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LloydDixonmargin.Thusaphaselagof117°atfccorrespondstoa63°phasemargin.Fig.1showsthegain-phaseplotsoftwoaver-agecurrentmodecontrolloopsandtheirtransientresponsetoastepchangeinloadcurrent.Bothloopscrossoverat10kHz.Bodeplotsaresimplifiedgain-phaseplotsthatareconvenientfordepictingthefrequencycharac-teristicsofafeedbackloop.The1POLEloophasoneactivepolefrom10HztooverlookHz.Thusthephaselagatfcis90°andthephasemarginisalso90°.Notethatwiththe-1slopeassociatedwithasinglepole,thegaindoesnotriserapidlyatlowerfrequencies.ThelowerSummary:Thispapersurveysmanypracticalaspectsofcontrolcircuitdesign.Topicsinclude:differentapproachesforachievingNyquiststabilitycriteria,transientresponsevs.phasemargin,lowfrequencyaccuracyvs.theshapeoftheBodeplot,whyitisdangeroustodependonconditionalstability,achievingthemaximumcrossoverfrequencyofaswitchedloop.,otherfactorswhichmaylimitloopbandwidth,sourcesoferror,andwheretoplacethegainneededforcorrection.Otherpracticalaspectsincludelargesignalbehaviorandhowtominimizeoffseterrordelaysofcompensationcapacitorsinthefeedbackpath.ControlLoopBasicsThecontrolloopinahighperfonnanceswitchingpowersupplyrequireshighgain,toachievegoodregulation,andhighbandwidth,toachieverapidresponsetosuddenchangesoflineorload.Controlloopgaininevitablydeclinesathighfrequency,limitingbandwidth.Thefrequencyatwhichtheloopgaincrossesthrough1(OdB)isdefinedasthecrossoverfrequency,fc.Thedeclin-ingloopgainathighfrequencyisaccompaniedbyphaselag(inadditiontothenonnal180°phaseshiftassociatedwithnegativefeedback).Theamountofadditionalphaselagisafunctionofhowrapidlythegaindropswithfrequency.AccordingtoNyquist'stabilitycriteria,theloopwillbeunstableiftheadditionalphaselagexceeds180°atthecrossoverfrequency.Iftheexcessphaselagatfcisveryclosetothe180°limit,theloopwillbestablebutthetransientresponsewillexhibitunder-dampedoscillations(ringing).Acleantransientresponserequiresconsiderablylessthan180°excessphaselagatfc.Theamountlessthan180°isreferredtoasphaseFig.1-Gain-PhasePlotandTransientResponseControlLoopDesigngain/bandwidthwillhurtpowersupplycharacteris-ticssuchasregulationandaudiosusceptibility.Notealsothatthetransientresponseisslower.Theareabetweenthe[LOADstepchangeandthe1POLEresponsecurverepresentsachargedeficitintheoutputcapacitorthatisnevermadeup,exceptbytheoutputvoltageultimatelysaggingandcallingformorecurrent,Inthe2POLEloop,ifthetwopoleswereactivethroughandwellabovefc(IOkHz),thephaselagatfcwouldapproach180°-almostzerophasemargin.Thetransientresponsewouldexhibitsevereringing.Butthe2polesarenotactiveaboveSkHz.Azerointhecontrolamplifiercausesa-2to-1slopetransitionatfc/2,Thisresultsin117°phaselag,oraphasemarginof63°,BelowSkHz(fc/2),the-2slopecausesthegaintoriserapidlyatlowerfrequencies,optimizinggain-bandwidth,Withthe-1/-2zerotransitionatfc!2,transientresponseiscriticallydamped-seeFigure1.Theinitialchargedeficitisnotonlysmaller,theslightovershootcompensatespreciselyfortheinitialdeficit.Thisischaracteristicofacriticallydampedsystem.Atfrequencieswellbelowthezeroatfc!2,phaselagdoesapproach1800,andthephasemarginvanishes.Thisisdesirablebecauseitisassociatedwiththe-2slopewhichrisesrapidlyatlowerfrequencies,improvinggain-bandwidth.180°phaselagisalsoacceptable.Adequatephasemarginisrequiredonlyatfc.Thecircuitwillnotmisbehavewith180°phaselagatfrequenciesbelowfc,aslongasthereisadequatephasemarginatfc.Twopolesareshownoccurringnear10Hz:thepowercircuitinductancewithitsseriesresistance,andthecontrolcircuitoperationalamplifierreach-ingitsgainlimit.Loopgainflattensout,andexcessphaselagdiminishestoward0°.ConditionalStability:Thecriteriaforloopstabilitypennitsthephaseshifttoactuallyexceed180°(negativephasemargin)atfrequenciesbelowfc.Thisiscalledconditionalstabilitybecauseitdependsuponthestrictrequirementthatthephaseshiftbe1800(preferablymuchless)attheunitygaincrossoverfrequency.Thus,the180°limitationappliesonlyatfcwherethegainis1,notatanylowerfrequencyeventhoughthegainmaybehigh.Itisusuallynotwisetodependuponconditionalstability.Underconditionssuchasstartuporlargerapidloadchanges,operationalamplifiersintheloopmaybedrivenintothestops.Thisisequiva-lenttoatemporaryreductioninloopgainandcrossoverfrequency.Ifphaseshiftexceeds1800atthetemporarilylowerfc,thecircuitwilllikelycommencealargesignaloscillationfromwhichitcannotrecover.Insummary,thegaincharacteristicshouldhavea-I(Ipole)slopeasittraversestheunitygaincrossoverfrequency.Azeroatfc/2willresultina-2(2pole)slopeatlowerfrequencies.Thiswillprovideoptimumbandwidthandcriticallydampedtransientresponse.Thephaselagapproaches180°atlowerfrequencies,butthisisquiteacceptable.However,itisunwisetogobeyond180°anddependonconditionalstability.Withinadequatephasemargin,theringingfre-quencywillbeatornearfc.Crossoverfrequency:Theunitygaincrossoverfrequency,fc,isusuallythebeststartingpointforoptimumcontrolloopdesign,workingbacktowardlowerfrequenciestoobtainthebestpossiblegain-bandwidth.Theoretically,fcofalinearclosedloopsystemcouldbeatanyfrequency,providedthecriteriaforadequatephasemarginarefulfilled.Inpractice,itbecomesnecessary