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Chapter 2Modeling of DCM DC/DC ConverterCharacteristics at the CCM/DCM boundaryAll converters may operate in DCM at light loadSteady-state output voltage becomes strongly load-dependentDynamics in DCM mode is different to CCM modeWe need equivalent circuits that model both the steady-state and small signalac models of converters operating in DCM.Then DCM modeCCMDCMBuck converterOutput to input ratio of Buck converterIn DCMDerivation of DCM averaged switch model:buck-boost exampleDefine switch terminal quantities v1,i1,v2,i2,as shown Let us find the averaged quantities ,for operation in DCM,and determine the relations between themStage 1 Switch is on and diode is offInductor current increase linearlyStage 2 Switch is off and diode is onInductor transfers energy to outputThe stage is ended once inductor current reduce to zero.Stage 3 Both Switch and diode are off Capacitor output energy to loadDCM waveformsPeak inductor current:d2(t)=?Average inductor voltage:In DCM,the diode turns off when the inductor current reaches zero.Hence,i(0)=i(Ts)=0,and the average inductor voltage is zero.This is true even during transients.Solve for d2:Average switch network port voltagesAverage v1(t)waveform:use:Similar analysis for v2(t)waveform leads toAverage switch network port currentsAverage i1(t)waveform:The integral q1 is the area under the i1(t)waveform during first subinterval.Use trianglearea formula:Similar analysis for i2(t)waveform，we gotInput port:Averaged equivalent circuitwhereThe loss-free resistor(LFR)Output port:Averaged equivalent circuitIn a lossless two-port network without internal energy storage:instantaneous input power is equal to instantaneous output power.Power balance in lossless two-port networks1.A two-port lossless network 2.Input port obeys Ohms Law3.Output is(nonlinear)dependent current source4.Power entering input port is transferred to output portAveraged switch model:DCM buck-boost exampleOriginal circuitAveraged modelSolution of averaged model:steady stateSwitch network input port power:Switch network output port power:Two-port network energy conservation:Let L short circuit C open circuit稳态工作点时的物理量大写字母表示稳态工作点时的物理量大写字母表示For the buck-boost converter,we haveThenSteady state input to output ratioSince the output voltage is negative for buck boost converter,minus is selected in above equationwhereAveraged transistor waveform obey Ohms lawAveraged diode waveform behaves as dependent current sourceInput port power is equal to output powerAveraged models of two-port switch network of DCM convertersDCM buck,boost modelBuckBoost+Steady-state model:DCM buck,boostLet L short circuit C open circuitBuckBoostConversion ratio of DCM converterSmall-signal ac modeling of the DCM switch networkPerturb and linearize:We getLinearization by Taylor seriesGiven the nonlinear equationExpand in three-dimensional Taylor series at the quiescent operating point:Input portOutput portSimilarly DC termsSmall-signal aclinearizationSmall-signal DCM switch model parametersTable Small-signal DCM switch model parametersSmall-signal ac model of DCM buck-boost converterSmall-signal ac models of the DCM buck and boost converters DCM buck,boost,and buck-boost converters exhibit a single-pole systemSimplification of DCM small-signal modelBuck,boost,and buck-boost converter models all simplify toTransfer functions of DCM convertercontrol-to-outputTransfer functions of DCM converterline-to-outputParameters of DCM converter transfer functions1.What is relationship between CCM model and DCM model?2.Is possible to get the model to cover both model?3.Is it possible to use the result from CCM to describe DCM model?Questions?Defining the switch network inputs and outputsinput vectorcontrol inputoutput vectorDependent variables Define According to power conservationThereforeDefinition of general conversion ratioDefinition is suited to both CCM and DCMFor CCM Switch network output CCM switch outputs during subinterval 1:For CCM operation,this equation is satisfied with =d.CCM switch outputs during subinterval 2:It is valid not only in CCM,but also in DCM.Meaning of valid not only in CCM,but also in DCM.A generalization of the CCM duty cycle d.CCM case,In DCM case,depend on the switch network independent inputsConcept of conversion ratio properties:Switch network output1 stage2 stageSwitch network output is equal to weighted sum of switch network outputs in both stages For CCM For general caseis not constant,depend on the switch network independent inputsEvaluation of in DCM caseDivided bySolve for DCM Buck converter averaged modelDCM Buck converterElimination of dependent quantitiesLossless switch network:thereforeDCM switch conversion ratio1.A general result for DCM2.In DCM,switch conversion ratio is a function of not only the transistor duty cycle d,but also the switch independent terminal waveforms i2 and v1.3.The switch network output not only depends on d,but also on independent variables in the switch network.It contains built-in feedback.4.Replace d of CCM expression with to obtain a valid DCM expressionCCM Buck converter average modelFrom CCM Buck average model to DCM Buck average modelCCM Buck converter average modelDCM Buck converter average modelPerturbation and linearizationSteady-state components:For DCM Buck:DCM Buck steady-state solutionDCM buck small-signal equationsSolve for derivative:The gains are found by evaluation of derivatives at the quiescent operating pointSmall-signal model of DCM buck converterControl-to-output transfer functionControl-to-output transfer functiondefineIfthenMagnitude ofGeneralized Averaged Switch Modeling for DCMdivide into linear sub-circuit and switch networklinearSwitch networkSystem state equationsAverage:Linear subcircuit equationSwitch network inputSwitch network outputSwitch network outputswitch network dependent outputs averaged over one switching periodNow attempt to write the converter state equations in the same formused for CCM state-space averaging model.This can be doneprovided that the above equation can be manipulated into the formwhere ys1(t)is the value of ys(t)in the CCM converter during subinterval 1 ys2(t)is the value of ys(t)in the CCM converter during subinterval 2 is called the switch conversion ratioSystem averaged state equationsSuppose ys1(t)is switch network output during subintervals 1 in CCM modeys2(t)is switch network output during subinterval 2 in CCM modeaveraged state equationsThe time-invariant network equations predict that the converter state equations for the first subinterval areFor subinterval 1Now equate the two expressions got with different waysFor the converter operating in CCM for subinterval 1 the state space equationEquate the state equation expressionsderived via the two methodsBy using equation subinterval 2,we can solve for and Now plug the results back into averaged state equationsBy simplification The appearance is similar to that of CCM,d(t)is replaced byLarge signal equationNonlinear equationSmall signal equivalent circuit got from CCM can be used with consideration of effects ofAveraged state equationsCCMSwitch conversion ratioIf it is true that1.CCM equations can be used directly,simply by replacing the duty cycle d(t)with the switch conversion ratio(t).Steady-state relations are found by replacing D with 0Small-signal transfer functions are found by replacing d(t)with(t).The switch conversion ratio is a generalization of the CCM dutycycle d.In general,may depend on the switch independent inputs,that is,converter voltages and currents.Perturb and linearizeIntroduce perturbationDC model0=AX+BUY=CX+EUwhereSmall-signal ac modelSmall signal model of Make derivatives for A generalized canonical modelThe switch conversion ratio is a generalization of the CCM duty cycle d.In general,may depend on the switch independent inputs,that is,converter voltages and currents.So feedback may be built into the switch network.Summary of Chapter 21.In the discontinuous conduction mode,the average transistor voltageand current obey Ohms law.2.The average diode voltage and current obey a power source characteristics,with power equal to the power effectively dissipated by Re.In the averaged equivalent circuit,the diode is replaced with a dependent current source.3.The two-port lossless network consisting of an effective resistor anda dependent current source.3.The large-signal averaged model can be solved under equilibriumconditions to determine the quiescent values of the converter currentsand voltages.4.A small-signal ac model for the DCM switch network can be derived by perturbing and linearizing the loss-free resistor network.The result has the form of a two-port y-parameter model.5.Relationship between CCM model and DCM model is clarified.
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