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UBC Theses and Dissertations

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

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IMPEDANCE CALCULATION OF CABLES USING SUBDIVISIONS OF THE CABLE CONDUCTORS by Kodzo Obed Abledu B.Sc.(Hons.), U n i v e r s i t y of Science and Technology, Kumasi, 1976 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n THE FACULTY OF GRADUATE STUDIES (Department o f E l e c t r i c a l Engineering) We accept t h i s t h e s i s as conforming to the re q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1979 (c) Kodzo Obed Abledu, 1979 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree a t 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 agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department n f £ U F C T K ( ^ * U - 6 K G t r ^ R i H q The U n i v e r s i t y o f B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date S€ P T e/Wfe<S« ^ , ABSTRACT The impedances of cables =are some of the parameters needed f o r v a r i o u s s t u d i e s i n cable systems. In t h i s work, the impedances of cables are c a l c u l a t e d using the s u b d i v i s i o n s of the conductors ( i n c l u d i n g ground) i n the system. Use i s a l s o made of a n a l y t i c a l l y derived ground r e t u r n formulae to speed up the c a l c u l a t i o n s . The impedances of most l i n e a r m a t e r i a l s are c a l c u l a t e d w i t h a good degree of accuracy but m a t e r i a l s w i t h h i g h l y n o n l i n e a r p r o p e r t i e s , l i k e s t e e l p i p e s , give l a r g e d e v i a t i o n s i n the r e s u l t s when they are represented by the l i n e a r model used. The method i s used to study a t e s t case of induced sheath currents i n bonded sheaths and i t gives very good r e s u l t s when compared w i t h the measured values. i i i T A B L E OF C O N T E N T S A B S T R A C T i i T A B L E OF C O N T E N T S i i i L I S T OF T A B L E S • V L I S T OF I L L U S T R A T I O N S v i A C K N O W L E D G E M E N T S v i i i L I S T OF S Y M B O L S i x 1 . I N T R O D U C T I O N 1 1 . 1 A B r i e f R e v i e w o f M e t h o d s f o r t h e C a l c u l a t i o n o f C a b l e I m p e d a n c e s 1 o 1 . 2 S k i n a n d P r o x i m i t y E f f e c t s 1 . 3 A B r i e f E x p l a n a t i o n o f t h e M e t h o d o f S u b c o n d u c t o r s 4 1 . 4 S c o p e o f t h e T h e s i s 5 2 . T H E O R Y AND T H E F O R M A T I O N AND S O L U T I O N OF E Q U A T I O N S 6 2 . 1 S u b d i v i s i o n s o f t h e C o n d u c t o r s ^ 2 . 2 A s s u m p t i o n s ^ 2 . 3 L o o p I m p e d a n c e s o f S u b c o n d u c t o r s ^ 2 . 4 F o r m a t i o n o f I m p e d a n c e M a t r i x 2 . 5 B u n d l i n g o f t h e S u b c o n d u c t o r s i n t h e I m p e d a n c e M a t r i x * ' ' " 2 . 6 R e d u c t i o n o f t h e L a r g e I m p e d a n c e M a t r i x 2 . 7 T h e C h o i c e a n d C o n s t r a i n t o n t h e R e t u r n P a t h 1 5 2 . 8 I n c l u d i n g t h e C o n s t r a i n t o n t h e C u r r e n t i n t h e M a t r i x S o l u t i o n - 1 3 3 . R E T U R N P A T H I M P E D A N C E , 2 0 3 . 1 R e t u r n i n N e u t r a l C o n d u c t o r s O n l y 2 0 3 . 2 R e t u r n i n G r o u n d O n l y 2 0 3 . 3 R e t u r n i n G r o u n d a n d N e u t r a l C o n d u c t o r s 2 2 3 . 4 U s e o f A n a l y t i c a l E q u a t i o n s f o r G r o u n d R e t u r n I m p e d a n c e • • . 2 2 i v 3.5 Model Using Ground Return Formulae D i r e c t l y with the Subconductors - Model II 25 3.6 Representing the Ground as One Undivided Conductor -Model I I I 26 3.7 Mutual Impedance between a Subconductor and Ground with Common Return i n Another Subconductor 2 8 3.8 Comparison of Model III with the Transient Network Analyzer C i r c u i t 31 4. RESULTS 33 4.1 Comparison of the Method of Subconductors with Standard Methods 33 4.2 Comparison of Ground Return Formulae 37 4.3 Comparison of Results from the D i f f e r e n t Models . . . . . . 47 4.4 Reproduction of Test Results 49 4.5 Pipe Type Cables 54 5. CONCLUSIONS 63 LIST OF REFERENCES 64 V L I S T OF T A B L E S T A B L E 4.1 V a r i a t i o n o f I m p e d a n c e w i t h t h e N u m b e r o f S u b d i v i s i o n s 34 4.2 V a r i a t i o n o f t h e I m p e d a n c e o f t h e C i r c u i t o f F i g . 4.3 w i t h t h e N u m b e r o f S u b d i v i s i o n s , S h o w i n g t h e I n c l u s i o n o f B o t h S k i n a n d P r o x i m i t y E f f e c t s i n t h e C a l c u l a t i o n s 37 4.3 S e l f I m p e d a n c e o f G r o u n d R e t u r n P a t h a s C a l c u l a t e d U s i n g S u b d i v i d e d G r o u n d a n d O t h e r F o r m u l a e 4.4 M u t u a l I m p e d a n c e B e t w e e n Two U n d e r g r o u n d C o n d u c t o r s ^ 4.5 C o m p a r i s o n o f V a r i o u s M o d e l s 0 4.6 I n d u c e d C u r r e n t s i n B o n d e d S h e a t h s ~ , - > 4.7 I m p e d a n c e o f P i p e T y p e C a b l e s f o r V a r i o u s D e g r e e s o f M a g n e t i c - - -S a t u r a t i o n J ~ o 4.8 Z e r o S e q u e n c e I m p e d a n c e M e a s u r e m e n t s o n T h r e e C a b l e s E n c l o s e d i n a P i p e w i t h P i p e R e t u r n J 4.9 I m p e d a n c e s o f C a b l e s i n M a g n e t i c P i p e s R e p r e s e n t e d a s T w o C o n c e n t r i c P i p e s o f D i f f e r e n t P e r m e a b i l i t i e s ^ v i LIST OF ILLUSTRATIONS FIGURE 1.1 Current distribution in solid round conductors due to skin and proximity effects 3 1.2 Current distribution in the subdivided conductors of the model 5 2.1 Subdivision of the main conductors 6 2.2 Circuit of two subconductors with common return 7 2.3 Geometry of Subconductors £, k, q. 8 2.4 Illustration of the reduction process. 15 2.5 A: two-w;ir'e return' c i r c u i t 1'6 2.6 A two-wire c i r c u i t with, common r e t u r n i n a t h i r d _ conductor 16 3.1 Subdivision of ground into layers of subconductors 21 3.2 Model with, only subconductors and ground return . 2 6 3.3. Model with ground represented as only one conductor 2 7 3.4 A circuit of two conductors with common ground return 29 3.5 A circuit of one conductor and the ground with common return in another conductor 2 9 4.1 A return ci r c u i t of two conductors far apart - . . . 33 4.2 Variation of impedance with the number of subconductors 35 4.3 A return circuit of two conductors very close together 35 4.4 Variation of the impedance of a buried conductor with depth of burial 3'8 4.5 Cross sections of buried conductors for ground return impedance calculations • 4 0 4.6 Comparison of calculated self impedances of a ground return loop • 43 4.7 Comparison of calculated mutual impedances between two buried conductors 46 4.8 Elec t r i c a l layout of the induced sheath current test 50 4.9 Circuit diagram of the induced sheath current test 51-v i i 4.10 V a r i a t i o n of magnetic p e r m e a b i l i t y of s t e e l pipe w i t h current i n the pipe 57 4.11 Shape of magnetizing curve during one c y c l e 57 4.12 L i n e a r i z e d magnetizing curves 62 ACKNOWLEDGEMENTS I would l i k e to express my thanks to my supervisor, Dr. H.W. Dommel, for h i s help throughout t h i s work and for the timely suggestions and corrections he made. Also, I wish to convey my gratitude to Mr. Gary Armanini of B r i t i s h Columbia Hydro and Power Authority, f o r making his report and test r e s u l t s available for use i n t h i s work. I am also very g r a t e f u l to the Government of the Republic of Ghana for financing my education at the University of B r i t i s h Columbia. For reading through and correcting the s c r i p t s , I would l i k e to thank Ms. Marilyn Hankey of the Faculty of Commerce. The typing i s b e a u t i f u l l y done by Mrs. Shih-Ying Hoy of the Department of E l e c t r i c a l Engineering; I do appreciate i t very much. ix L i s t of Symbols B f l u x density D„ distance between conductors £ and q £q D pipe diameter P e =2.71828 f frequency g subscript denoting ground GMR geometric mean radius h depth of b u r i a l of conductor i , I current I pipe current p j complex operator = /-I i,j,k,£,n subscripts km kilometres m = /(jyu/p) m metres M inductance q subscript denoting return path r,R resistance, radius v,V voltage X reactance Z impedance K Q, Bessel functions log Common logarithm (base 10) £n Natural logarithm (base e) Hz hertz U absolute permeability of free space = 4irx 10 ^ H/m X y =viQy , p e r m e a b i l i t y y r r e l a t i v e p e r m e a b i l i t y tj) f l u x ¥ f l u x l i n k a g e = ir = 3 . 1 4 1 5 9 2 6 . . . y = 0 . 5 7 7 2 1 5 7 . . . E u l e r s C o n s t a n t ft o h m (JJ =2lTf 1 C h a p t e r 1 I N T R O D U C T I O N 1 . 1 A B r i e f R e v i e w o f M e t h o d s f o r t h e C a l c u l a t i o n o f C a b l e I m p e d a n c e s F o r t h e a n a l y s i s o f t r a n s m i s s i o n l i n e s y s t e m s , o n e o f t h e b a s i c i n p u t p a r a m e t e r s i s t h e i m p e d a n c e o f t h e l i n e s . F a u l t s t u d i e s , s u r g e p r o p a g a t i o n s t u d i e s a n d t h e c a l c u l a t i o n o f m u t u a l i n d u c t i o n e f f e c t s b e t w e e n l i n e s a n d p a r a l l e l a d j a c e n t c o n d u c t o r s ( s u c h a s o t h e r l i n e s , p i p e s a n d f e n c e s ) a l l r e q u i r e r e a s o n a b l y a c c u r a t e i m p e d a n c e d a t a . U n d e r g r o u n d c a b l e s y s t e m s h a v e b e e n a n a l y s e d b y m a n y a u t h o r s . T h e w o r k o f D . M . S i m m o n s [ 1 ] r e s u l t e d i n t h e p u b l i c a t i o n o f s t a n d a r d c h a r t s w h i c h a r e f o u n d i n m a n y h a n d b o o k s a n d w h i c h a r e o f t e n u s e d i n i m p e d a n c e c a l c u l a t i o n s f o r d i s t r i b u t i o n s y s t e m s . F o r s i n g l e - c o r e d ( c o a x i a l ) c a b l e s , S c h e l k u n o f f [ 2 ] h a s d o n e a c o m p r e h e n s i v e a n a l y s i s a n d h i s r e s u l t s a r e w i d e l y u s e d . C a r s o n [ 1 0 ] , P o l l a c z e k [ 1 9 ] , W e d e p o h l a n d W i l c o x [ 9 ] h a v e a l s o d e r i v e d e q u a t i o n s f o r t h e i m p e d a n c e o f u n d e r g r o u n d c a b l e s w i t h g r o u n d r e -t u r n . S m i t h a n d B a r g e r [ 3 ] , L e w i s , e t a l . [ 4 , 5 ] a n d o t h e r s h a v e c a l c u l a t e d t h e i m p e d a n c e s o f c o n c e n t r i c n e u t r a l c a b l e s u s e d i n d i s t r i b u t i o n s y s t e m s . I n m a n y o f t h e f o r m u l a e u s e d i n i m p e d a n c e c a l c u l a t i o n s , f a c t o r s a r e u s e d t o c o r r e c t f o r t w o i m p o r t a n t e f f e c t s , n a m e l y : s k i n a n d p r o x i m i t y e f f e c t s . A n o t h e r a p p r o a c h h a s b e e n u s e d b y C o m e l l i n i , e t a l . [ 7 ] a n d L u c a s a n d T a l u k d a r [ 2 2 ] , w h o h a v e c a l c u l a t e d t h e i m p e d a n c e s o f t r a n s m i s s i o n l i n e s b y d i v i d i n g a l l t h e c o n d u c t o r s ( i n c l u d i n g g r o u n d ) i n t o s m a l l e r c o n d u c t o r s o f s p e c i f i c s h a p e , w h i c h a u t o m a t i c a l l y a c c o u n t s f o r t h e t w o e f f e c t s m e n t i o n e d a b o v e . T h i s a p p r o a c h i s a l s o u s e d i n t h i s t h e s i s . C a b l e s w i t h s e c t o r 2 s h a p e d c o n d u c t o r s o r c o n d u c t o r s o f a n y i r r e g u l a r c r o s s s e c t i o n o r o f n o n -u n i f o r m p r o p e r t i e s a c r o s s t h e c r o s s s e c t i o n c a n e a s i l y b e h a n d l e d w i t h t h i s m e t h o d . 1.2 S k i n a n d P r o x i m i t y E f f e c t s T h e r e s i s t a n c e o f a t r a n s m i s s i o n l i n e t o d i r e c t c u r r e n t i s e a s i l y d e t e r m i n e d f r o m t h e p h y s i c a l d i m e n s i o n s o f t h e w i r e a n d t h e t y p e o f m a t e r i a l b e c a u s e d i r e c t c u r r e n t i s u n i f o r m l y d i s t r i b u t e d a c r o s s t h e c r o s s s e c t i o n o f t h e w i r e . I n t h e c a s e o f a l t e r n a t i n g c u r r e n t , t h e r e e x i s t s a n o n u n i f o r m d i s t r i b u t i o n o f c u r r e n t o v e r t h e c r o s s s e c t i o n o f a c o n d u c t o r w h i c h i s c a u s e d b y t h e v a r i a t i o n o f c u r r e n t i n t h e c o n d u c t o r . T h i s p h e n o m e n o n i s c a l l e d " s k i n e f f e c t " . A n o t h e r p h e n o m e n o n , c a l l e d " p r o x i m i t y e f f e c t " , a r i s e s d u e t o t h e p r e s e n c e o f o t h e r c u r r e n t - c a r r y i n g c o n d u c t o r s c l o s e b y . C h a n g i n g c u r r e n t s i n t h e s e n e i g h b o u r i n g c o n d u c t o r s c a u s e s a d i s t o r t i o n i n t h e c u r r e n t d i s t r i -b u t i o n i n t h e f i r s t c o n d u c t o r . T h i s d i s t o r t i o n , u n l i k e t h a t d u e t o s k i n e f f e c t , i s n o t s y m m e t r i c a l a r o u n d t h e a x i s o f s y m m e t r y o f t h e c o n d u c t o r ( i f t h e c o n d u c t o r i s c i r c u l a r ) . I n a t w o - w i r e l i n e , f o r i n s t a n c e , m o r e c u r r e n t t e n d s t o f l o w e i t h e r o n t h e s i d e s o f t h e c o n d u c t o r s w h i c h f a c e e a c h o t h e r o r o n t h e o p p o s i t e s i d e s . T h e p h e n o m e n a o f s k i n a n d p r o x i m i t y e f f e c t s i n r o u n d c o n d u c t o r s a r e i l l u s t r a t e d i n F i g u r e 1.1. 3 ^ > Current D i s t r i b u t i o n s (a) Skin Effect (b) Prox imi ty Ef fec t F i g u r e 1.1 C u r r e n t d i s t r i b u t i o n i n s o l i d r o u n d c o n d u c t o r s d u e t o s k i n a n d p r o x i m i t y e f f e c t s T h e u n e v e n c u r r e n t d i s t r i b u t i o n a c r o s s t h e c r o s s s e c t i o n o f t h e c o n d u c t o r s c a u s e s a d d i t i o n a l p o w e r l o s s a b o v e t h a t p r o d u c e d b y a n e q u i v a l e n t a m o u n t o f d i r e c t c u r r e n t , t h u s i n c r e a s i n g t h e e f f e c t i v e a . c . r e s i s t a n c e o f t h e c o n d u c t o r . T h e h i g h e r c u r r e n t d e n s i t y t o w a r d s t h e c o n d u c t o r s u r f a c e r e d u c e s t h e s e l f l i n k a g e i n t h e i n n e r p a r t o f t h e c o n d u c t o r . T h i s d e c r e a s e s t h e i n t e r n a l i n d u c t a n c e o f t h e c o n d u c t o r . T h e e x t e n t t o w h i c h t h e a b o v e e f f e c t s a l t e r t h e v a l u e s o f t h e c o n d u c t o r i m p e d a n c e d e p e n d s o n h o w p r o n o u n c e d t h e y a r e . S k i n e f f e c t v a r i e s v e r y m u c h w i t h t h e s i z e o f t h e c o n d u c t o r a n d w i t h t h e f r e q u e n c y - i t i n c r e a s e s w i t h b o t h - w h i l e p r o x i m i t y e f f e c t d e p e n d s m a i n l y o n t h e g e o m e t r y ( b e i n g m o r e p r o n o u n c e d f o r c l o s e r s p a c i n g s b e t w e e n c o n d u c t o r s ) . B e s s e l f u n c t i o n s a r e u s e d t o c a l c u l a t e i n c r e a s e s i n r e s i s t a n c e t o a l t e r n a t i n g c u r r e n t d u e t o s k i n e f f e c t a n a l y t i c a l l y . T h i s m e t h o d i s w i d e l y f o u n d i n t h e l i t e r a t u r e , e x p e c i a l l y w h e n c o a x i a l c a b l e s a n d c y l i n -d r i c a l c o n d u c t o r s a r e b e i n g a n a l y z e d [ 1 , 9 ] . O n t h e o t h e r h a n d , p r o x i m i t y e f f e c t i s m o r e d i f f i c u l t t o a n a l y z e . C h a r t s a n d c o r r e c t i n g f a c t o r t a b l e s [17] d e r i v e d f r o m o t h e r w i s e c o m p l i c a t e d f o r m u l a e [ 1 8 ] a r e c u s t o m a r i l y u s e d i n m o s t h a n d c a l c u l a t i o n s t o c o r r e c t f o r t h i s e f f e c t . I n g e n e r a l , u n d e r g r o u n d c a b l e s a r e l a r g e i n s i z e a n d a r e u s u a l l y l a i d c l o s e t o g e t h e r . T h i s m a k e s b o t h s k i n a n d p r o x i m i t y e f f e c t s i m p o r t a n t i n t h e c a l c u l a t i o n o f i m p e d a n c e s , e v e n a t p o w e r f r e q u e n c y a n d e s p e c i a l l y a t h i g h e r f r e q u e n c i e s a s n e e d e d f o r s w i t c h i n g s u r g e s t u d i e s . 1 . 3 A B r i e f E x p l a n a t i o n o f t h e M e t h o d o f S u b c o n d u c t o r s I n t h e w o r k o f E n r i c o C o m e l l i n i e t a l . [ 7 ] i t i s s h o w n t h a t i t i s p o s s i b l e t o t a k e b o t h e f f e c t s i n t o a c c o u n t s i m u l t a n e o u s l y i n c a l c u l a t i n g t h e i m p e d a n c e o f a n y t r a n s m i s s i o n l i n e . T h i s i s d o n e b y d i v i d i n g t h e m a i n c o n d u c t o r s i n t o s m a l l e r s u b c o n d u c t o r s o f c y l i n d r i c a l s h a p e , b y f i n d i n g t h e s e l f a n d m u t u a l i m p e d a n c e s o f t h e s e s u b c o n d u c t o r s , a n d b y b u n d l i n g t h e m t o g i v e t h e i m p e d a n c e o f t h e m a i n c o n d u c t o r s . T h e w o r k d e s c r i b e d i n t h i s t h e s i s i s b a s e d o n t h e a b o v e r e f e r e n c e . D i v i d i n g t h e c o n d u c t o r s i n t o p a r a l l e l c y l i n d r i c a l s u b c o n d u c t o r s s e e k s t o a p p r o x i m a t e t h e c u r r e n t d i s t r i b u t i o n s s h o w n i n F i g u r e 1.1 b y t h o s e o f F i g u r e 1.2. Conduc to r s Approximate Current Distr ibut ions (a) Skin Effect (b) Proximity Effect F i g u r e 1 . 2 C u r r e n t d i s t r i b u t i o n i n s u b d i v i d e d c o n d u c t o r s o f t h e m o d e l O b v i o u s l y t h e a c c u r a c y t o b e e x p e c t e d f r o m s u c h a r e p r e s e n t a t i o n w i l l d e p e n d o n t h e d e g r e e o f d i s c r e t i z a t i o n , a n d h e n c e o n t h e n u m b e r o f s u b c o n d u c t o r s . 1 . 4 S c o p e o f T h i s T h e s i s T h e t h e o r y f o r c a l c u l a t i n g t h e i m p e d a n c e s f r o m s u b c o n d u c t o r s i s d e v e l o p e d u s i n g a f i c t i t i o u s ' r e t u r n p a t h ' w h i c h a l l o w s m o r e f l e x i b i l i t y i n t h e m o d e l . • A n a l y t i c a l l y d e r i v e d g r o u n d r e t u r n f o r m u l a e a r e t h e n i n c o r -p o r a t e d i n t o t h e m o d e l t o r e d u c e t h e n u m b e r o f s u b c o n d u c t o r s , s t o r a g e a n d c o m p u t i n g t i m e . P i p e t y p e c a b l e s a r e m o d e l l e d b y t r e a t i n g t h e s t e e l p i p e a s c o n c e n t r i c l a y e r s o f p i p e m a t e r i a l , w i t h e a c h l a y e r h a v i n g a d i f f e r e n t p e r m e a b i l i t y d e p e n d i n g o n t h e d e g r e e o f s a t u r a t i o n . 6 C h a p t e r 2 T H E O R Y AND T H E F O R M A T I O N A N D S O L U T I O N OF E Q U A T I O N S 2.1 S u b d i v i s i o n s o f t h e C o n d u c t o r s I n t h e m o d e l d e s c r i b e d b e l o w , e a c h c o r e o f t h e c a b l e i s c o n s i d e r e d a m a i n c o n d u c t o r , a s i s t h e s h e a t h , a n d i f p r e s e n t , t h e n e u t r a l c o n d u c t o r a n d t h e a r m o u r . E a c h m a i n c o n d u c t o r i s d i v i d e d i n t o a n u m b e r o f p a r a l l e l c y l i n d r i c a l s u b c o n d u c t o r s ( F i g u r e =2.1) . T h e c h o i c e o f a c y l i n d r i c a l s h a p e f o r t h e s u b c o n d u c t o r s m a k e s t h e d e r i v e d i n d u c t a n c e f o r m u l a e s i m p l e . O t h e r s h a p e s f o r t h e s u b c o n d u c t o r s h a v e b e e n t r i e d b y L u c a s a n d T a l u k d a r [22] b u t t h e r e s i s t a n c e v a l u e s c a l c u l a t e d b y t h e m s h o w a l a r g e d e v i a t i o n f r o m m e a s u r e d v a l u e s a t h i g h e r f r e q u e n c i e s , w h i c h s u g g e s t s t h a t f u r t h e r r e s e . a r c h i s n e e d e d b e f o r e t h e s e o t h e r s h a p e s c a n b e u s e d w i t h c o n f i d e n c e . F i g u r e 2.1 S u b d i v i s i o n o f m a i n c o n d u c t o r s 2.2 A s s u m p t i o n s I t i s a s s u m e d t h a t : i ) E a c h s u b c o n d u c t o r i s u n i f o r m a n d h o m o g e n e o u s t h r o u g h o u t i t s l e n g t h ; i i ) T h e m a g n e i t c p e r m e a b i l i t y o f a s u b c o n d u c t o r i s c o n s t a n t t h r o u g h o u t 7 t h e w h o l e c y c l e o f a l t e r n a t i n g c u r r e n t , b u t m a y b e d i f f e r e n t f r o m t h a t o f a n y o t h e r s u b c o n d u c t o r ; i i i ) T h e r e i s u n i f o r m c u r r e n t d i s t r i b u t i o n i n e a c h s u b c o n d u c t o r ; a n d i v ) A l l s u b c o n d u c t o r s a r e p a r a l l e l . 2 . 3 L o o p I m p e d a n c e s o f S u b c o n d u c t o r s T o d e r i v e t h e l o o p i m p e d a n c e s o f t h e s u b c o n d u c t o r s , f i r s t c o n s i d e r t h e t w o l o o p s f o r m e d b y a n y t w o s u b c o n d u c t o r s , % a n d k , w i t h a c o m m o n r e t u r n p a t h , q , i n F i g u r e 2 . 2 . T h e r e t u r n p a t h c a n e i t h e r b e o n e o f t h e s u b c o n d u c t o r s o r a f i c t i t i o u s c o n d u c t o r c h o s e n f o r t h e v o l t a g e m e a s u r e m e n t s a n d t h e c a l c u -l a t i o n o f i n d u c t a n c e s . F i g u r e 2 . 2 L o o p s f o r m e d b y t w o s u b c o n d u c t o r s w i t h a c o m m o n r e t u r n W r i t i n g t h e l o o p e q u a t i o n s f o r s u b c o n d u c t o r Z g i v e s : ( 2 . 1 ) w h e r e = v o l t a g e d r o p p e r u n i t l e n g t h o f s u b c o n d u c t o r = r e s i s t a n c e p e r u n i t l e n g t h o f s u b c o n d u c t o r R^ = r e s i s t a n c e p e r u n i t l e n g t h o f r e t u r n p a t h i , i , = c u r r e n t s i n s u b c o n d u c t o r s £ a n d k r e s p e c t i v e l y JO AC M = m u t u a l i n d u c t a n c e b e t w e e n l o o p s f o r m e d b y s u b c o n d u c t o r s J6K. £ a n d k w i t h a c o m m o n r e t u r n i n q N = n u m b e r o f s u b c o n d u c t o r s ( N = 2 , f o r £ , k i n F i g u r e 2 . 2 ) F o r a c s t e a d y - s t a t e c o n d i t i o n s , t h e i n s t a n t a n e o u s v o l t a g e v a n d c u r r e n t i i n ( 2 . 1 ) a r e r e p l a c e d b y t h e p h a s o r v a l u e s V a n d I a n d b y t h e p h a s o r v a l u e j c o l . F i g u r e 2 . 3 : G e o m e t r y o f s u b c o n d u c t o r s ( £ , k , q ) . F i g u r e 2 . 3 s h o w s t h e c r o s s s e c t i o n o f t w o s u c h s u b c o n d u c t o r s a n d t h e r e t u r n p a t h . T o d e r i v e t h e i n d u c t a n c e f o r m u l a e , c o n s i d e r c u r r e n t I i n s u b c o n d u c t o r £ a n d r e t u r n i n g i n q . T h e f l u x d e n s i t y B a t r a d i u s r o u t s i d e £ i s : B = yi .2irr (2.2) T o t a l f l u x p e r u n i t l e n g t h i n t h e e l e m e n t a l c y l i n d e r o f t h i c k n e s s <5r 6<j> = B * S r yi 2Trr 6 r (2.3) F l u x l i n k a g e o f l o o p k ^ d u e t o c u r r e n t I i n i i s : r = D yi 2irr d r £ k r = D 2* Kk (2.4) F l u x l i n k a g e o f l o o p k ^ d u e t o r e t u r n c u r r e n t I i n q i s : k q yldr _ _yl_ 2irr 2TT r D. (2.5) w h e r e r q ( e q u i v ) q ( e q u i v ) -t h e e q u i v a l e n t r a d i u s o f s u b c o n d u c t o r q f o r i n d u c t a n c e r = r q ( e Q u l y ) c a l c u l a t i o n ( = r ^ e ^ r q ^ ^ , t h e g e o m e t r i c m e a n r a d i u s ) T h e t w o f l u x e s a r e a d d i t i v e ; h e n c e t h e t o t a l f l u x l i n k a g e i s : * £ k - 2 V t o &q  } £ k Z T T r e x p ( - y /4) q r q ' (2.6) T h e m u t u a l i n d u c t a n c e ( M £ ^ ) b e t w e e n l o o p s & a n d k i s t h e r e f o r e M £ k £ k y 2T: £ n Iq kg + .IS. (2.7) w h e r e y r ^ = r e l a t i v e p e r m e a b i l i t y o f . t h e r e t u r n p a t h . T o d e r i v e t h e s e l f i n d u c t a n c e o f l o o p £ w e c o n s i d e r t h e s a m e c u r r e n t p a t h . F l u x l i n k a g e o f l o o p £ ^ d u e t o c u r r e n t I i n £ i s : £ q u l d r u l r = r 2 i r r 2TT £ n £ ( e q u i v ) 13_ £ ( e q u i v ) ( 2 . 8 ) F l u x l i n k a g e o f l o o p £ ^ d u e t o c u r r e n t I r e t u r n i n g i n q i s : D £ q r = r p i , p i „ • £ — d r = ^ — £ n 2 i r r 2TT q ( e q u i v ) £ q k q ( e q u i v ) t ( 2 . 9 ) H e n c e t h e t o t a l f l u x l i n k a g e i s : D * = ~ £ n £ £ 2 I T . / £ q , r £ . 7 ^ e x p ( - r - ) £ Z T T ^ e x p ( ) ( 2 . 1 0 ) w h e r e u . a n d p a r e t h e r e l a t i v e p e r m e a b i l i t i e s o f s u b c o n d u c t o r s £ a n d r £ r q r e s p e c t i v e l y T h e s e l f i n d u c t a n c e ( M ^ £ ) o f l o o p £ i s M££ = h i n = 27 £ n D £ q D £ q £ q y r £ , y r q _ ( 2 . 1 1 ) 2 . 4 : F o r m a t i o n o f I m p e d a n c e M a t r i x W r i t i n g t h e l o o p e q u a t i o n s , u s i n g ( 2 . 1 ) , ( 2 . 7 ) a n d ( 2 . 1 1 ) , f o r a t h e s u b c o n d u c t o r s g i v e s a s e t o f l i n e a r e q u a t i o n s 11 V 11 In V. kl km J l l l l l l l n Z l n l l Z l n l n J k l l l Zkmll ' • • Zkmln ^ l k l * " Zllkm Z l n k l ZInkm "klkm kmkm 11 In k l km (2.12a) i.e. [V] = [ Z b ± g ] [I] (2.12b) th where V.. refers to the voltage on the i subconductor of the J i .th . , „ 2 main conductor. I., is the current in the i * " * 1 subconductor of the j*"* 1 main conductor. Z.. is the mutual impedance between the loops formed by jimn the i * " * 1 and n*"*1 subconductors of the j 1"* 1 and m1"^  main conductors respectively. The partitioning of the matrix [Z^g] i n (2.12a) groups the equa-tions of the subconductors within each main conductor together. The resistances and inductances in the impedance matrix [Z^g] are constant, but since the current division among subconductors changes with frequency, skin and proximity effects are accounted for. Normally the large impedance matrix i s not of direct interest. Instead, the matrix giving voltages on the main conductors in terms of the 12 c u r r e n t s i n t h e s e m a i n c o n d u c t o r s i s n e e d e d , w h i c h c a n b e o b t a i n e d f r o m t h e l a r g e i m p e d a n c e m a t r i x ] b y r e d u c t i o n . M a t h e m a t i c a l l y , t h i s i s e q u i v a l e n t t o s o l v i n g t h e a l g e b r a i c e q u a t i o n s ' ( 2 . 1 2 a ) w i t h t h e c o n d i t i o n s o f ( 2 . 1 3 ) a n d ( 2 . 1 4 ) s h o w n b e l o w . P r a c t i c a l l y , i t i s e q u i v a l e n t t o r e d i s -t r i b u t i o n o f c u r r e n t s i n a l l s u b c o n d u c t o r s t o a c h i e v e a c u r r e n t d i s t r i b u t i o n w h i c h , w h e n m u l t i p l i e d b y t h e i m p e d a n c e s o f ( 2 . 1 2 a ) , s a t i s f i e s t h e v o l t a g e c o n d i t i o n o f ( 2 . 1 3 ) . T h e v o l t a g e s o n t h e s u b c o n d u c t o r s f o r m i n g a n y m a i n c o n d u c t o r a r e e q u a l ; h e n c e : V . , = V . 0 = = V . = V . ( 2 . 1 3 ) J l J 2 j n j A l s o , t h e c u r r e n t i n a n y m a i n c o n d u c t o r i s t h e s u m o f t h e c u r r e n t s i n t h e s u b c o n d u c t o r s i n t o w h i c h ! . i t i s d i v i d e d . T h u s : I . = I . . . + I + . . . + I . ( 2 . 1 4 ) J J l J 2 j n 2 • 5 B u n d l i n g o f t h e S u b c o n d u c t o r s i n t h e I m p e d a n c e M a t r i x E x p r e s s i n g t h e v o l t a g e s o n t h e m a i n c o n d u c t o r s i n t e r m s o f t h e c u r r e n t s t h e y c a r r y i s a c c o m p l i s h e d b y t h e u s e o f e q u a t i o n s ( 2 . 1 3 ) a n d ( 2 . 1 4 ) i n ( 2 . 1 2 a ) . C o n s i d e r t h e f i r s t m a i n c o n d u c t o r ; a s s u m e i t i s s u b d i v i d e d i n t o n s u b c o n d u c t o r s a s s h o w n i n ( 2 . 1 2 a ) . ( a ) S u b t r a c t i n g t h e f i r s t e q u a t i o n ( i . e . r o w - 1 i n 2 . 1 2 a ) f r o m t h e s u b s e q u e n t e q u a t i o n s ( i . e . r o w - 2 t o r o w - n ) o f t h a t m a i n c o n d u c t o r l e a v e s t h e l e f t - h a n d s i d e o f t h e o t h e r e q u a t i o n s e q u a l t o z e r o ( - i l l u s t r a t e d i n ( 2 . 1 5 ) ) . 13 ( b ) B y w r i t i n g I i n s t e a d o f 1 ^ ^ i n t h e f i r s t e q u a t i o n o f t h e f i r s t c o n d u c t o r ( i . e . r o w - 1 ) , a n e r r o r o f a d d i n g ^ilYlJ'H + ••• + ^ i m ^ l n ^ t o f i r s t e q u a t i o n h a s b e e n m a d e s i n c e I - ^ I ^ l + ^ 1 2 + • • • + ^ j _ n ' C o r r e s p o n d i n g e r r o r s a r e i n t r o d u c e d i n t o a l l t h e o t h e r e q u a t i o n s . T h e s e e r r o r s a r e r e m o v e d b y s u b t r a c t i n g t h e f i r s t c o l u m n o f t h e w h o l e m a t r i x [Z , . ] f r o m t h e s u b s e q u e n t ( n - 1 ) c o l u m n s o f t h a t b i g J m a i n c o n d u c t o r . T h e s e t w o s t e p s a r e i l l u s t r a t e d i n e q u a t i o n ( 2 . 1 6 ) . ( c ) T h e s a m e s t e p s , ( a ) a n d ( b ) , a r e c a r r i e d o u t o n t h e o t h e r m a i n c o n d u c t o r s . T h e s e g i v e a s e t o f l i n e a r e q u a t i o n s ( 2 . 1 5 ) e x p r e s s -: i n g t h e v o l t a g e s o n t h e c o n d u c t o r s i n t e r m s o f t h e t o t a l c u r r e n t s t i l i n t h e s e c o n d u c t o r s a n d i n t h e 2 n d t o n - s u b c o n d u c t o r s . " v l ' Z l l l l ? 1 1 1 2 - ? l l l n j | Z l l k l • • C l l k m V 0 S . 2 1 1 ; - ' ; ; • hi 0 ? l n l n < ( 2 . 1 5 ) \ Z k l l l > z k l k l ? k l k m \ 0 • * 0 k m l l ' f ^ k m k l ' • * ° k m k m T h e s y m b o l " 5 " d e n o t e s t h e e l e m e n t s w h i c h h a v e b e e n c h a n g e d i n ( 2 . 1 5 ) d u e t o t h e o p e r a t i o n s ( a ) a n d ( b ) , a n d t h e g e n e r a l t e r m i s : C = Z — Z — Z k m q n k m q n k l q n k m q l ( 2 . 1 6 ) ( f o r m , n ^ l ) 14 2 . 6 R e d u c t i o n o f t h e L a r g e I m p e d a n c e M a t r i x T h e e q u a t i o n s ( 2 . 1 5 ) a r e r e a r r a n g e d f o r t h e r e d u c t i o n p r o c e s s b y e x c h a n g i n g t h e p o s i t i o n s o f r o w s a n d c o l u m n s i n s u c h a w a y t h a t t h e " b u n d l e d " e q u a t i o n s o f ( 2 . 1 5 ) c o r r e s p o n d i n g t o t h e m a i n c o n d u c t o r s c o m e f i r s t , a s s h o w n i n ( 2 . 1 7 ) 1 V , k 0 = 0 J l l l l J k l l l J l l k l J k l k m 1 2 k m ( 2 . 1 7 ) o r i n a b b r e v i a t e d f o r m , V 0 A C , - 1 , B D F r o m ( 2 . 1 8 ) V = ( A - B D C ) I ( 2 . 1 9 a ) H e n c e t h e d e s i r e d i m p e d a n c e m a t r i x [ Z c ] i s [ Z „ ] = [ A - B D - 1 C ] ( 2 . 1 9 b ) ( 2 . 1 8 ) R e f e r e n c e [ 8 ] p r o v i d e s a m o r e e f f i c i e n t w a y o f f i n d i n g [ Z c ] f r o m ( 2 . 1 7 ) . U s i n g G a u s s i a n e l i m i n a t i o n o n t h e m a t r i x ( 2 . 1 7 ) , s t a r t i n g f r o m t h e l a s t r o w a n d g o i n g u p u n t i l t h e s u b m a t r i x [ B ] , a s s h o w n i n ( 2 . 1 8 ) , h a s j u s t b e e n r e d u c e d t o z e r o , a c h i e v e s t h e r e d u c t i o n . 1 5 T h e d e s i r e d i m p e d a n c e m a t r i x [ Z c ] ' c o r r e s p o n d s t o t h e s u b m a t r i x s t o r e d i n [ A * ] i n ( 2 . 2 0 ) . 1 V , k . 0 _ 0 1 2 ^ I k m J ( 2 . 2 0 ) A n i l l u s t r a t i o n o f t h i s f i n a l r e d u c t i o n s t a g e i s s h o w n i n F i g u r e 2 . 4 . F i g u r e 2 . 4 I l l u s t r a t i o n o f t h e r e d u c t i o n p r o c e s s 2 . 7 T h e C h o i c e a n d t h e C o n s t r a i n t o n t h e R e t u r n P a t h O b v i o u s l y , t h e g e o m e t r y a n d l o c a t i o n o f t h e r e t u r n p a t h i n F i g u r e 2 . 2 w i l l i n f l u e n c e t h e v a l u e s o b t a i n e d f o r t h e i n d u c t a n c e s . T h e i n f l u e n c e o f t h e r e t u r n p a t h i s r e m o v e d b y r e q u i r i n g t h a t t h e c u r r e n t t h r o u g h t h i s p a t h s h o u l d b e z e r o . T o i l l u s t r a t e t h i s , c o n s i d e r t h e c i r c u i t s h o w n i n F i g u r e 2.5. F i g u r e 2 . 5 A t w o - w/i r e r e t u r n , c i r c u i t W r i t i n g t h e l o o p e q u a t i o n f o r F i g . 2 . 5 g i v e s : V l = ( R 1 + h + ( R 2 + ^ X 2 2 - X 1 2 > h <2'21> s i n c e 1^=1 V l " ( R 1 + R 2 + J ( X 1 1 + X 2 2 ~ 2 X 1 2 } ) h <2'22> I n t r o d u c i n g a f i c t i t i o u s r e t u r n p a t h g i v e s a c o n f i g u r a t i o n o f F i g u r e . 2 . 6 . F o r e q u i v a l e n c e o f t h e t w o c i r c u i t s w e r e q u i r e : F i g u r e 2 . 6 A, • t w o - w . i r e . c i r c u i t w i t h , c o m m o n r e t u r n i n a t h i r d c o n d u c t o r 17 The loop equations of Figure,2.6 are: V a = (% + J ^ - X ^ ) ) ^ + <Rq + j ( X q q - X l q ) ) I q + ( X 1 2 - X 2 q ) I f e V, = (R 2 + i ( X 2 2 - X 2 q ) ) I b + (R q + i ( X q q - X 2 q ) ) I q + ( X 1 2 - X l q ) I a > (2.24) Imposing the constraint that the current i n the return path i s zero means that I q = 0 = I a + I b (2.25) I a = - I b (2.26) From equations (2.24) and (2.25) V a - V b = [R x + j ( X 1 1 - X l q ) - ( X 1 2 - X l q ) ] I a - [ R 2 + j ( X 2 2 - X 2 q ) - ( X 1 2 - X 2 q ) ] h (2-27) Using (2.26) i n (2.27) gives V a _ V b = [ R 1 + R2 + J ( X H + X 2 2 " 2 X 1 2 ) ] X a ( 2 # 2 8 ) which i s i d e n t i c a l to (2.22) derived using Figure 2.5. Therefore, i t i s t h e o r e t i c a l l y possible to choose a return path of any convenient shape and l o c a t i o n for the inductance c a l c u l a t i o n s as ... long as a zero current constraint i s imposed on such a path. Nevertheless, considerations discussed i n section 3.6 would require that the return path should be c y l i n d r i c a l i n shape, have a small radius, and be placed at a small distance below the earth surface not far from the cables and other conductors. 18 2.8 Including the Constraint on the Current i n the Matrix Solution Equation (2.20) gives the voltages on the main conductors (measured with respect to the return path) i n terms of the currents i n these conductors. In p r a c t i c e , however, voltages are measured with respect to the l o c a l ground (or n e u tral conductor or sheath). I f the constraint on the current i s introduced, t h i s changes (2.20) into the form: V. k-1 V. J l l J k - l , l k J L_- k l ••• k Since £ I o = 0 1=1 iL J l k J k - l , k J kk k-1 -I k-1-(2.29) ( 2 . 3 0 ) and I - - l r l 2 - . . . - I t ^ This gives: V. k-1 z i r z i k Z k - l , l Z k - l , k Z k l Z k k Z l k-1 Z l k Z k - l , k - l Z k - l , k Z k , k - l " Z k k k-1 ( 2 . 3 1 ) If conductor k represents the l o c a l ground (or neutral conductor or the sheath) with respect to which a l l voltages are measured, then sub-t r a c t i n g the equation for V from the other equations accomplishes t h i s rC and gives: V - V 1 k V - V k-1 k r- * Z l l J k l l k-1 J k - l , k - l k-1 (2.32) w h e r e Z . . = Z. . + Z, , - 2Z., 13 I J k k i k 19 T h e Z m a t r i x i s t h e i m p e d a n c e m a t r i x w h i c h i m p l i e s a l o c a l g r o u n d ( o r n e u t r a l c o n d u c t o r o r s h e a t h ) . 2 0 C h a p t e r 3 R E T U R N P A T H I M P E D A N C E I n p r a c t i c e , t h e r e a r e t h r e e c a s e s f o r t h e r e t u r n p a t h i n a n y t r a n s m i s s i o n s y s t e m : ( i ) r e t u r n i n n e u t r a l c o n d u c t o r s ( i n c l u d i n g p i p e s a n d g r o u n d w i r e s o n l y ) ; ( i i ) r e t u r n i n g r o u n d o n l y ; o r ( i i i ) r e t u r n i n g r o u n d a n d n e u t r a l c o n d u c t o r s . 3 . 1 R e t u r n i n N e u t r a l C o n d u c t o r s O n l y E a c h s h e a t h , p i p e o r n e u t r a l c o n d u c t o r i s r e p r e s e n t e d a s a s e p a r a t e c o n d u c t o r , a s a r e t h e c o r e s , a n d i s d i v i d e d a s i n S e c t i o n 2 . 1 . T h e f o r m u l a e o f e q u a t i o n s ( 2 . 7 ) a n d ( 2 . 1 1 ) a r e u s e d t o f o r m t h e i m p e d a n c e m a t r i x o f t h e s u b c o n d u c t o r s . T h e n e u t r a l s o r s h e a t h s c a n b e " e l i m i n a t e d " i n t h e r e d u c t i o n p r o c e s s , i f s o d e s i r e d , t o o b t a i n t h e i m p e d a n c e m a t r i x w h i c h r e l a t e s t h e v o l t a g e s f r o m p h a s e t o n e u t r a l t o t h e p h a s e c u r r e n t s . I n f a c t , a n y o t h e r c o n d u c t o r c a n a l s o b e e l i m i n a t e d i n t h e r e d u c t i o n p r o c e s s , i f s o d e s i r e d , p r o v i d e d t h a t t h e r e i s z e r o v o l t a g e o n i t , o r t h a t i t i s c o n n e c t e d i n p a r a l l e l w i t h a n o t h e r c o n d u c t o r . 3 . 2 R e t u r n i n G r o u n d O n l y , M o d e l 1 : T h e g r o u n d i s c o n s i d e r e d a s a s e p a r a t e c o n d u c t o r a n d i s s u b d i v i d e d i n t o l a y e r s o f s u b c o n d u c t o r s a s s h o w n i n F i g u r e 3 . 1 . T h e d i a m e t e r s o f t h e s u b c o n d u c t o r s i n a n y l o w e r l a y e r a r e c h o s e n t o b e t w i c e t h a t o f t h e p r e v i o u s l a y e r . T h i s c h o i c e a p p e a r s r e a s o n a b l e b e c a u s e t h e c u r r e n t d e n s i t y i n t h e g r o u n d d e c r e a s e s a s o n e m o v e s f a r t h e r a w a y f r o m t h e c a b l e s . R e a s o n a b l e r e s u l t s w e r e o b t a i n e d b y u s i n g a d e p t h e q u a l t o 3 3 0 0 / , r p / f m e t r e s . (a) ( b ) F i g u r e 3 . 1 S u b d i v i s i o n s o f g r o u n d i n t o l a y e r s o f s u b c o n d u c t o r s 22 p = g r o u n d r e s i s t i v i t y i n ftm, a n d f = f r e q u e n c y i n H z . T h e g r o u n d , a s a s y s t e m o f s u b c o n d u c t o r s , i s e l i m i n a t e d i n t h e r e d u c t i o n p r o c e s s a s s h o w n i n s e c t i o n 2 . 8 , t h u s l e a v i n g t h e g r o u n d r e t u r n i m p l i c i t l y i n c l u d e d i n t h e r e d u c e d i m p e d a n c e m a t r i x . T h e d i f f e r e n c e b e t w e e n t h e r e s u l t s o f t h e a r r a n g e m e n t s i n F i g u r e s 3* 1 ( a ) a n d ( b ) i s d i s c u s s e d i n s e c t i o n 4 . 2 . 3 . 3 R e t u r n i n G r o u n d a n d N e u t r a l c o n d u c t o r s I f t h e r e t u r n i s t h r o u g h b o t h g r o u n d a n d n e u t r a l c o n d u c t o r s , t h e s y s t e m i s m o d e l l e d a s a s e t o f k c o n d u c t o r s w h i c h a r e s u b d i v i d e d i n t o s u b -c o n d u c t o r s . T h e g r o u n d i s o n e s u c h c o n d u c t o r a n d i t i s e l i m i n a t e d i n t h e r e d u c t i o n p r o c e s s . T h e n e u t r a l s c a n b e r e t a i n e d o r e l i m i n a t e d a s d e s i r e d . 3 . 4 U s e o f A n a l y t i c a l E q u a t i o n s f o r G r o u n d R e t u r n I m p e d a n c e T o r e p r e s e n t t h e g r o u n d r e t u r n a d e q u a t e l y , i t m u s t b e d i v i d e d i n t o a l a r g e n u m b e r o f s u b c o n d u c t o r s . I n s y s t e m s w h e r e m a n y c a b l e s o r c o n d u c t o r s m u s t b e c o n s i d e r e d , i t i s b e t t e r t o c a l c u l a t e t h e g r o u n d r e t u r n i m p e d a n c e d i r e c t l y t o r e d u c e t h e a m o u n t o f s t o r a g e a n d c o m p u t i n g t i m e . E q u a t i o n s f o r g r o u n d r e t u r n c i r c u i t s o f 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 w e r e d e r i v e d b y J . R . C a r s o n [ 1 1 ] a n d a r e w i d e l y u s e d i n t h e p o w e r i n d u s t r y . T h e s e e q u a t i o n s a s s u m e t h a t t h e c o n d u c t o r s a r e l o c a t e d i n a i r o v e r f l a t e a r t h w h i c h i s i n f i n i t e i n e x t e n t a n d h a s a n u n i f o r m r e s i s t i v i t y . W h e n t h e s e s a m e e q u a t i o n s a r e a p p l i e d t o u n d e r g r o u n d c a b l e s , u s e f u l a p p r o x i -m a t i o n s t o t h e t r u e v a l u e s o f g r o u n d r e t u r n i m p e d a n c e s c a n b e o b t a i n e d . 23 W i t h o v e r h e a d l i n e s , i m a g e c o n d u c t o r s w h i c h l i e b e l o w t h e g r o u n d a r e u s e d i n t h e c a l c u l a t i o n s . H o w e v e r , w h e n t h e s e e q u a t i o n s a r e a p p l i e d t o u n d e r -g r o u n d c a b l e s , t h e s e i m a g e s n o w l i e a b o v e t h e g r o u n d s u r f a c e a t h e i g h t s e q u a l t o t h e d e p t h o f b u r i a l . I n a l a t e r p a p e r [ 1 0 ] , C a r s o n s h o w e d t h a t f o r c o n d u c t o r s b u r i e d u n d e r g r o u n d , t h e v a r i a t i o n o f g r o u n d r e t u r n i m p e d a n c e w i t h d i s t a n c e b e l o w t h e e a r t h ' s s u r f a c e i s r e l a t i v e l y s m a l l f o r t h e u s u a l d e p t h s o f b u r i a l ( i . e . a b o u t 1 . 0 m) a n d t h a t t h e g r o u n d r e t u r n i m p e d a n c e ( Z g ) c a n b e c a l c u l a t e d a s : Z = ( 1 + C ) Z ° ( 3 . 1 ) S g o w h e r e Z^ = t h e g r o u n d r e t u r n i m p e d a n c e i f t h e e a r t h w e r e t o e x t e n d i n d e f i n i t e l y i n a l l d i r e c t i o n s a r o u n d t h e c o n d u c t o r s o t h a t c i r c u l a r s y m m e t r y e x i s t s , C = a c o r r e c t i o n f a c t o r w h i c h a c c o u n t s f o r t h e f a c t t h a t t h e c o n d u c t o r i s l o c a t e d n e a r g r o u n d s u r f a c e R e f e r e n c e [ 1 0 ] g i v e s C a s : 2 K 0 ( j m ) 2 l o g ( l / m ) f o r s m a l l m ( 3 . 2 ) a n d r e f e r e n c e [ 1 2 ] g i v e s Z a s : ° _ m& K o ( m r ) , „ g 2 r r r K L ( m r ) w h e r e m = / ( 3 . 4 ) K5, 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 r = i n t e r n a l r a d i u s o f t h e e a r t h ( i . e . o u t e r r a d i u s o f t h e c o n d u c t o r i n s u l a t i o n ) 24 p = ground re s i s t i v i t y to = 2 i T f , f=frequency y = magnetic permeability of ground Carson's formula for overhead conductors cannot be used for c a l -culating the self impedance of underground conductors. Equations (3.1) is used for this purpose, but the mutual impedances are calculated using the overhead formula - which is known to give good approximations for buried conductors at power frequencies [ 2 0 ] . Equations for calculating the self and mutual impedances of under-ground conductors have also been derived by F. Pollaczek involving i n f i n i t e series [19]. Closed-form approximations to the self and mutual impedances of underground conductors valid for a wide range of values of the parameters involved have been derived by Wedepohl and Wilcox [9]. These equations (3.5), given below, are accurate up to frequencies of approximately 160 KHz for separations of approximately 1.0m between the conductors, and to approximately 1.7 MHz i f the separation is only 30 cm. Thus very accurate approximations can be obtained for most practical cases of cables la i d in the same trench to quite high frequencies. These equations are: Zs = ^ { -An M + T " T n^} fl M (3.5a) 2 3 (YmD } , Z i k = *f£ i - £ n + J " f m£ } ft/m (3.5b) where Z s, Z-y^  are self and mutual impedances of ground return path respectively, (ft/m) Y = Eulers constant = 0.5772157 h = depth of burial of conductor (metres) I = sum of depths of burial of conductors i and k (metres) 25 r = o u t e r r a d i u s o f c o n d u c t o r ( m e t r e s ) D M = d i s t a n c e b e t w e e n c o n d u c t o r s i a n d k ( m e t r e s ) IK. m = / j t j u / p p = e a r t h r e s i s t i v i t y i n fim E q u a t i o n s ( 3 . 5 ) a r e v a l i d f o r t h e r a n g e | m r | < 0 . 2 5 f o r s e l f i m p e d a n c e a n d |mD_^J < 0 . 2 5 f o r m u t u a l i m p e d a n c e . F o r t h e r a n g e | m D . , | > 0 . 2 5 r e f e r e n c e [ 9 ] s u g g e s t s t h e i n t e g r a t i o n : I K . J i k 2TT - £ / ( a 2 + m 2 ) -V / ( a 2 + m 2 ) - £ / ( a 2 - r m 2 ) + - e | a | + / ( a 2 + m 2 ) 2 / ( a 2 + m 2 ) e x p ( j a x ) d x ( 3 . 6 ) w h e r e x = h o r i z o n t a l d i s t a n c e b e t w e e n c o n d u c t o r s i a n d k V= m o d u l u s o f t h e d i f f e r e n c e o f t h e d e p t h s o f b u r i a l o f c o n d u c t o r s i a n d k . 3 . 5 M o d e l U s i n g G r o u n d R e t u r n F o r m u l a e D i r e c t l y w i t h t h e S u b c o n d u c t o r s M o d e l I I : A v e r y s i m p l e m o d e l w h i c h u s e s t h e a n a l y t i c a l g r o u n d r e t u r n f o r m u l a e t r e a t s e a c h s u b c o n d u c t o r a s a n i n s u l a t e d c o n d u c t o r w i t h t h e r e t u r n l o o p t h r o u g h t h e g r o u n d ( F i g u r e 3 . 2 ) a n d u s e s t h e a v a i l a b l e g r o u n d r e t u r n f o r m u l a e t o c a l c u l a t e t h e s e l f a n d m u t u a l i m p e d a n c e s . E q u a t i o n s ( 3 . 5 ) m a y b e u s e d i n t h i s c a s e . I f t h e r e s u l t s a r e n e e d e d f o r p o w e r f r e q u e n c y o n l y , e q u a t i o n s ( 3 . 7 ) a n d ( 3 . 8 ) b e l o w , w h i c h a r e f o u n d i n m a n y h a n d b o o k s [ 2 4 , 2 5 ] , m a y b e u s e d . F i g u r e 3 . 2 M o d e l w i t h o n l y s u b c o n d u c t o r s a n d g r o u n d r e t u r n T h e i m p e d a n c e s a t 6 0 H z o f t h e c i r c u i t i n F i g u r e 3 . 2 a r e : z i i = R i + R g + J ( ° - 1 7 3 6 l o s £MR7 + °-4892) fi/km <3-7) Z . . = R + j ( 0 . 1 7 3 6 l o g + 0 . 4 8 9 2 ) ft/km ( 3 . 8 ) l k g ° i k w h e r e Z ^ a n d Z ^ a r e t h e s e l f a n d m u t u a l i m p e d a n c e s r e s p e c t i v e l y , a n d R g = r e s i s t a n c e o f g r o u n d r e t u r n p a t h ( = 0 . 0 5 9 2 ft/km) GMR^ = g e o m e t r i c m e a n r a d i u s o f c o n d u c t o r i (m) R^ = r e s i s t a n c e o f c o n d u c t o r i ( f t / k m ) D k = d i s t a n c e b e t w e e n c o n d u c t o r i a n d k (m) 3 . 6 R e p r e s e n t i n g t h e G r o u n d a s O n e U n d i v i d e d C o n d u c t o r M o d e l I I I I n v i e w o f t h e f a c t t h a t t h e e q u a t i o n s u s e d f o r g r o u n d r e t u r n i m p e d a n c e c a l c u l a t i o n s m a y b e i n a c c u r a t e a t h i g h f r e q u e n c i e s a n d f o r w i d e s e p a r a t i o n s b e t w e e n c o n d u c t o r s i f a p p r o x i m a t i o n s a r e u s e d , o r c o s t l y t o o b t a i n i f i n f i n i t e s e r i e s a r e u s e d , i t w o u l d b e a d v a n t a g e o u s i f m o s t o f 27 t h e e l e m e n t s o f t h e m a t r i x [Z , . 1 o f e q u a t i o n ( 2 . 1 2 a ) c o u l d b e c a l c u l a t e d bxg w i t h t h e s i m p l e r e q u a t i o n s ( 2 . 7 ) a n d ( 2 . 1 1 ) . T h i s i n v o l v e s t h e i n t r o d u c t i o n o f a f i c t i t i o u s r e t u r n p a t h w i t h r e s p e c t t o w h i c h t h e i n d u c t a n c e s a r e c a l c u -l a t e d . T h e g r o u n d i s t h e n c o n s i d e r e d a s o n e a d d i t i o n a l c o n d u c t o r ( n o t i. s u b d i v i d e d i n t h i s c a s e ) a n d t h e m u t u a l i m p e d a n c e s b e t w e e n t h e g r o u n d a n d t h e s u b c o n d u c t o r s a r e c a l c u l a t e d a s s h o w n b e l o w . W i t h t h i s a p p r o a c h , e d d y c u r r e n t s w h i c h w o u l d c i r c u l a t e i n t h e g r o u n d i f c u r r e n t f l o w s i n t o c o n d u c t o r 1 a b o v e g r o u n d a n d r e t u r n s t h r o u g h c o n d u c t o r 2 a b o v e g r o u n d , a r e i g n o r e d . I n r e f e r e n c e [ 2 0 ] i t h a s b e e n s h o w n t h a t t h i s e f f e c t i s n e g l i g i b l e u p t o 1 K H z f o r t h e c a s e o f a 5 0 0 k V o v e r h e a d l i n e . I n t h e l o w e r f r e q u e n c y r e g i o n , t h i s a p p r o a c h g i v e s v e r y a c c u r a t e a n s w e r s , b u t a t h i g h e r f r e q u e n c i e s t h e r e s u l t s m u s t b e i n t e r p r e t e d w i t h s o m e c a u t i o n , u n l e s s i t c a n b e s h o w n t h a t 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 i s m u c h m o r e p r o n o u n c e d t h a n e d d y c u r r e n t e f f e c t s i n t h e g r o u n d . T h e m a i n a d v a n t a g e o f t h i s m o d e l i s t h a t t h e m o r e c o m p l i c a t e d g r o u n d r e t u r n f o r m u l a e m u s t o n l y b e u s e d i n o n e r o w a n d o n e c o l u m n o f t h e m a t r i x i n ( 2 . 1 2 a ) . F i g u r e 3 . 3 M o d e l w i t h g r o u n d r e p r e s e n t e d a s o n l y o n e c o n d u c t o r 28 I n a d d i t i o n , t h e r e i s f r e e d o m a s t o t h e c h o i c e o f l o c a t i o n o f t h e f i c t i t i o u s r e t u r n p a t h . T h e d i s t a n c e s t o b e u s e d i n e q u a t i o n ( 3 . 5 ) c a n b e n e a r l y h a l v e d b y c e n t r a l l y l o c a t i n g t h i s p a t h . T h i s r e d u c e s t h e v a l u e s o f t h e p a r a m e t e r | m D ^ | i n e q u a t i o n ( 3 . 5 ) , t h e r e b y g i v i n g m o r e a c c u r a t e a p p r o x i m a t i o n s , a n d a l s o d e l a y s t h e u s e o f t h e m o r e c o m p l i c a t e d f o r m u l a ( 3 . 6 ) f o r m u c h h i g h e r f r e q u e n c i e s . W r i t i n g t h e l o o p e q u a t i o n s f o r t h e c i r c u i t o f F i g u r e 3 . 3 g i v e s : ~ V " 1 v„ = N V g _ J l l I N J g l Z 1 N Z l g Z N N Z N g z M z g N g g i l !_ g. N ( 3 . 9 ) i n w h i c h Z . r e f e r s t o t h e m u t u a l i m p e d a n c e b e t w e e n s u b c o n d u c t o r i a n d i g g r o u n d w i t h c o m m o n r e t u r n i n q . T h i s c a n n o t b e c a l c u l a t e d d i r e c t l y b y e q u a t i o n ( 3 . 5 ) . E q u a t i o n ( 3 . 5 ) i s v a l i d o n l y w h e n t h e c o m m o n r e t u r n i s t h e g r o u n d . T h e n e x t s e c t i o n 3 . 7 s h o w s h o w Z i s d e r i v e d u s i n g e q u a t i o n s ( 3 . 5 ) . 3 . 7 T h e M u t u a l I m p e d a n c e B e t w e e n a S u b c o n d u c t o r a n d G r o u n d w i t h C o m m o n  R e t u r n i n A n o t h e r S u b c o n d u c t o r C o n s i d e r F i g u r e ( 3 . 4 ) i n w h i c h t h e c o m m o n r e t u r n i s t h e g r o u n d . T h e l o o p i m p e d a n c e s m a y b e w r i t t e n a s : J H g Z 9 1 L 2 1 s J 1 2 g J 2 2 g ( 3 . 1 0 ) 2 9 v» ( g r o u n d r e t u r n ) *1 t F i g u r e 3 . 4 Two c o n d u c t o r s w i t h c o m m o n g r o u n d r e t u r n A l l t h e i m p e d a n c e t e r m s i n ( 3 . 1 0 ) c a n b e c a l c u l a t e d b y u s i n g C a r s o n ' s o r W e d e l p o h l ' . s e q u a t i o n s . N o w c o n s i d e r a s i m i l a r c i r c u i t i n F i g u r e ( 3 . 5 ) i n w h i c h t h e c o m m o n r e t u r n i s c o n d u c t o r 2 . : » > V n ¥ Q ( g r o u n d ) if ( r e i u r n ) F i g u r e 3 . 5 C i r c u i t o f o n e c o n d u c t o r a n d t h e g r o u n d w i t h c o m m o n r e t u r n i n a s e c o n d c o n d u c t o r 30 T h e l o o p e q u a t i o n s m a y b e w r i t t e n a s : - 1 1 2 J l g 2 J g g 2 . ( 3 . 1 1 ) T h e t h i r d s u b s c r i p t s i n e q u a t i o n s ( 3 . 1 0 ) a n d ( 3 . 1 1 ) d e n o t e t h e c o m m o n r e t u r n . T h e t e r m Z . =Z . , „ i s t h e o n e o f i n t e r e s t h e r e . T h e c i r c u i t s o f F i g u r e s ( 3 . 4 ) l g 2 g l 2 a n d ( 3 . 5 ) a r e e q u i v a l e n t i f a n d V a = \ ~ V 2 V b = -V2 I 2 - - ( I g + I X ) ( 3 . 1 2 ) ( 3 . 1 3 ) ( 3 . 1 4 ) S u b s t i t u t i n g t h e s e i n t o e q u a t i o n ( 3 . 1 0 ) g i v e s • v x - v2" - " V 2 -Z l l g Z 1 2 g - Z 1 2 g Z 1 2 g Z 2 2 g - Z 2 2 g I , 1 - I - I , g 1 ( 3 . 1 5 ) V - V 1 2 Z l l g + Z 2 2 g ~ 2 Z 1 2 g Z 2 2 g Z 1 2 g Z 2 2 g Z 1 2 g J 2 2 g g ( 3 . 1 6 ) F r o m ( 3 . 1 2 ) a n d ( 3 . 1 3 ) , i t i s e v i d e n t t h a t e q u a t i o n s ( 3 . 1 1 ) a n d ( 3 . 1 6 ) a r e i d e n t i c a l , h e n c e : Z l g 2 Z 2 2 g Z 1 2 g ( 3 . 1 7 ) T h e m u t u a l i m p e d a n c e s ( Z ^ g ) r e q u i r e d i n e q u a t i o n ( 3 . 9 ) c a n t h e r e -f o r e b e c a l c u l a t e d u s i n g ( 3 . 1 7 ) , ( n o t e t h a t Z . = Z . w h e r e q i s t h e f i c t i t i o u s i g i g q c o m m o n r e t u r n ) . 31 T h u s u s i n g e q u a t i o n ( 3 . 1 7 ) i n f o r m i n g t h e m a t r i x o f e q u a t i o n ( 3 . 9 ) r e s u l t s i n t h e u s e o f t h e g r o u n d r e t u r n f o r m u l a e f o r o n l y o n e r o w a n d o n e c o l u m n . I f t h e i n f i n i t e s e r i e s ( o r i n f i n i t e i n t e g r a l ) f o r m s o f t h e g r o u n d r e t u r n i m p e d a n c e m u s t b e u s e d , m o d e l I I I w i l l b e f a s t e r t h a n m o d e l I I s i n c e t h e i n f i n i t e s e r i e s w o u l d h a v e t o b e e v a l u a t e d f o r e v e r y e l e m e n t o f t h e m a t r i x [Z , . ] i n t h e l a t t e r c a s e , b i g 3 . 8 C o m p a r i s o n o f M o d e l I I I w i t h t h e T r a n s i e n t N e t w o r k A n a l y z e r C i r c u i t s I t s h o u l d b e n o t e d t h a t t h e p r o c e d u r e i n M o d e l I I I i s r e l a t e d t o t h e m e t h o d u s e d i n r e p r e s e n t i n g t h r e e p h a s e t r a n s m i s s i o n l i n e s o n t h e T r a n s i e n t N e t w o r k A n a l y s e r ( T N A ) [ 2 6 ] , w h e r e t h e i m p e d a n c e o f t h e g r o u n d r e t u r n i s d e c o u p l e d f r o m t h e p h a s e s . I t i s t h e n i n c l u d e d a s a n e x t r a c o n d u c t o r a n d t h e r e f o r e e l i m i n a t e s t h e n e e d t o m o d e l t h e g r o u n d r e t u r n i n e v e r y l i n e . O n a t h r e e p h a s e l i n e , A V a Z a a Z a b Z A V b = b a Z b b Z AV c Z c a Z c b z a c b e c c ( 3 . 1 8 ) A s s u m i n g t h a t t h e l i n e i s t r a n s p o s e d , t h e a v e r a g e m u t u a l i m p e d a n c e i s ; z = f (z , + z, +z ) m 3 a b b e c a ( 3 . 1 9 ) S c j u a t i o n ( 3 . 1 8 ) c a n b e w r i t t e n a s : ~AV a A V b = AV c -z a a m Z a b - Z m Z a c - Z m , -z b a m Z b b - Z m b e - Z m - Z c a m Z c b - Z m Z c c - Z m m m m I + 1 , + i a b c 32 where I a + 1^ + I c = Ig i s the current i n the extra conductor, ground i n th i s case. The ground return formula with i t s pronounced frequency dependence i s then only used for i n the l a s t column. A l l other elements Z a^-Z m , Z , -Z are calculated with ground ignored. Furthermore, i f the l i n e i s ab m . 0 . 0 transposed, the diagonal elements Z^-Z^, etc., become equal to the p o s i t i v e sequence impedance, and a l l off-diagonal elements Z -Z , etc., become zero. '33 C h a p t e r 4 R E S U L T S 4.1 C o m p a r i s o n o f t h e M e t h o d o f S u b d i v i s i o n s w i t h S t a n d a r d M e t h o d s T h i s s e c t i o n s h o w s h o w s k i n a n d p r o x i m i t y e f f e c t s a r e t a k e n i n t o a c c o u n t b y s u b d i v i d i n g t h e c o n d u c t o r s . T h e i m p e d a n c e o f a r e t u r n c i r c u i t o f t w o c o n d u c t o r s p l a c e d t w o m e t r e s a p a r t , a s s h o w n i n F i g u r e 4.1, i s c a l c u l a t e d . T h e d . c . r e s i s t a n c e o f e a c h c o n d u c t o r i s 0.0417 fi/km a n d t h e f r e q u e n c y i s 60 Hz. 2000.0mm F i g u r e 4 . 1 A r e t u r n c i r c u i t o f t w o c o n d u c t o r s f o r a p a r t T h e l a r g e s e p a r a t i o n b e t w e e n t h e t w o c o n d u c t o r s m a k e s p r o x i m i t y e f f e c t n e g l i g i b l e . T h e i n c r e a s e i n r e s i s t a n c e d u e t o s k i n e f f e c t i s c o r r e c t e d f o r b y u s i n g B e s s e l f u n c t i o n s . T h e U B C / B P A l i n e c o n s t a n t s p r o g r a m [ 1 6 ] i s u s e d f o r t h i s . T h e c o r r e c t e d v a l u e o f t h e i m p e d a n c e i s : Z= 0 . 0 8 8 7 + j 0 . 7 9 0 1 fi/km T h i s i s t a k e n a s t h e e x a c t r e f e r e n c e v a l u e . B y u s i n g v a r i o u s n u m b e r s o f s u b d i v i s i o n s t h e i m p e d a n c e s s h o w n i n T a b l e 4 . 1 a r e o b t a i n e d . F i g u r e 4 . 2 s h o w s t h e i m p e d a n c e v a r i a t i o n s a s a f u n c t i o n o f t h e n u m b e r o f s u b d i v i s i o n s . I t i s s e e n t h a t t h e e x a c t r e f e r e n c e v a l u e s a r e 34 T a b l e 4 . 1 V a r i a t i o n o f I m p e d a n c e w i t h t h e N u m b e r o f S u b d i v i s i o n s N o . o f S u b d i v i s i o n s R ft/km X n/km X i n t e r n a l ft/km E r r o r s R X 1 0 . 0 8 3 3 0 . 7 9 1 3 0 . 0 3 7 7 6 . 1 % 0 . 2 % 7 0 . 0 8 5 5 0 . 8 0 1 6 0 . 0 4 8 0 3 . 6 % 1 . 4 % 1 9 0 . 0 8 7 8 0 . 7 9 4 4 0 . 0 4 0 8 1 . 0 % 0 . 5 % 3 7 0 . 0 8 8 3 0 . 7 9 2 3 0 . 0 3 8 7 0 . 5 % 0 . 3 % 6 1 0 . 0 8 8 5 0 . 7 9 1 4 0 . 0 3 7 8 0 . 2 % 0 . 2 % R e f e r e n c e 0 . 0 8 8 7 0 . 7 9 0 1 0 . 0 3 6 5 0 . 0 % 0 . 0 % a p p r o a c h e d a s t h e n u m b e r o f s u b d i v i s i o n s i s i n c r e a s e d . H o w e v e r , i t i s b e s t t o k e e p t h e n u m b e r o f s u b d i v i s i o n s a s l o w a s p o s s i b l e . N i n e t e e n s u b d i v i s i o n s m a y b e a p p r o p r i a t e i f a n e r r o r o f o n e p e r c e n t i s t o l e r a b l e . S u b s t a n t i a l s a v i n g s i n s t o r a g e a n d c o m p u t i n g t i m e r e s u l t f r o m k e e p i n g t h e n u m b e r o f s u b d i v i s i o n s d o w n . T h e t w o c o n d u c t o r s f o r m i n g t h e r e t u r n c i r c u i t o f F i g u r e . 4 . 1 a r e b r o u g h t c l o s e t o g e t h e r , a s s h o w n i n F i g u r e 4 . 3 . V a r i o u s n u m b e r s o f s u b -d i v i s i o n s a r e u s e d o n t h e t w o c o n d u c t o r s . T h e i m p e d a n c e s c a l c u l a t e d a r e c o m p a r e d w i t h c a l c u l a t i o n s d o n e u s i n g s t a n d a r d m e t h o d s w h i c h i n v o l v e t h e u s e o f p u b l i s h e d c h a r t s a n d t a b l e s t o c o r r e c t f o r p r o x i m i t y e f f e c t a s s h o w n i n C h a p t e r 2 o f r e f e r e n c e [ 1 7 ] . 3 5 0-032 -N U M B E R O F S U B C O N D U C T O R S B r o k e n l i n e s a r e t h e r e f e r e n c e v a l u e s . • ' F i g u r e 4 . 2 V a r i a t i o n o f i m p e d a n c e w i t h t h e n u m b e r o f s u b c o n d u c t o r s l< i 27.02 mm F i g u r e 4 . 3 A r e t u r n c i r c u i t o f t w o c o n d u c t o r s v e r y c l o s e t o g e t h e r A c c o r d i n g t o r e f e r e n c e [ 1 7 ] , t h e a . c . r e s i s t a n c e o f t h e r e t u r n c i r c u i t a b o v e i s r = R ' x ~y (A . D 36 w h e r e R ' = a . c . r e s i s t a n c e c o r r e c t e d f o r s k i n e f f e c t o n l y . R " / R ' = p r o x i m i t y e f f e c t r e s i s t a n c e r a t i o . A s i m i l a r e q u a t i o n h o l d s f o r t h e i n d u c t a n c e . F r o m t h e C h a r t s a n d T a b l e s o f r e f e r e n c e [ 1 7 ] , t h e p r o x i m i t y e f f e c t r e s i s t a n c e a n d i n d u c t a n c e r a t i o s a r e c a l c u l a t e d t o b e 1 . 1 8 a n d 0 . 9 5 r e s p e c t i v e l y . T h e i m p e d a n c e o f t h e c i r c u i t i n F i g u r e 4 . 3 , w h e n c o r r e c t e d f o r s k i n e f f e c t o n l y , i s : Z = 0 . 0 8 8 7 + j 0 . 1 4 1 0 ft/km A p p l y i n g t h e a b o v e p r o x i m i t y c o r r e c t i o n f a c t o r s g i v e s a v a l u e o f Z = 0 . 1 0 4 8 + j 0 . 1 3 4 0 ft/km. T h i s i s u s e d a s t h e r e f e r e n c e v a l u e i n T a b l e 4 . 2 w h i c h c o m -p a r e s i t w i t h t h o s e o b t a i n e d f r o m v a r i o u s s u b d i v i s i o n s . I n u s i n g c o n v e n t i o n a l m e t h o d s o f i m p e d a n c e c a l c u l a t i o n s , c o r r e c t i o n s f o r p r o x i m i t y e f f e c t a r e m a d e f o r t w o o r t h r e e c o n d u c t o r s b y t h e u s e o f " e s t i m a t i n g c h a r t s " a n d c o r r e c t i n g " f a c t o r t a b l e s " d e r i v e d f r o m o t h e r w i s e c o m p l i c a t e d f o r m u l a e [ 1 8 ] , a s m e n t i o n e d a b o v e . O n t h e o t h e r h a n d , t h e m e t h o d o f s u b d i v i s i o n s i s n o t l i m i t e d t o o n l y t w o o r t h r e e c o n d u c t o r s o r c o m m o n , f o r m s o f c o n d u c t o r a r r a n g e m e n t ( f l a t o r d e l t a s p a c i n g ) a s t h e c h a r t s r e f e r r e d t o a b o v e s e e m t o b e . A l s o , i n u s i n g s u b d i v i s i o n s , t h e c u r r e n t d i v i s i o n a m o n g t h e p a r a l l e l c o n d u c t o r s n e e d n o t b e k n o w n a p r i o r i . W h e r e m a n y c o n d u c t o r s a r e i n v o l v e d a n d t h e c u r r e n t d i v i s i o n i s n o t k n o w n ( a s i s t h e c a s e i n m o s t c a b l e s y s t e m s i n c i t i e s w h e r e m a n y c a b l e s a n d p i p e s r u n p a r a l l e l i n t h e s a m e o r a d j a c e n t d u c t s ) , t h i s m e t h o d o f s u b d i v i d i n g t h e c o n d u c t o r s i s v e r y u s e f u l , a n d g i v e s 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 . 37 T a b l e 4 . 2 V a r i a t i o n o f t h e I m p e d a n c e o f t h e C i r c u i t o f F i g . 4 . 3 w i t h t h e N u m b e r o f S u b d i v i s i o n s , S h o w i n g t h e I n c l u s i o n o f B o t h S k i n a n d P r o x i m i t y E f f e c t s i n t h e C a l c u l a t i o n s . N o . o f R X X i n t e r n a l ft/km E r r o r i n S u b d i v i s i o n s ft/km ft/km R X ** 1 0 . 0 8 3 3 0 . 1 4 2 2 0 . 0 3 7 7 2 0 . 5 % 6 . 1 % 7 0 . 0 9 6 6 0 . 1 4 7 4 0 . 0 4 2 9 7 . 8 % 1 0 . 0 % 1 9 0 . 1 0 1 0 0 . 1 3 9 4 0 . 0 3 4 9 3 . 6 % 4 . 0 % 3 7 0 . 1 0 2 2 0 . 1 3 7 0 0 . 0 3 2 5 2 . 5 % 2 . 2 % 6 1 0 . 1 0 2 6 0 . 1 3 6 1 0 . 0 3 1 6 2 . 1 % 1 . 6 % S K I N * 0 . 0 8 8 7 0 . 1 4 1 0 0 . 0 3 6 5 1 5 . 4 % 5 . 2 % R e f e r e n c e V a l u e 0 . 1 0 4 8 0 . 1 3 4 0 0 . 0 2 9 5 0 . 0 % 0 . 0 % * • C o r r e c t e d f o r s k i n e f f e c t o n l y . * * N o c o r r e c t i o n f o r b o t h s k i n a n d p r o x i m i t y e f f e c t s . 4 . 2 C o m p a r i s o n o f G r o u n d R e t u r n . F o r m u l a e F o r m u l a e f o r c a l c u l a t i n g t h e s e l f a n d m u t u a l i m p e d a n c e s o f l o o p s w i t h g r o u n d r e t u r n h a v e b e e n g i v e n b y m a n y a u t h o r s , i n c l u d i n g C a r s o n [ 1 0 , 1 1 ] , P o l l a c z e k [ 1 9 ] , W e d e p o h l a n d W i l c o x [ 9 ] , a n d K a l y u z h n y i a n d L i f s h i t s [ 1 3 ] . M o s t o f t h e s e f o r m u l a e a r e g i v e n i n t h e f o r m o f i n f i n i t e s e r i e s a n d a r e n o t a l w a y s e a s y t o u s e . T h e v a r i a t i o n o f g r o u n d r e t u r n i m p e d a n c e w i t h f r e q u e n c y , a s g i v e n b y s o m e o f t h e f o r m u l a e , a r e c o m p a r e d i n t h i s s e c t i o n . A c c o r d i n g t o J . R . C a r s o n [ 1 0 ] , t h e v a r i a t i o n o f t h e g r o u n d r e t u r n i m p e d a n c e w i t h t h e d e p t h o f b u r i a l . ' . ' . o f a c o n d u c t o r i s m i n i m a l , a n d c a n b e c a l c u l a t e d f o r m o s t f r e q u e n c i e s b y u s i n g e q u a t i o n ( 3 . 1 ) . T h i s i s v e r i f i e d b y u s i n g M o d e l I t o c a l c u l a t e t h e i m p e d a n c e o f a b u r i e d c o n d u c t o r w i t h g r o u n d r e t u r n f o r v a r i o u s d e p t h s o f b u r i a l . T h e r e s u l t s o f t h i s a r e s h o w n i n F i g u r e 4.4. 1.171 M 6 9 1-167 • < in o z < IMPI 1 1 1-5 -1-0 - 0 - 5 0 8 0-5 • 10 DEPTH OF B U R I A L ( m ) t-5 F i g u r e 4.4 V a r i a t i o n o f t h e i m p e d a n c e o f a b u r i e d c o n d u c t o r w i t h d e p t h o f b u r i a l A v e r y s i m p l e a n d u s e f u l f o r m o f t h e g r o u n d r e t u r n i m p e d a n c e h a s b e e n g i v e n b y W e d e p o h l a n d W i l c o x ( e q u a t i o n s ( 3 . 5 ) ) , w h i c h i s a n a p p r o x i m a -t i o n o f t h e i n f i n i t e s e r i e s f o r m o f s o l u t i o n . K a l y u z h n y i a n d L i f s h i t s [ 1 3 ] a l s o d e r i v e a f o r m u l a , t h e f i n a l ',. r e s u l t s o f w h i c h , t h o u g h v e r y d i f f e r e n t f r o m t h e m o r e c o n v e n t i o n a l o n e s , 3 9 a r e c l a i m e d t o a g r e e v e r y c l o s e l y w i t h e x p e r i m e n t a l l y m e a s u r e d d a t a . K a l y u z h n y i a n d L i f s h i t s g i v e t h e s e l f i m p e d a n c e o f g r o u n d r e t u r n ( Z e ) a s : Z e = ^ [ A n - 2 — 2TT I y P r n - J- (4.2) w h e r e y = 0 . 5 7 7 2 1 5 7 - E u l e r ' s c o n s t a n t . . r = r a d i u s o f b u r i e d c o n d u c t o r o v e r i n s u l a t i o n (m) p = < W P p = r e s i s t i v i t y o f g r o u n d (ftm) h = d e p t h o f b u r i a l o f c o n d u c t o r (m) E q u a t i o n ( 4 . 2 ) g i v e s a r e a l p a r t ( R e ) o f : R e = 2 ?r 2 f . 1 0 _ 7 Q/m (4.3) w h i c h i s q u i t e d i f f e r e n t f r o m t h a t o b t a i n e d f r o m C a r s o n ' s e q u a t i o n s w h i c h a p p r o x i m a t e s [ 1 7 ] t o : R e = T T 2 f . 1 0 7 tt/m (4.4) C a r s o n ' s e q u a t i o n s ( o r a p p r o x i m a t i o n s o f t h e m ) h a v e b e e n u s e d f o r m a n y y e a r s b y s e v e r a l a u t h o r s a n d o t h e r s i n v o l v e d w i t h a n a l y s i n g t h e c o n d u c t i o n o f e l e c t r i c c u r r e n t t h r o u g h t h e g r o u n d . T h e v e r y m a r k e d d e v i a t i o n f r o m C a r s o n ' s e q u a t i o n s g i v e n b y K a l y u z h n y i a n d L i f s h i t s i s , t h e r e f o r e , w o r t h y o f i n v e s t i g a t i o n . I n o r d e r t o d e t e r m i n e w h i c h f o r m u l a b e s t a p p r o x i m a t e s t h e b e h a v i o u r o f t h e e a r t h , t h e i m p e d a n c e s o f t h e c i r c u i t s o f F i g u r e 4 . 5 a r e c a l c u l a t e d b y u s i n g t h e s u b d i v i d e d g r o u n d r e p r e s e n t a t i o n s o f F i g u r e 3 . 1 ( i . e . M o d e l I ) . I n t h i s c a l c u l a t i o n , t h e g r o u n d i s d i v i d e d i n t o f i v e l a y e r s o f 6 2 s u b c o n d u c t o r s . I n T a b l e 4 . 3 a n d F i g u r e 4 . 6 , t h e r e s u l t s o b t a i n e d f r o m t h e s e l f i m p e d a n c e Oo012m ( a ) F i g u r e 4.5 C r o s s s e c t i o n s o f b u r i e d c o n d u c t o r s f o r g r o u n d r e t u r n i m p e d a n c e c a l c u l a t i o n s c a l c u l a t i o n s u s i n g M o d e l I a n d b y u s i n g t h e v a r i o u s f o r m u l a e a r e c o m p a r e d . T h e r e i s c l o s e a g r e e m e n t i n t h e r e s i s t a n c e v a l u e s a s c a l c u l a t e d u s i n g t h e a b o v e m e t h o d o f s u b d i v i d i n g t h e g r o u n d , C a r s o n ; ' s e q u a t i o n s , a n d W e d e p o h l ' s e q u a t i o n s , b u t t h e r e s u l t s o b t a i n e d u s i n g K a l y u z h n y i a n d L i f s h i t s e q u a t i o n d e v i a t e f r o m t h e o t h e r s . T h e r e a c t a n c e v a l u e s c a l c u l a t e d f r o m t h e f i r s t t h r e e m e t h o d s d e v i a t e m o r e w i d e l y f r o m e a c h o t h e r t h a n t h e c a l c u l a t e d r e s i s t a n c e v a l u e s . T h e d e v i a t i o n s o f r e a c t a n c e r a n g e b e t w e e n 7% a t p o w e r f r e q u e n c y t o 12% a t 1 . 0 M H z . b e t w e e n t h e r e s u l t s o f C a r s o n ' s a n d W e d e p o h l ' s f o r m u l a e w h i l e t h e r e s u l t s f r o m t h e s u b d i v i s i o n s m e t h o d l i e s o m e w h e r e i n b e t w e e n ( s e e F i g u r e 4 . 6 b ) f o r a l l f r e q u e n c i e s . A v e r y m a r k e d d e v i a t i o n i n r e a c t a n c e ( o v e r 5 0 % a t p o w e r f r e q u e n c y ) i s o b t a i n e d f r o m t h e f o r m u l a o f K a l y u z h n y i a n d L i f s h i t s . D i s c r e p a n c i e s i n t h e v a r i o u s c a l c u l a t i o n s a r i s e f r o m t h e f a c t t h a t a l l t h e m e t h o d s a r e a p p r o x i m a t i o n s t o t h e r e a l c a s e a n d a l s o b e c a u s e t h e " i n t e r s t i c e s " b e t w e e n t h e s u b c o n d u c t o r s a r e n e g l e c t e d i n F i g u r e 3 . 1 a . T o e v a l u a t e t h e i n f l u e n c e o f t h i s l a t t e r a p p r o x i m a t i o n , m o s t o f t h e i n t e r s t i c e s o f F i g u r e 3 . 1 a w e r e f i l l e d w i t h s u b c o n d u c t o r s : f o r a f u l l e r r e p r e s e n t a t i o n o f t h e g r o u n d c r o s s s e c t i o n . R e s u l t s o b t a i n e d u s i n g t h e l a t t e r s h o w o n l y a s l i g h t i m p r o v e m e n t o v e r t h e r e s u l t s o f F i g u r e 3 . 1 a . F o r e x a m p l e , a t 6 0 H z , t h e i m p e d a n c e o f t h e c o n d u c t o r i n F i g u r e 4 . 5 a w i t h g r o u n d r e t u r n i s c a l c u l a t e d t o b e Z = 0 . 0 5 9 7 + j 0 . 9 2 3 6 ft/km w h e n t h e g r o u n d r e p r e s e n t a t i o n o f F i g u r e 3 . 1 a i s u s e d . B y u s i n g t h e g r o u n d r e p r e s e n t a t i o n o f F i g u r e 3 . 1 b , t h e c a l c u l a t e d i m p e d a n c e i s : Z = 0 . 0 5 8 9 + j 0 . 9 1 8 1 ft/km. T h e l a t t e r r e p -r e s e n t a t i o n o f t h e g r o u n d o n l y g i v e s a n i m p r o v e m e n t o f l e s s t h a n 1% i n t h e r e s u l t s w h e n c o m p a r e d w i t h t h e f o r m e r . T h e r e f o r e t h e g r o u n d r e t u r n r e p -r e s e n t a t i o n g i v e n i n F i g u r e 3 . 1 a w o u l d b e a d e q u a t e f o r t h i s p u r p o s e . F r e - R E S I S T A N C E ( f t / k m ) R E A C T A N C E ( f t / k m ) q u e n c y ( H z ) S u b d l - . v i s i o n s Wede— p o h l C a r s o n .. K a l y u -z h n y i S u b d i -v i s i o n s W e d e -p o h l C a r s o n K a l y u -z h n y i 2 . 2 . 0 0 2 0 . 0 0 2 0 . 0 2 0 . 0 0 4 0 . 0 3 9 0 . 0 4 1 0 . 0 3 8 0 . 0 6 2 4 . 5 . 0 0 5 0 . 0 0 5 0 . 0 5 0 . 0 0 9 0 . 0 7 7 0 . 0 8 1 0 . 0 7 5 0 . 1 1 9 9 . 0 . 0 0 9 0 . 0 0 9 0 . 0 9 0 . 0 1 8 0 . 1 5 0 0 . 1 6 0 0 . 1 4 5 0 . 2 3 0 1 5 . 0 . 0 1 5 0 . 0 1 5 0 . 0 1 5 0 . 0 3 0 0 . 2 4 4 0 . 2 6 0 0 . 2 3 6 0 . 3 7 4 3 0 . 0 . 0 3 0 0 . 0 3 0 0 . 0 3 1 0 . 0 5 9 0 . 4 7 5 0 . 5 0 3 0 . 4 6 0 0 . 7 2 3 6 0 . 0 . 0 6 0 0 . 0 5 9 0 . 0 6 2 0 . 1 1 8 0 . 9 2 4 0 . 9 7 9 0 . 9 1 4 1 . 3 9 1 2 0 . 0 . 1 2 0 0 . 1 1 9 0 . 1 2 4 0 . 2 3 7 1 . 8 0 1 . 9 1 1 . 7 3 2 . 6 8 5 0 0 . 0 . 4 9 9 0 . 4 9 7 0 . 5 1 8 0 . 9 8 7 7 . 0 3 7 . 4 9 6 . 7 9 1 0 . 3 I K 1 . 0 0 0 . 9 9 8 1 . 0 4 1 . 9 7 1 3 . 6 1 4 . 5 1 3 . 2 1 9 . 7 5 K 5 . 0 5 5 . 0 5 5 . 2 0 9 . 8 7 6 3 . 0 6 7 . 6 6 0 . 7 8 8 . 3 1 0 K 1 0 . 2 1 0 . 2 1 0 . 4 1 9 . 7 1 2 1 . 0 1 3 1 . 0 1 1 7 . 0 1 6 8 . 0 5 OK 5 2 . 2 5 3 . 1 5 2 . 5 9 8 . 7 5 5 5 . 0 6 0 1 . 0 5 3 5 . 0 7 3 8 . 0 0 . 1 M 1 0 6 . 0 1 0 9 . 0 1 0 5 . 0 1 9 7 . 4 1 0 6 4 . 1 1 5 5 . 1 0 2 6 . 1 3 8 9 . 0 . 5 M 5 5 6 . 0 6 1 1 . 0 5 3 0 . 0 9 8 7 . 0 4 7 6 7 . 5 2 0 7 . 4 6 2 2 . 6 3 7 1 . l . O M 1 1 6 . 1 3 2 0 . 1 0 6 4 . 1 9 7 4 . 9 0 3 8 . 9 8 8 1 . 8 8 1 0 . 1 1 8 7 0 . T a b l e 4 . 3 S e l f I m p e d a n c e o f G r o u n d R e t u r n P a t h a s C a l c u l a t e d U s i n g S u b d i v i d e d G r o u n d , a n d O t h e r F o r m u l a e 43 F i g u r e "4.6 C o m p a r i s o n o f c a l c u l a t e d s e l f i m p e d a n c e s o f a g r o u n d r e t u r n l o o p . 44 M u t u a l I m p e d a n c e o f G r o u n d R e t u r n P a t h T a b l e 4 . 4 a n d F i g u r e 4 . 7 s h o w t h e r e s u l t s o f t h e m u t u a l i m p e d a n c e s c a l c u l a t e d f o r t h e t w o b u r i e d c o n d u c t o r s o f F i g u r e 4 . 5 b . D e v i a t i o n s o f a b o u t 1% i n t h e r e s i s t a n c e a n d a b o u t 15% i n t h e r e a c t a n c e a r e o b t a i n e d a t p o w e r f r e q u e n c y b e t w e e n t h e r e s u l t s o f s u b d i v i s i o n s a n d t h o s e o b t a i n e d f r o m W e d e p o h l ' s e q u a t i o n s . T h e m u t u a l i m p e d a n c e v a l u e s c a l c u l a t e d b y u s i n g s u b d i v i s i o n s a n d b y u s i n g C a r s o n ' s o v e r h e a d l i n e e q u a t i o n s [ 1 1 , 1 6 ] a r e s i m i l a r f o r m o s t o f t h e f r e q u e n c i e s u s e d ; t h u s i t s e e m s C a r s o n ' s o v e r h e a d l i n e e q u a t i o n s m a y b e u s e d f o r c a l c u l a t i n g t h e m u t u a l i m p e d a n c e s b e t w e e n b u r i e d c o n d u c t o r s [ 2 0 ] . D e s p i t e t h i s c l o s e a g r e e m e n t i n t h e r e s u l t s , i t s h o u l d b e r e m e m b e r e d t h a t C a r s o n ' s e q u a t i o n s [ 1 1 ] w e r e d e r i v e d f o r c o n d u c t o r s l o c a t e d a b o v e g r o u n d . T a b l e 4 . 4 M u t u a l I m p e d a n c e B e t w e e n Two U n d e r g r o u n d C o n d u c t o r s F r e -R E S I S T A N C E ( f t / k m ) R E A C T A N C E ( f t / k m ) q u e n c y H z S u b d i -v i s i o n s C a r s o n W e d e -p o h l S u b d i -v i s i o n s C a r s o n * W e d e -p o h l 2 . 2 0 . 0 0 2 0 . 0 0 2 0 . 0 0 2 0 . 0 2 4 0 . 0 2 4 0 . 0 2 6 4 . 5 0 . 0 0 5 0 . 0 0 4 0 . 0 0 4 0 . 0 4 6 0 . 0 4 5 0 . 0 5 2 9 . 0 0 . 0 0 9 0 . 0 0 9 0 . 0 0 9 0 . 0 8 8 0 . 0 8 7 0 . 0 9 9 1 5 . 0 0 . 0 1 5 0 . 0 1 5 0 . 0 1 5 0 . 1 4 2 0 . 1 3 9 0 . 1 6 1 3 0 . 0 0 . 0 3 0 0 . 0 3 0 0 . 0 3 0 0 . 2 7 1 0 . 2 6 6 0 . 3 0 8 6 0 . 0 0 . 0 6 0 0 . 0 5 9 0 . 0 5 9 0 . 5 1 6 0 . 5 0 6 0 . 5 9 0 1 2 0 . 0 0 . 1 1 9 0 . 1 1 8 0 . 1 1 9 0 . 9 7 9 0 . 9 5 9 1 . 1 3 5 0 0 . 0 0 . 4 . 9 9 0 . 4 9 0 0 . 4 9 7 3 . 6 3 3 . 5 5 4 . 2 5 I k 0 . 9 9 9 0 . 9 7 8 0 . 9 9 6 6 . 8 2'- 6 . 6 6 8 . 0 6 5 k 5 . 0 3 4 . 8 4 5 . 0 4 2 9 . 0 2 8 . 3 3 5 . 2 1 0 k 1 0 . 1 9 . 6 0 1 0 . 1 5 3 . 6 5 2 . 4 6 6 . 0 5.0k<- 5 1 . 8 4 6 . 4 5 2 . 5 2 1 6 . 0 2 1 3 . 0 2 7 8 . 0 1 0 0 k 1 0 5 . 0 9 0 . 7 1 0 8 . 0 3 8 6 . 0 3 8 5 . 0 5 0 9 . 0 5 0 0 k 5 9 9 . 0 4 1 3 . 0 5 9 4 . 0 1 3 7 6 . 0 1 4 7 1 . 0 1 9 8 3 . * O v e r h e a d l i n e e q u a t i o n s u s e d . cc-i -t- -ID a o-o «nO" X---Wedepohl 100 1000 1 0 0 0 0 F R E Q U E N C Y C Hz ) ,100000 1000000 + — Subdivisions o — C a r s o n (0/H) X---Wedepohl no 100 idoo loooo iooooo 1006000" F R E Q U E N C Y ( Hz ) I Figure 4.7 Comparison of calculated mutual impedances between two buried conductors. 47 4 . 3 C o m p a r i s o n o f R e s u l t s f r o m t h e D i f f e r e n t M o d e l s T h e d a t a i s t a k e n f r o m r e f e r e n c e [ 4 ] . T h r e e d i s t r i b u t i o n c a b l e s ( 1 / 0 AWG a l u m i n u m c o r e d c a b l e s w i t h r e d u c e d n e u t r a l s ) a r e l a i d i n a f l a t f o r m a t i o n 8 i n c h e s a p a r t . T h e c o r e a n d s h e a t h r e s i s t a n c e s a r e 0 . 1 8 8 2 a n d 0 . 4 7 0 7 o h m s / 1 0 0 0 ' f t r e s p e c t i v e l y w i t h t h e i n s i d e a n d o u t s i d e d i a m e t e r s o f t h e i n s u l a t i o n a s 5 1 5 m i l s a n d 9 5 5 m i l s r e s p e c t i v e l y . T h e l i s t e d v a l u e s o f t h e z e r o ( 0 ) , p o s i t i v e ( 1 ) a n d n e g a t i v e ( 2 ) s e q u e n c e i m p e d a n c e m a t r i x e l e m e n t s a t 6 0 H z i n t h e r e f e r e n c e a r e : t W • 1 0 ~ 0 . 4 8 3 + j 0 . 2 3 6 - 0 . 0 0 3 + j 0 . 0 0 1 0 . 0 - j O . 0 0 3 s y m m e t r i c - 0 . 0 0 7 + j 0 . 0 0 8 0 . 1 9 9 + j 0 . 0 9 6 0 . 0 1 0 + j 0 . 0 0 4 _ ft/1000 f t . B y u s i n g M o d e l I I a n d t h e g r o u n d r e t u r n i m p e d a n c e f o r m u l a e o f e q u a t i o n s ( 3 . 7 ) a n d ( 3 . 8 ) , t h e s e q u e n c e i m p e d a n c e s c a l c u l a t e d a r e : [ Z 0 1 2 ] = 1 0 " 0 . 4 8 3 + J 0 . 2 3 1 - 0 . 0 0 3 + j 0 . 0 0 1 0 . 0 ' - j O . 0 0 3 s y m m e t r i c - 0 . 0 0 7 + j 0 . 0 0 8 0 . 1 9 8 + j 0 . 0 8 9 O . O l O + j O . 0 0 2 ft/lOOOft. I f t h e g r o u n d r e t u r n i m p e d a n c e e q u a t i o n s ( 3 . 5 ) d e r i v e d b y W e d e p o h l a n d W i l c o x a r e u s e d i n M o d e l I I , t h e f o l l o w i n g i m p e d a n c e m a t r i x i s o b t a i n e d . 0 2 1 0 0 . 5 0 6 + J 0 . 2 1 9 - 0 . 0 0 2 + j 0 . 0 0 1 0 . 0 - j O . 0 0 3 s y m m e t r i c - 0 . 0 0 7 + j 0 . 0 0 8 0 . 1 9 8 + j 0 . 0 8 3 0 . 0 1 0 + j 0 . 0 0 2 ft/lOOOft. 48 T h e m a x i m u m d e v i a t i o n b e t w e e n t h e s e q u e n c e i m p e d a n c e s a n d t h e r e f e r e n c e v a l u e s i s l e s s t h a n 3% i n t h e p o s i t i v e s e q u e n c e . B y u s i n g t h e m o d e l o f s e c t i o n 3 . 6 w h e r e t h e g r o u n d i s r e p r e s e n t e d a s o n l y o n e c o n d u c t o r ( M o d e l I I I ) , t h e f o l l o w i n g i m p e d a n c e s a r e o b t a i n e d , w i t h e q u a t i o n s ( 3 . 5 ) b e i n g u s e d f o r t h e g r o u n d r e t u r n i m p e d a n c e c a l c u l a t i o n s : 0 2 1 . 0 r 0 . 5 1 0 + j 0 . 2 2 5 s y m m e t r i c - 0 . 0 0 2 + j 0 . 0 0 1 - 0 . 0 0 7 + j 0 . 0 0 8 ft/1000 f t . 2 | _ 0 . 0 + J 0 . 0 0 3 0 . 1 9 8 + J 0 . 9 0 0 0 . 0 1 0 + j 0 . 0 0 2 _ [ Z 0 1 2 ] = L T h e m a x i m u m d e v i a t i o n i s 3 . 6 % f r o m t h e r e f e r e n c e : i n t h e z e r o s e q u e n c e . B y d i v i d i n g t h e g r o u n d r e t u r n p a t h i n t o s u b c o n d u c t o r s a n d u s i n g n o n e o f t h e a n a l y t i c a l l y d e r i v e d g r o u n d r e t u r n f o r m u l a e ( M o d e l I ) , t h e f o l l o w -i n g a r e c a l c u l a t e d f o r c o m p a r i s o n p u r p o s e s . [ Z 0 1 2 ] = 1 0 " 0 . 4 8 6 + j 0 . 2 3 1 - 0 . 0 0 3 + j 0 . 0 0 1 - 0 . 0 - t - j O . 0 0 3 2 1 s y m m e t r i c - 0 . 0 0 7 + j 0 . 0 0 8 0 . 1 9 8 + j 0 . 9 0 0 O . O l O + j O . 0 0 2 ft/1000 f t . T h e m a x i m u m d e v i a t i o n i n t h i s c a s e i s 1 . 5 % i n t h e p o s i t i v e s e q u e n c e m a g n i t u d e . T h e s u m m a r y o f t h e c o m p a r i s o n i s p r e s e n t e d i n T a b l e 4 . 5 . T a b l e 4 . 5 T h e C o m p a r i s o n o f V a r i o u s M o d e l s M o d e l C o m p u t i n g T i m e ( s ) M a x D e v i a t i o n % M a t r i x S i z e I 1 0 . 3 1 . 5 1 0 1 x 1 0 1 I I 0 . 7 3 3 . 0 3 9 x 39 I I I 0 . 7 8 3 . 6 4 0 x . 4 0 49 4 . 4 R e p r o d u c t i o n o f T e s t R e s u l t s A t e s t w a s c o n d u c t e d a t B r i t i s h C o l u m b i a H y d r o a n d P o w e r A u t h o r i t y [ 2 1 ] o n a c a b l e s y s t e m w h e r e t h e i n d u c e d c u r r e n t s i n b o n d e d s h e a t h s w e r e m e a s u r e d f o r v a r i o u s v a l u e s o f p h a s e c u r r e n t s . T h e s e t e s t r e s u l t s a r e r e -p r o d u c e d i n t h i s s e c t i o n . T h e i m p e d a n c e s n e e d e d t o c a l c u l a t e t h e i n d u c e d s h e a t h c u r r e n t s a r e c a l c u l a t e d u s i n g M o d e l I I . T h e t e s t i s c a r r i e d o u t o n t w o t h r e e - p h a s e c a b l e s i n t h e s a m e d u c t b a n k . T h e s h e a t h s o f b o t h c a b l e s a r e b o n d e d t o g e t h e r a t t h e i r e n d s a n d g r o u n d e d t h r o u g h h i g h r e s i s t a n c e s ( s e e F i g u r e 4 . 8 ) . T h e u n b a l a n c e i n t h e p h a s e c u r r e n t s , t h e c i r c u l a t i n g c u r r e n t s i n t h e s h e a t h s , a n d t h e c u r r e n t i n a p a r a l l e l n e u t r a l c o n d u c t o r a r e m e a s u r e d . T a b l e 4 . 6 i s t a k e n f r o m r e f e r e n c e [ 2 1 ] a n d i t g i v e s b o t h t h e m e a s u r e d v a l u e s a n d t h e p r e d i c t e d v a l u e s o f t h e i n d u c e d s h e a t h c u r r e n t s o b t a i n e d u s i n g t h e i m p e d a n c e s c a l c u -l a t e d f r o m u s u a l h a n d b o o k m e t h o d s [ 3 , 2 4 ] . W r i t i n g t h e l o o p e q u a t i o n s a r o u n d t h e l o o p f o r m e d b y t h e b o n d e d s h e a t h s g i v e s : ° = I Z S l , k \ + I Z S 2 , k \ < 4 ' 4 > f o r k = A l , B l , C l , A 2 , B 2 , C 2 , S I , S 2 , N 0 = x s i + X S 2 ( 4 . 5 ) * A 1 + ^ 1 + x c i ( 4 . 6 ) V X A 2 + I B 2 + I C 2 ( 4 . 7 ) D u e t o t h e s y m m e t r y i n t h e c a b l e s a n d t h e s p i r a l l i n g o f t h e c o r e s , m o s t o f t h e m u t u a l i m p e d a n c e s a b o v e a r e e q u a l , f o r e x a m p l e : 50 ^ I2>70 f t , : j|-F i g u r e 4.8 E l e c t r i c a l l a y o u t o f . ; t h e i n d u c e d s h e a t h c u r r e n t t e s t . F i g u r e 4.9 C i r c u i t d i a g r a m o f i n d u c e d s h e a t h c u r r e n t t e s t . 52 Z A 1 S 1 Z B 1 S 1 Z C 1 S 1 Z A 1 S 2 = Z B 1 S 2 = Z C 1 S 2 H e n c e e q u a t i o n s ( 4 . 4 ) a n d ( 4 . 5 ) r e d u c e t o : 0 - Z s i S l ^ l + ( Z A 1 S 1 + Z A 1 S 2 ) ( I A 1 + I B 1 + I C 1 )  + ( Z A 2 S 1 + Z A 2 S 2 ) ( I A 2 + I B 2 + I C 2 ) + ( Z S 2 N +W * N ( 4 ' 8 ) hl = Z~Z; I ( Z A 1 S 1 + Z A 1 S 2 ) I 1 + ( Z A 2 S 1 + Z A 2 S 2 > I 2 + ( Z S 2 N + Z S 1 N ) I N ] ( 4 ' 9 ) T h e i n d u c e d s h e a t h c u r r e n t c a n t h e r e f o r e b e c a l c u l a t e d u s i n g e q u a t i o n ( 4 . 9 ) . T h e v a l u e s c a l c u l a t e d f r o m u s i n g s u b d i v i s i o n s i n t h e i m p e d a n c e c a l c u l a t i o n s a r e a l s o p r e s e n t e d i n T a b l e 4 . 6 . T h e m e a s u r e m e n t s w e r e m a d e f o r t w o c a s e s i n w h i c h t h e b o n d s b e t w e e n t h e s h e a t h s w e r e g r o u n d e d t h r o u g h g r o u n d r e s i s t o r s R ^ a n d R ^ > a n d w h e n t h e y w e r e n o t g r o u n d e d . T h e r e a r e i m p r o v e m e n t s i n t h e r e s u l t s a s c a l c u l a t e d b y t h e m e t h o d o f s u b d i v i s i o n s c o m p a r e d w i t h t h e u s u a l h a n d b o o k m e t h o d s [ 3 , 5 , 2 1 ] . W h i l e t h e l a t t e r g i v e a n a v e r a g e t o t a l d e v i a t i o n i n b o t h p h a s e a n d m a g n i t u d e o f 14% b e t w e e n t h e c a l c u l a t e d a n d m e a s u r e d r e s u l t s f o r t h e u n g r o u n d e d c a s e , t h e f o r m e r p r o d u c e s o n l y a n a v e r a g e t o t a l d e v i a t i o n o f 8 % . S i m i l a r r e s u l t s a r e a l s o e v i d e n t i n t h e s e c o n d s e t o f m e a s u r e m e n t s . T h e i m p r o v e m e n t s a r e m a i n l y d u e t o t h e i n c l u s i o n o f p r o x i m i t y e f f e c t s w h e n s u b d i v i s i o n s o f t h e c a b l e c o n d u c t o r s a n d s h e a t h s a r e u s e d i n t h e i m p e d a n c e c a l c u l a t i o n s . I t s h o u l d b e p o i n t e d o u t t h a t t h e r e a r e a s m a n y a s n i n e c o n d u c t o r s ( i n c l u d i n g s h e a t h s a n d n e u t r a l ) i n t h e c a b l e s y s t e m u n d e r c o n s i d e r a t i o n . T a b l e 4.6 I n d u c e d C u r r e n t s i n B o n d e d S h e a t h s M E A S U R E D C U R R E N T S ( A M P S ) C A L C U L A T E D % D e v i a t i o n s M a g n i t u d e M a g . & P h a s e * * _|_ 1 1 1 2 x s i hi:. * + * + U n g r c u n d e d B o n d s 1 25.01 4° 53.0 191° 23.0|_0° 10.8|255° 13.0 260° 11.8 261° 20 10'; 23 12 2 25.1 [21° 41.4 198° 17.0[0° 9.6 j 269° 10.6 270° 10.0|272° 10 5 11 6 3 23.2[32° 44.9 198° 18.4[rj° 10.0 269° 10.8 273° 10.2 275° 8 2 10 5 4 22.0 j28° 47.4|198° 20.7[0° 10.0 269° 11.0 272° 10.5 274° 10 5 11 6 5 35.0[14° 66.9)191° 29.4 [0° 14.9|255° 16.2 263° . 15.4 265° 8 3 11 7 6 30.6|353° 65.9 191° 30.4[0° 13.5(248° 15.2 256° 14.4|258° 13 7 16 11 A v e r a g e % D e v i a t i o n s 12 5 14 8 B o n d s G r o u n d e d 1 28.4[21° 56.9 191° 22.8|_0° 13.5|262° 13.5 265° 12.9|267° 0 5 1 7 2, 28.3|21° 55.1 194° 19.9\0° 12.5|262° 13.2 263° 12.6|269° 6 1 8 3 3 29.5[21° 52.4 198° 19.9 [0 ° 12.2j262° 13.0 270° 12.4|271° 7 1 10 5 4 23.0[32° 50.4 194° 21.8[0° 10.7|269° 11.5 271° 11.0|272° 7 3 8 4 5 26.4J32° 60.9 198° 25.9 [0° 12.91262° 13.8 273° 13.l|275° 7 2 11 o 7 6 31.7|28° 55.1 198° 18.0[0° 13.0j 262 ° 13.7 273° 13.0|274° 5 1 9 5 A v e r a g e % D e v i a t i o n s i 5 2 8 5 * T a k e n f r o m R e f e r e n c e [21] + c a l c u l a t e d u s i n g s u b d i v i s i o n s . 54 T h e m a i n d e v i a t i o n s o c c u r i n t h e p h a s e a n g l e s . T h i s m a y b e d u e t o t h e s l o w c o n v e r g e n c e o f t h e r e a c t a n c e t o t h e t r u e v a l u e ( w h e n s u b d i v i s i o n s a r e u s e d ) a s i l l u s t r a t e d i n T a b l e s 4 . 1 a n d 4 . 2 . T h i s c a u s e s f u r t h e r l a g g i n g i n a l l t h e p h a s e a n g l e s c a l c u l a t e d . T o c o n c l u d e t h i s s e c t i o n , i t c a n b e s a i d t h a t o n l y a s l i g h t i m p r o v e m e n t i n i m p e d a n c e c a l c u l a t i o n s c a n b e o b t a i n e d f o r s y s t e m s h a v i n g m a n y c o n d u c t o r s c l o s e t o g e t h e r i f s u b d i v i s i o n s o f c o n d u c t o r s a r e u s e d . T h e r e a l a d v a n t a g e o f s u b d i v i s i o n s w o u l d b e f o r c a l c u l a t i n g i m p e d a n c e s a t h i g h e r f r e q u e n c i e s f o r s w i t c h i n g s u r g e s t u d i e s o r s t u d i e s o f o t h e r t r a n s i e n t s i n d i s t r i b u t i o n s y s t e m s . A n o t h e r a d v a n t a g e o f t h e m e t h o d o f s u b c o n d u c t o r s i s t h e f a c t t h a t o t h e r t y p e s o f c a b l e s c a n b e a n a l y z e d f o r w h i c h h a n d b o o k . f o r m u l a e m a y n o t b e r e a d i l y a v a i l a b l e , e . g . , f o r s e c t o r - s h a p e d c o n d u c t o r s . 4 . 5 P i p e T y p e C a b l e s C a b l e s a r e s o m e t i m e s e n c l o s e d i n p i p e s w h i c h a c t a s d u c t s f o r c o n d u c t i n g c o o l i n g l i q u i d s o r i n s u l a t i o n g a s e s o r a s p r o t e c t i o n a g a i n s t m e c h a n i c a l o r c h e m i c a l d a m a g e . C o m m o n c a s e s a r e f o u n d i n o i l a n d g a s f i l l e d c a b l e i n s t a l l a t i o n s . P i p e m a t e r i a l s m a y b e p l a s t i c , a l u m i n u m o r s t e e l . P l a s t i c o r n o n - m e t a l l i c p i p e s p o s e n o p r o b l e m s i n i m p e d a n c e c a l c u l a t i o n s a n d a l u m i n u m p i p e s a r e a l s o e a s i l y t r e a t e d d u e t o t h e i r c o n s t a n t p e r m e a b i l i t y . S t e e l p i p e s a r e h i g h l y n o n l i n e a r a s f a r a s t h e i r m a g n e t i c p r o p e r t i e s a r e c o n c e r n e d . I n t h i s s e c t i o n , t h e n o n l i n e a r i t y i s n o t t a k e n i n t o a c c o u n t a n d t h e w h o l e s t e e l p i p e i s a s s u m e d l i n e a r a n d h a v i n g a c o n s t a n t p e r m e a b i l i t y . T h e c a b l e c o n d u c t o r s a n d p i p e a r e d i v i d e d i n t o s u b c o n d u c t o r s a n d e a c h s u b c o n d u c t o r i s 55 a s s i g n e d t h e p e r m e a b i l i t y o f i t s m a t e r i a l . T h e i m p e d a n c e i s t h e n c a l c u -l a t e d a c c o r d i n g t o t h e m e t h o d p r e s e n t e d i n C h a p t e r 2 . T h e d a t a i s t a k e n f r o m l a b o r a t o r y e x p e r i m e n t s c o n d u c t e d a t C o n s o l i d a t e d E d i s o n C o m p a n y o n c a b l e s e n c l o s e d i n s t e e l p i p e s f o r v a r i o u s d e g r e e s o f m a g n e t i c s a t u r a t i o n o f t h e p i p e s - a n d r e p o r t e d i n r e f e r e n c e s [ 1 4 ] a n d [ 1 5 ] . T h e r e f e r e n c e s g i v e p i p e c u r r e n t s a n d m a g n e t i c p e r m e a b i l i t i e s a t d i f f e r e n t c u r r e n t l e v e l s i n t h e p i p e , i n a d d i t i o n t o t h e m e a s u r e d i m p e d a n c e v a l u e s . T h e v a r i a t i o n o f p i p e p e r m e a b i l i t y w i t h c u r r e n t i s s h o w n i n F i g u r e 4 . 1 0 . T a b l e 4 . 7 p r o v i d e s t h e r e s u l t s o f s i n g l e p h a s e m e a s u r e m e n t s o n a n u n s h i e l d e d c a b l e i n a 5 i n c h s t e e l p i p e w i t h p i p e r e t u r n . T h e v a l u e s c a l c u l a t e d a n d t h e p e r c e n t a g e d e v i a t i o n s i n t h e r e a c t a n c e a r e s h o w n a l o n g s i d e t h e m e a s u r e d v a l u e s . F r o m T a b l e 4 . 7 i t i s s e e n t h a t m o s t o f t h e d e v i a t i o n s i n t h e c a l c u l a t i o n s a r e i n t h e r e a c t a n c e . T h i s i s b e c a u s e t h e m o d e l o f C h a p t e r 2 a s s u m e s t h a t t h e c o n d u c t o r s a r e a l l l i n e a r w h e r e a s i n t h e a c t u a l c a s e t h e m a g n e t i z a t i o n c u r v e ( o r B - H c u r v e ) o f t h e s t e e l p i p e i s n o n l i n e a r . F i g u r e 4 . 1 1 s h o w s a n i l l u s t r a t i o n o f t h e n o n l i n e a r i t y o f t h e B - H c u r v e . I t i s e v i d e n t t h a t a r o u n d t h e m o r e l i n e a r p o r t i o n s o f t h e B - H c u r v e ( i . e . , a r o u n d c u r r e n t s o f a b o u t 2 0 0 A i n t h i s p i p e a n d r e l a t i v e p e r m e a b i l i t y o f 1 0 0 0 ) , t h e d e v i a t i o n s i n t h e r e a c t a n c e a s c a l c u l a t e d a r e s m a l l . T h i s m a y b e b e c a u s e t h e a s s u m e d l i n e a r m o d e l f i t s t h i s r e g i o n m o s t . D e v i a t i o n s i n t h e r e s u l t s h e r e m a y b e d u e t o s a t u r a t i o n e f f e c t s o n t h e i n n e r s u r f a c e o f t h e m a g n e t i c p i p e - s i n c e m o r e c u r r e n t f l o w s o n t h e i n s i d e o f t h e p i p e . T a b l e 4 . 7 I m p e d a n c e o f P i p e T y p e C a b l e s f o r V a r i o u s D e g r e e s o f M a g n e t i c S a t u r a t i o n P i p e C u r r e n t ( A ) R e l a t i v e P e r m e a -b i l i t y M e a s u r e d R X m i c r o - o h m s ; / f t C a l c u l a t e d R | X m i c r o - o h m s / f t E r r o r s i n X 1 0 0 1 5 0 2 0 0 3 0 0 4 8 0 9 8 0 3 5 0 0 7 4 0 0 7 6 2 . 0 9 8 0 . 0 1 0 1 8 . 0 9 4 2 . 0 7 8 4 . 0 4 8 4 . 0 1 5 6 . 0 8 1 . 0 2 0 1 2 2 3 2 1 8 2 0 3 1 8 0 1 4 1 8 8 8 0 1 5 1 1 5 5 1 5 4 1 4 7 1 4 1 1 2 4 9 3 8 0 2 0 1 2 2 3 2 1 8 2 0 3 1 8 0 1 4 1 8 8 8 0 1 1 7 1 3 7 1 4 0 1 3 3 1 1 9 9 2 6 2 5 5 2 3 % 12% 9% 10% 15% 2 6 % 3 3 % 3 1 % F i g u r e 4 . 1 1 S h a p e o f m a g n e t i z i n g c u r v e d u r i n g o n e c y c l e 58 T h e r e s u l t s o f z e r o s e q u e n c e I m p e d a n c e c a l c u l a t i o n s f o r t h r e e c a b l e s i n t h e p i p e a r e g i v e n i n T a b l e 4.8 a l o n g s i d e t h e m e a s u r e d v a l u e s . L a r g e r d e v i a t i o n s i n t h e c a l c u l a t e d r e a c t a n c e s a r e o b t a i n e d e s p e c i a l l y w h e n l a r g e c u r r e n t s f l o w i n t h e p i p e . I t s h o u l d b e r e c a l l e d t h a t a s f a r a s a . c . i s c o n c e r n e d , t h e m a g n e t i c p e r m e a b i l i t y o f a n o n l i n e a r m a t e r i a l i s n o t c o n s t a n t t h r o u g h o u t t h e c y c l e b u t c h a n g e s w i t h t h e i n s t a n t a n e o u s v a l u e o f t h e c u r r e n t . B y a s s u m i n g c o n s t a n t p e r m e a b i l i t y f o r t h e s t e e l p i p e , a n i n h e r e n t e r r o r i s b e i n g m a d e . G o o d r e s u l t s m a y b e o b t a i n e d o n l y i f t h e a s s u m e d p e r m e a b i l i t y a p p r o x i m a t e s t h e a c t u a l v a l u e s f o r m o s t o f t h e r a n g e t h r o u g h w h i c h t h e c u r r e n t ( o r f l u x ) d e n s i t y i n t h e p i p e v a r i e s . F u r t h e r m o r e , i t i s k n o w n t h a t t h e i n n e r p o r t i o n s o f t h e p i p e c a r r y m o r e c u r r e n t t h a n t h e m i d d l e o r o u t e r p a r t s . T h u s d i f f e r e n t p o i n t s o f t h e p i p e c r o s s s e c t i o n e x p e r i e n c e d i f f e r e n t d e g r e e s o f s a t u r a t i o n . T h e i n n e r s u r f a c e m a y b e s a t u r a t e d w h i l e t h e o u t e r i s n o t . A n a t t e m p t i s m a d e t o m o d e l t h i s b y d i v i d i n g t h e p i p e i n t o c o n c e n t r i c l a y e r s a n d a s s i g n i n g d i f f e r e n t p e r m e a b i l i t i e s t o t h e v a r i o u s l a y e r s . T a b l e 4.9 g i v e s a s u m m a r y o f t h e r e s u l t s w h e n t h i s i s a p p l i e d t o t h e s a m e d a t a a s a b o v e . B e t t e r r e s u l t s a r e o b t a i n e d f r o m t h e c a l c u l a t i o n s , e s p e c i a l l y w h e n t h e p i p e i s n o t i n s a t u r a t i o n . T o p r o d u c e t h e r e s u l t s i n T a b l e 4.9, t h e T a b l e 4 . 8 Z e r o S e q u e n c e I m p e d a n c e M e a s u r e m e n t s o n T h r e e C a b l e s E n c l o s e d i n a P i p e w i t h P i p e R e t u r n . P i p e R e l a t i v e M e a s u r e d C a l c u l a t e d E r r o r s C u r r e n t P e r m e a - R R R X ( A ) b i l i t y ( m i c r o - D h m s / f t ) ( m i c r o - o h m s / f t ) 1 0 0 7 6 2 . 0 1 9 7 1 3 4 1 7 8 8 4 10% 37% 1 5 0 9 8 0 . 0 2 0 4 1 4 0 1 9 9 1 0 3 2% 2 6 % 2 0 0 1 0 1 8 . 0 2 0 0 1 3 9 1 9 4 1 0 7 3% 2 3 % 3 0 0 9 8 6 . 0 1 8 9 1 3 3 1 8 0 1 0 0 5% 2 5 % 5 0 0 7 6 7 . 0 1 6 5 1 2 4 1 5 8 8 6 4% 3 1 % 9 7 0 4 8 8 . 0 1 4 1 1 0 6 1 2 1 6 0 14% 57% 3 6 0 0 . 1 5 4 . 0 76 7 1 7 0 3 2 •8% 5 5 % 8 0 0 0 7 6 . 0 5 7 5 7 5 5 2 5 4% 5 5 % T a b l e 4 . 9 I m p e d a n c e o f C a b l e s i n M a g n e t i c P i p e s R e p r e s e n t e d a s T w o C o n c e n t r i c P i p e s o f D i f f e r e n t P e r m e a b i l i t i e s . P i p e M e a s u r e d C a l c u l a t e d E r r o r s C u r r e n t R X X R X ( A ) ( u f l / f t ) ( y f i / f t ) 1 0 0 2 0 1 1 5 1 2 0 3 1 5 0 1% .1% 1 5 0 2 2 3 1 5 5 2 3 0 1 8 1 3 % :.17% 2 0 0 2 1 8 1 5 4 2 2 7 1 9 2 4% 2 5 % 3 0 0 2 0 3 1 4 7 2 1 2 1 8 5 4% 2 6 % 4 8 0 1 8 0 1 4 1 1 8 8 1 6 0 4% 13% 9 8 0 1 4 1 1 2 4 1 4 6 1 1 4 4% 8% 3 5 0 0 8 8 9 3 8 9 6 7 1% 2 8 % 7 4 0 0 8 0 8 0 8 1 5 7 1% 2 9 % 1 0 0 * 1 9 7 1 3 4 1 8 6 1 1 8 6% 12% 1 5 0 * 2 0 4 1 4 0 2 1 4 1 4 9 5% 6% 2 0 0 * 2 0 0 1 3 9 2 1 1 1 5 9 6% 14% 3 0 0 * 1 8 9 1 3 3 1 9 6 1 5 3 4% 15% 5 0 0 * 1 6 5 1 2 4 1 7 2 1 2 8 4% 3% 9 7 0 * 1 4 1 1 0 6 1 2 9 8 3 9% 2 2 % 3 6 0 0 * 7 6 7 1 7 3 3 6 4% 4 9 % 8 0 0 0 * 5 7 5 7 5 6 2 8 2% 5 1 % * T h r e e c a b l e s i n p i p e . 61 p i p e i s d i v i d e d i n t o o n l y t w o c o n c e n t r i c l a y e r s . T h e i n n e r l a y e r i s a s s i g n e d a l o w e r p e r m e a b i l i t y t o r e f l e c t i t s b e i n g m o r e s a t u r a t e d . N o o t h e r c r i t e r i o n i s u s e d i n a s s i g n i n g t h e s e v a l u e s ; t h u s , f o r e x a m p l e , w h e n t h e p i p e c u r r e n t i s 9 8 0 A a n d t h e r e l a t i v e p e r m e a b i l i t y g i v e n i s 4 8 4 , t h e i n n e r l a y e r i s a s s i g n e d a l o w e r v a l u e o f 4 4 0 a n d t h e o u t e r l a y e r a s s i g n e d 5 2 0 . S i m i l a r c h o i c e s o f a b o u t 4 0 b e l o w a n d a b o v e t h e c o r r e s p o n d i n g p e r m e a b i l i t y v a l u e s l i s t e d i n T a b l e s 4 . 7 a n d 4 . 8 a r e m a d e f o r t h e o t h e r r e s u l t s s h o w n i n T a b l e 4 . 9 . A s c a n b e s e e n f r o m t h e r e s u l t s i n T a b l e s 4 . 7 , 4 . 8 a n d 4 . 9 t h e d e v i a t i o n s i n t h e c a l c u l a t e d v a l u e s o f t h e r e a c t a n c e d r o p f r o m a s h i g h a s 2 6 % w h e n c o n s t a n t p e r m e a b i l i t y i s a s s u m e d t o 8% w h e n t w o v a l u e s o f p e r m e a b i l i t y a r e u s e d f o r t h e s t e e l p i p e w h e n o n l y o n e c a b l e i s i n t h e p i p e a n d t h e p i p e c u r r e n t i s 9 8 0 A . A s i m i l a r d r o p f r o m 5 7 % t o 2 2 % i s a l s o o b t a i n e d f o r t h r e e c a b l e s i n t h e p i p e . H o w e v e r , i t s h o u l d b e p o i n t e d o u t t h a t t h e m e t h o d o f a s s i g n i n g p e r m e a b i l i t i e s t o t h e p i p e a s g i v e n a b o v e i s r a t h e r a r b i t r a r y a n d b e t t e r r e s u l t s c o u l d b e e x p e c t e d i f t h e d i f f e r e n t l a y e r s o f t h e p i p e a r e a s s i g n e d p e r m e a b i l i t y v a l u e s b a s e d o n s o m e d e f i n i t e c r i t e r i o n , s u c h a s t h e a v e r a g e c u r r e n t d e n s i t i e s i n t h e l a y e r s . F u r t h e r m o r e , a n y s u c h l i n e a r i z a t i o n , n o m a t t e r h o w w e l l i t i s d o n e , w o u l d s t i l l b e i n e r r o r s i n c e t h e p e r m e a b i l i t y v a r i a t i o n s d u r i n g a n y c y c l e i s o f t h e f o r m o f t h e l i n e a r i z e d s k e t c h s h o w n i n F i g u r e 4 . 1 2 a , w h e r e a s t h e r e p r e s e n t a t i o n a b o v e s e e k s t o p u t i t i n t h e f o r m o f F i g u r e 4 . 1 2 b . F i g u r e 4 . 1 2 L i n e a r i z e d m a g n e t i z a t i o n c u r v e s 631 C h a p t e r 5 C O N C L U S I O N S T h e m e t h o d o f s u b d i v i s i o n s o f c o n d u c t o r s f o r i m p e d a n c e c a l c u l a t i o n s h a s b e e n u s e d t o c a l c u l a t e t h e i m p e d a n c e o f c a b l e s . I t i s s h o w n t h a t t h e a c c u r a c y o f t h e c a l c u l a t i o n d e p e n d s o n t h e n u m b e r o f s u b c o n d u c t o r s u s e d . T h e i n t r o d u c t i o n o f g r o u n d r e t u r n f o r m u l a e s p e e d s u p t h e c a l c u l a t i o n s . T h i s m e t h o d t a k e s a c c o u n t o f b o t h s k i n a n d p r o x i m i t y e f f e c t s a n d w i l l t h e r e f o r e b e h i g h l y s u i t e d t o d i s t r i b u t i o n s y s t e m s w h e r e m a n y c o n d u c t o r s r u n p a r a l l e l i n c l o s e p r o x i m i t y . I t w i l l b e v e r y u s e f u l e s p e c i a l l y a t h i g h e r f r e q u e n c i e s w h e r e i m p e d a n c e v a l u e s a r e n e e d e d f o r s w i t c h i n g s u r g e s t u d i e s . A t e s t c a s e o f c i r c u l a t i n g c u r r e n t s i n b o n d e d s h e a t h s h a s b e e n s t u d i e d w i t h t h e m o d e l . T h e c a l c u l a t e d r e s u l t s c o m e c l o s e t o t h e f i e l d t e s t r e s u l t s . T h e m e t h o d i s a l s o s u i t e d f o r m o d e l l i n g c o n d u c t o r s w i t h n o n u n i f o r m p r o p e r t i e s a c r o s s t h e c r o s s s e c t i o n . C a b l e s e n c l o s e d i n m a g n e t i c p i p e s a r e a l s o m o d e l l e d w i t h p i p e l a y e r s a s s i g n e d d i f f e r e n t v a l u e s o f p e r m e a b i l i t i e s . G e n e r a l l y b e t t e r r e s u l t s a r e o b t a i n e d w h e n d i f f e r e n t l a y e r s o f p i p e m a t e r i a l a r e a s s i g n e d d i f f e r e n t p e r m e a b i l i t y v a l u e s . 6 4 L I S T OF R E F E R E N C E S [-1] D . M . S i m m o n s ^ " C a l c u l a t i o n o f t h e E l e c t r i c a l P r o b l e m s o f U n d e r g r o u n d C a b l e s " , T h e E l e c t r i c J o u r n a l , v o l . 2 9 , J a n u a r y - D e c e m b e r , 1 9 3 2 . [2] S . A . S c h e l k u n o f f , " T h e E l e c t r o m a g n e t i c T h e o r y o f C o a x i a l T r a n s m i s s i o n L i n e s a n d C y l i n d r i c a l S h e l l s " , B e l l S y s t e m T e c h . J o u r . , v o l . 1 3 , p p . 5 2 2 - 7 9 , 1 9 3 4 . [ 3 ] D . R . S m i t h a n d J . V . B a r g e r , " I m p e d a n c e a n d C i r c u l a t i o n C u r r e n t C a l c u l a t i o n s f o r UD M u l t i - W i r e C o n c e n t r i c N e u t r a l C i r c u i t s " , I E E E  C o n f e r e n c e R e c o r d o f 1 9 7 1 C o n f e r e n c e o n U n d e r g r o u n d D i s t r i b u t i o n , S e p t . 1 9 7 1 , p p . 9 9 2 - 1 0 0 0 . [ 4 ] W . A . L e w i s a n d R . C . E n d e r , D i s c u s s i o n o f 3 , i b i d , p p . 1 0 0 1 - 1 0 0 4 . [ 5 ] W . A . L e w i s , G . D . A l l e n a n d J . C . W a n g , " C i r c u i t C o n s t a n t s f o r C o n c e n t r i c N e u t r a l U n d e r g r o u n d D i s t r i b u t i o n C a b l e s o n a P h a s e B a s i s " , P a p e r . A 77 2 4 3 - 9 , I E E E W i n t e r P o w e r M e e t i n g 1 9 7 7 . [ 6 ] G . D . A l l e n a n d W . A . L e w i s , " T h e E l e c t r i c a l C h a r a c t e r i s t i c s o f U n d e r -g r o u n d D i s t r i b u t i o n C o n c e n t r i c - N e u t r a l C a b l e s " , B o o k p u b l i s h e d b y A . B . C h a n c e C o m p a n y , C e n t r a l i a , M i s s o u r i , 1 9 7 6 . [ 7 ] E . C o m e l l i n d , A . I n v e r n i z z i , G . M a n z o n i , " A C o m p u t e r P r o g r a m f o r : D e t e r m i n i n g E l e c t r i c a l R e s i s t a n c e a n d R e a c t a n c e o f a n y T r a n s m i s s i o n L i n e " , I E E E T r a n s , o n P o w e r A p p a r . & S y s t . v o l . P A S - 9 2 , p p . 3 0 8 - 3 1 4 , 1 9 7 3 . [ 8 ] H . W . D o m m e l a n d W . S . M e y e r , " C o m p u t a t i o n o f E l e c t r o m a g n e t i c T r a n s i e n t s " , i b i d , v o l . 6 2 , N o . 7 , J u l y 1 9 7 4 . [ 9 ] L . M . W e d e p o h l a n d D . J . W i l c o x , " T r a n s i e n t A n a l y s i s o f U n d e r g r o u n d P o w e r - T r a n s m i s s i o n S y s t e m s " , P R O C . I E E , v o l . 1 2 0 , N o . 2 , F e b . 1 9 7 3 , p p . 2 5 3 - 2 6 0 . [ 1 0 ] J . R . C a r s o n , " G r o u n d R e t u r n I m p e d a n c e : U n d e r g r o u n d W i r e w i t h E a r t h R e t u r n " , B e l l S y s t . T e c h . J o u r . , v o l . 8 , 1 9 2 9 , p p . 9 4 - 9 8 . .[.11] J . R . C a r s o n , " W a v e P r o p a g a t i o n i n O v e r h e a d W i r e s w i t h G r o u n d R e t u r n " , B e l l S y s t . T e c h . J o u r . , v o l . 5 , 1 9 2 6 , p p . 5 3 9 - 5 5 4 . [ 1 2 ] G . B i a n c h i a n d G . L u o n i , " I n d u c e d C u r r e n t s a n d L o s s e s i n S i n g l e - C o r e S u b m a r i n e C a b l e s " , I E E E T r a n s o n P o w e r A p p a r . & S y s t . , v o l . P A S - 9 5 , N o . l , J a n / F e b , 1 9 7 6 , p p . 4 9 - 5 7 . [ 1 3 ] V . F . K a l y u z h n y i a n d M . „ Y u , L i f s h i t s , " A M e t h o d f o r D e t e r m i n i n g t h e P a r a m e t e r s o f ' U n d e r g r o u n d - C o n d u c t o r - E a r t h ' C i r c u i t s " , E l e c t r i c h e s t r o , N o . 7 , 1 9 7 4 , p p . 2 8 - 3 6 . ] J . H . N e h e r , " P h a s e S e q u e n c e I m p e d a n c e o f P i p e T y p e C a b l e s " , I E E E T r a n s . o n P o w e r A p p a r . & S y s t . , A u g u s t 1 9 6 4 , p p . 7 9 5 - 8 0 4 . ] E . R . T h o m a s a n d R . H . K e r s h a w , " I m p e d a n c e o f P i p e T y p e C a b l e " , i b i d , v o l . P A S - 8 4 , N o . 1 0 , O c t . 1 9 6 5 , p p . 9 5 3 - 9 6 1 . ] " L i n e C o n s t a n t s o f O v e r h e a d L i n e s " , - U s e r s m a n u a l , B P A , J u n e 1 9 7 2 . ] E d i t h C l a r k e , " C i r c u i t A n a l y s i s o f A C P o w e r S y s t e m s " , v o l . I I , G e n e r a l E l e c t r i c C o m p a n y , 1 9 7 1 . ] A . H . M . A r n o l d , " P r o x i m i t y E f f e c t i n S o l i d a n d H o l l o w R o u n d C o n d u c t o r s " J o u r , o f I E E , P a r t I I , v o l . 8 8 , 1 9 4 1 , p p . 3 4 9 . ] F . P o l l a c z e k , " S u r l e c h a m p p r o d u i t p a r u n c o n d u c t e u r s i m p l e i n f i n e m e n t l o n g p a r c o u r c e p a r u n c o u r a n t a l t e r n a t i f " , " R e v u e G e n e r a l D e L 1 e l e c t r i c i t e " T o m e X X I X , N o . 2 2 , M a y 1 9 3 1 . ] H . W . D o m m e l , " E L E C 5 5 3 - A d v a n c e d A n a l y s i s o f P o w e r S y s t e m s , C l a s s n o t e s " U n i v . o f B . C . , 1 9 7 8 . ] G a r y A r m a n i n i , " A P r o p o s e d G r o u n d i n g a n d ' B o n d i n g S c h e m e f o r U n d e r g r o u n d D i s t r i b u t i o n S y s t e m s " , B . C . H y d r o R e p o r t . ] R . L u c a s a n d S . T a l u k d a r , " A d v a n c e s i n F i n i t e E l e m e n t T e c h n i q u e s f o r C a l c u l a t i n g C a b l e R e s i s t a n c e s a n d I n d u c t a n c e s " , I E E E T r a n s . o n P o w e r  A p p a r - a n d S y s t . , v o l . P A S - 9 7 , N o . 3 , M a y / J u n e 1 9 7 8 . ] B r a n k o D . P o p o v i c , " I n t r o d u c t o r y E n g i n e e r i n g E l e c t r o m a g n e t i c s " , T e x t b o o k A d d i s o n - W e s l e y P u b l i s h i n g C o . , M a y 1 9 7 3 . ] " E l e c t r i c a l T r a n s m i s s i o n a n d D i s t r i b u t i o n R e f e r e n c e B o o k " , F o u r t h E d i t i o n , W e s t i n g h o u s e E l e c t r i c C o . 1 9 6 4 . ] " U n d e r g r o u n d S y s t e m s R e f e r e n c e B o o k " , N E L A P u b l i c a t i o n 0 5 0 , 1 9 3 1 . ] J o h n G . K a s s a k i a n , " T h e E f f e c t s o f N o n - T r a n s p o s i t i o n a n d E a r t h R e t u r n F r e q u e n c y D e p e n d e n c e o n T r a n s i e n t s d u e t o H i g h S p e e d R e c l o s i n g " , I E E E T r a n s , v o l . P A S - 9 5 , M a r c h / A p r i l 1 9 7 6 , p p . 6 1 0 - 6 1 8 . 

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