1964, High-frequency Propagation On Nontransposed Power Line.pdf

The question concerning the pioper value for the zero-sequence impedance is difficult to answer. Effects of the ground return path are such that zero-sequence resistance and inductance both contain frequency terms, as a number of texts show. Therefore, during the transient solution, when frequencies other than the system base frequency exist, the zero-sequence impedance should properly be modified to account for these effects. A closed-form analytical solution for these effects would be very complex. Even during the process of a numerical solution on a digital computer, this is extremely difficult, since one does not know the transient frequencies and thus cannot make the proper corrections. A possible approach is to use average values for the zerosequence terms, say at 60 cycles as is done on the ANACOM, then record the transient line response. A study of the resulting

waveshapes to determine the dominant frequencies could be made, the zero-sequence parameters recalculated, and the problem solved again. Some such trial-and-error approach may thus lead to more exact results. Our current method is to use 60-cycle values, as supplied by transmission-line design engineers, and to accept the results as being satisfactory for the present. In regard to Dr. Robert's third comment, the authors are aware that the representation of the ground mode is an approximation because of the method employed in calculating the characteristic impedances and the propagation constants. It is felt that the usage of a single velocity of propagation for the ground mode is sufficiently accurate for most practical situations as mentioned earlier in discussing the approximations used in the papers.

High-Frequency Propagation on Nontransposed Power Line Michihiro Ushirozawa Summary: The high-frequency characteristics on nontransposed power lines have been calculated by the use of the principle of natural mode propagation, and it has been shown that some comparison between calculated and measured values on the vertically arranged 3-phase 2-circuit power line is possible.

Recently there has been a trend toward adoption of a nontransposed power line for EHV (extra-high-voltage) transmission. When installing power line carrier assemblies on such a line, it is necessary to pay due attention to the highfrequency transmission circuit, because in a comparatively long power line, the electrical unbalance between each phase of the line and the ground is so emphasized that the highfrequency characteristics differ greatly depending upon the selection of the coupling phase and the coupling method. In calculating the high-frequency characteristics of a power line, a formula1 has frequently been used where the power line is considered as a symmetrical line. However, a satisfactory reply cannot be expected with regard to the characteristics of the nontransposed line. By approximating such a line to an overhead parallel multiconductor line, the author has derived general formulas of computation and prepared a number of attenuation charts' classified by the coupling methods which are important in designing power line carrier assemblies. From these charts, line attenuation can be estimated immediately if some conditions are given, such as line length, coupling method, and frequency to be used. In a series of field tests conducted on numerous nontransposed power lines, the author compared the calculated values with the actually

Propagation Theory of Parallel Multiconductor Lines The transmission equation of parallel multiconductor lines,

asshowniFig.1,canbegiveninthefollowigequations. -- [V] = [ZJ [i] C)

()

- - [i] = [Y] [v] x

[v]= ((vv,2,. .,Vn))

[il =((ii,i2. ,in))

where (( )) indicates a single column matrix, and v1, v2, ., and i1, i2, ., i,, represent, respectively, line voltages and line currents at an optional point x of the parallel nconductor power line. The [Z] and [Y] in equation 1 are the impedance and admittance matrices of the line (square matrix of the order n). As is widely known, the propagation constants of the multiconductor line can be sought as eigenroots of the product of [Z] and [Y]. (3) VIZ[Y] - [UI=l[Y] [Z] -/i,2[U]1=0 with -y propagation constants and [U] the unit matrix.

__x__

_______________

measured values.

NOVEMBER 1964

l

2_

2

The purpose of this paper is to present the high-frequency 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 Distribution Committee and approved by the IEEE Technical Operations Committee for presenltation at the IEEE WXinter Power Meet-2 ing, New York. N. Y., February 2-7, 1964. Manuscript submittedt November 4, 1963; made available for printing March 30, 1964. MICHIHIO USHIROZAWA is with the Central Research Institute of Electric Power Industry, Kitatama-Gun, Tokyo, Japan.

(2)

V1

| v2

~

n

~

n

// ,,/' // 777777T/7// j// /// )~///'/ Fig. 1.

/////

Parallel multiconductor line

Ushirozawa -High-Frequency on Nontransposed Power Lines

1137

Since the characteristic equation of formula 3 is of the order n, it has n propagation constants. To be exact, this shows that the n of parallel multilines can be converted into the n of independent natural mode circuits. Now, in order to convert equation 1 into the natural modes, consider the primary conversion, as shown in equation 4. [v] = [Al Ivlj Ei] = [B] [i'] (4) [i'] =[B] -1[i] [v'] =[A] -I[v] The [v'] and [i'] in equation 4 indicate the high-frequency voltages and currents of the n of the natural modes (corresponding to the number of propagation constants) in a single column matrix as in equation 5.

"inverse conversion matrices of voltage and current," respectively. Substituting equation 1 for 4, one obtains the following equations:

[V']=((V',V2',. .,vn')) [i'] =((i,',i2,'.* *,jt))

Equation 6 is a propagation equation which corresponds to the natural mode circuits, namely, mode voltages and mode currents. Differentiating equation 6 further, equation 8 is

(5) The [v'] and [i'] in equation 5 are called "mode voltage" and "mode current," as distinct from [v] and [i] in equation 2. The [A] and [B] are the matrices which connect line voltages with mode voltages, and line currents with mode currents. Let the former be the "voltage conversion matrix," the latter the "current conversion matrix," and [A]-' and [B]-1 the

8.6M

-

Fig. 2. Configuration of EHV Tokyo Eastern line Conductors: 2x330 mm2

i0Gm >\ XG2 /\x. >

(D

X

)t.

sF

lOOM

4

x-\ <

(2)

.

E

<

I

0<

7

32

/

; GROUND

5

I:< s

3_ 42Xe9 2: / -2 ID

5010200OnC

FREQUENCY (KC)

1138

[Z'] = [A] '[Z] [B] [Y'] = [B] -1[Y] [A]

(7)

obtained as the one equivalent to equation 3. [A] '[Z] [Y] [A] = [-s2] [B] -'[Y] [Z] [B] =y[,'2]

(8)

The propagation constants [y,2 [i]and [,yiw] Of equation 8 have a value entirely equivalent to the voltages and currents that can be sought from equation 3. From the relationship shown in equation 8, it will be understood that each element of matrix [A] can be obtained by the

use of the following equation.

[[Z][Y]--yj[U]]((1,A2i,

..

.,Ani))=O

(9)

A2f .., A n))indicates a single column matrix in .1 first element is 1. If i= 1, 2, .. ., n and one at-

solve it, all the elements of the [A] matrix can be As for [A] it can be obtained as its reverse

matrix. (5)Next, [y2] is a diagonal matrix and is entirely the same even if a transposed matrix is taken. Therefore, by a transposition of equation 8, the following two equations 10 and 11 result: [B]'-[Y] [Z] [B] = [A] *[Y] [Z] [A] -'*= [yi2] (10) where the notation * shows the transposed matrix. From equation 10,

[WI']= [Z'] /[Y']

(12) and propagation constants 'y are indicated by equation 13.

[I)

CONDUCTIVITY

L

O~IL Xt-5 977 d t 5t /

where [Z'] and [Y'] are the diagonal matrix.

(11) In other words, the current conversion matrix can be obtained by transposing the voltage conversion matrix. Now, using [Z'], [Y'] in equation 7, characteristic impedances W1 of the natural mode circuits are

10-3 (MHO/

=

(6)

[B] =[A]1*, [B] =[A]*

°

<

07

(6)

/o

50

-aW['= [Y'][v']

+-2 ACSR in 0.4-meter double where ((1, which the Overhead ground wires: tempts to 120 mm2 ACSR obtained.
(4)

I4.6M

I (l) L.6m

- -[v'] = [Z'] [i']

.v0 (41 *o (5)

~~~~NO.OF MODE

(13)

From these relations,

O

6 * (2) . °

500

[PY7] = [Z'] [Y']

MN)

Fig. 3. Attenuation constants and characteristic impedance of natural modes 0 Currents flow in the normal direction 0 Currents flow in the reverse direction

[Wl = [yi] -1 [Z'] -yi] -'[A I-1[Z] [B] (14) As mentioned previously, solution of the parallel n-conductor lines can be made by the following process: As eigenroots of the [Z] * [Y] matrix in the existing line, seek the number n of yf. Next, if conversion matrix and characteristic impedance are sought with ei taken as a medium, then the solution can be made in a group of natural mode circuits. Accordingly, to find high-frequency voltage and current characteristics on the existing lines, it would suffice to proceed in the following way: First of all, convert the existing line into a natural mode

Ushirozawa-High-Frequency on Nontranaposed Power Lines

NOVEMBER 1964

circuit. Next, calculate mode voltage and mode current characteristics in a 4-terminal network on the mode circuit. After that, convert it into line voltages and line currents of the existing line.

Fig. 4. Equivalent circuit with non coupled phases grounded at the

[ Y] =jw[C] [R] = [Ro] + [Re] = [R lfx

where [L] = [C] = [Ro] = [Rel =

(15)

inductance matrix capacitance matrix conductor resistance matrix

resistance matrix due to skin effect of the ground

vrelocity In order to simplify the calculation, [R] is taken as approximately proportional to fX (x = constant, f= frequency). The aforementioned formulas will now be applied to the vertically arranged 3-phase 2-circuit power line which is generally seen in Japan. The EHV Tokyo Eastern line has the following characteristics: Conductor-2X330 mm2 (square millimeters) ACSR (aluminum cable steel-reinforced); overhead ground wire-two ACSR wires of 120 mm2; line voltage-275 kv; line length-116 km (kilometers). Fig. 2 shows a standard line arrangement. If one phase is regarded as a single-line equivalent, the total number of lines will be eight, including ground wires. This should be solved precisely. For the sake of simplicity, however, calculation will be made on the six lines neglecting the ground wires. (There is another method in which ground wires can be omitted by assuming that their electric potential is the same as the ground.) = angular

co

Table 1.

Line Voltage

Mode Voltage

Vi

Vi'

0.2616

v,' V3' V4'

V6' V6'

[A]'

Inverse Conversion Matrix of Voltages: (Tokyo Eastern Line) V2

0.1595

V3

0.1174

V4

0.1174

0.1935 -0.0507 -0.1826 -0.1826 0.0652 0.0449 -0.1088 0.0652 0.0277 O.00i3 0.0099 -0.0277 0.0476 0.1857 -0 .2155 -0 .0476 0.0752 -0.0752 0.2056 0.3130 1

V6

0.1595

-0.0507

V6

0.2616 0.1935

0.0449 -0.1088 -0.0099 -0.0013 0.2155 -0 .1857 -0 .2056 -0.3130

N2

|_

line terminals

Line Constants on a Nontransposed Power Line As a power line involves structurally many complex factors, it is necessary to approximate the power line to a fixed model so that the power line has an average height above the ground and is parallel to the ground. Further, ignoring an increase in inductance as a result of

skin effect of the ground-to-power line conductors and also ignoring leakage of insulators, equation 15 is obtained. [Z] = [R] ±iw[LI

Zo

I

SENDING END

Fig.

5.

Equivalent

with noncoupled phases open circuit

at the line terminals

o"'

i 42 Wn 4'3

: N _> i N

RECEIVING END

.-

Z_ N

2'

1

w W2 T2

N '2"1NA Zo

E

N , L _ _____n______ _ RECEIVING SENDING END

.N

END

By the Carson-Pollaczek method, if the ground conductivity a= 10-3 ihos/meter, the value Rg in the area from the power line carrier wave band to medium wave band and under the power line arrangement as shown in Fig. 2, increases with frequency as f2"3, as a result of skin effect of the ground. Conductor resistance Ro increases with frequency as fl"2. But it is smaller in value by one to two figures than Re. Therefore even if Rois assumed to be proportional to f2'3 to relate it to equation 15, it is possible that no serious error will arise. Fig. 3 shows the results of attenuation constant and characteristic impedance calculated in such a manner. The element A7,j of voltage conversion matrix [A] can be obtained from equation 9. In this case, the first-row elements AI, in matrix [A] can be put as 1. From its reverse matrix, [A]-' can be obtained, and [B] and [BI-' can be determined by using equation 11. Tables I and II show the inverse conversion matrices of voltages and currents. From this table, it can be understood that the six natural modes in the 3-phase 2-circuit power line consist of three modes where the currents of conductors placed at the same 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 the power line carrier frequency band, then characteristic impedance W, is given by an approximate equation 16, from equations 10 and 14. (16) [Wi] =g[A] -1[L] [B] Phase constants fi will be 03

a 2

( 17) 2~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(7

V1+A2

where 3=cw/g (g = 3 X0l km/second) Table 11. Inverse Conversion Matrix of Currents: [Bf'1 (Tokyo Eastern Line) Line Current

curren i,'

i4'

it i

i6'

.,2

1

1 1 1 1

NOVEMBER 1964

I 0.811 I~'0.8980 1 -1.5377 -0.5082 1.9046 -3.0422 4.4836 -16.4081 -1.3617 -0.4398 0.3296 0.7890

4 0.8101 -1.5377 1.9046 16.4081 0.4398 -0.3296

1 0.8980 -0.5082 -3.0422 -4.4836 1.3617 -0.7890

1

1

1 -1 -1 -1

(18)

Attenuation and Impedance of the Carrier Frequency Band The high-frequency transmission circuit, which usually

takes one or two phases of the power line as its coupling can be expressed by a combination of natural mode cirphase, CU1tS.6 The equivalent circuit is shown in Fig. 4 for the case of noncoupled phases grounded at the line terminals and in Fig. 5 for the case of noncoupled phases open at the line terminals.

Ushirozawa-Hiqh-Frequency on Nontranaposed Power Lines

1139

For noncoupled grounded conductors, the mode circuits are in parallel with the power source or load and are in series for the noncoupled open conductors. N2, NT' (i= 1,2, ..., 6) shows a ratio at which the voltages (currents) impressed on the coupled phases are converted into mode voltages (mode currents), or a reverse ratio, namely, the winding ratio of an ideal transformer which is inserted into the converting portion. It can be determined by using

Fig. 7. Measured attenuation values of various couplingsonthe Tokyo Western line (Comparison of average values between noncoupled

Fig. 6 shows some representative coupling methods illustrating the impressing of high-frequency voltages and currents on the terminal of line. 1. GROUNDING OF NONCOUPLED CONDUCTORS When the length 1 of the line is taken as infinite in Fig. 4, the input impedance Z, of the line, as seen from the power source or the load, will be

grounded open)

equation 4.

1

Z

(19)

t

=1

If the ratio Psi between the sending-end voltage vi (i= 1, 2, . force ^ . . .electromotive Er n) of each mode circuit and the of the power source (Fig. 4) is defined as a transmission coefficient of the sending end, then the ratio Psi will be

PsVi

(20)

' N=

where 1 12

(21)

25 20

15 /

z +

/ /

E3

//

>E2 -

/,

0

O

100

M23

200 300

FRE0UENCY(KC)

400

500

consideration the primary transmission wave only, for the sake of simplicity. Therefore, high-frequency voltage of the transmission wave at the receiving end iS ~~~~primuarv p (23) Vr = [Pr] [T] [Ps]E where

[T] = [6-5ilJ

(24)

i=1, 2, ..., 6 [T]: diagonal matrix [Ps] = ((Ps1, P82, . P96))

(25)

where (( )) is the single column matrix. P76r6)) [Pr] = ((PL PTS,

(26)

j

(27)

e

-

In case the length of line is limited, the traveling wave to be transmitted on the line repeats reflections as a result of the mismatching of impedance at the terminal ends of line. Therefore, if a frequency-attenuation curve is sought, fluctuation related to the length of line will appear at regular intervals. Since calculation for obtaining the width of fluctuation will become very complex in the case of a nontransposed power line, it is attempted to seek an average attenuation taking into COUPLING I16COUPLING

I+.

Mi2COUPLING2

('2

5) * -t La 2a

I) (lo *6

X

I

(28)

2Vr

B. with grounding of noncoupled conductors Attenuation will be (6 4 B= -20 1ogio Z0A2

f - t ~~~~~~~~2. T L-

* * * >-lo . To >-lo-lo a-zf= LNi'2I

N12 -e W

(decibels)

(29)

1

__ __ __ __

KEICOUPLING D00PLNGSGlcoupLINGMI2coupUiN&

I -

/ . 1

R \

/

*/ ~___I.'

_J

r-®j vI <

I~~~ ~V/ r __1

_ (B)

OPENING;OF NONCO1JPLED CSONDUCTORS Similarly, the input impedance Zf of the line is

Fig. 6. High-fre-

I

i~(A)

<

B = 20 log1,

.~~~~~~~~~~~~~~=

.

2

Generally, attenuation B between the sending end and receiving end can be given as

_

IEicoU PI

1140

/

/

5

On the other hand, if the ratio P,i between the load voltage From equations 2, 22, and 23, VT and the receiving-end voltages vi" of each mode circuit (Fig. 4) is defined as a transmission coefficient at the receiving end, 6 2 E then the ratio will be VT rPsi-'Yi E Z... TI Wei VT 2Nj ==1 (22) Pri =

//

/

0

cond uctors t

and

[

voltage 1quency and current at ~~~~~~various coupling methods

A-Line ' voltage (grounded non~~coupled conductors)

(open noncoupled conductors)

(30)

6=1

A transmission coefficient of the sending end and of the receiving end PSi and PTI Will be

hr whr

N1'W1

2N1Z (Z

A'

A'

A'=Z0+Zf

(32)

Considering the primary transmission wave only, the receiving-end voltage VT is

Ushirozawva-Hitgh-Frequency on Nontransposed Power Lines

NOVEMBER 1964

E

6

rT

° v A".,,.J

Ni/2Wie-il

Fig. 9. Horizontal configuration of single-

(33)

circuit power line

(I)

Ni2Wi e

--

(2)*

d - -

(5)

h

Accordingly, attenuation Bo, upon opening of the noncoupled conductors, will be Bo -20 log1o A2J

- d

(34)

(db)

i =1

Calculated and Measured Values of High-Frequency

It can be understood from the calculation results using Characteristics equations 29 and 34.that the difference in coupling method and Fio shows characteristics classified by the phase rslinagetvrainiatnuio. wilgreat variation inattenuation.Some Some,.. 7 method, attenuation on the Tokyo Western line (275-kv 2-circuit coupling folw concrete examples follow. nontransposed power line; line length, 77 km; configuration of the line is the same as Tokyo Eastern line). In the figures, 1. In the phase-to-ground, or E, coupling, attenuation will in abbreviations for the coupling method, the noncoupled be greater in the lower phase than in the upper and middle open phases are indicated by a dash, while those with no dash phases. show noncoupled phases grounded; the phase numbers are interstraddling 2. In the 2-phase-to-ground, or D, coupling l 1 . 1 ll .. . l ll in Fg Fig. 2.2. elgven in pparentheses circuit, attenuation will be small when middle and middle n in givenoin Fig. As shown in 7, the manner in which the attenuation phases are coupled, while attenuation wilbe lag ih characteristics differ, depending upon the coupling method coupling between upper and upper phases and lower and lower and coupling phase, agrees with the aforementioned calcula-

~ ~ concreteil ~ rexamples

phases.

tion results.

3. In the phase-to-phase, or I, coupling straddling intercircuit, the order in terms of small attenuation values is as follows: (a) coupling between upper phases; (b) coupling between middle phases; and (c) coupling between lower phases. 4. In the phase-to-phase, or M, coupling of the same circuit, attenuation value will be small in the upper-middle phase coupling and middle-lower phase coupling; while attenuation will be large in the upper-lower phase coupling. 5. Irrespective of coupling method and coupling phase, the

attenuthatn valuthe

.un groundeddconduce s e

issmallerat noncoupled

n

u

go

Fig. 8 shows representative examples of seeking attenuation B0, Bo, and impedance Z,, Zf with E and M coupling. If the length of the power line, frequency, coupling method, and coupling phases can be known, then the average attenuation on the line can be assumed by the figures. Also, Fig. 8 shows a comparison between measured and calculated values on the Tokyo Western line and the Minami-Kawagoe line (275-kv 3phase 2-circuit nontransposed power line, 13 km long; configuration of the line is the same as is shown on the Tokyo Eastern line). Next, the high-frequency characteristics7 in a medium-wave band were measured on part of the Northern Osaka line (275kv 2-circuit nontransposed power line, 33 km long). From this test, it became clear that the power line carrier frequency is conspicuous in the medium wave band, too. Calculation of Horizontal 3-Phase Single-Circuit Power

INPUT IMPEDANCE (0)

30

~ Es'

21

20Zs

L1 359_;3213~40>< E
20

El

10

In the

lLUES --

E2

9 5

z // w; MINAMI-KAWAGOE LINE) Es f kc e: km I! 'l 2 iI Q 0

o

-5I

Line

ED CALCULAT VA

E:

lO0

3

zD

'~CALCULATED ,2CALCULATED

Z 21 F

cb 7 k ,/fg,2w,7v

M ~MX/ /)<3;23,/SURED 3M1MEA'E

Y//7

///

M >'5Z

MINAMI-KAWAGOE ONE)

f: kc

2

3

&:km

-K-

f 2 e

(B)

NOVEMBER 1964

Fig. 8. Attenuation

(TOKYO-WESTERN LINE)

curves A-Calculated and measured

~~~~~[A]'-Z

0.33

0.32 033 1 1.03 1 1 0.17 -0 .32 0.17 ' [B] 'zs 1 -2.09 0 -0.50 1 0 -1 10 .50

(35)

Calculating values of attenuation constants a and characteristic impedance VV are as follows:

values

a, =1.69 X10-2f"3N (per km) 2B-Calculated _L4 and CY,=0.98X104yf"' N (per kin) ~~~measured values of M coupling cas=1.02X10'f"'t N (per kin) of Ecoupling

L

line,

line"0

.

Z0

power

calculations which were made on the Senju-Niigata (154-kv horizontal 3-phase 1-circuit nontransposed power line; diameter, 6 meters; height, 8 meters; conductor ACSR 170 mm2) by using the suggested methods.

e

20(AINPUT IMPEII,~INCEKI) 201- INPUT IMPEDANCE(fl) I M2 23 M1s 7DZs '52 A607

of a vertical 3-phase 1-circuit

V ALthe

IET

10'

5

case

the coupling method and relations between coupling phase and attenuation characteristics are almost the same as those of 3-phase 2-circuit power line. In the horizontal 3-phase 1-circuit power line, Fig. 9, the arrangement of phase conductors has a special symmetry. E,Hec,fruacabeba Hence,calculating formulas can be replaced by a slightly more simple form, and a number of solving methods8'9 have so far been announced. However j the author will give here the results of numerical

Ushirozawa-Hit/h-Frequency on ANontransposed Power Lines

W = 179 ohms W,= 58 ohms

W2=207 ohms

(36) 1141

10

Fig. 10. Attenuation curves of horizontal

M,3,.

30 20

M/

20

-

, El'

m7;< Ei / E

-

powe r i ~~~~single-circuitpoe line

10. RESEARCH ON POWER LINE CARRIER COMMUNICATION, Committee Report. Journal, Technical Laboratory, Central Research Institute of Electric Power Industry, Tokyo, Japan, vol. 8, 1958, p. 299.

~~~~~~~~~Dislcussion

< E L M,2/S/ // Mt2 / g f kc. Qkm 0 -23 !51'103 2 3

C. Perz (The Hydro-Electric Power Commission of Ontario, ~~~~~~~~~~M. Toronto, Ont., Canada): The author is to be commended for

07 _ 05 _

Je

slo3

Fig. 10 shows the calculated attenuation curv es. The order in terms of small attenuation is as follows: ill coupling of center phase to outer phase, E coupling of center phase, and Eoulngofote pae.Inte as f utrphset outer phase coupling, attenuation is smaller than the E coupling when the product of frequency and line length is small. As the product increases in value, however, its attenatin bcoes reaerthan that of E coupling,

tenuationbecomesgreater Conclusions The author haUs derived an approximate method for calculating high-frequency propagation characteristics on the nontrnspoedpowe wer lin ine andhasshow nd hs shwn is reatioshipwith nontanspsed itsreItionhip ith the measured values. As a result, it has become clear that, on the nontransposed power line which has a comparatively long

length, high-frequency characteristics will differ greatly depending upon variations in the coupling method and coupling phase, as a result of unbalance between phases of the line and betwTeen each phase and the ground. In the power line carrier communications, therefore, it will be necessary for us to understand well the characteristics

and to make proper selection of coupling phases.

References 1. TELE:COMMUNICATION TECHNICS FOR ELECTRIC POWER INDUSTRY (book). Institute of Electrical Engineers, Tokyo, Japan, 1957, p. 218. 2. PROPAGATION CHARACTERISTICS OF POWER LINE CARRIER WAVE ON POWER TRANSMISSION LINE, M. Ushirozawa. Report No. 600231,61008, Central Research Institute of Electric Power Industry, Tokyo, Japan, 1961. 3. RADIO INTERFERENCE DUE TO THE POWER TRANSMISSION LINE. Committee Report, Institute of* Electrical Engineers, Japan, 1957, p. 244. 4. THE GENERAL SOLUTION OF THE MULTICONDUCTOR CIRCUITS WITH NONRECIPROCAL LINE CONSTANTS UNDER THE STEADY STATES, H. Noda. Bulletin, Electro-technical Laboratories, Tokyo, Japan, vol. 15, no. 8, 1951, p. 470. 5. THE CALCULATION OF ATTENUATION CONSTANTS FOR RADIO NOISE ANALYSIS OF OVERHEAD LINES, L. 0. Barthold, G. E. Adams. AIEE Transactions, pt. III (Power Apparatubs and Systems), vol. 79, Dec. 190 P.951 6. POWER LINE CARRIER WAVE PROPAGATION ON POWER TRANSMISSION LINE, R. Sato, M. Akiyama. JoNrnwal, Institute of Electrical Engineers, Japan, vol. 79, 1959, p. 1558. 7. HIGH-FREQUENCY PROPAGATION TEST ON EHV OSAKA LINE, M. Ushirozawa, F. Nishiyama. Report No. 6100i3, Central Research Institute of Electric Power Industry, 1961. 8. PREPARATION OF SINUSOIDAL OSCILLATIONS ALONG A THREE WIRE WITH HORIZONTAL SPACING, M. V. Kostenko. Electrichestvo, MOSCOW, USSR, no. 8, 1959, P. 8. 9. WAVE PROPAGATION ALONG THREE-PHASE POWER LINES AND TELEPHONE LINES WITH POWER INDUCTION, 0. A. Petterson. Ericsson Technichs, Stockholm, Sweden, vol. 15, 1959. 1142

his analytical method dealing with propagation of electromagnetic waves along real power transmission lines. His paper is very valuable and important because it provides solutions of 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 power frequency and its harmonics the long-line-equivalent

circuit parameters can be calculated from propagation equations. Switching surges, power line carrier, and radio interference produced by corona on EHV lines extend the frequency range to about 2 megacycles. However, no simple method seems to exist which could be satisfactorily applied to the variety of problems. Depending on the earth resistivity and frequency, simplified methods are being employed, methods based on the same fundamental principles but characterized by

~~~~~different assumptions and resulting limitations.

A brief review of various analytical methods seems to be necessary for an obJective discussion of the particular method proposed by Dr.

Ushirozawa. The following is a solution for conductor voltages and current associated with a plane propagating in the x direction and satisfying equations 1 in wave the paper: {v} = e-( Ill [YI)(iii) {vs} = e- IY]x{v1} {j} = -([2 [z] (lIi){j0} e- V']r{j0}

(37) (38)

where {vo } and { io} are single column matrices of voltages and curnsfrx=.Frhroe {vus} = [Y] 1 2[Z] 112{{i} = [ZoI {io} (39) [Z1iasqremtxanitaybcoidedsthsug impedance matrix of the system of conductors.

Equations 37-39 seem to be simple in form, but generally they cannot be used directly because all matrices appear as arguments of functions.

Also, the elements of the elementary

matrices [z] and [y] depend on frequency, and their values are difficult to compute even if simplified assumptions are adopted. The following are the most commonly used assumptions: 1 ekg n ipaeetcretlse r elce (th seconkoate authr' dlaequatiorrns 15). r ngece (h eodo h uhrseutos1) 2. Earth losses can be represented by a matrix whose elements are self- and mutual resistances in series with the corresponding

inductive reactance elements of line conductors (the first of the author's equations 15). 3. Internal flux of conductors is neglected.

4. The electromagnetic field associated with conductor currents and voltages is approximated by plane waves perpendicular to conductors; therefore, there are no radiation losses. 5. Elements of capacitance matrix [C] are derived from Maxwell potential coefficients, whereas elements of inductance matrix [L] can be computed following classical methods developed by Carson1 or Pollaczek.2 A method of reduction of matrix functions in the solution of various aspects of surge propagation on power lines has been developed by Professor S. Hayashi of Kyoto University. In

his book,3 extensive and detailed applications of the Sylvester expansion theorem and Laplace transformation to transient phenomena are described. Various simplifications illustrated by experimental data lead to an insight into the effects of multivelocity waves and attenuation. A negligibly small error is made in evaluating velocities of propagation if the losses are neglected. Conversely, in the derivation of voltage and current

Ushirozawa-High-Frequency on Nontransposed Pouer Lines

NOVEMBER 1964

of matrix [R] [C], whereas the mode current components, columns of matrix [B], are the eigenvectors of its transpose. The two sets of eigenvectors are orthogonal; hence there is no exchange of energy between the modes. Matrix [C] is assumed to be independent of frequency and of earth resistivity and the author proposes a further simplification by neglecting the earth effect on the inductance matrix [L]. Therefore, the mode components of a given line and the mode attenuation coefficients both depend only on losses in the phase conductors and earth. Hence, in Dr. Ushirozawa's analysis the values of elements of matrix [R] are of primary importance. Can the author explain in some detail the method of computing the values of elements in matrix [ARe], equation 40? Also, the variation of exponent "x" with frequency and earth resistivity (author's equation 15) is of great interest. The application of Dr. Ushirozawa's method to actual problems requires numerical computation of matrix [AR,] for various earth resistivities and for a range of frequencies. Normally, this computation is more difficult and tedious than the calculation of modes and, therefore, additional information on the subject will be much appreciated. Does the author know of any simple method of computing the inductance correction matrix [ALe]? This matrix is needed for the evaluation of 8i or mode velocities. It seems that 6i calculated from the author's equation 17 has little practical meaning because the effect of matrix [ARe] on /3i is very small. For ai=2 decibels/mile the decrease of velocity caused by losses is less than 1% at 50 kc. However, measurements made in Russia and Canada indicate that this decrease is about 10% and that this large value can be attributed to the change in inductance, and to the additional lumped capacity caused by steel towers (less than 2%). The difference between mode velocities is quite important in some practical applications. The author uses non-normalized eigenvectors, columns of matrices [A] and [B]. Would it not be preferable to normalize these matrices? This could simplify the equivalent circuits in Figs. 4 and 5. I have calculated Wi in equation 36 for normalized matrices in equations 7 and 12:

components associated with constant attenuations, a single velocity of propagation may be assumed. These assumptions result in significant simplifications of computations. It is interesting that in the original method of Professor Hayashi no direct use is being made of the concept of eigenvectors. The simplest method of analyzing propagation of highfrequency signals is obtained if the effect of losses is initially neglected in equation 39: Matrix [Z0] is symmetric and its elements are real. This leads to natural modes of propagation characterized by mode impedances equal to the eigenvalues of matrix [Zo]. Because of the symmetry of the surge impedance matrix there is only one set of auto-orthogonal eigenvectors common to voltages and currents. Therefore, the actual phase voltages and currents can both be resolved into uncoupled components. The fact that these independent components do not interact allows a specific attenuation to be attributed to each natural mode. Provided that the depth of penetration of earth is small with respect to physical dimensions of the line, this method is very useful. Its application to horizontal singlephase lines is particularly simple and leads to meaningful solutions of practical problems confirmed by experiments.4-8 The introduction of earth losses in the elements of matrix [z], i.e., in inductance and resistance elements, greatly complicates the analyses. The self- and mutual terms of inductance and resistance both depend on frequency, earth resistivity, and line geometry. The computation of these elements is very tedious and the calculated values are only approximate. The effect of poorly conducting earth on the attenuation and velocity of propagation has been investigated by Russian scientists.9 Their findings bring to light some of the important aspects of attenuation and propagation velocities. The earth contribution to attenuation is proportional to f2, at low frequencies and for high resistivity of the soil. At medium frequencies and for normal earth resistivity this contribution is proportional to frequency f, and at high frequencies and for high soil conductivities it varies as fll2. The effect of earth on the inductance follows a different pattern. The increase in inductance is nearly constant at low frequencies and for low conductivities, and gradually approaches the value corresponding to perfectly conducting earth at higher frequencies and conductivities. The variation with frequency and earth resistivity of elements in the impedance matrix [z] has different effects on the surge impedance matrix [Z01 and on the propagation matrices ['] or [-y'], equations 37 and 38. The most important is the propagation matrix because its eigenvectors are the mode components of voltages and currents associated with independent propagation constants. A general expression for the square of the propagation matrix [-y2 iS [^y] 2 = [z] [y] = (iw[L] + [R] ) (jw[C] ) (40) =(jw[Lc] +jw[ALe + [Rc]+[[ARe]) (jw[C]) Subscript "c" denotes matrices of the system of parallel conductors suspended above a perfectly conducting ground, whereas subscript "e" denotes correction-factor matrices derived, for example, by Carson's method.' All the component matrices in equation 40 are symmetric; therefore, the propagation matrix [e'], equation 38, is the transpose of [-y]. The elements of the propagation matrix [-] are complex and the analysis leads to complex eigenvalues and complex mode components (eigenvectors). However, the absolute values of elements of matrix jw [L] are normally much larger than those of the corresponding elements of matrix [R]. This applies to frequencies of several kilocycles and higher, where the simplification of analysis proposed by Hayashi' can be successfully used. An examination of equation 40 indicates that by neglecting matrix [RI, a set of independent modes of voltages and currents can be associated with constant velocities of propagation derived from the eigenvalues of matrix [L] [C]. Results of such a simplifled analysis of surges on a real line were recently published by A. J. McElroy and H. i\I. Smith.'10 The difference between the modal propagation velocities is caused mainly by the effect of earth on the inductance matrix [ALe], equation 40, and therefore, it is not surprising that the analytical results were found to agree

REFERENCES 1. WAVE PROPAGATION IN OVERHEAD WIRES WITH GROUND RETURN, J. R. Carson. Bell System Technical Journal, NeW YOrk, N. Y., vol. 5, OCt. 1926, PP. 539-55. 2. UBER DAS FELD LINER UNENDLICH LANGEN WECHSELSTROMDURCHFLJOS5ENEN EINFACHLEITU-NG, F. Pollaczek. Elektrische Nachrichten Technik, Berlin, Germany, 1926, vol. 3, no. 9, pp. 33959 (also in French translation by J. B. Pomey, Revue Generale de l,'tlectricitt, Paris, France, 1931, vol. 29, no. 22, pp. 551-67).

modes are characterized by specific attenuation. The mode voltage components, columns of matrix [A], are the eigenvrectors

Barthold. IEEE Transactions on Power Apparatus and Systems, VOl. 83, JUlY 1964, PP. 665-71.

well with experimental findings. In the analysis of Dr. Ushirozawa the mode components are associated with the eigenvalues of matrix [R] [C3, and hence the

NOVEMBER 1964

Ushirozawa

-Hitih-Frequency

0.571 921

0.396 511

LO. 571 921

0.396 511

[A]= 0.588 628 -0.828 719 0.585 634

0.707 107 0 -0. 707 107]

0.707 1071 0 0.416 291 -0. 707 107j 0.416 291

[B] = 0.560 415 -0.808 333

L0.585 634

W1 = 564 ohms; W2 = 386 ohms; W3= 429 ohms It is interesting to compare these values with the eigenvalues derived directly from the surge impedance matrix [Z0] assuming a perfectly conducting ground: 460.0 62.8 30.7 [Zo] = 62.8 460.0 62.8 ohms L 30.7 62.8 460.0]

Z(')=563 ohms; Z(')=385 ohms; Z(')=429 ohms

Has the aulthor any comments on the relation between Wj of a lossy line and Z(V) of a lossless line? Dr. Ushirozawa's analysis presented in his paper will greatly assist in the applications of analytical methods to the solution of many current important problems. His answers to my questions would be of further assistance.

3. SURGES ON TRANSMISSION SYSTEMS (book), S. Hayashi. DenkiShoin, Inc., Kyoto, Japan, 1955. 4. RADIO-FREQUENCY PROPAGATION ON POLYPHAsE LINES, L. 0.

on Nontransposed Power Lines

1143

k m

leSi

o 3~ ~ ~ ~ ~ ~ ~ ~ ~ 2Fig. 11 (left). Distance Snk between conductor m and the image of conductor k, and angle 0

0.5

_

A: \

03

f -'

_

21 3

0.0

\Mk

m77777r

0 ~~~~~~~~~~~~0.05

m'

k

Fig. 12 (right). Curve of p(yo) versus so

5. THE PROPAGATION OF HIGH FREQUENCIES ON OVERHEAD LINES, L. 0. Barthold, J. Clade. Progress Report 420-3, CIGRE, Paris, France, June 1964, pp. 19-39. 6. NATURAL MODES OF POWER LINE CARRIER ON HORIZONTAL THREE-PHASE LINES, M. C. Perz. IEEE Transactions on Power Apparatus and Systems, vol. 83, July 1964, pp. 679-86. 7. A METHOD OF ANALYSIS OF POWER LINE CARRIER PROBLEMS ON THREE-PHASE LINES, M. C. Perz. Ibid., pp. 686-91. 8. EXPERIMENTAL EVALUATION OF POWER-LINE CARRIER PROPAGATION ON A 500-KV LINE, D. E. Jones, B. Bozoki. Ibid., Jan. 1964, pp. 16-23. 9. A METHOD FOR COMPUTING THE HIGH-FREQUENCY PARAMETERS OF OVERHEAD ELECTRIC-POWER TRANSMISSION LINES, V. N. Orlov, V. V. Sidelnikov. Telecommunications and Radio Engineering, pt. I (Telecommunications), no. 7, July 1962, pp. 60-70. (English translation published by IEEE.) 10. PROPAGATION OF SWITCHING-SURGE, WAVEFRONTS ON EHV TRANSMISSION LINES, A. J. McElroy, H. M. Smith. AIEE Transactions Pt. III (Power Apparatus and Systems), vol. 81, 1962 (Feb. 1963 section), pp. 983-98.

f =4xlO

007-

0

0!o

0

-I

4 mho45k:

.\ 0203

0507!1

2

S >~ ~ ~ ~ ~ ~ ~ ~ 2=voQ -!\ 3 5 710

20

50

-4

Ranges of y calculated on the line configuration as shown in Fig. 2 in the paper are indicated in Fig. 12, if the frequency range is 50r450 kc and earth conductivity is 10-4_10- mhos/ meter.

Therefore, there is a limit to values of so depending on the frequency, earth conductivity, and geometrical arrangement of the conductors of a power line. I have attempted to approximate the P( p) curve, which is limited by the s value, to a straight line on both logarithmic co-ordinates. In accordance with that part of the P(sp) curve approximated to the straight line, resistance AR, in equation 41 happens to be proportional to freor quency f2ln which is C linefhici2

The proportional to f2/ in Fig. 12 was used in the calculation, and its calculated results were in good agreement with the experimental findings. As shown in Fig. 12, it appears that the earth contribution to attenuation is proportional to frequency at low frequency and for low conductivity of the earth, and proportional tof1'2at high frequency and for high earth conductivity. Therefore, I could not quite understand Messrs. Orlov and Sidelnikov's assertion that attenuation varies as f2 at low frequency and for low conductivity of earth. The resistance AR, is proportional to f5 in equation 15 in the Michihiro Ushirozawa: I wish to thank Mr. Perz for his paper. This means that exponent "x" of frequency f varies with valuable comments. It is very difficult to analyze propagation the frequency range and also with the value of the earth conproperties of high-frequency waves on a power transmission line ductivity. because of the variety of conditions affecting the propagation. Moreover, the method described may be applicable to the Therefore, as Mr. Perz points out, the methods now being correction factor AL, of the inductance, but also have identical employed, although based on the same fundamental principles, limitations. are simplified according to purpose by making certain assumpI agree with Mr. Perz in that j,i calculated from equation 17 tions, so that the use of the simplified methods is limited, dein the paper has little practical meaning because the effect of pending on earth resistivity and frequency ranges. AR, on fi is very small. Equation 17 is a formula for the deOriginally, I wished to devise an analyzing method to find termination of the multipropagation velocity of natural modes some coupling method appropriate for a power line carrier system when AL, is taken into consideration. Hence, it seems that on an EHV 2-circuit vertical power line which exists generally propagation analysis should be performed by stricter means. in Japan. Hence, I had analyzed the propagation characThis kind of analysis, however, complicates equations 29 and teristics by taking account of the line configuration and of the 34, which are used for seeking line attenuation values with power line carrier frequency range, which is from 50 kc to 450 kc various coupling methods; therefore, I have neglected AL, in in Japan, and then compared the results with experimental calculation. lines. on some obtained EHV power findings The results of experimental estimation on the Shiobara test Mr. Perz has asked for the method of computing values of line in Japan (3-phase 2-circuit power line, 1.4 km long) indicate the element in matrix [ARJ, equation 4 of the paper. that the propagation velocities of modes 2-5, which correspond By Carson's method, the general expression of the resistance to the modes shown in Fig. 3 in the paper, are approximately as a result of skin effect on the ground is 99% of light velocity; those of modes 6 and 1 are approximately and 88%, respectively, of light velocity. 93% (41) =4wP(yP) ARe Two-phase-to-ground coupling is mainly applied to the signal transmission circuit for power line fault locating equipment. (42) 2\/ rS2 /X10i =2 This coupling method is usually used in conjunction with transmission circuits for telephone or carrier relaying equipment in where Japan. In this case, if a line fault should occur at a point w = angular velocity that is approximately 50 km from the end of the line, it is known P(yo) = a function having variable so, that a high-frequency a-c pulse of approximately 400 kc sent erearth conductivity (mhos/meter) f= frqec k)from the locating equipment at the end of the line does not return to the sending end. I believe that this may be the result of the = distance between conductor m and the image of conductor difference between modal propagation velocities. k (see Fig. 11) Of course, the use of normalized eigenvectors, columns of 6 matrices [A] and [B], is preferable to the use of non-normalized Fig. 12 indicates a curve of P(9o) versus so when the angle ones. However, for the purposes of this paper, I felt it was in Fig. 11 equals zero. The maximum angle 6Em for the 2-circuit not necessary to employ normalized eigenvectors since the power line is less than about 30 degrees, and the PGco) curve is calculations were made with a digital computer. very close to that given when C equals zero.

ft

ARB

1144

Ushirozawa-High-Frequ<ency on Nontransposed Power Lines

NOVEMBER 1964

When

Mr. Perz also indicates that values of modal surge impedance

Wi are nearly equal to the eigenvalues which are derived directly

Mo-Mn#Xi

from the surge impedance matrix [Z0] assuming a perfectly conducting ground. An illustration for this is as follows: From equation 7 in the paper, modal surge impedances Wi of the lossy line can be shown to be IW il

Iw] V[ ' [ Y'i = [Z] / [,Y = [-y] -1[A] -1[Z] [B]

U2 2

=

(44)

[-y,2]

U22+ U3 2= 2U,I+ U22 =1 ~~~~~~~~~~~~~~~U12+

(45)

Mo-Mm=XA

(46)

U1+U3=0 U2=O

From equation 15 in the paper

=[C[ =g[L] =[L] ico

io

[at i-j3i] Jf3d

ri~ ail

[A] -qg[L] [B] = LI3 -- j

[RI [C] = [M]

LMn

(50)

U1', U2', and U3' of matrix [B] are equal to those when Mm' is converted into Mm, as in equation 49. Calculating the values of the elements of matrix [MI] in the example indicated in the paper,

F1.2065

[A] -'[Zo][B]

Since ca<
AIo Mm IMm' Mo'

(49)

Normalized eigenvectors

where g is the light velocity. Substituting into equation 45,

[Wi]

(2MmI/) 2

=

When

[Wi] =L yi] [A] -1[Y] -1[B]

jo

2(Mo'-X1)2+(2Mm')2

U1 =U3

Substituting into equation 43,

[y]

2

(43)

By using equation 8 in the paper,

[A] -1[Z] [yi2][A] -'[Y]

(Mot- X)

u2

Mn Mm'

Mm Mo FMo Mm' Mn

[C] [RI [M'] = IMm oM' Mm =

[MIl=1.0863

1.0148 1.1126

1.0148

0.9611 1.0863

1.2065]

XKX102

(51)

where K is constant, K=f /gXl0-2. From equation 51, Mm = 1.0148KX 10 -

Mm'= 1.0863KX102

(52)

The difference between values of Mm and Mm' is small. Therefore, the elements of both eigenvectors of matrices [A] and [B] are composed values close to each other.

This means that

(48)

LMn Mm' Mo As seen in equation 48, the difference betweeen [M] and [M'] results merely from exchanging Mm and Mn'. Normalized eigenvectors U1, U2, and U3 of matrix [A] corresponding to eigenvalues Xi of matrix [Z] [Y] are shown as follows:

[A]h[B] Hence,

(53) equation 47

becomes

(54) [W1] -[A] -1[Zo] [B] -[A] -lfZo] [A] -[Z(?)] It can be seen from equation 54 that Wi is nearly equal to Z(i). This, I believe, means that the value of resistance components [R] on a lossy line is very small compared with that of reactance components [coL].

Switching-Surge Insulation Level of Porcelain Insulator Strings D. E. Alexander,

Senior Member IEEE

E. WT. Boehne, Fellow IEEE

Summary: A quantitative research report is made here, based on tests of EHV (extra-high-voltage) insulator strings. Impulse, switching-surge, and a-c tests on vertical, horizontal, and 45-degree V-strings were carried out, using both positive

and negative unidirectional impulses and 250- to 3,250-uosec (microsecond) switching surges. The sw-itching-surge and a-c

test. Interesting and useful data pertaining to the design of EHV lines were collected. ______________________________

-Paper 64-38, recommended by the IEEE Transmission and Dis-

the late Dr. Motta, to whom the center is dedicated (see

tribution Committee and approved IEEE Technical Meettions Committee for presentation at by the the IEEE Winter Power Operaing, New York, N. Y., February 2-7, 1964. Manuscript submitted November 6, 1963; made available for printing December 5, 1963. D. B. ALEXANDER iS with the I-T-E Circuit Breaker Company, Victor, N. Y.; and E. W. BOEENE is with the I-T-E Circuit nreaker Company, Philadelphia, Pa.

inlsulator

1, 1963.

tet

an we wet unde une ASA(Ameican AS (America wer testsweremadeboth mae bot drry ad

Srtandlardls Association) specifications. Researchl was instru-

mented in the CESI high-voltage laboratory in Milan, Italy. The full name of this laboratory is: Centro Elettrotecniceo Sperimzentale Italiano Giacinto Motta, the last two words referring to

Figs. 1 through 3). The study included radio-influence (RI) measurements, determination of voltage distribution along the

strings, photographs., and cathode-ray records of each

NOVEMBER 1964

The material in this paper was originally presented as CP63-363 at the IEEE Winter General Meeting, New York, N. Y., January 27-February

Alexander, Boehwne-Porcelain Insulator Strings

1145