数字芯片相关课件：lecture note2

Digitaltheory,digitalcomponentsandtheirmodelingOutlineReviewofStransformShannon’ssamplingtheoremZtransformDigitalPWManditsmodelingADCanditsmodelingWhyStransformH(t)isanimpulseresponseofasystem.(alinearsystem)Howtocalculatetheoutputy(t)intime-domain?Convolution:ConvolutionRecallhowtodoconvolutionConvolutionissue1.Complicated2.Needintegration3.IntegrationrangeRegardingconvolution,youareencouragedtogoovertheconcept.RelationshipbetweenLaplaceandconvolutiontheconvolutionofsignalsfandg,denotedh=f*g,isthesignalintermsofLaplacetransforms:H(s)=F(s)XG(s)LaplacetransformturnsconvolutionintomultiplicationRelationship(Cont’)let'sshowthatL(f*g)=F(s)G(s):changeorderofintegration:ThebenefitsoflaplaceConvertsfromtimedomaintoS-domainTurnsconvolutionintomultiplicationSimplifiescalculationMostimportant:ifdenotes=jw,turnsS-domaintofrequencydomain,whichisagreatfavorablefeatureinengineeringanalysis.LapaceandFourierFouriertransformaimsforperiodicalsignals.Integrationrangefrom(-∞,∞).(hassomestrongconstraints||f(x)||limitGenerally,Laplacetransformisforfunctionsthataresemi-infiniteorpiecewisecontinuous.f(t)=0;t<0(Constraint)IfweknowtheLaplacetransform,thentogettheFouriertransform,weevaluatetheLaplacetransformfors=jω,thatisalongtheimaginaryaxisinthes-plane.TheFourierTransformisusedtodescribesignals,suchasforcesandvibrations.TheLaplaceTransformisusedtodescribesystems,asthepole-residueformulationisverypracticaltocharacterizethepropertiesofthesystem.referencesFormoreinformationaboutLaplaceandFourier,referthesignalsandsystems-aclassictextbookbyOppenheim

Whyneedsampling?InputsignalsandoutputsignalsfromOPAMPareallanalogous.WhyneedsamplingReasons:SignalprocessedbyComputerY(k)=0.2*X(k)+0.8*X(k-1)Unlikeanalogcircuit,whichcanhandlethesignalscontinuously,thecomputer,executingcodesinsequence,takestimetohandlesignals.Soitcanonlyhandlethesignalsindiscretetime.

ResultSamplingisusuallydonetoconvertananalogsignaltoadigitalsignal,sowecanusedigitaltechniquestodosomeprocessingonit.ThisiswhatisdoneforrecordingonCDsanditisdonetoyourvoicefortransmissionoverradioonyourcellPhoneTypicalcomputercontrolsystemSamplinganditsrelatedissuesRegardingthesampling,threeissueshavetobedealtwithintheory:(giveanexample,sine+squarewaveforms)Cansampledsignals(discretesignals)representstheoriginalsignals,inotherword,haveanyinformationlostaftersampling.Ifyes,haveanythesamplingrequirementsputonthesamplingororiginalsignals.Ifalltherequirementsaremet,howtoreconstructthesignalsformthesampleddata.Answertoquestion1WecanuseFouriertransformtoidentifywhethertheoriginalsignalshavesamespectrumornot.Formoredetailedderivation,plsrefertoanyrelatedtextbookAnswertoquestion1(Cont’)Theresulttellsus:Badthing:thespectrumisnotidenticalGoodthing:whenk=0,theshapeofthespectrumissame,andtheamplitudeisonly1/Toftheoriginalone.FilterIfωmax>

1/2ωswhatwillhappen?AliasingissueThisphenomenonwecalledaliasing.Question:howtoavoidtheproblem?(1)limitspectrumoftheoriginalsignal,usinglow-passfilter(anti-aliasingfilter)(2)increasethesamplingfrequency(ωs)SpectrumOverlapAnswertoquestion2Iftheoriginalanalogsignalcouldbereconstructedcompletelyfromthesamplingsignal,thesamplingfrequencyfsmustbesatisfied(Shannonsamplingtheorem):Butinpracticalapplication,thesamplingfrequencyusuallyfs>10fmaxAnswertoquestion3Howtoreconstruct?Sincespectrumofsampledsignalconsistsofbasebandspectrumandspectralimagesshiftedatmultiplesof2π/T,reconstructionmeanstokeepthebasebandimageandgetridoftheotherimagesConcept:low-passfiltertopassbasebandwhileremovingimagesXS(j)N-NS-S2S-2SIdeallowpassfilterAnswertoquestion3(cont’)UsingsteptoapproximatetheoriginalsignalThemathematicexpressionofxh(t):zero-orderholdThetransferfunctionofthezero-orderholdcanbeobtainedfromtheimpulseresponse:Thefrequencyresponseofthezero-orderholdDAChasthisfeature.Thismethodisverysimpleandpractical.x*(t)x(t)xh(t)Ttg(t)SummaryonSamplingWhywehavetostudysamplingWhatisthemaindifferencebetweenanalogsignalanddiscretesignalHowcanwereconstructoriginalsignalfromthediscretizedsignal.Nextquestion:haveanyanalysistooltoanalyzethedigitalsystem.Z-TransformExpressionofthesampledsignalUsingtheLaplacetransform:DefineWehavetheZ-transform:Indeed,theZ-transformistotallysameastheS-transform.ThepurposeofZtransformistomakeexpressionsimple.Toavoidtranscendentfunction.Z-Transform(cont’)HowtoconvertatransferfunctioninS-domaintotheoneinZ-domain?Fromthedefinition(impulseinvariant)UsingaS->ZtablePartial-fractionexpansionapproachResiduesapproachUsingapproximation(onlyforcontroller)

ImpulseinvariantmethodThemainideaisthat:theimpulseresponseofanalogsystemissameasthatofitscorrespondingdiscretesystem.

ForexampleWhoseimpulseresponseisItiseasytoget:Partial-fractionexpansionapproachForexampleResiduesapproachesApproximationapproachesWidelyusedinpracticalengineeringfieldsAsweknow,inS-domain1/Smeansintegrationintimedomain.

y(k)=blueshadedarea+redshadedareay(k)=y(k-1)+blueshadedareay(k)=y(k-1)+Ts/2[u(k)+u(k-1)]Bilinear/trapezoidtransformApproach2-BackwarddifferenceOnlyuseu(k)toapproximateApproach3Onlyuseu(k-1)toapproximateThesethreeapproachesareonlyformappingcontrollersorfiltersinSdomainintoZ-domainduetotheirapproximationQuestionF(s)=(s-a)/(s+a)what’sthefunctionofthistransferfunction.IfweuseimpulseinvariantmethodtogetcorrespondingF(z),Anexample:aVoltageSourceInverterHalf-bridgeInverterAnexample(Cont’)Itiseasytodescribetheplantbyusingdifferentialequations:BasedonthismodelandusingLaplacetransformation,thetransferfunctionbetweentheinvertervoltageVOCandtheoutputcurrentIO,GIOVOCcanbefoundtobe:HowaboutthemodelofPWMThedifferencebetweenDigitalsystemandPWMmodulationConventionaldigitalsystemPWMsystemTsisfixedZOHisanapproximatingmethodTistime-variantApproximatingmethodisnotZOHUptodate,thereisnoverytheoreticalmethodtoderivethePWMmodel!AccuratePWMmodelisabasisforthepowerelectronicsconversion.MyownunderstandingonPWMmodelingtransferfunctionThepurposeofthePWMmodelingistofindtherelationshipbetweeninputm(s)andoutputy(s)UsinglargesignalapproachDiscretizingmodulationsignalOutputsignalCont’ApplyingStransform,wecanobtainAtlowfrequencys=jw,w<1/3Ts,thedelayunitcanbeapproximatedUsingfirstorderby:(Theoutputisnon-linearsystem)SubstituteFromtheviewofstatistics,Dchangesfrom0to1Scheme-leadingedgePWMScheme3-symetricPWMModelingusingsmallsignalapproach Small-signalmodelingisacommonanalysistechniqueinPowerelectronicswhichisusedtoapproximatethebehaviorofnonlinearcircuitwithlinearequations.ThislinearizationisformedabouttheDC

biaspointofthecircuit(thatis,thevoltage/currentlevelspresentwhennosmallsignalisapplied),andcanbeaccurateforsmallexcursionsaboutthispoint.Stepstoderivethesmallsignalmodel:1.Describethebehaviorofthecircuitusingnonlineardifferentialequation;2.ChooseDCbiaspoint;3.Applyinputdisturbancetothesystem4.Usefirstordertoapproximatethesystem.PWMmodeling-smallsignalapproach

PWMModulator:AnalogimplementationNaturallysampledimplementationofaPWMmodulator.PWMdynamicresponse(seethederivation!)ModelingondifferentmodulationschemesDPWMmodulator-dynamicresponseDPWM:trailingedgeimplementationEquivalentstructureDPWMmodulator-dynamicresponseDigitalPWM:smallsignalanalysisAnalogPWM:smallsignalanalysisDPWMmodulator-dynamicresponse(Cont’)UnityareaDiracImpulseperturbationsCorrectionpulsesDiracimpulseapproximationofcorrectionpulsesDPWMmodulator-dynamicresponse(Cont’)Theinputperturbationscanbe,inparticular,unityareaDiracimpulsesappliedatthemodulatorinput.Consideringoneoftheseimpulsestobeappliedattimezero,wecanimmediatelyfindthat,intheaboveapproximation,itgeneratesatimetranslatedimpulseattheoutput:whoseareaisequaltothemodulatorsmallsignalgain(i.e.theinverseofthesaw-toothslope).DPWMmodulator-dynamicresponse(Cont’)AnygenericdiscretetimesampledsignalcanbeexpressedasasumofweightedDiracpulses,suchas:therefore,itisnowpossibletoexpresstheLaplacetransformofthegenericmodulatoroutputasafunctionofthesampledinputsignal’sone.Sinceanyinputpulseistranslatedintoatimeshifted,scaledarea,correctionimpulsewecanwrite:WecannowcomputetheLaplacetransformofbothsidesoftheaboveexpression,exploitingtherulefortimetranslationandthebasicpropertyoftheDiracpulsetohaveaunityLaplacetransform.DPWMmodulator-dynamicresponse(Cont’)Consequently,wefindthefollowingrelation:Wherewhich,bytheway,happenstobetheequivalenttotheZ-transformofthesequence.ItisnowpossibletorelatetheLaplacetransformofthesampleddatasequence,MS(s),withtheoriginalsignal’sone,M(s).Wecanwrite:DPWMmodulator-dynamicresponse(Cont’)Ifweassume,asusual,thattheinputsignalspectrumislimitedinbandwidthbelowtheNyquistfrequency,andifweneglecttheoutputsignalfrequencycontentabovethesamefrequency,thenwecansay:And,consequently,thatrepresentsthetransferfunctionbetweenthemodulatorinputandoutputsignals.Asimilarprocedurecanbeappliedtoother,morecomplex,modulatororganizations.DPWMmodulator-dynamicresponse(Cont’)DigitalPWM:leadingedgeimplementationDPWMmodulator-dynamicresponse(Cont’)DigitalPWM:symmetricpulseimplementationDPWMmodulator-dynamicresponse(Cont’)Thetransferfunctionswejustfoundcorrespondtoanoninstantaneousbehaviorofthedigitalmodulator.Ascanbeseenbycomputingarg(PWM(jw))therewillalwaysbeaphaseshiftbetweentheinputandoutputsignal,whoseentityis,ingeneral,afunctionofthesteadystateduty-cyclevalue.Forexample,inthecaseofthesingleupdate,trailingedgeimplementationwecanfind:Similarly,forthesymmetricpulseimplementationwefind:whichisaremarkableresult,asitdoesnotdependontheparticularsteady-statevalueoftheduty-cycle,D.PWMupdateControldelay:Ts/2DigitalPWM:multi-sampledimplementationDigitalPWM:multi-sampledimplementationCommentsonmulti-sampledimplementationTheequivalentdelayisequaltotheonefoundfortheconventionaltrailingedgeimplementation,reducedbytheso-calledmulti-samplingeffect.Itisinterestingtoobservethat,asNtendstoinfinity,theequivalentdelaytendstozero,whichisconsistentwithacontinuoustime,naturallysampledimplementationofthemodulator,wherethesampleandholdeffectisnotpresent.Multi-samplingpresentssomelimitationsaswell,namely: -needforproperfilteringoftheswitchingnoise; -needfornonconventionalhardware;ControlDelayWherethecontroldelaycomesfrom?Digitalimplementation1.Sensing2.ADCconversion3.Operationalamplifier4.Calculation5.PWMupdateSensingdelayADCconversionTypicalprocessinginDSPTd≈TsHaveanywaytoreducethecomputationdelaytime?Method1Ref:DongshengYang,etalAReal-TimeComputationMethodwithDualSamplingModestoImprovetheCurrentControlPerformancesoftheLCL-TypeGrid-ConnectedInverter,IEEEtransactionsonIndustrialElectronics(tobepress)杨水涛等DSP-BasedMultiple-LoopControlStrategyforUPSInvertersWithEffectiveControlDelayElimination,电工技术学报2008Method2MultisamplingMethod3MultisamplingmultiPWMupdateMethod4:dualupdateComputationdelaytime:AnexampleonDualupdatedPWM20kVA400HzAPFPrincipleoftheAPFPWMupdateschemeDualupdateschemeAnexample(cont’s)2022/10/21Probleminmultisamplingmultiupdateissue2022/10/21Experimentalresults图3窄脉宽现象通道2：三角载波通道3：驱动波形通道4：驱动波形窄脉宽2022/10/21TheReasonThereasontogeneratenarrowPWM2022/10/21Limitslope图5窄脉宽消除原理2022/10/21ExperimentalResults图6正常的PWM波输出通道2：三角载波通道3：驱动波形通道4：驱动波形Example-2Experiments10usdelay20usdelay50usdelayADCmodelingQuantizationConversiontimeQuantizationErrorε(recap)SignaltoNoiseRatioSNR:SignaltoNoiseRatioRatioofsignalpowertonoisepower

Ifthevarianceofthesignalandnoiseareknown,SNRcandefineas:QuantizationNoiseQuantizationerrorεwithprobabilitydensityp(ε)canbeapproximatedasuniformdistribution

QuantizationNoisecont’dQuantizationnoisereducesSignal-Noise-Ration(SNR)ofADCEstimationofSNRwithRootMeanSquare(RMS)ofinputsignal(Vin_RMS)andofnoisesignal(Vqn_RMS)

EstimationofthemeansquareofinputsignalVin_RMS:SNRcalculationADCresolution:Fora12-bitADC,theSNR=78.26dB,whichisaccurateenoughforrealengineeringapplications.QuestionInactivepowerfilterapplications,theharmoniccurrentsshouldbeaccuratelyextractedfirstfromtheloadorsource.Theaccuracyoftheharmonicssensingisakeyinthesystemcompensationperformance.Ifwewantasensingresolutionreachingupto0.3Afora300AAPF,howmanybitsADCshouldwechoose?ADCconversiontimeUsuallyADCconversiontimeisveryfastinpowerelectronicsapplicationsincomparisonwiththeswitchingfrequency.inTI28xxDSPs,conversiontimeforeachchannelis80ns,whichisneglectable.Butonethingshouldbementioned:S/HCxRVinCx》CChapt3:DesignmethodologiesofdigitalcontrollerInstructorHaibingHuhuhaibing@Lab:A4-417OutlineModelingonSinglephaseDC/ACinverterAdesignexampleRoots-LocusMethodFrequencydomaindesignDirectDigitaldesignStatespacedesign

ModelingonSinglephaseDC/ACinverterPlantWithRincrease,rootsoftheplantwillbemuchclosertoimaginaryaxe.Therefore,forsingle-phaseinverterdesign,wedesignthecontrollerunderno-loadconditionDualloopcontrol-AnalogimplementationAssumingvoltagechangeismuchslowerthancurrentloop.Inthiscase,voltagecanbeconsideredasadisturbance.Orwecanuseoutputvoltagefeed-forwardtodecouplethesystem.DigitalcontrollerIfsamplingfs>>fsw,(fsw,controlbandwidth),thereisnodifferencebetweenanaloganddigitalsystems.DelayintroducedindigitalsystemAccordingtopreviousanalysis,thecomputationdelayisinevitablefordigitalcontroller.BasedonthedifferentPWMupdatescheme,thedelaytimecanbeTs/2

or

Ts.Thisistheside-effectintroducedbydigitalsystem.Ifsamplingfs>>fsw,fromthispointofview,wecandesigncontrollerinanalogdomainbyaddingthedelayunit,andthentransfertoZdomain.DetailedDC/ACmodelCurrentloopdesignDeterminecross-overfrequency(1/5-1/10fs)Determinephasegain(>30)Example-SinglephaseinverterSpecificationsL=4.3mHr=0.01oHmC=2.2uFUdc=400VSwitchingfrequency=20kHzPWM：DualupdateCurrentcontrollerDesignPIcontrollerCross-overfrequency2kHzPhasemargin45degreeComputationdelay:Td=25usTransferFunctionOpenlooptransferfunction:Basedonthesetwoequations,thecontrollerparameterscanbeachievedas：NextstepistodosimulationusingtheseparametersCurrentloopSimulationpuredelayunitgridDothesimulationSimulationindiscretedomainUnitdelayDiscretePIVoltageloopdesignTransferfunction1Cross-overfrequency1kHzPhasemargin40degreeSimulationverificationHowtomapfromsimulationtoDSPUnitfeedbackParameterscalculatedbasedonunitfeedback10bit/12bitADCDigitalPWMimplementationCont’AssumingADC12bit(4096)ADCinputrange3.3VDSPfrequency150MHzVoltageconditioningcircuit:[-400V,400]->[03.3V]Currentconditioningcircuit:[-40A,40A]->[03.3V]Cont’Forcurrent40A->2048feedbackcoef=2048/40Forvoltage400V->2048feedbackcoef=2048/400

NewPIparametersCont’NewPIparametersNewreferenceOriginalPWMDigitalPWMinDSPMatchPIparametersinDSPSimulationConsideringADCquantizationandnoiseSeethesimulationOutlineModelingonSinglephaseDC/ACinverterRoots-LocusMethodFrequencydomaindesignDirectDigitaldesignStatespacedesign

IntroduceofRoots-LocusMethodThedesignbytheroot-locusmethodisbasedonreshapingtherootlocusofthesystembyaddingpolesandzerostothesystem’sopen-looptransferfunctionandforcingtherootlocitopassthroughdesiredclosed-looppolesinthesplane.Thecharacteristicoftheroot-locusdesignisbasedontheassumptionthattheclosed-loopsystemhasapairofdominantclosed-looppoles.(Zerosandadditionalpolesaffecttheresponsecharacteristic).First,let’sreviewtheroots-locusinsplant.Assumethatthecontrolblockofasystemisasbelow:Gc(s)meansthecompensator,G(s)meansthecontrolobject,H(s)meansthefeedbackloop.Letdrawtheroots-locusnext.IntroduceofRoots-LocusMethodTheopentransferfunctionofsystemwithoutcompensatorisIfthetransferfunctioniswrittenasbelow:Usingmatlabcaneasilydrawtherootslocusjustlikewhatshowedintherightside.Therootslocusisdrawnbytheopentransferfunction,butitshowstherootsofcloseloopsystemwhosetrackisvaryingwiththeopenloopgainK.Thedirectionofarrowshowedinthepictureindicatesthedirectionofrootslocus.HeretheMAXKis1230.IntroduceofRoots-LocusMethodWehavementionedthatroots-locusmethodisawaytoaddpolesorzeros(orpolesandzeros)tothesystem,buthowwillthepolesandzerosaffectthesystem?Again,usetheaforementionedexampleandaddpolesorzerostothesystem,seewhatchangeitwillhappen.Ifapoleisadded.HeretheMAXKis658.AddedPoleIfazeroisadded.HeretheKis5790.AddedZeroIntroduceofRoots-LocusMethodFromtheexample,wecanseethataddingpolesorzeroswillhavedifferenteffectonthesystem.Ageneralconclusionisshowedbelow:EffectsoftheAdditionofPoles.Theadditionofapoletotheopen-looptransferfunctionhastheeffectofpullingtherootlocustotheright,tendingtolowerthesystem’srelativestabilityandtoslowdownthesettlingoftheresponse.EffectsoftheAdditionofZeros.Theadditionofazerototheopen-looptransferfunctionhastheeffectofpullingtherootlocustotheleft,tendingtomakethesystemmorestableandtospeedupthesettlingoftheresponse.So,ifitisneeded,wecanaddpolesorzerostothesystemwhichiscalledcompensatorintherightsideblock.Commonlyusedcompensators(orcontrollers)arelead,lag,andlag-leadcompensators,whichwillbeintroducedindetailslater.DesignProcedureFromtheroots-locus,wecanseetherootsofcloseloopsystemisvaryingwiththeopenloopgainK.So,settingthegainisthefirststepinadjustingthesystemforsatisfactoryperformance.Usually,settingthegaincannotfullysatisfythedemand,thus,secondstep,wecanaddcompensatorslikelead,lag,orlead-lag.Thepoleandzeroplacementisshowedasbelow:Compensator’stransferfunctionis:Lagnetworka>b(b)Leadnetworka<bAlsocanbewrittenas:whereLag-leadnetworkcanbeachievedbycombiningthesetwoer.DesignProcedureParametersList2kW2mH30uF25Ω380V220V(RMS),50Hz10kHz0.20.01TopologyofsingleinverterAgain,weuseanexampletointroducethedesignmethodwhichisshowedintherightside.Still,weneedtodesigntwoloops(currentandvoltageloop).Themainstepsarewrittenasbelow:First,fromtheperformancespecifications,determinethedesiredlocationforthedominantclosed-looppoles.Second,drawingtheroot-locusplotoftheuncompensatedsystem,ascertainwhetherornotthegainadjustmentalonecanyieldthedesiredclosed-looppoles.(ifnot,goingtothethirdstep.)Third,choosingacompensatoraccordingtothepracticalsystem,whichwillbeintroducedindetails.Forth,determinetheopen-loopgainofthecompensatedsystem.DesignProcedureNext,wewillreviewthecharacteristicsofroot-locuswhichwillbeusedinstepthree.Asshowedintherightpicture,ifs1isintheroot-locus,itshouldhavethecharacteristicsasbelow.HereKmeanstheopen-loopgainAccordingtothesecharacteristics,thestepthreecanbeintroducedindetail.Stepthree:1)Afterdeterminethedominantpoles,calculatetheangledefinedasϕ.2)choosingthepropercompensatoraccordingtotheanglecharacteristics.3)accordingtheangleneededtocompensate,determinethepoleandzeroofthecompensator.LeadcompensatorLagcompensatorAnExampleFirst,designofthecurrentloop.Theroot-locusofcurrentopen-loopcanbeachievedusingmatlab,andfortheroot-locus,thegainadjustmentcanmeetpracticaldemandsasshowedintherightside.Inthiscondition,thepolesandgainareasbelow:LeadorLagcompensatorcanalsobeadoptedifonewanttomakethecurrentloopperformancebetter.CurrentLoopVoltageLoopAfterdesignthecurrentloop,wecometodesignthevoltageloop.Tomakethesystemhavebetterperformanceandlessstaticerror,acompensatorwillbeadopted.First,drawtheroot-locusofvoltageloop.Seethedetailsoftheredcircuit.Stophere,wemustchooseproperdominantclose-polesforthesystem.Usually,thedesiredlocationisdeterminedfromtheperformancespecification.Here,tomakethesystembetterresponseandlessstaticerror,choosedominantpolesasbelow:DesignProcedureofCompensatorCalculatetheangleϕ.Theopen-loop’spolesandzerosarelistedinrightside:PolesZeros0-666670-8000080000-33285±2551i80000Tomaketheangleisequalto(2k+1)π,thustheanglecontributedbycompensatedisnegative50°.(-50°)So,lagcompensatorisadopted.DesignProcedureofCompensatorNext,determinethepoleandzeroofcompensator.Accordingtothecertainangle,onceoneofthetwoisdetermined,theotherwillbedetermined.Here,wedeterminepoletobe-300,thenzerois-4100.Laststep,calculateopen-loopgaintodetermineKc.CalculatetheequationintherightsidetogetKc.Here,Kcis25.Tothisend,wehavegotthecompensatorasbelow:Someadditionalremarks:Usually,thecompensatordesignedusingroots-locuswillensuregoodstability,butitmaynotensurelessstaticerror.Ifaparticularstaticerrorconstantisspecified,itisgenerallytousethefrequency-responseapproach.Here,alsodrawtheBodeplotoftheopen-loop.SimulationResultsHeregivessomesimulationresultsgotbymatlab\simulink.Simulationwaves.(a)isthesteadywaveofnon-load(b)isthewavefromfullloadtonon-load(Redlinevrefmeansthevoltagereferencewhileblacklinevomeanstheoutputvoltage)(a)(b)Sofar,thewholedesignprocedureofroot-locushasbeendone.Fromthesimulationresults,wecanseethesystemhasperfectstabilityanddynamicresponse.Butifwedonotgetproperparameters,weneedchoosethezeroandpoleagain.OutlineModelingonSinglephaseDC/ACinverterRoots-LocusMethodFrequencydomaindesignDirectDigitaldesignStatespacedesign

DesignmethodologiesofdigitalPIcontrollerPIcontrollerismostwidelyusedcontrollerinPowerElectronics,itsdesignhasbeenacommonissue.HerewillshowthedesignmethodsusingtheBodeplot.

Oftenithastwomainwaysaslistedbelow:1)DesignthePIcontrollerinAnalogdomainusingBodeplot;2)TransfertheAnalogcoefficientstoDiscretizingdomainwithacertainmethod.DesignoftheAnalogPIControllerusingtheBodeplotisaessentialprocedureinthewholemethod,properdesigncanmakethesystemmorestableaswellasbetterperformance(fastresponseandsmallstaticerror).Discretizingmethodalsohaseffectonthesystem,sochoosingaproperdiscretizingmethodisimportant.DigitalControlDiagramTypicalorganizationofasingle-phaseinverter(StronglineisPowercircuitwhilelightlineiscontrolloops)Tointroducethedesignmethod,anexampleischosenasbackground.Therightsidediagramisasingle-phaseinverterincludingthemainpowercircuitandthedigitalcontrolloops.Itcanbeseenfromthediagram,twocontrollooparebeenusedtogetbetterperformance.ItiscalledMulti-loopcontrolorganization,theinnerloopisoftenacurrentloop,theexternalloopisoftenavoltageloop,sowehavetodesignbothcontrollersforthetwoloops.AnExampleParametersList2kW2mH30uF25Ω380V220V(RMS),50Hz10kHz0.20.01Therightpictureisthemaincircuitofsinglephaseinverter.Here,wechooseasevereconditiontodesignthecontrollerparameterswheretheloadresistanceislargeenough,thismeansnon-loadcondition.Themainparametersofthecircuitislistedintherighttable.Inpracticaldesign,someESRwillbeconsideredincludingtheinductance’sESRandthecapacitor’sESR.DesignProcedureTypicalorganizationofasingle-phaseinverter(StronglineisPowercircuitwhilelightlineiscontrolloops)Sincebothloopneedtobedesigned,weshoulddesignstepbystep.Usually,Theresponsespeedofcurrentloopisfastthanvoltageloop,soitisreasonabletodealtheoutputvoltageasadisturbancewhenmodelingthecurrentloop.Inthiscase,thefirststepismodelingthecurrentloopanddesigncurrentcontroller.Currentloopofasingle-phaseinverter(StronglineisPowercircuitwhilelightlineiscontrolloops)DesignofinnerCurrentloop

Fromtheformerdiagram,thecurrentcontrolloopcanbeyielded.Currentcontrolloopofasingle-phaseinverterCurrentloopblockdiagramcontrolblockinsdomainTherightsideiswholetransferfunctionwithPIcontroller.Theopentransferfunctionisasfollow.

DesignProcedureofPIParametersThewaytodesignPIparametersinfrequencydomainisasfollow:1)DrawtheBodeplotofopenl