parameters and choose 400 kHz as a reasonable frequency for lightning surge studies, then we obtain the results of table
4.5 for the test example above, assuming T/D = 0.333 for skin effect correction and internal inductance calculation with
the tubular conductor formula, R’ = 0.53609
dc /km, and = 100 m.
Table 4.5 - Exact line parameters at 400 kHz

Traduciendo...
mode Z surge-mode () wave velocity (m/s) R’ ( /km)
1 1027.6-j33.9 285.35 597.4

2 292.0-j0.5 299.32 7.9
3 361.9-j0.5 299.37 8.2
4 311.1-j0.5 299.32 8.0
The differences are less than 0.5% in surge impedance and wave speed for the aerial modes 2 to 4, and not more than
5% for the ground return mode 1. These are small differences, considering all the other approximations which are made
in lightning surge studies. If series resistances are included by lumping them in 3 places, totally erroneous results may
be obtained if the user forgets to check whether R/4 # Z surge in the ground return mode. For the very short line length
of 90 m in this example, this condition would still be fulfilled here.
Using constant parameters at a particular frequency is of course an approximation as well, and some users may
therefore prefer frequency-dependent models. For very short line lengths, such as 90 m in the example, most frequency-
dependent models are probably unreliable, however. It may therefore be more sensible to use the simple model
Page 113
described here, for which answers are reliable, rather than sophisticated models with possibly unreliable answers.
A somewhat better lossless line model for lightning surge studies than the preceding one has been suggested
by V. Larsen [92]. To obtain this better model, the line parameters are first calculated in the usual way, at a certain
frequency which is typical for lightning surges (e.g., at 400 kHz). The resistances are then set to zero when the matrix
product [Z’ ][Y’

phase ] is phase
formed, before the modal parameters are computed. With this approach, [T ] will always be i
real. Table 4.6 shows the modal parameters of this better lossless model. They differ very little from those in Table
4.5.
Table 4.6 - Approximate modal parameters at 400 kHz with R=0
mode Z surge-mode () wave velocity (m/s)
1 1026.3 285.50
2 292.0 299.32
3 362.0 299.37
4 311.1 299.32
In particular, the wave velocity of the ground return mode 1 is now much closer to the exact value of Table 4.5. The
transformation matrix which goes with the modal parameters of Table 4.6 is
0. 0795 0. 115 &0.2231 0

0.55 2 &0.1 0. 910 &0.70711
[6K] '
0.55 2 &0.1 0. 910 0.70711
0. 335 &0. 7371 &0.739 7 0
In this case [T ] is
v no longer to [T ]; Eq.i (4.80) must be used instead.
4.1.5.3 Approximate Transformation Matrices for Transient Solutions
The transformation matrices [T ] and

v [T ] are theoretically
i complex, and frequency-dependent as well. With
a frequency-dependent transformation matrix, modes are only defined at the frequency at which the transformation
matrix was calculated. Then the concept of converting a polyphase line into decoupled single-phase lines (in the modal
domain) cannot be used over the entire frequency range. Since the solution methods for transients are much simpler
if the modal composition is the same for all frequencies, or in other words, if the transformation matrices are constant
with real coefficients, it is worthwhile to check whether such approximate transformation matrices can be used without
producing too much error. Fortunately, this is indeed possible for overhead lines [66, 78].
Guidelines for choosing approximate (real and constant) transformation matrices have not yet been worked out
at the time when these notes are being written. The frequency-dependent line model of J. Marti discussed in Section
4.2.2.6 needs such a real and constant transformation matrix, and wrong answers would be obtained if a complex
transformation matrix were used instead. Since a real and constant transformation matrix is always an approximation,
Traduciendo...
50
Page 114
its use will always produce errors, even if they are small and acceptable. The errors may be small in one particular
frequency region, and larger in other regions, depending on how the approximation is chosen.
One choice for an approximate transformation matrix would be the one used in the lossless approximations
discussed in Section 4.1.5.2. This may be the best choice for lightning surge studies.
For switching surge studies and similar types of studies, the preferred approach at this time seems to be to
calculate [T ]vat a particular frequency (e.g., at 1 kHz), and then to ignore the imaginary part of it. In this approach, [T ] v
should be predominantly real before the imaginary part is discarded. One cannot rely on this when the subroutine
E
returns the eigenvectors, since an eigenvector multiplied with e or any other j50
constant would still be a proper
eigenvector. Therefore, the columns of [T ] should

v be normalized in such a way that its components lie close to the real
axis. One such normalization procedure was discussed by V. Brandwajn [79]. The writer prefers a different approach,
which works as follows:
1. Ignore shunt conductances, as is customarily done. Then [Y’ ] is purely

phase imaginary. Use Eq. (4.85)
to find the diagonal elements of the modal shunt admittance matrix Y’ mode-k-preliminary .
-
2. In general, these "preliminary" modal shunt admittances will not be purely imaginary, but j C’ mode-k e
jk j k/2
instead. Then normalize [T ] by multiplying
v each column with e . With this normalized
transformation matrix, the modal shunt admittances will become j C’ mode-k , or purely imaginary as in
the phase domain.

3. To obtain the approximate (real and constant) transformation matrix, discard the imaginary part of
the normalized matrix from step 2.
4. Use the approximate matrix [T v-approx. ] from step 3 to find modal series impedances and modal shunt
admittances from Eq. (4.84) and (4.85) over the frequency range of interest. If [T ] is needed, use
i
V &
[6K&CRRTQZ ] ' [6 ( .9 )
X&CRRTQZ ]
5. If the line model requires nonzero shunt conductances, add them as modal parameters. Usually, only
conductances from phase to ground are used (with phase-to-phase values being zero); in that case,
the modal conductances are the same as the phase-to-ground conductances if the latter are equal for
all phases.
An interesting method for finding approximate (real and constant) transformation matrices has been suggested
by Paul [66]. By ignoring conductor resistances, and by assuming that the Carson correction terms R’ + j X’ in Eq. ii ii
(4.7) and R’ + jik X’ in Eq.ik (4.8) are all equal (all elements in the matrix of correction terms have one and the same
value), the approximate transformation matrix [T i-approx. ] is obtained as the eigenvectors of the matrix product
111...1
[% ) .......
RJCUG]
111...1
with all elements of the second matrix being 1. To find [T v-approx. ], Eq. (4.96) would have to be used. Wasley and
51
Page 115
Selvavinayagamoorthy [93] find the approximate transformation matrices by simply taking the magnitudes of the
complex elements, with an appropriate sign reflecting the values of their arguments. They compared results using these
approximate matrices with the exact results (using complex, frequency-dependent matrices), and report that fairly high
accuracy can be obtained if the approximate matrix is computed at a low frequency, even for the case of double-circuit
lines.
If the M-phase line is assumed to be balanced (Section 4.1.3.2), then the transformation matrix is always real
Traduciendo...
and constant, and known a priori with Eq. (4.58) and Eq. (4.59). Two identical and balanced three-phase lines with zero
sequence coupling only have the real and constant transformation matrix of Eq. (4.65).
4.2 Line Models in the EMTP
The preceding Section 4.1 concentrated on the line parameters per unit length. These are now used to develop
line models for liens of a specific length.
For steady-state solutions, lines can be modelled with reasonable accuracy as nominal -circuits, or rigorously
as equivalent -circuits. For transient solutions, the methods become more complicated, as one proceeds from the simple
case of a single-phase lossless line with constant parameters to the more realistic case of a lossy polyphase line with
frequency-dependent parameters.
4.2.1 AC Steady-State Solutions
Lines can be represented rigorously in the steady-state solution with exact equivalent -circuits. Less accurate
representations are sometimes used, however, to match the model to the one used in the transient simulation (e.g.,
lumping R in three places, rather than distributing it evenly along the line, or using approximate real transformation
matrices instead of exact complex matrices). For lines of moderate "electrical" length (typically # 100 km at 60 Hz),
nominal -circuits are often accurate enough, and are probably the best models to use for steady-state solutions at power
frequency. If the steady-state solution is followed by a transient simulation, or if steady-state solutions are requested
over a wide frequency range, then the nominal -circuit must either be replaced by a cascade connection of shorter
nominal -circuits, or by an exact equivalent -circuit derived from the distributed parameters.
4.2.1.1 Nominal M-Phase -Circuit
For the nominal M-phase -circuit of Fig. 3.10, the series impedance matrix and the two equal shunt
susceptance matrices are obtained from the per unit length matrices by simply multiplying them with the line length, as
shown in Eq. (4.35) and (4.36). The equations for the coupled lumped elements of this M-phase -circuit have already
been discussed at length in Section 3, and shall not be repeated here.
Nominal -circuits are fairly accurate if the line is electrically short. This is practically always the case if
complicated transposition schemes are studied at power frequency (60 Hz or 50 Hz). Fig. 4.25 shows a typical example,
with three circuits on the same right-of-way. In this case, each of the five transposition sections (1-2, 2-3, 3-4, 4-5, 5-6)
would be represented as a nominal 9-phase -circuit. While a nominal -circuit would already be reasonably accurate
52
Page 116
for the total line length of 95 km, nominal -circuits are certainly accurate for each transposition section, since the
longest section is only 35 km long. A comparison between measurements on the de-energized line L3 and computer
results is shown in table 4.7 [80]. The coupling in this case is predominantly capacitive.
Traduciendo...
(KI Transposition scheme for three adjacent circuits
Table 4.7 - Comparison between measurements and EMTP results (voltages on energized line L1 =
372 kV and on L2 = 535 kV)
phase measurement EMTP results
Induced voltages on de-energized line L3 if open at A 30 kV 27.5 kV

both ends B 15 kV 13.8 kV
C 10 kV 7.8 kV
Grounding currents if de-energized line L3 is A 11 A 10.5 A

grounded at right end B 5A 3.2 A
C 1A 1.5 A
Because nominal -circuits are so useful for studying complicated transposition schemes, a "CASCADED PI"
option was added to the BPA EMTP. With this option, the entire cascade connection is converted to one single -
circuit, which is an exact equivalent for the cascade connection. This is done by adding one "component" at a time, as
shown in Fig. 4.26. The "component" may either be an M-phase -circuit, or other types of network elements such as
53
Page 117
shunt reactors or series capacitors. Whenever component k is added, the nodal admittance matrix
(KI Schematic illustration of cascading operation for K th component
for nodes 1, 2, 3 is reduced by eliminating the inner nodes 2, to form the new admittance matrix of the equivalent for
the cascaded components 1, 2, ... K. This option keeps the computational effort in the steady-state solution as low as
possible by not having to use nodal equations for the inner nodes of the cascade connection, at the expense of extra
computational effort for the cascading procedure.
4.2.1.2 Equivalent -Circuit for Single-Phase Lines
Lines defined with distributed parameters at input time are always converted to equivalent -circuits for the
steady-state solution.
For lines with frequency-dependent parameters, the exact equivalent -circuit discussed in Section 1 is used,
with Eq. (1.14) and (1.15). The same exact equivalent -circuit is used for distortionless and lossless line models with
constant parameters.
In many applications, line models with constant parameters are accurate enough. For example, positive
sequence resistances and inductances are fairly constant up to approximately 1 kHz, as shown in Fig. 4.20. But even
with constant parameters, the solution for transients becomes very complicated (except for the unrealistic assumption
of distortionless propagation). Fortunately, experience showed that reasonable accuracy can be obtained if L’ and C’
are distributed and if
)
4'4 ý ( .97)
is lumped in a few places as long as R << Z . In the

surgeEMTP, R/2 is lumped in the middle and R/4 at both ends of an
otherwise lossless line, as shown in Fig. 4.27, and as further discussed in Section 4.2.2.5. For this transient
15
representation, the EMTP uses the same assumption in the
Traduciendo...
#
The EMTP should probably be changed to by pass this option if only steady state solutions are reSuested,
either at one freSuency or over a range of freSuencies.
Page 118
(KI Line representation with lumped resistances
steady-state solution, to avoid any discrepancies between ac steady-state initialization and subsequent transient
simulation, even though experiments have shown that the differences are extremely small at power frequency. By using
equivalent -circuits for each lossless, half-length section in Fig. 4.27, and by eliminating the "inner" nodes 1, 2, 3, 4,
an equivalent -circuit (Fig. 1.2) was obtained by R.M. Hasibar with
4
<UGTKGU ' 4cos TJ & 0.5 % 0.03125 4 sin TJ % L sinTJ cosTJ@
<
4
(0.375 % 2<)
<
4 4
(&2 & 0.125 )sin TJ % L sinTJcosTJ
1 < < ( .9 )
;UJWPV '
2 <UGTKGU
where
) )
J ' NGPIVJ . %
)
.
<'
)
%
)
4 ' NGPIVJ @ 4 ( .99)
4.2.1.3 Equivalent M-Phase -Circuit
To obtain an equivalent M-phase -circuit, the phase quantities are first transformed to modal quantities with
Eq. (4.84) and (4.85) for untransposed lines, or with Eq. (4.58) and (4.59) for balanced lines. For identical balanced
three-phase lines with zero sequence coupling only, Eq. (4.65) is used. For each mode, an equivalent single-phase -
circuit is then found in the same way as for single-phase lines; that is, either as an exact equivalent -circuit with Eq.
(1.14) and (1.15), or with Eq. (4.98) and (4.99) for the case of lumping R in three places. These single-phase modal
-circuits each has a series admittance Y series-mode and two equal shunt admittances 1/2 Y shunt-mode . By assembling these
admittances as diagonal matrices, the admittance matrices of the M-phase -circuit in phase quantities are obtained from
[;UGTKGU] ' [6K] [;UGTKGU&OQFG] [6K]V ( .100)
and
55
Page 119
1
2 [;UJWPV] ' 1
2 [6K] [;UJWPV&OQFG] [6K]V ( .101)
While it is always possible to obtain the exact equivalent M-phase -circuit at any frequency in this way,
Traduciendo...
approximations are sometimes used to match the representation for the steady-state solution to the one used in the
transient solution. One such approximation is the lumping of resistances as shown in Fig. 4.27. Another approximation
is the use of real and constant transformation matrices in Eq. (4.100) and (4.101), as discussed in Section 4.1.5.3.
4.2.2 Transient Solutions
Historically, the first line models in the EMTP were cascade connections of -circuits, partly to prove that
computers could match switching surge study results obtained on transient network analyzers (TNA’s) at that time. On
TNA’s, balanced three-phase lines are usually represented with decoupled 4-conductor -circuits, as shown in Fig. 4.28.
This representation can easily be derived from Eq. (4.44) by rewriting it as
F8# ) ) )
& ' (< ( .102)
FZ U& < O)+# % < O(+# % +$ % +%)
(KI Four conductor B circuit used on

TN# s
for phase A, and similar for phases B and C. The first term in Eq. (4.102) is Z’ I (or branch A1-A2
pos A in Fig. 4.28),
while the second term is the common voltage drop caused by the earth and ground wire return current I + I + I A B C
(branch N1-N2 in Fig. 4.28). Note, however, that Fig. 4.28 is only valid if the sum of the currents flowing out through
a line returns through the earth and ground wires of that same line. For that reason, the neutral nodes N2, N3, ... must
be kept floating, and only N1 at the sending end is grounded. Voltages with respect to ground at location i are obtained
by measuring between the phase and node N . In meshed
i networks with different R/X-ratios, this assumption is probably
not true. For this reason, and to be able to handle balanced as well as untransposed lines with any number of phases,
M-phase -circuits were modelled directly with M x M matrices, as discussed in Section 4.1.2.4. Voltages to ground
are then simply the node voltages. Comparisons between these M-phase -circuits, and with the four-conductor -
circuits of Fig. 4.28 confirmed that the results are identical.
Page 120
The need for travelling wave solutions first arose in connection with rather simple lightning arrester studies,
where lossless single-phase line models seemed to be adequate. Section 1 briefly discusses the solution method used
in the EMTP for such lines. This method was already known in the 1920’s and 1930’s and strongly advocated by
Bergeron [81]; it is therefore often called Bergeron’s method. In the mathematical literature, it is known as the method
of characteristics, supposedly first described by Riemann.
It soon became apparent that travelling wave solutions were much faster and better suited for computers than
cascaded -circuits. To make the travelling wave solutions useful for switching surge studies, two changes were needed
from the simple single-phase lossless line: First, losses had to be included, which could be done with reasonable
accuracy by simply lumping R in three places. Secondly, the method had to be extended to M-phase lines, which was
achieved by transforming phase quantities to modal quantities. Originally, this was limited to balanced lines with built-
in transformation matrices, then extended to double-circuit lines, and finally generalized to untransposed lines. Fig. 4.29
compared EMTP results with results obtained on TNA’s, using the built-in transformation matrix for balanced three-
phase lines and simply lumping R in three places.
Traduciendo...
Fig. 4.29 - Energization of a three phase line. Computer simulation results (dotted line) superimposed on 8 transient
57
Page 121
network analyzer results for receiving end voltage in phase B. Breaker contacts close at 3.05 ms in phase A, 8,05 ms
in phase B, and 5.55 ms in phase C (t=0 when source voltage of phase A goes through zero from negative to positive)
[82]. Reprinted by permission of CIGRE
While travelling wave solutions with constant distributed L’, C’ and constant lumped R produced reasonable
accurate answers in many cases, as shown in Fig. 4.29, there were also cases where the frequency dependence,
especially of the zero sequence impedance, could not be ignored. Choosing constant line parameters at the dominant
resonance frequency sometimes improved the results. Eventually, frequency-dependent line models were developed
by Budner [83], by Meyer and Dommel [84] based on work of Snelson [85], by Semlyen [86], and by Ametani [87].
A careful re-evaluation of frequency-dependence by J. Marti [88] led to a fairly reliable solution method, which seems
to become the preferred option as these notes are being written. J. Marti’s method will therefore be discussed in more
detail.
4.2.2.1 Nominal -Circuits

Nominal -circuits are generally not the best choice for transient solutions, because travelling wave solutions
are faster and usually more accurate. Cascade connections of nominal -circuits may be useful for untransposed lines,
however, because one does not have to make the approximations for the transformation matrix discussed in Section
4.1.5.3. On the other hand, one cannot represent frequency-dependent line parameters and one has to accept the
spurious oscillations caused by the lumpiness. Fig. 4.30 shows these oscillations for the simple case of a single-phase
line being represented with 8 and 32 cascaded nominal -circuits. The exact solution with distributed parameters is
shown for comparison purposes as well. The proper choice of the number of -circuits for one line is discussed in [89],
as well as techniques for damping the spurious oscillations with damping resistances in parallel with the series R-L
branches of the -circuit of Fig. 4.28.
Traduciendo...
Page 122
Fig. 4.30 - Voltage at receiving end of a single phase line if a dc voltage of 10 V is connected to the sending end at t=0
(line data: R=0.0376 /mile, L=1.52 mH/mile, C=14.3 nF/mile, length-320 miles; receiving end terminated with shunt
inductance of 100 mH)
The solution methods for nominal -circuits have already been discussed in Section 3.4. With M-phase
nominal -circuits, untransposed lines (or sections of a line) are as simple to represent as balanced lines. In the former
case, one simply uses the matrices of the untransposed line, whereas in the latter case one would use matrices with
averaged equal diagonal and averaged equal off-diagonal elements.
4.2.2.2 Single-Phase Lossless Line with Constant L’ and C’

The solution method for the single-phase case has already been explained in Section 1. The storage scheme
for the history terms is the same as the one discussed in the next Section 4.2.2.3 for M-phase lossless lines, except that
each single-phase line occupies only one section in the table, rather than M section for M modes. Similarly, the
initialization of the history terms for cases starting from linear ac steady-state initial conditions is the same as in Eq.
(4.108).
59
Page 123
Traduciendo...
(KI Linear interpolation
(KI Effect of linear interpolation on sharp peaks. Dotted line: )t

9.32 ...zs to make J integer multiple of )t. Solid line: )t 10 zs (J not integer
multiple of )t)
Page 124
The solution is exact as long as the travel time is an integer multiple of the step size t. If this is not the case,
then linear interpolation is used in the EMTP, as indicated in Fig. 4.31. Linear interpolation is believed to be a
reasonable approximation for most cases, since the curves are usually smooth rather than discontinuous. If
discontinuities or very sharp peaks do exist, then rounding to the nearest integer multiple of t may be more sensible
than interpolation, however. There is no option for this rounding procedure in the EMTP, but the user can easily
accomplish this through changes in the input data. Fig. 4.32 compares results for the case of Fig. 4.30 with sharp peaks
with and without linear interpolation. The line was actually not lossless in this case, but the losses were represented
in a simple way by subdividing the line into 64 lossless sections and lumping resistances in between and at both ends.
The interpolation errors are more severe if lines are split up into many sections, as was done here. If the line were only
split up into two lossless sections, with R lumped in between and at both ends, then the errors in the peaks would be
less (the first peak would be 18.8, and the second peak would be -15.4).
The accumulation of interpolation errors on a line broken up into many sections, with of each section not
being an integer multiple of t, can easily be explained. Assume that a triangular pulse is switched onto a long, lossless
line, which is long enough so that no reflections come back from the remote end during the time span of the study (Fig.
4.33). Let us look at how this pulse becomes distorted through interpolation as it travels down the line if
Traduciendo...
(a) the line is broken up into short sections of travel time 1.5 t each, and
(b) the line from the sending end to the measuring point is represented as one section ( = k @ 1.5 t, with k = 1,
2, 3,...).
(KI Single phase lossless line energized with triangular

pulse
At any point on the line, the current will be
1
K' X
<
and between points 1 and 2 separated by (Fig. 4.33),
X (V % J) ' X (V)
This last equation was used in Fig. 4.34, together with linear interpolation, to find the shape of the pulse as it travels
Page 125
down the line. The pulse loses its amplitude and becomes wider and wider if it is broken up into sections of travel time
1.5 t each. On the other hand, the pulse shape never becomes as badly distorted if the line is represented as one single
section.
What are the practical consequences of this interpolation error? Table 4.8 compares peak overvoltages from
16
a BPA switching surge study on a 1200 kV three-phase line , 133 miles long. Each section was split up into two
lossless half-sections, with R lumped in the middle and
Table 4.8 - Interpolation errors in switching surge study with t = 50 µs
Peak overvoltages (MV)

Run Line model
phase A phase B phase C
1 single section 1.311 1.191 1.496
2 7 sections 1.276 1.136 1.457
3 single section with 1.342 1.167 1.489

rounded
at both ends, as explained in Section 4.2.2.4. Run no. 1 shows the results of the normal line representation as one
section. Run no. 2 with subdivision into 7 sections produces differences of 2.6 to 4.7%. In run no. 3 the zero and
positive sequence travel times = 664.93

0 µs and = 445.74 1µs were rounded to 650 and 450 µs, respectively, to make
them integer multiples of t = 50 µs. These changes could be interpreted as a decrease in both L' and C' of 2.25%, 0 0
and as an increase in both L' and1 C' of 0.96%,

1 with the surge impedances remaining unchanged. Since line parameters
are probably no more accurate than ±5% at best anyhow, these implied changes are quite acceptable. With rounding,
a slightly modified case is then solved without interpolation errors. Whether an option for rounding to the nearest
integer multiple of t should be added to the EMTP is debatable. In general, rounding may imply much larger changes
in L', C' than in this case, and if implemented, warning messages with the magnitude of these implied changes should
be added as well. In Table 4.8, runs no. 3 to 1 differ by no more than 2.3%, and the interpolation error is therefore
acceptable if the line is represented as one section. Breaking the line up into very many sections may produce
unacceptable interpolation errors, however.
If the user is interested in a "voltage profile" along the line, then a better alternative to subdivisions into
Traduciendo...
sections would be a post-processor "profile program" which would calculate
$
The problem of interpolation errors is basically the same for single phase and M phase lines therefore, a
three phase case is presented here for which data was already available. Choosing a step size )t which makes
the travel time J an integer multiple of )t is more difficult for three phase lines, however, because there are two
travel times for the positive and zero seSuence mode on balanced lines (or three travel times for the 3 modes on
untransposed lines).
Page 126
(KI Pulse at incremental distances down the line
Page 127
Traduciendo...
voltages and currents at intermediate points along the line from the results at both ends of the line. Such a program is
easy to write for lossless and distortionless lines. Luis Marti developed such a profile algorithm for the more
complicated frequency-dependent line models, which he merged into the time step loop of the EMTP [90]. This was
used to produce moves of travelling waves by displaying the voltage profile at numerous points along the line at time
intervals of t.
Fig. 4.34(a) suggests a digital filtering effect from the interpolation which is similar to that of the trapezoidal
rule described in Section 2.2.1. To explain this effect, Eq. (1.6) must first be transformed from the time domain
1 1
XM(V) & KMO(V) ' XO(V&J) % KOM(V&J)
< <
into the frequency domain,
1 1
+' 8M & +MO ' 8O % +OM @ G&LTJ ( .103)
< <
For simplicity, let us assume that voltage and current phasors V and I atmnode mmkare known, and that we want to find
I = V /Z
k - I atkm node k. Without interpolation errors, Eq. (4.103) provides the answer. If interpolation is used, and if
for the sake of simplicity we assume that the interpolated value lies in the middle of an interval t, then Eq. (4.103)
becomes
)V )V
1 1 &LT J% &LT J&
+KPVGTRQNCVGF' 8O%+OM @ @ G %G ( .10 )
< 2
Therefore, the ratio of the interpolated to the exact value becomes
+KPVGTRQNCVGF )V
' cos T ( .105)
+GZCEV 2
which is indeed somewhat similar to Fig. 2.10 for the error produced by the trapezoidal rule.
Single-phase lossless line models can obviously only approximate the complicated phenomena on real lines.
Nonetheless, they are useful in a number of applications, for example
(a) in simple studies where one wants to gain insight into the basic phenomena,
(b) in lightning surge studies, and
(c) as a basis for more sophisticated models discussed later.
For lightning surge studies, single-phase lossless line models have been used for a long time. They are
probably accurate enough in many cases because of the following reasons:
(1) Only the phase being struck by lightning must be analyzed, because the voltages induced in the other
phases will be much lower.
(2) Assumptions about the lightning stroke are by necessity very crude, and very refined line models are
therefore not warranted.
Page 128
(3) The risk of insulation failure in substations is highest for backflashovers at a distance of approx. 2
km or less. Insulation co-ordination studies are therefore usually made for nearby strokes. In that
case, the modal waves of an M-phase line "stay together," because differences in wave velocity and
distortion among the M waves are still small over such short distances. They can then easily be
combined into one resultant wave on the struck phase. There seems to be some uncertainty, however,
about the value of the surge impedance which should be used in such simplified single-phase
representations. It appears that the "self surge impedance" Z ii-surge of Eq. (4.87a) should be used. For
nearby strokes it is also permissible to ignore the series resistance. Attenuation caused by corona may
be more important than that caused by conductor losses. At the time of writing these notes, corona
is still difficult to model, and it may therefore be best to ignore losses altogether to be on the safe side.
4.2.2.3 M-Phase Lossless Line with Constant L’ and C’
Additional explanations are needed for extending the method of Traduciendo...

Section 1 from single-phase lines to M-phase
lines. In principle, the equations are first written down in the modal domain, where the coupled M-phase line appears
as if it consisted of M single-phase lines. Since the solution for single-phase lines is already known, this is
straightforward. For solving the line equations together with the rest of the network, which is always defined in phase
quantities, these modal equations must then be transformed to phase quantities, as schematically indicated in Fig. 4.35.
(KI Transformation between phase and modal domain on a three phase line
For simplicity, let us assume that the line has 3 phases. Then, with the notations from Fig. 4.35, each mode
is described by an equation of the form of Eq. (1.6), or
1
K C& C(V) ' X C(V) % JKUV C& C(V&JC)
<C
1
K D& D(V) ' X D(V) % JKUV D& D(V&JD) ( .10 )
<D
Page 129
1
K E& E(V) ' X E(V) % JKUV E& E(V&JE)
<E
where each history term hist was computed and stored earlier. For mode a, this history term would be
1
JKUV C& C(V&JC) ' & X C(V&JC) & K C& C(V&JC) ( .107)
<C
and analogous for modes b and c. These history terms are calculated for both ends of the line as soon as the solution
has been obtained at instant t, and entered into a table for use at a later time step. As indicated in Fig. 4.36, the history
terms of a three-phase line would occupy 3 sections of the history tables for modes a, b, c, and the length of each section
would be increased / t, with increased being the travel time of the particular mode increased to the nearest integer multiple
17
of t . Since the modal travel times , , differ afrom
b each
c other, the 3 sections in this table are generally of different
length. This is also the reason for storing history terms as modal values, because one has to go back different travel
times for each mode in picking up history terms. For the solution at time t, the history terms of Eq. (4.106) are obtained
by using linear interpolation on the top two entries of each mode section.
Traduciendo...
%
# single phase line would simply occupy one section, whereas a six phase line would occupy six sections in
this table.
Page 130
(KI Table for history terms of transmission lines
After the
solution in each time step, the entries in the tables of Fig. 4.36 must be shifted upwards by one location, thereby
throwing away the values at the oldest point at t- increased . This is then followed by entering the newly calculated history
terms hist(t) at the newest point t. Instead of physically shifting values, the EMTP moves the pointer for the starting
address of each section down by 1 location. When this pointer reaches the end of the table, it then goes back again to
the beginning of the table ("wrap-around table") [91].
The initial values for the history terms must be known for t = 0, - t, -2 t, ... - increased . The necessity for
knowing them beyond t = 0 comes from the fact that only terminal conditions are recorded. If the conditions were also
given along the line at travel time increments of t, then the initial values at t = 0 would suffice. For zero initial
conditions, the history table is simply preset to zero. For linear ac steady-state conditions (at one frequency ), the
history terms are first computed as phasors (peak, not rms),
7
Traduciendo...
Page 131
1
*+56MO ' & 8O & +OM ( .10 a)
<
j
where V and
m I aremkthe voltage and current phasors at line end m (analogous for HIST ). With HIST =
mk *HIST* · e ,
the instantaneous history terms are then
JKUVMO(V) ' **+56MO*@cos(TV%") , YKVJ V'0, &)V, &2)V, ...
(4.108b)
Eq. (4.108) is used for single-phase lines as well as for M-phase lines, except that mode rather than phase quantities
must be used in the latter case.

Eq. (4.106) are interfaced with the rest of the network by transforming them from modal to phase quantities
with Eq. (4.78a),
RJCUG RJCUG RJCUG

K& ' ;UWTIG X % JKUV& ( .109a)
with the surge admittance matrix in phase quantities,
&
<C 0 0
&
;UWTIG ' 6K 0< D 0 6K V ( .109b)
&
0 0< E
and the history terms in phase quantities,
JKUV C& C
RJCUG
JKUV ' 6K JKUV D& D ( .109c)
&
JKUV E& E
For a lossless line with constant L' and C', the transformation matrix [T ] will always
i be real, as explained in the last
paragraph of Section 4.1.5.2. It is found as the eigenvector matrix of the product [C'][L'] for each particular tower
configuration, where [L'] and [C'] are the per unit length series inductance and shunt capacitance matrices of the line.
For balanced lines, [T] is known

i a priori from Eq. (4.58), and for identical balanced three-phase lines with zero
sequence coupling only it is known a priori from Eq. (4.65).
The inclusion of Eq. (4.109) into the system of nodal equations (1.8a) for the entire network is quite
straightforward. Assume that for the example of Fig. 4.35, rows and columns for nodes 1A, 1B, 1C follow each other,
as do those for nodes 2A, 2B, 2C (Fig. 4.37). Then the 3 x 3 matrix [Y ] enters into
surge two 3 x 3 blocks on the diagonal,
phase
as indicated in Fig. 4.37, while the history terms [hist 1-2 ] = [hist ,1A-2A
hist , hist1B-2B
] of Eq. (4.109c)
1C-2C enter into rows
1A, 1B, 1C, on the right-hand side with negative signs. Analogous history terms for terminal 2 enter into rows 2A, 2B,
Page 132
2C on the right-hand side. While [Y ] is entered

surge into [G] only once outside the time-step loop, the history terms must
be added to the right-hand side in each time step.
Traduciendo...
(KI Entries for a three phase line into system of eSuations
M-phase lossless line models are useful, among other things, for
(a) simple studies where one wants to investigate basic phenomena,
(b) in lightning surge studies, where single-phase models are no longer adequate, and
(c) as a basis for more sophisticated models discussed later.
Lightning surge studies cannot always be done with single-phase models. For simulating backflashovers on
lines with ground wires, for example, the ground wire and at least the struck phase must be modelled ("2-phase line").
Since it is not always known which phase will be struck by the backflashover, it is probably best to model all three
phases in such a situation ("4-phase line"). An example for such a study is discussed in Section 4.1.5.2, with 4-phase
lossless line models representing the distribution line, and single-phase lossless line models representing the towers.
Not included in the data listing are switches (or some other elements) for the simulation of potential flashovers from
the tower top (nodes D) to phases A, B, C.
4.2.2.4 Single and M-Phase Distortionless Lines with Constant Parameters
Distortionless line models are seldom used, because wave propagation on power transmission lines is far from
distortionless. They have been implemented in the EMTP, nonetheless, simply because it takes only a minor
modification to change the lossless line equation into the distortionless line equation.
Page 133
A single-phase transmission line, or a mode of an M-phase line, is distortionless if
) )
4 )
' ( .110)
. ) %)
Losses are incurred in the series resistance R’ as well as in the shunt conductance G’. The real shunt conductance of
an overhead line is very small (close to zero), however. If its value must be artificially increased to make the line
distortionless, with a resulting increase in shunt losses, then it is best to compensate for that by reducing the series
resistance losses. The EMTP does this automatically by regarding the input value R’ INPUT as an indicator for the total
losses, and uses only half of it for R’,
)
4 ) ) ) 1 4 +0276
' ' ( .111)
) ) 2 )
. % .
With this formula, the ac steady-state results are practically identical for the line being modelled as distortionless or with
R lumped in 3 places; the transient response differs mainly in the initial rate of rise. From Eq. (4.111), the attenuation
constant becomes
4 ) +0276 %
)
"' ( .112)
2 )
.
The factor 1/2 can also be justified by using an approximate expression for the attenuation constant for lines with low
attenuation and low distortion [48, p. 257],
) ) ) )
4 )
"' +0276 %) % +0276 . ( .113)
2 . 2 %)
which is reasonably accurate if R’ << L’ and G’ << C’. This condition is fulfilled on overhead lines, except at very
Traduciendo...
low frequencies. Eq. (4.112) is then obtained by dropping the term with G’ INPUT and by ignoring the fact that the waves
are not only attenuated but distorted as well.
If a user wants to represent a truly distortionless line where G’ is indeed nonzero, then the factor 1/2 should
of course not be used. The factor 1/2 is built into the EMTP, however, and the user must therefore specify R’ INPUT twice
as large as the true series resistance in this case.

-ý
With known, an attenuation factor e is calculated (ý = length of line). The lossless line of Eq. (1.6) is then
changed into a distortionless line by simply multiplying the history term of Eq. (1.6b) with this attenuation factor,
1 &"ý
JKUVMO(V&J) ' & XO(V&J) & KOM(V&J) @ G ( .11 )
<
The surge impedance remains the same, namely %L’/C’.
For M-phase lines, any of the M modes can be specified as distortionless. Mixing is allowed (e.g., mode 1
could be modelled with lumped resistances, and modes 2 and 3 as distortionless).
70
Page 134
Better results are usually obtained with the lumped resistance model described next, even though lumping of
resistances in a few places is obviously an approximation, whereas the distortionless line is solved exactly if the travel
time is an integer multiple of t.
4.2.2.5 Single and M-Phase Lines with Lumped Resistances
Experience has shown that a lossy line with series resistance R’ and negligible shunt conductance can be
modelled with reasonable accuracy as one or more sections of lossless lines with lumped resistances in between. The
simplest such approach is one lossless line with two lumped resistances R/2 at both ends. The equation for this model
is easily derived from the cascade connection of R/2 - lossless line - R/2, and leads to a form which is identical with that
of Eq. (1.6),
1
KMO(V) ' XM(V) % JKUVMO(V&J) ( .115)
<OQFKHKGF
except that the values for the surge impedance and history terms are slightly modified. With Z, R and calculated from
Eq. (4.99),
4
<OQFKHKGF ' < %
2
and
1 4
JKUVMO(V&J) ' & XO(V&J) % (<& )KOM(V&J)
<OQFKHKGF 2
This model with R/2 at both ends is not used in the EMTP. Instead, the EMTP goes one step further and lumps
resistances in 3 places, namely R/4 at both ends and R/2 in the middle, as shown in Fig. 4.27. This approach was taken
because the form of the equation still remains the same as in Eq. (4.115), except that
4
<OQFKHKGF ' < % ( .11 )
18
now. The history term becomes more complicated , and contains conditions from both ends of the line at t - ,
< 4
JKUVMO(V&J) ' & XO(V&J) % (<& )KOM(V&J)
<OQFKHKGF
& The eSuation at the bottom of p. 391, left column, in [50] contains an error. I and I should notwbe y
computed from ES. (7b) instead, use I (1/<) e (tw J) hi (t J) with
w the notation
w yof [50], where < is
< y puruqp of ES. ( .11 ). For I , exchange
y subscripts k and m.
71 Traduciendo...
Page 135
4/ 4
& XM(V&J) % (<& )KMO(V&J) ( .117)
<OQFKHKGF
Users who want to lump resistances in more than 3 places can do so with the built-in three-resistance model,
by simply subdividing the line into shorter segments in the input data. For example, 32 segments would produce lumped
resistances in 65 places. Interestingly enough, the results do not change much if the number of lumped resistances is
increased as long as R << Z. For example, results in Fig. 4.30 for the distributed-parameter case were practically
identical for lumped resistances in 3, 65, or 301 places. Fig. 4.29 shows as well that TNA results are closely matched
with R lumped in 3 places only.
One word of caution is in order, however. The lumped resistance model gives reasonable answers only if R/4
<< Z, and should therefore not be used if the resistance is high. High resistances do appear in lightning surge studies
if the parameters are calculated at a high frequency, e.g., at 400 kHz in Table 4.5, where R’ = 597.4 /km in the zero
sequence mode. Lumping R in 3 places would still be reasonable in the case discussed there where each tower span of
90 m is modelled as one line, since 13.4 is still reasonably small compared with Z = 1028 . If it were used to model
19
a longer line, say 90 km, then R/4 = 13,400 , which would produce totally erroneous results . In such a situation it
might be best to ignore R altogether, or to use the frequency-dependent option if higher accuracy is required.
For M-phase lines, any of the M modes can be specified with lumped resistances. Mixing is allowed (e.g.,
mode 1 could be modelled with lumped resistances, and modes 2, ... M as distortionless). The lumped resistances do
not appear explicitly as branches, but are built into Eq. (4.115) (4.116) and (4.117) for each mode. Should a user want
to add them explicitly as branches, e.g., for testing purposes, then they would have to be specified as M x M - matrices
[R] in phase quantities, which could easily be done with the M-phase nominal -circuit input option by setting L =
0 and C = 0. All modes would have to use the lumped resistance model in this set-up, that is, mixing of models would
not be allowed in it.
4.2.2.6 Single and M-Phase Lines with Frequency-Dependent Parameters

The two important parameters for wave propagation are the characteristic impedance
) )
4 % LT.
<E ' ( .11 )
) )
) % LT%
and the propagation constant
) ) ) ) ( .119)
( ' (4 % LT. )() % LT% )
Both parameters are functions of frequency, even for constant distributed parameters R’, L’, G’, C’ (except for lossless
and distortionless lines). The line model with frequency-dependent parameters can handle this case of constant
'
The UBC version of the EMTP stops with an error message if 4/ <. It would be advisable to add a
warning message as well as soon as 4/ gets fairly large (e.g. 0.05 <).
72
Page 136
20
distributed parameters , even though it has primarily been developed for frequency-dependent series impedance
parameters R’( ) and L’( ). This frequency-dependence of the resistance and inductance is most pronounced in the zero
sequence mode, as seen in Fig. 4.20. Frequency-dependent line models are therefore important for types of transients
which contain appreciable zero sequence voltages and currents. One such type is the single line-to-ground fault.
To develop a line model with frequency-dependent parameters which fits nicely into the EMTP, it is best to
use an approach which retains the basic idea behind Bergeron’s method. Let us therefore look at what the expression
v + Zi used by Bergeron looks like now, as one travels down the line. Since the parameters are given as functions of
frequency, this expression must first be derived in the frequency domain. AtTraduciendo...
any frequency, the exact ac steady-state
solution is described by the equivalent -circuit of Eq. (1.13), or in an input-output relationship form more convenient
here,
cosh((ý) <Esinh((ý)
8M 8O
' 1 ( .120)
+MO sinh((ý) cosh((ý) &+OM
<
which can be found in any textbook on transmission lines. Assume that we want to travel with the wave from node m
to node k. Then the expression V + Z I is obtained

c by subtracting Z times the second
c row from the first row in Eq.
(4.120),
8M & <E+MO ' (8O % <E+OM) @ G &(ý ( .121a)
or rewritten as
+MO ' 8M/<E & (8O/<E % +OM) @ G &(ý ( .121b)
with a negative sign on I since

km its direction is opposite to the travel direction. Eq. (4.121) is very similar to Bergeron’s
method; the expression V + Z I encountered

c when leaving node m, after having been multiplied with a propagation
ý same when arriving at node k. This is very similar to Bergeron’s equation for the distortionless line,
factor e , -the
-ý
except that the factor is e there, and that Eq. (4.121) is in the frequency domain here rather than in the time domain.
Before proceeding further, it may be worthwhile to look at the relationship between the equations in the
frequency and time domain for the simple case of a lossless line. In that case,
. ) ) ) &(ý &LTJ
<E ' , ( ' LT . % , CPF G .G
)
%
Anybody familiar with Fourier transformation methods for transforming an equation from the frequency into the time
-j
domain will recall that a phase sift of e in the frequency domain will become a time delay in the time domain.
Furthermore, Z is cnow just a constant (independent of frequency), and Eq. (4.121) therefore transforms to
This case differs from the line with lumped resistances inasmuch as the resistance becomes truly distributed
now.
73
Page 137
XM(V) & <EKMO(V) ' XO(V&J) % <EKOM(V&J)
which is indeed Bergeron’s equation (1.6).
For the general lossy case, the propagation factor
&(ý &"ý &L$ý

#(T) ' G 'G @G
with = + j , contains an attenuation factor e as well as -aý phase shift e , which are both functions
-j ý
of frequency.
To explain its physical meaning, let us connect a voltage source V source to the sending end m through a source impedance
which is equal to Z ( ),cto avoid reflections in m (Fig. 4.38). In that case, V + Z I = V m c mk source . Furthermore, let us
assume that the receiving end k is open. Then from Eq. (4.121),
8M ' 8UQWTEG @ #(T) ( .122)
(KI Voltage source connected to end m through matching

impedance
Traduciendo...
that is, the propagation factor is the ratio (receiving end voltage) / (source voltage) of an open-ended line if the line is
21
fed through a matching impedance Z ( ) toc avoid reflections at the sending end . If V source = 1.0 at all frequencies from
dc to infinity, then its time domain transform v source (t) would be a unit impulse (infinitely high spike which is infinitely
narrow with an area of 1.0), and the integral of v source (t) would be a unit step. Setting V source = 1.0 in Eq. (4.122) shows
that A( ) transformed to the time domain must be the impulse which arrives at the other end k, if the source is a unit
impulse. This response to the unit impulse,
C(V) ' KPXGTUG (QWTKGT VTCPUHQTO QH 6#(T)> ( .123)
will be attenuated (no longer infinitely high), and distorted (no longer infinitely narrow). Fig. 4.39 shows these
responses for a typical 500 kV line of 100 miles length. They were obtained
One could also connect a matching impedance < (T) fromi node k to ground to avoid reflections at the
receiving end as well. In that case, the left hand side of ES. ( .122) becomes 2V rather than V . Note
w that the w
(ý
ratio e starts from 1.0 and becomes less than 1.0 as the line length (or freSuency) is increased. This is in
contrast to the open circuit response V /V 1.0/cosh((ý)
wy more familiar to power engineers, which increases
with length or freSuency (Ferranti rise).
Page 138
(a) zero sequence mode (b) positive sequence (c) positive sequence
mode with same mode with
scale as (a) expanded scale
Fig. 4.39 - Receiving end response v (t) = a(t)

k for the network of Fig. 4.38 if v source (t) = unit impulse [94]. Reprinted
by permission of J. Marti
from the inverse Fourier transformation of A( ) = exp(- ý) calculated by the LINE CONSTANTS supporting routine
at a sufficient number of points in the frequency domain. The amplitude of the propagation factors A( ) for the case
of Fig. 4.39 is shown in Fig. 4.40.
The unit impulse response of a lossless line would be a unit impulse at t = with an area of 1.0. In Bergeron’s
method, this implies picking up the history term v /Z + i mat t - with

mk a weight of 1.0. In the more general case here,
history terms must now be picked up at more than one point, and weighted with the "weighting function" a(t). For the
example of Fig. 4.39(a),
(a) zero seSuence mode (b) positive seSuence mode

Traduciendo...
(KI Propagation factor #(T) for the line of Fig. .39 [9 ]. 4eprinted by
permission of J. Marti
75
Page 139
history terms must be picked up starting at = 0.6 ms

min back in time, to approx. = 2.0 ms back inmax
time. The value
min is the travel time of the fastest waves, while is the travel
max time of the slowest waves. Each terms has its own
weight, with the highest weight of approx. 5400 around = 0.7 ms back in time. Mathematically, this weighting of
history at the other end of the line is done with the convolution integral
JOCZ
JKUVRTQRCICVKQP ' & KO&VQVCN(V&W)C(W)FW ( .12 )
mJOKP
which can either be evaluated point by point, or more efficiently with recursive convolution as discussed later. The
expression i m-total in Eq. (4.124) is the sum of the line current i and of a current
mk which would flow through the
characteristic impedance if the voltage v were applied
m to it (expression I + V /Z in themkfrequency
m c domain).
With propagation of the conditions from m to k being taken care of through Eq. (4.124), the only unresolved
issue is the representation of the term V /Z ink Eq.

c (4.121b). For the same 500 kV line used in Fig. 4.39, the magnitude
and angle of the characteristic impedance Z are shown

c in Fig. 4.41. If the shunt conductance per unit length G’ were
ignored, as is usually done, Z would
c become infinite at = 0. This complicates the mathematics somewhat, and since
G’ is not completely zero anyhow, it was therefore decided to use a nonzero value, with a default option of 0.03 µs/km.
As originally suggested by E. Groschupf [96] and further developed by J. Marti [94], such a frequency-dependent
impedance can be approximated with a Foster-I R-C
Page 140
Traduciendo...
77
Page 141
network. Then the line seen from node k becomes a simple R-C network in parallel with a current source histpropagation
(Fig. 4.42(a)). One can then apply the trapezoidal rule of integration to the capacitances, or use any other method of
implicit integration. This transforms each R-C block into a current source in parallel with an equivalent resistance.
Summing these for all R-C blocks produces one voltage source in series with one equivalent resistance, or one current
source in parallel with one equivalent resistance (Fig. 4.42(b)). In the solution of the entire network with Eq. (1.8), the
frequency-dependent line is then simply represented again as a constant resistance R equiv to ground, in parallel with a
current source hist + hist

RC propagation , which has exactly the same form as the equivalent circuit for the lossless line.
To represent the line in the form of Fig. 4.42 in the EMTP, it is necessary to convert the line parameters into
a weighting function a(t) and into an R-C network which approximates the characteristic impedance. To do this, Z and c
are first calculated with the support routine LINE CONSTANTS, from dc to such a high frequency where both A( )
= exp(- ý) becomes negligibly small and Z ( ) becomes
c practically constant. J. Marti [94] has shown that it is best to
approximate A( ) and Z ( ) byc rational functions directly in the frequency domain. The weighting function a(t) can
then be written down explicitly as a sum of exponentials, without any need for numerical inverse Fourier transformation.
Similarly, the rational function approximation of Z ( ) produces

c directly the values of R and C in the R-C network in
Fig. 4.42.
Traduciendo...
Page 142
(a) with 4 C network (b) with eSuivalent resistance after

applying implicit integration
(KI FreSuency dependent line representation seen from line end k
The rational function which approximates A( ) has the form
(U%\ )(U%\ )...(U%\P)

&UJOKP M
#CRRTQZ(U) ' G ( .125)
(U%R )(U%R )...(U%RO)
with s = j and n < m. The subscript "approx" indicates that Eq. (4.125) is strictly speaking only an approximation to
-j min
the given function A( ), even though the approximation is very good. The factor e is included to take care of the
fact that a(t) in Fig. 4.39 is zero up to t = ; this avoids

min fitting exponentials through the portion 0 # t # where the min
-j
values are zero anyhow (remember that a time shift - in the time domain is a phase shift e in the frequency domain).
All poles p and

i zeros z in Eq.
i (4.125) are negative, real and simple (multiplicity one). With n < m, the rational
function part of Eq. (4.125) can be expanded into partial fractions,
(U%\ )(U%\ )...(U%\P) M M MO

M ' % ...% ( .12 )
(U%R )(U%R )...(U%RO) U%R U%R U%RO
The corresponding time-domain form of Eq. (4.15) then becomes
&R V&JOKP %M&G

R V&JOKP ...%MO&
GRO V&JOKP HQT V$JOKP
CCRRTQZ(V) ' M G
'0 HQT V JOKP ( .127)

Traduciendo...
This weighting function a approx (t) is used to calculate the history term hist propagation of Eq. (4.124) in each time step.
Because of its form as a sum of exponentials, the integral can be found with recursive convolution much more efficiently
79
Page 143
-pi(t- min)
than with a point-by-point integration. If we look at the contribution of one exponential term k e i ,
4 &RK V & JOKP FW

UK(V) ' K(V&W)MKG ( .12 )
mJOKP
then s(t)i can be directly obtained from the value s(t - t) known from
i the preceding time step, with only 3
multiplications and 3 additions,
UK(V) ' E @ UK(V&)V) % E @ K(V&JOKP) % E @ K(V&JOKP&)V) ( .129)
as explained in Appendix V, with c , c , c 1being

2 constants
3 which depend on the particular type of interpolation used
for i.
The characteristic impedance Z ( ) isc approximated by a rational function of the form [94]
(U%\ )(U%\ )...(U%\P)

<E&CRRTQZ(U) ' M ( .130)
(U%R )(U%R )...(U%RP)
with s = j . All poles and zeros are again real, negative and simple, but the number of poles is equal to the number of
zeros now. This can be expressed as
M M MP
<E&CRRTQZ(U) ' M % % ... ( .131)
U%R U%R U%RP
which corresponds to the R-C network of Fig. 4.42, with
4'M
MK 1
4K ' , %K ' , K'1,...P ( .132)
RK MK
Rather than applying the trapezoidal rule to the capacitances in Fig. 4.42, J. Marti chose to use implicit integration with
22
Eq. (I.3) of Appendix I , with linear interpolation on i. For each R-c block
XK FXK
K' % %K
4K FV
which has the exact solution
&"K)V 1 V &"K V&W

XK(V) ' G @ XK(V&)V) % G K(W)FW ( .133)
%KmV&)V
This method is identical to the recursive convolution of #ppendix V applied to ES. ( .131). Whether
recursive convolution is better than the trapezoidal rule is still unclear.
Page 144
with = 1/(R
i C ).i By
i using linear interpolation on i, the solution takes the form of
XK(V) ' 4GSWKX&K @ K(V) % GK(V&)V) ( .13 )

Traduciendo...
with e (t
i - t) being known values of the preceding time step (formula omitted for simplicity), or after summing up over
all R-C blocks and R ,0
X(V) ' 4GSWKX @ K(V) % G(V&)V) ( .135a)
with
P P
4GSWKX ' 4 % j 4GSWKX&KCPF G ' j GK ( .135b)
K' K'
which can be rewritten as
1
K(V) ' X(V) % JKUV4% ( .13 )
4GSWKX
The equivalent resistance R equiv enters into matrix [G] of Eq. (1.8), whereas the sum of the history terms hist + RC
hist propagation enters into the right hand side.

The key to the success of this approach is the quality of the rational function approximations for A( ) and
Z c( ). J. Marti uses Bode’s procedure for approximating the magnitudes of the functions. Since the rational functions
have no zeros in the right-hand side of the complex plane, the corresponding phase functions are uniquely determined
from the magnitude functions (the rational functions are minimum phase-shift approximations in this case) [94]. To
illustrate Bode’s procedure, assume that the magnitude of the characteristic impedance in decibels is plotted as a function
of the logarithm of the frequency, as shown in Fig. 4.43 [94]. The basic principle is to approximate the given curve by
straight-line segments which are either horizontal or have a slope which is a multiple of 20 decibels/decade. The points
where the slopes change define the poles and zeros of the rational function. By taking the logarithm on both sides of
Eq. (4.130), and multiplying by 20 to follow the convention of working with decibels, we obtain
20log*<E&CRRTQZ(U)* ' 20logM % 20log*U%\ *...% 20log*U%\P*
& 20log*U%R *@@@& 20log*U%RP* ( .137)
For s = j , each one of the terms in this expression has a straight-line asymptotic behavior with respect to . For
instance, 20 log *j + z * becomes

1 20 log z for <<1 z , which is 1constant, and 20 log for >> z , which is a straight i
line with a slope of 20 db/decade. The approximation to Eq. (4.137) is constructed step by step: Each time a zero corner
(at = z ) is iadded, the slope of the asymptotic curve is increased by 20 db, or decreased by 20 db each time a pole
corner (at = p) is added.

i The straight-line segments in Fig. 4.43 are only asymptotic traces; the actual function
becomes a smooth curve without sharp corners. Since the entire curve is traced from dc to the
Page 145
Fig. 4.43 - Asymptotic approximation of the magnitude of Z (ω) [94]. c
Reprinted by permission of J. Marti
highest frequency at which the approximated function becomes practically constant, the entire frequency range is
approximated quite closely, with the number of poles and zeros not determined a priori. J. Marti improves the accuracy
Traduciendo...
further by shifting the location of the poles and zeros about their first positions. Fig. 4.44 shows the magnitude and
phase errors of the approximation of A( ), and Fig. 4.45 shows the errors for the approximation of Z ( ) for the line c
used in Fig. 4.39.
L. Marti has recently shown [95] that very good results can be obtained by using lower-order approximations
with typically 5 poles and zeros rather than the 15 poles and zeros used in Fig. 4.44 and 4.45. Furthermore, he shows
that positive and zero sequence parameters at power frequency (50 or 60 Hz) can be used to infer what the tower
geometry of the line was, and use this geometry in turn to generate frequency-dependent parameters. With this
approach, simple input data (60 Hz parameters) can be used to generate a frequency-dependent line model internally
which is fairly accurate.
Page 146
(a) zero sequence mode (b) positive sequence mode

(15 zeros and 20 poles) (13 zeros and 17 poles)
Fig. 4.44 - Errors in approximation of A( ) for line of Fig. 4.39 [94]. Reprinted by permission of J.
Marti
For M-phase lines, any of the M modes can be specified as frequency-dependent, or with lumped resistances,
or as distortionless. Mixing is allowed. A word of caution is in order here, however: At the time of writing these notes,
the frequency-dependent line model works only reliably for balanced lines. For untransposed lines, approximate real
and constant transformation matrices must be used, as explained in Section 4.1.5.3, which seems to produce reasonably
Traduciendo...
Page 147
(a) zero sequence mode (b) positive sequence mode

(15 zeros and poles) (16 zeros and poles)
Fig. 4.45 - Errors in approximation of Z ( ) forc line for Fig. 4.39 [94]. Reprinted by permission of
J. Marti
accurate results for single-circuit lines, but not for double-circuit lines. Research by L. Marti into frequency-dependent
transformation matrices in connection with models for underground cables will hopefully improve this unsatisfactory
state of affairs.
Page 148
Traduciendo...
Page 149
Fig. 4.46 - Comparison between voltages at phase b for [94]:

(a) Field test oscillograph
(b) BPA’s frequency-dependence simulation
(c) New model simulation
Reprinted by permission of J. Marti
Field test results for a single-line-to-ground fault from Bonneville Power Administration have been sued by
various authors to demonstrate the accuracy of frequency-dependent line models [84]. Fig. 4.46 compares the field test
results with simulation results from an older method which used two weighting functions a and a [84], and
1 from 2the
newer method described here. The peak overvoltage in the field test was 1.60 p.u., compared with 1.77 p.u. in the older
method and 1.71 p.u. in the newer method. Constant 60 Hz parameters would have produced an answer of 2.11 p.u.
Traduciendo...
Page 150
70&'4)4170& %#$.'5
There is such a large variety of cable designs on the market, that it is difficult, if not impossible, to develop
one computer program which can calculate the parameters 4 , L , C for any type of cable.
For lower voltage ratings, the cables are usually unscreened and insulated with polyvinyl chloride. #n
example of a three phase 1 kV cable with neutral conductor and armor is shown in Fig. 5.1.
(KI #rmored 1 kV cable (1 stranded conductor, 2 insulation, 3 bedding, flat

steel wire armor, 5 helical steel tape, plastic outer sheath). 4eprinted by permission from
Siemens Catalog 19 0
Traduciendo...
(KI 12 to 35 kV distribution cable with concentric neutral conductors (1 stranded
conductor, 2, conductive layers, 3 plastic insulation , 5 conductive tape, concentric
neutral conductors, 7 helical copper tape, 10 inner sheath, 11 plastic outer sheath).
4eprinted by permission from Siemens Catalog 19 0
#t the distribution voltage level, the cables are usually screened with concentric neutral conductors, as
shown in Fig. 5.2.
Page 151
#t the transmission voltage level, two types of cables are in widespread use today, namely the pipe type
cable (Fig. 5.3) and the self contained cable (Fig. 5. ). In the pipe type cable, three paper insulated oil impregnated
cables are drawn into a steel pipe at the construction site. The helical skid wires make it easier to pull the cables.
#fter evacuation, the pipe is filled with oil and pressurized to a high pressure of approx. 1.5 kPa. Pipe type cables
are used for voltages from 9 to 3 5 kV, with 550 kV cables under development. The typical
(KI Pipe type oil filled cable [1 ]. l 1979 John Wiley Sons,
Ltd. 4eprinted by permission of John Wiley Sons, Ltd
self contained oil filled cable is a single core cable (Fig. 5. ). Its stranded core conductor has a hollow duct which
is filled with oil and kept pressurized with low pressure bellow type expansion tanks. Underground and submarine
self contained cables are essentially identical, except that underground cables do not always have an armor.
52
(KI Single core self contained cable (C stranded core conductor

with oil filled duct, I paper insulation, S metallic sheath, B
bedding, # armor, P plastic sheath). Details of conductive layers
left out
Gas insulated systems with compressed SF gas are

$ used for compact substation designs. The busses in such
substations consist of tubular conductors inside a metallic sheath, with the conductors held in place by plastic spacers
at certain intervals (Fig. 5.5). SF busses are

$ in use in lengths of up to 300 m. # similar design can be used for
cables, but SF cables

$ are still experimental, with the sheath usually being corrugated. In EMTP studies, such
relatively short
(a) Single phase (b) Three phase
(KI SF bus $
busses can often be ignored, or represented as a lumped capacitance. Only in studies of fast transients with high
freSuencies must SF busses

$ be represented as transmission lines. Since the single phase geometry is essentially
53
Page 153
similar to that of a self contained cable, and since the three phase geometry is similar to that of a pipe type cable,
no special programs are needed to handle SF busses or$cables, except that the three phase arrangement of Fig.
5.5(b) has no electrostatic screens as in the case of a pipe type cable of Fig. 5.3.
Fig. 5.1 to 5.5 are only a few examples for the large variety of cable designs. The support routine C#BLE
CONST#NTS was developed by #. #metani essentially for the coaxial single core cable design of Fig. 5. and
5.5(a), and later expanded for the pipe type cable of Fig. 5.3 and for the three phase SF busses of Fig. 5.5(b).
$ #t
this time, there is no support routine for the types of lower voltage cables shown in Fig. 5.1 and 5.2, but calculation
methods applicable to such non coaxial arrangements are briefly discussed in Section 5.7.
5KPING %QTG %CDNGU
Traduciendo...
The cable parameters of coaxial arrangements, as in Fig. 5. , are derived in the form of eSuations for
coaxial loops [150, 152]. In Fig. 5. , loop 1 is formed by the core conductor C and the metallic sheath S as return,
loop 2 by the metallic sheath S and metallic armor # as return, and finally loop 3 by the armor # and either earth
or sea water as return.
5GTKGU +ORGFCPEGU
The series impedances of the three loops are described by three coupled eSuations
F8
FZ <
) <
) 0 +
F8 ) ) )
& ' < < < + (5.1)
FZ
0<
) <
) +
F8
FZ
The self impedance < of loop 1 consists of 3 parts,
<< i „q ‡† < < €

i „q …tqg†t u€…‡xg†u …tqg†t u€ (5.2)
with
< i „q ‡† internal impedance (per unit length) of tubular core conductor with return path outside
the tube (through sheath here)
< i „q …tqg†t u€…‡xg†u € impedance (per unit length) of insulation between core and sheath, and
< …tqg†t u€ internal impedance (per unit length) or tubular sheath with return path inside the tube
(through core conductor here).
Similarly,
<< …tqg†t ‡†< < € g„y „ u€

…tqg†t g„y „Âu€…‡xg†u (5.3)
and
Page 154
< <!! g„y „ ‡† < <€

g„y „ qg„†t u€…‡xg†u qg„†t (5. )
with analogous definitions as for ES. (5.2). The coupling impedances < < and < < are negative ! !
because of opposing current directions (I in negative direction in loop 1, I in negative direction

! in loop 2),
<<< …tqg†t y‡†‡gx (5.5a)
< < !< ! g„y „ y‡†‡gx (5.5b)
with < …tqg†t y‡†‡gx mutual impedance (per unit length) of tubular sheath between the inside loop 1 and the
outside loop 2, and
< g„y „ y‡†‡gx mutual impedance (per unit length) of tubular armor between the inside loop 2 and the
outside loop 3.
Finally, < < 0! because! loop 1 and loop 3 have no common branch.
The simplest terms to calculate are the impedances of the insulation, which are simply
z T
<) ln (5. )
KPUWNCVKQP ' LT
2B S
"
with z permeability of insulation (z 2 10 H/km),
r outside radius of insulation,
S inside radius of insulation, ‡ in identical units (e.g. in mm)
If the insulation is missing, e.g., between armor and earth, then < u€…‡xg†u0.€
The internal impedance and the mutual impedance of a tubular conductor with inside radius S and outside
radius r (Fig. .5) are a function of freSuency, and are found with modified Bessel functions [1 9].
< †‡hq u€Dm/2BSD {I (mS) K (mr) K (mS) I (mr)} (5.7a)
< †‡hq ‡† Dm/2BrD {I (mr) K (mS) K (mr) I (mS)} (5.7b)
< †‡hq y‡†‡gxD/2BSrD (5.7c)
with D I (mr) K (mS) I (mS) K (mr) (5.7d)

Traduciendo...
The parameter
O ' LTz/D (5.7e)
is the reciprocal of the complex depth of penetration (OVE4LINE) p defined earlier in ES. ( .5).
# subroutine SKIN for calculating the impedance < †‡hq ‡† of ES. (5.7b) was developed at BP# for the
support routine LINE CONST#NTS, and later modified at UBC to TUBE for the calculation of < †‡hq u€and < †‡hq
y‡†‡gxas well. #ll arguments of the modified Bessel functions I , I , K , K are complex numbers with a phase angle
of 5E because of ES. (5.7e). In such a case, the following real functions of a real variable can be used instead:
DGT(Z) % LDGK(Z) ' + (Z L)
55
Page 155
)
DGT(Z) % LDGK(Z) ' L+
) (Z L) (5. )
MGT(Z) % LMGK(Z) ' - (Z L)

) )
MGT(Z) % LMGK(Z) ' & L- (Z L)
These functions are evaluated numerically with the polynomial approximations of ES. (9.11.1) to (9.11.1 ) of [1 9].
% %
For arguments x # , the absolute error is 10 , whereas for arguments x , the relative error is 3 10 .
To avoid too large numbers in the numerator and denominator for large arguments of x, the expressions f(x) and g(x)
in ES. (9.22.9) and (9.11.10) of [1 9] are multiplied with exp ( 1 j/%2 x). If both arguments mS and mr have
absolute values greater than , then in addition to the above multiplication, the K and K functions are further
multiplied by exp (2mS) to avoid indefinite terms 0/0 for very large arguments.
When the support routine C#BLE CONST#NTS was developed, subroutine TUBE did not yet exist, and
#. #metani chose slightly different polynomial approximations for the functions I , I , K , K in ES. (5.7). He uses
ES. (9. .1) to (9. . ) of [1 9] instead, with the accuracy being more or less the same as in the polynomials used in
subroutine TUBE.
Simpler formulas with hyperbolic cotangent functions in place of ES. (5.7) were developed by M. Wedepohl
[150], which also give fairly accurate answers as long as the condition (r S)/(r S) 1/ is fulfilled. This was
verified by the author for the data of a 500 kV submarine cable.
The only term which still remains to be defined is < in ES. (5. ). This
qg„†tis the earth or sea return
impedance of a single buried cable, which is discussed in more detail in Section 5.3.
Submarine cables always have an armor, while underground cables may only have a sheath. The armor
often consists of spiralled steel wires, which can be treated as a tube of eSual cross section with z 1, without too „
much error (153]. # more accurate representation is discussed in [151].
ES. (5.1) is not yet in a form suitable for EMTP models, in which the voltages and currents of the core,
sheath, and armor must appear, in place of loop voltages and currents. The transformation is achieved by
introducing the terminal conditions
VVV i „q …tqg†t II i „q
VV …tqg†tV g„y „ and I I I …tqg†t i „q
V !V g„y „ I !I I I g„y „ …tqg†t i „q (5.9)
where V voltage
i „q from core to ground,
V …tqg†tvoltage from sheath to ground,
V g„y „ voltage from armor to ground.
By adding row 2 and 3 or ES. (5.1) to the first row, and by adding row 3 to the second row, we obtain
5
Traduciendo...
Page 156
<
) ) )
F8EQTG/FZ EE < EU < EC
+EQTG
) ) )
& F8UJGCVJ/FZ ' < +UJGCVJ (5.10)
UE < UU < UC
F8CTOQT/FZ <) ) ) +CTOQT
CE < CU < CC
with < <ii 2< < 2< < , ! !!
< <i…
< < 2< …i
<, ! !!
< <ig< < < <gi, …g g… ! !!
< <……
2< < , ! !!
< <gg (5.10b)
Some authors use eSuivalent circuits without mutual couplings, in place of the matrix representation of ES.
(5.10) with self impedances (diagonal elements) and mutual impedances (off diagonal elements). For example, [150]
shows the eSuivalent circuit of Fig. 5. for a single core cable without armor, which is essentially the same as the
TN# four conductor representation of overhead lines in Fig. .2 .
(KI Three conductor B circuit suitable for TN# s
5JWPV #FOKVVCPEGU
For the current changes along the cable of Fig. 5. , the loop eSuations are not coupled,
& F+ /FZ ' ()

) % LT% ) )8
& F+ /FZ ' ()

) % LT% ) ) 8
(5.11)
& F+ /FZ ' ()

) % LT% ) )8
G and
u C areuthe shunt conductance and shunt capacitance per unit length for each insulation layer. If there is no
insulation (e.g., armor in direct contact with the earth), then replace ES. (5.11) by
V u0 (5.12)
The shunt capacitance of tubular insulation with inside radius S and outside radius r is
57
Page 157
2Bg gT
%) '
T (5.13)
ln
S
with g absolute permittivity or dielectric constant of free space (g defined in ES. ( .22)) and g relative „
„ 5.1
permittivity or relative dielectric constant of the insulation material. Typical values for g are shown in Table
[5 ].
6CDNG 4elative permittivity and loss factor of insulation material [5 ]. 4eprinted by permission of Springer
Traduciendo...
Verlag and the authors
4elative Permittivity at Loss Factor tan* at 50

Insulation Material 20EC Hz and 20EC
butyl rubber 3.0 to .0 0.05
insulating oil 2.2 to 2. 0.001 to 0.002
oil impregnated paper 3.3 to .2 0.003 to 0.00
polyvinyl chloride 3.0 to .0 0.02 to 0.10
polyethylene 2.3 0.0002
crosslinked polyethylene 2. 0.000
The shunt conductance G is ignored in the support routine C#BLE CONST#NTS, which is probably
reasonable in most cases. It cannot be ignored, however, if buried pipelines are to be modelled as cables, as
explained in Section 5. . If values for G are available for cables, it is normally in the form of a dielectric loss angle
* or loss factor tan*. Then
G TC tan* (5.1 )
Typical values for tan* are shown in Table 5.1. In the literature on electromagnetics, the shunt conductance is
usually included by assuming that g in ES.„ (5.13) is a complex number g g jg , with ES.„ (5.13) rewritten as
LT2Bg
) ) % LT% ) ' (g ) & Lg)))
T (5.15)
ln
S
For cross linked polyethylene, both g and g are more or less constant up to 100 mHz [1 ], with the typical values
of Table 5.1. For oil impregnated paper insulation, both g and g vary with freSuency. Measured values between
10 kHz and 100 mHz [15 ] showed variations in g of approximately 20 , whereas g varied much more. Fig. 5.7
shows the variations which can be expressed as a function of freSuency with the empirical formula
Page 158
0.9
gT ' 2.5 % (5.1 )
(1%LT @ 10
&)
(KI Measured values of , and , for a cable with oil

impregnated paper insulation at 20EC [15 ]. 4eprinted by
permission of IEE and the authors
The support routine C#BLE CONST#NTS now assumes g 0 and g being constant, but it could easily be
changed to include empirical formulas based on measurements, such as ES. (5.1 ). #t this time, formulas based
on theory are not available because the freSuency dependent behavior of dielectrics is too complicated. Except for
very short pulses ( 5 zs), the dielectric losses are of little importance for the attenuation [15 ], and using a constant
g with g 0 should therefore give reasonable answers in most cases. Traduciendo...
#gain, ES. (5.11) is not yet in a form suitable for EMTP models. With the conditions of ES. (5.9), they
are transformed to
;
) &;
) 0
F+EQTG/FZ 8EQTG
) ; ) %; ) )
& F+UJGCVJ/FZ' &; &; 8UJGCVJ (5.17)
F+CTOQT/FZ 0 &; ) ; ) %; ) 8CTOQT
where ; G jTC
u . u u
2CTCNNGN 5KPING %QTG %CDNGU
There are not many cases where single core cables can be represented with single phase models. # notable
exception is the submarine cable system, where the individual cables are laid so far apart (to reduce the risk of
anchors damaging more than one phase) that coupling between the phases can be ignored. In general, the three
59
Page 159
single core cables of a three phase underground installation are laid close together so that coupling between the
phases must be taken into account.
If we start out with loop analysis, then it is apparent that it is only the most outer loops (armor with earth
return, or sheath with earth return in the absence of armor) through which the phases become coupled. The magnetic
field outside the cable produced by loop 1 and 2 in Fig. 5. is obviously zero, because the field created by I in the
core is exactly cancelled by the
(KI Three single core cables
returning current I in the sheath, etc. The first two eSuations in (5.1) are therefore still valid, whereas the third
eSuation now has coupling terms among the three phases a, b, c, or
<) C < ) C 0 000 000

)
< C<
) C< ) C 000 000
) C< ) C 00<
) 00<
)
0 < CD CE
<
) D< ) D 0 000
<
) ' ) D< ) D< ) D 000
NQQR < (5.1 )
) ) D 00<
)
0 < D< DE
)
< E<
) E 0
U[OOGVTKE
)
< E<
) E< ) E
0 <
) E< ) E
with < , <gh, < being
gi the
hi mutual impedances between the three outer loops of Fig. 5. . By using ES. (5.9) for
the transformation from loop to phase (core, sheath, armor) Suantities, the matrix in ES. (5.1 ) becomes
Traduciendo...
5 10
Page 160
[<
) ) )
UGNH&C] [<OWVWCNC&D] OWVWCNC
[< &E]
) [<
) ) (5.19)
[<
RJCUG] ' UGNH&D] [< OWVWCND&E]
U[OOGVTKE [< )
UGNH&E]
The 3 x 3 submatrices [< ] etc. on gthe diagonal are identical to the matrix in ES. (5.10a) for each cable by itself,
…qxr
whereas the 3 x 3 off diagonal matrices have identical elements, e.g.,
<
) ) )
CD < CD < CD
) <
) ) ) (5.20)
[< CD
OWVWCNC&D] ' CD < CD <
<) ) )
CD
CD < CD <
The only elements not yet defined are the mutual impedances < , < , < of theghouterghearth hireturn loops, which
are discussed in more detail in Section 5.3. If one of the cables does not have an armor, its self submatrix is
obviously a 2 x 2 matrix and its mutual submatrix is a 2 x 3 matrix. For cables without sheath and armor, the
submatrices become 1 x 1 and 1 x 3, respectively.
There is no coupling among the three phases in the shunt admittances. Therefore, the shunt admittance
matrix for the three phase system is simply
[;
) 0
C] 0
) 0 [;
) (5.21)
[;
RJCUG] ' D] 0
0 0 [; )
E]
where [; ] is gthe 3 x 3 matrix of ES. (5.17) for phase a, etc.
The screening effect of the sheath and armor depends very much on the method of grounding. For example,
if cable a is operated at 100 # between core and ground, with sheath and armor ungrounded and open circuited, then
the full 100 # will flow in the outer loop (loop currents I 100, I 100, I 100 in Fig. 5. ). This
! will produce
maximum induced voltages in the conductors of a neighboring cable b. How much nuisance this induction effect
creates depends again on the method of grounding within cable b itself. If cable b is operated between core and
ground (loads connected from core to ground), and if its sheath and armor are ungrounded and open circuited, then
the induced voltage will drive a circulating current through the core, ground and load impedances. If cable b is
operated between core and sheath (loads connected from core to sheath), then there will be no circulating current
in that loop, because according to ES. (5.20), the induced voltages are identical in core and sheath. There would
be a circulating current through the sheath and armor in parallel with earth return if the sheath (and armor) is
grounded at both ends.
If both the sheath and armor in the current carrying cable a are grounded at both ends, then the voltage
induced in the conductors of the neighboring cable b would be small. For the practical example of a 500 kV ac
5 11
Page 161
submarine cable at 0 Hz, 1 of the core current would return through the sheath, 7. through the armor, and
only 5. through the outermost loop with ground or sea water return. The induction effect in neighboring cables
would then be only 5. compared to the case with ungrounded sheath and armor. The algebraic sum is larger than
100 because there are phase shifts among the three currents (I 1 e v "&E, I g„y
7. e„ v %'E , I qg„†t
5. e ). v&$E
…tqg†t
'CTVJ 4GVWTP +ORGFCPEGU

Traduciendo...
In ES. (5. ), the impedance of the loop formed by the outermost tubular conductor and the earth (or sea
water) as return path is needed. This shall be called the self earth return impedance. For the matrix of ES. (5.1 ),
the mutual earth return impedance < between uw

the loop formed by the outermost tubular conductor and the earth
return path of cable i, and the analogous loop of cable k, is needed as well.
The four main methods of installing cables are as follows [1 ]:
(a) The cable is laid directly in the soil, in a trench which is filled with a backfill consisting of either the
original soil or of other material with lower or more stable thermal resistivity.
(b) The cable is laid in ducts or troughs, usually of earthenware or concrete.
(c) The cable is drawn into circular ducts or pipes, which allows additional cables to be installed without
excavation.
(d) The cable is installed, in air, e.g. in tunnels built for other purposes.
In cases (a), (b) and (c) the cable is clearly buried underground, and formulas for buried conductors must
therefore be used. In case (a), the radius 4 of the outermost insulation is simply the outside radius of the cable. In
cases (b) and (c) it should be the inside radius of the duct if the duct has a similar resistivity as the soil, or the outside
radius if it is a very bad conductor, or possibly some average radius if it is neither a good nor a bad conductor. What
to do in case (d) is somewhat unclear. 4easonable answers might be obtained by representing the tunnel with an
eSuivalent circular cross section of radius 4. #nother alternative is to assume that the tunnel floor is the surface of
the earth, and then use the earth return impedance formula for overhead conductors. This would ignore current
flows in the earth above the tunnel floor.
$WTKGF %QPFWEVQTU KP 5GOK +PHKPKVG 'CTVJ
Exact formulas for the self and mutual earth return impedances of buried conductors were first derived by
Pallaczek [29]. In these formulas, the earth is treated as semi infinite, extending from the surface downwards and
sideways to infinity. If the horizontal distance between cable i and cable k is x, and if cable i and k are buried at
depth h and y, respectively (Fig. 5.9), then the mutual earth return impedance is [150]
) DO 4 exp6&(J%[) " %O
< 6- (OF)&- (O&)% exp(L"Z)F">
OWVWCN ' m&4
2B *"*% " %O
(5.22)
The assistance of N. Srivallipuranandan and L. Marti in research for this section is gratefully acknowledged.
5 12
Page 162
where
d %x (h y) direct distance between cables i and k,
D %x (h y) distance between cable i and image of cable k in air,
m reciprocal of depth of penetration for earth from ES. (5.7e),
" integration constant.
(KI Geometric configuration of two

cables
Traduciendo...
The self earth return impedance is obtained from ES. (5.22) by choosing the x,y coordinate on the surface of the
outermost insulation, e.g., x 4 and y H,
< (same
qg„†t as ES. (5.22), with y h, x 4) (5.23)
with 4 outside radius of outermost insulation. The permeability z of earth and air are assumed to be identical in
these eSuations. Furthermore, they are written in a slightly different form than in Pollaczek s original paper, but
they are in fact identical.
While the K terms in ES. (5.22) are easy to evaluate, the integral terms in both (5.22) and (5.23) cannot
be calculated that easily. Wedepohl [150] gives an infinite series, which has been compared by Srivallipuranandan
[1 ] with a direct numerical integration method based on 4omberg extrapolation. Both results agreed to within
0.1 . Since the function under the integral is highly oscillatory, direct numerical integration is not easy, and the
series expansion is therefore the preferred approach.
The support routine C#BLE CONST#NTS does not use the exact Pollaczek formula. #metani recognized
that the integral terms in ES. (5.22) and (5.23) become identical with Carson s earth return impedance if the
numerator exp { (h y)%" m } is replaced by exp { (h y)*"*}. #ccepting this approximation, which is valid
for *"* *m*, he can then use Carson s infinite series or asymptotic expansion discussed in Section .1.1.1. Fig.
5 13
Page 163
5.10 and 5.11 show the errors in #metani s results from support routine C#BLE CONST#NTS, as well as the errors
of Wedepohl s approximate formulas [150] for self impedance,
<
) DO
6& ln
(O4
% 0.5 & OJ>
GCTVJ '
(5.2 )
2B 2 3
and for mutual impedance
<
) DO
6&ln
(OF
% 0.5 &
2
Oý>
OWVWCN '
(5.25)
2B 2 3
with ( Euler s constant, and
ý sum of the depths of burial of the two conductors.
Wedepohl s approximations are amazingly accurate up to 100 kHz (error 1 ), and then become less
accurate as the
freSuency
increases (25
error at 1 mHz)
where the
condition *m4*
0.25 or *md*
0.25 is no
longer fulfilled.
Traduciendo...
(KI 4elative errors in self earth return impedance formulas for

buried conductors (4 . mm, D 100 Sm) [1 ]. 4eprinted by
permission of N. Srivallipuranandan
51
Page 164
(KI 4elative error in mutual earth return impedance formulas for

buried conductors (d 0.3 m, h 0.75 m, y 0.75 m) [1 ].
4eprinted by permission of N. Srivallipuranandan
Semlyen has recently developed a very simple formula based on complex depth (OVE4LINE) p 1/m
[15 ], analogous to ES. ( .3) for the case of overhead lines. For the self earth return impedance, the formula is
<
) LTz
ln(4 %
1
)
GCTVJ ' (5.2 )
2B O4
while a similar formula for the mutual impedance has not yet been found. The error of ES. (5.2 ) is plotted in Fig.
5.10. Considering the extreme simplicity of this formula as compared to Pollaczek s formula, it is amazing to see
how reasonable the results from this approximate formula are.
$WTKGF %QPFWEVQTU KP +PHKPKVG 'CTVJ
In some cases, it may be reasonable to assume that the earth is infinite in all directions around the cable.
This assumption can be made when the depth of penetration in the earth
5 15
Page 165
2 DGCTVJ SO
FGCTVJ ' ' 503 (O) (5.27)
*O* H(*\)
Traduciendo...
becomes much smaller than the depth of the burial. For submarine cables, where D is typically 0.2 Sm, this is
probably more or less true over the entire freSuency range of interest, whereas for underground cables it would only
be true above a few MHz or so. Bianchi and Luoni [151] have used this infinite earth assumption to find the sea
return impedance of submarine cables.
The self earth return impedance for infinite earth is easily obtained from the tubular conductor formula
(5.7a), by letting the outside radius r go to infinity. Then with S 4,
DO - (OT)
<) (5.2 )
GCTVJ ' 2B4 -1(O4)
The mutual earth return impedance was derived in [1 ] as
D - (OF)
<) (5.29)
OWVWCN ' 2B4K 4M - (O4K) - (O4M)
1XGTJGCF %QPFWEVQTU
If the cable is installed in air, or laid on the surface of the ground, then the earth return impedances are the
same as those discussed for overhead lines in Section . The support routine C#BLE CONST#NTS uses Carson s
formula in that case. For a cable laid on the surface of the ground, the height is eSual to 4. #metani has tried a
special formula of Sunde for conductors on the surface of the ground, but the answers were found to be very
oscillatory around the seemingly correct answer. Sunde s formula was therefore not implemented.
/WVWCN +ORGFCPEG $GVYGGP 1XGTJGCF %QPFWEVQT CPF $WTKGF %QPFWEVQT
There is inductive coupling between the loop of an overhead conductor with earth return and the loop of
a buried conductor with earth return. The mutual impedance between these two loops is needed, for example, for
studying the coupling effects in pipelines from overhead lines, as discussed in Section 5. . This case was treated
by Pollaczek as well, with
<
) 4 exp6& J*"*& [ " %O >
exp(L"Z) F"
OWVWCN 'm&4 (5.30)
*"* % " %O
#s in the case of buried conductors, #metani uses an approximation for this integral by replacing y%" m with
y*"*. With this approximation, the formula becomes identical with Carson s eSuations, with the height of the buried
conductor having a negative value. In connection with a pipeline study [15 ], it was verified that Carson s formula
and Pollaczek s formula give identical results at 0 Hz. #t higher freSuencies, the differences would probably be
similar to those shown in Fig. 5.11.
51
Page 166
2KRG 6[RG %CDNG
Compared to the geometry of the single core cable of Fig. 5. , the geometry of the pipe type cable of Fig.
5.3 is more complicated. It is therefore more complicated to obtain the impedances of a pipe type cable, mainly for
two reasons,
(a) The single core cables inside the pipe are not concentric with respect to the pipe.
(b) The steel pipe is magnetic, and subject to current dependent saturation effects.
The analysis is somewhat simplified by the fact that the depth of penetration into the pipe is less than the
pipe thickness at power freSuency and above. #t 0 Hz, it is 1.5 mm from ES. (5.27), with typical values of D
$
0.2 10 Sm and z 00, whereas
„ a typical pipe thickness for a 230 kV cable is . mm. For transient studies
with freSuencies above power freSuency, the pipe thickness can therefore be assumed to be infinite, or eSuivalently,
the earth return can be ignored. Table 5.2 shows the current returning in the earth for a single phase to ground
6CDNG Earth return current in a 230 kV pipe type cable for single phase fault (z 00) „
f current in earth
(Hz) (percent of core current)
0. 9 .50
Traduciendo...
31.00
0 0. 5
00 0.00
fault in a 230 kV pipe type cable, with the pipe being in contact with the earth. To arrive at these values, it was
assumed that the core of the faulted phase was in the center of the pipe, and that the two unfaulted phases can be
ignored. With these assumptions, the impedance formulas of Section 5.1 can be used. If the two unfaulted phases
were included, the earth return current would probably be even less because some current would return through the
shield tapes and skid wires of the unfaulted phases. The relative permeability z influences the„ values of Table 5.2
with z 50,
„ of the current would return through the earth at 0 Hz, or 0.02 with z 1 00. „
+PHKPKVG 2KRG 6JKEMPGUU 0Q 'CTVJ 4GVWTP
If the depth of penetration is less than the pipe thickness, then no voltage will be induced on the outside of
the pipe (< ‚u‚q y‡†‡gx0 from ES. (5.7c)), and conseSuently, the loop current pipe/earth return will be practically
zero. In that case, the pipe is the only return path. The configuration is then essentially the same as that of three
single core cables in Fig. 5. , except that the pipe replaces the earth as the return path.
If we assume that each phase consists of three conductors (e.g., core, shield tapes represented as sheath,
skid wires represented as armor), then the loop impedance matrix is the same as in ES. (5.1 ). Coupling will only
exist among the three outermost loops of each armor (skid wires) with return through the pipe. What is needed then
5 17
Page 167
is a formula for the self impedances < , < , < !!g

of the loops
!!h formed
!!i by each armor (skid wires) and the pipe,
and a formula for the mutual impedances < , < , < between
gh hitwo such
ig loops.
The support routine C#BLE CONST#NTS finds these impedances with formulas first derived by
Tegopoulos and Kriezis [159], and later used by Brown and 4ocamora [1 0]. In these formulas it is assumed that
the current is concentrated in an infinitesimally small filament at the center of each single core cable. This model
can be applied to conductors of finite radius if proximity effects are negligible, either because of symmetrical
positioning within the pipe, or because the conductor radius is small compared to the distance to other conductors
or the pipe wall. In pipe type cables, neither condition is met since the conductors are relatively large and lie on the
bottom of the pipe. The pipe type cable impedances from C#BLE CONST#NTS are therefore not completely
accurate, but no better analytical models are available at this time. Brown and 4ocamora, who proposed the
formulas originally, recommend methods based on the subdivision into partial conductors discussed in Section 5.7,
for more accurate impedance calculation [1 1]. Hopefully, a support routine based on the subdivision method will
become available some day.
The self impedance < , etc.

!!gof the loop between the armor (skid wires) and the pipe consists again of three
terms, as in ES. (5. ). The first term < g„y „ ‡† is the same as in ES. (5.7b), with the assumption that proximity
effects can be ignored. The second term for the insulation becomes more complicated than ES. (5. ), because of
the eccentric geometry,
) z S FK
< ln 1& (5.31)
KPUWNCVKQP ' LT
2B 4K S
with S, 4 andu d u
defined in Fig.
5.12. The third
term for the
internal impedance
of the pipe, with
return on the
inside, replaces
< inqg„†t
ES. (5. ):
Traduciendo...
(KI Geometry of pipe type cable (S,r inside and outside

radius of pipe 4 , 4 outside
u w radius of single core cables i, k d , u
d woffset from center)
51
Page 168
4 P
) z - (OS) FK -P(OS)
< % 2j (5.32)
RKRG KP ' LT 2B OS- (OS) S PzT-P(OS) & OS-
)P(OS)
P'
with m from ES. (5.7e), and z z z permeability

„ of the pipe,
K umodified Bessel function of the second kind of order i
K derivative
u of K . u
For the concentric case with d 0, ES.u (5.32) becomes identical with ES. (5.2 ).
The mutual impedance < , etc. gh

between two outermost loops formed by armor (skid wires) and pipe is
z S - (OS)
<) 6ln % zT
OWVWCN ' LT2B OS- (OS)
FK%FM &2FKFMcos¾KM
4 FKFM -P(OS) 1
%j ( )Pcos(P¾KM)(2zT & )> (5.33)
S PzT-P(OS)& OS-
) P
P' P(OS)
Except for replacing < with <qg„†t ‚u‚q u€, and for using < y‡†‡gx from ES. (5.33) instead of (5.22), all
calculations remain the same as in Section 5.2, including the transformation from loop to phase Suantities. If the
cables inside the pipe do not have an armor (skid wires) or a sheath (shield tapes), then some of the matrices will
be reduced to 2 x 2, or 1 x 1, as discussed in Section 5.2. In practice, the shield tapes and skid wires can probably
be represented as one single sheath.
The magnetic properties of the steel pipe are easily taken into account by using the proper values for the
relative permeability z in ES.

„ (5.32) and (5.33). Unfortunately, z depends on the „current because of saturation
effects, as shown in Fig. 5.13 [192]. To model saturation effects accurately is not simple, because even at one
freSuency, say at 0 Hz, the permeability would not remain constant over one cycle. # two slope saturation curve
was tried in [1 1], with the conclusion that reasonably accurate answers can be obtained with a constant value of z .„
The sensitivity of the results
with respect to z can „then
be checked by re running
the case with one or more
different values of z .„
(KI 4elative permeability as a function of

pipe current (I pipe
‚ current, D pipe ‚
diameter) [192]. l 19 IEEE
5 19
Page 169
Since the shield tapes and skid wires are in contact with the pipe wall, the values of the capacitances between
Traduciendo...
the shield tapes/ skid wires of the three phases and between them and the pipe are immaterial. They are shorted out.
ES. (5.21) can therefore be used directly for the shunt admittance matrix. The support routine C#BLE
CONST#NTS does not assume this contact with the pipe in the beginning, however, and is therefore more general.
For this general case, a potential coefficient matrix is found first,
[2
) [2
) ) )
C] CC] [2 CD] [2 CE]
) [2
) ) ) )
[2 % 2 (5.3 )
RJCUG] ' D] DC] [2 DD] [2 DE]
[2 ) [2 ) ) )
E] EC] [2 ED] [2 EE]
where [P ], [P
g ], [Ph] are the
i 3 x 3 matrices of each single core cable found by inversion of ES. (5.17) with G
0,
) ) &
% &% 0
) ) ) ) )
[2 &% &% (5.35a)
C] ' % %%
) ) )
0 &% %
%%
or [1 3]
) ) ) ) )
2 %2 %2 2 %2 2
) ) ) ) ) )
[2C ] ' 2 2 (5.35b)
%2 %2 2
) ) )
2 2 2
with P 1/C
u . u (5.35c)
The dielectric between the armors (skid wires) and the pipe is represented by the second term in ES. (5.3 ). Each
of the submatrices [P ] and

uu [P ] in theuwsecond term is a 3 x 3 matrix with 9 eSual elements,
) 1 S FK
2KK ' ln 1& (5.3 a)
2Bg gT 4K S
1 S
2KM) ' ln
2Bg gT (5.3 b)
FK% FM & 2FKFMcos¾KM
with the essential terms in ES. (5.3 ) being the same expressions appearing in ES. (5.31) and (5.33). The admittance
matrix is then found by inverting [P ], ‚tg…q
[;
) ) &
RJCUG] ' LT[2 RJCUG]
(5.37)
5 20
Page 170
(KPKVG 2KRG 6JKEMPGUU YKVJ 'CTVJ 4GVWTP
#t lower freSuencies, there is mutual coupling between the inner and outer surface of the pipe. The induced
voltage on the outer surface will then produce a circulating current through the pipe and earth return. This extra loop
must be added to the loop impedance matrix of ES. (5.1 ),
as in Eq. (5.18), 0
with elements defined 0
in Section 5.4.1 -Z’m
0
[Z’ loop ] = 0 (5.38a)
-Z’m
0
0
-Z’m
0 0 -Z’ 0
m 0 -Z’ 0 0
m -Z’ Z’ m s
Traduciendo...
with
Z’m= Z’ pipe-mutual from Eq. (5.7c), (5.38b)
Z’s = Z’ pipe-out + Z’ insulation + Z’ earth

. (5.38c)
The first two terms in Eq. (5.38c) are found from Eq. (5.7b) and (5.6) (Z’ insulation = 0 if pipe in contact with earth), and
Z’earth
is the earth-return impedance discussed in Section 5.3. Transforming Eq. (5.38a) to phase quantities produces
same matrix as 0 Z’ Z’ ....Z’ Z’ e
for infinite 0 Z’ Z’ ....Z’ Z’e

[Z’ phase ] = pipe thickness . .............. . (5.39a)
. .............. .
Z’ Z’ ....Z’ Z’e
0 0 ... 0 Z’ Z’ ...Z’
e Z’
e e s
with Z ’s from Eq. (5.38c)
Z e’ = Z ’s - Z ’m (5.39b)
Z’ = Z s’ - 2Z ’m
The last row and column in Eq. (5.39a) represent the pipe quantities, while the first 9 rows and columns refer to core,
sheath (shield tapes), armor (skid wires) of phases a, b, and c.
If the pipe is in contact with the earth, then the shunt admittance matrix is the same as in Section 5.4.1. If it
is insulated, then the potential coefficient matrix of Eq. (5.34) must be expanded with one extra row and column for the
pipe, and the same element
5 21
Page 171
1 TRKRG&KPUWNCVKQP
2)' ln (5. 0)
2Bg gT TRKRG&QWVUKFG
must be added to this expanded matrix,
same as in 0 P’ P’ ....P’
Eq. (5.34) 0 P’ P’ ....P’
[P’ phase ] = . + .............. (5.41)
. P’ P’ ....P’
0 0 ... 0
The admittance matrix is then again found by inversion with Eq. (5.37).
5.5 Building of Conductors and Elimination of Grounded Conductors

Conductors are sometimes connected together ("bundled"). For example, the concentric neutral conductors
in the cable of Fig. 5.2 are in contact with each other, and therefore electrically connected. In a pipe-type cable, the
shield tapes and skid wires are in contact with the pipe. In a submarine cable, the sheath is often bonded to the armor
at certain intervals, to avoid voltage differences between the sheath and armor.
In such cases, the connected conductors 1,...m can be replaced by (or bundled into) one equivalent conductor,
by introducing the bundling conditions
I 1+ I +...I
2 = Im; V =equiv
V = ...1 V = V2 m equiv (5.42)
into the equations for the series impedance and shunt admittance matrices. The bundling procedure for reducing the
equations from m individual to one equivalent conductor is the same as Method 1 of Section 4.1.2.2 for overhead lines,
and is therefore not explained again. It is exact if the conductors are continuously connected with zero connection
resistance (as the neutral conductors in Fig. 5.2), and accurate enough if the connections are made at discrete points with
negligible resistance (as in bonding of the sheath to the armor), as long as the distance between the connection points
is short compared to the wavelength of the highest frequency in the transient simulation.
As in the case of overhead lines with ground wires, some conductors in a cable may be grounded. For example,
the steel pipe of a pipe-type cable can usually be assumed grounded, because its asphalt mastic coating is not an electric
insulation. Also, neutral conductors may be connected to ground at certain Traduciendo...

intervals, or at both ends. If a conductor
i is grounded, then the condition is simply
V i= 0 (5.43)
and conductor i can then be eliminated from the system of equations in the same way as described in Section 4.1.2.1.
Again, the elimination is only exact if the conductor is grounded continuously with zero grounding resistance, and
accurate enough if the distance between discrete grounding points is short compared to the wavelength of the highest
frequency.
An example of bundled as well as grounded conductors would be a single-core submarine cable which has its
sheath bonded to the armor. Since the asphalt coating of the armor is not an electric insulation, the armor is in effect
5 22
Page 172
in contact with the sea water, and both sheath and armor are therefore grounded conductors. By eliminating both of
them, the submarine cable can be represented by single-phase equations for the core conductor, with the current return
combined in sea water, armor and sheath. For an overhead line, the equivalent situation would be a single-phase line
with two ground wires.
The case of segmented ground wires in overhead lines discussed in Section 4.1.2.5(b) can exist in cables as
well. For example, if the sheath is grounded at one end, but open and ungrounded at the other end, then the sheath could
be eliminated in the same way as segmented ground wires, provided the cable length is short compared to the wavelength
of the highest frequency. The support routine CABLE CONSTANTS does not have an option for such eliminations.
The user must represent the sheath as an explicit conductor, instead, with one end connected to ground. This offers the
advantage that the induced voltage at the other end can automatically be obtained, if so desired.
5.6 Buried Pipelines
Pipelines buried close to power lines can be subjected to hazardous induction effects, especially during single-
line-to-ground faults. To study these effects, one can include the pipeline as an additional conductor into the
transmission line representation (Fig. 5.14(a)). For steady-state analysis, one can also use the single-phase
representation of Fig. 5.14(b), with an impressed voltage
steady-state analysis, one can also use the single-phase representation of Fig. 5.14(b), with an impressed voltage
(a) polyphase representation (b) single phase representation
(KI Pipeline representation (g ground wire, a, b, c phase

conductors, p pipeline)
F8KPFWEGF ) ) ) )
& '< (5. )
FZ RC+C % < RD+D % < RE+E % < RI+I
There is no capacitive coupling between the power line and the pipeline if it is buried in the ground.
As explained later, nominal -circuits can only be used for very short lengths of pipeline (typically # 0.3 km
at 60 Hz). The single-phase representation is therefore preferable for steady-state analysis, because the distributed
parameters of Fig. 5.14(b) are more easily converted into an exact equivalent -circuit than the polyphase parameters
of Fig. 5.14(a). This results in the active equivalent -circuit of Fig. 5.15, with Y series and Y shunt being the usual
5 23
Page 173
Traduciendo...
parameters obtained from Eq. (1.14), while I induced is an active current [158],
& F8KPFWEGF/FZ
+KPFWEGF ' (5. 5)
<
)
RR
(KI #ctive eSuivalent B circuit
The correctness of the active -circuit can easily be shown. Starting from the differential equations
F8
& '< ) )
FZ RR % < RR +KPFWEGF
F+ )
& ';
FZ RR8
the introduction of a modified current
I modified = I + I induced
transforms the differential equations into the normal form of the line equations, with the assumption that I induced does not
change along the line (dI modified /dx = dI/dx),
&
F8
'<
)
FZ RR +OQFKHKGF
F+OQFKHKGF )
& ';
FZ RR8
The solution for a line between nodes k and m is given in Eq. (1.13), except that the current is now I modified , or rewritten,
+MO % +KPFWEGF ;UGTKGU % (1/2);UJWPV & ;UGTKGU 8M

'
+OM & +KPFWEGF & ;UGTKGU ;UGTKGU % (1/2);UJWPV8O
This is exactly the same equations which comes out of the equivalent circuit of Fig. 5.15.
52
Page 174
With this single-phase approach, the currents in the power line are assumed to be known, e.g., from the usual
type of short-circuit study. It is also assumed that they are constant over the length of the exposure to the pipeline, and
that the pipeline runs parallel to the power line (mutual impedances constant). If either assumption is not true, then the
power line-pipeline system must be split up into shorter sections as is customarily done in interference studies. The
effect of the pipe on the power line zero sequence impedance is usually ignored, but could be taken into account.
In both representations of Fig. 5.14, the mutual impedances between the pipe and the overhead conductors, as
well as the self impedance of the pipe with earth return, are needed. The mutual impedances are obtained with the
formulas discussed in Section 5.3.4. At 60 Hz, Carson’s formula will give practically identical results as the more
complicated formula of Pollaczek.
The self impedance Z’ ofppthe pipeline consists of the same three terms shown for the armor in Eq. (5.4). The
first two terms are calculated with Eq. (5.7b) and (5.6), while R’ is found from
earth the equations discussed in Section
5.3.
Traduciendo...
For the shunt admittance Y’ = G’pp + j C’, the capacitive part is calculated in the usual way with Eq. (5.13).
In contrast to the underground cable, the shunt conductance G’ of the pipeline can no longer be ignored. The insulation
around pipelines is electrically poor, either originally or because of puncturing during the laying operation. The loss
angle in Eq. (5.14) is so large on pipelines insulated with glass-fiber/bitumen that G’ becomes much larger than C’
at power frequency, and if one part of the shunt admittance is ignored it should be C’ rather than G’. On PVC-insulated
pipelines, G’ may still be smaller than C’, though.
If the shunt resistance of the insulation is relatively small, then the grounding resistance of the pipe should be
2
connected in series with it [170], or
1
) )' (5. )
4
) )
ITQWPFKPI
KPUWNCVKQP % 4
where R’ insulation = resistance of pipe insulation,
R’grounding = grounding resistance.
A useful formula for the grounding resistance is [170]
ý ý
(2J) % %
DGCTVJ 2ý 2 2
4 ) 2ln % ln (5. 7)
ITQWPFKPI ' B &
ý ý
(2J) % &
2 2
with earth = earth resistivity (e.g., in m),
h = depth of burial of pipe

ý = length of pipe
If the sheath, armor, or pipe of an underground cable or the ground wire of an overhead line is grounded,
then it has been standard practice to ignore the grounding resistance (V 0). #n alternative would be to use a
finite shunt admittance ; 1/4 s„ ‡€pu€s, as recently suggested [1 ].
5 25
Page 175
D = outside diameter of pipe.
Grounding grids must generally be analyzed as three-dimensional problems, even if they consist of only one
pipe. The grounding resistance from Eq. (5.47) is therefore no longer an evenly distributed parameter, but depends on
the length. Fortunately, the dependence of G’ on length is very small for typical values of G’ insulation [158]. In the region
of measured values for G’ between 0.1 S/km for newly-layed pipelines and 0.3 S/km for older pipelines with glass
fiber/bitumen insulation [170], the dependence of G’ on length is practically negligible, as shown in Fig. 5.16. Treating
G’ as an evenly distributed parameter is therefore a reasonable approximation.
(KI Shunt conductance of buried pipe
Traduciendo...
Because of G’ >> C’, the wavelength of buried pipelines is significantly shorter than that of underground
cables, as shown in Table 5.3 [170]. Therefore, a nominal -circuit of a circuit which includes a buried pipeline should
not be longer than approximately 0.8 km for
Table 5.3 - Wavelength of pipeline at 50 Hz [170]
G’ (S/km) wavelength (km)
0.1 41.3
1.0 13.1
10.0 4.13
52
Page 176
steady-state analysis, or approximately 0.08 km for switching surge studies [158].
Fig. 5.17 shows a comparison between measured and calculated voltages and currents in a pipeline, induced
by currents in a neighboring power line, with the pipeline representation as discussed here [158].
(KI Induced voltages and currents in a buried pipeline
5.7 Partial Conductor and Finite Element Methods
The support routine CABLE CONSTANTS uses analytical formulas which are essentially only applicable to
configurations with axial symmetry. The formulas for the nonconcentric configuration in pipe-type cables (Section 5.5)
are only approximate, and the authors of these formulas themselves suggest improvements along the lines discussed
here.
To find the impedances and capacitances for conductor systems with arbitrary shapes (e.g., for the cable of Fig.
5.1), numerical methods can be used in place of analytical formulas, which are either based on subdivisions into partial
conductors or on finite element methods. There is no support routine yet in the EMTP which uses these numerical
methods. The principle of these methods is therefore only outlined very briefly.
5 27
Traduciendo...
Page 177
5.7.1 Subdivision into Partial Conductors
With this method, each conductor is subdivided into small "partial" conductors ("subconductors" in [162],
"segments" in [164]), as shown in Fig. 5.18. Various shapes can be used for the partial conductors, with rectangles
being the preferred shape for strip lines in
(KI Subdivision of the main conductors into partial

conductors
integrated circuits (Fig. 5.19).
(KI Subdivision of strip lines into partial conductors

of rectangular shape [1 ]. Copyright 1979 by International
Business Machines Corporation reprinted by permission
In deriving the equations for the system of partial conductors, uniform current density is assumed within each
partial conductor. Then the voltage drops along a system of n partial conductors at one frequency are described by the
phasor equations
F8 /FZ 4 +
. . @ @.P
F8 /FZ 4 +
. . @ @.P
& @ ' @ % LT @ (5. )
@ @@@@
@ @ @
.P .P @ @ .PP
F8P/FZ 4P +P
The diagonal resistance matrix contains the dc resistances, and the full inductance matrix contains the self and mutual
inductances of the partial conductors. The formulas for the matrix elements depend on the shape of the partial
conductor, but they are well known.
To obtain the frequency-dependent impedance of a cable system, the matrices [R] and [L] are first computed.
At each frequency, the complex matrix [Z] = [R] + j [L] is formed, and reduced to the number of actual conductors with
Bundling Method 1 of Section 5.5. For example, if partial conductors 1,...50 belong to the core conductor, and partial
52
Page 178
conductors 51,...120 to the sheath, then this bundling procedure will reduce the 120 x 120-matrix to a 2 x 2-matrix,
which produces the frequency-dependent impedances
<EE(T) <EU(T)
.
<EU(T) <UU(T)
This numerical method works well as long as the conductors are subdivided into sufficiently small partial
conductors. The size of these partial conductors must be of the same order of magnitude as the depth of penetration.
5.7.2 Finite Element Methods
Finite element methods are more powerful than partial conductorTraduciendo...

methods in one sense, inasmuch as it is not
necessary to assume uniform current density within each element. However, it is very difficult to handle open-boundary
conditions with finite element methods, that is configurations where the magnetic field diminishes gradually as one
moves away from the conductors, with no clearly defined boundary of known magnetic vector potential reasonably close
to the conductors. In situations where a boundary is clearly defined, e.g., in pipe-type cables at high frequency where
the depth of penetration becomes much less than the wall thickness, finite element methods can be quite useful.
With finite element methods, the region inside and outside of the conductors is subdivided into small elements,
usually of triangular shape. Fig. 5.20(a) shows the example of a stranded conductor inside a pipe of radius R as the b
return path (clearly defined boundary with zero magnetic field A = 0 outside the pipe and zero derivative along the two
edges of the "wedge"). Because of axial symmetry, it is sufficient to analyze the "wedge" shown in Fig. 5.20(a). This
wedge region is then subdivided into triangular elements as shown in Fig. 5.20(b), with longer triangles as one moves
away from the conductor.

The magnetic vector potential A is assumed to vary linearly along the edges and inside of each triangle,
A = ax + by + c, (5.49)
when a first-order method is used (higher-order methods exist as well). The unknowns are essentially the values of A
in the node points. If they were shown in the z-direction of a three-dimensional picture, then the triangles would appear
in a shape similar to a geodesic dome, with the roof height being the value of A. The equations for finding A are linear
algebraic
5 29
Page 179
(a) Stranded conductor inside pipe of radius 4h
Traduciendo...
(b) Subdivision of region into triangular elements
Fig. 5.20 - Analysis of stranded conductor with finite element method [171]. Reprinted by permission of Yin Yanan
equations with a sparse matrix, but the number of node points or the number of equations is usually quite high (146
equations for the example of Fig. 5.20). Once the magnetic vector potential is known in the entire region, the
impedances can be derived from it.
For readers interested in finite element methods for cable impedance calculations, the papers by Konrad, Weiss
and Csendes [165, 166, 167] are a good introduction.
5.8 Modal Parameters

Once the series impedance and shunt admittance matrices per unit length [Z’ ], [Y’ ] are known,
phase the phase
derivation of modal parameters is exactly the same as described in Section 4.1.5 for overhead lines. They could be used,
for example, to develop exact equivalent -circuits for steady-state solutions as explained in Section 4.2.1.3.
5 30
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For transient simulations, it is more difficult to use modal parameters, as compared to overhead lines, because
the transformation matrix [T] can

i no longer be assumed to be constant as for a single-circuit overhead line. Fig. 5.21
shows the variation of the elements in the third column of [T ] for a typical
i three-phase arrangement of 230 kV single-
core cables with core conductor and sheath in each [155]. Especially around the power frequency of 50 or 60 Hz, the
variations are quite pronounced.
(KI Magnitude of the elements of column 3 of [T ]u
Above a few kHz, the loop between core conductor and sheath becomes decoupled from the outer loop between sheath
and earth return, because the depth of penetration on the inside of the sheath for loop 1 becomes much smaller than the
sheath thickness. In that case, Z tube-mutual ~ 0. This makes the transformation matrix constant above a few kHz, as evident
from Fig. 5.21. For a single-phase single-core cable with sheath and armor, the three modes are identical with the 3
loops described in Eq. (5.1) at high frequency where Z’ ~ 0 and

12 Z’ ~ 0. The23transformation matrix between loop and
phase quantities of Eq. (5.9),
100 1 00
[6K]
&' 110 CPF [6K] ' &1 1 0 (5.50)
111 0 &1 1
5.9 Cable Models in the EMTP
Co-author: L. Marti
As of now (Summer 1986), there are no specific cable models in the BPA EMTP. The only way to simulate
cables is to fit cable data into the models available for overhead lines. It has long been recognized, of course, that this
is only possible in a limited number of cases. A method specifically developed for cables, as discussed in Section
5.9.2.3, will hopefully be implemented in late 1986 or early 1987. It has already been tested extensively in the UBC
Traduciendo...
5 31
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EMTP.
5.9.1 Ac Steady-State Solutions

In principle, there is no difficulty in representing cables as nominal or equivalent -circuits in the same way
as overhead lines (Section 4.2.1). If nominal -circuits are used, it should be realized that the wavelength of
underground cables is shorter than on overhead lines. If a nominal -circuit should not be longer than 100 km at 60 Hz
for overhead lines, the limit is more typically 30 km for underground cables. If a pipeline is modelled, the limit can be
as low as 1 km, as discussed in Section 5.6.
Underground cables are often very short compared to the length of connected overhead lines. In such cases,
the (complicated) series impedances have very little effect on the results because the system sees the cable essentially
as a shunt capacitance. The cable can then be modelled as a simple lumped capacitance.
5.9.2 Transient Solutions
The accurate representation of cables with frequency-dependent impedances and frequency-dependent

transformation matrices is discussed in Section 5.9.2.3. Situations where simpler models should be accurate enough
are discussed first.
5.9.2.1 Short Cables
If a rectangular wave pulse travels on an overhead line and hits a relatively short underground cable, then the
cable termination is essentially seen as a lumped capacitance. The voltage then builds up exponentially with a time
constant of T = Z overhead •C cable

, shown in Fig. 5.22(a). If the cable is modelled somewhat more accurately as a lossless
distributed-parameter line, then the voltage build-up has the staircase shape of Fig. 5.22(b), with the average of the
sending and receiving end curve being more or less the same as the continuous curve in Fig. 5.22(a). As long as the
travel time [] of the cable is short compared to the time constant T, reasonably accurate results can be obtained if the
cable is represented as a lumped capacitance.
(a) Cable represented as lumped (b) Cable represented as
5 32
Page 182
capacitance lossless transmission line

------ sending end of cable
...... receiving end of cable
Fig. 5.22 - Voltage build-up in a cable connected to an overhead line
Nominal -circuit representations have often been suggested as approximate cable models. They obviously
represent the capacitance effect correctly, but the pronounced frequency-dependence in the series impedance is ignored.
Traduciendo...
Nominal -circuits give reasonable answers probably only in those cases in which the simpler lumped capacitance
representation is already accurate enough.
5.9.2.2 Single-Phase Cables
There are situations where single-phase representations are possible. An example is a single-phase submarine
cable in which the sheath and armor are bonded together, with the armor being in contact with the sea water. In such
a case, the sheath and armor can be eliminated from Eq. (5.10), which results in the reduced single-phase equation
F8E
& '< )
FZ EQTG +E
with Z’ core
being the impedance of the core conductor with combined current return through sheath, armor and sea water.
Coupling to the cables of the other two phases can be ignored because the three cables are layed relatively far apart, to
reduce the risk of anchors damaging more than one phase in the same mishap.
When the equations have been reduced to single-phase equations, then it is straightforward to use the
frequency-dependent overhead line model described in Section 4.2.2.6.
Sometimes it is not necessary to take the frequency-dependence in the series impedances into account. For
example, single phase SF -busses

6 have been modelled quite successfully for fast transients with two decoupled lossless
single-phase lines, one for the inside coaxial loop and a second one for the outside loop between the enclosure and the
earth-return. The coupling between the two loops through the enclosure is negligible at high frequencies because the
depth of penetration is much less than the enclosure wall thickness. The only coupling occurs through reflections at
the terminations. Agreement between simulation results from such simple models and field tests has been excellent
[169].
5.9.2.3 Polyphase Cables [155]
The simple overhead line models with constant parameters discussed in Section 4.2.2 are of limited use for
underground cables for two reasons:
(a) The transformation matrix [T] is frequency-dependent

i up to a few kHz, though a constant [T] would be i
acceptable for transients which contain only high frequencies (e.g., lightning surge studies).
(b) The modal parameters (e.g., wave velocity and attenuation) are more frequency-dependent than on overhead
lines, as shown in Fig. 5.23 for three single-core cables with core and sheath [150].
5 33
Page 183
(a) #ttenuation
Traduciendo...
(b) Velocity
Fig. 5.23 - Modal parameters as a function of frequency [150]. Reprinted by permission of IEE and
the authors
To derive an accurate model for an n-conductor cable system between nodes k and m, we can start from the
phasor equation (4.121) for the overhead line, if we replace that scalar equation, which was written for one phase or
mode, by a matrix equation for the n conductors,
[;E][8M] & [+MO] ' [#]6[;E][8O] % [+OM]> (5.51)
53
Page 184
-1
with [Y ]c = [Z ] =c characteristic admittance matrix in phase quantities,
[A] = e =-j propagation

ký factor matrix.
Eq. (5.51) is transformed to modal quantities, with
[+] ' [6K] [+OQFG] (5.52a)
and
[8] ' [6K V]& [8OQFG] (5.52b)
which yields
[+MO&OQFG] ' [;E&OQFG][8M&OQFG] & [#OQFG]6[;E&OQFG][8O&OQFG] % [+OM&OQFG]>
(5.53)
with both [Y c-mode ] and [A mode ] being diagonal matrices,
[;E&OQFG] ' [6K]

& [;E] [6K V]&
(5.5 a)
[#OQFG] ' [6K]

& [#] [6K]
(5.5 b)
The diagonal element of [A mode ] is obtained from the i-th eigenvalue of the product
i jY’ k jZ’ k, phase phase
&ý8K
#OQFG&K ' G (5.5 c)
and [T]i is the matrix of eigenvectors of the same product [Y’ ] [Z’ ]. Eq.
phase(5.53)phase
consists of n decoupled (scalar)
equations, with one equation for each mode.

Transforming these scalar equations into the time domain is the same procedure as described in Section 4.2.2.6
for the overhead line. For mode i, the second term in Eq. (5.53) is found with the same convolution integral as in Eq.
(4.124),
JOCZ
JKUVRTQRCICVKQP' & KO&VQVCN(V&W)C(W)FW HQT GCEJ OQFG (5.55)
mJOKP
with the current i m-total being the sum of the line current i andmkof a current which would flow through the characteristic
impedance of mode i if the voltage v of mode

m i were connected across it. Only known history terms appear in Eq.
(5.55), and hist propagation can therefore be found by n recursive convolutions for the n modes, in the same say as in Section
4.2.2.6. The modal propagation factors are very similar in shape to those of an overhead line, as shown for A mode-3 ()
in Fig. 5.24.
Traduciendo...
5 35
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(KI Magnitude of propagation factor for mode 3 of a conductor

system (three single core cables with core and sheath in each)
With
propagation of the conditions from m to k being taken care of through Eq. (5.55), the only unresolved issue in the modal
domain equations is the representation of the term Y V in Eq.

c k (5.53). Again, the frequency dependence of Y is similar c
to that of an overhead line, as shown in Fig. 5.25, and can be represented with the same type of Foster-I R-C network
shown in Fig. 4.42(a), and reproduced here as Fig. 5.26. By applying the trapezoidal rule of integration to the
capacitances, or by using recursive convolution as discussed in Appendix V, the R-C
(KI Magnitude of characteristic admittance for mode 3 (same

conductor system as in Fig. 5.2 )
53
Page 186
Traduciendo...
(a) 4 C network (b) ESuivalent resistance after applying implicit integration to C
(KI 4epresentation of one mode seen from side k
network is converted into an equivalent conductance G equiv in parallel with a known current source hist + hist RC propagation .
After the network solution at each time step, the current flowing through the characteristic impedance represented by
the R-C network must be calculated for both ends of the cable from G equiv v + hist ,RCbecause this term is needed after
the elapse of travel time to form the expression i m-total needed in Eq. (5.55).
From Fig. 5.26(b), it can be seen that each mode is now represented by the scalar, algebraic equation
ikm(t) = G v (t) + (hist +RChist

equiv k propagation ) (5.56)
with an analogous equation for i (t) at mk

the other end. If the transformation matrix were constant and real, then Eq.
(5.56) could very easily be transformed back to phase quantities,

t
[i km-phase (t)] = [T ][G
i equiv ][T i] [v k-phase ] + [T ][hist]
i
as explained in Eq. (4.109) for the overhead line. As shown in Fig. 5.21, the transformation matrix [T] of cables is ivery
much frequency-dependent, and the transformation back to phase quantities now requires convolutions based on Eq.
(5.52),
V
[KRJCUG(V)] ' [VK(V&W)] [KOQFG(W)]FW (5.57a)
m&4
V
[XOQFG(V)] ' [VK(V&W)]V [XRJCUG(W)]FW (5.57b)
m&4
where [t ]i is a matrix obtained from the inverse Fourier transform of the frequency-dependent matrix [T ]. Similar to i
the curve fitting used for the modal characteristic impedances, each element of [T ] is again approximated
i by rational
functions of the form
5 37
Page 187
O MK
6 z< (T) ' M %j (5.5 )
K'
LT % RK
with k ,0 k and
i p being
i real constants which, when transformed into the time domain, becomes
O
Vz< (V) ' M j MK exp(&RKV) W(V) (5.59)
F(V) %
K'
With the simple sum of exponentials in Eq. (5.59), recursive convolution can be applied again (Appendix V). Then,
the convolution integrals in Eq. (5.57) can be split up into a term containing the yet unknown voltages and currents at
time t, and the known history terms which can be updated recursively,
[KRJCUG(V)] ' [V ] [KOQFG(V)] % [JKUVEWTTGPV] (5. 0a)
[XOQFG(V)] ' [V ]V [XRJCUG(V)] % [JKUVXQNVCIG] (5. 0b)
with [t ]0 being a real, constant n x n-matrix. With Eq. (5.60), the transformation of the modal equations (5.56) to phase
quantities is now fairly simple,
[KMO&RJCUG(V)] ' [)RJCUG] [XM&RJCUG(V)] % [JKUVRJCUG] (5. 1a)

Traduciendo...
with
[)RJCUG] ' [V ] [)RJCUG] [V ]V (5. 1b)
and the history term
[hist ]phase
= [hist current ] + [t ]{[G
0 equiv ][hist voltage ]
+ [hist ]RC+ [hist propagation ]} (5.61c)
Since the form of Eq. (5.61a) is identical to that of Eq. (4.109) for the overhead line with constant [T ], adding thei model
to the EMTP is the same as described there. The extra effort lies essentially in the evaluation of the two extra history
vectors [hist current ] and [hist voltage ]. After the network solution at each time step, Eq. (5.60) is used to obtain the modal
quantities from the phase quantities.
The principle of the frequency-dependent cable model is fairly simple, but its correct implementation depends
on many intricacies, which are described in [155]. In particular, it is important to normalize the eigenvectors in such
a way that the elements of [T] as well

i as the modal surge admittances Y c-mode-i both become minimum phase shift
functions. This is achieved by making one element of each eigenvector a real and constant number through the entire
frequency range. Furthermore, standard eigenvalue/eigenvector subroutines do not produce smooth curves of [T ] and i
[Y c-mode ] as a function of frequency, because the order in which the eigenvalues are calculated often changes as one
moves from one frequency point to the next. This problem was solved by using an extension of the Jacobi method for
complex symmetric matrices. Symmetry is obtained by reformulating the eigenproblem
53
Page 188
[;
) )
RJCUG] [< RJCUG] [Z] ' 8[Z]
in the form
[*] [T] ' 8[T] (5. 2a)
where
[*] ' [.]V [<

)
RJCUG] [.] (5. 2b)
and
[Z] ' [.] [T] (5. 2c)
with [L] being the lower triangular matrix obtained from the Choleski decomposition of [Y’ ] [157]. The Choleski
phase
decomposition is a modification of the Gauss elimination method, as explained in Appendix III. One can also replace
[L] in Eq. (5.62) with the square root of [Y’ ] obtained

phase from
[; ) &
(5. 3)
RJCUG] ' [:] [7 ] [:]
1/2
where [ ] is the diagonal matrix of the square roots of the eigenvalues, and [X[ is the eigenvector matrix of [Y’ ]. phase
Both approaches are very efficient if G’ is ignored, or if tan is constant for all dielectrics in the cable system, because
1/2
[L] or [Y’ ]phase
must then only be computed once for all frequencies.
For parallel single core cables layed in the ground (not in air), [Y’] is diagonal if loop equations are used. In
1/2
that case it is more efficient to find the eigenvalues and eigenvectors for [Y’ ][Z’ ], whereloopboth [L]
loop and [Y’ ] loop
become the same diagonal matrix with %Y’ as itsloop-i

elements. The conversion back to phase quantities is trivial with
Eq. (5.50).
The reason why the Jacobi procedure produces smooth eigenvectors is that the Jacobi algorithm requires an
initial guess for the solution of the eigenvectors. This initial guess is readily available from the solution of the
eigenproblem of the preceding frequency step; consequently, the order of the eigenvectors from one calculation to the
next is not lost.
Figure 5.27(a) shows the magnitude of the elements of row 3 of the eigenvector matrix [T ] for the same 6-
i
conductor system as in Fig. 5.24, when standard eigenvalue/eigenvector routines are used. Fig. 5.27(b), on the other
i
hand, shows the same elements of [T ] calculated with the modified Jacobi algorithm.
As an application for this cable model, consider the case of three 230 kV single-core cables (with core and
sheath), buried side by side in horizontal configuration, with a length of 10 km. A unit-step voltage is applied to the
core of phase A, and the cores of phases B and C as well as all three sheaths are left ungrounded at both ends. The unit-
Traduciendo...
step function was approximated as a periodic rectangular pulse of 10 ms duration and a period of 20 ms with a Fourier
series containing 500 harmonics,
5 39
Page 189
X(V) ' C %j 6CKcos(TKV) % DKsin(TKV)>

K'
The wave front of this approximation is shown in Fig. 5.28. Choosing a Fourier series
(KI Fourier series approximation of unit step
approximation for the voltage source offered the advantage that exact answers could be found as well, by using ac
steady-state solutions with exact equivalent -circuits (Section 4.2.1.3) at each of the 500 frequencies, and by
superimposing them. Fig. 5.29 and 5.30 show the EMTP simulation results in the region of the third pulse,
superimposed on the exact answers. The two
(a) Standard eigenvalue/eigenvector subroutines
50
Page 190
Traduciendo...
(b) Modified Jacobi algorithm
Fig. 5.27 -
Magnitude of the elements of row 3 of [T ] (same
i 6-conductor system as in Fig. 5.24)
curves are indistinguishable in this third pulse region where the phenomena have already become more or less periodic.
This shows that the EMTP cable model is capable of producing highly accurate answers. The insert on the right-hand
side of Fig. 5.29 shows the response to the first pulse, where the EMTP simulation results differ slightly from the exact
answers, not because of inaccuracies in the model but because the EMTP starts from zero initial conditions while the
exact answer assumes periodic behavior even for t < 0.
(KI Step response, receiving end voltage of core (phase #)
51
Page 191
(KI Step response, receiving end voltage of sheath (phase #)
Traduciendo...
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64#05(14/'45
The first representation of transformers in the EMTP was in the form of branch resistance and inductance
matrices [4] and [L]. The support routine XFO4ME4 was written to produce these matrices from the test data of
single phase two and three winding transformers. Stray capacitances are ignored in these representations, and they
are therefore only valid up to a few kHz.
# star circuit representation for N winding transformers (called saturable transformer component in the
BP# EMTP) was added later, which uses matrices [4] and [L] with the alternate eSuation
[.]& [X] ' [.]& [4] [K] % [FK/FV] ( .1)
in the transient solution. This formulation also became useful when support routines BCT4#N and T4ELEG were
developed for inductance and inverse inductance matrix representations of three phase units. #n attempt was made
to extend the star circuit to three phase units as well, through the addition of a zero seSuence air return path
reluctance. This model has seldom been used, however, because the zero seSuence reluctance value is difficult to
obtain.
Saturation effects have been modelled by adding extra nonlinear inductance and resistance branches to the
inductance or inverse inductance matrix representations, or in the case of the star circuit, with the built in nonlinear
magnetizing inductance and iron core resistance. # nonlinear inductance with hysteresis effects (called pseudo
nonlinear hysteretic reactor in the BP# EMTP) has been developed as well. #n accurate representation of
hysteresis and eddy current effects, of skin effect in the coils, and of stray capacitance effects is still difficult at this
time, and some progress in modelling these effects can be expected in the years to come.
Surprisingly, the simplest transformer representation in the form of an ideal transformer was the last
model to be added to the EMTP in 19 2, as part of a revision to allow for voltage sources between nodes.
6TCPUHQTOGTU CU 2CTV QH 6JGXGPKP 'SWKXCNGPV %KTEWKVU
If a disturbance occurs on the high side of a step up transformer, then the network behind that transformer,
plus the transformer itself, is usually representation as a voltage source behind 4 L branches. Since the transformer
inductances tend to filter out the high freSuencies, such a low freSuency 4 Traduciendo...
L circuit appears to be reasonable.
To explain the derivation of such Thevenin eSuivalent circuits, the practical example of Fig. .1 shall be
used [ 0], where the feeding network consists of three generators and two three winding transformers. The
transformer short circuit reactances are X 0.117 BF

p.u., X 0.115 p.u., X 0.2
BT1 p.u., and the generator
FT
reactance is X 0.13p 5 p.u., all based on 100
Page 193
three winding
transformers
The th generator was disconnected

for acceptance testing.
(KI Network configuration for various field tests at CEMIG, Brazil [ 0]
MV# at 0 Hz. With the well known eSuivalent star circuit for three winding transformers (see Section .3.2), the
power plant in Fig. .1 can be represented with the positive and zero seSuence networks of Fig. .2. For simplicity,
resistances are ignored, but they could easily be included. It is further assumed here that the zero seSuence reactance
values of the transformer are the same as the positive seSuence values, which is only correct for three phase banks
built from single phase units, but not Suite correct for three phase units (if the zero seSuence values were known,
(a) Positive seSuence (negative (b) <ero seSuence

seSuence identical, except that
voltage sources are shorted)
Page 194
Traduciendo...
(KI ESuivalent circuits for the power plant (reactance values in p.u. based
on 100 MV# at 0 Hz)
then those values could of course be used in Fig. .2(b)). Furthermore, the generator is modelled as a symmetrical
voltage source E behind X . Notepthat the delta connected windings act as short circuits for zero seSuence currents
in Fig. .2(b), while the generators are disconnected to force I 0. The zero seSuence
’q„ parameters of the
generators are therefore irrelevant in this example.
The networks of Fig. .2 can now be reduced to the three Thevenin eSuivalent circuits of Fig. .3, which
in turn can be converted to one three phase Thevenin eSuivalent circuit as shown in Fig. . . This three phase
circuit is used in the EMTP for the representation of the power plant, with the data usually converted from p.u. to
actual values seen from the 3 5 kV side (X X 99.90

‚ S, X 33.17
€qs S, or X 77. 5 S, ’q„
X 22.25 … y
S at 0 Hz). The symmetrical voltage sources E , E , E behind

g the
h coupled
i inductances in Fig. . are the open
circuit voltages of the power plant on the 3 5 kV side. In the transient simulation, the matrix [X] is obviously
replaced by the inductance matrix [L].
(KI Thevenin eSuivalent circuits in seSuence Suantities
(KI Three phase Thevenin eSuivalent circuit in phase Suantities
+PFWEVCPEG /CVTKZ 4GRTGUGPVCVKQP QH 5KPING 2JCUG 6YQ CPF 6JTGG 9KPFKPI 6TCPUHQTOGTU
Transformers can only be represented as coupled [4] [L] branches if the exciting current is not ignored.
The derivations are fairly simple, and shall be explained with specific examples.
Page 195
6YQ 9KPFKPI 6TCPUHQTOGTU
#ssume a short circuit reactance of 10 , short circuit losses of 0.5 , and an exciting current of 1 , based
on the ratings V , S„g†u€s

„g†u€s of the transformer. The excitation losses are ignored, but could be taken into account as
explained in Section . . If the given Suantities are < , load losses

‚‡ P , and power rating
x ……S , then
„g†u€s the resistance
and reactance part of the short circuit impedance are
4RW ' 2NQUU / 5TCVKPI ( .2a)
:RW ' < RW ( .2b)

RW & 4
Since the load losses do not give any information about their distribution between windings 1 and 2, it is best to
assume
1
4 RW ' 4 RW ' 4RW Traduciendo... ( .2c)
2
If the winding resistances are known, and not calculated from P , then 4 and x4……
may of course
‚‡ be different,
‚‡
and 4 4‚‡4 is then

‚‡ used in‚‡ES. ( .2b). With the T circuit representation found in most textbooks, the p.u.
impedances are then as shown in Fig. .5. The short circuit impedance 0.005 j0.10 p.u. is divided into two eSual
parts, and the magnetizing reactance j99.95 p.u., which is purely imaginary when excitation losses are ignored, is
chosen to give an input impedance of 100 p.u. from one side, with the other side open, to make the exciting current
0.01 p.u. (the resistance 0.0025 p.u. is so small compared to 100 p.u. that it can be ignored in finding the value
j99.95). The eSuations with the branch impedance matrix in p.u. are then
(KI T circuit representation of transformer
8 RW 0.0025 0 100 99.95 + RW

' %L ( .3a)
8 RW 0 0.0025 99.95 100 + RW
for steady state solutions, or
8 K FK /FV
' [4] % [.] ( .3b)
8 K FK /FV
Page 196
for transient solutions, with [4] being the same matrix as in ES. ( .3a), and [L] 1 / T [X]. Most EMTP studies
are done with actual values rather than with p.u. values. In that case, the matrix in ES. ( .3) must be converted to
actual values, with
0.0025 8 0 100 8 99.95 8 8

1
[<] ' %L S (.)
5TCVKPI 0 0.0025 8 99.95 8 8 100 8
where S „g†u€sapparent power rating of transformer,
V , V voltage ratings of transformer.
ES. ( . ) gives the [4] and [X] matrices of coupled branches in S, as reSuired by the EMTP, with the correct turns
ratio V /V . If all Suantities are to be referred to one side, say side 1, then simply set V V in ES. ( . ).
It is important to realize that the branch impedance matrix [<] in ES. ( . ) does not imply that the two
coupled branches be connected as shown in the T circuit of Fig. .5. If it were indeed limited to that connection,
one could not represent a three phase bank in wye/delta connection, because both sides would always be connected
from node to ground or to some other common node. Instead, [<] simply represents two coupled coils (Fig. . ).
The connections are only defined through node name assignments. For example, if three single phase transformers
are connected as a three phase bank with a grounded wye connection on side 1 and a delta connection on side 2, then
the first transformer could have its two coupled branches from node H# to ground and from L# to LB, the second
transformer from HB to ground and LB to LC, and the third transformer from HC to ground and LC to L#. This
connection will also create the correct phase shift automatically (side 2 lagging behind side 1 by 30E for balanced
positive seSuence operation in this particular case).
Traduciendo...
(KI Two coupled coils
+NN %QPFKVKQPKPI QH +PFWEVCPEG /CVTKZ
The four elements in the [X] matrix of ES. ( .3) contain basically the information for the exciting current
(magnetizing reactance X 100 yp.u.), with the short circuit reactance being represented indirectly through the
small differences between X and X , and between X and X . If all four values were rounded to one digit behind
…t „†
the decimal point (X X X . 100 p.u.), then the short circuit reactance would be completely lost X
0). In most studies, it is the short circuit reactance rather than the magnetizing reactance, however, which influences
the results. It is therefore important that [X] be calculated and put into the data file with very high accuracy
Page 197
(typically with at least 5 or digits), to make certain that the short circuit reactance
:
: UJQTV ' : & UGGP HTQO UKFG 1 ( .5)
:
is still reasonably accurate. It is highly recommended to calculate X from ES. ( …t

.5),„†to check how much it differs
from the original test data. For a transformer with 10 short circuit reactance and 0. exciting current, the values
…t „†
of < , < , < would have to be accurate to within v0.001 to achieve an accuracy of v10 for X ! This
accuracy problem is one of the reasons why < , < , < cannot be measured directly in tests if this data is to contain
the short circuit test information besides the excitation test information. Mathematically, [X] is almost singular and
therefore ill conditioned, the more so the smaller the exciting current is. Experience has shown that the inversion
of [X] inside the EMTP does not cause any problems, as long as very high accuracy is used in the input data.
Problems may appear on low precision computers, however. The author therefore prefers inverse inductance matrix
representations, as discussed in Section .3.
6JTGG 9KPFKPI 6TCPUHQTOGTU
The impedance matrix of single phase three winding transformers can be obtained in a similar way with the
well known star circuit used in Fig. .2. In that circuit, the magnetizing reactance is usually connected to the star
point, but since its unsaturated value is much larger than the short circuit reactances, it could be connected to either
the primary, secondary or tertiary side as well. #ssuming that the exciting current for the example of Fig. .2 is
1 measured from the primary side, with excitation losses ignored, the magnetizing reactance in the star point would
then be 100.00 5 p.u. Then
100 100.00 5 100.00 5

[:] ' 100.00 5 100.12 0 100.00 5 R.W. (.)
100.00 5 100.00 5 100.12 0
The particular connection would again be established through the node names at both ends of the branches. For
example, the three branches could be connected from node H# to ground, L# to LB, and T# to TB. To convert
ES. ( . ) to actual values, divide all elements by the power rating S , and
„g†u€s multiply the first row and column with
voltage rating V , the second row and column with V , and the third row and column with V . !
The [4] and [X] matrices can either be derived by hand, or they can be obtained from the support routines
XFO4ME4, BCT4#N, or T4ELEG in the BP# version of the EMTP. The latter two support routines were
developed for three phase units, but can be used for single phase units as well.
+PXGTUG +PFWEVCPEG /CVTKZ 4GRTGUGPVCVKQP QH 5KPING 2JCUG 6YQ CPF 6JTGG 9KPFKPI 6TCPUHQTOGTU
If the exciting current is ignored, then the only way to represent transformers is with matrices [4] and [L] ,
which are handled by the EMTP as described in Section 3. .2. The author prefers this representation over all others,
because the matrices [4] and [L] are not ill conditioned, and because any value of exciting current, including zero,
Traduciendo...
Page 198
can be used. The built in star circuit in the BP# version of the EMTP uses this representation internally as well.
For three phase transformers, the conversion of the test data to [4] and [L] matrices is best done with the
support routine BCT4#N. For single phase units and for three phase transformers where < . < , the conversion
’q„ ‚…
is fairly simple, and can easily be done by hand, as explained next.
6YQ 9KPFKPI 6TCPUHQTOGTU
First separate the short circuit impedance into its resistance and reactance part with ES. ( .2). The [4] and
[TL] matrices in p.u. can then be written down by inspection from the eSuivalent circuit of Fig. .5 (after the
magnetizing inductance has been removed),
1 1
&
4 RW 0 :RW :RW
[4RW] ' CPF [T.RW]& ' ( .7)
0 4 RW 1 1
&
:RW :RW
The inverse branch reactance matrix [TL ] is the‚‡well known node admittance matrix of a series branch with p.u.
reactance X . For
‚‡ the example of Fig. .5, with exciting current ignored, the p.u. matrices would be
0.0025 0 10 &10
[4RW] ' , [T.RW]& ' (.)
0 0.0025 &10 10
The matrices in ES. ( .7) are converted to actual values with
4 RW8 0
1
[4] ' KP S ( .9a)
5TCVKPI 0 4 RW8
1 1
&
8 88
5TCVKPI
[T.]& ' KP 5 ( .9b)
:RW 1 1
&
88 8
with S „g†u€sapparent power rating
V , V voltage ratings.
ES. ( .9) contains the correct turns ratio V /V . If all Suantities are to be referred to one side, say side 1, then
simply set V V in ES. ( .9). To obtain [L] , the matrix in ES. ( .9) is simply multiplied with T.
#s already mentioned in Section 3.1.2, the two coupled branches described by ES. ( .9) can also be
represented as six uncoupled branches. Ignoring the resistances for the sake of this argument, and setting
Page 199
5TCVKPI
;'
L:RW8
8
V'
8
produces the steady state branch eSuations (3.3) and the alternate representations with uncoupled branches of Fig.
3.3.
6JTGG 9KPFKPI 6TCPUHQTOGTU
Separating 4 and X is more complicated now. Therefore, 4 shall be ignored in the following explanations.
4esistances can be included, however, if the support routines BCT4#N or T4ELEG are used (see Section .10.2
Traduciendo...
and .10.3). The starting point is the well known star circuit of Fig. .7. Its reactances are found from the p.u.
short circuit reactances X BF‚‡ , X BT‚‡ ,
(KI Star circuit for three winding

transformer with p.u. values based on voltage
ratings, or with actual values referred to one
side
X FT‚‡
, based on the voltage ratings and one common power base S . Since the power
hg…q transfer ratings S between BF
H L, S between
BT H T, and S betweenFTL T are usually not identical, a power base conversion is usually needed.
If we choose S 1.0hg…q
(in same units as power ratings S , S , S ), then BF BT FT
1 :*.RW :*6RW :.6RW

:*RW ' % &
2 :*. 5*6 5.6
1 :.6RW :*.RW :*6RW

:.RW ' % & ( .10)
2 5.6 5*. 5*6
1 :*6RW :.6RW :*.RW

:6RW ' % &
2 5*6 5.6 5*.
Page 200
For the example used in Section .1, with X 0.117 p.u.,BFX 0.115 p.u., X 0.2 1 BT
p.u. based on 100 FT
MV#, these star circuit reactances based on 1 MV# would be
:*RW ' &0.0000 5 , :.RW ' 0.001215 , :6RW ' 0.001195
Next, the well known star delta transformation is used to convert the star circuit of Fig. .7 into the delta
circuit of Fig. . ,
(KI Delta circuit
which gives us the susceptances
:6RW
$*.RW '
:
:.RW
$*6RW ' ( .11a)
: Traduciendo...
:*RW
$.6RW '
:
YKVJ : ' :*RW :.RW % :.RW :6RW % :*RW :6RW ( .11b)
For the numerical example,
$*.RW ' 9. , $*6RW ' 90 .371 , $.6RW ' &33. 95
Note that the susceptances in ES. ( .11a) are not the reciprocals of the short circuit reactances X used in ES. ( .10).
The p.u. matrix [TL ] based

‚‡ on S 1.0 is easily
hg…qobtained from Fig. . with the rules for nodal admittance
matrices as
Susceptance B is used here for the reciprocal of reactance X. This is not strictly correct, because
susceptance is the imaginary part of an admittance (which implies B 1/X).
Page 201
$*.RW % $*6RW &$*.RW &$*6RW
&$*.RW $*.RW % $.6RW &$.6RW ( .12)

[T.RW]& '
&$*6RW &$.6RW $*6RW % $.6RW
or for the numerical example,
1793. 55 & 9. &90 .371
[T.RW]& ' & 9. 55.9 9 33. 95 DCUGF QP 1 /8#

&90 .371 33. 95 70. 7
The matrix [TL ] in‚‡actual values is found as
1UV TQY CPF EQNWOP QH ( .12) OWNVKRNKGF YKVJ 1/8*
2PF TQY CPF EQNWOP QH ( .12) OWNVKRNKGF YKVJ 1/8. ( .13)

T. & ' KP 5
3TF TQY CPF EQNWOP QH ( .12) OWNVKRNKGF YKVJ 1/86
This matrix will contain the correct turns ratios. If all Suantities are to be referred to one side, say side H, then
simply set V VF V in ES.

T ( .13).
B Since the p.u. values are based on 1 MV#, the voltages in ES. ( .13) must
be in kV.
/CVTKZ 4GRTGUGPVCVKQP QH 5KPING 2JCUG 0 %QKN 6TCPUHQTOGTU
The newer support routines BCT4#N and T4ELEG are not limited to the particular case of two or three
coils, but work for any number of coils. If each winding is represented as only one coil , then transformers with
more than three coils will seldom be encountered, but if each winding is represented as an assembly of coils, then
transformer models for more than three coils are definitely needed. Breaking one winding up into an assembly of
coils may well be reSuired for yet to be developed high freSuency models with stray capacitances.
To explain the concept, only single phase N coil transformers are considered in this section. The extension
to three phase units is described in Section .5. For such an N coil transformer, the steady state eSuations with a
branch impedance matrix [<] are
coils.#[7coil is an aassemblage
] Since of successive
winding may convolutions
either be represented of a or
as one conductor, whereas
as more coils, the amore
winding is anterm
general assembly
coil is of
used here.
10
Traduciendo...
Page 202
8 < < ... < 0 +
8 < < ... < 0 +
. .
' ( .1 )
. . .. .. .
. .
80 <0 <0 ... <00 +0
The matrix in ES. ( .1 ) is symmetric. Its elements could theoretically be measured in excitation tests: If coil k is
energized, and all other coils are open circuited, then the measured values for I and V ,...V produce
w column kHof
the [<] matrix,
<KM ' 8K / +M ( .15)
Unfortunately, the short circuit impedances, which describe the more important transfer characteristics of
the transformer, get lost in such excitation measurements, as mentioned in Section .2. It is therefore much better
to use the branch admittance matrix formulation
[+] ' [;] [8] ( .1 )
which is the inverse relationship of ES. ( .1 ). Even though [<] becomes infinite for zero exciting current, or ill
conditioned for very small exciting currents, [;] does exist, and is in fact the well known representation of
transformers used in power flow studies. Furthermore, all elements of [;] can be obtained directly from the standard
short circuit test data, without having to use any eSuivalent circuits. This is especially important for N 3, because
the star circuit saturable transformer component in the BP# EMTP) is incorrect for more than three coils.
For an intermediate step in obtaining [;], the transfer characteristics between coils are needed. Let these
transfer characteristics be expressed as voltage drops between coil i and the last coil N,
TGFWEGF TGFWEGF TGFWEGF

8 &80 < < ... < +
0 &
8 &80 <
TGFWEGF TGFWEGF
< ... <
TGFWEGF +
0 &
. .
' ( .17)
. . . . . . .
. .
80& &80 < < ... < +0&
0 & 0 & 0 & 0&
„qp‡iqp
with [< ] again being symmetric. Since the exciting current has negligible influence on these transfer
!
characteristics, it is best to ignore the exciting current altogether. Then the sum of the p.u. currents (based on one
common base power S , and on the transformer voltage ratings of the N coils) must be zero, or
hg…q
!
From here on it is best to work with p.u. Suantities, or with Suantities referred to one side, to avoid carrying
the turns ratios through all the derivations.
11
Page 203
j +M RW ' 0 ( .1 )
M '
The p.u. values of the matrix elements in ES. ( .17) can then be found directly from the short circuit test data, as
first shown by Shipley [10 ]. For a short circuit test between i and N, only I in ES. ( .17) is nonzero,
uÂ‚‡ and V H
‚‡ 0. Then the i th row becomes
TGFWEGF
8K RW ' < KK RW +K RW ( .19)
Traduciendo...
The impedance in this eSuation is the short circuit impedance between coils i and N by definition,
TGFWEGF UJQTV
<KK RW ' < K0 RW ( .20)
„qp‡iqp
based on one common base power S . Thehg…q
off diagonal element < uwÂ‚‡ is found by relating rows i and k of ES.
( .17) to the short circuit test between i and k. For this test, I I , and V 0,wÂ‚‡
with all other
uÂ‚‡ currents
wÂ‚‡being
zero. Then rows i and k become
TGFWEGF TGFWEGF
8K RW & 80 RW ' < KK RW & < KM RW +K RW ( .21a)
TGFWEGF TGFWEGF
&80 RW ' < MK RW & < MM RW +K RW ( .21b)
„qp‡iqp „qp‡iqp
or after subtracting ES. ( .21b) from ( .21a), with < wu < uw ,

8K RW ' < KK RW % < MM RW & 2< KM RW +K RW ( .21c)
…t „†
By definition, the expression in parentheses of ES. ( .21c) must be the short circuit impedance < uwÂ‚‡ , or
TGFWEGF 1 UJQTV UJQTV UJQTV

<KM RW ' < ( .22)
2 K0 RW % <M0 RW & < KM RW
based on one common base power S . This hg…q

completes the calculation of the matrix elements of ES. ( .17) from
the short circuit test data, which is normally supplied by the manufacturer.
ES. ( .17) cannot be expanded to include all coils, since all matrix elements would become infinite with the
exciting current being ignored. To get to the admittance matrix formulation ( .1 ), ES. ( .17) is first inverted,
TGFWEGF TGFWEGF
[; RW ] ' [< RW ] & ( .23)
12
Page 204
In this inverse relationship, the voltage V HÂ‚‡of the last coil already exists, and all terms associated with it can be
collected into a N th column for V . TheHÂ‚‡

N th row is created by taking the negative sum of rows 1,...N 1 based
on ES. ( .1 ). This results in the full matrix representation
+ RW ; RW ; RW ... ; 0 RW 8 RW
+ RW ; RW ; RW ... ; 0 RW 8 RW
. .
' ( .2 a)
. . . . . . .
. .
+0 RW ;0 RW ;0 RW ... ;00 RW 80 RW
with
TGFWEGF
;KM RW ' ; KM RW HTQO 'S. ( .23) HQT K, M # 0&1 ( .2 b)
;K0 RW ' ;0K RW ' &j

0&; TGFWEGF
HQT Kû0
KM RW ( .2 c)
' 0& M
;00 RW ' &j ;K0 RW ( .2 d)
K'
To convert from p.u. to actual values, all elements in ES. ( .2 ) are multiplied by the one common base power S , hg…q
and each row and column i is multiplied with 1/V .u
For transient studies, the resistance and inductance parts must be separated, in a way similar to that of
„qp‡iqp
Section .3. This is best accomplished by building [< ] only from the reactance part of the short circuit test data,
which is
Traduciendo...
UJQTV UJQTV
: KM RW & 4K RW % 4M RW ( .25)
KM RW ' <
…t „†
with < uwÂ‚‡ p.u. short circuit impedance (magnitude),
4 4uÂeither
‚‡ p.u.
wÂload
‚‡ losses in short circuit test between i and k, or sum of p.u. winding
resistances.
The winding resistances then form a diagonal matrix [4], and
[.]& ' LT[;] ( .2 )
with [;] being purely built from reactance values jTL. Both [4] and [L] are used in ES. ( .1) to represent the N
coil transformer.
Support routine BCT4#N uses this procedure for obtaining [4] and [L] from the transformer test data,
with two additional refinements:
a. If the winding resistances are not given, but the load losses in the short circuit tests are known,
then the resistances can be calculated from ES. ( .2) for N 2, and from the following three
13
Page 205
eSuations for N 3,
NQUU
4 RW % 4 RW ' R RW
NQUU
4 RW % 4 RW ' 2 RW ( .27)
NQUU
4 RW % 4 RW ' 2 RW
Strictly speaking, ES. ( .2) and ( .27) are not Suite correct, because the load losses contain stray
losses in addition to the I 4 losses, but the results should be reasonable. For transformers with
or more coils there is no easy way to find resistances from the load losses, and coil resistances
must be specified as input data if N $ .
b. #dditional branches can be added to represent the exciting current, as described in Section . .
To short derivations for a numerical example, let us first use the two winding transformer of Fig. .5, with
exciting current ignored. The resistance and reactance part is already separated in this case, with 4 0.005 and ‚‡
„qp‡iqp
X ‚‡
0.10. The reduced reactance matrix of ES. ( .17) is just a scalar in this case, jX ‚‡ j0.10, and its
„qp‡iqp
inverse is the reciprocal ; ‚‡ j10. #dding a second row and column with ES. ( .2 ) produces
1 1 10 &10
[T.RW]& '
L L &10 10
which, together with 4 4 0.0025,

Â‚‡ is the same result shown in ES. ( . ).
Â‚‡
For the example of the three winding transformer used after ES. ( .10), the reduced reactance matrix
(without the factor j) is
TGFWEGF 0.1150 0.1195

[: RW ]' DCUGF QP 100 /8#
0.1195 0.2 10
which, after inversion, becomes
TGFWEGF 1 17.93 & . 9

[; RW ]' DCUGF QP 100 /8#
L & . 9 .5599
or after adding the third row and column with ES. ( .2 ),
17.93 5 & . 9 &9.0 372

1
[;RW] ' & . 9 .559 9 0.33 95 DCUGF QP 100 /8#
L
&9.0 372 0.33 95 .70 77 Traduciendo...
Page 206
which is the same answer as the one given after ES. ( .12), except for minor round off errors and for a change in
base power from 1 MV# to 100 MV#. The star circuit eSuivalent circuit of a three winding transformer is therefore
just a special case of the general method for N coils discussed here.
/CVTKZ 4GRTGUGPVCVKQP QH 6JTGG 2JCUG 0 %QKN 6TCPUHQTOGTU
The first attempt to extend single phase to three phase transformer models was the addition of a zero
seSuence reluctance to the eSuivalent star circuit ( saturable transformer element in the BP# EMTP). This was
similar to the approach used on transient network analyzers, where magnetic coupling among the three core legs is
usually modelled with the addition of extra delta connected winding to a three phase bank consisting of single phase
units. To relate the available test data to the data of the added winding is unfortunately difficult, if not impossible.
For example, a two winding three phase unit is characterized by only two short circuit impedances (one from the
positive seSuence test, and the other from the zero seSuence test). #dding delta connected windings to single phase
two winding transformers would reSuire three short circuit impedances, however, because this trick converts the
model into a three winding transformer. #dding extra delta connected windings becomes even more complicated
for three phase three winding units, not only in fitting the model data to the test data, but also because a four winding
model would be reSuired for which the star circuit is no longer valid [109]. It was therefore reasonable to develop
another approach, as described here.
The extension from single phase to three phase units turned out to be much easier than was originally
thought. Conceptually, each coil of a single phase units becomes three coils on core legs I, II, III in a three phase
unit (Fig. .9).
(a) Three legged core (b) Five legged core (c) Shell type design
design design
(KI Three phase transformers
In terms of eSuations, this means that each scalar Suantity < or ; must be replaced by a 3 x 3 submatrix of the form
15
Page 207
<U <O <O
<O <U <O ( .2 )
<O <O <U
where < is…the self impedance of the coil on one leg, and < is the mutual
y impedance to the coils on the other two
Traduciendo...
legs ". #s in any other three phase network component (e.g., overhead line), these self and mutual impedances are
related to the positive and zero seSuence values,
1
<U ' <\GTQ % 2<RQU
3
1
<O ' <\GTQ & <RQU ( .29)
3
2TQEGFWTG HQT 1DVCKPKPI =4? CPF =.?
By simply replacing scalars by 3 x 3 submatrices of the form ( .2 ), the [4] and [L] matrix representation
of a three phase transformer is found as follows:
1. Set up the resistance matrix [4]. If the winding resistances are known, use them in [4]. If they are to be
calculated from load losses, use ES. ( .2) for N 2, or ES. ( .27) for N 3. For N $ , there is no easy
way to calculate the resistances. Use positive seSuence test data in these calculations, and assume that the
three corresponding coils on legs I, II, III have identical resistances.
2. Find the short circuit reactances from ES. ( .25) for positive seSuence values. Use the same eSuation for
zero seSuence values, provided the zero seSuence test between two windings does not involve another
winding in delta connection. In the latter case, the data must first be modified according to Section .5.2.
„qp‡iqp
3. Build the reduced reactance matrix [X ‚‡ ] from ES. ( .20) and ( .22), by first calculating the positive
and zero seSuence values separately from the positive and zero seSuence short circuit reactances, and by
replacing each diagonal and off diagonal element by a 3 x 3 submatrix of the form ( .2 ). The elements
of this matrix are calculated with ES. ( .29).
Since the 3 x 3 submatrices contain only 2 distinct values X and X , it is…not necessary
y to work with 3 x
3 matrices, but only with pairs (X, X ). D. Hedman

… y derived a balanced matrix algebra for the
multiplication, inversion, etc., of such pairs [110], which is used in the support routines BCT4#N and
T4ELEG.
"
From Fig. .9 it is evident that the mutual impedance between legs I and II is slightly different from the one
between legs II and III, etc. Data for this unsymmetry is usually not available, and the unsymmetry is therefore
ignored here. To take it into account would reSuire that a three phase two winding transformer be modelled as a
six coil transformer (Section . ), with 15 measured short circuit impedances.
Page 208
„qp‡iqp „qp‡iqp „qp‡iqp

. Invert [X ‚‡ ] to obtain [B ‚‡ ], again using Hedman s balanced matrix algebra, and expand [B ‚‡ ]
to the full matrix [B ] with

‚‡ ES. ( .2 ).
5. Since the reactances were in p.u. based on one common S , the inverse inductance matrix [L] in actual
hg…q
values 1/H is obtained from [B ] by multiplying

‚‡ each element B with T S / VV , where V and
uwÂ‚‡ V
hg…q uw u w
are the voltage ratings of coil i and k. For the conversion of p.u. resistances to actual values in S, multiply
4 with
uÂ‚‡ V / S . u hg…q
/QFKHKECVKQP QH <GTQ 5GSWGPEG &CVC HQT &GNVC %QPPGEVKQPU
The procedure of Section .5.1 cannot be used directly for the zero seSuence calculation of transformers
with three or more windings if one or more of them are delta connected. #ssume that a three winding transformer
has wye connected primary and secondary windings, with their neutrals grounded, and a delta connected tertiary
winding. In this case, the zero seSuence short circuit test between the primary and secondary windings will not only
have the secondary winding shorted but the tertiary winding as well, since a closed delta connection provides a short
circuit path for zero seSuence currents. This special situation can be handled by modifying the short circuit data for
an open delta so that the procedure of Section .5.1 can again be used. With the well known eSuivalent star circuit
of Fig. .7, the three test values supplied by the manufacturer are ( pu in the subscript dropped to simplify
notation),
ENQUGF') :* %
:. :6
: *. ( .30a)
:. % :6
:*6 ' :* % :6 KP R.W. XCNWGU Traduciendo... ( .30b)
:.6 ' :. % :6 ( .30c)
which can be solved for X , X , BX : F T
:* ' :*6 & :.6 :*6 & : ENQUGF )

:.6 ( .31a)
*.
:. ' :.6 & :*6 % :* KP R.W. XCNWGU ( .31b)
:6 ' :*6 & :* ( .31c)
#fter this modification, the short circuit reactances X X , X X and

B X X are
F used
B as input
T data,Fwith T
winding T no longer being shorted in the test between H and L.
The modification scheme becomes more complicated if resistances are included. For instance, ES. ( .30a)
becomes
(4. % L:.) (46 % L:6)

<*.ENQUGF)' /000
4* % L:* % /000KP R.W. XCNWGU ( .32)
/ / (4. % 46) % L(:. % :6)
with .< . )being

BFix …qpÂ the value supplied by the manufacturer, and 4 , 4 , 4 being
B theFwinding
T resistances. This
17
Page 209
leads to a system of nonlinear eSuations, which is solved by Newton s method in the support routine BCT4#N. It
works for three winding transformers with wye/wye/delta and with wye/delta/delta connections so far, which
should cover most practical cases.
'ZEKVKPI %WTTGPV
The exciting current is very much voltage dependent above the knee point of the saturation curve 8
f(i). Fig. .10 shows a typical curve for a modern high voltage transformer with grain oriented steel, with the knee
point around 1.1 to 1.2 times rated flux [11 ]. The value of the incremental inductance d8/di is fairly low in the
saturated region, and fairly high in the unsaturated region. The exciting current in the unsaturated region can easily
be included in the [L] or [L] representations. Extra nonlinear branches are needed to include saturation effects,
and extra resistance branches to include excitation losses.
(KI Typical saturation curve [11 ]. l 19 1

IEEE
.KPGCT 7PUCVWTCVGF 'ZEKVKPI %WTTGPV
For single phase units and for three phase units with five legged core or shell type design (Fig. .9(b) and
(c)), the linear exciting current is very small and can often be ignored. If it is ignored, then the [L] matrix
representation described in Section .3 to .5 must be used. # (small) exciting current must always be included,
however, if [L] matrices are used, as explained in Section .2. For three phase units with three legged core design,
the exciting current is fairly high in the zero seSuence test (e.g., 100 ), and should therefore not be neglected.
Traduciendo...
The exciting current has an imaginary part, which is the magnetizing current flowing through the
magnetizing inductance L . It yalso has a smaller real part (typically 10 of the imaginary part), which accounts for
excitation losses. These losses are often ignored. They can be modelled reasonably well, however, with a shunt
conductance G inyparallel with the magnetizing inductance L . The p.u. magnetizing

y conductance is
Page 210
2GZE
)O RW ' ( .33)
5TCVKPI
and the reciprocal of the p.u. magnetizing reactance is
1 +GZE
' & )O RW ( .3 )
:O RW +TCVKPI
with Pq i excitation loss in excitation test,
Iq i magnitude of exciting current in excitation test,
S „g†u€s power rating, and
I „g†u€s current rating.
To assess the relative magnitudes of G and 1/X

y , let us take
y the values from the example of Section .2 as typical
(X …t
10 „†
, 4 0.5 , I 1 ). …t
Furthermore,
„† assume
qi that the excitation loss V G at rated voltage is 25 y
of the load loss I 4 at rated

…t „† current (a typical ratio for power transformers). Then G yÂ‚‡ 0.00125 and I /I q i „g†u€s
0.01. The reciprocal of the p.u. magnetizing reactance is therefore close to the value of the p.u. exciting current,
1 +GZE
. ( .35)
:O RW +TCVKPI
with the error being less than 1 in the numerical example.
How to include the linear exciting current in the model depends on whether an [L] or [L] matrix
representation is used, and whether the transformer is a single phase or a three phase unit.
5KPING 2JCUG 6TCPUHQTOGTU
In the [L] matrix representation, the magnetizing inductance L will already have
y been included in the
model. Usually, the T circuit of Fig. .5, or the star circuit of Fig. .7 with L connected to star point
y S, is used
in the derivation of [L]. Since L is much

yÂ‚‡larger than L , it
…t „†Â‚‡ could be placed across the terminals of the high,
low or tertiary side with eSual justification. #lternatively, 2L yÂ‚‡ could be connected to both high and low side,
which would convert the T circuit of the two winding transformer into a B circuit, or 3L could be connected
yÂ‚‡to
all 3 sides in the case of a three winding transformer. The conversion of L into actual values
yÂ‚‡ is done in the usual
way by using the voltage rating for that side to which the inductance is to be connected. For example, connecting
the p.u. inductance 3L to allyÂ‚‡

3 sides would mean that the actual values of these 3 inductances are
8*
.* ' 3.O RW
5TCVKPI
19
Page 211
8.
.. ' 3.O RW
5TCVKPI
86
.6 ' 3.O RW
Traduciendo...
5TCVKPI
In the [L] matrix representation, the internal nodes of the T or star circuit are not available, and the magnetizing
inductance must therefore be connected across one or all external terminals, as discussed above. Connecting it
across side i is the same as adding 1/L to the iyth diagonal element of [L] . This makes [L] nonsingular, and it
could therefore be inverted if the user prefers [4] and [L] matrices. This inversion option is available in the support
routine BCT4#N, even though this writer prefers to work with [L] because [L] is more or less ill conditioned as
discussed in Section .2.2.
While L does
y not create extra branches, but disappears instead into the [L] or [L] matrix, one or more
extra resistance branches are needed to model excitation losses with G yÂ‚‡ from ES. ( .33). #gain, G yÂ‚‡ can either
be added to one side, or 1/2 G yÂ‚‡ to both sides of a two winding transformer and 1/3 G yÂ‚‡ to all three sides of a
three winding transformer. The conversion to actual values is again straightforward, and 4 1/G is then used y y
as input data for the extra resistance branch.
6JTGG 2JCUG 6TCPUHQTOGTU
The inclusion of the linear exciting current for three phase units is basically the same as for single phase
units, except that G and y1/X from ES.

y ( .33) and ( .3 ) are now calculated twice, from the positive as well as
from the zero seSuence excitation test data. The reciprocals of the two magnetizing inductances,
$RQU ' 1/.O&RQU , $\GTQ ' 1/.O&\GTQ
are converted to a 3 x 3 matrix
$U $O $O
$O $U $O
$O $O $U
where
1
$U ' ($\GTQ % 2$RQU)
3
1
$O ' ($\GTQ & $RQU) ( .3 )
3
which is added to the 3 x 3 diagonal block in [L] of the high, low, or some other side. #lternatively, 1/N times
the p.u. 3 x 3 matrix could be added to the 3 x 3 diagonal blocks of all sides of an N winding transformer, after
conversion to actual values with the proper voltage ratings. #fter these additions, [L] becomes nonsingular and
20
Page 212
can therefore be inverted for users who prefer [L] matrices. Support routine T4ELEG builds an [L] matrix directly
from both the short circuit and excitation test data, as briefly described in Section .10.3.
To include excitation losses, three coupled resistance branches must be added across the terminals of one
side. The diagonal and off diagonal elements of this resistance matrix are
1 1 2
4U ' %
3 )O&\GTQ )O&RQU
1 1 1
4O ' & ( .37)
3 )O&\GTQ )O&RQU
The excitation test for the positive seSuence is straightforward, and the data is usually readily available.
Some precautions are necessary with the zero seSuence test data, if it is available, or reasonable assumptions must
be made if unavailable.
If the transformer has delta connected windings, the delta connections should be opened for the zero
seSuence excitation test. Otherwise, the test really becomes a short circuit test between the excited winding and the
delta connected winding. On the other hand, if the delta is always closed in operation, any reasonable value can be
used for the zero seSuence exciting current (e.g., eSual to positive seSuence exciting current), because its influence
Traduciendo...
is unlikely to show up with the delta connected winding providing a short circuit path for zero seSuence currents.
If the zero seSuence exciting current is not given by the manufacturer, a reasonable value can be found as
follows: Imagine that one leg of the transformer (# in Fig. .11) is excited, and estimate from physical reasoning
how much voltage will be induced in the corresponding coils of the other two legs (B and C in Fig. .11). For the
three legged core design of Fig. .11, approximately one half of flux 8 returns through phases
5 B and C, which
means that the induced voltages V and V 6will be close

7 to 0.5 V (with reversed polarity).
5 If k is used for this
factor 0.5, then
+GZE&\GTQ 1 % M
' ( .3 )
+GZE&RQU 1 & 2M
(KI Fluxes in three legged core type design
21
Page 213
ES. ( .3 ) is derived from
8# ' <U +# ( .39a)
8$ ' 8% ' <O +# ( .39b)
with < , …
< being
y the self and mutual magnetizing impedances of the three excited coils. With
<O <\GTQ & <RQU

8$ ' 8% ' 8# ' 8# ' &M8# ( . 0)
<U <\GTQ % 2<RQU
and < , ‚<…inversely

’q„ proportional to I q i ‚ …,qIi ’q„ , ES. ( .3 ) follows. Obviously, k cannot be exactly 0.5,
because this would lead to an infinite zero seSuence exciting current. # reasonable value for I q i ’q„ in a three legged
core design might be 100 . If I qi‚… were 0.5 , k would become 0. 9 2 , which comes close to the theoretical
limit of 0.5. Exciting the winding on one leg with 100 kV would then induce voltages of 9. kV (with reversed
polarity) in the windings of the other two legs.
For the five legged core type design of Fig. .9(b), maybe 2/3 of approximately (1/2)8 would return 5
through legs B and C. In that case, k would be 1/3, or I q i ’q„ q/Ii ‚ … .
The excitation loss in the zero seSuence test is higher than in the positive seSuence test, because the fluxes
8 5, 8 , 68 in 7the three cores are now eSual, and in the case of a three legged core type design must therefore return
through air and tank, with additional eddy current losses in the tank. Neither the value of the zero seSuence exciting
current nor the value of the zero seSuence excitation loss are critical if the transformer has delta connected windings,
because excitation tests really become short circuit tests in such cases.
The modification of [L] for magnetizing currents and the addition of resistance branches for excitation
losses create a model which reproduces the original test data very well. Table .1 compares the test data, which was
used to create the model with the support routine BCT4#N, with steady state EMTP solutions in which this model
was used to simulate the test conditions (e.g., voltage sources were connected to one side, and another side was
shorted, to simulate a short circuit test). In this case, the three winding resistances were specified as input data, and
an [L] matrix with 10 digit accuracy was used to minimize the problem of ill conditioning. The excitation data was
specified as being measured from the primary side, but 1/L and shunt conductance
y G were placed across the
y
tertiary side, for reasons explained in Section . .2. BCT4#N modifies L and 4 in this situation,
y to yaccount for
the influence of the short circuit impedance between the primary and tertiary side. For the zero seSuence short
Traduciendo...
circuit impedance between the primary and secondary side, the modifications of Section .5.2 were applied to
account for the effect of the delta connected tertiary winding.
6CDNG Data for three phase three winding transformer in ;yd connection
T;PE OF TEST TEST D#T# SIMUL#TION 4ESULTS
22
Page 214
pos. seSuence exciting current ( ) 0. 2 0. 2 1 (in phase #)

excitation test 0. 2 0 (in phase B)
0. 230 (in phase C)
excitation loss (kW) 135.73 135.731
zero seSuence exciting current ( ) 0. 2 0. 2 0 in all phases

excitation test
excitation loss (kW) 135.73 135.731
short circuit test ‚… .7 .7 0

impedances, with < ( ) (300)
three plane MV#
base in paren ‚… . .0
thesis < !( ) (7 )
‚… 5.31 5.310
< (! ) (7 )
’q„ 7.3 319 7.3 31

< ( ) (300)
’q„ 2 .25 1 3 2 .25 0

< !( ) (300)
’q„ 1 .552 2 1 .552

< (! ) (300)
) With open delta on side 3 (values were unavailable from test since they are unimportant if delta is closed
in operation, as explained in text, the positive seSuence values were used for zero seSuence as well).
) With closed delta on side 3.
) These values were calculated from the original test data given as 4 and X in percent with an accuracy of
2 digits after the decimal point.
5CVWTCVKQP 'HHGEVU
For the transient analysis of inrush currents, of ferroresonance and of similar phenomena it is clearly
necessary to include saturation effects. Only the star circuit representation in the BP# EMTP ( saturable transformer
component ) accepts the saturation curve directly, while the [L] and [L] representations reSuire extra nonlinear
inductance branches for the simulation of saturation effects.
Nonlinear inductances of the form of Fig. .10 can often be modelled with sufficient accuracy as two slope
piecewise linear inductances. Fig. .12 shows two and five slope piecewise linear representations from a practical
case [ 0] for the system shown before in Fig. .1. The simulation results (Fig. .13) are almost identical, and agree
reasonably well with field test results (Fig. .1 ). The slope in the saturated region above the knee is the air core
inductance, which is almost linear and fairly low compared with the slope in the unsaturated region. Typical values
for air core inductances are 2L (L short

…tcircuit
„† …tinductance)
„† for two winding transformers with separate
windings [111], or to 5 times L for autotransformers.

…t „† In the unsaturated region, the values can be fairly high
on very large transformers (see Fig. .10).
While it makes little difference to which terminal the unsaturated inductance is connected,
23
Page 215
Traduciendo...
(KI Two slope and five slope piecewise linear inductance
(KI Superimposed EMTP simulation results with two and five slope piecewise linear
inductance
Page 216
Traduciendo...
(KI Comparison between simulation and field test results
it may make a difference for the saturated inductance, because of its low value. Ideally, the nonlinear inductance
should be connected to a point in the eSuivalent circuit where the integrated voltage is eSual to the iron core flux.
To identify that point is not easy, however, and reSuires construction details not normally available to the system
analyst. For cylindrical coil construction, it can be assumed that the flux in the winding closest to the core will
mostly go through the core, since there should be very little leakage. This winding is usually the tertiary winding
in three winding transformers, and in such cases it is therefore best to connect the nonlinear inductance across the
tertiary terminals. Fig. .15 shows the star circuit derived by Schlosser [112] for a transformer with three cylindrical
windings (T closest to core, H farthest from core, L in between), where the integrated voltage in point # is eSual
to the flux in the iron core. The reactances of 0.5 S between # and T is normally not known, but it is so small
compared to 7.12 S between S and T, that the nonlinear inductance can be connected to T instead of #, with little
error. Fig. .15 also identifies a point B at which the integrated voltage is eSual to yoke flux. <ikherman [113]
suggests to connect another nonlinear inductance to that point B to represent yoke saturation. Since .9 S between
H and B is small compared to 22 S between H and S, this second nonlinear inductance could probably be connected
to H without too much error. The knee point and the slope in the saturated region of this second nonlinear
25
Page 217
(KI 4eactances (in S) of a three winding transformer

(from [112], which provides the data for 5 cylindrical
windings the two windings farthest from the core are
ignored here)
inductance are higher than those of the first nonlinear inductance (Fig. .1 ). Since it is already difficult to obtain
saturation curves for the core, this secondary effect of yoke saturation is usually ignored. Dick and Watson [11 ]
came to similar conclusions about the proper placement of the nonlinear inductance when they measured saturation
curves on a three winding transformer. Table .2 compares the air core inductance ( slope in saturated region)
#
obtained from laboratory tests with values obtained from the star circuit if the nonlinear inductance is connected to
the tertiary T, or to the star point S. The authors also show a more accurate eSuivalent circuit which would be useful
if yoke saturation or unsymmetries in the three core legs are to be included. If L is connected to T, then
y the
differences are less than v5 , whereas the differences become very large for the connection to S. Unfortunately,
the built in saturation curve in the BP# star circuit representation ( saturable transformer component ) is always
connected to the star point. This model could become more useful if the code were changed so that L could be y
connected to any terminal.
Traduciendo...
#
This star circuit also had a zero seSuence inductance of 1.33 p.u. connected to the high side (see Section
. .2.2).
Page 218
(KI Nonlinear inductances connected to H (yoke saturation)

and L (core saturation) of a two winding transformer. 4eprinted
with permission from [113], Copyright 1972, Pergamon Journals Ltd
The proper placement of the nonlinear inductance may or may not be important, depending on the
circumstances. For example, if the transformer of Table .2 with L in S were energized

y from the high side, then
the amplitude of the inrush current would be correct. If it were energized from the tertiary side, however, then the
$
amplitude of the inrush current would be 5 too low for high levels of saturation . If details of the transformer
construction are not known, then it is not easy to decide where to place L . In the example
y of Fig. .12 .1 , no
construction details were known, and L was simply

y placed across the high voltage terminals. In spite of this,
simulation results came reasonably close to field test results.
5KPING 2JCUG 6TCPUHQTOGTU
If the [L] model of Section .3 or . is used without the corrections for linear exciting current described
in Section . .1, then the nonlinear inductance is simply added across the winding closest to the core. If the [L]
model of Section .2 is used, or if [L] has already been corrected for the linear exciting current, then a modified
nonlinear inductance must be added in which the unsaturated part has been subtracted out (Fig. .17). This modified
nonlinear inductance has an infinite slope below the knee point.
6CDNG Comparison between measured and calculated air core inductances. l 19 1 IEEE
air core inductance (p.u.)
excited flux measured calculated error calculated error

winding at test with L in
y T() with L in
y S()
$
Inrush current approximately proportional to 1/L gu„ i „q for flux above knee point if unsaturated L L y gu„
i „q .
27
Page 219
Traduciendo...
H H 0.19 0.207 .5 0.19 0.0

L 0.12 0.129 .0 0.120 3.2
T 0.07 0.07 0.0 0.120 5 .0
L H 0.127 0.129 1. 0.120 5.5

L 0.131 0.125 . 0.11 11.0
T 0.07 0.07 2. 0.120 5 .0
T H 0.07 0.07 0.0 0.120 5 .0

L 0.07 0.07 0.0 0.120 5 .0
T 0.07 0.07 0.0 0.173 12 .0
Measured by integrating the voltage at that terminal. The measured short circuit inductances were L 0.073 BF
p.u., L 0.1305
BT p.u., L 0.0 93 p.u.,FTwhich produces the star circuit inductances of L 0.0775 p.u., L B F
0.0037 p.u., L 0.0530 Tp.u.
6JTGG 2JCUG 6TCPUHQTOGTU
Usually only the positive seSuence saturation curve (or the saturation curve for one core leg) is known.
Then it is best to connect the same nonlinear inductance across each one of the three phases (e.g., across the tertiary
terminals T# TB, TB TC, TC T#). This implies that the zero seSuence values are the same as the positive seSuence
values, which is probably a reasonable assumption for the five legged core and shell type construction.
For the three legged core design, the zero seSuence flux returns outside the windings through an air gap,
structural steel and the tank. Fig. .1 shows the measured zero seSuence magnetization curve for the transformer
described in Table .2 [11 ]. Because of the air gap, this curve is not nearly as nonlinear as the core saturation
curve of Fig. .10. It is therefore reasonable to approximate it as a linear magnetizing inductance. In [11 ] it is
shown that this zero seSuence magnetizing inductance should be connected to the high side. With the [L] model,
this is accomplished by setting B 0 and

‚…using B 1/L in ES. ( ’q„
.3 ), and by’q„
adding the 3 x 3 matrix with
%
B…
B B /3 to
y the 3 x
’q„3 diagonal block of the high side . This buries the zero seSuence magnetizing
inductance in [L] . The positive seSuence (core leg) nonlinear inductance (Fig. .10 for the example taken from
[11 ]) can then again be added across each one of the phases.
%
By setting B 0,‚ [L]
… will remain singular. This causes no problems if the inverse inductance is used.
Users who prefer [L] matrices would have to add another 3 x 3 matrix with B 2B /3 and B B … /3 to one‚ … y ‚…
of the sides, with B 1/L‚ ,…where L ‚is…the linear (unsaturated)
‚… positive seSuence magnetizing inductance.
Page 220
(KI Subtraction of linear (unsaturated) part in
saturation curve (value of # eSual in both curves)
Traduciendo...
(KI <ero seSuence magnetization curve [11 ]. l 19 1 IEEE
*[UVGTGUKU CPF 'FF[ %WTTGPV .QUUGU
The excitation losses obtained from the excitation test are mostly iron core losses, because the I 4 losses
are comparatively small for the low values of the exciting current. These iron core losses are sometimes ignored,
but they can easily be approximated with the linear shunt conductance of G of ES. ( .33). y
# linear shunt conductance G cannot

y represent the iron core losses completely accurately. These losses
consist of two parts,
2KTQP&EQTG ' 2J[UVGTGUKU % 2GFF[ EWTTGPV ( . 1)
namely of hysteresis losses P and of

t‘…†q„q…u… eddy current losses P . In
qpp‘Âi‡„„q€† the excitation tests, these two parts
cannot be separated, and only the sum P u„ € i „q is obtained. Before discussing more accurate representations, it is
useful to have some idea about the ratio between the two parts. 4ef. [51], which may be somewhat outdated, gives
ratios of
2J[UVGTGUKU/2GFF[ EWTTGPV ' 3 HQT UKNKEQP UVGGN
29
Page 221
2J[UVGTGUKU/2GFF[ EWTTGPV ' 2/3 HQT ITCKP&QTKGPVGF UVGGN
while a more recent reference [125] Suotes a typical ratio of 1/3. On modern transformers, hysteresis losses are
therefore much less important than they used to be before the introduction of grain oriented steel.
It is generally agreed that eddy current losses are proportional to 8 and to f [51], at least in the low
freSuency range, which seems to change to f in the #high freSuency range because of skin effect in the laminations.
FreSuency dependent eddy current representations were discussed in [115], where 4 is replaced by a ynumber of
parallel 4 L branches. It is doubtful whether this sophistication is needed, however, because the reduction caused
by a proportionality change from f to f at high# freSuencies is probably offset by other types of loss increases (e.g.,
by increases in coil resistance due to skin effect, etc.). #t any rate, laboratory tests would first have to be done to
verify the correctness of the freSuency dependence proposed in [115]. In such tests it may be difficult to separate
eddy current and hysteresis losses. If we accept a proportionality with 8 and f , then a constant resistance 4 does y
model these losses very well, because P V

qpp‘Âi‡„„q€† RG5 /4 and
y V RGS T8 RGS for sinusoidal excitation.
Hysteresis losses are a nonlinear function of flux and freSuency,
2J[UVGTGUKU ' M (8)C @ (H)D ( . 2)
In [51], a is said to be close to 3 for grain oriented steel, and b 1. In [11 ], a 2.7 and b 1.5. If a b
2 were used, then the sum of hysteresis and eddy current losses could be modelled by the constant resistance 4 or y
conductance G of yES. ( .33). This is a reasonable first approximation [125], especially if one considers that
hysteresis losses are only 25 of the total iron core losses in transformers with grain oriented steel. Fig. .19(a)
shows the nonlinear inductance of a current transformer, which was used by C. Taylor to duplicate field test results
in a case where the secondary current was distorted by saturation effects [117]. Fig. .19(b) shows 8 as a function
of the exciting current in the transient simulation, if iron core losses are modelled with a constant resistance 4 y
0 S. It can be seen that 4 not onlyy creates the typical shape of a normal magnetization curve (with lower d8/di
Traduciendo...
coming out of the origin, compared to 8 f(i) in Fig. .19(a)), but also creates minor loops with reasonable shapes.
30
Page 222
(a) Nonlinear magnetizing (b) Loops created by constant 4y

of current transformer for hysteresis and eddy
current losses
(KI Saturation in current transformer [117]. 4eprinted by permission of

C.W. Taylor
&
If the flux current loop for sinusoidal excitation is available, then 4 can also be calculated
y from
X
4O ' ( . 3)
)K
as an alternative to ES. ( .33), with )i being half of the horizontal width of the loop at 8 0 (Fig. .20), and v
T8 .ygES. ( . 3) is derived from realizing that at 8 0 all the current must flow through the parallel resistance
4 and
y that the voltage reaches its peak value T8 yg at 8 0 because of the 90E phase shift between voltage and
flux.
If more values of )i are used at various points along the 8 axis, together with the corresponding values for
v d8/dt, then a resistance 4 can be constructed

y which becomes nonlinear. This parallel combination of nonlinear
resistance and nonlinear inductance has been proposed by L.O. Chua and K.#. Stromsmoe [11 ] to model flux
current loops caused by hysteresis and eddy current effects. They give convincing arguments why this representation
is reasonable. In particular, they did make comparisons between simulations and laboratory tests, not only for a
small audio output transformer with laminated silicon steel, but for a supermalloy core inductor as well. Fig. .21
shows the nonlinear inductances and resistances for this audio output transformer [11 ]. Fig. .22 compares the
laboratory test results with simulation results [11 ] (first row laboratory results, second row simulation results). Fig.
&
The author is reluctant to call it hysteresis loop because the losses associated with this loop are the sum of
hysteresis and eddy current losses, with the latter actually being the larger part in transformers with grain
oriented steel.
31
Traduciendo...
Page 223
.22(a) is a family of flux current loops for 0 Hz sinusoidal flux linkage of various amplitudes. Fig. .22(b) shows
two loops, one with a sinusoidal flux linkage and the second with a sinusoidal current. Fig. .22(c) is a family of
loops obtained at 0 Hz for various amplitudes of sinusoidal current. Fig. .22(d) shows a family of loops for
sinusoidal flux linkages at 0, 120, and 1 0 Hz. In all cases, the agreement between measurements and simulation
results is excellent. The minor loops in Fig. .22(e) were obtained with a 0 Hz sinusoidal current superimposed
on a dc bias current. #gain, there appears to be excellent agreement.
(KI Flux current loop
The major drawback of this core loss representation with a linear or nonlinear resistance is its inability to
produce the correct residual flux when the transformer is switched off. This was one of the motivations for the
development of more sophisticated hysteresis models, but even these models do not seem to produce the residual flux
very accurately. This writer believes that there are no models available at this time which can predict residual fluxes
reliably, and that reasonable assumptions should therefore be made. There is no difficulty with the linear or
nonlinear 4 representation
y in starting a transient simulation with a residual flux if its value is provided as input data,
as explained in Section . . .
32
Page 224
Traduciendo...
(KI Model for exciting current with parallel, nonlinear resistances and inductances
[11 ]. l 1970 IEEE
The more sophisticated models mentioned above use pre defined trajectories or templates in the 8, i plane
to decide in which direction the curve will move if the flux either increases or decreases [11 , 119]. The techniSue
of [119] has been implemented in the BP# EMTP ( pseudolinear hysteretic reactor ) but a careful comparison with
the simpler 4 representations

y (either linear or nonlinear) has not yet been done. More research may be needed
before reliable hysteresis models become available. Such models may be based on the duality between magnetic and
electric circuits, which would then reSuire the dimensions of the iron core as input data [121], or they may be based
on the physics of magnetic materials [120].
33
Page 225
(KI Comparison between measured and simulated flux current loops [11 ]. l 1970 IEEE
4GUKFWCN (NWZ
'
4esidual flux is the flux which remains in the iron core after the transformer is switched off . It has a major
influence on the magnitude of inrush currents. Starting an EMTP simulation from a known residual flux is relatively
easy, with simple as well as with sophisticated hysteresis models. To find the residual flux from a simulation is more
complicated, and the results still seem to be unreliable at this time, even with sophisticated hysteresis models. Until
this situation improves, it might be best to use a typical value for the residual flux as part of the input data.
Unfortunately, not much data is available on residual flux. # recent survey by CIG4E [122] has not added much
to it either, except for the Suotation of 2 maximum values of 0.75 and 0.90 p.u. This survey does contain a
reasonable amount of information about values of air core inductances and saturation curves, however.
The UBC version of the EMTP starts the simulation from a nonzero residual flux with the following
approach, in connection with piecewise linear inductances (see also Section 12.1.3): #t t 0, the starting point
# lies at 8 and
„q…up‡gx i 0, and the simulation moves along a slope of L (unsaturated value), as shown in Fig. .23.
The slope is changed to L (saturated value) in point B as soon as 8 $ 8 . #t the same time, a value 8
w€qq is
…‰u†it
calculated which will bring the characteristic back through the origin when the slope is changed back to L as soon
as 8 # 8 . Thereafter,
…‰u†it the normal 8/i curve will be followed. More details, in particular the problem of
overshoot (8 slightly larger than 8 w€qqwhen going into saturation), are discussed in Section 12.1.3.3. For typical
saturation curves, such as the one shown in Fig. .10, the linear slope is almost infinite in that case, the first move
into saturation practically lies on the given 8/i curve, rather than somewhat higher as in Fig. .23.
Traduciendo...
'
There seems to be some confusion in terminology between residual and remanent flux. It appears that
remanent flux is the flux value at i 0 in the hysteresis curve under the assumption of sinusoidal excitation.
In the BP# version, this branch type has been generalized from 2 to n slopes ( pseudolinear inductor ), but
it appears that is no longer accepts residual flux as input data.
Page 226
(KI Starting from residual flux
The simple hysteresis model of a nonlinear L in parallel

y with a resistance 4 cannot be used
y to predict the
residual flux after the transformer is switched off. The energy stored in L will simply be ydissipated in 4 in this y
model, with an exponential decay in current and flux to zero values. The flux value at the instant of switching could
possibly be close to the residual flux, but this has never been checked. #lso, this value would only be meaningful
if the transformer is switched off by itself, without lines or other eSuipment connected to it.
#WVQVTCPUHQTOGTU
If an autotransformer is treated the same way as a regular transformer, that is, if the details of the internal
connections are ignored, the models discussed here will probably produce reasonably accurate results, except at very
low freSuencies. #t dc, the voltage ratio between the low and high side of a full winding transformer will be zero,
whereas the voltage ratio of the autotransformer of Fig. .2 becomes 4 /4 (dc voltage divider
CC C effect).
For a more accurate representation, series winding I and common winding II should be used as building
blocks, in place of high side H and low side L. This reSuires a re definition of the short circuit data in terms of
windings I and II. Since most autotransformers have a tertiary winding, this winding T shall be included in the re
definition.
First, the voltage ratings are
8+ ' 8* & 8.
8++ ' 8. (.)
8+++ ' 86
The test between H and L provides the reSuired data for the test between I and II directly, since II is shorted
Traduciendo...
35
Page 227
and since the voltage applied to H is actually applied to I (b and c are at the same potential through the short circuit
connection). Only the voltage ratings are different, and the conversion from H to I is simply
8*
<+ ++ ' <*. KP R.W. XCNWGU ( . 5)
8* & 8.
No modifications are needed for the test between II and III,
<++ +++ ' <.6 KP R.W. XCNWGU (.)
(KI #utotransformer with tertiary winding
For the test between H and T, the modification can best be explained in terms of the eSuivalent star circuit of Fig.
.7, with the impedances being < , < , <C, CC

based
CCCon V , V , V in this
C CCcase. With
CCC III short circuited, 1 p.u. current
(based on V VCCC
) will flow
T through < . This current
CCCwill also flow through I and II as 1 p.u. based on V , or B
converted to bases V , V C, I (V
CCVC)/V andBI V /VF . With
B theseCC
currents,
F Bthe p.u. voltages become
8* & 8.
8+ ' <+ % <+++ KP R.W. XCNWGU ( . 7)
8*
8.
8++ ' <++ % <+++ KP R.W. XCNWGU (.)
8*
Converting V and
C V to physical
CC units by multiplying ES. ( . 7) with (V V ) and ES. ( . ) Bwith VF , adding F
them, and converting the sum back to a p.u. value based on V produces theB measured p.u. value
8* & 8. 8.
<*6 ' <+ % <++ % <+++ KP R.W. XCNWGU ( . 9)
8* 8*
ESs. ( . 5), ( . ) and ( . 9) can be solved for < , < , < since < <C < and
CC < CCC
<<, C CC C CC CC CCC CC CCC
Page 228
8*8. 8* 8.
<+ +++ ' <*. % <*6 & <.6 KP R.W. XCNWGU ( .50)
(8* & 8.) 8* & 8. 8* & 8.
The autotransformer of Fig. .2 can therefore be treated as a transformer with 3 windings I, II, III by
simply re defining the short circuit impedances with ESs. ( . 5), ( . ) and ( .50). This must be done for the
positive seSuence tests as well as for the zero seSuence tests. If the transformer has a closed delta, then the zero
seSuence data must be further modified as explained in Section .5.2, after the re definition of the short circuit data.
Traduciendo...
+FGCN 6TCPUHQTOGT
#n ideal transformer was not added to the BP# EMTP until 19 2. The ideal transformer has no impedances
and simply changes voltages and current from side 1 to side 2 (Fig. .25) as follows:
X 1 K
' 'P ( .51)
X P K
(KI Ideal transformer
It is handled in the system of nodal eSuations (1. a) or (1.20) by treating current i as a variable, and by adding the
eSuation
PXM & PXO & (XL & Xý) ' 0 ( .52)
The matrix of the augmented system of eSuations, with an extra column for variable i , and an extra row for ES.
( .52), then has the form of Fig. .2 .
37
Page 229
(KI #ugmented [G] matrix
The ideal transformer can also be simulated with resistance branches and one extra node extra, as shown in Fig.
.27, because these branches augment the matrix in the same way as shown in Fig. .2 . In both approaches it is
important that node extra (or ES. ( .52)) is eliminated after nodes k, m, j, ý, to assure that the diagonal element
becomes nonzero during the elimination process.
Traduciendo...
(KI 4esistance modelling of ideal transformer
If the transformer is unloaded (i 0), the elimination process will fail with a zero diagonal element. The
UBC version would stop in that case with an appropriate error message, while the BP# version will first print a
warning, and then continue after automatic connection of a very large resistance to the node where the zero diagonal
Page 230
element has been encountered. This problem is related to the treatment of floating subnetworks (see next Section
.9).
(NQCVKPI &GNVC %QPPGEVKQPU
Most transmission autotransformers have delta connected tertiary windings for the suppression of third
harmonics. FreSuently, nothing is connected to such tertiary windings. In that case, and in similar cases, the delta
windings have floating potential with respect to ground (Fig. .2 ): only the voltages across the windings a b, b c,
c a are defined, but not the voltages in a, b, or c with respect to ground. Since the EMTP solves for node voltages
with respect to ground, the Gauss elimination will fail with a zero diagonal element.
(KI Floating delta

connection
To prevent the solution algorithm from failing, one can either ground one of the nodes (e.g., node a), or
connect stray capacitances or large shunt resistances to one or all 3 nodes. Connecting identical branches to each
of the 3 nodes has the cosmetic advantage that the voltages in a, b, c will be symmetrical, rather than one of them
being zero. The BP# version connects a large shunt resistance automatically, with an appropriate warning, whenever
a zero or near zero diagonal element is encountered. For example, if the zero diagonal is encountered at node c,
then a large resistance will be connected from c to ground which will make v 0. i
&GUETKRVKQP QH 5WRRQTV 4QWVKPGU CPF 5CVWTCDNG 6TCPUHQTOGT %QORQPGPV
Except for the Saturable Transformer Component in the BP# EMTP, which is an input option specifically
for transformers, all other transformer representations discussed here use the general branch input option for B
circuits (with C 0), and possibly additional linear or nonlinear, uncoupled resistance and inductance branches for
the representation of the exciting current. There are three support routines XFO4ME4, T4ELEG, and BCT4#N,
which convert the transformer data into impedance or admittance matrices, as well as a support routine CONVE4T
for the conversion of saturation curves V f(I ) into Traduciendo...

RGS RGS8 f(i). These support routines, as well as the built in
saturable transformer component, are briefly described here.
39
Page 231
5WRRQTV 4QWVKPG :(14/'4
This support routine for single phase transformers is somewhat obsolete, and has been superseded by
support routine BCT4#N. For two winding transformers, it uses essentially the approach of Section .3.1 to form
an admittance matrix
1 1
&
<RW <RW
[;RW] '
1 1
&
<RW <RW
without first separating 4 and L as in ES. ( .7). One half of 1 / jX yÂ‚‡ from ES. ( .35) is then added to ; Â‚‡ and
; , which
Â‚‡ makes the matrix nonsingular. #fter its inversion, and conversion from p.u. to actual values, the 2 x
2 branch impedance matrix is obtained. By not separating 4 and L, this impedance matrix has nonzero off diagonal
resistances, which would produce wrong results at extremely low freSuencies when the magnitude of 4 becomes
comparable with the magnitude of TL (in one particular example, 4 . TL at f 0.002 Hz). #t dc, an off diagonal
resistance would imply a nonzero induced voltage in the secondary winding, which should really be zero in a full
winding transformer.
For three winding transformers, the approach of Section .3.2 is used. First, the impedances of the
eSuivalent star circuit are found with ES. ( .10), which is then converted to the delta circuit with ES. ( .11) to obtain
the 3 x 3 admittance matrix [; ] of ES.‚‡( .12). #gain, there is no separation between 4 and L, and complex
impedances < are used in place of X in all these eSuations. One third of 1 / jX yÂ‚‡ from ES. ( .35) is then added to
; , Â‚‡
; Â‚‡ and ; !!Â‚‡, followed by matrix inversion and conversion to actual values. #gain, nonzero off diagonal
resistances will appear in the branch impedance matrix, as already discussed for the two winding transformer.
Except for errors at extremely low freSuencies, which is caused by not separating 4 and L, the model
produced by XFO4ME4 is useful if the precautions for ill conditioned matrices discussed in Section .2.2 are
observed.
5WRRQTV 4QWVKPG $%64#0
This support routine works for any number of windings, and for single phase as well as for three phase
units. It uses the approach of Section . and .5 to produce the [4] and [L] matrices of coupled branches.
BCT4#N has an option for inductance matrices [L] as well, in cases where the exciting current is nonzero. Because
of the ill conditioning problem (Section .2.2), the author prefers to work with [L] instead of [L], however.
Impedance matrices produced by BCT4#N and XFO4ME4 differ mainly in the existence of off diagonal
resistance values in the latter case, which should make the model from BCT4#N more accurate than that from
XFO4ME4 at very low freSuencies.
5WRRQTV 4QWVKPG 64'.')
This support routine was developed by V. Brandwajn at Ontario Hydro, concurrently with the development
Page 232
of BCT4#N at UBC. It builds the impedance matrix ( .1 ) of N winding single phase or three phase transformers
directly from short circuit and excitation test data, without going through the reduced impedance matrix described
in Section . . The exciting current must always be nonzero, and for very small values of exciting current, the
matrices are subject to the ill conditioning problem described in Section .2.2.
Traduciendo...
4ecall that ES. ( .1 ) is valid for three phase transformers as well, if each element is replaced by 3 x 3
submatrix as discussed in Section .5. With this in mind, the imaginary parts of the diagonal element pairs (X , … uu
X y) uu
of the excited winding i are first calculated from the current of the positive and zero seSuence excitation tests.
If excitation losses are ignored, then X in per unit

uu is simply the reciprocal of the per unit exciting current. With
positive and zero seSuence values thus known, the pair of self and mutual reactances is found from ES. ( .29). For
the other windings, it is reasonable to assume that the p.u. reactances are practically the same as for winding i,
since these open circuit reactances are much larger than the short circuit impedances. This will produce the
imaginary parts of the other diagonal elements . The real part of each diagonal element is the resistance of the
particular winding.
With the diagonal element pairs known, the off diagonal element pairs (< , < ) are calculated from ES.
… uw y uw
( .5), except that real values X are replaced by complex values <,
UJQTV
<KM ' <MK ' (<KK & < KM ) <MM ( .53)
These impedances are first calculated for positive and zero seSuence, and then converted to self and mutual
impedances with ES. ( .29).
#s pointed out in Section .2.2, the elements of [<] must be calculated with high accuracy otherwise, the
short circuit impedances get lost in the open circuit impedances. The lower the exciting current is, the more eSual
the p.u. impedances < , < uu

andww< becomeuwamong themselves in ES. ( .5). Experience has shown that the positive
seSuence exciting current should not be much smaller than 1 for a single precision solution on a UNIV#C
computer (word length of 3 bits) to avoid numerical problems. On computers with higher precision, the value could
obviously be lower. On large, modern transformers, exciting currents of less than 1 are common, but this value
can usually be increased for the analysis without influencing the results. Since these ill conditioning problems do
not exist with [L] , support routine BCT4#N should make T4ELEG unnecessary, after careful testing of both
routines has been carried out.
5WRRQTV 4QWVKPG %108'46
Often, saturation curves supplied by manufacturers give 4MS voltages as a function of 4MS currents. The
If it is known that the magnetizing impedance should be connected across a particular terminal, then the
diagonal elements are modified to account for the differences caused by the short circuited impedances between
the terminals.
Page 233
support routine CONVE4T changes V /I curves into

RGSflux/current
RGS curves 8 f(i) with the following
simplifying assumptions:
1. Hysteresis and eddy current losses in the iron core are ignored,
2. resistance in the winding is ignored, and
3. the 8/i curve is to be generated point by point at such distances that linear interpolation is
acceptable in between points.
For the conversion it is necessary to assume that the flux varies sinusoidally at fundamental freSuency as
a function of time, because it is most likely that the V /I curve has

RGSbeen
RGSmeasured with a sinusoidal terminal
voltage. With assumption (2), v d8/dt. Therefore, the voltage will also be sinusoidal and the conversion of VRGS
values to flux values becomes a simple re scaling:
84/5 2
8' ( .5 )
T
The re scaling of currents is more complicated, except for point i at the end of6 the linear region # B (Fig. .29):
K$ ' +4/5&$ 2 ( .55)
The following points i , i 7,...8 are found recursively: #ssume that i is the next value
9 to be found. #ssume further
that the sinusoidal flux just reaches the value 8 at its maximum,
9
Traduciendo...
8 ' 8' sin TV ( .5 )
(KI 4ecursive conversion of a V /I curve into a 8/i

RGScurve
RGS
Within each segment of the curve already defined by its end points, in this case # B and B C and C D, i is known
as a function of 8 (namely piecewise linear), and with ES. ( .5 ) is then also known as a function of time. Only the
CONVE4T was developed with the assistance of C.F. Cunha, CEMIG, Belo Horizonte, Brazil.
Page 234
last segment is undefined inasmuch as i is still9 unknown. Therefore, i f(t,i ) in the last segment.
9 If the integral
needed for 4MS values,
B
2
(' K F(TV) ( .57)
Bm
is evaluated segment by segment, the result will contain i as an unknown

9 variable. With the trapezoidal rule of
integration (reasonable step size 1E), F has the form
( ' C % DK' % EK' ( .5 )
with a, b, c known. Since F must be eSual to I RGS 9 by definition, ES. ( .5 ) can be solved for the unknown value
i9. This process is repeated recursively until the last point i has been found.
H
If the 8/i curve thus generated is used to re compute a V /I curve, RGS

it will
RGS match the original V /I RGS RGS
curve, except for possible round off errors. #s an example, support routine CONVE4T would convert the table of
per unit 4MS exciting currents as a function of per unit 4MS voltages,
V RGS (p.u.) I RGS

(p.u.)
0 0
0.9 0.005
1.0 0.0150
1.1 0.0 01
with base power 50 MV# and base voltage 35.1 kV, into the following flux/current relationship:
8 (Vs) i (#)
0 0
21 .22 0. 235
23 2. 2.723
2 20.71 7.2 7
This 8/i curve is then converted back into a V /I curve RGS

as an accuracy check. In this case, the V
RGS RGS and I RGS
values were identical with the original input data.
Very often, the V /I curve is only given around the knee point, and not for high values of saturation.
RGS RGS
In such cases, it is best to do the conversion first for the given points, and then to extrapolate on the 8/i curve with
the air core inductance.
Traduciendo...
5CVWTCDNG 6TCPUHQTOGT %QORQPGPV
This built in model was originally developed for single phase N winding transformers. It uses the star
circuit representation of Fig. .30. The primary branch with 4 , L is handled as an uncoupled 4 L branch between
nodes BUS1 , and star point S, whereas each of the other windings 2,...N is treated as a two winding transformer
(first branch from S to BUS2 , second branch from BUS1 to BUS2 w, with k 2,...N).
w The eSuations for each of
Page 235
these two winding transformers are derived from the cascade connection of an ideal transformer with an 4 L branch
(Fig. .31). This leads to
PM PM 4M
& 0
FKUVCT/FV P P XUVCT .M KUVCT
1
' & ( .59)
FKM/FV .M PM XM 4M KM
& 1 0
P .M
(KI Star circuit representation of N winding transformers
Fig. .30 Star circuit representation of N winding transformers
(KI Cascade connection of ideal transformer and 4 L branch
which is the alternate eSuation ( .1) with an inverse inductance matrix [L] . In the particular case of ES. ( .59),
the product [L] [4] is symmetric, which is not true in the general case.
The input data consists of the 4, L values of each star branch, and the turns ratios, as well as information
for the magnetizing branch. For three winding transformers, the impedances of the star branches are usually
available in utility companies from the data files kept for short circuit studies. If these values are in p.u., they must
Page 236
be converted to actual values by using the proper voltage rating V for each ofw the star branches k 1,...N. If the
short circuit impedances are known, then the star branch impedances can beTraduciendo...
calculated from ES. ( .10).
The saturable transformer component has some limitations, which users should be aware of:
1. It cannot be used for more than three windings, because the star circuit is not valid for N 3. This is
more an academic than a practical limitation, because transformers with more than three windings are
seldom encountered.
2. The linear or nonlinear magnetizing inductance, with 4 in parallel,y is connected to the star point, which
is not always the best connecting point, as explained in Section . .
3. Numerical instability has occasionally been observed for the three winding case. It is not believed to be a
programming error. The source of the instability has never been clearly identified, though it is felt that it
is caused by the accumulation of round off errors. V. Brandwajn ran a case in 19 5 in which the instability
disappeared when the ordering of the windings was changed (e.g., first winding changed to low side from
high side).
. While the saturable transformer component has been extended from single phase to three phase units
through the addition of a zero seSuence reluctance parameter, its usefulness for three phase units is limited.
Three phase units are better modelled with inductance or inverse inductance matrices obtained from support
routines BCT4#N or T4ELEG.
(TGSWGPE[ &GRGPFGPV 6TCPUHQTOGT /QFGNU
#t this time, no freSuency dependent effects have yet been included in the transformer model. There are
basically three such effects:
a. FreSuency dependent damping in the short circuit impedances,
b. freSuency dependence in the exciting current, and
c. influence of stray capacitances at freSuencies above 1 to 10 kHz.
CIG4E Working Groups [ , 1 ] have collected some information on the freSuency dependent L/4 ratios
of short circuit impedances (Fig. 2.17). #s explained in Section 2.2.3, this freSuency dependence can easily be
modelled with parallel resistances, which matches the experimental curves reasonably well (Fig. 2.19). When
dealing with matrices [L] or [L] , resistance or conductance matrices [4 ] or [G ] could

‚ be added
‚ automatically by
the program, with the user simply specifying the factor k in
1
[4R] ' M [.] , QT [)R] ' [.]& ( . 0)
M
FreSuency dependent effects in the exciting current were modelled with parallel 4 L branches in [115], as
discussed in Section .3.3. Whether the linear freSuency dependence in these parallel 4 L branches can be separated
easily from the nonlinear saturation effects would have to be verified in laboratory experiments.
For transient studies which involve freSuencies above a few kHz, capacitances must be added to the 4 L
models. #s suggested in [123], capacitances should be included
Page 237
a. between the winding closest to the core, and the core,
b. between any two windings, and
c. across each winding from one end to the other.
In reality, inductances and capacitances are distributed, but reasonably accurate results, as seen from
terminals, can be obtained by lumping one half of the capacitance at each end of winding for effects (a) and (b), and
by lumping the total capacitance in parallel with the winding for effect (c), as shown in Fig. .32. Each of these
capacitances can be calculated from the geometry of the transformer design. Obviously, the internal voltage
distribution across a winding, which is of such great concern to the transformer design, cannot be obtained with the
simple model of Fig. .32. Fig. .33 compares measured impedances of a transformer (500 MV#, 7 5/3 5/17.25
kV) and calculated impedances with a model where the capacitances were added according to Fig. .32. The
agreement is Suite good. Similar suggestions for the addition of capacitances have been made by others (e.g., [12 ]).
Traduciendo...
(KI #ddition of capacitances to 4 L model

(subscripts a, b, c refer to the three effects mentioned in
text)
(a) X1 excited, H1, ;1, ;2 (b) X1 excited, H1, ;1, ;2

grounded open circuited
(KI FreSuency response of single phase autotransformer with tertiary winding (marking of terminals
according to North #merican standards: H1 high voltage terminal, X1 low voltage terminal, ;1, ;2
terminals at both ends of tertiary winding) [123]. l 19 1 IEEE
Page 238
5+/2.' 81.6#)' #0& %744'06 5174%'5
Most of the simple sources are either voltage or current sources defined as a time dependent function f(t),
v(t) ' f(t), or i(t) ' f(t) (7.1)
FreSuently used functions f(t) are built into the EMTP. There is also a current controlled dc voltage source for
simplified HVDC simulations, which is more complicated than ES. (7.1). In addition to the built in functions, the
BP# version of the EMTP allows the user to define functions through user supplied FO4T4#N subroutines, and
to declare T#CS output variables as voltage or current source functions. The UBC version of the EMTP does not
have these two options, but allows the user to read f(t) step by step in increments at )t. This option has rarely been
used, however.
Note that f(t) 0 for a current source implies that the source is disconnected from the network (i 0),
whereas for a voltage source it implies that the source is short circuited (v 0).
%QPPGEVKQP QH 5QWTEGU VQ 0QFGU
If a voltage or current source is specified at a node, it is assumed to be connected between that node and
local ground, as shown in Fig. 7.1. # voltage source of v(t) 1.0 V means that the potential at that node is 1.0
V with respect to local ground, whereas a current source of 1.0 # implies that 1.0 # flows from the local ground
into that node.
Traduciendo...
(a) Voltage source (b) Current source (c) Current source
between node from local between two nodes
and local ground ground into node
(KI Source connections
71
Page 239
%WTTGPV 5QWTEGU $GVYGGP 6YQ 0QFGU
Current sources between two nodes, e.g., a current leaving node B and entering into node # as shown in
Fig. 7.1(c), must be specified as two current sources, namely as
i#(t) ' f(t) , and i$(t) ' & f(t) (7.2)
8QNVCIG 5QWTEGU $GVYGGP 6YQ 0QFGU
Until recently, voltage sources could not be connected between two nodes. With the addition of ideal
transformers to the BP# EMTP in 19 2 (Section . ), voltage sources between two nodes are easy to set up now.
In Fig. .25, simply ground node ý, connect the voltage source from node j to ground, and use a transformer ratio
of 1:1. This will introduce a voltage source between nodes k and m. # special input option has been provided for
using the ideal transformer for this particular purpose.
The UBC EMTP and older versions of the BP# EMTP do not accept voltage sources between nodes. One
could use the eSuivalent circuit of Fig. .27 for the ideal transformer, however, which turns into the circuit of Fig.
7.2. This representation works in the transient solution part of the UBC EMTP, provided the branches of Fig. 7.2
are read in last. In that case, the node extra will be forced to the bottom of the eSuations as shown in Fig. .2 .
The steady state subroutine in both versions, as well as the transient solution in the BP# version, use optimal re
ordering of nodes, which may not force the row for node extra far enough down to assure nonzero diagonal
elements during the Gauss elimination. Using Fig. 7.2 may therefore not always work, unless minor modifications
are made to the re ordering subroutine.
(KI ESuivalent circuit for voltage source

v(t) between nodes k and m
In all versions, a voltage source in series with a (nonzero) impedance can always be converted into a current
source in parallel with that impedance. The current source between the two nodes is then handled as shown in ES.
72
Page 240
Traduciendo...
(7.2). The conversion from a Thevenin eSuivalent circuit (v in series with <) to a Norton eSuivalent circuit (i in
parallel with <) is especially simple if the impedance is a pure resistance 4, as shown in Fig. 7.3.
(KI Conversion of v(t) in series with 4 into i(t)

v(t)/4 in parallel with 4
Converting a voltage source in series with an inductance L into a current source with parallel L is slightly more
complicated. L is again connected between nodes k and m, in the same way as 4 in Fig. 7.3. The definition of the
current source depends on the initial conditions, however. For example, if
v(t) ' VOCZ cos(Tt % N) (7.3)
and if the case starts from zero initial conditions, then
VOCZ
i(t) ' [sin(Tt % N) & sinN] (7. a)
TL
If the case starts from linear ac steady state conditions, with that voltage source being included in the steady state
solution, then
VOCZ
i(t) ' cos(Tt % N & 90E) (7. b)
TL
/QTG 6JCP 1PG 5QWTEG QP 5COG 0QFG
If more than one voltage source is connected to the same node, then the EMTP simply adds their functions
f (t),...f (t)P to form one voltage source. This implies a series connection of the voltage sources between the node
and local ground, as shown in Fig. 7. (a).
73
Page 241
(a) Series connection of (b)Traduciendo...

Parallel connection of
sources current sources
(KI Multiple voltage or current sources on same node
If more than one current source is connected to the same node, then the EMTP again adds their functions
f (t),...f (t)P to form one current source. This implies a parallel connection of the current sources, as shown in Fig.
7. (b).
Source functions can be set to zero by using parameters t 56#46 and T 5612 . The EMTP sets f(t) 0 for t
T 56#46 and for t $ T 5612 . By using more than one source function at the same node with these parameters, more
complicated functions can be built up from the simple functions, as explained in the UBC User s Manual and in the
BP# 4ule Book.
If voltage and current sources are specified at the same node, then only the voltage sources are used by the
EMTP, and the current sources are ignored. Current sources would have no influence on the network in such a case,
because they would be directly short circuited through the voltage sources.
$WKNV KP 5KORNG 5QWTEG (WPEVKQPU
Commonly encountered source functions are built into the EMTP. They are:
(a) Step function (type 11). In cases which start from zero initial conditions, the step function is
approximate in the sense that the EMTP will see a finite rise time from f(0) 0 to f()t) F , asOCZ
shown
in Fig. 7.5.
Page 242
(a) Starting from zero (b) Starting from initial

initial conditions value FOCZ
(KI Step function
(b) 4amp function (type 12) with f(t) as shown in Fig. 7. . The value of the function rises linearly from
T 56#46 to T 56#46 T to a value of F , and then

OCZremains constant until it is zeroed at t $ T 5612 .
(KI 4amp function
# modified ramp function (type 13) has the same rise to F at T OCZ 56#46 T as in Fig. 7. , but decays or rises with
a linear slope thereafter. By setting T 56#46 0 and T 0, this becomes a step function with a superimposed linear
decay or rise.
(c) Sinusoidal function (type 1 ) with Traduciendo...
f(t) ' FOCZ cos(Tt % N) if T56#46 # 0 (7.5a)
or
f(t) ' FOCZ cos(T(t & T56#46) % N) if T56#46 0 (7.5b)
with f(t) ' 0 for t T56#46
This is probably one of the most used source functions. Note that the peak value F must be specified,
OCZ
rather than the 4MS value. To start a case from linear ac steady state conditions, or to obtain a seSuence of steady
75
Page 243
state solutions at a number of freSuencies, use T 56#46 0 to indicate to the EMTP that this sinusoidal source should
be used for the steady state solution. The value of T 56#46 is immaterial as long as its value is negative, and the
complex peak phasor used for that source is then

LN
V or I F e OCZ (7. )
(d) Impulse function (type 15) of the form
&" V & e &" V

f(t) ' k e (7.7)
This function has been provided for the representation of lightning or switching impulses, as used in standard impulse
tests on transformers and other eSuipment. # typical lightning impulse voltage is shown in Fig. 7.7 [12 ], and a
typical switching impulse voltage is shown in Fig. 7. [12 ]. There is no simple relationship between the time
constants 1/" and a/" in ES. (7.7) and the virtual front time T (or time to crest T ) and the virtual time
ETto half
value T . Table 7.1 shows the values for freSuently used waveshapes, as well as values for k which produce a
maximum value of f 1.0OCZ

in ES. (7.7). The time at which the maximum occurs is found by setting the derivative
df/dt 0 from ES. (7.7) and solving for t . Inserting tOCZ

into ES. (7.7) then produces f . Note that 1/" and
OCZ OCZ
1/" in Table 7.1 are in zs, whereas the EMTP input is usually in s.
(KI General shape of lightning impulse voltage (IEC definitions: T virtual front time,
typically 1.2 zs v 30 T virtual time to half value, typically 50 zs v 20 ). 4eprinted with
permission from [12 ], l 19 , Pergamon Books Ltd
7
Traduciendo...
Page 244
(KI General shape of switching impulse voltage (IEC definitions: T time to crest, ET
typically 250 zs v 20 T virtual time to half value, typically 2500 zs v 0 , T time F
above 90 ). 4eprinted with permission from [12 ], l 19 , Pergamon Books Ltd
In impulse testing, the capacitance of the test object is usually much smaller than the capacitance of the
impulse generator. It is then permissible to regard the impulse generator as a voltage source with the function of
ES. (7.7). In cases where the impulse generator is discharged into lines, or into other test objects with impedances
which can influence the wave
6CDNG 4elationship between T , T , and " , " . 4eprinted with permission from [12 ], l
19 , Pergamon Books Ltd
T /T (zs) T ET
/T (zs) 1 (zs) 1 (zs) k to produce
" " f OCZ
1.0
1.2/5 3. 0. 0 2.01
1.2/50 .2 0. 05 1.037
1.2/200 2 0.3 1 1.010
250/2500 2 77 10 1.175
250/2500 3155 2.5 1.10
shape, it may be better to simulate the impulse generator as a capacitance and resistance network, as shown in Fig.
7.9 for a simple single stage impulse generator. The initial voltage across C would be nonzero, and the switch
closing would simulate the gap firing. Fig. 7.10 compares measurements against EMTP simulation results for the
waveshape of a multistage impulse generator, where the generator was modelled as a network of capacitances,
resistances in inductances [127]. The spark gaps were represented as time dependent resistances based on Toepler s
formula.
(a) Circuit type a (b) Circuit type b
(KI Single stage impulse generators
77
Page 245
Traduciendo...
(a) Measurement (b) Simulation results (1 exact with nonsimultaneous

firing of spark gaps, 2 simultaneous firing, 3
simultaneous firing with gaps as ideal switches)
(KI Waveshape of a multistage impulse generator [127]. l 1971 IEEE
%WTTGPV %QPVTQNNGF FE 8QNVCIG 5QWTEG
This source provides a simplified model of an HVDC converter station [12 ], and produces simulation
results which come reasonably close to field tests [129]. The current dependent voltage source is connected between
two nodes (cathode and anode), as indicated in Fig. 7.11. The current can only flow in one direction (from anode
to cathode). This is simulated internally with a switch on the anode side, which opens to prevent the current from
going negative and closes again at the proper voltage polarity. Spurious voltage oscillations may occur between the
anode and cathode side after the switch opens, unless the damping circuits across the valves are also modelled. Good
results were obtained in [12 ] when an 4C branch was added between the anode and cathode (4 900 S and C
0.15 zs in that case).
The current regulator is assumed to be an amplifier with two inputs (one proportional to current bias I , $+#5
and the other proportional to measured current i), and with one output e which determines
" the firing angle. The
transfer function of the regulator is
K (1 % sT )
G(s) ' (7. )
(1 % sT ) (1 % sT )
with limits placed on the output e in accordance

" with rectifier minimum firing angle, or inverter minimum extinction
angle.
Page 246
The current controlled dc voltage source is a function of e ,"
vFE ' k % k e " (7.9)
as shown in Fig. 7.12. The current regulator output e , minus a bias

" value (10V in Fig. 7.12) is proportional to
cos". The inverter normally operates at minimum extinction angle at the limit e , and the rectifier normally
"OKP
operates on constant current control between the limits. The user defines steady state limits for v , which are FE
converted to limits on e with" ES. (7.9). If the converter operates at the maximum limit e (or at the minimum"OCZ
limit e ),"OKP
either in initial steady state or later during the transient simulation, it will be back off the limit as soon
as the derivative de /dt "becomes negative (or positive) in the differential eSuation
de " di de "
(T % T ) ' K (I$+#5 & i) & kT & TT &e" (7.10)
dt dt dt
The value for d e /dt "is zero in ES. (7.10) as long as the converter operates at the limit.
Traduciendo...
(KI Current controlled dc voltage source
79
Page 247
(KI 4elationship between v and e (k 150 000,

FE "
k 15 000)
5VGCF[ 5VCVG 5QNWVKQP
Steady state dc initial conditions are automatically computed by the program with the specified value v (0). FE
Since the steady state subroutine was only written for ac phasor solutions, the dc voltage is actually represented as
v FE
v (t) cos(Tt)
FE with a very low freSuency of f 0.001 Hz. Practice has shown that this is sufficiently close
to dc, and still makes reactances TL and susceptances TC large enough to avoid numerical problems in the ac steady
state solution. When the current controlled dc voltage source was added to the EMTP, voltage sources between two
nodes were not yet permitted. For the steady state solution, a resistance 4 is therefore
GSWKX connected in series with
the voltage source, which is then converted into a current source in parallel with 4 . This
GSWKX produces accurate
results if the user already knows what the initial current i (0) is, because
FE the specified voltage source of the rectifier
is automatically increased by 4 i (0),

GSWKX FE and that of the inverter is decreased by 4 i (0).FEThe
GSWKX program user
should check, however, whether the computed current i does indeed

FE agree with what the user thought it would be.
This nuisance of having to specify i (0), without

FE knowing whether it will agree with the computed value, could be
removed by using the methods described in Section .3, if this HVDC model is used often enough to warrant the
program changes. The value of 4 is the

GSWKX same as the one used in the transient solution (Section 7.5.2).
The normal steady state operation of an HVDC transmission link, measured somewhere at a common point
(e.g., in the middle of the line) is indicated in Fig. 7.13. For the converter operating between the limits on constant
current control (which is normally the rectifier), I is automatically

$+#5 computed to produce the characteristic # #
of Fig. 7.13,
7 10 Traduciendo...
Page 248
e" (0)
I$+#5 ' i(0) % , if e " e "OCZ (7.11)
K "OKP e
with i(0), e (0)

" being the dc initial conditions. For the converter operating at maximum or minimum voltage (which
is normally the inverter), the current setting I 5'66+0) must be given as part of the input, which defines the point where
the converter backs off the limit and goes into constant current control. I is again automatically
$+#5 computed, which
in this case is
e" (0)
I$+#5 ' I5'66+0) % if e " (0) ' e "OKP (7.12)
K "OCZ or e
I 5'66+0) is typically 15 lower than the current order I 14&'4 at the steady state operating point for inverters (or 15
higher for rectifiers).
6TCPUKGPV 5QNWVKQP
In the transient solution, the dynamics of the current controller in the form of ES. (7.9) and (7.10) must
obviously be taken into account. First, rewrite the second order differential eSuation (7.10) as two first order
differential eSuations,
dx di
e " % Tx % P ' K (I$+#5 & i) & KT (7.13a)
dt dt
de "
x' (7.13b)
dt
with the new variable x and with the new parameters
T ' T% T (7.13c)
P ' TT (7.13d)
#fter applying the trapezoidal rule of integration to ES. (7.13a) and (7.13b) (replacing x by [x(t )t) x(t)]/2 and
dx/dt by [x(t) v(t )t)]/)t, etc.), and after eliminating x(t), one linear algebraic eSuation between e (t) and i(t) is "
obtained. Inserting this into ES. (7.9) produces an eSuation of the form
vFE(t) ' v (t) & 4GSWKX i(t) (7.1 )
which is a simple voltage source v (t) in series with an internal resistance 4 . This
GSWKX Thevenin eSuivalent circuit
is converted into a current source i (t) in parallel with 4 (Fig. 7.1 ).

GSWKX
7 11
Page 249
Traduciendo...
(KI Normal operation of HVDC

transmission link
(KI Norton eSuivalent circuit
The eSuivalent resistance 4 remains

GSWKX constant for a given step size )t,
2T
kK1 %
)t
4GSWKX ' (7.15)
2T P
1% %
)t ()t)
whereas the current source i (t) depends on the values e (t )t) and "x(t )t) of the preceding time step. #fter the
complete network solution at each time step, with the converter representation of Fig. 7.1 , the current is calculated
with ES. (7.1 ), and then used to update the variables e and x. "
If e "hits one of the limits e or e , "OCZ

it is kept "OKP
at the appropriate limit in the following time steps, with
x and dx/dt set to zero. B.C. Chiu has recently shown, however, that simply setting x and dx/dt to zero at the limit
does not represent the true behavior of the current controller [130]. The treatment of limits should therefore be
revised, if this current controlled dc voltage source remains in use. Backing off the limit occurs when the derivative
7 12
Page 250
de "/dt calculated from ES. (7.10) becomes negative in case of e e , or positive

" in case
"OCZof e e . " "OKP
The switch opens as soon as i(t) 0, and closes again as soon as V #01&' v $1661/ , to assure that current
can only flow in one direction. This updating of the current source i (t) from step to step is not influenced by the
switching actions.
Traduciendo...
7 13
Page 251
6*4'' 2*#5' 5;0%*410175 /#%*+0'
Co author: V. Brandwajn
The details with which synchronous machines must be modelled depend very much on the type of transient
study. Most readers will be familiar with the simple representation of the synchronous machine as a voltage source
E behind a subtransient reactance X . This prepresentation is commonly used in short circuit studies with steady
state phasor solutions, and is also reasonably accurate for transient studies for the first few cycles of a transient
disturbance. Switching surge studies fall into that category. #nother well known representation is E behind X p
for simplified stability studies. Both of these representations can be derived from the same detailed model by making
certain assumptions, such as neglecting flux linkage changes in the field structure circuits for E behind X , and p
in addition, assuming that the damper winding currents have died out for E behind X .p
The need for the detailed model described here arose in connection with subsynchronous resonance studies
in the mid 1970 s. In such studies, the time span is too long to allow the use of simplified models. Furthermore,
the torsional dynamics of the shaft with its generator rotor and turbine rotor masses had to be represented as well.
Detailed models are now also used for other types of studies (e.g., simulation of out of step synchronization). To
cover all possible cases, the synchronous machine model represents the details of the electrical part of the generator
as well as the mechanical part of the generator and turbine. For studies in which speed variations and torsional
vibrations can be ignored, an option is provided for by passing the mechanical part of the UBC EMTP.
The synchronous machine model was developed for the usual design with three phase ac armature windings
on the stator and a dc field winding with one or more pole pairs on the rotor. For a reversed design (armature
windings on the rotor and field winding on the stator), it is probably possible to represent the machine in some
eSuivalent way as a machine with the usual design. Even though the model was developed with turbine driven
generators in mind, it can be used for synchronous motors as well (e.g., pumping mode in a pumped storage plant).
The model cannot be used for dual excited machines (one field winding in direct axis and another field
winding in Suadrature axis) at this time, thought it would be fairly easy to change the program to allow for it. Since
such machines have not yet found practical acceptance, it was felt that the extra programming was not justified.
Induction machines can also not be modelled with it, though program changes could again be made for that purpose.
For these and other types of machines, the universal machine of Section 9 should be used.
While the eSuations for the detailed machine model have been more or less the same in all attempts to
incorporate them into the EMTP, there have been noticeable differences in how their solution is interfaced with the
rest of the network. K. Carlsen, E.H. Lenfest and J.J. LaForest were probably the first to add a machine model to
the EMTP, but the resulting M#NT4#P program [97] was not made available to users outside General Electric
Co. M.C. Hall, J. #lms (Southern California Edison Co.) and G. Gross (Pacific Gas Electric Co.), with the
assistance of W.S. Meyer (Bonneville Power #dministration), implemented the first model which became available
Traduciendo...
The synchronous machine model is the UBC EMTP is experimental and has not been released.
Page 252
to the general public. They opted for an iterative solution at each time step, with the rest of the system, as seen from
the machine terminals, represented by a three phase Thevenin eSuivalent circuit [9 ]. To keep this compensation
approach efficient, machines had to be separated by distributed parameter lines from each other. If that separation
did not exist in reality, short artificial stub lines had to be introduced which sometimes caused problems. V.
Brandwajn suggested another alternative in which the machine is basically represented as an internal voltage source
behind some impedance. The voltage source is recomputed for each time step, and the impedance becomes part of
the nodal conductance matrix [G] in ES. (1. ). This approach depends on the prediction of some variables, which
are not corrected at one and the same time step in order to keep the algorithm non iterative. While the prediction
can theoretically cause numerical instability, it has been refined to such an extent by now that the method has become
Suite stable and reliable. Whether an option for repeat solutions as correctors will be added someday remains to be
seen. Numerical stability has been more of a problem with machine models partly because the typical time span of
a few cycles in switching surge studies has grown to a few seconds in machine transient studies, with the step size
)t being only slightly larger, if at all, in the latter case.
$CUKE 'SWCVKQPU HQT 'NGEVTKECN 2CTV
Since there is no uniformity on sign conventions in the literature, the sign conventions used here shall first
be summarized:
(a) The flux linkage 8 of a winding, produced by current in the same winding, is considered to have
the same sign as the current (8 Li, with L being the self inductance of the winding).
(b) The generator convention is used for all windings, that is, each winding k is described by
F8M(V)
XM(V) ' &4M KM(V) & ( .1)
FV
(with the load convention, the signs would be positive on the right hand side).
(c) The newly recommended position of the Suadrature axis lagging 90E behind the direct axis in the
machine phasor diagram is adopted here [99]. In Park s original work, and in most papers and
books, it is leading, and as a conseSuence the terms in the second row of [T] of ES. ( .7b) have
negative signs there.
The machine parameters are influenced by the type of construction. Salient pole machines are used in hydro
plants, with 2 or more (up to 50) pole pairs. The magnetic properties of a salient pole machine along the axis of
symmetry of a field pole (direct axis) and along the axis of symmetry midway between two field poles (Suadrature
axis) are noticeably different because a large part of the path in the latter case is in air (Fig. .1a). Cylindrical rotor
machines have long cylindrical rotors with slots in which distributed field windings are placed (Fig. .1b). They
are used in thermal plants, and have 1 or 2 pole pairs. For cylindrical rotor machines the magnetic properties on
the two axes differ only slightly (because of the field windings embedded in the slots), and this difference
( saliency ) can often be ignored. The word saliency is used as a short expression for the fact that the rotor has
Page 253
different magnetic properties on the two axes.
Traduciendo...
(a) Salient pole machine (b) Cylindrical rotor machine
(KI Cross sections of synchronous machines (d direct axis, S Suadrature axis)

[101]. 4eprinted by permission of I. Kimbark
The machine model in the EMTP always allows for saliency if saliency is ignored, the same eSuations will still be
used, except that certain parameters will have been set eSual at input time (X X , etc.). ƒ p
The electrical part of the synchronous machine is modelled as a two pole machine with 7 coupled windings :
2 three armature windings (connected to the power system),
f one field winding which produces flux in the direct axis (connected to the dc source of the
excitation system),
g one hypothetical winding in the Suadrature axis to represent slowly changing fluxes in the
Suadrature axis which are produced by deep flowing eddy currents (normally negligible in salient
pole machines)
D one hypothetical winding in the direct axis to represent damper bar effects,
3 one hypothetical winding in the Suadrature axis to represent damper bar effects.
For machines with more than one pole pair, the electrical eSuations are the same as for one pole pair, except that
the angular freSuency and the torSue being used in the eSuations of the mechanical part must be converted as follows:
#nother, more modern approach is to measure the freSuency response from the terminals, which can then be
used to represent the machine with transfer functions between the terminals, without assuming a given number of
lumped windings a priori. One can also use curve fitting techniSues to match this measured response with that
from a series and parallel combination of 4 L branches [100]. The end results in the latter case is basically the
same model as described here, except that damper bars are sometimes represented by more than one winding,
and that the data is obtained from freSuency response tests.
Page 254
T &RQNG&OCEJKPG
TCEVWCN ' ( .2a)
R/2
R
6CEVWCN ' 6 &RQNG&OCEJKPG ( .2b)
2
where p/2 is the number of pole pairs.
The behavior of the 7 windings is described by two systems of eSuations, namely by the voltage eSuations
F
[X] ' &[4] [K] & [8] ( .3)
FV
with
[i] [i !,ris,8iQ, i , i , i , i ],
[8] [8 , 8 , 8 , 8 , 8 , 8 !, 8 ],r s 8 Q
[v] [v , v , v , v , 0, 0, 0]
! (zero
r in last 3 components because g , D , and 3 windings are short
circuited)
[4] diagonal matrix of winding resistances 4 , 4 , 4 , 4 , 4 , 4g , 4 g(subscript

g r a for
s armature),
8 Q
and by the flux current relationship
. . ... . 3
. . ... . 3
[8] ' [.] [K] YKVJ [.] ' (.)
. .. .. Traduciendo...
.3 .3 ... .33
To make the eSuations manageable, a number of idealized characteristics are assumed, which are reasonable for
system studies. These assumptions for the ideal synchronous machine [7 , p. 700] are :!
(1) The resistance of each winding is constant.
(2) The permeance of each portion of the magnetic circuit is constant (corrections for saturation effects
will be introduced later, however).
(3) The armature windings are symmetrical with respect to each other.
() The electric and magnetic circuits of the field structure are symmetrical about the direct or
Suadrature axis.
(5) The self inductance of each winding on the field structure (f, g, D, 3) is constant.
() The self and mutual inductances of the armature windings are a constant plus a second harmonic
sinusoidal function of the rotor position $ (second harmonic component zero if saliency ignored),
with the amplitude of the second harmonic component being the same for all self and mutual
!
For a detailed design analysis of synchronous machines, many of these idealizations cannot be made. Since
they imply that the field distribution across a pole is a fundamental sinusoid, harmonics produced by the
nonsinusoidal field distribution in a real machine could not be studied with the ideal machine implemented in the
EMTP.
Page 255
inductances.
(7) The mutual inductance between any winding on the field structure and any armature winding is a
fundamental sinusoidal function of the rotor position $.
() Effects of hysteresis are negligible.
(9) Effects of eddy currents are negligible or, in the case of cylindrical rotor machines, are represented
by the g winding.
Then,
. ' .U % .Ocos2$, UKOKNCTHQT. , .

. ' . ' /U % .Ocos(2$ & 120E) UKOKNCT HQT . , .
. H ' .H ' /CHcos$ UKOKNCT HQT . H, . H
. & ' .& ' /C&cos$ UKOKNCT HQT . &, . & ( .5)
. I ' .I ' /CIsin$ UKOKNCT HQT . I, . I
. 3 ' .3 ' /C3sin$ UKOKNCT HQT . 3, . 3
.HH, .II, .&&, .33, /H&, /I3 EQPUVCPV (PQV HWPEVKQPU QH $)
with $ being the angular position of the assumed two pole rotor relative to the stator ($ gi†‡gx$ ‚ xq ygitu€q / p/2),
which is related to the angular freSuency,
F$
T' (.)
FV
Some authors (e.g., Kimbark [101]) use a different sign for M in ES. ( .5).
… With the sign used here, the numerical
value will be negative.
The solution of the two systems of eSuations ( .3) and ( . ) is complicated by the fact that the inductances
in ES. ( . ) are functions of time through their dependence on $ in ES. ( .5). While it is possible to solve them
directly in phase Suantities , most" authors prefer to transform them from phase Suantities to d, S, 0 Suantities
because the inductances become constants in the latter reference frame. This transformation projects the rotating
fluxes onto the field axis, from where they appear as stationary during steady state operation. The transformation
was first proposed by Blondel, and further developed by Doherty, Nickle and Park in North #merica, it is now often
called Park s transformation. The transformation is identical for fluxes, voltages, and currents, and converts phase
Suantities 1, 2, 3 into d, S, 0 Suantities, with Suantities on the field structure remaining unchanged,
[8FS ] ' [6]& [8] KFGPVKECN HQT [X], [K] ( .7a)
with
Traduciendo...
"
If space harmonics in the magnetic field distribution had to be taken into account, then L and L in ES.
( .5) would have added th, th, and higher harmonics terms, and L etc. would have radded 3rd,
5th,...harmonics terms. In that case, solutions in phase Suantities would probably be the best choice.
Page 256
[8FS ] ' [8F, 8S, 8 , 8H, 8I, 8&, 83 ] , CPF
4emain unchanged
2 2 2
cos$ cos($&120E) cos($%120E) 0 0 0 0
3 3 3
2 2 2
sin$ sin($&120E) sin($%120E) 0 0 0 0
3 3 3
1 1 1
[6]& ' 0000 ( .7b)
3 3 3
0 0 0 1000
0 0 0 0100
0 0 0 0010
0 0 0 0001
ES. ( .7) is an orthogonal transformation it therefore follows that
&
[6] ' [6] VTCPURQUGF (.)
The matrices [T] and [T] are normalized here. This has the advantage that the power is invariant under
transformation, and more importantly, that the inductance matrix in d, S, 0 Suantities is always symmetric. The
lack of symmetry with unnormalized Suantities can easily lead to confusion, because it is often removed by rescaling
of field structure Suantities which in turn imposes unnecessary restrictions on the choice of base values if p.u.
Suantities are used. #uthors who work with unnormalized transformations use a factor 2/3 in the first two rows of
ES. ( .7b), and 1/3 in the third row. In many older publications the position of the Suadrature axis is assumed 90E
ahead of the direct axis, rather than lagging 90E behind d axis as here, and the second row of ES. ( .7b) has
therefore negative signs there.
Transforming ES. ( .3) to d, S, 0 Suantities yields
&T8S
%T8F
0
F
[XFS ] ' &[4] [KFS ] & [8FS ] % 0 ( .9)
FV
0
0
0
which is almost identical in form to ES. ( .3), except for the speed voltage terms T8 and T8 (voltage induced
ƒ p
Page 257
in armature because of rotating field poles). They come out of ES. ( .3) by keeping in mind that [T] is a function
of time,
Traduciendo...
F
[6]& F 6[6] [8FS ]> ' [8FS ] % [6]& F [6] [8FS ]
FV FV FV
Transforming ES. ( . ) yields flux current relationships which can be partitioned into two systems of eSuations for
the direct and Suadrature axis, and one eSuation for zero seSuence,
8F .F /FH /F& KF
8H ' /FH .HH /H& KH ( .10a)
8& /F& /H& .&& K&
3 3
YJGTG /FH ' /CH , /F& ' /C&
2 2
8S .S /SI /S3 KS
8I ' /SI .I I /I3 KI ( .10b)
83 /S3 /I3 .33 K3
3 3
YJGTG /SI ' /CI , /S3 ' /C3
2 2
CPF 8'.K ( .10c)
Most elements of these inductance matrices with constant coefficients have already been defined in ES. ( .5), except
for
direct axis synchronous inductance L L M 3/2

p L, … … y
Suadr. axis synchronous inductance L L M 3/2

ƒ L, … … y ( .11)
zero seSuence inductance L L 2M . … …
&GVGTOKPCVKQP QH 'NGEVTKECN 2CTCOGVGTU#
#
The assistance of S. Bhattacharya and ;e <hong liang in research for this section is gratefully
acknowledged.
Page 258
# set of resistances and of self and mutual inductances is needed in the two systems of eSuations ( .9) and
( .10), which are not directly available from calculations or measurements. #ccording to IEEE or IEC standards
[102, 103] the known Suantities are
armature resistance 4 ,g
armature leakage reactance X ,R
zero seSuence reactance X,
transient reactances X p, X , ƒ
subtransient reactances X p, X , ƒ
transient short circuit time constants T p, T , ƒ
subtransient short circuit time constants T p, T . ƒ
#ll reactances and time constants must be unsaturated values, because saturation is considered separately, as
explained in Section . . This is the reason why short circuit time constants are preferred as test data over open
circuit time constants, because the measurement of the latter is influenced by saturation effects [10 ]. Fortunately,
one set of time constants can be converted precisely into the other set [10 ], as explained in #ppendix VI in ES. (VI.
1 c) and (VI. 21),

Traduciendo...
) )) :F :F :F
6 6F ) % 1 & % 6F ))
F% 6 F' ) ) ))
:F :F :F
) )) :F
6 ) 6F ))
F 6 F ' 6F )) ( .12)
:F
for the direct axis, and identically for the Suadrature axis by replacing subscript d with S.
The number of known parameters is less than the number of resistance and inductance values in ES. ( .9)
and ( .10), and some assumptions must therefore be made before the data can be converted. Since the procedure
for data conversion is the same for the direct and Suadrature axis parameters, only the direct axis will be discussed
from here on.
Winding D is a hypothetical winding which represents the effects of the damper bar sSuirrel cage. We can
therefore assume any number of turns for it, without loss of generality. In particular, we can choose the number
of turns in such a way that
/F& ' /FH ( .13a)
in ES. ( .10a), and similarly
/S3 ' /SI ( .13b)
in ES. ( .10b). Many authors represent the flux current relationships with an eSuivalent star circuit, which reSuires
Page 259
all three mutual inductances in ES. ( .10a) to be eSual. This is achieved by modifying (rescaling) the field structure
Suantities as follows:
3 1
8HO ' M @ 8H , CPF KHO ' @ KH
2 3 ( .1 a)
M
2
/CH
YKVJ M' ( .1 b)
/H&
(identical for 8 and

8 i ). Then
8 ES. ( .10a) becomes
8F .F /O /O KF
8HO ' /O .HHO /O KHO ( .15a)
8&O /O /O .&&O K&O
with
3 3 3
/O ' M /CH , .HHO ' M .HH , .&&O ' M .&& ( .15b)
2 2 2
and
3 3
4HO ' M 4H , 4&O ' M 4& ( .1 )
2 2
Fig. .2 shows the eSuivalent star circuit for the direct axis, with the speed voltage term and resistances added to
make it correct for ES. ( .9) as well. Modifying the field structure Suantities is the same as changing the number
of turns in the field structure windings. It does not influence the data conversion, but it is simpler to carry out if the
modified form of ES. ( .15a) is used. The correct turns ratio is then introduced later from the relationship between
rated no load excitation current and rated terminal voltage.
The best data conversion procedure seems to be that of Canay [10 ]. It uses the four eSuations which define
the open and short circuit time constants, as derived in #ppendix VI.2, to find 4 , 4 , L and L ry 8y rry 88y (m
Traduciendo...
dropped in #ppendix VI to simplify the notation). Usually, only one pair of time constants (either T , T or T , p p p
T p) plus X , X pare known

p in that case, the other pair must first be found from ES. ( .12). Solving the four
eSuations for the four field structure Suantities presupposes that the mutual inductance M in ES. ( .15a) yis already
known. Its value has traditionally been found from leakage flux calculations. While turns ratios have been
Page 260
(KI ESuivalent circuit for direct axis with modified field structure Suantities
unimportant so far, they must be considered in the definition of leakage flux, since it is the actual flux N, rather than
the flux linkage 8 NN (N number of turns) which is involved. In defining the leakage flux we must either use
actual flux Suantities, or flux linkages with turns ratios of 1:1. The leakage flux linkage produced by i is then p
8ý ' .F KF & .FH KF , RTQXKFGF 0F:0H ' 1:1 ( .17)
Let us assume that all field structure Suantities are referred to the armature side, which implies N :N 1:1 in the gr
original eSuations ( . ) with phase Suantities, with the mutual inductance being M (cos $ 1.0 if magnetic
gr axis
of phase 1 armature winding lined up with direct axis). #fter transforming to d, S, 0 Suantities, the mutual
inductance in ES. ( .10a) between d and f changes to %3/%2 M , which implies

gr that the ratio is now N :N %3/%2 pr
: 1. To convert back to a ratio of 1:1, the second row and column in ES. ( .10a) must be multiplied with %3/%2,
which changes the mutual inductance to 3/2 M . Then thegrleakage flux linkage produced by i with a 1:1 ratio p
becomes
3
8ý ' .F KF & /CH KF
2
or for the leakage inductance,
3
.ý ' .F & /CH ( .1 )
2
Unfortunately, this eSuation is still not enough for finding M in the modified
y matrix of ES. ( .15a) because of the
unknown factor k in ES. ( .1 b). To find k, an additional test Suantity must be measured which has not yet been
10
Page 261
Traduciendo...
prescribed in the IEEE or IEC standards. It has therefore been common practice to assume k 1, which implies
M r8
M . Withgrthis assumption, the results for armature Suantities will be correct, but the amplitude of the fast
oscillations in the field current will be incorrect, as pointed out by Canay [10 ] and others. Fig. .3 shows the
measured field current after a three phase short circuit [10 ], compared with EMTP simulation results with k 1,
and with the correct value of k. Note that the d branch in the star circuit of Fig. .2 is the leakage inductance only
if all Suantities are referred to the armature
(KI Field current after three phase short circuit [10 ]. 4eprinted by permission
of IEE and the author
side and if k 1. If the factor k is known, then the d branch with field structure Suantities referred to the
armature becomes Canay s characteristic inductance
3
.E ' .F & M /CH ( .19)
2
The data conversion of the modified Suantities on the direct axis can now be done as follows: If k is
unknown, assume k 1, find M from ES.y( .1 ),
11
Page 262
/O ' .F & .ý ( .20a)
and realize that the fast oscillations in the field current will have a wrong amplitude, but the armature Suantities will
be correct. If the characteristic inductance is known (which can be calculated from k), find M from ES. ( .19), y
/O ' .F &.E ( .20b)
and the fast oscillations in the field current will be correct. Then use the conversion procedure of #ppendix VI.
to obtain the field structure Suantities 4 , 4 , L ,ryL 8y rry 88y ( m dropped in #ppendix VI), which will be rescaled
according to ES. ( .1 ). It is not necessary to undo the rescaling if one is only interested in Suantities on the
armature side, because scaling of field structure Suantities does not influence the armature Suantities. If the
conversion was done with p.u. Suantities, which will usually be the case, then multiply all resistances and inductances
with V „g†qp/S„g†qp
to obtain physical values (V „g†qprated line to line 4MS armature voltage, S rated apparent
„g†qp
power) for wye connected machines, followed by another multiplication with a factor 3 for delta connected machines.
The data conversion for the Suadrature axis Suantities is the same as that for the direct axis, except that one
Traduciendo...
does not have to worry about correct amplitudes in the oscillations of the current i . This current cannots be
measured, because the g winding is a hypothetical winding which represents eddy or damper bar currents. It is
therefore best to use k 1 and
/O SWCFTCVWTG CZKU ' .S & .ý ( .20c)
on the Suadrature axis.
4ather than undo the rescaling of ES. ( .1 ) by using t 1 / (%3/%2 k) with the procedure described after
ES. ( .22), it makes more sense to choose a factor t which introduces the correct turns ratio between physical values
of the armature and field structure Suantities. To find this factor, we must look at the open circuit terminal voltage
produced by the no load excitation current i . For

r € Âx gp open circuit, i i i 0,p and,ƒin steady
8 state operation,
d8 ƒ/dt 0, which leads to
XS ' T /O KHO
Since we know that the modified current must be t times the actual current,
KHO ' V @ KH
and since v isƒeSual to %3 V RGS (assuming symmetrical voltages in the three phases), we can find t from
3 8RJCUG
V' ( .21)
T /O KH&PQ NQCF
with V ‚tg…qrated 4MS line to ground voltage for wye connection, and line to line voltage for delta connection,
ir € Âx gprated no load excitation current which produces rated voltage at the terminal.
12
Page 263
Sometimes the no load excitation current is not known. Then any system of units can be used for the field structure.
One possibility is to set t 1 (field structure Suantities referred to the armature side). #nother possibility is to say
that a field voltage *v * 1.0

r should produce the rated terminal voltage. Then
*XH* 1.0
KH&PQ NQCF ' '
4H V 4HO
which, when inserted into ES. ( .21) gives
T /O
V' ( .22)
3 8RJCUG 4HO
Once t is known, the inductances are converted by multiplying the second and third row and column of the inductance
matrix in ES. ( .15a) with t, and by multiplying 4 and 4 with

ry t . The8y
Suadrature axis inductances and resistances
are also multiplied with t or t , respectively.
Sometimes, generators are modelled with less than windings on the field structure (D winding on the direct
axis missing, and/or g or 3 winding on the Suadrature axis missing). In such cases, the EMTP still uses the full
7 winding model and simply disconnects the unwanted winding by setting its off diagonal elements in the
inductance matrix to zero, and its diagonal element to an arbitrary value of TL 1 S. Its resistance is arbitrarily
set to zero. The inductances and resistances of the other windings on the same axis are calculated from ES. (VI. )
and (VI.5) (#ppendix VI), e.g., for a missing g winding,
/SI ' 0 , /I3 ' 0 , 4I ' 0 , T.II ' 1
and
/ .33
.33 ' , 43 ' , YJGTG / ' .S & .ý
)) ))
.S&. S
6S
$CUKE 'SWCVKQPU HQT /GEJCPKECN

Traduciendo... 2CTV
There are many transient cases where the speed variation of the generator is so small that the mechanical
part can be ignored. Simulating short circuit currents for a few cycles falls into that category. In that case, T in
ES. ( . ) and in the other eSuations is constant, and the angular position $ of the rotor needed in ES. ( .7) and ( .9)
is simply
$(V) ' $(0) % TV ( .23)
with $ and T being angle and speed on the electrical side.
For other types of studies it may be necessary to take the speed variations into account. The simplest model
13
Page 264
for the mechanical part is the single mass representation as used in stability studies,
F$ F$
, %& ' 6VWTDKPG & 6IGP ( .2 a)
FV FV
and
F$
'T ( .2 b)
FV
with J moment of inertia of rotating turbine generator mass,
$ rotor position,
T speed
D damping coefficient for viscous and windage friction (linear dependence on speed is a
crude approximation),
T †‡„hu€q torSue input to turbine,
T sq€ electromagnetic torSue of generator.
ES. ( .2 ) is valid for Suantities referred to the electrical or the mechanical side with the conversion from one to the
other being$
,OGEJ
,Gý '
(R/2)
R
$Gý ' $OGEJ ( .25)
2
&OGEJ
&Gý '
(R/2)
6OGEJ
6Gý '
R/2
With voltages given in V, and power in W, the unit for the torSue T becomes N m, for the damping coefficient
D it becomes N m / rad/s and for the moment of inertia J it becomes kgm (kg as mass).
Instead of the moment of inertia, the kinetic energy at synchronous speed is often given, which is identical
for the mechanical and electrical side,
1 1
'' ,OGEJ T ,Gý T ( .2 )
2 OGEJ ' 2
Gý
The inertia constant h (in seconds) is the kinetic energy E (e.g., in kWs) divided by the generator rating S „g†u€s(e.g.,
$
Subscript mech is used for the mechanical side, and subscript eý for the electrical side.
in kV#),
'
J' ( .27)
5TCVKPI
The relationship between the inertia constant h and the acceleration time T of the turbineg generator is
6C 2TCVKPI
J' ( .2 )
25TCVKPI
with P „g†u€s rated power of turbine generator (e.g., in MW),
S „g†u€s rated apparent power of generator (e.g., in MV#).
# single mass representation is usually adeSuate for hydro units, where turbine and generator are close
together on a stiff shaft. It is not good enough, however, for thermal units, if subsynchronous resonance or similar
problems involving torsional vibrations are being studied. In such cases, a number of lumped masses must be
represented. Usually to 20 lumped masses provide an adeSuate model . The model in the%EMTP allows any
number of lumped masses n $ 1, and automatically includes the special case of n 1 in ES. ( .2 ). Each major
element (generator, high pressure turbine, etc.) is considered to be a rigid mass connected to adjacent elements by
massless springs. Fig. . shows a typical 7 mass model.
The shaft/rotor system is assumed to be linear, which is reasonable for the small amplitudes of typical
torsional vibrations. The n spring connected rotating masses are then described by the rotational form of Newton s
second law,
(KI Mechanical part of a steam turbine generator with 7 masses (HP high pressure
turbine, IP intermediate pressure turbine, LP#, LPB, LPC low pressure turbine stages #,
B, C, GEN generator, EXC exciter)
F F
[,] [2] % [&] [2] % [-][2] ' [6VWTDKPG] & [6IGP GZE] ( .29a)
FV FV
and
%
There are studies where the lumped mass representation is no longer adeSuate, and where continuum models
must be used.
15
Page 266
F2
' [T] ( .29b)
FV
where
[J] diagonal matrix of moments of inertia (J ,...J in Fig. . ),%
[2] vector of angular positions (2 ,...2 in Fig. %

. , with 2 $), $
[T] vector of speeds,
[D] tridiagonal matrix of damping coefficients,
[K] tridiagonal matrix of stiffness coefficients,
[T †‡„hu€q
] vector of torSues applied to the turbine stages (T †‡„hu€q$0 and T 0 in
†‡„hu€q% Fig. . ),
[T sq€ q i ] vector of electromagnetic torSues of generator and exciter (components 1 to 5 0 in Fig. . ).

Traduciendo...
The moments of inertia and the stiffness coefficients are normally available from design data. The spring
action of the shaft section between masses i 1 and i creates a torSue which is proportional to the angle twist 2 2. uu
The proportionality factor is the stiffness coefficient or spring constant K . This spring action
u u torSue acts in
opposite directions on masses i 1 and i,
6URTKPI K& ' & 6URTKPI K ' -K& K 2K& &2K ( .30)
If these torSues are included in ES. ( .29), they create the term [K][2]. From ES. ( .30) it can be seen that [K] has
the following form
- &-
&- - %- &-
[-] ' &- - %- &-
. . . . .
&-P& P -P& P
Two damping effects are included with the damping coefficients, namely the self damping D of mass i u
(mostly friction between the turbine blades and steam), and the damping D created in the shaft
u u when the shaft
between masses i 1 and i is twisted with the speed T T . The damping

u utorSue acting on mass i is therefore
F2K F2K F2K& F2K F2K%

6FCORKPI K ' &K % &K& K & % &K K% & ( .31)
FV FV FV FV FV
From ES. ( .31) it can be seen that [D] has the same structure as [K], except that the diagonal element is now Du u
DD u uu ,
Page 267
& %& &&
&& & %& %& &&

[&] '
. . . . .
&&P& P &P& P%&P
It is very difficult to obtain realistic values for these damping coefficients. Fortunately, they have very little
influence on the peak torSue values during transient disturbances. However, for estimating the low cycle fatigue one
must consider the damping terms, which, unfortunately until now, have often been derived from unsuitable models
[107]. The damping of torsional oscillations is measured by observing the rate of decay when the shaft system is
excited at one of its natural freSuencies (modes). It is difficult to convert these modal damping coefficients into the
diagonal and off diagonal elements of [D].
For the vector of turbine torSues it is best to assume that the turbine power P †‡„hu€qT T †‡„hu€qremains
constant. This implies that the torSue decreases with an increase in speed, which is more reasonable than constant
torSue because if the turbine were to reach the same speed as the steam (or water) jet the torSue on the blades would
obviously drop to zero. It is possible to include the effects of the speed governor through T#CS modelling, but
usually the time span of transient simulations is so short that the governor effects will normally not show up.
The vector of electromagnetic torSues and the rotor position of the generator provide the link between the
eSuations of the mechanical and electrical part, with
R
2OGEJ&IGP @ ' $Gý ( .32a)
2
R
6OGEJ&IGP ' (8F KS & 8S KF)
2 ( .32b)
&XH KH % 4KGZE
H
6OGEJ&GZE ' ( .32c)
TOGEJ Traduciendo...
where it is assumed that (ES. .29) is written for the mechanical side. If it is written for the electrical side, then the
conversion of ES. ( .25) must be used. The term i 4 in ES. ( r.32c)

qi represents the losses incurred in the exciter
machine the negative sign for v comes

r from the source convention of ES. ( .1). If there is no exciter machine, as
in the case of rectifier excitation systems, then mass no. 7 in Fig. . would obviously be deleted, and ES. ( .32c)
would not be needed. T yqit q i is not in the BP# EMTP.
5VGCF[ 5VCVG 4GRTGUGPVCVKQP CPF +PKVKCN %QPFKVKQPU
Transient studies with detailed turbine generator models practically always start from power freSuency
17
Page 268
steady state conditions. The EMTP goes therefore automatically into an ac steady state solution whenever the data
file contains turbine generator models.
In some versions of the BP# EMTP, the machine is represented as a symmetrical voltage source at its
terminals. This approach is only correct if the generator feeds into a balanced network. In that case, the generator
currents are purely positive seSuence. In an unbalanced network, there are negative and zero seSuence currents,
which would see the generator as a short circuit. This is incorrect, because generators do have nonzero negative and
zero seSuence impedances < and€qs

< . In the ’q„
M39 version of the BP# EMTP, the user can specify unsymmetrical
voltage sources at the terminals. This is still not good enough, however, unless the user adjusts the negative and zero
seSuence components of the terminal voltage iteratively until V < I and V

€qs< I , with
€qs I€qs
, I being ’q„ ’q„ ’q„ €qs ’q„
the currents from the steady state network solution. The UBC EMTP does include negative and zero seSuence
impedances correctly, as explained next.
The negative seSuence impedance can be calculated as part of the data conversion. Its imaginary part is very
close to
:F)) % :S))
:PGI . ( .33)
2
and this value can be used without too much error if the negative seSuence impedance is needed for preliminary
calculations. The real part 4 is larger€qsthan the armature resistance 4 because of double gfreSuency circulating
currents in the field structure circuits its value is difficult to guess, and is therefore best taken from the data
conversion. For calculations internal to the UBC EMTP, the correct values from the data conversion are always
used. The zero seSuence impedance < 4 kX is’q„part of the

g input ’q„
data, but its value becomes immaterial
if the generator step up transformer is delta connected on the generator side and if the disturbance occurs on the line
side.
The positive seSuence representation can be a voltage source behind any impedance, as long as it produces
the desired values for the terminal voltage V and the current
‚… I . Knowing only
‚ …V may reSuire a preliminary
‚…
steady state solution with a voltage source V at the terminal,

‚… to find I . Then the value‚ …
needed for the voltage
source behind the impedance is
8UQWTEG ' 8RQU % < +RQU
#ny value can be chosen for <, but < < simplifies programming
€qs for the following reason: The EMTP solves
the network in phase Suantities, and assumes that all phase impedance matrices are symmetric. Only < < will €qs
produce a symmetric phase impedance matrix, however. Changing the program to handle unsymmetric matrices just
for generators would have reSuired a substantial amount of re programming.
The generator positive seSuence representation is then a voltage source behind < while the negative
€qs and
zero seSuence representations are passive impedances < and < , respectively
€qs (or zero voltage sources or short
’q„
circuits behind < and

€qs< ). Converted
’q„ to phase Suantities, these 3 single phase seSuence representations become
Traduciendo...
Page 269
a three phase symmetrical voltage source (E , E , E ) behind a !3 x 3 impedance matrix, as shown in Fig. .5(a),
with
<U <O <O
[<] ' <O <U <O ( .3 )
<O <O <U
1 1
YJGTG <U ' (<\GTQ % 2 <PGI) , <O ' (<\GTQ & <PGI)
3 3
To be able to handle any type of connection, including delta and impedance grounded or ungrounded wye
connections, the voltage sources behind [<] are converted into current sources in parallel with [<], as indicated in
Fig. .5(b), with
(a) Thevenin eSuivalent circuit (b) Norton eSuivalent circuit
(KI Turbine generator representation for steady state solution
))
+ '
))
+ ' [<]& ' ( .35)
+ ' ))
This is done because the EMTP could not handle voltage sources between nodes until recently and even after such
voltage sources are allowed now, this Norton eSuivalent circuit is at least as efficient. For armature winding 1, the
internal voltage source is
19
Page 270
' )) ' 8RQU % <PGI +RQU ( .3 )
with V ,‚ I…being
‚ … the unnormalized positive seSuence values. Unnormalized values are a more convenient input
form for the user, because with the unnormalized transformation of ES. ( .3 ) the positive seSuence values are
identical with the phase values V and I for armature winding 1 for balanced network conditions. For armature
windings 2 and 3, the internal voltages are
)) ' C' )) )) ' C' ))
' CPF ' ( .37)
vE
a e ). Traduciendo...
In the UBC EMTP, either V and I‚ …

can be specified
‚… as input data, in which case E is calculated from
ES. ( .3 ), or E can be specified directly. Specifying E is not as unusual as it may seem, because short circuit
programs use essentially the same generator representation (E behind X ). If users want pto specify active and
reactive power at the terminals, or active power and voltage magnitude, then the load flow option described in
Section 12.2 can be invoked, which will automatically produce the reSuired V and I . ‚… ‚…
The UBC EMTP connects the generator model of Fig. .5(b) to the network for the steady state solution,
which will produce the terminal voltages and currents at fundamental freSuency. For unbalanced network conditions,
this solution method is not Suite correct because it ignores all harmonics in the armature windings and in the network.
Experience has shown, however, that such an approximate initialization is accurate enough for practical purposes.
Fig. . shows simulation results for a generator feeding into a highly unbalanced load resistance (4 1.0 S, 4 5 6
1.0 S, 4 0.05 S),

7 with an initialization procedure which ignores the harmonics on the armature side, and
considers only the second harmonics on the field structure side, as discussed in Section . .2. The final steady state
is practically present from the start. The mechanical part is totally ignored in the steady state solution, because it
is assumed that the generator runs at synchronous speed. #gain, for unbalanced conditions this is not Suite correct
(KI Steady state behavior of generator with highly

unbalanced load
because, in that case, the constant electromagnetic torSue has oscillations superimposed on it which produce torsional
20
Page 271
vibrations whose effects are ignored.
#fter the steady state solution at fundamental freSuency has been obtained, the terminal voltages and
currents are converted to unnormalized symmetrical components to initialize the machine variables:
8\GTQ 111 8
8RQU ' 1 8
1CC , KFGPVKECN HQT EWTTGPVU ( .3 )
3
8PGI 1 CC 8
vE
(a e ). The inverse transformation is
8 111 8\GTQ
8 ' 1 CC 8RQU , KFGPVKECN HQT EWTTGPVU ( .39)
8 1CC 8PGI
+PKVKCNK\CVKQP YKVJ 2QUKVKXG 5GSWGPEG 8CNWGU

v(‚ …
If the positive seSuence voltage is obtained as a peak (not 4MS) phasor value *V *e , then from ES.
‚…
( .39),
X (V) ' *8RQU* cos(TV % (RQU)
X (V) ' *8RQU* cos(TV % (RQU & 120E) ( . 0)
Traduciendo...
X (V) ' *8RQU* cos(TV % (RQU % 120E)
Inserting these voltages into the transformation of ES. ( .7) produces
3
XF&RQU(V) ' *8RQU* sin((RQU & *)
2
3
XS&RQU(V) ' *8RQU* cos((RQU & *) ( . 1)
2
where * is the angle between the position of the Suadrature axis and the real axis for phasor representations (Fig.
.7). This angle is related to the rotor position $ on the electrical

qý side by
B
$Gý ' TV % * % ( . 2)
2
21
Page 272
The positive seSuence values v and pv in ES.ƒ ( . 1) are dc Suantities and hence do not change as a function of time
the argument (t) can therefore be dropped. From ES. ( . 1) it is evident that v and v can be combined
p into aƒ
complex expression
3
XS&RQU % LXF&RQU ' 8RQU G&L* ( . 3)
2
with V being V isv(‚a …

‚ … the complex Suantity *V *e . While‚ … phasor of freSuency
‚… T in the network solution
v*
reference frame, V e in‚ ES.
… ( . 3) becomes a dc Suantity in the d, S axes reference frame. Similarly,
3
KS&RQU % LKF&RQU ' +RQU G&L* (.)
2
v"‚ …
with I being
‚ … the complex current *I *e with respect
‚… to the real axis. With V and I ‚… ‚…
(KI Definition of *
known, we still need the angle * to find the d, S values. To calculate *, use ES. ( .9) and ES. ( .10), with i 8
isi 0, asQ well as d8 /dt 0 and d8p /dt 0 (no currents ƒin damper windings and all d, S Suantities constant
in symmetrical operation with positive seSuence values only),
XF&RQU ' &4C KF&RQU & T.SKS&RQU ( . 5a)
XS&RQU ' &4CKS&RQU % T.FKF&RQU % T/FHKH&RQU ( . 5b)
Traduciendo...
ES. ( . 5a) and ( . 5b) can be rewritten as a complex eSuation relative to the Suadrature axis
22
Page 273
XS&RQU % LXF&RQU ' &(4C % LT.S) (KS&RQU % LKF&RQU) % 'S&RQU ( . 5c)
where E ƒ ‚ … is a Suantity whose position on the Suadrature axis is important, but whose magnitude
'S&RQU ' (T.F & T.S) KF&RQU % T/FHKH&RQU
v*
is immaterial here. By inserting ES. ( . 3) and ES. ( . ) into ES. ( . 5c), and by multiplying with (%2/%3)e , all
dc Suantities become phasors in the network solution reference frame again. The angle * is then obtained from the
phasor eSuation
2
)XS&RQU GL* ' 8RQU % (4C % LT.S) +RQU (.)
3
(KI Calculation of angle *
With * known, the initial values v (0), v (0),

p ‚i…
(0), i (0)ƒ ‚are
… foundp ‚from
… ES.ƒ(‚ .…3) and ( . ). #s
mentioned before, the remaining currents i , i , i are

8 s zero
Q from the positive seSuence effects, except for i , whose r
initial value is calculated from ES. ( . 5b),
XS&RQU(0) % 4CKS&RQU(0) & T.FKF&RQU(0)

KH&RQU(0) ' ( . 7)
T/FH
ir(0)
‚ …is used to initialize v r
XH&RQU(0) ' &4H KH&RQU(0) (.)
23
Page 274
The initial value of the torSue produced by the positive seSuence Suantities is needed in the mechanical part.
It is calculated from the fluxes 8 p ‚ …(0) L i (0)p M

p ‚ i…(0) and 8 pr r ‚ … ƒ ‚ …(0) L i (0)ƒ as
ƒ‚…
6IGP&RQU(0) ' 8F&RQU(0) KS&RQU & 8S&RQU(0) KF&RQU(0) ( . 9)
The initial positive seSuence torSue can also be calculated from energy balance considerations (TT power
delivered to the network losses in the armature windings),

Traduciendo...
3 ( 3
T6IGP&RQU(0) ' 4G6+RQU8 *+RQU* 4C ( .50)
2 RQU> %2
(division by 2 because the phasors are peak values).
+PKVKCNK\CVKQP YKVJ 0GICVKXG 5GSWGPEG 8CNWGU&
If the network is balanced in steady state, then there are no negative seSuence values. This part of the
initialization can therefore be skipped if the negative seSuence (peak) phasor current
L"PGI
+PGI ' *+PGI* G ( .51)
obtained from the steady state solution is negligibly small.
Negative seSuence currents in the armature windings create a magnetic field which rotates backwards at a
relative speed of 2T with respect to the field structure. Second harmonic currents are therefore induced in all
windings on the field structure, which the EMTP takes into account in the initialization. These second harmonic
currents induce third harmonics in the armature windings, which in turn produce fourth harmonics in the field
structure windings, etc. Fortunately, these higher harmonics decrease rapidly in magnitude. They are therefore
ignored. Calculating the field structure harmonics or order higher than 2 could be done fairly easily, but the
calculation of the armature harmonics of order 3 and higher would involve solutions of the complete network at these
higher freSuencies. While the network solutions for harmonics may be added to the EMTP someday, this addition
does not appear to be justified for this particular purpose.
First, the negative seSuence current must be defined on the direct and Suadrature axis. By starting with the
negative seSuence currents in the three armature windings,
K (V) ' *+PGI* cos(TV % "PGI)
K (V) ' *+PGI* cos(TV % "PGI % 120E) ( .52)
K (V) ' *+PGI* cos(TV % "PGI & 120E)
&
The negative seSuence currents in the BP# EMTP can be incorrect (see beginning of Section . ).
Page 275
these d, S axes values are obtained through the transformation ( .7),
3
KF&PGI(V) ' & *+PGI* sin("PGI % * % 2TV)
2
3
KS&PGI(V) ' *+PGI* cos("PGI % * % 2TV) ( .53)
2
While the positive seSuence d, S axes currents are dc Suantities, the negative seSuence d, S axes currents are second
harmonics. This is important to keep in mind when we represent them with a phasor of freSuency 2T,
3
+FS&PGI ' +PGI GL* ( .5 )
2
with the understanding that
KF&PGI(V) ' &+O 6+FS&PGI GL TV>
KS&PGI(V) ' 4G 6+FS&PGI GL TV> ( .55)
For the initialization of i and i p, the negative

ƒ seSuence values at t 0 from ES. ( .55) must be added to the
Traduciendo...
respective positive seSuence values from ES. ( . ) to obtain the total initial values i (0) and i (0). The negative
p ƒ
seSuence d, S axes voltages are not needed in the initialization, but they could be obtained analogously to the
currents.
The second harmonic currents in the field structure windings are found by using the d, S axes phasor current
of freSuency 2T as the forcing function. The procedure is outlined for the direct axis it is analogous for the
Suadrature axis. From ES. ( .1),
F
XH&PGI ' &4HKH&PGI & (/FHKF&PGI % .HHKH&PGI % /H&K&&PGI)
FV
F
X&&PGI ' &4&K&&PGI & (/F&KF&PGI % /H&KH&PGI % .&&K&&PGI) ( .5 )
FV
The voltages on the left hand side are zero because the damper winding is always shorted, and the dc voltage source
supplying the field winding is seen as a short circuit by second harmonic currents. With zero voltages, and knowing
that all currents are second harmonics, ES. ( .5 ) can be rewritten as two phasor eSuations
25
Page 276
4H % L T.HH L T/H& +H&PGI L T/FH

'& +FS&PGI ( .57)
L T/H& 4& % L T.&& +&&PGI L T/F&
which are solved for the two phasors I , I . Their

r €qsinitial
8 €qs values are found on the basis of ES. ( .55) as
KH&PGI(0) ' & +O 6+H&PGI>
K&&PGI(0) ' & +O 6+&&PGI> ( .5 )
The value i (0) is then added to i (0) from

r €qs r ‚ ES.
… ( . 7) for the total initial field current, whereas i (0) is already 8 €qs
the value of the total damper current.
The phasor currents I ands €qs

I for the Q
Suadrature
€qs axis are obtained analogous to ES. ( .5 ), by replacing
subscripts f and D with g and 3. Their initial values on the basis of ES. ( .55) are then
KI&PGI(0) ' 4G 6+I&PGI>
+3&PGI(0) ' 4G 6+3&PGI> ( .59)
which are the total initial values since the respective positive seSuence values are zero.
The negative seSuence phenomena produce torSues which influence the initialization of the mechanical part.
4ecall that the electromagnetic torSue on the electrical side is 8 i 8 i . With pboth
ƒ fluxes
ƒ p and currents consisting
of positive and negative seSuence parts, the total torSue can be expressed as the sum of three terms,
6IGP ' 6IGP&RQU % 6IGP&PGI % 6IGP&RQU PGI ( . 0)
The positive seSuence torSue was already defined in ES. ( . 9), and the negative seSuence torSue is
6IGP&PGI ' 8F&PGI KS&PGI & 8S&PGI KF&PGI ( . 1)
The third term
6IGP&RQU PGI ' 8F&RQUKS&PGI % 8F&PGIKS&RQU & 8S&RQUKF&PGI & 8S&PGIKF&RQU ( . 2)
is an oscillating torSue produced by the interaction between positive and negative seSuence Suantities, with an
average value of zero. That it is purely oscillatory can easily be seen since all positive seSuence values in ES. ( . 2)
are constant, and all negative seSuence values oscillate at a freSuency of 2T. This and other oscillating terms are
ignored in the initialization of the mechanical part, where torsional vibrations are not taken into account.
Traduciendo...
The negative seSuence torSue of ES. ( . 1) consists of a constant part, which must be included in the
initialization of the mechanical part, and of an oscillating part with freSuency T which is ignored. To find the
constant part, the fluxes are first calculated as phasors,
Page 277
7F&PGI ' .F+FS&PGI % /FH+H&PGI % /F&+&&PGI ( . 3a)
7S&PGI ' .S+FS&PGI % /SI+I&PGI % /S3+3&PGI ( . 3b)
With the definition of ES. ( .55) for the instantaneous values of currents and fluxes, and after some manipulations
of the eSuations, it can be shown that the constant part is
6IGP&PGI&EQPUVCPV ' 4G67CXGTCIG>+O6+FS&PGI> & +O67CXGTCIG>4G6+FS&PGI>
( . a)
with
1
7CXGTCIG ' (7F&PGI % 7S&PGI) ( . b)
2
The oscillatory part is not needed, but could be calculated from
7S&PGI & 7F&PGI

6IGP&PGI&QUEKNNCVQT[ ' +O +FS&PGI GL TV ( . 5)
2
Identical values for the constant part are obtained from energy balance considerations (TT power delivered to
network losses in all windings),
1
T6IGP&PGI&EQPUVCPV ' 6&3*+PGI* (4PGI&4C) % *+H&PGI* 4H % *+I&PGI* 4I
2
% *+&&PGI* 4& % *+3&PGI* 43 > (.)
Because 3rd and higher order harmonics are ignored in the armature windings, and th and higher order
harmonics are ignored in the windings on the field structure, the initial torSue values are not exact. They are good
approximations, however, as can be seen from Table .1. This table compares the values obtained from the
initialization eSuations with the values obtained from a transient simulation (Fig. .9), for the severely unbalanced
case described in Fig. . . The constant torSue from the initialization procedure is almost identical with the average
torSue of the transient simulation (difference 1. ). Fig. .9 further shows that the initial torSue from ES. ( .73)
can be Suite different from the average torSue. Table .1 also compares the values for the 2nd and th harmonics
(not needed in the initialization, though). The values for the 2nd harmonic agree Suite well, but not the values for
the smaller th harmonic. This is to be expected, because the th harmonic torSue is influenced by 3rd harmonic
currents in the armature windings, which are ignored in this initialization procedure. The average value in Fig. .9
lies not exactly halfway between the maximum and minimum values because the th harmonic is phase shifted with
respect to the second harmonic.
27
Page 278
Traduciendo...
(KI TorSue obtained with transient simulation for case described in Fig. .
6CDNG Electromagnetic torSue for test case of Fig. . ()t .29 3 zs)
From Fourier #nalysis

Between 0 ms and
100 ms (MNm) 4elative Error
TorSue Component From ESuations ()
(MNm)
pos 2.953 from ( . 9)

or ( .50)
average neg 0.017 from ( . )

AAAAAAAAAAA or ( . )
sum 2.970 3.019 1. 2
2nd harmonic . 73 .3 0.
th harmonic 0.2 2 0.517 9.32
+PKVKCNK\CVKQP YKVJ <GTQ 5GSWGPEG 8CNWGU'
The initial zero seSuence values are easy to obtain, either from the d, S, 0 transformation of ES. ( .7), or
from the symmetrical component transformation of ES. ( .3 ). Physically, both are the same Suantities, except that
'
The zero seSuence currents in the BP# EMTP can be incorrect (see beginning of Section . ).
Page 279
the d, S, 0 transformation is normalized and the symmetrical component transformation chosen in ES. ( .3 ) is not.
Since the d, S, 0 Suantities are normalized,
1
K (0) ' (KC(0) % KD(0) % KE(0)) ( . 7a)
3
or
K (0) ' 3 4G 6+\GTQ> ( . 7b)
The zero seSuence Suantities do not produce any torSue, and therefore do not influence the initialization of
the mechanical part.
+PKVKCNK\CVKQP QH VJG /GEJCPKECN 2CTV
The links between the electrical and the mechanical part are the angle $ (0) from ES. ( . qx
2), which is
converted to the mechanical side with ES. ( .25), and the electromagnetic torSues T and T from ES. ( sq€
.32). qi
For the generator torSue, the constant part is
6IGP&EQPUVCPV ' 6IGP&RQU(0) % 6IGP&PGI&EQPUVCPV (.)
on the electrical side, which is converted with ES. ( .25) to the mechanical Traduciendo...
side. Since torsional vibrations coming
from the oscillating torSues of ES. ( . 2) and ES. ( . 5) in unbalanced cases are ignored in the steady state
initializations, the oscillating term is left off in ES. ( . ). For the exciter torSue, the oscillating terms are ignored
as well. Then,
1
T6GZE&EQPUVCPV ' &XH&RQU(0) KH&RQU(0) *+
% H&PGI* 4GZE ( . 9)
2
with I from
r €qs ES. ( .57).
Without torsional vibrations, the speeds of all turbine generator masses are one and the same, and the
acceleration of each mass is zero. Then ES. ( .29) simplifies to
[-] [2] ' [6VWTDKPG] & [6IGP GZE] & T[&UGNH] ( .70)
with T being the synchronous angular speed and [D ] being …qxr

the diagonal matrix of self damping terms D. The sum u
of the turbine torSues must, of course, eSual the sum of the electromagnetic and speed self damping torSues, so that
there is zero accelerating torSue initially,
j 6VWTDKPG&K ' 6IGP&EQPUVCPV % 6GZE&EQPUVCPV

j %&T
K ( .71)
The initial angles in the BP# EMTP can be incorrect in unbalanced cases, because the negative seSuence
torSues are not included in ES. ( . ) and ( . 9). If Table .1 is typical, these torSues are very small, however.
29
Page 280
ES. ( .71) is used to find the sum of the turbine torSues first, and then to apportion the total to the individual stages
from the percentage numbers to be supplied in the input (e.g., 30 of torSue in high pressure stage, 2 in
intermediate pressure stage, etc.). The right hand side of ES. ( .70) is then known, as well as the angle of the
generator mass from ES. ( .32a). [K] is singular. #ssume the generator to be mass no. k (with 2 known) then w
the remaining initial angles may be found in 2 ways:

!
(1) Multiply the diagonal element K withww
an arbitrary large number (e.g., 10 ), and reset the k th right hand
side value to this number times 2 . Thisw will, in effect, change the k th eSuation to variable 2 specified w
value of 2 . w
Then solve the system of linear eSuations ( .70), preferably with a subroutine for tri diagonal
matrices (reSuired in the time step loop anyhow).
(2) Starting to the left of generator mass k, find the angles of the lower numbered masses recursively from
K&
j 4*5
( .72a)
2K& ' 2K % , K ' M,...1
-K& K
(4HS right hand side terms of ES. ( .70)), and starting to the right of generator mass k, find the angles
of the higher numbered masses recursively from
j 4*5
K% ( .72b)
2K% ' 2K % , K ' M,...P
-K K%
These recursive eSuations are derived by summing up rows 1,...i, or by summing up rows i,...n in ES.
( .70) in either case, most terms on the left hand side cancel out because of the special structure of [K], as shown
after ES. ( .30).
While the oscillating terms of T and T

sq€are ignored
q i in finding the initial angles of the mechanical part,
they must be included in initializing the electromagnetic torSues for solving the differential eSuations in the time step
loop. This is best done using
6IGP&VQVCN(0) ' 8F(0) KS(0) & 8S(0) KF(0) ( .73)
where the currents are
KF(0) ' KF&RQU(0) % KF&PGI(0)
Traduciendo...
KS(0) ' KS&RQU(0) % KS&PGI(0)
and the initial fluxes are calculated from ES. ( .10). Similarly,
T6GZE&VQVCN(0) ' &XH&RQU(0) KH(0) % 4GZE 6KH(0)> ( .7 )
30
Page 281
where
KH(0) ' KH&RQU(0) % KH&PGI(0)
6TCPUKGPV 5QNWVKQP
The numerical methods for the transient solution part are based on [13]. The basic idea is to reduce the
machine eSuations to a three phase Thevenin eSuivalent circuit, similar to that of Fig. .5 for the steady state
initialization. The eSuivalent circuit for the transient solution differs from Fig. .5 mainly in two aspects:
(a) The impedance matrix [<] of Fig. .5 becomes a resistance matrix [4], after integrating the
machine eSuations ( .9) with the trapezoidal rule of integration, and after reducing the seven
eSuations for all windings to three eSuations for the armature windings.
(b) The sinusoidal voltage sources E of Fig. .5 become instantaneous voltage sources which must
be updated from step to step.
The updating procedure for the voltage sources reSuires the prediction of certain variables from the known
solution at t )t to the yet unknown solution at t. Different prediction methods have been tried over the years, and
their behavior with respect to numerical stability has gradually improved. Some earlier versions of the T;PE 59
synchronous machine model produced too much numerical noise [131], but beginning with version M3 , the
prediction methods are Suite stable and the simulation results are fairly reliable now [132]. Further progress with
respect to numerical stability can only be achieved if the overall EMTP algorithm is changed from a direct to an
iterative solution in each time step.
$TKGH 1WVNKPG QH 5QNWVKQP /GVJQF
#ssume that the solution at t )t is already known, and that the solution at t has to be found next. Then
the method works roughly as follows:
(1) Predict the generator rotor angle $(t) (first predicted variable).
(2) #pply the trapezoidal rule of integration to the 4 L branches of Fig. .2, in the direct axis as well
as in the Suadrature axis. Conceptually, this converts each 4 L branch into an eSuivalent
resistance in parallel with a known current source, as indicated in Fig. .10(a) and (b). The zero
seSuence consists of only one branch (Fig. .10(c)).
31
Page 282
Traduciendo...
(KI 4esistive networks resulting from trapezoidal rule of integration (u speed

voltages defined in ES. ( .75))
(3) 4educe the e and S axis resistive networks of Fig. .10 to one eSuivalent resistance in series with
one eSuivalent voltage source as shown in Fig. .11. For this reduction, assume that v (t) v (t r r
)t), which is exact if the excitation system is not modelled, or use some other prediction (e.g.
linear extrapolation).
(a) Direct axis (b) 3uadrature axis (c) <ero seSuence
(KI 4esistive networks
Unfortunately, the speed voltages
WF ' &T 8S
32
Page 283
WS ' T 8F ( .75)
at time t are also unknown, but since fluxes can never change abruptly, their values can be
predicted reasonably well. With predicted values for u (t), u (t) and v(t)
p (2nd,ƒ 3rd and th r
predicted variable), the reduction is straightforward. Conceptually, branches M, f, D for the d axis
in Fig. .10(a) are paralleled, and then connected in series with the c branch (analogous for the S
axis).
() Convert the 3 resistive Thevenin eSuivalent circuits for d, S, 0 Suantities to phase Suantities. If
this were done directly, then the resulting 3 x 3 resistance matrix would be time dependent as well
as unsymmetric. To obtain a constant, symmetric matrix, the eSuivalent resistances of the d and
S axis are averaged, as indicated in Fig. .12, and the saliency terms 4 4 / 2 i (t) and 4 4p ƒ p ƒp
/ 2 i (t) ƒare combined with the voltage sources e and e into one
p voltage
ƒ source.
Traduciendo...
(KI Modified resistive networks
This can only be done at the expense of having to use a predicted value for i (t) and i (t) (5th
p and ƒ
th predicted variable). #fter conversion to phase Suantities, the d, S, 0 networks become one
three phase network, with three source voltages behind a symmetric, constant resistance matrix
33
Page 284
[4 qƒ‡uˆ].
(5) Solve the complete network, with the machine representation of Fig. .12(d). In the EMTP,
current sources in parallel with [4 qƒ‡uˆ] are used in place of voltage sources in series with [4 qƒ‡uˆ].
() From the complete network solution in phase Suantities, extract the generator voltages and convert
them to d, S, 0 Suantities. Calculate the armature currents in d, S, 0 Suantities and the field
structure currents, and use them to find the electromagnetic torSues of the generator and exciter
from ES. ( .32) at time t. Then solve the eSuations of the mechanical part.
(7) Compare the predicted values of $, u , u , i p, ƒi with

p ƒ the corrected values from the solution of step
( ), and repeat steps ( ) and (7) if the difference is larger than the acceptable tolerance. When
returning to step ( ), it is assumed that the terminal voltages in phase Suantities remain the same.
() 4eturn to step (1) to find the solution at the next time step.
Some of the implementation details, which have been omitted from this brief outline, are discussed next.
Variations of the iteration and prediction methods are described in Section .5. .
6TCPUKGPV 5QNWVKQP QH VJG 'NGEVTKECN 2CTV
Consider the eSuations for the direct axis first, which are obtained from rows 1, , of ES. ( .9) and from
ES. ( .10a) as
XF 4C KF .F /FH /F& FKF/FV WF

XH ' & 4H KH& /FH .HH /H& FKH/FV% 0 ( .7 a)
0 4F K& /F& /H& .&& FK&/FV 0
with u being
p the speed voltage from ES. ( .75), or in short hand notation,
FK
[X] ' &[4] [K] & [.] % [W] ( .7 b)
FV
Because of numerical noise problems in pre M32 versions of the BP# EMTP, this eSuation is integrated with the
damped trapezoidal rule of Section 2.2.2 , with a damping resistance matrix [4 ] in parallel with [L],
‚
1%" 2 Traduciendo...
[4R] ' @ [.] ( .77)
1&" )V
where " is the reciprocal of the damping factor defined in ES. (2.21). For " 1 there is no damping, while "
0 is the critically damped case. In the present version of the BP# EMTP, a default value of (1 ")/(1 ") 100
is used.
The (unreleased) UBC version with synchronous machines uses the normal trapezoidal rule. By setting "
1 in the input, the BP# EMTP would use the normal trapezoidal rule as well.
Page 285
#pplying the damped trapezoidal rule of ES. (2.20) for v L di/dt to ES. ( .7 ), with v replaced by [u]
[v] [4][i], results in
1%"
[X(V)] ' [W(V)]%[JKUV(V&)V)]&6[4]% [.]>[K(V)] ( .7 a)
)V
with the known history term
1%"
[JKUV(V&)V)] ' 6&"[4]% [.]> [K(V&)V)] & "[X(V&)V)] % "[W(V&)V)] ( .7 b)
)V
ES. ( .7 a) described a voltage source [u(t)] [hist(t )t)] behind a resistance matrix
1%"
[4EQOR] ' [4] % [.] ( .79)
)V
Subscript comp is used because such eSuivalent resistive networks are called resistive companion models in
network theory [133].
For interfacing the synchronous machine eSuations with the network eSuations, the field structure Suantities
are eliminated from ES. ( .7 ). Dropping subscript comp and using subscripts d, f, D again, the field structure
currents can be expressed from the last two rows of ES. ( .7 a) with [4 i y‚ ] from ES. ( .79) as
&
KH(V) 4HH 4H& JKUVH(V&)V) XH(V) 4FH
' & & KF(V) ( . 0)
K&(V) 4H& 4&& JKUV&(V&)V) 0 4F&
which, when inserted into the first row of ES. ( .7 a), produces a single eSuation for the d axis,
XF(V) ' GF & 4F KF(V) ( . 1)
with the reduced companion resistance
4HH 4H& & 4FH

4F ' 4FF & [4FH 4F&] ( . 2a)
4H& 4&& 4F&
the voltage source
TGF
GF ' WF(V) % JKUV ( . 2b)
F (V & )V)
and the reduced history term
4HH 4H& & JKUVH&XH(V)

TGF
JKUV
F ' JKUVF & [4FH 4F&] ( . 2c)
4H& 4&& JKUV&
35
Page 286
Traduciendo...
By predicting the speed voltage u (t) T(t)8p (t), and by assuming

ƒ that v (t) v (t )t), the simpler r
resistive network of Fig. .11(a) is obtained, with a voltage source e behind the companion
p resistance 4 . If 4 p
2L/)t in all 4 L branches of Fig. .10(a), then
))
2. F
4F . ( . 3a)
)V
Therefore, ES. ( . 1) essentially represents the trapezoidal rule solution of a voltage source behind the subtransient
reactance X . In
p publications based on [13]. 4 is called a .p
If the dynamic behavior of the excitation system is to be simulated as well, then using v (t) v (t )t) r r
implies a one time step delay in the effect of the excitation system on the machine. Such a delay is usually
acceptable, because )t is typically much smaller than the effective time constant between the input and output of the
excitation systems. #lternatively, some type of prediction could be used for v (t). r
The derivations for the S axis are obviously analogous to those just described for the d axis, and lead to the
single eSuation for the S axis,
XS(V) ' GS & 4S KS(V) (.)
with the voltage source
TGF
GS ' WS(V) % JKUV ( . 5)
S (V & )V)
Here, only the speed voltage u (t) T(t)8ƒ (t) must be predicted
p because the voltage v (t) is zero. The S axis s
resistive network is shown in Fig. .11(b). #gain, if 4 2L/)t in all 4 L branches of Fig. .10(b), then
))
2. S
4S . ( . 3b)
)V
Therefore, ES. ( . ) essentially represents the trapezoidal rule solution of a voltage source behind the subtransient
reactance X . In
ƒ publications based on [13], 4 is called a .ƒ
The eSuations for the zero seSuence Suantities (row 3 in ES. ( .9) and ES. ( .10c)) are also integrated with
the damped trapezoidal rule, which leads to
X (V) ' JKUV (V & )V) & 4 K (V) ( . a)
with the companion resistance
1%"
4 ' 4C % . ( . b)
)V
and the known history term
Page 287
1%"
JKUV (V&)V) ' . & "4C K (V&)V) & "X (V&)V) ( . c)
)V
The zero seSuence resistive network of ES. ( . a) with e hist (t )t) is shown in Fig. .11(c). In publications
based on [13]. 4 is called a . !!
The reduced generator eSuations ( . 1), ( . ) and ( . a) can be solved in one of two ways:
(1) Find a three phase Thevenin eSuivalent circuit (resistive companion model) for the network seen
from the generator terminals, and solve it together with the generator eSuations.
(2) #dd the reduced generator eSuations to the network eSuations, and solve them simultaneously.
The first approach was used in [9 ]. It has the advantage that iterations can easily be implemented for the
correction of predicted values. However, generator must be separated by distributed parameter lines with travel time
for reasons of numerical efficiency, so that an independent three phase Thevenin eSuivalent circuit can be generated
Traduciendo...
for each generator (otherwise, M generators would have to be interfaced with one 3 x M phase Thevenin eSuivalent
circuit). If there are no such lines in reality, artificial stub lines with J )t must be used to separate the generators.
This can result in incorrect answers. For this reason, the first approach has been abandoned in the EMTP.
With the second approach, there is no restriction on the number of generators connected to the network,
or even to the same bus. However, it does reSuire the prediction of certain variables, which makes this approach
more sensitive to the accumulation of prediction errors. It is the only method retained in the present BP# EMTP,
and only this method is discussed here. To solve the generator eSuations with the network eSuations, the generator
resistive networks of Fig. .12 in d, S, 0 Suantities must be converted to phase Suantities, which produces a time
dependent and unsymmetric 3 x 3 resistance matrix. To accommodate such matrices would have reSuired a complete
restructuring of the basic (non iterative) solution algorithm of the EMTP. Instead, an average resistance
4CX ' (4F % 4S)/2 ( . 7)
is used on both axes. This reSuires saliency terms i (4 4 )/2 on the

p p ƒd axis and i (4 4 )/2 on the S axis,
ƒ ƒwhich
p
are added to the known voltage sources e , e by using

p ƒ predicted values for i , i (Fig. .12). For
p ƒgenerators with
X pX , theseƒ saliency terms are practically negligible. For the IEEE benchmark model [7 ] with different values
of X 0.135
p p.u. and X 0.200 p.u.,
ƒ the companion resistances are 4 3.5 p.u. and 4 5.3103 p.u.
p ƒ
for )t 200 zs. These values are practically identical with 2L /)t 3.5 10 p.u. andp 2L /)t 5.3052 p.u., ƒ
as mentioned in ES. ( . 3). The voltage drop across the saliency terms (4 4 )/2 would bep 20
ƒ of the voltage drop
across (4 4 p)/2 with

ƒ )t 200 zs.
With the average resistance of ES. ( . 7), the modified eSuations in d, S, 0 Suantities become
XF(V) GF&OQF 4CX KF(V)
XS(V) ' GS&OQF & 4CX KS(V) ( . a)
X (V) G 4 K (V)
37
Page 288
where
4F & 4S
GF&OQF ' GF & KF(V) ( . b)
2
4F & 4S
GS&OQF ' GS % KS(V) ( . c)
2
Predicted values i , ipare

ƒ used in the last two eSuations, and the voltage sources behind resistances are then converted
into current sources in parallel with the resistances,
1
KF&UQWTEG ' GF&OQF ( . 9a)
4CX
1
KS&UQWTEG ' GS&OQF ( . 9b)
4CX
1
G &UQWTEG ' G ( . 9c)
4
Finally, the d, S, 0 Suantities are converted to phase Suantities with a predicted angle $(t), which produces a resistive
companion model with current sources
1
cos$ sin$
2
K &UQWTEG KF&UQWTEG
2 cos($&120E) sin($&120E) 1
K &UQWTEG
' KS&UQWTEG ( .90)
3 2
K &UQWTEG K &UQWTEG
1
cos($%120E) sin($%120E)
2
and parallel with
4U 4O 4O
[4GSWKX] ' 4O 4U 4O Traduciendo... ( .91a)
4O 4O 4U
where
4U ' (4 % 24CX)/3, 4O ' (4 & 4CX)/3 ( .91b)
Since this model is identical with the resistive companion model of ES. (3. ) for coupled inductances, generators are
interfaced with the network eSuations as if they were coupled inductances. The matrix [4 qƒ‡uˆ] enters the nodal
conductance matrix [G] of ES. (1. ) once and for all outside the time step loop, while the parallel current sources
are updated from step to step.
#fter the complete network solution has been obtained at time t, the generator phase voltages are converted
Page 289
to d, S, 0 Suantities,
XF cos$ cos($&120E) cos($%120E) X
XS ' 2 sin$ sin($&120E) sin($%120E) X ( .92)

3
X 12 12 12 X
and the armature currents are found from
KF ' (GF&OQF & XF)/4CX
KS ' (GS&OQF & XS)/4CX ( .93)
K ' (G & X )/4
The field structure currents are recovered from ES. ( . 0) for the d axis, and from an analogous eSuation for the
S axis. Finally, the fluxes 8 , 8 arep ƒcalculated from ES. ( .10a) and ( .10b), and the electromagnetic torSues from
ES. ( .32), which are then used to solve the mechanical eSuations as described next.
6TCPUKGPV 5QNWVKQP QH VJG /GEJCPKECN 2CTV
The mechanical part is described by ES. ( .29), which can be rewritten as
FT
[,] % [&] [T] % [-] [2] ' [6PGV] ( .9 a)
FV
with the speeds of the system of masses
F2
[T] ' ( .9 b)
FV
and the net torSue
[6PGV] ' [6VWTDKPG] & [6IGP GZE] ( .9 c)
The torSue [T sq€ q i ] provides the only direct link with the electrical part, with another indirect link through 2 sq€
$ which had to be predicted to solve the electrical part.
#pplying the trapezoidal rule (or central difference Suotients) to ES. ( .9 a) and ( .9 b) yields
[T(V)] & [T(V&)V)] [T(V)] % [T(V&)V)] [2(V)] % [2(V&)V)]

[,] % [&] % [-]
)V 2 2
39
Traduciendo...
Page 290
[6PGV(V)] % [6PGV(V&)V)]
' ( .95)
2
and
[T(V)] % [T(V & )V)] [2(V)] & [2(V & )V)]

' ( .9 )
2 )V
4eplacing [2(t)] in ES. ( .95) with the expression from ES. ( .9 ) produces
2 )V
[,] % [&] % [-] [T(V)] ' [6PGV(V)] % [JKUV(V&)V)]
)V 2
( .97a)
2 )V
[JKUV(V&)V)] ' [,] & [&] & [-] [T(V&)V)] & 2[-][2(V&)V)]
)V 2
% [6PGV(V&)V)] ( .97b)
Normally, it is assumed that the turbine power is constant. In that case, the torSue on each mass i is
calculated by using predicted speeds T, u
2VWTDKPG K
6VWTDKPG K ' ( .9 )
TK
If constant turbine torSue is assumed, then ES. ( .9 ) is skipped. With the turbine torSues from ES. ( .9 ), and with
the electromagnetic torSues at time t already calculated in the electrical part, ES. ( .97a) can be solved directly for
the speeds [T(t)]. The matrices
2 )V 2 )V
[#] ' [,] % [&] % [-] CPF [$] ' [,] & [&] & [-]
)V 2 )V 2
are tridiagonal, and remain constant from step to step. They are triangularized once and for all before entering the
time step loop, with a Gauss elimination subroutine specifically written for tridiagonal matrices, which saves storage
as well as computer time. Inside the time step loop, the information in the triangularized matrix is used to apply the
elimination to the right hand sides, followed by backsubstitution ( repeat solution, as explained in Section III.1).
It is worth noting that the form of ES. ( .9 ) is the same as the system of branch eSuations for coupled 4 L
C branches. In that analogy, the matrix [J] is eSuivalent to a matrix [L] of uncoupled inductances, the matrix [D]
Page 291
to a matrix [4] of coupled resistances, and the matrix [K] to an inverse capacitance matrix [C] of coupled
capacitances. [T ] would
€q† be eSuivalent to the derivatives [dv/dt] of the applied branch voltages.
2TGFKEVKQP CPF %QTTGEVKQP 5EJGOGU
The synchronous machine code in the EMTP has undergone many changes, especially with respect to the
prediction and correction schemes. The presently used schemes, as well as variations of it, are summarized here.
2TGFKEVKQP QHT CPF$
The speeds of all masses are predicted with linear extrapolation,
[T(V)] ' 2 [T(V & )V)] & [T(V & 2)V)] Traduciendo... ( .99)
Since the speeds change slowly, in comparison with the electrical Suantities, this prediction should be accurate
enough. Predicted speeds are needed in two places, namely for the prediction of speed voltages (see Section
.5. .3), and for the calculation of turbine torSues from ES. ( .9 ). The accuracy of the predicted generator rotor
speed T issq€
more important because there is no speed voltage correction in the present iteration scheme. The
accuracy of the turbine rotor speeds prediction is less important, because the torSue calculations ES. ( .9 ) are
corrected in the iteration scheme of Section .5.1, if constant turbine power is assumed (default option in UBC
EMTP, only option in BP# EMTP). If constant turbine torSue is assumed, then the turbine speed predictions are
not needed at all.
Fig. .13 shows the speed and the electromagnetic torSue of a generator by itself (no turbine connected to
it), which runs unloaded at synchronous speed and is then switched into a resistance load at t 0 [13 ] (data in
Table .2). The generator slows down very Suickly in this case. The curves were obtained with the UBC EMTP
without iterations (no return from step 7 to in Section .5.1), and verified with a th order 4unge Kutta Merson
method (agreement to within digits). Both the UBC and BP# EMTP had bugs in the speed calculation, which were
not noticed before in cases of small speed changes. They were corrected after J. Mechenbier proved their existence
by using principles of energy conservation as suggested by H. Boenig and S. 4anade [13 ].
The angle $ of the generator rotor must be predicted so that the d, S, 0 networks of Fig. .12 can be
converted to phase Suantities for the complete network solution in step 5 of Section .5.1. There is no correction
for this conversion in the present iteration scheme. The angle $ is also needed for converting the voltage solution
back from phase to d, S, 0 Suantities in step of Section .5.1 here, corrected values are obtained from the solution
of the mechanical part if steps ( ) and (7) are iterated.
Page 292
Traduciendo...
(KI Speed and electromagnetic torSue of an unloaded generator when switched

into a resistance load. (a) Speed, (b) Electromagnetic torSue.
In the UBC EMTP, the predicted value for $ is calculated from the predicted speed T with the trapezoidal
sq€
rule of integration ( .9 ),
)V
$(V) ' $(V & )V) % 6TIGP(V & )V) % TIGP(V)> ( .100)
2
Page 293
M3 and later versions of the BP# EMTP use a predictor formula suggested by Kulicke [135], which is based on
the assumption that $ is a fourth order polynomial of t,
$ ' C% CV % CV% CV% CV ( .101)
By using three known values of $ at t )t, t 2)t, t 3)t, and two known values of the speed
F$
T' 'C % 2C V % 3C V % C V ( .102)
FV
at t )t, t 2)t, the coefficients a ,...a can be found from

" the solution of 5 linear eSuations. This is a Hermite
interpolation formula and leads to the predictor formula [7 p. 1 , P in Table 5.1]
$(V) ' &96$(V&)V)&$(V&2)V)> % $(V&3)V) % )V6TIGP(V&)V) % TIGP(V&2)V)> ( .103)
6CDNG Generator test case no. 1 [13 ]
4atings: 1 0 MV# (three phase), 15 kV (line to line), wye connected.
4eactances: X p1.7 p.u., X 0.2 5 p.u.,

p X 0.1 5 p.u. p
X ƒ1. p.u., X 0.1 5 p.u. ƒ(no g winding)
X R0.15 p.u., X 0.1 p.u.I
Time constants: T p5.9 s, T 0.030 s p
T ƒ‚0.075 s
4esistances: 4 0.00109
g p.u.
&
4 10
x gpS in steady state (no effect added because some versions
cannot handle isolated generator)
4 1x gp
S after switching at t 0
Moment of inertia: J 999.9 7 (N m)s . One pole pair.

v $E
Terminal voltage: V 12.2 7 e kV (peak) in steady state in phase 1 (symmetrical in 3
phases).
Step size )t 200 zs. f 0 Hz
With coefficients a ,...a known,

" a predictor formula for T d$/dt for use
sq€ in the speed voltages could be written
down with ES. ( .102) as well,
)VT(V) ' 1 )VT(V&)V) % 17)VT(V&2)V) & 27$(V&)V) % 2 $(V&2)V)
% 3$(V&3)V) ( .10 )
Traduciendo...
3
Page 294
The BP# EMTP uses the predicted speed from ES. ( .99), though. It is not clear whether the th order predictor
of ES. ( .103) is really superior to the predictor of ES. ( .99).
#XGTCIKPI QH F CPF S #ZKU %QORCPKQP 4GUKUVCPEGU
Instead of an average resistance (4 4 A/2p in Fig.ƒ .12, the M39 version of the BP# EMTP uses 4 on p
both d and S axes. To compensate for this, a term (4 4 ) i is added

ƒ p to the ƒvoltage source on the S axis, and no
compensating term is needed on the d axes. Whether this method is better than the averaging procedure of Fig. .12
is unclear. Both procedures are special cases of a class of averaging methods discussed in [13 ].
2TGFKEVKQP QH K K FS
The armature currents i , i must

p ƒ be predicted so that the saliency terms i (4 4 )/2 and i (4
p p4
ƒ )/2 can be ƒƒp
combined with the known voltage sources e , e (Fig.

p ƒ .12). No correction is made for this in the present iteration
scheme. Note that the saliency terms are practically zero if X X . In the UBC
ƒ version
p and in BP# versions
until M32, the predicted currents i , i arep also

ƒ used to find predicted speed voltages, as described in the next section.
In the UBC EMTP, linear extrapolation is used,
K(V) ' 2K(V & )V) & K(V & 2)V) ( .105)
where i is either i or i .pThe BP#

ƒ version uses a linear three point predictor formula which smoothes numerical
oscillations. With the current known at t )t, t 2)t and t 3)t as
(KI Linear prediction with smoothing
indicated in Fig. .1 , averaged values are first found at the two midpoints by linear interpolation
1 K(V & 3)V) % K(V & 2)V)

K(V & 2 )V) '
2 2
1 K(V & 2)V) % K(V & )V)

K(V& 1 )V) '
2 2
Then a straight line is drawn through the two midpoints, with a slope of
Page 295
K(V & )V) & K(V & 3)V)

UNQRG '
2 )V
to predict the current i(t),
5 1 3
K(V) ' K(V&)V) % K(V&2)V) & K(V&3)V) ( .10 )
2
Traduciendo...
This linear prediction with smoothing is conceptually similar to fitting a straight line through three points in the least
sSuares sense. Such a straight line least sSuare fitting would have the same slope, but a value at t 2)t of {i(t 3)t)
i(t 2)t) i(t )t)}/3 instead of {i(t 3)t) 2i(t 2)t) i(t )t)}/ in Fig. .1 , which would yield a
predictor
1 2
K(V) ' K(V&)V) % K(V&2)V) & K(V&3)V) ( .107)
3 3 3
Which predictor performs best is difficult to say. #ll predictor formulas discussed in this section depend
solely on past points, and not on the form of the differential eSuations for i , i . ES. ( .7p),ƒ and an analog eSuation
for the S axis, were tried at one time as Euler predictor formulas, but they performed worse than the predictors
discussed here. It might be worth exploring other predictor formulas, because the accuracy of the solution depends
primarily on the prediction of i , i , especially

pƒ if the speed voltages are calculated from i , i as well, as discussed
pƒ
in the next section. One could use ES. ( .103), for example, by replacing $ with i and T with di/dt calculated from
ES. ( .7 ).
Fig. .15 shows the current in phase 1 after a three phase short circuit of a generator with unrealistically
low armature resistance 4 0.0001

g p.u. The data for this case is summarized in Table .3. Since speed changes
were ignored, the only predicted values are i , i , as well

p ƒ as speed voltages in the BP# EMTP.
Page 296
Traduciendo...
(KI Current in phase 1 after a three phase short circuit. 4 0.0001 p.u. g
Page 297
6CDNG Generator test case no. 2
4atings: 00 MV# (three phase), 1 kV (line to line), wye connected
4eactances: X p0.92 p.u., X 0.1 p.u.,p X 0.1 1 p.u. p
X ƒ0.7 p.u., X 0.1 1 p.u. (no

ƒ g winding)
X RX 0.0 2I p.u.
Time constants: T p. s, T 0.05 s p
T ƒI0.05 s
"
4esistances: 4 10
g p.u. (unrealistically low value)
!
4 10
g p.u. (more realistic value)
4 1x gp
S
Moment of inertia: J 4 (constant speed)

v % !' E
Terminal voltage: V .92 e kV (peak) in steady state in phase 1 (symmetrical in
3 phases).
Step size )t 200 zs. f 0 Hz
Disturbance: Three phase short circuit at terminals at t 0
In such a case with low damping, the errors caused by the prediction do accumulate noticeably if the simulation runs
over 5000 steps to t 1 s.ygThe errors are decreased if the complete network solution is iterated (not yet available
as an option in the production codes of the EMTP). For comparison purposes, the exact solution is shown as well,
which was found for i , i ,pwith

ƒ the eigenvalue/eigenvector method discussed in #ppendix I.1, and then transformed
to phase Suantities with $ from ES. ( .23). Fig. .1 shows the results if the armature resistance is changed to a
more realistic value of 4 0.001g p.u. #s can be seen, the answers are now closer to the exact solution.
2TGFKEVKQP QH 5RGGF 8QNVCIGU
Starting with M32 of the BP# EMTP, the speed voltages u , u of ES. p( ƒ.75) are predicted in the same way
as i p, iƒ with ES. ( .10 ). In some of these versions, the prediction was done in a synchronously rotating reference
frame, and then converted directly to phase Suantities without going through d, S axes parameters. This has been
abandoned in Feb. 19 , and the speed voltages are now again predicted in d, S Suantities because the latter turned
out to be superior when applied to test case no. 1 of Table .2.
Traduciendo...
Page 298
(KI Current in phase 1 after a three phase short circuit. 4 0.001 p.u. g
Page 299
In pre M32 versions of the BP# EMTP, and in the (unreleased) UBC version with synchronous machines,
the speed voltages u T8 pand u T8 are

ƒ not predicted
ƒ pexplicitly. Instead, the predicted currents i (t), i (t) p ƒ
and the predicted speed T (t) aresq€

used to calculate the speed voltages from ES. ( .10a) and ( .10b). The field
structure currents appearing in these eSuations are expressed as a function of i with ES. ( . 0),
p which leads to the
expression
Traduciendo...
TGFWEGF
8F(V) ' . F KF(V) % 8F& ( .10 )
with the known reduced inductance
4HH 4H& & 4FH

TGFWEGF
. F ' .F & [.FH .F&] ( .109)
4H& 4&& 4F&
and the known flux 8 for pzero current (i 0), p
4HH 4H& & JKUVH&XH(V)

8F& ' [.FH .F&] ( .110)
4H& 4&& JKUV&
„qp‡iqp
The reduced inductance L p is practically identical with L if 4 2L/)t.
p For the IEEE benchmark model
[7 ] with )t 200 zs, TL 0.135129 p.u.

p compared to TL 0.135 p.u. In publications
p based on [13],
„qp‡iqp „qp‡iqp
TL p is called a and TL ƒ is called a .
+VGTCVKQP 5EJGOGU
Up to now, the complete network solution is direct, without iterations. The iteration scheme of Section
.5.1 does not repeat the network solution, and predicted values are therefore never completely corrected. There
is only one exception, namely the three phase short circuit at the generator terminals with zero fault resistance. In
that case the terminal voltages are always zero, and going back to step in the iteration scheme of Section .5.1
should be a complete correction of all predicted values.
It is doubtful whether the predictors can be improved much more. Further improvements can probably only
be made if the network solution is included in the iterations as well. This could be a worthwhile option, not only
for machines, but for other nonlinear or time varying elements as well.
5CVWTCVKQP
Saturation effects in synchronous machines can have an influence on load flow, on steady state and transient
stability, and on electromagnetic transients. While transformer saturation usually causes more problems than
machine saturation (e.g., in the creation of so called temporary overvoltages ), there are situations where saturation
in machines must be taken into account, too.
To model machine saturation rigorously is very difficult. It would reSuire magnetic field calculations, e.g.
Page 300
by finite element methods [1 1], which is already time consuming for one particular operating condition, and
practically impossible for conditions which change from step to step. #lso, the detailed data for field calculations
would not be available to most EMTP users. #n approximate treatment of saturation effects is, therefore, commonly
accepted. The modelling of saturation effects is discussed in four parts,
(a) basic assumptions,
(b) saturation effects in steady state operation, and
(c) saturation effects under transient conditions, and
(d) implementation in the EMTP
$CUKE #UUWORVKQPU
The data which is normally available is the open circuit saturation curve (Fig. .17), which shows the
terminal voltage as a function of the field current for open circuited armature windings (no load condition). In the
transient simulation, a flux current relationship is reSuired, rather than V f(i ). This is easily obtained
r from ES.
( .9), which becomes
XS ' T 8F ( .111a)
Traduciendo...
(KI Open circuit saturation curve
for balanced, open circuit steady state conditions, where both 8 and the transformer
ƒ voltages d8 /dt, d8 /dt are zero.
p ƒ
Since v isƒ eSual to %3V RGS , this eSuation can be rewritten as
3 8 &4/5 ' T 8F ( .111b)
where V RGS is the 4MS terminal voltage of armature winding 1 (line to ground 4MS voltage for wye connected
machines). It is therefore very simple to re label the vertical axis in Fig. .17 from voltage to flux values with ES.
( .111).
The saturation effects in synchronous machines do not produce harmonics during balanced steady state
50
Page 301
operation, because the open circuit saturation curve describes a dc relationship between the dc flux of the rotating
magnets (poles) and the dc field current reSuired to produce it. The magnitude of the dc flux determines the
magnitude of the induced voltages in the armature, while the shape of the flux distribution across the pole face
determines the waveshape of the voltage. If the distribution is sinusoidal, as assumed in the ideal machine
implemented in the EMTP, then the voltage will be sinusoidal as well. In reality, the distribution is distorted with
space harmonics, and it is this effect which produces the harmonics in synchronous machines.
There are many different ways of representing saturation effects [1 2], and it is not completely clear at this
time which one comes closest to field test results. More research on this topic is needed. #t this time, the
representation of saturation effects in the EMTP is based upon the following simplifying assumptions:
1. The flux linkage of each winding in the d or S axis can be represented as the sum of a leakage flux (which
passes only through that winding) and of a mutual flux (which passes through all other windings on that axis
as well), as illustrated in Fig. .1 ,
8 ' 8ý % 8O ( .112)
where
8 ýleakage flux unaffected by saturation,
8 ymutual flux subjected to saturation effects.
In reality, the leakage fluxes are subjected to saturation effects as well because they pass partly through iron
[1 0], but to a much lesser degree than the mutual flux. Saturation effects are therefore ignored in the leakage
fluxes. The data is not available anyhow if only one saturation curve (open circuit saturation curve) is given.
In terms of eSuivalent circuits, this assumption means that only some of the inductances are nonlinear (shunt
branch in star point in Fig. .2), while the others remain constant.
2. The degree of saturation is a function of the total air gap flux linkage 8 , y‡
8O ' H(8O&W) ( .113a)
with
8O&W ' 8 ( .113b)

OF&W % 8 OS&W
and
8OF&W ' /FW(KF % KH % K&) 8OS&W ' /SW(KS % KI % K3) ( .113c)

Traduciendo...
51
Page 302
(KI Leakage fluxes and mutual flux
where subscript m indicates mutual, and u indicates unsaturated values. In ES. ( .113c) it is important to
use the proper mutual inductances for the representation of the mutual flux. This leads back to the data
conversion problem discussed in Section .2. If Canay s characteristic reactance X is not known, then
i assume
k 1 in ES. ( .1 b), and use
from ES. ( .20a) and ( .20c). In this case, the eSuivalent star circuit of Fig. .2 shows the correct separation
/FW ' .F & .ý /SW ' .S & .ý
into the mutual inductance M M or yM for the

p‡ mutualƒ‡
flux (subject to saturation) and into the leakage
inductances for the leakage fluxes (linear d , f , and D branches). If Canay s characteristic reactance is used,
then Fig. .2 can no longer be used, as explained in Section . .5.
3. Only one flux, namely the total air gap flux, is subjected to saturation. The saturated mutual fluxes 8 , 8 yp yƒ
on both axes are found from their unsaturated values by reducing them with the same ratio (similar triangles
in Fig. .19),
(KI Unsaturated and saturated mutual fluxes
8O 8O
8OF ' 8OF&W @ 8OS ' 8OS&W @ ( .11 )
8O&W 8O&W
52
Page 303
This assumption is based on the observation that there is only one mutual flux, which lines up with the pole axis
Traduciendo...
if 8 is
yƒ very small, and which will shift to one side of the pole as 8 increases (Fig. yƒ
.20).
(a) flux from i aloner (8 0) yƒ (b) flux from i and

r stator currents 8 û 0 yƒ
(KI Flux in turbogenerator [1 1]. l 19 1 IEEE
. Saturation does not destroy the sinusoidal distribution of the magnetic field over the face of the pole, and all
inductances therefore maintain their sinusoidal dependence on rotor position according to ES. ( .5).
5. Hysteresis is ignored, while eddy currents are approximately modelled by the g winding, and maybe to some
extent with the D and 3 windings. More windings could be added, in principal, to represent eddy currents
more accurately.
5CVWTCVKQP KP 5VGCF[ 5VCVG 1RGTCVKQP
#t this time, the saturation effects are only modelled correctly in the ac steady state initialization if the
terminal voltages and currents are balanced. More research is needed before saturation can be represented properly
in unbalanced cases.
#s explained in Section . , the initialization of the machine variables follows after the phasor steady state
solution of the complete network. The initialization for balanced (positive seSuence) conditions is described in detail
in Section . .1, and only the modifications reSuired to include saturation effects will be discussed here.
The nonlinear characteristic of Fig. .17 makes it impossible to use the initialization procedure of Section
. .1 in a straightforward way. To get around this problem, it is customary to use an eSuivalent linear machine
in steady state analysis which gives correct answers at the particular operating point and approximate answers in the
neighborhood. This eSuivalent linear machine is represented by a straight line through the operating point and the
origin (dotted line in Fig. .21). Whenever the operating point moves, a new straight line through the new operating
point must be used.
The concept of the eSuivalent linear machine is used in the EMTP as follows.
1. Obtain the ac steady state solution of the complete network. From the terminal voltages and currents of the
53
Page 304
machine (positive seSuence values), find the internal machine variables with the method of Section . .1.
#ssume that the machine operates in the unsaturated region at this time.
(KI Linearization for steady state analysis
2. Determine the total magnetizing current

Traduciendo...
/SW
KO ' (KF % KH % K&) % (KS % KI % K3) ( .115a)
/FW
with i ,8 is iQbeing zero for balanced conditions. ES. ( .115) assumes turns ratios of N :N :N 1:1:1 (samep r 8
for Suadrature axis). If any other turns ratios are used, the first term would be
/FW KOF ' /FW KF % /FH&W KH % /F&&W K& ( .115b)
and the second term
/SW KOS ' /SW KS % /SI&W KI % /S3&W K3 ( .115c)
with
/FW KO ' (/FW KOF) % (/SW KOS) ( .115d)
Find the operating point on the nonlinear characteristic of Fig. .21. If this point lies in the linear region, then
the initialization is complete. Otherwise:
3. Calculate the ratio K from Fig. .21,
Page 305
#$
-' ( .11 a)
#%
and multiply the unsaturated mutual inductances with that ratio to obtained the saturated values of the eSuivalent
linear machine,
/F ' - @ /FW /S ' - @ /SW ( .11 b)
Use these values to repeat the initialization procedure of Section . .1. Then re calculate the magnetizing
current from ES. ( .115). If it agrees with the previously calculated value within a prescribed tolerance, then
the initialization is finished. If not, repeat step 3. Convergence is usually achieved with 1 to 2 iterations of
step 3.
In the BP# EMTP steady state solution, machines are now represented as voltage sources at the terminals,
and the terminal currents are obtained from that solution. With terminal voltages and currents thus known, their
positive seSuence components can be calculated and then used to correct the internal variables for saturation effects.
Since this correction does not change the terminal voltages and currents, the complete network solution does not have
to be repeated in step 3.
This will also be true in future versions of the EMTP, where the machine will be represented as symmetrical
voltage sources behind an impedance matrix. #gain, the terminal voltages and currents and their positive seSuence
components will be known from the steady state solution.
In unbalanced cases, the present representation will produce negative seSuence values, while the future
representation will produce correct values. How to use these negative seSuence values in the saturation corrections
has not yet been worked out. Since they produce second harmonics in the direct and Suadrature axes fluxes, it may
well be best to ignore saturation effects in the negative seSuence initialization procedure of Section . .2 altogether.
The eSuivalent linear machine produces correct initial conditions for the different model used in the transient
simulation, as can easily be verified if a steady state solution is followed by a transient simulation without any
disturbance.
5CVWTCVKQP WPFGT 6TCPUKGPV %QPFKVKQPU
The eSuivalent linear machine described in Section . .2 cannot be used in the transient solution, because
the proper linearization for small disturbances (as they occur from step to step) is not the straight line 0 C in Fig.
Traduciendo...
.21 ( linear inductance ), but the tangent to the nonlinear curve in point C ( incremental inductance ).
The saturation effect enters the transient solution discussed in Section .5 in two places, namely through
the speed voltages and through the transformer voltages. Consider the direct axis eSuations ( .7 ) first, which can
be rewritten as
55
Page 306
8ýF 8OF &T8ýS &T8OS

F 8ýH F 8OF
[X] ' &[4][K] & & % 0 % 0 ( .117)
FV FV
8ý& 8OF 0 0
for the d, f, D Suantities if each flux linkage is separated into its leakage flux and the common mutual flux,
8F ' 8ýF % 8OF
8H ' 8ýH % 8OF ( .11 )
8& ' 8ý& % 8OF
assuming turns ratios of N :N :N

p r 81:1:1 (analogous for Suadrature axis). Only the terms with 8 and 8 in ES. yp yƒ
( .117) are influenced by saturation, and only those terms are therefore discussed.
Consider first the speed voltage term T8 in ES. ( .117),

yƒ which is properly corrected for saturation by
simply using the correct saturated value 8 . The saturation

yƒ correction has already been described in ES. ( .11 ),
and is conceptually the same as the one used in ES. ( .11 ) for the steady state solution. Since the transient solution
works with predicted values of speed voltages T8 and T8 ,p as explained

ƒ in Section .5. . , they are used directly
in ES. ( .117) (not split up into two terms).
Next consider the transformer voltages [d8 /dt] in ES.

yp ( .117), where incremental changes ( incremental
inductances ) are important. By using the tangent of the nonlinear characteristic in the last solution point, one can
linearize the flux current relationship to
8O ' 8MPGG % /UNQRG KO ( .119)
(KI Linearization around last solution point
with M …x ‚qbeing an incremental inductance (Fig. .22). This eSuation can be used over the next time step, because
Page 307
Traduciendo...
the fluxes change only very slowly with typical step sizes of 50 to 500 zs. In the EMTP implementation, the
problem is even simpler because the saturation curve is represented as a two slope piecewise linear curve. In that
case, the linearization of ES. ( .119) changes only at the instant where the machine goes into saturation, and at the
instant when it comes out again. With ES. ( .113) and ( .115), the unsaturated total flux is
8O&W ' /FW KO ( .120)
which, inserted into ES. ( .119), produces
8O ' 8MPGG % D8O&W ( .121a)
with the ratio between incremental inductance M …x ‚qand linear (unsaturated) inductance M , p‡
/UNQRG
D' ( .121b)
/FW
#fter saturation has been defined for the total flux, it must be separated into d and S components again.
With assumption (3) from Section . .1, and with Fig. .19,
8OF ' 8MPGG&F % D8OF&W
8OS ' 8MPGG&S % D8OS&W ( .122a)
where
8MPGG&F ' 8MPGGcos$ 8MPGG&S ' 8MPGGsin$ $ ' tan& (8OS&W/8OF&W)
( .122b)
If ES. ( .117), and the analogous eSuation for the Suadrature axis, are solved with the trapezoidal rule of
integration, then the transformer voltage term affected by saturation,
[XOF] ' & F[8OF]/FV
is transformed with ES. ( .122) into
2D
[XOF(V)] ' & 6[8OF&W(V)] & [8OF&W(V & )V)>
)V
2
& 6[8MPGG&F(V)] & [8MPGG&F(V&)V)> & [XOF(V&)V)] ( .123)
)V
This eSuation shows how the transformer voltages must be corrected for saturation effects:
(a) multiply all mutual inductances by the factor b, and
57
Page 308
(b) add correction terms to account for the variation of the knee fluxes [8 w€qq ],
p and [8 w€qq ]ƒ
+ORNGOGPVCVKQP KP VJG '/62
Saturation effects were modelled for the first time in the M27 version of the BP# EMTP, based on the
concept of two independent saturation effects, one in the direct axis and the other in the Suadrature axis. This was
replaced with a newer model in the M32 version, which was essentially the model discussed here. It was not Suite
correct, however, because the correction terms in ES. ( .123) related to the knee fluxes were not included. The
model described here was implemented for the first time in the DCG/EP4I version to be released in 19 .
The open circuit saturation curve is approximated as a two slope piecewise linear characteristic (0 1 and
1 2 in Fig. .22). The number of linear segments could easily be increased, but a two slope representation is usually
adeSuate.
5VGCF[ 5VCVG +PKVKCNK\CVKQP
The initialization procedure is only correct for balanced networksTraduciendo...

at this time. The extension to unbalanced
cases is planned for the future. Until this is done, some transients caused by incorrect initialization can be expected
in unbalanced cases. Hopefully, they will settle down within the first few cycles.
The initialization follows the procedure of Section . .2. For the reactances X , X , which consistpof a ƒ
constant leakage part and a saturable mutual part,
:F ' :ý % T/F :S ' :ý % T/S ( .12 )
unsaturated values M , Mp‡are first

ƒ‡ used to obtain the internal machine variables with the method of Section . .1.
If the resulting magnetizing current lies in the saturated region, then the mutual reactances M , M in ES. ( .12 ) p ƒ
must be corrected with ES. ( .11 ). The calculation of the internal machine variables is then repeated with saturated
reactances one or more times, until the changes in the magnetizing current become negligibly small.
With the two slope piecewise linear representation implemented in the EMTP, the ratio K needed in ES.
( .11 ) is
/UNQRG KO % 8MPGG
-' ( .125)
/FW KO
with the meaning of the parameters shown in Fig. .22, and with i calculated from
y ES. ( .115).
6TCPUKGPV 5QNWVKQP
Saturation effects in the time step loop are modelled according to Section . .3. The coefficient b of ES.
( .121b) is set to 1.0 in the unsaturated region, and to M /M in the saturated

…x ‚q region.
p‡ Whenever the solution
moves from one region into the other, it is reset accordingly.
This coefficient b affects the values in the eSuivalent resistance matrix [4 qƒ‡uˆ] of ES. ( .91a) and in the
history term matrix of ES. ( . 2c). To include this coefficient, the inductance matrix of ES. ( .7 ) is split up into
Page 309
.OF&W .OF&W .OF&W .ýF
[.] ' D .OF&W .OF&W .OF&W % .ýH ( .12 )
.OF&W .OF&W .OF&W .ý&
(analogous for Suadrature axis). Whenever b changes, [L] is recalculated and then used to recalculate [4 qƒ‡uˆ] and
the history term matrix of ES. ( . 2c). With the two slope representation implemented in the EMTP, there are only
two values of b, and the matrices could therefore be precalculated outside the time step loop for the two values of
b 1 and b M /M . The…x
major effort lies in the re triangularization of the network conductance matrix [G]
‚q p‡
of ES. ( .1 ), however, which contains [4 qƒ‡uˆ] and therefore changes whenever the machine moves into the
saturated region, or out of it.
#n additional modification is reSuired in the calculation of the history terms with ES. ( .7 b). #s shown
in ES. ( .123), the knee fluxes 8 (t) and 8 (tw€qq

)t) must noww€qq
be included. Since the trapezoidal rule of
integration is not very good for the calculation of derivatives, the knee fluxes are included with the backward Euler
method. First, the knee fluxes 8 w€qq p(t) and 8 w€qq (t)
ƒ are predicted, using the three point predictor of ES. ( .10 ).
Then the trapezoidal rule expression
2
6[8MPGG&F(V)] & [8MPGG&F(V & )V)]>
)V
is replaced with the backward Euler expression
1 RTGFKEVGF
6[8 ( .127)
)V MPGG&F (V)] & [8MPGG&F(V & )V)]>
and the voltage term [v (t )t)]

yp is replaced by a voltage term which excludes the knee flux.
5CVWTCVKQP 'HHGEVU YKVJ %CPC[ U %JCTCEVGTKUVKE 4GCEVCPEG
If saturation is ignored, then it does not matter whether Canay s characteristic reactance is used or not,
Traduciendo...
because it only affects the data conversion part. With saturation included, however, the nonlinear inductance can
only be identified as the shunt branch M in Fig. y.2 if k 1 in ES. ( .1 b). If Canay s characteristic reactance
is known, then k û 1. This factor k must then be removed again from the rotor Suantities in ES. ( .15a), by
multiplying the second and third row and column with its reciprocal value,
8F KF
.F E/O E/O
3 2
8H KH
2 ' E/O E .HHO E /O 3 ( .12 )
3 E/O E /O E .&&O 2
8& K&
2 3
59
Page 310
where
1
E' ( .129)
M
and where M , Ly andrry

L 88y are the modified parameters straight out of the data conversion routine of #ppendix
VI. . #s explained in the text between ES. ( .17) and ( .1 ), the factor %3/%2 in ES. ( .12 ) is needed to produce
turns ratios of N :N :N
p r 81:1:1. Only with turns ratios of 1:1 can the fluxes be separated into their main and
leakage parts. The circuit of Fig. .23, which is eSuivalent to ES. ( .12 ), has the correct separation into the mutual
inductance cM 3/2
y M subjected
gr to saturation (for the mutual flux), and into the linear leakage
(KI ESuivalent circuit for direct axis with identity of leakage and main fluxes restored
from Fig. .2
inductances in the three branches d, f, D. For the Suadrature axis, Fig. .2 can still be used, with M being the y
nonlinear inductance, because Canay s characteristic reactance cannot be measured on that axis (current split between
g and 3 windings unmeasurable because both windings are hypothetical windings).
Most EMTP users will not know Canay s characteristic reactance because it is not supplied with the standard
test data. Therefore, it has not yet been included in the saturation model in the EMTP, e.g. in the form of Fig. .23,
because of lower priority compared to other issues. When it is implemented, one would have to decide whether the
inductance c M cMy , whichy is mutual to both f and D windings, should be constant or saturable as well.
0
Traduciendo...
Page 311
70+8'45#. /#%*+0'
Co author: H.K. Lauw
The universal machine was added to the EMTP by H.K. Lauw and W.S. Meyer [137,1 0], to be able to
study various types of electric machines with the same model. It can be used to represent 12 major types of electric
machines:
(1) synchronous machine, three phase armature
(2) synchronous machine, two phase armature
(3) induction machine, three phase armature
() induction machine, three phase armature and three phase rotor
(5) induction machine, two phase armature
() single phase ac machine (synchronous or induction), one phase excitation
(7) same as ( ), except two phase excitation
() dc machine, separately excited
(9) dc machine, series compound (long shunt) field
(10) dc machine, series field
(11) dc machine, parallel compound (short shunt) field
(12) dc machine, parallel field.
The user can choose between two interfacing methods for the solution of the machine eSuations with the rest
of the network. One is based on compensation, where the rest of the network seen from the machine terminals is
represented by a Thevenin eSuivalent circuit, and the other is a voltage source behind an eSuivalent impedance
representation, similar to that of Section .5, which reSuires prediction of certain variables.
The mechanical part of the universal machine is modelled Suite differently from that of the synchronous
machine of Section 9. Instead of a built in model of the mass shaft system, the user must model the mechanical part
as an eSuivalent electric network with lumped 4, L, C, which is then solved as if it were part of the complete electric
network. The electromagnetic torSue of the universal machine appears as a current source in this eSuivalent network.
$CUKE 'SWCVKQPU HQT 'NGEVTKECN 2CTV
#ny electric machine has essentially two types of windings, one being stationary on the stator, the other
rotating on the rotor. Which type is stationary and which is rotating is irrelevant in the eSuations, because it is only
the relative motion between the two types which counts. The two types are:
(a) #rmature windings (windings on power side in BP# 4ule Book). In induction and (normally) in
synchronous machines, the armature windings are on the stator. In dc machines, they are on the rotor,
where the commutator provides the rectification from ac to dc.
(b) Windings on the field structure ( excitation side in BP# 4ule Book). In synchronous machines the field
91
Page 312
structure windings are normally on the rotor, while in dc machines they are on the rotor, either in the form
of a short circuited sSuirrel cage rotor, or in the form of a wound rotor with slip ring connections to the
outside. The proper term is rotor winding in this case, and the term field structure winding is only used
here to keep the notation uniform for all types of machines.
These two types of windings are essentially the same as those of the synchronous machine described in
Section .1. It is therefore not surprising that the system of eSuations ( .9) and ( .10) describe the behavior of the
universal machine along the direct and Suadrature axes as well. The universal machine is allowed to have up to 3
armature windings, which are converted to hypothetical windings d, S, 0a ( a for armature) in the same way as in
Section .1. The special case of single phase windings is discussed in Section 9.3. The field structure is allowed
to have any number of windings D1, D2,...Dm on the direct axis, and any number of windings 31, 32,...3n on
the Suadrature axis, which can be connected to external circuits defined byTraduciendo...
the user. In contrast to Section , the
field structure may also have a single zero seSuence winding 0f ( f for field structure), to allow the conversion of
three phase windings on the field structure (as in wound rotor induction machines) into hypothetical D, 3, 0
windings.
With these minor differences to the synchronous machine of Section in mind, the voltage eSuations for
the armature windings in d, S Suantities become
XF 4C 0 KF 8F &T8S
F
'& & % (9.1a)
XS 0 4C KS FV 8S %T8F
with T being the angular speed of the rotor referred to the electrical side, and in zero seSuence,
X C ' &4C K C & F8 C/FV (9.1b)
The voltage eSuations for the field structure windings are
X& 4& K& 8&
X& 4& K& 8&
. . . F .
'& & (9.2a)
. . . FV .
. . . .
X&O 4&O K&O 8&O
X3 43 K3 83
X3 43 K3 83
. . . F .
'& & (9.2b)
. . . FV .
. . . .
X3O 43O K3O 83O
92
Page 313
and
&8H
XH' & 4HKH& (9.2c)
FV
The flux current relationships on the two axes provide the coupling between the armature and field structure sides,
8F KF
8& .F /F& /F& ... /F&O K&
8& /F& .& /& & ... /& &O K&
. ' /F& /& & .& ... /& &O . (9.3a)
. . . . . . .
. /F&O /& &O /& &O ... .&O .
8&O K&O
8S KS
83 .S /S3 /S3 ... /S3P K3
83 /S3 .3 /3 3 ... /3 3P K3
. ' /S3 /3 3 .3 ... /3 3P . (9.3b)
. . . . . . .
. /S3P /3 3P /3 3P ... .3P .
83P K3P
with both inductance matrices being symmetric. The zero seSuence fluxes on the armature and field structure side
are not coupled at all,
Traduciendo...
8C' .CKC (9.3c)
8H' .HKH (9.3d)
The universal machine has been implemented under the assumption that the self and mutual inductances in
ES. (9.3a) and (9.3b) can be represented by a star circuit if the field structure Suantities are referred to the armature
side, as shown in Fig. 9.1. This assumption
93
Page 314
(KI Star circuit representation of coupled windings in

direct axis (analogous in Suadrature axis)
implies that there is only one mutual (or main) flux which links all windings on one axis (8 in Fig. .1 ), andOthat
the leakage flux of any one winding is only linked with that winding itself. Strictly speaking, this is not always true.
For example, part of the leakage flux of the field winding (8 in Fig. .1 ) could
NH go through the damper winding
as well, but not through the armature winding, which leads to the modified star circuit of Fig. .23 (synchronous
machines) or Fig. 9.2 (induction machines). The data for such models with uneSual mutual inductances is seldom
available, however (e.g., Fig. .23 reSuires Canay s characteristic reactance, which is not available from standard
test data). The star circuit is therefore a reasonable assumption in practice. #t any rate, the code could easily be
changed to work with the self and mutual inductances of ES. (9.3) instead of the star circuit of Fig. 9.1.
With the star circuit representation of Fig. 9.1, the flux current eSuations (9.3a) can be simplified to
8F ' .ýF KF % 8OF
) )
8 (9. a)
& ' .) ý& K & % 8OF
. .
) )
8
&O ' .) ý&O K &O % 8OF
with
) )
8OF ' /F (KF % K (9. b)
& ...% K &O)
where the prime indicates that field structure Suantities have been referred to the armature side with the proper turns
O
ratios between d and D1, d and D2,...d and D . #ll referred mutual inductances are eSual to M in this F
representation, and the referred self inductances of ES. (9.3a) are related to the leakage inductances of the star
9
Traduciendo...
Page 315
branches by
.F ' .ýF % /F
) )
. (9.5)
&' . ý& % /F
) )
.
&O ' . ý&O % /F
The voltage eSuations (9.1) and (9.2) are valid for referred Suantities as well, if 4 , i ,... are replaced
&&by 4 , &
i&,... The Suadrature axis eSuations are obtained by replacing subscripts d, D in the direct axis eSuations with S,
3.
In the BP# EMTP 4ule Book, the turns ratios are called reduction factors, and the process of referring
Suantities to the armature side is called reduction (referring Suantities from one side to another is discussed in
#ppendix IV.3).
&GVGTOKPCVKQP QH 'NGEVTKECN 2CTCOGVGTU
By limiting the universal machine representation to the star circuit of Fig. 9.1, the input parameters are
simply the resistances and leakage inductances of the star branches and the mutual inductance, e.g., for the direct
axis,
4C , .ýF
) )
4 . ý&
&,
) )
4 . ý&O
&O ,
/F ,
(analogous for the Suadrature axis), and for the zero seSuence on the armature and field structure side,
.C,
) )
4 . H
H,
If the armature leakage inductance L is known

ý instead of the mutual inductance, then find M from ES. (9.5),
/F ' .F & .ýF , /S ' .S & .ýS
95
Page 316
If neither L nor
ý M is known, then use a reasonable estimate. The BP# EMTP 4ule Book recommends
.ýF ' 0.1 .F , .ýS ' 0.1 .S (9. )
which seems to be reasonable for round rotor synchronous machines, while for salient pole machines the factor is
closer to 0.2 than to 0.1. Compared to the large value of the magnetizing inductance of transformers, the value of
the mutual (or magnetizing) inductance M , M from
F ES.S (9. ) (90 of self inductance) is relatively low because
of the air gap in the flux path.
Compared to the (m 1) (m 2) / 2 inductance values in ES. (9.3a), the star circuit has only m 2
Traduciendo...
inductance values. For the most common machine representation with 2 field structure windings, ES. (9.3a) reSuires
values, compared to values for the star circuit. This means that the star circuit is not as general as ES. (9.3a),
but this is often a blessing in disguise because available test or design data is usually not sufficient anyhow to
determine all self and mutual inductances (see reSuirement of obtaining an extra inductance value X in Section .2).E
#s already discussed for the synchronous machine in Section .2, the resistances and self and mutual
inductances (or the star branch inductances here) are usually not available from calculations or measurements. If
the universal machine is used to model a synchronous machine, then the data conversion discussed in Section .2
can be used (input identical to synchronous machine model in version M32 and later).
For three phase induction machines, the data may be given in phase Suantities. If so, ES. ( .11) must be
used to convert them to d, S, 0 Suantities,
.F ' .S ' .U & /U

.Q ' .U % 2/U
with L Uself inductance of one armature winding,
M Umutual inductance between two armature windings (BP# 4ule Book uses opposite sign for M ). U
L Oin ES. ( .11) is zero for an induction machine, where the saliency term defined in ES. ( .5) does not exist. The
same conversion is used if the rotor windings are three phase. The mutual inductance between stator and rotor
follows from ES. ( .10).
3
/F&&K ' /C&&K
2
(same for S axis), with M cos $Cbeing

&K the mutual inductance between armature winding 1 and rotor winding Di
(i 1,...m), as defined in ES. ( .5). Note that the factor %3/%2 changes the turns ratio if the turns ratio between
phase 1 and the rotor winding is 1:1, it changes to %3:%2 in d, S, 0 Suantities (see also Section .2). This extra
factor must be taken into account when rotor Suantities are referred to the stator side.
For modelling three phase induction machines, a modified universal machine with its own data conversion
routine has recently been developed by Ontario Hydro [13 ]. It uses the standard NEM# specification data to find
the resistances and self and mutual inductances of the eSuivalent circuit. It is beyond the scope of this treatise to
describe the conversion routine in detail. Essentially, the field structure (which is the rotor in the induction machine)
Page 317
has two windings to represent the rotor bars as well as the eddy currents in the deep rotor bars of large machines,
or the double sSuirrel cage rotor in smaller machines. Since there is no saliency, d and S axis parameters are
identical. The assumption of eSual mutual inductances (or the star circuit) is dropped, and the eSuivalent circuit of
Fig. 9.2 is used instead. Not surprisingly, this eSuivalent circuit is identical with that of the synchronous machine
in Fig. .23, because a synchronous
(KI ESuivalent circuit of induction machine with deep rotor bars
machine becomes an induction machine if the field winding is shorted. In contrast to the standard universal machine,
saturation is included in the leakage inductance branch of the armature as well, and another nonlinear inductance is
added between the star point and the star branches of the field structure windings. Fig. 9.3 shows comparisons
Traduciendo...
between measurements and simulation results with this modified universal machine model [13 , 139], for a case of
a cold start up of an induction motor driven heat transfer pump (1100 hp, 00 V). Excellent agreement with the
field test results is evident for the whole start up period, which proves the validity of the modified universal machine
model over the whole range of operation.
97
Page 318
(a) active power input
(b) phase current

(4MS values)
(c) terminal voltage
(4MS values)
(KI Comparison between field test and simulation results for starting up anTraduciendo...
induction motor with a heat transfer
pump [13 , 139]. 4eprinted by permission of G.J. 4ogers and D. Shirmohammadi
Page 319
6TCPUHQTOCVKQP VQ 2JCUG 3WCPVKVKGU
ES. (9.1) to (9.3) completely describe the universal machine in d, S, 0 Suantities, irrespective of which type
of machine it is. To solve these machine eSuations together with the rest of the network, they must be transformed
to phase Suantities. It is in this transformation where the various types of machines must be treated differently.
Fortunately it is possible to work with the same transformation matrix for all types, by simply using different matrix
coefficients.
For the case of a three phase synchronous machine, the transformation has already been shown in ES. ( .7).
If this eSuation is rewritten for the armature Suantities only, then
8F 8
8S ' [6]& 8 KFGPVKECN HQT [X], [K] (9.7a)
8C 8
with
cos$ cos($&120E) cos($%120E)
2 sin$ sin($&120E) sin($%120E)

[6]& ' (9.7b)
3 1 1 1
2 2 2
being an orthogonal matrix, which means that
&
[6] ' [6] VTCPURQUGF (9.7c)
The rotor position $ is related to the angular speed T of the rotor by
T ' F$/FV (9.7d)
The transformation matrix [T] can be rewritten as a product of two matrices [137],
[6]& ' [2($)]& [5]& (9. a)
with
In [137] and [139], [T] is called [T] similarly, [P] and [S] are called [P] and [S] there.
99
Page 320
cos$ &sin$ 0
[2($)]& ' sin$ cos$ 0 (9. b)
0 0 1
2 1 1
& &
3 Traduciendo...
1 1
[5]& ' 0& (9. c)
2 2
1 1 1
3 3 3
each being orthogonal again,
& &
[2] ' [2] [5] ' [5] VTCPURQUGF (9. d)
VTCPURQUGF ,
The first transformation with the [S] matrix replaces the three phase coils (displaced by 120E in space) by the three
eSuivalent coils d and S (perpendicular to each other) and 0 (independent by itself). This is the same transformation
matrix used for ", $, 0 components in ES. ( . ), except for a sign reversal of the $ Suantities. The second
transformation with [P] makes the d, S axes rotate with the same speed as the field poles, so that they become
stationary when seen from the field structure. The field structure Suantities are not transformed at all.
This approach with two transformations can be applied to any type of machine. For a three phase induction
machine with a three phase wound rotor, both the armature and field structure Suantities are transformed with [S]
to get eSuivalent windings on the d and S axes, while the transformation with [P] is only applied to the armature
side. For direct current machines, there is not transformation at all for both the armature and field structure side.
For two phase armature windings displaced by 90E in space, the windings are already on the d, S axes.
Therefore
10
[5 ]& ' (9.9a)
0 &1
and
cos$ &sin$
[2 ]& ' (9.9b)
sin$ cos$
with the zero seSuence winding missing.
For single phase armature windings, there is only flux along one axis, or
9 10
Page 321
[5 ]& ' 1 (9.10a)
and
[2 ]& ' cos$ (9.10b)
with both the Suadrature axis and zero seSuence winding missing.
The EMTP uses only one transformation matrix [S] and [P] for all cases, and makes the distinction by
resetting the coefficients c , c , c in these matrices,
E % E % E 2/3 &E / &E /
[5]& ' 0 &E &E / 2 E / 2 (9.11)
E/ 3 E/ 3 E/ 3
with
c 1 for three phase ac windings, and c 0 otherwise,
c 1 for two phase ac windings, and c 0 otherwise,
c 1 for single phase ac windings and dc machines, and c 0 otherwise.
Since [S] in ES. (9.11) degenerates into 2 x 2 and 1 x 1 matrices for two phase and single phase windings, its
inverse cannot be found by inversion. Using ES. (9. d) instead of inversion works in all cases, however. The
Traduciendo...
matrix in ES. (9.11) is slightly different from that in [137], because it is assumed here that only phases 1, 2 exist
for two phase machines, and only phase 1 exists for single phase machines. In [137], phase 1 is dropped for two
phase machines, and phases 1 and 2 are dropped for single phase machines.
For ac armature windings, [P] of ES. (9. b) is used, realizing that the zero seSuence does not exist in the
two phase case, and that the zero seSuence as well as the S winding do not exist in the single phase case. For dc
armature windings, there is not second transformation with [P] .
/GEJCPKECN 2CTV
In contrast to the synchronous machine model, the universal machine does not have a built in model for the
mechanical part. Instead, the user must convert the mechanical part into an eSuivalent electric network with lumped
4, L, C, which is then treated by the EMTP as if it were part of the overall electric network. The electromagnetic
torSue of the universal machine appears as a current source injection into the eSuivalent electric network.
Table 9.1 describes the eSuivalence between mechanical and electrical Suantities. For each mass on the
shaft system, a node is created in the eSuivalent electric network, with a
6CDNG ESuivalence between mechanical and electrical Suantities
9 11
Page 322
Mechanical Electrical
T (torSue acting on mass) [Nm] i (current into node) [#]
T (angular speed) [rad/s] v (node voltage) [V]
2 (angular position of mass) [rad] S (capacitor charge) [C]
J (moment of inertia) [kgm ] C (capacitance to ground) [F]
K (stiffness coefficient or spring [Nm/rad] 1/L (reciprocal or inductance) [1/H]

constant)
D (damping coefficient) [Nms/rad] 1/4 (conductance) [S]
(1 Nm 0.7375 lb ft 1 kgm 23.73 lb ft )
capacitor to ground with value J for the moment of inertia. If there is damping proportional to speed on this mass,
a resistor with conductance D is put in parallel with the capacitor (D in ES. ( .31)).
K If there is a mechanical torSue
acting on that mass (turbine torSue on generators, mechanical load on motors), a current source is connected to that
node (positive for turbine torSue, negative for load torSue). If there are two or more masses, inductors are used to
connect adjacent shunt capacitors, with their inductance values being eSual to 1/K (reciprocal of stiffness coefficient
of the shaft coupling between two masses). If there is damping associated with this shaft coupling, then the inductor
is paralleled with a resistor whose conductance value is D (D in ES. ( .31)). KThe

M electromagnetic torSue is
automatically added to the proper node as a current source by the EMTP.
Fig. 9. summarizes the eSuivalence between the mechanical and electric components. 4epresenting the
mechanical system by an eSuivalent electric network can provide more
Traduciendo...
9 12
Page 323
(KI ESuivalence between mechanical and electric components
flexibility than the built in model of the synchronous machine of Section . With this approach it should be easy to
incorporate gear boxes, distributed parameter modelling of rotors, etc. The EMTP further provides for up to three
9 13
Page 324
universal machines sharing the same mechanical system.
5VGCF[ 5VCVG 4GRTGUGPVCVKQP CPF +PKVKCN %QPFKVKQPU

Traduciendo...
The steady state representation of the ac type universal machine is based on the assumption that the network
to which it is connected is balanced and linear. Only positive seSuence Suantities are used in the initialization, and
negative and zero seSuence Suantities are ignored if there are unbalances. The initialization procedure could
obviously be extended to handle unbalanced conditions as well, along the lines discussed in Section , but this
extension has been given low priority so far.
6JTGG 2JCUG 5[PEJTQPQWU /CEJKPG
For three phase synchronous machine representations, any positive seSuence voltage source behind any
positive seSuence impedance can be used, as long as it produces the desired terminal voltages and currents when
solved with the rest of the network. For simplicity, a three phase symmetrical voltage source directly at the terminals
is used for the steady state solution. If the current (or active and reactive power output) from that solution is not
what the user wants, then the power flow iteration option of the EMTP can be used, which will iteratively adjust the
magnitude and angle of the three phase voltage source until the desired active and reactive power output (or some
other prescribed criteria) have been achieved. Once the terminal voltages and currents are known, the rest of the
electrical machine variables are initialized in the same way as described in Section . .1.
If the excitation system is represented by an electric network (rather than constant v ), then the EMTP H
performs a second ac steady state solution for the excitation systems of all universal machines, with the field currents
i Hbeing treated as current sources I cos(T t),H with THbeing an angular

H freSuency which is so low that i is dc for H
practical purposes. This trick is used because the EMTP cannot find an exact dc steady state solution at this time
(the network topology for dc solutions is different from that of ac steady state solution inductances would have to
be treated as closed switches, capacitances as open switches, etc.).
From the initialization of the electrical variables, the electromagnetic torSue T OGEJ IGPon the mechanical side
is known from ES. ( .32b) as well. These torSues are used as current sources i(t) T cos(T t)OGEJ
OGEJ IGP in the
eSuivalent networks which represent the mechanical systems of all universal machines, with T OGEJagain being an
angular freSuency so low that i(t) is practically dc. The EMTP then performs a third ac steady state solution for the
initialization of the mechanical system Suantities. Note that this three step initialization procedure is direct, and does
not reSuire either predictions or iterations.
6YQ 2JCUG 5[PEJTQPQWU /CEJKPG
#rmature currents in two phase machines with eSual amplitudes and displacements of 90E produce a rotating
magnetic field in the same way as symmetrical three phase armature currents displaced by 120E. #s long as this
condition is met (which is the balanced or positive seSuence condition for two phase machines), the initialization is
identical with the three phase case after proper conversion to d, S, 0 Suantities. If the phase Suantities are
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Page 325
K (V) ' *+* cos(TU V % ")

K (V) ' *+* cos(TU V % " & 90E) (9.12)
with T being
U the (synchronous) freSuency of the supply network, then the d, S, 0 Suantities are obtained with [S ]
and [P ] from ES. (9.9) with T T as U
KF ' *+* sin(" & *)

KS ' *+* cos(" & *) (9.13)
where * is the angle between the position of the Suadrature axis and the real axis of the ac phasor representation.
ES. (9.13) is indeed identical with ES. ( . 1) for the balanced three phase machine, except for a factor of %3/%2
there.
5KPING 2JCUG 5[PEJTQPQWU /CEJKPG
Converting a single phase armature current
K (V) ' *+* cos(TUV % ") (9.1 )
into d, S, 0 Suantities results in

Traduciendo...
1 1
KF(V) ' *+* sin(" & *) & *+* sin(2TUV % " % *)
2 2
KS ' 0 K' 0 (9.15)
with the first term being the dc Suantity analogous to the positive seSuence effect in three phase machines, and the
second double freSuency term analogous to the negative seSuence effect in ES. ( .53) in three phase machines. This
is a mathematical expression of the fact that an oscillating magnetic field in a single phase armature winding can be
represented as the sum of a constant magnetic field rotating forward at synchronous speed (angular speed 0
relative to field winding) and a constant magnetic field rotating backwards at synchronous speed (angular speed
2T relative to field winding).
Since only the first term in ES. (9.15) is used in the initialization now, the initial conditions are not totally
correct, and it may take many time steps before steady state is reached. The steady state torSue includes a pulsating
term very similar to Fig. .9 for the case of an unbalanced three phase synchronous machine. #s an alternative to
universal machine modelling, the three phase synchronous machine model of Section could be used for single phase
machines, by keeping two armature windings open circuited. Unfortunately, the initialization with negative seSuence
Suantities described in Section . .2 is not yet fully correct in the BP# EMTP either, as explained in the beginning
of Section . , though it has been implemented in an unreleased version of the UBC EMTP.
&% /CEJKPGU
The initialization of dc machine Suantities is straightforward, and follows the same procedure outlined in
9 15
Page 326
Section 9.5.1. In d, S, 0 Suantities, balanced three phase ac Suantities appear as dc Suantities. Therefore, there is
essentially no difference between the eSuations of a balanced three phase synchronous generator and a dc machine.
6JTGG 2JCUG +PFWEVKQP /CEJKPG
In balanced steady state operation, the angular speed T of the rotor (referred to the electrical side with ES.
( .25)) differs from the angular freSuency T of the supply

U network by the p.u. slip s,
TU & T
U' (9.1 )
TU
The network sees the induction machine as a positive seSuence impedance whose value depends on this slip s. The
negative and zero seSuence impedances are of no interest if the initialization is limited to balanced cases.
Fig. 9.5 shows the well known eSuivalent circuit for the balanced steady state behavior of a three phase
induction machine, which can be found in many textbooks. Its impedance can
(KI Conventional eSuivalent circuit for steady state behavior of

induction machines (subscript a for armature side, subscript r for rotor
side)
easily be calculated, and with the relationship between leakage, self and mutual inductances
.CC ' .ýC % /
) )
. (9.17a)
TT ' . ýT % /
becomes
Traduciendo...
(LTU/)
<RQU ' 4C % LTU.CC &
)
4 T )
(9.17b)
% LTU. TT
U
This single phase impedance is used in phases 1, 2, 3 for the steady state solution, provided there is only one
winding on the field structure (rotor).
For the general case of m windings on the field structure, the calculation is slightly more complicated. First,
91
Page 327
let us assume that the armature currents are
K (V) ' *+* cos(TUV % ")
K (V) ' *+* cos(TUV % " & 120E)
K (V) ' *+* cos(TUV % " & 2 0E)
in balanced operation. Transformed to d, S, 0 Suantities, the currents become
3
KF (V) ' *+* sin(UTUV % " & *)
2
3
KS(V) ' *+* cos(UTUV % " & *) (9.1 a)
2
KQ(V) ' 0
which can be represented as a phasor of slip freSuency sT , projectedUonto the S, d axes,
3
+SF ' +RJCUG G&L* (9.1 b)
2
with I *I*e being the L"

RJCUG (peak) phasor current in the ac network solution reference frame, and with the
understanding that
LUTUV
KS(V) ' 4G 6+SF G >
LUTUV
KF(V) ' +O 6+SF G > (9.1 c)
#ll d, S Suantities vary with the slip freSuency sT , and can

U therefore be represented as phasors in the same way as
the armature currents.
To obtain the impedance, the rotor currents must first be expressed as a function of armature currents.
Since all rotor voltages are zero, ES. (9.2a) can be rewritten as
F
0 ' & [4T ] [KT ] & [8T] (9.19)
FV
with
[8T ] ' [.TC] KC % [.TT][KT] (9.20)
from ES. (9.3a) (subscript r for rotor or field structure Suantities, and a for armature Suantities). Since there
is no saliency in three phase induction machines, ES. (9.19) and (9.20) are identical for the d and S axes, except
9 17
Page 328
C i inF one case, and i in theS other case. The submatrices [L ] and [L ] Traduciendo...
that i is are TC
obtained from
TT the matrix of ES.
(9.3a) by deleting the first row [L ] is the first

TC column and [L ] the m x m matrix
TT of what is left. If the rotor
windings are not shorted, but connected to an 4 L network, then [4 ] and [L ] must be
T modified
TTto include the
resistances and inductances of this connected network (for connected networks with voltage or current sources see
Section 9.5. ). Since [i ] and iT can both

C be represented as phasors with ES. (9.1 ), the flux in ES. (9.20) is also
a phasor which, after differentiation, becomes
F
[7T] ' LUTU [.TC] +SF % LUTU[.TT] [+T] (9.21)
FV
Inserting this into ES. (9.19) produces the eSuation which expresses the rotor currents as a function of the armature
current phasor,
[+T] ' & 6[4T] % LUTU [.TT]>& LUTU [.TC] +SF (9.22)
To obtain the direct axis rotor currents as complex phasor Suantities, use Im{I } on the right hand
S F side of ES.
(9.22), while the use of 4e{I } willSFproduce the Suadrature axis rotor currents.
The next step in the derivation of the impedance is the rewriting of the armature eSuations (9.1a) in terms
of phasor Suantities. Since
F
7SF ' LUTU 7SF
FV
ES. (9.1a) becomes
8SF ' & 4C +SF & LUTU 7SF & LT7SF
or with sT TUT fromUES. (9.1 ),
8SF ' & 4C +SF & LTU 7SF (9.23)
With the flux from the first row of ES. (9.3a)
7SF ' .CC +SF % [.CT] [+T ] (9.2 )
V
where [L ] CT
[L ] , andTC
with ES. (9.22), ES. (9.23) finally becomes
8SF ' & 6(4C%LTU.CC) & LTU[.CT] 6[4T] % LUTU[.TT]>& LUTU [.TC]> +SF
Therefore, the positive seSuence impedance is
<RQU ' 4C% LTU.CC& LTU[.CT] 6[4T] % LUTU[.TT]>& LUTU[.TC] (9.25)
If there is only one winding on the rotor, then it can easily be shown that the impedance of ES. (9.17b) is identical
91
Page 329
with that of ES. (9.25), by using the definitions of ES. (9.17a).
To summarize: The three phase induction machine is represented as three single phase impedances < from RQU
ES. (9.25) in the three phases 1, 2, 3. #fter the ac network solution of the complete network, the armature currents
are initialized with ES. (9.1 b), and the rotor currents with ES. (9.22). The calculation with ES. (9.22) is done
twice, with the imaginary part of I to obtain

SF the direct axis Suantities, and with the real part of I to obtain the SF
Suadrature axis Suantities.
#s mentioned before, the initialization works only properly for balanced cases at this time. If initialization
for unbalanced cases is to be added some day, then the procedures of Section . .2 and . .3 for the synchronous
machines should be directly applicable, because negative and zero seSuence currents see the field winding as short
circuits. Therefore, there is no difference between synchronous and induction machines in the negative and zero
seSuence initialization.
6YQ 2JCUG +PFWEVKQP /CEJKPG
#s already discussed in Section 9.5.2 for the two phase synchronous machine, the eSuations for balanced
Traduciendo...
operation of a two phase machine are identical on the d, S axes with those of the three phase machine. The only
difference is the missing factor %3/%2 in the conversion from phase Suantities to d, S Suantities.
5KPING 2JCUG +PFWEVKQP /CEJKPG
The problem is essentially the same as discussed in Section 9.5.3 for the synchronous machine. Only
positive seSuence values are used now, and the second term in
1 1
KF(V) ' *+* sin(UTU V % " & *) & *+* sin((TU % T)V % " % *) (9.2 )
2 2
is presently ignored.
&QWDN[ (GF +PFWEVKQP /CEJKPG
If the rotor (field structure) windings are connected to an external network with ac voltage and/or current
sources, then the EMTP will automatically assume that their freSuency is eSual to the specified slip freSuency sT,
and ignored the freSuency values given for these sources.
Feeding the rotor windings from sources reSuires two modifications to the procedure of Section 9.5.5. In
these modifications, it is assumed that the external network is represented by a Thevenin eSuivalent circuit, with
voltage sources [V ] behind

6JGX an impedance matrix [< ] defined at 6JGX
slip freSuency.
First, the rotor impedance matrix [4 ] jsT [LT ] must beUmodified

TT to include the external impedances,
OQF
[< (9.27)
TT ] ' [4T] % LUTU[.TT] % [<6JGX]
This modification must be done twice, for the direct axis Suantities and for the Suadrature axis Suantities. Since
9 19
Page 330
OQF
[< ]6JGX
is in general different for the two axes, [< TT ] is no longer the same on both axes.
Secondly, the left hand side of ES. (9.19) is no longer zero, but [V ]. This will change
6JGX ES. (9.22) into
OQF
[+T] ' & [< (9.2 )
TT ]& 6 [86JGX] % LUTU [.TC] +SF>
OQF
#gain, this calculation must be done twice. For the direct axis, use Im{I } and the directSFaxis values [< TT ] and
OQF
[V 6JGX
], for the Suadrature axis 4e{I } and Suadrature
SF axis values [< TT ] and [V ].6JGX
With these two modifications, the steady state model of the induction machine is no longer a passive
impedance < , but

RQUbecomes a three phase voltage source [E ] behind
UQWTEG three single phase impedance branches
OQF OQF
< (9.29)
RQU ' 4C % LTU.CC & LTU[.CT] [< TT ]& LUTU [.TC]
The voltage source is found by calculating the direct axis contribution,
OQF
'F ' LTU [.CT] [< (9.30a)
TT&F]& [86JGX&F]
and the Suadrature axis contribution,
OQF
'S ' LTU [.CT] [< (9.30b)
TT&S]& [86JGX&S]
and then transforming to phase Suantities,
2
'UQWTEG& ' GL* ('S % L'F )
3
(9.30c)
YKVJ 'UQWTEG& ' 'UQWTEG& @ G&L E
CPF 'UQWTEG& ' 'UQWTEG& @ G%L E
Once the ac steady state solution of the complete network has been obtained, the d, S, 0 armature currents
are initialized with ES. (9.1 b), while the rotor currents are initialized with ES. (9.2 ).
Traduciendo...
6TCPUKGPV 5QNWVKQP YKVJ %QORGPUCVKQP /GVJQF
For the transient solution with the compensation method, the machine differential eSuations (9.1) and (9.2)
are first converted to difference eSuations with the trapezoidal rule of integration. Then ES. (9.1) becomes
9 20
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XF(V) 4C 0 0 KF(V) 8F(V) &T(V)8S(V) JKUVF
XS(V) ' & 0 4C 0 KS(V) & 2 8S(V) % %T(V)8F(V) % JKUVS (9.31a)

)V
X C(V) 0 0 4C K C(V) 8 C(V) 0 JKUV C
with the history terms known from the preceding time step,
JKUVF XF(V&)V) 4C 0 0 KF(V&)V) 8F(V&)V)
JKUVS ' & XS(V&)V) & 0 4C 0 KS(V&)V) % 2 8S(V&)V) %

)V
JKUV C X C(V&)V) 0 0 4C K C(V&)V) 8 C(V&)V)
&T(V&)V)8S(V&)V)
%T(V&)V)8F(V&)V) (9.31b)
The field structure eSuations (9.2) on the direct axis become
X& (V) 4& K& (V) 8& (V) JKUV&
. . . . .
2
. '& . . & . % . (9.32a)
)V
. . . . .
X&O(V) 4&O K&O(V) 8&O(V) JKUV&O
with the known history terms
JKUV& X& (V&)V) 4& K& (V&)V) 8& (V&)V)
. . . . .
2
. '& . & . . % . (9.32b)
)V
. . . . .
JKUV&O X&O(V&)V) 4&O K&O(V&)V) 8&O(V&)V)
On the Suadrature axis, they are identical in form to ES. (9.32), except that subscripts D1,...Dm must be replaced
by 31,...3n. Finally for ES. (9.2c),
2
XQH(V) ' & 4QH KQH(V) & 8QH % JKUVQH (9.33a)
)V
with
9 21
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Traduciendo...
2
JKUVQH ' & XQH(V&)V) & 4QH KQH(V&)V) & 4QH KQH(V&)V) % 8QH(V&)V) (9.33b)
)V
#s explained in Section 12.1.2, the network connected to the armature side of the machine can be
represented by the instantaneous Thevenin eSuivalent circuit eSuation
X (V) X& K (V)
X (V) ' X& % 4GSWKX K (V) (9.3 )
X (V) X& K (V)
with a sign reversal for the current compared to Section 12.1.2, to change from a load to source convention.
Similarly, if external networks are connected to the field structure windings, they will also be represented by
Thevenin eSuivalent circuits with eSuations of the form
X& (V) X& & K& (V)
. . .
. ' . % 4&&GSWKX . (9.35a)
. . .
X&O(V) X&O& K&O(V)
X3 (V) X3 & K3 (V)
. . .
. ' . % 43&GSWKX . (9.35b)
. . .
X3P(V) X3P& K3P(V)
and
XQH (V) ' XQH&Q % 4QH&GSWKX KQH(V) (9.35c)
The external network connected to the first three field structure windings is represented by a three phase
Thevenin eSuivalent circuit (Section 12.1.2.3), whereas the external networks connected to the rest of the field
structure windings are represented by single phase Thevenin eSuivalent circuits (Section 12.1.2.1). This limitation
results from the fact that the BP# EMTP could handle M phase Thevenin eSuivalent circuits only for M # 3 at the
time the Universal Machine was first implemented. In practice, this limitation should not cause any problems
because the field structure windings are usually connected to separate external networks. #n exception is the three
phase wound rotor of induction machines, which is the reason why a three phase eSuivalent circuit was chosen for
the first three rotor windings.
The solution of the machine eSuations is then roughly as follows:
9 22
Page 333
(1) Solve the complete network without the universal machines. Extract from this solution the Thevenin
eSuivalent open circuit voltages of ES. (9.3 ) and (9.35), as well as the open circuit voltages of the network
which represents and mechanical system.
(2) Predict the rotor speed T(t) with linear extrapolation.
(3) Transform ES. (9.3 ) from phase to d, S, 0 Suantities with ES. (9.7) if the armature windings are ac
windings,
XF XF&Q KF
XS ' XS&Q % 4RJCUG&GSWKXKS (9.3 a)
XQC XQC&Q KQC
where Traduciendo...
XF&Q X &Q
XS&Q ' [6]& X &Q CPF [4RJCUG&GSWKX] ' [6]& [4GSWKX] [6] (9.3 b)
XQC&Q X &Q
For dc armature windings, the Thevenin eSuivalent circuit is already in the form of ES. (9.3 a) without
transformation.
() Substitute ES. (9.3 a) into ES. (9.31a), and substitute ES. (9.35) into ES. (9.32). This eliminates the
voltages as variables. Then solve the resulting linear eSuations for the m n currents by Gauss
elimination, after the fluxes are first replaced by linear functions of currents with ES. (9.3). Using the star
circuit of Fig. 9.1 instead of the more general inductance matrix of ES. (9.3) simplifies this solution process
somewhat.
(5) Calculate the electromagnetic torSue on the electrical side,
6GN(V) ' KS(V) 8F(V) & KF(V) 8S(V) (9.37)
and convert it to T (t) on the mechanical side with ES. ( .25) if the mechanical system is not modelled
OGEJ
as a one pole pair machine. Use T (t) as a current

OGEJ source in the Thevenin eSuivalent circuit which
represents the mechanical system and solve it to obtain the speed (as an eSuivalent voltage). Up to 3
universal machines can share the same mechanical system, because the EMTP uses an M phase
compensation method for M # 3 (see Section 12.1.2.3).
() If the speed calculated in (5) differs too much from the predicted speed, then return to step (3). Otherwise:
(7) Update the history terms of ES. (9.31b), (9.32b) for d and S axes, and (9.33b) for the next time step.
() Transform the armature currents from d, S, 0 Suantities to phase Suantities with ES. (9.7) (only if the
windings are ac windings).
(9) Find the final solution of the complete network by super imposing the effects of the armature currents, of
9 23
Page 334
the field structure currents (if they have externally connected networks) and of the current representing the
electromagnetic torSue in the network for the mechanical system, with ES. (12. ) of Section 12.
(10) Proceed to the next time step.
Since the variables of the mechanical system usually change much slower than the electrical variables,
because of the relatively large moment of inertia of practical machines, the prediction of the speed is fairly good.
#s a conseSuence, the number of iterations typically lies between 1 and 3.
Interfacing the solution of the machine eSuations with the solution of the electric network through
compensation offers the advantage that the iterations are confined to the machine eSuations only. Furthermore, if
a small tolerance is used for checking the accuracy of the speed, the solution is practically free of any interfacing
error.
The only limitation of the compensation method is the fact that the universal machines must be separated
from each other, and from other compensation based nonlinear elements, through distributed parameter lines with
travel time. Stub lines can be used to introduce such separations artificially, but such stub lines create their own
problems. Because of this limitation, a second solution option has been developed, as described in the next section.
6TCPUKGPV 5QNWVKQP YKVJ #TOCVWTG (NWZ 2TGFKEVKQP
In the transient solution of the synchronous machine of Section , essentially voltage sources behind
resistances 4 and
C average subtransient inductances (L L )/2 are Fused, with
S the trapezoidal rule applied to the
inductance part. The voltage sources contain predicted currents and the predicted speed.
The prediction based interface option for the universal machine also uses voltage sources with elements of
prediction in them, but just behind resistances 4 , with no

C inductance part (Fig. 9. ). If we think of 4 as belonging C
to the electric network and not to the machine, then ES.
Traduciendo...
(KI Thevenin eSuivalent

circuit for universal machine
(9.1) becomes a simple relationship between armature voltages and fluxes,
XF 8F &T8S
XS ' & F 8S
% %T8F (9.3 )
FV
XQC 8QC 0
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The fluxes always change smoothly, in contrast to the voltages which can suddenly jump in case of short circuits.
Therefore, the fluxes are chosen as variables suitable for prediction. Furthermore, the fluxes 8 , 8 of induction F S
machines vary sinusoidally with slip freSuency during steady state operation, whereas the fluxes seen from a
synchronously rotating reference frame (rotating at the supply freSuency T ) would remain
U constant. Because of this,
the fluxes seen from a synchronously rotating reference frame are predicted, rather than 8 , 8 . This reSuires aF S
transformation of ES. (9.3 ) from the d, S axes to the synchronously rotating reference frame [1 0]. #lternatively,
one can forget about the original transformation from phase Suantities to the d, S axes altogether, and transform the
phase Suantities directly to the ds, Ss axes of the synchronously rotating reference frame. That means that d$/dt
T must be replaced by T , which

U leads to
XFU 8FU &TU8SU
XSU ' & F 8SU

% %TU8FU (9.39)
FV
XQC 8QC 0
The only difference with ES. (9.3 ) is the replacement of rotor speed T by the ac supply freSuency T . This simple U
change works only for the voltage eSuations for the flux current relationships the synchronously rotating reference
frame cannot be used because that would make the inductances time dependent rather than constant.
The fluxes 8 , 8FU, 8SUonQCthe synchronously rotating axes are now predicted linearly,
&8FU&RTGF 8FU(V&)V) 8FU(V&2)V)
8SU&RTGF ' 2 8SU(V&)V) & 8SU(V&2)V) (9. 0)
8QC&RTGF 8QC(V&)V) 8QC(V&2)V)
and the backward Euler method (see #ppendix I.9) is then applied to ES. (9.39),
XFU(V) 8FU&RTGF & 8FU(V&)V) &TU8SU&RTGF

XSU(V) ' & 1 8SU&RTGF & 8SU(V&)V)% %TU8FU&RTGF (9. 1)
)V
XQC(V) 8QC&RTGF & 8QC(V&)V) 0
With all Suantities on the right hand side known (either from the preceding time step or from prediction), the terminal
voltages are now known, too, and can be transformed back to phase Suantities with
X (V) cos(TUV) sin(TUV) 0 XFU(V)
X (V) ' &sin(TUV) cos(TUV) 0 XSU(V) (9. 2)
X (V) 0 0 1 XQC(V)
The representation of the universal machine as three voltage sources v (t), v (t), v (t) behind resistances 4 C
9 25
Traduciendo...
Page 336
in the complete electric network is only used on the armature side, whereas compensation based interfaces are still
maintained for the field structure windings and for the mechanical system. With this in mind, the solution process
works roughly as follows:
(1) With the universal machine represented as voltage sources behind 4 (implementedC as current sources in
parallel with 4 inCthe EMTP), solve the complete electric network. Extract from this solution the Thevenin
eSuivalent open circuit voltages of ES. (9.35) if there are any external networks connected to the field
structure windings (see Section 9. for details about three phase compensation on the first three windings,
and single phase compensation on the rest). Extract as well the open circuit voltages of the network which
represents the mechanical system.
(2) Execute steps (2) to (9) of the compensation based procedure described in the preceding Section 9. , except
that the armature currents i , i , i (and i , i , i after

F Stransformation
QC with [T ]) are now known from step
(1) and used directly in place of the Thevenin eSuations (9.3 ) for the armature part. The calculations for
the other parts remain unchanged.
(3) 4otate the armature fluxes 8 , 8 , 8Ffrom

S theQC
d, S axes to the synchronously rotating ds, Ss axes
8FU cos(TUV&$) &sin(TUV&$) 0 8F
8SU ' sin(TUV&$) cos(TUV&$) 0 8S (9. 3)
8QC 0 0 1 8QC
and use them to predict the voltage sources for the next time step with ES. (9. 0) to (9. 2). Note that no
predictions for the speed and angle are needed here.
() Proceed to (1) to find the solution at the next time step.
Experience has shown [1 0] that this interfacing option is as accurate as the compensation based interface
of Section 9. . It also reSuires less computation time. Its numerical stability can be partly attributed to the backward
Euler method in ES. (9. 1). #s shown in #ppendix I.9, the backward Euler method is identical to the trapezoidal
rule of integration with critical damping, and is therefore absolutely numerically stable. However, ES. (9. 1)
involves predictions as well, and the comparison is therefore not completely correct.
5CVWTCVKQP
Saturation effects are only represented for the main flux (M in Fig. 9.1),
F except for the special induction
machine model of Ontario Hydro, which includes saturation effects in the leakage fluxes as well.
The saturation curve of the universal machine is approximated as two piecewise linear segments for the d
axis, the S axis, or for both (Fig. 9.7). By using the star circuit of Fig. 9.1,
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Page 337
Traduciendo...
(KI Piecewise linear inductance
the piecewise linear representation can easily be implemented. Whenever the flux lies above the knee point value
8 MPGG
, the relationship of ES. (9. b) in the form of
8OF ' /F KOF (9. a)
is simply replaced by
8OF ' 8UCV % /F&UCV KOF (9. b)
on the direct axis, and analogous on the Suadrature axis.
4esidual flux can be represented as well. In that case, the characteristic of Fig. 9. is used. If the absolute
value of the flux is less than 8 , then

TGUKFWCN the M branch
F is open circuited,
KOF ' 0 KH *8OF* 8TGUKFWCN (9. 5a)
8OF ' 8TGUKFWCN % /F KOF KH 8TGUKFWCN # *8OF* # 8MPGG (9. 5b)
and
8OF ' 8UCV % /F&UCV KOF KH *8OF* 8MPGG (9. 5c)
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(KI 4esidual flux
T h e
decoupled approach of d and S axis saturation works reasonably well for salient pole synchronous machines and
for dc machines with a definite field coil in one axis. However, when both the armature and field structures are
round with no pronounced saliency, as in most induction machines and in round rotor synchronous machines, then
this decoupled approach leads to unacceptable results. Therefore, a total saturation option is available, which uses
a solution method very similar to that discussed in Section . .
Traduciendo...
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59+6%*'5
#ny switching operation in a power system can potentially produce transients. For the simulation of such
transients, it is necessary to model the various switching devices, such as
circuit breakers,
load breakers,
dc circuit breakers,
disconnectors,
protective gaps,
thyristors, etc.
So far, all these switching devices are represented as ideal switches in the EMTP, with zero current (4 4) in the
open position and zero voltage (4 0) in the closed position. If the switch between nodes k and m is open, then
both nodes are represented in the system of nodal eSuations, whereas for the closed switch, both k and m become
one node (Fig. 10.1). It is
(a) open (b) closed
(KI 4epresentation of switches in the EMTP
possible, of course, to add other branches to the ideal switch, to more closely resemble the physical behavior, e.g.,
to add a capacitance from k to m for the representation of the stray capacitance or the 4 C grading network of an
actual circuit breaker. The characteristics of the arc in the circuit breaker are not yet modelled, but work is in
progress to include them in future versions.
Switches are not needed for the connection of voltage and current sources if they are connected to the
network at all times. The source parameters T 56#46 and T 5612 can be used in place of switches to have current
sources temporarily connected for T 56#46 #t#T 5612 , as explained in Section 7. For voltage sources, this definition
would mean that the voltage is zero for t T 56#46 and for t T 5612 , which implies a short circuit rather than a
disconnection. Therefore, switches are needed to disconnect voltage sources.
Switches are also used to create piecewise linear elements, as discussed in Section 12.
$CUKE 5YKVEJ 6[RGU

Traduciendo...
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There are five basic switch types in the EMTP, which are all modelled as ideal switches. They differ only
in the criteria being used to determine when they should open or close.
6KOG %QPVTQNNGF 5YKVEJ
This type is intended for modelling circuit breakers, disconnectors, and similar switching devices, as well
as short circuits. The switch is originally open, and closes at T %.15' . It opens again after T 12'0 (if t ), either
OCZ
as soon as the absolute value of the switch current falls below a user defined current margin, or as soon as the
current goes through zero (detected by a sign change), as indicated in Fig. 10.2 For the simulation of circuit
breakers, the latter criterion for opening should normally be used. The time between closing and opening can be
delayed by a user defined time delay.
(a) current going through zero (b) current less than margin
(KI Opening of time controlled switch
The closing takes place at the time step nearest to T %.15' in the UBC version (Fig. 10.3(a)), and at the time
step where t $ T %.15' for the first time in the BP# version (Fig. 10.3(b)).
(a) UBC version (b) BP# version
(KI Closing of time controlled switch
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T %.15' 0 signals to the EMTP that the switch should be closed from the very beginning. If the simulation
starts from automatically calculated ac steady state conditions, then the switch will be recognized as closed in the
steady state phasor solution.
The BP# EMTP has an additional time controlled switch type (T#CS controlled switch type 13), in which
the closing and opening action is controlled by a user specified T#CS variable from the T#CS part of the EMTP.
With that feature it is easy to build more complicated opening and closing criteria in T#CS.
Traduciendo...
)CR 5YKVEJ
This switch is used to simulate protective gaps, gaps in surge arresters, flashovers across insulators, etc.
It is always open in the ac steady state solution. In the transient simulation, it is normally open, and closes as soon
as the absolute value of the voltage across the switch exceeds a user defined breakdown or flashover voltage. For
this checking procedure, the voltage values are averaged over the last two time steps, to filter out numerical
oscillations. Opening occurs at the first current zero, provided a user defined delay time has already elapsed. This
close open cycle repeats itself whenever the voltage exceeds the breakdown or flashover voltage again, as indicated
in Fig. 10.
(KI 4epetition of close open operation for gap switch
It is well known that the breakdown voltage of a gap or the flashover voltage of an insulator is not a simple
constant, but depends on the steepness of the incoming wave. This dependence is usually shown in the form of a
voltage time characteristic (Fig. 10.5), which can be measured in the laboratory for standard impulse waveshapes.
Unfortunately, the waveshapes of power system transients are usually very irregular, and voltage time characteristics
can seldom be used, therefore. #nalytical methods based on the integration of a function
V
F' (v(t) & v )M dt (10.1)
mV
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(KI Voltage time characteristic of a gap
could easily be implemented. In ES. (10.1), v and k are Qconstants, and breakdown occurs at instant t where the
integral value F becomes eSual to a user defined value [ ]. For k 1, this is the eSual area criterion of D. Kind
[172]. Neither the voltage time characteristic nor ES. (10.1) has been implemented so far.
The BP# EMTP has an additional gap switch type (T#CS controlled switch type 12), in which the
breakdown or flashover is controlled by a firing signal received from the T#CS part of the EMTP (Section 13).
With that feature, voltage time characteristics or criteria in the form of ES. (10.1) can be simulated in T#CS by
skilled users.
&KQFG 5YKVEJ
This switch is used to simulate diodes where current can flow in Traduciendo...
only one direction, from anode m to
cathode k (Fig. 10. ). The diode switch closes whenever v $ v (voltage

O values
M averaged over two successive time
steps to filter out numerical oscillations), and opens after the elapse of a user defined time delay as soon as the
current i becomes
OM negative, or as soon as its magnitude becomes less than a user defined margin.
(KI Diode switch
In the ac steady state solution, the diode switch can be specified as either open or closed.
6J[TKUVQT 5YKVEJ 6#%5 %QPVTQNNGF
This switch is the building block for HVDC converter stations. It behaves similarly to the diode switch,
except that the closing action under the condition of v $ v onlyO takesMplace if a firing signal has been received from
the T#CS part of the EMTP (Section 13).
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/GCUWTKPI 5YKVEJ
# measuring switch is always closed, in the transient simulation as well as in the ac steady state solution.
It is used to obtain current, or power and energy, in places where these Suantities are not otherwise available.
The need for the measuring switch arose because the EMTP does not calculate currents for certain types
of branches in the updating procedure inside the time step loop. These branches are essentially the polyphase coupled
branches with lumped or distributed parameters. The updating procedures could be changed fairly easily to obtain
the currents, as an alternative to the measuring switch.
5VCVKUVKECN &KUVTKDWVKQP QH 5YKVEJKPI 1XGTXQNVCIGU
Since circuit breakers can never close into a transmission line exactly simultaneously from both ends, there
is always a short period during which the line is only closed, or reclosed, from one end, with the other end still open.
Travelling waves are then reflected at the open end with the well known doubling effect, and transient overvoltages
of 1 p.u. at the receiving end are therefore to be expected. In reality, the overvoltages can be higher for the
following reasons:
(a) the line is three phase with three different mode propagation velocities,
(b) the network on the source side of the circuit breaker may be fairly complicated, and can therefore
create rather complicated reflections,
(c) the line capacitance may still be charged up from a preceding opening operation ( trapped charge
in reclosing operations),
(d) the magnitude of the overvoltage depends on the instant of closing (point on waveshape),
(e) the three poles do not close simultaneously (pole spread).
In the design of transmission line insulation, it would make little sense to base the design on the highest
possible switching surge overvoltage, because that particular event has a low probability of ever occurring, and
because the line insulation could not be designed economically for that single high value. Furthermore, it is
impossible or very difficult to know which combination of parameters would produce the highest possible
overvoltage. Instead, 100 or more switching operations are usually simulated, with different closing times and
possibly with variation of other parameters, to obtain a statistical distribution of switching surge overvoltages. This
is usually shown in the form of a cumulative freSuency distribution (Fig. 10.7).
Traduciendo...
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(KI Cumulative freSuency distribution of receiving end

overvoltages from 100 digital computer and TN# simulations [1 ].
4eprinted by permission of CIG4E
For the left most curve in Fig. 10.7, an overvoltage of 1. p.u. or higher would have to be expected in 5
of the switching operations. Insulation design for withstanding a certain overvoltage often refers to a 2 probability.
The withstand voltage of insulators does not only depend on the peak value, but on the waveshape as well. For
irregular waveshapes, as they occur in switching surges, it is very difficult to take the waveshape into account, and
it is therefore usually ignored.
The BP# EMTP has special switch types for running a large number of cases in which the opening or
closing times are automatically varied. The output includes statistical overvoltage distributions, e.g., in the form
of Fig. 10.7. There are two types, one in which the closing times are varied statistically, and the other in which they
are varied systematically. How well these variations represent the true behavior of the circuit breaker is difficult to
say. Before the contacts have completely closed, a discharge may occur across the gap and create electrical closing
slightly ahead of mechanical closing ( prestrike ). There is very little data available on prestrike values, however.
10
Page 345
5VCVKUVKEU 5YKVEJ
The closing time T %.15' of each statistics switch is randomly varied according to either a Gaussian (normal)
distribution, or a uniform distribution, as shown in Fig. 10. . #fter each variation, for all such switches, the case
is rerun to obtain the peak overvoltages. The mean closing time T and the standard deviation F are specified by the
Traduciendo...
user. In addition to closing time variations of each individual switch, a random delay can be added, which is the
same for all statistics switches, and which always follows a uniform distribution.
(KI Probability distribution for the closing time T %.15' of the statistics switch. f(T)
density function, F(T) cumulative distribution function
There is also an option for dependent slave switches, in which the closing time depends on that of a master
switch,
T%.15'&UNCXG ' T%.15'&OCUVGT % TTCPFQO (10.2)
with
T %.15' OCUVGT statistically determined closing time of a master statistics switch,
T TCPFQO random time delay defined by a mean time and standard deviation.
This slave switch may in turn serve as a master switch for another slave switch. Slave switches are usually used to
model circuit breakers with closing resistors. The first contact to close would be the master switch, with the next
one or more contacts to close being slave switches.
Statistics switches can also be used for random openings, instead of closings, but this option is less
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important. In realistic simulations, the current interruption only occurs at the first current zero after T 12'0 , and there
are only a few combinations of phase seSuences in which the three poles of a three phase circuit can interrupt. It
may be just as easy to simulate these combinations directly, rather than statistically.
5[UVGOCVKE 5YKVEJ
Each systematic switch has its closing time systematically varied, from T to T in eSualOKP
increments
OCZ of
)T. If this is done for the three poles of a three phase circuit breaker, it can result in a very large number of cases
which have to be run automatically, as indicated in Fig. 10.9.
Traduciendo...
(KI Three dimensional space for three closing times T %.15' # ,T %.15' $ ,
T%.15' %
#gain, there is an option for dependent slave switches, in which the closing time is
T%.15'&UNCXG ' T%.15'&OCUVGT % T1((5'6 (10.3)
where T 1((5'6 is now a constant, rather than a random variable as in ES. (10.2). #s in the case of statistics switches,
slave switches are used to model the second (or third,...) contact to close in circuit breakers with closing resistors.
Slave switches do not increase the dimension of the vector space shown in Fig. 10.9 for three master switches.
5QNWVKQP /GVJQFU HQT 0GVYQTMU YKVJ 5YKVEJGU
There is more than one way of handling changing switch positions in the transient solution part of the
EMTP. For the ac steady state solution part, the problem is simpler, because the eSuations are only solved once.
In that case, it is best to use 2 nodes for open switches, and 1 node for closed switches, as shown in Fig. 10.1.
10
Page 347
In some programs, the switch is represented as a resistance 4, with a very large value if the switch is open
and a very small value if the switch is closed. #s explained in Section 2.1, very large values of 4 do not cause
numerical problems in solution methods based on nodal eSuations, but very small values can cause numerical
problems. This approach was therefore not chosen for the EMTP. The calculation of the switch current is trivial
in this approach, with
KMO ' (XM & XO ) /4 (10. )
The compensation method described in Section 12.1.2 provides another approach for handling switches.
To represent M switches, an M phase Thevenin eSuivalent circuit would be precomputed with an eSuation of the
form
[XM] & [XO] ' [XM& ] & [XO&Q] & [46JGX] [KMO] (10.5)
The switch currents, which are needed for the superposition calculation (ES. (12. ) in Section 12.1.2), are simply
[i MO
] 0 if all switches are open or
[KMO] ' [46JGX]& {[XM& ] & [XO& ] } (10. )
if all switches are closed. If only some switches are closed, then [4 ] in ES. (10.6JGX
) is a submatrix obtained from
the full matrix after throwing out the rows and columns for the open switches. The switch currents are automatically
obtained in this approach, and there should not be any numerical problems. The compensation based method is not
used in the EMTP now, though it may be chosen in future versions for the inclusion of arc characteristics. It was
used in a predecessor version of the EMTP developed by the author in Munich. The treatment of switches in the
UBC EMTP, as discussed next in Section 10.3.1, is essentially the same as the compensation based method, even
though the programming details are different.
# third approach is to change the network connections whenever a switch position changes. #s indicated
in Fig. 10.1, there are two nodes whenever the switch is open, and only a single node whenever the switch is closed.
This approach has been implemented in the EMTP, in two different ways.
0GVYQTM 4GFWEVKQP VQ 5YKVEJ 0QFGU
In the UBC EMTP, and in an older version of the BP# EMTP, nodes which have switches connected are
eliminated last, as indicated in Fig. 10.10. Before entering the time step
Traduciendo...
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(a) initial reduction (b) completion of trian

gularization whenever
switches change position
(KI Matrix reduction for nodes with switches
loop, normal Gauss elimination is used on those nodes with unknown voltages (subset #) which do not have switches
connected to them. For the rest of the nodes of subset # with switches, the Gauss elimination is stopped at the
vertical line which separates the non switch nodes from the switch nodes. This creates the reduced matrix illustrated
in Fig. 10.10(a). #ll switches are assumed to be open in this calculation.
Whenever a switch position changes in the time step loop, this reduced matrix is first modified to reflect
the actual switch positions. If the switch between nodes k and m is closed, then the two respective rows and columns
are added to form one new row and column using the higher node number between k and m, and the other row and
column for the lower node number is discarded. If the switch is open, no changes are made in the reduced matrix.
#fter these modifications, the triangularization is completed for the entire matrix of subset #, as indicated in Fig.
10.10(b). In repeat solutions, the addition of rows for closed switches must be applied to the right hand sides as
well. In the backsubstitution, the voltage of the discarded lower node number is set eSual to the voltage of the
retained higher node number.
Using this reduced matrix scheme has the advantage that the triangularization does not have to be done again
for the entire matrix whenever switch positions change. Instead, re triangularization is confined to the lower part.
This scheme works well if the network contains only a few switches. If there are many switches, as in HVDC
converter station simulations, then this method becomes less and less efficient, and straightforward re
triangularization may then be the best approach, as described in Section 10.3.2. When the method was first
programmed, only two rows and columns could be added. This has led to the restriction that a node with unknown
voltage can only have one switch connected to it in this scheme, because two closed switches connected to one node
would reSuire the addition of three rows and columns (to collapse three nodes into one). This restriction no longer
applies to newer BP# versions which use the method of Section 10.3.2.
The current calculation for closed switches in the time step loop uses the row of either node k or m in the
reduced matrix (where the switch was assumed to be open) after the right hand sides have been modified by the
downward operations with the upper part of the triangular matrix. In effect, this sums up the currents through the
branches connected to k or m, which must be eSual to the switch current. In the ac steady state solution, the switch
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Page 349
currents are no calculated at all, but simply set to zero at t 0. This is obviously incorrect, but the values will be
correct at )t, 2)t,...t . OCZ Traduciendo...
%QORNGVG 4G 6TKCPIWNCTK\CVKQP
In newer versions of the BP# EMTP, the reduction scheme discussed in the preceding section is no longer
used. Instead, the matrix is built and triangularized completely again whenever switch positions change, or when
the slope of piecewise linear elements changes. The current is calculated from the original row or either node k or
m, with all switches open, with the proper right hand side.
With this newer scheme, any number of switches can be connected to any node, as long as the current in
each switch is uniSuely defined. # delta configuration of closed switches, or two closed switches in parallel, would
therefore not be allowed. #lso, a switch cannot connect two voltage sources together, which is unrealistic anyhow
because it would create an infinite current. The switch currents are now calculated in the ac steady state solution
as well, and switch currents are therefore correct at all times, including at t 0.
5YKVEJ %NQUKPI
When the EMTP prints a message that a switch is closed after T seconds, T will always be an integer
multiple of )t, because the EMTP cannot handle variable step sizes so far. The actual closing time T will therefore
differ somewhat from the user specified time T %.15' , as explained in Fig. 10.3.
The network will already have been solved, with the switch still open, when the decision is made to close
the switch at time T. #s shown in Fig. 10.11, all voltages and currents at t T are therefore the preclosing
values. #fter the network solution at t T, the matrix is rebuilt and re triangularized for the closed switch position,
and in the transition from T to T )t, it is assumed that all variables change linearly with finite slope, rather than
abruptly.
(KI Switch closing or opening at time T
In many cases, the linear transition with a finite slope indicated in Fig. 10.11 is a reasonable assumption. For
example, if the voltage v were the voltage across a capacitor, then v could not change abruptly anyhow. On the other
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hand, if it were the voltage across an inductance it could indeed jump, as indicated by the dotted line in Fig. 10.11.
Such voltage jumps are very common in HVDC converter stations. The exact method for handling such jumps would
be the addition of a second post change solution at T after the pre change solution at T , without advancing in
time. #s explained in #ppendix II, methods are now known to re initialize at T , but they have not yet been
implemented in the EMTP.
5YKVEJ 1RGPKPI
The treatment of switch opening in the solution is similar to that of switch closing. #gain, the network will
already have been solved, with the switch still closed, when the decision is made to open the switch at time T. To
explain the transition from T to T )t, Fig. 10.11 can again be used: all voltage and currents at t T will be the
pre change values, and after these values have been obtained, the matrix will be rebuilt and re triangularized for
the post change configuration. #ll variables are then assumed to vary linearly rather than abruptly in the transition
from T to T )t.
#s already explained in Section 2.2.2,, not re initializing the variables at T with a second post change
solution creates numerical oscillations in the voltages across inductances. They can be prevented with the re
Traduciendo...
initialization method of #ppendix II, which has not yet been implemented in the EMTP, or with the damping
resistances discussed in Section 2.2.2. For many years it was thought that the numerical oscillations occur only
because the current is never exactly zero when the switch opens, with a residual energy L()i) /2 left in the
inductance. It is now known that they also occur if )i 0. Decreasing )t will not cure the oscillations either.
There are cases where the numerical oscillation, in place of the correct sudden jump, can serve as an
indicator of improper modelling. #n example is transient recovery voltage studies, where a sudden jump in voltage
would indicate that the proper stray capacitances are missing from the model. Fig. 10.12 shows a simple example:
both switches I and II in the network of Fig. 10.12(a) are closed at t 0 to charge the capacitor. Switch II opens
when the capacitor is charged up and when the current is more or less zero. Fig. 10.12(b) shows the numerical
oscillations in the voltage v on the feeding network side. By adding a stray capacitance to the left side of the switch,
as illustrated in Fig. 10.12(c), the transient recovery voltage on the feeding side would no longer have the unrealistic
jump, as shown in Fig. 10.12(d).
#TE 2JGPQOGPC KP %KTEWKV $TGCMGTU
When the contacts of a circuit breaker open, they draw an electric arc which maintains the current flow until
interruption takes place at current zero. In high voltage circuit breakers, the arc resistance is negligibly small if
normal load currents or high short circuit currents are
10 12
Page 351
(a) network
(b) voltage across inductance
Traduciendo...
(c) modified network
10 13
Page 352
(d) voltage across inductance
(KI Capacitor charging and discharging
interrupted. In the interruption of small inductive currents (e.g., in switching off an unloaded transformer), the arc
resistance is higher because of the falling arc characteristic, and may be important in deciding whether current
interruption is successful or not. Immediately after current interruption, a transient recovery voltage builds up across
the contacts, which can lead to reignition if it exceeds the dielectric strength which re appears as the gap between
the contacts is being de ionized.
There is no circuit breaker arc model in the EMTP now, but work is in progress to add one. Static arc
models are not good enough, and differential eSuations describing the arc must be used instead. Most experts
working on current interruption problems use a modification of an eSuation first proposed by Mayr, of the form
FI 1 K
' &I (10.7)
FV J(I) 2(I)
where
g arc conductance,
i arc current,
J(t) conductance dependent time constant,
P(g) conductance dependent heat dissipation.
The parameters J(t) and P(g) are dependent on the characteristics of the particular circuit breaker. # detailed
investigation into the usefulness of various arc eSuations is presently being done by CIG4E Working Group 13.01
10 1
Page 353
Traduciendo...
( Practical #pplication of #rc Physics in Circuit Breakers ).
If high freSuency oscillations develop in the arc current prior to interruption, as they sometimes do in
switching off small inductive currents or in other current chopping situations, then reignition may occur within 1/
cycle after current interruption (the term restrike is used to describe resumption of current conduction if it occurs
1/ cycle or longer after current interruption, which most likely occurs in the interruption of capacitive currents).
For deciding whether reignition occurs, the arc eSuation of ES. (10.7) cannot be used. Instead, the transient
recovery voltage is compared again the dielectric strength, which increases as a voltage is compared against the
dielectric strength, which increases as a function of time, and if it exceeds it, then reignition occurs. For the
breakdown itself, Toepler s eSuation can be used, which is of the form [173]
1 V
I' K(W) FW (10. )
MU m
where
k constant,
s gap spacing
i current in gap (starting from an extremely small value).
v voltage across gap.
10 15
Page 354
574)' #44'56'45 #0& 2416'%6+8' )#25
To protect generators, transformers, cables, SF basses, and other devices against levels of overvoltages
which could permanently destroy their non self restoring insulation, surge arresters are installed as close as possible
to the protected device. Short connections are important to avoid the doubling effect of travelling waves on open
ended lines, even if they are short busses. Surge arresters have normally not been used for the protection of
transmission lines, because one can easily recover from insulator flashovers with fast opening and reclosing of circuit
breakers (self restoring insulation). Some utilities are studying the possibility of using surge arresters on
transmission lines, too, to limit switching surge overvoltages.
Protective gaps are seldom used nowadays, except in the protection of series capacitor stations.
2TQVGEVKXG )CRU
Traduciendo...
Protective gaps are crude protection devices. They consist of air gaps between electrodes of various shapes.
Examples are horns or rings on insulators and bushings, or rod gaps on or near transformers. They do protect
against overvoltages by collapsing the voltage to practically zero after sparkover, but they essentially produce a short
circuit which must then be interrupted by circuit breakers. #lso, their voltage time characteristic (Fig. 10.5) rises
steeply for fast fronts, which makes the protection against fast rising impulses Suestionable.
Protective spark gaps are still used to protect series capacitors. There, the sparkover does not increase the
transmission line current, but actually reduces it because the line impedance increases when the series capacitor is
by passed. Since the spark gap is unable to interrupt the current, a by pass circuit breaker must be closed to
extinguish the arc in the spark gap (Fig. 11.1). This by pass breaker must be opened again if the series capacitor
is to be re inserted. In the future, protective spark gaps may be replaced by metal oxide surge arresters.
(KI Series capacitor protection scheme
11 1
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Protective gaps are simulated in the EMTP with the gap switch discussed in Section 10.1.2.
5WTIG #TTGUVGTU
There are two basic types of surge arresters, namely silicon carbide surge arresters, and metal oxide surge
arresters. Until about 10 years ago, only silicon carbide arresters were used, but the metal oxide arrester is Suickly
replacing the older type to the extent that some manufacturers produce only metal oxide arresters now.
5KNKEQP %CTDKFG 5WTIG #TTGUVGT
Silicon carbide arresters consist of a silicon carbide resistor with a nonlinear v i characteristic, in series with
a spark gap (Fig. 11.2). The spark gap connects the arrester to the system when the overvoltage exceeds the
sparkover voltage, and the resistor limits the follow current and enables the arrester to reseal (interrupt the current
in the gap). To facilitate resealing, so called active spark gaps have been designed in which an arc voltage builds
up after some time. # resistor block in series with the gap is not very high (typically cm), and to produce the
desired sparkover voltage and nonlinear resistance for a particular voltage level, many such blocks are stacked
together in a series connection. To achieve reasonably uniform
Traduciendo...
(KI Nonlinear characteristic of a 220 kV silicon carbide surge arrester
voltage distribution along the stack, parallel 4 C grading networks are used, which are normally ignored in
simulations.
Silicon carbide arresters are modelled in the EMTP as a nonlinear resistance in series with a gap which has
a constant sparkover voltage. In reality the sparkover voltage depends on the steepness of the incoming wave, as
shown in Fig. 11.3 [17 ]. Since surges in a system have very irregular shapes, rather than the linear rise used in
11 2
Page 356
the measurements of Fig. 11.3, the steepness dependence of the sparkover voltage is not easy to implement, as
already discussed in Section 10.1.2. The nonlinear resistance in series with the gap is either solved with the
compensation method (Section 12.1.2), or with the piecewise linear representation (Section 12.1.3).
In silicon carbide surge arresters with current limiting gaps, a voltage builds up across the gap after 200
to 00 zs, which is best modelled as an inserted ramp type voltage source [175], as shown in Fig. 11. . This ramp
voltage source is not part of the EMTP arrester model now, but it can easily be added as an extra voltage source,
after one trial run to determine when sparkover occurs. This gap voltage is only important in switching surge
studies. In lightning surge studies, it can be ignored because of the time delay of 200 to 00 zs. Useful IEEE
guidelines for modelling silicon carbide arresters are found in [175].
a medium voltage
b high voltage, lightning surge protection
c high voltage, lightning and switching surge protection
(KI #rrester sparkover voltage time characteristic for wavefronts with linear rise [17 ].
4eprinted by permission of Plenum Publishing Corp. and Brown Boveri Oerlikon
It is doubtful whether very sophisticated models with dynamic characteristics, such as the type 9 modern
style SiC surge arrester based on [17 ] in the BP# EMTP, are useful, because it would be almost impossible to
obtain the reSuired data. Brauner [177] has developed a model with dynamic characteristics with special reference
to GIS insulation coordination, which appears to reSuire less data than the type 9 arrester.
11 3
Traduciendo...
Page 357
(KI #rrester gap characteristic
/GVCN 1ZKFG 5WTIG #TTGUVGT
Metal oxide or zinc oxide surge arresters are highly nonlinear resistors, with an almost infinite slope in the
normal voltage region, and an almost horizontal slope in the overvoltage protection region, as shown in Fig. 11.5.
They were originally gapless, but some manufacturers
(KI Voltage current characteristic of a 1200 kV gapless metal oxide surge

arrester [1 3]. l 19 2 IEEE
have re introduced gaps into the design. Its nonlinear resistance is represented by a power function of the form
S
X
K'R (11.1)
XTGH
where p, v and
TGHS are constants (typical values for S 20 to 30). Since it is difficult to describe the entire region
with one power function, the voltage region has been divided into segments in the BP# EMTP, with each segment
defined by its own power function. In the UBC EMTP, only one function is allowed so far. For voltages
11
Page 358
substantially below v , theTGH

current is extremely small (e.g., i p 0.5 p 10 for v/v 0.5), and a linear TGH
representation is therefore used in this low voltage region. In the meaningful overvoltage protection region, two
segments with power functions (11.1) are usually sufficient.
The static characteristic of ES. (11.1) can be extended to include dynamic characteristics similar to hysteresis
effects, through the addition of a series inductance L, whose value can be estimated once the arrester current is
approximately known from a trial run [10]. # metal oxide surge arrester model for fast front current surges with
time to crest in the range of 0.5 to 10 zs was proposed and compared against laboratory tests by Durbak [17 ]. The
basic idea is to divide the single nonlinear resistance into m parallel nonlinear resistances, which are separated by
low pass filters, as illustrated in Fig. 11. for two parallel nonlinearities, which is usually sufficient in practice. The
4 L circuit is the low pass filter which separates the two nonlinear resistances defined by i (v ) and i (v ). The QQ
Traduciendo...
inductance L represents
Q the small but finite inductance associated with the magnetic fields in the immediate vicinity
of the surge arrester, while 4 is used only

Q to damp numerical oscillations (see Section 2.2.2). C is the stray
capacitance of the surge arrester. The model of Fig. 11. can easily be created from existing EMTP elements. If
three such models were connected to phases a, b, c, then the six nonlinear resistances would have to be solved with
the compensation method with a six phase Thevenin eSuivalent circuit.
(KI Two section surge arrester model for fast

front surges [17 ]
# somewhat different model (Fig. 11.7) has been proposed by Knecht [179]. It consists of a nonlinear
resistance 4(v), a more or less constant capacitance C, and a linear, but freSuency dependent impedance <(T).
No IEEE guidelines have yet been published for the modelling of metal oxide surge arresters. The energy
absorbed in them is an important design factor, and should therefore be computed in whatever type of model is used.
Since energy absorption may change as the system is expanded, it is important to check whether ratings which were
appropriate initially may possibly be exceeded in future years. Energy absorption capability is probably more of a
limitation for switching surges than for lightning surges. The sharp change from the almost vertical to the almost
horizontal slope, which limits overvoltages almost ideally at the arrester location, could produce oscillations with
overshoot at locations some distance from the arrester, especially in substations with long bus runs. This may be
11 5
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another factor worth watching for.
Metal oxide surge arresters are generally solved with the compensation method in the EMTP, with iterations
using Newton s method as explained in Section 12.1.2. The piecewise linear representation is less useful because
the highly nonlinear characteristic of ES. (11.1) is not easily described by piecewise linear segments.
(KI #lternative surge arrester model
If the surge arrester is eSuipped with a shunt spark gap, as illustrated in Fig. 11. , then it is still represented
as a nonlinear resistance in the solution process except that the function for
Traduciendo...
(KI Metal oxide surge

arrester with shunt spark gap
that resistance will change abruptly from 4 (i) 4 (t) before sparkover to 4 (i) after sparkover. If the surge arrester
is eSuipped with a series spark gap, then a very high resistance is added to 4 (i) 4 (i) to represent the series gap
before sparkover.
11
Page 360
51.76+10 /'6*1&5 +0 6*' '/62
The basic theory behind the solution methods for the transient simulation and for the ac steady state phasor
solution has already been explained in Section 1. Extensions of the basic theory for more complicated network
elements have mostly been discussed in the sections dealing with these elements. What remains to be explained here
are the various options for handling nonlinearities, the load flow option, and methods for initializing variables with
more than one freSuency component.
+PENWUKQP QH 0QPNKPGCT 'NGOGPVU
The most common types of nonlinear elements are nonlinear inductances for the representation of
transformer and shunt reactor saturation, and nonlinear resistances for the representation of surge arresters.
Nonlinear effects in synchronous and universal machines are handled in the machine eSuations directly, and are
therefore not described here.
Usually, the network contains only a few nonlinear elements. It is therefore sensible to modify the well
proven linear methods more or less to accommodate nonlinear elements, rather than to use less efficient nonlinear
solution methods for the entire network. This has been the philosophy which has been followed in the EMTP. Three
modification schemes have been used over the years, namely
(1) current source representations with time lag )t (no longer used),
(2) compensation methods, and
(3) piecewise linear representations.
%WTTGPV 5QWTEG 4GRTGUGPVCVKQP YKVJ 6KOG .CI)V
#ssume that the network contains a nonlinear inductance with a given flux/current characteristic 8(i), and
that the network is just being solved at instant t. #ll Suantities are therefore known at t )t, including flux 8(t )t),
which is found by integrating the voltage across the nonlinear inductance up to t )t. Provided )t is sufficiently
small, one could use 8(t )t) to find a current i(t )t) from the nonlinear characteristic, and inject this as a current
source between the two nodes to which the nonlinearity is connected for the solution at instant t. In principle, any
number of nonlinearities could be handled this way.
Fig. 12.1 shows a current limiting device where this simple method was used for the two
12 1 Traduciendo...
Page 361
(a) circuit diagram (branch 1 3 (b) characteristics of

feeding network, branches saturable reactors
between 3 and current (both identical)
limiting device, shunt 4 in
load, switch in short
circuit)
(KI Current limiting device
nonlinear inductances. The simulation results are plotted in Fig. 12.2. The numerical oscillations around t 1.
cycles seem to be caused by the time lag )t, since they disappear with more sophisticated techniSues in Fig. 12.11.
Since this method is very easy to implement, it may be useful in special cases, provided that the step size
)t is sufficiently small. It is not a built in option in any of the available EMTP versions, however.
(KI Simulation results with current source representation
%QORGPUCVKQP /GVJQF
In earlier versions of the EMTP, the compensation method worked only for a single nonlinearity in the
network, or in case of more nonlinearities, if they were all separated from each other through distributed parameter
lines. It appears that the type 93 nonlinear inductance in the BP# EMTP still has this restriction imposed on it, but
for most other types, more nonlinearities without travel time separation are allowed now.
The extension of the compensation method to more than one nonlinearity was first implemented for metal
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oxide surge arresters, and later used for other nonlinear elements as well.
In compensation based methods, the nonlinear elements are essentially simulated as current injections, which
are super imposed on the linear network after a solution without the nonlinear elements has first been found. There
are rare situations where a network solution without the nonlinearity is impossible, as in the case of Fig. 12.3. With
the nonlinear branch removed, the current injected into node 1 from the current source would not have any path to
return to neutral. The EMTP would stop with the error message diagonal element of node 2 very small (matrix
singularity). # remedy would be to represent the nonlinearity as a parallel combination of a (normal) linear branch
and of a (modified) nonlinear branch. # related problem occurs if the nonlinear branch is disconnected from the
Traduciendo...
network, as in Fig. 12. . When the
(KI Unsolvable network if nonlinear

branch removed
(KI Disconnected nonlinear branch
EMTP tries to calculate the Thevenin eSuivalent resistance for the nonlinear branch by injecting current into node
m, a zero diagonal element will be encountered in the nodal conductance matrix, and the EMTP will stop with the
error message diagonal element in node m too small. The remedy in this case is the same: represent the nonlinear
branch as a linear branch in parallel with a (modified) nonlinear branch. The BP# manual also suggests the insertion
of high resistance paths where needed, but warns that the resistance values cannot be arbitrarily large.
1PG 0QPNKPGCT 'NGOGPV
Let us assume that the network contains only one nonlinear element between nodes k and m, as indicated
in Fig. 12.5. The compensation theorem states that this nonlinear branch can be
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(KI One nonlinear element connected to linear

network
excluded from the network, and be simulated as a current source i instead, which leaves node k and enters node m
if the nonlinear element is treated as a load and not as a source. The current i must fulfill two
MOeSuations, namely
the network eSuations of the linear part (instantaneous Thevenin eSuivalent circuit between nodes k and m),
XMO ' XMO& & 46JGX KMO (12.1)
(subscript 0 indicates solution without the nonlinear branch, v v v ), and theMO

relationship
M Oof the nonlinear
branch itself,
XMO ' H (KMO , FKMO/FV , V ,...) (12.2)
The value of the Thevenin resistance 4 in ES.6JGX

(12.1) is pre computed once before entering the time step
loop, and re computed whenever switches open and close.
The network eSuations (1. b) can be rewritten as Traduciendo...
[)##] [8#] ' [M#] (12.3)
with [k ]# being the known right hand side from ES. (1. b). To find the Thevenin resistance, a current of 1 # must
be injected into node k, and drawn out from node m. Therefore, replace [k ] with a vector# whose components are
all zero, except for 1.0 in row k and 1.0 in row m. Then perform one repeat solution with this right hand side
vector, which will produce a vector [r ]. This6JGX

vector is the difference of the k th and m th columns of the inverse
matrix [G ] ##
. Then
46JGX ' T6JGX&M & T6JGX&O (12. )
If one of the voltages, say in node m, is known (voltage source, or grounded node), then 1.0 does not appear in [k ]#
because node m belongs to set B of the nodes rather than to set #. In that case,
46JGX ' T6JGX&M (12.5)
If the voltages in both nodes k and m are known, then
12
Page 364
46JGX ' 0 (12. )
If the solution fails because of matrix singularity, it is likely that one of the situations illustrated in Fig. 12.3 and 12.
has been encountered, and remedies discussed there should then be used.
The solution with compensation proceeds as follows in each time step:
(1) Compute the node voltages [v ] without

# the nonlinear branch, with a repeat solution of ES. (12.3). From
this vector, and from the other known voltages [v ], extract $the open circuit voltage v v v . MO M O
(2) Solve the two scalar eSuations (12.1) and (12.2) simultaneously for i . If ES. (12.2)
MO is given analytically,
then the Newton 4aphson method is usually used (example: zinc oxide arrester models). If ES. (12.2) is
defined point by point as a piecewise linear curve, then the intersection of the two curves must be found
through a search procedure, as indicated in Fig. 12. for a nonlinear resistance.
(KI Simultaneous solution of two eSuations
(3) Find the final solution by superimposing the response to the current i , MO
[X#] ' [X#& ] & [T6JGX] KMO (12.7)
Superposition is permissible as long as the rest of the network is linear.
Step (1) is the normal solution procedure for linear networks. Step (2) takes little extra time because it
involves only two scalar eSuations. Step (3) reSuires N additional multiplications and additions if N number of
voltages in set #. Therefore, the extra work of steps (2) and (3) is rather small compared to repeated refactorizations
of [G ],##which would be reSuired for general, nonlinear networks.
6YQ QT /QTG 0QPNKPGCT 'NGOGPVU 5GRCTCVGF D[ 6TCXGN 6KOG
Lines with distributed parameters decouple the network eSuations for the two ends. This is not astonishing
Traduciendo...
because the phenomena at one end are not immediately seen at the other end, but travel time J later. Nonlinear
elements decoupled by distributed parameter lines can therefore be solved independent of each other, because each
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Page 365
has its own area Thevenin eSuivalent eSuation (12.1) decoupled from the others. The [r ] vector of a particular
6JGX
nonlinearity will only have nonzero entries for the nodes of its own area. Therefore, all [r ] vectors can be merged
6JGX
into a single vector, at the expense of another vector which contains the area number for each component [50]. This
is schematically indicated in Fig. 12.7.
(KI Disconnected subnetworks I, II, III, IV
6YQ QT /QTG %QPPGEVGF 0QPNKPGCT 'NGOGPVU
The compensation method can also be used to simulate the effect of M nonlinear branches with current
sources. Then, M vectors [r 6JGX ],...[r 6JGX / ] must be pre computed (and re computed whenever switches change
position). The first vector is found by inserting 1.0 and 1.0 into the appropriate locations for the first nonlinear
element, and then performing a repeat solution. This procedure is repeated for the 2nd,...M th nonlinear element.
The Thevenin eSuivalent resistance becomes an M x M matrix [4 ] in this case. The first column of this matrix
6JGX
is created by calculating the differences r r 6JGX OKfor

6JGX MK all M nonlinear elements i 1,...M from [r 6JGX ], the
second column by doing the same from [r 6JGX ], etc.
In the solution process, step (1) in Section 12.1.2.1 remains identical, but step (2) now reSuires the solution
of M nonlinear eSuations
[XMO] ' [XMO& ] & [46JGX] [KMO] (12. a)
Step (3) uses M vectors [r 6JGX ],...[r 6JGX / ] in place of one vector,
[X#] ' [X#& ] & [T6JGX& ],...[T6JGX&/] [KMO] (12. b)
with [i ] MO
being a vector with M components. If there are N voltages in set #, then N x M multiplications and
additions are reSuired in ES. (12. ). #s M becomes large, this effort may become larger than simply re solving
12
Page 366
[)##] [X#] ' [M# % (KMO CFFGF KP CRRTQRTKCVG RNCEGU) ] (12.9)
with one repeat solution, because the N x M matrix in ES. (12. ) is full, whereas sparsity methods are used in
performing a repeat solution with the triangularized matrix of ES. (12.9). In the BP# EMTP, repeat solutions of
ES. (12.9) are used if M 1, whereas ES. (12. ) is used in the UBC EMTP.
Traduciendo...
The M phase compensation method can be combined with the advantages of element separation through
travel time discussed in Section 12.1.2.2. For example, three surge arresters in phases #1, B1, C1 at the sending
end of a line and three surge arresters in phases #2, B2, C2 at the receiving end are best solved as two disconnected
groups, each with M 3, rather than as one group with M , though the latter approach would work as well if
the program allows for M 3. The merging procedure discussed in Section 12.1.2.2 is essentially the same, except
that each vector is replaced by M vectors.
0QPNKPGCT +PFWEVCPEG
The simultaneous solution of the network eSuation with the nonlinear eSuation, as illustrated in Fig. 12. ,
is straightforward if the nonlinear branch is a nonlinear resistance defined by v f(i ), or if it isMO

a time varying
MO
resistance with v 4(t)MO

i . For nonlinearMO
inductances, this solution process is not so direct because the
nonlinear characteristic is now in the form
8 ' H(K) (12.10)
with the flux 8 being the integral over the voltage v v v , M O
V
8(V) ' 8(V&)V) % X(W)FW (12.11)
m
V& V )
In the EMTP, this problem is solved by using the trapezoidal rule of integration on ES. (12.11), which converts the
flux 8(t) into a linear function of v(t),
)V
8(V) ' X(V) % JKUV(V&)V) (12.12a)
2
)V
JKUV(V&)V) ' 8(V&)V) % X(V&)V) (12.12b)
2
Inserting ES. (12.12a) into ES. (12.10) produces a resistance relationship, by first shifting the origin by hist (t
)t), and then rescaling the 8 axis into a v axis with a multiplication factor of 2/)t. This v(i) characteristic is solved
with the network eSuation in the same way as for any other nonlinear resistance.
#s an alternative, the network eSuation v v 4 i could also beQconverted

6JGX into a flux current
relationship, by using v 2(8 hist)/)t from ES. (12.12).
12 7
Page 367
0GYVQP 4CRJUQP /GVJQF
If an M phase Thevenin eSuivalent circuit must be used, in cases where the M nonlinear elements are not
separated by travel time, then a system of nonlinear eSuations must be solved. The Newton 4aphson iteration
method is the best approach for systems of nonlinear eSuations. It includes the scalar case (one nonlinear eSuation)
as well.
To illustrate the method, assume that the nonlinear elements are nonlinear resistances. Then ES. (12. a),
rewritten here for convenience as
[XMO] & [XMO&] % [46JGX] [KMO] ' 0 (12.13a)
must be solved, whereby [i ] can beMO

replaced by a diagonal matrix [f(v )], whose elements are the
MOi v
characteristics of the M nonlinear resistances (e.g., as defined in ES. (11.1)),
[KMO] ' [ H(XMO)] (12.13b)
Experience has shown that convergence is faster if ES. (12.13) is solved for voltages rather than for currents.
#pplying the Newton 4aphson method to ES. (12.13) produces
[46JGX] FHMO% [7] [)XMO] ' [XMO& ] & [XMO] & [46JGX] [H(XMO)] (12.1 )
FXMO
where the matrix on the left hand side ( Jacobian matrix ) and the right hand side are evaluated with approximate
Traduciendo...
answers from the last iteration step h 1. The improved solution is found by solving the system of linear eSuations
for [)v ], MO
with
[XMOJ ] ' [XMO J& ] % [)XMO] (12.15)
In ES. (12.1 ), [df /dv ] MO

is a diagonal
MO matrix of the derivatives of the i v characteristics, which destroys the
symmetry of the Jacobian matrix. To maintain symmetry, the following modification can be used: multiply the
Jacobian matrix with the inverse matrix [df /dv ] , and

MO solve
MO the eSuations for the variables [)x],
FHMO&
[46JGX] % [)Z] ' [XMO& ] & [XMO] & [46JGX] [H(XMO)] (12.1 a)
FXMO
The Jacobian matrix is now symmetric, and the diagonal elements of [df /dv ] are simplyMO
the reciprocals
MO of
df MO
/dv . #fter
MO [)x] has been found, the voltage corrections are
)ZMO
)XMO ' (12.1 b)
FHMO/FXMO
This modification is used in the UBC EMTP.
In the BP# EMTP, symmetry is achieved by working with the inverse matrix [4 ] . Multiplying ES.
6JGX
(12.13a) with this inverse matrix and applying the Newton 4aphson method to it produces
12
Page 368
[46JGX]
& % FHMO [)XMO] ' [46JGX] & { [XMO&] & [XMO] } & [H(XMO)]
(12.17)
FXMO
If the inverse matrix exists, then this procedure is as straightforward as ES. (12.1 ). [4 ] can be singular, 6JGX
however, if nonlinear elements are directly connected to voltage sources. In the scalar case, 4 6JGXwould become
zero, as shown in ES. (12. ), whereas the respective row and column in [4 ] becomes zero in the M phase case.
6JGX
This has to be treated in a special way in ES. (12.17), whereas no special cases arise with ES. (12.1 ).
To start the iterations with either ES. (12.1 ) or (12.17), an initial guess for the voltages is needed. Since
currents in nonlinear elements tend to change less from step to step than voltages, it is best to use the old currents
[i MO
(t )t)] from the preceding time step and the new open circuit voltages [v (t)] to obtain anMO
initial voltage guess
from ES. (12.13a). This voltage guess is used for [v ] in ES. (12.1
MOa) or (12.17), as well as for [df /dv ] and MO MO
[f(v MO
)]. This procedure seems to reSuire the least number of iterations, and has therefore been implemented in the
UBC EMTP.
0WOGTKECN 2TQDNGOU
#s long as the EMTP works with a fixed step size )t, numerical problems can arise with nonlinear elements.
If )t is too large, artificial negative damping or hysteresis can occur, as illustrated in Fig. 12. (solution proceeds
from 1 to 2 to 3 in consecutive steps). This can cause
(KI #rtificial negative damping
numerical instability. Since the dotted nonlinear characteristic would give identical answers, it is obvious that the
shape of the characteristic between sampled points does not enter into the solution, that is, the nonlinear characteristic
is only used in a spotty way. Piecewise linear resistances and inductances, as discussed in the next section, appear
to be more stable numerically (or possibly absolutely stable), but they may cause overshooting problems.
#nother problem is related to automatic ac steady state initialization. Since nonlinear elements are
approximated as linear elements in the ac phasor solution, a sudden jump can occur at t 0 between the linear and
nonlinear representations. For nonlinear inductances, the problem can be minimized through proper voltage source
Traduciendo...
rotations, as discussed in Section 12.1.3.3. The problem will be resolved when the superposition of harmonics
(Section 12. ) becomes available to the users.
2KGEGYKUG .KPGCT 4GRTGUGPVCVKQP
12 9
Page 369
2KGEGYKUG .KPGCT +PFWEVCPEG
#s discussed in Section . .2, the saturation characteristics of modern transformers can often be represented
accurately enough as a piecewise linear inductance with two slopes (Fig. 12.9). Such a piecewise linear inductance
can be simulated with two linear inductances L and
(KI Piecewise linear inductance with

two slopes
L Rin parallel (Fig. 12.10), provided that the flux in L is always computed
R by integrating the voltage v v M O
independent of the switch position. The switch is close whenever *8* $ 8 5#674#6+10 , and opened again as soon as
*8* 8 5#674#6+10 . Fig. 12.11 shows the simulation results for the current limiting device of Fig. 12.1 if two slope
piecewise linear inductances are
(KI Switched inductance

implementation of two slope piecewise linear
inductance
12 10
Page 370
Traduciendo...
(KI Simulation results for case of Fig. 12.1 with

two slope piecewise linear inductances
used. The numerical oscillations around 1. cycles present in Fig. 12.2 have now disappeared.
Using a switch to make the changeover from an inductance value of L to L in [G] of ES. (1. ) is simply
a programming trick, which has been used in the UBC EMTP and in older versions of the BP# EMTP, and has
sometimes been called switched inductance (or switched resistance ). In newer versions of the BP# EMTP, [G]
is changed directly and re triangularized whenever the solution moves from one straight line segment of a piecewise
linear inductance to another segment ( pseudolinear inductance or resistance in BP# EMTP 4ule Book). In this
direct matrix change approach, the recursive updating of the history term of ES. (2. ) would be wrong whenever
the slope changes. It is therefore better to use the non recursive formula (2.7), where the branch current must first
be determined from ES. (2. ) with the inductance value of the old slope, while ES. (2.7) reSuires the inductance
value of the new slope at instants of changeover.
The two slope piecewise linear inductance in the UBC EMTP has an option for starting the simulation from
a user specified residual flux 8 , which

TGUKFWCN overrides any internally calculated flux. With this option, the piecewise
linear characteristic 1 2 with slope L is used to point 2 where the slope is switched to L (Fig. 12.12). The flux 8
at the switching point is precalculated in such a way that the simulation will move directly into the normal two slope
characteristic thereafter. This procedure works well if the saturation is driven high enough to reach at least point
3. If not, some special tricks are used, which are described in more detail in the UBC User s Manual (parameter
IFLUX on time card).
In addition to the normal piecewise linear inductance, the BP# EMTP also has one with hysteresis behavior
( type 9 pseudo nonlinear hysteretic reactor ), as illustrated in Fig. 12.13. Moving along any linear segment is
still described by the same differential eSuation v L di/dt used for any other linear inductance. Therefore, the
representation in the transient solution part of the EMTP is the simple eSuivalent resistance 2L/)t in parallel with
a current source known from the history in the preceding time step (Fig. 2. ). The eSuivalent resistance must be
changed, however, whenever the simulation moves from one segment into another. The fact that
12 11
Page 371
(KI 4epresentation of residual flux in UBC

EMTP
Traduciendo...
(KI Piecewise linear inductance with hysteresis
the linear segment does (in general) not pass through the origin is automatically taken care of by the history terms.
Starting from a residual flux is permitted. This representation with hysteresis can be tricky to use, and the reader
is therefore referred to [119] for more details.
2KGEGYKUG .KPGCT 4GUKUVCPEG
Either the switching approach or the direct matrix change approach for nonlinear inductances works
eSually well for nonlinear resistances. History terms are of course not needed in this case. Each linear segment with
a slope of 4 dv/di is represented in the EMTP as a voltage source v -0'' in series with a resistance 4, or a current
source v -0'' /4 in parallel with a resistance 4 (Fig. 12.1 ).
12 12
Page 372
(a) piecewise linear (b) voltage source (c) current source

segment representation representation
(KI Piecewise linear resistance
0WOGTKECN 2TQDNGOU
With the direct matrix change approach, there is no reason to limit the shape of the nonlinear characteristic
to only two slopes. Newer versions of the BP# EMTP therefore permit essentially any number of piecewise linear
segments ( type 9 pseudo linear reactor and type 99 pseudo linear resistance in the BP# EMTP 4ule Book).
While multi slope piecewise linear elements are more useful than two slope elements, they can also create special
problems which do not exist with two slope elements, especially for the nonlinear resistance: if the piecewise linear
resistance is used to model a silicon carbide surge arrester with a spark gap, then the EMTP does not automatically
know which segment it should jump to after sparkover (Fig. 12.15). The user must therefore specify the segment
number as part of the input data (e.g., segment 2 3 in Fig. 12.15). This may reSuire a trial run, unless the network
seen from the surge
(KI Jump after sparkover ( generally

unknown network characteristic at instant of
sparkover) Traduciendo...
arrester location is relatively simple . For the 2 slope resistance in the UBC EMTP shown in Fig. 12.15, this
problem does not arise.
#ll piecewise linear representations cause overshoots, because the need for changing to the next segment
If only single phase lossless lines were connected to the surge arrester, then the slope of the network
eSuation would simply be 4 6JGX' 1/< .
UWTIG
12 13
Page 373
is only recognized after the last point (c in Fig. 12.1 ) has gone outside its proper range. The simulation will
therefore follow the dotted line into the next segment, rather
(KI Overshoot in
piecewise linear
representation
than the specified solid line at point x. Caution is therefore needed in the choice of )t to keep the overshoot small.
The overshoot is usually less severe on piecewise linear inductances because the flux, being the integral over the
voltage, cannot change very Suickly. The proper cure for the overshoot problem would be an interpolation method
which moves the solution backwards by a fraction of )t to point x in Fig. 12.1 , and then restarts the solution again
at that point with )t. The points along the time axis would then no longer be spaced at eSual distances. This method
is used in the transients program NETOM#C [15].
Both the piecewise linear representation and the compensation method suffer from the fact that nonlinear
inductances are approximated as linear inductances in the ac phasor solution, at least until the superposition of
harmonics discussed in Section 12. .2 has become available to most users. The problem should not occur with
nonlinear resistances which represent surge arresters. The voltages across these nonlinear resistances should be low
enough in the steady state solution to either draw negligibly small currents (metal oxide arresters), or be below
sparkover voltage (silicon carbide arresters). Transformers and shunt reactors do saturate in normal steady state
operation, however, and a jump from the linear to the nonlinear characteristic will therefore occur at t 0 (Fig.
12.17). Whether the jump occurs in 8 or i, or in both depends
(KI Jump between steady state and transient

solution
12 1
Page 374
Traduciendo...
to some extent on the type of representation. This problem can be minimized by rotating the voltage sources in such
a way that one of the three flux phasor 7 , 7 , 7 has

# zero
$ angle
% (Fig. 12.1 ). For balanced network conditions,
one flux would be at zero value at t 0, and the other
(KI Flux phasors (8(0) 4e{7})
two fluxes would be . of their peak value. Hopefully, this would be below the knee point of the saturation
curve. Note that fluxes and voltages are 90E out of phase when doing this rotation (checking that one of the currents
is close to zero at t 0 will verify the correctness of the rotation).
.QCF (NQY 1RVKQP
# load flow (power flow) option was added to the EMTP in 19 3 by F. 4asmussen (Elkraft, Denmark).
It adjusts the magnitudes and angles of sinusoidal sources iteratively in a seSuence of steady state solution, until
specified active and reactive power, or specified active power and voltage magnitude, or some other specified
criteria, are achieved. This will create the initial conditions for the subseSuent transient simulation.
Without the load flow option, the steady state conditions are obtained by solving the system of linear nodal
eSuations (1.21) only once. These eSuations are
[;##] [8#] ' [+#] & [;#$] [8$] (12.1 )
with user specified magnitudes and angles for the voltage sources [V ] and for the
$ current sources [I ]. The resulting#
power flows may or may not be what the user wants. There are many cases, however, where the details of the initial
power flows in the network do not influence the results of the transient simulation. For example, the switching surge
overvoltages on the line in the network of Fig. 12.19 are not influenced by the power flow pattern within the feeding
12 15
Page 375
(KI Network configuration for switching surge study
network, as long as the feeding network does not contain nonlinear elements. The only important parameters are
Traduciendo...
the (freSuency dependent) impedance of the feeding network (which does not depend on power flows anyhow), and
the pre closing voltage V . This

5 is true because a linear network can always be represented by a Thevenin eSuivalent
circuit which reSuires only these two parameters. The value of the open circuit voltage V is normally specified
5 by
the user (e.g., 5 above rated voltage), rather than obtained from a load flow solution. #ny combination of source
voltages V , V , V which produces the same V would create

5 identical overvoltages. One could therefore simply
VTKCN
assume eSual source voltages V V V , make one trial steady state solution to get V 5 , and then multiply the
VTKCN
voltages with the factor V /V 55 for the final simulation. # load flow solution is not needed in this case.
The best methods for load flow solutions are based on the Newton 4aphson method. When 4asmussen
added a load flow option in 19 3, he was aware of that, but he could not afford the tremendous programming effort
involved in its implementation. Instead, he developed a simpler method, which would serve his needs and at the
same time reSuire as little program changes as possible. This led to the method discussed next, which is somewhat
similar to the Gauss Seidel methods used in the early days of load flow program development. #n improved
approach, which also reSuires a minimum of program changes, is discussed in Section 12.2.2. It is clear, however,
that one would eventually have to use Newton 4aphson methods, and re program the steady state solution routine
completely, if further improvements are needed.
4CUOWUUGP U .QCF (NQY /GVJQF
Nodes at which the user specifies active power P and reactive power 3 (or some other combination of P,
3, voltage magnitude, and voltage angle) are treated as voltage sources in the direct solution of the system of linear
eSuations (12.1 ). For a network with 100 nodes, in which P, 3 is specified at 9 nodes, and where one node is the
slack node (*V*, 2 specified), the solution of the 90 eSuations (12.1 ) amounts essentially to a reduction of the
network to 10 voltage source nodes. #fter this solution, 4asmussen calculates the current at P, 3 nodes from the
eSuations of set B which have been left out in ES. (12.1 ),
12 1
Page 376
P
+M ' j ;MO 8O HQT CNN PQFGU M QH UGV $, GZEGRV HQT UNCEM PQFG (12.19)
O'
and then the power from
2M & L3M ' .M 8M

(
(12.20)
The calculated values of P , 3 are

M then
M compared with the values specified by the user. Based on these differences,
corrections are made to the angle 2 and magnitude

M *V * of each voltage
M V, M
2M & 2M&URGEKHKGF
)2M ' @ 2.5 (
1 (12.21)
(*2M* % *2M&URGEKHKGF*)
2
and
3M & 3M&URGEKHKGF
)*8M* ' @ 2000 (
1 (12.22)
(*3M* % *3M&URGEKHKGF*)
2
F is a deceleration factor which decreases from 1.0 to 0.25 in 500 iterations, with the formula
1000 & J
(' (12.23)
1000
(h iteration step). Once the voltages have been corrected, another direct solution of ES. (12.1 ) is obtained. This
cycle of calculations is repeated until )2 and )*VM* become sufficiently

M small.
The method of 4asmussen is comparable to the Gauss Seidel load flow solution method applied to the
reduced system
TGFWEGF
[; $$ ] ' [8$] ' [+$] (12.2 a)
Traduciendo...
where
[;
TGFWEGF
] ' [;$$] & [;$#] [;##]
& [;#$]
$$ (12.2 b)
and where [I ] was

# assumed to be zero to simplify the explanations. For the 100 node example, the performance
of 4asmussen s method would therefore have to be compared against the Gauss Seidel method for a 10 node system
(with one slack node), and not for a 100 node system. Since the Gauss Seidel method converges faster for smaller
systems, the reduction implied in the 4asmussen method is an advantage over straightforward Gauss Seidel methods.
12 17
Page 377
If the standard Gauss Seidel load flow method were applied to the reduced system, the corrected voltages
would be found from
TGFWEGFJ ' 2M&URGEKHKGF & L3M&URGEKHKGF TGFWEGF J&

; MM 8M & j ; MO 8O
8M
( J& PQFGU QH UGV $ GZEGRV -
for each node k of subset B (except for the slack node). # slightly modified method uses eSuations in the form in
which they are normally written for Newton s methods,
TGFWEGF
(& 3M & $ MM *8M* ) )2M ' 2M & 2M&URGEKHKGF (12.25)
TGFWEGF )*8M*
(2M % ) MM *8M* ) ' 3M & 3M&URGEKHKGF (12.2 )
*8M*
The coefficients on the left hand side are often called H and L inMM
the loadMM
flow solution literature. By comparing
ES. (12.25) and (12.2 ) with ES. (12.21) and (12.22), one can see that 4asmussen basically assumed fixed values
for H and
MM L , independent
MM of the type of network and the system of units used (p.u., V#, kV#, or MV#). The
influence of the chosen system of units seems to be more or less eliminated by using relative values )P/P and )3/3
in ES. (12.21) and (12.22). The method of 4asmussen may be sensitive to the type of network being studied.
Convergence may be slow, as for any Gauss Seidel related method, even if the reduced system is small.
%WTTGPV 5QWTEG +VGTCVKQP /GVJQF
From stability studies it is known that much better convergence can be obtained by representing the P, 3
nodes as current sources in the reduced network of ES. (12.2 ). The current sources are obtained from the voltage
solution of the preceding iteration step,
J 2M&URGEKHKGF&L3M&URGEKHKGF
+M ' (12.27)
( J&
8M
for all nodes k of set B except for slack node
With [I ] $thus known, ES. (12.2 ) is solved directly for an improved voltage solution [V ] (except for the slack
$
node). For the 100 node example, 99 eSuations would have to be solved, compared to 90 eSuations in 4asmussen s
method. However, convergence is potentially much faster. For the single phase 5 node test system of Fig. 12.20,
it took 9 iterations to converge to an accuracy of *)P* / *P*, *)3* / *3* # 10 for all nodes. 4asmussen s method
was not run for this case, but it would probably reSuire many more iterations.
The assistance of Dr. Mansour, Li 3uang Si and I.I. Dommel in running the experiments for this section is
gratefully acknowledged.
12 1
(KI Single phase 5 node test system (node 1 slack node, nodes
2, 3, , 5 P, 3 nodes)
The test case supplied by BP# for the load flow option is shown in Fig. 12.21. It consists of a three phase
generator (terminals #1, B1, C1) with series resistances, which feeds through a delta/wye connected transformer
and through coupled inductances into a three phase voltage source (slack nodes). P, 3 is specified at the three
generator terminals #1, B1, C1. The entire system is balanced, and is therefore eSuivalent to a 2 node single phase
(positive seSuence) network with only one P, 3 node and one slack node. 4asmussen s method takes 133 iterations
to converge to accuracy defined in the BP# test case data. When the current source iteration method was first tried
on it (with high hopes), it failed unexpectedly. The reason turned out to be the floating delta connection of the
transformer, which makes the admittance matrix on the generator side singular (or extremely ill conditioned). The
sum of the three currents I I I becomes

# slightly
$ nonzero
% (because of round off errors) in the iteration
process, and this extremely small zero seSuence current will be injected into an infinite zero seSuence impedance
on the delta side, which produces a large zero seSuence voltage. This causes the method to diverge.
(KI BP# test case for load flow option
To resolve this problem, one can connect shunt impedances to nodes #1, B1, C1 to make the matrix
nonsingular. Since node 1 is a synchronous machine, and since such a machine should properly be represented as
current sources in parallel with the negative seSuence impedances to handle unbalanced cases (Section . ), a natural
12 19
Page 379
choice would be
)) ))
: S
<UJWPV ' L F% : (12.2 )
2
With this shunt impedance, and with a modification of ES. (12.27) to account for the current in this impedance,
4M&URGEKHKGF & L 3M&URGEKHKGF

8 J&
J'
+M %
( J& (12.29)
8M <UJWPV
the current source iterations converge in 5 iterations, which is faster than 4asmussen s method, but much slower
than expected from the experience with the 5 node test system of Fig. 12.20.
It is known from stability studies that shunt impedances speed up convergence if they are determined in such
Traduciendo...
a way that they would produce the specified power at the rated voltage, or
*8TCVGF*
<UJWPV ' & (12.30)
2URGEKHKGF & L3URGEKHKGF
where P, 3 is negative for loads and positive for generation. With this shunt impedance, the current source iteration
method does indeed converge Suickly in 5 iterations. Fig. 12.22 shows
(a) 4eal part of V#N
12 20
Page 380
(b) Imaginary part of V#N
(KI Voltages as a function of iteration step for test case of Fig. 12.21
the real and imaginary part of voltage V for 4asmussen

# s method, and for the current source iteration method with
< UJWPV
from ES. (12.29) and from ES. (12.30). From a convergence standpoint, < from
UJWPV ES. (12.30) is obviously
best, but if one wants to represent the synchronous machine properly in unbalanced cases, < from
UJWPV ES. (12.29)
should be used.
#. ;an started implementing this method in Ontario Hydro in 19 5/ . # few issues remain to be resolved.
One is the treatment of P, *V* nodes where active power and voltage magnitude are specified. If one is willing to
pre calculate the internal impedance of the network as seen from each P, *V* node, then one can construct an
approximate Thevenin eSuivalent circuit after the solution at each iteration step. With V MPQYPand I being
MPQYP the
known values at a P, *V* node, and with < having6JGX

been pre calculated once and for all, the open circuit voltage
Traduciendo...
V 6JGXof the Thevenin eSuivalent circuit (Fig. 12.23) is simply
86JGX ' 8MPQYP & <6JGX +MPQYP (12.31)
12 21
Page 381
(a) Circuit (b) 4elationship between V and 3
(KI Thevenin eSuivalent circuit
If we assume that this Thevenin eSuivalent circuit is now correctly defined by V 6JGXand < ,6JGX
and that V and I are
allowed to change, then we obtain a relationship between 3 and *V*,
3 ' $ (*86JGX* *8* cos" & *8* ) % ) *86JGX* *8* sin" (12.32)
where " is the angle between V and V, and G jB 1/< . Since the angle " is
6JGX more related to active power
6JGX
flows than reactive power flows, it is reasonable to assume that " does not change if the 3 *V* relationship is
altered. With ", G, B and *V * known,

6JGXand with the specified node voltage magnitude being used for *V*, ES.
(12.32) can be solved for 3. This value of 3 is then used in calculating the current for the next iteration step from
ES. (12.29). For the 5 node test system of Fig. 12.20, this method converged in 1 iterations when nodes and 5
were treated as P, *V* nodes, with nodes 2 and 3 remaining P, 3 nodes.
The treatment of P, 3 nodes and P, *V* nodes in three phase unbalanced cases is still under development.
To obtain realistic answers, the user cannot specify power or voltage magnitude values more or less arbitrarily at
each one of the three phases. Instead, one must know how the load or generator reacts to unbalanced conditions.
#s explained in Section . , synchronous machines must be modelled as symmetrical voltage sources behind (or
symmetrical current sources in parallel with) a 3 x 3 impedance matrix calculated from < < and < . To RQU PGI \GTQ
obtain results which are physically possible, the same representation would have to be used in the load flow
iterations. Similar models valid for unbalanced conditions would have to be developed for the universal machine,
and for other devices which appear in the EMTP as loads or generators.
5VGCF[ 5VCVG 5QNWVKQPU YKVJQWV *CTOQPKEU
The linear ac steady state phasor solution at one freSuency has already been described in Section 1, and the
models of the various elements which must be used in that solution have been discussed in the respective sections.
The routine for the steady state solution was added by J.W. Walker, originally to obtain ac steady state initial
conditions automatically. Later, it became a useful tool on its own, e.g., for studying complicated coupling effects
between circuits on the same right of way (example in Fig. .25). The EMTP therefore has an option to terminate
12 22
Traduciendo...
Page 382
the run after the steady state solution, and to tabulate the phasor values in as much detail as the user may wish.
6JGXGPKP 'SWKXCNGPV %KTEWKVU
If a large network is to be solved repeatedly, with only a few parameters varied each time, then it may be
best to generate an M phase Thevenin eSuivalent circuit for the large network first, as illustrated in Fig. 12.2 . The
parameters of the Thevenin eSuivalent circuit can easily be obtained with the EMTP from M 1 steady state
solutions of the large network as follows:
(KI Parameter variation in branches 1, ... M
(1) 4emove the branches 1,...M (which may be coupled among themselves) from the large network. Obtain a
steady state solution, and record the open circuit branch voltages across the node pairs 1a 2a, 1b 2b,... at the
locations where the M branches were removed. If the branches are all connected from node to ground, then
these branch voltages are simply node voltages. This first steady state solution produces the open circuit
voltage vector of the M phase Thevenin eSuivalent circuit,
8
C& C
8
D& D
[86JGX] ' . (12.33)
.
8
/& /
(2) Find the impedance matrix
<CC <CD ... <C/
[<6JGX] ' <DC <DD ... <D/ (12.3 )
</C </D ... <//
of the Thevenin eSuivalent circuit column by column with M steady state solutions. First, short circuit all
voltage sources in the large network (easiest way to do this is to set their amplitudes to zero simply removing
12 23
Page 383
them from the data file would create open circuits), and cancel all current sources (set amplitudes to zero or
remove them from data file). To obtain column k of [< ], connect one current source of 1.0 # (4MS) to
6JGX
node 1k, and a second current source of 1.0 # (4MS) to node 2k, and ask for a steady state solution. Then
the elements of the k th column of [< ] are simply

6JGX the 4MS branch voltages,
<CM 8
C& C
<DM 8 D& D
. ' . (12.35)
. .
</M 8 Traduciendo...
/& /
With [V ] 6JGX
and [< ] known, the large (unchanged) part of the network is described by the M phase
6JGX
Thevenin eSuivalent circuit of Fig. 12.25, with its branch eSuation
[8] ' [86JGX ] & [<6JGX ] [+] (12.3 )
(KI M branch Thevenin eSuivalent circuit
where [V] and [I] are branch voltages and currents. If these branches are passive, with a branch impedance matrix
[< DTCPEJ
] whose values are to be varied repeatedly, then
[8] ' [<DTCPEJ] [+] (12.37)
which can be solved with ES. (12.3 ) for the currents,
[+] ' 6 [<6JGX ] % [<DTCPEJ] >

& [86JGX ]
(12.3 )
EMTP users may want to write their own program to solve ES. (12.3 ), rather than use the EMTP for it.
12 2
Page 384
Such a Thevenin eSuivalent circuit was used by BP# to study resonance problems on a shunt compensated
transmission line which is switched off at both ends, but which is still capacitively coupled to parallel circuits on the
same right of way. Because of complicated transposition schemes, the complete network is fairly large, whereas
the Thevenin eSuivalent circuit of the network seen from the three shunt reactor connection points #, B, C has only
a 3 x 3 [< ] 6JGX
matrix. Fig. 12.2 shows the results of this study, in which the inductance of the shunt reactors was
varied in small steps from 5. 5 to .00 H.
(KI 4esonance in shunt reactors

Traduciendo...
Thevenin eSuivalent circuits are in principle only valid at the freSuency at which they are calculated. In the
preceding example it would be known, however, that an open ended line is seen by the shunt reactor as a capacitance
up to some freSuency way above 0 Hz, and that the coupling to energized lines is capacitive as well. The Thevenin
impedance is therefore < 1/(jTC),

6JGX or in the matrix case
[%]
& ' LT [<6JGX ]
The capacitance matrix representation would then be useful for transient studies (up to a certain freSuency) as well.
(TGSWGPE[ 5ECP
The first addition to the steady state solution routine was a loop to vary the freSuency automatically from
f OKP
to f , either
OCZ in linear steps of )f or on a logarithmic scale. #t each freSuency, the solution is obtained in the
same way as before. This option has become known as freSuency scan. Instead of getting voltages and currents
as a function of time, their magnitudes and angles are obtained as a function of freSuency. This option is very useful
12 25
Page 385
for finding the freSuency dependent impedance of a network seen from a particular location. To obtain the
impedance, all voltage sources are short circuited and all current sources are removed. # current source of 1 # is
then added across the two nodes between which the impedance is to be obtained. The branch voltage will be eSual
to the impedance. Fig. 12.27 shows an example, where the impedance between two phases was computed with the
freSuency scan option of the EMTP, as well as indirectly measured with a phase to phase fault (time response
Fourier transformed to freSuency response), as part of a study to investigate potential subsynchronous resonance
problems.
(KI Comparison between impedance calculated with

freSuency scan and measured impedance [1 ]. l 19
IEEE
&KHHGTGPV (TGSWGPEKGU KP &KUEQPPGEVGF 2CTVU
The BP# EMTP is capable of finding the steady state solution in networks with sources having different
freSuencies, provided that the network is disconnected into subnetworks, with each subnetwork only containing
sources with the same freSuency. The need for this capability arose primarily in connection with universal machine
initialization (Section 9.5). For example, the armature windings of a synchronous machine and the connected power
system must be solved for ac conditions, whereas the field circuit reSuires a dc solution.
The same capability can be used to handle trapped charge on an isolated line, and HVDC links. In the latter
case, the converters are either represented as impedances or current sources on the ac side, and as voltage sources
on the dc side. This ignores the current harmonics on the ac side and the voltage harmonics on the dc side, but it
does produce reasonable initial conditions for the transient simulation. Fig. 12.2 shows simulation results for a dc
transmission line with six pulse converters, which were connected through converter transformers to ac networks
Traduciendo...
12 2
Page 386
(KI 4ectifier and inverter voltage, with simulation

starting from approximate initial conditions
represented as three phase Thevenin eSuivalent circuits. #t least in this case, the final steady state was reached
almost immediately.
The theory behind this single solution with multiple freSuencies is very simple. #ssume that there are two
subnetworks 1 and 2 with freSuencies f and f . Since they must be disconnected, ES. (1.20) has the form
[; ] 0 [8 ] [+ ]
' (12.39)
0 [; ] [8 ] [+ ]
and all that is reSuired is that freSuency f be used in forming [; ] and f in forming [; ].
For dc solutions, an inductance branch becomes a short circuit and the two nodes therefore collapse into
one node. To solve dc conditions exactly would therefore reSuire program modifications, which have been regarded
as a low priority item until now. Instead, dc sources are represented as ac sources of the form VcosTt or IcosTt,
with T being very low (typically f 10 Hz). Inductances are then very low impedances, rather than short circuits.
5VGCF[ 5VCVG 5QNWVKQP YKVJ *CTOQPKEU
Steady state harmonics in high voltage transmission systems are primarily produced by transformer (and
possibly shunt reactor) saturation, by HVDC converter stations, and by large rectifier loads (e.g., aluminum
reduction plants). In rectifiers and inverters, the magnitude of harmonics is reasonably well known, and these
harmonics can therefore be represented as given current or voltage sources in harmonic load flow programs
specifically designed for harmonics studies. In contrast, harmonics generated by transformer saturation depend
critically on the peak magnitude and waveform of the voltage at the transformer terminals, which in turn are
influenced by the harmonic currents and the freSuency dependent network impedances.
Transient simulations with the EMTP will contain harmonics effects either from transformer saturation of
from converters. If the simulation is carried out long enough to let the transients settle down to steady state
conditions, then the waveforms will contain the harmonics with reasonable accuracy up to a certain order, depending
12 27
Page 387
on the step size )t. # Fourier analysis program is available as a support routine in the EMTP to analyze such
waveforms. This approach is discussed first. There are cases, however, where it would be desirable to have the
harmonics already included in the steady state initial conditions, because steady state harmonics do sometimes have
an effect on the transients. This is discussed next in Section 12. .2.
*CTOQPKEU HTQO .KPGCT CE 5VGCF[ 5VCVG 5QNWVKQP HQNNQYGF D[ 6TCPUKGPV 5KOWNCVKQP

Traduciendo...
# simple method for obtaining saturation generated harmonics is to perform a transient simulation with the
EMTP which starts from approximate linear ac steady state conditions. For the initial ac steady state solution, the
magnetizing inductances of transformers are represented by their unsaturated values. In the transient simulation, the
only disturbances will then be the deviations between the linear and nonlinear magnetizing inductance representations.
The transients caused by these deviations will often settle down to the distorted steady state within a few cycles.
This simple method works only well if the final distorted steady state is reached Suickly in a few cycles.
Such is the case in the example cited in Section . .2 (Fig. .13 and .1 ), where steady state was reached within
approximately 3 cycles. For lightly damped systems, it may take a long time before the final steady state is reached.
Fig. 12.29 shows the voltages at
(KI Voltages with harmonic distortion on a 500 kV

line (simulation starts from approximate linear steady state)
both ends of a 500 kV line with shunt reactors which go into saturation at 0.92 p.u. of rated flux at the sending end
and at 1.05 p.u. of rated flux at the receiving end. Because of low damping, the steady state is reached only after
a long time. It is such cases where the steady state solution method described in the next section is useful.
12 2
Page 388
*CTOQPKEU HTQO 5VGCF[ 5VCVG 5QNWVKQPU
The method described in this section [1 5] has been implemented in Ontario Hydro s EMTP by #. ;an in
19 5, as part of joint work undertaken by the EMTP Development Coordination Group (DCG) and EP4I. It should
become available to users of the DCG/EP4I version of the EMTP in 19 or 19 7.
To obtain the harmonics directly from phasor eSuations, the nonlinear inductances must be replaced by
current sources, which contain the fundamental freSuency component as well as the harmonic freSuency components
(Fig. 12.30). The network itself is then linear, and the voltages at any freSuency are therefore easily found by
solving the system of linear eSuations (12.1 ). The nonlinear effects are represented as current sources in the vector
[I #]. The complete solution is found with two iterative loops. First power flow iterations are used to obtain an
approximate solution at fundamental freSuency, while the second distortion iterations take the higher harmonics
into account and correct the fundamental freSuency solution as well.
Traduciendo...
(KI 4eplacing nonlinear inductances

by current sources. (a) Network with nonlinear
inductance, (b) network with current source
2QYGT (NQY +VGTCVKQPU
In the power flow iterations, an approximate linear ac steady state solution is found which represents the
V 4/5
/I curves
4/5 of the nonlinear inductances correctly, but does not include harmonic distortion. For the nonlinear
inductance, say at node m in Fig. 12.30, the original data may already be in the form of a V /I curve, as shown
4/5 4/5
in Fig. 12.31. If not, it is straightforward to convert the 8/i curve into a V /I curve, with the support routine
4/5 4/5
CONVE4T (Section .10. ). To start the iteration process, a guess for the 4MS voltage V is used to find theO4MS
current I (Fig.
O 12.31). This current, with the proper phase shift of 90E with respect to V , is inserted into the O
current vector [I ] in#ES. (12.1 ), and a new set of voltages [V ] is then found by
# solving the system of linear
eSuations. This solution process is repeated, until the prescribed error criterion for the current I is satisfied. NoteO
that the admittance matrix [;] in ES. (12.1 ) remains constant for all iteration steps therefore, [;] is only
triangularized once outside the iteration loop. Inside the iteration loop, the downward operations and
12 29
Page 389
backsubstitutions are only performed on the right hand side, by using the information contained in the triangularized
matrix ( repeat solutions ).
(KI V /I characteristic
4/5 4/5of a nonlinear
inductance
In these power flow iterations at fundamental freSuency, the V /I curve is used as an approximation
4/5 4/5
to the curve relating the fundamental freSuency current I to the fundamental freSuency voltage V . If V 4/5 were
eSual to V , then I would

4/5 contain harmonics, which are ignored. The approximation does provide a good starting
point, however, for the following distortion iterations, in which harmonics are included.
If the network contains nodes of the load flow option type, e.g., active power P and reactive power 3
specified rather than current I, then the adjustments to achieve constant power can easily be incorporated into this
iterative loop by using ES. (12.29), or a similar eSuation, at the beginning of each iteration step.
&KUVQTVKQP +VGTCVKQPU
The power flow iterations produce a steady state solution at fundamental freSuency only, without
harmonic distortion. To obtain the harmonics, the 4MS voltages found from the power flow iterations are used
LN
in an initial estimate for the flux. Since v d8/dt, and assuming that the peak voltage phasor is *V*e , or
X(V) ' *8* cos(T V % N) (12. 0)
as a function of time (T angular fundamental freSuency), it follows that the flux is
*8*
8(V) ' sin(T V % N) (12. 1)
T
With 8(t) known, one full cycle of the distorted current i(t) is generated point by point with the 8/i curve (Fig.
12.32). If hysteresis is ignored, then it is sufficient to produce one Suarter of a cycle of i(t), since each half cycle
Traduciendo...
wave is symmetric, and since the second half is the negative of the first half of each cycle.
Subscripts 1, 2,... are used in this section to indicate the order of the harmonic (1 fundamental).
12 30
Page 390
The distorted current i(t) in each nonlinear inductance is then analyzed with the support routine Fourier
#nalysis, which produces the harmonic content expressed by
M
K(V) ' j *+P* sin(TPV % NP) (12. 2)
P'
with
TP ' PT (12. 3)
being the angular freSuency of the n th harmonic. Experience has shown that it is usually sufficient to consider the
fundamental and the odd harmonics of order 3 to 15, and to ignore the higher and even harmonics. #t each harmonic
considered (including the fundamental), the harmonic component from ES. (12. 2) is entered into [I ] with its proper#
magnitude and angle for all nonlinear inductances, and the voltages at that harmonic freSuency are then found by
solving the system of linear eSuations (12.1 ). Known harmonic current sources from converters and other harmonic
producing eSuipment are added into the vector [I ]. #
(KI Generating i(t) from 8(t)
Taking the fundamental and the odd harmonics 3, 5, 7, 9, 11, 13 and 15 into consideration reSuires
solutions of that system of eSuations, with [;] obviously being different for each of the harmonic freSuencies. For
lumped inductances L and capacitances C, it is clear that values T L and T C mustPbe used asPreactances and
susceptances in building [;]. Lines can be modelled as cascade connections of nominal B circuits, as long as the
number of B circuits per line is high enough to represent the line properly at the highest harmonic freSuency. It is
safer, however, to define the line data as distributed parameters, and to generate the exact eSuivalent B circuit at each
freSuency, as explained in Sections .2.1.2 and .2.1.3.
Once the voltages have been found for the fundamental and for the harmonics, an improved flux function
LN LN
8(t) can be calculated for each nonlinear inductance from the peak voltage phasors *V *e , *V *e , etc.,
12 31
Page 391
M *8P*
8(V) ' j sin(TPV % NP) (12. )
P'
TP
With 8(t) known, i(t) is again generated point by point as shown in Fig. 12.32, and then analyzed with the support
Traduciendo...
routine Fourier #nalysis to obtain an improved set of harmonics expressed as ES. (12. 2). These are then again used
to find an improved set of harmonic voltages. This iterative process is repeated until the changes in the harmonic
currents are sufficiently small. Experience has shown that 3 iterations are usually enough to obtain the harmonic
currents with an accuracy of v5 .
&KUETGRCPEKGU DGVYGGP *CTOQPKEU KP 5VGCF[ 5VCVG CPF 6TCPUKGPV 5QNWVKQP
The method described in the preceding section turns the EMTP into a harmonics load flow program. If it
is used that way, without a transient simulation following the steady state solution, then the problems discussed in
this section do not apply.
If the method is used as an improved initialization procedure for a subseSuent transient simulation, then
discrepancies can appear between the results from the steady state and transient solutions. These discrepancies were
not expected at first. They are cased by the unavoidable discretization error of the trapezoidal rule, which is used
for lumped inductances and capacitances in the EMTP. In the steady state solution for the n th harmonic, correct
reactance values T L would

P normally be used, while the transient simulation would see a somewhat larger reactance
kT L,
P with
)V
tan TP
2
M' (12. 5)
)V
TP
2
as explained in Section 2.2.1. The susceptance is also too large by the same factor k (Section 2.3.1). While a small
)t can keep the correction factor k of ES. (12. 5) reasonably close to 1.0 (e.g., )t 50 zs leads to a correction
factor of k 1.0015 at the 7th harmonic, or to an error of 0.15 ), it can never be avoided completely. Even small
errors can shift the resonance freSuencies of the network. Fig. 12.33 compares the impedance at the location of the
nonlinear inductance in the problem of Fig. .13, as it would be seen by a steady state phasor solution and by a
transient solution with the correction factor of ES. (12. 5). To emphasize the difference in Fig. 12.33, the line was
modelled as a cascade connection of three phase nominal B circuits,
12 32
Page 392
(KI FreSuency response with and without correction factor k ()t 200
zs)
rather than with distributed parameters. Since the EMTP uses other, more accurate, method for solving the eSuations
of distributed parameter lines, the differences would be much less with distributed parameter representations.
In the transient simulation, the discretization correction factor of ES. (12. 5) is unavoidable, and the answers
will therefore be slightly incorrect. In such situations, it may be best to introduce the same correction factor into
the initialization with the steady state solution method of Section 12. .2, to Traduciendo...
avoid discrepancies between initial
conditions and transient simulations. With this modification, the discrepancies between the initialization procedure
of Section 12. .2 and subseSuent transient simulations of an otherwise undisturbed network become practically
negligible.
Fig. 12.3 shows the transient simulation results for the same case used for Fig. .13, except that the
initialization procedure of Section 12. .2 was now used. It can be seen that the initial conditions must have contained
more or less correct harmonics because no disturbance is noticeable after t 0. Fig. 12.35 shows similar results
12 33
Page 393
(KI Same case as in Fig. .13, except simulation

starts from steady state with harmonics
for the case used in Fig. 12.29, with the initialization procedure of Section 12. .2. The improvement from the
inclusion of harmonics in the initialization is Suite evident in the second example.
Traduciendo...
(KI Same case as in Fig. 12.29, except simulation

starts from steady state with harmonics
12 3
Page 394
(GTTQTGUQPCPEG
#n attempt was made to apply the method of Section 12. .2 to ferroresonance cases, but with little success.
In ferroresonance phenomena, more than one steady state solution is possible. It depends very much on the initial
conditions and on the type of disturbance which one of these possible steady states will be reached. The method of
Section 12. .2 is therefore not useful for ferroresonance studies. The EMTP can be used for the simulation of
ferroresonance phenomena, however, though it will not give any insight into all possible steady state conditions.
In that sense, EMTP simulations are somewhat similar to transient stability simulations, which also do not give global
answers about the overall stability of the system.
12 35
Page 395
Traduciendo...
64#05+'06 #0#.;5+5 1( %10641. 5;56'/5 6#%5
Co #uthor: S. Bhattacharya
The program part T#CS (acronym for Transient #nalysis of Control Systems) was developed 10 years ago
by L. Dub¾. In 19 3/ , Ma 4en ming did a thorough study of the code, and made major revisions in it,
particularly with respect to the order in which the blocks of the control system are solved [1 7]. More improvements
will be made in the future by L. Dub¾ and others. Because changes are expected anyhow, and because L. Dub¾ was
not available for co authoring this section, the general philosophy of the solution method in T#CS and possible
alternatives are emphasized more than details of implementation.
T#CS was originally written for the simulation of HVDC converter controls, but it soon became evident
that it had much wider applications. It has been used for the simulation of
(a) HVDC converter controls,
(b) excitation systems of synchronous machines,
(c) current limiting gaps in surge arresters,
(d) arcs in circuit breakers,
and for other devices or phenomena which cannot be modelled directly with the existing network components in the
EMTP.
Control systems are generally represented by block diagrams which show the interconnections among
various control system elements, such as transfer function blocks, limiters, etc. Fig. 13.1 is a typical example. #
block diagram representation is also used in T#CS because it makes the data specification by the user simple. #ll
signals are assigned names which are defined by alphanumeric characters (blank is included as one of the
characters). By using the proper names for the input and output signals of blocks, any arbitrary connection of blocks
can be achieved. #mazingly, there is no uniform standard for describing the function of each block in an
unambiguous way, except in the case of linear transfer functions [1 9]. Users of the EMTP should be aware of this.
The control systems, devices and phenomena modelled in T#CS and the electric network are solved
separately at this time. Output Suantities from the network solution can be used as input Suantities in T#CS over
the same time step, while output Suantities from T#CS can become input Suantities to the network solution only over
the next time step. T#CS accepts as input network voltage and current sources, node voltages, switch currents,
status of switches, and certain internal variables (e.g., rotor angles of synchronous machines). The network solution
accepts output signals from T#CS as voltage or current sources (if the sources are declared as T#CS controlled
sources), and as commands to open or close switches (if the switch is a thyristor or a T#CS controlled switch).
13 1
Page 396
(KI Typical block diagram representation of a control
system
The present interface between the network solution and T#CS, and possible alternatives to it, are explained
first. The models available in T#CS are described next, followed by a discussion of the initialization procedures.
Traduciendo...
+PVGTHCEG DGVYGGP 6#%5 CPF VJG 'NGEVTKE 0GVYQTM
To solve the models represented in T#CS simultaneously with the network is more complicated than for
models of power system components such as generators or transformers. Such components can essentially be
represented as eSuivalent resistance matrices with parallel current sources, which fit directly into the nodal network
eSuations (1. ). The eSuations of control systems are Suite different in that respect. Their matrices are
unsymmetric, and they cannot be represented as eSuivalent networks.
Because of these difficulties, L. Dub¾ decided to solve the electric network (briefly called NETWO4K from
here on) and the T#CS models (briefly called T#CS from here on) separately. This imposes limitations which the
users should be aware of. #s illustrated in Fig. 13.2, the NETWO4K solution is first advanced from (t )t) to t
as if T#CS would not exist directly. There is an indirect link from T#CS to NETWO4K with a time delay of )t,
inasmuch as NETWO4K can contain voltage and current sources defined between (t )t) and t which were computed
as output signals in T#CS in the preceding step between (t 2)t) to (t )t). NETWO4K also receives commands
for opening and closing switches at time t, which are determined in T#CS in the solution from (t 2)t) to (t )t).
In the latter case, the error in the network solution due to the time delay of )t is usually negligible. First, )t for this
type of simulation is generally small, say 50 zs. Secondly, the delay in closing a thyristor switch is compensated
by the converter control, which alternately advances and retards the firing of thyristor switches to keep the current
constant in steady state operation. With continuous voltage and current source functions coming from T#CS, the
time delay can become more critical, however, and the user must be aware of its conseSuences. Cases have been
documented where this time delay of )t can cause numerical instability, e.g., in modeling the arc of circuit breakers
with T#CS [1 ].
13 2
Page 397
(KI Interface between NETWO4K and T#CS solution
Once NETWO4K has been solved, the network voltages and currents specified as input to T#CS are known
between (t )t) and t, and are then used to bring the solution of T#CS from (t )t) to t. No time delay occurs in
this part of the interface, except that T#CS itself has built in delays which may not always be transparent to the user.
If the EMTP is re written some day to eliminate the time delay from T#CS to NETWO4K, two approaches
(and possibly others) could be used:
(a) Predict the output from T#CS at time t, and use the predicted values to solve the NETWO4K and then
T#CS from (t )t) to t. Use the output from T#CS as corrected values, and repeat the solution of the two
parts again from (t )t) to t. If the differences between the predicted and corrected values are still larger
than a specified tolerance, then do another iteration step with a repeat solution, until the values have
converged to their final values. This approach is conceptually easy to implement, but its usefulness depends
on the convergence behavior. Two or three iteration steps, on average, would probably be acceptable. This
method would make it possible to add other corrections in NETWO4K and T#CS where only predictions
are used now (e.g. correction of predicted armature currents in synchronous machines).
Traduciendo...
(a) before reduction (b) after reduction
13 3
Page 398
(KI Form of difference eSuations for a control system. [Z +06'40#. ]

internal variables, [Z ] output
176 signals, and [W] input signals
b) Do not solve the eSuations in T#CS completely, but reduced them to an input output relationship at time
t, by eliminating variables which are internal to T#CS. This approach has been used successfully in a
stability program for the representation of excitation systems [72]. #ssume that the trapezoidal rule of
integration (or any other implicit integration method) is applied to the differential eSuations of the control
system. #ssume further that the variables are ordered in such a way that the internal variables [x +06'40#. ]
come first, then the output signals [x ] which176

become input to NETWO4K (v in an excitation system)
H
and finally the input signals [u] which come from the output of NETWO4K (v 6'4/+0#. in an excitation
system). Then the eSuations would have the form of Fig. 13.3(a). By eliminating the internal variables
[x +06'40#. ] with Gauss elimination, the reduced system of eSuations in the bottom rectangle of Fig. 13.3(b)
is obtained, which has the form
)
[#176 ] [Z176 ] % [#+0 ] [W] ' [D ] (13.1)
or in the case of an excitation system,
)
C176 XH % C+0 X6'4/+0#. ' D
In the latter case, this eSuation would have to be incorporated into the synchronous machine model of
Section . Limiters can be handled as well with this approach, as explained in [72].
Method (b) could be implemented in a number of different ways. For control systems which can be
represented by one transfer function, the implementation would be very simple, because T#CS already produces an
eSuation of the form of ES. (13.1), as explained later in ES. (13.7). For more complicated systems, the existing
code of T#CS could be used to solve the eSuations of system twice, e.g., in the case of the excitation system, for
2 predicted values of v 6'4/+0#. . The two solutions v (t) and

H v (t) would
H create 2 points in the v 6'4/+0#. v plane,
(
and a straight line through them would produce ES. (13.1) indirectly.
6TCPUHGT (WPEVKQP $NQEM YKVJ 5WOOGT
The transfer function block (Fig. 13. ) is used to describe a relationship between input U(s) and output X(s)
in the Laplace domain,
:(U) ' )(U) 7(U) (13.2a)
where the transfer function is a rational function of order n,
0 % 0 U...% 0OU O
)(U) ' - YKVJ O # P (13.2b)
& % & U...% &PU P
13
Page 399
Traduciendo...
The Laplace operator is replaced by jT to obtain the steady state freSuency response at any angular
freSuency T, including dc. For transient solutions, s is replaced by the differential operator d/dt, which converts
ES. (13.2) into a linear n th order differential eSuation
FZ FPZ FW F OW
&Z% & ...% &P '- 0W % 0 ...% 0O (13.3)
FV FV P FV FV O
(KI Transfer function block
In T#CS, the n th order differential eSuation is re written as a system of n first order differential eSuations
by introducing internal variables for the derivatives of u and x
FZ FZ FZP&
Z' , Z' , ... ... ZP '
FV FV FV
FW FW FWO&
W' , W' , ... ... WO ' (13. )
FV FV FV
With these internal variables, ES. (13.3) becomes an algebraic eSuation
& Z % & Z ... % &P ZP ' - (0 W % 0 W ... % 0O WO) (13.5)
To eliminate these internal variables again, the differential eSuations (13. ) are first converted into difference
eSuations with the trapezoidal rule of integration,
2 2
ZK(V) ' ZK& (V) & { ZK(V&)V) % ZK& (V&)V) } HQT K ' 1,... P (13. a)
)V )V
2 2
WL(V) ' WL& (V) & 6WL(V&)V) % WL& (V&)V)> HQT L ' 1,... O (13. b)
)V )V
where x x and u u.
Expressing x asP a function of x in ES.

P (13.5) with ES. (13. ), and then again expressing x as a function P
of x etc.,
P until only x is left, and using the same procedure for u, produces a single output input relationship of
the form [1 9]
13 5
Page 400
EZ(V) ' - FW(V) % JKUV(V & )V) (13.7)
This is the eSuation which is used in the transient solution of the control system. #fter the solution at each time step,
n history terms must be updated to obtain the single term hist for the solution over the next time step. If recursive
formulas are used, then
JKUV (V) ' -F W(V) & E Z(V) & JKUV (V&)V) % JKUV (V&)V)
... ' ...
... ' ...
JKUVK(V) ' -FKW(V) & EKZ(V) & JKUVK(V&)V) % JKUVK% (V&)V)

... ' ...
... ' ...
Traduciendo...
JKUVP(V) ' -FPW(V) & EPZ(V)

YKVJ (13. )
JKUV ' JKUV
The coefficients c, d are

K Kcalculated once at the beginning from the coefficients N, D of the transfer K K
function, with the recursive formula
K 2 2 P 2
EK ' EK& % (&2)K6( )K &K % (K% )K% &K% ...% ( )P &P > (13.9)
K) ()V K ) ( )V K ) ()V
where ( )LKis the binomial coefficient, and where the starting value is
P K
2
E' j &K , YKVJ E ' E (13.10)
K' )V
The formulas for d areKidentical, if D is replaced by N, and if the upper limit is m rather than n.
Instead of a single input signal u, T#CS accepts the sum of up to five input signals u ,...u , as illustrated
K
in Fig. 13.5 (subscripts 1,2,... are no longer used to indicate internal variables of a block from here on). To model
a summer by itself, a zero order transfer function block is used with K N D 1. This zero order transfer
function is contained in ES. (13.7) as a special case, with K c d 1 and hist 0.
13
Page 401
(KI Transfer function with summer (W W W W W

W)
If the control system consists solely of interconnected transfer function blocks and summers, then the entire
system is described by using an eSuation of the form (13.7) for each one of the blocks. For the example of Fig.
13. , there would be four eSuations
EC &-CFC &-CFC 0 -CFC 0 Z JKUVC
&-DFD 0 0 ED 0 &-DFD Z JKUVD

'
&-EFE EE 0 0 0 0 Z JKUVE
0 0 &-FFF EF 0 0 W JKUVF
which is the same form as in Fig. 13.2(a), with
[Z+06'40#. ] ' [Z Z Z ]
[Z176 ] ' [Z ]
Traduciendo...
[W] ' [W W ]
13 7
Page 402
(KI Control system with linear transfer functions
In T#CS, this system of eSuations
[#ZZ ] [:] % [#ZW ] [W] ' [JKUV] (13.11)
is solved by first performing a triangular factorization on {[# ]. [# ]} before

ZZ entering
ZW the time step loop. In each
time step, the unknown variables [x] are then found by
(a) assembling the right hand side [hist],
(b) performing a downward operation on it,
(c) doing a backsubstitution to obtain [x], and
(d) updating the history terms of each block.
The solution procedure is very similar to the one indicated in Fig. 13.3(b), except that the elimination does
not stop on the vertical line which separates the columns of [x +06'40#. ] and [x ],176
but continues to the diagonal
(indicated by dots in Fig. 13.3(b)). Since the matrix is unsymmetric here, both the upper and lower triangular matrix
coming out of the triangularization must be stored, in contrast to the matrix in NETWO4K where only the upper
triangular matrix is stored. Since the matrix is sparse, optimal ordering techniSues are used to minimize the number
of fill in elements. Only the nonzero elements are stored with a compact storage scheme similar to the one discussed
in #ppendix III. Whether pivoting is needed is unclear to the authors.
.KOKVGTU
There are two types of limiters, the windup limiter with clipped output ( static limiter in the EMTP 4ule
Book) and the non windup limiter with clamped output ( dynamic limiter in the EMTP 4ule Book). The windup
limiter can be visualized as a measuring instrument in which the needle (position output signal) can only be seen
within a limited window, but the needle is allowed to move freely (wind up) outside the window (Fig. 13.7(a)). In
the non windup limiter, the needle is restrained from moving outside the window (Fig. 13.7(b)). In both cases, the
movement of the needle is described by differential eSuations. The eSuation describing the limiting function has the
13
Traduciendo...
Page 403
same form in both cases, but the criteria for backing off are different.
(a) Windup limiter (b) Non windup limiter
(KI Limits in a measuring instrument
9KPFWR .KOKVGT
The output x of the windup limiter of Fig. 13. is
-W , KH ZOKP -W ZOCZ
Z ' 6 ZOKP , KH -W # ZOKP (13.12)
ZOCZ , KH -W $ ZOCZ
Either one of the three eSuations is still a linear algebraic eSuation of the form of ES. (13.7), with c d 1, hist
0 inside the limits, and c 1, d 0, hist (x or x ) at the limit. The properOKP

OCZ way of handling this limiter
is to change the linear eSuations (13.11) at instants when x hits the limit and when it moves off the limit again. This
reSuires occasional re triangularizations, which are no different in principle from those reSuired in NETWO4K
whenever switch positions change or when the solution in piecewise linear elements moves from one segment to
another.
(a) Block diagram representation (b) Limiting action
13 9
Page 404
(c) Clipping circuit implementation for

K 1 (diode # conducts when v v , +0 OCZ
diode B conducts when v v +0 OKP
(KI Windup limiter

Traduciendo...
If there are only a few limiters, one could also use the compensation method described in Section 12.1.2.
#ssume that the control system contains only two limiters which limit x and x . In that
K case,
M one could precalculate
column i and column k of the inverse matrix of [# ] with twoZZrepeat solutions before entering the time step loop.
In each time step, the variables would first be calculated as if no limits exist. Call this solution [x ]. If both
YKVJQWV x K
and x M YKVJQWV
YKVJQWV are outside their limits, then the necessary corrections )hist and )hist Kin the right hand
M side of ES.
(13.11) to produce limited values are found by solving the two eSuations
ZK&NKOKV ZK&YKVJQWV DKK DKM)JKUVK

' % (13.13a)
ZM&NKOKV ZM&YKVJQWVDMK DMM)JKUVM
If only x is outside
K YKVJQWV its limits, then
ZK&NKOKV ' ZK&YKVJQWV % DKK )JKUVK

)JKUVM ' 0 (13.13b)
The final solution is found by superposition,
DKDM
DKDM
)JKUVK
[Z] ' [ZYKVJQWV] % . . (13.13c)
)JKUVM
..
..
The coefficients b in ES. (13.3) are the elements of column i and k of [< ] . UU
#t the time when T#CS was first written, both the re triangularization procedure and the compensation
method were regarded as too costly, and the simpler method discussed in Section 13. was introduced instead. It
suffered initially from unnecessary time delays, which have now been mostly removed with the recent code changes
of Ma 4en ming in version M3 . Whether re triangularization or compensation will be used in future versions to
remove the remaining time delays remains to be seen.
In comparing Fig. 13. (a) with the piecewise linear representation of network elements discussed in Section
13 10
Page 405
12.1.3, one notices that the slope in the saturated region is always zero (hard limit) rather than finite (soft limit).
In rewriting T#CS, it may be worth considering whether soft limits would be a useful enhancement. In Fig. 13. (c)
the limits become soft if the internal resistances of the diodes and dc voltage sources are taken into account, or if
resistors are specifically added for that purpose. The eSuation for soft limits, with the notation from Fig. 13.9,
would be
-W KH ZOKP -W ZOCZ
:'6 ZOKP % -OKP(W&WOKP) KH -W # ZOKP (13.1 )
ZOCZ % -OCZ(W&WOCZ) KH -W $ ZOCZ
These eSuations have again the form of ES. (13.7), and soft limits can therefore be implemented in the same way
as hard limits. #s a matter of fact, the hard limit would become a special case of the soft limit of ES. (13.1 ) by
simply setting K or OCZ

K to zero.OKP
Traduciendo...
(KI Soft limits
0QP 9KPFWR .KOKVGT
In the windup limiter, the output of a transfer function block is just clipped, without affecting the dynamic
behavior of the transfer function block on the input side itself. In a non windup limiter, this is no longer true. Here,
the dynamic behavior of the transfer function block is changed by the limiting action.
Before describing the limiting action with eSuations, it is important to understand that non windup limiters
should only be used with first order transfer functions. For second and higher order transfer functions, it is no
longer clear which variables should be limited. Take a second order transfer function G(s) 1/ 2 as an example.
It can easily be shown [190] that backing off the limit will occur in three different ways in this case, depending on
whether the internal variables dx/dt or d x/dt , or both, are forced to remain at zero after the limit is hit. This
ambiguity can only be removed if the user defines the problem as two cascaded first order transfer function blocks,
13 11
Page 406
with the proper limits on each of them. Even for the first order transfer function, the meaning of the limiting
function is confused if it has any zeros (N û 0) [191]. It is because of these ambiguities why the limiter in the
current controlled dc voltage source described in Section 7. .2 may be incorrect.
To make the definition of non windup limiters uniSue, they should only be allowed on first order transfer
functions with no zeros of the form
-
)(U) ' (13.15)
1 % U6
The eSuations are
FZ
Z%6 ' -W KH ZOKP Z ZOCZ
FV
Z ' ZOKP KH Z # ZOKP CPF (-W & Z) 0 (13.1 )
Z ' ZOCZ KH Z $ ZOCZ CPF (-W & Z) 0
(a) Block diagram

representation
(b) Limiting action (c) Circuit implementation
(KI Non windup limiter
For operation within the limits, the differential eSuation is converted to the algebraic eSuation (13.7). That
eSuation, and the eSuations valid at the limit, are all linear algebraic eSuations, as in ES. (13.12) for the windup
limiter. The non windup limiter can therefore be handled in exactly the same way as the windup limiter, either with
re triangularization or with the compensation method. While the windup limiter has the coefficients c d 1, and
Traduciendo...
13 12
Page 407
hist 0 inside the limits, the non windup limiter has
F'-
26
E'1%
)V
26
JKUV ' -W(V & )V) & 1 & Z(V & )V)
)V
For deciding when to back off, the derivative T dx/dt Ku x must be used, rather than Ku.
Changing from a hard to a soft limit would also be possible with the non windup limiter. In the
implementation of Fig. 13.10(c), the limits would become soft if 4 û 0, where 4 can either be the internal
resistance of the <ener diode, or a resistor specifically added to create soft limits.
From the limited (unpublished) information available to the authors, it appears that T#CS handles the non
windup limiters with a pseudo compensation method, in which corrections are made to the right hand sides [hist]
in ES. (13.11) a priori at the beginning of each time step. #s explained above ES. (13.13), a correct implementation
of the compensation method reSuires a complete solution of the control system without limiters, followed by
superposition of correction terms for which elements of [# ] are needed.

ZZ This does not seem to be done in T#CS,
and the treatment of limiters is therefore somewhat suspicious. Since T#CS does reset the variable to its limit value
whenever it exceeds its limits, the answers are probably correct, except that the procedure is unable to eliminate the
time delays in closed loops discussed in Section 13. .
The pseudo compensation method also seems to create subtle differences in the way it backs off the limit.
It seems to use the eSuation
6
62Z(V) & ZNKOKV> ' - W(V) & Z(V)
)V
in the first step after backing off, which would be the backward Euler formula if the factor 2 were missing, while
ES. (13.7) would back off with
6 W(V) % W(V&)V) Z(V) % ZNKOKV

6Z(V) & ZNKOKV> ' - &
)V 2 2
.KOKVGT +ORNGOGPVCVKQP YKVJ 2QUUKDNG 6KOG &GNC[
With the code changes of Ma 4en ming in 19 3/ , the variables are now ordered in such a way that most
of the time delays which were caused by limiters no longer exist in version M3 and later versions. For example,
the open loop control system of Fig. 13.11 was originally solved in the seSuence S , G (s), G (s), S , S and finally
G (s), because of a rule that transfer function blocks feeding into special or supplemental device blocks S should be
solved first. This has been changed, and the blocks are now solved in their functional order S , G (s), G (s), S ,
13 13
Page 408
G (s), S . With this order, it is simple to observe the limits on the output x , without having to re triangularize the
matrix or without having to use the compensation method, because x is limited first before any other variables are
computed. In the system of eSuations (13.11), this means that the eSuation for G (s) must be the last one, with
enforcement of the limits on x being done in the backsubstitution. Ma 4en ming observes correctly [1 7] that there
is no difference between n th order and zero order transfer functions, or between windup and non windup limiters
in this simple ordering scheme.
# more complicated example for the new ordering rule is shown in Fig. 13.12, with 10 transfer function
Traduciendo...
blocks of which three have limits. The first four blocks G , G , G , and G form one set of eSuations which are
disconnected from the others. This first set of eSuations is solved simultaneously, with rows in ES. (13.7) ordered
G , G , G , and G . With this order, the output of G is the first variable to be found in the backsubstitution. By
keeping it within its limits at that point, the properly limited value will be used in the rest of the backsubstitution in
finding the outputs of G , G , and G . Using the known output of G , the output of G is found from one single
eSuation, and knowing the output of G , the output of G is found from another single eSuation and then kept within
its limits. Finally, the eSuations for G , G , G , and G for another independent set of eSuations, and if ordered in
that seSuence, the limits on the output of G can again be easily observed because it is the first variable found in the
backsubstitution. So in spite of feedback loops and limiters, the control system of Fig. 13.12 is now solved
simultaneously without the time delays observed in pre M3 versions. Note that this ordering scheme developed
for easy implementation of limiters may not completely minimize the fill ins in the triangularization, but this is a
small price to pay for the proper implementation of limiters.
(KI Simple open loop control system
Time delays cannot be avoided completely with the new ordering scheme. Fig. 13.13 shows an example
where two limiters are within the same loop. In this case, T#CS inserts a time delay of )t (if not explicitly done
so by the user) and the solution is then no longer simultaneous. Note that with re triangularization or with the
compensation method, the solution of that system would again become simultaneous.
13 1
Page 409
(KI Control system with feedback loops
(KI Two limiters within same loop
Traduciendo...
5KIPCN 5QWTEGU
T#CS has signal sources built into it, similar to the voltage and current sources in NETWO4K. They serve
as input signals to transfer function blocks and other blocks. In the system of eSuations (13.11), they are handled
as known values in vector [u].
4esident sources are signal sources with reserved names, which are available by simply referring to their
names. 4esident sources can also be used as voltage or current sources in NETWO4K through the T#CS
NETWO4K interface. They are
TIMEX simulation time in seconds (0 in steady state),
ISTEP number of time step,
DELT#T step size in seconds,
F4E3H< network freSuency in Hz of first sinusoidal source,
OMEG#4 2BF4E3H<,
<E4O 0.0,
MINUS1 1.0,
PLUS1 1.0,
13 15
Page 410
INFT; 4 (very large number which still fits computer system),
PI B.
There are also signal sources with data specifications supplied by the user, such as
single rectangular pulse,
sinusoidal function,
repetitive pulse,
repetitive ramp function,
node voltage from NETWO4K,
switch current from NETWO4K,
special NETWO4K variables (e.g., rotor angle of machines),
switch status,
any voltage or current source defined in NETWO4K.
5RGEKCN &GXKEGU
Transfer function blocks, limiters and signal sources are not enough to model realistic control systems.
Other building blocks have therefore been added to T#CS under the heading of Special Devices ( Supplemental
variables and devices in the EMTP 4ule Book). They make T#CS extremely versatile, but they do not fit neatly
into the control system eSuations (13.7). They are therefore solved seSuentially, rather than simultaneously as for
the transfer function blocks, with the user controlling the seSuence. In Fig. 13.11, the special device S would be
solved after G has been solved, and S would be solved after G has been solved. The solution would still be
simultaneous in this case. In general, the seSuence of calculations is more complicated, with non simultaneous
solutions through time delays. For details, the reader should consult the EMTP 4ule Book.
#ll special devices can either be designated as input devices, as output devices, or as internal devices by
the user. To make the solution as much simultaneous as possible, the user should keep the number of internal
devices as low as possible, and use input or output devices instead whenever possible. The rules for the designation
are as follows:
(a) Input devices: #ll inputs must either be T#CS signal sources or output from other input devices. They are
essentially used to pre process signals before they enter transfer function blocks (e.g., S in Fig. 13.11).
(b) Output devices: Their output must not be used as input to any other block, except to other output devices.
They are essentially used to post process control system outputs for its own sake, or before passing them
on as voltage or current sources or switching commands to NETWO4K (e.g. S in Fig. 13.11).
(c) Internal devices: They are inside the control system (e.g., S in Fig. 13.11).
The behavior of the special devices is either defined through user supplied FO4T4#N expressions, or with
built in types.
Traduciendo...
(1464#0 &GHKPGF 5RGEKCN &GXKEGU
13 1
Page 411
The FO4T4#N expression can be more or less as general as allowed by the FO4T4#N IV language itself
(for details see the EMTP 4ule Book). #lgebraic operators ( etc.), relational operators (.E3. etc.), logical
operators (.#ND. etc.), FO4T4#N intrinsic functions (SIN, EXP, etc.) and special functions defined in the EMTP
4ule Book can be used. In the example
#NGLE DEG(#T #N(CNT4L BI#S2)) 3 .2,
the output signal is #NGLE, while the input signals are CNT4L and BI#S2. #T #N is the arctangent function,
while DEG is a special function for converting radians to degrees.
# FO4T4#N expression of the form
V#4I#BLE V#4I#BLE {#rithmetic Expression}
is not allowed, because it gives rise to sorting problem within T#CS.
$WKNV +P 5RGEKCN &GXKEGU
There are 17 built in special devices at this time, for which the user supplies the parameters only. They
are
(a) accumulator and counter,
(b) controlled integrator,
(c) digitizer,
(d) freSuency sensor,
(e) input IF component,
(f) instantaneous min/max,
(g) level triggered switch,
(h) min/max tracking,
(i) multi operation time seSuenced switch,
(j) point by point user defined nonlinearity,
(k) pulse transport delay,
(l) relay operated switch,
(m) 4MS value,
(n) sample and track hold,
(o) signal selector,
(p) simple derivative (backward Euler),
(S) transport delay.
Details about their characteristics can be found in the EMTP 4ule Book.
+PKVKCN %QPFKVKQPU
The ac steady state solution for the electric network is found first, before T#CS variables are initialized.
#ll variables from NETWO4K are therefore available for the automatic initialization in T#CS, but not the other way
13 17
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around. This may cause problems, e.g., if a T#CS output defines a sinusoidal voltage source in NETWO4K whose
initial amplitude and phase angle, supplied by the user, could differ from the values coming out the T#CS
initialization. #n iterative steady state solution between NETWO4K and T#CS would resolve such discrepancies,
but they are probably so rare that such an iteration scheme cannot be justified. #n error message (possibly with
termination) would be useful, however.

Traduciendo...
The automatic initialization of T#CS variables is complicated and not foolproof at this time, and
improvements are likely to be added in the future. The present initialization procedure is therefore only described
in broad terms.
The input and output signals of transfer function blocks are usually dc Suantities in steady state. For dc
Suantities, the output input relationship of the transfer function (13.2b) becomes
0
ZFE ' - WFE (13.17)
&
which is the same form as ES. (13.7) for the transient solution, with c 1, d N /D , and hist 0. If the entire
control system consists of transfer function blocks only, a system of eSuations can be formed, similar to ES. (13.11),
and solved for the unknown T#CS variables [x ]. This is essentially

FE what T#CS does automatically now. The
variables [x ] FE
are not needed directly, but only indirectly for initializing [hist] in ES. (13.11) before entering the time
step loop.
Unfortunately, control systems are more complicated. #ny sophisticated control system has integrators G(s)
K 1/ . Their steady state output must now be supplied by the user, but these values are not always easy to find.
For example, the output of an unbounded integrator with nonzero input is a continuously increasing ramp function.
In practice, integrators are always bounded within upper and lower limits. Therefore, the steady state output of a
bounded integrator is either at its minimum or maximum value, which T#CS could distinguish from the sign of the
input signal. # realistic steady state eSuation of a bounded integrator for nonzero input would therefore be
ZOKP KH WFE 0
ZFE ' 6 (13.1 )
ZOCZ KH WFE 0
Evaluation of the steady state output value of a bounded integrator with zero input or of an unbounded integrator is
impossible from the knowledge of its input alone.
Further complications are introduced because T#CS signal sources are not restricted to dc Suantities in
steady state. They could be pulse trains, sinusoidal functions, and other periodic functions. To automate the
initialization procedure for all such eventualities is therefore still an unresolved issue. #t this time, the user must
supply initial conditions in complicated cases and for most special devices.
13 1
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#22'0&+: + 07/'4+%#. 51.76+10 1( 14&+0#4; &+(('4'06+#. '37#6+105
Computers are influencing network theory by demanding methods of analysis adapted to the solution of
computer sized problems, as stated by F.H. Branin [2], but very little of this influence has shown up yet in
textbooks on electric circuits and networks, not even in most of the recently published books. In this appendix, an
attempt is made to summarize some of the numerical solution techniSues for solving ordinary differential eSuations,
which one might consider in developing a general purpose program, such as the EMTP. Since power system
networks are mostly linear, techniSues for linear ordinary differential eSuations are given special emphasis.
+ %NQUGF (QTO 5QNWVKQP
Let us assume that the differential eSuations are written in state variable form, and that the eSuations are
linear,
FZ
'[#][Z]%[I(V)], (I.1)
FV
with a constant sSuare matrix [#], and a vector of known forcing functions [g(t)]. There is no uniSue way of writing
Traduciendo...
eSuations in state variable form, but it is common practice to choose currents in inductances and voltages across
capacitances as state variables. For example, ES. (I.1) could have the following form for the network of Fig. I.1:
(KI + Energization of an 4 L C network
FK 4 1
& & 1
FV . . K X(V)
' %. (I.2)
FXE 1 XE
0 0
FV %
With Laplace transform methods, especially when one output is expressed as a function of one input, the
system is often described as one n order VJ

differential eSuation, e.g., for the example of Fig. I.1 in the form
VJ
Such an n order differential eSuation can of course always be rewritten as a system of n first order differential
eSuations, by introducing extra variables x dx /dt, x dx /dt, to x dx /dt, for the higher
P orderP derivatives,
I1
Page 414
with x x. In the example, with x i and x di/dt,
4 . FZ 1 FX(V)
& 0 Z
FV % FV
( ' ( %
FZ
Z
1 0 FV 0 1 0
which, after pre multiplication with
& 01
4.
' 1 4
10 &
. .
produces another state variable formulation for this example. While [#] of this formulation differs from that in ES.
(I.2), its eigenvalues are the same.
The closed form solution of ES. (I.1), which carries us from the state of the system at t )t to that at t, is
V
[Z(V)]'G=#?)V@[Z(V&)V]% G=#? V&W [I(W)]FW, (I.3)
mV&)V
=#?)V
where the matrix e is called the transition matrix. ES. (I.3) contains the case where [x(t)] is simply desired
as a function of t by setting )t t. The computational task lies in finding this transition matrix. Since there is no
closed form solution for the matrix exponential e , the way=#?)V

out is to transform this matrix to a diagonal matrix,
whose elements can easily be evaluated by using the eigenvalues 8 of [#] and the matrix of eigenvectors (modal
matrix) [M] of [#]. and then to transform back again. #n efficient method for finding eigenvalues appears to be the
34 transformation due to J.G.F. Francis [3], and for finding eigenvectors the inverse iteration scheme due to
J.H. Wilkinson [ ], which has also been described in modified form by J.E. Van Ness [5]. With [7] and [M]
=#?)V
known, where [7] is the diagonal matrix of eigenvalues 8, e is diagonalized
K ,
[/]& G=#?)V[/] ' [G 7)V]
8K)V
Once the diagonal elements e have been found, this can be converted back to give
G=#?)V ' [/] [G 7)V] [/& ] (I. )
where
7)V 8K)V
[e ] diagonal matrix with elements e ,
Traduciendo...
[M] eigenvector (modal) matrix of [#], and
=#?)V
If [M] diagonalizes [#], it will also diagonalize e . The matrix exponential is defined as the series of ES.
P
(I.13), and then one simply has to show that [M] not only diagonalizes [#], but all positive powers [#] as well.
P P
Since [#] [M][7][M] it follows that [#] ([M][7][M] )([M][7][M] )...([M][7][M] ) of [#]
P
[M][7 ][M] . Therefore, [M] [#] [M] [7 ] isP again diagonal.
P
I2
Page 415
8 Keigenvalues of [#].
With ES. (I. ), ES. (I.3) becomes
[Z(V)] ' [/] [G 7)V] [/]& [Z(V&)V)] % V

[/] [G 7 V&W ] [/]& [I(W)] FW
(I.5)
mV&)V
The convolution integral in ES. (I.5) can be evaluated in closed form for many types of functions [g(t)].
For the network of Fig. I.1, the eigenvalues can be obtained by setting the determinant of [#] 8[U] to zero
( [U] identity matrix),
4 1
& &8
/00000000000 & /00000000000
. .
'0
1
&8
%
or
4 4 1
8' & v ( )& (I. )
2. 2. .%
If 4 2%(L/C), then the system is underdamped , and the argument under the sSuare root will be negative, giving
a pair of complex eigenvalues
4 1 4
8 ' " v L$ YKVJ " ' & ,$' &( ) (I.7)
2. .% 2.
For a specific case, let us assume that 4 1S, L 1H, C 1F. then
1 3
8'& vL ' GvL E
2 2
and
1 1 GL E &1
1 2
[/] ' , [/]& '
2 G&L E GL E L3 &G&L E 1
If we set )t t to obtain the state variables simply as a function of time and of initial conditions, then ES. (I.5)
becomes
If 4 2sSrt(L/C), then the system is overdamped, giving two real eigenvalues. The critically damped
case of 4 2 sSrt(L/C) seldom occurs in practice it leads to two identical eigenvalues. This latter case of
multiple eigenvalues may reSuire special treatment, which is not discussed here.
I3
Page 416
1 2
G"V (cos$V & sin$V & G"Vsin$V
K(V) 3 3 K(0)
' Traduciendo... @
XE(V) 2 1 XE(0)
G"Vsin"V G"V(cos$V % sin$V)
3 3 (I. )
1
G" V&W [cos($(V&W)) & sin($(V&W))]X(W)
2 V
% 3 FW
3 m
G" V&W sin($(V&W))X(W)
with " and $ as defined in ES. (I.7). If we were to assume that the voltage source is zero and that v (0) 1.0 p.u., E
then we would have the case of discharging the capacitor through 4 L, and from ES. (I. ) we would immediately
get (realizing that i(0) 0 ),
2
K(V) ' & G"V sin$V
3
1
XE(V) ' G "V (cos$V % sin$V)
3
Could such a closed form solution be used in an EMTP? For networks of moderate size, it probably could.
J.E. Van Ness had no difficulties finding eigenvalues and eigenvectors in systems of up to 120 state variables [5].
If the network contains switches which freSuently change their position, then its implementation would probably
become very tricky. Combining it with Bergeron s method for distributed parameter lines, or with more
sophisticated convolution methods for lines with freSuency dependent parameters, should in principle be possible.
Where the method becomes almost unmanageable, or useless, is in networks with nonlinear elements. #nother
difficulty would arise with the state variable formulation, because ES. (I.1) cannot as easily be assembled by a
computer as the node eSuations used in the EMTP. This difficulty could be overcome, however, since there are
ways of using node eSuations even for state variable formulations, by distinguishing node types according to the
types of branches (4, L, or C) connected to them.
Where do Laplace transform methods fit into this discussion since they provide closed form solutions as
well? To Suote F.H. Branin [2], ...traditional methods for hand solution of networks are not necessarily best for
use on a computer with networks of much greater size. the Laplace transform techniSues fit this category and should
at least be supplemented, if not supplanted, by numerical methods better adapted to the computer He then goes
on to show that essentially all of the information obtainable by Laplace transforms is already contained in the
eigenvalues and eigenvectors of [#]. It is surprising that very few, if any, textbooks show this relationship. The
Laplace transform of ES. (I.1) is
U[:(U)] & [Z(0)] ' [#]@[:(U)] % [)(U)] (I.9a)
Page 417
or rewritten
( U [7] & [#] ) @ [:(U)] ' [Z(0) % [)(U)] (I.9b)
From which the formal solution in the s domain is obtained as
[:(U)] ' ( U[7] & [#] )& @ ( [Z(0)] % [)(U)] ) (I.10)
The computational task in ES. (I.10) is the determination of the inverse of (s[U] [#]). The key to doing this
efficiently is again through the eigenvalues and eigenvectors of [#]. With that information, the matrix (s[U] [#])
is diagonalized,
[/]& @ ( U [7] & [#] ) @ [/] ' U [7] & [7] (I.11)
and then the inverse becomes
( U [7] & [#] )& ' [/] @ ( U [7] & 7 )& @ [/]& (I.12)
in which the inverse on the right hand side i now trivial to calculate since (s[U] [7]) is a diagonal matrix (that is,
one simply takes the reciprocals of the diagonal elements). To Suote again from F.H. Branin [2], ...one of the more
Traduciendo...
interesting features of this method is the fact that it is far better suited for computer sized problems than the
traditional Laplace transform techniSues involving ratio of polynomials and the poles and zeros thereof. In
particular, the task of computing the coefficients of the polynomials in a network function P(s)/3(s) is not only time
consuming but also prone to serious numerical inaccuracies, especially when the polynomials are of a high degree.
The so called topological formula approach [25] to computing these network functions involves finding all the trees
of a network and then computing the sum of the corresponding tree admittance products. But the number of trees
may run into millions for a network with only 20 nodes and 0 branches. #nd even if this were not enough of an
impediment, the computation of the roots of the polynomials P(s) and 3(s) is hazardous because these roots may be
extremely sensitive to errors in the coefficients. In the writer s judgment, therefore, the polynomial approach is just
not matched to the network analysis tasks which the computer is called upon to handle. The eigenvalue approach
is much better suited and gives all of the theoretical information that the Laplace transform methods are designated
to provide. For example, the eigenvalues are identical with the poles of the network functions. Moreover, any
network function desired may be computed straightforwardly and its sensitivity obtained, either with respect to
freSuency or with respect to any network parameter. Finally, even the pole sensitivities can be calculated...
+ 6C[NQT 5GTKGU #RRTQZKOCVKQP QH 6TCPUKVKQP /CVTKZ
The matrix exponential e can be=#?)V

approximated by a power series, derived from a Taylor series
expansion,
I5
Page 418
)V )V )V
G=#?)V ' [7] % )V@[#] % [#] % [#] % [#] % ... (I.13)
2! 3! !
This series is, in effect, the definition of the matrix exponential.
Using ES. (I.13), necessarily with a finite number of terms, appears to offer a way around the computation
of eigenvalues. However, the method runs headlong into another kind of eigenvalue problem which limits its
usefulness: namely, that when the matrix [#] has a large eigenvalue (which means a small time constant), the
integration step )t must be kept small in order to permit rapid convergence of ES. (I.13) [2]. This refers to the
problem encountered in stiff systems , where there are large differences between the magnitudes of eigenvalues,
and where the largest eigenvalues produce ripples of little interest to the engineer, who is more interested in the
slower changes dictated by the smaller eigenvalues, as indicated in Fig. I.2. The method of using ES. (I.13) becomes
numerically unstable, for a given finite
(KI + 4esponse of a stiff system
number of terms if )t is not sufficiently small to trace the small, uninteresting ripples. It is, therefore, not a practical
method for an EMTP. It exhibits the same proneness to numerical instability as the 4unge Kutta method discussed
in Section I.5, which is not too surprising, since this method becomes identical with the fourth order 4unge Kutta
method if 5th and higher order terms are neglected in ES. (I.13), at least if the forcing function [g(t)] in ES. (I.1)
is zero ( autonomous system ), as further explained in Section I.5. Since this method is not practical, more details
such as the handling of the convolution integral in ES. (I.3) are not discussed.
Traduciendo...
+ 4CVKQPCN #RRTQZKOCVKQP QH 6TCPUKVKQP /CVTKZ
# rational approximation for the matrix exponential, which is numerically stable and therefore much better
than ES. (I.13), is due to E.J. Davison [ ],
This was pointed out to the writer by K.N. Stanton when he was at Purdue University (now President of
ESC# Corp. in Seattle)
Page 419
)V )V )V
G=#?7V . ( [7] & [#] % [#] & [#] )& @
2 12
(I.1 )
)V )V )V
( [7] % [#] % [#] % [#] )
2 12
# lower order rational approximation, which is also numerically stable for all )t, neglects the second and high order
terms in ES. (I.1 ).
)V )V
G=#?)V . ( [7] & [#] )& @ ( [7] % [#] ) (I.15)
2 2
This is identical with the trapezoidal rule of integration discussed in the following section.
Would it be worthwhile to improve the accuracy of the EMTP, which now uses the trapezoidal rule, with
the higher order rational approximation of ES. (I.1 )? This is a difficult Suestion to answer. First of all, the EMTP
is not based on state variable formulations, and it is doubtful whether this method could be applied to individual
branch eSuations as easily as the trapezoidal rule (see Section 1). Furthermore, if sparsity is to be exploited, much
of the sparsity in [#] could be destroyed when the higher order terms are added in ES. (I.1 ). By and large,
however, the writer would look favorably at this method if the objective is to improve the accuracy of EMTP results,
even though it is somewhat unclear how to handle the convolution integral in ES. (I.3).
+ 6TCRG\QKFCN 4WNG QH +PVGITCVKQP
Since this is the method used in the EMTP, the handling of the forcing function [g(t)] in ES. (I.1), or
analogously the handling of the convolution integral in ES. (I.3), shall be discussed here. Let ES. (I.1) be rewritten
as an integral eSuation,
V
[Z(V)] ' [Z(V&)V)] % ( [#] [Z(W) % [I(W)] ) FW (I.1 )
mV&)V
which is still exact. By using linear interpolation on [x] and [g] between t )t and t, assuming for the time being that
[x] were known at t (which, in reality, is not true, thereby making the method implicit ), we get
)V )V
[Z(V)] ' [Z(V&)V)] % [#] @ ( [Z(V&)V)] % [Z(V)] ) % @ ( [I(V&)V)] % [I(V)] ) (I.17)
2 2
Linear interpolation implies that the areas under the integral of ES. (I.1 ) are approximated by trapezoidals (Fig.
I.3) therefore the name trapezoidal rule of integration . The method is identical with using central difference
Suotients in ES. (I.1),
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Page 420
Traduciendo...
[Z(V)] & [Z(V&)V)] [Z(V&)V)] % [Z(V)] [I(V&)V)] % [I(V)

' [#] % (I.1 )
)V 2 2
(KI + Trapezoidal rule of integration
and could just as well be called the method of central difference Suotients . ES. (I.17) and (I.1 ) can be rewritten
as
)V )V )V
( [7] & [#] ) @ [Z(V)] ' ( [7] % [#] ) @ [Z(V&)V)] % ( I(V&)V)] % [I(V)] ) (I.19)
2 2 2
which, after premultiplication with ([U] )tC[#]/2) , shows that we do indeed get the approximate transition matrix
of ES. (I.15).
Working with the trapezoidal rule of integration reSuires the solution of a system of linear, algebraic
eSuations in each time step. If )t is not changed, and as long as no network modifications occur because of
switching or nonlinear effects, the matrix ([U] )tC[#]/2) for this system of eSuations remains constant. It is
therefore best and most efficient to triangularize this matrix once at the beginning, and again whenever network
changes occur, and to perform the downward operations and backsubstitutions only for the right hand side inside
the time step loop, using the information contained in the triangularized matrix. The solution process is broken up
into two parts in this scheme, one being the triangularization of the constant matrix, the other one being the repeat
solution process for right hand sides (which is done repeatedly inside the time step loop). this concept of splitting
the solution process into one part for the matrix and a second part for the right hand side is seldom mentioned in
textbooks, but it is very useful in many power system analysis problems, not only here, but also in power flow
iterations using a triangularized [;] matrix, as well as in short circuit calculations for generating columns of the
inverse of [;] one at a time. For more details, see #ppendix III.
It may not be obvious that the trapezoidal rule applied to the state variable eSuations (I.1) leads to the same
answers as the trapezoidal rule first applied to individual branch eSuations, which are then assembled into node
eSuations, as explained in Section 1. The writer has never proved it, but suspects that the answers are identical.
For the example of Fig. I.1, this can easily be shown to be true.
The trapezoidal rule of integration is admittedly of lower order accuracy than many other methods, and it
is therefore not much discussed in textbooks. It is numerically stable, however, which is usually much more
Page 421
important in power system transient analysis than accuracy by itself. Numerical stability more or less means that
the solution does not blow up if )t is too large instead, the higher freSuencies will be incorrect in the results (in
practice, they are usually filtered out), but the lower freSuencies for which the chosen )t provides an appropriate
sampling rate will still be reasonably accurate. Fig. I. illustrates this for the case of a three phase line energization.
This line was represented as a cascade connection of 1 three phase nominal B circuits. The curve for )t 5E
(based on f 0 Hz, i.e., )t 231. zs)
Traduciendo...
(KI + Switching surge overvoltage at the receiving end in a three phase open ended
line
cannot follow some of the fast oscillations noticeable in the curve for )t 0.5E, but the overall accuracy is not too
bad. The error between the exact and approximate value at a particular instant in time is obviously not a good
measure by itself for overall accuracy, or for the usefulness of a method for these types of studies. In Fig. I. , an
error as large as 0. p.u. (at the location of the arrow, assuming that the curve for )t 0.5E gives the exact value)
is perfectly acceptable, because the overall shape of the overvoltages is still represented with sufficient accuracy.
# physical interpretation of the trapezoidal rule of integration for inductances is given in Section 2.2.1. This
interpretation shows that the eSuations resulting from the trapezoidal rule are identical with the exact solution of a
lossless stub line, for which the answers are always numerically stable though not necessarily as accurate as desired.
+ 4WPIG -WVVC /GVJQFU
These methods can be used for any system of ordinary differential eSuations,
FZ
[ ] ' [ H ([Z], V) ] (I.20)
FV
There are many variants of the 4unge Kutta method, but the one most widely used appears to be the following
fourth order method: Starting from the known value [x(t )t), the slope is calculated at the point 0 (Fig. I.5(a)),
I9
Page 422
[)Z ]
' [ H( [Z(V&)V) ], V&)V ) ] (I.21a)
)V
which is then used to obtain an approximate value [x ] at midpoint 1,
1
[Z ] ' [ Z(V&)V) ] % [ )Z ] (I.21b)
2
(a) (b) (c) (d)
(KI + Fourth order 4unge Kutta method
Now, the slope is recalculated at midpoint 1 (Fig. I.5(b)),
[ )Z ] V
' [ H( [Z ], V&) ] (I.21c)
)V 2
and this is used to obtain a second approximate value [x ] at midpoint 2,
1
[ Z ] ' [ Z(V&)V) ] % [ )Z ] (I.21d)
2
Traduciendo...
Then the slope is evaluated for a third time, now at midpoint 2 (Fig. I.5(c)),
[ )Z ] )V
' [ H ( [Z ], V& )] (I.21e)
)V 2
which is used to get an approximate solution in point 3 at time t,
[ Z ] ' [ Z(V&)V) ] % [ )Z ] (I.21f)
Finally, the slope is evaluated for a fourth time in point 3,
I 10
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[ )Z ]
' [ H ( [Z ], V ) ] (I.21g)
)V
From these four slopes in 0, 1, 2, 3 (Fig. I.5(d)), the final value at t is obtained by using their weighted averages,
)V [)Z ] [ )Z ] [ )Z ] [ )Z ]
[Z(V)] ' [Z(V&)V)] % @( %2 %2 % )
)V )V )V )V
(I.22)
The mathematical derivation of the 4unge Kutta formula is Suite involved (see, for example, in [7]).
Intuitively, it can be viewed as an exploration of the direction field at a number of sample points (0,1,2,3 in Fig.
I.5). There are variants as to the locations of the sample points, and hence as to the weights assigned to them. There
are also lower order 4unge Kutta methods which use fewer sample points.
#s already mentioned in Section I.2, the fourth order 4unge Kutta method of ES. (I.21) and (I.22) is
identical with the fourth order Taylor series expansion of the transition matrix if the differential eSuations are linear,
at least for autonomous systems with [g(t)] 0 in ES. (I.1). In that case, ES. (I.1) becomes
[ )Z ] )V
' [#] [Z(V&)V)] , [ Z ] ' ( [7] % [#] ) [Z (V&)V)]
)V 2
With these values, the second slope becomes
[)Z ] )V
' ( [#] % [#] ) @ [Z (V&)V)]
)V 2
and
)V )V
[Z ] ' ( [7] % [#] % [#] ) @ [Z(V&)V)]
2
Then the third slope becomes
[)Z ] )V )V
' ( [#] % [#] % [#] ) @ [Z(V&)V)]
)V 2
and
)V )V
[Z ] ' ( [7] % )V [#] % [#] % [#] ) @ [Z(V&)V)]
2
If the slopes are calculated at a number of points and graphically displayed as short lines, then one gets a
sketch of the direction field , as indicated in Fig. I.5(d).
I 11
Traduciendo...
Page 424
from which the fourth slope is calculated as
[)Z ] )V )V
' ( [#] % )V [#] % [#] % [#] ) @ [Z(V&)V)]
)V 2
Finally, the new value is obtained with ES. (I.22) as
)V )V )V
[Z(V)] ' ( [7] % )V [#] % [#] % [#] % [#] ) @ [Z(V&)V)]
2 2
which is indeed identical with the Taylor series approximation of the transition matrix in ES. (I.13).
If [#] is zero in ES. (I.1), that is, if [x] is simply the integral over the known function [g(t)], then the fourth
order 4unge Kutta method is identical with Simpson s rule of integration, in which the curve is approximated as a
parabola going through the three known points in t )t, t )t/2, and t (Fig. I. ).
(KI + Simpson s rule
The 4unge Kutta method is prone to numerical instability if )t is not chosen small enough. It becomes
painfully slow in the case of problems having a wide spread of eigenvalues. For the largest eigenvalue (or,
eSuivalently, its reciprocal, the smallest time constant) controls the permissible size of )t. But the smallest
eigenvalues (largest time constants) control the network response and so determine the total length of time over which
the integration must be carried out to characterize the response. In the case of a network with a 1000 to 1 ratio of
largest to smallest eigenvalue, for instance, it might be necessary to take in the order of 1000 times as many
integration steps with the 4unge Kutta method as with some other method which is free of the minimum time
constant barrier [2}. This problem is indicated in Fig. I.2: Though the ripples may be very small in amplitude,
they will cause the slopes to point all over the place, destroying the usefulness of methods based on slopes.
+ 2TGFKEVQT %QTTGEVQT /GVJQFU
These methods can again be used for any system of ordinary differential eSuations of the type of ES.
(I.20).To explain the basic idea, let us try to apply the trapezoidal rule to ES. (I.20), which would give us
I 12
Page 425
)V
[Z J ] ' [Z(V&)V)] % ( [ H ( [Z(V&)V)], V&)V ) ] % [ H ( [Z J& ], V ) ] ) (I.23)
2
In the linear case discussed in Section I. , this eSuation could be solved directly for [x]. In the general (time varying
or nonlinear) case, this direct solution is no longer possible, and iterative techniSues have to be used. This has
already been indicated in ES. (I.23) by using superscript (h) to indicate the iteration step at the same time, the
argument t has been dropped to simplify the notation. The iterative techniSue works as follows:
1. Use a predictor formula, discussed further on, to obtain a predicted guess [x ] for the solution at time
t.
J
2. In iteration step h (h 1,2,...), insert the approximate solution [x ] into the right hand side of ES. (I.23)
J
to find a corrected solution [x ].
Traduciendo...
J J
3. If the difference between [x ] and [x ] is sufficiently small, then the integration from t )t to t is
completed. Otherwise, return to step 2.
ES. (I.23) is a second order corrector formula. To start the iteration process, a predictor formula is needed
for the initial guess [x ]. # suitable predictor formula for ES. (I.23) can be obtained from the midpoint rule,
[Z (I.2 )
V] ' [Z(V&2)V)] % 2)V [ H ([Z(V&)V)], V&)V) ]
or from an extrapolation of known values at t 3)t, t 2)t, and t )t,
3
[Z )V( [ H([Z(V&)V)], V&)V) ] % [ H([Z(V&2)V)], V&2)V) ] ) (I.25)
V] ' [Z(V&3)V)] % 2
The difference in step 3 of the iteration scheme gives an estimate of the error, which can be used
(a) to decide whether the step size )t should be decreased (error too large) or can be increased (error very
small), or
(b) to improve the prediction in the next time step.
It is generally better to shorten the step size )t than to use the corrector formula repeatedly in step 2 above. In using
the error estimate to improve the prediction, it is assumed that the difference between the predicted and corrected
values changes slowly over successive time steps. This past experience can then be used to improve the prediction
with a modifier formula. Such a modifier formula for the predictor of ES. (I.25) and for the corrector of ES. (I.23)
would be
9
[Z ( [Z(V&)V)] & [Z (I.2 )
KORTQXGF] ' [Z ] % 10 V&)V ] )
Besides the second order methods of ES. (I.23) to ES. (I.2 ), there are of course higher order methods.
Fourth order predictor corrector methods seem to be used most often. #mong these are Milne s method and
Hamming s method, with the latter one usually more stable numerically. The theory underlying all predictor
I 13
Page 426
corrector methods is to pass a polynomial through a number of points at t, t )t, t 2)t, ..., and to use this polynomial
for integration. The end point at t is first predicted, and then once or more often corrected. Obviously, the
convergence and numerical stability properties of the corrector formula are more important than those of the
predictor formula, because the latter is only used to obtain a first guess and determines primarily the number of
necessary iteration steps. The predictor and corrector formula should be of the same order in the error terms. There
are different classes of predictors: #dams Bashforth predictors (obtained from integrating Newton backward
interpolation formulas), Milne type predictors (obtained from an open Newton Cotes forward integrating formula),
and others. Note that those formulas reSuiring values at t 2)t, or further back, are not self starting 4unge Kutta
methods are sometimes used with such formulas to build up enough history points.
It is Suestionable whether non self starting high order predictor corrector formulas would be very useful
for typical power system transient studies, since waves from distributed parameter lines hitting lumped elements look
almost like discontinuities to the lumped elements, and would therefore reSuire a return to second order predictor
correctors each time a wave arrives. In linear systems, the second order corrector of ES. (I.23) can be solved
directly, however, and is then identical with the trapezoidal rule as used in the EMTP.
+ &GHGTTGF #RRTQCEJ VQ VJG .KOKV 4KEJCTFUQP 'ZVTCRQNCVKQP CPF 4QODGTI +PVGITCVKQP
The idea behind these methods is fairly simple. Instead of using higher order methods, the second order
trapezoidal rule (either directly with ES. (I.17) for linear systems, or iteratively with ES. (I.23) for more general
systems) is used more than once in the interval between t )t and t, to improve the accuracy. #ssume that the normal
step size )t is used to find [x ] at t from [x(t )t)], as indicated in Fig. I.7. Now repeat the integration with the
trapezoidal rule with half
Traduciendo...
(KI + 4ichardson
extrapolation
the step size )t/2, and perform two integration steps to obtain [x ]. With the two values [x ] and [x ], an
intelligent guess can be made as to where the solution would end up if the step size were decreased more and more.
This extrapolation towards )t 0 (4ichardson s extrapolation) would give us a better answer
I1
Page 427
1
[Z(V)] ' [Z ] % ( [Z ] & [Z ] ) (I.27)
3
The accuracy can be further improved by repeating the integration between t )t and t with , ,1 ,...
intervals. The corresponding extrapolation formula for )t 6 0 is known as 4omberg integration.
Whether any of these extrapolation formulas are worth the extra computational effort in an EMTP is very
difficult to judge. Some numerical analysts seem to feel that these methods look very promising. They offer an
elegant accuracy check as well.
+ 0WOGTKECN 5VCDKNKV[ CPF +ORNKEKV +PVGITCVKQP
The writer believes that the numerical stability of the trapezoidal rule has been one of the key factors in
making the EMTP such a success. It is therefore worthwhile to expound on this point somewhat more.
The trapezoidal rule belongs to a class of implicit integration schemes, which have recently gained favor
amongst numerical mathematicians for the solution of stiff systems , that is, for systems where the smallest and
largest eigenvalues or time constants are orders of magnitude apart [70]. Most power systems are probably stiff in
that sense. While implicit integration schemes of higher order than the trapezoidal rule are freSuently proposed, their
usefulness for the EMTP remains Suestionable because they are numerically less stable. # fundamental theorem due
to DahlSuist [71] states:
Theorem: Let a multistep method be called # stable, if, when it is applied to the problem [dx/dt] 8[x],
4e(8) 0, it is stable for all )t 0.
Then: (i) No explicit linear multistep method is # stable.
(ii) No implicit linear multistep method of order greater than two is # stable.
(iii) The most accurate # stable linear multistep method of order two is the trapezoidal rule.
To illustrate the problem of numerical stability, let us assume that a fast oscillation somewhere in the network
produces ripples of very small amplitudes, which do not have any influence on the overall behavior of the network,
similar to those shown in Fig. I.2. Such a mode of oscillation could be described by [72]
FZ
% Z ' 0, YKVJ Z(0) ' 0, FZ/FV(0) ' 10& (I.2 )
FV
with its exact solution being
x 10 sin(t) (I.29)
The amplitude of 10 shall be considered as very small by definition. ES. (I.2 ) must be rewritten as a system of
first order differential eSuations in order to apply any of the numerical solution techniSues,
FZ /FV 01 Z
FZ /FV ' &1 0 Z (I.30)
I 15
Traduciendo...
Page 428
with x x and x dx/dt. The exact step by step solution with ES. (I.3) is
Z (V) Z (V&)V)
' G=C?)V (I.30a)
Z (V) Z (V&)V)
with
01
[#] ' (I.30b)
&1 0
#pplication of the trapezoidal rule to ES. (I.30) gives
)V
1& )V
Z (V) Z (V&)V)
1
' (I.31)
Z (V) )V )V Z (V&)V)
1% &)V 1&
It can be shown that
Z
(V) % Z (V) ' Z (V&)V) % Z (V&)V)
in ES. (I.31) for any choice of )t. Therefore, if the solution is started with the correct initial conditions x (0)
x (0) 10 , the solution for x will always lie between 10 and 10 , even for step sizes which are much larger
than one cycle of oscillation. In other words, the trapezoidal rule cuts across oscillations which are very fast but
of negligible amplitude, without any danger of numerical instability.
Explicit integration techniSues, which include 4unge Kutta methods, are inherently unstable. They reSuire
a step size tailored to the highest freSuency or smallest time constant (rule of thumb: )t # 0.2 T ), even though this
OKP
mode may produce only negligible ripples, with the overall behavior determined by the larger time constants in stiff
systems. #pplying the conventional fourth order 4unge Kutta method to ES. (I.30) is identical to a fourth order
Taylor series expansion of the transition matrix, as mentioned in Section I.5, and leads to
)V )V )V
1& % )V&
Z (V) 2 2 Z (V&)V)
' (I.32)
Z (V) )V )V )V Z (V&)V)
&)V% 1& %
2 2
Plotting the curves with a reasonably small )t, e.g., samples/cycle, reveals that the 4unge Kutta method of ES.
(I.32) is more accurate at first than the trapezoidal rule, but tends to lose the amplitude later on (Fig. I. ). This is
not serious since the ripple is assumed to be unimportant in the first place. If the step size is increased, however,
to )t /2/B cycles,
I1
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(KI + Numerical solution of d x/dt x 0 (a) exact,

(b) 4unge Kutta, (c) trapezoidal rule
then the amplitude will eventually grow to infinity. This is illustrated in table I.1 for )t 1 cycle.
Traduciendo...
6CDNG + Numerical solution of ES. (I.2 ) with )t 1 cycle
t in cycles 1 2 3 5
exact 0 0 0 0 0 0
trapezoidal 0.5 @10 0.9 @10 0.9 @10 0. 3@10 0.0 @10 0.53@10
rule
4unge Kutta 0.00 0.32 1 590 00 2, 00,000
4ef. 72 explains that the trapezoidal rule remains numerically stable even in the limiting case where the time
constant T in an eSuation of the form
T dx /dt K x x (I.33)
becomes zero. For T 0, the trapezoidal rule produces
K x (t) x (t) { K x (t )t) x (t )t) } (I.3 )
which is the correct answer as long as the solution starts from correct initial conditions K x (0) x (0) 0. Even
a slight error in the initial conditions,
K x (0) x (0) g
will not cause serious problems. Since ES. (I.3 ) just flips the sign of the expression from step to step, the error
g would only produce ripples v g superimposed on the true solution for x .
Semlyen and Dabuleanu suggest an implicit third order integration scheme for the EMTP, in which second
order interpolation (parabola) is used through two known points at t 2 )t and t )t, and through the yet unknown
solution point at t [73]. #pplying this scheme to ES. (I.30) produces
Z (V) CD Z (V&)V) EF Z (V&2)V)

' % (I.35)
Z (V) &D C Z (V&)V) &F E Z (V&2)V)
with
I 17
Page 430
0
C ' (1 & )V ) / det
1
13
D' )V / det
12
5
E' )V / det
1
)V
F'& / det
12
25
det ' 1 % )V
1
ES. (I.35) gives indeed higher accuracy than the trapezoidal rule, but only as long as the step size is reasonably
small, and as long as the number of steps is not very large. #fter 0 cycles, with a step size of samples/cycle, ES.
(I.35) would produce peaks which have already grown by a factor of 20,000. This indicates that the choice of the
step in ES. (I.35) is subject to limitations imposed by numerical stability considerations, whereas the trapezoidal rule
is not. # step size of samples/cycles is not too large for fast oscillations which have no influence on the overall
behavior. The trapezoidal rule simply filters them out. High order implicit integration schemes are therefore not
as useful for the EMTP as one might be thought to believe from recent literature on implicit integration schemes for
stiff systems.
+ $CEMYCTF 'WNGT /GVJQF
The major drawback of the trapezoidal rule of integration of Section I. is the danger of numerical
oscillations when it is used as a differentiator, e.g., in
v L di / dt (I.3 )
Traduciendo...
with current i being the forcing function. # sudden jump in di/dt, which could be caused by current interruption in
a circuit breaker, should create a sudden jump in the voltage v. Instead, the trapezoidal rule of integration produces
undamped numerical oscillations around the correct answer, as explained in Section 2.2.2. These oscillations can
be damped out by adding a parallel resistor 4 across the

R inductance. Section 2.2.2 shows that critical damping is
achieved if 4 2L/)t.
R In that case, the damped trapezoidal rule of ES. (2.20) transforms ES. (I.3 ) into
.
X(V) ' [K(V) & K(V&)V)] (I.37)
)V
which is simply the backward Euler method. Therefore, the critically damped trapezoidal rule and the backward
Euler method are identical.
In general, the undamped trapezoidal rule is better than the backward Euler method, because the latter
method produces too much damping. It is a good method, however, if it is only used for a few steps to get over
instants of discontinuities (see #ppendix II).
I1
Page 431
#22'0&+: ++ 4' +0+6+#.+<#6+10 #6 +056#065 1( &+5%106+07+6;
The numerical oscillations which occur in the voltages across inductances at points of discontinuities in di/dt,
or in currents through capacitances at points of discontinuities in dv/dt, oscillate around the correct answer. These
numerical oscillations can therefore be eliminated from the output if the output is smoothed, e.g., for the voltage
across an inductance,
1
v.(t)UOQQVJGF ' [v.(t) % v.(t&)t)] (II.1)
2
If this smoothing algorithm is not just applied to the output, but added directly into the trapezoidal rule solution
method, then we obtain
i(t) & i(t&)t)

v.(t)UOQQVJGF ' L (II.2)
)t
which is simply the backward Euler method (#ppendix I.9). B. Kulicke [15] recognized that such a backward
difference Suotient can be used to restart the solution process smoothly after a discontinuity, with the correct jumps
in v .across L, or in i through
% C. The backward Euler method does have absolute numerical stability, but it is not
as accurate as the trapezoidal rule. It is therefore only used to restart the solution with new initial conditions. B.
Kulicke also recognized that it is best to use half the step size with the backward Euler method to make the matrix
[G] needed for that backward difference solution identical with the matrix [G] of ES. (I. ), which is needed for the
trapezoidal rule solution after the discontinuity anyhow. In what follows, Kulicke s method of re initialization is
explained in detail for the inductance the derivations for the capacitance eSuations are analogous. There are three
steps in Kulicke s method, namely
(a) interpolation to obtain variables at the point of discontinuity,
(b) network solution at )t/2 after the discontinuity for the sole purpose of re initialization,
(c) re initialization of history terms at the point of discontinuity.
These three steps are then followed by the normal trapezoidal rule solution method.
(a) Interpolation
#ssume that current is to be interrupted at current zero in a circuit breaker. The EMTP solution will give
us answers at points 1, 2, 3 (Fig. II.1), with current zero crossing being discovered at point 3. Kulicke then uses
linear interpolation to locate the zero crossing at point 0, and then calculates the values of all variables and history
terms at that point 0, again with linear interpolation. The solution is then restarted at point 0, with the same )t as
before, but the uniform spacing along the time axis will be disturbed at that point, which would have to be recognized
in the output. For Kulicke s method to work, e.g., by re solving the network in point 3 with the switch open, is
unclear at this time, and may reSuire more work than linear interpolation. Interpolation would also help to eliminate
Traduciendo...
overshooting of knee points in piecewise linear elements.
II 1
Page 432
(KI ++ Linear interpolation to locate point of

discontinuity
(b) Network solution at t )t/2 Q
Let us call the instant of discontinuity t , withQthe argument t used for Suantities
Q before the jump, and t Q
for Suantities after the jump (Fig. II.2). Let us also look at the jump in di/dt across an inductance, which is caused
by the switch opening. Since no jump can occur in this
(KI ++ Voltage and current at point of discontinuity
current, we know that
i(t Q
) i(t ) Q (II.3)
If we now use the backward difference Suotient of ES. (II.2) to solve the network at t )t/2, then we obtain
Q
)t )t )t
i(tQ % )' v.(tQ % ) % i(tQ&) (II. )
2 2L 2
which is the same as ES. (1.3a), except that the history term is now simply i(t ). For capacitance,the
Q analogous
eSuation would be
II 2
Page 433
)t 2C )t 2C
iE(tQ % )' v(tQ % )& v(tQ&) (II.5)
2 )t 2 )t
again with a modified history term of 2Cv(t )/)t in this Q

case. The solution at t )t/2 is therefore found
Q in the
usual way with ES. (1. b), after [G] has been modified to account for switch opening or for whatever caused the
discontinuity, and after it has been re triangularized. Notice that this matrixTraduciendo...
change and re triangularization process
is reSuired anyhow, even if Kulicke s re initialization method is not used. The only difference for this extra solution
is in the right hand side, since the history term is now i(t ) instead of hist
Q from ES. (1.3b), with an analogous
modification of the capacitance history term.
(c) 4e initialization of history terms at tQ
The extra network solution at t )t/2 isQmade for the sole purpose of re initializing variables at t . For Q
the inductance, assuming a linear change in current between t and t )t/2, the
Q voltageQat t simply becomes Q
v(t
. Q) v(t )t/2) . Q (II. )
which would then be used in ES. (1.3b) to calculate the history term reSuired for the next, normal solution at t )t, Q
for which the triangularized matrix has already been obtained in step (b).
Similarly, assuming a linear change of voltage across capacitances, the current at t simply become Q
i%
(t )Qi (t )t/2)% Q (II.7)
II 3
Page 434
#22'0&+: +++ 51.76+10 1( .+0'#4 '37#6+105 /#64+:

4'&7%6+10 #0& +08'45+10 52#45+6;
The fastest direct method for solving a system of linear eSuations
; V ; V ... ; V I HH
; V ; V ... ; V I HH
..............................
; VH ; V ... ; V HI HH H H (III.1)
for the unknown voltages V , ... V , with given

H current I , ... I , is Gauss elimination,
H which in more or less
modified forms is also called triangularization, triangular factorization, LU decomposition, Gauss Banachiewicz,
Gauss Doolittle, Crout, etc.
Gauss Jordan elimination or diagonalization takes more operations for the solution of linear eSuations, but
for matrix inversion the differences in speed between Gauss and Gauss Jordan seem to become negligible, since both
!
methods reSuire essentially N operations. The Gauss Jordan method has therefore been chosen for the inversion
of small, but full matrices associated with coupled branches. For solving the complete network with ES. (1. b) or
ES. (1.21), Gauss elimination with sparsity techniSues is used, as discussed in Section III. .
# comparison of operation counts between these two basic methods is shown in Table III.1. Choleski s
method is a modification of Gauss elimination for positive definite, symmetric matrices, whereby the sSuare root is
taken of the diagonal elements to make the lower triangular matrix eSual to the transpose of the upper triangular
Traduciendo...
matrix. This reSuires extra calculations which are difficult to justify. It has been claimed, however, that Choleski s
method works better for ill conditioned matrices, probably because the sSuare root operation brings numbers closer
$ " !
together in orders of magnitude, e.g., 10 and 10 would become 10 and 10 . The writer has never tested this
claim, and suspects that it applies only to (obsolete) fixed point arithmetic.
6CDNG +++ Number of operations for direct solution of a system of linear eSuations
1. Full matrices
Operations count for N eSuations in N unknowns. Taken from [75]
#. Unsymmetric matrix storage reSuirement N for matrix
Number of operations
g
Method Process Mult. #dd./Sub. Div.
Gauss elimination for matrix N !N N !N N N

(triangularization) 33 32
1 repeat
solution N NN O
III 1
Page 435
Gauss elimination for matrix N !N N N !N 2N N

h
using scalar products 32 3 3
1 repeat
solution N NN O
! !
Gauss Jordan for matrix NN NNN N
(diagonalization) 22 2 2
1 repeat
solution N NN O
HH
B. Symmetric matrix storage reSuirement / for matrix
Number of operations
Method Process g Mult. #dd./Sub. Div.

! !
Gauss elimination for matrix N N 2N NN N
(triangularization) 23
1 repeat
solution N NN O
! !
Choleski (triangular for matrix N N 2N NN N(N
ization) 23 sSuare roots)
1 repeat O
solution NN NN
Gauss Jordan for matrix N !N N !N N N

(diagonalization) 33 32
1 repeat
solution N NN O
a) In the process for matrix only the elements of the matrix are transformed. In the process 1 repeat
solution, the transformation process is extended to the given vector [I] in the system of eSuations [;][V]
[I] and then [V] is found. If [I] changes only and [;] remains unchanged, then only the process 1 repeat
solution is used.
b) #lso called Gauss Banachiewicz and in slightly modified form Gauss Doolittle (advantageous only for
desk calculators and for digital computers with scalar product as a single operation).
2. Sparse matrices for network solutions
Impressive savings in storage reSuirements and number of operations possible. See Section III. .
Because [;] in ES. (III.1) has usually strong diagonal elements, pivoting is not used in the solution
routines of the EMTP and its support programs.
If ES. (III.1) is solved repeatedly with the same matrix [;], but with different right hand sides [I], then it
Traduciendo...
is best to split the elimination process into two parts, one for the matrix , and the other for repeat solutions . This
situation occurs in the transient simulation over successive time steps as long as the network does not change because
of switching operations or nonlinear effects. #s shown in Table III.1, the number of operations is much less for
repeat solutions than for a complete solution involving the process for the matrix. The savings are even more
III 2
Page 436
pronounced with the sparsity techniSues discussed in Section III. .
+++ )CWUU 'NKOKPCVKQP
Most readers are probably familiar with this method, which will be explained for the following example:
2x 3x x ! 20
x 5x 2x " 5
2x 5x x x ! " 3 (III.2)
x x 2x 3x !Â " 5
5VGR Leave the first row unchanged , and add such multiples of the first row to rows 2, 3, that zeros are
produced in column 1 of these rows:
unchanged
2 3 1 0 20
add 3 times row 1
0 3 2 15
0 7 23 add ( 1) times row 1
0 0 3 1 add ( 2) times row 1
this information must be saved if repeat solutions are to be

performed later
5VGR : Leave the second row unchanged , as well as row 1, and add such multiples of the second row to rows 3,
that zeros are produced in column 2 of these rows:
2 3 1 0 20 unchanged
0 3 2 15
0 0 1 2 7 add 2 times row 2
0 0 3 1 add 0 times row 2
save information if repeat solutions are to be performed later.
5VGR Leave the third row unchanged , as well as rows 1 and 2, and add such multiples of the third row that
zeros are produced in column 3 of that row:
In the transient simulation part of the EMTP, this row is divided by the diagonal element before proceeding
with the other row modifications.
ibid.
III 3
Page 437
2 3 1 0 20
unchanged
0 3 2 15
Traduciendo...
0 0 1 2 7
0 0 0 5 10 add ( ) times row 3
save information if repeat solutions are to be performed later.
#fter these downward operations of steps 1 to 3, the matrix has become triangularized , with an upper triangular
matrix ,
2x 3x x ! 20
x 3x 2x ! " 15
x! 2x " 7 (III.3)
5x " 10
and the unknowns can now easily be found backwards by backsubstitution : First, find xo , then x from row 3, !
etc., with the result
x "2
x !3
x7
x1
The determinant is obtained as a byproduct in the downward operations: It is the product of the diagonal
elements in the triangular matrix of ES. (III.3),
det {[#]} 2C C1C5 0
In the transient simulation, the system of linear eSuations is solved repeatedly with no change in the matrix,
but with changes in the right hand sides . In that case, the downward operations are only repeated for the vector
of the right hand side (process repeat solution in Table III.1), using the multiplication factors indicated on the right
side in steps 1 to 3, which can conveniently be stored in the columns where the zeros are created. This produces
the lower triangular matrix
1 2
2 0
#s an example, assume that a repeat solution of ES. (III.2) is sought with right hand sides of
III
Page 438
1
1
2
12
By repeating step 1 for the right hand sides, we obtain
2
1
10
and after step 2
1
2
5
10
and finally after step 3
1 Traduciendo...
2
5
10
which, after backsubstitution with the upper triangular matrix of ES. (III.3), produces the results
x "2
x !1
x 9/
x 19/
If the matrix is symmetric, then the lower triangular matrix need not be recorded for repeat solutions. The
information is already contained in the upper triangular matrix, since the rows of the upper triangular matrix divided
by its negative diagonal element are eSual to the columns of the lower triangular matrix. Symmetry is exploited in
this way by the EMTP in the transient simulation part.
If the inverse of [;] in ES. (III.1) were known, then it appears to be more straightforward to make repeat
solution with simple matrix multiplications,
[V] [;] [I] (III. )
This notational elegance is deceiving, however, because it ignores the computational burden of obtaining the inverse
matrix [;] in the first place. #s it turns out, the numerical process for inverting the matrix takes us right back to
the elimination techniSues for solving linear eSuations. Essentially, the inverse of a matrix is found by applying the
solution process to the N columns of the unit matrix as right hand sides, which amounts to N repeat solutions, or
III 5
Page 439
N !operations. On the other hand, the elimination process for matrix in Table III.1 reSuires only N /3 operations, !
with the number of operations in a repeat solution and in the multiplication of ES. (III. ) both being N . Therefore,
systems of linear eSuations should never by solved by using the inverse, because triangularization of the matrix takes
only 1/3 of the operations reSuired for matrix inversion. There are only three excused for using the inverse, namely
(a) in cases where N is so small that computer time is insignificant,
(b) in cases where the matrix is used so often that the time spent for its one time inversion is negligible
compared with the numerous multiplications with ES. (III. ), as in the case of updating history terms of
couple branches in ES. (3.9), with N usually being small as well, and
(c) in cases where the inverse matrix is needed explicitly, as in the computation of the capacitance matrix from
the potential coefficient matrix (ES. ( .23) in Section .1.1.2), or in calculating ()t/2)[L] of couple
branches (ES. (3. ) in Section 3.2).
+++ )CWUU ,QTFCP 'NKOKPCVKQP D[ &KCIQPCNK\CVKQP
This method is used for the inversion of small, full matrices of coupled branches in the EMTP, and for
matrix inversion in the support routine LINE CONST#NTS, in a version which exploits the symmetry of the matrix.
The writer chose it over inversion based on Gauss elimination many years ago because it reSuires basically the same
number of operations, namely N /2 and! because it is easier to program in a way which works for matrix inversion
as well as for matrix reduction. Gauss Jordan elimination is very similar to Gauss elimination, except that in step
1 one does not only produce zeros in the column below the diagonal element, but above the diagonal as well. The
solution is available immediately after the downward operation there are no linear eSuations, with the example of
ES. (III.2).
5VGR Divide first row by ; , and add such multiples of the modified first row to all other rows that zeros are
1 3/2 1/2 0 10
0 3 2 15
0 7 23
0 0 3 1
5VGR Divide second row by ; , and add such multiples of the modified second row to all other rows that zeros
are produced in column 2 of these rows:

Traduciendo...
1 0 5/ 3/ 35/
0 1 3/ 1/2 15/
0 0 1 2 7
0 0 3 1
5VGR Divide third row by ; , and add

!! such multiples of the modified third row to all other rows that zeros are
III
Page 440
1 0 0 1/2 0
0 1 0 1 9
0 0 1 2 7
0 0 0 5 10
5VGR Divide fourth row by ; , and add""such multiples of the modified fourth row to all other rows that zeros
are produced in column of these rows:
1 0 0 0 1
0 1 0 0 7
0 0 1 0 3
0 0 0 1 2
This final step gives the solution, since the matrix has now been transformed into a unit matrix.
x1
x7
x !3
x "2
+++ 5WDTQWVKPGU 4'&7%6 CPF %:4'& HQT /CVTKZ +PXGTUKQP 4GFWEVKQP CPF 5QNWVKQP
QH 'SWCVKQPU YKVJ 5[OOGVTKE /CVTKEGU
By applying the Gauss Jordan process simultaneously to N right hand sides in the form of a unit matrix,
the inverse matrix will be produced. The unit matrix need not be stored as such, because the nontrivial values
generated in each step can conveniently be stored in the columns in which the zeros are created. #fter the final step,
the original matrix will have been changed to its inverse in its own place. Since the matrices reSuiring inversion are
all symmetric in the EMTP, Shipley s version of the Gauss Jordan process is used [ 3], which takes advantage of
symmetry. In that process, the original matrix is replaced by its negative inverse. The subroutines 4EDUCT for
real matrices and CX4ED for complex matrices use this version for matrix inversion as well as for matrix reduction.
In the reduction option, the last rows and columns M 1, ... N are eliminated, and operations in certain parts of the
matrix are skipped, which in effect changes the process from Gauss Jordan to Gauss elimination. The subroutine
4EDUCT has been changed in UBC in 19 2 to solve linear eSuations with symmetric matrices by Gauss elimination
as well. The process works as follows, keeping in mind that the matrix is symmetric and that only elements in and
below the diagonal are processed, since a a . In step

uw j, where
wu j is counted backwards from N, N 1, to M 1,!
!
Eliminations are done backwards, eliminating X first, then
H X , etc., so thatHthe last rows and columns can
be eliminated in the matrix reduction option.
III 7
Page 441
process row j as follows: Traduciendo...
PGY QNF
; LM ' F@; (III.5)
LM , M ' 1, ... 0, GZEGRV HQT L
with
1 PGY
F'& CPF ; LL ' F
QNF (III. )
; LL
PGY QNF QNF PGY

; KM ' ; KM % ; KL @ ;LM , M (III.7)
' 1, ... K, GZEGRV HQT L
and
PGY QNF
; KL ' F@; KL (III. )
If M 0, this will produce the negative inverse. If 0 M N, this will produce a reduced matrix of order M.
The case of matrix reduction may warrant further explanations. Let the components in the vectors be
partitioned into 2 subsets 1 and 2. With corresponding partitioning of the matrices we get
[; ] [; ] [8 ] [+ ]
'
[; ] [; ] [8 ] [+ ]
or [; ][V ] [; ][V ] [I ] (III.9)
[; ][V ] [; ][V ] [I ] (III.10)
The objective is to arrive at a reduced system of eSuations for subset 1. The procedure used in the subroutines is
that of ES. (III.5) to (III. ), but may be easier to understand with the following matrix eSuations:
Solve ES. (III.10) for [V ],
[V ] [; ] [; ][V ] [; ] [I ] (III.11)
and insert this into ES. (III.9), which yields the reduced system of eSuations
„qp‡iqp
[; ][V ] [I ] [D ][I ] (III.12)
with the reduced matrix

„qp‡iqp
[; ] [; ] [; ][; ] [; ] (III.13)
and the distribution factor matrix
[D ] [; ][; ] (III.1 )
The name distribution factor matrix for [D ] comes from the fact that, when multiplied with the currents [I ] at
the eliminated nodes, it distributes their effects to the retained nodes 1, as can be seen from the right hand side of
ES. (III.12). This distribution factor matrix is never needed in the EMTP because reduction is only used in cases
III
Page 442
where [I ] 0 the subroutine 4EDUCT could easily be modified to produce [D ] as well as the reduced matrix,
†
however, by simply omitting three FO4T4#N statements. The transpose [D ] is the screening factor matrix
mentioned in ES. ( . 1) of Section .1.2.5
+++ )CWUU 'NKOKPCVKQP YKVJ 5RCTUKV[ 6GEJPKSWGU
Sparsity has been exploited intuitively for a long time. In the days of hand calculations, any body solving
the three eSuations
3x 2x x ! 7
xx 10
x x! 2
would have picked the second and third eSuations first, e.g., to express x and x as a function of !x , and to insert
these expressions into the first eSuation to find x . This is essentially the same ordering scheme which is used in
computer programs today.

Traduciendo...
Sparsity techniSues have been used in power system analysis since the early 19 0 s by W.F. Tinney and
his co workers [1 1] in the U.S.#., by H. Edelmann [1 2] in Germany, and by J. Carpentier [1 3] in France, and
by others. There is an extensive list of references on the subject, and improvements are still being made [1 ]. The
following explanations do not cover all the details, but they should be sufficient to understand how sparsity is sued
in the EMTP.
+++ $CUKE +FGC
Let us assume that we have to solve the node eSuations for the network of Fig. III.1, and let us use an X
to indicate nonzero entries in the nodal admittance matrix of ES. (III.1). Then the node eSuations
(KI +++ Simple network
have the form
III 9
Page 443
8 +
::::: 8 +
/000000000000
:: /000000000000
: : 8 ' + (III.15)
: :
8 +
: :
8 +
#fter triangularization, the eSuations will have the following form:
)
8 +
*: : : : :* +
)
8
*: : : :*
)
*: : : * 8 ' + (III.1 )
*: :*
8 +
)
*:*
8 )
+
The triangular matrix is now full, in contrast to the original matrix which was sparse. The fill in is, of course,
produced by the downward operations in the elimination process. This fill in depends on the node numbering, or
in other words, on the order in which the nodes are eliminated. To show this, let us exchange numbers on nodes
1 and 5 (Fig. III.1), and solve the problem again. It will be the same problem and we will get the same solution
because assigning numbers to the nodes is really arbitrary. The node eSuations now have the form
8 +
: : 8 +
/000000000000
: : /000000000000
: : 8 ' + (III.17)
::
8 +
:::::
8 +
Traduciendo...
which becomes after triangularization:
)
8 +
*: :* )
+
8
*: :*
)
*: :* 8 ' + (III.1 )
*: :*
8 )
+
*:*
8 )
+
III 10
Page 444
(KI +++ Nodes 1 and 5 re

numbered
Now there is no fill in at all (in general, there will be some fill in). This saving was achieved by just numbering the
nodes in a slightly different order, or in other words, sparsity can be preserved by using good ordering.
The simplest good ordering scheme is as follow: Number nodes with only 1 branch connected first, then
number nodes with 2 branches connected, then nodes with 3 branches connected, etc. Better ordering schemes are
discussed in [1 1], with Scheme 2 probably being the best compromise between time spent on finding a near optimal
order and the savings achieved through sparsity. Scheme 2 is used in the steady state and transient solution part of
the BP# EMTP. The UBC EMTP uses re ordering only in the transient solution part.
(KI +++ Comparison of the numerical effort between matrix

inversion and ordered triangular factorization for typical power
networks [1 5]. l 1973 IEEE
Exploitation of sparsity is extremely important in large power systems because it reduces storage
reSuirements and solution times tremendously. The curves in Fig. III.3, taken from a tutorial paper by Tinney and
!
Meyer [1 5], clearly show this. The solution time for full matrices is proportional to N . For sparse power systems
it increases about linearly. Typically, the number of series branches is about 1. x (number of nodes) and the
III 11
Page 445
Traduciendo...
number of matrix elements in the upper triangular matrix is about 2.5 to 3 times the number of nodes in steady state
eSuations. The node eSuations (1. ) for the transient solution are usually sparser because distributed parameter lines
do not contribute any off diagonal elements.
Fig. III. shows the steady state nodal admittance matrix of a single phase (positive seSuence) network with
127 nodes and 153 branches, before triangularization in the lower triangular part, and after triangularization in the
upper triangular part, with optimal ordering base on Scheme 2. The fill in elements are indicated by the symbol O ,
whereas X indicates the original elements. Because of fill in, the number of off diagonal elements in the upper
triangular matrix grows from 153 to 229, but this is still very sparse compared with 001 elements in a full matrix.
III 12
Page 446
Traduciendo...
(KI +++ Nonzero pattern of symmetric matrix before triangularization shown below
diagonal and after triangularization shown above diagonal, for a network with 127 nodes
+++ 4QY D[ TQY 'NKOKPCVKQP YKVJ 5VCVKE 5VQTCIG
While there are many variations of the basic Gauss elimination and associated sparse storage schemes, the
best choice for power system analysis seems to be row by row elimination with static storage. This is the scheme
used in the EMTP. Its two basic concepts are:
(a) The non zero pattern of the triangularized [;] or [G] matrix need not be known in advance (even though
III 13
Page 447
it could be obtained as a by product of the re ordering subroutine), but the nodes must be re numbered near
optimally for minimum fill in.
(b) #s each row of the upper triangular matrix is built, it is stored away once and for all and never changed
again (static storage). Since the nodal matrices are symmetric in the steady state as well as in the transient
solution part, the lower triangular matrix is not needed.
Concept (b) rules out the textbook approach to elimination shown in Section III.1, in which zeros are produced
column by column in the lower triangular matrix, because the resulting reduced matrices change with fill in elements
from one elimination step to the next.
4ow by row elimination on the matrix elements with static storage works basically as follows (Fig. III.5):
(KI +++ 4ow by row elimination
1. Set elimination step k 0.
2. Increase k by 1.
3. Stop process if k N.
. Build row k of [;] from branch tables in a one dimensional working row array (or transfer data into
working row if [;] is already available). Use either a full working row scheme or a packed working row
scheme, as discussed in Section III. .3
5. In the working row, eliminate ; (if nonzero)
w by adding the appropriate multiple of row 1 of the already
existing part of the upper triangular matrix. Then eliminate ; (if nonzero)w in an analogous way, then ; w!
etc., up to ; . Note
w w that rows of the existing part of the upper triangular matrix are only recalled from
Traduciendo...
storage, but not modified (static storage).
. Store the diagonal element in a table of length N (or its reciprocal on computers where division takes more
time than multiplication), and add the nonzero elements ;m to the right of it (m k) in compact form to
the existing part of the upper triangular matrix, e.g., with the row pointer/column index scheme of Section
III. . . Since the matrices in the EMTP are symmetric, only the upper triangular matrix has to be stored.
7. 4eturn to step 2.
If there is only one solution, as in the steady state initialization, then the right hand side [I] is processed
together with [;] as if it were an extra (N 1) th column.
For the repeat solutions in the transient solution part, the downward operations are made with the rows of
the upper triangular matrix, since the elements of row k of the upper triangular matrix, divided by the negative
diagonal element, are the multiplication factors usually stored in column k of the lower triangular matrix. If
III 1
Page 448
;4EC(K) is reciprocal of the diagonal element, if 4(I) is right hand side, if ;U(J) are the off diagonal elements of
row k of the upper triangular matrix, and if MU(J) are the column indices m of these elements, then elimination step
k of a repeat solution would be more or less similar to the following Fortran statements:
# 4(K) ;4EC(K)
DO 10 J (beginning of row), (end of row)
M MU(J)
10 4(M) 4(M) ;U(J) #
+++ 9QTMKPI 4QY
For the working row discussed in step of the preceding section, and indicated in Fig. III.5, a full row can
be used with a one dimensional array of dimension N, in which zero elements are actually represented by zero
values. In the eliminations of step 5 and in storing nonzero elements in step of the preceding section, each element
must be checked whether it is nonzero. This costs extra computer time, which is the price one has to pay for the
simplicity of the full working row scheme, where fill ins fall naturally into their proper location during the
elimination process of step 5 (preceding section). # full working row scheme can be used in situations where the
extra time of checking for zeros is not very important in the total computer time. This is more or less the case in
the transient solution part of the EMTP, where the [G] matrix is only triangularized occasionally, namely at the
beginning of the time step loop and whenever switches change their position. Therefore, the UBC EMTP and older
versions of the BP# EMTP use a full working row scheme. No additional storage is needed for that row, because
the one dimensional array needed in the time step loop for the right hand side is available at the time of
triangularization.
For utmost speed, packed working row schemes should be used, especially if the matrix is re triangularized
freSuently. This situation arose with the simulation of HVDC systems, where the switches representing the valves
open and close after every 20 steps or so. Newer versions of the BP# EMTP therefore use a packed working row
scheme, which is essentially the same as the one described in [1 ] in table IX. In spite of the necessity of additional
indexing tables, enough storage space and computer time is saved to justify the additional complications it entails.
+++ 4QY 2QKPVGT %QNWOP +PFGZ 5VQTCIG 5EJGOG
Before discussing the compact storage of the upper triangular matrix, it should be remembered that a
separate array is used for the working row, e.g.,
4E#L 4OW(N) for a full working row and real (not complex)matrix elements,
4E#L 4OW(M) for a packed working row with M N according to [1 ]

INTEGE4 NEXT(M), KOL(M)
# row pointer/column index scheme in the form discussed next, or in a similar form, seems to offer the best
choice in terms of ease of access and economy of space. Note that such a scheme cannot only be used for storing
the triangular matrix in compact form, but also for storing the original [;] matrix in compact form.
The diagonal elements (or their reciprocals on computers where division takes noticeably longer than
III 15
Traduciendo...
Page 449
multiplication) are stored in a one dimensional array of dimension N, say in ;DI#G (Fig. III. ). The nonzero off
diagonal elements of the upper triangular matrix are stored row by row in another one dimensional array of
dimension larger than N, (typically 3.5N), say in ;U, with the starting address of each row available from a row
pointer table of length N !, say in KST#4T. The address of the last entry in a row is simply the starting address
of the following row minus 1, which explains why an (N 1) th entry is needed in KST#4T. Obviously, the column
numbers get lost when elements are packed into ;U. Therefore, an extra column index table, say MU, is reSuired,
as indicated in Fig. III. . The overhead burden of this extra table,
table of length table of length table ;U for elements of upper

N for diagonal N 1 for row triangular matrix and table MU
elements (or pointer for associated column indices
their reciprocals)
;DI#G KST#4T ;U MU
1 (real) 1 (integer) 1 (real) (integer)

. . .
. . .
. . .
N N
N1 . 3.5N
(KI +++ 4ow pointer/column index storage scheme
of this extra table, which becomes less for complex elements in the steady state solution because only ;U would have
to be replaced by two tables but MU would still be a single table, is trivial for large matrices when the total storage
reSuirements are compared with the alternative of storing a full matrix in a two dimensional array. Experience has
shown that the number of words for compact storage is proportional to N (as is the computer time), whereas the
number of words reSuired for storing a full matrix is proportional to N (Table III.2). If N 1000 in Table III.2,
6CDNG +++ Storage reSuirement for upper triangular matrix and for vectors needed for repeat solutions in case of
symmetric matrices [1 7]
real elements complex elements
compact storage 9. N 15.2 N
full storage N N 2N NNN

2
then 9, 00 words would be needed for a real matrix and 15,200 words for a complex matrix, compared with 502,500
words and 1,005,000 words, respectively, for storing a full matrix. The savings are 9 and 9 .5 respectively.
III 1
Page 450
+++ 5RGEKCN 6GEJPKSWGU HQT 5[OOGVTKE /CVTKEGU
The matrices [;] for the steady state solution as well as [G] for the transient solution are both symmetric.
Symmetry can be exploited in two ways:
(a) The lower triangular matrix need not be stored for repeat solutions, and
(b) only elements to the right of the diagonal must be processed in step 5 of Section III. .2, which cuts the
operation count in the elimination process for the matrix approximately in half. No savings can be gained
in repeat solutions, however.
Point (a) has been discussed in Section III. .2 it results from the fact that a column of the lower triangular
matrix is eSual to the respective row of the upper triangular matrix, divided by the negative diagonal element. This
Traduciendo...
point is exploited in the EMTP.
Point (b) is true because the multiplication factors needed for the elimination of ; , ; , ... in step 5 of
w w
Section III. .2 are already available in the dotted column of Fig. III.5. This is not exploited in the EMTP, since it
is difficult to access this dotted column directly without some additional indexing tables. # simple way out of this
problem would be to store elements of the dotted column, as they are created, in a separate table for the rows of the
lower triangular matrix, but that would defeat the advantage of point (a) above.
There is some advantage in dividing the rows of the upper triangular matrix by the diagonal element, in the
loop where they are stored in step ( ) of Section III. .2. This way, N multiplications are saved in the
backsubstitution of the repeat solution, at the expense of one extra multiplication for each off diagonal element in
the triangular matrix. When the EMTP was first written, it was assumed that the matrix will only be triangularized
occasionally (before entering the time step loop and whenever switches change their position). Therefore, division
by the diagonal elements was chosen to keep the operation count in the repeat solutions inside the time step loop as
low as possible. In simulating HVDC systems, the savings in the time step loop may become less than the extra
operations needed for multiplying the off diagonal elements with the reciprocal of the diagonal element.
With division by the diagonal elements, the matrix process is only modified in step ( ) of Section III. .2.
In step (2) it must be realized of course that the elements of the upper triangular matrix are no longer ; , but wy
; /;wy. ww
The process for repeat solutions with the storage scheme of Section III. .2 works roughly as follows.
Downward operations:
1. Set elimination step k 0.
2. Increase k by 1.
3. Go to backsubstitution if k N.
. Get compact row k of the triangular matrix from storage, and
(a) save k th component of right hand side, # 4(K),
(b) multiply k th component with reciprocal of diagonal element, 4(K0 # ;4EC(K)
(c) modify all components of right hand side for which entries exist in row k of the
triangularized matrix (diagonal element excluded):
DO XX J (beginning of row), (end of row)
III 17
Page 451
M MU(J)
4(M) 4(M) ;U(J)#
Backsubstitutions (4 will be replaced by solution vector):
1. Set counter k N.
2. Decrease k by 1.
3. Stop process if k 1.
. Get compact row k of the triangular matrix from storage, and find the solution for k th component
with the following loop (diagonal element excluded):
# 4(K)
DO XX J (beginning of row), (end of row)
M MU(J)
XX # # ;U(J)4(J)
4(K) #
Traduciendo...
III 1
Page 452
#22'0&+: +8 #%67#. 8#.7'5 8'4575 2'4 70+6 37#06+6+'5
The use of per unit Suantities has been customary for so many years in the electric power industry, that it
is not always recognized that actual values can be used just as easily, and that the per unit system may have outlived
its usefulness. This writer sees no advantages in working with per unit Suantities, and feels much more comfortable
with actual values.
The widespread use f per unit Suantities probably started with network analyzers in the 1930 s. For power
flow and short circuit studies on network analyzers, per unit Suantities offered two advantages, namely scaling of
impedances to values available on the analyzer, and the possibility of representing transformers as simple series
impedances as long as their turns ratio was identical to the ratio of the base voltages on the two sides. Somewhat
similar arguments for per unit Suantities could be made in the early days of digital computers with fixed point
arithmetic, where the order of magnitude of intermediate and final results had to be about the same. On modern
computers with floating point arithmetic, there is no reason, however, why one shouldn t work directly with actual
values.
+8 2GT 7PKV 3WCPVKVKGU
# per unit Suantity is the ratio of the actual value of a Suantity to the base value of the same Suantity [7 ,
p. 2]. It has been customary to use one common base power S (apparent power)
hg…qfor the entire system (typically
100 MV#), and a different base voltage for each voltage level (e.g., V 115 kV and V 230hg…q
kV in a hg…q
115/230 kV system) as the base values. Then the per unit Suantities in a single phase network are
8DCUG
+R W ' +CEVWCN @
5DCUG
8DCUG
8R W ' (IV.1)
8DCUG
8 DCUG
;R W ' ;CEVWCN @
5DCUG
5DCUG
<R W ' <CEVWCN @
8 DCUG
It may be safest to use these single phase eSuations for three phase networks as well. In wy connected eSuipment,
S hg…q
would be the single phase base power of one winding (e.g., 100/3 MV#) and Vase would be the base voltage
across each winding, namely the phase to ground base voltage (e.g., 113//3 kV and 230//3 kV). In delta connected
eSuipment, S would
hg…q again be the single phase base power of each winding (e.g., 100/3 MV#), whereas the base
voltage V across
hg…q each winding would now be the phase to phase base voltage (e.g., 115 kV and 23 kV).
The following, well known formulas with three phase base values were developed for positive seSuence
Traduciendo...
IV 1
Page 453
power flow studies, where the distinction between wye and delta connections gets lost in the conversion from three
phase representations to eSuivalent single phase representations for balanced operation:
8DCUG&RJCUG&VQ&RJCUG
+R W ' +CEVWCN @ 3
5DCUG&VJTGG&RJCUG
8CEVWCN
8R W ' (PWOGTCVQTCPF FGPQOKPCVQTGKVJGT DQVJ RJCUG&VQ&RJCUGQT DQVJRJCUG&VQ&ITQWPF)
8DCUG
(8DCUG&RJCUG&VQ&RJCUG)
;R W ' ;CEVWCN @ (IV.2)
5DCUG&VJTGG&RJCUG
1
<R W '
;R W
ES. (IV.2) cannot only be used for the conversion of positive seSuence parameters, but for negative and zero
seSuence parameters as well, as shown in the example of Section IV.3.
Per unit Suantities, as ratios of actual to base values, are meaningless if the base values are not listed as part
of the data as well. For example, the positive seSuence series impedance of an overhead line is fully described by
three actual values,
) )
4
RQU % LT. RQU ' 0.05 % L . 0 S/MO, H ' 0 *\
or if 4 and
‚…L are independent
‚… of freSuency, by two values,
4 0.05
‚ … S/km, L 1.0 1 mH/km‚ …
On the other hand, the record for per unit Suantities consists of 5 values,
# #
4 jTL
‚ … 9. 5C10 j75.
‚ …1C10 p.u., f 0 Hz, S 100 MV# (three phase), V 230 kV hg…q hg…q
(phase to phase).
With 4 and
‚…L , the freSuency
‚… could be dropped from the record, but the time base should then be added,
# %
4 9.‚ …
5C10 p.u., L 20.0 C10 p.u.,
‚… S 100 MV# (three phase),hg…q
V 230 kV (phase to phase), hg…q
thg…q
1s.
#dding the time base may seem superfluous, but there are stability programs which use cycles (of 0 Hz) as a time
#
base, in which case L 12.03C10
‚ … p.u., t 1/ 0 s. hg…q
+8 %QPXGTUKQP HTQO 1PG $CUG VQ #PQVJGT
If per unit data is to be exchanged among utilities and manufacturers, then it is important to include the base
IV 2
Page 454
values, especially if one party customarily uses base values which are different from those used by the other party.
For example, a transformer manufacturer lists the short circuit input impedance in per unit based on the voltage and
power nameplate ratings of the transformers,
<ROCPWHCEVWTGT
W ' <CEVWCN @ 5PCOGRNCVG
(IV.3)
(8PCOGRNCVG)
In a particular case, these base values might be 0 MV# (three phase) and 2 Traduciendo...
1.5 kV (phase to phase). Utility
companies generally use different base values (e.g., S 100 MV#, V 230 kV). By solving
hg…q ES. (IV.3) for
hg…q
< gi†‡gxand using ES. (IV.2) to get back to per unit Suantities, one obtains
8PCOGRNCVG
WVKNKV[ OCPWHCEVWTGT 5DCUG
<R W '< RW @( ) @ (IV. )
8DCUG 5PCOGRNCVG
Obviously, the per unit Suantity of the manufacturer will be Suite different from the one used by the utility company.
In general, the formulas for conversion from base 1 to base 2 are
DCUG DCUG 5DCUG 8DCUG

+R W '+ RW
@ @
5DCUG 8DCUG
DCUG DCUG 8DCUG
8RW '8 RW @ (IV.5)
8DCUG
DCUG DCUG 5DCUG 8DCUG
; RW '; RW @ @( )
5DCUG 8DCUG
+8 #EVWCN 8CNWGU 4GHGTTGF VQ 1PG 5KFG 6TCPUHQTOGT
The advantage of representing transformers as simple series impedances with per unit Suantities, as long
as their turns ratio is identical to the ratio of the base voltages, exists with actual values as well, if the Suantities on
one side are referred to the other side. In the example of Fig. IV.1, Suantities on the low voltage side are referred
to the high voltage side with
15
+JKIJ ' +NQY @
2 1.5
(IV. )
2 1.5
8JKIJ ' 8NQY @
15
15
;JKIJ ' ;NQY @ ( )
2 1.5
2 1.5
<JKIJ ' <NQY @ ( )
15
IV 3
Page 455
(KI +8 Generator with step up transformer. Generator data: X X 10 based on p ƒ

rating of 13. kV and 1 0 MV#. Transformer data: X X based on rating‚ of
… ’q„
15/2 1.5 kV and 250 MV#
(a) positive (b) negative (c) zero
(KI +8 Positive, negative, and zero seSuence networks seen from high side
This conversion to the high side is advantageous if the generator and step up transformer are to be replaced by a
Thevenin eSuivalent circuit seen from the high side. With the data of Fig. IV.1, the positive, negative and zero
Traduciendo...
seSuence networks of Fig. IV.2 are obtained as follows: For the transformer,
2 1.5
:CEVWCN ' 0.0 S ' 1 . S (UGGP JKIJ UKFG)
250
and for the generator,
13.
:CEVWCN ' 0.10 S ' 0.105 S (UGGP JKIJ UKFG)
10
or
2 1.5
:CEVWCN ' 0.105 ( ) S ' 27. 2 S (UGGP JKIJ UKFG)
15
Note that the delta connection provides a short circuit for the zero seSuence currents (Fig. IV.2(c)). With X ‚…
X €qs
.0 S, X 1 . S, the final’q„
three phase Thevenin eSuivalent circuit of Fig. IV.3 is obtained by
converting the seSuence reactances to self and mutual reactances with ES. (3. ). The amplitude of the Thevenin
IV
Page 456
voltages is set eSual to the voltage seen on the high side for the particular operating condition, which may be 230
kV phase to phase in a particular case.
3x3 reactance matrix
:U :O :O
:O :U :O
:O :O :U
(KI +8 Three phase Thevenin eSuivalent circuit. Symmetric voltage sources V , V , V with 4MS amplitude
5 6 7
of 230/%3 kV x 3 .9…S, X 9.1 S y
One could also use per unit Suantities for the Thevenin eSuivalent circuits of Fig. IV.2, with the transformer
ratings as base values. In that case, X 0.0 p.u. for the transformer, and with ES. (IV.5),
X 0.10 C (250/1 0) C (13. /15) p.u. 0.117 p.u.
for the generator. Then, X X 0.1975

‚ … p.u., X
€qs0/0 p.u., which leads to X
’q„X .0 S, ‚… €qs
X ’q„
1 . S with S 250 MV# (three phase) and V 2 1.5 kV (phase to phase). hg…q
hg…q
+8 #FXCPVCIGU QH #EVWCN 8CNWGU
This writer prefers actual values over per unit Suantities for the following reasons:
(1) Confusion may arise with per unit Suantities because the base values are not always clearly stated. This
confusion cannot arise with actual values.
(2) The data record is shorter for actual values, as shown in the last paragraph of Section IV.1, even if S in hg…q
the per unit record is left off, with the understanding that it is always 100 MV#.
(3) #ctual values are fixed characteristics of a piece of eSuipment, independent of how this eSuipment is being
used. This is not true for per unit Suantities: If a 500 kV shunt reactor is temporarily used on a 3 5 kV
circuit, its per unit values based on 500 kV would have to be converted to a base of 3 5 kV.
() Since the ratio of transformer voltage ratings is not always eSual to the ratio of base voltages, one has to
allow for off nominal turns ratios (uneSual 1:1) with per unit Suantities anyhow. If one has to allow for
any ratio, then a ratio of 1:1.05 for per unit Suantities is neither easier nor more difficult to handle than a
ratio of 15 kV:2 1.5 kV for actual values. Therefore, one might as well use actual values. Furthermore,
the simple series impedance representation of transformers with per unit ratios of 1:1 (Fig. 3.3(c) with t
1.0) can seldom be used in EMTP studies. For example, a three phase bank of single phase transformers
in wye delta connection would reSuire a 2x2 [;] matrix model forTraduciendo...
each transformer, or alternatively, an
eSuivalent circuit representation with uncouple reactances as shown in Fig. 3.3(b). The case of t 1.0
offers no advantage whatsoever in that six branch circuit.
IV 5
Page 457
(5) If test data is available in per unit Suantities, e.g., for generators or transformers, then conversions are even
necessary for per unit values, since the base values do in general not agree with the nameplate ratings.
Therefore, one might as well convert to actual values. Furthermore, the EMTP does this conversion in
most cases anyhow, e.g., in the main program in the case of generators, or in supporting routines in the
case of transformers.
() #ll digital computers use floating point arithmetic nowadays, and therefore accept numbers over a wide
range of magnitudes. Therefore, the numbers do not have to be of the same order of magnitude, and a turns
ratio of 15 kV:2 1.5 kV causes no more problems than a turns ratio of 1:1.05.
Sometimes the Suestion is raised whether solutions with per unit values aren t possibly more accurate than
solutions with actual values. Many years ago on computers with fixed point arithmetic, per unit values may indeed
have produced more accurate than the other. To show this, let us look at the steady state solution of a single phase
network with nodal eSuations,
[; ] [V gi†‡gx
gi†‡gx ] [I ] gi†‡gx (IV.7)
where [I ] gi†‡gx
is given, and [V ] is
gi†‡gx to be found. In general, the network will have two or more voltage levels,
which will be taken into account in [; ] with

gi†‡gx the proper transformer turns ratios. To convert ES. (IV.7) to per
unit Suantities, the base voltages are first defined in the form of a diagonal matrix,
8DCUG
8DCUG
[8DCUG] ' @ (IV. )
@
8DCUG 0
with the possibility of each node having its own base voltage. In reality of course, all nodes within one voltage level
would have the same base value. With S being the same for the entire network, the current and voltage vectors
hg…q
in per unit and actual values are related by
[I ‚]‡(1/S )[V ][Ihg…q

] hg…q hg…q (IV.9)
[V gi†‡gx] [V ][V hg…q

] ‚‡ (IV.10)
Premultiplying ES. (IV.7) with [V ]/S , and

hg…qreplacing
hg…q [V ] with ES. (IV.10) will produce the per unit eSuations
gi†‡gx
[; ][V
‚ ‡ ] [I ]‚ ‡ ‚‡ (IV.11)
with
[; ] ‚(1/S
‡ )[V ][; hg…q hg…q ][V hg…q
gi†‡gx ] (IV.12)
Therefore, the conversion from actual to per unit values consists of the transformation of the coefficient matrix
[; ] into
gi†‡gx [; ] with
‚ ‡ ES. (IV.12). This transformation is very simple since [V ] is a diagonal matrix:
hg…q#side
from dividing all elements by the constant S , each hg…q

row i (i 1, 2, ... N) is multiplied by V hg…q, uand each row k
(k 1, 2, ...N) is multiplied by V hg…q.wThis is essentially a scaling operation.
IV
Page 458
This scaling operation has no influence on the solution process if pivoting is not used, but it may influence
the accumulation of round off errors. This influence on round off errors is difficult to assess. For a system of linear
eSuations, the following can however be said [77, p.39]: If scaling is done in such a way that it changes only the
% %
exponent of the floating point number (e.g., by using S 12 MV# or 2,V
hg…q hg…qÂ12 kV or 2 and V hg…qÂ
' Traduciendo...
512 kV or 2 on a computer using base 2 for the exponents), and if the order of eliminations is not changed, then
the scaled (per unit) coefficients will have precisely the same mantissas, and all intermediate and final results will
have precisely the same number of significant digits. Therefore, it is reasonable to assume that scaling will neither
improve nor degrade the accuracy of the solutions. M.D. Crouch of Bonneville Power #dministration has shown
that this assumption is correct for power flow solutions with bit precision.
+8 2GT 7PKV 8QNVCIG YKVJ #EVWCN +ORGFCPEGU
Sometimes, overvoltage studies are made with impedances in actual values, but with voltage source
amplitudes scaled to 1.0 p.u. or similar values,
V ‚V‡ gi†‡gx/ V hg…q
This produces overvoltages expressed in per unit, which is often preferred in insulation co ordination studies. If
there are no nonlinear elements in the network, then this approach is Suite straightforward. #ctual values can be
obtained from the per unit values by multiplying per unit voltages and currents with V , and per unit power
hg…q with
V .
hg…q
Some care is reSuired, however, if the network contains nonlinearities. For nonlinear resistances or
inductances defined point by point with pairs of values v, i or R, i both values of each pair must be divided by Vhg…q
in the input data. If the nonlinearities are defined by their piecewise linear slopes 4 , 4 , ... or L , L , ..., and by
the knee point v , v , ... or R , R , ..., only these knee point values must be divided by V in the input data. ˆg…q
Pivoting is generally not used in the EMTP, except in some subroutines for the inversion of small matrices
of couple branches.
IV 7
Page 459
#22'0&+: 8 4'%745+8' %1081.76+10
Consider the convoluted integral
4
U(V) ' H(V&W)G&R W&6 FW (V.1)
m6
to be found at time t, with s(t )t) already known from the preceding time step. This known value can be expressed
as
4
U(V&)V) ' GR)V H(V&W)G&R W&6 FW (V.2)
m
6% V )
by simply substituting a new variable u u )t into

PGYES. (V.1). #t the same time, the integration in ES. (V.1)
can be done in two parts,
U(V) ' )
6% V H(V&W)G&R W&6 FW %
4
H(V&W)G&R W&6 FW
m6 m6%)V
which becomes
Traduciendo...
U(V) ' )
6% V H(V&W)G&R W&6 FW % G&R)V @ U(V&)V) (V.3)
m6
with ES. (V.2). Therefore, s(t) is found recursively from s(t )t) with a simple integration over one single time step
)t. If we assume that f varies linearly between t T )t and t T, then [9 ]
s(t) c C s(t )t) c C f(t T) c C f(t T )t) (V. )
with the three constants
E ' G&R)V
1 1
E ' & (1 & G&R)V) (V.5)
R )V R
1 1
E '& G&R)V % (1 & G&R)V)
R )V R
V1
Page 460
#22'0&+: 8+ 64#05+'06 #0& 57$64#05+'06 2#4#/'6'45 1(

5;0%*410175 /#%*+0'5
The derivations are the same for the direct and Suadrature axis. They will therefore only be explained for
the direct axis. Furthermore, it is assumed that field structure Suantities have been rescaled (in physical or p.u.
Suantities) in such a way that the mutual inductances among the three windings d, f and D are all eSual, as explained
in Section .2, except that the subscript m (fore modified) is dropped from ES. ( .15a), to simplify the notation.
The eSuations with this simpler notation are then
8F .F / / KF
8H ' / .HH / KH (VI.1)
8& / / .&& K&
and
F8H /FV 4H 0 KH XH
& ' % (VI.2)
F8& /FV 0 4& K& 0
In the past, it has often been assumed that the damper windings can be ignored for the transient effects,
which are associated with the open circuit or short circuit time constants T or T . In earlier
p EMTP
p versions, this
assumption was made for the definition of the transient reactance X with ES. (VI.
p ), while for the definition of the
time constants the damper winding effects were always included. In later EMTP versions, the definition of the time
constants as well as of the transient reactance takes damper winding effects into account.
8+ 6TCPUKGPV 2CTCOGVGTU YKVJ 1PN[ 1PG 9KPFKPI QP VJG (KGNF 5VTWEVWTG
If there is no damper winding, or if the damper winding were to be ignored, then there is only the field
winding f on the field structure . The field current i can then ber eliminated from the second row of ES. (VI.1)
8H /
KH ' & KF
.HH .HH
which, when inserted into the first row, produces Traduciendo...
This is true for the direct axis. In the Suadrature axis, the analogous assumption is that either the g or the
3 winding is missing.
IV 1
Page 461
/ /
8F ' (.F & ) KF % 8H (VI.3)
.HH .HH
The flux 8 cannot

r change instantaneously after disturbance, and can therefore be regarded as constant at first. The
transient inductance which describes the flux/current relationship in the armature immediately after the disturbance
is therefore
) /
8 (VI. )
F ' .F & .HH
The open circuit time constant T , which

p describes the rate of change of flux 8 for open circuit
r conditions
(i p0) is obtained from ES. (VI.2) as
T pL / 4 rr r (VI.5)
#s shown in the next section, the definitions of both L and T changep in the presence
p of a damper
winding.
8+ 5WDVTCPUKGPV CPF 6TCPUKGPV 6KOG %QPUVCPVU YKVJ 6YQ 9KPFKPIU QP VJG (KGNF 5VTWEVWTG
The open circuit time constants are found by solving the eSuations for the currents i , i . By substituting
r8
the last two rows of ES. (VI.1),
8H / .HH / KH
' KF % (VI. )
8& / / .&& K&
into ES. (VI.2), and by setting i 0 for the

p open circuit condition, we get
FKH
FV &.&& / 4H 0 KH XH
K
' % (VI.7)
FK& .HH.&& & / / &.HH 0 4& K& 0
FV
The field winding voltage v is ther forcing function in this eSuation, while the open circuit time constants must be
the negative reciprocals of the eigenvalues of the matrix relating the current derivatives to the currents in ES.
(VI.7) . They are therefore found by solving
.&& 1 &4& .HH 1 4H 4& /

&4H % % & '0
.HH .&& & / 6 .HH .&& &/ 6 (.HH .&& &/ )
The theory is explained in #ppendix I.1, where it is shown that there will be two modes of the oscillations
defined by terms multiplied with e and e 8(8† eigenvalues).
8†
Since the eigenvalues are real and negative
here, their negative reciprocals define the two time constants.
IV 2
Page 462
for T. The results are Traduciendo...
)
6 FQ 1 .HH .&& 1 .HH .&& /
)) ' % v & % (VI. )
6 FQ 2 4H 4& 2 4H 4& 4H 4&
with the positive sign of the root for T , and pnegative sign for T . For some pderivations, the sums and differences
of these two time constants are more useful because of their simpler form,
) )) .HH .&&
6 % (VI.9a)
FQ % 6 FQ ' 4H 4&
) )) .HH .&& & /

6 (VI.9b)
FQ 6 FQ '
4H 4&
For the short circuit time constants, i in ES.p (VI. ) is no longer zero. Instead, we express it as a function
of i ,r 8i and 8 withp the first row of ES. (VI.1),
/ / /
/ KF ' 8F & KH & F& (VI.10)
.F .F .F
which, when inserted into (VI. ) and (VI.2), produces
FKH / F8F
XH% @
FV &.&&U /U 4H 0 KH .F FV
K
' % (VI.11)
FK& .HHU .&&U & / /U &.HHU 0 4& K& / F8F
U @
FV .F FV
with subscript s added to define the inductances modified for short circuit conditions,
L rr…
L M /L rr, L L M /Lp , M M M /L
88… 88 p … p (VI.12)
Taking v and
r d8 /dtpas the forcing functions, we obtain the short circuit time constants as the negative reciprocals
of the eigenvalues of the matrix in ES. (VI.11). Since this eSuation has the same form as ES. (VI.7), we can
immediately give the answer as
)
6F 1 .HHU .&&U 1 .HHU .&&U / U
' % v & % (VI.13)
))
6F 2 4H 4& 2 4H 4& 4H 4&
with the positive sign of the root for T , and pthe negative sign for T . #gain, their
p sums and differences are easier
to work with,
IV 3
Page 463
) )) .HHU .&&U
6 % (VI.1 a)
F% 6 F' 4H 4&
) )) .HHU .&&U & / U

6 (VI.1 b)
F6 F' 4H 4F
There is also a useful relationship between the open and short circuit time constants,
))
) )) ) )) . F
6 (VI.1 c)
F6 F' 6 FQ 6 FQ
.F
which can easily be derived from ES. (VI.9b) and (VI.1 b) by using the definition for L given later in pES. (VI.1 ).
It is not Suite correct to treat d8 /dt inp ES. (VI.11) as a forcing function, unless 4 is ignored. Only
g for 4 g
0 are the fluxes known from the first two rows of ES. ( .9) as
8 p8 (0)cosTt,
p 8 8 (0)sinTtƒ ƒ
with v 0,
p v 0 because
ƒ of the short circuit. In practice, 4 is not zero, but very
g small. Then the fluxes are still
"†Traduciendo...
known with fairly high accuracy if 8 (0) is replaced
p by 8 (0)e , where
p
1 T 4C 1 1
"' ' ( % ) (VI.1 d)
6C 2 )) ))
: F : S
is the reciprocal of the time constant for the decaying dc offset in the short circuit current [105]. If 4 were g
unrealistically large, then the time constants could no longer be defined independently for each axis, and the data
conversion would be much more complicated than the one described in Section VI. .
8+ 5WDVTCPUKGPV CPF 6TCPUKGPV 4GCEVCPEGU YKVJ 6YQ 9KPFKPIU QP VJG (KGNF 5VTWEVWTG
The subtransient reactance can easily be defined by knowing that the fluxes 8 , 8 cannot change r 8
immediately after the disturbance. By treating them as constants, we can express i , i as a functionr of
8 i with ES. p
(VI. ), which after insertion into the first row of ES. (VI.1), produces
.HH%.&&&2/ /
8F ' .F & / KF % [(.&&&/)8H % (.HH&/)8&] (VI.15)
.HH .&&&/ .HH.&&&/
By definition, the term relating 8 to i must

p bepthe subtransient inductance,
)) .HH % .&& & 2/

. (VI.1 )
F ' .F & /
.HH .&& & /
To obtain the definition of the transient reactance is more complicated. For many years people have simply
assumed that the damper winding currents have already died out after the subtransient period is over, and have used
IV
Page 464
ES. (VI. ). Canay has recently shown, however, that this assumption can lead to noticeable errors [10 ], and that
the data conversion is just as easy without this simplification. For the data given in the first IEEE benchmark model
for subsynchronous resonance [7 ], 0 of the current associated with the transient time constant T flows in the p
field winding, and another 20 in the damper winding after a short circuit (values obtained while verifying the
theory for this section). Ignoring the damper winding for the definition of X would therefore
p produce errors in the
field structure as well as in the armature currents.
#dkins [105] and others derive the transient reactance with Laplace transform techniSues. First, ES. (VI.2)
is solved for the currents, after replacing the fluxes with ES. (VI. ), which leads to the s domain expression for their
sum,
&U/ (4H%U.HH%4&%U.&&&2U/)
/ +H(U)%+&(U) ' +F(U) % H 8H(U)
(4H%U.HH) (4&%U.&&)&U /
where f(V )ris some function of the field voltage which is not of interest here. Inserting this into the first row of ES.
(VI.1) produces
U/ (4H%U.HH%4&%U.&&&2U/)
7F(U) ' .F & +F(U) % H 8H(U)
(4H%U.HH) (4&%U.&&)&U /
with the expression in parentheses being the operational inductance L (s), p
7 (s)
p L (s) I (s)
p f(Vp (s)) r (VI.17)
Through some lengthy manipulations it can be shown that it has the simple form
) ))
(1%U6
.F(U) ' .F F) (1%U6 F )
) ))
(VI.1 )
(1%U6
FQ) (1%U6FQ)
The basic definition of L and pL in the IEEE

p and IEC standards is
) ))
1 1 1 1 U6F 1 1 U6F
' %( & ) %( & ) (VI.19a)
.F(U) .F ) .F ) )) ) ))
. F 1%U6F . F . F 1%U6F
Traduciendo...
in the s domain, or
1 1 1 1 &V 6)F 1 1 &V 6))F

' %( & )G %( & )G
.F(V) .F ) .F )) ) (VI.19b)
. F . F . F
IV 5
Page 465
!
in the time domain . The transient reactance can therefore be found by expanding 1/L (s) from ES. (VI.1p) into
partial fractions,
) ) ) )) ) )) ) )) )) ))
1 1 1 (6 U 6F 1 (6 U 6F
F&6 FQ) (6 F&6 FQ) F &6 FQ) (6 F &6 FQ)
' & @ @ & @ @ (VI.20)
.F(U) .F .F ) ) )) ) .F )) )) ) ))
6 1%U 6 F 6 1%U 6 F
F (6 F&6 F ) F (6 F &6 F)
and by eSuating the coefficient of the second term in ES. (VI.19a), which describes what is read off the oscillogram
in the short circuit test, with the coefficient of the second term in ES. (VI.20), which describes the mathematical
model. Then, with the help of ES. (VI.1 c), we obtain
) .F )) .F .F ) ))
6F %6 % )'6 FQ (VI.21)
) F (1 & ) )) FQ % 6
. F . F . F
for the definition of the transient reactance or inductance.
Laplace transform techniSues are downgraded in #ppendix I for EMTP implementation, but for the type
of analytical work just described they are Suite useful. The transient reactance can also be derived using the
eigenvalue/eigenvector approach of ES. (I.5). The starting point for that approach is ES. (VI.11), which has the
general form
FZ
' [#] [Z] %[I(V)]
FV
of ES. (I.1), with the solution
7V] [/]& [Z(0)] % V 7 V&W ] [/]& [I(W)] FW

[Z(V)] ' [/] [G [/] [G (VI.22)
m
If we treat the variables as deviations from the pre short circuit steady state values, then the initial conditions for
these deviation variables are zero, and the first term in the above solution with [x(0)] drops out. This is in line
with the usual practice of assuming zero initial conditions in Laplace transform techniSues. What is of interest then
is the expression under the integral. To obtain it, we must first find the eigenvector matrix [M] of
&.&&U4H /U4&
1
[#] ' (VI.23)
.HHU.&&U & / /U4H &.HHU4&
U
which is
!
These definitions are used to read the inductance and time constant values from the oscillograms of the
short circuit test.
IV
Page 466
Traduciendo...
/U .HHU )
&6 F
4H 4H
[/] ' (VI.2 a)
.&&U )) /U
&6 F
4& 4&
with its inverse
/U ) .HHU
6
F&
4& 4H
1
[/]& ' (VI.2 b)
) .HHU ) )) )) .&&U /U
(6 ) (6 6
F& F&6 F ) F&
4H 4& 4H
That [M][M] unit matrix can easily be verified by knowing that T L /4 L /4 T from
p ES.88…
(VI.1
8 a). rr… r p
The forcing function vector [g(t)] is
/U & .&&U F8F

/
[I(V)] ' (VI.25)
) )) /U & .HHU FV
.F6
F6 F 4H4&
† ‡ TpÉ † ‡ TpÄ
The matrix with exponentials in ES. (VI.22) contains the two diagonal elements e and e . Since we are
only interested in the part associated with the transient time constant T , we ignore the
p parts containing T and p
obtain
.HHU )) /U
&6 F
4H 4H
7 V&W ] [/]& ' 1 & V&W 6)F & V&W 6))F
[/] [G G % [C 2Z OCVTKZ]G (VI.2 )
) )) /U .&&U
6 ))
F&6 F &6 F
4& 4&
Then
KH & VTCPUKGPV RTQFWEV QH OCVTKZ CPF XGEVQT

V & V&W 6)FF8F
' @ G FW
K& & VTCPUKGPV HTQO (8+.2 ) CPF (8+.25) m FV
which produces the 0 /20 split in the two field structure currents for the IEEE benchmark case mentioned at the
beginning of this section, when numerical values are inserted. Since
1 /
KF ' 8F & (KH % K&)
.F .F
the sum of i and

r i , after
8 multiplication with M/L , will give us
8 the transient part of i associated with T p pÉ
IV 7
Page 467
) ) ) ))
1 (6 V & V&W 6)FF8F
F&6 FQ) (6 F&6 FQ)
KF&VTCPUKGPV ' & G FW (VI.27)
.F 6
) ) )) m FV
F (6 F%6F )
By comparing the coefficient in front of the integral with the coefficient of the second term in ES. (VI.20), we can
see that the eigenvalue/eigenvector approach does indeed produce the same definition of the transient inductance as
the Laplace transform method.
8+ %CPC[ U &CVC %QPXGTUKQP
#ssume that M has been found from either ES. ( .20a) or ( .20b) (subscript m dropped here), and that
the four time constants T , T , T

p , T are
p known.
p Ifponly one pair of time constants as well as X , X are p p
known, the other pari can be found from ES. ( .12). We then obtain the two time constants of the f branch and
Traduciendo...
D branch of Fig. .2,
.H .&
6 ' ,6 ' , YKVJ .H ' .HH & /, .& ' .&& & / (VI.2 )
4H 4&
by solving the two eSuations
) )) /&.F ) )) .F
6 %6 ' (6 % (6 (VI.29a)
FQ % 6 FQ) / F%6F ) /
T T T T (L p p /M)
‚g„gxxqxÂGr8 (VI.29b)
with L being the inductance of M, L , L inr parallel,

‚g„gxxqxÂGr8 8 which can be shown with ES. (VI.1 ) to be
L ‚g„gxxqxÂGr8
MLL p p (VI.29c)
ES. (VI.29a) is obtained by multiplying ES. (VI.9a) with (1 M/L ) and thenpsubtracting it from ES. (VI.1 a), while
ES. (VI.29b) is obtained from ES. (VI.9b) with the definition of L from ES. (VI.1
p ). Once T and T are known,
the inductance of M, L in parallel

r is found,
/(6 &6 )
.RCTCNNGN /H '
) )) / (VI.30)
6 )6
FQ % 6 FQ & (1 % .RCTCNNGN /H&
This eSuation is derived from rewriting ES. (VI.9a) as
6 6 ) ))
/( % )'6 &6
.H .& FQ % 6 FQ & 6
IV
Page 468
and rewriting ES. (VI.9b) as
6 6 /
/( % )'( & 1) 6
.H .& .RCTCNNGN /H&
which produces M/L after

r subtracting the second from the first eSuation. #fter addition of 1 to M/L and division r
by M the reciprocal of L follows. Then

‚g„gxxqxÂGr
L r(L C M)
‚g„gxxqxÂGr / (M L )
‚g„gxxqxÂGr (VI.31a)
L 8(L C L‚g„gxxqxÂGr
‚g„gxxqxÂGr8 ) / (L L
‚g„gxxqxÂGr )
‚g„gxxqxÂGr8 (VI.31b)
and
4 L/T
r , 4r L /T , L L8 M, L L
8 M rr r 88Â 8 (VI.32)
Table VI.1 compares the results from the approximate data conversion of [7 ], from the data conversion
which ignores the damper winding in the definition of L by using ES.

p (VI. ) instead of (VI.21) [10 ], and from
Canay s data conversion. The approximate data conversion produces an incorrect model with X 0.15 instead p
of 0.1 9 (transient short circuit currents too large) and with T too large while T isp too small. The data p
conversion with the wrong definition of L produces

p an incorrect model with X 0.1 2 instead pof 0.1 9 (transient
short circuit currents 19 too large), but with correct time constants T and T . The iterative
p method
p mentioned
in [7 ] is correct and produces the same answers as Canay s conversion, except that no procedure is given there on
how to perform the iterations.
To double check whether Canay s data conversion is indeed correct, a system of seven eSuations of the form
[ di /pƒdt ] [#] [i ] [B] [v ] pƒ r
was set up which describes the three phase short circuit condition. The values of Table VI.1 were first used to find
the matrix [#]. Then the eigenvalues of [#] were determined. The reciprocals of four of the eigenvalues differ from
the time constants T , T p, T , Tp by no

ƒ more
ƒ than 0.05 for realistic values of 4 0.00 p.u., the reciprocal g
of one eigenvalue agrees with T of ES. (VI.1 d) to within 0.1 . Unrealistically large values of 4 would produce
g g
errors for reasons explained in Section VI.2 for 4 0.0 p.u., the
g error would still be only for T and 1 or ƒ
less for the other time constants.
Traduciendo...
8+ 0GICVKXG 5GSWGPEG +ORGFCPEG
Negative seSuence currents in the armature produce a magnetic field which rotates in opposite direction to
the field rotation, thereby inducing double freSuency currents in the field structure windings. The negative seSuence
impedance can therefore be obtained by setting s j2T in ES. (VI.1 ), and adding the armature resistance 4 to it, g
) ))
(1 % L T6
F) (1 % L T6 F)
<F&PGI ' 4C % LT.F (VI.33)
) ))
(1 % L T6
FQ) (1 % L T6 FQ)
IV 9
Page 469
6CDNG 8+ Data conversion for direct axis data from [7 ] (X 1.79 p.u., Xp 0.1 9 p.u., X 0.135pp.u., p
X R0.13 p.u., T .3 s, T 0.032
p s, f 0 Hz). p
#pprox. Wrong L p Canay
Conversion results
X rr(p.u.) 1. 999 1.703 1.721
X 88
(p.u.) 1. 57 1. 700 1. 55
4 (p.u.)
r 0.00105 0.0020 0.001 07
4 (p.u.)
8 0.00371 0.0020 5 0.00 070
Implied model parameters
X p(p.u.) from (VI.21) 0.15 0.1 1 0.1 9
T p(s) from (VI. ) 5. .3 .3
T p(s) from (VI. ) 0.0252 0.032 0.032
T p(s) from (VI.13) 0. 7 0.33 0. 000
T p(s) from (VI.13) 0.0219 0.030 0.0259
T rL /4 (s)rr r .300 2.1 3.2
T 8L /4 (s)88 8 1.192 2.1 1.0 5
For conversion of [10 ] to work, X had to be

R reduced by 1. .
and analogous for the Suadrature axis. Then
< €qs
. (< p €qs < ƒ €qs) /2 (VI.3 )
with 4 4e{<
€qs } and X Im{<
€qs }. €qsÂ €qs
If there is only one winding on the field structure, say only the 3 winding on the S axis, then
))
1 % L 2T6 S
<S&PGI ' 4C % LT.S (VI.33a)
))
1 % L 2T6 SQ
with
T ƒ(L / L ) Tƒ ƒ ƒ (VI.35b)
ES. (VI.35a) follows from (VI.33) by setting T 0 and Tƒ 0, and ES. (VI.35b)
ƒ from T L / 4 , with ƒ QQ… Q
L QQ…
defined by ES. (VI.12) and T L / 4 . ƒ QQ Q
IV 10
Traduciendo...
Page 470
#22'0&+: 8++ +06'40#. +/2'&#0%' 1( 564#0&'& %10&7%6145
For power line carrier problems, reasonably accurate attenuation constants are very important. 4eplacing
a stranded conductor by one tubular conductor of eSual cross section is not good enough for such purposes. Instead,
the internal impedance formula from [39] should be used
) ) 2.25 TzQzTk
4 S/O (VII.1)
KPVGTPCN ' T. KPVGTPCN '
T@ B@ (2%P)@ 2
or with D/(BCr) 4
) ) .5 @ 5@ 10&
4 TzT 4) S/O (VII.2)
KPVGTPCN ' T. KPVGTPCN ' 2%0
where
4 dc resistance of one of the outer strands of a stranded conductor (S/m)
zT relative permeability
zQ CBC10 (H/m)
T angular freSuency
D conductor resistivity (Sm)
r radius of each outer strand (m)
n number of outer strands
The factor 2.25 was found experimentally from field plotting in an electrolytic tank. The formula give reasonably
accurate results at freSuencies above 2 5 kHz for the most commonly used stranded conductors with the number of
outer strands either being , 12, 1 or 2 .
Fig. VII.1 compares measured attenuation constants with those calculated with the above formula. In [39]
it is shown that the measured attenuation constants come from the aerial mode which has a slightly slower wave
velocity than the other aerial mode. That mode was chosen on the same basis here. However, input data were used
which differ slightly from those given in [39]:
(1) Phase conductor 150 mm #ldrey was assumed to have 37 strands (1 on the outside), as defined in DIN
201, with conductor diameter 15. mm, strand diameter 2.25 mm, and conductor dc resistance
0.223 S/km (latter from Brown Boveri handbook).
(2) The relative permeability of the steel earth wire was assumed to be 50 to 100 (a Siemens handbook says that
these are typical values, with the actual value depending on the current density).
VII 1
Page 471
Traduciendo...
(KI 8++ Comparison between measured and calculated attenuation constants
VII 2
Page 472
4'('4'0%'5
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Traduciendo...
Page 474
=? % )CT[ #RRTQEJG %QORNGVG FG NC 2TQRCICVKQP OWNVKHKNCKTG GP JCWVG HTGSWGPEG RCT WVKNKUCVKQP FGU OCVTKEGU
EQORNGZGU %QORNGVG CRRTQCEJ VQ OWNVKEQPFWEVQT RTQRCICVKQP CV JKIJ HTGSWGPE[ YKVJ EQORNGZ OCVTKEGU
KP (TGPEJ 'F( $WNNGVKP FG NC &KTGEVKQP FGU 'VWFGU GV 4GEJGTEJGU 5GTKG $ PQ RR
=? # &GTK ) 6GXCP # 5GON[GP CPF # %CUVCPJGKTC 6JG EQORNGZ ITQWPF TGVWTP RNCPG C UKORNKHKGF OQFGN
HQT JQOQIGPGQWU CPF OWNVK NC[GT GCTVJ TGVWTP +''' 6TCPU 2QYGT #RR 5[UV XQN 2#5 RR
#WI
=? . / 9GFGRQJN CPF # ' 'HVJ[OKCFKU 9CXG RTQRCICVKQP KP VTCPUOKUUKQP NKPGU QXGT NQUU[ ITQWPF # PGY
EQORNGVG HKGNF UQNWVKQP 2TQE +'' XQN RR ,WPG
=? # 5GON[GP )TQWPF TGVWTP RCTCOGVGTU QH VTCPUOKUUKQP NKPGU CP CU[ORVQVKE CPCN[UKU HQT XGT[ JKIJ
HTGSWGPEKGU +''' 6TCPU 2QYGT #RR 5[UV XQN 2#5 RR /CTEJ
=? +''' 9QTMKPI )TQWR 'NGEVTQOCIPGVKE GHHGEVU QH QXGTJGCF VTCPUOKUUKQP NKPGU 2TCEVKECN RTQDNGOU
UCHGIWCTFU CPF OGVJQFU QH ECNEWNCVKQP +''' 6TCPU 2QYGT #RR 5[UV XQN 2#5 RR
/C[ ,WPG
=? 5 $WVVGTYQTVJ 'NGEVTKECN EJCTCEVGTKUVKEU QH QXGTJGCF NKPGU 'N 4GUGCTEJ #UUP 6GEJ 4RV 4GH 1 6
=? 4 * )CNNQYC[ 9 $ 5JQTTQEMU CPF . / 9GFGRQJN %CNEWNCVKQP QH GNGEVTKECN RCTCOGVGTU HQT UJQTV CPF
NQPI RQN[RJCUG VTCPUOKUUKQP NKPGU 2TQE +'' XQN RR &GE
=? 2 5KNXGUVGT +ORGFCPEG QH PQPOCIPGVKE QXGTJGCF RQYGT VTCPUOKUUKQP EQPFWEVQTU +''' 6TCPU 2QYGT
#RR 5[UV XQN 2#5 RR /C[
=? # ' -GPPGNN[ ( # .CYU CPF 2 * 2KGTEG 'ZRGTKOGPVCN TGUGCTEJ QP UMKP GHHGEV KP EQPFWEVQTU #+''
6TCPU RV ++ XQN RR
=? * , 9GUV CPF # . %QWTVU %WTTGPV FKUVTKDWVKQP YKVJKP CU[OOGVTKECN 7*8 DWPFNGU # NCDQTCVQT[
EQPHKTOCVKQP QH CPCN[VKECN UVWFKGU +''' 6TCPU 2QYGT #RR 5[UV XQN 2#5 RR ,WN[ #WI
=? 4 $ 5JKRNG[ CPF & 9 %QNGOCP # PGY FKTGEV OCVTKZ KPXGTUKQP OGVJQF #+'' 6TCPU RV + %QOOWP
CPF 'NGEVTQPKEU XQN RR
=? ' %NCTMG %KTEWKV #PCN[UKU QH # % 2QYGT 5[UVGOU XQN 0GY ;QTM ,QJP 9KNG[ 5QPU
=? & ' *GFOCP 2TQRCICVKQP QP QXGTJGCF VTCPUOKUUKQP NKPGU + 6JGQT[ QH OQFCN CPCN[UKU ++
'CTVJ EQPFWEVKQP GHHGEVU CPF RTCEVKECN TGUWNVU +''' 6TCPU 2QYGT #RR 5[UV XQN 2#5 RR
/CTEJ CPF RR ,WPG
=? . / 9GFGRQJN CPF 4 ) 9CUNG[ 9CXG RTQRCICVKQP KP RQN[RJCUG VTCPUOKUUKQP U[UVGOU TGUQPCPEG GHHGEVU
FWG VQ FKUETGVGN[ DQPFGF GCTVJ YKTGU 2TQE +'' XQN RR 0QX
=? )GPGTCN'NGEVTKE %Q 6TCPUOKUUKQP .KPG 4GHGTGPEG $QQM M8 CPF #DQXG 0GY ;QTM ( 9GKFPGT
5QP 2TKPVGTU
Page 475
=? ' 9 -KODCTM 'NGEVTKECN 6TCPUOKUUKQP QH 2QYGT CPF 5KIPCNU ,QJP 9KNG[ 5QPU 0GY ;QTM
=? +''' 9QTMKPI )TQWR 'NGEVTQUVCVKE GHHGEVU QH QXGTJGCF VTCPUOKUUKQP NKPGU +''' 6TCPU 2QYGT #RR
5[UV XQN 2#5 RR /CTEJ #RTKN
=? * 9 &QOOGN &KIKVCN EQORWVGT UQNWVKQP QH GNGEVTQOCIPGVKE VTCPUKGPVU KP UKPING CPF OWNVKRJCUG PGVYQTMU
+''' 6TCPU 2QYGT #RR 5[UV XQN 2#5 RR #RTKN
=? 9GUVKPIJQWUG 'NGEVTKE %QTR 'NGEVTKECN 6TCPUOKUUKQP CPF &KUVTKDWVKQP 4GHGTGPEG $QQM 2KVVUDWTIJ
9GUVKPIJQWUG 'NGEVTKE %QTR
=? ' X 4\KJC 5VCTMUVTQOVGEJPKM 6CUEJGPDWEJ HWGTTraduciendo...

'NGMVTQVGEJPKMGT XQN ++ $GTNKP 9KNJGNO 'TPUV W 5QJP
KP )GTOCP
=? 5KGOGPU (QTOGN WPF 6CDGNNGPDWEJ HWGT 5VCTMUVTQO +PIGPKGWTG 'UUGP )KTCTFGV KP )GTOCP
=? * *CRRQNFV CPF & 1GFKPI 'NGMVTKUEJG -TCHVYGTMG WPF 0GV\G 'NGEVTKE RQYGT RNCPVU CPF PGVYQTMU KP
)GTOCP 5RTKPIGT $GTNKP
=? ) 6 *G[FV CPF 4 % $WTEJGVV FKUEWUUKQP VQ 5KZ RJCUG OWNVK RJCUG RQYGT VTCPUOKUUKQP U[UVGOU (CWNV
CPCN[UKU D[ 0 $ $JCVV GV CN +''' 6TCPU 2QYGT #RR 5[UV XQN 2#5 RR /C[ ,WPG
=? * 'FGNOCPP $GTGEJPWPI GNGMVTKUEJGT 8GTDWPFPGV\G #PCN[UKU QH KPVGTEQPPGEVGF RQYGT U[UVGOU KP
)GTOCP 5RTKPIGT $GTNKP
=? * 9 &QOOGN CPF 9 5 /G[GT %QORWVCVKQP QH GNGEVTQOCIPGVKE VTCPUKGPVU 2TQE +''' XQN RR
,WN[
=? . / 9GFGRQJN #RRNKECVKQP QH OCVTKZ OGVJQFU VQ VJG UQNWVKQP QH VTCXGNNKPI YCXG RJGPQOGPC KP RQN[RJCUG
U[UVGOU 2TQE +'' XQN RR &GE
=? / % 2GT\ 0CVWTCN OQFGU QH RQYGT NKPG ECTTKGT QP JQTK\QPVCN VJTGG RJCUG NKPGU +''' 6TCPU 2QYGT #RR
5[UV XQN RR FKUEWUUKQP RR
=? / % 2GT\ # OGVJQF QH CPCN[UKU QH RQYGT NKPG ECTTKGT RTQDNGOU QP VJTGG RJCUG NKPGU +''' 6TCPU 2QYGT
#RR 5[UV XQN RR FKUEWUUKQP RR
=? . / 9GFGRQJN 'NGEVTKE EJCTCEVGTKUVKEU QH RQN[RJCUG VTCPUOKUUKQP U[UVGOU YKVJ URGEKCN TGHGTGPEG VQ
DQWPFCT[ XCNWG ECNEWNCVKQPU CV RQYGTNKPG ECTTKGT HTGSWGPEKGU 2TQE +'' XQN RR 0QX
=? 1 0KIQN #PCN[UKU QH TCFKQ PQKUG HTQO JKIJ XQNVCIG NKPGU ++ 2TQRCICVKQP VJGQT[ +''' 6TCPU 2QYGT
#RR 5[UV XQN RR FKUEWUUKQP RR
=? . / 9GFGRQJN CPF , 0 5CJC 4CFKQ KPVGTHGTGPEG HKGNFU KP OWNVKEQPFWEVQT QXGTJGCF VTCPUOKUUKQP NKPGU
2TQE +'' XQN RR 0QX
=? 4 2GNKUUKGT 2TQRCICVKQP QH GNGEVTQOCIPGVKE YCXGU QP C OWNVKRJCUG NKPG 4GXWG )GPGTCNG FG 'NGEVTKEKVG
XQN RR #RTKN CPF /C[ KP (TGPEJ
=? , # $TCPFCQ (CTKC CPF , 4 $QTIGU FC 5KNXC 9CXG RTQRCICVKQP KP RQN[RJCUG VTCPUOKUUKQP NKPGU C IGPGTCN
UQNWVKQP VQ KPENWFG ECUGU YJGTG QTFKPCT[ OQFCN VJGQT[ HCKNU +''' 6TCPU 2QYGT 5[UVGOU XQN 294&
Page 476
RR #RTKN
=? % 4 2CWN 5QNWVKQP QH VJG VTCPUOKUUKQP NKPG GSWCVKQPU HQT NQUU[ EQPFWEVQTU CPF KORGTHGEV GCTVJ 2TQE
+'' XQN RR (GD
=? $ 6 5OKVJ , $Q[NG $ )CTDQY ; +MGDG 8 -NGPC CPF % /QNGT /CVTKZ 'KIGPU[UVGO 4QWVKPGU
'+52#%- )WKFG 5GEQPF 'FKVKQP .GEVWTG 0QVGU KP %QOR 5EK XQN 5RTKPIGT 8GTNCI 0GY ;QTM
=? - ' #VMKPUQP #P +PVTQFWEVKQP VQ 0WOGTKECN #PCN[UKU ,QJP 9KNG[ 5QPU 0GY ;QTM R
=? +''' 9QTMKPI )TQWR 5YKVEJKPI 5WTIGU 2CTV +8 %QPVTQN CPF TGFWEVKQP QP CE VTCPUOKUUKQP NKPGU +'''
6TCPU 2QYGT #RR 5[UV XQN 2#5 RR #WI
=? 4 9KNNQWIJD[ GFKVQT 5VKHH &KHHGTGPVKCN 5[UVGOU 2NGPWO 2TGUU 0GY ;QTM
=? ) ) &CJNSWKUV # URGEKCN UVCDKNKV[ RTQDNGO HQT NKPGCT OWNVKUVGR OGVJQFU 0QTFKUM 6KFUMTKHV HQT
+PHQTOCVKQPU $GJCPFNKPI XQN RR
=? * 9 &QOOGN CPF 0 5CVQ (CUV VTCPUKGPV UVCDKNKV[ UQNWVKQPU +''' 6TCPU 2QYGT #RR 5[UV XQN
2#5 RR
=? # 5GON[GP CPF # &CDWNGCPW # U[UVGO CRRTQCEJ VQ CEEWTCVG UYKVEJKPI VTCPUKGPV ECNEWNCVKQPU DCUGF QP
UVCVG XCTKCDNG EQORQPGPV OQFGNNKPI +''' 6TCPU 2QYGT #RR 5[UV XQN 2#5 RR
/CTEJ #RTKN
=? +''' 6CUM (QTEG (KTUV DGPEJOCTM OQFGN HQT EQORWVGT UKOWNCVKQP QH UWDU[PEJTQPQWU TGUQPCPEG +'''
6TCPU 2QYGT #RR 5[UV XQN 2#5 RR 5GRV 1EV
=? * &QOOGN &KIKVCNG 4GEJGPXGTHCJTGP HWT GNGMVTKUEJG 0GV\G FKIKVCN EQORWVGT OGVJQFU HQT PGVYQTM
UQNWVKQPU KP )GTOCP #TEJKX H 'NGMVTQVGEJPKM XQN RR
=? +''' 5VCPFCTF &KEVKQPCT[ QH 'NGEVTKECN CPF 'NGEVTQPKEU 6GTOU 5GEQPF 'FKVKQP RWDNKUJGF D[ +'''
9KNG[ +PVGTUEKGPEG 0GY ;QTM Traduciendo...
=? ) ' (QTU[VJG CPF % $ /QNGT %QORWVGT 5QNWVKQP QH .KPGCT #NIGDTCKE 'SWCVKQPU 2TGPVKEG *CNN
'PINGYQQF %NKHHU 0 ,
=? 2 % /CIPWUUQP 6TCXGNNKPI YCXGU QP OWNVK EQPFWEVQT QRGP YKTG NKPGU # PWOGTKECN UWTXG[ QH VJG GHHGEVU
QH HTGSWGPE[ FGRGPFGPEG QH OQFCN EQORQUKVKQP +''' 6TCPU 2QYGT #RR 5[UV XQN 2#5 RR
/C[ ,WPG
=? 8 $TCPFYCLP /QFKHKECVKQPU QH 7UGT U +PUVTWEVKQPU HQT /#46+ 5'672 '/62 0GYUNGVVGT XQN PQ
RR #WI
=? * 9 &QOOGN # ;CP 4 , 1TVK\ FG /CTECPQ # $ /KNKCPK %CUG UVWFKGU HQT GNGEVTQOCIPGVKE
VTCPUKGPVU 7PKX QH $TKVKUJ %QNWODKC 8CPEQWXGT %CPCFC /C[
=? . $GTIGTQP &W %QWR FG $GNKGT GP *[FTCWNKSWG CW %QWR FG (QWFTG GP 'NGEVTKEKV¾ &WPQF 2CTKU
'PINKUJ VTCPUNCVKQP 9CVGT *COOGT KP *[FTCWNKEU CPF 9CXG 5WTIGU KP 'NGEVTKEKV[ #5/' %QOOKVVGG
9KNG[ 0GY ;QTM
=? %+)4' 9QTMKPI )TQWR 6JG ECNEWNCVKQP QH UYKVEJKPI UWTIGU + # EQORCTKUQP QH VTCPUKGPV PGVYQTM
Page 477
CPCN[UGT TGUWNVU 'NGEVTC 0Q RR 0QX
=? # $WFPGT +PVTQFWEVKQP QH HTGSWGPE[ FGRGPFGPV NKPG RCTCOGVGTU KPVQ CP GNGEVTQOCIPGVKE VTCPUKGPVU
RTQITCO +''' 6TCPU 2QYGT #RR 5[UV XQN 2#5 RR ,CP
=? 9 5 /G[GT CPF * 9 &QOOGN 0WOGTKECN OQFGNNKPI QH HTGSWGPE[ FGRGPFGPV VTCPUOKUUKQP NKPG
RCTCOGVGTU KP CP GNGEVTQOCIPGVKE VTCPUKGPVU RTQITCO +''' 6TCPU 2QYGT #RR 5[UV XQN 2#5 RR
5GRV 1EV
=? , - 5PGNUQP 2TQRCICVKQP QH VTCXGNNKPI YCXGU QP VTCPUOKUUKQP NKPGU (TGSWGPE[ FGRGPFGPV RCTCOGVGTU
+''' 6TCPU 2QYGT #RR 5[UV XQN 2#5 RR ,CP (GD
=? # 5GON[GP CPF # &CDWNGCPW (CUV CPF CEEWTCVG UYKVEJKPI VTCPUKGPV ECNEWNCVKQPU QP VTCPUOKUUKQP NKPGU
YKVJ ITQWPF TGVWTP WUKPI TGEWTUKXG EQPXQNWVKQPU +''' 6TCPU 2QYGT #RR 5[UV XQN 2#5 RR
/CTEJ #RTKN
=? # #OGVCPK # JKIJN[ GHHKEKGPV OGVJQF HQT ECNEWNCVKPI VTCPUOKUUKQP NKPG VTCPUKGPVU +''' 6TCPU 2QYGT
#RR 5[UV XQN 2#5 RR 5GRV 1EV
=? , 4 /CTVK #EEWTCVG OQFGNNKPI QH HTGSWGPE[ FGRGPFGPV VTCPUOKUUKQP NKPGU KP GNGEVTQOCIPGVKE VTCPUKGPV
UKOWNCVKQPU +''' 6TCPU 2QYGT #RR 5[UV XQN 2#5 RR ,CP
=? %+)4' 9QTMKPI ITQWR 6JG ECNEWNCVKQP QH UYKVEJKPI UWTIGU +++ 6TCPUOKUUKQP NKPG TGRTGUGPVCVKQP
HQT GPGTIK\CVKQP CPF TG GPGTIK\CVKQP UVWFKGU YKVJ EQORNGZ HGGFKPI PGVYQTMU 'NGEVTC 0Q RR
,CP
=? . /CTVK 8QNVCIG CPF EWTTGPV RTQHKNGU CPF NQY QTFGT CRRTQZKOCVKQP QH HTGSWGPE[ FGRGPFGPV VTCPUOKUUKQP
NKPG RCTCOGVGTU / # 5E 6JGUKU 7PKX QH $TKVKUJ %QNWODKC 8CPEQWXGT %CPCFC #RTKN
=? ( * $TCPKP 6TCPUKGPV CPCN[UKU QH NQUUNGUU VTCPUOKUUKQP NKPGU 2TQE +''' XQN RR 0QX
=? 8 .CTUGP QH '(+ KP 6TQPFJGKO 0QTYC[ RGTUQPCN EQOOWPKECVKQP
=? 4 ) 9CUNG[ CPF 5 5GNXCXKPC[CICOQQTVJ[ #RRTQZKOCVG HTGSWGPE[ TGURQPUG XCNWGU HQT VTCPUOKUUKQP NKPG
VTCPUKGPV CPCN[UKU 2TQE +'' XQN RR #RTKN
=? , 4 /CTVK 6JG RTQDNGO QH HTGSWGPE[ FGRGPFGPEG KP VTCPUOKUUKQP NKPG OQFGNNKPI 2J & VJGUKU 6JG
7PKXGTUKV[ QH $TKVKUJ %QNWODKC 8CPEQWXGT %CPCFC #RTKN
=? . /CTVK .QY QTFGT CRRTQZKOCVKQP QH VTCPUOKUUKQP NKPG RCTCOGVGTU HQT HTGSWGPE[ FGRGPFGPV OQFGNU
+''' 6TCPU 2QYGT #RR 5[UV XQN 2#5 RR 0QX
=? ' )TQUEJWRH 5KOWNCVKQP VTCPUKGPVGT 8QTICPIG CWH .GKVWPIUU[UVGOGP FGT *QEJURCPPWPIU )NGKEJUVTQO WPF
&TGJUVTQO 7DGTVTCIWPI 5KOWNCVKQP QH VTCPUKGPV RJGPQOGPC KP U[UVGOU YKVJ *8&% CPF CE VTCPUOKUUKQP
NKPGU KP )GTOCP 2J & VJGUKU 6GEJPKECN 7PKXGTUKV[ &CTOUVCFV )GTOCP[
=? - %CTNUGP ' * .GPHGUV CPF , , .C(QTGUV /CPVTCR /CEJKPG CPF 0GVYQTM 6TCPUKGPVU 2TQITCO 2TQE
+''' 2QYGT +PFWUVT[ %QORWVGT #RRN %QPH RR
=? ) )TQUU CPF / % *CNN 5[PEJTQPQWU OCEJKPG CPF VQTUKQPCN F[PCOKEU UKOWNCVKQP KP VJG EQORWVCVKQP QH
Traduciendo...
4
Page 478
GNGEVTQOCIPGVKE VTCPUKGPVU +''' 6TCPU 2QYGT #RR 5[UV XQN 2#5 RR ,WN[ #WI
=? +''' 9QTMKPI )TQWR 4GRQTV 4GEQOOGPFGF RJCUQT FKCITCO HQT U[PEJTQPQWU OCEJKPGU +''' 6TCPU
2QYGT #RR 5[UV XQN 2#5 RR 0QX
=? 4 2 5EJWN\ 9 & ,QPGU CPF & 0 'YCTV &[PCOKE OQFGNU QH VWTDKPG IGPGTCVQTU FGTKXGF HTQO UQNKF
TQVQT GSWKXCNGPV EKTEWKVU +''' 6TCPU 2QYGT #RR 5[UV XQN 2#5 RR /C[ ,WPG
=? ' 9 -KODCTM 2QYGT 5[UVGO 5VCDKNKV[ 5[PEJTQPQWU /CEJKPGU &QXGT 2WDNKECVKQPU 0GY ;QTM
TGRTKPV QH GFKVKQP D[ , 9KNG[ 5QPU
=? +''' 6GUV 2TQEGFWTGU HQT 5[PEJTQPQWU /CEJKPGU 5VCPFCTF DGKPI TGXKUGF KP VQ KPENWFG
VTKCN RTQEGFWTG HQT UVCPFUVKNN HTGSWGPE[ VGUVKPI
=? +'% 4GEQOOGPFCVKQPU HQT 4QVCVKPI 'NGEVTKE /CEJKPGT[ 2WDN #
=? + / %CPC[ &GVGTOKPCVKQP QH OQFGN RCTCOGVGTU QH U[PEJTQPQWU OCEJKPGU 2TQE +'' XQN RV
$ RR /CTEJ
=? $ #FMKPU CPF 4 ) *CTNG[ 6JG )GPGTCN 6JGQT[ QH #NVGTPCVKPI %WTTGPV /CEJKPGU #RRNKECVKQP VQ
2TCEVKECN 2TQDNGOU %JCROCP CPF *CNN .QPFQP
=? * 9 &QOOGN &CVC EQPXGTUKQP QH U[PEJTQPQWU OCEJKPG RCTCOGVGTU '/62 0GYUNGVVGT XQN 0Q
RR #RTKN
=? + / %CPC[ * , 4QJTGT - ' 5EJPKTGN 'HHGEV QH GNGEVTKECN FKUVWTDCPEGU ITKF TGEQXGT[ XQNVCIG CPF
IGPGTCVQT KPGTVKC QP OCZKOK\CVKQP QH OGEJCPKECN VQTSWGU KPNCTIG VWTDQIGPGTCVQT UGVU +'''
5VCVG QH VJG #TV 5[ORQUKWO 6WTDKPG )GPGTCVQT 5JCHV 6QTUKQPCNU 2WDN 0Q 6*11 294
=? 4 $ 5JKRNG[ & %QNGOCP CPF % ( 9CVVU 6TCPUHQTOGT EKTEWKVU HQT FKIKVCN UVWFKGU #+'' 6TCPU 2V
+++ XQN RR (GD
=? . ( $NWOG GVCN 6TCPUHQTOGT 'PIKPGGTKPI PF 'FKVKQP ,QJP 9KNG[ 5QPU 0GY ;QTM R
=? & ' *GFOCP 6JGQTGVKECN GXCNWCVKQP QH OWNVKRJCUG RTQRCICVKQP +''' 6TCPU 2QYGT #RR 5[UV XQN
2#5 RR 0QX &GE
=? & 2QXJ CPF 9 5EJWN\ #PCN[UKU QH QXGTXQNVCIGU ECWUGF D[ VTCPUHQTOGT OCIPGVK\KPI KPTWUJ EWTTGPV
+''' 6TCPU 2QYGT #RR 5[UV XQN 2#5 RR ,WN[ #WI
=? - 5EJNQUUGT #P GSWKXCNGPV EKTEWKV HQT 0 YKPFKPI VTCPUHQTOGTU FGTKXGF HTQO C RJ[UKECN DCUKU $TQYP
$QXGTK 0CEJTKEJVGP XQN RR /CTEJ CPF #RRNKECVKQP QH VJG GSWKXCNGPV EKTEWKV QH CP
0 YKPFKPI VTCPUHQTOGT $TQYP $QXGTK 0CEJTKEJVGP XQN RR ,WPG KP )GTOCP
=? / -J <KMJGTOCP /CIPGVK\KPI EJCTCEVGTKUVKEU QH NCTIG RQYGT VTCPUHQTOGTU 'NGMVTKEJGUVXQ 0Q RR
KP 4WUUKCP 'PINKUJ VTCPUNCVKQP KP 'NGEVTKE 6GEJPQNQI[ 7554
=? ' 2 &KEM CPF 9 9CVUQP 6TCPUHQTOGT OQFGNU HQT VTCPUKGPV UVWFKGU DCUGF QP HKGNF OGCUWTGOGPVU
+''' 6TCPU 2QYGT #RR 5[UV XQN 2#5 RR ,CP
=? , #XKNC 4QUCNGU CPF ( . #NXCTCFQ 0QPNKPGCT HTGSWGPE[ FGRGPFGPV VTCPUHQTOGT OQFGN HQT
GNGEVTQOCIPGVKE VTCPUKGPV UVWFKGU KP RQYGT U[UVGOU +''' 6TCPU 2QYGT #RR 5[UV XQN 2#5 RR
Page 479
0QX
=? # )QNFUVGKP 6TCPUHQTOGT R KP )GTOCP %JCRVGT KP 6GEJPKM 0Q GFKVGF D[ 6J $QXGTK CPF
6J 9CUUGTTCD (KUEJGT (TCPMHWTV )GTOCP[
=? % 9 6C[NQT 5KOWNCVKQP QH EWTTGPV VTCPUHQTOGTU +PVGTPCN 4GRQTV $QPPGXKNNG 2QYGT #FOKPKUVTCVKQP
(GD
Traduciendo...
=? . 1 %JWC CPF- # 5VTQOUOQG .WORGF EKTEWKV OQFGNU HQT PQPNKPGCT KPFWEVQTU GZJKDKVKPI J[UVGTGUKU
NQQRU +''' 6TCPU %KTEWKV 6JGQT[ XQN %6 RR 0QX
=? , ) (TCOG 0 /QJCP CPF 6 .KW *[UVGTGUKU OQFGNKPI KP CP GNGEVTQOCIPGVKE VTCPUKGPVU RTQITCO
+''' 6TCPU 2QYGT #RR 5[UV XQN 2#5 RR 5GRV
=? 0 )GTOC[ 5 /CUVGTQ CPF , 8TQOCP 4GXKGY QH HGTTQTGUQPCPEG RJGPQOGPC KP JKIJ XQNVCIG RQYGT
U[UVGOU CPF RTGUGPVCVKQP QH C XQNVCIG VTCPUHQTOGT OQFGN HQT RTGFGVGTOKPKPI VJGO %+)4' 4GRQTV 0Q
2CTKU
=? . -TCJGPDWJN $ -WNKEMG CPF # 9GDU 5KOWNCVKQPUOQFGNN GKPGU /GJTYKEMNWPIUVTCPUHQTOCVQTU \WT
7PVGTUWEJWPI XQP 5CVVKIWPIUXQTICPIGP UKOWNCVKQP OQFGN QH CP 0 YKPFKPI VTCPUHQTOGT HQT VJG CPCN[UKU
QH UCVWTCVKQP RJGPQOGPC KP )GTOCP 5KGOGPU (QTUEJWPIU WPF 'PVYKEMNWPIU $GTKEJVG 5KGOGPU
4GUGCTEJ CPF &GXGNQROGPV 4GRQTVU XQN RR
=? ' %QNQODQ CPF ) 5CPVCIQUVKPQ 4GUWNVU QH VJG GPSWKTKGU QP CEVWCN PGVYQTM EQPFKVKQPU YJGP UYKVEJKPI
OCIPGVK\KPI CPF UOCNN KPFWEVKXG EWTTGPVU CPF QP VTCPUHQTOGT CPF UJWPV TGCEVQT UCVWTCVKQP EJCTCEVGTKUVKEU
'NGEVTC 0Q RR /C[
=? 6 #FKGNUQP # %CTNUQP * $ /CTIQNKU CPF , # *CNNCFC[ 4GUQPCPV QXGTXQNVCIGU KP '*8
VTCPUHQTOGTU OQFGNKPI CPF CRRNKECVKQP +''' 6TCPU 2QYGT #RR 5[UV XQN 2#5 RR
,WN[
=? 4 % &GIGPGHH # OGVJQF HQT EQPUVTWEVKPI VGTOKPCN OQFGNU HQT UKPING RJCUG 0 YKPFKPI VTCPUHQTOGTU
2CRGT # RTGUGPVGF CV +''' 2'55WOOGT /GGVKPI .QU #PIGNGU %CNKH ,WN[
=? ) 9 5YKHV 2QYGT VTCPUHQTOGT EQTG DGJCXKQT WPFGT VTCPUKGPV EQPFKVKQPU +''' 6TCPU 2QYGT #RR
5[UV XQN 2#5 RR 5GRV 1EV
=? ' -WHHGN CPF 9 5 <CGPIN *KIJ 8QNVCIG 'PIKPGGTKPI 2GTICOQP 2TGUU 1ZHQTF
=? ( 9 *GKNDTQPPGT (KTKPI CPF XQNVCIG UJCRG QH OWNVKUVCIG KORWNUG IGPGTCVQTU +''' 6TCPU 2QYGT #RR
5[UV XQN 2#5 RR 5GRV 1EV
=? 9 ( .QPI # UVWF[ QP UQOG UYKVEJKPI CURGEVU QH C FQWDNG EKTEWKV *8&% VTCPUOKUUKQP NKPG +'''
6TCPU 2QYGT #RR 5[UV XQN 2#5 RR /CTEJ #RTKN
=? & , /GNXQNF 2 4 5JQEMNG[ 9 ( .QPI CPF 0 ) *KPIQTCPK 6JTGG VGTOKPCN QRGTCVKQP QH VJG 2CEKHKE
*8&% +PVGTVKG HQT FE EKTEWKV DTGCMGT VGUVKPI +''' 6TCPU 2QYGT #RR 5[UV XQN 2#5 RR
,WN[ #WI
=? $ % %JKW # EQPXGTVGT OQFGN HQT VJG FKIKVCN UKOWNCVKQP QH VTCPUKGPVU KP CE FE VTCPUOKUUKQP U[UVGOU
Page 480
/ # 5E VJGUKU 7PKXGTUKV[ QH $TKVKUJ %QNWODKC 8CPEQWXGT %CPCFC /C[
=? 8 $TCPFYCLP +PHNWGPEG QH PWOGTKECN PQKUG QP VJG UVCDKNKV[ QH 6[RG 5/ OQFGN '/62 0GYUNGVVGT
XQN PQ RR (GD
=? 8 $TCPFYCLP +PXGUVKICVKQP CPF KORTQXGOGPV QH NQPI VGTO UVCDKNKV[ HQT VJG 6;2' U[PEJTQPQWU
OCEJKPG OQFGN '/62 0GYUNGVVGT XQN PQ RR 0QX
=? & # %CNCJCP 0WOGTKECN EQPUKFGTCVKQPU HQT KORNGOGPVCVKQP QH PQPNKPGCT VTCPUKGPV EKTEWKV CPCN[UKU
RTQITCO +''' 6TCPU %KTEWKV 6JGQT[ XQN %6 RR ,CP
=? , /GEJGPDKGT 5KOWNCVKQP QH U[PEJTQPQWU OCEJKPGU KP ECUGU YKVJ NCTIG URGGF EJCPIGU '/62
0GYUNGVVGT XQN PQ RR 1EV
=? $ -WNKEMG 5KOWNCVKQP RTQITCO 0'61/#% OQFGNNKPI QH U[PEJTQPQWU CPF KPFWEVKQP OCEJKPGU KP
)GTOCP 5KGOGPU (QTUEJWPIU W 'PVYKEMNWPIU $GTKEJVG XQN RR
=? 4 4COCPWLCO # OGVJQF QH KPVGTHCEKPI 1NKXG U OQFGN QH U[PEJTQPQWU OCEJKPG KP CP GNGEVTQOCIPGVKE
VTCPUKGPVU RTQITCO '/62 0GYUNGVVGT XQN PQ RR #WI
=? * - .CWY CPF 9 5 /G[GT 7PKXGTUCN OCEJKPG OQFGNNKPI HQT VJG TGRTGUGPVCVKQP QH TQVCVKPI GNGEVTKE
OCEJKPGT[ KP CP GNGEVTQOCIPGVKE VTCPUKGPVU RTQITCO +''' 6TCPU 2QYGT #RR 5[UV XQN 2#5 RR
,WPG
=? ) , 4QIGTU CPF & 5JKTOQJCOOCFK +PFWEVKQP OCEJKPG OQFGNNKPI HQT GNGEVTQOCIPGVKE VTCPUKGPV
RTQITCO 1PVCTKQ *[FTQ +PVGTPCN /GOQTCPFWO
=? & 5JKTOQJCOOCFK 7PKXGTUCN OCEJKPG OQFGNNKPI KP 'NGEVTQOCIPGVKE 6TCPUKGPV 2TQITCO '/62

Traduciendo...
'/62 0GYUNGVVGT XQN PQ RR #RTKN CPF 6TCPU 'PIKPGGTKPI 1RGTCVKPI &KXKUKQP
%CPCFKCP 'NGEVTKECN #UUQEKCVKQP XQN
=? * - .CWY +PVGTHCEKPI HQT WPKXGTUCN OWNVK OCEJKPG U[UVGO OQFGNNKPI KP CP GNGEVTQOCIPGVKE VTCPUKGPVU
RTQITCO +''' 6TCPU 2QYGT #RR 5[UV XQN 2#5 RR 5GRV
=? 9 ( 6KPPG[ CPF , 9 9CNMGT &KTGEV UQNWVKQPU QH URCTUG PGVYQTM GSWCVKQPU D[ QRVKOCNN[ QTFGTGF
VTKCPIWNCT HCEVQTK\CVKQP 2TQE +''' XQN RR 0QX
=? * 'FGNOCPP 1TFGTGF VTKCPIWNCT HCEVQTK\CVKQP QH OCVTKEGU 2TQE 2QYGT 5[UVGOU %QORWVCVKQP
%QPHGTGPEG 5VQEMJQNO
=? , %CTRGPVKGT 1TFGTGF GNKOKPCVKQPU 2TQE 2QYGT 5[UVGOU %QORWVCVKQP %QPHGTGPEG .QPFQP
=? 9 ( 6KPPG[ 8 $TCPFYCLP CPF 5 / %JCP 5RCTUG XGEVQT OGVJQFU +''' 6TCPU 2QYGT #RR 5[UV
XQN 2#5 RR (GDT
=? 9 ( 6KPPG[ CPF 9 5 /G[GT 5QNWVKQP QH NCTIG URCTUG U[UVGOU D[ QTFGTGF VTKCPIWNCT HCEVQTK\CVKQP
+''' 6TCPU #WVQOCVKE %QPVTQN XQN #% RR #WI
=? 9 ( 6KPPG[ 5QOG GZCORNGU QH URCTUG OCVTKZ OGVJQFU HQT RQYGT PGVYQTM RTQDNGOU 2TQE TF 2QYGT
5[UVGOU %QORWVCVKQP %QPHGTGPEG 4QOG
=? 9 ( 6KPPG[ CPF 9 . 2QYGNN %QORCTKUQP QH OCVTKZ KPXGTUKQP CPF URCTUG VTKCPIWNCT HCEVQTK\CVKQP HQT
Page 481
UQNWVKQP QH RQYGT PGVYQTM RTQDNGOU 2CRGT RTGUGPVGF CV C 4QOCPKC 7 5 %QPHGTGPEG QP 2QYGT 5[UVGOU
$WEJCTGUV
=? $ / 9GGF[ 'NGEVTKE 2QYGT 5[UVGOU 6JKTF 'FKVKQP ,QJP 9KNG[ CPF 5QPU %JKEJGUVGT 'PINCPF
=? *CPFDQQM QH /CVJGOCVKECN (WPEVKQPU 'FKVGF D[ / #DTCOQYKV\ CPF + # 5VGIWP RWDN D[ 7 5 &GRV
QH %QOOGTEG
=? . / 9GFGRQJN CPF & , 9KNEQZ 6TCPUKGPV CPCN[UKU QH WPFGTITQWPF RQYGT VTCPUOKUUKQP U[UVGOU
U[UVGO OQFGN CPF YCXG RTQRCICVKQP EJCTCEVGTKUVKEU 2TQE +'' XQN RR (GD
=? ) $KCPEJK CPF ) .WQPK +PFWEGF EWTTGPVU CPF NQUUGU KP UKPING EQTG UWDOCTKPG ECDNGU +''' 6TCPU
2QYGT #RR U[UV XQN 2#5 RR ,CP (GD
=? 5 # 5EJGNMWPQHH 6JG GNGEVTQOCIPGVKE VJGQT[ QH EQCZKCN VTCPUOKUUKQP NKPGU CPF E[NKPFTKECN UJKGNFU $GNN
5[UVGO 6GEJPKECN ,QWTPCN XQN RR
=? & 4 5OKVJ CPF , 8 $CTIGT +ORGFCPEG CPF EKTEWNCVKPI EWTTGPV ECNEWNCVKQPU HQT 7& OWNVK YKTG
EQPEGPVTKE PGWVTCN EKTEWKVU +''' 6TCPU 2QYGT #RR 5[UV XQN 2#5 RR /C[ ,WPG
=? 1 $TGKGP CPF + ,QJCPUGP #VVGPWCVKQP QH VTCXGNNKPI YCXGU KP UKPING RJCUG JKIJ XQNVCIG ECDNGU 2TQE
+'' XQN RR ,WPG
=? . /CTVK 5KOWNCVKQP QH GNGEVTQOCIPGVKE VTCPUKGPVU KP WPFGTITQWPF ECDNGU YKVJ HTGSWGPE[ FGRGPFGPV
OQFCN VTCPUHQTOCVKQP OCVTKEGU 2J & VJGUKU 7PKXGTUKV[ QH $TKVKUJ %QNWODKC 8CPEQWXGT %CPCFC 0QX
=? # 5GON[GP FKUEWUUKQP VQ 1XGTJGCF NKPG RCTCOGVGTU HTQO JCPFDQQM HQTOWNCU CPF EQORWVGT RTQITCOU
D[ * 9 &QOOGN +''' 6TCPU 2QYGT #RR 5[UV XQN 2#5 R (GD
=? 2 9 *QTPDGEM 0WOGTKECN /GVJQFU 3WCPVWO 2WDNKUJGTU 0GY ;QTM
=? * 9 &QOOGN CPF , * 5CYCFC 6JG ECNEWNCVKQP QH KPFWEGF XQNVCIGU CPF EWTTGPVU QP RKRGNKPGU CFLCEGPV
VQ CE RQYGT NKPGU 4GRQTV VQ $ % *[FTQ CPF 2QYGT #WVJQTKV[ HQT %'# %QPVTCEV PQV
TGNGCUGF [GV
=? , # 6GIQRQWNQWU CPF ' ' -TKG\KU 'FF[ EWTTGPV FKUVTKDWVKQP KP E[NKPFTKECN UJGNNU QH KPHKPKVG NGPIVJ FWG
VQ CZKCN EWTTGPV 2CTV ++ UJGNNU QH HKPKVG VJKEMPGUU +''' 2#5 RR /C[
=? ) 9 $TQYP CPF 4 ) 4QECOQTC 5WTIG RTQRCICVKQP KP VJTGG RJCUG RKRG V[RG ECDNGU 2CTV +
7PUCVWTCVGF RKRG +''' 6TCPU 2QYGT #RR 5[UV XQN 2#5 RR ,CP (GD
=? 4 ) 4QECOQTC CPF ) 9 $TQYP 5WTIG RTQRCICVKQP KP VJTGG RJCUG RKRG V[RG ECDNGU 2CTV ++
&WRNKECVKQP QH HKGNF VGUVU KPENWFKPI VJG GHHGEVU QH PGWVTCN YKTGU CPF RKRG UCVWTCVKQP +''' 6TCPU 2QYGT
#RR 5[UV XQN 2#5 RR /C[ ,WPG
=? 2 FG #TK\QP CPF * 9 &QOOGN %QORWVCVKQP QH ECDNG KORGFCPEGU DCUGF QP UWD FKXKUKQP QH
EQPFWEVQTU +''' 6TCPU 2QYGT &GNKXGT[ XQN 294& RR ,CP

Traduciendo...
=? # #OGVCPK # IGPGTCN HQTOWNCVKQP QH KORGFCPEG CPF CFOKVVCPEG QH ECDNGU +''' 6TCPU 2QYGT 5[UV
XQN 2#5 RR /C[ ,WPG
Page 482
=? 9 6 9GGMU . . 9W / ( /E#NNKUVGT # 5KPIJ 4GUKUVKXG CPF KPFWEVKXG UMKP GHHGEV KP TGEVCPIWNCT
EQPFWEVQTU +$/ ,QWTPCN QH 4GUGCTEJ CPF &GXGNQROGPV XQN PQ RR 0QX
=? # -QPTCF 6JG PWOGTKECN UQNWVKQP QH UVGCF[ UVCVG UMKP GHHGEV RTQDNGOU CP KPVGITQFKHHGTGPVKCN
CRRTQCEJ +''' 6TCPU 8QN /#) RR ,CP
=? # -QPTCF +PVGITQFKHHGTGPVKCN HKPKVG GNGOGPV HQTOWNCVKQP QH VYQ FKOGPUKQPCN UVGCF[ UVCVG UMKP GHHGEV
RTQDNGOU +''' 6TCPU 8QN /#) RR ,CP
=? , 9GKUU CPF < , %UGPFGU # QPG UVGR HKPKVG GNGOGPV OGVJQF HQT OWNVKEQPFWEVQT UMKP GHHGEV RTQDNGOU
+''' 6TCPU 8QN 2#5 RR 1EV
=? 0 5TKXCNNKRWTCPCPFCP 5GTKGU KORGFCPEG CPF UJWPV CFOKVVCPEG OCVTKEGU QH CP WPFGTITQWPF ECDNG
U[UVGO / # 5E 6JGUKU 7PKXGTUKV[ QH $TKVKUJ %QNWODKC 8CPEQWXGT %CPCFC
=? ' 2 &KEM 0 (WLKOQVQ ) . (QTF CPF 5 *CTXG[ 6TCPUKGPV ITQWPF RQVGPVKCN TKUG KP ICU KPUWNCVGF
UWDUVCVKQPU RTQDNGO KFGPVKHKECVKQP CPF OKVKICVKQP +''' 6TCPU 2QYGT #RR 5[UV XQN 2#5 RR
1EV
=? * $QEMGT CPF & 1GFKPI +PFWEGF XQNVCIGU KP RKRGNKPGU ENQUG VQ JKIJ XQNVCIG NKPGU
'NGMVTK\KVCVUYKTVUEJCHV KP )GTOCP XQN RR
=? ;KP ;CPCP #P CRRNKECVKQP QH HKPKVG GNGOGPV OGVJQF ('/ VQ VJG ECNEWNCVKQPU QH VTCPUOKUUKQP NKPG
RCTCOGVGTU +PVGTPCN 4GRQTV 7PKX QH $TKVKUJ %QNWODKC 8CPEQWXGT %CPCFC #RTKN
=? & -KPF 6JG GSWCN CTGC ETKVGTKQP HQT KORWNUG XQNVCIG UVTGUU QH RTCEVKECN GNGEVTQFG EQPHKIWTCVKQPU KP CKT
'6< # KP )GTOCP XQN RR
=? ( *GKNDTQPPGT FKUEWUUKQP KP = ?
=? 7 $WTIGT 5WTIG CTTGUVGTU YKVJ URCTM ICRU %JCRVGT KP 5WTIGU KP *KIJ 8QNVCIG 0GVYQTMU GFKVGF D[ -
4CICNNGT 2NGPWO 2TGUU 0GY ;QTM
=? +''' 9QTMKPI )TQWR /QFGNNKPI QH EWTTGPV NKOKVKPI UWTIG CTTGUVGTU +''' 6TCPU 2QYGT #RR 5[UV
XQN 2#5 RR #WI
=? & 2 %CTTQN 4 9 (NWIWO , 9 -CND CPF * # 2GVGTUQP # F[PCOKE UWTIG CTTGUVGT OQFGN HQT WUG KP
RQYGT U[UVGO VTCPUKGPV UVWFKGU +''' 6TCPU 2QYGT #RR 5[UV XQN 2#5 RR /C[ ,WPG
=? ) $TCWPGT 5KOWNCVKQP QH VJG RGTHQTOCPEG QH OGVCN ENCF UWDUVCVKQPU CPF QRGP CKT UWDUVCVKQPU WPFGT
NKIJVPKPI UWTIGU KP )GTOCP 2J & FKUUGTVCVKQP &CTOUVCFV )GTOCP[
=? & 9 &WTDCM <KPE QZKFG CTTGUVGT OQFGN HQT HCUV UWTIGU '/62 0GYUNGVVGT XQN PQ RR ,CP
=? $ -PGEJV 5QNKF UVCVG CTTGUVGTU %JCRVGT KP 5WTIGU KP *KIJ 8QNVCIG 0GVYQTMU GFKVGF D[ - 4CICNNGT
2NGPWO 2TGUU 0GY ;QTM
=? , % (NQTGU ) 9 $WEMNG[ CPF ) /E2JGTUQP 6JG GHHGEVU QH UCVWTCVKQP QP VJG CTOCVWTG NGCMCIG
TGCEVCPEG QH NCTIG U[PEJTQPQWU OCEJKPGU +''' 6TCPU 2QYGT #RR 5[UV XQN 2#5 RR
Page 483
/CTEJ
=? / 8 - %JCTK < , %UGPFGU 5 * /KPPKEJ 5 % 6CPFQP CPF , $GTMGT[ .QCF EJCTCEVGTKUVKEU QH
U[PEJTQPQWU IGPGTCVQTU D[ VJG HKPKVG GNGOGPV OGVJQF +''' 6TCPU 2QYGT #RR 5[UV XQN 2#5
Traduciendo...
RR ,CP
=? 4 ) *CTNG[ & , 0 .KOGDGGT ' %JKTTKEQ\\K %QORCTCVKXG UVWF[ QH UCVWTCVKQP OGVJQFU KP U[PEJTQPQWU
OCEJKPG OQFGNU +'' 2TQE XQN RV $ RR ,CP
=? # 5YGGVCPC 0 -WPMNG 0 *KPIQTCPK 8 6CJKNKCPK &GUKIP FGXGNQROGPV CPF VGUVKPI QH M8 CPF
M8 ICRNGUU UWTIG CTTGUVGTU +''' 6TCPU 2QYGT #RR 5[UV XQN 2#5 RR ,WN[
=? / $ *WIJGU 4 9 .GQPCTF 6 ) /CTVKPKEJ /GCUWTGOGPV QH RQYGT U[UVGO UWDU[PEJTQPQWU FTKXKPI
RQKPV KORGFCPEG CPF EQORCTKUQP YKVJ EQORWVGT UKOWNCVKQPU +''' 6TCPU 2QYGT #RR 5[UV XQN
2#5 RR
=? * 9 &QOOGN # ;CP 5JK 9GK *CTOQPKEU HTQO VTCPUHQTOGT UCVWTCVKQP +''' 6TCPU 2QYGT
&GNKXGT[ XQN 294& RR #RTKN
=? 4 ' /E%QVVGT * # 5OQNNGEM 5 , 4CPCFG 9 * -GTUVKPI #P KPXGUVKICVKQP QH VJG
HWPFCOGPVCN HTGSWGPE[ KORGFCPEG QH C UKPING RJCUG FKUVTKDWVKQP NCVGTCN +''' 6TCPU 2QYGT &GNKXGT[
XQN 294& RR ,CP
=? / 4GP OKPI 6JG EJCNNGPIG QH DGVVGT '/62 6#%5 XCTKCDNG QTFGTKPI '/62 0GYUNGVVGT XQN PQ
RR #WI
=? , # .KOC 0WOGTKECN KPUVCDKNKV[ FWG VQ '/62 6#%5 KPVGT TGNCVKQP '/62 0GYUNGVVGT XQN 0Q
RR ,CP
=? . &WD¾ CPF * 9 &QOOGN 5KOWNCVKQP QH EQPVTQN U[UVGOU KP CP GNGEVTQOCIPGVKE VTCPUKGPVU RTQITCO YKVJ
6#%5 2TQE +''' 2+%# %QPH RR /C[
=? 0 &TCXKF %QORCTKUQP QH XCTKQWU TGRTGUGPVCVKQPU QH C NKOKV HWPEVKQP KP C UGEQPF QTFGT U[UVGO
JCPFYTKVVGP PQVGU
=? +''' 6CUM (QTEG %QPXGPVKQPU HQT DNQEM FKCITCO TGRTGUGPVCVKQPU +''' 6TCPU 2QYGT 5[UV XQN
2945 RR #WI
=? , * 0GJGT 2JCUG UGSWGPEG KORGFCPEG QH RKRG V[RG ECDNGU +''' 6TCPU 2QYGT #RR 5[UV XQN
RR #WIWUV

Emtp Theory Book

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19/6/2020 EMTP THEORY BOOK

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$TCPEJ QH 5[UVGO 'PIKPGGTKPI

1TKIKPCN $2# EQPVTCEV 0Q &' #% $2

! UJQTV EKTEWKV QT HCWNV CPCN[UKU

! RQYGT QT NQCF HNQY CPCN[UKU

! UVCDKNKV[ CPCN[UKU CPF

! CPCN[UKU QH GNGEVTQOCIPGVKE VTCPUKGPVU

DGIKPPGT VJGTG KU VJG FKHHKEWNV[ QH FGXGNQRKPI CP WPFGTUVCPFKPI

'/62 UKOWNCVKQPU JQYGXGT

,WN[ $2# '/62 6JGQT[ $QQM KP 92

7PKXGTUKV[ QH $TKVKUJ %QNWODKC NQECVGF KP 8CPEQWXGT $ % %CPCFC KP /QTG KPHQTOCVKQP

-YCPI [K )GT EQPVKPWGU KP 9GUV .KPP

,WN[ $2# '/62 6JGQT[ $QQM KP 92

WP\KRRKPI VJGUG DGEQOG CPF -D[VGU

%Q %JCKTOGP QH %CPCFKCP #OGTKECP '/62 7UGT )TQWR

&T 9 5EQVV /G[GT &T 6UW JWGK .KW

+0641&7%6+10 61 6*' 51.76+10 /'6*1& 75'& +0 6*' '/62

some useful information in the description of what exists today.

eSual to the injected current i :

K (V) % K (V) % K (V) % K (V) ' K (V) (1.1)

(KI Details of a larger network around node no. 1

currents, i , etc., as functions of the node voltages. For the resistance,

with the central difference eSuation

X(V) % X(V&)V) K(V) & K(V&)V)

This can be rewritten, for the case of Fig. 1.1, as

linear, ordinary differential eSuations.

L , C inductance and capacitance per unit length ,

x distance from sending end,

have the well known solution due to d #lembert:

K ' ((Z & EV) & H(Z % EV)

X ' <((Z & EV) % <H(Z % EV) (1.5a)

c velocity of wave propagation (constant).

X % <K ' 2<((Z & EV) (1.5b)

must also remain constant. With travel time

preceding time steps.

[)] [X(V)] ' [K(V)] & [JKUV] (1. a)

Brackets are used to indicate matrix and vector Suantities.

with [G] n x n symmetric nodal conductance matrix,

[v(t)] vector of n node voltages, Traduciendo...

[i(t)] vector of n current sources, and

[hist] vector of n known history terms.

[)##][X#(V)] ' [K#(V)] & [JKUV#] & [)#$][X$(V)] (1. b)

system of linear eSuations is solved for [v (t)], using

which was written by J.W. Walker.

ES. (1.1) now becomes

obvious. For the resistance,

for the inductance,

and for the capacitance,

+ ' LT%(8 & 8 ) (1.12)

if the eSuivalent B circuit representation of Fig. 1.2 is used, with

(KI ESuivalent B circuit for ac steady state

and sometimes eSually useful,

where ( is the propagation constant,

( ' (4) % LT.)) ()) % LT%)) (1.15)

For the lossless case with 4 0, and G 0, ES. (1.1 ) simplifies to

B circuit is usually called the nominal B circuit,

<UGTKGU ' ý@(4) % LT.))

into ES. (1.9),

+0641&7%6+10 61 6' 51.76+10 /'61& 75'& +0 6*' '/62