Series Impedance and Shunt Admittance Matrices of an Underground Cable System
SERIES IMPEDANCE AND SHUNT ADMITTANCE MATRICES OF AN UNDERGROUND CABLE SYSTEM
by
Navaratnam Srivallipuranandan B.E.(Hons.), University of Madras, India, 1983
A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE
in
THE FACULTY OF GRADUATE STUDIES (Department of Electrical Engineering)
We accept this thesis as conforming to the required standard
THE UNIVERSITY OF BRITISH COLUMBIA, 1986
C Navaratnam Srivallipuranandan, 1986
November 1986 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.
Department of
The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3
Date
6 n/8'i} SERIES IMPEDANCE AND SHUNT ADMITTANCE MATRICES
OF AN UNDERGROUND CABLE
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.
(ii) Table of Contents
Abstract — - - - li Table of Contents - iij List of Table - •• - V List of Figures - -:— VI List of Symbols - Viii
Acknowledgement , - •- ix
1. INTRODUCTION .—.:. 1
2. SERIES IMPEDANCE AND SHUNT ADMITTANCE MATRICES 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. COMPARISON OF INTERNAL IMPEDANCE FORMULAE 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
(iii) 4. EARTH RETURN IMPEDANCE 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. LAMINATED TUBULAR CONDUCTORS 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. TEST CASES 6.1 Single-Core Cable 86 6.2 Three-Phase Cable 91 6.3 Shunt Admittance Matrix 93 7. CONCLUSION 94 APPENDIX A 96 APPENDIX B : 98 APPENDIX C 103 APPENDIX D 106 REFERENCES : 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 F2 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'(
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 Ill
(Vii) LIST OF SYMBOLS
E — electric field strength, H = magnetic field strength / = frequency, IT = 3.1415926 (o = 27rX /, angular frequency — 1, complex operator
-7 fi0 = 4JTX 10 , absolute permeability of free space
/xr, — relative permeability of the medium i t*i ~ /iox/iri> 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 »
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 K = V{R + joiL)/(G + j 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 - (Z21| (zd • • • 1*8*1 Z = (1.2a) " l*/w] and (nil py 1*2*1 Y = (1.2b) PAWI Submatrix [Z„\ along the diagonal of matrix [Z\ is the self impedance of cable i which can be written as Z w Zc.s. Z„ = z (1.3) sic{ zs.s. where Zc.c = self impedance of the core of cable i" Zc.s. = Zs;c.= mutual impedance between the core and the sheath of cable » .7j s = self impedance of the sheath of cable i The off-diagonal submatrix 2,; is the mutual impedance between the cable i and cable j which can be written as (1.4) where Zee = mutual impedance between the core of the cable i and core of the cable j mutual impedance between the core of the cable i and the sheath of the cable j Zci - mutual impedance between the sheath of cable i and the core of the cable z..t. = j ZSi = mutual impedance between the sheath of the cable t and the sheath of the cable j Evaluation of all these elements; of submatrices [Z„\ and \Ztj\ is, in general, not easy. The best approach seems to be the one proposed by Wedepohl [22] based on the earlier work done by Schelkunoff [6]. Both authors find these impedances from the longitudinal voltage drops in the core and sheath, [Z = V/J). These longitudinal voltage drops can be obtained from the potential difference V, - between the core and the sheath and the potential difference - 3 - between the sheath and the earth, as shown in Figure 1.1. The potential differences V1 and V2 can be expressed as a function of the loop currents /j and J2 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,)/, - Zsikm 72 (1.5) where Zcrf = 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 EMTP 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 ADMITTANCE 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. Figure 2.1 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 r0 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]_ = -IV dx where'V and / are n-dimensional vectors of voltages and currents, respectively, at a distance z along the longitudinal axis of the cable system. All voltages and currents are phasor values at a particular angular frequency a>. The series impedance matrix Z is given by \Zn\ (*IN1 l*«l 1*2=1 \ZW\ Z = (2.3) \ZNN] 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, Zr -r . Z' c - ? • l*,l (2.5) Z'r -c. Z * .s. 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 Submatrix |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 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 ai follows: = Eik, - Ec,c - joL„h (2.10a) dx r d\ 2 = Et - Eikt - jo>L„/2 (2.10b) ~dx 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 r0 and r, being the outer and inner radii of the insulation. The term L„ can be defined similarly. The parameters jtaLc, and ji*Llt 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 (2-12) dV~ ~ Zikm Ztht + Zm2 + Zti dx 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 '/.' (2.13) d\'2 Zm Z2 dx where the self impedance of loop 1 consists of 3 parts = Z\ Zcre + Zinl + 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 (2-15) — - dx Zskt ^««2 ^« ' ^«ifn 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) where D = /j(m6)A',. (ma) - /,(ma) AT,(m6), p = resistivity of the conductor, m = intrinsic propagation constant of the conductor of equation (2.17), /<>,/, «= modified Bessel functions of the first kind and of zero and first order, respectively. /f0,W, = the modified Bessel functions of the second kind and of rero and first ordf-r. respectively. Using these formv.'ae the elements of the submatrix [Z.,\ can be found. 2.2.4 Submatrix |ZJ The off-diagonal submatrix [ZtJ\ which represents the mutual impedances between cable i and cable j can be best explained if we consider a transmission system consisting of only two cables « and j as shown in Figure 2.6. Before we analyze the elements of the submatrix, 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. - 17 - 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 r d\ x dV? dVf Z;J (2.19) dx dx dx ZS5 dVf d\r? If the voltages —— and —— are expressed in terms of the current in the primary circuit as dx dx dV? d\r° anc = = I0Z0s l ~.— IoZoi> then equation (2.19) simplifies to dx dx £1 z»z Os 1 - ZQZIQ (2.20) dx Zn ZQJ and the shielding factor of the grounded shielding conductor is then given by Z,i Z0i n = l (2.21) z„zss 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• dXr O ductor, then Therefore Z0l = Z0s, which makes the shielding factor to be equal dx dx ' to (2.22) - 19 - 2.2.4.4 Elements of Submatrix (Zt;). Keeping in mind the shielding effect described above, we will now derive the elements of the submatrix \Z,}]. Again, loop currrents are used, as has been done before for determining the elements df the submatrix jZ„j. Considering the cable system shown in Figure 2.3, we can define the following loop currents for the ith cable i) loop current /',, whose path consists of the core's external surface and sheath's internal surface ii) loop current 73, whose path consists of the sheath's external surface and earth. Similarly, the loop currents l[ and I'2 can be defined for cable j. If we consider the loop currents F2 and I{, there will be no induced voltage in loop 2 of cable i due to the loop current I\ as the net field produced by the loop current 1\ is zero out• side the cable j [6], Using the law of reciprocity of mutual impedances, it can be deduced that there will be no induced voltage in the loop 1 of cable j due to the loop current V2. Hence, relating the loop currents with the potential differences between the conductors we obtain: 0 'dV\/dx z\ Z'm 0 r\ x Z'2 0 diydx z m Zsi ft (2.23) d\'\/dx 0 0 z\ Am n d\"2/dj 0 ZSI Zin Zk Most impedance terms in equation (2.23), have already been defined except the term Zj.Sy, which is the mutual impedance between the earth return loops 2 of cables »' and j. If the cables are buried in an homogeneous infinite earth, where the penetration depth of the return current in the earth is less than the depth of burial, then the value of ZSiS> is given by [9]: pm"lK (ms) ' - y _ \_ 0 ' (2 24) where a = distance between the centres of the cables, r,,Tj = external radii of the caMcs i and j, and m = intrinsic propagation constant of the earth. For a homogeneous semi-infinite earth, where the penetration depth in the earth is more than the depth of burial, the mutual impedance Zt.Sf is given by (22]: - 20 - _ jap. K0(m/?) - /C0(mZ) + J (2.25) 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\ = /{ = I' n = ih + Hr (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 + 2z; + Z'm + Z'2 dV\,/dx z\ r2 Zss Hr dVJdx z'm + z\ z\ Zss Ilk (2.27) Z{ .+ 2ZL dV{,ldx Zss Zss + z>Zln2 + Z'2 Hr s,s} 7 dV{hldx Z' + Zsi Zss 2 zi z2 «*. From equation 2.27, the impedance submatrix [Z,}] defined earlier in equation (2.5) is given by - 21 - |2,l = (2.28) It is interesting to note, that the mutual impedance between the coreof cable i and the core of cable j and the mutual impedance botween the core of cable i and sheath of cable j are the same. This raises the question whether the shielding effect is properly represented in the equations. It is indeed implicitly taken care of in the formulation witb loop currents. This can be illustrated with the help of a conductor w placed in close proximity to a cable buried in the earth, as shown in Figure 2.8 Earth Conductor W Figure 2.8 - Transmission System Consisting of a Single Conductor and a Cable. Fov the system shown in Figure 2.8, the voltages and currents oil the cable can be written as: (2.29a) — %cc Ic ZCSIS + ZCU,IVI dx dV sh (2.29b) ~ Zci Ic + Z,t /, + ZlV) /„ dx where — the self impedance of the core and sheath of the cable, respectively, Zet,Ztt — the mutual impedance between the core and the sheath, and . Zca..Zia. = the mutual impedances between the core and conductor w and between the sheath and conductor w, respectively. - 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: ZSUI (2.30) ^ss Substituting the value of 7S into equation (2.29a) we obtain dV, Zc$' Zs. Z-^ L + dx z" z.. ~z~ Za ' Zsw 1 - z„ - IwZCu (2.31) zSi Zls ' Zcv, 7 7 The term 1 - is the shielding factor of the sheath for the field produced by conduc• Za Zcu, tor w. With Zcw = Zsw, it can be simplified to 1 - which is the same as equation (2.22). Zss Hence, the shielding effect is implicitly taken care of in the equations. 2.3 Shunt Admittance Matrix Y for N Cables in Parallel 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 Capacitive 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 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 4 t*=4.66 1(r (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(to) 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). - " + (, + yJx.o-T" 12311 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) and 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') = £«, + (2.36a) 1 + O)2^ (ts-e„)oiT I = Im(£') = (2.36b) 1 + o)2^ 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 XLPE 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 0 (2.37a) dl2 0 n v2 dx ?here Yi = Gx + jBA — admittance of insulation between core and sheath, Y2 = 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 Yx -Yr (2.37b) -Y, Yx + Y2 .v»*. dx Hence, the submatrix \Y„] is given by: Y ~Yx t (2.38) -Yx Yx + Y2 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 (2.39) R,eart h 3 where the grounding resistance /?eirth i given by g 2 Pearth 2J_ + fa V(2ff) + (t/2) + 1/2 R earth (2.40) 4;r D lnV(2H)2 + (1/2)2 - 1/2 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 /?eirttin equation (2.40) is a function of length. In [20], it is shown that the change in the value of C2.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. COMPARISON OF INTERNAL IMPEDANCE FORMULAE 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. \Io{rnb)Kl(ma) + 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 70,/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>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-10 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'tl(|m|r) (3.2a) 70(mr)= /o(V7T^lT)= B„(\m\r)+ jBei(|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 2jir/,(mr) where r = radius of the conductor -32 - m = intrinsic propagation constant of the conductor /0,/1=modified 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) (35) 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> = -^-coth(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) 2 where Rc is the dc resistance given by p/xr 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 TUBE 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 - Table 3.1 Internal Impedance of a Solid Conductor (p = 17 10~'Om and r = 0.0234m) FREQUENCY TUBE WEDEPOHL SEMLYEN (Hz) RESISTANC; E (Q/km) .01 0.0098825 0.0098775 0.0098825 ; . 1 0.0098825 0.0098776 0.0098825 1 0.0098858 0.0098809 0.0098874 10 0.0102067 0.0102034 0.0103294 100 0.0203380 0.0211463 0.0190556 1,000 0.0582719 i 0.0592378 0.0561595 10,000 0.1786977 0.1797193 0. 1763397 100,000 0.5596756 0.5607150 0.5572406 1 ,000,000 1.7644840 1.7655290 1.7620250 10,000,000 5.5744390 5.5754860 5.5719720 INDUCTANC3 E (jiH/km) .01 50.000000 51.800000 50.000000 . 1 49.999920 51.799920 49.999750 49.991580 51.792250 49.974780 1 ; 10 49.181730 !i 51.042710 47.836480 100 27.567070 i 28.354910 25.930690 1,000 8.853760 ; 8.868063 8.798590 10,000 2.803902 2.804328 2.802123 100,000 0.886793 0.886806 0.886736 1,000,000 0.280432 0.280433 0.280430 10,000,000 0.088680 0.088680 0.088680 The cpu times were found to be: TUBE 0.042 ms Wedepohl 0.038 ms Semlyen 0.022 ms - 34 - figure 3.1(a) and (b) Impedance of a Solid Conductor ae a Function of Frequency. - 35 - 5- WEDEPOHL SEMLYEN 2 ' - x Ui fi ----.: w -1--1 c ro *—« i / CO 0) / cc / -5- \ / -7-4 10~11 2 10_1 1 10' 102 105 104 10s 106 10' 12 8 WEDEPOHL SEMLYEN ^ 4 o L— uj o o c —4 o T3 -8 1/ 10~* 10"1 1 10' 102 103 104 105 106 107 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]: 38a Zt = -^cotb(m(6-o)) - -^—{^ + ( ) Zi = |^coth(m(6-a))+^(3/a + l/6} (3.8b) - 37 - Zah = ^|LcoBeft(m(6--c)) (3.8c) 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-:' <»;»») = m2// (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: Zt = ^coiHm{b.a))--^ (3,la) Zt = ^cotHm(b-a))+^^ {3.1, b) Z<» = -7~TCoseA(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 Z4t 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 = 0.0385m,6 = 0.0413m) FREQUENCY TUBE WEDEPOHL SCHELKUNOFF BIANCHI (Hz) RESs I STANCE ($2/k.m ) .01 0. 299163 0.299163 0.288640 0.299163 . 1 0. 299163 0.299163 0.288640f 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 .8928 4 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- TUBE WEpEPOHL SCHELKUNOFF BIANCHI v o c _p or O.H 10" 10" 10' 102 103 10' 105 10' 10 10- TUBE E WEDEPOHL \ x SCHELKUNOFF jt BIANCHI aj i- o c _o "o T> C 0.1+- 10" 10" 10' 10J 103 10* 10' 10' 10 Frequency [Hz] Figure S.4(a),(b) • Impedance Z, of a Tubular Conductor (with Internal Return) as a Function of Frequency. - 40 - WEDEPOHL SCHELKUNOFF ^ 2 BIANCHI o " 0 tt> o \ . - ' " c \ (0 co \ /' "to tu _-> A cc z 10~2 10"' 1 101 102 10J 104 10s 106 107 WEDEPOHL SCHELKUNOFF e£ 2 BIANCHI ko_ k_ Ul tv> 0 o c <0 u "D c -2 I .Mill,, IHj I 1,1,11V ! I llllll, I .IIIItT, I I I Mill, I I llllll, I I II.Ill, I llllllf 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 impedance term 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 Z4 - 42 - 0.4 WEDEPOHL SCHELKUNOFF 0.1- \ ui CP o c CD - -o.H TJ C - — 0.3 f 111 (Mill I I I lllll, TT!!I, 1 1 lllTHt ' T 1 MTIir, I I 1 llllll' 1 i i inn: ' i i mill i linn 10~2 10" 1 10' 102 10s 10" 10* 106 107 \2 4/r>3 Frequency [Hz] Figure S.6 Errors in Wedepohl's and Sehetkunoff'e Formulae for Ztl - 43 - Table S.S Mutual Impedance (Ze6, of a Tubular Conductor (p = 2.1 10"7/?m, o = 0.0385m ,6 = 0.0413m) FREQUENCY TUBE WEDEPOHL SCHELKUNOFF (Hz) RESISTANC :E (8/km) .01 0.29916343 0.29916343 0.29934775 . 1 0.29916343 0.29916343 0.29934776 1 0.29916343 0.29916343 0.29934776 10 0.29916338 0.29916338 0.29934771 100 0.29915838 0.29915838 0.29934270 1 ,000 0.29865873 0.29865846 0.29884247 10,000 0.25299441 0.25297170 0.25312757 100,000 -0.06964902 -0.06962398 -0.06966688 1,000,000 0.00001930 0.00001929 0.00001930 INDUCTANC3 E (uH/km) .01 -2.3386052 -2.3391813 -2.3406226 . 1 -2.3386052 -2.3391813 -2.3406226 1 -2.3386052 -2.3391813 -2.3406226 10 -2.3386052 -2.3391810 -2.3406223 100 -2.3385803 -2.3391563 -2.3405975 1,000 -2.3361094 -2.3366832 -2.3381230 10,000 -2.1095191 -2.1099033 -2.1112033 100,000 -0.0097380 -0.0097146 -0.0097206 1 ,000,000 0.0000082 0.0000082 0.0000082 The errors in Wedepohl's approximation are practically negligible for both resistance and inductance up to a frequency of 10MHz. The errors in Schelkunoff's approximation are around 3.5% up to a frequency of 10kHz and decreases for higher frequencies. Schelkunoff's approximation gives the same values as Wedepohl's approximation for the inductance term. The errors in Bianchi's approximation are negligible for frequencies up to 1kHz but increase thereafter for the resistance term. The error in the inductance is 3.8% for the whole frequency range. - 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<5 1 STANCE (jB/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 149755 1000000 3. 518632 3.518953 3. 529473 • 3.63192 6 10000000 11 .10564 11 .10605 11 .11652 ; 1T 1.4852 . 7 INIHJCTANC E (uH/ 'km) .01 51759: 4.51977 i *• 51977 I 4.6783 6 . 1 : 51759 j 4.51977 i 4. 51977 i 4. 67836 4. ! 1 4. 51759 1 4.51977 ! 4.5197 7 4.6783 6 10 4. 51759 4.51977 4. 51977 4. 67836 100 4. 51756 4.51975 4. 51975 4. 67834 1000 4. 51510 4.51728 4. 51728 4. 67578 10000 4. 28865 4.29052 4. 29052 4. 441 06 100000 1 .7645 3 1 .76463 1. 76463 1 .8265 5 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 io4 io5 10s io7 Frequency [Hz] Figure S.7(a),(b) - Impedance Zb of a Tubular Conductor (with External Return) as a Function of Frequency. - 46 - s / 2-- V /\ O k- UJ 0 BIANCHI 10~2 10"' 1 10' 102 10S 10" 10* 10* 107 o fc— fc— UJ 0 a> u c - to WEDEPOHL u 3 TJ SCHELKUNOFF c -2 BIANCHI — A ] II. i i i . i II,i| i i i . nn, • J • i in., .ii.i.ii, i . , i mi, III nm, .MI mil i i 11 in 10~2 10"1 1 10' 102 10S 104 10S 10* 107 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. - 48 - The routines for Wedepohl's approximation - WEDAP, Semlyen's approximation - SEMAP, Schelkunoff's approximation - SCHAP, Bianchi's approximation - BNCAP and TUBEC were written by the 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 Earth Return Impedance in a Homogeneous Infinite Earth 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: pm K0(mR) Z -- (4.2) 2na A'j(ma) 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: pmIK0(mR) E = - (1.3) 2za Kt(ma) 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 impedance between two cables of radii o and 6 respectively, buried in a homogeneous infinite earth is given by: 2 pm K0(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 Earth Return Impedance in a Homogeneous Semi-Infinite Earth 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 - 'Y / w H / 0) o c 10 1 —I- to to cu az -3 I I I! I I ! I 1 T T1!| •-' ! '1 - T TTTT i s 3 2 10 10" 10" 10~ 10" 10"' 10' 25 15 cu o c CO \ o T3 C -15 -25 -f 1 1 I 1 Mll| I 1 I I I Mil 1 U I I I I I Hl| 1 1 I 1 I Ml| 1 1 llllll, 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: 2 2 g exp[-(o + m )\h-y\] - eXP[-(^ + m )\h + y\\ E__ — | 2(p- + m~) + r exp(-|ftlfyl VV+ttr) exp(jd>x )d 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 = Vis + (/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 } mt - (* •+ ml)P- ~ ^F(mZ,\x |,/)) (4.10) where 2 F(mZ,\x\,l)= ffcVl^T ^==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 dif• 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.. 2 2 2 exp [-(* + m )|A-j,| ] - exp[-(cA2 + m )\h + y\] expO^x ) (4.12) ?(c*2 + mz) If h = y then the first exponential term within the closed brackets { } will become 1 and the integration now becomes 2 2 2 jl-exp[-(cr( x)d 10 6- T A o> o c V JO -2 V) o> -6 -10 I I I ill 1 "' '^f 1 T I 1 1 I 1111| 104 10' 2 3 10 10 50 30- £ \ 10 o \ o c o o ~10 D \ / -30 -50 7— ••T-",,T "I m I I I 1 I I I I i i i i i i i 4 10' 10* 1CV 10 Interval 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 taken as 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 NUMERICAL EQUATION 4.10 (Hz) INTEGRATION RESISTANCE (8/krn ) .01 0.00000986985 0.00000986986 . 1 0.00009870394 0.00009870399 1 0.00098721243 0.00098721145 10 0.00987746990 0.00987752230 100 0.09894307000 0.09894471000 1 ,000 0.99462669000 - 0.99467682000 10,000 10.0999510000 10.1014090000 100,000 105.026550000 105.062940000 1,000,000 1119.21050000 1119.73660000 10,000,000 10416.7130000 10410.0480000 REACTANCE (fi/km .01 0.00014814024 0.00014814024 . 1 0.00133672137 0.00133672130 1 0.01192043120 0.01192028400 10 0.10477298900 0.10472980000 100 0.90245276590 I 0.90245110000 1 ,000 7.57239400000 ! 7.57234170000 10,000 61.8788630000 . 61.8625200000 100,000 461.037952000 460.988500000 1,000,000 3018.40132100 3017.08610000 10,000,000 13049.3294800 13031.2620000 = — ln(Z/R) + J (4.15) 2x fhere Z - Vi2+ (A+y)z, - 59 - m = intrinsic propagation constant of the earth. The integral part of equation (4.15) can be further simplified to (4.16) o U| + \U2 +• m2 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 S5 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. )000 4000 r«CQUCNCT (Hi) 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 m2 with the exponent term exp(—| h+y\ Vc42+m2) 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: pm Z, = .lnhHEl +0 5 _ ±mh (4.17a) 2n 2 3 pm •In— L + 0.5 ml (1.17b) 2ir 2 3 where 7 = Euler's constant, h = depth of burial of the conductor, I = sum of the depths of burial of the conductors, R = Vi2 + (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 = "^Mr + 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 WITH WITHOUT FREQUENCY DISPLACEMENT DISPLACEMENT (Hz) CURRENT CURRENT RESISTANCE (8/ki .01 0.00000986985 0.00000986986 . 1 0.00009870393 0.00009870393 1 0.00098720986 0.00098720982 10 0.00987747671 0.00987747340 1 00 0.09894371230 0.09894339595 1 000 0.99468700004 0.99465537187 10 000 10.1056633504 10.1024602926 100 000 105.572646568 105.239909564 1 oob ooo 1 172.74925588 1136.35180546 10 000 000 15170.9605539 11593.2569450 INDUCTANCE ("H/ nm) 2.82478692844 2.82478689712 2.59451983737 2.59451979297 2. 36423415099 2.36423411661 — • o 2. 13388974446 2.13388971332 1 .90335982756 1.90335979916 1.67224525374 1.67224520020 1 .43930244333 1.43930156514 1 .200803841 19 1.20078329139 0.94712614357 0.94694242963 0.66822941718 OOOOOO — OOOOOO O oo • ooo • OOOOO • o — 0.64980341214 - 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. oi_ k. UJ CD o -5- c to CO CD CC -15 -25 i i i IIIMI— ni]—i i i inn,—i i i Mini—i i i inn, 1 i i mii| nil 1 i i mill i i i mil! 10~2 10" 1 10' 102 10J 104 105 106 io7 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 limits of 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 25% 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 iii 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 FREQUENCY POLLACZEK WEDEPOHL AMETANI SEMLYEN (Hz) RE; > I STANCE (£2/1cm ) .01 0.0098825 0.0098825 0.0098775 0.0098825 .01 0.0000099 0.0000099 0.0000099 0.0000099 . 1 0.0000987 0.0000987 0.0000987 0.0000987 1 0.0009872 0.0009872 0.0009867 0.0009870 10 0.0098775 0.0098775 0.0098618 0.0098692 100 0.0989434 0.0989457 0.0984519 0.0986840 1000 0.9946554 0.9948561 0.9795263 0.9865784 10000 10.102460 10.119288 9.6564680 9.8575313 100000 105.23991 106.59172 93.498270 98.315053 1000000 1136.3518 1236.6438 913.03214 974.99123 10000000 11593.257 17765.288 10826.058 9498.8394 INI XJCTANCE (mHy'km ) .01 2.8247869 2.8247838 2.8247917 2.7016048 . 1 2.5945198 2.5945167 2.5945418 2.4713467 1 2.3642341 2.3642310 2.3643105 2.2410895 10 2.1338897 2.1338866 , 2.1341378 2.0108352 100 1.9033598 1.9033564 1.9041501 1.7805898 1000 1.6722452 1.6722386 1.6747414 1.5503729 10000 1.4393016 1.4392629 1.4471064 1.3202459 100000 1 .2007833 1.2004118 1.2245157 1.0904032 1000000 0.9469424 0.9429813 1.0126228 0.8614597 10000000 0.6682294 0.6267975 0.7962689 0.6353565 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 5 s 7 2 J 10' 10 10 10 10" 10" 1 10' io 10 Frequency [Hz] Figure 4.6(a),(b) Earth Return Self Impedance as a Function of Frequency - 66 - 25 WEDEPOHL 15- AMETANI / SEMLYEN / 5- ui o -C — 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 mii| 1 1 1 iiui| 1 1 1 mir 10"2 10"' 1 10' 102 10J 104 105 106 107 25 WEDEPOHL 15- AMETANI SEMLYEN 5- LU CD o c -5 ra > o T3 C — -15 — 25 ~f 1 1 T mrn r-i TTTTTTI 1 i 11 ini[ 1 i i itui] r-rri-TTTTi r—r-rrrmj r iitini] r i i rim\ r-rrmrt} 10~2 10"' 1. 10' 102 10J 104 10S 106 107 Frequency [Hz] Figure 4- 7 Errors in Earth Return Self Impedance - 67 - 2. Cables are buried at normal depths (l-2m) but are used at high frequencies (100kHz and above). In such situations, the depth of penetration in the earth is given by: d = 503.3\/pe«u// (4.19) (where d is in m, / is in fi—m and / is in Hz) and becomes smaller than the depth of burial. The second possibility does not arise normally in power system studies. For a typical underground transmission system, with an earth resistivity of 10 J?—m, and a burial depth of lm, the frequency at which the penetration of the return current in the earth becomes less than lm is 3MHz or higher. In power systems, one rarely encounters such frequencies. If such cases do arise, however, the infinite earth return impedance formulae given by equation (4.2) and (4.7) could be used to find the earth return impedances. It is interesting to know whether equation (4.10) for the semi-infinite case is still valid if the burial depth is large. This can be checked as follows: If the earth resistance is assumed to be 10 17— m, then the depth of penetration given by equation (4.19) will be less than 5.5 m, for frequencies 0.1 MHz and higher. Hence, if a cable is buried at a depth of 5.5m, then the values obtained for earth return self and mutual impedances, using the equations (4.2) or (4.7), should be the same as those obtained using equation (4.10) for frequencies above 0.1MHz. Figure 4.8 shows the differ• ence in the value of resistance obtained by Pollaczek's formula and equation (4.2) for the earth return self impedance in the frequency range 10kHz to 1MHz. The difference decreases from 19% to less than 1% while the depth of penetration decreases from 15.9m to 1.59in. Hence, it appears that Pollaczek's formula is correct even at large depths of burial even though it is better to use the equations (4.2) and (4.7) for the case of infinite earth [9,22]. Subroutine TUBE has an option for finding the infinite earth return self impedance, but it cannot be used for finding the mutual impedance. The routine TUBEC developed by the author has options for finding both self and mutual impedances, and can take displacement currents into account as well. 4.7 Mutual Impedance Between a Cable Buried in the Earth and an Overhead Line or Vice Versa Another impedance of interest to power engineers as well as to communication engineers is the mutual impedance between an underground cable and an overhead line or vice versa. The electric field strengths in air due to a current carrying conductor buried in the earth - 68 - 20 12- cu u c CD "* 4 - CU u c -4 CO to CO cc -12 -20 + -i—i—• III 104 10s 106 Frquency [Hz] 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: 2 expj-/i| c*| 2 0 -dV •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 Tubular 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, m zb h — 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 .1) lm %aa 'm-l) (5 dx — (ZTb 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 z (5.2) . ~ bi 'm-l dx Therefore we can find a relationship between I„ and /„_, from equations (5.1) and (5.2), (5.3) lm + Zli ' - 72 - Now let us consider the electric Held strength on the outer surface of the mth layer. On m one hand it is — z^Im, and on the other hand it is —(Zb tIm — 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 remaining layers one by one until we obtain the impedance of the complete laminated conduc• tor made up of m layers. (Zlb? Ab = ZU - „ , , • = 2, • • • m (5.6) Z'at + Ab 5.1.2 Internal Impedance with Internal Return Similarly we can find the internal impedance of a laminated tubular conductor with m current returning inside. Let Z™, Zb h 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), /, z \ a (5.9) h Zt\ + z- Now, consider the electric field strength on the inner surface of the first layer. On one hand it an o n tne — is —{ — za\lo)t d 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 z aa ~ 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 Application 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 CASE i: Core and Sheath not Coated Axis Core J Shcalh v/ssssss//////////////////////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. ZMS can be obtained from equation (2.11), and Zih, (sheath - with internal return) from equation (3;.la). 5.2.2 CASE ii: 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 - TV _ (5.15) Zll + z,sk The total impedance Z can then be written as 2 (Zspm ) Z — Z:rc + Zins + Zip, (5.16) zspc Zik, 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), = -(zj, + ZM dx oT's = ~(ZnIi:+ Z2I2) (5.17a,b) ~di~ where V, = potential difference between the core and paint = potential difference between the paint layer and sheath v2 Zi Zms "** Zsp, — Z2i Zspm zm Z2 — Zipt + z$k, Since the paint layer and the sheath are at the same potential we have V2 = 0. Hence from equation (5.17b), - 76 - I2 = (-Zm/Z2)/1 (5.18) Substituting the value of I2 into equation (5.16a) gives dV, (5.19) Therefore the impedance for surge propagation between core and sheath is given by z = z, - — or 7-7 _L 7 J- 7 _ (^sym which is identical with equation 5.16. 5.2.3 CASE iii: Only Core Coated Axis 1, Sheath ////////////////////////////// Earth | Figure 5.5 Core Alone Coated The surface for surge propagation in this case consists of the surface of the laminated conductor (made up of core and the coating) and the inner surface of the sheath. The impedance for surge propagation between core and sheath is therefore given by (5.20) Z = z$ + Znt + Za, where Z,A, and Zint are the same impedances explained earlier. z$ is the internal impedance of the laminated conductor with external return. This can be obtained from equation (5.6), where layer 1 is the core and layer 2 is the paint on core, 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 CASE iv: Both Core and Sheath Coated Axis 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 surge propagation 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 (5.24) z zcpc + z,RS + Zipt where all the impedance values have been explained earlier. - 78 - Results 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 -ORC. R IRCR -IRSH ORSC H Sheath w/////r7/777777777777777777> \/77//7777777777777777//777777/, Earth j 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 Stainless Steel Coating The skin depth for a particular material is given by 6 = V2p/M/i (5.25) -79 - Table 6.1 Resistivity and Relative Permeability of Coating Materials RELATIVE TYPE MATERIAL PERMEABILITY RESISTIVITY Core and Sheath Aluminum 1.0 2.62E-08 Paint (a) Stainless Steel 1500.0 4.70E-07 (b) Supermalloy 100 000.00 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 SKIN DEPTH (Hz) (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. — 6U " 100 CASE 1 10 CASC 2 E CASE_3 CASE 4 c in a) rr 0.1 0.01 0.001 1CT 10" 10' 103 103 10' 10s 10' 10' Frequency [Hz] \ O.B- \ \ CAST 1 w CASE.. 2 0.6 x C_ASE_3 CASE ^ c o o o.* ~o c 0-2. 0-+ 10" • 10- 10' 10J 105 10' •10s 10* 10' Frequency [Hz] 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 Coating The high resistivity and high permeability of supermalloy make its skin depth very small even at low frequencies, as shown in Table 5.3. Table 5.3 Skin Depth of Supermalloy FREQUENCY SKIN DEPTH (Hz) (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 - \ CASE 1 <£2 ^\ \ \ CAse 2 3 \ i- ~ V LAIL3. CASE i c o 5y X 2 \\ •o c J 5 5 10"2 10" 1 10' 10 10 10." 10 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 CASE 1 10- CASE 2 u c D tn ca 0.1. 0.01. 0.001J 1CT1 10"' 10' 102 103 10' 105 10s 10' Frequency [Hz] 20- \ 15- X \ E CASE 1 CASE 2 CASE_3 0 + e 10" 10" 10' .10* 10J 10" 10* 10 10' Frequency [Hz] 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 The internal impedance formulae for tubular conductors and the earth return impedance formulae were discussed in detail in Chapters 3 and 4, respectively. These impedances make up the elements of submatrices [Z„] and [Zi}\. In this chapter the values of these submatrices are obtained for a specific underground cable system, using the exact formulae as well as approximations. The approximate formulae proposed by Wedepohl [22] agree very closely with the exact formulae and take very little cpu time. They also provide simple expressions for hand calcula• tion pruposes. Therefore only the approximate formulae proposed by Wedepohl [22] are con• sidered in this chapter. The results are also compared with values obtained from the Cable Constants routine in the EMTP, which was developed by Ametani [27]. 6.1 Single Core Cable The impedances of a single core cable are given by a 2X2 matrix of the form (6.1) m = z. where all elements were defined earlier in Chapter 3. With the data of the test case described in Appendix A, the values of these elements were obtained from the exact formulae, from Wedepohl's approximation formulae and from Ametani's Cable Constant routine, as tabulated in Table 6.1. Figures 6.1 depicts the errors in Ametani's and Wedepohl's approximate formu• lae for the impedance Z.c. Figures 6.2 and 6.3, respectively, show the errors for the impedances Zs. and Zss The errors in Ametani's and Wedepohl's formulae are not significant at low fre• quencies, but for higher frequencies they cannot be neglected. We also notice that the errors in Zcc, ZCi and ZS! all have similar values. They are essentially created by the errors in the earth return formulae used by Ametani and Wedepohl, as shown in Figure 4.7. As mentioned earlier, Wedepohl's approximate earth return formula is valid only if the condition \ mZ \ <0.25 is satis• fied. This is only true at low frequencies. At high frequencies, the intrinsic propagation con• stant m, becomes larger, and this causes the errors in the results. The errors also increase if the separation between the cables becomes larger. The reasons for the errors in Ametani's earth return impedance formulae at high frequen• cies has already been discussed in Chapter 4. - 87 - Table 8.1 Impedances of a Single Core Underground Cable. Zee (fi/km) EX; AME1''A N I WEDE :POHL FREQ R X R X R X (Hz) 1 .010873 .016082 .010873 .016083 .010868 .016093 10 .020084 .146299 .020069 .146315 .020081 .146416 100 .119303 1 .30456 .118805 I. 30504 . 120114 I. 30506 1000 1.05509 1 1 .4759 1.03996 II. 4916 1.05626 II. 4759 10000 10.4803 99.6843 10.0343 100.175 10.4983 99.6822 100000 108.240 839.848 96.4973 854.761 109.593 839.614 Zcs (£2/ km) 1 .000987 .015097 .000987 .015098 .000987 .015097 10 .009878 .136501 .009862 .136516 .009878 .136501 100 .098954 1.22016 .098465 1.22066 .098956 1.22016 iooo .995717 10.7494 .980590 10.7651 .995919 10.7494 10000 10.2001 92.8295 9.75414 93.3201 10.2170 92.8272 100000 106.430 775.524 94.6876 730.438 107.782 775.291 Zss im) i Wl 1 .300151 .015083 .300150 .015083 .300151 .015083 10 .309041 .136354 .309025 .136369 .309041 .136354 100 .398112 1.21869 .397612 1.21919 .398115 1 .21869 1000 1.29438 10.7347 1 .27925 10.7504 1.29458 10.7347 10000 10.4531 92.6969 10.0071 93.1875 10.4700 92.6946 100000 106.361 775.518 94.6180 790.432 107.713 775.285 - 88 - 50 30- / 10- 7 u c « -10 cn io cu rr AMETANI -30 WEDEPOHL — 50 \ i i i Mini i i i nun —n TTTTTTTJ—i TI itiiTf nr TTHTTT; T-TTTTTTT] r i i mit| "i T T mn| " i i ritin 10~2 10" 1 10' 102 103 104 105 106 107 20 12 4- 01 o -4 c \ to u ca cu AMETANI CC -12 WEDEPOHL -20 1 III Mini i i i mill i i i IIIIII i i i mm i i i iiuii—i i 11mil i i 11iui| i i 11mil i i 11mi 10"2 10"' 1 101 102 105 104 10s 106 107 Frquency [Hz] Figure 6.1 - Errors in Ametani's and Wedepohl's Approximations in Zcc - 89 - 50 30- / 10- / 0) u c -10- tncu tc AMETANI -30- WEDEPOHL — 50 1 T ITTIfH ' I '1 MllTir I 'T I 1 im]'~ T'TTTTmr T"T T^TIITf I'TlTnil}' 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 -4- CO \ *-> o ra CD CC AMETANI -12- WEDEPOHL — 20 | i i 11 —i i i IIIIII I i i Mini 1 i i mill 1 i . IIIIII—i i i Mini 1 i I Mini—i i i utii| ill Mini 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 - 50 30- / 10 / LU co u « -10 +—< m cc AMETANI -30- WEDEPOHL 1 —50 | III IIIIII i i i IIIIII—i i i IIIUI i i 11uii| i i 11mil I i 11uii|—> i i IIIIII I I I IIIIII—i i i inn TO"2 10"' 1 10' 102 103 10" 105 106 107 20 CD u c to o co CD AMETANI at -12 WEDEPOHL —20 I III niM| i i i iiiii, rrm] 1 iiiinij i i IIIIII; I IIIIIMI I I IIIIII) I IIIIIIII I iimiq 10"2 10"' 1 10' 102 103 10" 10S 106 107 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 \Zn\ |Z12] \ZK) 1*1- (0.2) \2A \z*\ 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 |Zt;] of the form: Z - .f . Z r r • (0.3) Zj.ry Zs.s. 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. Table 6.2 Mutual Impedance between Two Cables with Burial Depth of 0.7om and Separation of 0.80m. Zij (fi/kin) EX; ^CT AME*]rAN I WEDI3POH L FREQ RES REA RES REA RES REA 1 .000987 .012562 .000987 .012563 .000987 .012562 10 .009877 .111152 .009862 . 111167 .009878 .111151 100 .098943 .966670 .098452 .967169 .098946 .966668 1000 .994644 8.21457 .979517 8.23027 .994856 8.21452 10000 10.1015 67.5095 9.65569 68.0000 10.1193 67.5070 100000 105.154 525.238 93.4318 540.150 106.592 525.000 - 92 - 50 30 / O )0 / UJ cu o li -to- cn cn tn AMETANI -30 H WEDEPOHL -50 1 ni 1 i i IIIIII 11 MM; 1 \ i i mi, 1—i t mil, 1—t 11 nn' — TTTT. i—r^rrrrrrr- 2—i n: 7 10~ 10"' 1 10' 102 103 104 10S 106 io 50 30- O 10- UJ cu o c -10 \ AMETANI 0> CC WEDEPOHL -30 50 -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"" 10-2 10" 1 10' 102 10S 104 10S 106 107 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 Matrix 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 EMTP [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 EMTP 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 - APPENDIX A Test Examples for Buried Cables Earth d, x x GA '- » 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 = 0.75m, depth of burial of each cable di,d2,dt X12 = 0.30m, horizontal distance between cables 1 and 2 X ja = 0.30m, horizontal distance between cables 2 and 3 0.0234m, radius of the core *2 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 8 Peon 1.7X10" J?-m resistivity of the core material 2.1xi0"7/?-m resistivity of the sheath material P- r core? r shea.tb» /'fexrtb'/'rilr = 1.0, relative permeability of the core, sheath, earth, and air respectively. - 98 - APPENDIX 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: dE dllz 3Ez t = (\lp + »"toe)E,, — — iliifiH, rd<*> dz rd(j> 3E, dE, 3//r dHz = (l/> + i ifdlrHJ 3H,\ 1 (9[rE4) ^ dEt\ (B-1) — 1— — ' = + i«e)Er. TT f = ~ r ( dr d ) 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 dlrllA . - (1/p + i -(l/p + itoe)Er (B.2b) a, dE2 dE 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 r Tz r exponential variation of the quantities E,,EZ and as E,eC *, Eze~ and H^,e~ ', 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 2 where m = I ' co2/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 T2 is negligibly small by comparison with m2. Therefore, we can write equation (B.4) as d2H* i dH, sT = mXH* (B.5) dr' dr The solution for Ht of equation (B.5) is in the form of Bessel functions given by: Alx{mr) + BK^mr) (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, (B.7a) dx _d_ (B.7b) (x*Kn)= -x*Kn.x dx 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 = _ /»/»("») (B10b) 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 [irimajK^mb) + /v"0(mo)/,(m6) 2naD 7 - Pm ^/0(m6)K'i(r7ja) + AT0(m6)/1(ma) j Z> ~ 2xbD 7 = P 2nabD (B.13) Schelkunoff stated these results in the following two theorems. 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 - APPENDIX 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 •It) 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 = —+-fjrdrJ v(C.l ) 2nr r 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 Substituting for H from equation (C.l) yields dtp = udr-!-+±Jjrdr (C.3) 2xr r •* Now let us write Kirchhoff's voltage law around the rectangle ABCD of unit length and width , : r anc tne dr. The net resistive voltage drop is — '|~^~J^ * induced voltage is jmd 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 (C.4) — p dr + jtauH dr = 0 dr Substituting for//from equation (C.1), we have pdJ •dr + jtAfidr — + -fjrdr = 0 (C.5) dr 2itr Multiplying this equation by ~- and differentiating with respect to r we have par d2J dJ_ _ jvuJ + = 0 (C.6) dr2 rdr p If we substitute m* for Jfa>/i, then equation (C.4) can be written as P - 105 - £ «•»««*. (C7) and equation (C:6) can be written as d2J . dJ lJ = 0 (C.8) dr' r dr m 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 (Cll) 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 (Cll) as B = ~ 2nal Using the equation E = pJ, the solution for the electric field strength at any point in the soil is found to be pml K0{mr) . E~~2*a K^ma) (C13) 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 - APPENDIX 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. p A y . (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 V2E - 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 2 V E - 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 2 (D.7) a m E++ + plmfIS(x — a)6(y — h) 3z By' where Pi For the earth, equation (D.2) can be written as 2 2 a £_ + a E_+ = m|E_. (D.8) dx2 • here 2 2 2 m = — CO p i2 I P2 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 (D.10) dx dx Horizontal component of H is continuous at the surface dE++ a£_+ aE_+ (D.il) Pidy P2&y p-z^y' 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 - 2 d E+ + 2 2 -6 E^ + = mfE++ + /j,m /exp(-^a)%-A) (D.12) — £+ + | y-0 E-+ | Y'-0 — £Q (D:14) 1 * j i d£_+ dE+ • 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: 2 2 2 -6 E^+ — 0 E+ + + 0EO = m E++ + p^rn? Iexp{-j6a)sin{0h) (D.16) i.e., 2 2 (0 + Cf)E++ = 0EO - Plm lexp(-jea)sin(0h) (D.17) where Cf = 02 + m2 Similarly, taking the Fourier sine transformation of equation (D.13) with respect to y' with 0' as parameter, we have 2 2 2 -0 E_+ + fl' E.+ + /J*E0 = m £_+ (D.18) Hence + Cf)£_+ = £'E0 (D.19) 2 : where C'| = t? + m2 . Now, taking the inverse Fourier sine transformation of equation (D.17) with respect, to 0 we have = E0exp( —C,y) L -•^ /exp(-^a)|exp(-C1|/i-j/|)-exp(-C,|/l+y|)j (D.20) Similarly, taking the inverse Fourier sine transformation of eqution (D.19) with respect tofl' 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) (D.22) ui (C, + :—<7a) u2 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(-C2| h +y |)] 2C, exp{-C1\h+y\) exp(j6\ i-a| )d0 (D.23) C. + — C2 Similarly, substituting E0 in equation (D.21) and taking the inverse transformation with respect to 6, we obtain the value of E. + as follows m Pi fl "r exp{— C^y — C1h}exp(j6\ x — a\)d6 ~ J (D.25) E-+ = 2n Ci + —C2 U2 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, 2 2 d £+_ d E+ 2 = m,E,. y^O, (D.26) dx5 By - Ill - H y ? (x.y) x _i Earth • (x,y') . p.- W) Figure D.2 - Current Carrying Filament Buried in the Earth. 32E__ 82E__ , , . (D.27) 1 + — = m|£__ + — )5 — ax ay p2m2fS(x a (y h ) 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 p2m2I " exp{ — Cly — C2h) E+. = •xp{j6\x-d\ )d0 (D.28) 2TT -»- —C2 ?2 [exp(-C2| h'-y'\ )-exp(-C2| h'+y'\ ] p2m2I £•_ _ .= - 2n 2C, exp(-C2|/i'-rV|) exp(jO \ x-a'\ )d8 (D.29) C, + —C2 »2 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 2 i \ c exp{—ayx 9 V^-+m }exy p . . . , • ' _ • . K0(mr)= J \ (jis)d3 (D.30) 2Vr+m2 where r = and where K0 is the modified Bessel function of the 2nd kind and of the zeroth order, we can write E+ + and E__ as follows: pm,/2 ( E„ = + / ;= (jtf|x-o»| )rfg \A2 + rnf + \/V + 2 where /?, ~ V {T - af+(h - y?, Zx = V(z-a) + (h+yf pm2I ( m — — —jA'o( 2^2) KG(m2Z2) 2 r exp{-|j/+/i V^+m2 }exP( + / . 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