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Propagation of the wave front on untransposed overhead and underground transmission lines Lee, Kai-Chung 1977

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PROPAGATION OF THE WAVE FRONT ON UNTRANSPOSED OVERHEAD AND UNDERGROUND TRANSMISSION L I N E S  by  it  LEE  KAI-CHUNG  1  B . S c , U n i v e r s i t y o f W i s c o n s i n , 1973 M . S c , 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 , 1975  A T H E S I S SUBMITTED I N P A R T I A L FULFILMENT OF THE REQUIREMENT FOR THE DEGREE OF MASTER OF A P P L I E D  in  the  SCIENCE  Department of  Electrical  We a c c e p t t h i s  thesis  to the r e q u i r e d  THE U N I V E R S I T Y  Engineering  conforming  standard  OF B R I T I S H COLUMBIA  April,  (c)-  as  1977  Lee K a i - C h u n g , 1977.  In presenting this thesis in partial  fulfilment o f the requirements f o r  an advanced degree at the University of B r i t i s h Columbia, I agree  that  the Library shall make it freely available f o r r e f e r e n c e and s t u d y . I further agree that permission for extensive copying o f t h i s  thesis  f o r scholarly purposes may be granted by the Head of my Department o r by his representatives.  It  i s understood that copying o r p u b l i c a t i o n  o f this thesis f o r financial gain shall not be allowed without my written permission.  Department of The University of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5  ABSTRACT  The propagation of the switching surge wave front on multiphase power l i n e s was investigated by modal analysis and conventional Fourier Transformation.  A 500 kV untransposed, three-phase transmission l i n e ,  for  which f i e l d test r e s u l t s were a v a i l a b l e , was chosen as a test case. Phase A of t h i s test l i n e was excited from a double exponential voltage source and the voltage response at the receiving end was calculated and measured i n a l l three phases.  The calculated voltage a r r i v a l time mat-  ched c l o s e l y the measured value, and was very close to the time taken by electromagnetic waves i n a i r at a speed of 0.3 km/us.  The calculated v o l -  tage response curves also came close to the measured r e s u l t s  (errors within  TABLE OF CONTENTS ABSTRACT . . •  i i  TABLE OF CONTENTS .  i i : L  L I S T OF I L L U S T R A T I O N S  v  ACKNOWLEDGEMENTS  vi-  INTRODUCTION  CHAPTER I 4.  1  COMPUTATION OF L I N E CONSTANTS  Introduction  2. T r a n s m i s s i o n  3 Line Data  3  3. L i n e P a r a m e t e r C a l c u l a t i o n  6  4. C a l c u l a t i o n s of S k i n E f f e c t i n Conductors  10  5. O u t p u t f r o m L i n e C o n s t a n t s  13  Program  6. P o s i t i v e a n d Z e r o S e q u e n c e P a r a m e t e r s  CHAPTER I I  COMPUTATION OF TRANSFERIFUNCTION FOR  19  FREQUENCY  RESPONSE OF TEST L I N E 1. I n t r o d u c t i o n  22  ^2. O u t l i n e o f t h e T h e o r y U s e d i n t h e  Transfer  F u n c t i o n Program 3. I n c l u s i o n  of Boundary  22 C o n d i t i o n s a t Sending End  . . . .  4., T r a n s f e r F u n c t i o n f o r T e s t L i n e  CHAPTER I I I  28 .  32  TIME RESPONSE OF TEST L I N E THROUGH FOURIER TRANSFORMATION  iL. Introduction  36  12. N u m e r i c a l F o u r i e r T r a n s f o r m a t i o n o f I n p u t f r o m Time t o F r e q u e n c y Domain  i i i  Voltage 36  3. O u t p u t V o l t a g e i n F r e q u e n c y Domain  40  ,4' O u t p u t V o l t a g e i n T i m e D o m a i n b y N u m e r i c a l Inverse Fourier Transformation "5. N u m e r i c a l A s p e c t s  of F o u r i e r  . . . . .  41  Transformation  Program  43  CHAPTER I V D U P L I C A T I O N OF F I E L D TESTS  .  D o u b l i n g E f f e c t on Open-Ended L i n e !?• C o m p a r i s o n W i t h F i e l d M e a s u r e m e n t s  49 and O t h e r  Simulation Results  CHAPTER'V  50  CONCLUSIONS  54  V  APPENDIX 1  Transfer Function Program A L i s t i n g s  APPENDIX 2  Phase Smoothing Program,.  APPENDIX 3  Fourier Transformation Program.. L i s t i n g s  BIBLIOGRAPHY  Listings  .  55 61 62  6  iv  8  /  L I S T OF ILLUSTRATIONS Figure  Page  1.  O v e r a l l scheme o f p r o g r a m m e s u s e d  2.  Transmission  l i n e geometry  4  3.  L i n e parameter c a l c u l a t i o n  7  4.  S k i n e f f e c t on r e s i s t a n c e and i n t e r n a l i n d u c t a n c e e a c h b u n d l e d c o n d u c t o r by G a l l o w a y ' s f o r m u l a and tubular conductor formula  5.  6.  7.  8.  9.  10.  . .  2  of 11  Elements of the r e s i s t a n c e m a t r i x of the t e s t l i n e , w i t h G a l l o w a y ' s f o r m u l a and t h e f o r m u l a f o r t h e tubular conductor, f o r skin e f f e c t . . .  17  Elements of the r e a c t a n c e m a t r i x of the t e s t l i n e , w i t h G a l l o w a y ' s f o r m u l a and t h e f o r m u l a f o r t h e tubular conductor, for s k i n e f f e c t  18  Change i n sequence r e s i s t a n c e due t o i n conductor bundle r e s i s t a n c e  change 20  Transmission conditions  boundary  line configuration with  23  Magnitude of t r a n s f e r f u n c t i o n s w i t h s k i n e f f e c t c a l c u l a t i o n b y G a l l o w a y ' s f o r m u l a and b y t u b u l a r conductor formula  33  Phase of t r a n s f e r f u n c t i o n s (Identical results with skin effect calculation by G a l l o w a y ' s f o r m u l a and by t u b u l a r c o n d u c t o r formula)  34  11.  Linear  i n t e r p o l a t i o n of  input voltage  12.  Linear  i n t e r p o l a t i o n of  output voltage  13.  Input voltage  i n t i m e domain i n t i m e domain  and c a l c u l a t e d o u t p u t v o l t a g e H(u) in Fig.  = 1.0  .  .  38  . .  38  with  Z.0°  13 f r o m 0.1  45  14.  Same t e s t a s  t o 15 ms  15.  Output v o l t a g e  16.  O u t p u t v o l t a g e a t r e c e i v i n g end o f t r a n s m i s s i o n l i n e w i t h f i e l d m e a s u r e m e n t a vn d o t h e r s i m u l a t i o n r e s u l t s . . .  a t r e c e i v i n g end o f  transmission  46 line  .  . 52  55  ACKNOWLEDGEMENT  I  would l i k e  Dommel f o r s u p p l y i n g  t o t h a n k my t h e s i s  this  providing  thesis.  P r o f e s s o r Hermann W.  t h e t o p i c , f o r v a l u a b l e c r i t i c i s m and a d v i s e ,  countless hours of d i s c u s s i o n s this  supervisor,  Also, I  and  for  d u r i n g t h e r e s e a r c h w o r k a n d w r i t i n g up  of  I would l i k e  the L i n e ConstantsProgram  t o show a p p r e c i a t i o n t o D r .  and an i n i t i a l  Dommel  for  v e r s i o n of the F o u r i e r  T r a n s f o r m program. I D.  would a l s o  like  to thank the t h e s i s  Wvong f o r p r o o f - r e a d i n g t h i s I  v e r s i o n of  am a l s o  g r a t e f u l t o M r . K i n g K.  the e a r l y p a r t of  The f i n a n c i a l s u p p o r t  Malcome  suggestions.  Tse f o r p r o v i d i n g the and h e l p f u l d i s c u s s i o n s  initial about  the  the research.  o f The 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  is  g r a t e f u l l y acknowledged. Special  thanks are expressed  E n g i n e e r i n g main o f f i c e semi-legible  to Mrs.  for producing t h i s  Semmens i n t h e  Electrical  e x c e l l e n t l y typed t h e s i s  f r o m my  handwriting.  Finally, less  and o f f e r i n g f u r t h e r  the T r a n s f e r F u n c t i o n Program  programme d u r i n g  also  thesis  co-reader, Professor  I  am e n d e b t e d t o my p a r e n t s a n d my w i f e f o r t h e i r  encouragement and p a t i e n c e .  vi  tire-  1  INTRODUCTION The a t t e n u a t i o n a n d d i s t o r t i o n o f wave f r o n t s multiphase  transission  purpose  t h i s w o r k was u s e f u l f o r  of  lines  or underground  c a b l e s was  on s i n g l e  investigated.  1  of  line,  System i n Japan.  Trank  In  transmission  the f i e l d  tests,  l i n k of  This  t h e Tokyo  the sending  The  untransposed  2*  f o r w h i c h t e s t r e s u l t s were a v a i l a b l e . '  t h e 500 kV A z u m i  The  surge i n s u l a t i o n c o o r d i n a t i o n study.  s o l u t i o n m e t h o d s w e r e a p p l i e d t o t h e s p e c i f i c c a s e o f a 500 kV  overhead  circuit  line is  Electric  part  Power  end o f  t h e l i n e was e n e r —Ct i t —Ct 91 g i z e d w i t h a d o u b l e e x p o n e n t i a l s u r g e wave o f t h e f o r m v ( t ) = k ( e -e" ), a s a r e p r e s e n t a t i o n o f s u r g e phenomenon! o n a l i n e e g . l i g h t n i n g s u r g e , 1  3 from an i m p u l s e g e n e r a t o r computer  simulation,  forms  input voltages,  of  waves,  through  this  double  a series  resistance  such as  single  e x p o n e n t i a l decay wave,  was  first  is  illustrated in Fig.  used to g i v e  m e t r y and c o n d u c t o r Transfer  After  the t r a n s f e r  c h a r a c t e r i s t i c s as (TFP)  The L i n e C o n s t a n t s  a f u n c t i o n of  functions  end)  analysis  Program  (LCP)  from the tower  frequency.  geo-  Then,  f o r a l l frequency  the  t i m e f o r any f o r m of  d e n o t e r e f e r e n c e numbers  input  at  points.  obtained,  (FTP)was u s e d t o f i n d t h e v o l t a g e  the F o u r i e r Transform Program  * The s u p e r s c r i p t s  triangular  the  at d i s c r e t e frequency p o i n t s were  r e c e i v i n g end as a f u n c t i o n o f  other  was u s e d t o o b t a i n t h e o u t p u t v o l t a g e  ( 8 3 . 2 1 2 km f r o m s e n d i n g  the F o u r i e r Transform Program  In  were used f o r  the d i s t r i b u t e d l i n e parameters  Function Program.  t h e r e c e i v i n g end  1.  the  studied.  The way i n w h i c h c o m p u t e r p r o g r a m m e s thesis  In  e x p o n e n t i a l i n p u t wave as w e l l as  s t e p w a v e a n d d e l t a wave w e r e a l s o  in this  of 415fi.  at  the  voltage.  , l i n e a r i n t e r p o l a t i o n between i n the  bibliography.  successive  d a t a p o i n t s were used i n t h e t i m e domain as w e l l as  quency domain.  For  i n the  t h e 500 kV l i n e u s e d a s a n e x a m p l e , a d e n s i t y o f  p o i n t s p e r d e c a d e on t h e f r e q u e n c y s c a l e gave s u f f i c i e n t l y a c c u r a t e The c o m p u t a t i o n o f r e c e i v i n g end w i t h a n y o n e o f t a k e a b o u t 80 s .  20 results.  i n three phases at  the phases energized at the sending  the  end w o u l d  CPU t i m e o n t h e UBC IBM 3 7 0 / 1 6 8 c o m p u t e r s y s t e m a t a  of a p p r o x i m a t e l y GC$30. all  the v o l t a g e response  fre-  F o r a more g e n e r a l c a s e o f  three input voltages  three phases, only a s l i g h t m o d i f i c a t i o n i n the Transfer  g r a m it w o u l d b e r e q u i r e d t o o b t a i n t h e  results.  Tower g e o m e t r y and conductor c h a r a c t e r i s t i c s  vk  Line Constants  Program  4 S e r i e s impedance m a t r i x as a f u n c t i o n of frequency, ^constant shunt c a p a c i t a n c e m a t r i x . Transfer Function  Program  \  Transfer f u n c t i o n at discrete frequencies  F o u r i e r Transform  Program  it  Output v o l t a g e s i n time domain f o r a l l 3 phases  TIT— I END!  Fig.  1.  O v e r a l l scheme o f p r o g r a m  cost  ~" u s e d .  Function  on  Pro-  3  CHAPTER  I  COMPUTATION OF L I N E CONSTANTS Introduction  1'.)  The p r o g r a m is  a modified version  w h i c h was u s e d f o r t h e c a l c u l a t i o n o f l i n e of the L i n e Constants  Program  parameters  w r i t t e n b y H. W.  4 5 Dommel,'  . It  the constant  c a l c u l a t e s the frequency-dependent shunt  series  impedance m a t r i x  c a p a c i t a n c e m a t r i x f o r overhead l i n e s from the given  geometry and c o n d u c t o r c h a r a c t e r i s t i c s a t s p e c i f i e d f r e q u e n c y p o i n t s . the a n a l y s i s  of underground  by a c a b l e c o n s t a n t s  cables,  program.  t h i s program  The v a l u e o f  2.  at the s p e c i f i e d frequency  t h e impedance and  t h e 500 kV A z u m i - T r a n k  For  capacitance  to o b t a i n the  transfer  points.  Transmission L i n e Data The t r a n s m i s s i o n l i n e u s e d a s a n e x a m p l e f o r  i s p a r t of  tower  would have t o be r e p l a c e d  m a t r i c e s i s needed f o r t h e T r a n s f e r F u n c t i o n Program, functions  and  transmission  l i n k of  this  simulation  t h e Tokyo  study  Electric  1 2 Power System i n J a p a n ground w i r e s . aluminum c a b l e s tics  .  This  Each phase  is  (see F i g .  2).  a r e l i s t e d i n T a b l e 1.  t h e German s t a n d a r d tail  '  DIN  is  a three-phase untransposed  a bundle conductor w i t h four  steel-reinforced  The t o w e r g e o m e t r y a n d c o n d u c t o r  characteris-  The c o n d u c t o r c h a r a c t e r i s t i c s w e r e t a k e n  48204 b e c a u s e  i n the d e s c r i p t i o n of  l i n e w i t h two  the f i e l d  from  t h e y were n o t d e f i n e d i n enough d e 4 5 tests ' .  4  ©....llrn.-.^  O } 2 ground wires  10m(AH) } 3 phase conductors  25m (AH) i (AH) average height // -1 > ,V >>> >/i i a > in i i rt in II HI a 111 r m,m r i  nn / Ground r e s i s t i v i t y = 200 fj-m  a)  Tower geometry (height i s average height above ground, not height at tower l o c a t i o n ) .  b)  Bundle conductors of each phase  c)  O  Aluminum strands  ®  Steel  strands  26 A l . / 7 s t . Steel-reinforced aluminum cable used for ground wires and phase conductors Fig.  2  Transmission l i n e geometry  5  TABLE  I  TRANSMISSION L I N E DATA  General  data Length of transmission l i n e  = 8 3 . 2 1 2 km..  Average h e i g h t above ground three  phase conductors  of  (flat  Average h e i g h t above ground  configuration)  = 25 m  of  ground w i r e s  '  = 35 m  E a r t h r e s i s t i v i t y (presumably farmland)  = 200 ft.m —8  Resistivity  of aluminum  = 3.21  R e l a t i v e p e r m e a b i l i t y o f a l u m i n u m (y )  x 10  Q.m  =1.0 —8  P e r m e a b i l i t y of aluminum ( y y ) o  Details Type:  = 4?r x 10  r  f o r ground w i r e S t e e l - r e i n f o r c e d aluminum c a b l e , as  shown i n F i g .  T o t a l no. of aluminum s t r a n d s of  =  Steel core diameter  Type:  = 1 5 . 7 mm  N  r e s i s t a n c e a t 20°C  of phase  = 0 . 2 6 2 ft/km  conductor  S t e e l - r e i n f o r c e d aluminum c a b l e as  shown i n F i g . 2 c , w i t h  c o n d u c t o r s i n e a c h p h a s e a s shown i n F i g . T o t a l no. of aluminum s t r a n d s No. o f aluminum s t r a n d s of  16  = 5 . 8 5 mm  Outside diameter of conductor D.C.  2c.  =.26  No. of aluminum s t r a n d s i n o u t e r l a y e r conductor  Details  H/m  2b =  26  =  16  i n outer layer  conductor  Steel core diameter  = 8.1  Outside diameter of conductor  = 2 1 . 7 mm  D.C.  = 0.136  r e s i s t a n c e a t 20°C  mm  ft/km  6  3.  L i n e Parameter C a l c u l a t i o n 6 2 (i)  Series  impedance m a t r i x - C a r s o n ' s f o r m u l a '  i s used  c a l c u l a t i n g the impedances of the conductor e a r t h r e t u r n l o o p s . ductivity is flat  assumed t o be u n i f o r m and t h e e a r t h p l a n e i s  and p a r a l l e l  to the conductors.  Also,  for  Earth  con-  assumed t o be  s p a c i n g s between conductors  are  a s s u m e d t o be l a r g e c o m p a r e d w i t h c o n d u c t o r r a d i i ,  that i s , proximity effects  are ignored.  t^^]  Z  i i ^  The e l e m e n t s o f t h e i m p e d a n c e m a t r i x  =  (  R  i i  +  A  R  i i  }  +  j  (  2  u  l  °  n  GMB^  A X  +  ii> i  and  Z..  =  f l  /  are given  as  k m  = 1,  (1-1)  ...N  Z.. S. . In - i + AX  = AR. . + j ( 2 u l 0  .)n/km  1  j  =1,  i = 1,  ...N;  ...N;  i  f j,  (1-2)  th where R ^  = r e s i s t a n c e of i on s k i n  h^  c o n d u c t o r i n fi/km ( s e e s e c t i o n  4.  effect)  = average height  above ground th  S^_. = d i s t a n c e b e t w e e n i  of i  th  c o n d u c t o r i n m,  c o n d u c t o r and ground  image  of  th j s..  conductor i n m (see F i g .  = d i s t a n c e between 1 ^ (see F i g .  GMR^  a n d j "* 1  3), conductors  1  in m  3),  = g e o m e t r i c mean r a d i u s  of i*"*  1  c o n d u c t o r i n m,  03  = angular  frequency,  AR  = c o r r e c t i o n terms i n r e s i s t a n c e f o r e a r t h r e t u r n  effect,  AX  = c o r r e c t i o n terms i n r e s i s t a n c e f o r e a r t h r e t u r n  effect.  C a r s o n ' s c o r r e c t i o n t e r m s AR a n d AX a r e f u n c t i o n s o f t h e  angle  7  Fig.  3 Line parameter c a l c u l a t i o n  8  <f>_ (see F i g . 3) and of the parameter  a = ks IP  where  r-4 k = 4TT/5 X 10 .f^h^ for self S  impedance  | S . . for mutual impedance  p = earth r e s i s t i v i t y i n ft.m f = frequency i n Hz For numerical c a l c u l a t i o n s , Carson's i n t e g r a l for A R and A X has been deve4 loped into an i n f i n i t e series , which i s used for a < -5, -4 ir 2 AR' = 4wl0 -hr - b..a cos $ + b [ ( c - lna)a cos o I Z Z 3 4 + b^a cos 3<f> - d^a cos 4<j> 0  - b a ^ cos 5c() + b , [ ( c c  J  + b^a -  7  D  cos 7<f> - d a  8  2  (j>a  s i n 2$]  - lna)a^ cos 6<|) + $a^sin 6<j>]  £  O  cos 8<f>  D  . . . .  2<f> +  0  }  (l-3a)  AX' = 4colO~ {y (0.6159315 - lna) + b ^ cos <> } - d a 4  2  2  cos 2cp  3 4 4 + b^a cos 3cp - b^[(c^ - lna)a cos 4c() + <j>a s i n 4<j>] + b^a^ cos 5<j>- d a^ cos 6<j> + b a'' cos 7<J> -> 6 ' 8 8 b g [ ( C g - lna)a cos 8<j> + <f) a s i n 8 <j>] 7  + . . . . where  }  (l-3b)  b. , c . and d. are constants given be i l l y  =  b„ Z  = - i - for even subscripts ±o  ' ^ t ! ,<™ _ with sign 0  /2 -g for odd. subscripts i  b^  1  .  i  = 1.2,3,4; 9,10,11,12; . . . 5 , 5 ... =  j  6  7  f  8  ;  1  3  f  l  6  j  l  f  l  6  ;  9  and  c l  C  and  = 1.3659315  2  =  l-2  C  I  +  +  1T2>  1 > 2  d . = -?- b . I , 4 I  Note that from eqtns  (l-3a)  and ( l - 3 b ) ,  e a c h 4 t e r m s i n i = 1,4 f o r m a  r e p e t i t i v e group i n t h e i n f i n i t e s e r i e s . 7 2 F o r a > 5, t h e a p p r o x i m a t i o n f o r m u l a e g i v e n by B u t t e r w o r t h ' used,  instead of the i n f i n i t e AV A  K  =  / o <t> a c  K  s  ,cos a  is  series  c o s 3t)> , 3 c o s 5j> 3 3 a a rcj) _ /2 c o s 2<j> c o s 3<j> 2 3 a a +  +  +  -4 45 c o s 7ck 4colO n / ^ "a7 ^ 7Tv2 -4 3 c o s 5d> 4 5 c o s 7<K 4colO 5 7 a a v2 U  ;  /  T  (l-4b) Note t h a t t h e i n f i n i t e s e r i e s will  only converge  terms a r e h i g h l y (ii)  for  R and  a f t e r a b o u t 10 o r m o r e t e r m s i f a > 3 .  o s c i l l a t i n g i n that  integrals  The f i r s t few  case.  Shunt c a p a c i t a n c e m a t r i x  the inverse  X derived from Carson's  [C] - The c a p a c i t a n c e m a t r i x  [C]is  of the p o t e n t i a l c o e f f i c i e n t matrix [ P ] . [C] = [ P ] "  1  The m a t r i x e l e m e n t o f [ P ] c a n e a s i l y b e o b t a i n e d f r o m t h e t o w e r geometry, ?  P.. i i  = 2C  and  P  = 2c  where  r^ = r a d i u s c  2 x  10~  ,  4  2h. In — r i  km/F v  (1-5)  In  km/F  (1-6)  of conductor i n m  = v e l o c i t y o f l i g h t i n km/s  Eqtns example)  x 10 " "~  (1-5) and (1-6) a r e v a l i d as l o n g as ^ ( 0 . 0 2 m i n t h e  i s much s m a l l e r t h a n s p a c i n g s b e t w e e n c o n d u c t o r s  (14 m i n t h e  10  example). dependent is  Note that  the elements of  on t h e t o w e r geometry  the shunt capacitance m a t r i x are  and a r e n o t dependent  an a p p r o x i m a t i o n w h i c h i s v a l i d f o r  where e a r t h c o r r e c t i o n terms 4.  Calculations  of  The s k i n e f f e c t formula.  up t o a p p r o x i m a t e l y 1 MHz, g are not yet important .  Conductors  i n the e a r t h r e t u r n i s  W h i l e the e a r t h r e t u r n s k i n e f f e c t has  parameters,  skin  frequencies. the s u r f a c e  effect  As of  i n the conductors  frequency  increases,  the conductor.  This  frequencies  for capacitances  Skin Effect i n  on f r e q u e n c y .  only  This  accounted f o r by  Carson's  a m a j o r i n f l u e n c e on  must a l s o be c o n s i d e r e d  at  higher  t h e c u r r e n t f l o w s m o r e and m o r e  can be d e s c r i b e d by t h e n o m i n a l  line  on  depth  9 of p e n e t r a t i o n of  current  as  (6)  given  by  •n-f y  6 = /  where  C  r e s i s t i v i t y of  P= c  conductor m a t e r i a l i n  y = absolute magnetic p e r m e a b i l i t y i n f = frequency Since  the current  H/m  i n Hz  i n confined to the s u r f a c e of  quencies,  the conductor r e s i s t a n c e  increases  decreases  w i t h frequency  4).  matrix  fi.m  (see F i g .  In  the conductor  and t h e i n t e r n a l  eqtn  (1-1),  at high  fre-  inductance  the s e l f  inductance  element _, = 2 x l(f  L  a  2h. In  H/icm  (1-8)  i is  the r e s u l t a n t  of  t h e i n t e r n a l and e x t e r n a l i n d u c t a n c e , r,. ' 2h. L . . = 2 x 10 I n 7 ^ — + 2 x 10 In — IX GMR. r.  i.e.  1  The f i r s t side  the conductor,  ductance) lines  and s e c o n d  is  term i n eqtn (1-9)  is  (1-9)  due t o t h e f l u x i n s i d e and  r e s p e c t i v e l y . Note that the f i r s t  s m a l l compared t o t h e second  at low frequencies  H/km  x  1  term ( i n t e r n a l  term f o r h i g h v o l t a g e  and v a n i s h e s c o m p l e t e l y a t h i g h  outin-  overhead  frequencies.  '11 S k i n e f f e c t on r e s i s t a n c e and i n t e r n a l i n d u c t a n c e o f each b u n d l e d c o n d u c t o r by G a l l o w a y ' s f o r m u l a and t u b u l a r c o n d u c t o r formula  R e s i s t a n c e R(X2/km) I n t e r n a l r e a c t a n c e X^(fi/km) I n d u c t a n c e L (H)  6  Fig. 4  12  Thus,  t h e s k i n e f f e c t on t h e t o t a l  the e n t i r e frequency range.  inductance i s normally n e g l i g i b l e  However,  over  t h e s k i n e f f e c t on r e s i s t a n c e i s q u i t e  pronounced. The s k i n e f f e c t o n t h e c o n d u c t o r r e s i s t a n c e a n d i n t e r n a l r e a c t a n c e was c a l c u l a t e d i n t w o w a y s . as a s o l i d  W i t h t h e f i r s t method, t h e c o n d u c t o r was t r e a t e d  t u b u l a r a l u m i n u m c o n d u c t o r o f t h e same c r o s s - s e c t i o n a l a r e a a s t h e  actual conductor.  The s t e e l c o r e was c o m p l e t e l y i g n o r e d .  mended a s a r e a s o n a b l e  approximation.  The f o r m u l a f o r t h e i n t e r n a l impedance  of a t u b u l a r conductor o f nonmagnetic m a t e r i a l i s ^ i n t e r n a l = ^conductor dc  +  T h i s was r e c o m *  10 4 '  ^^internal  dc •1 j -  n_  > ( b e r ( m r ) + j b e i (mr) )+$> ( k e r ( m r ) + 1 k e i (mr) ) ) ( W ^ + r (hk rVpr' ' ftr ) ) ( b reK r„' _ t( m/•„.-. r ) +\34.11V^J.I b e i ' (mr))+cj> e r ' ( mfmrHikpi r ) + j k e i ' (m  2>  mr(.l S  (1-10) w  h  e  r  *  e  "  =  b e r ' ( m q ) + , j b e i ' (mq) ker'(mq)+jkei'(mq)  R,  = d.c.resistance  o f c o n d u c t o r i n ft/km  r  = outside radius  o f conduct®r>in m  q  = outside radius  of s t e e l core i n m  Mc  s  (mr)  k q/2 = r( - ^ - j )  t  2 /ks, l/2 = (.,_ )  \  (mq)  1-s.'  N  9  1-s and  8irl0~ f 4  The e x p r e s s i o n s kei(...)  b e r (...) + j b e i ( . . . ) , b e r ' ( . . . )  and k e r ' ( . . . )  + j kei'(...)  + j bei(...),  are modified Bessel  can be e v a l u a t e d by p o l y n o m i a l approximation  11 4 ' .  ker(...)+j  functions,  which  An e m p i r i c a l f o r m u l a  13  for  c o n d u c t o r r e s i s t a n c e a n d c o n d u c t o r i n t e r n a l r e a c t a n c e was  developed  12 by G a l l o w a y  , w h i c h i s based on measurements  t h i s approach, (16 s t r a n d s  current is  assumed t o be c o n f i n e d t o t h e o u t e r l a y e r  i n o u t e r l a y e r i n t h e example)  internal reactance  (X^)  i n the e l e c t r o l y t i c tank.  are then equal.  .  Internal resistance  With t h i s  f o r m u l a , we  In  strands  (R )  and  £  obtain  K /coup R  jr  0  L _  =  ° where  /Z  CJ = a n g u l a r y  r  N  /  K  M  (  = p e r m e a b i l i t y of  = outer radius  n  = no.- o f  K  =2.25,  conductor m a t e r i a l  ;  L  )  of  strands  conductor i n  (fi-m)  (m)  i n outer layer  f a c t o r due t o  (H/m)  (see F i g .  2c)  stranding  f o r t h e i n t e r n a l i m p e d a n c e c a l c u l a t e d w i t h t h e a b o v e two 4.  1  frequency  r  a r e shown i n F i g .  _  (2+n)  ir  = r e s i s t i v i t y of conductor m a t e r i a l  Results  1  approaches  A t lower f r e q u e n c i e s , where s k i n e f f e c t i s not  yet  4 prominent,  the r e s u l t s  a cross-over  for tubular conductors  p o i n t a r o u n d 130 H z ,  the r e s u l t s  are f a i r l y accurate from Galloway's  p r o b a b l y more r e l i a b l e s i n c e t h a t f o r m u l a t a k e s dividual thus  strands  of  results 5_i  are usually  Output  the outer l a y e r i n t o account.  4 conductors  formula  are  The d o t t e d l i n e i n F i g . It  of overhead l i n e s a r e seldom measured  4  s h o u l d be n o t i c e d as  the  computed  s u f f i c i e n t l y accurate.  from L i n e Constants  For  After  the skin e f f e c t i n the i n -  i n d i c a t e s t h e p r e d i c t e d f i e l d measurement v a l u e s .  that l i n e parameters  .  the transmission  Program l i n e of F i g .  per bundle i n each of  2,  t h e r e a r e 14 c o n d u c t o r s ,  t h e t h r e e p h a s e s and 2 g r o u n d w i r e s  T h u s , i n i t i a l l y we h a v e a 14 x 14 s e r i e s  i m p e d a n c e a n d a 14 x 14  shunt  i.e. above.  14  capacitance matrix P  V  P  1,1  '2,1  1,14  1,2 2,2  2,14 (1-12)  V  and  P  14  " dV /dx  P  14,1  "  1  dV /dx 2  14,14j 14x14  14,2  '1,2 '2,1  J  ,  '  2 , 2 ••• '  Z J  <14  V  1,14  I  2,14'  n  1 2 (1-13)  •  dV /dx 1 4  Z  14,l  Z  14,2  I  14,14 •  These  14  *  14 x 14 m a t r i c e s c a n b e r e d u c e d t o t h e d e s i r e d 3x3 m a t r i c e s b y  sidering  the bundling  zero v o l t a g e phase A as  c o n d i t i o n i n b o t h ground w i r e s .  1,2,3,4;  ground w i r e s  c o n d i t i o n i n the 3 phase bundle c o n d u c t o r s ,  as  p h a s e B as  13,14,  5,6,7,8;  If  and  we d e n o t e c o n d u c t o r s  phase C as  then f o r ground w i r e s  con-  9 , 1 0 , 1 1 , 1 2 and  the in  both  13 a n d 1 4 , we h a v e  dV  13 — = 0 dx  (1-14)  dV  14 = 0 dx  and  v V  13  14  and f o r b u n d l i n g  =  0  =  °  i n p h a s e A , we  (1-16)  have  15  dV^ dx I  dx  1  +  I  2  I  +  dV,  dV,  dx"  dx  dx (1-16)  3  I,  +  \ =v =v =v = v 2  3  4  A  (1-17) Q  With eqtns  + Q  x  (1-15)  the desired  2  + Q  and  3  + Q  4  (1-17),  =  Q  eqtn  3x3 p o t e n t i a l m a t r i x  A  P  AA  P  AB  P  AC  V  B  P  BA  P  BB  P  BC  P  CB  . CA P  shunt c a p a c i t a n c e m a t r i x  version  [C]  3 x  3  ^^3x3"  =  (1-13)  [C]  Similarly,  P  Q  %  cc.  is  (1-18)  3x3  then o b t a i n e d by s i m p l e m a t r i x  eqtns  (1-14)  and  (1-16)  3x3  series  in-  can be used  to  impedance -  [Z]. A  dV /dx B  =  _dV /dx c  Thus, w i t h C a r s o n ' s conductors,  the three  Z  AA  Z  A3  Z  BA  Z  BB  - CA  Z  CB  Z  z.  I  AC BC  A (1-19)  B T  J  3x3  f o r m u l a and w i t h one o f  - C-i t h e two s k i n  f o l l o w e d by m a t r i x r e d u c t i o n f o r b u n d l i n g  we o b t a i n t h e 3 x 3 s e r i e s  i m p e d a n c e a n d 3x3  effect  formulae  and g r o u n d  wires,  shunt capacitance m a t r i c e s  for  phases. The 3x3 s h u n t c a p a c i t a n c e m a t r i x o b t a i n e d  of  with  A  to 3 equations w i t h the desired  "dV /dx  for  can be reduced to 3 e q u a t i o n s  [P]"''^,  The 3 x 3  matrix  (1-12)  V  V  reduce eqtn  A  the test  example i s  shown i n T a b l e 2 .  from the tower  the elements  of  geometry  t h e 3><3 s y m m e t r i c  16  series impedance matrix are shown as a function o f frequency i n F i g s . 5 and 6. Note that the differences between both skin effect formulae hardly show up at high frequencies on a logarithmic scale.  TABLE 2. Capacitance Matrix of Three Phase Test Line  [C] =  0.06889  -0.01183  -0.00347  -0.01183  0.07080  -0.01183  -0.00347  -0.01183  0.06889  yF/km  17  Elements of the r e s i s t a n c e m a t r i x of the t e s t l i n e , w i t h G a l l o w a y ' s f o r m u l a and t h e f o r m u l a f o r t u b u l a r conductor, for skin e f f e c t  Resistance 10 7 -  (fi/km)  J  — 10  10  7  Fig.  ~3  103  5  10  2f  ' 5  10 Frequency(Hz)  18  Elements of the reactance m a t r i x of the t e s t l i n e , w i t h G a l l o w a y ' s f o r m u l a and t h e f o r m u l a f o r t u b u l a r conductor, for skin e f f e c t  Reactance  u)L(fl/km)  10  10  2  lo Fig. 6  3  10  4  5 10 Frequency(Hz)  19  6.  P o s i t i v e and Z e r o Sequence  Parameters  T h e r e a r e 3 modes o f TEM p r o p a g a t i o n o n t h e 3 p h a s e E a c h mode i s  d e c o u p l e d f r o m t h e o t h e r and has  s t i c s i m p e d a n c e and p r o p a g a t i o n line  was  constant  (see  transposed, which i t i s not,  its  test  line.  own i n d i v i d u a l c h a r a c t e r i -  later  eqtn  t h e n two o f  (2-8)).  If  the  test  t h e 3 modes w o u l d b e  c h a r a c t e r i z e d b y p o s i t i v e s e q u e n c e p a r a m e t e r s w h i l e t h e t h i r d one w o u l d b e c h a r a c t e r i z e d by z e r o sequence p a r a m e t e r s . untransposed  line  to a transposed  sequence p a r a m e t e r s , w h i c h w i l l of b o t h approaches  For sequence are given  Z  s  averages  g i v e us  some i n s i g h t  i n t o the o v e r a l l  calculation (tubular  zero effect  conductor  (Z  line,  and Z  pos  t h e f o r m u l a e r e l a t i n g p o s i t i o n and  zero  ) to the s e r i e s  impedance m a t r i x  zero  elements  by Z  where Z  given  formula).  the transposed  impedances  by i d e a l i z i n g the  o n e , we c a n l o o k a t t h e p o s i t i v e a n d  f o r conductor s k i n e f f e c t  f o r m u l a and G a l l o w a y ' s  Thus,  and Z of  where Z.. i s IJ  m  = Z  pos  - Z  s  = Z  zero  s  (1-20)  m  + 2Z  are the s e l f  (1-21)  m  and m u t u a l impedances, w h i c h i n t u r n a r e  the  t h e d i a g o n a l and o f f - d i a g o n a l e l e m e n t s r e s p e c t i v e l y , Z  s  =  Z  m -  1 / 3 ( Z  1  /  3  (  Z  AA AB  an e l e m e n t o f  The i m p e d a n c e s ' Z  +  Z  BB  +  Z  CC  )  (  1  "  2  2  )  +  Z  BC  +  Z  CA  )  (  1  "  2  3  )  [Z]„ „ 3x3 J  i n eqtns  ( 1 - 1 6 ) and  p e d a n c e s m a t r i x e l e m e n t s shown i n F i g s . quence r e s i s t a n c e s  a r e shown i n F i g .  7.  5 and6. In F i g .  (1-17) are the s e r i e s  im-  The p o s i t i v e a n d z e r o  se-  7,  the r e s i s t a n c e of  b u n d l e c o n d u c t o r o b t a i n e d w i t h t h e t u b u l a r c o n d u c t o r f o r m u l a and  the  Galloway's  20  istance  ,..„ . (fi/km)  Change i n sequence resistance due to change i n conductor bundle resistance  Fig. 7  21  formula are also resistance as  from G a l l o w a y ' s  the value  resistance  shown f o r  comparison. formula i s  (AR ) c  o g  ).  However,  zero sequence  resistance. (e.g.  of  the 2 e l i m i n a t e d ground w i r e s .  9% h i g h e r  i n the p o s i t i v e sequence  resistance  is  o n l y about  an i n c r e a s e  about 1%.  conductor,  also in  a t 50 K H z ) Also note  17% w h e r e a s  4)  conductor se-  resistance in  resistance  the (AR  due t o t h e a d d i t i o n a l  effect  t h a t the zero sequence  resis-  sequence  resistance  Fig.  in positive  the d i f f e r e n c e s i n conductor  than the p o s i t i v e  formulae i s  The i n c r e a s e  The d i f f e r e n c e i n z e r o s e q u e n c e  s l i g h t l y higher  effect  formula.  ,(see  the  show up w i t h e x a c t l y t h e same v a l u e  is  t a n c e i s much h i g h e r  frequencies,  t w i c e as h i g h  shows up w i t h t h e same v a l u e a s  b e t w e e n t h e two f o r m u l a e do n o t  skin  about  from the t u b u l a r conductor  quence r e s i s t a n c e , ( A R ^  increase  At higher  resistance.  A t 50 K H z ,  caused by the d i f f e r e n c e  the increase  i n the zero  the  in  sequence  )  22  CHAPTER  II  COMPUTATION OF TRANSFER FUNCTION FOR FREQUENCY RESPONSE OF TEST L I N E 1..  Introduction A f t e r knowing  v  the s e r i e s  admittance m a t r i x w i t h zero ij 3x3  [ Y  ]  is  t h e waves  end and t h e o u t p u t o n a n y o n e o f  energised  For  (see F i g .  the chosen  t e s t e x a m p l e , one  end.  That i s , < t < 3  T  8).  the p e r i o d of i n v e s t i g a t i o n time t  t h e mode w i t h h i g h e s t  is  which performs  an expanded v e r s i o n of a program  M i c a Dam t r a n s m i s s i o n O u t l i n e of  line  (see Appendix  the t h e o r y used  P r o p a g a t i o n o f waves  - [ f £ ] =  [f?3  the  necessary  of B.C.  Hydro's  Program  on m u l t i p h a s e l i n e w i t h c o n s t a n t  + IG] [ v ]  Tse"*"^  listings).  i n the Transfer Function  TR] [ i ]  IL] IC]  is  w r i t t e n by K.K.  1 f o r program  d e s c r i b e d by t h e w e l l - k n o w n g e n e r a l t r a n s m i s s i o n  - r f § =  from  wave v e l o c i t y . - .  f o r a t e r m p r o j e c t i n EE 5 s 3 o n t h e f r e q u e n c y r e s p o n s e  is  which  T  The T r a n s f e r F u n c t i o n P r o g r a m  g..  the  t o be found f o r a p e r i o d o f t i m e d u r i n g  the t r a v e l time of  calculations  at  phase,  i n t h e 3 - p h a s e a t t h e r e c e i v i n g end o f  t e s t l i n e was  one  the three phases  r e f l e c t e d a t t h e r e c e i v i n g end have n o t y e t r e t u r n e d b a c k  the sending  where T i s  shunt  t h e t r a n s f e r f u n c t i o n b e t w e e n t h e i n p u t on  The o u t p u t v o l t a g e 8 3 . 2 1 2 km l o n g  „ and t h e  ^ ^ 3 x 3  t h e r e c e i v i n g end c a n b e f o u n d . d e s i g n a t e d A,  [Z..]_  conductance  "  of the 3-phase t e s t l i n e , phase a t the sending  impedance m a t r i x  parameters  l i n e equation  ^  17  '  18  (2.1) (2.2)  t=0  R  4  '6  A £  V ' = V - i " R,. A ,g A o  lllillllMiillUlUllUjIill  Boundary  conditions  i n frequency  A t s e n d i n g end o V = V A g  domain  o I. R,. A  °  o  o  A t r e c e i v i n g end •  Single  £  . V, = H . ( w ) V . j  Fig.  8 Transmission  3  line  input t r i p l e  output  system  j - A, B o r C g  ;  c o n f i g u r a t i o n s w i t h boundary  conditions  24  where  [v]  is  t h e 3x1 column m a t r i x of phase  voltages  [i]  is  t h e 3x1 column m a t r i x of p h a s e  currents  [L]  is  t h e 3x3 i n d u c t a n c e m a t r i x  [R]  is  t h e 3x3 r e s i s t a n c e m a t r i x  [C]  is  t h e 3x3 c a p a c i t a n c e m a t r i x  [G] i s  t h e 3x3 c o n d u c t a n c e m a t r i x  (N.B F o r o v e r h e a d t r a n s m i s s i o n always  lines,  [G]  neglected). However,  the above eqtns  (2-1)  w i t h f r e q u e n c y dependent l i n e parameters. i n the form of  steady s t a t e phasor  (2-2)  Instead,  equations  are not useable f o r we h a v e t o u s e  i n the frequency  lines  equations  domain  ™  (2-3).  rf^i  = m  iv]  (2-4)  [Z] =  [R] +  ju[L]  -  = series  i m p e d a n c e m a t r i x i n ft/km a s o b t a i n e d n u m e r i c a l l y f r o m  Chapter [Y]  1.  = ju>[c]  [V]  = shunt  c a p a c i t a n c e m a t r i x i n ft/km a l s o  and  are the v e c t o r s of phase v o l t a g e s  [I]  respectively, Eqtn  (2-3)  A  i -  .  dx  where  and  - i»  - ^  where  i s v e r y s m a l l and i s p r a c t i c a l l y  [ZY]  I  -  m  [Z]  A  [ZY] [Y]  [V]  currents,  get  18  v  [ £ u [Y]  and p h a s e  1  values.  can be d i f f e r e n t i a t e d w . * ' . t . x t o  =  [Z]  i n the form of phasor  from Chapter  [V] (2-5)  25  T h e 3x3 m a t r i x i n e q t n i.e.  there i s  (2-5)  has  non-zero  c o u p l i n g between the phases.  off-diagonal  T h e e a s i e s t way t o s o l v e 18  coupled equations  is  elements  t o d e c o u p l e them by m o d a l a n a l y s i s  these  16 '  .  With  this  19 approach,  [ZY']  is  transformed to a diagonal matrix w i t h  [ M ] " [ Z Y 1 • [M]  =  where  [A] = d i a g o n a l  matrix  and  [M] = m o d a l m a t r i x , i . e . [M] ^ = i n v e r s e o f [M]  1  Both  [M]  and  [M]  [M], (2-6)  [A]  columns  of  eigenvectors  of  ^ were o b t a i n e d w i t h s u b r o u t i n e s  [ZY]  f r o m t h e UBC 20  Computing 'DCEIGN' [ZY],  was  C e n t r e Programme L i b r a r y , n a m e l y w i t h computes  and  the eigenvalues  'CDINVT'  computes  and e i g e n v e c t o r s  the inverse  t e s t e d f o r a c c u r a c y by r u n n i n g  results  transformation is  [v  m6de  Thus, 3 second from the o t h e r ,  'CDINVT'  the complex m a t r i x  the modal m a t r i x  [M].  'DCEIGN' 19  digits.  described  With  [M]  ^ known,  The to  by ( 2 _ 7 )  (2-5)  with  1  =  [M] [ZY][V]  m o d e  =  [M]  =  [M]" [ZY][M][V °  =  [A][V °  [M]  ^  gives  - 1  _ 1  [ZY][V]  1  m  m  d e  d e  ]  ]  order d i f f e r e n t i a l namely,  .  the phase  ] = [MrVi  P r e - m u l t i p l y i n g eqtn 2 [M]" ^-] dx V ]  of  of  and  a t e s t e x a m p l e w i t h known a n s w e r s  a g r e e d up t o 7 s i g n i f i c a n t  mode v o l t a g e  'DCEIGN'  equations  >  from eqtn  (2-7)  from eqtn  (2-6)  are o b t a i n e d , each  decoupled  26  ,2„mode  d.v  r  0  0  mode 1  •o  mode 2  dx" ,2„mode  d.v  0  2  X  dx'  2.  ,2„mode d v3 0  0  X  dx"  (2-8)  mode 3  3  w h e r e X^, X^ a n d X^ are t h e 3 e i g e n v a l u e s o f  [ZY].  The 3 p h a s e v o l t a g e s h a v e now b e e n t r a n s f o r m e d i n t o 3 m o d a l q u a n t i t i e s , w h i c h d e s c r i b e t h e i n d e p e n d e n t l y d e c o u p l e d modes o f TEM p r o p a g a t i o n . c a n b e t r a n s f o r m e d b a c k t o p h a s e v o l t a g e s w i t h t h e mode t o p h a s e  They  relation-  ship derived from equation X2-7), [V] =  [M][V ° m  d e  (2-9)  ]  The g e n e r a l s o l u t i o n t o t h e s e c o n d o r d e r l i n e a r e q u a t i o n o f t h e m o d a l v o l t a g e c o m p o n e n t s i s w e l l known  component c a n be t r e a t e d as i f For distance x =  V  1,'A  mode 2,'A_  mode  L 3 , H —> v  -mode 1+  e  mode 2+  e  „mode -3+  e  mode 1+ A  „mode 2+ .mode V 3+  because each modal 23 2 A 25  i t were a h y p o t h e t i c a l s i n g l e phase l i n e .  £ km away f r o m t h e s e n d i n g  -mode ~  22  differential  e  end  -fk.  l , „modefx. 1 - + V^_ • e 1 -  -/A  '£ , „mode 2- + V _ • e  2 _  2  -/L  -  Y  3  ^  e" 4 Y  I , +  +  mode  v  '  . e  3.^.  mode _  + V^°  d e  .  e /_ Y  (2-10)  '  27  where  *?•propagation specific  constant  f o r steady state behaviour  at  frequency  TT  ^mode  F o r w a r d m o d a l v o l t a g e waves a t x = 0 o f A, B and C respectively, receiving  V™° „_=  t r a v e l l i n g from the sending  respectively, sending  t r a v e l l i n g f r o m t h e r e c e i v i n g end t o  t h e n we c a n a s s u m e t h a t t h e l i n e i s  v  (2-10)  is  mode _ 1-  mode _ 2"  thus reduced  mode  2  TT  the  end.  i n f i n i t e l y long.  can then n e g l e c t the backward r e f l e c t e d v o l t a g e wave,  v  B and C  we a r e o n l y i n t e r e s t e d i n t h e a t t e n u a t i o n a n d d i s t o r t i o n o f  t h e wave f r o n t ,  Eqtn  the  end.  r e f l e c t e d modal v o l t a g e waves a t x = 0 of A,  d e  If  end t o  mode _  e ^ l *  V  e  ' 1  V  e~V  V2+  mode  n  to  .mode  =  i.e.  3-  mode 1+  mode ,£  We  mode  e  3+  mode 1+  mode 2+  T J  '3'  e  "Y £|  T  3  mode  3+  J  (2-11) o r s i m p l y b y m a t r i x n o t a t i o n , we h a v e  [v•modej  =  1  e  -[Y]-£.  where  [y]  = 3x3  e and  again t v  =  mode  ]  ryodej  (2-12)  "T  JO  d i a g o n a l m a t r i x w i t h d i a g o n a l elements [H(co)] , t r a n s f e r f u n c t i o n m a t r i x o f  The r e s u l t s  a 2  n  ^  Y3  the transmission  system  =  f o r w a r d v o l t a g e wave (N.B.  Y  thus  at x =  0.  obtained are also v a l i d f o r the voltage  a t t h e r e c e i v i n g end o f a n o p e n - e n d e d  l i n e of  finite  length.  response For  times  less  28  than 3  times t r a v e l  time,  the open-ended  results  obtained from eqtn  further  detail  i n Chapter  From e q t n  4,  This  section  (2-12),  can be t r a n s f o r m e d  m  ode 0  the modal v o l t a g e s  the sending  = [M]e  Note that from eqtn alone without  *  £  V  3.  Inclusion For  as  Y  ]  e"  [  Y  ]  £  the chosen  end  8.  rM]  _ 1  in  d e  relationship  from eqtn  (2-9)  from eqtn  (2-12)  a t t h e r e c e i v i n g end  end [ V°]  [V™° ]  [V ] i n  terms  as  [V°]  (2-13) cannot be c a l c u l a t e d from  [y]  [V°] Conditions  test line  Boundary  at Sending  case,  conditions  End.  t h e l i n e was  -  e n e r g i z e d on p h a s e  f o r phase v o l t a g e s  ( d i s t a n c e x = 0 denoted by s u p e r s c r i p t V  [V°] =  ^  the  i.e.  of Boundary  shown i n F i g .  the sending  l  discussed  b y t h e mode t o p h a s e  (2-13), phase v o l t a g e s  [M],  [ ]  _  twice  a t t h e r e c e i v i n g end  +  of phase v o l t a g e s 1  is  Vmode  the phase v o l t a g e s  at  effect  ]  e" •ly.  [M]  are simply  1.)  T h u s , we c a n e x p r e s s  [V ]  doubling  to the phase v o l t a g e s  rv*] = [ M ] [v =  (2-12).  line results  o)  and c u r r e n t s  are  A at  then  I?R° (2-14)  V. V I  and  H°] =  (2-15)  0 0  Substitution  of  eqtn  (2-3)  i n t o eqtn  (2-7)  differentiated w.r.t.x  gives  )  29  dV  -I-  mode I-  =  [M]  •'. [ Z ] I I ]  1  0  ^ IA]II°] 12  l l  A  A  21  A  22  A  31  A  32  A  l  A  "  3  0  A  23  A  33  0  "11 =  Again,  I  differentiating  l  21  A  31  (2-16)  eqtn (2-12) w . r . t . x  gives  mode (2-17) E q u a t i n g R.H.S. o f e q t n (2-16) and  (2-17),we  get  A, ll l  [] Y  e  -  x [ Y ]  [v™  ] - 1°  o d e  l  21  l  31  (2-18)  H o w e v e r , a t t h e s e n d i n g end we h a v e x = 0 a s b o u n d a r y c o n d i t i o n , e q t n therefore  gives I „mode, [V ] = r  +  r  -.-1 [y]  T  i  or  (2-18)  rv™ ] = i° d e  6 o  o I  A ,11  A  l  21  l  31  o l  x  o  ll 21  I  Y J 3  4  31  =  I  A  /  y  2  A /  y  3  2 1  31  (2-19)  30  From t h e phase-mode voltage  IV ]  r e l a t i o n s h i p of  eqtn  (2-9)  we g e t  the sending  end  phase  as  rv°] = M i v 7  d e  ] M  M U  M  21  M  31  12  M  M  22  M  M  32  13  A / u  Y  l  A /  Y  2  A  Y  3  21  23  3  /  1  33  o  A  (2-20)  (For a d e t a i l e d p i c t u r e of Now, we c a n e v a l u a t e V  g  ~  lo  T R  T h e n , we c a n g e t  the boundary  the f i r s t "  row o f  ^ l l ^ l ^ l  the sending  +  M  conditions, eqtn  12 21 A  see F i g .  (2-20). 2  / Y  +  M  13 31 A  / Y  A  where Thus, at  "  Z  &q  M  1  1  A  1  /  Y  = M A / U  substituting  the sending  1  U  eqtn  +  L  Y  l  M  +  1  2  (2-21)  J  =  _§_ . Z eq  / Y  1  2  M ^ / Y , ,  A  21  A  3  +  M  +  i n t o eqtn  l l  A  1  A  2  end a r e o b t a i n e d  .mode, +  3 • )  end c u r r e n t i n p h a s e A as  V 1° =  8).  1  1  M  3  1  A  3  3  1  /  Y  A  3  1  / Y  (2-19)  +  3  3  R  +  g / Z Q  R  Q  f o r 1 ° , the modal  ( 2  eq  2 1  >  (2-22) voltages  as /  Y  / y  / Y  l 2 3  (2-23)  31  Using end  t h e mode-phase  (x=0)  r e l a t i o n s h i p of eqtn  phase v o l t a g e s  in  the 3 phase l l  A  [M]  eq Also,  for  Y  mode  ] = e"  £  t  Y  IV™°  ]  A  2  1  / Y  2  A  3  1  / Y  3  e"V  /  21  eq A  3  1  Y  / y  / Y  the modal v o l t a g e  V  [M]  V eq  A  e  -Y ^ 3  A  should  are  (2-12)  (2-25)  2 3  l l 2  1  31  /  Y  i n the 3 phase  / Y  / y  are  l (2-26)  2  3.  b e r e a l i z e d t h a t i f we e x c i t e p h a s e k o f  then the output phase v o l t a g e s  components  l  t h e r e c e i v i n g end p h a s e v o l t a g e s  A  It  (2-28)  from eqtn  l l  A  sending  as  ]  d e  A  a g a i n , we o b t a i n t h e  l  t h e r e c e i v i n g e n d a t x = £,  [V  Finally,  /  (2-9)  the 3 phase,  (k = A , B ,  or  are  r-  [M]  A  l k  /  A  2k  / y  2  A  3k  / y  3  eq  This  is  t h e f o r m u l a t h a t we u s e  anaoption line  to s p e c i f y which of  Y  l (2-27)  i n t h e F o u r i e r T r a n s f o r m Programme.  t h e 3 phases a r e to be e n e r g i z e d f o r  It the  has test  case. T h u s , we o b t a i n t h e t r a n s f e r  seen from F i g .  8.  function  [H(CJ)] f o r  the t e s t  case  as  32  H ( ) A  LH(io)] =  U  H (o))  T,  B  (2-28)  V  H (to) c  4.  Transfer  function f o r test  The m a g n i t u d e s  line  and phases o f t h e t r a n s f e r f u n c t i o n s f o r t h e t e s t  case of F i g . 8 a r e p l o t t e d i n F i g s . proaches  9 and 10, r e s p e c t i v e l y , f o r both a p -  used i n e v a l u a t i n g t h e s k i n e f f e c t i n t h e c o n d u c t o r s .  The d i f -  f e r e n c e b e t w e e n t h e t w o s k i n e f f e c t f o r m u l a e o n l y show u p i n t h e m a g n i t u d e spectrum i n t h e low frequency more o r l e s s  (0 - 1 0 0 H z ) r e g i o n .  The r e s u l t s  coincide  a t f r e q u e n c i e s above 100 H z .  The p h a s e o f t h e t r a n s f e r f u n c t i o n i n c r e a s e s m o n o t o n i c a l l y ( a s 9 shown i n F i g . 9  ).  This  can e a s i l y b e e x p l a i n e d f o r t h e s i n g l e phase  case  where t h e t r a n s f e r f u n c t i o n becomes  ) = <T  mi  H(U  where (N.B.  =  ^  / / T » 1 2 . . T £ / ( R +  =  e -  \ j , , n  ^  L  )  j  )  C  (2-29)  H = length of l i n e Compare w i t h e q t n ( 2 - 1 2 )  f o r 3 decoupled modes).  and i m a g i n a r y p a r t s o f y & u s i n g b i n o m i a l +  (jwe)  1/2  Y ~ [(jo)!)  172  +|(ja)L)-  - 1/2  R' ,Cs = - (jj)  +  1/2  ] •  1/2 1  j^aC) ^  (J  A  £  a  for real  , a >> b .  a  Y = V(R+ja)L)ja>C = (R+ju)L) .  Expanding  expansion  1/2 1/2 , 1 - 1 / 2 , "b + . (a + B ) ' = a 2  Thus,  a  W C  )  1 / 2  , R « jwL  1 / 2  + jg  t h e magnitude and phase o f t h e t r a n s f e r f u n c t i o n f o r t h e s i n g l e  line are respectively -a  i|'H(u) I = e "  l  (2-30) phase  1/2 = e *2 V  (2-31)  34  Phase  (rad) Phase of t r a n s f e r f u n c t i o n s (Identical results with skin effect calculation by G a l l o w a y ' s f o r m u l a and by t u b u l a r c o n d u c t o r formula)  Frequency(Hz) Fig.10  35  and  L  « -  H(u)  where R i n c r e a s e s frequency for sequency fer  Ag = - £ O J ( L C )  a p p r e c i a b l y w i t h frequency where L decreases  z e r o sequence and s t a y s more o r l e s s  and where C s t a y s c o n s t a n t ;  function increases  decreases  almost  with frequency  statement covers  only  similar  phase  angle?6 i s  phase  angle  phase  f r o m 0°  beyond  = k +  angle  is  angle  Therefore,  2TT ( s e e A p p e n d i x is  used  This  zero.  with  positive of  the  the  trans-  magnitude  function  special  logic  1 f o r FORTRAN  to guarantee  that  can be a c h i e v e d by Whenever  setting  the c a l c u l a t e d  1  1, angle values  ensured  the  o f ± IT r a d f r o m t h e p r e d i c t e d e x t r a p o l a -  a n g l e v a l u e , k w i l l be i n c r e m e n t e d by  and a l l f o l l o w i n g p h a s e the phase  to 360°.  initially  f a l l s out of the range  k:  for  w i t h a FORTRAN t r i g o n o m e t r i c  f u n c t i o n o f co.  where k i s  slightly  filter.  'PHASEPRO' program  a continuous  up a c o u n t e r v a l u e k,  ted  the range  The s e p a r a t e  the phase  t  to a low pass  be i n c l u d e d t o extend the a n g l e s  listings).  That i s ,  constant  l i n e a r l y w i t h f r e q u e n c y , whereas  The c a l c u l a t i o n o f a n g l e s  to  (2-32)  1 / 2  a r e i n c r e a s e d by k(2ir)  t o be c o n t i n u o u s  .  and m o n o t o n i c a l l y  This  way,  increasing.  36  CHAPTER 3 . TIME RESPONSE OF TEST L I N E THROUGH FOURIER TRANSFORMATION Introduction  1.  After transfer voltage v  (t)  V  (u).  the frequency response of  functions,  the output voltage  transformed  plying V  g  [H( )] V  iL  W  t  g  the beginning,  the inverse Fourier  voltages  v.(t), A  v_(t) B  t h e n compared w i t h t h e f i e l d .  give  then o b t a i n e d by  multi-  Transfer  i.e.  transformation  and v „ ( t ) C  26  the f r e q u e n c y domain t o  [H(co)] o b t a i n e d f r o m t h e 2.  voltage  (  is  used to o b t a i n the  i n the time domain.  are applied to the t e s t  o b t a i n e d by Groschupf  the input  input  (GO)  Finally,  techniques  function  described i n Chapter  [V ( o)] =  known i n t h e f o r m o f  i n t h e f r e q u e n c y domain i s  (u) w i t h t h e t r a n s f e r  Function Program  At  from t h e t i m e domain i n t o  The o u t p u t v o l t a g e  is  can be c a l c u l a t e d f o r any g i v e n  (v ) by F o u r i e r T r a n s f o r m a t i o n .  is  the l i n e  case of F i g .  8.  t e s t measurements^  The p r o g r a m s  from the time to the frequency domain,  The above  _  a n d v i c e v e r s a , was  )  described  The o b t a i n e d r e s u l t s  the F o u r i e r  1  output  and w i t h s i m u l a t i o n  used f o r  3  are  results  Transformations  adopted from a  27 version  i n i t i a l l y w r i t t e n b y H. W. Dommel  (see Appendix  3 f o r program  '  listings). 2I-.  Numerical Fourier Transformation For a given  input voltage v  general obtain the input v o l t a g e  of (t)  input voltage  from time to  i n t h e t i m e d o m a i n , we c a n  i n t h e f r e q u e n c y d o m a i n V^(to) w i t h  frequency in the  O  following Fourier Transformation  formula  oo  oo  A (to) = /„, v  o  (t)  cos  ut dt  (3-2)  s i n wt d t  (3-3)  OO  B(oj) = /w h e r e A(co)  a n d B(co)  v (t) e  a r e t h e r e a l and i m a g i n a r y  parts  of V  (co), O  respectively,  37  V ( t o ) = A(co) + j B(co)  .  g  If eqtns  we a s s u m e t h a t t h e i n p u t v o l t a g e  i s zero f o r time t ^ 0,  then  (3-2) and (3-3) can be s i m p l i f i e d t o  A (a)) = / o B(co) where  (3-4)  v„ ( t ) c o s cot d t  T  (3-5)  g  = J * v ( t ) s i n cot d t  (3-6)  T  B  O  &  (o,T) i s the time i n t e r v a l i n which v ( t ) i s  C a s e 1.  Input v o l t a g e If  interval  defined point by p o i n t  the input voltage  (o,T) a t c l o s e l y  i s d e f i n e d p o i n t by p o i n t i n t h e i n t e g r a t i o n  spaced  time i n t e r v a l s , then  assume l i n e a r i n t e r p o l a t i o n b e t w e e n p o i n t s interval  non-zero.  e  (t^,t ),  g  = v  +  2  A  (t - t ) ,  t  t ^  x  of eqtn  (3-7) i n t o eqtn t  A  12  ( a ) )  =  {  [  l  v  +  v  = (v,  ^ H  Then,  f o r an  2" 1 : At V  sin  2  _  _  t  l  )  ]  c  o  s  u  t  d  t  (  v  i —  Atto  t  ?~ l v  7'At  *  1  cot,  v  1  / 2 t c o s cot d t t^  -  1 . . 1 ^ 2 — ( t s m cot H cos totjl co co 1  2  v  t  2 c o s tot d t +  1 = - s i n ut [(y  +  - t  2  gives  ^  v  (3-7)  2  1 (  ?~ /  (3-5)  t  4  - - ^ " V - s m cot 1 ^  V l  V  _ V  7— At  1  = (  2 ~ l t V  2  t  & At = t  J  Substitution  (see F i g . 11).  to  we h a v e v -v  2  v (t)  i t i s reasonable  r  V  +  ,  c (COS  2  _ V  1  -J^-tJ 2  ., 2)t 9  1^  V  -  2  _ V  _ £ _ 1  COS 0 ) t 1 )  1 <  _  i  3  _  8  )  38  Input voltage  Fig.11  Fig.12  39  v -v 2  or f i n a l l y ,  A.^(u)  ^I 2  =  v  s  """  n  u  2  t  ~  V  l "^ s  n  u  l  t  —~"(cos  +  ut  - cos u t ^ ) ]  2  Atu Similarly, (t^,t ), 2  f o r t h e i m a g i n a r y v o l t a g e component i n t h e i n t e r v a l  we o b t a i n t B  12  ( a ) )  {  =  [  v  l  +  V  finally  (v  x  1 u  B-„ (u)  2" 1  -  2  t  _  t  V  iq  )  ]  S  ±  n  u  t  d  2.  cos  t  V  s i n ut dt +  | | 2  t  2" 1 *  V  )^ 2.  _ V  l  t  ^2  t  Y  X  L  V  ut„ + v,  1  1  cos  ut- H  1  t s i n ut dt  ' 1 . ,,'2 cos ut + - s i n u t ) | ^  cos u t | ^ + " ^ - ( - t  v  = — [-v_  12  V  (  - * V  =  2" 1 At V  2  =  or  (3-9)  2" l ——(sin Atu v  ut. -  to 2  sin  ut-)]  1  (3-10) The c a l c u l a t i o n s o f A ^ ( u )  and B ^ ( u )  2  intervals  are repeated f o r a l l time  2  to cover the whole r e g i o n  (o,T).  The r e a l and i m a g i n a r y p a r t of  the v o l t a g e i n t h e f r e q u e n c y domain a t a s p e c i f i c f r e q u e n c y v i s t h e sum o f t h e s e V (u)  then  simply  parts = A(u) + jB(u)  g  N  = k  !  Q  Vk+l  (  a  0  +  j B  k k+l 5  ( ( 0 )  '  W  h  e  r  e  N  =  T At"  The a b o v e c a l c u l a t i o n s m u s t b e made o v e r t h e e n t i r e f r e q u e n c y r a n g e a t t h e same f r e q u e n c y p o i n t s a t w h i c h t h e t r a n s f e r f u n c t i o n s been c a l c u l a t e d .  Output v o l t a g e i n t h e f r e q u e n c y domain i s  thus  have  obtained  at a l l transfer function frequencies. Case  2.  Input voltage defined a n a l y t i c a l l y F o r some t y p e s o f i n p u t v o l t a g e s v  (t).  A ( u ) a n d B ( u ) a r e known  28 analytically  .  Take a s i n g l e e x p o n e n t i a l decay i n p u t v o l t a g e as an  40  example, v-(t)  = e"  a t  ,  t * 0  '  We c a n d i r e c t l y e v a l u a t e A(to) a n d b y V^'(a)) = A (to) + =  (3-12)  B(to)  jB(to)  —~— a+jco  (3-13)  T h u s , we c a n o b t a i n t h e r e a l a n d i m a g i n a r y v o l t a g e  components  f r e q u e n c y domain by t h e e x a c t F o u r i e r T r a n s f o r m a t i o n . we c a n o m i t t h e f i r s t  p a r t o f o u r p r o g r a m >u  in  With t h i s  the technique,  and o b t a i n t h e o u t p u t v o l t a g e  the f r e q u e n c y domain by m u l t i p l y i n g e q t n (3-13) w i t h the c o r r e s p o n d i n g fer  functions,  =  a  [H(g)] ^  (3-14)  Output V o l t a g e i n Frequency From e q t n  domain  (3-1),  Domain  we h a v e t h e o u t p u t v o l t a g e  J  =  [H(to)]  V  (to)  For inverse Fourier section D),  we u s e  It  angle of  frequency  must be r e a s o n a b l y  (see points.  smooth  to  of the t r a n s f e r f u n c t i o n  is  results.  h a s b e e n shown t h a t t h e m a g n i t u d e  smooth  (3-1)  t r a n s f o r m a t i o n back to the time domain,  l i n e a r i n t e r p o l a t i o n between c o n s e c u t i v e  obtain satisfactory  (see eqtns  ( 2 - 3 1 ) and  (2-32).  the t r a n f e r f u n c t i o n provided i t i s  This  is  also  true for the  extended beyond  2TT r a d ( s e e  phase Figs.  10). From e q t n s  ponents  frequency  from eqtn  Thus, t h e o u t p u t v o l t a g e f r e q u e n c y components  9 and  i n the  as [V>  fairly  trans-  i.e.  [V ] 3.")  in  of  frequencies  (3-1  and  the input v o l t a g e .  ( 3 - 1 0 ) , we o b t a i n t h e r e a l a n d i m a g i n a r y T h e y become h i g h l y  o s c i l l a t i n g at  and a r e n o t s u i t a b l e f o r l i n e a r i n t e r p o l a t i o n .  com-  higher  Therefore,  the  41  r e a l and i m a g i n a r y v o l t a g e components values.  a r e converted t o magnitude and phase  T h e p h a s e a n g l e i s a g a i n e x t e n d e d b e y o n d 2TT b y t h e same  l o g i c as described i n Chapter  smoothing  T h u s , we c a n w r i t e t h e o u t p u t v o l t a g e i n  2.  £ the f r e q u e n c y domain V (to) a s V (to) £  14.  =  (3-15)  S(u)^R(u))  Output v o l t a g e i n t i m e domain b y n u m e r i c a l i n v e r s e F o u r i e r  From t h e g i v e n o u t p u t v o l t a g e obtain the output voltages  i n t h e f r e q u e n c y domain  i n t h e t i m e domain b y i n v e r s e F o u r i e r  Transformation £ [V (to) ] , we Transforma-  tion v From e q t n  £ / \  (t) =  1  fS  —  TT  /  £ , v  j tJ o t  ( 3 - 1 5 ) , we g e t v r*.\ ( t ) = — f°° S ofS ( c\ o ) 3 e( !i  1  J  TT  Similar  ,  V (to) e  T T  dto  0  U  T +  R  a ( 3i- 1n6 )  ) J dto  0  t o s e c t i o n B, f o r F o u r i e r T r a n s f o r m a t i o n ,  the inverse  F o u r i e r T r a n s f o r m a t i o n a l s o uses l i n e a r i n t e r p o l a t i o n between a d j a c e n t points  i n t h e frequency domain,  voltages  as w e l l as f o r t h e phase angles  e x p l a i n e d i n s e c t i o n C, t h i s curves  f o r t h e magnitudes  i n contrast  (R^, R ) • 2  i s p e r m i s s i b l e because  to thehighly  (S^, S ) o f t h e o u t p u t 2  (see F i g . 1 2 ) . As S and R a r e smooth  o s c i l l a t i n g r e a l and imaginary  A(to) a n d B ( t o ) . S i n c e o n l y a r e a l v o l t a g e component e x i s t s the c o n t r i b u t i o n t o t h e output v o l t a g e  components  i n t h e time domain,  from t h e i n v e r s e F o u r i e r  Transforma-  t i o n o f t h e f r e q u e n c y i n t e r v a l [ t o ^ , to,,] i s v^(t)  = TT  or  S(to)  cos (tot + R) dto,  0  v j '(t) = - . 12  where  r  JC2 U  S(to)  c o s (tot + R)dto  (3-17)  IT <B^.  S(to) = S  S -S +-f—^(o> - u-) 1 Ato 1  (3-18)  n  R —R R(to)  = R. + - f — - ( t o 1 AtO l  where  -  w.) 1  (3-19)  to^<to<to , & Ato=u -o)^ 2  2  42  Substituting the magnitude and phase eqtns (3-18) and (3-19) into eqtn (3-13) we get \  i2  v  ( t )  S ~~ S  -  l[  =  /  2  I  s  i  -f^ -V (u  +  s -s 0  2  cO  1 Ato R„-R  A  / 2  ] c  o  ^  s  +  R  +  i  J  K ^ ^ i > ^  s -s  n  [ ( S J - - T  to^  H- ~"R  +  —  1  to]  •  - R  1  Ato R  1  ?  (a + bio) cos (c + sto)dto  U  where constants a , b , c and s are constant for a s p e c i f i c  frequency i n t e r v a l ,  s -s 9  a =• S, - - ? — a), 'i Ato r i  (3-20)  1  2" 1 b = ~ S  S  ( R_-R  3  -  2  1  1  c = R. - - | — ^ 1 Ato 2' 1 s =— + t R  ) (3-22) 1  R  (3-23)  Ato  Thus, we obtain v  £  12  (t) = a/ 2 cos (c + sto)dto + b/ 2 tocos(c + sio)diO to^ io ^ U  W  to2 to 2 t, 1 — s i n (c + sto) I + — [tosin(c + to) + - cos (c+sto) ] s  or  ' 10-^  S  S  s  "--j.  finally v^(t)  = sin(c+sto2)  (-|+i02  "s"^  -  s i n  (  c + s t o  + - [cos(c+sto )-cos(c+sio )] s Z -0  1  1  i)  (^3  +  0  3  \ 1$  (3-24)  The c a l c u l a t i o n with eqtn (3-24) i s repeated for a l l frequency  a i n t e r v a l s to cover the frequency region over which the output voltage V (to)  43 r  is  defined.  The o u t p u t v o l t a g e a t a n y s p e c i f i c t i m e i s  contributions  from a l l frequency  v (t) £  w h e r e co' 5.  is  =J  v j  Q  the l a s t  ( t »  2  (3-25) point.  of F o u r i e r Transformation  There are s e v e r a l aspects which deserve Fourier Transformation ,1.  Program  S u i t a b i l i t y of  reasonable"smoothness"  to ensure  Program special attention in  reasonably  accurate  linear interpolation in numerical integration - A  of  input voltage v  (t)  and o u t p u t v o l t a g e  in  frequency  t o p e r m i t l i n e a r i n t e r p o l a t i o n between  jacent data points.  the magnitude  2.  Therefore,  are used to avoid  components  as  the h i g h l y  described i n section  Density  of d a t a p o i n t s  Too d e n s e d a t a p o i n t s w i l l  result  accuracy  requirement reasonably  time domain,  the density  interpolation is  Number  computer  costs  well for  A density  the t e s t case for  must be d e f i n e d transfer  — It  is  of  studied.  the input v o l t a g e v  f u n c t i o n magnitudes  Thus 7 t o 8 decades  of  decrease  the r e q u i r e d no. substantially  frequency data p o i n t s ,  20  satisfies  i n f r e q u e n c y d o m a i n o v e r w h i c h H(co)  easy to judge  loops.  drastically, while  (on a l o g a r i t h m s c a l e )  of d a t a p o i n t s  of decades  frequency  assumed between  on i t s wave shape and c a n r e a d i l y b e d e t e r m i n e d b y t h e p r o g r a m 3.  output  i n the numerical i n t e g r a t i o n  i n l o s s of accuracy.  p e r decade i n t h e f r e q u e n c y domain  the  ad-  3.  - Linear  increase  and p h a s e a n g l e o f  o s c i l l a t i n g r e a l and i m a g i n a r y  a d j a c e n t f r e q u e n c y and t i m e d a t a p o i n t s  sparse data points w i l l  the  results.  d o m a i n Vw(co) m u s t b e g u a r a n t e e d  voltage  the  interval  frequency data  Numerical Aspects  t h e n t h e sum o f  points the  In (t)  the depends  user. and V  of decades  at high  too  as  (co) the  frequencies.  s t a r t i n g at f  =1 Hz start  will  44  ensure reasonable accuracy without i n c r e a s i n g the t e s t case s t u d i e d .  computer c o s t s  I n t e g r a t i o n between f = 0 and.f 6  frequency data points  start is  p o l a t i o n between 0 and f a t any d e c a d e . fairly f  This is  s  t  a  r  done s e p a r a t e l y ,  f  t o o much  start  .where  the  again assuming l i n e a r  a l l o w a b l e as  3.  cause a p p r e c i a b l e r  l o n g as  the output v o l t a g e V  r  the F o u r i e r Transformation Program  where t h e t r a n s f e r f u n c t i o n i s H(to)  = 1  set  data  (to) r e m a i n s frequency  deviations,  I n p u t v o l t a g e wave f o r m - An e f f i c i e n t  accuracy of  inter-  T h e r e f o r e , we c a n s t a r t o u r f r e q u e n c y  c o n s t a n t and l i n e a r i n t e r p o l a t i o n from z e r o t o t h e s t a r t i n g  . ' does n o t start  for  to  a n d s i m p l e way t o c h e c k is  the  t o r u n i t i n a t e s t mode  1,  0  L  a n d t o c h e c k how c l o s e l y t h e o u t p u t v o l t a g e i n t h e t i m e d o m a i n a g r e e s w i t h the input v o l t a g e v is  (t). In our t e s t case, g a double e x p o n e n t i a l of the form v  g  where and  a  2  (t)  = e'V  -1 s  = 3.27  x  6  10 s  At =  (see F i g s . step widths  14,  for  In F i g .  13, i n  the  of  J  scale)  t i m e >10us, t h e i n p u t v o l t a g e v i f o r which step widths  ms  Ato = 20 p t s / d e c a d e chosen.  (log  exponential a decay, A t = 0.1  and  13 a n d 1 4 ) .  from  0.05us  In F i g .  a single  (3-26)  _ 1  Ato = 20 p t s / d e c a d e  were chosen.  (t) g  e x a c t l y the output v o l t a g e thus obtained  t i m e i n t e r v a l f r o m 0 t o 7us  were  3 x 10  our t r a n s f o r m a t i o n program  tially  e"V  = 0.17  T h i s i n p u t v o l t a g e matches  and  -  t h e known i n p u t v o l t a g e v  (log  scale)  of  (t)  is  essen-  45  I n p u t v o l t a g e and c a l c u l a t e d o u t p u t v o l t a g e w i t h voltage(p.u.)  H(co)=1.0 ,0° z  1.0  A t = 0 . 0 5 /as Ato = 20 p o i n t s / d e c a d e s  0.8  f  f  = 1 Hz start end  = 10 MHz  0.6  0.4  0.2  7 Time^us) Fig.13  46  voltage(p.u.) Same test as i n Fig.13 from 0.1 to 15 ms  ,o  5  15 Time(ms) Fig.14  47  4.  Numerical problems with step function inputs - No problem of  numerical i n s t a b i l i t y were encountered when the f i e l d tests of the test l i n e were simulated. t i a l wave (see  The input voltage v (t) eqtn 3 - 2 6 ) .  i n this, case i s a double exponen-  The computations were numerically stable for  large values of decay constants,  namely  and  > 1 0 . However, many cases  were run with step function inputs for checking purposes to debug the programme, and to gain confidence before the duplication of f i e l d tests could be attempted.  Serious numerical i n s t a b i l i t y problems were encountered with  pure step function inputs, which were then overcome by replacing the step —at function with an exponentially decaying function e  .  The decay parameter  a i s chosen i n such a way that this function i s p r a c t i c a l l y equal to a step function over the time span of i n t e r e s t . For an input voltage step function v  g  (t)  = 1  (3-27)  the voltage i n the frequency domain i s V (co) - / " 1 e g o  J u t  dt  - 1 — 1 cot I = -.— e - • JCO  = For  1  - ~  00  o  Mm e -  (3-28)  J t o t  time, t -»• , the second term of eqtn ( 3 - 2 8 ) i s highly o s c i l a 0 0  ting and i s non-zero, which causes numerical i n s t a b i l i t y i n the Fourier Transformation Programme.  However, t h i s problem can be remedied by i n t r o -  ducing a slow decay into the input voltage v (t) as i n eqtn ( 3 - 1 2 ) , namely v (t) = 1 • e ~ , at  t > 0  The input voltage i n the frequency domain now becomes  from eqtn  (3-12)  48  ,r  V  s  / \ (to)  o  r°°  -  = /  =  -  a  e '  1 e  a+-j to  t  -e  -Ji t o t . ,  dt  -(q+jto)t|°° 'o 1  a+jto  Now,  f o r time to  lim t->«>  oi+jto  t h e second  -at -itot e ~ e J  t e r m i s no l o n g e r  o s c i l l a t i n g due t o t h e  ..—•at presence of t h e decay f a c t o r e V_(to)  =  Furthermore,  i n i t , and goes t o z e r o as t-*», o r  *  the single  e x p o n e n t i a l decay v o l t a g e  i s b e t t e r than  •  —at a c u t - o f f step a smoother Thus, of  function voltage  (rectangular pulse)  a m p l i t u d e and phase angle  fewer data p o i n t s  inasmuch  as l « e  spectrum than a r e c t a n g u l a r  pulse.  p e r d e c a d e a r e r e q u i r e d t o a c h i e v e t h e same  degree  accuracy. The p r o b l e m o f n u m e r i c a l i n s t a b i l i t y w i t h a s t e p f u n c t i o n  is  has  t h e r e f o r e e a s i l y solved by i n t r o d u c i n g the decay f a c t o r  experiments lity.  showed t h a t  a > 10 w i l l  Note t h a t f o r the case  decaying  input voltage v (/•*.\ t) = e  1 _  a  t  a.  voltage  Numerical  b e good enough t o e n s u r e n u m e r i c a l  stabi-  a = 10, t h e d e v i a t i o n of t h e e x p o n e n t i a l l y /  g  from the i d e a l For  step voltage  i s n e g l i g i b l e f o r t h e time span of i n t e r e s t  here.  t  = lOys max __ . . -10x10 v (t) = e g  = That  0.9999  i s , t h e maximum d e v i a t i o n i s l e s s  at the upper l i m i t r  r  t of the study. max  than 0.01% from the step  input  voltage  49  CHAPTER  IV  DUPLICATION OF F I E L D TESTS 1.  Doubling effect In  on o p e n - e n d e d  the a n a l y s i s  line.  l e a d i n g to eqtn.  ( 2 - 1 2 ) , we h a v e n e g l e c t e d t h e  r e f l e c t e d v o l t a g e wave and o b t a i n e d t h e m o d a l v o l t a g e f o r t h e line  a t a d i s t a n c e I from the sending  infinite  end.  ,.mode , . „mode . , -ySL V^ (to) = V (oo) e  , s (4-1)  +  This  expression is  not d i r e c t l y usable f o r the t e s t case  we now h a v e a t r a n s m i s s i o n l i n e  of  finite  t h e r e c e i v i n g e n d t e r m i n a l ( s e e F i g . 8). modal q u a n t i t i e s a r e analogous  i s welloknown.  s i n g l e phase  V  to the e q u a t i o n of a s i n g l e phase  where ,.  T h u s , we c a n u s e  case f o r v o l t a g e s  mode o  „ + Z  yinode  ].mode = - — — o Z o  s . \ s i n h yl  I  V  m  o  d  and  e  ,  ,  T  s e n d i n g end x = o, and  * o mode m o c  , £.  F o r an open-ended l i n e ,  = °.  .mode , „ mode V + Z I o o o T  6  TT  mode %  1 0  ^  - e  _  Y  £  14 24 29 ' ' ; i n the modal domain, // o\ (4-2;  c o s h yl .  d e  mode  (4-3)  ,  (V ° o m  d e  .^  T h e n we B b t a i n f r o m ( 4 - 2 ) ,  .  ( c o s h yl  mode  , ,  ,  the  .  a r e t h e m o d a l v o l t a g e and  ,  = V„  TT  V" £  the  mode . , „ ,, s i n h Y & °„ ^  and V^  0  finite  I  TT  at  line,  a r e the modal v o l t a g e and c u r r e n t a t  e  1^  I™ ^  and t h e open-ended  t h e w e l l known s o l u t i o n f o r  + I^° *>  r e n t a t t h e r e c e i v i n g end x =  i.e.  line  and c u r r e n t s  mode , = V. c o s h yl %  o  open-ended  The e q u a t i o n s f o r t h e d e c o u p l e d  where the comparison between the i n f i n i t e line  length which i s  since  and  cur-  (4-3)  . .  + sinh-fYJl)  yl  + Z  I ° o m  o  d e  ).  (4-4)  50 F r o m t = o t o t < 2rr, n o r e f l e c t i o n h a s e n d , and t h e c o n d i t i o n s  a t the sending  of an i n f i n i t e l i n e during v  mode  o  (This r e l a t i o n s h i p  end a r e  t h e r e f o r e t h e same a s t h o s e  o  i s no l o n g e r  t r u e a t the sending  of eqtn  (4-5) i n t o e q t n  (4-4), we g e t f o r t h e  end, m  m  e  ~Y^  T h i s i s t w i c e the o b t a i n e d r e c e i v i n g end v o l t a g e a t l o c a t i o n x = %.  Thus,  o f the open-ended  there i s a doubling  Comparison w i t h f i e l d measurements  the t e s t  f o r the  infinite line  e f f e c t i n the r e c e i v i n g  l i n e i n comparison w i t h the i n f i n i t e  The o u t p u t v o l t a g e for  e n d a f t e r t >_ 2T,  t r u e a t t h e r e c e i v i n g e n d f o r t _> 3T.)  e  2^.»  receiving  time p e r i o d ,  ^mode _ 2 ~ Y £ y ° d e _ ^ y ° d e 1 o +  voltage  from the  }  With s u b s t i t u t i o n receiving  come b a c k  jmode  =  0  and i s n o l o n g e r  this  yet  and o t h e r s i m u l a t i o n  end  line.  results  a t t h e r e c e i v i n g e n d i s p l o t t e d i n F i g . 15  case w i t h the v o l t a g e  doubling  e f f e c t taken i n t o  account.  The a r r i v a l t i m e o f t h e f i r s t p a r t o f t h e v o l t a g e wave c o i n c i d e s  closely 3  w i t h the time t a k e n b y e l e c t r o m a g n e t i c waves i.e.  (TEM p r o p a g a t i o n  277 y s f o r 83.212 km a t a w a v e v e l o c i t y o f 3 k m / u s .  line,  this  sequence  ) i n air  On a  f i r s t p a r t o f t h e wave w o u l d b e a s s o c i a t e d w i t h t h e  parameters,  transposed positive  and t h e s e c o n d p a r t o f t h e wave w o u l d c o r r e s p o n d t o  the z e r o sequence wave.  I t can b e observed  z e r o s e q u e n c e mode i s s l o w e r  t h a t t h e wave v e l o c i t y . o f t h e  than t h a t o f the p o s i t i v e sequence  mode.  A l s o t h e s k i n e f f e c t c a l c u l a t i o n wilth G a l l o w a y ' s f o r m u l a gave s l i g h t l y h i g h e r r e s i s t a n c e s (ve e g . ,> A R ^ 0.73 ft a t 50 K H z , _^ 0.67 ft a n d AR zero — 6  p  see  ,  o  s  c h a p t e r 1, s e c t i o n E ) t h a n t h e f o r m u l a f o r t u b u l a r c o n d u c t o r s .  The  51  o u t p u t v o l t a g e b a s e d on G a l l o w a y ' s  formula w i l l  that obtained w i t h the t u b u l a r conductor Fig.  15.  Galloway's  f o r m u l a , as  d o u b l e e x p o n e n t i a l wave f r o n t c o n t a i n s Galloway's  formula is For  Ametani  12  probable  more a c c u r a t e .  comparison  and-simulation  The s i m u l a t i o n r e s u l t s compare  purposes, studies  causes of d i s c r e p a n c i e s  increase  in  40% r i s e  of  of  26  where  Sections 4  and 5 ) .  results  from  are included i n F i g . described i n this  results  16. thesis  ( w i t h i n 8%).  between s i m u l a t i o n and f i e l d  (200  0, • m) —  u  the zero sequence  Also,  a homogenous e a r t h i s again  cause  Some  measurements  An  increase 30,31  parameters  o n l y an  the zero  and se-  approximation  some d i f f e r e n c e s i n  conductor —  We a s s u m e d a c o n d u c t o r  for increase  of  increase  conductor  temperature to  impedance  temperature of  resistance  20°C.  appreaciably  120°C).  a d i f f e r e n c e b e t w e e n n u m e r i c a l and measured v a l u e s  t h a n 8% i s w e l l w i t h i n a c c e p t a b l e a c c u r a c y studies.  1,  the  components  the f i e l d measurement  by Groschupf  increase  temperature w i l l  However, less  Chapter  because  calculations.  Temperature  An i n c r e a s e (eg.  (See  is  phenomena:  of a s t r a t i f i e d e a r t h which w i l l  2)  high frequency  from  measurements  a t t e n u a t i o n a n d d e c r e a s e wave v e l o c i t y o f  quence v o l t a g e wave.  l i n e parameter  This  than  can be seen  to f i e l d  expected.  of uniform e a r t h r e s i s t i v i t y  xn e a r t h r e s i s t i v i t y w i l l thereby  closer  f a v o r a b l y w i t h the f i e l d measurement  Assumption  This  o b t a i n e d w i t h the methods  may b e due t o t h e f o l l o w i n g 1)  formula.  formula gives results  than the t u b u l a r conductor  be s l i g h t l y s m a l l e r  criteria  for  these  of  types  52  Output voltage at receiving end of transmission l i n e  Output voltage (p.u.)  t=0 415 „  83.212km output voltage  1.0  0.8  0.6  tubular conductor  0.4  " Galloway's formula  0.2  -0.2  J  53  Output voltage at receiving end of transmission l i n e with f i e l d measurement and Groschupf's simulation results  t=0  Output voltage (p.u.) 1.0  415fi v. xnput  83.212km A B C  output voltage  54 CHAPTER V CONCLUSIONS The a t t e n u a t i o n and d i s t o r t i o n o f wave f r o n t s overhead  transmission  case of a Japanese example because voltage  500 kV t h r e e - p h a s e  multiphase  c a b l e s was, s t u d i e d .  o v e r h e a d l i n e was  were a v a i l a b l e f o r  A specific  c h o s e n as a  this  line.  a g r e e d v e r y w e l l w i t h the f i e l d measurement  Results  of another  The  results  and  thesis  be u s e f u l f o r s w i t c h i n g s u r g e i n s u l a t i o n c o - o r d i n a t i o n s t u d i e s systems.  simu-  investigator.  obtained w i t h t h e i r technique developed i n t h i s  power t r a n s m i s s i o n  test  The  a t t h e o p e n - e n d e d r e c e i v i n g e n d was s i m u l a t e d .  with simulation results  will  or underground  f i e l d measurements  response  lation results  lines  on  For instance,  in  t h e t e c h n i q u e c o u l d be used  t o c a l c u l a t e t h e wave f r o n t w h i c h c o u l d h i t a t r a n s f o r m e r a t t h e  receiving  end o f  front  the l i n e .  It  s h o u l d be r e a l i z e d , however,  would be m o d i f i e d by the t r a n s f o r m e r i t s e l f .  t h a t t h i s wave  T h i s wave f r o n t m o d i f i e d b y  the t r a n s f o r m e r would be a w o r t h w h i l e t o p i c f o r f u t u r e r e s e a r c h . on t h e r i s e  time of  may b e more o r l e s s electric utility  the i n c i d e n t v o l t a g e wave, stressed.  industry  32  This  is  the transformer  Depending  insulation  a problem of current concern i n  the  APPENDIX 1 TRANSFER FUNCTION PROGRAM  MlCnTGAM  TeH*!h*L  SVSTfc"  P l . f u S  _j o o i _ . 00 0?  '.0"7  !  u0 U B i)0 OP  1]  uO 1 J y 0 15-  , J) , 0  (1.  L * ^  A ( 3 , 3 ) , -1 (' 3 , 4 1 ,  I. « • »  O ^ f G A . f l E C A Y , A . ? " 2 , Z ( 3 , 3 ) .71 ( 3 , 3 )  I.'* n .f-CY f 5El  * * *.  c  r.  iNPJl  5  (3  5 . 0 o "> 6 . u 1 0 -  .iff  s  v.  _ u a i ? 0 0 1 <i  vE  t  1 2 . 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' , / )  10B 1  (17  1 08 c  * « * . * *  -_L0i7_  c c 1 1 0  PROPAGATION  CONSTANT  ZS-C-U.S. C D saR.T-c.Ecx  rn  ES  : —  109 -110  a o oooo ooo 0 0 I")  ooo ooo 0 0 0-  1 1 1 000 112 PRTvT  IP.FSCn  -XXX  ooo tU_0_  P002  R  c'  57 L  ;3> o O c o o •-1 ir-  •J  o  at  o c O  o o o c  o o 3  o o ~  o o  o o o  c> O o o  o o o  o o o  o o  : o - C < 0  o c. o ci  o o C' c o o  o c =  d o q ^ o  o c o  o o o  o o =>  o co  o o o o o c i o o o c .o o c o rvi ••*•> - J o o .T} o  o o o  o o c o o o  c o  o o c| O  o o O  o o  o o <q O o o c. 3 O Cl o  O O c: o o  d o o a Q o c- o r . rvi c  o o c o *3 •:  o o o  c >: o  H  I  —• -«  Z  >  cn cn x . x aJ uJ cr "N » - O Li_ H H U . CO —• *-* o z D u < x x »* t u UJ 3 —• ^ a * z or O J rO •-• U J  r  "  o c  c  . c  a  >  u  a  rv  _ • LT 1/  _• (  -  -  r v r-1 O a u CT X  — I  » < c- t r < >— i c i !=••  <i  rr? c  a oc. o o o  o o o c ci o  a c c  oj ci  u  u  u  U  U  Ul/^  u  u  o  - |  J  U  'J U u  ! _l r\> o o a. • c o ^ o • • u I — O £Z IT  "  LU < it u , c LO t V H i — *s UJ "* X X tI UJ UJ  Or. oo o C3 O  <  • U 'JJ G £ UJ C I II C i II II  > >  ^ m x rco n x « t a l o o c o o o  =  f\' <  tC UJ  >- oi  11 Ci  c . n) o  u  MI CT C — r^i a.' • £ -xj « o a o o o a o c:  .-- r\j  IT. U  «— CT t-J :  <r c ru r v — —  l~  LP * U . (-J i^l o uj ip a ~. aj r < H ir| ( V H - t- ZJ a o o a. >j IT' ^ -Z Li — n m ~ 2\ o  t- *-t rr_ fX t_J <_1 U. lA.  3 »— v ^ X K CT ?. o 3 ^ a o o  II IU  H  11  II  M  U. U (I I I >» x U, , b 1  II  s »H M *si o di X  > ^ >  rO  -58-  o o  o o  _> c c c  o o 3  O O o •  o i o • o  t_ O O O O O ' c? o o • • • 1  r- r— i — r - ;  Kl  i  c c c  o O O c O O o <\j o O  "T>  X  X  O O  o  i o o G G O < c- O C ' . o o o O O O -  i o  C  i •:  <M Kt X - X X <4 —  ;  (j- a  o o c •  o o • c o => o ' • • •  o;  J  cr  c  • r - -o ov o <J (J- o o  l  o  o  o  o  o  d  o o c o o o  c  d  I  o  •  nn  a-o r - a l o q o o <r f\j (\j f\| (\J AJ A l m  — c ru  o  r\i r y A .  AJ <V f\l  o  • o  o  •  o  •  o  *  ' O CT-  • ct  r\J AJ A l A l A l  X  • Kt UJ * I CO K l I -r * I uJ ro I  c  I r  UJ  •  <jj •  II  LT-  ^  1  -t  -  *0  «i  x  _r —• - J •' o <; a : > J II II J J e t *- *-j. , r-. 3 i a- w w 11 u j . - - fx X  X  ;  uu A A j * - ijj •« —' CO —< Ct  tj _  r u u u u u  [  .^t 2]  ! _J r>  --. 4  _j  -  o  — U  -  d  o  e  d  ,  J_  Z  c  >- a  II _  _• « _)  1  ._'  rr - i (X — . . .  _  _> o  4  S  q  4 3  (T. q UJ  u u l u  II  II  II  Kr K l  \a  Oi o  o ' d  r  — r i ro 3  o  d  1  II  L7. LC Ui u_ 5" Q . cr ct — rv K — -< o o d Q. Q > > =f > >  •—  (  »-q  rJ (  J J O [ -a «i •< -4 < r_ u j u l U J 1 M a « o: H o c o o O  -i —  >~  _ a _ « _ < • _ •  ^-  w rfl 1 . o c a. o . > : > ; > >  >  I r- it -  o  1  Jj  5* ;» > >  -  *  If  us  J U O : <i «a : — - • u x •  a  O q  a —* _o  *  KJ  c  o  ift cl ;  59  ©  o  o  ©  o  o  ©  ©  o = O O O O O C . O C © O © © © C  © © O © © © C ©  © ©  -rxjc  o  ©  rvj ©  o  1  o  ©• -  : ifi -Cl r » a) » c fi c I r\i f\> r . rv f\; A i\j r\j f^. m K I ^ ru<^'\jrvjr\'vru<Ajrv'Vifvr\ 1  ©  o  o  _0  =r X ct) I-OKIKIKIKI <"vi r u ;\i rvj r\*  cc •-» — Kl d CL  -* rt n K> K r to a.  :  1 !  U>  IX ' Kl rv :  co c.  1 -< K : nj J _ L U,  •-< _ UJ  : « rv • CO CL» U i :>  X. O J »—  "  i 1  © => nJ Kl Kl H C CL O. -  5  J  O  CO o. LU > w UJ X r_ »-< cr • — { w C :  __ a .  a: O | o a. I _t a : a UJ  U . L . cr. i a- K I . _. o • 1  c  U  cl  u '  u.' uj >  K.  ;  i/ .  o_ ~t : * f  :t  J  ' •  -t > r g _r a. > >  c it r K*.  K> _.' K l 3 K l i-t a. » rv, >c: r « - ~> o UJ *•  A1 u : -j "  «o  a w-ta : •— ce *v CO  i  i <  rvi * - rvj » I <  • H s: »- : r -  o  UJ <_ i  i  | II  • it  . r ui  D >-i C J -  / a -i  _r  T Cl f t v-i a C. i O ( o »- . LD tO S.  b_ W H  • C CJ i_> t o  X  U  I  _ 1: co co < I— _.' VJ T co 2 o , •  f\ U*  K| IT.  ©  ©  un i  in  in  IT.  ©  o  ©j C !  J O  t - H ~z to n —i —* »CO « t *J CE CO lOt Ct  60 (C  [fl  I _ '_ a J I_J o a © LP Kt -O KJ •O •O CC -C q ^ — xKl o  O O O  JS  c •> O ! K l sr ^ ; (M A J OJ f  o ©  o o  o c  o o  o  ©  q  o  I . t I _ J l_i t*J -Aj © fvj if; rvi 3 s I P a- c — © Aj P J — K l  o  ©  ©  o  ©  ©  ©  o  o  q  ©  o  a  o  o  I I * I l U l u u. uc.• c* _r> © if © A J A i X» x - 6- n r- r~ r - .o -c• 4) -i) © A 3 3 K* A j A J  o  q O  ©  ©  ©  ©  o  © I! _ O AJ AJ  ©  d  o © I I _ _ Kl — X! IT i X I > O- A i i i r uj • LP  I/}  rO II  J-  (_) o  II  1  Z 2  Z  in c c c © o  -OK c p o c o © © o © e o o © ©  2  «  tO CO  W  *  (O  «  UJ  !  4  1 •  > '  a  1  UJ  H  Cr  X  1  \  - J  O~z  j  •—  {  .-_» <->  j  >-  UJ X  I  UJ  1  < 1 1  •t  !  UJ  a  in  _  i—* <j>  i  z  \ c  <  _ I  1  (-1 < a  0  UJ  _ri z O Q_ > uU UJ  o  ©  q  o a © © ©J l II i l U i UJ UJ UJ i  ©  u d  o  o  UJ © r~ O •o ©  UJ U J *iJ U J uj} ^ * m AI a — A J CT- A l d X — K l 43 i n -i a - o A } © o o © d ' a  o  <l  ©  q  ©  o  u^ ir. A : d iT> © c r » LT- — £ i f i 1/ . T .1 C i  IT  ©  O  C  O  ©  . o  a  •  ©  ©  C  © K J -a © c X tl r-j J". i/l  ©  UJ u O K  I  UJ  r  ^r.  d ©  ©  a  ©  © © •  ©  ©  © ©  +  +; +  UJ  x| c - <r A : u sO A) -c i/-.  n — C rji  I  UJ =J  t-  U J UJ  o  rX '  rX  u"tn o  l I UJ U i K l K" U c r~- a: w M — AJ © U*> x| t n —• Ai •T i n m -O 1  ,iJ ui) U J ^ • r- X K l IT- C J — w, c: IT — X i n ir. =3 i n ••0 L P o  ©  ©  C  ©  o  q  l u  l|  +  .-  c ->  UJ  A  c  ~C p ^2 •3 n 1/ ©  c  © © d © © C- © c C c c' •»- -*• ••i -•- •t- t UJ UJ J uJ a , u j ' O J U J U J u Aj, a o ctj x n fx . X . Kf r © u " x r v r\« r - a ] w> A j -4 «: a A J .-n C O I T - x AJ X - A J X , — 4J Aj — a - e-- c Ki — ^J X t~> i/i C 'V X — u7 ft- f j c — — — — A i Aj A J K i K i K l u c  d- ©  c  o  o  e  d ©  ©  d -  r\j  ©  ©  ©  c  »-  _J _r o »— U UJ X  Ui  uJ t u_ a l a  2  UJ  z o »r_ <-> UJ X U J  If  «  N  co 7  -T IJI  d  o  ©  C r. I*  r -J  © © cl  CT A J — if © ("W 0> K l A J O C* Kl X A j 1/ un CT a K I K I n a a c  ©  ©  q  o i U n o a  q  d  o o c l l W UJ a J cr K i .(?• KI m © in r IT- u N K A," — cr sr a  3  ol  ©  ©  ©  q  td © l uj u, Al r*» ur-j •o 3 '  ©  c:  l  'JJ  — 0" C O  ©  uJ S) x c  ©  l U.  <  ©  d  ©  O  rv «  o  _• •3  q ="©*=i'o'©*  ©  ©  — — X  .n  "O  ©  ' ©  ©  l LL  I I UJ U J uJ -O C IT' C "3 K l C Kl O O m xj o in A J Si V A ; u~ c- ^ i-» a ^ Cf. — K. -c x a © — f rv P. cr x i r» Lf • i P i n m u"j ^ — - j j j  ©  I  l f l u u; t7- c A ; r\j K; A j T *~ r- IT. — — A*  UJ ur- © __ U" © © c 4> r-~ rJ. N C -o a d ^ 3  o  — u> i~  l 'dJ Kl rAJ \S> ©  ©1  O  LTt ©  i <_  AJ  © l UJ =3  U".  •-.-•JO  =  -i  A: -  ©  x  —'  l M. X. ^ Kl Kt X i CT-  U  o  A  c_>  d  I  A J UT| SJ C  UJ tr st: O-  1  © . o  o  K i x» td i/i o © 4) ^ uj r^. t c yrjj  ov a  u- UJ uJ A J — rv r- '.o A J A t sX X> •o c 1/1 irt C X- x> O 3/ d 3 a 3  II  q  ij.' U.i ac AJ 3ir> © r~ r c ©  a  —  Kl  It rv rr x ro  —  II  • A J A J  a ru K> B a ; -3 u i i n -O  al Lo X> — oo o  O  nn r n  o  - o :r j"» -O f a a ^ ^ ^ q iV M r v f\j oh  ! K,  d d ui  __• UJ UJ -ii U J _J IL,' _u UJ U. uJ o r- « ™ c uo K i - c K , c rtj — C -1 ^ S ^rVKiLT © LP. CT 3 a i - « K l IT- -C £> IT ^ o -o r » x x- .o X ' X) r - o © c: ©•  i o  2!  aJ u j j J U x - cr oc t n A J A |  J* K* O ,£> iO K « N  ©  o  t  d  ©  © ©  © ©  © ©  +  +  UJ  UJ  © a -»-' uj •C' — A j •ZT K. a Oin r- -3 Kt Kv K l in  LP  ©  © © i © © d c <i -*• + -»i u; U J lu •u •^j C P ~J f^- o AJ © •C K l d i n -£> r--  ©  ©  0  o  c  C  ©  c c  UJ X K;  U  U"  P  ©  C  +  r4; AJ A m IT' •X X  A; A. A in J"  ©  + UJ  xPJ  Kl SI <J>  r-  c c -f- 4 U- u C in c AJ c «~ c O c 1  X  ©  c  APPENDIX 2 PHASE SMOOTHING PROGRAM MICHIGAN  TERMINAL  SYSTEM  CC  PROGRAM REA| *p  0001 0002 0003  -l  FOURIER(INPUT.OUTPUT) — 4(1500),PHT(150U),U(1000),FREG,TEXT(10)  3.000  n w r i f r , <STFT,-. , u m o , • t T . »  a " ooo 1.050  REAL*«  M  r 'A  n  r  H  t  r A N F H , A 01 n, n T F F , S . A 1 . W1  t  TFRE(3).TFIM(3)  5,000-  D E C A Y , V 0 U T 0 , 0-T , T R I S E . D T I N , P O U T 0  000 7  STElGs'i.DO  0 0 08  «1=0.no  POOR  A  5.200 5.100 S.eOU  1 = n. n n REAn (?, l nil »olTE(6,110)1  00 1 0 001 1  n  1.200  XTF(3),PTF(3)  RE A I.* A  5.900 '6,200 -7,OOO  — SEAO(3,33)FREO,TFRE(11,TFIM(1),TFRE(2),TFIM(2),TFRE(3),TFIM(3)  0013 00 15  f n u  0016  T=2 FORMAT(///10X,"PHASE  110 -1 1-1  _ Q 0 1.7-  f  » f ; < , n l  ..,i.fFH,fc.S»11  :  SPECIFIED  WHICH  PHASE  9.640  TO  BE  9 . 7 0 0 10.000  SMOOTHED  11 . 0 0 0  A«=TcRF(I)  0018 ooio  b >•• = T F T •-; 11 ) S I i D S o P T ( A M *AR+ Bw*B'i)  0020  12.00  —  15.000 -16.000 -  . . J E L { H W . L . T . . . 0 . U Q - ) - S 2 s 6 , 263-1 a S 3 0 - 7 . l - 7 - 9 . S f l « S 2 -  17.000  0023  Cl=nMFGA-wi  002"  A*P=(RTEIG*C1-S2+A1>/6,28318530 7  0025  lyp.iL-P^lr.MOI.S.lliPl  0026  KPs AKP  Id.000 19.000  '.  20.000 21 . 0 0 0  AK P =KP  0027  - 22.OOd-  ..S2 = A K P * 6 , 2 3 3 1 8 5 3 0 - 7 1 7.95BflO-t-S2-  _.!!02a_  2 3 . 000  0029  STE1G=IS2-Al)/Cl  0030  A1 = S 2  00 3 1  »• i s n ' - T ' i A WRTTF(7,.33)HERrZ,XTF(l),PTF(l),XTF(2),PTF(2),XTF(3),PTF(3)  2 4,000  -RITE(7,33JFREO,TFRE(I),TFIM(I),S1,S2  _0.0 3 3  _51. - F - 0 . R ± i A . I - X E - 1 _ 3 - , - 5 ^ 2 E a 2 - . 3 , X - 1 S . . - 5 , E 4 - S . . 5 . , E - L Z . - 5 ' , E - L & . - S ) GO  U  AflPTIflNS * 0 P T1 u N s  *S.T.A.rj.S T-IC S * ERRORS  IN  »,  u i T f » F m i  NA.F M  SOURCE  32.000 -3-3,0«035.000  3  3 6 . 0 00  IP , F^C'?T  IN. E F F E C T *  •STATISTICS* NO  To  END (rfrcf^t^.  a s . ooo 32.200  0032  0035  0  14.000  S2=DARC0S(AW/S1)  0021 -0022-  < . 4 I) 0 9.500  EXCITED',13)  F O R M A T (.13)  •I'  8.000 8.200  OMEC.»sfr.2"3lH5307O0*FPEU  0014  EXECUTION  .ooo  Pool  2,000  C2.S?. OMEGA,PrilNER,PHI O L O . H , S I . C 1 » F L 0 » T N , T  REAL.*  003  PAGE  11:45:13  a u i  0005  0.012  03-23-77  MAIN  G(H336)  HE»l'*«  0000 -.00 0 6 .  n  FORTRAN  , LISTINGS  s  R  j S-'i'SP'TF i  MAIN  STATEMENTS  , =  "if,  MBITF CK  L I NECNT  =  60  35,PROGRAM  NO—0-I-AG-NnS.T.lC-S—GE.NEK AJ-E-D  lOMl.NflMAR SIZE  s  33226  .  MAIN if-r.fn  t{  TN T u F  AHrwF  r O M P 11 a T T QMS „  TERMINATED  ON $R  -L040  exgniTniv  7= TFJA?A*  23*.SOtiRCE*  BEEIMS  PHASE  EXCITED  1  3= TFREIMJAPAN'(2)  6=*SINK«  H  APPENDIX 3 FOURIER TRANSFORMATION PROGRAM  MICHIGAN  SYSTS-x CC:  FORTRAN  RF'1  *»  A( 1 5 0 0 ) , P H I  PFA|.R  12:03:25  n5O0),U(1O0O),FREi3(2OO),TEXT(1O)  r?,"i?,n"cKA,PMl'4t'*,PHlPLP.H,81,Cl,Fl.O».TN,T  S » l  »A  0 0 0<4  RF M  *P  00«5  Kft(  *«  0 0 op  Rt»L*!>  T U i i T S , 7 ^ R R n Y C 1 000 ) , C T O u T 2 » [ ) T I N 2  0007  KF/«l*p  AI  00 0R  CALL  POOR  13-23-77  FOURIERCI.NPUT, OUTPUT)  . PROGRAM  ono l ooo? 9003  MAIN  G(U133ei)  LISTINGS  P"!!l- f .liPl  11 , U T  f  Aft A * P .  • A i f i, • API ti P T F F  f  r  f  S  Q00  2  000 000  3  A1 .w 1  f  0(10 _  . _  ooa ooo  N,RtN,TTIN,TFIN,TINPUT,fLuATN >( ' S E T  M 1 N U S 7 E » P = 0N ' ,  lh)  o 0 f)  T- TSFSS.r.-On  001 0  REirM?  0011  If  00 12  1 n)  ooo 1 1 000  K ( , P T , I B , I Z , O M I N , r > T , T M A X , { T E x T ( I ) , I « l , 10 )  »os,  i > R l T h ( h , ! 0 IK O P T , I S , I Z , Q M I N , 3 T , T . M « X . ( i t » ( ,( 3 1 3 , 1 X , 3 E 1 1 , 3 , 1 O A u } 1 0  f  01)15 0 0 1 H  I F ( T P . Ef. . 0 1  0015 6  .INPUT  FnR  0 0 0  13  Ooi)  1 «  000  -in.  f Ye X T H ) . 1 = 1 . 1II) 0 , 1 (, A it ) Fl'S-"AY ( 1 H  lh  1/4 o o c  MS  INCIDENT.  0 0 0  1 7 Oon  I - » S T B * 17 + 1  0017  12  1 5 uoo  STT'P  » " T TF f p . r l  0016  000  1 9 000  WAVE._  00 1 8  « ! • : = . feRLi I)  20  (•0 0  00 19  «IM =3  2)  000  OOP"  T F T •- =  _2_2_  Ci (I i)  002;  25  Onu  u022  T T T ' =4 0 0 O . n - i i o A L P H A 1 = A1 ' i / T T T'4  24  00 0  O.V3  A L P"i A2 = H  .27  -0*  i-./rri •  R T T r(o, * P 0 ) A L  0 02 4  »00  0025  O  H  .  .  .  ..  A 1 , A L P H A ?  FUt- :-AT ( F X , " I N R U T  VOLTAGE  TIME  CONSTANTS:  ALPHAI=',EI3,S,•  ALPHA2=I  1 002h  IF ( r E x R . F o , n  0 0 27  I F ( TF V P . F Q . ? ) A L P H A = A L P niLl  CHANGE  (" I  K  P'  A|.PHA = 4LPHA1 T nil  ST A TE  ,  A2  4  M  E N T S  ..IF.  D.l E M  NSIQN  I S  CHANGED  * * * * * * * * * * * * * *  » ' I T F ( 6 , t* 1 p.) A l R H 4  0 02* 61o  0029  .00,50  F O R • A T ( / / ' * » * T 1 ^ E  CONSTANT  IF f T , _ G T . 1 5 Q 0 )  T O 97  IF(TZ.GT,2O0)  0031  Gil GO  TO  I N  OECAY  =  ' , E 1 3 , 5 , ' * « « * * ' )  25  oca  2c  000  27  000  _2i  COG  28  UOO  29  ooo  30  000  31  0 0 0  32  ooc  l i  97  0  fl P  3-  000  S 1 = 1Z  35  ooo  h:(i.Ofl  3o  ooo.  0034  S 2 = 2 . 3 u25,"509299«oPO/S 1  37  ooo  0035  OP  36  OOO  _3Ji  I) 0 u ooo  0032  .0033 .  1 1  = 1 . I Z  f?i •': (K) =  003<i 11  0037  f  -EXP(H»S2)  Msr + l.Dp.  ^EARING .._ . . I f  4 1  A^PLTTilPF.  A NO  ( K U O T . F P . 1 ) . GO  PHASE  SPECTRA  T O . 1-L ...  OR  PRESETTING  TH E  M  * * * * * * * * * * * * * * *  ..:  4 1 0 (I o 12 Ii?.  O?0  c  «2  oao  T.  TS  01!  0 0 39 0040  RFAI)  12  J12. 1 * 0  IN  * = 2,T *  k F A0 ( 3 , 33 )  * ( » ) I PH I  tn )  U2  0 10  i42  100  42  200  41  0 0 0  '4 4  000  4(,  900 I) Oil  A(*1:1 Pr-T(x) = 0.00  C C 004 ,  0 0 0  C C  THIS  INPUT  T H F TRANSFER  F U N C T I O N  3J  F.JRMAT(51)(,2(El3.h 5)(lJ_.  004 3  ]3  FPRMAT(2E10.0)  U04ij  I ?  CPNTIM'.IE  0042  /  IF(tfll  L J  0 00  I L P " 4 , A L P H A I , A L P H A ? , A I ? , V M ( 1 0 C O  M r  POOt  (i n n  n t C t v , v P ' . l T 0 , n T , T R I S F , D T l N » P U I i T 0 , T F R E ( 1 5 0 O ) , I F I M ( i 5 0 0 )  FTNC  PAGE  1  ANK(A(K)},LT,0)  __  EXP(—GAMMA*L)  _  G O T O 200 ^  AND  -BETA*L  47  .Ut  ooo 000_.  49  ooo  50  000  5 1 /  '111  f  M  MICHIGAN  TERMINAL  SYSTEM  0016 on a s 0049 C  READING  r  THTS  0051  5«.O00  .  5^  INPUT  IS  57.000 . 5 6 . 000 .  SxIPPFn  60 . 000  G O TO 2 1  FOR  OP T O N  3  _  _  .  .  ;  Sl-O.DO C  SI  IS  F TRs T  VOLTAGE  VALUE  . 6 4 , 000 6 5 . 0 00  N=1 DTIN;,050-06 T 1 MP,.Tr-OTT't  0055 905b .  6 6.000 h 7 . 0 Oil  wRTTF(6.8091PT  0057  eo9  0058 -0.059-  FORMAT!' F L 0  0060  A  T .•!  =  66,000  IN  INPUT  TIME  -IE U;..~T , 2 0 0 )GO  —1.7  FROM  STEP  0  TO  5  MS  69.000  s ' , E 1 3 . « )  -70 , 0 0 0 -  TO_22  71 . 0 0 0  N  72.000  TIMPIITSTINPUT + DTIN  006 1  7 3 , 0 0 0  l l p a l. Y I N 1 = T T N P U T . S3SfA|.PHAl+AI.MriA2)/(ALPHA1-ALPHA2)  0062 C  ._0.0.61-  ALPHA12  0065  INPUT  VOLTAGE  IS  SET  TO  7 « . 0 00 74.200  POSITIVE  A!2r-<:2. FORMATt//! (ALPH*WALPHA2)/ALPHAl-ALPHA2)s  819  .75.0  r C  IS PRFSFT TNSTFAU V O L '' A f . F S ? s2 = S?*DEXP(-AI.PMAoTIMPUT)  TMPIIT  OF  77 . 000  78.pop  READ-IN  79.000 80,000  s?=FLOATN*0TIN/TRISE  C  i _-  "l> ( K O P T . E 0 . 7 J G O TO H=(Sl+S2)«OTIN/2.+H  0067 0068 Q0h9  s  0070  U(N)=R2  0071  N5N • 1  _0_O.7-2_  _G0_T.Q-.X7 22  .81.00082.000  97  6 3 . Cv0 84.000  i ; s 2  65.COO 86.000 - 6 . 7 . .0.0.0 TO  I F ( M i I. T . 2 ) GO  86.  95  0074  9  N - N - 1  92.000  W B I T E (ft , 6ft1 ) H  0077  8 0 ! _ E oRMA T Ci "FIRST THE  0079  100  OF  INPUT  VOLTAGE  VOLTAGE I ^ U T AT 7.FR0 T T<? f 6 . 81 o 1 A 1 2 WRITE r e , 7 ? )  0080  - 9 3 . 0 0 0-  U S E - V O L T A G E _ A R E *—-Q F _ J _ - . . U . T _ _ _ _ » E X 5 . 5 )  ENTRY  FRED  IS  IS  PRINTED  94 . 0 0 0  Oil  F ( * = 0 ) = T I ME  OF  95.000 STEP  =  96 . 0 0 0 9 7.000  (11(1),1 = 1,N>  9 6.000  VOUTOzt.DO/DECAY 0062  . — 9 9 . 0 00.-  -is.fi-1-T-E-r.Q...S.l-U  _-.0.8__ 611  0083 0084  812  0085  55  1 00 . 00  FORMAT(///'TIME OF INPUT VOLTAGE*) /.RITE f 6 , 8 1 2 ) ( T A R R A Y ( I I ) » n = 1 ,20 0)  1 02 . .'00  FORMAT ( 1O F ! 1.31 FOK-AT(////'OUTPUT  VOLTAGE  IN  FREO  DOMAIN  AT  FREOsO.  ' . 2 E 1 5 . 5 )  0088  32  "  l P  FORMATC'O'?24X  HNrT nMi T  '  '  'INPUT  FUNCTION  FRFn,,FNrv H7l f  RFAI  10 3 . 0 0 0 104,000  _*.*ft.*.* * * *.**..**ft**_  - C - . T - R A N S F-Oa-1 _IJJ. _ _ ! _ _ _ ! - C.T4 O N . - _ R .0 M - _ T . I - F _ j r j _ _ f . R E Q U t N C - Y _ - O M . A I N 0087  o  1 0 1 . 0C0  Hsl.O/H  0066  0 , 0 0 0  91,000  TIN?=TMAXIN/2.  0076  000  69.0 00  T M A v l M S T iriPi.iT  . 1)075  00-  76.000  I.E11.5)  S5=S2*(0EKP(-ALPHA1.T1NPUT)-DEXP(-ALPHA?.TINPUT))  0066  61.000 62,000 6 3.000  INPUT  U.C1 J=S»...  0051  noo  So.000  t*************************-******************** 5R.000  FUNCTION  IF(KDPT.ro.3)  C  _M.?.a_  16  .  H=O.Do  16  0053  0073  TO  PHI(Ki=0.D0  15  0050  P002  53.000 GO  GO T O 1 8 n o I 5 K = l • T-R A<K)=,.00  Hi  PAGE  12|03:25 -52.OO0-  IE (TBl.ANK ( H ) , L T . 0 )  0017  0 06 3  03-23-77  MAIN  G ( H 3 3 6 )  _16 ..RE»D . (i. 1.3.)H  _0.01.5-  0052  FORTRAN  IN  FREQUENCY V  DOM A  IMAGINARY  106.000  IN',19X,'TRANSFEk AMPi  -10 5.0 0 0-  TTIIPf (AP.SOI UTE.  107,000 108,000  64  v.  EI  <-  a ° - - __ o o  o  ro  o —  —  O o o  C c o  ci rvj  O a o  O o o  LO  -to  o O o  O O O '  C c c  O O O :  o O o  o a o  o o o o o c : cr. c c C' o o o O O O O O i -  c  q o o o o o CJ c • c o o c o o o o o o  o c C" O o  •  r - c o r r - o  it  o  o c a o _ > g o o o o o O O O O O f  » • » • •  — .*V  T  ^3  =J  1/1 J J CW =J  o o o  a a o  o o o o o  i  o cj  o  o a o o r*j ^  o • c o o o  _.- o a o • d o o c o o o j O (\l ' J ^ o  •j*> ir» 'X i r LO u i  O  :C X CT" C> — l/> L P L P X .  d c| f\jj  —  <J _. _5  UJ  s i  *  -T. <t C CJ *  UJ  <t  s: o  UJ  a  f-O tVt  >  o o  »-/ ui o  •c  <T —• ro ro LO CO  H •~i —i aJ  <  rO w 21 •<  CO  (_>  Z <t rx V— cc  o •«£ 0=1  «  L_  2 UJ CO  >-  i-i *„  —•*  UJ  w  a oc uj rst I >i (_» UJ  x-  f3 Ui or U-i  •  JT  P-. in  1  X  c  !  —* (_>  z.  1  tf)  II  c c c  o  •  II u  tT3  U.'  C".  Ci c  n II u. Cr c — v —  _J  z  U  X  I I N  c * ^~ >_ *— v_ M C  o o  -— cc  »-. •  *— 1L.  • ft t - ti_ C 1 » -1 + V i_' t— *~^— ._ it « o Q. U II M U, 2 *—i • _c X" * c —  o  + .. •  a  II  a.  . o  , _  •tj fO  or CL  co w  o o i _i C  Z\  _)  UJ  _  c  II t*VJ It j II i — II C J r u f\> =) H U {/. U I  - —<  u  z  t—1 3T  a X c o  - A J fO o o- tr a a o o D O o  ^  CT C o  LT  o c  o o o  rt O a a- o o <_ o o o  c — o c —« —• o o o  <r UJ  c  a _ ^ r'l t t » Si LTl » * — to < U J : co — «. <*4 r\j r o x * • K» (V i » — :rj •t II K t •S; -3! ^ <_ C/J it a c T.  UJ j; o « ru — •a  _i c c  II  j  IT. o  _• » <i ; u. f  _ _ _ j <r. Q. c J - u .  rC ro  >-  cr  <r lu  '  X"  x  t  + -6  u  cr  <x  a  u r-  c <_-  _t *-»  _n » •O "-• • u. <r : UJ t - «•— »X II >~ II C L IJU It 1 i — -- ... n. o — Or. :< (T. 3 -<3 O i—i »— <\  -J o  —  <J  i  (V  r\j <t X a. i I < ;  C + — U J x  QL  •5  •  X  3 IT- X h ct o  ol o  o  o o  o  o! o  o  •  o  oi o  o  ro  MICHIGAN 0  1  SYSTEM  TERMJNAL  3d  FORTRAN  G(41336)  MAIN  03-23-77  12103125  S2=0AeCnS(AK/Si)  PAGE  161  nno  0 135  IF f B - . L T . O . P I M  01 3 6 0117  C 1 snMF.GA-w 1  165.000  AKP=r<!TFir,*n-S?*A1Wh,?flX1«5^07  1  0133  AKP=OKP+SIGNCO.5,AKP)  167.000  013=)  K Pr A K P  16 8 . 0 0 0  n l <• Oiui  AKPsKP  169  >  S2=6.28318530717958-S2  161.000 0 n 0  6 6  S? = A h p « 6 , 2 8 3 1 6 5 3 0 7 1 7 9 5 8 0 0 + S2  170.000  STE  17 1 , 0 0 0 17? 000  oiuu  WlmMfGA  01«5  C  C S 2 - 4 ] ) / C l  A1X<5?  _.c _.: OUT  Cl  AS  51  17U.00O  VOLTAGE  " N O  t, P H I  AND  PROGRAMME  ARE  175 onn 4 » j . u  PRINTED  176.000 17 6 . 0 0 0 17 7 n0n  ABE  SHIFTED  LIME  117  TRANSFER. FUNCTION  0H7  PTFsphT(ISTORE) 51  TO  176.000  HFRrZ,AW,nw,Sl,S2,A(IST0RE),PMI(IST0RE)  STILL  t TF=.A ( I S T O R E )  0 1 49  FORMULA  F  INCREMENTAL  0116 oma  BY  SI  WRITE(6,5^1 A_  OBTAINED  TSTnR =TSTnPF+1  ISTOKE  c c.  nF  AMPLIT11OF  r c  173,000  S 1 / T MA X I N  C 1=5=1 . . .  c c  1r  AS  READ  IN  FROM  17  INPUT  FILE  —  ! 8 2 . 0 0 0  F 0 - ' ' « r i F 1 3 . 5 . ? F 1 2 . 3 . F 1 5 . 5 . E l 3 . S . E 1 7 . 5 . F 1 3 . 5 . 3 x . 2 F 1 3  1  1  5 )  1*5. &  YQL.TA.GE  PHASE  AS  A£1ER. /»171  53 GO uu  0 155  TO  18«,000  0156  38  TO  19  36  TRANSFER 57  190,000  OMDECi0KDEC*10,O0 GO  c  FUNCTION  FROM  FREQUENCY  TO  TIME  OOMAlN  « * * * « * * « * « « * *  CONTINUE 18  19  HELTA  T  OF  OUTPUT  01SR  19  1  rOllTl=?60,E-6  0159  I=TOUTl  0160  DTOUT.T  FO"MAT(<0niiTP|'T  FUNCTION  t  OFI  T A T * > . F 1 2 5 . • . TNT T I A !  =  1,  2 0 1 0 0 0  050  201,100 T.F.  AT. Z F ^ O  FRED  201.150 2 0 1,200  c  201.250  0162  TFO=.0896522 c  TFo=  20 1 , 5 0 1 ' 2 0 1 7 p 11  0370565  GALLOWAV'S  FORMULA  ON  •SKIN >  OECMEAsE  T . F ,  c  2 0 1.760 20 1.820 2 0 1 80 0  c 0163  A ( 1 ) =TFO*A12«(1,00/ALPHAl-1.00/ALPHA2)  nil  PHI(li=0,O0  0 165  WRTTFC6,55)A(iJ.PHIf1) KI:p  0 166 016*  TIME  201.020  c  0 1 67  t  201  c  1  00  9 . 0 0 0  201.000  TFfl=n.999603  c  0  7 , 0 0 0  2 0 0 , 0 0 0 V 2 f T l  1E12.5) C  ij 0 0  OH,0  19  WRITE(6,58) 58  c  4 , 0 0 0  196,000  DTOi!T = , 2 5 0 - 0 6  c  19.3,0 00 19 5  DTOUT  0157  0161  1 , 0 0 0  1 9 ? , 0 0 11  nUTPUl  r. C  0 0 0  189,000  .  0151  o n 0  187.000  ISTnHF. = I S T O R E * l  c 0153  r  18 6 , 0 0 0  wRlTEf6,51)HERTZ,A*,B«,Sl,S2,XTF,PTF,A(IST0PE),PHICIST0RE)  0152  8 3  181.000  PrilfIsT0RF)sS2+PHl(lSTORE) _C_ _i_i..P.H.l.. « a » _ . B E C Q * E S Ji.U.lPJtT  0151  9 , 0 0 0  1 8 0 , 0 0 0 181,000  A( 1R T O H F ) = S 1 * A f I S T O R E )  0150  63  AW=0,00 Bw=n,nn  {  0 0 0  » H 2 o m  TG=  P004  20  2 . 0 0 0  2 0 3 . 0 0 0 20 4 0 0 0 205,000 206,000 207,000  — .  O c a  o o -3  c c o -  o o 0  o o o  o o o  o c o  _a a - o a c —  —, f\j r o cj' _r.  AJ  AJ AJ  A_  AJ  A_ AJ  o c o  o c o  o c o  o e o  o c o  o o o  c o o a c c. c- a o o c o  c o o  o c o  H c r~-l t o o- o « — — r\) A J A , r\J A j A j A J f\J A J A J r y o j f\j ( \ r\i o j rvj r u r\j r\j r\j  r\j m  a\  o o C  cl  o o o o c o a c. o c C: Ct © O w  C f\J ! * l 7 Lfl r« Ki y i M rn t\, r u r j r \ fV f\J \ M  (\  o  o o  o o  O  O  a o o q o c a c-co O O O  (\i ^  r\i  ^3  r y fvi  m -c T  ^  3  AJ  Ai  (V  o c o  o o a  c c o  o cf o o c o o o c j cz> o  o c a  o o c c z> o  cJ o cj o c o  cr  N co a - - i\i K K i l ^ l/l O > ^ i / i i n i r j i _n L A ufl t o L O i ^ 3 r\j r\i o j rvj (\j A J AJ AJ A i A i rvi fxi  o o o  a o o  o o o  c o cl o oJ  cr c - -i A J r o -c o -o - o A i AJ A J A i A.'  c  3  CJ  oLA ur-  cr o-i IT LO c* or- r -  O  r - r - r*-|  <r> <r u i  Ki K i K, X) ' O £Cj A J A. A j C -C o I I I ! Aj Aj A J U Li U ' II II II A.' AM A J U U J 1  1  •C -X3  i n. ! 'ii ! *  a. •-  U_  — _J +  -  i— o  '-_> •  O  O  Si o  c  M  3CI A J  tC  ^ AJ  -t  f-  X,  x U  >-—.__;__.—  co x i . ' — t it — a_ < c w)  U 'Ji  AJ AJ A_ L. t-> w U- U-i  II — A JLu  d  2 CO Z\ (O C || Aj CT,  O C II II AJ O ccj 1  C 'I _i I  c rJ  o O  O C! o  o  o  o  c  o! o  o  o O  c > t_> O • t-j  *r u s c  «  H  UJ C  O'f  c  ^-j Lr. -£)  :  ai  » - II •—  c  <l C  Ui  < O  C  jJ <  tl> l i • — —i 3 : CO t_> 'JO ( II  -O  o  w  a —  1  LA  d  o  cc eC Oj  II: II LO II < • Lt, (_) II CL a — co l_J c e o II w c i __: < II II •- c- _' — a . c/> x o a u_ tn II i O C « l 0-1  AJ K i  c  c  co 3  o  ^  O  Ol o  AJ C AJ o -=\ o  Kl C Aj A i o oi  ~c  tfi C o AJ AJ o c  t ? CT o o A j AJ A J O O Oi  K) 3",  LA  -iJ H  (O cr- o — — l AJ A J A J AJ  UJ  AJ  MICHIGAN  TERMINAL  SYSTEM  -200-  _0222..  00  FORTRAN  MAIN  G ( H 3 3 6 )  ! = *,r»  210-  210  0 22U 022S 0226  220  PHI  f11 = 0 . 0 0  r-o  m  26o,000 267.000 26H,OOO  16  269.000  S 1 = 1 .riO  0227  S2=0,00  0229  GO.  022<S  TO  270.000  53  27 1 , 0 0 0 -  -  272.000  END  .OPTIONS  IN  EFFECT*  • OPTIONS  TN F F F F T T *  . S T A T I S T I C S *  In,EBCDIC,SOURCE,NOLIST.NODECK,LOAD,NOMAP NAMF  S O U R C E  • S T A T I S T I C S *  NO  POOb  -265.00 0 -  A(I)=n,no  0223  RAGE  12:03:25  03-23-77  S  • ,  MATH  S T A T E M E N T S  D I A G N O S T I C S  I I N E P T = 229,PROGRAM  _  __) SIZE  79861  G E N E R A T E D  N O - t P - R O . R . S . I H . . ._ A I N No  S T A T E M E N T S  F Y F C I I T T O N  $R  2= .SOURCE*  -LOAD  EXECUTION 9  f  TI"E  T H E  ABOVE  C O M P I L A T I O N S ,  6=*SINK*  7 - F . N D A L P H A 1 28  O^OoF-06  O - 5 0 F - 0 1  3-TFJAPAN2  0  tOnF.,00  L I N E 1 / 1 0 0 0 MS IMPI/T I N P U T - . V O L T - A G E . - T I I " E _ - C y n S . T - A N T . S : . _ A L P - H A 1.-5  «**TTMF INPUT  IN  BEGINS  30  JAPAN  F L A G G E D  T F R M T N A T F O  CONSTANT  TIME  STEP  •VOLTAGE  TM  AREA  0  OFTAy  FROM  o  OF  Tn  5  MS  INPUT  0 . . . t . 7 _ l _ ! . 0 _ _ _ l . 3 _ . 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