A-general-unified-approach-to-modeling-switching-c

'fAGENERALWIFIEDAPPBDACBTOMODELLINGSWITCHING-CONVERTERPaireRSTAGESR.D.MiddlebrookandSlobodanCukCaliforniaInstituteofTechnologyPasadena.CaliforniaABSTRACTAmethodformodellingswitching-converterpowerstagesisdeveloped,whosestartingpointistheunifiedstate-spacerepresentationoftheswitchednetworksandwhoseendresultiseitheracompletestate-spacedescriptionoritsequivalentsmall-signallow-frequencylinearcircuitmodel.Anewcanonicalcircuitmodelisproposed,whosefixedtopologycontainsalltheessentialinput~utputandcontrolpropertiesofanydc-to-deswitchingconverter,regardlessofitsdetailedconfiguration,andbywhichdifferentconverterscanbecharacterizedintheformofatablecon-venientlystoredinacomputerdatabanktopro-videausefultoolforcomputeraideddesignandoptimization.Thenewcanonicalcircuitmodelpredictsthat,ingeneral,switchingactionintro-ducesbothzerosandpolesintothedutyratiotooutputtransferfunctioninadditiontothosefromtheeffectivefilternetwork./1.INTRODUCTION1.1BriefReviewofExistinp;Hodellinp;Techniques1Iimodellingofswitehingconvertersingeneral,andpowerstagesinparticular,twomainapproaches-onebasedonstate-spacemodellingandtheotherusinganaveragingtechnique-havebeendevelopedextensively,buttherehasbeenlittlecorrelationbetweenthem.Thefirstapproachremainsstrictlyinthedomainofequation1IIIDipulations,andhencereliesheavilyonnumericalmethodsandcomputerized1m:plementat1ons.Itsprimaryadvantageisintheunifieddescriptionofallpowerstagesregardlessofthetype(buck,boost,buck-boostoranyothervariation)throughutilizationoftheexactstate-spaceequationsofthetwoswitchedmodels.Ontheotherhand,theapproachusinganaveragingtechniqueis11IThisworkwassupportedbySubcontractNo.A72042-RHBEfromTRWSystemsGroup,underNASAPrimeContractNAS3-l9690ModelingandAnalysisofPowerProcessingSystems.18-PESC76RECORDbase~onequivalentcircuitmanipulations,resultinginasingleequivalentlinearcircuitmodelofthepowerstage.Thishasthedistinctadvantageofprovidingthecircuitdesignerwith'physicalinsightintothebehaviouroftheoriginalswitchedcircuit,andofallowingthepowerfultoolsoflinearcircuitanalysisandsynthesistobeusedtothefullestextentindesignofregulatorsincorporatingswitchingconverters.1.2ProposedNewState-SpaceAveragingApproachThemethodproposedinthispaperbri'dgesthegapearlierconsideredtoexistbetweenthestate-spacetechniqueandtheaveragingtechniqueofmodellingpowerstagesbyintroductionofstate-spaceaveragedmodelling.Atthesamet1me~offerstheadvantagesofbothexistingmethods-thegeneralunifiedtreatmentofthestate-spaceapproach,aswellasanequivalentlinearcircuitmodelasitsfinalresult.Furthermore,itmakescertaingenerali%.ationspossible,whichotherwisecouldnotbeachieved.Theproposedstate-spaceaveragingmethod,outlinedintheFlowchartofFig.1.allowsaunifiedtreatmentofalargevarietyofpowerstagescurrentlyused,sincetheaveragingstepinthestate-spacedomainisverys1mpleandclearlydefined(compareblockslaand2a).Itmerelyconsistsofaveragingthetwoexactstate-spacedescriptionsoftheswitchedmodelsoverasinglecycleT.wherefs.liTistheswitchingfrequency(block2a)..Hencethereisnoneedforspeciallmow...bowinmassagingthetwoswitchedcircuitmodelsintotopologicallyequivalentformsinordertoapplycircuit-orientedproceduredirectly.asrequiredin[1](blocklc).Nevertheless,throughahybridmodellingtechnique(block2c).thecir-cuitstructureoftheaveragedcircuitmodel(block2b)canbereadiiyrecognizedfromtheaveragedstate-spacemodel(block2a).Henceallthebenefitsofthepreviousaveragingtechniqueareretained.Eventhoughtb1~out~linedprocessmightbepreferred.onecanproceedfromblocks2aand2bintwoparallelbutcom-pletelyequivalentdirections:onefollowingpathastrictlyintermsofstate-spaceequations,andtheotheralongpathbintermsofcircuitmodels.Ineithercase,aper~urbationandlinearizationJlol,-S!'«'f~alo_sI'/~,dlud-.D+;,f'.J)~';,x.X+x,i-r.;.~-IJ+§D~I-D,tI'I.1-tI,,,tur6al,on',J60,,Iot,-s!,o'Pdtnodrl,i.Arx+~rJ;:J'eXAariA..d~.wII..;Ii1/db..ci'tf.,d-Dcr,ole.r..t'e,rI5t.I'SI'I.sen;I,....i-I,,.Ir,.t:'ftt.n.1T#':;-A,xt-t,'}x-A.x+-t.'j/P-c.rx1J';if'-cixslalt-s!'eavUOJI:!x-(i~!It,u',oaOl1!t.,,i.1S(Qt.-s/,,'a~1st.adysl.l.f«-)motl,AX+I~-Oy-.rx;,,...i,(.,.I/IHIUsl1nol),tlel:i-Ax.&~+[(A,-A.)X~-Y]Ji-erx+(ei-cf)Xd,..v.'d,ri.-.fio.'o{/,',uval....slol,-,vu,'io.sin'circulIrteor.I,form(J/,I.-opo«realiz,!'on)eireu,Y..,'Is,-.1.1TI':eir(uif1..011'1'/01;0.',putro,lionona(i,or/zollo.(;..orizotion,orred,dprocoduroFig.1.Flowchartofaveragedmodellingapproachesprocessrequiredtoincludethedutyratiomodulationeffectproceedsinaverystraightfor-wardandformalmanner,thusemphasizingthecorner-stonecharacterofblocks2aand2b.Atthisstage(block2aor2b)thesteady-state(dc)andlinetooutputtransferfunctionsarealreadyavailable,asindicatedbyblocks6aand6brespectively,whilethedutyratioto.outputtransferfunctionisavailableatthefinal-stagemodel(4aor4b)asindicatedbyblocks7aand7b.Thetwofinalstagemodels(4aand4b)thengivethecompletedescriptionoftheswitching.converterbyinclusionofbothindependentcon~troIs,thelinevoltagevariationandthedutyratiomodulation.Eventhoughthecircuittransformationpathbmightbepreferredfromthepracticaldesignstandpoint,thestate-spaceaveragingpathaisi