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Matrix analysis of steady state, multiconductor, distributed parameter transmission systems 1965
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Title | Matrix analysis of steady state, multiconductor, distributed parameter transmission systems |
Creator |
Dowdeswell, Ian J.D. |
Publisher | University of British Columbia |
Date Created | 2011-09-29T20:52:47Z |
Date Issued | 2011-09-29 |
Date | 1965 |
Description | Problems concerning transmission lines have been solved in the past by treating the line in terms of lumped parameters. Pioneering work was done by L. V. Bewley and S. Hayashi in the application of matrix theory to solve polyphase multiconductor distributed parameter transmission system problems. The availability of digital computers and the increasing complexity of power systems has renewed the interest in this field. With this in mind, a systematic procedure for handling complex transmission systems was evolved. Underlying the procedure is the significant concept of a complete system which defines how the parametric inductance, capacitance, leakance and resistance matrices must be formed and used. Also of significance is the use of connection matrices for handling transpositions and bonding, together with development of the manipulation of these matrices and the complex (Z) and (T) matrices. In the numerical procedure, methods were found to transform complex matrices into real matrices of twice the order and to determine the coefficients in the general solution systematically. The procedure was used to deal with phase asymmetry and mixed end boundary conditions. |
Subject |
Electric Lines Matrices |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | Eng |
Collection |
Retrospective Theses and Dissertations, 1919-2007 |
Series | UBC Retrospective Theses Digitization Project [http://www.library.ubc.ca/archives/retro_theses/] |
Date Available | 2011-09-29T20:52:47Z |
DOI | 10.14288/1.0093743 |
Degree |
Master of Applied Science - MASc |
Program |
Electrical and Computer Engineering |
Affiliation |
Applied Science, Faculty of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
URI | http://hdl.handle.net/2429/37721 |
Aggregated Source Repository | DSpace |
Digital Resource Original Record | https://open.library.ubc.ca/collections/831/items/1.0093743/source |
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MATRIX ANALYSIS OF STEADY STATE, MULTI-CONDUCTOR, DISTRIBUTED PARAMETER TRANSMISSION SYSTEMS by IAN J. D. DOWDESWELL B . A . S c , U n i v e r s i t y of B r i t i s h Columbia, 1961 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 E n g i n e e r i n g ¥e a c c e p t t h i s t h e s i s as conforming t o the r e q u i r e d s t a n d a r d Members of the Department of E l e c t r i c a l E n g i n e e r i n g THE UNIVERSITY OF BRITISH COLUMBIA November > 1965 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an advanced degree a t t h e 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 a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r - m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e 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 £jLAsn^JL I ^ W J U A - H ^ The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada Date '3 . x u ; < i s ABSTRACT Problems concerning transmission lines have been solved in the past by treating the line in terms of lumped parameters. Pioneering work was done by L. V» Bewley and S. Hayashi in the application of matrix theory to solve polyphase multi- conductor distributed parameter transmission system problems. The ava i labi l i ty of d ig i ta l computers and the increasing complexity of power systems has renewed the interest in this f i e l d . With this in mind, a systematic procedure for handling complex transmission systems was evolved. Underlying the pro- cedure is the significant concept of a complete system which defines how the parametric inductance, capacitance, leakance and resistance matrices must be formed and used. Also of significance is the use of connection matrices for handling transpositions and bonding, together with development of the manipulation of these matrices and the complex (Z) and (T) matrices. In the numerical procedure, methods were found to transform complex matrices into real matrices of twice the order and to determine the coefficients in the general solution systematically. The pro- cedure was used to deal with phase asymmetry and mixed end boundary conditions. i i TABLE OP 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 Acknowledgement • v i i 1. INTRODUCTION 1 2. GENERAL DIFFERENTIAL EQUATIONS FOR MULTI-CONDUCTOR SYSTEMS 2 3. THE DIFFERENTIAL EQUATIONS FOR STEADY STATE ANALYSIS 6 4. SOLUTION OF THE DIFFERENTIAL EQUATIONS 4.1 Characteristic Root and Characteristic Vector Analysis 10 4.2 The General Solution 13 5. BOUNDARY CONDITIONS 16 6. TRANSPOSITION MATRICES AND THE COMPLEX CHARACTERISTIC MATRIX 6.1 The Transposition Matrix 23 6.2 Expansion of Complex Matrices to Real Matrices of Twice the Order ..•»••••••••.••.•••••...•....... 27 6.3 Transposition and Connection Matrices f o r Multiple Section Lines 30 7. THE (Z) & (Y) MATRICES FOR A COMPLETE SYSTEM •; 7.1 The (Z). & (Y) Matrices ....... 35 7.2 Properties of the (Z) & (Y) Matrices 36 7.3 Restrictions on the Use of the Distributed Parameters ..................................... 37 8. EXAMPLES OF APPLICATION AND RESULTS 8.1 The Overhead Transmission System 42 8.2 Results 48 i i i Page 8.3 The Underground Transmission System 49 8.4 Results 50 9. CONCLUSIONS 58 APPENDIX A. The Parameters R, L and C 60 A.l Assumptions ».. 60 A.2 The Resistance, RT 60 A.3 The Inductance, L* 61 A.4 The Capacitance, CT ............................ 64 APPENDIX B. A Flowsheet of Solution Procedure .......... 67 APPENDIX C. Data Sheets C l . l Overhead Conductor System, No Load Voltage and Power, Transposed and Untransposed 68 C1.2 Overhead Conductor System, No Load Current, Transposed and Untransposed 68 C1.3 Overhead Conductor System, Pull Load Voltage and Power, Transposed and Untransposed ............ 69 C1.4 Overhead Conductor System, Pull Load Current, Transposed and Untransposed 69 C2.1 Underground Conductor System, No Load Voltage and Power 70 C2.2 Underground Conductor System, No Load Current . 70 C2.3 Underground Conductor System, F u l l Load Voltage and Power 71 C2.4 Underground Conductor System, Fu l l Load Current 71 REFERENCES . 72 iv LIST OF ILLUSTRATIONS Figure; Page 2.1 Part of a Mutually Coupled Circuit of (n+l) Conductors . . . . . . 2 5.1 Single Section of a Doubly Bonded Cable Transmis- sion System with Six Independent Conductors .... 19 6.1 Partially Transposed Transmission Line 32 6.2 Two Sections of Transmission Line .............. 35 8.1 . A Three-Section, Six-Conductor with Ground, Over- head System 43 8.2 Ground Wire Current, 11 I , amps vs Distance along the Line, x, metres . . . f 4 4 8.3 Power Consumption, P, m.w. at No Load vs Distance along the Line, x, metres ....•.....•••••.•••».. 44 8.4 Reactive Capacitative Power Consumption, Q, m.v.a. at No Load vs Distance along the Line, x, metres 45 8.5 Current Phase Angle Differences at Sending End for both Transposed & Untransposed Systems ......... 45 8.6 A-phase Current, I I I , amps vs Distance along the Line, x, metres • 46 8.7 A-phase'Current Argument, (i , degrees vs Distance along the Line, x, metres 46 8.8 A-phase Voltage, |V I, kv vs Distance along the. Line, x, metres • 47 8.9 A-phase Voltage Argument, [ v . degrees vs Distance along the Line, x, metres 47 8.10 An Underground, three phase cable system with separate ground wire and sheaths around each conductor ..*.».........•••...•.........•..•.... 52 8.11 No Load Power, P, megawatts vs Distance along the Line, x, metres • 53 8.12 No Load Reactive Power, Q, m.v.a. vs Distance along the Line, x, metres ...................... 53 8.13 Conductor Current Phase Angle Differences at the Sending End •«.. 54 v Figure Page 8.15 No Load and F u l l Load Sheath Current Phase Angle Differences at the Receiving End ............... 54 8.14 Conductor Current, T i l , amps vs Distance along the Line, x, metres ............................ 55 8.16 No Load and F u l l Load Sheath Current, f l j , amps vs Distance along the Line x, metres ........... 56 8.17 Sheath Voltage Phase Angle Differences at the Sending End 56 8*18 No Load Sheath Voltage; |V.|, volts vs Distance along the Line, x, metres • 57 8*19 F u l l Load Sheath Voltagej |V,|, volts vs Distance along the Line, x, metres .......».e.•••»•••«..• 57 A t tl A group of n+1 Current Carrying Conductors ..... 62 A«,2 Cross-section of part of a system of conductors where one conductor completely encloses another 65 v i ACKNOWLEDGEMENT The author would l i k e to express his sincere gratitude to his supervising professors Dr. F. Noakes and Dr. I. N. Yu for guidance, perseverance and continued i n s p i r a t i o n throughout the course of the research. The author also wishes to thank the members of the Department of E l e c t r i c a l Engineering. The author i s indebted to the National Research Council of Canada for f i n a n c i a l support of the research. v i i 1 1. INTRODUCTION The purpose of this thesis i s to develop a procedure for solving the problem of polyphase, distributed parameter transmission systems, during steady state operation. H i s t o r i - i c a l l y , this problem has been attacked by treating the lin e configuration i n terms of lumped c i r c u i t parameters, obtained through transformations from the distributed parameters, lead- ing to various closed form solutions. Early work i n the development of matrix methods for analysis was done by L«V. Bewley ( l ) . The approach taken was to analyze the lossless polyphase l i n e and to expand the analysis to include lines with losses. From t h i s , t r a v e l l i n g wave solutions were developed which led to a study of surges by matrix methods. L.A» Pipes (2) followed Bewley's approach but used Laplace transform methods. P a r a l l e l developments were made by S. Hayashi (3) who extended the analysis to transient phenomena, including t r a v e l l i n g wave properties of surges. The increasing complexity and interconnection of modern power systems., together with the f l e x i b i l i t y and a v a i l a - b i l i t y of d i g i t a l computers, makes the use of matrix methods both imperative and p r a c t i c a l . ¥ith this i n mind, a systematic mathematical and numerical procedure for handling the complex system i s evolved i n this thesis. The ra t i o n a l i z e d M.K.S. system of units i s used throughout. 2 2. GENERAL DIFFERENTIAL EQUATIONS FOR MULTI-CONDUCTOR SYSTEMS Consider a system of (n + l) parallel conductors mutually coupled electrostatically and electromagnetically. By definition, this is a complete system i f and only i f the sum of the currents over the whole system is zero, n + 1 i i = 0 2-1 i = 1 This definition precludes radiation effects, but this is an acceptable approximation at low frequencies. Fig 2.1 Part of Mutually Coupled Circuit of (n + l) conductors. 3 Of the (n + l ) conductors, n w i l l be defined as being independent; the (n + l ) t h conductor becomes the reference conductor for voltages and the"return" path for unbalanced currents. Depending on the physical arrangement of the trans- mission system, this reference conductor would normally be taken as a ground conductor or an equivalent earth conductor ( 4 ) . The voltage and current equations for the i t h conduc- tor may be written £ v . ^ y . " 1ST = ITt + V i 2~2 2> i , 2>q. n " "Sic = TT + 1 i 2 ~ 3 where v. = potential of conductor i with respect to some l a r b i t r a r y reference vp. = t o t a l f l u x linkages per unit length of conduc- tor i due to currents i n a l l conductors R̂ = series resistance per unit length of conductor i i . = current i n conductor i I q^ = charge per unit length on conductor i and i ^ = leakage current per unit length from conductor i The system i s assumed to be linear i n the following analysis. Let p be the d i f f e r e n t i a l operator p = 5 / t Associated with each unit length of conductors i and j are Z! . = R. + pL. . xx 1 ^ 11 Z!. = pL. . 13 * i j T! . = G.. + pC.. x i 11 • 11 I! . = G. . + pC. . 10 13 * 13 4 where IL = series resistance of conductor i L.. = self inductance coefficient of conductor i 1 1 L.. = mutual inductance coefficient of conductors 1 J i & j G i i = s e ^ capacitance coefficient of conductor i C . = mutual capacitance coefficient between conduc— 1 J tors i & j G.. = leakance from the ith conductor to the arbi— 1 1 trary reference G.. = leakance between conductors i & j The differential equations of the ith conductor become " T x = Z i l A i + Z i 2 *2 + + Z i i * i + — + Z i , n + 1 in+l 2-4 3 i . - T i = T i l V l + T i 2 V2 + + T i i V i + — + T i , n + 1 v n + l 2-5 In matrix form, the 2(n + l) equations for the (n + l) conductors may be written = ( Z ' ( P ) ) (i) 2-6 - l ^ 1 = (T'(p)) (v) 2-7 where (v) and (i) are column vectors, ( Z 1 ) and (Y 1) are square matrices which are functions of time (the differential operator p). By differentiation with respect to x and substitution^ equations 2-6 and 2-7 may be combined to give 5 4̂ M = (Z'(p)) (I f ( p ) ) (v) 2-8 where (Z 1) and ( Y T ) are both independent of x. 3. THE DIFFERENTIAL EQUATIONS FOR STEADY STATE ANALYSIS C o n s i d e r the system of conductors o p e r a t i n g under a-c s t e a d y s t a t e c o n d i t i o n s such t h a t the v o l t a g e s of the (n+l) conductors a t a p o s i t i o n x , w i t h r e s p e c t t o some a r b i t r a r y r e f e r e n c e , are g i v e n by 3-1 a j(»t+0o) KJ n+l n+l/ S i n c e the system i s l i n e a r , the c u r r e n t response w i l l have the same form w i t h d i f f e r e n t phase a n g l e s 0^, 0^> ••• * ^ n + i - / I , j (at^.) ( I ' ) 3-2 1 2 e 3(«t+02) I n + 1 e J ( • * * » « > . where ( V ) and ( l T ) are phasor v e c t o r s . S u b s t i t u t i o n o f these phasor v e c t o r s i n t o e q u a t i o n s 2-6 and 2-7 r e s p e c t i v e l y ^ w i t h the o p e r a t o r p r e p l a c e d by jtt y i e l d s 7 - (V') = (Z'(«)) (!') 3-3 - 3! ( T 1 ) = (!'(•)) (V') 3-4 Equations 3-3 and 3-4 are written in terms of voltages with respect to some arbitrary reference. For a complete system, some reference within the system, such as a "ground" conductor may be used. If the (n+l)th conductor is chosen as the refer- ence conductor, then the voltage phasor vector becomes and 3-5 3-6 i = 1 Applying these constraints to equation 3-3 yields the reduced system of equations for n independent conductors. - (V) = (Z ( « ) ) (I) 3 - 7 where Z^) = ZL(.) + - « J - * A + l f , < « ) i» 0 = 1» 2, n 8 Since ( Y ' ( « ) ) i n equation 3-4 i s not a f u n c t i o n of x - (v«) = £ ( i ' ( • ) ) - 1 ( ! • ) L e t ( Y ( « ) ) ""1 be the reduced form of ( Y T (w)) """"/then, -(v) = C Y ( « ) ; - 1 fe (I) and the cur r e n t equation f o r n independent conductors may be w r i t t e n - fe ( D = ( I (•)) (V) 3 - 8 This a n a l y s i s i n d i c a t e s t h a t the r e d u c t i o n must be achieved with the ( Y* ( » ) ) matrix i n i t s i n v e r s e form. The leakance matrix (Gr) may be separated i n t o two p a r t (a) the leakance e m p i r i c a l l y d e r i v e d from the l o s s e s due to the supporting mechanism (towers, conduits etc.) of the t r a n s m i s s i o n system^ (b) the leakance due to the geometrical c o n f i g u r a t i o n of the conductors and to the c o n d u c t i v i t y of the surrounding media. I f p a r t (b) alone i s considered, then since the f i e l d d i s t r i b u t i o n and leakage c u r r e n t d i s t r i b u t i o n are the same f o r any g i v e n l i n e a r system of conductors ( 5 ) , where (P') i s the p o t e n t i a l c o e f f i c i e n t matrix f o r the (n+l) conductors <r i s the c o n d u c t i v i t y of the medium surrounding the conductors and e i s the p e r m i t t i v i t y of the medium surrounding the conductors. 9 Therefore, the matrix ( P ) for the reduced system of n independent conductors will have elements of the form 13 13 n+ljn+1 i,n+l n+l , j 1 i, 3 = 1 » 2, •»»t n and ( I («•)) = ( 0 - + j » e ) (P)" 1 Note that for most transmission systems, 6-/«e <£. 1 Finally, the reduced equations 3-7 and 3-8 may be combined as before to give ^ 2 (V) = ( £ ( • ) ) (!(•)) ( ? ) i ( A ( o ) ) ) (V) d x 3-9 d 2 ( D = (!(•)) ( Z ( « ) ) ( I ) £ (B(«)) ( I ) ^ 2 dx 3-10 10 4. SOLUTION OF THE DIFFERENTIAL EQUATIONS 4.1 C h a r a c t e r i s t i c Root and C h a r a c t e r i s t i c Vector A n a l y s i s (6) Equation 3-9, the v o l t a g e equation, may be w r i t t e n 0 4-1 This has a n o n - t r i v i a l s o l u t i o n i f and only i f the determinant det | (A) - d 2 / d x 2 ( u ) | = 0 4-2 where (u) i s the i d e n t i t y m a t r i x . This determinantal equation i s the c h a r a c t e r i s t i c equation whose s o l u t i o n y i e l d s the charac- t e r i s t i c r o o t s . Consider again equation 3-9. There are n o r d i n a r y second order l i n e a r d i f f e r e n t i a l equations with constant c o e f f i — d 2 c i e n t s which are homogeneous i n —5- . dx^ Hence the form of the s o l u t i o n i s — N & X — Y x V i = I> ( Cil,e r + °i*e 9 i = 1, 2, n 4-3 where the C's and C ' s are the complex constants of i n t e g r a t i o n and y'^ = v/X^ , where the X's are the c h a r a c t e r i s t i c roots of the determinantal equation 4—2. 11 There are 2n constants of integration in the above form of the solution but i t w i l l be shown that only 2n of these constants are independent. Substitution of the general solution, equation 4-3 into the equation 3-9 yields n equations of the form 2 r=l r=J + ^ * (c. e w r + C! e o r ) ^ 0 r i r l r ' y x - v x n y x Sr . „ . *r x ^ r - A ( c _ J r i l i r + 0 ! e~* r*) i r ' n + ^ A, J C _ e o r + C' e ° r ) in nr nr v x —^ x Collecting terms in e and e r r=T we have r=i ( ( * C i r - A i l ° l r - A i 2 C 2 r ' — " A - C - ~ > in nr + <<* r " A i i ) C i r - A i l ° i r " Ai2°2r ~ — " A i n C A r 4-4 Hence each of the coefficients in equation 4-4 is individually equal to zero for a non—trivial solution, (i = 1, 2, . . . , n). 2 2 This provides n equations f o r the unprimed constants, C,and n equations in the primed constants, c', as both i and r vary. For the unprimed constants, C r 7=0 / A l l r A21 A 12 L ln A22 " Kr"" A 2 n V n l = 0 , r = 1,2, a • • , n 4-5 12 Similar equations may be written for the constants, f and since these systems of equations are homogeneous, there are (n—l) independent relations between the constants C and also between the constants C for each choice of r» This leaves 2n independent constants to be found from the boundary conditions. It is apparent from the above discussion that the determination of the relationship between the constants, C yields the characteristic vectors, with one vector corresponding to each choice of X. This may be shown explicitly by rewriting equations 4—5 as follows, A l l - X r A 21 h i A12 # # * * A l n A 2 2 " X r A -X nn r, (c , / C \ C =(0) l r nr nr 0 o / C 2r nr V / /A, , - X 11 r A 21 A, n t • - • • A \ 12 A 2 2 " V ^-l,n-rXr/ D 2r \ D n - l , r / - - N \ L2n Ln-l,n/ 4-6 where D. = C. /C and C ^ 0. i r i r nr nr This analysis indicates a method of determining the characteris- t i c vectors numerically. The same vectors hold for both the constants C and C . 13 4.2 The General Solution The voltage solution may be written as J no ^ jr no ^> j r r=l r=l Y. = C . >~D. e + 0 1 ^ D- e ~ r > r = l,2,.»*n no j r no ^> ;~ 4-7 where D = 1 . and C .. C T. are unknown constants to be nr * nj* no determined from the boundary conditions. By a similar analysis, the current solution^ may be written as _ n v x n -V x I. = F . ̂ >"G. e° r + F 1 . > G. e , r = l,2,...n 3 nj ^ j r nj jr ' -r=l r 3 4 - 8 where G = 1 . and F .. F* , are unknown constants. It will nr ' nj* nj be shown that the y ' s are the same for both voltage and current solutions. The constants C . i C . and F ., F'. are not indepen- nj nj nj nj dent but are related through equations 3-7 and 3-8. This solution may also be written in the alternative hyperbolic form using the hyperbolic sine and cosine. The general solution for voltage and current may be written in matrix form ( V ) = ( D ) U C n r e * r X ) + ( C n r e * ^ 4-9 C D = ( O f + (* n re"* r X)} 4-10 where (D) and (G) are square matrices containing characteristic vectors as columns. For example, the f i r s t column of (D) is the characteristic vector which satisfies ( (A) - X 1 (U)) ( D x ) = 0 and hence is associated with the characteristic value X-̂ . Note that the entries D (and G ) for r = 1, 2, ...,n wi l l be nr nr y ' ' unity. 14 The matrices (C e ) represent column vectors nr ( 0 e " & X ) t \ nr / /C i e nl C n 2 6 4-11 V e-V/ » T i n ' nn As stated previously, the voltage and current solutions are related through equations 3—7 and 3-8$ these equations imply that the current solution may be obtained from the voltage solu- tion and vice versa. Let the current solution be known, then by rearrange- ment of equation 3-8 v = -(y(a))"1 fj - (T) = - (H-))"1 « » ) { ( » l l r . * r X y r ) • — Y X. y x - y x By equating coefficients of e r and e r respectively, we obtain ( D ) ( c n r ) =-(i ( . ) ) - l ( o ) ( » „ y r ) - ( ' W ) ( « ) ( u ) 4-12 15 A N D ( D X C N R ) = (T(<°>) ' ( O K A ) 4" 1 3 = (*(•>) (0(*Aryr> where leakance has been ignored and ( !(«))( P ( « ) ) = (U) . Thus the general form for the voltage and current solutions may be written ( ? ) = - ( P ( . ) ) ( 0)j( F n r t f r e * r X ) - (Kj/*1*)] ' 4-14 ( l ) = ( O J ( ^ n / r X ) + ( S A / X r X ) ] . r=l,2,..,n 4-15 In a similar manner the current constants may be determined in terms of the voltage constants by use of equation 3-7 using the properties of duality, <n = ( » ) j ( C n / r X ) + ( C A / ^ ) ] , r=l,2,..,n ( T ) . - f z W ) - 1 ^ ) ^ / ^ ) - ( ^ / ^ l r=l,2,..,n 4-17 For equations 4—16 and 4—17, the voltage solutions w i l l be defined as the "primary" solution; the current solu- tion is a "derived" solution. Conversely, for equations 4—14 and 4-15, the current solution i s the primary solution from which the voltage is derived. This latter form will be chosen to illustrate the following analysis of the boundary conditions. 4-16 16 5. BOUNDARY CONDITIONS Consider the boundary c o n d i t i o n s f o r the complete; • conductor system. There must be 2n such c o n d i t i o n s which may be s p e c i f i e d as c o n s t r a i n t s on the v o l t a g e , c u r r e n t or both at the boundaries. For such an n-conductor system, there are i n general 4n c o n d i t i o n s at the boundaries, 2n at each end of the l i n e . These are < T > » - - i - ( T . > and ( V ) _ I ( Y r ) ( * ) x = 0 = ( I r ) , sending end c o n d i t i o n s , , r e c e i v i n g end c o n d i t i o n s . A l t e r n a t i v e l y , the o r i g i n of x may be d e f i n e d at the sending end, i n which case the r e c e i v i n g end i s designated by x = I, ; i n both cases, x i n c r e a s e s from the sending end to the r e c e i v i n g end. Of the 4n boundary c o n d i t i o n s , 2n must be known i n order to o b t a i n a unique system s o l u t i o n . Several s p e c i a l cases may be considered, ( i ) ( V g ) and ( I g ) or ( V r ) and ( I r ) ( i i ) ( V s ) or ( l g ) and ( V r ) or ( I r ) ( i i i ) ( V g ) or ( I g ) and ( Zr ) 1 7 ( i v ) any 2n c o n d i t i o n s of ( V g ) , ( I ) , ( V r ) and ( I p ) where ( Z ) i s the r e c e i v i n g end impedance m a t r i x d e f i n e d by ( O X X r ) . The p r i m a r y c u r r e n t form of s o l u t i o n i s used to i l l u s t r a t e these s p e c i a l c a s e s . ( i ) ( V r ) and ( I r ) known, "n" e q u a t i o n s may be w r i t t e n f o r ( I ") . and n e q u a t i o n s f o r ( V ) , -(*<•))( <0[(*nrXr> " ( P 'nry r)i " ( V r ) 0 1 ( G ) [ ( P n r K r ) - 0 ' n r * r ) | ="(!(•))( T r ) The 2n e q u a t i o n s i n 2n unknowns may be r e w r i t t e n i n the form fcO (u ) p n r nr - ( G ) - 1 ( K « ) ) ( V ) where (u^) = 0 •y2 r 5-1 5-2 0 An e x p l i c i t s o l u t i o n may be o b t a i n e d f o r the column v e c t o r s ( F } a n d ( F ' ^ , ^ n r / \ nr / ' ( F n r ) " ^ ( ^ ( I , ) " ( u ^ X d ) - 1 ( Y ( . ) ) ( V r ) ^ 18 L e t ( X S ) H C O X U K ) ) " 1 •'• ( F n r ) = *[USrXxXlr) -UX'WXOJ and ( P A r ) = *[(D)fXxXl r) + (xXK»)X¥r)^ which i s the simplest form f o r numerical s o l u t i o n u s i n g a d i g i t a l computer. and or (')[( F e nr s ) known. are + ( * ' )\ N nr 'J \ nr / / \ nr > j / ( O ( u ) ( l i e " * * ) ( lJe*<) which may be solved as i n case ( i ) . ( i i b ) ( I and ( v s ) known. The matrix equation i s / ( u ) ( u ) \ ( P )\ \ nr / (pi ) \ nr / which may be solved as i n case ( i ) . ( i i i ) ( I ) and ( ) known. From ( I g ) , u r ^ I ) 5-3 - ( G ) - 1 C K . ) ) ( 0 5-4 F e nr ) + ( p A r e ^ ^ . ( I . ) 19 From ( V ) - ( P ( « ) ) C O K F n A ) or in matrix form (Z r)(G) + P(tt) ( 6)(ny)jC8 r)(G) - P(o>) (G)(U g ) j \ ( F A r ) (iv) Bonding of cables. K l . ) 5-5 1 1 i L O ; * \ ' 2 / \ i \ i \ i i i i ( / / i A D i 5 \ '3 J i / l / ' I i i i oc-O Fig. 5-1 Single Section of a Doubly Bonded Cable Transmission System with Six Independent Conductors. Consider a section of a doubly bonded cable transmission system as shown, where conductors 1, 2, and 3 are the cable cores, conductors 4, 5, and 6 are the respective sheaths which are open circuited at x = 0 and x = -L. The sheaths are doubly bonded to 20 the equivalent ground conductor, g , at x = some point between the terminals. This ground conductor will be used as the voltage reference conductor. If the earth is to be considered as interacting with this system, then a further equivalent earth conductor would be necessary—this would be used as the reference conductor, giving a system of seven independent conductors. Consider this system of conductors where the core cur- rents and the c6,re voltages at the load are specified. In this case the double-bonding junction must be treated as a "new" boundary and the given section of the transmission system must be treated as two sub-sections. If the load end sub-section is designated as sub-section 0 , — - 0, and the remaining sub—section as (?) , ~ x2 ~ ̂ ' then for section© the system of equations to be solved for the boundary conditions • ( V]_ 92 3 ) x - 0 ^ ( ^r ) 1 ' r e c e i v i n g e n d conditions ( ?4,5,6 ) X l = -L = ( A ) l = ( ° ) > "ending end * 1 1 conditions ( T i ) x = 0 = ( I r ) , j = 1, 2, 6 and , receiving end conditions i s -(p(«oX<0(u*) (rows 1,2,&3) -(p(.)XoX^._,l:<1) (rows 4,5,&6) (all rows) ! (p(.>x8x°«) 1 _(rows_l,J,&3]_ (p(-)XoX"x»Wi: (rows 4j5,&6) (al l rows) / \ v nr' 1 / \ 1 f „. \ F' ) 1 • nr'1 \ / 5-6 21 which may be expanded as 9.. I T . it i l l * » t ' 6 > > > I 3 6i (ti where £ i > i s the ( i , j ) t h element of the product (P(<O))(G) and g^^ i s the ( i , j ) element of the c h a r a c t e r i s t i c v e c t o r matrix ( G ). The system of equations 5-6 can be sol v e d f o r the constant v e c t o r s ( P ) n a n d ( F ' ) , f o r sub-section© . Hence * nr ' 1 v nr / 1 w the remaining unknown cur r e n t s and vo l t a g e s at the l e f t - h a n d boundary of s u b - s e c t i o n ( 7 ) , x 1 = -$^t can be found. The core v o l t a g e s and cur r e n t s are continuous and the boundary c o n d i t i o n s f o r s u b — s e c t i o n (z) are <Vx,=0 1 ( V r ) 2 " (fj) x,=-L • J = 1. 2 . & and ^ i , 2 , 3)x 2 = o M . ; ) 2 22 The system of e q u a t i o n s /- P(a)(G)(Uy) ' ( a l l rows) *<•> ( 0 ) ( U y ) \ / \ =/(V r) ( a l l rows; \ / \ / V (G) (rows 1,2,&3) (G) (Ue~ ^ ^ 2 ) (rows 4,5,&6) (G) (rows 1,2,&3) 1 (G)(Ue + ^ 2 ) / (rows 4,5,&6)/ < Fnr>2 (P' ) v n r ; 2 ( I C ) . r 2 5-7 can he s o l v e d f o r the c o n s t a n t v e c t o r s (P ) 0 and (P* ) 0 nr 2 nr 2 Knowing the c o n s t a n t v e c t o r s f o r each s u b - s e c t i o n of the t r a n s m i s s i o n l i n e , the complete s o l u t i o n f o r the complete s e c t i o n can be d e t e r m i n e d u s i n g e q u a t i o n s 4-14 and 4—15. I f the sheath bondings f o r each s u b — s e c t i o n are connected, a d d i t i o n a l c o n s t r a i n t s are imposed on the system. I n t h i s case> the c o n s t r a i n t e q u a t i o n i s 2 ( 1 . ) 0=4 J 1 3=4 5-8 Si n c e the l o a d boundary c o n d i t i o n s f o r the case are s p e c i f i e d , t h e n the c o n n e c t i o n of the bonding causes a c o n s t r a i n t to be im- posed of sub-section© by sub-section®. Such a c o n s t r a i n t can be handled by u s i n g a t r a n s p o s i t i o n m a t r i x as s p e c i f i e d i n the nex t c h a p t e r . However, s i n c e the e n t r i e s i n a t r a n s p o s i t i o n m a t r i x are u n i t y , t h i s r e p r e s e n t s a ' ' l o s s l e s s " i n t e r n a l boundary or t r a n s p o s i t i o n p o i n t . T h i s i s s a t i s f a c t o r y as a f i r s t a p p r o x i m a t i o n . 23 For a "lossy" transposition, where voltage and current magnitude and/or phase angle for a given conductor does change, the corresponding entry in the transposition matrix w i l l be in general a complex number with absolute value different from unity. Such an entry, ^^y m a y be represented by t. . = e ± ( / + ^ ) where p is the attenuation factor, and r\ gives the phase angle shift. 24 6. TRANSPOSITION MATRICES AND THE COMPLEX CHARACTERISTIC MATRIX 6*1 The Transposition Matrix* Consider a transmission line with multiple sections where at each or any junction two or more of the conductors have their physical locations in space interchanged. Such an interchange is indicated in the resistance, ( R ) , inductance, ( L ) and capaci- tance, ( C ) coefficient matrices by a corresponding interchange of the appropriate rows and columns. For example, the interchange of two conductors (i&j) at one junction may be made in the ( R ), ( L ) and ( C ) matrices by using the transformation matrix ( E r) which is formed from the identity matrix ( U ) by interchanging the i and j rows or columns i j E 1 0 0 1 i 0 - 1 - 6-1 1 I i 0 - — - Premultiplication of any matrix ( A (say), by the matrix ( E^ ) , (which is compatible with matrix ( A ) ,) causes the i ^ * 1 and j ^ * 1 rows of ( A ) to be interchanged, and postmultiplica- tion causes the corresponding columns of ( A ) to be interchanged, 25 i . e . ^ E r > ) ( A V ) y i e l d s ( A ^ w i t h i ^ * 1 and j ^ * 1 rows interchanged, ( A ^ ( E r ) y i e l d s ( A ) with i ^ * 1 and j ^ * 1 columns interchanged, and ( E r ) ( A ^ E p ) y i e l d s ( A ^ w i t h i ^ * 1 and j ^ * 1 rows and columns interchanged. Consider some p r o p e r t i e s of the set of matrices ( E ^ } By i n s p e c t i o n , / these matrices are symmetric, (O = ( E r ) t 6 ~ 2 I f the matrix ( E ^ of rank r , i s p a r t i t i o n e d as f o l l o w s , ( O ( V.) 0 o ( u r . p , where ( u\ ) i s an i d e n t i t y matrix of rank i , and ( J ) i s a matrix of rank (j - i - l ) , ( J ) f0 (0) (0) (U._.^(0)| (o) i & where the symbol 0 represents the s c a l a r zero, .-. ( j ) 2 = / l 0\ = (Vi_i) <Vi-2> ,0 and ( E ) ( J ) 26 6-3 a n d ( E ) i s o rthogonal. For t h i s matrix ( E ) the r d i s t r i b u t i v e law holds, i . e . ( E r ) ( R ) ( E r ) + ( E r ) ( L ) ( E r ) ( E r ) ( z ) ( E r ) and hence, K)(A)(Er) Thus the t r a n s f o r m a t i o n i s v a l i d on the product, ( A ) j without r e v e r t i n g to the separate p a r t s of the matrix ( A ) . From these p r o p e r t i e s of the t r a n s p o s i t i o n matrix, i t unchanged, but not the c h a r a c t e r i s t i c v e c t o r s . The f a c t t h a t the c h a r a c t e r i s t i c r o o t s of the d i f f e r e n t s e c t i o n s of the l i n e are the same i s to be expected from p h y s i c a l c o n s i d e r a t i o n s of the t r a n s - m i s s i o n l i n e as a whole, since the geometrical c o n f i g u r a t i o n of the conductors i s unchanged. The r e l a t i o n s h i p between c h a r a c t e r i s t i c v e c t o r s f o r two s e c t i o n s of l i n e may be found from the d e f i n i n g equations of the c h a r a c t e r i s t i c v e c t o r s f o r each s e c t i o n , maybe seen that the o p e r a t i o n ( E ) ( A ) ( E ) i s a s i m i l a r i t y t r a n s - formation which leaves the c h a r a c t e r i s t i c roots ( A ) ( x ) = X ( x ) , 27 Hence the r e q u i r e d r e l a t i o n s h i p which depends on the o r t h o g o n a l i t y of ( E r ) i s ( x ) = ( K r X y ) . Consider the p h y s i c a l s i t u a t i o n where at a j u n c t i o n , more than one p a i r of conductors i s interchanged. I m p l i c i t here i s the assumption that the necessary elementary t r a n s f o r m a t i o n matrices of the type (E^) may interchange at d i f f e r e n t times the same rows and columns more than once. t h t h For example, i n order to interchange the i , j , and ~th *th "th k conductors c y c l i c a l l y , the i and j rows and columns must "th be interchanged, f o l l o w e d by an interchange of the k and " o l d " i ^ * 1 rows and columns. Consider n such interchanges at one j u n c t i o n and the a s s o c i a t e d elementary matrices ( ) T{E^) , ..., ( E ) . The r e l a t i o n s h i p between the o r i g i n a l s e c t i o n of t r a n s m i s s i o n l i n e w i t h a s s o c i a t e d c h a r a c t e r i s t i c matrix ( A') i s given by ( A ' ) = ( E n ) ( E n _ 1 ) . . . . ( E 2 ) ( E 1 ) ( A ) ( E 1 ) ( E 2 ) . . . ( E n ) = ( « ) ( A X X ) The r e l a t i o n s h i p s between the c h a r a c t e r i s t i c roots and v e c t o r s of the matrices ( A ) and ( A' ) are v a l i d i f i t can be proved t h a t ( «> ) = ( X ) , i . e . t h a t such a t r a n s f o r m a t i o n i s a s i m i l a r i t y t r a n s f o r m a t i o n . I t w i l l a l s o be shown t h a t ( <o) i s an orthogonal a t r i x , ( «)(<o ) ̂ = ( U ) . = ( K ) ( V ! > ( " 2 X > i » - 1 = ( E , ) - 1 ^ ) - 1 ( E ^ ) - 1 ^ ) - 1 28 Also, ( « O t = ( B 1 ) t ( B 2 ) t . . • • ( E n _ l ) t ( E n ) t = ( E l ) ( E 2 ) ' * • * ( E n - l X E n ) = ( X ) **. («)(«) .j. = ( u ) as required. Note that (<o) is not symmetric. Thus the characteristic roots of ( A) and ( A' ) are the same and the characteristic vectors of these two matrices are related by (A)(x) = A ( X ) ( A ' ) ( i O = A ) where ( A» ) = ( X ) _ 1 ( A ) ( X ) ( x ) = ( X ) ( y ) or ( y ) = ( <o )(x ) . Although the ( A ) matrices are complex symmetric matrices, only real elementary transformations have been used. 6.2 Expansion of Complex Matrices to Real Matrices of Twice the Order. (8, 9, 10, l l ) In the numerical determination of the characteristic roots of the complex matrix ( A ) , i t is found convenient to expand this matrix into a real matrix of twice the order, i.e. i f ( A ) is of rank n, then ( A ) e X p a n ^ e a W 1 1 l be of rank 2n. Hence for the expanded matrix there wi l l be twice as many characteristic roots as for the original matrix. It wi l l be shown that the 2n roots of ( A) eXpComprise the n roots of ( A ) and n conjugates of these roots* 29 The usual way of expanding a complex matrix ( T ) into the real matrix ( S jwhich doubles the number of rows and columns, is v o' to expand each element, a + jb, of ( T)as f o i l ows, a + jb ~ f a b\ \-b a/ A more convenient form of the matrix ( S Q) is ( S ) = / ( A ) (B)\ -(B) ( A ) / where ( A ) is a matrix comprising the real parts of each complex element of ( T ) in the same order, and ( B ) is the corresponding matrix for the imaginary part. Now, i t may be shown that the matrices ( S Q ) and ( S ) are similar by use of transformation matrices of the type ( E ) of the last section. i.e. -(S ) = ( P ) - 1 * S o)(P ) = ( P)(S)(P) where ( P ) = ( E 1 ) ( B 2 ) . . . . ( V l M B n ) and (P)(P) t = (U ) Hence since similar matrices have the same characteris- t i c polynomials and characteristic roots, either of the above forms may be used. The relationship between the characteristic values, X, of the complex matrix ( A ) = ( B ) + j ( C ) , and the charac- t e r i s t i c roots of the expanded real matrix ( A ) m a y be shown as follows. The characteristic equations of the complex matrix ( A ) and i t s conjugate matrix ( A ) * are det \ ( A ) - X ( U ) j = 0 30 and det j (A)* - X* ( U ) \ = 0 The characteristic equation of the following expanded complex matrix has twice the rank of the original complex matrix ( A ) , det /( A ) - X ( U ) 0 \ = 0 \ 0 ( A ) * - X*(U)/ or det [ (A ) - X ( U ) j . det [(A)* - X* ( U )\ = 0 Therefore the characteristic roots of this expanded complex matrix are the roots of ( A ) and their conjugates. Consider the similarity transformation ( T ) ~*( K ) (T ) where ^ ( T ) = ^ fl -j' For ( K ) = f{ B ) + j ( C ) 0 0 ( B ) - j ( C )y this transformation becomes (T)-\K)(T) = Z / 1 1\ /(B) + 3(C) -j(B) +(C.)\ =/(B)(G)\ =(A e W 7 2 \ j - j / \ ( B ) - j ( C ) j ( B ) - (0)/ \(-C)(B)/ which is real. The expanded matrix ( A)gXp can also be reduced to the form '(B) - ( C ) N .( 0 ) ( B )/ by a further similarity transformation using the matrix T Q where 1 O i These expanded matrices being similar, they have the same characteristic roots and these roots w i l l be the roots of (A) and their conjugates, as required for the roots of a real matrix. Since this process yields 2n characteristic roots and only the n roots associated with the complex matrix ( A ) are required, then i t becomes necessary to separate these n required roots from the 2n roots obtained. One approach to the selection of the required roots is to determine the complex characteristic polynomial by evaluating det | ( A ) - X( U )| = 0. Only n of the roots would satisfy this equation; the remaining roots must be discarded. Such a numerical process was devised using the numerical evaluation of (12) the characteristic polynomial attributable to A.M. Danilevsky. ' As an alternative approach, i t w i l l be shown that i f the rank of the original complex matrix, ( A ) is small (n^lO say) then i t is possible to determine the required roots by inspection. Since the roots must appear as conjugate pairs, then the real parts of the roots, Re(X^) w i l l be repeated. For the imagin- ary parts, Im(X_̂ ) there w i l l be a change in sign. Hence the problem becomes one of separating from the 2n known imaginary parts, the n required imaginary parts. This may be achieved by comparing the aggregate of the n imaginary parts of the roots to the imaginary part of the trace of the complex matrix (A) since Im (tr ( A) ) = ± Im (X.) i=l 32 6.3 Transposition and Connection Matrices for Multiple Section Line s. Consider the partially transposed transmission line shown in Fig. 6.1 where the geometrical configuration of the line is the same for both sections and losses due to the trans- position i t s e l f are negligible. * J — 1 e t> . z ' \ r o 7 ,. - , , ft. — , . ,, L \ ft 0 0 . 3 7 s A 0 / \ < 1 • — 1 D 0 ® 0 Fig. 6.1. Partially Transposed Transmission Line. The relationships between the two sections are: (i) (x) -1 (Zl)(x) = (z2) and ( X ) " 1 ( 1 ^ ( 1 ) = ( I 2 ) ( i i ) (X)'1(J1)( Z l ) ( X ) = ( T 2 ) ( Z 2 ) ( i i i ) the characteristic roots of the two sections are the same 33 ( i v ) the c h a r a c t e r i s t i c v e c t o r s of the two s e c t i o n s are r e l a t e d by ( x ) " 1 ( d l i ) - ( d 2 i ) - ( « 2 i ) (x)-l(h) - ( o 2 ) where ^ ^ ) » ( s ) represent one c h a r a c t e r i s t i c v e c t o r of the v o l - tage and cur r e n t s o l u t i o n s r e s p e c t i v e l y , and ( D ) , ( G ) r e p r e - sent the square arrays of the n c h a r a c t e r i s t i c column v e c t o r s of the v o l t a g e and c u r r e n t s o l u t i o n s r e s p e c t i v e l y . For the two s e c t i o n s of l i n e , the vol t a g e and cur r e n t s o l u t i o n equations 4-14 and 4-15 become (V - - ( P 1 ( - ) ) ( ^ ) ( u » ) [ ( » l f n r . , , ' X l ) - ( F i ( n r . ^ I t ) | ( f 1 ) - M ( ' I l / 1 ) *("l,n/ V l)l f o r s e c t i o n Q and ( Y 2 ) = -(P2(.)X °tX*4('2,»SlXi) - K n / ^ )| f o r s e c t i o n ^ . But, ( p 2 ( « ) ) = ( x ) " 1 ! ^ ^ ) ^ ) = ( x ) t ( p 1 ( « ) ) ( x ) and ( G 2 ) = ( X) _ 1 ( G x ) = ( x ) t ( G ^ ••• ( * 2 i 0 ) ) ( Q 2 ) = ( X ) t ( P l ( w ) ) ( G l ) Hence the c u r r e n t and v o l t a g e s o l u t i o n s f o r s e c t i o n (D become or and or 3 4 ( V 2 ) = - ( l ) t ( P 1 ( . ) ) ( G 1 ) ( u r i ( ( F 2 i n r . ^ ) . (,.̂ .-̂ 2)1 * r x 2 \ . /„, " * r X and ( l 2 ) = ( x ) T ( G ! ) ( F 2 f n r e 5 r 2 ) + ( * 2 f e r 2 nr Since the sequence in which the propagation constants, ^ r are taken is unchanged by the transformation, the matrix (u^) is the same for both sections. At the transposition boundary, x 2 = 0 x± = -l± and ( v 2 ) = ( v ± ) x 2 = 0 x l = 1 1 ,nr ( x ) t ( G l ) ( x ) t ( G l ) \/(* 2 , n j v-(x) t (P^-OXGjXuy) (x) t ( p ^ ^ D t ^ f u ^ t F ^ ^ J < * > t / ( ° l > W Y F 2 , n r ) N V ^ C ) ) ( G . X U J J ) ( P . U ) ) ( G ^ O J * ) / \ ( P 2 > N R ), 6 . 4 This system of equations may be solved for ( F 2 n r ) and ( F l in terms of ( F, ) and (F' ) . The f i r s t v 2,nr/ v l,nr' v l,nr/ n equations represent current continuity and the last n equations represent continuity of the space derivative of current. 35 Note t h a t since the matrices ( U f l ) and ( U e ~ ̂ ) are d i a g o n a l , they commute and may be combined to form Since equation 6-4 provides a r e l a t i o n s h i p between the constants of i n t e g r a t i o n f o r the two s e c t i o n s of the t r a n s - mission l i n e , then the a n a l y s i s can be extended to give s i m i l a r r e l a t i o n s h i p s between a l l s e c t i o n s of a m u l t i p l e s e c t i o n l i n e . a 1 ' C ©•• —— —— "'• ™ ' o - 0 4V / 4. r Z V ^ 0 L o o —-—-\ u n V r 0 s-W sr V r. 3 ' \ 3 " A D / V fe © F i g . 6.2. Two Secti o n s of Transmission L i n e . The form of the system of equations 6-4 i n d i c a t e s an a l t e r n a t i v e and simpler method of c o n s i d e r i n g the t r a n s m i s s i o n l i n e s e c t i o n s . For the t r a n s m i s s i o n l i n e shown i n F i g . 6.2, the se c t i o n s are i d e n t i c a l and hence have the same c h a r a c t e r i s t i c equation, roots and v e c t o r s ; only the constants of i n t e g r a t i o n are d i f f e r e n t . At the t r a n s p o s i t i o n boundary, the r e s p e c t i v e c u r r e n t s and v o l t a g e s are r e l a t e d through a connection m a t r i x ^ 1 3 36 This connection matrix is identical to the matrix ( x ) ^ as observed in equation 6-3 i.e. ( X ) t ( Tx ) = ( T 2 ) and ( X ) t ( V x ) = ( V 2 ) This is an invariant power transformation as required since Use of the connection matrix facilitates solution for multiple section transmission lines by greatly reducing the complexity of the numerical analysis. 37 7 THE ( Z ) AND ( T ) MATRICES FOR A COMPLETE SYSTEM 7.1 The (Z) & (Y) M a t r i c e s Under the c o n s t r a i n t s t h a t the v o l t a g e r e f e r e n c e i s . w i t h i n the system, and t h a t the sum of the c u r r e n t s over a l l c o n d u c t o r s i s z e r o , i t has been shown t h a t the m a t r i c e s ( Z 1 ) and ( Y' ) of rank ( n + l ) reduce to the m a t r i c e s ( Z ) and ( Y ) of rank n. For the ( Z ) m a t r i x , Hence , Z = Z1 + Z 1 -, n - Z' , - Z' , r s r s n+l,n+l r,n+l n+l,s Z = R + R + j « ( L + L , , , - , - L .n - L ) r r r n d v r r n+l,n+l r,n+l n + l , r ' D' D • ' , D ' . D' n+l t x \ r x in , D ) , .i«M. /1 „ r x , i n+l ,x l n+l ,x \ _ r r r n + l , n+l r,n+l n + l , and s i n c e D ; > n + 1 = D ^ r Z = (R + R ) + Jf± l n ( ^ ± i ) 2 7-1 r 2 7 1 D r r ' D n + l , n + l where D^j i s the geometric mean d i s t a n c e between c o n d u c t o r s i and j and D ^ i s the geometric mean r a d i u s of conductor i . S i m i l a r l y , z = R + j-fc | n °r tn+l' DA+l fs 7 _ 2 2 1 1 D r s - D n + l , n + l and s i n c e D^ = ^ j ^ , m a t r i x ( Z ) i s symmetric. The reduced m a t r i x ( Y ) may be found from the reduced form of the p o t e n t i a l c o e f f i c i e n t m a t r i x ( P ) where P = P' + P' _ p' - P' r r r r n+l,n+l r,n+l n + l , r 1 Ir, ^ r . n + l ^ 7 - r r n + l , n+l 38 P = ^ In r t n + l n + l ? s ? _ 4 D r r ' D n + l , n + l and s i n c e D. . = D . . . the m a t r i x ( P ) i s symmetric. The reduced m a t r i x , ( 1 ) i s g i v e n by C T ) = j« ( P ) ~ 1 = j« ( C ) 7-5 Si n c e the m a t r i x ( P ) i s symmetric, ( Y ) i s symmetric. 7.2 P r o p e r t i e s of the ( Z ) and ( I ) M a t r i c e s . A s s o c i a t e d w i t h the v o l t a g e e q u a t i o n s we have ( A ) = ( Z X I ) , and f o r the c u r r e n t e q u a t i o n s ( B ) = ( Y ) ( Z ) S i n c e ( Z ) = (Z ) t and ( l ) = ( l ) t , (Z)(I) - ( Z ) t ( l ) t = ( ( T)(Z» t ( A ) = ( B ) T 7-6 Thus ( A ) and ( B ) are s i m i l a r and hence have the same c h a r a c t e r i s t i c v a l u e s . Since the m a t r i c e s ( D ) and ( G ) are the c h a r a c t e r - i s t i c v e c t o r m a t r i c e s c o r r e s p o n d i n g to the m a t r i c e s ( A ) and ( B ) r e s p e c t i v e l y , we may w r i t e ( D ) " 1 ( A ) ( D ) = ( U X ) 7-7 and ( G ) - 1 ( A )(G ) = ( UX ) 7-8 The t r a n s p o s e of e q u a t i o n 7-8 g i v e s t ( B ) t ( G ) t ( G ) t ( B ) t G ) t 1 = ( ™ ) t - ( ™ ) 39 and from equation 7-6, ( G ) t ( A ) ( G ) " 1 = ( U X ) 7-9 Hence, from equations 7-7 and 7-9, we o b t a i n ( G ) t ( D ) = ( U ) 7-10 which i s a s u f f i c i e n t but not necessary c o n d i t i o n . I t may als o be seen that i f t h i s c o n d i t i o n i s s a t i s f i e d , the matrices ( G ) and ( D ) commute. Since the vo l t a g e and cur r e n t forms of s o l u t i o n are r e l a t e d to the matrices ( A ) and ( A ) ̂ r e s p e c t i v e l y , there e x i s t s a matrix ( T ) such t h a t ( T ) ~ 1 ( A ) ( r ) = ( A ) t 7-11 Hence equation 7-8 becomes ( G ) - 1 ( T ) - 1 ( A ) ( T ) ( & ) - ( " O 7-12 and from equations 7-7 and 7-12 ( D ) = ( - O ( G ) . 7-13 7-3 R e s t r i c t i o n s on the Use of the D i s t r i b u t e d Parameters. Fundamental to any d e r i v a t i o n or use of the d i s t r i b u t e d parameters i s the assumption t h a t there i s a r e l a t i o n s h i p with (14) Maxwell's electromagnetic equations The a p p l i c a t i o n of c i r c u i t concepts to electromagnetic f i e l d phenomena, i s r e s t r i c t e d to those frequencies where the wavelength i s f a r gr e a t e r than the p h y s i c a l dimensions of the c i r c u i t . This c o n d i t i o n i s s a t i s f i e d f o r power systems ope r a t i n g at low f r e q u e n c i e s . 40 For a t r a n s m i s s i o n l i n e i n a medium with homogeneous d i e l e c t r i c , the d i s t r i b u t i o n of leakage c u r r e n t i n the space surrounding the conductors f o l l o w s the same p a t t e r n as the e l e c t r i c f l u x d i s t r i b u t i o n and thus the conductance matrix ( G ) has the same form as the capacitance matrix ( C ) with c o n d u c t i v i t y i n place of d i e l e c t r i c constant ( 5 ) ; ( G ) + j < o ( c ) o c ( < r d + j « e d ) where the s u b s c r i p t "d" denotes d i e l e c t r i c . Hence i g n o r i n g the conductance ( G ) i m p l i e s t h a t the displacement c u r r e n t i s f a r gr e a t e r than the conduction c u r r e n t i n the d i e l e c t r i c , i . e . <s 1 toed There i s a f u r t h e r c o n t r i b u t i o n to the matrix ( G ) due to the supporting mechanism of the conductor system. This can only be expressed e m p i r i c a l l y . In the p h y s i c a l model used, the e f f e c t of r e s i s t a n c e ( R ) but not of conductance ( G ) was i n c l u d e d . This i m p l i e s that w i t h i n the conductor the displacement c u r r e n t s are n e g l i g i b l e compared to the conduction currents c where the s u b s c r i p t "c" denotes conductor and Q* i s f i n i t e . Since there i s a component of e l e c t r i c f i e l d i n the d i r e c t i o n of propagation to f o r c e the cu r r e n t through the conductors, then the e l e c t r i c and magnetic f i e l d d i s t r i b u t i o n s must be d i s t u r b e d which i n t u r n a f f e c t s the o r i g i n a l inductance 41 and capacitance parameters. However, t h i s may be ne g l e c t e d i f the a,xial e l e c t r i c f i e l d components w i t h i n the homogeneous d i e l e c t r i c are small compared to the transverse components d « 1 r c P r o x i m i t y e f f e c t i n v o l v e s a l l the parameters but has the most a p p r e c i a b l e e f f e c t on the inductance and the capacitance. I f the s e p a r a t i o n between conductors, D, i s much g r e a t e r than the conductor r a d i u s , T'Q i.e. -JJ <5s. 1 the e f f e c t i s n e g l i g i b l e . These r e s t r i c t i o n s are a p p l i c a b l e to the mathematical model developed i n p r e c e d i n g chapters. Increased s o p h i s t i c a t i o n of the model would r e q u i r e more s t r i n g e n t r e s t r i c t i o n s . 42 r 8 EXAMPLES OF APPLICATION AND RESULTS Two examples of a p p l i c a t i o n were c o n s i d e r e d to t e s t the v a l i d i t y of the t h e o r y . The f i r s t was an a e r i a l d o u b l e - l i n e t h r e e phase t r a n s m i s s i o n system w i t h an overhead ground w i r e . The second was a t h r e e phase sheathed c a b l e underground t r a n s m i s - s i o n system w i t h a s e p a r a t e ground w i r e . I t was assumed t h a t homogeneous media surrounded the t r a n s m i s s i o n system i n b o t h c a s e s . E f f e c t s of the e a r t h on d i s t r i b u t i o n parameters were i g n o r e d . The r a t i o n a l M.K.S. system of u n i t s was used i n the c a l c u l a t i o n s . Leakance was i g n o r e d i n b o t h examples. 8.1 The Overhead T r a n s m i s s i o n System The overhead t r a n s m i s s i o n l i n e i n P i g u r e 8.1 c o n s i s t s of s i x h o l l o w aluminum c o n d u c t o r s , 1-6, w i t h i n s i d e r a d i u s 0.00622 metres and o u t s i d e r a d i u s 0.0145 metres, and a copper ground c o n d u c t o r , g, w i t h r a d i u s 0.00636 metres. The g e o m e t r i c a l c o n f i g u r a t i o n of t h i s system i s shown i n F i g u r e 8.1. B oth t r a n s p o s e d and u n t r a n s p o s e d systems were c o n s i d e r e d . T r a n s p o s i t i o n p o i n t s are shown i n F i g u r e 8.1. The system was assumed t o operate a t a c o n s t a n t t emperature. The t r a n s m i s s i o n l i n e has a c a p a c i t y of 200 M.V.A. a t 230 KV phase to phase and o p e r a t e s a t 60 c y c l e s per second. The l o a d i s assumed to have a 0.8 l a g g i n g power f a c t o r w i t h t h r e e phase T—connected b a l a n c e d impedances. S o l u t i o n s f o r f u l l l o a d and no l o a d c o n d i t i o n s were found f o r c u r r e n t , v o l t a g e and power. 43 1 4 , z 1 < I •• . 3 " 6 4 L O A D • t r a n s p o s i t i o n p o i n t s . 9* 4-5" ̂ etves 5 9 vrietres ^etyes Fig. 8.1 A three section, six conductor with ground, overhead system. I t _ _ _ untjranapoiedi £.TAl>sfS>sej4J.. x - fo i l load 0 oo , load. 44 2 . 6 6 4 0 0 R«ceiuin<] End 599 C O O W . Sending End Fig. 8 . 2 ire. COVT t*t , III Dis tance Aloftg t h e L ine , x. , wttirts 3996 OO X Sending E«̂ d Fit). B-3 Pokier Consumption } P } w.u. «.t No Load us. Distance ftlong the Line. , x , we.U«s, 45 1 3 3 2 0 0 Receiving End 3 9 9 C O O 5C S e n d i n g Er\ct F i g . 8.4- Reac/tiue. Co-pac'itatiue Power Consumption w\v>a. at No Loo-d. us Distance Along the Li * e x } wetves. \ 1c Fig. ©.5 Current Phase Angle Differences at Sending End | o r both Transposed and. Untransposed Systems. 4 6 13.1 50 "ioo 2oo 100 1 i i 1 1 i ! I *—— t 1 ! * 1 1— no 1 f f uC >oicL. on A, i : -™ 1 i I i p o i « 4 . & I 1 1 1 i j 1 i i 133200 " 2 G 6 40O F i g . 8. 6 3996OO ie cc- pKo.se. C owent } |I,J , ©.vnps. us Distance. Along the b«e ,Xj metves. 170 <6o I S O 1 4 0 * 9 3 9 5 9 1 9 o 1 | > o«t«L i > — 1 • 1 — — • —- H no (( > ad. Reeei 3996OO X. Sewd.i*g End. F i Q . 8 . 7 CL- phase C u w e n t Avou^ent , / l a . > deo,Y«es os D i s t ance A lov*a tKe L i^C , X, Wietv«x, 140 l i e S f i A d ' ^ g E n d . F , q . 8 . 8 355 C e o X Receiving E " ^ a - p k o s e V o l t a g e } | , kv os F\q. 8.^ a-pKa.se. Voltage Avgu^en t 3 /V̂ , cleg-zees us Distance Alomg t^e L i v i e , x> welrves. 48 8.2 R e s u l t s I n P i g u r e 8.2, the ground w i r e c u r r e n t v a r i a t i o n a l o n g the t r a n s m i s s i o n l i n e from the r e c e i v i n g end i s p l o t t e d . For the u n t r a n s p o s e d system, the c u r r e n t v a r i e s l i n e a r l y w i t h d i s - tance as might he e x p e c t e d , and l i t t l e v a r i a t i o n i s e v i d e n t between no l o a d and f u l l l o a d c o n d i t i o n s . F o r the t r a n s p o s e d system, the c u r r e n t v a r i a t i o n i s c y c l i c w i t h maxima and minima o c c u r r i n g a t the t r a n s p o s i t i o n p o i n t s f o r b o t h no l o a d and f u l l l o a d , a l t h o u g h the g e n e r a l t r e n d i s an i n c r e a s e i n c u r r e n t from the r e c e i v i n g end towards the sending end. The t r e n d can be e x p l a i n e d on the b a s i s of the d i f f e r e n t v o l t a g e s o c c u r r i n g a t the t r a n s p o s i t i o n p o i n t s . The r e v e r s a l of c u r r e n t magnitude a t the t r a n s p o s i t i o n p o i n t s i s a t t r i b u t a b l e to the e f f e c t of the m e c h a n i c a l l y abrupt t r a n s p o s i t i o n on c u r r e n t c o n t i n u i t y . F i g u r e 8.3 and F i g u r e 8.4 show the v a r i a t i o n of power and r e a c t i v e power r e s p e c t i v e l y of the t r a n s m i s s i o n system a t no l o a d . The r e a l power i n c r e a s e s r a p i d l y a t the sending end; the c a p a c i t a t i v e r e a c t i v e power i n c r e a s e s l e s s r a p i d l y . F i g u r e 8.5 shows the c u r r e n t phase angle d i f f e r e n c e s a t no l o a d and f u l l l o a d . A l t h o u g h the t h r e e phase c u r r e n t phasors are b a l a n c e d a t f u l l l o a d , they are q u i t e unbalanced a t no l o a d due to the e f f e c t on the c h a r g i n g c u r r e n t s of the a s y m m e t r i c a l geometry of the t r a n s m i s s i o n l i n e . F i g u r e 8^6 and F i g u r e 8.7 show the a-phase c u r r e n t v a r i a t i o n i n magnitude and phase r e s p e c t i v e l y a l o n g the t r a n s m i s - s i o n l i n e * As e x p e c t e d , the no l o a d c u r r e n t d i s t r i b u t i o n i n - c r e a s e s l i n e a r l y from the r e c e i v i n g end. A t f u l l l o a d the 49 c u r r e n t decreases towards the sending end, which i s an i n d i c a t i o n of the compensation e f f e c t of the changing c u r r e n t on the load c u r r e n t . L i t t l e d i f f e r e n c e was observed at f u l l l o a d between the phase angle v a r i a t i o n along the l i n e of the transposed and untransposed systems. A d i f f e r e n c e can be seen however, under no l o a d c o n d i t i o n s . The a-phase vol t a g e magnitude v a r i a t i o n and phase angle s h i f t are i n d i c a t e d i n Figure 8.8 and F i g u r e 8.9 r e s p e c t i v e l y . Under f u l l l o a d, the v o l t a g e d i f f e r e n c e between the transposed and untransposed systems was s l i g h t , and f o r no load, no d i f f e r - ence was d e t e c t a b l e . S i m i l a r statements can be made f o r phase angle s h i f t s along the t r a n s m i s s i o n l i n e . The data f o r the above graphs i s i n c l u d e d i n Appendix C l . In a d d i t i o n , i t may be seen that at f u l l l o a d, while the power consumption i n c r e a s e s towards the sending end, the i n d u c t i v e r e a c t i v e power decreases. 8 . 3 The Underground Transmission System The underground t r a n s m i s s i o n system i n Figure 8.10 c o n s i s t s of three sheathed conductors and a ground wire. The s o l i d copper conductors, 1-3, have a ra d i u s of 0 . 0 1 3 2 metres and the aluminum sheaths, 4-6, have an i n s i d e r a d i u s of 0 . 0 2 3 9 metres and an outside r a d i u s of 0.0247 metres. The s o l i d copper ground wire has a r a d i u s of 0.00318 metres. The geometrical c o n f i g u r a - t i o n of the system i s shown i n F i g u r e 8.10. The system was assumed to operate at a constant tempera- ture i n a medium with r e l a t i v e d i e l e c t r i c constant, e = 4.0. 50 The t r a n s m i s s i o n l i n e has a c a p a c i t y of 10 M.V.A. at 13.2 KV phase to phase and o p e r a t e s a t 60 c y c l e s per second. The l o a d i s assumed to have a 0.9 l a g g i n g power f a c t o r w i t h t h r e e - phase Y-connected b a l a n c e d impedances. S o l u t i o n s f o r f u l l l o a d and no l o a d c o n d i t i o n s were found f o r c u r r e n t , v o l t a g e and power. 8.4 R e s u l t s I n c o n t r a s t to the overhead t r a n s m i s s i o n system, the ground w i r e c u r r e n t of the underground system i s independent of l o a d and d e c r e a s e s i n magnitude s l i g h t l y from the sending end t o the r e c e i v i n g end. A s i m i l a r s m a l l change was observed i n the phase a n g l e . These r e s u l t s may be seen i n Appendix C.2. F i g u r e 8.11 and F i g u r e 8.12 show t h a t the v a r i a t i o n of power and r e a c t i v e power r e s p e c t i v e l y a l o n g the t r a n s m i s s i o n l i n e a t no l o a d i s l i n e a r . The conductor c u r r e n t phase a n g l e s are symmetric f o r f u l l l o a d c o n d i t i o n s but not f o r no l o a d c o n d i t i o n s , as shown i n F i g u r e 8.13. The phase angle d r i f t a l o n g the l i n e i s n e g l i g i - b l e f o r no l o a d and i s s l i g h t f o r f u l l l o a d , as i n d i c a t e d i n the d a t a of Appendix C.2. The conductor c u r r e n t v a r i a t i o n a l o n g the l i n e from the r e c e i v i n g end, as shown i n F i g u r e 8.14, i n c r e a s e s f o r b o t h no l o a d and f u l l l o a d . I n F i g u r e 8.15, the sheath c u r r e n t phase angle d i f f e r - ences are shown. These are the same f o r b o t h no l o a d and f u l l and t h e r e i s no phase angle d r i f t a l o n g the l i n e f o r any phase. 51 The variation of the sheath current along the line is shown in Figure 8.16. From Appendix C.2 i t may be seen that conductor voltage variation along the line in both magnitude and phase is small at no load, but increases slightly at f u l l load. The three phase voltages are always balanced. Figure 8.17 shows the phase angle differences of the sheath voltages. Considerable imbalance is apparent at no load but is less severe at f u l l load. The phase angle d r i f t along the line is small. In Figure 8.18 and Figure 8.19 the sheath voltage variations along the line for no load and f u l l load respectively are shown. Linear increase from the receiving end is observed at f u l l load, but the increase is not linear at no load. The data for the above graphs is included in Appendix C.2. In addition i t may he seen that at f u l l load, while the power consumption increases towards the sending end, the inductive reactive power decreases. V Pig. 8.10 An underground, three phase cable system with separate ground wire and sheaths around each conductor. O 5 0 0 F i g , 8 . I I N o L o a d Distavvee Along tKe L'i*e , x , v*t.l*ts. Peeeiuinej E n d A o o o F i a . 8 . 1 2 No LOOLCI R e a c t i v e Pou>ev , Q,, my.a.. os D i s t a n c e Alono, the L ine , , weUes. 54 Fia.8.13 Co-ndoctov CuYYen.t Phase Anale Dif (evewces at -the Seeding ElY\d.. / ' \ / ! » »• ( / ^ - - J l /i- " loa .*2 F i 9 6 . 1 5 " N o L o a d ' a n d F u l l L o a d S k e a k k C u v v e n t P k a s e . A n g l e D i ^ e v t v \ e e » a t i h e . R c c e A u i ^ a , E n d ' 55 4 4 0 435 <4 20 IS 10 ]V\0 lo<xc<,, End.. tooo I TOO "Xooo F~i C|. 9 . I 4 - C o f t d u c t o Y C u T Y e-ot ,IX,|} a w p s u s . D i s t a n c e Along t h e Lir . e , j metves. 56 IIJ 15 ! ^ ! 1 ! 1 T 1 ! i j . . . ! . _ . . IO 1 •^"^ i ; ! - - - 1 j s | i • ; I i i | i » o Rect i u'iv\< looo 2LOOO ZSOO St F.g . 8 .16 N o L o a d OVA Full 1— oacl S k e a t k , avnps us. D i s t ance Alo^a tKe Li vie. ,sc, wetv«$. F i 3 . 8 . 1 7 S h e e t k Vol taac PHase A.ujle D i f f tre-ftces at the Sending End . 51 HI ! ! ! ! 16.0 i i i I . i i | 1 | i i I i ; i sro j i i ! i 1 2S i i I j I i M s ! O 1600 >6"00 aooo iffoo •x % End F i 3 . 8 . 1 8 M o L o a d S h e a t h V o l t a a e > 1 V 4 I , v o l t s us D i s t o v w c e A l o n e ] t h e L i v > t ^ x . , v * e . t v e s , 4oo loo loo | • ! 1 i 1 i I ^ i 1 f —,—,— I I i O soo IOOO isroo tooo *ffoo X. 3ooo Rece'txnno, E n d . Sending E n d Fig 8.19. Full Load Sheath Voltjck̂ a JVJ ,. uolt« us. Distance A\ong the .L'»me j ^ C " , " wet.ys$ 58 9 CONCLUSIONS An accurate mathematical procedure was developed to be used in the analysis and design of multiconductor transmission systems under various loading or boundary conditions. The va l id i ty of the theory was substantiated using two numerical examples. The results of these two analysis are given in the report. The sequence in which the parts of the numerical analy- sis must be performed is shown in Appendix B. It is apparent that given the conductor and geometrical specifications of a particular transmission system, a variety of terminal or boundary conditions can be analysed for that system without repeating the steps which lead to the general solution. In developing this procedure, an important theoretical concept was evolved; the concept of a complete system. Consider the example of the overhead conductor system, which comprises seven conductors, including the ground wire but excluding any earth effects. Only six of these are independent. By choice, the ground conductor was used as a voltage reference, but the magnetic, electric and loss effects due to this conductor, which may not be ignored, appear in the system parametric matrices. The reduced system resistance matrix for example is not a diagonal matrix since the resistance of the ground wire appears as a component of a l l matrix elements. Had the earth effect been included in the model as an equivalent earth conductor, then there would have been eight conductors, seven of which would have been independent. The 59 ground wire or the equivalent earth conductor would be chosen as a reference conductor. Hence definition of the complete system requires the specification of a closed system of conductor one of which w i l l be used as a voltage reference conductor. This approach to transmission line analysis suggests that i t is ideally suited to time shared machine aided design. The optimum boundary terminations or the best locations for the transpositions, for example, could be arrived at by using a computer to verify an analyst's heuristic reasoning. Future research into this f i e l d should include analysis of the f u l l significance of the location of the characteristic values in the complex plane with respect to propagation and attenuation. Further development w i l l lead to the superposition of analysis of the same system at various frequencies for trans- ient studies or for carrier wave transmission studies. A more precise formulation and method for finding the complex trans- position matrices which occur at lossy transposition boundaries w i l l also be required, particularly where optimum solutions are to be found. 60 APPENDIX A THE PARAMETERS R, L AND C. A . l Assumptions I n the d e r i v a t i o n ^ ^ ' of the m a t r i c e s ( R ) , ( L ) and ( C ) , i t w i l l be assumed t h a t s k i n e f f e c t s , p r o x i m i t y e f f e c t s and s a t u r a t i o n may be i g n o r e d . I t w i l l be assumed a l s o t h a t d e r i v a t i o n u s i n g s t a t i c f i e l d s w i l l n o t i n v a l i d a t e the a p p l i c a t i o n to a s l o w l y changing or q u a s i — s t a t i c system. The r a t i o n a l i z e d M.K.S. system of u n i t s w i l l be used. I n i t i a l l y , a l l the parameters f o r a system of (n -K l ) co n d u c t o r s w i l l be r e l a t e d t o some a r b i t r a r y e x t e r n a l r e f e r e n c e , but i n the f i n a l form,the r e s t r i c t i o n t h a t the sum of the c u r r e n t s w i t h i n the system i s z e r o w i l l be a p p l i e d , and a l l v o l t a g e s w i l l ;be r e f e r r e d t o some conductor w i t h i n the system ( e . g . a ground c o n d u c t o r , or an e q u i v a l e n t e a r t h c o n d u c t o r ) . A, 2 The R e s i s t a n c e , R"1"., The m a t r i x of r e s i s t a n c e per u n i t l e n g t h i s where R^ = jo^ /A^, t h A. = c r o s s s e c t i o n a l a r e a of the i conductor 61 and Ji t = + at) = res i s t iv i ty of conductor i at a. temperature t ° C where q = res i s t iv i ty of the conductor material at a o _ temperature t C and o< = thermal coefficient of re s i s t iv i ty . A-3 The Inductance, L̂ " Consider the group of ( n + l ) conductors shown in F ig . A . l The axes are set up through conductor 11 o. about which the flux linkages are to be computed. The point X is some remote point where magnetic effects may be considered to be negligible. The total number of linkages produced by flux which crosses the x — axis between the origin and the point X is given by D T I n + 1 . x + . . . . + I n + j a , n + 1 A-2 where \i = permeability of the surrounding medium r =• radius of conductor j D. = distance between conductor j and the point X D. . = distance between conductors i and j I. = current in i conductor J A similar expression can be written for the flux linkages surround- ing the remaining conductors. 62 O © c b O a X ' x P i g . A.l Group of n + 1 Current C a r r y i n g Conductors. For two p a r a l l e l c y l i n d r i c a l conductors of a r b i t r a r y c r o s s - s e c t i o n , the t o t a l f l u x linkage about one of the conductors i s g iven by (15), where I i s the current i n the conductor i s the G.M.D., the geometric mean di s t a n c e between the conductors ^ i i """S G.M.R., the geometric mean radi u s of the conductor. A-3 S3 Replacing the distances D and r of equation A-2 by the geometric mean distances as defined in equation A-3 gives / i n i In ^ °S +1? X \ / Ta\ 2n I DL D a b \ , n + 1 \ / aa , D In - f t . Dba V f n + 1 Wn ax D n + 1, a 1-3 n + 1, x D n + l , n + l For a linear system, the inductance coefficient, may be defined as a L.. « I3 1 J ^ I . = 0 l i ^ J I . = 0 1 i i j -Ji 2it 2TX i n D J3 1 D. A-5 A-6 and hence the inductance coefficient matrix (L ) becomes / L11 L12 J21 1, n + 1 \ n + 1, 1 .... L n + l t n + ± A-7 A-4 The Capacitance, C For a system Df ( n + l ) p a r a l l e l conductors p o t e n t i a l c o e f f i c i e n t s are d e f i n e d by the equation a 'P P , aa ab a, n + 1 \ /^a \ n + 11 ba \ P n + 1, a** Pn + 1, n + l / \°-n + 1/ where the P , are the co e f f i c i e n t s of p o t e n t i a l S o l v i n g f o r the charges Q. we have 3 I fia\ 'C c , aa ab 'a, n + 1 'ba Ai + J n + l , a*** ** n + 1, n + 1 Where the C , are the capacitance coefficient For the system of (n + l ) conductors, P. . A V . i ^ d l 2%e D . n -1' x D. J y J 65 where the D's represent the distances between the points denoted by the subscripts, and e is the permittivity of the medium surrounding the conductors. Clearly, the matrices of potential, capacitance and inductance coefficients are symmetric* Since, for the potential coefficient matrix P > P > 0 rr rs ^ then for the capacitance coefficient matrix C > 0 rr * and C S 0 rs For a system of conductors containing coaxial cables where one conductor is completely enclosed by another as shown in Fig A.3, then because conductor j is shielded by conductor i . C, = C. , = 0 J O Fig. A.2 Cross—section of part of a system of conductors where one conductor completely encloses another. 66 For the system, ( Q ) = ( c ) ( v ) , then 03 1A. V. 3 V = 0 m m ^ 3 In p a r t i c u l a r , f o r V\ = 0, C.... becomes the capacitance between two c o n c e n t r i c c y l i n d e r s . A l s o , Qi 1 J V 3 V = 0 m m ^ 3 v. 3 V = 0 m m 4 3 and since the m a t r i x ( C ) i s symmetric, then C. . = C . . 13 3 i In terms of the p o t e n t i a l c o e f f i c i e n t s , P ^ = Pj^. a n d P. 0 = P.. imply t h a t C ., = C .9 =0, and P. . = P. . = P. . i m p l i e s 3>c 3^ 3*" !3 3 1 1 1 ;. . = c. . = - c . . 13 31 33 In t h i s d i s c u s s i o n , i t has been assumed t h a t the p e r m i t t i v i t y of the medium surrounding the conductors i s constant. 67 APPENDIX B A flowsheet of the solution procedure for (n + l ) conductor system. Procedure Part I Compute E, L, C, G coefficient matrices from transmission line configuration. Checks Manual calculation of selected elements. Form reduced Z and Y matrices. Find characteristic values of characteristic equation (A) - X (U) =0 where (A) = (Z)(T) Trace (A) =' i=l Find characteristic vectors D - voltage vectors G - current vectors (D)"1(A)(D) = (X U) (G)" 1(A) t(G) = (X U) Part II Solve for 2n unknown constants using known boundary conditions and the connection matrices at the transpositions. The voltage form of the solution must give the same results as the current form of solu- tion Generate required output from the particular solution. Part I of the procedure gives the general solution for the given transmission l ine ; Part II provides the particular solutions for the specified sets of boundary conditions and connection matrices at the transpositions. 68 Appendix C l . l Overhead Conductor System, No Load Voltage and Power Distance 1V I T, a< 'Vh' K P/Q (km) (kv) (kv) (kv) (deg) (deg) (deg) (m.v.a.) UNTRANSPOSED 0 132.8 132.8 132.8 0 -117.2 117.2 0 0.06/ 133.2 130.8 127.6 127.7 0.58 -117.1 117.3 - 46.2 0.46/ 266.4 125.0 121.9 122.2 0.50 -116.8 117.5 - 87.1 1.52/ 399.6 115.6 112.8 113.5 0.30 . -116.3 119.1 -118.1 TRANSPOSED 0.46/ 266.4 124.8 120.0 122.0 0.30 -116.7 118.0 - 87.0 1.51/ 399.6 115.7 113.4 112.7 0.04 -115.1 118.6 -117.9 Appendix C1.2 Overhead Conductor System, No Load Current. Distance 11J T, a l ^ 1 K 1̂ (km) (amp) (amp) (amp) (deg) (deg) (deg) (degVampxlO 4 UNTRANSPOSED 0 0 0 0 0 0 0 0 -47.7/ 133.2 58.5 62.0 59.6 92.2 28.7 -152.8 41.0 -47.4/ 266.4 115.0 121.0 117.0 92.3 28.8 -152.7 81.0 -47.0/ 399.6 169.0 178.0 172.0 92.5 28.6 -152.5 118.0 TRANSPOSED -27.7/ 266.4 177.0 120.0 118.0 91.2 29.7 -150.8 31.0 -89.2/ 399.6 170.0 174.0 174.0 90.5 28.6 -150.6 42.0 69 Appendix C1.3 Overhead Conductor System, Pull Load Voltage and Power, Distance 1V 1 T. A ^b 1 ^ 0 P/Q (km) (kv) (kv) (kv) (deg) (deg) (deg) (m.v.a UNTRANSPOSED 312.9/ 0 132.8 132.8 132.8 0 -117.2 117.2 234.7 324.2/ 133.2 156.1 153.0 148.0 7.6 -108.7 126.8 266.2 333.5/ 266.4 176.8 175.0 165.5 13.5 -102.2 134.7 264.8 341.0/ 399.6 193.9 193.4 181.0 18.5 - 96.9 141.2 230.6 TRANSPOSED 333.5/ 266.4 176 a 172.0 169.1 14.5 -102.0 133.5 264.9 341.0/ 399-6 190.9 191.0 186.5 20.4 - 97.3 139.6 230.5 Appendix C1.4 Overhead Conductor System, Pull Load Current Li. Distance 11 I L a I h l (km) (amp) (amp) 0 500.0 500.0 133.2 458*0 458.0 266.4 409.0 408.0 399.6 357.0 358.0 266*4 410.0 412.0 399.6 364.0 359.0 u i 'Li ; ix c a h (amp) (deg) (deg) UNTRANSPOSED 500.0 143.1 23.1 462.0 148.9 29.4 415.0 156.3 37.2 372.0 166.0 47.5 TRANSPOSED 414.0 156.7 37.1 365.0 166.7 46.6 - 86.9 - 90.6 - 97.1 -107.0 - 96.8 -107.0 (I /II I g g (deg) (deg)/ampxlO^ 0 -47.7/ 41.6 -47.6/ 82.8 -47.2/ 122.5 - 3 6 . 0 / 19.7 , -77.0/ 69-4 70 Appendix C2.1 Underground Conductor System, No Load Voltage and Power System Distance Voltage: Magnitude and Argument for Real and conductors 1-6, LY/|V| ^ower^ 6 L (deg) (deg) (deg) (deg) (deg) (deg) P/Q. (m) (kv) (kv) (kv) (kv) (kv) (kv) (nuy.a.) -120.0/ 120.0/ 0.0/ 0.0/ 0.0/ 0.0/ 76.20 76.20 0.0 0.0 0.0 0.0 -120.0/ 120.0/ -7.6/ -54.6/ -125.6/ 0.61/ 76.20 76.20 2.52 3.41 1.41 -0.18 -120.0/ 120.0/ -6.1/ -51.6/ -120.5/ 1.22/ 76.20 76.20 4.75 6.85 3.13 -0.37 • -120.0/ 120.0/ -4.5/ -48.7/ -116.4/ 1.83/ 76.205 76.205 6.69 10.35 5.20 -0.55 -120.0/ 120.0/ -2.65/ -45.7/ -112.9/ 2.44/ 76.21 76.21 8.35 14.00 7.61 -0.73 -120.0/ 120.0/ -0.55/ -42.7/ -110.1/ 3.05/ 76.21 76.21 9.73 17.70 10.41-0.92 -120.4/ 119.6/ 1.9/ -40.1/ -107.7/ 3,.66/ 76.21 76.21 10.84 21.60 13.60 -1.10 Appendix C2.2 Underground Conductor System, No Load Current, Distance Current: Magnitude and Argument for - conductors 1-7, / i / l l l L (deg) (deg) (deg) (deg) (deg) (deg) (deg) (m) ^ A T ( I T (A) (I) (I) (AT (A) 0.0/ 0 76.20 0.0/ 500 76.20 0.0/ 1000 76.20 0.0/ 1500 76.205 0.0/ 2000 76.21 0.0/ 2500 76.21 0.04/ 3000 76.21 0 0 . 0 / 0 . 0 0 . 0 / 0 . 0 0 . 0 / 0 . 0 - 1 4 . 7 / 1 7 . 8 -II6.9 / 1 2 2 . 1 / 1 6 . 9 1 7 . 0 8 4 . 5 / 5 . 1 5 5 0 0 5 . 7 / 2 . 7 5 - 9 8 . 6 / 2 . 9 4 1 4 2 . 6 / 2.76 - 1 4 . 6 / 1 4 . 8 - 1 1 6 . 9 1 2 2 . 0 / 1 4 . 1 14 .2 8 4 . 4 / 5 . 2 5 1 0 0 0 5 . 7 / 5 . 5 0 - 9 8 . 6 / 5 . 8 7 1 4 2 . 6 / 5 . 5 0 - 1 4 . 6 / 1 1 . 8 5 - 1 1 6 . 9 1 2 2 . 1 / 1 1 . 2 7 1 1 . 3 5 8 4 . 2 5 5 . 3 5 1 5 0 0 5 . 7 / 8 . 2 3 - 9 8 . 6 / 8.80 1 4 2 . 6 / 8.28 - 1 4 . 7 / 8 . 8 8 - 1 1 6 . 9 / ' 1 2 2 . 1 / 8 . 4 4 8 . 5 1 8 4 . 1 5 . 4 5 2 0 0 0 5 . 7 / 1 0 . 9 9 - 9 8 . 6 / 1 1 . 7 5 1 4 2 . 6 / 11.05 - 1 4 . 7 / 5 . 9 2 - 1 1 6 . 9 / 1 2 2 . 1 / 5 . 6 3 5 . 6 8 8 4 . 0 / 5 . 5 5 2 5 0 0 5 . 7 / 1 3 . 7 1 - 9 8 . 6 / 14.67 1 4 2 . 6 / 13.80 - 1 4 . 7 / 2.96 -116 . 9 / 1 2 2 . 1 / 2.82 2 . 8 4 8 3 . 9 / 5 . 6 6 3 0 0 0 5 . 7 / 1 6 . 4 5 - 9 8 . 6 / 17.60 1 4 2 . 6 / 16 .55 0 . 0 / 0 . 0 0 . 0 / 0 . 0 / 0 . 0 0 . 0 8 3 . 7 5 / 5 . 7 6 71 Appendix C2.3 Underground Conductor System, Fu l l Load Voltage and Power Distance Voltage: Magnitude and Argument for conductors l-6,.JV/lVl L (m) 0 500 1000 1500 2000 2500 3000 (deg) (deg) kv 0.0/ ' 76.20 0.0/ 76.23 0.0/ 76.26 0.0/ 76.29 0.0/ 76.33 0.0/ 76.36 0.2/ 76.39 kv -120.0/ 76.20 -120.0/ 76.23 -120.0/ 76.26 -120.0/ 76.29 -120.0/ 76.33 -120.0/ 76.36 -119.75 76.4 System Real and Reactive Power (deg) (deg) (deg) (deg) P/Q kv v v v (m.v.a.) 120.0/ 0.0/ 0.0/ 0.0/ 90.05/ 76.20 . 0.0 0.0 0.0 43.82 120.0/ 61.8/ -47.2/ 176.15/ 90.67/ 76.23 62.4 65.5 63.0 43.70 120.0/ 62.1/ -47.0/ 176.20/ 91.29/ 76.26 124.8 131.1 124.9 43.59 120.0/ 62.2/ -46.9/ 176.40/ 91.91/ 76.29 187.0 197.1 188.9 43.48 120.0/ 62.3/ -46.8/ 176.60/ 92.53/ 76.33 250.0 262.5 251.3 43.36 120.0/ 62.4/ -46.6/ 176.75/ 93.15/ •76.36 312.0 328.5 314.5 43.25 120.15/ 62.6/ -46.4/ 176.95/ 93.77/ 76.4 374.0 395.0 376.6 43.13 Appendix 02.4 Underground Conductor System, F u l l Load Current Distance Current: L (m) 0 500 1000 1500 2000 2500 3000 Magnitude and Argument for Conductors 1-7, (deg) A (deg) A -26.00/ -146,00/ 439.0 439.0 -25.80/-145.65/ 440.0 440.0 -25.60/ -145.40/ 443.0 442.0 -25.45/ -145.10/ 446.0 445.0 -25.2^ -144.90/ 448.0 446.5 -25.10/-144.60/ 450.0 448.0 -24.9/ -144.25/ 452.0 450.0 (deg) A (deg) A (deg) A' (deg) A 94.00/-14.7/ 439.0 17.75 94.30/ -14.7/ 440.0 14.80 94.60/ -14.7/ 441.0 11.90 94.90/-14.7/ 443.0 8.90 95.20/-14.7/ 445.0 5.68 95.40/ -14.7/ 447.0 2.84 95.80/ 0.0/ 449.0 0.0 -117.0/'12 2.0/ 16.90 17.00 -117.0/ 122.0/ 14.10 14.20 -117.0/ 122.0/ 11.30 11.30 -117.0/ 122.0/ 8.45 8.40 -117.0/122.0/ 5.64 5.67 -117.0/122.01 2.82 2.84 0.0/ 0.0/ 0.0 0.0 (deg) A 84.5/ 5.15 84.3/ 5.25 84.251 5.35 84.10/ 5.45 84.00/ 5.55 83.91 5.66 83.75/ 5.76 72 REFERENCES 1. Bewley, L.V., Travelling Waves on Transmission Systems. John Wiley & Sons, Inc., New York, 1951. 2. Pipes, L.A., Matrix Methods for Engineering. Prentice-Hall. Inc., Englewood C l i f f s , N.J., 1963. 3. Hayashi, S., Surges on Transmission.Systems. Denki-Shoin Inc., Kyoto, Japan, 1955. 4. Carson, J.R., Ground Return Impedance: Underground Wire with Earth Return. Bell Systems Technical Journal, Vol. 8, 1929, pp. 94-98. 5. Weinhach, M.P., Electric Power Transmission. MacMillan Co., New York, 1948. 6. Marcus, M., Basic Theorems in Matrix Theory. U.S. Dept. of Commerce N.B.A., App. Math. Series, page 3, definition 1.7. 7. Carson, J.R., The Guided & Radiated Energy in Wire Transmission. J.A.I.E.E., 43, 906-913,. October, 1924. 8. Brenner, J.L., Expanded Matrices from Matrices with Complex Elements. SIAM Review. Vol. 3. #2. April. 1961. P. 165. 9. Marcus, M., Basic Theorems in Matrix Theory. U.S. Dept. of Commerce N.B.S., App. Math., Series page 12, Theorem 3.8. 10. Gott, E., A Theorem on Determinants. SIAM Review, Vol. 2, No. 4, October, I960, p. 288. 11. Afriat, S.N.. Composite Matrices. Quart. J. Math., Oxford, Ser. (2), 5, 1954, pp. 81-98. 12. Faddeva, V.N., Computational Methods of Linear Algebra. Dover Pub. Inc., 1959, Ch. 3, Sec. 24, pp. 166-176. 13. Kron, G., Tensors for Circuits. Dover Pub. Inc., 1959,.Ch. 3, pp. 22. 14. Ramo, S., and Whinnery, J.R., Fields & Waves in Modern Radio. "John Wiley & Sons Inc., New York, I960. 15. Woodruff, L*F„', Electric Power Transmission. J. Wiley & Sons, New York, 1946. 16. Zabdrszky, J., & Rittenhouse, J.W., Electric Power Transmis- sion. The Ronald Press, New York, 1954.
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