‘ ——————-~_ \ THS EN HIGH VOLTAGE TRANSMISSION LINE The”: foe- i‘he Daqme of M. S. MICHEGAN STATE UNWERSITY Gobincfia P. Purkayastha 1956 -‘A- “"_‘. k -V . —- , LIBRA 2 Y 5"- Michiga’z Sta it: University are “a v5 “L... This is to certify that the thesis entitled "THE ECONO‘KY OF BUNDLITTG CONDUCTOR IN HIGH VOLTAGE TRANSMISSION LINE" presented by Gobinda P. Purkayastha has been accepted towards fulfillment of the requirements for L degree in __E_:_E;s_. 0—169 The subject or Power Transmission by bundle conductor has received considerable attention during the last few years. The bundle conductor is the use or more than one conductor per phase in a three-phase, high voltage tranSmission line. The Power transmission by use of two conductors per phase in a horizontal configuration in overhead line is the subject matter of this paper. This paper investigates a definite relation to the bundle in comparison with a single conductor per phase with reSpect to inductance, capacitance, surface voltage gradient, disruptive critical voltage and corona loss. The characteristics of the lines are studied with different aspects, and appropriate formulas have been established for the bundle conductor of two conductors per phase. Several problems are solved,and a comparative study has been made between the single and two conductor bundle per phase by plotting curves. This paper reveals that when two overhead conductors, a few inchesapart, are used for each phase of a transmission line circuit instead of one of larger cross section, the inductive reactance is reduced by 20% and capaci- tive increases by 155 for a particular spacing of the conductor. As the in- ductance of the line decreases and the capacitance increases with greater spacing of sub-conductors, the use or bundle conductor is the gain in the appreciably lower value of surge impedance. It has been shown that the double conductor line has the capacity to carry 1.2 times the allowable load of that of a single conductor line. This paper shows the gain in lower surface voltage gradient attributed to bundle and the decrease or 5% through use of the bundle has been shown. The maximum surface gradient occurs at the middle phase due to non-uniform charge density on the conductor periphery of the bundle conductor circuit. -2- A precise computation of its value has been determined by Eaxwell's coefficient. The reduction in the mavimum surface gradient is based on the value of single conductor surface gradient. As the surface voltage gradient directly affects the corona-starting voltage, critical voltage is 15% higher for a bundle. With the same amount of material, much higher VOltage can be used without corona loss when the transmission line is built with two small conductors per phase properly ar- ranged than with a single conductor of equivalent area. This is a great advantage of the bundle over a single conductor circuit. The economy of the bundle has been established on this point in case of a high voltage transmis- sion line. The corona losses have been found to be much smaller when the sub-conduc- tor spacing of the two-conductor bundle is less than 15 inches. The multiple shielding effects of a subconductor reduce the electric field intensity on the individual conductors, which in turn reduces corona. According to theo- retical calculation, it is found that the corona losses are about the same when the spacing is greater than 15 inches. THE ECONOMY OF BUNDLING CONDUCTOR IN HIGH VOLTAGE TRANSMISSION LINE By Gobinda P, Purkayastha A THESIS Submitted to the School of Graduate Studies of Michigan State University of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Electrical Engineering 1956 gbtgi’ bi AC KN OWL ED GM ENT The author wishes to express his heartful gratitude to Dr. J. A. Strelzoff, Professor of Electrical Engineer- ing of Michigan State University for his guidance and valuable counsel. Also, my grateful thanks to the authorities of the Michigan State University Library for their help in procuring materials for this work even from out-of—state institutions. The help rendered by Mr. L. R. Housman, a graduate student of the Electrical Engineering Department in proof- reading of this thesis cannot be left unaccounted. ii AB STRACT A general survey of the theoretical study has been carried out of the Bundle-conductors for Transmission line use. The dif- ferent aSpects of bundle-conductors are studied and an appropriate formula is obtained for inductance, capacitance, voltage-gradient, corona loss, and disruptive critical voltage. Several problems are solved for line characteristics, and graphs are shown in every case. A comparative study has been made between the single and two conductor bundle per phase by plotting curves. iii CHAPTER II III IV V VI VII TABLE OF CONTENTS INTRODUCTION............................................. THE INDUCTIVE REACTANCE OF THO COKDUCTOdS PER PHASE...... CAPACITIVE SUSCEPTANCE OF THO CONDUCTOHS PER PHASE....... SUHFACE VOLTIGE GRADIENT OF BUNDLE CONDUCTOR............. DISHUPTIVE CRITICAL VOLTAGE OF THE BUNDLE................ CORONA LOSS MEAS REMENT.................................. APPLICATION.............................................. CONCLUSIONOOOOOOOIO...OOOOOOOOOOOOOOOCOOOOOO...000.00.... APPENDICESOOOOOOOOOOOOOOOOOOOOOOOOOOOO.0.0.0.0....0.0.0.9...O... Appendix 2 - Appendix 5 - Appendix 1 -- Inductive Reactance........................ Capacitive Susceptance..................... Appendix 3 -- Surface Voltage Gradient................... Appendix h - Disruptive Critical Voltage................ Corona LOSS................................ BIBLIOGRAPHY.................................................... iv Page vii \II 11 SCOPE OF THIS PAPER 1) The bundle of only two conductors per phase has been studied. 2) Only horizontal flat configuration of the bundle, i.e., all are in the same plane, has been considered and only on overhead lines. 3) For varying m from t to 2b inches, all characteristics of the line are obtained. h) Conductors are considered smooth and round in all cases except for corona calculation. 5) The equivalent cross-section of the conductor is constant: diameter for l conductor per phase is l.hl inch; diameter for 2 conductors per phase is 1.0 inch. 0) Ground-wire effect is neglected in capacitance and surface charge calculation. 7) The formulas for x, b, g, e0, and P (consult list of symbols) has been established for the two conductor bundle per phase. 8) Each characteristic as a function of m is illustrated by working out one problem in each case. 9) For verification of the theoretical calculation, some examples about the line construction cost is quoted,1 but the economy regarding cost of line is out of the scope of this paper. 10) Nothing has been stated about the most suitable subconductor Spacing as it is beyond the scope of the present paper. 1 H. L. Deloney and w. L. Rush, "Bundle Conductor for Transmission Line Capacity Increase, Electircal world, December, 1955. V CD 0 ll n'x the the the the the the the the the the the LIST Oi“ SYMB OLS frequency in cycles per second maximum surface gradient in volts/inch disruptive critical voltage in kv applied voltage (working) to neutral in kv inductive reactance in ohms/mile capacitive susceptance in mho/mile total current per phase in amps potential difference in volts charge in coulombs capacitance in farads distance between conductors of different phases (position 1, 2, and 3) in feet the the - the the the the the the the the the the the distance between conductors of the same phase in inches height above ground in feet distance between conductor r and image of s in feet radius of the conductor in inches diameter of the conductor in inches flux density in webers/sq. meter self inductance in henry mutual inductance in henry corona loss for 3-phase in kw/mile irregularity factor barometric pressure in cm temperature in degrees centigrade air density factor iaxwell's coefficient the the corona function permittivity in farads/m vi INTRODUCTION During the last few years, the subject of Power Transmission by bundle-conductor circuits has received considerable attention. The term bundle conductor, employed in this paper refers to the use of two conductors instead of one for each phase. The two conductors are therefore electrically in parallel and form one effective conductor of large cross—section. The purpose of this paper is to investigate a definite relation to the bundle in comparison with a single conductor per phase with respect to inductance, capacitance, voltage-gradient, disruptive critical voltage, and corona loss. This paper reveals when two overhead conductors, a few inches apart, are used for each phase of a transmission line circuit instead of one of larger cross section, that the inductive reactance is reduced by 20% and capacitive increases by 15% for a particular spacing of the conductors. The gain in lower surface voltage-gradient is attributed to bundle and the decrease has been shown by 5% by the use of bundle. The disruptive critical voltage is higher for a bundle because multiple shielding effects of sub-conductors reduce the electric field in- tensity on the individual conductors which in turn reduces the corona loss. The scope of this paper does not include any vertical configura- tion of conductors. Only horizontal two conductors per phase vii separated by 8 to 2b inches are studied. Conductors are considered smooth but for stranded and hollow conductors a correction factor is given. Formulas have been established for the calculation of inductive reactance, capacitive susceptance, surface voltage-gradient, dis- ruptive critical voltage for the three phase horizontal configuration. Formulas and estimating curves given in this paper may be of interest to the transmission engineer but this only supplements previous work. viii CHAPTER I THE INDUCTIVE REACTANCE OF Two CONDUCTOhS Psi PHASE When a 3-phase bundle two conductor transmission line is com- pletely transposed so that each conductor occupies each of the tower position for equal distances, the phase being rotated in cyclic order, the total inductance to neutral of any conductor will be the sum of the inductances in each position. The average inductance per mile of all conductors to neutral will be the same. A formula for the average inductance to neutral per mile of each conductor of a perfectly transposed 3-phase transmission line having two conductors per phase has been derived in Appendix 1, by the method of Geometric Mean Distance (G.M.D.) From Equation (10) Appendix 1 2 - (s ) LC = 0.080h7 + 0.7hll3 loglo 2/d .___SE§__ millihenry/mile where S d = The geometric mean distance between conductors of the gm different phases in feet. mgmd = The geometric mean distances between the conductor of the same phase in feet. Geometric Mean Distance for two conductonsper phase with sub-conductor spacing m and phase distance S in feet has been derived in equation (28) Appendix 28. (G.M.D.)2 for the horizontal flat configuration with two conductors per phase, as given in equation (28): 3 2 (sgmdfi = sz. 3/? 6/ [1- 1 The total inductance to neutral of any conductor in any position is the sum of self and mutual inductances. The formula for self and mutual inductance of non—magnetic wires in air has been taken from "Bulletin of the Bureau of Standards." From equations (6) and (7) in Appendix 1 of this paper L1 = 2L! [loge IJLJ - 3 ] d E M = 2L! [loge 25L” - 1 ] Now the average 60 cycle reactance per conductor from Equation (ll) Appendix 1 2 . Xc = 0.3031; + 0.27% logic-f; *(Sam_d).. ohm/mile m Since two conductors of the phase are in parallel, the average in- ductance per phase is the one-half of the inductance per conductor. ’1 2 . . . X = ‘2' [0.030% + 0.27% loglo 31;— <5amd> ohm/rune d 1n mch m This value of reactance is on the assumption of solid, smooth round wire. The correction for Stranded and hollow conductor are given in Tables I and II. It has been illustrated by Problem 1 and curve 1 in this paper that an overhead double-conductor line has approximately 20% less TAB LE I COhRECTION FOR STRANDED CONDUCTOHS FOh REACTANCE CALCULATION Conductors Number of Strands Per Phase 7 19 39 451 1 0.0086 0.0033 0.0017 0.0010 2 0.00h3 0.0017 0.0009 0.0005 3 0.0029 0.0011 0.0006 0.000h b 0.0021 0.0008 0.000h 0.0003 5 0.0017 0.0007 0.0003 0.0002 TABLE II COHfiECTION FOR HOLLOW CONDUCTORS FOR REACTANCE CALCULATION Conductors Ratio of Internal to External Diameter Per Phase 0.2 0.3—7 0.6* 0.8 1.0 1 -0.0022 -0.0075 -0.01hS -0.0222 -0.0303 2 -0.0011 -0.0038 —0.0073 -0.0111 -0.0152 3 -0.0007 -0.0025 -0.00h8 -0.007h -0.0101 b -0.0005 -0.0019 -0.0036 -0.0056 -0.0076 5 «0.000h -0.0015 -0.0029 -0.00hh -0.0061 reactance than a single conductor line of the same phase of spacing S and of equivalent cross-section. It has also been shown in Problem 1(b) that for the same reactance of both bundle and single conductor, the diameter of a single conductor has to increase five times for particular spacing of sub-conductor. CHAPTER II CAPACITIVE SUSCEPTANCE OF THU CONDUCTORS PflR PHASE Assuming a 3-phase bundle of a two conductor line is completely transposed so that each conductor occupies each of the tower position for equal distance, the phase being rotated in cyclic order, the total capacitance to neutral can be found by Geometric Mean Distance method, by means of the approximate average value of the charge when the charges on the line are unequal. For horizontal flat configuration, a formula for the average capacitance to neutral per mile of each conductor of a perfectly transposed 3-phase transmission overhead line having two Conductors per phase has been derived in equation (2b) of Appendix 2. From equation (2h) 1 18 x 109 1n (Sm. d>2 mr farad The capacitive susceptance for 60 cycle to neutral from equation (25) 1h.66 x 10"6 - loglo (ngd)2 mr be mho/mile Since two conductors are parallel, the average b per phase is two times that per conductor. From equation (27) 29.28 b (per phase) = loglo ZHKngd)2 micromho/mile d m where d in inch The equation for b is computed by assuming uniform charge density. The error due to this approximation is of the order of 1 to 2%. It has been illustrated by Problem 2 and curve 2 in this paper that an overhead double conductor line has nearly 20% greater susceptance than that of a single conductor line of the same S and of equivalent cross-section. CHJPTER III SURFACE} VOLTAGE WDIENT OF BUNDLE CONDUCTOR With the tendency toward higher a.c. transmission line voltages, due to greater distance of power transmission, the type of line con-. ductor to be used becomes of increasing importance. is conductor surface voltage gradient directly affects corona loss, and corona starting voltage, the ultimate problem is of loss. This problem has a solution if a bundle conductor, i.e., two or three conductors per phase are used instead of one. A good physical picture of the phenomenon associated with flux conditions in a bundle conductor at the same potential is necessary for understanding the effects of bundle conductor two per phase. Figure Figure 2 illustrates the lines of force and equipotential lines, two conductors per phase of a 3-phase line. Figure 2 indicates the charge distribution among conductors at the instant of time when the voltage on the center phase is one and that of the end phases -0.S. Figure 2 also indicates that the lines of force are distorted between conductors, resulting in a variable voltage gradient around the con- ductor periphery. The analytical method used in the calculation of conductor surface gradient is outlined in the Appendix 3. The maximum surface gradient occurs at the middle phase, and the expression for g is from equation (38) Appendix 3 2r 2r2 _ Vb (l +‘E— ' m2 + . . . .) g ’ q 1 volt/inch 2x2'303r (loglo ‘ _._. 10g €> mr 3 where g is the maximum surface gradient at the mid-phase in volt/inch m is distance between conductor of the same phase in inches r is the radius of the conductor in inches Vb is the applied voltage on the mid-phase. As the charge density is not uniform around the wire of the bundle conductor circuit, a precise computation of its value is determined by the equation given in a paper1 by H. B. Dwight. Problem 3 in this paper is an example of g with varying sub-conductor distance. Curve 3 illustrates that the greatest reduction in maximum gradient is with a sub-conductor Spacing of 12 inches. The equation is deduced on the following assumptions: a) Conductors are arranged horizontally, i.e they are in the '3 same plane. 1H. B. Dwight, "The Direct Method of Calculation of Capacitance of Conductor," Trans. A.I.E.E., vol. 52, 1933. b) Conductors are perfectly smooth and round with m Spacing for other conductors of the same phase. c) The distance between phases S is considered from the center of the conductor of one phase to another. d) The term voltage gradient designates the maximum voltage- gradient at the conductor surface. e) The ground effect is neglected. The charge distribution on the conductors of one and two conductors per phase are shown in Figures 3 and h, with applied voltage +1 on center phase and -0.5 on the end phases. -0.025 -0.031 +0.05 +0.05 -0.031 -0.029 Figure 3 -0.0L5 +0.085 -0.0h5 Figure h The equations for the calculation of the above charges in bundle conductors each having self and mutual capacitance with each of the other conductors and with certain applied voltage to ground, the only way of calculation is by Maxwell coefficient. LTwo-Dimentional Fields in Electrical Engineering, Bewly & Macmilan, l9h8. lO _ 7 r-— -- F- - VJ. P11 P12 P13 P14 P15 P16 Q1 V2 P21 P22 P23 P24 P25 P26 Q2 V3 P31 P32 P33 P34 P35 P36 + Q3 V4 P41 P42 P43 P44 P45 P46 Q4 V5 P51 P52 P53 P54 P55 P56 Q5 LY6._ 31 P62 P63 P64 P65 P1! Lqe— where V is the voltage applied P is Maxwell‘s coefficient or proportionality factor Q is the charge on conductor For abbreviation the equation can be rewritten: . -1 [Q] = [P] [V] It may be shown in general, by Green's Theorem that Prs = Psr (Electricity and Magnetism, Jeans) and they are all positive. The numerical value of the potential coefficient has been worked out by the method of image considering ground as zero potential for the bundle of two conductors per phase, in Appendix 38. The general equation deduced for P in Appendix 3B is as follows: P11=P22 =~ . . =Pee = 2103?“ P213...+..P56 =210g2€2 where h is the height above ground, and 2h is the distance from the conductor concerned to its own image. Drj is the distance between conductor r to the image of con- ductor j and so on. CHAPTER IV DISRUPTIVS CRITICAL VOLTAGi OF THE BUNDLE The discuptive critical voltage is the voltage at which corona starts to form on transmission line conductors in fair weather. This is the limit of operating voltage on the line. It depends on the potential gradient at the surface of the conductor. In Chapter III, g, the maximum stress at the conductor surface has been derived and g, corresponding to e0, may be called the disruptive gradient. e0 greatly depends upon the conductor radius, if r is smaller, then eO will be larger. As the surface gradient of the center phase is the highest for horizontal configuration, a formula is developed in Appendix h for ac for the middle phase conductor which has the lowest critical voltage. For a perfectly smooth polished conductor for which there is only one per phase e0 =gxerox§10ge kv ale: where g is the critical gradient 8 is the distance between phase conductors r is the radius of the conductor. If each phase consists of 2-conductor of Spacing m from equation (h?) then _ ngoé; S2 k eo-ml+-2_I: ogefi? V m 11 12 a is considered constant for conductors used in H.V. transmission line and value of g for wires taken under a variety of conditions on 1 the outdoor line are given in Table III. According to Peek, g is 53.6 kv/inch. Since g = 53.6 kV/inch e = 12 rM S 52 e --;i—€%—- 10S10"‘ kv 1 +._£ mr m e0 is determined by the equation (L7) in Appendix 5, and M0 is taken as 0.96 for solid wires. TaBLE III DISRUPTIVE ChITICAL VOLTAGE GRADIETT FOR WIRES (Values corrected to 76 cm. Barometer and 250) Spacing in cm. Radius in cm. go kv/cm.max. 152 0.08t 31.3 229 0.08h 31.6 550 0.0th 36.5 122 0.16h 26.8 2th 0.161; 29 .0 366 0.16t 25.6 h88 0.16h 25.3 9l.h 0.259 28.7 183 0.259 26.5 275 0.259 26.0 397 0.259 26.2 91.h 0.h63 28.7 183 o.t63 30.h 21b O.h63 30.5 275 0.h63 31.0 F. W} Peek, Dielectric Phenomenon in High Voltage Engineering, Third edition, McGraw—hill Book Co., 1929. CHAPTER V CORONA LOSS MBaSUREMENT In bundle conductors, the voltage-gradient is not uniform around the conductor periphery. A correlation must be established between corona-loss and the variable voltage gradient. For theoretical calcu- lation, relative losses can be determined by working with maximum conductor surface voltage-gradient or for two conductors per phase grouping; the mean value of maximum and average voltage-gradient is used. ‘The maximum voltage gradient is considered in this paper. The loss on a transmission line varies depending upon the temperature, and weather conditions. Some various factors like mois- ture, frost, fog, sleet, rain, and snow have an appreciable effect upon the loss. A number of formulas has been worked out for estimating the corona loss values of which Peterson's formula is widely used. A theoretical formula modified by empirical correction was developed by Peterson which is applicable for calculating corona loss for the lower value of the losses as well as higher values. The corona loss is considered to be the loss due to charging current flowing rapidly through the corona-envelope. The drop in voltage through the envelope is assumed as being the integral of the potential gradient from the point where the gradient in air exceeded 53.6 kv/inch to a point on the conductor at some higher value. 13 1h For the portion of the cycle where the voltage is too low to cause air-breakdown, no loss is assumed to occur. 1 As a result of this theory, the following formula is evolved. From equation (E8) Appendix 5 0.0000337 e 2 PC = 5.? n QC!) kw (logic ; )2 where PC is the corona loss in kw per mile per conductor f is the frequency 8 is the distance between phases r is the radius of the conductor in inches eo is the disruptive voltage in kv (VC' is the value taken from the curve A and is a function of the ratio en/eo. For a smooth round conductor, the value of e0, as calculated in Appendix t, can be used with a slight modification for air density. 2 S2 90 = 123 M08 A? 10510 '5; kv to neutral 1 +.§£ m The air density 5= 1 at 2500., 76 cm. barometer. At any other condition, it may be calculated by the following equation: 1Joseph Carroll, "Corona Loss Measurement for the Design of Transmission Line," Trans. i.I.E.E., vol. 52, 1933. where b barometric pressure in cm. t temperature in degrees Centigrade M0 is the irregularity factor and varies from 0.98 to 0.93 for roughend wires en is the voltage to neutral in kv (applied). The equation (h8) in Appendix 5 is true for one or two conductors per phase. For six conductors, the total loss is six times that of Pc- For all practical purposes, the losses on the bundle conductor, however, can be calculated on the basis of those of the single con- 1 ductor. * According to theoretical calculation in Problem 5, the corona loss are about the same when the spacing is 15 inches. From the test2 experiment on the bundle of two conductors, it has been observed that losses are much smaller than that of a single conductor per phase. The losses on a single conductor per phase is about ten times as high as those of the bundle. 1 0. Gerber, "Corona Losses of Single and Bundle-Conductors," C.I.G.R.E., paper h03, 1950. 2 F. Cahen, "Results of Test Carried out at the 500 kv Experimental Station of Cherilly (France) on Corona-Behaviour of Bundle Conductors," Trans. A.I.E.E., vol. 67, 19h8. -._4 _.___.___- -_-_— .a-q ,, i i o . v _*__ . __.__« CHiPTER VI APPLICATION Problem 1 Assume a 3-phase line of Spacing between phases to feet horizontal flat configuration. Assuming that each phase conductor consists of two sub-conductors having one inch diameter each and Spaced m feet apart horizontally. a) Calculate the phase inductive reactance as a function of m and plot. b) Calculate and plot the diameter of a single conductor as a function of m which is equivalent to the bundle in the sense that it has equal inductive reactance. Solution a) Assuming m varies from 8 to 2h inches 2) conductor diameter are such as to keep the total conductor area per phase constant. 2-conductor per phase -l diameter one inch of each sub-conductor l-conductor per phase -— diameter l.h1 inches (for equivalent cross-section) CD CD 0 Hit—:- Figure 5 4"— S —" 17 18 Reactance from Appendix 28 for two conductors per phase 1 X =-; [0.030h + 0.279hl 10210 2b 3/5— a ] + 0.1397 10glo 3/2—'§ 6/ [1_(§)2]2(1_(g§)2] ohms/mile where S and m are in feet and d is in inches; S/d is constant. The first part of the X is constant and equal to: % {0.030h + 0.279h1 loglo 2t x 1.25 x gg_ ] 1 = O.hh The second part of X is a function of m and can be calculated: m in m in inghgs feet loglo 3/2‘ Sfim .8 3/h 1.8325 12 1 1.699 16 S/h 1.6021 20 5/3 1.147771 2h 2 1.h176 The value for \6/[1-(.1r39)2]‘2 [1433?] is practically constant and equals to 0.955 for m varying from 8 to 2b inches. The value of the second part is now tabulated below: m in inches 0.1397 logic 3/5— % 5/[1-(§)2]2[1-(%§)2] 8 0.2113 12 0.225 16 0.2111 20 0.196 2h 0.182 19 Now the phase inductive reactance is shown as a function of m. m = 8 inches X (reactance) 0.hh + 0.2h3 = 0.663 in ohms/mile m = 12 inches X = 0.hh + 0.225 = 0.665 m = 16 inches X = O.hh + 0.21h = 0.65h m = 20 inches X = O.hh + 0.196 = 0.636 m = 2h inches X = O.hh + 0.182 = 0.622 It is found from the result as the Spacing between the sub-conductor increases, the reactance decreases. The relation between m and X has been plotted in curve 1. b) For the single conductor per phase of diameter l.h1 inches reactance for phase Spacing to feet is calculated as follows: Figure 6 i) As the (1.14.1). of the figure is 3/2‘5, from Appendix 1 X [0.0303h + 0.279111 loglo 2b 351% ] 0.851 ohms/mile This value will remain constant if S and d are constant. ii) To calculate the diameter of a single conductor from the result of the part a) considering equivalent reactance in both the cases. x in ohms/mile 0.683 - 0.0303b 0.665 - 0.0303h 0.65u - 0.0303h 0.636 - 0.0303u 0.622 - 0.0303h from a) 20 0.0303h + 0.279h1 log10 2h 3/2" S/d 0.279tl log10 1200/dl 0.279hl loglo 1200/d2 0.279h1 loglo 1200/d3 0.279h1 loglo 1200/d4 0.2?9h1 log10 1200/d5 S u ‘.-3 u S n 5 u inches (7." 12 inches 16 inches 20 inches 2b inches vs d 5.5 inches Q. H II d2 = 6.3 inches - 6.9 inches 0.. (I) I d4 = 8 inches 6.7 inches 0.. 0‘ II III- I I'll- -‘I _ ll _ 1. —.—-—. a 5 —_ - ‘----o n—.—.—.—+-¢ _.- ._—.. .——.—f——..—o —-—- _L—._ .— I." fill: '11) Q .i ll 22 Problem 2 Assume a 3-phase line of Spacing 20 feet lxuizontal flat configuration. Assuming that each phase conductor consists of two sub-conductors having one inch diameter each and Spaced m feet apart horizontally. a) Calculate the phase capacitive reactance as a function of m and plot. b) Calculate also for the single conductor per phase of equivalent cross-section. Solution a) Assuming m varies from 6 to 2h inches Conductor diameter are such as to keep the total conductor area per phase constant. 2-conductor per phase -- diameter one inch each l-conductor per phase -- diameter 1.hl inches Considering Figure 5 in Problem 1, S is the phase distance and equal to 20 feet. Capacitive susceptance from Appendix 2 for two conductors per phase 29.28 b = loglo 214 3/73 + loglo 3/2— 5:2 mho/mile 29.28 s s _—-m 2 2 m 2 logic 21 3/2‘ a + logic 3/2' in 6/[1-(3) 1 [14-53) 1 In the above equation, S and m are in feet and d is in inches; S/d is constant. 23 First part of the denominator of the expression is constant and equal to: loglo 2t 3/2’ S = 2.632h Second part of the denominator of b is a function of m and can be calculated. m in m in ipghgs feet 8 2/3 12 l 16 b/3 20 5/3 28 2 The value for logic 3/2‘ s/m 73? ell-($31 log 37.5 log 25 log 18.75 log 15 log 12.5 1-(§)2] is practically constant and equal to 0.96 for m varying from 8 to 2b inches. The value for the second part of the denominator is now tabulated below: m in inches 0'3 log... 3/2“ s/m 6/[1-<§)31211—(§321 1.556 1.380 1.255 1.155 1.08 l“ 2h Now the capacitive susceptance is shown as a function of m as below: m in inches 8 12 16 20 2h b in_L; mho/mile f 29.28/L.188 = 7.05 29.28/h.01 = 7.35 29.28/3.88 s 7.6 29.28/3.78 = 7.75 29.28/3.7l = 7.8 It is found from the above result as the Spacing between sub- conductor increases, b increases. m and b has been shown in curve 2. b) Single conductor per phase, Reference to Figure 6 G.M.D. 3/2' 3 Nature of the relation between diameter of which is l.hl inches. of the Spacing between phases ° b _ ' 1h.66 1h.66 . . loglo .213. 3”— S loglo 300 = 5.9 mho/mile. o—Q—o— --.-_...._ ._ _.-._ - w _ n _ T _ _ _ _ -._+ ,_., _ o . IOI III *IIOI . fi --——-0- -.—.-— o a o o 0-—.——-——v— — o c. l l .._——9- r—o—t— I 0“. 0—0 Problem 3 Assuming a 3-phase transmission line of Spacing between phases 32 feet horizontal flat configuration and considering each phase consists of two bundle conductors having one inch diameter of each. a) Calculate the maximum surface voltage gradient of the middle-phase as a function of m which varies from 8 to 12 inches. b) Plot the surface voltage gradient reduction in per cent of bundles based on value of single conductor of equivalent area. Solution a) Assuming m varies from 8 to 12 inches. Conductor diameter are such as to keep the total conductor area per phase constant. 2-conductor per phase -- diameter one inch of each l-conductor per phase -- diameter l.hl inches Maximum surface gradient occurs at the middle-phase. Then from Appendix 3, equation (38» V’b(l+g-1:-2r2+oooooo) m m2 volt/inch 2 x 2.303 r (log 10 - l 1082) mr 3 where m is the intragrOUp distance in inches r is the radius of bundle of each in inches S is the phase-conductor distance in inches 27 The numerator of the expression g is calculated, taking Vb as one volt, the maximum value at particular instant when the voltages of the other phases are -O.5 and neglecting the higher order term in the parenthesis. m in inches (1 + 2rzm - 2r2Zm2 + .....) .8 ' 1.15 12 1.08 16 1.06 20 1.05 2h 1.08 The denominator is again a function of m and can be calculated with the values of S equal to 32 feet. S/d is constant for varying m. m in inches 2 x 2.303r(19310 S/Zfi; - 1/3 1082) 8 5.0h 12 b.82 16 b.7 20 b.55 2h h.h2 The maximum surface voltage gradient of the middle-phase as a function of m is given below: g (maximum gradient) m in inches g m for Vb = 1 volt 8 0.228 16 0.226 12 0.22h 20 0.231 2h 0.23b 28 b) g of the middle conductor with flat Spacing one conductor per phase, from equation Appendix 3. V6 g = 2.303r (log S - % log 2) The radius of the conductor equals 0.705 inch g - Considering vb as 1 kv (1000 v) m in inches 12 16 20 2h 2.303 X 0.705 (log Maximum gradient for middle phase of bundle in volt/inch 3FHF .709 228 22h 226 231 23h Maximum gradient for middle phase ' 1 log 2) 3 volt/inch Per ce reduct nt gradient ion of bundle of single conduct- based on the value or in volt/inch 238 23b 23b 23b 23b of sin gle conductor 2.6 b.26 3.h2 1.28 0 Voltage gradient is decreased by more than h% for Spacing 16 inches based on the value of single conductor per phase of equivalent area . l' ; l .. -5-.. .. . . a I . ll I’ll'T lull." a e n o o o - -.-._- u. m _ _ ....... . llllllll' +IOIIIIO IOI . one. _ M _ a col ."w _ — . FLIIEI. a .— . a _ . .il‘n‘lll.‘ H .Mth. ~‘l. .o.o“ .0. 0.-.. . .I. ..«H 11. c..» . o.-.aououu ; o W O _ . + . . . o ..¢oo 0 0| 0,.“ I . _.....m....._..........._*.-..F» *"..r.... — ._ .—.. __ ”.1“4 . l ..m._.._.." . ”.l. a: _ l :................yn+-..i+Tn+u_ , _. . 301- o a. at. n l 9 OI. ” o. _ 1 . _ I” o . w a . H . . H . a n a o + . w . — . 1 a a _ O . . . . ~ . . - 0.. *‘L . . +ai1h'm flklhc IOIIPAIVI _ a. u 1‘. 0 .. co o col .>.9.¢ lo. I .n _ _ . , I ..l oll . . __ _ a . . . . . _. .m. ~ ._. «Jim .t 7+3. 4-“th -LL- 30 W a) Considering the Problem 1, calculate the disruptive critical voltage of bundle of two conductors per phase for m, the spacing between conductors of same phase, varying from 8 to 2b inches. 0 b) Plot a curve showing the relation between single and bundle conductor in respect of eO for varying m. Given MO = 0.96 5 = 0.966 Datas: S = LO feet, r = 0.5 inch for bundle and 0.705 inch for one. Solution The lower disruptive critical voltage of mid-phase conductor can be expressed by the equation (h?) in Appendix h. 12 M é; 2 80 = "QEEEfiF" loglo .§_ (1) l +-__ mr m The equation (1) is a function of m only and S, m, r are in inches m in inches 123r Mo /1 +-g% loglo Sg/mr 5 51.5 b.76 12 53.6 b.58 16 Sb.6 b.bs 20 SS.h b.36 2b 55.6 b.28 31 The equation for ac is based upon the value of g, the surface gradient of the conductor which Peek1 took as 53.6 kv/inch. Peek indicated that g is constant for conductors of diameter that would be used for H.V. transmission line. But subsequent investigation2 raises considerable doubt as to whether g is constant in the range of low losses. The indications are that g might well increase with decreas- ing radius shown in Problem 3. This factor can be taken into consider- ation and would be regarded as a factor favorable to bundle conductor. The calculation of Problem h on the basis of 53.6 kv/inch for g e0 in kv/corona m in inches starting voltage 5 2b; 12 2th 16 2&2 20 2L1 2h 2h0 eo for single conductor per phase is given by the equation (L7a) e0 123 MorSloglo S/r kv 123 x .96 x .956 x 0.705 log %—$—O-; 228 kv. The gain on the corona-starting voltage of bundle over single con- ductor per phase is 8%. lF. W. Peek, Dielectric Phenomenon in High Voltage Epgineering, Third-edition, McGrawbhill.Book 00., 1929, 2 M. Themoshok, "Relative Surface Voltage Gradient of Grouped Conductors," Trans. A.I.E.E, vol. 67, l9h8. o o u o _ h b I O . o o . . u . lrllt. null null 11" _ v t 0 o Q o n - —_ .—.—-.—..——.——.— In... _ o--. -‘- -——-— a I ' I I O .--.—-. H _.—--—o -—-n —.—‘ — --+ l o I -r .—.._—c—-9 ' n -o I Q——§- ‘1 u-— t I - -—o I l . Fo—t- ,l’.| [1110 1 ‘wlil 11!. n: plllfil,.. I I’l‘x‘ilr‘fitb (1‘ i . ‘1 t 33 Problem 5 a) Calculate the corona loss of the bundle of two conductors per phase having one inch diameter of each. Assume 3-phase line, Spacing between phases LO feet, horizontal flat configuration. Voltage between lines is h80 kv and f 60 cycle. Temperature is 200 C. and barometer 72.2 cm. Plot the loss for 3-phase as a function of m, sub-conductor Spacing. b) Calculate the corona loss at the same temperature and pressure of the single conductor per phase of equivalent area, Solution a) The loss is given by the equation (hS) of Appendix 5 of Peterson formula: 0.0000337-3} en2 c z . S U?%') kw/mile per conductor (loglO-;)2 fig; == The ratio of en/eo, the value of which is given in the curve h e is the voltage to neutral and equal to h80/l.73 = 278 kv. The air density factor 5 3.926 = 3.92 x 72.2 273 + t 273 + 20 0.966 The numerator of the equation is constant for particular f and' e and the value is calculated below: I], 0.0000337-"f-en2 = 3.37 x 60 x (275)2 10'5 = 156 3h The denominator is also constant for particular S and r which equals 3 ix (logic g) = (10g10‘%4%3)2 = (logic 960)2 (2.98)2 = 6.95 _ 156 Pc “-579 (PC. 217-5 (PC.- Now for same MO and e; we can take the data from the Problem h m in inches eO in kv en/eo 8 2b; 278/2h5 = 1.13h 12 2th 278/2uh = 1.1h 16 2&2 278/2h2 = 1.15 20 2&1 278/2b1 = 1.152 2h 2b0 278/2b0 = 1.156 The value of en/eo taken from the curve h and tabulated en/eo CPc' 1.13h 0.053 1.1L 0.062 1.15 0.07 1.152 ' 0.078 1.156 0.085 Final results can be calculated thus: 35 m in inches P (corona loss for 3-phase 17.5 x!§§fl_per mile) 8 5.h5 12 6.38 16 7.26 20 6.3 2t 9.8 The minimum loss in fair weather condition corresponds to a spac- ing of 6 inches. Beyond 16 inches losses keep on increasing more rapidly. At 2h inches, they are almost double. b) Calculation of loss for the Single conductor per phase is exactly the same as above: r = 0.705 inch b80 2 e0 = 228 kv from Problem h; (loglo 0.705 ) = 19.5 en/eo = 278/228 = 1.218 From curve h, the value of (Pb' corresponding to 1.218 is 0.11 PC = 19.5 x 0.11 The total loss for 3-phase equals to: P = 3 x 19.5 X 0.11 = 6.h3 kw per 3—phase mile According to theoretical calculation, the corona loss is about the Same when the Spacing is 15 inches. But corona loss is smaller when the Spacing is less than 15 inches. Though there is not much difference between single and bundle so far as loss is concerned, but it is an advantage that the corona starting voltage is much higher in case of bundle. Illl. I 6 leI 100'! O'Ill ' .III #———4 C | I . 6 v a 6 n I I 6 6 6 1 ~6— 6_6..__ .6-.. 6- I 6 I 6 6- I 6 6 4 --. -6 4 I | ’ f 6 -6 .- . A .._—6 6-- .6 1.66 >T——. l _1_ 6 66- J] 1 L 7 e ° 1 I O .._. -.. 6.6—6. .._.- v 74 A 1-.- . 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TABLE 1112 Line Length 220 kv 220 kv in km 592 mm2 (single) 2 x 328 mm2 (bundle) 200 100 (base value) 98 hOO 100 93 600 100 90 800 100 87 1000 100 86 1Comparison of Lines Cost with Single and Double-Conductor, C.I.G.R.E., paper 1105, 1950. 21bid. CHAPTER VII CONCLUSION An overhead double-conductor line has approximately 20% less reactance than a single conductor line of the same weight of conduct- ing metal. If the voltage control is such that it can operate successfully with a certain per cent of current times reactance, on either 50 or 60 cycles frequency, the allowable load on the double conductor line is 1.? times that of a single conductor line, In many usual cases, eSpecially where there is not complete control of the voltage by synchronous condensers, the reactance is the most important item in determining the power rating of the line for both the voltage drop and the stability limit of the load which depends principally on the reactance. Therefore, in many instances, without increasing the weight of the conductor metal, a line can be built for about one- fifth greater power rating at very little increase in cost where ice load is absent by using double conductor construction. The "Bundled" 336,hOO C.N., A.C.S.R. circuit costs approximately $600 per mile more than the single 666 C.M., A.C.S,R. circuit (which is approximately 5% more based on total circuit cost) whereas it has approximately 30% more actual capacity based on impedance drop limitations. The capacitance of an overhead double conductor line is 20% greater than that of a single Conductor line of the same weight of 1H. L. Deloney and hfi L, Rush, "Bundle Conductor for Trans- mission Line Capacity IncreaseJ'Electrical world, December, 1955. 38 39 conductor metal (Graph 2). This is an advantage in the case of power- networks where synchronous condensers are used almost entirely with strong field currents. An increase of the line capacitance of 20% means a definite saving of the amount for synchronous condensers. Another advantage of the use of bundle-conductor is the gain in the appreciably lower value of surge impedance as the inductance of the line decreases and the capacitance increases with greater Spacing of sub-conductors. with the lower surge impedance, there is a possibility of greater stability and a larger capacity of the line. With the same amount of material, much higher voltage can be used without corona loss when the transmission line conductor is built with two small conductors per phase properly arranged than with a single conductor. It is due to the multiple shielding effects of the sub-conductor which reduce the electric field intensity on the individual conductor. It has been calculated that the surface voltage- gradient in case of bundle conductor reduced by h% for a particular Spacing based on the value of single conductor of equivalent area. APPENDICES APPENDIX 1 Inductance and 60 Cycle Reactance of Bundle Conductor Circuit When a 3-phase bundle two conductor transmission line is com- pletely transposed so that each conductor occupies each of the tower position or its equivalent for equal distances, the phases being rotated in cyclic order, the total inductance to neutral of any con— ductor will be the sum of the inductances in each position. The phase position on the tower will be designated by I, II, III and the conductor positions 1,2; l',2'; l",2" respectively. Starting with phase A in position I, the conductor of the phase A which first occupies the position I-l will in turn occupy each of the six position for equal distances one-sixth of the total length of each conductor. Let I I t/2 1, U0 Assuming I _ 2 I I 1 . /3“ . , E (. .2. + J .2...) —2- a current in phase III For since waves of current of frequency f, the total inductive current per phase length (total) of each conductor length of conductor in each position .3?) a2 current in phase II “NFJ n tvha 'voltage drop in any conductor will be the sum of the voltage dr0ps ill the conductor for 6 positions and may be eXpressed in terms of self'inductance L and mutual inductance M. 'P b1 QHQ CO Figure 7 Position I-l VI--l = 2TPF[ %' (L1 +1"11.2) +% a3 (5111' + 1412') +11;- a (9111" + M12")] There Will be two equations one for each of conductor (1) position in phase position I. There will be also two equations when phase a occupies position II and phase position III. I ' I V . I VII-.l' = 21T‘FE §(L1 +M'1'2') + _2_ a2(l'11'1.+141'2.)+'§ a (1511'). + I\‘11'2)] (2) VIII-l" = “If [ %(L1 + M1"2") 4*; 32(1'v11"1 + Mfg) at; a (riff 231311"th Adding for 6-positions, the total drop V is obtained V = 2“? ;2[" [6L1 + 2 (£112 + 241.2! + IJ11"2' ‘ (Mli' + Ml'l" + I"11"]. + “12' + M1'2' + M1"2 +141; + Ml-g +1415) + Mag! + Mg'g- +1425” (t) From equations (1), (2), (3), and (h), it is seen that there will be 12 mutuals between conductors in the same phase positions and total 3 for the 3-phase positions. The number of mutuals between conductors of different phases will be 12. The total inductance Lt; V , ,V I’t = 2Wf.I_ = [0L1 4’ 2 Z3Maa ‘ ldwab] (5) 2 142 where Maa mutual inductance between conductor of same phase, mutual inductance between conductor of different phases . The formula in absolute units for self and mutual inductance of non-magnetic wires in air follows from those given in the "bulletin of the Bureau of Standards" when the length L':2>.d and S hL' L1 = 2L' [ loge '—a— - -E ] self inductance (6) , 2L' , ., h = 2L' [ loge -—§- - l ] mutual inductance (Z) heplacing L' by L/6 in (6) and (7) and substituting in (S) and called Maa and Mab by Geometric mean distance of m and 8 respectively, the final equation becomes: L = 6 x 2 x [Ln.7— - +-—v-— -Ln *7 - l t 3 .d 1: 1 O [ mm 1 - 12 x 2L [ Ln 2L - 1 ] 6 6 ngd 2 hL 2L 6 x (ngd) l x = L [2 1n 7 X (iriflnd X L, L2 + -2- ] (t) “h L2 - , 2 1 . Lt - L [2 ln 5 + 2 ] abhenries (9) Expressing in practical units and d and M in same units, average inductance to neutral per conductor is: 2 Lcon = 0.080147 + 0.7h113 loglog 52ml. as mgmd = m (10) Tflfli/mile and average 60 cycle reactance: per conductor X con _ o 3 3 + o 79 Oglo d T— OImS 1“]. e 1:3 Reactance per phase of two conductor 2‘512 [ 0.0303h + 0.279hl log10.€§.gagfliz_ ohms/mile (12) x = 1h nan where d in inches ML .LI‘PENDIX 2 Capacitance and 60 Cycles Reactance of Bundle-Conductor Circuit Considering a 3-phase transmission line with one conductor per phase, completely transposed so that each conductor occupies each of the tower positions for equal distances, the phase being rotated in cyclic order, the total capacitance to neutral can be found out very easily by Geometric mean distance method, by means of the approximate average value of the charge when the charges on the line are unequal. A case which occurs in this case is that a symmetrical flat Spacing, arranged cab, the middle conductor a has its potential vector along the axis of reference. Assume that the charges qa, qb, qc are 120 degrees apart in phase and they are of the same magnitude in all sections of the line. “ho e; ta 6 O b I Figure 8 qc= l q,a+jCI/~ ‘ (13) qb= 3- %a-j cog, (no The potential difference produced by these assumed charges hS Eab=18x109 : qu 1n BBQ-‘3 (15) j=a 3a =1sx109( q/aln'15:+ quln§+q,cln2-g) =K[%a1n-§-+(§q,a+jqzq)1n§+(-§q,a+j‘bq)1n21 where K e 15 x 109 ab “-3- ealn 332:3 @1153] (16) m :.g E + j /%_E I Equating the reals to the reals and imaginaries to the imaginaries and solving ~‘I ‘1.”‘9- 'mn.;¥.fl‘w ‘ud.-‘ E . , qa = 18 x 109 1n 5~ cou per meter (17) B/Er 52 qq = 36 x 109 1n g§ cou per meter (18) In the next transposition sections where phase a replaces b, b replaces c and c replaces a, the charge on a will be equal to the charges on b in the first section considered, shifted forward through an angle of 120 degrees. Likewise in the remaining section qa will be equal to the value qC in the first section shifted backward through an angle of 120 degrees. Designating sections 1, 2, 3 E qal ‘ 18 x 109 1n 5 (19) 372" 1 E [3'13 q212‘“2 KlnL'JgKlnfl‘S )(‘§+3/:§'_) 3/2 r “F .._ Efi + BE J(/-3—E 31L ) hKl D M 1n 23 11K 1n§§ LK1 L6 q33=('% ES +j .133 )Q-l-j-IL) ~ K 1n _ 2x 1n 2s 2 2 3/2r "? E 3E /3E /3—E = D + :1 -a'<“““2‘"um a -_-—-—s hK ln hK ln :__ n"; ‘ LK 1n 3 2r T 3/2r The average of qa over the whole transposed line is equal to (21) qa (average) 3 (22) E 1 q 1 1 9 12 1 ~ + 12 1 l§§ f 3 10 1137'??? n I" E l 2§ + l 8 =72? [ n 1‘ n3/2r ] S l 23 n 3/2r 1n r E [ 1n 312 s = 'K r (lnfl _ 22/3)(1 2E+1 256) r 1n n r n = E K 1n 3/§_S cou per meter r (23) l C = 18 x 109 1n 3CE:S Farads/to neutral 2L) r (' The term In 3/2—8 is equal to the geometric mean of the three Spacings. The capacitive susceptance to neutral is ‘3‘ “he (5) . 2 15 x 109 1n sgmd r . -e b = 2Tr¥‘ x 60 x 160,900 _ 1h.66 x 10 , / ,1 con .18 X 1011 X 2.303 logic Scmd - logic Spmd m1o m1 e (26) ) h? For bundle conductor of two per phase, the term is identical with the inductive reactance but reciprocal of the external inductance to neutral. ,, -6 2 x lb.66 x 10 29.28 b (per phase) = 2b S d 2 = loglo T (—%—) 2h Srm 2 micro mho/mile where d(diameter) in inches (27) Note: It is computed by assuming uniform charge density. The error due to this approximately is the order of l or 2% in transmission line equation. b8 APPnlh'DIX 2 (8) Equation for the Geometric Mean Distance of the unsymmetrical 3—phase line with two conductors oer phawe. = case for the hogicontal flat configuration: Figne 9 Geometric Mean Distance of the conductors of the different phases: U) (0 ll gmd -[ 12/é4<2s)22 1 U1: .._, II 2 1 m 2 s .3/11— b/[l-(E) ]2[1-(§5)2] Let for abbreviation Z = 6/7 1_‘§ 2 2 l- m 2 [ (S) 1 [ (fig) 1 Then from Appendix I, , 2213 = 3 [0.03011 + .2791 loglo T 3/2' ] + 0.1397 loglo ME; 2 . 2 ohms/mile (29) From Appendix 2 29.28 logic 211 3/5 g + 10810 3/5 .3. z m mho/mile (30) h? 50 APPENDIX 3(A) Surface Voltage Gradient of Bundle of Two Conductor Per Phase The dielectric flux density at a point x meters from the axis of the charged wire: D = q/X. coulombs/sq.m (31) This is true because we have a total flux of q coulombs per meter length of the wire passing radially through the curved surface of a circular cylinder one meter long having radius x meters. The area of the surface is 2 “'1 1:1. The force per unit charge in the field, at radius x which is the same as the field intensity or voltage-gradient is d3 = 1 dB 36 x 109 x Q, ' a; E a; = 2 ~32— volts/m g0 = 18 x 1011 8).:— volts/cm (32) Therefore, the gradient at the surface of an isolated conductor is 18 x 101-1 _3_ ---------------- (32) r where Q in cou/ and r radius in inches From Appendix: 2 equation (17) Eb qb = 18 x 1011 1n S Cou/cm 3/2‘2 ._ Eb m1n § - ; 1n 2) 3 (33) 51 The value of qb in equation (33) may be substituted in equation (32) giving the followins: The voltage gradient at the surface of the middle conductor with flat spacing; one conductor per phase to neutral: g _ 18 x 1011 Eb _ Eb o _ - . s 1 S _ l n r (18 x 1011(1n F - 3 ln 2) 2.303? (logic ; 3 logloc) (3b) This gradient is in volts per inch when r in inches. For a trans- mission circuit which has one conductor per phase, the assumption that the potential gradient is uniform around the surface of the conductor is justifiable, when the ratios of the diameter to Spacing is very large. When the charge density is not uniform around the wire in case of bundle conductor circuit, a computation of its value may be needed for precise computation of corona. A convenient way to compute the charge density and the voltage-gradient at any part of the surface is used the following equation from paper1 by H. B. Dwight, equation 6. For the charge density at angle 9 due to qa, the wire‘s own charge and qb, the charge on a neighboring parallel wire at an axial distance p. q = q/a qlb (3 2I1r - 'fl'r r r2 rn (.5 0086 +733 cos2e+ ....+—fi cosne) (35) The angle 9 is measured from the line joining the centers of the two wires. 1 H. B. Dwight, "The Direct method of Calculation of Capacitance of Conductor," Trans. 5.I.3.d. vol. h3, 192b, S2 The effect of charges qc: qd and so forth, on other parallel wires is 23 given by series involving qb but "/// angle 9 is measured from the approxi- P mate line of centers in each case. Figure 10 In the case of double conductor line, where there are two con- ductors per phase, qb is almost exactly in phase with qa especially for the middle phase which has the greatest gradient. For conductors of d inches diameter and m inches apart the effect of the conductor is to multiply the gradient at e = 180 degrees by l + 2r/m-2r2/m2 +_ +(higher power neglected) (36) The charge on one of the middle conductors of a double-conductor line is one-half of the charge per phase and Bi 0 b ‘ 2K(log EES' - g log 2) (37) Substituting this equation (37) in equation (32) and adding the correction for distortion of the charge density and putting GSS = mr The maximum gradient obtained for the middle phase 2r 2r2 vb(l+-}-n-_Tn-§+oooo) = (38) 2 x 2.303r (log —§L- - 1 log 2) mr 3 53 APPENDIX 3(5) Maxwell's Potential Coefficient by Image Method A transmission circuit consists usually of several conductors and ground. For a three phase line, the two conductors per phase, there is a total of six conductors and ground. For calculation of potential coefficient by image method increases the conductor numbers to 12 if the effect of ground is replaced by the six images of the bundle conductors of two per phase. The equations for the calculation of charges with certain applied voltages to ground can be expressed by the equation _ -1 [Q] = [P] [V] (38a) where Q1 is the charge on conductor P is the Maxwell's coefficient V is the applied voltage. By the laws of electrostatics. the gradient or intensity at a distance r from a line charge Q in a medium of permittivity is gush—52.1.1.1. = 2‘1 (39) w 2Wr€ r The potential reckoned from a distance R is I‘ dYr = __ 2? = 2? R Suppose a conductor of radius r at a height h above ground at a potential v with respect to ground. The ground plane may be regarded Sh as a zero potential surface, and the field will be the same as through the effects of the ground replaced by the image of the conductor at a . depth below the ground surface and having a charge -Q. From Figure 11 below) and from equation (hO) 0.1 = P1163]. = Q1 2 loge '7: (hCa) (€=1) Now the potential at a point distance d from #1 and the distance D from the image #1' is again by equation (LO) U2 = #2111 = Q]. 2 loge g— (hOb) The equation (38a) can be rewritten in terms of Q, p, and v. e e ph 55 - --—--1 R1 P11 P12 ' ‘ " P16 PVT Q2 P21 P22""‘ P26 v2 Q3 P31 P32 ' ' ' P36 V3 = (L00) Q4 P41 P42 ‘ ‘ ‘ P46 V4 Qs P61 P52 ' ‘ ‘ PS6 vs Q6 p61 P62 - - - pee vs L. —1 in“ h- .- The equation (hOc) directly determines the charges on a three phase transmission line with two conductor per phase. The potential co- efficient P11, p12, etc., for a horizontal flat configuration can be taken from equation (hOa) and (hOb) for self and mutual reSpectively. Self P11 2 loge 2h/r Mutual P12 = 2 loge D/d The Figure 12 shows the arrangement of the conductors under consider- ation with proper distance-mark and the images are shown for the determination of 36 coefficient. 1 2 "' W s : 7' s+ 7/ //////// ////‘////7//////////// /////////I///// 01.9 3; Figure 12 II II II II loge loge loge loge loge loge loge Se 2 h/r D1'2/m Di's/S Dz'a/S-m Dl4'/Swn D15'/2S D16'/2S+m (hOd) The value of Djk; the distance between the centers of the jth conductor h ) S, and and the image of the kth conductor is given in terms of m. /(2h)2 + m2 /(2h)2 + 53 =/Qh02+(3m92 = K21)? + <25)2 = /(2h-"3) + (28m)2 DB's a /(2h)2 +(S—m)2 (hCe) The equation (hCe) is substituted in equation (hOd) which gives the coefficients (Maxwell's) in electrostatic units per cm. length of line. From equation (hOc) the charge on individual conductor of the 6—conductor then calculated out by assuming some applied voltage. AS the two conductors of the bundle are electrically parallel, the voltage 58 APPENDIX h Disruptive Critical Voltage of a Bundle-Conductor (Two) In Transmission Line For a transmission circuit which has one conductor per phase, the assumption that the potential gradient is uniform around the surface of the conductor is justifiable, when the ratios of diameter to Spacing between phases is very small (Appendix 3). However, a conductor of a multiple-conductor circuit is relatively close to the other conductor of the same phase. An appreciable field distortion may result which will lower the surface gradient. The following analyses determines the disruptive critical voltage assuming that the field due to the conductors of the other phases is negligible and the field produced by the other conductors of the same phase may be considered uniform. Let E01 = The field produced by the other conductors of the same phase in the region of conductor 1, conductor 1 may be any conductor in any phase. The intensity at a distance m from a line charge of Q per_unit length is: E = .33 (bl) 2 (h?) b01 = 22/[ m12 cos 912] where mlg = the distance between conductor 1 and the second of the same phase. 01 = the angle between a line joining the centers of 2 conductor 1 and 2 and the reference axis. 59 At the surface of a conducting cylinder in a uniform field, the maximum intensity is 2E01 and the total maximum intensity at the surface when the conductor is charged, is obtained by adding 2301 and fl‘ 1 la 0 U o . 2e/r and tne m Ximum intenSity Em 18, em = 2301 + 3% (£13) where Q = charge on conductor r = radius of conductor Substituting (b2) in (L3) and replacing r by d/2 the equation (ht) is obtained: Eml = hQ[ / (m%é— C05 812)2 +‘(%§§” Sin 612)2 *‘é ] (Eh) For a three-phase system the charge on the other two phase conductors are —Q/2 when the charge on conductor 1 is +Q, the potential of con- ductor l of phase 1 becomes:1 v1 = 2.“; loge 311.} = 22 log Lag (b5) where h is the height above ground Substituting the value of Q determined from equation (LL) in equation (hS) and 53.6kv/inch for Eml from Table III; the following expression is obtained for the disruptive critical voltage in effective kv to neutral of conductor 1, _ 2/(5 s )(s s ) V1 = 2 ml x 1n dlnlz l _ (hi3) h[\/(fi1—2' COS 9312);: + (all: Sln 812)2 +%] 53.6 x ln 2/(313314)(515316) dmie l 2 1 . 2[\/63;; cos 912) + (5:; Sin 912)2 + QIH 1Two-Dimentional Fields in Electrical Engineering, Bewly and Macmilan, l9h8. It is clear from equation (he) that in any arrangement, the con- ductors of the central phase have a lower disruptive critical voltage than those of the two outside phases as the numerator of the argument of the logarithm of equation has a smaller value for the center phase conductors than it has for the outside phase conductors. Introducing the irregularity and air density factors, Mo anch. respectively and multiplying numerator and denominator of the equation (he) by d; e the lowest corona voltage of the system 0, 2 (S 5 Ms s 7 d x 53.6 x 2.303 Mo S 1‘3ng f 13 13m 14 15 80 = — 2 2L /(—9—.cos 612) + Lil. Sin 912)2 + l ] m12 m12 2rx535x23031’6—10 2.23 2(l+g-I-;- =815=816 m _ 123r MOS 32 (t7) 9° ’ 1 2r logic a?“ *“5 where ac line to line corona-starting voltage in kv where S = the distance between phase conductor m = the distance between the conductor of the same phase r = the radius of the conductor Mo = the irregularity factor oq the air-density factor 1 e0 for one conductor per phase is given by the following equation : 3 e0 = 123 MOrS loglo ; hv (ma) 1 Electric Power Transmission and Distribution by Wbodruff. 61 APPENDIX 5 Corona Loss heasurement in nundle Conductor The formulae worked out for estimating corona-loss are all of empirical correlation the test values. Peek's formula is applicable for the higher value of the losses but Peterson formula which is applicable for the lower value as well as higher values, of losses are widely used. The loss can be expressed by the following equation. 0.0000337 e 2 . Pc = 'f T] GFE') (hh) (logic é )2 r where PC is the corona loss in kw per mile per conductor. / The ratio of en/eo is a function org the value of which can be taken from the curve . The Peek's loss formula is given below for reference: PC = 13-9- (f+ 25) [E (en-e0)2 x lO—E> kw/mile (119) For a mile of the whole line all three wires, the loss is three times of the equation (hB). For bundle of two, the total loss is six times the value of the equation (DE). The equations (h8) are on the basis of the charge density uniform on the conductor periphery. 62 BIBLIOGRAPHY Cahen, F., "desults of Test Carried Out at the SOOkv Experimental Station of Cherilly (France) on Corona-Behaviour of Bundle Conductors," Trans. A.I.E.E., vol. él, 19h8. Carroll, Joseph, "Corona Loss Measurement for the Design of Transmission Line," Trans. A.I.E.E. vol. 52, 1933. Clarke, 8., "Three Phase Multiple-Conductor Circuits," Trans. A.I.I_‘;.I_‘:., V010 2.1-1.) 19320 I'Comparison of Lines Cost with Single and Double—Conductor," C.I.G.h.E. paper hCS, 1950. Deloney, H. L. and Hush, N. L., "Bundle Conductor for Transmission Line Capacity Increase," Electrical world, December 1955. Dwight, H. 8., "The Direct Method of Calculation of Capacitance of Conductor," Trans. a.I.E.E., vol. h3, l92h. Dwight, H. 8., "Double Conductors for Transmission Line," Trans. $.1.E.E., vol. El, 1932. Dwight, h. B., and Scheidler, F. E., "Transmission Line Capacitance and Surface-Voltage-Gradient," Trans., A.I.E.E., vol. 71, l9Sh. Gerber, 0., "Corona Losses of Single and Bundle-Conductors,"C.I.G.R.E. paper h03, 1950. Kluss, 3., "Capacitance of Multiple-Conductor System," (German), Elektrotech_§, Feb. 1950. Peek, F. W., Dielectric Phenomenon in High Voltage Engineering, Third edition, McGraw-hill book Co., 1929. Quilico, G., "The General Electrical Problems of Multiple Conductor Over-head Lines," C.I.G.R.E., paper 219, 1950. Themoshok, M., "Relative Surface Voltage Gradient of Grouped Conductors," Trans. ;;OIOE£O, v01. :22, 191-180 Two-Dimefiional Fields in Electrical Engineering, Bewly & I-Iacmilan l9hb. woodruff, Electric Power Transmission and Distribution. Zaborszky and Rittenhouse, electric Power Transmission. ENCR. Uh