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1 HistoricalPerspective RobertDMiddlebrook PhD,Standford,1955 CalTech Professor, SlobodanCúk PhDCalTech,1976 CalTech Prof, Modelaswitchedsystemasan averaged,timeinvariantsystemwith where A.R.BrownandR.D.Middlebrook, SampleddataModelingofSwitchingRegulators PESC1981 LinearCircuitModelingUsingStateSpace Inswitchposition1
2 LinearCircuitModelingUsingStateSpace Inswitchposition1 Whichcanbewritten,instatespace,formas Or,generally, Inthesecondswitchposition,wewillhaveanew(linear)circuitwith SwitchingSignal InaPWMconverterwithtwo switchpositions,thetwolinear circuitscombineaccordingtoa switchingfunction s(t) 1 0 where
3 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,
4 ApproximateSteadyStateWaveforms ApproximateSteadyStateWaveforms
5 ApproximateSteadyStateWaveforms TheAveragingApproximation Ifwaveformscanbeapproximatedaslinear sotheaverageslopeis or,rearranging
6 TheAveragedSystem Thisequationisnowthemodelofanew,equivalent linearsystem where whichhasaveragedbehavioroveroneswitching period Thisapproximationisperhaps valid,if Statewaveformsaredominantlylinear Dynamicsofinterestareat BuckStateSpaceAveraging Inswitchposition1 Inswitchposition2
7 BuckAveragedModel So,ouraveragemodelis = Averaging:Discussion
8 DiscreteTimeNatureofPWM DiscreteTimeNatureofPWM
9 DiscreteTimeNatureofPWM HistoricalPerspective RobertDMiddlebrook PhD,Standford,1955 CalTech Professor, SlobodanCúk PhDCalTech,1976 CalTech Prof, Modelaswitchedsystemasan averaged,timeinvariantsystemwith where DennisJohnPackard PhD,CalTech 1976 Modelaswitchedsystemasadiscretetime systemwith where A.R.BrownandR.D.Middlebrook, SampleddataModelingofSwitchingRegulators PESC1981
10 LargeSignalModelingofSMPS DiscreteTimeModeling Everysubcircuit isapassive,linearcircuit Passive,linearcircuitscanbesolvedinclosed form Canmodelstatesatdiscretetimeswithout averaging Onlyassumptionsrequired IndependentinputsareDCorslowlyvarying
11 SolutiontoStateSpaceEquation Closedformsolutiontostatespaceequation Multiplybothsidesby Lefthandsideis SolutiontoStateSpaceEquation Cannowbesolvedbydirectintegration Rearranging
12 MatrixExponential MatrixexponentialdefinedbyTaylorseriesexpansion Wellknownissuewithconvergenceinmanycases N C.Moler andc.v.loan, Nineteendubiouswaystocomputetheexponentialofamatrix, SIAMReview,vol.20,pp ,1978. PropertiesoftheMatrixExponential Matrixexponentialalwaysexists i.e.summationwillalwaysconverge Exponentialofanymatrixisalwaysinvertible, with
13 FirstOrderTaylorSeriesExpansion Linearrippleapproximation Validonlyifswitchingfrequencymuchfaster thansystemmodes SimplificationforSlowVaryingInputs IfAisinvertibleand
14 ApplicationtoSwitchingConverter ApplicationtoSwitchingConverter
15 ApplicationtoSwitchingConverter ApplicationtoSwitchingConverter
16 ApplicationtoSwitchingConverter GeneralForm Generally,for separateswitchingpositions Equationisintheformofadiscretetimesystemwith Again,theeffectofchangingmodulation(i.e. )ishiddeninnonlinearterms Find bysmallsignalmodeling
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