references cited, the subject of negative- stations were equipped with two potential tion was expected. The assumption ofsequence polarization has been covered transformers connected in open delta. the calculated risk of wrong operations hasquite thoroughly. Also, the authors have Space limitations precluded the installations been fully justified and the improvementdescribed the various applications in which of a third potential transformer so that in over-all system operation has been trulynegative-sequence polarized ground relays potential polarization could not be achieved. gratifying.may be utilized to advantage and have It appeared that only negative-sequencecompared their characteristics with relays polarizing could be accomplished eco-using other forms of polarization, which nomically. W. A. Elmore and J. L. Blackburn: Wemakes this paper a very valuable source of From calculations, it was determined appreciate the interesting discussions. Theyinformation when the engineer is faced that overcurrent relay co-ordination could contribute materially to the paper by add-with an unusual situation. This discussion be obtained using zero-sequence current ing practical experiences and by lendingsuggests such a situation and the solution and that for negative-sequence polarizing, emphasis that negative sequence is anotherof its problems. at two of the stations, the negative-sequence valuable tool for relay engineers.The problem was caused by pole top voltage would be low enough under some In comment on the last paragraph of

fires on an ungrounded 55-kv delta-con- conditions to give marginal operation. A Mr. Hendrickson's discussion, the sus-nected transmission network. The faults review of the combinations of conditions ceptibility of the relay to false tripping iswere generally arcing single line-to-ground under which operation would be question- extremely low and we are not aware offaults that were very difficult to locate able revealed that they were far less any operation problems in this connection.even by extensive switching. The ensuing frequent than those where operation would Several factors are involved to assure aoutages frequently covered large areas be ensured. On this basis, it was decided high security to these relays: (1) the nega-that were disproportionate to the magnitude to install negative-sequence polarized tive-sequence directional unit is used inof the fault. It was believed that a better ground relays at all points where zero- the sense of a fault detector in conjunctionway of isolating faults than by manual sequence current was not available as a with fault measuring or discriminationswitching could be found. polarizing source. Although wrong relay units; (2) the relays are designed to have

Previous studies had shown that the cost operations occasionally could be expected negligible response to offset transients;of grounding the system properly and at the two points mentioned, correct opera- (3) 3-phase loss of potential or 3-phaseinstalling conventional zero-sequence cur- tions on the rest of the system would short-circuiting of the current transformersrent polarized directional overcurrent greatly improve the over-all operation of does not provide negative-sequence quan-ground relays was prohibitive. The system the system. tities; and (4) lightning and switchinginvolved was composed of nine inter- Since the first installation, a small surges generally are not significant pro-connected major stations and approxi- grounding bank has been installed at the ducers of negative sequence.mately 325 miles of transmission lines, the far end of the system from the original wye. Unequal loss of potential or current andlongest of which was approximately 40 connected banks, not for polaizing, but unequal pole closing can produce negativemiles and the shortest approximately 7 for additional zero-sequence current for sequence and could cause directional unitmiles long. An investigation and inventory better relay co-ordination near the end of operation. In these cases the fault measur-of the stations in the system involved, the system. Twenty-two negative-sequence ing or discriminating units and thesedisclosed that there were two stations near polarized overcurrent ground relays have inherent, or built-in, time delays wouldone end of the system with transformers been installed to date, and in the past 4 provide security at least until the secondwhose 55-kv windings were connected in years, 20 operations have been logged, contingency of a fault occurred. We areungrounded Y and the neutrals were with no improper operations. One of these not aware of any field problems in this areaaccessible, and that all but one of the other occurred at a point where marginal opera- with the present relays.

VIU* I1 . 9 n 1 1 I T . * the subject a little mysterious, particularlyfo tage In uctionl Ill ara lel | ransmissionl from a practical standpoint. This paper

will review the fundamentals and study

ircu its the positive-, negative-, and zero-sequenceinduction that can exist between circuits.Of particular interest is the negative-se-

J. L. BLACKBURN quence induction as negative-sequenceMEMBER AIEE directional relays are valuable tools in

avoiding mutual induction problems inground relaying. These relay applica-

Summary: General formulas are developed impedance can be as high as 50 to 70% of tions are covered in detail in a companionwith calculation examples for determining the zero-sequence self-impedance of the paper.'the positive-, negative-, and zero-sequence lines. The conductor configuration shouldvoltage induced in transmission circuits be symmetrical around a vertical axis toparalleled either on the same tower or on avoid differences in the induced voltages Fundaments of Mutualadjacent towers. For nontransposed lines in the two circuits. Induction-Positive andthe induced voltages are a function of all Negative Sequencethree sequence currents. The positive-and negative-sequence mutual impedance _HE MUTUAL COUPLING between Electromagnetic inductionbetween par-is less than 10% of the self-impedance of .Uthe lines while the zero-sequence mutual *3-phase transmissionlines onthe same alleled circuits exists because the flux

tower or on adjacent towers in the same established by the current in one line-- ~~~right-of-way must be considered during cuts the conductors of the adjacent or

Paper 62-1073, recommended by the AIEE Relays fault calculations and in some cases in the pa.rallel line. Consider two 3-phase par-

Operations Department for presentation at the design of the protective relay system. allel lines either on the same tower or onAIEK Summer General Meeting, Denver, Colo,June 17-22, 106f2. Manuscript submitted March The fundamentals of mutual coupling, or adjacent towers, as shown in Fig. 1,12, 1962; made available for prnting April6, electromagn.etic induction, have been with only positive- and negative-sequence

L. BACKBRN i wih th WesingouseElecric stated in textbooks for many years. currents flowing. The net flux availableCorporaton,Newak, N. J.However, many enginleers appear to find from positive- or negative-sequence cur-

FEBRUARY 1963 Blackburn-Voltage Induction in Parallel Transmission Circuits 921

b Xb' and is cut by the flux established by ,_)¢(_

a c ' c current flowing in the line. Conse- '-'quently, the mutual impedance ZOM be- Ox, b, c a', b', c'tween parallel lines can be as high as50 to 70% of the self-impedance of theline. The methods of calculating theseself- and mutual zero-sequence impedances 777777777777777777

777777 7777777777777. ,77777 is well documented for transposed cir- EQUIVALENTcuits and is tabulated for various cir- DEPTH OF EARTH

Fig. 1. One typical conAguration for two cuits with and without ground wires in RETURN-AVzr2800 FT.3-phase parallel iion reference 3. Zero-sequence calculations I-2r'1OO '

are based on numerous assumptionsincluding transposed lines and it is Fig. 2. Zero-sequence equivalent for the

rent in either circuit to cut the adjacent remarkable how close the calculated fault two parallel lines of Fig. Icircuit will be minimized because the currents agree with values even withcurrents are equal in magnitude and sym- untransposed lines.metrically spaced 120 degrees apart in 1I = 0, and equation 52 reduces to equa-time phase. Actually, there would be General Formulas for Mutual tion 57 when I,=12= 0.no induced voltage in the parallel circuit Induction The induced voltages or mutual im-from the positive- and negative-sequence pedances are complex quantities and ex-currents if it were physically possible to General formulas for determining the cept for the transposed zero-sequencespace each of the three conductors of the induced positive-, negative-, and zero- equation (equation 57), their values mayparallel circuit equidistant from each of sequence voltages in two parallel circuits have angles anywhere between 0the three conductors of the line itself. are developed in the Appendixes as in- and 360 degrees. This is because theIn other words, if a, b, c are the conductors dicated in Table I. induced voltages result from three 120-of one line and a'b'c' the conductors of the Where the lines are not transposed, degree electrically spaced currents andother line, then the spacing d between the the induced voltage of any given se- their relative physical proximity to thetwo circuits must be such that daa'=dab'= quence is a function of all three sequence other three conductors in the adjacentdac' and dba'=dbb'=dbc' and dcal=d,a,- currents, as shown in equations 18, 33, circuits.dec^ With this, the induced positive- and 52. Transposing both circuits (1) Examining the general equations 18,and negative-sequence voltages are both reduces the induction and (2) results in 33, and 52, it is observed that all of thezero. A study of this requirement will the induced voltage of any given sequence distances in the real term are involved inshow that it is physically impossible to being a function only of the correspond- the square root denominator of the imag-arrange the conductors to meet the require- ing sequence current. This is shown in inary term. For the positive and nega-ments. The nearest approach would be equations 25, 37, and 55. Thus, ex- tive sequence voltage equations 18 andto have each circuit exist as a triaxial pressions for mutual impedance can be 33, the real term of I, induced componentcable. obtained as given in equations 26, 38, and of V2' is equal to but of opposite sign to the

Thus, with practical separations of the 57. Equation 57 is of similar derivation as real term of the I2 induced component ofconductors and circuits there will be the GMR-GMD (geometric mean radius- V1'. Correspondingly, the real term of I2induced positive- and negative-sequence geometric mean distance) concepts in for V2' is equal but of opposite sign tovoltages related to the degree of dis- symmetrical components.2 3 the real term of It for V1'. The imagi-symmetry between the circuits. Prac- All of the sequence voltages are derived nary term of I1 for V2' is equal to thetically, the positive- and negative-se- for the phase A component. The phase imaginary term of I2 for V1', and corre-quence induction between the circuits is B and C components exist and can be spondingly, the imaginary term 12 for V2'very small and the mutual impedances are obtained by the fundamental relations equals the A1 imaginary term for VI'.less than 10% of the self-impedance of of the method of symmetrical components. The zero-sequence components of the twothe line. Usually they do not exceed 3 Comparing the induced voltage ex- voltages V1' and V2' are equal except forto 7%. These values are for nontrans- pressions, the transposed cases are iden- the opposite sign of the real component.posed lines. Transpositions can reduce tical with the untransposed cases when In examining zero-sequence equationthe values by 50% or more. the currents of the other sequences are 52 it is observed that II and 12 com-

zero. Thus, equation 18 reduces to ponents are equal except that the realequation 25 with I1=I=0, equation functions have opposite signs. The zero-

Fundamentals of Mutual 33 reduces to equation 37 when I2= sequence current factor is the symmetricalInduction-Zero SequenceZero-sequence current is equal and in-TalI

phase in each of the three conductors blIand returns via the earth and/or the ~~ground wires. This means that the three Apeno Induced Voltage Linles Equation Nos.conductors of each circuit can be re-placed by an equivalent single conductor, I . Negative sequence .....Untransposed .........1-18as shown in Fig. 2. With the return II .Negative sequence ... . Transposed........ 1926III . Positive sequence .......Untransposed............ ...27-33mathematically averaging 2,800 feet IV . Positive sequence .....Transposed ........34-38from the conductors for 100 meter-ohms v . Negative sequence .....Untransposed wvith ground . .......39-47earth resistivity, it can be seen that the vi . zero sequence .....Untransposed.............48-52parallel circuit iS in the magnetic field viZr euneTasoe.35

922 Blackburn- Voltage Induction in Parallel Transmission Circuits FEBRUARY 1963

component formula for the mutual im- duced voltage is found to be 51% of ZO. ductors resulting from currents flowing inpedance as explained earlier in connection If these lines are not transposed most the a, b, c conductors. The formulaswith equation 57 for transposed lines. of the positive- and negative-sequence also can be used to calculate the induced

induced voltage is caused by the zero- voltages in the a, b, c set resulting fromTYPIcAL EXAMPLES USING sequence current. With the assumptions currents flowing in a'b'c' by interchang-

THE FORMULAS of Io only and I,=12=lo the mutual im- ing the conductor labels a and a', b and b',Two examples of the calculations of the pedances are 9.1% and 7.73% of X1 re- c and c'. Calculations for a typical ex-

induced voltages on parallel lines are spectively. These lines are unusually ample are given in Appendix X. Forgiven in Appendixes VIII and IX. The close together and, again, increased calculating the induced voltage in Line

spacings would reduce these values. 2, Line 1 conductors are a, b, c and Line 2fiers wistforelatdble circuitline onpasinglbe- Grouping the lines a-b-c-a'-b'-c' instead conductors are a', b', c'. To determinetower with relatively close spacings be-'tween the two circuits. The positive- and of the preferred a-b-c-c'-b'-a' changes the the induced voltage in Line 1, phases a,negative-sequence mutual reactance is induced voltages but not the percentages, b, c in Line 1 are changed to a', b', c'-j 0 0168 ohm per mile, assuming that assuming Io only and 1 =12=Io. respectively and the phases used in Lineboth lines are transposed. This is only EFFECT OF GROUND WIRES 2, corresponding to a, b, c, and the for-2.34% of the self-reactance of either line. mulas used as developed.By contrast, the zero-sequence mutual im- The presence of ground wires tends to If the two sets of conductors are sym-pedance with the lines transposed is 0.286 reduce the induced voltages. This is metrical around a vertical axis, then the+j 1.78 ohms per mile which is 63.8% of studied in connection with the induced induced voltages in the two lines will bethe zero-sequence self-impedance, with negative-sequence voltage in Appendix V. equal. This is shown in part B of Ap-the physical spacings as given. The I1, 12, and Io expressions of equation pendix X. For either induction in Line

If the lines are not transposed, the posi- 47 are the same as the corresponding 1 or in Line 2, the spacings to the current-tive, negative-, and zero-sequence in- ones of equation 18, Appendix I, except inducing conductors in the adjacent lineduced voltages are complex. In order to that equation 47 has additional factors for are the same.compare these, it was first assumed that the ground wire return currents. These Where the conductors are not sym-the zero-sequence current predominates, ground wire currents, Ix and ly, are metrical around the vertical axis, as inThis was accomplished by assuming 10 assumed to flow in a ditection opposite to Part A of Appendix X, the voltages in-only with I2= 1 = 0. The second assump- Ia, I and I.. duced in Line 2 from Line 1 currents aretion was to let I1=12= Io. With lo only, In the example of Appendix VIII, two different from the voltages induced in Linethe induced positive- and negative- ground wires X and Y from left to right, 1 from Line 2 currents. A different volt-sequence voltages are almost 1/2, and 24 feet apart and 13 above the middle age exists to cause circulation of currentswith 1,=12=1o, the voltages are approxi- conductors b and b' were added, and the between the lines. For the nontrans-mately double that of the transposed case. induced negative-sequence voltage cal- posed horziontal lines of the Appendix X

Increasing the spacings between these culated per -equation 47. Values for Ix example, the positive-sequence voltagetwo circuits reduces the mutual voltage. and I are very difficult to calculate for difference is a function of I, and J0, and theFor example, if the spacing is doubled the untransposed circuits, but a good negative-sequence voltage difference ofZ2m =j 0.0054 ohm per mile for transposed approximation can be obtained by assum- I2 and lo. The I, and I2 current factorslines and V2'= (-0.0047-j 0.00261) ing the circuits transposed, in whicb case are very small so that the effect essentially1I-j 0.00542+(-0.00564+j 0.00326)Ie Ix and Il are a fraction of theIa current is a function of Io which exists duringfor nontransposed lines. flowing. In the example 61.5% of the ground faults. The zero-sequence volt-

It is interesting to note that if the right- current will flow in the two ground wires age difference is a function of only I, andhand set of conductors are a', b', c' from assuming the ground wires are the same 12, so that a zero-sequence circulatingtop to bottom, instead of c', b', a', the type of conductors as the phase conduc- current can exist on phase faults, such aspositive- and negative-sequence induced tors. If the right-hand, or Y, wire is an ab or ca fault, where I, and 12 arevoltages are different but their total omitted, then 43% of the current returns not 180 degrees apart.mutual impedance values with either 1o in the X ground wire. These were deter- Generally, this unbalance effect hasonly or I1=12=Io are the same or 1.25% mined from reference 3. Table II not caused trouble but a case is knownand 4.4% of X, respectively. The trans- summarizes several calculations based on where failure to trip occurred for anposed values increase slightly from 2.34% equation 47, assuming I1121I0. internal phase fault on a line using pilotto 2.5% of XI. The zero-sequence value wire relays having a very sensitive zero-remained unchanged. The a, b, c-c' INDUCED VOLTAGE UNBALANCES N THE sequence setting. The circulating zero-b', a' top-to-bottom arrangement of the PARALLELED CIRCUITS sequence current provided a blockingconductors should be used since there are The formulas in the Appendixes give indication to the relays overcoming theother reasons why a'b', c' top-to-bottom the induced voltages in the a', b', c' con- positive sequence. The solution is toconductor spacing is not desirable. Thesewill be developed later in connection withAppendix X. Table IIThe second example in Appendix LX ___________________________________

shows calculations of theinducedvoltages Ground Wire Vt' (c', b', a' Per Cenat V,' (a', b', c' Per Centwith flat horizontal spacings, on separate C:urrent Top to Bottom) of Xl Top to Bottom) of Xltowers. With the lines transposed, theinduced positive- and negative-sequence 1Il-iv-o.O.0-o.0234j0.0213.....4.4% ...0.0a068+j0.0309.....4.4%IX-IU =1/2IO .-0.0053-1 0.0008......0.75%.....-0.0033+10.0032.....0.64%voltages are reduced. The positive- and Iz=0.431o . l-0.0147-J 0.0111......2.56%....-0.0023+j0.0183.....2.56%0negative-sequence mutual impedanlces are IV/ - 0. 1851I01.09%o of X1 and the zero-sequence in- Ix =IVi 1/210 does not aPPear PraCtiCal bUt iS included fOr comparative purposes.

FEBRUARY 1963 Blackburn-Voltage Induction in Parallel Transmission Circuits 923

desensitize the relay or, preferably, to Vb'=IaZab'+IbZbb'+IbZcb' (2) Correspondingly,space symmetrically the two sets of And the mutual voltage induced in (Za,'+a'Zbb'+aZb,') a -10.2794conductors around the vertical axis. conductor c', resulting from current flowing (Log dba'+a2 Log dbb'+a Log db,') (16)This is why it was previously stated that in conductors a, b, c, isthe preferred arrangement of the conduc- (Zca'+a"Zcb'+aZcc')= 0.2794tors in Appendix VIII is abc and c'b'a' Vc"=IaZac'+IbZbc'+IcZCC (3) (Logdca'+alLogd,'+aLogdecc) (17)from top to bottom and in Appendix IX, The negative-sequence voltage induced Substituting equations 15, 16, and 17 inabc-c'b'a'. in conductors a', b', c' from currents Ia, equation 8, usng a- -0.5+j0.866 and

It should be noted that in the example ba,h in conductors a, b, c is as- - 0.5-i 0.868, and reducing and com-of Appendix IX the voltages induced in v,'=!( V'+a'Vb'+aVC') bining results inthe left line from currents in the right line 3 VP'= 0.0931I, [OW Log dacdbbdca'will be different from those induced in the 1 dab'dba'dec'right line from the left line currents in all = [Ia(Zaa'+aZab'+aZact)+cases because the separations of the in- I+Log -- +divudal conductors in the two circuits are Ib(Zba,+aZbb'+aZbc)+ /Lgdab'dac'dba'dbb'dca'dcc'not the same. This would be true for Ic(Zca'+a2Zcb'+aZcc')l (4) 0d0931 [0868Log acdba'dcb'parallel lines of different configurations. From symmetrical components, L dab'dbc'dca'

Ia=I1+I2+10 (5) daadbb'dec I

Conclusions Ib=a'I1+ah2+IO (6) j Log dabdca'dbc'dc'dcb']+The fundamentals of electromagnetic Ic = al +a'12+Io (7) 0.0931IoL0.866 Log datd cdcc-

induction have been reviewed and the Substituting equations 5, 6, and 7 in ,,,general formulas for the calculations of equation 4 and reducing, gives j Log daa'dbadca' 1 (18)the positive-, negative-, and zero-sequence 1/dab'dac'dbb'dbc'dcb drc 'induced voltages in parallel transmission V2'=-Ii[(Zaa'+a2Zab'+aZa2')+ where V,' is in volts/mile at 60 cycles.lines have been developed and applied 3to typical examples for transposed and a(Zba'+a2Zbb'+aZbe')+nontransposed lines. These induced volt- a(Zca'+a'Zeb'+aZcc')l+ Appendix 11. Negative-Sequenceages are complex and they are a function 1 Induced Voltage For Twoof all three sequence currents. The posi- 3 12[(Zaw'+taZab'+aZac')+ Transposed Parallel Linestive- and negative-sequence mutual im- a(Zaa'+a2Zbb'+aZbc')+pedances are less than 10% of the self- a,,(Za'+aZ,b'+aZec')+ Assume that the two lines are transposedimpedance of the line and thus are negligi- a(ZCG+aZCb+GZCC)J + as shown in the following configurations:ble; however, the zero-sequence factor can 1 I Obe as high as 50 to 70% of the zero-se- 3I0t(Z441+a'Z.t+aZa')+ abo Oc Co Oa bI O'tquence self-impedance and must be con- (ZI,'+a'Zbb'+GZbc')+ cO 0,' bor' a IObsidered in fault studies and relay applica- (Zca`+a'Zcb'+aZcc')J (8) 1st Section 2nd Section 3rd Section

tions.Wihteetasoiin,temulWhen the conductor configurations of Carson's formula' for mutual impedance With these transpositions, the mutual

the two lines are symmetrical about a between two paral3el conductors with a voltages induced in conductors a, b, c'the to lms arsymmtrsc1 abut a commo earh reurn, t 60cycls, "resulting from currents Ia, lj,, c in con-vertical axis, the voltages induced in the common eth return, at 0 cycles, i

ductors a, b, c are as follows and similartwo parallel lines are equal. Non- Zaa=00954+j 0.2794 Log , ohms/mile to those of equations 1, 2, and 3:symmetry produces different voltages daa' 1which can cause relaying problems. It (9) V'3-(IaZaa'+IbZa'+IcZca')+is hoped that this study has contributed D 1to a better understanding of voltage in- Zab'= 0.0954+j 0.2794 Log DI ohms/mile UaZ-'+IbZcb+IcZab')+duction in transmission circuits. da 3

1 (IaZcc'+IbZac'+IcZbc') (19)3Appendix 1. Negative-Sequence Zac'=0.0954+j0.2794Log De- ohms/mile 1

Induced Voltage For Two da,' v'= (IaZab'+IbZbb'+IcZcb')+Untransposed Parallel Lines (11)3

Expanding results in 1- (IaZbc'+IbZcc'+IcZac')+The conductors of one line are a, b, c, 3

and of the other line a', b', c'. The dis- Zaa'=0.0954+j 0.2794 Log D,- 1tance d between the various conductors is j.74Lgda l) ~ tactIZa+co)(0expressed by d,,' dab', dac' etc.; the j3.74Lgd, 1)-(hZa+tZa+Za)(0impedances by Zaa', Zac'. etc. a'Z,b'-0.09M4a'+j0.2794a' LogD,- 1

For any conductor arrangements, thej .74 Lgdb(1) Vc-(aZc+ZcIZc+mutual voltage induced in conductor a' .74 o a'(3 , IZc+bb'IZc)resulting from current flowing in con- aZac' = 0.0954a+j 0.2794a Log Dc- 1ductors a, b, and c is j0.2794a Log d,c' (14) - (IaZba'+IbZca'+IcZaa')+

Correspondingly, the mutual voltage - (IaZca'+IbZ,a/+IcZbb') (21)induced in conductor 5', resulting from (Zaa'+a'Zab'+aZac') =-j 0.2794current flowing on conductors a, b, c, is (Log d,,'+a' Log dab'+a Log d,¢') (15) The induced negative-sequence voltage

924 Blackburn-Voltage Induction in Parallel Transmission Circuits FEBRUARY 1963

in conductors a', b', c' from currents Ia, 1 in conductors a', b', c' resulting fromIb, Ic in conductors a, b, c is VI' 3 [Ia(Zaa'+aZab'+a2Zac')+ currents Ia, Ib, I, is

I1 bZa'ab'aZc' 1V= - ( Va'+a2 V&'+a Vc') IV(ZbI+aZbb±aZbC)± l' -( Va'+a Vb'+aIVc') (34)

3 Ie(Zca'+aZe&'+a2Zcc')1 (28)

V2'=1 [Ia(Zaa'+Zub'+Zcc')+ Substituting equations 5, 6, and 7 in VI'= -a(Zaa"+Zbb'+Zcc)+9 equation 28, and reducing gives 9

Ii(Zba'+Zcb'+Zac')+ 1 Ib(Za'+Zcb'+Zac')+Ic(ZCa'+Zab'+Zbbc') + VI1 - I [(Zaa'+aZab'+a2Zac')+ Ic(Zcea'+Zab'+Zbc')I +

1 1- [a'Ia(Zab'+Zbc'+Zca')+ a'(Zba'+aZbb'+a2Zbc')+ - [Ia(a)(Zab'+Zbc'+Zca')+9 9

a'Ib(Zaa +Zbb +Zcc')+ a(Za'+aZcb'+aZcc ')]+ aIb(Zbb +Zcc +Zaa')+a +Ic(Zc+'+Zac'+Zba') + 12aIc(Zcb'+Zac'+ZDa')] +

- [a1a(Zac'+Zsa'+Zcb')+ + 9 [a2Ia(Zac'+Zba'+Zcb')+

a1b(Zbc'+Zca+Zasb)+ a'(Z,a'+aZcb'+aZcc')JI+ a21(Zbsc'+Zca'+Zab')+aIc(Zaa'+Zbb'+Zcc') (22) 1 a2Ic(Zc'+Z5aa'+ZbM')1 (35)

Substtutinequaions5, 6,and 7in 3O1(Zaa'+aZab'+a2Zac')+Substituting equations 5, 6, and 7 in3reducing, By substituting equations 5, 6, and 7equation 22 and combining and reducing, (Zta'+aZbb'+a2Zbc')+ in equation 35, and combining and reducing,the positive- and zero-sequence current (Z'+aZ'+a2Zc')1 (29) the negative- and zero-sequence currentfactors all cancel out since 1+a+a2 =0. factors all cancel out since 1+a+a$ -O.Therefore, Applying Carson's formulas' in a manner Thus,

1 similar to that shown in equations 9 1V2 - I2[(Zaa'+Z'+Zcc')+ through 17, new values for the impedance VI -- IIE(Zaa'+Zbb'+Zee')+

3 combinations are obtained as follows: 3

a(+Zc51+Za 'Zc') ] (3) (Zaa'+aZab'+a'1Zac') = -O.2794 a(Zab`+ZbC'+Zea')+(Log daa1+a Logdab'++al Log dac') (30) a'(Z '+Zsa'+Z¢')] (36)

It will be noted that equation 23 is the By applying Carson's formulas in thesame as the 12 component of equation 8. (Zba'+aZ4b'+alZbc') -j 0.2794 same manner as shown in connection withAnother form of equation 23 is (Log dba1+a Log dwb'+a' Log d4') (31) equations 9, 10, and 11 and reducing and

combining one obtainsV2'=- Id(Zaa'+a'Zab'+aZac')+ (Z4ca'+aZc+adZd)= -jO0.2794 =0d,,'d'18

3 (Log d4a+a Log dcb'+a" Log d,,') (32) W=0.09311, 0.866Log

( a1+(Zea'+a2Zcb¢t)+aZcc'j (24Substituting equations 30, 31, and 32 daadbb'dcba(Zc01+a9Zcs'+aZ~c')] (24) in equation 29 and reducing and combiniing, j Lo09dbdc

Substituting equations 15. 16, and 17 remembering that a--0.5+j O.86W and J Vdab'dac'dbadbc'dca'dcb'Jin equation 24, combining and reducing a= 0.5 j0.866, gives volts/mile at 60 cycles (37)

gives~~~~~~~~~~~~~Lgda514'db5'da Finally,0.2794 F dac'da 'dcb ' VIO=0.0931l0.866 Log daN'dba'db 1 r _______

3 dab'db86dca' 31VI0da68dLog,,,-IjLab8d6 Log a-

i Lo daa'dbb'dcc' ZIM 0.0931 8 'bdac'da'Vdsdaa'dba,'dscdca'dcjLog

- /dab'dac'dba'dbc'dca'dcb'l d25 j Log (38)0.0931120.866 Log dab'dbadcc' j L (38Finally, o dac'd,b'dca'

0.2794 , where ZiM is the positive-sequence mutual23'0.2794 L dac'daa'dcb'_ daa'dbc'dcb + impedance in ohms/phase/mile at 60

Z =-= Ldabdadbc5'd5a' Vdab'dac'dba'dbbdcadcc cycles.Note that equation 37 is the same as the

Logjd 'dbd' 1 (26) 0.0931I[o0.866 Log dab'ddCb positive-sequence current term of equation\1dab'dacdba'dbc'dcaLdcbog dac'doc'dcc' 33 of the untransposed line equation, Ap-V'dab'dac'dba'dbc'dca'dcb' JaaLa dca'pendix III.

where Z2M units are in ohms per phase/ j Log V3 pmile at 60 cycles. (3da3dac)dbb'dbcdcbdrc'

where the unit is volts/mile at 60 cycles. Appendix V. Negative-Appendix Ill. Positive-Sequence Sequence Induced Voltage for

Induced Voltage For Two Two Untransposed ParallelUntransposed Parallel Lines AppendixV Positive-Sequence LInes with Ground Wires

Induced Voltage for TwoThe positive-sequence voltage induced Transposed Parallel Lines Consider the circuit generally spaced as

in conductors a', b', c' from currents Ia, shown: The distances betwveen variousID, and Ic in conductors a, b, c is Assume that the lines are transposed as conductors are 4,v', dab~', d4ss, etc., so that

outlined in Appendix II. The voltages derivations apply to varying configurations.1,-(V'a in2V')2duced in conductors a', b', c' resultingV1'=-V5'±Vs'+2V~')(27) from current flowing in conductors a, b, z(gl) t

c are given in equations 19, 20, and 21 a0 cSubstituting equations 1, 2, and 3 in respectively. bo05'

equation 27 results in The induced positive-sequence voltage c0 a

FEBRUARY 1963 Blackburn-Voltage Induction in Parallel Transmission Circuits 925

The mutual voltages in conductor a', o ~ dac'd&c'dcc' j 0.2794 Log daa'dab'dac'dba'dbb'dbc'Xb', c' respectively, resulting from current 0.093110 g d dca'dc'dcc (51)flowing in phase conductors a, b, c andreturning in ground wires x and y, are j Log daa'dba 'dcs ' Equation 51 further reduces to

Va'=IaZaa' IbZ 0a'+IcZcaI- F\dab'daA'dba'dbc'dcb'dcc r dca'dcb'dcc'IzZxaI'Zva' (39) dxc Vo, 0.09311=0.866 Logdb-dbb,dbc

Vb' =IaZab'+IbZbO'+IcZcb'- j0L[gdLoagd5a'da 1

I:z;ZZb '-IVZVb' (40) j Log dza' 1 0.0931Iy i Log

ve' = IaZac'+IbZbc'+IcZIc'g 0.093112 0.866 Log dd'd'b'dc'Jr,z1 Zyl(41) Fdyc,' dya' -i0912086Lgdea'dbb'dbc'

The induced negative-sequence voltage dyb0y\idLo'dg c dodaadab'dac'in conductors a', b', c' from currents Ia, volts/mile at 60 cycles (47) #\dbadabdc'dca'dc'daCcIb, Ic in conductors a, b, c and -I. and-II in the ground wires x and y is Io 0.2862+j 0.8382

v2=1(vn +asvb/+avc/) Appendix VI. Zero-Sequence3 Induced Voltage for Two Log da'1 Untransposed Parallel Lines[Ia(Zaa'+aZab'+aZac')+ volts/mile at 60 cycles (52)3

The zero-sequence voltage induced inIb(Zba+a'hZabb+aZbct)+ conductors a', b', c' from currents Ia Appendix Vii. Zero-SequenceIc(Zca'+a2ZctO'+aZcc')- Ic in conductors a, b, c is Indix Volt Foro Two

Ix(ZZa'+a2Zxb'+aZxc')- 1 Induced Voltage for TwoIl/(ZYa'+a2Zyb'+aZyc')] (42) Vo'=3( Va'+ Vb'+ Ve') (48) Transposed Parallel Lines

The first three terms of equation 42 are Substituting equations 1, 2, and 3, gives, Assume that the lines are transposed asthe same as equation 4 and thus reduce outlined in Appendix II. The voltagesto equation 18. The last two terms with 1induced in conductors a', b', c' resultingIx and Iy can be reduced applying Carson's 703 [Ia(Zaa'+Zab'+Zac')+ from current flowing in conductors a, b, cformula similar to that in equations 9-11. fare given in equations 19, 20, and 21

lb(Zba+Zbb'+Zbc')+ ~~respectively.(Zza'+a'Zzb'+aZzc') = -j 0.2794 Ic(Zca'+Zcb'+Zcc')J (49) The induced zero-sequence voltage in(Log dza'+a' Log dxb'+a Log dxc') (43) . . . . conductors a', b', c' resulting from currentsSubstituting equations 5, 6, and 7 in I I, Ic i

(Zya'±a2Zvs'+aZyc') =-j 0.2794 equation 49 and reducing results in a, b, c S

(Log da'+a2 Log dyb +a Log dy,') (44) 1 V =( Va'+ VD'+ Vc') (53)Vs' II[(Zaa0'+Zab'+Zac')+ 3

Further substitution of a-0.5+j 0.866 3and a= -0.5-i 0.866 in equation 43 and a2(Zba'+Zbb'+Zbc')+ Vo"' Ia(ZaG'+Z55'+Zcc')+44 gives a(Zca'+Zcb'+Zcc')]+(Zza'+a2Zx'+aZzc') = -j 0.2794 1 Ib(Zba +ZcbD+Zac')+

(Log dZa'-0.5 Log dZb1-j 0.866 3 2aaa+ab+ac)+ Ic(Zca'+Z00'+Z&c')]+Log dzb'-0.5 Log dze'+j 0.866 a(Zaa'+Zbb'+Zbc')+ 1[a(Zab'+Zbc'+Zca')+

Log dzc') a2(Zca'+Zcb'+Zcc')] +

0.2794(+0.866 Log -- 1Io[(Zaa'+Zab1+Zac')+ Ic(Zcb1+Zac'+Zba')1 +

dra (Zba +Zbb +zbc )+ - [1a(Zac +Zba +Zeb )+j\Ldzbgdzzc) (45) ((aZca,Z+b'Z+c)] (50) )15(Zbc'+Zca'+Za&5)+

(ZYa'+a2Zy'+aZyc') = 0.2794 By substituting Carson's formula2 for Ic(Zaa'+Zab'+Zcc')1 (54)mutual impedance, as given in equations(0866 Lo dyc' Log

da(46 9, 10, 11, etc., and reducing and collecting By substituting equations 5, 6, and 7

06J(46) terms, equation 50 becomes in equation 54, and combining and reducing,d \ b dyc the positive- and negative-sequence current

Equation 42 then reduces by applying Vo'=3=I [-jO.2794 Log daa'dab'dac'+ factors all cancel out since 1+a+as=0.equation 18 and substituting equations 45 Thus,and 46 to obtain the following j 0.2794/2 Log dba'd5b5d5c'd,'dcb'dcc'- t 1

F dac'5tdca'd 0.866 (0.2794) Log dba6'4b 4c'+ V0 3 0Za+b+ZcZa+V21 091xL.6Log d -dt 0.866(0.2794) Log dca'dc0'dcc'I+ Z&c'+Zca't+Zac'+Z54'+Zc511 (55)

daatdbctdobt 1 ~~-12[-j 0.2794 Log d001d01,'d001+ Applying Carson's formulas' in the same

3manner as shown in connection withLog dab'dac'dba'dbb'deczdcc j 0.2794/2 Log dba'dbb'dbC'dca'dtdcrdc'+ equations 9, 10, and 11, the impedance

0.0931I2F0.866 Lgdac'4ba'dcbt .6(.29)Lg'btbto expression of equation 55 becomes

Laotc 0.866(0.2794) LOg dea'dc51dcc'1+ VOM 1095+.24Lo -

dAaa'db5'deCC' 1IO=- 3005+0274Lg1j

/aLog

c tde dctc dbt + - Io[9X0.0954+Fj 9X0.0954 Log D.,- j 0.2794 Log daa'+0.0954+

926 Blackburn--Voltage Induction in Parallel Transmission Circuits FEBRUARY 1963

j 0.2794 Log D, -j 0.2794 Log dbb'+ Log 24X24X24 1 (-0.0078+j 0.0045)1o (18)0.0954+j 0.2794 Log D.- V/22.5X 18X22.5X22.5X 18X22.5 Vo' =(-0.0078+j 0.0045)Ii+(0.0078+i 0.2794 Log d,A'] and so forth for I2 volts/mile (25) j 0.0045)I2+(0.286+j 1.78)Io (52)

the remaining 6 terms] (56) -00931[0- 24 1] For comparisons, assume Io only,I I

19 ~~~~~~~~~~~~.03 jLog 22.52X18J12 12=0. Then,Zo 3 (9)(.0954)+j (0.2794) Log D tsmile VI'=0.009Io /30 degrees, 1.25% of XI

0.2794/3 Log (daa'dab'dac'4db'dbb'X 0.0168 V2' O.OO9Io /150 degrees, 1.25% of XId~c'da'dc,'de')"'.286+3 08382 Z,M= -ji 0.0168 ohms/mile or ~~XdbC'dcatdcb'dcc') 002862+j..8382 072 V.'= 1.80 Io /80.9 degrees, 63.8 % of Z.

Log ________________De 100=2.34% of the positive-sequence-;/daa'dab'dac'dba'dbl'dbc'dca'dcb'dec' reactance For I, = I2 = Io:

ohms/mile at 60 cycles (57) Similarly, the induced positive-sequence V1' (0.0234-j 0.0213)IIvoltage and mutual impedance can be -0.0318II /-42.3 degrees, 4.4% of Xi

This equation is derived similarily as the calculated from equations 37 and 38, andGMR-GMD concepts in symmetrical com- the zero-sequence mutual impedance from V2 =(-0.0234-j 0.0213)12ponents, see references 2 and 3. equation 57. A summary of these values =0.0318I2 /222.3 degrees, 4.4% of XIis as follows:

ZiM= -i 0.0168 ohms/mile (38) VO'(0.286+j 1.789)IoAppendix VINI. Calculationsd f Z2M = -j 0.0168 ohms/mile (26) = 1.811. /80.9 degrees, 64.3% of Zothe Positive-, Negative-, and Zero-

Zom = 0.286+j 1.78 = 1.80 /80.9 degrees If the right-hand set of conductors areSequence Induced Voltages in ZOM=0.286+i1.78-1.80 /80.9 degreeS a', b', C' from top to bottom instead ofTwo Parallel Lines on the Same ohms/mile (57) c', b', a' as previously studied, the results_ ~~~~~~~~~~~~~~~~~~~aredifferent and are aS fOlloWS:Tower As indicated above ZiM and Z2M are

2.34% of the positive- and negative- Nontransposed lines:The configuration and spacings are sequence reactance (XI =X2) of the line V j0 8I+ 0 4 j0 04

shown in the sketch. For these lines compared with ZOm which is 63.8% of the V'=3 O18Ii+(-0.0146+3O0.084)12+zero-sequence impedance ZO. (0.0078+j 0.0045)IO (33)

dac'=dCa'18 feetB. For Nontransposed Lines V2'=(0.0146+jO.0084)Il+j 0.018h2+

dab'= dba'= dbc'= deb'= 22.5 feetThe induced negative-sequence voltage (-0.0078+j 0.0045)Io (18)

daa'=dbb'd,cc=24 feet in either line resulting from current in the VO'=(-0.0078+j 0.0045)Ii+(0.0078+Ra 0.259, Xa = 0.435, Xd for 3 conductors other line is from equation 18. j 0.0045)I2+(0.286+j 1.78)1I (52)

0.285 V2W=0.09311IF0.866 Log 18X24X88- Comparison for Io only, l=Ii= =0Go..8~C' 22.5X22.5X24 v'oo9I/3

0.4 18' Log 24 X22.5X22.5 1 VI'=0.009Io /30 degrees, 1.25% of XI8' /22.5X 18 X22.5X24X 18X24 V2' =0.009Io /150 degrees, 1.25% of XI

18X225X225 0' =1.801.g /80.9 degrees, 63.8% of Zom8' 0.09311210.866L%II04'8. g22.5X22.5X18 For I,=lI=Io:

Bypeiu acltosh osatf Log24X24X24 VI' (-0.0068+j 0.0309)Ii =0.0316IhBy previous calculationss, the constants of /22.5X18X22.5X22.5X18X22.5 /102.6 degrees, 4.4% of Xithe line, assuming that it is transposed, are

F 18X22.5X24 V2' = (0.0068+j 0.0309)2 =0.0316I2Z, = Z2 = 0.259+j 0.720 = 0.765 +0.0931o 0.866 g 22.5X24X22.5 /77.4 degrees, 4.4% of XI

/70.2 degree ohms/mile, each line 24 X22.5X 18 ] Vo'=(0.286+j 1.789)Io = 1.811I

Zo=0.545+j2.76=2.82 /78.8 degree V/22.5X18X24X22.5X22.5X24 /80.9 degrees, 64.3% of ZSMohms/mile, each line volts/mile (18)

Zorn=0.286+j 1.78=1.80 /80.9 degree The corresponding induced positive- Appendix IX. Calculations ofohms/mile, mutual and zero-sequence voltages are calculated the Positive-, Negative and

from equations 33 and 52. All induced Zero-Sequence Induced VoltagesParalleled Zo of both circuits =0.415+ voltages in volts/mile are summarized. . T n o Sp.j2.26=2.3 /79.6 degree ohms/mile In TWO Parallel Lines on Separate

V,' = -j0168II+(0.0156-j O.009)I2+ TowersZom_6 Xomn6 t-38 0008i004)o(3=6.8 -=41 -=.8 (007+ 004)I (33 The configuration and spacings areoo~ ~ X X, V,1 '( -0.0156-j3 0.009)I1-3 0.01681,+ shown in Fig. 3. For the lines inl Fig. 3

A. For Transposed LinesThe induced negative-sequence voltage in

either line resulting from current in the a b Ccc'b a 'other line is from equation 25, or the mutual 0-13 '-0=-13'--0 - 35 t -.---0. -22 '-0-22 -.Oimpedance from equation 26 is

V2'=0 0931F0 866 Lg18X22.5X22.5 Fig. 3. ConFiguration and spacings for positive-, negative-, end zero-sequence induced voltageLo 2.5X22.5X18 in two parallel lines on separate towers

FEBRUARTY 1963 Blackburn- Voltage Induction in Parallel Transmission Circuits .927

daa'- 105 dba'=92 dca'=79 J Lo 105X70X35 1+ Vo'=(0.286+j 1.29)Io=1.321o

dab'- 83 dbb'-70 dcb'=57 L V83X61X92X48X79X57J /77.5 degrees, 48.5% of Z,

dac'- 61 dbc'-48 dec'=35 0.0931Io[0.866 Log 61 X48X35

(all distances in feet) 105L3X79X57 Appendix X. Comparison of theFor line a.b.c j Log

1 Induced Voltages for Twor fe a83X61X70X48X57X35 ConIe urationsZ1 -0.306+j 0.784 =0.84 /68.7 degrees volts/mile (18)

ohms/mile Similarly, using equations 33 and 52, Consider two nontransposed line con-

Zo-0.592+j 2.675=2.72 /77.5 degrees the induced positive- and zero-sequence figurations for a flat horizontal parallelvoltages can be calculated. All three line with the same general spacings as

ohms/mile voltages are summarized: illustrated:

Z.,,,=0.286+j 1.36=1.39 /78.8degrees V ='-(-0.00057+j 0.0085)Ii+ A. Unsymmetrical Around the Verficalohms/mile (-0.0082+j 0.0018)It+ Ai

x-=51% -=51% -°=3.41- 3.24 (0.041-j 0.0575)Io (33) Line lXI Zo Xl Zi V2'=(0.0082+j 0.0018)I1+ a b c a' b' c'A. For Transposed Lines (0.00057+j 0.0085)I2+ 0 13' 0 13' 0 35'~ 0 13' 0 13 0

(-0.041-i 0.0575)1o (18) da4'=61 dbaj'=48 dca'-35Transpositions per the sequence outlined Vo'=(-0.0236-j 0.0356)Ih+ dab' = 74 d"' =61 d,b,' =48

in Appendix II. The induced negative- (0.0236-j 0.0356)I2+sequence voltage in the a'b'c' line resulting 0 dac' = 87 dbeI 74 dcct= 61from current in the abc line is, from equa- (0.286+j 1.36)Io (52)tion 25, For comparisons: lo only I1 =12=0 Voltage induced in Line 2 from currents in

Line 1 (equations 33, 18, 52).'-0.093112 0.866 Log 61X92X57 VI'=0.0711o /-54. 5 degrees, 9.1% of XI VI'=(-0.00157-j 0.00593)I,+

83 X48X79

105X70X35 V2 '=0.07110 /234.5 degrees, 9.1% of XI (0.00555-j 0.00311)I2+iLog- (-0.02082+j 0.0423)IoV/83X61 X92X48X79X57J VO'= 1.391o /78.2 degrees, 51% of ZO -0.0082+V2't=(-0.00555-j 0.0311)Ii+

volts/mile (25) For 11=I2=1o (0.00157-j 0.00593)I2+=0.09312[0.866X0.0071X j 0.0911 V1' =(0.0322-j 0.0472)Ii = 0.0571I1 (0.02082+j 0.0423)Io

= (0.00057+j 0.0085)I2 volts/mile / -55.7 degrees, 7.3% of XiO= (-0.0262-j 0.0392)11+

Z,,=0.00057+j 0.0085 ohm/mile or V,'=(-0.0322-] 0.0472)12=0.0571I2 (0.0262-j 0.0392)I2+0.0085+ /235.7 degrees, 7.3% of Xi (0.286+j 1.41)1,

074100=1.09% of positive-0.784 Vo'=(0.286+j 1.29)Io=1.321o Line 1 Line 2

sequence reactance /77.5 degrees, 48.5% of Zo at b' c' a b c

Substitution in equations 38, and 57 If the right-hand set of conductors are 0 13' 0 13' 0 35'. 0 13' 0 13' 0gives the positive- and zero-sequence a', b', c' from left to right instead of c'mutual impedances. A summary of these b', a' as previously studied, the results are daa'"61 4a'=74 dca'=87values for the three sequences is as follows: different. For the nontransposed lines d,,'=48 dbb'=61 de'=74Zjm - -0.00057+j .0085=0.0085 V1'-(-0.0025-j 0.00795)I1+ da,='35 dbc'=48 dc'-61

/93.8 degrees ohms/mile (38) (0.00705-j 0.00474)I2+'=0+8-.0293-+.0643'I, f33) Voltage induced in Line 1 from currentsZim =0.00057+] .0085 =0.0085 '* ' ' i Line 2

/86.2 degrees ohms/mile (26) V,'=(-0.0071-] 0.00474)1+ V1'=(0.00157-j 0.00593)hi+ZoM=0.286+j 1.36 = 1.39 (0.0025-j 0.00795)I2+ (0.00555-jO.00311)I,+

/78.8 degrees ohms/mile (57) (.0293+j 0.0643)Io (18) (0.0262-j 0.0392)I+Zim and ZIM are 1.09% of the positive- VO' (-0.0236]j 0.0356)Ii+ V2' = (-0.00555-j 0.00311)I +

and negative-sequence reactances (X1-X2) (0.0236-j 0.0356)I2+ (-0.00157-] 0.00593)I+of the line compared with ZOM which is (0.286+j 1.36)Io (52) (- 0.0 2663.8% of the zero-sequence impedance (ZO). ( 0262-j O.0392)Io

Comparisons: for Io only I,=I2= 0B. For Nontransposed Lines Vo'= (0.0208+j 0.0493)II+The induced negative-sequence voltage VV'= 0.0711o/114. 5 degrees, 9.1% ofX1 (-0.0208+e 0.0423)r+

En a'b'c' resulting from current in abc, V,'=0.0711I /65.5 degrees, 9.1% of X1 (0.286+j 1.41)1.from equation 18 is -

V,'= 1.39!, /78.2 degrees, 51% of XO There will be circulating currents be-F.t0.031IrO.66 og61 X<70X79_ tween the two lines, which is proportional

_ 83X92X35 For h=12=10 to the difference between the two inducedvoltages given previously, thus subtracting

]Log ~~105X48X57 -+ V1'u( 0.0246+] 0.0516)I1-0O.OS71I, the induced voltage in line 2 from theV83X61X92X70X79X35J+ /115.5 degrees, 7.3%of X1 induced voltage in line 1.

°.O31I[O 66 og61 X92 X57 VKs -(0.0246+] 0.0516)12 =0.057112 V1' difference = -0.00314IX +83X48X79 /64.5 degrees, 7.3% of X1 (-0.047+] 0.0815)1,

928 Blackburn-Voltage Induction in Parallel Transmission Circuits FEBRUARY 1963

VW' difference= +0.00314I2+ Line 1 Line 2 VO'=( -0.0262-j 0.0392)Il+(0.047+j 0.0815)Io a' b' c' c b a (0.0262-jO.0392)I,+(0.286+j 1.41)Io

Vo' difference=(-0.047 -j 0.0815)I,+ O 13' 013' 0 35' 0 13' 0 13' 0 There is no difference between the induced

(0.047-j 0.0815), daa'=87 dba'=74 dccz'=61 voltages in the two lines.dab'=74 dbb'=61 dcb'=48

with I -Is, the zero-sequence voltaged 7

difference reduces to V.' difference - dac' = 61 dbc' = 48 dc0'= 35 References=jO.1631i. Voltage induced in either Line 1 from

curent'min 2,or m Lne2 fom urrnt 1. NBOATIVR-StQuBcNcB DiRBcTrIONAL GtowNDcurrent in Line 2, or in Line 2 from current RELAYING, W. A. Elmore, J. L. Blackbur AIEXB. Symmetrical Around the Vertical m Line 1, from equations 33, 18, 52, is Transadioxs, see pages 913-21 of this issue.

"a~~ ~ ~ ~ ~~~~1= .063~(-.09+ .06) 2. SYXMBTRIcAL CompoNENqTa (book), C. F.LxinILine2nV,'=j 0.00623I+(-0.0052+i 0.0 1 Wagner, R. D. Evans. McGraw-Hill BookLine I Line 2 +(0.0262 -j 0.0392)1o Company, Inc., New York, N. Y., 1933.

a b c c' b' a' V2 =(0.00592+j 0.00161)I + S. SPBED ZBRO SEQUINCE IMPEDANCE CALCULA-Tlom Font TRANSMuSSsON LINEsS, J. L. Blackburn0 13' 0 13' 0 35' 0 13' 0 13' 0 j 0.00623I,+( -0.0262-i 0.0392)I. Eectrical World, vol. 163, March 1960, pp. 60-61.

P * * ~ * " I e . can cause incorrect operation of direc-rotection or wircuits witn aeries tional and distance relay elements andin some instances can even cause incorrectoperation of the differential type of linewa7pacltors relaying such as phase comparison.

JOHN BERDY This section will briefly review this subjectASSOCIATE MEMBER AIEE and in addition discuss the effects of the

series capacitor protective gaps.

Summary: An analysis of the protective pensation can provide improved stability EFFECT OF SERIES CAPACITORSproblems introduced by series capacitors =imits, improved voltage regulation, a de- Asre aaio a fetteoealeads to the conclusion that conventional s, p o o ra a onomic A series capacitor may affect the opera-line protective schemes will, in most in- sied load divsion, or a more economic tion of directional-type protective relaysstances, have limited application on series- utilization of the extra-high-voltage sys- by affecting the phase position of thecompensated lines. A line relaying scheme tem (higher loadings). voltages and currents as seenbytherelays.is presented which has almost universal Of equal importance to system design When a series capacitor is located betweenapplication on compensated lines. This We eiscpctri oae ewe

scheme, designed specifically for lines with and operation is the protective relaying a relay terminal and a fault, the relay cur-series capacitors, will meet the speed, selec- of the series-compensated system, for, in rents or voltages may be reversed from thetivity, and sensitivity requirements of most instances, the protection of circuiuts positions they usually assume for a faultpresent-day systems. with series capacitors is not as simple and on a system whose reactances are induc-

straightforward as the protection of other tive. For instance, consider the equiv-circuit configurations. Depending on its alent system shown in Fig. 1. For dis-

THE APPLICATION of series capaci- location and on the degree of compensa- cussion purposes, assume that the systemI tors to high-voltage transmission lines tion, the series capacitor may introduce voltages are m phase prior to the fault

has once again been receiving considerable various relay problems which could re- and during the initial stages of the faultattention during the past few years. For sult in increased fault-clearing times and/ and that the system angle is 60 degrees,the most part, the renewed interest in this or a loss of selectivity. It is the purpose inductive. Fig. 1 shows the vector dia-subject has been in the area of extra-high- of this paper to discuss the protective gramof the system voltages and currentsvoltage transmission. With a few excep- problems associated with lines having on this inductive system for a 3-phasetions, most of the extra-high-voltage sys- series capacitors and to discuss the ap- fault (F) near terminal A. The vectortems being planned for construction in the plicability of standard protective schemes diagram also shows the voltage VA whichUnited States and in Canada are being to such lines. In addition, this paper will would appear at the terminals of a relaydesigned to use some degree of series com- present a relaying scheme which waspensation. specifically designed for the protection IFI f 2The benefits of series capacitor com- of lines with series capacitors and which ; A-

pensation from a system design and oper- meets the speed, selectivity, and sensi- F 2

ating point of view are well known to the tivity requirements of present-day sys-system planning engineer. Series corn- tems. VA

Paper 62-1093, recommended by the AIEE Relays Effect of Series Capacitors on\Committee ad approved by the AIEE Technical Protective ReLays\Operations Department for presentation at the\AIRE Summer General Meeting, Denver, Colo., ..IFJune 17-22, 1962. Manusrpt submitted Dc- The adverse effects of series capacitors 1Fcember 11, 1961; made avallable for printing, on protective relay operation have been 1

Ma . 92 previously described in the literature.1 Fig. 1. V/ector dibgram for an InductiveJ. BunnY Is with the General Electric COmpany, I a ensonta sre aaiossseSchenectady, N. Y.Ithsbeshw thtsrecaciosytm

FEBRUARY 1963 Berdy~Protection of Circuits with Series Capacitors 929