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Analysis,simulationandexperimentalstudyofchaoticbehaviourinparallel-connectedDC–DCboostconvertersAmmarN.Natsheha,J.GordonKettleborougha,JamalM.Nazzalb,*aDepartmentofElectronicandElectricalEngineering,LoughboroughUniversity,Loughborough,LeicestershireLE113TU,UKbFacultyofEngineering,Al-AhliyyaAmmanUniversity,PostCode19328Amman,JordanAccepted5July2007CommunicatedbyProfessorG.IovaneAbstractThepaperdescribesanexperimentalstudyofthebifurcationbehaviourofamodularpeakcurrent-modecontrolledDC–DCboostconverter.Theparallel-input/parallel-outputconvertercomprisestwoidenticalboostcircuitsandoper-atesinthecontinuous-currentconductionmode.Acomparisonismadebetweentheresultsobtainedfromanexperi-mentalconverterwiththoseobtainedfrombifurcationdiagramsgeneratedfrompreviousworkandwaveformsfromanewMATLAB/SIMULINKsimulationpresentedinthispaper.Anothercomparisonismadebetweenthemodularconverterdiagramswiththoseofthesingleboostconverter.2007ElsevierLtd.Allrightsreserved.1.IntroductionDC–DCswitch-modeconvertersarenon-lineardevicesandrecentlyithasbeenobservedthattheymaybehaveinachaoticmanner.Thishasgeneratedmuchinterest,particularlyinsingle-stagetopologiessuchasthebuck,boost,andbuck-boostconverters[1–7].TheconnectionofseveralDC–DCconvertersinparallel,withtheloadsharedbetweenmodules,reducescurrentstressontheswitchingdevicesandincreasessystemreliability.However,despitethegrowingpopularityofthesemod-ularconverters,theirbifurcationphenomenahaverarelybeenstudied.Theworkpresentedin[8]describesthebifur-cationphenomenainaparallelsystemoftwoboostconvertersunderamaster-slavecurrent-sharingcontrolscheme.Oneconverterhasvoltagefeedbackcontrol,whiletheotherhasanadditionalinnercurrentloopwhichadjuststhevolt-agefeedbackloop,toensureequalsharingoftheloadcurrent.Thispaperinvestigateschaoticbehaviourinatwo-moduleparallelinput/parallel-outputboostconverteroperatingunderapeakcurrent-modecontrolscheme,whereeachmodulehasitsowncurrentfeedbackloop.Theconverterconsistsoftwoidenticalboostcircuitsandoperatesinthecontinuous-currentconductionmode.AcomparisonismadebetweenwaveformsobtainedfromanexperimentalconverterandbothaMATLAB/SIMULINKmodel0960-0779/$-seefrontmatter2007ElsevierLtd.Allrightsreserved.doi:10.1016/j.chaos.2007.07.020*Correspondingauthor.Tel.:[phone];fax:[phone].E-mailaddress:[e-mail](J.M.Nazzal).Chaos,SolitonsandFractals39(2009)2465–2476www.elsevier.com/locate/chaosandresultsobtainedfrombifurcationdiagramsgeneratedfromanonlinearmappinginclosedformusingMATLAB[9].2.BifurcationandchaostheoryBifurcationtheory,originallydevelopedbyPoincare,isusedtoindicatethequalitativechangeinbehaviorofasys-temintermsofthenumberandthetypeofsolutions,underthevariationofoneormoreparametersonwhichthesys-temdepends[10].Inbifurcationproblems,inadditiontostatevariables,therearecontrolparameters.Therelationshipbetweenanyofthesecontrolsparametersandanystatevariableiscalledthestate-controlspace.Inthisspace,locationsatwhichbifur-cationsoccurarecalledbifurcationpoints.Bifurcationsofanequilibriumorfixed-pointsolutionareclassifiedaseitherFig.1.Simplifiedcircuitdiagramforthetwo-moduleconverter.Fig.2.ModulecurrentvoltagewaveformsforthecircuitofFig.1.2466A.N.Natshehetal./Chaos,SolitonsandFractals39(2009)2465–2476staticbifurcations,suchassaddle-node,pitchfork,ortranscriticalbifurcations;orasdynamicbifurcationswhicharealsoknownastheHopfbifurcationthatexhibitsperiodicsolutions.Forthefixed-pointsolutions,thelocalstabilityofthesystemisdeterminedfromtheeigenvaluesoftheJacobianmatrixofthelinearizedsystem.Ontheotherhand,withperiodic-solutions,thesystemstabilitydependsonwhatisknownastheFloquettheoryandtheeigenvaluesoftheMonodromymatrixthatareknownintheliteratureasFloquetorcharacteristicmultipliers.ThetypesofbifurcationaredeterminedfromthemannerinwhichtheFloquetmultipliersleavetheunitcircle.Therearethreepossiblewaysforthistohappen[10]:(i)IftheFloquetmultiplierleavestheunitcirclethrough+1,thenthreepossiblebifurcationsmayoccur:transcrit-ical,symmetry-breaking,orcyclic-foldbifurcation.(ii)IftheFloquetmultiplierleavesthrough1,theperiod-doubling(Flipbifurcation)occurs.(iii)IftheFloquetmultipliersarecomplexconjugateandleavetheunitcirclefromtherealaxis,thesystemexhibitssecondaryHopfbifurcation.Toobservethesystemdynamicsunderalltheabovepossiblebifurcations,acompletediagramknownasabifurca-tiondiagramisconstructed.Abifurcationdiagramshowsthevariationofoneofthestatevariableswithoneofthesystemparameters,otherwiseknownasacontrolparameter.3.MathematicalmodellingandnumericalanalysisAsimplifieddiagramfortheproposedconverterisshowninFig.1.Itconsistsoftwopeakcurrent-modecontrolledDC–DCboostconverterswhoseoutputsareconnectedinparalleltofeedacommonresistiveload.Eachconverterhasitsowncurrentfeedbackloopcomprisingacomparatorandaflip-flop.Eachcomparatorcomparesitsrespectivepeakinductorcurrentwithareferencevalue,todeterminetheon-timeoftheswitch.ThecurrentandvoltagewaveformsforthecircuitofFig.1areshowninFig.2.SwitchesS1andS2arecontrolledsuchthattheycloseatthesametime.Whentheyareclosed,diodesD1andD2are