Hopf-bifurcation-and-chaos-in-a-free-running-curre

448IEEETRANSACTIONSONCIRCUITSANDSYSTEMS—I:FUNDAMENTALTHEORYANDAPPLICATIONS,VOL.47,NO.4,APRIL2000HopfBifurcationandChaosinaFree-RunningCurrent-ControlledC´ukSwitchingRegulatorC.K.Tse,SeniorMember,IEEE,Y.M.Lai,Member,IEEE,andH.H.C.Iu,StudentMember,IEEEAbstract—Anautonomousfree-runningC´ukconverterisstudiedinthispaper.AnalysisofthedescribingnonlinearstateequationsshowsthatthesystemlosesstabilityviaasupercriticalHopfbifurcation.Theboundaryofstabilityisderivedandlocaltrajectoriesofmotionstudied.Cycle-by-cyclesimulationsoftheactualsystemrevealthetypicalbifurcationfromastableequi-libriumstatetochaos,vialimitcycles,andquasi-periodicorbits.Experimentalmeasurementsconfirmthebifurcationscenarios.Theoccurrenceofsuchkindsofbifurcationinautonomousdc/dcconvertershasbeenrarelyknowninpowerelectronics.IndexTerms—Bifurcation,chaos,C´ukconverter,dc/dccon-verter,powerelectronics.I.INTRODUCTIONPOWERelectronicsisafieldrichinnonlineardynamics[1],andengineerswhoworkinthisfieldarefrequent“chaosobservers”whethertheyknowitornot.Infact,duringtheearlydevelopmentandtestingstagesofpowerelectronicssystems,amultitudeofnonlinearphemenona,suchassubharmonics,quasi-periodicityandchaos,arealmostinvariablyencountered.Becausepowerelectronicsengineersareprimarilyconcernedwithonlytheregularperiodic(fixedpoint)operation,theytendtoavoidany“strange”operationbyadjustingcomponentsorpa-rametersthattheybelievearecausingproblemstotheirsystems.Veryoftensuchadjustmentprocessesaredoneinatrial-and-errormanner.However,toachievemorereliabledesigns,weneedabetterunderstandingofthecircuitoperationunderallpossiblepracticalconditions.Moreover,wemayopenupnewpossibilitiesinoperatingpowerelectronicssystemsifenoughunderstandingisgainedofthemanyunusedoperatingregimes.Self-oscillatingorfree-runningcurrent-controlledswitchingconvertersareoftenusedinlow-costswitchingpowersupplies,sincetheyrequirenoexternalclocksandtheirconstructionsarerelativelysimple.Incontrasttotheirnonautonomouscounter-partsforwhichchaosisobservedevenforthesimplestfirst-orderdiscontinuous-modeconverters[2],[3],free-runningcon-vertersoforderbelowthreecannotexhibitchaos.Theessentialfeatureofanautonomousswitchingconverteristheabsenceofanyexternaldrivingsignal,whichismandatoryinthenonau-ManuscriptreceivedMarch11,1999;revisedAugust19,1999.ThisworkwassupportedbytheHongKongResearchGrantsCouncilunderacompetitiveearmarkedresearchGrantPolyU109/96E.ThispaperwasrecommendedbyAs-sociateEditorC.W.Wu.TheauthorsarewiththeDepartmentofElectronicandInformationEngi-neering,HongKongPolytechnicUniversity,Kowloon,HongKong,China.(e-mail:[e-mail]).PublisherItemIdentifierS1057-7122(00)02915-9.tonomouscaseforperiodicswitchingofthepowerswitch.Untilnow,muchhasbeenreportedonthebifurcationandchaoticbe-haviorofnonautonomousswitchingconverters[2]–[8],whileverylittleworkhasbeenperformedinidentifyingbifurcationandchaosinautonomousswitchingconverters.InthispaperwestudythedynamicsoftheC´ukconverterwhichiswidelyusedinpowerelectronics.Beingoffourth-order,theC´ukconvertercanoperatechaoticallyinfree-running(autonomous)mode.Wewillreportinthispaperthebifurcationbehaviorofafree-run-ningcurrent-controlledC´ukconverter.Inparticular,ourstudywillcoverthefollowingaspects:1)derivationofdescribingstateequation;2)studyofstabilityoftheequilibriumstateandidenti-ficationofHopfbifurcationbasedonthedescribingstateequa-tion;3)computersimulationsofthecircuitsrevealingthebifur-cationfromstableequilibriumstate(fixedpoint),throughlimitcyclesandquasi-periodicorbits,andeventuallytochaos;and4)experimentalverificationofthebifurcationscenario.II.SYSTEMDESCRIPTIONSpecifically,thesystemunderstudyconsistsofaC´ukcon-verterwhoseoperationisbasedonafree-runninghystereticcurrent-modecontrol.Inactualimplementationtheswitchisturnedonandoff,inahystereticfashion,whenthesumoftheinductorcurrentsfallsbeloworrisesaboveacertainpresethys-tereticband[9],[10].TheaveragevalueandwidthofthispresetbandareadjustedbyafeedbackSchmitttriggercircuit.Also,theoutputvoltageisfedbacktosettheaveragevalueofthehystereticband,forcingthecontrolvariabletoberelatedbythefollowingcontrolequation:(1)whereandinductorcurrents;outputvoltage;controlfunction.Forexample,asimpleproportionalcontroltakestheformof(2)whereisthegainfactor.Thisequationhasthefollowingequiv-alentform,assumingregulatedoutput:(3)whereandarethecontrolparameters.Fig.1showsasim-plifiedschematicofthesystem.1057–7122/00$10.00©2000IEEETSEetal.:HOPFBIFURCATIONANDCHAOSINAFREE-RUNNINGCURRENT-CONTROLLEDC´UKSWITCHINGREGULATOR449Fig.1.C´ukconverterunderhystereticcurrent-modecontrol.A.DerivationofAutonomousStateEquationsThesystemcanberepresentedbythefollowingstate-spaceequationswherewhentheswitchisturnedon,andwhentheswitchisoff:(4)Thestate-spaceaveragedmodelhasthesameformasabove,withreplacedbythedutycyclewhichisthefractionoftheswitchingperiodforwhichtheswitchisturnedon.Sinceisrelatedtobyalinearalgebraicequation,thesystemreducesitsorderbyone.Specifically,when(3)istakenintoaccount,thesystemcanbereducedtothefollowingthird-ordersystem:(5)whereisthedut