Analysis of the control loop for a DC-DC Converter

3–AnalysisofthecontrolloopforaDC-DCConverter3.1–CircuitmodelingFigure3.1showstheschematicdiagramofabuckdc-dcconverteroperatingwithclosedloop.ThereoneseesthatthemodulatingvoltageforthePWM(vCTRL)isobtainedwithaPIcontrollerthatprocessestheerrorbetweenthereference(vREF)andtheactualoutputvoltageofthedc-dcconverter(v0).Themodulatingvoltageiscomparedtoasawtoothcarrier(vST)togeneratethePulse-WidthModulated(PWM)trainofpulses(vGS)thatisusedtocontrolthemainswitchofthebuckconverter.+vd-+v0-DRLLCR+-Controller∑vrefverrvSTvCTRLvGS+-Fig.3.1–Buckdc-dcconverterandcontrolcircuit.Figure3.2showstheblockdiagram(smallsignal)ofthesystem.++Hs()D0vvdST_0$vd$vCTRL$vi0$v0GsC()-+GsS()SensorController$_vref0$errFig.3.2–Blockdiagram(smallsignal)ofthesystem3.2–Basicdefinitions3.2.1–Looptransferfunctionandclosed-looptransferfunctionFortheanalysisofthecircuitperformanceandthedesignofthecontroller,itisessentialtodefinethelooptransferfunctionandtheclosedlooptransferfunctionsfortheoutputvoltageasafunctionofthereferencevoltagearegivenby:)()()(0_sHvvsGsLTFSTdC=(3.1))()(1)()()(0_0_sHvvsGsHvvsGsGSTdCSTdCCL+=(3.2)Theclosedlooptransferfunctionfortheoutputvoltageasafunctionoftheinputvoltageoftheconverterisgivenby:00_00)()(1)()(ˆ)(ˆDsHvvsGsHDsvsvSTdCd+=(3.3)3.2.2–BodePlotsThisisthemaintoolthatwillbeusedinthedesignofthecontrollerforthepowerelectronicconvertersinthiscourse.Figure3.3showssomeexamplesofasymptoticalBodeplotsrelatedtothebuckconverterwithunitfeedback.TheplantisrepresentedbyasecondordersystemduetotheLCoutputfilter.ThecontrollerisassumedtobeaPIpresentingthefollowingtransferfunction:GsKssC()=+1ττ(3.4)OneseesinFig.3.3(a)howthegainandthephaseofthecontroller(GC(s))andplant(H(s))varywithfrequency.Figure3.3(b)showsthelooptransferfunction(LTF(s)),whichisobtainedbyaddingthetwopreviousplots.Thereoneseesthatthecrossoverfrequencyorbandwidth(ωx)increasesduetothegainofthePIcontroller.Lookingattheplotrelatedtotheclosedlooptransferfunction,oneseesthatitpresentsagainofapproximately0dBforlowfrequencieswhere1)()()(=sLTFsHsGC.Conversely,forhighfrequencieswhereGsHsC()()1,thegainoftheclosedlooptransferfunctionisveryclosetothatofthelooptransferfunction.ωωC1/τω0°-90°-180°GC(s)H(s)ωωC1/τω0°-90°-180°GCL(s)LTF(s)Gain(db)Gain(db)PhasePhaseωx(a)(b)PMGMFig.3.3-BodePlots(a)ControllerandPlant;(b)Loopandclosed-looptransferfunctionByusingtheBodeplotsonecaneasilydesignthecontrollerthatwillresultinadesiredperformance.ThiscanbedonebyimposinganadequatevalueforthePhaseMargin(PM)forthelooptransferfunction(LTF(s))whichisdefinedasthevaluegivenbythedifferencebetweenthephaseofthelooptransferfunctionatthecrossoverfrequency(ωx)and-180°.TheGainMargin(GM)isdefinedasthedifferencebetween0dBandthegainofthelooptransferfunctionatthefrequencywherethephasebecomes-180°.Astablesystempresentspositivephaseandgainmargins.3.3–Classicalcontrollers3.3.1-Proportional(P)Figure3.4showsthetransferfunction,bodeplotandanimplementationwithoperationalamplifierofaproportionalcontroller.ωω0°Gain(db)Phase(c)K(a)-+RiRf(b)KRRfi=−Fig.3.4–PController;(a)Transferfunction;(b)Bodeplot;(c)ImplementationItpresentsafiniteerrorsteady-stateerror(eSS).Oneshouldkeepinmindthatthehigherthegain(K)thesmallertheerrorbutthelessstablethesystembecomesduetothereductionofthephasemargin.3.3.2-ProportionalplusIntegral(PI)Figure3.5showsthetransferfunction,bodeplotandanimplementationwithoperationalamplifierofaproportionalplusintegralcontroller.(c)Kss1+ττ(a)-+Ri(b)KRRfi=−ω1/τω0°-90°Gain(db)PhaseRfCτ=RCf10/τ0.1/τFig.3.5-PIController;(a)Transferfunction;(b)Bodeplot;(c)ImplementationThemaincharacteristicsofaPIcontrollerarethatitdoesnotpresentasteadystateerror.However,itreducesthephasemarginofthesystemwhatcancausetheinstabilityofasystemwithasecondorderplant.However,thisreductioncanbechosenduringthedesignofthecontrollerbytheappropriatechoiceofitstimeconstant(τ),resultinginasystemwithenoughdamping.ThezeroofthePIcanalsobecalculatedinordertocompensateforthelargesttimeconstantoftheplant.3.3.3-ProportionalplusDerivative(PD)Figure3.6showsthetransferfunction,bodeplotandanimplementationwithoperationalamplifierofaproportionalplusderivativecontroller.(c)()Ks1+τ(a)-+RiRf(b)KRRfi=−ω1/τω0°90°Gain(db)PhaseCiτ=RCii10/τ0.1/τFig.3.6-PDController;(a)Transferfunction;(b)Bodeplot;(c)ImplementationThemaincharacteristicsofaPDcontrollerarethatitincreasesthecrossoverfrequency,resultinginafasterresponse,andabiggerphasemargin.Ontheotherhand,thesystembecomessusceptibletohigh-frequencynoisebecauseofthehighgainatthehighfrequencyregion.3.3.4-ProportionalplusIntegralplusDerivative(PID)Figure3.7showsthetransferfunction,bodeplotandanimplementationwithoperationalamplifierofaproportionalplusintegralplusproportionalcontroller.(c)KsTsTid11++(a)(b)KRCRCRCiiffif=−+ω1/Tdω0°90°Gain(db)PhaseTRCRCiiiff=+-90°1/Ti-+RiCiRfCfTRCRCRCRCdiiffiiff=+Fig.3.7-PIDController;(a)Transferfunction;(b)Bodeplot;(c)ImplementationItpresentsthecharacteristicsofaPIcontrollerinthelowfrequencyrangeandimprovessteadystateperformance.Intheregioncloseto1/TditbehavesasaPDcontroller,improvingthestabilityandthe