IMPEDANCE CALCULATION OF CABLES USING SUBDIVISIONS OF THE CABLE CONDUCTORS by Kodzo Obed A b l e d u B.Sc.(Hons.), U n i v e r s i t y o f S c i e n c e and Technology, Kumasi, 1976 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES (Department o f E l e c t r i c a l E n g i n e e r i n g ) We a c c e p t t h i s t h e s i s as conforming to t h e r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1979 (c) Kodzo Obed A b l e d u , 1979 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 agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree that permission f o r extensive copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be 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 n o t be a l l o w e d w i t h o u t my written permission. Department n f £ U F C T K ( ^ * U - The U n i v e r s i t y o f B r i t i s h 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date S€PTe/Wfe<S« 6 K G t r ^ R i H q Columbia ^ , ABSTRACT The impedances o f c a b l e s =are some o f t h e parameters needed f o r v a r i o u s s t u d i e s i n c a b l e systems. I n t h i s work, the impedances o f c a b l e s a r e c a l c u l a t e d u s i n g the s u b d i v i s i o n s o f t h e c o n d u c t o r s ( i n c l u d i n g ground) i n t h e system. Use i s a l s o made o f a n a l y t i c a l l y d e r i v e d ground r e t u r n formulae t o speed up t h e c a l c u l a t i o n s . The impedances o f most l i n e a r m a t e r i a l s are c a l c u l a t e d w i t h a good degree o f a c c u r a c y b u t m a t e r i a l s w i t h h i g h l y nonlinear p r o p e r t i e s , l i k e s t e e l pipes, give large d e v i a t i o n s i n the r e s u l t s when they a r e r e p r e s e n t e d by t h e l i n e a r model used. The method i s used t o study a t e s t case o f i n d u c e d sheath c u r r e n t s i n bonded sheaths and i t g i v e s v e r y good r e s u l t s when compared w i t h t h e measured v a l u e s . i i i TABLE OF CONTENTS ABSTRACT TABLE OF i CONTENTS i LIST OF TABLES LIST OF ILLUSTRATIONS i i i • V v i ACKNOWLEDGEMENTS v i i i LIST i 1. OF SYMBOLS INTRODUCTION 1.1 x 1 A B r i e f Review Impedances of Methods for the Calculation of Cable 1 o 2. 3. 1.2 Skin 1.3 A Brief 1.4 Scope THEORY and P r o x i m i t y Effects Explanation of of t h e Method of Subconductors 4 the Thesis AND T H E 5 F O R M A T I O N AND S O L U T I O N 2.1 Subdivisions of 2.2 Assumptions 2.3 Loop 2.4 Formation of Impedance 2.5 Bundling of the Subconductors 2.6 Reduction of the Large 2.7 The C h o i c e 2.8 Including OF E Q U A T I O N S 6 the Conductors ^ ^ Impedances of Subconductors ^ Matrix i n t h e Impedance Impedance and C o n s t r a i n t the Constraint * ' ' on the Current Path 15 i n the Matrix Solution- , 3.1 Return i n Neutral Conductors 3.2 Return i n Ground Only 3.3 Return i n Ground and N e u t r a l 3.4 Use of A n a l y t i c a l Equations " Matrix on t h e R e t u r n RETURN PATH IMPEDANCE Matrix 13 20 Only 20 20 Conductors f o r Ground 22 Return Impedance • • . 2 2 iv 3.5 3.6 3.7 3.8 4. 25 R e p r e s e n t i n g the Ground as One U n d i v i d e d Conductor Model I I I 26 Mutual Impedance between a Subconductor and Ground w i t h Common Return i n Another Subconductor 28 Comparison o f Model I I I w i t h the T r a n s i e n t Network A n a l y z e r Circuit RESULTS 4.1 5. Model U s i n g Ground Return Formulae D i r e c t l y w i t h the Subconductors - Model I I 31 33 Comparison of the Method o f Subconductors w i t h Standard Methods 33 4.2 Comparison o f Ground Return Formulae 37 4.3 Comparison o f R e s u l t s from the D i f f e r e n t 4.4 Reproduction of Test Results 49 4.5 P i p e Type Cables 54 CONCLUSIONS LIST OF REFERENCES Models . . . . . . 47 63 64 V LIST OF TABLES TABLE 4.1 Variation of 4.2 Variation Number o f Proximity o f t h e I m p e d a n c e o f t h e C i r c u i t o f F i g . 4.3 S u b d i v i s i o n s , Showing the I n c l u s i o n of Both Effects i n the Calculations 4.3 Self Impedance Impedance of with Ground Ground and Other Formulae 4.4 Mutual Impedance Between 4.5 Comparison 4.6 Induced 4.7 Impedance of Various Currents of Pipe Return Path of Subdivisions 34 with the S k i n and 37 as C a l c u l a t e d Using Two U n d e r g r o u n d Subdivided Conductors ^ Models i n Bonded Type t h e Number 0 Sheaths Cables ~,-> for Various Degrees of Magnetic Saturation 4.8 4.9 - ~ J Zero Sequence Impedance Measurements i n a Pipe with Pipe Return Impedances Concentric - on Three Cables of Cables i n Magnetic Pipes Represented Pipes of D i f f e r e n t P e r m e a b i l i t i e s Enclosed J o a s Two ^ vi LIST OF ILLUSTRATIONS FIGURE 1.1 Current d i s t r i b u t i o n i n s o l i d round conductors due to skin and proximity effects 3 Current d i s t r i b u t i o n i n the subdivided conductors of the model 5 2.1 Subdivision of the main conductors 6 2.2 C i r c u i t of two subconductors with common return 7 2.3 Geometry of Subconductors 8 2.4 I l l u s t r a t i o n of the reduction process. 2.5 A two-w;ir'e r e t u r n ' c i r c u i t 2.6 A two-wire c i r c u i t conductor 1.2 £, k, q. 15 : with, common r e t u r n 1'6 in a third _ 16 3.1 Subdivision of ground into layers of subconductors 21 3.2 Model with, only subconductors and ground return 3.3. Model with ground represented as only one conductor 27 3.4 A c i r c u i t of two conductors with common ground return 29 3.5 A c i r c u i t of one conductor and the ground with common return .26 in another conductor 29 4.1 A return c i r c u i t of two conductors far apart 4.2 Variation of impedance with the number of subconductors 35 4.3 A return c i r c u i t of two conductors very close together 35 4.4 V a r i a t i o n of the impedance of a buried conductor with depth of b u r i a l Cross sections of buried conductors for ground return impedance calculations • 4.5 - . . . 33 3'8 40 4.6 Comparison of calculated s e l f impedances of a ground return loop • 43 4.7 Comparison of calculated mutual impedances between two buried conductors 46 4.8 E l e c t r i c a l layout of the induced sheath current test 50 4.9 C i r c u i t diagram of the induced sheath current test 51- vii 4.10 V a r i a t i o n o f magnetic p e r m e a b i l i t y o f s t e e l p i p e w i t h c u r r e n t i n the p i p e 57 4.11 Shape o f m a g n e t i z i n g curve d u r i n g one c y c l e 57 4.12 L i n e a r i z e d m a g n e t i z i n g curves 62 ACKNOWLEDGEMENTS I would l i k e t o express my thanks t o my s u p e r v i s o r , Dr. H.W. Dommel, f o r h i s h e l p suggestions throughout t h i s work and f o r the t i m e l y and c o r r e c t i o n s he made. A l s o , I w i s h t o convey my g r a t i t u d e to Mr. Gary Armanini of B r i t i s h Columbia Hydro and Power A u t h o r i t y , f o r making h i s r e p o r t and t e s t r e s u l t s a v a i l a b l e f o r use i n t h i s work. I am a l s o very g r a t e f u l to the Government of the R e p u b l i c o f Ghana f o r f i n a n c i n g my e d u c a t i o n For reading a t the U n i v e r s i t y o f B r i t i s h through and c o r r e c t i n g the s c r i p t s , I would to thank Ms. M a r i l y n Hankey o f the F a c u l t y o f Commerce. b e a u t i f u l l y done by Mrs. Engineering; I Columbia. Shih-Ying do a p p r e c i a t e like The t y p i n g i s Hoy o f the Department o f E l e c t r i c a l i t v e r y much. ix L i s t o f Symbols B flux density D„ £q D P d i s t a n c e between conductors £ and q e =2.71828 p i p e diameter f frequency g s u b s c r i p t denoting GMR geometric mean r a d i u s h depth o f b u r i a l o f conductor i, I I p current pipe current j complex o p e r a t o r i,j,k,£,n /-I subscripts km kilometres m = /(jyu/p) m metres M = ground inductance q s u b s c r i p t denoting r,R resistance, radius v,V voltage X reactance Z impedance return path K , Bessel log Common l o g a r i t h m £n Natural logarithm Hz hertz U a b s o l u t e p e r m e a b i l i t y o f f r e e space = 4irx 10 ^ H/m Q functions (base 10) (base e) X y y tj) =viQy r , relative permeability permeability flux ¥ flux linkage ir =3.1415926... y =0.5772157... ft ohm (JJ =2lTf = Eulers Constant 1 Chapter 1.1 A Brief Review For input propagation lines of Methods the analysis parameters studies which of D.M. a r e found calculations Schelkunoff f o r the Calculation transmission line the lines. conductors reasonably cable Simmons systems [1] h a s done systems. the basic surge effects lines, pipes between and data. analysed by many are often analysis of used For single-cored a comprehensive one o f studies, i n the publication and which Impedances induction as other been Cable Fault impedance have resulted for distribution (such accurate i n many h a n d b o o k s [2] of of systems, and the c a l c u l a t i o n of mutual a l l require work INTRODUCTION i s the impedance Underground The of and p a r a l l e l adjacent fences) 1 authors. standard i n charts impedance (coaxial) and h i s r e s u l t s cables, are widely used. Carson derived turn. the equations Smith used effects. and of many Another [3], of the formulae used [ 2 2 ] , who h a v e has been of specific shape, which approach used and others used effects, i n this also ground r e - have calculated namely: ground) systems. calculations, the impedances accounts have i n distribution i n impedance (including automatically i s also [4,5] [9] cables with used by C o m e l l i n i , calculated a l l the conductors and Wilcox underground cables f o r two i m p o r t a n t approach Wedepohl et a l . concentric neutral dividing This of Lewis, by above. [19], f o r the impedance to correct Talukdar Pollaczek and Barger impedances In are [10], into skin and et a l . of factors proximity [7] and Lucas transmission lines smaller conductors f o r t h e two e f f e c t s thesis. Cables with mentioned sector 2 shaped conductors uniform or properties conductors across the of any cross irregular section cross s e c t i o n or of non- can e a s i l y be h a n d l e d with this method. Skin 1.2 and Proximity The r e s i s t a n c e of determined because the from direct wire. In distribution by the the of "skin case of of over the in the direct wire current, cross the of to distributed alternating phenomenon, of in neighbouring these bution in effect, other the is and across there s e c t i o n of conductor. a This to or the flow conductor. circular). either opposite The conductors symmetrical is on current the the type of cross exists a is easily material section of nonuniform conductor which phenomenon is is caused called in the causes This around In conductors a sides a distortion axis two-wire of the of arises c l o s e by. distortion, the effect", line, Changing in the unlike symmetry for conductors due of due the instance, which face the currents current that to to d i s t r i skin conductor of Figure s k i n and 1.1. proximity effects in round (if more current each other sides. phenomena illustrated called "proximity current-carrying first not conductor tends are uniformly current presence on transmission line effect". Another the is current variation a p h y s i c a l dimensions current the Effects conductors 3 ^> Current Distributions ( a ) Skin Effect Figure (b) P r o x i m i t y E f f e c t Current 1.1 distribution proximity The conductors amount the of uneven causes direct self internal linkage more current, in distribution power thus much w i t h both - of extent impedance the while pronounced for current the inductance conductor with current additional higher The very solid round conductors due to skin and loss across above i n c r e a s i n g the the that cross s e c t i o n of produced effective by a.c. an the equivalent resistance of conductor. The the in effects to inner the part which of towards of the the conductor conductor. This surface reduces decreases the conductor. depends size density the above o n how the proximity effects pronounced conductor alter they and w i t h effect depends closer spacings between mainly the are. the on values Skin effect frequency the conductors). of - geometry i t the varies increases (being Bessel to alternating functions current widely found drical conductors effect is more derived [17] in in most hand close in the at higher 1.3 thesis seeks of the is to Figure correct makes cables both of Method the Enrico work take of into both any of into the of these main the above Dividing the conductors 1.2. the cables and cylin- hand, are in proximity factor tables customarily size proximity and are effects frequency surge et a l . account and used usually important especially studies. [7] it is shown simultaneously This of is Subconductors is done by subconductors, The and work in that by by the main finding bundling described i t calculating dividing c y l i n d r i c a l shape, conductors. on approximate power of method other [18] resistance This correcting large switching subconductors impedances are in effect. at Comellini effects this and transmission line. smaller impedance based of the and skin even the formulae for Explanation and m u t u a l give to for impedance self complicated coaxial On Charts needed to as [1,9]. analyze. increases analytically. e x p e c i a l l y when impedances, frequencies calculate skin effect underground c a l c u l a t i o n of conductors to This to analyzed otherwise together. possible the difficult general, A Brief to being calculations In is due used literature, are from In laid the are in the them to this reference. current into parallel cylindrical distributions shown in subconductors Figure 1.1 by those Conductor s Approximate Current Distributions (b) Proximity Effect (a) Skin Effect Figure 1.2 Obviously w i l l depend on Current d i s t r i b u t i o n of the model the accuracy the degree This Thesis of to be in subdivided expected discretization, from such and hence conductors a on representation the number of subconductors. 1.4 Scope of The developed in the porated theory using model. into computing calculating fictitious •Analytically the model to 'return derived reduce the the impedances path' ground number which return of from allows subconductors more formulae are subconductors, type layers permeability of cables pipe depending on are modelled material, the degree with of by treating each layer saturation. the steel having a is flexibility then incor- storage time. Pipe concentric a for pipe as different and 6 Chapter T H E O R Y AND T H E 2.1 Subdivisions In a main and the armour. the described below, Each main subconductors values at higher before these makes 2.1 present, tried show suggests can be used w i t h Subdivision the cable a number considered conductor of parallel a cylindrical formulae by Lucas simple. and Talukdar a large that i s the neutral into inductance c a l c u l a t e d by them shapes Figure have been of The c h o i c e o f the derived which core i s divided (Figure=2.1). frequencies, other each and i f conductor f o r the subconductors resistance values EQUATIONS Conductors as i s the sheath, the subconductors shapes F O R M A T I O N A N D S O L U T I O N OF the the model conductor, cylindrical for of 2 deviation further Other [22] b u t from rese.arch shape measured i s needed confidence. of main conductors Assumptions 2.2 It i) ii) Each i s assumed subconductor The magneitc that: i s uniform permeability of and homogeneous throughout a subconductor i s constant i t s length; throughout 7 the whole that i i i ) There iv) 2.3 A l l Loop To the two path, or a of in is of uniform current are derive loop the formed by current, but may be different from subconductor; of Figure alternating other Impedances fictitious lation any of subconductors loops q, cycle distribution in each subconductor; and parallel. Subconductors impedances any 2.2. two The conductor the subconductors, subconductors, return chosen of path for the % and k, can e i t h e r voltage be first with one a consider common return of the subconductors measurements and the calcu- inductances. Figure 2.2 Loops formed common Writing the loop by two subconductors with a return equations for subconductor Z gives: (2.1) where = voltage R^ i drop per unit length of = r e s i s t a n c e per unit length of = resistance per unit length of , i , = currents in subconductor subconductor return subconductors path £ and k respectively AC JO M = mutual inductance between loops formed by subconductors J6K. £ and N current phasor = number For ac in (2.1) value jcol. i k with of return steady-state are Figure 2.3 path. To subconductor common subconductors Figure the a £ and shows (N=2, by the Geometry the derive returning in for conditions, replaced 2.3: return the in cross of £, the phasor k in Figure 2.2) instantaneous values subconductors s e c t i o n of inductance q. q two formulae, V and voltage I and v and by the subconductors and (£,k,q). such consider current I in The flux B Total flux per density radius length * linkage in the elemental of loop k^ due to q(equiv) of loop k^ due to calculation (=r^e The two yldr _ _yl_ 2irr 2TT I in i i s : (2.4) 2 return r current I in q is: D. (2.5) q(equiv)- q( Q ly) the u equivalent ^ r q ^ ^ , fluxes are the radius of subconductor geometric mean additive; hence ZTT r q for inductance radius) the total flux linkage &q *£k - The mutual t o } inductance M£k where 2V £k yr^=relative <5r * Kk dr kq r current r=D £k e thickness (2.3) yi linkage of 6r 2irr r=D cylinder Sr yi where £ is: (2.2) 2Trr r = r outside .2irr 6<j> = B Flux r yi = unit Flux B at £k (M£^) y between £n 2T: permeability Iq loops kg of .the q exp(-y /4) rq & and k is + .IS. return path. i s : (2.6) ' therefore (2.7) To current derive the self linkage of £q r=r linkage loop t h e same , pi . r£ = ~ ££ 2 and p (2.8) I returning i n q i s : £q pi „ (2.9) q(equiv) k t linkage i s : £q 7 ^ £ £n IT./ , r£ . exp( - r - ) are the relative rq i n £ i s : £(equiv) q(equiv) flux I 13_ £n 2TT £^ d u e t o c u r r e n t D u ul • £ — d r = ^— £n 2irr 2TT the total * where £ we c o n s i d e r £^ due t o c u r r e n t £(equiv) £q r=r loop uldr 2irr of D Hence of loop path. Flux Flux inductance ^ exp ( ) (2.10) ZTT permeabilities of subconductors £ and respectively The self inductance ( M ^£) of D M 2.4: = ££ h i Formation of Writing the subconductors n = 27 £n £ Impedance the loop gives £q D loop £q £ i s y r£ , y rq_ (2.11) q Matrix equations, using a s e t of l i n e a r (2.1), equations (2.7) and (2.11), for a 11 V J 11 Z In llln llll Z lnll Z lnln lnkl llkm 11 Inkm In Z ^lkl*" Z (2.12a) V. kl J Z km i.e. [V] = [ Z where b ± g "klkm klll kmkm Z kmll kl ' • • kmln km (2.12b) ] [I] th V.. refers to the voltage on the i subconductor of the Ji .th . , „ 2 main conductor. I., 1 1 i s the current i n the i * " * subconductor of the j*"* main conductor. Z.. jimn i s the mutual impedance between the loops formed by 1 1 1 1 1 the i * " * and n*"* subconductors of the j " * and m"^ main conductors respectively. i The p a r t i t i o n i n g of the matrix [ Z ^ g ] n (2.12a) groups the equa- tions of the subconductors within each main conductor together. The resistances and inductances i n the impedance matrix [ Z ^ g ] are constant, but since the current d i v i s i o n among subconductors changes with frequency, skin and proximity effects are accounted f o r . Normally the large impedance matrix i s not of d i r e c t i n t e r e s t . Instead, the matrix giving voltages on the main conductors i n terms of the 12 currents the i n these large impedance equivalent of (2.13) tribution which, to of below. equations the impedances to of can be obtained Mathematically, '(2.12a) Practically, i n a l l subconductors voltages i t with the this (2.12a), a current satisfies i s conditions i s equivalent achieve from to redis- distribution the voltage on the subconductors forming any main conductor are hence: = V.0 J2 the current subconductors I. J = = V. = V. jn j i n any main into which!.it = I... + I + J l J2 Bundling currents in ] by reduction. shown by which (2.13). V., J l 2•5 i s needed, the algebraic currents The Also, solving and (2.14) of conductors matrix when m u l t i p l i e d condition equal; main of (2.13) conductor i s divided. . . . the currents i n the Thus: + I. (2.14) jn the Subconductors i n the Impedance Expressing the voltages they i s accomplished by carry i s t h e sum o f on the main Matrix conductors the use of i n terms equations of (2.13) the and (2.14) (2.12a). Consider subconductors (a) the f i r s t as shown Subtracting subsequent leaves (- i n main conductor; assume i t i s subdivided into n (2.12a). the f i r s t equations the left-hand illustrated i n equation ( i . e . row-1 i n 2.12a) ( i . e . row-2 side (2.15)). of to row-n) the other of that equations from main equal the conductor to zero 13 (b) By w r i t i n g conductor to first I instead ( i . e . row-1) , equation Corresponding 1^^ an e r r o r has been errors i n the f i r s t made of since are introduced into a r e removed by whole matrix [Z, the subsequent from subtracting of ^12 + the + ^ilYlJ'H I-^I^l errors . ] bigJ equation adding These main a l l the other the f i r s t (n-1) ^j_n' + equations. column columns ^im^ln^ + ••• ••• + first of of the that conductor. These (c) of two s t e p s The same are i l l u s t r a t e d i n equation steps, conductors. (a) These : i n g the voltages and ( b ) , give (2.16). are c a r r i e d out on the other a set of linear on the conductors equations i n terms of main (2.15) express- the t o t a l currents til in " these l ' v conductors Z 0 l l l l ? and i n t h e 2nd t o n- 1112- ? S.211 0 ? \ Z due to ; - ' ; ; l l k l Z • • C llkm V hi • lnln< (2.15) > k l l l z k l k l ? klkm \ * 0 (2.15) | • 0 The lllnj subconductors. kmll ' symbol " 5" f denotes the operations C kmqn = Z kmqn (a) —Z (for the elements and ( b ) , klqn — Z m,n^l) ^kmkl ' • *°kmkm which have been and the general kmql changed i n term i s : (2.16) 14 2.6 Reduction The exchanging equations in of the Large equations (2.15) the positions of (2.15) Impedance of Matrix are rearranged rows and columns corresponding f o r the reduction i n such to the main a way t h a t conductors come the by "bundled" f i r s t , as shown (2.17) 1 V, k 0 J l l l l J l l k l J k l l l J klkm = 12 0 or process (2.17) km i n abbreviated form, V A B 0 C D (2.18) From Hence (2.18) ,-1, V=(A-BD C)I the desired [Z„] impedance = Reference (2.17). Using (2.19a) [A - [8] Gaussian last row and going been reduced BD-1C] [Zc] i s (2.19b) provides a more elimination up u n t i l to zero, matrix efficient on t h e m a t r i x the submatrix achieves the [B], reduction. way o f finding (2.17), as shown [Zc] starting i n (2.18), from from the has just 15 The stored in d e s i r e d impedance m a t r i x [A*] in [Zc]'corresponds to the submatrix (2.20). 1 V, k . (2.20) 0 12 _ 0 An i l l u s t r a t i o n of Figure 2.7 The ^ 2.4 Choice Illustration and Obviously, 2.2 of w i l l the path influence return s h o u l d be Figure 2.5. this path is zero. final reduction of the Constraint the geometry the values removed To the on and obtained by I reduction the l o c a t i o n of for requiring illustrate Return this, the that km stage J is in Figure process Path the return inductances. the consider shown current the path The in influence through circuit Figure shown this in 2.4. Figure 2.5 Writing V l = ( R t w o - w/i r e r e t u r n , A the loop 1 equation for F i g . 2.5 gives: + h + c i r c u i t ( R 2 ^X22-X12> + 2 h 21 <' > s i n c e 1^=1 V Introducing For l " ( R 1 + R 2 + a fictitious equivalence Figure J ( X 11 return + X 22 ~ path o f t h e two c i r c u i t s 2.6 A, • t w o - w . i r e . in a t h i r d 2 X 12 gives 2 h } ) a configuration of Figure.2.6. we r e q u i r e : c i r c u i t with, conductor 22 <' > common r e t u r n 17 The loop e q u a t i o n s V a = (% o f Figure,2.6 a r e : + J ^ - X ^ ) ) ^ + <R + i(X + (R + q j ( X q q -X l q ))I + (X q 1 2 -X 2 q ) I f e > V, = ( R 2 2 2 -X 2 q ))I b q + i(X -X q q 2 q ))I q + (X 1 2 -X l q ) I (2.24) a Imposing the c o n s t r a i n t t h a t the c u r r e n t i n the r e t u r n path i s zero means t h a t I I q = - I a From e q u a t i o n s V = 0 = I I + (2.25) b (2.26) b (2.24) and (2.25) - V a a b = [R + j ( X x ( X 1 X - 12- 2q Using 1 - X ) ] l q ) - ( X 1 2 - X l q ) ] I a - [R + 2 j(X 2 2 -X 2 q ) (2 27) h - (2.26) i n (2.27) g i v e s V a _ V b = [ R 1 + R 2 + ( X J H + X 22" 2 X 12 ) ] X ( a 2 # 2 8 ) which i s i d e n t i c a l to (2.22) d e r i v e d u s i n g F i g u r e 2.5. Therefore, of any convenient i t i s t h e o r e t i c a l l y p o s s i b l e t o choose a r e t u r n path shape and l o c a t i o n f o r the i n d u c t a n c e c a l c u l a t i o n s as ... long as a zero c u r r e n t c o n s t r a i n t i s imposed on such a path. considerations discussed should Nevertheless, i n s e c t i o n 3.6 would r e q u i r e t h a t the r e t u r n path be c y l i n d r i c a l i n shape, have a s m a l l r a d i u s , and be p l a c e d a t a s m a l l d i s t a n c e below the e a r t h s u r f a c e n o t f a r from the c a b l e s and other conductors. 18 2.8 I n c l u d i n g the C o n s t r a i n t on the C u r r e n t i n the M a t r i x S o l u t i o n Equation (2.20) g i v e s the v o l t a g e s on the main conductors w i t h r e s p e c t to the r e t u r n path) i n terms of the c u r r e n t s i n these (measured conductors. In p r a c t i c e , however, v o l t a g e s are measured w i t h r e s p e c t to the l o c a l ground (or n e u t r a l conductor or s h e a t h ) . I f the c o n s t r a i n t on the c u r r e n t i s i n t r o d u c e d , t h i s changes (2.20) i n t o the form: J ll J lk J k-l,l J k-l,k (2.29) V. k-1 V. J k L_- ••• k l J k-1 -I kk k-1- k Since £ I o 1=1 and I - - l = ( 2 . 3 0 ) 0 iL r l - 2 ...-I ^ t This gives: z i r z Z i k lk-1 Z lk ( 2 . 3 1 ) V. k-1 Z k-l,l Z kl Z Z kk k-l,k Z k-l,k-l Z Z k,k-l" kk k-l,k k-1 Z I f conductor k r e p r e s e n t s the l o c a l ground (or n e u t r a l conductor or the sheath) w i t h r e s p e c t to which a l l v o l t a g e s are measured, then sub- tracting the e q u a t i o n f o r V from the other equations accomplishes rC and g i v e s : this 19 V 1 -V r- k * lk-1 l l Z (2.32) V where Z -V k J . . = Z. . + Z, , IJ kk neutral Z matrix conductor or kl J k - l , k - l k-1 2Z., ik 13 The (or k-1 is the impedance m a t r i x sheath). which implies a local ground 20 Chapter In practice, transmission RETURN PATH there are three return i n neutral (ii) return i n ground only; (iii) return i n ground and n e u t r a l Return i n Neutral Each conductor, sheath, (2.7) subconductors. process, i f voltages from provided with Model 1: results phase there i n any to and ground wires only); Only conductor a r e used to or sheaths obtain i s zero pipes conductors. and i s d i v i d e d to neutral i n Ground of to i s represented as i n S e c t i o n form 2.1. the impedance can be " e l i m i n a t e d " the impedance the phase matrix on i t , or In that i t The i f formulae the reduction relates any so separate of i n the fact, process, as a matrix which currents. i n the reduction voltage the other desired, i s connected i n parallel Only, i s considered subconductors i n any lower choice decreases were path conductor. subconductors This for the return (including or neutral and (2.11) The ground layers ground pipe so d e s i r e d , another layer. Conductors The n e u t r a l s that Return cases or can also be eliminated 3.2 into conductors as a r e the c o r e s , equations conductor IMPEDANCE system: (i) 3.1 of 3 appears as shown layer by using i n Figure are chosen reasonable a s o n e moves obtained as a separate farther a depth 3.1. from equal to and i s subdivided The d i a m e t e r s to be twice because away conductor that the current the density the cables. 3300/,rp/f of of previous i n the Reasonable metres. the (a) ( b ) Figure 3.1 Subdivisions of ground into layers of subconductors 22 p = ground f = frequency The reduction the 3.3 of ftm, If is in a system in the of subconductors, section reduced arrangements Ground the in is eliminated leaving matrix. 3* 1 ( a ) the The and (b) in ground the return difference is between discussed in The as is through a set ground is of one k both such Analytical To represent the number subconductors. conductors impedance must be directly Equations were industry. derived These over flat When these to earth same the of Equations considered, to reduce for by J.R. which is i t the ground equations values for and retained return is are i t or systems better amount of to circuits of where many cables or the ground return and overhead used assume that the conductors are of ground to return in the located cables, impedances time. transmission a n d h a s an u n i f o r m underground divided computing are widely applied desired. be and are the must [11] extent in it Carson in as sub- Impedance calculate storage the into eliminated eliminated adequately, In conductors, subdivided is Ground Return return infinite equations true ground can be and n e u t r a l which conductor of a large neutrals ground conductors 3.4 Use The conductors process. mations thus Figures reduction lines 2.8, impedance and N e u t r a l return modelled conductors . into and Hz. shown in in 4.2. Return system the in as as included results section ground, process implicitly resistivity can be .. power in air resistivity. useful approxi- obtained. 23 With in overhead the ground equal cables, to the a earth's 1.0 of and Z paper is that S which l i e below when these equations l i e above the now the are ground ground applied surface at are to used under- heights burial. [10], variation surface m) images later the conductors However, these depth underground, about image calculations. In the lines, of Carson ground relatively the = (1+C) = the ground showed return small return for that for conductors impedance with the depths usual impedance (Zg) buried distance of can be below burial calculated Z° g (3.1) o where Z^ ground extend = a Reference [10] gives C 2K0 and reference [12] where K5, m& 2rrr that is in if the earth a l l directions circular factor conductor which symmetry accounts located near were around modified r = the exists, for ground the fact that surface (jm) Z 2 log(l/m) for small m (3.2) as: Ko(mr) KL(mr) ,„ m = / are to as: gives ° _ g so correction the impedance indefinitely conductor C return (3.4) Bessel internal conductor functions radius of the insulation) earth (i.e. outer radius (i.e. of the as: 24 p = ground to = 2iTf, resistivity f=frequency y = magnetic permeability of ground Carson's formula for overhead conductors cannot be used for c a l culating the s e l f impedance of underground conductors. Equations (3.1) i s used for this purpose, but the mutual impedances are calculated using the overhead formula - which i s known to give good approximations for buried conductors at power frequencies [ 2 0 ] . Equations for c a l c u l a t i n g the s e l f and mutual impedances of underground conductors have also been derived by F. Pollaczek involving i n f i n i t e series [19]. of underground Closed-form approximations to the s e l f and mutual impedances conductors v a l i d for a wide range of values of the parameters involved have been derived by Wedepohl and Wilcox [9]. These equations (3.5), given below, are accurate up to frequencies of approximately 160 KHz for separations of approximately 1.0m between the conductors, and to approximately 1.7 MHz i f the separation i s only 30 cm. Thus very accurate approximations can be obtained for most p r a c t i c a l cases of cables l a i d i n the same trench to quite high frequencies. These equations are: Zs = ^ { -An M (YmD Z i k = *f£ i- £ n + T2 " T3 n^} fl M } + J , " f m£ } (3.5a) ft/m (3.5b) where Z , Z-y^ are s e l f and mutual impedances of ground return path respectively, s (ft/m) Y = Eulers constant = 0.5772157 h = depth of b u r i a l of conductor (metres) I = sum of depths of b u r i a l of conductors i and k (metres) 25 r = outer D M IK. radius of conductor = d i s t a n c e between (metres) conductors i and k (metres) m = /jtju/p p = earth Equations |mD_^J (3.5) resistivity are valid < 0.25 for mutual For the range i n fim f o r the range |mr| ik impedance and impedance. |mD., | > 0 . 2 5 r e f e r e n c e [9] IK. -£/(a2+m2) J < 0.25 for self suggests /(a2+m2) -V + 2TT |a|+/(a +m ) 2 the integration: -£/(a2-rm2) -e exp(jax)dx 2/(a +m ) 2 2 2 (3.6) where x = horizontal modulus V= of 3.5 Model Model II: formulae loop A very treats through formulae be Using used to may b e used. Return i each Formulae subconductor (Figure case. If of Directly the depths i and k of burial and uses and mutual the results which with the Subconductors the a n a l y t i c a l ground as an i n s u l a t e d 3.2) and ( 3 . 8 ) below, conductors and k. simple model which uses calculate the self (3.7) the difference conductors Ground the ground i n this equations of d i s t a n c e between conductor the available impedances. a r e needed a r e found with the ground Equations f o r power return return return ( 3 . 5 ) may frequency i n many h a n d b o o k s only, [24, 25], Figure 3.2 The z impedances i i = Z.. l where Z^ GMR^ R^ D k R i + Z^ + J ( ° - j(0.1736 1 7 3 of 6 l o are the self circuit s £MR7 + + ground return in Figure °- 4892 3.2 are: fi/km ) 0.4892) and m u t u a l = r e s i s t a n c e of = d i s t a n c e between view the of between ground mean radius 3 7 <-) ft/km the fact be conductors series are of One that i (3.8) the i and it (=0.0592 i k and ft/km) (m) (m) Conductor equations at high approximations used, respectively, (ft/km) Undivided inaccurate if path conductor conductor as impedances return conductor Ground c a l c u l a t i o n s may infinite and °ik geometric In the log = III separations subconductors 60 Hz r e s i s t a n c e of Model if g at = Representing obtain R only g 3.6 impedance + = R k and Rg Model with would be used for ground frequencies are used, or advantageous and return for costly if most wide to of 27 the elements of the matrix [Z, . 1 of equation (2.12a) (2.11). This could be calculated bxg with of the simpler a fictitious lated. return The ground subdivided the equations i n this subconductors With ground is path then case) this conductor 2 above ground, that effect but caution, unless more The ground return matrix i n to which i t are ignored. main region, frequencies c a n be shown than eddy must of only 1 In KHz this skin model be used conductor between which would above effects this introduction are calcu- ( n o t i. the ground and below. ground reference for circulate in and returns [20] i t the case of approach the results that current advantage formulae 1 the the inductances impedances currents up t o involves as one a d d i t i o n a l conductor frequency at higher pronounced into eddy i s negligible the lower answers, respect considered approach, flows In with a r e c a l c u l a t e d as shown current line. and and the mutual i f this (2.7) must gives a 500 kV very be i n t e r p r e t e d i n the ground. t h e more overhead with is 3.3 Model with ground i n one row and one column represented as only one some much complicated of (2.12a). Figure shown accurate i n the conductors that through has been effect i s the conductor the 28 In the addition, fictitious return there path. can be nearly halved values of parameter the by ~ V 1 to be (3.5), in delays for the the use of circuit l l Z 1N IN Z NN the of choice of used equation equations J distances |mD^| also " the this frequencies. loop to locating for the as centrally and Writing freedom The approximations, much h i g h e r is path. equation This thereby more of (3.5) reduces the g i v i n g more accurate complicated formula Figure Z in location 3.3 (3.6) gives: l g = v„ N V in which ground ig with equation the Z. g _ refers to The the mutual in Equation next z gl common r e t u r n (3.5). ground. J q. (3.5) Z gN z M Ng gg il !_ g. impedance between This is s e c t i o n 3.7 cannot valid shows be only how Z (3.9) N subconductor i and calculated directly when is the common r e t u r n derived using by is equations (3.5). 3.7 The M u t u a l Impedance B e t w e e n a S u b c o n d u c t o r Return i n Another Subconductor Consider The loop impedances Figure may J (3.4) i n which be w r i t t e n Hg the and Ground w i t h common r e t u r n is the Common ground. as: J 12g (3.10) Z L J 91 2 1 s 22g 29 t (ground return) *1 v» Figure A l l Carson's or the consider return is Two c o n d u c t o r s return impedance Wedelpohl'.s Now common 3.4 terms in (3.10) ¥ if Q common ground can be c a l c u l a t e d by Figure (3.5) using equations. a similar conductor circuit in i n which the 2. » : > Vn with (ground) (reiurn ) Figure 3.5 Circuit with of common one conductor return in a and the second ground conductor 30 The loop equations may b e w r i t t e n -112 J as: lg2 (3.11) J The third subscripts The term and (3.5) are equivalent gg2. i n equations (3.10) and (3.11) Z. =Z . , „ i s t h e o n e o f i n t e r e s t lg2 gl2 and Va = \ Vb = -V I - 2 ~ V The c i r c u i t s return. of Figures (3.4) (3.12) 2 (3.13) + g Substituting (3.14) IX) these •v - v" x t h e common i f 2 - ( I here. denote 2 into Z l l g equation Z 12g (3.10) gives 12g Z Z I, 1 22g (3.15) - " V 1 V 2 - -Z - V 2 Z llg Z 22g + Z -Z 12g 22g~ 2 Z 12g Z 22g -I 22g Z g -I, 1 12g (3.16) From (3.12) identical, and ( 3 . 1 3 ) , 12g i s evident J that 22g The lg2 mutual Z 22g Z equations (3.11) and (3.16) are (3.17) 12g impedances be c a l c u l a t e d u s i n g (3.17), (Z^g) (note required that Z. i g common g hence: Z fore i t Z return). i n equation (3.9) can there- =Z. q i s the f i c t i t i o u s i where g q 31 Thus (3.9) one results column. ground i n the use of If return II since of the matrix 3.8 using equation series impedance must Comparison of It the method would III should be noted that (TNA) [26], where decoupled from therefore e l i m i n a t e s the need the phases. On a t h r e e A V A V = Z ca z 3 A V Z ab Z bb Z cb ab the procedure b AV c aa = only one row and forms be f a s t e r phase Network for of than every the model element then Analyzer i n Model III of the ground i n c l u d e d as an e x t r a the ground return Circuits i s related transmission lines the impedance Z Z Transient return i s conductor i n every cc impedance i s ; (3.19) ) as: , -z Z ab Z bb Z cb m -Z ca mutual ca m ba (3.18) the average m -Z m ac -Z be m -Z m -Z Z m m m m m m -Z -Z Z cc I +1, a and line. ac be to on the z be -z a w i l l the Transient transposed, can be w r i t t e n ~AV integral) to be evaluated = f (z , + z, + z m (3.18) i s III for equation line, ba the line formulae of case, to model aa c that i s Z a b It phase AV Scjuation model used i n representing three Network Analyser Assuming with the matrix infinite have i n the l a t t e r Model return (or be used, the i n f i n i t e series . ] big i n forming the ground the i n f i n i t e [Z, (3.17) +i b c 32 where I a + 1^ + I t h i s case. c = I g i s the c u r r e n t i n the e x t r a conductor, ground i n The ground r e t u r n formula w i t h i t s pronounced frequency i s then o n l y used for i n the l a s t column. dependence A l l o t h e r elements Z ^ - Z a Z , -Z a r e c a l c u l a t e d w i t h ground i g n o r e d . Furthermore, ab m . 0 . 0 m , i f the l i n e i s t r a n s p o s e d , the d i a g o n a l elements Z ^ - Z ^ , e t c . , become e q u a l to the p o s i t i v e sequence impedance, and a l l o f f - d i a g o n a l elements Z -Z , e t c . , become z e r o . '33 Chapter Comparison 4.1 This account of two The by of section subdividing conductors d.c. the Method shows the placed r e s i s t a n c e of RESULTS 4 of how Subdivisions s k i n and conductors. two each metres apart, is Standard proximity The conductor with effects impedance as shown of in are taken return Figure and 0.0417 fi/km a Methods the into circuit is 4.1, calculated. frequency is 60 Hz. 2000.0mm Figure The 4.1 large A return separation effect negligible. The for using Bessel functions. used The This by for taken subdivisions value of of Z= 0.0887 as the the Figure number between increase in The the two two conductors conductors r e s i s t a n c e due UBC/BPA l i n e to for apart makes proximity skin effect constants is program corrected [16] is this. corrected is c i r c u i t of the + impedance j exact impedances 4.2 shows subdivisions. It 0.7901 fi/km reference shown the is i s : in value. Table By 4.1 using are impedance variations seen the that exact various numbers of obtained. as a function reference of values the are 34 Table 4.1 Variation No. may keep Errors 1 0.0833 0.7913 0.0377 6.1% 0.2% 7 0.0855 0.8016 0.0480 3.6% 1.4% 19 0.0878 0.7944 0.0408 1.0% 0.5% 37 0.0883 0.7923 0.0387 0.5% 0.3% 61 0.0885 0.7914 0.0378 0.2% 0.2% Reference 0.0887 0.7901 0.0365 0.0% 0.0% as t h e number of subdivisions i f an e r r o r i n storage subdivisions The close divisions in Subdivisions n/km be a p p r o p r i a t e of of Subdivisions subdivisions compared t h e Number internal ft/km of brought with X X t h e number savings use Impedance R ft/km approached to of of two c o n d u c t o r s together, i s increased. However, i t as low as p o s s i b l e . Nineteen subdivisions one p e r c e n t time 2 of forming as shown result i s tolerable. from the return i n Figure o n t h e two c o n d u c t o r s . calculations published Chapter and computing X keeping i s best Substantial t h e number of down. are used with of R charts done using and t a b l e s reference [17]. to 4.3. The standard correct circuit of Various impedances methods Figure.4.1 numbers of sub- calculated which for proximity involve effect are as are the shown 35 0-032 - NUMBER Broken Figure 4.2 OF lines Variation SUBCONDUCTORS are the reference of impedance with values. •' t h e number of subconductors i l< 27.02 mm Figure 4.3 A return According circuit above circuit to reference of [17], two c o n d u c t o r s very the a . c . resistance close of together the return i s r = R' x ~y (A.D 36 where R'=a.c. resistance R"/R' A and Tables ratios the reference calculated circuit in j the 0.1340 pares i t ft/km. In for This using effect complicated formulae of referred to division among many the is and case in most parallel in the is seem to the conductors conductors is correction as the from methods two cable same or systems in adjacent useful, and Charts r e s i s t a n c e and The skin effect inductance impedance only, of i s : the in three Table of Z=0.1048 4.2 which only need two the or from + com- division where many method reasonably is accurate of otherwise the as the the not known and or charts current a priori. cables of use hand, spacing) known corrections conductors subdivisions, be the other three delta not this by derived On or current calculations, conductors tables" using ducts), a value in impedance (flat cities gives value above. to gives subdivisions. "factor Also, and for of or arrangement involved From the respectively. reference limited be. effect factors as mentioned not ratio. inductance. various for only. ft/km p a r a l l e l conductors are very 0.95 corrected correcting [18], the proximity 0.1410 made conductor above j used are subdivisions common, f o r m s when resistance for and conventional charts" of 1.18 obtained "estimating method + skin effect effect the proximity those proximity 4.3, = 0.0887 above with be for holds [17], to Figure Z Applying = proximity similar equation of are corrected Where (as pipes subdividing results. is run the 37 Table 4.2 Variation with of of Both Skin X 0.0966 0.1474 0.0429 7.8% 10.0% 19 0.1010 0.1394 0.0349 3.6% 4.0% 37 0.1022 0.1370 0.0325 2.5% 2.2% 61 0.1026 0.1361 0.0316 2.1% 1.6% 0.0887 0.1410 0.0365 15.4% 5.2% 0.1048 0.1340 0.0295 0.0% 0.0% of return [19], these Ground effect have been Wedepohl to as g i v e n Return. only. skin use. given and W i l c o x are given b y many [9], of and mutual authors, effects. of impedances of ground the formulae, of i n c l u d i n g Carson and Kalyuzhnyi i n the form The v a r i a t i o n b y some and p r o x i m i t y Formulae for calculating the self formulae easy for skin No c o r r e c t i o n f o r b o t h Formulae frequency, R 7 * 4.3 Inclusion in 6.1% Comparison always Error 20.5% *•Corrected not X internal ft/km Fig. i n the Calculations. 0.0377 ** of Effects of the 0.1422 Value Most Showing 0.0833 Reference Pollaczek the C i r c u i t 1 SKIN ground of Subdivisions, X ft/km ** with of and P r o x i m i t y R ft/km No. of Subdivisions 4.2 the Impedance t h e Number and L i f s h i t s infinite return series [10,11], [13]. and are impedance a r e compared loops i n this with section. According with for the for I J.R. depth most Model to Carson of burial.'.'.of frequencies to by calculate various [10], depths of variation a conductor using the the equation impedance burial. of The is of ground minimal, (3.1). and This a buried results the of is can be are with shown impedance calculated verified conductor this return by using ground in return Figure 4.4. 1.171 M69 1-167 < • in IMPI o z < 1 1 Figure A been given tion of 0 DEPTH OF simple Wedepohl infinite and of which, useful and W i l c o x series K a l y u z h n y i and results 10 0-5 BURIAL(m) V a r i a t i o n of the impedance w i t h depth of b u r i a l 4.4 very by the -0-5 -1-0 1-5 though form Lifshits very form of the (equations of [13] • 8 of t-5 a buried ground (3.5)), conductor return which is impedance an has approxima- solution. also different derive from the a formula, more the final conventional ',. ones, 39 are claimed Kalyuzhnyi to and Ze where y very Lifshits = ^ = radius p = P y - of Equation which (4.2) is Re Carson's by = 2?r2f electric [17] = a . of ground data. return (Ze) as: constant conductor over insulation (m) of ground real 10_7 of part (ftm) conductor (Re) (m) of: (4.3) Q/m from that obtained from Carson's equations which to: TT2f authors current equations impedance measured (4.2) Euler's burial different equations several of gives quite approximates self experimentally <WP = depth Re with - J- r n buried p = resistivity h the -2— [An I 2TT closely give 0.5772157 = ..r agree . 10 7 (or approximations and others through g i v e n by (4.4) tt/m the involved ground. Kalyuzhnyi of them) with The have been analysing very and L i f s h i t s i s , used the marked for many conduction deviation therefore, from worthy years of Carson's of inves tigation. In of the earth, by using In this In Table the order the to impedances subdivided calculation, 4.3 determine and the Figure which of ground the circuits of representations ground 4.6, formula best the is divided results into approximates Figure of Figure five obtained 4.5 3.1 layers from are the the calculated (i.e. of self behaviour 62 Model I). subconductors. impedance Oo012m (a Figure 4.5 ) Cross s e c t i o n s of b u r i e d conductors r e t u r n impedance c a l c u l a t i o n s for ground calculations There is above method of but from more MHz.between 4.6 at b) from for power the results values from evaluate the of Figure the of the a slight 60 H z , is the impedance over of results when resentation equations, and compared. and the Wedepohl's Lifshits used. the + By equation in 12% power frequency i n between of in case and neglected approximation, subconductors: for from also in most a of fuller Figure 3.1 a. conductor in Figure 4.5 a with ft/km when 0.9236 Z the 3.1 the ground = 0.0589 gives an former. a would + using the 0.9181 Therefore the adequate fact because 3.1 the that the a. To interstices show only example, at return is representation of of Figure The less ground for 50% the ground ft/km. of (over representation For representation j Figure Lifshits. latter ground improvement be the the and Figure of j (see Kalyuzhnyi at while reactance calculations arise real to formulae results only Figure at deviation are methods values. obtained i s : ground three calculated resistance formula the latter using compared w i t h given to first somewhere the various the Results the be Z=0.0597 of from with the c a l c u l a t e d impedance resentation marked this section. a is are calculated using and Wedepohl's subconductors of 7% very approximations cross Figure between l i e the filled to the are the method obtained in than Carson's A a were calculated 3.1 of frequencies. improvement the other subdivisions influence ground as using Kalyuzhnyi c a l c u l a t e d from range results between 3.1 each reactance frequency) "interstices" obtained Carson;'s formulae others. the methods various ground, Discrepancies a l l the the of a l l using subdividing widely 1.0 by resistance values reactance deviations and the the The results I in the The deviate Model c l o s e agreement equations, deviate using this 3.1 latter than 1% return b, rep- in the rep- purpose. RESISTANCE Frequency (Hz ) Subdl-. visions REACTANCE (ft/km) Subdivisions Wedepohl Carson Kalyuzhnyi (ft/km) Wede— pohl Carson .. K a l y u zhnyi 2.2 .002 0.002 0.02 0.004 0.039 0.041 0.038 0.062 4.5 .005 0.005 0.05 0.009 0.077 0.081 0.075 0.119 9.0 .009 0.009 0.09 0.018 0.150 0.160 0.145 0.230 15.0 .015 0.015 0.015 0.030 0.244 0.260 0.236 0.374 30.0 .030 0.030 0.031 0.059 0.475 0.503 0.460 0.723 60.0 .060 0.059 0.062 0.118 0.924 0.979 0.914 1.39 120.0 .120 0.119 0.124 0.237 1.80 1.91 1.73 2.68 500.0 .499 0.497 0.518 0.987 7.03 7.49 6.79 10.3 IK 1.00 0.998 1.04 1.97 13.6 14.5 13.2 19.7 5K 5.05 5.05 5.20 9.87 63.0 67.6 60.7 88.3 10K 10.2 10.2 10.4 19.7 121.0 131.0 117.0 168.0 5 OK 52.2 53.1 52.5 98.7 555.0 601.0 535.0 738.0 0.1M 106.0 109.0 105.0 197.4 1064. 1155. 1026. 1389. 0.5M 556.0 611.0 530.0 987.0 4767. 5207. 4622. 6371. l.OM 116. 1320. 1064. 1974. 9038. 9881. 8810. 11870. Table 4.3 S e l f Impedance o f Ground R e t u r n Path as C a l c u l a t e d U s i n g S u b d i v i d e d Ground, and O t h e r Formulae 43 Figure "4.6 Comparison of ground r e t u r n calculated loop. self impedances of a 44 Mutual Impedance Table calculated 1% in the for of Ground 4.4 the and two Figure buried r e s i s t a n c e and frequency between the Return Path 4.7 show the conductors about results 15% of in results of the of the mutual impedances Figure 4.5b. reactance are obtained at those obtained from subdivisions and Deviations of about power Wedepohl's equations. The by using the used mutual Carson's frequencies for Despite Carson's impedance overhead used; thus c a l c u l a t i n g the this close equations values line it equations seems mutual [11,16] Carson's impedances agreement in [11] derived were c a l c u l a t e d by the for are overhead between results, using i t subdivisions similar line for most equations buried conductors should be remembered conductors located above and of may be [20]. that ground. Table 4.4 Mutual Impedance Between Two Underground Conductors * RESISTANCE Frequency Subdi- Hz visions REACTANCE (ft/km) Carson Wede- Subdi- pohl visions (ft/km) Carson* Wedepohl 2.2 0.002 0.002 0.002 0.024 0.024 0.026 4.5 0.005 0.004 0.004 0.046 0.045 0.052 9.0 0.009 0.009 0.009 0.088 0.087 0.099 15.0 0.015 0.015 0.015 0.142 0.139 0.161 30.0 0.030 0.030 0.030 0.271 0.266 0.308 60.0 0.060 0.059 0.059 0.516 0.506 0.590 120.0 0.119 0.118 0.119 0.979 0.959 1.13 500.0 0.4.99 0.490 0.497 3.63 3.55 4.25 Ik 0.999 0.978 0.996 6 . 8 2'- 6.66 8.06 5k 5.03 4.84 5.04 29.0 28.3 35.2 10k 10.1 9.60 10.1 53.6 52.4 66.0 5.0k<- 51.8 46.4 52.5 216.0 213.0 278.0 100k 105.0 90.7 108.0 386.0 385.0 509.0 500k 599.0 413.0 594.0 1376.0 1471.0 1983. Overhead line equations used. cc- i- t- ID X---Wedepohl a oo 100 1000 10000 FREQUENCY C Hz ,100000 1000000 ) «nO" + — Subdivisions o—Carson (0/H) X---Wedepohl no Figure 4.7 100 idoo loooo FREQUENCY ( Hz iooooo ) 1006000" I Comparison of calculated mutual impedances between two buried conductors. 47 4.3 Comparison The (1/0 of data AWG a l u m i n u m formation 0.4707 the Different i s taken from r e f e r e n c e [4] . T h r e e cables with apart. ohms/1000'f t listed impedance reduced The c o r e respectively a s 515 m i l s The sequence from cored 8 inches insulation Results values matrix of are l a i d resistances the inside cables i n a flat are 0.1882 and o u t s i d e diameters and of the respectively. the zero elements distribution neutrals) and sheath with and 955 m i l s Models at (0), 60 Hz positive (1) and negative i n the reference (2) are: 0 ~0.483+j0.236 tW • 1 -0.003+j0.001 0.0 By and using Model (3.8), symmetric II -0.007+j0.008 0.199+j0.096 -jO.003 and t h e ground the sequence ft/1000 return impedances impedance ft. 0.010+j0.004_ formulae of equations (3.7) calculated are: 0 "0.483+J0.231 [ Z 012]= 1 -0.003+j0.001 0.0' If are the ground used return i n Model impedance II, 0.198+j0.089 equations the following (3.5) impedance O.OlO+jO.002 derived matrix is 2 0.506+J0.219 -0.002+j0.001 0.0 ft/lOOOft. -0.007+j0.008 -jO.003 0 0 symmetric -jO.003 by Wedepohl and Wilcox obtained. 1 symmetric -0.007+j0.008 0.198+j0.083 ft/lOOOft. 0.010+j0.002 48 The maximum values section is deviation less 3.6 than where the following the ground between 3% i n the ground impedances return the sequence positive is are impedance the 0 1 2 obtained, r 0.510+j 0.225 ]=L -0.002+j0.001 The |_0.0 maximum none ing of are the with ] = 1 dividing The calculated for is 3.6% the from the reference: i n ground return path derived ground return comparison Table for ft. into the zero sequence. subconductors formulae (Model and I), using the follow- 1 -0.007+j0.008 ft/1000 0.198+j0.900 in this of the 4.5 case i s 1.5% comparison The is Comparison Time (s) I used purposes. Computing Model being III), symmetric -t-jO.003 summary (Model 0.010+j0.002 _ 2 deviation The (3.5) of ft/1000 0.198+J0.900 -0.003+j0.001 maximum equations model conductor -0.007+j0.008 analytically -0.0 one the 1 "0.486+j0.231 012 using reference symmetric 0 [ Z only the calculations: +J0.003 deviation By as By and 2 .0 2 sequence. represented 0 [Z impedances 10.3 ft. O.OlO+jO.002 in the positive presented of Various Max Deviation in sequence Table Models Matrix Size % 1.5 101 x 101 II 0.73 3.0 39 x 39 III 0.78 3.6 40 40 x. 4.5. magnitude. 49 4.4 Reproduction A [21] on test a cable measured for produced in sheath the phase conducted values section. are test The grounded is values the lated of from sheaths of the British induced phase Columbia currents currents. impedances both it two cables both sheath u s u a l handbook are in to and bonded test Power Authority sheaths were results calculate the are II. three-phase cables are bonded together (see Figure 4.8). in the in at The sheaths, the same their ends unbalance and the Table 4.6 is measured values and the using the impedances currents re- induced measured. the methods Hydro These needed circulating currents gives the on resistances conductor induced Writing of The high a parallel neutral and the c a r r i e d out sheaths currents, [21] at c a l c u l a t e d using Model through reference Results system where currents and in was this bank. Test various The duct of obtained taken in current from predicted calcu- [3,24]. loop equations Sl,k \ around the loop formed by the bonded gives: ° = for k= Al, 0 x = si + *A1 V Due to the the mutual X A2 Z I + + symmetry B l , X A2, Z S2,k B2, \ C2, <4'4> SI, S2, N (4.5) + B2 in impedances Cl, I S2 ^1 I + + ci x I the above (4.6) (4.7) C2 cables are and equal, the for spiralling example: of the cores, most of 50 ^ Figure I2>70 f t 4.8 E l e c t r i c a l layout , of.;the : induced j|- sheath current test. Figure 4.9 Circuit diagram of induced sheath current test. 52 Hence Z A1S1 Z A1S2 Z = equations 0 hl Z B1S1 B1S2 (4.4) I ( Z The equation induced (4.9). = Z C1S2 and (4.5) reduce -ZsiSl^l = Z~Z; C1S1 Z + ( Z A1S1+ZA1S2) ( I A1 + ( Z A2S1+ZA2S2) ( I A2 + ( Z S2N W A1S1 + Z A1S2)I1 sheath The v a l u e s f o r two c a s e s through ground resistors There of subdivisions the latter 14% between the former are also mainly cable give a r e improvements produces evident only be p o i n t e d + Z C2 ) A2S2>I2 n using total deviation an average and sheaths out that and neutral) a r e used there i n the cable were phase deviation effects of N ) I ) ] ( ' 9 ) i n the measurements not were grounded grounded. [3,5,21]. method While and magnitude 8%. 4 using Similar subdivisions results of calculations. conductors consideration. of case, The improvements when as n i n e under 8 f o r the ungrounded i n the impedance system The methods i n both a r e a s many S1N the sheaths i n the second s e t of measurements. proximity + Z ' 4 as c a l c u l a t e d by the results total 4.6. they i n the results S2N subdivisions between d when + ( Z be c a l c u l a t e d i n Table t h e u s u a l handbook the i n c l u s i o n of conductors sheaths a the c a l c u l a t e d and measured due t o should from t h e bonds R ^ and R ^> an average + I ) ( presented i n which compared w i t h B2 C1 + I can therefore calculated were + I (ZA2S1 + current calculations are also B1 + I *N + impedance made to: are the It (including Table MEASURED CURRENTS 4.6 Induced Currents i n Bonded Sheaths CALCULATED (AMPS) % Deviations Magnitude * 1 Ungrc unded 1 1 x * si hi:. 10.8|255° 13.0 260° 2 Mag. & Phase _|_ * + * + 11.8 261° 20 10'; 23 12 10.6 270° 10.0|272° 10 5 11 6 Bonds 1 25.01 4° 53.0 191° 23.0|_0° 2 25.1 [21° 41.4 198° 17.0[0° 3 23.2[32° 44.9 198° 18.4[rj° 10.0 269° 10.8 273° 10.2 275° 8 2 10 5 4 22.0 j28° 47.4|198° 20.7[0° 10.0 269° 11.0 272° 10.5 274° 10 5 11 6 5 35.0[14° 66.9)191° 29.4 [0° 14.9|255° 16.2 263° . 15.4 265° 8 3 11 7 6 30.6|353° 65.9 191° 30.4[0° 13.5(248° 15.2 256° 13 7 16 11 12 5 14 8 9.6 j 269° Average Bonds 14.4|258° % De v i a t i o n s Grounded 1 28.4[21° 56.9 191° 22.8|_0° 13.5|262° 13.5 265° 12.9|267° 0 5 1 7 2, 28.3|21° 55.1 194° 19.9\0° 12.5|262° 13.2 263° 12.6|269° 6 1 8 3 3 29.5[21° 52.4 198° 19.9 [0 ° 12.2j262° 13.0 270° 12.4|271° 7 1 10 5 4 23.0[32° 50.4 194° 21.8[0° 10.7|269° 11.5 271° 11.0|272° 7 3 8 4 5 26.4J32° 60.9 198° 25.9 [0° 13.8 273° 13.l|275° 7 2 11 6 31.7|28° 55.1 198° 18.0[0° 12.91262° 13.0j 262 ° 13.7 273° 13.0|274° 5 5 1 2 9 8 7 5 5 Average % De v i a t i o n s i * Taken from Reference [21] + calculated using subdivisions. o 54 The to the are in slow used) a l l phase improvement conductors real advantage for distribution 4.5 may Pipe Type cooling mechanical or to and the angles. true 4.2. This value This it of can be can be said may (when causes be for studies or advantage cables be due subdivisions further available, of of lagging for for slight systems having are used. impedances other method analyzed e.g., a conductors studies the only for calculating of can be that obtained subdivisions would sometimes liquids or enclosed in insulation c h e m i c a l damage. of The at higher transients in subconductors which is handbook sector-shaped . conductors. pipes gases Common c a s e s which or are as act as ducts protection found in o i l for against and gas filled installations. Pipe non-metallic materials pipes pipes are also pipes are highly this steel if Another readily are conducting In phase Cables Cables cable section, surge types be 4.1 calculations switching not Tables subdivisions other the reactance together systems. that formulae of in calculated. this close occur the in impedance many fact angles conclude in frequencies of illustrated To the deviations convergence as the main easily section, pipe conductors is and pose may no assumed pipe plastic, problems treated in aluminum impedance or steel. Plastic calculations and or aluminum due to their constant permeability. Steel far as their magnetic properties concerned. nonlinearity is not nonlinear the be as linear are and divided having into taken a into constant subconductors account and permeability. and each are the whole The cable subconductor is 55 assigned lated from the permeability according to laboratory enclosed of pipes-and pipe the currents pipe, pipe the in in results addition steel pipe deviations of to with pipe the in assumes the conductors that the is 4.11 i l l u s t r a t i o n of It curve ( i . e . , is around permeability of small. may This Deviations inner of the in surface pipe. evident the of B-H that the the the the here in magnetic an is the of because the the be - in s t e e l pipe nonlinearity of linear 200A i n this in the to model current in of actual a 5 as inch the Chapter case 2 the Figure curve. and fits saturation s i n c e more in values. portions pipe of provides nonlinear. B-H reactance linear due is in percentage model the the more 4.7 cable the give levels deviations the on variation measured the taken saturation Table unshielded is references The 4.10. calcu- Company current c a l c u l a t e d and This of The values. alongside assumed pipe on data magnetic [15]. Figure then Edison different most about may at is The of l i n e a r whereas around of and that deviations be b e c a u s e results a l l curve) currents 1000), shown seen are curve an [14] values reactance. magnetization shows (or are i t are The degrees impedance shown 2. Consolidated measurements return. calculations is impedance Chapter various measured current 4.7 at references the reactance From Table for The in permeabilities s i n g l e phase with in conducted in and m a g n e t i c material. presented steel pipes reported permeability its method experiments cables the the of of the B-H relative calculated this region effects flows are on on most. the the inside Table 4.7 Impedance Degrees Pipe Current (A) Relative Permeability of of Pipe Type Magnetic Measured R X micro-ohms; Cables for Various Saturation Calculated R | X micro-ohms /ft Errors X /ft 100 762.0 201 151 201 117 23% 150 980.0 223 155 223 137 12% 200 1018.0 218 154 218 140 9% 300 942.0 203 147 203 133 10% 480 784.0 180 141 180 119 15% 980 484.0 141 124 141 92 26% 3500 156.0 88 93 88 62 33% 7400 81.0 80 80 80 55 31% in Figure 4.11 Shape of magnetizing curve during one cycle 58 The in the pipe deviations large results are in It changes constant Good the of points The pipe of to the the the pipe may by be not results i n only of is as of measured three values. cables Larger especially when is concerned, the magnetic constant throughout the cycle the range an saturated the to the is applied i t saturation. pipe To the the from the produce which the known that or outer is data as results (or inner parts. Thus An layers 4.9 made. approximates the not. Table being current of portions different saturation. attempt and gives flux) a is assigning summary above. calculations, the is degrees concentric same assuming permeability outer layers. By error different into various obtained is middle while to inherent through the current. assumed Furthermore, than a.c. the section experience are the obtained not value if the dividing this far steel pipe, current permeabilities is the most cross this as material obtained more surface for are calculations for pipe. varies. r e s u l t s when pipe Impedance alongside 4.8 instantaneous for pipe Better the be carry model different of may the Table recalled that the values in inner made in permeability actual the in sequence a nonlinear with results density flow of zero calculated reactances should be permeability but given the currents of in especially when Table the 4.9, Table 4.8 Zero Sequence Cables Pipe Current (A) Relative Permeab i l i t y Impedance Enclosed in Measurements a Pipe with Pipe on Three Return. Calculated Measured R (micro- Dhms/ft) R (micro- ohms/ft) Errors R X 100 762.0 197 134 178 84 10% 37% 150 980.0 204 140 199 103 2% 26% 200 1018.0 200 139 194 107 3% 23% 300 986.0 189 133 180 100 5% 25% 500 767.0 165 124 158 86 4% 31% 970 488.0 141 106 121 60 14% 57% 3600. 154.0 76 71 70 32 •8% 55% 8000 76.0 57 57 55 25 4% 55% Table 4.9 Impedance as Pipe * Two of Cables Concentric Current Measured R X (A) (ufl/ft) in Pipes Magnetic of Calculated X Pipes Different Represented Permeabilities. Errors R X (yfi/ft) 100 201 151 203 150 1% 150 223 155 230 181 3% 200 218 154 227 192 4% 25% 300 203 147 212 185 4% 26% 480 180 141 188 160 4% 13% 980 141 124 146 114 4% 8% .1% :.17% 3500 88 93 89 67 1% 28% 7400 80 80 81 57 1% 29% 12% 100* 197 134 186 118 6% 150* 204 140 214 149 5% 200* 200 139 211 159 6% 14% 300* 189 133 196 153 4% 15% 500* 165 124 172 128 4% 3% 970* 141 106 129 83 9% 22% 3600* 76 71 73 36 4% 49% 8000* 57 57 56 28 2% 51% Three cables in pipe. 6% 61 pipe i s divided assigned other the into a lower current inner layer 520. Similar values Table is used i s to reflect assigned and the a of lower about 4.7 layers. i t s these relative value of 40 b e l o w and 4 . 8 The inner b e i n g more values; for a r e made for the No example, given 440 and t h e o u t e r and above i s saturated. thus, permeability layer is 484, layer the assigned corresponding the other when permeability results shown in 4.9. deviations can be seen i n the 26% w h e n constant are for used current cables is the steel current variations Figure of to i s assumed to one c a b l e A s i m i l a r drop from 57% t o is and 4 . 9 from the pipe also the as h i g h two v a l u e s in 22% i s 4.8 drop 8% w h e n only i t should the pipe values densities no m a t t e r 4.7, the reactance p i p e when could be expected permeability of i n Tables of permeability and the obtained as for pipe three pipe. permeabilities results the results permeability 980A. i n the from calculated values However, form 980A choices concentric i n assigning l i s t e d i n Tables As in two permeability criterion is pipe only i n the how w e l l during 4.12a, Figure based i t any is 4.12b. as g i v e n i f out above layers. done, is definite the s t i l l form the representation the method rather layers better the pipe are assigned such as the average of any such linearization, since the l i n e a r i z e d above assigning and be i n e r r o r of of arbitrary criterion, Furthermore, would of that is the different o n some cycle whereas be p o i n t e d seeks to the permeability sketch put i t i n shown the Figure 4.12 L i n e a r i z e d m a g n e t i z a t i o n curves 63 1 Chapter The has been used accuracy The method of to run introduction therefore parallel frequencies test ground method return takes be highly test with the model. of formulae speeds of both skin to d i s t r i b u t i o n It values will circulating currents of It that subconductors the used. up t h e c a l c u l a t i o n s . systems useful where i n bonded come effects many surge sheaths close to at higher studies. has the and conductors especially for switching results calculations i s shown and p r o x i m i t y be very a r e needed The c a l c u l a t e d f o r impedance cables. on t h e number account suited impedance case of conductors been field results. properties method across modelled Generally are of of the impedance i n close proximity. The also calculate where A studied subdivisions the c a l c u l a t i o n depends This will of CONCLUSIONS 5 assigned the cross with better i s also pipe for modelling section. layers results different suited Cables assigned are obtained permeability enclosed different when values. conductors i n magnetic values different with of layers nonuniform pipes are permeabilities. of pipe material 64 LIST [-1] [2] OF REFERENCES D . M . Simmons^ " C a l c u l a t i o n o f t h e E l e c t r i c a l P r o b l e m s o f C a b l e s " , The E l e c t r i c J o u r n a l , v o l . 2 9 , January-December, Underground 1932. S.A. Schelkunoff, Transmission Lines "The Electromagnetic and C y l i n d r i c a l S h e l l s " , Bell Theory of Coaxial System Tech. Jour., v o l . 13, pp. 522-79, 1934. [3] D.R. Smith and J . V . Barger, "Impedance and C i r c u l a t i o n Current C a l c u l a t i o n s f o r UD M u l t i - W i r e C o n c e n t r i c N e u t r a l C i r c u i t s " , IEEE Conference Record o f 1971 Conference on Underground D i s t r i b u t i o n , Sept. 1971, p p . 992-1000. [4] W.A. Lewis [5] W.A. Lewis, Neutral A [6] 77 and R.C. Ender, 243-9, G.D. A l l e n ground A.B. G.D. A l l e n Underground and J . C . Wang, Distribution IEEE Winter Power and W.A. Lewis, Distribution Chance Discussion of 3, ibid, Meeting 1001-1004. Constants on a Phase for Concentric Basis", Paper Centralia, . 1977. "The E l e c t r i c a l C h a r a c t e r i s t i c s of Concentric-Neutral Company, "Circuit Cables pp. Cables", Missouri, Book Under- published by 1976. [7] E. C o m e l l i n d , A. I n v e r n i z z i , G. M a n z o n i , " A Computer P r o g r a m f o r : Determining E l e c t r i c a l Resistance and Reactance o f any Transmission L i n e " , IEEE T r a n s , o n Power Appar. & S y s t . v o l . P A S - 9 2 , p p . 308-314, 1973. [8] H . W . Dommel ibid, and W . S . Meyer, v o l . 6 2 , No. 7, July "Computation of Electromagnetic Transients", 1974. [9] L . M . Wedepohl and D . J . W i l c o x , " T r a n s i e n t A n a l y s i s o f Underground P o w e r - T r a n s m i s s i o n S y s t e m s " , PROC. I E E , v o l . 1 2 0 , N o . 2 , F e b . 1 9 7 3 , pp. 253-260. [10] J . R . C a r s o n , "Ground Return Impedance: Underground Wire w i t h Return", B e l l Syst. Tech. J o u r . , v o l . 8 , 1929, pp. 94-98. Earth .[.11] J.R. Bell Return", [12] G. C a r s o n , "Wave P r o p a g a t i o n i n O v e r h e a d W i r e s w i t h Syst. Tech. J o u r . , v o l . 5 , 1926, pp. 539-554. B i a n c h i a n d G. Submarine No.l, [13] Luoni, Cables", Jan/Feb,1976, IEEE pp. "Induced Trans Currents o n Power and Losses Appar. & Syst., Ground i n Single-Core v o l . PAS-95, 49-57. V . F . K a l y u z h n y i and M . „ Y u , L i f s h i t s , "A Method f o r D e t e r m i n i n g t h e Parameters of 'Underground-Conductor-Earth' Circuits", Electrichestro, No. 7, 1974, p p . 2 8 - 3 6 . ] J . H . Neher, "Phase Sequence Impedance o f P i p e Type on Power A p p a r . & S y s t . , A u g u s t 1 9 6 4 , p p . 7 9 5 - 8 0 4 . ] E.R. vol. ] "Line ] Edith Clarke, "Circuit Analysis E l e c t r i c Company, 1 9 7 1 . ] A.H.M. Arnold, "Proximity Effect i n Solid and Hollow Jour, of IEE, Part I I , v o l . 88, 1941, pp. 349. ] F . P o l l a c z e k , " S u r l e champ p r o d u i t p a r u n c o n d u c t e u r s i m p l e i n f i n e m e n t long parcource p a r un courant alternatif", " R e v u e G e n e r a l De L 1 e l e c t r i c i t e " Tome X X I X , N o . 2 2 , M a y 1 9 3 1 . ] H.W. Dommel, Univ. ] Gary Thomas a n d R . H . K e r s h a w , " I m p e d a n c e o f P i p e PAS-84,No. 10, Oct. 1965, pp. 953-961. Constants o f Overhead "ELEC 553 - Lines", - Type Users m a n u a l , B P A , o f AC Power Advanced Cables", Analysis Systems", o f Power IEEE Cable", Trans. ibid, June 1972. v o l . II, Round General Conductors" Systems, Classnotes" of B . C . , 1978. Armanini, Distribution "A Proposed Systems", Grounding B . C . Hydro and 'Bonding Scheme for Underground Report. ] R. L u c a s a n d S . T a l u k d a r , " A d v a n c e s i n F i n i t e Element T e c h n i q u e s f o r C a l c u l a t i n g C a b l e R e s i s t a n c e s a n d I n d u c t a n c e s " , IEEE T r a n s . o n Power Appar- and S y s t . , v o l . PAS-97, No. 3 , May/June 1978. ] B r a n k o D. P o p o v i c , " I n t r o d u c t o r y E n g i n e e r i n g A d d i s o n - W e s l e y P u b l i s h i n g C o . , May 1 9 7 3 . ] " E l e c t r i c a l Transmission and D i s t r i b u t i o n E d i t i o n , Westinghouse E l e c t r i c Co. 1964. ] "Underground ] John G. K a s s a k i a n , "The E f f e c t s o f N o n - T r a n s p o s i t i o n a n d E a r t h R e t u r n Frequency Dependence on T r a n s i e n t s due t o High Speed R e c l o s i n g " , IEEE Trans, v o l . PAS-95, M a r c h / A p r i l 1976, p p . 610-618. Systems Reference Book", Electromagnetics", Reference Book", Textbook Fourth NELA P u b l i c a t i o n 0 5 0 ,1 9 3 1 .
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Impedance calculation of cables using subdivisions of the cable conductors Abledu, Kodzo Obed 1979
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Title | Impedance calculation of cables using subdivisions of the cable conductors |
Creator |
Abledu, Kodzo Obed |
Date Issued | 1979 |
Description | The impedances of cables are some of the parameters needed for various studies in cable systems. In this work, the impedances of cables are calculated using the subdivisions of the conductors (including ground) in the system. Use is also made of analytically derived ground return formulae to speed up the calculations. The impedances of most linear materials are calculated with a good degree of accuracy but materials with highly nonlinear properties, like steel pipes, give large deviations in the results when they are represented by the linear model used. The method is used to study a test case of induced sheath currents in bonded sheaths and it gives very good results when compared with the measured values. |
Subject |
Electric cables Impedance (Electricity) |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-03-11 |
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 | 10.14288/1.0100270 |
URI | http://hdl.handle.net/2429/21770 |
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 | Unknown |
Aggregated Source Repository | DSpace |
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