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Effect of ground conductivity and permittivity on the mode propagation constants of an overhead transmission… Doench, Claus 1966

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THE EFFECT OF GROUND CONDUCTIVITY AND PERMITTIVITY ON THE MODE PROPAGATION CONSTANTS OF AN OVERHEAD TRANSMISSION LINE by CLAUS." DOENCH B.A.Sc, University of Toronto, 1962. A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in the Department of Electrical Engineering We accept this thesis as conforming to the required standard Research Supervisor Members of the Committee Head of the Department Members of the Department of Electrical Engineering THE UNIVERSITY OF BRITISH COLUMBIA June, 1966 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 'of the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r . r e f e rence and s t u d y . 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 ex-t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n . o f t h i s t h e s i s f o r f i n a n -c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f Et-£CT£(CAt. ETtja/A)&E£ftJ6i The U n i v e r s i t y o f - B r i t i s h C o l umbia Vancouver 8, Canada Date "JOVE It i%C> ABSTRACT A general analytical method to derive the distributed circuit parameters and mode propagation constants for an n-conductor transmission line i s developed. The analysis uses electromagnetic f i e l d concepts and the results are interpreted in terms of distributed circuit parameters. The procedure involves transforming the problem of the n-conductor line above a ground with finite conductivity into that of an n-conductor above a ground with infinite conductivity. Correction factors are added to account for the f i n i t e conductivity of the ground. The distributed circuit parameters thus calculated are used to calculate the mode propagation constants over a frequency range from 10 Hz to 1 MHz for values of ground conductivity varying between 1 mho/m and 10~^ mho/m and relative permittivity varying between 10 and 50. Numerical results for the distributed circuit parameters and mode propagation constants for a typical 500 kV single circuit transmission line and various ground conditions are given. The results show that one mode has a higher attenuation and a lower velocity than either of the other two modes, suggesting the zero sequence mode for a completely balanced system. i i TABLE OE CONTENTS Page Abstract i i Table of Contents i i i List of Illustrations v List of Tables v i i List of Symbols v i i i Acknowledgement x i 1. Introduction 1 2. Assumptions and Approaches ,. 6 2.1 General Description of the System 6 2.2 Basic Assumptions 7 2.3 Circuit Parameters ,. 8 3. Derivation of the Hertz Vector due to Current in a Wire 9 3.1 Hertz Vector due to a Current Element in a Medium of Infinite Extent 10 3.2 Hertz Vector due to a Current Element above a Ground with Infinite Conductivity 10 3.3 Hertz Vector due to a Current Element above a Ground with Finite Conductivity 11 3.4 Complete Hertz Vector for a Current in a Straight Conductor above the Ground 16 4. Evaluation of Transmission Line Resistance and Inductance Including Ground Effects 17 4.1 Derivation of the Series Impedance Formulae ... 17 4.2 Evaluation of and 18 5. Derivation of the Shunt Admittance 25 5.1 Derivation of Shunt Admittance Formulae 25 5.2 Evaluation of the Maxwell Potential Coefficients K. . and L , 26 i i i Page 6. Derivation of Propagation Constants and A,B,C,D Parameters of an n-conductor Transmission System .. 31 6.1 Calculation of the Propagation Constants 31 6.2 Formation of the Impedance Matrix Z 32 6.3 Formation of the Admittance Matrix Y 34 6.4 The A,B,C,D Parameters of the System ......... 35 7. Numerical Results for a 500 kV Transmission Line .. 36 7.1 Description of the Transmission Line 36 7.2 Ideal Parameters 37 7.3 Skin Effect 38 7.4 Impedance Correction ......................... 38 7.5 Admittance Correction 40 7.6 Conclusion 42 8. Calculation of the Mode Propagation Constants 49 9. Conclusions 68 Appendix A The Field due to an Isolated Oscillating Current Element 71 Appendix B The General Vector in Cylindrical Coordinates 74 Appendix C Derivation of the Internal Impedance of Cylindrical Conductors 78 Appendix D Solution of the Integral in Equation 4-30 . 82 References 86 i v LIST OF ILLUSTRATIONS Figure Page 2- 1 Distributed Circuit Parameters of an n-conductor 3- 1 Current Element Orientation 9 3- 2 Cylindrical Coordinate System for the External Field 9 4- 1 Two-Conductor Configuration 17 7-1 Impedance Correction Factor Q with 9 =0, 7-2 Impedance Correction Factor P with 9 =0, S - 0 43 7-3 Impedance Correction Factor Q with n = 10 .... 44 7-4 Impedance Correction Factor P with n = 10 .... 44 7-5 Impedance Correction Factor Q with n = 50, 9 = 0 45 7-6 Impedance Correction Factor P with n = 50, 9 = 0 45 7-7 Potential Coefficient Correction Factor N with Q = 0, d = 10~5 mho/m 46 7-8 Potential Coefficient Correction Factor M with 9 = 0, <4 = 10-5 mho/m 46 7-9 Potential Coefficient Correction Factor N with 9 - 0, £ = 10-4 mho/m 47 7-10 Potential Coefficient Correction Factor M with 9 = 0, S = 10-4 mho/m 47 7-11 Potential Coefficient Correction Factor N with © = 0, <£ = 10-5 mho/m 48 7- 12 Potential Coefficient Correction Factor M with 9 = 0, <d = IO"3 mho/m 48 8- 1 Mode 3 Propagation Velocity without Permittivity Correction 56 8-2 Mode 3 Propagation Velocity, no Capacitance Correction with n = 10 56 v Figure Page 8-3 Mode 3 Propagation Velocity, no Capacitance Correction with n = 30 57 8-4 Mode 3 Propagation Velocity, no Capacitance Correction with n = 50 ., 57 8-5 Mode 3 Propagation Velocity, Approximate Capacitance Correction with n = 10 58 8-6 Mode 3 Propagation Velocity, Approximate Capacitance Correction with n = 30 58 8-7 Mode 3 Propagation Velocity, Approximate Capacitance Correction with n = 50 59 8-8 Mode Attenuation with Infinitely Conducting Ground 60 8-9 Mode Attenuation with no Permittivity Correction 61 8-10 Mode Attenuation, no Capacitance Correction with n = 10 62 8-11 Mode Attenuation, no Capacitanoe Correction with n = 30 63 8-12 Mode Attenuation, no Capacitance Correction with n = 50 64 8-13 Mode Attenuation, Approximate Capacitance Correction with n = 10 * 65 8-14 Mode Attenuation, Approximate Capacitance Correction with n = 30 66 8-15 Mode Attenuation, Approximate Capacitance Correction with n = 50 67 v i LIST OF TABLES Table Page 7.1 Potential Coefficient Correction Terms 42 -5 8.1 Attenuation i n Nepers/Mile f o r d = 10 mho/m. and f =.100 kHz 52 8.2 Attenuation in Nepers/Mile for d = 10"-'mho/m. and f = 10 kHz 53 8.3 Attenuation in Nepers/Mile for d = 10 mho/m. and f = 1 kHz 54 v i i Symbol List a 2 = -e 0w(n-l)/y + j a i = conductor radius b = 1 / r 2 = n- - j d / e 0 t o b i^ = horizontal conductor spacing c = Euler's constant = 0.57722 C,C^^ = capacitance matrix, elements E,E = electric f i e l d intensity (vector, components) G, Gr^ ^ = conductance matrix, elements H, H = magnetic f i e l d intensity (vector, components) h = conductor height 1^  = phasor current I , 1 = phasor current vector (mode, line) i = current vector J,J = current density (vector, components) J"n(x) = Bessel function of the f i r s t kind of order n K , ] ^ = potential coefficient matrix, elements k = propagation constant - jm L,L i^ = inductance matrix, elements m = propagation constant m o ' m l = P r°P aSation constant (in air , in the ground) M. ., IT. . = real and imaginary parts of the correction terms ^ 3 for the elements of the potential coefficients matrix n = relative ground permittivity P. .,Q. . = real and imaginary parts of the correction terms ^ for the elements of the impedance matrix R = radial distance in spherical coordinates v i i i radial distance from the current element radial distance from the image of the current element resistance matrix, elements radial distance in cylindrical coordinates distance from conductor i to the image of conductor j distance from conductor i to conductor j elemental length quantity defined in Appendix B phasor voltage phasor voltage vectors (mode, line) voltage vector modal velocity admittance matrix, elements impedance matrix, elements internal conductor impedance attenuation constant phase constant 2 2 propagation constant = u + m propagation constant (in a i r , in the ground) complex permittivity permittivity (of free space, of the ground) magnetic permeability of free space Hertz vector, components radial distance i n cylindrical coordinates radial distance from conductor 1 radial distance from the image of conductor conductivity ix r 2 = m2/m2 0 = scalar function of position co = angular frequency Y = e c = 1.7811 x ACKNOWLEDGEMENT The author would like to express his gratitude to his supervising professor, Dr. Y.N. Yu who guided and inspired him throughout the course of the research. The author also wishes to thank members of the Department of Electrical Engineering and especially Dr. M.M.Z. Kharadly for his help and advice and Mr. H.R. Chinn for the use of his computer program in plotting curves. The author i s indebted to the University of British Columbia, the National Research Council of Canada and the British American Oil Company for financial support of the research. Thanks are due to Mr. A. MacKenzie for drawing the graphs and to Miss B. Rydberg for typing the thesis. x i 1 THE EFFECT OF GROUND CONDUCTIVITY AND PERMITTIVITY ON THE MODE PROPAGATION CONSTANTS OF AN OVERHEAD TRANSMISSION LINE. Chapter 1, Introduction. The advent of extra-high-voltage transmission has renewed the interest in the study of energy propagation along transmission lines. These studies are associated with fault current calculations, system stability, switching and restrik-ing overvoltages, and the propagation characteristics of carrier waves along power lines. A necessary prerequisite for investi-gations of the above problems is a thorough knowledge of the multiconductor transmission line distributed circuit parameters. Although these parameters may be calculated from the conductor conductivity, the frequency and the geometry of the system, they are also dependent on the ground conductivity, permittivity, and permeability. While an exact calculation of the parameters is impossible, due to irregularities in the shape of the ground surface and the lack of uniform conductivity, a f a i r l y accurate calculation i s possible by replacing the ground with a plain or multi-layer homogeneous media. After the distributed circuit parameters have been found the propagation problem is solved from the transmission line equations by meeting the boundary conditions. The f i r s t important engineering study of the effect of ground with f i n i t e conductivity on the electromagnetic propagation produced by current-carrying conductors above the ground i s due to J.R. Carson^ 1^ 2^. Applying electromagnetic 2 fi e l d theory, he calculated the f i e l d due to an alternating current in a straight i n f i n i t e l y long wire above and parallel to a plain and electrically homogeneous ground. The derivation contains four basic assumptionsj 1. The ground relative permeability i s unity. 2. The ground i s electrically homogeneous. 3. The polarization currents may be neglected. 4. The current is propagated without attenuation at the speed of light. The f i r s t three assumptions greatly simplify the results. The fourth assumption implies reasonably efficient energy transmission, as indicated in reference (2). In the same publication Carson derived resistance and inductance correction factors. They are valid only for frequencies below 10 kHz because of the omission of the ground relative permittivity in the derivation. (6) Rudenberg^ ' analysed the same problem. He used a model of the transmission system which has a semi-circular ground surface, with the axis coincidental with that of a conduotor. In addition to Carson's assumptions he assumed that the current distribution in the ground is a function of the radial distance from the conductor. The resistance and inductance correction terms are expressed in terms of the zero and f i r s t order Bessel functions of the third kind. At low frequencies these functions have small arguments. Using only the f i r s t term of the power series expansions of the Bessel functions similar results to those given hy Carson are obtained. Rudenberg did not derive the mutual inductance correction terms and he neglected the ground permittivity effects. Carson"s results were extended hy W i s e ^ ^ 1 ^ ^ " 1 ^ . His f i r s t paper removed the restriction of assuming a ground relative permeability of unity. His subsequent works included the effect of polarization currents, gave correction terms for the line admittance, and extended Carson*s impedance correction terms to a broader frequency range, Carson's (7) impedance correction terms were also extended by Sunde v , / to include the effect of a multi-layer ground. The investigation of multi-conductor transmission, using matrices, was started by Bewley^. His method was f i r s t to consider an ideal lossless line and then to expand (3) (20) the analysis to include losses. R i c e w ' and Pipes v ' followed Bewley's approach but added the use of Laplace transform methods. A major contribution has been the work of Hayashi w / who extended analysis to include transient phenomena. He also developed new techniques in matrix calculus to f a c i l i t a t e the solutions, for example his extension of Sylvester's expansion theorem. Although his analysis included conductor skin effects at higher frequencies, he used ground impedance correction from Rudenberg5s model which neglected ground permittivity effects, A systematic mathematical procedure for handling the multi-conductor transmission line problem was recently published by Wedepohlv . In two subsequent publications v M ; he gave numerical procedures including the correction of the impedance parameters following Wise's methods, A matrix analysis of 4 m u l t i - c o n d u c t o r t r a n s m i s s i o n s y s t e m s w i t h v a r i o u s b o u n d a r y c o n d i t i o n s was d o n e h y D o w d e s w e l l i n h i s t h e s i s , h o w e v e r h i s (23) w o r k was r e s t r i c t e d t o s t e a d y s t a t e a n a l y s i s a t p o w e r f r e q u e n c y x . (12) A r i s m u n a n d a r ' a l s o d e r i v e d a d m i t t a n c e c o r r e c t i o n t e r m s w i t h t h e a s s u m p t i o n t h a t t h e l i n e s a r e c l o s e t o t h e g r o u n d s u r f a c e , f r o m w h i c h h e c a l c u l a t e d c h a r a c t e r i s t i c i m p e d a n c e s f o r s w i t c h i n g s u r g e s t u d i e s . T h i s t h e s i s d e r i v e s c o r r e c t i o n f a c t o r s f o r t h e d i s t r i b u t e d c i r c u i t p a r a m e t e r s i n c l u d i n g t h e e f f e c t s o f c o n d u c -t i v i t y a n d r e l a t i v e p e r m i t t i v i t y o f t h e g r o u n d , a n d g i v e s a s y s t e m a t i c m e t h o d o f c a l c u l a t i n g t h e mode p r o p a g a t i o n c o n s t a n t s o f a m u l t i - c o n d u c t o r t r a n s m i s s i o n l i n e . A g e n e r a l d e s c r i p t i o n o f t h e s y s t e m a n d t h e a s s u m p t i o n s r e q u i r e d i n t h e d e r i v a t i o n a r e g i v e n i n C h a p t e r 2. The e l e c t r o m a g n e t i c f i e l d s u r r o u n d i n g a c u r r e n t - c a r r y i n g c o n d u c t o r a b o v e a p l a i n g r o u n d i s d e r i v e d i n C h a p t e r 3 i n t e r m s o f a n E - t y p e H e r t z v e c t o r , v a l i d f o r t h e f r e q u e n c y r a n g e f r o m 10 H z t o 1 M H z . The d i s t r i b u t e d c i r c u i t i m p e d a n c e p a r a m e t e r s f o r t h e t r a n s m i s s i o n l i n e a r e d e r i v e d i n C h a p t e r 4 a n d t h e d i s t r i b u t e d c i r c u i t a d m i t t a n c e p a r a m e t e r s i n C h a p t e r 5, f r o m t h e e l e c t r o m a g n e t i c f i e l d s o l u t i o n g i v e n i n C h a p t e r 3* T h e s e d i s t r i b u t e d c i r c u i t p a r a m e t e r s i n c l u d e t h e e f f e c t s o f g r o u n d r e l a t i v e p e r m i t t i v i t y a n d c o n d u c t i v i t y . The d e r i v a t i o n f o l l o w s c l o s e l y t h a t o f C a r s o n ^ , W i s e ^ 1 1 ^ a n d (7) S u n d e v . The d e t a i l e d s o l u t i o n o f t h e t r a n s m i s s i o n l i n e (a) e q u a t i o n , f o l l o w i n g W e d e p o h l ' s s o l u t i o n v , i s g i v e n i n C h a p t e r 6. T h e d i s t r i b u t e d c i r o u i t p a r a m e t e r s f o r a t y p i c a l 500 kV t r a n s m i s s i o n l i n e f o r v a r i o u s f r e q u e n c i e s a n d g r o u n d 5 c o n d i t i o n s a r e t a b u l a t e d i n C h a p t e r 7 . C h a p t e r 7 a l s o i n c l u d e s g r a p h s g i v i n g t h e m a g n i t u d e o f t h e d i s t r i b u t e d c i r c u i t p a r a m e t e r c o r r e c t i o n f a c t o r s o v e r t h e d l t a t e d f r e q u e n c y r a n g e f o r v a r i o u s g r o u n d c o n d i t i o n s , A d e t a i l e d d e s c r i p t i o n o f t h e s o l u t i o n , w i t h n u m e r i c a l r e s u l t s , o f t h e w a v e e q u a t i o n f o r t h e t y p i c a l t r a n s m i s s i o n l i n e a r e i n c l u d e d i n C h a p t e r 8. The r a t i o n a l i z e d MES s y s t e m o f u n i t s i s u s e d t h r o u g h o u t t h e t h e s i s , u n l e s s s p e c i f i e d . 6 Chapter 2 . Assumptions and Approaches. 2 . 1 General Description of the System. The system under study consists of a number of separate conductors and ground return lines situated above and parallel to a plane ground. No geometric symmetry i s assumed in the transverse plane. The transmission line equations for a line with n conductors may be written in terms of the distributed circuit parameters in the general form as " • " § 2 = R i + L T t = ( R + L p ) i = Z i --|^=Gv + C-|^=(G + Cp)v = Yv 2 - 1 2 - 2 where Z and Y are n x n symmetric matrices, and v and i column matrices with n elements. Each element of the Z and Y matrices includes the effect of ground return the system shown in f i g , 2 - 1 , Conductor i R.. L.. L. . AA>V^ T W^ nrr«V (4) and i s illustrated in Conductor j J^j ^ Pig. 2-1 Distributed Circuit Parameters of an n-conductor Line. 7 2.2 B a s i c A s s u m p t i o n s . The e f f e c t o f f i n i t e g r o u n d c o n d u c t i v i t y a n d r e l a t i v e p e r m i t t i v i t y o n t h e t r a n s m i s s i o n o f e n e r g y a l o n g a t r a n s m i s s i o n l i n e c a n h e d e r i v e d i n t e r m s o f f i e l d c o n c e p t s . I n e n g i n e e r i n g , h o w e v e r , c i r c u i t c o n c e p t s a r e m o r e p r a c t i c a l , s i n c e c u r r e n t a n d v o l t a g e c a n h e e a s i l y m e a s u r e d . H e n c e t h e p r o b l e m i s t o a n a l y s e t h e f i e l d a n d e x p r e s s t h e r e s u l t s i n t e r m s o f t h e d i s t r i b u t e d c i r c u i t p a r a m e t e r s . I n d e v e l o p m e n t o f t h e a n a l y s i s i n t h i s t h e s i s t h e g r o u n d i s c o n s i d e r e d h o m o g e n e o u s w i t h a r e l a t i v e p e r m e a b i l i t y o f u n i t y . The e x t e n s i o n o f t h e t h e o r y t o m u l t i - l a y e r g r o u n d s w i t h d i f f e r e n t p e r m e a b i l i t i e s , d o e s n o t r e q u i r e a n y new c o n c e p t s a l t h o u g h i t b e c o m e s m u c h m o r e i n v o l v e d a l g e b r a i c a l l y . The e l e c t r o m a g n e t i c f i e l d s u r r o u n d i n g t h e t r a n s m i s s i o n l i n e c o n s i s t s p r i m a r i l y o f t h e TM mode w i t h some d e p a r t u r e f r o m t h i s d u e t o t h e l o s s e s i n t h e g r o u n d , A H e r t z v e c t o r o f t h e t y p e i s u s e d t o d e s c r i b e t h e f i e l d . I t h a s t w o c o m p o n -e n t s , o n e o r i e n t e d i n t h e d i r e c t i o n o f p r o p a g a t i o n a n d t h e o t h e r v e r t i c a l l y i n t o t h e g r o u n d . I n t h e c o n s t r u c t i o n o f t h e f i e l d s o l u t i o n t h e f o l l o w i n g a s s u m p t i o n s a r e m a d e : p 1, A t t h e f r e q u e n c y o f i n t e r e s t W ' E L I i n a i r i s m u c h l e s s t h a n u n i t y . T h i s a s s u m p t i o n i s v a l i d f o r f r e q u e n c i e s i n t h e r a n g e f r o m 1 0 H z t o 1 M H z , 2, I n t h e s o l u t i o n o f t h e wave e q u a t i o n a f i r s t o r d e r a p p r o x i m a t i o n i s c o n s i d e r e d s u f f i c i e n t l y a c c u r a t e f o r p r a c t i c a l p u r p o s e s i n v i e w o f t h e p h y s i c a l i r r e g u l a r i t i e s o f t h e t r a n s m i s s i o n l i n e . 8 3^ The p r i n c i p l e o f s u p e r p o s i t i o n i s u s e d t o f i n d t h e c o m p l e t e s o l u t i o n f o r t h e s y s t e m . 4. A i r c o n d u c t i v i t y i s c o n s i d e r e d n e g l i g i b l e . 2.3 C i r c u i t P a r a m e t e r s . The i d e a l c i r c u i t p a r a m e t e r i s d e r i v e d f i r s t b y a s s u m i n g t h a t t h e g r o u n d h a s i n f i n i t e c o n d u c t i v i t y . F o r a g r o u n d o f f i n i t e c o n d u c t i v i t y t h e c i r c u i t p a r a m e t e r i s e x p r e s s e d a s t h e sum o f t h e i d e a l p a r a m e t e r p l u s a c o r r e c t i o n t e r m f o r t h e g r o u n d c o n d i t i o n . T h i s a n a l y s i s t r a n s f o r m s a n n - c o n d u c t o r l i n e o v e r a g r o u n d o f f i n i t e c o n d u c t i v i t y t o a n n - c o n d u c t o r l i n e o v e r a g r o u n d o f i n f i n i t e c o n d u c t i v i t y . 9 Chapter 3. Derivation of the Hertz Vector due to Current in a Straight Wire. The Hertz vector for a ourrent along an infinite straight wire parallel to a f l a t ground is obtained hy inte-grating the elemental Hertz vectors for current elements along the wire from minus in f i n i t y to plus i n f i n i t y . Fig. 3-1 Current Element Orientation. X Fig. 3-2 Cylindrical Coordinate System for the External Field. 10 3-1 Hertz Vector due to a Current Element In a Medium of  Infinite Extent. The Hertz vector due to a current element Idz in an isotropic homogeneous medium of infinite extent i s derived in Appendix A, and is given hy \ = 0 3-1 TV = 0 3-2 -rr Ie 0 dz X = 4*mQ2R 5-4 3-2 Hertz Vector due to a Current Element above a Ground with  Infinite Conductivity. The total f i e l d above ground due to a current element and i t s image, as shown in f i g . 3-1, consists of a primary f i e l d , plus a secondary f i e l d due to the f i n i t e conductivity of the ground. Let the total f i e l d be z ,loz Moz J J 1 2 where "IT Q Z i s the primary f i e l d , and"!To z i s the secondary f i e l d vector. The suffix "o" designates the f i e l d vector above the ground surface. Prom equation 3-4 we have -m R.. -m R, ITI _ -Itouldz e 0 1 e 0 * , , ^ = 4 ^ R l " R2 3 where R-^ and R2 are the distances from the point of interest to the current element and i t s image respectively. 11 3-3 H e r t z V e c t o r d u e t o a C u r r e n t E l e m e n t a b o v e a G r o u n d w i t h  F i n i t e C o n d u c t i v i t y . When t h e g r o u n d h a s f i n i t e c o n d u c t i v i t y some e n e r g y w i l l b e d i s s i p a t e d , a n d t h e f i e l d may b e d e s c r i b e d b y means o f t w o H e r t z v e c t o r c o m p o n e n t s , i n t h e y a n d z d i r e c t i o n s . E q u a t i o n B-15 g i v e s t h e g e n e r a l s o l u t i o n i n t h e c o o r d i n a t e s y s t e m o f f i g . 3-2. H o w e v e r , t h e p a r t i c u l a r s o l u t i o n m u s t m e e t t h e b o u n d a r y c o n d i t i o n s w h i c h may b e s t a t e d a s f o l l o w s : a . The f i e l d v a n i s h e s a t a n i n f i n i t e d i s t a n c e f r o m t h e s o u r c e . b . A t t h e s o u r c e o n l y 7TZ e x i s t s a n d i s f i n i t e , s i n c e t h e c u r r e n t e l e m e n t i s o r i e n t a t e d i n t h e z d i r e c t i o n . c . The t a n g e n t i a l E a n d H f i e l d s a r e c o n t i n u o u s a c r o s s t h e g r o u n d s u r f a c e . A f u r t h e r r e s t r i c t i o n i s i m p o s e d d u e t o t h e g e o m e t r y o f t h e s y s t e m ; d . The f i e l d s a r e s y m m e t r i c a l a b o u t t h e y - z p l a n e . The f o r m o f t h e f i e l d f u n c t i o n s may b e c o n s t r u c t e d f r o m e q u a t i o n B-15, A p p e n d i x B , b y a p p l y i n g t h e a b o v e f o u r r e s t r i c t i o n s . Due t o ( a ) a n d ( b ) o n l y B e s s e l f u n c t i o n s o f t h e f i r s t k i n d c a n a p p e a r i n t h e s o l u t i o n , a n d d u e t o ( b ) IT c a n c o n t a i n t h e z e r o o r d e r f u n c t i o n b u t TT c a n n o t . Due t o ( d ) t h e s o l u t i o n s c a n c o n t a i n t e r m s i n c o s P o n l y . A s s u m e a s a f i r s t a p p r o x i m a t i o n t h a t o n l y t h e l e a d i n g t e r m o f t h e s u m m a t i o n o v e r t h e p o s s i b l e v a l u e s o f P i s o f s i g n i f i c a n c e . F i n a l l y i n a c c o r d a n c e w i t h e q u a t i o n 3-6 l e t t h e v e c t o r a b o v e t h e g r o u n d b e t h e sum o f t w o f u n c t i o n s . T h e H e r t z v e c t o r f o r t h e r e g i o n y » h may now b e g i v e n b y 12 3-7 3-8 TT 0 Z = J ~ [ f 0 ( u ) e ( y" h )*o + g Q( u) e-^o] J o(ru) d u o eo ir i z = j gx(u) e y * l J Q(ru) du o TT o y = J c o s V p0(u) e~y*o J^ru) du 3-9 o oo TT l y = J" cos V p-^u) e y y l J^ru) du 3-10 where the suffix "1" designates the f i e l d below the ground surface. There now remains the evaluation of the functions fQ» SQt g±t P 0 and p 1 using condition (c). Condition (c) may be stated in terms of the Hertz vector components by means of equations A-10 and A-3 as follows' ,(7) a a y OZ + » '.fig-0 z o 7 x H>v + ^ o z d y 0 z • " " f t , + 3-11 + ^ 1 * m "37 0 z = m. 2 T^T, o T 02 2 — = m i T 3-12 3-13 3.14 Equation 3-11 may be written as integrating both sides yields 13 However, « and are zero when x becomes infinite and hence o 1 f(z,y,u) i s zero. Integrating both sides of equation 3-14 yields "o^oy = " " l ^ l y + S( z»y' u) Again both 71^  and Tl^y are zero when x becomes infinite, hence g(z,y,u) i s zero. With these results equations 3-11 through 3-14 reduce to the following equations TF + T^ - i y + T^ 5 5 m 2 TT = m 2 TT, 3-16 o oz 1 l z y mo T F = m i T F " 3 " 1 7 mo ^oy = m l ""iy 5-18 The various f i e l d functions may now be evaluated in terms of the function f Q(u) they are * - X, - h r « o ( u ) = F T X 1 fo^> e 5-19 * 0 + *: 2 u Y o r 2 -h* 7 ^ S i ( u ) = v °+ f o ( u ) e " ° 5-20 2 u y n ( i - r 2 ) - h r p ( u) = 2 f ( u) e 0 3-21 14 where P l(u) = r 2 P 0(u) 3-22 T 2 = m2/m2 3-23 The function fQ(u-) may he determined hy letting = X and mn = m , the solution must then he the same as l o l o for a uniform medium of infinite extent. Then -m R-, f (u) e ( y - h ) * o j (ru) du = ^ T % ° d z 3-24 4**1*1 Then hy transforming the left-hand-side of equation 3-24 to the same origin as the right-hand-side and letting f (u) = B -—— 3-25 ° ° o where B i s a constant, and hy making the substitution * 2 = u 2 + m2 3-26 equation 3-29 becomes ? v / v -m R-[ B e" y *° J O ( R / » J - N | , A * O - M f ^ 3 - 2 7 o 4* mQ R x Assuming that mQ is a small quantity in air compared with ¥ and restricting the solution to frequencies below 1 MHz equation 3-27 becomes oo [ B e~ y'*° J ( r X ) d * = - M i ^ z 3 _ 2 Q o o o o 4 f l ; m2 R i ( 1 Q ) which is Lipschitz's integral . Hence 15 u 3-29 Using the same reasoning -m R2  .jcoiile dz 4«m 2R 2 ) e -(y+h)v 0 J Q(ru) du 3-30 Then comhining equation 3-30 with 3-19 to obtain the second part of equation 3-7 -m R. -y o . .jioule dz f " y * o g Q( u) e J 0 ( r u ) d u = ~ o r 2 y + J f ° ( u ) 4«m 2R 2 e J (ru) du 3-31 The last term of 3-31 corresponds to the secondary f i e l d due to f i n i t e ground conductivity J ' z = J y1 + V 0 f o ( u ) e d » 3-32 Then the complete Hertz vector in the region 0— y ^ h i s 7T = 0 ox 3-33 ~ 0 0 -(y+h)tf TT C 0 S y i M ( i - r V e 0 J ( r u ) o y i 4«m2 ( v i + y o ) ( r 2 V 1 + Y O ) 1 du f f = -itouldz ' o z 4«m 2 o - m o R l " mo R 2 e e R. R. r ~l: 2ue J 177 (y+h)y 3-34 T17 do J (ru)du 3-35 The current at any point z on the line is given hy - tfz I = I e 3-36 16 where i s the propagation constant i n the z d i r e c t i o n . Noting that - cos y uJ-^ru) = J 0 ( r u ) 3-37 equations 3-34 and 3-35 become - *Z c*? 2jco|iloe dz ^ -(y+h)tf TT oz 4«m o j toil I e dz 0 r o  ? 4itm T z f u ( l - r 2 ) e " ^ ° ° ^ T , , J (ir 0 + ^ H r 2 ^ + V 0 ) J o ( r u ) e ^ e " m o R 2 , f 2 u e - ( y + h ) y o du 3-38 _ J Q(ru)du 3-39 3-4 Complete Hertz Vector f o r a Current i n a Straight  Conductor above Ground The complete Hertz vector f o r the current-carrying conductor i s now obtained from integration of the Hertz vectors due to the current elements along the l i n e from minus i n f i n i t y to plus i n f i n i t y . The components of the Hertz vector are TT = o 3-40 co 4«m, I r u ( l - r g ) e 2 - ( y + b ) ^ r o J(ru)dudz jwul e - t f z 4«m, OA I - C O O O 3-41 - t f z -m Rn -m R~ e 0 1 e 0 2> L Rn R, + ~ -(y+h)ro "j 1 ^ + y0 J o ( n i ) d u J dz 3-42 17 Chapter 4. Evaluation of Transmission Line Resistance and  Inductance Including Ground Effects. 4.1 Derivation of the Series Impedance Formulae. onductor j i I I t I I I III Ground Fig. 4-1 Two-Conductor Configuration. Fig. 4-1 shows two typical conductors of an n-conductor transmission system. The total electric f i e l d on the surface of conductor i i s the sum of the electric fields due to a l l n conductors. From equation A-10 the electric f i e l d on the i-th conductor due to the j-th conductor in the z direction i s given hy E. = -m2 Tf. + Hr-1Z O ']Z £ z b y ' iz 4-1 Then the complete electric f i e l d on the surface of conductor i due to a l l conductors is n E. = Y7 -m2fr, +4-iz Z_i o jz l y dv, V " &z 4 - 2 18 The fields inside and outside conductor i must he equal at the boundary. If the internal impedance of conductor i i s then z.I. e 1 1 0 n 1=1 " m o ^ z + "Ji o y o z _ 4-3 where z^ is derived in Appendix C. The second term on the right-hand-side of equation 4-1 may be expressed as a gradient of the scalar potential, V^, where C 7 iL ["HZ IZLfl = / L A ^ z | _ ^ y + Z J i=l 4-4 Prom the general transmission line equation E-8 the voltage equation in phasor form can be written & 3 J=i 13 Jo 4-5 Hence by combining equations 4-3 and 4-5 the following formulae can be established for the self and mutual impedances, respectively ; 4-6 Z. . = z. + m TT. ix i o iz ho* z Z„ . = m2*^ iO o " j z 4-7 where j =1, 2, n except i , and a l l functions are evaluated on the surface of conductor i . 4.2 Evaluation of Z ^ and Z^. The evaluation of these impedances requires the evaluation of the infinite integrals in the TT Z function given 19 by equation 3-42. To evaluate TTz the assumption must be made that attenuation along the line is negligible, then * = mo where 4-8 m0 = J /e^to 2 = jk This is an ideal value for *X but is necessary due to the following two considerations: 1. To assume that the attenuation is not zero on an inf i n i t e l y long line amounts to assuming a source of infini t e energy and makes the integral i n f i n i t e . 2. To assume a propagation velocity less than that of light makes the integral extremely d i f f i c u l t to evaluate. Substituting jk for mQ and in equation 3-42 If = I e ^ k z e ^ k z z o J - oo oo / -jkR, -jkR 2  •1cou ( e e, _ 4*m2 \ R l R2 ) dz 4-9 oo - tfn(h+y) J 2ue J (ru)du •icou o v ; 4«m2 * o + ^ 1 o ° From equation 4-6 we have z i i = z ± + h + j 2 a ' 1 0 where T r - j k R - - j k R 2 n 20 To evaluate 1^, let P l = R l + z P 2 = R2 + z 4-13 where R l = \P\ + z* a n d R2 = /"I + B* 2 'ij v " i "d 4-14 where b ^ is the horizontal distance between the i t h and jth conductors, assuming this distance to be much greater than the radius of conductor i , h^ is the height of the it h conductor above ground and h^ that of jth conductor. The general case is considered because i t i s required in the evaluation of Z-^y When J equals i , b ^ becomes the radius of conductor i , and Pi. = a i ' ^2 = 2 h i ' where a i is the radius of conductor i , and the height h^ is assumed to be much greater than the radius a^. With the change of variable from z to and P 2 we have P P dP-L = ^ : dz, dP 2 = g | dz 4-16 and P-j^  oo , P 2 <*o when z *-~ <*> p ! a ! f p 2 " 2 W h e n Z ~~ " *° P2 P2 where a ± = -1 , a g = _| 4 _ 1 7 21 for a very large z. Then T - lW x l ~ 4* a ± l 4* e ' dP, 4-18 j «2 where the integrals are defined as the exponential integrals Hence equation 4-18 can he written as (18) T - ^ m x l " 4* l i ( e ') - l i ( e ) 4-19 where oo - l K e " * ) = c + ln(t) + n ] n n=l and c i s Euler's constant. Por a small value of t the f i r s t two terms in the above expression predominate and a l l other terms can be neglected,' then - J M in -2-" 2* />± 4-20 Equation 4-20 gives the self reactance per unit length of line conductor when the ground has infi n i t e conductivity. To evaluate I 2 let the order of integration be changed, and I 2 becomes - r (h+y) o - oo Since J Q(ru) and cos(kz) are even functions, and sin(kz) is an odd function of kz 22 oo o o — OO JV^2 JQ(ru)dz = 2 Jcos(kz) JQ(ru)dz o = 0 , k > u = 2cos(x \/x? - k 2 ) Hence f o r k— u T - i M L 2 " at ue 2cos V Q x T o ' ^ 0 ( h + y ) COS 5 Q X k ^ u du 4-22 4-23 To change the lower l i m i t of the above integration, substitute X2. = u2 - *2, du - Isgla then = .1cou ?t J o i cos 0 , X dV / y y x - ^ 0 ( h + y ) V _ ML ( - y 0 ) e :cos g.x It d* 4-24 To f i n d results, s i m i l a r to Carson's but including the effect of r e l a t i v e ground permittivity, i n accordance with referenoe (9) l e t s' = 1 -je w(n-l) 4-25 since and 2 2 2 2 y1 = u + m = u + jcofid — e1nw' y 2 2 2 0 = u - e aw O 0 23 hence * 1 " * o = 1^<4 B2 4-26 Further l e t y' = /wJI3y, h" = /co|i<3'h, x' = A>u<£x, v = . 4-27 Then oo I 2 = ^ [ (/v2 - i s 2 - v) e ^ ' + y ^ o o s x'v dv * s o 4^28 F i n a l l y l e t y M = sy', h" = ah', x" = sx' f v = su 4-29 then oo T2 = f f + ^ - u ) e - u ( h " + y J , ) c o s x»u du 4-30 o Equation 4-30 i s the same equation as given hy (2) Carson v except f o r s (the ground permi t t i v i t y correction f a c t o r ) . In his case s = 1 because he assumed that the r e l a t i v e permittivity of the ground equals one. Then and Z i i = z i + If ^  fj* + f <rii + »il> 4-32 The l a s t term of equation 4-32 i s the correction factor f o r the f i n i t e ground conductivity and r e l a t i v e permit-t i v i t y . The exact solutions f o r and are given i n Appendix D, and numerical re s u l t s are included i n chaper 7. Using the same procedure as above, we have f o r Z. . X J 24 4-33 where S*^  i s the distance from conductor i to the image of conductor j , and i s the distance from conductor i to conductor Chapter 5« Evaluation of the Shunt Admittance 25 5.1 Derivation of Shunt Admittance Formulae. Prom equation 2-1 the transmission line equation of the n-conductor system can he written C , , V, 5-1 d=l where i = 1,2,3,4, n. Making the same assumptions as in equation 4-8, and noting that = -;jk, differentiation yields n ^ C^Vj = J I ± i = 1,2,(3,4,) n. ^ 5-2 Then solving this system of equations for the voltages, we have n V i = S H 1 = 1.2.:..., n. 5-3 d=i where K. . i s Maxwell's potential coefficient. Substituting these results into equation 4-4 yields k 2>1Tiv bTTiz £ K I, = w i J o y hence " k I i L ^ y J Thus the potential coefficients can be derived from the f i e l d components. 26 5.2 Evaluation of the Maxwell Potential Coefficients K 1 J L  and K^j. The same assumptions are made here regarding the evaluation of the i n f i n i t e integrals as stated i n the preceding chapter f o r equation 4-8. Prom equations 5-4, 3-41 and 3-42 where In = 1 4«e oo o oo -jkR, -jkR e e R 2 J dz 5-5 5-6 F 2 s -(y+h)*, 2 " 4«e 0 J J * 0 + J Q ( r u ) dudz 5-7 — oO O I, = » P -(y+h)Y u ( l - r 2 ) e ° 3 2«e 1 * f -jkz d C -r 2)e  T r^O A,,** —rr -r-— \ e J -rr \ b J (ru; dudz 0 j k d y J dzj ( v + v ) ( Y 2 y v,} o — OO o> JL O -L O 5-8 1^ and I 2 have been evaluated i n chapter 4. I. + In = l n i — 2 - 2 5 T l n ^ T + ^ T 7 ( P + = L N - + - ^ P ) o ' 1 o 2weQ """/^ ' 7 t e-5-9 1^ i s evaluated as follows -(y+h)tf I, = 3 " 2*e ~ v 2 -(y+h)tf ^ o ^ J (JT, + y n ) ( r 2 V ^ ) J o -1- 0 -1 0 - CO e - j k z d_ j ( r u ) d z d u dz o 5-10 oo - -(y+h)tf .! ruy o(i-r 2)e  2 3 t e o ^ k J ) r\ X O X O OO oo 1 h ^ - * k z e _ ; ) k z J 0 ( r u ) l + \ j k e " J ^ J o ( r u ) d z |du -oo-oo 5-11 27 Since the f i r s t term in the bracket i s zero and the second term has been evaluated i n equation 4-22, equation 5-11 becomes oo 9 -(y+h) "X ^ , r u( l-*r) e °cos Ox I = - - J - 5— 2_ d u 5-12 2 - 2 k2, 2 v . v ~ 2 rv2 Noting that Y ; = u -  , udu = Y " 0 0 0 and Y Q -Y or^ = X Q + X^Y *i+* 0) the integral breaks into two parts, and T _ 1 o J v . 1 I o d " ^ T ( Q - ^ P ) + n T - ^ ^ I 2 + ^ -(M+dN) 5-13 "0 0 o where ?^ -(y+h)"X C O S X X _0 r 2 0 o J y + JLy° d y o 5" 1 4 Then 1 2hH 1 K i i - sbrln "4 + =fc ( M i i + ; ) N i i ) 5 - 1 5 0 1 0 and The' last terms in equations 5-15 and 5-16 give the correction factors for f i n i t e ground conductivity and relative permittivity to Maxwell's potential coefficients. An approximation of M and N is given in section 5.3 of this chapter and the numerical results are included in chapter 7 . 28 5.3 Evaluation of M.. and N . . 3J LL Making the same substitutions as in 4-25, 4-26 and 4-27, equation 5-14 becomes oo M + jN = °J g 3f"T " d v 5-17 o A + 3 8 2 2 1 (12) Next let js = a , = b, then o A + 3 8 V 2 f* -v(h'+y') c o a_, v M + = K gV c o s x vdv 5-18 J j / v + a + bv The numerical computation of this integral i s involved due to the large number of variables. An approximate solution w i l l be developed below. For large values of v the integrand vanishes, hence 5-18 may be written as v_ M . ,-ivr ~ 1 e"* v^ h' + y t ^cosx'v-,,, r- i n M + 3N = \ i. g g < dv 5-19 J Jv + a + bv o » If within the range of values from 0 to v Q, I 2av| < |v + a I 2 5-20 then f r 2 • ~2 ^- • ~ v- + a * Q-v + a - ^ _ 5 _ 2 1 The limitations of this assumption w i l l be discussed in chapter 7. Substituting 5-21 into 5-19 ( H ) 29 oo M a. HIM f (v + a) e"Y^^\hosxW, ^ J (v + r-,)(v + r 2 ) ( b + l ) d v O 5-22 where r i - f ( 1 - / ^ r f i ^ r2 = f ( 1 + / ^ T I = H H ) 5-23 Next let g± = h? + y' - dx' = loirs' He'*6 = R'e^ 9 5-24 g 2 = h 1 + y 1 + Jx' = f^3 Re 3 9 = R'e^6 co M + jlT = 2 ( b ^ ( r 2 - r i ) [ * ^ ] [ e " S l T + e " S 2 V ] d v o 5-25 Equation 5-25 includes four separate integrals of similar form. Let C O - S i v and 1 f e~ S l [1 = 2 ( h + l ) ( r 2 - r 1 ) J v T r 7 d v 5~ 2 6 v + = w 5-27 then g^r, oo ~ w g-jT, - r 2 e f e 1 " r 2 e / " s l r l X l = 2(h+l)(r 2-r 1) J " T " d w " 2 ( b + l ) ( r 2 - r 1 ) l i ( e r 1 5-28 The other three integrals in 5-25 have similar solutions. Therefore 30 ) + r , e 1 2 l i ( e 1 2 ) M + J* = 2 ( b + l ) ( r 2 - r 1 ) p 2 e g i r i l i ( e " S i r i ) + r x - r ^ ^ l i C e " * 2 * 1 ) + ^ e ^ l K e " " 2 ' 2 ) ] 5 - 2 9 The numerical r e s u l t s of the evaluation of M and N as given hy equation 5 - 2 9 are included i n chapter 7 . 31 Chapter 6. Derivation of Propagation Constants and A.B.C.D  Parameters of an n-conductor Transmission System. 6.1 Calculation of the Propagation Constants. The transmission line equations for an n-conductor ( 8 ) system may he written in matrix and phasor form as follows I = Z(u) Y(ii)) V 6-1 V 2 " — p = T(u) Z(u) I 6-2 where V and I are column vectors of n terms and Z and Y P P are n x n symmetric. matrices. The suffix p denotes the line coordinate system. The systems of second order differential equations represented hy equations 6-1 and 6-2 can he solved hy trans-forming them into a new coordinate system wherein the trans-formed ZY or YZ matrix becomes a diagonal matrix. Then equations 6-1 and 6-2 become * 2 i c w 2. where the suffix c denotes the mode coordinate system. The eigenvalues of the system may be denoted by ^ , 1 = 1,2,3, • ••• n, and are obtained from | ZY - I 2 | = 0 or [YZ - IY 21 = 0 6—5 32 since (ZY) t = Y tZ t = YZ Here I is a unit diagonal matrix. The solutions of equations 6-3 and 6-4 are V = e C, + e A C 0 6-6 Xi 1 2 * i Z " ^ i Z I = e J" D 1 + e £>2 6-7 where Y i represents the n propagation constants in the n-modes of the system. The solution in the line coordinate system, designated hy the suffix p, i s now determined as follows 7 = 5 7 and I = S I 6-8 p c p c where R and S represent n x n transformation matrices, each consisting of n columns of eigenvectors of the system. Each eigenvector i s determined from the corresponding eigenvalue hy means of the following equations (ZY - tf2 I) R(i) = 0 6-9 (YZ - ) f 2 I) S(i) = 0 6-10 where R(i) is the i t h column of the R matrix, and S(i) the ith column of the S matrix. In general RtS = = D 6-11 where D i s a diagonal matrix, where the elements may he complex. 6.2 Formation of the Impedance Matrix Z. The impedance matrix Z has two types of elements, the diagonal and the off-diagonal elements. The diagonal 'elements are given hy equation 4-32 and the off-diagonal elements 33 by equation 4-33* ^et the diagonal elements be Z i i = R i i + ^ L i i 6 ~ 1 2 where R i i - smt + f p i i " 5 where R ^ x , i s the in t e r n a l AC resistance of the conductor or the r e a l part of z i i n equation 4-32, and "jf^F^ i s the r e a l part of the ground correction factor. Let z i = R i n t + J w L i n t 6 " 1 4 then 2h. where ^ Q.^ i s the imaginary part of the ground correction factor. The off-diagonal elements are given by equation 4-33 z i d = ^ l n ^ + f <P« + *V 6"16 Now i f we write then hi • f hi 6-18 and hi - h 1 0 • $hi For. a l l off-diagonal terms z i d = z J i 6-20 34 6.3 Formation of the Admittance Matrix Y. The general form of the potential coefficients was derived in chapter 5. There i s no distinction i n form between diagonal and off-diagonal Elements. If i s written as K, . = K! . + A E . , 6-22 i j i j where is the correction term for the ground effect. Then the capacitance matrix of the transmission system may be written as C T - AC = (KT + AK)" 1 = Z'"1 - K'"1 AKK'-1 6-23 Defining gives AC = C'AKC = — C» (M + jN)0' 6-24 Hence the corrected capacitance matrix is C = C - + = C'(I - i-(M + jH)C») 6-25 The admittance matrix Y can be written Y = jtoC = jwC'(I - ~ ( M + JN)C ) = G + jwC" 6-26 where G = uO'HC' 6-27 and C" = C * (I - ~MC ) 6-28 ff£ 6.4 The A.B.C.D Parameters of the System. 35 By analogy to the single conductor case, the behaviour of an n-conductor transmission line can be described by the following equation; 6-29 where the parameters AjB,C and D are n x n matrices. The column vectors and Ig are voltages and currents at a distance IL from the sending end, and V and I those at the s s sending end. The A,B,C,D and the characteristic impedance matrices are derived in reference (8) and are — — V A B s t I C D I. s _ t _ A = B = C = R(cosh Y ± Z ) R [sinh-. — S -1 R _ 1Z Tsinh: ¥ ±&~ L \ . S - 1 ! D = S(cosh X. JL ) S -1 Z = R XT 1 R _ 1Z o i 6-30 6-31 6-32 6-33 6-34 36 Chapter 7. Numerical Results f o r a 500 kV Transmission Line. 7.1 Description of the Transmission Line. The ground effect on the distributed c i r c u i t parameters fo r an overhead transmission l i n e i s dependent on l i n e geometry, ground conductivity, ground r e l a t i v e permittivity, and frequency. The l i n e geometry can have a large number of va r i a t i o n s . In order to i l l u s t r a t e the significance of the formulae developed i n Chapters 4, 5 and 6 a t y p i c a l 500 kV l i n e i s chosen. For the given geometry the ground conductivity, r e l a t i v e permit-t i v i t y , and frequency are varied. The 500 kV l i n e i s chosen because recent developments i n long distance energy transmission have calle d f o r more detailed study of the properties of l i n e s at t h i s voltage l e v e l than has previously been ava i l a b l e . The single c i r c u i t l i n e without overhead ground wires consists of: A bundle of four conductors at the corners of an 18 inch square, per phase. Conductor size 583.2 MCM ACSR Conductor DC resistance ........ 0.1764 ohm/mile at 50°C Conductor diameter (including stranding factor) 0.948 i n Average phase spacing 40 f t . Average conductor height ....... 54 f t . The l i n e conductors are at equal heights above the ground i n a f l a t array. The ground conditions were considered to vary from dry rock to wet marsh land. This represents a ground r e l a t i v e p e r m i t t i v i t y range from 10 to 50 and a ground 37 conductivity range from 10""'' to 1 mho/m. 7.2 Ideal Parameters. From equation 4-33 I S ' _ Mi i n - l i h/m 7-1 Inserting the lengths in the ahove equation and converting the units to milli-henries per mile gives the following inductance matrix, not including internal conductor inductance 1.63 .339 .167 .339 1.63 .339 mh/mile 7-2 .167 .339 1.63 Inverting this matrix and multiplying by a known constant gives the capacitance matrix in units of micro-farads per mile C .0183 -.00302 -.00148 ^.00302 .0189 -.00302 -.00148 -.00302 .0183 u-f/mile 7-3 with the inductance in henries per meter and the potential coefficients in darafs per meter the matrices are, 5.10 1.06 .520 L = 2 x 10" 7 1.06 5.10 1.06 _.520 1.06 5.10_ "5.10 1.06 .520~ K = 18 x 10 9 1.06 5.10 1.066 .520 1.06 5.10 h/m 7-4 darafs/m 7-5 38 7 . 3 Skin E f f e c t . For the conductor with a radius of .479 i n and with a bundle spacing of 18 i n , the proximity effect i s considered n e g l i g i b l e . The i n t e r n a l conductor impedance per bundle i s taken as one quarter of the i n t e r n a l impedance per conductor and i s calculated from C-16, C-17, or C-21, C-22 or C-26 and C-27 depending on the magnitude of ^cou^a. 7.4 Impedance Correction. The impedance correction terms depend on the frequency, ground conductivity, and r e l a t i v e p e r m i t t i v i t y . Figs. 7.1 and 7.2 show the correction factors f o r the diagonal elements of the impedance matrix, according to the formulae, = I i ± + 2Q 1 1 7-6 E i l = Eii + f Pii ™ assuming no p e r m i t t i v i t y correction. The correction factors fo r the r e l a t i v e p e r m i t t i v i t y of 10 are shown i n figs.; 7 . 3 and 7.4. Since the difference i n values between the diagonal elements and the off-diagonal elements i s not great the curves of the off-diagonal elements have been omitted from the remaining figures. Figs. 7 . 3 through 7.6 show the correction factors fo r r e l a t i v e p e r m i t t i v i t i e s of 10 and 40. By comparing f i g . 7 . 3 through 7.6 with f i g s . 7.1 and 7.2 i t may be seen that f o r conductivities above 10~ mho/m the r e l a t i v e p e r m i t t i v i t y has n e g l i g i b l e e f f e c t , but the effect becomes quite pronounced fo r 3 9 conductivities below 10"^ mho/m. For example, the correction matrices without ground permittivity correction at a frequency of 10 kHz are 1 2 3 1 2 3 1 2 1 R = 1 2 3 1 2 3 1 2 3 ohm/mile 7 - 8 1 2 1 1 2 3 1 2 3 7 6 2 3 . 6 0 5 . 5 5 8 I = .605 . 6 2 3 .605 mh/mile 7 - 9 . 5 5 8 .605 .623 when<^ = 10 mho/meter. For the same frequency butd* = 10' mho/meter 60.8 58.0 50.6 R = 58.0 60.8 58.0 ohm/mile 7 - 1 0 50.6 58.0 60.8 . 1 3 7 .126 . 0 9 9 1 = .126 . 1 3 7 . 1 2 6 mh/mile 7 - 1 1 . 0 9 9 .126 . 1 3 7 For ground relative permittivity of 10 and at the frequency of 10kHz. 2 7 9 2 7 9 2 7 8 R = 2 7 9 2 7 9 2 7 9 ohm/mile 7 - 1 2 2 7 8 2 7 9 2 7 9 40 .698 .677 .629 1 = .677 .698 .677 .629 .677 .698 mh/mile 7-13. when A = 10"^ mho/meter, for the same frequency hut ^  - 10*" 2 mho/meter R = 61.2 58.4 51.0 58.4 61.2 58*4 ohm/mile 7-14 51.0 58.4 61.2 .137 .126 .099 .126 .137 .126 mh/mile 7-15 .099 .126 .137 I = From f i g s i 7.1 through 7.6 i t may he seen that for —2 conductivity below 10~ mho/m and a frequency above 1 kHz the ground relative permittivity should be included in the calculation. 7.5 Admittance Correction. The admittance correction terms were evaluated by the two methods described i n Chapter 5. First the integral in equation 5-14 was evaluated using numerical methods and then the correction terms were obtained from equation 5-29. Figs. 7.7 through 7.12 show a comparison of the correction terms obtained by the two methods. The discrepancy between the two sets of values Increases with decrease in ground conductivity. For the range of conductivities greater than 10'Wm and the range of relative permittivities between 10 and 50 the maximum •41 discrepancy i s of the order of 10%. This is considered tolerable (16) • • • for propagation calculations v ' in view of the relatively small magnitude of the correction terms as compared with the potential coefficients. The difference in the magnitude between the diagonal elements and the off-diagonal elements i s of the order of 2% at the lowest conductivities and thus the off-diagonal elements are not included i n figs. 7 . 7 through 7.12. —'3 For conductivities of 10 mho/m or greater a l l the correction factors become negligibly small. In calculating the propagation constants a correction of 3% or greater i n the diagonal elements is considered significant. A comparison of exact and approxi-mate values of the correction factors and their magnitude are shown i n figs . 7 . 7 through 7.12. This requires correction factors to be calculated for conductivities less than 10 ' mho/m and frequencies greater than 100 Hz. The potential coefficient terms are corrected according to the formula i K i j = K i j + 1 8 x 109(2Mi;) + J2N±;J) 7-16 To illustrate the magnitude, the potential coefficient correction terms for the diagonal elements at frequencies of 1 kHz, 10 —3 kHz and 100 kHz and for conductivities of 10* mho/m and 10""^  mho/m for a relative permittivity of 10 are shown i n table 7.1. 42 Table 7.1 Potential coefficient correction terms. 2M 2U f = 1 kHz = 1 0 " 3 mho/m <d> = 10"^ mho/m 0.03 0.07 fA = 10 kHz £ = 1 0 * 3 mho/m ^ = 1 0 " 5 mho/m 0.003 0,43 0.007 0.34 f = 100 kHz 6> = 1 0 " 3 mho/m <i= 10"^ mho/m 0.028 0.66 0.047 0.17 7.6 Conclusion. It has been found that the impedance should always be corrected for ground conductivity. In addition when c^ /we is less than 180, the impedance should be corrected for the ground relative permittivity and the capacitance should be corrected for both ground conductivity and relative permittivity. When oVwe is greater than 180, these correction terms are small, less than 3% approximately, and can be neglected. FIG 7-1 IMPEDANCE CORRECTION FACTOR Q WITH 9*0. <f*Q FREQUENCY HZ FIG 7-2 IMPEDANCE CORRECTION FACTOR P WITH 6*0. <f*0 •5 l& FREQUENCY HZ ;44 FIG 7-3 IMPEDANCE CORRECTION FACTOR Q WITH rt«/0 •6 •7 •6* *5 P '4 -3 '2 -1 2 3 4 S 10 10 10 10 10 FREQUENCY HZ FIG 7-4 IMPEDANCE CORRECT/ON FACTOR P WITH ft=/0 10 . . . . . 9 = 0 6-3*5° • \\ c^jof^^ — -\ •vvr ""V • i j 10 /o 3 4 10 10 FREQUENCY HZ 10 45 FIG 7-5 IMPEDANCE CORRECTION FACTOR Q WITH />«50. B*0 <» 2 <n3 <J ,„5 10 10 10 10 10 FREQUENCY HZ FIG 7-6 IMPEDANCE CORRECTION FACTOR P WITH n=50. 0=0 8 7 •6 •5 •4 •3 •2 -1 10 ,• • s^t-ib1 &*10 tf./o'^^ , 1 , .L 1 . • l,„. J 1,-1 0 10 10 10 10 FREQUENCY HZ 10 10 .46 FIG 7- 7 -5 POTENTIAL COEFFICIENT CORRECTION FACTOR N WITH 9*0. 0**10 mho/m -201 nsSO n =40 n*30 n = 20 n*10 10 10 FREQUENCY HT FIG 7-8 POTENTIAL COEFFICIENT CORRECTION FACTOR M WITH 9 - 0. d-1(T mho/ m 'SO '40 '30 M 20 '10 '00 10 — — — — EXACT APPROXIMATE / • " \ / / v / ' * \ /' • \ \ \ \ <\ M MM** ^ > n mW 1 d - ^ i i i — 1 _ _ ~*"^*^^*^^^ i i n "20 n-3Q n*40 n*5Q 10 10 10 FREQUENCY; HZ 10 10 47 '•' ::y:'rp:^ FIG 7-9 POTENTIAL COEFFICIENT CORRECTION FACTOR N WITH 0 • 0. tt»fo4 mho/m -20 ' 10 •oo 10 E M c r APPROXIMA Ti /> \ \ JKSoV A ,„ 1 I „•,' .-.1. .„J_ „ 1 J - i i n*50 n'40 n*30 n-20 />-/o 10 FIG 7-10 10 10" 10 FREQUENCY HZ 10 70" POTENTIAL COEFFICIENT CORRECTION FACTOR M WITH 8 = 0. <M0 mho/m -30\ 20 M 10 '00 — . — ' — _ — EXACT APPROXIMATE * ' . . . . . . Ft ft \\ ^ \ v> 1 1 • • i —- l l n-10 n-20 n»40 10 10 3 4 10 10 FREQUENCY HZ 10 48 FIG 7- // - 3 POTENTIAL COEFFICIENT CORRECTION FACTOR N WITH O*Q. 0**10mho/m 'OS 04 •03 - ):^y •.;.-..<:-,: :;02 N -01 01 -02 ff *\ // \\ //^ \ . . ..',< \ U \ \ • 1 ' 1 - 1 1 . _J L , i i n*20 n-SO 10 FIG 7-12 10 10 10 FREQUENCY HZ 10 15 • 10 M •OS • • • r IT** 1 1 •• I—™j n*70 n-30 n<40 n-SO 10 10 10 10 FREQUENCY HZ 10 10 49 Chapter 8. Calculation of the Mode Propagation Constants. The correction factors given in Chapter 7 are inserted into the transmission line equations, 6-1, and the mode propa-gation constants are calculated from equation 6-5. Over the frequency range from 10 Hz to 1 MHz two of the mode propagation constants are nearly equal and over the range from 100 kHz to 1 MHz a l l three propagation constants are nearly equal. To increase the accuracy of the calculation both the impedance and the admittance matrices are treated as follows Z = ^ Z ' 8-1 Then let Y = j2«ewY' 8-2 A = ZY = - {/eiiZ'Y1 = co2eiiA» 8-3 The real part of the diagonal elements of the matrix A' are nearly equal to -1 while the imaginary part of the diagonal elements and the off-diagonal elements are relatively small in magnitude, let A' = A" - I 8-4 then from | A « l|= 0 8-5 one has iJA" - (ff / 2 + 1)1 | = 0 or | A ! I - # / . 2 I I = 0 8-6 50 Then the eigenvalues of A" w i l l not he as close together as those of A'. The propagation constants can now he obtained from, a j + J ^  = ^  i = / ( A u ^ 2 - 1) 8-7 where is attenuation i n nepers per meter and i s the phase constant in radians per-meter. The mode velocity is obtained as follows v i = p\" m/sec 8-8 In terms of the velocity of light in free space, the normalized mode velocity is v- = 2itf/p i ]/7JT' 8-9 The velocities at modes 1 and 2 are very close to unity under a l l conditions, and a l l three mode velocities are close to unity for a perfectly conducting grounds However, for f i n i t e ground conductivity and relative permittivity the velocity of mode 3 varies from 0.48 to nearly 1 per unit (. Fig. 8-1 shows the velocity of mode 3 without per-mittivity correction for conductivities from 10~^ to .1 mho/m. Figs. 8-2 through 8^-4 show the velocity of mode 3 with impedance correction but without capacitance correction, for relative permittivities from 10 to 50 over the same range of conductivities as in f i g . 8-1. Figs. 8-5 through 8-7 show the velocity of mode 3 with capacitance correction over the same range of conductivities and permittivities. From the figures i t can be seen that relative permittivity has l i t t l e effect on the 51 velocity of mode 3 for a conductivity greater than 10 mho/m, and the capacitance correction only affects the calculation where the conductivity is less than 10 mho/m. In figs. 8-1 through 8-7 i t can he seen that the permittivity correction of both impedance and capacitance tends to increase the velocity of mode 3 at high frequencies. Pig. 8-8 shows the mode attenuation constants, i n nepers per mile, for a perfectly conducting ground, in which case the attenuation is due entirely to internal conductor a.c. resistance. Pig. 8-9 shows the mode attenuation constants for a range of ground conductivity from 10~^ to -1 mho/m without permittivity correction of the impedance and without capacitance correction. Pigs. 8-10 through 8-12 show mode attenuation constants over the same conductivity range and over a range of relative permittivities from 10 to 50, with the impedance corrected for conductivity and relative permittivity, hut without capacitance correction. Pigs. 8-13 through 8-15 show the mode attenuation constants with conductivity and relative permittivity correction for the impedance and the capacitance over the same ranges of relative permittivity and conductivity. The attenuation of mode 3 is most affected hy changes in ground conductivity, the attenuation of mode 2 is affected to a lesser degree and the attenuation of mode 1 i s least affected. Prom figs. 8-9 through 8-15 i t can he seen that for conductivities greater than 10 mho/m the relative permittivity has l i t t l e effect on the mode attenuation constants. For lower conductivities both the impedance correction and the 52 capacitance correction have an increasing effect on mode attenuation for frequencies greater than 100 Hz. Tables 8-1 through 8-3 show a comparison of variations in the attenuation for a conductivity of 10~^ mho/m and at frequencies of 100. kHz, ,.10 kHz and 1 kHz. Table 8.1 Attenuation i n nepers/mile for ^  = 10"^ mho/m and f = 100 kHz Mode 3 Mode 2 Mode 1 £To permittivity correction 3.6 x 10" 1 1.5 x 10~3 1.3 x 10"3 Impedance correction n = 10 n = 20 n = 30 n = 40 n = 50 7.0 x 10"1 7.2 x 10"1 7.3 x 10"1 7.4 x 10"1 7.5 x 10" 1 2.7 x 10~3 4.0 x 10~3 5.2 x 10~3 6.5 x 10~3 7.6 x 10~3 1.4 x 10~3 1.4 x 10"3 1.4 x 10"3 1.5 x IO"3 1.5 x IO" 3 Capacitance correction n = 10 n = 20 n = 30 n = 40 n = 50 4.6 x 10"1 5.8 x 10"1 6.2 x IO""1 6.6 x IO" 1 6.6 x 10"1 2.7 x 10~3 4.0 x 10~3 5.2 x 10~3 Si5 x 10"3 7.6 x 10"3 1.5 x IO"3 ti it »t i» S 3 Table 8.2 Attenuation in nepers/mile f o r ^ = 10~^ mho/m and f = 10 kHz Mode 3 Mode 2 Mode 1 Wo permittivity-correction 3.7 X i o - 2 3.3 x l(f •4 4.0 x 10" •4 Impedance correction n = 10 4.4 X i o - 2 3.4 x 10" •4 4.0 x 10" •4 n = 20 5.2 X IO"2 II i t n = 30 5.7 X IO"2 It i t n = 40 6.0 X IO"2 It i t n = 50 6.3 X IO"2 t l t i Capacitance correction n = 10 3.3 X IO"2 3.4 x 10" •4 4.0 x 10" -4 n = 20 4.0 X IO"2 t i i t n = 30 4.6 X IO'2 i t t i n = 40 5.1 X IO"2 II t i n = 50 5.4 X ICT2 t i i t From table 8-3 i t can be: seen that capacitance correction is significant for low conductivities even at a frequency 1 kHz for the attenuation of mode 3. For the attenua-tion calculation, conductivity and relative permittivity correction of impedance and capacitance matrices i s required for conductivities below 10 mho/m and frequencies above _3 100 Hz. For conductivities above 10 mho/m no relative per-mittivity correction of the impedance and no capacitance correction (2) are required and Carson's formulaev ' are sufficient to obtain 54 the ground correction terms for the circuit parameters. Tahle 8.3 Attenuation in nepers/mile for<^ = 10"^ mho/m and f = 1 kHz Mode 3 Mode 2 Mode 1 ISTo permittivity correction 3.6 x 10~3 1.15 x 10"4 1.4 x 10"4 Impedance correction n = 10 n = 20 n = 30 n = 40 n = 50 3.3 x 10~3 3.4 x 10~3 3.4 x 10~3 3.5 x 10~3 3.6 x 10""3 1.15 x 10"4 it ti ti II 1.4 x 10"4 II II ti II Capacitance correction n = 10 n = 20 n = 30 n = 40 n = 50 7.8 x 10~3. 4j5 x 10~3 3.5 x 10~3 3.4 x 10~3 3.4 x 10"3 1.15 x 10"4 II II II II 1.4 x 10"4 it it ti tt An insight into the behaviour of the mode propagation constants may be obtained by comparing the transmission line with a line that i s transposed at short intervals. Then a l l the off-diagonal terms in both the impedance and admittance matrices would be equal and the diagonal elements in each matrix would also be equal. The transformation matrices, S and R, would be equal and would be the matrix, used for transformation 55 to symmetrical components. Thus the mode 3 propagation constants can he compared with the zero sequence propagation constants in a balanced system. To illustrate this similarity the R and S matrices for a frequency of 100 kHz, a ground conductivity of 10~4 mho/m and a relative permittivity of 10 are .484 - j.161 -.763 + j.106 .592 - j.003 R = -.787 + j.177 .115 - j.102 .566 + j.014 8-10 .299 + j.00 .619 + j.oo .570 + j.00 .223 + j.246 -.750 - j.246 .517 + j.oo S t = -.603 - j.H5 - . 1 4 3 + j.126 .766 + j.00 8-11 .571 + j.002 .570 j.017 .591 + j.00_ .275 + j.246 -'.833 — j.147 .543 + j.020 R"1 = -.656 - j.064 -.125 + j.067 .808 + j.00 8-12 .568 - j.005 .572 + j.005 .593 - j.011 FIG 6-1 MODE 3 PROPAGATION VELOCITY WITHOUT PERMITTIVITY CORRECTION 10 102 103 IO4 W5 10 . FREQUENCY HZ FIGB-2 MODE 3 PROPAGATION VELOCITY. NO CAPACITANC&>CQRRECTION WITH n 57 FIG 8-3 MODE 3 PROPAGATION VELOCITY. NO CAPACITANCE CORRECTION WITH n-30 10 10 10 10 FREQUENCY HZ FIG 8-4 MODE 3 PROPAGATION VELOCITY. NO CAPACITANCE CORRECTION WITH n*50 10 10 10 10 FREQUENCY HZ 10 FIG 8-5 FREQUENCY HZ v :'V^f- :--V v 60 61 FIG 6-9 MODE A T T E N U A T I O N W I T H N O P E R M I T T I V I T Y C O R R E C T I O N iot — i — — . ' • ,-" i V — i — ~ ~ r r~ 64 FIG 8-12 MODE ATTENUATION. NO CAPACITANCE CORRECTION WITH n-50 70i i — - i i - i FREQUENCY HZ 66 FIG 8-74 MODE ATTENUATION. APPROXIMATE CAPACITANCE CORRECTION WITH n «30 . i o v — ; i — - — — i — r — — i 1 67 68 Chapter 9. Conclusions. The methods of C a r s o n ^ and Wise^"^ have been extended to calculate the distributed circuit parameters for a multi-conductor transmission line. The correction terms for the distributed circuit parameters have been derived for variations in line geometry, ground conductivity, ground relative permittivity, and for a frequency range up to 1 MHz. Their corrected parameters have been used in the transmission line equation to calculate the mode propagation velocities and attenuation constants. A practical example has been investigated in detail to show the variations in the distributed circuit parameters and the mode propagation constants due to various ground condi-tions. For a typical 500 kV transmission line the impedance should always he corrected for ground conductivity. When <<^/toe i s less than 180, the impedance should, in addition^ be corrected for the relative permittivity of the ground and the capacitance should be corrected for both the ground conductivity and the relative permittivity of the ground. When equals 180, the maximum error is approximately 3$ in Carson's correction of the impedance parameters and the uncorrected potential coefficients. The i n i t i a l rate of rise of recovery voltage in circuit breakers clearing faults in power systems can be evaluated solely in terms of the characteristic impedance of the transmission line and the energizing conditions. The former depends in turn on the corrected impedance and admittance 69 matrices and the natural frequency of the energizing system. Thus i t may he possible to investigate the effect of ground conditions on the i n i t i a l rate of recovery voltages for circuit breakers opening under fault conditions. The program developed for calculating the mode propagation constants in this thesis may be extended to include terminal conditions. This allows the investigation of the various permutations to find the optimum condition for energizing and terminating a line for carrier communicationv. In the calculation of the mode propagation constants in this thesis the conventional formulae for the internal impedance of the line conductors have been used with the assumption .that the equivalent radius i s the maximum radius over the strands. For bundled conductors further investigations would be required to determine the internal impedance more accurately. Since the internal impedance affects the diagonal elements of the impedance matrix i t has a direct effect on the mode propagation constants. Further developments in this f i e l d should be directed to find the effects of corona and tower footing resistance on the transmission system parameters. Corona disturbs the electric f i e l d adjacent to the conductors and may have a noticeable effect on the parameters. The effect i s non-linear since i t is voltage dependent. The tower footing resistance may become important for lines with overhead ground conductors where the tower spacing approaches a quarter of the wave length of the impressed signal. 70 The procedure developed in this thesis i s solely for overhead transmission lines. In the present form i t i s unsuitable for the calculation of propagation constants of underground systems, which presents quite a different problem due to their geometric configuration with respect to the ground. 71 Appendix A. The Field due to an Isolated Oscillating Current  Element. In an isotropic homogeneous medium Maxwell's f i e l d equations are stated a s ^ 4 ^ V*E = -jcouH A-l VxH = J + 3 we E A-2 C where, the time dependence i s sinusoidal and the conductivity acc where e = e - j c to df the medium is counted for hy the complex permittivity e , C Define a vector TT such that H = jcoe V X T A-3 c Then from equation A f l and,, A-2 V x E = to2ec^VxTT = -m2 VxlT A-4 hence E = -m2 TT +V(0) A-5 where 0 is a scalar function of position. Then from equation A-2 jweVx (VxTT) = J + 3toec(-m2TT +^/(0)) A-6 Dividing by jwe and expanding the f i r s t term of equation A-6 gives - V 2 7 T + V - V - T T = - * 2 T T A-7 C let V(0) =VV."TT A-8 (19) a Lorentz type condition, and equation A-7 becomes 72 V21T = m 2 7 T - ^ A-9 and E = -m27T + W . T A-10 Let the solution of equation A-9 be -mrR A e R dv A - l l o where v Q is a small closed surfaoe at the point (x,y,z) in Cartesian coordinates. Then -m R V 2T=V 2 £ ^ - f T-*v r r 2 " m R — i I [nTAe -m R 0 n = J [ — R — + A* 'V 2 (|)Jdv vo 2 r -m R = m o Substituting A-12 into A-9 gives T + Ae" V 2 (J) dv A-12 p -m R p -j J = - jwec 1 Ae V (f) dv . A-13 o The solution of A-13 i s 0 when R ^  0 J = 4 A-14 jwecA4it when R = 0 The current flows only in the z dlreotion, henoe J and J * y are both zero at r equals zero. When the element has a current I and a cross section a, J = i at R = 0 A-15 z a Hence A = , * a A-16 4«0we a 73 a n d ^ -m R Ie . 4 « 3(oeaR o Assuming v Q i s small and independent o f r, then -m R where z 4 « d w e aR o v = sa o and s i s the elemental length which i s replaced hy dz. Hence -m R TT _ Ie dz A _ 1 7 "z ~ 4*jwe R A ± f TT X = 0 A-18 TV = 0 A-19 •7 74 Appendix B. The General IT Vector in Cylindrical Coordinates, Fig, B-l Cylindrical coordinate system for the internal f i e l d The wave equation for the H ^ vector in the coordinate system shown in f i g . B-l i s The solution i s obtained by separating the variables, let TT Z = R(/>) $ ( 0)Z(a) B-2 Then 1 ^ "/? c ^ R l ^ 1 d l . 1 Vz R f a/3 L r ^ J + < ^ 2 T 7 * <f $  ^ 2 Z 2 p _ = m ±i—'3 Since the f i r s t two terms of B-3 are independent of zt the equation becomes U^Z B-4 wheret 2 . 2 . 2 = u + m and u i s a constant. B-5 75 The solution of B-4 i s Z = k±e + A 2e Q B-6 Then B-3 reduces to R ^ K c 3 f J + $ ^ 0 2 + U / ° = 0 B-7 Since the second term of the equation i s independent of r, B-7 can he written as + p 2$ = 0 B-8 0 0 * * 2 where p i s a constant. The solution of B-8 i s $ - B 1 cos P 0 + B 2 sin P 0 B-9 Equation B-7 now hecomes ^D*ifl + [(^)2-p2]R=o which can he transformed to the form </» »> ^ u 7 i^q] + [ > u ) 2 - P 2 ] * = o B - l l This i s Bessel*s equation of order P. The solution has the following form R = C Z {f> u) B-12 where Z^(^u) indicates the generalized form the the Bessel function. The exact types of Bessel functions required for the solution depend on the boundary conditions of the particular problem. Then the complete solution for "TT2 i s the sum of a l l 76 possible solutions and may be writen as TTZ = ^ ( B l p c o s P 0 + B 2 psinP0)* J ( A - J u ) P=0 e - Y z + A 2 ( u ) e y z ) B-13 xC Zp(/>u)du The wave equation in the Cartesian coordinate system can be separated into three components. Each component satisfies (7) IT B-14 where TT may be either "IT „, ~T„ or TT . . Thus the solutions x ' y z w i l l be similar functions. For the coordinate system shown in f i g . B - 2 . The solution for T and TT w i l l be similar and z y of the general form shown in equation B-15. This is the form of the solution used in the ground effect calculations in Chapter 3. y Fig. B -2 Cylindrical Coordinate System for the External Field 77 oo oo ^ ( B i p c o s P V + B 2 p s i n P y ) J ( k j ^ e - * 7 + A 2 ( u ) e * y ; P=0 x Z p(ru) du B-15 78 Appendix C. Derivation of the Internal Impedance of Cylindrical Conductors. In this Appendix we shall he concerned only with the fields inside the conductors. Consider a solid cylindrical conductor with the axis along the z-axis of the coordinate system shown in f i g . B-l. The direction of propagation i s along the positive z-axis. The electromagnetic f i e l d inside the conductor consists of the TM mode only. A Hertz vector of the TT type in the z direction is used to describe the f i e l d . Due to the circular symmetry of the system the f i e l d components are inde-pendent of 0, and hence the Hertz vector is also independent of 0. The solution of the wave equation for"iT 2 i s given in equation B-13. However as TV „ is independent of 0 and must be z fini t e at the origin only the f i r s t term of the solution exists and the required Bessel function is of the f i r s t kind of order zero. Therefore IT = A e ~ * z J (/>u) C-1 z o ' Using equations A-3 and A-10 the f i e l d components E g and H^ are E = (-m2 +* 2)TT„ C-2 H0 = jtoecuAe" u) C-3 Let the external radius of the cylinder be a, then at the boundary Y » I e Hv = ° - _ <M 79 C o n s e q u e n t l y I A = j w e c 2 * a u G " 5 v 2 2 W h e n o i s v e r y l a r g e , Y i s n e g l i g i b l e a s c o m p a r e d t o m . T h e n f o r p r a c t i c a l p u r p o s e s u 2 = - m 2 C - 6 a n d o v „ - m 2 I J Q ( / > u ) e - y z E = f t C 7 z j w e 2 2 « a u J 1 ( a u ) L e t E g a t r a d i u s a b e g i v e n b y E„ = z , I e ~ * 2 C - 8 z i o w h e r e z i i s t h e e q u i v a l e n t i n t e r n a l i m p e d a n c e o f t h e c y l i n d r i c a l c o n d u c t o r . T h e n - m J Q ( a u ) z i = j w e c 2 « a u J 1 ( a u ) F o r a c o n d u c t o r ' c ~ ~ 3 co e_ = - < ~ C - 1 0 a n d w h e r e T h e n m = j 0 , V C - l l a' = Jujie? C - 1 2 3 1 , 3 c c ' J o ( 3 1 < 5 a ' a ) 2 « a d J 1 ( d 1 , 5 a ' a ) » 4 = . ' C - 1 3 The n u m e r i c a l e v a l u a t i o n o f z ^ may b e d i v i d e d i n t o t h r e e r e g i o n s , d e p e n d e n t o n t h e v a l u e o f a ' a . 80 1) When a*a < 0.1 (13) For small values off a'a the following expansion holds, 2 i = i ^ * ( « ' a ) 2 A ( a ' a ) 4 , ( a ' a ) 6  « 8 + 192 " J 3072 ' ' • ' I w a 2 j C-14 l e t Z i = R i n t + 3 X i n t C-15 Then f o r small values of a'a * a o and which are the formulae obtained when a uniform current d i s t r i -bution i s assumed. 2 j When a'a > 10 For large values of a'a z i may be written as z = « ' U +• .1) 0-18 1 2j2*a<5 Let the skin depth d be defined as d =M, c*i? Then 5 i = 2*adJ which i s Rayleigh's formula. Then R i n t = 2*ad<S C " 2 1 L ^ = O^^J.. C-22 J i n t ~ 2«ad^ 8 1 3) When 0 .1 ^ a'a — 10 Equation C-13 may be changed into a more conyentient form f o r numerical computation by l e t t i n g J 0(d 1' 5a'a) = Ber(a'a) + JBei(a'a) C-23 The f i r s t order Bessel function i s obtained by d i f f e r e n t i a t i n g equation C-23 J 1(j 1 , 5a'a) = j 0 ' 5 JBer'Co'a) + jBei'(a'a)] C-24 Then the i n t e r n a l impedance z^ becomes _ a' r»Bei(a'a) + .1Ber(a'a) "I n 9 r . z i ~ 2*a<* |_Ber'(a'a) + jBei'(a'a)J W 5 and the resistance and inductance are TJ - «' [Ber(a'a)Bei' (a'a) - Bei(a'a)Ber'(a'alj * i n t " \ — Ber' 2(a'a) + B e i ' 2 ( a ' a ) ] C " 2 6 u. fBer(a'a)Ber' (a'a) + Bei(a'a)Bei' (a'a)] c _ 2 7 a a ' L Ber , 2(a'a) + Bei' 2(a'a) J L i n t 2«aa power series The numerical values of the functions Ber, Bei, Ber', tabi (15) ( 1 7 ) and Bei' are tabulated v , or they may be calculated from 82 Appendix D. Solution of the Integral in Equation 4-50. The integral is O O r x = M i it Qv? + j - u)e- u( h" +y">cos x»u du D-l where and o x" = X ' s = sju>\id* X y" = y' s = sj(4i^*y h" = h's = s J cop-ci'h 2 s = 1 + je oco(n-l) D-2 D-5 Equation D-l has at least two possible solutions, (2) one suggested by J.R. Carsonv ' for small values of R" and the other obtained by repeated partial integration for large values of R". The f i r s t solution i s I = P + JQ R+e^  ^ 7 y [ E L ( 3 R « e 3( 0 + ^ ) + G(dR«'e^(e + ^ ) \ + R"e _ 1 re2J(e-cf) + e-2j(0+<P| D - 4 R" L J Where K^(x) i s a Bessel function of the second kind and the function G(x) is defined a s ^ P/ \ V ( - l ) n + 1 x 2 n 2 2 1 1- 1 nl.(n-l)I . . 0( X) = ^ ^ - ( 2 n + l ) ' f 2 n - l ) n=l 83 From D-4 and D-5 P = 0.125it(l-s 3 r+s 1 : L) + 0.51n(|7^(s l r+s 3 i) - 0.5cf ( l - s 1 : L - s 3 r ) - 0.59(s 2 r +s 4 i) + ^ = ( r 3 r - r l r - r 3 1 - r 1 1 ) + 0 . 5 ( r 2 r + r 4 i ) D-6 Q = 0.25-0.125«(s l r-s 3 i) -* O.^U+s^+s^) - 0 .5©(s 4 r-s 2 i) 0 . 5 1 n ( f | r ) ( l - s 3 r + a l i ) + ^ V r 3 r + r 3 i - r l i ) + ° « ^ r 2 i - r 4 r ) where ^ s l r n=l 0 0 *n+l n=l 00 >2r " n=l 0 0 \n+l s 2 i n=l 00 vn+1 3r n=l 0 0 sn+l s 3 i n=l 00 s4r = ZI 2nl"(2n!l): (^)4nsin4n©cos4n<f D - l 4 n=l D-7 = Z 2ni ^ 2 n - l ) i (^) 4 n" 2QOs(4n-2)QcoB(4n-2)<f . D-8 s l i = Z 2J"(2n-l)» (^)4n"2cos(4n-2)©sin(4n-2)cT D-9 s-~ = ZI 2n^2n-l)r (^) 4 n" 2sin(4n-2)9co S(4n-2)<T D-10 = ZI 2J1V2Z1U ( ^ ) 4 n - 2 B i n(4n-2)0sin(4n-2 ) c T D - l l B - = Z2ni /(2n+l)f (^)4nc°s4n©cos4n <f D-12 = ZI 2 ^ ^ ^ ! ) : ( Si) 4 ncos4nesin4n<T D-13 8 4 3 4 i = Yx 2ni~(L-l)< (^) 4 nsin4n@sin4ncf D-15 n=l Cx3 l r Z n=l CO (or, o\t2 9 4 (n-l!)p„4n-3/ -, \n+l 12n z21 1_2 R i^l} c o s ( 4 n _ 3 ) © c o s ( 4 n _ 3 )<f ( 4 n - 3 ) r ( 4 n - l ) D-16 l i 7 (or, ?)i2 o4(n-l) n , ,4x1-3/ - i \n+l ^ 2 n ~ 2 ^ 2 9 S Lrii cos(4n-3)©sin(4n-3)o r ( 4 n - 3 ) r ( 4 n - l ) D-17 • 2 r 2i n=l co ph+1 n=l j=l z[z* n=l Lj=i _ 1 4 n _ 1 4n D-18 5 i = i ^ T T T ( ^ ) 4 r l - 2 o o s ( 4n -2 ) e a i n ( 4 n - 2 D-19 3r ^ ( - l ) n + 1 ( 2 n - l ) j 2 2(4n-2) R„4n-l n=l ( 4 n - l ) J (4n+l) cos(4n-l)©cos (4n-l)<f D-20 3i n=l 7 ( - l ) n + 1 ( 2 n - l ) . ' 2 2 4 n~ 2R" 4 n- 1 \ (4n-l) I 2 (4n+l) c o s (4 n-l) ©s in (4n-l) cf D-21 ' 4 r GO -2n+l 11=14=1 co f2n+l 4n+2 ^ 2 ^ ^ ( ^ ) 4 n c o s 4 n 9 c o S 4 n c T D-22 •41 n=l > 1 ZI Z J - 4nT2 2 ^ ^ n T ^ c o s 4 n @ s i n 4 n <*" D " 2 3 (9) 8 5 The second solution is <$o r p = - cos2Q, cos22T _J= cos(2n-l)Q 0 0 B ( 2 r^ 1 ) < f 2 fei R"^11--1- L H"2 ' /2 fe] + (-l) n + 1sin(2n-l)cf]| D-24 Q = OOB2Q BlnfrT + J . V oos(2n-l)Q L^g ^ U t f v n=l + (-l) nsin(2n-l)<f] D-25 For numerical computation the solutions are divided into two ranges 1) When R"-£ 5.0 In this range use equations D-6 through D-23 2) When R" > 5.0 In this range use equations D-24 and D-25. (2) These equations reduce to J.R. Carson's solutions when SQ is equal to one and cT is equal to zero. 86 References. 1. Carson, J.R., Wave Propagation Over Parallel Wires; the Proximity Effect, Philosophical Magazine, Vol. XLI, April, 1921, pp. 607-633. 2. Carson, J.R., Wave Propagation in Overhead Wires with Ground Return, B.S.T.J., Vol. V, October, 1926, pp. 539-554. 3. Rice, S.O., Steady State Solution of the Transmission Line Equations, B.S.T.J., Vol. XX, April, 1941, pp. 131-178. 4. Bewley, L.V., Travelling Waves on Transmission Systems, Dover, 1963. 5. Hayashi, S n Surges in Transmission Systems, Denki-Shoin, Japan, 1955. 6. Rudenberg, R., Transient Performance of Electric Power Systems, Wiley, 1951. 7. Sunde, E.D., Earth Conduction Effects in Transmission Systems, Van Nostrand, 1949. 8. Wedepohl, L.M., Application of Matrix Methods to the Solution of Travelling Wave Phenomena in Poly-phase Systems, Proc. I.E.E., Vol. 110, No. 12, December, 1963, pp. 2200-2212. 9. Wise, W.H., Effect<of Ground Permeability on Ground Return Circuits, B.S.T.J., Vol. X, July, 1931, pp. 474-484. 10. Wise, W.H., Propagation of High Frequency Currents in the Ground Return Circuits, Proc. I.R.E,, Vol. 22, No. 4, April, 1934, pp. 522-527. 11. Wise, W.H., Potential Coefficients for Ground Return Circuits, B.S.T.J., Vol. XXVII, 1949, pp. 365-371. 12. Arismunandar, A., Capacitance Correction Factors for Transmission Lines to Include the Finite Conductivity and Dielectric Constant of the Earth, Trans. I.E.E.E., No. 63-1030, Special Supplement, 1963, pp. 436-456. 13. King, R.W.P., Fundamental Electromagnetic Theory, Dover, 1963. 14. Collin, E.C., Field Theory of Guided Waves, McGraw H i l l , I960. 15.. McLachlan, N»W., Bessel Functions for Engineers, Oxford, 1955. 87 16. Hedman, D.E., Propagation on Overhead Transmission Lines; I. Theory of Modal Analysis, II. Earth Conduction Effects and Practical Results. Trans. I.E.E.E., Vol. 83, July, 1965r P P » 665-670. 17. Dwight, H.B., Mathematical Tables, Second Edition, Dover, 1958. 18. Jahnke, E., and Emde, P., Tables of Functions, Dover, 1945. 19. Bowman, P., Introduction to Bessel Functions, Dover, 1958, 20. Pipes, L.A., Matrix Methods for Engineering, Prentice-Hall, 1963. 21. Wedepohl, L.M., Shorrocks, W.B., and G-alloway, R.H., Calculation of Electric Parameters for Short and Long Poly-phase Transmission Lines, Proc. I.E.E., Vol. I l l , No. 12, December, 1964, pp. 2051-2059. 22. Wedepohl, L.M., Electric" Characteristics of Poly-phase Transmission Systems with Special Reference to Boundary-value Calculations at Power-line Carrier Frequencies, Proc. I.E.E., Vol. 112, November, 1965, pp. 2130-2112. 23. Dowdeswell, I.J.D., Matrix Analysis of Steady State, Multi-Conductor, Distributed Parameter Transmission Systems, M.A.Sc. Thesis, University of British,Columbia, November, 1965. 

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