HistoricalPerspective RobertDMiddlebrook PhD,Standford,1955 CalTech Professor,19551998 SlobodanCúk PhDCalTech,1976 CalTech Prof,19771999 Modelaswitchedsystemasan averaged,timeinvariantsystemwith where A.R.BrownandR.D.Middlebrook, SampleddataModelingofSwitchingRegulators PESC1981 LinearCircuitModelingUsingStateSpace Inswitchposition1
LinearCircuitModelingUsingStateSpace Inswitchposition1 Whichcanbewritten,instatespace,formas Or,generally, Inthesecondswitchposition,wewillhaveanew(linear)circuitwith SwitchingSignal InaPWMconverterwithtwo switchpositions,thetwolinear circuitscombineaccordingtoa switchingfunction s(t) 1 0 where
SMPSStateSpace Intraditionalstatespacemodelingoflinearsystems withu(t) containingacontrolinput.whena andb areconstant, thisisalinearsystem.however,wehave or,equivalently whichisnonlinear:howdowedealwithit? ConvertingtoLinearSystem Assumethatoursystemmodel canbeapproximatedbysomelinearsystem whichremovesthenonlinearityofthesystem Nonlinearitiescamefromswitching Expectthatswitchingdynamicswillbelost Note:Thissystemisnowlinearin and,butnotinour controlsignal,
ApproximateSteadyStateWaveforms ApproximateSteadyStateWaveforms
ApproximateSteadyStateWaveforms TheAveragingApproximation Ifwaveformscanbeapproximatedaslinear sotheaverageslopeis or,rearranging
TheAveragedSystem Thisequationisnowthemodelofanew,equivalent linearsystem where whichhasaveragedbehavioroveroneswitching period Thisapproximationisperhaps valid,if Statewaveformsaredominantlylinear Dynamicsofinterestareat BuckStateSpaceAveraging Inswitchposition1 Inswitchposition2
BuckAveragedModel So,ouraveragemodelis = Averaging:Discussion
DiscreteTimeNatureofPWM DiscreteTimeNatureofPWM
DiscreteTimeNatureofPWM HistoricalPerspective RobertDMiddlebrook PhD,Standford,1955 CalTech Professor,19551998 SlobodanCúk PhDCalTech,1976 CalTech Prof,19771999 Modelaswitchedsystemasan averaged,timeinvariantsystemwith where DennisJohnPackard PhD,CalTech 1976 Modelaswitchedsystemasadiscretetime systemwith where A.R.BrownandR.D.Middlebrook, SampleddataModelingofSwitchingRegulators PESC1981
LargeSignalModelingofSMPS DiscreteTimeModeling Everysubcircuit isapassive,linearcircuit Passive,linearcircuitscanbesolvedinclosed form Canmodelstatesatdiscretetimeswithout averaging Onlyassumptionsrequired IndependentinputsareDCorslowlyvarying
SolutiontoStateSpaceEquation Closedformsolutiontostatespaceequation Multiplybothsidesby Lefthandsideis SolutiontoStateSpaceEquation Cannowbesolvedbydirectintegration Rearranging
MatrixExponential MatrixexponentialdefinedbyTaylorseriesexpansion Wellknownissuewithconvergenceinmanycases N C.Moler andc.v.loan, Nineteendubiouswaystocomputetheexponentialofamatrix, SIAMReview,vol.20,pp. 801 836,1978. PropertiesoftheMatrixExponential Matrixexponentialalwaysexists i.e.summationwillalwaysconverge Exponentialofanymatrixisalwaysinvertible, with
FirstOrderTaylorSeriesExpansion Linearrippleapproximation Validonlyifswitchingfrequencymuchfaster thansystemmodes SimplificationforSlowVaryingInputs IfAisinvertibleand
ApplicationtoSwitchingConverter ApplicationtoSwitchingConverter
ApplicationtoSwitchingConverter ApplicationtoSwitchingConverter
ApplicationtoSwitchingConverter GeneralForm Generally,for separateswitchingpositions Equationisintheformofadiscretetimesystemwith Again,theeffectofchangingmodulation(i.e. )ishiddeninnonlinearterms Find bysmallsignalmodeling