IEEEJOURNALOFSOLID-STATECIRCUITS,VOL.SC-Q,NO.6,DECEMBER1974347StanleyF.MoyerwasborninHamburg,Pa.,onSeptember8,1931.HegraduatedfromCapitolRadioEngineeringInstitute,Washington,D.C.in1956.Hehasalsoat-tendedAlbrightCollege,Reading,Pa.jandLafayetteCollege,Easton,Pa.In1956hejoinedBellLaboratories,Read-ing,Pa.,wherehehasworkedonthedevelopmentofgermaniumdiffusedbasetransistors,silicontransistors,andfield-effecttransistors.Since1966,hehasworkedonthedevelopmentofsiliconmonolithicintegratedcircuits.[51R.A.StehlinandG.W.Niemann,“Complementarytran-sistor-transistorlogic(CTzL)—anapproachtohigh-speedmicropowerlogic,”IEEEJ.Solid-StateCircuits,vol.SC-7,pp.153–160,Apr.1972.(S’64)-M’61)wasborninGlenville,W.Vs.,onMarch14,1937.HereceivedtheB.S.EE.degreefromWestVirginiaUniversity,Morgantown,in1959,theM.S.degreefromMassachusettsInsti-tuteofTechnology,Cambridge,in1961,andthePh.D.degreeinelectricalengineeringfromLehighUniversity,Bethlehem,Pa.,in1968.HehasbeenaTeachingAssistantatM.I.T.andservedtwoyearsasanarmyofficerintheOrdnanceGuidedMissileSchool,Huntsville,Ala.In1962hebecametories,Reading,aMemberPa.HehasoftheTechnicalbeenengagedStaff,BellinLabora-characterizationPaulC.Davisandmodelingoftransistorsandinthedesignoflinearanddigitalintegratedcircuitry.Dr.DavisisamemberofEtaKapDaNuandTauBetaPi,andanassociatememberofSigmaXi.“-VeikkoR.SaariwasborninBrooklyn,N.Y.,onApril24,1928.HereceivedtheB.S.andM.S.degreesinphysicsfromtheUniversityofMinnesota,Minneapolis,in1951and1956,respectively.Since1956hehasbeenworkingonthedesignofsolid-statecircuitsatBellLab-oratories,Holmdel,N.J.Mr.SaariisamemberoftheAmericanAssociationofPhysicsTeachers.RelationshipBetweenFrequencyResponseandSettling‘TimeofOper;tion~AmplifiersB.YESHWANTKAMATH,STUDENTMEMBER,IEEE,ROBERTG.MEYER,PAULR.GRAY,MEMBER,IEEEMEMBER,IEEE,ANDAbstracf—Theeffectsofpole-zeropairs(doublets)onthefre-quencyresponseandsettlingtimeofoperationalamplifiersareexploredusinganalyticaltechniquesandcomputersimulation.Itisshownthatdoubletswhichproduceonlyminorchangesincircuitfrequencyresponsecanproducemajorchangesinsettlingtime.Theimportanceofdoubletspacingandfrequencyareexamined.Itisshownthatsettlingtimealwaysimprovesasdoubletspacingisreducedwhereastheeffectof.doubletfrequencyisdifferentfor0.1and0.01percenterrorbands.Finallyitisshownthatsimpleanalyticalformulascanbeusedtoestimatetheinfluenceoffre-quencydoubletsonamplifiersettlingtime.I.~MANYINTRODUCTIONapplicationsofoperationalamplifiers,thesettlingtimeisanimportantparameter[1].TheIsettlingtimeisthetimetakenfortheoutputoftheamplifiertosettletowithin0.1or0.01percentaftertheManuscriptreceivedMay13,1974;revisedAugust8,1974.ThisworkwassupportedbytheJointServicesElectronicsProgramunderContractF44620-71-C-@187.TheauthorsarewiththeDepartmentofElectricalEngineeringandComputerSciencesandtheElectronicsResearchLaboratory,IJniversityofCalifornia,Berkeley,Calif.9472Y3.applicationofaninputstep.Thistestisusuallydoneinaunity-gainconfigurationwitha1O-Vstepinputandthustheerrorbandsontheoutputare10and1mV,re-spectively,for0.1and0.01percentaccuracy.ThesetestsareimportantinspecifyingtheoperationalamplifierforuseinsuchapplicationsasA/DandD/Aconverters.Thesettlingtimeofanamplifieriscomposedoftwodistinctperiods[1].Thefirstperiodiscalledtheslewtimeduringwhichtheamplifieroutputmakesthetransi-tionfromtheoriginaloutputvoltagetothevicinityofthenewvalue.Duringthisperiodtheamplifieractsinagrosslynonlinearfashionandthelengthofthisperiodisgenerallydeterminedbythecurrentavailabletochargetheamplifiercompensationcapacitance.Thesecondpor-tionofthesettlingtimeistheperiodafterslewlimitingwhentheamplifieroutputisnearitsfinalvalueandthecircuitactsinaquasi-linearfashion.Ithasbeenshown[2],[3]thatthisperiodissignificantlyaffectedbythepresenceofpole-zeropairs(calleddoublets)intheam-plifiertransferfunction.Inhigh-speedoperationalampli-fiers,theslewtimecanbeveryshortandthissecondAuthorized licensed use limited to: University of Electronic Science and Tech of China. 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Restrictions apply. 348IEEEJOURNALOFSOLID-STATECIRCUITS,DECEMBER1974Fig.1.Operationalamplifiermodelforcalculationresponse.oftransientperiodinthequasi-linearregioncandominatetheset-tlingtimeoftheamplifier.Thecircuitbehaviorclescribedaboveisdifficulttoevaluateinpractice.Directmeasurementsoftheseas-pectsofamplifierperformanceareclifficulttoperformandtheeffectsofdifferentparametersinthecircuitaredifficulttoisolate.Inthispapertheeffectofcloubletsonthefrequencyresponseandsettlingtimeofafastopera-tionalamplifierisinvestigatedusinganalyticaltech-niquesandcomputersimulation.II.THEORYslewrateoftheamplifierthroughthecompensationcapacitor.Theycanalsohaveaverystrongeffectonsettlingtimeifthephasemarginisallowedtobecometoosmall.However,inanoptimumdesign,thesepoleswillproduceonlyaslightlyunderdampedresponse,andthisresponsewillbeshortindurationcomparedtothetransientsintroducedbythedoublet.Acloselyspaceddoubletisassmneclandthedoubletfrequencyisassumedmuchlessthantheamplifierunity-gainfrequency.ThecircuitofFig.1isanalyzedintheAppendixwhereitisshownthattheresponsetoastepinputofamplitudeVis:V..,(t)=V(l–k,exp[–co.ot]+k,exp[–(t/r,)]),fort>T,where(1)(2)‘r2?—1Q.(3)Doubletsinanamplifiertransferfunctioncanarisefromanumberofcauses.Forexample,bypassingonesideofanactiveloaclathighfrequencies[4]leadstotheintroductionofapole-zeropairwithanoctavesepa-ration.Theuseofafeedforwardcapacitortobypassalateralp-n-ptransistor[2]cancausethecreationofadoubletwith&30percentmismatchbetweenpoleandzerofrequencies.Previousanalysesoftheeffectsofadoublethaveas-sumedlinearoperationofthecircuitinquestion[1]–[3].However,theapplicationofa1O-Vsteptoanopera-tionalamplifierresultsinquitenonlinearoperationdur-ingtheslewingperiod,andalinearanalysiscannotbejustifiedforthiscase.Amodelwhichrepresentsthesitu-ationinapracticalamplifierisshowninFig.1.Theam-plifierisconnectedinaunity-gainfeedbackloop.Thedifferentialinputstageismodeledasshownwithmaxi-mumavailablecurrent10andatransfercharacteristicslopeofg,mforvaluesof]V~llessthan(10/0,,,}.ForvaluesofIV~I>(lO/g,~),theinputstagelimitsatacur-rentl..Thisisthemajorsourceofnonlinearityinthecircuitduringthetransientresponse.Adoubletisas-sumedpresentintheamplifierfrequencyresponseto-getherwiththedominantpoleproducedbythecompen-sationcapacitor.TheseareshownschematicallyinFig.1.Sincethisportionofthecircuitisassumedtoactlinearly,theprecisecletailsofthecircuitarrangementarenotcritical,However,toconformtothepracticalexampleconsideredlater,acurrentIIisassumedfedfromtheinputstagetothecircuitproducingthedoublet,andacurrentIzisassumedfedfromthiscircuittotheoutput.Theuseofvoltagevariablesdoesnotchangetheresultsoftheanalysis.Inthisanalysis,thehigherfrequencypolesnearanclbeyondcrossoverhavebeenneglected.ThesepolesarecertainlyimportantsincetheylimitthebandwidthandslewingperiodW.doubletzerofrequencyUPdoubletpolefrequencyCOCOAX(amplifierdominantpole)bandwidthAopen-looplowfrequencygain.T,sunity-gainEquation(1)indicatesthepresenceofaslowsettlingcomponentintheoutputwithatimeconstant(l/~,)andamagnitudekzV.Equation(2)indicatesthatforagivenfractionaldoubletspacing(forexample,~.=1.3Or)themagnitudeoftheslowsettlingcomponentisproportionaltothedoubletfrequency,andthemagnitudeoftheinputstep.However,(3)indicatesthatthetimeconstantoftheslowsettlingcomponentisinverselyproportionaltothedoubletfrequency.Thuslowerfrequencydoubletswillgivearesponsewhichpersistsforalongerperiodbutwithasmalleramplitude.Higherfrequencydoubletswillgivearesponsewhichdiesoutfasterbuthasalargeramplitude.Therelativeimportanceofthesetrendswilldependontheparticularsituation.Ifsettlingto0.1percentonlyisimportant,alowerfrequencydoubletmaygiveaslow-settlingcomponentwhichisalwayswithintheerrorbandandthusdoesnotdegradeperformance.Forsettlingto0.01percentthesamedoubletmaycausegreatincreasesinsettlingtime.Ahigherfrequencydoubletislikelytoproducesettlingtimedegradationinallcasesbutitseffectwilldieawaymuchsooner[3].III.COMPUTERSIMULATIONTheanalyticalexpressionsderivedaboveuseagreatlysimplifiedmodeloftheoperationalamplifier.Inordertocheckthevalidityoftheseapproximationsandtofurtherinvestigatetheseeffects,computersimulationswereperformedonthePA772operationalamplifier[2].ThisamplifierwaschosenbecauseitisahighspeedamplifierwithafastsettlingtimeandtheeffectsofAuthorized licensed use limited to: University of Electronic Science and Tech of China. 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ICAMATHetd:FREQUENCYRESPONSEANDSETTLINGT1i%f10349artificiallyintroduceddoubletsareeasytodistinguish.Theamplifieritselfhasadoubletduetotheuseoffeed-forwardbutadoubletcompressionschemewhichisanintegralpartoftheamplifiermakestheeffectofthisdoubletundetectable.Inordertoobservetheeffectofadoubletontheresponseoftheamplifier,onehalfoftheactiveloadwasbypassedwithacapacitorinasimilarfashiontotheLM118[4].Byvaryingthemagnitudeofthecapacitorandaresistorinserieswithit,doubletiwereintroducedatdifferentfrequencieswithdifferentpole-zeroseparations.Theopen-loopunity-gainfre-quencywaskeptconstantinallcasesatOJCO=6.1MHzbysmalladjustmentsinthevalueofthecompensationcapacitor.ThecompensationcapacitorusedwaslargerLhantheoneintheoriginal,pA772toallowfortheseadjustments.TheprogramusedwastheSPICEprogramdevelopedattheUniversityofCalifornia,Berkeley.SPICEhasageneralnonlineartransientanalysiscapabilitywithcom-prehensivelarge-signaldevicemodels.Thecomputedopen-loopgainandphaseresponseforthebasicampli-fierareshowninFig.2(a)and(b)togetherwiththeresponsewhenadoubletat18kHzisincluded.Evenwiththeadditionofsuchadoubletwith32percentmis-matchbetweenpoleandzerofrequencies,thechangeinthefrequencyresponseissmall.Forasmallerdoublet(suchas8percentpole-zeromismatch)theeffectonfrequencyresponseisindiscernibleonthesescales.InFig.2(c)anexpandedgraphisshownofphaseversusfrequencyfortheamplifierwithandwithoutdoublets.Thedoubletsusedherewereat18kHzwithpole-zeromismatchesof32,20,and8percent.Itisapparentthatthe8percentdoublethasverylittleeffectonthefre-quencyresponse,butitwillbeshowntohaveasignifi-canteffectonthesettlingtime.InFig.3(a)thecomputedtransientresponseisshownwitha1O-Vinputstep(–5to+5V)fortheoriginalcircuitinaunity-gainvoltagefollowerconfiguration.Onthisscale,theadditionofdoubletsproducesanalmostimperceptiblechangeinresponse.InFig,3(b),thisre-sponseisshownonanexpandedscaletoallowobserva-tionof0.1percent(10mV)settlingtime.Thisisseentobeabout0.43,psforthecaseofnodoublet,withabout0.2psofthisbeingslewingtimeandtherestbeingcausedbyovershoot.Becauseoftheneglectofhigherfrequencypoles,thisovershootisnotpredictedbytheanalysisintheAppendix.AlsoshowninFig.3(b)areresponsecurveswithdou-bletsaddedtotheamplifier.Itisapparentthatthedoubletat180kHzwith32percentspacingproducestheworstdegradationofsettlingtimewithavalueof1.56ps.The32percentdoubletat18kHzgivesasettlingtimeof0.80~s,eventhoughitpersistsmuchlongerduetoitslongertimeconstant.Themuchsmallerinitialvalueoftheslowsettlingcomponentduetothe18kHzdoubletresultsinitsdecayingbelow10mVsoonerthanisthecaseforthe180kHzdoublet.120-Gain(dB)NOdoublet1.32Doubletot18kHz80-40-0IHzIIkHzFrequenty1MHzIIOMHZI(a),.I!izIkiiz‘IMHzhlHzFrequency(b)-90Phase(degree)1-954DoubletOt18kHz1.32OoubletspacingNodoublet1.081.2-1oo“1IH:‘T1II(’HzFrequency(c)IMHz10idHzFig.2.Computedopen-loopfrequencyresponseoftheamplifierwithandwithoutdoublets.(a)Gainversusfrequency.(b)Phaseversusfrequency.(c)ExpandedPha*versusfreWencY.areshownevenfurtherex-pandedinFig.3(c)sothat0.01percent(1mV)settlingtimescanbedetermined.Inthiscase,foranygivenfractionaldoubletspacing,thelowerfrequencydoublethasbyfartheworsteffectonsettlingtime.Thispossi-bilitywaspointedoutinSectionII.Eventhoughthehigherfrequencydoubletsgivelargerinitialamplitudesoftheslowsettlingcomponent,theveryslowdecayforthelowerfrequencydoubletsmakesthemmuchmoreThegraphsofFig.3(b)significantforl-mVsettlingtime,Theworstcaseisforthe32percentdoubletseparationatafrequencyof18kHzgivingal-mVsettlingtimeof14.14ps.TheseresultsareillustratedinFig.4wherecomputedsettlingtimesareshownasafunctionofdoubletspacingAuthorized licensed use limited to: University of Electronic Science and Tech of China. Downloaded on March 20,2010 at 03:08:30 EDT from IEEE Xplore. Restrictions apply. %5QIEEEJOURNALOFSOLID-STATECIRCUITS,DECEMBER1974I5.0L0.01\\0.51.32Do,blelal180kHz1.32Doubletof18kHz%Nod~~bl~,1.0Time(1,5(~sec)(2,0(a)(b)I\\1,0NodoubletL1.512.0Time[pee}(c)Fig.3.Computedunity-gaintransientresponseoftheampIifiertoa1O-Vinputstepfrom—5to+5V,(a)Linearvoltagescale.(b)and(c)ExpandedvoltagescaJe.DoubletSettlhgtoImVat18kHz180kHz180kHz18kHzr1.2Doubletspacing,@zIUp11.310mV11.1Fig.4.Computedunitygainsettlingtimeversusdoubletspacingfortwodifferentdoubletfrequenciesandsettlingaccuracies,Inputstepfrom–5to+-5V.Authorized licensed use limited to: University of Electronic Science and Tech of China. Downloaded on March 20,2010 at 03:08:30 EDT from IEEE Xplore. Restrictions apply. KAMATHf?~Ctl.:FREQUENCYRESPONSEANDSETTLINGTIMETABLEIDoubletSpacingcomputerDoubletat18kHzDoubletat180kHzNoDoublet1.081.21.321.081.21.328.096.536.150.8250.7080.654ktv,(mV)predicted8.147.296.880.8120.7260.683computer2.466.259.9724.667.2106predicted2.285.819.1924.262.998.710-mVSettlingTime(J&s)computer0.440.480.800.771.341,560.43predicted0,470.510.570.721.341.56l-mVSettlingTime(W)computer7.312.014.22.623.013.050.56351(2)predicted6.712.815.22.593.013.14(18and180kHz).Thisshowsclearlythatfor10-mVsettling,the180-kHzdoubletgivestheworstdegradationwhereasforl-mVsettlingthelongtimeconstantassociatedwiththe18-kHzdoubletcausesthistobetheworstcase.Thesetestswererepeatedfornegativestepsfrom+5to–5Vandsimilarresultswereobtained.ThecomputedresultsaresummarizedinTableI.Theconsistenttrendtowardssmallersettlingtimeisap-parentasthedoubletspacingisreduced.AlsoshowninTableIarevaluesof.,andk2Vcalculatedfrom(2)and(3)togetherwithvaluesdeterminedfromthecom-putedresponses.Thevaluesof~zweresimplydeter-minedfromthetailoftheexponentialdecayandthevaluesofk2Vweredeterminedbyextrapolatingbackfromtheregionoftheexponentialdecay.Quitecloseagreementisseeninthevaluesdeterminedfromthetwomethods.Finally,TableIshowscalculatedvaluesofsettlingtimeusing(1).Theseshowgoodagreementwiththecomputedvalues.fortwodifferentdoubletfrequenciesFinally,ithasbeenshownquitegenerallythatthesimpleformulasderivedherecanbeusedtopredicttheinfluenceoffrequencydoubletsonoperationalamplifiersettlingtime.APPENDIXThecircuitofFig.1istobeanalyzedforastepinputofmagnitudeVwhereingeneralv>I,/g..(4)IV.CONCLUSIONSIthasbeenshownthatfrequencydoubletsinthetransferfunctionofanoperationalamplifiermaycauseseveredegradationofsettlingtimewhileonlycausingminorchangesinthefrequencyresponseoftheampli-fier.Theeffectofthedoubletdependsbothonitsfre-quencyandthepole-zerospacing.Reductionofthepole-zerospacingofthedoubletreducesitseffect,butpole-zerofrequencymismatchesassmallas8percentcancausesignificantincreasesinsettlingtime.Theeffectofthedoubletfrequencyonsettlingtimeismorecomplex.For0.01percentsettling,alowerfre-quencydoubletcanpotentiallycausetheworstperform-ancedegradationbecauseofitslongtimeconstant.How-ever,for0.1percentsettling,thesmalleramplitudeofalowerfrequencydoubletmaymeanthatitisalwayswithintheerrorbandandhaslittleeffect.Inthatcasea‘higherfrequencydoubletwouldcausemoresettlingtimedegradationbecauseofitslargeramplitude,eventhoughitdecayedfaster.Inthissituation,theinitialvoltageVdappliedtothedifferentialinputstageissufficienttocauselimitingandaconstantcurrent10issuppliedtothedoubletandout-putstage.Computersimulationshowsthatinpracticethisoccurs.Theperiodwhentheinputstageislimitingrepresentstheslewingperiodoftheamplifierandthefeedbackloopiseffectivelyopenduringthisperiod.WhentheoutputvoltageVo,,tapproacheswithinIo/gmofV,thedifferentialstageentersitslinearregionandtheamplifiersettlestoitsfinalvaluewiththefeedbackloopclosed.Assumptionsmadearethatthedoubletiscloselyspacedandthedoubletfrequencyismuchlessthantheamplifierunity-gainfrequency,i.e.,~.<
T,V.”,(t)EV(l–k,’exp[–co..(t–T,)]+kz’exp[–u.(fwhere~,~(%/(0=)–11(OJJ’CI).)–1–T,)])(18)B.YeshwantKamath(S’72)wasborninMangalore,India,onAugust26,1948.HereceivedtheB.Tech.degreeinelectricalengineeringfromtheIndianInstituteofTechnology,Madras,India,in1969,andtheM.S.E.E.degreefromSyracuseUniver-sity,Syracuse,N.Y.,in1971.Heispres-entlyattheUniversityofCalifornia,Berkeley,studyingtowardsthePh.Ddegreeinelectricalengineering.Hisinterestsareinbioelectronicsandin-gratedcircuits.[(1–:‘‘“=1–(C+Jco.)‘2[1+(a.o/uJ–11fort>T,.(21)Equation(18)canbewrittenasVo”t(t)NV(l–k,exp[–co.ot]+kzexp[–coct]),(22)RobertG.Meyer(S’64–M’68),foraphotographandbiography,pleaseseep.166oftheAugust1974issueofthisJOURNAL.Since0.<
