Voltage Mode Boost Converter Small Signal …

1 ApplicationReportSLVA274A May2007 sinherentRightHalfPlane-zero(RHP-zero).T heboostconverter sdouble-poleandRHP-zeroaredependantonthe inputvoltage,outputvoltage,loadresistanc e,inductance,andoutputcapacitance, ;HighOutputCurrent(2A) ;HighOutputCurrent(2A) ;LowOutputCurrent(Io= ) (VIN= ,Vo= ,Io=2A,C=100 F 2tantalumcapacitor).. (VIN= ,Vo= ,Io=2A,C=100 F 2 CeramicCapacitor).. May2007 +-VCrC2020zeroRHP1 ZdoOdvsQs1s1s1Gd v Gw+ w+ w- w+ @=-(1)IN2o2 INdoVV)D1(VG=- (2)Cr1C1Z =w(3) p -2oINzeroRHPVVL2 Rfor2oINL2zeroRHPVVLRL)rR()D1( - - w-(4)oroIN2L0 VVCL1RR)D1(rCL1 -+ w p oINOVVLC21f(5)Q[w0rLL)1C R)rc (6) ,thesmall-signaltransferfunctionfromthed utycycle(D)oftheswitchtotheboostconverte routput(vo)incontinuousconductionmode(CC M) (SLVA061)[1].]

2 , , May2007 RevisedJanuary2009 SubmitDocumentationFeedback-40dB/dec-180 Double-polefGainPhase0 0dB-90 oINoVVLC21f p=+20dB/dec-90 fGainPhase0 x fRHP_zero10 x fRHP_zero2oINzero_RHPVVL2Rf p= Odependsontheinputvoltage(VIN)andtheoutp utvoltage(Vo)aswellasinductance(L)andout putcapacitance(C).Figure3showsaBodeploto ftheRHP-zero, RHP-zerodependsonloadresistance(R)andind uctance(L)aswellasinputvoltage(VIN)andou tputvoltage(Vo). RHP-zero,thegainincreasesat20dB/decfrom May2007 p=-(7) m mp=(8)-40dB/dec-20dB/dec180 -90 fGainPhase0 +20 margin becomes Small ornegative by RHP-zero impactDC gainoINoVVLC21f p=2oINzero_RHPVVL2Rf p= :L=10 H,Co=100 FVIN=3 VVo= (R= )Thefrequencyof Oand RHP-zeroarecalculatedusingEquation4andEq uation5asfollows.

3 AsshowninFigure4,eventhoughtheadditionof theRHP-zeroisatahigherfrequencythantheco nverterdoublepole,theRHP-zerophasedropst artsadecadeearlier,andthereforenegativel yimpactsthepotentialphasemarginoftheconv erter ,the Oand RHP-zerodependonVIN,Vo,R(Io),L, (smallerloadresistanceR)hasasignificanti mpacton RHP-zero,bringingitcloserto o, May2007 {}{}iCLC22 CLCC1 OdvRV)rR(r)rR'D(R'D)s(PR'D)rR)(rsL()1sCr )(rR()s(XDc)s(XDA)AsI(cD)s(v)s(G ++++++-++= + -=DD=-(9)P(s)+s2LC R)rc 2)sNJL R)rc )rLC R)rc 2)D RrcC R)rc Nj)rL R)rc )D R D R)rc (10) )s(G)s(G)s(G)s(Gerrorffc_FBdv =(11)+-Error AmpFBVINVOrLLdovCrCFeedback and ffc network)(sGerror)( sGdv)( sESR(rcinFigure1)andtheinductorsDCR(rL) , ,Figure5showstheblockdiagramofatypicalbo ostconverter,consistingofthepowerstage,f eedbackandfeed-forwardcompensationnetwor k, (s)isgivenas;Where,Gdv(s);Transferfuncti onofpowerstageofboostconverterGFB_ffc(s) ;Transferfunctionoffeedbacknetworkandfee dforwardcompensationnetworkGerror(s); May2007 RevisedJanuary2009 ConverterSmallSignalControlLoopAnalysisU singtheTPS610305 SubmitDocumentationFeedback2 MethodforStabilizingtheTPS61030-40dB/dec - 180 fGainPhase0 GO- 90 - 270 Black; double pole systemBlue; RHP-Zero systemRed.

4 Feed forward compensation(1-zero, 1-pole system)Phase Boost contribution bycompensation circuitfpfzReflected RHP-zero impactReflected Phase boostcontributionfofBWfRHP-zeroMffozeroR HP -zBWf2f +20dB/dec+20dB/decoINoVVLC21f p=2oINzero_RHPVVL2Rf p= ,thefollowingconditionsmustbemet:1. RHP-zeromustbehigherinfrequencythan O(byratioMtobeexplainedlater)inordertopr eventtheRHP-zero sphasedropfromaffectingthepowerstage ( BW)mustbesetatorlowerthanthefrequencyoft hemaximumphase-boosteffectgeneratedbyfee d-forwardcompensationnetworkwithzeroseta t BWissethigherthanthefrequencyofthemaximu mphaseboost,thephase-boosteffectbecomess mallwhilegainincreases, ,assumethatthemaximumphase-boostfrequenc yis2 May2007 [STEP1]SelectOutputCapacitorVo_ripple_ES R+Io ESR(12)f1VV1 CICTIVoINoonocap_ripple_o - = =(13)f1VV1 VICoINcap_ripple_ooripplemin_ - =(14)BWdip_otrandip_otrantranmin_f41 VidtViC =(15) [STEP2]DetermineInductanceDCRDC lossL2inrI (byignoringripplecomponent)

5 ,andsetthemaximumacceptableinductorDClos sto30%ofrL_MAX+30% Ptotal*lossI2in(16)Where,Ptotal_loss+Pou t 1h*1 , (Equation12)andtheoutputcapacitorripple( Equation13). , ,ifthetransientduration(dt)andloadcurren t(itran)isknown,Equation15canbeusedtodet erminetheminimumrequiredcapacitancetopre ventoutputvoltagedip(VO-dip)duetoaloadtr ansient:Wheredtisapproximatedto1/4 sESR,theinductor sDCR(rL)hasasignificantimpactontheboostc onverterpowerstage' , , , ,SW(N-MOSFET)loss,Diode(P-MOSFET)loss, ,thedesignercanignoretheinductor sAClosses,computetheinductor ,settingatargetefficiency ,SLVA274A May2007 [STEP3]SeparatingtheDouble-PoleandRHP-Ze roFrequenciesMffozeroRHP -(17)2oINVVMRCL (18)2omin_INminMAXVVMRCL =(19) [STEP4]MaximizePhaseBoostEffectofFeedFor wardCompensationNetworkG(s)+1)sCi R1)Ri 1)R1R2)sCi R1)Ri)R1 RiR2 (20) z+12pCi R1)Ri (21)R1R2Ci = 10 pFRi = 100 kWFBVoInternal ffc p+12pCi R1R2)RiR2)R1 RiR1)R2(22)

6 RHP-zeroand o, to100m allowsavalueofM= ,thusprovidingalittlephasedampingbetween thetwofrequencies,morefrequencyseparatio nisrequired(M 15).NoteM=10orM=15arestartingpointsandfi nalstabilityshouldbeverifiedbytransientt estinginthelab(asmentionedin[STEP5]).Whe re,M=10(onedecade) ,wecanget:WhenloadresistanceRisatitsmini mum(meaningIoisMAX)andtheinputvoltageisa titsminimum,the oand RHP-zeroareclosest(asshowninEquation4and Equation5) , (ffc) ,zerofrequencyandpolefrequencyequationsa reshowninEquation20throughEquation22, May2007 RevisedJanuary2009 SubmitDocumentationFeedback100 45 045 90 )))f)(AFB(Phasepw)))f)(AFB(Gainpw()2R1RC 21fiz+ p = =() () () + + + p = =f - Hzp ZBWf2f (23)30oG10ffoBW @(24) + = + =2R1R2R5 VVlog202R1R2R5 Glog20)dB(GIN2odoo(25)2Z30 GoINf210VV21C1Lo p (26)2zoMAX_INminf210VV21C1L30oG p =(27) ,R2=200k ,Ci=10pF,andRi=100k.

7 ,fp=58kHz, ,assumethemaximumfrequencyforphaseboostf romthefeed-forwardnetworkis2 :Notingthatthedouble-polehasa 40-dB/dectheoreticalroll-off,but 30dB/decwasusedtoapproximatethedampingef fectsoftheoutputcapacitor'sESRandtheindu ctor' :AsshowninEquation5 o;therefore, BW, , May2007 [STEP5]EvaluatingtheStabilityExperimenta lly[STEP 1] Select Cout and ESR[STEP 3] Maximum Inductanceby separation of double - pole and RHP - zero[STEP 4] Minimum Inductanceby optimum phase Boost effectCheck L > HmNO[STEP 5] Evaluation;(1) Bode- plot(2)Load transient testSpec;V (min)IN- V (MAX)INV /IooVo_rippleTransientYESresult[STEP 2] Define DCR of , ,therisetimeofthecurrentstepshouldbefast erthan1/ ,butalsothelargesignalresponsebehaviorth atoccursinnormalapplications, , May2007 RevisedJanuary2009 SubmitDocumentationFeedback3 Example1;HighOutputCurrent(2A)WithTantal umCapacitorRi = 100 kWCi = 10 pF+-Error AmpR1R2 SWFBVBATGNDVINVoCrCrLTPS61030 Internal ffcL15 m DCRW200 F/25 - = - =(28)F200kHz1041mV250A2f41 ViCBWdip_otrantanmin_m= = =(29)Go(dB)+20 log V2oVi 5 R2R1)R2 + (dB)(30))MAX( m= =(31)(min) p m= p =(32)rL_MAX+PInductance_lossI2in_MAX+ A2+33 mW(33).

8 HighOutputCurrent(2A) :VIN= ( )Vo_ripple=5mV(bycapacitance)Targeteffic iency90%(Ploss= ,Iin= )Vo_transient=250mV(assumetransientIo=0 Ato2A,and BW=10kHzforcalculationpurposebelow)Calcu latingtheminimumcapacitanceusingthecrite riaoftheoutputvoltageripplefromEquation1 4; ;RipplecausedbyESRisVo_ripple_ESR=Io ESR=2A 25m = ;FromtheresultofEquation28andEquation29, 200 ,thefeedbacknetworkconsistsofR1= andR2=200k .TPS61030hasinternalfeedforwardcompensat ionnetwork(RiandCi) May2007 ( ) ,()()GainAwfq( ) ,()()GainAwfq( ) ,()()PhaseAwfq( ) ,()()PhaseAwfq( ) ,()()PhaseAwfq( ) ,()()fqCalculation condition is ;Vin 3= 2 104- = 106 =R22 105 =Ri1 105 =Ci1 1011- =Example1;HighOutputCurrent(2A) ,Pinductance_loss=30% PLOSS=30% ,theallowableinductancerangeis1 ,theallowableDCR(rL)oftheinductorislesst han33m.

9 Figure11showsaMathcad -calculatedBodeplotofthetransferfunction includingtheboostpowerstage,feedbacknetw ork,feedforwardcompensationnetwork, H, (M)achievedusingtheseinductancevaluesare asfollows:L= HisM=8L= HisM=11L= HisM=15 Thecalculationsconsolidatetheinductor sDCRandtheONresistanceofinternalSWintoon e50m May2007 RevisedJanuary2009 SubmitDocumentationFeedbackfRHP -zero=29kHzfo= L= HmGainPhasefRHP -zero=51kHzfo= (meetM>10)50degL= L= HmGainPhasefRHP -zero=95kHzfo= (meetM>10) L= HmGainPhaseSmall ringingVout= (50mV/div) ringingNo ringingI (1A/div) ;HighOutputCurrent(2A) H(frequencyseparationratioM=8), H,whichisM=11, ,M> . (VIN= ,Vo= ,Io=2A,C=100 F 2tantalumcapacitor)SLVA274A May2007 RevisedJanuary2009 ConverterSmallSignalControlLoopAnalysisU singtheTPS6103013 SubmitDocumentationFeedback4 Example2;HighOutputCurrent(2A)WithCerami cCapacitorRi = 100 kWCi = 10 pF+-Error AmpR1R2 SWFBVBATGNDVINVoCrCrLTPS61030 Internal ffcL15 mW200 F/5 mCeramicmWH2VV15 RCL2omin_INminceramic_MAXm= =(34)(min) W p m= p =(35)Example2;HighOutputCurrent(2A) , > ,Fromthecalculationresults,therangeofind uctanceis1 Hto2 May2007 ( ) ,()()GainAwfq( ) ,()()GainAwfq( ) ,()()PhaseAwfq( ) ,()()PhaseAwfq( ) ,()()PhaseAwfq( ) ,()()fqCalculation condition is ;Vin 3=Vo3.

10 6=R 2 104- = 103- = 106 =R22 105 =Ri1 105 =Ci1 1011- = ;HighOutputCurrent(2A)WithCeramicCapacit orFigure14showsaMathcad-calculatedBode-p lotofthetransferfunctionwithESR=5m (assumingcapacitorESRandconnectionresist ancetotal5m ).ThephasecurveissteeperinFigure14thanFi gure11, H, Hwereusedtodemonstratehowstabilityimprov eseventhoughL= H(M=8)andL= H(M=11)donotsatisfytheM>= forL= H,and30 forL= H(M=15)inductorprovidesa45 May2007 RevisedJanuary2009 ConverterSmallSignalControlLoopAnalysisU singtheTPS6103015 SubmitDocumentationFeedback15degL= L= HmGainPhasefRHP-z=29 kHzfo= kHzM =8L= L= HmGainPhaseffRHP-zeroRHP-zero=51 kHzfo= kHzM =1147degL= L= HmGainPhase=95 kHzfo= kHzM =15(meetM>=15)RingingLittle ringingNo ringingV = (50mV/div)I(1A/div).