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 i n the Department of E l e c t r i c a l Engineering We accept t h i s thesis as conforming to the required standard Research Supervisor Members of the Committee Head of the Department Members of the Department of E l e c t r i c a l Engineering THE UNIVERSITY OF BRITISH COLUMBIA June, 1966 In p r e s e n t i n g this thesis in partial f u l f i l m e n t 'of t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y Columbia, I agree that the Library f o r . r e f e rence and s t u d y . tensive by copying o f t h i s shall for scholarly t h e Head o f my D e p a r t m e n t o r by h i s understood cial gain that shall Department o f "JOVE permission It i%C> ETtja/A)&E£ftJ6i Columbia for ex- p u r p o s e s may be g r a n t e d this thesis n o t be a l l o w e d w i t h o u t my w r i t t e n Et-£CT£(CAt. available representatives. copying o r p u b l i c a t i o n . of The U n i v e r s i t y o f - B r i t i s h V a n c o u v e r 8, C a n a d a Date make i t f r e e l y I f u r t h e r agree that thesis of British It for is finan- permission. ABSTRACT A general a n a l y t i c a l method to derive the distributed c i r c u i t parameters and mode propagation constants f o r an nconductor transmission l i n e i s developed. The analysis uses electromagnetic f i e l d concepts and the results are interpreted i n terms of distributed c i r c u i t parameters. The procedure involves transforming the problem of the n-conductor line above a ground with f i n i t e conductivity into that of an nconductor above a ground with i n f i n i t e conductivity. Correction factors are added to account f o r the f i n i t e conductivity of the ground. The distributed c i r c u i t parameters thus calculated are used to calculate the mode propagation constants over a frequency range from 10 Hz to 1 MHz f o r values of ground conductivity varying between 1 mho/m and 10~^ mho/m and r e l a t i v e permittivity varying between 10 and 50. Numerical results f o r the distributed c i r c u i t parameters and mode propagation constants f o r a t y p i c a l 500 kV single c i r c u i t transmission l i n e and various ground conditions are given. The results show that one mode has a higher attenuation and a lower v e l o c i t y than either of the other two modes, suggesting the zero sequence mode f o r a completely balanced system. ii TABLE OE CONTENTS Page Abstract i i Table of Contents i i i L i s t of I l l u s t r a t i o n s v L i s t of Tables vii L i s t of Symbols viii Acknowledgement xi 1. Introduction 2. Assumptions and Approaches 3. 1 ,. 2.1 General Description of the System 6 2.2 Basic Assumptions 7 2.3 C i r c u i t Parameters ,. 5. 8 Derivation of the Hertz Vector due to Current i n a Wire 4. 6 9 3.1 Hertz Vector due to a Current Element i n a Medium of I n f i n i t e Extent 10 3.2 Hertz Vector due to a Current Element above a Ground with I n f i n i t e Conductivity 10 3.3 Hertz Vector due to a Current Element above a Ground with F i n i t e Conductivity 11 3.4 Complete Hertz Vector f o r a Current i n a Straight Conductor above the Ground 16 Evaluation of Transmission Line Resistance and Inductance Including Ground E f f e c t s 17 4.1 Derivation of the Series Impedance Formulae ... 17 4.2 Evaluation of 18 and 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, iii 26 Page 6. Derivation of Propagation Constants and A,B,C,D Parameters of an n-conductor Transmission System .. 31 7. 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 Numerical Results f o r 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 40 7.6 Conclusion Correction 42 8. Calculation of the Mode Propagation Constants 49 9. Conclusions 68 Appendix A Appendix B Appendix C The F i e l d due to an Isolated O s c i l l a t i n g Current Element 71 The General Coordinates 74 Vector i n C y l i n d r i c a l Derivation of the Internal Impedance of C y l i n d r i c a l Conductors Appendix D 78 Solution of the Integral i n Equation 4-30 . 82 References 86 iv LIST OF ILLUSTRATIONS Figure Page 2- 1 Distributed C i r c u i t Parameters of an n-conductor 3- 1 Current Element Orientation 3- 2 C y l i n d r i c a l Coordinate System f o r the External Field 9 9 4- 1 Two-Conductor Configuration 7-1 Impedance Correction Factor Q with 9 = 0 , 7-2 Impedance Correction Factor P with 9 = 0 , S - 0 17 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 Impedance Correction Factor P with n = 50, 9 = 0 7-6 45 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 7-9 9 = 0, <4 = 10-5 mho/m Potential Coefficient Correction Factor N with 9 - 0, £ = 10-4 mho/m Potential Coefficient Correction Factor M with 7-10 46 47 9 = 0, S = 10- mho/m 47 Potential Coefficient Correction Factor N with © = 0, <£ = 10-5 mho/m 48 Potential Coefficient Correction Factor M with 9 = 0, <d = IO" mho/m 48 4 7-11 7- 12 3 8- 1 Mode 3 Propagation Velocity without Permittivity Correction 56 8-2 Mode 3 Propagation Velocity, no Capacitance Correction with n = 10 v 56 Figure 8-3 8-4 8-5 8-6 8-7 8-8 8-9 8-10 8-11 8-12 8-13 8-14 8-15 Page Mode 3 Propagation Velocity, no Capacitance Correction with n = 30 57 Mode 3 Propagation Velocity, no Capacitance Correction with n = 50 ., 57 Mode 3 Propagation Velocity, Approximate Capacitance Correction with n = 10 58 Mode 3 Propagation Velocity, Approximate Capacitance Correction with n = 30 58 Mode 3 Propagation Velocity, Approximate Capacitance Correction with n = 50 59 Mode Attenuation with I n f i n i t e l y Conducting Ground 60 Mode Attenuation with no Permittivity Correction 61 Mode Attenuation, no Capacitance Correction with n = 10 62 Mode Attenuation, no Capacitanoe Correction with n = 30 63 Mode Attenuation, no Capacitance Correction with n = 50 64 Mode Attenuation, Approximate Capacitance Correction with n = 10 * 65 Mode Attenuation, Approximate Capacitance Correction with n = 30 66 Mode Attenuation, Approximate Capacitance Correction with n = 50 67 vi LIST OF TABLES Table 7.1 8.1 8.2 8.3 Page Potential Coefficient Correction Terms 42 -5 Attenuation i n Nepers/Mile f o r d = 10 mho/m. and f =.100 kHz 52 Attenuation i n Nepers/Mile f o r d = 10"-'mho/m. and f = 10 kHz 53 Attenuation i n Nepers/Mile f o r d = 10 mho/m. and f = 1 kHz 54 vii Symbol L i s t a a = -e w(n-l)/y 2 0 + j = conductor radius i b = 1/r 2 = n- - jd/e to 0 b^ = horizontal conductor spacing c = Euler's constant = 0.57722 C,C^^ = capacitance matrix, elements E,E = e l e c t r i c 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, l i n e ) i = current vector J,J = current density (vector, components) J" (x) = Bessel function of the f i r s t kind of order n K,]^ = potential c o e f f i c i e n t matrix, elements k = propagation constant - jm i n L,L ^ = inductance matrix, elements m = propagation constant i m o' l m = M. ., IT. . ^ 3 P °P Sation constant ( i n a i r , i n the ground) r a = r e a l and imaginary parts of the correction terms f o r the elements of the potential c o e f f i c i e n t s matrix n = r e l a t i v e ground permittivity P. .,Q. . ^ = r e a l and imaginary parts of the correction terms f o r the elements of the impedance matrix R = r a d i a l distance i n spherical coordinates viii r a d i a l distance from the current element r a d i a l distance from the image of the current element resistance matrix, elements r a d i a l distance i n c y l i n d r i c a l coordinates distance from conductor i to the image of conductor j distance from conductor i to conductor j elemental length quantity defined i n Appendix B phasor voltage phasor voltage vectors (mode, l i n e ) voltage vector modal v e l o c i t y admittance matrix, elements impedance matrix, elements internal conductor impedance attenuation constant phase constant 2 propagation constant = propagation constant u 2 + m ( i n a i r , i n the ground) complex permittivity permittivity (of free space, of the ground) magnetic permeability of free space Hertz vector, components r a d i a l distance i n c y l i n d r i c a l coordinates r a d i a l distance from conductor 1 r a d i a l distance from the image of conductor conductivity ix r 2 = m /m 2 2 0 = scalar function of position co = angular frequency Y = e c = 1.7811 x ACKNOWLEDGEMENT The author would l i k e to express h i s 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 E l e c t r i c a l Engineering and especially Dr. M.M.Z. Kharadly f o r h i s help and advice and Mr. H.R. Chinn f o r the use of h i s computer program i n p l o t t i n g curves. The author i s indebted to the University of B r i t i s h Columbia, the National Research Council of Canada and the B r i t i s h American O i l Company f o r f i n a n c i a l support of the research. Thanks are due to Mr. A. MacKenzie f o r drawing the graphs and to Miss B. Rydberg f o r typing the thesis. xi 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 i n the study of energy propagation along transmission l i n e s . These studies are associated with f a u l t current calculations, system s t a b i l i t y , switching and r e s t r i k ing overvoltages, and the propagation characteristics of c a r r i e r waves along power l i n e s . A necessary prerequisite f o r i n v e s t i - gations of the above problems i s a thorough knowledge of the multiconductor transmission l i n e distributed c i r c u i t 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 i s impossible, due to i r r e g u l a r i t i e s i n 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 p l a i n or multi-layer homogeneous media. After the distributed c i r c u i t parameters have been found the propagation problem i s solved from the transmission l i n e 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. C a r s o n ^ ^ ^ . 1 2 Applying electromagnetic 2 f i e l d theory, he calculated the f i e l d due to an alternating current i n a straight i n f i n i t e l y long wire above and p a r a l l e l to a p l a i n and e l e c t r i c a l l y homogeneous ground. The derivation contains four basic assumptionsj 1. The ground r e l a t i v e permeability i s unity. 2. The ground i s e l e c t r i c a l l y homogeneous. 3. The polarization currents may be neglected. 4. The current i s propagated without attenuation at the speed of l i g h t . The f i r s t three assumptions greatly simplify the results. The f o u r t h assumption implies reasonably efficient energy transmission, as indicated i n reference (2). same publication Carson derived resistance and correction f a c t o r s . In the inductance They are v a l i d only f o r frequencies below 10 kHz because of the omission of the ground r e l a t i v e permittivity i n 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 d i s t r i b u t i o n i n the ground i s a function of the r a d i a l distance from the conductor. inductance correction terms are expressed The resistance and i n terms of the zero and f i r s t order Bessel functions of the t h i r d kind. low frequencies these functions have small arguments. At 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 p e r m i t t i v i t y e f f e c t s . Carson"s results were extended hy W i s e ^ ^ ^ ^ " ^ . 1 1 His f i r s t paper removed the r e s t r i c t i o n of assuming a ground r e l a t i v e permeability of unity. His subsequent works included the effect of polarization currents, gave correction terms f o r the l i n e admittance, and extended Carson*s impedance correction terms to a broader frequency range, Carson's (7) impedance correction terms were also extended by S u n d e v,/ to include the effect of a multi-layer ground. The investigation of multi-conductor using matrices, was started by B e w l e y ^ . transmission, His method was f i r s t to consider an i d e a l l o s s l e s s l i n e and then to expand (3) (20)' followed the analysis to include losses. R i c e ' and Pipes w v Bewley's approach but added the use of Laplace transform methods. A major contribution has been the work of H a y a s h i extended analysis to include transient phenomena. developed new w / who He also techniques i n 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 h i s analysis included conductor skin effects at higher frequencies, he used ground impedance correction from Rudenberg s model which neglected ground permittivity e f f e c t s , 5 A systematic mathematical procedure f o r handling the multiconductor transmission Wedepohl v . In two l i n e problem was recently published subsequent publications v M ; by he gave numerical procedures including the correction of the impedance parameters following Wise's methods, A matrix analysis of 4 multi-conductor conditions t r a n s m i s s i o n systems w i t h v a r i o u s was d o n e h y w o r k was r e s t r i c t e d Dowdeswell to steady in his state thesis, analysis at boundary however power his frequency x (23) (12) Arismunandar terms w i t h surface, for the surge This distributed also assumption that f r o m w h i c h he switching tivity ' the calculated admittance lines are correction close characteristic to derives correction parameters and r e l a t i v e factors i n c l u d i n g the permittivity of the for the effects ground, of of a multi-conductor transmission line. of the system and t h e i n Chapter a current-carrying i n Chapter frequency Chapter 3* effects of derivation (7) v . equation, Chapter 500 6. 5, f r o m t h e the follows The circuit relative closely detailed circuit that of solution distributed kV t r a n s m i s s i o n l i n e field for the various valid for include Wise^ 1 1 (a) , in parameters transmission v the circuit is parameters ^ in the and c o n d u c t i v i t y . s o l u t i o n cirouit derived solution given C a r s o n ^ , of is are derived parameters permittivity surrounding distributed admittance electromagnetic constants description field transmission line f o l l o w i n g Wedepohl's The The a derivation a p l a i n ground 1 MHz. These d i s t r i b u t e d ground the an E-type Hertz vector, 4 and the d i s t r i b u t e d i n Chapter Sunde above f r o m 10 H z t o for in electromagnetic conductor impedance parameters Chapter The 2. 3 i n terms of range A general assumptions required conduc- and g i v e s 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 given ground impedances s y s t e m a t i c method o f are the studies. thesis circuit derived The and line given in for typical frequencies a and ground . 5 conditions graphs are giving correction tabulated the magnitude factors over the ground c o n d i t i o n s , transmission line are system of u n i t s specified. of 7. Chapter the d i s t r i b u t e d dltated f r e q u e n c y A detailed description with numerical results, MES i n Chapter of circuit range the t h e wave e q u a t i o n f o r included i s used of 7 also i n Chapter throughout the 8. for includes parameter various solution, the The thesis, typical rationalized unless 6 Chapter 2 . 2.1 Assumptions and Approaches. General Description of the System. The system under study consists of a number of separate conductors and ground return l i n e s situated above and p a r a l l e l to a plane ground. No geometric symmetry i s assumed i n the transverse plane. The transmission l i n e equations f o r a l i n e with n conductors may be written i n terms of the distributed c i r c u i t parameters i n the general form as " • " § 2 = R i + L 2-1 T t = ( R + L p ) i = Z i - - | ^ = G v + C - | ^ = ( G + Cp)v = Yv 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 (4) and i s i l l u s t r a t e d i n the system shown i n f i g , 2 - 1 , Conductor i R.. L.. TW^ AA>V^ Conductor j J^j L. . nrr«V ^ Pig. 2-1 Distributed C i r c u i t Parameters of an n-conductor Line. 7 Basic 2.2 Assumptions. The permittivity line on t h e voltage circuit c a n he field circuit is field concepts. Hence the the r e s u l t s and energy along a c o n c e p t s a r e more p r a c t i c a l , development The In engineering, current problem i s the relative transmission since i n terms of consists from t h i s of the analysis extension the to and analyse distributed in this thesis electromagnetic f i e l d primarily the other v e r t i c a l l y describe the ground. In It transmission departure A Hertz field. of the some ground, the concepts algebraically. TM mode w i t h i n the grounds a n y new surrounding i n the d i r e c t i o n into s o l u t i o n the the losses i s used to one o r i e n t e d of require the permeability theory to m u l t i - l a y e r p e r m e a b i l i t i e s , does not due t o type of becomes much more i n v o l v e d The field conductivity c o n s i d e r e d homogeneous w i t h a r e l a t i v e although i t ents, of e a s i l y measured. with different the ground parameters. of u n i t y . line finite i n terms and e x p r e s s In ground of transmission of c a n he d e r i v e d however, the effect vector h a s two compon- p r o p a g a t i o n and the the construction f o l l o w i n g assumptions are of of the made: p 1, At the less frequency than unity. frequencies 2, In the practical the This s o l u t i o n of the transmission is assumption i s v a l i d f r o m 1 0 Hz t o wave considered purposes W'ELI i n a i r interest i n the range approximation i s of of i n view line. sufficiently the for 1 MHz, equation a f i r s t of much order accurate physical for irregularities 8 3^ The principle complete Air 4. 2.3 Circuit The assuming that ground of as the sum o f the ground of superposition solution for conductivity is the i s used to find the system. considered negligible. Parameters. ideal circuit the finite ground parameter i s has infinite conductivity the the This circuit analysis line over a ground of finite line over a ground of infinite first conductivity. i d e a l parameter plus condition. derived transforms conductivity For parameter i s a correction to conductivity. an an by a expressed term for n-conductor n-conductor 9 Chapter 3. Derivation of the Hertz Vector due to Current i n a Straight Wire. The Hertz vector f o r a ourrent along an i n f i n i t e straight wire p a r a l l e l to a f l a t ground i s obtained hy integrating the elemental Hertz vectors f o r current elements along the wire from minus i n f i n i t y to plus i n f i n i t y . F i g . 3-1 Current Element Orientation. X F i g . 3-2 C y l i n d r i c a l Coordinate System f o r the External F i e l d . 10 3-1 Hertz Vector due to a Current Element In a Medium of I n f i n i t e Extent. The Hertz vector due to a current element Idz i n an isotropic homogeneous medium of i n f i n i t e extent i s derived i n Appendix A, and i s given hy \ = 0 3-1 TV = 0 3-2 -rr X 3-2 Ie dz 0 = 4*m R 5-4 2 Q Hertz Vector due to a Current Element above a Ground with I n f i n i t e Conductivity. The t o t a l f i e l d above ground due to a current element and i t s image, as shown i n 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 t o t a l f i e l d be z 1 where "IT vector. QZ ,l oz M oz J 2 i s the primary f i e l d , and"!T oz J i s the secondary f i e l d The s u f f i x "o" designates the f i e l d vector above the ground surface. Prom equation 3-4 we have ITI ^ _ -Itouldz 4^ = where R-^ and R 2 -m R.. e 0 R l -m e 1 " 0 R R, * 2 , , 3 are the distances from the point of interest to the current element and i t s image respectively. 11 Hertz 3-3 Vector Finite due t o ground has f i n i t e be d i s s i p a t e d , and t h e two H e r t z vector gives the fig. 3-2. However, a. with conditions The in the particular coordinate energy at of Equation system s o l u t i o n must meet w h i c h may b e s t a t e d a s f i e l d vanishes some the y and z d i r e c t i o n s . general solution i n the conductivity f i e l d may b e d e s c r i b e d b y means components, B-15 boundary Element above a Ground Conductivity. When t h e will a Current of the follows: an i n f i n i t e distance from the source. b. At c. A further the source o n l y 7T exists Z the current element i s The t a n g e n t i a l E and H f i e l d s the ground and i s orientated finite, i n the are z since direction. continuous across surface. restriction is imposed due t o the geometry of the system; d. The The fields form of f r o m e q u a t i o n B-15, restrictions. the first y » (a) B, and zero by a p p l y i n g the the only plane. constructed above o r d e r f u n c t i o n b u t TT cannot. only of functions. h may now b e g i v e n by in the P is let The cos P only. l e a d i n g term of of significance. the vector Hertz four Bessel functions and due e q u a t i o n 3-6 two y-z solution, possible values sum o f the f u n c t i o n s may b e (b) can c o n t a i n terms accordance with be t h e Appendix approximation that over the field kind can appear i n solutions first s y m m e t r i c a l about the Due t o can contain the the are to for (b) IT Due t o (d) Assume a s the the a summation Finally above the vector of in ground region 12 TT 0Z =J~[f ( o e u ) 0 ( y " h ) *o + g ( ) - ^ o ] J (ru) Q u e o eo ir = j y x u 3-7 3-8 g (u) e * l J ( r u ) du iz d Q o TT TT oy =J c o s V p (u) e~ *o ly o oo = J" cos V p-^u) e y 0 l y y J ^ r u ) du 3-9 J ^ r u ) du 3-10 where the s u f f i x "1" designates the f i e l d below the ground surface. f» Q SQt There now remains the evaluation of the functions g±t P 0 and p 1 using condition (c). Condition (c) may be stated i n terms of the Hertz vector components by means ,(7) of equations A-10 and A-3 as follows' a ay OZ o 7x » '.fig0 z + H>v + ^ o z d y 0 z 3-11 + • ""ft, + 3-12 m "37 2 ^TT,0 2 — = o T 0 m2 i z = m. T Equation 3-11 may be written as integrating both sides yields 3-13 3.14 ^ 1 * 13 However, « and o are zero when x becomes i n f i n i t e and hence 1 f(z,y,u) i s zero. Integrating both sides of equation 3-14 yields "o^oy Again both 71^ ""l^ly = S( »y' ) + z u and Tl^y are zero when x becomes i n f i n i t e , hence g(z,y,u) i s zero. With these results equations 3-11 through 3-14 reduce to the following equations TF m + T^ iy - + T^ 5 o TToz = m 1 TT, lz 2 3-16 2 m o T m o ^oy = = F m m 5 y i T F " 3 l ""iy " 1 7 5-18 The various f i e l d functions may now be evaluated i n terms of the function f ( u ) they are Q « o ( u = ) * - X, F* 0 T + X*: o ^ > f 1 2uY r -hr 5-19 e -h* 2 o Si 7 ^ = v °+ ( u ) f o ( u ) e " ° 5-20 2uy (i - r ) -hr 2 n p ( ) u = 2 f ( ) e 0 u 3-21 14 P l r (u) = 2 P (u) 3-22 0 where T = m /m 2 2 3-23 2 The function f (u-) may he determined hy l e t t i n g Q = X and m = m , the solution must then he the same as l o l o for a uniform medium of i n f i n i t e extent. Then n -m R-, f (u) e - ( y h ) * o j (ru) du = ^ % T ° 3-24 d z 4**1*1 Then hy transforming the left-hand-side of equation 3-24 to the same o r i g i n as the right-hand-side and l e t t i n g f (u) = ° B -—— 3-25 °o where B i s a constant, and hy making the substitution * = u 2 + m 2 3-26 2 equation 3-29 becomes ? [ B v e" y /v *° -m RJ O ( R / » J - N | , * A O - M o 4* Assuming that m Q with ¥ f ^ 3 m Q - 2 7 Rx i s a small quantity i n a i r compared and r e s t r i c t i n g the solution to frequencies below 1 MHz equation 3-27 becomes oo [ B e~ '*° J ( r X ) d* o o o o y which i s Lipschitz's integral = -Mi^z 2 4 f l ; m (1Q) . R i Hence 3 _ 2 Q 15 u 3-29 Using the same reasoning -m R .jcoiile dz 4«m R -(y+h)v ) e 2 2 3-30 J ( r u ) du 0 Q 2 Then comhining equation 3-30 with 3-19 3-7 part of equation f -y . " o* o -m .jioule 4«m R y g ( ) e u Q to obtain the second ( J 0 r u ) = ~ d u R. dz 2 2 o r 2 y J + f The l a s t term of 3-31 ° ( u ) e 3-31 J (ru) du corresponds to the secondary f i e l d due to f i n i t e ground conductivity J 'z y J = + 1 V o f 0 ( u ) e d 3-32 » Then the complete Hertz vector i n the region 0— y ^ = 0 7T ox ~ TT i o y ff ' = o z -itouldz 4«m o 2 C 0 3-33 y i M ( i - S 4«m - o l m e R. R h is 2 ( v 0 r 0 V e -(y+h)tf y )(r V 2 i + o r " o 2 m R e R. J 1 ~ + 0 Y O J ( r u ) ) 1 3-34 (y+h)y l: 2ue 177 T17 du J (ru)du o d 3-35 The current at any point z on the l i n e i s 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 equations 3-34 uJ-^ru) = and 3-35 0 become - *Z dz o TT 4«m oz f Tz dz r 4itm ^ o j toil I e o 0 3-4 c*? 2jco|il e 3-37 J (ru) e ? u ( l - r (ir 0 J ^ -(y+h)tf ) e " ^ ° ° 2 + ^ e" o m R 2 H r 2 ,f2ue- ^ ^+ ( y + h ) y , T , V ) o 3-38 J ( r u du ) 0 o _ J (ru)du Q 3-39 Complete Hertz Vector f o r a Current i n a S t r a i g h t Conductor above Ground The complete Hertz v e c t o r f o r the c u r r e n t - c a r r y i n g conductor i s now obtained from i n t e g r a t i o n of the Hertz v e c t o r s due to the c u r r e n t elements a l o n g 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 . TT The components o f the Hertz v e c t o r are = o 3-40 co I 4«m, jwul e r r u(l - 2 g -(y+b)^r ) e J(ru)dudz 3-41 OA I -tfz 4«m, - -tfz L e -m R 0 -m n e 1 R 0 n > R, R~ 2 CO ~ 1 "j -(y+h)ro OO + o ^ + y0 J o ( n i ) d u J dz 3-42 17 Chapter 4. 4.1 Evaluation of Transmission Line Resistance and Inductance Including Ground Effects. Derivation of the Series Impedance Formulae. onductor j i II t I I I III Ground F i g . 4-1 Two-Conductor Configuration. F i g . 4-1 shows two t y p i c a l conductors of an n-conductor transmission system. The t o t a l e l e c t r i c f i e l d on the surface of conductor i i s the sum of the e l e c t r i c f i e l d s due to a l l n conductors. From equation A-10 the e l e c t r i c f i e l d on the i - t h conductor due to the j-th conductor i n the z direction i s given hy E. 1Z = -m 2 Tf. + HrO ']Z £z by ' iz 4-1 Then the complete e l e c t r i c f i e l d on the surface of conductor i due to a l l conductors i s n E.i z = Z_i Y 7 -m fr, o jz 2 + 4- dv, l y V" &z 4 - 2 18 The f i e l d s inside and outside conductor i must he equal at the boundary. I f the internal impedance of conductor i i s then n z.I. e " o^z m 10 1 1=1 + oy "Ji where z^ i s derived i n Appendix C. oz _ 4-3 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 iL ["HZ IZLfl 7 = 4-4 /LA ^z|_^y J i=l Prom the general transmission l i n e equation E-8 the voltage + Z equation i n phasor form can be written & 3 13 J=i 4-5 Jo Hence by combining equations 4-3 and 4-5 the following formulae can be established for the s e l f and mutual impedances, respectively ; Z. . = z. + m ix i o Z„ . = iO where j =1, 2, TT. iz ho* 4-6 z om"*^ jz 4-7 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 i n f i n i t e integrals i n the T T function given Z 19 by equation 3-42. To evaluate TTz the assumption must be made that attenuation along the l i n e i s negligible, then * = o m where 4-8 = J /e^to m 0 = jk 2 This i s an i d e a l value f o r *X but i s necessary due to the following two considerations: 1. To assume that the attenuation i s not zero on an i n f i n i t e l y long l i n e amounts to assuming a source of i n f i n i t e energy and makes the integral i n f i n i t e . 2. To assume a propagation v e l o c i t y less than that of l i g h t makes the integral extremely d i f f i c u l t to evaluate. Substituting jk f o r m Q oo Ifz = Ioe ^ k e^ z J - oo / k J 2ue _ 4*m \ l -jkR e, R 2 2 ) n 2 where R - tf (h+y) J (ru)du •icou o 4«m * o ^1 o ° From equation 4-6 we have ii i n equation 3-42 -jkR, •1cou ( e z 2 oo z and v dz 4-9 ; + = ± z T + h + j r 2 a -jkR- -jkR n 2 ' 1 0 20 To evaluate 1^, l e t l P P = l R + = R 2 z 4-13 + z 2 where R l \P\ = + z * a n d R = /"I * + B 2 'ij v 2 "d " i 4-14 where b ^ i s the horizontal distance between the i t h and j t h conductors, assuming this distance to be much greater than the radius of conductor i , h^ i s the height of the i t h conductor above ground and h^ that of j t h conductor. The general case i s considered because i t i s required i n the evaluation of Z-^y When J equals i , b ^ becomes the radius of conductor i , and Pi. = where a i a i ' ^2 = 2 i' h i s the radius of conductor i , and the height h^ i s assumed to be much greater than the radius a^. With the change of variable from z to and P 2 we have P dP-L = ^ : dz, and P-j^ p oo ! a ! 2 ,P f P dP = <*o 2 " p 2 ± = -1 , a g = W _| dz 4-16 when z 2 P 2 where a P g| h e n Z ~~ *-~ <*> " *° 2 4 _ 1 7 21 for a very large z. T x - l ~ Then lW a 4* ± l 4* e j ' dP, 4-18 «2 where the integrals are defined as the exponential integrals (18) Hence equation 4-18 can he written as T x - l " ^ li(e m 4* ') - l i ( e where ) 4-19 oo -lKe"*) = c + ln(t) + n=l and c i s Euler's constant. n ] n Por a small value of t the f i r s t two terms i n the above expression predominate and a l l other terms can be neglected,' then -JM " in 2* -2/> 4-20 ± Equation 4-20 gives the s e l f reactance per unit length of l i n e conductor when the ground has i n f i n i t e conductivity. To evaluate I changed, and I 2 l e t the order of integration be becomes 2 -r o (h+y) - oo Since J ( r u ) and cos(kz) are even functions, and sin(kz) i s Q an odd function of kz 22 oo oo JV^ J(ru)dz = 2 Jcos(kz) J(ru)dz 2 Q Q — OO o = 0 , = k> \/x? 2cos(x 2cos V Q T u - k x ' T o Hence f o r k— u 2 ) k ^ u 4-22 ^ (h y) 0 + COS ue - iML 2 " at 5 X Q 4-23 du To change the lower l i m i t o f the above i n t e g r a t i o n , X . = u - *, du - Isgla 2 2 2 then = .1cou ?t cos o 0 ,i X dV J /y ( _ ML - y x -^ y ) e ( h + y ) 0 0 V :cos g.x It To f i n d r e s u l t s , s i m i l a r to Carson's but the substitute d* 4-24 including e f f e c t of r e l a t i v e ground p e r m i t t i v i t y , i n accordance w i t h referenoe (9) l e t s' = 1 - je w(n-l) 4-25 since 2 y 1 and y 2 0 O = u 2 + m 2 = u 2 + jcofid — = u - e aw 2 2 0 enw' 1 23 hence *1 " *o 1^<4 = B 4-26 2 Further l e t y' = /wJI3y, h" = /co|i<3'h, x' = A>u<£x, v = . 4-27 oo Then I = ^ [ 2 * (/v - i s 2 - v) e ^ ' + y ^ o o s x'v dv 2 o s 4^28 Finally l e t y M = sy', then x" = sx' h" = ah', v = su f 4-29 oo 2 T = f f + o ^ - u ) e- u ( h " y + J , ) cos x»u du 4-30 Equation 4-30 i s the same equation as g i v e n hy (2) Carson v factor). except f o r s (the ground p e r m i t t i v i t y correction In h i s case s = 1 because he assumed that the r e l a t i v e p e r m i t t i v i t y o f the ground equals one. Then and Z ii = z i + I f ^ fj* + f <ii r + »il> 4-32 The l a s t term o f equation 4-32 i s the c o r r e c t i o n f a c t o r f o r the f i n i t e ground c o n d u c t i v i t y tivity. The exact s o l u t i o n s for and and r e l a t i v e permitare given i n Appendix D, and numerical r e s u l t s are included i n chaper 7. U s i n g 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 conductor i s the distance from conductor i to 25 Chapter 5« 5.1 Evaluation of the Shunt Admittance Derivation of Shunt Admittance Formulae. Prom equation 2-1 the transmission l i n e 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 i n equation 4-8, and noting that = -;jk, d i f f e r e n t i a t i o n yields n ^ C^Vj = J I i = 1,2,(3,4,) ± ^ n. 5-2 Then solving t h i s system of equations f o r the voltages, we have n V i =S H 1 = 1.2.:..., n. 5-3 d=i where K. . i s Maxwell's potential c o e f f i c i e n t . Substituting these results into equation 4-4 yields 2>1T v i k £ K I, = w i J bTT iz oy hence " k I i L^y J Thus the potential c o e f f i c i e n t s can be derived from the f i e l d components. 26 5.2 E v a l u a t i o n o f the Maxwell P o t e n t i a l C o e f f i c i e n t s K 1 J L and K ^ j . The same assumptions are made here r e g a r d i n g the e v a l u a t i o n of the i n f i n i t e i n t e g r a l s as s t a t e d i n the preceding chapter f o r equation 4-8. Prom equations 5-4, 3-41 and 3-42 5-5 where oo In = 1 4«e e o " 4«e -jkR e R 2 oo F 2 -jkR, 2 J 0 1 * J ( r u ) dudz + 0 5-7 Q O » u ( l - r )e )e f -jkz d Cu(l-r * I,3 = 2«e 5-6 -(y+h)*, s J — oO J dz —rr -r-— \ e P 22 b -rr \ J -(y+h)Y ° ° J (ru; dudz dzj v v 2y v, o> JL O -L O have been evaluated i n chapter 4. j k d y 0 J ( + ) ( Y o 5-8 } — OO 1^ and I I. i + I 2 = - 2 5 T l n^ T — 2 n l n ^ T 7 + '1 o ( P = + o 2we L Q T r ^ O A,,** """/^ ' - - 7te + N ^ P ) 5-9 1^ i s evaluated as f o l l o w s ~ I, = 3 " 2*e v ^ e o^ J o (JT, + y ) ( r V ^ ) .! o^ 2 n -1 - 0 oo 2 3 t e -(y+h)tf -(y+h)tf 2 - 1 0 2 J ) X O X O d_ j dz o ( r u ) d z d u - CO 5-10 OO e o r\ J -(y+h)tf ruy (i-r )e k -jkz _ ; ) k z oo J ( r u ) l + \h j^k e- "* ^ J ( r u ) d z |du 1 0 J k z o -oo-oo 5-11 27 Since the f i r s t term i n the bracket i s zero and the second term has been evaluated i n equation 4-22, equation 5-11 becomes oo , I 5— = - - J - Noting that -(y+h) "X 9 r u(l-*r) e ^ °cos O x 2_ d u Y ; = u- - k ,, udu = Y 2 " and 2 2 0 Y 2 -Y r^ Q 5-12 o v = 0 0 rv2 ~ 2 . v X + X^Y *i+* ) Q 0 the i n t e g r a l breaks into two parts, and T _ 1 o " ^T -^ ( Q P ) + n T - ^ ^ I 0 "0 where . 2 ^? -(y+h)"X J Then y K ii - + sbr "4 2h 0 1 o d + ^-(M+dN) o JLy°_0 r2 1 I 5-13 X X COS ln H 1 and v J 0 d y o o 5 " 14 1 + =fc ( M ii + ; ) N ii ) 5 - 1 5 0 The' l a s t terms i n equations 5-15 and 5-16 give the correction factors f o r f i n i t e ground conductivity and r e l a t i v e permittivity to Maxwell's potential c o e f f i c i e n t s . An approximation of M and N i s given i n section 5.3 of t h i s chapter and the numerical results are included i n chapter 7 . 28 5.3 Evaluation of M.. and N . . 3J LL Making the same substitutions as i n 4-25, 4-26 and 4-27, equation 5-14 becomes oo °J g M + jN = A A o o 2 2 Next l e t j s = a , M + 1 f* = 3f"T + 38 + 38 = b, " 5-17 d v V2 then (12) -v(h'+y') _, K g dv j/ v + a + bv coa c o s x v v 5-18 V J The numerical computation of t h i s i n t e g r a l 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 +. 3N ,-ivr M = \~ J i.1g e"* g^<' v Jv o » h + a 5-19r- ^cosx'v-,,, dv + bv +yt i n If within the range of values from 0 to v , Q I 2av| < |v + a I 2 5-20 then f rv- •*~ Q^2 2 + a v + a• ~- ^ _ 5 _ 2 1 The limitations of t h i s assumption w i l l be discussed i n chapter 7. Substituting 5-21 into 5-19 ( H ) 29 oo M a. HIM e" ^^\hosxW, f (v + a) ^ J Y (v + r-,)(v + r ) ( b + l ) d v 2 O 5-22 where r i - f ( 1 -/^rfi^ r 2 = f ( 1 + /^ I HH T = ) 5-23 Next l e t g ± = h? + y' - dx' = loirs' He'* = R ' e ^ g 2 = h 6 9 5-24 + y 1 + Jx' = f^3 1 Re = R'e^ 3 9 6 co M + jlT = 2 ( b ^ ( r 2 - r i ) [ * ^ o ] [ e " S l T + e " S 2 ] V d v 5-25 Equation 5-25 includes four separate integrals of similar form. Let CO [ - S iS l v 1 f e~ d v 1 = 2(h l)(r -r ) J v T r 7 + 2 5 1 ~ 2 6 and v + g^r, - r l 5-27 oo ~ 2 f = 2(h+l)(r -r ) J " T " r then X = w g-jT, w e 1 e 2 1 1 " 2 / " l l " 2(b l)(r -r ) r d w e s r l i ( e + 2 1 5-28 The other three integrals i n 5-25 have similar solutions. Therefore 30 M + J* = 2 ( b l ) ( r - r ) p e + 2 1 2 g i r i li(e" S i r i -r^^liCe"* * ) + ^ e ^ l K e " " 2 1 )) +r r , e + 2 ' 2 x 1 2 li(e ) ] 1 2 ) 5-29 The numerical r e s u l t s o f the e v a l u a t i o n o f M and N as given hy equation 5 - 2 9 a r e i n c l u d e d i n chapter 7 . 31 Chapter 6. 6.1 Derivation of Propagation Constants and A.B.C.D Parameters of an n-conductor Transmission System. Calculation of the Propagation Constants. The transmission l i n e equations f o r an n-conductor (8) system may he written i n matrix and phasor form as follows I V = Z(u) Y(ii)) V " 2 — 6-1 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 l i n e coordinate system. The systems of second order d i f f e r e n t i a l equations represented hy equations 6-1 and 6-2 can he solved hy transforming them into a new coordinate system wherein the transformed ZY or YZ matrix becomes a diagonal matrix. Then equations 6-1 and 6-2 become * i 2 c w 2. where the s u f f i x 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 2 [YZ - IY 1= 0 6—5 32 since (ZY) t = Y Z t t = YZ Here I i s a unit diagonal matrix. The solutions of equations 6-3 and 6-4 are V Xi i C, + e A 1 = e * "i D " + e^ Z I where Y = e J 1 C 0 6-6 2 i £> Z 6-7 2 represents the n propagation constants i n the n-modes of the system. The solution i n the l i n e coordinate system, designated hy the suffix p, i s now determined 7 = 5 7 p c and as follows I = S I p c 6-8 where R and S represent n x n transformation matrices, each consisting of n columns of eigenvectors of the system. eigenvector i s determined Each from the corresponding eigenvalue hy means of the following equations (ZY - tf I) R(i) = 0 6-9 (YZ - ) f 6-10 2 2 I) S ( i ) = 0 where R(i) i s the i t h column of the R matrix, and S(i) the i t h column of the S matrix. RS t = In general = 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. 'elements are given hy equation 4-32 The diagonal and the off-diagonal elements 33 by equation 4-33* ^ e t the d i a g o n a l Z i i = i i R + ^ elements be i i L 6 ~ 1 2 where R i i -mt f i i s + p " 5 where R ^ x , i s the i n t e r n a l AC r e s i s t a n c e of the conductor o r the r e a l part of z i n equation 4-32, and " j f ^ F ^ i s the r e a l i part of the ground c o r r e c t i o n f a c t o r . z i = int R J + w L Let int 6 " 1 4 then 2h. where ^ Q.^ i s the imaginary part of the ground c o r r e c t i o n factor. The z i d o f f - d i a g o n a l elements are g i v e n by equation 4-33 = ^ l ^ f <« *V n + P + 6 " 16 Now i f we w r i t e then hi • f hi - 6 18 and hi - h • $hi 1 0 For. a l l o f f - d i a g o n a l terms z id = z J i 6-20 34 6.3 Formation of the Admittance Matrix Y. The general form of the potential coefficients was derived i n chapter 5. There i s no d i s t i n c t i o n i n form between diagonal and off-diagonal Elements. If i s written as K, . = K! . ij where + A E . 6-22 , ij i s the correction term f o r the ground e f f e c t . Then the capacitance matrix of the transmission system may be written as C T - AC = (K + AK)" = Z'" - K'" AKK' T 1 1 1 -1 6-23 Defining gives AC = C'AKC = — C» (M + jN)0' 6-24 Hence the corrected capacitance matrix i s 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 - ~ M C ) ff£ 6-28 35 6.4 The A.B.C.D Parameters of the System. By analogy to the single conductor case, the behaviour of an n-conductor transmission l i n e can be described by the following equation; — — V A s B t 6-29 I C s_ D I. t_ where the parameters AjB,C and D are n x n matrices. The and Ig are voltages and currents at a column vectors 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 i n reference (8) and are A = R(cosh Y Z ) R - 1 B = [ ± sinh-. 6-30 R Z 6-31 S- 1 ! 6-32 _1 — C = S Tsinh: ¥ &~ ± L \ D = S(cosh Zo = R Xi T 1 . X. JL ) S R Z _1 -1 6-33 6-34 36 Chapter 7. 7.1 Numerical R e s u l t s f o r a 500 kV T r a n s m i s s i o n L i n e . D e s c r i p t i o n of the Transmission L i n e . The ground e f f e c t on the d i s t r i b u t e d c i r c u i t parameters for an overhead t r a n s m i s s i o n l i n e i s dependent on l i n e geometry, ground c o n d u c t i v i t y , ground r e l a t i v e p e r m i t t i v i t y , and frequency. The l i n e geometry can have a l a r g e number of v a r i a t i o n s . In order to i l l u s t r a t e the s i g n i f i c a n c e 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. the For g i v e n geometry the ground c o n d u c t i v i t y , r e l a t i v e permit- t i v i t y , and frequency are v a r i e d . The 500 kV l i n e i s chosen because recent developments i n l o n g d i s t a n c e energy t r a n s m i s s i o n have c a l l e d f o r more d e t a i l e d study of the p r o p e r t i e s of l i n e s at t h i s v o l t a g e l e v e l than has p r e v i o u s l y been a v a i l a b l e . The s i n g l e c i r c u i t l i n e without overhead ground wires c o n s i s t s o f : A bundle of f o u r conductors a t the corners of an 18 i n c h square, per phase. Conductor s i z e 583.2 MCM ACSR Conductor DC r e s i s t a n c e ........ 0.1764 ohm/mile a t 50°C Conductor diameter ( i n c l u d i n g stranding factor) 0.948 i n Average phase spacing 40 f t . Average conductor h e i g h t ....... 54 f t . The l i n e conductors are at equal h e i g h t s above the ground i n a f l a t a r r a y . The ground c o n d i t i o n s were considered to v a r y from dry r o c k to wet marsh l a n d . 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 t o 50 and a ground 37 conductivity range from 10""'' to 1 mho/m. 7.2 Ideal Parameters. From equation 4-33 S' I _ Mi i n - l i h/m 7-1 Inserting the lengths i n the ahove equation and converting the units to m i l l i - h e n r i e s per mile gives the following inductance matrix, not including internal conductor inductance 1.63 .339 .167 .339 1.63 .339 .167 .339 1.63 mh/mile 7-2 Inverting t h i s matrix and multiplying by a known constant gives the capacitance matrix i n units of micro-farads per mile C .0183 -.00302 -.00148 ^.00302 .0189 -.00302 -.00148 -.00302 .0183 u-f/mile 7-3 with the inductance i n henries per meter and the potential coefficients i n darafs per meter the matrices are, L = 2 x 10" K = 18 x 1 0 7 9 5.10 1.06 .520 1.06 5.10 1.06 _.520 1.06 5.10_ "5.10 1.06 .520~ 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 w i t h a r a d i u s of a bundle spacing of 18 i n , the negligible. The taken as one q u a r t e r of the and i n t e r n a l impedance per from C-16, C-17, depending on the magnitude of 7.4 Impedance or C-21, impedance c o r r e c t i o n and correction terms depend on the relative permittivity. show the the impedance m a t r i x , a c c o r d i n g to the = I E of the il f a c t o r s f o r the i ± = ii E + 2Q + f P the Figs. 7.3 formulae, ™ The conductivities correction factors shown i n figs.; 7 . 3 i n v a l u e s between the elements i s not through 7.6 show the above 10~ n e g l i g i b l e e f f e c t , but the and 7.2 mho/m the 40. great the i t may correction By be and diagonal elements have been omitted from the w i t h f i g s . 7.1 and 7-6 f o 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 through 7.6 F i g s . 7.1 ii off-diagonal off-diagonal frequency, 11 permittivity correction. Since the d i f f e r e n c e figures. and d i a g o n a l elements of r e l a t i v e p e r m i t t i v i t y of 10 are elements and or C-26 ^cou^a. 7.2 7.4. C-22 conductor Correction. ground c o n d u c t i v i t y , f o r the with p r o x i m i t y e f f e c t i s considered C-27 assuming no i n and i n t e r n a l conductor impedance per bundle i s i s calculated The .479 curves remaining factors comparing f i g . seen t h a t for relative permittivity e f f e c t becomes q u i t e 7.3 has pronounced f o r 39 conductivities below 10"^ mho/m. For example, the correction matrices without ground permittivity correction at a frequency of 10 kHz are R = I = when<^ = 10 123 123 121 123 123 123 121 123 123 7623 .605 .558 .605 .623 .605 .558 .605 .623 mho/meter. ohm/mile 7-8 mh/mile 7-9 For the same frequency butd* = 10' mho/meter R = 1 = 60.8 58.0 50.6 58.0 60.8 58.0 50.6 58.0 60.8 .137 .126 .099 .126 .137 .126 .099 .126 .137 ohm/mile 7-10 mh/mile 7-11 For ground r e l a t i v e permittivity of 10 and at the frequency of 10kHz. R = 279 279 278 279 279 279 278 279 279 ohm/mile 7-12 40 1 = .698 .677 .629 .677 .698 .677 .629 .677 .698 mh/mile 7-13. when A = 10"^ mho/meter, f o r the same frequency hut ^ - 10*" 2 mho/meter R = I = 61.2 58.4 51.0 58.4 61.2 58*4 51.0 58.4 61.2 .137 .126 .099 .126 .137 .126 .099 .126 .137 ohm/mile 7-14 mh/mile 7-15 From f i g s i 7 . 1 through 7.6 i t may he seen that f o r —2 conductivity below 10~ mho/m and a frequency above 1 kHz the ground r e l a t i v e permittivity should be included i n the calculation. 7.5 Admittance Correction. The admittance correction terms were evaluated by the two methods described i n Chapter 5. F i r s t the i n t e g r a l i n 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 i n ground conductivity. For the range of conductivities greater than 10'Wm and the range of r e l a t i v e p e r m i t t i v i t i e s between 10 and 50 the maximum •41 discrepancy i s of the order of 10%. This i s considered tolerable (16) • • • for propagation c a l c u l a t i o n s ' i n view of the r e l a t i v e l y v small magnitude of the correction terms as compared with the potential c o e f f i c i e n t s . The difference i n 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 o f f diagonal elements are not included i n f i g s . 7 . 7 through 7.12. —'3 For conductivities of 10 mho/m or greater a l l the correction factors become n e g l i g i b l y small. In calculating the propagation constants a correction of 3% or greater i n the diagonal elements i s considered s i g n i f i c a n t . A comparison of exact and approxi- mate values of the correction factors and their magnitude are shown i n f i g s . 7 . 7 through 7.12. This requires correction factors to be calculated f o r conductivities less than 10 ' mho/m and frequencies greater than 100 Hz. The potential coefficient terms are corrected according to the formula i K ij = K ij + 1 8 x 10 (2M 9 i;) + J2N ) ±;J 7-16 To i l l u s t r a t e the magnitude, the potential coefficient correction terms f o r the diagonal elements at frequencies of 1 kHz, 10 —3 kHz and 100 kHz and f o r conductivities of 10* mho/m and 10""^ mho/m f o r a r e l a t i v e permittivity of 10 are shown i n table 7.1. 42 Table 7.1 Potential c o e f f i c i e n t correction terms. 2U 2M f = 1 kHz = 1 0 " mho/m 3 <d> = 1 0 " ^ mho/m 0.03 0.07 £ = 1 0 * mho/m 0.003 0.007 ^ = 1 0 " mho/m 0,43 0.34 6> = 1 0 " mho/m 0.028 0.047 <i= 1 0 " ^ mho/m 0.66 0.17 fA = 10 kHz 3 5 f = kHz 100 3 7.6 Conclusion. It has been found that the impedance should always be corrected for ground conductivity. is less than 180, In addition when c^/we the impedance should be corrected for the ground r e l a t i v e p e r m i t t i v i t y and the capacitance should be corrected f o r both ground conductivity and r e l a t i v e p e r m i t t i v i t y . When oVwe i s 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 2 10 10 •6 3 4 10 10 FREQUENCY HZ S 10 10 FIG 7-4 IMPEDANCE CORRECT/ON FACTOR P WITH ft=/0 9=0 . . . . . •7 6-3*5° •6* *5 P \\ • c^jof^^ '4 -3 — \ - '2 -1 •vvr ""V •• 10 /o i 3 j 4 10 10 FREQUENCY HZ 10 45 FIG 7-5 IMPEDANCE CORRECTION FACTOR Q WITH <» 10 2 10 <n 10 <J 10 3 FREQUENCY 8 />«50. B*0 ,„ 10 5 HZ 10 FIG 7-6 IMPEDANCE CORRECTION FACTOR P WITH n=50. 0=0 ,• 7 •6 •5 s^t-ib • 1 •4 •3 &*10 •2 tf./o'^^ -1 0 10 , 1, .L 1. • l,„. 10 J 1,-1 10 FREQUENCY 10 HZ 10 10 .46 FIG 7- 7 -5 POTENTIAL COEFFICIENT CORRECTION FACTOR N WITH 9*0. 0**10 m -201 nsSO n =40 n*30 n = 20 n*10 10 10 FREQUENCY HT FIG 7-8 POTENTIAL COEFFICIENT CORRECTION FACTOR M 'SO EXACT APPROXIMATE — — — m WITH 9 - 0. d-1(T m — '40 /•"\ // / '30 /' • M 20 M ^ v* \ '\\ \\ <\ > n mW MM** '10 ~*"^*^^*^^^ '00 10 1 10 d - ^ i i i—1__ 10 10 FREQUENCY; HZ 10 i i n "20 n-3Q n*40 n*5Q 10 47 '•' :y:'rp:^ : FIG 7-9 POTENTIAL COEFFICIENT CORRECTION FACTOR N WITH 0 • 0. -20 ' tt»fo 4 mh EMcr APPROXIMA Ti 10 /> \\ JKSoV A •oo n*50 n'40 n*30 n-20 10 ,„ 1 10 .-.1. .„J_ „ I „•,' 10 1 10" 10 FREQUENCY HZ J i - />-/o i 10 70" FIG 7-10 POTENTIAL COEFFICIENT CORRECTION FACTOR M WITH 8 = 0. <M0 mho -30\ EXACT APPROXIMATE — . — ' — _ — 20 * M ...... ' ft \\ ^\ v> n-10 Ft 10 n-20 n»40 • '00 10 10 i —- 3 4 10 10 FREQUENCY HZ l l 1 10 1 • 48 FIG 7- // -3 POTENTIAL COEFFICIENT CORRECTION FACTOR N WITH O*Q. 0**10mho/m 'OS ff 04 . •.;.-..<-,: : N \\ \ //^ •03 - ):^y *\ // . ..',< \U\\ :;02 -01 n*20 01 -02 10 •1' 1 1 - 1 . _J L , i 10 10 10 FREQUENCY HZ 10 FIG 7-12 n-SO i 15 • • • n*70 • 10 r M IT** n-30 n<40 n-SO •OS 1 10 10 I—™j 1 •• 10 10 FREQUENCY HZ 10 10 49 Chapter 8. Calculation of the Mode Propagation Constants. The correction factors given i n Chapter 7 are inserted into the transmission l i n e equations, 6-1, and the mode propagation 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 Y = j2«ewY' 8-2 Then l e t A = ZY = - {/eiiZ'Y 1 = coeiiA» 2 8-3 The r e a l 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 r e l a t i v e l y small i n magnitude, l e t A' = A" - 8-4 I then from l|= 0 |A« 8-5 one has iJA" - ( f f + 1)1 | = 0 / 2 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 where j + J^ = ^ i =/ ( A u ^ 2 - 1) i s attenuation i n nepers per meter and constant i n radians per-meter. 8-7 i s the phase The mode v e l o c i t y i s obtained as follows v i = p\" m /sec 8-8 In terms of the v e l o c i t y of l i g h t i n free space, the normalized mode v e l o c i t y i s v- = 2itf/p /7JT' 8-9 i ] The v e l o c i t i e s at modes 1 and 2 are very close to unity under a l l conditions, and a l l three mode v e l o c i t i e s are close to unity f o r a perfectly conducting grounds for f i n i t e ground conductivity However, and r e l a t i v e permittivity the v e l o c i t y of mode 3 varies from 0.48 to nearly 1 per unit . ( Fig. 8-1 shows the v e l o c i t y of mode 3 without per- m i t t i v i t y correction f o r conductivities from 10~^ to .1 mho/m. Figs. 8-2 through 8^-4 show the v e l o c i t y of mode 3 with impedance correction but without capacitance correction, f o r r e l a t i v e p e r m i t t i v i t i e s from 10 to 50 over the same range of conductivities as i n f i g . 8-1. Figs. 8-5 through 8-7 show the v e l o c i t y of mode 3 with capacitance correction over the same range of conductivities and p e r m i t t i v i t i e s . From the figures i t can be seen that r e l a t i v e permittivity has l i t t l e effect on the 51 v e l o c i t y of mode 3 for a conductivity greater than 10 and the capacitance mho/m, correction only a f f e c t s the calculation where the conductivity i s less than 10 mho/m. In f i g s . 8-1 through 8-7 i t can he seen that the permittivity correction of both impedance and capacitance tends to increase the v e l o c i t y of mode 3 at high frequencies. Pig. 8-8 shows the mode attenuation constants, i n nepers per mile, f o r a perfectly conducting ground, i n which case the attenuation i s due e n t i r e l y to i n t e r n a l 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. attenuation constants Pigs. 8-10 through 8-12 show mode over the same conductivity range and over a range of r e l a t i v e p e r m i t t i v i t i e s from 10 to 50, with the impedance corrected for conductivity and r e l a t i v e permittivity, hut without capacitance correction. Pigs. 8-13 through 8-15 show the mode attenuation constants with conductivity and r e l a t i v e p e r m i t t i v i t y correction f o r the impedance and the capacitance over the same ranges of r e l a t i v e permittivity and conductivity. The attenuation of mode 3 i s most affected hy changes i n ground conductivity, the attenuation of mode 2 i s affected to a lesser degree and the attenuation of mode 1 i s least affected. Prom f i g s . 8-9 through 8-15 i t can he seen that f o r conductivities greater than 10 mho/m the r e l a t i v e 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 f o r frequencies greater than 100 Hz. Tables 8-1 through 8-3 show a comparison of variations i n 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 f o r ^ = 10"^ mho/m and f = 100 kHz Mode 3 £To permittivity correction Mode 2 Mode 1 3.6 x 10" 1 1.5 x 10~ 3 1.3 x 10" 3 n = 10 7.0 x 10" 1 2.7 x 10~ 3 1.4 x 10~ 3 n = 20 7.2 x 10" 1 4.0 x 10~ 3 1.4 x 10" 3 n = 30 7.3 x 10" 1 5.2 x 10~ 3 1.4 x 10" 3 n = 40 7.4 x 10" 1 6.5 x 10~ 3 1.5 x IO" 3 n = 50 7.5 x 10" 1 7.6 x 10~ 3 1.5 x IO" 3 n = 10 4.6 x 10" 1 2.7 x 10~ 3 1.5 x IO" 3 n = 20 5.8 x 10" 1 4.0 x 10~ 3 ti n = 30 6.2 x IO"" 5.2 x 10~ 3 it n = 40 6.6 x IO" 1 3 »t n = 50 6.6 x 10" 1 3 i» Impedance correction Capacitance correction 1 Si5 x 10" 7.6 x 10" S3 Table 8.2 Attenuation i n nepers/mile f o r ^ = 10~^ mho/m and f = 10 kHz Mode 2 Mode 3 Wo permittivitycorrection 3.7 X i o - 2 n = 10 4.4 X i o - 2 n = 20 5.2 X IO" n = 30 5.7 X n = 40 6.0 X IO" IO" n = 50 6.3 X IO" n = 10 3.3 X n = 20 4.0 X n = 30 4.6 X n = 40 5.1 X n = 50 5.4 X IO" IO" IO' IO" ICT Mode 1 x l (•f4 •4 4.0 x 10" 3.4 x 10"•4 •4 4.0 x 10" 3.3 Impedance correction 2 II it 2 It it 2 It it tl ti 2 Capacitance correction •4 3.4 x 10" 2 -4 4.0 x 10" 2 ti it 2 it ti 2 II ti ti it 2 From table 8-3 i t can be: seen that capacitance correction i s s i g n i f i c a n t f o r low conductivities even at a frequency 1 kHz f o r the attenuation of mode 3. For the attenuation calculation, conductivity and r e l a t i v e 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 r e l a t i v e per- m i t t i v i t y correction of the impedance and no capacitance correction (2) are required and Carson's formulae ' are s u f f i c i e n t to obtain v 54 the ground correction terms f o r the c i r c u i t parameters. Tahle 8.3 Attenuation i n nepers/mile for<^ = 10"^ mho/m and f = 1 kHz permittivity correction ISTo Mode 1 Mode 2 Mode 3 3.6 x 10~ 3 1.15 x 10" 4 1.4 x 10" 4 n = 10 3.3 x 10~ 3 1.15 x 10" 4 1.4 x 10" 4 n = 20 3.4 x 10~ 3 it II n = 30 3.4 x 10~ 3 ti II n = 40 3.5 x 10~ 3 ti ti n = 50 3.6 x 10"" II II Impedance correction 3 Capacitance correction 1.15 x 10" n = 10 7.8 x 10~ . n = 20 4j5 x 10~ 3 II it n = 30 3.5 x 10~ 3 II it n = 40 3.4 x 10~ 3 II ti n = 50 3.4 x 10" 3 II tt 3 4 1.4 x 10" 4 An insight into the behaviour of the mode propagation constants may be obtained by comparing the transmission l i n e with a l i n e that i s transposed at short intervals. Then a l l the off-diagonal terms i n both the impedance and admittance matrices would be equal and the diagonal elements i n each matrix would also be equal. The transformation matrices, S and R, would be equal and would be the matrix, used f o r 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 i l l u s t r a t e t h i s s i m i l a r i t y the R and S matrices for a frequency of 100 kHz, a ground conductivity of 10~ 4 mho/m and a r e l a t i v e permittivity of 10 are .484 R = - j.161 -.787 + j.177 .299 + j.00 .223 + j.246 S t = R" 1 = -.603 - j.H5 -.763 + j.106 .115 .566 + j.014 -.750 - j.246 .517 + j.oo -.143 + j.126 .766 + j.00 .275 + j.246 -'.833 .568 j.003 .570 + j.00 .570 - j.102 - .619 + j.oo .571 + j.002 -.656 - .592 — j.017 .591 + j.00_ j.147 .543 + j.020 j.064 -.125 + j.067 j.005 .572 + j.005 .808 + j.00 .593 - j.011 8-10 8-11 8-12 FIG 6-1 MODE 3 PROPAGATION VELOCITY WITHOUT PERMITTIVITY CORR 10 10 2 10 IO FREQUENCY HZ 3 . 4 W 5 10 FIGB-2 MODE 3 PROPAGATION VELOCITY. NO CAPACITANC&>CQRRECTIO 57 FIG 8-3 MODE 3 PROPAGATION VELOCITY. NO CAPACITANCE CORRECTIO 10 10 10 10 FREQUENCY HZ FIG 8-4 MODE 3 PROPAGATION VELOCITY. NO CAPACITANCE CORRECTI 10 10 10 10 FREQUENCY HZ 10 FIG 8-5 FREQUENCY HZ v :'V^f- --V : v 60 61 FIG iot MODE 6-9 A T T E N U A T I O N W I T H — i — — . ' • ,-" N O i V— P E R M I T T I V I T Y C O R R E C T I O N i—~~r r~ 64 FIG 8-12 MODE ATTENUATION. NO CAPACITANCE CORRECTION WITH n- 70i i — - i i FREQUENCY HZ - i 66 FIG 8-74 MODE ATTENUATION. APPROXIMATE CAPACITANCE CORRECTION WITH n . 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 c i r c u i t parameters f o r a multi-conductor transmission l i n e . The correction terms f o r the distributed c i r c u i t parameters have been derived f o r variations i n l i n e geometry, ground conductivity, ground r e l a t i v e permittivity, and f o r a frequency range up to 1 MHz. Their corrected parameters have been used i n the transmission l i n e equation to calculate the mode propagation v e l o c i t i e s and attenuation constants. A p r a c t i c a l example has been investigated i n d e t a i l to show the variations i n the distributed c i r c u i t parameters and the mode propagation constants due to various ground conditions. For a t y p i c a l 500 kV transmission l i n e the impedance should always he corrected for ground conductivity. <<^/toe i s less than 180, When the impedance should, i n addition^ be corrected for the r e l a t i v e permittivity of the ground and the capacitance should be corrected f o r both the ground conductivity and the r e l a t i v e permittivity of the ground. 180, When equals the maximum error i s approximately 3$ i n Carson's correction of the impedance parameters and the uncorrected potential coefficients. The i n i t i a l rate of r i s e of recovery voltage i n c i r c u i t breakers clearing f a u l t s i n power systems can be evaluated s o l e l y i n terms of the c h a r a c t e r i s t i c impedance of the transmission l i n e and the energizing conditions. The former depends i n 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 f o r c i r c u i t breakers opening under f a u l t conditions. The program developed f o r calculating the mode propagation constants i n this thesis may be extended to include terminal conditions. This allows the investigation of the various permutations to find the optimum condition f o r energizing and terminating a l i n e f o r c a r r i e r communicationv. In the c a l c u l a t i o n of the mode propagation constants i n t h i s thesis the conventional formulae f o r the i n t e r n a l impedance of the l i n e 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 i n t e r n a l impedance more accurately. Since the i n t e r n a l impedance affects the diagonal elements of the impedance matrix i t has a d i r e c t effect on the mode propagation constants. Further developments i n t h i s f i e l d should be directed to f i n d the effects of corona and tower footing resistance on the transmission system parameters. f i e l d adjacent Corona disturbs the e l e c t r i c to the conductors and may have a noticeable effect on the parameters. i s voltage dependent. The effect i s non-linear since i t The tower footing resistance may become important f o r l i n e s with overhead ground conductors where the tower spacing approaches a quarter of the wave length of the impressed s i g n a l . 70 The procedure developed i n t h i s thesis i s solely for overhead transmission l i n e s . In the present form i t i s unsuitable f o r the calculation of propagation constants of underground systems, which presents quite a different problem due to t h e i r geometric configuration with respect to the ground. 71 Appendix A. The F i e l d due to an Isolated O s c i l l a t i n g 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 df the medium i s aaccounted f o r hy the complex permittivity e , cc C where e = e -j c to Define a vector TT such that H = jcoe V T c A-3 X Then from equation A f l and,, A-2 V x E = to e ^V TT = -m V x l T 2 x A-4 2 c hence E = -m TT 2 +V(0) A-5 where 0 i s a scalar function of position. Then from equation A-2 jweVx ( V x T T ) = J + 3toe (-m TT 2 c +^/(0)) A-6 Dividing by jwe and expanding the f i r s t term of equation A-6 gives - V 7 T + V-V-TT = - * TT 2 2 A-7 C let V(0) =VV."TT A-8 (19) a Lorentz type condition, and equation A-7 becomes 72 V 1T = 7T-^ A-9 E = -m7T + W . T A-10 2 2 m and 2 Let the solution of equation A-9 be -m R r dv A e R A-ll o where v Q i s a small closed surfaoe at the point (x,y,z) i n Cartesian coordinates. Then -m R V T=V2£^-fT-*v 2 r r 2 " I [nTAe —i mR -m R 0 n = J [ — R — + * 'V (|)Jdv A v 2 o 2 = m T + r -m R Ae" V (J) dv A-12 (f) dv . A-13 2 o Substituting A-12 into A-9 gives -m R Ae V o The solution of A-13 i s J = - jwe p 1 c J =4 0 p -j when R ^ 0 A-14 jwe A4it when R = 0 c The current flows only i n the z dlreotion, henoe J * are both zero at r equals zero. and J y When the element has a current I and a cross section a, J i at R = 0 z = a A-15 Hence A = ,* 4«0we a a A-16 73 ^ a n d o Assuming v Q -m R Ie . 4«3(oeaR i s small and independent o f r , then -m R z 4«dwe aR o where v o = sa and s i s the elemental length which i s replaced hy dz. Hence -m R _ Ie dz " z ~ 4*jwe R TT TT X TV •7 A A _ 1 7 ± f = 0 A-18 = 0 A-19 74 Appendix B. The General I T Vector i n C y l i n d r i c a l Coordinates, Fig, B - l C y l i n d r i c a l coordinate system f o r the internal f i e l d The wave equation f o r the H ^ vector i n the coordinate system shown i n f i g . B - l i s The solution i s obtained by separating the variables, TT Z = R(/>) $ let (0)Z(a) B-2 Then 1 "/? c ^ R l ^ a/ L ^ J 3 r 1 <f d $ l .. 1 ^Vz Z T7 * 2 2 p_ = m <^2 Since the f i r s t two terms of B-3 are independent of z R f ^ + t ±i—'3 the equation becomes U^Z B-4 where t 2 = u. 2+ .m2 and u i s a constant. B-5 75 The solution of B-4 i s Z = ke + A e ± B-6 Q 2 Then B-3 reduces to ^ R Kc3fJ + $ ^02 + U = 0 /° B-7 Since the second term of the equation i s independent of r , B-7 can he written as + p $ = 0 B-8 2 0 0 * * 2 where p i s a constant. $ - B 1 The solution of B-8 i s cos P 0 + B 2 sin P B-9 0 Equation B-7 now hecomes ^D*ifl [ ^ - ] + ( )2 p2 R=o which can he transformed to the form </» »> ^ u 7 i^q] + [ > u) 2 -P2] * =o B-ll 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 f o r the solution depend on the boundary conditions of the particular problem. Then the complete solution f o r "TT i s the sum of a l l 2 76 possible solutions and may be writen as TTZ = ^(B l p c o s P 0 + B sinP0)* J P=0 (A-Ju)e - 2p xC Y z + A (u)e y z 2 ) B-13 Zp(/>u)du The wave equation i n the Cartesian coordinate system can be separated into three components. Each component s a t i s f i e s (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 f o r T and TT z w i l l be similar and y of the general form shown i n equation B-15. This i s the form of the solution used i n the ground effect calculations i n Chapter 3. y F i g . B-2 C y l i n d r i c a l Coordinate System f o r the External F i e l d 77 oo ^(B oo i p c o s P V + B sinPy) 2 p J ( k j ^ e - * 7 + A (u)e* ; y 2 P=0 x Z ( r u ) du p B-15 78 Appendix C. Derivation of the Internal Impedance of C y l i n d r i c a l Conductors. In t h i s Appendix we s h a l l he concerned only with the f i e l d s inside the conductors. Consider a s o l i d c y l i n d r i c a l conductor with the axis along the z-axis of the coordinate system shown i n f i g . B - l . The d i r e c t i o n 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 i n the z d i r e c t i o n i s used to describe the f i e l d . type Due to the c i r c u l a r symmetry of the system the f i e l d components are independent of 0, and hence the Hertz vector i s also of 0. The solution of the wave equation for"iT equation B-13. 2 independent i s given i n However as TV „ i s independent of 0 and must be z f i n i t e at the o r i g i n only the f i r s t term of the solution exists and the required Bessel function i s of the f i r s t kind of order zero. Therefore IT = Ae~* J (/>u) o ' z z Using equations A-3 and A-10 C-1 the f i e l d components E g and H^ are E = (-m +* )TT„ 2 H0 = jtoe uAe" u) c Let C-2 2 C-3 the external radius of the cylinder be a, then at the Y » boundary I e Hv = ° - _ <M 79 Consequently I A v Wheno for jwe 2 *au = 2 i s v e r y l a r g e ,Y practical is negligible and = -m 2 o -m I J 2 E z Eg at radius z i = Q v „ (/>u)ey C 7 1 2 a be g i v e n by C-8 2 internal impedance o f t h e c y l i n d r i c a l Then z a z t jwe 2«auJ (au) -m For Then C-6 f i s the equivalent conductor. 2 a s compared t o m . 2 E„z = z ,i I o e ~ * where 5 purposes u Let " G c i = J (au) Q jwe 2«auJ (au) 1 c conductor e_ = - < ~ ' c ~ ~ 3 co C - 1 0 m = j V C - l l a' = Jujie? C - 1 2 and 0 , where Then »4 = 3 1 , 3 cc'Jo(3 2«ad The n u m e r i c a l three regions, dependent . J (d 1 1 1 < 5 a'a) ' , 5 C-13 a'a) e v a l u a t i o n o f z ^ may b e d i v i d e d on the value of a ' a . into 80 1) When a*a < 0.1 For s m a l l v a l u e s off a'a the f o l l o w i n g expansion h o l d s , i 2 i = ^ * («'a) « 2 8 (a'a) A 192 + , 4 " J (a'a) 3072 (13) 6 ''•' I w a 2j C-14 let i= Z R int 3X + C-15 i n t Then f o r s m a l l v a l u e s of a'a o *a and which a r e the formulae obtained when a uniform c u r r e n t d i s t r i - b u t i o n i s assumed. 2j When a'a > 10 For l a r g e v a l u e s o f a'a z z = may be w r i t t e n as « ' U +• .1) 0-18 2j2*a<5 1 Let i the s k i n depth d be d e f i n e d as d =M, c*i? Then 5 i = 2*adJ which i s Rayleigh's formula. R int = Then 2*ad<S L ^ = O^^J.. i n t ~ 2«ad^ J C " 2 1 C-22 81 When 0 . 1 ^ 3) a'a — 10 Equation C-13 may be changed i n t o a more conyentient form f o r numerical computation by l e t t i n g J ( d ' a ' a ) = Ber(a'a) + JBei(a'a) 1 C-23 5 0 The f i r s t order B e s s e l f u n c t i o n i s obtained by d i f f e r e n t i a t i n g equation C-23 J (j a'a) 1,5 = 1 j ' JBer'Co'a) + 0 5 jBei'(a'a)] C-24 Then the i n t e r n a l impedance z^ becomes z _ a' and the r e s i s t a n c e TJ - «' int W n jBei'(a'a)J 5 9 r . and inductance are [Ber(a'a)Bei' (a'a) - Bei(a'a)Ber'(a'alj *int " L r»Bei(a'a) + .1Ber(a'a) "I ~ 2*a<* |_Ber'(a'a) + i Ber' (a'a) + B e i ' ( a ' a ) ] 2 \ — u. 2«aa a a ' 2 C " fBer(a'a)Ber' (a'a) + Bei(a'a)Bei' (a'a)] _ c L Ber (a'a) + Bei' (a'a) ,2 2 2 6 2 7 J The numerical v a l u e s of the f u n c t i o n s Ber, B e i , Ber', and B e i ' are tabi tabulated power s e r i e s (15) (17) v , or they may be c a l c u l a t e d from 82 Appendix D. Solution of the Integral i n Equation 4-50. The i n t e g r a l i s OO x = r Mi it Qv? + j - u)e- ( " y">cos x»u du u h D-l + o where x" = X ' s = sju>\id* X D-2 y" = y' s = s j ( 4 i ^ * y h" = h's = s J cop-ci'h and 2 s je co(n-l) o 1 + = D-5 Equation D-l has at least two possible solutions, (2) one suggested by J.R. Carson ' f o r small values of R" and the v other obtained by repeated p a r t i a l integration f o r large values of R". The f i r s t solution i s I = P + JQ R+e^ ^ 7 y [ E ( L R « e 3 3 ( 0 + ^ ) + G( R«'e^ d (e + ^ ) \ + R"e _ 1 r 2J(e-cf) -2j(0+<P| e R" Where K^(x) + L e J D - 4 i s a Bessel function of the second kind and the function G(x) i s defined a s ^ P/ \ 0( ) X = V ^ n=l (-l) x 2 - nl.(n-l)I ^-(2n l)'f2n-l) n + 1 + 2 n 211 1 . . 83 From D-4 and D-5 P = 0.125it(l-s +s ) + 0.51n(|7^(s +s ) - 0.5cf ( l - s 3r - 0.59(s 2 r + 1:L lr s ) +^=(r 4 i -r 3 r l r -r 3 1 -r 1 1 3i ) + 0.5(r 2 r + r ) 4 i 1 : L -s ) 3 r D-6 Q = 0 . 2 5 - 0 . 1 2 5 « ( s - s ) -* O . ^ U + s ^ + s ^ ) - 0 . 5 © ( s - s ) lr 0.51n( |r)(l-s f 3i 3 r + 4r a ) + ^ V l i r 3 r + r 3 i - r l i ) + 2i °«^ 2i- 4r r r D-7 where s = lr ^ Z 2ni ^ 2 n - l ) i (^) " QOs(4n-2)QcoB(4n-2)<f n=l 4n 0 s 2 . D-8 0 *n+l li = Z J"(2n-l)» n=l 2 (^) " cos(4n-2)©sin(4n-2)cT D-9 (^) " sin(4n-2)9co (4n-2)<T D-10 (^) - Bin(4n-2)0sin(4n-2)cT D-ll 4n 2 00 s - ~ " ZI 2 n ^ 2 n - l ) r >2r n=l = 0 s B 2i 3r = 4n 2 S 0 \n+l ZI 2J1V2Z1U n=l 4n 2 00 = Z2ni vn+1 /(2n+l)f (^) c°s4n©cos4n <f D-12 sn+l 3 i = ZI 2 ^ ^ ^ ! ) : n=l ( i) cos4nesin4n<T D-13 4n n=l 00 s S 4n 00 s 4 r = ZI 2nl"(2n!l): n=l (^) sin4n©cos4n<f 4n D-l 4 ) 84 3 4i = Yx 2ni~(L-l)< D-15 (^) sin4n@sin4ncf 4n n=l Z Cx3 lr (or, o\t 4(n-l!)p„4n-3/ -, \n+l 2 9 12n 21 _2 R (4n-3)r(4n-l) n=l CO 7 z i^l} 1 c o _ )© s ( 4 n 3 c o s ( 4 n _ 3 )<f D-16 (or, ?)i o4(n-l) ,,4x1-3/ - i \n+l 2 n li 2 ^2n~2^ 9 S (4n-3)r(4n-l) n=l cos(4n-3)©sin(4n-3)o r D-17 co ph+1 _1 4n •2r n=l 2i Lrii j=l D-18 z[z* _1 4n 5 i = i ^ T T T ( ^ ) 4 r - 2 l o o s ( n-2 ) e i n ( 4 n - 2 4 a D-19 n=l Lj=i ^(-l) n + 1 (2n-l)j2 3r (4n-l)J n=l 7(-l) 3i 2 n + 1 \ 2 R (4n+l) (2n-l).' (4n-l) I (4n-2) „4n-l D-20 2 ~ R" - 2 4 n cos(4n-l)©cos (4n-l)<f 2 4 n (4n+l) 1 c o s (4 n-l) ©s in (4n-l) cf D-21 n=l GO -2n+l '4r 4n+2 11=14=1 ^ ^^(^) 2 4 n cos4n9co 4ncT S D-22 co f2n+l •41 ZI Z n=l > 1 J - 4nT2 2^^nT^ c o s 4 n @ s i n 4 n <*" D " 2 3 (9) The second solution i s <$o /2 fe] _J= cos(2n-l)Q H" ' - cos2Q, cos22T 2 p = + (-l) Q = n+1 s i n ( 2 n - l ) c f ]2| OOB2Q BlnfrT + 8 J. v fei R"^ -- - V 11 1 oos(2n-l)Q r 00B(2r L ^ 5 1)<f D-24 L^g^Utf n=l + (-l) sin(2n-l)<f] n 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 t h i s range use equations D-24 and D-25. (2) These equations reduce to J . R . Carson's solutions SQ i s equal to one and cT i s equal to zero. when 86 References. 1. Carson, J.R., Wave Propagation Over P a r a l l e l Wires; the Proximity E f f e c t , Philosophical Magazine, Vol. XLI, A p r i l , 1921, pp. 607-633. 2. Carson, J.R., Wave Propagation i n 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, A p r i l , 1941, pp. 131-178. 4. Bewley, L.V., Travelling Waves on Transmission Systems, Dover, 1963. 5. Hayashi, S Surges i n Transmission Systems, Denki-Shoin, Japan, 1955. 6. Rudenberg, R., Transient Performance of E l e c t r i c Power Systems, Wiley, 1951. 7. Sunde, E.D., Earth Conduction Effects i n Transmission Systems, Van Nostrand, 1949. 8. Wedepohl, L.M., Application of Matrix Methods to the Solution of Travelling Wave Phenomena i n 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 C i r c u i t s , B.S.T.J., Vol. X, July, 1931, pp. 474-484. n 10. Wise, W.H., Propagation of High Frequency Currents i n the Ground Return C i r c u i t s , Proc. I.R.E,, Vol. 22, No. 4, A p r i l , 1934, pp. 522-527. 11. Wise, W.H., Potential Coefficients f o r Ground Return C i r c u i t s , B.S.T.J., Vol. XXVII, 1949, pp. 365-371. 12. Arismunandar, A., Capacitance Correction Factors f o r Transmission Lines to Include the F i n i t e Conductivity and D i e l e c t r i c 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. C o l l i n , E.C., F i e l d Theory of Guided Waves, McGraw H i l l , I960. 15.. McLachlan, N»W., Bessel Functions f o r Engineers, Oxford, 1955. 87 16. Hedman, D.E., Propagation on Overhead Transmission Lines; I. Theory of Modal Analysis, II. Earth Conduction Effects and P r a c t i c a l Results. Trans. I.E.E.E., Vol. 83, July, 1965 P P » 665-670. r 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 f o r Engineering, Prentice-Hall, 1963. 21. Wedepohl, L.M., Shorrocks, W.B., and G-alloway, R.H., Calculation of E l e c t r i c Parameters f o r 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, MultiConductor, Distributed Parameter Transmission Systems, M.A.Sc. Thesis, University of British,Columbia, November, 1965.
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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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Title | Effect of ground conductivity and permittivity on the mode propagation constants of an overhead transmission line |
Creator |
Doench, Claus |
Publisher | University of British Columbia |
Date Issued | 1966 |
Description | A general analytical method to derive the distributed circuit parameters and mode propagation constants for an n-conductor transmission line is developed. The analysis uses electromagnetic field 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 finite 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 10⁻⁵ 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. |
Subject |
Electric conductivity Electric lines |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-09-29 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0093745 |
URI | http://hdl.handle.net/2429/37722 |
Degree |
Master of Applied Science - MASc |
Program |
Electrical and Computer Engineering |
Affiliation |
Applied Science, Faculty of Electrical and Computer Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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