Session 8A4

Series-Impedance Of Overhead Transmission Systems

Abstract: This paper is a contribution to the series- -impedance evaluation of overhead transmission sys- tems. A generalization of Carson's solution is devel- oped by including the effect of overhead cylindrical conductors having finite outer radius.

Introduction

This paper is a contribution to the development of methods for series-impedance evaluation of overhead cylindrical cables and overhead transmission lines that take into account skin effect in the soil and the proximity of the earth/air plane boundary.

The classical approach is due to Carson [l] where use is made of filiform current distributions to model overhead conductors above a homogeneous earth having finite conductivity with neglect of the displacement current density inside the soil - quasi steady state solution. This type of solution has been used for overhead lines [4, 51 and for overhead cables [g].

The discussion of the quasi steady state assump- tion by several authors [6, 7, 81 shows that, for working frequencies below the range 1 - 10 MHz, Carson's approach is usually accurate enough for typical values of the earth conductivity.

In the present paper, a generalization of Carson's solution is developed by leaving aside the assumption of a filiform current distribution in overhead conduc- tors and assuming cylindrical overhead conductors of finite outer radius. As for Carson's solution the Droximity of the earth/air plane boundary is taken into account and a homogeneous earth with finite con- ductivity and the quasi steady state conditions are assumed. The self-impedance of the earth return path and the error in Carson's approach are evaluated. It is shown how the above error depends on the field penetration depth inside the soil, the height of the conductor above the soil and the conductor radius.

Earth Field Determined by a Finite Radius Overhead Cylindrical Conductor

The solution for a finite radius cylindrical conductor above earth (Fig. 1) will be established assuming :

P I x

earth Fig. 1 - Cross section of an overhead finite

radius cylindrical conductor.

- earth and air fill semi-infinite half-spaces, the

V. M. Malo Machado and J. F. Borges da Silva Centro de Electrotecnia da Universidade Tecnica c'e Lisboa lnstituto Superior Tecnico Department of Electrical Engineering 1096 Lisboa Codex, Portugal

earth/air surface being a plane (Ss). L-rth and air are both linear and homogeneous media, the air being of permeability uo and the earth being of finite conductivity os and permeability ks.

- the displacement current density is neglected for all space and the field is assumed to depend only from the transversal coordinates.

- sinusoidal time dependence of angular frequency w is assumed for the field. Inside the cylindrical con- ductor an axial current is assumed with an intensity given by its phasor I. The phasor of the earth current, Is, is therefore

I = - I . (1)

Under these assumptions, an expression for the transversal coordinate dependence (x,y), Fig. 1, of the magnetic vector potential axial component, A, inside the earth may be built in a way similar to the one developed in [ Z ] as an integral in the real varia- ble 2 , regular for x + - -

+ m

A(x,y) =lm F(a).exp[xm+jay] da , x < 0 ( 2 )

Re [m] < 0 .

In tke above expression the function F(a) is to be determined by imposing boundary conditions on the solution, q being a complex parameter given by

q = (fi/ds).exp(-jn/4) , bS = (wusos/2)-1/2 . (3)

where 6, is the field penetration depth inside earth at the given working frequency.

The corresponding solution for the air may be built by adding two linearly independent terms [2]

A(x,y) = A'(x,y) + A''(X,y) ( 4 )

The first term, A ' , takes care of boundary condi- tions on the earth/air surface, Ss, and will be written

A'(x,y) = / U(a).exp[-lalx+jay] da , x > 0 (5)

and the second term, A", with singularities located at the conductor axis, will take care of boundary condi- tions on the conductor surface, Sc, and is written, to this end, as

A " ( r , Q ) = A;(r,@) ( 6 )

with A ( r , Q ) = E (r).exp(jp@) , A. = uoI/(Zn) , (6')

where ( r , ~ ) represent polar coordinates centered on the conductar axis (Fig. 1) and

+-

-a

+ m

p=- -

c r-lp'/2 1 PPO E (r) = (7) I CO ln(l/r) , CO = 1 , p=o .

In the above expression the coefficients cp, p= 21. t2, ... may be adjusted to have the solution satisfy the imposed boundary conditions.

670

The terms of the above series (61, A;, in rectan- gular coordinates, taking the Fourier transform with respect to y (see Appendix l), may be obtained by where is given by (9,) using the following expression

Tm,p = a ( m ) a(p) (17)

To complete the description of (151, the integral for the Im,p(~o,Ro) factor is given below (see Appen- dix 2)

I ~ , ~ ( x ~ , R ~ ) = -& /' b k ( b + ~ 1 . e x p ( Z b / R o ) d b . ( 1 8 1

+ " A;((x-h,y) =La Gp(a).exp[la((x-h)+jay] da (8)

where (x-h) < 0 was assumed in order to impose bound- ary conditions at the earth/air surface, x=0,

Ro xo -_ -cp.a(p).[-la~l('p'-1)/2 , ap6 0

G (a) = (9) 0 , ap> 0 where k=)ml+lpl. and, the arguments, see ( 3 ) and

Fig. 1. are (19) xo = qrc , R = rc/h [ ( I1 1-1 ) !I * C Z O

where a ( 4 ) = I ( 9 ' ) I 1 , 1=0. respectively giving a measure of the skin effect

inside the conducting earth and the proximity effect of the earth to the conductor with a finite radius rc

the earth/air surface, S s , are given by at height h above earth/air surface. The integrals (18) may be evaluated using the series development indicated in Appendix 3 .

The boundary conditions for the magnetic field at

1 A(O-.y) = A(OC,y)

(10 )

Using (2). (4), ( 5 ) , (6) and ( 8 ) . they lead to a result which will be expressed in the form

A(x,y) = Ab(x,y) + As(x,y) , x > 0 (11)

where Ab(x,y) is the field in the air due to the current distribution in the cylindrical conductor and its image on the earth/air separation plane assuming a perfectly conducting soil, i.e., taking into account ( a ) , A is given by

and A,(x,y) correspondes to additional contributions due to the imperfectly conducting soil

. exp[-la( (X+h)+Jay] da . (13') In order to impose boundary conditions at the con-

ductor surface Sc it is necessary to formulate (13) in polar coordinates centered on the conductor axis (see Appendix Z ) , by using a Fourier series development for A in the angle 0 around the conductor axis

where R and Cp are respectively the normalized radial coordinate (Fig. 1) and the normalized coefficient of order p (7). defined by

The term Ab(x,y), given by (121, may be obtained in polar coordinates in the form of Fourier series development in a way similar to that described in [lo] for N=2 (two cylindrical conductors) with symmetric current distribution inside the conductors, or, in general, using the procedure described in [ll] .

The phasor of the magnetic vector potencial axial component, A, inside the conductor is given by:

where (r,0) are the polar coordinates around the con- ductor axis, n is the voltage drop per unit length of the conductor along the axial direction taking the conducting earth as reference [3] and Jm(qcr) is the Bessel function of first kind, order m and argument

pc and ac being respectively the conductor permeabili- ty and conductivity, and O m , m=O, fl, i 2 ,... , are coefficients to be determined by imposing boundary conditions at the conductor surface Sc.

(Fig. l), may now be expressed as Boundary conditions mentioned above for r=r

Using properties of Fourier series developments, the boundary conditions (221, considering the field in rhe air given by (11) in polar coordinates and the field inside the cylindrical conductor given by (20), lead to:

Quantities appearing in eq. (23) are defined as 1/12ml , mp >,os m + 0

, m = O , p # O (16' ) follows: , m = O , p = O and 5 = I :I2 l o , mp<O

as found in Appendix 2. Further, we have

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with B = (-1 )m+p-l [ (m+p-l) !]a (m a(p )/2(m+P) ( 2 6 )

where a ( i ) is given by ( 9 ' ) , and with Tm,p and Im,p(k,Ro) expressed respectively by (17) and (18).

m,p

In equation (23') - ko = ln(2/Ro) + Io,o(~o,Ro) +E cP so,P' (27) p=l

So,p being given by (25) for m=O.

The coefficients C,, m=1,2,. .. are evaluated im- posing the boundary conditions (22). Numerical evalu- ation requires truncating the series development at; the Mth term. In this way, the formulation described by (23) becomes a system of M independent linear equations in C, to be solved using Gauss's elimination method. As a consequence of geometric system symmetry relative to the y=O plane. Fig. 1, we find that

c-m = Cm' m=1,2, ... . ( 2 8 )

Series-Impedance

The series-impedance, Z , of the transmission system is defined as in [3] by

n = z I (29)

where n, I are respectively the phasor of the voltage drop per unit length along the axial direction and the phasor of the current flowing in the cylindrical conductor. The definition (29) means that 2 may be evaluated from equation (23') taking into account (27) where the coefficients Cp are determined by solving the system of equations (23). The series-impedance Z is usually assumed to be decomposed into terms:

(30)

where Zps is the series-impedance for the case of a perfectly conducting soil neglecting the proximity effect of the earth/air surface,

2 = 2 + z + z PS s P

Zic being the internal impedance of the cylindrical conductor, Z, is the contribution due to the presence of the soil with finite conductivity neglecting the proximity effect of the earth/air surface

Zs = j w p Io, ( xo, R o ) / ( 2 n ) (32)

and Z is the correction term corresponding to the proxi:ity effect between the cylindrical conductor and the earth/air surface, taking into account the presence of a conductor with finite radius

(33) cP so,P. z = j w p o / ( 2 T )

The result for Carson's approximation may be obtained from this general solution, by taking Zp = 0 as a consequence of Cp = 0, p=1,2,. .., being valid if Rolxol << 1, and noting that Io,o~xo,Ro~, given by (le), is equivalent to Carson's integral from [I]. This means that Carson's solution is an approximation to this general solution obtained by taking only the zeroth order term of the Fourier series development around the conductor axis.

p= 1

Numerical Results

Numerical results were obtalned for a perfectly conducting cylindrical conductor. The system of eqs. (23) was evaluated taking the limit for Q, (24)

lim Qm = 1, ( 3 4 ) lqcrcl + -

and considering M=10 at most in order to attain suitable accuracy in the majority of practical situations.

Figures 2 and 3 show how the series-impedance depends on the soil skin effect and the proximity of the earth/air surface.

The skin effect parameter being defined by

Xo = (xo( =firc/6s. (35)

where xo is given by (19) and ( 3 ) . In order to preserve the quasi-steady state assumption, valid for a working frequency below the range 1 ~ 10 MHz [6], the upper limit Xo=l was taken and, typical values for the earth conductivity assumed.

The proximity effect parameter, R,. is defined by (19). The series-impedance is represented by the resistance per unit length, R s , and by the inductance per unit length, L

Rs = Re[Z] , L = Im[Z]/u (36)

which are displayed as p.u. values using respectively as units the steady state resistance per unit length of a cylindrical conductor with radius rc and with conductivity equal to the earth conductivity a s , RdF, and the steady state internal inductance per unit length of an isolated cylindrical conductor, Lio.

a) b)

Fig. 2 - Impedance elements (36) versus X (35) taking R (19) as a parameter for an o?erhead per- f8ctly conducting cylindrical conductor.

.

20

15 0

4- -1

10

5 0 0.4 0.8

a) b)

Fig. 3 - Impedance elements (36) versus R (19) taking X (35) as a parameter for an o?erhead per- f& tly conducting cy1 indrical conductor.

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It is found that the resistance increases and the inductance decreases as the skin effect in the soil becomes stronger, i.e.. when the field penetration depth inside the conducting earth decreases.

On the other hand, the proximity effect tends to confine the field configuration and thereby determines a tendency of the resistance to increase and of the inductance to decrease. The behaviour of the inductance with respect to the proximity effect, is due, mainly, to the logarithmic term of (31).

a) b)

Fig. 4 - Resistance (a) and inductance (b) errors in Carson's approach versus X (35) taking R (19) as a parameter for an gverhead perfgctly conducting cylindrical conductor.

The error in Carson's result for an overhead per- fect conductor of negligible cros-section, concerning the resistance of the earth return path, ER. is repre- sented in Fig. 4-a), and the error for the inductance, EL, is represented in Fig. 4-b). The errors vanish for almost steady state conditions ( X * 0) and in the absence of the proximity effect, i.e., for an almost isolated cylindrical conductor (rc <<h).

From numerical results it may be concluded that Carson's classical solution may be applied with less than 1% error for the series-impedance value if the condition,

X R < 0.08, (37) 0 0

combining the skin effect and proximity effect param- eters, is verified.

Conclusions

A generalized solution for the field determined by an overhead cylindrical conductor of finite radius was established assuming the quasi steady state hypothe- sis, taking into account the field penetration in the soil and the presence of the earth/air plane boundary.

Numerical results were obtained for a perfect cy- lindrical conductor above earth for frequencies below the range 1 - 10 MHz. It was shown how the series- -impedance depends on the field penetration depth in- side the soil, the height of the conductor above earth and the conductor radius.

Comparison was made with results obtained from Carson's solution. Results presented in this paper may be used to check the accuracy of this classical solution when used to evaluate the series-impedance of overhead transmission line systems. It is found that for GRo 4 0.1 the classical solution may be used-with an error for the series-impedance less than 1%. It is therefore possible to evaluate the series-impeda6ce matrix of overhead transmission lines [4] and overhead cables [9] in the majority of practical situations with enough accuracy by simply using Carson's classi- cal solution.

Acknowledgements

The authors wish to acknowledge the support given to this work by the Instituto Nacional de InvestigaSHo Cientifica.

References

[I] - J.R. Carson, "Wave propagation in overhead wires with ground return", Bell System Tech., Vol. 5, pP. 539-554, Oct.1926. V.M. Machado, J.F. Borges da Silva, "Series- -impedance of underground transmission sys- tems", IEEE Trans. on Power Delivery, Vo1.3, NQ2, pp. 417-424, April 1988.

V.M. Machado, J.F. Borges da Silva, "Series- -impedance of underground cable system", - Trans. on Power Deliverx, Vo1.3, Ne4, pp.1334- -1340, Oct. 1988.

A.Deri, G. Tevan, "Mathematical verification of Dubanton's simplified calculation of overhead transmission line parameters and its physical interpretation", Archiv fur Elekt., Vol. 63, pp.191-198. 1981.

A. Semlyen, D. Shirmohammedi. "Calculation of induction and magnetic field effects of three ohase overhead lines above homoeeneous earth", IEEE Trans., Vol. PAS-101. NQ 8, - pp. 2747-2754, Aug. 1982.

L.M.Wedepoh1. A.E. Efthymiadis, "Wave propaga- tion in transmission lines over lossy ground: a new, complete field solution", Proc. IEE, Vo1.125, pp. 505-517, June 1978. A. Semlyen. "Ground return parameters of trans- mission lines, an asymptotic analysis for very high frequencies", IEEE Trans., Vol .PAS-100, NO 3, pp. 1031-1038, March 1981.

R.G. Olsen, T.A. Pankaskie, "On the exact, Carson and image theories for wires at or above the earth's interface", IEEE Trans., Vol.PAS- -102, NQ 4, pp. 769-778. April 1983.

[g] - V.M. Machado, J.A. Brandso Faria, J.F. Borges da Silva, "Ground return effect on wave propa- gation parameters of overhead power cables", IEEE/PES 1989 Transmission and Distribution Conference, New Orleans, 89TD 357-5 PWRD, April 1989.

P.G. Heyda, "Electromagnetic induction in a system of conductors carrying alternating currents", Proc. IEE, Vo1.113, NQ8, pp.1373- -1375, Aug. 1966.

V.M Machado, PenetraGHo do campo electro- magnCtico em sistemas de condutores cilindricos circulares, Ph.D. Thesis, Instituto Superior TBcnico - U.T.L., Lisboa, 1987.

G.N. Watson, A treatise on the theory of Bessel functions. Cambridge University Press, 1966, 2nd edition. A. Sommerfeld. Partial differential equations in Physics. New York: Academic Press, Inc., 1949, pp. 84-123.

Appendix 1

The pth term of the Fourier series expassion (6) and (6') using the Fourier transform with respect to y. may be written as

+ - (a,x) exp(jay) da (38)

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where, taking into account (7), for p > 0

jcp = =!, w-' exp[aw-a(x-h)] dw (39)

making w=(x-h)-jy and -!being the path of Fig. 5, for x-h < 0.

Im

closed

for a > O I \for a < 0 path 4

Fig. 5 - w complex plane. Applying the theorem of the residues and Cauchy's

theorem, the following result is obtained

G (a,x) = G (a) exp[lal(x-h)] (40)

where - c .aP-'/(p-l)!/~ if a < o

G (a) = (41) I o if a $ O

Using a similar procedure for p < 0, the result expressed by (9) is obtained. The result for p=O can be found by integration using the result for p=-1 and remembering that

2 ln(l/r) = ln(l/w) + ln(l/w*) , w = r exp(j0) (42)

Appendix 2

Equation (13) may be obtained in cylindrical coordinates around the conductor axis by using the definition of the m th coefficient of the Fourier series development (14). taking into account (9) and

(43)

where a(p) is given by (17'). Making the substitution (x-h)+jy = r exp(j+), the integral form Ym(a.r) is expressed by

Y,(a,r) = &lnexp[ar exp(ju@)-jm+]d+ for p#O (44)

where u=+l if p > 0 and U=-1 if p < 0.

II

The exponential function may be decomposed into a power series development convergent in the neighbour- hood of the origin, giving the result for Y-(a,r)

Following a similar way for p=O, the result (16) is obtained.

Appendix 3

The integral form (18) may be evaluated by using the substitution referred to in [Z] :

(46)

Choosing the appropriate integration path, follow-- ing a way similar to the one indicated in [2] and making k=)ml+(pl, we obtain

b = x cosa.

where k+2

step 2

In eq. (48), n =O for k even and n =1 for k odd,

16 (k+2)! -k!) if n#k+2

(49) (k+n) ! (k-n) ! ( (k+n+2) (k-n+2) 1 if n=k+2, Ok,n =

Hn(x) is the Hankel function of second kind of order n and argument x according to the definition given in [2, 131, and

. ,3n/2 ( 5 0 ) 6,(xo,Ro) = $1 V2 exp(2xocosa/Ro+jna) do .

The integral (50) may be evaluated by using the following power series development, rapidly convergent for small values of Ixo/Ro( :

(m+n) odd

or by using the numerical Romberg method for large values of I x ~ / R ~ ~ .

B i o g r a p h i e s

Y . M. Ma16 Machado uas b o r n i n P o r t o . P o r t u g a l , on May 6. 1953. He r e c e i v e d h i s degree o f E l e c t r i c a l E n g i n e e r from t h e E n g i n e e r i n g F a c u l t y o f t h e U n i v e r s i t y o f P o r t o (FEUP) and a Ph.0. degree i n E l e c t r i c a l E n g i n e e r i n g f r o m t h e I n s t i t u t o S u p e r i o r TCcnico ( I S T / U l L ) . L isbon. i n 1988. He has been t e a c h i n g s i n c e 1974 and now he h i s P r o f e s s o r of "Theory of E l e c t r i c a l E n g i n e e r i n g " a t t h e Dept. o f E l e c t r i c a l E n g i n e e r i n g o f t h e IST/UTL. H i s main r e s e a r c h a r e a has been t h e s e r i e s - i n p e d a n c e e v a l u a t i o n i n l i n e a r t r a n s m i s s i o n l i n e s t a k i n g i n t o account s k i n and p r o x i m i t y e f f e c t s .

J. F. Borges da S i l v a , uas b o r n i n L isbon, P o r t u g a l , on May 10. 1934. He r e c e i v e d h i s degrees i n E l e c t r i c a l E n g i n e e r i n g f rom t h e I n s t i t u t o S u p e r i o r TCcnico o f t h e T e c h n i c a l U n i v e r s i t y of L i s b o n ( I S l / U T L ) . i n c l u d i n g a Ph.0. i n 1971. He has been s i n c e on t h e F a c u l t y o f t h e E l e c t r i c a l E n g i n e e r i n g Depar tment o f t h e I S l / U l L . becoming a f u l l p r o f e s s o r i n 1979. L a t e l y . h i s a a i n r e s e a r c h i n t e r e s t s have been t h e p h y s i c a l and m a t h e m a t i c a l problems i n v o l v e d i n m o d e l l i n g l i n e a r and n o n - l i n e a r phenomena i n a uay s u i t a b l e f o r use i n computer s i m u l a t i o n of e l e c t r i c a l n e t u o r k s .

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