UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Optimization of conductor shapes and configuration of conductor bundles for high voltage transmission Christensen, Gustav Strom 1960

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1960_A7 C4 O6.pdf [ 4.67MB ]
Metadata
JSON: 831-1.0105039.json
JSON-LD: 831-1.0105039-ld.json
RDF/XML (Pretty): 831-1.0105039-rdf.xml
RDF/JSON: 831-1.0105039-rdf.json
Turtle: 831-1.0105039-turtle.txt
N-Triples: 831-1.0105039-rdf-ntriples.txt
Original Record: 831-1.0105039-source.json
Full Text
831-1.0105039-fulltext.txt
Citation
831-1.0105039.ris

Full Text

OPTIMIZATION OF CONDUCTOR SHAPES AND CONFIGURATION OF CONDUCTOR BUNDLES FOR HIGH VOLTAGE TRANSMISSION by G u s t a v S t r o m C h r i s t e n s e n B. S c . ( E n g . ) , U n i v e r s i t y o f A l b e r t a , 1958 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n t h e Department o f E l e c t r i c a l E n g i n e e r i n g We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e s t a n d a r d s r e q u i r e d f r o m c a n d i d a t e s f o r t h e d e g r e e o f M a s t e r o f A p p l i e d S c i e n c e . Members o f t h e Department o f E l e c t r i c a l E n g i n e e r i n g THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1960 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 requirements 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 Columbia, 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 study. I f u r t h e r agree 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 copying of 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 granted 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 understood t h a t copying or p u b l i c a t i o n of 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 allowed without my w r i t t e n p e r m i s s i o n . Department o f The U n i v e r s i t y of B r i t i s h Columbia, Vancouver $, Canada. Date ABSTRACT T h i s t h e s i s d i s c u s s e s t h e m i n i m i z a t i o n o f t h e e l e c t r i c f i e l d i n t e n s i t y on e l e c t r i c power t r a n s m i s s i o n l i n e c o n d u c t o r s . Two s e p a r a t e c a s e s a r e c o n s i d e r e d . One, t r a n s m i s s i o n l i n e s w i t h one c o n d u c t o r p e r p h a s e , and two, t r a n s m i s s i o n l i n e s w i t h s e v e r a l c o n d u c t o r s p e r p h a s e . N o n - c i r c u l a r c o n d u c t o r s a r e c o n s i d e r e d f o r t h e c a s e when o n l y one c o n d u c t o r i s u s e d f o r e a c h p h a s e . Some c o n t o u r s s u b j e c t t o c o n f o r m a l mapping a r e i n v e s t i g a t e d t o d e t e r m i n e i f c o n d u c t o r s o f s u c h shapes p r o v i d e l e s s v a r i a t i o n i n e l e c t r i c s u r f a c e c h a r g e d e n s i t y t h a n c i r c u l a r c o n d u c t o r s . A l s o a p e r -t u r b a t i o n method i s d e v e l o p e d w h i c h e s s e n t i a l l y c o n s i s t s i n p e r t u r b i n g t h e b o u n d a r y o f a c i r c u l a r c y l i n d e r i n s u c h a manner t h a t t h e e l e c t r i c s u r f a c e c h a r g e d e n s i t y on t h e c y l i n d e r i s u n i f o r m when i t i s p l a c e d p a r a l l e l t o l i n e c h a r g e s . U s i n g t h i s method i t i s f o u n d t h a t a s o l u t i o n e x i s t s o n l y i n one p a r t i c u l a r c a s e . I n c o n s i d e r i n g , s e v e r a l c o n d u c t o r s p e r p h a s e , t h a t i s , b u n d l e d c o n d u c t o r s , t h e b u n d l e c o n f i g u r a t i o n i s changed f r o m t h e s y m m e t r i c a l f o r m n o r m a l l y u s e d u n t i l t h e same maximum e l e c t r i c f i e l d i n t e n s i t y e x i s t s on a l l c o n d u c t o r s o f t h e same p h a s e . T r a n s m i s s i o n l i n e s w i t h t h r e e and f o u r c o n d u c t o r b u n d l e s a r e t r e a t e d . The optimum c o n f i g u r a t i o n was d e t e r -mined by a i d o f t h e Alwac I I I - E d i g i t a l computer and t h e r e -s u l t s o b t a i n e d a r e i n d i c a t e d . i i TABLE OP CONTENTS page A b s t r a c t i i Acknowledgement . . . . . . . . . . v i 1. I n t r o d u c t i o n 1 2. Some B a s i c P r o b l e m s 4 2-1. L i n e C h a r g e s and D i p o l e L i n e s 4 2- 2, C o n d u c t i n g C y l i n d e r and L i n e C h arge . . . . 5 3. V a r i a t i o n o f E l e c t r i c F i e l d I n t e n s i t y w i t h Phase A n g l e and Phase S p a c i n g 10 3- 1. D e t e r m i n a t i o n o f L i n e C h a r g e s . . . . . . . 10 3- 2. D e t e r m i n a t i o n o f t h e Phase A n g l e a t w h i c h Maximum E l e c t r i c F i e l d I n t e n s i t y O c c u r s . 12 4. N o n - c i r c u l a r C o n t o u r s 15 4- 1. C a s s i n i a n O v a l 15 4-2. E l l i p t i c C y l i n d e r and two L i n e C h a r g e s . . . 17 4- 3. A p a r t i c u l a r n o n - c i r c u l a r C y l i n d e r and two L i n e C h a r g e s 22 5. A P e r t u r b a t i o n Method . . 29 5- 1. C o n d u c t i n g C y l i n d e r 29 5-2. C o n d u c t i n g C y l i n d e r and L i n e C h arge . . . . 31 5-3. C o n d u c t i n g C y l i n d e r and two L i n e C h a r g e s . . 40 5- 4. N u m e r i c a l C o m p u t a t i o n s 46 6. B u n d l e d C o n d u c t o r s 52 6- 1. T r a n s m i s s i o n L i n e w i t h t h r e e C o n d u c t o r s p e r Phase . . . 53 6-2. T r a n s m i s s i o n L i n e w i t h f o u r C o n d u c t o r s p e r Phase . . . . . . . . . . 63 6- 3. Some G e n e r a l C o n s i d e r a t i o n s . 65 7. C o n c l u s i o n s 67 7- 1. Recommendations f o r F u t u r e Work 67 R e f e r e n c e s . . . . . 69 A p p e n d i x I . . 73 A p p e n d i x I I 77 i i i LIST OP ILLUSTRATIONS Figure page 2-1. Line Charge to follow 4 2-2. Dipole Line to follow 5 2- 3. Line Charge and Conducting Cylinder . . . to follow 5 3- 1. Ungrounded three-phase Transmission Line to follow 10 3- 2. Conducting Cylinder and two Line Charges 12 4- 1. Cassinian Oval to follow 15 4-2. Charge D i s t r i b u t i o n on Cassinian . . . . to follow 17 4-3. Circular Cylinder and two Line Charges to follow 17 4-4. E l l i p t i c Cylinder and two Line Charges to follow 17 4-5. Charge Density Variation to follow 21 4-6. Circular Cylinder and Line Charge . . . . to follow 22 4-7. Deformed Cylinder and Line Charge . . . . to follow 22 4-8. Charge Density Variation to follow 25 4- 9. Circular Cylinder and two Line Charges to follow 27 4-10. Deformed Cylinder and two Line Charges to follow 27 4-11. Conductor Shape as a Function of Spacing (Single Phase) to follow 28 4-12. Conductor Shape as a Function of Spacing (Three Phases) . to follow 28 5- 1. Conducting Cylinder 30 5-2. Conducting Cylinder and Line Charge 31 5-3. Conducting Cylinder and two Line Charges 41 5-4. Conducting Cylinder and two Line Charges 45 i v L I S T OP ILLUSTRATIONS F i g u r e page 2-1. L i n e C h arge t o f o l l o w 4 2-2. D i p o l e L i n e t o f o l l o w 5 2- 3. L i n e C h arge and C o n d u c t i n g C y l i n d e r . . . t o f o l l o w 5 3- 1. Ungrounded 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 t o f o l l o w 10 3- 2. C o n d u c t i n g C y l i n d e r and two L i n e C h a r g e s 12 4- 1. C a s s i n i a n O v a l t o f o l l o w 15 4-2. C h a r g e D i s t r i b u t i o n on C a s s i n i a n . . . . t o f o l l o w 17 4-3. C i r c u l a r C y l i n d e r and two L i n e C h a r g e s t o f o l l o w 17 4-4. E l l i p t i c C y l i n d e r and two L i n e C h a r g e s . . . t o f o l l o w 17 4-5. C h arge D e n s i t y V a r i a t i o n t o f o l l o w 21 4-6. C i r c u l a r C y l i n d e r and L i n e Charge . . . . t o f o l l o w 22 4-7. Deformed C y l i n d e r and L i n e Charge . . . . t o f o l l o w 22 4-8. C h arge D e n s i t y V a r i a t i o n . t o f o l l o w 25 4- 9. C i r c u l a r C y l i n d e r and two L i n e C h a r g e s t o f o l l o w 27 4-10. D e formed C y l i n d e r and two L i n e C h a r g e s t o f o l l o w 27 4-11. C o n d u c t o r Shape as a F u n c t i o n o f S p a c i n g ( S i n g l e P h a s e ) t o f o l l o w 28 4-12. C o n d u c t o r Shape as a F u n c t i o n o f S p a c i n g ( T h r e e P h a s e s ) t o f o l l o w 28 5- 1. C o n d u c t i n g C y l i n d e r 30 5-2. C o n d u c t i n g C y l i n d e r a n d L i n e Charge 31 5-3. C o n d u c t i n g C y l i n d e r and two L i n e C h a r g e s 41 5-4. C o n d u c t i n g C y l i n d e r and two L i n e C h a r g e s . . . . . 45 i v F i g u r e page 5-5. Conducting C y l i n d e r and two L i n e Charges 47 5- 6. Charge D e n s i t y V a r i a t i o n -fco f o l l o w 49 6- 1. Three-phase Transmission L i n e to f o l l o w 54 6-2. Charge per U n i t Length versus the Parameter b to f o l l o w 58 6-3. F i e l d I n t e n s i t y versus the Parameter b to f o l l o w 62 6-4. Three Conducting C y l i n d e r s and two Line Charges t o , f o l l o w 60 6-5. V a r i a t i o n of b Q ^ with Phase and Bundle Spacing* *. . to f o l l o w 62 6-6. Thiee Phase Transmission L i n e to f o l l o w 63 6-7. Charge per U n i t Length versus Bundle Spacing . . . . t o f o l l o w 65 6-8. V a r i a t i o n of s0A . with Phase Spacing . . f . ? . to f o l l o w 65 I - l . Three Conducting C y l i n d e r s to f o l l o w 73 v ACKNOWLEDGEMENT The work done i n t h i s t h e s i s i s s u p p o r t e d by t h e N a t i o n a l R e s e a r c h C o u n c i l o f Canada,, T h i s t h e s i s was w r i t t e n under t h e s u p e r v i s i o n o f D r 0 F r a n k Noakes who p r o v i d e d t h e a u t h o r w i t h c o n t i n u a l g u i d a n c e and en-couragement w h i c h i s g r a t e f u l l y a c k n o w l e d g e d h e r e . The a u t h o r w i s h e s t o e x p r e s s h i s s i n c e r e g r a t i t u d e f o r a d v i s e r e c e i v e d f r o m Dr,, E.V. Bohiij i n p a r t i c u l a r i n c o n n e c t i o n w i t h t h e d e -v e l o p m e n t o f t h e p e r t u r b a t i o n method d e s c r i b e d i n t h i s t h e s i s . A l s o t h e a u t h o r w i s h e s t o acknowledge e n l i g h t e n i n g d i s c u s s i o n s h e l d w i t h D r . A „ D 0 Moore, D r . G\> Walker and Dr„ J c F o S z a b l y a i n c o n n e c t i o n w i t h t h e work done i n t h i s t h e s i s . The a u t h o r i s i n d e b t e d t o The N o r t h e r n E l e c t r i c Company f o r a f e l l o w s h i p g r a n t e d f o r t h e s e s s i o n 1958-1959 and t o the N a t i o n a l R e s e a r c h C o u n c i l o f Canada f o r a S t u d e n t s h i p g r a n t e d f o r t h e s e s s i o n 1959=-196Q„ v i OPTIMIZATION OP CONDUCTOR SHAPES AND CONFIGURATION OF CONDUCTOR BUNDLES FOR HIGH VOLTAGE TRANSMISSION 1. INTRODUCTION I n o r d e r t o f a c i l i t a t e economic e l e c t r i c power t r a n s -m i s s i o n o v e r v e r y l o n g d i s t a n c e s t r a n s m i s s i o n l i n e v o l t a g e s have been i n c r e a s e d a p p r e c i a b l y s i n c e t h e s e c o n d W o r l d War 0 1 2 3 I n some c a s e s v o l t a g e s up t o 400 k ? . a r e employed 9 ' , and e x t e n s i v e e x p e r i m e n t a l t e s t l i n e s have been c o n s t r u c t e d i n o r d e r t o s t u d y t h e p o s s i b i l i t y o f e m p l o y i n g even h i g h e r t r a n s -m i s s i o n voltages„ C o r o n a i s one o f t h e phenomena i n v e s t i -g a t e d by a i d o f t h e s e t e s t l i n e s s and s e v e r a l p a p e r s ^ ' ^ 9 ^ y have been p u b l i s h e d i n d i c a t i n g t h e r e s u l t s o b t a i n e d f r o m s u c h s t u d i e s o C o r o n a i s one o f t h e more i m p o r t a n t e f f e c t s w h i c h must be c o n s i d e r e d i n c o n n e c t i o n w i t h h i g h v o l t a g e e l e c t r i c power t r a n s m i s s i o n . T h e r e a r e t w o main r e a s o n s t h a t t h i s e f f e c t s h o u l d be k e p t a t a mlnimum 0 One, t h e e l e c t r i c power l o s s i n c u r r e d , and two, t h e h i g h f r e q u e n c y e l e c t r o m a g n e t i c r a d i a -t i o n a r i s i n g f r o m C o r o n a d i s c h a r g e s 0 The e l e c t r i c power l o s s due t o C o r o n a has been i n v e s t i -g a t e d b o t h t h e o r e t i c a l l y and e x p e r i m e n t a l l y ^ 9 ^ 9 9 and c a n be a p p r e c i a b l e f o r l o n g t r a n s m i s s i o n l i n e s D A t t h e p r e s e n t t i m e t h e h i g h f r e q u e n c y r a d i a t i o n r e s u l t i n g f r o m C o r o n a i s b e i n g i n v e s t i g a t e d r a t h e r e x t e n s i v e l y ^ 9 ^ 9 ^ ^ ' 1 1 0 The r e a s o n b e i n g t h a t t h i s r a d i a t i o n i n t e r f e r e s w i t h p u b l i c c o m m u n i c a t i o n 2 n e t w o r k s , e s p e c i a l l y i n s p a r s e l y p o p u l a t e d a r e a s where s i g n a l s t r e n g t h s g e n e r a l l y a r e low. As has b e e n p o i n t e d o u t C o r o n a i s a d e t r i m e n t a l e f f e c t when a s s o c i a t e d w i t h e l e c t r i c power t r a n s m i s s i o n . I t t h e r e -f o r e f o l l o w s t h a t i n g e n e r a l t r a n s m i s s i o n l i n e s ystems s h o u l d be d e s i g n e d s u c h t h a t C o r o n a d i s c h a r g e s do n o t o c c u r on t h e l i n e s u n d e r n o r m a l o p e r a t i n g c o n d i t i o n s . B a s i c a l l y C o r o n a s t a r t s a t t h e s u r f a c e o f a c o n d u c t o r when t h e e l e c t r i c f i e l d i n t e n s i t y j u s t o f f t h e s u r f a c e r e a c h e s t h e d i e l e c t r i c s t r e n g t h o f t h e a i r s u r r o u n d i n g t h e c o n d u c t o r . The d i e l e c t r i c s t r e n g t h o f a i r v a r i e s w i t h d e n s i t y , b u t has a v a l u e o f 29.8 kv.peak/cm a t 25° C. and 76 cm Hg. p r e s s u r e . Prom e l e c t r o s t a t i c s i t i s w e l l known t h a t t h e e l e c t r i c f i e l d i n t e n s i t y j u s t o f f a c o n d u c t i n g s u r f a c e i s p r o p o r t i o n a l t o t h e s u r f a c e c h a r g e d e n s i t y a t t h e p a r t i c u l a r p o i n t u n d e r c o n s i d e r a t i o n . Hence, by p r o v i d i n g a l a r g e r s u r f a c e a r e a f o r a g i v e n amount o f c h a r g e t o r e s i d e on t h e e l e c t r i c f i e l d i n -t e n s i t y a t t h e s u r f a c e may be d e c r e a s e d . A l a r g e s u r f a c e a r e a c a n be o b t a i n e d i n two ways f o r a t r a n s m i s s i o n l i n e system. One, by u s i n g one c o n d u c t o r p e r p h a s e , b u t o f s u f f i c i e n t c r o s s -s e c t i o n t o p r o v i d e t h e r e q u i r e d s u r f a c e a r e a , o r two, by u s i n g more t h a n one c o n d u c t o r p e r p h a s e , i . e . , u s i n g b u n d l e d c o n -d u c t o r s . The l a t t e r a l t e r n a t i v e has t h e added a d v a n t a g e o f g i v i n g l e s s i n d u c t i v e r e a c t a n c e p e r phase t h a n t h e f o r m e r , and has b e e n a d o p t e d t o some e x t e n t f o r h i g h v o l t a g e t r a n s m i s s i o n l i n e s ' ' . T h i s t h e s i s t r e a t s b o t h t h e c a s e o f one c o n d u c t o r p e r 3 phase and s e v e r a l c o n d u c t o r s p e r p h a s e . C o n d u c t o r s i n t h e f o r m o f c i r c u l a r c y l i n d e r s a r e g e n e r a l l y u s e d f o r t r a n s m i s s i o n l i n e s ; t h e s e may be smooth or s t r a n d e d . U s i n g one c i r c u l a r c o n d u c t o r p e r p h a s e r e s u l t s i n a n o n - u n i f o r m e l e c t r i c f i e l d d i s t r i b u t i o n a r o u n d t h e c i r c u m f e r e n c e o f e a c h phase c o n d u c t o r . I n t h i s t r e a t m e n t an a t t e m p t i s made t o f i n d c o n d u c t o r shapes so t h a t t h i s f i e l d v a r i a t i o n i s m i n i m i z e d , o r so t h a t t h e m a x i -mum f i e l d i n t e n s i t y o c c u r r i n g i s d e c r e a s e d . W i t h s e v e r a l c i r c u l a r c o n d u c t o r s p e r phase t h e r e s u l t i n g maximum f i e l d i n -t e n s i t y on e a c h c o n d u c t o r o f t h e same phase b u n d l e i s g e n e r a l l y 12 n o t e q u a l . T h i s i s due t o u n e q u a l c a p a c i t a n c e a s s o c i a t e d w i t h e a c h c o n d u c t o r o f a b u n d l e and t o t h e c h a r g e r e s i d i n g on t h e o t h e r > p h a s e s . The c a p a c i t a n c e i s o f c o u r s e due t o t h e p r e s e n c e o f e a r t h , t h e two o t h e r p h a s e s and c o n d u c t o r s o t h e r t h a n t h e one c o n s i d e r e d o f t h e same p h a s e . I n t h i s t h e s i s t h e optimum p o s i t i o n f o r t h e i n d i v i d u a l b u n d l e c o n d u c t o r s i s i n d i c a t e d f o r two c a s e s . The optimum p o s i t i o n i s .defined as t h e one a t w h i c h t h e maximum f i e l d i n t e n s i t y i s a p p r o x i -m a t e l y e q u a l on a l l c o n d u c t o r s i n one phase b u n d l e . 4 2 . SOME BASIC PROBLEMS E l e c t r i c t r a n s m i s s i o n l i n e s are g e n e r a l l y s u f f i c i e n t l y long t h a t they may be considered i n f i n i t e i n one dimension. A l s o the e l e c t r i c f i e l d component p a r a l l e l to the l e n g t h of the l i n e conductors i s i n general completely n e g l i g i b l e with r e s p e c t to the f i e l d component normal to the s u r f a c e of the conductors. Hence, f o r a l l p r a c t i c a l purposes i t i s s u f f i -c i e n t to c o n s i d e r the e q u i v a l e n t e l e c t r o s t a t i c case, t h a t i s , the t r a n s m i s s i o n l i n e i s kept at a c e r t a i n p o t e n t i a l with r e -spect to some s p e c i f i e d r e f e r e n c e and c a r r i e s an i n t r i n s i c e l e c t r o s t a t i c charge per u n i t l e n g t h . T h i s case may be t r e a t e d as a two dimensional problem, and thus the p o t e n t i a l decrease due to c u r r e n t flow along the t r a n s m i s s i o n l i n e i s neglected, however, s u f f i c i e n t accuracy i s obtained by t h i s approach. The two s e c t i o n s f o l l o w i n g w i l l d e s c r i b e some simple, but im-p o r t a n t concepts. 2 - 1 . Line Charges and D i p o l e L i n e s A l i n e charge designates a c e r t a i n amount of e l e c t r i c charge, q coul./m, concentrated along an i n f i n i t e s t r a i g h t l i n e . The magnitude of q designates the s t r e n g t h of the l i n e . By Gauss Law the e l e c t r i c f i e l d i n t e n s i t y E at a d i s t a n c e r meters from such a l i n e charge p l a c e d i n a medium of p e r m i t t i -2 2 v i t y e Q c o u l . /new. m may be found from 2 i t r e Q E = q coul./m or P i g . 2-3 Line Charge and Conducting Cylinder 5 E = 2 l t | r volts/m = volts/in (2-1) o where k = 2ne i n r a t i o n a l i z e d m.k.s. u n i t s which are used o throughout. See P i g . 2-1. The e l e c t r o s t a t i c f i e l d i s con-s e r v a t i v e , that i s , the work done on u n i t charge i n moving i t from one po i n t to another i s independent of path. Hence, gradient ( P o t e n t i a l ) = - ( E l e c t r i c f i e l d i n t e n s i t y ) . Let V designate the p o t e n t i a l at some p o i n t , then grad V = - E, and since by symmetry V i s independent of the angle 6 the poten-t i a l d i f f e r e n c e between two points P and P^ i s r V - V]_ = - J | |£ = | (In r x - In r) v o l t s . (2-2) r l A d i p o l e l i n e c o n s i s t s of two l i n e charges each of strength q, but of opposite p o l a r i t y placed a short distance, s, apart. Using equation 2-2 i t fo l l o w s that the p o t e n t i a l due to a d i p o l e l i n e i s Q i n ( r 2 + + cos 6)2 V = | 2g j- v o l t s . (2-3) In ( r 2 + J - - |2£ cos 6 ) 2 From F i g , 2-2 i t i s evident that r ^ = r + | j- cos 6 and s 13 rg = r - 77 cos 6. Then equation 3-1 becomes V « £ In (1 + f cos 6 ) A | s c ; s 6 v o l t s . (2-4) The q u a n t i t y (^ .) i s denoted the strength of the d i p o l e l i n e . 2T2. Conducting Cylinder and Line Charge Consider the i s o l a t e d system shown i n F i g . 2-3. The conducting c y l i n d e r c a r r i e s a charge of -q coul,/m, and the l i n e c h a r g e has a s t r e n g t h o f +q coul„/m. The s i m p l e s t way t o d e t e r m i n e t h e p o t e n t i a l o f t h e c y l i n d e r i s by t h e a i d o f image t h e o r y . T h i s b a s i c a l l y means t h a t an a r r a n g e m e n t o f d i s c r e t e l i n e c h a r g e s must be f o u n d s u c h t h a t t h e b o u n d a r y c o n d i t i o n s o f t h e p r o b l e m a t hand a r e s a t i s f i e d . I n t h i s c a s e t h e b o u n d a r y c o n d i t i o n i s t h a t t h e p o t e n t i a l on t h e c y l i n -d e r s u r f a c e i s c o n s t a n t . The s o l u t i o n t o t h i s p r o b l e m i s w e l l known. Two l i n e c h a r g e s p l a c e d a t p o i n t s P^ and Pg r e s p e c t i v e l y as shown i n F i g . 2-3 w i l l t o g e t h e r p r o d u c e an e q u i p o t e n t i a l s u r f a c e o f r a d i u s a and c e n t r e 0. The p o t e n t i a l due t o t h e s e two l i n e c h a r g e s a t p o i n t P i s f r o m e q u a t i o n ( 2 - 2 ) . 1 2 2 (s + r - 2 r s cos 6 ) 2 o i s t r - <srs c o s o / v V = - £ I n —A 5 ^ ( 2 - 5 ) p k 4 0 _2 1 ~2 P k / a 4 2 0 a-( ^ + r - 2 r c o s 6 ) 2 s C o n s i d e r i n g t h e p l a n e A - A i n F i g . 1-3 i t i s e v i d e n t t h a t t h i s p l a n e i s a t z e r o p o t e n t i a l and a c t u a l l y s e r v e s as r e -f e r e n c e f o r Vp. F o r r = a e q u a t i o n (2-5) becomes Vp = - § I n (-) = c o n s t a n t (2-6) j& a F u r t h e r i t i s n o t e d t h e e q u i p o t e n t i a l s u r f a c e as r e q u i r e d e n -c l o s e s t h e same c h a r g e p e r u n i t l e n g t h as t h e c o n d u c t i n g c y l i n -d e r . From symmetry c o n s i d e r a t i o n s i t f o l l o w s t h a t t h e l i n e c h a r g e s a t P^ and Pg t o g e t h e r a l s o p r o d u c e s an e q u i p o t e n t i a l 14 15 s u r f a c e o f r a d i u s , a, and c e n t r e , Pg ' . Hence E q u a t i o n (2-6) r e p r e s e n t s t h e p o t e n t i a l o f a c o n d u c t i n g c y l i n d e r w i t h c h a r g e - q c o u l . / m p l a c e d p a r a l l e l t o a s i m i l a r c o n d u c t i n g c y l i n d e r w i t h charge +q coul./m. 2 T> 2 Prom P i g . 2-3 the distance s = 2m + — and m = - — 1 ® I - i<§>2 Then s (1-6) y i e l d s Vp = - J In D 2 2 S u b s t i t u t i o n f o r s i n Equation 2a (2-7) For a s i n g l e phase transmi s s i o n l i n e D i s the distance bet-ween centres of the conductors. However, i n general D » a so that equation (2-7) becomes This i s equivalent to assuming the image charge at point P^ i n F i g . 1-3 i s p o s i t i o n e d at point 0, and f o r D>10a the error i n Vp i s l e s s than 1$. The e l e c t r i c f i e l d i n t e n s i t y at the surface of the con-ducting c y l i n d e r shown i n F i g . 2-3 may be found as f o l l o w s . Consider Equation (2-5). and •| In ( s 2 + r 2 - 2rs cos 6) = i n s - | i n (1 - f e ^ ) ( l - § e^' 6) 1 a 4 2 a 2 2* In (~2 + r ™ 2r — cos 6) 8 1 -* = In r - 7j In (1 . iL- e 3 6 ) ( 1 „ a 2 sr sr 3 But i n general In ( l - z) = -s Therefore z 3 C O f o r z < 1, X 2 2 v~ i* ii X In (s + r - 2rs cos 6) = In s - )_ (—) — cos n6 and 4 2 0 - 0 2 i In (\ + r 2 - 2r f - cos 6) = In r - Y ( ~ r ) n ~ cos n6 f o r f 1 and — < 1, sr S u b s t i t u t i o n i n Equation (1-5) then y i e l d s eo Vp = - J In i + J I ± cos n6 (£) n - ( ^ - ) n * k r k Y n | s' x s r ' (2-8) Let o denote surface charge density i n coul./m . Then the e l e c t r i c f i e l d i n t e n s i t y E must s a t i s f y the equation E = -0— eo at the boundary between a conducting medium and a d i e l e c t r i c medium of p e r m i t t i v i t y e Q. The e l e c t r i c f i e l d i n t e n s i t y i s normal to a conducting medium, hence grad (V ) 6 p'r=a dV (E) becomes (-j-^) = - (E) v 'r=a dr r=a v r=a Then from Equation (2-8) I = a. (-r=a k a <>© n-1 - ( E ) . _ „ - S (i + 2 I a — cos n6) or ^ r = a ~ ~ ka 1 + 2 Y ( — )n cos n6 j volts/m 1 8 (2-9) when a = - r^— 1 1 + 2 Y ( — ) n cos n6i coul./m* 2ita s 1 (2-10) which i s the e l e c t r i c charge density on the surface of the conducting c y l i n d e r shown i n F i g . 1-3. Equation (2-9) may be put i n t o a d i f f e r e n t , but equivalent form as f o l l o w s . The expansion 1 z •s = 1 + z + 7?— + ... i s v a l i d f o r 1 — z £ < 1. Let z = Be*'6 where 0 t R < 1 then ^ = 1 + Re^ 6 + R 2 e J 2 6 + o o o 1 - Re 16 Consider the expression 1 1 1 - R cos 6 = 2 2 1 - 2 R cos 6 + R 2 (1 - Re^ 6) (1 - Re~^ 5) = | (1 + Re^ 6 +...)+ | (1 + Re""J6 + ...) or CO 1 - R c o s 6 * - 1 + I R n cos n6. (2-11) 1 - 2 R cos 6 + R 1 Equation (2-11) may be w r i t t e n 2 oo R cos 6 - R = £ R n c o g n 6 ( 2 _ 1 2 ) 1 - 2R cos 6 + R 1 When a d d i t i o n of Equations (2-11) and (2-12) y i e l d s i n 2 ^ " = 1 + 2 Z R n cos nfi (2-13) 1 - 2 R cos 6 + R 1 Comparison of Equations (2-9) and (2-13) shows that with R = — 1 . ( * ) 2 ( E ) r = a = " k ~ 2 7^2 ( 2 ~ 1 4 ) ;r=a Ka x „ 2 - cos 6 + (~r s s which i s the expression u s u a l l y given f o r the case of a l i n e 13 charge placed p a r a l l e l to a conducting c y l i n d e r This p a r t i c u l a r problem has been t r e a t e d i n d e t a i l i n s p i t e of the f a c t that i t s s o l u t i o n appears i n p r a c t i c a l l y every textbook on e l e c t r o m a g n e t i s m ^ ' H o w e v e r , t h i s problem forms the basis f o r most of the work to f o l l o w and i t was therefore thought worthwhile to consider t h i s simple case rather thoroughly„ 10 3. VARIATION OP ELECTRIC FIELD INTENSITY WITH PHASE ANGLE AND PHASE SPACING In a three-phase system the charge on each conductor v a r i e s w i t h time, hence i n order to determine the maximum f i e l d i n t e n s i t y occurring on any one conductor t h i s time de-pendence must be taken i n t o account. Further one would ex-pect the phase angle at which t h i s maximum f i e l d i n t e n s i t y occurs to be dependent on the geometric c o n f i g u r a t i o n of a transmiss i o n l i n e system. Hence the object here i s to de-termine these r e l a t i o n s . F i g . 3-1 shows a three-phase system w i t h ungrounded n e u t r a l . For the purpose of computing the p o t e n t i a l s of the conductors i n terms of the unknown charges, the conductors and the geometry of the system one can make the f o l l o w i n g approximation. The spacing s i s s u f f i c i e n t l y large that the p o t e n t i a l s e x t e r n a l to the conductors can be produced by l i n e charges p o s i t i o n e d at the centre of the conductors. This a c t u a l l y i m p l i e s that one assumes the p o t e n t i a l c o n t r i b u t i o n due to one phase conductor to be constant over the surface of the other two. The er r o r committed by using t h i s procedure f o r a s i n g l e phase l i n e has been mentioned already. See sec-t i o n 2-2. 3-1. Determination of Line Charges The three-phase transmission l i n e shown i n F i g . 3-1 i s assumed ungrounded and s u f f i c i e n t l y f a r removed from ground that t h i s has no inf l u e n c e on the e l e c t r i c f i e l d c o n f i g u r a t i o n Phase 1 Phase 2 Phase 3 V^ = V cos ojt V 2 = V cos (w-t-120) Vg = V cos(cot-240) F i g . 3-1 Ungrounded three-phase Transmission Line Phase 1 Phase 2 Phase 3 *1 F i g . 3-2 Conducting C y l i n d e r and two Line Charges 11 around the conductors. Hence one can w r i t e q 2 + q . 2 + q . 3 = 0 (3-1) f o r any phase angle. A l s o , making use of the approximation mentioned above and Equation (2-2). V 0 1 = In £ + £i In - + £3 l n 2s 21 k a k s k a = ^ ( l n J - l n 2) + ™ (In I - l n 2) (3-2) where k = 2ite o S i m i l a r l y V 2 3 = IT ( l n f + l n f } + IT ( l n 2 " l n f } (3™3) A d d i t i o n of Equation (3-2) and (3-3) y i e l d s , 3 q 2 s q 2 V 2 1 + V 2 3 = T T l n f - IT l n 2° But V 2 1 + V 2 3 = V c o s ~ 1 2 ° ) " V c o s wt + V cos (cat - 120) - V cos (wt - 240) = 3 cos (wt - 120). Hence one may solve f o r q 2 and k V cos (cot - 120) _ , /„ „\ q 0 = - — ™ f — — L coulo/m (3-4) ^ l n I - i l n 2 Further V12 = ^ (In f + In 2) + ^ ( l n I + l n 2) (3-5) and v l 3 = TT ( l n S + l n 2 ) + IT l n T ^3"6) 2 q l „_ s . „ x . *2 2s 12 But + = ^ c o s Then s u b s t i t u t i n g f o r q 2 from Equa-t i o n (3-4) and s o l v i n g f o r q^. 1 k(V In |cos art - V3~2 In 2 s i n wt) q, = - = — - j — — — o r - eoul./m (3-7) 1 (In - - i In 2) In ±1 a 3 a Then one may f i n d q^ from Equation (3-1) q.3 = - ( q 2 + ^ j . k j v In | cos (wt - 240) + V3~ 2 In 2 s i n (wt - 240)) _ r-— — o ~ — coul./m ( i n f - | In 2 ) ( i n §S) (3-8) 3-2. Determination of the Phase Angle at which Maximum E l e c t r i c F i e l d I n t e n s i t y Occurs The r e l a t i o n s h i p between the maximum f i e l d i n t e n s i t y on phase conductor 1, phase angle wt and spacing s w i l l f i r s t be determined. As shown i n F i g . 3-2 phase conductors 2 and 3 are considered l i n e charges of strength q 2 and q^ f o r t h i s purpose. See Appendix I . Then using Equation ( I - l ) one can determine the surface charge de n s i t y on conductor 1 from the equation ° ! <«•«« - 5sr - i § f <t>n <">* n 6 - £ \ <fe>n c ° s n 6 <3-9> C l e a r l y the maximum charge d e n s i t y w i t h respect to 6 occurs at 5 = 0° when co co Y ( * ) n = and Y ( f - ) n .= ^ ± ^ v s ' s-a ^ v2s' 2s - a Further the maximum value of o^ wit h respect to wt i s found from do, ^dwV = 0 6-0 dq. a ^ 2 a dq. ~ 2na dwt na s-a dwt na 2s-a d&)t S u b s t i t u t i o n of the values of q^, q^ and q^ from Equations (3-4), (3-7) and (3-8) y i e l d s tan (art) 1 •* • B a, ,„ &o a - i n In S£ In — + a In 2 1 1 In 2 a 32 2s a 3 2 + 1 32 In — -S™9» 3* a In 2 , s 2s-a a 2s-a 1 32 (3-10) To obtain a rep r e s e n t a t i v e value of art l e t a = 1 inch and s = 25 f t . Then tan (art)^ = -.073 or a)t = 4.2° Prom Equation (3-7) i t i s noted that f o r a = 1 inch s = 25 f t . occurs at *1 max tan (wt) ^  = - ^ l n 2 — or wt = -4.0° 32 i n 300 Hence the charge on conductor 2 and 3 have n e g l i g i b l e i n -fluence on the phase angle of o"^ . An equation s i m i l a r to (3-10) may be derived f o r the centre conductor. Using the same procedure as f o r conductor '. tan (wt), 1 i„ 2 s 2 1 x 1 Z" 2a 3 2 ! n 2 3 x - T l n T + 1 s + a. 2a l n 2 i n ! + a Q, 3 ~ £L In 2 l n 2 a s - a l n SL a 3 a l n - (3-11) s - a a (s + a) 3 2 (s - a) For a = 1 i n c h s =25 f e e t one f i n d s tan ( w t ) 2 = -1.76 or (wt)g = 119.6( 14 I t may be noted that i s i n phase with Vg. See Equation (3-2). For p r a c t i c a l t r a n s m i s s i o n l i n e s l n ( - ) l i e s i n the narrow 19 range 5.3 to 6.4 . Therefore, t h i s q u antity cannot cause much v a r i a t i o n i n the phase angles (ojt)-^ and (cot);,. The spacing between l i n e s i s g e n e r a l l y determined by the i n s u l a -t i o n l e v e l r e q u i r e d , that i s , the magnitude of the overvoltages expected to occur i n the system. The usual spacings range from 10 to 12 inches f o r each 10 kv rms. between l i n e s w i t h a 22 minimum spacing of about 20 inches. Hence considering the expressions determined f o r (oot)^ and (ootjg i t i s evident that f o r a l l p r a c t i c a l purposes one can assume that the maximum f i e l d i n t e n s i t y and charge per u n i t length occurs on any one con-ductor when i t s phase voltage'has i t s peak value. This pro-cedure i s followed i n the a n a l y s i s c a r r i e d out i n chapter 4 and 12 13 19 5. I t has, been pointed out by several authors ' ' that the maximum charge occurs on the centre phase f o r the co n f i g u -r a t i o n shown i n F i g . 3-1, This f a c t can be v e r i f i e d from equations (3-4), (3-7) and (3-8). 15 4. NON-CIRCULAR CONTOURS In t h i s chapter some s p e c i f i c , n o n - c i r c u l a r conductor shapes are considered, the object being to determine i f such conductors used i n a three-phase transmission l i n e would pro-vide a f i e l d c o n f i g u r a t i o n around each conductor of greater u n i f o r m i t y than i s obtained with c i r c u l a r conductors. For t h i s purpose the phase conductors adjacent to the one con-si d e r e d are t r e a t e d as l i n e charges; 4-1. C a s s i n i a n Oval Refer to F i g . 4-1. Two p o s i t i v e l i n e charges each of strength ^ coul./m are placed at points (s,0) and (-s,0) r e s p e c t i v e l y . I t f o l l o w s that the p o t e n t i a l at any point P i s given by the equation 1 1 V p = 2k l n ( r 2 + s 2 - 2rs cos 6 ) 2 + |g l n ( r 2 + s 2 + 2rs cos 6 ) 2 + K o (4-1) where k = 2ite and K i s a constant reference p o t e n t i a l . Also o o the e q u i p o t e n t i a l s produced by these two l i n e charges are gov-erned by the equation 2 2 4 r l r 2 = C where c i s a constant. S u b s t i t u t i o n of the values f o r r ^ and y i e l d s r 4 + s 2 - 2 r 2 s 2 cos 26 = c 4 (4-2) 20 21 which i s the equation of a Cassinian Oval ' f o r c > a. From image theory (see s e c t i o n 2-2) any one of these contours P i g . 4-1 Cassinian Oval P i g . 4-2 Charge D i s t r i b u t i o n on Cassinian Oval 16 may be considered a conducting surface. Hence one can f i n d the surface charge d i s t r i b u t i o n on a conductor having t h i s shape by determining grad from Equation (4-1). Further, when a conductor i s placed s i n g l y i n space and i s kept at a c e r t a i n p o t e n t i a l w i t h respect to some reference the surface charge on the conductor w i l l assume a s p e c i f i c d i s t r i b u t i o n depending on the shape of the conductor. This charge d i s t r i b u t i o n i s termed "the f r e e charge d e n s i t y " i n what follows. In p o l a r co-ordinates tgrad V P p (^£)2 + (1 ^£)2 1 2 Then using Equation (4-1) o p q /r - s cos 5 r + s cos 6\ N „ = 2 k V o + o > (4-3) dr (4-4) 1 o'p q /S sin 5 s s i n 6\ r 06^ 2k (4-5) •1 4 2 Hence evalu a t i n g Equation (4-3) by a i d of (4-4) and (4-5) grad V _ _2£. ' k c 2 (4-6) The f r e e charge density, o(6) = e grad V hence o(6) = o coul./m 2 2ite (4-7) Choosing the values s = 0.75 cm c = 1.00 cm one may p l o t the contour shown i n F i g . 4-1 by using the equation (4-8) r = 1 1 s 2 cos 26 + ( s 4 c o s 2 26 - s 4 + c 2 ) ^ ' 2 A l s o , the f r e e charge d e n s i t y was computed f o r t h i s contour 17 by a i d of Equation (4-7). See F i g . 4-2 . From the charge d i s t r i b u t i o n obtained i t appears that a conductor shape as shown i n F i g . 4-1 might be s u i t a b l e f o r the centre conductor of a three-phase transmission l i n e with f l a t spacing, that i s , w i t h the other two phase conductors placed e q u i d i s t a n t from the centre conductor on the p o s i t i v e and negative y-axis r e -s p e c t i v e l y . However, the Cassinian Oval w i l l not be considered any f u r t h e r here. 4-2. E l l i p t i c C y l i n d e r and two Line Charges. 25 13 Consider the f u n c t i o n ' w = z + — (4-9) z Here w i s seen to be an a n a l y t i c f u n c t i o n of z w i t h s i n g u l a r i -t i e s only at z = 0 and z =oo. Further TO-1-? ^° except at z = - d. Hence a mapping performed by using (4-9) would not be conformal at these two p o i n t s . See Appendix I I . Refer to the c i r c u l a r contour, z = ae** , shown i n F i g . 4-3. I f t h i s contour i s mapped onto the W-plane by a i d of Equation (4-9) i t i s evident W = a e j 6 + ^Lr (4-10) ae J By equating r e a l and imaginary parts and e l i m i n a t i n g 6 from (4-10) i t i s found that F i g . 4-3 C i r c u l a r C y l i n d e r and two Line Charges u F i g . 4-4 E l l i p t i c C y l i n d e r and two Line Charges 18 which i s the equation of an e l l i p s e i n the W-plane. See F i g . 4-4. Here d i s chosen such that d < a, tha t i s , the two points (-d,0) and (d,0) are i n s i d e the c i r c u l a r contour i n the Z-plane. Hence the mapping i s conformal f o r a l l points >" a i n the re g i o n zj 5> a. Also i t i s noted from Equation (4-9) that the mapping i s one to one between the Z-plane f o r and the W-plane f o r the region external to the e l l i p t i c con-tour given by Equation (4-11). F i g . 4-3 shows a conducting c y l i n d e r w i t h charge q coul./m placed p a r a l l e l to two l i n e charges each of strength coul./m. This system i s analogous to the one t r e a t e d i n s e c t i o n 2-2 and can also be solved by image theory. P l a c i n g two image l i n e 2 ft2 charges at po i n t s P-L(0, ) and P2(0» - ~ ) r e s p e c t i v e l y i n F i g . 4-3 the complex p o t e n t i a l , Q = V + jU, at point P i n the Z—plane i s given by -2 =,2 .a .a z - j ~ z + j — (4-12) s where K i s a constant reference p o t e n t i a l and k = 2ize . Here o o the complex p o t e n t i a l i s used only f o r convenience. The r e a l p o t e n t i a l V = Re (Q ) i . e . P P p 4k , r 2 + s 2 - 2rs sin6 , , r 2 + s 2 + 2rs sin§ l n . n + l n — — \ 2 a 4 _ a 2 . K r + —TT - 2r — sin6 £ s s 2 a 4 _ a 2 . . r + -TT + 2r — sm6 d s s or (V ) = i - 2 l n 5. + K N p'r=a 2k a o ( 4-13) (4-14) therefore V i s a constant on the c i r c u l a r contour as requi r e d . P 19 Mapping the system shown i n F i g . 4-3 i n t o the W-plane by a i d of the f u n c t i o n (4-9) i t f o l l o w s that one obtains an e l l i p t i c c y l i n d e r placed p a r a l l e l to two l i n e charges i n the W-plane. See F i g . 4-4. From Equation (4-14) the p o t e n t i a l on the c i r c u l a r c y l i n d e r i n F i g . 4-3 i s constant. This means that the e l l i p t i c c y l i n d e r i n the W-plane has the same constant p o t e n t i a l since that boundary c o n d i t i o n i s unchanged by a con-formal transformation. See Appendix I I . Further, the two image l i n e charges placed at points P^ and Pg i n the Z-plane may w e l l be p o s i t i o n e d outside the e l l i p t i c contour i n the W-plane, e.g., when s»a / x 2 . . d 2s d s a However, these l i n e charges are not s i n g u l a r i t i e s f o r the regi o n z ^ a which i s considered here. As pointed out e a r l i e r the mapping i s one to one f o r the exte r n a l areas, hence the l i n e charges at points Pg and P^ i n the Z-plane map i n t o the points Pg and P^ i n the W-plane. Using Equation (II-8) which i s r e w r i t t e n here f o r con-venience E = E w z dz dw one can then determine the charge density on the e l l i p t i c (H-8) grad (Q p) c y l i n d e r shown i n F i g . 4-4. For t h i s purpose dV — £ must be determined from Equation (4-13). I t i s evident dr t hat the expression obtained w i l l be analogous to the one de-r i v e d i n s e c t i o n 2-2 f o r the case of a c i r c u l a r c y l i n d e r and one l i n e charge. See Equation (2-14). 20 Then d V T > a I 1 " (f> 2 1 <l>2 / P\ q / s s dr 'poa 2ka L + ( a ) 2 _ 2 | g i n 6 1 + ( a } 2 + 2 | g i n 6 \ s s s s or E z=aeJ 6 = f a " I 4 " % 2 ~ ( 4 " 1 5 ) s + a + 2a s cos 26 Further, d i f f e r e n t i a t i n g Equation (4-9) with respect to w o 1 _ 5L2. d_ dz dw 2 dw z Hence 2 T (4-16) dz dw a ( a 4 + d 4 - 2 a 2 d 2 cos 26)^ S u b s t i t u t i o n of Equations (4-15) and (4-16) i n Equation (II-8) then y i e l d s the e l e c t r i c f i e l d i n t e n s i t y normal to the surface of the e l l i p t i c c y l i n d e r E 4 4 \ s - a 4 I 7 2 2 17 s + a + 2a s cos 26 ( a 4 + d 4 - 2 a 2 d 2 cos 2 6 ) 2 / The surface charge de n s i t y thus becomes _ q_ / s 4 - a 4 \ / a . \ ° " ~ 2 * I s 4 + a 4 + 2 a 2 s 2 cos 26 ' * ( a 4 +• d 4 - 2 a 2 d 2 cos 26 ) 2 I coul./m 2 (4-17) Also i t can be seen from Equations (4-15) and (4-16) that the free charge d e n s i t y on the e l l i p t i c c y l i n d e r i s given by CL, « S : r coul./m 2 (4-18) ( a 4 + d 4 - 2 a 2 d 2 cos 26)2 2 2 To consider a numerical case l e t d = 1.40 cm ., a = 2 cm. 009 a .008 6 i n degrees P i g . 4-5 Charge Density V a r i a t i o n 21 p and s = 4.68 cm. The values chosen f o r d and s are such that o*w evaluated from Equation (4-17) has equal magnitude at 6 = 0 ° and 6 = 90°. P i g . 4-5 shows a as a f u n c t i o n of the ° w angle 6. In order to determine the improvement obtained i n the charge d i s t r i b u t i o n by using an e l l i p t i c c y l i n d e r i n place of a c i r c u l a r c y l i n d e r one must i n v e s t i g a t e an equivalent system of a c i r c u l a r c y l i n d e r and two l i n e charges. The c r i t e r i a used here to determine the dimensions of t h i s system are.the f o l l o w i n g : i ) Equal perimeters of the e l l i p t i c and the c i r c u l a r c y l i n d e r i i ) A l l l i n e charges must be the same distance from the geo-metric centre of the conducting c y l i n d e r s . 26 The perimeter, S, of an e l l i p s e i s given by 2ll ! S = b j (1 - | e 1 2 s i n 2 o ) ^ da (4-19) 0 where b = major a x i s e-^  = e c c e n t r i c i t y . Prom equation (4-11) and the values already chosen f o r the constants one f i n d s 2 b = 2.70 and e^ = .768. Hence expanding the r a d i c a l i n Equation (4-19) and i n t e g r a t i n g the f i r s t three terms S = 13.08 cm. Therefore the radius of the equivalent c i r c u l a r c y l i n d e r i s r i = 1 i i ^ = 2 - 0 8 c m -22 Further, from Equations (4-9) and (4-11) and the values chosen o above f o r d , a and s the distance O'P^ i n F i g . 4-4 i s °'P3 " ( 4 * 6 8 " C m = 4.38 cm which i s the distance s i n F i g . 4-3 f o r the equivalent system. Ev a l u a t i o n of the change den s i t y on the equivalent c i r c u l a r c y l i n d e r from Equation (4-15) with a = 2.08 cm and s = 4.38 cm y i e l d s the curve shown i n F i g . 4-5. Comparing the curves shown i n F i g . 4-5 i t i s c l e a r that the e l l i p t i c c y l i n d e r provides the be t t e r charge d i s t r i b u t i o n . One can, f o r instance, compare the q u a n t i t i e s a - o" . f o r ' max min the two cases. Thus f o r the c i r c u l a r c y l i n d e r a - a . -J max min .454 as opposed to .112 f o r the e l l i p t i c c y l i n d e r . Hence i t would be of advantage to use an e l l i p t i c c y l i n d e r f o r the centre conductor i n a transmission l i n e system with f l a t spacing. 4-3. A p a r t i c u l a r n o n - c i r c u l a r Cylinder and two Line Charges. The case of a c i r c u l a r c y l i n d e r placed p a r a l l e l to a l i n e charge was t r e a t e d i n s e c t i o n 2-2. From Equation (2-9) the maximum f i e l d i n t e n s i t y occurs at 6 = 0°, and i t was f e l t t hat t h i s f i e l d i n t e n s i t y could be decreased by f l a t t e n i n g the c i r -c u l a r c y l i n d e r i n the region 6 = 0 ° . See F i g . 2-3. To i n v e s t i g a t e t h i s the f u n c t i o n given by (4-9) was used to map the c i r c u l a r contour, z = j y Q + ae^ , shown i n F i g . 4-6 i n t o the W-plane. The equation governing t h i s contour i n the W-plane becomes w = j y + a e j 6 + ^ (4-20) j y o + a e J U F i g . 4-6 C i r c u l a r C y l i n d e r and Line Charge I 0« V ^ \ W-plane / u r l i ,-q. P i g . 4-7 Deformed C y l i n d e r and Line Charge 23 or separating r e a l and imaginary parts c d a coso u = a coso + 2 2 a + y Q + 2ay Q sin6 A2 t • c x (4-21) d (a sm6 + y ) v = a sin6 + y 5 o ~ ' a + y~ + 2 a y ~ s i n S O " o To determine the shape of a p a r t i c u l a r contour l e t a = 2.0, d 2 =0.50 and y =0.50 o computing the u and v co-ordinates from (4-21) one obtains the curve shown i n P i g . 4-7, and i t i s evident that a c y l i n d e r i s obtained which i s deformed i n the des i r e d manner. Prom s e c t i o n 4-2 i t f o l l o w s that one can obtai n a conformal mapping of the system shown i n F i g . 4-6 by a i d of the f u n c t i o n (4-9). Also the general d i s c u s s i o n given i n s e c t i o n 4-2 r e -garding the mapping f u n c t i o n and the mapping i t s e l f a p p l i e s to t h i s s e c t i o n as w e l l . Refer to F i g . 4-6. The case of a conducting c y l i n d e r placed p a r a l l e l to a l i n e charge has already been t r e a t e d i n s e c t i o n 2-2. Hence the complex p o t e n t i a l , Q = V + jU, at a point P i s z + j ( s - y ) 2 P = S l n 2 — + K o < 4- 2 2 ) • /a \ z + o ( — - y 0 ) where K Q i s a constant reference p o t e n t i a l and k = 2ite o. Then mapping t h i s system i n t o the W-plane one obtains the conf i g u r a -t i o n shown i n F i g . 4-7. I t f o l l o w s that the charge density on the transformed c y l i n d e r can be determined from Equation ( I I - 8 ) . Hence evaluating 24 E z ~ vdz ' vdz ; from Equation (4-22) E 2 2 s - a z=jy H-ae**0" k a s 2 + a 2 + 2as sin6 (4-23) Further, from Equation (4-9) _2 dz dw 2 2 z - d J o but i n t h i s case z = j y Q + ae J , therefore dz dw J6x2 ( j y 0 .+ ae J ) ( j y 0 + a e J 6 ) 2 - d 2 when dz dw 2 2 a + y^ + 2ay sin8 o w o ( a 2 + y Q 2 + 2ay Q s i n 6 ) 2 - 2 d 2 ( a 2 c o s 26-y Q 2-2y oa sin6) + d 4 (4-24) S u b s t i t u t i o n of Equations (4-23) and (4-24) i n Equation (II-8) y i e l d s E w ka 2 2 a + y Q + a y Q sin6 2 2 2 2 2 2 (a + y Q + 2ay Q sin6) - 2d (a cos 26-y Q -2y Qa sin6) + d 41 2 = 2 o 2 s — a 2 2 s + a + 2as sin5 (4-25) Hence the surface charge d e n s i t y can be determined from 25 2na 2 2 a + y Q + 2ay Q sin6 ( a 2 + y Q 2 + 2ay Q s i n 6 ) 2 - 2 d 2 ( a 2 c o s 2 6 -y Q 2-2y oa sin6) + d 4 = 2 o 2 s — a 2 2 c s + a + 2as smo (4-26) To consider a s p e c i f i c numerical example l e t a = 2 cm, 2 2 y Q = .0625 cm, d = .0625 cm , s = 60 cm. Using these values i n Equation (4-26) one obtains the curve shown i n F i g . 4-8. Then, to determine the charge d e n s i t y v a r i a t i o n on the equiva-l e n t c i r c u l a r c y l i n d e r (see s e c t i o n 4-2) one must f i n d the distance 0'P| and the perimeter of the deformed contour. See F i g . 4-7. The length O'PJ can be found from Equation (4-9). Thus j2 O'P' = s - y - — 1 Jo s - y Q or O'P^ = 59.94 Also from F i g . 4-6 OP-j^  = 60 cm. Therefore O'P^ = 0P 1 i n t h i s case. The f o l l o w i n g procedure was used to determine the p e r i -meter, S, of the deformed contour. From Equation (4-20) .2 3* w = oy, + a e ^ + d< j y Q + ae' l e t |w = B then evaluating the r i g h t hand side of t h i s equation .0057 a . 0045' • ' • » • --90 -70 -50 \-30 -10 10 30 50 70 90 6 i n degrees P i g . 4-8 Charge Density V a r i a t i o n R = 26 1 2 2 2 \ — I 2 2 2 Tjd +a co s 26-y Q -2y a sin6\2 a + y + 2y a sin§ + 2d O O d d _ . c a + y f t • + 2y a sin6 (4-27) The perimeter, S, can then be found from 271 1 S = o p 2 ,dRx2 R + W 2 d6 (4-28) The i n t e g r a l (4-28) was evaluated numerically by a i d of Le-27 gendre-Gauss Quadrature and the ALWAC I I I - E d i g i t a l computer. 2 Thus f o r the values chosen above f o r a, y , d and s S = 12.57345 cm. as compared to the perimeter S of the o r i g i n a l c i r c u l a r con-c t o u r , S = 2Tta = 12.56637 cm. c Hence the d i f f e r e n c e between S and S i s n e g l i g i b l e . Therefore c the charge d e n s i t y on the equivalent c y l i n d e r was computed from Equation (4-23) w i t h a = 2.0 cm s = 60.0 cm. The curve ob-t a i n e d i s shown i n P i g . 4-8. Comparison of the two curves shown i n F i g . 4-8 i n d i c a t e s that both the maximum and minimum f i e l d i n t e n s i t y on the deformed contour i s l e s s than on the c i r c u l a r c y l i n d e r . S p e c i f i c a l l y , the d i f f e r e n c e between the two f i e l d i n t e n s i t i e s at 6 = -90° i s 3.5$ f o r t h i s case. Therefore, using a c y l i n d e r of the shape shown i n F i g . 4-11 f o r the outer conductors i n a t h r e e -phase system with f l a t spacing would provide some improvement i n the surface charge d i s t r i b u t i o n . In order to represent an a c t u a l three-phase system w i t h f l a t spacing an a d d i t i o n a l l i n e charge must be placed p a r a l l e l 27 to the c y l i n d e r i n P i g . 4-6. Therefore, consider a c i r c u l a r c y l i n d e r w i t h charge q coul./m placed p a r a l l e l to two l i n e charges each of strength —1> coul./m as shown i n P i g . 4-9. This system can be mapped i n t o the W-plane using the same procedure employed f o r a c i r c u l a r c y l i n d e r and one l i n e charge. Further, the charge density on the deformed conductor shown i n F i g . 4-10 i s given by the equation f(w) 2%& 2 2 / a + y Q + 2ay Q sin6 X (a 2+ y Q^+ 2ay Q s i n 6 ) 2 - 2 d 2 ( a 2 c o s 2 & - y Q 2 - 2 y o a sin6)+ d 4 / 2 2 s 2 - i a 2 S " a ' 4 1 (4-29) 2 2 0 . , . 2 1 2 . K s + a + 2as sine s + ^ a + as smo j Numerical computations based on Equation (4-26) i n d i c a t e 2 tha t the two parameters y Q and d should be such that the charge density on the deformed contour v a r i e s i n the manner shown i n F i g . 4-8. The reason being that t h i s provides the best charge d i s t r i b u t i o n obtainable with the p a r t i c u l a r contour t r e a t e d i n 2 t h i s s e c t i o n . These values of y and d are therefore con-o sidered optimum. I t was thought of i n t e r e s t to determine the optimum values 2 of y Q and d f o r a s e r i e s of phase spacings, s. For t h i s pur-pose a programme was w r i t t e n f o r the ALWAC I I I - E d i g i t a l com-puter such that Equations (4-26) and (4-29) could be evaluated 2 f o r various values of a, y , d and s. Thus, having chosen a 2 p a r t i c u l a r phase spacing, s, and ra d i u s , a, y Q and d could be P i g . 4-9 C i r c u l a r C y l i n d e r and two Line Charges v V W-plane V / u < 1 > i P i g . 4-10 Deformed Cy l i n d e r and two Line Charges 28 determined by t r i a l and e r r o r . The v a r i a t i o n s obtained are i n d i c a t e d i n P i g s . 4-11 and 4-12, and i t w i l l be noted that only one value of a was used. 2 Considering the values obtained f o r y Q and d i t i s c l e a r that the deformation required f o r large values of s i s very s m a l l , f o r both the s i n g l e and three-phase system. F i g , 4=11 Conductor Shape as a Function of Spacing (Single Phase) .14 0 30 60 90 120 150 180 210 240 Spacing s i n cm P i g . 4=12 Conductor Shape as a Function of Spacing (Three Phases) 29 5. A PERTURBATION METHOD I t was shown i n the previous chapter that using the non-c i r c u l a r conductors considered there d i d provide some improve-ment i n the surface charge d i s t r i b u t i o n on a l l phase conductors, however, these shapes are of course not optimum. The optimum conductor shapes being defined here as those p r o v i d i n g uniform charge d i s t r i b u t i o n around the perimeter of each conductor i n a three-phase system. For t h i s reason a more d i r e c t method of attack was developed. This method c o n s i s t s e s s e n t i a l l y of applying a f i r s t order p e r t u r b a t i o n to the boundary of a c i r -c u l a r c y l i n d e r placed p a r a l l e l to l i n e charges. At the same time the shape obtained must s a t i s f y the c o n d i t i o n of uniform surface charge d e n s i t y on the deformed c y l i n d e r . The three sections f o l l o w i n g describe i n d e t a i l the method and the r e -s u l t s obtained, 5-1. Conducting Cylinder The c i r c u l a r c y l i n d e r shown i n Fig„ 5-1 i s placed s i n g l y i n space and i s kept at a p o t e n t i a l V with respect to some reference. Consider the p o t e n t i a l at point P on the con-ductor surface. The p o t e n t i a l at that point due to a small charge element at point N i s where o =• surface charge density i n coul„/m k = 2ite r l * a2 + a 2 - 2 a 2 cos(a - 6) dS = ad6 30 For convenience the reference p o t e n t i a l has here been assumed equal f o r a l l points on the c y l i n d e r and i d e n t i c a l l y zero. y P F i g . 5-1 Conducting Cylinder Then i n t e g r a t i o n of (5-1) gives the p o t e n t i a l at point P due to the t o t a l charge on the conductor, or 2% d6 (5-2) but l n r, l n 2 2 2a - 2a cos(a-6) = l n a + In 2 cos(a-6) 1 2 Also a i s constant i n t h i s case and can therefore be removed from under the i n t e g r a l s i g n , hence (5-2) becomes 2lt 2% V p = It / a l n a d 6 + 1 / a l n [ 2 - 2 cos(a-6)|d6 (27taa) l n a (5-3) since the second i n t e g r a l i s equal to zero. (See Equation (5-35)). The quantity (2nao) i s the charge q per u n i t length of the conducting c y l i n d e r , therefore 31 (5-4) This expression i s i d e n t i c a l to the one g e n e r a l l y obtained f o r a c i r c u l a r c y l i n d e r by other methods, and f o r the purpose at hand does i n f a c t correspond to the case of a l i n e charge of f i n i t e l y f a r removed from i t . Hence determining the poten-t i a l i n t h i s manner gives the correct answer f o r t h i s simple case. 5-2. Conducting C y l i n d e r and Line Charge F i g . 5-2 shows a conducting c y l i n d e r with charge q coul./m place p a r a l l e l to a l i n e charge of strength -q coul./m. The p o t e n t i a l at point P i s strength -q coul./m placed p a r a l l e l to the c y l i n d e r , but i n -P F i g . 5-2 Conducting Cylinder and Line Charge (5-5) S where a — surface charge density i n coul./m' k = 2ne o 2 2 2 (5-6) = r 6 + r a 32 2 2 2 „ Tn = r + s - 2r s cos a d» ct X dS = ( r 6 2 + r ' 2 ) 2 d6 s > a Also the reference p o t e n t i a l has been assumed i d e n t i c a l l y zero. F i r s t consider the conductor shown i n F i g . 5-2 a c i r c u l a r c y l i n d e r of radius a. To evaluate (5-5) one can s u b s t i t u t e f o r a(6) from Equation (2-10) and i t f o l l o w s the p o t e n t i a l at point P i s V = f l n (-) (See Equation (2-6)) (5-7) I? ^ Now l e t the surface of the c i r c u l a r c y l i n d e r be perturbed by a small amount so that the polar radius at the angle 6 becomes r 6 = a + h 6 ( h 6 « a) where hg i s a f u n c t i o n of 6. ^ h i s n o t a t i o n i s used throughout i n the d i s c u s s i o n f o l l o w i n g . Hence, s u b s t i t u t i n g f o r r and r 6 i n (5-6) P p o v1 = (a + h f i) + (a + h a) - 2(a + hg)(a + h f f) cos(a-6) 2 2 2 r 2 = (a + h f f) + s - 2s(a + h f l) cos a (5-8) Expanding (5-8) and r e t a i n i n g only terms of f i r s t order i n h Ct and hg (5-8) becomes 2 r l 2 - 2 cos(a-6) a2 + a ( h a + hg) (5-9) 2 2 2 r 0 = a + 2ah + s - 2as cos a - 2sh cos a d* CL Ct S u b s t i t u t i n g Equation (5-9) i n (5-5) 33 2Y = £ l n ( a 2 + 2ah + s 2 - 2as cos a - 2sh cos a) -p k a a L In 2 - 2 cos(a-6)| | a + a ( h a + hg) dS (5-10) At t h i s point i t w i l l be re q u i r e d i ) 2V = l n — = constant ' p . k a i i ) o = constant w i t h respect to the angle 6 i i i ) r2n h Rd6 = 0 o (5-11) i v ) q = charge per u n i t length i s constant. Also from (5-6) 1 dS = ( r 6 2 + r £ 2 ) 2 d6 (a + hfi) 1 + i J i _ 2 + x + 2 a + h E + = (a + hg)d6 to f i r s t order. dh c where h£ = w ~ d6 (5-12) I t then f o l l o w s j a dS = q or 4 a = 2rca S u b s t i t u t i n g (5-12) and (5-13) i n Equation (5-10) one obtains 2V = ^ l n ( a + 2ah + s - 2as cos a - 2ash cos a) -p k a a ' 27cak 2% J In 2 -0 2 cos(a-6) (a + hg)d6 -2 ^ J d n a)(a + h 6)d6 0 2TC fek / ( l n < a + h a + V] ( a + Vd6 0 34 (5-14) The three i n t e g r a l s w i l l now be considered i n d i v i d u a l l y i ) I f l n [ 2 - 2 cos(a-6)]j (a + h 6)d6 = 0 * ' 2TC 1 ln|2 - 2 cos(a-6)] hg d6 (5-15) 0 2rc J a l n [ 2 - 2 cos(a-6)] d6 = 0 (See Equation (5-35)) 0 2ll 2Tt i i ) / ( l n a)(a + h g)d6 = / a ( l n a)d6 since 0 = 2na l n a (5-16) J7C since J ( l n a)hg d6 = 0 (See Equation (5-11)) 0 2TC i i i ) J Jln(a + h a + hg) 5 Tt (a + hg)d6 = I a ln(a + h a + hg)d6 + 2TC 0 J hg In(a + h a + hg)d6 (5-17) 0 h + h-But l n ( a + h + h.) = In a + l n ( l + — -) G O €t h + h- , h + h. „ . a o i / _ a 6\2 , = l n a + - — - TJA ) + ... (5-18) since i n general 2 3 ) f o r ,| z| < 1 Therefore r e t a i n i n g only the f i r s t term i n the s e r i e s (5-18) and s u b s t i t u t i n g f o r l n ( a + hg + h a) i n (5-17) z2 3 l n ( l + z) = (z - 7 j - + —2TC J ( l n ( a + h a 0 ' 271 4- hg) (a + hg)d6 = J , a l n a d6 + 0 2lt , . 2TC 0 6 J 0 2lt 35 hg l n a dfi + o a or 2n a l n ( a + h + h K ) ( a + h K)d6 = 2ita l n a + 2ith l6 r r he + n J h § l n a d6 + J h - 6 ( — £)d6 = 0 a s i n c e  h. a d6 (5-19) from Equation (5-11). Also i s a constant i n t h i s i n t e g r a -t i o n and a l l terms of second order i n hg i s n e g l e c t e d . F u r t h e r , the f i r s t term i n Equation (5-14) can be w r i t t e n 2 2 l n ( a + 2ah + s - 2as cos a - 2sh cos a) = CC OL 2 l n a + l n 1 + 2 -T + ( f ) 2 - | ^ cos a - ^ | h a cos al (5-20) a Hence r e w r i t i n g Equation (5-14) by a i d of (5-15), (5-16), (5-19) and (5-20) 2V = f l n 2uak , ~ h a /S\2 2s 2s h a 2q , 1 + 2 + (~) - f^- cos a - — — cos a + P In a 2-K l n 2 - 2 cos(a-6)|hg d6 - J In a 3, i n a - f k k a (5-21) h 2 g h But h « a , t h e r e f o r e the two terms 2 — and - — — cos a can a ' a a a be n e g l e c t e d i n the f i r s t term of (5-21). F u r t h e r , s u b s t i -t u t i n g f o r V from (5-11) and c a n c e l l i n g common terms and f a c t o r s Equation (5-21) becomes 2 l n - = l n a , /S\2 2s \ a 1 + (—) - r — cos al - — EL CV I cL 2% 1 2na l n - 2 cos(a-6)jhg d6 (5-22) Also l n T / s \2 2s 1 + ( f ) - — cos a ct c t = 2 l n - + In a 36 1 + (f)2 - r*cos« therefore (5-22) can be w r i t t e n /It i n 0 h + i -a 2 u 2 - 2 cos(a-6) hg d6 = a l n n /9-\2 0 a 1 + (—) - 2— cos a s s (5-23) which i s an i n t e g r a l equation of the second k i n d with the symmetric k e r n e l , K(a,6) = l n 2 - 2 cos(a-6) The i n t e g r a l E q u a t i o n 2 8 ' 2 9 ' 3 0 (5-23) can be solved d i -i n r e c t l y by expanding h f t, hg and l n terms of the orthogonal f u n c t i o n s , e^na e*~*'r^. Therefore -I /&\2 n& 1 + (—) - 2— cos a s s +00 h = Y A e J n a a n jm6 — &o + co l n l n , , /a\2 0 a 1 + (—) - 2— cos a s s h c = YA e — O O + C O /_ n 2 - 2 cos(a-6) + O0 + C O jna e=j*o (5-24) (5-25) (5-26) (5-27) — oo — c o The c o e f f i c i e n t s , f , i n (5-26) can be found as f o l l o w s In 1 + <f) 2-f*=os .)- l n 1 - £ 1 - £ e ~ J a s s c o I ±(f) n .***- I |(f)n e ~ J n a (5-28) since (^)< 1. Further, the c o e f f i c i e n t s K n r i n Equation (5-27) are defined by 37 K nr 2TC 2TC = f e ^ r 6 d6 ~ f l n j 2 - 2 cos(a-6)| e ~ J n a d a (5-29) 211 I 2%i I I The i n t e g r a l , I = J l n 0 2TC 2 - 2 cos(a-6) e"*'11" da w i l l f i r s t be determined. For t h i s purpose l e t ,2TI a-6 1 = 2 J |ln s i n 0 e ^ n a da 2TC = 2 | ( l n s i n 2=2) e' da + 2 / ( l n s i n ^)e' da (5-30) 0 Let b = e J , then s u b s t i t u t i o n of the exponential form of the sine f u n c t i o n i n (5-30) y i e l d s 6 I « 2 J e - J n a 0 In 2 + j f + j f - j f + l n ( l - b e"^) da + 2TE 2/ e ^ n a 0 l n 2 - j f + j f - of + l n ( l - be"J a) da Carrying out the i n t e g r a t i o n s i n d i c a t e d an s i m p l i f y i n g 2we-JN5 I = - n 27t + 2 J e 0 -0n<* l n ( l - be~ J ( X) da + 2 e " j n a l n ( l - b e ^ a ) da (5-31) Neither of the two i n t e g r a l s i n (5-31) e x i s t s at a = 6 since b = e ^ , however, considering the two i n t e g r a l s alone one can w r i t e .6 2TC e J n a l n ( l - b e ^ a ) da + j e~^na l n ( l - be"J a) da = 0 6 l i m e ~ j n a l n ( l - b e ~ j a ) da + Jo 2rc l i m / e -jn a l n ( l - be" 3 a) da 38 (5-32) Then i f one can l e t g-^ O and p-"-0 a f t e r the i n t e g r a t i o n has been c a r r i e d out and the r e s u l t remains f i n i t e the proper 39 40 value of the i n t e g r a l s has been obtained ' . Thus c a r r y i n g out p a r t i a l i n t e g r a t i o n of (5-32) and s i m p l i f y i n g the r e s u l t l i m n I e 2lt im n I 0 -i l m | e P " -6-p l i m n-1 g - 0 b -0 J ' n a l n ( l - a e " ^ ) da + ~ J n a l n ( l - a e ~ j a ) da = e - j ( n - l ) a l n ( l _ a e - j a } d a + 2n p — 0 b l i m n-1 f e ~ J ( n - 1 ) a l n ( l - ae~J a) da 6-p Which f o r n = 1 i s i d e n t i c a l l y zero. I t f o l l o w s that f o r any value of n (n / 0) 6 2tt [ e- J' n a l n ( l - be"J a) da + f e ^ n a l n ( l - b e ^ a ) da = 0 0 6 Therefore Equation (5-31) becomes 2TC e ^ n 6 I = - n (5-33) and s u b s t i t u t i n g f o r I i n Equation (5-29) 2TC 1 K n r - ~ 2TI J i r S e ~ J n 6 j j o e _ ^ — d 6 This i n t e g r a l i s non-zero only when r = n, hence K = - -nr n (5-34) 39 and Equation (5-27) becomes l n 2 - 2 cos(a-6) C O = 1 - 1 e J n ( a - 6 ) + £ - - e - J n ( a - 6 ) (5-35) 1 n 1 n Consider Equation (5-23) which can now be solved by a i d of (5-24), (5-25), (5-28) and (5-35). Thus +oo + - C O 1_ 271 2TC oo + 0 0 . . oo -oo m £ _ I eJn(a-6) + £ _ 1 e-jn(a-6) 1 n 1 n oo n - a I I(f) n e J n a - a I |(f) n e ^ n a n s d6 = (5-36) but the i n t e g r a l i s non-zero only when m = n, therefore + C O i- n oo - a oo C O e + Y - — eJna + I - — e-3na = Y n l n C O £ e^ 1 1" - a I ±{±)n e " j n a n*s n' s' (5-37) 1 " " 1 Equation (5-37) c o n s t i t u t e s 2n + 1 equations i n 2n + 1 unknowns which are the complex F o u r i e r c o e f f i c i e n t s of h f t. Thus the nth, -nth and Oth equations are or s o l v i n g f o r A ^ or s o l v i n g f o r A -n n a/a\n An " n - = " nV - ~ n - l V . A — n _ a/a_\n -n ~ n ~ ~ n s' A = - ^ ( £ ) n -n n-1 N s' A = 0 o (5-38) (5-39) (5-40) 40 Prom Equations (5-38) and (5-39) 1 ~ -1 ~ 1 - 1 V -29 41 This s t a t e of a f f a i r s i s p e r m i s s i b l e ' i f J l n 0 and a N2 - * 1 + - 2 f cos a e 3 a da = 0 2TC l n 0 1 + - 2 - cos a e " J U da = 0 s s I that i s , i f e~^ a i s orthogonal to the r i g h t hand side of Equa-t i o n (5-23); however, Equation (5-28) c l e a r l y shows that these i n t e g r a l s cannot be equal t o zero. Therefore one can s t a t e that Equation (5-23) has no s o l u t i o n . Thus i t appears that there i s no s o l u t i o n to the problem of a c y l i n d r i c a l conductor and one l i n e charge. That i s , no c y l i n d r i c a l conductor shape e x i s t s which has uniform surface charge d e n s i t y i n the presence of a l i n e charge. The method employed here i s a f i r s t order p e r t u r b a t i o n , and i t seems reasonable to expect that such an approach would provide the s o l u t i o n f o r s » a i f such a s o l u t i o n d i d i n f a c t e x i s t . 5-3. Conducting C y l i n d e r and two Line Charges Even though no s o l u t i o n appears to e x i s t f o r the unsym-m e t r i c a l problem t r e a t e d i n the l a s t s e c t i o n i t seems l i k e l y t h at a s o l u t i o n does e x i s t f o r the system c o n s i s t i n g of a con-ducting c y l i n d e r and two l i n e charges placed symmetrically w i t h respect to the c y l i n d e r . Such a system i s therefore i n v e s t i -gated i n the present s e c t i o n . 41 P i g . 5-3 shows a conducting c y l i n d e r with charge q coul./m placed p a r a l l e l to two l i n e charges, each of strength c o u l . / i The p o t e n t i a l at point P i s m. VP = § k l n r 2 + f k l n r 3 - J f l n r l d S S 2 2 2 where r„ = r + s + 2r s cos a o a a (5-41) ~ 2 — A * — _ .. ' i "? / Q -,u — o v •» f a ' P i g . 5-3. Conducting C y l i n d e r and two Line Charges and the other symbols have the values i n d i c a t e d by Equation (5-6). I n i t i a l l y l e t the conductor shown i n F i g . 5-3 be a c i r -c u l a r c y l i n d e r of radius a. The p o t e n t i a l at point P can then be determined from Equations (5-41) and (4-15). I t fo l l o w s V then becomes P V = % l n 1 p k a (See Equation (4-14)) (5-42) The boundary of the c i r c u l a r c y l i n d e r i s now perturbed by the method developed i n s e c t i o n 5-2., Hence the equations de-termined i n th a t s e c t i o n f o r the expression s r l d s 42 can be u t i l i z e d d i r e c t l y here. Therefore consider 1 2 2 2 l n r Q = l n ( r + s + 2r s cos a ) o a a (5-43) where r = a + h a a hence s u b s t i t u t i n g f o r r i n (5-43), and n e g l e c t i n g a l l but f i r s t order terms i n h a on expansion 2 2 l n r = l n ( a + 2ah + s + 2 as cos a + 2 sh cos a) (5-44) o Ct CC Combining Equations (5-44), (5-21) and (5-41) the f o l l o w i n g expression i s obtained s\2 2 VP = 8 l n a + k ln k l n a + 2k l n 1 + 2 - 2 + ( 1 ) 2 + cos a - 2^ -2 c o s a L a a a a a h 0 h 1 + 2 - 2 + (£)<* + 2| cos a + 2| -2 cos o - M n a - ^ l n a - f — -k k k a 2nak 2it l n 2 - 2 cos(a-6) hg d& (5-45) Sine h <C< a the terms i n —2 are neglected i n the log a r i t h m i c ct a expressions i n (5-45). Further, s u b s t i t u t i n g f o r V from P Equation (5-42) and c a n c e l l i n g common terms and f a c t o r s one ob-t a i n s 2TC la + h J In 2 - 2 c o s ( a - 6 ) ] h 6 de = J l n 1 + ( f )2 - 2f cos a] + 1 + ( f ) 2 + 2| cos a s s (5-46) which a l s o i s an i n t e g r a l equation of the second k i n d with symmetric k e r n e l , 43 K (6,a) = l n 2 - 2 cos (a-6) d6= I t f o l l o w s that (5-46) can be solved by the method employed f o r (5-23) i n s e c t i o n 5-2. Hence the s e r i e s expansions deve-loped there can be used f o r s o l u t i o n of (5-46). However, con-s i d e r l n (1 + (|) 2 + 2| cos a) = i n ( l + f e J a ) ( l + f e-J a) = CO CO Z ( - D n + 1 £ ( f ) n e J n a + I ( - l ) n + 1 i ( f ) n (5-47) Since •§ < 1 s Then s u b s t i t u t i n g from (5-24), (5-25), (5-28), (5-35) and (5-47) i n Equation (5-46) - c o -J I - o o / \ 1 1 CO . o o I f + f ( - l ) n + 1 ' i ( f ) n e - J n a (5-48) X X I t i s seen t h a t the i n t e g r a l i n (5-48) i s non-zero only when m = n. Further a l l odd terms i n n cancel on the r i g h t hand side of t h i s equation. Therefore + oo co A ^ . ZK + I - 3* e J n < X + 51 - ±=£ e^ n a = - ao 1 1 CO CO _ a I 1_ ( | ) 2 n e 32na,„ ft £ l_(£)2n e - j 2 n a ( 5 _ 4 9 ) Which determines the complex Fo u r i e r c o e f f i c i e n t s of h . r a From (5-49) the 2nth and Oth c o e f f i c i e n t s can be deter-mined from * > 44 A 2n 1 / a x i l A 2 n ~ 2n~ ~ " a 2n V or s o l v i n g f o r A 2 N A 2 n = " < ^ T ) ( | ) 2 n (5-50) A = 0 (5-51) o Further a l l the odd c o e f f i c i e n t s determined by (5-49) can be represented by the f o l l o w i n g equation * 2 n - l " % T " 0 <5"52> hence A 3 = A 5 = * * * = A2n-1 = 0 but, f o r n = 1 Equation (5-52) becomes A l A l " l 1 - 0 and t h i s represents a s p e c i a l case because A^ i s indeterminate 29 and n : 1 i s an eigenvalue of the k e r n e l . However, from the r i g h t hand side of Equation (5-49) i t f o l l o w s J I 1_ ( | ) 2 n ( e 3 2 n a + e - j 2 n a } 0-j« d a = Q 0 1 Therefore A^ may be assigned any value c o n s i s t e n t w i t h the physics of the problem. In t h i s case a cosine v a r i a t i o n , t h a t i s , a v a r i a t i o n of the form C cos 6 (C = constant) i s e v i d e n t l y not d e s i r a b l e , hence A^ i s given the value zero. Thus from (5-50) the F o u r i e r expansion of h i s given by oo h a = -a Y. sf lT ( f ' 0 0 3 2 n a ( 5 _ 5 3 ) 45 and the equation of the contour required r = a + h a a oo = a - a Y. 2n-l Ns ( ~ ) 2 n c o s 2nd (5-54) Hence i n order the surface charge density i s uniform on the conducting c y l i n d e r shown i n P i g . 5-3 the shape of the c y l i n d e r must be that given by Equation (5-54). I t was thought of i n t e r e s t to t r e a t though b r i e f l y one a d d i t i o n a l case i n t h i s s e c t i o n . P i g . 5-4. Conducting C y l i n d e r and two Line Charges. Consider P i g . 5-4. Here the two l i n e charges have both been d i s p l a c e d from the h o r i z o n t a l a x i s by a constant angle 8. The p o t e n t i a l at point P i s VP • k ln 2 2 s + r - 2r s cos(a-B) tt tt 2 2 s + r + 2r s cos(a+8) CL Cl 1 E l n r l d S (5-55) 46 Since the form of t h i s equation i s equivalent to (5-41) the s o l u t i o n i s w r i t t e n without f u r t h e r explanation. Therefore the equation determining the complex F o u r i e r c o e f f i c i e n t s of h^ i n t h i s case becomes (See Equation (5-48)) + eo co A . 0 0 . ? A n e J Q a + I - -H e * n a + £ - ±=S e " J n a = i— n V n V n - oo 1 1 CO . CO a j - l^a^n eJn(a-B) _ a j - l^a^n e-jn (a-0) 2 ^ xx s 2 ^ xx s + a Y_ (-l)n+1 I ( 2 : ) n e0h( a +P) + £ £ i ( - ) n e ~ J n ^ a + B ^ (5-56) 2 ^ u s 2 ^ ^ or w r i t i n g the nth equation A _ SL i (*)n r e-J nP _ J N $ ) A n n ~ 2 n V ^ E E ; or s o l v i n g f o r A n = ( f ) n s i n nP ( 5- 5 7 ) Thus i s not f i n i t e , and f u r t h e r i t i s noted that e""J i s not orthogonal to the r i g h t hand side of Equation (5-56). Hence t h i s problem has no s o l u t i o n unless 8 = 0 when F i g . 5-4 i s i d e n t i c a l to F i g . 5-3. 5-r4. Numerical Computations The c o n f i g u r a t i o n s shown i n F i g s . 5-2 and 5-3 were i n -v e s t i g a t e d w i t h a view towards a three-phase system w i t h f l a t spacing. Hence i t i s only p o s s i b l e to s p e c i f y an optimum shape f o r the centre phase conductor/ In order to v e r i f y t h a t Equation (5-54) does i n f a c t give the c o r r e c t shape f o r that case the surface charge de n s i t y was computed numerically f o r the c y l i n d e r shown i n F i g . 5-3. The equation used f o r 47 t h i s purpose can be derived as f o l l o w s . Tangent F i g . 5-5 Conducting Cylinder and two Line Charges R e f e r r i n g to F i g . 5-5 the p o t e n t i a l at poi n t P i s VP « k l n r 3 + k l n r 2 - / 4^ l n r i d S s Hence the e l e c t r i c f i e l d i n t e n s i t y on the surface i n the r f f l d i r e c t i o n i s given by r - s cos a r - s cos a i _ _ £_ _a 2 k „ 2 2 k 2 + r 3 r 2 r r ~ - TR c o s ( a - 6 ) \ 4^ -—— ds <5-58> J s r l and i n the a d i r e c t i o n E = -a , r a^a q s s i n a q s s i n 6 , = 2 k ~~2 2k" 2 + 1 r 3 . . r 2 ^ L _ a _ d s ( 5 _ 5 9 ) S r l 48 Let the angle 8 be such th a t . i n §g „ . . . . „ . . £ then the normal f i e l d i n t e n s i t y , E n, at point P on the surface i s E = E s i n 8 - E cos 8 n r a a But the surface charge d e n s i t y , a(6) = 2e Q E n, the f a c t o r , 2, occurring because E^ i s on the surface i t s e l f ; t herefore sub-s t i t u t i n g f o r E and E from (5-58) and (5-59) r & a r s i n 8 + s sin(a+8) 9 r s i n 8 - s s i n (B+a) / v 2q a v v 2q a 0 ( a ) = _ § 4 i ~ 2 r 3 r2 „ r v s i n 8 - r R sin(a-6 + 8) + f f e ( ~ " ^2 ~ d S ( 5- 6 0' Js r l where o(6) has been taken out from under the i n t e g r a l sign be-cause i t i s assumed constant f o r the purpose of numerical com-putations . I t may be noted that f o r a c i r c u l a r c y l i n d e r of radius a Equation (5-60) reduces to 4 4 s - a •x -x 2 L s"* + a"* + 2a s" cos 26 Comparison of Equations (4-15) and (5-61) shows that the same expression i s obtained f o r , o, i n both cases. However, t h i s i s of course not true i n general, since (5-60) a c t u a l l y con-s t i t u t e s an i n t e g r a l equation i n o. The i n t e g r a l i n (5-60) was evaluated by a i d of Legendre Gauss Quadrature. This numerical method of I n t e g r a t i o n i s based on the equation ° ° f f c 4 , , 4 3 , Z.ii <6-61> 49 b (5-62) where 27 m number of points considered i n the i n t e r v a l (b,c). a. l predetermined c o e f f i c i e n t s . predetermined value of the independent v a r i a b l e . R •m(f) E r r o r i n c u r r e d by approximating an i n t e g r a l by (5-61). The i n t e r v a l (0, 2TC) was devided i n t o ten parts and equation (5-61), with m s 12, was then a p p l i e d to each part i n t u r n . A d d i t i o n of the r e s u l t s thus gave the t o t a l value of the i n t e -g r a l i n (5-60). The numerical example chosen here i s a = 0.50 cm s = 10 cm, thus from Equation (5-54) r =s 0.50 - (.0025 cos 2a + .00000208 cos 4a + ...) Thus r e t a i n i n g only the f i r s t two terms r f l i s determined to at l e a s t f i v e s i g n i f i c a n t f i g u r e s and t h i s was considered s u f f i -c i e n t f o r the purpose at hand. Therefore Equation (5-60) was evaluated by a i d of the ALWAC I I I - E d i g i t a l computer w i t h r a = 0.50 - .0025 cos 2a. F i g . 5-5 shows c(a) as a f u n c t i o n of the angle a both f o r the deformed c y l i n d e r and the equivalent c i r c u l a r c y l i n d e r . I t f o l l o w s that Equation (5-54) does give the c o r r e c t shape f o r the centre conductor. I t was thought of i n t e r e s t to make an a n a l y t i c a l compari-son of the contours given by Equations (5-54) and (4-11). The p o l a r equation of an e l l i p s e i s given by the w e l l known formula .02015 a bo u cd .01985 .01980 I . . . . . . . . — 0 10 20 30 40 50 60 70 80 90 6 i n degrees P i g . 5-6 Charge Density V a r i a t i o n 1 5 0 b (1 - e 2 ) 2 r = ' 1 ( l - e^ 2 s i n 2 6 ) 2 1 , /, 2x2/, , 1 2 . 2 C 3 4 . 4 , . x = b (1 - e 1 ) (1 f •g- s i n & + Q ei s i n o •••) where b = major a x i s e^ = e c c e n t r i c i t y or i n s p e c t i o n of the s e r i e s above i n d i c a t e s C O r = A + V B cos 2n6 o j~ n where'A and B^ are constants. Although i t does not appear p o s s i b l e to evaluate these constants i t i s seen that the form of (5-62) i s equivalent to tha t of (5-54). For comparison l e t the conductor shown i n F i g . 5-3 be an e l l i p t i c c y l i n d e r w i t h minor a x i s at 6 = 0. The numerical example chosen here i s the same as above, namely a = 0.50 cm 2 s = 10 cm and d = .00125. From Equation (4-11) i t i s seen 2 that t h i s p a r t i c u l a r value of d r e s u l t s i n a minor a x i s of .4975 cm. Hence using Equation (4-17) the charge de n s i t y on the e l l i p t i c c y l i n d e r was computed. This computation i n d i -cated that the charge de n s i t y on the e l l i p t i c conductor agrees wi t h that obtained by using Equation (5-54) to w i t h i n f i v e s i g n i f i c a n t f i g u r e s . This concludes the work done i n t h i s t h e s i s on determin-ing the optimum shape f o r the conductors of a three-phase system. From the method employed here i t appears t h a t the optimum shapes i n nea r l y a l l cases cannot be determined d i -r e c t l y . However, i t i s seen that some improvement can be 51 obtained by using the conductor shapes considered i n the l a s t two chapters. On the other hand i t i s noted the required change i n shape i s very small f o r a l l systems w i t h large spacing between phase conductors. 52 6. BUNDLED CONDUCTORS Using s e v e r a l conductors f o r each phase, that i s , bundled conductors i n s t e a d of one conductor f o r each phase o f f e r s some d i s t i n c t advantages, such as decreased i n d u c t i v e reactance, i n -creased c a p a c i t i v e reactance and a lower voltage gradient at the surface of the transmission l i n e conductors. These f e a -tures become qu i t e important as transmission voltages and distances increase, and s i n c e , as mentioned e a r l i e r , t h i s i s the trend i n e l e c t r i c power transmission i t f o l l o w s that the 6 9 12 35 c h a r a c t e r i s t i c s of such systems have been i n v e s t i g a t e d . ' ' ' Prom the point of view of minimizing corona on a t r a n s -mission l i n e the voltage gradient i s the more important f a c t o r to be considered. Several papers have been published r e c e n t l y which deal w i t h the determination of voltage gradients on three-phase systems using bundled conductors. For most cases the c o n f i g u r a t i o n i n v e s t i g a t e d i s t h a t of f l a t spacing, and i t f o l l o w s that the centre phase conductors then i n h e r e n t l y have the l a r g e r charge per u n i t length as was a l s o found to be the case when only one conductor per phase was used. See s e c t i o n 3-2. For t h i s reason the voltage gradient on the centre phase conductor i s g e n e r a l l y regarded the l i m i t i n g f a c t o r f o r such a transmission l i n e system. 8' 3^ Also the charges on the i n d i -v i d u a l conductors of the centre phase are g e n e r a l l y found to be very n e a r l y equal provided these conductors are arranged symmetrically with respect to the outer phases. 34 As has been pointed out i t would be of advantage to 53 have a l l p arts of a transmission l i n e system stressed e l e c t r i -c a l l y to the same degree. In order to achieve t h i s o b j e c t i v e i t has been suggested that l a r g e r s i z e d conductors be used f o r the centre phase than f o r the outer phases. As a matter of f a c t t h i s procedure has been used i n one inst a n c e . See d i s -cussion of Ref. 34. For a transmission l i n e using two con-ductors per phase, one conductor placed d i r e c t l y above the other, the charges per u n i t length are equal on these two con-34 ductors to w i t h i n .lfo. Hence the maximum gradients w i l l a l s o be approximately equal, and one can thus s p e c i f y a l l con-ductors to be of the same s i z e f o r the outer phases. How-ever, when three or four conductor bundles are used the con-ductors i n the outer phases must be given a s p e c i f i c c o n f i g u -r a t i o n i n order to obtain equal maximum voltage gradient on a l l conductors of one phase. The object of t h i s chapter i s to determine t h i s c o n f i g u r a t i o n f o r three and four conductor bundles. 6-1. Transmission Line with three Conductors per Phase The three-phase system t r e a t e d i n chapter 3 was ungrounded, therefore i t was per m i s s i b l e to use Equation (3-1) to determine the charge per u n i t length of each phase conductor. However, the systems t r e a t e d i n t h i s chapter are assumed to be grounded and a d i f f e r e n t approach i s then required i n tha t Equation (3-1) i s i n v a l i d f o r l i n e s w i t h grounded n e u t r a l . On the other hand, f o r the purpose of computing p o t e n t i a l s i t w i l l again be assumed that n e g l i g i b l e e r r o r r e s u l t s from assuming the charge 5 4 on each conductor to be concentrated at i t s centre. I t was shown i n s e c t i o n (2^-2) t h a t two equal and opposite l i n e charges placed a c e r t a i n distance apart w i l l together produce a plane of zero p o t e n t i a l midway between the two l i n e charges. Hence i f the surface of the ground i s regarded a p e r f e c t l y conducting plane at zero p o t e n t i a l i t f o l l o w s that an overhead transmission l i n e and ground may be replaced by an equivalent system c o n s i s t i n g of the transmission l i n e and i t s image. Thus the e f f e c t of ground on the e l e c t r i c f i e l d c o n f i g u r a t i o n around the transmission l i n e conductors i s taken i n t o account. I t w i l l be noted that t h i s e f f e c t was neglected i n the system t r e a t e d i n chapter 3. F i g . 6-1 shows a three-phase transmission l i n e w ith f l a t spacing and three conductors per phase. Also the image con-ductors are included, and these are p o s i t i o n e d a distance, H, below the ground plane equivalent to the height of the a c t u a l conductors above ground. Let the phases be denoted A, B and C and the i n d i v i d u a l conductors 1, 2, 3 etc. as i n d i c a t e d . Then the equation determining the p o t e n t i a l of the nth con-ductor can be w r i t t e n m = 9 q_ s • V = Y l n -SS n V k s m = 1 nm (6-1) where q^ = charge i n coul./m on conductor m (s' ) / = distance between conductor n and the image of conductor m :m distance between conductor n and conductor m Phase A Phase B Phase C P i g . 6-1 Three-phase Transmission Line 55 (s* ) = distance bet-ween conductor n and i t s own image nm n=m *» (s ) = radius of conductor n v nm n=m k = 2ite . o I f one then l e t s n take on the values 1 to 9 i n succession i t fo l l o w s that nine simultaneous equations are obtained. P u t t i these equations i n matrix form P l l P12 P 1 3 P14 P 1 5 P16 P17 P18 P19 P21 P22 P 2 3 P24 P 2 5 P26 P27 P28 P29 _P91 P92 P 9 3 P94 P 9 5 P96 P97 P98 P99 *1 *2 *9 (6-2) where q s' (P ) / = In -ss nm (P ) * nm n=m q_ s 1 3 In -SB k (6-3) (6-4) I t f o l l o w s that P = P . nm mn The c o e f f i c i e n t s P_ were introduced by Maxwell and are there-nm " 13 37 39 fore termed Maxwell's p o t e n t i a l c o e f f i c i e n t s U»°'»'J . The charges q^ to q^ can be determined by i n v e r s i o n of Equation (6-2) so th a t 56 *1 ^2 C3 C, c. c, c. c. c, c. C3 c. 3d where J V 2 0 e 0 o © )j (6-5) (c ) = ( - i ) 2 n ^ nm n=m x ' A M (6-6) (6-7) and M , NI and A denote minors and determinant r e s p e c t i v e l y nn nm t-1 r J i n Equation (6-2). The c o e f f i c i e n t s (C ) and (C ) / ^ v nm n=m v nm'n^m are termed Maxwells c o e f f i c i e n t s of s e l f - i n d u c t i o n and mutual i n d u c t i o n r e s p e c t i v e l y . Further (C ) i s the d i r e c t capaci. nm n=m c tance between conductor n and ground while -(C ) / i s the nm n^ m 13 37 d i r e c t capacitance between conductors n and m Consider phase A i n F i g . 6-1. Conductors 1, 2 and 3 are placed e q u i l a t e r a l l y and are a l l kept at the same p o t e n t i a l above ground, namely the phase voltage V^. Then i f the d i s -•france H i s s u f f i c i e n t l y l a r g e the e f f e c t due to ground w i l l be very s l i g h t and the d i f f e r e n c e i n the magnitude of the charges q.^ , ^2 a n d ^3 d u e *° ^ n e P r e s e n c e °^ ground w i l l be quite s m a l l . However, i t i s evident that the d i r e c t capacitances to the conductors of the other phases w i l l be l a r g e r f o r 57 conductor 3 than f o r 1 and 2. Hence conductor 3 w i l l neces-s a r i l y c a r r y a l a r g e r charge per u n i t length than 1 and 2. I t f o l l o w s t h a t an i d e n t i c a l argument holds f o r conductor 9 i n phase C. From the c o n f i g u r a t i o n shown i n F i g . 6-1 i t seems reason-able to expect that the charge on conductors 3 and 9 w i l l de-crease i f the distances s^g, s23» s98 a n d s97 a r e decreased s l i g h t l y . To i n v e s t i g a t e t h i s a programme was w r i t t e n f o r the ALWAC I I I - E d i g i t a l computer. The p o t e n t i a l c o e f f i c i e n t s i n Equation (6-2) were then a l l evaluated and arranged i n matrix form by a i d of t h i s programme, and a standard subroutine f o r matrix i n v e r s i o n was u t i l i z e d to compute the c a p a c i t i v e c o e f f i c i e n t s i n Equation (6-5). Further, the phase voltages were given the values V^ = 1 cos cot V f i = 1 cos(wt - 120) V c = 1 cos (art - 240) so that on determining q^ to qg from Equation (6-5) the i n -stantaneous values of the charge i n coul./m-volt were obtained. Refer to F i g . 6-1. The general procedure followed i n o p t i m i z i n g the distances s^g, Sgg, Sgg and Sg^ f o r d i f f e r e n t phase spacings, D, and bundle s p a c i n g s , i c , was as f o l l o w s : i ) The conductor diameter, d, was chosen i i ) The phase spacing, D, was chosen i i i ) The bundle spacing, c, was chosen i v ) The height above ground, H, was chosen 1 1 v) The distances b = s 2 g - ( | ) 2 2 and b = 2 ,ds2 s98 ~ v 2 ; 58 were decreased simultaneously u n t i l the charge q_ was smaller than q^ and q 2 . The r e s u l t of such a s e r i e s of computations appears i n F i g . 6-2. In t h i s p a r t i c u l a r case d = 1 i n c h D = 20.0 f t . c = 1.50 f t . H = 40 f t . and i t w i l l be noted that a l i n e a r v a r i a t i o n with the para-meter b i s obtained. This was found to be true i n p r a c t i c a l l y a l l cases f o r the range of b i n v e s t i g a t e d here. Also the values i n d i c a t e d f o r q^, q 2 and q^ are the magnitudes of these charges. The phase angle between them was found to be l e s s than one degree and t h i s was consequently neglected. At the same time the charges were found to l a g the phase voltage by approximately s i x degrees. In order to determine the v a r i a t i o n of the optimum value of b with phase spacing, D, and bundle spacing, c, a s e r i e s of computations was c a r r i e d out. The numerical values chosen f o r t h i s purpose were as f o l l o w s : d D c H 1.0 i n c h 20.0 f t . 1.25 f t . 40.0 f t . 1.0 " 20.0 it 1.50 it 40.0 II 1.0 " 25.0 •i 1.25 it 40.0 it 1.0 " 25.0 n 1.50 II 40.0 it 1.0 " 30.5 it 1.25 •i 40.0 it 1.0 " 30.5 it 1.50 it 40.0 it Thus a curve s i m i l a r to the one shown i n F i g . 6-2 was obtained f o r each set of values l i s t e d above. .088 0) a CM a o o ce o .086 ,084 ,082 ,080 .90 1.00 1.10 1.20 1.30 1.40 1.50 b i n f t . P i g . 6-2 Charge per Unit Length versus the Parameter b CD CM ~ 2.40 - p o > I •p \ (0 • f > r H O J> fl •rl >> -P •rl « a CO +» M 2.00 *d i—i a> •H 2.30 2.20 2.10 3 max .90 1.00 1.10 1.20 1.30 1.40 1.50 b i n f t . P i g . 6-3 F i e l d I n t e n s i t y versus the Parameter b 59 The optimum value of b i s defined here as that value at which the maximum e l e c t r i c f i e l d i n t e n s i t i e s on conductors 1, 2 and 3 are approximately equal. Hence having obtained the v a r i a t i o n of the charges with the parameter b as o u t l i n e d above one can compute the e l e c t r i c f i e l d on each conductor f o r a s e r i e s of values of b and thus determine the optimum p o s i t i o n f o r conductor 3. For the purpose of computing the e l e c t r i c f i e l d i n t e n s i t y on conductors 1, 2 and 3 the f o l l o w i n g assumptions are made, i ) The spacings s^g, s 2 g and s^ 2 are s u f f i c i e n t l y large that the two conductors adjacent to the one considered can be represented as l i n e charges. See Appendix I . i i ) The height H above ground i s such that the image charges have no e f f e c t on the surface charge d i s t r i b u t i o n on any of the conductors, i i i ) The phases B and C are s u f f i c i e n t l y d i s t a n t that each can be represented as a l i n e charge placed at the geometric centre of each phase. F i g . 6-4 shows three conducting c y l i n d e r s placed p a r a l l e l to two l i n e charges. This system i s equivalent to the three-phase transmi s s i o n l i n e i n F i g . 6-1 when one introduces the s i m p l i f y i n g assumptions o u t l i n e d above. The charge per u n i t l e n g t h of the l i n e charges i s -q^ coul./m and -q^ coul./m r e -s p e c t i v e l y , and these are equ!al i n magnitude to the t o t a l charge on phases B and C. Hence from Equations (2-9) and ( I - l ) the e l e c t r i c f i e l d i n t e n s i t i e s E^, E 2 and Eg at the sur-face of conductors 1, 2 and 3 r e s p e c i t v e l y are E 60 q 2q 0 0 2q 0 0 0 f i g -I ( — ) n cos n(a+90-6 9) + 1 SB1 2qn co X cx * 2 • E " I <^>"— » < - « » " E 2 £ (4>n •» "<«-l>i> + 2q C O •B £ (-£-) n cos n(a-90+B 9) + a X SB2 2q 0 0 __C J" ( _ a _ } n c o g n ( a _ 9 0 + p 3 ) ( 6 _ 1 0 ) X C2 E 3 = ^ q 2q 0 0 E " E 1 ' I c ° s n ( 6 - 1 8 0 + P l ) X3 2q 0 co 2q„ co tr- 2- I ( ^ ) n cos 11(6-180-8, ) + I (-^-J 1 1 cos n6 + ka x s 2 3 1 K.a 1 s f i 3 2qn co E T I (^-)n cos n6 .(6-11) K a x s C 3 where k = 2TteQ sBn a n d SCn r e P r e s e n " k * n e distances from l i n e charges q^ and q^ to the centres of conductors 1, 2 and 3 with n = 1, 2 or 3. At t h i s p o i n t some f u r t h e r approximations are made. Since the i n s t a n t of maximum charge on phase A i s considered here the charges q^ and q^ are approximately equal to - ^ q 3 coul./m. Also the f i r s t order terms i n • are such that sBn — — « —2L- where m and n = 1 9 2 or 3 and m ^ n. sBn smn Phase A Phase B Phase C P i g . 6-4 Three Conducting Cylinders and two Line Charges 61 Hence since i t i s only d e s i r e d to obtain a f i r s t order approxi-mation to the e l e c t r i c f i e l d i n t e n s i t y the l a s t two terms can be neglected i n each of the Equations (6-8), (6-9) and (6-10). Therefore r e t a i n i n g only the f i r s t terms i n each of the remain-ing s e r i e s the f o l l o w i n g expressions are obtained E, = 1 ka 2 q 3 2q ^ — cos(a+8 1) - cos (a+90) '31 '21 q.o 2q, 2q E 2 = ka- " k i ^ cos(a-90) - ^ 2 - C o s ( a - P l ) E 3 ~ ka 2q, 2q 0 - cos(6-180+p 1) - 008(6-180-^) 13 23 i (6-12) (6-13) (6-14) I t w i l l be noted that Equation (6-10) a c t u a l l y represents the e l e c t r i c f i e l d i n t e n s i t y on a conducting c y l i n d e r with charge q^ coul./m placed i n two uniform e l e c t r i c f i e l d s of i n t e n s i t y ks and ks. volts/m r e s p e c t i v e l y . I t fol l o w s that a s i m i l a r 31 ""21 statement can be made about Equations (6-11) and (6-12). The angles at which the maximum f i e l d i n t e n s i t y occurs on conductors 1, 2 and 3 can now be determined by d i f f e r e n t i a t i o n . Let these angles be denoted a m ^ , a n < l 6 mg r e s p e c t i v e l y , then from the equations a* - ° i t f o l l o w s 5cT and = 0 a m l = a r c * a n = arctan '31 ' <1 / 2 V S 3 1 s 2 i s21 x2 4 ' S 2 l \ 2 q 3 " 1  S21 2 s 2 s 32 s 2 i / 2 s12x2 ^ ( s 3 2 ~ "4" ) S12 (6-15) (6-16) 62 6 m3 = arctan 32 s 12 s (6-17) Therefore s u b s t i t u t i n g a m l , a m 2 and 5 m 3 i n Equations (6-10) (6-11) and (6-12) r e s p e c t i v e l y one can compute the maximum f i e l d i n t e n s i t y on each conductor., The r e s u l t of such a se-r i e s of computati ons i s shown i n F i g . 6—3. The curves shown there are based on the values of b, q^, q 2 a n d <lg i n d i c a t e d i n F i g . 6-2. Also comparison of F i g s . 6-2 and 6-3 i n d i c a t e s that the point at which q_2 = q^ gives very nearly the optimum value of the parameter b. The assumptions made i n d e r i v i n g Equations (6-12), (6-13) and (6-14) introduced e r r o r s i n the e l e c t r i c f i e l d i n t e n s i t i e s computed. To i n v e s t i g a t e the magnitude of these errors con-s i d e r a system w i t h phase spacing of 20 f t . and bundle spacing of 15 inches. The the f i r s t order terms due to the presence 3 1 of phase B=^- q x = .075q and the second order terms due to o the presence of the adjacent bundle conductors = q x y^=.03q. In t h i s p a r t i c u l a r case q = .08 hence the e r r o r = ,01 = T^& since E = 2.0. max The procedure o u t l i n e d above f o r computing the maximum e l e c t r i c f i e l d was then used to compute E, , E 0 and * 1 max9 2 max Eo m o v f o r each set of numerical values given by Equation (6-8). Thus a set of curves was obtained each being s i m i l a r i n form to F i g . (6-3). Hence, by a i d of these curves the optimum value of b could be p l o t t e d as a f u n c t i o n of phase spacing, D, w i t h bundle spacing, c, as a parameter. See 1.40 1.30 1.20 f 1.10 1.00 .90 c = 1.50 f t . :] c = 1.25 f t . .80 15.0 20.0 25.0 30.0 Phase Spacing i n f t . 35.0 F i g . 6-5 V a r i a t i o n of b .j. with Phase and Bundle Spacing 63 P i g . 6-5. The point at which the charge per u n i t length of conductor 3, q^, i s equal to the charge per u n i t length of con-ductor 2, q 2 , i s included f o r comparative purposes, i t i s e v i -dent that f o r p r a c t i c a l purposes one can assume E„ = o max E 2 max w h e n *2 = V 6-2. Transmission Line w i t h four Conductors per Phase Consider the three-phase transmission l i n e shown i n P i g . 6-6. In t h i s case four conductors are used f o r each phase and the object i s to determine the optimum p o s i t i o n f o r the conductors i n the outer phases. For t h i s purpose the charge per u n i t length on each conductor must be determined, and i t f o l l o w s that the procedure o u t l i n e d i n d e t a i l i n s e c t i o n 6-1 can be used here as w e l l . Therefore the p o t e n t i a l of the nth conductor i s m=12 q s' \~ I r " r <6-18> m=l nm where a l l symbols have the same meaning as i n Equation (6-1). In t h i s case n = 1 to 12 so that twelve simultaneous equa-t i o n s are obtained. I n v e r t i n g these equations one f i n d s the charge per u n i t length on the nth conductor m=12 q„ = I C Y (6-19) m=1 nm m This corresponds to one of the Equations given by (6-5) i n the previous s e c t i o n . Considering the capacitance associated w i t h each con-ductor i n phase A, P i g . 6-6, i t i s evident that the charge Phase A Phase B Phase C ffl P i g . 6-6 Three-phase Transmission Line 64 per u n i t length of conductors 2 and 4 w i l l be greater than that of 1 and 3 f o r the c o n f i g u r a t i o n shown. S i m i l a r l y the charge per u n i t length of conductors 9 and 12 i s greater than that on 10 and 11 i n phase C. However,, i n t h i s case i t i s not r e a d i l y seen how the c o n f i g u r a t i o n should be changed i n order to equalize the charges on the conductors i n the outer phases. Therefore Equations (6-18) and (6-19) were evaluated by a i d of the ALWAC I I I - E d i g i t a l computer. The programme used f o r t h i s purpose was w r i t t e n such that the c o n f i g u r a t i o n of the outer phases could be v a r i e d to some extent from that shown i n P i g . 6-6. Using t h i s procedure i t was found that decreasing the distance Sg^ symmetrically, that i s 9 decreasing s£ 2 a n d i n -creasing s ^ by the same amount had the d e s i r e d e f f e c t . I t fo l l o w s that the distance Sg ^ w a s decreased simultaneously. The e f f e c t obtained by i n t r o d u c i n g t h i s v a r i a t i o n i n the con-f i g u r a t i o n of the outer phases i s shown i n F i g . 6-7. I t w i l l be noted that the v a r i a t i o n obtained i n the charges are l i n e a r f o r the range of s 2 ^ i n v e s t i g a t e d . In order to obtain the v a r i a t i o n of the optimum value of s24 w:"-*n P n a s e spacing a s e r i e s of computations were c a r r i e d out. The numerical values chosen f o r t h i s purpose were the f o l l o w i n g . See F i g . 6-6, d D c H 1.0 i n c h 25.0 f t . 1.50 f t . 1.50 f t . 40,0 f t . 1.0 " 30.0 " 1.50 " 1.50 " 40.0 " 1.0 " 35.0 1.50 " 1.50 " 40.0 " 3 o o a • r t 0) b0 U .069 .067 ,065 .063 .061 1.10 1.20 1.30 1.40 1.50 1.60 Spacing s 2 4 i n f t . P i g . 6-7 Charge per u n i t Length versus Bundle Spacing a a • r l -P P< O CM (0 1.50 1.40 I 1.30 I 1.20 1.10 I* c = 1.50 f t , 22.5 25 30 Phase Spacing i n f t . 35 F i g . 6-8 V a r i a t i o n of S g ^ ^, with Phase Spacing 65 The r e s u l t obtained from these computations i s shown i n F i g . 6-8. In t h i s case the optimum c o n f i g u r a t i o n i s based on equal charge on the conductors of the outer phases rather than on: equal .maximum' e l e c t r i c f i e l d i n t e n s i t i e s . However, as shown i n sjactinn 6-1 both c r i t e r i a give approximately the same answer. 6-3. Some General Considerations I t was mentioned i n the i n t r o d u c t i o n to t h i s chapter that the centre phase conductors have a l a r g e r charge per u n i t length than those of the outer phases. This f a c t was v e r i f i e d i n the computations c a r r i e d out here. Thus f o r the system w i t h three conductor bundles the charge was found to be 8 - 10$ greater on the centre phase than on the outer phases. Fur-ther i n d i s p l a c i n g conductors 3 and 9 (See F i g . 6-1) i n the manner i n d i c a t e d i n s e c t i o n 6-1 the charge on each centre phase conductor was no t i c e d to decrease by approximately For the system using four conductor bundles the d i f f e r e n c e i n charge on the centre and outer phases was about 7$. However, d i s p l a c i n g conductors 2, 4, 9 and 12 (See F i g . 6-6) as de-sc r i b e d i n s e c t i o n 6-2 had p r a c t i c a l l y no e f f e c t on the charge on the centre phase conductors. When the p o s i t i o n of con-ductor 3 i n F i g . 6-1 i s changed by decreasing s ^ g and S g g symmetrically and keeping the phase spacing, D, constant the f l u x l i n k i n g phase A w i l l i n c r ease. However, as pointed out above the charge on the centre phase conductors decreases by using t h i s procedure. I f ins t e a d one decreased the phase 66 spacing, D, simultaneously w i t h s ^ 3 and s 2 3 , thus keeping con-ductor 3 f i x e d i n p o s i t i o n , but d i s p l a c i n g 1 and 2 i t f o l l o w s the i n d u c t i v e reactance of phase A decreases, while the charge on the centre phase conductors increases. Hence the pro-cedure that should by used w i l l depend upon which e f f e c t i s the more important i n a p r a c t i c a l case. I t i s evident that the c a p a c i t i v e reactance i s a l s o changed by a l t e r i n g the bundle c o n f i g u r a t i o n , however, t h i s e f f e c t i s considered to be n e g l i -g i b l e here since the a c t u a l change i n bundle spacing i s q u i t e s m a l l . This concludes the work done on bundled conductors i n t h i s t h e s i s . The methods used to determine the optimum con-f i g u r a t i o n f o r the outer phase bundles i n a three-phase system wi t h f l a t spacing are of course not general. However, to de-termine the optimum c o n f i g u r a t i o n a n a l y t i c a l l y d i d not appear p o s s i b l e due to the complexity of the problem. 7. CONCLUSIONS 67 The i n v e s t i g a t i o n c a r r i e d o ut i n t h i s t h e s i s r e g a r d i n g t h e optimum shape f o r t r a n s m i s s i o n l i n e c o n d u c t o r s i n d i c a t e s t h a t a d i r e c t a n a l y t i c a l s o l u t i o n apparently does n o t e x i s t i n most c a s e s . However, some improvement i n t h e e l e c t r i c f i e l d c o n f i g u r a t i o n c a n be o b t a i n e d i n t h e f o l l o w i n g way f o r a t h r e e -p hase s y s t e m w i t h f l a t s p a c i n g . An e x a c t s o l u t i o n was ob-t a i n e d f o r t h e c e n t r e c o n d u c t o r i n s e c t i o n 5-3. Hence u s i n g t h a t c o n d u c t o r f o r t h e c e n t r e phase and t h e d e f o r m e d c o n d u c t o r t r e a t e d i n s e c t i o n 4-3 f o r t h e o u t e r p h a s e s t h e f i e l d i n t e n s i t y on t h e o u t e r c o n d u c t o r s i s d e c r e a s e d w h i l e i t i s made u n i f o r m on t h e c e n t r e c o n d u c t o r . S u c h an a r r a n g e m e n t p r o v i d e s t h e b e s t s o l u t i o n o b t a i n e d h e r e . A c o n d u c t o r o f a d i f f e r e n t shape t h a n t h a t c o n s i d e r e d i n s e c t i o n 4-3 m i g h t w e l l p r o v i d e a b e t t e r s o -l u t i o n , however, i f t h e c o n t o u r s c o n s i d e r e d c a n n o t be s u b -j e c t e d t o c o n f o r m a l mapping an a n a l y t i c a l a p p r o a c h i s v e r y cum-bersome. The r e s u l t s o b t a i n e d f r o m t h e c o m p u t a t i o n s c a r r i e d o u t f o r t r a n s m i s s i o n l i n e s w i t h b u n d l e d c o n d u c t o r s i n d i c a t e t h a t t h e maximum e l e c t r i c f i e l d i n t e n s i t y on t h e o u t e r phase c o n d u c t o r s c a n be e q u a l i z e d by s m a l l changes i n t h e b u n d l e c o n f i g u r a t i o n n o r m a l l y u s e d . However, i n o r d e r t o make u s e o f t h i s f a c t i n p r a c t i c e , as o u t l i n e d i n t h e i n t r o d u c t i o n t o c h a p t e r 6, one woul d have t o d e v i c e a method f o r d e t e r m i n i n g t h e c o n d u c t o r d i a m e t e r a c t u a l l y r e q u i r e d . The r e a s o n b e i n g t h a t t h e c h a r g e p e r u n i t l e n g t h v a r i e s w i t h t h e c o n d u c t o r d i a m e t e r u s e d . 68 7-1. Recommendations f o r Future Work I t appears that using an experimental method would be the eas i e s t way to determine the optimum shape f o r a conductor placed p a r a l l e l to a l i n e charge. An e l e c t r o l y t i c tank would probably provide the best means f o r t h i s purpose since one then would be able to determine the f i e l d v a r i a t i o n d i r e c t l y from the p o t e n t i a l contours obtained. However, i n that case some p l i a b l e , conducting m a t e r i a l would be required f o r the conductors themselves. The change in c u r r e d i n i n d u c t i v e and c a p a c i t i v e reactance by a l t e r i n g the bundle c o n f i g u r a t i o n was not computed i n chap-t e r 6. However, even though these changes are sma l l , the change i n i n d u c t i v e reactance might require c o n s i d e r a t i o n i n a long t r a n s m i s s i o n l i n e . Further, as mentioned above, the conductor diameter a c t u a l l y required f o r the outer phases must be determined. The equation determining the gradient on each conductor could be used f o r t h i s purpose, however, one would at the same time r e q u i r e the charge on each conductor as a f u n c t i o n of i t s diameter. This procedure appears q u i t e cumbersome, but i t might be p o s s i b l e to introduce some s i m p l i f y i n g assumptions. 69 REFERENCES 1. Gloyer, H., and Vogelsang, T., "380 KV. Transmission Lines i n Western Germany", The I n t e r n a t i o n a l Conference  on Large E l e c t r i c Systems (C.I .G.R.-E.), V o l . I l l , No. 401, 1958, pp. 1-13. 2. Akopjan, A.G., Burgsdorv, V.V., Butke v i t c h , V.V., G e r t z i k , A.K., Gruntal, V.L., Rokotjan, S.S., and Sovalov, S.A., "The Development of 400-500 KV. Systems i n the Soviet Union", The I n t e r n a t i o n a l Conference on Large E l e c t r i c Systems (C.I.G.R.E.), V o l . I l l , No. 410, 1958, pp. 1-29. 3. von G e i j e r , G., Jancke, G., and Holmgreen, B., "Choice of I n s u l a t i o n Level i n Networks Containing Long High Voltage Power L i n e s " , The I n t e r n a t i o n a l Conference  on Large E l e c t r i c System's (C.I.G.R.E.), V o l . I l l , No. 414, 1958, pp. 1-19. 4. Lloyd, B.L., and Naeff, 0., "Corona Loss Measurements on Bundle Conductors - 500 KV. Test P r o j e c t of the American Gas and E l e c t r i c Company", The Transactions  of the American I n s t i t u t e of E l e c t r i c a l Engineers, V o l . 76, December 1957, pp. 1164-1172. 5. Bogdanova, N.B., G e r t z i k , A.K., Emeljanov, N.P., Kolpakova, A.I., Markovitch, I.M., Popkov, V.I., Sovalov, S.A., S l a v i n , G.A., "Some Results of Studies conducted i n the Soviet Union on Extra-Long Distance 600 KV. Transmission Systems", The I n t e r n a t i o n a l Conference  on Large E l e c t r i c Systems (C.I.G.R.E.), V o l . I l l , No. 411, 1958, pp. 1-18. 6. Cahen, F., "Results of Tests C a r r i e d out at the 500 KV. Experimental S t a t i o n at C h e v i l l y , France, e s p e c i a l l y on Corona Behavior of Bundle Conductors", Trans-actions of the American I n s t i t u t e of E l e c t r i c a l En-gineers, V o l . 67, Part I I . 1948, pp. 1118-27. 7. Peek J r . , F.W., D i e l e c t r i c Phenomenon i n High-Voltage En-gi n e e r i n g, New York, McGraw-Hill, 1929. 8. M i l l e r , J r . , C.J., "Mathematical P r e d i c t i o n of Radio and Corona C h a r a c t e r i s t i c s of Smooth, Bundles Con-ductors", Transactions of the American I n s t i t u t e  of E l e c t r i c a l Engineers, V o l . 75, Part I I I , 1956, pp. 1029-37. 9. Adams, G.E., "An A n a l y s i s of the Radio-Interference Cha-r a c t e r i s t i c s of Bundled Conductors", Transactions  of the American I n s t i t u t e of E l e c t r i c a l Engineers, V o l . 76, Part I I I , 1957, pp. 1569-85. 70 10. L i a o , T.W., and L a f o r e s t , J . J . , " R e l a t i o n s h i p between Corona and Radio Noise on Transmission Lines", Transactions of the American I n s t i t u t e of E l e c - t r i c a l Engineers, V o l . 78, October 1959. pp. 706-712. 11. L i a o , T.W., "Radio Influence Voltages caused by Surface Imperfections on S i n g l e and Bundle Conductors' 1^ Transactions of the American I n s t i t u t e of E l e c - t r i c a l Engineers, V o l . 78, December 1959. pp. 1038-46. 12. Temoshok, M., " R e l a t i v e Surface Voltage Gradients of Grouped Conductors", Transactions of the American  I n s t i t u t e of E l e c t r i c a l Engineers, V o l . 67, Part I I , 1948, pp. 1583-89. 13. Weber, E., Electromagnetic F i e l d s , V o l I . , New York, John Wiley and Sons Inc. 1957. 14. Peck, E.R., E l e c t r i c i t y and Magnetism, New York, McGraw-H i l l Book Company Inc., 1953. 15. Attwood, S.S., E l e c t r i c and Magnetic F i e l d s , New York, John Wiley and Sons, 1949. 16. Stewart, C.A., Advanced Calculus, London, Methuen and Co. L t d . , 1940. 17. Dwight, H.B. "The D i r e c t Method of C a l c u l a t i o n of Capa-citance of Conductors", Transactions of the Ameri-can I n s t i t u t e of E l e c t r i c a l Engineers, V o l . 43, 1924, pp. 1034-39. 18. Dwight, H.B., and S c h e i d l e r , F.E., "Capacitance and Sur-face Voltage Gradient of Transmission L i n e s " , Transactions of the American I n s t i t u t e of E l e c -t r i c a l Engineers, V o l . 71. Part I I I . 1952. pp. 563-566. 19. Dwight, H.B. E l e c t r i c a l Elements of Power Transmission L i n e s , New York, McMillan Company, 1953. 20. Batemann, H., P a r t i a l D i f f e r e n t i a l Equations, Cambridge, The U n i v e r s i t y Press, 1932. 21. Edwards, J . , I n t e g r a l C a l c u l u s , London, McMillan and Co., 1930. 22. C h r i s t i e , C.V., E l e c t r i c a l Engineering, New York, McGraw-H i l l Book Co., 1952. 23. C h u r c h i l l , R.V., I n t r o d u c t i o n to Complex V a r i a b l e s and  A p p l i c a t i o n s , McGraw-Hill Book Co.. 1948. 71 24. Copson, E.T., Theory of Functions of a Complex V a r i a b l e , Oxford, The Clarendon Press, 1955. 25. Rauscher, M., I n t r o d u c t i o n to Aeronautical Dynamics, New York, John Wiley and Sons, 1953. 26. B y e r l y , W.E., Elements of the I n t e g r a l C a l c u l u s, New l o r k , G.E. Stechart and Co., 1926. 27. Hildebrand, P.B., I n t r o d u c t i o n to Numerical A n a l y s i s , New York, McGraw-Hill Book Co., 1956. 28. Page, C. H., P h y s i c a l Mathematics, New York, D. Van Nostrand Co., 1955. 29. Margenau, H., and Murphy, G.M., The Mathematics of Phy-s i c s and Chemistry, New York, D. Van Nostrand Co., 1957. 30. Sehmeidler, W., Integralgleichungen mit Anwendungen i n Physik und Technik, L e i p z i g , Akademische Verlags-g e s e l l s c h a f t , 1955. 31. Neumann, E.R., Ueber die Konforme Abbildung Komplimentarer Gebiete, Mathematische Annalen, V o l . 116, 1938/ 1939, pp. 664-695. 32. M o r r i s , R.M., Two-Dimensional P o t e n t i a l Problems, Pro-ceedings of the Cambridge P h i l o s o p h i c a l Society, V o l . 33, 1937, pp. 474-484. 33. Wrinch, D.M., Some Problems of Two-Dimensional E l e c t r o -s t a t i c s , P h i l o s o p h i c a l Magazine, Ser. 6, V o l . 48, J u l y - Dec. 1924, pp. 692-703. 34. Dwight, H.B., Surface Voltage Gradient on Power Trans-m i s s i o n L i n e s , Transactions of the American I n - s t i t u t e of E l e c t r i c a l Engineers, V o l . 76, 1957, pp. 1217-1220. 35. Gross E.T.B., and Stensland, L.R., C h a r a c t e r i s t i c s of Twin Conductor Arrangements, Transactions of the  American I n s t i t u t e of E l e c t r i c a l Engineers, V o l . 77, Part I I I , 1958, pp. 721-725. 36. Reichman, J . , Bundled Conductor Voltage Gradient C a l c u l a -t i o n , Transactions of the American I n s t i t u t e of  E l e c t r i c a l Engineers, V o l . 78, 1959, pp. 598-607. 37. Clarke, E., C i r c u i t A n a l y s i s of A-C Power Systems, V o l . I , New York, John Wiley and Sons Inc., 1950. 72 38. Adams, G.E., Voltage Gradients on High-Voltage Trans-mission L i n e s , Transactions of the American I n - s t i t u t e of E l e c t r i c a l Engineers, V o l . 74, 1955, pp. 5-11. 39. Woods, F.S., Advanced Ca l c u l u s, New York, Ginn and Com-pany, 1934. 40. Hildebrand, F.B., Advanced Calculus f o r Engineers, New York, P r e n t i c e H a l l Inc., 1949. 41. Courant, R., and H i l b e r t , D«, Methods of Mathematical Physics, New York, Interscience P u b l i s h e r s Inc., 1953. APPENDIX I . THREE PARALLEL CONDUCTING CYLINDERS Equation (2-10) 73 17 ° 2na C O 1 + 2 Y. ( ~ ) n cos n6 coul./m 2 (2-10) 1 s r e f e r s to the case of a conducting c y l i n d e r w i t h charge -a, coul./m placed p a r a l l e l to a l i n e charge of strength +q coul./m. I f the charge on the conducting c y l i n d e r and the l i n e charge both are p o s i t i v e i t f o l l o w s that Equation (2-10) becomes a ~ 27ta CO 1 ~ 2 Y. ( f " ) n c o s nS coul./m ( I - l ) 1 s Refer to the i s o l a t e d system shown i n P i g . I - l . The three conducting c y l i n d e r s may be considered an ungrounded three—phase transmission l i n e w i t h charges and p o t e n t i a l s as i n d i c a t e d . The p o t e n t i a l s V^, Vg and Vg are s p e c i f i e d w i t h respect to the ungrounded n e u t r a l of the equivalent s t a r connection. I t i s desired to determine the surface charge d e n s i t y on each conductor, and f o r t h i s purpose an " i t e r a t i v e " method w i l l be employed as f o l l o w s . At the outset the three conductors are assumed to have uniform surface charge d e n s i -a l 4 2 q 3 / 2 t i e s °1 = 2 i a » °2 = 2^ a n d °3 = 2wa" c o u l ° / m r e s p e c t i v e l y . Then a f i r s t c o r r e c t i o n , 0-^2' ^a * o u n d * O T "^ne charge d e n s i t y of conductor 1 due to the presence of the uniform charge q 2 on conductor 2. S i m i l a r l y a f i r s t c o r r e c t i o n , o*^ g, i s found f o r conductor 1 due to the presence of the uniform charge on conductor 3. Thus the f i r s t s u b s c r i p t r e f e r s to 74 the conductor to which the c o r r e c t i o n i s to be a p p l i e d and the second s u b s c r i p t to the conductor g i v i n g r i s e to the correc-t i o n . I t f o l l o w s that one s i m i l a r l y must f i n d o"2^ , °23 a n d 0*31' egg• Then having determined the f i r s t c o r r e c t i o n one may f i n d the second c o r r e c t i o n > ^{3 e t c . which depend on the f i r s t corrections ov,^, 0 2g e* c« Consider a d i f f e r e n t i a l l i n e element of charge at point N on conductor 2. This element i s a l i n e charge of strength q 2 / / x a d-8 coul./m and from Equation ( I - l ) i t f o l l o w s dcNo = - ( f e r a dB 9) n =co 12 ~ ~ 2™ *aia a dP2> £ C 0 S n ( 6 - a»> But cos n a, d 5 ^ and s i n n a 0 _1 n 1 + k =00 I 1 n = 1n + k - 1 k ( | ) k cos k 6 2 (1-2) (1-3) k = co n , v n + k - 3| ( | )k s i n k 8 2 is. (1-4) d s k = 1 Equations (1-3) and (1-4) are then s u b s t i t u t e d i n Equation (1-2) above and the r e s u l t i n g expression i s i n t e g r a t e d w i t h respect to 8. C o l l e c t i n g terms one obtains <12 0 0 '12 1 NB' 0 0 s n6 I t f o l l o w s that a s i m i l a r procedure used f o r conductor 3 r e -s u l t s i n q „ 00 H 3 J13 28' then ' l - K - S t ' (f )n cos -6 - g I (fc>» cos n6 (1-5) CO oo 75 For conductors 2 and 3 one also f i n d s q2 q l *y /axn Q = ozr- - — L (-) cos nB, oo 2 ~ 2na na * j *s' o o L3 V" t&\Ti c — L (-) cos n6 C O a 3 "~ 2lta H l V" \ n o 2 V / a x n Q (1-6) (1-7) 1 1 From F i g . 1-1 ( T C - 6) = 8 9 or 6 = (it - 8 0 ) . Hence *1 C O q „ c o ^3 I (|)n cos n 8 2 - — I {=£) n cos n 8 2 Proceeding i n the same manner as before t ( f ) n d o i 2 = n a q 0 n s=co , s' c o s n^2 ~ tea V~ } C°S n P 2 n = 1 n = 1 m = ©2 2 _ ( f ) m cos m(6 - a 2) m = 1 adB, Tta q , n = o o u l v /a \n o X (g.) cos nB. n = 1 (1-8) q . 2 n = c o Tea n = 1 £ ( J ) n cos n8 3 m = c o I] (f) m cos m (6 - ct«J adB, 7ta (1-9) m = 1 S u b s t i t u t i n g Equations (1-3) and (1-4) i n (1-8) and (1-9), and i n t e g r a t i n g t h i s expression w i t h respect to 8 2 4i '12 where n a ^3 Tta A, cos 6 + A„ cos 26 + ... + A cos n6 + ... +! l & n B^ cos 6 + B 2 cos 26 + ... + B n cos n6 + ...+ In n=co k=co A = n (?)*" D|)n cos n8 2 ~ (* + * - D(f) k cos k B 2 dB 2 0 S' n = l ' s k=l a\n f(J) f + (f>2 r + ^(f)2 + ... + ( f ) k (n +kk - ^(f)" + B n = + ( | ) " ( - f <?>! + <!>'< tii + k - 1\ /a\k o o o ^ * a\2/n + 1\/a\2 2 + /-a\ k x( J ) (-) + ... + 76 and m has been replaced by n. S i m i l a r expressions may be de-r i v e d f o r o^g 0"^ o^g etc. w i t h i n c r e a s i n g l y complex ex-pressions r e s u l t i n g . I t i s noted that the lowest order terms i n (—) due to the s second c o r r e c t i o n i s of 3rd order, i . e . , (—) . For a prac-s t i c a l t r a n s m i s s i o n l i n e a = 1 inc h s = 30 f e e t hence (—)^ i s s completely n e g l i g i b l e . However, consider Equations (1-5), (1-6) and (1-7). These were derived on the assumption that the charge on conductors other than the one considered could be represented as l i n e charges at the centre of the c i r c u l a r c y l i n d e r s . Hence one then has a good estimate of the error i n c u r r e d i n the charge density by t r e a t i n g the adjacent con-ductors as l i n e charges. 77 APPENDIX I I . CONFORMAL MAPPING OP HARMONIC FUNCTIONS Let z = x + j y and w = u + j v represent two complex planes which w i l l be denoted the Z-plane and W-plane i n the d i s c u s s i o n f o l l o w i n g . Consider the f u n c t i o n w = f ( z ) . For a l l points df (z) where f (z) i s a n a l y t i c and ^ z ' £ 0 the f u n c t i o n f (z) i s s a i d to give a conformal mapping of those points from the Z-plane 23 onto the W-plane . Hence one can map a region of the Z-plane onto the W-plane conformally i f the mapping f u n c t i o n f ( z ) s a t i s f i e s the conditions s p e c i f i e d above. A f u n c t i o n which s a t i s f i e s Laplace equation i s termed a harmonic f u n c t i o n . Let V represent a p o t e n t i a l f u n c t i o n , then i n a charge f r e e region , d V n * x 2 + V Consider a transformation of t h i s equation to u and v co-ordinates where u = u(x,y ) , v = v(x,y) and w = u + j v . Then ..13 one can wrxte _ bV £u £V ^v £x ~ >>u c\x <yr a 2v _bv tt£+^L(^)2.$v d 2v , h 2X )2 . v 6 X2 + d u 2 W + dr dx2 + ^ 2 > t 1 1 " 1 ' S i m i l a r l y 5 ^ - ^ u + ^ u2 W j + V + d v2 V U I 2 ) A d d i t i o n of Equations ( I I - l ) and (II-2) y i e l d s A 2 + -\ 2 ~ V T T2 + A 2' u x °y ^u u v ( ^ ) 2 + (S*>2| (H-3 , ) Since the Cauchy Riemann equations sta t e 78 = and $H = -ox by by b* Hence Laplace equation i s i n v a r i e n t to conformal mappings, th a t i s , harmonic functions remain harmonic when subjected to con-formal transformations. However, i t i s c l e a r that one r e -quires the mapping f u n c t i o n to s a t i s f y the conditions l a i d down above since dw (|f)2 + <^>2 dz Any conducting surface kept at a p o t e n t i a l V i s an equi-p o t e n t i a l surface. Hence i f the conducting surface i s p o s i -t i o n e d i n the Z-plane i t f o l l o w s that V(x,y) = c (II-4) where c i s a constant. Under a conformal transformation from the Z-plane to the W-plane the co-ordinates x and y are represented as functions of u and v, therefore i n the W-plane V(x,y) = V x(u, v ) , y(u,v) j = c (H-5) Therefore the boundary c o n d i t i o n given by Equation (II-4) i n the Z-plane remains unchanged i n the W-plane. The e l e c t r i c f i e l d i n t e n s i t y i n the Z-plane i s given by Also i n the W-plane the e l e c t r i c f i e l d i n t e n s i t y i s given by E + j E „ - 51 - -j 2£ « - ( M ) * (H-7) u " J v bu J bv *n where Q i s a complex p o t e n t i a l f u n c t i o n with Q = V + jU and U being the harmonic conjugate to V„ Equation (II-7) may also be w r i t t e n 79 U " V Hence i t f o l l o w s E E dz dw (II-8) Equation (II-8) obviates the n e c e s s i t y of evaluating the poten-t i a l f u n c t i o n i n the W-plane i f one i s i n t e r e s t e d only i n the magnitude of the e l e c t r i c f i e l d i n t e n s i t y . Only a b r i e f d i s c u s s i o n of conformal mapping has been given here, however, the features mentioned are considered the more important ones. See References 13,23*24. 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0105039/manifest

Comment

Related Items