S 1 e . 1 . fThe systematic treatment of the equiv-teac y-; tate Lquivai ent Circuits or alent circuits of commutator machineswill be based upon the present study and

ynchronous an nduction achines will form the subject matter of a futureSynchronous Induction ivua~~~~~.umuuu~~ publication.GABRIEL KRON The Primitive Machine at Rest

MEMBER AIEEIt will be assumed that the equations

Synopsis: Equivalent circuits are developed When rotating machines are inter- and equivalent circuits of all machinesfor the various types of salient-pole syn- connected with each other or with trans- may be derived as special cases of thosechronous and asymmetrical induction ma- mission lines and other stationary struc- of a "primitive" machine. In thischines, expressed along the physical direct . machine:and quadrature axes (cross-field theory). tures, additionalphysicalreferenceframesThe torques again are found by a direct must be resorted to. Such additional 1. One member has a salient pole andmeasurement of power. The networks of frames are for instance asymmetrical windings, the other memberall types of machines are found as special is smooth and has symmetrical windings.cases of that of the "primitive" machine. 3. The axes rotate with a speed different 2. Depending whether the salient pole isAll equivalent circuits along the physical from that of the rotor. moving or stationary, two such "primitive"axes are derivable from those along the 4. The axes execute small oscillations machines will be distinguished (Figure 1),hypothetical symmetrical component axes superimposed upon a steady rotation. namely, the "primitive synchronous" ma-(revolving-field theory) by a real transfor-. '*-

eation. Although the latter types of net- 5. There are angular displacements (con- chine and the "primitive induction" ma-works have already been published pre- stant or variable) between the various axes chine. The equations and equivalent cir-viously, they are reproduced here for com- of the same frame. cuits of both types of primitive machinesviously, they are reproduced here for com- are identical, differing only by the sign ofparison with the new networks and for a With each of these physical axes hypo- the speed term v and of the torque T.unified tensorial treatment of rotating ma-chines and their eqiiivalent circuits. thetical symmetrical component axes 3. There exist two independent sets of

may be associated. axes at right angles in space, one along theThis paper restricts itself to the sys- salient pole (direct axis d) and one at right

The Two Basic Reference Frames tematic development of the steady-state angles to it (quadrature axis q).equivalent circuits of single machines 4. Along each axis four windings are as-

N the theory of single, isolated rotating when their equations are expressed along twsr on the rotor.machines possessing no commutators the first two types of axes. While the

two basic reference frames usually are existence of equivalent circuits along the When the machine is at rest, each axisassutmed. hypothetical symmetrical component axes may be represented by an arrangement of1. The axes are rigidly connected to the are widely recognized, it is not so well inductive coils representing the magneticsalient-pole structure. appreciated that equivalent circuits of structure (Figure 2e). See also reference2. Hypothetical "syrmimetrical component" equal simplicity may be constructed for 4 for greater detail. The electrical struc-axes are associated with the foregoing any machine along the more physical ture is represented by the resistances andphysical axes. direct and quadrature axes. impressed voltages in series.

d d Pe Equationsof the Primitive Machine

PO CIA2d [2dit :. The transient equations of the primi-;?00, i i,d D Sid L tive induction machine are (reference 1,

Os A > r uqpage 129)e-I= Z where Z is given int7 , s a r ' ~~~~~~~~~~Fora synchronous machine p0 has

2 2 l2dX 1 1t 1 l ~~~~~~~~~~~oppositesigns, the subscript s becomes f_ fXy { s t/ - -Areq- ~~~~~~~~~~~Paper 48-27, recommended by the ATEE rotating

02q0uq-xq2iq / q 2machinery committee and approved by the AIEE//_ _Z////" % 2 ~~~~~~~~~~~~~~technicalprogram committee for presentation at/ / /t_ t ~~~~~~~~~~~~~~~~~~~theAIEE winter general meeting, Pittsburgh, Pa.,

January 26-30, 1948. Manuscript submittedSeptember 5, i947; made available for prinltingDecember 24, i947.

a) THE PRIMWITIVE b) THE PRIMITIVE Figure 1. The primi- GA.BRIEL KRON is consulting engineer, GeneralSYNCHRONOUS MJACHINE INDUCTION MACHINE tive rotating machine ..Electric Company, Schenectady, N. Y.

1948, VOLUME 67 Kron- Synchronous and Indluction Machines 175

2I .2334.4X 2 3 4 |- ~~~~~~~~~~~~~~~~~~~~~~~p/rlp/r,| Xg p/r3 |p/r4 |a) TWO STATOR AND TWO ROTOR WINDINGS 'l 2S3 (4

jjf~~~~~~~~~i~~~~~2L~ ~~~~~~~ 9'Is~~~~~~XI4-- 2s X3sX4'P Is

/ -S

02 t f5 Figure 5. The poly- ZS, X, and X, matrices will be specialAIR 1- -1 phase primitive ma- cases of Zr and will differ from it as just

~ARttAMATURE + GAP| FIELD --- chine indicated.jxot j fg jXft (d). Polyphase syn- During steady state, when the fre-

chronous machine quency of currents in all axes is not funda-fT t I,t t if tt (b). Polyphase in- mental frequency (unity) but some other

ec2 z tB JxJ rf I t t duction machine frequencyf, then in the transient Zf+Vu Wt it l 1 fy NJ F Torque= real of iB* 1. p becomesjfw

(d) 2. pG becomes vw3. coL and wM become X

rt r so Accordingly Zr of equation 2 becomesROTOR AIR STATOR dr qr

jXrt JXfrl iXSI )ss Zr=dr rr+jfXdr VXtlrqr -VXdr rr+jfXqr

e#. _____f__ _ __ _ _ __ _ _ Equivalent Circuits AlongT__V f, tY t Symmetrical Components(b)

To bring the matrix Z of Figure 3 toARMATURE AIR-GAP FIELD symmetrical form two transformations.

are introduced in succession.jX jX jX jxf rf 1. Symmetrical components are introduced'IIV 2 by

ef ~~i(xoq-X d jXod j(xoq-Xad ixad S r1 2

ttt1 ]0 LJ S2 CsTI1 (7r0v XtjxU, _ jxl2 jxI jXe rf where

o) REACTION MACH1INE b) SLIP COUPLING f b

roa jxc jxf rt sr v rf . CS= Cs-1 (8>

*s- As a result each Z, X, and e is transformed:

eb B j< Bb C, Cs(stv) .rThe*Zrcys

ie b ad______

ib xob I ad-- 2lC

aV jXJ2 2. The jXfre c is r e to uny b

ixwhesre~

C) SYNCHRONOUS MACHINE d) SYNCHRONOUS MACHINE

backward axes) d frl brToqerd of Bf*+ibBb* er=Lj.,..e.J (5) nrb f+v (11)s

From now on whenever new referenceSimilarly the impressed voltage vector frames will be introduced only this Zr As a result each Z, X, and e is multiplied a

is ~~~~~~~~~~(ander) matrices will be calculated. The Zrt = nrlZr t ~~~(12)ds2 q2 d3 q1 d q dr2 2rr e"Z n=Cst (13)

e=| edS2 es2 | edSl eq7sI | edrl e | edT2 eDrr | (3) Rememberingthat ncontainsnov,thefinal matrices are

Represented as a compound matrix, it fr brbecomes rf +( rkd (XdrXqr"4

r. ixS-v r,-v r22

S2 Si r1 r2 b, j(XdrXar) Yr( nXdjXar)e=| 3si |eSI|ersI|er2I| (4) _____2___)___|___f+v__J__S|

1948, VOLUME 67 Kron-Synchronous and Induction Machines 177

ARMATURE AIR-GAP pacitor motor differs from the primitiVer02 machine in two respects:

, ,2 'j

.1. Its direction of rotation is from q to d,jx2d j j hencethefaxisrotatesalso fromqtod.a~od ad

.7 kd rf ral I ( X J 2. The ratio of turns on the q, axis is a.._~~~~~~~ ~ ~

~~~~~~ ~~~~~tSe.l f X.A

rac r f < Equivalent Circuits Along*b2 Bbjx'akq_ b jXagq Physical Axes

-r0N Once the equations of the primitive~v Jx;2 JXel machine have been brought to a sym-

metrical form along some particular axes,a) EQUIVALENT fIELD IMPEDANCES b) SLIP COUPLING it is comparatively easy to keep the sym-

metrical form along other arbitrary axes.r0 jx r xe A return to the original d and q axes may

jI sbe accomplished in two successive steps

jet i~~~~~~~~ Bfres , . JXaq ed + Jeq -,1 1. The inverse tratisformation C8-1 re-4J(X0djXaq) * d(s.+ v) q turns to the original physical axes

jS r;-.ef V_(ef JXod d' q'b ed-ljeq b S.

V2 Daq \/r l(s-v) J \/2b 1C j (16)_ ra sv X.,

c) SYNCHRONOUS MACNINE d) SYNCHRONOUS MACHINE Figure 8 (below). Induction machines (revolv-AT SYNCHRONOUS SPEED BELOW SYNCHRONOUS SPEED ing field theory)

X Torque = redl o~~~~~~~~fjrBJ*+ i,rB,*Figure 7. Synchronous machines (forward and Torque= real of fbackward axes). Field circuits are eliminated ROTOR STATOR ROTOR STATOR

Torque=reda of ifBf*+ ibBb* rr rr jxI-V jXr JXsd rSd I JXr JXsd rsd

edr +jeqr edr-jeur jifX- ferIt=| fr- ef reJS|(15) I < r Jdf A)ed -Jeq/aThe corresponding equivalent circuits

-j _B 2on the two primitive machines are shown efr br Xd sq2sd ebs brBmb aZsq/2a edJeq/on Figure 4. The torques are found by eb2is. -Zthe product of the resultant rotor (or rr rrarmature) currents iand fluxesE. When I+V Jr JXsd rsd [+v JXr jxd rsdthe machine structure is smooth, the a) INSTRUMENT SELSYN b) GAPAGITOR MOTORwindings are symmetrical and only poly- .rr. jX, jXs r JXr jx5 rsphase currents and voltages exist, then tfrBf fs_I_dsij=-.jid and only the forward meshes eds+jeqs rf 1dsfexist. The corresponding polyphase (\f. u) r m V f |Xmprimitive equivalent circuits are shown ' 1,edsin Figure 5. edr*Bb sr B jxActual Machines 1'f UV) i1br JXm fV2s f

r X rs +tThe equivalent circuits of the various r fxrixs jxr jxs rstypes of synchronous and induction c),UNBALANCED IND. MOTOR d)SINGLE -PHASE IND. MOTORmachines shown on Figures 6-8 are rr2Xrt jx 5 r5special cases of the primitive machine andfi6are found from the latter by simply open- Itfr jxke5s fing certain meshes. These networks al- r B Jm eready havre been published in references I Fi7u * S I14 e) DOUBLE SQUIRREL-CAGE IND. MOTOROn synchronous machines it iscustom- rrx xsr- Jr j5r

ary to eliminate all field axes and con- k*tJ7JX ,5~j~J5rsider only the armature axes. The cor- S : jst| r| . iresponding simplified circuits are shown er 4@ Bf jxrm}| Xe5 IfiBfrBgJXm in Figure 7. Note that the phase dis- j| ifrj [1J||placement between the excitation and theinfinite bus is still the angle 6. The ca- I') DOUBLY - fED INDQ MOTOR g) POLYPHASE IND.MOTOR

178 Kron-Synchronous and Induction Machines AWEE TRANSACTIONS

bat the resultant Z is not quite sym- * *metrical. TORQUE *REAL OF dBq +

d2. The latter may be brought to a sym- a f fmetrical form by allowing jij to flow in the 8 0 f fnetwork in place of 0. That is, the addi- AIRtional transformation (changing the time -ARMATURE GAP FIELDscale in the network along the q axis) is JKd JXdtZ

17i7 ~~~jxoto .dai jd xfC 2 = ~ ~~~~~~~~~~~~~~~~ G d a a -* |te O-i ga p. a t j B g Xdf fO dfz-j (17) 1dat~~~~~~..~~~*goa1V2 if idft ;If,DRSC

The two transformations may be com- ..Z fef tqfebined into one as_ed Oq 2 Bqi IfL f CUADRI

d q r a2 j% Xiqf.edq~~~~~~~~~~Cn=CA-IC2=.\>, l (|| 18) jxf 1 jXai jXqfI jxqftCn

~ a/b32 -1a) SYNCHRONOUS MACHINE (DIRECT AND QUADRATURE AXES)

Transforming Z,/' and e," of equationst s Sp d14 and 15 asROTOR AIR STATOR qGAP SAO

e"'f = cnt*eff (20) rrp 1e"'=C51 *e~~~~~~~~~~~~~~~~~~ dA d~ r Adst~ jxjdIgthe resultant matrices are f-u pdrm d ' MAIN

fOdr2+JbOqrt -v r, j Bd jxad f AXISq .. __ # s \ i##~f- 'VP ............................... f dst

d qf

d fr d r (1r ivrdrt+jferk(22)d +Jdr Cql ROSSf2-v2 f2_v2 9 qhrii B

fen+jvettingupthe yesultant j"' (22)o- Figure90.Tyncheprimi machines(directalong physical axes)In setting up the resultant Z"' analo- Figure 10. Synchronous machines (direct and quadrature axes)

gously to Figure 3 it should be remem- Torque= redl of i'B *+(jj7>Bd)'bered thatr Jx rat j j xfr1. The X"' matrices contain no r terms. aJ r

2. The matrices with a subscript s con- Isq t jdi tid2 iftain no v terms. V ) ad vrI2 r.1 vrrT | XadThe corresponding equivalent circuits e Wu

of the two types of primitive machines are tq qq8.q JXoq IP-v ..q jxaqshown in Figure 9. For a polyphase ji 1machine (where i =-jjd hence jiq=id)the equivalent circuit reduces to the same - - jVSt jxj2 jX11form as that given for symmetrical com-ponents, Figure 5, since if= N/2 id. Hence a) REACTION MACHINE b) SLIP COUPLINGon Figure 5 each i may be considered asrepresenting either if or id. The same rf f Stu JJxstatement applies to e. 7 i7 1. d ds?For the various types of synchronous jOq {t jBd JXod t i' sed-puegk vr OAtcircuits are givren in Figures 10 through W 1 512. It should be noted that *d (t B -ved+Jseq xi B ijXag -I1. In the symmetrical component circuits

r0 jx2tween the f and b axes for the asymmetrical vmember only.2. In the physical circuits (Figures 9-12) c) SYNCHRONOUS MACHIlNE d) SYNCHIRONOUS MACHINSEmutual couplings exist between the d and qaxes for the symmetrical member only. AT SYNCHRONOUS SPEEO BELOW SYNCHRONOVS. SPEEO

1948, VOLUME 67 Kron-Synchronous and Induction Machines 179

ra2 equivalent circuit to that of any other bytv JXj2 J means of a C.

jx

,

XjXk _x djXaI I joad ad rkdxf fjd2 rjI r.

cdi s l7R I iXod rXi d Transformations Between Machines-A_ v. \and Their Equivalent Circuits

Xoq - hrq J aq Having shown that all rotating ma-s

r. X_,2 X,,, chines form a "group" and that all theirra2t iX12 XjI equivalent circuits form another group,the question arises in what relations are

O) EQUIVALENT FIELD IMPEDANCES b) SLIP COUPLING the two groups to each other.It may be said in general that while the

r,a px ra jx same currents flow in a machine and in itsS+_V .1 equivalent circuit, nevertheless differentjXod sed-jveq XjBd Ixtd voltages are impressed on them. That is,jeq -r B6d aixod se-Uj d ef jxodrj in goingfrom the equations of a machine

a d s rf to its equivalent circuit the voltages are8 transformed by n1 = n"' but the currents

ed q q -ved Iseq , q are not. If the subscript m represents aXaq S2-u2 jxaqI machineandnanetworkthen

ra jX-t r0 jx en=n'lem, (25)S tv ea'= n, e,'

C) SYNCHRONOUS MACHINE d) SYNCHRONOUS MACHINE Z=fn-l'Zm (26)Za'nOf'Zax:AT SYNCHIRONOUS SPEEt) BELOW SYNCHRONOUS1 -t*

and the equations of a machine and itsequivalent circuit are, respectively,

Summary of Transformations Figure 11. Synchronous machines (direct and em =Zm2quadrature axes). Field circuits are eliminated eZa,}(21. In considering the equations of Torque = real of idBq* +(jij1jd)* * _ IZ

actual rotating machines it is possible to ec = zf= Zmo (28)pass from the physical d and q axes to the ea =Za'2 )hypothetical f and b axes and back again Figure 12 (below). Induction machines Hence the tensors of the group of equiv-bv the transformation C, and its inverse (cross-field theory) alent circuits are "intermediary tensors"

f b Torque = real of jd,B,*+(jjr%Bd)* having their indices in two different refer-d

C ROTOR STATOR ROTOR- STATORd q r2 r1

r

jXr jXsd rsd r JXr JXsd rsd-1 a- (23) idrN'2 b 1 -j (dr+Jueqr ( d jdl tidr did ed

2. In considering the equations not of 1-u T 21a IX-V2 | t Iactual machines but of their equivalent i r Iqscircuits, it is possible to pass from the vedrt1 eqr IqT jX q eqs t qr ;eiiphysical d and q axes to the hypothetical 1-u2 8 Bq Jxmq teqs I Bq Xm t jeq/f and b axes and back again by the trans-formation C,, (Equation 18) and its in- rr jxr jxsq rsq .IL jxr jxsq rsq -JXcverse a a a

f b a) INSTRUMENT SELSYN b) CAPACITOR MOTOR1 d| 1 1

a/2ql 1 1 -1t f-~~~~rrjx -x ILr jx, jx5 r,\/2ql Jxr() Jxrfed+eqr B tid dd

Hence, just as all actual rotating ma- uedr+|-jfeqr zLjqr 11qsi i jq eqs Jf ,|fchines form a "group" (allowing to pass 2 z t q EJXm g 'ti Bq mfrom the equations of any one machine tothat of any other machine by means of a rr jxr iX r rr jxrnonsingular C) similarly their equivalent f-u f I-i)circuits also form a group and it is pos-sible to passfromthe equation of any one c) UNBALANCED IND. MOTOR d) SINGLE-PHASE IND. MOTOR

180 Kron-Synchronlous and Induction Machines AJEE TRANSACTIONS

ence frames and the two groups are simply 2. When expressed along the physical di- Referencessubgroups of a more extended group. rect and quadrature axesThe transformation matrices n = nc np = CsnsCs 1 (30) 1. A SHORT COURSE IN TENSOR ANALYSIS FOR

changing from the network to its machine ELECTRICAL ENGINEERS, Gabriel Kron. Johnchanging fromtha e networktowts: machindm qm Wiley and Sons, Inc., New York, N. Y., 1942.and back again are as follows:

dn fl -jV 2. EQUIVALENT CIRCUIT OF THE CAPACITOR1. When expressed along symmetrical ri = MOTOR, Gabriel Kron. General Electric Reviewcomponents qn jv f (Schenectady, N. Y.), September 1941, page 51 1.

dn qn 3. EQUIVALENT CIRCUIT OF THE SALIENT-POLEb. LSYNCHRONOUS MACHENE, Gabriel Kron. Generalf, bm -1 1 dm f____________Iv31 Electric Review (Schenectady, N. Y.), December

_n|_2_-2 -j 1941, page 679.

bn f+v | 4. EQUIVALENT CIRCUIT OF THE PRIMITIVE Ro-It has to be remembered, though, that TATING MACHINE, Gabriel Kron. General Electricfn bn (29) at first the time axis of i1in the network Review (Schenectady, N. Y.), March 1946, page 43.

I/_m|l/(f-v) I equations has to be changed byj as shown 5. THE EQUIVALENT CIRCUIT OF TEES CAPACITOR

ns b 11(f+V) in equation 17 before n-may be used. MOTOR, Sheldon S. L. Chang. AIEE TRANSAC-TIONS, volume 66, 1947, pages 631-40.

No Discussion

1948, VOLUME 67 Kron-Synchronous andl Induction Machines 181