Series impedance and shunt admitance matrices of an underground cable system

Prévia do material em texto

SERIES IMPEDANCE AND SHUNT A D M I T T A N C E MATRICES 
OF 
A N UNDERGROUND C A B L E SYSTEM 
by 
Navaratnam Srivallipuranandan 
B.E.(Hons.), University of Madras, India, 1983 
A THESIS SUBMITTED IN P A R T I A L F U L F I L L M E N T OF T H E 
R E Q U I R E M E N T S F O R T H E D E G R E E O F 
M A S T E R OF APPLIED S C I E N C E 
in 
T H E F A C U L T Y O F G R A D U A T E STUDIES 
(Department of Electrical Engineering) 
We accept this thesis as conforming 
to the required standard 
T H E UNIVERSITY O F BRITISH COLUMBIA, 1986 
C Navaratnam Srivallipuranandan, 1986 
November 1986 
In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the 
requirements for an advanced degree at the University 
of B r i t i s h Columbia, I agree that the Library s h a l l make 
i t f r e e l y available for reference and study. I further 
agree that permission for extensive copying of t h i s thesis 
for scholarly purposes may be granted by the head of my 
department or by h i s or her representatives. I t i s 
understood that copying or publication of t h i s thesis 
for f i n a n c i a l gain s h a l l not be allowed without my written 
permission. 
Department of 
The University of B r i t i s h Columbia 
1956 Main Mall 
Vancouver, Canada 
V6T 1Y3 
Date 
6 n /8'i} 
SERIES I M P E D A N C E A N D S H U N T A D M I T T A N C E M A T R I C E S 
O F A N U N D E R G R O U N D C A B L E 
ABSTRACT 
This thesis describes numerical methods for the: evaluation of the series 
impedance matrix and shunt admittance matrix of underground cable 
systems. In the series impedance matrix, the terms most difficult to 
compute are the internal impedances of tubular conductors and the 
earth return impedance. The various form u hit- for the interim!' 
impedance of tubular conductors and for th.: earth return impedance 
are, therefore, investigated in detail. Also, a more accurate way of 
evaluating the elements of the admittance matrix with frequency 
dependence of the complex permittivity is proposed. 
Various formulae have been developed for the earth return 
impedance of buried cables. Using the Polhiczek's formulae as the 
standard for comparison, the formula of Ametani and approximations 
proposed by other authors are studied. Mutual impedance between an 
underground cable and an overhead conductor is studied as well. The 
internal impedance of a laminated tubular conductor is different from 
that of a homogeneous tubular conductor. Equations have been 
derived to evaluate the internal impedances of such laminated tubular 
conductors. 
( i i ) 
Table of Contents 
Abstract — - - - l i 
Table of Contents - i i j 
List of Table - •• - V 
List of Figures - -:— VI 
List of Symbols - V i i i 
Acknowledgement , - •- i x 
1. I N T R O D U C T I O N .—.: . 1 
2. SERIES I M P E D A N C E A N D S H U N T A D M I T T A N C E M A T R I C E S 
2.1 Basic Assumptions 6 
2.2 Series Impedance matrix [Z] for N Cables in Parallel 7 
2.2.1 Submatrix [Z„] 9 
2.2.2 Skin Effect 13 
2.2.3 Internal Impedance of Solid and Tubular Conductors 14 
2.2.4 Submatrix [Z,-yJ - •• 15 
2.2.4.1 Proximity Effect 16 
2.2.4.2 Proximity Effect of a Single-Phase Circuit of Two Identical 
Conductors 17 
2.2.4.3 Shielding Effect of the Sheath 17 
2.2.4.4 Elements of Submatrix [Z„] 19 
2.3 Shunt Admittance Matrix [K]; for N Cables in Parallel 22 
2.3.1 Leakage Conductance and Capacitive Suceptance 22 
2.3.2 Frequency Dependence of the Complex Permittivity 23 
2.3.3 Submatrix 27 
2.4 Conclusion 29 
3. C O M P A R I S O N O F I N T E R N A L I M P E D A N C E F O R M U L A E 
3.1 Exact Formulae for Tubular Conductors -•- 30 
3.2 Internal Impedance of a Solid Conductor ..— 31 
3.3 Internal impedance of a Tubular Conductor ; 36 
3.4 Conclusion — -•- 47 
( i i i ) 
4. E A R T H R E T U R N I M P E D A N C E 
4.1 Earth Return Impedance of Insulated Conductor 50 
4.2 Earth Return Impedance in a Homogeneous Infinite Earth 50 
4.3 Earth Return Impedance in a Homogeneous Semi-Infinite Earth 52 
4.4 Formulae used by Ametani, Wedepohl and Semlyen 57 
4.5 Effect of Displacement Current Term and Numerical Results 61 
4.6 Cable Buried at Depth Greater than Depth of Penetration 64 
4.7 Mutual Impedance between a Cable Buried in the Earth and an Over-
head Line or vice versa 67 
4.8 Conclusion - 69 
5. L A M I N A T E D T U B U L A R C O N D U C T O R S 
5.1 Internal Impedance of a Laminated Tubular Conductor 70 
5.1:1 Internal Impedance with External Return 70 
5.1.2 Internal Impedance with Internal Return 72 
5.2 Application to Gas-Insulated Substations 73 
5.2.1 Case i: Core and Sheath not Coated 74 
5.2.2 Case ii: Only Sheath Coated 74 
5.2.3 Case iii: Only Core Coated 76 
5.2.4 Case iv: Both Core and Sheath Coated 77 
5.2.5 Stainless Steel Coating 79 
5.2.6 Supermalloy Coating 82 
5.2.7 Comparison between Stainless Steel and Supermalloy Coatings 
82 
5.3 Conclusion 85 
6. T E S T C A S E S 
6.1 Single-Core Cable 86 
6.2 Three-Phase Cable 91 
6.3 Shunt Admittance Matrix 93 
7. C O N C L U S I O N 94 
A P P E N D I X A 96 
A P P E N D I X B : 98 
A P P E N D I X C 103 
A P P E N D I X D 106 
R E F E R E N C E S : 114 
(Iv); 
List of Tables 
3.1 Internal Impedance of a Solid Conductor 33 
3.2 Internal Impedance Za of a Tubular Conductor 38 
3.3 Mutual Impedance (Za(,) of a Tubular Conductor with Current Return-
ing Inside 43 
3.4 Internal Impedance Zy of a Tubular Conductor with Current Returning 
Outside -. , 44 
4.1 Solution of PoIIaczek's Equation by Numerical Integration and Using 
Infinite Series 58 
4.2 Earth Return Self Impedance with and without Displacement Current 
Term .' 1 61 
4.3 Earth Return Self Impedance as a Function of Frequency 64 '• 
5.1 Resistivity and Relative Permeability of Coating Materials 79 
5.2 Skin Depth of Stainless Steel 79 
5.3 Skin Depth of Supermalloy 82 
6.1 Impedances of Single Core Underground Cable 87 
6.2 Mutual Impedance between Two Cables with Burial Depth of 0.75m 
and Separation of 0.30m 91 
(V) 
List of Figures 
1.1 Potential Difference V, between Core and Sheath and F 2 between 
Sheath and Earth 3 
2.1 Basic Single Core Cable Construction 7 
2.2 Loop Currents in a Single Core Cable 9 
2.3 Potential Difference between Two Concentric Conductors 10 
2.4 Three Conductor Representation of a Single Core Cable : 10 
2.5 Sheath with Loop Currents Ix and I2 ... •- 15 
2.6 Two Cable System , -. 16 
2.~! Circuit Arrangement of Primary, Secondary and Shielding Conductors, 
with Shielding Conductor Grounded at Both Ends".. 18 
2.8 Transmission System Consisting of a Single Conductor and a Cable 
; 21 
2.9 Cross-Section of a Coaxial Cable 23 
2.10 (a),(b) - Measurements of e'(<o) and ("(<*) of an OiMmpregnated Test 
Cable at 20°cC 24 
2.11 Values of e'(o>) and «"((•)) Obtained from the Empirical Formula 
26 
2.12 Polarization-Time Curve of a Dielectric Material 27 
3.0 Loop Currents in a Tubular Conductor 30 
3.1 (a),(b) - Impedance of a Solid Conductor as a Function of Frequency 
3.2 (a),(b) - Errors in Wedepohl's and Semlyen's Formulae for a Solid 
Conductor 35 
3.3 Cross-Section of a Tubular Conductor 30 
3.4 (a),(b) - Impedance Za of a Tubular Conductor (with Internal Return): 
as a Function of Freqency 39 
3.5 Errors in Wedepohl's, Schelkunoff's and Bianchi's Formulae for Za 
40 
3.6 Errors in Wedepohl's and Schelkunoff's Formulae for Zab 42 
3.7 (a),(b) - Impedance Zb of Tubular Conductor (with External Return) 
as a Function of Frequency ..' 45 
3.8 Errors in Wedepohl's, Schelkunoff's and Bianchi's Formulae for Zb 
:.. „ , 46 
4.1 Electric Field Strength at Point P 51 
4.2 Error in Replacing a Conductor of Finite Radius by a Filament Con-
ductor —-- 53 
4.3 Solution of Real and Imaginary Part of Equation (4.9),for a 
Freqency of 1MHz. > - - «... 56 
4.4 Relative Error in the Evaluation of Carson's Formulae with an 
Asymptotic Expansion 59 
(vi) 
4.5 Error in the Earth Return Self-Impedance if the Displacement 
Current is Ignored 62 
4.6 (a),(b) - Earth Return Self Impedance as a Function of Freqency 
, 65 
4.7 Errors in Earth Return Self Impedance 60 
4.8 Differences in Resistance Values of Semi-Infinite and Infinite Earth 
Return Formulae : 68 
5.1 (a),(b) - Numbering of Conductor Layers to Find the Internal 
Impedances of a Laminated Tubular Conductor 71 
5.2 Representation of the nt th Layer 71 
5.3 Core and Sheath not coated 74 
5.4 Inner Surface of the Sheath only, Coated 75 
5.5 Core Alone Coated 76 
5.6 Core as well as Inner Surface of the Sheath Coated 77 
5.7 Dimensions of the Bus Duct in a Gas-Insulated Substation <• 78 
5.8 (a),(b) - Variation of Resistance; and Inductance with Frequency for 
the Four Cases; Stainless Steel Coating, Thickness, 0.1mm 80 
5.9 (a),(b) - Variation of Resistance and Inductance with Freqency for the 
Four Cases; Stainless Steel Coating, Thickness 0.5mm 81 
5.10 (a),(b) - Variation of Resistance and Inductance with Freqency for 
the Four Cases; Supermalloy Coating, Thickness 0.01mm 83 
5.11 (a),(b) - Variation of Resistance and Inductance with Frequency for 
the Four Cases; Supermalloy Coating, Thickness 0.05mm 84 
6.1 Errors in Ametani's and Wedepohl's Approximations in Zcc 88 
6.2 Errors in Ametani's and Wedepohl's Approximations in Zgc 89 
6.3 Errors in Ametani's and Wedepohl's Approximations in Zef 90 
6.4 Errors in Ametani's and Wedepohl's Approximations in the Mutual 
Impedance between Two Cables 92 
A.l Three-Phase Cable Set-up for the Study 96 
A. 2 Basic Construction of Each Single Core Cable 96 
B. l The Relative Directions of the Field Components in a Coaxial 
Transmission Line 98 
B. 2 Loop Currents in a Tubular Conductor 101 
C. 1 Representation of a Buried Conductor in an Infinite Earth 104 
D. l Current Carrying Filament in the Air 106 
D.2 Current Carrying Filament Buried in the Earth I l l 
( V i i ) 
L I S T O F SYMBOLS 
E — electric field strength, 
H = magnetic field strength 
/ = frequency, 
IT = 3.1415926 
(o = 27rX /, angular frequency 
— 1, complex operator 
fi0 = 4JTX 10 - 7, absolute permeability of free space 
/xr, — relative permeability of the medium i 
t*i ~ / io x / i r i> total permeability of the medium « 
7 = Euler's constant 
p, = resistivity of a particular medium t 
€,((o) = €, — complex dielectric constant or permittivity of a particular 
medium » 
<j> = flux density 
/ = current 
J — current density 
c,exp = exponential 
In = natural logarithm 
( joy/ty 1 
m, — |:- — o> fi^t- | , known as intrinsic propagation constant of a particu-
lar medium t. If the displacement currents are ignored, then the 
( joi / i , I 
value of m is equal to I I . Displacement currents are ignored unless li Pi J 
otherwise specified 
/„ = modified Bessel function of the 1st kind and of the nth order 
Kn — modified Bessel function of the 2nd kind and of the nth order 
K — characteristic impedance 
F = propagation constant 
T = relaxation time of a dielectric 
R = resistance per unit length 
L = inductance per unit length 
G = conductance per unit length 
C — capacitance per unit length 
Y = shunt admittance matrix 
Z = series impedance matrix 
(viii) 
Acknowledgements 
I would like to acknowledge my appreciation to Dr. H. W. Dommel for his 
encouragement and supervision throughout the course of this research. 
I would also like to thank Mrs. Guangqi Li, Mr. Luis Marti and Mr. C. E. 
Sudhakar for their many valuable suggestions. 
Thanks are due to Mrs. Nancy Simpson for typesetting the manuscript. 
The financial support of Bonneville Power Administration, Portland, Oregon, 
U.S.A., and from my brother "Anna" is gratefully acknowledged. 
(ix) 
-1-
1 . INTRODUCTION 
Underground cables are used extensively for the transmission and distribution of electric 
power. Although expensive when compared with overhead transmission, laying cables under-
ground is often the only choice in urban areas. As the urban areas expand, the cable circuits 
tend to increase in length. At present, cable circuits are being employed which have lengths of 
the order of 100km. With an increase in system lengths and higher system voltages, the 
induction effects on nearby communication circuits are becoming more important. Also, for 
the power system itself, the steady-state and transient behaviour of underground cables must 
be known. For interference studies as well as for power system studies, methods for finding 
cable parameters over a range of frequencies are, therefore, needed. 
The transmission characteristics of an underground cable circuit or submarine cable cir-
cuit are determined by their propagation constant T, and characteristic impedance /<", which 
may be calculated for an angular frequency w from the following equations 
F = V(K + j<aL)(G + juC) (1.1a) 
K = V{R + joiL)/(G + j<aC) (1.1b) 
Where R,L,G and C are the four fundamental line parameters, i.e., resistance, inductance, 
conductance and capacitance per unit length. These cable parameters are therefore the basic 
data for all interference and power system studies. 
Cables are principally classified based on 
i) their location, i.e., aerial, submarine and underground 
ii) their protective finish, i.e., metallic (lead, aluminium) or non-metallic (braid) 
iii) the type of insulation, i.e., oil-impregnated paper, cross linked polyethylene (XLPE) etc. 
iv) the number of conductors, i.e., single conductor, two conductors, three conductors and so 
on. 
In this thesis, a single conductor aluminium cable with a concentric lead sheath and with 
insulation of either oil impregnated paper or XLPE is studied in detail (refer to Appendix A). 
The series impedance and shunt admittance matrices of a cable system made up of N 
cables, can be written as 
- 2 -
Z = 
(Z 2 1 | ( zd • • • 1*8*1 
a n d 
Y = 
(nil p y 
" l*/w] 
1*2*1 
PAWI 
(1.2a) 
(1.2b) 
S u b m a t r i x [Z„\ a l o n g the d i a g o n a l o f m a t r i x [Z\ is t h e se l f i m p e d a n c e o f c a b l e i w h i c h c a n 
be w r i t t e n as 
Z„ = 
Zw Zc.s. 
zsic{ zs.s. 
(1.3) 
w h e r e 
Zc.c = se l f i m p e d a n c e o f the c o r e o f c a b l e i" 
Zc.s. = Zs;c.= m u t u a l i m p e d a n c e b e t w e e n t h e c o r e a n d the s h e a t h o f c a b l e » 
.7j s = se l f i m p e d a n c e o f the s h e a t h o f c a b l e i 
T h e o f f - d i a g o n a l s u b m a t r i x 2,; is t h e m u t u a l i m p e d a n c e b e t w e e n t h e c a b l e i a n d c a b l e j w h i c h 
c a n be w r i t t e n as 
(1.4) 
w h e r e 
Zee = m u t u a l i m p e d a n c e b e t w e e n t h e c o r e o f t h e c a b l e i a n d c o r e o f t h e c a b l e j 
Zci - m u t u a l i m p e d a n c e b e t w e e n t h e c o r e o f t h e c a b l e i a n d t h e s h e a t h o f t h e c a b l e j 
z..t. = m u t u a l i m p e d a n c e b e t w e e n t h e s h e a t h o f c a b l e i a n d t h e c o r e o f t h e c a b l e j 
ZSi = m u t u a l i m p e d a n c e b e t w e e n t h e s h e a t h o f t h e c a b l e t a n d t h e s h e a t h o f the c a b l e j 
E v a l u a t i o n o f a l l t h e s e e l e m e n t s ; o f s u b m a t r i c e s [Z„\ a n d \Ztj\ i s , i n g e n e r a l , n o t e a s y . T h e 
b e s t a p p r o a c h s e e m s t o be t h e o n e p r o p o s e d b y W e d e p o h l [22] b a s e d o n t h e e a r l i e r w o r k d o n e 
b y S c h e l k u n o f f [6]. B o t h a u t h o r s f i n d t h e s e i m p e d a n c e s f r o m t h e l o n g i t u d i n a l v o l t a g e d r o p s in 
t h e c o r e a n d s h e a t h , [Z = V/J). T h e s e l o n g i t ud i n a l v o l t a g e d r o p s c a n be o b t a i n e d f r o m t h e 
p o t e n t i a l d i f f e r e n c e V , - b e t w e e n t h e c o r e a n d t h e s h e a t h a n d t h e p o t e n t i a l d i f f e r e n c e 
- 3 -
between the sheath and the earth, as shown in Figure 1.1. 
The potential differences V1 and V 2 can be expressed as a function of the loop currents /j 
and J 2 with the help of Schelkunoff's theorems. (Appendix B) 
For finding the elements of the shunt admittance matrix, it is a usual practice (22,23,27] 
to assume the permittivity of the insulation to be a real constant. In reality, the value of the 
permittivity is a frequency-dependent complex value, the real part of which accounts for the 
susceptance term and the imaginary part accounts for the conductance term. Therefore, it is 
necessary to find a general expression for the permittivity as a function of frequency. 
Chapter 2 discusses, in detail, the following topics: 
i) Formation of series impedance and shunt admittance matrices, 
ii) Wedepohl's approach for finding the elements of the series impedance matrix, 
iii) Frequency-dependence of the permittivity constant 
iv) Proximity effect and shielding effect in the evaluation of the mutual impedance subma-
trix Z,r 
As explained in Appendix B, the potential differences V, and V2 can be evaluated in terms 
of loop currents /, and I2 using SchelkunofTs theorem. For example, the potential difference Vl 
can be written as 
- 4 -
V, = (Zctt + ZtHl + Z , A , ) / , - Z s i k m 7 2 (1.5) 
where 
Z c r f = internal impedance of the core with current return outside, 
Z,ns — impedance of the insulation between the core and sheath 
Ztkl = internal impedance of the sheath with current return inside 
ZsKm — mutual impedance between the loops 1 and 2. 
The formulae developed by Schelkunbff to evaluate these internal impedances, Zcrt,Zsh, 
and mutual impedance, ZtKm, which take the skin effect into account, are exact and given in 
terms of modified Bessel functions. These exact expressions are not suitable for hand calcula-
tion purposes. There have been several attempts to obtain approximations to these classical 
formulae in order to make them suitable for hand calculations, (22, 23, 24]. 
Some of these approximate formulae are compared wjjth the exact formulae of (6] in 
Chapter 3, in terms of accuracy and computer time. The errors which are caused by neglecting 
the displacement current are discussed in Chapter 3 as well. 
Generally, the earth acts as a return path for part of the current in the underground or 
submarine cable system. The cable parameters are very much influenced by the earth return 
impedance. These impedances are obtained from the axial electric field strength in the earth 
due to the return current in the ground. 
In Chapter 4 the following topics are discussed: 
(i) Earth return impedances of cables buried in an infinite earth, where the depth of penetra-
tion of the return current in the ground is smaller than the depth of burial, or in iother 
words, where the distribution of return current in the ground is circularly symmetrical. 
(ii) Earth return impedances of cables buried in a semi-infinite earth, where the depth of 
penetration of the return current in the ground is larger than the depth of burial. 
(iii) Error introduced in the answers if the displacement current is neglected in the computa-
tion. 
(iv) Approximations proposed by Wedepohl and Semlyen and a comparison of their equations 
with the classical formula of Pollaczek [lj in terms of accuracy and computer time. 
"I (v) Ametani's (27] cable constant routine in the E M T P program. 
(vi) Mutual impedances between a buried conductor and an overhead conductor. 
In Chapter 5, we turn our attention from cables made up of homogeneous conductors to 
cables whose core and sheath are made up of laminated conductors of different materials. A 
practical application of this type of conductor is proposed by Harrington [32]. He suggested 
- 5 -
that transient sheath voltages in gas-insulated substations can be reduced by coating the con-
ductor and sheath surfaces with high-permeability and high-resistivity materials. Formulae for 
the internal impedances of such laminated conductors are derived and used to show the damp-
ing effect as a function of frequency. 
Chapter 6 concludes this thesis by comparing the values of the cable parameters obtained 
for a particular three-phase cable system shown in Appendix A, by using the exact formulae of 
Schelkunoff [6] and Pollaczek [l], as well as Ametani's approach [27] and Wedepohl's approxi-
mation [22]. 
- 6 -
2. SERIES IMPEDANCE AND SHUNT A D M I T T A N C E MATRICES 
2.1 Basic Assumptions 
The transmission characteristics of a conducting system such as an underground cable cir-
cuit or a submarine cable circuit are determined by its propagation constant T and characteris-
tic impedance K, which can be calculated for the angular frequency a) from the formula 
T = V(R + {juiL)(G + jcoC) 
K =; V(/? + jo>L)/(G + jmC) (2.1) 
where R,L,G and C are the four fundamental line parameters - resistance, inductance, conduc-
tance and capacitance, all per unit length. Determination of these parameters in a cable is not 
easy, but involves rather difficult analysis. Pioneering work in the calculation of underground 
cable parameters has been done by Wedepohl and Wilcox (22], based on the earlier work of 
Schelkunoff (6). 
The first step in defining the electrical parameters of an underground or submarine cable 
system is to set up the equations which describe the electric and magnetic fields. A complete 
set of such equations would constitute a perfect mathematical model. If these equations could 
be solved without any approximations then the response of the model would be indistinguish-
able from that of the real system, which it represents. In practice, however, this ideal situation 
cannot be realized. For example, it is not possible to perfectly represent the electrical proper-
ties such as resistivity, permeability and permittivity of the earth which form the return path 
for the currents flowing in the cable. Rigorous representation of such factors would lead to a 
set of very complex equations which may be very difficult to solve. In practice, therefore, some 
simplifying assumptions are made. The assumptions made in this thesis are: 
1. The cables are of circularly symmetric type. The longitudinal axes of cables which form 
the transmission system are mutually parallel and also parallel to the surface of the 
earth. It is implied in this assumption, that the cable has longtitudinal homogeneity. In 
other words, the electrical constants do not vary along the longtitudinal axes. 
2. The change in electric field strength along the bngitodinal axes of the cables are negligi-
ble compared to the change in radial electric field strength. This assumption permits the 
solution of the field equations in two dimensions only. 
3 The electric field strength at any point in the earth due to the carrents flowing in a cable 
is not significantly different from the field that would result if the net current were con-
centrated in an insulated filament placed at the centre of the cable and the volume of the 
cable were replaced by the soil. 
- 7 -
4. Displacement currents in the air, conductor and earth can be ignored. 
Assumptions 3 and 4 are justified up to high frequencies (1MHz], as will be demonstrated 
later in Chapter 3 and Chapter 4. 
2.2 Series Impedance Matrix [Z\ for N Cables in Parallel 
Let us assume that the transmission system consists of N cables. Each cable has a cross 
section of the type shown in Figure 2.1, representative of a typical high voltage (H.V.) under-
ground cable. 
Figure2.1 Basic Single Core Cable Construction 
The core consists of a tubular conductor C with the duct being filled with oil. In the case 
of solid conductors, the inner radius r 0 would be zero. The insulation between the core and the 
sheath is usually oil-impregnated paper, surrounded by a metallic tubular sheath 5, and insula-
tion around the sheath. 
In such systems, there are n = 2N metallic conductors. The soil in which the cables arc ; 
buried constitutes the ( n - f l)th conductor which is chosen as the reference for the conductor 
voltages. Such a transmission system can be described by the two matrix equations 
4^= -ZI (2.2a) 
dx 
(2.2b) 
- 8 -
d]_ 
dx 
= -IV 
where'V and / are n-dimensional vectors of voltages and currents, respectively, at a distance z 
along the longitudinal axis of the cable system. A l l voltages and currents are phasor values at 
a particular angular frequency a>. The series impedance matrix Z is given by 
Z = 
\Zn\ 
l*«l 1*2=1 
(*IN1 
\ZW\ 
\ZNN] 
(2.3) 
Each submatrix, \ZU\ assembled along the leading diagonal is a square matrix of dimension 2 
representing the self impedances of cable i , by itself, 
*c,c, *£.s; (2.4) 
where 
Zc.c. = self impedance of the core of cable 1 
Z,.s. = Zt.e. = mutual impedance between core and sheath of cable » 
Zs.St = self impedance of the sheath of cable » 
The off-diagonal submatrix |Z1;] represents the mutual impedances between cable »" and cable j. 
This submatrix is also a square matrix of dimensions 2, 
l*,l 
Zr - r . Z' c - ? • 
Z' r -c. Z * .s. 
(2.5) 
inhere 
Z£.c. = mutual impedance between core of cable t and core of cable j 
ZCiij — mutual impedance between core of cable 1 and sheath of cable j 
Z,.t. = mutual impedance between sheath of cable 1 and core of cable j 
Zt.$i = mutual impedance between sheath of cable 1 and sheath of cable j 
Similarly, the shunt admittance matrix Y"can be defined as: 
- 9 -
Y = (2.6) 
•where the submatrices jV„j and |Y"y) can be defined in a similar way, as described in section 2.3. 
2.2.1 S u b m a t r i x |Z„J 
The elements of the submatrix \Z„] can be determined by considering a single cable whose 
longitudinal cross section is as shown in Figure 2.2. The longitudinal voltage drops in such a 
cable are best described by two loop equations, with loop 1 formed by the core and sheath (as 
return) and loop 2 formed by the sheath and earth (as return). 
insulation 2 
sheath 
insulation 1 
core 
Figure 2.2 Loop Currents in a Single Core Cable. 
It has been shown by Carson [4] that the change in the potential difference between j and 
(/ + 1) of a concentric cylindrical system as shown in Figure 2.3 is given by 
——• + E} — Ej + x —ju>n4>, (27) 
where 
Ej = longitudinal electric field strength of the outer surface of the conductor j 
E ; * j = longitudinal electric field strength of the inner surface of conductor (j'+l) 
- 10 -
Axis 
Conductor j 
Conductor (j+l) 
Figure 2.S Potential Difference between Two Concentric Conductors. 
V. - potential difference between the j and the (j+l) conductor 
<t>, = magnetic flux through the area described by the contour ABCD. 
Since part of the current in the cable can return through the earth, the cable must be 
represented by 3 conductors (core, sheath, earth), as shown in Figure 2.4. 
Axis 
Sheaih 
Insulation 2 
Earth 
Figure 2.4 Three Conductor Representation of a Single Core Cable. 
- 11 -
The values of longitudinal electric field strengths Ecre,Esk,,Eihe (i.e., on the external sur-
face of the core, internal surface of the sheaih and the external surface of the sheath respec-
tively) can be expressed as 
Ecre = Z[rt /„ (2.8a) 
Eikt = -*,»./, + ZskmI2 i(2.8b) 
Eikc = Zikc I2 - Ztkm /, (2.8c) 
The electric field strength along the surface of the earth can be written as 
Ec = -ZeiI2 (2.9) 
where 
Zcrc = internal impedance per unit length of the core's external surface with current return-
ing through a conductor outside the core. 
Zskt = internal impedance per unit length of the sheath's internal surface with current 
returning through a conductor inside the tubular sheath. 
Zskm = mutual impedance per unit length of the sheath which gives the voltage drop on the 
external surface of the sheath, when current passes through the internal surface or 
vice versa. 
Zike — internal impedance per unit length of the sheath's external surface with current 
returning through a conductor outside the tubular sheath. 
Zes = self impedance of the earth's return path. 
Equation (2.7) can be then be written for the contours ABGD and EFGH a i follows: 
dx 
d\r2 
~dx 
= Eik, - Ec,c - joL„h (2.10a) 
= Et - Eikt - jo>L„/ 2 (2.10b) 
In equation (2.10a), the total magnetic flux through the area described by the contour ABCD is 
Lcs /j where 
L» = itl*(r^) (211) 
and r 0 and r, being the outer and inner radii of the insulation. The term L„ can be defined 
similarly. The parameters jtaLc, and j i * L l t are the impedances Znl and Zn2 of the respective 
insulations. 
- 12 -
Substituting the values for the electric field strengths from equation (2.8) and (2.9) into 
equation (2.10), we have 
dz 
dV~ 
dx 
~ Zikm Ztht + Zm2 + Zti (2-12) 
The matrix equation (2.12) relating the potential differences between the concentric 
cylindrical conductors and the loop currents can also be obtained from Figure 2.2. directly, i.e.. 
d\\ 
dx zx Zm '/.' 
d\'2 Zm Z2 (2.13) 
dx 
where the self impedance of loop 1 consists of 3 parts 
Z\ = Zcre + Z i n l + Zitix, 
and similarly for loop 2 
Z2 ~ Zsh. 4- Z,n2 + Zti, 
while the mutual impedance between loop 1 and loop 2 is 
Zm = ZShm 
Equations (2.12) or (2.13) are not yet in the usual form in which the voltages and currents 
of the core and the sheath are related to each other. They can be brought ijjto such a form by. 
considering the appropriate terminal conditions namely 
v2 = Vtk I2 = Iik + /„ (2-14) 
where 
Vcr = voltage from the core to the local ground, 
Vsll = voltage from the sheath to the local ground, 
IC1 = total current flowing in the core, 
Isk = total current flowing in the sheath. 
Substituting the values for voltages V,,V2 and currents 7,,/2 from equation (2.14) into equation 
(2.12), and adding rows 1 and 2, we obtain 
- 13 -
dV„ ' ZCre + Z,nl + Zsh, + Ztkc + Icr 
dx Z,n2 Zes — 1Zlkm Ze$ ~~ ztkm 
dx Zskt ^««2 ^ « — ' ^«i fn -
(2-15) 
The impedance matrix given in equation (2.15) is the self impedance submatrix [Z„\ for cable t. 
It can be seen that the elements of the impedance submatrix \Z„] are obtained from the 
internal impedances of tubular conductors • and from the earth return impedance. These 
impedances are frequency dependent because of the skin effect, which is discussed in the next 
section. 
2.2.2 Skin Effect 
In the derivation of formulae for resistance and inductance of conductors, it is often 
assumed that the current density is constant over the cross section of the conductor. This 
assumption is justified only if 
(i) the resistivity is uniform over the cable cross section, and if 
(ii) the conductor radius is small compared to the depth of penetration 
However, as the size or permeability of the conductor increases or as the frequency increases 
(resistivity still being uniform), the current density varies with the distance from the axis of the 
conductor, current density being maximum at the surface of the conductor and minimum at 
the centre. 
The reason for skin effect is as follows: 
In a long conductor of uniform resistivity the direction of current is everywhere parallel to 
the axis, and the voltage drop per unit length is the same for all the parallel filaments 
into which the conductor may be imagined to be subdivided,since these filaments are 
electrically in parallel. The voltage drop in each filament consists of a resistive com-
ponent proportional to and in phase with the current density in the filament, and an 
inductive component, equal to joi times the magnetic flux linking the filament. There is 
more flux linking the central filament of a round conductor than linking the filaments at 
the surface, because the latter are surrounded only by the external flux, whereas the 
former is surrounded also by all the internal flux. The greater the flux linkage and the 
inductive drop, the smaller must be the current density and the resistive drop in order for 
the total drop per unit length to be the same. Hence, the current density is least at the 
centre of the conductor and greatest at the surface [7]. 
- 14 -
The ac resistance, which is defined as the power lost as heat, divided by the square of the 
current, is increased by the skin effect, because the increase in loss caused by the increase in 
current in the outer parts of the conductor is greater than the decrease in loss caused by the 
decrease in current in the inner parts. The inductance, defined as flux linkage divided by 
current, is decreased by skin effect because of the decrease in internal flux. 
2.2.3 Internal Impedance of Solid and Tubular Conductors 
As mentioned in the previous section, the voltage drop per unit length is the same in all 
the parallel filaments into which the conductor can be subdivided because all filaments are 
electrically in parallel. The ratio between this voltage drop and the sum of al! filament 
currents is the internal impedance. For a solid conductor of radius a and resistivity p. the 
internal impedance is given by [7] 
pm I0{ma) 
2ira/,(m(i) 
where 
/„./, = modified Bessel functions of the first kind and of zero and first order, respectively. 
m = the intrinsic propagation constant of the conductor of equation (2.16). 
The derivation of the internal impedance formula for tubular conductors is more complex 
due to the boundary conditions. For example, if we consider the sheath in Figure 2.2. the loop 
current /,. passes through the inner surface of the sheath and returns internally and the loop 
current I2 passes through the outer surface of the sheath and returns externally. This is illus-
trated in Figure 2.5. 
Therefore, we have to consider the magnetic field strengths on both surfaces (which are 
then the boundary conditions) while solving the Maxwell's field equations to determine the for-
mulae for the internal impedances. A detailed analysis of this problem had been done by Schel-
kunoff [6]: his formulae which are relevant to this thesis are summarized in Appendix B. 
As shown in Figure 2.5, the return path for the current flowing in a tubular conductor 
may be provided either inside or outside the tube, or partly inside and partly outside. We 
designate Z„ as the internal impedance of the inner surface of the tube with internal return, 
and Zb as the internal impedance of the outer surface of the tube with external return, and Zab 
as the mutual impedance between one surface of the conductor to the other. The values of 
Za,Zb and Zai are given as follows: 
Za = ^•[/ 0(ma)K,(mJ) + AT0(ma )/,(*.&)] 
- 15 -
Figure 2.5 • Sheath with Loop Currents /, and /j. 
Z = P 
st 2xabD 
(2.17:.,b,c) 
w h e r e 
D = /j(m6)A',. (ma) - / , ( m a ) AT,(m6), 
p = r e s i s t i v i t y o f t h e c o n d u c t o r , 
m = i n t r i n s i c p r o p a g a t i o n c o n s t a n t o f t h e c o n d u c t o r o f e q u a t i o n (2.17), 
/<>,/, «= m o d i f i e d B e s s e l f u n c t i o n s o f t h e f i r s t k i n d a n d o f z e r o a n d f i r s t o r d e r , r e s p e c t i v e l y . 
/ f 0 ,W, = t h e m o d i f i e d B e s s e l f u n c t i o n s o f t h e s e c o n d k i n d a n d o f r e r o a n d first o rd f - r . 
r e s p e c t i v e l y . 
U s i n g t h e s e f o r m v . ' a e t h e e l e m e n t s o f t h e s u b m a t r i x [Z.,\ c a n b e f o u n d . 
2.2.4 S u b m a t r i x | Z J 
T h e o f f - d i a g o n a l s u b m a t r i x [ZtJ\ w h i c h r e p r e s e n t s t h e m u t u a l i m p e d a n c e s b e t w e e n c a b l e i 
a n d c a b l e j c a n be b e s t e x p l a i n e d i f w e c o n s i d e r a t r a n s m i s s i o n s y s t e m c o n s i s t i n g o f o n l y t w o 
c a b l e s « a n d j as s h o w n i n F i g u r e 2.6. B e f o r e w e a n a l y z e t h e e l e m e n t s o f the s u b m a t r i x , we 
- 16 -
Earth 
Cable j 
Figure 2.6 Tvuo Cable System 
will briefly discuss the influence of proximity effects and shielding effects on these elements. 
2.2.4.1 Proximity Effect 
Skin effect is caused by the non-uniformity of current density in a conductor. This 
current, density is a function of distance from the axis, but not of direction from the axis. 
However, in parallel conductor transmission, in addition to the self-magnetic field (field gen-
erated by the current flowing through the conductor), there will be magnetic fields generalcd 
by currents iD adjacent conductors. These fields interact and result in distortion in the ovr;:.l! 
symmetric field distribution. The effects of the distortion of symmetry are known -j\ |>r«.^ ii:ii: \ 
effects, which in most cases affect the distributed parameters of the transmission system {3.3]. 
A.H.M. Arnold [13], has given a comprehensive treatment on proximity effect resistance 
ratios for single-phase and three-phase circuits. He has given equations and tubulated func-
tions of i (defined below) for determining the proximity effect resistance ratios R'zr' in a 
single-phase circuit of two identical tubular conductors, with solid conductors being a special 
case. The ratio R'/R' is defined as the ratio of the effective ac resistance with proximity effect 
taken into account to the effective oc resistance wbcu the conductors are far apart, such that 
the proximity efTect is negligible. Further, factors to be applied to the ratio /?'//?'while consid-
ering a three-phase circuit with symmetrical triangular spacing or with flat spacing arc also 
given in the same reference. 
- 1 7 -
2.2.4.2 Proximity Effect of a Single-Phase Circuit of Two Identical Conductors 
The ratio R'/R', defined earlier, for a tubular conductor with the solid conductor being a 
special case, depends upon three variables, i.e., t/d,d/a and x defined as 
t/d = ratio of thickness t of the tube to its outside diameter d.(t/d = 0.5 for solid conductor). 
d/s = ratio of outside diameter d of a conductor to distance s between the axes of the conduc-
tors. 
x = 2nV2ft{d-t)/p 
x can be further simplified to [13], 
x — 
1.52-\/f/Rdc where 
= the dc resistance of the conductor in Dim. 
The proximity effect resistance ratio is then given by 
/?'//?'=- (2.18) 
where A. B and C are functions of x and t/d which can be determined from tables given in [13]. 
Similarly, proximity effect inductance ratios of a single-phase cable can be obtained as well. 
Also, both proximity resistance and inductance ratios for a 3-phase system can be obtained 
from the single-phase proximity resistance and inductance ratios. 
For the example chosen in this thesis d/s is less than 0.35. For such a value, the proxim-
ity effect can be ignored up to frequencies of 1MHz, [13, 33]. Hence, proximity effects are 
ignored here. 
2.2.4.3 Shielding Effect of the Sheath 
Another factor of importance in determining the mutual impedance, is the shielding effect 
of the cable sheath, which is normally grounded at both ends. Consider a primary circuit 0. a 
parallel exposed secondary circuit x and a shielding conductor * whose ends arc grounded as 
d\-'° 
shown in Figure 2.7- Let be the induced voltage in the exposed circuit duc to the mag-
dx 
netic coupling without any shielding conductor and let — —be the induced voltage in the 
dx 
dV? i 
shielding circuit. The current in the ground shielding conductor is then — , where Z., 
dx Zss 
is the self impedance of the shielding conductor with earth return, Now, voltage induced in the 
exposed circuit by the current in the shielding conductor is — , where Z.z is the mutual 
Figure 2.7 • Circuit Arrangement of Primary, Secondary 
and Shielding Conductors, with Shielding Conductor Grounded at Both Ends 
impedance between the shielding and the exposed circuit. Therefore, the net voltage induced in 
the exposed circuit is 
d\rx dV? dVf Z;J 
dx dx dx Z S 5 
(2.19) 
dVf d\r? 
If the voltages — — and — — are expressed in terms of the current in the primary circuit as 
dx dx 
dV? d\r° 
= I0Z0s
 a n cl ~.— = IoZoi> then equation (2.19) simplifies to 
dx dx 
£ 1 
dx 
1 - Os z»z Zn ZQJ ZQZIQ (2.20) 
and the shielding factor of the grounded shielding conductor is then given by 
Z,i Z0i 
n = l z„z 
(2.21) 
ss Ox 
If the shielding and the secondary conductor are exposed to the same field, which is the case 
for a shielding wire very close to a telephone line, and for the cable sheath around the core con-
ductor, then 
dX rO 
dx dx ' 
Therefore Z0l = Z0s, which makes the shielding factor to be equal 
to 
(2.22) 
- 19 -
2.2.4.4 E l e m e n t s o f S u b m a t r i x ( Z t ; ) . 
K e e p i n g in m i n d t h e s h i e l d i n g e f f e c t d e s c r i b e d a b o v e , w e w i l l n o w d e r i v e t h e e l e m e n t s o f 
t h e s u b m a t r i x \Z,}]. A g a i n , l o o p c u r r r e n t s a r e u s e d , as h a s b e e n d o n e b e f o r e f o r d e t e r m i n i n g 
t h e e l e m e n t s d f t h e s u b m a t r i x j Z „ j . C o n s i d e r i n g t h e c a b l e s y s t e m s h o w n i n F i g u r e 2 .3 , we c a n 
d e f i n e t h e f o l l o w i n g l o o p c u r r e n t s f o r t h e i t h c a b l e 
i) l o o p c u r r e n t / ' , , w h o s e p a t h c o n s i s t s o f t h e c o r e ' s e x t e r n a l s u r f a c e a n d s h e a t h ' s i n t e r n a l 
s u r f a c e 
i i) l o o p c u r r e n t 7 3 , w h o s e p a t h c o n s i s t s o f t h e s h e a t h ' s e x t e r n a l s u r f a c e a n d e a r t h . 
S i m i l a r l y , t h e l o o p c u r r e n t s l[ a n d I'2 c a n be d e f i n e d f o r c a b l e j . 
If w e c o n s i d e r t h e l o o p c u r r e n t s F2 a n d I{, t h e r e wi l l be n o i n d u c e d v o l t a g e in l o o p 2 o f 
c a b l e i d u e t o t h e l o o p c u r r e n t I\ as t h e n e t f i e l d p r o d u c e d b y t h e l o o p c u r r e n t 1\ is z e r o o u t -
s ide t h e c a b l e j [6], U s i n g the law o f r e c i p r o c i t y o f m u t u a l i m p e d a n c e s , i t c a n be d e d u c e d t h a t 
t h e r e wi l l be no i n d u c e d v o l t a g e in t h e l o o p 1 o f c a b l e j d u e t o t h e l o o p c u r r e n t V2. H e n c e , 
r e l a t i n g the l o o p c u r r e n t s w i t h t h e p o t e n t i a l d i f f e r e n c e s b e t w e e n t h e c o n d u c t o r s we o b t a i n : 
'dV\/dx z\ Z'm 0 0 r\ 
diydx zxm Z'2 0 Zsi ft 
d\'\/dx 0 0 z\ Am n 
d\"2/dj 0 Z S I Zin Zk 
(2.23) 
M o s t i m p e d a n c e t e r m s in e q u a t i o n (2 .23) , h a v e a l r e a d y b e e n d e f i n e d e x c e p t the t e r m 
Zj. S y, w h i c h is t h e m u t u a l i m p e d a n c e b e t w e e n t h e e a r t h r e t u r n l o o p s 2 o f c a b l e s »' a n d j . If t h e 
c a b l e s are b u r i e d in a n h o m o g e n e o u s i n f i n i t e e a r t h , w h e r e t h e p e n e t r a t i o n d e p t h o f t h e r e t u r n 
c u r r e n t in t h e e a r t h is less t h a n t h e d e p t h o f b u r i a l , t h e n t h e v a l u e o f Z S i S > is g i v e n b y [9]: 
pm"lK0(ms) ' -y _ \_ ' (2 24) 
w h e r e 
a = d i s t a n c e b e t w e e n t h e c e n t r e s o f t h e c a b l e s , 
r,,Tj = e x t e r n a l r a d i i o f the c a M c s i a n d j , a n d 
m = i n t r i n s i c p r o p a g a t i o n c o n s t a n t o f t h e e a r t h . 
F o r a h o m o g e n e o u s s e m i - i n f i n i t e e a r t h , w h e r e t h e p e n e t r a t i o n d e p t h i n t h e e a r t h is m o r e 
t h a n t h e d e p t h o f b u r i a l , the m u t u a l i m p e d a n c e Zt.Sf is g i v e n b y (22]: 
- 20 -
_ jap. K0(m/?) - /C0(mZ) + J 
where 
d,,d} = depth of burial of cables » and j , 
m = intrinsic propagation constant of the earth, 
z = V « ' + (d, + d,y 
s = horizontal separation between cables »' and j. 
If we measure the voltages with respect to ground, then we can write 
v2 = V\h 
n = vu 
and 
V = /' 
I\ = 
/{ = 
n = 
I' 
ih + Hr 
(2.25) 
(2.26) 
Substituting the values given by equation (2.26), into equation (2.23), and adding rows 1,2 and 
rows 3,4, we obtain the series impedance matrix for two cable system, as 
(2.27) 
From equation 2.27, the impedance submatrix [Z,}] defined earlier in equation (2.5) is 
given by 
dV\,/dx z\ + 2z; + r2 Z'm + Z'2 Zss Hr 
dVJdx z'm + z\ z\ Zss Ilk 
dV{,ldx Zss Zss 
s,s} 
Z{ .+ 2ZL + z>2 Zln + Z'2 Hr 
dV{hldx Z s i Zss Z'2 + zi z2 7«*. 
- 21 -
| 2,l = 
It is i n t e r e s t i n g t o n o t e , t h a t t h e m u t u a l i m p e d a n c e b e t w e e n t h e c o r e o f c a b l e i a n d t h e 
c o r e o f c a b l e j a n d t h e m u t u a l i m p e d a n c e b o t w e e n t h e c o r e o f c a b l e i a n d s h e a t h o f c a b l e j a r e 
t h e s a m e . T h i s r a i s e s t h e q u e s t i o n w h e t h e r t h e s h i e l d i n g e f fec t is p r o p e r l y r e p r e s e n t e d in t h e 
e q u a t i o n s . It is i n d e e d i m p l i c i t l y t a k e n c a r e o f i n t h e f o r m u l a t i o n w i t b l o o p c u r r e n t s . T h i s c a n 
be i l l u s t r a t e d w i t h t h e h e l p o f a c o n d u c t o r w p l a c e d i n c lose p r o x i m i t y t o a c a b l e b u r i e d in t h e 
e a r t h , as s h o w n in F i g u r e 2.8 
Earth 
(2 .28) 
Conductor W 
Figure 2.8 - Transmission System Consisting of a Single Conductor and a Cable. 
F o v t h e s y s t e m s h o w n in F i g u r e 2 .8 , t h e v o l t a g e s a n d c u r r e n t s o i l the c a b l e c a n be w r i t t e n 
as: 
dx 
dVsh 
dx 
— %cc Ic ZCSIS + ZCU,IVI 
~ Zci Ic + Z,t /, + ZlV) /„ 
(2 .29a ) 
( 2 . 2 9 b ) 
w h e r e 
Zet,Ztt 
Zca..Zia. 
— t h e s e l f i m p e d a n c e o f t h e c o r e a n d s h e a t h o f the c a b l e , r e s p e c t i v e l y , 
— t h e m u t u a l i m p e d a n c e b e t w e e n t h e c o r e a n d the s h e a t h , a n d . 
= t h e m u t u a l i m p e d a n c e s b e t w e e n t h e c o r e a n d c o n d u c t o r w a n d b e t w e e n t h e s h e a t h 
a n d c o n d u c t o r w, r e s p e c t i v e l y . 
- 22 -
Suppose that the cable sheath is not grounded at the ends, but used as the return pa;b 
for the current flowing in the core. Then there is no magnetic field outside the sheath, and no 
voltage will therefore be induced in conductor u>. Since this induced voltage is Z^L. + Zs. /,.. 
with Isk — ~ / £ T . it follows that Z^ — Zsw must be true. On the other hand, if the sheath is 
grounded at the ends, then there will be a circulating current through the sheath and earth, 
and Vsk becomes zero. Hence, the value of 7$ can be found from equation (2.29b) as: 
^ss 
ZSUI (2.30) 
Substituting the value of 7S into equation (2.29a) we obtain 
dV, 
dx 
Z-^ 
z" z.. 
z„ - zSi 
L + 
1 -
Zc$' Zs. 
~z~ 
Za ' Zsw 
Zls ' Zcv, 
IwZCu (2.31) 
The term 1 - 7 7 
Za Zcu, 
is the shielding factor of the sheath for the field produced by conduc-
tor w. With Zcw = Zsw, it can be simplified to 1 -
Zss 
which is the same as equation (2.22). 
Hence, the shielding effect is implicitly taken careof in the equations. 
2.3 Shunt A d m i t t a n c e M a t r i x Y for N Cables in Paral le l 
In a manner similar to the series impedance matrix Z, the shunt admittance matrix Y can 
be expressed in terms of two submatrices [Y„] and [Y,J. Since the soil acts as an electrostatic 
shield between the cables, the off-diagonal submatrix [>',,] will be a null matrix. Hence, we only 
have to derive the submatrix [Yj,|. Before we obtain the elements for the submatrix [}'„], the 
admittance of insulation will be discussed first. 
2.3.1 Leakage Conductance and Capaci t ive Susceptance 
Figure 2.9 shows a cross-section of a coaxial cable, with insulation between core and 
sheath, and between sheath and earth. Let us assume that the insulation has a relative permit-
tivity of 
The admittance Y per unit length of the insulation is defined as 
j<i)27reo€* 
y = G + jB = — 
In r2/r1 
- 23 -
Figure 2.9 - Croaa-Section of a Coaxial Cable. 
77^ 7 V ~ J 
jti)27rf„( 
(2.32) 
In TJT , In rjr, 
The first term in the left hand side of equation (2.32), is the leakage conductance of the 
insulation. It is the result of the combined effects of leakage current through the insulation 
and of the dielectric loss [7]. The second term is known as the capacitive susceptance of the 
insulation between the two conductors (core and sheath, or sheath and earth). 
2.3.2 Frequency Dependency of the Complex Permittivity 
Generally, the dielectric constant i is assumed to be a real constant with the imaginary 
part of t neglected due to its relatively small value, (of the order of 10" 4 compared to the real 
value [7,13,22.27]). However, the complex dielectric constant i is not a constant as its name 
implies. It depends on a number of factors such as the frequency of the applied field, the tem-
perature and the molecular structure of the dielectric substance [15]. 
Let us consider two commonly used, insulating materials for power cables, namely cross-
linked polyethylene (XLPE) and oil-impregnated paper. The values of i and i for XLPE are 
approximately constant for at least up to frequencies of \00MHz [16]. Typically, they have 
values of 
i = 2.33 
t*=4.66 1(r 4 (2.33) 
- 24 -
Unfortunate!}', little is mentioned in the literature about the frequency dependency of the 
permittivity in the case of oil-impregnated paper (18]. Recently Johanscn and Breien [17] pub-
lished the measured values of i and t for the oil-impregnated paper for a frequency range of 
1Hz to 10Q.4/7/;. The value of i was found to vary by 20%, whereas, the value of ' varies by 
200% for the same frequency range, at a temperature of 20°c. 
Figure 2.10(a),(b) Measurements of t ( t o ) 
and t*(co) 
j * on an Oil-Impregnated Test Cable at 2(f c. 
Figure 2.10(a) and (b) show the experimental data obtained for I and t for the frequency range 
10* to 108Hz only. Based on this experimental data, the authors [17] developed an empirical 
formula for t'(oi). 
- " + (, + y J x . o - T " 1 2 3 1 1 
Figures 2.11 (a) and (b) show the plot for i and ('obtained from the empirical formula, 
which closely match the experimental data of Figure 2.10(a) and (b). 
- 26 -
Frequency [Hz] 
Figure 2.11 Values of t\ui) a n d t"(o>) 
Obtained from the Empirical Formula. 
- 26 -
A general formula for tbe complex permittivity of any material as a function of frequency 
is given by Bartnikas [15]. According to Bartnikas, when a dielectric is subjected to an ac field, 
at low electric field gradients, its electrical response will depend upon a number of parameters 
such as the frequency of the applied field, the temperature and the molecular structure of the 
•dielectric substance. Under some conditions, no measurable phase difference between the 
dielectric displacement; D, and voltage gradient E will occur, and consequently the ratio DIE 
will be defined by a constant equal to the real value of the permittivity, c'. When a dc field E 
is suddenly applied across a dielectric, tbe dielectric will almost instantaneously, or in a very 
short time, attain a finite polarization value. This polarization value will be almost instantane-
ous, since it will be determined by the electronic and atomic polarizability effects. The limiting 
value of the real dielectric constant <'for this polarization is defined as <«, so that the resulting 
dielectric displacement is Dm or imE. The slower processes, due to the dipolc oriontatiou or 
ionic migration, will give rise to a polarization which will attain its saturation value consider-
ably more gradually because of such effects as the inertia of the permanent dipoles. The static 
dielectric displacement vector, Dit in this case is equal to isE, where t, is the static value of 
the real dielectric constant, e. 
Figure 2.12 - Polarization-Time Curve of a Dielectric Afaterial 
In the idealized polarization time curve depicted in Figure 2.12, Pf is the achieved satura-
tion value of tbe polarization resulting from permanent dipoles or from any other displacement 
of free charge carriers. Depending upon the temperature and the chemical and physical struc-
ture of the material, the saturation value, Pt may be achieved in a time that may vary any-
where from a few seconds to several days. If we denote the time-dependent portion of P% as 
P(t), the equation of the curve in Figure 2.12 can be represented by a form characteristic of the 
- 27 -
charging of a capacitor 
P(t) = Ps [l - exp(-f/r)] (2.35) 
where T is the time constant of the charging process. The time constant, T is a measure of the 
time lag and is referred to as the relaxation time of the polarization process. 
Now, the real and imaginary parts of the permittivity of an insulating material can be 
expressed as a function of frequency in terms of tm.e, and T as 
t = Rc(e') = £«, + 
I = Im(£') = 
1 + O ) 2 ^ 
(ts-e„)oiT 
1 + o) 2 ^ 
(2.36a) 
(2.36b) 
In summary, frequency dependence of the permittivity t is complicated, although, for 
some insulating materials, such as (XLPE), i is practically constant. The changes are very sig-
nificant, i.e., of the order of 102, for oil-impregnated paper. Typical values for the real and 
imaginary parts of t for X L P E are given in equation (2.33). The real and imaginary values of 
t for oil-impregnated paper can be obtained from equation (2.34), based on the reference [17]. 
A general formula for ''-he complex permittivity of any material is given by equation (2.37a). 
2.3.3 Submatrix \YU\ 
We shall now determine the self admittances of the cable system shown in Figure 2.2. 
The loop equations for the current changes along loops 1 and 2 will be: 
di, 
dx 
dx 0 
dl2 0 n v2 (2.37a) 
?here 
Yi = Gx + jBA — admittance of insulation between core and sheath, 
Y 2 = G2 + jB2 — admittance of insulation between sheath and earth, 
Vj = voltage between the core and sheath, 
V2 — volt age between the sheath and earth. 
Substituting the values for currents IltI2 and voltages VltV2 from equation (2.14) into 
(2.36) and subtracting row 2 from row 1, we obtain: 
- 28 -
dlc, 
dx 
dx 
Yx -Yr 
-Y, Yx + Y2 . v » * . (2.37b) 
Hence, the submatrix \Y„] is given by: 
Yt ~Yx 
-Yx Yx + Y2 (2.38) 
Recently, Dommel and Sawada [19,20] suggested that the admittance matrix should 
include the effect of the grounding resistance as well if the insulation between the sheath and 
earth is electrically poor, as in oil or gas pipelines. In such cases, the leakage current flows 
through the series connection of the insulation resistance and the finite grounding resistance. 
For conduction effects in pipelines they, therefore, use 
1 
' insulation R, earth 
(2.39) 
where the grounding resistance /?e i r th i 3 given by 
R earth 
Pearth 
4;r 
2J_+ fa V(2ff) g + (t/2)2 + 1/2 
D lnV(2H)2 + (1/2)2 - 1/2 
(2.40) 
with 
Pearth
 = earth resistivity, 
H = depth at which the cable is buried, 
/ — length of the cable. 
Strictly speaking, G2 in equation (2.39) is no longer an evenly distributed parameter 
because /? e i r t tin equation (2.40) is a function of length. In [20], it is shown that the change in 
the value of C 2.with the length is practically negligible, and treating G2 as an evenly distri-
buted parameter is therefore a reasonable assumption. 
2.4 Conclusion 
First, the series impedance and shunt admittance matrices were defined. These matrices 
are made up of self and mutual impedance (or admittance) submatrices. Elements of the self 
impedance submatrices were obtained from the internal impedance formulae for tubular con-
ductor and from the earth return self impedance formula. The elements of (he mutual 
impedance submatrices were obtained from the earth return mutual impedance formulae. The 
shielding effect of the sheath and the proximity effects between the conductors were studied 
next to assess their influence on the elements of the mutual impedance submatrix. Finally, the 
- 29 -
self admittance submatrix and mutual admittance submatrix were defined. Since the earth 
acts as an electrostatic shield, the elements of the mutual admittance submatrix are zero. The 
permittivity c* of the insulation which is needed to evaluate the elements of the self admittance 
submatrix, is frequency dependent and complex. An empirical formula for finding the real and 
imaginary parts of the permittivity t * as a function of frequency was shown, which can then be 
used to find the elements of the self admittance submatrix. 
- 30 -
3. C O M P A R I S O N O F I N T E R N A L I M P E D A N C E F O R M U L A E 
lu the previous-chapter the scries impedance matrix was assembled from the internal 
impedances of tubular conductors and from the earth return self and mutual impedances of 
tubular conductors. The formulae for the internal impedances and earth return impedances 
are given in terms of modified BesscI functions, which can be expressed as an infinite scries for : 
small arguments and as an asymptotic series for large arguments. Before computers became 
available, exact calculations were almost impossible and approximate formulae were therefore 
developed. Such approximations had been proposed by several authors (G,22,2'5',"2 l]. In this 
chapter, approximate formulae for the internal impedances are compared with i lie exact for-
mulae in terms of cpu time and in terms of accuracy. The earth return formulae are discussed 
in the next chanter. 
3.1 Exact Formulae for Tubular Conductors 
Axis 
I 
17/ 
Figure 3.0 - Ix>op Currents in a Tubular Conductor 
The internal impedances of a tubular conductor as given in equation (2.17) are as follows: 
Z. = f/otmoJK-.M) + KjmaVAmb)] (3.1a) 
Zt = -^L. \ I o { r n b ) K l ( m a ) + K^mbVJma)] (3.1b) 
Z.» = (3-lc) 
2ncbD 
where D - /j(ml»)K',(ina) - Il(ma)Kl{mb) 
- 31 -
The argumeDts for the modified Bessel functions 7 0 , / i and K0,Ki of the first kind and 
second kind are complex because the intrinsic propagation m of the conductor is ^/w/y/p) . 
For more exact- calculations, m should be {j<a/i/p— <>>2/i<) , where the first term under the 
square root accounts for the conduction current and the second term accounts for the displace-
ment current. The displacement current can be ignored in good conductors as it is negligible 
compared to the conduction current up to frequencies of 10MHz. For example, copper conduc-
tors with p = 1.7 10~8/?m resistivity would have a displacement current at 10MHz which is 11 
orders of magnitude smaller than the conduction current, and even smaller than that below 
10MHz. However, in the earth, the displacement current has an influence on the impedance as 
the earth's conductivity is of the cider IO - 1 0 smaller than the conductivity of a good conductor 
such as copper. Displacement currents are therefore taken into account in the earth return 
impedances discussed in the next chapter. 
Subroutine TUBE originally developed by H. W. and I. I. Dommel [30] (and later modified 
by L. Marti [29]) and Amctani's Cable Constants program implemented in BPA's EMTP, 
assume that the displacement currents can be neglected. This fact enables us to express the 
modified Bessel functions of the 1st and 2nd kind in terms of Kelvin functions. For example, 
the complex functions K0{mr) and I0(mr) can be expressed as real and imaginary parts as fol-
lows: 
A'„(mr)= K0(VJ\m]r) = Kj\m\r)+ jA' t l(|m|r) (3.2a) 
7 0(mr)= /o(V7T^lT)= B„(\m\r)+ jB e i(|m|r) (3.2b) 
where B„ and £?,, are Kelvin functions of the first kind, and K~¥ and Kcl are those of the second 
kind. To evaluate the Kelvin functions, infinite series and asymptotic series can be used for 
small real arguments and large real arguments, respectively. Subroutine TUBE and Ametani's 
routine use such series with a sufficient number of terms to guarantee high accuracy, 
3.2 Internal Impedance of a Solid Conductor 
The exact, formula for the internal impedance of a solid conductor of radius r follows from 
Zh of equation (3.1) by setting o = 0 and 6 = r: 
^1^1 3 
2 j i r / , (mr) 
where 
r = radius of the conductor 
- 3 2 -
m = intrinsic propagation constant of the conductor 
/ 0 , / 1 = modif ied Bessel functions of the first kind and of zero and first order, respectively. 
Wedepohl [22] suggested an approximation to this exact formula, given by 
z _ pmcoth(fcmr) + p(l-l/2k) 
This approximation was developed by first considering the equation 
= pmcoth(fnr) ( 3 5 ) 
27rr 
This equation is known to exhibit similar properties as the exact equation given by Equation 
(3.3). For example, at high frequencies, the impedance term Zx tends to be pm/2xr which is a 
well known skin effect formula [22], while at lower frequencies it represents pure resistance 
although not, in fact, equal to the required value p/xr2. Equation (3.5) can be improved to take 
account of the dc resistance more precisely by writing 
Z > = - ^ - c o t h ( W ) + ^ - f * ) (3.6) 
2nr nr 
where k is an arbitrary constant. The second term on the right hand side of this equation 
corrects the impedance at direct current. The value of A; chosen to give the correct resistive 
component is 0.777. 
There is another interesting formula derived by Semlyen in the discussion of reference 24, 
where the internal impedance of a solid conductor is given as 
Zx = y/R? + Za (3.7) 
where Rc is the dc resistance given by p/xr2 and, Za is the impedance at very high frequencies 
given by pm/2-r. Table 3.1 shows the values of resistance Rx = Re{Zx} and inductance 
Lx = — 7m{Z,} obtained from subroutine T U B E and from Wedepohl's and Semlyen's approxi-(i) 
mation formulae. 
Figure 3.1(a) and (b) show the resistance and inductance as a function of frequency. The 
errors in the values of resistance and inductance in Wedepohl's formula and Semlyen's formula 
are plotted in Figure 3.2(a) and (b). 
From the table and figures we can see that Wedepohl's formula has an error of 1-3% in 
the frequency range 100Hz to 1kHz for the resistive part, and 4% error up to frequency of 
1kHz in the inductive part. Semlyen's formula has an error of 4-7% for the frequency range 
60Hz to 20kHz, in the resistive part, and an error of 4-11% for the frequency range 20Hi to 
300Hz in the inductive part. 
- 33 -
T a b l e 3.1 
Internal Impedance of a Solid Conductor 
(p = 17 10~'Om and r = 0 . 0 2 3 4 m ) 
FREQUENCY 
(Hz) 
TUBE WEDEPOHL SEMLYEN 
RESISTANC ;E (Q/km) 
.01 
. 1 
1 
10 
100 
1,000 
10,000 
100,000 
1 ,000,000 
10,000,000 
0.0098825 
0.0098825 
0.0098858 
0.0102067 
0.02033800.0582719 
0.1786977 
0.5596756 
1.7644840 
5.5744390 
0.0098775 
0.0098776 
0.0098809 
0.0102034 
0.0211463 
i 0.0592378 
0.1797193 
0.5607150 
1.7655290 
5.5754860 
0.0098825 ; 
0.0098825 
0.0098874 
0.0103294 
0.0190556 
0.0561595 
0. 1763397 
0.5572406 
1.7620250 
5.5719720 
INDUCTANC 3E (jiH/km) 
.01 
. 1 
1 
10 
100 
1,000 
10,000 
100,000 
1,000,000 
10,000,000 
50.000000 
49.999920 
49.991580 
49.181730 
27.567070 
8.853760 
2.803902 
0.886793 
0.280432 
0.088680 
51.800000 
51.799920 
51.792250 
! i 51.042710 
i 28.354910 
; 8.868063 
2.804328 
0.886806 
0.280433 
0.088680 
50.000000 
49.999750 
49.974780 
; 47.836480 
25.930690 
8.798590 
2.802123 
0.886736 
0.280430 
0.088680 
The cpu times were found to be: 
TUBE 
Wedepohl 
Semlyen 
0.042 ms 
0.038 ms 
0.022 ms 
- 34 -
figure 3.1(a) and (b) Impedance of a 
Solid Conductor ae a Function of Frequency. 
- 35 -
5-
2 ' 
U i 
w -1-1 
c 
ro 
*—« 
CO 
0) 
cc -5-
-7-4 
11 
12 
WEDEPOHL 
SEMLYEN 
-
f i x----.: 
-
i / / 
/ \ / 
10~2 10_1 1 10' 10 2 10 5 10 4 10 s 10 6 10' 
8 
^ 4 
o 
L— 
uj o 
o 
c 
— 4 
o 
T3 
-8 
WEDEPOHL 
SEMLYEN 
1/ 
10~* 10" 1 1 10' 10 2 10 3 10 4 10 5 10 6 10 7 
Frequency [Hz] 
Figure 3.2(a) and (b) Errors in Wedepohl's and 
Stmlycn'» Formulae for a Solid Conductor. 
- 36 -
To verify that the displacement current is indeed negligible, a modified subroutine 
TUBEC was written which takes displacement currents into account. Within the accuracy 
given in Table 3.1. TUBE and TUBEC produced identical mults. 
3.3 Internal Impedance of a Tubular Conductor; 
Figure S.S • Cross Section of a Tubular Conductor 
There are three impedances associated with a tubular conductor of the type shown in Fig-
ure 3.3: 
1) Internal impedance Za, which gives the voltage drop on the inner surface, when a unit 
current returns through a conductor inside the tube. 
2) Internal impedance Zt, which gives the voltage drop on the outer surface when a unit 
current returns through a conductor outside the tube. 
3) Mutual impedance Zai of the tubular conductor which gives the voltage drop on the outer 
surface when a unit current returns through a conductor inside the tube or vice versa. 
The formulae for Z^,Zb and Z,t, originally developed by Schelkunoff, are given in equation (3.11. 
These formulae are given in terms of modified Bessel functions, and are obviously not suitable 
for hand calculations. Schelkunoff therefore approximated these exact formulae by replacing 
the modified Bessel functions J0,IX,K0 and K", by their asymptotic expressions and performing 
the necessary division as far as the second term. 
Schelkunoff's approximations are as follows [6]: 
Zt = -^cotb(m(6-o)) - -^—{^ + ( 3 8 a ) 
Z i = | ^ c o t h ( m ( 6 - a ) ) + ^ ( 3 / a + l/6} (3.8b) 
- 37 -
Zah = ^|LcoBeft(m(6--c)) (3. 8 c) 
Wedepohl and Wilcox, give a similar approximation [22] but with a different approarh. 
The magnetic intensity H and the electric current density / in a tubular conductor can be 
related by the equations: 
f + f-:' <»;»») 
= m 2 / / (3.9b) 
If the tube is thin compared with its mean radius, equation (3.9a) can be written as: 
dH 2H , 
Equation (3.9b) and (3.10) lead to a second-order differential equation in H. Now, solving 
for H from this second-order differential equation and following the same procedure as given in 
Appendix A for the exact formulae, we obtain the following equations: 
Z t = ^ c o i H m { b . a ) ) - - ^ ( 3 , l a ) 
Z t = ^ c o t H m ( b - a ) ) + ^ ^ { 3.1, b ) 
Z < » = -7~T C o s e A ( m(6 - a ) ) (3.11c) 
7t(a + 0) 
There is another approximation derived by Bianchi [23]. If the difference between the 
radii b and a is very much less than either of the radius o and 6, or in other words, if 
(6 — a)«a,b then the impedances can be expressed as 
Z ° = Z> = / ™ coth(m(6-a)) (3.12a) 
Z°> = - 7^«>seA(m(6-a)) • (3.12b) 
The impedance Zti obtained by Bianchi in equation (3.12b) is the same as that obtained by 
Wedepohl in equation (3.11c). 
Using the exact formula (subroutine TUBE) Wedepohl's approximate formula, 
Schelkunoff's approximate formula and Bianchi's approximate formula, the values of resistance 
and inductance terms of the impedances ZtZt and Z 4 t were obtained for a typical tubular con-
ductor. Table 3.2 shows the resistance and inductance values as a function of frequency for the 
internal impedance Za. Figures 3.4(a) and (b) show the resistance and inductance for the fre-
quency range 0.01 Hz to 10MHz. To highlight the differences in the results, the errors in the 
- 38 -
Table S.2 
Internal Impedance Zt of i Tubular Conductor with 
Current Returning Inside (p = 2 . 1 \0~7Dm,a = 0 . 0 3 8 5 m , 6 = 0 . 0 4 1 3 m ) 
FREQUENCY TUBE WEDEPOHL SCHELKUNOFF BIANCHI 
(Hz) 
RES s I STANCE ($2/k .m) 
.01 0. 299163 0.299163 0.288640 0.299163 
. 1 0. 299163 0.299163 0.288640 f 0.299163 
1 0. 299163 0.299163 0.288640: 0.299163 
10 0. 299163 0.299163 0.2886401 0.299163 
100 0. 299169 0.299169 0.288646 0.299169 
1000 0. 299761 0.299762 0.289224 0.299741 
10000 0. 354424 0.354480 0.343956 0.352539 
100000 1. 18036 1.18068 1.17015 1 .14975 
1000000 3. 75275 3.75312 3.74259 3.63192 
10000000 11 .8915 11.8919 11.8813 11.4852 
INI 5UCTANCE ( MH/ 'km) 
.01 4. 84605 4.84848 4.84848 4.67836 
. 1 4. 84605 4.84848 4.84848 4.67836 
1 4. 84605 4.84848 4.84848 4.67836 
10 4. 84605 4.84848 : 4.84848 4.67836 
100 4. 84603 4.84846 4.84846 4.67834 
1000 4. 84339 4.84581 4.84581 4.67578 
10000 4. 60048 4.60256 1 4.60256 4.44106 
100000 1 . 89284 1.89296 1.89296 1.82654 
1000000 0. 59905 0.59906 0.59906 0.57804 
10000000 0. 18944 0.18944 0.18944 0.18279 
approximate formulae are plotted in Figure 3.5. From these figures and the table it can be 
seen that 
1. Wedepohrs formula has almost no error up to a frequency of 1MHz for both the resis-
tance and inductance. 
2. Schelkunoff's approximation has an error of 2-4% up to a frequency of 10kHz in the resis-
tive part. The inductance value obtained by SchelkunofTs approximation is the same as 
that obtained by Wedepohl's approximation. 
- 39 -
10-
v o c _p 
or 
TUBE 
WEpEPOHL 
SCHELKUNOFF 
BIANCHI 
O . H 
10" 
10-
10" 10' 102 103 10' 105 10' 10 
E 
\ x j t 
aj i -o c _o 
"o 
T> C 
TUBE 
WEDEPOHL 
SCHELKUNOFF 
BIANCHI 
0.1+-
10" 10" 10' 10J 103 
Frequency [Hz] 
10* 10' 10' 
Figure S.4(a),(b) • Impedance Z, of a Tubular Conductor 
(with Internal Return) as a Function of Frequency. 
10 
- 40 -
^ 2 
o 
" 0 
tt> o c 
(0 
c o 
" t o 
tu _-> 
cc z 
WEDEPOHL 
SCHELKUNOFF 
B I A N C H I 
\ . - ' " 
\ 
\ / ' 
A 
10~2 10"' 1 101 102 10J 104 10s 106 107 
e£ 2 
o k_ 
k_ 
U l 0 tv> o c 
<0 
u 
"D 
c - 2 
WEDEPOHL 
SCHELKUNOFF 
B I A N C H I 
I . M i l l , , IHj I 1,1,11V ! I l l l l l l , I . I I I I t T , I I I M i l l , I I l l l l l l , I I I I . I l l , I l l l l l l f 
10"2 10"1 1 10' 102 105 10" 105 10s 107 
Frequency [Hz] 
Figure S.5 -Errors in Wedepohl'*, Sehelkunoff'e and Bianchi'a Formulae 
forZt. 
- 41 -
3. Bianchi's approximation formulae is good for frequencies less that 10 kHz in the case of 
resistance, where the error is almost zero, but beyond that frequency the error increases. 
In the inductance the error is around 3% for the whole frequency range. 
The cpu time for the routines were found to be 
TUBE 0.059 ms 
Wedepohl 0.037 ms 
Schelkunoff 0.036 ms 
Bianchi 0.033 ms 
Similar comparisons were made for the mutual impedance Zab and the impedance Zt. 
Table 3.3 gives the values of resistance and inductance of the impedanceterm Z..^ obtained 
from routine TUBE, from Wedepohl's approximation formula and from Schelkunoff's approxi-
mation formula at different frequencies. Since the approximations proposed by Wedepohl and 
Bianchi are identical, only Wedepohl's approximation was considered. 
The errors in the approximate formulae are shown in Figure 3.6. It can be seen from Fig-
ure 3.6 that the errors in both Wedepohl's and Schelkunoff's approximations are less than 0.5^ 
up to a frequency of 1MHz. Wedepohl's approximation is closer to the exact formula. 
The cpu times were found to be: 
TUBE 0.056 ms 
Wedepohl/Bianchi 0.038 ms 
Schelkunoff 0.037 ms 
In a similar manner, the resistance and inductance components of Zt were calculated. 
Table 3.4 shows the values of these parameters using the exact formula and various approxi-
mations at different frequencies. 
Figures 3.7(a) and (b) show the resistance and inductance as a function of frequency for 
the range 0.01 Hz to 10MHz. The errors in the approximations are plotted in Figure 3.8. We 
note the following for the impedance Z 4 
- 42 -
0.4 
0.1-
ui 
CP o c 
CD 
- - o . H 
T J 
C 
— 0.3 f 111 (Mill I I I l l l l l , 
W E D E P O H L 
S C H E L K U N O F F 
\ 
TT!!I, 1 1 l l l T H t ' T 1 MTIir, I I 1 l l l l l l ' 1 
\2 4/r>3 
i i inn: - ' i i mill i linn 
10~2 10" 1 10' 102 10 s 10" 10* 106 107 
Frequency [Hz] 
Figure S.6 
Errors in Wedepohl's and Sehetkunoff'e Formulae for Ztl 
- 43 -
Table S.S 
Mutual Impedance (Z e 6, of a Tubular Conductor 
(p = 2.1 10" 7/?m,o = 0.0385m ,6 = 0.0413m) 
FREQUENCY 
(Hz) 
TUBE WEDEPOHL SCHELKUNOFF 
RESISTANC :E (8/km) 
.01 
. 1 
1 
10 
100 
1 ,000 
10,000 
100,000 
1,000,000 
0.29916343 
0.29916343 
0.29916343 
0.29916338 
0.29915838 
0.29865873 
0.25299441 
-0.06964902 
0.00001930 
0.29916343 
0.29916343 
0.29916343 
0.29916338 
0.29915838 
0.29865846 
0.25297170 
-0.06962398 
0.00001929 
0.29934775 
0.29934776 
0.29934776 
0.29934771 
0.29934270 
0.29884247 
0.25312757 
-0.06966688 
0.00001930 
INDUCTANC 3E (uH/km) 
.01 
. 1 
1 
10 
100 
1,000 
10,000 
100,000 
1 ,000,000 
-2.3386052 
-2.3386052 
-2.3386052 
-2.3386052 
-2.3385803 
-2.3361094 
-2.1095191 
-0.0097380 
0.0000082 
-2.3391813 
-2.3391813 
-2.3391813 
-2.3391810 
-2.3391563 
-2.3366832 
-2.1099033 
-0.0097146 
0.0000082 
-2.3406226 
-2.3406226 
-2.3406226 
-2.3406223 
-2.3405975 
-2.3381230 
-2.1112033 
-0.0097206 
0.0000082 
T h e e r r o r s i n W e d e p o h l ' s a p p r o x i m a t i o n a re p r a c t i c a l l y n e g l i g i b l e f o r b o t h r e s i s t a n c e a n d 
i n d u c t a n c e u p t o a f r e q u e n c y o f 10MHz. 
T h e e r r o r s i n S c h e l k u n o f f ' s a p p r o x i m a t i o n a r e a r o u n d 3.5% u p t o a f r e q u e n c y o f 10kHz 
a n d d e c r e a s e s f o r h i g h e r f r e q u e n c i e s . S c h e l k u n o f f ' s a p p r o x i m a t i o n g i v e s t h e s a m e v a l u e s 
as W e d e p o h l ' s a p p r o x i m a t i o n f o r t h e i n d u c t a n c e t e r m . 
T h e e r r o r s i n B i a n c h i ' s a p p r o x i m a t i o n a r e n e g l i g i b l e f o r f r e q u e n c i e s u p t o 1kHz bu t 
i n c r e a s e t h e r e a f t e r f o r t h e r e s i s t a n c e t e r m . T h e e r r o r i n t h e i n d u c t a n c e is 3.8% f o r t h e 
w h o l e f r e q u e n c y r a n g e . 
- 44 -
Table S-4 
Internal Impedance Zk of a Tubular Conductor 
tvith Current Returning Ouleide 
(p = 2.1 10~ 7/?m,a = 0.0385m ,6 = 0.0413m) 
FREQUENCY TUBE WEDEPOHL SCHELKUNOFF BIANCHI 
(Hr) 
RE< 51 STANCE ( j B / J ;m) 
.01 0. 299163 0.299163 0. 309686 0. 299163 
.1 0. 299163 0.299163 0. 309686 0. 299163 
1 0. 299163 j 0.299163 0. 309686 0. 299163 
10 0. 299163 0.299163 0. 309686 0. 299163 
100 0. 299169 0.299169 0. 309691 0. 299169 
1000 0. 299720 0.299721 0. 310243 0. 299741 
10000 0. 350679 0.350729 0. 361252 0. 352539 
100000 1. 120643 1.120912 ; i . 131444 
; T . 
149755 
1000000 3. 518632 3.518953 3. 529473 • 3. 631926 
10000000 11 .10564 11 .10605 11 .11652 1 1 .48527 
INI HJCTANCE (uH/ 'km) 
.01 51759: 4.51977 i *• 51977 I 4. 67836 
. 1 : 4 . 51759 j 4 . 5 1 9 7 7 i 4. 51977 i 4. 67836 1 4 . 51759 1 4 . 5 1 9 7 7 ! 4. 51977 
! 4. 67836 
10 4 . 51759 4 . 5 1 9 7 7 4 . 51977 4. 67836 
100 4 . 51756 4 . 5 1 9 7 5 4. 51975 4. 67834 
1000 4 . 51510 4.51728 4. 51728 4. 67578 
10000 4 . 28865 4 . 2 9 0 5 2 4 . 29052 4. 441 06 
100000 1 . 76453 1 .76463 1. 76463 1 . 82655 
1000000 0. 55844 0.55844 0. 55844 0. 57804 
10000000 0. 17660 0.17660 0. 17660 0. 18280 
- 45 -
°icr2 io"' 1 io' 101 103 io 4 io 5 10s io 7 
Frequency [Hz] 
Figure S.7(a),(b) - Impedance Zb of a Tubular 
Conductor (with External Return) as a Function of Frequency. 
- 46 -
2-
O 
k-
UJ 0 
<o o c to 
to _•> 
s 
/ 
V 
/\ 
-
s 
/ 
V 
/\ 
WEDEPOHL 
SCHELKUNOFF 
B I A N C H I 
o 
fc— fc— 
UJ 
0 
a> 
u c -to 
u 
3 
TJ c -2 
10~ 2 1 0 " ' 1 10 ' 1 0 2 1 0 S 1 0 " 10* 10* 1 0 7 
WEDEPOHL 
SCHELKUNOFF 
B I A N C H I 
— A ] II. i i i . i II,i| i i i . nn, • J • i in., .i i . i . i i , i . , i mi, III nm, .MI mil i i 11 in 
1 0 ~ 2 1 0 " 1 1 1 0 ' 1 0 2 1 0 S 1 0 4 1 0 S 10* 1 0 7 
Frequency [Hz] 
Figure S.8 - Errors in Wedepohl's Schelkunoff's 
and Bianchi's Formulae for Zt 
- 47 -
The cpu times were found to be: 
TUBE 0.058 ms 
Wedepohl 0.038 ms 
Schelkunoff 0.037 ms 
Bianchi 0.033 ms 
3.4 Conclusion 
The exact or classical formulae for finding the internal impedances for a tubular conduc-
tor were given by Schelkunoff. Since these formulae were not suitable for hand calculation pur-
poses, approximations for these classical formulae were developed by many authors, including 
Schelkunoff himself. In this chapter, the accuracy and the cpu time taken by these approxima-
tions were compared with the classical formulae for the tubular conductor of a typical cable. 
The displacement current term is neglected in subroutine TUBE and in Ametani's cable 
constant routine, which use the exact formulae. Though not discussed in detail, a modified 
subroutine TUBEC was developed which takes the displacement current into account. In all 
cases, the results from TUBEC and TUBE were identical within the accuracy shown in the 
tables. 
Subroutines were then written for the approximate formulae of Wedepohl, Schelkunoff 
and Bianchi, and the values obtained from these approximations were compared with the 
values obtained from the classical formula. Wedepohl's approximation formulae were indeed 
very good if the conductor thickness is small compared to its mean radius. The approximation 
proposed by Schelkunoff is similar to that proposed by Wedepohl except for the 2nd term. 
Schelkunoff approximated the modified Bessel functions in the exact formulae by the asymp-
totic series and retained only two terms, which produces reasonably accurate results as long as 
the argument is larger than 8 [34]. In the example, the argument terms | ma | and | mb | did 
not reach the value 8 up to frequencies of 2kHz. Hence, approximations based on the asymp-
totic series would obviously produce errors at low frequencies. 
Bianchi's approximation is only good at low frequencies less than 1kHz, for the resistance 
and has an acceptable error of 3-4% in the inductance up to frequencies of 10MHz. It should 
also be noted that all the above approximations are valid only if the thickness of the conductor 
is smaller than the mean radius of the tubular conductor. 
- 4 8 -
The routines for Wedepohl's approximation - WEDAP, Semlyen's approximation -
SEMAP, Schelkunoff's approximation - SCHAP, Bianchi's approximation - BNCAP and TUBEC 
were written bythe author. If these routines were written by a more experienced programmer, 
they might consume less cpu time than shown earlier. Even then, the cpu time for the exact 
formula (TUBE) would not be much more than that of the approximations. Therefore, the 
exact formula with subroutine TUBE is recommended for computer solution. For hand calcula-
tion or for calculations with electronic calculators, Wedepohl's formulae are recommended. 
- 49 -
4. EARTH RETURN IMPEDANCE 
The self and mutual impedance of conductors with earth return are of importance in stu-
dies of inductive interference in communication circuits from nearby overhead lines or under-
ground cables. Also they are important in the calculation of voltages in power lines or com-
munication circuits due to lightning surges or other transients [14]. Generally, the earth acts 
as a potential return path for currents in the underground, aerial or submarine cables. The 
values of cable constants therefore depend on the earth return impedances. In practical situa-
tions, the earth's electrical characteristics such as resistivity, permeability and permittivity are 
not constant. However, simulation results came reasonably close to fied test results if a homo-
geneous earth is assumed. The equations for self and mutual impedances are therefore 
developed with that assumption. 
The impedances are obtained from the axial electric field strength in the earth due to the 
return current in the ground, which in turn, can be obtained from Maxwell's equations. If a 
cable is assumed buried in an infinite earth, (where the depth of penetration of the return 
current in the ground is smaller than the depth of burial, or, in other words, the distribution of 
return current is circularly symmetrical) the electric field strength can be easily derived, 
because only the earth medium must be considered in Maxwell's equations (Appendix C). On 
the other hand, if the earth is treated as semi-infinite, (where the depth of penetration of the 
return current is larger than the depth of burial so that the depth of penetration of the return 
current is not circularly symmetrical) the problem of finding the axial electric field strength in 
the earth is quite complex, because both air and earth media must be considered in Maxwell's 
equations (Appendix D). The solutions for the electric field strengths, for both infinite and 
semi-infinite earth are derived, assuming first that the conductor is a filament with negligible 
radius. In the case where the return current in the ground is circularly symmetric, exact equa-
tions are still easy to derive for cables of finite radius. It is quite difficult, however, to extend 
the equations for the filament conductor to a conductor of finite radius in the case where the 
earth return current distribution is not circularly symmetric. In this chapter, we discuss the 
conditions under which these equations derived for the filament conductor can be extended to a 
conductor of finite radius. Furthermore, the effect of neglecting the displacement current term 
is discussed as well. 
Ametani's Cable Constant routine in EMTP uses different formulae for these impedances. 
His approach, as well as the approximations proposed by Wedepohl and Semylen are discussed 
and compared with the exact equations. Another impedance of interest is the mutual 
impedance between a buried conductor and an overhead conductor. This topic is also covered 
here. 
- 50 -
4.1 Earth Return Impedance of Insulated Conductor 
The simplest underground cable consists of a conductor laid at depth d below the surface 
of the ground with insulation around it which forms a concentric dielectric cylinder of external 
radius a. The earth then forms the return path. The ground return self impedance Zti is 
defined as the ratio of the axial electric field strength at the external surface of the insulation 
to the current flowing in the cable. The earth return mutual impedance between the loops of 
two buried, insulated conductors is defined as the ratio of the axial electric field strength at 
the external surface of the insulation to the current flowing in the other conductor and vice 
versa. 
As a preliminary step, the self and mutual impedances will be first found on the assump-
tion that the cables are buried in an earth which is homogeneous and infinite in extent. 
Clearly, this situation does not arise in practical applications, although it is a reasonable 
approximation if the cable is buried at great depth, or if it is at modest depth but the frequen-
cies are so high that the return current will flow very close to the cable. Furthermore, this 
treatment will be found useful in justifying the simplifying assumption 2 in section 2.1 of 
Chapter 2, and will also be helpful in interpreting the results for the more general case of earth 
return impedance in a homogeneous semi-infinite earth. 
4.2 E a r t h Return Impedance in a Homogeneous Infinite E a r t h 
The calculation of earth return impedance in an infinite earth is relatively easy since 
there is no surface discontinuity as in the semi-infinite case (earth and air). 
With the assumptions mentioned]in Chapter 2 (except for assumption 3), it is shown in 
Appendix C that the electric field strength at a radial distance r from a cable of insulation 
outer radius a carrying current / which returns through the earth can be written as: 
E = . _ pml Ko(mr) ( 4 ,) 
ITXQ. A'j(mo) 
where 
p = resistivity of the earth 
m = intrinsic propagation constant of the earth. 
The earth return self impedance per unit length of the cable is obtained from equation 
(4.1) by substituting r = a together with the general relation E = —ZI. It also follows from Zb 
of equation (3.1b), if we regard the earth as a tubular conductor with inside radius a whose 
outside radius b goes to infinity. 
- 51 -
The mutual impedance between the cable and a filamentary insulated conductor at a 
radial separation R is obtained by substituting r = R. The mutual impedance between two 
cables with finite radii over their insulation is different from the case of a cable of finite radius 
and a filament conductor. It can, however, be deduced from the mutual impedance between 
the cable and a filamentary type conductor by invoking the law of reciprocity of mutual 
impedance [9]. 
We have already shown in equation (4.1) that the mutual impedance between a cable of 
radius o and a filament conductor separated radially by a distance R is given by: 
Z --pm 
K0(mR) 
2na A'j(ma) (4.2) 
Hence, from the law of reciprocity of mutual impedance, the electric field strength on the 
surface of the cable due to a current / in the filament can be written as: 
E = -
pmIK0(mR) 
2za K t ( m a ) 
(1.3) 
Now consider a cable of radius 6 carrying a current / as shown in Figure 4.1. The electric 
field strength at a point P at a radial distance R is given by: 
Earth 
Figure 4-1 - Electric Field Strength al point P 
pm I'K0(mR) 
27rbKl(mb) 
The same field strength will be experienced at point P, due to a filament conductor placed 
along the axis of the cable and carrying a current f, but now the equation will be: 
pmfK^mR) 
2nb(\/mb) 
(as 6 - 0 , Ki(mb)-\/mb) 
- 52 -
From equations (4.4) and (4.5) we obtain the value of f to be equal to f/mb Kx(mb). 
Therefore, the electric field strength in the soil outside a cable of radius b carrying current / is 
indistinguishable from that of a current filament placed along the axis of the cable and carry-: 
ing a current f/mbK^mb). 
Hence, the electric field strength at the surface of the cable of radius o, which is at a 
radial distance R from a cable of radius b is found from equation (4.3) by substituting for I the 
value of f. Therefore, we obtain the field to be: 
_ Pm*fK0(mR) . 
27imaK1(ma)mbK1(mb) ' ' ' 
Hence, the mutual impedancebetween two cables of radii o and 6 respectively, buried in a 
homogeneous infinite earth is given by: 
pm2K0(mR) - • (''•") 27tma Ki(ma)mb K}(mb) 
Now we can deduce an interesting result. The series expansion of A'j(r) shows that as 
z-0, K1(x)~l/x. Therefore, for small values of \ma\ and |mi|, or in the limiting case of 
o,6-0 (filament conductors), the self and mutual impedances obtained from equation (4.2) and 
(4.7) will be given by: 
Zs = -^Koima) ... (4.8a) 
Zm = ^ ~K0(mR) (4.8b) 
It is interesting to know the error if the cable of finite radius is replaced by a filament 
conductor placed along its axis. For p = 10O— m (low earth resistivity), a = b — 7.5cm (large 
radii), and a separation of 30 cm, values which perhaps represent a worst case, the errors in 
the resistance and inductance from equation (4.8b), as compared with equation (4.7), is plotted 
in Figure 4.2. 
From Figure 4.2 we see that the approximate formulae have an error of less than 29o, up 
to \ma\ =? |m6| = 0.1. This happens at a frequency of IMHz. For much lower values, the 
error is practically negligible. This result is important as it will be used in justifying the exten-
sion of formulae for filament conductors to cables of finite radii. 
4 . 3 E a r t h R e t u r n Impedance in a Homogeneous S e m i - I n f i n i t e E a r t h 
The self and mutual impedances of cables buried in semi-infinite homogenous earth are 
deduced from the electric field strength in the ground due to a buried filament conductor. 
- 53 -
w H 
0) 
o 
c 
1 0 1 
—I-
t o 
t o c u az 
-3 
10 
25 
15 
cu 
o 
c 
CO 
o 
T 3 C 
-15 
/ 
' Y 
/ 
10"s 10" 10~3 
I I I! I I ! I 1 T T1!| •-' ! 
10"2 10"' 
\ 
'1 - T TTTT i 
10' 
-25 -f 1 1 I 1 Mll| I 1 I I I M i l 1 U I I I I I H l | 1 1 I 1 I Ml| 1 1 l l l l l l , 
10" 10" 10" 10" 10" 10- 10' 
ma 
Figure 4-2 - Error in Replacing a Conductor of Finite Radius 
by a Filament Conductor 
- 54 -
The electric field strength in the ground due to a buried insulated filament carrying a 
current which returns through the soil was first deduced by Pollaciek [l). In fact, he derived 
formulae for four cases: 
(1) The electric field strength in the air due to a current-carrying conductor in the air. 
(2) The electric field strength in the earth due to a current-carrying conductor in the air. 
(3) The electric field strength in the air due to a current-carrying conductor in the earth. 
(4) The electric field strength in the earth due to a current-carrying conductor in the earth. 
The mathematical derivation in all four cases become complicated by the plane of discon-
tinuity at the earth's surface. Pollaczek [l] does not discuss the derivations and only mentions 
that they have been obtained through the reciprocity of Green's functions. 
Recently Mullineux [10,11,12] obtained expressions for the fields produced in air and earth 
due to an overhead conductor by using double integral transformation. This technique is 
equally applicable to buried cable systems [22], and is used in Appendix D to obtain the four 
types of fields. 
Comparisons between filamentary type conductors and conductors of finite radii for the 
infinite earth give every reason to expect that these formulae in Appendix D for filament con-
ductors will be accurate enough for cables of finite radii provided that the condition | ma | <0.1 
is satisfied. Hence, for the case of a buried cable at depth h, the electric field strength in the 
soil resulting from a net current I flowing in the cable is given by equation (D.32(c)) in Appen-
dix D as: 
E__ — | e x p [ - ( o
g + m2)\h-y\] - e X P [ - ( ^ + m2)\h + y\\ 
2(p- + m~) 
+ r exp(-|ftlfyl VV+ttr) exp(jd>x )d<t> (4.9) 
where . ' ; " . 
x = horizontal distance between the filament and the point at which the field is being deter-
mined, 
y = depth at which the field is being determined, 
p = resistivity of the earth, and 
m = intrinsic propagation constant of the earth. 
The first integral term is identified as Jfc:0(mr?) - rTo(mZ)j [1,22]; where R = Vi" + (h-yf 
and Z = V i s + (/i + y)2. The second integral part can be numerically evaluated [28] or 
- 55 -
expanded into an infinite series [22]. The series expansion is in the form of modified Bessel 
functions. Therefore, equation (4.9) can be written in the series expansion form as follows: 
= -^~\KdmR) - K0(mZ) -f —K^mZ) + ^ "*^Ki\mZ) 
In \ Z mZ 
- } (* •+ ml)P-mt ~ ^ F(mZ,\x |,/)) (4.10) 
where 
F(mZ,\x\,l)= f f c V l ^ T 2 ^==r\e-mZtdt (4.11) 
1/2 I Vl-t2 J 
Now the earth return self and mutual impedance terms may be extracted from equation (4.10) 
or from numerically integrating equation (4.9). The self impedance term is obtained by choos-
ing the coordinates z and y to correspond to the location of the external surface of the cable 
and the mutual impedance by simply inserting the coordinates of the second cable axis . 
The numerical integration of equation (4.9) is quite difficult as the solution is highly oscil-
latory. For example. Figure 4.3 shows the solution for the resistance and inductance of the 
earth return self impedance at a frequency of 1MHz, for the cable discussed in Appendix A. 
For such cases, special numerical integration techniques have to be applied [28]. The numerical 
integration routine available in the UBC system library DCADRE [26], which uses a cautious 
adaptive Romberg extrapolation technique, has been used in obtaining the solution for the 
equation (4.9). The solution converges fast for the case when the two cables are buried at d i f -
ferent depths below the earth, but in the case when the cables are at the same depth or in the 
case of finding the earth return self impedance, the convergence is rather slow. To explain this 
phenomena, consider the first integral part on the right hand side of equation ( 4 . 9 ) , i.e.. 
e x p [-(*2 + m 2 ) |A- j , | ] - exp[-(cA2 + m2)\h + y\] 
?(c*2 + mz) expO^x )</<•> ( 4 . 1 2 ) 
If h = y then the first exponential term within the closed brackets { } will become 1 and the 
integration now becomes 
j l - e xp[-(cr+m 2 ) | / i+y | j (<p2+m2)\h+y\ Uxp{j<t>x)d<t> 
- 56 -
10 
6-
T 
o> o c 
JO 
V) o> 
- 2 
-6 
-10 
10' 
50 
£ 
o o c o 
30-
10 
o ~ 1 0 D 
- 3 0 
- 5 0 
10' 
A 
V 
1 "' '^f 1 T I 1 1 I 1 1 1 1 | 
10 2 10 3 
I I I i l l 
\ 
\ 
\ / 
7— ••T-",,T "I m I I I 1 I I I I i 
10* 1CV 
Interval 
104 
i i i i i i 
104 
Figure 4-8 - Solution of Real and Imaginary Part of Equation (4-9), 
for a Frequency of 1MHz . 
- 57 -
Even though the exponential term within the closed brackets { } approaches zero very 
fast, we are left with 
which causes the slow convergence. 
On the other hand, if h^y then both the first and second exponential terms within the 
brackets { } of equation (4.12) will approach zero and hence the convergence is faster. 
Due to the slower convergence, the cpu time taken to compute the self impedance or the 
mutual impedance in case of two cables buried at the same depth is relatively high. For the self 
impedance, the computer cost varied between $.50 and $1.00, for one particular frequency. 
However, the results obtained by applying the DCADRE integration routine to equation (4.9) 
and the results obtained from equation (4.10) are almost identical, as shown in Table 4.1 for 
the mutual impedance between two cables with the following data: 
depth of cable 1, y = 0.75m 
depth of cable 2, h = 0.76m 
radial distance between the two cables = 0.5m 
earth resistivity peyAh = 100/7— m. 
Hence, from Table 4.1 we can see that the equation (4.10), which is the series expansion 
for the classical equation (4.9) is accurate enough for practical purposes. Therefore, equation 
(4.10) is takenas the standard equation for finding the self and mutual impedances of buried 
cables in a semi-infinite earth, and the results obtained by the other formulae proposed by 
Semlyen [24], Ametani [27], and Wedepohl [22] are compared with respect to it. 
4.4 Formulae Used by Ametani, Wedepohl and Semlyen. 
Ametani's approach is implemented in BPA's Cable Constant routine of EMTP, and is 
based on Carson's formulae for overhead conductors. The self and mutual impedances of over-
head conductors with earth return effects can be derived from equation ((D.32(b)) Appendix C) 
together with the relation E — — ZI. 
In the case of overhead conductor at a height h from the ground, the electric field 
strength in air at a point (whose height is y and which is at horizontal distance x from the con-
ductor) due to current / flowing in the conductor is given by 
- 58 -
Table 4.1 
Solution of Pollaczek'o Equation by Numerical 
Integration and Using In finite Series 
FREQUENCY 
(Hz) 
NUMERICAL 
INTEGRATION 
EQUATION 4.10 
RESISTANCE (8/kr n) 
.01 
. 1 
1 
10 
100 
1 ,000 
10,000 
100,000 
1,000,000 
10,000,000 
0.00000986985 
0.00009870394 
0.00098721243 
0.00987746990 
0.09894307000 
0.99462669000 -
10.0999510000 
105.026550000 
1119.21050000 
10416.7130000 
0.00000986986 
0.00009870399 
0.00098721145 
0.00987752230 
0.09894471000 
0.99467682000 
10.1014090000 
105.062940000 
1119.73660000 
10410.0480000 
REACTANCE (fi/km 
.01 
. 1 
1 
10 
100 
1 ,000 
10,000 
100,000 
1,000,000 
10,000,000 
0.00014814024 
0.00133672137 
0.01192043120 
0.10477298900 
0.90245276590 
7.57239400000 
61.8788630000 
461.037952000 
3018.40132100 
13049.3294800 
0.00014814024 
0.00133672130 
0.01192028400 
0.10472980000 
I 0.90245110000 
! 7.57234170000 
. 61.8625200000 
460.988500000 
3017.08610000 
13031.2620000 
= — 
2x 
ln(Z/R) + J (4.15) 
fhere 
Z - V i 2 + (A+y ) z , 
- 59 -
m = intrinsic propagation constant of the earth. 
The integral part of equation (4.15) can be further simplified to 
o U | + \ U 2 +• m2 
(4.16) 
Equation (4.16) is widely known as Carson's formula. Strictly speaking, this formula is only 
valid for the case of overhead conductors. Ametani used this correction term in finding the 
earth return self and mutual impedances of buried conductors, instead of the second integral 
term used in equation (4.9). 
Carson's formula given by equation (4.16) can be numerically integrated [28] or can be 
expanded into an infinite series [5], in terms of r = | mZ \. Ametani chooses the latter 
approach. His procedure uses 2 different series, one when r S 5 and the other one when r>5. 
Recently, Shirmohamadi of Ontario Hydro [28] and L. Marti at UBC discovered that the error 
between the numerical evaluation and the asymptotic expansion is as high as 5-8% as shown in 
Figure 4.4, for the values of r between 5-10, depending on the value of 8. 
) 0 0 0 4 0 0 0 
r«CQUCNCT ( H i ) 
Figure 4-4 - Relative Error in the Evaluation of Carson's Formula 
with an Asymptotic Expansion. 
- 60 -
Shirmohamdi avoids this error by using Gauss-Legendre quadrature technique for the direct 
evaluation of equation (4.13) in this region of r, while L. Marti [24] avoids these errors by 
extending both the asymptotic and infinite series, and by using a switchover criterion which 
depends on the geometry of the line (ie. on the value of 0). 
The use of Carson's formula for underground systems will be reasonably accurate at low 
frequencies because the value of m 2 with the exponent term exp(—| h+y\ Vc4 2+m 2) in equation 
(4.9) is very small compared to the value of 4>z at low frequencies, and, therefore, can be 
ignored. At high frequencies, however, that term becomes quite significant and, therefore, can-
not be ignored. For this reason, the resistance and inductance obtained by Ametani's method 
has an error in the order of 10% or more for frequencies above lKHz when compared with 
Pollaczek's equation, i.e., equation (4.9). 
Wedepohl and Wilcox [22], who proposed the infinite series expansion form of equation 
(4.9) gave an approximation to the infinite series expansion (equation (4.10)) which is valid only 
if the condition |mZ| <0.25 is satisfied. Their closed-form approximation for the self and 
mutual earth return impedances are given by: 
Z, = 
pm 
2n 
pm 
2ir 
.lnhHEl + 0 5 _ ± m h 
2 3 
•In— L + 0.5 ml 
2 3 
(4.17a) 
(1.17b) 
where 
7 = Euler's constant, 
h = depth of burial of the conductor, 
I = sum of the depths of burial of the conductors, 
R = V i 2 + (h-yf and 
m = intrinsic propagation constant of the earth. 
Semlyen and Wedepohl [24] developed another interesting formula for the self impedance 
of a cable of radius r in terms of complex depth which is nothing but 1/m, defined here as p. 
Accordingly, the self impedance term is given by 
Zs = " ^ M r + V'r) (4.18) 
- 61 -
4.5 Effect of Displacement Current and Numerical Results 
So far. the effect of displacement currents has been ignored. As shown in Chapter 2 . 
there is no noticeable error in the internal impedances of tubular conductor if displacement 
currents are ignored. Hut unlike in good conductors, the displacement currents in the earth 
arc noticeable, at least at high frequencies. The displacement current term can be easily incor-
porated in equation (4-10) by snbsti :ing Vm"-t> J/i( for in. Table 4.2 shows the values of 
resistance and inclu;'. :nce with and without the displacement currents. 
Tabic 4.2 
Earth Return Self Impedance with and without 
Displacement Current Term 
FREQUENCY 
(Hz) 
WITH 
DISPLACEMENT 
CURRENT 
WITHOUT 
DISPLACEMENT 
CURRENT 
RESISTANCE (8/ki 
.01 
. 1 
1 
10 
1 00 
1 000 
10 000 
100 000 
1 oob ooo 
10 000 000 
0 . 0 0 0 0 0 9 8 6 9 8 5 
0 . 0 0 0 0 9 8 7 0 3 9 3 
0 . 0 0 0 9 8 7 2 0 9 8 6 
0 . 0 0 9 8 7 7 4 7 6 7 1 
0 . 0 9 8 9 4 3 7 1 2 3 0 
0 . 9 9 4 6 8 7 0 0 0 0 4 
1 0 . 1 0 5 6 6 3 3 5 0 4 
1 0 5 . 5 7 2 6 4 6 5 6 8 
1 1 7 2 . 7 4 9 2 5 5 8 8 
1 5 1 7 0 . 9 6 0 5 5 3 9 
0 . 0 0 0 0 0 9 8 6 9 8 6 
0 . 0 0 0 0 9 8 7 0 3 9 3 
0 . 0 0 0 9 8 7 2 0 9 8 2 
0 . 0 0 9 8 7 7 4 7 3 4 0 
0 . 0 9 8 9 4 3 3 9 5 9 5 
0 . 9 9 4 6 5 5 3 7 1 8 7 
1 0 . 1 0 2 4 6 0 2 9 2 6 
1 0 5 . 2 3 9 9 0 9 5 6 4 
1 1 3 6 . 3 5 1 8 0 5 4 6 
1 1 5 9 3 . 2 5 6 9 4 5 0 
INDUCTANCE ("H/ nm) 
o
 
—
 
o
o
-
o
o
o
-
O
O
O
O
O
-
O
O
O
O
O
O
—
 
• 
o
 
O
O
O
O
O
O
O
 —
 —
 
— 2 . 8 2 4 7 8 6 9 2 8 4 4 
2 . 5 9 4 5 1 9 8 3 7 3 7 
2 . 3 6 4 2 3 4 1 5 0 9 9 
2 . 1 3 3 8 8 9 7 4 4 4 6 
1 . 9 0 3 3 5 9 8 2 7 5 6 
1 . 6 7 2 2 4 5 2 5 3 7 4 
1 . 4 3 9 3 0 2 4 4 3 3 3 
1 . 2 0 0 8 0 3 8 4 1 19 
0 . 9 4 7 1 2 6 1 4 3 5 7 
0 . 6 4 9 8 0 3 4 1 2 1 4 
2 . 8 2 4 7 8 6 8 9 7 1 2 
2 . 5 9 4 5 1 9 7 9 2 9 7 
2 . 3 6 4 2 3 4 1 1 6 6 1 
2 . 1 3 3 8 8 9 7 1 3 3 2 
1 . 9 0 3 3 5 9 7 9 9 1 6 
1 . 6 7 2 2 4 5 2 0 0 2 0 
1 . 4 3 9 3 0 1 5 6 5 1 4 
1 . 2 0 0 7 8 3 2 9 1 3 9 
0 . 9 4 6 9 4 2 4 2 9 6 3 
0 . 6 6 8 2 2 9 4 1 7 1 8 
- 62 -
The errors in the answers obtained by neglecting the displacment current arc shown in 
Figure 4.5. The error in the resistance is less than 3% up to a frequency of 1MHz and 
increases to 209o in the frequency range 1MHz- lOMHr. 
o i_ k. 
UJ 
CD 
o c 
to 
CO 
CD 
CC 
- 2 5 
- 5 -
-15 
i i i IIIMI— n i ] — i i i i n n , — i i i M i n i — i i i inn, 1 i i mii| nil 1 i i mill i i i mil! 
10~ 2 1 0 " 1 10' 10 2 10 J 10 4 10 5 10 6 io 7 
Frequency [Hz] 
Figure 4-5 - Error in the Earth Return Self Impedance 
if the Displacement Current is Ignored 
- 63 -
Hence, we can neglect the effect of displacement current terms up to a frequency of 1MHz 
which is well within the limitsof practical interest. 
The earth return self impedance is compared next for the following approaches, with the 
displacement current term neglected: 
1. Pollaczck's original formula, 
2. Wedepohl's approximations, -
3. Ametani's approach, 
4. Semlyen's approximation. 
The value of resistance and inductance at different frequencies are tabulated in Table 4.3. 
Figures 4.6(a) and 4.6(b) illustrate the variation of resistance and inductance in the frequency 
range 0.001 Hz to 10MHz. The errors are plotted in Figure 4.7 for the same frequency range. 
Wedepohl's approximation gives an error of less than 1% up to a frequency of 100kHz, for the 
resistive part, thereafter it increases steadily. It is around 2 5 % at a frequency 1MHz. The 
error in the inductive part is almost zero up to a frequency of 1MHz. The reason for the 
noticeable error in the resistance, beyond a frequency of 100kHz, is that the condition 
| mZ | <0.25 is violated. Semlyen's approximation is good at low frequencies for the resistive 
part but at high frequencies the error is higher. It has an error of around 4% in the case of 
inductive part over the whole frequency range. As mentioned earlier, the error i i i Ametani's 
procedure is not significant at low frequencies, but increases from 2 % to 20% iu (.-be frequency 
range 10kHz to 1MHz as shown in Figure 4.7. 
Three routines were developed for the calculation of earth return self and mutual 
impedances with Pollaczek's original formula. The part which is .difficult to evaluate is the 
integral term F(mZ,\x\,l) (equation (4.11)) in equation (4.10).. This part can either be 
expanded into a series with a suitable number of terms and each term can then be integrated, 
or a suitable library subroutine for numerical integration can be used. Routine SEARTH 
developed by the author uses the first approach by considering 15 terms. Routine LEARTH 
developed by Luis Marti uses the UBC library subroutine DCADRE for the evaluation of the 
integral. One more routine, namely CEARTH was developed by the author, which can take the 
displacement current into account. This routine also uses the UBC library subroutine DCA-
DRE to evaluate the function discussed earlier. Routines SEARTH and LEARTH give identical 
answers but differ in the cpu time. The former one takes 3.4 ms while LEARTH takes 2.9 ms 
for the evaluation of resistance and inductance at a particular frequency. Routine CEARTH 
takes 23.0 ms for the same evaluation. The routine for Ametani's approach takes 5.30 ms, 
while the routines for Wedepohl's and Semlyen's approximations take 0.30 ms and 0.24 ms of 
cpu time, respectively. 
Table 4.S 
Earth Return Self Impedance 
as a Function Frequency 
F R E Q U E N C Y P O L L A C Z E K W E D E P O H L A M E T A N I S E M L Y E N 
(Hz) 
R E ; > I S T A N C E (£2/1 cm) 
. 0 1 0 . 0 0 9 8 8 2 5 0 . 0 0 9 8 8 2 5 0 . 0 0 9 8 7 7 5 0 . 0 0 9 8 8 2 5 
. 0 1 0 . 0 0 0 0 0 9 9 0 . 0 0 0 0 0 9 9 0 . 0 0 0 0 0 9 9 0 . 0 0 0 0 0 9 9 
. 1 0 . 0 0 0 0 9 8 7 0 . 0 0 0 0 9 8 7 0 . 0 0 0 0 9 8 7 0 . 0 0 0 0 9 8 7 
1 0 . 0 0 0 9 8 7 2 0 . 0 0 0 9 8 7 2 0 . 0 0 0 9 8 6 7 0 . 0 0 0 9 8 7 0 
1 0 0 . 0 0 9 8 7 7 5 0 . 0 0 9 8 7 7 5 0 . 0 0 9 8 6 1 8 0 . 0 0 9 8 6 9 2 
1 0 0 0 . 0 9 8 9 4 3 4 0 . 0 9 8 9 4 5 7 0 . 0 9 8 4 5 1 9 0 . 0 9 8 6 8 4 0 
1 0 0 0 0 . 9 9 4 6 5 5 4 0 . 9 9 4 8 5 6 1 0 . 9 7 9 5 2 6 3 0 . 9 8 6 5 7 8 4 
1 0 0 0 0 1 0 . 1 0 2 4 6 0 1 0 . 1 1 9 2 8 8 9 . 6 5 6 4 6 8 0 9 . 8 5 7 5 3 1 3 
1 0 0 0 0 0 1 0 5 . 2 3 9 9 1 1 0 6 . 5 9 1 7 2 9 3 . 4 9 8 2 7 0 9 8 . 3 1 5 0 5 3 
1 0 0 0 0 0 0 1 1 3 6 . 3 5 1 8 1 2 3 6 . 6 4 3 8 9 1 3 . 0 3 2 1 4 9 7 4 . 9 9 1 2 3 
1 0 0 0 0 0 0 0 1 1 5 9 3 . 2 5 7 1 7 7 6 5 . 2 8 8 1 0 8 2 6 . 0 5 8 9 4 9 8 . 8 3 9 4 
I N I X J C T A N C E (mHy 'km) 
. 0 1 2 . 8 2 4 7 8 6 9 2 . 8 2 4 7 8 3 8 2 . 8 2 4 7 9 1 7 2 . 7 0 1 6 0 4 8 
. 1 2 . 5 9 4 5 1 9 8 2 . 5 9 4 5 1 6 7 2 . 5 9 4 5 4 1 8 2 . 4 7 1 3 4 6 7 
1 2 . 3 6 4 2 3 4 1 2 . 3 6 4 2 3 1 0 2 . 3 6 4 3 1 0 5 2 . 2 4 1 0 8 9 5 
1 0 2 . 1 3 3 8 8 9 7 2 . 1 3 3 8 8 6 6 , 2 . 1 3 4 1 3 7 8 2 . 0 1 0 8 3 5 2 
1 0 0 1 . 9 0 3 3 5 9 8 1 . 9 0 3 3 5 6 4 1 . 9 0 4 1 5 0 1 1 . 7 8 0 5 8 9 8 
1 0 0 0 1 . 6 7 2 2 4 5 2 1 . 6 7 2 2 3 8 6 1 . 6 7 4 7 4 1 4 1 . 5 5 0 3 7 2 9 
1 0 0 0 0 1 . 4 3 9 3 0 1 6 1 . 4 3 9 2 6 2 9 1 . 4 4 7 1 0 6 4 1 . 3 2 0 2 4 5 9 
1 0 0 0 0 0 1 . 2 0 0 7 8 3 3 1 . 2 0 0 4 1 1 8 1 . 2 2 4 5 1 5 7 1 . 0 9 0 4 0 3 2 
1 0 0 0 0 0 0 0 . 9 4 6 9 4 2 4 0 . 9 4 2 9 8 1 3 1 . 0 1 2 6 2 2 8 0 . 8 6 1 4 5 9 7 
1 0 0 0 0 0 0 0 0 . 6 6 8 2 2 9 4 0 . 6 2 6 7 9 7 5 0 . 7 9 6 2 6 8 9 0 . 6 3 5 3 5 6 5 
4.6 Cables Burled at Depth Greater than Depth of Penetration 
If the depth of burial is greater than the earth return current's depth of penetration, or 
other words, if the distribution of return current is circularly symmetrical, then the cable c 
be considered to be buried in an infinite earth. In practice, this can arise in two situations: 
1. Cables arc buried at large depths below the ground, 
\ 
- 05 -
POLLACZEK 
WEDEPOHL 
AMETANI 
SEMLYEN 
0_,j •• , , , , „ „ ....t r-^ >..nr 
1 0" J 10" 1 10' io2 10 
Frequency [Hz] 
Figure 4.6(a),(b) 
Earth Return Self Impedance 
as a Function of Frequency 
10' 105 10s 107 
- 66 -
ui 
25 
15-
5-
o 
-C <o - 5 . 
CO 
in 
CD 
or 
-15 
WEDEPOHL 
AMETANI 
SEMLYEN / 
/ 
— 25 "1—1 i 1111111—1 1 11111 n 1 1 11 IIIII 1 1 1111111—1 1 1 . .1.11 1 1 . null—1 1 1 m i i | 1 1 1 i i u i | 1 1 1 m i r 
10" 2 10"' 1 10' 10 2 10 J 10 4 10 5 10 6 10 7 
25 
15-
5-
LU 
CD 
o 
c - 5 ra 
o 
T3 
C 
— -15 
WEDEPOHL 
AMETANI 
SEMLYEN 
> 
— 2 5 ~f 1 1 T mrn r-i TTTTTTI 1 i 11 ini[ 1 i i itui] r - r r i - T T T T i r—r-rrrmj r i it ini] r i i rim\ r - r r m r t } 
10~ 2 10"' 1. 10' 10 2 10 J 10 4 10 S 10 6 10 7 
Frequency [Hz] 
Figure 4- 7 
Errors in Earth Return Self Impedance 
- 6 7 -
2. C a b l e s are b u r i e d a t n o r m a l d e p t h s ( l - 2 m ) b u t a re u s e d a t h i g h f r e q u e n c i e s ( 1 0 0 k H z a n d 
a b o v e ) . 
In s u c h s i t u a t i o n s , t h e d e p t h o f p e n e t r a t i o n i n t h e e a r t h is g i v e n b y : 
d = 5 0 3 . 3 \ / p e « u / / (4 .19) 
( w h e r e d is in m , / is i n fi—m a n d / is i n H z ) 
a n d b e c o m e s s m a l l e r t h a n the d e p t h o f b u r i a l . 
T h e s e c o n d p o s s i b i l i t y d o e s n o t a r ise n o r m a l l y i n p o w e r s y s t e m s t u d i e s . F o r a t y p i c a l 
u n d e r g r o u n d t r a n s m i s s i o n s y s t e m , w i t h a n e a r t h r e s i s t i v i t y o f 10 J ? — m , a n d a b u r i a l d e p t h o f 
l m , t h e f r e q u e n c y a t w h i c h the p e n e t r a t i o n o f t h e r e t u r n c u r r e n t in t h e e a r t h b e c o m e s less 
t h a n l m is 3 M H z o r h i g h e r . In p o w e r s y s t e m s , o n e r a r e l y e n c o u n t e r s s u c h f r e q u e n c i e s . If s u c h 
c a s e s d o a r i s e , h o w e v e r , the in f in i te e a r t h r e t u r n i m p e d a n c e f o r m u l a e g i v e n b y e q u a t i o n (4.2) 
a n d (4.7) c o u l d be u s e d t o find the e a r t h r e t u r n i m p e d a n c e s . 
It is i n t e r e s t i n g t o k n o w w h e t h e r e q u a t i o n (4 .10) f o r t h e s e m i - i n f i n i t e c a s e is s t i l l v a l i d if 
the b u r i a l d e p t h is l a r g e . T h i s c a n be c h e c k e d as f o l l o w s : 
If the e a r t h r e s i s t a n c e is a s s u m e d t o be 10 17— m, t h e n t h e d e p t h o f p e n e t r a t i o n g i v e n b y 
e q u a t i o n (4.19) wi l l be less t h a n 5.5 m , f o r f r e q u e n c i e s 0.1 M H z a n d h i g h e r . H e n c e , i f a 
c a b l e is b u r i e d at a d e p t h o f 5 . 5 m , t h e n t h e v a l u e s o b t a i n e d f o r e a r t h r e t u r n se l f a n d 
m u t u a l im p e d a n c e s , u s i n g t h e e q u a t i o n s (4 .2) o r (4.7), s h o u l d be t h e s a m e as t h o s e 
o b t a i n e d u s i n g e q u a t i o n (4.10) f o r f r e q u e n c i e s a b o v e 0 . 1 M H z . F i g u r e 4.8 s h o w s t h e d i f f e r -
e n c e in the v a l u e o f r e s i s t a n c e o b t a i n e d b y P o l l a c z e k ' s f o r m u l a a n d e q u a t i o n (4.2) f o r t h e 
e a r t h r e t u r n se l f i m p e d a n c e in t h e f r e q u e n c y r a n g e 1 0 k H z t o 1 M H z . T h e d i f f e r e n c e 
d e c r e a s e s f r o m 1 9 % t o less t h a n 1 % w h i l e t h e d e p t h o f p e n e t r a t i o n d e c r e a s e s f r o m 1 5 . 9 m 
t o 1 .59 in . H e n c e , it a p p e a r s t h a t P o l l a c z e k ' s f o r m u l a is c o r r e c t e v e n a t l a r g e d e p t h s o f 
b u r i a l e v e n t h o u g h it is b e t t e r t o use t h e e q u a t i o n s (4.2) a n d (4.7) f o r t h e c a s e o f i n f i n i t e 
e a r t h [9,22]. 
S u b r o u t i n e T U B E has a n o p t i o n f o r finding t h e in f in i te e a r t h r e t u r n se l f i m p e d a n c e , b u t it 
c a n n o t be u s e d for f i n d i n g the m u t u a l i m p e d a n c e . T h e r o u t i n e T U B E C d e v e l o p e d b y the 
a u t h o r has o p t i o n s f o r finding b o t h se l f a n d m u t u a l i m p e d a n c e s , a n d c a n t a k e d i s p l a c e m e n t 
c u r r e n t s i n t o a c c o u n t as we l l . 
4.7 M u t u a l I m p e d a n c e B e t w e e n a C a b l e B u r i e d i n t h e E a r t h a n d a n O v e r h e a d L i n e 
o r V i c e V e r s a 
A n o t h e r i m p e d a n c e o f i n t e r e s t t o p o w e r e n g i n e e r s as we l l as t o c o m m u n i c a t i o n e n g i n e e r s 
is t h e m u t u a l i m p e d a n c e b e t w e e n a n u n d e r g r o u n d c a b l e a n d a n o v e r h e a d l ine o r v i c e v e r s a . 
T h e e l e c t r i c field s t r e n g t h s in a i r d u e t o a c u r r e n t c a r r y i n g c o n d u c t o r b u r i e d in the e a r t h 
- 68 -
c u u c 
CD 
CU u c 
CO 
to 
CO 
cc 
2 0 
1 2 -
"* 4 -
-4 
- 1 2 
- 2 0 + 
104 10s 
Frquency [Hz] 
- i — i — • I I I 
106 
Figure 4.8 
Differences in Resistance Values of Semi-Infinite 
and In finite Earth Return Formulae 
or the field Zs_+ in earth due to a current-carrying conductor in the air is given by Equations 
(D.32(b)) and (D.32(d)), respectively, in Appendix D. In both cases the mutual impedance is 
given by: 
0 e x p j - / i | c*| -dV<t>2+ m 2j 
exp(j>| x | )d<f> 
| c6| + V(t>2+m2 
•where 
A = height of the conductor in air, 
d = depth of burial of the buried conductor, 
| i | = the horizontal distance between the conductors, 
m = intrinsic propagation constant of the earth. 
This integral can be evaluated in terms of infinite series in somewhat the same way as 
was done for E + + in [4,27] and for E__ in [22]. 
- 69 -
4.8 Conclusion 
To summarize, the self and mutual impedances of conductors with earth return were 
derived for two situations, namely for ' • 
1. Cables buried in infinite earth, and for 
2. Cables buried in semi-infinite earth. 
The impedances were obtained from the axial electric field strengths in the earth due to return 
currents in the ground. These electric field strengths were derived from Maxwell's equations, 
for filamentary type conductors of negligible radius. Since we were interested in cables of finite 
radius, the solutions for filamentary type conductors were extended to cables of finite radius. 
The solutions for the earth return impedance with semi-infinite earth is in infinite 
integral form. Wedepohl [22] transformed this infinite integral equation into an equation con-
sisting of Bessei functions. It was found that the values obtained from the numerical integra-
tion of the infinite integral and from Wedepohl's transformation were very close. Ametani's 
approach for finding earth return impedances which is implemented in Cables Constants rou-
tine in the BPA's EMTP and other approximations suitable for hand calculations were com-
pared for typical cable data. Ametani's approach gave erroneous results at high frequencies 
due to an erroneous assumption. Wedepohl's approximation was found to give reasonably 
accurate answers and is well suited for hand calculations. 
At the end of the chapter, the evaluation of mutual impedance between a buried conduc-
tor and overhead conductor, and vice versa, is briefly discussed. 
- 70 -
5. Laminated Tubular Conductors 
In Chapter 3, formulae for internal impedances of homogeneous tubular conductors were 
derived. These formulae are used in this chapter to obtain the impedances of cables whose 
core and sheath are made up of laminated conductors of different materials. A practical appli-
cation of this type of conductor was recently proposed by Harrington [32]. He suggested that 
the transient sheath voltage rise in a gas-insulated substation can be reduced by coating the 
conductor and sheath surfaces with high-permeability materials, thereby increasing the 
impedance of the surfaces for surge propagation, which in turn will damp out high frequency 
transients. 
5.1 Internal Impedances of a Laminated Tu b u l a r Conductor 
The internal impedances needed for laminated conductors are the same as those needed 
for homogeneous conductors, namely: 
1. The internal impedance z0(J of the laminated tubular conductor which gives the voltage 
drop on the inner surface when unit current returns through a conductor inside the tube. 
2. The internal impedance zbb of the laminated tubular conductor which gives the voltage 
drop on the outer surface when unit current returns through a conductor outside the 
tube. 
5 . 1 . 1 Internal Impedance with External Return 
Let us first number the layers consecutively with the inner most layer being number 1 as 
shown in Figure 5.1. For the analysis, we start with the mth outer most layer shown in Figure 
5.2. 
Let 
= internal impedance of the mth layer with current returning inside, 
= internal impedance of the mth layer with current returning outside, 
Z^ = mutual impedance between the two surfaces, 
zbmh — internal impedance of all m layers when the current return is external 
For the very first layer, we note that Zb[^ = z$\ If we use concentric loop currents as before in 
Chapter 3, then the loop current 7m_, of the first m —1 layers combined, returns on the inner 
surface of the mth layer, while loop current Im flows on the outer surface. Using Schelkunoff's 
theorem 2 from Appendix B, the electric field strength along the inner surface of the mth layer 
becomes 
- 71 -
Axis 
Layer (m-l) 
Layer m 
Figure 5.1 Numbering of Conductor Layers to Find the 
Internal Impedances of a Laminated Tubular Conductor 
r m 
Figure 5.2 Representation of the mth Layer 
dV 
dx 
— (ZTb lm %aa 'm-l) (5.1) 
But the inner surface of the mth layer is the outer surface of the first m - l layers combined for 
which the electric field strength is given by 
. ~ zbi 'm-l 
dx 
Therefore we can find a relationship between I„ and /„_, from equations (5.1) and (5.2), 
lm + Zli ' 
(5.2) 
(5.3) 
- 72 -
Now let us consider the electric Held strength on the outer surface of the mth layer. On 
one hand it is — z^Im, and on the other hand it is —(ZbmtIm — Z^/ m _i) using Schclkunoff's 
theorem 2. Thus we have the following identity: 
m m m ^ m ~ * 
Zbb = %bb ~ %ab ~~j . ' ' (5.4) 
m 
Substituting for 7m_i//m from equation (5.3), we obtain 
• m _ 7m J^i! '/<:«;» 
Zbb ~ *bb m , m - x ( o - 5 ) 
•^aa + f»6 
which gives the internal impedance of all m layers of the laminated tubular conductor, with 
current return on the outside. Starting with the first layer where z$ = Zby, we add the 
remaininglayers one by one until we obtain the impedance of the complete laminated conduc-
tor made up of m layers. 
(Zlb? 
Z'at + Ab 
Ab = ZU - „ , , • = 2, • • • m (5.6) 
5 .1 .2 I n t e r n a l I m p e d a n c e w i t h I n t e r n a l R e t u r n 
Similarly we can find the internal impedance of a laminated tubular conductor with 
current returning inside. Let Z™, Zbmh and Z™b related to the same internal impedances defined 
in the previous section. Let z*a be the internal impedance of all m layers when the current 
return is internal. Also, we note that for the very last layer, i.e. layer m in Figure 5.1, 
ZTa = zTn- Using Schelkunoff's theorem 2, we find the electric field strength along the outer 
surface of the 1st layer as 
= ~(Zb\h ~ Z,\lo) (5-7) 
But the outer surface of the 1st layer is the inner surface of the rn —1 outer layers combined, 
for which the1 electric field strength can be written as 
= (5-8) 
Therefore, we can find a relationship between I0 and Ix from equations (5.7) and (5.8), 
/, za\ 
h Zt\ + z-
(5.9) 
Now, consider the electric field strength on the inner surface of the first layer. On one hand it 
is —{ — za\lo)t a n d o n t n e other hand it is — (Z^/i — Z}J0) using Schellkunoffs theorem 2. 
Therefore we have the following identity, 
- 73 -
= ZL ~ Z^-j- (5.10) 'o 
Substituting for / j / / 0 from equation (5.9) we obtain 
i _ . _ (^ )2 
zaa ~ Zaa _] 2 ,5.11) 
^bb
 +
 zat 
which gives the internal impedance of all m layers of the laminated tubular conductor, with 
current return on the inside. Starting with the last layer, i.e. layer m, where z™ = Z™, v.v add 
the remaining layers one by one until the impedance of the complete conductor made up of m 
layers is obtained, 
IZ' )2 
= -Zla ~ _, '* • » = m - l , m - 2 > - l . (5.12) 
^lb + zaa 
5.2 A p p l i c a t i o n to Gas-Insulated Substations 
The equations for the internal impedance of laminated conductors will now be used to 
obtain the surge propagation characteristics in a gas-insulated substation with conductor 
coatings. 
Gas-insulated substations are subjected to transient sheath voltage rises whenever switch-
ings or fault surges occur. These surges propagate along the outer surface of the inner conduc-
tor and the inner surface of the sheath, as if the two surfaces were cylindrical wave guides, as 
well as along the outer surface of the sheath and the ground. The impedances of these surfaces 
play an important role in attenuating the surges, and thereby the transient sheath voltage rise. 
Since these surface impedances depend on the resistivity and the magnetic permeability of the 
material, it has been proposed by Harrington [32], to coat these surfaces with material of high 
resistivity and high permeability for surge suppression purposes. The coating should be such 
that its thickness is less than its current penetration depth at power frequency (60Hz or 50Hz), 
so that the resistance is not changed during steady-state operation. In addition to the base 
case without coatings, three different coating configurations are examined. The four cases con-
sidered are as follows: 
Core and sheath not coated. 
i. Only the inner surface of the sheath coated. 
ii. Only the outer surface of the inner core coated. 
v. Both the outer surface of the inner core as well as the inner surface of the sheath coated. 
For each of these cases, the formulae for the impedances for surge propagation arc 
derived. 
5.2.1 C A S E i : C ore and Sheath not Coated 
Axis 
Core 
S h c a l h v/ssssss//////////////////////J^ J 
Earth \ % , J 
Figure 5.3 - Core and Sheath not Coated 
This is the simplest of all the cases where the impedance for surge propagation in loop 1 
is given by 
Z = Z:rc + Z,ns + Ziht (5.13) 
where Zcre ( core - with external return) can be obtaind from equation (3.3) if the core conduc-
tor is solid, or from equation (3.1b) if it is tubular. Z M S can be obtained from equation (2.11), 
and Zih, (sheath - with internal return) from equation (3;.la). 
5.2.2 C A S E i i : Only Sheath Coated 
In this case the surface for the surge propagation consists of the outer surface of the core 
conductor and the inner surface, of the laminated conductor made up of coating paint layer and 
sheath. Hence, the impedance for surge propagation between core and sheath is given by 
Z = Zcre + Z,ns + zll (5.14) 
where Zcre and Z,ni are the same as explained for case i. zll is the internal impedance of the 
laminated conductor with internal return. This is obtained from equation (5.12} where layer 1 
is the sheath and layer 2 is the coating material (superscipt "sp" denotes the paint layer on 
sheath and superscript "sh" denotes the sheath). Hence 
- 75 -
Axis 
Core 
Paint 
Sheath 
Earth 
Figure 5.4 Inner Surface of the Sheath only Coated. 
sp\2 
y*V - T V _ 
Zll + z, sk (5.15) 
The total impedance Z can then be written as 
Z — Z:rc + Zins + Zip, 
(Zspm )2 
zspc Zik, 
(5.16) 
where Zsp, (sheath coated with paint layer - with internal return) and ZsK, (sheath - with inter-
nal return) can be obtained from equation (3.1a). Zipm (mutual between sheath conductor and 
paint layer) is found from equation (3.1c), and Zspe (sheath coated with paint layer - with exter-
nal return) from equation (3.1b). Equation (5.16) can also be derived from the loop equations 
of the loops 1,2 (figure 5.4), 
dx 
oT's 
~di~ 
= -(zj, + ZM 
= ~(ZnIi:+ Z2I2) (5.17a,b) 
where 
V, = potential difference between the core and paint 
= potential difference between the paint layer and sheath 
Zms "** Zsp, 
— Z2i Zspm 
v2 
Zi 
zm 
Z2 — Zipt + z$k, 
Since the paint layer and the sheath are at the same potential we have V2 = 0. Hence from 
equation (5.17b), 
- 7 6 -
I2 = ( - Z m / Z 2 ) / 1 
S u b s t i t u t i n g t h e v a l u e o f I2 i n t o e q u a t i o n (5 .16a) g i v e s 
dV, 
(5 .18) 
(5 .19) 
Therefore the impedance for surge propagation between core and sheath is given by 
z = z, - — 
or 
7 - 7 _L 7 J - 7 _ ( ^ s y m 
w h i c h is i d e n t i c a l w i t h e q u a t i o n 5 .16 . 
5.2.3 C A S E i i i : O n l y C o r e C o a t e d 
Axis 
Sheath 
Earth | 
/ / / / / / / / / / / / / / / / / / / / / / / / / / / / / / 
1, 
Figure 5.5 Core Alone Coated 
T h e s u r f a c e f o r s u r g e p r o p a g a t i o n i n t h i s c a s e c o n s i s t s o f t h e s u r f a c e o f t h e l a m i n a t e d 
c o n d u c t o r ( m a d e u p o f c o r e a n d t h e c o a t i n g ) a n d t h e i n n e r s u r f a c e o f t h e s h e a t h . T h e 
i m p e d a n c e f o r s u r g e p r o p a g a t i o n b e t w e e n c o r e a n d s h e a t h is t h e r e f o r e g i v e n b y 
Z = z$ + Znt + Z a , (5 .20) 
w h e r e Z , A , a n d Zint a r e t h e s a m e i m p e d a n c e s e x p l a i n e d e a r l i e r . z$ is t h e i n t e r n a l i m p e d a n c e o f 
t h e l a m i n a t e d c o n d u c t o r w i t h e x t e r n a l r e t u r n . T h i s c a n be o b t a i n e d f r o m e q u a t i o n (5.6), w h e r e 
l a y e r 1 is t h e c o r e a n d l a y e r 2 is t h e p a i n t o n c o r e , 
Substituting ztf into equation (5.20) gives 
(Z )2 
Z = Zcre - 7 + Z,ns + Ztk, (5.22) 
where Zcre is the same as explained earlier. Z^ (core coated with paint layer - with external 
return) can be obtained from equation (3.1b). Zcpm (mutual between paint layer and core con-
ductor) from equation (3.1c), and ZCfi (core coated with paint layer - with internal return) from 
equation (3.1 a) 
5 . 2 .4 C A S E i v : B o t h C o r e a n d S h e a t h C o a t e d 
A x i s 
Paint 
Sheath 
Earth 
Figure 5.6 Core aa well as Inner Surface 
of the Sheath Coated 
In this case, the outside of the core and the inside of the sheath arc coated. The surfaces 
for surgepropagation consist of.the surface of the laminated conductor 1 (made up of the core 
and coating paint layer on the core conductor) and the inner surface of the laminated conduc-
tor 2 (made up of the sheath and the coating paint layer on the sheath conductor). The 
impedance for surge propagation between core and sheath is given by 
Z = z% + Zml + z?. 
Using the values of z$ and z\\ from equations (5.15) and (5.21) Z can be written as 
z zcpc + z,RS + Zipt (5.24) 
where all the impedance values have been explained earlier. 
- 78 -
R e s u l t s 
Two coating materials arc considered, i.e., stainless steel and supermalloy. These routing 
materials would be applied iu the form of paints. The bus duct of the gus-"insuiatccl substation 
is assumed to have the dimensions given in Figure 5.7. 
Axis 
. 
Sheath w/////r7/777777777777777777> 
C 
\/77//7777777777777777//777777/, 
Earth j 
-ORCR 
IRCR 
-IRSH 
ORSH 
Figure 5,7 Dimensions of the Bus Duct in a Gas-htsulatcd Substation 
Inner radius of the core, IRCR = 10.0mm 
Outer radius of the core, ORCR = 65.0mm 
Inner radius of the sheath, IRSH = 350mm 
Outer radius of .the sheath, ORSH = 380mm 
The values of relative permeability and resistivity of the coating materials (stainless steel and 
supermalloy) and of the core and sheath material (aluminium) are given in Table 5.1. 
5 . 2 . 5 S t a i n l e s s S t e e l C o a t i n g 
The skin depth for a particular material is given by 
6 = V2p /M/ i (5.25) 
- 7 9 -
Table 6.1 
Resistivity and Relative Permeability of Coating Materials 
T Y P E MATERIAL 
RELATIVE 
PERMEABILITY RESISTIVITY 
Core and Sheath Aluminum 1.0 2.62E-08 
Paint 
(a) 
(b) 
Stainless Steel 
Supermalloy 
1500.0 
100 000.00 
4.70E-07 
0.00E-07 
Using the values of relative permeability and resistivity for "stainless steel, the skin depth at 
various frequencies was calculated and tabulated in Table 5.2. 
Table 5.2 
Skin Depth of Stainless Steel 
FREQUENCY 
(Hz) 
SKIN DEPTH 
(mm) 
10.0 2.81723 
60.0 1.15013 
100.0 ; 0.89089 
1 000.0 0.28172 
10 000.0 0.08909 
Since the thickness of the coating should be very much smaller than the skin depth cf the 
material at normal operating frequency, coating thickness of 0.1mm and 0.5mm were assumed 
to be practical values. Figures 5.8(a) and (b) show the variation in resistance and inductance 
for a coating thickness of 0.1mm and Figures 5.9(a) and (b) for a coating thickness of 0.5mm. 
— 6 U " 
100 
10 
E 
c 
in 
a) rr 0.1 
0.01 
0.001 
C A S E 1 
CASC 2 
CASE_3 
C A S E 4 
1CT 10" 10' 103 103 10' 
Frequency [Hz] 
10s 10' 10' 
x 
O.B-
0.6 
c o 
o o.* 
~o 
c 
0-2. 
0-+ 
10" 
\ 
\ \ 
w CAST 1 CASE.. 2 
C_ASE_3 
CASE ^ 
• 10- 10' 10J 105 10' 
Frequency [Hz] 
•10s 10* 10' 
Figure 5.8(a),(b) Variation of Resistance and Inductance 
•with Frequency for the Four Cases; 
Stainless Steel Coaling, Thickness 0.1mm. 
- 82 -
5.2.6 Supermalloy Coa t i n g 
The high resistivity and high permeability of supermalloy make its skin depth very small 
even at low frequencies, as shown in Table 5.3. 
T a b l e 5.3 
Skin Depth of Supermalloy 
FREQUENCY 
(Hz) 
SKIN DEPTH 
(mm) 
1.0 1.23281 
10.0 0.38985 
60.0 0.15915 
100.0 0.12328 * 
1 000.0 0.03898 
10 000.0 0.01233 
Since the coating thickness should be smaller than the skin depth at normal operating fre-
quency, it would be necessary to keep the coating thickness to less than 0.1mm. Figures 
5.10(a) and (b) show the variation of resistance and inductance with frequency for a coating 
thickness of 0.01 mm and Figures 5.11(a) and (b) for a coating thickness of 0.05mm respec-
tively. 
5.2.7 Comparison between Stainless Steel and Supermalloy Coatings 
For the case of stainless steel, we note from the figures 5.10(a) that there is no noticeable 
difference in the resistance up to a frequency of 100Hz for all four cases if the coating thickness 
is 0.1mm. Beyond that it increases sharply for cases 2, 3 and 4 as compared to the base case. 
When the coating thickness is increased-to 0.5mm, the differences are pronounced at frequen-
cies as low as 1Hz, as shown in Figure 5.11(a). This indicates that the coating thickness should 
not be increased beyond 0.1mm, since it would change the resistance at steady state operating 
frequency (50Hz or 60Hz) too much, and thereby increase the losses as well as the operating 
temperature. 
Due to the high permeability of stainless steel, the inductance is very high for cases 2, 3 
and 4 as compared to case 1, as shown in Figures 5.10(b) and 5.11(b). However, the increase in 
inductance should not cause any problems in bus ducts which are very short compared to the 
length of transmission lines. 
- 83 -
\ 
£ ^ \ \ C A s e 2 
\ 
c o 
y 2 •o c 
\ 5 X \ 
CASE 1 
<2 \ 
i - 3~ V LAIL3. 
CASE i 
10"2 10" 1 10' 10J 105 10." 105 10' 10 
Frequency [Hz] 
Figure 5.10 (a) and (b) Variation of Resistance and Inductance 
with Frequency for the Four Cases; 
Supermalloy Coating, Thickness 0.01mm. 
- 84 -
1000CH 
1000-
100-
E 
u c D 
tn 
ca 
10-
0.1. 
0.01. 
0.001J 
CASE 1 
CASE 2 
1CT1 10"' 
2 0 -
10' 102 103 
Frequency [Hz] 
10' 105 10s 1 0 ' 
\ 
15-
E 
<u 10-
u c o 
"o 3 -
"D 
C 
X \ 
\ 
CASE 1 
CASE 2 
CASE_3 
CASE * 
0 + 
10" 10" 10' .10* 10J 10" 
Frequency [Hz] 
10* 10e 10' 
Figure 5.1l(a),(b) Variation of Resistance and Inductance 
with Frequency for the Four Cases; 
Supermalloy Coating, Thickness 0.05mm. 
- 85 -
In the case of supermalloy, due to its higher resistivity and very lur^e permeability the 
coating thickness should not be increased beyond 0.01mm for the same reasons explained ear-
lier for stainless steel. 
The practicality of stainless steel or supermalloy coatings for surge suppression has been 
questioned by Boggs and Fujimoto [32]. Such coatings may, be cost effective. Using highly 
resistive materials such as steel for the entire sheath has been considered as well. This would 
be feasible with single-point ground, which would prevent currents from circulating through the 
sheath, thereby avoiding sheath losses. However, single-point grounding has adverse implica-
tions for transient ground rise, however. If switching surges are produced, transient overvol-
tages would appear at many points within the gas-insulated substation. 
5.3 Conclusions 
The internal impedances of tubular laminated conductors have been derived. These equa-
tions are used to find the internal impedances of bus ducts in gas-insulated substations whose 
core and/or sheath are coated with high-resistivity paints for the suppression of surges. 
- 86 -
6. TEST CASES 
T h e i n t e r n a l i m p e d a n c e f o r m u l a e f o r t u b u l a r c o n d u c t o r s a n d t h e e a r t h r e t u r n i m p e d a n c e 
f o r m u l a e w e r e d i s c u s s e d i n d e t a i l i n C h a p t e r s 3 a n d 4, r e s p e c t i v e l y . T h e s e i m p e d a n c e s m a k e 
u p t h e e l e m e n t s o f s u b m a t r i c e s [ Z „ ] a n d [Zi}\. In t h i s c h a p t e r t h e v a l u e s o f t h e s e s u b m a t r i c e s 
a r e o b t a i n e d f o r a s p e c i f i c u n d e r g r o u n d c a b l e s y s t e m , u s i n g t h e e x a c t f o r m u l a e as we l l as 
a p p r o x i m a t i o n s . 
T h e a p p r o x i m a t e f o r m u l a e p r o p o s e d b y W e d e p o h l [22] a g r e e v e r y c l o s e l y w i t h t h e e x a c t 
f o r m u l a e a n d t a k e v e r y l i t t le c p u t i m e . T h e y a l s o p r o v i d e s i m p l e e x p r e s s i o n s f o r h a n d c a l c u l a -
t i o n p r u p o s e s . T h e r e f o r e o n l y t h e a p pr o x i m a t e f o r m u l a e p r o p o s e d b y W e d e p o h l [22] a re c o n -
s i d e r e d in th is c h a p t e r . T h e r e s u l t s a re a lso c o m p a r e d w i t h v a l u e s o b t a i n e d f r o m t h e C a b l e 
C o n s t a n t s r o u t i n e in t h e E M T P , w h i c h w a s d e v e l o p e d b y A m e t a n i [27]. 
6.1 S i n g l e C o r e C a b l e 
T h e i m p e d a n c e s o f a s ing le c o r e c a b l e a re g i v e n b y a 2X2 m a t r i x o f t h e f o r m 
m = 
w h e r e al l e l e m e n t s were d e f i n e d e a r l i e r in C h a p t e r 3 . W i t h t h e d a t a o f t h e t e s t c a s e d e s c r i b e d 
in A p p e n d i x A , t h e v a l u e s o f t h e s e e l e m e n t s w e r e o b t a i n e d f r o m t h e e x a c t f o r m u l a e , f r o m 
W e d e p o h l ' s a p p r o x i m a t i o n f o r m u l a e a n d f r o m A m e t a n i ' s C a b l e C o n s t a n t r o u t i n e , as t a b u l a t e d 
in T a b l e 6 .1 . F i g u r e s 6.1 d e p i c t s t h e e r r o r s in A m e t a n i ' s a n d W e d e p o h l ' s a p p r o x i m a t e f o r m u -
lae f o r the i m p e d a n c e Z.c. F i g u r e s 6.2 a n d 6 .3 , r e s p e c t i v e l y , s h o w t h e e r r o r s f o r the i m p e d a n c e s 
Zs. a n d Zss T h e e r r o r s in A m e t a n i ' s a n d W e d e p o h l ' s f o r m u l a e a re n o t s i g n i f i c a n t at low f re -
q u e n c i e s , b u t f o r h i g h e r f r e q u e n c i e s t h e y c a n n o t be n e g l e c t e d . W e a l s o n o t i c e t h a t the e r r o r s in 
Zcc, ZCi a n d ZS! a l l h a v e s i m i l a r v a l u e s . T h e y are e s s e n t i a l l y c r e a t e d b y the e r r o r s in the e a r t h 
r e t u r n f o r m u l a e u s e d b y A m e t a n i a n d W e d e p o h l , as s h o w n in F i g u r e 4 .7 . A s m e n t i o n e d e a r l i e r , 
W e d e p o h l ' s a p p r o x i m a t e e a r t h r e t u r n f o r m u l a is v a l i d o n l y i f t h e c o n d i t i o n \ mZ \ <0.25 is s a t i s -
fied. T h i s is o n l y t r u e at low f r e q u e n c i e s . A t h i g h f r e q u e n c i e s , t h e i n t r i n s i c p r o p a g a t i o n c o n -
s t a n t m , b e c o m e s l a r g e r , a n d th is c a u s e s t h e e r r o r s in the r e s u l t s . T h e e r r o r s a lso i n c r e a s e i f 
t h e s e p a r a t i o n b e t w e e n t h e c a b l e s b e c o m e s l a r g e r . 
T h e r e a s o n s f o r the e r r o r s in A m e t a n i ' s e a r t h r e t u r n i m p e d a n c e f o r m u l a e at h i g h f r e q u e n -
c ies h a s a l r e a d y been d i s c u s s e d in C h a p t e r 4 . 
z. (6.1) 
- 87 -
T a b l e 8.1 
Impedances of a Single Core Underground Cable. 
Zee (fi/km) 
E X ; AME1 ' ' A N I W E D E : P O H L 
F R E Q 
(Hz) 
R X R X R X 
1 
10 
100 
1000 
10000 
100000 
. 0 1 0 8 7 3 
. 0 2 0 0 8 4 
. 1 1 9 3 0 3 
1 .05509 
1 0 . 4 8 0 3 
1 0 8 . 2 4 0 
. 0 1 6 0 8 2 
. 1 4 6 2 9 9 
1 . 3 0 4 5 6 
1 1 . 4 7 5 9 
9 9 . 6 8 4 3 
8 3 9 . 8 4 8 
. 0 1 0 8 7 3 
. 0 2 0 0 6 9 
. 1 1 8 8 0 5 
1 .03996 
1 0 . 0 3 4 3 
9 6 . 4 9 7 3 
. 0 1 6 0 8 3 
. 1 4 6 3 1 5 
I . 30504 
I I . 4916 
1 0 0 . 1 7 5 
8 5 4 . 7 6 1 
. 0 1 0 8 6 8 
.020081 
. 120114 
1 .05626 
1 0 . 4 9 8 3 
1 0 9 . 5 9 3 
. 0 1 6 0 9 3 
. 1 4 6 4 1 6 
I . 30506 
I I . 4759 
9 9 . 6 8 2 2 
8 3 9 . 6 1 4 
Zcs (£2/ km) 
1 
10 
100 
i o o o 
10000 
100000 
. 0 0 0 9 8 7 
. 0 0 9 8 7 8 
. 0 9 8 9 5 4 
. 9 9 5 7 1 7 
10 .2001 
1 0 6 . 4 3 0 
. 0 1 5 0 9 7 
.136501 
1 .22016 
1 0 . 7 4 9 4 
9 2 . 8 2 9 5 
7 7 5 . 5 2 4 
. 0 0 0 9 8 7 
. 0 0 9 8 6 2 
. 0 9 8 4 6 5 
. 9 8 0 5 9 0 
9 . 7 5 4 1 4 
9 4 . 6 8 7 6 
. 0 1 5 0 9 8 
. 1 3 6 5 1 6 
1 .22066 
10 .7651 
9 3 . 3 2 0 1 
7 3 0 . 4 3 8 
. 0 0 0 9 8 7 
. 0 0 9 8 7 8 
. 0 9 8 9 5 6 
. 9 9 5 9 1 9 
1 0 . 2 1 7 0 
1 0 7 . 7 8 2 
. 0 1 5 0 9 7 
.136501 
1 .22016 
1 0 . 7 4 9 4 
9 2 . 8 2 7 2 
7 7 5 . 2 9 1 
i 
Zss Wl im) 
1 
10 
100 
1000 
10000 
100000 
.300151 
.309041 
. 3 9 8 1 1 2 
1 .29438 
10 .4531 
106 .361 
. 0 1 5 0 8 3 
. 1 3 6 3 5 4 
1 .21869 
1 0 . 7 3 4 7 
9 2 . 6 9 6 9 
7 7 5 . 5 1 8 
. 3 0 0 1 5 0 
. 3 0 9 0 2 5 
. 3 9 7 6 1 2 
1 . 2 7 9 2 5 
10 .0071 
9 4 . 6 1 8 0 
. 0 1 5 0 8 3 
. 1 3 6 3 6 9 
1 .21919 
1 0 . 7 5 0 4 
9 3 . 1 8 7 5 
7 9 0 . 4 3 2 
.300151 
.309041 
. 3 9 8 1 1 5 
1 .29458 
1 0 . 4 7 0 0 
1 0 7 . 7 1 3 
. 0 1 5 0 8 3 
. 1 3 6 3 5 4 
1 . 2 1 8 6 9 
1 0 . 7 3 4 7 
9 2 . 6 9 4 6 
7 7 5 . 2 8 5 
- 88 -
50 
30-
1 0 -
u c 
« -10 
cn 
io 
cu rr 
- 3 0 
/ 
7 
AMETANI 
WEDEPOHL 
— 5 0 \ i i i Mini i i i nun —n T T T T T T T J — i T I i t i i T f nr T T H T T T ; T-TTTTTTT] r i i mit| " i T T m n | " i i r i t i n 
10~ 2 1 0 " 1 10' 10 2 10 3 10 4 10 5 10 6 10 7 
20 
12 
4-
01 
o 
c to 
u 
ca cu CC 
-4 
-12 
\ 
AMETANI 
WEDEPOHL 
-20 1 III Mini i i i mi l l i i i IIIIII i i i m m i i i i i u i i — i i 11mil i i 11iui| i i 11mil i i 11mi 
10" 2 10"' 1 10 1 10 2 10 5 10 4 10 s 10 6 10 7 
Frquency [Hz] 
Figure 6.1 - Errors in Ametani's and Wedepohl's Approximations in Zcc 
- 89 -
50 
30-
0) 
u c 
tn 
cu 
tc 
10-
-10-
- 3 0 -
/ 
/ 
AMETANI 
WEDEPOHL 
— 50 1 T ITTIfH ' I '1 M l l T i r I 'T I 1 i m ] ' ~ T ' T T T T m r T"T T^TIITf I ' T l T n i l } ' 1 I 1 11'ltT) 1 "TTTTnTJ I I T'H'ffl 
10"2 10"' 1 10' 102 103 104 10s 106 107 
20 
12 
cu o c 
CO 
*-> o ra 
CD 
CC 
- 4 -
-12-
\ 
AMETANI 
WEDEPOHL 
— 20 | i i 1 1 — i i i IIIIII I i i M i n i 1 i i m i l l 1 i . I I I I I I — i i i M i n i 1 i I M i n i — i i i u t i i | i l l M i n i 
10"2 10"' 1 10' 102 10J 104 105 106 107 
Frquency [Hz] 
Figure 6.2 - Errors in Ametani 'a and Wedepohl's Approximations in Za 
- 90 -
LU 
5 0 
30-
10 
co 
u 
« - 1 0 
+—< 
m 
cc 
- 3 0 -
/ 
/ 
AMETANI 
WEDEPOHL 
—50 | I I I I I I I I I i i i I I I I I I — i i i I I I U I i i 11uii| i i 1 1 m i l I i 11uii|—> i i I I I I I I I I I I I I I I I — i i i i n n 1 
TO" 2 1 0 " ' 1 1 0 ' 1 0 2 1 0 3 10" 1 0 5 1 0 6 1 0 7 
2 0 
CD 
u 
c 
to 
o 
co 
CD 
at - 1 2 
AMETANI 
WEDEPOHL 
—20 I I I I n i M | i i i i i i i i , 
1 0 " 2 1 0 " ' 1 
rrm] 1 i i i i n i j i i I I I I I I ; I I I I I I M I I I I I I I I I ) I I I I I I I I I I i i m i q 
1 0 ' 1 0 2 1 0 3 10" 1 0 S 1 0 6 1 0 7 
Frquency [Hz] 
Figure 6.3 - Errors in Ametani's and Wedepohl's Approximations in Z, 
- 91 -
6.2 Three-Phase Cable 
In the case of a three-phase cable system, the series impedance matrix is given by 
1*1-
\Zn\ |Z 1 2] \ZK) 
\2A \z*\ 
(0.2) 
where [Z„] is the self impedance submatrix of cable i as given by equation (0.1). The mutual 
impedances between cable i and cable j are represented by submatrix |Z t ;] of the form: 
Z - .f . Z r r • 
Zj.ry Zs.s. 
(0.3) 
As shown in Chapter 2, all four.elements of submatrix are equal to each other. The values 
from the exact formula (4.10) and from Ametani's and Wedepohl's approximate formulae are 
tabulated in Table 6.2 for the three-phase cable system described in Appendix A. 
T a b l e 6.2 
Mutual Impedance between Two Cables with 
Burial Depth of 0.7om and Separation of 0.80m. 
Z i j (fi/kin) 
E X ; ^ C T AME*] rANI WEDI 3POHL 
FREQ RES REA RES REA RES REA 
1 
10 
100 
1000 
10000 
100000 
.000987 
.009877 
.098943 
.994644 
10.1015 
105.154 
.012562 
.111152 
.966670 
8.21457 
67.5095 
525.238 
.000987 
.009862 
.098452 
.979517 
9.65569 
93.4318 
.012563 
. 111167 
.967169 
8.23027 
68.0000 
540.150 
.000987 
.009878 
.098946 
.994856 
10.1193 
106.592.012562 
.111151 
.966668 
8.21452 
67.5070 
525.000 
- 92 -
5 0 
3 0 
O ) 0 
UJ 
cu o 
li -to-
cn 
cn 
tn 
- 3 0 H 
- 5 0 —i—n: TTTT. i—r^rrrrrrr-
10~2 10"' 
5 0 
3 0 -
O 1 0 -
UJ 
cu o c 
0> 
CC 
- 1 0 
- 3 0 
/ 
/ 
AMETANI 
WEDEPOHL 
ni 1 i i I I I I I I 1 11 MM; 1 \ i i mi, 1—i t mil, 1—t 11 nn' 
1 10' 10 2 10 3 10 4 10 S 10 6 io 7 
\ 
AMETANI 
WEDEPOHL 
5 0 -f 1 i i mii| 1 i i inn, 1 i i mii| 1 i i iiiiii 1 i ' mii; 1 i mm, 1 i i i nn; 1 , 1 1 ""1 ' 1 11"" 
1 0 - 2 10" 1 10' 10 2 10 S 10 4 10 S 10 6 10 7 
Frequency [Hz] 
Figure 6.4 - Errors in Ametani's and Wedepohl's Approximations 
in the Mutual Impedance between Two Cables. 
- 93 -
The errors in Ametani's and Wedepohl's approximate formulae are plotted in Figure 6.4. 
The reasons for the errors are essentially the same as those discussed in Section 6.1. 
6.3 Shunt Admittance Matr ix 
The elements of the shunt admittance matrix obtained from Ametani's Cable Constant 
routine in the EMTP shows that the relative permittivity t is assumed to be real and con-
stant. As explained earlier in Chapter 2, the relative permittivity is complex as well as 
frequency-dependent, but this data is usually difficult to obtain. A real, constant permittivity 
should give reasonable answers in many cases. 
- 94 -
7. CONCLUSION 
Various formulae proposed in the literature for the series impedance and shunt admit-
tance matrices of underground cable systems have been compared in this thesis. The elements 
of the series impedance matrix are evaluated from formulae for the internal impedance of tubu-
lar conductors and from formulae for the earth return impedance. Exact equations for the 
internal impedance of tubular conductors were first derived by Schelkunoff [6j. They are given 
in terms of modified Bessel functions, and are therefore not suitable for hand calculations. 
Since then closed-form approximations suitable for hand calculations have been proposed by 
many authors, including Schelkunoff. A comparison of these approximate formulae shows that 
the formulae proposed by Wedepohl [22] give answers which are usually accurate enough for 
engineering purposes. With computers being almost universally available nowadays, approxi-
mate formulae are no longer that important, however, and programming the exact formulae 
may therefore be the best approach. 
The displacement current term is usually neglected in the formulae for the internal 
impedances of conductors. It is shown that it can indeed be neglected for frequencies up to 
10MHz. The shielding effect of grounded sheaths is explained as well, and it is shown that it is 
implicitly accounted for in the mutual impedances. 
The permittivity of the insulating material is needed for the elements of the shunt admit-
tance matrix. Its value is frequency dependent as well as complex. In some cases, (e.g., cross-
linked polyethylene), the permittivity can be assumed to be constant and real up to very high 
frequencies, while in other cases (e.g., oil-impregnated paper) the changes with frequency are 
quite significant. Two insulating materials, namely cross-linked polyethylene and oil-
impregnated paper, are discussed in detail because they are the materials most often used in 
power cables. A general formula for the complex permittivity of insulation materials is given 
by Bartnikas [15], based on the relaxation time of the dielectric material. Ametani's Cable 
Constants routine in the E M T P [27] assumes that the permittivity is real and constant which 
may not always be accurate enough. 
The earth return impedance formula derived by Pollaczek [l] for the case of a semi-
infinite earth is valid only for filamentary type conductors of negligible radius. This formula 
can be used for a conductor of finite radius a, if the condition | mo | <0.1 holds. This condition 
is satisfied up to a frequency of 1 MHz even for a worst case low earth resistivity of 10 fl— m. 
Hence Pollaczek's formula is recommended as the accurate formula. Values obtained from 
various approximate formulae and from Ametani's Cable Constants routine in the E M T P were 
compared against Pollaczek's formula. The results agree closely at low and medium frequen-
cies but significant differences arise at high frequencies. 
- 95 -
Equations for the internal impedances of a laminated tubular conductor have been derived 
from the equations for homogeneous tubular conductors. They are used to study the increase 
in the surface impedances of bus ducts in gas-insulated substations if the conductors are coated 
with high-resistivity magnetic material. This coating technique has been proposed by Harring-
ton [32] for reducing the transient sheath voltage rise during switching operations, although 
others have criticized it as impractical, [discussion 32] 
- 96 -
A P P E N D I X A 
Test Examples for Buried Cables 
Earth 
d, 
G A x'- x» 
Figure A.l - Three-Phaae Cable Setup for the Study 
Each cable is of a single core type with dimensions as given below 
conducting sheath 
central 
conductor 
insulation 
Figure A.2 - Basic Construction of each tingle core cable 
di,d2,dt 
X12 
X ja 
= 0.75m, depth of burial of each cable 
= 0.30m, horizontal distance between cables 1 and 2 
= 0.30m, horizontal distance between cables 2 and 3 
*2 
Peon 
P- r core? r shea.tb» 
/ ' f ex r tb ' / ' r i l r 
0.0234m, radius of the core 
0.0385m, inner radius of the sheath 
0.0413m, outer radius of the sheath 
0.0484m, outside radius of the cable 
100 fl—m, resistivity of the earth 
1.7X10"8J?-m resistivity of the core material 
2.1xi0"7/?-m resistivity of the sheath material 
= 1.0, relative permeability of the core, sheath, earth, and air respectively. 
- 98 -
A P P E N D I X B 
Internal Impedances of a Tubular Conductor 
Based on tbc work of Schelkunoff [6], the derivation of the internal impdance formulae for 
tubular conductors is summarized here. 
B.l Circularly Symmetric Magnetic Fields 
In polar coordinates, Maxwell's equations assume the following form: 
dllz 
rd<*> 
3//r 
dz 
dz 
dHz 
= (\lp + »"toe)E,, 
= (l/> + i<ai)Et, 
3Ez 
rd(j> 
3E, 
dz 
dEt 
dE, 
— — iliifiH, 
= —ioifill^ 
ifdlrHJ 3H,\ 1 (9[rE4) ^ dEt\ 
— 1 — — ' = + i«e)E r. T T f = ~ 
r ( dr d<p ) r { dr 3<t> ) 
(B-1) 
where H and E are electric and magnetic field strengths, respectively. Here we are interested 
in the circular magnetic field around conductors, with its lines of force forming a system of 
coaxial circles. Such circular magnetic fields are associated with currents flowing in isolated 
wires, as for example in a single vertical antenna, or between the conductors of a coaxial cable, 
as shown in Figure B.l. 
Figure B.l - The relative directions of the field components 
in a coaxial transmission line. 
- 99 -
From equation (B.l) we see that when the quantities are independent of the angle <f>, one of the 
independent subsets composed of the 1st and 3rd equation on the left of equation (B.l), 
together with the 2nd equation on the right, define the circular magnetic field strengths as fol-
lows: 
dlrllA 
. - (1/p + i<at)rE, (B.2a) or 
a, 
dE2 dE 
-(l/p + itoe)Er (B.2b) 
iattH4. (B.2c) dr dz 
It has been shown by Schelkunoff that H^,Er and Ez have components which vary exponentially 
along the longitudinal axis of the cable, i.e., along the z axis in Figure B.l. If we express the 
exponential variation of the quantities E,,EZ and as E,eCr*, Eze~Tz and H^,e~r', then the 
quantities E,,E2 and H# are functions of r only. Substituting these values into equation (B.2) 
we obtain 
Er = f • H4 (B.3a) 
dE, 
iuuHt = —— + TE, (B.3b) 
dr^—A. = (Up + iu>f)rE, (B.3c) 
dr 
where the quantity T is called the longitudinal propagation constant. Now solving for H# from 
Equation (B.3), we obtain 
where m2 = I ' co 2/ze|. This quantity m is called the intrinsic propagation constant of 
I P 
the conductor material. For solid conductors, the term u>2ut which accounts for the displace-
ment current is negligibly small compared to the conduction current. Hence we can neglect it 
up to quite high frequencies. The intrinsic propagation constants of metals are relatively large 
quantities even at low frequencies as shown in Table B.l for copper. 
- 100 -
Table B.l 
Propagation Constant of Commerical Copper 
p = 1.7 X 10" 8tt-m 
iH,) y/a){i/p = | m | 
0 0.0 
1 21.40 
10 87.67 
100 214.00 
10,000 2140.00 
1,000,000 21400.00 
100,000,000 214000.00 
On the other hand, the longitudinal propagation constant T is relatively very small, even at 
high frequencies. For example, if air is the dielectric between the conductors T will be of the 
order of (l/3)ia>10~10. Hence, even at high frequencies T 2 is negligibly small by comparison with 
m2. Therefore, we can write equation (B.4) as 
d2H* i dH, 
dr' dr sT
 = mXH* 
The solution for Ht of equation (B.5) is in the form of Bessel functions given by: 
Alx{mr) + BK^mr) 
(B.5) 
(B.6) 
Since we are interested in longitudinal voltage drops, we must find the longitudinal elec-
tric field stength first. This can be obtained from equation (B.3) and (B.6) along with the fol-
lowing rules of differentiation for modified Bessel functions of any order n, 
dx 
_d_ 
dx 
(x*Kn)= -x*Kn.x 
(B.7a) 
(B.7b) 
The solution for the longitudinal electric field strength then becomes 
- 101 -
E, = pm\AI0(mr) - BK0{mr)\ (B.8) 
In a tubular conductor whose inner and outer radii are a and b, respectively, coaxial 
return path for the current may be either outside or inside the tube or partly inside and partly 
outside. We designate Zt as the internal impendance of the tubular conductor with internal 
return and Zb as the internal impedance with external return. If the return path is partly 
internal and partly external, we have in effect a two-phase transmission line with a distributed 
transfer impeduace Zab between the two loops of internal and external return. 
In order to determine these impedances, let us assume that a total current (/, + /.) is 
flowing in the tubular conductor, with part /„ returning inside and part Ib returning outside. 
Figure B.2 - Loop Currents in a Tubular Conductor 
Since the total current enclosed by the inner surface of the conductor is — Ia and that enclosed 
by the outer surface is Ib (70 + Ib — /„), the magnetic field strengths at these two surfaces take 
the values ( — Ia/2na) and (Ib/2nb) respectively. Hence from equation (B.6) we have 
A/,(7»io)+ M,(ma) = -lJ2na (B.9a) 
A li(mb) + BKx(mb) = Ib/2r.b (B.9b) 
From these two equations the values of A and B can be evaluated as 
2-naD 2nbD 
= _ /»/»("») ( B 1 0 b ) 
2naD 2xbD 
where 
- 102 -
D = r1(mb)k1(ma) - I^majk^mb) (B.II) 
Substituting these values in (B.7) and using the identity /„ (z)/C"i(z) + K0(x)Ix(x) = 1/x,, 
we obtain the longitudinal electric field strength at any point on the conductor. However, we 
are interested in its' values at the surfaces as they constitute the surfaces of propagation. 
Hence, equating r successively to a and 6 we obtain 
E,(a) = ZJa + ZabIb (B.12a) 
E,(b) = ZcbIa + ZbIb (B.12b) 
where 
2naD 
7 - Pm 
Z> ~ 2xbD 
7 = P 
[irimajK^mb) + /v"0(mo)/,(m6) 
^/ 0(m6)K'i(r7ja) + AT0(m6)/1(ma) j 
2nabD 
Schelkunoff stated these results in the following two theorems. 
(B.13) 
Theorem 1 
If the return path is wholly external (Ia = 0) or wholly internal (/;, = 0), the longitudinal 
electric field strength on that surface of a tubular conductor which is nearest to the return 
path equals to the corresponding surface impedance per unit length multiplied by the total 
current flowing in the conductor and the field strength on the other surface equals to the 
transfer impedance per unit length multiplied by the total current. 
Theorem 2 
If the return path is partly external and partly internal, the separate components of the 
field strength due to the two parts of the total current are calculated by the above theorem 
and added to obtain the total field strength. 
- 103 -
A P P E N D I X C 
Calculation of Earth Return Impedances 
in an Infinite Homogeneous Earth 
If the return current distribution in the ground is circularly symmetrical, then we refer to 
such a case as infinite earth. This happens in practice when the cables are either buried at 
large depth or when the frequency is very high. In both cases, the penetration depth d given 
by 503 • I t ) m, becomes smaller than the depth of burial. Then only the earth medium 
must be considered, which simplifies the solution. If the cables are buried close to the earth's 
surface on the other hand, which is usually the case, then the distribution of current in the 
ground is no longer symmetrical (at least at low frequencies), and the magnetic field both in air 
and earth must then be considered which makes the solution more complicated. 
Consider a cable lying along the Z-axis of the cartesian coordinate reference frames as 
shown in Figure C . l . Let the positive direction be along the Z-axis, and let the conductor 
carry a current I flowing in the positive direction returning through the ground. Let the radius 
over the outer insualtion be a. From Ampere's Law (neglecting the displacement current term) 
the magnetic field strength H at a radius r & o is given by 
2-nrH = I + J 2nrJdr 
i.e. 
T 
H = —+-fjrdr (C.l) 
2nr rJ v 
where J is the current density in the ground. 
Suppose that the earth is subdivided into concentric cylindrical shells of radius r and 
thickness dr in which the current density / and magnetic field strength / /are constant. Then 
the magnetic flux per unit length of such a shell is given by 
d<f> = BdA = ulldr (C.2) 
Substituting for H from equation (C.l) yields 
dtp = udr -!-+±Jjrdr 
2xr r •* 
(C.3) 
Now let us write Kirchhoff's voltage law around the rectangle ABCD of unit length and width 
dr. The net resistive voltage drop is — ',|~^:~J^r a n c * t n e induced voltage is jmd<l> or jtaiilldr. 
- 104 -
Figure C.1 - Representation of a Buried Conductor 
in an In finite Earth. 
Since the sum of these two voltages must be zero, we obtain 
dJ 
— p dr + jtauH dr = 0 
dr 
Substituting for//from equation (C.1), we have 
pdJ 
dr 
•dr + jtAfidr — + -fjrdr 
2itr 
= 0 
(C.4) 
(C.5) 
Multiplying this equation by ~- and differentiating with respect to r we have 
par 
d2J + dJ_ _ jvuJ 
dr2 rdr p 
= 0 (C.6) 
If we substitute m* for J f a > / i, then equation (C.4) can be written as 
P 
- 105 -
£ «•»««*. (C7) 
and equation (C:6) can be written as 
d2J . dJ 
m 
lJ = 0 (C.8) 
dr' r dr 
Equation (C.8) is immedately recognizied as a Bessel equation whose solution is of the form 
/ - AI0{mr) + BK0{mr) (C.9) 
We note that I0(x) approaches infinity as z approaches infinite. However, we cannot permit a 
solution of J to increase indefinitely as r approaches infinity and we must conclude that ,4=0 
PI-
Hence, equation (C.9) becomes 
/ = BK0(mr) (C.10) 
Using equation (C.7) we find a solution for the magnetic field strength H as 
m2H = -BKAmr)m ( C l l ) 
Now applying the boundary condition that H = I/2ira in the ground immediately adjacent to 
the cable, we obtain the value for the constant B from equation ( C l l ) as 
B = ~ 2nal<!(ma) ( C ' 1 2 ) 
Using the equation E = pJ, the solution for the electric field strength at any point in the 
soil is found to bepml K0{mr) . 
E~~2*a K^ma) ( C 1 3 ) 
The earth return self impedance as well as the mutual impedance between two buried cables 
can be deduced from this equation (see Chapter 4). 
- 106 -
A P P E N D I X D 
Calculation of Earth Return Impedances 
in a Semi-Infinite Homogeneous Earth 
The limited conductivity of the ground path for the return currents as well as conductor 
skin effects result in the frequency dependence of the line parameters. The parameters of a 
transmission line over a ground of perfect conductivity are given by textbook formulae, but the 
earth return effects and skin effects need special treatment. While a complete solution of the 
actual problem is impossible, on account of the uneven surface under the line and the lack of 
conductive homogeneity in the earth, a solution of the problem, where the actual earth is 
replaced by a plane homogeneous semi-infinite solid, gives reasonably accurate answers. The 
same applied to the underground case, too. 
The first step in finding the earth return impedances is to derive the respective longitudi-
nal electric field strengths. Let us first consider au overhead line and derive the electric field 
strengths in air and in earth. 
A y p 
. (cc.h) 
X 
Earth 
Q* (x.y) 
Figure D.l - Current-Carrying Filament in the Air 
Let medium 1, denoted by subscript 1, correspond to air and medium 2, denoted by subscript 
2, correspond to earth. Let point P{a,h) correspond to the current-carrying filament lying 
- 107 -
along the Z-axis of the Cartesian coordinate system. Let E+ + (x,y;a,h) be the electric field 
strength in air at a point Q(x,y) and E- + (z,y';a,h) be the electric field strength in the earth at 
a point Q'(x,y'). Note that the y axis is positive in the air and that the y' axis is positive in the 
earth, as shown in Figure D.l. 
From Maxwell's theory, the general equation for electromeganetic wave propagation is 
given by 
V 2E - V(V£) = - tovjtf (D.l) 
where p,fi and e correspond to the respective medium to which this equation is applied. Using 
assumption 4 (Chapter 2), we can say that VE — 0 in both air and earth. Hence, equation 
(D.l) can be written as 
V 2E - CO [It (D.2) 
Now let us define the fields which we would like to derive as follows 
E++ = E+ + z = Electric field strength in the air due to the current-carrying filament in the air 
E-+ — E- + z = Electric field strength in the earth due to the current-carrying filament in the. 
air 
If we assume that a sinusoidal current I of angular frequency' o> is passing through a filament 
concentrated at the point (a,h) in the x—y plane as illustrated in Figure D.l, then the current 
density is zero everywhere in the air except at the point (a,h) where it is infinite. Such an 
idealized situation can be represented by the Dirac delta function 5(x-^a) defined as 
in such a way that 
/ 5(x-a)dx = 1 ! (D.4) 
— CB 
which implies that if f(x) is continuous at x = 0 and bounded elsewhere [12] 
J f(x)5(x-a)dx = f(a) (D.5) 
Hence in the air, the current density can be expressed in the form 
- 108 -
I6(x-a)S(y-h) (D.6) 
Now keeping this result in mind and noting that we are interested only in the electric field 
9EZ ' -strength along the Z-axis and = 0 (using assumption 2 from Chapter 2) equation (D.2) 
can be written for the case of air as 
d2E+ + d2E.+ 2 
3z a By' m
2E++ + plmfIS(x — a)6(y — h) (D.7) 
where 
Pi 
For the earth, equation (D.2) can be written as 
a2£_ + a2E_+ 
dx2 
• here 
m2 = 
= m|E_. (D.8) 
I P2 
2 2 — CO p i2 
The solutions for E++ and E_+ should be obtained in such a way that they satisfy the follow-
ing boundary conditions: 
1. Continuity of E at the surface. 
Lim £++ = Lim E_+ — Lim £ _ + = /?0(say) 
V - * 0 y — 0 jr'-+o 
Vertical component of B is continuous at the surface 
dx dx 
Horizontal component of H is continuous at the surface 
dE++ a£_+ aE_+ 
(D.10) 
Pidy P2&y p-z^y' 
(D.il) 
The solutions for E+ + and E_ + can be found by using integral transform techniques. Taking 
the Fourier complex transformation of equations (D.7), (D.8), (D.9) and (D.ll) with respect to 
x, with 6 as the parameter, we obtain the following equations: 
- 109 -
d 2 E + + 
-62E^ + = m f E + + + /j,m 2/exp(-^a)%-A) (D.12) 
<f 2 £.. 
-62E_ + + j ^ - mf E.+ - (D.13) 
£ + + | y-0 — E-+ | Y ' - 0 — £ Q 
1 dE+* j i d £ _ + 
(D:14) 
• 1 OD _ + I 
Hi dy fi2 dy' 
Now taking the Fourier sine transformation of equation (D.12) with respect to y with 0 as the 
parameter, we have: 
-62E^+ — 02 E+ + + 0EO = m 2 E + + + p^rn? Iexp{-j6a)sin{0h) (D.16) 
i.e., 
(02 + C f ) E + + = 0EO - Plm2lexp(-jea)sin(0h) (D.17) 
where Cf = 02 + m 2 
Similarly, taking the Fourier sine transformation of equation (D.13) with respect to y' 
with 0' as parameter, we have 
- 0 2 E _ + + fl'2E.+ + /J*E 0 = m 2 £ _ + (D.18) 
Hence 
+ Cf)£_ + = £'E0 (D.19) 
where C'| = t?2 + m 2 : . 
Now, taking the inverse Fourier sine transformation of equation (D.17) with respect, to 0 
we have 
= E0exp( —C,y) 
-•^ L/exp(-^a)|exp(-C 1 | / i-j/|)-exp(-C,|/ l+y | ) j (D.20) 
Similarly, taking the inverse Fourier sine transformation of eqution (D.19) with respect to fl' we 
have 
F_+ = E0exp(-C2y) (D.20) 
By taking the derivative of Z? +* with respect to y and the derivative of E_+ with respect to y. 
- no -
and substituting in equation (D.l 5), we obtain the value of E0 as follows: 
— P\fn f /exp( — j6a)exp{—Cxh) 
ui 
(C, + :—<7a) 
u2 
(D.22) 
Substituting E0 in equation (D.20) and taking the inverse complex transformation with respect 
to 0, we obtain the value of E + + as follows: 
.2/ - [expf-C,| h-y | )-exp(-C 2| h +y |)] 
2 C , 
exp{-C1\h+y\) 
C. + — C2 
exp(j6\ i - a | )d0 (D.23) 
Similarly, substituting E0 in equation (D.21) and taking the inverse transformation with respect 
to 6, we obtain the value of E. + as follows 
Pimfl "r exp{— C^y — C1h}exp(j6\ x — a\)d6 
E-+ = ~ J 
2n 
Ci + — C 2 
U2 
(D.25) 
Now that we have derived the equations for the electric field strengths in the air and in the 
earth due to an overhead conductor, we will turn our attetion to the case of an underground 
conductor. 
Electric Field Strength in the Air and in the Earth 
due to Current Carrying Filament Buried in the Earth. 
Let 
= E--Z~ Electric field strength in the earth due to the current-carrying filament buried 
in the earth. 
E+. = is + _/= Electric field strength in the air due to the current-carrying filament buried in 
the earth. 
Similar to equations (D.7) and (D.8), Maxwell's equations for electromagnetic wave propagation 
in air and earth respectively, for this case are given by, 
d 2 £ + _ d2E+ 
dx5 By 2 = m , E , . y ^ O , 
(D.26) 
- I l l -
H y 
? (x.y) 
x 
_ i 
• (x,y') . 
p.- W) 
Earth 
Figure D.2 - Current Carrying Filament Buried in the Earth. 
32E__ 82E__ , , . 
+ — = m | £ _ _ + p2m2fS(x — a)5(y — h ) ax1 ay (D.27) 
Similar to the procedure used in the derivation of fields and we solve for and 
such that they satisfy the boundary conditions given by equation (D.9) through (D.l 1). 
Hence we have 
E+. = 
p2m2I " e x p { — Cly — C2h) 
2TT 
-»- — C 2 
?2 
•xp{j6\x-d\ )d0 (D.28) 
£•_ _ .= -
p2m2I 
2n 
[ e x p ( - C 2 | h'-y'\ )-exp(-C 2| h'+y'\ ] 
2C, 
exp(-C 2 | / i ' - r V | ) 
C, + —C2 
»2 
exp(jO \ x-a'\ )d8 (D.29) 
If we assume that the relative permeability of air and earth are the same, i.e., uTl = u,2 then 
- 112 -
ril = fi2 and we can show that the equations derived for E+ + ,E_ + ,E^_ and E+_ are the same 
as those derived by Pollaczek [lj. 
Now using the standard results 
i \ c exp{—aV^-+m2}exp . . . , • ' _ • . 
K0(mr)= J y x 9 \ y(jis)d3 (D.30) 2Vr+m2 
where r = and whereK0 is the modified Bessel function of the 2nd kind and of the 
zeroth order, we can write E+ + and E__ as follows: 
E „ = 
pm,2/ ( 
+ / ; = ( j t f | x - o » | )rfg \A2 + rnf + \ / V + 
where /?, ~ V {T - af+(h - y?, Zx = V ( z - a ) 2 + (h+yf 
pm2I ( 
— — j A ' o ( m 2 ^ 2 ) — KG(m2Z2) 
r exp{-|j/+/i V^+m22}exP( + / . (j*l z - a |)«« 
— \A2 + m? + \A2 + ™ 2 
(D.31(a,b)) 
where /?2 .=• V ( z - o ] ' + (A'=7P, Z 2 = V(z-a') 2 + (A'+^j2 
Further using assumption 4 in Chapter 2, we can neglect the displacment current up to a 
fairly high frequency, and also noting the fact that the resistivity of air, i.e., p, is very large, we 
can conclude that the term mf ~0. This produces the final equations: 
7(011 aI ( 
E++ = —UniZM 
+ / ^ - ^ ^ " ^ ^ e x n l ^ l z - a l ^ 
— \ e\ + y/e2 + m2 
E_+ = - joidol - exp{-A I 0\ -y'Ve2 + rn2} J w exp(;6l| z-a] )d8 2n |*| + \/e2+m2 
- 113 -
2TT 
1 ( 
-\l<0(m2R2)-K{ 
,{m2Z2) 
e x p l - b ' + A ' l V<?2+m|} 
\e\ + V^ +^ I 
exp(j'tl| i - a ' | </0 
jco/ip/ ; exp{-y | 6\-h\/e2 + m22} 
E + _ = / e x p ^ | x - a | ) d * 
2* — |*|. + V^+m| 
(D.32(a.b,o,d)) 
- 114 -
[6 
[7 
[8 
[9 
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V. F. PoIIaczek, Uber die Induktionswirkungen ciner Wech selstromeinfachleitung, E.N.T., 
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- 115 -
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