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Impedance calculation of cables using subdivisions of the cable conductors Abledu, Kodzo Obed 1979

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IMPEDANCE CALCULATION OF CABLES USING SUBDIVISIONS OF THE CABLE CONDUCTORS  by  Kodzo Obed A b l e d u B.Sc.(Hons.), U n i v e r s i t y o f S c i e n c e and Technology, Kumasi, 1976  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE  in THE FACULTY OF GRADUATE STUDIES (Department o f E l e c t r i c a l E n g i n e e r i n g )  We a c c e p t t h i s t h e s i s as conforming to t h e r e q u i r e d  standard  THE UNIVERSITY OF BRITISH COLUMBIA September, 1979 (c) Kodzo Obed A b l e d u , 1979  In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an advanced degree a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree that permission f o r extensive copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by t h e Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my written permission.  Department n f  £ U F C T K ( ^ * U -  The U n i v e r s i t y o f B r i t i s h 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date  S€PTe/Wfe<S«  6 K G t r ^ R i H q  Columbia  ^ ,  ABSTRACT  The impedances o f c a b l e s =are some o f t h e parameters needed f o r v a r i o u s s t u d i e s i n c a b l e systems.  I n t h i s work, the impedances o f c a b l e s a r e c a l c u l a t e d u s i n g the s u b d i v i s i o n s o f t h e c o n d u c t o r s ( i n c l u d i n g ground) i n t h e system. Use i s a l s o made o f a n a l y t i c a l l y d e r i v e d ground r e t u r n formulae t o speed up t h e c a l c u l a t i o n s . The impedances o f most l i n e a r m a t e r i a l s are c a l c u l a t e d w i t h a good degree o f a c c u r a c y b u t m a t e r i a l s w i t h h i g h l y nonlinear p r o p e r t i e s , l i k e s t e e l pipes, give large d e v i a t i o n s i n the r e s u l t s when they a r e r e p r e s e n t e d  by t h e l i n e a r model used.  The method i s used t o study a t e s t case o f i n d u c e d sheath c u r r e n t s i n bonded sheaths and i t g i v e s v e r y good r e s u l t s when compared w i t h t h e measured v a l u e s .  i i i  TABLE  OF  CONTENTS  ABSTRACT TABLE  OF  i CONTENTS  i  LIST  OF  TABLES  LIST  OF  ILLUSTRATIONS  i i  i  • V v i  ACKNOWLEDGEMENTS  v i i i  LIST  i  1.  OF  SYMBOLS  INTRODUCTION 1.1  x  1  A B r i e f Review Impedances  of Methods  for the Calculation of  Cable 1 o  2.  3.  1.2  Skin  1.3  A Brief  1.4  Scope  THEORY  and P r o x i m i t y  Effects  Explanation  of  of  t h e Method  of  Subconductors  4  the Thesis  AND T H E  5  F O R M A T I O N AND S O L U T I O N  2.1  Subdivisions  of  2.2  Assumptions  2.3  Loop  2.4  Formation  of  Impedance  2.5  Bundling  of  the Subconductors  2.6  Reduction  of  the Large  2.7  The C h o i c e  2.8  Including  OF E Q U A T I O N S  6  the Conductors  ^ ^  Impedances  of  Subconductors  ^  Matrix i n t h e Impedance  Impedance  and C o n s t r a i n t the Constraint  *  '  '  on the Current  Path  15  i n the Matrix  Solution-  ,  3.1  Return  i n Neutral  Conductors  3.2  Return  i n Ground  Only  3.3  Return  i n Ground  and N e u t r a l  3.4  Use of A n a l y t i c a l Equations  "  Matrix  on t h e R e t u r n  RETURN PATH IMPEDANCE  Matrix  13  20  Only  20 20  Conductors f o r Ground  22 Return  Impedance  •  •  . 2 2  iv  3.5  3.6  3.7  3.8  4.  25  R e p r e s e n t i n g the Ground as One U n d i v i d e d Conductor Model I I I  26  Mutual Impedance between a Subconductor and Ground w i t h Common Return i n Another Subconductor  28  Comparison o f Model I I I w i t h the T r a n s i e n t Network A n a l y z e r Circuit  RESULTS 4.1  5.  Model U s i n g Ground Return Formulae D i r e c t l y w i t h the Subconductors - Model I I  31  33  Comparison of the Method o f Subconductors w i t h Standard Methods  33  4.2  Comparison o f Ground Return Formulae  37  4.3  Comparison o f R e s u l t s from the D i f f e r e n t  4.4  Reproduction of Test Results  49  4.5  P i p e Type Cables  54  CONCLUSIONS  LIST OF REFERENCES  Models  . . . . . .  47  63  64  V  LIST  OF  TABLES  TABLE 4.1  Variation  of  4.2  Variation Number o f Proximity  o f t h e I m p e d a n c e o f t h e C i r c u i t o f F i g . 4.3 S u b d i v i s i o n s , Showing the I n c l u s i o n of Both Effects i n the Calculations  4.3  Self  Impedance  Impedance  of  with  Ground  Ground  and Other  Formulae  4.4  Mutual  Impedance  Between  4.5  Comparison  4.6  Induced  4.7  Impedance  of Various  Currents of  Pipe  Return  Path  of  Subdivisions  34 with the S k i n and 37  as C a l c u l a t e d Using  Two U n d e r g r o u n d  Subdivided  Conductors  ^  Models  i n Bonded Type  t h e Number  0  Sheaths  Cables  ~,->  for Various  Degrees  of Magnetic  Saturation 4.8  4.9  -  ~  J  Zero Sequence Impedance Measurements i n a Pipe with Pipe Return Impedances Concentric  -  on Three  Cables  of Cables i n Magnetic Pipes Represented Pipes of D i f f e r e n t P e r m e a b i l i t i e s  Enclosed J  o  a s Two ^  vi  LIST OF ILLUSTRATIONS FIGURE 1.1  Current d i s t r i b u t i o n i n s o l i d round conductors due to skin and proximity effects  3  Current d i s t r i b u t i o n i n the subdivided conductors of the model  5  2.1  Subdivision of the main conductors  6  2.2  C i r c u i t of two subconductors with common return  7  2.3  Geometry of Subconductors  8  2.4  I l l u s t r a t i o n of the reduction process.  2.5  A two-w;ir'e r e t u r n ' c i r c u i t  2.6  A two-wire c i r c u i t conductor  1.2  £, k, q.  15  :  with, common r e t u r n  1'6 in a third _ 16  3.1  Subdivision of ground into layers of subconductors  21  3.2  Model with, only subconductors and ground return  3.3.  Model with ground represented as only one conductor  27  3.4  A c i r c u i t of two conductors with common ground return  29  3.5  A c i r c u i t of one conductor and the ground with common return  .26  in another conductor  29  4.1  A return c i r c u i t of two conductors far apart  4.2  Variation of impedance with the number of subconductors  35  4.3  A return c i r c u i t of two conductors very close together  35  4.4  V a r i a t i o n of the impedance of a buried conductor with depth of b u r i a l Cross sections of buried conductors for ground return impedance calculations •  4.5  - . . . 33  3'8 40  4.6  Comparison of calculated s e l f impedances of a ground return loop • 43  4.7  Comparison of calculated mutual impedances between two buried conductors  46  4.8  E l e c t r i c a l layout of the induced sheath current test  50  4.9  C i r c u i t diagram of the induced sheath current test  51-  vii  4.10  V a r i a t i o n o f magnetic p e r m e a b i l i t y o f s t e e l p i p e w i t h c u r r e n t i n the p i p e  57  4.11  Shape o f m a g n e t i z i n g curve d u r i n g one c y c l e  57  4.12  L i n e a r i z e d m a g n e t i z i n g curves  62  ACKNOWLEDGEMENTS  I would l i k e t o express my thanks t o my s u p e r v i s o r , Dr. H.W.  Dommel, f o r h i s h e l p  suggestions  throughout t h i s work and f o r the t i m e l y  and c o r r e c t i o n s he made.  A l s o , I w i s h t o convey my g r a t i t u d e  to Mr. Gary Armanini of B r i t i s h Columbia Hydro and Power A u t h o r i t y , f o r making h i s r e p o r t and t e s t r e s u l t s a v a i l a b l e f o r use i n t h i s work.  I am a l s o very  g r a t e f u l to the Government of the R e p u b l i c o f  Ghana f o r f i n a n c i n g my e d u c a t i o n  For reading  a t the U n i v e r s i t y o f B r i t i s h  through and c o r r e c t i n g the s c r i p t s , I would  to thank Ms. M a r i l y n Hankey o f the F a c u l t y o f Commerce. b e a u t i f u l l y done by Mrs. Engineering;  I  Columbia.  Shih-Ying  do a p p r e c i a t e  like  The t y p i n g i s  Hoy o f the Department o f E l e c t r i c a l  i t v e r y much.  ix  L i s t o f Symbols  B  flux density  D„ £q D P  d i s t a n c e between conductors £ and q  e  =2.71828  p i p e diameter  f  frequency  g  s u b s c r i p t denoting  GMR  geometric mean r a d i u s  h  depth o f b u r i a l o f conductor  i, I I p  current pipe current  j  complex o p e r a t o r  i,j,k,£,n  /-I  subscripts  km  kilometres  m  = /(jyu/p)  m  metres  M  =  ground  inductance  q  s u b s c r i p t denoting  r,R  resistance, radius  v,V  voltage  X  reactance  Z  impedance  return  path  K ,  Bessel  log  Common l o g a r i t h m  £n  Natural logarithm  Hz  hertz  U  a b s o l u t e p e r m e a b i l i t y o f f r e e space = 4irx 10 ^ H/m  Q  functions (base 10) (base e)  X  y y tj)  =viQy r  ,  relative  permeability permeability  flux  ¥  flux  linkage  ir  =3.1415926...  y  =0.5772157...  ft  ohm  (JJ  =2lTf  =  Eulers  Constant  1  Chapter  1.1  A Brief  Review  For input  propagation lines  of Methods  the analysis  parameters  studies  which  of  D.M.  a r e found  calculations Schelkunoff  f o r the Calculation  transmission line the lines.  conductors  reasonably  cable  Simmons  systems  [1]  h a s done  systems.  the basic surge  effects  lines,  pipes  between and  data.  analysed  by many  are often  analysis  of  used  For single-cored  a comprehensive  one o f  studies,  i n the publication  and which  Impedances  induction  as other  been  Cable  Fault  impedance  have  resulted  for distribution  (such  accurate  i n many h a n d b o o k s  [2]  of  of  systems,  and the c a l c u l a t i o n of mutual  a l l require  work  INTRODUCTION  i s the impedance  Underground The  of  and p a r a l l e l adjacent  fences)  1  authors.  standard i n  charts  impedance  (coaxial)  and h i s r e s u l t s  cables, are widely  used.  Carson derived turn. the  equations Smith  used  effects. and  of  many  Another  [3],  of  the formulae  used  [ 2 2 ] , who h a v e  has been  of  specific  shape,  which  approach  used  and others  used  effects,  i n this  also  ground  r e -  have  calculated  namely:  ground)  systems.  calculations,  the impedances  accounts  have  i n distribution  i n impedance  (including  automatically  i s also  [4,5]  [9]  cables with  used by C o m e l l i n i ,  calculated  a l l the conductors  and Wilcox  underground  cables  f o r two i m p o r t a n t  approach  Wedepohl  et a l .  concentric neutral  dividing  This  of  Lewis,  by  above.  [19],  f o r the impedance  to correct  Talukdar  Pollaczek  and Barger  impedances  In are  [10],  into  skin  and  et a l . of  factors  proximity  [7]  and  Lucas  transmission  lines  smaller  conductors  f o r t h e two e f f e c t s  thesis.  Cables  with  mentioned  sector  2  shaped  conductors  uniform  or  properties  conductors  across  the  of  any  cross  irregular  section  cross  s e c t i o n or  of  non-  can e a s i l y  be h a n d l e d  with  this  method.  Skin  1.2  and  Proximity  The  r e s i s t a n c e of  determined because the  from  direct  wire.  In  distribution by  the  the  of  "skin  case of  of  over  the in  the  direct  wire  current,  cross  the  of  to  distributed  alternating  phenomenon,  of  in  neighbouring  these  bution  in  effect,  other  the  is  and  across there  s e c t i o n of  conductor.  a  This  to  or  the  flow  conductor.  circular).  either  opposite  The  conductors  symmetrical is  on  current  the the  type  of  cross  exists  a  is  easily material  section  of  nonuniform  conductor  which  phenomenon  is  is  caused  called  in  the  causes This  around In  conductors  a  sides  a distortion  axis  two-wire of  the  of  arises  c l o s e by.  distortion,  the  effect",  line,  Changing  in  the  unlike  symmetry for  conductors  due  of  due  the  instance, which  face  the  currents  current  that  to  to  d i s t r i skin  conductor  of  Figure  s k i n and 1.1.  proximity  effects  in  round  (if  more  current  each  other  sides.  phenomena  illustrated  called "proximity  current-carrying  first  not  conductor  tends  are  uniformly  current  presence  on  transmission line  effect".  Another  the  is  current  variation  a  p h y s i c a l dimensions  current the  Effects  conductors  3  ^>  Current  Distributions  ( a ) Skin Effect Figure  (b) P r o x i m i t y E f f e c t  Current  1.1  distribution  proximity  The conductors amount the  of  uneven  causes direct  self  internal  linkage  more  current,  in  distribution power  thus  much w i t h both  -  of  extent  impedance the  while  pronounced  for  current the  inductance  conductor  with  current  additional  higher  The  very  solid  round  conductors  due  to  skin  and  loss  across  above  i n c r e a s i n g the  the  that  cross  s e c t i o n of  produced  effective  by  a.c.  an  the  equivalent  resistance  of  conductor.  The the  in  effects  to  inner the  part  which  of  towards  of  the  the  conductor  conductor.  This  surface  reduces  decreases  the  conductor.  depends size  density  the  above  o n how the  proximity  effects  pronounced  conductor  alter  they  and w i t h  effect  depends  closer spacings  between  mainly  the  are. the on  values Skin  effect  frequency the  conductors).  of  -  geometry  i t  the varies increases (being  Bessel to  alternating  functions  current  widely  found  drical  conductors  effect  is  more  derived  [17] in  in  most  hand  close  in  the  at  higher  1.3  thesis  seeks of  the is  to  Figure  correct  makes  cables both  of  Method  the  Enrico  work  take of  into  both  any  of  into  the  of  these  main  the  above  Dividing  the  conductors  1.2.  the  cables  and  cylin-  hand,  are  in  proximity  factor  tables  customarily  size  proximity  and  are  effects  frequency  surge  et  a l .  account  and  used  usually important  especially  studies.  [7]  it  is  shown  simultaneously  This  of  is  Subconductors  is  done  by  subconductors, The  and  work  in  that  by  by  the  main  finding  bundling  described  i t  calculating  dividing  c y l i n d r i c a l shape,  conductors.  on  approximate  power  of  method  other  [18]  resistance  This  correcting  large  switching  subconductors  impedances  are  in  effect.  at  Comellini  effects  this  and  transmission line.  smaller  impedance based  of  the  and  skin  even  the  formulae  for  Explanation  and m u t u a l  give  to  for  impedance  self  complicated  coaxial  On  Charts  needed  to  as  [1,9].  analyze.  increases  analytically.  e x p e c i a l l y when  impedances,  frequencies  calculate  skin effect  underground  c a l c u l a t i o n of  conductors  to  This  to  analyzed  otherwise  together.  possible  the  difficult  general,  A Brief  to  being  calculations  In is  due  used  literature,  are  from  In laid  the  are  in  the  them  to  this  reference.  current  into  parallel cylindrical  distributions  shown  in  subconductors  Figure  1.1  by  those  Conductor s  Approximate Current Distributions (b) Proximity Effect  (a) Skin Effect Figure  1.2  Obviously w i l l  depend  on  Current d i s t r i b u t i o n of the model  the  accuracy  the  degree  This  Thesis  of  to  be  in  subdivided  expected  discretization,  from  such  and hence  conductors  a  on  representation  the  number  of  subconductors.  1.4  Scope  of  The developed in  the  porated  theory  using  model. into  computing  calculating  fictitious  •Analytically the  model  to  'return derived  reduce  the  the  impedances  path' ground number  which return of  from  allows  subconductors more  formulae  are  subconductors,  type  layers  permeability  of  cables pipe  depending  on  are  modelled  material, the  degree  with of  by  treating  each  layer  saturation.  the  steel  having  a  is  flexibility then  incor-  storage  time.  Pipe concentric  a  for  pipe  as  different  and  6  Chapter T H E O R Y AND T H E  2.1  Subdivisions  In a main and  the armour.  the  described below,  Each main  subconductors  values  at higher  before  these  makes  2.1  present,  tried show  suggests  can be used w i t h  Subdivision  the cable  a number  considered  conductor  of  parallel  a cylindrical  formulae  by Lucas  simple.  and Talukdar  a large that  i s  the neutral  into  inductance  c a l c u l a t e d by them  shapes  Figure  have been  of  The c h o i c e o f  the derived  which  core  i s divided  (Figure=2.1).  frequencies,  other  each  and i f  conductor  f o r the subconductors  resistance values  EQUATIONS  Conductors  as i s the sheath,  the subconductors  shapes  F O R M A T I O N A N D S O L U T I O N OF  the  the model  conductor,  cylindrical for  of  2  deviation  further  Other [22] b u t  from  rese.arch  shape  measured i s  needed  confidence.  of main  conductors  Assumptions  2.2  It i) ii)  Each  i s assumed subconductor  The magneitc  that: i s uniform  permeability  of  and homogeneous  throughout  a subconductor  i s constant  i t s  length;  throughout  7  the  whole  that i i i )  There  iv)  2.3  A l l  Loop  To the  two  path, or  a  of  in  is  of  uniform  current are  derive  loop  the  formed  by  current,  but  may  be  different  from  subconductor;  of  Figure  alternating  other  Impedances  fictitious  lation  any  of  subconductors  loops  q,  cycle  distribution  in  each  subconductor;  and  parallel.  Subconductors  impedances  any  2.2.  two  The  conductor  the  subconductors,  subconductors,  return  chosen  of  path  for  the  % and k,  can e i t h e r voltage  be  first  with one  a  consider  common  return  of  the  subconductors  measurements  and  the  calcu-  inductances.  Figure  2.2  Loops  formed  common  Writing  the  loop  by  two  subconductors  with  a  return  equations  for  subconductor  Z  gives:  (2.1)  where  = voltage  R^ i  drop  per  unit  length  of  =  r e s i s t a n c e per  unit  length  of  =  resistance per  unit  length  of  ,  i ,  =  currents  in  subconductor subconductor  return  subconductors  path  £ and k  respectively  AC  JO  M  = mutual  inductance  between  loops  formed  by  subconductors  J6K. £ and N  current phasor  = number For  ac  in  (2.1)  value  jcol.  i  k  with  of  return  steady-state are  Figure  2.3  path.  To  subconductor  common  subconductors  Figure  the  a  £ and  shows  (N=2,  by  the  Geometry  the  derive  returning  in for  conditions,  replaced  2.3:  return  the in  cross  of  £,  the  phasor  k  in  Figure  2.2)  instantaneous values  subconductors  s e c t i o n of  inductance q.  q  two  formulae,  V  and  voltage I  and  v  and  by  the  subconductors  and  (£,k,q).  such  consider  current  I  in  The  flux  B  Total  flux  per  density  radius  length  *  linkage  in  the  elemental  of  loop  k^  due  to  q(equiv)  of  loop  k^  due  to  calculation  (=r^e  The  two  yldr  _ _yl_  2irr  2TT  I  in  i i s :  (2.4)  2  return  r  current  I  in  q  is:  D.  (2.5) q(equiv)-  q( Q ly) the  u  equivalent  ^ r q ^ ^  ,  fluxes  are  the  radius  of  subconductor  geometric  mean  additive;  hence  ZTT  r  q  for  inductance  radius)  the  total  flux  linkage  &q  *£k -  The  mutual  t  o  }  inductance  M£k where  2V  £k  yr^=relative  <5r  * Kk  dr  kq  r  current  r=D  £k  e  thickness  (2.3)  yi  linkage  of  6r  2irr r=D  cylinder  Sr  yi  where  £ is:  (2.2)  2Trr  r = r  outside  .2irr  6<j> = B  Flux  r  yi  =  unit  Flux  B at  £k  (M£^)  y  between  £n  2T:  permeability  Iq  loops  kg  of .the  q  exp(-y  /4)  rq  & and k  is  + .IS.  return  path.  i s :  (2.6) ' therefore  (2.7)  To current  derive  the self  linkage  of  £q r=r  linkage  loop  t h e same  ,  pi  . r£  = ~ ££ 2  and p  (2.8)  I  returning  i n q i s :  £q  pi „  (2.9)  q(equiv)  k  t  linkage i s :  £q 7 ^ £  £n IT./  , r£ . exp( - r - )  are the relative  rq  i n £ i s :  £(equiv)  q(equiv) flux  I  13_  £n  2TT  £^ d u e t o c u r r e n t  D  u  ul  • £ — d r = ^— £n 2irr 2TT  the total  *  where  £ we c o n s i d e r  £^ due t o c u r r e n t  £(equiv)  £q  r=r  loop  uldr 2irr  of  D  Hence  of loop  path.  Flux  Flux  inductance  ^  exp (  )  (2.10)  ZTT  permeabilities  of  subconductors  £ and  respectively  The  self  inductance  (  M  ^£) of  D  M  2.4:  =  ££ h i  Formation  of  Writing the  subconductors  n  =  27  £n  £  Impedance  the loop gives  £q  D  loop  £q  £ i s  y  r£  ,  y  rq_  (2.11)  q  Matrix  equations,  using  a s e t of l i n e a r  (2.1),  equations  (2.7)  and (2.11),  for a  11  V  J  11  Z  In  llln  llll  Z  lnll  Z  lnln  lnkl  llkm  11  Inkm  In  Z  ^lkl*"  Z  (2.12a) V. kl  J  Z  km  i.e.  [V] = [ Z  where  b ± g  "klkm  klll  kmkm  Z  kmll  kl  ' • • kmln  km  (2.12b)  ] [I]  th V.. refers to the voltage on the i subconductor of the Ji .th . , „ 2 main conductor. I.,  1  1  i s the current i n the i * " * subconductor of the j*"* main  conductor. Z.. jimn  i s the mutual impedance between the loops formed by 1  1  1  1  1  the i * " * and n*"* subconductors of the j " * and m"^ main conductors respectively.  i  The p a r t i t i o n i n g of the matrix [ Z ^ g ]  n  (2.12a) groups the equa-  tions of the subconductors within each main conductor together.  The resistances and inductances i n the impedance matrix [ Z ^ g ] are  constant, but since the current d i v i s i o n among subconductors changes  with frequency, skin and proximity effects are accounted f o r .  Normally the large impedance matrix i s not of d i r e c t i n t e r e s t . Instead, the matrix giving voltages on the main conductors i n terms of the  12  currents the  i n these  large  impedance  equivalent of  (2.13)  tribution which,  to  of  below.  equations  the impedances  to  of  can be obtained  Mathematically,  '(2.12a)  Practically,  i n a l l subconductors  voltages  i t  with  the  this  (2.12a),  a current satisfies  i s  conditions  i s equivalent  achieve  from  to  redis-  distribution the  voltage  on the subconductors  forming  any main  conductor  are  hence:  = V.0 J2  the current  subconductors  I. J  =  = V. = V. jn j  i n any main  into  which!.it  = I... + I + J l J2  Bundling  currents in  ] by reduction.  shown  by  which  (2.13).  V., J l  2•5  i s needed,  the algebraic  currents  The  Also,  solving  and (2.14) of  conductors  matrix  when m u l t i p l i e d  condition  equal;  main  of  (2.13)  conductor  i s divided.  . . .  the currents  i n the  Thus:  + I.  (2.14) jn  the Subconductors  i n the Impedance  Expressing  the voltages  they  i s accomplished by  carry  i s t h e sum o f  on the main  Matrix  conductors  the use of  i n terms  equations  of  (2.13)  the and  (2.14)  (2.12a).  Consider subconductors  (a)  the f i r s t  as shown  Subtracting subsequent leaves (-  i n  main  conductor;  assume  i t  i s subdivided  into  n  (2.12a).  the f i r s t equations  the left-hand  illustrated i n  equation  ( i . e . row-1 i n 2.12a)  ( i . e . row-2 side  (2.15)).  of  to  row-n)  the other  of  that  equations  from  main  equal  the  conductor to  zero  13  (b)  By w r i t i n g conductor to  first  I  instead  ( i . e . row-1) , equation  Corresponding  1^^  an e r r o r  has been  errors  i n the f i r s t  made  of  since  are introduced  into  a r e removed  by  whole  matrix  [Z,  the subsequent  from  subtracting  of  ^12  +  the  +  ^ilYlJ'H  I-^I^l  errors  . ] bigJ  equation  adding  These  main  a l l the other the f i r s t (n-1)  ^j_n'  +  equations.  column  columns  ^im^ln^  +  ••• •••  +  first  of  of  the  that  conductor.  These  (c)  of  two s t e p s  The same  are i l l u s t r a t e d i n equation  steps,  conductors.  (a)  These  : i n g the voltages  and ( b ) , give  (2.16).  are c a r r i e d out on the other  a set of  linear  on the conductors  equations  i n terms  of  main  (2.15)  express-  the t o t a l  currents  til in  "  these  l '  v  conductors  Z  0  l l l l  ?  and i n t h e 2nd t o n-  1112-  ?  S.211  0  ?  \  Z  due to  ; - '  ; ;  l l k l  Z  •  •  C  llkm  V hi  •  lnln<  (2.15)  >  k l l l  z  k l k l  ?  klkm  \  *  0  (2.15)  |  •  0  The  lllnj  subconductors.  kmll '  symbol  " 5"  f  denotes  the operations  C  kmqn  = Z  kmqn  (a)  —Z  (for  the elements and ( b ) ,  klqn  — Z  m,n^l)  ^kmkl '  • *°kmkm  which have been  and the general  kmql  changed  i n  term i s :  (2.16)  14  2.6  Reduction  The exchanging equations in  of  the Large  equations  (2.15)  the positions of  (2.15)  Impedance  of  Matrix  are rearranged  rows  and columns  corresponding  f o r the reduction i n such  to the main  a way t h a t  conductors  come  the  by  "bundled"  f i r s t ,  as  shown  (2.17)  1  V, k 0  J  l l l l  J  l l k l  J  k l l l  J  klkm  =  12  0  or  process  (2.17)  km  i n abbreviated  form,  V  A  B  0  C  D  (2.18)  From Hence  (2.18)  ,-1, V=(A-BD C)I  the desired [Z„]  impedance =  Reference (2.17).  Using  (2.19a)  [A -  [8]  Gaussian  last  row and going  been  reduced  BD-1C]  [Zc]  i s  (2.19b)  provides  a more  elimination  up u n t i l  to zero,  matrix  efficient  on t h e m a t r i x  the submatrix  achieves  the  [B],  reduction.  way o f  finding  (2.17),  as shown  [Zc]  starting  i n (2.18),  from  from  the  has  just  15  The stored  in  d e s i r e d impedance m a t r i x  [A*]  in  [Zc]'corresponds  to  the  submatrix  (2.20).  1  V, k . (2.20) 0  12  _ 0  An  i l l u s t r a t i o n of  Figure  2.7  The  ^  2.4  Choice  Illustration  and  Obviously, 2.2 of  w i l l the  path  influence  return  s h o u l d be  Figure  2.5.  this  path  is  zero.  final reduction  of  the  Constraint  the  geometry  the  values removed To  the  on  and  obtained by  I  reduction  the  l o c a t i o n of for  requiring  illustrate  Return  this,  the that  km  stage  J  is  in  Figure  process  Path  the  return  inductances. the  consider  shown  current the  path The  in  influence  through  circuit  Figure  shown  this in  2.4.  Figure 2.5  Writing  V  l  =  ( R  t w o - w/i r e r e t u r n ,  A  the loop  1  equation  for F i g . 2.5 gives:  +  h  +  c i r c u i t  ( R  2  ^X22-X12>  +  2  h  21  <' >  s i n c e 1^=1  V  Introducing For  l  "  ( R  1  +  R  2  +  a fictitious  equivalence  Figure  J  ( X  11  return  +  X  22 ~  path  o f t h e two c i r c u i t s  2.6  A, • t w o - w . i r e . in  a  t h i r d  2 X  12  gives  2  h  } )  a configuration  of  Figure.2.6.  we r e q u i r e :  c i r c u i t  with,  conductor  22  <' >  common  r e t u r n  17  The loop e q u a t i o n s  V  a  = (%  o f Figure,2.6 a r e :  + J ^ - X ^ ) ) ^  + <R  + i(X  + (R  +  q  j (  X  q q  -X  l q  ))I  + (X  q  1 2  -X  2 q  ) I  f e  >  V, = ( R  2  2 2  -X  2 q  ))I  b  q  + i(X  -X  q q  2 q  ))I  q  + (X  1 2  -X  l q  )  I  (2.24)  a  Imposing the c o n s t r a i n t t h a t the c u r r e n t i n the r e t u r n path i s zero means t h a t  I  I  q  = - I  a  From e q u a t i o n s  V  = 0 = I  I  +  (2.25)  b  (2.26)  b  (2.24) and (2.25)  - V  a  a  b  =  [R  + j ( X  x  ( X  1  X  - 12- 2q Using  1  - X  ) ]  l  q  ) - ( X  1  2  - X  l  q  ) ] I  a  -  [R + 2  j(X  2 2  -X  2 q  )  (2 27)  h  -  (2.26) i n (2.27) g i v e s  V  a  _  V  b  =  [ R  1  +  R  2  +  ( X  J H  + X  22"  2 X  12  ) ]  X  (  a  2  #  2  8  )  which i s i d e n t i c a l to (2.22) d e r i v e d u s i n g F i g u r e 2.5.  Therefore, of any convenient  i t i s t h e o r e t i c a l l y p o s s i b l e t o choose a r e t u r n path  shape and l o c a t i o n f o r the i n d u c t a n c e  c a l c u l a t i o n s as ...  long as a zero c u r r e n t c o n s t r a i n t i s imposed on such a path. considerations discussed should  Nevertheless,  i n s e c t i o n 3.6 would r e q u i r e t h a t the r e t u r n path  be c y l i n d r i c a l i n shape, have a s m a l l r a d i u s , and be p l a c e d a t a  s m a l l d i s t a n c e below the e a r t h s u r f a c e n o t f a r from the c a b l e s and other conductors.  18  2.8  I n c l u d i n g the C o n s t r a i n t on the C u r r e n t i n the M a t r i x S o l u t i o n  Equation  (2.20) g i v e s the v o l t a g e s on the main conductors  w i t h r e s p e c t to the r e t u r n path) i n terms of the c u r r e n t s i n these  (measured conductors.  In p r a c t i c e , however, v o l t a g e s are measured w i t h r e s p e c t to the l o c a l ground (or n e u t r a l conductor  or s h e a t h ) .  I f the c o n s t r a i n t on the c u r r e n t i s  i n t r o d u c e d , t h i s changes (2.20) i n t o the form:  J  ll  J  lk  J  k-l,l  J  k-l,k  (2.29)  V.  k-1  V.  J  k  L_-  •••  k l  J  k-1 -I  kk  k-1-  k Since  £  I  o  1=1  and  I - - l  =  ( 2 . 3 0 )  0  iL  r  l  -  2  ...-I ^ t  This gives: z  i r  z  Z  i k  lk-1  Z  lk ( 2 . 3 1 )  V.  k-1  Z  k-l,l  Z  kl  Z  Z  kk  k-l,k  Z  k-l,k-l  Z  Z  k,k-l" kk  k-l,k  k-1  Z  I f conductor k r e p r e s e n t s the l o c a l ground (or n e u t r a l conductor or  the sheath) w i t h r e s p e c t to which a l l v o l t a g e s are measured, then sub-  tracting  the e q u a t i o n f o r V  from the other equations accomplishes rC  and g i v e s :  this  19  V  1  -V  r-  k  *  lk-1  l l  Z  (2.32) V  where  Z  -V  k  J  . . = Z. . + Z, , IJ kk  neutral  Z  matrix  conductor  or  kl  J  k - l , k - l  k-1  2Z., ik  13  The (or  k-1  is  the  impedance m a t r i x  sheath).  which  implies a  local  ground  20  Chapter  In  practice,  transmission  RETURN PATH  there  are three  return  i n neutral  (ii)  return  i n ground  only;  (iii)  return  i n ground  and n e u t r a l  Return  i n Neutral  Each conductor,  sheath,  (2.7)  subconductors. process,  i f  voltages  from  provided with  Model  1:  results  phase  there  i n any  to  and ground  wires  only);  Only  conductor  a r e used  to  or sheaths  obtain  i s zero  pipes  conductors.  and i s d i v i d e d  to neutral  i n Ground  of  to  i s represented  as i n S e c t i o n  form  2.1.  the impedance  can be " e l i m i n a t e d "  the impedance the phase  matrix  on i t ,  or  In  that  i t  The  i f  formulae the  reduction  relates any  so  separate  of  i n the  fact,  process,  as a  matrix  which  currents.  i n the reduction  voltage  the other  desired,  i s connected  i n parallel  Only,  i s considered  subconductors i n any lower  choice  decreases were  path  conductor.  subconductors This  for the return  (including  or neutral  and (2.11)  The ground  layers  ground  pipe  so d e s i r e d ,  another  layer.  Conductors  The n e u t r a l s  that  Return  cases  or  can also be eliminated  3.2  into  conductors  as a r e the c o r e s ,  equations  conductor  IMPEDANCE  system:  (i)  3.1  of  3  appears  as shown  layer  by using  i n Figure  are chosen  reasonable  a s o n e moves  obtained  as a separate  farther a depth  3.1.  from  equal  to  and i s  subdivided  The d i a m e t e r s  to be twice  because away  conductor  that  the current  the  density  the cables. 3300/,rp/f  of  of  previous i n the  Reasonable  metres.  the  (a)  ( b ) Figure  3.1  Subdivisions  of  ground  into  layers  of  subconductors  22  p = ground f  =  frequency  The reduction  the  3.3  of  ftm,  If is  in  a system in  the  of  subconductors,  section  reduced  arrangements  Ground  the  in  is  eliminated  leaving  matrix.  3* 1 ( a )  the  The  and  (b)  in  ground  the  return  difference is  between  discussed  in  The  as  is  through  a set  ground  is  of one  k  both  such  Analytical  To  represent  the  number  subconductors.  conductors impedance  must  be  directly  Equations were  industry.  derived These  over  flat  When  these to  earth same the  of  Equations  considered, to  reduce  for by  J.R.  which  is  i t  the  ground  equations  values  for  and  retained  return  is  are i t or  systems  better  amount  of  to  circuits  of  where  many  cables  or  the  ground  return  and  overhead used  assume  that  the  conductors  are  of  ground  to  return  in  the  located  cables,  impedances  time.  transmission  a n d h a s an u n i f o r m  underground  divided  computing  are widely  applied  desired.  be  and  are  the  must  [11]  extent  in  it  Carson  in  as  sub-  Impedance  calculate  storage  the  into  eliminated  eliminated  adequately,  In  conductors,  subdivided is  Ground Return  return  infinite  equations true  ground  can be  and n e u t r a l  which  conductor  of  a large  neutrals  ground  conductors  3.4  Use  The  conductors  process.  mations  thus  Figures  reduction  lines  2.8,  impedance  and N e u t r a l  return  modelled  conductors .  into  and  Hz.  shown in  in  4.2.  Return  system  the  in  as  as  included  results  section  ground,  process  implicitly  resistivity  can be  ..  power in  air  resistivity. useful  approxi-  obtained.  23  With in  overhead  the  ground equal  cables, to  the  a  earth's 1.0  of  and  Z  paper  is  that  S  which  l i e below  when  these  equations  l i e  above  the  now  the  are  ground  ground  applied  surface  at  are to  used under-  heights  burial.  [10],  variation  surface  m)  images  later  the  conductors  However,  these  depth  underground,  about  image  calculations.  In  the  lines,  of  Carson  ground  relatively the  =  (1+C)  =  the  ground  showed  return  small return  for  that  for  conductors  impedance  with  the  depths  usual  impedance  (Zg)  buried  distance of  can be  below  burial  calculated  Z° g  (3.1)  o  where  Z^  ground  extend  = a  Reference  [10]  gives  C  2K0  and  reference  [12]  where  K5,  m& 2rrr  that  is  in  if  the  earth  a l l directions  circular  factor  conductor  which  symmetry accounts  located near  were  around  modified  r  =  the  exists, for  ground  the  fact  that  surface  (jm)  Z  2  log(l/m)  for  small  m  (3.2)  as:  Ko(mr) KL(mr)  ,„  m = /  are  to  as:  gives  ° _ g  so  correction  the  impedance  indefinitely  conductor C  return  (3.4)  Bessel  internal conductor  functions  radius  of  the  insulation)  earth  (i.e.  outer  radius  (i.e.  of  the  as:  24 p = ground to =  2iTf,  resistivity  f=frequency  y = magnetic permeability of ground  Carson's formula for overhead conductors cannot be used for c a l culating the s e l f impedance of underground  conductors.  Equations (3.1) i s  used for this purpose, but the mutual impedances are calculated using the overhead formula - which i s known to give good approximations for buried conductors at power frequencies [ 2 0 ] .  Equations for c a l c u l a t i n g the s e l f and mutual impedances of underground conductors have also been derived by F. Pollaczek involving i n f i n i t e series [19]. of underground  Closed-form approximations to the s e l f and mutual impedances conductors v a l i d for a wide range of values of the parameters  involved have been derived by Wedepohl and Wilcox [9].  These equations (3.5),  given below, are accurate up to frequencies of approximately 160 KHz for separations of approximately 1.0m between the conductors, and to approximately 1.7 MHz  i f the separation i s only 30 cm.  Thus  very accurate approximations  can be obtained for most p r a c t i c a l cases of cables l a i d i n the same trench to quite high frequencies. These equations are:  Zs = ^  { -An  M (YmD  Z  i k = *f£  i-  £ n  + T2 " T3 n^} fl M } +  J  , " f m£ }  (3.5a)  ft/m  (3.5b)  where Z , Z-y^ are s e l f and mutual impedances of ground return path respectively, s  (ft/m) Y = Eulers constant = 0.5772157 h = depth of b u r i a l of conductor  (metres)  I = sum of depths of b u r i a l of conductors i and k (metres)  25 r  = outer  D  M  IK.  radius  of  conductor  = d i s t a n c e between  (metres)  conductors  i  and k  (metres)  m = /jtju/p p = earth  Equations |mD_^J  (3.5)  resistivity  are valid  < 0.25 for mutual  For  the range  i n fim  f o r the range  |mr|  ik  impedance  and  impedance.  |mD., | > 0 . 2 5 r e f e r e n c e  [9]  IK.  -£/(a2+m2) J  < 0.25 for self  suggests  /(a2+m2)  -V  +  2TT  |a|+/(a +m ) 2  the  integration:  -£/(a2-rm2)  -e  exp(jax)dx  2/(a +m )  2  2  2  (3.6)  where  x = horizontal modulus  V=  of  3.5  Model  Model  II:  formulae loop  A very treats  through  formulae be  Using  used  to  may b e  used.  Return  i  each  Formulae  subconductor (Figure  case.  If  of  Directly  the depths  i  and k of  burial  and uses  and mutual  the results which  with  the  Subconductors  the a n a l y t i c a l ground  as an i n s u l a t e d  3.2)  and ( 3 . 8 ) below,  conductors  and k.  simple model which uses  calculate the self  (3.7)  the difference  conductors  Ground  the ground  i n this  equations  of  d i s t a n c e between  conductor  the available  impedances.  a r e needed a r e found  with  the  ground  Equations  f o r power  return return  return  ( 3 . 5 ) may  frequency  i n many h a n d b o o k s  only,  [24,  25],  Figure  3.2  The  z  impedances  i i  =  Z.. l  where  Z^  GMR^ R^ D k  R  i  +  Z^  +  J ( ° -  j(0.1736  1  7  3  of  6  l o  are  the  self  circuit  s £MR7 +  +  ground  return  in Figure  °-  4892  3.2  are:  fi/km  )  0.4892)  and m u t u a l  =  r e s i s t a n c e of  =  d i s t a n c e between  view  the  of  between  ground  mean  radius  3  7  <-)  ft/km  the  fact be  conductors series  are  of  One  that  i  (3.8)  the  i  and  it  (=0.0592 i  k  and  ft/km)  (m)  (m)  Conductor  equations at  high  approximations  used,  respectively,  (ft/km)  Undivided  inaccurate if  path  conductor  conductor  as  impedances  return  conductor  Ground  c a l c u l a t i o n s may  infinite  and  °ik  geometric  In  the  log  =  III  separations  subconductors  60 Hz  r e s i s t a n c e of  Model  if  g  at  =  Representing  obtain  R  only  g  3.6  impedance  +  = R k  and  Rg  Model with  would  be  used  for  ground  frequencies are  used,  or  advantageous  and  return  for  costly if  most  wide to of  27  the  elements  of  the matrix  [Z,  .  1 of  equation  (2.12a)  (2.11).  This  could be  calculated  bxg with of  the simpler  a fictitious  lated.  return  The ground  subdivided the  equations  i n this  subconductors With  ground  is  path then  case)  this  conductor  2 above  ground,  that  effect  but  caution,  unless  more  The ground  return  matrix  i n  to which  i t  are ignored.  main  region,  frequencies  c a n be shown  than  eddy  must  of only  1  In KHz this  skin  model  be used  conductor  between  which would  above  effects  this  introduction are  calcu-  ( n o t i.  the ground  and  below.  ground  reference for  circulate in  and returns [20]  i t  the case of  approach  the results  that  current  advantage  formulae  1  the  the inductances  impedances  currents  up t o  involves  as one a d d i t i o n a l  conductor  frequency  at higher  pronounced  into  eddy  i s negligible  the lower  answers,  respect  considered  approach,  flows  In  with  a r e c a l c u l a t e d as shown  current  line.  and  and the mutual  i f  this  (2.7)  must  gives  a 500 kV very  be i n t e r p r e t e d  i n the  ground.  t h e more  overhead  with is  3.3  Model with  ground  i n one row and one column  represented  as only  one  some  much  complicated of  (2.12a).  Figure  shown  accurate  i n the conductors  that  through  has been  effect  i s  the  conductor  the  28  In the  addition,  fictitious return  there path.  can be  nearly  halved  values  of  parameter  the  by  ~ V  1  to  be  (3.5),  in  delays  for  the  the  use  of  circuit  l l  Z  1N  IN  Z  NN  the  of  choice of used  equation  equations  J  distances  |mD^|  also  "  the  this  frequencies.  loop  to  locating  for  the  as  centrally  and  Writing  freedom  The  approximations, much h i g h e r  is  path.  equation  This  thereby  more  of  (3.5)  reduces  the  g i v i n g more  accurate  complicated formula  Figure  Z  in  location  3.3  (3.6)  gives:  l g  =  v„ N  V  in  which  ground  ig  with  equation the  Z.  g _  refers  to  The  the  mutual in  Equation next  z  gl  common r e t u r n  (3.5).  ground.  J  q.  (3.5)  Z  gN  z  M  Ng  gg  il  !_ g.  impedance between This  is  s e c t i o n 3.7  cannot  valid shows  be  only how  Z  (3.9)  N  subconductor  i  and  calculated directly  when is  the  common r e t u r n  derived  using  by is  equations  (3.5).  3.7  The M u t u a l Impedance B e t w e e n a S u b c o n d u c t o r Return i n Another Subconductor  Consider The  loop  impedances  Figure may  J  (3.4)  i n which  be w r i t t e n  Hg  the  and  Ground w i t h  common r e t u r n  is  the  Common  ground.  as:  J  12g (3.10)  Z  L  J  91 2 1  s  22g  29  t  (ground  return)  *1  v»  Figure  A l l Carson's  or  the  consider  return  is  Two c o n d u c t o r s return  impedance  Wedelpohl'.s  Now common  3.4  terms  in  (3.10)  ¥  if  Q  common  ground  can be  c a l c u l a t e d by  Figure  (3.5)  using  equations.  a similar  conductor  circuit  in  i n which  the  2.  »  :  > Vn  with  (ground)  (reiurn )  Figure  3.5  Circuit with  of  common  one  conductor  return  in  a  and  the  second  ground  conductor  30  The  loop  equations  may b e w r i t t e n  -112  J  as:  lg2 (3.11)  J  The  third  subscripts  The  term  and  (3.5) are equivalent  gg2.  i n equations  (3.10)  and (3.11)  Z. =Z . , „ i s t h e o n e o f i n t e r e s t lg2 gl2  and  Va  = \  Vb  = -V  I  -  2  ~ V  The c i r c u i t s  return.  of Figures  (3.4)  (3.12)  2  (3.13)  +  g  Substituting  (3.14)  IX)  these  •v - v" x  t h e common  i f  2  - ( I  here.  denote  2  into  Z  l l g  equation  Z  12g  (3.10)  gives  12g  Z  Z  I, 1  22g  (3.15) -  "  V  1  V  2  -  -Z  -  V  2  Z  llg  Z  22g  + Z  -Z  12g  22g~  2 Z  12g  Z  22g  -I  22g  Z  g  -I, 1  12g (3.16)  From  (3.12)  identical,  and ( 3 . 1 3 ) ,  12g  i s evident  J  that  22g  The  lg2  mutual  Z  22g  Z  equations  (3.11)  and (3.16)  are  (3.17)  12g  impedances  be c a l c u l a t e d u s i n g  (3.17),  (Z^g) (note  required that  Z. i g  common  g  hence:  Z  fore  i t  Z  return).  i n equation  (3.9) can there-  =Z.  q i s the f i c t i t i o u s  i  where g  q  31  Thus (3.9) one  results  column.  ground  i n the use of If  return  II  since  of  the matrix  3.8  using equation  series  impedance must  Comparison of  It the method  would  III  should be noted  that  (TNA)  [26],  where  decoupled  from  therefore  e l i m i n a t e s the need  the phases.  On a t h r e e  A V  A V  = Z  ca  z  3  A V  Z  ab  Z  bb  Z  cb  ab  the procedure  b  AV  c  aa  =  only  one row and  forms  be f a s t e r  phase  Network  for  of  than  every  the model  element  then  Analyzer  i n Model  III  of  the ground  i n c l u d e d as an e x t r a the ground  return  Circuits  i s related  transmission lines  the impedance  Z Z  Transient  return  i s  conductor  i n every  cc impedance i s ;  (3.19)  )  as:  , -z  Z  ab  Z  bb  Z  cb  m -Z  ca  mutual  ca  m  ba  (3.18)  the average  m  -Z m  ac  -Z  be  m -Z m  -Z  Z  m  m  m  m  m  m  -Z  -Z  Z cc  I  +1, a  and  line.  ac be  to  on the  z  be  -z  a  w i l l  the Transient  transposed,  can be w r i t t e n  ~AV  integral)  to be evaluated  = f (z , + z, + z  m (3.18)  i s  III  for  equation  line,  ba  the line  formulae  of  case,  to model  aa  c that  i s  Z  a b  It  phase  AV  Scjuation  model  used i n representing three  Network Analyser  Assuming  with  the matrix  infinite  have  i n the l a t t e r  Model  return  (or  be used,  the i n f i n i t e series . ] big  i n forming  the ground  the i n f i n i t e  [Z,  (3.17)  +i b  c  32  where I  a  + 1^ + I  t h i s case.  c  = I g i s the c u r r e n t i n the e x t r a conductor,  ground i n  The ground r e t u r n formula w i t h i t s pronounced frequency  i s then o n l y used  for  i n the l a s t  column.  dependence  A l l o t h e r elements Z ^ - Z a  Z , -Z a r e c a l c u l a t e d w i t h ground i g n o r e d . Furthermore, ab m . 0 . 0  m  ,  i f the l i n e i s  t r a n s p o s e d , the d i a g o n a l elements Z ^ - Z ^ , e t c . , become e q u a l to the p o s i t i v e sequence impedance, and a l l o f f - d i a g o n a l elements Z  -Z  , e t c . , become z e r o .  '33  Chapter  Comparison  4.1  This account of  two  The  by  of  section  subdividing  conductors  d.c.  the  Method  shows the  placed  r e s i s t a n c e of  RESULTS  4  of  how  Subdivisions  s k i n and  conductors.  two  each  metres  apart, is  Standard  proximity  The  conductor  with  effects  impedance as  shown  of  in  are  taken  return  Figure  and  0.0417 fi/km  a  Methods  the  into  circuit is  4.1,  calculated.  frequency  is  60 Hz.  2000.0mm Figure  The  4.1  large  A  return  separation  effect  negligible.  The  for  using Bessel  functions.  used  The  This  by  for  taken  subdivisions  value  of  of  Z=  0.0887  as  the  the  Figure number  between  increase  in The  the  two  two  conductors  conductors  r e s i s t a n c e due UBC/BPA l i n e  to  for  apart  makes  proximity  skin effect  constants  is  program  corrected  [16]  is  this.  corrected  is  c i r c u i t of  the  +  impedance  j  exact  impedances  4.2  shows  subdivisions.  It  0.7901  fi/km  reference shown  the is  i s :  in  value. Table  By 4.1  using are  impedance  variations  seen  the  that  exact  various  numbers  of  obtained.  as  a  function  reference  of  values  the are  34  Table  4.1  Variation  No.  may  keep  Errors  1  0.0833  0.7913  0.0377  6.1%  0.2%  7  0.0855  0.8016  0.0480  3.6%  1.4%  19  0.0878  0.7944  0.0408  1.0%  0.5%  37  0.0883  0.7923  0.0387  0.5%  0.3%  61  0.0885  0.7914  0.0378  0.2%  0.2%  Reference  0.0887  0.7901  0.0365  0.0%  0.0%  as  t h e number  of  subdivisions  i f  an e r r o r  i n storage  subdivisions  The close  divisions  in  Subdivisions  n/km  be a p p r o p r i a t e  of  of  Subdivisions  subdivisions  compared  t h e Number  internal ft/km  of  brought  with  X  X  t h e number  savings  use  Impedance  R ft/km  approached to  of  of  two c o n d u c t o r s together,  i s increased.  However,  i t  as low as p o s s i b l e .  Nineteen  subdivisions  one p e r c e n t time  2 of  forming  as shown  result  i s  tolerable.  from  the return  i n Figure  o n t h e two c o n d u c t o r s .  calculations  published  Chapter  and computing  X  keeping  i s  best  Substantial  t h e number  of  down.  are used with  of  R  charts  done  using  and t a b l e s  reference  [17].  to  4.3. The  standard correct  circuit  of  Various impedances methods  Figure.4.1  numbers  of  sub-  calculated  which  for proximity  involve effect  are  as  are the shown  35  0-032 -  NUMBER Broken  Figure  4.2  OF  lines  Variation  SUBCONDUCTORS  are the reference  of  impedance  with  values.  •'  t h e number  of  subconductors  i  l<  27.02 mm Figure  4.3  A return  According circuit  above  circuit  to reference  of  [17],  two c o n d u c t o r s  very  the a . c . resistance  close  of  together  the  return  i s  r  = R'  x  ~y  (A.D  36  where  R'=a.c.  resistance  R"/R'  A and  Tables  ratios the  reference  calculated  circuit  in  j  the  0.1340  pares  i t  ft/km.  In for  This  using  effect  complicated  formulae  of  referred  to  division  among  many the  is  and  case  in  most  parallel  in  the is  seem to  the  conductors  conductors  is  correction as  the  from  methods two  cable same  or  systems  in  adjacent  useful,  and  Charts  r e s i s t a n c e and The  skin effect  inductance  impedance  only,  of  i s :  the  in  three  Table  of  Z=0.1048  4.2  which  only  need  two  the or  from  +  com-  division  where  many method  reasonably  is  accurate  of  otherwise the  as  the  the  not  known  and  or charts  current  a priori.  cables of  use  hand,  spacing)  known  corrections  conductors  subdivisions, be  the  other  three  delta  not  this  by  derived  On  or  current  calculations,  conductors  tables"  using  ducts),  a value  in  impedance  (flat  cities  gives  value  above.  to  gives  subdivisions.  "factor  Also,  and  for  of  or  arrangement  involved  From the  respectively.  reference  limited  be.  effect  factors  as mentioned not  ratio.  inductance.  various  for  only.  ft/km  p a r a l l e l conductors  are  very  0.95  corrected  correcting  [18],  the  proximity  0.1410  made  conductor  above  j  used  are  subdivisions  common, f o r m s  when  resistance  for  and  conventional  charts"  of  1.18  obtained  "estimating  method  +  skin effect  effect  the  proximity  those  proximity  4.3,  = 0.0887  above  with  be  for  holds  [17],  to  Figure  Z  Applying  = proximity  similar equation of  are  corrected  Where (as  pipes  subdividing results.  is  run the  37  Table  4.2  Variation with of  of  Both  Skin  X  0.0966  0.1474  0.0429  7.8%  10.0%  19  0.1010  0.1394  0.0349  3.6%  4.0%  37  0.1022  0.1370  0.0325  2.5%  2.2%  61  0.1026  0.1361  0.0316  2.1%  1.6%  0.0887  0.1410  0.0365  15.4%  5.2%  0.1048  0.1340  0.0295  0.0%  0.0%  of  return  [19],  these  Ground  effect  have  been  Wedepohl  to  as g i v e n  Return.  only.  skin  use.  given  and W i l c o x  are given  b y many [9],  of  and mutual authors,  effects.  of  impedances  of  ground  the formulae,  of  i n c l u d i n g Carson  and Kalyuzhnyi  i n the form  The v a r i a t i o n  b y some  and p r o x i m i t y  Formulae  for calculating the self  formulae  easy  for skin  No c o r r e c t i o n f o r b o t h  Formulae  frequency,  R  7  *  4.3  Inclusion  in  6.1%  Comparison  always  Error  20.5%  *•Corrected  not  X internal ft/km  Fig.  i n the Calculations.  0.0377  **  of  Effects  of the  0.1422  Value  Most  Showing  0.0833  Reference  Pollaczek  the C i r c u i t  1  SKIN  ground  of  Subdivisions,  X ft/km  **  with  of  and P r o x i m i t y  R ft/km  No. of Subdivisions  4.2  the Impedance  t h e Number  and L i f s h i t s  infinite return  series  [10,11], [13].  and are  impedance  a r e compared  loops  i n this  with section.  According with for  the  for  I  J.R.  depth  most  Model  to  Carson  of  burial.'.'.of  frequencies to  by  calculate  various  [10],  depths  of  variation  a conductor  using  the  the  equation  impedance  burial.  of  The  is  of  ground  minimal,  (3.1).  and  This  a buried  results  the  of  is  can be  are  with  shown  impedance  calculated  verified  conductor this  return  by  using  ground in  return  Figure  4.4.  1.171 M69 1-167  <  •  in  IMPI  o z <  1  1  Figure  A been  given  tion  of  0  DEPTH  OF  simple  Wedepohl  infinite  and  of  which,  useful  and W i l c o x  series  K a l y u z h n y i and results  10 0-5 BURIAL(m)  V a r i a t i o n of the impedance w i t h depth of b u r i a l  4.4  very by  the  -0-5  -1-0  1-5  though  form  Lifshits very  form  of  the  (equations of  [13]  •  8  of  t-5  a buried  ground  (3.5)),  conductor  return which  is  impedance an  has  approxima-  solution.  also  different  derive  from  the  a  formula,  more  the  final  conventional  ',.  ones,  39  are  claimed  Kalyuzhnyi  to and  Ze  where  y  very  Lifshits  = ^  = radius  p =  P  y  -  of  Equation  which  (4.2)  is  Re  Carson's by  =  2?r2f  electric  [17]  =  a  .  of  ground  data.  return  (Ze)  as:  constant  conductor  over  insulation  (m)  of  ground  real  10_7  of  part  (ftm)  conductor  (Re)  (m)  of:  (4.3)  Q/m  from  that  obtained  from  Carson's  equations  which  to:  TT2f  authors  current  equations  impedance  measured  (4.2)  Euler's  burial  different  equations  several  of  gives  quite  approximates  self  experimentally  <WP  = depth  Re  with  - J-  r n  buried  p = resistivity h  the  -2—  [An I  2TT  closely  give  0.5772157  =  ..r  agree  .  10  7  (or  approximations  and  others  through  g i v e n by  (4.4)  tt/m  the  involved ground.  Kalyuzhnyi  of  them)  with The  have  been  analysing  very  and L i f s h i t s  i s ,  used  the  marked  for  many  conduction  deviation  therefore,  from  worthy  years  of Carson's  of  inves tigation.  In of  the  earth,  by  using  In  this  In  Table  the  order the  to  impedances  subdivided  calculation, 4.3  determine  and  the  Figure  which  of  ground  the  circuits  of  representations  ground 4.6,  formula best  the  is  divided  results  into  approximates  Figure of  Figure  five  obtained  4.5  3.1  layers  from  are  the  the  calculated  (i.e. of  self  behaviour  62  Model  I).  subconductors.  impedance  Oo012m  (a  Figure  4.5  )  Cross s e c t i o n s of b u r i e d conductors r e t u r n impedance c a l c u l a t i o n s  for  ground  calculations There  is  above  method  of  but  from  more  MHz.between  4.6 at  b)  from for  power  the  results  values  from  evaluate  the  of  Figure  the  of  the  a  slight  60 H z ,  is  the  impedance  over of  results  when  resentation  equations, and  compared.  and  the  Wedepohl's  Lifshits  used.  the  +  By  equation  in  12%  power  frequency  i n between  of  in  case and  neglected  approximation,  subconductors: for  from  also  in most  a  of  fuller  Figure  3.1  a.  conductor  in  Figure  4.5  a with  ft/km  when  0.9236  Z  the 3.1  the  ground  = 0.0589 gives  an  former. a would  +  using  the  0.9181  Therefore  the  adequate  fact  because 3.1  the  that  the a.  To  interstices  show  only  example,  at  return  is  representation  of  of  Figure  The  less ground  for  50%  the  ground  ft/km. of  (over  representation  For  representation j  Figure  Lifshits.  latter  ground  improvement  be  the  the  and  Figure  of  j  (see  Kalyuzhnyi  at  while  reactance  calculations arise real  to  formulae  results  only  Figure  at  deviation  are  methods values.  obtained  i s :  ground  three  calculated resistance  formula  the  latter  using  compared w i t h given  to  first  somewhere  the  various  the  Results  the  be Z=0.0597  of  from  with  the  c a l c u l a t e d impedance  resentation  marked  this  section.  a is  are  calculated using  and Wedepohl's  subconductors  of  7%  very  approximations  cross  Figure  between  l i e  the  filled  to  the  are  the  method  obtained  in  than  Carson's  A  a were  calculated 3.1  of  frequencies.  improvement  the  other  subdivisions  influence  ground  as  using Kalyuzhnyi  c a l c u l a t e d from  range  results  between  3.1  each  reactance  frequency)  "interstices"  obtained  Carson;'s  formulae  others.  the  methods  various  ground,  Discrepancies a l l  the  the  of  a l l  using  subdividing  widely  1.0  by  resistance values  reactance  deviations  and the  the  The  results  I in  the  The deviate  Model  c l o s e agreement  equations, deviate  using  this  3.1  latter  than  1%  return  b, rep-  in  the  rep-  purpose.  RESISTANCE  Frequency (Hz  )  Subdl-. visions  REACTANCE  (ft/km)  Subdivisions  Wedepohl  Carson  Kalyuzhnyi  (ft/km)  Wede— pohl  Carson  .. K a l y u zhnyi  2.2  .002  0.002  0.02  0.004  0.039  0.041  0.038  0.062  4.5  .005  0.005  0.05  0.009  0.077  0.081  0.075  0.119  9.0  .009  0.009  0.09  0.018  0.150  0.160  0.145  0.230  15.0  .015  0.015  0.015  0.030  0.244  0.260  0.236  0.374  30.0  .030  0.030  0.031  0.059  0.475  0.503  0.460  0.723  60.0  .060  0.059  0.062  0.118  0.924  0.979  0.914  1.39  120.0  .120  0.119  0.124  0.237  1.80  1.91  1.73  2.68  500.0  .499  0.497  0.518  0.987  7.03  7.49  6.79  10.3  IK  1.00  0.998  1.04  1.97  13.6  14.5  13.2  19.7  5K  5.05  5.05  5.20  9.87  63.0  67.6  60.7  88.3  10K  10.2  10.2  10.4  19.7  121.0  131.0  117.0  168.0  5 OK  52.2  53.1  52.5  98.7  555.0  601.0  535.0  738.0  0.1M  106.0  109.0  105.0  197.4  1064.  1155.  1026.  1389.  0.5M  556.0  611.0  530.0  987.0  4767.  5207.  4622.  6371.  l.OM  116.  1320.  1064.  1974.  9038.  9881.  8810.  11870.  Table  4.3  S e l f Impedance o f Ground R e t u r n Path as C a l c u l a t e d U s i n g S u b d i v i d e d Ground, and O t h e r Formulae  43  Figure  "4.6  Comparison of ground r e t u r n  calculated loop.  self  impedances  of  a  44  Mutual  Impedance  Table calculated 1%  in  the  for  of  Ground  4.4 the  and two  Figure buried  r e s i s t a n c e and  frequency  between  the  Return  Path  4.7  show  the  conductors  about  results  15% of  in  results  of  the  of  the  mutual  impedances  Figure  4.5b.  reactance  are  obtained  at  those  obtained  from  subdivisions  and  Deviations  of  about  power Wedepohl's  equations.  The by  using  the used  mutual  Carson's  frequencies for  Despite Carson's  impedance  overhead  used;  thus  c a l c u l a t i n g the this  close  equations  values  line it  equations  seems  mutual  [11,16]  Carson's  impedances  agreement  in  [11]  derived  were  c a l c u l a t e d by  the  for  are  overhead  between  results,  using  i t  subdivisions  similar line  for  most  equations  buried  conductors  should be  remembered  conductors  located  above  and of  may  be  [20]. that  ground.  Table  4.4  Mutual  Impedance Between  Two  Underground  Conductors  *  RESISTANCE  Frequency  Subdi-  Hz  visions  REACTANCE  (ft/km)  Carson  Wede-  Subdi-  pohl  visions  (ft/km)  Carson*  Wedepohl  2.2  0.002  0.002  0.002  0.024  0.024  0.026  4.5  0.005  0.004  0.004  0.046  0.045  0.052  9.0  0.009  0.009  0.009  0.088  0.087  0.099  15.0  0.015  0.015  0.015  0.142  0.139  0.161  30.0  0.030  0.030  0.030  0.271  0.266  0.308  60.0  0.060  0.059  0.059  0.516  0.506  0.590  120.0  0.119  0.118  0.119  0.979  0.959  1.13  500.0  0.4.99  0.490  0.497  3.63  3.55  4.25  Ik  0.999  0.978  0.996  6 . 8 2'-  6.66  8.06  5k  5.03  4.84  5.04  29.0  28.3  35.2  10k  10.1  9.60  10.1  53.6  52.4  66.0  5.0k<-  51.8  46.4  52.5  216.0  213.0  278.0  100k  105.0  90.7  108.0  386.0  385.0  509.0  500k  599.0  413.0  594.0  1376.0  1471.0  1983.  Overhead  line  equations  used.  cc-  i-  t- ID  X---Wedepohl  a oo  100  1000  10000 FREQUENCY C Hz  ,100000 1000000  )  «nO"  + — Subdivisions o—Carson  (0/H)  X---Wedepohl  no Figure 4.7  100  idoo  loooo  FREQUENCY  ( Hz  iooooo )  1006000" I  Comparison of calculated mutual impedances between two buried conductors.  47  4.3  Comparison  The (1/0  of  data  AWG a l u m i n u m  formation 0.4707  the Different  i s taken  from  r e f e r e n c e [4] . T h r e e  cables with  apart.  ohms/1000'f t  listed  impedance  reduced  The c o r e  respectively  a s 515 m i l s  The sequence  from  cored  8 inches  insulation  Results  values  matrix  of  are l a i d  resistances  the inside  cables  i n a  flat  are 0.1882  and o u t s i d e  diameters  and of  the  respectively.  the zero  elements  distribution  neutrals)  and sheath  with  and 955 m i l s  Models  at  (0),  60 Hz  positive  (1)  and negative  i n the reference  (2)  are:  0 ~0.483+j0.236  tW  •  1  -0.003+j0.001 0.0  By and  using Model (3.8),  symmetric  II  -0.007+j0.008 0.199+j0.096  -jO.003  and t h e ground  the sequence  ft/1000  return  impedances  impedance  ft.  0.010+j0.004_  formulae  of  equations  (3.7)  calculated are:  0 "0.483+J0.231  [ Z  012]=  1  -0.003+j0.001 0.0'  If are  the ground used  return  i n Model  impedance  II,  0.198+j0.089  equations  the following  (3.5)  impedance  O.OlO+jO.002  derived matrix  is  2  0.506+J0.219 -0.002+j0.001 0.0  ft/lOOOft.  -0.007+j0.008  -jO.003  0 0  symmetric  -jO.003  by Wedepohl  and Wilcox  obtained. 1  symmetric -0.007+j0.008 0.198+j0.083  ft/lOOOft. 0.010+j0.002  48  The  maximum  values section  is  deviation  less  3.6  than  where  the  following  the  ground  between  3% i n  the  ground  impedances  return  the  sequence  positive is  are  impedance  the  0 1 2  obtained,  r 0.510+j 0.225  ]=L  -0.002+j0.001  The  |_0.0  maximum  none ing  of are  the  with  ] = 1  dividing  The  calculated  for  is  3.6%  the  from  the  reference: i n  ground  return  path  derived  ground  return  comparison  Table  for  ft.  into  the  zero  sequence.  subconductors  formulae  (Model  and I),  using the  follow-  1  -0.007+j0.008  ft/1000  0.198+j0.900  in  this  of  the  4.5  case  i s  1.5%  comparison  The  is  Comparison  Time (s)  I  used  purposes.  Computing Model  being  III),  symmetric  -t-jO.003  summary  (Model  0.010+j0.002 _  2  deviation  The  (3.5)  of  ft/1000  0.198+J0.900  -0.003+j0.001  maximum  equations  model  conductor  -0.007+j0.008  analytically  -0.0  one  the  1  "0.486+j0.231 012  using  reference  symmetric  0  [ Z  only  the  calculations:  +J0.003  deviation  By  as  By  and  2  .0  2  sequence.  represented  0  [Z  impedances  10.3  ft.  O.OlO+jO.002  in  the  positive  presented  of  Various  Max Deviation  in  sequence  Table  Models  Matrix Size  % 1.5  101  x  101  II  0.73  3.0  39 x  39  III  0.78  3.6  40  40  x.  4.5.  magnitude.  49  4.4  Reproduction  A [21]  on  test  a cable  measured  for  produced  in  sheath  the  phase  conducted  values  section. are  test  The  grounded  is  values  the  lated  of from  sheaths  of  the  British induced  phase  Columbia currents  currents.  impedances  both  it  two  cables  both  sheath  u s u a l handbook  are  in  to  and  bonded test  Power  Authority  sheaths  were  results  calculate  the  are  II.  three-phase  cables  are  bonded  together  (see  Figure  4.8).  in  the  in at  The  sheaths,  the  same  their  ends  unbalance and  the  Table  4.6  is  measured  values  and  the  using  the  impedances  currents  re-  induced  measured.  the  methods  Hydro  These  needed  circulating currents  gives  the  on  resistances  conductor  induced  Writing  of The  high  a parallel neutral and  the  c a r r i e d out  sheaths  currents,  [21]  at  c a l c u l a t e d using Model  through  reference  Results  system where  currents  and  in  was  this  bank.  Test  various  The duct  of  obtained  taken  in  current  from  predicted calcu-  [3,24].  loop  equations  Sl,k  \  around  the  loop  formed  by  the  bonded  gives:  °  =  for  k=  Al,  0  x  =  si +  *A1  V Due  to  the  the  mutual  X  A2  Z  I  + +  symmetry  B l ,  X  A2,  Z  S2,k  B2,  \  C2,  <4'4> SI,  S2,  N (4.5)  +  B2  in  impedances  Cl,  I  S2  ^1 I  +  +  ci  x  I  the above  (4.6) (4.7)  C2  cables are  and  equal,  the for  spiralling example:  of  the  cores,  most  of  50  ^  Figure  I2>70 f t  4.8  E l e c t r i c a l layout  ,  of.;the  :  induced  j|-  sheath  current  test.  Figure  4.9  Circuit  diagram  of  induced  sheath  current  test.  52  Hence  Z  A1S1  Z  A1S2  Z  =  equations  0  hl  Z  B1S1  B1S2  (4.4)  I ( Z  The equation  induced  (4.9).  =  Z  C1S2  and (4.5) reduce  -ZsiSl^l  = Z~Z;  C1S1  Z  +  ( Z  A1S1+ZA1S2)  ( I  A1  +  ( Z  A2S1+ZA2S2)  ( I  A2  +  ( Z  S2N W  A1S1  + Z  A1S2)I1  sheath  The v a l u e s  f o r two c a s e s  through  ground  resistors  There of  subdivisions  the  latter  14%  between  the  former  are  also  mainly cable  give  a r e improvements  produces  evident  only  be p o i n t e d  + Z  C2  )  A2S2>I2  n  using  total  deviation  an average  and sheaths out that  and neutral)  a r e used  there  i n the cable  were  phase  deviation  effects  of  N  ) I  )  ]  (  '  9  )  i n the  measurements  not  were  grounded  grounded.  [3,5,21].  method While  and magnitude  8%.  4  using  Similar  subdivisions  results  of  calculations.  conductors  consideration.  of  case,  The improvements  when  as n i n e  under  8  f o r the ungrounded  i n the impedance  system  The  methods  i n both  a r e a s many  S1N  the sheaths  i n the second s e t of measurements. proximity  + Z  '  4  as c a l c u l a t e d by the  results  total  4.6.  they  i n the results  S2N  subdivisions  between  d when  + ( Z  be c a l c u l a t e d  i n Table  t h e u s u a l handbook  the i n c l u s i o n of  conductors  sheaths  a  the c a l c u l a t e d and measured  due t o  should  from  t h e bonds  R ^ and R ^>  an average  + I  )  (  presented  i n which  compared w i t h  B2  C1  + I  can therefore  calculated  were  + I  (ZA2S1  +  current  calculations are also  B1  + I  *N  +  impedance made  to:  are the It  (including  Table  MEASURED CURRENTS  4.6  Induced  Currents  i n Bonded  Sheaths  CALCULATED  (AMPS)  %  Deviations  Magnitude  * 1  Ungrc unded  1  1  x  *  si  hi:.  10.8|255°  13.0 260°  2  Mag. & Phase  _|_  *  +  *  +  11.8 261°  20  10';  23  12  10.6 270°  10.0|272°  10  5  11  6  Bonds  1  25.01 4°  53.0 191° 23.0|_0°  2  25.1 [21°  41.4 198°  17.0[0°  3  23.2[32°  44.9 198°  18.4[rj°  10.0 269°  10.8 273°  10.2 275°  8  2  10  5  4  22.0 j28°  47.4|198°  20.7[0°  10.0 269°  11.0 272°  10.5 274°  10  5  11  6  5  35.0[14°  66.9)191°  29.4 [0°  14.9|255°  16.2 263° . 15.4 265°  8  3  11  7  6  30.6|353°  65.9 191°  30.4[0°  13.5(248°  15.2 256°  13  7  16  11  12  5  14  8  9.6 j 269°  Average Bonds  14.4|258°  % De v i a t i o n s  Grounded  1  28.4[21°  56.9 191°  22.8|_0°  13.5|262°  13.5 265°  12.9|267°  0  5  1  7  2,  28.3|21°  55.1 194°  19.9\0°  12.5|262°  13.2 263°  12.6|269°  6  1  8  3  3  29.5[21°  52.4 198°  19.9 [0 °  12.2j262°  13.0 270°  12.4|271°  7  1  10  5  4  23.0[32°  50.4 194°  21.8[0°  10.7|269°  11.5 271°  11.0|272°  7  3  8  4  5  26.4J32°  60.9 198°  25.9 [0°  13.8 273°  13.l|275°  7  2  11  6  31.7|28°  55.1 198°  18.0[0°  12.91262° 13.0j 262 °  13.7 273°  13.0|274°  5 5  1 2  9 8  7 5 5  Average  % De v i a t i o n s i  *  Taken  from Reference  [21]  + calculated  using subdivisions.  o  54  The to  the  are in  slow  used) a l l  phase  improvement  conductors  real  advantage for  distribution  4.5  may  Pipe  Type  cooling  mechanical  or  to  and  the  angles. true  4.2.  This  value  This  it  of  can be can be  said  may  (when  causes  be  for  studies  or  advantage  cables  be  due  subdivisions  further  available,  of  of  lagging  for  for  slight  systems  having  are  used.  impedances  other  method  analyzed  e.g.,  a  conductors  studies the  only  for  calculating  of  can be  that  obtained  subdivisions  would  sometimes  liquids  or  enclosed  in  insulation  c h e m i c a l damage.  of  The  at  higher  transients  in  subconductors  which  is  handbook  sector-shaped  .  conductors.  pipes  gases  Common c a s e s  which  or  are  as  act  as  ducts  protection  found  in  o i l  for  against  and  gas  filled  installations.  Pipe non-metallic  materials  pipes  pipes  are  also  pipes  are  highly  this  steel  if  Another  readily  are  conducting  In  phase  Cables  Cables  cable  section,  surge  types  be  4.1  calculations  switching  not  Tables  subdivisions  other  the  reactance  together  systems.  that  formulae  of  in  calculated.  this  close  occur  the  in  impedance  many  fact  angles  conclude in  frequencies  of  illustrated  To  the  deviations  convergence  as  the  main  easily  section,  pipe  conductors  is and  pose  may no  assumed pipe  plastic,  problems  treated  in  aluminum  impedance  or  steel.  Plastic  calculations  and  or  aluminum  due  to  their  constant  permeability.  Steel  far  as  their  magnetic  properties  concerned.  nonlinearity  is  not  nonlinear the  be  as  linear  are  and  divided  having into  taken a  into  constant  subconductors  account  and  permeability. and  each  are the  whole  The  cable  subconductor  is  55  assigned lated from  the  permeability  according  to  laboratory enclosed  of  pipes-and  pipe the  currents pipe,  pipe the  in  in  results  addition  steel  pipe  deviations  of  to  with  pipe  the  in  assumes  the  conductors  that  the  is  4.11  i l l u s t r a t i o n of  It curve  ( i . e . ,  is  around  permeability  of  small.  may  This  Deviations inner of  the  in  surface pipe.  evident  the of  B-H  that  the  the  the  the  here  in  magnetic  an  is  the  of  because  the  the  be -  in  s t e e l pipe  nonlinearity  of  linear  200A i n  this  in  the  to  model  current  in  of  actual  a 5  as  inch  the  Chapter case  2  the  Figure  curve.  and  fits  saturation  s i n c e more  in  values.  portions pipe  of  provides  nonlinear.  B-H  reactance  linear  due  is  in  percentage  model  the  the  more  4.7  cable  the  give  levels  deviations the  on  variation  measured  the  taken  saturation  Table  unshielded  is  references  The  4.10.  calcu-  Company  current  c a l c u l a t e d and  This  of  The  values.  alongside  assumed  pipe  on  data  magnetic  [15].  Figure  then  Edison  different  most  about  may  at  is  The  of  l i n e a r whereas  around of  and  that  deviations  be b e c a u s e results  a l l  curve)  currents  1000),  shown  seen  are  curve  an  [14]  values  reactance.  magnetization shows  (or  are  i t  are  The  degrees  impedance  shown  2.  Consolidated  measurements  return.  calculations  is  impedance  Chapter  various  measured  current  4.7  at  references  the  reactance  From Table  for  The  in  permeabilities  s i n g l e phase  with in  conducted  in  and m a g n e t i c  material.  presented  steel pipes  reported  permeability  its  method  experiments  cables the  the  of  of  the  B-H  relative  calculated this  region  effects flows  are  on  on  most.  the  the  inside  Table  4.7  Impedance Degrees  Pipe Current (A)  Relative Permeability  of  of  Pipe  Type  Magnetic  Measured R X micro-ohms;  Cables  for  Various  Saturation  Calculated R  |  X  micro-ohms  /ft  Errors X  /ft  100  762.0  201  151  201  117  23%  150  980.0  223  155  223  137  12%  200  1018.0  218  154  218  140  9%  300  942.0  203  147  203  133  10%  480  784.0  180  141  180  119  15%  980  484.0  141  124  141  92  26%  3500  156.0  88  93  88  62  33%  7400  81.0  80  80  80  55  31%  in  Figure  4.11  Shape  of  magnetizing  curve  during  one  cycle  58  The in  the  pipe  deviations large  results are in  It  changes  constant Good the  of  points The  pipe of  to  the  the  the  pipe  may by  be  not  results i  n  only  of  is  as  of  measured  three  values.  cables  Larger  especially  when  is  concerned,  the  magnetic  constant  throughout  the  cycle  the  range  an  saturated the  to  the  is  applied  i t  saturation.  pipe  To  the  the  from  the  produce  which  the  known  that  or  outer  is  data  as  results  (or  inner  parts.  Thus  An  layers 4.9  made.  approximates  the  not.  Table  being  current  of  portions different  saturation.  attempt and  gives  flux)  a  is  assigning summary  above.  calculations, the  is  degrees  concentric  same  assuming  permeability  outer  layers.  By  error  different  into  various  obtained  is  middle  while  to  inherent  through  the  current.  assumed  Furthermore, than  a.c.  the  section experience  are  the  obtained  not  value  if  the  dividing  this  far  steel pipe,  current  permeabilities  is  the  most  cross  this  as  material  obtained  more  surface  for  are  calculations for  pipe.  varies.  r e s u l t s when  pipe  Impedance  alongside  4.8  instantaneous  for  pipe  Better the  be  carry  model  different of  may  the  Table  recalled that  the  values  in  inner  made  in  permeability  actual  the  in  sequence  a nonlinear  with  results  density  flow  of  zero  calculated reactances  should be  permeability but  given  the  currents  of  in  especially  when  Table  the  4.9,  Table  4.8  Zero  Sequence  Cables  Pipe Current (A)  Relative Permeab i l i t y  Impedance  Enclosed  in  Measurements  a Pipe  with  Pipe  on  Three  Return.  Calculated  Measured R (micro- Dhms/ft)  R (micro- ohms/ft)  Errors R  X  100  762.0  197  134  178  84  10%  37%  150  980.0  204  140  199  103  2%  26%  200  1018.0  200  139  194  107  3%  23%  300  986.0  189  133  180  100  5%  25%  500  767.0  165  124  158  86  4%  31%  970  488.0  141  106  121  60  14%  57%  3600.  154.0  76  71  70  32  •8%  55%  8000  76.0  57  57  55  25  4%  55%  Table  4.9  Impedance as  Pipe  *  Two  of  Cables  Concentric  Current  Measured R X  (A)  (ufl/ft)  in  Pipes  Magnetic of  Calculated X  Pipes  Different  Represented Permeabilities.  Errors R  X  (yfi/ft)  100  201  151  203  150  1%  150  223  155  230  181  3%  200  218  154  227  192  4%  25%  300  203  147  212  185  4%  26%  480  180  141  188  160  4%  13%  980  141  124  146  114  4%  8%  .1% :.17%  3500  88  93  89  67  1%  28%  7400  80  80  81  57  1%  29% 12%  100*  197  134  186  118  6%  150*  204  140  214  149  5%  200*  200  139  211  159  6%  14%  300*  189  133  196  153  4%  15%  500*  165  124  172  128  4%  3%  970*  141  106  129  83  9%  22%  3600*  76  71  73  36  4%  49%  8000*  57  57  56  28  2%  51%  Three  cables  in  pipe.  6%  61  pipe  i s divided  assigned other the  into  a lower  current  inner  layer  520.  Similar  values Table  is  used  i s  to  reflect  assigned  and the a  of  lower about  4.7  layers. i t s  these  relative value  of  40 b e l o w  and 4 . 8  The  inner  b e i n g more  values;  for  a r e made  for  the  No  example,  given  440 and t h e o u t e r and above  i s  saturated.  thus,  permeability  layer  is  484,  layer  the  assigned  corresponding  the other  when  permeability  results  shown  in  4.9.  deviations  can be seen  i n the  26% w h e n  constant  are  for  used  current cables  is  the  steel  current  variations Figure of  to  i s assumed  to  one c a b l e  A s i m i l a r drop  from  57% t o  is  and 4 . 9  from  the pipe  also  the  as h i g h  two v a l u e s  in  22% i s  4.8  drop  8% w h e n  only  i t  should  the pipe  values  densities  no m a t t e r  4.7,  the reactance  p i p e when  could be expected  permeability  of  i n Tables  of  permeability  and the  obtained  as  for  pipe three  pipe.  permeabilities results  the results  permeability  980A.  i n the  from  calculated values  However,  form  980A  choices  concentric  i n assigning  l i s t e d i n Tables  As  in  two  permeability  criterion is  pipe  only  i n the  how w e l l during 4.12a,  Figure  based  i t any  is  4.12b.  as g i v e n i f  out  above  layers. done, is  definite  the  s t i l l form  the representation  the method  rather  layers  better  the pipe  are  assigned  such as  the  average  of  any such  linearization, since  the l i n e a r i z e d  above  assigning and  be i n e r r o r of  of  arbitrary  criterion,  Furthermore,  would of  that  is  the different  o n some  cycle  whereas  be p o i n t e d  seeks  to  the  permeability  sketch  put  i t  i n  shown the  Figure  4.12  L i n e a r i z e d  m a g n e t i z a t i o n  curves  63  1  Chapter  The has  been  used  accuracy The  method  of  to  run  introduction  therefore parallel  frequencies  test  ground  method  return  takes  be highly  test  with  the model.  of  formulae  speeds  of both  skin  to d i s t r i b u t i o n It  values  will  circulating currents  of  It  that  subconductors  the  used.  up t h e c a l c u l a t i o n s .  systems useful  where  i n bonded come  effects many  surge  sheaths  close  to  at  higher  studies.  has  the  and  conductors  especially  for switching  results  calculations  i s shown  and p r o x i m i t y  be very  a r e needed  The c a l c u l a t e d  f o r impedance  cables.  on t h e number  account  suited  impedance  case of  conductors  been  field  results.  properties  method  across  modelled  Generally are  of  of  the impedance  i n close proximity.  The  also  calculate  where  A studied  subdivisions  the c a l c u l a t i o n depends  This will  of  CONCLUSIONS  5  assigned  the cross  with  better  i s also  pipe  for modelling  section.  layers  results  different  suited  Cables  assigned  are obtained  permeability  enclosed  different  when values.  conductors  i n magnetic  values  different  with  of  layers  nonuniform pipes  are  permeabilities. of  pipe  material  64  LIST  [-1]  [2]  OF  REFERENCES  D . M . Simmons^ " C a l c u l a t i o n o f t h e E l e c t r i c a l P r o b l e m s o f C a b l e s " , The E l e c t r i c J o u r n a l , v o l . 2 9 , January-December,  Underground 1932.  S.A. Schelkunoff,  Transmission  Lines  "The Electromagnetic  and C y l i n d r i c a l S h e l l s " ,  Bell  Theory  of Coaxial  System Tech.  Jour.,  v o l . 13, pp.  522-79, 1934. [3]  D.R.  Smith  and J . V .  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