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OPTIMIZATION AND C O N F I G U R A T I O N  OF CONDUCTOR  SHAPES  OF CONDUCTOR BUNDLES  VOLTAGE  FOR H I G H  TRANSMISSION  by Gustav B. S c . ( E n g . ) ,  A  THESIS THE  Strom  Christensen  U n i v e r s i t y o f A l b e r t a , 1958  S U B M I T T E D I N P A R T I A L F U L F I L M E N T OF REQUIREMENTS MASTER in  FOR T H E D E G R E E  OF A P P L I E D  OF  SCIENCE  t h e Department of  Electrical  We  accept  standards degree  this  Engineering  thesis  as conforming  required from of Master  candidates  of Applied  tothe f o r the  Science.  Members o f t h e D e p a r t m e n t of E l e c t r i c a l Engineering  THE  U N I V E R S I T Y OF B R I T I S H  April,  1960  COLUMBIA  In the  presenting  requirements  f o r an  of  B r i t i s h Columbia,  it  freely available  agree that for  that  copying  gain  shall  Department  advanced degree a t  or  not  his  University s h a l l make  f o r reference  and  study.  I  for extensive be  copying  granted  representatives.  by  of Columbia,  of  the  It i s  of t h i s t h e s i s  a l l o w e d w i t h o u t my  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 $, C a n a d a . Date  the  of  Library  publication be  fulfilment  the  p u r p o s e s may  o r by  in partial  I agree that  permission  scholarly  Department  this thesis  further this  Head o f  thesis my  understood  for financial  written  permission.  ABSTRACT  This field Two  thesis  intensity  separate  with  on  conductor  per  phase,  conductors  per  phase.  conductor to  of  the  uniform this  when  particular  bundled the  boundary  to  less  i t i s found  that  a  cylinder  d e n s i t y on to  line  solution  in  i f  electric Also  a  consists i n such  the  only  perin  a  manner  cylinder  charges.  exists  when  determine  variation  essentially  case  with  contours  conductors.  circular  surface charge  is  Using in  one  case.  considering,several conductors,  the  form  field  phase.  Transmission  mined  by  a i d of  sults  obtained  bundle  intensity  treated. the are  conductors  normally  electric  are  investigated  which  lines  transmission lines  Some  circular  a  conductors.  transmission  phase.  i t i s placed parallel  symmetrical  bundles  two,  provide  of  electric  considered f o r the  are  i s developed  electric  method  In  the  are  mapping  shapes  and  One,  f o r each  d e n s i t y than  method  perturbing that  such  charge  turbation  i s used  conformal  conductors surface  conductors  the  transmission line  considered.  one  subject  power  are  Non-circular only  electric  m i n i m i z a t i o n of  cases  one  several  discusses the  The Alwac  phase,  configuration used  exists  lines  per  with  on  indicated.  i i  the  i s changed same  a l l conductors  three  optimum III-E  until  and  four  configuration  digital  that i s ,  computer  from  maximum of the  same  conductor was  deter-  and  the  re-  TABLE  OP  CONTENTS  page  Abstract  i i  Acknowledgement  .  . . . . . . . . .  v i  1.  Introduction  1  2.  Some B a s i c  4  2-1. 2- 2, 3.  6.  Bundled 6- 1. 6-2. 6- 3.  7.  10  C a s s i n i a n Oval E l l i p t i c C y l i n d e r a n d two L i n e C h a r g e s A p a r t i c u l a r n o n - c i r c u l a r C y l i n d e r and two L i n e C h a r g e s Method  I  Appendix  II  12  . . .  15 17 22  . .  29  Conducting Cylinder Conducting C y l i n d e r and Line Charge . . . . C o n d u c t i n g C y l i n d e r a n d two L i n e C h a r g e s . . Numerical Computations  29 31 40 46  Conductors  52  Transmission per Phase Transmission per Phase Some G e n e r a l  Line with . . . Line with  three  Conductors 53  four Conductors . . . . . . . . . . Considerations .  Recommendations  f o r Future  Work  . . . . . .  63 65 67  References Appendix  10  15  Conclusions 7- 1.  4 5  Phase  Contours  A Perturbation 5- 1. 5-2. 5-3. 5- 4.  Intensity with  . . . .  Determination o f L i n e Charges . . . . . . . Determination o f t h e Phase Angle a t which Maximum E l e c t r i c F i e l d I n t e n s i t y O c c u r s .  Non-circular 4- 1. 4-2. 4- 3.  5.  L i n e Charges and D i p o l e L i n e s Conducting C y l i n d e r and Line Charge  Variation of Electric Field Angle and Phase Spacing 3- 1. 3- 2.  4.  Problems  .  67 69 73 77  i i i  LIST OP  ILLUSTRATIONS  Figure  page  2-1.  Line Charge  to f o l l o w  4  2-2.  D i p o l e Line  to f o l l o w  5  2- 3.  L i n e Charge and Conducting C y l i n d e r  to f o l l o w  5  3- 1.  Ungrounded three-phase Transmission Line  to f o l l o w  10  3- 2.  Conducting C y l i n d e r and two  ...  Line  Charges  12  4- 1.  C a s s i n i a n Oval  4-2.  Charge D i s t r i b u t i o n on C a s s i n i a n  4-3.  C i r c u l a r C y l i n d e r and two Charges E l l i p t i c C y l i n d e r and two  4-4.  . . . .  to f o l l o w  15  to f o l l o w  17  to f o l l o w  17  to f o l l o w  17  Line Line  Charges 4-5.  Charge D e n s i t y V a r i a t i o n  to f o l l o w  21  4-6.  C i r c u l a r C y l i n d e r and L i n e Charge . . . .  to f o l l o w  22  4-7.  Deformed C y l i n d e r and L i n e Charge . . . .  to f o l l o w  22  4-8.  Charge Density V a r i a t i o n  to f o l l o w  25  4- 9.  C i r c u l a r C y l i n d e r and two Charges Deformed C y l i n d e r and two Charges  to f o l l o w  27  to f o l l o w  27  to f o l l o w  28  to f o l l o w  28  4-10. 4-11. 4-12.  Line Line  Conductor Shape as a Function Spacing ( S i n g l e Phase)  of  Conductor Shape as a F u n c t i o n  of  Spacing (Three Phases)  .  5- 1.  Conducting C y l i n d e r  30  5-2.  Conducting C y l i n d e r and Line Charge  31  5-3.  Conducting C y l i n d e r and two  L i n e Charges  41  5-4.  Conducting C y l i n d e r and  Line Charges  45  iv  two  LIST  OP  ILLUSTRATIONS  Figure  page  2-1.  Line  2-2.  Dipole  2- 3.  Line  3- 1.  Ungrounded Line  three-phase  Conducting  Cylinder  3- 2.  Charge Line  Charge  and  Conducting  Cylinder  . . .  to  follow  4  to  follow  5  to  follow  5  to  follow  10  Transmission  and  two  Line  Charges  12  4- 1.  Cassinian  Oval  4-2.  Charge  4-3.  Circular Cylinder Charges  and  two  Line  4-4.  Elliptic  and  two  Line  Distribution  Cylinder  on  Cassinian  . . . .  Charges  to  follow  15  to  follow  17  to  follow  17  follow  17  to  follow  21  . . . t o  4-5.  Charge  4-6.  Circular  Cylinder  and  Line  Charge  . . . .  to  follow  22  4-7.  Deformed  Cylinder  and  Line  Charge  . . . .  to  follow  22  4-8.  Charge  . t o  follow  25  4- 9.  Circular Cylinder Charges  and  to  follow  27  Deformed C y l i n d e r Charges  and  to  follow  27  to  follow  28  to  follow  28  4-10.  4-11.  4-12.  Density Variation  Density  Variation two two  Line Line  Conductor Spacing  Shape as a F u n c t i o n (Single Phase)  of  Conductor  Shape  of  Spacing  (Three  as  a Function  Phases)  5- 1.  Conducting  Cylinder  30  5-2.  Conducting  Cylinder  and  Line  5-3.  Conducting  Cylinder  and  two  Line  Charges  5-4.  Conducting  Cylinder  and  two  Line  Charges  iv  Charge  31 41 . . . . .  45  Figure  page  5-5.  Conducting Cylinder  5- 6.  Charge D e n s i t y  6- 1.  Three-phase T r a n s m i s s i o n L i n e  6-2.  Charge p e r U n i t L e n g t h v e r s u s Parameter b  6-3. 6-4. 6-5.  and  two  Line  Variation  F i e l d Intensity versus Parameter b  Q  ^  Thiee Phase T r a n s m i s s i o n L i n e  6-7.  Charge p e r U n i t L e n g t h Bundle Spacing  I-l.  V a r i a t i o n of s Spacing . . f  .  0A  . ?  49  to f o l l o w  54  to f o l l o w  58  to  follow  62  to,follow  60  to f o l l o w  62  to  follow  63  follow  65  to follow  65  to follow  73  the  and  .  6-6.  6-8.  -fco f o l l o w  and  w i t h Phase  B u n d l e Spacing* *.  47  the  Three Conducting C y l i n d e r s two L i n e C h a r g e s V a r i a t i o n of b  Charges  versus . . . . t o w i t h Phase  .  Three Conducting C y l i n d e r s  v  ACKNOWLEDGEMENT  The Research This  work  done  Council  i nthis  t h e s i s was w r i t t e n  couragement  from  which  velopment  held  with  for  for  acknowledged  gratitude  f o radvise  1959=-196Q„  vi  and en-  the dethesis.  discussions  J Fo  Szablya  Electric  Company  c  thesis.  t o The N o r t h e r n  o f Canada  Frank  received  i n this  and Dr„  i n this  0  The a u t h o r  with  enlightening  f o r the session  Council  here.  described  D r . G\> W a l k e r  of Dr  guidance  i n connection  method  t h e work done  granted  Research  thesession  Moore,  i s indebted  a fellowship  National  0  with  author  continual  wishes t o acknowledge  Dr. A„D  connection  by t h e N a t i o n a l  the supervision  with  i n particular  of the perturbation  the author  The  the author  h i s sincere  E.V. Bohiij  Also  in  under  i s gratefully  t o express  Dr,,  i s supported  o f Canada,,  N o a k e s who p r o v i d e d  wishes  thesis  1958-1959  and t o  f o ra Studentship  the  granted  OPTIMIZATION AND C O N F I G U R A T I O N  OP CONDUCTOR  OF CONDUCTOR  VOLTAGE  1.  In  order  mission have  over  been  long  BUNDLES  FOR H I G H  TRANSMISSION  INTRODUCTION  to facilitate very  SHAPES  economic  distances  increased appreciably  electric  transmission  power line  transvoltages  since t h e second World 1 2  In  some c a s e s  extensive order  gated  up t o 400 k ? . a r e employed  experimental  t o study  mission  have  voltages  lines  the possibility  voltages„  Corona  by a i d o f these been  test  published  have  been  o f employing  lines  ' , and  constructed i n even h i g h e r  indicating  and s e v e r a l  s  the results  0  3  9  i s one o f t h e phenomena  test  War  trans-  investi-  papers^'^ ^ 9  obtained  from  y  such  studieso Corona be  i s one o f t h e more i m p o r t a n t  considered  i n connection  transmission. should  be k e p t  incurred, tion  a t a mlnimum  from  electric  both  high voltage  a r e two main 0  a n d two, t h e h i g h  arising The  gated  There  with  Corona power  theoretically  be a p p r e c i a b l e f o r l o n g time t h e high frequency  reasons  electric  that  frequency  this  power  power  loss radia-  0  due t o C o r o n a  has been  and e x p e r i m e n t a l l y ^ transmission lines radiation resulting D  extensively^ ^ ^^'  being  that  interferes  9  with  9  9  ^  9  9  investiand can  At the present from Corona i s  investigated rather radiation  must  effect  electromagnetic  being  this  which  One, t h e e l e c t r i c  discharges loss  effects  1 1 0  public  The  reason  communication  2 networks,  especially  strengths  generally  As  has  be  follows  designed  lines starts  at  that  the  of  a i r varies with  at  25°  and  to  76  cm  surface  charge  given  tensity area  can  One,  by  more  surface  be  to  than  ductors. giving has  The  less  been  lines  '  one  adopted '  This  to  at  has  the  It  there-  systems  should  occur  on  the  Corona  electric  dielectric dielectric  a value  of  29.8  known t h a t  the  reside be  per  a  on  field strength strength  kv.peak/cm  the  ways per  for  a  reactance extent  surface  electric A  but  surface  per  point  large  of  area,  phase  for high  in-  surface line  system.  or  using  two,  bundled  by  con-  advantage the  voltage  former,  the  case  of  one  of and  transmission  . thesis treats both  for  cross-  added than  area  sufficient  i . e . , using the  under  field  transmission  phase,  phase,  electric  is proportional  larger  decreased.  a l t e r n a t i v e has  some  surface  the  particular  providing  required  conductor  inductive  effect  Basically  The  conducting  i n two  latter  not  when t h e  reaches  but  to  may  the  line  do  conductor.  conductor  provide one  by  charge  obtained  using  section  Hence,  at  detrimental  conditions.  density  of  the  discharges  i t i s well  off a  amount  signal  pressure.  electrostatics  consideration. a  the  Hg.  just  where  transmission.  conductor  density,  intensity  the  a  power  surface  a i r surrounding  Prom field  of  is a  transmission  operating  o f f the  of  C.  Corona  Corona  surface  just  out  electric  that  areas  low.  i n general  normal  the  intensity  are  with  such  under  sparsely populated  been pointed  when a s s o c i a t e d fore  in  conductor  per  3 phase form  and s e v e r a l of circular  lines;  these  conductor  cylinders  p e r phase  In  this  treatment  so  that  this  field  circular tensity  used  the circumference  variation  i s minimized,  intensity  occurring  i s decreased.  conductors  p e r phase  the resulting  on each  conductor  o f t h e same p h a s e  one  circular  electric  o f each  i s made t o f i n d  i nt h e  f o r transmission  Using  i n a non-uniform  an attempt  field  Conductors  or stranded.  results  around  p e r phase.  are generally  may b e s m o o t h  distribution  mum  conductors  phase  field  conductor.  conductor  o r so t h a t With  shapes t h e maxi-  several  maximum bundle  field  in-  i s generally  12 not  equal  with the  each  than  i s due t o u n e q u a l  conductor  of a bundle  of earth,  capacitance  and t o t h e charge  The c a p a c i t a n c e i s o f c o u r s e t h e two o t h e r  phases  optimum  position  f o r two c a s e s .  t h e one a t w h i c h  mately  f o rthe individual  equal  on a l l c o n d u c t o r s  field  residing  bundle  intensity  i n one p h a s e  on  due t o t h e  In this  The optimum p o s i t i o n  t h e maximum  associated  and conductors  t h e o n e c o n s i d e r e d o f t h e same p h a s e .  indicated as  This  other>phases.  presence  the  .  other  thesis  conductors i s  i s .defined i s approxi-  bundle.  4 2. Electric  SOME BASIC PROBLEMS  transmission  long that they  may  be  lines  are  considered  electric  the  conductors i s i n general  respect  to the  conductors. cient the  to consider  to  due  the line  dimensional flow  however, s u f f i c i e n t two  portant  i s kept  reference  along  Gauss Law  Dipole  thus the  the  case,  of  suffithat i s ,  an  intrinsic  c a s e may  potential line  by  re-  is  this  be  treated  decrease  neglected,  approach.  some s i m p l e ,  but  im-  Lines a c e r t a i n amount o f along  magnitude of q d e s i g n a t e s the  electric  2  c o u l . /new.  field  an the  intensity  charge p l a c e d may Q  be = q  found  from  coul./m  infinite strength  electric straight of the  E at a d i s t a n c e  i n a medium o f  2  m  2itre E or  This  i s obtained  charge designates  meters from such a l i n e  Q  surface  concepts.  The  e  carries  transmission  c h a r g e , q coul./m, c o n c e n t r a t e d  vity  and  sections following w i l l describe  A line  By  with  purposes i t i s  length.  the  accuracy  of  negligible  electrostatic  p r o b l e m , and  2 - 1 . L i n e C h a r g e s and  line.  completely  length  at a c e r t a i n p o t e n t i a l with  charge per u n i t  to current  The  equivalent  dimension.  to the  component n o r m a l t o t h e  some s p e c i f i e d  electrostatic a two  component p a r a l l e l  Hence, f o r a l l p r a c t i c a l  transmission  spect  as  field  sufficiently  i n f i n i t e i n one  A l s o the line  field  generally  line. r  permitti-  P i g . 2-3  L i n e Charge and Conducting C y l i n d e r  5  E =  2 l t  |  volts/m  r  o  =  volts/in  (2-1)  where k = 2ne i n r a t i o n a l i z e d m.k.s. u n i t s which a r e used o throughout.  See P i g . 2-1.  The e l e c t r o s t a t i c f i e l d i s con-  s e r v a t i v e , t h a t i s , t h e work done on u n i t charge i n moving i t from one p o i n t t o another i s independent of p a t h . gradient  Hence,  (Potential) = -(Electric f i e l d intensity).  designate  Let V  t h e p o t e n t i a l a t some p o i n t , t h e n grad V = - E, and  s i n c e by symmetry V i s independent o f t h e a n g l e 6 t h e p o t e n t i a l d i f f e r e n c e between two p o i n t s P and P^ i s r V - V_ = - J  |  ]  r  |£  =  | (In r  x  - In r) volts.  (2-2)  l  A d i p o l e l i n e c o n s i s t s o f two l i n e charges each o f s t r e n g t h q, b u t of o p p o s i t e p o l a r i t y p l a c e d a s h o r t s, a p a r t .  U s i n g equation  due t o a d i p o l e l i n e i s in (r + + V = | 2g 2  Q  In ( r + 2  J -  -  distance,  2-2 i t f o l l o w s t h a t t h e p o t e n t i a l cos 6)2 j- v o l t s .  |2£ cos  (2-3)  6 ) 2  From F i g , 2-2 i t i s e v i d e n t t h a t r ^ = r + |j- cos 6 and s rg = r - 77 cos 6.  Then e q u a t i o n  V « £ I n (1 + f cos 6 ) A |  3-1 becomes s  c  ;  s  6  13  volts.  (2-4)  The q u a n t i t y (^.) i s denoted t h e s t r e n g t h of t h e d i p o l e 2T2.  line.  Conducting C y l i n d e r and L i n e Charge C o n s i d e r t h e i s o l a t e d system shown i n F i g . 2-3.  The  c o n d u c t i n g c y l i n d e r c a r r i e s a charge o f - q coul,/m, and t h e  line to  charge  has  determine  image  the  radius  a  charges  V  p  and at  =  - £ k  Considering plane  ference  k  this  points  together  point  P  this  at  centre  o  P  In  2  the  + t  —  plane  From  charges  at  same  are  (2-6) charge  of  rr  and  is well  arrangement  of  the  boundary In  Pg an  this the  equipotential  cylin-  -  2r  =  a  o6 )  equation (-) a  unit  radius,  coul./m  a,  the  and  also  p o t e n t i a l of  placed  parallel  two  of line  i t is  evident  re-  (2-6)  surface  as  the  produces  Pg  to  as  becomes  that  an  the  encylin-  line  equipotential  15 '  .  Hence  conducting a  required  conducting  i t follows  a  that  constant  as  v  (2-5)  1 6)2  cos  length  centre,  these  in  /  (2-5)  =  considerations together  surface  a c t u a l l y serves  equipotential per  shown  2  i n F i g . 1-3  - § In j&  to  line  (2-2). 1 5^ a_2 -  0  0  Two  r e s p e c t i v e l y as  2<srs r s cc oo ss  2 r A  known.  p o t e n t i a l due  --  + -  the  Pg  2  r =  charge  represents -q  of  p o t e n t i a l on  14 surface  way  aid  that  the  satisfied.  the  p o t e n t i a l and  For  symmetry P^  an  such  equation  A  A  zero  and  The  / a 4 ( ^ s~2  i t i s noted  der.  hand  produce  4  f o r Vp.  the  found  simplest  i s by  means t h a t  problem  i s from (i ss  i s at  closes  The  cylinder  i s that  P^  0.  Vp  Further  the  be at  coul„/m.  constant.  placed will  must  condition  s o l u t i o n to  2-3  +q  basically  problem  boundary  The  Fig.  the  is  of  p o t e n t i a l of  charges  of  surface  charges  strength  This  line  conditions  der  the  theory.  discrete  case  a  similar  Equation  cylinder  with  conducting  c y l i n d e r w i t h charge +q coul./m.  2 T> 2 Prom P i g . 2-3 t h e d i s t a n c e s = 2m + — and m = - — 1 ® 2 2 S u b s t i t u t i o n f o r s i n Equation Then s I - i<§>2 (1-6)  yields Vp = - J I n  (2-7)  D  2a  For a s i n g l e phase t r a n s m i s s i o n l i n e D i s t h e d i s t a n c e ween c e n t r e s of t h e c o n d u c t o r s . so t h a t e q u a t i o n  However, i n g e n e r a l  betD»a  (2-7) becomes  T h i s i s e q u i v a l e n t t o assuming the image charge a t p o i n t P^ i n F i g . 1-3 i s p o s i t i o n e d a t p o i n t 0, and f o r D > 1 0 a t h e e r r o r i n Vp i s l e s s than 1$. The e l e c t r i c  f i e l d i n t e n s i t y a t t h e s u r f a c e o f t h e con-  d u c t i n g c y l i n d e r shown i n F i g . 2-3 may be found as f o l l o w s . Consider  Equation  (2-5).  •| I n ( s + r 2  2  - 2 r s cos 6)  = i n s - | i n (1 - f e ^ ) ( l - § e^' ) 6  and 1 a 2* I n (~2 +  2  4  1  8  = In r But i n g e n e r a l  a ™ 2r —  r  7j  In  2  cos 6)  -* (1 . iL36 sr e  I n ( l - z) = -s  ) ( 1  „ a sr 2 3 z 3  for  Therefore X  CO  2 2 v~ I n (s + r - 2 r s cos 6) = I n s - )_  i* i i X  (—)  — cos n6  z < 1,  and 4  2  i In (\  + r  f  for  2  1  0 - 0  - 2r f - cos 6) = I n r and  - Jk  2  ( ~ r ) ~ cos n6 n  — < 1, sr  S u b s t i t u t i o n i n E q u a t i o n (1-5) t h e n Vp* =  Y  eo  I n ir + Jk  IY  yields  ±n cos n6 | ( s' £)  n  - s( ^r -')  L e t o denote s u r f a c e charge d e n s i t y i n coul./m electric f i e l d intensity  (2-8)  n  x  . Then the  E must s a t i s f y the e q u a t i o n E = - — o 0  e  at the boundary between a c o n d u c t i n g medium and a d i e l e c t r i c medium of p e r m i t t i v i t y  e . Q  The e l e c t r i c f i e l d i n t e n s i t y medium, hence grad (V ) p'r=a  (E)'r=a  Then from E q u a t i o n -  dV (-j-^) dr r=a = - (E)r=a v  (2-8)  ( E )I . _ „ =- a. S  r=a  becomes  v  6  i s normal t o a c o n d u c t i n g  (i(k a  +  2  <>©  n-1  I  a—  cos  n6)  or ^r=a  1 + 2  ~ ~ ka  when a = - r^— 1 1 + 2 2ita s  Y 1  Y 1 (—)  (—) 8  n  n  cos n6 j v o l t s / m  (2-9)  (2-10)  cos n6i coul./m*  which i s the e l e c t r i c charge d e n s i t y on the s u r f a c e of the c o n d u c t i n g c y l i n d e r shown i n F i g . 1-3. E q u a t i o n (2-9) may be put i n t o a d i f f e r e n t , form as f o l l o w s .  The  but e q u i v a l e n t  expansion  z + ... i s v a l i d f o r •s 1 = 1 + z + 7?— 1 — z £  <  1.  Let  z = Be*'  where  6  ^ 1 - Re  0t R < 1  = 1 + Re^ + R 6  then  J  2 e  2  +  6  o o o  16 Consider the expression 1 - R cos 6 1 - 2 R cos 6 + R  1 2 (1 - R e ^ )  = 2  1 2 (1 - Re~^ )  6  5  = | (1 + R e ^ + . . . ) + | (1 + Re""J + ...) 6  6  or CO  * - 1 + I R 1 - 2 R cos 6 + R 1 E q u a t i o n (2-11) may be w r i t t e n 1  R  c  o  s  6  2 R cos 6 - R 1 - 2R cos 6 + R  cos n6.  n  (2-11)  oo  £ n 1  =  R  c o g  n  6  ( 2  _  1 2 )  When a d d i t i o n of E q u a t i o n s (2-11) and (2-12) y i e l d s i  n "  2  =1 + 2  1 - 2 R cos 6 + R  ^ Z R  n  cos nfi  (2-13)  1  Comparison o f E q u a t i o n s (2-9) and (2-13) shows t h a t w i t h R = — 1 .(*) ( E )  r=a = " k r=a Ka  ~  2  2  7^2  ( 2  ~  1 4 )  - cos 6 + (~r s s which i s t h e e x p r e s s i o n u s u a l l y g i v e n f o r t h e case of a l i n e 13 charge p l a c e d p a r a l l e l t o a c o n d u c t i n g c y l i n d e r ;  x  „  2  T h i s p a r t i c u l a r problem has been t r e a t e d i n d e t a i l i n s p i t e o f t h e f a c t t h a t i t s s o l u t i o n appears i n p r a c t i c a l l y every t e x t b o o k on e l e c t r o m a g n e t i s m ^ ' H o w e v e r ,  this  problem forms t h e b a s i s f o r most of t h e work t o f o l l o w and i t was t h e r e f o r e thought w o r t h w h i l e t o c o n s i d e r t h i s s i m p l e case r a t h e r thoroughly„  10 3. VARIATION OP ELECTRIC FIELD INTENSITY WITH PHASE ANGLE AND PHASE SPACING In a three-phase system t h e charge on each conductor v a r i e s w i t h t i m e , hence i n order t o determine t h e maximum f i e l d intensity  o c c u r r i n g on any one conductor t h i s time de-  pendence must be t a k e n i n t o account.  F u r t h e r one would ex-  p e c t t h e phase angle a t which t h i s maximum f i e l d  intensity  occurs t o be dependent on t h e geometric c o n f i g u r a t i o n of a t r a n s m i s s i o n l i n e system. termine these  Hence t h e o b j e c t here i s t o de-  relations.  F i g . 3-1 shows a three-phase system w i t h ungrounded neutral.  F o r t h e purpose o f computing t h e p o t e n t i a l s o f t h e  conductors i n terms of t h e unknown charges, t h e conductors and t h e geometry  of t h e system one can make t h e f o l l o w i n g  approximation.  The s p a c i n g s i s s u f f i c i e n t l y l a r g e t h a t t h e  p o t e n t i a l s e x t e r n a l t o the conductors can be produced by l i n e charges p o s i t i o n e d a t t h e c e n t r e o f t h e c o n d u c t o r s . a c t u a l l y i m p l i e s t h a t one assumes t h e p o t e n t i a l  This  contribution  due t o one phase conductor t o be c o n s t a n t over t h e s u r f a c e of t h e o t h e r two.  The e r r o r committed by u s i n g t h i s  f o r a s i n g l e phase l i n e has been mentioned a l r e a d y .  procedure See s e c -  t i o n 2-2. 3-1. D e t e r m i n a t i o n o f L i n e Charges The three-phase t r a n s m i s s i o n l i n e shown i n F i g . 3-1 i s assumed ungrounded and s u f f i c i e n t l y f a r removed from ground t h a t t h i s has no i n f l u e n c e on t h e e l e c t r i c f i e l d  configuration  Phase 2  Phase 1  V^ = V cos ojt  V  2  = V cos (w-t-120)  Phase 3  Vg = V cos(cot-240)  F i g . 3-1 Ungrounded three-phase T r a n s m i s s i o n L i n e  Phase 1  Phase 2  Phase 3  *1  F i g . 3-2 Conducting C y l i n d e r and two L i n e Charges  11 around t h e c o n d u c t o r s .  Hence one can w r i t e q  f o r any phase a n g l e .  q.  +  2  =  21 0 1  = ^  (ln  J  (3-1)  (2-2).  I n £ + £i I n - + £3 a k s k  k  = 0  3  A l s o , making use of t h e a p p r o x i m a t i o n  mentioned above and E q u a t i o n V  q.  +  2  2s a  l n  - l n 2) + ™  I  (In  - l n 2)  (3-2)  where k = 2ite  o  Similarly V  2 3 = IT  f  ( l n  +  f  l n  IT  } +  ( l n  "  2  l n  f  }  (3™3)  A d d i t i o n of E q u a t i o n (3-2) and (3-3) y i e l d s , 3 q V  V  21  +  V  23  =  V  23 = TT  2  s  q  f - IT  l n  2  l n  2  °  But V  21  +  ~  c o s  1 2  °) "  V  c o s  wt + V cos (cat - 120)  - V cos (wt - 240) = 3 cos (wt - 120). Hence one may s o l v e f o r q q  and  2  k V cos (cot - 120) _ , = -—™ f — — coulo/m  /„ „\ (3-4)  L  0  ^  ln  I -i  ln 2  Further 12 = ^ and  V  l l 3 = TT 2 q  v  (In „_  ( l n  f  + I n 2) + ^  S  s . +  l n  2 )  +  „  x  (ln  T  . *2  IT  l n  I 2s  + l n 2)  (3-5)  ^"  3 6)  12 But  +  ^  =  Then s u b s t i t u t i n g f o r q  c o s  t i o n (3-4) and s o l v i n g f o r q^. 1 k(V I n |cos art - V3~2 I n 2 s i n wt) q, = =—-j— — — o r ( I n - - i I n 2) I n ±1 a 3 a 1  2  from Equa-  eoul./m  (3-7)  Then one may f i n d q^ from E q u a t i o n (3-1) q. = - ( q + ^ 3  j .  2  k j v I n | cos (wt - 240) + V 3 ~ I n 2 s i n (wt - 240)) _ r-— —o~— ( i n f - | I n 2 ) ( i n §S) 2  coul./m (3-8)  3-2. D e t e r m i n a t i o n o f t h e Phase Angle a t which Maximum E l e c t r i c F i e l d I n t e n s i t y Occurs The r e l a t i o n s h i p between t h e maximum f i e l d i n t e n s i t y on phase conductor 1, phase angle wt and s p a c i n g s w i l l f i r s t be determined.  As shown i n F i g . 3-2 phase c o n d u c t o r s 2 and 3  are c o n s i d e r e d l i n e charges of s t r e n g t h q purpose.  See Appendix I .  and q^ f o r t h i s  2  Then u s i n g E q u a t i o n ( I - l ) one can  determine t h e s u r f a c e charge d e n s i t y on conductor 1 from t h e equation ° ! <«•«« - 5sr - i § f  <t> <">* n  n 6  - £  \  <fe>  n  c  °  s  n 6  C l e a r l y t h e maximum charge d e n s i t y w i t h r e s p e c t t o 6 occurs a t 5 = 0°  when  co  Y ^  co v  (*) s'  n =  s-a  and  Y ^  ( f - ) .= ^ ± 2s' 2s - a n  v  F u r t h e r t h e maximum v a l u e o f o^ w i t h r e s p e c t t o wt i s found from  <-> 3  9  do, ^dwV 6-0 = 0  dq.  dq. a na 2s-a d&)t  a ^2 na s-a dwt  ~ 2na dwt  S u b s t i t u t i o n o f t h e v a l u e s of q^, q^ and q^ from Equations ( 3 - 4 ) , (3-7) and (3-8) y i e l d s t a n (art)  1  -  •*i •n  ,I„n S£ &o  a,  B  In — +  a  1 1 In 2 a 32 2s a 3 , + In — 1 2s-a S™9» 3* 32 2  a  In 2  In 2 a 2s-a 1 32  s a  (3-10)  To o b t a i n a r e p r e s e n t a t i v e v a l u e o f art l e t a = 1 i n c h and s = 25 f t . Then  t a n (art)^ = -.073  o r a)t = 4.2°  Prom E q u a t i o n (3-7) i t i s noted t h a t f o r a = 1 i n c h s = 25 f t . occurs a t *1 max t a n (wt) ^ = -  ^  l n  2  —  or wt = -4.0°  32 i n 300 Hence the charge on conductor 2 and 3 have n e g l i g i b l e i n f l u e n c e on t h e phase angle o f o"^. An e q u a t i o n s i m i l a r t o (3-10) may be d e r i v e d f o r t h e centre conductor. 1  t a n (wt),  2 !n  3 2  -T For  l n  i„  2  Z"  2 3  x  T  +  2a i s + a.  s  1 x 1  +  n !  Q,  1  t a n ( w t ) = -1.76  (s + a)  3  a  ~  3  2  3  a  s - a  (s - a)  s = 2 5 f e e t one f i n d s or (wt)g = 119.6  a s - a  ln 2  £L  In 2  2a l n 2  a = 1 inch 2  U s i n g t h e same procedure as f o r conductor '.  (  ln  SL a  l n - (3-11) a  14 I t may be noted t h a t  i s i n phase w i t h Vg.  See E q u a t i o n  (3-2). F o r p r a c t i c a l t r a n s m i s s i o n l i n e s l n ( - ) l i e s i n t h e narrow 19 range 5.3 t o 6.4  .  T h e r e f o r e , t h i s q u a n t i t y cannot cause  much v a r i a t i o n i n t h e phase angles  (ojt)-^ and (cot);,.  s p a c i n g between l i n e s i s g e n e r a l l y determined  The  by t h e i n s u l a -  t i o n l e v e l r e q u i r e d , t h a t i s , t h e magnitude of t h e o v e r v o l t a g e s expected t o occur i n t h e system.  The u s u a l spacings range  from 10 t o 12 i n c h e s f o r each 10 kv rms. between l i n e s w i t h a 22 minimum s p a c i n g o f about 20 i n c h e s . e x p r e s s i o n s determined  Hence c o n s i d e r i n g t h e  f o r (oot)^ and (ootjg i t i s e v i d e n t t h a t  f o r a l l p r a c t i c a l purposes one can assume t h a t t h e maximum f i e l d i n t e n s i t y and charge p e r u n i t l e n g t h occurs on any one cond u c t o r when i t s phase v o l t a g e ' h a s i t s peak v a l u e .  This pro-  cedure i s f o l l o w e d i n t h e a n a l y s i s c a r r i e d out i n chapter 4 and 12 13 19 5.  I t has, been p o i n t e d out by s e v e r a l a u t h o r s  '  '  that  the maximum charge occurs on t h e c e n t r e phase f o r t h e c o n f i g u r a t i o n shown i n F i g . 3-1, equations  T h i s f a c t can be v e r i f i e d from  ( 3 - 4 ) , (3-7) and ( 3 - 8 ) .  15 4. NON-CIRCULAR CONTOURS In t h i s c h a p t e r some s p e c i f i c , n o n - c i r c u l a r conductor shapes a r e c o n s i d e r e d , t h e o b j e c t b e i n g t o determine conductors used i n a three-phase  i f such  t r a n s m i s s i o n l i n e would p r o -  v i d e a f i e l d c o n f i g u r a t i o n around each conductor o f g r e a t e r u n i f o r m i t y than i s obtained w i t h c i r c u l a r conductors.  For  t h i s purpose t h e phase conductors a d j a c e n t t o t h e one cons i d e r e d a r e t r e a t e d as l i n e  charges;  4-1. C a s s i n i a n Oval R e f e r t o F i g . 4-1.  Two p o s i t i v e l i n e charges each o f  s t r e n g t h ^ coul./m a r e p l a c e d a t p o i n t s (s,0) and (-s,0) respectively.  I t f o l l o w s t h a t t h e p o t e n t i a l a t any p o i n t  P i s g i v e n by t h e e q u a t i o n 1 V  p  =  2k  l  n  (  r  + s  2  2  1  - 2 r s cos 6 ) + | g l n ( r 2  2  + s +K  where k = 2ite  and K  o  o  2  o  + 2 r s cos 6 ) (4-1)  i s a constant reference p o t e n t i a l .  the e q u i p o t e n t i a l s produced  Also  by these two l i n e charges a r e gov-  erned by t h e e q u a t i o n r  where c i s a c o n s t a n t . and r  4  l  2 r  2  2  =  4 C  S u b s t i t u t i o n of the values f o r r ^  yields + s  2  - 2r s 2  2  cos 26 = c  2  (4-2)  4  20 21 which i s t h e e q u a t i o n of a C a s s i n i a n Oval  '  f o r c > a.  From image t h e o r y (see s e c t i o n 2-2) any one o f these contours  P i g . 4-1 C a s s i n i a n Oval  P i g . 4-2 Charge D i s t r i b u t i o n on C a s s i n i a n Oval  16 may be c o n s i d e r e d a c o n d u c t i n g s u r f a c e .  Hence one can f i n d  the s u r f a c e charge d i s t r i b u t i o n on a conductor h a v i n g t h i s shape by d e t e r m i n i n g  grad  from E q u a t i o n ( 4 - 1 ) .  F u r t h e r , when a  conductor i s p l a c e d s i n g l y i n space and i s kept a t a c e r t a i n potential  w i t h r e s p e c t t o some r e f e r e n c e t h e s u r f a c e charge on  the conductor w i l l assume a s p e c i f i c d i s t r i b u t i o n depending on t h e shape o f t h e conductor.  T h i s charge d i s t r i b u t i o n i s termed  "the f r e e charge d e n s i t y " i n what In p o l a r  co-ordinates  tgrad V P p  (^£)2  follows.  1 (1 ^ £ ) 2 2  +  (4-3)  Then u s i n g E q u a t i o n (4-1) op q / r - s cos 5 „ = V o + dr  N  2  r + s cos 6\ o  k  1 o'p r 06^  q  2k  /S s i n 5  (4-4)  >  s s i n 6\  •1  4  (4-5)  2  Hence e v a l u a t i n g E q u a t i o n (4-3) by a i d of (4-4) and (4-5) _ _2£. ' kc  grad V  The f r e e charge d e n s i t y , o(6) = e o(6) =  (4-6)  2  2ite  grad V  o coul./m  Choosing t h e v a l u e s s = 0.75 cm  hence (4-7)  2  c = 1.00 cm  one may  p l o t t h e contour shown i n F i g . 4-1 by u s i n g t h e e q u a t i o n r =  s  2  cos 26 + ( s c o s 4  2  26 - s  4  1 1 + c )^' 2  2  A l s o , t h e f r e e charge d e n s i t y was computed f o r t h i s  (4-8) contour  17 by a i d of E q u a t i o n  (4-7).  See F i g .  4-2  .  From the charge  d i s t r i b u t i o n o b t a i n e d i t appears t h a t a conductor shape as shown i n F i g .  4-1  might be s u i t a b l e f o r the c e n t r e conductor  of a three-phase t r a n s m i s s i o n l i n e w i t h f l a t s p a c i n g , t h a t i s , w i t h t h e o t h e r two phase conductors  p l a c e d e q u i d i s t a n t from  the c e n t r e conductor on the p o s i t i v e and n e g a t i v e y - a x i s r e spectively.  However, the C a s s i n i a n Oval w i l l not be  considered  any f u r t h e r h e r e . 4-2.  E l l i p t i c C y l i n d e r and  two L i n e Charges. 25  C o n s i d e r the f u n c t i o n  13 '  w = z + — z  (4-9)  Here w i s seen t o be an a n a l y t i c f u n c t i o n of z w i t h s i n g u l a r i t i e s o n l y a t z = 0 and z  =oo.  Further  TO- -? ^° 1  except a t z = - d.  Hence a mapping performed by u s i n g  would not be conformal  a t these two p o i n t s .  R e f e r t o the c i r c u l a r c o n t o u r , 4-3.  (4-9)  See Appendix I I .  z = ae** , shown i n F i g .  I f t h i s contour i s mapped onto the W-plane by a i d of  Equation  (4-9) i t i s e v i d e n t W  =  ae  j 6  + ^Lr ae  (4-10)  J  By e q u a t i n g r e a l and imaginary p a r t s and e l i m i n a t i n g 6 from (4-10) i t i s found t h a t  F i g . 4-3 C i r c u l a r C y l i n d e r and two L i n e Charges  u  F i g . 4-4 E l l i p t i c C y l i n d e r and two L i n e Charges  18 which i s t h e e q u a t i o n 4-4.  of an e l l i p s e i n t h e W-plane.  See F i g .  Here d i s chosen such t h a t d < a, t h a t i s , t h e two  p o i n t s (-d,0) and (d,0) a r e i n s i d e t h e c i r c u l a r contour i n the Z-plane.  Hence t h e mapping i s conformal f o r a l l p o i n t s  i n the r e g i o n  z j 5> a.  A l s o i t i s noted from E q u a t i o n (4-9) >" a  t h a t t h e mapping i s one t o one between t h e Z-plane f o r and t h e W-plane f o r t h e r e g i o n e x t e r n a l t o t h e e l l i p t i c t o u r g i v e n by E q u a t i o n  con-  (4-11).  F i g . 4-3 shows a c o n d u c t i n g  c y l i n d e r w i t h charge q coul./m  p l a c e d p a r a l l e l t o two l i n e charges each of s t r e n g t h  coul./m.  T h i s system i s analogous t o t h e one t r e a t e d i n s e c t i o n 2-2 and can a l s o be s o l v e d by image t h e o r y . P l a c i n g two image l i n e 2 2 charges a t p o i n t s P- (0, ) and P (0» - ~ ) r e s p e c t i v e l y i n ft  L  2  F i g . 4-3 t h e complex p o t e n t i a l , Q = V + j U , a t p o i n t P i n t h e Z—plane i s g i v e n by .a-2  (4-12)  =, .a z + j — s 2  z - j ~  where K o i s a constant r e f e r e n c e p o t e n t i a l and k = 2ize o . Here the complex p o t e n t i a l i s used o n l y f o r convenience. The r e a l potential V  P  = Re (Q ) i . e . P  , l n r + s . - 2 r s s i n 6 + ,l,n —r — + s + 2 r s sin§ 4k 2 a _ a . . 2 a _ a . r + -TT + 2 r — sm6 r + —TT - 2 r — s i n 6 \ d s £ s s s ( -13) 2  p  2  4  2  n  2  4  2  2  K  4  or (V N  ) = i p'r=a 2k  therefore V  P  2 l n 5. + K a o  i s a constant  on t h e c i r c u l a r  (4-14) contour as r e q u i r e d .  19 Mapping t h e system shown i n F i g . 4-3 i n t o t h e W-plane by a i d of t h e elliptic  f u n c t i o n (4-9) i t f o l l o w s t h a t one o b t a i n s an  c y l i n d e r p l a c e d p a r a l l e l t o two l i n e charges i n t h e  W-plane.  See F i g . 4-4.  on t h e c i r c u l a r  From E q u a t i o n (4-14) t h e p o t e n t i a l  c y l i n d e r i n F i g . 4-3 i s c o n s t a n t .  that the e l l i p t i c  T h i s means  c y l i n d e r i n t h e W-plane has t h e same c o n s t a n t  p o t e n t i a l s i n c e t h a t boundary c o n d i t i o n i s unchanged by a c o n formal transformation.  See Appendix I I .  F u r t h e r , t h e two  image l i n e charges p l a c e d a t p o i n t s P^ and Pg i n t h e Z-plane may w e l l be p o s i t i o n e d o u t s i d e t h e e l l i p t i c  contour i n t h e  W-plane, e.g., when s » a /x  2 d  .  s  .d s a 2  However, t h e s e l i n e charges a r e not s i n g u l a r i t i e s region  z  ^ a which i s c o n s i d e r e d h e r e .  f o r the  As p o i n t e d out  e a r l i e r t h e mapping i s one t o one f o r t h e e x t e r n a l a r e a s , hence the  l i n e charges a t p o i n t s Pg and P^ i n t h e Z-plane map i n t o  the  p o i n t s Pg and P^ i n t h e W-plane. U s i n g E q u a t i o n ( I I - 8 ) which i s r e w r i t t e n here f o r con-  venience E w =  E  z  dz dw  (H-8)  one can t h e n determine t h e charge d e n s i t y on t h e e l l i p t i c c y l i n d e r shown i n F i g . 4-4. For t h i s purpose dV — £ must be determined from E q u a t i o n ( 4 - 1 3 ) . dr  grad (Q ) p  I t i s evident  t h a t t h e e x p r e s s i o n o b t a i n e d w i l l be analogous t o t h e one der i v e d i n s e c t i o n 2-2 f o r t h e case of a c i r c u l a r c y l i n d e r and one l i n e charge.  See E q u a t i o n ( 2 - 1 4 ) .  20 Then /  d V  dr  I /  aq  T >P\  'poa  2ka L \  " ( fs> a _ | s s  1  +  2  (  )  2  2  g  i  <l> s  1  n  6  1  +  (  a 2 }  +  2  s  |  g i n 6  2  s  or E  z=aeJ  = fa "I 4 " % 2 ~ s + a + 2a s cos 26  6  ( 4  "  1 5 )  F u r t h e r , d i f f e r e n t i a t i n g E q u a t i o n (4-9) w i t h r e s p e c t t o w 1 _ 5L2. dw  o  d_ dz z2 dw  Hence dz dw  a (a  4  + d  4  2 (4-16)  T  - 2a d 2  2  cos 2 6 ) ^  S u b s t i t u t i o n o f E q u a t i o n s (4-15) and (4-16) i n E q u a t i o n ( I I - 8 ) t h e n y i e l d s t h e e l e c t r i c f i e l d i n t e n s i t y normal t o t h e s u r f a c e of t h e e l l i p t i c  cylinder  \ s4 - a4 4 I 7 2 2 17 s + a + 2a s cos 26  E  (a  4  + d  4  - 2a d 2  cos 2 6 ) /  2  2  The s u r f a c e charge d e n s i t y thus becomes _ q_ / °" ~ * I s 2  s - a + 2a s 4  + a  4  4  2  4  2  \/ cos 26 '  a  (a  4  +• d  4  .  - 2a d 2  2  \ *  cos 2 6 ) I  coul./m  2  (4-17)  2  A l s o i t can be seen from E q u a t i o n s (4-15) and (4-16) t h a t t h e f r e e charge d e n s i t y on t h e e l l i p t i c  c y l i n d e r i s g i v e n by  CL, «  coul./m  S (a  4  + d  4  :  - 2a d 2  2  r  2  (4-18)  cos 2 6 ) 2 2  To c o n s i d e r a n u m e r i c a l case l e t d  2 = 1.40 cm ., a = 2 cm.  009  a  .008  6 i n degrees P i g . 4-5 Charge D e n s i t y V a r i a t i o n  21 p  and s = 4.68 cm.  The v a l u e s chosen f o r d  and s a r e such  t h a t o* e v a l u a t e d from E q u a t i o n (4-17) has equal magnitude a t w  6 = 0 ° and 6 = 90°.  P i g . 4-5 shows a as a f u n c t i o n of t h e ° w  angle 6. In order t o determine  t h e improvement o b t a i n e d i n t h e  charge d i s t r i b u t i o n by u s i n g an e l l i p t i c c y l i n d e r i n p l a c e o f a c i r c u l a r c y l i n d e r one must i n v e s t i g a t e an e q u i v a l e n t system of a c i r c u l a r c y l i n d e r and two l i n e charges. used here t o determine  t h e dimensions  The c r i t e r i a  o f t h i s system a r e . t h e  following: i ) Equal p e r i m e t e r s of t h e e l l i p t i c and t h e c i r c u l a r c y l i n d e r i i ) A l l l i n e charges must be t h e same d i s t a n c e from t h e geometric centre of the conducting  cylinders. 26  The p e r i m e t e r , S, o f an e l l i p s e i s g i v e n by 2ll ! S = b j (1 - | e s i n o ) ^ da 0 2  1  2  (4-19)  where b = major a x i s e-^ = e c c e n t r i c i t y . Prom e q u a t i o n (4-11) and t h e v a l u e s a l r e a d y chosen f o r t h e c o n s t a n t s one f i n d s 2 b = 2.70 and e^  = .768.  Hence expanding t h e r a d i c a l i n E q u a t i o n (4-19) and i n t e g r a t i n g the f i r s t t h r e e terms S = 13.08 cm. Therefore the radius of the equivalent c i r c u l a r c y l i n d e r i s r  i  =  1  i i ^= 2  0 8  c m  -  22 F u r t h e r , from Equations (4-9) and (4-11) and t h e v a l u e s chosen o above f o r d , a and s t h e d i s t a n c e O'P^ i n F i g . 4-4 i s °' 3 " * " = 4.38 cm P  ( 4  6 8  C m  which i s t h e d i s t a n c e s i n F i g . 4-3 f o r t h e e q u i v a l e n t E v a l u a t i o n of t h e change d e n s i t y on t h e e q u i v a l e n t  system.  circular  c y l i n d e r from E q u a t i o n (4-15) w i t h a = 2.08 cm and s = 4.38 cm y i e l d s t h e curve shown i n F i g . 4-5. Comparing t h e curves shown i n F i g . 4-5 i t i s c l e a r t h a t the e l l i p t i c  c y l i n d e r provides  t h e b e t t e r charge d i s t r i b u t i o n .  One can, . for ' f o r i n s t a n c e , compare t h e q u a n t i t i e s a max - o"min the two c a s e s . Thus f o r t h e c i r c u l a r c y l i n d e r a - a . max min J  .454 as opposed t o .112 f o r t h e e l l i p t i c  cylinder.  Hence i t  would be o f advantage t o use an e l l i p t i c c y l i n d e r f o r t h e c e n t r e conductor i n a t r a n s m i s s i o n l i n e system w i t h f l a t s p a c i n g . 4-3. A p a r t i c u l a r n o n - c i r c u l a r C y l i n d e r and two L i n e Charges. The case of a c i r c u l a r c y l i n d e r p l a c e d p a r a l l e l t o a l i n e charge was t r e a t e d i n s e c t i o n 2-2.  From E q u a t i o n (2-9) t h e  maximum f i e l d i n t e n s i t y occurs a t 6 = 0°, and i t was f e l t  that  t h i s f i e l d i n t e n s i t y c o u l d be decreased by f l a t t e n i n g t h e c i r c u l a r c y l i n d e r i n the region 6 = 0 ° .  See F i g . 2-3.  To i n v e s t i g a t e t h i s t h e f u n c t i o n g i v e n by (4-9) was used t o map t h e c i r c u l a r c o n t o u r , z = j y + ae^ , shown i n F i g . 4-6 Q  i n t o t h e W-plane.  The e q u a t i o n  g o v e r n i n g t h i s contour i n t h e  W-plane becomes w = jy + ae  j 6  + jy  o  ^ + ae  (4-20) J U  F i g . 4-6 C i r c u l a r C y l i n d e r and L i n e Charge  V  ^ \  I  0«  r  l  W-plane  /  i  u  ,-q.  P i g . 4-7 Deformed C y l i n d e r and L i n e Charge  23 or s e p a r a t i n g r e a l and imaginary u = a coso + c  a  2  d a coso 2 + y + 2ay Q  parts sin6  Q  2 t • c x d (a s m 6 + y ) v = a sin6 + y 5 o ~ ' + y~ + y ~ S O "o To determine t h e shape of a p a r t i c u l a r contour l e t  (4-21)  A  a  a = 2.0,  d  2  =0.50  2 a  and y  s i n  =0.50 o  computing t h e u and v c o - o r d i n a t e s  from (4-21) one o b t a i n s t h e  curve shown i n P i g . 4-7, and i t i s e v i d e n t t h a t a c y l i n d e r i s o b t a i n e d which i s deformed i n t h e d e s i r e d manner. Prom s e c t i o n 4-2 i t f o l l o w s t h a t one can o b t a i n a conformal mapping o f t h e system shown i n F i g . 4-6 by a i d o f t h e f u n c t i o n (4-9).  A l s o t h e g e n e r a l d i s c u s s i o n g i v e n i n s e c t i o n 4-2 r e -  g a r d i n g t h e mapping f u n c t i o n and t h e mapping i t s e l f a p p l i e s t o t h i s s e c t i o n as w e l l . R e f e r t o F i g . 4-6.  The case of a conducting  cylinder  p l a c e d p a r a l l e l t o a l i n e charge has a l r e a d y been t r e a t e d i n s e c t i o n 2-2.  Hence t h e complex p o t e n t i a l , Q = V + j U , a t a  point P i s 2  = S  P  l  n  z + j(s - y ) 2 — o • /a \ z + o(— - y ) +  K  < 4  2 2 )  0  where K  Q  i s a constant  r e f e r e n c e p o t e n t i a l and k = 2 i t e . o  Then  mapping t h i s system i n t o t h e W-plane one o b t a i n s t h e c o n f i g u r a t i o n shown i n F i g . 4-7. on t h e t r a n s f o r m e d (II-8).  I t f o l l o w s t h a t t h e charge d e n s i t y  c y l i n d e r can be determined from  Hence e v a l u a t i n g  Equation  24 E z from E q u a t i o n E  ~ dz ' v  v  dz  ;  (4-22)  z=jy H-ae**" 0  k  s  a  2  s2 - a2 + a + 2as s i n 6  (4-23)  2  F u r t h e r , from E q u a t i o n (4-9) _2  dz dw  z2 - d 2  but i n t h i s case z = j y + a eJ o , t h e r e f o r e J  Q  dz dw  (jy  0  (jy  0  J6x2 .+ a e ) J  + aeJ ) 6  2  - d  2  when dz dw 2 a 2 + y^ o + 2ayo w  (a  2  + y  2 Q  + 2ay  s i n 6 ) - 2 d ( a c o s 2 6 - y - 2 y a sin6) + d 2  Q  sin8  2  2  2  Q  4  o  (4-24) S u b s t i t u t i o n o f E q u a t i o n s (4-23) and (4-24) i n E q u a t i o n  (II-8)  yields E w  ka a  2 2 (a + y + 2 a y Q  Q  2  sin6)  + y 2  2 Q  - 2d  + ay 2  Q  sin6  2 2 (a cos 2 6 - y - 2 y a s i n 6 ) + d 41 2 Q  Q  = — oa s 2 2 s + a + 2as s i n 5 2  2  Hence t h e s u r f a c e charge d e n s i t y can be determined from  (4-25)  25 2na a2 + y 2 + 2 a y Q  (a  + y  2  2 Q  + 2ay  = — ao s s2 + a 2 + 2as 2  Q  sin6)  2  - 2d (a cos 2  2  Q  sin6 2 6 - y - 2 y a sin6) + d 2  Q  4  o  2  (4-26)  s mco  To c o n s i d e r a s p e c i f i c n u m e r i c a l example l e t a = 2 cm, 2 2 y  Q  = .0625 cm, d  = .0625 cm , s = 60 cm.  Using these values  i n E q u a t i o n (4-26) one o b t a i n s t h e curve shown i n F i g . 4-8. Then, t o determine t h e charge d e n s i t y v a r i a t i o n on t h e e q u i v a l e n t c i r c u l a r c y l i n d e r (see s e c t i o n 4-2) one must f i n d t h e d i s t a n c e 0'P| and t h e p e r i m e t e r of t h e deformed contour. F i g . 4-7.  See  The l e n g t h O'PJ can be found from E q u a t i o n ( 4 - 9 ) .  Thus O'P' = s - y 1 o J  j2 - — s - y  Q  or O'P^ = 59.94 A l s o from F i g . 4-6 OP-j^ = 60 cm. T h e r e f o r e O'P^ = 0 P  1  i n this  case.  The f o l l o w i n g procedure was used t o determine t h e p e r i meter, S, of t h e deformed c o n t o u r . w  l e t |w  = oy, + a e ^ +  From E q u a t i o n (4-20)  .2 d<  jy  Q  3* + ae'  = B t h e n e v a l u a t i n g t h e r i g h t hand s i d e of t h i s e q u a t i o n  .0057 a  . 0045' -90  -70  -50  • \-30  ' -10  10  • 30  6 i n degrees P i g . 4-8 Charge D e n s i t y V a r i a t i o n  » 50  • 70  90  I R =  2  a  + yO  2  26 12 2 2 \— Tjd +a co s 2 6 - y -2y a s i n 6 \ 2  2  Q  + 2y O a sin§ + 2d  d  _  d  + y  a  ft  .  c  • + 2y a s i n 6 (4-27)  The p e r i m e t e r , S, can t h e n be found from 271  S =  p R  2  +  o The  integral  ,dRx2 W  1 d6  2  (4-28) was  (4-28)  e v a l u a t e d n u m e r i c a l l y by a i d of Le27  gendre-Gauss Quadrature  and the ALWAC I I I - E d i g i t a l computer. 2  Thus f o r the v a l u e s chosen above f o r a, y , d S = 12.57345  and s  cm.  as compared t o t h e p e r i m e t e r S  of the o r i g i n a l c i r c u l a r  con-  c tour,  S  = 2Tta = 12.56637  cm.  c Hence t h e d i f f e r e n c e between S and S  i s negligible.  Therefore  c the charge d e n s i t y on the e q u i v a l e n t c y l i n d e r was E q u a t i o n (4-23) w i t h a = 2.0 t a i n e d i s shown i n P i g .  cm  s = 60.0  cm.  computed from  The curve  ob-  4-8.  Comparison of the two  curves shown i n F i g . 4-8  indicates  t h a t b o t h the maximum and minimum f i e l d i n t e n s i t y on the deformed contour i s l e s s t h a n on t h e c i r c u l a r c y l i n d e r .  Specifically,  t h e d i f f e r e n c e between the two f i e l d i n t e n s i t i e s a t 6 = i s 3.5$  f o r t h i s case.  shape shown i n F i g . 4-11  -90°  T h e r e f o r e , u s i n g a c y l i n d e r of the f o r the o u t e r conductors  i n a three-  phase system w i t h f l a t s p a c i n g would p r o v i d e some improvement i n t h e s u r f a c e charge d i s t r i b u t i o n . In o r d e r t o r e p r e s e n t an a c t u a l three-phase  system w i t h  f l a t s p a c i n g an a d d i t i o n a l l i n e charge must be p l a c e d p a r a l l e l  27 t o t h e c y l i n d e r i n P i g . 4-6.  Therefore, consider a c i r c u l a r  c y l i n d e r w i t h charge q coul./m p l a c e d p a r a l l e l t o two l i n e charges  each of s t r e n g t h —1> coul./m as shown i n P i g . 4-9.  system can be mapped i n t o t h e W-plane u s i n g t h e same  procedure  employed f o r a c i r c u l a r c y l i n d e r and one l i n e charge. the charge d e n s i t y on t h e deformed conductor  This  Further,  shown i n F i g . 4-10  i s g i v e n by the e q u a t i o n f  (w)  2%&  2 2 a + y + 2ay  /  Q  (a + y ^+ 2 a y 2  Q  / X  2  sin6  s i n 6 ) - 2 d ( a c o s 2 & - y - 2 y a sin6)+ d 2  Q  Q  2  2  2  Q  2  s - i a " ' 2 2 . , . 2 1 2 . s + a + 2as s i n e s + ^ a + as s m o S  2  a  2  4  0  (4-29)  1  K  Numerical  computations  t h a t t h e two parameters y  Q  4  o  j  based on E q u a t i o n (4-26) i n d i c a t e 2 and d  s h o u l d be such t h a t t h e charge  d e n s i t y on t h e deformed contour v a r i e s i n t h e manner shown i n F i g . 4-8.  The reason b e i n g t h a t t h i s p r o v i d e s the best charge  d i s t r i b u t i o n o b t a i n a b l e w i t h t h e p a r t i c u l a r contour t r e a t e d i n 2 this section. These v a l u e s of y and d are t h e r e f o r e cono s i d e r e d optimum. I t was thought of i n t e r e s t t o determine the optimum v a l u e s 2 of y  Q  and d  f o r a s e r i e s of phase s p a c i n g s , s.  For t h i s  pose a programme was w r i t t e n f o r t h e ALWAC I I I - E d i g i t a l  pur-  com-  p u t e r such t h a t Equations  (4-26) and (4-29) c o u l d be e v a l u a t e d 2 f o r v a r i o u s v a l u e s of a, y , d and s. Thus, h a v i n g chosen a 2  p a r t i c u l a r phase s p a c i n g , s, and r a d i u s , a, y  Q  and d  c o u l d be  P i g . 4-9 C i r c u l a r C y l i n d e r and two L i n e Charges  v V  W-plane  V  /  <  u  >i  1  P i g . 4-10 Deformed C y l i n d e r and two L i n e Charges  28 determined  4-12,  by t r i a l and  error.  The v a r i a t i o n s o b t a i n e d are i n d i c a t e d i n P i g s . 4-11  and  and i t w i l l be noted t h a t o n l y one v a l u e of a was 2  used.  C o n s i d e r i n g the v a l u e s o b t a i n e d f o r y  Q  and d  i t i s clear that  the d e f o r m a t i o n r e q u i r e d f o r l a r g e v a l u e s of s i s v e r y s m a l l , f o r both the s i n g l e and three-phase  system.  F i g , 4=11 Conductor Shape as a F u n c t i o n of Spacing ( S i n g l e Phase)  .14  0  30  60  90  120  150  180  210  240  Spacing s i n cm P i g . 4=12  Conductor Shape as a F u n c t i o n of Spacing (Three Phases)  29 5. A PERTURBATION METHOD I t was  shown i n the p r e v i o u s chapter t h a t u s i n g the  c i r c u l a r conductors  c o n s i d e r e d t h e r e d i d p r o v i d e some improve-  ment i n the s u r f a c e charge d i s t r i b u t i o n on a l l phase however, these shapes are of course not optimum. conductor  non-  The  conductors, optimum  shapes b e i n g d e f i n e d here as those p r o v i d i n g u n i f o r m  charge d i s t r i b u t i o n around the p e r i m e t e r of each conductor a three-phase  system.  of a t t a c k was  developed.  in  For t h i s reason a more d i r e c t method T h i s method c o n s i s t s e s s e n t i a l l y of  a p p l y i n g a f i r s t order p e r t u r b a t i o n t o the boundary of a c u l a r c y l i n d e r p l a c e d p a r a l l e l to l i n e charges.  cir-  At the same  time the shape o b t a i n e d must s a t i s f y the c o n d i t i o n of u n i f o r m s u r f a c e charge d e n s i t y on the deformed c y l i n d e r .  The  three  s e c t i o n s f o l l o w i n g d e s c r i b e i n d e t a i l the method and the r e sults  obtained,  5-1. Conducting The  Cylinder  c i r c u l a r c y l i n d e r shown i n Fig„ 5-1  i s placed singly  i n space and i s kept a t a p o t e n t i a l V w i t h r e s p e c t to some reference.  C o n s i d e r the p o t e n t i a l a t p o i n t P on the con-  ductor surface.  The p o t e n t i a l at t h a t p o i n t due t o a s m a l l  charge element a t p o i n t N i s  o =• s u r f a c e charge d e n s i t y i n coul„/m  where  k = 2ite a + a l * dS = ad6 2  r  2  - 2a  2  c o s ( a - 6)  30 For convenience the r e f e r e n c e p o t e n t i a l has here been assumed equal f o r a l l p o i n t s on the c y l i n d e r and i d e n t i c a l l y y  zero.  P  F i g . 5-1 Conducting C y l i n d e r Then i n t e g r a t i o n of (5-1) g i v e s the p o t e n t i a l a t p o i n t P due t o the t o t a l charge on the c o n d u c t o r , or 2%  (5-2)  d6 but l n 2a  l n r,  2  - 2a  2  cos(a-6) 1 2 cos(a-6) 2  = l n a + In  A l s o a i s c o n s t a n t i n t h i s case and can t h e r e f o r e be removed from under the i n t e g r a l s i g n , hence (5-2) becomes 2lt V  p  =  It /  a  (27taa)  2% l n  a  d 6  + 1  /  a  l  n  [  2  -  2  cos(a-6)|d6  ln a  s i n c e the second i n t e g r a l i s equal t o z e r o . (5-35)).  (5-3) (See E q u a t i o n  The q u a n t i t y (2nao) i s the charge q per u n i t l e n g t h  of the c o n d u c t i n g c y l i n d e r , t h e r e f o r e  31 (5-4) This expression i s i d e n t i c a l a circular  t o the one g e n e r a l l y o b t a i n e d f o r  c y l i n d e r by other methods, and f o r the purpose a t  hand does i n f a c t correspond  t o the case of a l i n e charge of  s t r e n g t h -q coul./m p l a c e d p a r a l l e l to the c y l i n d e r , but i n f i n i t e l y f a r removed from i t .  Hence d e t e r m i n i n g the  poten-  t i a l i n t h i s manner g i v e s the c o r r e c t answer f o r t h i s  simple  case. 5-2.  Conducting Fig.  5-2  C y l i n d e r and L i n e Charge  shows a c o n d u c t i n g c y l i n d e r w i t h charge q coul./m  p l a c e p a r a l l e l t o a l i n e charge of s t r e n g t h -q coul./m.  The  p o t e n t i a l at p o i n t P i s P  Fig.  5-2  Conducting  C y l i n d e r and L i n e Charge (5-5)  S  where a — s u r f a c e charge d e n s i t y i n coul./m' k = 2ne 2  = r6  2  o + r  a  2  (5-6)  32 T  n  2  2 2 „ = r + s - 2r s cos a ct X  d»  dS = ( r s  + r ' ) d6  2  2  6  2  >a  A l s o t h e r e f e r e n c e p o t e n t i a l has been assumed i d e n t i c a l l y  zero.  F i r s t c o n s i d e r the conductor shown i n F i g . 5-2 a c i r c u l a r c y l i n d e r of r a d i u s a.  To e v a l u a t e  f o r a(6) from E q u a t i o n  (5-5) one can s u b s t i t u t e  (2-10) and i t f o l l o w s t h e p o t e n t i a l a t  point P i s = f l n (-)  V I?  (See E q u a t i o n  (2-6))  (5-7)  ^  Now l e t the s u r f a c e of t h e c i r c u l a r c y l i n d e r be p e r t u r b e d by a s m a l l amount so t h a t the p o l a r r a d i u s a t t h e angle 6 becomes r  = a + h  6  ( h « a)  6  6  where hg i s a f u n c t i o n of 6.  ^ h i s n o t a t i o n i s used throughout  i n the d i s c u s s i o n f o l l o w i n g .  Hence, s u b s t i t u t i n g f o r r  r  i n (5-6)  6  P  v  p  o  = (a + h )  1  r  and  + (a + h )  fi  2 2  = (a + h )  a  2  ff  + s  - 2 ( a + h g ) ( a + h ) cos(a-6) ff  (5-8)  2  - 2 s ( a + h ) cos a fl  Expanding (5-8) and r e t a i n i n g o n l y terms of f i r s t order i n h  Ct  and hg (5-8) becomes r  r  l  2  cos(a-6)  a  2  + a ( h + hg) a  (5-9) 2  0  d*  2-2 =  a  2  + 2ah  CL  + s  2  S u b s t i t u t i n g Equation  - 2as cos a - 2sh  Ct  (5-9) i n (5-5)  cos a  33 2Y p = £ k ln(a  + 2aha + s  2  In 2 - 2  L  2  - 2as cos a - 2sha cos a) -  cos(a-6)| | a  + a ( h + hg) a  dS  (5-10)  At t h i s p o i n t i t w i l l be r e q u i r e d i )' 2V p . = k  o = c o n s t a n t w i t h r e s p e c t t o the angle 6  ii) iii)  l n —a = c o n s t a n t  2n  r  (5-11)  h d6 = 0 R o  i v ) q = charge per u n i t l e n g t h i s c o n s t a n t . A l s o from (5-6) dS = ( r  2 6  + r£ ) 2  (a + h ) fi  1 x  1 2  d6 i J i _ 2 a + h  +  2  +  + +  d6  E  = (a + hg)d6 t o f i r s t  order.  (5-12)  dh where h£ = ~ c  w  I t then f o l l o w s j a  4  dS = q  or  a = 2rca  Substituting  (5-12) and (5-13) i n E q u a t i o n (5-10) one o b t a i n s  2Vp = ^k l n ( a + 2aha + s  - 2as cos a - 2ash a cos a) ' -  2%  27cak 2 ^  J  I n 2 - 2 cos(a-6)  0 J 0  d n a)(a  +  h )d6 6  (a + hg)d6 -  34 2TC  fek /  ( < ln  a  +  a  h  V]  +  0  ( a  V  +  (5-14)  d6  The t h r e e i n t e g r a l s w i l l now be c o n s i d e r e d i n d i v i d u a l l y  I f [ ln  i)  0  2  -  cos(a-6)]j (a + h ) d 6 =  2  6  *  '  2TC  l n | 2 - 2 cos(a-6)]  1 0  hg d6  (5-15)  2rc  since  J 0  a l n [ 2 - 2 cos(a-6)] d6 = 0  2ll  ii) /  (See E q u a t i o n (5-35))  2Tt  ( l n a)(a + h )d6 = / a ( l n a)d6 0 = 2na l n a g  (5-16)  J7C  J 0  since  ( l n a)hg d6 = 0  (See E q u a t i o n (5-11)) 5 Tt  2TC  Jln(a + h  iii) J  a  + hg) (a + hg)d6 = I a ln(a + h  hg I n ( a + h  + hg)d6 +  0  2TC  J  a  a  + hg)d6  (5-17)  0 But  ln(a+ h G  h + h.) = I n a + l n ( l + —  + h-)  O  €t  h „  .  = ln a + since i n general 2 l n ( l + z) = (z - 7 j 2 z  +  + ha  -  , h + h. i/_a 6\2 ,  o  - TJA  —  )  +  ...  (5-18)  3 3 — ) f o r ,| z| < 1  T h e r e f o r e r e t a i n i n g o n l y the f i r s t term i n t h e s e r i e s (5-18) and s u b s t i t u t i n g f o r l n ( a + hg + h ) i n (5-17) a  2TC  J 0  271  (ln(a+h '  4a  hg)  (a + hg)d6 = J 0  , a l n a d6 +  2lt  ,  .  J  0  35  2lt  2TC 6  o  hg l n a dfi +  a d6  0  or 2n ln(a  + h r  from  h  J  since  a  + h ) ( a + h ) d 6 = 2ita l n a + 2ith 6 a K  he + h. n he  r  +J  l n a d6  §  Equation  (5-11).  h- (—  Also  ln(a  2  + 2ah  CC  2 ln a + ln  1  the f i r s t  +  i s a constant  -T  +  (  f  Equation  )  2  i n this  integra-  o r d e r i n hg i s n e g l e c t e d . i n Equation  - 2as cos a - 2 s h  (5-14) c a n be w r i t t e n  cos a) =  - | ^ cos a - ^ | h a  a  cos al  (5-14) by a i d o f ( 5 - 1 5 ) ,  (5-20)  (5-16),  (5-20)  , ~ a = f ln 1 + 2 h  2V  term  a  OL  2  Hence r e w r i t i n g (5-19) and  2  + s  £)d6 = 0  6  t i o n and a l l t e r m s o f s e c o n d Further,  (5-19)  l K  +  /S\2 (~)  2s 2s - f^- cos a - —  h  a —  2q , cos a + P In a  2-K  2uak  ln  2-2  cos(a-6)|hg  d6 - J I n a  3, i n a - f k k a But  (5-21)  h « a , t h e r e f o r e t h e two a '  be n e g l e c t e d i n t h e f i r s t tuting factors  for V  from  Equation  , 2 ln - = ln 1 + a 1 2na  2%  h terms 2 — a  term  2  of (5-21).  (5-11) and c a n c e l l i n g  g  and - — a  h — cos a c a n a  Further,  substi-  common terms and  (5-21) becomes /S\2 2s \ (—) - r — cos al -  EL  ln  CV  I cL  a —  - 2 cos(a-6)jhg  d6  (5-22)  36 Also  ln  / s \2 2s 1 + (f ) cos a  = 2 l n - + In a  T  —  ct  ct  1  + (f) - r* « 2  cos  t h e r e f o r e (5-22) can be w r i t t e n It  in 2 - 2  h a + i 2 u/ 0  /9-\2  a l n 1 + (—) s n  c o s ( a - 6 ) hg d6 =  a - 2— cos a s  (5-23)  0  which i s an i n t e g r a l e q u a t i o n of t h e second k i n d w i t h t h e symmetric k e r n e l , K(a,6) = l n 2 - 2  cos(a-6)  The i n t e g r a l E q u a t i o n  2 8  '  2 9  '  (5-23) can be s o l v e d d i -  3 0  -I /&\2 n& r e c t l y by expanding h , hg and l n 1 + (—) - 2— cos a i n s s terms of t h e o r t h o g o n a l f u n c t i o n s , e^ e*~*' ^. T h e r e f o r e ft  r  na  h h  a c  +00  Y A n  =  + co  =  YA  , , /a\2 a - 2— cos a l n 1 + (—) s s  n  (5-24)  a  jm6  (5-25)  CO  (5-26)  /_ n  + O0  — oo  e  OO +  0  cos(a-6)  J  — &o  —  ln 2 - 2  e  +CO  —  jna  e  =j*o  (5-27)  co  The c o e f f i c i e n t s , f , i n (5-26) can be found as f o l l o w s In 1 +  <f) -f*=os . ) - l n 1 - £  1 - £ e~J s  2  s  I ± ( f ) .***n  co  I  a  |(f)  n  e~J  n a  (5-28)  s i n c e (^)< 1. Further, the c o e f f i c i e n t s K by  n r  i n E q u a t i o n (5-27) a r e d e f i n e d  37 2TC  K  =  nr  211  f  e  I  2TC  ^  r 6  d6 2%~  i  f  l n j 2 - 2 cos(a-6)| e ~ J d a I I  (5-29)  n a  2TC  The i n t e g r a l , I = J 0 be d e t e r m i n e d .  ln 2-2  cos(a-6) e"*' " da w i l l  first  11  F o r t h i s purpose l e t  ,2TI  | l n s i na-6  1 = 2 J  e^  da  n a  0  2TC  = 2 | ( l n s i n 2=2) e'  da + 2 /  (ln sin  da (5-30)  ^)e'  0  Let b = e  , t h e n s u b s t i t u t i o n of t h e e x p o n e n t i a l form of t h e  J  s i n e f u n c t i o n i n (5-30) y i e l d s 6 In 2 + j f + j f - j f + l n ( l - b e " ^ ) I « 2 J e-J 0 n a  da +  2TE  2/  e^  n a  l n 2 - j f + j f - of + l n ( l - b e " J ) a  da  0 C a r r y i n g out t h e i n t e g r a t i o n s I = -  2we-J  n + 2 J e0< *l n ( l - b e ~ 0  N5  n 27t  2  e"  i n d i c a t e d an s i m p l i f y i n g  j n a  J(X  ) da +  (5-31)  l n ( l - b e ^ ) da a  N e i t h e r of t h e two i n t e g r a l s i n (5-31) e x i s t s a t a = 6 s i n c e b = e ^ , however, c o n s i d e r i n g  t h e two i n t e g r a l s a l o n e one can  write .6 e  2TC  J  n  a  0 lim  l n ( l - b e ^ ) da + j 6 a  e~  o  J  j n a  e~^  na  l n ( l - b e ~ ) da + j a  l n ( l - b e " J ) da = a  38 2rc / e - j n a l n ( l - b e " ) da  lim  (5-32)  3 a  Then i f one can l e t g-^O  and p-"-0 a f t e r the i n t e g r a t i o n has  been c a r r i e d out and t h e r e s u l t remains f i n i t e the proper 39 40 v a l u e of t h e i n t e g r a l s has been o b t a i n e d  '  .  Thus  carrying  out p a r t i a l i n t e g r a t i o n of (5-32) and s i m p l i f y i n g the r e s u l t e J'  I  lim n  n a  l n ( l - a e " ^ ) da +  a  l n ( l - a e ~ ) da =  2lt  n I| l iim m n P  e~ J  n  j a  -i-6-p  " 0  lim g-0  n-1 b -  e  -j(n-l)a  l n ( l  _  -ja  a e  }  d a  +  0  2n  lim p—0  n-1 f ~J b 6-p  (  n  e  -  1  )  a  l n ( l - a e ~ J ) da a  Which f o r n = 1 i s i d e n t i c a l l y z e r o . v a l u e of n (n / 0) 6 2tt [ e- ' l n ( l - b e " J ) da + f e ^ 0 6 J  na  a  I t f o l l o w s t h a t f o r any  ln(l - be^ )  n a  a  da = 0  T h e r e f o r e E q u a t i o n (5-31) becomes I = -  e^  2TC  n  and s u b s t i t u t i n g f o r I i n E q u a t i o n 2TC K  nr - ~  jijrrSo ee_~^J—  (5-33)  (5-29)  n 6  1  2TI  n 6  d 6  J  T h i s i n t e g r a l i s non-zero o n l y when r = n, hence K  nr  = - n  (5-34)  39 and E q u a t i o n (5-27) becomes C O  2 - 2 cos(a-6) = 1 - 1  ln  1  e n  J  -  n ( a  +  6 )  £ 1  - -  n  e  -J  n  (  a  -  6  )  (5-35)  C o n s i d e r E q u a t i o n (5-23) which can now be s o l v e d by a i d o f (5-24), (5-25), (5-28) and (5-35). +oo + -  Thus  C O  1_  2TC  .  + 0 0  271  -oo oo  - a I  . oo £ _ I Jn(a-6) e  1  m  n  £  +  n  _ 1 -jn(a-6) e  n  1  d6 =  oo  I(f) n s  n  eJ  n a  - a  I  |(f)  e^  n  (5-36)  n a  but t h e i n t e g r a l i s non-zero o n l y when m = n, t h e r e f o r e +  C O  i- n  e  +  C O  - a £ 1  oo Y  eJ + oo I -— na  - —  Y  n  l  na  =  n  C O  e^ " - a  I 1  11  n*s " "  e-3  ±{±) n' s'  n  e"  (5-37)  j n a  E q u a t i o n (5-37) c o n s t i t u t e s 2n + 1 equations i n 2n + 1 unknowns which are t h e complex F o u r i e r c o e f f i c i e n t s of h . ft  Thus t h e  n t h , - n t h and Oth equations a r e A  n a/a\n n " n = " nV -  or s o l v i n g f o r A ^ - ~  . or s o l v i n g f o r A -n  A A  -n ~ =  -n o  =  (5-38)  n-lV  A—n _ a/a_\n n ~ ~ n s' -  0  ^ ( N£ )  n - 1 s'  n  (5-39) (5-40)  40 Prom E q u a t i o n s (5-38) and 1 ~  1-1V  -1 ~  (5-39)  -  T h i s s t a t e of a f f a i r s i s p e r m i s s i b l e J 0  l n  1 +  ln  1 +  a2 N  * - 2 f cos a e  29 41 ' i f da = 0  3 a  and 2TC  s  0 t h a t i s , i f e~^ tion  a  - 2 - cos a e " s I  J U  da = 0  i s o r t h o g o n a l t o the r i g h t hand s i d e of Equa-  (5-23); however, E q u a t i o n (5-28) c l e a r l y  i n t e g r a l s cannot be equal t o z e r o . t h a t E q u a t i o n (5-23) has no  shows t h a t t h e s e  T h e r e f o r e one can s t a t e  solution.  Thus i t appears t h a t t h e r e i s no s o l u t i o n t o the problem of a c y l i n d r i c a l conductor and one l i n e charge.  That i s , no  c y l i n d r i c a l conductor shape e x i s t s which has u n i f o r m s u r f a c e charge d e n s i t y i n the presence of a l i n e charge.  The method  employed here i s a f i r s t order p e r t u r b a t i o n , and i t seems r e a s o n a b l e t o expect t h a t such an approach would p r o v i d e the solution for s » a  i f such a s o l u t i o n d i d i n f a c t  exist.  5-3. Conducting C y l i n d e r and two L i n e Charges Even though no s o l u t i o n appears t o e x i s t f o r the unsymm e t r i c a l problem t r e a t e d i n the l a s t s e c t i o n i t seems l i k e l y t h a t a s o l u t i o n does e x i s t f o r the system c o n s i s t i n g of a cond u c t i n g c y l i n d e r and two l i n e charges p l a c e d s y m m e t r i c a l l y w i t h r e s p e c t t o the c y l i n d e r .  Such a system i s t h e r e f o r e  gated i n t h e p r e s e n t s e c t i o n .  investi-  41 Pig.  5-3 shows a c o n d u c t i n g c y l i n d e r w i t h charge q coul./m c o u l . / im.  p l a c e d p a r a l l e l t o two l i n e charges, each of s t r e n g t h The p o t e n t i a l a t p o i n t P i s V P  = § k  l  n  2  r  +  f k  l  n  3 - J  r  f  l n  r  l  (5-41)  d S  S 2 2 2 r„ = r + s + 2 r s cos a o a a  where  ~ 2 —  _  A * —  —  '  ..  /  •»  v  o  Pig.  i f  a  Q  "? -, '  u  5-3. Conducting C y l i n d e r and two L i n e Charges  and t h e o t h e r symbols have t h e v a l u e s i n d i c a t e d by E q u a t i o n (5-6). I n i t i a l l y l e t t h e conductor shown i n F i g . 5-3 be a c i r c u l a r c y l i n d e r o f r a d i u s a.  The p o t e n t i a l a t p o i n t P can  then be determined from E q u a t i o n s (5-41) and (4-15). follows V  P  It  t h e n becomes  Vp = % k l n a1  (See E q u a t i o n  (4-14))  (5-42)  The boundary o f t h e c i r c u l a r c y l i n d e r i s now p e r t u r b e d by t h e method developed i n s e c t i o n  5-2.,  Hence t h e equations de-  termined i n t h a t s e c t i o n f o r the expression r  s  l  d  s  42 can be u t i l i z e d d i r e c t l y h e r e . ln r  o Q  = ln(r a  2  + s  2  + 2r  a  Therefore 1 2  s cos  consider (5-43)  a)  where r  a  hence s u b s t i t u t i n g f o r r f i r s t order terms i n h 2 = l n ( a + 2ah  ln r o  a  = a + h  a  i n (5-43), and n e g l e c t i n g a l l but on  expansion  2 + s  + 2 as cos a + 2 sh  Ct  Combining E q u a t i o n s  cos a)  (5-44)  CC  (5-44), (5-21) and (5-41) t h e f o l l o w i n g  expression i s obtained  =8  2V P  k  l n  a  l n  a  +  +  k  ln  2k  l n  L  1 + 2-2 a  + (s\2 1) + a  h 1 + 2-2  h + (£)<* + 2 | cos a + 2 | -2 cos o  2  a  cos a - 2^ - 2 a a  cos  a  0  - M n a - ^k l n a - fk — k a 2it 2nak  ln 2 - 2  (5-45)  cos(a-6) hg d&  S i n e h ct<C< a t h e terms i n —2 a are n e g l e c t e d i n the l o g a r i t h m i c e x p r e s s i o n s i n (5-45). F u r t h e r , s u b s t i t u t i n g f o r V from P  E q u a t i o n (5-42) and c a n c e l l i n g common terms and f a c t o r s one obtains a  l  +  h J  2TC  In 2 - 2  cos(a-6)]h  de = J l n 1 + ( f ) - 2 f cos a] + 2  6  1 + (sf ) + 2 s| cos a 2  (5-46) which a l s o i s an i n t e g r a l e q u a t i o n of t h e second k i n d w i t h symmetric k e r n e l ,  43 cos(a-6)  K (6,a) = l n 2 - 2  I t f o l l o w s t h a t (5-46) can be s o l v e d by t h e method employed for  (5-23) i n s e c t i o n 5-2.  Hence t h e s e r i e s expansions deve-  l o p e d t h e r e can be used f o r s o l u t i o n of (5-46).  However, con-  sider ln  (1  + ( | ) + 2 | cos a ) = 2  in ( l  +  f J  a  e  ) ( l  +  f  e-J ) =  eJ  n a  a  CO  Z  CO  (-D  n + 1  £(f)  n  I  +  (-l)  n + 1  i(f)  (5-47)  n  S i n c e •§ s< 1 Then s u b s t i t u t i n g from (5-24), (5-25), (5-28), (5-35) and (5-47) i n E q u a t i o n  -J I -  -co  f  1  / \  oo  .  CO  I  (5-46) d6=  1  oo  +  X  f  (-l)  X  n + 1  'i(f)  n e  -J  n  a  (5-48)  I t i s seen t h a t t h e i n t e g r a l i n (5-48) i s non-zero o n l y when m = n.  F u r t h e r a l l odd terms i n n c a n c e l on t h e r i g h t hand  s i d e of t h i s e q u a t i o n . + oo - ao  _ a  A  co  ZK  + I - 3*  ^ eJn<X  1  CO  I  Therefore  (  + 51 - ±=£ e ^ CO  1_ | ) 2 n 2na,„ e3  ft  .  £  na  1  l_ £)2n - j 2 n a (  e  = ( 5  _  4 9 )  Which determines t h e complex F o u r i e r c o e f f i c i e n t s of h . r  a  From (5-49) t h e 2nth and Oth c o e f f i c i e n t s can be d e t e r mined from  *> A A  2n  1  2 n ~ 2n~ ~ "  or s o l v i n g f o r  A  /axil  V  2n  a  44  2 N  A  2n = " < ^ T ) ( | )  Ao  (5-50)  2 n  =0  (5-51)  F u r t h e r a l l the odd c o e f f i c i e n t s determined r e p r e s e n t e d by the f o l l o w i n g  equation  " % T  *2n-l  by (5-49) can be  "  <" >  0  5  52  hence A  3  =  A  5  =  ***  =  A  2n-1  =  0  b u t , f o r n = 1 E q u a t i o n (5-52) becomes A A  l " l  l 1  -  0  and t h i s r e p r e s e n t s a s p e c i a l case because A^ i s i n d e t e r m i n a t e 29 and n : 1 i s an e i g e n v a l u e of the k e r n e l  .  However, from the  r i g h t hand s i d e of E q u a t i o n (5-49) i t f o l l o w s J  I  0  1  1_  (  |)2n  ( e 3  T h e r e f o r e A^ may  2na  +  e  -j2na  }  0  -j«  d a  =  Q  be a s s i g n e d any v a l u e c o n s i s t e n t w i t h the  p h y s i c s of t h e problem.  I n t h i s case a c o s i n e v a r i a t i o n , t h a t  i s , a v a r i a t i o n of the form C cos 6  (C =  constant)  i s e v i d e n t l y not d e s i r a b l e , hence A^ i s g i v e n t h e v a l u e z e r o . Thus from (5-50) the F o u r i e r expansion of h oo h = -a Y. s f l T f ' (  a  0 0 3  2 n a  i s g i v e n by ( 5 _ 5 3 )  45 and the e q u a t i o n of the contour r e q u i r e d r  a  = a + h = a - a  a oo  Y.  (s~ )  2n-l  2 n  c o s  N  2nd  (5-54)  Hence i n o r d e r t h e s u r f a c e charge d e n s i t y i s u n i f o r m on the c o n d u c t i n g c y l i n d e r shown i n P i g . 5-3 the shape of the c y l i n d e r must be t h a t g i v e n by E q u a t i o n (5-54). I t was thought of i n t e r e s t t o t r e a t though b r i e f l y a d d i t i o n a l case i n t h i s  Pig.  5-4.  one  section.  Conducting C y l i n d e r and two L i n e Charges.  C o n s i d e r P i g . 5-4.  Here the two l i n e charges have both been  d i s p l a c e d from t h e h o r i z o n t a l a x i s by a c o n s t a n t angle  8.  The p o t e n t i a l a t p o i n t P i s  V P  •k  ln 2 2 s + r - 2r s  1 E  l n  s  tt tt  r  2  + r l  d S  2 CL  cos(a-B)  + 2r s cos(a+8) Cl  (5-55)  46 S i n c e the form of t h i s e q u a t i o n i s e q u i v a l e n t t o (5-41) the s o l u t i o n i s w r i t t e n without f u r t h e r explanation.  Therefore  the e q u a t i o n d e t e r m i n i n g the complex F o u r i e r c o e f f i c i e n t s of h^ i n t h i s case becomes (See E q u a t i o n (5-48))  + eo  ?i— A n eJ - oo  Q a  n  a 2  +  co  IV  1  .  - -H n  CO  e*  .  0 0  +  n a  .  £V 1  - ±=S n  "J  e  n  a =  CO  j - l ^ a ^ n J n ( a - B ) _ a j - l ^ a ^ n -jn(a-0) ^ xx s 2 ^ xx s e  e  a Y_ (-l) 2 ^  I(2:) us  n+1  +  A  or w r i t i n g the n t h  _  A  0h( P) + £ £  e  2  n  a + B  ^  (5-56)  n  N  e  n =  ^  (  f  ) ns i n  n  E  E  ;  P  i s not f i n i t e ,  ( 5  J  t h i s problem has no s o l u t i o n u n l e s s 8 = 0 i d e n t i c a l to F i g . 5-r4. N u m e r i c a l  -  and f u r t h e r i t i s noted t h a t e""  o r t h o g o n a l t o the r i g h t hand s i d e of E q u a t i o n  The  ~J ^ n  e  SL i (*) r -J P _ J $ )  n n ~ 2 n V or s o l v i n g f o r  Thus  ^ ^  n  equation  A  A  i(-)  a+  n  (5-56).  Hence  5-3.  Computations  c o n f i g u r a t i o n s shown i n F i g s . 5-2  and 5-3 were i n flat  Hence i t i s o n l y p o s s i b l e to s p e c i f y an optimum  shape f o r the c e n t r e phase c o n d u c t o r / that Equation  In order to v e r i f y  (5-54) does i n f a c t g i v e the c o r r e c t shape f o r  t h a t case the s u r f a c e charge d e n s i t y was for  i s not  when F i g . 5-4 i s  v e s t i g a t e d w i t h a view towards a three-phase system w i t h spacing.  5 7 )  the c y l i n d e r shown i n F i g . 5-3.  The  computed n u m e r i c a l l y e q u a t i o n used f o r  47 t h i s purpose can be d e r i v e d as  follows.  Tangent  F i g . 5-5  Conducting C y l i n d e r and two L i n e Charges  R e f e r r i n g t o F i g . 5-5 the p o t e n t i a l a t p o i n t P i s V P  «k  l  n  r  k  +  3  ln  / 4^  -  2  r  l n  i  r  d S  s Hence t h e e l e c t r i c f i e l d i n t e n s i t y on the s u r f a c e i n t h e r  ffl  d i r e c t i o n i s g i v e n by r  i _  - s cos a „  2k  2  2k  r  \ 4^ -—— r  J  2 3  r  r - s cos a _ £_ _ a  r  ~  cos(a-6)  - R T  s  +  2  <->  ds  55 8  l  r  and i n t h e a d i r e c t i o n E a  = ,  r  q =  2k  a^a s sin a r  1S  q  ~~2  .  3  ^ L _ a  _ r  l  s sin 2  2k"  .  r  d s  6  , +  2 ( 5  _  5 9 )  48 L e t t h e a n g l e 8 be such t h a t  §g „ . . . . „ . .  .in  £  t h e n t h e normal f i e l d i n t e n s i t y , E , a t p o i n t P on t h e s u r f a c e n  is E  n  = E  sin 8 - E  r a  a  cos 8  But t h e s u r f a c e charge d e n s i t y , a(6) = 2 e E , t h e f a c t o r , 2, o c c u r r i n g because E^ i s on t h e s u r f a c e i t s e l f ; t h e r e f o r e substituting for E and E from (5-58) and (5-59) r & a r s i n 8 + s sin(a+8) r s i n 8 - s s i n (B+a) / v 2q a 2q a ( ) _ § 4i~ 2 Q  v  0  a  v  n  9  =  r  „ r v + f f e( ~ J  3  r  sin 8 - r "  s  R  s i n ( a - 6 + 8)  ^2 r  2  ~  d  S  ( 5  6 0  '  l  where o(6) has been t a k e n out from under t h e i n t e g r a l s i g n because i t i s assumed c o n s t a n t f o r t h e purpose o f n u m e r i c a l computations . I t may be n o t e d t h a t f o r a c i r c u l a r c y l i n d e r of r a d i u s a E q u a t i o n (5-60) reduces t o s4 - a4 • x , Z.ii2 L ° ° f f c s"* 4x +, a"* ,-4 + 2a s" cos 26  <- >  3  6  61  Comparison of E q u a t i o n s (4-15) and (5-61) shows t h a t t h e same e x p r e s s i o n i s o b t a i n e d f o r , o, i n both c a s e s .  However, t h i s  i s o f course not t r u e i n g e n e r a l , s i n c e (5-60) a c t u a l l y s t i t u t e s an i n t e g r a l  con-  e q u a t i o n i n o.  The i n t e g r a l i n (5-60) was e v a l u a t e d by a i d o f Legendre Gauss Quadrature.  T h i s n u m e r i c a l method o f I n t e g r a t i o n i s  based on t h e e q u a t i o n  49 (5-62) b where 27 number of p o i n t s c o n s i d e r e d i n the i n t e r v a l  m a.  predetermined  l  (b,c).  coefficients.  predetermined v a l u e of the independent  variable.  E r r o r i n c u r r e d by a p p r o x i m a t i n g an i n t e g r a l by (5-61).  R•m(f)  The i n t e r v a l (0, 2TC) was d e v i d e d i n t o t e n p a r t s and e q u a t i o n (5-61), w i t h m s 12, was then a p p l i e d t o each p a r t i n t u r n . A d d i t i o n of t h e r e s u l t s thus gave the t o t a l v a l u e of the i n t e g r a l i n (5-60). The n u m e r i c a l example chosen here i s a = 0.50 r  =s 0.50  cm  s = 10 cm,  thus from E q u a t i o n (5-54)  - (.0025 cos 2a + .00000208 cos 4a +  Thus r e t a i n i n g o n l y the f i r s t two terms r  f l  ...)  i s determined t o a t  l e a s t f i v e s i g n i f i c a n t f i g u r e s and t h i s was c o n s i d e r e d s u f f i c i e n t f o r the purpose a t hand.  T h e r e f o r e E q u a t i o n (5-60) was  e v a l u a t e d by a i d of the ALWAC I I I - E d i g i t a l computer w i t h r  a  = 0.50  - .0025 cos 2a.  F i g . 5-5 shows c(a) as a f u n c t i o n of the angle a b o t h f o r the deformed c y l i n d e r and the e q u i v a l e n t c i r c u l a r c y l i n d e r .  It  f o l l o w s t h a t E q u a t i o n (5-54) does g i v e the c o r r e c t shape f o r the centre conductor. I t was thought of i n t e r e s t t o make an a n a l y t i c a l  compari-  son of the contours g i v e n by E q u a t i o n s (5-54) and (4-11).  The  p o l a r e q u a t i o n of an e l l i p s e i s g i v e n by the w e l l known f o r m u l a  .02015  a  bo  u  cd  .01985  .01980 I 0  . 10  . 20  . 30  . 40  . 50  . 60  . 70  6 i n degrees P i g . 5-6 Charge D e n s i t y V a r i a t i o n  . 80  — 90  1 b (1 - e r  )  2  2  '  =  5 0  1  ( l - e^  sin 6) 1 , /, 2x2/, , 1 = b (1 - e ) (1 •g2  2  f  1  where  2  2  . 2 3 4.4,. x s i n & + Qi o •••) C  e  s  i  n  b = major a x i s e^ = e c c e n t r i c i t y  or  i n s p e c t i o n o f t h e s e r i e s above i n d i c a t e s C O  r = Ao + where'A  Vj ~ Bn cos 2n6  and B^ a r e c o n s t a n t s .  A l t h o u g h i t does not appear  p o s s i b l e t o e v a l u a t e these c o n s t a n t s i t i s seen t h a t t h e form of  (5-62) i s e q u i v a l e n t t o t h a t o f ( 5 - 5 4 ) . For  comparison l e t t h e conductor shown i n F i g . 5-3 be an  e l l i p t i c c y l i n d e r w i t h minor a x i s a t 6 = 0.  The n u m e r i c a l  example chosen here i s t h e same as above, namely a = 0.50 cm 2  s = 10 cm and d  = .00125.  From E q u a t i o n (4-11) i t i s seen 2  that t h i s p a r t i c u l a r value of d .4975 cm. the  r e s u l t s i n a minor a x i s o f  Hence u s i n g E q u a t i o n (4-17) t h e charge d e n s i t y on  e l l i p t i c c y l i n d e r was computed.  T h i s computation i n d i -  c a t e d t h a t t h e charge d e n s i t y on t h e e l l i p t i c conductor agrees w i t h t h a t o b t a i n e d by u s i n g E q u a t i o n (5-54) t o w i t h i n f i v e significant  figures.  T h i s c o n c l u d e s t h e work done i n t h i s t h e s i s on d e t e r m i n ing  t h e optimum shape f o r t h e conductors o f a three-phase  system.  From t h e method employed here i t appears t h a t t h e  optimum shapes i n n e a r l y a l l cases cannot be determined d i rectly.  However, i t i s seen t h a t some improvement  can be  51 o b t a i n e d by u s i n g t h e conductor two c h a p t e r s .  shapes c o n s i d e r e d i n the l a s t  On the other hand i t i s noted t h e r e q u i r e d  change i n shape i s v e r y s m a l l f o r a l l systems w i t h l a r g e s p a c i n g between phase c o n d u c t o r s .  52 6. BUNDLED CONDUCTORS U s i n g s e v e r a l conductors conductors  f o r each phase, t h a t i s , bundled  i n s t e a d of one conductor f o r each phase o f f e r s some  d i s t i n c t advantages, such as decreased i n d u c t i v e r e a c t a n c e , i n creased c a p a c i t i v e r e a c t a n c e  and a lower v o l t a g e g r a d i e n t a t  the s u r f a c e of the t r a n s m i s s i o n l i n e c o n d u c t o r s . t u r e s become q u i t e important  These  as t r a n s m i s s i o n v o l t a g e s  fea-  and  d i s t a n c e s i n c r e a s e , and s i n c e , as mentioned e a r l i e r , t h i s i s the t r e n d i n e l e c t r i c power t r a n s m i s s i o n i t f o l l o w s t h a t the 6 9 12 c h a r a c t e r i s t i c s of such systems have been i n v e s t i g a t e d . ' '  35 '  Prom the p o i n t of view of m i n i m i z i n g corona on a t r a n s m i s s i o n l i n e the v o l t a g e g r a d i e n t i s the more important f a c t o r to be c o n s i d e r e d .  S e v e r a l papers have been p u b l i s h e d r e c e n t l y  which d e a l w i t h the d e t e r m i n a t i o n of v o l t a g e g r a d i e n t s on threephase systems u s i n g bundled c o n d u c t o r s .  For most cases the  c o n f i g u r a t i o n i n v e s t i g a t e d i s t h a t of f l a t s p a c i n g , and i t f o l l o w s t h a t the c e n t r e phase conductors  t h e n i n h e r e n t l y have  the l a r g e r charge per u n i t l e n g t h as was  a l s o found to be  case when o n l y one conductor per phase was 3-2.  used.  the  See s e c t i o n  For t h i s r e a s o n the v o l t a g e g r a d i e n t on the c e n t r e phase  conductor i s g e n e r a l l y regarded  the l i m i t i n g f a c t o r f o r such a  transmission l i n e system. ' ^  A l s o the charges on the  8  v i d u a l conductors  3  indi-  of the c e n t r e phase are g e n e r a l l y found t o  be v e r y n e a r l y equal p r o v i d e d these conductors  are  arranged  s y m m e t r i c a l l y w i t h r e s p e c t t o the outer phases. 34 As has been p o i n t e d out i t would be of advantage t o  53 have a l l p a r t s of a t r a n s m i s s i o n l i n e system s t r e s s e d c a l l y t o t h e same degree.  I n order to achieve t h i s  electriobjective  i t has been suggested t h a t l a r g e r s i z e d c o n d u c t o r s be used f o r the c e n t r e phase t h a n f o r t h e o u t e r phases.  As a matter of  f a c t t h i s procedure has been used i n one i n s t a n c e . c u s s i o n o f Ref. 34.  See d i s -  F o r a t r a n s m i s s i o n l i n e u s i n g two con-  d u c t o r s p e r phase, one conductor p l a c e d d i r e c t l y above t h e o t h e r , t h e charges per u n i t l e n g t h a r e equal on t h e s e two con34 d u c t o r s t o w i t h i n .lfo.  Hence t h e maximum g r a d i e n t s w i l l  a l s o be a p p r o x i m a t e l y e q u a l , and one can thus s p e c i f y a l l d u c t o r s t o be o f t h e same s i z e f o r t h e o u t e r phases.  con-  How-  ever, when t h r e e o r f o u r conductor bundles a r e used t h e cond u c t o r s i n t h e o u t e r phases must be g i v e n a s p e c i f i c  configu-  r a t i o n i n o r d e r t o o b t a i n equal maximum v o l t a g e g r a d i e n t on a l l c o n d u c t o r s o f one phase.  The o b j e c t of t h i s c h a p t e r i s  t o determine t h i s c o n f i g u r a t i o n f o r t h r e e and f o u r conductor bundles. 6-1. T r a n s m i s s i o n L i n e w i t h t h r e e Conductors p e r Phase The three-phase system t r e a t e d i n chapter 3 was ungrounded, t h e r e f o r e i t was p e r m i s s i b l e t o use E q u a t i o n (3-1) t o determine the charge p e r u n i t l e n g t h o f each phase c o n d u c t o r .  However,  t h e systems t r e a t e d i n t h i s c h a p t e r a r e assumed t o be grounded and a d i f f e r e n t approach i s then r e q u i r e d i n t h a t E q u a t i o n (3-1) i s i n v a l i d f o r l i n e s w i t h grounded n e u t r a l .  On t h e o t h e r  hand, f o r t h e purpose of computing p o t e n t i a l s i t w i l l a g a i n be assumed t h a t n e g l i g i b l e e r r o r r e s u l t s from assuming t h e charge  54  on each conductor t o be c o n c e n t r a t e d a t i t s c e n t r e . I t was  shown i n s e c t i o n (2^-2) t h a t two equal and  l i n e charges p l a c e d a c e r t a i n d i s t a n c e a p a r t w i l l  opposite  together  produce a p l a n e of zero p o t e n t i a l midway between the two charges.  line  Hence i f the s u r f a c e of the ground i s regarded  a  p e r f e c t l y c o n d u c t i n g plane a t zero p o t e n t i a l i t f o l l o w s t h a t an overhead t r a n s m i s s i o n l i n e and ground may  be r e p l a c e d by  an e q u i v a l e n t system c o n s i s t i n g of the t r a n s m i s s i o n l i n e i t s image.  Thus the e f f e c t of ground on the e l e c t r i c  and  field  c o n f i g u r a t i o n around the t r a n s m i s s i o n l i n e conductors i s t a k e n i n t o account.  I t w i l l be noted t h a t t h i s e f f e c t was  i n the system t r e a t e d i n chapter F i g . 6-1  neglected  3.  shows a three-phase  transmission l i n e with f l a t  s p a c i n g and t h r e e conductors per phase.  A l s o the image con-  d u c t o r s are i n c l u d e d , and these are p o s i t i o n e d a d i s t a n c e , H, below the ground plane e q u i v a l e n t t o the h e i g h t of the a c t u a l conductors above ground.  L e t the phases be denoted A, B and  C and the i n d i v i d u a l conductors  1, 2, 3 e t c . as i n d i c a t e d .  Then the e q u a t i o n d e t e r m i n i n g the p o t e n t i a l of the n t h cond u c t o r can be w r i t t e n V  n  =  m = 9  q_  V m = 1  k  Y  s• l n -SS s nm  (6-1)  where q^ = charge i n coul./m on conductor  m  ( s ' ) / = d i s t a n c e between conductor n and the image of conductor :m  m  d i s t a n c e between conductor n and conductor  m  Phase A  Phase B  P i g . 6-1 Three-phase T r a n s m i s s i o n  Phase C  Line  55 (s*nm )n=m = d i s t a n c e bet-ween conductor n and i t s own image *» (s nm ) n=m = r a d i u s of conductor n v  k = 2ite . o I f one t h e n l e t s n t a k e on t h e v a l u e s 1 t o 9 i n s u c c e s s i o n i t f o l l o w s t h a t n i n e simultaneous equations a r e o b t a i n e d .  Putti  these e q u a t i o n s i n m a t r i x form P  P  l l  P  12 13 14 15 16 17 18 19 P  P  P  P  P  P  P  *1  21 22 23 24 25 26 27 28 29 P  P  P  P  P  P  P  *2  P  (6-2)  _ 91 9 2 9 3 9 4 9 5 9 6 9 7 9 8 9 9 P  P  P  where (P  ) / =  P  P  q  In  q_  (P ) * nm n=m It follows that  P  3  P  P  *9  P  s'  -ss  (6-3)  nm s I n -SB 1  (6-4)  k  P  = P . nm mn The c o e f f i c i e n t s P_ were i n t r o d u c e d by Maxwell and a r e therenm " 13 37 39 f o r e termed Maxwell's p o t e n t i a l c o e f f i c i e n t s »°'»' . U  J  The charges q^ t o q^ can be determined by i n v e r s i o n o f E q u a t i o n (6-2) so t h a t  56 3 C,  *1  C  ^2  C  c.  c.  c,  c.  c,  c.  3  c. J  V  2 0  e  (6-5)  0  o  ©  3d  )j  where  (c  )  = (-i)  nm n=m  ^  2 n  '  x  (6-6)  A M  (6-7)  and M nn , NInm and -A denote minors and d e t e r m i n a n t r e s p e c t i v e l y i n E q^ u a t i o n ( 6 - 2 ) . The c o e f f i c i e n t s (Cnm )n=m and (C ) / nm'n^m t  1  r  v  are  J  v  termed Maxwells c o e f f i c i e n t s of s e l f - i n d u c t i o n and mutual  induction respectively.  F u r t h e r (C  ) i s the d i r e c t capaci. nm n=m tance between conductor n and ground w h i l e -(C ) / i s the nm n^m 13 37 c  d i r e c t c a p a c i t a n c e between conductors n and m C o n s i d e r phase A i n F i g . 6-1.  Conductors 1, 2 and 3 are  p l a c e d e q u i l a t e r a l l y and are a l l kept a t t h e same p o t e n t i a l above ground, namely the phase v o l t a g e V^.  Then i f the d i s -  •france H i s s u f f i c i e n t l y l a r g e the e f f e c t due t o ground w i l l  be  v e r y s l i g h t and the d i f f e r e n c e i n the magnitude of the charges q.^, ^2 small.  a n d  ^3  d u e  *° ^  n e  P  r e s e n c e  °^ ground w i l l be q u i t e  However, i t i s e v i d e n t t h a t the d i r e c t c a p a c i t a n c e s  to t h e c o n d u c t o r s of the o t h e r phases w i l l be l a r g e r f o r  57 conductor 3 than f o r 1 and 2.  Hence conductor 3 w i l l  neces-  s a r i l y c a r r y a l a r g e r charge per u n i t l e n g t h than 1 and 2.  It  f o l l o w s t h a t an i d e n t i c a l argument h o l d s f o r conductor 9 i n phase C. From t h e c o n f i g u r a t i o n shown i n F i g . 6-1  i t seems r e a s o n -  a b l e t o expect t h a t the charge on conductors 3 and 9 w i l l crease i f the d i s t a n c e s s ^ g , 23» 9 8 s  slightly.  s  a n d  s  97  a r e  de-  decreased  To i n v e s t i g a t e t h i s a programme was w r i t t e n f o r  t h e ALWAC I I I - E d i g i t a l computer.  The p o t e n t i a l  coefficients  i n E q u a t i o n (6-2) were then a l l e v a l u a t e d and arranged i n m a t r i x form by a i d of t h i s programme, and a s t a n d a r d s u b r o u t i n e f o r m a t r i x i n v e r s i o n was u t i l i z e d t o compute the c a p a c i t i v e c o e f f i c i e n t s i n Equation (6-5).  F u r t h e r , the phase v o l t a g e s  were g i v e n the v a l u e s V^ = 1 cos cot V  fi  = 1 cos(wt -  120)  V  c  = 1 cos (art -  240)  so t h a t on d e t e r m i n i n g q^ t o qg from E q u a t i o n (6-5) the i n stantaneous v a l u e s of the charge i n c o u l . / m - v o l t were o b t a i n e d . R e f e r t o F i g . 6-1.  The g e n e r a l procedure f o l l o w e d i n  o p t i m i z i n g the d i s t a n c e s s ^ g , S g g , Sgg and S g ^ f o r d i f f e r e n t phase s p a c i n g s , D, and bundle s p a c i n g s , i c , was i ) The conductor diameter, d, was i i ) The phase s p a c i n g , D, was i i i ) The bundle s p a c i n g , c, was  chosen  chosen chosen  i v ) The h e i g h t above ground, H, was chosen 1 v) The d i s t a n c e s b = s g - ( | ) and b = 2  as f o l l o w s :  2  2  s  98  2  ~  v  ,ds2 2 ;  1  58 were decreased s i m u l t a n e o u s l y u n t i l the charge q_ s m a l l e r t h a n q^ and  was  q . 2  The r e s u l t of such a s e r i e s of computations appears i n F i g . 6-2.  I n t h i s p a r t i c u l a r case  d = 1 inch  D = 20.0 f t .  c = 1.50  ft.  H = 40 f t .  and i t w i l l be noted t h a t a l i n e a r v a r i a t i o n w i t h the p a r a meter b i s o b t a i n e d .  T h i s was found t o be t r u e i n p r a c t i c a l l y  a l l cases f o r t h e range of b i n v e s t i g a t e d h e r e . v a l u e s i n d i c a t e d f o r q^, q charges.  2  A l s o the  and q^ are t h e magnitudes  of t h e s e  The phase angle between them was found t o be l e s s  t h a n one degree and t h i s was c o n s e q u e n t l y n e g l e c t e d .  At the  same time the charges were found t o l a g the phase v o l t a g e  by  a p p r o x i m a t e l y s i x degrees. I n o r d e r t o determine the v a r i a t i o n of the optimum v a l u e of b w i t h phase s p a c i n g , D, and bundle s p a c i n g , c, a s e r i e s of computations was c a r r i e d out.  The n u m e r i c a l v a l u e s chosen f o r  t h i s purpose were as f o l l o w s : d 1.0  inch  D  c  H  20.0 f t .  1.25 f t .  40.0 f t .  1.0  "  20.0 it  1.50  it  40.0  1.0  "  25.0 •i  1.25  it  40.0 it  1.0  "  25.0  n  1.50  1.0  "  30.5 it  1.25  •i  40.0 it  1.0  "  30.5 it  1.50  it  40.0 it  II  II  40.0 it  Thus a curve s i m i l a r to the one shown i n F i g . 6-2 was o b t a i n e d f o r each s e t of v a l u e s l i s t e d  above.  .088  0)  a  .086  CM  a  ,084  o  o ,082 ce  o  ,080 .90  1.00  1.10  1.20  1.30  1.40  1.50  b in ft. P i g . 6-2 Charge per U n i t Length v e r s u s t h e Parameter b  CD  CM  ~  -p  o I  2.40  > •p  \  2.30  3  max  (0  •f> rH  O  J>  2.20  fl •rl >> -P •rl  «  2.10  a CO  +» M  *d i—i a> •H  2.00 .90  1.00  1.10  1.20  1.30  1.40  1.50  b in ft. P i g . 6-3 F i e l d I n t e n s i t y v e r s u s t h e Parameter b  59 The optimum v a l u e of b i s d e f i n e d here as t h a t v a l u e a t which t h e maximum e l e c t r i c f i e l d i n t e n s i t i e s 2 and 3 a r e a p p r o x i m a t e l y e q u a l .  on conductors 1,  Hence h a v i n g o b t a i n e d t h e  v a r i a t i o n o f t h e charges w i t h t h e parameter b as o u t l i n e d above one can compute t h e e l e c t r i c f i e l d on each conductor f o r a s e r i e s o f v a l u e s of b and thus determine t h e optimum p o s i t i o n f o r conductor 3. For  t h e purpose of computing t h e e l e c t r i c f i e l d  intensity  on conductors 1, 2 and 3 t h e f o l l o w i n g assumptions a r e made, i ) The s p a c i n g s s^g, s g and s ^ a r e s u f f i c i e n t l y l a r g e t h a t 2  the  2  two conductors a d j a c e n t t o t h e one c o n s i d e r e d can be  r e p r e s e n t e d as l i n e c h a r g e s .  See Appendix I .  i i ) The h e i g h t H above ground i s such t h a t t h e image charges have no e f f e c t on t h e s u r f a c e charge d i s t r i b u t i o n  on any  of t h e c o n d u c t o r s , i i i ) The phases B and C a r e s u f f i c i e n t l y d i s t a n t t h a t each can be r e p r e s e n t e d as a l i n e charge p l a c e d a t t h e geometric c e n t r e of each phase. F i g . 6-4 shows t h r e e c o n d u c t i n g c y l i n d e r s p l a c e d p a r a l l e l t o two l i n e c h a r g e s .  T h i s system i s e q u i v a l e n t t o t h e t h r e e -  phase t r a n s m i s s i o n l i n e i n F i g . 6-1 when one i n t r o d u c e s t h e s i m p l i f y i n g assumptions o u t l i n e d above.  The charge p e r u n i t  l e n g t h o f t h e l i n e charges i s - q ^ coul./m and - q ^ coul./m r e s p e c t i v e l y , and t h e s e a r e equ!al i n magnitude t o t h e t o t a l charge on phases B and C.  Hence from E q u a t i o n s (2-9) and  ( I - l ) the e l e c t r i c f i e l d i n t e n s i t i e s  E^, E  2  and Eg a t t h e s u r -  f a c e o f conductors 1, 2 and 3 r e s p e c i t v e l y a r e  60 q  2q  f i g  I1  E  2q  ( — ) B1  2  0 0 0  cos n ( a + 9 0 - 6 ) + 9  co  n  • E 2q  n  S  X *  2q  0 0  cx  I  "  <^>"—  »<-«» " E  2  £  (4> •» "<«-l>i> n  +  C O  •B £  (-£-)  X  a  2q __C  S  cos n ( a - 9 0 + B ) +  n  9  B2  0 0  J" _ a _ n X C2 q 2q =E ^ " E ' I (  }  c o g  n ( a  _  9 0 +  p ) 3  ( 6  _  1 0  )  0 0  E  c  1  3  2q co trI ka  X3  0  (s ^ )  2  x  2 3  co  2q E T Ka  n  I  (^-) s  n  x  C 3  n  °  s  n(6-180  + P l  )  2q„ cos 11(6-180-8, ) + 1 K.a  co 1  I  s(-^-J f i 3  11  cos n6 +  cos n6  .(6-11)  where k = 2Tte s  Bn  a  n  d  S  Q  Cn  r e  P  r e s e n  "k *  n e  d i s t a n c e s from l i n e charges q^ and  q^ t o t h e c e n t r e s o f conductors 1, 2 and 3 w i t h n = 1, 2 or 3. At t h i s p o i n t some f u r t h e r a p p r o x i m a t i o n s a r e made.  Since  the i n s t a n t o f maximum charge on phase A i s c o n s i d e r e d here the charges q^ and q^ a r e a p p r o x i m a t e l y equal t o - ^ q A l s o t h e f i r s t order terms i n • s  — — « — 2 L - where m and n = 1 Bn mn s  s  9  a r e such t h a t Bn  2 or 3 and m ^ n.  3  coul./m.  Phase A  Phase B  P i g . 6-4 Three Conducting C y l i n d e r s and two L i n e Charges  Phase C  61 Hence s i n c e i t i s o n l y d e s i r e d t o o b t a i n a f i r s t order a p p r o x i mation t o t h e e l e c t r i c f i e l d i n t e n s i t y t h e l a s t two terms can be n e g l e c t e d i n each of t h e E q u a t i o n s ( 6 - 8 ) , (6-9) and ( 6 - 1 0 ) . T h e r e f o r e r e t a i n i n g o n l y t h e f i r s t terms i n each o f t h e remaining  s e r i e s the f o l l o w i n g expressions are obtained  2q 2q ^ — cos(a+8 ) cos (a+90) '31 '21 q.o 2q, 2q 2 = ka- " k i ^ cos(a-90) - ^ 2 os(a- ) 3  E, = 1 ka E  1  C  E  3 ~ ka  2q,  -  cos(6-180+p ) -  2q  1  13  P l  0  008(6-180-^)  23  (6-12) (6-13) (6-14)  i  I t w i l l be noted t h a t E q u a t i o n (6-10) a c t u a l l y r e p r e s e n t s t h e e l e c t r i c f i e l d i n t e n s i t y on a c o n d u c t i n g c y l i n d e r w i t h charge q^ coul./m p l a c e d i n two u n i f o r m e l e c t r i c f i e l d s o f i n t e n s i t y ks  and 31  ks. ""21  volts/m r e s p e c t i v e l y .  I t follows that a similar  statement can be made about E q u a t i o n s (6-11) and (6-12). The a n g l e s a t which t h e maximum f i e l d i n t e n s i t y occurs on conductors 1, 2 and 3 can now be determined by d i f f e r e n t i a t i o n . Let  t h e s e a n g l e s be denoted a ^ ,  a n <  m  l 6 g r e s p e c t i v e l y , then m  from t h e e q u a t i o n s  5cT  a* - ° it a  ml  and  = 0  follows =  a r c  *  ' <12  '31  a n  s2 1 2i 4 ' 2  / 2 V 3 1  S  s  s 32 s i 12x2 ~ "4" 2  / 2 ^ 32 ( s  2l\ 3 "21 2s S  (6-15)  2  x  S  = arctan  21  q  s  )  S  12  (6-16)  62 s 12  m3 = a r c t a n  6  (6-17)  s 32  Therefore s u b s t i t u t i n g a  m l  , a  m 2  and 5  m 3  i n E q u a t i o n s (6-10)  (6-11) and (6-12) r e s p e c t i v e l y one can compute t h e maximum f i e l d i n t e n s i t y on each conductor.,  The r e s u l t of such a s e -  r i e s of computati ons i s shown i n F i g . 6—3. t h e r e a r e based on t h e v a l u e s of b, q^, q Fig.  6-2.  the  The curves shown a  2  n  d <lg i n d i c a t e d i n  A l s o comparison of F i g s . 6-2 and 6-3 i n d i c a t e s t h a t  p o i n t a t which q_ = q^ g i v e s v e r y n e a r l y t h e optimum v a l u e 2  of t h e parameter b. The assumptions made i n d e r i v i n g E q u a t i o n s (6-12), and (6-14) i n t r o d u c e d e r r o r s i n t h e e l e c t r i c f i e l d computed.  (6-13)  intensities  To i n v e s t i g a t e t h e magnitude o f t h e s e e r r o r s con-  s i d e r a system w i t h phase s p a c i n g o f 20 f t . and bundle s p a c i n g of  15 i n c h e s . The t h e f i r s t o r d e r terms due t o t h e presence 3 1 of phase B=^- q x = .075q and t h e second o r d e r terms due t o o  the presence of t h e a d j a c e n t bundle conductors = q x y^=.03q. In t h i s p a r t i c u l a r case q = .08 hence t h e e r r o r = ,01 = T^& s i n c e E max = 2.0. The procedure o u t l i n e d above f o r computing t h e maximum e l e c t r i c f i e l d was t h e n used t o compute E, , E and * 1 max 2 max 0  9  Eo  m  o  v  (6-8).  f o r each s e t o f n u m e r i c a l v a l u e s g i v e n by E q u a t i o n Thus a s e t o f curves was o b t a i n e d each b e i n g s i m i l a r  i n form t o F i g . ( 6 - 3 ) .  Hence, by a i d o f t h e s e curves t h e  optimum v a l u e o f b c o u l d be p l o t t e d as a f u n c t i o n o f phase s p a c i n g , D, w i t h bundle s p a c i n g , c, as a parameter.  See  1.40  1.30  1.20  f  c = 1.50 f t .  1.10  1.00  :] c = 1.25 f t .  .90  .80 15.0  20.0  25.0  30.0  35.0  Phase Spacing i n f t . F i g . 6-5 V a r i a t i o n of b  .j. w i t h Phase and Bundle Spacing  63 P i g . 6-5.  The p o i n t a t which t h e charge p e r u n i t l e n g t h of  conductor 3, q^, i s equal t o t h e charge p e r u n i t l e n g t h of cond u c t o r 2, q , i s i n c l u d e d f o r comparative purposes, i t i s e v i 2  dent t h a t f o r p r a c t i c a l purposes one can assume E„  =  o max E  2 max  w  h  e  n  *2 =  V  6-2. T r a n s m i s s i o n L i n e w i t h f o u r Conductors p e r Phase C o n s i d e r t h e three-phase t r a n s m i s s i o n l i n e shown i n P i g . 6-6.  I n t h i s case f o u r conductors a r e used f o r each phase  and t h e o b j e c t i s t o determine t h e optimum p o s i t i o n f o r t h e conductors i n t h e o u t e r phases.  F o r t h i s purpose t h e charge  per u n i t l e n g t h on each conductor must be determined, and i t f o l l o w s t h a t t h e procedure o u t l i n e d i n d e t a i l i n s e c t i o n 6-1 can be used here as w e l l . nth  conductor i s m=12 q  T h e r e f o r e t h e p o t e n t i a l of t h e  s'  \~ I r " r m=l  <- > 6 18  nm  where a l l symbols have t h e same meaning as i n E q u a t i o n ( 6 - 1 ) . In t h i s case n = 1 t o 12 so t h a t t w e l v e simultaneous equations are obtained.  I n v e r t i n g t h e s e e q u a t i o n s one f i n d s t h e  charge per u n i t l e n g t h on t h e n t h conductor q„ =  m=12 I C Y m=1 nm m  (6-19)  T h i s corresponds t o one o f t h e E q u a t i o n s g i v e n by (6-5) i n the p r e v i o u s s e c t i o n . C o n s i d e r i n g t h e c a p a c i t a n c e a s s o c i a t e d w i t h each cond u c t o r i n phase A, P i g . 6-6, i t i s e v i d e n t t h a t t h e charge  Phase A  Phase B  Phase C  ffl  P i g . 6-6 Three-phase T r a n s m i s s i o n  Line  64 per u n i t l e n g t h of conductors 2 and 4 w i l l be g r e a t e r t h a n t h a t of 1 and 3 f o r the c o n f i g u r a t i o n shown.  S i m i l a r l y the charge  per u n i t l e n g t h of conductors 9 and 12 i s g r e a t e r than t h a t on 10 and 11 i n phase C.  However,, i n t h i s case i t i s not  r e a d i l y seen how t h e c o n f i g u r a t i o n should be changed i n order t o e q u a l i z e t h e charges on the conductors i n t h e outer phases. T h e r e f o r e E q u a t i o n s (6-18) and (6-19) were e v a l u a t e d by a i d of the ALWAC I I I - E d i g i t a l computer.  The programme used f o r  t h i s purpose was w r i t t e n such t h a t the c o n f i g u r a t i o n of t h e outer phases c o u l d be v a r i e d t o some extent from t h a t shown i n P i g . 6-6. U s i n g t h i s procedure i t was found t h a t d e c r e a s i n g t h e d i s t a n c e Sg^ s y m m e t r i c a l l y ,  that i s  9  decreasing  s£  a n 2  d in-  c r e a s i n g s ^ by t h e same amount had t h e d e s i r e d e f f e c t . f o l l o w s t h a t t h e d i s t a n c e Sg ^ The e f f e c t o b t a i n e d  w a s  decreased  simultaneously.  by i n t r o d u c i n g t h i s v a r i a t i o n i n the con-  f i g u r a t i o n of t h e outer phases i s shown i n F i g . 6-7. be noted t h a t t h e v a r i a t i o n o b t a i n e d for  It  It will  i n t h e charges are l i n e a r  t h e range of s ^ i n v e s t i g a t e d . 2  I n order t o o b t a i n t h e v a r i a t i o n of t h e optimum v a l u e of s  2 4 "-* P w:  out.  n  n a s e  s p a c i n g a s e r i e s of computations were c a r r i e d  The n u m e r i c a l  following. d  values  chosen f o r t h i s purpose were t h e  See F i g . 6-6,  D  c  H  1.0 i n c h  25.0 f t .  1.50 f t .  1.50 f t .  40,0 f t .  1.0 "  30.0 "  1.50 "  1.50 "  40.0 "  1.0 "  35.0  1.50 "  1.50 "  40.0 "  .069  .067  ,065 3  o o a •rt  .063  0)  b0  U  .061 1.10  1.20  1.30 Spacing s  P i g . 6-7  1.40  1.50  1.60  in f t .  2 4  Charge per u n i t Length v e r s u s Bundle  Spacing  1.50  1.40 I  a a  1.30  I  c = 1.50 f t ,  •rl  -P P< O  1.20  CM (0  1.10  I* 22.5  25  30  35  Phase S p a c i n g i n f t . F i g . 6-8 V a r i a t i o n of S g ^  ^,  w i t h Phase S p a c i n g  65 The r e s u l t o b t a i n e d from these computations i s shown i n F i g . 6-8.  I n t h i s case t h e optimum c o n f i g u r a t i o n i s based on  equal charge on t h e conductors of t h e o u t e r phases r a t h e r t h a n on: e q u a l .maximum' e l e c t r i c f i e l d i n t e n s i t i e s . as shown i n sjactinn 6-1 b o t h c r i t e r i a  However,  give approximately the  same answer. 6-3. Some G e n e r a l C o n s i d e r a t i o n s I t was mentioned  i n the i n t r o d u c t i o n t o t h i s chapter that the  c e n t r e phase conductors have a l a r g e r charge p e r u n i t l e n g t h t h a n those o f t h e o u t e r phases.  T h i s f a c t was v e r i f i e d i n  t h e computations c a r r i e d out h e r e .  Thus f o r t h e system w i t h  t h r e e conductor bundles t h e charge was found t o be 8 - 10$ g r e a t e r on t h e c e n t r e phase t h a n on t h e o u t e r phases.  Fur-  t h e r i n d i s p l a c i n g conductors 3 and 9 (See F i g . 6-1) i n t h e manner i n d i c a t e d i n s e c t i o n 6-1 t h e charge on each c e n t r e phase conductor was n o t i c e d t o decrease by a p p r o x i m a t e l y  For  t h e system u s i n g f o u r conductor bundles t h e d i f f e r e n c e i n charge on t h e c e n t r e and o u t e r phases was about 7$.  However,  d i s p l a c i n g conductors 2, 4, 9 and 12 (See F i g . 6-6) as des c r i b e d i n s e c t i o n 6-2 had p r a c t i c a l l y no e f f e c t on t h e charge on t h e c e n t r e phase c o n d u c t o r s .  When t h e p o s i t i o n of con-  d u c t o r 3 i n F i g . 6-1 i s changed by d e c r e a s i n g s ^ g and S g g s y m m e t r i c a l l y and k e e p i n g t h e phase s p a c i n g , D, c o n s t a n t t h e f l u x l i n k i n g phase A w i l l i n c r e a s e .  However, as p o i n t e d out  above t h e charge on t h e c e n t r e phase conductors decreases by u s i n g t h i s procedure.  I f i n s t e a d one decreased t h e phase  66 s p a c i n g , D, s i m u l t a n e o u s l y w i t h s ^  3  and s  2 3  , thus k e e p i n g con-  d u c t o r 3 f i x e d i n p o s i t i o n , but d i s p l a c i n g 1 and 2 i t f o l l o w s the  i n d u c t i v e r e a c t a n c e of phase A d e c r e a s e s , w h i l e the charge  on the c e n t r e phase conductors i n c r e a s e s .  Hence the p r o -  cedure t h a t s h o u l d by used w i l l depend upon which e f f e c t i s the  more i m p o r t a n t i n a p r a c t i c a l c a s e .  I t i s evident that  the  c a p a c i t i v e r e a c t a n c e i s a l s o changed by a l t e r i n g the bundle  c o n f i g u r a t i o n , however, t h i s e f f e c t i s c o n s i d e r e d t o be n e g l i g i b l e here s i n c e the a c t u a l change i n bundle s p a c i n g i s q u i t e small. T h i s c o n c l u d e s the work done on bundled conductors i n this thesis.  The methods used t o determine t h e optimum con-  f i g u r a t i o n f o r the o u t e r phase bundles i n a three-phase system w i t h f l a t s p a c i n g are of course not g e n e r a l .  However, t o de-  termine the optimum c o n f i g u r a t i o n a n a l y t i c a l l y d i d not appear p o s s i b l e due t o the c o m p l e x i t y of the problem.  67 7.  The optimum a  investigation shape  direct  most  analytical  that  f o r the centre  conductor  the outer  on  the centre  solution  considered  lution, jected  i n section  Such A  conductor 4-3 m i g h t  mapping  while  Hence  intensity  an a n a l y t i c a l  uniform  provides  of a different  considered  using  conductor  i t i s made  well provide  three-  was o b -  the f i e l d  an arrangement  i f the contours  t o conformal  5-3.  field  fora  solution  phases  that  not exist i n  and t h e deformed  i s decreased  i n section  however,  phase  4-3 f o r t h e o u t e r  here.  indicates  i n t h e f o l l o w i n g way  conductor  conductor.  regarding the  i n the electric  An e x a c t  conductors  obtained  does  spacing.  f o r the centre  i n section  on  that  flat  thesis  conductors  some i m p r o v e m e n t  c a n be o b t a i n e d  system with  treated  out i n t h i s  apparently  solution  However,  configuration  tained  carried  f o r transmission line  cases.  phase  CONCLUSIONS  the best  shape  than  a better so-  cannot  be  approach  sub-  i s very  cum-  bersome. The  results  transmission maximum can  obtained  lines  electric  with  field  used.  practice, would  have  to device  changes i n order  a method  actually  required.  per  length varies  with  carried  indicate  on t h e o u t e r  t o make  The r e a s o n the conductor  conductors  configuration  use of t h i s to chapter  f o r determining being  outf o r  that the  phase  i n t h e bundle  i n the introduction  diameter unit  conductors  intensity  However,  as o u t l i n e d  the computations  bundled  be e q u a l i z e d by s m a l l  normally  from  the  fact i n  6, o n e  conductor  that the  diameter  used.  charge  68 7-1. Recommendations f o r F u t u r e Work I t appears t h a t u s i n g an e x p e r i m e n t a l method would be t h e e a s i e s t way t o determine t h e optimum shape f o r a conductor p l a c e d p a r a l l e l t o a l i n e charge.  An e l e c t r o l y t i c tank would  p r o b a b l y p r o v i d e t h e b e s t means f o r t h i s purpose s i n c e one t h e n would be a b l e t o determine t h e f i e l d v a r i a t i o n from t h e p o t e n t i a l contours o b t a i n e d .  directly  However, i n t h a t case  some p l i a b l e , c o n d u c t i n g m a t e r i a l would be r e q u i r e d f o r t h e conductors t h e m s e l v e s . The change i n c u r r e d i n i n d u c t i v e and c a p a c i t i v e r e a c t a n c e by a l t e r i n g the bundle c o n f i g u r a t i o n was not computed i n chapt e r 6.  However, even though t h e s e changes a r e s m a l l , t h e  change i n i n d u c t i v e r e a c t a n c e might r e q u i r e c o n s i d e r a t i o n i n a long transmission l i n e .  F u r t h e r , as mentioned above, t h e  conductor diameter a c t u a l l y r e q u i r e d f o r t h e o u t e r phases must be determined.  The e q u a t i o n d e t e r m i n i n g t h e g r a d i e n t on each  conductor c o u l d be used f o r t h i s purpose, however, one would a t t h e same time r e q u i r e t h e charge on each conductor as a f u n c t i o n of i t s d i a m e t e r .  T h i s procedure appears q u i t e cumbersome, b u t  i t might be p o s s i b l e t o i n t r o d u c e some s i m p l i f y i n g  assumptions.  69 REFERENCES 1.  G l o y e r , H., and V o g e l s a n g , T., "380 KV. T r a n s m i s s i o n L i n e s i n Western Germany", The I n t e r n a t i o n a l Conference on Large E l e c t r i c Systems (C.I .G.R.-E.), V o l . I l l , No. 401, 1958, pp. 1-13.  2.  Akopjan, A.G., Burgsdorv, V.V., B u t k e v i t c h , V.V., G e r t z i k , A.K., G r u n t a l , V.L., R o k o t j a n , S.S., and S o v a l o v , S.A., "The Development of 400-500 KV. Systems i n t h e S o v i e t Union", The I n t e r n a t i o n a l Conference on Large E l e c t r i c Systems (C.I.G.R.E.), V o l . 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Copson, E.T., Theory of F u n c t i o n s of a Complex V a r i a b l e , O x f o r d , The C l a r e n d o n P r e s s , 1955.  25.  Rauscher, M., I n t r o d u c t i o n to A e r o n a u t i c a l Dynamics, York, John W i l e y and Sons, 1953.  26.  B y e r l y , W.E., Elements of t h e I n t e g r a l C a l c u l u s , l o r k , G.E. S t e c h a r t and Co., 1926.  27.  H i l d e b r a n d , P.B., I n t r o d u c t i o n t o N u m e r i c a l A n a l y s i s , New York, M c G r a w - H i l l Book Co., 1956.  28.  Page, C. H., P h y s i c a l Mathematics, New York, D. Nostrand Co., 1955.  29.  Margenau, H., and Murphy, G.M., The Mathematics of Phys i c s and C h e m i s t r y , New York, D. Van N o s t r a n d Co., 1957.  30.  S e h m e i d l e r , W., I n t e g r a l g l e i c h u n g e n m i t Anwendungen i n P h y s i k und T e c h n i k , L e i p z i g , Akademische V e r l a g s g e s e l l s c h a f t , 1955.  31.  Neumann, E.R., Ueber d i e Konforme A b b i l d u n g Komplimentarer G e b i e t e , Mathematische Annalen, V o l . 116, 1938/ 1939, pp. 664-695.  32.  M o r r i s , R.M., Two-Dimensional P o t e n t i a l Problems, P r o ceedings of the Cambridge P h i l o s o p h i c a l S o c i e t y , V o l . 33, 1937, pp. 474-484.  33.  Wrinch, D.M., Some Problems of Two-Dimensional E l e c t r o s t a t i c s , P h i l o s o p h i c a l Magazine, S e r . 6, V o l . 48, J u l y - Dec. 1924, pp. 692-703.  34.  Dwight, H.B., S u r f a c e V o l t a g e G r a d i e n t on Power T r a n s m i s s i o n L i n e s , T r a n s a c t i o n s of t h e American I n s t i t u t e of E l e c t r i c a l E n g i n e e r s , V o l . 76, 1957, pp. 1217-1220.  35.  Gross E.T.B., and S t e n s l a n d , L.R., C h a r a c t e r i s t i c s of Twin Conductor Arrangements, T r a n s a c t i o n s of t h e American I n s t i t u t e of E l e c t r i c a l E n g i n e e r s , V o l . 77, P a r t I I I , 1958, pp. 721-725.  36.  Reichman, J . , Bundled Conductor V o l t a g e G r a d i e n t C a l c u l a t i o n , T r a n s a c t i o n s of the American I n s t i t u t e of E l e c t r i c a l E n g i n e e r s , V o l . 78, 1959, pp. 598-607.  37.  C l a r k e , E., C i r c u i t A n a l y s i s of A-C Power Systems, V o l . I , New York, John W i l e y and Sons I n c . , 1950.  New  New  Van  72 38.  Adams, G.E., V o l t a g e G r a d i e n t s on H i g h - V o l t a g e Transm i s s i o n L i n e s , T r a n s a c t i o n s of t h e American I n s t i t u t e o f E l e c t r i c a l E n g i n e e r s , V o l . 74, 1955, pp. 5-11.  39.  Woods, F.S., Advanced C a l c u l u s , New York, Ginn and Company, 1934.  40.  H i l d e b r a n d , F.B., Advanced C a l c u l u s f o r E n g i n e e r s , New York, P r e n t i c e H a l l I n c . , 1949.  41.  Courant, R., and H i l b e r t , D«, Methods of Mathematical P h y s i c s , New York, I n t e r s c i e n c e P u b l i s h e r s I n c . , 1953.  73 APPENDIX I . THREE PARALLEL CONDUCTING CYLINDERS 17 Equation  (2-10) C O  2na 1 + 2  °  Y. ( ~ ) cos n6 1 n  s  coul./m  (2-10)  2  r e f e r s t o t h e case of a c o n d u c t i n g c y l i n d e r w i t h charge -a, coul./m p l a c e d p a r a l l e l t o a l i n e charge of s t r e n g t h +q coul./m.  I f t h e charge on t h e c o n d u c t i n g c y l i n d e r and t h e  l i n e charge b o t h a r e p o s i t i v e i t f o l l o w s t h a t E q u a t i o n (2-10) becomes CO  ~ 27ta  a  1~2  Y. ( f " ) 1  nc o s  n  s  S  coul./m  (I-l)  R e f e r t o t h e i s o l a t e d system shown i n P i g . I - l .  The  t h r e e c o n d u c t i n g c y l i n d e r s may be c o n s i d e r e d an ungrounded three—phase t r a n s m i s s i o n l i n e w i t h charges and p o t e n t i a l s as indicated.  The p o t e n t i a l s V^, Vg and Vg a r e s p e c i f i e d w i t h  r e s p e c t t o t h e ungrounded n e u t r a l o f t h e e q u i v a l e n t s t a r connection.  I t i s d e s i r e d t o determine t h e s u r f a c e charge  d e n s i t y on each c o n d u c t o r , and f o r t h i s purpose an " i t e r a t i v e " method w i l l be employed as f o l l o w s .  At t h e o u t s e t t h e t h r e e  conductors a r e assumed t o have u n i f o r m s u r f a c e charge d e n s i l 2 3 /2 °1 = 2 i a » °2 = 2^ °3 = 2wa" °/ respectively. a  t  i  e  s  4  q  a  n  d  c o u l  Then a f i r s t c o r r e c t i o n , 0-^2' ^  a  *  o u n  d *  O T  m  "^  ne  charge d e n s i t y  of conductor 1 due t o t h e presence of t h e u n i f o r m charge 2 on conductor 2. S i m i l a r l y a f i r s t c o r r e c t i o n , o*^g, i s found f o r conductor 1 due t o t h e presence o f t h e u n i f o r m  q  charge on conductor 3.  Thus t h e f i r s t s u b s c r i p t r e f e r s t o  74 the conductor t o which t h e c o r r e c t i o n i s t o be a p p l i e d and t h e second s u b s c r i p t t o t h e conductor g i v i n g r i s e t o t h e c o r r e c tion.  I t f o l l o w s t h a t one s i m i l a r l y must f i n d o" ^, °23  a  n  d  2  0*31' egg•  Then h a v i n g determined t h e f i r s t c o r r e c t i o n one  may f i n d t h e second c o r r e c t i o n  > ^ { 3 e t c . which depend on  the f i r s t c o r r e c t i o n s ov,^, 0 g * « e  c  2  C o n s i d e r a d i f f e r e n t i a l l i n e element o f charge a t p o i n t N on conductor 2. q  2  / / x d-8 coul./m and from E q u a t i o n ( I - l ) i t f o l l o w s  a  *aia  dcNo = -  (fera  12 ~ ~ 2™ But cos n a, d  This element i s a l i n e charge o f s t r e n g t h  5  ^  a  dB )  (1-2)  d  P2> n = £1  k =00 1 + I 1  _1 n  =co  n  9  C 0 S  n + k - 1 (|) k  n ( 6  k  - »> a  cos k 6  2  (1-3)  and sin na  k = co n + , v k = 1  0  n s  d  k - 3| ( | ) s i n k 8 k  2  is.  (1-4)  E q u a t i o n s (1-3) and (1-4) a r e t h e n s u b s t i t u t e d i n E q u a t i o n (1-2) above and t h e r e s u l t i n g e x p r e s s i o n r e s p e c t t o 8.  i s integrated with  C o l l e c t i n g terms one o b t a i n s  <1  2  '12  0 0  1  NB'  0  0  s n6  I t f o l l o w s t h a t a s i m i l a r procedure used f o r conductor 3 r e sults i n  q „ 00 3 H  J  13  then  'l-K-St'  CO  28'  (f )  n  cos  -6  - g  I (fc>» cos n6  oo  (1-5)  75 For conductors 2 and 3 one a l s o f i n d s 2 = ozr2 ~ 2na q  l *y /axn L (-) cos nB, na * j *s'  q -  3 "~ 2lta  L  \  2  o  n  (-)  V  1  cos n6  (1-6)  /a n x  (1-7)  Q  1  From F i g . 1-1 ( T C - 6) = 8 *1  c  t&\Ti  C O  oo V"  l  oo V"  3  —  —  H  a  L  Q  C O  (|)  I  or  9  6 = (it - 8 ) . q„  cos n 8  n  2  Hence  0  co  ^3 I -—  {=£)  cos n 8  n  2  P r o c e e d i n g i n t h e same manner as b e f o r e d o  i  =  2  t ,( f s' ) n  cos  ~ tea q  V~ °  n  0  s=co  } C S  ^2 n = 1 n = 1 m = ©2 2 _ ( f ) cos m(6 - a ) m = 1 q, n =oo q. n =co l v /a \n o X (g.) cos nB. Tea n = £1( J ) n = 1 m =co m  na  n  adB, n P  2  Tta  (1-8)  m  2  2  u  cos n 8  n  adB, 3  7ta  m = I] 1 (f) cos m (6 - ct«J  (1-9)  S u b s t i t u t i n g Equations (1-3) and (1-4) i n (1-8) and ( 1 - 9 ) , and i n t e g r a t i n g t h i s expression with respect to 8  4i  '12  A,l cos 6 + A„& cos 26 + ... + A n cos n6 + ... +!  na  ^3 B^ cos 6 + B Tta where A  n  In  n=co  2  cos 26 + ... + B  (?)*" D|) cos n 8  =  2  ' n=l'  S  k=co  n  s  2  ~  k=l  (* +  * -  cos n6 +  n  ...+  D ( f ) cos k B k  2  2  +n k  (f) n =  +  k  o o o  k  ( | ) " ( - f <?>! x(  dB  f(J) f + (f>r + ^(f) + ... + ( - ^(f)" +  0 a\n  B  2  tii  J  +  + k - 1\ /a\k  ) (-)  a\2/n <!>'<  ^*  + 1\/a\2 2 +  + ... +  /-a\ k  2  76 and m has been r e p l a c e d by n. 0"^  r i v e d f o r o^g  S i m i l a r e x p r e s s i o n s may be de-  o^g e t c . w i t h i n c r e a s i n g l y complex ex-  pressions r e s u l t i n g . I t i s noted t h a t the lowest order terms i n (—) due t o t h e s second c o r r e c t i o n i s of 3 r d o r d e r , i . e . , (—) . For a p r a c s t i c a l t r a n s m i s s i o n l i n e a = 1 i n c h s = 30 f e e t hence (—)^ i s s completely n e g l i g i b l e . (1-6) and ( 1 - 7 ) .  However, c o n s i d e r Equations  These were d e r i v e d on t h e assumption t h a t  the charge on conductors  o t h e r than t h e one c o n s i d e r e d c o u l d  be r e p r e s e n t e d as l i n e charges cylinders.  (1-5),  a t the c e n t r e of the c i r c u l a r  Hence one then has a good e s t i m a t e of the e r r o r  i n c u r r e d i n t h e charge d e n s i t y by t r e a t i n g the a d j a c e n t d u c t o r s as l i n e  charges.  con-  77 APPENDIX I I .  CONFORMAL MAPPING OP HARMONIC FUNCTIONS  L e t z = x + j y and w = u + j v r e p r e s e n t two complex p l a n e s which w i l l be denoted t h e Z-plane and W-plane i n t h e d i s c u s s i o n following.  Consider the f u n c t i o n w = f ( z ) . For a l l points df (z) where f (z) i s a n a l y t i c and ^ ' £ 0 t h e f u n c t i o n f (z) i s s a i d t o g i v e a conformal mapping o f those p o i n t s from t h e Z-plane 23 z  onto t h e W-plane  .  Hence one can map a r e g i o n o f t h e Z-plane  onto t h e W-plane c o n f o r m a l l y  i f t h e mapping f u n c t i o n f ( z )  s a t i s f i e s t h e c o n d i t i o n s s p e c i f i e d above. A f u n c t i o n which s a t i s f i e s L a p l a c e e q u a t i o n i s termed a harmonic f u n c t i o n .  L e t V represent a p o t e n t i a l f u n c t i o n , then  i n a charge f r e e r e g i o n  , dV  n  *x V C o n s i d e r a t r a n s f o r m a t i o n o f t h i s e q u a t i o n t o u and v c o - o r d i n a t e s where u = u ( x , y ) , v = v ( x , y ) and w = u + j v . one can w r x.t.e1 3 2  +  Then  _ bV £u £V ^ v £x ~ >>u c\x <yr  a v _bv 2  tt£ ^L(^)2.$v 6 2  d  +  X  2  2  W  2 u  )2  d v,hX  +  r  +  d  dx  2  +  ^2  >  . t  1 1  v  " ' 1  Similarly 5 ^ - ^u  +  ^ 2 W  j  V  +  u  +  d 2 V  U  I  2  )  v  A d d i t i o n o f E q u a t i o n s ( I I - l ) and ( I I - 2 ) y i e l d s A 2 u  x  +  -\ 2 ~  °y  V  TT2  ^u  +  A 2' u  v  (^)  2  + (S*> | 2  S i n c e t h e Cauchy Riemann e q u a t i o n s s t a t e  (H-3,)  78 and $H = -  =  ox  by  by  Hence L a p l a c e  b*  e q u a t i o n i s i n v a r i e n t t o conformal mappings, t h a t  i s , harmonic f u n c t i o n s remain harmonic when s u b j e c t e d t o conformal transformations.  However, i t i s c l e a r t h a t one r e -  q u i r e s t h e mapping f u n c t i o n t o s a t i s f y t h e c o n d i t i o n s  laid  down above s i n c e  (|f) <^> 2  2  dw dz  +  Any  conducting  s u r f a c e kept a t a p o t e n t i a l V i s an e q u i -  potential surface.  Hence i f t h e conducting  surface i s p o s i -  t i o n e d i n t h e Z-plane i t f o l l o w s t h a t V(x,y) = c  (II-4)  where c i s a c o n s t a n t .  Under a conformal  transformation  from t h e Z-plane t o t h e W-plane t h e c o - o r d i n a t e s  x and y a r e  represented  as f u n c t i o n s o f u and v , t h e r e f o r e i n t h e W-plane  V(x,y) = V  x(u,v),  Therefore  y(u,v) j = c  (H-5)  t h e boundary c o n d i t i o n g i v e n by E q u a t i o n  (II-4) i n  the Z-plane remains unchanged i n t h e W-plane. The  e l e c t r i c f i e l d i n t e n s i t y i n t h e Z-plane i s g i v e n by  A l s o i n t h e W-plane t h e e l e c t r i c f i e l d i n t e n s i t y i s g i v e n by E  +  u "  j  J  E  v  „ -  51 -  bu  J  -j 2£ « - ( M ) * bv  *n  (H-7)  where Q i s a complex p o t e n t i a l f u n c t i o n w i t h Q = V + j U and U being t h e harmonic conjugate t o V„ a l s o be w r i t t e n  Equation  ( I I - 7 ) may  79 U  "  V  Hence i t f o l l o w s E  E  dz dw  Equation (II-8) obviates  (II-8) the n e c e s s i t y of e v a l u a t i n g the poten-  t i a l f u n c t i o n i n t h e W-plane i f one i s i n t e r e s t e d o n l y i n t h e magnitude of t h e e l e c t r i c f i e l d  intensity.  Only a b r i e f d i s c u s s i o n o f conformal mapping has been g i v e n here, however, the f e a t u r e s mentioned a r e the more i m p o r t a n t ones.  considered  See References 13,23*24.