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®IiiiiiiInstantaneoustorquecontrolPhilipC.KjaerABBCorporateResearch,V~ster~s,SwedenandJeremyJ.GribbleDERA,Pyestock,UK1iiiililiillii~ii~t~!!~ii~~ii~iii~!Ii~i~i~iiii~!i~i~!i~i~iii~!~i~i~iii~i~i~i~iii~iiii~i~!ii~i~i~i~!~i~I~~i~@~iii~!®Ii~i~iiiI~I~i~i~i~iii~!~i~iii~!i~!~i~iiThischapterdescribesthedevelopmentofanadvancedcontrolschemethatpermitshigh-gradecontroloftheswitchedreluctancemotor.'High-grade'inthiscontextmeansperformancethatwillenabletheswitchedreluctancemachinetobeconsid-eredforapplicationscurrentlyemployingvectorcontrolleda.c.drives,brushlessd.c.,orevend.c.commutatormotordrives.Thespecificfeaturesofhigh-gradecontrolinclude:(i)high-bandwidthtorquecontrol;(ii)lowtorqueripple;(iii)four-quadrantoperationoverawidespeedrangeincludingzerospeed;and(iv)seamlesstransitionfrommaximum-efficiencyoperationtomaximumfield-weakeningwithefficientuseofthecontrollervoltage.Thiscontrolstrategyisregardedasthefunctionalequivalentofa.c.vectorcontrol.Themotormodelisbasedontheassumptionsthat(i)thereisnomutualmagneticcouplingbetweenphases;(ii)iron(andother)lossesandphaseresistancearenegligible;and(iii)allelectromagneticparameterscanbecalculatedanalyticallyornumerically.Itisassumedthatthem-phasemachineisfedfromaclassicasymmetric,losslessconverterwith2mswitches.Theangulardisplacementinelectricaldegreesbetweenneighbouringphasesj-1andjisdefinedas:360°Opitch~NrOj=Oj_1--Opitch.(6.1)1Theworkinthischapterwascompletedin1997atGlasgowUniversity.Electroniccontrolofswitchedreluctancemachines99Bysuperposition,thetotaltorqueisexpressedasthesumofthemindividualphasetorques:mmT't°tal--ZT,j--Zr,(ij,Oj).(6.2)j=lj=lToachievethehigh-bandwidthtorquecontrolneededindemandingapplicationsrequiringgoodtransientperformance,itisnecessarytoemployinstantaneoustorquecontrol,andtoreplacethenotionofplaincommutationwithacontrolschemethatcarefullyprofilesthephasecurrentstoproducethedesiredtotalmotortorquebyco-ordinatingthetorquesproducedbyindividualphases.6.1.1FormulationoftorquecontrolproblemIftheindividualphasecurrentscanbecontrolledwithahighbandwidth,thentheproblemofchoosingtheappropriatecurrentwaveformstoensurelowtorquetipplebecomesthekeyissue.Putanotherway,theproductionoftorquebyindividualphasesmustbeco-ordinatedsothatthetotaltorquetracksthereferencevaluegeneratedbythepositionorspeedcontrolloops.Unfortunatelythereisnoinherentuniquedistributionoftorquebetweenphases,sothetorque-sharingfunctionsandtheassociatedcurrentorfluxwaveformsmustbedesigned.Fixingthetorquewaveformtosomeadhocfunctionisnotsatisfactorybecausetheassociatedcurrentandflux-linkagewaveformsdirectlyaffecttheohmiclossesandthefeedingvoltagerequired(Ilic-Spongetal.,1987a).Toselectanoptimaltorque-sharingfunctionwhoseprimaryobjectiveislowtippletorquecontrol,itisnecessarytospecifyasecondaryobjective.iiiiiiiN~%i~i''''~~~'~iiiiN%NNNNNNN#N%NNNNNN®iiNi®iiiNiNiiiiNiiiiiiiiiNi#iNN®iiNi~N~iNNiiiiThefirstreportedattempttoreducethetorquetipplewasthefour-phaseswitchedreluctanceservodrivepresentedbyByrneetal.(1985).Themachinewasdesignedtogiveasinusoidaltorque-per-phaseforaconstantcurrentlevel.Whenexcitedwithsinu-soidalhalf-wavecurrentstwooverlappingphasescouldproduceconstanttotaltorque,withtorqueripplereportedtobeaslowas5%.ItwasrecognizedbyIlic-Spongetal.(1987a)thatthetotaltorqueingeneralcouldbekeptripple-freebysharingitbetweenindividualphases,andtheterm'torquesharingfunctions'(TSF)wasintroduced.Theswitchedreluctancemachinediffersfromtheconventionalthree-phasea.c.machineinthatthetotaltorquemaybesharedmoreorlessarbitrarilybetweenphases,astheyarelargelyindependent.Thenonlineartorqueproblemhasreceivedspecialinterestfromcontrolengineers,forexampleIlic-Spongetal.(1987b),wherethefeedbacklinearizationcontroltech-niquewasappliedto'linearize'theswitchedreluctancemotor.Unfortunately,commu-tationwasneglected.Althoughthenonlinearityisnotofadynamicnature,foraperiodtheswitchedreluctancemotorwasapopular'problem'forcontrolengineers,andworksfollowingIlic-Spongetal.(1987b)canbefoundinCailleuxetal.(1993),PandaandDash(1996),RossiandTonielli(1994),andAmoretal.(1993a,b;1995).However,100Instantaneoustorquecontrolthesenonlinearcontroltechniquesrequireanalyticalelectromagneticmodels,whereasthereferencesciteduserathersimpleandinaccuratemodels.Therehavebeenmanyattemptstofixthetorque-per-phasewaveformstosomeadhocfunction(Ilic-Spongetal.,1987a;HusainandEhasani,1994a;Hung,1993;andGoldenbergetal.,1994).ExamplesareSchrammetal.(1992),Reayetal.(1993a,b),wherethetorque-per-phasewaschosentobetrapezoidal,andtheproportionsofthetrapeziawereoptimizedtoreducethepeakphasecurrent.However,thereislittlejustificationforatrapezoidalphasetorqueprofile.Furthermore,thistorquewaveformwillimposeunachievablerequirementsonthepowerelectronicconverter,asisclearfromthefollowingexample,whichassumesanonsaturablemotor:drOrdi0(1dL)didLdidO=OidO=05-2dOi2=.i.--(6.3)dOdOdOAtzerocurrentanonzerodr/d0willrequireinfinitedi/dO.Mostoftheproposedtorque-sharingfunctionsmeettheprimaryobjectiveoflowtorqueripple,butdonotrecognizetheneedforas