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INTRODUCTION:Switchingpowersuppliesuseclosed-loopfeedbacktoachievedesignobjectivesforlineandloadregulationanddynamicresponse.Fortunately,theclosed-loopsystemsusedinswitchingpowersuppliesareusuallynotverycom-plicated,permittingtheuseofsimpleanalyticaltechniquestoachieveloopstabilization.Asimpli-fiedversionoftheNyquiststabilitycriteriacanbeusedbecauseunitygaincrossoveroccursonlyonceinthegainvs.frequencycharacteristic.Bodeplotsprovideasimpleandpowerfulmethodofdis-playingandcalculatingtheloopgainparameters(seeAppendixB).Thispaperbeginswithaquickreviewofbasiccontrollooptheory.LinearControlLoopTheoryAsshowninFigure1,apowersupplyfeedbackloopcanbedescribedintermsofsmall-signallin-earequivalentgainblocks.The(s)appendedtocertaingainblocksindicatesthatthegainvariesasafunctionoffrequency.KEA(S)ErroramplifierwithcompensationKMODPulsewidthmodulatorKPWRPowerswitchingtopologyKLC(S)OutputpowerfilterKFBFeedbackAlthoughthepulsewidthmodulatorandpowerswitchingcircuitarereallynotlinearelements,theirstate-spaceaveragedlinearequivalentscanbeusedatfrequenciesbelowtheswitchingfrequency,fS.Open-loopandclosed-loopgain:Theopen-loopgain,T,isdefinedasthetotalgainaroundtheentirefeedbackloop(whethertheloopisactuallyopen,forpurposeofmeasurement,orclosed,innormaloperation).T(s)=KEA•KMOD•KPWR•KLC•KFB(1)Closed-loopgain,G,definestheoutputvs.controlinputrelationship,withtheloopclosed:5-1ControlLoopCookbookControlLoopCookbookLloydH.DixonFigure1.-FeedbackLoopG(s)=(2)Atlowfrequencies,open-loopgainTisnor-mallyverymuchgreaterthan1,sothatclosed-loopgainGapproachestheideal1/KFB.Athigherfre-quencies,Tdiminishes,mostlybecauseofthelow-passfiltercharacteristicKLC(S).ThefrequencywhereThasdiminishedto1(0dB)isdefinedasthecrossoverfrequency,fC.ReferringtoEq.2andFigure2,atfC(whereT=1,withassociated90°phaselag),theclosed-loopgainG(s)is3dbdown(with45°phaselag).Thus,theopen-loopcross-overfrequencyisalsotheclosed-loop“cornerfrequency”,whereG(s)rollsoff.Inapowersupplyvoltagecontrolloop,G(s)definesthepowersupplyoutputvs.thereferencevoltage.KFBisusuallyasimplevoltagedivider.Forexample,ifVREFis2.5V,a2:1divider(KFB=0.5,G=2)resultsinVOUT=5Volts.(RefertoAppendixA.)Inatwo-loopsystem(aswithcurrent-modecontrol,tobediscussedlater)theclosed-loopgainG(s)oftheinnerloopisoneelementoftheopen-loopgainT(s)oftheouterloop.“Gain”elementsasshowninFigure1neednothavethesameunitsfortheiroutputandinput(suchasVolts/Volt).IfFig.1isacurrentmodecon-trolloop,“Output”isacurrentsource,andKFBismostlikelyacurrentsenseresistor.KFB“gain”isthenexpressedinVolts/Amp,andclosedloopgainG(s)isactuallyatransconductance(Amps/Volt).PulsewidthmodulatorKMODhasitsgainexpressedasd/V(Dutycycle/Volt).Thisdiscrep-ancyin“gain”unitsisresolvedinthenextgainblock,KPWR,whosecharacteristicisV/d.Overallopen-loopgainT(s)determineshowmuchoutputerrorresultsfromadisturbanceintro-ducedatanypointintheloopcomparedtotheresultiftheloopwasopen.Projectthedisturbanceforwardtotheoutput(multiplybythegainbetweenthedis-turbanceandtheoutput),thendividebytotalopen-loopgain,T.Forexample,withnofeedback(openloop,constantdutycycle),a10%changeinVINresultsina10%VOUTchange.Withthefeed-backloopclosed,ifTis100atthefrequencyofthedisturbance(DCinthisexample),thentheVOUTchangeisonly0.1%(10%/100).NotethattheOutputaccuracydoesnotdependsignificantlyonopen-loopgainaccuracy.Intheexampleabove,ifTwas80insteadof100,VOUTwouldchangeby0.125%(10%³VIN/80),insteadof0.1%.However,outputaccuracydoesdependdirectlyontheaccu-racyofthefeedbackportionofthecontrolloop,KFB.Alternatively,adisturbancecanbeprojectedbacktothesummingpointattheinputoftheerroramplifier.Forexample,the1Volt“valley”voltageofthesawtoothrampappliedtothePWMcompara-toriseffectivelya1VoltDCoffsetor“disturbance”.IftheE/Againis1000,this1Verrorisequivalenttoa1mVerrorinthereferencevoltage,andtrans-latesintothesamepercentageerrorattheoutput.NyquistStabilityCriteria:ReferringtoFigure2,iftheopen-loopgainTcrosses1(0dB)onlyonce,thesystemisstableifthephaselagatthecrossoverfrequency,fC,islessthan180°(inadditiontothenormal180°phaseshiftassociatedwithanynegativefeedbacksys-tem).Letusdefinetheterm“phaselag”torefertoanyadditionalamountofphaselagbeyondthe180°inherentwithnegativefeedback.Ifthe(addi-T1+T1KFB5-2ControlLoopCookbookFigure2.-Open&ClosedLoopGaintional)phaselagatfCexceeds180°,theloopwilloscillateatfrequencyfC.The“phasemargin”istheamountbywhichthephaselagatfCislessthanthecriticalvalueof180°.The‘gainmargin’isthefactorbywhichthegainislessthanunity(0dB)atthefrequencywherethephaselagreaches180°.IfthephaselagatfCisonlyafewdegreeslessthan180°(smallphasemargin),thesystemwillbestable,butwillexhibitconsiderableovershootandringingatfre-quencyfC.Aphasemarginof45°providesforgoodresponsewithalittleovershoot,butnoringing.NotethatNyquist’s180°phaselimitappliesonlyatfC.AtfrequenciesbelowfC,thephaselagispermittedtoexceed180°,eventhoughtheopen-loopgainisverymuchgreaterthan1.Thesystemisthensaidtobeconditionallystable.ButiftheloopgaintemporarilydecreasessothatfCmovesdownintothefrequencyrangewherethephaselagexceeds180°,conditionalstabilit