浙江大学现代电力电子学课件

1、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

2、 signalac models of converters operating in DCM.Then DCM modeCCMDCMBuck converterRVDTLVVSg2)(oLII Output to input ratio of Buck converterIn DCMLight loadDerivation of DCM averaged switch model:buck-boost exampleDefine switch terminal quantities v1, i1, v2, i2, as shown Let us find the averaged quant

3、ities , for operation in DCM, and determine the relations between themTsTsTsTsvivi2211,Stage 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 S

4、witch 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

5、for d2:11( )( )ssst Tt TLTLsttsssdiLvtv dtLdti tTi tTTdtT 0)(TsLtvAverage switch network port voltagesAverage v1(t) waveform:use:Similar analysis for v2(t) waveform leads to1231dddAverage switch network port currentsAverage i1(t) waveform:The integral q1 is the area under the i1(t)waveform during fi

6、rst subinterval. Use trianglearea formula:Similar analysis for i2(t) waveform， we got222111212212( )( )( )1( )(,)2( )()( )sssssssTTsTiTTTeTv tv tdt Ti tfvvLv tR dv t Input port: Averaged equivalent circuitwhereThe loss-free resistor (LFR)Output port: Averaged equivalent circuitIn a lossless two-port

7、 network without internal energy storage: instantaneous input power is equal to instantaneous output power.Power balance in lossless two-port networksA two-port lossless network Input port obeys Ohms LawOutput is (nonlinear) dependent current sourcePower entering input port is transferred to output

8、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RVRVIVPportou

9、t222_)(V)-（)(DRRVVegRVDRVeg22)()(2_DRVPegportinFor 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 equation)(DRRVVeg)(DRRVVegSRTLK2whereAveraged transistor waveform obey Ohms lawAveraged

10、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-s

11、ignal 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:)(,)(,)()()()(21111tdtvtvftdRtvtissssTTeTT1122112112111121212112,( ),( )( ), ( ). vVvVd Ddfv V D

12、df V v DIi tf V V Dv tv tdvdvdf V V dd tddInput port)(,)(,)()()()(21111tdtvtvftdRtvtissssTTeTTOutput portSimilarly DC termsSmall-signal aclinearization)(,)(,)()()()()(2122212tdtvtvftvtdRtvtisssssTTTeTTSmall-signal DCM switch model parametersTable Small-signal DCM switch model parametersSmall-signal

13、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 model0)(TsLtv0)(tvLBuck, boost, and buck-boost converter models all simplify toTransfer functions

14、of DCM convertercontrol-to-outputCrRssdrRjsCrRsCrRsdjsv)/(1)()/(1)/(1)/()()( 222222CrRsrRjsdsvsGsvvdg)/(1)/()()( )(2220)(Transfer functions of DCM converterline-to-outputCrRsrRsvgCrRsrRsvgsvg)/(1)/()()/(1)/()()( 2222212CrRsrRgsvsvsGsdgvg)/(1)/()()( )(2220)(Parameters of DCM converter transfer functi

15、onsWhat is relationship between CCM model and DCM model?Is possible to get the model to cover both model?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 1122( )(

16、)( )( )TsTsTsTsv ti tv ti t2112( )( )( )( )TsTsTsTsv ti tv ti t2112( )( )( )( )( )TsTsTsTsv ti ttv ti tAccording to power conservationThereforeDefinition of general conversion ratioDefinition is suited to both CCM and DCMFor CCM )()(tdt Switch network output CCM switch outputs during subinterval 1:F

17、or CCM operation, this equation is satisfied with = d.CCM switch outputs during subinterval 2:ssTTstitvtitvt)()()()()(21121y00)(2tsy2112( )( )( )( )( )TsTsTsTsv ti ttv ti t00)(1 ()()()()()(2112ttitvttitvssssTTTTIt is valid not only in CCM, but also in DCM .12( )( )( )( )( )ssTssttttt yyyMeaning of 1

18、2( )( )( )( )( )ssTssttttt yyyvalid 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

19、 of switch network outputs in both stages )()()(21tydtdytyssTsSFor CCM For general caseis not constant, depend on the switch network independent inputs)(t)()(tdt Evaluation of in DCM caseDivided bySTtv)(2Solve for DCM Buck converter averaged modelDCM Buck converter)()()()(112dRtitvtveTTTsssssesssTeT

20、TeTTTtvdRtitvdRtitvtv)()()(1)()()()()(1212121ssTTetvtidR)()()(1121Elimination of dependent quantitiesLossless switch network:thereforeDCM switch conversion ratioA general result for DCMIn DCM, switch conversion ratio is a function of not only the transistor duty cycle d, but also the switch independ

21、ent terminal waveforms i2 and v1. The switch network output not only depends on d, but also on independent variables in the switch network. It contains built-in feedback.Replace d of CCM expression with to obtain a valid DCM expression12( )( )( )( )( )ssTssttttt yyyCCM Buck converter average modelFr

22、om CCM Buck average model to DCM Buck average modelTstit)(2（Tstvt)(1（Tstitd)(2（Tstvtd)(1（CCM Buck converter average modelDCM Buck converter average modelPerturbation and linearizationSteady-state components:For DCM Buck :DCM Buck steady-state solution0gVVRDRVRVDRVIDRegee0120)(11)(11)(11RRe/41120DCM

23、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 function)(1)()(sZIVIsZVsTeiSgSeigi)(1)()()(2sTsTIksdsiiiSS)(1)()1()1()()()()( )(20)(sTsTsCRIksCRsdsisdsv

24、sGiiSSsvvdg)1()()()(20)()()( sCRsdsisGsvsdsvvdgControl-to-output transfer function)(1)()()(sTsTsGsGiivdvdsRCRIksCRIksGSSSSvd1)1()(define1)(1)(sTsTii)()(sGsGvdvd1)(sTiIfthen2021111)(1)()(LCssRLsRCTLCssRLsRCRIVsZIVIsZVsTSgeiSgSeigiMagnitude of2011)(LCssRLsRCTsTi)(1)(sTsTiiGeneralized Averaged Switch M

25、odeling for DCMdivide into linear sub-circuit and switch networklinearSwitch networkSystem state equationsAverage:)()()()(tttdttdssyBuBxAxKFF)()()()(ttttssyEuExCyFF)()()(tttssuExCuutttftcss),(),()(uuyLinear subcircuit equationSwitch network inputSwitch network outputSwitch network outputswitch netwo

26、rk 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

27、 during subinterval 1 ys2(t) is the value of ys(t) in the CCM converter during subinterval 2 )(t)(1)tt（is called the switch conversion ratioSystem averaged state equationsSuppose )()( )()()(21tttttssTssyyyys1(t) is switch network output during subintervals 1 in CCM modeys2(t) is switch network outpu

28、t during subinterval 2 in CCM modeaveraged state equations)()()()(tttdttdssyBuBxAxKFF)()()()(ttttssyEuExCyFFThe 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 conv

29、erter 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 )(22tsyB)(2tssyE Now plug the results back into averaged state equationsBy simplification The appearance is similar to

30、 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 equationsCCM)(t)(tSwitch conversion ratioIf it is true thatCCM equations can be used directly, simply by replacing the duty

31、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,

32、that is, converter voltages and currents.)(tPerturb and linearizeIntroduce perturbationDC model0 = AX + BUY = CX + EUwhereSmall-signal ac model),()(csuutSmall signal model of Make derivatives for )(t( )( )( )TTsscctttk uk uTs( )( )Tc( )( )( ),( )( )( ),( )( )ssTsccTsssTsccTsscTsTstsTstscTsTstcTstdttdtdttdtuUuUuUuUuukuuukuA generalized canonical