电力电子系统建模及控制1-DC-DC变换器的动态模型

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20.High-performanceMaterials受到电力电子技术的深刻影响电力电子装置的分类（有功）电源：直流开关电源、逆变电源、不停电电源（UPS）、直流输电装置等无功电源：静止无功补偿装置（SVC）、静止无功发生装置（SVG）、有源电力滤波器、动态电压恢复装置（DVR）等传动装置：直流调速装置、各类电机的变频调速装置等。电力电子装置的应用范围十分广泛，粗略可分为：（有功）电源无功电源传动装置电力电子技术发展两条主线半导体功率器件的发展

从结型控制器件（晶闸管、功率GTR、GTO）到场控器件（MOSFET、IGBT、IGCT）的发展历程高频化、低功耗、场控化成为功率器件发展主要特征功率器件发展历程也是向理想电子开关逐步逼近的过程功率变换电路的发展：电路拓扑结构逐渐走向稳定器件和电路的日趋成熟，注意力转向电力电子装置的整体性能的优化问题，电力电子系统的问题比以往更加受到重视。电力电子系统问题控制系统动态性能、稳态的性能并联系统的控制问题：均流、输出纹波抑制技术组合或接连系统的分析与控制功率管理控制（效率的提高技术）热分析和设计电力电子装置对电能质量影响和改进电磁兼容分析和设计SOC,SOP功率集成系统级功率集成：硬件电路的标准化＋网络控制技术电力电子装置的技术指标电力电子装置需要满足静态指标和动态指标要求。

如直流开关电源指标：电源调整率、负载调整率、输出电压的精度、纹波、动态性能、变换效率、功率密度、并联模块的不均流度、功率因数和EMC。

影响技术指标因素：功率电路设计，系统控制的设计

功率电路设计：电路拓扑、磁设计、热设计、功率元件驱动功率电路设计影响的指标：变换效率、功率密度、纹波等技术

系统控制的设计影响的指标：电源调整率、负载调整率、输出电压的精度、动态性能、并联模块的不均流度等指标功率电路设计与系统控制的设计就如汽车的左、右轮，同等重要

通讯基础电源的系统为了达到所需的静态和动态指标，一般需要引入反馈控制。电力电子系统中包含功率开关器件或二极管等非线性元件，是非线性系统AC/DCDC/DCObjective:maintainv(t)equaltoanaccurate,andconstantvalueV.Therearedisturbancesdueto•inputvg(t)•loadRThereareuncertaintiesincomponentvaluesDC/DCconvertersystem

ObjectiveofmodelingtheconverterDesignacompensatorfortheconvertersystemtohavegoodcharacteristicsinbothstaticanddynamicsStabilityanalysisBuildingasimulationmodeltoevaluateaconvertersystemHowtomodeltheconverterorinverteristhegoalofthiscourse?ModelingofCCMDC/DCconverterModelingofDCMDC/DCconverterModelingofDC/DCconverterwithcurrentpeakcontrolCompensationdesignforDC/DCconvertersystemModelingofsinglephaseinverterModelingofThreephaseinverterorconverterContentsChapter1ModelingofCCMDC/DCconverterDeveloptoolsformodelingofCCMDC/DCconvertersystemsHowdoACvariationsinvg(t),R,ord(t)affecttheoutputvoltagev(t)?Whatarethesmall-signaltransferfunctionsoftheconverter?PurposeofthischapterCapturedominantbehaviorandignoreinsignificantphenomenaSimplifiedmodelyieldsphysicalinsight,allowingengineertodesignsystemtooperateinspecifiedmannerModelingDefinition:describingdynamicphysicalbehaviorbymathematicalmeansFormodeling,simplificationsareusedduetofollowingreasons:DC/DCconvertersystem(staticstate)ttSwitchfrequencyDandDm

areconstants,|Dm

|<<Dfrequency

ismuchsmallerthantheconverterswitchingfrequencyDC/DCconvertersystem(dynamicstate)SupposeinterferenceisintroducedasfollowsttContainsfrequencycomponentsat:•Modulationfrequencyanditsharmonics•Switchingfrequencyanditsharmonics•Sidebandsofswitchingfrequency|Dm|<<DThenharmonicsofmodulationfrequencyisverysmallSwitchingharmonicsandsidebandscomponentsarerelativelysmall.OutputvoltagespectrumwithsinusoidalmodulationofdutycycleInput-outputrelationshipDC/DCconverterInput-outputrelationshipconsideringACcomponentsDC/DCconverterForACcomponents,ithasthepropertyofthelinearsystemApproach:•RemoveswitchingharmonicsbyaveragingthewaveformoveroneswitchingperiodApproachofconvertermodelingPredicthowlow-frequencyvariationsindutycycleinducelowfrequencyvariationsintheconvertervoltagesandcurrents•Ignoretheswitchingripple•IgnorecomplicatedswitchingharmonicsandsidebandsAveragingtoremoveswitchingrippleAverageoveroneswitchingperiodtoremoveswitchingrippletExplanationDefinetThenh(t)h(t)isalowpassfilterFiltercharacteristicsh(t)tAmplitudefreq.characteristicsH(t)islowpassfilterwithbandwidthlessthanfS1Functionofaveragingoperator1Inductorequation:DividebyLandintegrateoveroneswitchingperiod:Left-handsideisthechangeininductorcurrent.Right-handsidecanbeexpressedasaverageinductorvoltagemultipliedbyTs/Lasfollows:ThenetchangeininductorcurrentoveroneswitchingperiodisequaltotheperiodTsmultipliedbytheaverageslopeEulerformulaHenceWithsimilarderivationwecangetEquationsforinductorandcapacitorAfteraveragingoperationtheinductanceequationandcapacitanceequationstillexistCCMBuck-boostconverterexampleStage1,(0,DTs)Stage2,(DTs,Ts)Stage1:Switchinposition1Inductorvoltageandcapacitorcurrentare:Stage2:Switchinposition2Inductorvoltageandcapacitorcurrentare:AveragetheinductorvoltagegAveragecapacitorcurrentAccordingtoInsertAveragingstateequationBuck-boostinputcurrentwaveformisAveragevalue:AveraginginputcurrentConverteraverageequationsDescribelow-frequencydynamicsoftheconverter•Ignoretheswitchingripple•IgnorecomplicatedswitchingharmonicsandsidebandsLowfrequency?1PerturbationandlinearizationConverteraveragedequations:Nonlinearbecauseofmultiplicationofthecontrolvariabled(t)withstatevariablessuchasi(t)andv(t).Constructsmall-signalmodel:LinearizeaboutquiescentoperatingpointIftheconverterisdrivenwithsomesteady-state,orquiescentinputsreachthequiescentvaluesI,V,andIg,givenbythesteady-stateanalysisasAftertransientshavesubsided,theinductorcurrent,capacitorvoltage,andinputcurrentIntroducetheperturbationtoinputvoltageanddutycyclePerturbationInresponse,theconverterdependentvoltagesandcurrentswillalsobeperturbedInductorequationafterperturbation:PerturbationofinductorequationNotethatd’(t)isgivenbyMultiplyoutandcollectterms:PerturbedinductorequationTheright-handsidecontainsthreeterms:•DCterms•First-orderacterms,alinearfunctionoftheacvariations•Second-orderacterms,nonlinearSinceI

isaconstant(dc)term,itsderivativeiszeroSimplificationThe2ndorderactermismuchsmallerthanthe1stordertermsThe2ndorder

termisignoredDCtermismovedawayduetoDCequationAfterneglectingthe2ndordertermandusingDCequation,wegetLinearizedinductorequationLinearequationwhichdescribessmallsignalacvariationsQuiescentvaluesD,D’,V,andVgaretreatedasconstantsAfterperturbationCapacitorequationAfterneglectingthe2ndordertermandusingDCequation,wegetsmall-signallinearizedcapacitorequation.BeforeperturbationSmall-signallinearmodelofCCMBuck-boostconverterOutputequationStateequationConstructequivalentcircuitcorrespondingtothesmallsignallinearequationsInductorloopequation方块：受控源圆：独立电源CapacitornodeequationInputportnodeequationCompleteequivalentcircuitEquivalentcircuitofotherconvertersMidsummaryAverageoveroneswitchingperiodtoremoveswitchingripple1tAftertheaverageoperationtheinductanceequationandcapacitanceequationstillexistMidsummaryBuck-boostConverteraveragedequationsDescribelow-frequencydynamicsoftheconverter•Ignoretheswitchingripple•IgnorecomplicatedswitchingharmonicsandsidebandsLowfrequency?Midsummary1Small-signallinearmodelofCCMBuck-boostconverterMidsummaryPerturbationandlinearizingEquivalentcircuitMidsummaryState-spaceaveragedmodelGiven:aPWMconverter,operatingincontinuousconductionmode,withtwosubintervalsduringeachswitchingperiod.Duringsubinterval1,whentheswitchesareinposition1,theconverterreducestoalinearcircuitdescribedbyDuringsubinterval2,whentheswitchesareinposition2,theconverterreducestoanotherlinearcircuitdescribedbySlopeofthestatevectorduringfirstsubintervalDuringsubinterval1,wehaveSotheelementsofx(t)changewiththeslopeSmallrippleassumption:theelementsofx(t)andu(t)donotchangesignificantlyduringthesubinterval.HencetheslopesareessentiallyconstantandareequaltoNetchangeinstatevectoroverfirstsubinterval:NetchangeofthestatevectorduringfirstsubintervalStatevectornowchangeswiththeessentiallyconstantslopeThevalueofthestatevectorattheendofthesecondsubintervalisSlopeofthestatevectorduringthe2ndsubintervalNetchangeinstatevectoroveroneswitchingperiodEliminatex(dTs),toexpressx(Ts)directlyintermsofx(0):Collectterms:derivativeofstatevectorUseEulerapproximation:Weobtain:Low-frequencycomponentsofoutputvectorRemoveswitchingharmonicsbyaveragingoveroneswitchingperiod:Collectterms:Averagedstateequations:quiescentoperatingpointTheaveraged(nonlinear)stateequations:Hence,theconverterquiescentoperatingpointisthesolutionof0=AX+BUY=CX+EUwhereA=DA1+D'A2B=DB1+D'B2C=DC1+D'C2E=DE1+D'E2X=equilibrium(dc)statevectorU=equilibrium(dc)inputvectorY=equilibrium(dc)outputvectorD=equilibrium(dc)dutycycleTheconverteroperatesinequilibriumwhenthederivativesof<x(t)>Tsiszero.

Equilibrium(dc)state-spaceaveragedmodel0=AX+BUY=CX+EU

A=DA1+D'A2B=DB1+D'B2C=DC1+D'C2E=DE1+D'E2SolutionforXandY:whereAveragedstatespaceequationsLargesignaldynamicalmodelNonlinearmodelPerturbationandlinearizationSubstituteintoaveragedstateequations:PerturbationandlinearizationLinearizedsmall-signalstateequationsUseDCequations:0=AX+BU,Y=CX+EUThesecond-order(nonlinear)termsaresmallerState-spaceaveragedsmallsignalACmodelCircuitAveragingandAveragedSwitchModelingAbovemethodsaremathematicalintensiveRatherthandoingmathematicaloperations,isitpossibletoderivethemodelbyperformingtheconvertercircuittransformation.SeparateswitchnetworkfromremainderofconverternonlinearlinearLinearsubcircuitNonlinearswitchnetworkBoostconverterexampleOriginalcircuitisdivideintolinearsubcircuitandswitchnetworkSimpledc-dcconvertercase,theswitchnetworkistwoterminalnetworkTheswitchnetworkterminalvariablesareterminalvoltagesandcurrents:v1(t),i1(t),v2(t),andi2(t).Twooftheseterminalvraibalesaretakenasindependentinputstotheswitchnetwork,andtheremainingtwovariablesarethenviewedasdependentoutputsoftheswitchnetwork.Definitionoftheswitchnetworkterminalvariablesisnotunique.DiscussionTwowaystodefinetheswitchnetworkBoostconverterexampleSincei1(t)andv2(t)coincidewiththeconverterinductorcurrentandoutputvoltageandarestatevariables,itisconvenienttodefinethesewaveformsastheindependentinputstotheswitchnetwork.v1(t)andi2(t)areselectedasdependentoutputs.+__+Obtainingatime-invariantnetwork:ModelingtheterminalbehavioroftheswitchnetworkReplacetheswitchnetworkwithdependentsources,whichrepresentsthedependentoutputoftheswitchnetworkBoostconverterexampleDefinitionofdependentsourcewaveformsThewaveformsofthedependentsourcesareidenticaltotheactualterminalwaveformsoftheswitchnetwork.ThedependentsourcenetworkisequivalenttoswitchnetworkThecircuitaveragingstepNowaverageallwaveformsoveroneswitchingperiod:Naturaltimeconstantsoftheconverteraremuchlargerthantheswitchingperiodduetolow-passfilter.AveragingovertheswitchingperiodTswillnotsignificantlyalteringthesystemresponse,whichremovestheswitchingharmonics,whilepreservingthelow-frequencycomponentsofthewaveforms.Inpractice,theonlyoperationneededforthisstepistoaveragetheswitchnetwork.Averagingstep:boostconverterexampleComputeaveragevaluesofdependentsourcesAveragethewaveformsofthedependentsourcesPerturbThecircuitbecomes:linearizeLinearizedcircuit-averagedmodelModelthetwoterminalswitchnetworkwithequivalentvoltageandcurrentsourcesAverageconverterwaveformsoveroneswitchingperiod,toremovetheswitchingharmonics.Actuallyaveragethewaveformsofthedependentsources.Perturbandlinearizethedependentsourcesnetwork,toobtainasmall-signalequivalentcircuitSummary:CircuitaveragingmethodAveragedswitchmodeling:CCMCircuitaveragingoftheboostconverter:inessence,theswitchnetworkwasreplacedwithaneffectiveidealtransformerandgenerators:BasicfunctionsperformedbyswitchnetworkFortheboostexample,wecanconcludethattheswitchnetworkperformstwobasicfunctions:•Transformationofdcandsmall-signalacvoltageandcurrentlevels,accordingtotheD’:1conversionratio•IntroductionofacvoltageandcurrentsourcescontrolledbydutycyclevariationsCircuitaveragingmodifiesonlytheswitchnetwork.Toobtainasmallsignalconvertermodel,weneedonlyreplacetheswitchnetworkwithitsaveragedmodel.Suchaprocedureiscalledaveragedswitchmodeling.Averagedswitchmodeling:Procedure1.Defineaswitchnetworkanditsterminalwaveforms.Forasimpletransistor-diodeswitchnetworkasinthebuck,boost,etc.,therearetwoportsandfourterminalvariabls:v1,i1,v2,i2.Theswitchnetworkalsocontainsacontrolinputd.2.Fortwoportstherearefourterminalquantities:v1,i1,v2,i2.Twoterminalquantitiesareselectedasindependentvariablesandtheothersasdependentvariables.3.Toderiveanaveragedswitchmodel,expressingtheaveragevaluesofdependentvariableswithindependentvariables.4.PerturbationandlinearizeBuckconverterCCMswitchcellResultingaveragedswitchmodel:CCMbuckconverterReplacementofswitchnetworkbydependentsourcesThreebasicswitchnetworksandsmall-signalacmodelsAllPWMCCMdc-dcconvertersperformthesamebasicfunctions:•Transformationofvoltageandcurrentlevels•Low-passfilteringofwaveforms•ControlofwaveformsbyvariationofdutycycleCanonicalmodel:•Astandardformofequivalentcircuitmodel,whichrepresentstheabovephysicalproperties•PluginparametervaluesforagivenspecificconverterThecanonicalcircuitmodelCanonicalcircuitmodelLine-to-outputtransferfunctionControl-to-outputtransferfunctionExample:manipulationofthebuck-boostconvertermodelintocanonicalformSmall-signalacmodelofthebuck-boostconverter•Moveindependentsourcestoinputsideoftransformers•Moveinductortooutputsideofthetransformers•CombinetwotransformersCanonicalcircuitmodelStep1Pushvoltagesourcethrough1:DtransformerMovecurrentsourcethroughD’:1transformerStep2Breakgroundconnectionofcurrentsource,andconnecttonodeAinstead.ConnectanidenticalcurrentsourcefromnodeAtoground,sothatthenodeequationsareunchanged.Step3Theparallel-connectedcurrentsourceandinductorcannowbereplacedbyaThevenin-equivalentnetwork:Step4Nowpushcurrentsourcethrough1:Dtransformer.Pushcurrentsourcepastvoltagesource,againby:Breakinggroundconnectionofcurrentsource,andconnectingtonodeBinstead.Notethattheresultingparallel-connectedvoltageandcurrentsourcesareequivalenttoasinglevoltagesource.Step5:finalresultPushvoltagesourcethrough1:Dtransformer,andcombinewithexistinginput-sidetransformer.Combineseries-connectedtransformers.Coefficientofcontrol-inputvoltagegeneratorVoltagesourcecoefficientis:Simplification,usingdcrelations,leadstoPushingthesourcespasttheinductorcausesthegeneratortobecomefrequency-dependent.CanonicalcircuitparametersforconvertersModelingth