Application of matrix methods to the solution of

travelling-wave phenomena in polyphase systems

L. M. Wedepohl, Ph.D., B.Sc.(Eng.), Graduate

Synopsis
The interest in travelling waves and surge phenomena in power systems has grown considerably because
of the relevance to power-line carrier communication and protection, fault location, switching of unloaded
lines and the recovery voltages on circuit-breakers under short-line fault conditions.
The paper summarizes the solution to the single-conductor wave equation, develops the solution for the
2-conductor case by classical methods, a n d then indicates the rapidly growing complexity of the problem
with increasing numbers of conductors when solving by classical methods. T h e 2-conductor case is then
solved by using the method of matrix algebra. These results are shown to be valid for any number of con-
ductors, and the concept of surge impedance and propagation coefficient for polyphase systems is introduced.
The single-phase equation is shown to be a particular case of the general equation and the similarity
between the results of the two cases is indicated. Examples of the matrix method are given, including a
proof that symmetrical components are a particular case of the general result. T h e paper concludes by
indicating that the method is particularly suitable for calculations carried out with a digital computer.

List of symbols (/) All solutions given are for the steady-state case, i.e. for
= self-impedance per unit length of ith conductor one specific frequency only. This has been done because
Z\f = mutual impedance per unit length between ith and there is no general solution to the wave equation for
yth conductors arbitrary waveshapes, even in the simplest case. Particular
= self-admittance per unit length of ith conductor transient solutions may be derived from the steady-state
Y\<? = mutual admittance per unit length between ith and solutions by making suitable approximations and using
yth conductors standard inversion techniques.
= voltage to earth of ith conductor
f
If == current in ith conductor
characteristic self-impedance of phase i
1 Introduction
7<0) Travelling waves and surge phenomena in power
= characteristic mutual impedance between phases i
Zf] andy systems are of importance in solving problems relating to
power-line carrier communication, protection of very long
= characteristic self-admittance of phase i lines, fault location, switching of unloaded lines and calcula-
Yf, = characteristic mutual admittance between phases i tion of recovery voltages on circuit-breakers under short-line
andy fault conditions.
= rth component voltage, forward travelling Up to the present, these problems have been usually solved
yic-) = rth component voltage, reverse travelling
by making simplifications and assumptions. However, these
= rth component current, forward travelling should only be used for the restricted conditions for which
= rth component current, reverse travelling they are valid. When, as is sometimes the case, they are used
= rth component impedance indiscriminately, the results obtained can be misleading.
y<rc) = rth component admittance
A significant advance in the solution of such problems
= rth propagation coefficient was made by Fallou1, who, with an assumption of complete
== proportionality constant between ith phase voltage symmetry of the conductors, applied the concept of sym-
Vr and rth component voltage
metrical components to the solution of travelling-wave
= proportionality constant between ith phase current phenomena. This work was applied to various problems by
and rth component current De Quervain and others.2"4 This method is limited, however,
Qir
in that it yields average values for surge impedances and
(a) If subscripts do not appear in the above quantities, they propagation coefficients, and this unfortunately masks
are matrices. important effects produced by asymmetry of conductors.
(b) K, is the transpose of matrix K, i.e. rows and columns are
interchanged. In investigating radio interference arising from overhead
(c) R is the complex conjugate of matrix K, i.e. each element power lines, Adams5 introduced the use of matrix algebra
of the former matrix is the complex conjugate of each methods in the analysis of the distribution of radio-frequency
element of the latter. currents in asymmetrical systems of conductors. It was valid,
(d) det K is the determinant of the elements of matrix K. for this investigation, to assume zero resistance in the con-
ductors and an infinitely conducting earth plane with no
(e) Superscripts (c) and O) have been used to identify com- earth wires. The results were therefore of a restricted nature,
ponent and phase quantities, respectively. This has but the method of analysis was important as a foundation
enabled the normally accepted symbols V, I, Z and Y to for the complete solution of the problem.
be retained for voltages, currents, impedances and
admittances, irrespective of their nature. The present paper deals with the complete solution for
various arrangements and for the generalized case. The
Paper 4333 P,firstreceived 6th April and in revised form 18th July 1963 application of the generalized solution in practice will, how-
Dr. Wedepohl is with A. Reyrolle and Co. Ltd. ever, need digital computing facilities.
2200 PROC. IEE, Vol. 110, No. 12, DECEMBER 1963
2 Review of cases of wave propagation in
simple systems
2.1 Single conductor in presence of an infinite earth 7-(0) _ ^ U _ /^ll
plane
The arrangement of a single conductor in the presence
of an infinite earth plane is shown schematically in Fig. 1, V£) is the difference between the two components in
eqn. 9, i.e.
(P) (P) z
n 'A* 1 %
.(P) V[c) = F< c +> e x p - y x x - V[c~ > e x p y , x . . (13)
ho
•AAMAA- The equations define completely the general wave equation
for the single-conductor system. It may be noted that each
component voltage produces an associated current, the two
being related by the so-called characteristic impedance Z[f.
Y,,-Ax v, ( p V p ) It may be also noted that the V^ component travels
from right to left in Fig. 1 in accordance with the negative
sign of the corresponding current in eqn. 10.
The arbitrary constants V{c+) and F/ 0 "' may be eliminated
x=0 x x=l
from a knowledge of the boundary or terminal conditions at
each end of the system. For example, if Vl0 and / 10 are the
Fig. 1 voltage and current at the end of the system where x = 0,
Equivalent circuit of single-conductor system and Vx r and /, r are the corresponding quantities when x = r,
which includes details of one element of the system. The ^io = ^ . [ ^ ( c + ) + ^, <c - ) ] (14)
fundamental equations relating to the element are ex
P - + > exp Ylr] (15)
0) Sn[V{ c + )
-V}c->] . . . . (16)

hr = [ ^ J - ' S i i t ^ e x p - Yir - Vf->expyir] (17)

YtfVr (2)
By elimination, the usual 4-pole equations of the transmission
Hence line are obtained:
d2Vip) V\r = vio c o s h Yir ~ zuho sinh yxr . . . . (18)
-£$- = ZtfYtfvp (3) h r = ~ [ZuT 1
Pio sinh Ylr + 710 cosh y,r . (19)
and ^=YtfZ[»l[» (4) The final concept to be considered in this Section is that
p
of reflection factor. Consider conditions at end r of the system
Solving for V\ \ of Fig. 1. The voltage and current are defined by eqns. 15
V[p) = A exp —yxx + B exp yxx . . . . (5) and 17.
The voltage and current are also related by the terminal
where A and B are arbitrary constants and impedance Z l l r , Vlr = ZnrIXr. By elimination, it is seen that
yi-VlW (6) exp yxr = - yxr (20)
p)
V[ may be seen to consist of two components: one where
travelling in the forward direction and the other in the Z llr -Zf?
reverse direction. ~
Although only one conductor is being considered, eqn. 5
will be rewritten to conform with the notation used for more K is known as the reflection factor, since it expresses the
complex cases at a later stage in the paper, namely reverse travelling wave as a fraction of the forward component.
y[p) = 5,,[K,(c+> exp -yxx + F,<c-> exp Ylx] . (7)
2.2 T w o conductors in the presence of an infinite earth
This shows the two components of V[ \ The reason for the p plane
arbitrary constant S1, j will become apparent for more complex Referring to Fig. 2 showing an element of the system,
cases. The above equation may be further simplified as the following equations relate voltage and current in each
follows: phase:
vy = snv\* (8) dVip)
(22)
where
VU!i = ytc+) e x p _ yiX yie-)
+ e x p yiX (9)
dV\p)
(23)
The current is derived from V[p) using eqn. 1:
dl\p)
(24)

. (10) (25)
PROC. IEE, Vol. 110, No. 12, DECEMBER 1963
2201
 The following equations may be derived by elimination: where
d V\ 2 p) yic) = yie+) exp _ yiX + y(c-) exp yiX
_i- = PIlfr+P12^> (26) V^ = V2<c+) exp - y2x + F f - > exp y2x
d2Vkp)
-jJ- = P2lV<r + P22Vy (27) + 4P 12 / > 21 ]}
2 p)
d l\
4i>12P21]}
-£ = puiy> + p2lil» (28)
d2H>p)
~ P22)2
- ^ = Pl2H» + P22l2»> (29) IP 21
Solving eqns. 22 and 23 for I[p) and lg> in terms of V[c\
(p) (P) (p) it is shown in Appendix 11.1 that the solution for the two
in- currents is given by the following expressions:

1\P) =

(p)
. . . (36)
(P) (P) (p) (p) (p)
1
2
det Z(p)
x=0 x=l _ I • • • (37)
c) 0
(p) V[ = x — Vl ^ exp yxx
2,,-Ax x — Vif~^ exp y2x
-WW
det Z*>
•A/WV^ The following points regarding the solution should be noted:
(a) Two propagation coefficients are present, each being
associated with a pair of forward and backward travelling
waves.
(b) Each type of travelling wave of voltage appears on
both phases in a ratio determined by the system parameters
only. However, the relationship between travelling waves of
different type is arbitrary.
(c) Each travelling wave of voltage is accompanied by a
corresponding current. The relationship between voltage and
current of a particular component type in each phase is a
constant and is determined by the system parameters only.
These constants have the character of surge impedances, and
their values to each component in each phase are
det Zip)
Yi[Zg -
Fig. 2
det Z(p)
Equivalent circuits of 2-conductor arrangement
a Overall relationships
y2[Sx2Z%IS22
bl Impedance elements
c Admittance elements 7(0 _ det Zip)
Z
21 ~

w
where det Z
Z (c) _
Pxx = (30)

Pn = " (31) These relationships are determined from eqns. 36 and 37.
The arbitrary constants are eliminated from a knowledge of
P2X = (32) the voltages and Qurrents at each end of the line, as for the
single phase case.
P22 = - (33) It may be seen that the line voltages and currents are
determined in terms of components in a fashion analogous
It is shown in Appendix 11.1 that the solution for the to symmetrical components in symmetrical 3-phase systems.
various voltages is Even for this case the labour in obtaining a solution is fairly
great and the need for an ordered solution is indicated.
(34) The results of this Section could have been expressed in
standard form, as was done for the case of a single cpnductor
(35) in the previous Section. However, the mathematical manipula-
2202 PROC. IEE, Vol. 110, No. 12, DECEMBER 1963
tion involved is tedious, and for this reason the matrix method These have the same form as eqn. 3 for the single-conductor
is introduced in the next Section. The equivalence of the case and have the simple solution
results is demonstrated before proceeding to the general case. VUA = j/(c+) e x p _ YiX + yu-t exp y{X # .
yu) = yic+) e x p _ yiX + yic-) exp yiX
3 Matrix solution to the 2-conductor problem
c+ c+) c+) )
The main difficulty in solving the 2-phase (and in where V[ \ V{ , V± and V£~ are arbitrary constants.
general multi-phase) problem is due to the fact that second- The matrix 5 is chosen in such a way that S~XPS is diagonal
order rates of change of voltage (and current) in each phase since then a direct solution of the above equations is possible,
are a function of the voltages (currents) in all phases (see solving for each component in turn. It is shown in Appendix
eqns. 26-29). Before a classical solution may be obtained, 11.2 that the diagonal relationship S~[PS — y2, where y 2 is a
an elimination process must be carried out. Eqns. 22-29 may diagonal matrix, is only possible if the following equation is
be rewritten in matrix form as follows: satisfied:
det (P - y2) = 0 (47)
(38)
dx This relationship is seen to be independent of the S matrix.
It is also shown in Appendix 11.2 that the two values of y2,
(39) and hence of y, obtained from eqn. 47 correspond to the
two values of the propagation coefficient for the 2-phase
(40) case obtained in Section 2.2. (The choice of the symbol y 2
dx2 for the diagonal matrix anticipated this result.)
The elements of column i of the S matrix are then deter-
1 mined by solving the set of homogeneous linear dependent
dx2 simultaneous equations given in matrix form by
. . . . (41)
and / (/>)
are column matrices corresponding to the
- yf)s0) = o (48)
voltages and currents of each phase, and Z ( p ) a n d Y^ are where S (/) is a column matrix whose elements correspond to
square matrices, the element at the intersection of row i and the elements of the /th column of S. Since the equations are
column j being the mutual impedance or admittance between dependent owing to the vanishing of the determinant of the
phases i and ;. It may be noted that both these matrices are coefficients, any one value may be chosen arbitrarily and the
bilateral since a passive network is considered, and for this remaining element solved in terms of this one. It is shown in
reason Z\*> = Z<*> and yO»> = Y(tp\ a relationship used Appendix 11.2 that the values of the elements 5"i i, S22, Sn a n d
above in deriving eqn. 41. P is a square matrix previously S2u determined by making use of eqn. 48, and the phase
used to simplify the analysis, its elements being defined by voltages, as determined from eqn. 43, are identical with the
eqns. 30-33, i.e. the element at the intersection of row / and solutions previously determined by classical methods. It is
column j is further shown that the phase currents are identical with those
determined by classical methods. The phase currents are
(42)
solved by inverting the matrix equation 38, i.e.
p,j =
1 1
The matrix solution is based on a linear transformation of • • (49)
dx dx
voltage and subsequent manipulation, so that second-order
differential relationships involve diagonal matrices only.
Mutual effects are thus eliminated, making a direct solution 4 General solution for the polyphase case
component for component possible. The complete analysis is The matrix approach to the solution of the polyphase
carried out in Appendix 11.2, and only the results are given wave problem was introduced in Section 3. The second-order
below. matrix differential equations involving voltage and current,
Component-voltage matrix K(c) is introduced, related to
= py(p) (50)
the phase-voltage matrix by dx2
(43)
and (51)
Substitution in eqn. 40 and rearrangement yields dx2
= y2V (c)
are related to component voltages by the linear trans-
(44)
dx2 formations :
y{p) _ (52)
2 /(p) = Q[(c) (53)
Matrix S is chosen in such a way that y is a diagonal
matrix of the form where the square matrices S and Q are as yet undefined.
Eqns. 52 and 53 express in general terms a relationship which
72 = is well known in work on symmetrical components, in which
case the phase voltages Vx, V2 and K3 are related to the
As a direct consequence of this relationship, matrix equa- sequence component voltages V+, V_ and Vo by eqn. 53,
tion 44 separates into two simple second-order equations : where
1 1 1

Also, S = Q in this particular case.

PROC. IEE, Vol. 110, No. 12, DECEMBER 1963 2203
 Substituting for V^ and 7 (p) in eqns. 50 and 51, the trans- S matrices are solved one column at a time by solving the
formed equations are obtained: system of homogenous dependent simultaneous equations
(P-yf)Sa) = o (66)
= y2F<c> (54)
dx2 2
(P, - y )G(/) = 0 (67)

= Q-xPtQI{c) = y' 2 / (c) (55) where (/) indicates that the elements of column i of the
dx2 respective matrices are to be considered. Since these systems
of equations are dependent (owing to the vanishing of the
The matrix method is based on the fact that, by a suitable determinant of the coefficients), one value in each column is
choice of S and Q, specified arbitrarily and the remaining elements are evaluated
y2 = S-lPS (56) accordingly.
It may be noted that the n values of yf are known to
and y' 2 = Q~lPtQ (57) mathematicians as eigenvalues and the corresponding S^ and
<2(/) as eigenvectors.
are diagonal matrices. The matrix Z® connecting component voltages and
Once y and y' have been determined, eqns. 54 and 55 currents is
are solved as a series of simple wave equations, since mutual
effects have been eliminated. For example, y(c) — z(c)/(c) (68)
where
d2V{*
= y\Vf (58) (69)
dx2 (c)
It is proved in Appendix 11.3 that Z is a diagonal
matrix, this being a proof of the fact that component voltages
(59) produce component currents of the same type only, i.e. there
is no mutual interaction between components.
etc., and By making use of well known theorems of matrix algebra,
V\c) = exp - y,JcF/c+) + exp . . . (60) it is possible to tabulate a list of properties relating to the
matrix equations of the polyphase-wave equation. These are
7(c) is determined in a similar fashion. as follows, with their derivation in Appendix 11.3:
By analogy with the work in Section 3 and Appendix 11.2, (a) The S and Q matrices are neither orthogonal nor unitary,
it may be seen that, for eqn. 56 to be valid, i.e. y2 diagonal, i.e. SS, =t D, QQ, # D, SS, ^ D, QQ, ^ D, where the
= 0 (61) inequalities signify that none of the matrix products are
diagonal in form.
In eqn. 61, S^ represents column i of the S matrix. Eqn. 61 (b) The S and Q matrices are mutually orthogonal, i.e.
represents in matrix form a system of homogenous simul- Q,S = S,Q = D, where D is a diagonal matrix.
taneous equations in which the unknowns are the n elements (c) The S and Q matrices do not form a unitary set, i.e.
of column i of the S matrix. S,Q =£ D and Q,S ^ D, where D is a diagonal matrix.
For such a system of simultaneous equations to have a id) The transformed impedance and admittance matrices are
non-trivial solution, it is necessary for the determinant of the diagonal, i.e. S~lZQ = Dz and Q~lYS — Dy where Dz
coefficients to be zero, i.e. and Dy are diagonal matrices. Also, the product of the
transformed impedance and admittance matrices com-
det (P - y?) = 0 (62) mutes and is equal to the propagation matrix y2, i.e.
A similar consideration of the solution of the Q matrix DzDy = DyDz = y2.
relating component to phase currents given in eqns. 53 and (e) The Z (c) and F ( c ) matrices, unlike the y 2 matrix, are
57 will show that, if a diagonal relation in the latter equation not unique. This is a consequence of the fact that the S
is to be achieved, it is necessary for the following determinental and Q matrices are not in themselves unique. It is shown
equation to hold: in Appendix 11.3 that, if S and Q are solutions for the
connection matrices, then so are S' and Q' where S' = SA,
det OP, -y'j) = O (63) and Q' = QA', A and A' being any diagonal matrices.
Since y' 2 in eqn. 57 is diagonal, it follows that the matrix The significance of these matrix equations is that the
Pt — y' 2 differs from matrix P, only by the modification to columns of the S and Q matrices are evaluated in terms
the elements in the principal diagonal. Thus of one arbitrarily chosen element, so that the matrices
are only unique insofar as the elements of a particular
Pt - y'2 = (P - y' 2 ), (64) column are in a fixed ratio.
It is well known that the determinant of a matrix is equal to if) The impedance of each phase to each component is
the determinant of the transposed matrix, so that unique, e.g. the impedance of phase 1 to component of
type 1 is [Sn ^i(c)]/[Qii/i(c)], and, in general, the impedance
det (P - y'2) = det (P - y'2), (65) of phase r to the ith component is [SriV}c)]l[QriI\c)], and
it is shown in Appendix 11.3 that these impedances are
By comparing eqns. 62 and 65, it may be seen that y 2 = y' 2 . unique.
The use of the symbol y 2 for the diagonalization of both
the voltage and current matrices anticipated this important
result, y2 does, in fact, define the propagation coefficient for 5 Polyphase surge impedance, propagation
each component, and for the solution to have a meaning it is coefficients and reflection factors
necessary to obtain the same set of values, either from a In earlier Sections the matrix solution of the polyphase
consideration of voltages or from a consideration of currents. wave equation was developed. In order to complete the
Once having determined the n values of y2, the Q and solution, the concepts of surge impedance, reflection factor
2204 PROC. IEE, Vol. 110, No. 12, DECEMBER 1963
and propagation coefficient are introduced in relation to commutative. It is readily seen that eqns. 18 and 19 are a
boundary conditions in order to complete the analogy between special case of eqns. 73 and 74.
the general solution and the simple solution to the single-
conductor wave equation developed in Section 2. 5.4 Interpretation of polyphase transmission matrices
In this Section it has been shown that the matrix
5.1 Polyphase surge impedance solution to the polyphase problem includes the single-
Surge impedance in the single-conductor case is conductor system as a special case.
generally defined as the ratio between voltage and current Polyphase surge impedance in a multiconductor system is
in an infinite line, or as the terminal impedance which gives a bilateral set of impedances which define the currents which
Zw — Z (0) would flow in each conductor of an infinitely long line as a
a reflection factor of zero, i.e. K = " = 0 and result of impressing voltages on each of the conductors. The
Z l l r = Z<°>(eqn.21). Au + ^n converse holds if currents were injected into each of the con-
Polyphase surge impedance will be defined in the latter ductors; the surge-impedance matrix would define the voltage
way, since the concept of reflection factor is developed at the of each conductor relative to the earth plane.
same time. Polyphase reflection factor defines the voltage on each
The analysis is carried out in Appendix 11.4, where it is conductor at the terminals of a system travelling in the reverse
shown that the polyphase surge impedance is direction in response to a set of incident voltages arriving in
Z«» = Sy-*S-W> (70) the forward sense at the terminals. It follows that the total
voltage expressed in matrix form at the terminals is
(0)
It is also shown that Z is a symmetrical matrix, i.e. V=(l +K)V^\ where V<f> is the incident (forward-
travelling) phase-voltage matrix. It may be noted that K is,
The validity of this expression in the single-phase case is in general, not a diagonal matrix, so that reflected voltages
evident, since the matrices become elements and may appear on conductors in which there are no incident
voltages. Exceptions are:
(a) all conductors short-circuited when K= — 1, so that
The polyphase surge admittance the reflected voltage on each conductor is equal and
y(0) = (71) opposite to the incident value, the total voltage being zero
(jb) all conductors open-circuited when K= 1, so that the
is also defined. This expression is particularly useful when reflected voltage is equal to the incident voltage on each
analysing surge impedances under a variety of boundary conductor.
conditions since it always yields a physically realizable net- The 4-pole matrix equations 73 and 74 for the multi-
work. Also, at high frequencies, owing to the dominance of conductor system may be useful in solving certain problems.
reactive elements in the Z and F matrices, the surge-impedance However, in many problems it may be more straightforward
matrix tends to become almost purely resistive; consequently to resolve phase quantities into components from a know-
resistive equivalent circuits may be used. Once having ledge of the S, Q and y matrices since each component has
obtained y ( 0 ) by calculation, the effect of differing termina- a definite propagation coefficient, whereas in eqns. 73 and 74
tions and through-connected lines may be evaluated by propagation is influenced by the relative strengths of com-
making use of a network analyser. The equivalent circuit ponents of each type, which, in turn, are functions of the
having been set up, a large number of conditions may be boundary conditions. No general conclusions may be drawn
investigated very rapidly. This technique is useful both in about the approach to be used—each problem has to be
evaluating surge impedances in relation to circuit-breaker examined before deciding which method is most convenient.
performance and certain aspects of power-line carrier trans-
mission, namely matching the effects due to line traps. These
topics will be considered in papers to follow. 6 Conductor systems with planes of
symmetry
5.2 Polyphase reflection factor It was shown in Section 4 that the solution to the
It is also shown in Appendix 11.4 that the polyphase general wave equation involved in the first instance the
reflection factor is solution of a polynomial equation of degree equal to the
number of conductors in the transmission system. Thus, in a
(72) single-circuit 3-phase transmission system, a cubic equation
This is identical with eqn. 21 for the single-conductor case, must be solved, while for a twin-circuit 3-phase system a
since under these circumstances the matrices in eqn. 72 sixth-degree equation arises. The problem is further com-
become elements. plicated by the fact that in the general case the coefficients of
the polynomial equation are complex. It is possible to solve
5.3 Polyphase propagation coefficient a cubic polynomial by trigonometrical methods, but the task
of solving a sixth-degree polynomial with complex coefficients
It is shown in Appendix 11.5 that the voltages and is almost impossible without the use of a digital computer.
currents when x = r (right-hand side) may be expressed in
Fortunately it is possible in most practical cases of twin-
terms of the values when x = 0 (left-hand side) as follows:
circuit systems to factorize the polynomial into the product
of two cubic equations. The method of approach is quite
general and is applicable to all systems which are partially
(73)
symmetrical, irrespective of the number of conductors.
= - Q sinh The technique is to factorize det (P — y2) into two lower-
^ (74) rank determinants, and since det (P — y2) = 0, each of the
lower-rank determinants must separately be zero. The method
As before, the equations are valid for the single-conductor is detailed in Appendix 11.5, where it is shown that, if the
case when the matrices become elements and are therefore number of conductors is even, the determinental equation
PROC. IEE, Vol. 110, No. 12, DECEMBER 1963 2205
 det (P — y2) = 0 may be reduced to the solution of two The determinental equation for y2 is of the form
lower-rank equations detCP^ — y2) = 0 and det(PB —y2) = 0.
In the reduced equations the rank of determinants PA and
PB is half that of P, so that, for example, in the case of a 0 = det (P - y2) = det
twin-circuit system, two cubic polynomials have to be solved
instead of one of sixth degree. In the case of an odd number
of conductors, it is shown that the reduced determinants have
rank (n — l)/2 and (n + l)/2, respectively, where n is the + Pn - y2) det ( P n - P 1 3 - y 2 )
= det 22
number of conductors in the system. - yy)
It may also be seen from Appendix 11.5 that the factoriza-
tion assumes a particular significance when considering the factorized by the method detailed in Section 6 for a system
distribution of voltages and currents for a given mode of with a plane of symmetry and an odd number of conductors.
propagation, i.e. the values of the elements of one column The three propagation coefficients are
of the S and Q matrices corresponding to a particular value
of y. It is shown that all voltages and currents corresponding = HPn + ^.22 + ^12 + V[(Pn ~ P22 + Pn
to the first factor occur in equal and opposite pairs in corre-
sponding image conductors, while the voltages and currents
corresponding to the second factor occur in equal and oppo- y 2 = ±{P n + P22 + P12 - - ^22 + P12)2
site pairs in corresponding image conductors. If the number + 8P 13 P 3I ]}
of conductors is odd, the polynomial of higher degree always
yields components of the former type. Also, in the case of an
odd number, it is evident that there is no current in the centre It may be noted that y\ has the simple form y\y\
conductor and the voltage to earth of this conductor is zero [Zff - Z$] [yff + Y$] after substituting for P n and P, 2
for components of the latter type. in terms of" Z (p) and Y(p\ In solving for the S and Q matrices,
It may be noted that, even when digital computers are the first element of each column will be arbitrarily specified
used for solving problems, a significant saving in time takes as being unity. It will be found that these matrices then
place when factorizing the characteristic polynomial in the have the simple form
manner detailed above. Furthermore, it is readily seen that
the solution of the equations of a twin-circuit system with a 1 1 1 1 1 1
plane of symmetry reduces to the solution of two single- ri
1 1 - 1 ,Q = 1 1 -1
circuit systems. This is valuable in the preparation of digital- 5,,
computer programmes, which need then only cater for the
S3, 0 G31 G32 0
case of a single-circuit system with instructions to carry out (These forms are determined by inspection according to the
reductions when dealing with twin-circuit systems. rules developed in Section 6.)
Only the centre element of the first two columns of each
matrix is unknown. For the S matrix, these may be
determined from the first set of three dependent simultaneous
equations, i.e.
7 Particular solutions for a 3-phase single-
circuit line with a plane of symmetry (P n - P, 2 S 2f + P12S3r = 0, or
7.1 Horizontal configuration ,. = S3r =
In order to complete the paper, the solution to the Similarly it may be shown that
partially symmetrical 3-phase system will be developed as an
illustration of the matrix method. In such a system, con-
ductors 1 and 3 are at an equal height above ground and
conductor 2 lies on a line perpendicular to the earth plane The impedances of each phase to each component may be
midway between 1 and 3. determined from eqn. 122. Thus, for example,
For the system, the impedance and admittance matrices
are of the general form ft = ( z n + z 12 + z,3<23i)/yi

A
7(P)
n nr
Zj?
In practice, it will be found that Sn and Qn are very nearly
unity and S32 and Q32 are very nearly equal to —2, so that
(These forms are valid if earth wires are included, provided components of types 1 and 2 are similar to Clarke zero and
that the earth wires are symmetrical with respect to conductor 2 a components, while type 3 components are in exact corre-
and the earth plane.) spondence with Clarke jS components.
The P matrix will have the general form
7.2 Hypothetical case of symmetrical configuration
Pn Pn When a fully symmetrical 3-phase system is considered,
p = P22 Pn the following additional simplifications occur:
31 22

where P,y = £ ZikYkj

2206 PROC. IEE, Vol. 110, No. 12, DECEMBER 1963
 where id) Work with preliminary programmes has yielded satis-
factory solutions, and this work will be developed and
Pn= . expanded.
(e) With the satisfactory development of programmes, many
interesting application problems can be examined without
The following values will be found for y2: complexity being a restriction.
y\ = (/) The solutions may be used to show the validity of
symmetrical-component theory in ideal configurations
of symmetry.

These are well known forms in symmetrical-component 9 Acknowledgments

analysis.
In solving for the elements of the columns of the S and The author wishes to thank Messrs. A. Reyrolle and
Q matrices, it will be found that all elements of column 1 of Co. for permission to publish the paper.
both matrices are unity, so that this component is a true Thanks are expressed to Mr. F. L. Hamilton (Engineer-in-
zero-sequence wave. However, when considering the remain- Charge Research) and Mr. J. B. Patrickson (Deputy
ing elements in both matrices, it will be found that the three Engineer-in-Charge Research) for their helpful discussion
and advice.
simultaneous equations corresponding to each column are all
identical and of the general form Slr + S2r + S2r = 0, and
similarly for the elements of a column of the Q matrix. This 10 References
is the only restriction on their value, so that any number 1 FALLOU, J. A.: 'Propagation des courants de haute frequence poly-
of component systems exist. Typical systems which satisfy phases le long des lignes aeriennes de transport d'energie affectees
de courts-circuits ou de defauts d'isolement', Bull. Soc. Franc. Elect.,
these requirements are positive- and negative-sequence com- 1932, Series 5, 2, p. 787
ponents and Clarke a and jS components. 2 DE QUERVAIN, A.: 'Carrier links for electricity supply and distri-
In evaluating the impedance of each phase to components bution', Brown Boveri Rev., 1948, 35, p. 116
3 GOLDSTEIN, A.: 'Propagation characteristics of power line carrier
of each type, the following well known results will be links', ibid., p. 266
obtained: 4 CHEVALLIER, A.: 'Propagation of high-frequency waves along an
electrically long symmetric three-phase line', Rev. gen. Elect., 1945,
54, p. 25
5 ADAMS, G. E. : 'Wave propagation along unbalanced h.v. trans-
mission lines', Trans Amer. Inst. Elect. Engrs, 1959, 78, Part HI,
p. 639
+ /• = 2 or 3
This illustrative Section has shown the way in which the 11 Appendixes
matrix method is used to solve a typical problem and has
further shown that well known symmetrical-component 11.1 Classical solution for 2-conductor case
systems are a particular case of the general solution. Rewriting eqns. 26 and 27 in operational form,

7.3 Significance in relation to symmetrical-component (D2 - PU)V<P) - Pl2V2» = 0 (75)

theory p) 2
- P 2 1 V[ + (D - P22)F<"> = 0 (76)
It has been shown in the previous Section that sym- 2 2 2
metrical components in a symmetrical system are in con- where D = d ldx . Therefore
formity with the general theory. A problem is then to assess [{D2 - Pn)(D2 - P22) - Pl2P2l]Vl» = 0
those circumstances in which it is permissible to use sym-
metrical components as an approximation to the actual or [D* - (Pn + P22)D2 + (PnP22 - Pl2P2l)]Vl» = 0
solution.
In general, it will be necessary to use the exact approach or (D2 - y2)(D2 - y
2
)V{p) = 0 (77)
when it is required to assess the difference in surge impedance where
between various conductors, mutual effects, and velocity and
attenuation factor for various modes of propagation of high- . . . (78)
frequency signals on power lines. In most cases it will be
and y\ = 2
obvious when the symmetrical-component approach gives P2Z) - V[(P,, - P22) ~ 4/>12P2l]}
sufficiently accurate results. These are problems in application • • • (79)
and fall outside the scope of the paper.
Hence
V\p) = Sn[V{c+) exp - yxx + F<c-> exp ylX]
8 Conclusions
(a) A completely generalized method of solving travelling- + SX2[V^ exp - y2x + K<c-> exp y2x] . (80)
wave phenomena on polyphase lines has been developed V[c+\ V[c~\ V?+\ V!f-\ Su and Sl2 are arbitrary
by the use of modern methods of matrix algebra. constants.
(6) Without the further simplification possible for particular
cases, the generalized solutions require digital-computer V? = (D2 - Pn)V[p)lPl2
facilities for evaluation.
= (y2 - Pu)Sn[V<°+> exp - Ylx + K,<c~> exp y,jc]/P12
(c) The equations are particularly suitable for such digital
computation, as the determination of the characteristic + hi ~ pn)Si2[Vic+) exp - y2x + V£~> exp y2x]lPn
and vectors of the matrices form part of many standard = S1X[V[ ^ exp - yxx + K,(c-> exp Ylx] +
C
codes. This means that the solution of polynomial
equations is an automatic process. S22[F2;c+> exp - y2x + K2(c-> exp y2x] . . , (81)
PROC. IEE, Vol. 110, No. 12, DECEMBER 1963 2207
where S2l = Sn(y2 - Pn)/P12 and hence
= Sn{(Pn - P22) - u - P22)1 V[c) = ,4 e x p - y x x + B e x p y ^ x . . . . (93)
12 • (82) V^ = D exp - y 2 * + E exp y2x . . . . (94)
2
and (y - Pu)Sl2lPl2 = 522 where A and B are arbitrary constants and the problem has
or Sl2 = 2
S22P12l(y -Pu) been reduced to the solution of two elementary wave
equations.
Rewriting eqn. 88, PS = Sy2 or PS - Sy1 = 0, and
expanding:
4P 1 2 P 2 1 ]}5 2 2
and 0 = [(Ai - y\)sn + P12S21 (Pn - rl)sl2 + ^125221
Sl2 _ 2Pl2{P22 - Pn
2
- P 2 2 ) + 4P 12 /> 21 ]} lP2lSn + (P22 - y2)S2l P2lSl2+(P22-y2)S22j^
~ P22)2 ~ For this matrix identity to be true it is necessary for each
element of the expanded matrix eqn. 95 to be zero.
= +{Pn~ P22 ~ ~ P22)2 + 2i (83) From this it may be seen that each column of eqn. 95
From eqns. 22 and 23, defines a set of homogeneous equations in two of the four
unknowns of the S matrix. For example, taking the left-hand
-dV^/dx = Z, ,/<"> + Z12/2"> column,
(Pn - y2)Sn + P12S2l = 0 (96)
Hence ^21^.1 + ( ^ 2 2 - 7 ^ 2 1 = 0 (97)
p)
/|« = [ - Z!gdV[ ldx + Z^ It is well known that, if the determinant of the coefficients
-Z'f] (84) of the unknowns is non-zero, the solution is Sn = S2l = 0 ,
which is trivial. Conversely, if the solution is to have a
If = [_ Zff meaning, the determinant of the coefficients must be zero, i.e.
(85) det(P-y2) (98)
= 0
From eqns. 80 and 81,
Similar reasoning from a consideration of 5"12 and 5"22 would
dV[p)/dx = - yiSn[V{
c+)
exp - yxx - F,(c->exp yxx] yield
- y2^i2[^ 2 (c+) e x P - 72* - y(2~' exp y2x] det(P-y2) = 0 (99)

dV(p)Jdx = - yiS2i[V[c+)exp - yxx - V[e^exp yxx] In fact, both y2 and y\ are contained in the equation
det (P - y2) = 0 (100)
- Y2S22[Vic+) ex
P - 7ix ~ ' / 2 c - ) exp y2x]
2
and substituting in eqns. 84 and 85, since this is a quadratic in y .
Substituting for the elements of P in eqn. 100 yields

(Pn ~ y2)(P22 - 72) - PnP2i = 0

. (86)
or y\ = ±[/>n + P 22 + V ( ^ n - J°22)2 + 4 ^ 2 1 ]
A = H(pn + P22) " V(Pn ~ P22)2 + 4^21]
Z(p) (87)
It is important to note that this result is not confined to
where det Z<p) = - Z\p2)2 the 2-phase case. Similar reasoning will show that eqn. 100
is valid for any number of phases. The P matrix is modified
yjc> = ylc+) e x p _ yiX _ yu:-) exp y ^x by subtracting y2 from each element along the principal
diagonal; the determinant of the modified matrix is equated
yg) = ylc+) e x p _ n x _ yu:-i exp ^ to zero, yielding an nth degree polynomial in y 2 . The result is
well known in mathematics, the various values of y 2 being
known as eigenvalues of the matrix P.
11.2 Transforming matrices to diagonal form The solution for the elements of S is now straightforward.
From eqn. 44 Owing to the vanishing of the coefficients of S in, for example,
eqns. 96 and 97, the values of Sn and iS21 are dependent
(88) and it is necessary to specify one element in order to obtain
the second. To conform with the analysis carried out in
If S is so chosen that S~lPS is diagonal, i.e. S~lPS = y2, Appendix 11.1, Sn is specified, so that from eqn. 96,
then d2V^\dx2 = y\ K(c) (89)
c)
y? 0 V\ Similarly,
i.e. (d2/dx2) , . . . . (90)
0 y2 V?
These values are identical with those obtained by classical
This permits a direct solution, since methods.
d2V[c)ldx2 = y\V[c) (91) Again it is important to note that the results are valid for
d2V^\dx2 = yjV2'c) (92) any number of phases. Once having obtained the n values
2208 PROC. IEE, Vol. 110, No. 12, DECEMBER 1963
of y 2 from a solution of the nth degree polynomial previously Returning to eqn. 102, a matrix product of the following
discussed, the elements of the S matrix are solved column type may be formed:
by column using one value of y 2 at a time and noting that,
because the system of simultaneous equations is dependent, (107)
it is necessary to arbitrarily specify one element in each and by considering y , 2
column.
Once having determined the values of elements of the
£ matrix, the phase voltages may be determined from eqn. 43, Transposing,
i.e.
y(P) _ gyle) Q(n),PSm = Q{n)S{m)yl (108)
v\» = s n si2 A exp — yxx + B exp yxx Subtracting eqn. 108 from eqn. 107,
vp s2X s22 D exp — y2x + E exp y2x 0=Q ( / ,),S ( m ) (y,2-y 2 ) (109)
and expanding,
In general, ym ^ yn, and therefore
y[p) = ^ n ( ^ exp - yxx + £exp
(110)
Sl2(D exp — y2x + £ exp y2x)
V(2P) = S2X{A exp — y{x + 5 exp y ^ ) + The special case of ym = yn, i.e. of two or more propagation
coefficients being equal, is unlikely to occur in practice, but
S22{D exp — y 2 x + E exp y 2 x does occur in theory when simplifying assumptions are made,
With the chosen values of S, this solution is identical with e.g. in 3-phase systems assumed to be completely symmetrical.
that obtained by classical methods (eqns. 34 and 35). This special case is dealt with in Section 7.
All possible products given by eqn. 110 are contained in
11.3 Properties of the wave equation
the matrix product QtS, and it may be concluded that this
product is diagonal in form, i.e.
Rearranging eqns. 88 and 89, PS = Sy2. Considering
the elements in column m, Q,S=D (Ill)
where D is some diagonal matrix.
(101) Complex conjugate products may be formed corresponding
to eqn. 107, namely
This may be written
Q(n),PS(m) = QMlS(m)yl (112)
(102)
Here S^ is column m of the S matrix. Multiplying eqn. 102
by S^t, which is a row matrix formed by transposing column Taking the complex transpose of both sides
n of the S matrix,
Q(n)tPS(m) = Qw,S(m)y2n (113)
S(n)lPS(m) = S(n)S(m)y%, (103)
Since ym ^ yn in the general case, it follows that
Similarly, by considering y 2 in place of y2, in eqn. 102, the Q(n)iS(m) ^ 0 and that the Q and S matrices do not form a
following equation may be derived: unitary set.
From dV^dx= - S~lZ^QI^ = - DZI& . . (114)
Transposing, dl^dx = - Q- Y(P>SV^ X
= - DyV^ . . (115)
(104) where D2 = - l
S~ Z^Q
Taking the difference between eqns. 103 and 104, Dy= -Q~lY^S
(105) Now d2V^/dx2 = D2DyV^ (116)
Since P ^ P, in the general case, the left-hand side, and 2 2
consequently the right-hand side, of eqn. 105 is not zero. and d l^ldx = DyDzlM (117)
This shows that the S matrix does not have the mathe- From eqn. 88,
matical property of orthogonality, i.e. the product of two
columns taken element by element is not zero. All possible DzDy = DyDz = y2 (118)
products of this type are contained in the matrix product
For Dz and Dy to form a product which is commutative
and furthermore which is diagonal, y 2 , it is necessary that
(106) they should in themselves be diagonal.
From eqn. 88, if A is any diagonal matrix,

and it may therefore be concluded that in the general case S-IPS = y2 = A A - y = Ay 2 A" ! . . . (119)
eqn. 106 is not a diagonal matrix. Similar reasoning will show
that, in general, the Q matrix is not orthogonal and further- since diagonal matrices are commutative.
more that neither the S nor Q matrices are unitary, the term Premultiplying eqn. 119 by A" 1 and post-multiplying by A,
'unitary' meaning that the product of one column of the S A - ^ - ' P S A = y2
(or Q) matrix with the complex conjugate of another column
taken element by element is zero. i.e. (SA^PGSA) = y 2
PROC. IEE, Vol. 110, No. 12, DECEMBER, 1963 2209
and OSO-'W) (120) Substituting in eqn. 114,
where S ' = SA (121)
This proves that if S is a solution to eqn. 88, so is S', i.e.
where S and S' are related by eqn. 121.
The surge impedance of each phase to a component of a (125)
certain type may be evaluated as follows, remembering that where V^ = K (c+) c
exp - yx - V^ ~^ exp yx
the current of a certain type can only be produced by a
voltage of the same type as is given by eqns. 114 and 115. Z<c> =y-lS~lZWQ (126)
The voltage and current in phase r and due to a com-
ponent of type s (having propagation coefficient ys) are Since y~l is diagonal, and from eqn. 114 Dz is diagonal, it
K(s) = SrsV<c) and Ir(s) = Qr{s)I$c\ From eqns. 38, 52 follows that Z (c) is diagonal.
and 53,
11.4 Surge impedance, reflection factor and propagation
coefficient
Consider a set of phase voltages travelling in the positive
i.e. (d/dx)SrsvP = - £ Z%QW sense (left to right) in a multiconductor system and encoun-
m= 11
tering a set of terminal impedances, Zn at the right-hand
Substituting V$ = e)
V{sc) exp -ypc, terminal when x = r.
From eqns. 52, 53, 60 and 125 we can write
/to = j(c) e x p _ ysX

Then

and
= e[exp ( - yr)/(c+> + exp
= I i.Z^Qms)lQrsys) I(rc) = exp ( - yr)I^c+) - exp
l
Also, the phase voltages at the terminal must be related to
the phase currents by
= S (122)
Z,.I^ (127)
The surge impedance in phase r and due to component of p)
Substituting for V[ and /r , (p)
type s is a function of the system impedances Zrm which
are constant, the propagation coefficient ys which is constant
and the ratio of elements in column s of matrix Q. Although
these elements are arbitrary, their ratio is not, since n — 1 /(c) = zrQI(rc)
of these elements were evaluated in terms of the nth, and Q exp (yr)/<c-> = - KQ exp ( - y/-)/ (c+) . . (128)
since they arise from the solution of a system of linear
l
simultaneous equations they must all be proportional to this Where K= [Zr + SZ^Q~^[Zr - SZ^Q~ ] . (129)
one element and hence their ratio is determined only by the
system parameters and the propagation coefficient. It may be seen that eqn. 128 defines the matrix of reflected
Consideration of eqn. 39 would show in a similar fashion phase currents in terms of the incident phase currents,
that Polyphase surge impedance is defined in terms of the set of
terminal impedances which gives reflection-free conditions,
(123) i.e. K = 0 in eqn. 129. Therefore
Z<°> - -' == 0
so that
or (130)
c)
From eqn. 126, Z< = y - , and therefore
(124)
y (131)
(0)
The proof that Z is symmetrical and hence bilateral is as
It may be noted that, when m = r, QmJQrs = SlllsISrs = 1, follows:
Z(p) = Zip) and Y(p) — Yip)
This shows that the surge impedance in the rth phase is = Qj lZMS, since Z<c) diagonal
always a function of the self-impedance and admittance of = SD~{Z^DQ~X because Q,S = D from eqn. 111
that phase, together with the mutual impedances whose
values are modified in accordance with eqn. 124.
The matrix Z<c) connecting component currents and Considering voltages instead of currents, it may be seen
voltages is defined in the following manner. Let that
y{e) = y(c+) e x p _ yx + y(c-) e x p yx
exp (yr)K(c-> = Kexp ( - yr)V^c+\
y(c) = /(c+) e x p - y x - y(c-) e x p yx where K= {Q[Z^]~lS-1 + Yr}^{Q[Z^]-'S-1 - Yr}
2210 PROC. IEE, Vol. 110, No. 12, DECEMBER 1963
and again it may be seen that, for reflection-free termination, 1 - 1 where 1 is the ///2 x «/2 unit matrix
the terminal admittance is Let K =
1 1
Y° = 1 0 0
032) 0 1 0
Note that Yr=(Zr) ~K 0 0 1
The polyphase wave equation may be defined in terms of n being the number of conductors.
the terminal conditions Vo, / 0 , Vr and lr in the following
manner:
Vo =
Substituting for P from eqn. 135,
= SZ<c>[/<c+> - lie-)]
0
0 Pa-Pb-y2\
Therefore
= det {Pa +Pb- y 2 ).det (P o - Pb - y2) = 0
i.e. det {Pa + Pb - y2) = 0 (136)
2
= S [exp ( - exp or det (Pa - Pb - y ) = 0 (137)
y - exp Eqns. 66 and 67 for solving for the P and Q matrices may
lr=Q [exp ( - yr)I^ + exp (yr)fc^] be simplified in the following way:
Substituting for / ( c + ) and /<*->,
Vr = SZ(c>{cosh (yr)[Z^]~lS~l VQ - sinh (yr)Q~ %} K-{PKK~lS0) = K-
/,. = ^{coshCyAOQ-^o - sinhCyOtZ^-'S- 1 ^}
Vr = SZ® cosh (yr)[ZV]-lS-lV0 (138)
-SZ& sinh (.yr)Q~lI0 2
where K is the same matrix used to simplify det (P — y )
= Scosh(yr)S-lV0-ZWQsinh(yr)Q-lI0 . (133)
. . . (139)
Ir = - Q sinh (yr)[ZW]-lS-lV0 + Q cosh (yr)Q-lI0
= -Q sinh (yr)G-1[Z<°>]-1Ko + Q cosh (yr)Q~ll0 Pb -y2 = 0 . (140)
i.e. S
. . . . (134) o lli)b
It should be noted that in these expressions, exp yr, (141)
exp -= yr, cosh yr and sinh yr are diagonal matrices.
where S'^ has been partitioned for comformable matrix
11.5 Systems with planes of symmetry multiplication in eqn. 140.
A partially symmetrical system is defined as one in It is evident from eqn. 140 that, if det (Pa + Pb - y?) = 0
which the conductors are arranged in pairs at the same and det (Pa — Pb — y?) =£ 0, S'0)b = 0 and S[i)c is obtained
height above ground and at equal horizontal distances on by solving the set of dependent simultaneous equations
either side of a reference plane perpendicular to the ground (142)
(Pa + Pb - yj)S'0)a = 0
plane. In a system with an odd number of conductors, the
odd conductor lies on the reference plane. Most double- Similarly, if the converse applies, 5(,)0 = 0 and S[j)b is obtained
circuit transmission systems are partially symmetrical, as are by solving the simultaneous equation det {Pa — Pb — y2) = 0
single-circuit horizontal systems.
Considering first the system with an even number of con- 0
Thus 5' = •• (143)
ductors, the Z(p\ y ( P ) and hence P matrices have the 0
following general form:
and S = KS'
zb Ya Yb
y(P) = 1 -1 sa o
Za 1 1 o 5;
Pa Pb
P = (135) (144)
Pb Pn s:
In solving the polyphase wave equation it is necessary to It is evident from the foregoing treatment that the problem
evaluate the n, propagation coefficients by using eqn. 62, i.e. of solving the ^-conductor partially symmetrical system is
det (P - y2) = 0. reduced to the solution of two ^/i-conductor unsymmetrical
It is well known that the determinant of a matrix product systems. All operations and manipulations may be carried
is equal to the product of the determinants of the matrices out independently on the reduced systems as if the other
taken in any order, i.e. did not exist, and only when the composite parameters are
det (P - y2) .= det %~ l(P - y2)K, required are the independent values combined by means of
matrix K.
where K is any regular matrix. The significance of this result is seen from the structure of
PROC. IEE, Vol. 110, No. 12, DECEMBER 1963 2211
the S matrix (eqn. 144). For the first n/2 values of y2, propa- As before, the elements of the S matrix as indicated in
gation takes place with voltages (and currents) equal and in eqn. 138, i.e. K~\P - yj) = 0
phase in each symmetrical pair of conductors, and equal and
in antiphase for the remaining n/2 values of y2. (Pa -yf) o 0
*U)b = 0
2
In a partially symmetrical system with an odd number of or (Pa + Pb- y)
conductors, the matrix P will take the general form (Pe-y2 S
U)c
where S'^ has been portioned for comformable multiplication.
It is evident from previous argument that S[^b = S[^c —
0 if det (/*,-/»* - y ? ) = 0
2
where Pa and Pb are square matrices of rank (n — l)/2, and S'0)a = 0 if det {Pa +Pb~ y ) = 0
Pc is a column matrix of (n — l)/2 elements, Pd is a row (Pe - y2)
matrix of (n — l)/2 elements and Pe is a single element. In so that
order to simplify the determinental equation det (P — y2), a Sa 0
matrix K is introduced as before. In this case S'= 0 Sb
1 1 0 1 -- 1 0 0 S'c
K = -1 1 0 1 1 0 where there are (n - l)/2 Sa and (n + l)/2 Sb and S'c.
0 0 1 0 0 2 Now S = KS'
In K, the first four unit matrices are of rank (n — l)/2 and 1 1 0 Sa 0
the final unit matrix is the unit element. K and K~l are 1 1 0 0 si
therefore arranged for conformable multiplication with P:
0 0 1 0 S'c
det (P - y2) = det K~\P - y2)K
Sa
(Pa -Pb-y2) 0 0 (147)
= det (Pa +Pb- y2) Pc o s:
2Pd (Pe ~ Y2) Two general forms of propagation exist: in the former
2 P. type, signals are equal and in antiphase on paired conductors
= det (P a -P 6 -y ).det 2 = 0 with no propagation in the odd conductor. This is evident
2P, (Pe-y ) from the fact that the odd conductor lying on the plane of
2 symmetry is on an equipotential for magnetic and electric
i.e. det ( P a - P 6 - y ) = (145)
fields and could be removed without affecting the paired
(Pa
-i_ \ -
(p +P
b y2) Pc conductors. In the latter type, signals are equal and in phase
or det =0 (146) in paired conductors and the centre conductor is included.
2Pd >.-y2)
An example of odd symmetry is given in Section 7, where it
In this case the solution of an nth order polynomial has is shown that, in the horizontal single-circuit line, propagation
been reduced to the solution of two polynomials of order between outer phases is independent of the presence of the
(n - l)/2 and (n + l)/2, respectively. centre conductor.

2212 PROC. IEE, Vol. 110, No. 12, DECEMBER 1963