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Series impedance and shunt admittance matrices of an underground cable system Navaratnam, Srivallipuranandan 1986

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SERIES I M P E D A N C E A N D SHUNT A D M I T T A N C E MATRICES OF A N UNDERGROUND  CABLE SYSTEM  by  Navaratnam  Srivallipuranandan  B.E.(Hons.), U n i v e r s i t y of M a d r a s , India, 1983  A THESIS S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F T H E REQUIREMENTS  FOR T H E DEGREE OF  M A S T E R OF APPLIED  SCIENCE  in T H E F A C U L T Y O F G R A D U A T E STUDIES (Department of Electrical Engineering)  We accept this thesis as conforming to the required standard  T H E UNIVERSITY  O F BRITISH C O L U M B I A ,  C Navaratnam Srivallipuranandan, 1986 November  1986  1986  In p r e s e n t i n g  t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of  requirements f o r an advanced degree a t the  the  University  o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make it  f r e e l y a v a i l a b l e f o r reference  and  study.  I  further  agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may department o r by h i s o r her  be granted by the head o f representatives.  my  It is  understood t h a t copying or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l gain  s h a l l not be  allowed without my  permission.  Department o f The U n i v e r s i t y o f B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T  1Y3  Date  6  n/8'i}  written  SERIES I M P E D A N C E  A N D SHUNT ADMITTANCE  OF A N UNDERGROUND  MATRICES  CABLE  ABSTRACT  This thesis describes numerical impedance matrix  methods for the: evaluation of the series  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. evaluating  the  elements  of the  Also, a more accurate  admittance  matrix  with  way of  frequency  dependence of the complex permittivity is proposed. Various impedance standard  formulae of buried  have cables.  been  developed  Using  for the  the Polhiczek's  earth  return  formulae  as the  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  conductor.  of  a  homogeneous  tubular  Equations  have  derived to evaluate the internal impedances of such laminated conductors.  (ii)  been  tubular  Table of Contents  Abstract  —  -  -  Table of Contents  l i  -  List of Table  iij  ••  -  List of Figures  -  -  V VI  -:—  List of Symbols Acknowledgement  -  -  Viii  ,  -  1.  INTRODUCTION  2.  SERIES I M P E D A N C E A N D S H U N T A D M I T T A N C E M A T R I C E S  •-  .—.:.  ix  1  2.1 2.2  Basic Assumptions Series Impedance matrix [Z] for N Cables in Parallel 2.2.1 Submatrix [Z„] 2.2.2 Skin Effect 2.2.3 Internal Impedance of Solid and Tubular Conductors 2.2.4 Submatrix [Z,-yJ 2.2.4.1 Proximity Effect 2.2.4.2 Proximity Effect of a Single-Phase Circuit of Two Identical Conductors 2.2.4.3 Shielding Effect of the Sheath 2.2.4.4 Elements of Submatrix [Z„] 2.3 Shunt Admittance Matrix [K]; for N Cables in Parallel 2.3.1 Leakage Conductance and Capacitive Suceptance 2.3.2 Frequency Dependence of the Complex Permittivity 2.3.3 Submatrix 2.4 Conclusion  ••  6 7 9 13 14 15 16 17 17 19 22 22 23 27 29  3. C O M P A R I S O N O F I N T E R N A L I M P E D A N C E F O R M U L A E 3.1 3.2 3.3 3.4  Exact Formulae for Tubular Conductors Internal Impedance of a Solid Conductor Internal impedance of a Tubular Conductor Conclusion —  (iii)  -•..— ; -•-  30 31 36 47  4. E A R T H 4.1  RETURN  IMPEDANCE  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 Overhead Line or vice versa  67  4.8  Conclusion  69  5. L A M I N A T E D 5.1  5.2  CONDUCTORS 70  5.1:1  Internal Impedance with External Return  70  5.1.2  Internal Impedance with Internal Return  72  Application to Gas-Insulated Substations Case i:  Core and Sheath not Coated  73 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  5.2.5  Stainless Steel Coating  79  5.2.6  Supermalloy Coating  82  5.2.7  Comparison between Stainless Steel and Supermalloy Coatings  Conclusion  6. T E S T 6.1 6.2 6.3  TUBULAR  Internal Impedance of a Laminated Tubular Conductor  5.2.1  5.3  -  77  82 85  CASES  Single-Core Cable Three-Phase Cable Shunt Admittance Matrix  86 91 93  7. C O N C L U S I O N  94  APPENDIX A  96  APPENDIX B  :  98  APPENDIX C  103  APPENDIX D  106  REFERENCES  :  114  (Iv);  List of Tables  3.1 3.2 3.3 3.4  Internal Impedance Internal Impedance Mutual Impedance ing Inside Internal Impedance Outside  4.1 4.2 4.3 5.1 5.2 5.3 6.1 6.2  of a Solid Conductor Z of a Tubular Conductor (Z (,) of a Tubular Conductor with Current Return-  33 38  a  a  43 Zy of a Tubular Conductor with Current Returning  -.  ,  Solution of PoIIaczek's Equation by Numerical Integration and Using Infinite Series Earth Return Self Impedance with and without Displacement Current Term .' 1 Earth Return Self Impedance as a Function of Frequency Resistivity and Relative Permeability of Coating Materials Skin Depth of Stainless Steel Skin Depth of Supermalloy Impedances of Single Core Underground Cable Mutual Impedance between Two Cables with Burial Depth of 0.75m and Separation of 0.30m  (V)  44  58 61 64 '• 79 79 82 87 91  List of Figures  1.1 2.1 2.2 2.3 2.4 2.5 2.6 2.~! 2.8 2.9 2.10 2.11  Potential Difference V, between Core and Sheath and F  between  2  Sheath and Earth Basic Single Core Cable Construction Loop Currents in a Single Core Cable Potential Difference between Two Concentric Conductors Three Conductor Representation of a Single Core Cable Sheath with Loop Currents I and I ... •Two Cable System , -. Circuit Arrangement of Primary, Secondary and Shielding Conductors, with Shielding Conductor Grounded at Both Ends".. Transmission System Consisting of a Single Conductor and a Cable ; x  2  Cross-Section of a Coaxial Cable (a),(b) - Measurements of e'(<o) and ("(<*) of an OiMmpregnated Test Cable at 20°cC Values of e'(o>) and «"((•)) Obtained from the Empirical Formula  :  3 7 9 10 10 15 16 18 21 23 24  26 2.12  Polarization-Time Curve of a Dielectric Material  27  3.0 3.1  Loop Currents in a Tubular Conductor (a),(b) - Impedance of a Solid Conductor as a Function of Frequency  30  3.2  (a),(b) - Errors in Wedepohl's and Semlyen's Formulae for a Solid Conductor  3.3 3.4 3.5  35  Cross-Section of a Tubular Conductor (a),(b) - Impedance Z of a Tubular Conductor (with Internal Return): as a Function of Freqency Errors in Wedepohl's, Schelkunoff's and Bianchi's Formulae for Z  30  a  39  a  40 3.6 3.7 3.8  Errors in Wedepohl's and Schelkunoff's Formulae for Z (a),(b) - Impedance Z of Tubular Conductor (with External Return) as a Function of Frequency ..' Errors in Wedepohl's, Schelkunoff's and Bianchi's Formulae for Z  4.3 4.4  45  b  „  :.. 4.1 4.2  42  ab  b  ,  Electric Field Strength at Point P Error in Replacing a Conductor of Finite Radius by a Filament Conductor —-Solution of Real and Imaginary Part of Equation (4.9), for a Freqency of 1MHz. > - «... Relative Error in the Evaluation of Carson's Formulae with an Asymptotic Expansion  (vi)  46 51 53 56 59  4.5  Error in the Earth Return Self-Impedance if the Displacement Current is Ignored  4.6  (a),(b) - Earth Return Self Impedance as a Function of Freqency  4.7  Errors in Earth Return Self Impedance  60  4.8  68  5.1  Differences in Resistance Values of Semi-Infinite and Infinite Earth Return Formulae : (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  5.7  Dimensions of the Bus Duct in a Gas-Insulated Substation  5.8  (a),(b) - Variation of Resistance; and Inductance with Frequency for the Four Cases; Stainless Steel Coating, Thickness, 0.1mm (a),(b) - Variation of Resistance and Inductance with Freqency for the Four Cases; Stainless Steel Coating, Thickness 0.5mm (a),(b) - Variation of Resistance and Inductance with Freqency for the Four Cases; Supermalloy Coating, Thickness 0.01mm  ,  5.9 5.10 5.11  65  77  the Four Cases; Supermalloy Coating, Thickness 0.05mm 6.2  Errors in Ametani's and Wedepohl's Approximations in Z  B. l  80 81 83 84  Errors in Ametani's and Wedepohl's Approximations in Z  A.l A. 2  < • 78  (a),(b) - Variation of Resistance and Inductance with Frequency for  6.1 6.3 6.4  62  cc  gc  Errors in Ametani's and Wedepohl's Approximations in Z Errors in Ametani's and Wedepohl's Approximations in the Mutual Impedance between Two Cables Three-Phase Cable Set-up for the Study Basic Construction of Each Single Core Cable ef  88 89 90 92 96 96  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 A i r  106  D.2  Current Carrying Filament Buried in the Earth  (Vii)  I l l  LIST O F E  — electric field strength,  H  =  magnetic field strength frequency,  SYMBOLS  /  =  IT  = 3.1415926  (o  =  27rX /, angular frequency  fi  =  4JTX 1 0 , absolute permeability of free space  /x,  — relative permeability of the medium i  t*i  ~ / o / r i > total permeability of the medium «  7  =  p,  = resistivity of a particular medium t  — 1, complex operator 0  r  - 7  i  €,((o) =  x  i  Euler's constant €, — complex dielectric constant or permittivity of a particular medium »  <j>  =  /  = current  J  — current density  flux density  c,exp =  exponential  In  =  natural logarithm  m,  —  ( joy/ty |:—  1 o> fi^t- | , known as intrinsic propagation constant of a particu-  lar medium t.  If the displacement currents are ignored, then the ( joi / i , I value of m is equal to I I . Displacement currents are ignored unless li Pi J otherwise specified /„  =  modified Bessel function of the 1st kind and of the n t h order  K  —  modified Bessel function of the 2nd kind and of the n t h order  K  —  characteristic impedance  F  =  propagation constant  T  =  relaxation time of a dielectric  R  =  resistance per unit length  L  =  inductance per unit length  n  G  = conductance per unit length  C  — capacitance per unit length  Y  = shunt admittance  Z  = series impedance matrix  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 L i , 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,  U.S.A., and from my brother "Anna" is gratefully acknowledged.  (ix)  Portland,  Oregon,  -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 underground 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, induction effects on nearby communication circuits are becoming more important.  the  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 circuit 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 K Where R,L,G  + j<aL)(G + j u C )  (1.1a)  = V{R + joiL)/(G + j<aC)  (1.1b)  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 ( X L P E ) 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 X L P E is studied in detail (refer to Appendix A). The  series impedance and shunt admittance matrices of a cable system made up of N  cables, can be written as  - 2 -  (Z | 21  (zd  1*8*1  • • •  Z =  (1.2a)  " l*/w] and  (nil  py  1*2*1  Y =  (1.2b) PAWI  Submatrix  [Z„\ a l o n g t h e d i a g o n a l o f m a t r i x [Z\ is t h e s e l f i m p e d a n c e o f c a b l e i w h i c h c a n  be w r i t t e n a s  w  Z..  sic  z..  Z  Z„ =  z  c  {  s  s  (1.3)  s  where =  Z. c  c  s e l f i m p e d a n c e o f t h e c o r e o f c a b l e i"  Z . . = Z .= c  .7j  s  s;c  =  s  mutual  i m p e d a n c e b e t w e e n the core a n d the s h e a t h of cable »  self i m p e d a n c e o f the s h e a t h o f c a b l e i  T h e off-diagonal  submatrix  2,  ;  is t h e  mutual  impedance between the cable i a n d cable j which  c a n be w r i t t e n a s  (1.4)  where  Zee  =  mutual  impedance between  the core of the cable i a n d core of the cable j  Z  -  mutual  impedance between  the core of the cable i a n d the sheath of the cable j  z.. . =  mutual  impedance between  the sheath of cable i a n d the core of the cable j  Z  mutual  impedance between  t h e s h e a t h o f the cable t a n d the s h e a t h of the cable j  ci  t  Si  =  E v a l u a t i o n o f a l l t h e s e e l e m e n t s ; o f s u b m a t r i c e s [Z„\ a n d \Z \ i s , i n g e n e r a l , n o t e a s y .  The  tj  b e s t a p p r o a c h s e e m s t o be t h e o n e p r o p o s e d b y W e d e p o h l  [22] b a s e d o n t h e e a r l i e r w o r k  b y S c h e l k u n o f f [6]. B o t h a u t h o r s f i n d t h e s e i m p e d a n c e s f r o m  the  the  drops can  core  potential  and  sheath,  difference  [Z = V,  -  V/J).  These  between  the  longitudinal core  and  the  voltage sheath  and  longitudinal  the  done  voltage drops  be o b t a i n e d potential  from  difference  in the  - 3 between the sheath and the earth, as shown in Figure 1.1.  The potential differences V  1  and V  2  can be expressed as a function of the loop currents /j  and J with the help of Schelkunoff's theorems. (Appendix B) 2  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 submatrix  Z,  r  As explained in Appendix B, the potential differences V, and V  2  of loop currents /, and I can be written as  2  can be evaluated in terms  using SchelkunofTs theorem. For example, the potential difference V  l  V, = (Z  ctt  +Z  tHl  + Z, ,)/, - Z A  s i k m  7  4 -  2  (1.5)  where Z  c r f  = internal impedance of the core with current return outside,  Z,  — impedance of the insulation between the core and sheath  Z  tkl  = internal impedance of the sheath with current return inside  Z  sKm  — mutual impedance between the loops 1 and 2.  ns  The formulae developed by Schelkunbff to evaluate these internal impedances, and mutual impedance, Z , tKm  Z ,Z , crt  sh  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 calculation 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)  E a r t h return impedances of cables 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 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 computation.  (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) (vi)  Ametani's (27] cable constant routine in the E M T P program. 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 conductor 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 damping 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 approximation [22].  -  6 -  2. SERIES I M P E D A N C E A N D S H U N T A D M I T T A N C E M A T R I C E S 2.1 Basic Assumptions The transmission characteristics of a conducting system such as an underground cable circuit or a submarine cable circuit are determined by its propagation constant T and characteristic 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, conductance 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 indistinguishable 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 properties 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 negligible 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 concentrated in an insulated filament placed at the centre of the cable and the volume of the cable were replaced by the soil.  - 74.  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 I m p e d a n c e M a t r i x [Z\ f o r N C a b l e s in P a r a l l e l 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.) underground 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 r would be zero. The insulation between the core and the 0  sheath is usually oil-impregnated paper, surrounded by a metallic tubular sheath 5, and insulation around the sheath. In such systems, there are n = 2N metallic conductors.  T h e 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^= dx  -ZI  (2.2a) (2.2b)  -  8  d]_ = dx  -  -IV  where'V and / are n-dimensional vectors of voltages and currents, respectively, at a distance z along the longitudinal axis of the cable system. A l l voltages and currents are phasor values at a particular angular frequency a>. The series impedance matrix Z is given by \Zn\  (*IN1  l*«l 1*2=1  \Z \ W  Z =  (2.3) \ZNN]  Each submatrix, \Z \ assembled along the leading diagonal is a square matrix of dimension 2 U  representing the self impedances of cable i , by itself, *£.s;  *c,c,  (2.4)  where Z.. =  self impedance of the core of cable 1  Z,. . =  Z.. =  c  c  s  t  Z. s  =  St  mutual impedance between core and sheath of cable »  e  self impedance of the sheath of cable »  The off-diagonal submatrix |Z ] represents the mutual impedances between cable »" and cable j. 1;  This submatrix is also a square matrix of dimensions 2,  l*,l  Z - . Z'c - ? • Z' - . Z * . . r  r  r  c  s  inhere mutual impedance between core of cable t and core of cable j  Z.. = £  c  Z  Ciij  mutual impedance between core of cable 1 and sheath of cable j  —  mutual impedance between sheath of cable 1 and core of cable j  Z,.. = t  Z. t  $i  =  mutual impedance between sheath of cable 1 and sheath of cable j  Similarly, the shunt admittance matrix Y"can be defined as:  (2.5)  - 9 -  Y =  (2.6)  •where the submatrices jV„j and |Y" ) can be defined in a similar way, as described in section 2.3. y  2.2.1 S u b m a t r i x |Z„J The elements of the submatrix \Z„] can be determined by considering a single cable whose longitudinal cross section is as shown in Figure 2.2. The longitudinal voltage drops in such a cable are best described by two loop equations, with loop 1 formed by the core and sheath (as return) and loop 2 formed by the sheath and earth (as return).  insulation 2 sheath insulation 1 core  Figure 2.2 Loop Currents in a Single Core Cable.  It has been shown by Carson [4] that the change in the potential difference between j and (/ + 1) of a concentric cylindrical system as shown in Figure 2.3 is given by ——• + E — Ej + x }  —ju>n4>,  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)  ;  (27)  -  10 -  Axis Conductor j  Conductor  (j+l)  Figure 2.S Potential Difference between Two Concentric  V. <t>, =  Conductors.  potential difference between the j and the (j+l) conductor magnetic flux through the area described by the contour ABCD.  Since part of the current in the cable can return through the earth, the cable must be represented by 3 conductors (core, sheath, earth), as shown in Figure 2.4.  Axis  Sheaih  Insulation 2 Earth  Figure 2.4 Three Conductor Representation of a Single Core Cable.  -  11 -  The values of longitudinal electric field strengths E ,E ,,E cre  sk  ihe  (i.e., on the external sur-  face of the core, internal surface of the sheaih and the external surface of the sheath respectively) can be expressed as E E  = Z  cre  ikt  E  /„  [rt  (2.8a)  = -*,»./, + Z I  2  i(2.8b)  = Z  /,  (2.8c)  skm  ikc  ikc  I -Z 2  tkm  The electric field strength along the surface of the earth can be written as E = -Z I c  ei  (2.9)  2  where Z  crc  =  internal impedance per unit length of the core's external surface with current returning through a conductor outside the core.  Z  skt  =  internal impedance per unit length of the sheath's internal surface with current returning through a conductor inside the tubular sheath.  Z  skm  =  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.  Z  ike  —  internal impedance per unit length of the sheath's external surface with current returning through a conductor outside the tubular sheath.  Z  es  =  self impedance of the earth's return path.  Equation (2.7) can be then be written for the contours A B G D and E F G H a i follows:  = E , - E , - joL„h ik  dx d\  c  (2.10a)  c  r  ~dx  2  = E -E t  ikt  - jo>L„/  (2.10b)  2  In equation (2.10a), the total magnetic flux through the area described by the contour A B C D is L  cs  /j where  » it *( ^)  L  =  l  r  (211)  and r and r, being the outer and inner radii of the insulation. The term L„ can be defined 0  similarly. The parameters jtaL , c  insulations.  and j i * L  l t  are the impedances Z  nl  and Z  n2  of the respective  -  12 -  Substituting the values for the electric field strengths from equation (2.8) and (2.9) into equation (2.10), we have  dz dV~  ~ Zikm  Z  tht  + Z  m2  + Z  (2-12) ti  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  z  Z  Z  Z  x  d\'  2  m  m  '/.'  2  (2.13)  dx where the self impedance of loop 1 consists of 3 parts Z\  =  Z  cre  + Z  i n l  +  Z , itix  and similarly for loop 2 Z2 ~ Z . sh  4- Z, 2 + n  Z, ti  while the mutual impedance between loop 1 and loop 2 is Z  m  Zh  =  S  m  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  v  2  = V  tk  I = I 2  ik  (2-14)  + /„  where V  cr  V  = voltage from the core to the local ground,  sll  = voltage from the sheath to the local ground,  I  = total current flowing in the core,  I  = total current flowing in the sheath.  C1  sk  Substituting the values for voltages V,,V and currents 7,,/ from equation (2.14) into equation 2  (2.12), and adding rows 1 and 2, we obtain  2  - 13 -  dV„ '  Z re  Z,  Z, 2  dx dx  +  C  n  Z ,+ Z  + nl  sh  Z  es  tkc  — 1Z  +  lkm  Icr Z $ ~~ z e  tkm  (2-15) Z kt s  ^««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 S k i n E f f e c t 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 component 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 The ratio between this voltage drop and the sum of al! filament  electrically in parallel.  currents is the internal impedance. For a solid conductor of radius a and resistivity p. the internal impedance is given by [7] pm  I {ma) 0  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 I passes through the outer surface of the sheath and returns externally. This is illus2  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 formulae for the internal impedances. A detailed analysis of this problem had been done by Schelkunoff [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 Z as the internal impedance of the outer surface of the tube with external return, and Z b  ab  as the mutual impedance between one surface of the conductor to the other. The values of Z ,Z a  b  and Z  are given as follows:  ai  Z = ^ • [ / ( m a ) K , ( m J ) + AT (ma )/,(*.&)] a  0  0  -  15 -  Figure 2.5 • Sheath with Loop Currents /, and /j.  Z  st  =  P  2xabD (2.17:.,b,c)  where  D = /j(m6)A',. (ma) -  /,(ma)  AT,(m6),  p = resistivity o f the conductor, m  =  i n t r i n s i c p r o p a g a t i o n c o n s t a n t o f t h e c o n d u c t o r o f e q u a t i o n (2.17),  /<>,/, «= m o d i f i e d B e s s e l f u n c t i o n s o f t h e f i r s t k i n d a n d o f z e r o a n d f i r s t o r d e r , r e s p e c t i v e l y .  /f ,W, 0  = the modified  Bessel  functions o f the second kind  a n d of rero  and  first  ordf-r.  respectively.  U s i n g t h e s e f o r m v . ' a e t h e e l e m e n t s o f t h e s u b m a t r i x [Z.,\ c a n b e f o u n d .  2.2.4  Submatrix The  and  |ZJ  off-diagonal submatrix  [Z \ w h i c h r e p r e s e n t s t h e m u t u a l i m p e d a n c e s b e t w e e n c a b l e i tJ  cable j c a n be best explained if we consider a transmission s y s t e m consisting o f only two  c a b l e s « a n d j a s s h o w n i n F i g u r e 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 P r o x i m i t y E f f e c t Skin effect is caused by the non-uniformity o f current density in a conductor.  This  current, density is a function o f distance from the axis, but not o f direction from the axis. However, in parallel conductor transmission, in addition to the self-magnetic field (field generated 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 functions 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 considering a three-phase circuit with symmetrical given in the same reference.  triangular spacing or with flat spacing arc also  - 17 2.2.4.2 P r o x i m i t y Effect of a Single-Phase C i r c u i t of T w o 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 conductors. x =  2nV2ft{d-t)/p  x can be further simplified to [13], x — 1.52-\/f/R  dc  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.  ity effect can be ignored up to frequencies of 1MHz,  For such a value, the proxim-  [13, 33]. Hence, proximity effects are  ignored here.  2.2.4.3  S h i e l d i n g E f f e c t of the S h e a t h  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  netic coupling without  dx  be the induced voltage in the exposed circuit duc to the  any shielding conductor and  let — — dx  be the induced voltage in the  i  dV? shielding circuit. The current in the ground shielding conductor is then  is the self impedance of the shielding conductor with earth return, Now, exposed circuit by the current in the shielding conductor is  mag-  — , where Z., Z  dx  ss  voltage induced in the  — , where Z.  z  is the mutual  Figure 2.7 • Circuit Arrangement of Primary, Secondary and Shielding Conductors, with Shielding Conductor Grounded at Both Ends  impedance between the shielding and the exposed circuit. Therefore, the net voltage induced in the exposed circuit is d\  r x  dV?  dVf  dx  dx  dx dVf  Z Z  S  ;J  (2.19)  5  d\r?  If the voltages — — and — — are expressed in terms of the current in the primary circuit as dx  dV? dx  dx  d\ ° r  = IZ 0  anc  l ~.—  =  IoZoi>  dx  0s  £ 1  1 -  dx  z»z Zn  then equation (2.19) simplifies to Os  ZQJ  (2.20)  ZQZIQ  and the shielding factor of the grounded shielding conductor is then given by n= l  Z,i Z  0i  z„z ss  (2.21)  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 conrO  ductor, then  dX  dx  dx '  Therefore Z  0l  = Z , which makes the shielding factor to be equal 0s  to  (2.22)  -  2.2.4.4 E l e m e n t s  of Submatrix  19  -  (Z ). t ;  K e e p i n g in m i n d the shielding effect d e s c r i b e d a b o v e , the  submatrix  \Z, ].  A g a i n , loop currrents are  }  the elements df the submatrix j Z „ j .  we will now derive the elements  u s e d , as has been d o n e before for  of  determining  C o n s i d e r i n g t h e c a b l e s y s t e m s h o w n in F i g u r e 2.3, we c a n  define the following loop currents for the i t h cable i)  loop current  /',, whose path consists of the core's external surface a n d sheath's  internal  surface ii)  loop current 7 , whose path consists of the sheath's external surface and earth. 3  S i m i l a r l y , t h e l o o p c u r r e n t s l[ If  we consider the  a n d I'  2  loop currents F  cable i due to the loop current side the cable j there  [6],  c a n be d e f i n e d for cable  I\  2  a n d I{,  there  j.  will be n o i n d u c e d voltage in loop 2 of  a s t h e n e t f i e l d p r o d u c e d b y t h e l o o p c u r r e n t 1\  is z e r o o u t -  U s i n g t h e law o f r e c i p r o c i t y o f m u t u a l i m p e d a n c e s , it c a n be d e d u c e d t h a t  w i l l be n o i n d u c e d v o l t a g e  in the  loop 1 of cable j  due to  the  loop current  Hence,  V. 2  r e l a t i n g the loop c u r r e n t s w i t h the p o t e n t i a l differences between the c o n d u c t o r s we o b t a i n :  'dV\/dx  z\ z  Z'  Z'  0 0  0  0  diydx  0  d\" /dj 2  Zj. ,  impedance  terms  in e q u a t i o n  Z  (2.23),  S I  0  r\  Z  ft  z\  Am  n  Zin  Zk  2  m  d\'\/dx  Most  m  x  si  have  already  been  (2.23)  defined  except  the  w h i c h is t h e m u t u a l i m p e d a n c e b e t w e e n t h e e a r t h r e t u r n l o o p s 2 o f c a b l e s »' a n d j .  Sy  If  cables are b u r i e d in a n h o m o g e n e o u s infinite e a r t h , where the p e n e t r a t i o n d e p t h of the c u r r e n t i n t h e e a r t h is l e s s t h a n t h e d e p t h o f b u r i a l , t h e n t h e v a l u e o f  y  _  pm"lK (ms) \_ 0  Z  S i S >  is g i v e n b y  the  return  [9]:  '  '  term  (2 24)  where a  =  r,,Tj  = external radii of the c a M c s i a n d j ,  m  = For  distance between the centres of the cables, and  intrinsic propagation constant of the  earth.  a h o m o g e n e o u s s e m i - i n f i n i t e e a r t h , w h e r e t h e p e n e t r a t i o n d e p t h i n t h e e a r t h is m o r e  t h a n the d e p t h of burial, the m u t u a l i m p e d a n c e  Z. t  Sf  is g i v e n b y  (22]:  -  _  jap.  20 -  K (m/?) - /C (mZ) + J 0  (2.25)  0  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  v = 2  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 dV\,/dx dVJdx  z\ + 2z; + r Z' + Z' z' + z\ z\ 2  m  m  dV{,ldx  Zss  dV{ ldx  Zss s,s  Z  Zss  h  2  Z{ .+  2ZL  }  s i  Z'  2  +  Zss  Hr  Zss  Ilk  + z>Zln 2  Z'  +  z  zi  2  2  (2.27)  Hr 7  «*.  From equation 2.27, the impedance submatrix [Z, ] defined earlier in equation (2.5) is }  given by  -  21  -  |2,l =  It  is i n t e r e s t i n g  core of cable j the  same.  equations.  to note, that the  the c o r e o f cable i a n d  the  a n d the m u t u a l i m p e d a n c e botween the core of cable i a n d s h e a t h of cable j  are  T h i s raises the It  be i l l u s t r a t e d  (2.28)  mutual  question whether  impedance  between  the shielding effect  is i n d e e d i m p l i c i t l y t a k e n c a r e o f i n t h e f o r m u l a t i o n with the  help of a c o n d u c t o r w  e a r t h , as s h o w n in F i g u r e  is p r o p e r l y r e p r e s e n t e d witb loop currents.  p l a c e d in close p r o x i m i t y  in  the  This can  to a cable buried in  the  2.8  Earth  Conductor  W  Figure 2.8 - Transmission System Consisting of a Single Conductor and a Cable. Fov  t h e s y s t e m s h o w n i n F i g u r e 2 . 8 , t h e v o l t a g e s a n d c u r r e n t s o i l t h e c a b l e c a n be w r i t t e n  as: — %cc Ic  dx  dV dx  sh  ~ Z  ci  Z I CS  S  CU  VI  I + Z, /, + Z c  t  (2.29a)  + Z ,I  lV)  (2.29b)  /„  where  Z ,Z et  tt  — t h e s e l f i m p e d a n c e o f t h e c o r e a n d s h e a t h o f the c a b l e , r e s p e c t i v e l y , — the m u t u a l  Z ..Z . ca  ia  =  the  mutual  i m p e d a n c e b e t w e e n the c o r e a n d the s h e a t h , a n d . impedances between  a n d c o n d u c t o r w,  respectively.  the  core and conductor w  and between  the  sheath  -  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. with I  sk  — ~ / T . it follows that Z^  — Z  £  + Z . /,.. s  must be true. On the other hand, if the sheath is  sw  grounded at the ends, then there will be a circulating current through the sheath and earth, and V  sk  becomes zero. Hence, the value of 7 can be found from equation (2.29b) as: $  Z  SUI  (2.30)  ^ss  Substituting the value of 7 into equation (2.29a) we obtain S  dV,  Z-^  dx  z  " z..  z„  -  7 7  1 -  c  z  s  ~z~ 1  -  Za ' Z Z  ls  Si  The term  Z $' Z .  L +  sw  IwZ u ' Zcv, C  (2.31)  is the shielding factor of the sheath for the field produced by conduc-  Za Z , cu  tor w. With Z  = Z ,  cw  sw  it can be simplified to  1 -  Zss  which is the same as equation (2.22).  Hence, the shielding effect is implicitly taken care of in the equations.  2.3  S h u n t A d m i t t a n c e M a t r i x Y for N C a b l e s in P a r a l l e l In a manner similar to the series impedance matrix Z, the shunt admittance matrix Y can  be expressed in terms of two submatrices [Y„] and [Y,J. Since the soil acts as an electrostatic shield between the cables, the off-diagonal submatrix [>',,] will be a null matrix. Hence, we only have to derive the submatrix [Yj,|. Before we obtain the elements for the submatrix [}'„], the admittance of insulation will be discussed first.  2.3.1  L e a k a g e C o n d u c t a n c e 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 permittivity of The admittance Y per unit length of the insulation is defined as j<i)27reo€* y  =  G  +  jB =  — In  r /r 2  1  -  23 -  Figure 2.9 - Croaa-Section of a Coaxial Cable.  77^7  V ~  J  jti)27rf„( In TJT  ,  (2.32) 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" compared to the real 4  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 temperature and the molecular structure of the dielectric substance [15]. Let us consider two commonly used, insulating materials for power cables, namely crosslinked polyethylene (XLPE) and oil-impregnated paper. The values of i and i for XLPE are approximately constant for at least up to frequencies of \00MHz [16]. Typically, they have values of  i = 2.33 t  *=4.66 1 ( r  4  (2.33)  - 24 -  Unfortunate!}', little is mentioned in the literature about the frequency dependency of the permittivity in the case of oil-impregnated paper (18]. Recently Johanscn and Breien [17] published 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 2 0 0 % for the same frequency range, at a temperature of 20°c.  Figure 2.10(a),(b) Measurements of t ( t o ) and  t*(co)  j * on an Oil-Impregnated Test Cable at 2(f c.  Figure 2.10(a) and (b) show the experimental data obtained for I and t for the frequency range 10* to 10 Hz only. Based on this experimental data, the authors [17] developed an empirical 8  formula for t'(oi).  - " + (, + y J x . o - T "  1 2 3 1 1  Figures 2.11 (a) and (b) show the plot for i and ('obtained from the empirical formula, which closely match the experimental data of Figure 2.10(a) and (b).  -  26 -  Frequency [Hz]  Figure 2.11 Values of t\ui)  a n d t"(o>)  Obtained from the Empirical Formula.  -  26 -  A general formula for tbe complex permittivity of any material as a function of frequency is given by Bartnikas [15]. According to Bartnikas, when a dielectric is subjected to an ac field, at low electric field gradients, its electrical response will depend upon a number of parameters such as the frequency of the applied field, the temperature and the molecular structure of the •dielectric substance.  Under some conditions, no measurable phase difference between the  dielectric displacement; D, and voltage gradient E will occur, and consequently the ratio DIE will be defined by a constant equal to the real value of the permittivity, c'. When a dc field E is suddenly applied across a dielectric, tbe dielectric will almost instantaneously, or in a very short time, attain a finite polarization value. This polarization value will be almost instantaneous, 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 D or i E. m  m  The slower processes, due to the dipolc oriontatiou or  ionic migration, will give rise to a polarization which will attain its saturation value considerably more gradually because of such effects as the inertia of the permanent dipoles. The static dielectric displacement vector, D  it  in this case is equal to i E, where t, is the static value of s  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, P is the achieved saturaf  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 structure of the material, the saturation value, P may be achieved in a time that may vary anyt  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) = P [l - exp(-f/r)]  (2.35)  s  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  (2.36a)  t = Rc(e') = £«, + 1 + O) ^ 2  I = Im( ') = £  (t -e„)oiT s  (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.  nificant, i.e., of the order of 10 , for oil-impregnated paper. 2  The changes are very sig-  Typical values for the real and  imaginary parts of t for X L P E are given in equation (2.33). The real and imaginary values of t  for oil-impregnated paper can be obtained from equation (2.34), based on the reference [17].  A general formula for ''-he complex permittivity of any material is given by equation (2.37a).  2.3.3 Submatrix \Y \ U  We The  shall now determine the self admittances of the cable system shown in Figure 2.2.  loop equations for the current changes along loops 1 and 2 will be: di,  0  dx  0 n  dl  2  v  (2.37a)  2  dx  ?here Yi = G  x  Y  2  2  = G  + jB — admittance of insulation between core and sheath, A  + jB  2  — admittance of insulation between sheath and earth,  Vj = voltage between the core and sheath, V  2  — volt age between the sheath and earth.  Substituting the values for currents  II lt  2  and voltages VltV2 from equation (2.14) into  (2.36) and subtracting row 2 from row 1, we obtain:  -  dl , dx  28 -  c  Yx -Yr -Y, Y + Y x  (2.37b)  . »*. v  2  dx Hence, the submatrix \Y„] is given by:  Y ~Yx -Yx Yx + Y  (2.38)  t  2  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,earth  where the grounding resistance /? h i given by 3  eirt  R earth  2J_ fa V(2ff) + (t/2) + 1/2 g  Pearth  2  (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, G in equation (2.39) is no longer an evenly distributed parameter 2  because /?  eirtt  in equation (2.40) is a function of length. In [20], it is shown that the change in  the value of C .with the length is practically negligible, and treating G as an evenly distri2  2  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 conductor and from the earth return self impedance formula.  The elements of (he mutual  impedance submatrices were obtained from the earth return mutual impedance formulae. The shielding effect of the sheath and the proximity effects between the conductors were studied next to assess their influence on the elements of the mutual impedance submatrix. Finally, the  -  29  -  self admittance submatrix and mutual admittance submatrix were defined.  Since the earth  acts as an electrostatic shield, the elements of the mutual admittance submatrix are zero. The permittivity c* of the insulation which is needed to evaluate the elements of the self admittance submatrix, is frequency dependent and complex. An empirical formula for finding the real and imaginary parts of the permittivity t * as a function of frequency was shown, which can then be used to find the elements of the self admittance submatrix.  - 30 -  3. C O M P A R I S O N  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 f o r  :  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].  chapter, approximate  In this  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 E x a c t F o r m u l a e f o r T u b u l a r  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:  f/otmoJK-.M) +  Z. =  Z  t  = -^L. \  Z.» =  I o { r n b ) K l  (m ) a  KjmaVAmb)]  (3.1a)  + K^mbVJma)]  (3.1b)  (3-lc)  2ncbD  where D - /j(ml»)K',(ina)  -  I (ma)K {mb) l  l  - 31 -  The  argumeDts for the modified Bessel functions 7 , / i and K ,Ki 0  0  of the first kind and  second kind are complex because the intrinsic propagation m of the conductor is ^/w/y/p) . For  more exact- calculations, m should be {j<a/i/p— <>>/i<) , where the first term under the 2  square root accounts for the conduction current and the second term accounts for the displacement 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 conductors with p = 1.7 10~ /?m resistivity would have a displacement current at 10MHz which is 11 8  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 I O such as copper.  - 1 0  smaller than the conductivity of a good conductor  Displacement currents are therefore taken into account in the earth return  impedances discussed in the next chapter. Subroutine T U B E 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 E M T P , 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 K {mr) 0  and I (mr) can be expressed as real and imaginary parts as fol0  lows: A'„(mr)= K (VJ\m]r) 0  7 (mr)= 0  = Kj\m\r)+  /o(V7T^lT)=  B„(\m\r)+  jA' (|m|r)  (3.2a)  tl  jB (|m|r)  (3.2b)  ei  where B„ and £?,, are Kelvin functions of the first kind, and K~ and K ¥  cl  are those of the second  kind. T o evaluate the Kelvin functions, infinite series and asymptotic series can be used for small real arguments and large real arguments, respectively. Subroutine T U B E 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 Z of equation (3.1) by setting o = 0 and 6 = r: h  ^1^1 2jir/,(mr)  where r = radius of the conductor  3  -32  -  m = intrinsic propagation constant of the conductor / , / m o d i f i e d Bessel functions of the first kind and of zero and first order, respectively. 0  1=  Wedepohl [22] suggested an approximation to this exact formula, given by z  _ pmcoth(fcmr)  p(l-l/2k)  +  This approximation was developed by first considering the equation =  pmcoth(fnr)  (  3  5  )  27rr  This equation is known to exhibit similar properties as the exact equation given by Equation (3.3). For example, at high frequencies, the impedance term Z tends to be pm/2xr which is a x  well known skin effect formula [22], while at lower frequencies it represents pure resistance although not, in fact, equal to the required value p/xr . Equation (3.5) can be improved to take 2  account of the dc resistance more precisely by writing Z = -^-coth(W)+ ^ - f * ) 2nr nr  (3.6)  >  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 Z = y/R? + Z x  (3.7)  a  where R is the dc resistance given by p/xr and, Z is the impedance at very high frequencies 2  c  given by pm/2-r.  a  Table 3.1 shows the values of resistance R = R {Z } and inductance x  e  x  L = — 7m{Z,} obtained from subroutine T U B E and from Wedepohl's and Semlyen's approxix  (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 2 0 H i to 300Hz in the inductive part.  - 33 -  T a b l e 3.1  Internal Impedance of a Solid Conductor (p = 1 7 10~'Om  FREQUENCY (Hz)  and r =  0.0234m)  WEDEPOHL  TUBE  SEMLYEN  RESISTANC;E (Q/km) .01 . 1 1 10 100 1,000 10,000 100,000 1 ,000,000 10,000,000  0.0098825 0.0098825 0.0098858 0.0102067 0.0203380 0.0582719 0.1786977 0.5596756 1.7644840 5.5744390  i  0.0098825 ; 0.0098825 0.0098874 0.0103294 0.0190556 0.0561595 0. 1763397 0.5572406 1.7620250 5.5719720  0.0098775 0.0098776 0.0098809 0.0102034 0.0211463 0.0592378 0.1797193 0.5607150 1.7655290 5.5754860  INDUCTANC3E (jiH/km) .01 .1 1 10 100 1,000 10,000 100,000 1,000,000 10,000,000  50.000000 49.999920 49.991580 49.181730 27.567070 8.853760 2.803902 0.886793 0.280432 0.088680  51.800000 51.799920 51.792250 !i 51.042710 i 28.354910 ; 8.868063 2.804328 0.886806 0.280433 0.088680  The cpu times were found to be: TUBE  0.042 ms  Wedepohl  0.038 ms  Semlyen  0.022 ms  ;  50.000000 49.999750 49.974780 47.836480 25.930690 8.798590 2.802123 0.886736 0.280430 0.088680  - 34 -  figure 3.1(a) and (b) Impedance of a Solid Conductor ae a Function of Frequency.  - 35 -  5-  WEDEPOHL SEMLYEN  '-  2  f  Ui w  c ro  i  ----.:  x  -1-1 -  *—«  i  CO 0)  /  /  / \ /  cc -5-7-4  11 10~  2  10  _1  1  10'  10  2  10  5  10  4  10  10  s  6  10'  12  WEDEPOHL  8  SEMLYEN ^  4  o L—  uj  o  o c —4  o T3  1/  -8  10~*  10"  1  1  10'  10  2  10  3  10  4  10  5  10  Frequency [Hz] Figure 3.2(a) and (b) Errors in Wedepohl's and Stmlycn'»  Formulae for a Solid  Conductor.  6  10  7  -  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 T u b u l a r Conductor;  Figure S.S • Cross Section of a Tubular Conductor  There are three impedances associated with a tubular conductor of the type shown in Figure 3.3: 1)  Internal impedance Z , which gives the voltage drop on the inner surface, when a unit a  current returns through a conductor inside the tube. 2)  Internal impedance Z , which gives the voltage drop on the outer surface when a unit t  current returns through a conductor outside the tube. 3)  Mutual impedance Z  ai  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^,Z and Z, , originally developed by Schelkunoff, are given in equation (3.11. b  t  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 J ,I ,K 0  X  0  and K", by their asymptotic expressions and performing  the necessary division as far as the second term. Schelkunoff's approximations are as follows [6]: Z = -^cotb(m(6-o)) - -^—{^ t  +  Z = | ^ c o t h ( m ( 6 - a ) ) + ^ ( 3 / a + l/6} i  (  3 8 a  )  (3.8b)  -  Zah =  37  -  ^|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-:'  +  =  m  2  /  <»;»») (3.9b)  /  If the tube is thin compared with its mean radius, equation (3.9a) can be written as: dH  2H  ,  Equation (3.9b) and (3.10) lead to a second-order differential equation in H.  Now,  solving  for H from this second-order differential equation and following the same procedure as given in Appendix A for the exact formulae, we obtain the following equations:  Z t  = ^  c  o  i  H  m  {  b  .  a  )  Z t  = ^  c  o  t  H  m  (  b  -  a  )  <»  Z  =  )  )  - - ^  +  (  ^ ^  {3  -7~TCoseA(m(6-a)) 7t(a + 0)  3,  l a  .1,  )  b )  (3.11c)  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 — a)«a,b  then the impedances can be expressed as  Z  Z  The  ° = >  °>  Z  =  /™  =  coth(m(6-a))  (3.12a)  -7^«>seA(m(6-a))  impedance Z  ti  6, or in other words, if  •  (3.12b)  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 Z Z t  t  and Z  4t  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 Z . a  Figures 3.4(a) and (b) show the resistance and inductance for the fre-  quency range 0.01 Hz to 10MHz. T o highlight the differences in the results, the errors in the  - 38 -  Table S.2 Internal Impedance Z of i Tubular Conductor with t  Current Returning Inside (p =  FREQUENCY (Hz)  TUBE  2.1  \0~ Dm,a 7  = 0.0385m,6 =  SCHELKUNOFF  WEDEPOHL  0.0413m)  BIANCHI  RESs I STANCE ($2/k.m) .01 . 1 1 10 100 1000 10000 100000 1000000 10000000  0. 299163 0. 299163 0. 299163 0. 299163 0. 299169 0. 299761 0. 354424 1. 18036 3. 75275 11 .8915  0.288640 0.288640 0.288640: 0.2886401 0.288646 0.289224 0.343956 1.17015 3.74259 11.8813  0.299163 0.299163 0.299163 0.299163 0.299169 0.299762 0.354480 1.18068 3.75312 11.8919  f  0.299163 0.299163 0.299163 0.299163 0.299169 0.299741 0.352539 1 .14975 3.63192 11.4852  INI 5UCTANCE ( H/'km) M  .01 . 1 1 10 100 1000 10000 100000 1000000 10000000  4. 84605 4. 84605 4. 84605 4. 84605 4. 84603 4. 84339 4. 60048 1 .89284 0. 59905 0. 18944  4.84848 4.84848 4.84848 4.84848 : 4.84846 4.84581 4.60256 1.89296 0.59906 0.18944 1  4.84848 4.84848 4.84848 4.84848 4.84846 4.84581 4.60256 1.89296 0.59906 0.18944  4.67836 4.67836 4.67836 4.67836 4.67834 4.67578 4.44106 1.82654 0.57804 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 resistance and inductance.  2.  Schelkunoff's approximation has an error of 2-4% up to a frequency of 10kHz in the resistive 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'  10  2  10  3  10'  10  10'  10  10*  10'  10'  10  5  10-  TUBE  E  WEDEPOHL  \ x jt aj  o c _o "o  SCHELKUNOFF BIANCHI  i-  T> C  0.1+-  10"  10"  10'  10  J  10  3  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 "  tt>  0  o c  \  . - ' "  \  (0  \ /' A  co "to tu  cc  _-> z  10~  2  10"'  1  10  10  1  2  10  10  J  10  4  10  s  10  6  7  WEDEPOHL  e£  SCHELKUNOFF  2  BIANCHI  o k_  k_ Ul  v t>  0  o c <0  u "D  c  -2  I .Mill,,  10"  2  IHj  10"  1  1  I 1,1,11V  ! I llllll,  10'  I .IIIItT,  10  2  I I I Mill,  10  5  I I llllll,  10"  I I II.Ill,  10  5  I  10  s  llllllf  10  7  Frequency [Hz]  Figure S.5 -Errors  in Wedepohl'*, Sehelkunoff'e forZ . t  and Bianchi'a  Formulae  3.  41 -  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 Z  ab  and the impedance Z . t  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 approximation 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 Figure 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 Z were calculated. t  Table 3.4 shows the values of these parameters using the exact formula and various approximations at different frequencies. Figures 3.7(a) and (b) show the resistance and inductance as a function of frequency for the range 0.01 Hz to 10MHz. The errors in the approximations are plotted in Figure 3.8. We note the following for the impedance Z  4  - 42 -  0.4  WEDEPOHL SCHELKUNOFF 0.1-  \  ui CP o c  CD  -  -o.H  TJ C  — 0.3  f  111  10~  2  (Mill  I  10"  I I lllll,  1  i i inn: ' i i mill  i linn  -  TT!!I,  10'  1 1 lllTHt  ' T 1 MTIir,  10 \2  2  10 r>3  I I 1 llllll' s  4/  1  10"  10*  10  6  Frequency [Hz]  Figure S.6 Errors in Wedepohl's and Sehetkunoff'e  Formulae for  Z  tl  10  7  - 43 -  Table S.S Mutual Impedance (Z , of a Tubular Conductor e6  (p = 2.1 10"/?m,o = 0.0385m ,6 = 0.0413m) 7  FREQUENCY (Hz)  TUBE  WEDEPOHL  SCHELKUNOFF  RESISTANC:E (8/km) .01 .1 1 10 100 1 ,000 10,000 100,000 1,000,000  0.29916343 0.29916343 0.29916343 0.29916338 0.29915838 0.29865873 0.25299441 -0.06964902 0.00001930  INDUCTANC3E .01 .1 1 10 100 1,000 10,000 100,000 1 ,000,000  0.29934775 0.29934776 0.29934776 0.29934771 0.29934270 0.29884247 0.25312757 -0.06966688 0.00001930  0.29916343 0.29916343 0.29916343 0.29916338 0.29915838 0.29865846 0.25297170 -0.06962398 0.00001929  -2.3386052 -2.3386052 -2.3386052 -2.3386052 -2.3385803 -2.3361094 -2.1095191 -0.0097380 0.0000082  (uH/km)  -2.3391813 -2.3391813 -2.3391813 -2.3391810 -2.3391563 -2.3366832 -2.1099033 -0.0097146 0.0000082  -2.3406226 -2.3406226 -2.3406226 -2.3406223 -2.3405975 -2.3381230 -2.1112033 -0.0097206 0.0000082  T h e errors in W e d e p o h l ' s a p p r o x i m a t i o n are practically negligible for b o t h resistance a n d inductance up to a frequency of The  10MHz.  errors in Schelkunoff's a p p r o x i m a t i o n are around  a n d decreases for higher frequencies.  3.5% u p  to  10kHz  a frequency of  Schelkunoff's a p p r o x i m a t i o n gives the s a m e values  as W e d e p o h l ' s a p p r o x i m a t i o n f o r the i n d u c t a n c e t e r m . The  errors  in  Bianchi's  approximation  increase thereafter for the whole frequency range.  are  resistance term.  negligible  for  frequencies  up  to  T h e e r r o r i n t h e i n d u c t a n c e i s 3.8%  1kHz for  but the  -  44 -  Table S-4  Internal Impedance Z of a Tubular Conductor k  tvith Current Returning Ouleide (p = 2.1 10~ /?m,a = 0.0385m ,6 = 0.0413m) 7  FREQUENCY (Hr)  TUBE  RE<51 STANCE .01 .1 1 10 100 1000 10000 100000 1000000 10000000  0. 299163 0. 299163 0. 299163 0. 299163 0. 299169 0. 299720 0. 350679 1. 120643 3. 518632 11 .10564  SCHELKUNOFF  WEDEPOHL  BIANCHI  ( j B / J;m)  0.299163 0.299163 j 0.299163 0.299163 0.299169 0.299721 0.350729 1.120912 3.518953 11 .10605  0. 309686 0. 309686 0. 309686 0. 309686 0. 309691 0. 310243 0. 361252 ; i . 131444 3. 529473 11 .11652  0. 299163 0. 299163 0. 299163 0. 299163 0. 299169 0. 299741 0. 352539 149755 • 3.631926 1 1.48527  ;T.  INIHJCTANCE (uH/ 'km) .01 .1 1 10 100 1000 10000 100000 1000000 10000000  :  51759: 51759 4. 4 . 51759 4 . 51759 4 . 51756 4 . 51510 4 . 28865 76453 1 . 0 . 55844 0 . 17660  4.51977 j 4.51977 4.51977 4.51977 4.51975 4.51728 4.29052 1 .76463 0.55844 0.17660 1  i *• 51977 4.51977 ! 4.51977 4 . 51977 4. 51975 4. 51728 4 . 29052 1. 76463 0 . 55844 0. 17660  i  I 4.67836 i !  4.67836 4.67836 4. 67836 4. 67834 4. 67578 4. 441 06 82655 1 . 0. 57804 0. 18280  -  °icr  2  io"'  1  io'  10  45  1  -  10  io  3  4  io  5  10  s  Frequency [Hz] Figure S.7(a),(b) - Impedance Conductor  (with External  Return)  Z of a b  as a Function  Tubular of  Frequency.  io  7  - 46 -  /  2--  s  V  /\ O  kUJ  <o  0  o c  to  WEDEPOHL SCHELKUNOFF  _•>  to  BIANCHI  10~  2  10"'  1  10'  10  2  10  S  10"  10*  10*  10  7  o  fc— fc—  UJ a> u  0  cto  WEDEPOHL  u  3  TJ  c  SCHELKUNOFF -2 BIANCHI  — A ] I I . i i i . i II,i| 10~ 10" 1 2  1  i i i . nn, • J • i in., .ii.i.ii, i . , i mi, III nm, .MI mil i i 11 in 10' 10 10 10 10 10* 10 2  S  4  S  Frequency [Hz]  Figure S.8 - Errors in Wedepohl's and Bianchi's  Schelkunoff's  Formulae for Z  t  7  - 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 conductor were given by Schelkunoff. Since these formulae were not suitable for hand calculation purposes, 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 approximations were compared with the classical formulae for the tubular conductor of a typical cable. The displacement current term is neglected in subroutine T U B E and in Ametani's cable constant routine, which use the exact formulae. Though not discussed in detail, a modified subroutine T U B E C was developed which takes the displacement current into account. In all cases, the results from T U B E C and T U B E 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 asymptotic 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 asymptotic 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.  - 48The  routines  for Wedepohl's approximation  - W E D A P , Semlyen's approximation -  SEMAP, Schelkunoff's approximation - SCHAP, Bianchi's approximation - B N C A P and T U B E C 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 ( T U B E ) would not be much more than that of the approximations. Therefore, the exact formula with subroutine T U B E is recommended for computer solution. For hand calculation 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 studies of inductive interference in communication circuits from nearby overhead lines or underground cables. Also they are important in the calculation of voltages in power lines or communication 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 situations, 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 homogeneous 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 equations 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 E M T P 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 E a r t h 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 Z is ti  defined as the ratio of the axial electric field strength at the external surface of the insulation to the current flowing in the cable. T h e 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 assumption 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 frequencies are so high that the return current will flow very close to the cable. Furthermore, this treatment will be found useful in justifying the simplifying assumption 2 in section 2.1 of Chapter 2, and will also be helpful in interpreting the results for the more general case of earth return impedance in a homogeneous semi-infinite earth.  4.2 E a r t h R e t u r n Impedance in a Homogeneous Infinite E a r t h The  calculation of earth return impedance in an infinite earth is relatively easy since  there is no surface discontinuity as in the semi-infinite case (earth and air). With the assumptions mentioned]in Chapter 2 (except for assumption 3), it is shown in Appendix C that the electric field strength at a radial distance r from a cable of insulation outer radius a carrying current / which returns through the earth can be written as:  E  =  . _  pml Ko(mr)  ( 4  ,)  ITXQ. A'j(mo)  where p  = resistivity of the earth  m  = intrinsic propagation constant of the earth. The  earth return self impedance per unit length of the cable is obtained from equation  (4.1) by substituting r = a together with the general relation E = —ZI. It also follows from Z  b  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 K (mR) 2na A'j(ma) 0  Z --  (4.2)  Hence, from the law of reciprocity of mutual impedance, the electric field strength on the surface of the cable due to a current / in the filament can be written as:  E = -  pmIK (mR) 0  (1.3)  2za K ( m a ) t  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'K (mR) 0  27rbK (mb) l  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 K (mb). x  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: _  P  m*fK (mR)  .  0  27imaK (ma)mbK (mb)  ' ' '  1  1  Hence, the mutual impedance between two cables of radii o and 6 respectively, buried in a homogeneous infinite earth is given by:  pm K (mR) 2  0  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, K (x)~l/x. 1  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:  Z = -^Koima) ...  (4.8a)  Z  (4.8b)  s  m  = ^~K (mR) 0  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 extension of formulae for filament conductors to cables of finite radii.  4 . 3 E a r t h R e t u r n Impedance in a Homogeneous Semi-Infinite E a r t h The self and mutual impedances of cables buried in semi-infinite homogenous earth are deduced from the electric field strength in the ground due to a buried filament conductor.  - 53 -  /  /  H  w  'Y  0) o c 1  1  0  to to cu  —I-  az -3  I I I! I  10"  10  I  10"  10~  10"  s  !  I 1  T T1!| •-' !  '1  -  T TTTT i  10"'  2  3  10'  25  15  cu o c  \  CO o T 3  C  -15  -25  -f  10"  1  1 I 1 Mll|  10"  I  1 I I I Mil  10"  1  U  I  I  I I I Hl|  10"  10"  1  1 I 1 I Ml|  1  1  llllll,  10-  ma  Figure 4-2 - Error in Replacing a Conductor of Finite Radius by a Filament Conductor  10'  - 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 conductors 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 Appendix D as:  E__ —  exp[-(o  |  x  m )\h-y\] 2  +  -  e  X  P  [-(^ +  m )\h 2  + y\\  2(p- + m ~ )  +  where  g  .  r exp(-|ftlfyl  '  ;  VV+ttr)  "  exp(jd>x )d<t>  (4.9)  .  = horizontal distance between the filament and the point at which the field is being determined,  y  = depth at which the field is being determined,  p  = resistivity of the earth, and  m  = intrinsic propagation constant of the earth.  The first integral term is identified as Jfc: (mr?) - rTo(mZ)j [1,22]; where R = V i " + (h-yf 0  and Z = V i  s  + (/i + y) . 2  The second integral part can be numerically evaluated [28] or  -  55 -  expanded into an infinite series [22]. The series expansion is in the form of modified Bessel functions. Therefore, equation (4.9) can be written in the series expansion form as follows:  = -^~\KdmR) - K (mZ) -f —K^mZ) + ^ "*^Ki\mZ) In \ Z mZ 0  -  }  (* •+ ml) -  ~ ^F(mZ,\x |,/))  mt  P  (4.10)  where  ffcVl^T  F(mZ,\x\,l)=  2  ^==r\e- dt  (4.11)  mZt  Vl-t  1/2 I  2  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 choosing 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 oscillatory. 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 [ 2 8 ] . The numerical integration routine available in the U B C system library D C A D R E [ 2 6 ] , which uses a cautious adaptive Romberg extrapolation technique, has been used in obtaining the solution for the equation ( 4 . 9 ) . The solution converges fast for the case when the two cables are buried at d i f ferent depths below the earth, but in the case when the cables are at the same depth or in the case of finding the earth return self impedance, the convergence is rather slow. T o explain this phenomena, consider the first integral part on the right hand side of equation ( 4 . 9 ) , i.e..  exp  [-(*  2  + m ) | A - j , | ] - exp[-(cA + m )\h + y\] 2  2  2  ?(c* + m ) 2  z  e x p O ^ x )</<•>  (4.12)  If h = y then the first exponential term within the closed brackets { } will become 1 and the integration now becomes  jl- xp[-( (<p c r +m + m)\h+y\ ) | / i + y | jUxp{j<t>x)d<t> 2  e  2  2  -  56 -  10  6-  T A o> o c  JO  V  -2  V) o> -6  -10  10'  1 "'  '^f  1  I  T  10  50  1  I  1  I  1111| 10  2  I  i l l  I  3  10  4  10  4  30-  £ 10  o o c o o D  ~  1 0  \  \  \ /  -30  -50  7— ••T-"T "I , ,  10'  m  I  I  I  1  I  I  I I i  1CV  10*  i  i  i  i  i i  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 D C A D R E 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  p  eyAh  =  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 F o r m u l a e U s e d by A m e t a n i , W e d e p o h l a n d 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 overhead 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 conductor) due to current / flowing in the conductor is given by  - 58 -  Table 4.1 Solution of Pollaczek'o Equation by Numerical Integration and Using In finite Series FREQUENCY (Hz)  NUMERICAL INTEGRATION  EQUATION 4.10  RESISTANCE (8/krn) .01 .1 1 10 100 1 ,000 10,000 100,000 1,000,000 10,000,000  0.00000986985 0.00009870394 0.00098721243 0.00987746990 0.09894307000 0.99462669000 10.0999510000 105.026550000 1119.21050000 10416.7130000  0.00000986986 0.00009870399 0.00098721145 0.00987752230 0.09894471000 0.99467682000 10.1014090000 105.062940000 1119.73660000 10410.0480000  REACTANCE (fi/km 0.00014814024 0.00133672137 0.01192043120 0.10477298900 0.90245276590 7.57239400000 61.8788630000 461.037952000 3018.40132100 13049.3294800  .01 .1 1 10 100 1 ,000 10,000 100,000 1,000,000 10,000,000  =  —  2x  fhere Z - Vi + 2  (A+y) , z  ln(Z/R)  +  J  0.00014814024 0.00133672130 0.01192028400 0.10472980000 I 0.90245110000 ! 7.57234170000 . 61.8625200000 460.988500000 3017.08610000 13031.2620000  (4.15)  -  59 -  m = intrinsic propagation constant of the earth. The integral part of equation (4.15) can be further simplified to  o U | + \ U +• m 2  2  (4.16)  Equation (4.16) is widely known as Carson's formula. Strictly speaking, this formula is only valid for the case of overhead conductors. Ametani used this correction term in finding the earth return self and mutual impedances of buried conductors, instead of the second integral term used in equation (4.9). Carson's formula given by equation (4.16) can be numerically integrated [28] or can be expanded into an infinite series [5], in terms of r = | mZ \. Ametani chooses the latter approach.  His procedure uses 2 different series, one when r S 5 and the other one when r > 5 .  Recently, Shirmohamadi of Ontario Hydro [28] and L. Marti at U B C 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 ( H i )  Figure 4-4 - Relative Error in the Evaluation of Carson's Formula with an Asymptotic Expansion.  - 60 -  Shirmohamdi avoids this error by using Gauss-Legendre quadrature technique for the direct evaluation of equation (4.13) in this region of r, while L. Marti [24] avoids these errors by extending both the asymptotic and infinite series, and by using a switchover criterion which depends on the geometry of the line (ie. on the value of 0). The use of Carson's formula for underground systems will be reasonably accurate at low frequencies because the value of m  with the exponent term exp(—| h+y\ Vc4 +m ) in equation  2  2  2  (4.9) is very small compared to the value of 4> at low frequencies, and, therefore, can be z  ignored. A t high frequencies, however, that term becomes quite significant and, therefore, cannot be ignored. For this reason, the resistance and inductance obtained by Ametani's method has an error in the order of 1 0 % or more for frequencies above l K H z when compared with Pollaczek's equation, i.e., equation (4.9). Wedepohl and Wilcox [22], who proposed the infinite series expansion form of equation (4.9) gave an approximation to the infinite series expansion (equation (4.10)) which is valid only if the condition |mZ| <0.25 is satisfied. Their closed-form approximation for the self and mutual earth return impedances are given by:  Z,  =  pm  . hHEl  +  ln  0  2  2n pm  •In—  2ir  L  5  _ ±  + 0.5  2  m  h  3  ml  (4.17a)  (1.17b)  3  where 7  = Euler's constant,  h  = depth of burial of the conductor,  I  = sum of the depths of burial of the conductors,  R  = Vi  m  = intrinsic propagation constant of the earth.  2  + (h-yf and  Semlyen and Wedepohl [24] developed another interesting formula for the self impedance of a cable of radius r in terms of complex depth which is nothing but 1/m, defined here as p. Accordingly, the self impedance term is given by  Zs  = "  ^  M  r + V'r)  (4.18)  -  61  -  4.5 Effect of Displacement Current and Numerical Results So far. the effect of displacement currents has been ignored. As shown in Chapter 2 . there is no noticeable error in the internal impedances of tubular conductor if displacement currents are ignored. Hut unlike in good conductors, the displacement currents in the earth arc noticeable, at least at high frequencies. The displacement current term can be easily incorporated in equation (4-10) by snbsti  :ing Vm"-t> /i( for in. Table 4.2 shows the values of J  resistance and inclu;'. :nce with and without the displacement currents.  Tabic 4.2  Earth Return Self Impedance with and without Displacement Current Term  WITH DISPLACEMENT CURRENT  FREQUENCY (Hz)  WITHOUT DISPLACEMENT CURRENT  R E S I S T A N C E (8/ki .01 . 1 1  1 10 100 1 oob 10 0 0 0  10 1 00 000 000 000 ooo 000  0.00000986985 0.00009870393 0.00098720986 0.00987747671 0.09894371230 0.99468700004 10.1056633504 105.572646568 1 172.74925588 15170.9605539  0.00000986986 0.00009870393 0.00098720982 0.00987747340 0.09894339595 0.99465537187 10.1024602926 105.239909564 1136.35180546 11593.2569450  • O O O O O O —  O O O O O -  ooo-  oo-  o —  — O O O O O O O — —  o  I N D U C T A N C E ("H/ nm) 2.82478692844 2.59451983737 2. 36423415099 2. 13388974446 1 .90335982756 1.67224525374 1 .43930244333 1 . 2 0 0 8 0 3 8 4 1 19 0.94712614357 0.64980341214  2.82478689712 2.59451979297 2.36423411661 2.13388971332 1.90335979916 1.67224520020 1.43930156514 1.20078329139 0.94694242963 0.66822941718  - 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  c  -5-  to CO CD  CC  -15  -25  i  10~  2  i  i  IIIMI—  10"  ni]—i  1  i i inn,—i  10'  i i M i n i — i i i inn,  10  2  10  1  i i mii|  J  10  nil 4  10  5  1  i i  mill 10  6  Frequency [Hz] Figure 4-5 - Error in the Earth Return Self Impedance if the Displacement Current is Ignored  ii i  mil!  io  7  - 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 2 5 % at a frequency 1MHz. error in the inductive part is almost zero up to a frequency of 1MHz.  The  The reason for the  noticeable error in the resistance, beyond a frequency of 100kHz, is that the condition | mZ | <0.25 is violated. Semlyen's approximation is good at low frequencies for the resistive part but at high frequencies the error is higher. It has an error of around 4 %  in the case of  inductive part over the whole frequency range. As mentioned earlier, the error i i i Ametani's procedure is not significant at low frequencies, but increases from 2 % to 2 0 % iu (.-be frequency range 10kHz to 1MHz Three  as shown in Figure 4.7.  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 L E A R T H developed by Luis Marti uses the UBC  library subroutine D C A D R E for the evaluation of the  integral. One more routine, namely C E A R T H was developed by the author, which can take the displacement current into account. This routine also uses the UBC DRE  library subroutine DCA-  to evaluate the function discussed earlier. Routines S E A R T H and L E A R T H give identical  answers but differ in the cpu time. The former one takes 3.4 ms while L E A R T H takes 2.9  ms  for the evaluation of resistance and inductance at a particular frequency. Routine C E A R T H 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  P O L L A C Z E K  WEDEPOHL  AMETANI  SEMLYEN  (Hz)  R E ; > I S T A N C E ( £ 2 / 1cm)  .01  0 . 0 0 9 8 8 2 5  0 . 0 0 9 8 8 2 5  0 . 0 0 9 8 7 7 5  0 . 0 0 9 8 8 2 5  .01  0 . 0 0 0 0 0 9 9  0 . 0 0 0 0 0 9 9  0 . 0 0 0 0 0 9 9  0 . 0 0 0 0 0 9 9  .1  0 . 0 0 0 0 9 8 7  0 . 0 0 0 0 9 8 7  0 . 0 0 0 0 9 8 7  0 . 0 0 0 0 9 8 7  1  0 . 0 0 0 9 8 7 2  0 . 0 0 0 9 8 7 2  0 . 0 0 0 9 8 6 7  0 . 0 0 0 9 8 7 0  10  0 . 0 0 9 8 7 7 5  0 . 0 0 9 8 7 7 5  0 . 0 0 9 8 6 1 8  0 . 0 0 9 8 6 9 2  100  0 . 0 9 8 9 4 3 4  0 . 0 9 8 9 4 5 7  0 . 0 9 8 4 5 1 9  0 . 0 9 8 6 8 4 0  0 . 9 7 9 5 2 6 3  0 . 9 8 6 5 7 8 4  1000  0 . 9 9 4 6 5 5 4  0 . 9 9 4 8 5 6 1  10000  1 0 . 1 0 2 4 6 0  1 0 . 1 1 9 2 8 8  9 . 6 5 6 4 6 8 0  9 . 8 5 7 5 3 1 3  1 0 0 0 0 0  105.23991  1 0 6 . 5 9 1 7 2  9 3 . 4 9 8 2 7 0  9 8 . 3 1 5 0 5 3  1 0 0 0 0 0 0  1 1 3 6 . 3 5 1 8  1 2 3 6 . 6 4 3 8  9 1 3 . 0 3 2 1 4  9 7 4 . 9 9 1 2 3  1 0 0 0 0 0 0 0  1 1 5 9 3 . 2 5 7  1 7 7 6 5 . 2 8 8  1 0 8 2 6 . 0 5 8  9 4 9 8 . 8 3 9 4  I N IX J C T A N C E  (mHy'km)  2 . 8 2 4 7 8 6 9  2 . 8 2 4 7 8 3 8  2 . 8 2 4 7 9 1 7  2 . 7 0 1 6 0 4 8  .1  2 . 5 9 4 5 1 9 8  2 . 5 9 4 5 1 6 7  2 . 5 9 4 5 4 1 8  2 . 4 7 1 3 4 6 7  1  2.3642341  2 . 3 6 4 2 3 1 0  2 . 3 6 4 3 1 0 5  2 . 2 4 1 0 8 9 5  10  2 . 1 3 3 8 8 9 7  2 . 1 3 3 8 8 6 6  2 . 1 3 4 1 3 7 8  2 . 0 1 0 8 3 5 2 1 . 7 8 0 5 8 9 8  .01  ,  100  1 . 9 0 3 3 5 9 8  1 . 9 0 3 3 5 6 4  1.9041501  1000  1.6722452  1 . 6 7 2 2 3 8 6  1 . 6 7 4 7 4 1 4  1 . 5 5 0 3 7 2 9  10000  1 . 4 3 9 3 0 1 6  1 . 4 3 9 2 6 2 9  1 . 4 4 7 1 0 6 4  1 . 3 2 0 2 4 5 9  1 . 2 0 0 7 8 3 3  1 . 2 0 0 4 1 1 8  1 . 2 2 4 5 1 5 7  1 . 0 9 0 4 0 3 2  1 0 0 0 0 0 0  0 . 9 4 6 9 4 2 4  0 . 9 4 2 9 8 1 3  1 . 0 1 2 6 2 2 8  0 . 8 6 1 4 5 9 7  1 0 0 0 0 0 0 0  0 . 6 6 8 2 2 9 4  0 . 6 2 6 7 9 7 5  0 . 7 9 6 2 6 8 9  0 . 6 3 5 3 5 6 5  1 0 0 0 0 0  4.6 C a b l e s B u r l e d 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 " 10  J  •• , , , , „ „ 10" 1  .... r-^>..nr 10' io 2 10 Frequency [Hz] Figure 4.6(a),(b) t  10'  Earth Return Self Impedance as a Function of Frequency  10  5  10  s  10  7  - 66 -  25  WEDEPOHL 15-  / /  AMETANI SEMLYEN  5-  ui o <o CO  -C  -5.  in CD  or -15  — 25 "1—1 10"  i  1111111—1  1  10"'  2  11111 n 1  1  1 11  IIIII  1 1 1111111—1 1  10'  10  1 . .1.11  10  2  11 10  J  11  . null—1  10  4  5  mii|  10  111 6  iiui|  10  1 7  25  WEDEPOHL  15-  AMETANI SEMLYEN 5LU CD o  c ra  >  -5  o  T3 C  —  -15  ~f 10~  11  —25  2  T mrn 10"'  r-i  TTTTTTI 1.  1 i 11 ini[ 10'  1 i i itui] 10  r-rri-TTTTi  2  10  r—r-rrrmj  r iitini]  r i i rim\  10  10  10  J  4  Frequency [Hz] Figure 4- 7 Errors in Earth Return Self Impedance  S  6  r-rrmrt} 10 7  11  mir  -  2.  67  -  C a b l e s are buried at normal depths ( l - 2 m ) b u t are used at high frequencies (100kHz a n d above).  I n s u c h s i t u a t i o n s , t h e d e p t h o f p e n e t r a t i o n i n t h e e a r t h is g i v e n b y :  d =  503.3\/pe«u//  (4.19)  ( w h e r e d is i n m , / is i n fi—m a n d /  is i n H z )  and becomes smaller than the depth of burial. The  s e c o n d possibility does n o t arise  normally  underground transmission system, with an earth lm,  the frequency  than  l m is 3 M H z  at which  the penetration  o r higher.  cases d o arise, however,  in power  system  studies.  For a  typical  r e s i s t i v i t y o f 10 J ? — m , a n d a b u r i a l d e p t h o f  of the return  current  in the earth  b e c o m e s less  In p o w e r s y s t e m s , o n e r a r e l y e n c o u n t e r s s u c h f r e q u e n c i e s .  the infinite  earth  return  impedance formulae  If s u c h  given by equation  (4.2)  a n d ( 4 . 7 ) c o u l d b e u s e d t o find t h e e a r t h r e t u r n i m p e d a n c e s . It  is i n t e r e s t i n g t o k n o w  t h e b u r i a l d e p t h is l a r g e . If t h e e a r t h equation cable  e q u a t i o n (4.10) for the semi-infinite  ( 4 . 1 9 ) w i l l be less t h a n at  c a s e is s t i l l v a l i d if  T h i s c a n be c h e c k e d as follows:  r e s i s t a n c e is a s s u m e d t o b e 1 0 17— m,  is b u r i e d  mutual  whether  a depth  impedances,  using  then the depth of penetration  5.5 m , f o r f r e q u e n c i e s 0.1  M H z a n d higher.  of 5.5m, then  the values obtained  the  (4.2)  equations  o r (4.7),  for earth  should  o b t a i n e d u s i n g e q u a t i o n (4.10) f o r f r e q u e n c i e s a b o v e 0 . 1 M H z .  given by  H e n c e , if a  return  be t h e s a m e  self a n d as  those  F i g u r e 4.8 s h o w s t h e d i f f e r -  ence i n t h e v a l u e o f r e s i s t a n c e o b t a i n e d b y P o l l a c z e k ' s f o r m u l a a n d e q u a t i o n (4.2) f o r t h e earth  return  self  decreases from to  1.59in.  impedance  in  1 9 % t o less t h a n  the  frequency  range  10kHz  to  1 % while t h e depth of penetration  H e n c e , it a p p e a r s t h a t  Pollaczek's formula  1MHz.  The  difference  decreases from 15.9m  is c o r r e c t e v e n a t l a r g e d e p t h s o f  b u r i a l e v e n t h o u g h it is b e t t e r t o u s e t h e e q u a t i o n s ( 4 . 2 ) a n d ( 4 . 7 ) f o r t h e c a s e o f i n f i n i t e earth  [9,22].  Subroutine T U B E cannot  be u s e d  author  has options  has an option for  for finding for  the  finding  mutual  both  finding  t h e infinite e a r t h r e t u r n self i m p e d a n c e , b u t it  impedance.  self a n d m u t u a l  T h e routine  T U B E C  developed  impedances, a n d c a n take  by the  displacement  c u r r e n t s i n t o a c c o u n t as w e l l .  4.7  Mutual Impedance  or Vice  Between  a C a b l e B u r i e d in the E a r t h  and an Overhead  Line  Versa  A n o t h e r impedance of interest is t h e m u t u a l T h e electric  impedance  field  strengths  between  t o p o w e r e n g i n e e r s as well a s t o c o m m u n i c a t i o n e n g i n e e r s an u n d e r g r o u n d cable a n d a n overhead  line o r vice v e r s a .  in air d u e t o a c u r r e n t c a r r y i n g c o n d u c t o r buried in the  earth  - 68 -  2 0  1 2 -  cu  u  c  CD  "*  4-  CU  u  c  -4  CO  to CO  cc  -12  -i—i—• I  +  - 2 0  10  10  4  s  I  I  10  6  F r q u e n c y [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:  0  e x p j - / i | c*| -dV<t>2+  mj 2  exp(j>| x | )d<f>  | c6| + V(t> +m 2  2  •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 C o n c l u s i o n T o 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 consisting of Bessei functions. It was found that the values obtained from the numerical integration 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 routine in the BPA's E M T P and other approximations suitable for hand calculations were compared 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. A t the end of the chapter, the evaluation of mutual impedance between a buried conductor 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 application 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  I n t e r n a l Impedances of a L a m i n a t e d T u b u l a r  Conductor  The internal impedances needed for laminated conductors are the same as those needed for homogeneous conductors, namely: The  1.  internal impedance z  0(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 z  bb  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 I n t e r n a l Impedance with E x t e r n a l R e t u r n 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^ z  b  = mutual impedance between the two surfaces,  m h  — internal impedance of all m layers when the current return is external  For the very first layer, we note that Z [^ = z$\ b  Chapter 3, then the loop current 7 _, m  surface of the mth  If we use concentric loop currents as before in  of the first m —1  layer, while loop current I  m  layers combined, returns on the inner  flows on the outer surface. Using  Schelkunoff's  theorem 2 from Appendix B, the electric field strength along the inner surface of the mth layer becomes  - 71 -  Axis  Layer (m-l) Layer m  Figure 5.1 Numbering of Conductor Layers to Find the Internal Impedances of a Laminated Tubular Conductor  r  m  Figure 5.2 Representation of the mth Layer  dV dx  —  (ZTb  lm  %aa  'm-l)  (5.1)  But the inner surface of the mth layer is the outer surface of the first m - l layers combined for which the electric field strength is given by  . ~  z  bi 'm-l  (5.2)  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. O n one hand it is  — z^I , and on the other hand it is —(Z I — Z ^ / _ i ) using Schclkunoff's m  m  b t  m  m  theorem 2. Thus we have the following identity: m Zbb  m %bb ~  =  m ^  %ab  m  ~~j  ~ *  m  Substituting for 7 _i// m  . '  (5.4)  '  from equation (5.3), we obtain  m  • m _ m J^i!  '/<:«;»  7  Zbb ~  *bb  ,  m  •^aa  m  -  (o-5)  x  f»6  +  which gives the internal impedance of all m layers of the laminated tubular conductor, with Starting with the first layer where z$ = Z y, we add the  current return on the outside.  b  remaining layers one by one until we obtain the impedance of the complete laminated conductor made up of m layers.  Ab = ZU - „  (Zlb?  Z'at  +  ,  (5.6)  , • = 2, • • • m  Ab  5.1.2 I n t e r n a l I m p e d a n c e w i t h I n t e r n a l  Return  Similarly we can find the internal impedance of a laminated current returning inside. Let Z™, Z  m b h  tubular conductor with  and Z™ related to the same internal impedances defined b  in the previous section. Let z* be the internal impedance of all m layers when the current a  Also, we note that for the very last layer, i.e. layer m in Figure 5.1,  return is internal. ZTa  =  z  Tn- Using Schelkunoff's theorem 2, we find the electric field strength along the outer  surface of the 1st layer as  = ~(Z \h  ~ Z,\lo)  b  (5-7)  But the outer surface of the 1st layer is the inner surface of the rn — 1 outer layers combined, for which the electric field strength can be written as 1  =  (5-8)  Therefore, we can find a relationship between I and I from equations (5.7) and (5.8), 0  /,  z\  h  Z \ + z-  x  a  (5.9)  t  Now, consider the electric field strength on the inner surface of the first layer. O n one hand it is —{ — z \lo)t a  a  n  d  o  n t  n  e  other hand it is — (Z^/i  Therefore we have the following identity,  —  Z}J ) using Schellkunoffs theorem 2. 0  -  = ZL  73  -  ~ Z^-j'o  (5.10)  Substituting for / j / / from equation (5.9) we obtain 0  i _ ._ aa ~ Z  z  aa  (^)2 _] ^bb  ,5.11)  2  +  z  at  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'  = -Zla ~  _, ^lb  2  '* +  • » = m-l,m-2 -l. >  (5.12)  aa  z  5.2 A p p l i c a t i o n t o G a s - I n s u l a t e d S u b s t a t i o n s The  equations for the internal impedance of laminated conductors will now  obtain the surge propagation  be used to  characteristics in a gas-insulated substation with conductor  coatings. Gas-insulated substations are subjected to transient sheath voltage rises whenever switchings or fault surges occur. These surges propagate along the outer surface of the inner conductor 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 considered are as follows: Core and sheath not coated. i.  Only the inner surface of the sheath coated.  ii.  Only the outer surface of the inner core coated.  v.  Both the outer surface of the inner core as well as the inner surface of the sheath coated.  For each of these cases, the formulae  for the impedances for surge propagation arc  derived.  5.2.1  C A S E i : C o r e and Sheath not  Coated  Axis Core  S h c a l h  Earth  v/ssssss//////////////////////J^ \  J  ,  %  Figure 5.3 - Core  and  Sheath  not  J  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, + Z ns  (5.13)  iht  where Zcre ( core - with external return) can be obtaind from equation (3.3) if the core conductor is solid, or from equation (3.1b) if it is tubular. Z  M  S  can be obtained from equation (2.11),  and Z , (sheath - with internal return) from equation (3;.la). ih  5.2.2  C A S E ii: O n l y Sheath C o a t e d 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 = Z  cre  + Z, + zll ns  where Zcre and Z,ni are the same as explained for case i.  (5.14)  zll is the internal impedance of the  laminated conductor with internal return. This is obtained from equation (5.12} where layer 1 is the sheath and layer 2 is the coating material (superscipt "sp" denotes the paint layer on sheath and superscript "sh" denotes the sheath). Hence  - 75 -  Axis Core  Paint Sheath Earth  Figure 5.4 Inner Surface of the Sheath only Coated.  sp\2 y*V  T V  -  _  (5.15)  + z,sk  Zll  The total impedance Z can then be written as  Z —Z  :rc  +Z  ins  + Z , ip  (Z  spm  )  2  (5.16)  z pc Z , s  ik  where Z , (sheath coated with paint layer - with internal return) and Z , (sheath - with intersp  sK  nal return) can be obtained from equation (3.1a). Z  ipm  paint layer) is found from equation (3.1c), and Z  spe  (mutual between sheath conductor and  (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 ( f i g u r e 5.4),  = -(zj,  dx  oT's  = ~(Z I + n  ~di~  i:  +  ZM  ZI) 2  (5.17a,b)  2  where V,  = potential difference between the core and paint  v  2  = potential difference between the paint layer and sheath Z s "** Z ,  Zi  m  z  — Zi  Z  — Z  m  2  2  ipt  sp  Z m sp  + z , $k  Since the paint layer and the sheath are at the same potential we have V = 0. Hence from 2  equation (5.17b),  -  I  2  =  (-Z  m  /Z )/ 2  Substituting the value of I  2  76  -  (5.18)  1  into equation (5.16a) gives  dV, (5.19)  Therefore the impedance for surge propagation between core and sheath is given by  z = z, - — or _  J- 7  _L 7  7 - 7  ( ^ s y m  w h i c h is i d e n t i c a l w i t h e q u a t i o n 5 . 1 6 .  5.2.3  C A S E  iii: O n l y C o r e  Coated  Axis  1,  Sheath / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / Earth | Figure 5.5 Core Alone Coated  The  surface for surge propagation in this case consists of the surface of the  conductor  (made  u p of core  and the coating)  a n d the  inner  surface  of the  laminated  sheath.  The  i m p e d a n c e f o r s u r g e p r o p a g a t i o n b e t w e e n c o r e a n d s h e a t h is t h e r e f o r e g i v e n b y  Z  =  where Z , , a n d A  z$ + Z  nt  Z  int  +  Z  a  (5.20)  ,  are the s a m e i m p e d a n c e s explained earlier.  the laminated conductor with external return.  z$  is t h e i n t e r n a l i m p e d a n c e o f  T h i s c a n be o b t a i n e d f r o m e q u a t i o n (5.6), w h e r e  l a y e r 1 is t h e c o r e a n d l a y e r 2 is t h e p a i n t o n c o r e ,  Substituting ztf into equation (5.20) gives  (Z Z = Z where Z  cre  cre  -  )  2  7  + Z,  + Z,  (5.22)  tk  ns  is the same as explained earlier. Z^ (core coated with paint layer - with external  return) can be obtained from equation (3.1b). Z  cpm  ductor) from equation (3.1c), and Z  Cfi  (mutual between paint layer and core con-  (core coated with paint layer - with internal return) from  equation (3.1 a)  5.2.4  C A S E  iv: B o t h  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 conductor 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% + Z  ml  + z?.  Using the values of z$ and z\\ from equations (5.15) and (5.21) Z can be written as  z  z  cpc  + z,  RS  + Z  ipt  where all the impedance values have been explained earlier.  (5.24)  - 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 -ORCR . IRCR  -IRSH C ORSH 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, O R C R  =  65.0mm  Inner radius of the sheath, IRSH  =  Outer radius of .the sheath, O R S H  =  350mm  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  Core and Sheath  Aluminum  PERMEABILITY  RESISTIVITY  1.0  2.62E-08  1500.0  4.70E-07  Paint (a)  Stainless Steel  (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 (Hz)  SKIN D E P T H (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 CASC 2  10  E  CASE_3 CASE 4  c  in a)  rr  0.1  0.01  0.001  1CT  10'  10"  10  10  3  10'  3  10  10'  s  10'  Frequency [Hz]  \  O.B-  \ \  w  x  CAST 1 CASE.. 2  0.6  C_ASE_3 CASE ^  c  o o ~o c  o.*  0-2.  0-+  10"  10'  • 10-  10  J  10  5  10'  •10  s  10*  Frequency [Hz] Figure  5.8(a),(b) Variation  of Resistance  •with Frequency for the Four Stainless  Steel Coaling,  Thickness  and  Inductance  Cases;  0.1mm.  10'  -  82 -  5.2.6 S u p e r m a l l o y C o a t i n g The high resistivity and high permeability of supermalloy  make its skin depth very small  even at low frequencies, as shown in Table 5.3.  T a b l e 5.3 Skin Depth of Supermalloy  FREQUENCY (Hz)  SKIN D E P T H (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 frequency, 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 C o m p a r i s o n between Stainless Steel a n d S u p e r m a l l o y  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 frequencies 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 -  \  <2 £  ^ \  i- ~  \  CASE 1 \  V  3  CAse  \  c o  y 5  •o c  X  2  2  L IL . A  \  3  CASE i  \  10"  2  10"  1  10'  10  J  10  5  10."  10  5  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.  10  -  84  -  1000CH  1000-  100-  E  CASE 1 10-  CASE 2  u  c D tn  ca 0.1.  0.01.  0.001 1CT J  1  10"'  10'  10  10  2  3  10'  10  10  5  s  10'  Frequency [Hz] 20-  \ X  15-  \  E  CASE 1 CASE 2  <u u  \  10-  c o  "o 3 "D C  0  CASE_3 CASE *  +  10"  10"  10'  .10*  10  J  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.  e  10'  - 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 earlier 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 implications for transient ground rise, however. If switching surges are produced, transient overvoltages would appear at many points within the gas-insulated substation.  5.3 C o n c l u s i o n s The internal impedances of tubular laminated conductors have been derived. These equations 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  formulae up the are  impedance formulae  were discussed in detail  elements  obtained  for tubular  i n C h a p t e r s 3 a n d 4, r e s p e c t i v e l y .  [Z \.  of submatrices [Z„] a n d  for  a  specific  conductors and the e a r t h return  In  i}  underground  cable  These impedances  this chapter the values of these  s y s t e m , using the  impedance  exact  make  submatrices  formulae  as  well  as  approximations. The  approximate  formulae tion  formulae  proposed by Wedepohl  a n d t a k e v e r y little c p u t i m e .  pruposes.  Therefore  sidered in this  chapter.  only the The  agree  are  formulae  proposed by W e d e p o h l  also c o m p a r e d w i t h values o b t a i n e d  C o n s t a n t s routine in the E M T P , w h i c h was d e v e l o p e d by A m e t a n i  6.1  Single C o r e  m = w h e r e all e l e m e n t s Appendix  Wedepohl's  A,  were defined earlier  the  values  approximation  in T a b l e 6.1.  F i g u r e s 6.1  l a e f o r t h e i m p e d a n c e Z. . c  Z. s  and  Z  ss  The  cc  and Z  Z  Ci  S!  return formulae  errors  of  these  formulae  This  stant  m,  from  are  the  conCable  [27].  were  W i t h the d a t a of the test case d e s c r i b e d obtained  becomes larger,  from  the  exact  formulae, as  a n d 6.3, respectively, show the errors for the and Wedepohl's  formulae  are  We  not  significant  from  tabulated  errors in A m e t a n i ' s a n d W e d e p o h l ' s a p p r o x i m a t e  all have s i m i l a r v a l u e s .  is o n l y t r u e a t  form  and from Ametani's Cable Constant routine,  F i g u r e s 6.2  used by A m e t a n i  m a t r i x of the  (6.1)  in C h a p t e r 3.  elements  depicts the  in A m e t a n i ' s  Wedepohl's approximate fied.  [22]  z.  q u e n c i e s , b u t for h i g h e r f r e q u e n c i e s t h e y c a n n o t be n e g l e c t e d . Z ,  exact  Cable  T h e i m p e d a n c e s o f a s i n g l e c o r e c a b l e a r e g i v e n b y a 2X2  in  very closely with the  T h e y also provide simple expressions for h a n d calcula-  approximate  results  [22]  formu-  impedances at  low  also notice t h a t the errors  T h e y are essentially c r e a t e d b y the e r r o r s in the  a n d W e d e p o h l , as s h o w n i n F i g u r e 4.7.  A s mentioned  e a r t h r e t u r n f o r m u l a is v a l i d o n l y i f t h e c o n d i t i o n low f r e q u e n c i e s .  At  high frequencies, the  a n d this c a u s e s the e r r o r s in the r e s u l t s .  the separation between the cables becomes  intrinsic  in  earth  earlier,  \ mZ \ <0.25 is propagation  satiscon-  T h e errors also increase  if  larger.  T h e reasons for the errors in A m e t a n i ' s e a r t h r e t u r n i m p e d a n c e f o r m u l a e cies has a l r e a d y been d i s c u s s e d in C h a p t e r  fre-  4.  at h i g h  frequen-  - 87 T a b l e 8.1 Impedances of a Single Core Underground Cable.  Zee (fi/km)  F R E Q (Hz) 1 10 100 1000 10000 100000  R .010873 .020084 .119303 1.05509 10.4803 108.240  WEDE: P O H L  AME1' ' A N I  EX; R  X .016082 .146299 1 .30456 1 1 .4759 99.6843 839.848  X  .010873 .020069 .118805 1.03996 10.0343 96.4973  .016083 .146315 I . 30504 I I . 4916 100.175 854.761  R  X  .010868 .016093 .020081 .146416 . 120114 I . 30506 1.05626 I I . 4759 10.4983 99.6822 109.593 839.614  Zcs (£2/ km) 1 10 100 iooo 10000 100000  .000987 .009878 .098954 .995717 10.2001 106.430  .015097 .136501 1.22016 10.7494 92.8295 775.524  i 1 10 100 1000 10000 100000  .300151 .309041 .398112 1.29438 10.4531 106.361  .015083 .136354 1.21869 10.7347 92.6969 775.518  .000987 .009862 .098465 .980590 9.75414 94.6876  Zss  .015098 .136516 1.22066 10.7651 93.3201 730.438  .000987 .009878 .098956 .995919 10.2170 107.782  .015097 .136501 1.22016 10.7494 92.8272 775.291  .300151 .309041 .398115 1.29458 10.4700 107.713  .015083 .136354 1 .21869 10.7347 92.6946 775.285  Wlim)  .300150 .309025 .397612 1 .27925 10.0071 94.6180  .015083 .136369 1.21919 10.7504 93.1875 790.432  - 88 -  50  30-  / 7  10-  u  c  «  -10  cn io cu  rr  AMETANI -30  —50  WEDEPOHL  i  \  10~  i  i Mini  10"  2  i  i  i  nun  —n  TTTTTTTJ—i  1  nr  T I itiiTf  10'  10  2  TTHTTT;  10  T-TTTTTTT]  r  10  3  i i  mit| 10  4  "i  T T  mn| " i  10  5  i  ritin  10  6  7  20  12  4-  01  o c  \  -4  to  u ca cu CC  AMETANI -12  WEDEPOHL  -20 1 10"  2  III  Mini  10"'  i i i mill  1  i i i  i i i mm  IIIIII  10  i i i iiuii—i  10  1  2  10  i  5  11mil 10  i i 4  11iui| 10  s  i i  11mil 10  i 6  i  11mi 10  7  Frquency [Hz]  Figure 6.1 - Errors in Ametani's  and Wedepohl's Approximations  in Z  cc  -  89 -  50  30-  /  /  100)  u  c  tncu  -10-  tc  AMETANI  -30-  WEDEPOHL  1 T ITTIfH ' I '1 M l l T i r  — 50  10"  I 'T I 1 i m ] ' ~ T ' T T T T m r  1  10"'  2  10'  T"T T^TIITf  10  2  I'TlTnil}'  10  1 I 1 11'ltT)  10  3  4  1 "TTTTnTJ  10  s  I  10  I T'H'ffl  10  6  7  20  12  cu o c  CO *->  \  - 4 -  o ra CD  CC  AMETANI  -12-  — 20  WEDEPOHL  |  i  10"  2  i 11  —  i i i IIIIII  10"'  1  I  i i Mini  1 i . IIIIII—i  1 i i mill  10'  10  2  1 i I M i n i — i  i i Mini  10  J  10  4  10  5  i i utii|  i  10  6  l  l Mini  10  7  Frquency [Hz]  Figure 6.2 - Errors in Ametani 'a and Wedepohl's Approximations in Z  a  - 90 -  50  30-  /  /  10  LU co u «  +—<  -10  m cc  AMETANI - 3 0 -  WEDEPOHL  —50  |  III  TO"  2  IIIIII  i i i  10"'  IIIIII—i  1  i  i  i i 11uii|  IIIUI  10'  10  2  i i  11mil  10  I  i 11uii|—> i i 10"  3  10  IIIIII 5  I  I I IIIIII—i  10  i i  inn  10  6  1  7  20  CD  u c to o co CD  at  AMETANI -12  WEDEPOHL  —20  I  10"  I I I niM| 2  10"'  i  i i iiiii,  1  rrm]  10'  1  iiiinij  10  i i I I I I I I ; I 2  10  I I I I I M I  3  10"  I I I I I I I I ) I  10  S  I I I I I I I I  10  6  I  i i m i q  10  7  Frquency [Hz]  Figure 6.3 - Errors in Ametani's and Wedepohl's Approximations in Z,  - 91 6.2 T h r e e - P h a s e C a b l e In the case of a three-phase cable system, the series impedance matrix is given by \Zn\ |Z ] \Z ) 12  K  (0.2)  1*1\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 |Z ] of the form: t;  Z- . Z Zj.ry Z . . .f  r  r •  (0.3)  s 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. T a b l e 6.2 Mutual Impedance between Two Cables with Burial Depth of 0.7om and Separation of 0.80m.  Zij  EX;^CT  (fi/kin) AME*]rANI  WEDI3POHL  FREQ  RES  REA  RES  REA  RES  REA  1 10 100 1000 10000 100000  .000987 .009877 .098943 .994644 10.1015 105.154  .012562 .111152 .966670 8.21457 67.5095 525.238  .000987 .009862 .098452 .979517 9.65569 93.4318  .012563 . 111167 .967169 8.23027 68.0000 540.150  .000987 .009878 .098946 .994856 10.1193 106.592  .012562 .111151 .966668 8.21452 67.5070 525.000  - 92 -  50  30  / O  /  )0  UJ cu  o  li -tocn cn AMETANI  tn H  - 3 0  WEDEPOHL  -50  10~  — i — n : TTTT.  2  10"'  i—r^rrrrrrr-  1  ni  10'  1  i i IIIIII  11  1  10  2  10  MM;  1  10  3  4  \ i i mi,  10  1—i t mil,  10  S  1—t 11  nn'  io  6  7  50  3 0 -  10-  O  UJ  cu o c  \  -10  AMETANI  0>  CC  WEDEPOHL  - 3 0  5 0 -f 10  - 2  1 i i mii| 1 i i inn, 10" 1  1 i i mii| 1 i i iiiiii 10' 10  2  1 i ' mii; 10 S  1 i mm, 10 4  1 i i i nn; 10  ""1 10  1 , 1  S  1  6  ' "" 10 11 1  F r e q u e n c y [Hz] Figure 6.4 - Errors in Ametani's and Wedepohl's Approximations in the Mutual Impedance between Two Cables.  7  -  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 A d m i t t a n c e M a t r i x 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 constant.  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 admittance 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 tubular 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  impedances of conductors.  in the formulae for the internal  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 admittance matrix. Its value is frequency dependent as well as complex.  In some cases, (e.g., cross-  linked polyethylene), the permittivity can be assumed to be constant and real up to very high frequencies, while in other cases (e.g., oil-impregnated paper) the changes with frequency are quite  significant.  Two  insulating  materials,  namely  cross-linked  polyethylene  and oil-  impregnated paper, are discussed in detail because they are the materials most often used in power cables.  A general formula for the complex permittivity of insulation materials is given  by Bartnikas [15], based on the relaxation time of the dielectric material.  Ametani's Cable  Constants routine in the E M T P [27] assumes that the permittivity is real and constant which may not always be accurate enough. The  earth return impedance formula derived by Pollaczek [l] for the case of a semi-  infinite earth is valid only for filamentary type conductors of negligible radius. This formula can be used for a conductor of finite radius a, if the condition | mo | <0.1 holds. This condition is satisfied up to a frequency of 1 M H z even for a worst case low earth resistivity of 10 fl— m. Hence Pollaczek's formula is recommended as the accurate formula. Values obtained from various approximate formulae and from Ametani's Cable Constants routine in the E M T P were compared against Pollaczek's formula. The results agree closely at low and medium frequencies 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 Harrington [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,  G  x A  '-  »  x  Figure A.l - Three-Phaae Cable Setup for the Study  Each cable is of a single core type with dimensions as given below  conducting sheath  central conductor insulation  Figure A.2 - Basic Construction of each tingle core cable  di,d ,dt = 0.75m, depth of burial of each cable 2  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 0.0385m, inner radius of the sheath  *2  0.0413m, outer radius of the sheath 0.0484m, outside radius of the cable 100 fl—m, resistivity of the earth 1.7X10" J?-m resistivity of the core material 8  Peon  2.1xi0" /?-m resistivity of the sheath material 7  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: dllz rd<*> 3//  dz dHz  r  dz ifdlrHJ  = (\lp + »"toe)E,,  =  (l/> +  i<ai)E , t  dE  3Ez  3E,  dE,  dz  — — iliifiH,  =  —ioifill^  1 (9[rE )  3H,\  — 1— r ( dr  t  rd(j>  4  — ' = d<p )  ^  + i«e)E . r  r {  dr  dE \ t  TT f = ~ 3<t> )  (B-1)  where H and E are electric and magnetic field strengths, respectively. Here we are interested in the circular magnetic field around conductors, with its lines of force forming a system of coaxial circles. Such circular magnetic fields are associated with currents flowing in isolated wires, as for example in a single vertical antenna, or between the conductors of a coaxial cable, as shown in Figure B.l.  Figure B.l - The relative directions of the field components in a coaxial transmission line.  - 99 From equation (B.l) we see that when the quantities are independent of the angle <f>, one of the independent  subsets composed of the 1st and 3rd equation on the left of equation (B.l),  together with the 2nd equation on the right, define the circular magnetic field strengths as follows: dlrllA . or  a,  - (1/p + i<at)rE,  (B.2a)  -(l/p  (B.2b)  dE  dE  dr  dz  2  + itoe)E  r  iattH .  (B.2c)  4  It has been shown by Schelkunoff that H^,E and E r  z  have components which vary exponentially  along the longitudinal axis of the cable, i.e., along the z axis in Figure B.l. If we express the exponential variation of the quantities E,,E and Z  as E,eC *, E e~ r  Tz  z  and H^,e~ ', then the r  quantities E,,E and H# are functions of r only. Substituting these values into equation (B.2) 2  we obtain E  =  r  iuuHt  ^—A. dr  f •  H  (B.3a)  dE, = —— + TE, dr  (B.3b)  = (Up + iu>f)rE,  (B.3c)  4  where the quantity T is called the longitudinal propagation constant. Now solving for H# from Equation (B.3), we obtain  where m = 2  I' I  co /ze|. 2  This quantity m  is called the intrinsic propagation constant of  P  the conductor material. For solid conductors, the term u> ut which accounts for the displace2  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" tt-m 8  y/a){i/p = | m |  iH,)  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~ . Hence, even at high frequencies T is negligibly small by comparison with 10  m. 2  2  Therefore, we can write equation (B.4) as  d H*  i dH, dr  2  dr'  The solution for H  sT  =  X*  m  H  (B.5)  of equation (B.5) is in the form of Bessel functions given by:  t  Al {mr) x  +  BK^mr)  (B.6)  Since we are interested in longitudinal voltage drops, we must find the longitudinal electric field stength first. This can be obtained from equation (B.3) and (B.6) along with the following rules of differentiation for modified Bessel functions of any order n, (B.7a)  dx _d_(x*K )= dx n  -x*K . n  x  The solution for the longitudinal electric field strength then becomes  (B.7b)  - 101 -  E, = pm\AI (mr) - BK {mr)\ 0  (B.8)  0  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 Z return and Z  b  t  as the internal impendance of the tubular conductor with internal  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 Z  between the two loops of internal and external return.  ab  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 I returning outside. b  Figure B.2 - Loop Currents in a Tubular Conductor  Since the total current enclosed by the inner surface of the conductor is — I and that enclosed a  by the outer surface is I (7 + I — /„), the magnetic field strengths at these two surfaces take b  0  b  the values ( — I /2na) and (I /2nb) respectively. Hence from equation (B.6) we have a  b  A/,(7»io)+  M , ( m a ) = -lJ2na  (B.9a)  A li(mb) + BK (mb) = I /2r.b  (B.9b)  x  b  From these two equations the values of A and B can be evaluated as  2-naD  _ /»/»("»)  =  2naD where  2nbD  2xbD  (  B  1  0  b  )  -  102 -  D = r (mb)k (ma) - I^majk^mb) 1  (B.II)  1  Substituting these values in (B.7) and using the identity /„ (z)/C"i(z) + K (x)I (x) = 1/x,, 0  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) = ZJ  a  + Z I ab  E,(b) = Z I cb  (B.12a)  b  + ZI  a  b  (B.12b)  b  where  2naD  > ~  7 Z  7  =  [irimajK^mb) + /v" (mo)/,(m6) 0  P ^/ (m6)K'i(r7ja) + AT (m6)/ (ma) j 2xbD m  0  0  1  P  2nabD  (B.13) Schelkunoff stated these results in the following two theorems.  Theorem 1 If the return path is wholly external (I = 0) or wholly internal (/;, = 0), the longitudinal a  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 E a r t h 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  (C.l)  H = —+-fjrdr 2nr r J  v  where J is the current density in the ground. Suppose that the earth is subdivided into concentric cylindrical shells of radius r and thickness dr in which the current density / and magnetic field strength / / a r e constant. Then the magnetic flux per unit length of such a shell is given by d<f> =  BdA = ulldr  (C.2)  Substituting for H from equation (C.l) yields  (C.3)  -!-+±Jjrdr dtp = udr 2xr r •*  Now let us write Kirchhoff's voltage law around the rectangle ABCD of unit length and width  dr. The net resistive voltage drop is  ' |~^ ~J^ ,  —  :  r a n c  *  t  n  e  induced voltage is jmd<l> or  jtaiilldr.  -  104 -  Figure C.1 - Representation of a Buried Conductor in an In finite Earth.  Since the sum of these two voltages must be zero, we obtain  dJ —p  dr + jtauH dr = 0  (C.4)  dr Substituting f o r / / f r o m equation (C.1), we have  pdJ dr  •dr  +  jtAfidr  Multiplying this equation by  — +  -fjrdr = 0  2itr  ~- and differentiating with respect to r we have  par  dJ 2  dr  2  If we substitute m* for  Jfa>/i  P  (C.5)  +  dJ_ _  jvuJ  rdr  p  = 0  , then equation (C.4) can be written as  (C.6)  -  £  105 -  «•»««*.  (C7)  and equation (C:6) can be written as  d J . dJ 2  dr'  mJ = 0 l  r dr  (C.8)  Equation (C.8) is immedately recognizied as a Bessel equation whose solution is of the form  / - AI {mr) + BK {mr) 0  (C.9)  0  We note that I (x) approaches infinity as z approaches infinite. However, we cannot permit a 0  solution of J to increase indefinitely as r approaches infinity and we must conclude that , 4 = 0  PIHence, equation (C.9) becomes / = BK (mr)  (C.10)  0  Using equation (C.7) we find a solution for the magnetic field strength H as  m H = -BKAmr)m  (Cll)  2  Now applying the boundary condition that H = I/2ira  in the ground immediately adjacent to  the cable, we obtain the value for the constant B from equation ( C l l ) as  B  =  ~ 2nal<!(ma)  ( C  '  1 2 )  Using the equation E = pJ, the solution for the electric field strength at any point in the soil is found to be  pml K {mr) ~~2*a K^ma)  .  0  E  (  C  1  3  )  The earth return self impedance as well as the mutual impedance between two buried cables can be deduced from this equation (see Chapter 4).  -  106 -  APPENDIX D Calculation of E a r t h Return Impedances in a Semi-Infinite Homogeneous E a r t h 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 longitudinal electric field strengths.  Let us first consider au overhead line and derive the electric field  strengths in air and in earth.  A  y  p  . (cc.h)  X  Earth  Q*  (x.y)  Figure D.l - Current-Carrying Filament in the Air  Let medium 1, denoted by subscript 1, correspond to air and medium 2, denoted by subscript 2, correspond to earth. Let point P{a,h) correspond to the current-carrying filament lying  -  107  -  along the Z-axis of the Cartesian coordinate system. Let E+ + (x,y;a,h) be the electric field strength in air at a point Q(x,y) and E-  +  (z,y';a,h) be the electric field strength in the earth at  a point Q'(x,y'). Note that the y axis is positive in the air and that the y' axis is positive in the earth, as shown in Figure D.l. From Maxwell's theory, the general equation for electromeganetic  wave propagation is  given by  V E - V(V£) =  - tovjtf  2  (D.l)  where p,fi and e correspond to the respective medium to which this equation is applied. assumption 4 (Chapter 2), we can say that VE  Using  — 0 in both air and earth. Hence, equation  (D.l) can be written as  V E  -  2  CO  Now  let us define the fields which we would like to derive as follows  E++  = E  +  z  E-+  — E-  +  z  +  (D.2)  [It  = Electric field strength in the air due to the current-carrying filament in the air =  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)  Hence in the air, the current density can be expressed in the form  (D.5)  -  108  -  I6(x-a)S(y-h) Now  (D.6)  keeping this result in mind and noting that we are interested only in the electric field  strength along the Z-axis and  9E  '  Z  -  = 0 (using assumption  2 from Chapter 2) equation (D.2)  can be written for the case of air as  d E+ +  d E.+  2  2  2  mE+  + p mfIS(x — a)6(y — h)  2  3z  By'  a  +  l  (D.7)  where  Pi  For the earth, equation (D.2) can be written as  a £_  a E_  2  2  +  +  m|E_.  =  dx  2  (D.8)  • here  m  2  The solutions for E++ and E_  =  —  I P2  CO  2  p2i  2  should be obtained in such a way that they satisfy the follow-  +  ing boundary conditions: 1.  Continuity of E at the surface. Lim  £++  E_+  = Lim  V-*0  y—  — Lim  0  £_ + =  /? (say) 0  jr'-+o  Vertical component of B is continuous at the surface  dx  (D.10)  dx  Horizontal component of H is continuous at the surface  dE++ Pidy The solutions for E+ + and E_  +  a£_  +  P2&y  aE_  +  p-z^y'  (D.il)  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:  -  d E  109 -  2  -6 E^  +  2  +  +  <f £..  = mfE  + /j,m /exp(-^a)%-A)  (D.12)  2  + +  2  -6 E_ + + 2  £+ + |  y-0  1 dE+* Hi Now  E-+  —  - mf E.+  ^  j  | 'Y  •j  0  —  (D.13)  (D:14)  £Q  i1 OD d £ __  +  +  dy  -  I  fi dy' 2  taking the Fourier sine transformation of equation (D.12) with respect to y with 0 as the  parameter, we have: -6 E^+ 2  — 0 E+ + + 0E  = m E  2  2  O  + +  + p^rn? Iexp{-j6a)sin{0h)  (D.16)  i.e., (0 + C f ) E 2  + +  where Cf = 0 +  = 0E m  2  - m lexp(-jea)sin(0h)  (D.17)  2  O  Pl  2  Similarly, taking the Fourier sine transformation of equation (D.13) with respect to y' with 0' as parameter, we have -0 E_ 2  +  +  fl' E.  + /J*E = m £ _  2  (D.18)  2  +  0  +  Hence + Cf)£_ where C'| = t? + m  = £'E  (D.19)  0  .  :  2  +  2  Now, taking the inverse Fourier sine transformation of equation (D.17) with respect, to 0 we have = E exp( —C,y) 0  -•^ /exp(-^a)|exp(-C |/i-j/|)-exp(-C,|/ +y|)j L  1  l  (D.20)  Similarly, taking the inverse Fourier sine transformation of eqution (D.19) with respect tofl'we have F_  +  = E exp(-C y) 0  By taking the derivative of Z? * with respect to y and the derivative of E_ +  (D.20)  2  +  with respect to y.  - no  -  and substituting in equation (D.l 5), we obtain the value of E  as follows:  0  P\fn f/exp( — j6a)exp{—C h)  —  x  u  i  (D.22)  (C, + :—<7 ) 2 a  u  Substituting E  0  in equation (D.20) and taking the inverse complex transformation with respect  to 0, we obtain the value of E  .2/  +  as follows:  +  [expf-C,| h-y  -  | )-exp(-C | h +y |)] 2  2C,  exp{-C \h+y\) 1  C. + — Similarly, substituting E  0  C  exp(j6\ i - a | )d0  (D.23)  2  in equation (D.21) and taking the inverse transformation with respect  to 6, we obtain the value of E. + as follows  Pi fl m  E-+ =  ~ 2n  " exp{— C^y — C h}exp(j6\ x — a\)d6  J  r  1  Ci +  —C  (D.25)  2  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 A i r and in the Earth due to Current Carrying Filament Buried in the Earth. Let = E-- ~ Z  Electric field strength in the earth due to the current-carrying filament buried in the earth.  E+.  = is _/= +  Electric field strength in the air due to the current-carrying filament buried in the earth.  Similar to equations (D.7) and (D.8), Maxwell's equations for electromagnetic wave propagation in air and earth respectively, for this case are given by,  d £ _ 2  +  dx  5  dE 2  +  2  By  = m,E,.  y^O,  (D.26)  -  H  Ill -  y ? (x.y) x _i  p.-  Earth  • (x,y') .  W)  Figure D.2 - Current Carrying Filament Buried in the Earth.  3 E__  ax  1  , , .  8 E__  2  2  +  ay  — = m | £ _ _ + p m fS(x 2  2  (D.27)  — a)5(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 E. =  pmI 2  +  2TT  pmI 2  £•_ _ .= -  2  2  " e x p { — C y — C h) l  2  -»- — C ?2  •xp{j6\x-d\  (D.28)  )d0  2  [ e x p ( - C | h'-y'\ ) - e x p ( - C | h'+y'\ ] 2  2  2C,  2n  exp(-C |/i'-rV|) 2  (D.29)  exp(jO \ x-a'\ )d8  C, + —C  2  »2 If we assume that the relative permeability of air and earth are the same, i.e., u  Tl  = u, then 2  -  i  r l  112 -  = fi and we can show that the equations derived for E+ ,E_ 2  +  +  ,E^_  and E+_ are the same  as those derived by Pollaczek [lj. Now using the standard results i \ K (mr)=  c exp{—aV^-+m }exp . . . , J \ (jis)d 2  y x  0  where r =  and where K  9  y  2Vr+m  •  ' _ • . (D.30)  3  2  is the modified Bessel function of the 2nd kind and of the  0  zeroth order, we can write E+ + and E _ _ as follows:  pm,/ ( 2  E „  =  +  /  ;  =  (jtf|x-o»|  \A + rnf + \ / V +  )rfg  2  where /?, ~ V {T - af+(h  - y?, Z  x  = V(z-a)  2  +  (h+yf  pm I ( 2  —  —jA'o( 2^2) m  K (m Z )  —  G  2  2  exp{-|j/+/i V^+m22}exP( + / . (j*l z - a |)«« r  — \A + m? + \A + ™ 2  where /?  2  .=• V ( z - o ] ' +  (A'7P, Z =  2  2  = V(z-a')  + (A'+^j  2  (D.31(a,b))  2  2  Further using assumption 4 in Chapter 2, we can neglect the displacment current up to a fairly high frequency, and also noting the fact that the resistivity of air, i.e., p, is very large, we can conclude that the term mf ~ 0 . This produces the final equations:  E+ +  =  7(011aI  (  —UniZM  + /  ^ - ^ ^ " ^ ^ e x n l ^ l z - a l ^  — \ e\ + y/e E_+  =  - joidol  2n  J-  exp{-A  I 0\ -y'Ve  |*| +  2  w  \/e +m 2  2  + 2  +  m  2  rn }exp(;6l| z - a ] )d8 2  -  1  (  2TT-\l<0(m2R2)-K{  113  -  ,{m Z ) 2  2  e x p l - b ' + A ' l V<? +m|} exp(j'tl| i - a ' | </0 2  \e\ + V^+^I jco/ip/ ; exp{-y | E _ +  =  /  * —  2  6\-h\/e  |*|. +  2  +  m } 2  (D.32(a.b,o,d))  2  exp^|x-a|)d* V^+m|  -  114  -  REFERENCES V. F. PoIIaczek,  Ubrr das Fcld ciner un endlich langon Wechsel = Stromdurch flossenen  Einfachleitung, E.N.T., Band 3, Heft 9, 1926, pp. 339-360. 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