2007_28 Salient Pole Synch Machine Simulation

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2007-28

Simulation of a Salient Pole Synchronous Machine with

Both Field Pole and Stator Core Saturation

WisconsinElectric

Machines &PowerElectronicsConsortium

University of Wisconsin-Madison

College of Engineering

Wisconsin Power Electronics Research

Center

2559D Engineering Hall1415 Engineering Drive

Madison WI 53706-1691

2007 Confidential

Research Report

T.A. Lipo, T. Komatsu*, K. Shinohara*

*KagoshimUa niversityKagkoshimaJ apan

Universityo f WisconsinMadisonW I, U.S.A.


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Simulationof a SalientPole Svnchronous achinewithBothFieldPoleandStatorCoreSaturation

T.A LipoUniversity f WisconsinMadisonWI, U.S.A.

Abstract Saturation effects n salient pole synchronousmachines are generally modelled by assuming that thesaturation effect is primarily limited to the field pole.However, saturation of the stator teeth and core can alsobecome saturated such as when operating as a generatorwith a leading power factor load. This paper addresseshemodeling and simulation of simultaneous saturation ofboth the stator and rotor taking into account the correctcomponents of flux linkage which contribute to theseeffects. Suitable saturation curves are computed usingfinite elementmethods.Simulation resultsare shown usingthe new saturation model.

I. INrRooucrtoNSaturationeffects n salientpole synchronousmachinesaretypically modelled by assuming that the saturation effect isprimarily limited to the field pole and the effect is, hence,a

function of the field flux linkages.However, situationsoccur inwhich saturation of the stator teeth and core also becomesaturatedsuchasoperationas a generatorwith a leadingpowerfactor load. The saturationeffect of the statorcannot be simplylumped with the field pole saturation since stator coresaturation is the vector sum of d- and q-axis stator fluxlinkageswhile the field pole is essentiallydependant nly ond-axis flux linkage components.Hence, dependingupon load,saturation of a synchronousmachine may have either statorsaturation, field pole saturation or simultaneoussaturationofboth components.This paper addressesmeans or modelingand simulation of simultaneous saturation of both the statorand rotor taking into account the correct components of fluxlinkage which contribute to these effects. Suitable saturationcuryes are computed using finite element methods. Typicalsimulation resultsof the improved model are shown.

Ii. d-q RernesENrArIoNoE e SYNcrnoNous MACHINEPark's model of a salient-pole synchronousmachine,

representedn the rotor reference rame, is shown in Figure 1.The quantities rs, rdp ro, arld r/r corespond to the stator, d-axis rotor amortisseur,q-axis bmortisseur and field windingresistanceall referred to the stator by the appropriate turnsratio, The quantities L*4 and Lô are the d- and q- axesmagnetizing inductances respectiv6ly. In Figure 1 the statorleakage nductance ypically expressed Llrhas beenseparatedinto two portions namely L7", arrdLp, to denote the leakage

T. KomatsuKagoshima niversityKagkoshima apan

inductanceswhich correspond o the end winding and the coreportions of the leakage flux linkages respectively. The twoportions ofthe stator eakage nductanceshave been separatedsince saturationof the stator core only involves that portion ofthe stator eakage lux which passeshrough the statorcore.

The equationscorresponding o the circuit ofFigure 1 are,d) ,_' q r = r r i , 1 r * t * o r l d r 'd)r,^v r r : r r i T r * - * - < r l r t r g ,

' , dXo,= r I -1 4q r ' q r ' g r d t. , dLa,vd, : rd r td r - d ,

. . d\ , ,vn= r?J1i* i (5)For purposesof simulation Eq. 5 is usually manipulated o theform

,r!r!!!t (6), : auLôrJr* -"J, dtwhere e, : a 6(L *d/ r fr)v , and rrl, is a selected basefrequency.The amortisseur oltagesvqr and vdr arezeroexceptfor special ases.

The flux linkages inFigure2,They arc,

) " r r = L l " r i u " + t r g " "7"0,= L,ur ior* Ld",Lq, = Ltqr iqr*X*,1) "o r : L to r in r *L^d)"1,= L117iy,+Xâ

K. ShinoharaKagoshima niversifyKagkoshimaapan

Eqs. 1-5 are defined by referring to

(8)(e)

(10)( 1 1 )

( 1 )

(2)

/ 1 \

(4)

(1 )


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The electromechanical orque produced by the machine is1 :"m q'"mdx :"qcsNdc t -

(r2)( 3 )(14)

(1s)

(16)

( 1 8 )

(1 )

(20)

(21)

(22)

L*q( iq r+ iq r )LâUar+ 6r+ i1r)Ltrr iqr+L^qL l r r id r+ *d

typically xpresseds 1]/?\ / p\,, : li)l)(L a, i , ),n,ia, )

1 . .

\AAz-

d-axis equivalent c i rcui t

.q-axis6quivalnt ircuit

Figure I d-q axisequivalentcircuit ofa salientpole synch-ronousmachine where the stator eakage nductancehasbeenseparated nto saturable core) and non-saturable endwinding) portions.

d-axisequivalent ircuit

"q T'ft axis eauivalenl circtjit

Figure 2 Flux linkagesused as statevariables.

(r7)where T6o4 is the load torque (negative if the "load"correspondso aprimemover orque)and /is combined nertiaof machineand oad.[. SIMULATION F A SYNCHRoNOUSacuINE UsTNG I-ux

LrNracgs AS VARTABLESSince he differentialequationsof the machines,Eqs. (1)-

(5), contain mixed variables either flux linkages or currentscould be eliminated from the differential equations by meansof the algebraic relations, Eqs. (7)-(15X15). Modelling ofsaturation is best achieved by retaining flux linkages as thestate variables. The current can be solved in terms of the fluxlinkagesas,

Finally, since he machine s generally ied to an externalload/prime mover, in its simplest form the equation whichdescribes coupling between the electrical and mechanicalsystemcan be written as

T T - ( 2 \ d 0 ,' e- ' Ioad \ i l " dt

x _ 1, _ "qs "qcst q t - -" I e s

, * Ld r - xd^'ds - ---T-" le s

' q r

'd r

x -x"q r "mqL,n,xdr-\^d

L td,L. f r -L*a

LurEliminating ô and Xô fromEqs. l8f{20) usingEq. 1a)and 15)an substitutinghb esult ntoEqs. 20H22),

t -.lr

t . r .- r , - / c s ^ " / s r' - q r I ' " q s t ' " q c s- u les - l esI Tn u / c s ^ u l , t ." d r t ' " d s 1 ' " d c s" l es " l es

L , ^ . L , ^7 l na ' - - i . -Ndrsl e s - l e s

Ll e sotl'd.

'd r

(23)

(24)

(2s)

+Lo,IL L/6s /cs/-x-\ r)(\+"+"+ln" xoo \.,l t lr t l


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Lts : Ltcs+ tesThe internalstatorcore lux linkages ,0* and 7"0^now be written in terms of the total stator and rotbrlinkages s

^ -L , r r (L t r r , Lâ* ) . , L , r r (Lô* ) ^ ,L r r (L ro* \ ,"0, ':4\ i,- t^ )"a'- 2^l t ,a,1"a, '11 )"t,

(26) uq.Ican ds

flux

l * l * l * lLrd Lr, Lrd, Lt,f,

l l l- T - T -L^, Lt, L,n,These resultscan be inserted nto the differential equationsdescribed y the circuit of Figure l. Upon solving or the time

vds],qs

Figure 3 Signal flow diagram or a salientpole synchronousmachine.

III. MoDELINGoF SATURATIoNIn many cases he saturationof the stator eeth for field poledominate in which case saturation can be taken into account

accurately by expressingthe air gap flux linkage as a non-linear function of the air gap MMF. While the air gap MMF isdifficult to determine under a loaded condition, the requiredrelationship can be establishedof the motor is operatedunderan open circuit in which case he MMF is clearly proportionalonly to field current since the stator current is, in this case,zero. If the open circuit voltage is plotted versus the fieldcurent the saturation urve of Figure 4(a) canbe established.The slope of a line drawn from the origin to a point on thestraight line (unsaturated)portion of the curve is equal to thestator d-axis mutual reactance.If the abscissaof Figure 4(a) is

derivative terms and integrating, he result is,lT r^ ILn, : l l un" TL()".""- I . ") - a,)"r . ldtJ L u l . r - o ' " < t J ' " ' J

ra,= [ua,*#,r ,,,*].dr).+,xn,fat G2)'n, fl"r,.f,1*,'n,,-(^n,.(?),'r)ll.33),0, II,o,.l#,â,,-()'0,.C)^;l\.34),o I#^{",. ,!fol*, ,,(r,, *), l}.

(35)_ (P\ r t ) -' , : [ ;Jl l) J(r,-rtoad)dt (36)

The flow of signals for simulation of a salient polesynchronousmachine s shown in Figure 3.

(3 )


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multiplied by the d-axis mutual reactanceand the ordinate by| / a , , the normalized curve of Figure 4(b) results n which theabscissa emainsproportional to MMF while having the unitsof webers.The slopeof the unsaturatedportion of the curve isnow clearly unity. The difference between the saturatedandunsaturatedvalues of air gap flux linkage can be defined asL?'.or as llustrated on the figure.

Figure 4 (a) Open circuit saturationcurve and (b) derivedcurve.

In general, the same approach can be used to model thesaturationofthe stator core. In this case he core flux linkage isnoted as 1,"" and the deviation from the unsaturatedvalue as41,.,. The value of Al,r, can now be plottedas a function ofthe unsaturated alue of statorcore flux linkages \cs(unsar\.Since saturation does not result in a phase shif in tliefundamental component of flux linkages and only decreasesthe amplitude,both the d- andq- componentsof saturated oreflux should be decreased y the samevalue. Thus,

(41)(42)

When combined with Eq. (12), the q-axis portion of theunsaturated alue offlux linkase is

L. . , - . . : L df f rs\ n t *Lt" I ) , , Iqcs\unsat)Lcr L\ \ / "7 t * ' 4r ' Crq )"n '1, Lo, , Alo"r l-7 - -= - l

" lq, "rq JSimilarly

xd,, u,, y=, l((:f) ** *, * tt,+,. t:,*) r ,

*P"y"^*)ld r "1.f, umd )where, for convenience, he following quantitieshavedefined,

xq, = Llr r iqs+ xqcs(sar l= Lt"r iqr+ Xqcs(unsar)-)"qcs

- cq

(43)

(44)been

(45)| , L t r , | , | , L t * IL rn L lr rL*q L t^ L tr rL q,I ,ltuc d

J - a L t r , | * I * L l r , 1 * L , r u tLra LtrrL*d Lt^ LlrrLld, LtrrLu,(46)

Saturationf the ieldpolecanbe mplementedn theusualfashion.Specif,rcally,he equation or field flux linkages,Eq.(11),with hehelpof Eq. 13),bemodified o the orm

( 37 ) L '4 i7 r : Lra+ Lt1,i,LLd",: $4e!1i.."'ucs(unsat)tAtr4cs 'fcs\urc!!nX*' " c s (unsa t )

Lcs(unsa t ) : f f i: -f(Lrrlunro4)

representshesaturationurve.Saturationftheq-axiscannowbe ncorporatedfEqs.(7)and 9) aremodified o the orm,

L*d I LLr,-;-#-( ids+ id) (41)- "lfr ' "mdWhen the freld pole is saturatedextra field current must be

(3g) added o account br saturation. q. @7)becomes(L aiJ,),t=4j"f,

+L(L ai ) - rft-fi 4,+ 4,)

md ' "lfr "lfr ' Lmd

(40)

(48)The proper value of L(Lmdif) is typically discerned rom theopencircuit saturation urv6, Fig. 4(a). Signal low mode lsofthe two saturationeffects are mplemented n Figure 5.

IV. FTNITE r-sr\4eNr NALysrsWhen saturation of the magnetic circuit of a synchronousmachine is lumped in terms of a single saturation curve, theinformation needto determine he deeree of saturation can be


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r;itu',r1111.;tu,ro,,[T I .)j-mz-i L rl::;L!:iel,:al

Figure5 Flow diagramor simulation f (a)stator oresaturation nd b) fieldpolesaturation.discernedrom heopencircuit saturation urveasdescribednSection II. However,since he two saturation ffectsunderconsiderationxist n series or the no load est, he ndividualsaturation urvescan not be determinedwithoutan arbitraryassumption oncerninghe split of the MMF dropsacrosshetwo iron portions.A preferable lternatives to employ initeelementanalysis,assuming hat the basicgeometryof themachines known.Figure6 shows he FEM modelused overi$, hesolution.Thesaturationurves anbe determined ysystematically irst setting the stator iron permeability oinfinisl and hen setting he rotor ron permeabilityo infinity.Theresultsof the calculations reshownn Figure7. The wosaturation urvesdeduced rom FigureFigure 7 is shown nFigure .

V. Marr-ae/Strvrur-rNxR-esurrsThe simulation low diagramsare cadily mplementednMatlab/Simulinkor similar computerprograms.Figure 9shows racesor a line startusing he flow diagrams f Figs.3

Figure6 Dimensions f onepoleof synchronous achinestudied sing inite elements.

, 0.6g; 0 4, u- l:. 0.2

I, 0 00.4

l_rnsat[Wb]Figure 7 Saturationcurvesproducedby finite elements.

0,20.1

F 0.12F 0.08

0 0.2 0.4 0.6 0.8l_wsatWb]

Figure 8 Core and Polesaturationcurves.and 5. In, general he time for acceleration ncreasesby about20o/ocompared o an unsaturated model and is less dampedwhen he machinepulls into synchronism.

VL CoNcr-usioNThis paper has presented a method for modelling both

stator core and field pole saturation. Simulation studiesindicate that the overall behavior is less damped than withsimple field pole saturation.Use of this new model should

0,8

ld rot

0.6

0.040

Figure.6 EM analysisesult


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500E 250i o -zso-500o 500! zso3 g 1 0timelsl

Figure 9 Torqueand Speedduring acceleration rom rest.prove particularly beneficial in motor drive applications inwhich large torque demands(q-axis stator current) are madeon the machine resulting in heavy stator core saturation evenwhen accompanied by a relatively modest value of fieldcunent.

REFERENCESD.W. Novotny and T.A. Lipo, "Dynamics and Vector Control of ACDrives", (book) Oxford Press,1996.

2007_28 Salient Pole Synch Machine Simulation

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