Patiek, K ......... 6032 Sollergren, B ........... 6045Paolucci, A ......... 5856 Stamm, H........... 5864Pasi, A ......... 5929 Stenkvist, K. E ........... 5766, 5865, 5866Paume, A........ 5922 Stigen, 0 ................................................ 5767Pedersen, A ........ 5857,6036 Szendy, K ............. 5845Pichon, A ........ 5922 Szpor, S .............................. 5662Plenener, R ................................... 6111Popolansk, F ...................................... 5825, 6107 Tajthy, T.5937Price, J. B............................................... 5930 Tedford, D. J.......................... 5546Prinz, H .............6037 Terry, A....................... 5924

Pugh, P. S.5763 Thompson, T. B .6009T6th, I. 5764Rabins, L..... 6038 Tuttle, C. H ........................................... 5811Rabus, W ...... 6039,6040Rfacz, L .....5764 Vajda, G................5937Ratzke, J. 5858 Vales,J .... 5938,6046Reimer, J .5859 Veverka, A....... 5747, 5748, 5768, 5867, 5939, 6047, 6125Rimkus, H .............................................. 5860 Vlnal, F ....... 6035Rippon, E. C ........ 6041 Vocho6, V ............................... 5769, 6126Roge, G ...... 5922 Vogel, F. J ......... 5868Rohats, N .......................... 3627 Vyskocil, V ........... 5825Roife, I. M ................. 6042 Walther, K....... 5770, 5940Ronovsky, J ... 5916 Waste, W

.......5771

Rossier, C ....... 5861, 5862, 5931, 6015 Waters, M...

5663Roth, A ....... 5932 Weed, H. R... 5941

Rudge, W. J .......... ........ 5751 Wellauer, M .5839Saint Paul, R ... 5761 Wentz, E. C ........................................ 5869,5906Sapozhnikov, A. V .................. ... 6123 W erth, E . ......................................... 5942Saraoja, E. K .. 6043 White, E. L ....... 4617, 5052, 5123, 6048, 6049, 6050ra a

.65 5804...22.ill..m.R.... ............ .... ..... .... .... .....566Savoisienne ......5...7......... 65, 5804, 5922 Willheim, R......................

5663Schafer, W. 5347 Williams E. G.5741Schaffer, H. E . 6044 Wittgenstein,

..........................5810, 6007

Scheda, F. A ..............5................................ 507Schleich, A ........................ 6027 Yamajo, T

..........6051

Schlosser, K ................................... 5818 Yamamoto, M . 6051Schneider, E ....... .............. 5934 Yamamura, Y

.........5870

Setala, A .. 6043Shargorodskii, V. L . 5834 Zaduk, H ................................................ 6052Silverstein, J ......................................... 6023 Zambardino, R........................................0...53Sirotinskii, L. I . 5935 Zappalorto, E. L............................... 6054Sk'ala, M ......... 5863, 5936, 6124 Zimmermann, C. P ................ 5763Slowikowski, S ......... 5550 Zoledziowski, S ................... 5550

E . , ,. . f r E . l.J EI D equations relating the voltages andquivalent Circuits ror ty.inurica - otor, currents of the stator and field windingsand the positional angle of the rotor.Reluctance, and Salient-Rotor Although -equivalent circuits have beendeveloped to represent these equations,they have not been in a form which

Sync ronous hzac ines permitted simple interconnection withthe conventional single-phase equivalentcircuit of the power system to which

GORDON R. SLEMON the machine is connected."MEMBER AIEE This paper presents a unified approach

to the analysis of polyphase synchronousSummary: This paper presents a unified the cvlindrical-rotor machine is generally machines in the balanced steady state.approach to the steady-state analysis of treated first.' An equivalent circuit The cylindrical-rotor machine, the re-polyphase synchr-onous machines of three .r.a,e. . luctance machine, and the salient-rotormajor types: cylindrical rotor, reluctance, consisting of a voltage source in series machine are considered in turn. Expres-and salient rotor. The major magnetic with the synchronous impedance is sions are derived for the distributedparameters of the machines are derived. developed. While this equivalent circuitSingle-phaseequivalentcircuits are de- is very useful in studies of operating magnetic fields, the flux linkages, andveloped for each of the machine types. performance, it differsfrom the equivalent tetru. Etiaetcrut rThese equivalent circuits may be incorpo- . 'rated directly into the single-phase network circuit of the induction machine evenrepresenting a power system. though the two machine types have

considerable structural and magnetic Machiery6-1CCom tmitteeand aprproved by the AIEEsimilarity. Technical Operations Department for presentation

iA AANY PAPERS and books have been Salient-pole synchronous machines at the JAIEE WintFerGenerayl Meetinog, New York,IVIwritten on the analysis of syn- have conventionally been analyzed by N.bmitt,Janua ery28-erur 219 62.Maeanusciptlefochronous machines. In the conventional the use of the two-reaction theory.2 Pritig Noebr2,16.educational approach to these machines, The result of this theory is a set of Toronto, ToronLtEo,Ont., Canada. niesiyo

AUGUST 1962 Slemon-Equivalent Circuits for Synchronous Machines 219

Let phase a carry a current for each stator phase is then given byia =m sin cot amperes (2) the following:

'~~~~~~~~~~~~~~~~~~~~~~t -7X/i henry (9 mprs 2/e Then, if the reluctances of the iron henrY (9)b < r*\ portions of the magnetic system are Let the rotor have Nf turns distributedneglected, the magnetic potential across sinusoidally about the axis of symmetrythe air gap at angle 0 is 0=B so that the number of conductors

N/Im per radian of periphery isFaO= sin wt cos 0 amperes (3) N121

-- -+--< 4 | nf= 2fjsin(o-,9)j conductors/radian (10)If phases b, c, also carry sinusoidal

currents of maximum value Im which If the rotor winding carries a currentare delayed by 2Tr/m, 4ir/m, ... , re- if the magnetic potential which it willspectively, the magnetic potential across produce across the air gap will bethe air gap will be

ST A TOR/ fVNimFr=-Nfcos(0-,) amperes (11)FO= 2 [sins t cos 0+sin (o+27r/m) 22Fig. 1. Structurm and windings oF cylindrical. cos (0+2r/m)+sin (cot+4ir/m) Comparison of equations 11 and 4

fotor synchronous machine cos (0+47r/m)+.. .] shows that the same magnetic field canrotor synchronous mmNine mlV8Im . a (4) be produced in the air gap by a constant=- sin (cwt- ) amperes (4)4 current if in the rotor winding as by aproduced for each of the three machine Let the effective length of the air gap, balanced polyphase set of stator currentstypes. These equivalent circuits have corrected for fringing around rotor and of the form given for phase a in equationthe following properties: the stator stator slots, be g. The flux density in 2 ifvoltage and current per phase are repre- the air gap is NsImsented on a single pair of terminals. if=m 2 N amperes (12'Therefore, the circuits can be incorpo- Be poFsa/g

mNj.lanrated directly into the equivalent single- =A-g sin (cot-0) and

phase representation of a polyphase 4gpower system and ordinary circuit anal- =Bmsin(ct-0)webers/meter (5) ---radians (132ysis may be used for the prediction of To find the flux linkage of stator wind-system performance. The circuits are ing a, first determine the flux linkage Equation 13 specifies that the rotoraUmodifications of the familiar equivalent a single turn at angle 0 returning at turn at an angular velocity of c radianscircuits for transformers and induction 0+7r per second and that its axis be positionedmachines. The important similarities O+ at 3= -T/2 at t=0 for the time referenceand differences of these machines can , i Botrdo chosen in equation 2. As seen from theeasily be demonstrated. Jo stator, the magnetic effect of the rotor

= - 2Bmtr cos (ct-0) webers (6) is identical to that of a stationary rotorThe Cylindrical-Rotor Synchronous where r is the air gap radius and t is the carrying a balanced polyphase set ofMachine axial length of the machine. From currents of frequency co radians per

equation 1 the number of phase a con- second in a set of distributed rotorFig. l shows schematically the mag- ductors is a band dO is (N8 sin OdO)/2. windings.

netic structure and windings of a cylin- Thus, the total flux linkage of winding In operation, both stator and rotordrical-rotor synchronous machine. The a is windings will carry currents. The mag-stator has m windings which are dis- netic potentials can be added algebrai-tributed at intervals of 2er/m radians r N8 sin Xo,d cally to givearound the periphery; the rotor has a J_a2 F-F,+Fjamperes (14)single distributed winding. To maintain 7ra constant shape for the radial distribu- 2 Let the resultant flux density betion of the magnetic field in the air gap Thus, under balanced current condi- B =Bm sin (cot+ a-0) webers/meter2 (15)for all values of stator and rotor currents, tions the magnetizing inductance as seenall windings are assumed to have their sttor pase is given aseen The flux linkage of the rotor windingturns distributed sinusoidally about the by a stator phase iS given from equation may be determined as in equations 6periphery. Fig. 1 shows only the central 5 and equation 7 as and 7 and is given byturn of each winding. Line=Aai

Suppose the stator is provided with a =-Nmw j-htry (8) )X?flf N1Bmfl sin (co+a-,8) webers (16)relatively large number of slots. Let - 82 g~hnyeach stator winlding have N8 turns. IFor phase a these turns are distributed The stator winding will have a flux Iinthe slots so that the number of phase linkage Xta due to flux which encircles t8-= cot (17)aconuctor per adianof prihr the stator condulctors, but will not cross

is closely approximated by the expression the air gap to penetrate into the rotor. \mp' 2 N,Bmtr sin (a-o) webers (18)This flux linkage can be found from the 2N, slot, tooth, air gap, and end connection In the steady state where Bin, at, and

na=2 sln@|conductors/radian (' dimensions, and the lageinductance ,Bo are constant, this flux linkage is a

220 Slemon- Equivalent Circuits for Synchronous Machines AUGUST 1962

STATOR IDEAL ROTOR source If. The magnitude of this source IMACHI NE STRANSORMAHIONE current is dependent on the field current

_ If if if and the current ratio of the machine R iwLRS IcoLI15 tLI I 1 ,,, R~ I given in Fig. 2. The angle of the current

I | LIJJ source depends on the position 00 of the E jwL Ems tE E e~~~ rotor at the chosen reference time S sM hf

s jwLms mf designated as t=O. If the angle a inequation 15 is designated as -r/2, thenthe phasor Ems in Fig. 3 will have anangle of zero and the phasor Im wiRl Fig. .3. SimplifIed equivalent circuit for

Fig. 2. Equivalent circuit of cylindrical-rotor have an angle of- r/2. From equations cylindrical-rotor synchronous machinesynchronous machine 11, 15, and 17 the angle of the source

Rdtios of ideal machine transformation current Iowill be ( deo+w/2). of the voltage Em.. The derivation ofFrequency-w:O The rotor angle go will depend on the this nonlinear circuit, the method ofVoltage-1: 0 torque T applied to the mechanical measuring its parameters, and a more

-'1Nf:1 system in the direction of ,. This torque elaborate equivalent circuit including theCurrent- mN8 can be expressed as effect of rotor saturation as provided inPower-1:0 dXf reference 4. Application of Th6venin's

=i do newton-meters (21) theorem to the circuit of Fig. 3 givesconstant. No voltage is induced in the the conventional equivalent circuit con-rotor winding. There will be additional where A is the t.otal flux linkage of the sisting of the synchronous impedanceflux linkage in the rotor due to leakage field winding. Sice the fie.ld current is Zs=Rs++j (Lt,+Lms) in series with aflux but this also will be constant. constant the leakage flux linkage, Xtf= source voltage Eo=jwLm.Ij.The electrical behavior of the machine Lf,if, is constant and the torque depends

can now be described by the equivalent only on the change of Xmf with 0. Sub- Reluctance-Type Synchronouscircuit of Fig. 2.4 In the rotor section stituting equations 16 and 17 into Machineof the circuit the induced rotor voltage equation 21 givesem, is zero in the steady state. Thuc, T=- ijNiBmtr cos (go- a) newton-meters Fig. 4 shows the magnetic structureif the rotor terminal voltage ef is constant, (22) and windings of an elementary polyphase

reluctance machine. The stator isi1= amperes (19) identical with that of Fig. 1. The rotor

Rf of the electrical variables of Fig. 3 by has no winding and has a nonuniformThe current ir in the rotor is equivalent Substituting air gap which is symmetrical about the

in its effect on the stator to a sinusoidal VN,. axes 0= # and 0=0+%T/2.current of effective value Ir and fre- mImN, (23) If the stator currents are balancedquency w. The phasor If adds vec- and the current in phase a istorially to the stator current 15 to produce andia=I sin (wt+ a) (26)the magnetizing current phasor Im. I+The induced voltage in the stator is Ems = 2 NB,,mtr (24) the magnetic potential across the air

aap is, by modification of equation 4,Ems wLm,Im volts (20) into equation 22, setting a,= -7r/2, a iThis is equivalent to the rate of change to give F= Nsin(lm +a-0) (27)

with time of the flux linkage in equationm

47. The stator resistance, R,, and leakage T-- Est 1IJI COS (BO+1r/2)inductance, L, per phase complete the newton-meters (25)stator section of the circuit.

In Fig. 2 the stator and rotor variables Thus the torque exerted by the machineare linked by an ideal machine trans- in the direction of positive , is equal toformation that is similar to an idea.l (-m/cX) times the power entering thetransformer in that its properties are current source If in Fig. 3. For zero bstated as ratios and it represents the torque the angle Po is normally equal tobehavior of a perfect machine that has -xr. When the angle go is advancedno resistance, leakage flux, or magnetiz- from -7r, the torque is negative indicat-ing current. The current ratio of this ing generator action; when it is retardedideal transformation is as given in from -7r, the torque is positive indicat- aequation 12 except for the factor x/2 iP.g motor action. \\ / lwhich is required to convert peak values The equivalent circuit of Fig. 3 may \jto effective values of current. The be used in conjunctio)n with equation 25 \\/voltage ratio states only that the rotor to p)redict the steady-state operating \ \\/induced voltage is zero independhen.t of characteristics of the cylinldrical-rotor\ ' g /the value of stator induced voltage, machine. If the iron reluctances of the \ e/The equiivalent circuit of Fig. 2 may machine can.not be neglected their effects STATOR

be replaced by the simpler form of Fig. can be included, accurately enough for3 in which the effect of the rotor is most purposes, by making the magnetiz- Fig. 4. Structure and windlngs of rluctance-represented by the sinusoidal current in.ginductancee, Lms, a nonlinear function type synchronous machim.

AUGUST 1962 Slemon- Equivalent Circuits for Synchronous Machines 221

3 The rate of change with time of Xma is The second admittance in equation 36'__N I the induced voltage ems in the stator. has as its inverse an impedance Z,. If|R IcaL | 2m Using phasor notation and letting both numerator and denominator of theis

JcLmo Im=Im sin (wt+ a), second term in equation 36 are multipliedr 1 through by /-,3o and simplified, the

Ems =[oLo-j.Ln/2(JoIo impedance Z7 may be obtained in the

E Ems iLm2COs 2po = [Lm2 sin 2(foo-a)+jw(Lmo- formLm2 cos 2(3o-"a))]Im (32) z c= w(Lmo+Lm2) cot ,0 +

-wALm2SIN200 where 2LM270=-N82t'rho (33)L 2-LM2 (37)

Fig. 5. Equivalent circuit for reluctance-type 8synchronous machine This results in the equivalent circuit

and of Fig. 6 which has only one parameterThe air gap flux density at angle 0 is Lm= N5trh2=- (34) that varies with rotor angle Bo.

approximated by 16 In developing the equivalent circuitsThe magnetizing impedance is there- of Figs. 5 and 6 for the reluctance

B6= Ao Fe (28) fore maximum and inductive at (Po- a) = machine, the time reference has beeng() 27r/2 and is minimum and inductive at chosen by specifying that Im= |Imiwhere g(e) is the effective length of the (do- a) =0. At other angles (o- a), sin (4t-7r/2). Often, it is more con-air gap at angle 0 Let us expand the magnetizing impedance includes a venient to specify the voltage, Em,,l/g(o) as a Fourier series. real part which may be positive or rather than the current, Im, as the1 negative indicating the flow of electric reference phasor. Fig. 7 is a vector- =ho+h2cos2(0-#3)+h4cos4(0-$)+... energy in or out of the machine. diagram of the magnetizing branch asg(o(29) o From equation 32 and a knowledge described by equation 32 with a = - r/2.

Of the stator resistance and leakage If Ems is used as the reference vectorSubstituting equation 29 into equation inductance, the equivalent circuit of and is assigned the angle zero, the con-

28 and using equation 17 to define , Fig 5 may be constructed. The magne- struction in Fig. 7 shows that ,o willmNImI tizing current is used as the time reference be replaced by 3. To determine the0=o sjm ho sin (wt+a-O)+ and its angle a is set to -7r/2. relationship between go and d,e, apply4 The torque of the machine may be the law of sines to triangles Oxz and-sin (3wt+et+20o-3O)- found by the use of an expression similar Oyz in turn giving2 to equation 21 using stator currents and w(Lao,-LM2) Ems-2sin(wt-a+2$o-0)+ flux linkages. However, from energy sin(r -s)2 considerations, the mechanical powerh4

sin (5wt+a+4o- 50)- output (Tw) must equal the sum of the and2 power inputs to the resistive element (L +LM2) Emsho 1 ~~~~~~~inthe magnetizing branch of Fig. 5 in (/+~)-sn(2-)

2 sin (3wt-a+46o-30)+ - (30) all phases. Thus, ( /2+c) (7/2-o)from which

The flux linkage of the stator winding T - IM2 (wLm2 sin 2#o) (L+L)a may be found, as before, by finding the . cot #'e = cot0, (38)flux linkage of a single turn at 0 and = M-Imm 2Lm2 sin 2,o newton-meters (Lmo- Lm2)0+7r using equation 6 and then employ- (35) Thus the impedance Zr in equation 37ing equation 7 to find the total flux With the chosen time reference, the and the corresponding branch in Fig. 6linkage. The result is torque is zero when 0 equals 0 or 7. may be expressed as

Advance in ,; produces generator actioii;N2man= hosin (t+a)- retardation of 'o produces motor action. z=,=o -cot gse+j (39)8r h The equivalent circuit of Fig 5 2. n2+

-2 sin (wt- a+2#o) (31) contains two parameters that vary with where 6 is the rotor position at the time2 rotor position t3o. An alternative circuit, that the induced voltage in phase a

which may allow simpler computation, is zero.sdUL may be obtained by representing theR, jwL15 Im magnetizing branch of Fig. 5 as two EI

s parallel admittances one of which is zIm I 'mr the admittance at f3 = 0 or ir. From X J-Lm2lmoEt jw( moLm) equation 32 with a=- ir/2 /*

jCDrCL4oLm2)t Em 2Lrn2 Is 1 L jL2/p 0~Xi/Jomim2

~~~jw(LmotILm3) m- -Fig. 6. Alternative form of equivalentcircuit for reluctnce-type synchronous Lms(1- /2A) (36) Fig. 7. Vector diagram showing variables

machine j(Lmo+Lme)(Lmo+Lm9/2I%) of Fig. 5222 Slemon-Equivaleni Circuits for Synchronous MCachines AUGUST 196S2

in that it has a winding on the rotor. isFor simplicity it will be assumed thatthe rotor winding is sinusoidally dis- R5 ;L5 |m j/a2

b 7 tributed as for the cylindrical-rotor 2rmachine. The effect of the rotor wind- mO l ca(L2 a7L2ing will be identical with its effect in Es Ems 2 L I

I the machine while2jL 3(L7/X ! /J \ \ \ the cylindrical-rotor machine while mo m2 | 2)the effect of rotor saliency will be that -C4L2 2 )GOT |of the reluctance machine. An equiva- 2 L

__A_L /_ lent circuit for the salient-pole syn-chronous machine may therefore be Fig. 9. Equivalent circuit of salient-rotot

ROTOR / \ f produced directly by using the current synchronous machinesource branch from Fig. 3 and themagnetizing impedance representation the voltage Ema as reference, itfrom Fig. 6. The result is shown in usingbe vown by as ofercuitFig.9 of F9ga 9 that

Since both the current source branchSTATOR ' and the resistance branch are dependent Ems

Fig. 8. Structure and winding. of salient- on rotor position, it is important that IseZ+w(Lmo-Lm?) (47)rotor synchronous machine the same reference angle be used in co -- I-determining that angle j3 for both.

In Fig. 9 the voltage Emg has been chosen Wurhen the angle is known, the fieldIt will be noted that the equivalent as the reference for time and has been current, IJ, can be found readily.

circuits of Figs. 5 and 6 are free of voltage assigned the angle zero. The angle ,e The equivalent circuit of Fig. 9 mayand current sources indicating that the is the rotor position at the time that be considered as a general circuit repre-reluctance machine acts as an energy this voltage is zero and rising. senting all three classes of synchronousconverter only when its terminal voltage The torque of the salient-rotor machine machines considered in this paper. Foris established by an external agency. is given by a combination of the expres- example, if the rotor saliency is zero!Also, the power factor limitations of sions in equations 25 and 41 the inductance Lm2 becomes zero andthis type of machine can be appreciated Fig. 9 becomes identical with the cylin-by direct examination of the circuit. T=- Ems I111 cos (e+7r/2)+ drical-rotor circuit of Fig. 3. In Fig. 10,The conventional expression for the wL a vector diagram showing variables oftorque of a reluctance machine may be LM21 Emsj 2 sin 203e (43) Fig. 9 is shown.developed readily from Fig. 6. Using w(Lmo2-Lm22) Jthe rotor angle ,B referred to Ems and In conventional synchronous machine Conclusionthe expression for Zr in equation 39 analysis the voltage back of synchronous The objective of this paper has been

2LM2E,, 1 reactance iS normally given the angle 6. ThobetvofhiparhsbenImr m (40) r i n g t to present a unified analytical approachc-(LmolLm22 -Cot oe+j) This angle is related to #, by to three major types of polyphase syn-

and = ,Ce+ (44) chronous machines. The starting pointhas been a description of the magnetic

T= If equatins 44 and 42 are substituted field within the machine and the culmi-Real part of (IrnrEms*) in-to equation 43 the expression for nation has been a general equivalentmLm2Ema2 torque becomes circuit. Standard methods of circuit2(Lmo2-L,02) sin 2ne T- n7i[E analysis may be used for the prediction

meters (41 ) - ms r sin + of machine or machine system per-(Lmer-L41) 2o1formance.

In the two-reaction theory of syn- (Lmd-Lmq) EmS l2 sin 25 (45) For brevity, the analysis has beenchronous machines the inductances Lmo 2wLmdLmq restricted to the balanced steady-stateand Lm2 and. replaced by the direct-axis Standard method.s of circuit analysis condition. The approach may, however,and quadrature-axis inductances given may be used with the equivalent circuit be extended to cover a variety of transientby of Fig. 9 for the prediction of the operat-Lmd Lmo+Lmt2 ing characteristics of the salient-rotor ELmq = Lmo-Lm2 (42) synchronous machine. Since the angle I l I

All expressionsdforethe reluctance appears twice in the circuit, the /\S

All expressions for the relutance analysis is simplest if this angle is known. / ( emachine can be written in terms of these If direct analytical Methos aedn eparameters to predict the field current and rotor e 2 /| mmS

angle fo)r a given set of terminal condi-Salient-Rotor Synchronous Machine tions, it is useful to note that the currents / Im

'mr and I.,. are always in quadrature /Fig. 8 shows the magnetic structure with each other as shown in the vector I

and windings of a salient-rotor syn- diagram of Fig. 10. If the stator currentchronous machine This machine is is resolved into components Fig. 10. Vector diagram showing variablessimilar inl magnetic structure to the of Fig. 9 for a leading-powher-ator motoringreluctance-type machine differing only =Is=I,p+jhq (46) mode of operationAUGUST 1962 Slemon-Equivalent Circuits for SynchronousAMachines ...223

problems. While only sinusoidally dis- circuits indicate that all these polvphase 2. SYNCHRONOUS MACHINES: PART I-AN Ex-TENSION OF BLONDEL'S TWO-REACTION THBORY,tributed windings have been considered, machines may be considered as variants PAT 11-STEADY-STATE POWER-ANGLE CHARAC-

the winding factors for other types of of each other. This equivalent circuit TBRISTIcS, R. E. Doherty, C. A. Nickle. AIEEthewinding factors,for other types of of each other. This equivalent circuit Transactions, vol. 45, June 1926, pp. 912-47.winding are well known and may readily approach may also be extended to include 3 EQUIVALENT CIRCUITS OF ELECTRIC MA-be incorporated into the analysis. polyphase commutator machines.4 CHINERY (book), G. Kron. John Wiley & Sons,The concepts utilized in this analysis Inc., New York, N. Y., 1951.

References ~~~~~4. EQUIVALENT CIRCUITS FoR TRANSFORMERSare essentially the same as those used in ReferencesINES INCUITS NOR-INARERsAND MACHINEDS INCLUDING NON-LINEAR EFFECTS,the analysis of polyphase induction 1. ELECTRIC MACHINERY (book), A. E. Fitz- G. R. Slemon. Proceedings, Institution of Elec-

gerald, Charles Kingsley, Jr. McGraw-Hill Book trical Engineers, London, England, vol. 100, pt.machinery, and the resultant equivalent Company, Inc., New York, N. Y., 1952. II, 1953, pp. 469-86.

Discussion siderations, leading to pertinent equations, With the angle ft already known, the re-as in the text, could they be determined sistanceexperimentally by conventional tests? If

RichardT. Smith(Tracor, Inc., Austin, Tex.): so, how can saturation be accounted for? (L2)(Lmo1Lm 2)COtRe)Dr. Slemon has developed an interesting al- It is agreed that the purpose of the paper 2Lm2ternative circuit representation of balanced, was to develop equivalent circuits, yet itsteady-state performanceof certain machines would be interesting to know the author's may be determined. The circuit thenthat will be helpful in teaching. His state- comments on these questions, which are consists of three known parallel impedancesments concerning the usefulness of these related to the paper rather indirectly. supplied from a known constant currentcircuits after they have been incorporated The author seems to imply that it is source, and standard methods of circuitinto power system networks may be some- necessary that the winding be sinusoidally analysis may be used to find the terminalwhat misleading. Since the equivalent distributed. This is an unnecessary re- voltage.circuits of all but the round-rotor machine striction since a close approximation to As Nasar and Saunders have mentioned,contain impedances that are functions of sinusoidal distribution is obtained by several authors have developed equivalentthe angle #,, and since this angle is in turn utilizing judiciously distributed windings. circuits for synchronous machines. Ina function of machine terminal conditions, What he really seems to say is that the most of these, the direct and quadratureit appears that these circuits cannot be fundamental component of the current axes have been represented separately, andhandled as directly as the author im- sheet is the predominant one. This is a mathematical transformation has been-plied. For example, a single synchronous true if the winding is properly designed. necessary to determine the terminal voltagemachine with known field current and There are two ways to achieve this, either and current of the machine in terms of the,circuit constants, loaded with a known by the use of sinusoidal distribution, which equivalent circuit variables. While theseload impedance, presents a problem in is almost never employed in the general circuits are important for analysis of thefinding the machine terminal voltage that purpose synchronous machine except on the internal behavior of the machine they arecannot be solved merely by employing field, or by the use of uniform-turns-per-slot not well suited to system studies where athe equivalent circuit of Fig. 9 with the types of windings, which utilize the dis- single-phase model, which can be connectedload impedance across its terminals. tribution and pitch of the coils to obtain directly to a single-phase network, is

the equivalent to a sinusoidal distribution. desired. The circuit used by Tarboux tointroduce the study of synchronous

S. A. Nasar and R. M. Saunders (Uni- REFERENCES machines is the best starting point in theversity of California, Berkeley, Calif.): 1. THE DIRECT- AND QUADRATURE-AXIS EQUIVA- opinion of the author. Fig. 2 extends thisWe share with the author some of his LENT CIRCnITS OAFE TransacHions, vol. 64, DEC by stating the transformation propertiesmisgivings about the transition between 1945, pp. 861-68. of the coupling between rotor and stator.induction machinery for which succinct 2. ALTERNATING CURRENT MACHINERY (book) The measurement of the parameters forand well-developed equivalent circuits in J. G. Tarboux. International Textbook Company: the equivalent circuit of Fig. 3 is discussedboth cross-field and double-revolving-field Scranton, Pa., 1947, first edition, chap. 33. in reference 4 of the paper. Briefly, thetheory variables have been employed stator leakage reactance, the magnetizingextensively. In the case of the synchronous reactance, and the ratio relating the directmachine it is difficult to develop equivalent Gordon R. Slemon: I am grateful to the field current, if, to the equivalent source,circuits and we have found this to be dis- discussers for their interest and for the If, are determined from an open-circuitturbing to some students. Therefore, the pertinent points they raised. Professor curve and a zero-power-factor curve. As apaper given here is a contribution to our Smith questions the direct applicability first approximation, saturation is accountedthinking in educational circles. of the circuits for network calculations. It for by making the magnetizing inductance,The development presented in the paper is true that calculations using the salient- Lm, dependent on the voltage, Em,. The

is indeed very logical and simple. It is pole equivalent circuit of Fig. 9 are more appropriate values are read directly fromthe first time that a unified approach complex that those for the round-rotor the open-circuit curve. If saturation isleading to readily usable circuits has been circuit of Fig. 3 because two separate included in this way the results are identicalpresented for synchronous machines. It mechanisms of torque production are with those obtained by the Potier methodmay be mentioned, however, that a more involved. Giveni the field current, If. of analysis. For the equivalent circuitcomplete, but rather complicated, circuit the load impedance, (RL+jXL), and the of Fig. 9 it is necessary to do a slip test towas developed by Rankin,' and, for that machine parameters, R,, Lr,, Lm,o, and determine the quadrature-axis reactance.matter, numerous circuits have been Lm2, the terminal voltage of the machine The effect of saturation in the salient-polepresented from time to time. Attention of may be found directly as follows. The machine may be included in a first approxi-the author is invited to the statement that angle 6e in equation 47 is dependent on mation by assuming that only the direct-the conventional circuit "differs from the the network impedances only and is inde- axis magnetizing reactance w (Lmo+Lm)equivalent circuit of the induction ma- pendent of the magnitude of the v-oltage is saturable. This parameter is made achine." While the statement is correct, Ems. Let function of the voltage Ems and the ap-there is a striking similarity between Fig. 2 1 propriate correction is also made to theof the paper and a circuit developed by G+jB = parameters in the branch carrying currentTarboux in Fig. 307.' Tarboux's illustra- RL+RS+J(XL+JLLa) linT in Fig 9.tion merely shows the essential resemblance Then, for unit value of the voltage Ems, The assumption of a winding withbetween the two; hence, the author's equation 47 becomes sinusoidally distributed turns was madeextension to all three types of synchronous 1 not only for convenience in analysis butmachines is an addition to the literature, also because it is my opinion that it repre-Although the parameters involved in the ct ai -B+c(Lmo-LM2) sents the best starting point. When the

circuits can be evaluated from design con- G properties of a machine with this ideal

224 Slemon-Equlivalent Circuits for Synchronouls M>achines AUGUST 1962