ReFerences 4. VOLTAGE REGULATOR, E. M. Sorensen. Inc., New York, N. Y., section on THYRATRONUnited States Patent 2,455,143. CIRCUITS, W. R. Kind, volume 5, page 16-03.

1. AN ELBCTRO MEBCHANICALLY STABILIZED 5. ELECTRONC INSTRUMENTS RADIATION SERIES, 8. PRINCIPLES OF SERVOMECHANISMS, Brown,MAINS SUPPLY UNIT, A. E. Maine. Electronic Greenwood, Holdam, MacRae. McGraw-Hill Campbell. John Wiley and Sons Inc., New York,Engineerixg (London, England), September 1949 Publishing Co., New York, N. Y., 1948. Page N. Y., 1948.pages 319-21. 469.

2. VOLTAGE REGULATOR SYSTEM, F. L. Moseley. 6. CATHODE FOLLOWER CIRCUITS, Walter Richter.United States Patent 2,453,451. Electronics (New York, N. Y.), May 1943, page 112.

3. VOLTAGE STABILIZERS, F. A. Benson. Elec- 7. ELECTRICAL ENGINEERS HANDBOOK, Harold No Dicsotronic Engineering (London, England), May 1949. Pender, Knox McIlwain. John Wiley and Sons No 11SCUSSlOf

A | * f | n * Tl ~~~~~~~~~~BasicEquationsApplication or tne 2-Reaction TheoryFrom the analysis of 2-phase salient

to Electric Motors pole alternators with one rotor circuit,the following expressions are obtained:

{Id=Lddid +Ldff (1)E. M. SAEBACGH -qLqqiq (2)

MEMBER AIEE4/f= Lffif+Ldfid (3)

Synopsis: The 2-reaction theory, intro- tained are similar to those given by the whereduced first by Blondel and later expanded cross-field theory. Pd=fiUX linkages of the direct axis windingby others, is generally applied to the analysis By the 2-reaction theory it is easy to pq=fiux linkages of the quadrature axisof synchronous machines. This theory,however, can be used to analyze and calcu- derive the basic differeintial equations windinglate the performance of all types of electric from the circuits of the machine con- A,f=-flux linkages of the field windingmachinery. In the following the 2-reaction sidered. From these differential equa- Ldd, Lqq and Lff=self inductances of thetheory is used: to derive the differential tion direct axis, the quadrature axis andequations of a-c motors; to develop the s the steady-state equations may be the field windingssteady state equations of a-c motors; to obtained. From the steady-state equa- Ldf=mutual inductance between the directdraw vector diagrams for a-c motors; to tions vector diagrams can be drawn al- axis and fieldsolve the steady state equations and find the though these are not necessary for the id, eqand if=currents in the direct, quadra-currents and torques. The advantages of solution of the problem. From the basic ture, and field circuitsthis method are: its simplicity; the ex-clusi mehof vageness d ambigity; aou equations the torque of the motor can The equations of the terminal voltagesclusion of vagueness and ambiguity about Alobotandargiebycertain signs; and it presents a unified alsobe obtained, are given by:method for the mathematical analysis of all In their simplest forms the differential delectric machines. equations are usually expressed in terms vd = -idRd --4d - 'lq (4)

of the self and mutual inductances of the dt dtcircuits of the machine. These induct- d da (5)N teaching courses on a-c machinery, ances can in turn be expressed in terms of V = - R- q± {di

* the writer found that the two methods reactances and the reactances in terms ofusually used for the calculation of the their components: the leakage and mag-performance of a-c machinery, namely the netizing reactances.method of symmetrical components1'2 In this paper the generalized 2-reactionand the cross field theory3'4 while simple theory will be applied to the following: Vand straight forward for certain types of Omachines, present difficulties when ap- Single-phase induction motor. fplied to other types.i While working Two-phase induction motor. 2with Parks equations5 he observed that if Two-phase induction motor usel as athe quadrature axis was assumed to lag capacitor motor.the direct axis, the equations thus ob- The capacitor motor.

The repulsion motor.

The series motor.Paper 51-319, recommended by the AIEE RotatingMachinery Committee and approved by the AIEE dTechnical Program Committee for presenttion at Assumptions dthe AIEE Fall General Meeting, Cleveland, Ohio,October 22-26, 1951. Manuscript submitted v

6pill,1951. md vial o pitn uut I h analysis th fet fhysteresis aa o o o Vdand eddy currents are neglected. The

E3. M. SABBAGH is a Professor of Electrical Engi- lose casd hseei_n dycrneering with Purdue University, Lafayette, Ind. lsecaedby hseel n dycr

Th witr isestoacnolegeth hlpgien rents can be taken into consideration by_adaned macrwlhieryt askoleg h hel e Th_urnsan r suehim by the graduate students in the School of Elec- the addition of a resistive component tootrical Engineering at Purdue University. The**-probiems contained herein were given to his class in the equivalent magnetizing reactance.) _advncemchieryasexercises fo)r the application Th urnsadvoltages aesumd Vq D

of the 2-reaction theory to asynchronous machines.R. S. Carlson and L. W. Anania gave the sollitions sinusoidal with time, and the magneto^for the 2-phase motor as acapacitor motor, and the motive forces are assumed to be sinu- Figure 1. Diagrammatical representation ofcapacitor motor. Richard K. W. Cheng gave thesolutions of the repulsion and series motors. soidally distributed in space. single-phase induction motors

1748 Sabbagh-Application of the 2-Reaction Theory to Electric M7kotors A1EE TRANSACTIONS

(SXmIf)q (SXm(Id+If ))q

If\ (-JXmVf) d (ifR If +Id

i,f / rJl (fX, PX_f+I)tf (JXmld)f (SIdXsId)q

(-SIqX)d (JlfXf)f Id IdRdqt (-J dXd)dS\

IIdRd <(-JXmlq)qV Xqq

\\(SldXdd)q

Figure 2 (left). Vector diagram of single-phase induction motors inV (-SXqqIq)d/ \ (SIdXdd)q terms of self and mutual reactances

/ (-JIdXdd)d (-JIqXqq)q Figure 3 (above). Vector diagram of single-phase induction motors in(-JldXdd)d (-JlqX qq)q terms of leakage and magnetizing reactances

(JIfXff)fput 90 degrees behind the direct axis in quadrature with each other. The directthe direction of rotation. axis winding d is in space phase with the

vf=irRf+-,V (6) The mechanical power given by the stator. The quadrature axis windingdt expression q is in phase quadrature with the stator.

where dLnm As these.two windings are short circuitedpM=1 ini dEm (7) Vd-V= n

Vd, Vq and vf=the terminal voltages of the 2m= - t ntm dtdirect axis winding, the quadrature n m d dotaxis windings and the field winding. O-idRd-t{d- d (13)

Rd, Rq and R,= the resistances of the three yields the torque equation dt dtcircuits. dt d dao

da/dt=the angular velocity of the rotor. drt=p O=-iRvqq d (14)Pmy (8) dt dtdas

In the analysis giving these equations,the so-called armature reaction is helping 1 x x dLnm Whe nt ale o andsub given in=--/ / X dcm (9) equations l and 2 are substituted inthe field, namely the armature current is 2 daequations 13 and 14 we getassumed to lead the generated voltage. n m ,

Hence, the direct axis current is leading where 0= -idRd- (Ldirc+Lddid)-LqQi,d (15)the quadrature axis current by 90 de- r =instantaneous torque dt dtgrees. Pm = instantaneous power di da

Equations 4 and 5 are the equation of After substitution equation 9 becomes 0 = -iqRq-Ld+(Ldir+Lddid) (16)generator circuits while equation 5 has anapplied voltage. Nq Nd * dir didEquatiedvons4and5clearlyshowhw r=Nq'diq-Ndqid synchronous watts (10) v=ifRf±+Lyd +Ldrd (17)Equations 4 and 5 clearly show how thevoltages in the direct and quadrature

7.04 Nv Nd The steady-state equations now are ob-axis are produced. In each circuit there =sdI Nq qi t1

are an induced transformer voltage, and synchronous speed LNd Nq tamed from 15, 16, and 17, and area generated speed voltage. pound feet (11) 0= -Id(Rd+jXdd)-jXmIf-SXqIQ (18)The transformer voltage in the direct When the effective values of currents are 0 = -q(Rq +jXqg) +SXmIf+SXddId (19)axis circuit is given bv -d4P/dIt and that used in equation 1 1, 1, and 2 the averagein the quadrature axis by -di#/dt. torque T is V=If(Rf+jXff)+jXmId (20)The generated voltage, caused by rota- truTi

tion is given, in the direct axis circuit, T [Nq Nd 1 whereby LNd Nq Xdd =jLdd

Ndq,& da synchronous watts (12) Xm =6wLfdNQ dt The applications of these equations to XQQ= Lqq

and is 180 degrees out of phase with a-c motors now will be undertaken. daxthe quadrature axis flux linkages 1//q. dtThe generated voltage in the quadra- Single-Phase Induction Motors= -

ture axis, due to motion, is given by

Nql,t'd dax Under running condition, this induction X =the angular velocity of the applied volt-mtrcnbe represented by the diagram agNd~ ~ ~ ~ ~ ~~~~~oo can .w3 the angular velocity of the rotorNd dt~~~~~~~~~shown in Figure 1.

and is in phase with 'frd. This is always Voltage V is applied to the stator. The The solution of equations 18, 19, and 20true in motors if the quadrature axis is rotor is represented by two windings in yields

1951, VOLUME 70 Sabbagh-Applicattion of the 2-Reactionz Theory to Electric Motors 1749

V 2 ^ V2 ing, the quadrature axis winding and the[RdR2 XddXqq(l-S)+ l _ field winding, and X. is the magnetizing

j(RdXqq+RqXdd)] (21) reactance of the field. The quantityV 1 3gffl l If+Id is the magnetizing current. The

Id=-[XMXaV(l S2)-jXjnRj (22) O term -jXm(If+Id) is the transformerD n voltage induced by the field flux in theV VIQQQQSt5OOOOOQ ) ts direct axis winding. The quantity -Sh-

ID=-(SXmRd) (23) f2 Xmisthespeedvoltagegeneratedbythequadrature flux in the direct axis winding.

where The term -SI,qXq is the speed voltage

D=RdR&Rf-RdX<qXff - generated by the quadrature leakage fluxRq(XddXff-Xm2) + in the direct axis winding, and -jIdXd isRfXddXqq(l- S2) +j [XddRqRf- the transformer voltage induced in theXqq(XddXfy-Xm2)(1 -S2)+ d direct axis winding by the direct axis

RdRqXff+RdRfXqq I leakage flux. Equation 28 refers to the

The torque equation is given by direct axis winding, equation 29 to theVd quadrature axis, and equation 30 to the

T=w[(LdfIf+LddId) * Iq-LqqIq Id] (24) 00000 °°°l field axis.

For aaThe vector diagram corresponding to

then O equations 28, 29, and 30 is given in Figureothen 3. Subscripts d, q, and f have been addedT=XmIf (25) to denote the windings to which the volt-

V2 Vq-V age belongs.=F SX2d[d d2(11-S2) ] (26)|SDI2Xmd[ dd( Figure 4. Diagrammatical representation of a

2-phase induction motor Application to 2-Phase Motor:where

jID 2 =fRd2Rf-RdXddXff-Rd(XddXsff-In the 2-phase motor there are two

Xm2)+RfXdd2(l -S2)12+ generated and applied in the direct axis is windingsf1 andf2 on the stator as shown

[XddRdRf-Xdd(XddX1f- equal to the ohmic voltage drop. diagrammatically in Figure 4. Winding

Xm2)(1 - S2)+Rd2Xff+RdRfXdd 32 Equations 19 and 20 make similar state- fi contributes flux which links with the(27) ments for the quadrature and field cir- direct axis winding on the rotor. Wind-

cuits. ing f2 sets up flux which links with theFrom equations 18, 19, and 20 a vector If the self reactances in equations 18, quadrature axis on the rotor. The flux

diagram of the currents and voltages may 19, and 20 are replaced by their equivalent linkages of the q axis winding now becomebe drawn as shown in Figure 2. The value of leakage and magnetizing react-subscript put at the end of each bracket ances, the following equations are ob- (31)encircling the voltage quantity is added tained. The quadrature axis winding on the otherfor convenience only to show to what hand contributes flux linkages to thecircuit the voltage belongs. Thus the IdRd -jIdXd -jXm(If+Id) - SX2Iq- stator winding f2, so that the flux linkagesvoltage (-jIdXdd)d, for example, is in- SXmIq (28) of this winding areduced in the direct axis circuit d, while 4R2=S(Id+If)Xm +SIdXd-jXI -jIqXm(SIdXdd)q signifies that this speed voltage (29) 1P2=Lfif2+iqMqf2 (32)is generated in the quadrature axis q. IIfXf The flux linkages of thefi winding areThe vector diagrams in this paper are Vj(Id±If)Xm+IfRf+X (30)

not drawn to scale. Equation 18 states where Xd, Xq, and Xf are respectively the ^6=Lffifl±Ld.ridthat the sum of the voltages induced, leakage reactance of the direct axis wind- and those of the direct axis are given

(-SXqqIq)d by

(JXmLd)l Figure 5 (left). Vector diagram of a two-phase induction motor in(JI2X22)2 V2 terms of self and mutual reactances

(SXml)q (-JXmIl)d Figure 6 (below). Equivalent circuit diagram of a 2-phase inductionmotor

(-JXqqIq)q 1\WI2 (SIdXdd)q Rs JXs R XR(JXmlq)2 I\ (JXmIq)2

VI /Lq (-JXmI2)q

(-SXmI2)d X

(-JIdXdd)d __ _ _ _ __ _ _ _ !_ _ _ _ _ _ _ _ _ _ _

1750 Sabbagh-AppVliccation of the 2-Reaction Theory to Electric Motors ALEE TRANSACTIONS

Figure 7(A) (left). (-SXddIq) dDiagrammatical rep-resentation of a 2- (-Amll)dphase induction

UA Xm1)Imotor as a capacitor

V motor 12\ R1}

(JXuI2)2 11R7 (-JXcIOI

(-JXqq lq)q

Figure 7(B) (right). JqX(SddId)Vector diagram of a Iq (SXddld)q2-phase induction SXm 12)d

I0000 motor as a capacitorq motor in terms of (JXddId) d

d self and mutual re-actances (XIi)I

When converted to the steady-state substitute in equations 45 to 48 the valuescondition these equations become, with given below

{d =Lddid +Ldfnif the rotor short circuitedZdd =Zqq 'RR±+jXR+JXm (55)

The differential equations of the motor 0 = -IdZdd-jXmIi - SXqqIq-SXmI2 (45) Z,f=Rs±jXs+JXm (56)can now be written as 0= -IqZqq-JXMI2SXddld SXMI (46) =-s (57)

dOtdda Sd1-Vd=-idRd_ d_ (33) V=IlZil +jXmL (47)dt dt (3) V=I1Z22+JXML (47) The value of I, is then found to be

vqiR d4-qdad= IZda jXm34 (48)RR +jS(XR +XM)

Vq =jQRq -A ±'ddt (34) When solved these equations yield I. =Vdt dt ~~~~~~~~~~~~~~~~~~~~~~~~(Rs+jXs) [RR ±j(XR +Xm)sI +d,(=l Rd +jXdd-jSXqq jXm(RR+jsXR) (58)

vfl=i,Rf+ dt (35) II=V RfRd-XffXdd+Xm2(1-S2)+ ord4'f2 SX,q,Xff+j[XffRd +XddRr-SXqqRf RR \

Vf2 = if2Rf2+d (36) (49) (+jXR jXm

For the 2-phase motor with constant gap I2 (50) Rs+X + R(

Mqf2=LdPM=ulI (37) Id=V jrRd-XrrXdd +Xm2 l-S + Fromm(1S) o j(XR+Xt) (59)

Rp = Rf2 =Rf (38) IdVRfRdXffXdd+±XM2(1 -SS2)±+Rp = Rf2= Rf ~~~~~SXqqXff+j [XffRd+XddRf -SXQqRf I From equation 59 the classical equivalent

circuit of a polyphase induction motorXffI=Xff2=Xff (39) (51) on a per phase basis is obtained.

under balanced condition of applied volt- Ix= -jId (52) By substituting equations 55 to 57 inequation 54 the torque equation in termsof leakage and magnetizing reactance is

Vf2=-Vfi = -jV (40) T=wM [IL *Iq-I2 IdI synchronous watts obtained as

Substitution of equations 37 to 40 in (53) 2VXequations 23 to 26 gives the following V2[RdXm(l -S) +RdXm(1 -S)lXm mdifferential equations: [RfRd-XffXdd+Xm2(1-S2)+ RR 2

dia di, SXqqXff]2±+[XfRd+XddRf- SXQqRf]2 LRs--XSXR-XSXT - XRXM +

Vd =-idRd-Lddd -Md - 2V2RdXm2(1-S) F RR RR 12dt dt ____________Xs +XM ±Rs(XR +Xm)

da do [RfRd-XffXdd +Xm2(1-S2)± L s s

Lqqiq _-Mi,2 (41) SXqqXff ]2+ [XffRd+XddRf- SXqqRf12 (60)dt dt

diQ di2 (54) This agrees with the value of torque ob-v,q =-iqRq Lqqz -1 d ± The vector diagram given in Figure 5 tamned from the equivalent circuit.

dao dao and representing equations 45 to 48Lddiddt±Mli1d (42) stands for the 2-phase motor and is Two-Phase Motor as a Capacitorsimilar to the one given in Figure 2 for the Motor

0} =,R+ di'+ ldid (43) single-phase motor. Here the subscriptsdl dt modifying the voltage terms indicate the The diagrammatic circuit of a 2-phase

di2 diq circuit to which this voltage belongs, motor used as a capacitor motor is shownVf2=i2Rf-ILffd +Mdt' (44) If it is desired to find the value of the in Figure 7(A). The four differentialcurrents in terms of the leakage reactances equations when translated to steady state

i1 and i2 are used for i,r1 and fi2. the magnetizing reactance and the slip, give the following:

1951, VOLUME 70 Sabbagh-Application of the 2-Reaction Theory to Electric Motors 1751

Vd =0= -IdZdd-jXmI- SXddIq-SXmI2 where current in the quadrature axis; SXm(I2+(61) D= 2Xm2RdRl +2Xm2XddXll- I.) is the speed voltage generated in the

*IqZqqJXmI2±SXddId±SXmI1 Xm4+Rl2Xdd2+Rd2X112-Rd2R12 direct axis by the quadrature axis flux;Vq(==-I)Z XMI2+SXddId+SXMII Xdd2X1l2+4RdR1XddXl- and SIqXR is the speed voltage generated

2RdR,XddXc-Rd2XllXc+ in the direct axis by the quadrature axisV - II [RI+j(Xn1 - Xc)] +JXmId (63) X11Xdd XC -XddXmXc±+S2Xm4- secondary leakage flux.

V=I2(R1+jX1j) +jXmIq () S2R2XddX2+S2X1-2XddX2X+ The vector diagrams are plotted from

When solved these equations give S2XddXm2XC-S2Xdd2XllXc+ these equations and are shown in Figurej[2RdXlIXm2+2RlXddXm2+ 8.

V 2RdXddRl2 -2RdXddXl,2 +2Rd2R1X1I1= - [RRd2 -RlXdd2 -2RdXddXll+ -2RXX1lXdd2 -XCRd2RI+XCXdd2R± Application to Capacitor Motor

DRdXm2+S2RlXdd2+j{ 2R1RdXdd + +2XcRdXddXll -XcXm2Rd-Rd2X,1-Xdd2Xll - S2XddXm2+ 2S2XddXm2Ri+2S2RlXllXdd2- A diagrammatical representation of theXddXm2+S2Xdd2XlL+SRdXm2}I S2XCXdd2R 1 (69) capacitor motor is shown in Figure 9.

(65) The vector diagram obtained from equa- The motor consists of two windings12 a[RdXm +S2R,Xdd2±RRd- tion 61 to 64 is given in Figure 7(B). The having different number of turns but as-

D torque obtained from equations 65 to 69 sumed displaced from each other by 90

RlXdd2-2RdXddXll+2RdXddXc+ is given by electrical degrees in space.j -SRdXm2+XddXm2+S2Xdd2X1I - In series with one winding is an elec-S2Xdd2XC S2XddXm2+2RdXddRl+ T = Xm(11'Iq-I2 Id) (70) trolytic condenser which also has a re-Rd2Xl1 -Xdd2Xll- sistance.

Rd2XC +Xdd2XC I] (66) In terms of leakage reactances and mag- The rotor is usually of the squirrel cage

Id=Y [XmRlXddnetizing reactances equations 61 to 64 type. Let the winding with the condenser

Id -XmR,Xdd-XmRdXll + give be called the S winding and the other theS2R,XddXm+SXmRdR1 + 0 = IdRR -jIdXR -jXR(I +Id)- R winding.j{ XmRdRi-XmXddXll - S2Xm + IqSXR-SXm(I2+Iq) (71) The differential equations for thisXm3+S2XddXmXll +SXmRdXll- motor are

SXmRdXc} 1 (67) 0=-I2RR -f12X5 Xm(I2±Iq)+IdSXR+SXm(I1+Id) (72) Vd =-idRd-d da (75)

a= - [SXmRdRi +S2XmXddRI- V = IR8 +j1I(Xs -Xc) +jXm(Ii +Id) (73) dt dt

XmRdXll +XmRdXc -XmR,Xdd + V = 12Rs +jI2Xs +jXm(Iq +I2) (74) vy =-iq--d da (76)j {-SXmXiiRd +Xm3+S2XmXddXll dt dt-S2XmXddXc -S2XM3+XmRdRl- Equation 71 gives the voltage equations d

XmXddXll+XrnXddXc} 1 (68) in the direct axis. The quantity I1+Id is vs =isRs±+ -s (77)dt

the magnetizing current in the directaxis; jXm(I1+Id) is the transformer volt- diR +d 1R (78)

V age induced in the direct axis by the di- dtrect axis flux; I2+IQ is the magnetizing and

JXCI|I 'd =idLdd+iSMdS (79)

(-SXRIq)d lq=iqLqq +iRMqR (80)/ I, 4I2 Ps =isLss+idMds (81)

Jxm(i +Id) I +Id -JXm(II+Id)d

S~Xm(Iq+2)d/1Rs JXsL't> IdJIiXi ~~(-JXRId)d q

k2Rs.l VV

J 2 Figure 8 (left).RJ kXs)JXm(h+lq 2 Vector diagram of a

JXm(k+lq)a ~~~~~~2-phase induction

|2iSXm(I2+ld)q motor as a capacitorSXm(If+ d) motor in terms of

>r, /L+I leakage and magne-(JXsIa)q 12 tizing reactances

(-JXRIq)q ~ ¢ SXRId 0Id

Iq q o

Figure 9 (right). Diagrammatical representa- r-JXm(Tq+12)q tion of a capacitor motorI

1752 Sabbagh-Appldication of the 2-Reaction Theoiy to Electric Motors AIEE TRANSACTIONS

(-SXqqIq)cd (JXdsId)s

X (SisXds/)qJXRIR4 IRRR / (-JXC 15)5 / VJX mR(IR+ Iq)R

(JXRRIRR V (|JXdIS Xa (-JXcls)

F.JXqqIq)q / }(JXqRIq) R / X; h1J

(-JXqRIR)J /Rq l R

/ /~~(-XdI) (SXddId)q /JJRXR I/R R(-SXqRIR)d

// / dIqIRXR;/qqIq)(JXqRIq) SR/ - SXmRUR+lnd (-SXqIq)d

Figure 10. Vector diagram of a capacitor motor in terms of self and JXms(Is+ ld)s (JXqIqq /)Zf4IXd)Q "--JXOI(15+ I4mutual reactances (-JXdId)d--. d a d

Iq

'PR =iRLRR +iqMqR (82) ==--[RdRQRR XaQXRRRd

By substitution the differential equations RqRXdRXqd±dq2+ Jm(I+

(-JXdId) RQXqRIRXdd)q qX d q

become S2XddXqqXRR +j { RqRdXRR + qJXSRdid di$dS RdRRXqq+RRRRXdd-XddXqqXRR-

Vd = - jdRd- Ldd - MdsdtLqqK -S2XddXqR2±XddXQR + 2d dt dt S2XddXqqXRR+SXdsXqRRMd](92)

MqRiR- (83) V Idt I=i[SRdRRXdsRSXddXqR(lS) Figure 11. Vector diagram of a capacitorVQ=utualgRreactancesXD motor in terms of leakage and magnetizing

R=iRLRR Mqdt dt J{ -SIRdXRRXdS± Xds2XqR(R1R2)- reactances.dac . dcx XddXsXqRR(R-S2)+

Lddid-g+Mdsis--- (84) XddXcXqR(1 -82)±RdRsXQR } ] (93).dt dl ~~~~~~~~~~~~~termsof leakage and magnetizing react-

did8 IS daRdRX RSRXdRXdXqqRR+Vd idRds±Ldds -+MdS- (85)iR--SXddX2RQSXddXqqR ac.Tefllwnusittossol

XddXqqRs±+XdXdRqXss±+

VR~~~idR XqqXssRd-XddRqXc-Xq2RdXc + Xdd = Xd ±Xdm (97)di=XRRR+LRR+ diq (86) j{ SXqRXdsRd-Xds2Xqq- XdSdtddt tS2XddXqqXS±S2XddXqqXC± (98)

In the steady state these equations give S2XRQXds2 -XddRqRs-FXiaRodRs-0 Rj IX S -RdRdXss±XdX SXss±RdRqXc - Xea±Xms(99

°L-I(djdd-idda

MdS-iqS -dot84XddXqqXc}] (94) a-2 (99)

IRSXQR (87)

0 MIdXdd+SISXds-Iq(Rq+JXqq) - with D given by whereIRJXQR (88) D=- RdRsXqR2±XddXssXqR2(1-S2) - Xd=leakage reactance of the direct axis

V =IdiXdS+Is(Rs+(XSX-Xc)) (89d) XddXcXqR2(l -S2)-RXdS2XqR2(1 - winding32)tRoRkXdS +RdRRXqq(Xs-Xe) ± Xdm = magnetizing reactance of the directv=IRsXqR+IR(RR±+XRR) (90) RRRsXddXqq(-qS2)-RRdRqRRRS+ axis windingRRRqXdd(Xs -Xc) ±RdRSXqqXRR--The resistance RS is the sum of the re- XddXqqXRR(Xss-Xc)(1 -S2)+ XmS = magnetizing reactance of the S wind-

sistance of the S winding and the resist- RdRqXRR(Xs,s Xc)±RqRsXddXRR + ingance of the condenser. XQqXRRXdS2(l S2)±jXqRdXqR2 XS=Xd+Xma (100)The vector diagram for these equations (XeSS-XC)-R-XddXqR2(ls-S2) -

is given in Figure 10 and except for scale RRRdRsXqS +RRXddXqq(XssR-Xd )Xq-=Xd +XmR (11)is similar to that given for the 2-phase (19-S2)-RdRRRQ(Xss-Xe)- a (102)induction motor used as capacitor motor. R,JRSRRXdd-RXRRXdS±+b=XQ±XhmWhen solved equations 87 to 90 give RdXqRR(XddX 2(l± =lXaaXer

RsXddXqqXRR( 1- S2) -RdRqRSXRR +the following results. RqXddXRR(Xs -Xd)- = Xq+XmR (103)

V [ XRXIRXR-XRR9Sd RRXqqXdS2(l-S2) (9S) The quadrature axis constants are

Id ~ ~~ ~ ~ ~ ~ ~~~~Rqd(S--[-X)dsRdRSXqqXRR+*-

D The torque equation now becomes measured from the R winding and henceis XQfRXdsRe+SXQRRRS+ the ratio of turns of the R winding and the

- XqhXRRXdSoS2XdSXR2±T= XddId*Iq XdsIs -IS I - q windingis assumed 1-to-1.WhereXdSXqR2±S2XQqXdSXBR + X,,RIR*Id (96)RRRPXdS±SXBR<XSS - X,= leakage reactance of the quadrature

SRTeXdRXC ]a (91) Equations 87 to 90 may be written in winding

1951, VOLUME 70 Sabbagh-Application of the 2-Reaction Theory to Electric Motors 1753

/ the stator number of turns, and the other dd di,V==-idRd-Ldd--Mdld--in space phase with the quadrature axis dt dt

and has N sin ,B turns. The rotor rotates . daas shown and hence causes the quadrature a(dqLiq±iiMiq)-g (118)axis to lead the direct axis in space.

)q1ja < Hence da/dt= °w=°=- diLq _MIdi,+vqO iqRq-Lq'd Mdj±1/ S . AThe basic differential equations are lodt

dV,j J TJI =ilRx Q1(109)(idLdd+ilMld)f' (119)v

dipR +(109) a dt

vi=1R1± /dtd4d Nd da When converted to steady state form

Vd0= -idRd -- (110) these equations yieldFigure 12. Schematic diagram of a repulsion dt Nq dt

motor d4-R Nq da V =IlR ±+jI1X11 +IdjXld +IUjX12 (120)dt Nd (1 Vd =0 -IdRd-jIdXdd-fllXld+

XnR=magnetizing reactance of the R In setting up the differential equations SaIQXqq+SaIiXiq (121)winding the ratio of the number of turns in the V=0 =IqRq-jIqXqq -jI1Xi4-

Xzm=magnetizing reactance of the quad- direct and quadrature axis should be s SXq.=magnetizing reactanceof the quad- -~~~~~~~IdXdd -- IlXld (122>rature winding taken into consideration. The flux link- a a

ages 4tq of the quadrature axis generate a

aN (104) speed voltage in the direct axis of value The solution of these equatikns for I,,NR Nd/Nq/1ilda1/dt. Similarly the flux link- Id, and I, gives the following results:

Substitution of these equations in ages 41d of the direct axis generate a speed Vequations 87 to 90 gives Ii [RdRq -XddXqq(l -S2)+

DId Xms j(RdXqq+RqXdd)] (123)0=- +±is Li "IdRd-jIdXd-

Ncos B d=-[Xdql-S2) -ISaRqXlq -(IR+Iq)SXmR-IqSXq (105) D

/ Id Xms jRqXId] (124>O IdSXd±+iJs±+ S IqR2

a) V~~~~~~~~~~~~~~~~~l[XXd(1 S)± RdXl-

IIIXC -(IR +Iq)jXMR ( 106) IA=D aXXd(1S -dXd

V=IsRs+Isj(Xs-Xc)+ IS+ S Id 3N sinB jRdXlal (125>

~ )(107) whr

V=IRRR+IRjXR +(IR +Iq)jXmR (108) Figure 13. Diagrammatical representation of D =RlRdRq-RlXdXqq( -S2) -+

a repulsion motor X11XjqRd - RqXllXdd+RdX2+where XR and X, are the leakage react- RqXld2+j{RlRdXqq+RlR,Xdd +ance of the R and the S windings respec- RdRqXll-XllXddXqq(l- S2)+tively. voltage in the quadrature axis given by Xlq2Xdd(l- S2) +Xld2Xqq(l-S2)-SFrom the foregoing equations the Nq/Nd#dda/dt, where Nd is the number -XldX,qRd+SaXlqX1dRq} (126>

avector diagram shown in Figure 11 is of turns on direct axis and Nq is thedrawn. The equations 105 to 108 and the number of turns on quadrature axis. The torque equation is given byvector diagram are exactly similar to those Here also because of the direction of rota-obtained by the cross-field theory. tion T=W(kd-IQ*-lq NIdd) (127)

Nd NqApplication to the Repulsion Motor d -tws= -S&. (112) =Xn Cos aIiXIq'-Xm sin aIlXId' (128)

In its simplest form the repulsion motor The flux linkages are given by wheremay be represented by the diagram of . Xm is the magnetizing reactance of theFigure 12. =Lll +Mdlid+M5siq (113) stator windingThe rotor is short circuited by brushes. fd=Lddid+Mdlil (114)

The brush axis is set at an angle ,B with thestator axis. Some of the rotor conductorsI Li +Mjiare locally short circuited by the bbrushesLLetwhich cover them. These conductors Nd/constitute the quadrature axis and local a=N(16current X~flows through them. The rest Nt(16of the rotor turns are in series and as such and substitute equations 113 to 116 in pare short circuited by the brushes, and eqations 109tol.Th flown I /constitute the direct axis circuit. equain tol111eThuflloinThe stator could be divided into two

circuits, one in space phase with the vil+Ldi1+ did+ dif (17 Figure 14. Schemnatic diagram of a seriesdirect axis and has N cos /3 turns, Nbeing dtiR+Lj- Mdt +M t~ (117)

1754 Sabbagh-Application of the 2-Reaction Theory to Electric Motors AJEE TRANSACTIONS

I=Nq (129) Figure 16. Equivalent circuit R1 J Xi Rd J XdI_- diagram of a series motor 7

Idl=Id Nd (130) JXm JmN1

ANALYSIS IN TERMS oF LEAKAGE AND SXmMAGNETIZING REACTANCES a

The values for self and mutual react-ances are given by

Lll=L,+llfl Id'=IN(131)Nd Application to Series MotorsId'= 1j(39

Nd (132) N1 In its simplest form the series motor

Nq ~Iq'=N (140) may be represented by the diagramsM,,=M,- sin ,B (133) N1 shown in Figure 14 and Figure 15.

N1 N( 2 The differential equations are as fol-Rd'= Rd N- (141) low:

Ldd =Ld+M1(N) (134)\N

Rq= Rq2(Ne (142) Vx=iiR1+d4 +i+Rd+ dt+Nd'qd- (153)

Lqq=Lq+MIr ehNq~(135) AN, dt dt N, dt

VI)/ X = N )tNj2and X I'=NX,(N' ) (143) v,=0 =-iR,-4q+ jNqdda (154)

If equations 131 to 135 are substitutedin equations 117 to 119, the following re- When equations 139 to 143 are sub- wheresults will be obtained for the steady-state stituted in equations 136 to 137, the fol- 4=Liiii±+M liq (155)condition: lowing equations will be obtained:

kd=Ldd$1 (156)Nd V = 1 [R1 +jX1 +jXm I +Id'jXm COS , +

V= Il[Rl+j(X +Xm) ] +IdNJXmCos ( + I,,jXm sin (3 (144) ,i Lq=Lq,iq±+M,il (157)

N Nq0. XIm[jXm Cos (3-SXm sin (]- Substitution and transformation intoI<iNXm sin (3 (136) Id' [Rd'+j(Xd'+Xm)I+ steady state give

Nd Ia'S(Xa' +Xm) (145) VI =IR1 +jX1iIi +jXq,Iq +IlRd +jXddIl+0 -I1N[jXm cos (-SXm sin (3I- 0 = -I UXm sin #+SXm Cos (3]- Nd NN, Id'S(Xd'+X) - '[R'+ SXe,-NdI +SX,,11I (158)

(Nd) j(X,,'+Xm)1 (146) N,, N1Id Rd±+jXd +jXm +N,

The solution of these equations for the 0 =-IqRq-jXq,Iq-jXqJXII+SI XddIlNd Nq[+(N)] currents gives Nd

N, v~1= R'q'-21- 2+(d'±,'X Solution for I, and Iq, gives-IrN[jmsn0+S.cs0DV= -17NI jXm sin 3+SXm cos (3] (147) I=-[-Rq+jXeq] (160)

D

NdNqSXd+NSXm - Id/=D[XmX(l -S2) Cos (3+ I = N[sqx jX ] (161)

I,,[R,+jXq +jXm(§n)] (138) SRq'Xm sin (-jRq'Xm cos (3] (148) -NL\ N i/1~~N I,, =[ X )sn(3where

I(r'=-[XmX(l- S2) sin -where Xi, Xd, and X,, are leakage react- D D= - qR,+Rd+SXalNdd-ances of the stator winding, the direct axis SRd'Xm COS -jRd'Xm sin ( (149) +,winding, and the quadrature axis winding. where XQq(Xnl+Xdd) +S2XddX,,q+XXq12l+Xm is the magnetizing reactance of the xN=Xd+Xm=X+Xm (150) jdX\q RI+Rd+SXelN +stator. Let XXNX ~'X 10)fX,,~R+dS ,,ly

D = RlRd'Rq' -RiX2(1-S2)- Rq(XI +Xdd)+Xq1SX(Rd'+Rq')(X1+Xm)X+ IN, Nd \Xm2 [Rd' sin2 (3+R,' cos2 (31+ (-Xdd - (162)

53 ~j[Ri(Rd ' +Rq')X±Rd'Rg'(xX + \Nd NQ,33 ~~Xm) 9X1+Xm)X2(1 -S2) + The torque equation is given byC ~~~Xm2X(1 -S2) +SXm2(R,' Rct')

V @:]In gJd The torque equation thlen becomes \Nd Ng,)I 0 T= IDI2 t~~~~~~-S(Rd' cos2 (+R,' sin2 () =NXl,, I 1

Figure 15. Diagrammatical representation of (Rd'+R,')X2(Rd-R,,) sn(co 3 = 1DI12 [,72+,,,I,,I N,a series motor (152) synchronous watts (164)

1951, VOLUME 70 Sabbagh-Applicattion of the 2-Reaction Theory to Electric Motors 1755

To give the currents and torque in Id =II (182) andterms of leakage and magnetizing react- ( Xm SXmances, rewrite the basic equations as aSXd+ J jXm Power output=ST= a (190)

d;,l4=v-

A /jB (183)vi=i,R, +- -Vd (165) A This result could be predicted from

dtwhere physical reasoning.

dlk6d Nd dax (166) A = R2(R1 ±Rd)- XXm -a2Xd X IISXm/a is the speed generated voltagedt N, 'dt (

or back electromotive force of the motor.

diq N,t doe (Xl +Xm)+±Rq'Xm-a2X When this is multiplied by the armaturevq=O qRq d -,Nd d (167) a current the developed power results.

dt Nd dtI Xm\ 2

But (Xd+ (I(1-2) (184)But a2 Conclusion

MqNM B= R,'(X1 +Xm) +a2RIKXd+2)+ The basic equations derived from the

/Nd\2 X 2-reaction theory are applied for theMd y) MI (169) aSXm(Xd+ 2)+ analysis of asynchronous motors. When

N, a2 the self and mutual reactances are re-

('NQ\m (170) (Rq+a2Rd)(Xd+ m) (185) placed by their equivalent values ofN a2 leakage and magnetizing reactances, the

Ld (Nd2\ The torque equation is given by equations thus obtained are those of theLq \NqJ so-called cross field theory. No vector

T=w|III 2m (186) diagram is necessary for the establish-0j=ij(Lj+Mj)+iqMQj (172) a ment of these equations.4'd=il(Ld+MW (173) Xm (Rq')2+(a2Xd +Xm)2

II/d=il(Ld+Md) (173)= V_2__References

t=iiq(L,±+Ma±+Mqi,i (174) a A2+B References

Letsynchronous watts (187) 1. Two-REACTION THEORY OF SYNCHRONOUS

MACHINES, R. H. Park. AIEE Transactions,

Ng. Equation 181 could be written volume 48, July section, 1929, pages 716-30.Q =-t,(,175) 2. THE CRoss-FIELD THEORY OF THE CAPACITOR

N( V MOTOR, A. F. Puchstein, T. C. Lloyd. ElectricalN2\a I A +jB Engineering (AIEE Transactions), volume 60,

Rg)'= RqN, ~~(176) 11= .AjB(188) February 1941, pages 58-62.

tNet (176) R,R'+j(a2Xd+Xm) 3. THEORY AND CALCULATION OF THE SQUIRRELN1 CAGE REPULSION MOTOR, H. R. West. AIEE

a= (177) - V Transactions, volume 43, June 1924, pages 1048-Nd ~ ~~ ~ ~ ~ ~ ~ ~~~.x 157.

R,+Rd +± +j X1 +Xd+XmX 4. PHYSICAL CONCEPTION OP SINGLE-PHASEThe steady-state equations may be a \ MOTOR OPERATION, R. Beach. Electrical Engineer-

found to be 2S2 Xm\2 ing, July 1924, pages 1048-55./ 1 ass Xd±+Y ±X2 5. SYMMETRICAL COMPONENTS As APPLIED TO

V = Is [R, +jXj +jXlnI ±I'jXm Vd (178) + ''± SINGLcE-PHASE INDUCTION MOTOR, F. W. Suhr.\, a2! X \n Electrical Engineering (AIEE Transactions), volume

-r X Xm1 Rq+ia2 Xd+± 2 64, September 1945, pages 651-56.Vd= -II Rd+jXd±j±S ± \ 6. ALTERNATING CURRENT MACHINERY (book),

a2 a (189) J. G. Tarboux. International Textbook Company,Scranton, Pa., 1947.

I tXM + 7 If R,! is infinite, the equivalent circuit of 7. ALTERNATING CURRENT MACHINES (book),Iq'S aXd I (179) T. C. McFarland. D. Van Nostrand Company,

a / the motor becomes as shown in Figure Inc., New York, N. Y., 1948.

0 =Il(SaXd± SXm \X - 16. 8. ALTERNATING CURRENT MACHINES (book),a mJ The component SXm/a the only com- A. F. Puchstein, T. C. Lloyd. John Wiley anda The component SXnla the only com- ~~~~~Sons, New York, N. Y., 1942, 655 pages.

Iq'[Rq'+ja2Xd+jXm] (180) ponent variable with speed represents 9. PRINCIPLES op ALTERNATING CURRENTthe developed power or power output, MACHINERY (book), R. R. Lawrence. McGraw-

These equations give since Hill Book Company, Inc., New York, N. Y.,1937.

RqP+j(a2Xd +Xm) xm 10. APPLICATION OF SYMMETRICAL COMPONENTSIx =V A+jB(181 ) T=-|Vd,2 (book), W. V. Lyon. McGraw-Hill Book Com-

A+jB a pany, Inc., New York, N. Y., 1937.

Discussion equation of all rotating machines as the ten- tions in a per-unit system. Its use is be-sor equation coming widespread in industry; it is hoped

- ~~~that the universities might employ it moreW. G. Heifron (General Electric Company, e =Ri+~p+-7kp5 (1) fully to better prepare their graduates.Schenectady, N. Y.): Dr. Sabbagh's dem- where y is + 1 depending on whether, in It seems to the writer that Dr. Sabbagh,onstration of the wide applicability of the this case, the d or q axis is considered. Thus in listing the advantages of the 2-reaction2-reaction theory is very interesting. The Dr. Sabbagh's paper is chiefly interesting in method, has not stressed the primary reasonfact that graduate students of Purdue Uni- its application to vector diagrams. Kron for the simplicity of the equations, which isversity worked up the solutions is not only gives equivalent circuits for these motors in that the inductances do not vary with time.to their credit but to Dr. Sabbagh's, for it references 1 and 2 of this discussion which To illustrate this advantage, per-unit equa-is a sign of his unulsual ability as ateacher to would be useful in developing the vector tions which are, more or less, phase equa-inspire his students to greater efforts. diagrams, et cetera. -tions ofthe single-phase and 2-phase motors

Gabriel Kron has written1 the general It is disappointing not to see the equa- are given below. The writer objects some-

1756 Sabbagh-App?lication of the 2-Reaction Theory to Electric Motors AJEE TRANSACTIONS

flent circuits rotating with the rotor and fpositioned with the centerlines of the two f1dissimilar rotor circuits. Conversely, thetransformation used here is still applicable

/od \ if the stator is salient and the rotor smooth./ OL t \ In such a case X1 and X2 would contain a

constant and a cos 2a term and a new re-actance X12 which also is a sin 2a termwould be created by the saliency. Chapter ( 52q lt } 2 of reference 3 of this discussion has a treat- r

ment of a similar case which shows that the f2 qmagnitude of the variable part of X1 and X2approximately equals the magnitude of X12.

If it is true that Xlf=X2X it follows thatX1l==X2 and the transformation to give {dfollows very simply. It is of interest toshow the transform which gives ed. Figure 2. Two-phase induction motor

Figure 1. Single-phase induction motor ed - 0 Riicosa-R2i2sina-cos aP4l, -sin aP412 (11) 41' = Xi5+Xr2f2if2

what to the use of the d and q notation for, We now find that R1 = R2 is a necessary = Xqsiq+Xqf2if2 (25)as used by Park,3 such a notation refers to condition for this transformation to be use-equivalent circuits that rotate with the ful, and not a particular case as Dr. Sab- The other equations, of course, follow byrotor, whereas Dr. Sabbagh's system has the bagh insinuates. With this satisfied, the same routine. Note, however, that it isequivalent circuits fixed in space in a modi- not necessary to restrict the equations byfied form of the a, ,B, 0 system popularized ed = -Rlid -cos aPVs - /'1p cos a + the relationships given in equations 37, 38,by Clarke4 and most recently used by Con- 4lp cos a-sin a42 - 42P sin a+ and 39 of the paper. Thus this same analy-cordia.5 In this matter, as well as whether 42p sin a (12) sis may be easily extended to cover the "two-the q axis leads the d, or vice versa, it ap- phase induction motor used as a capacitorpears to be every man for himself and an en- ed = -Rjid-P[W1 cos a±+42 sin al- motor" and "the capacitor motor."suing confusion of the reader. A suggestion [i/i sin az-4/u cos aC pa (13)that the notations mean what they mean in REFERENCESthe so-called "classical" papers is hereby ed = -Rsjd PId 'qP (1) 1. A SHORT COURSE IN TENSOR ANALYSIS, Gabrielmade by the writer. Kron. John Wiley & Sons, New York, N. Y., 1941.

Two-PHASE INDUCTION MOTOR2. EQUIVALENT CIRCUITS or ELECTRIC MACHIN-

SINGLE-PHASE INDUCTION MOTOR Again the applicability of the transform ERY, Gabriel Kron. John Wiley & Sons, New York,

For the coil locations shown in Figure 1 depends upon the two rotor circuits being N. Y., 1951.

of this discussion the phase equations are identical except for the phase relationships 3. See reference 3 of the paper.(that is, R1= R2, Xrlfl(m.) =Xr2f2(.), etc.). 4. PROBLEMS SOLVED BY MODIFIED SYMMETRICAL

= Xisi+Xlfif COS a (2) With these satisfied, and a constant air gap, COMPONENTS, Edith Clarke. General Electric Re-view (Schenectady, N. Y.), volume 41, 1938, pages

V12=X2i2+X2fif sin a (3) the equations for the circuits of Figure 2 are 488-95 and 545-49.i'=Xfrif+XIfi cos a+XIfiI sin a (4) $6qI=X'r=ifl+Xr1fsirl cos a+Xr2rlir2 sin a 5. SYNCHRONOUS MACHINES, THEORY AND PER-4'f=fff+XiilCOS a+2i i 4 FORMANCE, Charles Concordia. John Wiley and

(15) Sons, New York, N. Y., 1951.

Vf=R!if+P#,f (5) i/f2 = Xf2+ if2+ XrlIftrl sin a-VI =0= -Ri1-p4k1 (6) X,2f2i,2 cos a (16) E. M. Sabbagh: I would like to thank

Mr. Heffron for his interesting discussionV2=0= -Ri2-P4/2 (7) Vl/4=Xirs+Xrlflifs cos a+ of the above paper. His kind words are a

Xrlf2if2 sin a (17) source of encouragement and are greatlyThe general transformat=ons are+appreciated.Kd =K, cos a+K2 sin a (8) X2=Xi12+Xr2f8)1sin a- With regard to the d and q notations, IXr2f2if2 COS a (18) still feel that they are the correct ones toKQ=KI sin a-K2 cos a (9) Vf,=RRf1if+pv, (19) use. In Park's paper the armature is sta-

Vf1 ~~~~~~~~~~tionary and the field iS rOtating. ThiS iSInvestigating the application of equations

7 and 8 to equation 3 we see that if xlr=x2r Vf2 = Rf2if2+PQ4f2 (20) usually the case with large synchronousmachines. On the other hand if, as is some-equation 3 becomes Vi '0'-= Riri - I4ri (21) times the case with small synchronous ma-

*I/f=Xff+Xld (22 chines, the field is the stator and the arma-lf= Xffif Xif(d 10) V2=0= -Rir2-pN/r2 (22) ture is the rotor, the d and q windings

Such an equality is indicated in this case be- Using essentially the same transform as equivalent to the armature will be stationarycause of the nature of the rotor. But if the before in space and rotating with respect to themotor were an unusual design (say a single- rotor. The d winding is along the same axisphase reluctance motor) so that XlfHX2f, this kd = Xir1 COS a+Xir2 sin a+ as the field winding and the q winding is intype of transformation would not be appli- Xrifiifs cosO a+Xr2rsifl sin2 a+ quadrature with the d winding or along thecable. As a matter of fact there is no trans- Xrifiif2 sin a cos a - interpoles. This is exactly what has beenformnation which will eliminate trigonometric X i coa sin a (23) assumed in the paper. The saliency in thiscoefficients where both the rotor and the rcase is on the stator and the equations de-stator exhibit saliency. If the motor we are ,6 =Xj ±Xrlf 1iJl veloped are to cover this case. Thisconsidering was salient in the rotor and - Xddid+Xdflifl (24) assumption is indeed fortunate as the arma-smooth in the stator the desirable trans- tures of asynchronous machines are theformation would be one that gives equiva- Similarly, we obtain rotors and the fields, the stators.

1951, VOLUME 70 Sabbagh-Application of the 2-Reaction Theory to Electric Motors 1757