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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 INES by it1 LEE KAI-CHUNG 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 THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENT FOR THE DEGREE OF MASTER OF APPL IED SCIENCE i n t h e Depa r tment o f E l e c t r i c a l E n g i n e e r i n g We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BR IT ISH COLUMBIA A p r i l , 1977 (c)- Lee Kai -Chung , 1977. In presenting this thesis in partial fulfilment o f the requirements fo 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 reference and study. I further agree that permission for extensive copying o f t h i s t h e s i s f o r scholarly purposes may be granted by the Head of my Department or by his representatives. It i s understood that copying or 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 Brit ish Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 ABSTRACT The propagation of the switching surge wave front on multiphase power l ines was investigated by modal analysis and conventional Fourier Transformation. A 500 kV untransposed, three-phase transmission l ine , for which f i e l d test results were available, was chosen as a test case. Phase A of this test l ine was excited from a double exponential voltage source and the voltage response at the receiving end was calculated and measured in a l l three phases. The calculated voltage a r r i v a l time mat-ched closely the measured value, and was very close to the time taken by electromagnetic waves in 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 results (errors within TABLE OF CONTENTS ABSTRACT . . • i i TABLE OF CONTENTS . i i : L L I ST OF ILLUSTRATIONS v ACKNOWLEDGEMENTS v i -INTRODUCTION 1 CHAPTER I COMPUTATION OF L INE CONSTANTS 4 . I n t r o d u c t i o n 3 2 . T r a n s m i s s i o n L i n e D a t a 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 o f S k i n E f f e c t i n C o n d u c t o r s 10 5. Ou tpu t f r o m L i n e C o n s t a n t s P r o g r a m 13 6. P o s i t i v e and Zero Sequence P a r a m e t e r s 19 CHAPTER I I COMPUTATION OF TRANSFERIFUNCTION FOR FREQUENCY RESPONSE OF TEST L INE 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 Theo r y Used i n t h e T r a n s f e r F u n c t i o n P r o g r a m 22 3. I n c l u s i o n o f Boundary C o n d i t i o n s a t S e n d i n g End . . . . 28 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 . 32 CHAPTER I I I TIME RESPONSE OF TEST L INE THROUGH FOURIER TRANSFORMATION i L . I n t r o d u c t i o n 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 V o l t a g e f r o m Time t o F r e q u e n c y Domain 36 i i i 3. Ou tpu 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 ime Domain by N u m e r i c a l I n v e r s e F o u r i e r T r a n s f o r m a t i o n . . . . . 41 "5. N u m e r i c a l A s p e c t s o f F o u r i e r T r a n s f o r m a t i o n P r o g r a m 43 CHAPTER IV DUPLICATION OF F IELD TESTS . D o u b l i n g E f f e c t on Open-Ended L i n e 49 !?• C o m p a r i s o n W i t h F i e l d Measurements and O t h e r S i m u l a t i o n R e s u l t s 50 CHAPTER'V CONCLUSIONS V 54 APPENDIX 1 T r a n s f e r F u n c t i o n P r o g r a m A L i s t i n g s . 55 APPENDIX 2 Pha se Smooth ing Program,. L i s t i n g s 61 APPENDIX 3 F o u r i e r T r a n s f o r m a t i o n P r o g r a m . . L i s t i n g s 62 BIBLIOGRAPHY 6 8 i v / L I ST OF ILLUSTRATIONS F i g u r e Page 1. O v e r a l l scheme o f programmes u sed . . 2 2. T r a n s m i s s i o n l i n e geomet ry 4 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 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 o f 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 t u b u l a r c o n d u c t o r f o r m u l a 11 5. E l e m e n t s o f t h e r e s i s t a n c e m a t r i x o f t h e 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 t u b u l a r c o n d u c t o r , f o r s k i n e f f e c t . . . 17 6. E l e m e n t s o f t h e r e a c t a n c e m a t r i x o f t h e 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 t u b u l a r c o n d u c t o r , f o r s k i n e f f e c t 18 7. Change i n s equence r e s i s t a n c e due t o change i n c o n d u c t o r b u n d l e r e s i s t a n c e 20 8. T r a n s m i s s i o n l i n e c o n f i g u r a t i o n w i t h b o u n d a r y c o n d i t i o n s 23 9. M a g n i t u d e o f 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 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 f o r m u l a 33 10 . Pha se o f t r a n s f e r f u n c t i o n s ( I d e n t i c a l r e s u l t 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 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 f o r m u l a ) 34 1 1 . L i n e a r i n t e r p o l a t i o n o f i n p u t v o l t a g e i n t i m e domain . . 38 12 . L i n e a r i n t e r p o l a t i o n o f o u t p u t v o l t a g e i n t i m e domain . . 38 13 . 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 H (u ) = 1.0 Z.0° 45 14 . Same t e s t as i n F i g . 13 f r o m 0.1 t o 15 ms 46 15 . 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 . . 52 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 measurement and o t h e r s i m u l a t i o n r e s u l t s . . . 55 v ACKNOWLEDGEMENT I w o u l d l i k e t o t h a n k my t h e s i s s u p e r v i s o r , P r o f e s s o r Hermann W. Dommel f o r s u p p l y i n g 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 , and f o r c o u n t l e s s h o u r s o f d i s c u s s i o n s d u r i n g t h e r e s e a r c h work and w r i t i n g up o f t h i s t h i s t h e s i s . A l s o , I w o u l d l i k e t o show a p p r e c i a t i o n t o D r . Dommel f o r I p r o v i d i n g t h e L i n e C o n s t a n t s P r o g r a m and an i n i t i a l v e r s i o n o f t h e F o u r i e r T r a n s f o r m program. I w o u l d a l s o l i k e t o t h a n k t h e t h e s i s c o - r e a d e r , P r o f e s s o r Malcome D. Wvong f o r p r o o f - r e a d i n g t h i s t h e s i s and o f f e r i n g f u r t h e r s u g g e s t i o n s . I am a l s o g r a t e f u l t o M r . K i n g K. T se f o r p r o v i d i n g t h e i n i t i a l v e r s i o n o f t h e T r a n s f e r F u n c t i o n P r o g r a m and h e l p f u l d i s c u s s i o n s about t h e programme d u r i n g t h e e a r l y p a r t o f t h e r e s e a r c h . The f i n a n c i a l s u p p o r t 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 i s a l s o g r a t e f u l l y a c k n o w l e d g e d . S p e c i a l t h a n k s a r e e x p r e s s e d t o M r s . Semmens i n t h e E l e c t r i c a l E n g i n e e r i n g ma i n o f f i c e f o r p r o d u c i n g t h i s e x c e l l e n t l y t y p e d t h e s i s f r o m my s e m i - l e g i b l e h a n d w r i t i n g . F i n a l l y , I am endeb ted t o my p a r e n t s and my w i f e f o r t h e i r t i r e -l e s s encouragement and p a t i e n c e . v i 1 INTRODUCTION 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 on s i n g l e c i r c u i t m u l t i p h a s e t r a n s i s s i o n l i n e s o r unde r g r ound c a b l e s was i n v e s t i g a t e d . The p u r p o s e o f t h i s work was u s e f u l f o r 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 y . The s o l u t i o n methods we re 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 u n t r a n s p o s e d 1 2 * o v e r h e a d l i n e , 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 i s l i n e i s p a r t o f t h e 500 kV Azumi T r a n k t r a n s m i s s i o n l i n k o f t h e Tokyo E l e c t r i c Power Sy s tem i n J a p a n . I n t h e f i e l d t e s t s , t h e s e n d i n g end o f t h e l i n e was e n e r -—Ct i t —Ct 9 1 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 1 - e " ) , as a r e p r e s e n t a t i o n o f s u r ge phenomenon! on a l i n e e g . l i g h t n i n g s u r g e , 3 f r o m an i m p u l s e g e n e r a t o r t h r o u g h a s e r i e s r e s i s t a n c e o f 415f i . I n t h e compute r s i m u l a t i o n , t h i s d o u b l e e x p o n e n t i a l i n p u t wave as w e l l a s o t h e r f o rms o f i n p u t v o l t a g e s , s u c h as s i n g l e e x p o n e n t i a l decay wave , t r i a n g u l a r wave s , s t e p wave and d e l t a wave we re a l s o s t u d i e d . The way i n w h i c h compute r programmes we re u sed f o r t h e a n a l y s i s i n t h i s t h e s i s i s i l l u s t r a t e d i n F i g . 1. The L i n e C o n s t a n t s P r o g r a m (LCP) was f i r s t u sed t o g i v e t h e d i s t r i b u t e d l i n e p a r a m e t e r s f r o m t h e tower g e o -m e t r y 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 s a f u n c t i o n o f f r e q u e n c y . Then , t h e T r a n s f e r F u n c t i o n P r o g r a m . (TFP) was u sed t o o b t a i n t h e o u t p u t v o l t a g e a t t h e r e c e i v i n g end (83 .212 km f r o m s e n d i n g end) f o r a l l f r e q u e n c y p o i n t s . A f t e r t h e t r a n s f e r f u n c t i o n s a t d i s c r e t e f r e q u e n c y p o i n t s w e r e o b t a i n e d , t h e F o u r i e r T r a n s f o r m P r o g r a m (FTP)was u sed t o f i n d t h e v o l t a g e a t t h e r e c e i v i n g end as a f u n c t i o n o f t i m e f o r any f o r m o f i n p u t v o l t a g e . I n t h e F o u r i e r T r a n s f o r m P r o g r a m , l i n e a r i n t e r p o l a t i o n be tween * The s u p e r s c r i p t s d e n o t e r e f e r e n c e numbers i n t h e b i b l i o g r a p h y . s u c c e s s i v e d a t a p o i n t s we re u sed i n t h e t i m e doma in as w e l l a s i n t h e f r e -quency doma in . F o r t h e 500 kV l i n e u sed as an e xamp le , a d e n s i t y o f 20 p o i n t s p e r decade 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 r e s u l t s . The c o m p u t a t i o n o f t h e v o l t a g e r e s p o n s e i n t h r e e pha se s a t t h e r e c e i v i n g end w i t h any one o f t h e pha se s e n e r g i z e d a t t h e s e n d i n g end w o u l d t a k e abou t 80 s . CPU t i m e on t h e UBC IBM 370/168 computer s y s t e m a t a c o s t o f a p p r o x i m a t e l y GC$30. F o r a more g e n e r a l c a s e o f t h r e e i n p u t v o l t a g e s on a l l t h r e e p h a s e s , o n l y a s l i g h t m o d i f i c a t i o n i n t h e T r a n s f e r F u n c t i o n P r o -gram it w o u l d be r e q u i r e d t o o b t a i n t h e r e s u l t s . Tower geomet r y 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 vk L i n e C o n s t a n t s P r o g r a m 4 S e r i e s impedance m a t r i x as a f u n c t i o n o f f r e q u e n c y , ^constant s hun t c a p a c i t a n c e m a t r i x . T r a n s f e r F u n c t i o n P r o g r a m \ T r a n s f e r f u n c t i o n a t d i s c r e t e f r e q u e n c i e s F o u r i e r T r a n s f o r m P r o g r am it O u t p u t v o l t a g e s i n t i m e doma in f o r a l l 3 pha se s TIT— I END! F i g . 1. O v e r a l l scheme o f p r o g r a m ~" u s e d . 3 CHAPTER I COMPUTATION OF L INE CONSTANTS 1'.) I n t r o d u c t i o n The p r o g r a m w h i c h was u sed f o r t h e c a l c u l a t i o n o f l i n e p a r a m e t e r s i s a m o d i f i e d v e r s i o n o f t h e L i n e C o n s t a n t s P r o g r a m w r i t t e n b y H. W. 4 5 Dommel , ' . I t c a l c u l a t e s t h e f r e q u e n c y - d e p e n d e n t s e r i e s impedance m a t r i x and t he c o n s t a n t s h u n t c a p a c i t a n c e m a t r i x f o r o v e r h e a d l i n e s f r o m t h e g i v e n t owe r geomet ry 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 . F o r t h e a n a l y s i s o f u n d e r g r o u n d c a b l e s , t h i s p r o g r a m w o u l d have t o be r e p l a c e d by a c a b l e c o n s t a n t s p r o g r a m . The v a l u e o f t h e impedance and c a p a c i t a n c e 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 P r o g r am, t o o b t a i n t h e t r a n s f e r f u n c t i o n s a t t h e s p e c i f i e d f r e q u e n c y p o i n t s . 2. T r a n s m i s s i o n L i n e D a t a The t r a n s m i s s i o n l i n e u s ed as an example f o r t h i s s i m u l a t i o n s t u d y i s p a r t o f t h e 500 kV A z u m i - T r a n k t r a n s m i s s i o n l i n k o f t h e Tokyo E l e c t r i c 1 2 Power Sy s tem i n J a p a n ' . T h i s i s a t h r e e - p h a s e u n t r a n s p o s e d l i n e w i t h two g round w i r e s . E a ch pha se i s a b u n d l e c o n d u c t o r w i t h f o u r s t e e l - r e i n f o r c e d a luminum c a b l e s ( s ee F i g . 2 ) . The t owe r geomet ry 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 r e l i s t e d i n T a b l e 1. 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 we re t a k e n f r o m t h e German s t a n d a r d DIN 48204 b e c a u s e t h e y we re n o t d e f i n e d i n enough d e -4 5 t a i l i n t h e d e s c r i p t i o n o f t h e f i e l d t e s t s ' . 4 ©....llrn.-.^ O } 2 ground wires 10m(AH) 25m (AH) i } 3 phase conductors (AH) average height // -1 > ,V >>> >/i i a > in i r i i rt in II HI a 111 r m,m nn / Ground re s i s t i v i ty = 200 fj-m a) Tower geometry (height is average height above ground, not height at tower location). b) Bundle conductors of each phase O Aluminum strands ® Steel strands c) 26 A l . / 7 s t . Steel-reinforced aluminum cable used for ground wires and phase conductors F i g . 2 Transmission l ine geometry 5 TABLE I TRANSMISSION L INE DATA G e n e r a l d a t a L e n g t h o f t r a n s m i s s i o n l i n e = 83 .212 km.. A v e r a g e h e i g h t above g round o f t h r e e pha se c o n d u c t o r s ( f l a t c o n f i g u r a t i o n ) = 25 m A v e r a g e h e i g h t above g round o f g round w i r e s ' = 35 m E a r t h r e s i s t i v i t y ( p r e sumab l y f a r m l a n d ) = 200 ft.m —8 R e s i s t i v i t y o f a luminum = 3.21 x 10 Q.m R e l a t i v e p e r m e a b i l i t y o f a luminum (y ) = 1 . 0 —8 P e r m e a b i l i t y o f a luminum ( y o y r ) = 4?r x 10 H/m D e t a i l s f o r g round w i r e Type : S t e e l - r e i n f o r c e d a luminum c a b l e , a s shown i n F i g . 2 c . T o t a l n o . o f a luminum s t r a n d s =.26 No . o f a l uminum s t r a n d s i n o u t e r l a y e r o f c o n d u c t o r = 16 S t e e l c o r e d i a m e t e r = 5.85 mm O u t s i d e d i a m e t e r o f c o n d u c t o r N = 15 .7 mm D.C. r e s i s t a n c e a t 20°C = 0.262 ft/km D e t a i l s o f pha se c o n d u c t o r Type : S t e e l - r e i n f o r c e d a luminum c a b l e a s 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 ach pha se as shown i n F i g . 2b T o t a l n o . o f a luminum s t r a n d s = 26 No . o f a l uminum s t r a n d s i n o u t e r l a y e r o f c o n d u c t o r = 16 S t e e l c o r e d i a m e t e r = 8.1 mm O u t s i d e d i a m e t e r o f c o n d u c t o r = 21.7 mm D.C. r e s i s t a n c e a t 20°C = 0.136 ft/km 6 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 2 ( i ) S e r i e s impedance m a t r i x - C a r s o n ' s f o r m u l a ' i s u s ed f o r c a l c u l a t i n g t h e impedances o f t h e c o n d u c t o r e a r t h r e t u r n l o o p s . E a r t h c o n -d u c t i v i t y i s 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 assumed t o be f l a t and p a r a l l e l t o t h e c o n d u c t o r s . A l s o , s p a c i n g s be tween c o n d u c t o r s a r e assumed t o be l a r g e compared w i t h c o n d u c t o r r a d i i , t h a t i s , p r o x i m i t y e f f e c t s a r e i g n o r e d . The e l e m e n t s o f t h e impedance m a t r i x t ^ ^ ] a r e g i v e n as Z i i ^ = ( R i i + A R i i } + j ( 2 u l ° n GMB^ + A X i i > f l / k m i = 1, . . . N ( 1 -1 ) and Z . . = Z . . S. . = AR. . + j ( 2 u l 0 In - i 1 + AX .)n/km j = 1 , . . . N ; i = 1, . . . N ; i f j , ( 1 - 2 ) t h where R ^ = r e s i s t a n c e o f i c o n d u c t o r i n fi/km ( see s e c t i o n 4 . on s k i n e f f e c t ) t h h^ = a v e r a g e h e i g h t above g round o f i c o n d u c t o r i n m, t h S^ _. = d i s t a n c e be tween i c o n d u c t o r and g round image o f t h j c o n d u c t o r i n m ( see F i g . 3 ) , s . . = d i s t a n c e be tween 1 ^ and j1"*1 c o n d u c t o r s i n m ( see F i g . 3 ) , GMR^ = g e o m e t r i c mean r a d i u s o f i * " * 1 c o n d u c t o r i n m, 03 = a n g u l a r f r e q u e n c y , AR = c o r r e c t i o n t e rms 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 e f f e c t , AX = c o r r e c t i o n te rms 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 e f f e c t . C a r s o n ' s c o r r e c t i o n te rms AR and AX a r e f u n c t i o n s o f t h e a n g l e 7 Fig . 3 Line parameter calculation 8 <f>_ (see F i g . 3) and of the parameter a = ks IP r- -4 where k = 4TT/5 X 10 .f^h^ for self impedance S | S . . for mutual impedance p = earth re s i s t i v i ty in ft.m f = frequency in Hz For numerical calculations, Carson's integral for AR and AX has been deve-4 loped into an in f in i te series , which is used for a < -5, -4 ir 2 2 AR' = 4wl0 -hr - b..a cos $ + b 0 [ ( c 0 - lna)a cos 2<f> + (j>a s in 2$] o I Z Z 3 4 + b^a cos 3<f> - d^a cos 4<j> - b c a^ cos 5c() + b , [ ( c £ - lna)a^ cos 6<|) + $a^sin 6<j>] J D O 7 8 + b^a cos 7<f> - d Da cos 8<f> - . . . . } (l-3a) AX' = 4colO~4{y (0.6159315 - lna) + b ^ cos <}> - d 2 a 2 cos 2cp 3 4 4 + b^a cos 3cp - b^[(c^ - lna)a cos 4c() + <j>a sin 4<j>] + b^a^ cos 5<j>- d a^ cos 6<j> + b 7a'' cos 7<J> -> 6 ' 8 8 - b g [ ( C g - lna)a cos 8<j> + <f) a sin 8 <j>] + . . . . } (l-3b) where b. , cy. and d. are constants given be i l l /2 b^ = -g for odd. subscripts i b„ = - i - for even subscripts i Z ±o ' ^ t ! 0 , < ™ _ 1 = 1.2,3,4; 9,10,11,12; . . . with sign - . = 5 j 6 , 7 f 8 ; 1 3 f l 6 j l 5 f l 6 ; . . . 9 and c 2 = 1.3659315 C l = C l - 2 + I + 1T2> 1 > 2 and d . = -?- b . I , 4 I N o t e t h a t f r o m e q t n s ( l - 3 a ) and ( l - 3 b ) , e a ch 4 te rms i n i = 1,4 f o r m a r e p e t i t i v e g roup 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 ' i s u s e d , i n s t e a d o f t h e i n f i n i t e s e r i e s - 4 A V - / c o s <t> cos 3t)> , 3 co s 5j> 45 cos 7ck 4colO n / ^ a 3 + 3 + "7 ^ 7T- U a a a v2 r- - 4 ,cos cj) _ /2 co s 2<j> co s 3<j> 3 c o s 5d> 45 c o s 7<K 4colO A K = K a 2 + 3 5 7 ; / T a a a a v2 ( l - 4 b ) N o t e t h a t t h e i n f i n i t e s e r i e s f o r R and X d e r i v e d f r o m C a r s o n ' s i n t e g r a l s w i l l o n l y c o n v e r g e a f t e r abou t 10 o r more te rms i f a > 3. The f i r s t few te rms a r e h i g h l y o s c i l l a t i n g i n t h a t c a s e . ( i i ) Shun t c a p a c i t a n c e m a t r i x [C] - The c a p a c i t a n c e m a t r i x [ C ] i s t h e i n v e r s e o f t h e p o t e n t i a l c o e f f i c i e n t m a t r i x [ P ] . [C] = [ P ] " 1 The m a t r i x e l emen t o f [P] c an e a s i l y be o b t a i n e d f r o m t h e tower g e o m e t r y , i i " " ~ r ? , 2h . P . . = 2C x 10 I n — - km/F ( 1 - 5 ) i v and P = 2 c 2 x 1 0 ~ 4 I n km/F ( 1 - 6 ) where r^ = r a d i u s o f c o n d u c t o r i n m c = v e l o c i t y o f l i g h t i n km/s E q t n s ( 1 - 5 ) and ( 1 - 6 ) a r e v a l i d a s l o n g a s ^ ( 0 . 0 2 m i n t h e example ) i s much s m a l l e r t h a n s p a c i n g s be tween c o n d u c t o r s (14 m i n t h e 10 e x a m p l e ) . N o t e t h a t t h e e l e m e n t s o f t h e s hun t c a p a c i t a n c e m a t r i x a r e o n l y dependen t on t h e t ower geomet ry and a r e n o t dependen t on f r e q u e n c y . T h i s i s 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 f r e q u e n c i e s up t o a p p r o x i m a t e l y 1 MHz, g where e a r t h c o r r e c t i o n te rms f o r c a p a c i t a n c e s a r e n o t y e t i m p o r t a n t . 4 . C a l c u l a t i o n s o f S k i n E f f e c t i n C o n d u c t o r s The s k i n e f f e c t i n t h e e a r t h r e t u r n i s a c c o u n t e d f o r by C a r s o n ' s f o r m u l a . W h i l e t h e e a r t h r e t u r n s k i n e f f e c t has a m a j o r i n f l u e n c e on l i n e p a r a m e t e r s , 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 must a l s o be c o n s i d e r e d a t h i g h e r f r e q u e n c i e s . As f r e q u e n c y i n c r e a s e s , t h e c u r r e n t f l o w s more and more on t h e s u r f a c e o f t h e c o n d u c t o r . T h i s can be d e s c r i b e d by t h e n o m i n a l d e p t h 9 o f p e n e t r a t i o n o f c u r r e n t (6) a s g i v e n by 6 = / C •n-f y where Pc= r e s i s t i v i t y o f c o n d u c t o r m a t e r i a l i n fi.m y = a b s o l u t e m a g n e t i c p e r m e a b i l i t y i n H/m f = f r e q u e n c y i n Hz S i n c e t h e c u r r e n t i n c o n f i n e d t o t h e s u r f a c e o f t h e c o n d u c t o r a t h i g h f r e -q u e n c i e s , t h e c o n d u c t o r r e s i s t a n c e i n c r e a s e s and t h e i n t e r n a l i n d u c t a n c e d e c r e a s e s w i t h f r e q u e n c y ( see F i g . 4 ) . I n e q t n ( 1 - 1 ) , t h e s e l f i n d u c t a n c e m a t r i x e l e m e n t _ , 2h . L = 2 x l ( f a I n H/icm (1 -8 ) i i s t h e r e s u l t a n t o f 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 , i . e . r,.1' 2 h . L . . = 2 x 10 I n 7 ^ — + 2 x 10 I n — - H/km I X GMR. r . ( 1 -9 ) 1 x The f i r s t and second t e r m i n e q t n ( 1 - 9 ) i s due t o t h e f l u x i n s i d e and o u t -s i d e t h e c o n d u c t o r , r e s p e c t i v e l y . N o t e t h a t t h e f i r s t t e r m ( i n t e r n a l i n -d u c t a n c e ) i s s m a l l compared t o t h e s econd t e r m f o r h i g h v o l t a g e o v e r h e a d l i n e s a t l o w f r e q u e n c i e s 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 f r e q u e n c i e s . '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 f o r m u l a 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^(f i/km) I n d u c t a n c e L (H) 6 F i g . 4 12 Thu s , t h e s k i n e f f e c t on t h e t o t a l i n d u c t a n c e i s n o r m a l l y n e g l i g i b l e o v e r t he e n t i r e f r e q u e n c y r a n g e . However , 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 p r o n o u n c e d . The s k i n e f f e c t on t h e c o n d u c t o r r e s i s t a n c e and 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 two way s . W i t h t h e f i r s t me thod , t h e c o n d u c t o r was t r e a t e d as a s o l i d t u b u l a r a luminum 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 a c t u a l c o n d u c t o r . 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 . T h i s was r e c o m * mended as a r e a s o n a b l e a p p r o x i m a t i o n . The f o r m u l a f o r t h e i n t e r n a l impedance 10 4 o f a t u b u l a r c o n d u c t o r o f nonmagne t i c m a t e r i a l i s ' ^ i n t e r n a l = ^ c o n d u c t o r + ^ ^ i n t e r n a l dc dc •1 n _ 2>> ( be r ( m r ) + j b e i (mr) )+$> ( k e r ( m r ) + 1 k e i (mr) ) j - mr( . l S ) r K „ _ t /•„.-. \ 4.11V^J.I ( W ^ + r h rVpr' fmrHikpi ' ft ( b e r ' ( m r ) + 3 b e i ' (mr))+cj> ( k e r ' ( m r ) + j k e i ' (mr)) (1 -10 ) b e r ' (mq)+,jbei ' (mq)  w h e r e * = " k e r ' ( m q ) + j k e i ' ( m q ) R, = d . c . r e s i s t a n c e o f c o n d u c t o r i n ft/km Mc r = o u t s i d e r a d i u s o f c o n d u c t ® r > i n m q = o u t s i d e r a d i u s o f s t e e l c o r e i n m s r (mr) = ( - ^ - j ) k q / 2 1-s.' and 2 t \ /ks, N l / 2 (mq) = (.,_ 9 ) 1-s 8 i r l 0 ~ 4 f The e x p r e s s i o n s b e r ( . . . ) + j b e i ( . . . ) , b e r ' ( . . . ) + j b e i ( . . . ) , k e r ( . . . ) + j k e i ( . . . ) and k e r ' ( . . . ) + j k e i ' ( . . . ) a r e m o d i f i e d B e s s e l f u n c t i o n s , w h i c h 11 4 can be e v a l u a t e d by p o l y n o m i a l a p p r o x i m a t i o n ' . An e m p i r i c a l f o r m u l a 13 f o r c o n d u c t o r r e s i s t a n c e and 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 d e v e l o p e d 12 by G a l l o w a y , w h i c h i s b a sed on measurements i n t h e e l e c t r o l y t i c t a n k . I n t h i s a p p r o a c h , c u r r e n t i s 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 s t r a n d s (16 s t r a n d s i n o u t e r l a y e r i n t h e example ) . I n t e r n a l r e s i s t a n c e ( R £ ) and i n t e r n a l r e a c t a n c e (X^) a r e t h e n e q u a l . W i t h t h i s f o r m u l a , we o b t a i n K /coup R 0 jr = L _ N / K M ( 1 _ 1 ; L ) ° /Z r ir (2+n) where CJ = a n g u l a r f r e q u e n c y y = p e r m e a b i l i t y o f c o n d u c t o r m a t e r i a l (H/m) = r e s i s t i v i t y o f c o n d u c t o r m a t e r i a l (fi-m) r = o u t e r r a d i u s o f c o n d u c t o r i n (m) n = no.- o f s t r a n d s i n o u t e r l a y e r ( s ee F i g . 2c ) K = 2 . 2 5 , f a c t o r due t o s t r a n d i n g R e s u l t s f o r t h e i n t e r n a l impedance c a l c u l a t e d w i t h t h e above two a p p r o a c h e s a r e shown i n F i g . 4. A t l o w e r f r e q u e n c i e s , where s k i n e f f e c t i s n o t y e t 4 p r o m i n e n t , t h e r e s u l t s f o r t u b u l a r c o n d u c t o r s a r e f a i r l y a c c u r a t e . A f t e r a c r o s s - o v e r p o i n t a r ound 130 H z , t h e r e s u l t s f r o m G a l l o w a y ' s f o r m u l a a r e 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 t h e s k i n e f f e c t i n t h e i n -d i v i d u a l s t r a n d s o f t h e o u t e r l a y e r i n t o a c c o u n t . The d o t t e d l i n e i n F i g . 4 t hu s 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 . I t s h o u l d be n o t i c e d t h a t l i n e p a r a m e t e r s o f o v e r h e a d l i n e s a r e s e l dom measured as t h e computed r e s u l t s a r e u s u a l l y s u f f i c i e n t l y a c c u r a t e . 5_i O u t p u t f r o m L i n e C o n s t a n t s P r og r am F o r t h e t r a n s m i s s i o n l i n e o f F i g . 2, t h e r e a r e 14 c o n d u c t o r s , i . e . 4 c o n d u c t o r s p e r b u n d l e i n each o f t h e t h r e e pha se s and 2 g round w i r e s above . Thu s , i n i t i a l l y we have a 14 x 14 s e r i e s impedance and a 14 x 14 s h u n t 14 c a p a c i t a n c e m a t r i x V V 14 P P 1,1 1,2 ' 2 , 1 2,2 P P 14,1 14,2 1,14 2,14 14,14j 14x14 <14 (1 -12 ) and " d V 1 / d x " d V 2 / d x d V 1 4 / d x '2,1 ' 1 , 2 ' J 2 , 2 ••• ' Z 1 4 , l Z 1 4 , 2 , Z 1,14 J 2 , 1 4 ' 14 ,14 V 1 I n • 2 I 14 • * (1 -13 ) These 14 x 14 m a t r i c e s can be 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 by c o n -s i d e r i n g t h e b u n d l i n g c o n d i t i o n i n t h e 3 pha se b u n d l e c o n d u c t o r s , and t h e z e r o v o l t a g e c o n d i t i o n i n b o t h g round w i r e s . I f we d e n o t e c o n d u c t o r s i n phase A as 1 , 2 , 3 , 4 ; pha se B as 5 , 6 , 7 , 8 ; pha se C as 9 , 10 , 11 , 12 and b o t h g round w i r e s as 1 3 , 1 4 , t h e n f o r g round w i r e s 13 and 14 , we have dV — = 0 ( 1 -14 ) 13 dx dV 14 = 0 dx and v 1 3 = 0 V 1 4 = ° ( 1 -16 ) and f o r b u n d l i n g i n pha se A, we have 1 5 dx dV^ dx dV, dx " dV, dx dx (1 -16 ) I 1 + I 2 + I 3 + I, \ = v2 = v 3 = v 4 = v A Q x + Q 2 + Q 3 + Q 4 = Q A ( 1 -17 ) W i t h e q t n s ( 1 -15 ) and ( 1 - 1 7 ) , e q t n ( 1 -12 ) can be r e d u c e d t o 3 e q u a t i o n s w i t h t h e d e s i r e d 3x3 p o t e n t i a l m a t r i x [P ] " ' ' ^ , V A P A A P A B P A C Q A V B P B A P B B P B C % (1 -18 ) V . P C A P C B P c c . 3x3 The 3x3 s hun t c a p a c i t a n c e m a t r i x [C] i s t h e n o b t a i n e d by s i m p l e m a t r i x i n -v e r s i o n [ C ] 3 x 3 = ^ ^ 3 x 3 " S i m i l a r l y , e q t n s ( 1 -14 ) and (1 -16 ) c a n be u sed t o r e d u c e e q t n (1 -13 ) t o 3 e q u a t i o n s w i t h t h e d e s i r e d 3x3 s e r i e s impedance -m a t r i x [ Z ] . z . " d V A / d x Z A A Z A 3 d V B / d x = Z B A Z B B _dV c /dx - Z C A Z C B AC BC I A B T J 3 x 3 - C-i (1 -19 ) Thu s , w i t h C a r s o n ' s f o r m u l a and w i t h one o f t h e two s k i n e f f e c t f o r m u l a e f o r c o n d u c t o r s , 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 and g round w i r e s , we o b t a i n t h e 3x3 s e r i e s impedance and 3x3 s hun t c a p a c i t a n c e m a t r i c e s f o r t h e t h r e e p h a s e s . The 3x3 s hun 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 f r o m t h e tower geometry o f t h e t e s t example i s shown i n T a b l e 2. t h e e l e m e n t s o f t h e 3><3 s y m m e t r i c 16 series impedance matrix are shown as a function of frequency in 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 y F / k m 17 E l e m e n t s o f t h e r e s i s t a n c e m a t r i x o f t h e 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 c o n d u c t o r , f o r s k i n e f f e c t R e s i s t a n c e (fi/km) 1 0 J 7 -— 7 ~3 2f ' 5 10 10 103 10 10 F r e q u e n c y ( H z ) F i g . 5 18 E l e m e n t s o f t h e r e a c t a n c e m a t r i x o f t h e 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 c o n d u c t o r , f o r s k i n e f f e c t R e a c t a n c e u)L(fl/km) 2 3 4 5 10 10 l o 10 10 F r e q u e n c y ( H z ) F i g . 6 19 6. P o s i t i v e and Z e r o Sequence P a r a m e t e r s T h e r e a r e 3 modes o f TEM p r o p a g a t i o n on t h e 3 pha se t e s t l i n e . Each 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 i t s own i n d i v i d u a l c h a r a c t e r i -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 c o n s t a n t ( see l a t e r e q t n ( 2 - 8 ) ) . I f t h e t e s t l i n e was t r a n s p o s e d , w h i c h i t i s n o t , t h e n two o f t h e 3 modes w o u l d be c h a r a c t e r i z e d by p o s i t i v e sequence 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 be 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 . Thu s , by i d e a l i z i n g t h e g i v e n u n t r a n s p o s e d l i n e t o a t r a n s p o s e d one, we can l o o k a t t h e p o s i t i v e and z e r o sequence p a r a m e t e r s , w h i c h w i l l g i v e us some i n s i g h t i n t o t h e o v e r a l l e f f e c t o f b o t h a p p r o a c h e s f o r c o n d u c t o r s k i n e f f e c t c a l c u l a t i o n ( t u b u l a r c o n d u c t o r f o r m u l a and G a l l o w a y ' s f o r m u l a ) . F o r t h e t r a n s p o s e d l i n e , 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 z e r o sequence impedances (Z and Z ) t o t h e s e r i e s impedance m a t r i x e l e m e n t s po s z e r o a r e g i v e n by Z = Z - Z ( 1 -20 ) pos s m Z = Z + 2Z (1 -21 ) z e r o s m where Z and Z a r e t h e s e l f and m u t u a l impedance s , w h i c h i n t u r n a r e t h e s m a v e r a g e s o f 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 = 1 / 3 ( Z A A + Z B B + Z C C ) ( 1 " 2 2 ) Z m - 1 / 3 ( Z A B + Z B C + Z C A ) ( 1 " 2 3 ) where Z . . i s an e l e m e n t o f [ Z ] „ „ IJ J 3 x 3 The impedance s ' Z i n e q t n s ( 1 -16 ) and (1 -17 ) a r e t h e s e r i e s i m -pedance s m a t r i x e l e m e n t s shown i n F i g s . 5 and6 . The p o s i t i v e and z e r o s e -quence r e s i s t a n c e s a r e shown i n F i g . 7. I n F i g . 7, t h e r e s i s t a n c e o f t h e 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 G a l l o w a y ' s 20 Change in sequence resistance due to change in , . . „ . conductor bundle resistance istance (fi/km) Fig . 7 21 f o r m u l a a r e a l s o shown f o r c o m p a r i s o n . A t h i g h e r f r e q u e n c i e s , t h e c o n d u c t o r , r e s i s t a n c e f r o m G a l l o w a y ' s f o r m u l a i s abou t t w i c e as h i g h , ( s e e a l s o F i g . 4) as t h e v a l u e f r o m 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 . The i n c r e a s e i n c o n d u c t o r r e s i s t a n c e ( A R c ) shows up w i t h t h e same v a l u e as an i n c r e a s e i n p o s i t i v e s e -quence r e s i s t a n c e , ( A R ^ o g ) . However , t h e d i f f e r e n c e s i n c o n d u c t o r r e s i s t a n c e be tween t h e two f o r m u l a e do n o t show up w i t h e x a c t l y t h e same v a l u e i n t h e z e r o sequence r e s i s t a n c e . The d i f f e r e n c e i n z e r o sequence r e s i s t a n c e (AR ) i s s l i g h t l y h i g h e r ( e . g . 9% h i g h e r a t 50 KHz) due t o t h e a d d i t i o n a l e f f e c t o f t h e 2 e l i m i n a t e d g round w i r e s . A l s o n o t e t h a t t h e z e r o sequence r e s i s -t a n c e i s much h i g h e r t h a n t h e p o s i t i v e sequence r e s i s t a n c e . A t 50 KHz , t h e i n c r e a s e i n t h e p o s i t i v e sequence r e s i s t a n c e cau sed by t h e d i f f e r e n c e i n s k i n e f f e c t f o r m u l a e i s a bou t 17% whe rea s t h e i n c r e a s e i n t h e z e r o sequence r e s i s t a n c e i s o n l y abou t 1%. 22 CHAPTER I I COMPUTATION OF TRANSFER FUNCTION FOR FREQUENCY RESPONSE OF TEST L INE 1.. I n t r o d u c t i o n v A f t e r k now ing t h e s e r i e s impedance m a t r i x [ Z . . ] _ „ and t h e s h u n t a d m i t t a n c e m a t r i x w i t h z e r o c o n d u c t a n c e [ Y i j ] 3 x 3 " ^ ^ 3 x 3 o f t h e 3 -pha se t e s t l i n e , t h e t r a n s f e r f u n c t i o n be tween t h e i n p u t on one phase a t t h e s e n d i n g end and t h e o u t p u t on any one o f t h e t h r e e pha se s a t t h e r e c e i v i n g end c an be f o u n d . F o r t h e cho sen t e s t e x a m p l e , one p h a s e , d e s i g n a t e d A , i s e n e r g i s e d ( see F i g . 8 ) . The o u t p u t v o l t a g e i n t h e 3 -pha se a t t h e r e c e i v i n g end o f t h e 83.212 km l o n g t e s t l i n e was t o be f ound f o r a p e r i o d o f t i m e d u r i n g w h i c h t h e waves 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 f r o m the s e n d i n g end . Tha t i s , t h e p e r i o d o f i n v e s t i g a t i o n t i m e t i s T < t < 3 T where T i s t h e t r a v e l t i m e o f t h e mode w i t h h i g h e s t wave v e l o c i t y . - . The T r a n s f e r F u n c t i o n P r o g r am w h i c h p e r f o r m s t h e n e c e s s a r y c a l c u l a t i o n s i s an expanded v e r s i o n o f a p r og r am w r i t t e n by K.K. Tse"*"^ f o r a t e r m p r o j e c t i n EE 5 s 3 on t h e f r e q u e n c y r e s p o n s e o f B.C. H y d r o ' s M i c a Dam t r a n s m i s s i o n l i n e ( see A p p e n d i x 1 f o r p r o g r a m l i s t i n g s ) . g.. O u t l i n e o f t h e t h e o r y u sed i n t h e T r a n s f e r F u n c t i o n P r o g r am P r o p a g a t i o n o f waves 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 p a r a m e t e r s 17 18 i s 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 l i n e e q u a t i o n ' - r f § = IL] TR] [ i ] ^ ( 2 . 1 ) - [ f £ ] = IC] [f?3 + IG] [ v ] (2 .2 ) R ' 6 t=0 4 A £ V ' = V - i " R,. A ,g A o lllillllMiillUlUllUjIill Boundary c o n d i t i o n s i n f r e q u e n c y domain A t s e n d i n g end o o V = V - I. R,. A g A ° o o A t r e c e i v i n g end • S i n g l e i n p u t t r i p l e o u t p u t s y s t e m £ . V, = H.(w) V j - A , B o r C . j 3 g ; F i g . 8 T r a n s m i s s i o n l i n e c o n f i g u r a t i o n s w i t h bounda r y c o n d i t i o n s 24 where [ v ] i s t h e 3x1 co lumn m a t r i x o f pha se v o l t a g e s [ i ] i s t h e 3x1 co lumn m a t r i x o f pha se c u r r e n t s [ L ] i s t h e 3x3 i n d u c t a n c e m a t r i x [R] i s t h e 3x3 r e s i s t a n c e m a t r i x [C] i s 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 l i n e s , [G] 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 a l w a y s n e g l e c t e d ) . However , t h e above e q t n s ( 2 -1 ) and (2 -2 ) a r e n o t u s e a b l e f o r l i n e s w i t h f r e q u e n c y dependent l i n e p a r a m e t e r s . I n s t e a d , we have t o u se e q u a t i o n s i n t h e f o r m o f s t e a d y s t a t e p h a s o r e q u a t i o n s i n t h e f r e q u e n c y domain - ^ - i» ™ - r f ^ i = m iv] ( 2 - 3 ) . ( 2 -4 ) where [Z] = [R] + j u [ L ] = s e r i e s impedance m a t r i x i n ft/km as o b t a i n e d n u m e r i c a l l y f r o m C h a p t e r 1. [Y] = ju>[c] = s h u n t 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 f r o m C h a p t e r 1 [V] and [ I ] a r e t h e v e c t o r s o f pha se v o l t a g e s and pha se c u r r e n t s , r e s p e c t i v e l y , i n t h e f o r m o f p h a s o r v a l u e s . 18 E q t n ( 2 -3 ) can be d i f f e r e n t i a t e d w . * ' . t . x t o g e t A . i - - m [ £ u v dx = [Z] [Y] [V] A [ZY] [V] ( 2 -5 ) where [ZY] I [Z] [Y] 25 The 3x3 m a t r i x i n e q t n ( 2 -5 ) has n o n - z e r o o f f - d i a g o n a l e l e m e n t s i . e . t h e r e i s c o u p l i n g be tween t h e p h a s e s . The e a s i e s t way t o s o l v e t h e s e 18 16 c o u p l e d e q u a t i o n s i s t o d e c o u p l e them by moda l a n a l y s i s ' . W i t h t h i s 19 a p p r o a c h , [ZY'] i s t r a n s f o r m e d t o a d i a g o n a l m a t r i x w i t h [ M ] , [ M ] " 1 [ Z Y 1 • [M] = [ A ] ( 2 - 6 ) where [ A ] = d i a g o n a l m a t r i x [M] = moda l m a t r i x , i . e . co lumns o f e i g e n v e c t o r s o f [ZY] and [M] ^ = i n v e r s e o f [M] B o t h [M] and [M] ^ we re o b t a i n e d w i t h s u b r o u t i n e s f r o m t h e UBC 20 Comput ing C e n t r e Programme L i b r a r y , name ly w i t h ' DCE IGN ' and 'CD INVT ' ' DCE IGN ' computes t h e e i g e n v a l u e s and e i g e n v e c t o r s o f t h e comp lex m a t r i x [ Z Y ] , and ' CD INVT ' computes t h e i n v e r s e o f t h e m o d a l m a t r i x [M] . ' DCE IGN ' 19 was 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 a t e s t example w i t h known answer s . The r e s u l t s a g r e e d up t o 7 s i g n i f i c a n t d i g i t s . W i t h [M] ^ known, t h e pha se t o mode v o l t a g e t r a n s f o r m a t i o n i s d e s c r i b e d by [vm 6 d e] = [ M r V i ( 2 _ 7 ) P r e - m u l t i p l y i n g e q t n ( 2 - 5 ) w i t h [M] ^ g i v e s 2 [ M ] " 1 ^ - ] = [ M ] - 1 [ Z Y ] [ V ] dx V m o d e ] = [ M ] _ 1 [ Z Y ] [ V ] = [ M ] " 1 [ Z Y ] [ M ] [ V m ° d e ] > f r o m e q t n ( 2 - 7 ) = [ A ] [ V m ° d e ] f r o m e q t n (2 -6 ) Thus , 3 s e cond o r d e r d i f f e r e n t i a l e q u a t i o n s a r e o b t a i n e d , e a c h d e c o u p l e d f r o m t h e o t h e r , n a m e l y , 26 ,2„mode d . v r d x " ,2„mode d . v 2 d x ' ,2„mode d v3 0 0 mode 1 0 X 2 . •o mode 2 0 0 X 3 mode 3 ( 2 - 8 ) d x " where X^, X^ and X^ are t h e 3 e i g e n v a l u e s o f [ Z Y ] . The 3 pha se v o l t a g e s have now been 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 . They can be t r a n s f o r m e d b a c k t o pha se v o l t a g e s w i t h t h e mode t o p h a s e r e l a t i o n -s h i p d e r i v e d f r o m e q u a t i o n X 2 - 7 ) , (2-9) [V] = [ M ] [ V m ° d e ] The g e n e r a l s o l u t i o n t o t h e s econd o r d e r l i n e a r d i f f e r e n t i a l 22 e q u a t i o n o f t h e moda l v o l t a g e components i s w e l l known b e c a u s e e a c h moda l 23 2 A 25 component can be t r e a t e d as i f i t we re a h y p o t h e t i c a l s i n g l e pha se l i n e . ' ' F o r d i s t a n c e x = £ km away f r o m t h e s e n d i n g end - m o d e ~ V 1 , ' A - m o d e 1+ mode 2 , ' A _ mode 2+ mode L v 3 , H —> „mode - 3 + mode 1+ A „mode 2+ V .mode 3+ -fk. l , „mode- fx. e 1 - + V^_ • e 1 -- / A '£ , „mode e 2 - + V 2 _ • e 2 _ -/L I , mode e 3 + . e 3.^. e - Y ^ + v m o d e _ e " Y 4 + V ^ ° d e . e Y / _ (2-10) 27 where * ? •p r opaga t i on c o n s t a n t f o r s t e a d y s t a t e b e h a v i o u r a t s p e c i f i c f r e q u e n c y 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 r e s p e c t i v e l y , t r a v e l l i n g f r o m t h e s e n d i n g end t o t h e r e c e i v i n g e n d . V ™ ° d e „ _ = r e f l e c t e d moda l v o l t a g e waves a t x = 0 o f A, B and C r e s p e c t i v e l y , 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 s e n d i n g e n d . I f 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 and d i s t o r t i o n o f t h e wave f r o n t , t h e n we c an assume t h a t t h e l i n e i s i n f i n i t e l y l o n g . We can t h e n n e g l e c t t h e backwa rd r e f l e c t e d v o l t a g e wave , i . e . v mode _ mode _ mode _ n 1 - 2 - " 3-E q t n ( 2 -10 ) i s t h u s r e d u c e d t o mode 1+ mode mode v2 , £ = T Tmode V .mode 2+ V mode 3+ e ^ l * e ~ V e '3 ' e ' 1 e " Y 3 £ | V mode 1+ Tmode J2+ Tmode J3+ (2 -11 ) o r s i m p l y by m a t r i x n o t a t i o n , we have [v1 •modej = e - [ Y ] - £ . r y o d e j (2 -12 ) JO "T where [y ] = 3x3 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 e l e m e n t s Y 2 a n ^ Y3 e = [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 t h e t r a n s m i s s i o n s y s t em and a g a i n t v m o d e ] = f o r w a r d v o l t a g e wave a t x = 0. (N.B. The r e s u l t s t h u s o b t a i n e d a r e a l s o v a l i d f o r t h e v o l t a g e r e s p o n s e a t t h e r e c e i v i n g end o f an open -ended l i n e o f f i n i t e l e n g t h . F o r t i m e s l e s s 28 t h a n 3 t i m e s t r a v e l t i m e , t h e open -ended l i n e r e s u l t s a r e s i m p l y t w i c e t h e r e s u l t s o b t a i n e d f r o m e q t n ( 2 - 1 2 ) . T h i s d o u b l i n g e f f e c t i s d i s c u s s e d i n f u r t h e r d e t a i l i n C h a p t e r 4 , s e c t i o n 1.) F rom e q t n ( 2 - 1 2 ) , t h e moda l v o l t a g e s a t t h e r e c e i v i n g end [ V ™ ° d e ] can be t r a n s f o r m e d t o t h e pha se v o l t a g e s by t h e mode t o pha se r e l a t i o n s h i p rv*] = [ M ] [vm 0 o d e] = [ M ] e" •ly. V mode + f r o m e q t n (2 -9) f r o m e q t n (2 -12 ) Thu s , we c an e x p r e s s t h e pha se v o l t a g e s a t t h e r e c e i v i n g end [V ] i n t e rms o f pha se v o l t a g e s a t t h e s e n d i n g end [ V°] a s [V1] = [ M ] e _ l Y ] ^ r M ] _ 1 [ V ° ] ( 2 -13 ) N o t e t h a t f r o m e q t n ( 2 - 1 3 ) , pha se v o l t a g e s c a n n o t be c a l c u l a t e d f r o m [ y ] a l o n e w i t h o u t [ M ] , i . e . [ V £ ] * e " [ Y ] £ [V° ] 3. I n c l u s i o n o f Bounda ry C o n d i t i o n s a t S e n d i n g End . F o r t h e cho sen t e s t l i n e c a s e , t h e l i n e was e n e r g i z e d on pha se A as shown i n F i g . 8. Bounda ry c o n d i t i o n s f o r pha se v o l t a g e s and c u r r e n t s a t t h e s e n d i n g end ( d i s t a n c e x = 0 d e n o t e d by s u p e r s c r i p t o) a r e t h e n [V°] = V - I ? R ° and H°] = V. V I 0 0 ( 2 -14 ) (2 -15 ) S u b s t i t u t i o n o f e q t n ( 2 - 3 ) i n t o e q t n ( 2 - 7 ) d i f f e r e n t i a t e d w . r . t . x g i v e s ) 29 dV - I -mode I-= [M] 1 •'. [Z ] I I 0 ] ^ IA]II°] A l l A 1 2 A l 3 " A 2 1 A 2 2 A 2 3 0 A 3 1 A 3 2 A 3 3 0 = I "11 l 21 A31 A g a i n , d i f f e r e n t i a t i n g e q t n ( 2 -12 ) w . r . t . x g i v e s mode E q u a t i n g R.H.S. o f e q t n (2 -16 ) and ( 2 - 17 ) ,we g e t A, [Y] e - x [ Y ] [ v ™ o d e ] - 1° l l l l 21 l 31 (2 -16 ) (2 -17 ) (2 -18 ) However , a t t h e s e n d i n g end we have x = 0 a s bounda r y c o n d i t i o n , e q t n (2 -18 ) t h e r e f o r e g i v e s I A, r „mode , r -.-1 T o [ V + ] = [ y ] I A 11 l 21 l 31 o r r v ™d e ] = i ° i 6 o o l o I Y 3 J x l l 21 4 31 = I A 2 1 / y 2 A 3 1/ y 3 (2 -19) 30 From t h e phase-mode r e l a t i o n s h i p o f e q t n (2-9) we g e t t h e s e n d i n g end pha se v o l t a g e IV ] as rv°] = M i v 7 d e ] M U 1 2 M 13 A u / Y l M 2 1 M 22 M 23 A 2 1 / Y 2 M 3 1 M 32 M 33 A 3 1 / Y 3 A o ( Fo r a d e t a i l e d p i c t u r e o f t h e b o u n d a r y c o n d i t i o n s , s ee F i g . 8 ) . Now, we can e v a l u a t e t h e f i r s t row o f e q t n ( 2 - 2 0 ) . V g ~ TlRo " ^ l l ^ l ^ l + M 1 2 A 2 1 / Y 2 + M 1 3 A 3 1 / Y 3 ) • Then , we can g e t t h e s e n d i n g end c u r r e n t i n pha se A as ( 2 -20 ) 1 ° = V g / Z ( 2 - 2 1 > eq (2 -22 ) A " M 1 1 A 1 1 / Y L + M 1 2 A 2 1 / Y 2 + M 1 3 A 3 1 / Y 3 + R Q where Z&q = M U A U / Y l + M ^ / Y , , + M 1 3 A 3 1 / Y 3 + R Q Thus , s u b s t i t u t i n g e q t n (2 -21 ) i n t o e q t n ( 2 -19 ) f o r 1 ° , t h e m o d a l v o l t a g e s a t t h e s e n d i n g end a r e o b t a i n e d as . m o d e , = _ § _ . 1 + J Z eq A l l / Y l A 2 1 / y 2 A 3 1 / Y 3 ( 2 -23 ) 31 U s i n g t h e mode-phase r e l a t i o n s h i p o f e q t n ( 2 -9 ) a g a i n , we o b t a i n t h e s e n d i n g end (x=0) pha se v o l t a g e s i n t h e 3 pha se as [ M ] eq A l l / Y l A 2 1 / Y 2 A 3 1 / Y 3 (2-28) A l s o , f o r t h e r e c e i v i n g end a t x = £, t h e moda l v o l t a g e components a r e [V mode ] = e " £ t Y ] I V ™ ° d e ] eq A l l / Y l e " V A 2 1 / y 2 A 3 1 / Y 3 f r o m e q t n (2 -12 ) (2 -25) F i n a l l y , t h e r e c e i v i n g end pha se v o l t a g e s i n t h e 3 pha se a r e V V eq A l l / Y l [ M ] A 2 1 / Y 2 e - Y 3 ^ A 3 1 / y 3 . (2 -26) I t s h o u l d be r e a l i z e d t h a t i f we e x c i t e pha se k o f t h e 3 p h a s e , (k = A , B , o r t h e n t h e o u t p u t pha se v o l t a g e s a r e r -eq A l k / Y l [M] A 2 k / y 2 A 3 k / y 3 (2 -27 ) T h i s i s t h e f o r m u l a t h a t we u se i n t h e F o u r i e r T r a n s f o r m Programme. I t has a n a o p t i o n t o s p e c i f y w h i c h o f t h e 3 pha se s a r e t o be e n e r g i z e d f o r t h e t e s t l i n e c a s e . Thu s , we o b t a i n t h e t r a n s f e r f u n c t i o n [H (CJ ) ] f o r t h e t e s t c a s e as s een f r o m F i g . 8. 32 LH(io)] = H A ( U ) H B(o)) T, H c (to) V (2 -28 ) 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 The m a g n i t u d e s and pha se s 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 c a s e o f F i g . 8 a r e p l o t t e d i n F i g s . 9 and 10 , r e s p e c t i v e l y , f o r b o t h a p -p r o a c h e s u sed 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 be tween t h e two s k i n e f f e c t f o r m u l a e o n l y show up i n t h e m a g n i t u d e s p e c t r u m i n t h e l o w f r e q u e n c y (0 - 100 Hz) r e g i o n . The r e s u l t s c o i n c i d e more o r l e s s a t f r e q u e n c i e s above 100 H z . The pha se 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 (as 9 shown i n F i g . 9 ). T h i s c an e a s i l y be e x p l a i n e d f o r t h e s i n g l e pha se c a s e where t h e t r a n s f e r f u n c t i o n becomes / / T » 1 2 . . T \ j , , n (2 -29 ) where H = l e n g t h o f l i n e H(U) = <Tmi = ^ = e - £ / ( R + ^ L ) j a ) C (N.B. Compare w i t h e q t n (2 -12 ) f o r 3 d e c o u p l e d modes ) . E x p a n d i n g f o r r e a l 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 e x p a n s i o n 1/2 1/2 , 1 -1/2 , (a + B ) ' = a + 2 a "b + . , a >> b. Y = V(R+ja)L)ja>C = (R+ju)L) 1 / 2. ( j w e ) 1 / 2 , R « jwL Y ~ [ ( jo) ! ) 1 7 2 + | ( ja)L)- 1 / 2 ] • ( J W C ) 1 / 2 - 1/2 R' ,Cs 1/2 A = - (jj) + j ^ a C ) 1 ^ £ a + j g (2 -30 ) Thu s , t h e m a g n i t u d e 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 pha se 1/2 l i n e a r e r e s p e c t i v e l y -a l i|'H(u) I = e" = e *2 V (2 -31 ) 34 Phase ( r ad ) Phase o f t r a n s f e r f u n c t i o n s ( I d e n t i c a l r e s u l t 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 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 f o r m u l a ) F r e q u e n c y ( H z ) F i g . 1 0 35 and LH(u) « - Ag = - £ O J ( L C ) 1 / 2 ( 2 -32 ) where R i n c r e a s e s a p p r e c i a b l y w i t h f r e q u e n c y whe re L d e c r e a s e s s l i g h t l y w i t h f r e q u e n c y f o r z e r o sequence and s t a y s more o r l e s s c o n s t a n t f o r p o s i t i v e s equency and where C s t a y s c o n s t a n t ; T h a t t i s , t h e pha se a n g l 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 a l m o s t l i n e a r l y w i t h f r e q u e n c y , wherea s t h e m a g n i t u d e d e c r e a s e s w i t h f r e q u e n c y s i m i l a r t o a l ow p a s s f i l t e r . The c a l c u l a t i o n o f a n g l e s 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 s t a t e m e n t c o v e r s o n l y t h e r ange f r o m 0° t o 3 6 0 ° . T h e r e f o r e , s p e c i a l l o g i c t o be i n c l u d e d t o e x t e n d t h e a n g l e s beyond 2TT ( s ee A p p e n d i x 1 f o r FORTRAN l i s t i n g s ) . The s e p a r a t e 'PHASEPRO' p r o g r am i s u s ed t o g u a r a n t e e t h a t t h e phase ang l e ? 6 i s a c o n t i n u o u s f u n c t i o n o f co. T h i s c an be a c h i e v e d by s e t t i n g up a c o u n t e r v a l u e k, where k i s i n i t i a l l y z e r o . Whenever t h e c a l c u l a t e d phase a n g l e f a l l s o u t o f t h e r a n g e 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 -t e d pha se 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 1 k: = k + 1, and a l l f o l l o w i n g pha se a n g l e v a l u e s a r e i n c r e a s e d by k (2 i r ) . T h i s way, t h e pha se a n g l e i s e n s u r e d 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 i n c r e a s i n g . 36 CHAPTER 3. TIME RESPONSE OF TEST L INE THROUGH FOURIER TRANSFORMATION 1. I n t r o d u c t i o n A f t e r t h e f r e q u e n c y r e s p o n s e o f t h e l i n e i s known i n t h e f o r m o f t r a n s f e r f u n c t i o n s , t h e o u t p u t v o l t a g e c an be c a l c u l a t e d f o r any g i v e n i n p u t v o l t a g e ( v ) b y F o u r i e r T r a n s f o r m a t i o n . A t t h e b e g i n n i n g , t h e i n p u t v o l t a g e v ( t ) i s t r a n s f o r m e d f r o m t h e t i m e doma in i n t o t h e f r e q u e n c y domain t o g i v e V ( u ) . The o u t p u t v o l t a g e i n t h e f r e q u e n c y domain i s t h e n o b t a i n e d by m u l t i -p l y i n g V (u) w i t h t h e t r a n s f e r f u n c t i o n [H(co)] o b t a i n e d f r o m t h e T r a n s f e r g F u n c t i o n P r o g r a m d e s c r i b e d i n C h a p t e r 2 . i . e . [V i L ( t o) ] = [H(W)] V (GO) ( 3 _ 1 ) g F i n a l l y , t h e i n v e r s e F o u r i e r t r a n s f o r m a t i o n i s u sed t o o b t a i n t h e o u t p u t v o l t a g e s v . ( t ) , v _ ( t ) and v „ ( t ) i n t h e t i m e doma in . The above d e s c r i b e d A B C t e c h n i q u e s a r e a p p l i e d t o t h e t e s t c a s e o f F i g . 8. The o b t a i n e d r e s u l t s a r e t h e n compared w i t h t h e f i e l d t e s t measu rement s^ and w i t h s i m u l a t i o n r e s u l t s 26 o b t a i n e d by G r o s c h u p f . The p rog rams u sed f o r t h e F o u r i e r T r a n s f o r m a t i o n s f r o m t h e t i m e t o t h e f r e q u e n c y doma in , and v i c e v e r s a , was a d o p t e d f r o m a 27 v e r s i o n i n i t i a l l y w r i t t e n by H. W. Dommel ( see A p p e n d i x 3 f o r p r o g r a m ' l i s t i n g s ) . 2I-. 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 v o l t a g e f r o m t i m e t o f r e q u e n c y F o r a g i v e n i n p u t v o l t a g e v ( t ) i n t h e t i m e doma in , we c an i n g e n e r a l o b t a i n t h e i n p u t v o l t a g e i n t h e f r e q u e n c y domain V^(to) w i t h t h e O f o l l o w i n g F o u r i e r T r a n s f o r m a t i o n f o r m u l a oo oo A (to) = /„, v ( t ) co s u t d t ( 3 -2 ) o OO B(oj) = /- v e ( t ) s i n wt d t ( 3 -3 ) where A(co) and B(co) a r e t h e r e a l and i m a g i n a r y p a r t s o f V (co), r e s p e c t i v e l y , O 37 V g ( to ) = A(co) + j B(co) . ( 3 -4 ) I f we assume t h a t t h e i n p u t v o l t a g e i s z e r o f o r t i m e t ^ 0 , t h e n e q t n s ( 3 - 2 ) and ( 3 - 3 ) c an b e s i m p l i f i e d t o A (a)) = / T v„ ( t ) co s cot d t ( 3 -5 ) o g B(co) = J * T v B ( t ) s i n cot d t ( 3 -6 ) O & where (o ,T ) i s t h e t i m e i n t e r v a l i n w h i c h v e ( t ) i s n o n - z e r o . Case 1. I n p u t v o l t a g e d e f i n e d p o i n t b y p o i n t I f t h e i n p u t v o l t a g e 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 i n t e r v a l ( o ,T ) a t c l o s e l y spaced t i m e i n t e r v a l s , t h e n i t i s r e a s o n a b l e t o assume l i n e a r i n t e r p o l a t i o n be tween p o i n t s ( see F i g . 1 1 ) . Then , f o r an i n t e r v a l ( t ^ , t 2 ) , we have v - v v g ( t ) = v 2 + A t ( t - t x ) , t ^ t 4 t2 ( 3 -7 ) J & A t = t 2 - t 1 S u b s t i t u t i o n o f e q t n ( 3 -7 ) i n t o e q t n ( 3 -5 ) g i v e s t V 2 _ V 1 A 1 2 ( a ) ) = { 2 [ v l + ~ l t ( t _ t l ) ] c o s u t d t ( 3 _ 8 ) v ? ~ v / ^ v ? ~ v l t = (v, 7 — 2 cos tot d t + 7'- / 2 t co s cot d t 1 A t * A t t ^ = ( V l - - ^ " V - s m cot 1^ -^ V 2 " V 1 1 . . 1 ^ 2 H : — ( t s m cot H cos t o t j l A t co co 1 1 V 2 _ V 1 V 2 _ V 1 = - s i n ut2[(y1 + -J^-tJ -s i n cot, r , 2 1^ _ £ _ 1 < _ i + v 2 _ v i c ., — ( C O S 2)t 9 - C O S 0 ) t 1 ) Atto 3 8 Input voltage F i g . 1 1 F i g . 1 2 39 v 2 - v o r f i n a l l y , A . ^ ( u ) = ^ I v 2 s " " " n u t 2 ~ V l s " ^ n u t l + — ~ " ( c o s u t 2 - co s u t ^ ) ] Atu (3-9) S i m i l a r l y , 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 ( t ^ , t 2 ) , we o b t a i n t V 2 " V 1 B 1 2 ( a ) ) = { 2 [ v l + A t ( t _ t l ) ] S ± n u t d t V 2 " V 1 t V 2 " V 1 t = - * )^ 2. s i n u t d t + * t ^2 t s i n u t d t V 2 _ V i q | | 2 X Y V L ' 1 . ,,'2 = ( v x - cos u t | ^ + " ^ - ( - t co s u t + - s i n u t ) | ^ 1 v 2 " v l o r f i n a l l y B-„ (u) = — [-v_ cos u t „ + v , co s u t - H — — ( s i n u t . - s i n u t - ) ] 12 u 2. 1 1 1 A t u to 2 1 (3 -10 ) The c a l c u l a t i o n s o f A ^ 2 ( u ) and B ^ 2 ( u ) a r e r e p e a t e d f o r a l l t i m e i n t e r v a l s t o c o v e r t h e w h o l e 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 o f t h e 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 n s i m p l y t h e sum o f t h e s e p a r t s V g ( u ) = A ( u ) + j B ( u ) N T =k!Q Vk+l ( a 0 + j B k 5 k + l ( ( 0 ) ' W h e r e N = At" The above c a l c u l a t i o n s must be 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 have been c a l c u l a t e d . O u t p u t v o l t a g e i n t h e f r e q u e n c y domain i s t h u s o b t a i n e d a t a l l t r a n s f e r f u n c t i o n f r e q u e n c i e s . Case 2. I n p u t v o l t a g e d e f i n e d 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 ) and B (u ) a r e known 28 a n a l y t i c a l l y . 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 a s an 40 examp le , v - ( t ) = e " a t , t * 0 ' ( 3 -12 ) We can d i r e c t l y e v a l u a t e A(to) and B(to) by V^'(a)) = A (to) + jB(to) = — ~ — (3 -13 ) a+jco Thu s , we can o b t a i n t h e r e a l and i m a g i n a r y v o l t a g e components i n t h e f r e q u e n c y doma in 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 . W i t h t h i s t e c h n i q u e , we can o m i t t h e f i r s t p a r t o f o u r p r og r am >u and o b t a i n t h e o u t p u t v o l t a g e i n t h e 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 t h e c o r r e s p o n d i n g t r a n s -f e r f u n c t i o n s , i . e . [ V a ] = [H(g)] ^ ( 3 -14 ) 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 From e q t n ( 3 - 1 ) , we have t h e o u t p u t v o l t a g e i n t h e f r e q u e n c y domain as [V> J = [H(to)] V (to) f r o m e q t n ( 3 - 1 ) F o r i n v e r s e F o u r i e r t r a n s f o r m a t i o n b a c k t o t h e t i m e doma in , ( see s e c t i o n D ) , we u s e l i n e a r i n t e r p o l a t i o n be tween c o n s e c u t i v e f r e q u e n c y p o i n t s . Thu s , 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 must be r e a s o n a b l y smooth t o o b t a i n s a t i s f a c t o r y r e s u l t s . I t has been shown t h a t t h e m a g n i t u d e o f t h e t r a n s f e r f u n c t i o n i s f a i r l y smooth ( see e q t n s ( 2 -31 ) and ( 2 - 3 2 ) . T h i s i s a l s o t r u e f o r t h e phase a n g l e o f t h e t r a n f e r f u n c t i o n p r o v i d e d i t i s e x t e n d e d beyond 2TT rad(see F i g s . 9 and 1 0 ) . From e q t n s (3 -1 and ( 3 - 1 0 ) , we o b t a i n t h e r e a l and i m a g i n a r y com-p o n e n t s o f t h e i n p u t v o l t a g e . They become h i g h l y o s c i l l a t i n g a t h i g h e r f r e q u e n c i e s 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 . T h e r e f o r e , t h e 41 r e a l and i m a g i n a r y v o l t a g e components a r e c o n v e r t e d t o m a g n i t u d e and pha se v a l u e s . The pha se a n g l e i s a g a i n e x t e n d e d beyond 2TT by t h e same smoo th i n g l o g i c a s d e s c r i b e d i n C h a p t e r 2. Thu s , we can w r i t e t h e o u t p u t v o l t a g e i n £ t h e f r e q u e n c y domain V ( to) a s V £ ( t o ) = S(u)^R(u)) ( 3 -15 ) 14. Ou tpu t v o l t a g e i n t i m e domain by n u m e r i c a l i n v e r s e F o u r i e r T r a n s f o r m a t i o n £ From t h e g i v e n o u t p u t v o l t a g e i n t h e f r e q u e n c y domain [V ( to) ] , we o b t a i n t h e o u t p u t v o l t a g e s i n t h e t i m e domain by i n v e r s e F o u r i e r T r a n s f o r m a -t i o n £ / \ 1 fS T T £ , v j t o t , v ( t ) = — / V (to) e J dto TT 0 From e q t n ( 3 - 1 5 ) , we g e t !ir*.\ 1 f°° of \ 3 ( U T + R ) J a i n v ( t ) = — S S (co) e J dto ( 3 -16 ) TT 0 S i m i l a r 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 , t h e i n v e r s e 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 u se s l i n e a r i n t e r p o l a t i o n be tween a d j a c e n t p o i n t s i n t h e f r e q u e n c y doma in , f o r t h e m a g n i t u d e s (S^, S 2 ) o f t h e o u t p u t v o l t a g e s as w e l l as f o r t h e pha se a n g l e s (R^, R 2 ) • ( see F i g . 1 2 ) . A s e x p l a i n e d i n s e c t i o n C, t h i s i s p e r m i s s i b l e b e c a u s e S and R a r e smooth c u r v e s i n c o n t r a s t t o t h e h i g h l y 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 components A ( t o ) and 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 i n t h e t i m e doma in , t h e c o n t r i b u t i o n t o t h e o u t p u t v o l t a g e f r o m t h e i n v e r s e F o u r i e r T r a n s f o r m a -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 ) = - r S ( to ) cos ( t o t + R) d t o , TT 0 o r v j ' ( t ) = - . JCU2 S ( to ) co s ( t o t + R ) d t o ( 3 -17) 1 2 IT <B^. S -S where S ( t o ) = S n + - f—^ ( o> - u - ) (3 -18 ) 1 Ato 1 R —R R ( t o ) = R. + - f — - ( t o - w.) ( 3 -19 ) 1 lAtO 1 where to^<to<to2, & Ato=u2-o)^ 42 Substituting the magnitude and phase eqtns (3-18) and (3-19) into eqtn (3-13) we get \ S ~~ S H- ~"R v i 2 ( t ) - l[2 I s i + - f^ ( u -V ] c o s ^ + R i + J K ^ ^ i > ^ s 0 - s n s - s = / 2 [ ( S J - - T c O + — to] • to^  1 Ato 1 Ato R „ - R 1 R ? - R 1 A / U 2 (a + bio) cos (c + sto)dto where constants a,b,c and s are constant for a specific frequency interval , s 9 - s a =• S, - - ? — 1 a), (3-20) b = ' i Ato r i S2" S1 ~ ( 3 - 2 1 ) R _ - R 1 c = R. - - | — ^ (3-22) 1 Ato 1 R 2' R 1 s = — + t (3-23) Ato Thus, we obtain v £ (t) = a/U2 cos (c + sto)dto + b/W2 tocos(c + sio)diO 12 to^  io ^  to 2 t, 1 — sin (c + sto) I + — [tosin(c + Sto) + - cos (c+sto) ] s ' 10-^  S s "--j. to 2 or f ina l ly v ^ ( t ) = sin(c+sto2) (-|+i02 "s"^  - s i n ( c + s t o i ) (^ 3 + 0 3 \ 1$ + - [cos(c+sto0)-cos(c+sio1)] (3-24) s Z -1-The calculation with eqtn (3-24) is repeated for a l l frequency a intervals to cover the frequency region over which the output voltage V (to) r 43 i s d e f i n e d . The o u t p u t v o l t a g e a t any s p e c i f i c t i m e i s t h e n t h e sum o f t h e c o n t r i b u t i o n s f r o m a l l f r e q u e n c y i n t e r v a l v £ ( t ) = J Q v j 2 ( t » ( 3 -25 ) where co' i s t h e l a s t f r e q u e n c y d a t a p o i n t . 5. N u m e r i c a l A s p e c t s o f F o u r i e r T r a n s f o r m a t i o n P r og r am T h e r e a r e s e v e r a l a s p e c t s w h i c h d e s e r v e s p e c i a l a t t e n t i o n i n t h e F o u r i e r T r a n s f o r m a t i o n P r og r am t o e n s u r e r e a s o n a b l y a c c u r a t e r e s u l t s . ,1. S u i t a b i l i t y o f l i n e a r i n t e r p o l a t i o n i n n u m e r i c a l i n t e g r a t i o n - A r e a s o n a b l e " s m o o t h n e s s " o f i n p u t v o l t a g e v ( t ) and 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 Vw(co) must be g u a r a n t e e d 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 be tween a d -j a c e n t d a t a p o i n t s . T h e r e f o r e , t h e m a g n i t u d e and p h a s e a n g l e o f t h e o u t p u t v o l t a g e a r e u s ed t o a v o i d t h e h i g h l y 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 f r e q u e n c y components as d e s c r i b e d i n s e c t i o n 3. 2. D e n s i t y o f d a t a p o i n t s - L i n e a r i n t e r p o l a t i o n i s assumed between 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 i n t h e n u m e r i c a l i n t e g r a t i o n l o o p s . Too dense d a t a p o i n t s w i l l i n c r e a s e computer c o s t s d r a s t i c a l l y , w h i l e t o o s p a r s e d a t a p o i n t s w i l l r e s u l t i n l o s s o f a c c u r a c y . A d e n s i t y o f 20 p o i n t s p e r decade i n t h e f r e q u e n c y domain (on a l o g a r i t h m s c a l e ) s a t i s f i e s t h e a c c u r a c y r e q u i r e m e n t r e a s o n a b l y w e l l f o r t h e t e s t c a s e s t u d i e d . I n t h e t i m e d o m a i n , t h e d e n s i t y o f d a t a p o i n t s f o r t h e i n p u t v o l t a g e v ( t ) depends 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 u s e r . 3. Number o f decades i n f r e q u e n c y domain o v e r w h i c h H(co) and V (co) must be d e f i n e d — I t i s ea sy t o j u d g e t h e r e q u i r e d no . o f decades a s t h e t r a n s f e r f u n c t i o n m a g n i t u d e s d e c r e a s e s u b s t a n t i a l l y a t h i g h f r e q u e n c i e s . Thus 7 t o 8 decade s o f f r e q u e n c y d a t a p o i n t s , s t a r t i n g a t f =1 Hz w i l l s t a r t 44 e n s u r e r e a s o n a b l e a c c u r a c y w i t h o u t i n c r e a s i n g computer c o s t s t o o much f o r t h e t e s t c a s e s t u d i e d . I n t e g r a t i o n be tween f = 0 a n d . f .where t h e 6 s t a r t f r e q u e n c y d a t a p o i n t s s t a r t i s done s e p a r a t e l y , a g a i n a s sum ing l i n e a r i n t e r -p o l a t i o n be tween 0 and f s t a r f T h e r e f o r e , we can s t a r t o u r f r e q u e n c y d a t a a t any d e c a d e . T h i s i s a l l o w a b l e as l o n g as t h e o u t p u t v o l t a g e V (to) r e m a i n s f a i r l 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 f r o m z e r o t o t h e s t a r t i n g f r e q u e n c y f . ' does n o t c au se a p p r e c i a b l e d e v i a t i o n s , s t a r t r r 3. 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 and s i m p l e way t o c heck t h e a c c u r a c y o f t h e F o u r i e r T r a n s f o r m a t i o n P r o g r a m i s t o r u n i t i n a t e s t mode where t h e t r a n s f e r f u n c t i o n i s s e t t o 1, H(to ) = 1 L0 and t o c heck 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 domain a g r e e s w i t h t h e i n p u t v o l t a g e v ( t ) . I n ou r t e s t c a s e , t h e known i n p u t v o l t a g e v ( t ) g g i s a d o u b l e e x p o n e n t i a l o f t h e f o r m v ( t ) = e ' V - e"V ( 3 -26 ) g 3 -1 where = 0.17 x 10 s and a 2 = 3.27 x 1 0 6 s _ 1 T h i s i n p u t v o l t a g e matches e x a c t l y t h e o u t p u t v o l t a g e t h u s o b t a i n e d f r o m ou r t r a n s f o r m a t i o n p r o g r am ( see F i g s . 13 and 1 4 ) . I n F i g . 1 3 , i n t h e t i m e i n t e r v a l f r o m 0 t o 7us s t e p w i d t h s o f A t = 0 .05us and Ato = 20 p t s / d e c a d e ( l og J s c a l e ) we re c h o s e n . I n F i g . 14, f o r t i m e >10us, t h e i n p u t v o l t a g e v ( t ) i s e s s e n -i t i a l l y a s i n g l e e x p o n e n t i a l a d e c a y , f o r w h i c h s t e p w i d t h s o f A t = 0.1 ms and Ato = 20 p t s / d e c a d e ( l o g s c a l e ) we re c h o s e n . v o l t a g e ( p . u . ) 1.0 I n pu 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 H(co)=1.0 z ,0° 45 0 .8 At = 0 .05 /as Ato = 20 p o i n t s / d e c a d e s f - = 1 Hz s t a r t f = 10 MHz end 0.6 0 .4 0.2 7 T ime^us ) F i g . 1 3 4 6 voltage(p.u.) Same test as in Fig.13 from 0.1 to 15 ms 5 ,o Fig.14 15 Time(ms) 47 4 . Numerical problems with step function inputs - No problem of numerical ins tab i l i ty were encountered when the f i e ld tests of the test l ine were simulated. The input voltage v (t) in this, case is a double exponen-t i a l wave (see eqtn 3 - 2 6 ) . The computations were numerically stable for large values of decay constants, namely and > 10 . However, many cases were run with step function inputs for checking purposes to debug the pro-gramme, and to gain confidence before the duplication of f i e ld tests could be attempted. Serious numerical ins tab i l i ty 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 is chosen in such a way that this function is pract ica l ly equal to a step function over the time span of interest. For an input voltage step function v (t) = 1 ( 3 - 27 ) g the voltage in the frequency domain i s V (co) - / " 1 e - J u t d t g o -1 — 1 cot I 0 0 = -.— e - • JCO 1 o = - ~ Mm e - J t o t ( 3 -28 ) For time, t -»• 0 0 , the second term of eqtn ( 3 -28 ) i s highly osc i la -ting and i s non-zero, which causes numerical ins tab i l i ty in the Fourier Transformation Programme. However, this problem can be remedied by intro-ducing a slow decay into the input voltage v (t) as in eqtn ( 3 - 1 2 ) , namely v (t) = 1 • e~ a t , t > 0 from eqtn ( 3 -12 ) The input voltage in the frequency domain now becomes 48 , r / \ r°° - a t - i t o t . , V (to) = / e ' - e J d t s o = - 1 e - ( q + j t o ) t | ° ° a+-j to ' o 1 l i m - a t - i t o t e ~ e J a + j t o oi+jto t->«> Now, f o r t i m e t o t h e second t e rm 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 p r e s e n c e o f t h e decay f a c t o r e i n i t , and goes t o z e r o a s t-*», o r V _ ( t o ) = * F u r t h e r m o r e , t h e s i n g l e 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 t h a n • —at a c u t - o f f s t e p f u n c t i o n v o l t a g e ( r e c t a n g u l a r p u l s e ) i na smuch as l « e has a smoother a m p l i t u d e and phase a n g l e s p e c t r u m t h a n a r e c t a n g u l a r p u l s e . Thu s , f e w e r d a t a p o i n t s p e r decade a r e r e q u i r e d t o a c h i e v e t h e same d e g r e e o f a c c u r a c y . 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 v o l t a g e i s t h e r e f o r e e a s i l y s o l v e d by i n t r o d u c i n g t h e d e c a y f a c t o r a. N u m e r i c a l e x p e r i m e n t s showed t h a t a > 10 w i l l be good enough t o e n s u r e n u m e r i c a l s t a b i -l i t y . N o t e t h a t f o r t h e c a s e a = 10 , t h e d e v i a t i o n o f t h e e x p o n e n t i a l l y d e c a y i n g i n p u t v o l t a g e 1 / /•*.\ _ a t v g ( t ) = e f r o m t h e i d e a l s t e p v o l t a g e i s n e g l i g i b l e f o r t h e t i m e span o f i n t e r e s t h e r e . F o r t = l O y s max _ _ . . - 1 0 x 1 0 v g ( t ) = e = 0.9999 Tha t i s , t h e maximum d e v i a t i o n i s l e s s t h a n 0 .01% f r o m t h e s t e p i n p u t v o l t a g e a t t h e uppe r l i m i t t o f t h e s t u d y . r r max 49 CHAPTER IV DUPLICATION OF F IELD TESTS 1. D o u b l i n g e f f e c t on open -ended l i n e . I n the a n a l y s i s l e a d i n g t o e q t n . ( 2 - 1 2 ) , we have 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 moda l v o l t a g e f o r t h e i n f i n i t e l i n e a t a d i s t a n c e I f r o m t he s e n d i n g e n d . ,.mode , . „mode . , -ySL , s V^ (to) = V + (oo) e ( 4 -1 ) T h i s e x p r e s s i o n i s n o t d i r e c t l y u s a b l e f o r t h e t e s t c a s e s i n c e we now have a t r a n s m i s s i o n l i n e o f f i n i t e l e n g t h w h i c h i s open -ended a t t h e r e c e i v i n g end t e r m i n a l ( see F i g . 8). 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 moda l q u a n t i t i e s a r e a n a l o g o u s t o t h e e q u a t i o n o f a s i n g l e phase l i n e , where t h e c o m p a r i s o n be tween t h e i n f i n i t e l i n e and t h e open -ended f i n i t e l i n e i s w e l l o k n o w n . Thu s , we can u se t h e w e l l known s o l u t i o n f o r t h e 14 24 29 s i n g l e phase ca se f o r v o l t a g e s and c u r r e n t s ' ' ; i n t h e moda l d o m a i n , mode mode , „ mode . , „ // o\ V = V. c o sh yl + Z I ,, s inhY& ( 4 - 2 ; o % °„ ^ yinode ].mode = - — — s . \ s i nh yl + I ^ ° d e c o sh yl ( 4 - 3 ) o Z *> . o where I m o d e and V m o c * e a r e t he moda l v o l t a g e and c u r r e n t a t t h e o o ,. , , T mode , T Tmode , , , . ^ . s e n d i n g end x = o , and 1^ and V^ a r e t h e moda l v o l t a g e and c u r -r e n t a t t he r e c e i v i n g end x = £. F o r an open -ended l i n e , I ™ 0 ^ 6 = ° . Then we B b t a i n f r o m ( 4 - 2 ) and ( 4 - 3 ) .mode , „ T mode T Tmode , , . , . . V + Z I = V„ ( c o s h yl + s i n h - f Y J l ) o o o % T Tmode yl i . e . V " 1 0 ^ - e _ Y £ ( V m ° d e + Z I m ° d e ) . ( 4 - 4 ) £ o o o 50 From t = o t o t < 2rr, no r e f l e c t i o n has y e t come b a c k f r o m t h e r e c e i v i n g e n d , and t he c o n d i t i o n s a t t he s e n d i n g end a r e t h e r e f o r e t he same as t h o s e o f an i n f i n i t e l i n e d u r i n g t h i s t i m e p e r i o d , v mode = jmode } 0 o o ( T h i s r e l a t i o n s h i p i s no l o n g e r t r u e a t t h e s e n d i n g end a f t e r t >_ 2T, and i s no l o n g e r t r u e a t t he r e c e i v i n g end f o r t _> 3T.) W i t h s u b s t i t u t i o n o f e q t n (4-5) i n t o e q t n (4-4), we g e t f o r t he r e c e i v i n g e n d , ^mode _ 2 e ~ Y £ y m ° d e _ ^ y m ° d e e ~ Y ^ 1 o + T h i s i s t w i c e t he 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 f o r t h e i n f i n i t e l i n e a t l o c a t i o n x = %. Thus , t h e r e i s a d o u b l i n g e f f e c t i n t he r e c e i v i n g end v o l t a g e o f t h e open -ended l i n e i n c o m p a r i s o n w i t h t h e i n f i n i t e l i n e . 2^ .» C o m p a r i s o n w i t h f i e l d measurements and o t h e r s i m u l a t i o n r e s u l t s The o u t p u t v o l t a g e a t t he r e c e i v i n g end i s p l o t t e d i n F i g . 15 f o r t h e t e s t c a s e w i t h t he v o l t a g e d o u b l i n g e f f e c t t a k e n i n t o a c c o u n t . The a r r i v a l t i m e o f t he 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 c l o s e l y 3 w i t h t h e t i m e t a k e n by e l e c t r o m a g n e t i c waves (TEM p r o p a g a t i o n ) i n a i r , i . e . 277 y s f o r 83.212 km a t a wave v e l o c i t y o f 3 km/us. On a t r a n s p o s e d l i n e , t h i s f i r s t p a r t o f t h e wave w o u l d be a s s o c i a t e d w i t h t h e p o s i t i v e s equence p a r a m e t e r s , and t h e s e cond 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 t h e z e r o sequence wave . I t can be o b s e r v e d t h a t t he wave v e l o c i t y . o f t h e z e r o s equence mode i s s l o w e r t h a n t h a t o f t h e p o s i t i v e s equence 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 w i l t h 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 ( e g . , AR ^ 0.67 ft and AR ^ 0.73 ft a t 50 KHz , 6 v e > p o s _ z e r o — see 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 f o r m u l a w i l l be s l i g h t l y s m a l l e r t h a n t h a t 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 . T h i s can be seen f r o m F i g . 1 5 . G a l l o w a y ' s f o r m u l a g i v e s r e s u l t s c l o s e r t o f i e l d measurements t h a n 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 , as e x p e c t e d . T h i s i s b e c a u s e t h e 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 h i g h f r e q u e n c y components whe re G a l l o w a y ' s f o r m u l a i s more a c c u r a t e . (See C h a p t e r 1, S e c t i o n s 4 and 5 ) . F o r c o m p a r i s o n p u r p o s e s , t he f i e l d measurement r e s u l t s f r o m 12 26 A m e t a n i a n d - s i m u l a t i o n s t u d i e s by G r o s c h u p f a r e i n c l u d e d i n F i g . 16 . The s i m u l a t i o n r e s u l t s o b t a i n e d w i t h t he methods d e s c r i b e d i n t h i s t h e s i s compare f a v o r a b l y w i t h t he f i e l d measurement r e s u l t s ( w i t h i n 8% ) . Some p r o b a b l e cau se s o f d i s c r e p a n c i e s be tween s i m u l a t i o n and f i e l d measurements may be due t o t h e f o l l o w i n g phenomena: 1) A s s u m p t i o n o f u n i f o r m e a r t h r e s i s t i v i t y (200 0, • m) — An i n c r e a s e u 30 ,31 xn e a r t h r e s i s t i v i t y w i l l i n c r e a s e t h e z e r o sequence p a r a m e t e r s and t h e r e b y i n c r e a s e a t t e n u a t i o n and d e c r e a s e wave v e l o c i t y o f t h e z e r o s e -quence v o l t a g e wave . A l s o , a homogenous e a r t h i s o n l y an a p p r o x i m a t i o n o f a s t r a t i f i e d e a r t h w h i c h w i l l a g a i n c au se some d i f f e r e n c e s i n impedance l i n e p a r a m e t e r c a l c u l a t i o n s . 2) Tempe ra tu re o f c o n d u c t o r — We assumed a c o n d u c t o r t e m p e r a t u r e o f 20°C . An i n c r e a s e i n t e m p e r a t u r e w i l l i n c r e a s e c o n d u c t o r r e s i s t a n c e a p p r e a c i a b l y ( e g . 40% r i s e f o r i n c r e a s e o f t e m p e r a t u r e t o 1 2 0 ° C ) . However , a d i f f e r e n c e be tween n u m e r i c a l and measured v a l u e s o f l e s 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 c r i t e r i a f o r t h e s e t y p e s o f s t u d i e s . 52 Output voltage at receiving end of transmission l ine Output voltage (p.u.) 1.0 0.8 0.6 0.4 0.2 J -0.2 415 „ t=0 83.212km output voltage tubular conductor " Galloway's formula 53 Output voltage at receiving end of transmission l ine with f i e ld measurement and Groschupf's simulation results Output voltage (p.u.) 1.0 415fi t=0 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 on m u l t i p h a s e o v e r h e a d t r a n s m i s s i o n l i n e s o r unde r g r ound c a b l e s was, s t u d i e d . A s p e c i f i c c a se o f a J a p a n e s e 500 kV t h r e e - p h a s e o v e r h e a d l i n e was cho sen as a t e s t example b e c a u s e f i e l d measurements were a v a i l a b l e f o r t h i s l i n e . The v o l t a g e r e s p o n s e a t t h e open -ended r e c e i v i n g end was s i m u l a t e d . The s i m u -l a t i o n r e s u l t s a g r e e d v e r y w e l l w i t h t he f i e l d measurement r e s u l t s and w i t h s i m u l a t i o n r e s u l t s o f a n o t h e r i n v e s t i g a t o r . R e s u l t s o b t a i n e d w i t h t h e i r t e c h n i q u e d e v e l o p e d i n t h i s t h e s i s w i l l 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 i n power t r a n s m i s s i o n s y s t e m s . F o r i n s t a n c e , t h e t e c h n i q u e c o u l d be u sed 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 he r e c e i v i n g end o f t h e l i n e . I t s h o u l d be r e a l i z e d , h oweve r , t h a t t h i s wave f r o n t w o u l d be m o d i f i e d by t h e t r a n s f o r m e r i t s e l f . T h i s wave f r o n t m o d i f i e d by t h e t r a n s f o r m e r w o u l d 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 . Depend ing on t he r i s e t i m e o f t he i n c i d e n t v o l t a g e wave, t h e t r a n s f o r m e r i n s u l a t i o n may be more o r l e s s s t r e s s e d . T h i s i s a p r o b l e m o f c u r r e n t c o n c e r n i n t h e 32 e l e c t r i c u t i l i t y i n d u s t r y APPENDIX 1 TRANSFER FUNCTION PROGRAM LISTINGS M l C n T G A M T e H * ! h * L S V S T f c " F I W I ' J K G ( « M 3 » 0 'A IN 0 3 - 2 3 - 7 7 1 1 : 3 7 : u ? P A G E P O O I _ j o o i _ . 0 0 0 ? 10 « 3 J 0 i ' f> ' . 0 " 7 u 0 U B i)0 OP II f i ' 0 -i; 0 1 ] uO 1 J y 0 1 5 -« ( > 1 « ., ii II 1 s _ u a i ? 0 0 1 <i » o 1 5 0 02 0 0 0 ? l 0 0 2 ? d i V 3 0 0 2 S C O " P l . f u S OFT , C O M ' l , v c X C i r 5 L F < » i o C - J R R i , C - . I R R 2 . C I . ' R ~ I p > . 1 - > \ IT, - < 1 , * 1 I < • 3 1 I , 0 00 -2 . 0 1 0 3 . 0 1 0 r - i , w r A CO!-' S E A •if C O " -cn>-I .« « OLF * R I * » =>LF I'l r l / i ^ O ) . • ) ! * l F , r « k ' j , T A - l , V P T ( 3 ) . W l * G , P l » S e i . P H A S E X * 1 h C7 ' 3 , 41 , V D ( 41 , V P ( 3 ) ,X , C U « « v'U I . vi)K , v t X ^ F . VfcX 1 1 , Z l ( 3 , 3 ) . Z l 1 ( 3 , 3 ) F ' < f 4 1 . i 5 S ! ^ , T , S U , ' , ' l M H ' < ' ' ( 3 ) , f I ( 3 ) , v K ( j , 3 ) , V l ( 3 » 3 ) x . l h v * n f 3 , l ) . ' " , 3 ' ' ( 3 . 1 ) . V ' ' 3 0 ( 3 , l ) . V > « 0 ( 3 , l ) 5 . 0 o "> 6 . u 1 0 - 7 ,0 .10 <i . 0 0 3 v. a i o . i a . t n u . c * * *. f * = 3 .iff L * ^ I. « • » I.'* n 5 E l A Y = 1; A ( 3 , 3 ) , -1 (' 3 , 4 1 , r. ( 3 , J ) , 0 (1. 3 ) O ^ f G A . f l E C A Y , A . ? " 2 , Z ( 3 , 3 ) . 7 1 ( 3 , 3 ) , Y ( 3 . 3 ) . f - C Y i N P J l 5 " 1 1 . 0 0 - 1 1 2 . "I P 0 1 3 , 0 0 0 1 u . 1 0 0 1 S . 0 i 0 11, . f ini i C vE t 1 2 3 3 C T r i , " R I I T H F A F l ' S • P E i i v f ' . T s ' . l S . TS W . ' T S T E P P H A S E V O L T A G E T . ' 2 , 3 ) ' • C F H A T « i ) ( ) ? I S C - . K P t c X *:*tl I T T A N C t O A T R I X Y I S P U R E L Y l b . 2 0 0 1 7 . u 1 -.1 1 ft . <J 1 a ! 9 . 0 0 (I 2 0 . o- r. o , L K 1 - » t » 1 I'< yfbn f ? , l l Y f l , U , Y ( ? , l ) , Y { 2 , ? ) « Y ( 3 , l ) , Y ( 3 , 2 ) , Y ( 3 , 3 ) R E A D f 2 ( l ) / t l . l ) . 7 £ 2 » n . Z ( 2 . 2 ) . 7 C 3 . 1 ) , Z C 3 , 2 ) , 2 ( 3 , 3 ) h i S - S T ( o i l i 3 , "51 P F F O r 2 , ! ) Z T C l , n » Z I ( 2 . 1 ) . 7 T ( 2 < ? ) . Z I ( 3 , l ) , Z I ( 3 , 2 ) , Z I ( 3 , 3 ) P R I M A T ( / / / 1 H 1 , ' . . . T H E F 0 L L P » I ' . G ARE C O M P U T I N G ' * T F R E G U E NC Y ' , E 1 3 . 6 1 , ' " 7 1 1 _ 1 1 = - - , 2 i = r . - 1 1 = 0 . — — Y {2 Y ( 4 Y ( 3 -UZ Z ( 3 Z f J 71 ( Z l ( Z l ( 2 2 . 0 0 ' , 2 3 . 0 0 0 2 i . 0 o 0 2 ^ . o 0 0 2 7 . 0 1 <> 27 . 0 1 0 i M . O - i o 2 9 , 0 0 0 3 0 , 0 1 0 - J U - ^ i A J -CECf , 1 1 = 1. , 2 l = T . ? , > 1 = 0 . 3 , 1 ) = 0 . 3 , >> = • ' . 3 2 . 0 0 0 3 4 . 0 1 0 3 - . - J O O -3 5 . 0 0 0 3 i . - '10 C c c»e. c I F £ 0 , 9 9 9 9 9 9 1 GO TO 9 9 A T E FUI L 7 * Y FROM I N P U T L O - E * T R I A N G U L A R M A T R I X 3 8 . u 10 3 9 . 0 0 0 - U O . 0 O 0 ti 1 , 0 i 0 u 2 , J 0 0 0 3 . 0 0 0 0 0 < *i u 0 2 7 Oti 2 " 0 0 P o 0 1 3 1 D O 10 1 = 1 » -J DO 1 o J = 1,••• 7 ( 1 , J 1 = 7 <-!, I I Z I ( T , .! ) - l I (.1 , I 1 r 7 r t , i l s p r - P l * f 7 f T , 11 , 71 (T ,-T) 1 i. 4 . 0 1 0 ii -3 . 1) 0 J ^ . V V* U ii 7 , o n 0 u i j . r i A . i 0 0 3 1 0 0 3 2 0 0 3 3 0 0 3 i v f I , J l = Y (.!, n c-. - - T : - , - . ' P - i l - . I 2 F O S — A T ( / , ' T «E 0 " j . i I s 1 . ••! ^ E A I P A R T OF I H ^ E O A N C t M A T R I X I',/) S u . O O ( . 5 1 . 0 1 0 5 2 . - o o 5 3 . 0 o u 5 » , 0 -i 0 U l M I C H I G A N T E R M I N A L S Y S T E M F O R I R A N G ( « 1 5 5 6 ) a _ 5 M A I N 0 3 - 2 3 - 7 7 1 1 t 3 7 i u a P A G E P 0 0 2 - 0 0 3 7 . 0 0 3 8 0 0 39 F O R M A T - ( 3 ( S X , r ) | 3 . 5 ) ) -P R I N T 5 F O R M A T f / / ' T H E I M A G I N A R Y P A R T OF I M P E D A N C E M A T R I X I',/) n n y i T - i , M 00U 1 30 C P S h ' T a , ( Z I ( I , J ) , 0 = 1 , 3 ) U 0 « 2 0 0 " 3 0 n " a — C . . P R I N T O U T OF Y . . . . . . . -P R I N T 6 ' 6 F O R M A T ( / , ' T H £ S U S C E P T A N C E M A T R I X Y ' , / ) n n /i ,-i T a t , N . -So 57 5 6 - 5 — 6 0 0 1 6 2 6 3 6 U o o o -ooo ooo o o a ono ooo ooo — ooo ooo XKU)— R c' U 0 U 5 n A ii». uo P R I N T « , ( Y ( I , J i , J = l , 3 ) C C PRODUCT n F Z R Y C n o c n T - l R M 6 6 67 6 8 69 7 0 -IX ooo 0 0 0 0 0 0 -0 0 0 0 0 0 00 0 0 0 1 7 0 0 U R __0 _ 9.. 0 0 5 0 0 0 5 1 1)0 5 ? D O S O J = 1 » N A (1 , J l = 7. ( I , J ) -5.0 e c i . J I = Y . ( i , J i ... C A L L M Y ^ U L T ( A , H , C , 0 ) O O 6 0 I = l . N n n nfi .i= i . 'i 0 0 5 3 O 0 5 (I _0.0S5_ 0 0 5 6 0 0 5 7 0 0 ^ 8 7.11 ( T . J ) = n ( I , J ) 60 A ( I , J l - 2 I ( I , J ) C A L L . MY.MULT... ( A , B . C . . D . ) _ 0 0 7 0 I s l . w no 70 J = 1 , N - L B Z l C T . ,T) = - f C T . -XX-_ _ 0 5 9 _ 0 0 6 0 0 0 6 1 " O n ? c . . . « . . » . Y is P U R E L Y I M A G I N A R Y e R . i u . t - - 9 F O R M S T ( / , ' T M E R E a L P A R T O F Z * Y M A T R I X ' , / ) D O 9 0 I = 1 , N • J 1 . ( .71 I I ..1 ) , 1=1 . 3 1 : 9 0 PR I 72 73 - 7 « 75 76 _7_Z-i'OO 0 0 0 ooo ooo 0 0 0 0 0 0 7 8 7') — 8 0 K l 8 2 S 3 0 0 0 0 0 0 0 0 0 . 0 0 0 0 0 0 ooo 61 85 3 7 88 00 0 OOO 0 0 0 -0 0 0 00 0 ooo 0 0 6 3 OOoU _Q.0i5. . U 0 6 6 0 0 6 7 - , ;< 1 N T 8 8 F O R M A T ( / , ' T H E I M A G I N A R Y P A R T O F Z * Y M A T R I X ' i / ) D-Q._a.a_i S.1...N 80 P R I N T <4, CH t ( I , J ) , J s l , 3 ) C A L L n C E I G N ( 2 1 , 7 . 1 1 , N , . N , E R , E I , V R , V I , 1 E R M O R , 1 , 0 ) _C : : 90 91 - 9 2 9 3 C< 0 0 0 0 0 ooo-ooo ooo r. o o C C O M P L E X " E I G E N V A L U E S E R + J E I «. C O M P L E X E I G E N V E C T O R S V R + J V I C O B T A I N E D F R O M S U B R O U T I N E D C E I G N J : '• 0 0 6 8 O O u O mi 7 0 P R I N T ! | I I F O R M A T ( / , ' T H E E I G E N V A L U E S O F Z « Y M A T R I X ' , / ) r n \ o n T = t , M  U 0 7 1 0 0 7 g J i _ 7 - S _ EC (1 ) s O r . M . p u x ( F P (11 , E I ( I ) ) 100 P R I N T l ? , t i ! f l ) , E I ( I ) 1? FPS _ 4 T _ t S X . , J _ R E A U L . 2 1 . 3 . 5 , . 5 X . , . - U L M A . G l „ A R X ± - , . Q - | / $ . 5 1 -0 0 7 4 00 75 0 0 7K P R I N T 1 3 1 3 F O R M A T ( / , M M E P R O P A G A T I O N C O N S T A N T I S T H E S Q U A R E R O D ! ' , / ) n n < i T = I , N 9 6 97 — 9 8 9 9 100 1 01 0 0 0 0 0 0 0 0 0 -0 0 0 0 0 0 O n " 1 02 1 03 -1-0 u 1 05 1 0 B 1 (17 OOO 0 0 0 a o o-ooo ooo 0 0 I") - _ L 0 i 7 _ c * « * . * * P R O P A G A T I O N C O N S T A N T E S ZS-C-U.S. C D saR.T-c.Ecx rn : — c c 1 1 0 PRTvT I P . F S C n 1 0 8 1 0 9 - 1 1 0 1 1 1 1 1 2 -XXX ooo ooo 0 0 0 -0 0 0 ooo tU_0_ 57 L at o •J ;3> o o O c o o o o •-1 ir- c H o o o o o c> c o o o o O 3 ~ o CT C — r^i a.' • £ -xj a o o a o c : 3 »— v <r c ^ X r u r v K CT — — ?. o 3 ^ a o c . n) o o c o M I ^ m x r-« co n x « t a l o o o c o o o o O o o o o o o o o c . o o o o c i —• -« r Z > " > «— CT t-J : : o o o o d - C C ' c c q < 0 o o = Or. o-o o C3 O u a rv _ • ( _ • LT 1/ — I o o o o o ^ c o o c -o o o => o o o o o o c i o o o o o o c . o o c o o c o c rvi ••*•> - J o o .T} o o o o o c | a o- o o a o j c . o o c c o o c i o c c i u u u u U U U l / ^ u u o o o o o <q o o o o c. O O 3 O Cl cn cn x . x aJ uJ c r "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 t- *-t rr_ fX t_J <_1 U. lA. - - - 11 Ci r v r-1 O rr? - | a u c CT X > > II l~ I IU c . c a LP * U . 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AJ <V I o o o o o • • • • • * nn o ' O CT- c t f\ l r\J AJ A l A l A l K l i X • Kt I UJ * I CO K l I - r * I uJ ro a — - • —* u -_o x • 1 <jj • uu A A j * - ijj •« UJ • O q us II ^ LT- - -t * 0 « i X —' CO —< Ct Jj J U O : <i «a : _r —• - J •' o <; a: >J II II JJ ; e t *- *--j. , - r-. 3 i a- - w w 11 u j . - - fx X 1 5* ;» > > I r K J ! _J r> _j --. I f x > 2] >~ J_ 4 - _ a _ « _ < • _ • ^ - Z _ _• « _) c >- a ._' _ _> o q 4 S r u u u u u t j _ o — U ( - - - r J ( »-q Kr K l d o e d c * a * o [ w rfl 1 . o c a. o. > : > ; > > .^t - i — 1 rr - i (X — u u l u , J J O [ -a «i •< -4 < r_ u j u l U J 1 M a « o: H o c o o O 1 II II II II II L7. LC Ui u_ 5" . . . •— Q . cr ct 4 — rv K — -< (T. q o o d Q. Q-U J 3 > > =f > > I r - i t Oi o — r i r o 3 ift -c o c l ; \a o ' d r o d 59 © o o © o o © © o = O O O O O C © © O © © . 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IT i AJ X I > O- A i i i r uj • L P I /} o o © q LTt © © d i <_ © I © A J ' © Aj © o c d -I I * I I lU l u u. u- U c.• c* _r> © A J UT| © A J A i X» S J C x - 6- n = r~ r - .o -c• -i 3 4) -i) © A: X i 3 K* Aj AJ -O © © © O ©1 u- UJ uJ UJ UJ A J — rv tr u-r- '.o A J st: __ At sX1 X> O- © C X- x> 4> r-~ rJ. 3 a 3 -o a d © o a © © ©J l l II i l UJ U i UJ UJ UJ i =3 - O C I T ' C " O K l O O m x j _• AJ Si V A ; •3 ^ Cf. — K. f r v P . -.n Lf • i P i n m u"j o © © o © d uJ U J U J U J u S) C © O K x| x K J -a n c - <r c © c — A : u A X t l C sO A) x r-j r- -c U " . J " . i/ l j i i/-. = " © * = i ' o ' © * U J U J J uJ a, u j Aj, a o ctj x n u" x rv r\« r- a] A J .-n C O I T - x — a - e-- c K i i / i C 'V — — — — A i Aj © c o o e d I f « N co 7 d • A J A J K l u i uJ a l I t ij.' U.i r- Lo rv a- c © X> rr AJ 3-^ — x ir> © X) oo r- r~ r -©• o o c © © © © © © q l l l f l l M. 'dJ u u ; L L X. 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K P s A K P A K P = K P ..S2 = A K P * 6 , 2 3 3 18530 -71 7.95BflO-t-S2-S T E 1 G = I S 2 - A l ) / C l A1 = S2 »• i s n ' - T ' i A W R T T F ( 7 , . 3 3 ) H E R r Z , X T F ( l ) , P T F ( l ) , X T F ( 2 ) , P T F ( 2 ) , X T F ( 3 ) , P T F ( 3 ) - R I T E ( 7 , 3 3 J F R E O , T F R E ( I ) , T F I M ( I ) , S 1 , S 2 _51. -F -0 .R±iA. I -XE-1_3-,-5^2Ea2-. 3 , X-1S..-5 , E4-S . . 5 . , E - L Z . - 5 ' , E - L & . - S ) GO To 3 END (rfrcf^t^. I P , F ^ C ' ? T R j S-'i'SP'TF i " i f , MBITF CK l O M l . N f l M A R NO 0 0 3 2 _0.0 3 3 0 0 3 U 0 0 3 5 A f l P T I f l N S * 0 P T1 u N s IN. E F F E C T * N A M. F s MA IN , L I N E C N T = 60 • S T A T I S T I C S * S O U R C E S T A T E M E N T S = 3 5 , P R O G R A M S I Z E s *S.T.A.rj.S T-IC S * NO—0-I-AG-NnS.T.lC-S—GE.NEK AJ-E-D . E R R O R S IN MA IN 3 3 2 2 6 9 . 5 0 0 9 . 6 4 0 9 . 7 0 0 1 0 . 0 0 0 11 . 0 0 0 1 2 . 0 0 0 1 4 . 0 0 0 1 5 . 0 0 0 - 1 6 . 0 0 0 -1 7 . 0 0 0 I d . 0 0 0 1 9 . 0 0 0 2 0 . 0 0 0 21 . 0 0 0 - 2 2 . O O d -2 3 . 0 0 0 2 4 , 0 0 0 as. ooo 3 2 . 0 0 0 3 2 . 2 0 0 - 3 - 3 , 0 « 0 -3 5 . 0 0 0 3 6 . 0 00 » , n u i T f » F m i t{ i f - r . f n TN T u F A H r w F rOMP 11 a T T QMS „ E X E C U T I O N T E R M I N A T E D $R - L 0 4 0 7 = T F J A ? A * e x g n i T n i v B E E I M S 2 3 * . S O t i R C E * 3 = T F R E I M J A P A N ' ( 2 ) 6 = * S I N K « ON H P H A S E E X C I T E D 1 APPENDIX 3 FOURIER TRANSFORMATION PROGRAM LISTINGS M I C H I G A N SYSTS -x F O R T R A N G ( U 1 3 3 e i ) M A I N 1 3 - 2 3 - 7 7 o n o l ooo? 9003 C C : . P R O G R A M F O U R I E R C I . N P U T , O U T P U T ) R F ' 1 * » A ( 1 5 0 0 ) , P H I n 5 O 0 ) , U ( 1 O 0 O ) , F R E i 3 ( 2 O O ) , T E X T ( 1 O ) P F A | . R r ? , " i ? , n " c K A , P M l ' 4 t ' * , P H l P L P . H , 8 1 , C l , F l . 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( 31 3 , 1 X , 3 E 1 1 , 3 , 1 OAu } Ef. . 0 1 S T T ' P p . r l f Ye XT H ) . 1 = 1 . 1II) 1 1 12 13 1 « 1 5 -in. Q00 0 0 0 000 (i nn L J 0 00 0(10 o o a o o o o 0 f) o o o 0 0 0 0 0 0 Ooi) 0 0 0 u o o 0 0 0 0 0 1 6 0 0 1 7 00 1 8 00 19 O O P " 6 F l ' S - " A Y I - » S T B * . I N P U T F n R « ! • : = . « I M = 3 T F T •- = ( 1 H 17 + 1/4 feRLi . 2 7 0 , 1 (, A it ) 1 o o c MS I N C I D E N T . W A V E . _ I) - 0 *  1 7 l h 1 9 20 2 ) _2_2_ Oon 0 0 0 0 0 0 (•0 0 0 0 0 Ci (I i) 0 0 2 ; u 0 2 2 O . V 3 0 02 4 0 0 2 5 » 0 0 T T T ' = A L P H A A L P"i A R T T r FUt- :-AT 4 0 0 1 = A 2 = H ( o , ( F X 1 O . n - i i o 1 ' i / T T T'4 i - . / r r i • . . . .. * P 0 ) A L O H A 1 , A L P H A ? , " I N R U T V O L T A G E T I M E C O N S T A N T S : A L P H A I = ' , E I 3 , S , • A L P H A 2 = I 2 5 24 2 5 2 c 2 7 _ 2 i Onu 00 0 o c a 0 0 0 0 0 0 COG 0 0 2 h 0 0 27 0 0 2 * 0 0 2 9 .00,50 I F ( r E x R . F o , n A | . P H A = 4 L P H A 1 28 IF ( TF V P . 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U II M U, 2 CO (T. 3 -< 3 O i—i »— :< — *—i • _c X" , _ * c >- <\ tT3 •tj rC or fO ro _ J C L z U u X " cr >-z <x cr a o - AJ fO ^ LT rt O c — X o- t r a a- CT o o a a- o o c x c o - o C c o o <_ o —« —• o D O o o o o o o o t—1 3T o o o o o c c • c o o c c o o o o o o o o o o o c : o c a o _ > g C ' o o cr. c o o o o o O O O O O i - O O O O O f • » • » • • r - c o r r - o — .*V T 1/1 JJ i o o a o o o a o o c j o o o o o a o • o o c o d o r*j ^ o o o j o o o l o o o II co w Z\ _ c u ' II t*VJ It j II . o o o i — II C J r u f\> i _ i C =) H U {/. 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T F i r , * n - S ? * A 1 W h , ? f l X 1 « 5 ^ 0 7 1 6 1 . 0 0 0 1 6 5 . 0 0 0 1 6 6 0 n 0 0 1 3 3 013=) n l <• A K P = O K P + S I G N C O . 5 , A K P ) K P r A K P A K P s K P 1 6 7 . 0 0 0 16 8 . 0 0 0 1 6 9 0 0 0 { O i u i » H 2 o m S ? = A h p « 6 , 2 8 3 1 6 5 3 0 7 1 7 9 5 8 0 0 + S2 S T E TG= C S 2 - 4 ] ) / C l A1X<5? 1 7 0 . 0 0 0 17 1 , 0 0 0 1 7 ? 0 0 0 o i u u 0 1 « 5 _.c W l m M f G A C 1 r S 1 / T M A X I N _.: C 1=5=1 . . . 1 7 3 , 0 0 0 1 7 U . 0 0 O 1 7 5 o n n c c r A M P L I T 1 1 O F n F V O L T A G E O B T A I N E D BY F O R M U L A AND PROGRAMME ARE P R I N T E D OUT A S C l " N O SI 51 T S T n R F = T S T n P F + 1 4 » j . u — . 1 7 6 . 0 0 0 17 6 . 0 0 0 17 7 n 0 n c c c. I S T O K E I N C R E M E N T A L S H I F T E D TO L I M E 1 1 7 W R I T E ( 6 , 5 ^ 1 H F R r Z , A W , n w , S l , S 2 , A ( I S T 0 R E ) , P M I ( I S T 0 R E ) A_ t, P H I ABE S T I L L T R A N S F E R . F U N C T I O N AS READ IN FROM I N P U T F I L E 1 7 6 . 0 0 0 17 9 , 0 0 0 — 1 8 0 , 0 0 0 1 8 1 , 0 0 0 ! 8 2 . 0 0 0 1 8 3 r o n 0 0 1 1 6 0 H 7 o m a t TF=.A ( I S T O R E ) P T F s p h T ( I S T O R E ) 51 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 5 ) 0 1 4 9 0 1 5 0 _ C _ A( 1 R T O H F ) = S 1 * A f I S T O R E ) P r i l f I s T 0 R F ) s S 2 + P H l ( l S T O R E ) _ i_ i . .P .H . l . . « a » _ . B E C Q * E S J i .U. lPJtT Y Q L . T A . G E & P H A S E A S A £ 1 E R . / » 1 7 1 1 8 1 . 0 0 0 1 * 5 . 0 0 0 18 6 , 0 0 0 0 1 5 1 0 1 5 2 c w R l T E f 6 , 5 1 ) H E R T Z , A * , B « , S l , S 2 , X T F , P T F , A ( I S T 0 P E ) , P H I C I S T 0 R E ) 53 I STnHF. = I S T O R E * l 1 8 7 . 0 0 0 1 8 « , 0 0 0 1 8 9 , 0 0 0 0 1 5 3 . 0 1 5 1 0 1 5 5 GO TO 38 uu O M D E C i 0 K D E C * 1 0 , O 0 GO TO 36 1 9 0 , 0 0 0 19 1 , 0 0 0 1 9 ? , 0 0 11 0 1 5 6 c r. T R A N S F E R n U T P U l F U N C T I O N FROM F R E Q U E N C Y TO T IME O O M A l N « * * * « * * « * « « * * 57 C O N T I N U E 1 9 . 3 , 0 0 0 19 4 , 0 0 0 19 5 ij 0 0 0 1 5 7 0 1 S R C DTOUT 18 H E L T A T OF O U T P U T DTOi !T = , 2 5 0 - 0 6 r O l l T l = ? 6 0 , E - 6 1 9 6 , 0 0 0 19 7 , 0 0 0 1 O H , 0 00 0 1 5 9 0 1 6 0 0 1 6 1 I = T O U T l W R I T E ( 6 , 5 8 ) D T O U T . T 58 F O " M A T ( < 0 n i i T P | ' T F U N C T I O N V 2 f T l t OFI T A T * > . F 1 2 t 5 . • . TNT T I A ! T I M E = 1, 19 9 . 0 0 0 2 0 0 , 0 0 0 2 0 1 0 0 0 C c 1 E 1 2 . 5 ) T F f l = n . 9 9 9 6 0 3 2 0 1 . 0 0 0 2 0 1 . 0 2 0 2 0 1 0 5 0 c c c T . F . AT. Z F ^ O F R E D 2 0 1 , 1 0 0 2 0 1 . 1 5 0 2 0 1 , 2 0 0 0 1 6 2 c c T F O = . 0 8 9 6 5 2 2 T F o = 0 3 7 0 5 6 5 2 0 1 . 2 5 0 20 1 , 5 0 1 ' 2 0 1 7 p 11 c c c G A L L O W A V ' S F O R M U L A ON • S K I N > O E C M E A s E T . F , 2 0 1 . 7 6 0 2 0 1 . 8 2 0 2 0 1 8 0 0 0 1 6 3 0 1 n i l 0 165 A ( 1 ) = T F O * A 1 2 « ( 1 , 0 0 / A L P H A l - 1 . 0 0 / A L P H A 2 ) P H I ( l i = 0 , O 0 W R T T F C 6 , 5 5 ) A ( i J . P H I f 1 ) 2 0 2 . 0 0 0 2 0 3 . 0 0 0 2 0 4 0 0 0 0 166 0 1 67 0 1 6 * K I : p 63 A W = 0 , 0 0 B w = n , n n 2 0 5 , 0 0 0 2 0 6 , 0 0 0 2 0 7 , 0 0 0 O o c c o c a - 3 o -o o o o o o o o c c 0 o o o o o o o o o c c c e c o c o o o o o o o o a c o a\ c . c - a o c o c o o o c l o o o o o c o o a o o q o o a c. o c o o a c - c -C C: Ct © O w O O O O O o o o c o c f c c o c o o o a o o c j o o o o o c J c o c c c c j cz> o a z> o c o o a o c o o o o o c l o o o o o oJ _a a- o —, a c — A J A _ A J A J f\j ro cj' _r. c r~-l A J A _ A J r\j m A J to o- o « H — — r\) A J A , r\J ry o j f\j ( \ r\i o j Aj Aj A J f\J A J ( \ rvj r u r\j r\j r\j \ C f\J !* l 7 Lfl r « K i y i M r n M t\, ru r j r \ fV f\J r y f vi (\i ^ m -c N ^3 T ^ 3 ^ r\i A J A i ( V rvi fxi co cr a - - i\i K i 3 ^ i / i i n i r j i r\j r\i o j rvj (\j i n . ^ ! ' i i a. •-! * c co x o c 3 C J O o- cr o-i LA IT LO u- c* o-r - r - r -r - r - r*-| <r> <r u i K i K i K, X) 'O £Cj A J A.1 A j C - C o I I I ! 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I H . . ._ A I N No S T A T E M E N T S F L A G G E D I N T H E A B O V E C O M P I L A T I O N S , F Y F C I I T T O N T F R M T N A T F O  2 2 9 , P R O G R A M S I Z E 79861 $R - L O A D 2 = . S O U R C E * 6 = * S I N K * 7 - F . N D A L P H A 1 2 8 E X E C U T I O N B E G I N S f 9 30 0 t O n F . , 0 0 O ^ O o F - 0 6 O - 5 0 F - 0 1 3 - T F J A P A N 2 J A P A N L I N E 1/1000 MS I M P I / 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 0 . . . t .7_ l_ ! . 0___l .3_.A_PHA2_—0 .327 . 0 0 E + 0.7-. « * * T T M F C O N S T A N T TM O F T A y 0 1 7 0 O 0 E t O 3 * * « « * I N P U T T I " E T I M E S T E P F R O M o T n 5 M S = • V O L T A G E A R E A O F I N P U T = 0 . 5 0 0 0 E -• 0 . 9 6 3 6 1 E . 0 7 05 ( A L M H A 1 * A L P H A - n . n - 0 . - O 5 0 F . t 0 n . O . 9 6 1 9 E + 0 O ...-.0.. ?. 9.24 E.t 0.0. - 0 . 9 9 A 3 E + 0 0 - 0 . 9 9 9 U E + 0 0 - n . Q O q S F + O O 2 1 / A L P M A I - A . n m o B F + on P ~ A ? ) _ 0 - r > T ? / a n f > 0 0 , 1 0 0 0 IE+ 0 1 • o , 3 H 7 7 F + 0 0 •JL / I8P IF + 00 . 0 , 5 5 8 ^ + 00 - O t 6 ? 5 1 F t O 0 - 0 , 6 « 1 6 F t f l O - O . 7 2 9 6 F + 0O - 0 • 7 7 0 4.F 10 0 - 0 . R 3 U U F tOO -0 , 9 6 7 6 F tOO . - 0 . . ? . 9 . 3 5 E t 0 i L - 0 . 9 9 8 5 E t O O - 0 , 9 9 9 1 E + 00 - O . Q Q Q S F t O O - o , 8 5 9 U E + 00 • 0 . 9 7 2 5 F + 0 0 -0...9-9U.5E-+.00-. - O . 9 9 8 7 E + 0 O - 0 . 9 9 9 5 E + 0 0 - n , 9 9 9 S F +00 - O . 8 8 0 6 E + 0 0 - 0 , ° 7 6 6 E + 0 0 . . - . £ ) . 9 9 5 3 t + QQ_ - 0 . 9 9 8 9 E + 0 0 - 0 . 9 9 9 5 E tOO - n 9 9 9 5 F + 00 - 0 , R 9 8 6 E +00 - 0 . 9 8 0 1 F +00 -_0 .996.QE.t-0.Q_ - 0 , 9 9 9 OF + 00 - 0 . 9 9 9 5 E + 0 0 - n . 9 9 Q S F +00 • 0 . 9 1 3 9 E + 0 0 • 0 , 9 8 3 1 E t O O • - 0 . 9 9 6 5 E + 00 • O , 9 9 9 l E t 0 0 • 0 . 9 < / n 5 E t O O - n , 9 O 9 5 E + 0 0 - 0 . 9 2 6 9 E + 0 0 - 0 . 9 » 5 e E + 0 0 - 0 . 9 9 7 Q E t O . 0 _ - 0 . 9 9 9 2 E + 0 0 - 0 . 9 9 9 5 E + 0 0 - 0 . 9 9 9 5 F + 0 0 • 0 . 9 3 7 9 E + 0 0 • 0 . 9 R 7 8 E + 0 0 • o . 9 « 7 ; ; F t O . - 0 . 9 9 9 2 E + 0 0 • 0 , 9 9 9 5 f t o o • 0 . 0 9 9 5 F + 0 0 - O , 9 4 7 2 E t 0 0 - 0 . 9 8 9 6 E t O O ?0 . 9 9 7 8 E t . 0 0 . •0 . 9 9 9 3 F . +00 • o , 9 9 9 5 E • 0 0 . O t 9 9 9 5 F t 0 0 9 5 5 2 E t 0 0 9 9 1 1 E + 0 0 9 9 8 l E t O O . 9 9 9 4 E + 00 9 9 9 5 E + 0 0 9 9 9 S F +00 • 0 . 9 9 9 5 E + 0 0 - 0 . 9 9 9 4 E + 0 0 . 0 , 9 9 9 «£.tO'. l . 0 . 9 9 9 3 E + 0 0 - 0 . 9 9 9 2 E + 0 0 . o . 9 9 a i F + o o - o 0 . 9 9 9 5 E t 00 0 ; 9 9 9 u E * 00 .0.99.9.3 F.J-JIO.. 0 . 9 9 9 2 E + 0 0 0 . 9 9 9 ^ E + O 0 9 9 O 1 F + 0 0 ' • 0 . 9 9 9 5 E + 9 0 - 0 , 9 9 9 u E + 0 0 -0-..9._.9.3Et.O.O. - 0 . 9 9 9 2 F + 0 0 - o . 9 9 9 ? F t 0 0 - o . 9 9 q i F t n o • 0 . 9 9 9 5 E + 0 G • 0 . 9 9 9 u E +00 ;_.9_9_9..3E.tO.G. - 0 . 9 9 9 2 E + 0 0 • 0 , 9 9 9 1 E t O O • 0 . 9 9 9 IF + 00 > 0 , 9 , o - E t 0 0 • 0 , 9 9 9 u E t O O r _ Q . 9 . 9 ? i E t O O _ - 0 , 9 9 9 2 F t O O • 0 . 9 9 9 1 E t O O - 0 . 0 9 9 t F • 0 0 • 0 . 9 9 9 5 E t 0 0 • 0 , 9 9 9 4 E 100 • 0 . 9 9 9 3 . . t 0 0 • 0 . 9 9 9 2 E + 0 0 • 0 . 9 9 9 1 E 100 • 0 , 9 9 0 - 1 F + 00 • 0 . 9 9 9 5 E + 0 0 • 0 . 9 9 9 U E + 0 0 - 0 . 9 9 9 3 E + 0 0 -- 0 . 9 9 9 2 E + 0 0 - 0 . 9 9 9 1 F + 0 0 - n . 9 9 9 o r + o o • 0 , 9 9 9 1 f . +00 • 0 . 9 9 9 4 E + 0 0 • 0 . 9 9 9 3 E + 0 0 • 0 . 9 9 9 2 E + 0 0 • 0 . 9 9 9 I E + 0 0 . 0 . 9 9 9 n F + 0 0 - 0 . 9 9 9 4 f t 0 0 - 0.9991E+ 0 0 . 0 . 9 9 9 3 E + 0 0 - 0 . 9 9 9 2 E + 0 0 - 0 . 9 S 9 1 E + 0 0 .0.9990F t 0 0 , 9 9 9 ' J C + OO , 9 9 9 3 E t O O , 9 9 9 3 E t O O -, 9 9 9 2 t t 0 0 , 9 9 9 1 E t O O , 9 9 9 0 f t O O - 0 . 9 9 9 0 E t O O - 0 , 9 9 8 9 E t O O ..0..«5.8J}£.+.0 .a - 0 . 9 9 8 7 E t O O - 0 , 9 9 « 7 E t 0 0 .r^oOK-F + Of) • 0 . 9 9 9 0 E + 00 • 0 . 9 9 A 9 F . + 00 „Q,._°._.6£.+.0.Q. - 0 . 9 9 8 7 E + 0 0 - o , 9 9 A 7 f t o o • fl _ 3QP.-,Ft00 • o , 9 9 9 o E + 0 0 -0 . 9 9 8 9 F + 00 r_o.59-6aE.tao - O . 9 9 8 7 E t 0 0 - 0 , 9 9 8 6 E t O O • " . 9 9 B „ F » 0 0 - 0 . 9 9 9 0 t t O O • 0 . 9 9 8 9 E t O O -.0.9.9 88E.+.O.Q.. .0 , 9 9 « 7 E t o O • 0 . 9 9 8 6 E t O O - n q o H 5 F t n i ) - 0 , 9 9 9 0 E tOO -0 , 9 9 J j 9 E t'OO i 0 . . 9 9 a 8 _ . t . 0 O . • 0 . 9 9 8 7 F t O O - 0 . 9 9 8 6 F t O O • f l . 9 9 8 5 F t f l i l • 0 . 9 9 9 0 E t 0 0 • O . 9 9 8 9 E t O 0 •O . . 9 9 a _ E t 0 0 . • 0 . 9 9 8 7 E t O O • p . 9 9 _ 6 E t O O •n 9 9 A % F * 0 0 - 0 . 9 9 8 9 £ t O O - 0 , 9 9 8 9 E t O O . - 0.99BaEtoO-- 0 . 9 9 6 7 E + 0 0 - 0 . 9 9 8 6 E t O O - 0 , 9 9 K 5 F +00 • 0 . 9 9 I J 9 E + 00 - 0 . 9 9 6 9 E + 00 _ 0 . 9 9 8 _ E + 0 0 -• 0 . 9 9 8 7 E + 0 0 • 0 . 9 9 8 6 E t O O ,n 9 9 H ^ F +011 • 0 . 9 9 8 9 F . +00 • 0 , 9 9 8 8 E t O O . 0 . 9 9 8 - E + 0.0 •0 . 9 9 8 7 E + 00 • 0 . 9 9 8 6 E + 0 0 •0 . 9 9 » S F . O O , 9 9 8 9 E t O O , 9 9 8 8 F . t O O , 9 9 8 8 E . t O O . . , 9 9 8 7 E t 0 0 . 9 9 8 6 E + 0 0 9 9 A S F « 0 0 0 . 9 9 S 5 E + 0 0 - 0 . 9 9 8 5 E + 0 0 - 0 . 9 9 8 5 E + 0 0 - 0 . 9 9 8 5 E + 0 0 - 0 . 9 9 8 5 E + 0 0 - 0 . 9 9 8 4 E + 0 0 - 0 . 9 9 8 1 E + 0 0 - 0 . 9 9 8 U E + 0 0 - 0 . 9 9 8 U E + 0 0 - 0 . 9 9 8 4 E + 0 0 T I M E OF I N P U T V O L T A G E n 0 0 C O O P - 0 7 Q . 1 n O F - O h 1 5 0 F . 0 6 O 3 O 0 F - O 6 0 , 2 ' ; 0 r - 0 6 [ 1 , 300F -06 0 , 1 5 0 F - 0 6 n . d f l f l F - 0 , - 5 0 F . 0 6 BIBLIOGRAPHY 1. T. A m e t a n i , " S i n g l e P r o p a g a t i o n on a 500kV U n t r a n s p o s e d L i n e : F i e l d T e s t R e s u l t s " , CIGRE WG 13-05 13-74 (W.G. 05) 36 IWD, O c t . 1974. 2 . T. A m e t a n i e t a l , " S u r g e P r o p a g a t i o n on J a p a n e s e 500 kV U n t r a n s p o s e d T r a n s m i s s i o n L i n e " , P r o c . I E E , V o l . 121 , pp . 1 36 - 138 , 1974. 3. A . 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