tems should not be part of a mere everyday many of the methods have already been Perhaps the greatest handicap to over-office routine, performed by trained cal- worked out in detail. It is only a question come, before power engineers can evenculators, by punch-card machines or by of how fast the power companies them- start solving their integrated system prob-special electronic calculating devices. The selves will utilize the facilities already avail- lems by analytical methods, is their awe,analytical tool is already available and able along these lines. their fear of the "mysteries" of tensors.

Determination of Instantaneous Currents ia ia +io (1)1

ib =-i+io+ ( \/3/2)i: (2)

and Voltages by Means of Alpha, Beta, 1iC 2-ia+io-( \/3/2)i,B (3)

and Zero Components Simultaneous solution of the precedingequations gives

W. C. DUESTERHOEFT MAX W. SCHULZ, JR. EDITH CLARKE ia =( ()ib+ ) (4)ASSOCIATE AIEE NONMEMBER AIEE FELLOW AIEE 3 2

(ib-ic) 5

A PPLICATIONS of the method of In Part II, equations for instantaneous (i5alpha, beta, and zero components phase currents and voltages at the ter- (ia+i(+ic)

(conveniently written a,/3,O components) minals of an unloaded ideal synchronous io 3 (6)to the determination of phase currents machine during all types of short circuitsand voltages of fundamental frequency in are derived by means of a,0,0 compo- If i in equations 1-6 is replaced or e,unbalanced 3-phase power systems have nents. It is thought the method of de- equations relating instantaneous fluxbeen given.'-' The advantages of a,j3,O veloping the complete harmonic series for linkages (i6) or voltages (e), respectively,components over symmetrical compo- instantaneous currents and voltages, in the phases and in the a, /, and 0 cir-nents, in the determination of fundamen- given in the Appendix I, has not been cuits will be obtained. Equations 1-6tal-frequency phase quantities, is largely presented previously. are general equations which apply underrestricted to unsymmetrical 3-phase sys- In Part III, equations for eircuit- all conditions. No simplifying assump-tems in which the positive- and negative- breaker recovery voltages derived by tions need be made; the quantitiessequence impedances of the rotating Park and Skeats6 are redeveloped using involved may be expressed in per unit ofmachines of the system can be assumed a,/3,O components. The simplification in stated base quantities, or in any con-equal. This restriction is not present in development made possible by the ap- sistent system of units.the determination of instantaneous phase plication of a,/,0 components can be seenquantities by the method of a,3,0 com- readily by comparison with the originalponents. development. In developing equations involvingThe purpose of this paper is to present a synchronous machines4'5 the work is

method of analysis rather than to de- Part I. Equations for Use in greatly simplified if per unit quantities arevelop new equations. For this reason, Parts II and III used, and an ideal synchronous machinewell-known equations previously de- is assumed with saturation, hysteresis andrived by outstanding engineers have been NOTATION eddy currents in the iron neglected, theseselected for redevelopment here in order latter effects to be taken into account laterto show the simplifications made pos- Let i, 4, and e with appropriate sub- if of importance.sible by the application of a,/3,O com- scripts represent instantaneous armatureponents. Sincere admiration is hereby current, flux linkage, and voltage, re- UNIT OR BASE ARMATURE QIANTITIESexpressed for the ability of these engineers spectively; let subscripts a, b, c refer to Unit armature current and voltage areto carry through the original development phase a, b, c, respectively; let subscripts crest values of rated phase current andwithout the benefit of a,#,O components. a, /, 0.refer to , /, and 0 components, rated phase voltage, respectively; unitThe paper is divided into three parts. respectively; let subscripts d and q refer

In Part I, relations are established be- to the direct- and quadrature-aids com- Paper 51-226, recommended by the AIEE Trans-tween instantaneous phase quantities and ponents, respectively. mission and Distribution Committee and approved

by the AIEE Technical Program Committee fortheir az,/,0, components, and between a RLTOSB WENP EQU TIES presentation at the AIEE Summer General Meet-

RELATINSBETEEN PHSE QUATITIES ing, Toronto, Out., Canada, June 25-29, i951.and /3 components and direct- and quad- AND THEIR a,/3,0 COMPONENTS Mavalalefocrintinsu M? erary3,

,i 951iad

rature-axis components. Park's equa- 'aviblfrmtg y319.tions for an ideal synchronous machine,4'5 By definition, instantaneous phase w. c. DUESTERHOEFT and EDITH CLAR}E are both

with the University of Texas, Austin, Tex., andin terms of direct-axis, quadrature-axis, currents ta, tb, ic of normal phase order MAX W. SCHULZ, JR., is with the General Electricand zero-sequence components, are ex- abc at any point in a 3-phase system, in Company, Pittsfield, Mass.pressed in terms of z,/3,0 components for terms of their instantaneous a,/B O com- The authors wish to express their appreciation for

'' ~~the assistance given by Mr. Chwan-Chang; Lee inuse in Part II and Part III. ponents of current, are the preparation of this paper.

1248 Duesterhoeft, Schulz, Clarke-Instcantaneous Currents AJEE TRANSACTIONS

impedance is the ratio of rated phase ponent networks; 0 is the angular dis- in the voltage of the field exciter will havevoltage to rated phase current; unit placement in electrical radians of the negligible effect during the first cycle orspeed of the rotor is synchronous speed; direct-axis of the rotor from the axis of less required for currents and voltages tounit time is one electrical radian; unit phase a, measured in the normal direction reach their maximum values.armature linkage will generate unit of the rotor. Xd' and xq"t are per-unitarmature voltage at rated speed; unit direct- and quadrature-axis reactances, LINE-To-LINE SHORT CIRCUITarmature magnetomotive force is the respectively; e is per unit armature Let the fault be between phases b and c.magnetomotive force produced by rated voltage rise to machine terminals; i is per The conditions imposed by the fault are:positive-sequence armature currents; unit current flowing from machine ter- iaO; ib= -i.; eb=e,. These equationsunit permeance when multiplied by unit minals; and If= d-c field current in per substituted in equations 4 and 6, and inmagnetomotive force produces unit arma- unit of that base field current which will 5 with i replaced by e, give ia= 0; io=0;ture linkages. produce unit armature linkage at no load e, =0. The above equations allow the

RELATIOSBETWENDIRET-ANDand rated speed, with saturation neg- reduction of equation 12, with r =0, toRELATIONS BETWEEN DIRECT AND lected, and generate unit armature volt-QUADRATURE-AXIS COMPONENTS age. Incos O=p(x-y cos 20)il (16)AND ag AND A COMPONENTS During the first instant after a dis- If each side of equation 16 is integrated,

As zero-sequence components (written turbance, the speed of the rotor can be there results0 components in the a, /, 0 system) are assumed constant at rated speed becausethe same in both systems, it is unneces- of rotor inertia. At rated speed, the IJ (sin 0-sin 0O)=(x-ycos 20)ipsary to consider them in establishing rela- speed of the rotor in per unit of synchro- .

c

tions between the two systems of com- nous speed is unity; and the angle O, at The current ia, being zero immediatelyponents. If equations 1, 2, and 3 are any time t in electrical radians, may be mediately after the fault; hence, If sinsubstituted for ia, ib, and ic, respectively, writtenin the equations for id and iq in reference 5, t

appears in the preeding equation as

and the resultant equations solved for 0=00+t (15) inegia and i,s, the following equations are ob- .. . givestamied: where 00= Oat time t = 0, from which time (sin 0-sin O0)in electrical radians is measured. In (i y= (17)id= i, cos 0+ig sin 0 (7) the work which follows in Parts I and I (x -y cos 20)

i = -i sin 0+io cos 0 (8) constant rotor speed will be assumed, and Substitution of equation 17 and i"a = 0 inarmature resistance (r) in equations equation 11 gives

ia =id cos 0-i sin 0 (9) 11, 12, and 13 will be neglected to permit py sin 20Ir (sin 0-sin Go)i3 =id sin 0±iq cos 0 (10) ready solution without a differential ea= -If sin 0- (x -y cos 20)

analyzer. As armature resistance is 18If i in the preceding equations is re- small relative to reactance, neglecting it (1)

placed by i1 or e, equations relating direct- will not appreciably affect initial cur-and quadrature-axis components and a rents and voltages; for subsequent HARMONIC SERIES FOR CURRENTSand : components of flux linkage i or values, appropriate decrement factors Currents ig in equation 17 can be ex-voltage e, respectively, result. can be applied, as will be explained later. panded in a harmonic series in terms of x

and y by application of equations 99 andPARK'S EQUATIONS FOR AN IDEAL Part II. Short-Circuit Currents and 96 with Z given by equation 94, if a andSYNCHRONOUS MACHINE IN TERMS Voltages at the Terminals of an b in these equations are replaced by xOF az,/,0 COMPONENTS Unloaded Ideal Synchronous and y, respectively. When x and y are

The development of these equations Machine, Operated at Rated then replaced by their values in terms of(which are expressed in terms of direct- Speed Xd "f and x01' from equation 14 and in mul-and quadrature-axis components) by tiplied by V/3/2 to give ib=- (sinceusing a,/3,O components instead of phase Equations 11-14, with resistance neg- j=o= 0), the harmonic series at thequantities, and the expression of them in lected and constant rotor speed assumeds first instant is obtainedterms of a,#,0 components have been will be usedto determine phase currentsgiven.7 For the initial conditions follow- and voltages at the machine terminals ib =ic = [I/3I (xd "+ VxXd x)]qXing a short circuit at the terminals of an in terms of their harmonic components sounloaded machine, operating at rated during the first instant after various EZn-I sin (2n-1)0-(\/3/2) Xspeed, these equations in terms of a,f,0 types of short iruits. Subsequent ncomponents are values of current and voltage may be ob- [IdX sin 0o/Vxd x ] X

tained by application of appropriate dec- [n 1=-Isin-p(x+y cos20)a -pyX rement factors. Even harmonics (in- 1+2 Z cos 2nOj (19)

sin 20i13-nia (11) cluding d-c components) and natural- n

e, =Ir cos e-py sin 2OaiaPX frequency terms are attenuated by arma- where(x-ycos 2O)i,i -nips (12) ture time constants; odd harmlonics (in-_

cluding fundamental-frequency terms) (V/xi" -Vx:) (20)eo=-pxoio-nio (13) are attenuated by rotor timeeconstants. (v7+xd"f/'Xq")

(Xd"t+Xq") (Xe" .Xq"t) As capacitance of the generator is neg-x= ; = 2(14) lected, there will be no natural frequency Equation 19 checks the work of

terms. In determining maximum cur- Doherty and Nickle.9 As pointed out bywhere p=d/dt; r is per-unit armature re- rents and voltages, action of voltage these authors, it isinteresting to note thatsistance, assume the same in all com- regulators can be neglected as any change the coefficients in the harmonic series are

1951, VOLUME 70 Duesterhoeft, Schulz, Clarkue-Instantaneous Currents 1249

the constant Z, which is less than unity, p(x+ixo+y cos 20)ja -If sin 6 (26) and simplification, givetaken to increasingly higher powers for If each side of equation 26 is integrated plf[(x-y+ 'xo) sin 0-y cos 6o sinl 20]the higher harmonics. The first series in n t c i e =(

19 ive thfudaentl ad od-hr- and the condition is satisfied that i.a must (x+ 2xo+y cos 20)19 gives the fundamental and odd-har- b eoa =0(2monic terms; the second series gives the (32)d-c and even-harmonic terms. IfJ(Cos 6- Cos 0) (27) Equation 32 can be expanded in a series

HARMONIC SERIES OR VOLTAGES cos 2) in terms of x, y, and xo by application of

Voltagee,When i from equation 27 and i, equations 99 and 97, where Z is given byVoltage e,,,,, in equation 18, can be are substituted in equation 12 equation 94. When expressed in terms

simplified by integration of the first term Of Xd", Xq"/, and xo by a procedure similarand indicated differentiation, as in the e [=If Os a py sin 26(cos 0-cos Oo)] to that used to obtain ia in equation 29,following equation ex+ =If coss6-0L (x± 2xo+y cos 20) and the indicated differentiation per-

y sin 26(sin 0- sin Oo)1 (28) formede'a =PIf cos 0- _,-ycs2)21\I(d+XO Xf+loea~PfLcos (x-y cos 20) i HARMONIC SERIES FOR CURRENTS e 2- 1 (X + X0) (xq + 2X0)

PI(xV-y) cos O+y sin 60 sin 20 (21) The series for ia, equation 27, in terms Xd'++2xo+ V/(Xd"+ 2xO) (Xq"+ 2Xo)L (x-y cos 26) 1 of x, y, and xo can be obtained by using

Equation 21 can be expanded in har- Equations 100 and 96, with Z given by ) (2n-l)Z'-' cos (2n-1)0-monic series in terms of x and y by ap- equation 94 if a and b in these equations n=_plication of equations 100 and 97 with z are replaced (x+2x0) and -y, respec-given by equation 94, if a and b in these tively. W'hen x and y are replaced by 41f cos 6oZnZn cos 2n6 (33)equations are replaced by x and y, re- values in terms of Xd' and xq" from equa-

n=1

spectively. tion 14 ia and Z are determined. Sinceia=ia,+ io, and io = 2ia where Z is given by equation 30.

2(x-y)ZZn_1 cos (2n-1)60a=1 ian_ ___ +t2 PHASE VOLTAGES

--P-f x-y+V/x2-y2 From equations 1, 2, and 3 with i re-CO \ 5placed by e,

2pIf sin 0OEZn sin 2n/ (22) 3Ir Z cos (2n- 1)6n = 1 nn=1 ea =e,+eo=O (34)

When x and y in the fraction in equa- [Xd+ 2xo+V\/(Xd"+ 2'xO) (X0"±+ Ixo) 3ea V3eotion 22 are replaced by their values in r - 2 2 (35)terms of Xd' and xq' from equation 14 31 cos 6o 1+2 zn cos 2n] -

and numerator and denominator multi- 3ej V\_3e,3plied by V\Xd"jXq", and the indicated dif- 2V 2+1x0)(X"+15i) (29)ferentiation performed, ea = ea is 21S After e,a from equation 31 and e,5 from

e_ _= where equation 33 have been substituted inea --ea-2IfVXd"Xq"w = V,/x0t+ -Vxd"+±xOequations 35 and 36, the resultant equa-

-21f+/Xd Mxq(2n -1)Zni-1 sin (2n -1)6 ( q ±\ +30) tions can be simplified by application ofn=1

xqff+ 12X ~~+ NIXd "+ 12 X the equation:

(Xd"+V\IXdXQ" ) HARMONIC SERIES FOR VOLTAGES A sin 8+B cos 5 = V\A 2+B 2sin (6-y)co

+4If sin 6OEnZn cos 2n6 The a-component of voltage can be where-=l found from equation 25 if ia in this equa-

tion is replaced by (2/3)ia given by equa- 7 tan (B/A)eb=ec= -e0 (23) tion 29, and the indicated differentiation Thus

performed.where Z is given by equation 20. ea =p 1Oxi ebV3, VXd x +(xo)(d"+xq)+Xox

Xd + -2Xo+ V\(Xd + 2Xo) (Xq"+ Xo)LINE-TO-GROUND SHORT CIRCUIT -X0If x o

Let the fault be between phase a and Xd"+ 2Xo+V(Xd'+ 2XO) (Xt"+ 1Xo) (2n-1)Zf-i sin [(2n-1)6+-y]-ground. The conditions imposed by the o

fault are: te = 0;iu =id = 0. These equa- (2n 1)Z-I sin (2n-1)0+ 2\/31f cos oVXd"Xq"+ (IsO) (Xd" +x2") ±so2tions substituted in equations 4, 5, and 6 ________________

and in equation 1, with i replaced by e, n=l V/(xa" ± SO) (xq"+ so)give 2xoIf cos Oo X

i~=O; ia=2io; ea= -eO (24) /(d+x)x"2o n nsin (2n6+ -y) (37)

Equations 24 substituted in equations vn nsin 2n6 (31) n11 and 13 reduce these equations to ex-

* r * r * 1 :1 r * t~~~n= i wherepressionswhich can be solved for j where Z is given by equation 30.2Vd+x/)("+/2<= -eo=pxoio=pIxoJ Integration and indicated differentia- = tanl- V+(d'+°/)xoS+°=-Ir sin 6-P(x+y cos 26)jay (25) tion of the first term of e,s in equation 28, /x

1250 Duesterhoeft, Schulz, Clarke-Instantaneous Currents AJEE TRANSACTIONS

and Z is given by equation 30. The equa- ib=-3ia/2+±/\ip/2 (43) Part III. Circuit Breaker Recoverytion for e, differs from that for eb only in Voltagethat sign before -y is negative instead of When the terms in the numerators of

positive, equations 41 and 42 involving 0 (evenharmonics) are omitted, and these equa- Circuit breaker recovery voltage may

DOUBLE LINE-TO-GROUND FAULT tions are substituted in equation 43 be defined as the voltage appearing acrossthe poles of a circuit breaker after it has

Let the fault be between phases b and c . -3(x-y) cos +0±/3(x-y+2xo) sin 0 interrupted a current. The first an-and ground. The conditions imposed by b 2[X2-y2+2xo(x-y cos 20)] alytical treatment of the subject is giventhe fault are: ia= 0, and eb=e= 0. These (44) in a paper by Park and Skeats.6 Theyequations substituted in equation 1, divide the effects of recoverv voltage intoand in equations 4, 5, and 6 with i re- HARMONIC SERIES FOR VOLTAGES high- and low-frequency effects and treatplaced by e, give From equations 38 and 13 these two effects separately. High-fre-

quency effects depend upon the capaci-Q= -io e,s=0 ea=2eO =2eo= 2Pxoio=2pxoia (45) tance in the system. Two types of sys-

Equations 38 with equations 11, 12, and If equation 41 for i is expressed in its tems are considered for low-frequency ef-13 give harmonic series, then substituted in equa fects: (1) those in which the impedances of

tion 45 'and the differentiation perf stationary equipment are much greaterp(x+2xo+y cos 20)i,+pyX tion 45, and the diferentlation performe=, than the impedances of rotating machines,

sin 20i, = -If( sin 0) so that the effects of unequal reactances inpy sin 20ia+P(x-y cos 20)id =Ir(cos 0) = (3/2)ea, is the direct and quadrature axes of rotating

- 6xoxQ 'If machines are relatively unimportant;If both sides of the preceding equations ea Xq'(Xd+±2xo) + (2) those in which the impedances of

are integrated and the integration con- V\Xd 'tx0(Xd'+±2xo) (x,+±2xo) rotating machines predominate. It is thestants evaluated, there results m latter type which will be considered here,

s(2n-1)Z" sin (2n-1)0- the purpose being to check the final(x+2xo+y cos 20)ifa+YX equations in Appendices B, C, and D ofsin 2o =If(cos 0- cos 0) (39)n_ the reference paper6 by using o,3,0 com-

y sin 20i±+ (x -y cos 20)ij8 =If(sin 0- sin Go) 6Xd'XIf cos Go X ponents.(40) V\Xd Xq'(Xd +2Xo) (xa0+±2xo) The equations for fault current, de-

veloped in Part II, will be used to deter-Simultaneous solution of equations 39 mine the voltage across the poles of theand 40 for i,, and i# gives nZn sin 2n0+61f X circuit breaker for the first phase to clear

a= n=1 of a fault at the terminals of an unloadedIf[(x-y) cos a-x cos AO+cy cos 20 cos 00+ ideal synchronous machine operating at

y sin Oo sin 20] sin o nxZn cos 2na (46) rated speed. In the case of multiple polex2-y2+2xox-2xoy cos 20 2 .Jbreakers, the resistance of an arc formed

(41) n=1 by an uninterrupted phase current is as-

where sumed to be zero. Natural frequencyin= effects caused by capacitance will be neg-

If [(2xo+x-y) sin 0-(2xo+x) sin Oo- V\xd'(xq±+2xO)- VXq;'(Xd"+2Xo) lected.y sin Oo cos 20+y cos Oo sin 20] Z= External reactances can be added

2 2+22 20 v~~~~-\Xd (Xq +2xo) + -\Xq' (Xd +2xo)x2-y2+2xox-2xoy cos 20 V +72 to the machine reactances but, as the(42) (47) length of externally connected circuits

increases, the capacitance assumes greaterHARMONIC SERIES FOR CURRENTS THREE-PHASE FAUJLT importance. Resistance will be treated

To apply the equations of Appendix Ito The conditions imposed by the fault, in as in the reference paper;' its effect onequations 41 and 42, a and b must have a grounded or unground system are: the initial magnitudes of currents andthe following values in equations 100, 97, ia±ib+ic = 0, and ea = eb = e,. These equa- voltages is small and will be neglected,96, and 98, and in Z given by equation tions substituted in equation 3, and in but the attenuation of currents and volt-94 equations 4, 5, and 6 with i replaced by e, ages is included in the decrement factors.

give io = 0, ea 0, and eo = 0. When ea = 0 Use will be made of the method of super-a=x2-y2+2xox=Xd"Xq"+XO(Xd"+Xq") and e3 = 0 are substituted in equations 11 position, whereby a current is interruptedb =2xoy=XO(Xd'--x011) and 12, these equations will be the same as (made zero) by superposing an equal and

those for a double line-to-ground fault opposite current without changing theThe procedure for determining the with the terms containing x0 omitted. system. The voltage drop across the

complete harmonic series for ia and i,s, There ia= ai=4, can be written from switch terminals caused by the applica-and then substituting them in the equa- equation 41 by omitting the terms in the tion of a current equal and opposite to thetions for i0 and 4, and combining terms, is denominator containing x0. fault current will be the same as thesimilar to that used to obtain eb in equa- voltage across the switch caused by open-tion 37. It will not be given here; in- (x-y) cos 0-xcoso±+y cos (20-G) ing the switch. When faults involvingstead, the odd-harmonic terms of ib will sa 1f (x2_ y2) (48) ground in a grounded system are treated,be expressed in a form suitable for use in the voltage across the circuit breaker isPart III. r (xd"+Xo") cos Go- ] equal to the phase voltage to ground of theFrom equation 38, i = -4,!; therefore -IcosG - (Xd" -x0")cos (20-00) opened phase. In ungrounded faults or

from equation 2 L Xd"(2xd"'xq") j(49) ungrounded systems, the circuit breaker

1951, VOLUME 70 Duesterhoeft, Schulz, Clarke-Instantaneous Currents 1251

voltage will be the difference between two phase a remains open and phase c remains -3(x -y) cos 0+ x/3(x -y+2xo) sin 0=0phase voltages. grounded, ia'= O and T,'= 0; eb, Tb, and (62)

f , which were zero before ib' was ap- V3(x-y)FLUX LINKAGES AND VOLTAGES plied, will take on new values. In the tan 0= (- (63)The following equations relate sudden development given below, as all quanti- (x -y+2xo)

changes ing<,, equ,andra

with sudden ties are increments due to the applica- Let 4fb in equation 61 be written Tb=changes inia,iT,and i T. tion of ib', the primes will be omitted. N/D. Then eb=pPTb= (DpN-NpD)/D2.

With ia = 0, in equation 1, i= -io. To satisfy the condition of equation 62-r'=-ia'(x+y cos 20) -1i3'y sin 20 (50) Therefore from equations 2 and 3 at the instant phase b clears, N= 0; but

1=-ia y sin 20-io'(x-y cos 20) (51) pN is not zero. Therefore, eb = pN/D atib'= -3 a-+V3 (56) the instant phase b clears.

To'=-=ioxo (52) 2 2

(Xd+XQ ) (X iX,) -'0=-ioX =iaXO (57) eb =P 'b=(k)X-; y= 2(53)2 ' 2 If i in equations 2 and 3 is replaced by [3(x-y) sin 0+x/3(x-y+2xo) cos0] (64)T and T, = subtracted from Tb, there [XX-cs8A3sn8where primes indicate sudden changes Ia[2x+xo-y cos 20 + V3y sin 20]

from previous values; and the negative, where for current zero, sin 0, cos 0, cossigns with currents follow the convention "b = \V3*g (58) 20, and sin 20 must have the followingfor signs of the reference papers:4, f flux To is given by equation 51 in terms Of values determined by equation 63.linkages due to the main field are positive, Iad is gieby oequai 51 intermsdethose due to current in the positive direc- and is The procedure to determine V/3(x-y)tion are negative. Voltage rise to the i and i,is to substitute equations 50, 51, 3(xy)2±(x-y±2xo)2machine terminals in any circuit, a, is and 57 in the equation T,= 0; then from

written simultaneous solution of the resultant sin 20 =equation and equation 56, ia. and i, in 3(x-y)2+(x-y+2xo)2

ea =PT'a (54) terms of ib are obtained. When ia and

where STa included all linkages With cir- io are substituted in equation 51, To' in 0 x y+2xow idtak terms of ib' is determined. From equa- cos

tion 58, 'tb is obtained in terms ib; iS (x-y+2xo)1-3(x-y)2DECREMENT FACTORS then replaced by the negative of the fault cos 20 3(x-y)2+-3(x-y )2 (65)

current at the instant phase b clears,The equations for short-circuit cur- given by equation 55. With T1b known, When these trigonometric functions are

rents given in Part II are for initial cur- eb =pT,=circuit breaker recovery volt- substituted in equation 64, the resultantrents. By the time a circuit breaker age. equation simplified, and x and y replacedoperates to interrupt a fault current, the \Wrhen equations 50, 51, and 57 are sub- by their values in terms of Xd" and xe,"magnitude of the current will have stituted in TJ = 0, equation 3 with i re- from equation 53, eb is given by the fol-changed. Let placed by ', the resultant equation is lowing equation

k= decrement factor for odd harmonics ia(X+Ycos20+V3y2sin20+2xo)+)which may combine effects of rotor ix [ycsi 20+ V3(yx-icos+20o)1=0 eb =X(2Xq+X)+X(Xq+X)(X/2ktransient and subtransient time con- i()[y sin 20+\/3(x-y cos 20) 0 Xd'(2xz+Xo)2+xo(2x, "+xo)(xq'+2xo)stants with exciter response (59) (66)

ka =decrement factor for even harmonics Simultaneous solution of equations 59 Equation 66, which gives the voltage

time constant and 56 gives equations for ia and i,6 in across the switch terminals at the instantterms of ib'. When these equations are phase b clears, checks equation D-28 of

OPENING A DOUBLE LINE-TO-GROUND substituted in equation 51 and the re- the reference paper.6FAULT sultant equation multiplied by V/3, Tb isLet the fault involve phases b and c in a obtained. OPENING A 3-PHAsE FAULT TO GROUND

groundedsstem,and assume that phase b[X2y2+xxycos2IN A GROUNDED SYSTEMgrounded system, and_=__ _-_y_cs_20_ (60) Iwlbauthatphase - is theb opens before phase c, after armature 'b=(2x+xo -y cos 20+ V3y sin 20) fisIt will be assumed that phase a is thetransients have died away. The fault first to clear after d-c and second har-current ib, at the instant phase b clears, When ib,, given by equation 55, re- monic components of current have dis-is k, times that given by equation 44 with places ib' in equation 60, Tb becomes appeared. The fault current ia at the in-If replaced by unity for rated voltage be- stant phase a clears is kr times the firstfore the fault. Let ib' indicate the cur- yb - . terms of equation 48, with If replaced byrent to be superposed on existing condi- kr- -3(x-y) cos0+ 3(x -y+2xo) sin0] E, the line-to-neutral voltage before thetions. Then, 2 [2x+xo-y cos 20+ V/3y sin 20] fault occurs. The current ia' to be super-

(61) posed on existing conditions, therefore,-krH[3(x-y) cos0+ i

<,a3(x-y+2xo) sin 0] As the assumption is made6 that phase it 2[x2-y2+2xo(x-y cos 20)] (55) b clears at current zero, the value of volt- --krE(x-y) cos 0}age eb-=P'Ib at that instant will be de- ta = 2_ -(67)

Before phase b opens, which is before termined. At current zero, the numerator4't iS applied, e =e = 0 and a= 0. With of the fraction in equation 44 for fault As phases b and c remain grounded afterresistance neglected and eb= c=0, 'tb= current 4b must be zero. To satisfy this ia1'is applied, "tbt-= 'e' =0 If equationsV= 0; from equation 5, with i re- condition, the following equations are re- 2 and 3 with i replaced by t'I, are added

placed ', =0. After 4', is applied, as quired and then subtracted the following equa-

1252 Duesterhoeft, Schulz, Clarke-Instantaneous Currents ATEE TRANfSACTIONS

tions are obtained OPENING A 3-PHASE FAULT (NOT TO 3 [kr(x -y) cos 0-kax cos o+kaYXGROUND) IN AN UNGROUNDED SYSTEM COS 20 cos Oo+kay sin 20 sin 0o]

It will be assumed that phase a clears 2(x-y cos-20)'i't' =0 (69) first. Two cases will be considered: (1) 3 [k, (x-y) cos 0+kay sin 20 sin 0o]From equation 1 with i replaced by T' d-c and second harmonic components of 2 x-y cos 20

currents have disappeared when phase a 3*a'= *'+T0ot=34fo' (70) clears, and (2) they have not disappeared. 3ka cos 0oAs T'a was zero before ia' was applied, The current to be superposed on existingtat in equation 70 is the actual flux linkage condition in either case will be indicated The equation for Tba can be expanded in4'a after phase a clears. For convenience, by ia'; and Ta in terms of ia' will be de- series by equations 100 and 97 where Z isthe primes will be omitted in the develop- termined before substitution is made for given by 94, if x and y replace a and b, re-ment which follows, but it is understood ia'. All values of T and i used below are spectively, in these equations. When xthat all values of T and i are increments understood to be changes in T and i due and y are then replaced by their value indue to the application of ia'* to the application of ia'- terms of Xd" and xq' from equation 53, andFrom equations 50-52 and 68-70, the WVithout a neutral conductor or ground Tba differentiated, an expression for eba,

following equations are obtained return path, io =0 and T0o =0. There- the circuit breaker recovery voltage willxyc22 = -2x fore, ia = ia and la = Taa As phases b and be obtained.ha(x+ ~ cos 20) - i~y sin 20= 2'I'~ c are connected, e,=eb; and with resist-

-iay sin 20-i/3(x-y cos 20) =0 ance neglected, Tbb= Tc From equation eba= [-3k,x/(xD+Vx\/dXl X5 with i replaced by 'I, 'I'=0. Simulta- L T O (

Simultaneous solution of the preceding .'tion solution of equations 50 and 51 for n = co

equations gives ita in terms of i0; ig is not i with E'=O and "a='a, gives L Z(2n-1)Zni sin (2n-1)0IreqUired. la ihT n a=T,gvsn= 1

2ioxo(x-y-cos 20) ta-t =ia'a(Y os0)77)(iax2-y 2) (71) (X2-Y2) +6ka sin oEnZn cos 2n0 (82)n=i

ia' =i,,+io gra = -ia (x2-y2) (78)

2ioxo(x-y cos 20+io(X2-y2) (x -y cos 20) where

(x2-y2) With T' = T0-o,= = \aa/2=a/2. (V/Xd \VXO W)From equation 72, io in terms of ia' is Then X/Xd+V\/Xq

io= a '(XY-y2) (73) -- ia (x2-y2) As in the reference paper, the recovery[2xo(x-y cos 20)+x2-y2] 'ba ='Ta -'I'V =3'3]a (2) (79) voltage at current zero depends upon Oo,

From equations 70, 52, and 73 2 (x-y cos 20) 0 and xq"/Xd". Equation 82 should checkequation B-59 of the reference paper.6

Ta=3"'o= -3ioxo Case 1 However, as there are only two series inThe value of i0' to be substituted in'

-ia'3Xo(X2y2) (7 equation 79 is the same as that given in equation 82 and four in equation B-59 in22t~~(4) .addition to five other terms, the agree-[2xo(x-y cos 20)+x2-y2] equation 67; and if phase a clears at addt toeiv the terms t aee-' , ~~~~~mentbetween the two has not been es-

When ia' in equation 74 is replaced by its current zero, cos 0=0 and 0= ir/2 or tablished at the present writing. Whenvalue from equation 67 37r/2. If equation 67 is substituted for the development of equation 82 is com-

ia in equation 79, the differentiation per- pared with that of equation B-59, the ad-3k,Exo(x-y) cos 0 formed, and 0 replaced by 37r/2 vantages of a,j3,0 components and also of

[2xo(x-y cos 20) +x2_y2] (7 (3' (x -y) /3\ xq" the harmonic series of Appendix I areeba=PIba krE =1 apparent.

The assumption is made6 that phase a / + \ Xclears at current zero. From equations (80)48 and 67, this requires that cos 0=0; The equation for this case and that given Appendix I. Specialtherefore, 0= 7r/2 or 37r/2. After equa- in reference 6 agree. Trgn Serietion 75 is differentiated, and then 0 re- Trigonometric Seriesplaced by 37r/2, ea is given by the follow- Case 2ing equation at the instant phase a clears The value of ia' to be substituted in The following trigonometric functions will

equation 79 is the negative of i.given by be expanded in harmonic series.3xok E(x-y) qaon lt engtv agVn )ea =Pa2x(x+y +=2 _ equation 48 with appropriate decrement 12xo(xJy/ +X2 _y2 factors, and If replaced by unity. fi(0) (a-bcos20) (83)

WArhen x andy are replaced by their values [k,(x-y) cos 0-k0cos 0o+k0X b sin20((84)from equation 53, the magnitude of ea, the cos (20-00)] f2(O) =(a-b cos 20)()recovery voltage iS ta' =- (X2-y2) bo2

krEa(3xoxq") (81)_______(85Xd(X,#+2Xo) ~~(76) (8) f()(a-b cos 20) (5

~~x0 +2xo ~~Equation 81 checks equation B-4 of ref- sin 0

Equation 76 checks equation C-i5 of erence 6, when the same notation is used. f4(0) = (abco -0 (86)reference 6. If 0= ir/2 insteadof 3wr/2,the XVhen equation 81 is substituted in equa- (- o 8sign of ea inequation 76 would be nega- tion 79, with the last term in the numer- a (0) = abcos 20)8tive. ator expanded, "tba becomes 5 (- o 8

1951, VOLUME 70 Duesterhoeft, Schulz, Clarke-Instantcaneous Currents 1253

where a and b are scalars, and b <a. before the radical satisfies the condition The series for equation 86 can be obtainedConsider the following expression, ex- that the series in equation 93 will approach by multiplying both sides of equation 96 by

panded by long division, unity as b approaches zero, which means sin 0 and simplifying the resultant equation.that Z must be zero when b is zero. Thus,

1/(1-Z)=1++Z±2± . zn~+.._ sin09 1

Xa-Li Z m a-/a2-b2 a-b cos 20 V/a2-b22Lim Z =Lin.--=1+LZ b-.o b-o.0 b _o

n=1 Isa2-(a2-b2) b sin 0+2 Zn cos 2n0 sin 0

Let ==Lim -=Lim -=Ob->2o(+i2 2) b-o 2a_ n=

Z-Z/4, =Z&si=Z(cos 4±j sin 4+)r b-ob(a+Va-b) bL.02 n~=Z=-10=ZEi1'=Z(cos O+j sin o) If both numerator and denominator of 1co

+b and -b added to the numerator, and -Va2_b21

c

+a and -a added to the denominator, the L n

1-Z(cos c+j sin --=1 +±(/ ) equation for Z becomesco

n~~~~~~~~~~~~~~~~~~~~~~i=1aZ-l(cosn+ b sin(2n+1)0 ZZn sin(2n-1) 1

=1+ Zn= (on + b ni

jsin no) (a+b)-2V/a2-b2+(a-b) 1

Rationalizatioin of the left-hand side (a+b)-(a-b) /a2 -b2and equating real and imaginary terms, Va+b -V/a -b Frgives si//a + b+ /a-b (94) sin E-Z sin 0+ Z ZnX

Z_ sin v =EZ' sin no (88) The following equations involving Z willco1 -2Z cos 4 I+Z2 be found useful

Z4co 2Z b 2aZ sin (2n+1)0- Z sin(2n-1)01-ZCosZ osn4, (89); 1+Z =- ; n= 2

1-2Z COS O+Z2 i± 1z ono (9 +Z2 a b

2(b-aZ) 1+Z2 aZ 1If unity is subtracted from each side of 1_D22 ba- sin0-Zsin0+equation 89 b ' ~2b-aZ-\a2b

Z 1LZ Cos 4_Z2 b-aZ - _2b2(

1-2Z cos o+z2 2...JZn cos n (9) If 4, in equation 93 is replaced by 20, and both - Si (2n-1)0-n=l sides of the equation divided by a, the re- n=2

sultant equations when simplified by sub- c

From the addition of equations 89 and 90, stitutions from equation 95 becomesthere results

1 1Zn sin (2n-1) |00 ] 1~~~__ n=2

_____1-Z2 =1+2vZn a-b cos 20 a2_b2X1-2Z cos ,+Z2 / i _ X _ 1-Zo n12ZCO oZ2

n =1co i-

zn- sin (2n-1)0(91) L1+2 Zn cos 2nO (96) - a2-b2

Equations 88-91 are given by Bromwich;'= 2the work which follows is an extension. Equation 96 expresses equation 83 as a = n-2\ n-i xIf equation 89 is multiplied by Z2 and added series, where Z is given by equation 94. a+b+Jr a2- b2to equation 90, and both sides of the result- If 4 in equation 88 is replaced by 20, n=1ant equation divided by (1-Z2), there (1 +Z2) in the denominator of the fraction re- sin (2n - 1)0 (99)results placed by its value from equation 95, nu-

merator and denominator then divided by Z, The series for equation 87 can be obtainedZ Cos 4, Z2 1+Z2x and both sides of the resultant equation by multiplying both sides of equation 96 by

1-2Z cos o,Z2 1-Z2 1JZ2 multiplied by 2, there results cos 0.00

co The development is analogous to that

Zcos n4 (92) b sin 20 Zngiven in equation 99.2 zncOsnf (92) b sln 28=2 ) nsin 2n0 (97)A_ a-b cos 20 cos 0 2

a-b cos 20 a-b+V\a2-b2If each side of equation 91 is multiplied Equation 97 expresses equation 84 as a 0by (1+Z2)/(1 -Z2), the resultant equation series, where Z is given by equation 94.is When 4 in equation 92 is replaced by 20, )Zn cos (2n-1)0 (0l0)

and substitutions are made from equations21 1+Z2 94 and 95, the equation becomesn=

= X

1- co 1,bcs2 - Va/ b2 2a Equations 99 and 100 express equations 861-1+z2coszib~~ ~ ~~~~____2av- + 2a X_ and 87, respectively, as series where Z isa-b cos 20 Va2_b2 V>ai_bi given by equation 94.

[1+22 Zn cos n (93) Zzn cos 2n0 (98) References

Let 2Z/(1+Z2) =b/a. Then, Z= Equation 98 expresses equation 85 as a COMSPONENTS, Edith Clarke. GeneraSIEleTRICA(a-Val -b2)/b, where the negative sign series where Z is given by equation 94. Review (Schenectady, N. Y.), November and

1254 Duesterheeft, Schulz, Clarke-Instantaneous Currents ATEE TRANSACTIONS

December, 1938, volume 41, numbers 11 and 12, LINKAGES, R. H. Park. General Electric Review Edith Clarke. John Wiley and Sons, New York,pages 488-94 and 545-49. (Schenectady, N. Y.), volume 31, June 1928, pages N. Y., volume II, 1950.2. TWO-PHASE CO-ORDINATES OF A THREE-PHASE 332-34.CIRCUIT, Edward W. Kimbark. AIEE Transac- 5. Two REACTION THEORY OF SYNCHRONOUS MA- 8. AmERICAN STANDARD DEFINITIONS OF ELEC-tions, volume 58, 1939, pages 894-910. CHINES-GENERALIZED METHOD OF ANALYSIS- TRICAL TERMS, AIEE, 1942.

PART I, R. H. Park. AIEE Transactions, volume 9. SYNCHRONOUS MACHINES IV-SINGLE PHASE3. CIRCUIT ANALYSIS OF A-C POWER SYSTEMS, 48, July 1929, pages 716-30. CIRCUITS, R. E. Doherty, C. A. Nickle. AIEEEdith Clarke. Jobn Wiley and Sons, New York, 6. CIRCUIT BREAKER RECOVERY VOLTAGES, Transactions, volume 47, April 1928, pages 457-92.N. Y., volume 1, chapter X, 1943.R. H. Park, W. F. Skeats. AIEE Transactions, 10. THEORY OF INFINITE SERIES, T. J. I'a. Brom-

4. DEFINITION OF AN IDEAL SYNCHRONOUS MA- volume 50, Marcb 1931, pages 20439. wich. Macmillan and Company, Limited, London,CHINE AND FORMULA FOR THE ARMATURE FLUX 7. CIRCUIT ANALYSIS OF A-C POWER SYSTEMS, England, 1942.

Discussion which are "single axis" unbalances, they systems2 by means of the transient an-may be studied by replacing the 3-phase alyzer when single-phase circuits are usedmachine by a 2-phase machine. Dreyfus to reproduce 3-phase phenomena.'

Eric T. B. Gross (Illinois Institute of Tech- (1911, 1912, 1916), Biermanns (1915),nology, Chicago, Ill.): It has not been Rudenberg (1925) made valuable investi- REFERENCESrecognized until a few years ago that Clarke gations along these lines in Europe some 1 DAMPING AND RESONANCE IN POLYPHASBcomponents are very useful in many unbal- time ago and a close relation to the treat- GENERATORS, R. Willheim. Archiv fuer Elektro-anced 3-phase problems, and this paper ment with Clarke components should not technik, (Berlin, Germany), 1929, pages 593-611.indicates clearly some of the distinct ad- be surprising. See also reference 1. 2. TRANSIENT ANALYSIS OF THREE-PHASE POWERvantages of Clarke components. The com- In some cases, especially in connection SYSTEMS, PART I, Eric T. B. Gross, Leonard Rabins.plex operator a, so significant in symmetri- with the application of the network an- Journal, Franklin Institute (Philadelphia, Pa.),cal components sometimes introduces com- alyzer, symmetrical components can not be volume 251, 1951, pages 33341.plications. Since the unbalances of great used whereas Clarke components provide a 3. TRANSIENT ANALYSIS OF THREE-PHASE POWBRpractical importance are the short circuit of basis for the solution. One such example SYSrEMS, PARl IFa kleinar Rstitute (Philadelphia,"one phase" alone or "between two phases," concerns the study of transients in power Pa.), voltume 251, 1951, pages 521-37.

1951, VOLUME 70 Duesterhoeft, Schulz, Glarke-Instantaneous Currents 1255