逆变器的直流母线电压稳定方法－IAST－2014（减小直流侧电容容值的方法）

404IEEETRANSACTIONSONINDUSTRYAPPLICATIONS,VOL.50,NO.1,JANUARYFEBRUARY2014DCLinkVoltagStabilizationforRducdDCLinkCapacitorInrtrWookJinL,Mmbr,IEEE,andSungKiSul,Fllow,IEEEAbstract—Inconntionalmotordrisystmsusingpulswidthmodulation(PWM)inrtrs,larglctrolyticcapacitorsarusdforstabilizationofthdclinkoltag.Sincthlctrolyticcapacitorsarbulkyandrducrliabilityofthsystmdutoshortliftim,thrhabnmanyffortstoliminatorrducthlctrolyticcapacitorsinthmotordrisystm.Howr,thPWMinrtrwithrducddclinkcapacitorhasaproblmthatthdclinkoltagislssstablcompardtothconntionalinrtrbcausthcapabilityofstoringnrgyisalsorducd.Inthispapr,adclinkoltagstabilizationalgorithmusinganactidampingisproposdsothatthdclinkoltagcanbstabilizdwithrducddclinkcapacitor.Toachiloadsourcindpndntstabilization,asourcstatstimatorwhichstimatsbothsourcoltagandcurrntisalsoproposd.Thfluctuationofthdclinkoltagdutoastploadchangcanbalsosupprssdundrthtolrancrangusingthstimatdsourccurrnt.Thffctinssofthproposdmthodsisaluatdbyxprimntalrsults.IndxTrms—Actidamping,constantpowrload(CPL),dclinkcapacitor,dclinkoltagstabilization,lctrolyticcapacitor,pulswidthmodulation(PWM)inrtrs.I.INTRODUCTIONTHEDCLINKcapacitorsinthpulswidthmodulation(PWM)inrtractasannrgybuffrtostabilizthdclinkoltagandtokpitalmostconstant.Thus,lctrolyticcapacitorswithlargcapacitancprunitolumhabncommonlyusdasthdclinkcapacitors.Howr,thlctrolyticcapacitorsinthdclinkarbulkyandmakthinrtrlssrliablbcausofthirshortliftimxpctancy.Moror,thlargdclinkcapacitanccaussthlargrtotalharmonicdistortionofthinputsourccurrntwhnathrphaspassirctifirsuchasathrphasdiodrctifirisusdasafrontnd.Asarsult,thrhabnffortstorplacthlctrolyticcapacitorswithasmallfilmcapacitor[1]–[9].Howr,thdclinkoltagofthinrtrwithrducddclinkcapacitancmaybunstabl,rsultinginoroltagorManuscriptrcidOctobr16,2012;risdFbruary9,2013;accptdMarch19,2013.DatofpublicationJun18,2013;datofcurrntrsionJanuary16,2014.Papr2012IDC487.R1,prsntdatth2009IndustryApplicationsSocityAnnualMting,Houston,TX,USA,Octobr4–8,andapprodforpublicationinthIEEETRANSACTIONSONINDUSTRYAPPLICATIONSbythIndustrialDrisCommittofthIEEEIndustryApplicationsSocity.W.J.LiswithSamsungElctronics,Suwon442600,Kora(mail:lfins1@snu.ac.kr).S.K.SuliswithSoulNationalUnirsity,Soul151742,Kora(mail:sulsk@plaza.snu.ac.kr)Colorrsionsofonormorofthfigursinthispapraraailablonlinathttp:ixplor.i.org.DigitalObjctIdntifir10.1109TIA.2013.2268733undroltagfaultsifthsourcinductancislargcompardtothdclinkcapacitanc.Thisissubcomssignificantlysriousinthcasthatthinrtrissupplyingpowrtoconstantpowrloads(CPLs).ItiswllknownfromlitraturthatthCPLmayladthinstabilityofthsystmdutoitssocalld“ngatiimpdanc”charactristic[10]–[24].Whnthdclinkoltagisincrasing,thloadcurrntisgttingsmallrsoastomaintainconstantpowrtothload.Thisintractionbtwnthdclinksidandthloadsidcanrsultinoroltagorundroltagatthdclink.Thaformntiondinstabilityproblmcanbsoldbydsigningthinputfiltrwhosparamtrssatisfythstabilitycritrionprsntdin[10],[13],and[23].In[10]and[13],thinstabilitydutothCPLswasanalyzdusinglinarizationofthCPLs,whrasthinputadmittanchasbnanalyticallyxprssdin[23]forprdictionofdclinkinstabilitis.Howr,thrquirdfiltrinductanctosmoothnthsourccurrntmayblargrthanthaluthatmtsthstabilitycritrion.Ifthparamtrsofthinputfiltrcannotmtthcritrionforthisrason,addingsomdampingrsistorsatthfiltrinductorscanbanaltrnatisolutiontoincrasthffctidampingofthsystm.In[24],ariouspassidampingcircuits,whicharadddtothfiltrlmnts,arprsntd.Howr,thdampingrsistorsincrascostandphysicalsizofthsystm.Thus,thrhabnmanyrsarchstodalwiththisproblmwithoutaddingthdampingrsistorsnotonlyinthmotordriapplicationsbutalsointhdc–dcconrtrapplications[20]–[22].In[10]–[19],thfluctuatingdclinkoltagisactilystabilizdbychangingthdynamicimpdancsnfromthdclinktobpositi.Changingimpdancofthloadcanbachidbymodifyingthtorquproducingcurrntcommandsofthmotorbcausthoutputpowrisdirctlyrlatdtothtorqu[10]–[15].Howr,ifthfrquncyofthdclinkoltagfluctuationisbyondthcurrntcontrolbandwidth,itisnotasytochangthdynamicimpdanccorrctlybcausthrmaybarlatilylargtimlagbtwnthmotorcurrntcommandsandthralmotorcurrnts.Toorcomthisproblm,oltagcommands,instadofcurrntcommands,canbdirctlyprturbdinthcasthatthfrquncyofthdclinkoltagfluctuationisbyondthcurrntcontrolbandwidth.TholtagprturbationtchniqushabnwidlyusdforVfcontrolldprmanntmagntmotordris.SincthprmanntmagntmotorswithoutdamprwindingsdonotassurthsynchronizationofthmotorundropnloopVfcontrol,itmaybxpctdtoxhibitinstabilityofthrotormotion.In[26]and[27],thstabilityofthdriwasimprodbyincorporatingastabilizingcontrol00939994©2013IEEE.Prsonalusisprmittd,butrpublicationrdistributionrquirsIEEEprmission.Shttp:www.i.orgpublicationsstandardspublicationsrightsindx.htmlformorinformation.LEEANDSUL:DCLINKVOLTAGESTABILIZATIONFORREDUCEDDCLINKCAPACITORINVERTER405whichutilizsdclinkcurrntfdbackormotorcurrntfdback.Similarly,thinstabilityproblmsdutothrducdclinkcapacitorcanbmitigatdbymodifyingthoutputoltagcommands[16]–[18].In[16],itisassumdthatthinputsourcoltagisaconstantdcinordrtomployalinarizdcontrollaw.Howr,dutothisassumption,thffctinssofthactidampingisassurdonlywhrthsourcoltagisinariantbcausthchangdimpdancisdirctlyrlatdtothdclinkoltag.Inthdampingmthodin[18],thdclinkcapacitorcurrntismasurdandusdinmakingadampingsignal,whichisinjctdtotholtagrfrnc.Dutothlowpassfiltrusdtoattnuathighfrquncycomponntsinthdclinkoltag,itisnothardtosupprsshighfrquncyrsonancinthdclinkoltag.In[17],thhighpassfiltrddclinkoltagsignalisusdinmakingthdampingoltags;howr,thinfluncofthhighpassfiltrwasnotanalyzdsufficintly.Thhighpassorbandpassfiltr,whichistypicallymploydtogtthrsonanccomponntofthdclinkoltagin[10]–[18],cannotcompltlyattnuatthsourcoltagfluctuationrsultingfromthrctificationofththrphasgridspciallyifthrsonancfrquncyisclostothsixthharmonicfrquncyofthgrid.Thus,itishardforthosstabilizationcontrollrstoslctilysupprssthrsonanc.Inadditiontothaformntiondissu,thrapidloadchangscanalsorsultinthdclinkoroltagorundroltag[25].Ifthdiodrctifirfrontndisusdinthmotordri,thsystmcannotmanagthrgnratinrgyfromthmotor.Hnc,thrgnratingoprationofthmotorshouldbprohibitd.Inordrtoprntthmotorfromgnratingpowrbacktothdclink,thmotortorqurfrncisalwayskptpositiorqualtozro.Enthoughthrgnratdnrgyfromthmchanicalpowrcanbblockdbylimitingthmotortorqurfrnc,thnrgystordinboththmotorinductancandthfiltrinductanccanbdumpdintothdclinkcapacitor,rsultinginoroltagatthdclink.Forinstanc,thrwillbaoltagspikatthdclinkwhnthoutputpowrdcrassstpwissincthnrgystordinbothinductancsmostothdclinkrapidly.Onthothrhand,ifthoutputpowrincrassrapidly,thrwillbaoltagdipatthdclinksincthnrgystordinthcapacitorisnotnoughtomaintainthdclinkoltag.Thisphnomnonisbcausthcapabilityofstoringnrgyofthdclinkcapacitorissowakthatthnrgystordinthfiltrinductancandthmotorinductanccannotbmanagd.Thsimplstapproachtoprntthistransintdclinkoltagfluctuationisslowingdownthrsponsofthinrtr(orrducingthcontrolbandwidth).Inthatcas,howr,itisinitablthatorallprformancwouldbdgraddbcausofthrducdcontrolbandwidth.Inthispapr,anoldclinkoltagstabilizingmthodisproposd.Thstabilizationisbasicallybasdonthactidampingwhichactilychangsthdynamicimpdancofthsystmbydirctlymodifyingthinrtroutputoltag.Inordrtomakthstabilizationnotaffctdbythsourcoltagariation,thstimatdsourcoltagandthdclinkoltagarutilizdinthproposdcontrollaw.Tostimatthsourcstats(sourcoltagandsourccurrnt),aclosdloopstimatortoprdictthsourcstatsisalsoproposd.Fig.1.SimplifidmodlofthmotordrisystmmployingaPWMinrtr.Furthrmor,asimpldclinkoltagrgulationmthodtosupprssthtransintdclinkoltagfluctuationdutothrapidloadchangisalsoproposd.Inthproposdmthod,thinrtroutputoltagisdirctlymodifidusingboththprdictddclinkoltagandthsourccurrntfromthstimator.Usingthproposdmthods,thdclinkoltagcanbkptstablwithoutanyfaultfromboththinstabilitydutothCPLandthdclinkoltagfluctuationdutothrapidloadchang.Thispaprisorganizdinthfollowingmannr.SctionIIprsntsamodlofthconntionalmotordrisystmanddscribsthinstabilityproblmdutothCPL.InSctionIII,thalgorithmsusdtostabilizthdclinkoltagarproposd.ExprimntalrsultsarprsntdinSctionIVtorifythproposdalgorithms.II.INSTABILITYDUETOREDUCEDDCLINKCAPACITORAsimplifidmodlofthmotordrisystmmployingaPWMinrtrisshowninFig.1.Thinputsidsourccanbmodldasaoltagsourcandanquialntsrisimpdancnthoughthpowrsourciscomposdofathrphasdiodrctifirandathrphasgrid,bcausonlytwoofthsixdiodsofthrctifirconductandthothrsarblockdatanytim.Inthatcas,tholtagsourccanbrgarddasaquasidcsourcpractically.Thn,thPWMinrtrandthmotorloadarmodldasacurrntsourc(iin),whichrprsntsthmancurrntflowingfromthdclinktothinrtrduringaPWMpriod.LsandRsarthquialntsrisimpdancofthinputpowrsourcincludingthinputfiltr,andCdcisthdclinkcapacitanc.ThstatquationsofthsystmshowninFig.1ardscribdbyLsdisdt=s−Rsis−dc(1)Cdcddcdt=is−iin.(2)IfthmotorloadpullsaconstantpowrofPL,thinrtrcurrntiincanbxprssdasiin=PLdc=PLdc0+˜dc(3)406IEEETRANSACTIONSONINDUSTRYAPPLICATIONS,VOL.50,NO.1,JANUARYFEBRUARY2014whrdc0isamanaluofthdclinkoltagand˜dcisadclinkoltagdiation.If˜dcisrlatilysmallcompardtoitsmanalu,thinrtrcurrntcanblinarizdasiin=PLdc0+˜dc∼=PLdc0−PL2dc0˜dc.(4)From(4),itcanbshownthatthCPLactsasangatiimpdanc.Withthlinarizdinrtrcurrntin(4),thstatquationsof(1)and(2)hathfollowingcharactristicquation:s2+(RsLs−PLCdc2dc0)s+(2dc0−RsPLLsCdc2dc0)=0.(5)From(5),thstabilitycritrionforthdclinkcapacitorcanbobtaindasCdc>LsPLRs2dc0.(6)ItcanbnotdthatthgnratingsystmsarinhrntlystablnthoughthdclinkcapacitorisxtrmlyrducdsincPLisngatiingnratingapplications.Inmotoringapplications,howr,ifthcapacitancofthinrtrdosnotsatisfythcritrionin(6),thstatsofthsystmincludingthdclinkoltagandthsourccurrntarunstabloroscillating[10],[13].Ifthlctrolyticcapacitoratthdclinkisrplacdwithafilmcapacitorwithsmallcapacitanc,itishardlypossiblforthsystmtomtthiscritrion.Asarsult,thsmiconductorswitchssuchasIGBTsmaybdstroydfromoroltag.Inordrtoprntthissituation,aspcialalgorithmtostabilizthdclinkoltagisrquirdifthdclinkcapacitancisnotsufficintlylarg.Thproposdstabilizationmthodusdtosolthisproblmwillbxplaindinthnxtsction.III.PROPOSEDSTABILIZATIONMETHODThdclinkinstabilityproblmpriouslydiscussdcanbsoldbymanipulatingthinrtroutputcurrnt(iin)accordingtothdclinkoltagfluctuation.Onofthmansofmanipulatingthinrtroutputcurrntistomodifythtorquproducingcurrntcommandaccordingtothdclinkoltagdiationsincthtorquofthmotorisproportionaltothoutputpowr.Howr,ifthoscillationfrquncy,whichisrlatdtothsystmparamtrsofthinputinductancandthdclinkcapacitanc,istoohighforthcurrntcontrollrtognratthmanipulatingcurrnt,thinstabilityofthsystmdosnotdisapparnbymanipulatingthtorquproducingcurrntcommand.Gnrally,thrsonancfrquncyofthsystmwithrducddclinkcapacitorisquithighrthanthcurrntcontrolbandwidthbcausthfrquncyisininrsproportiontothsquarrootofthdclinkcapacitanc.Thrfor,manipulatingtholtagcommandofthinrtrismorffctithanmanipulatingthtorquproducingcurrntcommandforsupprssingthhighfrquncyoscillation.Fig.2.(a)Virtualrsistor(Rdamp)tostabiliztholtagofthdclinkcapacitor.(b)Implmntationofthirtualrsistancbyaddingthdampingcurrnt(idamp).A.DCLinkVoltagStabilizationMthodforCPLThbasicprinciplofthproposdstabilizationisthatthngatiimpdancofthCPLcanbnutralizdbyaddingsompositiimpdanc(ordamping).AsshowninFig.2(a),airtualdampingrsistorisadddbtwntholtagsourcandthdclinkcapacitorinordrtoincrasthdampingofthsystm.Sincthirtualdampingrsistorisnotaphysicalon,thlosssowingtothdampingargnratdnotatthirtualdampingrsistorbutatthmotorandthinrtrbcausthraldampingcurrntflowsthroughthinrtrandthmotor.Thimplmntationofthactidampingffctusingthirtualdampingrsistorcanbachidbyinjctingadditionalinrtrcurrnt(idamp)asshowninFig.2(b),whichisproportionaltothdiffrncbtwnthdclinkoltagandthstimatdsourcoltagasidamp=(dc−ˆs)Rdamp(7)whrˆsisthstimatdsourcoltag.ThstimationofthsourcoltagwillbxplaindinSctionIIIB.Bcausthtotalcurrntssnfromthdclinkcapacitorlookxactlythsamasifaphysicalrsistorisconnctdtothdclink,thoscillationofthdclinkoltagcanbffctilydampd.AsRdampdcrass,thdclinkoltagistightlyrgulatdtothsourcoltag.Inothrwords,thoscillationofthdclinkoltagissupprssdmorandmorbyslctingasmallrdampingrsistanc.Howr,alargrdampingcurrntrquirsalargrinrtroltag,soitisrcommnddtostRdamptobsmallnoughjusttoprntthinstabilityofthsystm.ThmaximumaluofRdampwhichcanstabilizthsystmcanbcalculatdfromthmodifidstatquationswhrthdampingcurrntisadddtothstatquationin(2).Byrplacingiinin(2)withiin+idamp,thmodifidstatquationbcomsCdcddcdt=is−iin−idamp=is−iin−(dc−ˆs)Rdamp.(8)LEEANDSUL:DCLINKVOLTAGESTABILIZATIONFORREDUCEDDCLINKCAPACITORINVERTER407In(8),thdclinkoltagdiation˜dc=dc−dc0,andthmanaluofthdclinkoltagdc0∼=ˆsnglctingthsourcrsistanc,whichistypicallyquitsmall.Thn,usingthlinarizdinrtrcurrntiinin(4),thmodifidstatquation(8)yildsCdcddcdt=is−PLdc0+PL2dc0˜dc−˜dcRdamp.(9)Using(1)and(9),anwcharactristicquationofthstabilizdsystmcanbcalculatdass2+(RsLs−PLCdc2dc0+1CdcRdamp)s+(Rdamp2dc0−RdampRsPL+Rs2dc0LsCdcRdampdc0)=0.(10)Fromthcharactristicquationin(10),astabilityconditionisdridasfollows:1Rdamp>PL2dc0−RsCdcLs.(11)Itcanbconcluddfrom(11)thatthmaximumdampingrsistanc(orminimumdampingadmittanc)bcomslargrasthloadpowrincrass.Thatmansthatthdclinkoltagshouldbmortightlyrgulatdwithasmallrdampingrsistancasthloadpowrincrass.Inaddition,itcanbalsoconcluddthatnodampingrsistancisrquirdifthrightsidof(11)isngatisincRdampcanbinfinitlylarginsuchcass.Thatisxactlythsamconclusionthatthsystmwhichmtsthcritriain(6)isinhrntlystabl.Aftrslctinganadquatdampingrsistancforthstabilization,thnxtimplmntationstpistoralizthdampingcurrnt(idamp)obtaindfrom(7).Injctionofthadditionaldampingcurrnt(idamp)tothloadmachincanbachidbyaddinganadditionaloltagctor�damptothoriginaloltagcommandctor�∗dqfromthcurrntcontrollrforthmotorcontrol.Thdampingoltagctor�dampwhichproducsthdampingcurrnt(idamp)canbdirctlycalculatdfromthfollowingquationsincthoutputpowrofthinrtrisqualtothinnrproductofthinrtroltagctorandthmotorcurrntctor:idamp=32�damp·�idqdc(12)whr�idqisthmotorcurrntctorinthd−qrotatingrfrncfram.Thrarinfinitsolutionsof�dampin(12)bcausthquationisnotascalarquationbuta2Dctorquation.Thsolutionslionthlinwhichisrticaltothcurrntctor�idq,asshowninFig.3(a).Thbstsolutionisthctorthathasthlastmagnitudbcaus�dampistobadddtothcurrntcontrollroutputanditisbasicallyaoltagdisturbanctothcurrntcontrollr.Thus,thdampingoltagctor�damponthaxiswhrthloadcurrntctorlisbcomsthoptimalsolution.Bcausthsolutionctorlisonthloadcurrntctor,itisconninttointroducanwrfrncframFig.3.(a)Sraloltagctors(blu)thatproducthsamdampingcurrnt(idamp)withthmotorcurrntctor(rd).(b)Spacctordiagramoftholtagoutputofthinrtrwhichiscomposdofthoriginaloltagctor(grn),thdampingoltagctor(blu),andthfinaloltagoutputctor(black).calldloadcurrntrfrncframwhrallloadcurrntslionthdaxisofthfram.Thn,inthloadcurrntrfrncfram,thctorquation(12)canbtransformdtoascalarquationasidamp=23dcidampiload=23dc(dc−ˆs)iloadRdamp(13)whriload=|�idq|isthmagnitudofthloadcurrntandthsuprscriptimansthloadcurrntrfrncfram.Aftrthadditionaldampingoltagtoinjctiscalculatdfrom(13),itisadddtothoriginaloltagcommandfromthcurrntcontrollrasshowninFig.3(b).ThblockdiagramofthproposdstabilizationalgorithmisdpictdinFig.4.InFig.4,thrartwotransformationblocksusdtochangthrfrncframoftholtagcommandctors.Thfirstonrotatsthrfrncframfromthsynchronousrfrncframtothcurrntrfrncfram.Sincthdampingoltagiscalculatdinthcurrntrfrncframandithasonlythdaxiscomponntofthcurrntrfrncfram,itisadddtothdaxisoltagcommandfromthcurrntcontrollrinthcurrntrfrncfram.ThscondframtransformationfinallyrotatsthrfrncframfromthcurrntrfrncframtothstationaryrfrncframforthPWMofthinrtr.408IEEETRANSACTIONSONINDUSTRYAPPLICATIONS,VOL.50,NO.1,JANUARYFEBRUARY2014Fig.4.Rfrncframtransformationincorporatdwiththproposddclinkstabilizationalgorithm.Fig.5.Simulationrsultsofdclinkoltagstabilization:dclinkoltag,motorcurrnts,andsourccurrnts(fromtoptobottom).TABLEIPARAMETERSOFTHEDRIVESYSTEMFig.5showsthsimulationrsultsofthproposdstabilizationalgorithm.Thsimulationwasprformdwithaprmanntmagntmotordrisystm,whosparamtrsarlistdinTablI.Boththdclinkoltagandthsourccurrntsfluctuatsignificantlywhilthstabilizationisdisabld(righthalfsidofthwaform).Withthproposdstabilization(lfthalfsidofthwaform),boththdclinkoltagandthsourccurrntsarwllstabilizd,althoughthmotorcurrntsaralittldistortddutothadddoltag�dampforstabilization.Howr,thmotorcurrntsarnmordistortdwhilthstabilizationisdisabldbcausofsignificantlyfluctuatingdclinkoltag.Fig.6.(a)Simplifidquialntmodlforconstructingastatstimator.(b)Blockdiagramofthproposdstatstimator.B.SourcStatEstimatorToachithactidampingffctasshowninFig.2(a),itisrquirdtoobtainthdclinkoltagdiationdutothrsonanc.Ifthsourcoltagispurdc,itispossibltoobtainthdclinkoltagdiationbymployingahighpassfiltrorabandpassfiltratthdclinkoltag.Howr,ifthsourcoltagisfromathrphasrctifir,itisnotasytodistinguishthdclinkoltagdiationfromthdclinkoltagcontainingthsourcoltagharmonicsrsultingfromthrctification,sincthrsonancfrquncyisoftnclostothsixthharmonicfrquncyofthgrid.Moror,bcausthtrminaloltagofthsourcincludstholtagdropsdutointrnalimpdanc,itisdifficulttomasurthxactsourcoltagitslf.Thus,asourcstatstimatorisprsntdinordrtostimatthsourcoltag.Thpurposofthisstatstimatoristoobtainboththsourcoltagandsourccurrnt,whicharncssaryinstabilizingthsystm.Fromthisstatstimator,thdclinkoltagdiationdutothsystminstability,whichisrquirdtocalculatthdampingcurrntin(7),isdirctlyobtaindbysubtractingthstimatdsourcoltagfromthdclinkoltag.InthsystmmodldpictdinFig.1,thstats(oltagandcurrnt)ofthsourcsidcanbstimatdbcaustharagdinrtroutputcurrnt(iin)isaknowncontrolariablandthsystmiscompltlyobsrabl.TharagdinrtrcurrntoronPWMpriod(iin)canbdirctlycalculatdfromthinrtroltagoutput(�dq)andthload(machin)currnt(�idq)asiin=32�dq·�idqdc.(14)Nglctingthsourcrsistancforsimplicity,thsystmmodlisillustratdinFig.6(a),andthstatspacsystmquationcanbdridas⎡⎣˙dc˙si˙s⎤⎦=⎡⎣001Cdc000−1Ls1Ls0⎤⎦⎡⎣dcsis⎤⎦+⎡⎣−1Cdc00⎤⎦iin.(15)LEEANDSUL:DCLINKVOLTAGESTABILIZATIONFORREDUCEDDCLINKCAPACITORINVERTER409Usingthsystmmodlandthsystminput(iin),itispossibltobuildaLunbrgrtypstimatorasshowninFig.6(b).Thstimatorquationisginby⎡⎣˙ˆdc˙ˆs˙ˆis⎤⎦=⎡⎣001Cdc000−1Ls1Ls0⎤⎦⎡⎣ˆdcˆsiˆs⎤⎦+⎡⎣−1Cdc00⎤⎦iin+L⎡⎣dc−[100]⎡⎣ˆdcˆsiˆs⎤⎦⎤⎦(16)whrL=⎡⎣L1L2L3⎤⎦isthgainmatrixofthstimator.ThcharactristicquationofthstimatorwiththgainmatrixLiss3+L1s2−1Cdc(L3+1Ls)s+L2CdcLs=0.(17)Thsourcstatstimatorshouldbmuchfastrthanthsourcoltagfluctuationinordrtostimatthsourcoltagwithlastrror.Thus,thbandwidthofthstimatorshouldbmuchhighrthansixtimsthgridfrquncy(g),sincthsourcoltagsnfromthdclinkhasalotofsixthharmonics.Forxampl,allthrpolsartoblocatdons=−bw(bw�6·g);thn,thcharactristicquationisqualto(s+bw)3=0.(18)Sinc(17)and(18)shouldbidntical,thgainmatrixbcomsL=⎡⎣3bwLsCdc3bw3Cdc2bw−1⎤⎦.(19)Inthstatstimator,onlythdclinkoltagandload(motor)currntsndtobmasurd,andbotharprrquisitforconntionalmotordrisystms.Hnc,thsourcoltag(s)andcurrnt(is)canbstimatdwithoutanyadditionalmasurmntfromthproposdstimator.ThdiscrttimprdictionstimatordridfromthcontinuoustimrsionisgininthAppndix.C.DCLinkVoltagStabilizationMthodforRapidLoadChangFromthsourcstatstimator,thsourccurrntis,whichcannotaryrapidlyduringonPWMpriodbcausofrlatilylargsourcinductanc,canbstimatd.Also,thinrtroutputcurrntiinduringthnxtPWMpriodisaknowncontrolariablfrom(14)assumingthloadcurrnt�idq[k]∼=�idq[k+1].Hnc,itispossibltoprdictthdclinkcapacitorcurrntusingthstimatdsourccurrntandthinrtroutputcurrntduringthnxtPWMpriod.Withthiscapacitorcurrnt,itisalsopossibltoprdictthdclinkFig.7.(a)Allowablinrtroltagband(shadd)toaoiddclinkorundroltagandamotorcurrntctor(rd).(b)CasI:tholtagcommandctorisonthband.(c)CasII(oroltag):thdaxisoltagcommandissmallrthanthlowrlimit.(d)CasIII(undroltag):thdaxisoltagcommandxcdsthupprlimit.oltagariationifthcapacitancofthdclinkcapacitorisknown.Thus,thdclinkoltagariationduringthnxtPWMpriodcanbdscribdasΔdc[k+1]=TCˆdc(iˆs[k+1]−iin[k+1])(20)whrΔdcisthdclinkoltagariationaftroncontrolpriodTandCˆdcisthstimatddclinkcapacitanc.Bcausthfilmcapacitorhasquittighttolrancofcapacitanclssthan20