PROPAGATION OF TRAVELLING WAVES ON TRANSMISSION LINES- FREQUENCY DEPENDENT PARAMETERS

J.K. Snelson The Hydro-Electric Power Commission of Ontario

Toronto, Canada

Abstract-This paper describes an extension of Bergeron's Method of Characteristics that is applicable to the analysis of transmission lines with frequency dependent parameters. This method can be inserted in existing general purpose programs that use Bergeron's Method. The application to single phase and multiphase transposed lines is discussed with examples and the theory is developed for multiphase untransposed lines.INTRODUCTION

Bergeron's Method applies to lossless lines where R and G are zero and L and C are independent of frequency. Subject to these limitations there are relationships between the conditions at each end of the line at time t and at time t - T which exist independent of the terminating networks.o KI NSU RG E I MPEDANCE

eKfO _

TRAVEL TIM E

-

Z

eM

TERMINAL K

TERMINAL M

General purpose computer programs for the calculation of transients in electromagnetic systems have been written in recent years. Many of these use Bergeron's Method of Characteristics for the handling of continuously distributed transmission lines. The program by Dommel1 is an example which can handle large systems with a wide variety of components, including non-linear elements. In these programs the handling of lossless transmission lines is easy but it is difficult to deal with lines with frequency dependent resistance and inductance. Hedman2 gives curves of resistance and inductance against frequency for typical transmission line configurations which show that the resistance can increase by a factor of 103 as the frequency varies from 60 Hz to 1 MHz. In addition Battisson et a13 using a Fourier Transform technique showed that the frequency dependence of transmission line parameters has significant effects on both the wave shape and the peak voltages of some switching surges. It therefore seemed desirable to develop a method of including frequency dependence in the general purpose programs based on Bergeron's Method. Budner4 described a method for doing this which has the disadvantage of considerably increasing the computation and storage requirements. The method described here can easily be included in programs based on Bergeron's Method and should require much less computation time and storage than the method of reference 4.

Fig. 1. Single Phase Lossless Line Forethe line of Figure 1 these relationships are:

ek(t)em(t)

-

Z@ik(t)

= em(t-r) + =

Z*im(t=T)Z.ik(t-7*)(2)

-

Z.im(t)

ek(t-r)

+

Equation 2 gives relationships between e and i at both ends of the transmission line which, provided the conditions a travel time earlier are known, enable the transmission line to be replaced by a current source in parallel with a resistance Z. This allows a solution to be obtained for the voltages and currents at time t in the network consisting of the ends of the transmission lines and the components connected to them. A method for solving this network is described by Dommell. The quantities eZ.i are known as characteristics and are directly related to the forward and backward travelling waves. In this paper the following nomenclature is used:

Fk = ek + Z ik = 2 x forward travelling wave at end k Bk = ek - Z ik = 2 x backward travelling wave at end k Equation 2 can be rewritten:(3)

BERGERON'S METHODA brief outline of Bergeron's Method is given here and ideas are introduced that will be used later in the paper. The theory of Bergeron's Method is developed in more detail by Dommel1. The transmission line equations are:-

Bk(t)B(t)

=

F

(t - r)(4)

Ox

-e

=

L A

&t

+ Ri

=

Fk (t -r)

_0g

ax

=

C

---

Ot

+

Ge

(1)

iK

R/4Z

R/2

R/4

;M

eKfwhere e and i are voltage and current in the line at a distance x and L,C,R and G are respectively the line series inductance, shunt capacitance, series resistance and shunt conductance per unit length.Paper 71C 26-PWR-IV-B, recommended and approved by the Power System Engineering Committee of the IEEE Power Society for presentation at the 1971 PICA Conference, Boston, Mass., May 24-26, 1971. Manuscript submitted January 11, 1971; made available for printing June 9, 1971.

0o4

j-4j

OM M----o

Fig. 2. Approximate Model For Line With Series Resistance

DommelI extends the basic equation (4) to include an approximation for series losses. The model used for the transmission lines is shown in Figure 2. This results in equations:85

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B

(t)z

FM(t

-

7)

+

Fm(t-r)

Bm)=7 Bmt Z+R/4

R4 Fk (t -r) + Z+R,'4

(5)

0.1

In these equations the impedance used in defining the characteristics of equation (3) is modified to (Z + R/4). Equation (4) is a particular case of equation (5) where R is zero. Physically equations (4) and (5) can be interpreted in terms of impulse responses in the following way. For the lossless case described by equation (4), if an impulse of forward (ie, into the line) travelling wave is injected at one end at time t = 0, then the backward (ie out of the line) travelling wave at the other end at time r is equal to it. For the transmission line model with series losses of Figure 2 and equation (5), if an impulse of forward travelling wave is injected at end m at time t = 0, then part of it is transmitted and contributes to Bk(r) and part of it is reflected by the resistor at the centre and contributes to Bm(r). The responses of Bk and Bm are shown in Figure 3.

WI

0

fI

(B)

BACKWARD

RESPONSE

0.010

I

TIME IN MILLISECONDS

2

3

A=

50

,lSEC

Fig. 4. Response Functions of Frequency Dependent 200-Mile Line

BK

t

of impulses must be carried out many times, once at each time step, but is only a simple calculation and so should not greatly increase the computation time. DERIVATION OF IMPULSE RESPONSE FUNCTIONSThis section derives from the transmission line equations expressions for the impulse response functions discussed in the previous section. As frequency dependence is included the problem is transformed into the frequency domain using Fourier Transforms. For the purposes of this paper the Fourier Transform of time function X(t) will be denoted by X (co). The Fourier Transform of equation 1 is:

0

(A)

FORWARD RESPONSE

T

_ TIME

Bm(B) BACKWARD RESPONSE

t

70 _r-- TIM E---Io-

I Mf m

Fig. 3. Response Functions For Lumped Resistance ModelThese impulse responses are approximations to the impulse responses of a line with continuously distributed resistance, and frequency dependent resistance and inductance. If the representation of continuously distributed resistance was improved by including a large number of smaller lumped resistances connected by short lossless transmission lines, then the reflection from the resistance in the centre of the line shown in Figure 3b would be replaced by many smaller reflections. These reflected pulses would not arrive only at time r but would occur over a range of time. The reflections from resistances near the sending end would start arriving soon after time zero. The reflections from resistors near the far end would arrive shortly before time 2r and would be smaller as the pulse will be attenuated as it must pass down the line twice. If the effects of frequency dependence were to be included then the travel time and attenuation of different frequency components would be different and this will further modify the shape of the responses. The principle effect would be, that the pulse of Figure 3a would be broadened. These effects result (as shown in the next section) in responses of the form shown in Figure 4. With responses of this form Bk(t) and Bm(t) are no longer determined by Fk(t - r), and Fm(t - r) alone. However, if Fk and Fm are considered to be made up of a series of impulses of varying amplitude but all of a duration At then Bk and Bm can be found by summing the effects at time t of a number of these impulses. The shape of the responses would only need to be determined once for each line and so could be a complex calculation without greatly increasing the computation time. The summing of the effects of a number 86

-(redx

=

jwLi.,JwOe

+

-i.i (6)

=

+

The complex surge impedance and propagation constant are defined by:Z=

s(R

+

jwL)/(jcC

+

G)

(7)

'Y

=

4(R+ jwL) (:JwC+ G)

(8)

The solution to equation 6 is known to be:e

=em

osh ( Yd) -

%Z sit+

(Yd)

ik = -

cosh (Yd)

i;/Zsinh(Yd)

(9)

determine the value of ZI to use in the definition of characteristics (equation 3). ZI must be the impedance the line represents to the outside network at the instant an impulse is applied. This response will be governed by the high frequency effects. Therefore we define:

as variables to forward and backward characteristics it is necessary to

Before a transformation can be made from currents and voltages

ZI

=

W- 0 o

is (Z)

(10)

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This limit exists for transmission lines because as c tends to infinity, Rav/Jand L tends to a constant Loo

The time step of u is chosen to be At (the time step of the main calculation) and so equation (15) becomes:

Therefore

k: j = 29lim

+ 0w L.

=

(11)

B/t) = E W1(n At)

(t

-

n At)

Z1 is a real number and so represents a pure resistance. Using Z1 as the surge impedance we define the transform of the characteristics as:Fk=

nnxln=n+

ek

+

Z1

k(12)

>

W2(nAt) Fk(t

-

not)(16)

Bk%

-

Zl Tk

n1=1

and similar expressions for Fm and Bmi The variables ek, ik, em Tm can be eliminated from equations 9 and 12 to give:

n92B

= t

(t)

=

EW1 (n At) Fk(

BkBwhere

A1 m i F

A2 Fk 2 k

= A1

Fk+ 2F

(13)

+

>

n-n3W2(nAt) Fm(t-

nAt)

1 =c

n=1where W1 = A1IAt and W2 = A2At. The choice of limits, n1, n2 and n3 is discussed in Appendix I. Equation (16) can replace equation (4) or (5) of a program based on Bergeron's Method and the rest of the program can remain the same. In developing these expressions no assumptions have been made as to the nature of the terminating networks and so this change need not restrict the generality of the program. This equation is the summation of the effects of impulses originating over a period of time that was suggested in the previous section. Provided the line parameters can be found at any frequency W1 and W2 can be calculated using equation (14) and a numerical inverse Fourier transform technique. Figure 4 shows the response functions of a typical 200-mile line calculated using these expressions. These show the general features predicted by the general argument of the previous section.

A

c - Y/2(Z1/Z - z/z1z) sin A~~~~~ Yd) + 1J2-/ eihi(Yd) YVd - - d)2 cosh Z1/Z + Z/z1 sinh-

(14)Equations 13 are in the frequency domain. To transform into the time domain it is necessary to remember that multiplication of functions in the frequency domain corresponds to convolution of the transformed functions in the time domain. Transformed into the time domain equations 13 become:

Bet) = fAi(U) Fm(t)k

-

u) du

+

fA2(u) Fk(t -)2

t

co

B3(t)

(u) Fk(t - u) du + f A2(u) Fm(t - u) du

,co

(15)

APPLICATION TO SINGLE PHASE LINESThe equations as developed in the previous section can be applied directly to single phase lines. There are two main difficulties, the determination of the line constants at a wide range of frequencies and finding the numerical inverse Fourier Transforms of the weighting functions. The line constants may be calculated using the formulae developed by Carson5, the graphs published by Hedman2 or they may be measured. The method does not require the variation with frequency to follow any particular analytical form. However, care must be taken to ensure that the constants are physically realizable or else the response functions may show physically impossible features. The results quoted in this paper were all obtained using the technique of Modified Fourier Transforms6'7'8 to obtain the inverse Fourier Transforms of the weighing functions. A brief discussion of this technique as used in this paper is given in Appendix I. Figure 5 shows the response of a 200-mile open ended line with zero shunt conductance and frequency dependent series resistance and inductance to a step function. The line constants used were calculated with a program based on Carson's method for the configuration of a 230-kv line when excited in the ground mode. This shows an initial response approximating to a square wave which rapidly changes to something approximating a damped sine wave.87

The significance of A1 and A2 can be seen by examining equation (13). If Fk is zero ie, ek + ik ZI = 0 and Fm is a Dirac impulse function then the Fourier transforms of Fk and Fm are 0 and 1 respectively. Then from equation (13) Bk = A1 and B. = A2. This gives in the time domain Bk = A1 and Bm = A2 Therefore Al is the backward characteristic at the remote end of the line and A2 is the backward characteristic at the sending end due to an impulse of forward characteristic. Certain properties of the response functions A1 and A2 can be deduced from this. For real systems no response can occur before the excitation is applied; therefore, A1 and A2 are zero for all negative time. There can be no response in a real system at the remote end at a time less than the length of the line divided by the speed of light. Therefore A1 is zero up to this time. The condition that ek + ikZ 1 = 0 implies that the line is terminated in an impedance ZI, the surge impedance. Therefore, the injected impulse is not reflected from the end and the response at the remote end consists of a single pulse. These considerations enable the infinite integrals of equation (15) to be replaced by integrals over the short time that A1 and A2 are sensibly non zero. In addition for computation purposes it is convenient to approximate the integrals by summations.

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aS

Z

(2 1-I 0

S

Fig. 5. Step Response of 200-Mile Line16 mH200 M ILE_ O _

TRANSMISSIONLINE

-

O

B

C

line parameters. This matrix diagonalizes such matrices and separates the problem into a number of modes. These consist of a ground mode and (m-i) line modes (where m is the number of phases). The line modes all have the same parameters which are relatively independent of frequency. The ground mode involves currents returning through the earth and its parameters are highly frequency dependent. With this transformation each mode can be solved as a separate single phase problem using the equations developed for the single phase case. As it is principally the ground mode that is frequency dependent it is a reasonable compromise between accuracy and computation time to consider the ground mode frequency dependent using equation ( 16) and to neglect the frequency dependence of the line modes using equations (4) or (5). The program developed by Dommel1 has been extended to include continuously transposed lines where the ground mode or both the ground mode and the line modes are frequency dependent. The modification has been done without restricting the generality of the original program. This modified program has been used for a number of studies. As an example of the use of the program Figure 6 shows the single phase energization of a 200-mile, 230-kv line from a source of 16 mH inductance both with and without frequency dependent ground mode parameters.

(A)

SYSTEM REPRESENTED

APPLICATION TO UNTRANSPOSED MULTIPHASE LINES The theory developed'so far cannot be directly applied to untransposed multiphase lines as modal transformations will be different at each frequency and so the modal transformations must be included in the frequency dependent part of the calculation. However, the ideas used in the single phase case can be applied to the multiphase untransposed case. Appendix II develops, using matrix algebra, the equivalent expressions for this case. This results in equations of the same form as equation (16) except that the quantities must be interpreted as matrix quantities. However, the amount of computation required to calculate the matrix response functions will be many times larger than for single phase or transposed lines. There is no theoretical reason why this theory should not be implemented but so far it has not been considered that the gain in accuracy of representation justifies the programming effort involved.

Z

PHASE A PHASE B AND C---

W EL

(C) FREQUENCY*

*(60 HZ VALUE_S)

PARAM ETERS

INDEPENDENT

COMPARISON BETWEEN NEW METHOD AND BUDNER'S METHOD

Budner's Method4 is mathematically similar to the method presented in this paper. It differs, however, in that equation (9) in the frequency domain is manipulated into the following form:0A

k

=

Y

+

Y2

em

im = Y101

em

+

Y2 ek

(17)

2

3

a 6 7 5 TIME IN MILLISECONDS4

9

10

I

12

Fig. 6. Single Phase Energisation of 200 Mile Line

APPLICATION TO CONTINUOUSLY TRANSPOSED MULTIPHASE LINESAn approximation which greatly simplifies the study of multiphase lines is to assume the line to be continuously transposed. In this case the series impedance and shunt conductance matrices both have equal diagonal elements and equal off diagonal elements. Dommell gives a simple transformation matrix which is independent of the actual88

The admittance functions Y I and Y2 are then transformed into the time domain to give a relationship expressing the currents in terms of convolution integrals of the admittance functions and the voltages. The physical interpretation of the admittance functions Y1 and Y2 is that they represent the transform of the currents at the sending and receiving ends when an impulse of voltage is applied with both ends subsequently short circuited. When transfprmed into the time domain these functions have many peaks as the pulse is reflected up and down the line by the short-circuited ends. Eventually the pulse dies away as it is attenuated by the losses in the line. Therefore, the range of time during which these admittance functions are non zero is many travel times of the line. This means that the admittance functions in the time domain can only be adequately described by a large number of points. Budner uses 2048 points in his paper for his numerical example. This number could

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possibly be reduced a little if the frequency dependence of the line mode were ignored. However, the number of points would still have to be large to describe the multiple peaks adequately. The method used in this paper uses in effect the impulse responses of the line when terminated in it's surge impedance at both ends. This prevents reflections and ensures that the response at each end consists of a single pulse which generally lasts less than two travel times of the line. These single pulses can be adequately described by relatively few points. The example shown in Figure 6b used 48 points to describe each response function. The two methods both solve the same equations by mathematically correct techniques. However, the new method is more convenient to numerically compute because the response functions can be represented by fewer points. This reduces the computation time as fewer points must be found by numerical inverse Fourier transformation and fewer multiplications are required to perform the convolution at each time step. It also reduces the amount of information that must be stored in the computer memory as fewer locations are needed to store the response functions, and fewer locations are needed to store the history of the line voltages and currents.

A useful check on the accuracy can be made for lines with zero shunt conductance. In this case a line with unit trapped charge in the steady state should remain charged with no change. In this case all currents are zero and all voltages are unity for all time. For these conditions equation (16) reduces to:n=E_

1

=

E

W(n,&t) ,

+ .2 w2(nAt)

(18)

n--=The difference between the sums of the response functions and unity is a measure of the error of the numerical inverse Fourier Transform calculation. For the cases tested with the values specified above, the error has usually been less than 1 per cent. There is always a small error as there are many approximations in the numerical calculation of W1 and W2. This error, if it is allowed to remain, can lead to trapped charge either growing or decaying exponentially over many time steps. This problem can be overcome for zero shunt conductance lines by scaling the response functions to make their sum equal to 1. An alternate technique might be to use the Fast Fourier Transformer; however, in this application it would have certain limitations due to the requirement that the number of points in the time domain must equal the number of points in the frequency domain. In calculating the response functions the number of points needed in the time domain is governed by the number of points needed to adequately define the response function. As the response function is a single pulse it can be defined by relatively few points eg, 48 points were used for the example shown in Figure 6. However, to obtain adequate accuracy in the calculation of these points requires the consideration of a large number of points in the frequency domain, eg, in the example shown in Figure 6, 500 points were used. If these numbers of points had to be equal as in the Fast Fourier Transform then either the number of points in the frequency domain must be reduced with a consequent loss of accuracy or many more points than necessary must be calculated in the time domain. In using the method described in this paper the value of the response functions must be known at intervals of one time step of the main calculation for one or two travel times of the line. When the time step is small, this, may lead to a much larger number of points being required than are needed to adequately describe the response functions. In this case the program can be arranged to calculate some of the points by numerical inverse Fourier transformation and to interpolate between these points to find the other points needed.APPENDIX II

CONCLUSIONSAn extension of Bergeron's Method of Characteristics to analyse frequency dependent lines has been presented. This method is relatively easy to insert in existing general purpose programs using Bergeron's Method. The application of the method to single phase and transposed multiphase lines has been discussed with examples of its use and the theory has been extended to cover untransposed multiphase lines.

ACKNOWLEDGEMENTS The author wishes to thank the Bonneville Power Authority for making the original program available.APPENDIX I

Method of Performing Numerical Inverse Fourier TransformThe results quoted in this paper were all obtained using the technique of Modified Fourier Transforms. This technique was developed by Day et al6978. Here the convergence of the Fourier Integrals is accelerated by the use of a complex frequency. In using the method certain parameters must be specified. These are the maximum angular frequency, the angular frequency step size and the real part of the complex frequency. Day et al6,7T8 investigated these factors in some detail. For this paper the following values have been used which follow closely the values recommended by Day et al.

Response Functions of Untransposed Multiphase Line

Maximum angular frequency= 40/r Angular frequency step size = 40/(500r) = 11(1.5r) Real part of frequency These values seem adequate to handle ground modes of propagation where high frequencies are considerably attenuated. For line modes where the frequency dependence is slight a considerably higher maximum angular frequency would be required. The range of time for which the response functions must be calculated depends to some extent on the degree of frequency dependence of the parameters. However, the response function at the receiving end should not be required outside the range 0.95T - 3r and at the sending end should not be required outside the range 0 - 2r (ie, in equation ( 16) nl need not be less than 0.95T/At, n2 need not be greater than 3r/At and n3 need not be greater than 2r/At).89

Wedepohl9 performed the analysis of untransposed long m-phase lines in the steady state. The following analysis uses many of the results of that paper. The transmission line equations can be written in the frequency domain as:

-2

=[RG

+

JL]jwc

[GR

+

C J