IMPEDANCE CALCULATION OF CABLES USING SUBDIVISIONS OF THE CABLE CONDUCTORS. Kodzo Obed Abledu

Transcription

1 IMPEDANCE CALCULATION OF CABLES USING SUBDIVISIONS OF THE CABLE CONDUCTORS by Kodzo Obed Abledu B.Sc.(Hons.), University of Science 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 of Electrical Engineering) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1979 (c) Kodzo Obed Abledu, 1979

2 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department n f U F C T K ( ^ * U - 6 K G t r ^ R i H q The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date S PTe/Wfe<S«^,

3 ABSTRACT 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 i t gives very good results when compared with the measured values.

4 i i i TABLE OF CONTENTS ABSTRACT i i TABLE OF CONTENTS i i i LIST OF TABLES V LIST OF ILLUSTRATIONS vi ACKNOWLEDGEMENTS v i i i LIST OF SYMBOLS i x 1. INTRODUCTION A Brief Review of Methods for the Calculation of Cable Impedances 1 o 1.2 Skin and Proximity Effects 1.3 A Brief Explanation of the Method of Subconductors Scope of the Thesis 5 2. THEORY AND THE FORMATION AND SOLUTION OF EQUATIONS Subdivisions of the Conductors ^ 2.2 Assumptions ^ 2.3 Loop Impedances of Subconductors ^ 2.4 Formation of Impedance Matrix 2.5 Bundling of the Subconductors i n the Impedance Matrix * ' ' " 2.6 Reduction of the Large Impedance Matrix 2.7 The Choice and Constraint on the Return Path Including the Constraint on the Current i n the Matrix Solution RETURN PATH IMPEDANCE, Return in Neutral Conductors Only Return in Ground Only Return in Ground and Neutral Conductors Use of Analytical Equations for Ground Return Impedance. 2 2

5 iv 3.5 Model Using Ground Return Formulae Directly with the Subconductors - Model II Representing the Ground as One Undivided Conductor - Model III Mutual Impedance between a Subconductor and Ground with Common Return in Another Subconductor Comparison of Model III with the Transient Network Analyzer Circuit RESULTS Comparison of the Method of Subconductors with Standard Methods Comparison of Ground Return Formulae Comparison of Results from the Different Models Reproduction of Test Results Pipe Type Cables CONCLUSIONS 63 LIST OF REFERENCES 64

6 V LIST OF TABLES TABLE 4.1 Variation of Impedance with the Number of Subdivisions Variation of the Impedance of the Circuit of Fig. 4.3 with the Number of Subdivisions, Showing the Inclusion of Both Skin and Proximity Effects i n the Calculations Self Impedance of Ground Return Path as Calculated Using Subdivided Ground and Other Formulae 4.4 Mutual Impedance Between Two Underground Conductors ^ 4.5 Comparison of Various Models Induced Currents i n Bonded Sheaths ~, - > 4.7 Impedance of Pipe Type Cables for Various Degrees of Magnetic Saturation J ~ 4.8 ero Sequence Impedance Measurements on Three Cables Enclosed o in a Pipe with Pipe Return J 4.9 Impedances of Cables i n Magnetic Pipes Represented as Two Concentric Pipes of Different Permeabilities ^

7 vi LIST OF ILLUSTRATIONS FIGURE 1.1 Current distribution in solid round conductors due to skin and proximity effects Current distribution in the subdivided conductors of the model Subdivision of the main conductors Circuit of two subconductors with common return Geometry of Subconductors, k, q Illustration of the reduction process A : two-w;ir'e return' circuit 1'6 2.6 A two-wire circuit with, common return in a third _ conductor Subdivision of ground into layers of subconductors Model with, only subconductors and ground return Model with ground represented as only one conductor A circuit of two conductors with common ground return A circuit of one conductor and the ground with common return in another conductor A return circuit of two conductors far apart Variation of impedance with the number of subconductors A return circuit of two conductors very close together Variation of the impedance of a buried conductor with depth of burial 3'8 4.5 Cross sections of buried conductors for ground return impedance calculations Comparison of calculated self impedances of a ground return loop Comparison of calculated mutual impedances between two buried conductors Electrical layout of the induced sheath current test Circuit diagram of the induced sheath current test 51-

8 vii 4.10 Variation of magnetic permeability of steel pipe with current in the pipe Shape of magnetizing curve during one cycle Linearized magnetizing curves 62

9 ACKNOWLEDGEMENTS I would like to express my thanks to my supervisor, Dr. H.W. Dommel, for his help throughout this work and for the timely suggestions and corrections he made. Also, I wish to convey my gratitude to Mr. Gary Armanini of British Columbia Hydro and Power Authority, for making his report and test results available for use in this work. I am also very grateful to the Government of the Republic of Ghana for financing my education at the University of British Columbia. For reading through and correcting the scripts, I would like to thank Ms. Marilyn Hankey of the Faculty of Commerce. The typing is beautifully done by Mrs. Shih-Ying Hoy of the Department of Electrical Engineering; I do appreciate it very much.

10 ix List of Symbols B flux density D distance between conductors and q q D P pipe diameter e = f g frequency subscript denoting ground GMR geometric mean radius h depth of burial of conductor i, I current I pipe current p j complex operator = /-I i,j,k,,n subscripts km m m M kilometres = /(jyu/p) metres inductance q subscript denoting return path r,r v,v X resistance, radius voltage reactance impedance K Q, Bessel functions log Common logarithm (base 10) n Natural logarithm (base e) Hz hertz U absolute permeability of free space = 4irx 10 ^ H/m

11 X y =viqy, permeability y r relative permeability tj) flux flux linkage = ir = y = Eulers Constant ft ohm (JJ =2lTf

12 1 Chapter 1 INTRODUCTION 1.1 A Brief Review of Methods for the Calculation of Cable Impedances For the analysis of transmission line systems, one of the basic input parameters is the impedance of the lines. Fault studies, surge propagation studies and the calculation of mutual induction effects between lines and p a r a l l e l adjacent conductors (such as other l i n e s, pipes and fences) a l l require reasonably accurate impedance data. Underground cable systems have been analysed by many authors. The work of D.M. Simmons [1] resulted in the publication of standard charts which are found in many handbooks and which are often used i n impedance calculations for distribution systems. For single-cored (coaxial) cables, Schelkunoff [2] has done a comprehensive analysis and his results are widely used. Carson [10], Pollaczek [19], Wedepohl and Wilcox [9] have also derived equations for the impedance of underground cables with ground return. Smith and Barger [3], Lewis, et a l. [4,5] and others have calculated the impedances of concentric neutral cables used in distribution systems. In many of the formulae used i n impedance calculations, factors are used to correct for two important effects, namely: skin and proximity effects. Another approach has been used by Comellini, et a l. [7] and Lucas and Talukdar [22], who have calculated the impedances of transmission lines by dividing a l l the conductors (including ground) into smaller conductors of specific shape, which automatically accounts for the two effects mentioned above. This approach is also used in this thesis. Cables with sector

13 2 shaped conductors or conductors of any irregular cross section or of nonuniform properties across the cross section can easily be handled with this method. 1.2 Skin and Proximity Effects The resistance of a transmission line to direct current is easily determined from the physical dimensions of the wire and the type of material because direct current is uniformly distributed across the cross section of the wire. In the case of alternating current, there exists a nonuniform distribution of current over the cross section of a conductor which i s caused by the variation of current i n the conductor. This phenomenon i s called "skin effect". Another phenomenon, called "proximity effect", arises due to the presence of other current-carrying conductors close by. Changing currents in these neighbouring conductors causes a distortion i n the current d i s t r i bution i n the f i r s t conductor. This distortion, unlike that due to skin effect, i s not symmetrical around the axis of symmetry of the conductor (if the conductor is c i r c u l a r ). In a two-wire l i n e, for instance, more current tends to flow either on the sides of the conductors which face each other or on the opposite sides. The phenomena of skin and proximity effects in round conductors are i l l u s t r a t e d in Figure 1.1.

14 3 ^> Current (a) Skin Effect Distributions (b) Proximity Effect Figure 1.1 Current distribution i n s o l i d round conductors due to skin and proximity effects The uneven current distribution across the cross section of the conductors causes additional power loss above that produced by an equivalent amount of direct current, thus increasing the effective a.c. resistance of the conductor. The higher current density towards the conductor surface reduces the self linkage i n the inner part of the conductor. This decreases the internal inductance of the conductor. The extent to which the above effects alter the values of the conductor impedance depends on how pronounced they are. Skin effect varies very much with the size of the conductor and with the frequency - i t increases with both - while proximity effect depends mainly on the geometry (being more pronounced for closer spacings between conductors).

15 Bessel functions are used to calculate increases in resistance to alternating current due to skin effect analytically. This method i s widely found in the literature, expecially when coaxial cables and c y l i n d r i c a l conductors are being analyzed [1,9]. On the other hand, proximity effect i s more d i f f i c u l t to analyze. Charts and correcting factor tables [17] derived from otherwise complicated formulae [18] are customarily used in most hand calculations to correct for this effect. In general, underground cables are large i n size and are usually l a i d close together. This makes both skin and proximity effects important i n the calculation of impedances, even at power frequency and especially at higher frequencies as needed for switching surge studies. 1.3 A Brief Explanation of the Method of Subconductors In the work of Enrico Comellini et a l. [7] i t is shown that i t is possible to take both effects into account simultaneously i n calculating the impedance of any transmission l i n e. This i s done by dividing the main conductors into smaller subconductors of c y l i n d r i c a l shape, by finding the self and mutual impedances of these subconductors, and by bundling them to give the impedance of the main conductors. The work described i n this thesis i s based on the above reference. Dividing the conductors into p a r a l l e l c y l i n d r i c a l subconductors seeks to approximate the current distributions shown in Figure 1.1 by those of Figure 1.2.

16 Conductor s Approximate Current Distributions (a) Skin Effect (b) Proximity Effect Figure 1.2 Current distribution i n subdivided conductors of the model Obviously the accuracy to be expected from such a representation w i l l depend on the degree of discretization, and hence on the number of subconductors. 1.4 Scope of This Thesis The theory for calculating the impedances from subconductors is developed using a f i c t i t i o u s 'return path' which allows more f l e x i b i l i t y in the model. Analytically derived ground return formulae are then incorporated into the model to reduce the number of subconductors, storage and computing time. Pipe type cables are modelled by treating the steel pipe as concentric layers of pipe material, with each layer having a different permeability depending on the degree of saturation.

17 6 Chapter 2 THEORY AND THE FORMATION AND SOLUTION OF EQUATIONS 2.1 Subdivisions of the Conductors In the model described below, each core of the cable i s considered a main conductor, as i s the sheath, and i f present, the neutral conductor and the armour. Each main conductor is divided into a number of p a r a l l e l c y l i n d r i c a l subconductors (Figure=2.1). The choice of a c y l i n d r i c a l shape for the subconductors makes the derived inductance formulae simple. Other shapes for the subconductors have been tried by Lucas and Talukdar [22] but the resistance values calculated by them show a large deviation from measured values at higher frequencies, which suggests that further rese.arch is needed before these other shapes can be used with confidence. Figure 2.1 Subdivision of main conductors 2.2 Assumptions It is assumed that: i) Each subconductor is uniform and homogeneous throughout i t s length; i i ) The magneitc permeability of a subconductor i s constant throughout

18 7 the whole cycle of alternating current, but may be different from that of any other subconductor; i i i ) There is uniform current distribution in each subconductor; and iv) A l l subconductors are p a r a l l e l. 2.3 Loop Impedances of Subconductors To derive the loop impedances of the subconductors, f i r s t consider the two loops formed by any two subconductors, % and k, with a common return path, q, i n Figure 2.2. The return path can either be one of the subconductors or a f i c t i t i o u s conductor chosen for the voltage measurements and the calculation of inductances. Figure 2.2 Loops formed by two subconductors with a common return Writing the loop equations for subconductor gives: (2.1)

19 where = voltage drop per unit length of subconductor = resistance per unit length of subconductor R^ = resistance per unit length of return path i, i, = currents i n subconductors and k respectively JO AC M = mutual inductance between loops formed by subconductors J6K. and k with a common return in q N = number of subconductors (N=2, for, k i n Figure 2.2) For ac steady-state conditions, the instantaneous voltage v and current i i n (2.1) are replaced by the phasor values V and I and by the phasor value jcol. Figure 2.3: Geometry of subconductors (, k, q ). Figure 2.3 shows the cross section of two such subconductors and the return path. To derive the inductance formulae, consider current I in subconductor and returning i n q.

20 The flux density B at radius r outside i s : B = yi.2irr (2.2) Total flux per unit length in the elemental cylinder of thickness <5r 6<j> = B * Sr yi 2Trr 6r (2.3) Flux linkage of loop k^ due to current I i n i i s : yi 2irr dr 2 * Kk (2.4) r=d k r=d Flux linkage of loop k^ due to return current I i n q i s : r = r kq q( e Q u ly) yldr 2irr yl_ 2TT r D. q(equiv)- (2.5) where r q(equiv) the equivalent radius of subconductor q for inductance calculation ( = r^e ^ r q ^ ^, the geometric mean radius) The two fluxes are additive; hence the total flux linkage i s : * k - 2V t o &q } k TT r exp(-y /4) q rq ' (2.6) The mutual inductance ( M ^) between loops & and k is therefore M k k y 2T: n Iqkg +.IS. (2.7) where y r^=relative permeability of.the return path.

21 To derive the self inductance of loop we consider the same current path. Flux linkage of loop ^ due to current I in i s : q r=r (equiv) uldr 2irr ul 2TT n 13_ (equiv) (2.8) Flux linkage of loop ^ due to current I returning in q i s : D q r=r q(equiv) pi, pi dr = ^ n 2irr 2TT q k q(equiv) t (2.9) Hence the total flux linkage i s : * = ~ n 2 IT./ D q, r. 7 ^ exp( - r - ) TT ^ exp ( ) (2.10) where u. and p are the relative permeabilities of subconductors and r rq respectively The self inductance ( M^ ) of loop i s D M = hi n = 27 n q D q q y r, y rq_ (2.11) 2.4: Formation of Impedance Matrix Writing the loop equations, using (2.1), (2.7) and (2.11), for a the subconductors gives a set of linear equations

22 11 V 11 J l l l l llln ^ l k l * " llkm 11 In lnll lnln lnkl Inkm In (2.12a) V. kl J klll "klkm kl km kmll ' kmln kmkm km i.e. [V] = [ b ± g ] [I] (2.12b) th where V.. refers to the voltage on the i Ji subconductor of the.th 2., main conductor. I., is the current in the i*"* 1 subconductor of the j*"* 1 main conductor... jimn is the mutual impedance between the loops formed by the i*"* 1 and n*"* 1 subconductors of the j 1 "* 1 and m 1 "^ main conductors respectively. The partitioning of the matrix [^g] i n (2.12a) groups the equations of the subconductors within each main conductor together. The resistances and inductances in the impedance matrix [^g] are constant, but since the current division among subconductors changes with frequency, skin and proximity effects are accounted for. Normally the large impedance matrix is not of direct interest. Instead, the matrix giving voltages on the main conductors in terms of the

23 12 currents in these main conductors i s needed, which can be obtained from the large impedance matrix ] by reduction. Mathematically, this is equivalent to solving the algebraic equations '(2.12a) with the conditions of (2.13) and (2.14) shown below. P r a c t i c a l l y, i t is equivalent to r e d i s tribution of currents in a l l subconductors to achieve a current distribution which, when multiplied by the impedances of (2.12a), s a t i s f i e s the voltage condition of (2.13). The voltages on the subconductors forming any main conductor are equal; hence: V., = V. 0 = = V. = V. (2.13) J l J2 jn j Also, the current i n any main conductor is the sum of the currents i n the subconductors into which!.it is divided. Thus: I. = I... + I I. (2.14) J J l J2 jn 2 5 Bundling of the Subconductors i n the Impedance Matrix Expressing the voltages on the main conductors in terms of the currents they carry is accomplished by the use of equations (2.13) and (2.14) in (2.12a). Consider the f i r s t main conductor; assume i t is subdivided into n subconductors as shown in (2.12a). (a) Subtracting the f i r s t equation ( i. e. row-1 in 2.12a) from the subsequent equations ( i. e. row-2 to row-n) of that main conductor leaves the left-hand side of the other equations equal to zero (- i l l u s t r a t e d i n (2.15)).

24 13 (b) By writing I instead of 1^^ in the f i r s t equation of the f i r s t conductor ( i. e. row-1), an error of adding ^ilylj'h + + ^im^ln^ to f i r s t equation has been made since I-^I^l + ^ ^j_ n' Corresponding errors are introduced into a l l the other equations. These errors are removed by subtracting the f i r s t column of the whole matrix [,. ] from the subsequent (n-1) columns of that b i g J main conductor. These two steps are i l l u s t r a t e d in equation (2.16). (c) The same steps, (a) and (b), are carried out on the other main conductors. These give a set of linear equations (2.15) express- : ing the voltages on the conductors i n terms of the total currents til i n these conductors and i n the 2nd to n - subconductors. " v l ' l l l l? 1112-? l l l n j l l k l C llkm V 0 S.211 ; - ' ; ; hi 0? lnln< (2.15) \ k l l l > z k l k l? klkm \ 0 * 0 kmll ' f ^kmkl ' * kmkm The symbol " 5" denotes the elements which have been changed in (2.15) due to the operations (a) and (b), and the general term i s : C = kmqn kmqn klqn kmql (2.16) (for m,n^l)

25 Reduction of the Large Impedance Matrix The equations (2.15) are rearranged for the reduction process by exchanging the positions of rows and columns i n such a way that the "bundled" equations of (2.15) corresponding to the main conductors come f i r s t, as shown in (2.17) 1 J l l l l J l l k l V, k J k l l l J klkm 0 = 12 (2.17) 0 km or in abbreviated form, V A B 0 C D (2.18),-1, From (2.18) V=(A-BD C)I (2.19a) Hence the desired impedance matrix [ c] i s [ ] = [A - BD - 1 C] (2.19b) Reference [8] provides a more efficient way of finding [c] from (2.17). Using Gaussian elimination on the matrix (2.17), starting from the last row and going up u n t i l the submatrix [B], as shown in (2.18), has just been reduced to zero, achieves the reduction.

26 15 The desired impedance matrix [ c]'corresponds to the submatrix stored i n [A*] in (2.20). 1 V, k. (2.20) 0 12 _ 0 ^ I km J An i l l u s t r a t i o n of this f i n a l reduction stage is shown i n Figure 2.4. Figure 2.4 Illustration of the reduction process 2.7 The Choice and the Constraint on the Return Path Obviously, the geometry and location of the return path i n Figure 2.2 w i l l influence the values obtained for the inductances. The influence of the return path is removed by requiring that the current through this path should be zero. To i l l u s t r a t e this, consider the c i r c u i t shown i n Figure 2.5.

27 Figure 2.5 A two - w/i r e r e t u r n, c i r c u i t Writing the loop equation for Fig. 2.5 gives: V l = ( R 1 + h + ( R 2 + ^ X22- X 12> h < 2 ' 21 > since 1^=1 V l " ( R 1 + R 2 + J ( X 11 + X 22 ~ 2 X 12 } ) h < 2 ' 22 > Introducing a f i c t i t i o u s return path gives a configuration of Figure.2.6. For equivalence of the two c i r c u i t s we require: Figure 2.6 A, two - w.ire. c i r c u i t with, common r e t u r n i n a t h i r d c o n d u c t o r

28 17 The loop equations of Figure,2.6 are: V a = (% + J ^ - X ^ ) ) ^ + <R q + j (X q q -X l q ))I q + (X 1 2 -X 2 q ) I f e V, = (R 2 + i(x 2 2 -X 2 q ))I b + (R q + i(x q q -X 2 q ))I q + (X 1 2 -X l q ) I a > (2.24) Imposing the constraint that the current in the return path is zero means that I q = 0 = I a + I b (2.25) I a = -I b (2.26) From equations (2.24) and (2.25) V a - V b = [R x + j(x 1 1 -X l q)-(x 1 2-X l q)]i a - [R 2+ j(x 2 2 -X 2 q ) - ( X12- X 2q ) ] h (2-27) Using (2.26) in (2.27) gives V a _ V b = [ R 1 + R 2 + J ( X H + X 22" 2 X 12 ) ] X a ( 2 # 2 8 ) which is identical to (2.22) derived using Figure 2.5. Therefore, it is theoretically possible to choose a return path of any convenient shape and location for the inductance calculations as... long as a zero current constraint is imposed on such a path. Nevertheless, considerations discussed in section 3.6 would require that the return path should be cylindrical in shape, have a small radius, and be placed at a small distance below the earth surface not far from the cables and other conductors.

29 Including the Constraint on the Current in the Matrix Solution Equation (2.20) gives the voltages on the main conductors (measured with respect to the return path) in terms of the currents in these conductors. In practice, however, voltages are measured with respect to the local ground (or neutral conductor or sheath). If the constraint on the current is introduced, this changes (2.20) into the form: J ll J lk V. k-1 J k-l,l J k-l,k k-1 (2.29) V. k J L_- kl J kk -I k-1- k Since I o = 0 1=1 il ( ) and I - - l r l I t^ This gives: z ir z ik lk-1 lk ( ) V. k-1 k-l,l k-l,k k-l,k-l k-l,k k-1 kl kk k,k-l" kk If conductor k represents the local ground (or neutral conductor or the sheath) with respect to which all voltages are measured, then subtracting the equation for V from the other equations accomplishes this and gives: rc

30 19 V -V 1 k r- * l l lk-1 (2.32) V -V k-1 k J k l J k - l, k - l k-1 where.. =.. +,, - 2., 13 IJ kk ik The matrix i s the impedance matrix which implies a l o c a l ground (or neutral conductor or sheath).

31 20 Chapter 3 RETURN PATH IMPEDANCE In practice, there are three cases for the return path in any transmission system: (i) return in neutral conductors (including pipes and ground wires only); (ii) return in ground only; or ( i i i ) return in ground and neutral conductors. 3.1 Return in Neutral Conductors Only Each sheath, pipe or neutral conductor i s represented as a separate conductor, as are the cores, and is divided as in Section 2.1. The formulae of equations (2.7) and (2.11) are used to form the impedance matrix of the subconductors. The neutrals or sheaths can be "eliminated" i n the reduction process, i f so desired, to obtain the impedance matrix which relates the voltages from phase to neutral to the phase currents. In fact, any other conductor can also be eliminated i n the reduction process, i f so desired, provided that there i s zero voltage on i t, or that i t is connected in p a r a l l e l with another conductor. 3.2 Return in Ground Only, Model 1: The ground is considered as a separate conductor and i s subdivided into layers of subconductors as shown in Figure 3.1. The diameters of the subconductors in any lower layer are chosen to be twice that of the previous layer. This choice appears reasonable because the current density in the ground decreases as one moves farther away from the cables. Reasonable results were obtained by using a depth equal to 3300/,rp/f metres.

32 (a) ( b ) Figure 3.1 Subdivisions of ground into layers of subconductors

33 22 p = ground r e s i s t i v i t y in ftm, and f = frequency i n Hz. The ground, as a system of subconductors, is eliminated i n the reduction process as shown in section 2.8, thus leaving the ground return i m p l i c i t l y included in the reduced impedance matrix. The difference between the results of the arrangements i n Figures 3* 1(a) and (b) is discussed i n section Return in Ground and Neutral conductors If the return is through both ground and neutral conductors, the system is modelled as a set of k conductors which are subdivided into subconductors. The ground is one such conductor and i t is eliminated i n the reduction process. The neutrals can be retained or eliminated as desired. 3.4 Use of Analytical Equations for Ground Return Impedance To represent the ground return adequately, i t must be divided into a large number of subconductors. In systems where many cables or conductors must be considered, i t i s better to calculate the ground return impedance directly to reduce the amount of storage and computing time. Equations for ground return c i r c u i t s of overhead transmission.. lines were derived by J.R. Carson [11] and are widely used in the power industry. These equations assume that the conductors are located in air over flat earth which is i n f i n i t e in extent and has an uniform r e s i s t i v i t y. When these same equations are applied to underground cables, useful approximations to the true values of ground return impedances can be obtained.

34 23 With overhead l i n e s, image conductors which l i e below the ground are used in the calculations. However, when these equations are applied to underground cables, these images now l i e above the ground surface at heights equal to the depth of b u r i a l. In a later paper [10], Carson showed that for conductors buried underground, the variation of ground return impedance with distance below the earth's surface is relatively small for the usual depths of burial ( i. e. about 1.0 m) and that the ground return impedance (g) can be calculated as: = (1+C) (3.1) S g o where ^ = the ground return impedance i f the earth were to extend indefinitely in a l l directions around the conductor so that circular symmetry exists, C = a correction factor which accounts for the fact that the conductor is located near ground surface Reference [10] gives C as: 2K 0 (jm) 2 log(l/m) for small m (3.2) and reference [12] gives as: _ m& Ko(mr), g 2rrr K L(mr) where m = / (3.4) K5, are modified Bessel functions r = internal radius of the earth ( i. e. outer radius of the conductor insulation)

35 24 p = ground resistivity to = 2iTf, f=frequency y = magnetic permeability of ground Carson's formula for overhead conductors cannot be used for calculating the self impedance of underground conductors. Equations (3.1) is used for this purpose, but the mutual impedances are calculated using the overhead formula - which is known to give good approximations for buried conductors at power frequencies [20]. Equations for calculating the self and mutual impedances of underground conductors have also been derived by F. Pollaczek involving infinite series [19]. Closed-form approximations to the self and mutual impedances of underground conductors valid 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 if the separation is only 30 cm. Thus very accurate approximations can be obtained for most practical cases of cables laid in the same trench to quite high frequencies. These equations are: s = ^ { -An M + T " T n^} fl M (3.5a) 2 3 (YmD }, ik = *f i - n + J " f m } ft/m (3.5b) where s, -y^ are self and mutual impedances of ground return path respectively, (ft/m) Y = Eulers constant = h = depth of burial of conductor (metres) I = sum of depths of burial of conductors i and k (metres)

36 25 r = outer radius of conductor (metres) D M = distance between conductors i and k (metres) IK. m = /jtju/p p = earth r e s i s t i v i t y in fim Equations (3.5) are valid for the range mr < 0.25 for self impedance and md_^j < 0.25 for mutual impedance. For the range md., > 0.25 reference [9] suggests the integration: IK. Jik 2TT - /(a 2 +m 2 ) -V /(a 2 +m 2 ) - /(a 2 -rm 2 ) -e + a +/(a 2 +m 2 ) 2/(a 2 +m 2 ) exp(jax)dx (3.6) where x = horizontal distance between conductors i and k V= modulus of the difference of the depths of burial of conductors i and k. 3.5 Model Using Ground Return Formulae Directly with the Subconductors Model II: A very simple model which uses the analytical ground return formulae treats each subconductor as an insulated conductor with the return loop through the ground (Figure 3.2) and uses the available ground return formulae to calculate the self and mutual impedances. Equations (3.5) may be used in this case. If the results are needed for power frequency only, equations (3.7) and (3.8) below, which are found in many handbooks [24, 25], may be used.

37 Figure 3.2 Model with only subconductors and ground return The impedances at 60 Hz of the c i r c u i t in Figure 3.2 are: z i i = R i + R g + J ( l o s MR ) fi/km < 3-7 ).. = R + j( log ) ft/km (3.8) l k g i k where ^ and ^ are the self and mutual impedances respectively, and Rg = resistance of ground return path (= ft/km) GMR^ = geometric mean radius of conductor i (m) R^ = resistance of conductor i (ft/km) D k = distance between conductor i and k (m) 3.6 Representing the Ground as One Undivided Conductor Model III In view of the fact that the equations used for ground return impedance calculations may be inaccurate at high frequencies and for wide separations between conductors i f approximations are used, or costly to obtain i f i n f i n i t e series are used, i t would be advantageous i f most of

38 27 the elements of the matrix [,. 1 of equation (2.12a) could be calculated bxg with the simpler equations (2.7) and (2.11). This involves the introduction of a f i c t i t i o u s return path with respect to which the inductances are calculated. The ground is then considered as one additional conductor (not i. subdivided i n this case) and the mutual impedances between the ground and the subconductors are calculated as shown below. With this approach, eddy currents which would circulate in the ground i f current flows into conductor 1 above ground and returns through conductor 2 above ground, are ignored. In reference [20] i t has been shown that this effect i s negligible up to 1 KHz for the case of a 500 kv overhead l i n e. In the lower frequency region, this approach gives very accurate answers, but at higher frequencies the results must be interpreted with some caution, unless i t can be shown that skin effect in the conductors is much more pronounced than eddy current effects in the ground. The main advantage of this model i s that the more complicated ground return formulae must only be used in one row and one column of the matrix i n (2.12a). Figure 3.3 Model with ground represented as only one conductor

39 28 In addition, there is freedom as to the choice of location of the f i c t i t i o u s return path. The distances to be used i n equation (3.5) can be nearly halved by centrally locating this path. This reduces the values of the parameter md^ in equation (3.5), thereby giving more accurate approximations, and also delays the use of the more complicated formula (3.6) for much higher frequencies. Writing the loop equations for the c i r c u i t of Figure 3.3 gives: ~ V " 1 J l l 1N l g v N = IN NN Ng N (3.9) V g _ Jgl z M gn z gg il!_ g. in which. refers to the mutual impedance between subconductor i and ig ground with common return i n q. This cannot be calculated directly by equation (3.5). Equation (3.5) is valid only when the common return i s the ground. The next section 3.7 shows how i s derived using equations (3.5). 3.7 The Mutual Impedance Between a Subconductor and Ground with Common Return i n Another Subconductor Consider Figure (3.4) in which the common return is the ground. The loop impedances may be written as: J H g J 12g (3.10) 91 L 2 1 s J 22g

40 29 t v» (ground return) *1 Figure 3.4 Two conductors with common ground return A l l the impedance terms in (3.10) can be calculated by using Carson's or Wedelpohl'.s equations. Now consider a similar c i r c u i t i n Figure (3.5) in which the common return i s conductor 2. :» > Vn Q (ground) if (reiurn ) Figure 3.5 Circuit of one conductor and the ground with common return in a second conductor

41 30 The loop equations may be written as: -112 J lg2 (3.11) J gg2. The third subscripts i n equations (3.10) and (3.11) denote the common return. The term. =., is the one of interest here. The c i r c u i t s of Figures (3.4) lg2 gl2 and (3.5) are equivalent i f V a = \ ~ V 2 (3.12) V b = -V 2 (3.13) and I 2 - -(I g + I X) (3.14) Substituting these into equation (3.10) gives v x - v 2 " l l g 12g 12g 22g I, 1 (3.15) - " V g - 22g -I -I, g 1 V - V 1 2 llg + 22g~ 2 12g 22g 12g (3.16) 22g 12g J 22g g From (3.12) and (3.13), i t is evident that equations (3.11) and (3.16) are i d e n t i c a l, hence: lg2 22g 12g (3.17) The mutual impedances (^g) required in equation (3.9) can therefore be calculated using (3.17), (note that. =. where q is the f i c t i t i o u s i g i g q common return).

42 31 Thus using equation (3.17) in forming the matrix of equation (3.9) results i n the use of the ground return formulae for only one row and one column. If the i n f i n i t e series (or i n f i n i t e integral) forms of the ground return impedance must be used, model III w i l l be faster than model II since the i n f i n i t e series would have to be evaluated for every element of the matrix [,. ] i n the latter case, big 3.8 Comparison of Model III with the Transient Network Analyzer Circuits It should be noted that the procedure i n Model III is related to the method used i n representing three phase transmission lines on the Transient Network Analyser (TNA) [26], where the impedance of the ground return is decoupled from the phases. It is then included as an extra conductor and therefore eliminates the need to model the ground return i n every l i n e. On a three phase l i n e, A V a aa ab ac A V b AV c = ba ca bb cb be z cc (3.18) Assuming that the line is transposed, the average mutual impedance i s ; z = f (z, + z, +z ) m 3 ab be ca (3.19) Scjuation (3.18) can be written as: ~AV a aa -z m ab - m - ac m m A V b =, -z ba m bb - m be - m m AV c - ca m cb - m - cc m m I +1, +i a b c

43 32 where I a + 1^ + I c = Ig is the current in the extra conductor, ground in this case. The ground return formula with its pronounced frequency dependence is then only used for in the last column. All other elements a^- m,, - are calculated with ground ignored. Furthermore, if the line is ab m transposed, the diagonal elements ^-^, etc., become equal to the positive sequence impedance, and all off-diagonal elements -, etc., become zero.

44 '33 Chapter 4 RESULTS 4.1 Comparison of the Method of Subdivisions with Standard Methods This section shows how skin and proximity effects are taken into account by subdividing the conductors. The impedance of a return c i r c u i t of two conductors placed two metres apart, as shown in Figure 4.1, i s calculated. The d.c. resistance of each conductor i s fi/km and the frequency is 60 Hz mm Figure 4.1 A return c i r c u i t of two conductors for apart The large separation between the two conductors makes proximity effect negligible. The increase in resistance due to skin effect is corrected for by using Bessel functions. The UBC/BPA line constants program [16] is used for this. The corrected value of the impedance i s : = j fi/km This is taken as the exact reference value. By using various numbers of subdivisions the impedances shown in Table 4.1 are obtained. Figure 4.2 shows the impedance variations as a function of the number of subdivisions. It is seen that the exact reference values are

45 34 Table 4.1 Variation of Impedance with the Number of Subdivisions No. of Subdivisions R ft/km X n/km X Errors internal ft/km R X % 0.2% % 1.4% % 0.5% % 0.3% % 0.2% Reference % 0.0% approached as the number of subdivisions i s increased. However, i t is best to keep the number of subdivisions as low as possible. Nineteen subdivisions may be appropriate i f an error of one percent is tolerable. Substantial savings i n storage and computing time result from keeping the number of subdivisions down. The two conductors forming the return c i r c u i t of Figure.4.1 are brought close together, as shown in Figure 4.3. Various numbers of subdivisions are used on the two conductors. The impedances calculated are compared with calculations done using standard methods which involve the use of published charts and tables to correct for proximity effect as shown in Chapter 2 of reference [17].

46 NUMBER OF SUBCONDUCTORS Broken lines are the reference values. ' Figure 4.2 Variation of impedance with the number of subconductors l< i mm Figure 4.3 A return c i r c u i t of two conductors very close together According to reference [17], the a.c. resistance of the return c i r c u i t above is r = R' x ~y (A.D

47 36 where R'=a.c. resistance corrected for skin effect only. R"/R' = proximity effect resistance ratio. A similar equation holds for the inductance. From the Charts and Tables of reference [17], the proximity effect resistance and inductance ratios are calculated to be 1.18 and 0.95 respectively. The impedance of the c i r c u i t i n Figure 4.3, when corrected for skin effect only, i s : = j ft/km Applying the above proximity correction factors gives a value of = j ft/km. This is used as the reference value i n Table 4.2 which compares i t with those obtained from various subdivisions. In using conventional methods of impedance calculations, corrections for proximity effect are made for two or three conductors by the use of "estimating charts" and correcting "factor tables" derived from otherwise complicated formulae [18], as mentioned above. On the other hand, the method of subdivisions is not limited to only two or three conductors or common, forms of conductor arrangement (flat or delta spacing) as the charts referred to above seem to be. Also, i n using subdivisions, the current division among the p a r a l l e l conductors need not be known a p r i o r i. Where many conductors are involved and the current division is not known (as is the case i n most cable systems in c i t i e s where many cables and pipes run p a r a l l e l in the same or adjacent ducts), this method of subdividing the conductors is very useful, and gives reasonably accurate results.

48 37 Table 4.2 Variation of the Impedance of the Circuit of Fig. 4.3 with the Number of Subdivisions, Showing the Inclusion of Both Skin and Proximity Effects i n the Calculations. No. of R X Subdivisions ft/km ft/km X Error in internal ft/km R X ** % 6.1% % 10.0% % 4.0% % 2.2% % 1.6% SKIN * % 5.2% Reference Value % 0.0% * Corrected for skin effect only. * * No correction for both skin and proximity effects. 4.2 Comparison of Ground Return. Formulae Formulae for calculating the self and mutual impedances of loops with ground return have been given by many authors, including Carson [10,11], Pollaczek [19], Wedepohl and Wilcox [9], and Kalyuzhnyi and Lifshits [13]. Most of these formulae are given in the form of i n f i n i t e series and are not always easy to use. The variation of ground return impedance with frequency, as given by some of the formulae, are compared i n this section.

49 According to J.R. Carson [10], the variation of the ground return impedance with the depth of burial.'.'.of a conductor i s minimal, and can be calculated for most frequencies by using equation (3.1). This is verified by using Model I to calculate the impedance of a buried conductor with ground return for various depths of burial. The results of this are shown in Figure M < in o z < IMPI DEPTH OF BURIAL(m) 8 10 t-5 Figure 4.4 Variation of the impedance of a buried conductor with depth of burial A very simple and useful form of the ground return impedance has been given by Wedepohl and Wilcox (equations (3.5)), which i s an approximation of the i n f i n i t e series form of solution. Kalyuzhnyi and Lifshits [13] also derive a formula, the f i n a l ',. results of which, though very different from the more conventional ones,

50 39 are claimed to agree very closely with experimentally measured data. Kalyuzhnyi and Lifshits give the self impedance of ground return (e) as: e = ^ [An -2 2TT I y P r n - J- (4.2) where y = Euler's constant..r = radius of buried conductor over insulation (m) p = <WP p = r e s i s t i v i t y of ground (ftm) h = depth of burial of conductor (m) Equation (4.2) gives a real part (Re) of: Re = 2?r 2 f. 10 _ 7 Q/m (4.3) which is quite different from that obtained from Carson's equations which approximates [17] to: Re = TT 2 f tt/m (4.4) Carson's equations (or approximations of them) have been used for many years by several authors and others involved with analysing the conduction of e l e c t r i c current through the ground. The very marked deviation from Carson's equations given by Kalyuzhnyi and L i f s h i t s i s, therefore, worthy of inves tigation. In order to determine which formula best approximates the behaviour of the earth, the impedances of the c i r c u i t s of Figure 4.5 are calculated by using the subdivided ground representations of Figure 3.1 ( i. e. Model I). In this calculation, the ground i s divided into five layers of 62 subconductors. In Table 4.3 and Figure 4.6, the results obtained from the self impedance

51 Oo012m (a ) Figure 4.5 Cross sections of buried conductors for ground return impedance calculations

52 calculations using Model I and by using the various formulae are compared. There is close agreement i n the resistance values as calculated using the above method of subdividing the ground, Carson;'s equations, and Wedepohl's equations, but the results obtained using Kalyuzhnyi and Lifshits equation deviate from the others. The reactance values calculated from the f i r s t three methods deviate more widely from each other than the calculated resistance values. The deviations of reactance range between 7% at power frequency to 12% at 1.0 MHz.between the results of Carson's and Wedepohl's formulae while the results from the subdivisions method l i e somewhere in between (see Figure 4.6 b) for a l l frequencies. A very marked deviation i n reactance (over 50% at power frequency) i s obtained from the formula of Kalyuzhnyi and L i f s h i t s. Discrepancies in the various calculations arise from the fact that a l l the methods are approximations to the real case and also because the "interstices" between the subconductors are neglected in Figure 3.1 a. To evaluate the influence of this latter approximation, most of the interstices of Figure 3.1 a were f i l l e d with subconductors: for a fuller representation of the ground cross section. Results obtained using the latter show only a slight improvement over the results of Figure 3.1 a. For example, at 60 Hz, the impedance of the conductor i n Figure 4.5 a with ground return is calculated to be = j ft/km when the ground representation of Figure 3.1 a is used. By using the ground representation of Figure 3.1 b, the calculated impedance i s : = j ft/km. The latter representation of the ground only gives an improvement of less than 1% in the results when compared with the former. Therefore the ground return representation given i n Figure 3.1 a would be adequate for this purpose.

53 Fre RESISTANCE (ft/km) REACTANCE (ft/km) quency (Hz ) Subdl-. visions Wede pohl Carson Carson.. Kalyuzhnyi Subdivisions Wedepohl Kalyuzhnyi IK K K OK M M l.om Table 4.3 Self Impedance of Ground Return Path as Calculated Using Subdivided Ground, and Other Formulae

54 43 Figure "4.6 Comparison of calculated self impedances of a ground return loop.

55 44 Mutual Impedance of Ground Return Path Table 4.4 and Figure 4.7 show the results of the mutual impedances calculated for the two buried conductors of Figure 4.5b. Deviations of about 1% in the resistance and about 15% in the reactance are obtained at power frequency between the results of subdivisions and those obtained from Wedepohl's equations. The mutual impedance values calculated by using subdivisions and by using Carson's overhead line equations [11,16] are similar for most of the frequencies used; thus i t seems Carson's overhead line equations may be used for calculating the mutual impedances between buried conductors [20]. Despite this close agreement in the results, i t should be remembered that Carson's equations [11] were derived for conductors located above ground.

56 Table 4.4 Mutual Impedance Between Two Underground Conductors quency Hz Fre RESISTANCE (ft/km) REACTANCE (ft/km) Subdivisionpohvisions Wede- Subdi Wedepohl Carson Carson* Ik ' k k k< k k * Overhead line equations used.

57 cci- t- - ID X---Wedepohl a o- o FREQUENCY C Hz ), «no" + Subdivisions o Carson (0/H) X---Wedepohl Figure 4.7 no 100 idoo loooo iooooo " FREQUENCY ( Hz ) I Comparison of calculated mutual impedances between two buried conductors.

58 Comparison of Results from the Different Models The data is taken from reference [4]. Three distribution cables (1/0 AWG aluminum cored cables with reduced neutrals) are l a i d in a flat formation 8 inches apart. The core and sheath resistances are and ohms/1000'f t respectively with the inside and outside diameters of the insulation as 515 mils and 955 mils respectively. The l i s t e d values of the zero (0), positive (1) and negative (2) sequence impedance matrix elements at 60 Hz i n the reference are: 0 ~0.483+j0.236 symmetric tw j j0.008 ft/1000 ft jo j j0.004_ By using Model II and the ground return impedance formulae of equations (3.7) and (3.8), the sequence impedances calculated are: 0 "0.483+J0.231 symmetric [ 012 ] = j j0.008 ft/loooft. 0.0' -jo j0.089 O.OlO+jO.002 If the ground return impedance equations (3.5) derived by Wedepohl and Wilcox are used in Model II, the following impedance matrix is obtained J0.219 symmetric j j0.008 ft/loooft jo j j0.002