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Effect of ground conductivity and permittivity on the mode propagation constants of an overhead transmission… Doench, Claus 1966

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THE EFFECT OF GROUND CONDUCTIVITY AND PERMITTIVITY ON THE MODE PROPAGATION CONSTANTS OF AN OVERHEAD TRANSMISSION LINE  by  CLAUS." DOENCH B.A.Sc, University of Toronto, 1962. A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF  MASTER OF APPLIED SCIENCE  i n the Department of E l e c t r i c a l Engineering  We accept t h i s thesis as conforming to the required  standard  Research Supervisor Members of the Committee  Head of the Department Members of the Department of E l e c t r i c a l Engineering THE UNIVERSITY OF BRITISH COLUMBIA June, 1966  In p r e s e n t i n g  this  thesis  in partial  f u l f i l m e n t 'of t h e  r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y Columbia,  I agree that  the Library  f o r . r e f e rence and s t u d y . tensive by  copying o f t h i s  shall  for scholarly  t h e Head o f my D e p a r t m e n t o r by h i s  understood cial  gain  that shall  Department o f  "JOVE  permission  It  i%C>  ETtja/A)&E£ftJ6i Columbia  for ex-  p u r p o s e s may be g r a n t e d  this  thesis  n o t be a l l o w e d w i t h o u t my w r i t t e n  Et-£CT£(CAt.  available  representatives.  copying o r p u b l i c a t i o n . of  The U n i v e r s i t y o f - B r i t i s h V a n c o u v e r 8, C a n a d a Date  make i t f r e e l y  I f u r t h e r agree that thesis  of British  It for  is  finan-  permission.  ABSTRACT A general a n a l y t i c a l method to derive the distributed c i r c u i t parameters and mode propagation constants f o r an nconductor transmission l i n e i s developed.  The analysis uses  electromagnetic f i e l d concepts and the results are interpreted i n terms of distributed c i r c u i t parameters.  The procedure  involves transforming the problem of the n-conductor  line  above a ground with f i n i t e conductivity into that of an nconductor above a ground with i n f i n i t e conductivity. Correction factors are added to account f o r the f i n i t e conductivity of the ground.  The distributed c i r c u i t parameters thus calculated  are used to calculate the mode propagation constants over a frequency range from 10 Hz to 1 MHz f o r values of ground conductivity varying between 1 mho/m and 10~^ mho/m and r e l a t i v e permittivity varying between 10 and 50. Numerical results f o r the distributed c i r c u i t parameters and mode propagation constants f o r a t y p i c a l 500 kV single c i r c u i t transmission l i n e and various ground conditions are given.  The results show that one mode has a higher attenuation  and a lower v e l o c i t y than either of the other two modes, suggesting the zero sequence mode f o r a completely balanced system.  ii  TABLE OE CONTENTS Page Abstract  i i  Table of Contents  i i i  L i s t of I l l u s t r a t i o n s  v  L i s t of Tables  vii  L i s t of Symbols  viii  Acknowledgement  xi  1.  Introduction  2.  Assumptions and Approaches  3.  1 ,.  2.1  General Description of the System  6  2.2  Basic Assumptions  7  2.3  C i r c u i t Parameters  ,.  5.  8  Derivation of the Hertz Vector due to Current i n a Wire  4.  6  9  3.1  Hertz Vector due to a Current Element i n a Medium of I n f i n i t e Extent  10  3.2  Hertz Vector due to a Current Element above a Ground with I n f i n i t e Conductivity  10  3.3  Hertz Vector due to a Current Element above a Ground with F i n i t e Conductivity  11  3.4  Complete Hertz Vector f o r a Current i n a Straight Conductor above the Ground 16  Evaluation of Transmission Line Resistance and Inductance Including Ground E f f e c t s  17  4.1  Derivation of the Series Impedance Formulae ...  17  4.2  Evaluation of  18  and  Derivation of the Shunt Admittance  25  5.1  Derivation of Shunt Admittance Formulae  25  5.2  Evaluation of the Maxwell Potential Coefficients  K. . and  L,  iii  26  Page 6.  Derivation of Propagation Constants and A,B,C,D Parameters of an n-conductor Transmission System .. 31  7.  6.1  Calculation of the Propagation Constants  31  6.2  Formation of the Impedance Matrix Z  32  6.3  Formation of the Admittance Matrix Y  34  6.4  The A,B,C,D Parameters of the System .........  35  Numerical Results f o r a 500 kV Transmission Line .. 36 7.1  Description of the Transmission Line  36  7.2  Ideal Parameters  37  7.3  Skin Effect  38  7.4  Impedance Correction .........................  38  7.5  Admittance  40  7.6  Conclusion  Correction  42  8.  Calculation of the Mode Propagation Constants  49  9.  Conclusions  68  Appendix A Appendix B Appendix C  The F i e l d due to an Isolated O s c i l l a t i n g Current Element  71  The General Coordinates  74  Vector i n C y l i n d r i c a l  Derivation of the Internal Impedance of C y l i n d r i c a l Conductors  Appendix D  78  Solution of the Integral i n Equation 4-30 . 82  References  86  iv  LIST OF ILLUSTRATIONS Figure  Page  2- 1  Distributed C i r c u i t Parameters of an n-conductor  3- 1  Current Element Orientation  3- 2  C y l i n d r i c a l Coordinate System f o r the External Field  9  9  4- 1  Two-Conductor Configuration  7-1  Impedance Correction Factor Q with 9 = 0 ,  7-2  Impedance Correction Factor P with 9 = 0 , S - 0  17  43  7-3  Impedance Correction Factor Q with n = 10 .... 44  7-4  Impedance Correction Factor P with n = 10 .... 44  7-5  Impedance Correction Factor Q with n = 50, 9 = 0 Impedance Correction Factor P with n = 50, 9 = 0  7-6  45 45  7-7  Potential Coefficient Correction Factor N with Q = 0, d = 10~5 mho/m 46  7-8  Potential Coefficient Correction Factor M with  7-9  9 = 0, <4 = 10-5 mho/m Potential Coefficient Correction Factor N with 9 - 0, £ = 10-4 mho/m Potential Coefficient Correction Factor M with  7-10  46 47  9 = 0, S = 10- mho/m 47 Potential Coefficient Correction Factor N with © = 0, <£ = 10-5 mho/m 48 Potential Coefficient Correction Factor M with 9 = 0, <d = IO" mho/m 48 4  7-11 7- 12  3  8- 1  Mode 3 Propagation Velocity without Permittivity Correction 56  8-2  Mode 3 Propagation Velocity, no Capacitance Correction with n = 10 v  56  Figure 8-3 8-4 8-5 8-6 8-7 8-8 8-9 8-10 8-11 8-12 8-13 8-14 8-15  Page Mode 3 Propagation Velocity, no Capacitance Correction with n = 30  57  Mode 3 Propagation Velocity, no Capacitance Correction with n = 50 .,  57  Mode 3 Propagation Velocity, Approximate Capacitance Correction with n = 10  58  Mode 3 Propagation Velocity, Approximate Capacitance Correction with n = 30  58  Mode 3 Propagation Velocity, Approximate Capacitance Correction with n = 50  59  Mode Attenuation with I n f i n i t e l y Conducting Ground  60  Mode Attenuation with no Permittivity Correction  61  Mode Attenuation, no Capacitance Correction with n = 10  62  Mode Attenuation, no Capacitanoe Correction with n = 30  63  Mode Attenuation, no Capacitance Correction with n = 50  64  Mode Attenuation, Approximate Capacitance Correction with n = 10 *  65  Mode Attenuation, Approximate Capacitance Correction with n = 30  66  Mode Attenuation, Approximate Capacitance Correction with n = 50  67  vi  LIST OF TABLES Table 7.1 8.1 8.2 8.3  Page Potential Coefficient Correction Terms 42 -5 Attenuation i n Nepers/Mile f o r d = 10 mho/m. and f =.100 kHz 52 Attenuation i n Nepers/Mile f o r d = 10"-'mho/m. and f = 10 kHz 53 Attenuation i n Nepers/Mile f o r d = 10 mho/m. and f = 1 kHz 54  vii  Symbol L i s t a a  = -e w(n-l)/y  2  0  + j  = conductor radius  i  b  = 1/r  2  = n- -  jd/e to 0  b^  = horizontal conductor spacing  c  = Euler's constant = 0.57722  C,C^^  = capacitance matrix, elements  E,E  = e l e c t r i c f i e l d intensity (vector, components)  G, Gr^^  = conductance matrix, elements  H, H  = magnetic f i e l d intensity (vector, components)  h  = conductor height  1^  = phasor current  I, 1  = phasor current vector (mode, l i n e )  i  = current vector  J,J  = current density (vector, components)  J" (x)  = Bessel function of the f i r s t kind of order n  K,]^  = potential c o e f f i c i e n t matrix, elements  k  = propagation constant - jm  i  n  L,L ^  = inductance matrix, elements  m  = propagation constant  i  m  o' l m  =  M. ., IT. . ^ 3  P °P Sation constant ( i n a i r , i n the ground) r  a  = r e a l and imaginary parts of the correction terms f o r the elements of the potential c o e f f i c i e n t s matrix  n  = r e l a t i v e ground permittivity  P. .,Q. . ^  = r e a l and imaginary parts of the correction terms f o r the elements of the impedance matrix  R  = r a d i a l distance i n spherical coordinates viii  r a d i a l distance from the current element r a d i a l distance from the image of the current element resistance matrix, elements r a d i a l distance i n c y l i n d r i c a l coordinates distance from conductor i to the image of conductor j distance from conductor i to conductor j elemental length quantity defined i n Appendix B phasor voltage phasor voltage vectors (mode, l i n e ) voltage vector modal v e l o c i t y admittance matrix, elements impedance matrix, elements internal conductor impedance attenuation constant phase constant 2  propagation constant = propagation constant  u  2  + m  ( i n a i r , i n the ground)  complex permittivity permittivity (of free space, of the ground) magnetic permeability of free space Hertz vector, components r a d i a l distance i n c y l i n d r i c a l coordinates r a d i a l distance from conductor 1 r a d i a l distance from the image of conductor conductivity  ix  r  2  = m /m 2  2  0  = scalar function of position  co  = angular frequency  Y  = e  c  = 1.7811  x  ACKNOWLEDGEMENT The author would l i k e to express h i s gratitude to his supervising professor, Dr. Y.N. Yu who guided and inspired him throughout the course of the research. The author also wishes to thank members of the Department of E l e c t r i c a l Engineering and especially Dr. M.M.Z. Kharadly f o r h i s help and advice and Mr. H.R. Chinn f o r the use of h i s computer program i n p l o t t i n g curves. The author i s indebted to the University of B r i t i s h Columbia, the National Research Council of Canada and the B r i t i s h American O i l Company f o r f i n a n c i a l support of the research. Thanks are due to Mr. A. MacKenzie f o r drawing the graphs and to Miss B. Rydberg f o r typing the thesis.  xi  1 THE EFFECT OF GROUND CONDUCTIVITY AND PERMITTIVITY ON THE MODE PROPAGATION CONSTANTS OF AN OVERHEAD TRANSMISSION LINE. Chapter 1, Introduction. The advent of extra-high-voltage transmission has renewed the interest i n the study of energy propagation along transmission l i n e s .  These studies are associated with f a u l t  current calculations, system s t a b i l i t y , switching and r e s t r i k ing overvoltages, and the propagation characteristics of c a r r i e r waves along power l i n e s .  A necessary prerequisite f o r i n v e s t i -  gations of the above problems i s a thorough knowledge of the multiconductor transmission l i n e distributed c i r c u i t  parameters.  Although these parameters may be calculated from the conductor conductivity, the frequency and the geometry of the system, they are also dependent on the ground conductivity, permittivity, and permeability. While an exact calculation of the parameters i s impossible, due to i r r e g u l a r i t i e s i n the shape of the ground surface and the lack of uniform conductivity, a f a i r l y accurate calculation i s possible by replacing the ground with a p l a i n or multi-layer homogeneous media.  After the distributed  c i r c u i t parameters have been found the propagation problem i s solved from the transmission l i n e equations by meeting the boundary conditions. The f i r s t important engineering study of the effect of ground with f i n i t e conductivity on the electromagnetic propagation produced by current-carrying conductors above the ground i s due to J.R. C a r s o n ^ ^ ^ . 1  2  Applying electromagnetic  2 f i e l d theory, he calculated the f i e l d due to an alternating current i n a straight i n f i n i t e l y long wire above and p a r a l l e l to a p l a i n and e l e c t r i c a l l y homogeneous ground.  The derivation  contains four basic assumptionsj 1.  The ground r e l a t i v e permeability i s unity.  2.  The ground i s e l e c t r i c a l l y homogeneous.  3.  The polarization currents may be neglected.  4.  The current i s propagated without attenuation at the speed of l i g h t . The f i r s t three assumptions greatly simplify the  results.  The  f o u r t h assumption implies reasonably  efficient  energy transmission, as indicated i n reference (2). same publication Carson derived resistance and correction f a c t o r s .  In the  inductance  They are v a l i d only f o r frequencies  below 10 kHz because of the  omission  of the ground r e l a t i v e  permittivity i n the derivation.  (6) Rudenberg^ ' analysed the same problem.  He used a  model of the transmission system which has a semi-circular ground surface, with the axis coincidental with that of a conduotor.  In addition to Carson's assumptions he assumed  that the current d i s t r i b u t i o n i n the ground i s a function of the r a d i a l distance from the conductor. inductance correction terms are expressed  The resistance and i n terms of the  zero and f i r s t order Bessel functions of the t h i r d kind. low frequencies these functions have small arguments.  At  Using  only the f i r s t term of the power series expansions of the Bessel functions similar results to those given hy Carson are obtained.  Rudenberg did not derive the mutual inductance correction terms and he neglected the ground p e r m i t t i v i t y e f f e c t s . Carson"s results were extended hy W i s e ^ ^ ^ ^ " ^ . 1  1  His f i r s t paper removed the r e s t r i c t i o n of assuming a ground r e l a t i v e permeability  of unity.  His subsequent works  included the effect of polarization currents, gave correction terms f o r the l i n e admittance, and extended Carson*s impedance correction terms to a broader frequency range,  Carson's (7)  impedance correction terms were also extended by S u n d e  v,/  to  include the effect of a multi-layer ground. The investigation of multi-conductor using matrices, was  started by B e w l e y ^ .  transmission,  His method  was  f i r s t to consider an i d e a l l o s s l e s s l i n e and then to expand (3) (20)' followed the analysis to include losses. R i c e ' and Pipes w  v  Bewley's approach but added the use of Laplace transform methods. A major contribution has been the work of H a y a s h i extended analysis to include transient phenomena. developed new  w /  who  He also  techniques i n matrix calculus to f a c i l i t a t e the  solutions, for example his extension of Sylvester's expansion theorem.  Although h i s analysis included conductor skin effects  at higher frequencies, he used ground impedance correction from Rudenberg s model which neglected ground permittivity e f f e c t s , 5  A systematic mathematical procedure f o r handling the multiconductor transmission Wedepohl  v  .  In two  l i n e problem was  recently published  subsequent publications  v  M  ;  by  he gave  numerical procedures including the correction of the impedance parameters following Wise's methods,  A matrix analysis of  4 multi-conductor conditions  t r a n s m i s s i o n systems w i t h v a r i o u s  was d o n e h y  w o r k was r e s t r i c t e d  Dowdeswell  to  steady  in his  state  thesis,  analysis  at  boundary however  power  his  frequency  x  (23)  (12) Arismunandar terms w i t h surface, for  the  surge  This distributed  also  assumption that  f r o m w h i c h he  switching  tivity  '  the  calculated  admittance  lines  are  correction  close  characteristic  to  derives  correction  parameters  and r e l a t i v e  factors  i n c l u d i n g the  permittivity  of  the  for  the  effects  ground,  of  of  a multi-conductor  transmission line.  of  the  system and t h e i n Chapter  a current-carrying i n Chapter frequency  Chapter  3*  effects  of  derivation  (7) v  .  equation, Chapter 500  6.  5, f r o m t h e  the  follows The  circuit  relative closely  detailed  circuit  that  of  solution  distributed  kV t r a n s m i s s i o n l i n e  field  for  the  various  valid  for  include  Wise^  1 1  (a)  ,  in  parameters  transmission  v  the  circuit  is  parameters  ^  in  the  and c o n d u c t i v i t y .  s o l u t i o n  cirouit  derived  solution given  C a r s o n ^ ,  of  is  are derived  parameters  permittivity  surrounding  distributed  admittance  electromagnetic  constants  description  field  transmission line  f o l l o w i n g Wedepohl's The  The  a  derivation  a p l a i n ground  1 MHz.  These d i s t r i b u t e d ground  the  an E-type Hertz vector,  4 and the d i s t r i b u t e d  i n Chapter  Sunde  above  f r o m 10 H z t o for  in  electromagnetic  conductor  impedance parameters Chapter  The  2.  3 i n terms of range  A general  assumptions required  conduc-  and g i v e s  c a l c u l a t i n g t h e mode p r o p a g a t i o n  given  ground  impedances  s y s t e m a t i c method o f  are  the  studies.  thesis  circuit  derived  The  and  line  given  in  for  typical  frequencies  a  and  ground  .  5 conditions graphs  are  giving  correction  tabulated  the magnitude  factors  over the  ground c o n d i t i o n s ,  transmission line  are  system of u n i t s  specified.  of  7.  Chapter  the d i s t r i b u t e d  dltated f r e q u e n c y  A detailed description  with numerical results,  MES  i n Chapter  of  circuit  range the  t h e wave e q u a t i o n f o r  included  i s used  of  7 also  i n Chapter  throughout the  8.  for  includes parameter various  solution, the  The  thesis,  typical  rationalized unless  6  Chapter 2 . 2.1  Assumptions and Approaches.  General Description of the System. The system under study consists of a number of  separate conductors and ground return l i n e s situated above and p a r a l l e l to a plane ground.  No geometric symmetry i s assumed  i n the transverse plane. The transmission l i n e equations f o r a l i n e with n conductors may be written i n terms of the distributed c i r c u i t parameters i n the general form as  " • " § 2  =  R  i  +  L  2-1  T t  =  (  R  +  L  p  )  i  =  Z  i  - - | ^ = G v + C - | ^ = ( G + Cp)v = Yv  2-2  where Z and Y are n x n symmetric matrices, and v and i column matrices with n elements.  Each element of the Z and Y matrices  includes the effect of ground return (4) and i s i l l u s t r a t e d i n the  system shown i n f i g , 2 - 1 ,  Conductor i  R..  L.. TW^  AA>V^  Conductor j J^j  L. . nrr«V  ^  Pig. 2-1 Distributed C i r c u i t Parameters of an n-conductor Line.  7 Basic  2.2  Assumptions. The  permittivity line  on t h e  voltage  circuit c a n he  field  circuit  is  field  concepts.  Hence the  the r e s u l t s  and  energy along a  c o n c e p t s a r e more p r a c t i c a l ,  development  The  In  engineering, current  problem i s the  relative  transmission  since  i n terms of  consists  from t h i s  of  the  analysis  extension  the  to  and  analyse  distributed  in this  thesis  electromagnetic f i e l d primarily the  other v e r t i c a l l y  describe  the  ground.  In  It  transmission  departure  A Hertz  field. of  the  some  ground,  the  concepts  algebraically.  TM mode w i t h  i n the  grounds  a n y new  surrounding  i n the d i r e c t i o n  into  s o l u t i o n the  the  losses  i s used to  one o r i e n t e d  of  require  the  permeability  theory to m u l t i - l a y e r  p e r m e a b i l i t i e s , does not  due t o  type  of  becomes much more i n v o l v e d  The  field  conductivity  c o n s i d e r e d homogeneous w i t h a r e l a t i v e  although i t  ents,  of  e a s i l y measured.  with different  the  ground  parameters.  of u n i t y .  line  finite  i n terms  and e x p r e s s  In ground  of  transmission of  c a n he d e r i v e d  however,  the  effect  vector  h a s two  compon-  p r o p a g a t i o n and the  the  construction  f o l l o w i n g assumptions are  of  of  the  made:  p 1,  At  the  less  frequency  than unity.  frequencies 2,  In  the  practical the  This  s o l u t i o n of  the  transmission  is  assumption i s v a l i d f r o m 1 0 Hz t o  wave  considered  purposes  W'ELI i n a i r  interest  i n the range  approximation i s  of  of  i n view line.  sufficiently the  for  1 MHz,  equation a f i r s t  of  much  order  accurate  physical  for  irregularities  8  3^  The  principle  complete Air  4. 2.3  Circuit The  assuming that ground  of  as the  sum o f  the  ground  of  superposition  solution for  conductivity  is  the  i s used to  find  the  system.  considered  negligible.  Parameters. ideal circuit the  finite  ground  parameter i s  has  infinite  conductivity  the  the  This  circuit  analysis  line  over a ground  of  finite  line  over a ground  of  infinite  first  conductivity.  i d e a l parameter plus  condition.  derived  transforms  conductivity  For  parameter i s  a correction  to  conductivity.  an  an  by a  expressed  term  for  n-conductor  n-conductor  9  Chapter 3.  Derivation of the Hertz Vector due to Current i n a Straight Wire. The Hertz vector f o r a ourrent along an i n f i n i t e  straight wire p a r a l l e l to a f l a t ground i s obtained hy integrating the elemental Hertz vectors f o r current elements along the wire from minus i n f i n i t y to plus i n f i n i t y .  F i g . 3-1  Current Element Orientation.  X  F i g . 3-2  C y l i n d r i c a l Coordinate System f o r the External F i e l d .  10 3-1  Hertz Vector due to a Current Element In a Medium of I n f i n i t e Extent. The Hertz vector due to a current element Idz i n an  isotropic homogeneous medium of i n f i n i t e extent i s derived i n Appendix A, and i s given hy \  = 0  3-1  TV  = 0  3-2  -rr  X 3-2  Ie  dz  0  = 4*m R  5-4  2  Q  Hertz Vector due to a Current Element above a Ground with I n f i n i t e Conductivity. The t o t a l f i e l d above ground due to a current element  and i t s image, as shown i n f i g . 3-1,  consists of a primary f i e l d ,  plus a secondary f i e l d due to the f i n i t e conductivity of the ground.  Let the t o t a l f i e l d be z  1 where "IT vector.  QZ  ,l  oz  M  oz  J  2 i s the primary f i e l d , and"!T  oz  J  i s the secondary f i e l d  The s u f f i x "o" designates the f i e l d vector above the  ground surface. Prom equation 3-4 we have ITI ^  _ -Itouldz 4^  =  where R-^ and R  2  -m R.. e 0  R  l  -m  e  1  "  0  R  R,  *  2  , , 3  are the distances from the point of interest  to the current element and i t s image respectively.  11 Hertz  3-3  Vector  Finite  due t o  ground  has f i n i t e  be d i s s i p a t e d , and t h e  two H e r t z  vector  gives  the  fig.  3-2.  However,  a.  with  conditions The  in  the  particular  coordinate  energy  at  of  Equation  system  s o l u t i o n must meet  w h i c h may b e s t a t e d a s  f i e l d vanishes  some  the y and z d i r e c t i o n s .  general solution i n the  conductivity  f i e l d may b e d e s c r i b e d b y means  components,  B-15  boundary  Element above a Ground  Conductivity.  When t h e will  a Current  of  the  follows:  an i n f i n i t e  distance from  the  source. b.  At  c.  A further  the  source  o n l y 7T  exists  Z  the  current  element i s  The  t a n g e n t i a l E and H f i e l d s  the  ground  and i s  orientated  finite,  i n the  are  z  since direction.  continuous  across  surface.  restriction  is  imposed due t o  the  geometry  of  the  system; d.  The The  fields  form of  f r o m e q u a t i o n B-15, restrictions. the  first  y  »  (a)  B,  and  zero  by a p p l y i n g the  the  only  plane. constructed  above  o r d e r f u n c t i o n b u t TT  cannot.  only of  functions.  h may now b e g i v e n  by  in  the  P is let The  cos P  only.  l e a d i n g term of of  significance.  the vector Hertz  four  Bessel functions and due  e q u a t i o n 3-6  two  y-z  solution,  possible values  sum o f  the  f u n c t i o n s may b e  (b)  can c o n t a i n terms  accordance with be t h e  Appendix  approximation that  over the  field  kind can appear i n  solutions  first  s y m m e t r i c a l about  the  Due t o  can contain the the  are  to  for  (b)  IT  Due t o  (d)  Assume a s the  the  a  summation  Finally  above the  vector  of  in  ground region  12 TT  0Z  =J~[f ( o  e  u )  0  ( y  "  h )  *o  +  g ( ) - ^ o ] J (ru) Q  u  e  o  eo  ir = j  y  x  u  3-7 3-8  g (u) e * l J ( r u ) du  iz  d  Q  o TT  TT  oy  =J  c o s V p (u) e~ *o  ly  o oo = J" cos V p-^u) e  y  0  l  y y  J ^ r u ) du  3-9  J ^ r u ) du  3-10  where the s u f f i x "1" designates the f i e l d below the ground surface. f» Q  SQt  There now remains the evaluation of the functions g±t P  0  and p  1  using condition (c).  Condition (c)  may be stated i n terms of the Hertz vector components by means ,(7) of equations A-10 and A-3 as follows'  a  ay  OZ  o 7x  »  '.fig0 z  +  H>v  +  ^ o z  d y  0 z  3-11  +  • ""ft,  +  3-12 m  "37 2 ^TT,0 2 — = o T  0 m2  i  z  = m.  T  Equation 3-11 may be written as  integrating both sides yields  3-13  3.14  ^ 1 *  13  However, « and o  are zero when x becomes i n f i n i t e and hence  1  f(z,y,u) i s zero.  Integrating both sides of equation 3-14  yields  "o^oy Again both 71^  ""l^ly  =  S( »y' )  +  z  u  and Tl^y are zero when x becomes i n f i n i t e , hence  g(z,y,u) i s zero.  With these results equations 3-11 through  3-14 reduce to the following equations  TF m  +  T^  iy  -  +  T^  5  o TToz = m 1 TT, lz 2  3-16  2  m  o T  m  o ^oy =  =  F  m  m  5  y  i T F "  3  l ""iy  "  1  7  5-18  The various f i e l d functions may now be evaluated i n terms of the function f ( u ) they are Q  « o  (  u  =  )  * - X, F* 0 T + X*: o ^ > f  1  2uY r  -hr 5-19  e  -h*  2  o  Si  7 ^  = v °+  ( u )  f  o  ( u ) e  "  °  5-20  2uy (i - r )  -hr  2  n  p ( ) u  =  2  f  ( ) e  0  u  3-21  14  P l  r  (u) =  2  P (u)  3-22  0  where T  = m /m 2  2  3-23  2  The function f (u-) may he determined hy l e t t i n g Q  = X and m = m , the solution must then he the same as l o l o for a uniform medium of i n f i n i t e extent. Then n  -m R-, f (u) e  -  ( y  h )  * o j (ru) du = ^  %  T  °  3-24  d z  4**1*1 Then hy transforming the left-hand-side of equation 3-24 to the same o r i g i n as the right-hand-side and l e t t i n g f (u) = °  B  -——  3-25  °o  where B i s a constant, and hy making the substitution *  = u  2  + m  2  3-26  2  equation 3-29 becomes ? [ B  v  e"  y  /v *°  -m RJ  O  (  R  / » J  -  N  | ,  *  A  O  -  M  o  4*  Assuming that m  Q  with ¥  f  ^  3  m  Q  -  2  7  Rx  i s a small quantity i n a i r compared  and r e s t r i c t i n g the solution to frequencies below  1 MHz equation 3-27 becomes oo  [ B e~ '*° J ( r X ) d* o o o o y  which i s Lipschitz's integral  =  -Mi^z 2 4 f l ; m  (1Q)  .  R i  Hence  3  _  2 Q  15 u  3-29  Using the same reasoning -m R .jcoiile dz 4«m R  -(y+h)v ) e  2  2  3-30  J ( r u ) du  0  Q  2  Then comhining equation 3-30 with 3-19 3-7  part of equation  f  -y . " o* o  -m  .jioule 4«m R  y  g ( )  e  u  Q  to obtain the second  (  J 0  r u  )  = ~  d u  R.  dz  2  2  o  r  2 y  J  +  f  The l a s t term of 3-31  °  (  u  )  e  3-31  J (ru) du  corresponds to the secondary f i e l d  due  to f i n i t e ground conductivity  J  'z  y  J  =  +  1  V  o  f  0  ( u )  e  d  3-32  »  Then the complete Hertz vector i n the region 0— y ^  = 0  7T  ox  ~  TT  i  o y  ff '  = o z  -itouldz 4«m o  2  C  0  3-33  y i M ( i -  S  4«m  - o l m  e R.  R  h is  2 (  v  0  r  0  V e  -(y+h)tf  y )(r V 2  i  +  o  r  " o 2 m  R  e R.  J  1  ~  +  0  Y  O  J ( r u )  )  1  3-34  (y+h)y l:  2ue  177  T17  du  J (ru)du o  d  3-35  The current at any point z on the l i n e i s given hy - tfz I = I e 3-36  16 where  i s the propagation constant i n the z d i r e c t i o n .  Noting that - cos y equations 3-34  uJ-^ru) =  and 3-35  0  become  - *Z  dz  o  TT  4«m  oz  f  Tz dz  r  4itm  ^  o  j toil I e o 0  3-4  c*?  2jco|il e  3-37  J (ru)  e  ?  u ( l - r (ir 0  J  ^  -(y+h)tf ) e " ^ ° °  2  + ^  e" o m  R 2  H  r  2  ,f2ue-  ^  ^+  ( y + h ) y  ,  T  ,  V ) o 3-38 J  (  r  u  du )  0  o  _  J (ru)du Q  3-39  Complete Hertz Vector f o r a Current i n a S t r a i g h t Conductor above Ground The complete Hertz v e c t o r f o r the c u r r e n t - c a r r y i n g  conductor i s now  obtained from i n t e g r a t i o n of the Hertz v e c t o r s  due to the c u r r e n t elements a l o n g the l i n e from minus i n f i n i t y to plus i n f i n i t y .  TT  The components o f the Hertz v e c t o r are  = o  3-40 co  I  4«m,  jwul e  r  r  u(l -  2  g  -(y+b)^r  )  e  J(ru)dudz  3-41  OA  I  -tfz  4«m, -  -tfz L  e  -m R 0  -m  n  e  1  R  0  n  > R,  R~  2  CO  ~  1  "j  -(y+h)ro  OO  +  o  ^  +  y0  J  o  (  n  i  )  d  u  J  dz  3-42  17 Chapter 4.  4.1  Evaluation of Transmission Line Resistance and Inductance Including Ground Effects.  Derivation of the Series Impedance Formulae.  onductor j  i II t I I I  III  Ground  F i g . 4-1  Two-Conductor Configuration.  F i g . 4-1 shows two t y p i c a l conductors of an n-conductor transmission system.  The t o t a l e l e c t r i c f i e l d on the surface  of conductor i i s the sum of the e l e c t r i c f i e l d s due to a l l n conductors.  From equation A-10 the e l e c t r i c f i e l d on the i - t h  conductor due to the j-th conductor i n the z direction i s given hy E. 1Z  = -m  2  Tf. + HrO ']Z £z  by  '  iz  4-1  Then the complete e l e c t r i c f i e l d on the surface of conductor i due to a l l conductors i s n  E.i z = Z_i Y 7  -m fr, o jz 2  +  4-  dv, l y  V"  &z  4 - 2  18 The f i e l d s inside and outside conductor i must he equal at the boundary.  I f the internal impedance of conductor i i s  then  n z.I. e  " o^z m  10  1  1=1  +  oy  "Ji  where z^ i s derived i n Appendix C.  oz _  4-3  The second term on the  right-hand-side of equation 4-1 may be expressed as a gradient of the scalar potential, V^, where  C iL ["HZ IZLfl 7  =  4-4  /LA ^z|_^y J i=l Prom the general transmission l i n e equation E-8 the voltage +  Z  equation i n phasor form can be written  & 3  13  J=i  4-5  Jo  Hence by combining equations 4-3 and 4-5 the following formulae can be established for the s e l f and mutual impedances, respectively  ;  Z. . = z. + m ix i o  Z„ . =  iO  where j =1,  2,  TT.  iz  ho*  4-6  z  om"*^ jz  4-7  2  n except i , and a l l functions are evaluated  on the surface of conductor i . 4.2  Evaluation of Z ^  and  Z^.  The evaluation of these impedances requires the evaluation of the i n f i n i t e integrals i n the T T function given Z  19 by equation 3-42.  To evaluate TTz the assumption must be made  that attenuation along the l i n e i s negligible, then * = o m  where  4-8 = J /e^to  m 0  = jk  2  This i s an i d e a l value f o r *X but i s necessary due to the following two considerations: 1.  To assume that the attenuation i s not zero on an i n f i n i t e l y long l i n e amounts to assuming a source of i n f i n i t e energy and makes the integral i n f i n i t e .  2.  To assume a propagation v e l o c i t y less than that of l i g h t makes the integral extremely d i f f i c u l t to evaluate. Substituting jk f o r m  Q  oo  Ifz = Ioe ^  k  e^  z  J - oo  /  k  J  2ue  _ 4*m \  l  -jkR  e,  R  2  2  )  n  2  where  R  - tf (h+y) J (ru)du  •icou o 4«m * o ^1 o ° From equation 4-6 we have ii  i n equation 3-42  -jkR,  •1cou ( e  z  2  oo  z  and  v  dz  4-9  ;  +  = ± z  T  +  h  +  j  r  2  a  -jkR-  -jkR n 2  '  1  0  20 To evaluate 1^, l e t l  P  P  =  l  R  +  = R  2  z  4-13 + z  2  where R  l  \P\  =  +  z  *  a  n  d  R  = /"I * + B  2  'ij  v  2 "d  " i  4-14 where b ^ i s the horizontal distance between the i t h and j t h conductors, assuming this distance to be much greater than the radius of conductor i , h^ i s the height of the i t h conductor above ground and h^ that of j t h conductor.  The general case  i s considered because i t i s required i n the evaluation of Z-^y When J equals i , b ^ becomes the radius of conductor i , and Pi. = where a  i  a  i '  ^2 =  2  i'  h  i s the radius of conductor i , and the height h^ i s  assumed to be much greater than the radius a^. With the change of variable from z to  and P  2  we  have P dP-L = ^ : dz, and P-j^ p  oo  !  a  !  2  ,P f  P  dP = <*o  2  "  p 2  ±  = -1 ,  a  g  =  W  _|  dz  4-16  when z  2  P  2  where a  P  g|  h  e  n  Z  ~~  *-~ <*> "  *°  2 4  _  1 7  21 for a very large z.  T x  -  l  ~  Then  lW  a  4*  ±  l  4*  e  j  '  dP,  4-18  «2  where the integrals are defined as the exponential integrals (18) Hence equation 4-18 can he written as T x  -  l  "  ^  li(e  m  4*  ') - l i ( e  where  )  4-19  oo  -lKe"*) = c + ln(t) +  n=l and c i s Euler's constant.  n ] n  Por a small value of t the f i r s t  two terms i n the above expression predominate and a l l other terms can be neglected,' then  -JM "  in  2*  -2/>  4-20  ±  Equation 4-20 gives the s e l f reactance per unit length of l i n e conductor when the ground has i n f i n i t e conductivity. To evaluate I changed, and I  2  l e t the order of integration be  becomes  2  -r o  (h+y) - oo  Since J ( r u ) and cos(kz) are even functions, and sin(kz) i s Q  an odd function of kz  22 oo  oo  JV^  J(ru)dz = 2 Jcos(kz) J(ru)dz  2  Q  Q  — OO  o = 0 , =  k> \/x?  2cos(x  2cos V  Q  T  u  - k  x '  T o  Hence f o r k—  u 2  )  k ^  u  4-22  ^ (h y) 0  +  COS  ue  - iML 2 " at  5 X Q  4-23  du  To change the lower l i m i t o f the above i n t e g r a t i o n ,  X . = u - *, du - Isgla 2  2  2  then =  .1cou  ?t  cos  o 0 ,i X  dV  J /y (  _ ML  -  y x -^ y ) e  (  h  +  y  )  0  0  V :cos g.x  It To f i n d r e s u l t s , s i m i l a r to Carson's but the  substitute  d*  4-24  including  e f f e c t of r e l a t i v e ground p e r m i t t i v i t y , i n accordance w i t h  referenoe (9) l e t s' = 1 -  je  w(n-l) 4-25  since 2  y 1  and y 2 0  O  = u  2  + m  2  = u  2  + jcofid —  = u - e aw 2  2  0  enw' 1  23 hence *1  " *o  1^<4  =  B  4-26  2  Further l e t y' =  /wJI3y,  h" = /co|i<3'h,  x' = A>u<£x,  v =  . 4-27  oo  Then  I  = ^ [  2  *  (/v - i s 2  - v) e ^ ' + y ^ o o s x'v dv  2  o  s  4^28  Finally l e t y  M  = sy',  then  x" = sx'  h" = ah',  v = su  f  4-29  oo 2  T  = f  f  +  o  ^  -  u  )  e-  u ( h  " y +  J , )  cos  x»u du  4-30  Equation 4-30 i s the same equation as g i v e n hy (2) Carson  v  factor).  except f o r s (the ground p e r m i t t i v i t y  correction  In h i s case s = 1 because he assumed that  the r e l a t i v e  p e r m i t t i v i t y o f the ground equals one. Then and Z  ii =  z  i  +  I f ^ fj* + f <ii r  +  »il>  4-32  The l a s t term o f equation 4-32 i s the c o r r e c t i o n f a c t o r f o r the f i n i t e ground c o n d u c t i v i t y tivity.  The exact s o l u t i o n s  for  and  and r e l a t i v e permitare given i n  Appendix D, and numerical r e s u l t s are included  i n chaper 7.  U s i n g the same procedure as above, we have f o r Z. . X J  24  4-33  where S*^ i s the distance from conductor i to the image of conductor j , and conductor  i s the distance from conductor i to  25 Chapter 5« 5.1  Evaluation of the Shunt Admittance  Derivation of Shunt Admittance Formulae. Prom equation 2-1 the transmission l i n e equation of  the n-conductor system can he written C,, V,  5-1  d=l where i = 1,2,3,4,  n.  Making the same assumptions as i n equation 4-8, and noting that  = -;jk, d i f f e r e n t i a t i o n yields n  ^  C^Vj = J I  i = 1,2,(3,4,)  ±  ^  n. 5-2  Then solving t h i s system of equations f o r the voltages, we have n V  i =S H  1  = 1.2.:..., n.  5-3  d=i where K. . i s Maxwell's potential c o e f f i c i e n t .  Substituting  these results into equation 4-4 yields  2>1T  v i  k  £ K I, = w i J  bTT  iz  oy  hence "  k  I  i  L^y  J  Thus the potential c o e f f i c i e n t s can be derived from the f i e l d components.  26 5.2  E v a l u a t i o n o f the Maxwell P o t e n t i a l C o e f f i c i e n t s K  1 J L  and K ^ j . The  same assumptions are made here r e g a r d i n g the  e v a l u a t i o n of the i n f i n i t e i n t e g r a l s as s t a t e d i n the preceding chapter f o r equation 4-8. Prom equations 5-4, 3-41 and 3-42 5-5  where  oo  In = 1 4«e  e  o  " 4«e  -jkR e R 2  oo  F 2  -jkR,  2  J  0  1  *  J ( r u ) dudz  +  0  5-7  Q  O  »  u ( l - r )e )e f -jkz d Cu(l-r  *  I,3 = 2«e  5-6  -(y+h)*,  s  J  — oO  J  dz  —rr -r-— \ e  P  22  b  -rr \  J  -(y+h)Y  ° °  J (ru; dudz  dzj v v 2y v, o> JL O -L O have been evaluated i n chapter 4. j k d y  0  J  (  +  )  (  Y  o 5-8  }  — OO  1^ and I I.  i  + I  2  =  - 2 5 T l n^ T  — 2  n  l  n  ^ T 7  +  '1  o  (  P  =  +  o  2we  L  Q  T r ^ O A,,**  """/^ ' -  -  7te  +  N  ^  P  )  5-9 1^ i s evaluated as f o l l o w s ~ I,  = 3 " 2*e  v  ^ e  o^ J o  (JT, + y ) ( r V ^ )  .! o^  2  n  -1  -  0  oo  2 3 t e  -(y+h)tf -(y+h)tf  2  -  1  0  2  J  ) X  O  X  O  d_ j dz o (  r  u  )  d  z  d  u  - CO  5-10 OO  e  o  r\  J  -(y+h)tf  ruy (i-r )e k  -jkz  _ ; ) k z  oo  J ( r u ) l + \h j^k e- "* ^ J ( r u ) d z |du 1  0  J  k  z  o  -oo-oo  5-11  27  Since the f i r s t term i n the bracket i s zero and the second term has been evaluated i n equation 4-22, equation 5-11 becomes oo  ,  I  5—  = - - J -  Noting that  -(y+h) "X  9  r u(l-*r) e  ^  °cos O x  2_ d u  Y ; = u- - k ,, udu = Y 2  "  and  2  2  0  Y  2  -Y r^  Q  5-12  o  v  =  0  0  rv2  ~ 2  . v  X + X^Y  *i+* )  Q  0  the i n t e g r a l breaks into two parts, and  T  _  1  o  " ^T -^ ( Q  P )  +  n T - ^ ^  I  0  "0  where  .  2  ^? -(y+h)"X  J Then  y  K  ii  -  +  sbr "4 2h  0  1  o d  + ^-(M+dN) o  JLy°_0 r2  1 I  5-13  X X  COS  ln H  1  and  v  J  0  d  y  o  o  5  "  14  1 +  =fc  ( M  ii  + ; ) N  ii  )  5  -  1  5  0  The' l a s t terms i n equations 5-15 and 5-16 give the correction factors f o r f i n i t e ground conductivity and r e l a t i v e permittivity to Maxwell's potential c o e f f i c i e n t s .  An  approximation of M and N i s given i n section 5.3 of t h i s chapter and the numerical results are included i n chapter 7 .  28 5.3  Evaluation of M.. and N . . 3J LL Making the same substitutions as i n 4-25, 4-26 and  4-27, equation 5-14 becomes oo  °J g  M + jN =  A A  o o  2 2 Next l e t j s = a ,  M +  1  f*  =  3f"T  + 38 + 38  = b,  "  5-17  d v  V2  then  (12)  -v(h'+y') _, K g dv j/ v + a + bv coa c o s x v  v  5-18  V  J  The numerical computation of t h i s i n t e g r a l i s involved due to the large number of variables.  An approximate  solution w i l l be developed below. For  large values of v the integrand vanishes, hence  5-18 may be written as v_ M +. 3N ,-ivr M = \~ J  i.1g e"* g^<' v  Jv  o »  h  + a  5-19r-  ^cosx'v-,,, dv + bv +yt  i n  If within the range of values from 0 to v , Q  I  2av| < |v + a  I  2  5-20  then f rv- •*~ Q^2  2  +  a  v  +  a• ~- ^ _  5  _  2 1  The limitations of t h i s assumption w i l l be discussed i n chapter 7.  Substituting 5-21 into 5-19 ( H )  29  oo M a. HIM  e" ^^\hosxW,  f (v + a)  ^  J  Y  (v + r-,)(v + r ) ( b + l )  d  v  2  O  5-22 where r  i  - f  (  1  -/^rfi^  r  2 = f  (  1  +  /^ I HH T  =  )  5-23 Next l e t g  ±  = h? + y' - dx' = loirs' He'* = R ' e ^  g  2  = h  6  9  5-24 + y  1  + Jx' = f^3  1  Re  = R'e^  3 9  6  co M + jlT =  2  (  b  ^ ( r  2  -  r  i  )  [  * ^  o  ]  [  e  "  S  l  T  +  e  "  S  2  ]  V  d  v  5-25 Equation 5-25 includes four separate integrals of similar form. Let CO  [  - S iS l v  1 f e~ d v 1 = 2(h l)(r -r ) J v T r 7 +  2  5  1  ~  2 6  and v + g^r,  - r  l  5-27  oo ~ 2 f = 2(h+l)(r -r ) J " T " r  then X  = w g-jT,  w  e  1  e  2  1  1  " 2 / " l l " 2(b l)(r -r ) r  d  w  e  s  r  l i ( e  +  2  1  5-28  The other three integrals i n 5-25 have similar solutions. Therefore  30  M  +  J* = 2 ( b l ) ( r - r ) p e +  2  1  2  g i r i  li(e"  S i r i  -r^^liCe"* * ) + ^ e ^ l K e " " 2  1  )) +r r , e +  2  '  2  x  1  2  li(e  ) ]  1  2  )  5-29  The numerical r e s u l t s o f the e v a l u a t i o n o f M and N as given hy equation 5 - 2 9 a r e i n c l u d e d i n chapter 7 .  31 Chapter 6.  6.1  Derivation of Propagation Constants and A.B.C.D Parameters of an n-conductor Transmission System.  Calculation of the Propagation Constants. The transmission l i n e equations f o r an n-conductor (8)  system may he written i n matrix and phasor form as follows I V  = Z(u) Y(ii)) V "  2  —  6-1 p  = T ( u ) Z(u) I  6-2  where V and I are column vectors of n terms and Z and Y P P are n x n symmetric.  matrices.  The suffix p denotes the  l i n e coordinate system. The systems of second order d i f f e r e n t i a l equations represented hy equations 6-1 and 6-2 can he solved hy transforming them into a new coordinate system wherein the transformed ZY or YZ matrix becomes a diagonal matrix. Then equations 6-1 and 6-2 become  * i 2  c  w  2.  where the s u f f i x c denotes the mode coordinate system. The eigenvalues of the system may be denoted by ^  , 1 = 1,2,3, • ••• n, and are obtained from | ZY - I  2  | = 0  or  2  [YZ - IY  1= 0  6—5  32 since  (ZY)  t  = Y Z t  t  = YZ  Here I i s a unit diagonal matrix.  The solutions of equations  6-3 and 6-4 are V  Xi  i  C, + e  A  1  = e * "i D " + e^ Z  I where Y  = e  J  1  C  0  6-6  2  i £> Z  6-7  2  represents the n propagation constants i n the n-modes  of the system.  The solution i n the l i n e coordinate system,  designated hy the suffix p, i s now determined 7 = 5 7 p c  and  as follows  I = S I p c  6-8  where R and S represent n x n transformation matrices, each consisting of n columns of eigenvectors of the system. eigenvector i s determined  Each  from the corresponding eigenvalue  hy means of the following equations (ZY - tf I) R(i) = 0  6-9  (YZ - ) f  6-10  2  2  I) S ( i ) = 0  where R(i) i s the i t h column of the R matrix, and S(i) the i t h column of the S matrix. RS t  =  In general  = D  6-11  where D i s a diagonal matrix, where the elements may he complex. 6.2  Formation of the Impedance Matrix Z. The impedance matrix Z has two types of elements,  the diagonal and the off-diagonal elements. 'elements are given hy equation 4-32  The diagonal  and the off-diagonal elements  33 by equation 4-33*  ^ e t the d i a g o n a l  Z  i i  = i i R  +  ^  elements be  i i  L  6  ~  1 2  where R  i i -mt f i i s  +  p  " 5  where R ^ x , i s the i n t e r n a l AC r e s i s t a n c e of the conductor o r the r e a l part of z  i n equation 4-32, and " j f ^ F ^ i s the r e a l  i  part of the ground c o r r e c t i o n f a c t o r .  z  i  = int R  J  +  w L  Let  int  6  "  1 4  then 2h.  where ^ Q.^ i s the imaginary part of the ground c o r r e c t i o n factor. The  z  i  d  o f f - d i a g o n a l elements are g i v e n by equation 4-33  =  ^  l  ^ f <« *V  n  +  P  +  6  "  16  Now i f we w r i t e  then  hi • f hi  -  6  18  and  hi - h  • $hi  1 0  For. a l l o f f - d i a g o n a l terms  z  id  =  z  J i  6-20  34 6.3  Formation of the Admittance Matrix Y. The general form of the potential coefficients was  derived i n chapter 5.  There i s no d i s t i n c t i o n i n form between  diagonal and off-diagonal Elements.  If  i s written as K, . = K! .  ij  where  +  A  E  .  6-22  ,  ij  i s the correction term f o r the ground e f f e c t .  Then  the capacitance matrix of the transmission system may be written as C  T  - AC = (K + AK)" = Z'" - K'" AKK' T  1  1  1  -1  6-23  Defining  gives AC = C'AKC = —  C» (M + jN)0'  6-24  Hence the corrected capacitance matrix i s C = C  -  +  = C'(I - i-(M + jH)C»)  6-25  The admittance matrix Y can be written Y = jtoC = jwC'(I - ~ ( M + JN)C ) = G + jwC"  6-26  where G =  uO'HC'  6-27  and C" = C * (I - ~ M C ) ff£  6-28  35  6.4  The A.B.C.D Parameters of the System. By analogy to the single conductor case, the behaviour  of an n-conductor transmission l i n e can be described by the following equation; —  —  V  A  s  B  t 6-29  I  C  s_  D  I.  t_  where the parameters AjB,C and D are n x n matrices.  The  and Ig are voltages and currents at a  column vectors  distance IL from the sending end, and V and I those at the s s sending end.  The A,B,C,D and the characteristic impedance  matrices are derived i n reference (8) and are A = R(cosh Y Z ) R - 1 B =  [  ±  sinh-.  6-30  R Z  6-31  S- 1 !  6-32  _1  —  C = S  Tsinh: ¥ &~ ±  L  \  D = S(cosh Zo = R Xi T  1  .  X. JL ) S R Z _1  -1  6-33 6-34  36 Chapter 7. 7.1  Numerical R e s u l t s f o r a 500 kV T r a n s m i s s i o n L i n e .  D e s c r i p t i o n of the Transmission L i n e . The ground e f f e c t on the d i s t r i b u t e d c i r c u i t parameters  for  an overhead t r a n s m i s s i o n l i n e i s dependent on l i n e  geometry,  ground c o n d u c t i v i t y , ground r e l a t i v e p e r m i t t i v i t y , and frequency. The l i n e geometry can have a l a r g e number of v a r i a t i o n s .  In  order to i l l u s t r a t e the s i g n i f i c a n c e of the formulae developed i n Chapters 4, 5 and 6 a t y p i c a l 500 kV l i n e i s chosen. the  For  g i v e n geometry the ground c o n d u c t i v i t y , r e l a t i v e permit-  t i v i t y , and frequency are v a r i e d .  The 500 kV l i n e i s chosen  because recent developments i n l o n g d i s t a n c e energy t r a n s m i s s i o n have c a l l e d f o r more d e t a i l e d study of the p r o p e r t i e s of l i n e s at  t h i s v o l t a g e l e v e l than has p r e v i o u s l y been a v a i l a b l e .  The  s i n g l e c i r c u i t l i n e without overhead ground wires c o n s i s t s o f : A bundle of f o u r conductors a t the corners of an 18 i n c h square, per phase. Conductor s i z e  583.2 MCM  ACSR  Conductor DC r e s i s t a n c e ........ 0.1764 ohm/mile a t 50°C Conductor diameter ( i n c l u d i n g stranding factor)  0.948 i n  Average phase spacing  40 f t .  Average conductor h e i g h t ....... 54 f t . The l i n e conductors are at equal h e i g h t s above the ground i n a f l a t a r r a y .  The ground c o n d i t i o n s were considered  to v a r y from dry r o c k to wet marsh l a n d .  This represents a  ground r e l a t i v e p e r m i t t i v i t y range from 10 t o 50 and a ground  37 conductivity range from 10""'' to 1 mho/m. 7.2  Ideal Parameters. From equation  4-33 S'  I  _ Mi i n - l i  h/m  7-1  Inserting the lengths i n the ahove equation and converting the units to m i l l i - h e n r i e s per mile gives the following inductance matrix, not including internal conductor inductance  1.63  .339  .167  .339  1.63  .339  .167  .339  1.63  mh/mile  7-2  Inverting t h i s matrix and multiplying by a known constant gives the capacitance matrix i n units of micro-farads per mile  C  .0183  -.00302  -.00148  ^.00302  .0189  -.00302  -.00148  -.00302  .0183  u-f/mile  7-3 with the inductance i n henries per meter and the potential coefficients i n darafs per meter the matrices are,  L = 2 x 10"  K = 18 x 1 0  7  9  5.10  1.06  .520  1.06  5.10  1.06  _.520  1.06  5.10_  "5.10  1.06  .520~  1.06  5.10  1.066  .520  1.06  5.10  h/m  7-4  darafs/m  7-5  38 7.3  Skin E f f e c t . For  the  conductor w i t h a r a d i u s of  a bundle spacing of 18  i n , the  negligible.  The  taken as one  q u a r t e r of the  and  i n t e r n a l impedance per  from C-16,  C-17,  depending on the magnitude of  7.4  Impedance  or C-21,  impedance c o r r e c t i o n and  correction  terms depend on the  relative permittivity.  show the  the  impedance m a t r i x , a c c o r d i n g to the = I  E  of the  il  f a c t o r s f o r the  i  ±  = ii E  +  2Q  +  f  P  the  Figs. 7.3  formulae,  ™ The  conductivities  correction  factors  shown i n figs.; 7 . 3  i n v a l u e s between the elements i s not  through 7.6  show the  above 10~  n e g l i g i b l e e f f e c t , but  the  and  7.2  mho/m the  40.  great the  i t may  correction By be  and  diagonal  elements have been omitted from the  w i t h f i g s . 7.1  and  7-6  f o r r e l a t i v e p e r m i t t i v i t i e s of 10 and through 7.6  F i g s . 7.1  ii  off-diagonal  off-diagonal  frequency,  11  permittivity correction.  Since the d i f f e r e n c e  figures.  and  d i a g o n a l elements of  r e l a t i v e p e r m i t t i v i t y of 10 are  elements and  or C-26  ^cou^a.  7.2  7.4.  C-22  conductor  Correction.  ground c o n d u c t i v i t y ,  f o r the  with  p r o x i m i t y e f f e c t i s considered  C-27  assuming no  i n and  i n t e r n a l conductor impedance per bundle i s  i s calculated  The  .479  curves remaining  factors  comparing f i g . seen t h a t  for  relative permittivity  e f f e c t becomes q u i t e  7.3  has  pronounced f o r  39  conductivities below 10"^ mho/m. For example, the correction matrices without ground permittivity correction at a frequency of 10 kHz are  R =  I =  when<^ = 10  123  123  121  123  123  123  121  123  123  7623  .605  .558  .605  .623  .605  .558  .605  .623  mho/meter.  ohm/mile  7-8  mh/mile  7-9  For the same frequency butd* = 10'  mho/meter  R =  1 =  60.8  58.0  50.6  58.0  60.8  58.0  50.6  58.0  60.8  .137  .126  .099  .126  .137  .126  .099  .126  .137  ohm/mile  7-10  mh/mile  7-11  For ground r e l a t i v e permittivity of 10 and at the frequency of 10kHz.  R =  279  279  278  279  279  279  278  279  279  ohm/mile  7-12  40  1 =  .698  .677  .629  .677  .698  .677  .629  .677  .698  mh/mile 7-13.  when A = 10"^ mho/meter, f o r the same frequency hut ^ - 10*"  2  mho/meter  R =  I  =  61.2  58.4  51.0  58.4  61.2  58*4  51.0  58.4  61.2  .137  .126  .099  .126  .137  .126  .099  .126  .137  ohm/mile  7-14  mh/mile  7-15  From f i g s i 7 . 1 through 7.6 i t may he seen that f o r —2 conductivity below 10~ mho/m and a frequency above 1 kHz the ground r e l a t i v e permittivity should be included i n the calculation. 7.5  Admittance  Correction.  The admittance correction terms were evaluated by the two methods described i n Chapter 5.  F i r s t the i n t e g r a l  i n equation 5-14 was evaluated using numerical methods and then the correction terms were obtained from equation 5-29.  Figs.  7.7 through 7.12 show a comparison of the correction terms obtained by the two methods.  The discrepancy between the two  sets of values Increases with decrease i n ground conductivity. For the range of conductivities greater than 10'Wm  and the  range of r e l a t i v e p e r m i t t i v i t i e s between 10 and 50 the maximum  •41 discrepancy i s of the order of 10%. This i s considered tolerable (16) • • • for propagation c a l c u l a t i o n s ' i n view of the r e l a t i v e l y v  small magnitude of the correction terms as compared with the potential c o e f f i c i e n t s .  The difference i n the magnitude between  the diagonal elements and the off-diagonal elements i s of the order of 2% at the lowest conductivities and thus the o f f diagonal elements are not included i n f i g s . 7 . 7 through 7.12. —'3  For conductivities of 10  mho/m or greater a l l the correction  factors become n e g l i g i b l y small.  In calculating the propagation  constants a correction of 3% or greater i n the diagonal elements i s considered s i g n i f i c a n t .  A comparison of exact and approxi-  mate values of the correction factors and their magnitude are shown i n f i g s . 7 . 7 through 7.12.  This requires correction  factors to be calculated f o r conductivities less than 10 ' mho/m and frequencies greater than 100 Hz.  The potential  coefficient terms are corrected according to the formula i K  ij  =  K  ij  +  1  8 x  10 (2M 9  i;)  + J2N ) ±;J  7-16  To i l l u s t r a t e the magnitude, the potential coefficient correction terms f o r the diagonal elements at frequencies of 1 kHz, 10 —3 kHz and 100 kHz and f o r conductivities of 10*  mho/m and  10""^ mho/m f o r a r e l a t i v e permittivity of 10 are shown i n table 7.1.  42  Table 7.1  Potential c o e f f i c i e n t correction terms. 2U  2M  f =  1  kHz  = 1 0 " mho/m 3  <d> = 1 0 " ^ mho/m  0.03  0.07  £ = 1 0 * mho/m  0.003  0.007  ^ = 1 0 " mho/m  0,43  0.34  6> = 1 0 " mho/m  0.028  0.047  <i= 1 0 " ^ mho/m  0.66  0.17  fA =  10  kHz 3  5  f =  kHz  100 3  7.6  Conclusion. It has been found that the impedance should always  be corrected for ground conductivity. is less than 180,  In addition when c^/we  the impedance should be corrected for the  ground r e l a t i v e p e r m i t t i v i t y and the capacitance  should be  corrected f o r both ground conductivity and r e l a t i v e p e r m i t t i v i t y . When  oVwe i s  greater than 180,  these correction terms are  small, less than 3% approximately, and can be neglected.  FIG 7-1 IMPEDANCE CORRECTION FACTOR Q WITH 9*0. <f*Q  FREQUENCY  HZ  FIG 7-2 IMPEDANCE CORRECTION FACTOR P WITH 6*0. <f*0  •5  l&  FREQUENCY  HZ  44  ;  FIG 7-3 IMPEDANCE CORRECTION FACTOR Q WITH rt«/0  2 10  10  •6  3 4 10 10 FREQUENCY HZ  S 10  10  FIG 7-4 IMPEDANCE CORRECT/ON FACTOR P WITH ft=/0 9=0  . . . . .  •7  6-3*5°  •6*  *5 P  \\  •  c^jof^^  '4 -3  —  \  -  '2 -1  •vvr ""V •• 10  /o  i  3  j  4  10 10 FREQUENCY HZ  10  45  FIG 7-5 IMPEDANCE CORRECTION FACTOR Q WITH  <» 10 2  10  <n 10  <J 10  3  FREQUENCY  8  />«50. B*0  ,„ 10  5  HZ  10  FIG 7-6 IMPEDANCE CORRECTION FACTOR P WITH n=50. 0=0 ,•  7 •6 •5  s^t-ib  •  1  •4  •3  &*10  •2  tf./o'^^  -1 0  10  , 1,  .L  1.  • l,„.  10  J  1,-1  10 FREQUENCY  10 HZ  10  10  .46  FIG 7- 7 -5 POTENTIAL COEFFICIENT CORRECTION FACTOR N WITH 9*0. 0**10 m -201  nsSO n =40 n*30 n = 20 n*10  10 10 FREQUENCY HT  FIG 7-8 POTENTIAL COEFFICIENT CORRECTION FACTOR M 'SO EXACT APPROXIMATE —  —  —  m WITH 9 - 0. d-1(T m  —  '40  /•"\  // /  '30  /' •  M 20  M  ^  v* \ '\\ \\  <\  > n mW  MM**  '10 ~*"^*^^*^^^ '00 10  1  10  d - ^  i  i  i—1__  10 10 FREQUENCY; HZ  10  i  i  n "20 n-3Q n*40 n*5Q  10  47 '•' :y:'rp:^ :  FIG 7-9 POTENTIAL COEFFICIENT CORRECTION FACTOR N WITH 0 • 0. -20 '  tt»fo  4  mh  EMcr  APPROXIMA Ti  10 />  \\  JKSoV A •oo  n*50 n'40 n*30  n-20  10 ,„ 1 10  .-.1. .„J_ „  I „•,'  10  1  10" 10 FREQUENCY HZ  J  i  -  />-/o  i  10  70"  FIG 7-10 POTENTIAL COEFFICIENT CORRECTION FACTOR M WITH 8 = 0. <M0 mho -30\ EXACT APPROXIMATE —  . — '  — _  —  20 *  M  ......  '  ft  \\  ^\  v> n-10  Ft 10  n-20 n»40  •  '00 10  10  i —-  3 4 10 10 FREQUENCY HZ  l l  1  10  1  •  48 FIG 7- //  -3  POTENTIAL COEFFICIENT CORRECTION FACTOR N WITH O*Q. 0**10mho/m 'OS ff  04  .  •.;.-..<-,: :  N  \\  \  //^  •03 - ):^y  *\  //  .  ..',<  \U\\  :;02  -01  n*20 01 -02 10  •1'  1  1  -  1  . _J  L ,  i  10  10 10 FREQUENCY HZ  10  FIG 7-12  n-SO  i  15  •  •  •  n*70  • 10 r  M  IT**  n-30 n<40 n-SO  •OS  1 10  10  I—™j  1 ••  10 10 FREQUENCY HZ  10  10  49 Chapter 8.  Calculation of the Mode Propagation Constants.  The correction factors given i n Chapter 7 are inserted into the transmission l i n e equations, 6-1, and the mode propagation constants are calculated from equation 6-5.  Over the  frequency range from 10 Hz to 1 MHz two of the mode propagation constants are nearly equal and over the range from 100 kHz to 1 MHz a l l three propagation constants are nearly equal.  To  increase the accuracy of the calculation both the impedance and the admittance matrices are treated as follows Z = ^ Z '  8-1  Y = j2«ewY'  8-2  Then l e t A = ZY = - {/eiiZ'Y  1  = coeiiA» 2  8-3  The r e a l part of the diagonal elements of the matrix A' are nearly equal to -1 while the imaginary part of the diagonal elements and the off-diagonal elements are r e l a t i v e l y small i n magnitude, l e t A' = A" -  8-4  I  then from l|= 0  |A«  8-5  one has iJA" - ( f f + 1)1 | = 0 / 2  or |A  ! I  -  #/.2  I  I=  0  8-6  50 Then the eigenvalues of A" w i l l not he as close together as those of A'.  The propagation constants can now he obtained  from, a  where  j  +  J^  = ^ i =/ ( A u ^  2  - 1)  i s attenuation i n nepers per meter and  constant i n radians per-meter.  8-7 i s the phase  The mode v e l o c i t y i s obtained  as follows v  i  =  p\"  m  /sec  8-8  In terms of the v e l o c i t y of l i g h t i n free space, the normalized mode v e l o c i t y i s v- = 2itf/p /7JT'  8-9  i ]  The v e l o c i t i e s at modes 1 and 2 are very close to unity under a l l conditions,  and a l l three mode v e l o c i t i e s are  close to unity f o r a perfectly conducting grounds for f i n i t e ground conductivity  However,  and r e l a t i v e permittivity the  v e l o c i t y of mode 3 varies from 0.48 to nearly 1 per unit . (  Fig.  8-1 shows the v e l o c i t y of mode 3 without per-  m i t t i v i t y correction f o r conductivities from 10~^ to .1 mho/m. Figs. 8-2 through 8^-4 show the v e l o c i t y of mode 3 with impedance correction but without capacitance correction, f o r r e l a t i v e p e r m i t t i v i t i e s from 10 to 50 over the same range of conductivities as i n f i g . 8-1. Figs. 8-5 through 8-7 show the v e l o c i t y of mode 3 with capacitance correction over the same range of conductivities and p e r m i t t i v i t i e s .  From the figures i t can  be seen that r e l a t i v e permittivity has l i t t l e effect on the  51 v e l o c i t y of mode 3 for a conductivity greater than 10 and the capacitance  mho/m,  correction only a f f e c t s the calculation  where the conductivity i s less than 10  mho/m.  In f i g s . 8-1  through 8-7 i t can he seen that the permittivity correction of both impedance and capacitance  tends to increase the v e l o c i t y  of mode 3 at high frequencies. Pig.  8-8 shows the mode attenuation constants, i n  nepers per mile, f o r a perfectly conducting ground, i n which case the attenuation i s due e n t i r e l y to i n t e r n a l conductor a.c. resistance.  Pig. 8-9 shows the mode attenuation  constants  for a range of ground conductivity from 10~^ to -1 mho/m without permittivity correction of the impedance and without capacitance  correction.  attenuation constants  Pigs. 8-10 through 8-12 show mode  over the same conductivity range and over  a range of r e l a t i v e p e r m i t t i v i t i e s from 10 to 50, with the impedance corrected for conductivity and r e l a t i v e permittivity, hut without capacitance  correction.  Pigs. 8-13 through 8-15  show the mode attenuation constants with conductivity and r e l a t i v e p e r m i t t i v i t y correction f o r the impedance and the capacitance  over the same ranges of r e l a t i v e permittivity and  conductivity.  The attenuation of mode 3 i s most affected hy  changes i n ground conductivity, the attenuation of mode 2 i s affected to a lesser degree and the attenuation of mode 1 i s least affected.  Prom f i g s . 8-9 through 8-15 i t can he seen  that f o r conductivities greater than 10  mho/m the r e l a t i v e  permittivity has l i t t l e effect on the mode attenuation  constants.  For lower conductivities both the impedance correction and the  52 capacitance correction have an increasing effect on mode attenuation f o r frequencies greater than 100 Hz.  Tables 8-1  through 8-3 show a comparison of variations i n the attenuation for a conductivity of 10~^ mho/m and at frequencies of 100. kHz, ,.10 kHz and 1 kHz. Table 8.1 Attenuation i n nepers/mile f o r ^ = 10"^ mho/m and f = 100 kHz Mode 3 £To permittivity correction  Mode 2  Mode 1  3.6 x 10"  1  1.5 x 10~  3  1.3 x 10"  3  n = 10  7.0 x 10"  1  2.7 x 10~  3  1.4 x 10~  3  n = 20  7.2 x 10"  1  4.0 x 10~  3  1.4 x 10"  3  n = 30  7.3 x 10"  1  5.2 x 10~  3  1.4 x 10"  3  n = 40  7.4 x 10"  1  6.5 x 10~  3  1.5 x IO"  3  n = 50  7.5 x 10"  1  7.6 x 10~  3  1.5 x IO"  3  n = 10  4.6 x 10"  1  2.7 x 10~  3  1.5 x IO"  3  n = 20  5.8 x 10"  1  4.0 x 10~  3  ti  n = 30  6.2 x IO""  5.2 x 10~  3  it  n = 40  6.6 x IO"  1  3  »t  n = 50  6.6 x 10"  1  3  i»  Impedance correction  Capacitance correction  1  Si5 x 10" 7.6 x 10"  S3  Table 8.2 Attenuation i n nepers/mile f o r ^ = 10~^ mho/m and f = 10 kHz Mode 2  Mode 3 Wo permittivitycorrection  3.7 X i o -  2  n = 10  4.4 X i o -  2  n = 20  5.2 X  IO"  n = 30  5.7 X  n = 40  6.0 X  IO" IO"  n = 50  6.3 X  IO"  n = 10  3.3 X  n = 20  4.0 X  n = 30  4.6 X  n = 40  5.1 X  n = 50  5.4 X  IO" IO" IO' IO" ICT  Mode 1  x l (•f4  •4 4.0 x 10"  3.4 x 10"•4  •4 4.0 x 10"  3.3  Impedance correction  2  II  it  2  It  it  2  It  it  tl  ti  2  Capacitance correction •4 3.4 x 10"  2  -4 4.0 x 10"  2  ti  it  2  it  ti  2  II  ti  ti  it  2  From table 8-3 i t can be: seen that capacitance correction i s s i g n i f i c a n t f o r low conductivities even at a frequency 1 kHz f o r the attenuation of mode 3. For the attenuation calculation, conductivity and r e l a t i v e permittivity correction of impedance and capacitance matrices i s required for conductivities below 10  mho/m and frequencies above _3  100 Hz. For conductivities above 10  mho/m no r e l a t i v e per-  m i t t i v i t y correction of the impedance and no capacitance correction (2) are required and Carson's formulae ' are s u f f i c i e n t to obtain v  54 the ground correction terms f o r the c i r c u i t  parameters.  Tahle 8.3 Attenuation i n nepers/mile for<^ = 10"^ mho/m and f = 1 kHz  permittivity correction  ISTo  Mode 1  Mode 2  Mode 3 3.6 x 10~  3  1.15 x 10"  4  1.4 x 10"  4  n = 10  3.3 x 10~  3  1.15 x 10"  4  1.4 x 10"  4  n = 20  3.4 x 10~  3  it  II  n = 30  3.4 x 10~  3  ti  II  n = 40  3.5 x 10~  3  ti  ti  n = 50  3.6 x 10""  II  II  Impedance correction  3  Capacitance correction 1.15 x 10"  n = 10  7.8 x 10~ .  n = 20  4j5 x 10~  3  II  it  n = 30  3.5 x 10~  3  II  it  n = 40  3.4 x 10~  3  II  ti  n = 50  3.4 x 10"  3  II  tt  3  4  1.4 x 10"  4  An insight into the behaviour of the mode propagation constants may be obtained by comparing the transmission l i n e with a l i n e that i s transposed at short intervals.  Then a l l  the off-diagonal terms i n both the impedance and admittance matrices would be equal and the diagonal elements i n each matrix would also be equal.  The transformation matrices, S and R,  would be equal and would be the matrix, used f o r transformation  55 to symmetrical components.  Thus the mode 3 propagation constants  can he compared with the zero sequence propagation constants in a balanced system.  To i l l u s t r a t e t h i s s i m i l a r i t y the R and  S matrices for a frequency of 100 kHz, a ground conductivity of 10~  4  mho/m and a r e l a t i v e permittivity of 10 are .484  R =  -  j.161  -.787 + j.177 .299 + j.00  .223 + j.246 S  t =  R"  1  =  -.603  -  j.H5  -.763 + j.106 .115  .566 + j.014  -.750  -  j.246  .517 + j.oo  -.143  + j.126  .766 + j.00  .275 + j.246  -'.833  .568  j.003  .570 + j.00  .570  -  j.102  -  .619 + j.oo  .571 + j.002  -.656  -  .592  —  j.017  .591 + j.00_  j.147  .543 + j.020  j.064  -.125 + j.067  j.005  .572 + j.005  .808 + j.00 .593  -  j.011  8-10  8-11  8-12  FIG 6-1 MODE 3 PROPAGATION VELOCITY WITHOUT PERMITTIVITY CORR  10  10  2  10 IO FREQUENCY HZ 3  .  4  W  5  10  FIGB-2 MODE 3 PROPAGATION VELOCITY. NO CAPACITANC&>CQRRECTIO  57  FIG 8-3 MODE 3 PROPAGATION VELOCITY. NO CAPACITANCE CORRECTIO  10  10  10 10 FREQUENCY HZ  FIG 8-4 MODE 3 PROPAGATION VELOCITY. NO CAPACITANCE CORRECTI  10  10  10 10 FREQUENCY HZ  10  FIG 8-5  FREQUENCY HZ  v :'V^f- --V :  v  60  61 FIG  iot  MODE  6-9 A T T E N U A T I O N  W I T H  — i — — . ' • ,-"  N O  i V—  P E R M I T T I V I T Y C O R R E C T I O N  i—~~r  r~  64  FIG 8-12 MODE ATTENUATION. NO CAPACITANCE CORRECTION WITH n-  70i  i  — - i  i  FREQUENCY HZ  -  i  66  FIG 8-74  MODE ATTENUATION. APPROXIMATE CAPACITANCE CORRECTION WITH n .  i o v —  ;  i  —  -  —  —  i  —  r  — — i  1  67  68 Chapter 9.  Conclusions.  The methods of C a r s o n ^ and Wise^"^ have been extended to calculate the distributed c i r c u i t parameters f o r a multi-conductor transmission l i n e .  The correction terms f o r  the distributed c i r c u i t parameters have been derived f o r variations i n l i n e geometry, ground conductivity, ground r e l a t i v e permittivity, and f o r a frequency range up to 1 MHz. Their corrected parameters have been used i n the  transmission  l i n e equation to calculate the mode propagation v e l o c i t i e s and attenuation  constants.  A p r a c t i c a l example has been investigated i n d e t a i l to show the variations i n the distributed c i r c u i t parameters and the mode propagation constants due to various ground conditions.  For a t y p i c a l 500 kV transmission l i n e the impedance  should always he corrected for ground conductivity.  <<^/toe i s  less than 180,  When  the impedance should, i n addition^ be  corrected for the r e l a t i v e permittivity of the ground and the capacitance  should be corrected f o r both the ground conductivity  and the r e l a t i v e permittivity of the ground. 180,  When  equals  the maximum error i s approximately 3$ i n Carson's correction  of the impedance parameters and the uncorrected  potential  coefficients. The i n i t i a l rate of r i s e of recovery voltage i n c i r c u i t breakers clearing f a u l t s i n power systems can be evaluated  s o l e l y i n terms of the c h a r a c t e r i s t i c impedance of  the transmission l i n e and the energizing conditions.  The  former depends i n turn on the corrected impedance and admittance  69  matrices and the natural frequency of the energizing system. Thus i t may he possible to investigate the effect of ground conditions on the i n i t i a l rate of recovery voltages f o r c i r c u i t breakers opening under f a u l t conditions. The program developed f o r calculating the mode propagation constants  i n this thesis may be extended to include  terminal conditions.  This allows the investigation of the  various permutations to find the optimum condition f o r energizing and terminating a l i n e f o r c a r r i e r communicationv. In the c a l c u l a t i o n of the mode propagation  constants  i n t h i s thesis the conventional formulae f o r the i n t e r n a l impedance of the l i n e conductors have been used with the assumption .that the equivalent radius i s the maximum radius over the strands.  For bundled conductors further investigations  would be required to determine the i n t e r n a l impedance more accurately.  Since the i n t e r n a l impedance affects the diagonal  elements of the impedance matrix i t has a d i r e c t effect on the mode propagation constants. Further developments i n t h i s f i e l d should be directed to f i n d the effects of corona and tower footing resistance on the transmission system parameters. f i e l d adjacent  Corona disturbs the e l e c t r i c  to the conductors and may have a noticeable  effect on the parameters. i s voltage dependent.  The effect i s non-linear since i t  The tower footing resistance may become  important f o r l i n e s with overhead ground conductors where the tower spacing approaches a quarter of the wave length of the impressed s i g n a l .  70 The procedure developed i n t h i s thesis i s solely for overhead transmission l i n e s .  In the present form i t i s  unsuitable f o r the calculation of propagation constants of underground systems, which presents quite a different problem due to t h e i r geometric configuration with respect to the ground.  71 Appendix A.  The F i e l d due to an Isolated O s c i l l a t i n g Current Element.  In an isotropic homogeneous medium Maxwell's f i e l d equations are stated a s ^ ^ 4  V*E  = -jcouH  A-l  VxH  = J + 3 we E  A-2  C  where, the time dependence i s sinusoidal and the conductivity df the medium i s aaccounted f o r hy the complex permittivity e , cc C  where e  = e -j c  to Define a vector TT such that H = jcoe V T c  A-3  X  Then from equation A f l and,, A-2  V x E = to e ^V TT = -m V x l T 2  x  A-4  2  c  hence E  = -m TT 2  +V(0)  A-5  where 0 i s a scalar function of position.  Then from equation  A-2 jweVx ( V x T T ) = J + 3toe (-m TT 2  c  +^/(0))  A-6  Dividing by jwe and expanding the f i r s t term of equation A-6 gives - V 7 T + V-V-TT =  - * TT  2  2  A-7  C  let V(0)  =VV."TT  A-8 (19)  a Lorentz type condition,  and equation A-7 becomes  72  V 1T =  7T-^  A-9  E = -m7T + W . T  A-10  2  2  m  and 2  Let the solution of equation A-9 be -m R r  dv  A e R  A-ll  o where v  Q  i s a small closed surfaoe at the point (x,y,z) i n  Cartesian coordinates.  Then -m R  V T=V2£^-fT-*v 2  r r 2 " I [nTAe  —i  mR  -m R  0  n  = J [ — R — + * 'V (|)Jdv A  v  2  o  2  = m T  +  r  -m R Ae"  V  (J) dv  A-12  (f) dv .  A-13  2  o Substituting A-12 into A-9 gives -m R Ae V o The solution of A-13 i s J = - jwe  p 1  c  J =4  0  p -j  when R ^ 0 A-14  jwe A4it when R = 0 c  The current flows only i n the z dlreotion, henoe J  *  are both zero at r equals zero.  and J  y  When the element has a current  I and a cross section a, J  i  at R = 0  z = a  A-15  Hence A = ,* 4«0we a a  A-16  73  ^  a n d  o  Assuming v  Q  -m R Ie . 4«3(oeaR  i s small and independent o f r , then -m R z  4«dwe  aR  o  where v o = sa and s i s the elemental length which i s replaced hy dz. Hence -m R  _ Ie dz " z ~ 4*jwe R  TT  TT  X  TV •7  A A  _  1 7  ± f  = 0  A-18  = 0  A-19  74 Appendix B.  The General I T Vector i n C y l i n d r i c a l Coordinates,  Fig, B - l C y l i n d r i c a l coordinate system f o r the internal f i e l d The wave equation f o r the H ^ vector i n the coordinate system shown i n f i g . B - l i s  The solution i s obtained by separating the variables, TT  Z  = R(/>) $  let  (0)Z(a)  B-2  Then 1  "/? c ^ R l ^  a/ L ^ J 3  r  1  <f d $ l  ..  1 ^Vz Z  T7 *  2  2  p_  = m  <^2 Since the f i r s t two terms of B-3 are independent of z R  f  ^  +  t  ±i—'3  the  equation becomes  U^Z  B-4  where  t  2 = u. 2+ .m2 and u i s a constant.  B-5  75  The solution of B-4 i s Z = ke  + A e  ±  B-6  Q  2  Then B-3 reduces to  ^  R  Kc3fJ  +  $  ^02  + U  = 0  /°  B-7  Since the second term of the equation i s independent of r , B-7 can he written as + p $ = 0  B-8  2  0  0  *  *  2 where p  i s a constant. $ - B  1  The solution of B-8 i s  cos  P 0  + B  2  sin P  B-9  0  Equation B-7 now hecomes  ^D*ifl [ ^ - ] +  (  )2 p2  R=o  which can he transformed to the form  </» »> ^ u 7  i^q]  +  [ > u)  2  -P2]  * =o B-ll  This i s Bessel*s equation of order P.  The solution has the  following form R = C Z {f> u)  B-12  where Z^(^u) indicates the generalized form the the Bessel function.  The exact types of Bessel functions required f o r the  solution depend on the boundary conditions of the particular problem.  Then the complete solution f o r "TT i s the sum of a l l 2  76  possible solutions and may be writen as  TTZ  = ^(B  l p  c o s P 0 + B sinP0)* J  P=0  (A-Ju)e -  2p  xC  Y z  + A (u)e  y z  2  )  B-13  Zp(/>u)du  The wave equation i n the Cartesian coordinate system can be separated into three components.  Each component  s a t i s f i e s (7)  IT  B-14  where TT may be either "IT „, ~T„ or TT . . Thus the solutions x 'y z w i l l be similar functions. For the coordinate system shown in f i g . B-2.  The solution f o r T  and TT z  w i l l be similar and y  of the general form shown i n equation B-15.  This i s the form  of the solution used i n the ground effect calculations i n Chapter  3. y  F i g . B-2  C y l i n d r i c a l Coordinate System f o r the External F i e l d  77  oo ^(B  oo i p  c o s P V + B sinPy) 2 p  J  ( k j ^ e - *  7  + A (u)e* ; y  2  P=0 x Z ( r u ) du p  B-15  78 Appendix C.  Derivation of the Internal Impedance of C y l i n d r i c a l Conductors.  In t h i s Appendix we s h a l l he concerned only with the f i e l d s inside the conductors.  Consider a s o l i d c y l i n d r i c a l  conductor with the axis along the z-axis of the coordinate system shown i n f i g . B - l . The d i r e c t i o n of propagation i s along the positive z-axis. The electromagnetic f i e l d inside the conductor consists of the TM mode only.  A Hertz vector of the TT  i n the z d i r e c t i o n i s used to describe the f i e l d .  type  Due to the  c i r c u l a r symmetry of the system the f i e l d components are independent of 0, and hence the Hertz vector i s also of 0.  The solution of the wave equation for"iT  equation B-13.  2  independent i s given i n  However as TV „ i s independent of 0 and must be z  f i n i t e at the o r i g i n only the f i r s t term of the solution exists and the required Bessel function i s of the f i r s t kind of order zero.  Therefore IT  = Ae~* J  (/>u) o '  z  z  Using equations A-3 and A-10  C-1 the f i e l d components E  g  and  H^  are E  = (-m  +* )TT„  2  H0 = jtoe uAe"  u)  c  Let  C-2  2  C-3  the external radius of the cylinder be a, then at the Y »  boundary I e  Hv = ° - _  <M  79 Consequently  I A  v  Wheno for  jwe 2 *au  =  2  i s v e r y l a r g e ,Y  practical  is negligible  and  = -m  2  o -m I  J  2  E  z  Eg at radius  z  i  =  Q  v „ (/>u)ey  C 7  1  2  a be g i v e n  by C-8  2  internal  impedance o f t h e c y l i n d r i c a l  Then  z  a  z  t  jwe 2«auJ (au)  -m  For  Then  C-6  f  i s the equivalent  conductor.  2  a s compared t o m .  2  E„z = z ,i I o e ~ * where  5  purposes u  Let  "  G  c  i  =  J (au) Q  jwe 2«auJ (au) 1  c  conductor e_ = - < ~ ' c ~ ~ 3 co  C - 1 0  m = j  V  C - l l  a' = Jujie?  C - 1 2  and 0  ,  where  Then »4  =  3  1 , 3  cc'Jo(3  2«ad  The n u m e r i c a l three  regions,  dependent  .  J (d 1  1  1 < 5  a'a)  '  ,  5  C-13  a'a)  e v a l u a t i o n o f z ^ may b e d i v i d e d on the value  of a ' a .  into  80 1)  When a*a < 0.1 For s m a l l v a l u e s off a'a the f o l l o w i n g expansion h o l d s , i 2  i  =  ^  * («'a)  «  2  8  (a'a)  A  192  +  ,  4  "  J  (a'a)  3072  (13)  6  ''•'  I  w a  2j  C-14  let i=  Z  R  int  3X  +  C-15  i n t  Then f o r s m a l l v a l u e s of a'a  o  *a and  which a r e the formulae  obtained when a uniform c u r r e n t d i s t r i -  b u t i o n i s assumed. 2j  When a'a > 10 For l a r g e v a l u e s o f a'a z  z  =  may be w r i t t e n as  « ' U +• .1)  0-18  2j2*a<5  1  Let  i  the s k i n depth d be d e f i n e d as  d =M,  c*i?  Then 5  i  =  2*adJ  which i s Rayleigh's formula.  R  int  =  Then  2*ad<S  L ^ = O^^J.. i n t ~ 2«ad^ J  C  "  2  1  C-22  81  When 0 . 1 ^  3)  a'a — 10  Equation C-13 may be changed i n t o a more conyentient form f o r numerical computation  by l e t t i n g  J ( d ' a ' a ) = Ber(a'a) + JBei(a'a) 1  C-23  5  0  The f i r s t order B e s s e l f u n c t i o n  i s obtained by d i f f e r e n t i a t i n g  equation C-23  J (j a'a) 1,5  =  1  j ' JBer'Co'a) + 0  5  jBei'(a'a)]  C-24  Then the i n t e r n a l impedance z^ becomes  z  _  a'  and the r e s i s t a n c e  TJ  -  «'  int  W  n  jBei'(a'a)J  5  9 r  .  and inductance are  [Ber(a'a)Bei' (a'a) - Bei(a'a)Ber'(a'alj  *int "  L  r»Bei(a'a) + .1Ber(a'a) "I  ~ 2*a<* |_Ber'(a'a) +  i  Ber' (a'a) + B e i ' ( a ' a ) ] 2  \ —  u.  2«aa a a  '  2  C  "  fBer(a'a)Ber' (a'a) + Bei(a'a)Bei' (a'a)] _ c  L  Ber (a'a) + Bei' (a'a) ,2  2  2 6  2 7  J  The numerical v a l u e s of the f u n c t i o n s Ber, B e i , Ber', and B e i ' are tabi tabulated power s e r i e s (15)  (17) v  , or they may be c a l c u l a t e d  from  82 Appendix D.  Solution of the Integral i n Equation 4-50.  The i n t e g r a l i s OO  x  =  r  Mi it  Qv? + j - u)e- ( " y">cos x»u du u  h  D-l  +  o  where  x" =  X  ' s = sju>\id* X D-2  y" = y' s = s j ( 4 i ^ * y h" = h's = s J cop-ci'h and 2 s  je co(n-l) o  1 + =  D-5  Equation D-l has at least two possible solutions, (2)  one suggested by J.R. Carson  ' f o r small values of R" and the  v  other obtained by repeated p a r t i a l integration f o r large values of R". The f i r s t solution i s I = P + JQ  R+e^  ^  7  y  [ E  (  L  R « e 3  3  (  0  +  ^  ) +  G( R«'e^ d  (e +  ^ ) \  + R"e  _ 1 r 2J(e-cf) -2j(0+<P| e  R" Where K^(x)  +  L  e  J  D - 4  i s a Bessel function of the second kind and the  function G(x) i s defined a s ^ P/ \ 0( ) X  =  V ^ n=l  (-l) x 2 - nl.(n-l)I ^-(2n l)'f2n-l) n + 1  +  2 n  211  1  . .  83 From D-4 and D-5 P = 0.125it(l-s +s ) + 0.51n(|7^(s +s ) - 0.5cf ( l - s 3r  - 0.59(s  2 r +  1:L  lr  s ) +^=(r 4 i  -r  3 r  l r  -r  3 1  -r  1 1  3i  ) + 0.5(r  2 r +  r ) 4 i  1 : L  -s ) 3 r  D-6  Q = 0 . 2 5 - 0 . 1 2 5 « ( s - s ) -* O . ^ U + s ^ + s ^ ) - 0 . 5 © ( s - s ) lr  0.51n( |r)(l-s f  3i  3 r +  4r  a ) + ^ V l i  r  3 r  +  r  3 i -  r  l i  )  +  2i  °«^ 2i- 4r r  r  D-7 where s  =  lr  ^ Z 2ni ^ 2 n - l ) i (^) " QOs(4n-2)QcoB(4n-2)<f n=l 4n  0  s  2  . D-8  0  *n+l li = Z J"(2n-l)» n=l 2  (^) " cos(4n-2)©sin(4n-2)cT  D-9  (^) " sin(4n-2)9co (4n-2)<T  D-10  (^) - Bin(4n-2)0sin(4n-2)cT  D-ll  4n  2  00 s  - ~ " ZI 2 n ^ 2 n - l ) r >2r n=l =  0  s  B  2i  3r  =  4n  2  S  0  \n+l ZI 2J1V2Z1U n=l  4n  2  00  =  Z2ni  vn+1  /(2n+l)f  (^) c°s4n©cos4n <f  D-12  sn+l 3 i = ZI 2 ^ ^ ^ ! ) : n=l  ( i) cos4nesin4n<T  D-13  4n  n=l 00  s  S  4n  00  s  4 r = ZI 2nl"(2n!l): n=l  (^) sin4n©cos4n<f 4n  D-l  4  )  84  3  4i  =  Yx 2ni~(L-l)<  D-15  (^) sin4n@sin4ncf 4n  n=l  Z Cx3  lr  (or, o\t 4(n-l!)p„4n-3/ -, \n+l 2  9  12n 21 _2 R (4n-3)r(4n-l) n=l CO 7  z  i^l}  1  c  o  _ )©  s ( 4 n  3  c o s  (  4 n  _ 3 )<f D-16  (or, ?)i o4(n-l) ,,4x1-3/ - i \n+l 2  n  li  2 ^2n~2^ 9 S (4n-3)r(4n-l)  n=l  cos(4n-3)©sin(4n-3)o r D-17  co ph+1  _1  4n  •2r n=l  2i  Lrii  j=l  D-18  z[z*  _1  4n  5  i = i ^  T  T  T  ( ^ ) 4 r - 2 l  o  o  s  ( n-2 ) e i n ( 4 n - 2 4  a  D-19  n=l Lj=i ^(-l)  n + 1  (2n-l)j2  3r  (4n-l)J  n=l 7(-l)  3i  2  n + 1  \  2  R  (4n+l)  (2n-l).'  (4n-l) I  (4n-2) „4n-l  D-20  2 ~ R" -  2  4 n  cos(4n-l)©cos (4n-l)<f  2  4 n  (4n+l)  1  c o s (4 n-l) ©s in (4n-l) cf D-21  n=l GO -2n+l '4r  4n+2  11=14=1  ^ ^^(^) 2  4  n  cos4n9co 4ncT S  D-22  co f2n+l •41  ZI  Z  n=l  > 1  J - 4nT2  2^^nT^  c o s 4 n @ s i n 4 n  <*"  D  "  2  3  (9) The second solution i s <$o /2 fe] _J= cos(2n-l)Q  H" ' - cos2Q, cos22T 2  p  =  + (-l)  Q =  n+1  s i n ( 2 n - l ) c f ]2|  OOB2Q BlnfrT  +  8  J. v  fei R"^ -- -  V  11  1  oos(2n-l)Q  r 00B(2r  L  ^  5  1)<f  D-24  L^g^Utf  n=l  + (-l) sin(2n-l)<f] n  D-25  For numerical computation the solutions are divided into two ranges 1)  When R"-£ 5.0 In this range use equations D-6 through D-23  2)  When R" > 5.0 In t h i s range use equations D-24 and D-25. (2) These equations reduce to J . R . Carson's solutions  SQ i s equal to one and cT i s equal to zero.  when  86 References. 1.  Carson, J.R., Wave Propagation Over P a r a l l e l Wires; the Proximity E f f e c t , Philosophical Magazine, Vol. XLI, A p r i l , 1921, pp. 607-633.  2.  Carson, J.R., Wave Propagation i n Overhead Wires with Ground Return, B.S.T.J., Vol. V, October, 1926, pp. 539-554.  3.  Rice, S.O., Steady State Solution of the Transmission Line Equations, B.S.T.J., Vol. XX, A p r i l , 1941, pp. 131-178.  4.  Bewley, L.V., Travelling Waves on Transmission Systems, Dover, 1963.  5.  Hayashi, S Surges i n Transmission Systems, Denki-Shoin, Japan, 1955.  6.  Rudenberg, R., Transient Performance of E l e c t r i c Power Systems, Wiley, 1951.  7.  Sunde, E.D., Earth Conduction Effects i n Transmission Systems, Van Nostrand, 1949.  8.  Wedepohl, L.M., Application of Matrix Methods to the Solution of Travelling Wave Phenomena i n Poly-phase Systems, Proc. I.E.E., Vol. 110, No. 12, December, 1963, pp. 2200-2212.  9.  Wise, W.H., Effect<of Ground Permeability on Ground Return C i r c u i t s , B.S.T.J., Vol. X, July, 1931, pp. 474-484.  n  10. Wise, W.H., Propagation of High Frequency Currents i n the Ground Return C i r c u i t s , Proc. I.R.E,, Vol. 22, No. 4, A p r i l , 1934, pp. 522-527. 11.  Wise, W.H., Potential Coefficients f o r Ground Return C i r c u i t s , B.S.T.J., Vol. XXVII, 1949, pp. 365-371.  12.  Arismunandar, A., Capacitance Correction Factors f o r Transmission Lines to Include the F i n i t e Conductivity and D i e l e c t r i c Constant of the Earth, Trans. I.E.E.E., No. 63-1030, Special Supplement, 1963, pp. 436-456.  13.  King, R.W.P., Fundamental Electromagnetic Theory, Dover, 1963.  14.  C o l l i n , E.C., F i e l d Theory of Guided Waves, McGraw H i l l , I960.  15.. McLachlan, N»W., Bessel Functions f o r Engineers, Oxford, 1955.  87 16. Hedman, D.E., Propagation on Overhead Transmission Lines; I. Theory of Modal Analysis, II. Earth Conduction Effects and P r a c t i c a l Results. Trans. I.E.E.E., Vol. 83, July, 1965 P P » 665-670. r  17. Dwight, H.B., Mathematical Tables, Second Edition, Dover, 1958. 18. Jahnke, E., and Emde, P., Tables of Functions, Dover, 1945. 19. Bowman, P., Introduction to Bessel Functions, Dover, 1958, 20. Pipes, L.A., Matrix Methods f o r Engineering, Prentice-Hall, 1963. 21. Wedepohl, L.M., Shorrocks, W.B., and G-alloway, R.H., Calculation of E l e c t r i c Parameters f o r Short and Long Poly-phase Transmission Lines, Proc. I.E.E., Vol. I l l , No. 12, December, 1964, pp. 2051-2059. 22.  Wedepohl, L.M., Electric" Characteristics of Poly-phase Transmission Systems with Special Reference to Boundary-value Calculations at Power-line Carrier Frequencies, Proc. I.E.E., Vol. 112, November, 1965, pp. 2130-2112.  23. Dowdeswell, I.J.D., Matrix Analysis of Steady State, MultiConductor, Distributed Parameter Transmission Systems, M.A.Sc. Thesis, University of British,Columbia, November, 1965.  

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