开关电源输入滤波器的设计

InputFilterDesignforSwitchingPowerSupplies:

WrittenbyMicheleSclocchiMichele.Sclocchi@ApplicationEngineer,NationalSemiconductorThedesignofaswitchingpowersupplyhasalwaysbeenconsideredakindofmagicandart,foralltheengineersthatdesignoneforthefirsttime.Fortunately,todaythemarketoffersdifferenttoolsthathelpthedesigners.NationalSemiconductorwasthefirstcompanytoofferthe“SimpleSwitcher”software,andon-linesimulationtoolthatallowsthedesignandsimulationofaswitchingpowersupply.Newultra-fastMOSFETsandsynchronoushighswitchingfrequencyPWMcontrollersallowtherealizationofhighefficientandsmallerswitchingpowersupply.Alltheseadvantagescanbelostiftheinputfilterisnotproperlydesigned.Anoversizedinputfiltercanunnecessarilyaddcost,volumeandcompromisethefinalperformanceofthesystem.Thisdocumentexplainshowtochooseanddesigntheoptimalinputfilterforaswitchingpowersupplyapplication.Theinputfilteronaswitchingpowersupplyhastwoprimaryfunctions.Oneistopreventelectromagneticinterference,generatedbytheswitchingsourcefromreachingthepowerlineandaffectingotherequipment.Thesecondpurposeoftheinputfilteristopreventhighfrequencyvoltageonthepowerlinefrompassingthroughtheoutputofthepowersupply.ApassiveL-Cfiltersolutionhasthecharacteristictoachievebothfilteringrequirements.Thegoalfortheinputfilterdesignshouldbetoachievethebestcompromisebetweentotalperformanceofthefilterwithsizeandcost.UNDAMPEDL-CFILTER:ThefirstsimplepassivefiltersolutionistheundampedL-Cpassivefiltershowninfigure(1).Ideallyasecondorderfilterprovides12dBperoctaveofattenuationafterthecutofffrequencyf0,ithasnogainbeforef0,andpresentsapeakingattheresonantfrequencyf0.：=2e1LCInputimpedanceSwitchingPowerSupplyFigure1:UndampedLCfilterCutofffrequency[Hz](resonancefrequency)：=2e1LCInputimpedanceSwitchingPowerSupplyFigure1:UndampedLCfilter20100 1-1(3 1-1C4 1-105Frequency,Hz20100 1-1(3 1-1C4 1-105Frequency,HzFigure2:TransferFunctionofL-CFilterfordifferertsdampingfactorsSecondOrderInputfilterooooo1123___Bd斯40Oneofthecriticalfactorsinvolvedindesigningasecondorderfilteristheattenuationcharacteristicsatthecornerfrequencyf.Thegainnearthecutofffrequencycouldbeverylarge,andamplifythenoiseatthatfrequency.Tohaveabetterunderstandingofthenatureoftheproblemitisnecessarytoanalyzethetransferfunctionofthefilter:Voufilter(S) 1Ffilter1(s)：= 一牛1ter() 1+s-一+L-C-s2RloadThetransferfunctioncanberewrittenwiththefrequencyexpressedinradians:1- 12.L1—L-C-°+j-°-RloadFfilter1^)：=s：=j-°1°0：=0 L-C八 八21+j-2・J—-——°0 %2CutofffrequencyinradiantDampingfactor(zeta)Thetransferfiinctionpresentstwonegativepolesat: _r.①(+_.,，＜—1Thedampingfactor。describesthegainatthecornerfrequency.For。＞1thetwopolesarecomplex,andtheimaginarypartgivesthepeakbehaviorattheresonantfrequency.Asthedampingfactorbecomessmaller,thegainatthecornerfrequencybecomeslarger,theideallimitforzerodampingwouldbeinfinitegain,buttheinternalresistanceoftherealcomponentslimitsthemaximumgain.Withadampingfactorequaltoonetheimaginarycomponentisnullandthereisnopeaking.Apoordampingfactorontheinputfilterdesigncouldhaveothersideeffectsonthefinalperformanceofthesystem.Itcaninfluencethetransferfunctionofthefeedbackcontrolloop,andcausesomeoscillationsattheoutputofthepowersupply.TheMiddlebrook'sextraelementtheorem(paper[2]),explainsthattheinputfilterdoesnotsignificantlymodifytheconverterloopgainiftheoutputimpedancecurveoftheinputfilterisfarbelowtheinputimpedancecurveoftheconverter.Inotherwordstoavoidoscillationsitisimportanttokeepthepeakoutputimpedanceofthefilterbelowtheinputimpedanceoftheconverter.(Seefigure3)Onthedesignpointofview,agoodcompromisebetweensizeofthefilterandperformanceisobtainedwithaminimumdampingfactorof1/寸2,whichprovidesa3dBattenuationatthecornerfrequency,andafavorablecontroloverthestabilityofthefinalcontrolsystem.mO0.011001-mO0.011001-103 1-104Frequency,Hz1-105Figure3:Otputimpedance0theinptfilter,andirputimpedance0theswitchngpcwesupply:thetwocuresshouldbewellseparated.PARALLELDAMPEDFILTER:Inmostofthecasesanundampedsecondorderfilterlikethatshowninfig.1doesnoteasilymeetthedampingrequirements,thus,adampedversionispreferred:SwitchingPowerSupplySwitchingPowerSupplyFigure4showsadampedfiltermadewitharesistorRdinserieswithacapacitorCd,allconnectedinparallelwiththefilter’f.scapacitorCThepurposeofresistorRdistoreducetheoutputpeakimpedanceofthefilteratthecutofffrequency.ThecapacitorCdblocksthedccomponentoftheinputvoltage,andavoidsthepowerdissipationonRd.ThecapacitorCdshouldhavelowerimpedancethanRdattheresonantfrequency,andbeabiggervaluethanthefiltercapacitor,tonoteffectthecutoffpointofthemainR-Lfilter.TheoutputimpedanceofthefiltercanbecalculatedfromtheparallelofthethreeblockimpedancesZ1,Z2,andZ3:1Zfilter2(s)：=—1 1 1一+ +Z1(s) Z2(s) Z3(s)s-L-(1+Rd-Cd.s)

s3-L-C-Cd.Rd+s2-L-(C+Cd)+s.Rd-Cd+1Thetransferfunctionis:Z…Ffilter2(s)：=7驾=3Z1+Zeq2.3 s3-L-C-Cd-Rd1+RdC.s+s2-L-(C+Cd)+Rd-Cd-s+1WhereZ。〃isZparallelwithZ.eq2.3 2 3Thetransferfunctionpresentsazeroandthreepoles,wherethezeroandthefirstpolefallclosetoeachotheratfrequencyg1/RdCd.Theothertwodominantpolesfallatthecutofffrequency,®o=l/^LC.Withoutcompromisingtheresults,thefirstpoleandFfilter2(s)：= ( )~. 7L-VC+Cd)-s2 L-C-Cd-Rd-s31+(T^s)+(17^)1+L-C(n+1)-s2+L-CQ-Rd-s3

+ Rd-C-n-s + Rd-Cd-sWhere Cd:=n-Cthezerocanbeignored,andtheformulacanbeapproximatedtoasecondorderone:(forfrequencieshigherthanw»1/RdCd,theterm(1+RdCds)»RdCds)Theapproximatedformulafortheparalleldampedfilterisidenticaltothetransferfunctionoftheundampedfilter;theonlydifferencebeingthedampingfactor。is匕.n+1LC2：= i n2-RdEL-CcalculatedwiththeRdresistance.Itisdemonstratedthatforaparalleldampedfilterthepeakingisminimizedwithadampingfactorequalto:。2opt:|(2+n)-(4+3。2opt:! 2-n2-(4+n)Combiningthelasttwoequations,theoptimumdampingresistancevalueRdisequalto:Rdopt2-Rdopt2-n2-(4+n)J(2+n)-(4+3-n)withn=4Cd：=4-CWiththeblockingcapacitorCdequaltofourtimesthefiltercapacitorC.Figures5and6showstheoutputimpedanceandthetransferfunctionoftheparalleldampedfilterrespectively.Frequency,HzFigure6:Transferfunctionoftheparallelpdalrfilter.Frequency,HzFigure6:Transferfunctionoftheparallelpdalrfilter.SERIESDAMPEDFILTER:AnotherwaytoobtainadampedfilteriswitharesistanceRdinserieswithaninductorLd,allconnectedinparallelwiththefilterinductorL.(figure7)Atthecutofffrequency,theresistanceRdhastobeahighervalueoftheLdimpedance.Figure7:SeriesdampedfilterSwitchinglPowerSupplylFigure7:SeriesdampedfilterSwitchinglPowerSupplylZfilter3(s)：=1 1 1Z1Zfilter3(s)：=1 1 1Z1(s)+Z2(s)+Z3(s) s~L1+工.s+

(n+1)ns2-L-C- n+1Z2Ffilter3(s)：=^^qzRd+sG+Ld)Rd+0+Ld)s+LCRd-s2+L-Ld。s3Theoutputimpedanceandthetransferfunctionofthefiltercanbecalculatedthesamewayastheparalleldampedfilter:s-L-(Rd+Ld-s)Rd+(L+Ld)-s+L-C-Rd-s+Rd-^-s+s2.LC-(n+Rd-^-s+s2.LC-(n+1) n+1Fromtheapproximatedtransferfunctionoftheseriesdampedfilter,thedampingfactorcanbecalculatedas:..=1.Rd 笠 •—2'(n+1)*Thepeakingisminimizedwithadampingfactor:whereLwhereLd：=n-L_:'n・(3+4.n).(1+2.n)°3oPt：= 2.(1+4n)2n2n:=一15Rd：=2《3opf(n+F,_s)：= er Zm/s).F,_s)：= er Zm/s).Zm(s) Zm(s)+Zm(s)Z叫(s)+Zm2(s)+冲s)+" &+s£+L》 Rd+sC+D+s2%&+L?q+L1旬+日［弓R•.+l)+以萄+L1-C1-.+动+s4-L1乌。弓母+s5-L1-L2-Li-C1-c2Thedisadvantageofthisdampedfilteristhatthehighfrequencyattenuationisdegraded.(Seefigure10).MULTIPLESECTIONFILTERS:Mostofthetimeamultiplesectionfilterallowshigherattenuationathighfrequencieswithlessvolumeandcost,becauseifthenumberofsinglecomponentsisincreased,itallowstheuseofsmallerinductanceandcapacitancevalues.(figure8)Figure8:Figure8:TwosectioninputfilterTheoutputimpedanceandthetransferfunctioncanbecalculatedfromthecombinationofeachblockimpedance:'Zm.(s)-Zm2(s) )Zfilter4(s)：=lZm")+Z^(s)+冲Zfilter4(s)：=Zm1(s)-Zm2(s)Zm1(s)+Zm2(s)+时‘)+哑s^Lr+L2^Rr+sELr^L2+L^+L2L3'+^L1■L)1C1■^Rj~+~Zm2(s)Rd+&C+L)+蔬氏+L?•弓+L1妇+s3［曲-C+L)+板订+L1-C1-C+功+s4L•L2-C1。场+s%弘。-CZm2(s)L1：=2 L2:=7-L1 Ld4:=~2C2:=4-C1Rd4:=:4.；Thetwo-stagefilterhasbeenoptimizedwiththefollowingratios:Thefilterprovidesanattenuationof80dBwithapeakfilteroutputimpedancelowerthan20.1-1041-105101001-1041-105101001-103Frequency,HzFigure9:Otputimpedance0theseiescbmpedfilter,ndtwestagecampedfilter.Figure10:Transferfuctionoftheseriesdampedfilter,andtwo-stagedarpedfilter.Theswitchingpowersupplyrejectsnoiseforfrequenciesbelowthecrossoverfrequencyofthefeedbackcontrolloop,andhigherfrequenciesshouldberejectedfromtheinputfilter.Tobeabletomeettheforwardfilteringwithasmallsolution,theinputfilterhastohavethecornerfrequencyaroundonedecadebelowthebandwidthofthefeedbackloop.CAPACITORANDINDUCTORSELECTION:Anotherimportantissueaffectingthefinalperformanceofthefilteristherightselectionofcapacitorsandinductors.ForhighfrequencyattenuationcapacitorswithlowESLandlowESRforripplecurrentcapabilitymustbeselected.Themostcommoncapacitorsusedarethealuminumelectrolytictype.ToachievelowESRandESLth