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 o f B r i t i s h C o l u m b i a , 1961 A THESIS SUBMITTED I N PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF A P P L I E D SCIENCE i n t h e Department o f Electrical ¥e a c c e p t this Engineering t h e s i s as c o n f o r m i n g required t o the standard Members o f t h e D e p a r t m e n t of E l e c t r i c a l Engineering THE UNIVERSITY OF B R I T I S H COLUMBIA November > 1965 In p r e s e n t i n g the this thesis Columbia, I agree that the Library 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 . mission f o r extensive representatives., cation of this w i t h o u t my w r i t t e n Department o f thesis for financial £jLAsn^JL 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 I ^ W J U A - H ^ Columbia that gain permission. '3 . x u ; < i s i t freely per- f o r scholarly by t h e Head o f my D e p a r t m e n t o r by I t i s understood thesis s h a l l make I f u r t h e r agree that copying o f t h i s p u r p o s e s may be g r a n t e d Date fulfilment of 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 o f British his in partial copying o r p u b l i - shall n o t be a l l o w e d ABSTRACT Problems concerning transmission lines have been solved in the past by treating the line i n terms of lumped parameters. Pioneering work was done by L . V» Bewley and S. Hayashi in the application of matrix theory to solve polyphase m u l t i conductor distributed parameter transmission system problems. The a v a i l a b i l i t y of d i g i t a l computers and the increasing complexity of power systems has renewed the interest i n this field. With this i n mind, a systematic procedure for handling complex transmission systems was evolved. Underlying the pro- cedure i s the significant concept of a complete system which defines how the parametric inductance, capacitance, resistance matrices must be formed and used. leakance and 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 r e a l matrices of twice the order and to determine the coefficients i n the general solution systematically. The pro- cedure was used to deal with phase asymmetry and mixed end boundary conditions. ii TABLE OP CONTENTS Page Abstract ............. i i Table of Contents List i i i v of I l l u s t r a t i o n s 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 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 Analysis 4.2 The General S o l u t i o n Vector 13 5. BOUNDARY CONDITIONS 6. TRANSPOSITION MATRICES AND THE COMPLEX CHARACTERISTIC MATRIX 7. 8. 10 16 6.1 The T r a n s p o s i t i o n M a t r i x 23 6.2 Expansion of Complex M a t r i c e s t o Real M a t r i c e s o f Twice the Order ..•»••••••••.••.•••••...•....... 27 6.3 T r a n s p o s i t i o n and Connection Section Lines Matrices f o r M u l t i p l e 30 THE (Z) & (Y) MATRICES FOR A COMPLETE SYSTEM •; 7.1 The (Z). & (Y) M a t r i c e s ....... 35 7.2 P r o p e r t i e s of the (Z) & (Y) M a t r i c e s 36 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 ..................................... 37 EXAMPLES OF APPLICATION AND RESULTS 8.1 The Overhead Transmission 8.2 Results System 42 48 iii Page 9. 8.3 The Underground Transmission System 49 8.4 Results 50 CONCLUSIONS APPENDIX A. 58 The Parameters R, L and C A.l Assumptions A.2 The Resistance, R A.3 The Inductance, L* A.4 The Capacitance, C 60 ».. 60 60 T 61 T ............................ 64 APPENDIX B. A Flowsheet of Solution Procedure .......... APPENDIX C. Data Sheets Cl.l C1.2 67 Overhead Conductor System, No Load Voltage and Power, Transposed and Untransposed 68 Overhead Conductor System, No Load Transposed and Untransposed 68 Current, C1.3 Overhead Conductor System, P u l l Load Voltage and Power, Transposed and Untransposed ............ 69 C1.4 Overhead Conductor System, P u l l Load Current, Transposed and Untransposed C2.1 69 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 C2.4 and Power 71 Underground Conductor System, F u l l Load Current 71 REFERENCES . 72 iv LIST OF ILLUSTRATIONS Figure; 2.1 Part of a Mutually Coupled C i r c u i t of (n+l) Conductors . . . . . . Page 2 5.1 Single Section of a Doubly Bonded Cable Transmission System with Six Independent Conductors .... 1 9 6.1 P a r t i a l l y Transposed Transmission Line 6.2 Two Sections of Transmission Line .............. 35 8.1 . A Three-Section, Six-Conductor with Ground, Overhead System 43 8.2 Ground Wire Current, 11 I , amps vs Distance along the Line, x, metres . . . f 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 4 5 8.5 Current Phase Angle Differences at Sending End f o r 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 . along the Line, x, metres 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 32 4 degrees vs Distance 47 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 l 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 tt vi 65 ACKNOWLEDGEMENT The author would l i k e to express h i s sincere gratitude to h i s s u p e r v i s i n g p r o f e s s o r s Dr. F. Noakes and Dr. I . N. Yu f o r guidance, perseverance and continued i n s p i r a t i o n the course of the r e s e a r c h . throughout The author a l s o wishes to thank the 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 author i s indebted to the N a t i o n a l Research C o u n c i l o f Canada f o r f i n a n c i a l support of the r e s e a r c h . vii 1 1. INTRODUCTION The purpose of t h i s t h e s i s i s to develop a procedure for s o l v i n g the problem of polyphase, d i s t r i b u t e d parameter t r a n s m i s s i o n systems, during steady state o p e r a t i o n . Histori- c a l l y , t h i s problem has been attacked by t r e a t i n g the l i n e c o n f i g u r a t i o n i n terms of lumped c i r c u i t parameters, obtained through transformations from the d i s t r i b u t e d parameters, l e a d ing to v a r i o u s c l o s e d form s o l u t i o n s . E a r l y work i n the development a n a l y s i s was to done by L«V. Bewley (l). of matrix methods f o r The approach taken was analyze the l o s s l e s s polyphase l i n e and to expand the a n a l y s i s to i n c l u d e l i n e s w i t h l o s s e s . From t h i s , travelling wave s o l u t i o n s were developed which l e d to a study of surges by matrix methods. but L.A» P i p e s ( 2 ) followed Bewley's used Laplace transform methods. made by S. Hayashi (3) who approach P a r a l l e l developments were extended the a n a l y s i s to t r a n s i e n t phenomena, i n c l u d i n g t r a v e l l i n g wave p r o p e r t i e s of surges. The i n c r e a s i n g complexity and i n t e r c o n n e c t i o n of modern power systems., together w i t h the f l e x i b i l i t y bility of d i g i t a l and availa- computers, makes the use of matrix methods both imperative and p r a c t i c a l . ¥ith t h i s i n mind, a systematic mathematical and numerical procedure f o r h a n d l i n g the complex in this thesis. throughout. system i s evolved The r a t i o n a l i z e d M.K.S. system of u n i t s i s used i 2 2. GENERAL DIFFERENTIAL EQUATIONS FOR MULTI-CONDUCTOR SYSTEMS Consider a system of (n + l ) p a r a l l e l conductors mutually coupled e l e c t r o s t a t i c a l l y and electromagnetically. By d e f i n i t i o n , this i s a complete system i f and only i f the sum of the currents over the whole system i s zero, n +1 ii = 0 2-1 i =1 This d e f i n i t i o n precludes radiation effects, but this i s an acceptable approximation at low frequencies. Fig 2.1 Part of Mutually Coupled C i r c u i t of (n + l ) conductors. 3 Of the independent; the (n + l ) conductors, n w i l l be d e f i n e d as (n + l ) t h conductor becomes the being reference conductor f o r v o l t a g e s and t h e " r e t u r n " path f o r unbalanced currents. Depending on the p h y s i c a l arrangement of the t r a n s - m i s s i o n system, t h i s reference conductor would normally as a ground conductor or may current equations ^y. " 1ST = ITt 2> i , 2 2 n + i 1 2 ~ 3 v. = p o t e n t i a l of conductor i w i t h respect to some a r b i t r a r y reference vp. = t o t a l f l u x l i n k a g e s per u n i t length of conduct o r i due to currents i n a l l conductors R^ = s e r i e s r e s i s t a n c e per u n i t l e n g t h of conductor i i. = current i n conductor i q^ = charge per u n i t length on conductor i l I i ^ = The analysis. TT = ~ Vi + 2>q. " "Sic and f o r the i t h conduc- be w r i t t e n £v. where taken an e q u i v a l e n t e a r t h conductor ( 4 ) . The v o l t a g e and tor be leakage c u r r e n t per u n i t l e n g t h from conductor i system i s assumed to be l i n e a r i n the f o l l o w i n g Let p be the d i f f e r e n t i a l operator p = 5 / A s s o c i a t e d with each u n i t l e n g t h of conductors i and j are Z! . = xx Z!. = R. + pL. . 1 ^ 11 pL. . *i j 13 T! . = G.. xi I! . = G. . + pC. . 10 11 13 + pC.. • 11 * 13 t 4 where IL = series resistance of conductor i L.. = self inductance coefficient of conductor i L.. = mutual inductance coefficient of conductors i & j 11 1 J G ii = C . s ^ e capacitance coefficient of conductor i = mutual capacitance coefficient between conduc— tors i & j G.. = leakance from the i t h conductor to the a r b i — trary reference G.. = leakance between conductors i & j 1 J 1 1 The d i f f e r e n t i a l equations of the i t h conductor become " T x = Z i l i A + Z i 2 *2 + + Z ii * i — + + Z i , n 1 in+l + 2-4 3i. - Ti = T il V l + T i2 2 V + + T ii V i + — + T i,n 1 n l v + + 2-5 In matrix form, the 2(n + l ) equations for the (n + l ) conductors may be written - l ^ 1 = (Z'( )) (i) 2-6 = (T'(p)) (v) 2-7 P where (v) and ( i ) are column vectors, ( Z ) and (Y ) are square 1 1 matrices which are functions of time (the d i f f e r e n t i a l operator p). By d i f f e r e n t i a t i o n with respect to x and equations 2-6 and 2-7 may be combined to give substitution^ 5 ^4 where (Z ) 1 and M = (Z'(p)) (I (p)) f ( Y ) a r e b o t h i n d e p e n d e n t o f x. T (v) 2-8 3. THE DIFFERENTIAL EQUATIONS FOR STEADY STATE ANALYSIS Consider a-c steady conductors state the system of conductors operating c o n d i t i o n s such t h a t the v o l t a g e s 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 under of the (n+l) arbitrary r e f e r e n c e , are g i v e n by a KJ 3-1 j(»t+0 ) o n+l/ n+l Since the system i s l i n e a r , the c u r r e n t response h a v e t h e same f o r m w i t h d i f f e r e n t phase a n g l e s j (at^.) - /I, where 1 2 I n will 0^, 0^> ••• * ^ + i n 3-2 (I' ) e 3(«t+0 ) 2 + 1 e J (•**»«>. ( V ) and ( l ) a r e p h a s o r v e c t o r s . T 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 2-6 and 2-7 r e s p e c t i v e l y ^ w i t h t h e o p e r a t o r equations p r e p l a c e d b y jtt y i e l d s 7 - (V') = (Z'(«)) (!') 3-3 - 3! ( T ) = (!'(•)) (V') 3-4 1 Equations 3 - 3 and 3 - 4 are written i n 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. I f the (n+l)th conductor i s chosen as the r e f e r - ence conductor, then the voltage phasor vector becomes 3-5 and 3-6 i = 1 Applying these constraints to equation 3 - 3 yields the reduced system of equations f o r n independent conductors. - where Z^) = ZL(.) + (V) = (Z ( « ) ) (I) - «J 3-7 - i» 0 = 1» 2, *A+lf,<«) n 8 Since (Y'(«)) i n e q u a t i o n 3-4 - (v«) = Let ( Y ( « ) ) "" be t h e r e d u c e d 1 form -(v) = and CY £ of i s not a f u n c t i o n of x ( i ' (•)) ( Y («); - the c u r r e n t equation f o r n independent T (w)) fe 1 - 1 (!•) """"/then, (I) conductors may be written - fe ( D = This analysis the indicates ( Y* ( » ) ) (a) t h e l e a k a n c e transmission (b) t h e l e a k a n c e If distribution any (Gr) may be s e p a r a t e d i n t o e m p i r i c a l l y d e r i v e d from (towers, part two p a r t t h e l o s s e s due t o system^ due t o t h e g e o m e t r i c a l c o n f i g u r a t i o n o f t h e (b) a l o n e and l e a k a g e of the surrounding media. i s considered, then current distribution since the f i e l d a r e t h e same f o r system o f c o n d u c t o r s ( 5 ) , (P') i s t h e p o t e n t i a l coefficient m a t r i x f o r the (n+l) conductors <r i s the c o n d u c t i v i t y o f t h e medium s u r r o u n d i n g t h e i s the p e r m i t t i v i t y o f t h e medium s u r r o u n d i n g t h e conductors and e conductors. with conduits e t c . ) of the and t o t h e c o n d u c t i v i t y given linear where 3-8 t h e r e d u c t i o n must be a c h i e v e d matrix s u p p o r t i n g mechanism conductors (V) matrix i n i t s i n v e r s e form. The l e a k a n c e the that ( I (•)) 9 Therefore, the matrix ( P ) f o r the reduced system of n independent conductors w i l l have elements of the form 13 13 n+ljn+1 i,n+l n+l,j i, 3 1 » 2, •»»t = 1 n and (I («•)) = ( 0 - + j» ) e (P)" Note that for most transmission systems, 1 6-/«e <£. 1 F i n a l l y , the reduced equations 3-7 and 3-8 may be combined as before to give ^ 2 (V) = (£(•)) (!(•)) (?) i (A(o))) (V) 3-9 d x d 2 ^ 2 dx (D = (!(•)) (Z(«)) (I) £ (B(«)) (I) 3-10 10 4. 4.1 SOLUTION OF Characteristic Equation THE DIFFERENTIAL EQUATIONS Root and 3-9, Characteristic Vector Analysis the v o l t a g e e q u a t i o n , may (6) be w r i t t e n 0 4-1 T h i s has a non-trivial s o l u t i o n i f and det | (A) o n l y i f the - d /dx 2 2 ( u) | determinant =0 4-2 where is the (u) i s the identity matrix. characteristic teristic This determinantal e q u a t i o n whose s o l u t i o n y i e l d s equation the charac- roots. Consider a g a i n e q u a t i o n 3-9. second o r d e r l i n e a r differential There a r e n o r d i n a r y equations d with constant coeffi— 2 c i e n t s w h i c h a r e homogeneous i n Hence the f o r m o f the — I> . solution i s & N Vi = —5dx^ (C i l , e r + —Y x X °i* e 9 i = 1, 2, n 4-3 where the and C's and C ' s are the y'^ = v/X^ , where the X's determinantal equation 4—2. are complex c o n s t a n t s the characteristic of integration r o o t s of the 11 There are 2n constants of integration i n 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 * (c. 0 r ^ r=l ir y x e Sr w r v x „ . e - *r +. C! ) x o r lr ^ rn- ' _ J yr x + 0i!r e~* r *)' il ir A ( c r=J + n ^ + r=T v x Collecting terms i n e ( r=i + ( * C <<* r " A ir - ii) ir C and e A —^ x - A - i2 2r A il°ir + C' e ° r ) nr o r we have r il°lr A, J C _ e i n nr C ' — " - in - ~ nr > A " i2°2r ~ — A " C A r 7=0 in Ar C 4-4 Hence each of the coefficients i n equation 4-4 is i n d i v i d u a l l y equal to zero for a non—trivial solution, (i = 1, 2, . . . , n ) . This provides n2 equations f o r the unprimed constants, equations i n the primed constants, C,and n2 c', as both i and r vary. For the unprimed constants, C /All A 21 V r A 12 A L 2 2 " r"" K A ln = 0 , r = 1,2, a • • ,n 2n nl 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 i s 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 e x p l i c i t l y by r e w r i t i n g equations 4—5 as follows, A ll - r X A21 A 12 A 22" r # # 11 A 21 A 0 X A hi /A, , - X nn r l r / Cnr\ 2r o / C nr V -X r, 12 22"V D ^-l,n-r r/ X / - - N A, n t • - • • A \ A C nr = ( 0 ) (c, * * ln 2r \ n-l,r/ D \ L 2n L n-l,n/ 4-6 where D. = C. /C and C ^ 0. ir i r nr nr This analysis indicates a method of determining the characterist i c vectors numerically. constants C and C . The same vectors hold for both the 4.2 13 The General Solution The voltage solution may be written as Y.J = Cno. ^>~D. j r no r=l j r + e no^ ^> - j r ~ no r=l ^> ;~ 0 1 D e > r = l,2,.»*n r 4-7 where D nr =1. and C .. C . n j * no T * are unknown constants to be determined from the boundary conditions. By a similar analysis, the current solution^ may be written as _ I. = 3 where G nr n v x n -V x F . ^>"G. e° + F . > G. e nj ^ jr nj jr r=l r3 r =1. ' , 1 r = l,2,...n ' - and F .. F* , are unknown constants. n j * nj 4-8 It w i l l be shown that the y ' s are the same f o r both voltage and current solutions. The constants C . i C . and F ., F'. are not indepennj nj nj nj dent but are related through equations 3-7 and 3-8. This solution may also be written i n the alternative hyperbolic form using the hyperbolic sine and cosine. The general solution f o r voltage and current may be written i n matrix form (V ) = (D)UC CD = (Of e* n r r X ) ( nr + C + * ^ e (* e"* )} rX nr 4-9 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) i s the characteristic vector which s a t i s f i e s ((A) - X 1 (U)) (D ) x = 0 and hence i s associated with the characteristic value X-^. that the entries D nr unity. (and G ) nr Note for r = 1, 2, ...,n w i l l be ' ' y 14 The matrices (C e nr ( 0 e" \ nr & X ) represent column vectors ) / t /Cn li e C n2 V » 4-11 6 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 solut i o n and vice versa. Let the current solution be known, then by rearrangement of equation 3-8 v = -(y(a))"1 f j - ( T ) = - (H-))" 1 « » ) { ( » l l r . * r X y r ) • —Y X. y x -y x By equating coefficients of e and e respectively, we obtain r ( )(c D n r ) r =-(i(.))- (o)(»„yr) l -('W)(«)(u) 4-12 15 ( X AND D C ) = ((<°>) ' ( O K A ) T NR 4 " 13 = (*(•>) (0(*Ary > r ( !(«))( P ( « ) ) where leakance has been ignored and = (U) . Thus the general form for the voltage and current solutions may be written -(P(.))(0)j( F tf e* ) (?)= n r r r X - (Kj/* *)] ' 1 4-14 (l)= (OJ(^ / ) r X n + ( A/ S X r X )] . r=l,2,..,n 4-15 In a similar manner the current constants may determined i n terms of the voltage constants by use of be equation 3-7 using the properties of duality, <n (T). = ( » ) j ( C n / r X ) + ( C A / ^ ) ] , r=l,2,..,n 4-16 -fzW)- ^)^/^) - ( ^ / ^ l 1 r=l,2,..,n For equations 4—16 and 4—17, 4-17 the voltage solutions w i l l be defined as the "primary" solution; the current solution i s a "derived" solution. and 4-15, Conversely, for equations 4—14 the current solution i s the primary solution from which the voltage i s derived. to i l l u s t r a t e This l a t t e r form w i l l be chosen the following analysis of the boundary conditions. 16 5. BOUNDARY CONDITIONS Consider the boundary conditions conductor system. There must be 2n s u c h c o n d i t i o n s w h i c h may be specified as c o n s t r a i n t s on t h e v o l t a g e , f o r t h e complete; • current or both a t the boundaries. For s u c h an n - c o n d u c t o r s y s t e m , t h e r e 4n c o n d i t i o n s are i n general a t t h e b o u n d a r i e s , 2n a t e a c h end o f t h e l i n e . These a r e < T > » - - i - ( .> T , and ( V ) I _ (Y s e n d i n g end c o n d i t i o n s , ) r , receiving ( *) x = 0 Alternatively, end, i n which = end c o n d i t i o n s . (I ) r the o r i g i n o f x may be d e f i n e d case the r e c e i v i n g a t the sending end i s d e s i g n a t e d by x = I, ; i n b o t h c a s e s , x i n c r e a s e s f r o m t h e s e n d i n g end t o t h e r e c e i v i n g end. Of t h e 4n b o u n d a r y order t o o b t a i n a unique conditions, system 2n must be known i n solution. Several special may be c o n s i d e r e d , (i) ( V g (ii) ( V s (iii) ( V g ) and ( I ) or ( l ) or ( I ) g g g or ( V r ) and ( V r ) and ( ) and ( I r ) ) or ( I r ) Z ) r cases 17 (iv) a n y 2n c o n d i t i o n s o f ( V where ( Z ) i s t h e r e c e i v i n g (O XX r (i) these (V and ( I -(*<•))( 0 1 ( G ) [ ( P n r K ) r and ( I cases. r ) known, f o r ( I ") . ), <0[(* X> " 0' * )| ="(!(•))( T ) nr - ) of s o l u t i o n i s used t o " n " e q u a t i o n s may be w r i t t e n and n e q u a t i o n s f o r ( V r ) . special ) r ) ,( I ) , ( V n r r r ( 'nry )i P " r ( r) V 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 nr -(G)- (K«))(V 1 nr r ) 5-1 where (u^) = 5-2 0 •y 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 t h e c o l u m n vectors ( F } a n d ( F ' ^ , ^ nr/ \ nr / ' ( nr) F " ^ ( ^ ( I , ) p ) e n d impedance m a t r i x d e f i n e d b y The p r i m a r y c u r r e n t f o r m illustrate g " (u^Xd)- 1 (Y(.))(V )^ r 18 Let (X ) H C O X U K ) ) " S 1 •'• ( n r ) = *[USrXxXl ) - U X ' W X O J F r and ( P ) = *[(D)fXxXl ) + (xXK»)X r)^ ¥ A r r which i s the simplest digital form f o r n u m e r i c a l solution using a computer. known. s) are + ( * ' )\ n r 'J N and or (')[( F e nr /(O (u) (lie"**) / (lJe*<) \ nr / \ nr > j u r ^ I ) 5-3 w h i c h may be s o l v e d as i n case ( i ) . (iib) ( I The and ( matrix /(u) v s ) known. equation i s (u) ( P \ \ )\ nr / (pi ) \ nr / -(G)- 1 CK.))(0 5-4 w h i c h may be s o l v e d as i n c a s e ( i ) . ( (iii) From F nr ) I ( I e and ( g ) known. ) , ) + ( p A r e ^ ^ .( I . ) 19 From ( V ) - (P(«))COK nA) F or i n matrix form Kl.) P(tt) (Z )(G) + r (6)(ny)jC8 )(G) r P(o>) (G)(U g ) j \ ( F A r ) 5-5 (iv) Bonding of cables. 1 i 1 ; * ' / 2 ( i \ 5 / \ \ \ \ i i i i / i i i J / l I O A i '3 L / ' i i D 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, between the terminals. g , at x = some point This ground conductor w i l l be used as the voltage reference conductor. I f the earth i s 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 currents 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. designated as sub-section 0 sub—section as (?) , I f the load end sub-section i s , — - 0, and the remaining ~ 2 ~ ^' then for section© the system of x equations to be solved f o r the boundary conditions • ( ]_ 2 3 ) x V 9 ( 4,5,6 * ? and ( i )x T ) = 0 X l 1 ^ = -L1 = ( ^r ) 1 ' -(p(«oX<0(*) u =(A)l=(°)> ( I ) , j = 1, 2, (rows 1,2,&3) _,l:<1 ) 8 _(rows_l,J,&3]_ (p(-)XoX"x»: Wi e n d conditions "ending end conditions 6 r 1 g r e c e i v i n ! (p(.>xx°«)/ is -(p(.)XoX^. - 0 , receiving end conditions \ / \ 1 v nr' 1 f „. \ F' ) • nr'1 1 (rows 4,5,&6) ( a l l rows) (rows 4j5,&6) ( a l l rows) \ / 5-6 21 w h i c h may be expanded as 9.. IT. * ill '6> where £ and i > it (ti » t >> I i s the ( i , j ) g^^ i s t h e ( i , j ) t 36i element o f the product h (P(<O))(G) element o f the c h a r a c t e r i s t i c vector m a t r i x ( G ). The constant system o f e q u a t i o n s 5-6 c a n be s o l v e d f o r t h e vectors ( P ) a n d ( F ' ) , f o r sub-section© . * nr ' 1 nr / 1 n the remaining unknown c u r r e n t s boundary of s u b - s e c t i o n ( 7 ) voltages for and c u r r e n t s 1 and v o l t a g e s = -$^t a t the l e f t - h a n d c a n be f o u n d . are continuous and t h e b o u n d a r y The c o r e conditions s u b — s e c t i o n (z) are <Vx,=0 1 ( r ) 2 " ( j) x,=-L ^i, ,3)x =o 2 and , x Hence w v 2 V f M . ; ) 2 • J = 1. 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 ) \ / ( a l l rows; \ / \ \ =/(V ) / r < nr>2 F (G) (G) (rows 1,2,&3) (rows 1,2,&3) 1 (G) (Ue~ ^ ^ ) (G)(Ue ^ V (rows 2 + 4,5,&6) 2 ) ( I r ) .2 C (P' ) nr 2 / v (rows 4,5,&6)/ ; 5-7 c a n he s o l v e d f o r t h e c o n s t a n t v e c t o r s (P ) and (P* ) 2 nr 2 0 nr 0 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 o f the transmission line, the complete s o l u t i o n f o r t h e complete section c a n 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 a n d 4—15. I f the sheath bondings f o r each s u b — s e c t i o n are c o n n e c t e d , a d d i t i o n a l c o n s t r a i n t s a r e i m p o s e d on t h e s y s t e m . In t h i s case> t h e c o n s t r a i n t e q u a t i o n i s 5-8 2(1.) 0=4 Since the l o a d boundary J 1 3=4 c o n d i t i o n s f o r t h e case a r e s p e c i f i e d , t h e n t h e c o n n e c t i o n o f t h e b o n d i n g c a u s e s a c o n s t r a i n t t o be i m p o s e d o f sub-section© b y sub-section®. S u c h a c o n s t r a i n t c a n be h a n d l e d b y 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 t h e n e x t c h a p t e r . However, s i n c e t h e e n t r i e s i n a t r a n s p o s i t i o n matrix are u n i t y , t h i s represents 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 point. 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 . This i s 23 For a "lossy" transposition, where voltage and current magnitude and/or phase angle for a given conductor does change, the corresponding entry i n the transposition matrix w i l l be i n general a complex number with absolute value different from unity. Such an entry, ^^y m a y be represented by t. . = e ± / ^ ( where p shift. + ) i s the attenuation factor, and r\ gives the phase angle 24 6. 6*1 TRANSPOSITION MATRICES AND MATRIX THE COMPLEX CHARACTERISTIC The Transposition Matrix* Consider a transmission line with multiple sections where at each or any j u n c t i o n two or more of the conductors have their physical locations i n space interchanged. Such an interchange i s indicated i n 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 i n the ( R ) , ( L ) and ( C ) matrices by using the transformation matrix ( E ) which i s formed from r the identity matrix ( U ) by interchanging the i and j rows or columns E 1 0 0 1 i j i 0 - 1 6-1 I i 1- 0 - Premultiplication of any matrix — (A - (say), by the matrix ( E^ ) , (which i s compatible with matrix ( A ) ,) causes the i^* 1 and j ^ * rows of ( A ) to be interchanged, and postmultiplica1 tion causes the corresponding columns of ( A ) to be interchanged, 25 ^E i.e. r > )(A ) ( A ^ yields V ( A ^(E ) with i ^ * y i e l d s ( A ) with i ^ * r and ( E ) ( A ^ E p ) yields ( A ^ r and j ^ * rows 1 interchanged, and j ^ * columns 1 with i ^ * 1 1 interchanged, and j ^ * rows and columns 1 1 interchanged. Consider By inspection, / these matrices are symmetric, (O If the matrix ( E ^ (O ( u\ ) matrix ( .) V E r )t 6 as follows, o f r a n k i , and ( J ) f0 (0) (U._.^(0)| (o) i & where t h e symbol 0 r e p r e s e n t s t h e s c a l a r .-. ( j ) = / l zero, 0\ = (Vi_i) 2 <Vi-2> ,0 E is a (j - i - l ) , (0) ( 2 (ur.p, ( J ) and ~ 0 i s an i d e n t i t y m a t r i x of rank = ( of rank r , i s p a r t i t i o n e d o where of the s e t of matrices ( E ^ } some p r o p e r t i e s ) (J) 26 6-3 and(E ) is orthogonal. t h i s matrix ( E ) the d i s t r i b u t i v e r For (E )(R)(E ) r and + r (E )(L)(E ) r law h o l d s , i . e . (E )(z)(E ) r r r hence, K)(A)(E ) r Thus t h e t r a n s f o r m a t i o n reverting to the separate p a r t s of the m a t r i x ( A ) . From t h e s e 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 maybe s e e n t h a t formation on t h e p r o d u c t , ( A ) j w i t h o u t i s valid the operation which leaves ( E ) ( A ) ( E ) is roots of the d i f f e r e n t s e c t i o n s same i s t o be e x p e c t e d f r o m p h y s i c a l mission the line as a whole, s i n c e conductors The sections trans- the c h a r a c t e r i s t i c r o o t s unchanged, b u t n o t the c h a r a c t e r i s t i c v e c t o r s . characteristic a similarity The f a c t of the l i n e considerations the geometrical that the are the of the t r a n s - configuration of i s unchanged. 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 of l i n e f o r two may be f o u n d f r o m t h e d e f i n i n g e q u a t i o n s o f t h e characteristic vectors f o r each s e c t i o n , (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 w h i c h depends on t h e o r t h o g o n a l i t y of ( E ) i s r ( x ) = (KrXy). more t h a n is the Consider the p h y s i c a l one of conductors pair assumption matrices o f the same rows and s i t u a t i o n where a t a i s interchanged. t h a t the n e c e s s a r y t y p e (E^) may elementary interchange columns more t h a n junction, Implicit here transformation at d i f f e r e n t times the once. th th example, i n o r d e r t o i n t e r c h a n g e t h e i , j , and ~th *th "th k c o n d u c t o r s c y c l i c a l l y , the i and j rows and columns must For "th be i^* interchanged, 1 rows and f o l l o w e d by o f the k and n such elementary interchanges a t one ) {E^) matrices ( r e l a t i o n s h i p between t h e , T original section j u n c t i o n and ..., ( E ) . n = relationships of t r a n s m i s s i o n 1 2 1 1 a t r i x , ( «)(<o ) ^ . . (E ) 2 between t h e characteristic and ( A' ) are v a l i d r o o t s and n vectors i f i t can be proved = ( 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 transformation. line («)(AXX) of the m a t r i c e s ( A ) ( «> ) n the The 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 m a t r i x ( A') i s g i v e n by (A') =(E )(E _ ) . . . .(E )(E )(A)(E )(E ). The "old" columns. Consider associated an i n t e r c h a n g e It will = a l s o be that similarity shown t h a t ( <o) i s an orthogonal ( U ) . = = ( K ) ( V ( E , ) - ! > 1 ^ ) - ( 1 " 2 X > i » - 1 ( E ^ ) - 1 ^ ) - 1 28 Also, ( « O = ( B t ) 1 = ( l)( E **. = (u) ( « ) ( « ) .j. ( B t E 2 2 )' ) . . • • (E _l) t ( n ) t E t n * • * ( n-lX n) E E = as required. Note that (<o) ( ) X i s 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 ' ) ( i O where ( A» (x) or ) = ( X ) = ( X )( _ 1 = A = A (X) ) ( A ) ( X ) y ) ( 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 i s found convenient to expand t h i s matrix into a real matrix of twice the order, i . e . i f ( A ) i s of rank n, then ( A ) p e X a n ^ W e a 1 1 l be of rank 2n. Hence f o r the expanded matrix there w i l l be twice as many characteristic roots as f o r the o r i g i n a l matrix. It will be shown that the 2n roots of ( A) pComprise the n roots of ( A ) eX 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, i s o' to expand each element, a + j b , of ( T)as f o i l ows, v a + jb ~f a b\ \-b a/ A more convenient form of the matrix ( S ) is Q (S) = / (A) (B)\ (A)/ -(B) where ( A ) i s a matrix comprising the real parts of each complex element of ( T ) i n the same order, and ( B ) i s the corresponding matrix f o r 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 l a s t section. i.e. -(S ) = ( P ) - * S ) ( P ) 1 o ) = ( E )(B ) where (P and (P)(P) 1 t 2 .... = ( P)(S)(P) (VlM ) B n = (U ) Hence since similar matrices have the same characterist 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 r e a l matrix ( A ) m a y as follows. be shown The characteristic equations of the complex matrix ( A ) and i t s conjugate matrix ( A ) * are det \ ( A ) - X ( U ) j = 0 30 j (A)* and det - The characteristic X* ( U ) \ = 0 equation of the following expanded complex matrix has twice the rank of the o r i g i n a l complex matrix ( A ) , /( A ) - X ( U ) det \ 0 \ = 0 ( A ) * - X*(U)/ 0 or det [ ( A ) - X(U )j . det Therefore the characteristic [(A)* - X* ( U )\ = 0 roots of this expanded complex matrix are the roots of ( A ) and their conjugates. Consider the s i m i l a r i t y transformation where ( T ) ~*( K)(T ) ^ ( For ( K ) = T ) f{ = fl ^ B ) + -j' j(C ) 0 0 (B ) - j( C ) y this transformation becomes (T)-\K)(T) = W Z/ 1\ /(B) + 3(C) - j ( B ) ( C . ) \ =/(B)(G)\ = ( A 7 2 \ j - j / \ ( B ) - j ( C ) j ( B ) - ( 0 ) / \(-C)(B)/ 1 + which i s r e a l . The expanded matrix ( A) gX p can also be reduced to the form '(B) .( 0 ) -(C) N ( B )/ by a further s i m i l a r i t y transformation using the matrix T 1 Oi Q where e These expanded matrices being similar, they have the same characteristic roots and these roots w i l l be the roots of (A) their conjugates, and 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 i s to determine the complex characteristic polynomial det | ( A ) - X( U )| = 0. by evaluating Only n of the roots would s a t i s f y this equation; the remaining roots must be discarded. numerical process was devised using the numerical the characteristic polynomial Such a evaluation of (12) attributable to A.M. Danilevsky. ' As an alternative approach, i t w i l l be shown that i f the rank of the o r i g i n a l complex matrix, ( A ) i s small ( n ^ l O say) then i t i s 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 i n 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 ( t r ( A ) ) = ± Im i=l (X.) 32 6.3 Transposition and Connection Matrices for Multiple Section Line s. Consider the p a r t i a l l y transposed transmission shown i n F i g . 6.1 where the geometrical line configuration of the line i s the same for both sections and losses due to the transposition i t s e l f are negligible. 1 * J — e o '\ r t> . z \ 7\ 0 . 3 7 - , , ,. 0 — , . ,, 0 A ft s / 0 ft. L <1 • — 1 D ® 0 F i g . 6.1. P a r t i a l l y Transposed Transmission Line. The relationships between the two sections are: (i) (x) - ( )(x) = (z) 1 (X)" and (ii) (iii) (X)' (J )( 1 Zl (1^(1) 1 1 2 Z l ) ( X ) = ( I ) 2 = (T )(Z ) 2 2 the c h a r a c t e r i s t i c roots of the two sections are the same 33 (iv) the c h a r a c t e r i s t i c related vectors o f the two s e c t i o n s a r e by ( x ) " ( d 1 l ) i - ( d ) 2 i or and - (x)- ( ) - (o ) represent one c h a r a c t e r i s t i c l or ^^ ) » where ( s) tage and c u r r e n t sent the square a r r a y s of the v o l t a g e solution s o l u t i o n s r e s p e c t i v e l y , and (P of the v o l - (G ) , column repre- vectors solutions respectively. t h e two s e c t i o n s of l i n e , and 4-15 (-))(^)(u»)[(» 1 vector (D) of the n c h a r a c t e r i s t i c e q u a t i o n s 4-14 - - (V 2 h and c u r r e n t For («2i) the v o l t a g e and c u r r e n t become l f n r . , , ' X l )-(F i ( n r .^ I t )| ( f ) - M ( ' / ) *("l,n/ )l 1 Il section Q for Vl 1 and ( Y ) = -(P (.)X ° X*4('2,»S ) - K n / ^ )| lXi 2 for ( p 2 t section^ . ( « ) ) = and (G ) = ( X) ••• ( * 2 2 2 i 0 ) ) ( ( Q x But, ) 2 ) Hence t h e c u r r e n t " _ 1 1 = ! ^ ^ (G ) x ( X ) ^ ) = ( x ) = (x) ) t ( P and v o l t a g e l ( w ) t ) ( t ( p 1 ( « ) ) ( x ) (G^ G l ) s o l u t i o n s f o r s e c t i o n (D become 34 (V ) = 2 and ( l 2 -(l) (P (.))(G )(uri((F t 1 1 ) = (x) (G!) T 2 f n r e i n r .^ ) (,.^.-^2)1 . *r 2\ x (F 2 5 r 2 . / „ , " e* r ) + (* nr r X 2 2 f Since the sequence i n which the propagation ^ r constants, are taken i s unchanged by the transformation, the matrix (u^) i s the same for both sections. At the transposition boundary, x and ( v 2 2 =0 = -l x ± ± ) x =0 = ( v ± ) x 2 l= 1 1 ,nr (x) ( t v -(x) t G l ) (x) ( t (P^-OXGjXuy) (x) <*>t/(°l> V ^ C ) ) t G l ) 2 n (p^^Dt^fu^tF^^J W (G.XUJJ) \/(* , j Y 2,nr) F (P.U)) (G^OJ*)/\(P 2 > N R N ), 6. 4 This system of equations may be solved for and (F 2n r ) ( Fl i n terms of ( F, ) and ( F ' ) . The f i r s t v 2,nr/ l,nr' l,nr/ v v n equations represent current continuity and the last n equations represent continuity of the space derivative of current. 35 Note t h a t diagonal, constants mission the matrices (Ufl) and ( U e ~ ^ ) are t h e y commute and may be combined t o f o r m Since the since e q u a t i o n 6-4 p r o v i d e s a r e l a t i o n s h i p between o f i n t e g r a t i o n f o r t h e two s e c t i o n s line, t h e n t h e a n a l y s i s c a n be e x t e n d e d t o g i v e 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 —— ™ "'• - 4V Z r ' / 4. n V r. sr / V A D ' \ 3 L o r V s-W o ^ 0u —-—-\ 0 line. 0 V o similar section 'C 1 a ©•• —— of the t r a n s - " 3 fe © Fig. 6.2. The alternative line equation, are different. currents method o f c o n s i d e r i n g F o r the transmission are i d e n t i c a l roots of Transmission Line. f o r m o f t h e system o f e q u a t i o n s 6-4 i n d i c a t e s an and s i m p l e r sections. sections Two S e c t i o n s line the transmission shown i n F i g . 6.2, t h e and hence have t h e same c h a r a c t e r i s t i c and v e c t o r s ; only the constants of i n t e g r a t i o n A t 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 and v o l t a g e s are r e l a t e d through a connection matrix^ 1 3 36 This connection matrix i s identical to the matrix observed i n equation i.e. (X) t and ( X ) t ( x ) ^ as 6-3 (T x ( V x ) = ( T ) = ( V ) 2 2 ) This i s an invariant power transformation as required since Use of the connection matrix f a c i l i t a t e s solution for multiple section transmission lines by greatly reducing the complexity of the numerical analysis. 37 7 7.1 THE ( Z ) AND ( T ) MATRICES FOR A COMPLETE SYSTEM The ( Z ) & ( Y ) M a t r i c e s Under t h e c o n s t r a i n t s t h a t t h e v o l t a g e r e f e r e n c e i s . w i t h i n t h e s y s t e m , a n d t h a t t h e sum o f t h e c u r r e n t s o v e r a l l c o n d u c t o r s i s z e r o , i t h a s been shown t h a t t h e m a t r i c e s ( Z ) 1 and ( Y' ) of rank of rank n. ( n + l ) r e d u c e t o t h e m a t r i c e s ( Z ) and ( Y ) For the ( Z ) matrix, Z rs = Z rr = R + R + j « ( L r n rr Z 1 rs + Z 1 -, - Z' , - Z' , n+l,n+l r,n+l n+l,s n Hence , in , ) D and s i n c e D ; d „ /1 , .i«M. = > n + 1 Z = D' rx rr D ^ (R +L,, ,-,-L . n+l,n+l r,n+l v , D• ' , n + l ,x n + l , n+l i - L ) n+l,r' n D' . nn++ll ,x x r,n+l l D' \\ _ rrx n+l, t r ln(^±i) rr' n+l,n+l + R ) + Jf± r 7-1 2 2 7 1 D D where D^j i s t h e g e o m e t r i c mean d i s t a n c e b e t w e e n c o n d u c t o r s i and j and D ^ i s t h e g e o m e t r i c mean r a d i u s o f c o n d u c t o r i . Similarly, z = R + j-fc | °r n+l' A+l s D n 2 1 1 t D f 7 _ 2 rs- n+l, +l D n and since D^ = ^ j ^ , matrix The r e d u c e d m a t r i x ( Z ) i s symmetric. ( Y ) may be f o u n d f r o m t h e 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 rr = P' + P' rr n+l,n+l 1 _ p' r,n+l Ir, ^ r . n + l ^ rr n + l , n+l ( P ) where - P' n+l,r 7 - 38 P ^ = In r n+l n+l s t D ? ? _ 4 rr' n+l,n l D + and s i n c e D. . reduced = matrix, D... the m a t r i x 7.2 i s symmetric. the m a t r i x (P) = j« ( P ) ~ i s symmetric, ( Z ) P r o p e r t i e s of the = 1 j« ( C ) and ( I ) M a t r i c e s . (A) f o r the current = ( Z X I ) (Y)(Z) Since ( Z) = (Z ) and ( l ) = (Z)(I) - ( Z ) characteristic ( B ) (B) ((T)(Z» = t ( B ) t 7-6 T a r e s i m i l a r and hence have t h e same values. vector matrices respectively, ( D) we may G are the character- to the m a t r i c e s ( A) and (D)" 1 (A)(D) = (UX) 7-7 (G ) - 1 ( A )(G ) = ( UX ) 7-8 = (™) ) t (( t ( G ) write o f e q u a t i o n 7-8 ( and corresponding and The t r a n s p o s e t = Since the matrices istic t ( l ) , ( l ) t (A) and have equations = (A) we , (B) Thus 7-5 ( Y ) i s symmetric. A s s o c i a t e d w i t h the v o l t a g e equations and The ( 1 ) i s g i v e n by CT ) Since (P) BB )) t (( t gives GG )) tt 1 t - ( ™ ) 39 and from equation 7-6, ( G ) Hence, f r o m equations ( A ) ( G ) " 7-7 ( G which i s a s u f f i c i e n t a l s o be seen t and 7-9, = 1 (UX) 7-9 we o b t a i n 7-10 ) ( D ) = (U ) t but not necessary that i f this condition. condition i s satisfied, I t may the matrices ( G ) and ( D ) commute. S i n c e t h e v o l t a g e and c u r r e n t forms o f s o l u t i o n a r e related exists t o the matrices a matrix ( T ( A ) and ( A ) ^ r e s p e c t i v e l y , ) such t h a t ( T ) ~ Hence e q u a t i o n 7-8 from 1 ( A ) ( r ) equations 7-7 1 ( T ) - and Restrictions ( A ) 7-11 t 1 ( A ) ( ) ( T & ) - ("O 7-12 7-12 = (-O(G). (D) 7-3 = becomes ( G ) and there 7-13 on t h e Use o f t h e D i s t r i b u t e d F u n d a m e n t a l t o any d e r i v a t i o n parameters i s the assumption that there Parameters. o r use o f t h e d i s t r i b u t e d i s a relationship with (14) Maxwell's e l e c t r o m a g n e t i c The field application phenomena, equations of c i r c u i t i s restricted wavelength i s f a r g r e a t e r than circuit. to those to electromagnetic f r e q u e n c i e s where t h e the p h y s i c a l This condition i s s a t i s f i e d a t low f r e q u e n c i e s . concepts dimensions of the f o r power systems o p e r a t i n g 40 For a transmission line i n a medium w i t h dielectric, the d i s t r i b u t i o n surrounding the conductors f o l l o w s electric has in flux distribution of leakage of d i e l e c t r i c where constant matrix ( C ) with greater ( G ) implies oc(<r that t h a n the c o n d u c t i o n i.e. d conductivity + j«e ) d Hence the d i s p l a c e m e n t current <s ( G ) (5); the s u b s c r i p t "d" denotes d i e l e c t r i c . conductance as t h e and t h u s t h e c o n d u c t a n c e m a t r i x +j<o(c) ( G ) i n t h e space t h e same p a t t e r n t h e same f o r m as t h e c a p a c i t a n c e place current homogeneous i n the i g n o r i n g the current isfar dielectric, 1 d toe There i s a f u r t h e r c o n t r i b u t i o n t o the m a t r i x supporting mechanism of the conductor expressed e m p i r i c a l l y . resistance This are ( R ) implies system. ) This due t o the can o n l y be I n t h e p h y s i c a l model u s e d , t h e e f f e c t o f but not of conductance that within ( G ( G ) was included. t h e c o n d u c t o r the d i s p l a c e m e n t n e g l i g i b l e compared t o t h e c o n d u c t i o n currents currents c where the s u b s c r i p t Since direction conductors, " c " d e n o t e s c o n d u c t o r and there i s a component of propagation to force then the e l e c t r i c Q* of e l e c t r i c the c u r r e n t is finite. field i n the through the and m a g n e t i c 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 w h i c h i n t u r n a f f e c t s t h e o r i g i n a l inductance 41 and capacitance the a,xial parameters. electric dielectric field are small If the appreciable « c effect i n v o l v e s a l l t h e p a r a m e t e r s b u t has effect on t h e i n d u c t a n c e effect -JJ <5s. capacitance. i s much g r e a t e r than 1 i s negligible. These r e s t r i c t i o n s model d e v e l o p e d of D, and t h e r a d i u s , T'Q i.e. the components 1 t h e s e p a r a t i o n between c o n d u c t o r s , conductor be n e g l e c t e d i f compared t o t h e t r a n s v e r s e r the most t h i s may components w i t h i n t h e homogeneous d Proximity However, in preceding are a p p l i c a b l e to the mathematical chapters. Increased sophistication the model w o u l d 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 Two the v a l i d i t y RESULTS e x a m p l e s o f a p p l i c a t i o n were c o n s i d e r e d of the t h e o r y . The f i r s t was an a e r i a l to test double-line t h r e e p h a s e t r a n s m i s s i o n s y s t e m w i t h an o v e r h e a d g r o u n d w i r e . The s e c o n d was a t h r e e phase s h e a t h e d c a b l e u n d e r g r o u n d s i o n system w i t h a separate ground w i r e . homogeneous m e d i a s u r r o u n d e d t h e cases. The calculations. The of s i x hollow L e a k a n c e was ignored 8.1 with inside g, w i t h r a d i u s 0.00636 m e t r e s . s y s t e m i s shown i n F i g u r e Both transposed T r a n s p o s i t i o n p o i n t s are assumed t o o p e r a t e 1-6, i n Pigure and at a constant operates l o a d i s assumed t o have a 0.8 geometrical 8.1. The considered. system was a c a p a c i t y o f 200 a t 60 c y c l e s per M.V.A. a t second. l a g g i n g power f a c t o r w i t h three impedances. Solutions for f u l l found f o r c u r r e n t , voltage The temperature. t r a n s m i s s i o n l i n e has phase T—connected b a l a n c e d a copper u n t r a n s p o s e d s y s t e m s were shown i n F i g u r e 8.1. phase t o phase and consists radius o u t s i d e r a d i u s 0.0145 m e t r e s , and c o n f i g u r a t i o n of t h i s The System aluminum c o n d u c t o r s , ground conductor, used i n the i n both examples. overhead t r a n s m i s s i o n l i n e 0.00622 m e t r e s and KV t r a n s m i s s i o n system i n both r a t i o n a l M.K.S. s y s t e m o f u n i t s was Overhead T r a n s m i s s i o n The 230 assumed t h a t E f f e c t s o f t h e e a r t h on d i s t r i b u t i o n p a r a m e t e r s were ignored. 8.1 I t was transmis- l o a d and no and power. l o a d c o n d i t i o n s were The 43 1 4 , z L O 1 < I •• A D . 3 " 6 4 •transposition points. ^etyes 9* 4-5" ^etves 59 F i g . 8.1 system. vrietres A three section, s i x conductor with ground, overhead 44 It ___ x 0 untjranapoiedi £.TAl>sfS>sej4J.. -foil load o o , load. 599 C O O 2.66400 R«ceiuin<] End Fig. W. Sending End 8.2 ire. Distance COVT t*t , III Aloftg t h e L i n e , x. , wttirts 3996 OO X Sending E«^d Fit). B-3 Pokier Consumption } P } w.u. «.t No L o a d Distance ftlong the Line. , x , we.U«s, us. 45 133200 399COO S e n d i n g Er\ct Receiving E n d F i g . 8.4- Reac/tiue. Co-pac'itatiue w\v>a. at No Loo-d. us Distance Power Consumption Along the L i * e x } wetves. \ 1c Fig. ©.5 5C C u r r e n t Phase Angle Differences at Sending E n d | o r both Transposed and. Untransposed Systems. 46 11 1 13.1 i i 50 i ! I *—— 1 t ! * 1 1— no 1>oicL. : 1 i I f uC on A, f i -™ i poi«4. & "ioo 1 I 1 1 2oo j i 100 i i 1 133200 F i g . "2G6 8 . 6cc- pKo.se. C o w e n t Distance. } 40O 3996OO |I,J , ©.vnps. ie us Along t h e b « e , X j metves. 1 | 170 > o«t«L <6o i ISO 1 4 0 *> • 1 9 3 1 — 95 9 1 — no (( ad. — • —- H > 3996OO 9 o Sewd.i*g End. Reeei FiQ. 8.7 CL- p h a s e os Cuwent Distance Avou^ent , / l a . A lov*a tKe L i ^ C , X, > deo,Y«es Wietv«x, X. 140 l i e 355 C e o S f i A d ' ^ g Receiving E n d . F , q . 8 . 8 F\q. 8.^ a-pkose Voltage } | , kv X E " ^ os a-pKa.se. Voltage Avgu^en t /V^ , cleg-zees 3 us Distance Alomg t^e Livie , x> welrves. 48 8.2 Results In Pigure 8.2, the ground wire the t r a n s m i s s i o n l i n e from the the u n t r a n s p o s e d system, the tance as m i g h t he b e t w e e n no expected, l o a d and system, the full current v a r i a t i o n r e c e i v i n g end i s plotted. load, although load conditions. For 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 the g e n e r a l t h e r e c e i v i n g end t r e n d i s an i n c r e a s e towards the sending end. The the transposition points. 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 mechanically no the minima full i n current from t r e n d can be occurring at r e v e r s a l of c u r r e n t magnitude e f f e c t of the and Figure 8.4 show t h e v a r i a t i o n o f power r e a c t i v e power r e s p e c t i v e l y o f t h e t r a n s m i s s i o n s y s t e m load. at a b r u p t 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 transposed l o a d and e x p l a i n e d on t h e b a s i s o f t h e d i f f e r e n t v o l t a g e s The dis- l i t t l e v a r i a t i o n i s evident 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 o c c u r r i n g a t the For current varies l i n e a r l y with and along The r e a l power i n c r e a s e s r a p i d l y a t t h e sending at end; 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 a t no l o a d and full phasors are b a l a n c e d a t no l o a d due asymmetrical shows t h e load. Although at f u l l and Figure v a r i a t i o n i n m a g n i t u d e and creases As expected, l i n e a r l y from the 8.7 charging c u r r e n t s of the line. show t h e a-phase phase r e s p e c t i v e l y a l o n g t h e no current are q u i t e u n b a l a n c e d geometry of the t r a n s m i s s i o n 8^6 differences the t h r e e phase l o a d , they t o t h e e f f e c t on t h e Figure sion line* c u r r e n t phase a n g l e current the transmis- load current d i s t r i b u t i o n i n - r e c e i v i n g end. At f u l l load the 49 c u r r e n t d e c r e a s e s towards the o f the compensation e f f e c t current. the Little phase a n g l e untransposed no load variation shift ence was are the voltage along data observed at f u l l the line o f the on indication the load l o a d between transposed and s e e n however, u n d e r magnitude v a r i a t i o n the slight, made f o r phase transmission f o r the an wire has tion of the be seen t h a t at f u l l towards the load, while sending end, the the inductive System radius a radius system system i n F i g u r e s h e a t h e d c o n d u c t o r s and sheaths, The line. underground t r a n s m i s s i o n outside differ- above g r a p h s i s i n c l u d e d i n A p p e n d i x increases copper conductors, and transposed decreases. of three aluminum respectively. can be Underground Transmission the 8.9 S i m i l a r statements 8.3 consists Figure l o a d , no power The and phase f o r no reactive The 8.8 and and I n a d d i t i o n , i t may power c o n s u m p t i o n ture current d i f f e r e n c e between the systems was detectable. The solid changing indicated i n Figure load, shifts C l . w h i c h i s an A d i f f e r e n c e can be a-phase v o l t a g e untransposed angle along end, conditions. Under f u l l and of the d i f f e r e n c e was systems. The angle sending 4-6, 1-3, o f 0.00318 m e t r e s . i s shown i n F i g u r e i n a medium w i t h The The of 0 . 0 2 3 9 solid copper geometrical and metres ground configura- 8.10. assumed to o p e r a t e relative The of 0 . 0 1 3 2 m e t r e s inside radius o f 0.0247 m e t r e s . s y s t e m was a ground w i r e . have a r a d i u s have an 8.10 dielectric at a constant constant, e = tempera4.0. 50 The 13.2 KV The t r a n s m i s s i o n l i n e has phase t o phase and operates l o a d i s assumed t o have a 0.9 phase Y - c o n n e c t e d b a l a n c e d found f o r c u r r e n t , voltage l o a d and three- and l o a d c o n d i t i o n s were power. d e c r e a s e s i n magnitude s l i g h t l y from the phase a n g l e . o f power and a t no 8.11 load i s and The l o a d and d a t a of Appendix The seen i n Appendix F i g u r e 8.12 not f o r no phase a n g l e is slight as show t h a t t h e shown. and i s no to the C.2. variation transmission are symmetric f o r l o a d c o n d i t i o n s , as d r i f t along for f u l l the l o a d , as shown i n F i g u r e 8.14, line shown is negligi- i n d i c a t e d i n the the line from increases f o r both load. I n F i g u r e 8.15, ences are end observed i n the conductor current v a r i a t i o n along full sending of C.2. t h e r e c e i v i n g end, there be c o n d u c t o r c u r r e n t phase a n g l e s F i g u r e 8.13. l o a d and the linear. l o a d c o n d i t i o n s but f o r no s m a l l change was 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 full A similar These r e s u l t s may Figure line overhead t r a n s m i s s i o n system, c u r r e n t of the underground system i s independent t h e r e c e i v i n g end. no second. Results ground w i r e ble c y c l e s per l a g g i n g power f a c t o r w i t h l o a d and no I n c o n t r a s t to the in a t 60 impedances. Solutions for f u l l 8.4 a c a p a c i t y o f 10 M.V.A. a t the sheath These a r e t h e phase a n g l e c u r r e n t phase a n g l e same f o r b o t h no drift along the line l o a d and f o r any differfull phase. 51 The variation of the sheath current along the line i s shown i n Figure 8.16. From Appendix C.2 i t may be seen that conductor voltage variation along the line i n both magnitude and phase i s small at no load, but increases s l i g h t l y 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 i s apparent at no load but i s less severe at f u l l load. The phase angle d r i f t along the line i s 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 i s observed at f u l l load, but the increase i s not linear at no load. The data f o r the above graphs i s included i n 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 500 Fig, 8 . I I No Load Distavvee Along tKe L'i*e , x , v*t.l*ts. Aooo Peeeiuinej E n d Fia. 8.12 N o LOOLCI Reactive Pou>ev , Q,, Distance Alono, the L i n e , my.a.. , weUes. os 54 Fia.8.13 /' !»»•( Phase Anale Dif (evewces at -the Seeding ElY\d.. Co-ndoctov / CuYYen.t \ /^--Jl /i "loa.*2 F i 9 6.15" No Load'and Cuvvent at ihe. Full Pkase. RcceAui^a, Load Angle End' S keakk Di^evtv\ee» 55 440 435 ] <4 V\0 lo<xc<,, 20 IS 10 End.. F~i C|. 9 . I 4- tooo CoftductoY Distance I TOO "Xooo C u T Y e - o t ,IX,| a w p s } Along t h e Lir.e , us. j metves. 56 IIJ 1 ! 1 ^ ! ! 15 T 1 i j ! ...!._.. 1 IO i •^"^ ; - - - ! j 1 | i s o • ; I i looo R e c t i u'iv\< F . g . 8 .16 No Load , avnps Fi . 3 8.17 Sheet k | i i » ZSOO 2LOOO St OVA Full 1 — oacl us. D i s t a n c e Voltaac D i f f tre-ftces at the Skeatk Alo^a tKe Li vie. ,sc, wetv«$. PHase A.ujle Sending E n d . 51 HI ! ! i 16.0 | 1 ! I i i i i ! i . | iI i j sro ; i i ! i i 1 I i i j I i 2S M s ! O 1600 F i . 3 Mo 8 . 18 Load Sheath Distovwce 4oo | A lone] >1V I, 4 volts us t h e Liv>t ^ x . , v*e.tves, ^ 1 f I —,—,— loo I i loo O % End i 1 i I i Voltaae •x iffoo 1 ! • aooo >6"00 soo Rece'txnno, E n d . Fig 8.19. IOOO isroo tooo *ffoo X. 3ooo Sending E n d Full Load Sheath Voltjck^a JVJ ,. uolt« us. Distance A\ong the .L'»me j ^ C " , " wet.ys$ 58 9 CONCLUSIONS An accurate mathematical procedure was developed to be used i n the analysis and design of multiconductor transmission systems under various loading or boundary conditions. The v a l i d i t y of the theory was substantiated using two numerical examples. The results of these two analysis are given i n the report. The sequence i n which the parts of the numerical analysis must be performed is shown i n 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, e l e c t r i c and loss effects due to this conductor, which may not be ignored, appear i n 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 i n 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 d e f i n i t i o n 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 i s i d e a l l y 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 v e r i f y 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 i n the complex plane with respect to propagation attenuation. and Further development w i l l lead to the superposition of analysis of the same system at various frequencies for transient studies or for carrier wave transmission studies. A more precise formulation and method for finding the complex transposition matrices which occur at lossy transposition boundaries w i l l also be required, p a r t i c u l a r l y where optimum solutions are to be found. 60 APPENDIX A THE PARAMETERS A.l R, L AND C. Assumptions In the d e r i v a t i o n ^ ^ ' ( L ) and proximity ( C ) , i twill of the matrices be assumed t h a t e f f e c t s and s a t u r a t i o n assumed a l s o that may be i g n o r e d . derivation using static i n v a l i d a t e the a p p l i c a t i o n to a slowly system. conductors w i l l but i n the f i n a l Itwill f i e l d s w i l l not changing or q u a s i — s t a t i c 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 form,the r e s t r i c t i o n that be a p p l i e d , t o some c o n d u c t o r w i t h i n c o n d u c t o r , o r an e q u i v a l e n t e a r t h The m a t r i x o f r e s i s t a n c e reference, and a l l v o l t a g e s currents will the system ( e . g . a ground conductor). 1 jo^ be u s e d . t h e sum o f t h e The R e s i s t a n c e , R""., where R^ = be a l l t h e p a r a m e t e r s f o r a s y s t e m o f ( n -K l ) t h e system i s z e r o w i l l ;be r e f e r r e d A, 2 effects, The r a t i o n a l i z e d M.K.S. s y s t e m o f u n i t s w i l l Initially, within skin (R ) , per u n i t length i s /A^, th A. = c r o s s s e c t i o n a l a r e a o f t h e i conductor 61 and Ji t = + at) = r e s i s t i v i t y of conductor i at a. temperature t ° C where = r e s i s t i v i t y of the conductor material at a o_ q temperature t and A-3 o< C = thermal coefficient of r e s i s t i v i t y . The Inductance, L^" Consider the group of ( n + l ) conductors shown i n F i g . A . l The axes are set up through conductor which the flux linkages are to be computed. 11 o. about The point X i s some remote point where magnetic effects may be considered to be negligible. The t o t a l number of linkages produced by flux which crosses the x — axis between the o r i g i n and the point X i s given by + . . . . + In I T + D n + 1. x j a , n + 1 A-2 where \i = permeability of the surrounding medium r D. =• radius of conductor j = distance between conductor j and the point X D . . = distance between conductors i and j I. = current i n i J conductor A similar expression can be written for the flux linkages surrounding the remaining conductors. 62 O b © c O X a Pig. A.l Group o f n + 1 C u r r e n t C a r r y i n g C o n d u c t o r s . For cross-section, is 'x two p a r a l l e l the t o t a l cylindrical flux conductors of a r b i t r a r y l i n k a g e a b o u t one o f t h e c o n d u c t o r s g i v e n by ( 1 5 ) , A-3 where I i s t h e c u r r e n t i n t h e c o n d u c t o r is t h e G.M.D., t h e g e o m e t r i c mean d i s t a n c e between t h e conductors ^ii """ S G.M.R., t h e g e o m e t r i c conductor. mean r a d i u s o f t h e S3 Replacing the distances D and r of equation A-2 by the geometric mean distances as defined i n equation A-3 gives Ta \ 2n I / i n i DL aa , In Wn + 1 D °\S, + 1? X ^ ab \ / n + 1 \ / D -ft D Vfn In . ba ax D n + 1, a n + 1, x 1-3D n+l, n+l For a linear system, the inductance coefficient, may be defined as a L.. « I = I. l 3 -Ji 0 A-5 in 2it J3 i ^ J D I. = 1 ^ 1 J 2TX D. i i j and hence the inductance /L 0 11 1 A-6 coefficient matrix (L ) becomes L 1, n + 1 12 21 J \ n + 1, 1 .... L n + l t n + ± A-7 A-4 The C a p a c i t a n c e , potential C F o r a system Df coefficients a r e d e f i n e d by t h e e q u a t i o n 'P a ( n + l ) P , ab aa parallel conductors a, n + 1 \ / ^ a \ ba n + 11 \ n P + 1, a * * n P where t h e P , a r e t h e c o e f f i c i e n t s S o l v i n g f o r the charges + 1, n + l / \°-n + 1/ of potential Q. we have 3 I 'Caa cab , fia\ 'a, n + 1 'ba Ai + J Where t h e C , n + l , are the capacitance F o r t h e system P. . A a*** V. 2%e ^ d n + 1, n + 1 coefficient o f (n + l ) c o n d u c t o r s , l i ** n D. -1' D. J y J x 65 where the D's represent the distances between the points denoted by the subscripts, and e i s the permittivity of the medium surrounding the conductors. Clearly, the matrices of potential, capacitance and inductance coefficients are symmetric* Since, f o r the potential coefficient matrix P rr > P > rs ^ 0 then for the capacitance coefficient matrix Cr r and C rs > * 0 0 S For a system of conductors containing coaxial cables where one conductor i s completely enclosed by another as shown in F i g A.3, then because conductor j i s shielded by conductor i. C, = C. , = 0 J O F i g . A.2 Cross—section of part of a system of conductors where one conductor completely encloses another. 66 F o r the system, (Q) = ( c ) ( v ) 1A. then V. 3 03 V m , = 0 m ^ 3 I n p a r t i c u l a r , f o r V\ two concentric = 0, C.... becomes the cylinders. capacitance between Also, Qi 1 J V V 3 = m 0 m ^ 3 V v. 3 m = 0 m 4 3 and since the m a t r i x (C) i s symmetric, then C. . = C . . 13 3i I n terms o f t h e P. 0 = P.. i m p l y 3>c ;. . = c.. = - potential coefficients, P ^ t h a t C ., = C . =0, 3^ 3*" 9 c.. In t h i s and discussion, constant. a n d P. . = P. . = P. . i m p l i e s !3 3 1 i t has 13 31 33 t h a t t h e p e r m i t t i v i t y o f t h e medium s u r r o u n d i n g is = Pj^. 1 1 been assumed the conductors 67 APPENDIX B A flowsheet of the solution procedure for (n + l ) conductor system. Procedure Checks Manual calculation Part I Compute E , L, C, G of selected elements. coefficient matrices from transmission line configuration. Form reduced Z and Y matrices. Part II Find characteristic values of characteristic equation (A) - X (U) = 0 where (A) = (Z)(T) Trace (A) =' Find characteristic vectors D - voltage vectors G - current vectors (D)" (A)(D) = (X U) Solve for 2 n 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 solution i=l 1 (G)" 1 (A) t (G) = (X U) Generate required output from the particular solution. Part I of the procedure gives the general solution for the given transmission l i n e ; 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 K P/Q (km) < (kv) ' h' (kv) 0 132.8 132.8 (deg) (deg) (deg) UNTRANSPOSED 132.8 0 -117.2 117.2 133.2 130.8 127.6 127.7 0.58 -117.1 117.3 266.4 125.0 121.9 122.2 0.50 -116.8 117.5 399.6 115.6 112.8 113.5 0.30 T, a V (kv) . -116.3 119.1 TRANSPOSED 266.4 124.8 120.0 122.0 0.30 -116.7 118.0 399.6 115.7 113.4 112.7 0.04 -115.1 118.6 (m.v.a.) 0 0.06/ - 46.2 0.46/ - 87.1 1.52/ -118.1 0.46/ - 87.0 1.51/ -117.9 Appendix C1.2 Overhead Conductor System, No Load Current. Distance 11J T, (km) l^ a (amp) (amp) K 1 (amp) (deg) 1^ (deg) (deg) (degVampxlO UNTRANSPOSED 0 0 0 0 0 0 0 133.2 58.5 62.0 59.6 92.2 28.7 -152.8 266.4 115.0 121.0 117.0 92.3 28.8 -152.7 399.6 169.0 178.0 172.0 92.5 28.6 -152.5 TRANSPOSED 266.4 177.0 120.0 118.0 91.2 29.7 -150.8 399.6 170.0 174.0 174.0 90.5 28.6 -150.6 0 -47.7/ 41.0 -47.4/ 81.0 -47.0/ 118.0 -27.7/ 31.0 -89.2/ 42.0 4 69 Appendix C1.3 Overhead Conductor System, P u l l Load Voltage and Power, Distance 1V 1 T. (km) ^b A (kv) 1 (kv) ^0 (deg) (kv) (deg) (deg) P/Q (m.v.a UNTRANSPOSED 0 132.8 132.8 132.8 0 -117.2 117.2 133.2 156.1 153.0 148.0 7.6 -108.7 126.8 266.4 176.8 175.0 165.5 13.5 -102.2 134.7 399.6 193.9 193.4 181.0 18.5 - 96.9 141.2 TRANSPOSED a 172.0 169.1 14.5 -102.0 133.5 190.9 191.0 186.5 20.4 - 97.3 139.6 266.4 176 399-6 Appendix C1.4 333.5/ 264.9 341.0/ 230.5 Overhead Conductor System, P u l l Load Current Distance 11 I 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 a 312.9/ 234.7 324.2/ 266.2 333.5/ 264.8 341.0/ 230.6 I h l 358.0 266*4 410.0 412.0 399.6 364.0 359.0 ui 'Li; ix Li. c (amp) a (deg) h (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 / I I I g g (deg)/ampxlO^ 0 -47.7/ 41.6 -47.6/ 82.8 -47.2/ 122.5 -36.0/ 19.7, -77.0/ 69-4 70 Appendix C2.1 Underground Conductor System, No Load Voltage and Power Distance Voltage: L (m) (deg) (kv) conductors 1-6, LY/|V| (deg) (deg) (deg) (deg) (kv) (kv) (kv) (kv) 0 0.0/ 76.20 -120.0/ 120.0/ 76.20 76.20 500 0.0/ 76.20 -120.0/ 120.0/ -7.6/ 76.20 76.20 2.52 1000 0.0/ 76.20 1500 2000 2500 3000 0.0/ 0.0 -120.0/ 120.0/ -6.1/ 76.20 76.20 4.75 0.0/ -120.0/ 120.0/ - 4 . 5 / 76.205 76.205 6.69 76.205 0.0/ -120.0/ 120.0/ -2.65/ 76.21 76.21 76.21 8.35 0.0/ -120.0/ 120.0/ -0.55/ 76.21 76.21 9.73 76.21 0.04/ -120.4/ 119.6/ 1.9/ 76.21 76.21 10.84 76.21 Appendix C2.2 System Real and Magnitude and Argument f o r 0.0/ 0.0 ^ower^ P/Q. (nuy.a.) 6 (deg) (kv) 0.0/ 0.0 0.0/ 0.0 -54.6/ -125.6/ 0.61/ 3.41 1.41 -0.18 -51.6/ 6.85 -48.7/ 10.35 -45.7/ 14.00 -42.7/ 17.70 -120.5/ 1.22/ 3.13 -0.37 • -116.4/ 1.83/ 5.20 -0.55 -112.9/ 2.44/ 7.61 -0.73 -110.1/ 3.05/ 10.41-0.92 -40.1/ -107.7/ 3,.66/ 21.60 13.60 -1.10 Underground Conductor System, No Load Current, Distance Current: L (m) (deg) ^AT Magnitude and Argument f o r conductors 1-7, / i / l l l (deg) (deg) (deg) (deg) (deg) (IT (A) (I) (I) (AT 0 . 0 / - 1 4 . 7 / -II6.9/122.1/ 0.0 17.8 16.9 17.0 (deg) (A) 0 0.0/ 0.0 0.0/ 0.0 500 5.7/ 2.75 -98.6/ 2.94 142.6/ -14.6/ -116.9 2.76 1 4 . 8 14.1 122.0/ 84.4/ 5.25 1000 5.7/ 5.50 -98.6/ 5.87 142.6/ -14.6/ -116.9 122.1/ 5.50 11.85 11.27 11.35 84.25 5.35 5.7/ 8.23 -98.6/ 1500 142.6/ -14.7/ 8.28 8.88 -116.9/'122.1/ 8.44 8.51 84.1 5.45 2000 5.7/ 10.99 -98.6/ 11.75 142.6/ -14.7/ -116.9/122.1/ 5.92 5.63 5.68 11.05 84.0/ 5.55 -98.6/ 2500 5.7/ 13.71 142.6/ -14.7/ -116.9/122.1/ 2.82 2 . 8 4 83.9/ 5.66 5.7/ 16.45 -98.6/ 3000 8.80 14.67 17.60 13.80 2.96 142.6/ 16.55 0.0/ 0.0 0.0/ 0.0 14.2 0.0/ 0.0 84.5/ 5.15 83.75/ 5.76 71 Appendix C2.3 Underground Conductor System, F u l l Load Voltage and Power System Real and Distance Voltage: Magnitude and Argument f o r Reactive conductors l-6,.JV/lVl Power L (deg) (deg) (deg) (deg) P/Q (deg) (deg) kv v v v (m.v.a.) kv (m) kv 0 500 1000 1500 2000 2500 3000 0.0/ ' -120.0/ 76.20 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 Appendix 02.4 Distance -120.0/ 76.23 -120.0/ 76.26 -120.0/ 76.29 -120.0/ 76.33 -120.0/ 76.36 -119.75 76.4 120.0/ 61.8/ 76.23 62.4 120.0/ 62.1/ 76.26 124.8 120.0/ 62.2/ 76.29 187.0 120.0/ 62.3/ 76.33 250.0 120.0/ 62.4/ •76.36 312.0 120.15/ 62.6/ 76.4 374.0 -47.2/ 65.5 -47.0/ 131.1 -46.9/ 197.1 -46.8/ 262.5 -46.6/ 328.5 -46.4/ 395.0 -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 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 2500 -25.2^ -144.90/ 448.0 446.5 -25.10/-144.60/ 450.0 448.0 3000 -24.9/ -144.25/ 452.0 450.0 95.80/ 449.0 2000 0.0/ 0.0 90.05/ 43.82 176.15/ 63.0 176.20/ 124.9 176.40/ 188.9 176.60/ 251.3 176.75/ 314.5 176.95/ 376.6 90.67/ 43.70 91.29/ 43.59 91.91/ 43.48 92.53/ 43.36 93.15/ 43.25 93.77/ 43.13 Magnitude and Argument f o r Conductors 1-7, (deg) (deg) (deg) (deg) (deg) A A A A' A 0 1500 0.0/ 0.0 Current: (deg) A 1000 0.0/ . 0.0 Underground Conductor System, F u l l Load Current L (m) 500 120.0/ 76.20 95.40/ -14.7/ 447.0 2.84 0.0/ 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 f o r 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. B e l l Systems Technical Journal, Vol. 8, 1929, pp. 94-98. 5. Weinhach, M.P., E l e c t r i c Power Transmission. MacMillan Co., New York, 1948. 6. Marcus, M., Basic Theorems i n Matrix Theory. U.S. Dept. of Commerce N.B.A., App. Math. Series, page 3, d e f i n i t i o n 1.7. 7. Carson, J.R., The Guided & Radiated Energy i n 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. A p r i l . 1961. P. 165. 9. Marcus, M., Basic Theorems i n 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. A f r i a t , 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. 14. Kron, G., Tensors f o r C i r c u i t s . Dover Pub. Inc., 1959,.Ch. 3, pp. 22. Ramo, S., and Whinnery, J.R., Fields & Waves i n Modern Radio. "John Wiley & Sons Inc., New York, I960. 15. Woodruff, L*F„', E l e c t r i c Power Transmission. J. Wiley & Sons, New York, 1946. 16. Zabdrszky, J., & Rittenhouse, J.W., E l e c t r i c Power Transmission. The Ronald Press, New York, 1954.
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Matrix analysis of steady state, multiconductor, distributed parameter transmission systems Dowdeswell, Ian J.D. 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 Issued | 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 |
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 | 10.14288/1.0093743 |
URI | http://hdl.handle.net/2429/37721 |
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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