1964, High-Frequency Propagation On Nontransposed Power Line PDF
1964, High-Frequency Propagation On Nontransposed Power Line PDF
1964, High-Frequency Propagation On Nontransposed Power Line PDF
impedance is difficult to answer. Effects of the ground return made, the zero-sequence parameters recalculated, and the
path are such that zero-sequence resistance and inductance both problem solved again. Some such trial-and-error approach
contain frequency terms, as a number of texts show. There- may thus lead to more exact results. Our current method is
fore, during the transient solution, when frequencies other than to use 60-cycle values, as supplied by transmission-line design
the system base frequency exist, the zero-sequence impedance engineers, and to accept the results as being satisfactory for the
should properly be modified to account for these effects. A present.
closed-form analytical solution for these effects would be very In regard to Dr. Robert's third comment, the authors are
complex. Even during the process of a numerical solution on a aware that the representation of the ground mode is an approxi-
digital computer, this is extremely difficult, since one does not mation because of the method employed in calculating the
know the transient frequencies and thus cannot make the proper characteristic impedances and the propagation constants. It
corrections. is felt that the usage of a single velocity of propagation for the
A possible approach is to use average values for the zero- ground mode is sufficiently accurate for most practical situations
sequence terms, say at 60 cycles as is done on the ANACOM, as mentioned earlier in discussing the approximations used in
then record the transient line response. A study of the resulting the papers.
High-Frequency Propagation on
Nontransposed Power Line
Michihiro Ushirozawa
Summary: The high-frequency characteristics on nontrans- Propagation Theory of Parallel Multiconductor Lines
posed power lines have been calculated by the use of the principle The transmission equation of parallel multiconductor lines,
of natural mode propagation, and it has been shown that some
comparison between calculated and measured values on the asshowniFig.1,canbegiveninthefollowigequations.
vertically arranged 3-phase 2-circuit power line is possible.
-- [V] = [ZJ [i]
Recently there has been a trend toward adoption of a non- C)
transposed power line for EHV (extra-high-voltage) trans- - - [i] = [Y] [v] ()
mission. When installing power line carrier assemblies on x
such a line, it is necessary to pay due attention to the high- [v]= ((vv,2,. .,Vn))
frequency transmission circuit, because in a comparatively [il =((ii,i2. ,in)) (2)
long power line, the electrical unbalance between each phase
of the line and the ground is so emphasized that the high- where (( )) indicates a single column matrix, and v1, v2,
frequency characteristics differ greatly depending upon the ., and i1, i2, ., i,, represent, respectively, line voltages
selection of the coupling phase and the coupling method. and line currents at an optional point x of the parallel n-
In calculating the high-frequency characteristics of a power conductor power line.
line, a formula1 has frequently been used where the power The [Z] and [Y] in equation 1 are the impedance and ad-
line is considered as a symmetrical line. However, a satis- mittance matrices of the line (square matrix of the order n).
factory reply cannot be expected with regard to the char- As is widely known, the propagation constants of the multi-
acteristics of the nontransposed line. conductor line can be sought as eigenroots of the product of
By approximating such a line to an overhead parallel multi- [Z] and [Y].
conductor line, the author has derived general formulas of VIZ[Y] - [UI=l[Y] [Z] -/i,2[U]1=0 (3)
computation and prepared a number of attenuation charts'
classified by the coupling methods which are important in with -y propagation constants and [U] the unit matrix.
designing power line carrier assemblies.
From these charts, line attenuation can be estimated im-
mediately if some conditions are given, such as line length,
coupling method, and frequency to be used. In a series of __x__
field tests conducted on numerous nontransposed power lines,
the author compared the calculated values with the actually
measured values. _______________
The purpose of this paper is to present the high-frequency 2 2_
characteristics of a vertically arranged 3-phase 2-circuit power __ __ __ -
line and of a horizontally arranged 3-phase 1-circuit line. _________________________
Paper 64-49, recommended by the IEEE Transmission and Dis- l | -
tribution Committee and approved by the IEEE Technical Opera- V1 v2 ~ ~
tions Committee for presenltation at the IEEE WXinter Power Meet-2 n n
ing, New York. N. Y., February 2-7, 1964. Manuscript submittedt
November 4, 1963; made available for printing March 30, 1964. 777777T/7// j// /// )~///'/
// ,,/' // /////
MICHIHIO USHIROZAWA is with the Central Research Institute of
Electric Power Industry, Kitatama-Gun, Tokyo, Japan. Fig. 1. Parallel multiconductor line
where
[L] = inductance matrix By the Carson-Pollaczek method, if the ground conductivity
[C] = capacitance matrix a= 10-3 ihos/meter, the value Rg in the area from the power
[Ro] = conductor resistance matrix line carrier wave band to medium wave band and under
[Rel = resistance matrix due to skin effect of the ground the power line arrangement as shown in Fig. 2, increases with
co vrelocity
= angular
frequency as f2"3, as a result of skin effect of the ground.
In order to simplify the calculation, [R] is taken as approxi- Conductor resistance Ro increases with frequency as fl"2.
mately proportional to fX (x = constant, f= frequency). But it is smaller in value by one to two figures than Re.
The aforementioned formulas will now be applied to the Therefore even if Rois assumed to be proportional to f2'3 to
vertically arranged 3-phase 2-circuit power line which is relate it to equation 15, it is possible that no serious error
generally seen in Japan. The EHV Tokyo Eastern line will arise.
has the following characteristics: Conductor-2X330 mm2 Fig. 3 shows the results of attenuation constant and char-
(square millimeters) ACSR (aluminum cable steel-reinforced); acteristic impedance calculated in such a manner.
overhead ground wire-two ACSR wires of 120 mm2; line The element A7,j of voltage conversion matrix [A] can
voltage-275 kv; line length-116 km (kilometers). Fig. 2 be obtained from equation 9. In this case, the first-row
shows a standard line arrangement. elements AI, in matrix [A] can be put as 1. From its reverse
If one phase is regarded as a single-line equivalent, the total matrix, [A]-' can be obtained, and [B] and [BI-' can be
number of lines will be eight, including ground wires. This determined by using equation 11. Tables I and II show
should be solved precisely. For the sake of simplicity, how- the inverse conversion matrices of voltages and currents.
ever, calculation will be made on the six lines neglecting the From this table, it can be understood that the six natural
ground wires. (There is another method in which ground modes in the 3-phase 2-circuit power line consist of three
wires can be omitted by assuming that their electric potential modes where the currents of conductors placed at the same
is the same as the ground.) height from the ground surface flow in the same direction, and
three other modes where they flow in a reverse direction.
Recently, G. E. Adams5 studied the same concept. If it is
considered that generally a,/f<3 l (3i; phase constants) in
Table 1. Inverse Conversion Matrix of Voltages: [A]' the power line carrier frequency band, then characteristic
(Tokyo Eastern Line) impedance W, is given by an approximate equation 16, from
Line Voltage equations 10 and 14.
Mode
Voltage Vi V2 V3 V4 V6 V6 [Wi] =g[A] -1[L] [B] (16)
Vi' 0.2616 0.1595 0.1174 0.1174 0.1595 0.2616 Phase constants fi will be
v,' 0.1935 -0.0507 -0.1826 -0.1826 -0.0507 0.1935
V3' 0.0449 -0.1088 0.0652 0.0652 -0.1088 0.0449
V4' O.00i3 0.0099 -0.0277 0.0277 -0.0099 -0.0013 a 2
V6'
V6'
0.1857 -0 .2155 -0 .0476
0.3130 0.2056 1
0.0476
0.0752 -0.0752
0.2155 -0 .1857
-0 .2056 -0.3130
03 V1+A2 2~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(7
( 17)
where
3=cw/g (g = 3 X0l km/second) (18)
Table 11. Inverse Conversion Matrix of Currents: [Bf'1 Attenuation and Impedance of the Carrier Frequency
(Tokyo Eastern Line) Band
Line Current The high-frequency transmission circuit, which usually
curren .,2 I 4 1 takes one or two phases of the power line as its coupling
I~'0.8980
1 0.811 0.8101 0.8980
-0.5082
1
1
phase,
CU1tS.6
can be expressed by a combination of natural mode cir-
i,' 1 -0.5082 -1.5377 -1.5377
it 1 -3.0422 1.9046 1.9046 -3.0422 1 The equivalent circuit is shown in Fig. 4 for the case of
i4' 1 4.4836 -16.4081 16.4081 -4.4836 -1
i 1 -1.3617 -0.4398 0.4398 1.3617 -1 noncoupled phases grounded at the line terminals and in Fig.
i6' 1 0.7890 0.3296 -0.3296 -0.7890 -1 5 for the case of noncoupled phases open at the line terminals.
~~~~~~~~~Dislcussion
07 _
< E L M,2/S/
// Mt2 ~~~~~~~~~~M.
C. Perz (The Hydro-Electric Power Commission of Ontario,
Toronto, Ont., Canada): The author is to be commended for
05 _ / his analytical method dealing with propagation of electro-
g f kc. Qkm magnetic waves along real power transmission lines. His paper
0 -23 !51'103 2 3 slo3 is very valuable and important because it provides solutions of
Je practical problems and demonstrates a good agreement between
calculated and experimental results.
The problem of wave propagation on a system of parallel
conductors above a lossy ground appears in many important
practical applications covering a wide frequency range. At
Fig. 10 shows the calculated attenuation curv es. The power frequency and its harmonics the long-line-equivalent
order in terms of small attenuation is as follows: ill coupling circuit parameters can be calculated from propagation
of center phase to outer phase, E coupling of center phase, and equations. Switching surges, power line carrier, and radio
Eoulngofote pae.Inte as f utrphset interference produced by corona on EHV lines extend the fre-
quency range to about 2 megacycles. However, no simple
outer phase coupling, attenuation is smaller than the E method seems to exist which could be satisfactorily applied to
coupling when the product of frequency and line length is the variety of problems. Depending on the earth resistivity
small. As the product increases in value, however, its at- and frequency, simplified methods are being employed, methods
tenatin bcoes reaerthan that of E coupling, based on the same fundamental principles but characterized by
tenuationbecomesgreater ~~~~~different assumptions and resulting limitations.
A brief review
Conclusions of various analytical methods seems to be necessary for an
obJective discussion of the particular method proposed by Dr.
The author haUs derived an approximate method for cal- Ushirozawa.
culating high-frequency propagation characteristics on the The following is a solution for conductor voltages and current
nontanspsed
nontrnspoedpowe
wer lin ine andhasshow
nd hs shwn is reatioshipwith
itsreItionhip ith associated with a plane
satisfying equations 1 in wave propagating in the x direction and
the paper:
the measured values. As a result, it has become clear that, on
the nontransposed power line which has a comparatively long {v} = e-( Ill [YI)(iii) {vs} = e- IY]x{v1} (37)
length, high-frequency characteristics will differ greatly de- {j} = -([2 [z] (lIi){j0} e- V']r{j0} (38)
pending upon variations in the coupling method and coupling
phase, as a result of unbalance between phases of the line and where {vo } and { io} are single column matrices of voltages and
betwTeen each phase and the ground. curnsfrx=.Frhroe
In the power line carrier communications, therefore, it will {vus} = [Y] 1 2[Z] 112{{i} = [ZoI {io} (39)
be necessary for us to understand well the characteristics [Z1iasqremtxanitaybcoidedsthsug
and to make proper selection of coupling phases. impedance matrix of the system of conductors.
Equations 37-39 seem to be simple in form, but generally
References they cannot be used directly because all matrices appear as
1. TELE:COMMUNICATION TECHNICS FOR ELECTRIC POWER INDUSTRY arguments of functions. Also, the elements of the elementary
(book). Institute of Electrical Engineers, Tokyo, Japan, 1957, matrices [z] and [y] depend on frequency, and their values are
p. 218. difficult to compute even if simplified assumptions are adopted.
2. PROPAGATION CHARACTERISTICS OF POWER LINE CARRIER WAVE The following are the most commonly used assumptions:
ON POWER TRANSMISSION LINE, M. Ushirozawa. Report No. 1 ekg n ipaeetcretlse r elce
600231,61008, Central Research Institute of Electric Power Industry, (th seconkoate authr'
dlaequatiorrns 15). r ngece
Tokyo, Japan, 1961. (h eodo h uhrseutos1)
3. RADIO INTERFERENCE DUE TO THE POWER TRANSMISSION LINE. 2. Earth losses can be represented by a matrix whose elements
Committee Report, Institute of* Electrical Engineers, Japan, 1957, are self- and mutual resistances in series with the corresponding
p. 244. inductive reactance elements of line conductors (the first of
4. THE GENERAL SOLUTION OF THE MULTICONDUCTOR CIRCUITS the author's equations 15).
WITH NONRECIPROCAL LINE CONSTANTS UNDER THE STEADY STATES, 3. Internal flux of conductors is neglected.
H. Noda. Bulletin, Electro-technical Laboratories, Tokyo, Japan,
vol. 15, no. 8, 1951, p. 470. 4. The electromagnetic field associated with conductor currents
5. THE CALCULATION OF ATTENUATION CONSTANTS FOR RADIO NOISE and voltages is approximated by plane waves perpendicular to
ANALYSIS OF OVERHEAD LINES, L. 0. Barthold, G. E. Adams. AIEE conductors; therefore, there are no radiation losses.
Transactions, pt. III (Power Apparatubs and Systems), vol. 79, Dec. 5. Elements of capacitance matrix [C] are derived from Maxwell
190 P.951 potential coefficients, whereas elements of inductance matrix
6. POWER LINE CARRIER WAVE PROPAGATION ON POWER TRANS- [L] can be computed following classical methods developed by
MISSION LINE, R. Sato, M. Akiyama. JoNrnwal, Institute of Electrical Carson1 or Pollaczek.2
Engineers, Japan, vol. 79, 1959, p. 1558. A method of reduction of matrix functions in the solution of
7. HIGH-FREQUENCY PROPAGATION TEST ON EHV OSAKA LINE, various aspects of surge propagation on power lines has been
M. Ushirozawa, F. Nishiyama. Report No. 6100i3, Central Research developed by Professor S. Hayashi of Kyoto University. In
Institute of Electric Power Industry, 1961. his book,3 extensive and detailed applications of the Sylvester
8. PREPARATION OF SINUSOIDAL OSCILLATIONS ALONG A THREE expansion theorem and Laplace transformation to transient
WIRE WITH HORIZONTAL SPACING, M. V. Kostenko. Electrichestvo, phenomena are described. Various simplifications illustrated
MOSCOW, USSR, no. 8, 1959, P. 8. by experimental data lead to an insight into the effects of multi-
9. WAVE PROPAGATION ALONG THREE-PHASE POWER LINES AND velocity waves and attenuation. A negligibly small error is
TELEPHONE LINES WITH POWER INDUCTION, 0. A. Petterson. made in evaluating velocities of propagation if the losses are
Ericsson Technichs, Stockholm, Sweden, vol. 15, 1959. neglected. Conversely, in the derivation of voltage and current
1142 Ushirozawa-High-Frequency on Nontransposed Pouer Lines NOVEMBER 1964
components associated with constant attenuations, a single of matrix [R] [C], whereas the mode current components,
velocity of propagation may be assumed. These assumptions columns of matrix [B], are the eigenvectors of its transpose.
result in significant simplifications of computations. It is The two sets of eigenvectors are orthogonal; hence there is no
interesting that in the original method of Professor Hayashi exchange of energy between the modes. Matrix [C] is assumed
no direct use is being made of the concept of eigenvectors. to be independent of frequency and of earth resistivity and the
The simplest method of analyzing propagation of high- author proposes a further simplification by neglecting the earth
frequency signals is obtained if the effect of losses is initially effect on the inductance matrix [L]. Therefore, the mode
neglected in equation 39: Matrix [Z0] is symmetric and its components of a given line and the mode attenuation coefficients
elements are real. This leads to natural modes of propagation both depend only on losses in the phase conductors and earth.
characterized by mode impedances equal to the eigenvalues of Hence, in Dr. Ushirozawa's analysis the values of elements of
matrix [Zo]. Because of the symmetry of the surge impedance matrix [R] are of primary importance.
matrix there is only one set of auto-orthogonal eigenvectors Can the author explain in some detail the method of comput-
common to voltages and currents. Therefore, the actual phase ing the values of elements in matrix [ARe], equation 40? Also,
voltages and currents can both be resolved into uncoupled the variation of exponent "x" with frequency and earth re-
components. The fact that these independent components do sistivity (author's equation 15) is of great interest. The applica-
not interact allows a specific attenuation to be attributed to tion of Dr. Ushirozawa's method to actual problems requires
each natural mode. Provided that the depth of penetration of numerical computation of matrix [AR,] for various earth re-
earth is small with respect to physical dimensions of the line, sistivities and for a range of frequencies. Normally, this com-
this method is very useful. Its application to horizontal single- putation is more difficult and tedious than the calculation of
phase lines is particularly simple and leads to meaningful solu- modes and, therefore, additional information on the subject will
tions of practical problems confirmed by experiments.4-8 be much appreciated.
The introduction of earth losses in the elements of matrix [z], Does the author know of any simple method of computing
i.e., in inductance and resistance elements, greatly complicates the inductance correction matrix [ALe]? This matrix is needed
the analyses. The self- and mutual terms of inductance and for the evaluation of 8i or mode velocities. It seems that 6i
resistance both depend on frequency, earth resistivity, and line calculated from the author's equation 17 has little practical
geometry. The computation of these elements is very tedious meaning because the effect of matrix [ARe] on /3i is very small.
and the calculated values are only approximate. For ai=2 decibels/mile the decrease of velocity caused by
The effect of poorly conducting earth on the attenuation and losses is less than 1% at 50 kc. However, measurements made
velocity of propagation has been investigated by Russian scien- in Russia and Canada indicate that this decrease is about 10%
tists.9 Their findings bring to light some of the important and that this large value can be attributed to the change in
aspects of attenuation and propagation velocities. The earth inductance, and to the additional lumped capacity caused by
contribution to attenuation is proportional to f2, at low fre- steel towers (less than 2%). The difference between mode
quencies and for high resistivity of the soil. At medium fre- velocities is quite important in some practical applications.
quencies and for normal earth resistivity this contribution is The author uses non-normalized eigenvectors, columns of
proportional to frequency f, and at high frequencies and for high matrices [A] and [B]. Would it not be preferable to normalize
soil conductivities it varies as fll2. these matrices? This could simplify the equivalent circuits in
The effect of earth on the inductance follows a different Figs. 4 and 5. I have calculated Wi in equation 36 for normal-
pattern. The increase in inductance is nearly constant at low ized matrices in equations 7 and 12:
frequencies and for low conductivities, and gradually approaches
the value corresponding to perfectly conducting earth at higher 0.571 921 0.396 511 0.707 107
frequencies and conductivities. [A]= 0.588 628 -0.828 719 0
The variation with frequency and earth resistivity of elements LO. 571 921 0.396 511 -0. 707 107]
in the impedance matrix [z] has different effects on the surge
impedance matrix [Z01 and on the propagation matrices ['] or 0.585 634 0.416 291 0.707 1071
[-y'], equations 37 and 38. The most important is the propaga- [B] = 0.560 415 -0.808 333 0
tion matrix because its eigenvectors are the mode components of L0.585 634 0.416 291 -0. 707 107j
voltages and currents associated with independent propagation
constants. A general expression for the square of the propaga- W1 = 564 ohms; W2 = 386 ohms; W3= 429 ohms
tion matrix [-y2 iS
It is interesting to compare these values with the eigenvalues
[^y] 2 = [z] [y] = (iw[L] + [R] ) (jw[C] ) derived directly from the surge impedance matrix [Z0] assuming
=(jw[Lc] +jw[ALe + [Rc]+[[ARe]) (jw[C]) (40) a perfectly conducting ground:
Subscript "c" denotes matrices of the system of parallel con- 460.0 62.8 30.7
ductors suspended above a perfectly conducting ground, whereas [Zo] = 62.8 460.0 62.8 ohms
subscript "e" denotes correction-factor matrices derived, for L 30.7 62.8 460.0]
example, by Carson's method.' All the component matrices
in equation 40 are symmetric; therefore, the propagation matrix Z(')=563 ohms; Z(')=385 ohms; Z(')=429 ohms
[e'], equation 38, is the transpose of [-y]. The elements of the Has the aulthor any comments on the relation between Wj
propagation matrix [-] are complex and the analysis leads to of a lossy line and Z(V) of a lossless line?
complex eigenvalues and complex mode components (eigen- Dr. Ushirozawa's analysis presented in his paper will greatly
vectors). However, the absolute values of elements of matrix assist in the applications of analytical methods to the solution of
jw [L] are normally much larger than those of the corresponding many current important problems. His answers to my questions
elements of matrix [R]. This applies to frequencies of several would be of further assistance.
kilocycles and higher, where the simplification of analysis
proposed by Hayashi' can be successfully used.
An examination of equation 40 indicates that by neglecting REFERENCES
matrix [RI, a set of independent modes of voltages and currents 1. WAVE PROPAGATION IN OVERHEAD WIRES WITH GROUND RE-
can be associated with constant velocities of propagation derived TURN, J. R. Carson. Bell System Technical Journal, NeW YOrk,
from the eigenvalues of matrix [L] [C]. Results of such a simpli- N. Y., vol. 5, OCt. 1926, PP. 539-55.
fled analysis of surges on a real line were recently published by 2. UBER DAS FELD LINER UNENDLICH LANGEN WECHSELSTROM-
A. J. McElroy and H. i\I. Smith.'10 The difference between the DURCHFLJOS5ENEN EINFACHLEITU-NG, F. Pollaczek. Elektrische
modal propagation velocities is caused mainly by the effect of Nachrichten Technik, Berlin, Germany, 1926, vol. 3, no. 9, pp. 339-
earth on the inductance matrix [ALe], equation 40, and therefore, 59 (also in French translation by J. B. Pomey, Revue Generale de
it is not surprising that the analytical results were found to agree l,'tlectricitt, Paris, France, 1931, vol. 29, no. 22, pp. 551-67).
well with experimental findings. 3. SURGES ON TRANSMISSION SYSTEMS (book), S. Hayashi. Denki-
In the analysis of Dr. Ushirozawa the mode components are Shoin, Inc., Kyoto, Japan, 1955.
associated with the eigenvalues of matrix [R] [C3, and hence the 4. RADIO-FREQUENCY PROPAGATION ON POLYPHAsE LINES, L. 0.
modes are characterized by specific attenuation. The mode Barthold. IEEE Transactions on Power Apparatus and Systems, VOl.
voltage components, columns of matrix [A], are the eigenvrectors 83, JUlY 1964, PP. 665-71.
NOVEMBER 1964 Ushirozawa -Hitih-Frequency on Nontransposed Power Lines 1143
m
leSi
m'
\Mk
k
o 3~ ~ ~ ~ ~ ~ ~ ~ ~ 2-
Fig. 11 (left). Distance Snk between
conductor m and the image of con-
ductor k, and angle 0
m77777r
0.0
007-
0.5
03
_
~~~~~~~~~~~~0.05
0!o
0
A:
0203
\
.\
_
f -'
4 mho45k:
0507!1 2
21 3
f =4xlO 0
-I
5. THE PROPAGATION OF HIGH FREQUENCIES ON OVERHEAD LINES, Ranges of y calculated on the line configuration as shown in
L. 0. Barthold, J. Clade. Progress Report 420-3, CIGRE, Paris, Fig. 2 in the paper are indicated in Fig. 12, if the frequency
France, June 1964, pp. 19-39. range is 50r450 kc and earth conductivity is 10-4_10- mhos/
6. NATURAL MODES OF POWER LINE CARRIER ON HORIZONTAL meter.
THREE-PHASE LINES, M. C. Perz. IEEE Transactions on Power Therefore, there is a limit to values of so depending on the
Apparatus and Systems, vol. 83, July 1964, pp. 679-86. frequency, earth conductivity, and geometrical arrangement of
7. A METHOD OF ANALYSIS OF POWER LINE CARRIER PROBLEMS the conductors of a power line. I have attempted to approximate
ON THREE-PHASE LINES, M. C. Perz. Ibid., pp. 686-91. the P( p) curve, which is limited by the s value, to a straight
8. EXPERIMENTAL EVALUATION OF POWER-LINE CARRIER PROPA- line on both logarithmic co-ordinates. In accordance with that
GATION ON A 500-KV LINE, D. E. Jones, B. Bozoki. Ibid., Jan. part of the P(sp) curve approximated to the straight line, re-
1964, pp. 16-23. sistance AR, in equation 41 happens to be proportional to fre-
C linefhici2
9. A METHOD FOR COMPUTING THE HIGH-FREQUENCY PARAMETERS quency f2ln
or
OF OVERHEAD ELECTRIC-POWER TRANSMISSION LINES, V. N. Orlov, The proportional to f2/ in Fig. 12 was used
which is
V. V. Sidelnikov. Telecommunications and Radio Engineering, pt. I in the calculation, and its calculated results were in good agree-
ment with the experimental findings. As shown in Fig. 12, it
(Telecommunications), no. 7, July 1962, pp. 60-70. (English transla-
tion published by IEEE.) appears that the earth contribution to attenuation is proportional
10. PROPAGATION OF SWITCHING-SURGE, WAVEFRONTS ON EHV ft
to frequency at low frequency and for low conductivity of the
TRANSMISSION LINES, A. J. McElroy, H. M. Smith. AIEE Trans- earth, and proportional tof1'2at high frequency and for high earth
actions Pt. III (Power Apparatus and Systems), vol. 81, 1962 conductivity. Therefore, I could not quite understand Messrs.
(Feb. 1963 section), pp. 983-98. Orlov and Sidelnikov's assertion that attenuation varies as
f2 at low frequency and for low conductivity of earth.
Michihiro Ushirozawa: I wish to thank Mr. Perz for his The resistance AR, is proportional to f5 in equation 15 in the
valuable comments. It is very difficult to analyze propagation paper. This means that exponent "x" of frequency f varies with
properties of high-frequency waves on a power transmission line the frequency range and also with the value of the earth con-
because of the variety of conditions affecting the propagation. ductivity.
Therefore, as Mr. Perz points out, the methods now being Moreover, the method described may be applicable to the
correction factor AL, of the inductance, but also have identical
employed, although based on the same fundamental principles,
are simplified according to purpose by making certain assump- limitations.
tions, so that the use of the simplified methods is limited, de- I agree with Mr. Perz in that j,i calculated from equation 17
pending on earth resistivity and frequency ranges. in the paper has little practical meaning because the effect of
Originally, I wished to devise an analyzing method to find AR, on fi is very small. Equation 17 is a formula for the de-
some coupling method appropriate for a power line carrier system termination of the multipropagation velocity of natural modes
on an EHV 2-circuit vertical power line which exists generally when AL, is taken into consideration. Hence, it seems that
in Japan. Hence, I had analyzed the propagation charac- propagation analysis should be performed by stricter means.
teristics by taking account of the line configuration and of the This kind of analysis, however, complicates equations 29 and
power line carrier frequency range, which is from 50 kc to 450 kc 34, which are used for seeking line attenuation values with
in Japan, and then compared the results with experimental various coupling methods; therefore, I have neglected AL, in
findings obtained on some EHV power lines. calculation.
Mr. Perz has asked for the method of computing values of The results of experimental estimation on the Shiobara test
the element in matrix [ARJ, equation 4 of the paper. line in Japan (3-phase 2-circuit power line, 1.4 km long) indicate
that the propagation velocities of modes 2-5, which correspond
By Carson's method, the general expression of the resistance
ARB as a result of skin effect on the ground is
to the modes shown in Fig. 3 in the paper, are approximately
99% of light velocity; those of modes 6 and 1 are approximately
ARe =4wP(yP) (41) 93% and 88%, respectively, of light velocity.
Two-phase-to-ground coupling is mainly applied to the signal
=2 2\/ rS2 /X10i (42) transmission circuit for power line fault locating equipment.
This coupling method is usually used in conjunction with trans-
where mission circuits for telephone or carrier relaying equipment in
w = angular velocity Japan. In this case, if a line fault should occur at a point
P(yo) = a function having variable so, that is approximately 50 km from the end of the line, it is known
erearth conductivity (mhos/meter) that a high-frequency a-c pulse of approximately 400 kc sent
f= frqec k)from the locating equipment at the end of the line does not return
= distance between conductor m and the image of conductor to the sending end. I believe that this may be the result of the
k (see Fig. 11) difference between modal propagation velocities.
Of course, the use of normalized eigenvectors, columns of
Fig. 12 indicates a curve of P(9o) versus so when the angle 6 matrices [A] and [B], is preferable to the use of non-normalized
in Fig. 11 equals zero. The maximum angle 6Em for the 2-circuit ones. However, for the purposes of this paper, I felt it was
power line is less than about 30 degrees, and the PGco) curve is not necessary to employ normalized eigenvectors since the
very close to that given when C equals zero. calculations were made with a digital computer.
1144 Ushirozawa-High-Frequ<ency on Nontransposed Power Lines NOVEMBER 1964
Mr. Perz also indicates that values of modal surge impedance When
Wi are nearly equal to the eigenvalues which are derived directly
from the surge impedance matrix [Z0] assuming a perfectly con- Mo-Mn#Xi
ducting ground. An illustration for this is as follows: 2
From equation 7 in the paper, modal surge impedances Wi of u2 (Mot- X)
the lossy line can be shown to be 2(Mo'-X1)2+(2Mm')2
IW il = Iw] V[ ' [ Y'i = [Z] / [,Y U2 2 = (2MmI/) 2
= [-y] -1[A] -1[Z] [B] (43)
U1 =U3
By using equation 8 in the paper,
[A] -1[Z] [yi2][A] -'[Y] [-y,2]
(44) ~~~~~~~~~~~~~~~U12+
U22+ U3 2= 2U,I+ U22 =1 (49)
Substituting into equation 43, When
[Wi] =L yi] [A] -1[Y] -1[B] (45) Mo-Mm=XA
From equation 15 in the paper U1+U3=0
[y] =[C[ =g[L] =[L] (46) U2=O (50)
ico jo io Normalized eigenvectors U1', U2', and U3' of matrix [B] are
where g is the light velocity. equal to those when Mm' is converted into Mm, as in equation 49.
Substituting into equation 45, Calculating the values of the elements of matrix [MI] in the
example indicated in the paper,
[at i-j3i] ri~ ail
[Wi] Jf3d
[A] -qg[L] [B] = LI3 -- j [A] -'[Zo][B] F1.2065 1.0148 0.9611
Since ca<<I3, ,B0, L0.9611
[MIl=1.0863 1.1126
1.0148
1.0863
1.2065]
XKX102 (51)
Summary: A quantitative research report is made here, test. Interesting and useful data pertaining to the design of
based on tests of EHV (extra-high-voltage) insulator strings. EHV lines were collected.
Impulse, switching-surge, and a-c tests on vertical, horizontal,
and 45-degree V-strings were carried out, using both positive ______________________________