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·~•094ISBN0·19·511644-5TheOxfordSeriesinElectricalandCompulerEngineeringAdelS.SedraSeriesEditorAllenandHolberg,CMOSAnalogCircuitDesign,2ndEditionBobrow,ElementaryLinearCircuitAnalysis,2ndEdi1ionBobrow,FundamentalsofElectricalEngineering,2ndEditionBurnsandRoberts,AnIntroductiontoMixed-Signal/CTestandMeasurementCampbeU,TheScienceandEngineeringofMicroelectronicFabrication,2ndEditionChen,Analog&DigitalControlSystemDesignChen,LinearSystemTheoryandDesign,3rdEditionChen,SystemandSig11alAnalysis,2ndEditionChen,DigitalSignalProcessingComer,DigitalLogicandStateMachineDesign,3rdEditionCooperandMcGillem,ProbabilisticMethodsofSignalandSystemAnalysis,3rdEditionDeCarloandLin,LinearCircuitAnalysis,2ndEditionDimitrijev,UnderstandingSemiconductorDevicesFortney,PrinciplesofElef:tmnics:Analog&DigitalFranco,ElectricCircuitsFundamental.sGranzow,DigitalTransmi,fsionLine.rGuruandHiziroglu,ElectricMachineryandTransformers,3rdEditionHooleandHoole,AModemShortCourseinEngineeringElectromo.gneticsJones,IntroductiontoOpticalFiberCommunicationSystemsKrein.ElementsofPowerElectronicsKuo,DigitalControlSystems,3rdEditionLathl,ModernDigitalandAru~logCommunicationsSystems,JrdEditionLathi,SignalProcessingandLinearSystemsLathl,LinearSystemsandSigru~lsMartin,DigitalintegratedCircuitDesignMcGillemandCooper,ContinuousandDiscreteSignalandSystemAnalysis,3rdEditionMiner,LinesandElecrromagneticFieldsforEngineersParhami,ComputerArithmeticRobertsandSedra,SPICE.2ndEditionRoulston,AnIntroductiontothePhysicsofSemiconduc:torDevicesSadiku,ElementsofElectroma1fnetics,3rdEditionSantina,Stubberud,andHostetter,DigitalControlSystemDesign,2ndEditionSarma,IntroductiontoElectricalEngineeringSchaumannandVanValkenburg,DesignofAnalogFiltersSchwarz,Elec:tmmagneticsforEngineersSchwarzandOldham,ElectricCilEngineering:AnIntroduction,2ndEditionSedraandSmith,MicroelectronicCircuits,4thEditionStefani,Savant,Shahian,andHostetter,DesignofFeedbackConrrolSystems,4thE.ditionVanValkenburg,AnalogFilterDesignWarnerandGrung,SemiconductorDeviceli:lectronicsWarnerandGrung,MOSFETTheoryandDesignWolovich,Automo.ricControlSystemsYariv,OpticalElectronicsinModemCommunications.5thEditionCMOSAnalogC~rcuitDes~gnSecondEditionPhillipE.AllenGeorgiaInstituteofTechnologyDouglasR.HolbergCygnalIntegratedProducts,Inc.NewYorkOxfordOXFORDUNIVERSITYPRESS2002tontentsPrefacexiiiChapter1IntroductionandBackground11.1AnalogIntegrated-CircuitDesign1.21.3Notation,Symbology,andTerminology1.4ExampleofAnalogVLSlMixed-SignalCircuitDesign1.5Summary15AnalogSignalProcessingProblemsReferences1691617Chapter2CHOSTechnologq182.1U2.3H2.52.6a.7BasicMOSSemiconductorFabricationProcessesTttepnJunction29TheMOSTransistor36PassiveComponents43OtherConsiderationsofCMOSTechnologyIntegratedCircuitLayout55Summary66Problems68References70481910vitiCONTENTS'·'..Cbnpter3CMOSDeviceHodeling723.1SimpleMOSLarge-SignalModel(SPICELEVELI)3.23.33.4OtherMOSLarge-SignalModelParameters79Small-SignalModelfortheMOSTransistor87ComputerSimulationModels92UUSPICESimulationofMOSCircuits3.7Summary109Problems110SubthresholdMOSModelReferences9799112Chapter4RnalogCMOSSubcircuits1134.14.24.3113MOSSwitchMOSDiode/ActiveResistorCurrentSinksandSources124126uCurrentMirrors4.SCurrentandVoltageReferencesuBandgapReference4.7Summary159Problems159References134143153166Chapter5CMOSAmplifiers167S.lS.25.35.4S.S5.&Inverters168DifferentialAmplifiersCascadeAmplifiers1BO199CurrentAmplifiers211OutputAmplifiers218High-GainAmplifierArchitectures22973•S.7Summary232Problems233ReferencesContents•242ChapterGCMOSOperationalAmplifiers2436.1&.2G.36.4DesignofCMOSOpAmps244CompensationofOpAmps253DesignofTwo-StageOpAmps269Power-SupplyRejectionRatioofTwo-StageOpAmps286&.SCascadeOpAmps2936.66.7SimulationandMeasurementofOpAmpsG.BSummary341Problems342MacromodelsforOpAmps310323References349Chapter7High-PerformanceCMOSOpHmps3517.17.27.37.47.57.67.7BufferedOpAmps352High-Speed/FrequencyOpAmpsDifferential-OutputOpAmpsMicropowerOpAmpsLow-NoiseOpAmpsLow-VoltageOpAmpsSummary368384393402415432Problems433References437Chapter8Comparators4398.18.2CharacterizationofaComparator4391\vo-Stage,Open-LoopComparators4451xxCONTENTS8.38.18.58.68.7OtherOpen-LoopComparators461ImprovingthePerformanceofOpen-LoopComparatorsDiscrete-TimeComparatorsHigh-SpeedComparatorsSummary488Problems488References464475483491Chapter9SwitchedCapacitorCircuits4929.19.2SwitchedCapacitorCircuitsSwitchedCapacitorAmplifiers507USwitchedCapacitorIntegrators5209.4.z-DomainModelsofTwo-PhaseSwitchedCapacitorCircuits9.5First-OrderSwitchedCapacitorCircuits9.6Second-OrderSwitchedCapacitorCircuits5509.79.8SwitchedCapacitorFiltersSummary600Problems600References493532544561611Chapter10Digital-AnalogandAnalog-DigitalConverters6121~.1IntroductionandCharacterizationofDigital-AnalogConverters10.2ParallelDigital-AnalogConverterslUExtendingtheResolutionofParallelDigital-AnalogConverters10.410.510.610.7SerialDigital-AnalogConverters623665Medium-SpeedAnalog-DigitalConverters1~.8High-SpeedAnalog-DigitalConvertersJU.!OversamplingConverters698635647IntroductionandCharacterizationofAnalog-DigitalConvertersSerialAnalog-DigitalConverters613667682652Contents10.10Summary713Problems715ReferencesHppendiXAHppendiX8HppendiXCIndex777729CircuitAnalysisforAnalogCircuitDesignCMOSDeviceCharacterization744TimeandFrequencyDomainRelationshipsforSecond-OrderSystems768733xlPREFACETheobjectiveofthesecondeditionofthisbookcontinuestobetoteachthedesignofCMOSanalogcircuits.Theteachingofdesignreachesfarbeyondgivingexamplesofcircuitsandshowinganalysismethods.Itincludesthenecessaryfundamentalsandbackgroundbutmustapplytheminahierarchicalmannerthatthenovicecanunderstand.ProbablyofmostimportanceistoteachtheconceptsofdesigninganalogintegratedcircuitsinthecontextofCMOStechnology.Theseconcept.~enablethereadertounderstandtheoperationofananalogCMOScircuitandtoknowhowtochangeitsperformance.Withtoday'scomputer-orientedthinking,itisvitaltomaintainpersonalcontrolofadesign,toknowwhattoexpect,andtodiscernwhensimulationresultsmaybemisleading.Asintegratedcircuitsbecomemorecomplex,itiscrucialtoknow"howthecircuitworks."Simulatingacircuitwithouttheunderstandingofbowitworkscanleadtodisastrousresults.Howdoesthereaderacquiretheknowledgeofhowacircuitworks?Theanswertothisquestionbasbeenthedrivingmotivationofthesecondeditionofthistext.Thereareseveralimportantstepsinthisprocess.Thefirstistolearntoanalyzethecircuit.ThisanalysisshouldproducesimpleresultsthatcanbeunderstoodandreappliedindifferentcircumStances.Thesecondistoviewanalogintegratedcircuitdesignfromahierarchicalviewpoint.Thismeansthatthedesignerisabletovisualizehowsubcircuitsareusedtoformcircuits,howsimplecircuitsareusedtobuildcomplexcircuits,andsoforth.Thethirdstepistosetforthproceduresthatwillhelpthenewdesignercomeupwithworkingdesigns.Thishasresultedintheinclusionofmany"designrecipes,"whichbecamepopularwiththefirsteditionandhavebeenexpandedinthesecondedition.ItisimportantthatthedesignerrealizethattherearesimplythreeoutputsoftheelectricaldesignofCMOSanalogcircuits.Theyare(1)aschematicofthecircuit.(2)decurrents,and(3)WILratios.Mostdesignftowsor''recipes"canbeorganizedaroundthesethreeoutputsveryeasily.Fifteenyearsago,itwasnotclearwhatimportanceCMOStechnologywouldhaveonanalogcircuits.However.ithasbecomeveryclearthatCMOStechnologyhasbecomethetechnologyofchoiceforanalogcircuitdesigninamixed-signalenvironment.This"choice"isnotnecessarilythatofthedesignerbutofindustrytrendsthatwanttousestandardtechnologiestoimplementanalogcircuitsalongwithdigitalcircuits.Asaresult,thefirsteditionofCMOSAnalogCircuitDesignfulfilledaneedforatextinthisareabeforetherewereanyothertextsonthissubject.Ithasfoundextensiveuseinindustryandhasbeenusedinclassroomsallovertheworld.Likethefirstedition,thesecondeditionhasalsochosennottoin·eludeBITtechnology.Thewisdomofthischoicewillbeseenastheyearsprogress.Thesecondeditionhasbeendevelopedwiththegoalofextendingthestrengthsofthefirstedition,namelyintheareaofanalogcircuitdesigninsightandconcept'!.xllllilYPREFACEThesecondeditionhasbeenalongtimeincomingbuthasresultedinauniqueblendingofindustryandacademia.Thisblendingba~occurredoverthepast15yearsinshortcoursestaughtbythefirstauthor.Over50shoncourseshavebeentaughtfromthefirsteditiontoover1500engineersaUovertheworld.Intheseshortcourses,theengineersdemandedtounderstandtheconceptsandinsighttodesigninganalogCMOScircuits,andmuchoftheresponsetothesedemandsha~beenincludedinthesecondedition.Inadditiontotheindustrialinputtothesecondedition.theauthorshavetaughtthismaterialatGeorgiaInstituteofTechnologyandtheUniversityofTexa.~atAustinoverthepast10-15year.;.Thisexperienceha.~providedinsightthathasbeenincludedinthesecondeditionfromtheviewpointofstudentsandtheirquestions.Also.theacademicapplicationofthismaterialhasresultedinalargebodyofproblem.~thathavebeengivenastest.~andhavenowbeenincludedinthesecondedition.Thefirsteditionhad335problems.Thesecondeditionhasover500problems,andmostofthosearenewtothesecondedition.Theaudienceforthesecondeditionisessentiallythesameasforthefirstedition.ThefirsteditionwasveryusefultothosebeginningacareerinCMOSanalogdesign-manyofwhomhavecommunicatedtotheauthorsthatthetexthasbeenareadyreferenceintheirdailywork.Thesecondeditionshouldcontinuetobeofvaluetobothnewandexperiencedengineersinindustry.Theprinciplesandconceptsdiscussedshouldneverbecomeoutdatedeventhoughtechnologychanges.Thesecondaudienceistheclassroom.TheoutputofqualifiedstudentstoenterthefieldofanalogCMOSdesignhasnotmetthedemandfromindustry.OurhopeisthatthesecondeditionwillprovidebothinstructorsandstudentswithatoolthatwillhelpfulfiIIthisdemand.Inordertohelpfacilitatethisobjective,bothauthorsmaintainwebsiteslhalpennitthedownloadingofshortcourselectureslides,shortcourseschedulesanddates,classnotes,andproblemsandsolutionsinpdfformat.Moreinformationcanbefoundatwww.aicdesign.org(P.E.Allen)andwww.holberg.org(D.R.Holberg).TheseNitesarecontinuallyupdated,andthereaderorinstructorisinvitedtomak.euseoftheinformationandteachingaidescontainedonthesesites.Thesecondeditionhasreceivedextensivechanges.ThesechangesincludethemovingofChapter4ofthefirsteditiontoAppendixBofthesecondedition.Thecomparatorchapterofthefirsteditionwasbeforetheopampchaptersandhasbeenmovedtoaftertheopampchapters.Inthe15yearssinceth-efirstedition,thecomparatorhasbecomemorelikeasenseamplifierandlesslikeanopampwithoutcompensation.ADUljorchangehasbeentheincorporationofChapter9onswitchedcapacitorcircuits.Therearetworeasonsforthis.Switchedcapacitorsareveryimportantinanalogcircuitsandsystemsdesign,andthisinformationisneededformanyoftheanalog-digitalanddigital-analogconvertersofChapter10.Chapter11ofthefirsteditionhasbeendropped.Therewereplanstoreplaceitwithachapteronanalogsystemsincludingphase-lockedloopsandVCOs,buttimedidnotallowthistoberealized.Theproblemsofthesecondeditionareorganizedintosectionsandhavebeendesignedtoreinforceandextendtheconceptsandprinciplesassociatedwithaparticulartopic.ThebierachicalorganizationoftilesecondeditionisillustratedinTable1.1-2.ChapterlpresentsthematerialnecessarytointroduceCMOSanalogcircuitdesign.ThischaptergivesanoverviewofthesubjectofCMOSanalogcircuitdesign,definesnotationandconvention,mak~abriefsurveyofanalogsignalprocessing,andgivesanexampleofanalogCMOSdesignwithemphasisonthehierarchialaspectofthedesign.Chapters2and3formthebasisforanalogCMOSdesignbycoveringthesubjectsofCMOStechnologyandmodeling.Chapter2reviewsCMOStechnologyasappliedtoMOSdevices,pnjunctions.passivecomponentscompatiblewithCMOStechnology,andothercomponent.~suchalithelateralandsubstrate:,,.'.PrefacexvBJTandlatchup.Thischapteralsoincludesasectionontheimpactofintegratedcircuitlayout.Thisportionofthetextshowsthatthephysicaldesignoftheintegratedcircuitisasimportantastheelectricaldesign,andmanygoodelectricaldesignscanberuinedbypoorphysicaldesignorlayout.Chapter3introducesthekeysubjectofmodeling,whichisusedthroughouttheremainderofthetexttopredicttheperformanceofCMOScircuits.ThefocusofthischapteristointroduceamodelthatisgoodenoughtopredicttheperfonnanceofaCMOScircuittowithin::t10%to±20%andwillallowthedesignerinsightandunderstanding.Computersimulationcanbeusedtomoreexactlymodelthecircuitsbutwillnotgiveanydirectinsightorunderstandingofthecircuit.ThemodelsinthischapterincludetheMOSFETlarge-signalandsmall-signalmodels,includingfrequencydependence.Inaddition,howtomodelthenoiseandtemperaturedependenceofMOSFETsandcompatiblepassiveelementsisshown.Thischapteralsodiscussescomputersimulationmodels.Thistopicisfartoocomplexforthescopeofthisbook,butsomeofthebasicideasarepresentedsothatthereadercanappreciatecomputersimulationmodels.OthermodelsforthesubthresholdoperationarepresentedalongwithhowtouseSPICEforcomputersimulationofMOSFETcircuits.Chapters4and5representthetopicsofsubcircuitsandamplifiersthatwillbeusedtodesignmorecomplexanalogcircuits,suchasanopamp.Chapter4coverstheuseoftheMOSFETasaswitchfollowedbytheMOSdiodeoractiveresistor.ThekeysubcircuiL~ofcurrentsinks/sourcesandcurrentmirrorsarepresentednext.The.o;esubcircuitspermittheillustrationofimportantdesignconceptssuchasnegativefeedback,designtradeotfs,andmatchingprinciples.Finally,thischapterpresentsindependentvoltageandcurrentreferencesandthebandgapvoltagereference.Thesereferencesattempttoprovideavoltageorcurrentthatisindependentofpowersupplyandtemperature.Chapter5developsvariousrypesofamplilien;.Theseamplifiersarecharacterizedfromtheirlarge-signalandsmall-signalperformance,includingnoiseWidbandwidthwhereappropriate.Thecategoriesofamplifiersincludetheinverter,differential.cascode.current,andoutputamplifiers.Thelastsectiondiscusseshowhigh-gainamplifien;couldbeimplementedfromtheamplifierblocksofthischapter.Chapters6,7,and8representexamplesofcomplexanalogcircuits.Chapter6introducesthedesignofasimpletwo-stageopamp.Thisopampisusedtodeveloptheprinciplesofcompensationnecessaryfortheopampto-beu.o;eful.Thetwo-stageopampisusedtoformallypresentmethodsofdesigningthistypeofanalogcircuit.Thischapteralsoexamine.,thedesignofthecascodeopamps.particularlythefolded-cascodeopamp.Thischapterconcludeswithadiscussionoftechniquestomeasureand/orsimulateopampsandmacromodels.Macromodelscanbeusedtomoreefficientlysimulateopampsathigherlevelsofabstraction.Chapter7presentsthesubjectofhigh-performanceopamps.Inthischaptervariousperformancesofthesimpleopampareoptimized,quiteoftenattheexpenseofotherperformanceaspects.Thetopicsincludebufferedoutputopamps,high-frequencyopamps,differentialoutputopamps,low-poweropamps,low-noiseopamps,andlow-voltageopamps.Chapter8presentstheopen-loopcomparator,whichisanopampwithoutcompensation.Thisisfollowedbymethodsofdesigningthistypeofcomparatorforlinearorslewingresponses.Methodsofimprovingtheperformanceofopen-loopcomparators,includingautozeroingandhysteresis,arepresented.Finally,thischapterdescribesregenerativecomparatorsandhowtheycanbecombinedwithlow-gain,high-speedamplifierstoachievecomparatorswithaveryshortpropagationtimedelay.Chapters9and10focusonanalogsystems.Chapter9iscompletelynewandpre.or,orrmsvaloeofthesignalLowercaseUppercaseUppercaseLoowercasc:UppercaseLowercaseUpperctieLowerc8StlExample.,.a..q.Q.Figure1.2-1showshowthedefinitionsinTable1.2-1wouldbeappliedtoaperiodicsignalsuperimposeduponadevalue.Thisnotationwillbeofhelpwhenmodelingthedevices.Forexample,considertheportionoftheMOSmodelthatrelatesthedrain-sourcecurrenttothevariousterminalvoltages.Thismodelwillbedevelopedintennsofthetotalinstantaneousvariables(in).Forbia.~ingpurposes,thedevariables(JIJ)willbeused;forsmall-signalanalysis,theacvariables(id)willbeused;andfinally,thesmall-signalfrequencydiscussionwillusethecomplexvariable(/d).Theseconditemtobediscussedhereiswhatsymbolsareusedforthevariouscomponents.(Mostofthesesymbolswillalreadybefamiliartothereader.However,inconsistenciesexistabouttheMOSsymbolshowninFig.1.2-2.)ThesymbolsshowninFigs.1.2-2(a)and1.2-2(b)areusedforenhancement-modeMOStransistorswhenthesubstrateorbulk(8)isconnectedtotheappropriatesupply.Mostoften,theappropriatesupplyisthemostpositiveoneforp-channeltransistorsandthemostnegativeoneforn-channeltransistors.Althoughthetransistoroperationwillbeexplainedlater,theterminalsarecalleddrain(D),gate(G),andsource(S).Ifthebulkisnotconnectedtotheappropriatesupply,thenthesymbolsshowninFigs.J.2-2(c)and1.2-2(d)areusedfortheenhancement-modeMOStransistors.ItwillbeimportanttoknowwherethebulkoftheMOStransistorisconnectedwhenitisusedincircuits.Figure1.2-3showsanothersetofsymbolsthatshouldbedefined.Figure1.2-3(a)representsadifferential-inputoperationalamplifieror,insomeinstances,acomparator,whichmaybaveagainapproachingthatoftheoperation.alamplifier.Figures1.2-3(b)and1.2-3(c)representanindependentvoltageandcurrentsource.respectively.Sometimes,thebatterysymbolisusedinsteadofFig.1.2-3(b).Finally,Figs.1.2-3(d)through1.2-3(g)representthefourtypesofidealcontrolledsources.FigureL.2-3(d)isavoltage-controlledvoltagesource(VCVS),Fig.1.2-3(e)isavoltage-controlledcurrentsource(VCCS),Fig.1.2-3(f)isacurrent-controlledvoltagesource(CCVS},andFig.1.2-3(g)isacurrent-controlledcurrentsource(CCCS).ThegainsofeachofthesecontrolledsourcesaregivenbythesymbolsAwGm,R,,andA1(fortheVCVS,VCCS,CCVS.andCCCS,respectively).Figure1.1-1Notationforsignals.I8INTRODUCTIONANDBACKGROUNDFiKUJ'I!1.2-1MOSdevicesymbols.(a)Enhancementn-channeltransistorwithbulkconnectedtomostnegativesupply.{b)Enhancementp-channeltransistorwithbulkconnectedtomostpositivesup-ply.(c),(d)Sameas(a)and(b)exceptbulkconnectionisnotconstrainedtnrespectivesupply.(ll)(cl/20+(d)vo~---····c(e)~l'E:J~·L(0(&IFigurel.l-3(a)Symbolforanoperationalamplifier.(b)Independentvoltagesource.(c)Jndependentcurrentsource.(d)Voltagecontrolledvoltagesource(VCVS).(e)Voltage-controlledcurrentsource(VCCS).(f)Current·controlledvoltagesow-ce(CCVS).(g)CUJ'I"Cnt-controlledcurrentsource(CCCS).1.3AnalogSignalProcessing9URNRLOGSIGNRLPROCESSINGBeforebeginninganin-depthstudyofanalogcircuitdesign.itisworthwhiletoconsidertheapplicationofsuchcircuits.Thegeneralsubjectofanalogsignalprocessingincludesmostofthecircuitsandsystemsthatwillbepresentedinthistext.Figure1.3-1showsasimpleblockdiagramofatypicalsignal-processingsystem.Inthepast,suchasignal-processingsystemrequiredmultipleintegratedcircuitswithconsiderableadditionalpassivecomponents.However,theadventofanalogsampled-datatechniquesandMOStechnologyhasmadeviablethedesignofageneral!iignalprocessorusingbothanaloganddigitaltechniquesonasingleintegratedcircuit[2].Thefirststepinthedesignofananalogsignal-processingsystemistoexaminethespec,ificationsanddecidewhatpartofthesystemshouldbeanalogandwhatpartshouldbedigital.Inmostcases,theinputsignalisanalog.Itcouldbeaspeechsignal,asensoroutput,aradarreturn,andsoforth.ThefirstblockofFig.1.3-1isapreprocessingblock.Typically,thisblockwillconsistoffilters,anautomatic-gain-controlcircuit.andananalog-to-digitalconverter(ADCorA/D).Often.verystrictspeedandaccuracyrequirementsareplacedonthecomponentsinthisblock.Thenextblockoftheanalogsignalprocessorisadigitalsignalprocessor.Theadvantagesofperformingsignalprocessinginthedigitaldomainarenumerous.Oneadvantageisduetothefactthatdigitalcircuitryiseasilyimplementedintbesmallestgeometryprocessesavailable,providingacostandspeedadvantage.Anotheradvantagerelatestotheadditionaldegreesoffreedomavailableindigitalsignalprocessing(e.g.,linear-phasefilters).Additionaladvantageslieintheabilitytoeasilyprogramdigitaldevices.Finally.itmaybenecessarytohaveananalogoutput.Intbiscase,apostprocessingblockisnecessary.Itwilltypicallycontainadigital-to..analogconvener(DACorDlA).amplification,andfiltering.Inasignal-processingsystem,oneimportantsystemconsiderationisthebandwidthofthesignaltobeprocessed.AgraphoftheoperatingfrequencyofavarietyofsignalsisgiveninFig.1.3-2.Atthelowendareseismicsignals,whichdonotextendmuchbelowtHzbecauseoftheabsorptioncharacteristicsoftheearth.Attheotherextremearemicrowavesignals.Thesearenotusedmuchabove30GHzbecauseofthedifficultiesinperformingeventhesimplestformsofsignalprocessingathigherfrequencies.ToaddressanyparticularapplicationareaillustratedinFigI.3-2atechnologythatcansupporttherequiredsignalbandwidthmustbeused.Figure1.3-3illustratesthespeedcapabilitiesofthevariousprocesstechnologiesavailabletoday.Bandwidthrequirementsandspeedarenottheonlyconsiderationswhendecidingwhichtechnologytouseforanintegratedcircuit(IC)addressinganapplicationarea.Otherconsiderationsarecostandintegration.ThecleartreodtodayistouseCMOSdigitalcombinedwithCMOSanalog(asneeded)wheneverpossiblebecausesignificantintegrationcanbeachieved,thusprovidinghighlyreliablecompactsystemsolutions.Analoginpul--.Preprocessing(fillcringandAIDconversion)AnalogII+DigilalsignalprocessorIDigillllII+IIIPostprocessingAnalog(D/Aconversion~OUiplltandfillc:riog)AnalogFigure1.3-1Atypicalsignal-processingsystemblockdiagram.10INTRODUCllONANDBACKGROUNDFigurel.J-2Frequencyofsignalsusedinsignal-processingapplications.Vidop,~~rii~:S.imicS011RadarAM-F~radlcTVAidiMicro•Teleommulicollonl10100IlLlOlLJ(XIkIMJOMlOOM10"""100HIOGSignolFnoquency(lbl1.4EXHMPLEOFRHHLUGYLSIMIXED-SIGHRLCIRCUITDESIGNAnalogcircuitdesignmethodologyisbestillustratedbyexample.Figure1.4-1showstheblockdiagramofafullyintegrateddigitalread/writechannelfordisk-driverecordingapplications.Thedeviceemployspartialresponsemaximumlikelihood(PRML)sequencedetectionwhenreadingdatatoenhancebit-error-rateversussignal-to-noiseratioperformance.Thedevicesupportsdataratesupto64Mbits/sandisfabricatedina0.8IJ.Mdouble-metalCMOSprocess.Inatypicalapplication.thisICreceivesafullydifferentialanalogsignalfromanexternalpreamplifier,whichsensesmagnetictransilionsonaspinningdisk-driveplatter.Thisdifferentialreadpulselsfirstamplifiedbyavariablegainamplifier(VGA)undercontrolofarealtimedigitalgain-controlloop.Afteramplification,thesignalispassedtoaseven-poletwozeroequiripple-phaselow-passfilter.Thezerosofthefilterarerealandsymmetricalabouttheimaginaryaxis.Thelocationsofthezerosrelativetothelocationsofthepolesareprogrammableandaredesignedtoboostfiltergainathighfrequenciesandthusnarrowthewidthofthereadpulse,Figurel.J-3Frequenciesthatcanbeproces5edbypresent-dayiCMOtechnologies.BiplaranInaBiJ>Olrdigi~logic~Msdigi~uscllol!icli!osrualo•OpticalbuA•10100U.JillteFigure2.3-1Physicalstructureofann-channelandp-channe1transistorinano-welltechnology.:.andsourceandareseparatedbyadistanceL(referredtoasthedevicelength).Atthesurfacebetweenthedrainandsourceliesagateelectrodethatisseparatedfromthesiliconbyathindielectricmaterial(silicondioxide).Similarly,thea-channeltransistorisformedbytwoheavilydopedo+regionswithinalightlydopedp-substrate.It,too,hasagateonthesurfacebetweenthedrainandsourceseparatedfromdlesiliconbyathindielectricmaterial(silicondioxide).Essentially,bothtypesoftransistorsarefour-tenninaldevicesasshowninFig.1.2-2(c,d).TheBterminalisthebulk,orsubstrate,whichcontainsthedrainandsourcediffusions.Forann-wellprocess,thep-bulkconnectioniscommonthroughouttheintegratedcircuitandisconnectedtoVss(themostnegativesupply).Mulliplen-wellscanbetilbricatedonasinglecircuit,andtheycanbeconnectedtodifferentpotentialsinvariouswaysdependingontheapplication.Figure2.3-2showsann-channeltransistorwithallfourterminalsconnectedtoground.Atequilibrium,thep-substrateandthen+sourceanddrainformapnjunction.Therefore,adepletionregionexistsbetweenthen+sourceanddrainandthep-substrate.Sincethesourceanddrainareseparatedbyback-to-backpnjunctions,theresistancebetweenthesourceanddrainisveryhigh(>1012fl).ThegateandthesubstrateoftheMOStransistorformtheparallelplatesofacapacitorwiththeSi02asthedielectric.Thiscapacitancedividedbytheareap·substrak:Figure2.3-2Crosssectionofann-channeltransistorwithalltenninalsgrounded.J8CMOSTECHNOLOGYofthegateisdesignatedasCu~·*Whenapositivepotentialisappliedtothegatewithrespecttothesourceadepletionregionisfonnedunderthegateresultingfrombolesbeingpushedawayfromthesilicon-silicondioxideinterface.Thedepletionregionconsistsoffixedionsthathaveanegativecharge.Usingone-dimensionalanalysis,thechargedensity,p,ofthedepletionregionisgivenbyP""q(-N,)(2.3-1)ApplyingthepointformofGauss'slaw,theelectricfieldresultingfromthischargeisE(x)P=-dxJe:=f-qN,.,-qN,--dx=--x+CSsJ(2.3-2)BsJwhereCistheconstantofintegration.Theconstant.C,isdeterminedbyevaluatingE(x)attheedgesofthedepletionregion(x=0attheSi-Si02interface;x=xdattheboundaryofthedepletionregioninthebulk).E(O)=-qN..,esiEo=--0+C=C(2.3-3)(2.3-4)(2.3-5)ThisgivesanexpressionforE{x):E(x)qN,=es;(xd-x)(2.3-6)Applyingtherelationshipbetweenpotentialandelectricfieldyields(2.3-7)IntegratingbothsidesofEq.(2.3-7)withappropriatelimitsofintegrationgives"'J'dt/>=-fz"qN,Bst(xd"'·x)dx=(2.3-8)0"ThesymbolCnormallybasunitsoffarads;however,inthefieldofMOSdevicesitoftenhasunitsoffaradsperunitarea(e.g.,F/m2).2.3qNAx~--=2sslTheMOSTransistortPs-tPF39(2.3-9)whererppistheequilibriumelectrostaticpotential(Fermipotential)inthesemiconductor,rPsisthesurfacepotentialoflbesemiconductor,andxdisthethicknessoflbedepletionregion.Forap-typesemiconductor,rPFisgivenas(2.3-10)andforann-typesemiconductortPFisgivenas(2.3-11)Equation(2.3-9)canbesolvedforx4assumingthatlt/J,-rp~~0toget(2.3-12)TheimmobilechargeduetoacceptorionsthathavebeenstrippedoflbeirmobileholesisgivenbyQ=-qN,.,h(2.3-13)SubstitutingEq.(2.3-12)intoEq.(2.3-13}gives(2.3-14)Whenthegatevoltagereachesavaluecalledthethresholdvoltage,designatedasVr.thesubstrateunderneaththegatebecomesinvened;thatis,itchangesfromap-typetoann-typesemiconductor.Consequently,ann-typechannelexistsbetweenthesourceanddrainthatallowscarrierstoftow.Inordertoachievethisinversion,lbesurfacepotentialmustincreasefromitsoriginalnegativevalue(rp,=1/Jp),tozero(t/1.,=0).andthentoapositivevalue(1/J,=-rpp).Thevalueofgate-sourcevoltagenecessaryto-causethischangeinsurfacepotentialisdefinedasthethresholdvoltage,Vr.Thisconditionisknownasstronginversion.Then-channeltransistorinthisconditionisillustratedinFig.2.3-3.Withthesubstrateatgroundpotential,thechargestoredinthedepletionregionbetweenthechannelunderthegateandthesubstrateisgivenbyEq.(2.3-14),whereq,,hasbeenreplacedby-rp,toaccountforthefactthatvas=VrThischargeQbOiswrinenas(2.3-15)Ifareverse-biasvoltagev8sisappliedacrossthepnjunction,Bq.(2.3-15)becomes(2.3-16)-40CMOSTECHNOLOGYFOXFOX------/{lnvct1edchat1nelp-substrate-::-dy:!--WyJ---!y!•y=.O~y=Ly.+-dyFigure2.3-3Crosssectionoflllln-channeltransistorwithsmallY~:~slllldVG$>V7'Anexpressionforthethresholdvoltagecanbedevelopedbybreakingitdownintoseveralcomponents.First,!:hetenn*tbusmustbeincludedtorepresentthedifferenceintheworkfunctionsbetweenthegatematerialandbulksiliconinthechannelregion.ThetenntPMsisgivenbytPMs=tPF(substrate)-q,,(gate)(2.3-17)-zq,,..-where,P,(metal)=0.6V.Second,agatevoltageof[(Q~o/C.,,))isrequiredtochangethesurfacepotentialandoffsetthedepletion-layerchargeQb·Lastly,!:hereisalwaysanundesiredpositivechargeQ.,presentintheinterfacebetweentheoxideandthebulksilicon,Thischargeisduetoimpuritiesandimperfectionsattheinterfaceandmustbecompensatedbyagatevoltageof-(b,JC0,.Thus,thethre.holdvoltagefortheMOStransistorcanbeexpressedasVr=4»Ms+(Qh)+(-Q--..)C.,,-2(/IF--C.,.(2.3-18)Thethresholdvoltagecanberewrittenas(2.3-19)whereV7ll=...MS-2~F-QwQ,.-COX""C-{2.3-20}•Historically.thistermhasbeenreferredtoasthemetal·fO·siliconwork:function.Wewillcontinuethetraditionevenwhenthegateterminalissomethingotherthanmelal(e.g.,polysilicon).2.3TheMOSTransistor41TABLE2.3-1SignsfortheQuantitiesintheThresholdVoltageEquationParametern-Channel(p-TypeSubstrate)p-Channel(n·TypeSubstrate)~t.iSMetala+Sigatep+Sigate..,+Q..+++a60oa.Vsa+'(+++andthebodyfactor,body-effectcoefficient,orbulk-thresholdparameter-yisdefinedas(2.3·21)Thesignsoftheaboveanalysiscanbecomeveryconfusing.Table2.3·1attemptstoclarifyanyconfusionthatmightarise[25].CALCULATIONOFTHETHRESHOLDVOLTAGEFindthethre.~holdvoltageandbodyfactor-yforann-channeltransistorwithann+silicongateift0,=200A,NA=3X1016em-3,gatedoping,N0=4X1019em-3,andifthenumberofpositivelychargedionsattheoxide-siliconinterfaceperareais1010em-2•&taMII.J.MFromBq.(2.3-10),rPF(subslrate}isgivenas163X10)tPp(substrate)=-0.0259In(=-0.377V1.45X1010Theequilibriumelectrostaticpotentialforthen+polysilicongateisfoundfromEq.(2.3-11)as.p,(gate)=0.0259ln(194X10)=0.563V1.45X1010Equation(2.3-17)gives.PMsasrflp(substrate)-q,,(gate)=-0.940Vice.r,Vol.ED-II,pp.324-345,July1964.28.H.Shichm1111andD.Hodges,"Modi:lingandSimulationofInsulated-GateField-EffectTransistorSwitchingCircuits,''IEEEJ.Solid-StateCin:uits,Vol.SC-13,No.3,pp.285-289.Sept.1968.l9.J.L.McCreary,"MatchingPropcnies,andVoltageandThmperatlllllDependenceofMOSCapacitors,''IEEEJ,SolidStateCircuils,Vol.SC-16,No.6.pp.608-616,Dec.1981.30.D.B.E.~treichandR.W.Dunon,"ModelingLatch-UpinCMOSIntegratedCircuitsandSystems,''IEEEThins.CAD,Vol.CAD-I,pp.157-162,Oct.1982.31.S.M.Sze.PhysicsofSemiconducwrDevices,2ndedNewYork:Wiley.1981,p.28.32.R.A.Blauschild,P.A.Tucci,R.S.Muller,andR.G.Meyer.'ANewTemperature-StableVoltageReference,'IEEEJ.SolitJ.StateCin:uil.l,Vol.SC-19,No.6,pp.767-774,Dec.1978.33.C.D.MotchenbQ{!herandF.0.Fitchen.Low·NoiseElectronicDesign.NewYork:Wiley.1973.34.J.L.McCrearyandP.R.Gray,"Ail-MOSChargeRedistributionAnalog-to-DigitalConversionThcbniques--PartI,''IEEEJ.Solid-StateCircuits,Vol.SC-10,No.6,pp.371-379.Dec.1975.JS.J.B.Shyu,G.C.Ternes,andF.Krummenacher,"RandomErrorEffectsinMatchedMOSCapacitorsandCurrentSoutces,"IEEEJ.Solid-StateCin:uits,Vol.SC-19,No.6,pp.948-955,Dec.1984.36.R.W.Brodersen,P.R.Gray.andD.A.Hodges,"MOSSwitched-CapacitorFilters,"Pro.:.IEEE.Vol.67,pp.61-75,Jan.1979.'!1.D.A.Hodges,P.R.Gray,andR.W.Brodersen,"Potcntia1ofMOSThchnologiesforAnalogIntegratedCircuirs,"IEEElSolid·SttlleCircuits,Vol.SC-8,No.3,pp.285-294,June1978.Chapter3CMOSDeviceModelingBeforeonecandesignacircuittobeintegratedinCMOStechnology,onemustfirsthaveamodeldescribingthebehaviorofallthecomponentsavailableforuseinthedesign.Amodelcantaketheformof~ru~thematicalequations,circuitrepresentations,ortables.Mostofthemodelingusedinthistextwillfocusontheactiveandpassivedevicesdiscussedinthepreviouschapterasopposedtohigher-levelmodelingsuchasmacromodelingorbehavioralmodeling.ltsbou\dbestressedattheoutsetthatamodelisjusttbatandnomore-itisnottberealthing!Inanidealworld,wewouldhaveamodeltbataccuratelydescribesthebehaviorofadevi(vas-Vr)(3.1-16)InFig.3.1-2,thenonsaturatedregionliesbetweentheverticalaxis(vos=0)andtheVvs=Vas-VrcUTVe.ThethirdregionoccurswhenVnsisgreaterthanvos(sat)orVas-V1•Atthispointthecurrenti0becomesindependentof"ns·Therefore,v05inEq.(3.1-1)isreplacedbyvDS(sat)ofEq.(3.1-11)toget0.vn5),whereVosistheactualdrain-sourcevoltageandnotv05(sat).ThesaturationregionmodelmodifiedtoincludechannellengthmodulationisgiveninEq.(3.1-18):0-VTG5v.T,.-:---valii-v,2.02.S=0.00vllSI(VGlQ-Y.,.lFigure3.1·3OutputcharacteristicsoftbeMOSdevice.TheoutputcharacteristicsoftheMOStransistorcanbedevelopedfromEqs.(3.1-14},(3.1-16),and(3.1-18).Figure3.1-3showsthesecharacteristicsplottedonanormalizedbasis.Thesecurveshavebeennormalizedtotheuppercurve,whereVa.ruisdefinedasthevalueofv0:;thatcausesadraincurrentofI00inthesaturationregion.TheentirecharacteristicisdevelopedbyextendingthesolidcurvesofPig.3.1-2horizontallytotherightfromthemaximumpoints.ThesolidcurvesofFig.3.1·3correspondto>.=0.IfA4:-0,thenthecurvesarethedashedlines.AnotherimportantcharacteristicoftheMOStransistorcanbeobtainedbyplottingi"versusVcsusingEq.(3.1-18).Figure3.1-4showsthisresult.ThischaracteristicoftheMOStransistoriscalledthetransconductancecharacteristic.WenotethatthetransconductancecharacteristicinthesaturationregioncanbeobtainedfromFig.3.1-3bydrawingaverticallinetotherightoftheparabolicdashedlineandplottingvaluesofi0versusv0s.Figure3.1-4isalsousefulforillu.~tratingtheeffectofthesource-bulkvoltage,v58•AsthevalueofVs8increases,Flgun:!3.1-4TransconduclancecharacteristicoftheMOStransistorasafunctionofthesource-bulkvoltage,l'ss-078CMOSDEVICEMODELINGthevalueofVrincreasesfortheenhancement,n-channeldevices(forap-channeldevice,IVrlincreasesasv88increases).V7alsoincreasespositivelyfor!hen-channeldepletiondevice,butsinceV7isnegative,thevalueofVrapproacheszerofromthenegativeside.Ifvs8islargeenough,Vrwillactuallybecomepositiveandthedepletiondevicebecomesanenhancementdevice.SincetheMOStransistorisabidirectionaldevice,determiningwhichphysicalnodeisthedrainandwhichthesourcemayseemarbitrary.Thisisnotreallythecase.Forann-channeltransistor,thesourceisalwaysatthelowerpotentialofthetwonodes.Forthep-channeltransistor,thesourceisalwaysatthehigherpotential.Itisobviousthatthedrainandsourcedesignationsarenotconstrainedtoagivennodeofatransistorbutcanswitchbackandforthdependingonthetenninlllvoltage~appliedtothetranSistor.Acircuitversionofthelarge-signalmodeloftheMOStransistorconsistsofacurrentsourceconnectedbetweenthedrainandwurceterminals.thatdependsonthedrain,source,gate,andbulkterminalvoltagesdefinedbythesimplemodeldescribedinthissection.Thissimplemodelhasfiveelectricalandprocessparametersthatcompletelydefineit.TheseparametersareK',Vr.'Y•Xand2411'·Thesubscriptnorpwillbeusedwhentheparameterreferstoann-channelorp-channeldevice,respectively.TheyconstitutetheLEVEL1modelparametersofSPICEfS].'JYpicalvaluesforthesemodelparametersaregivenillTable3.1-2.Thefunctionofthelarge-signalmodelistosolveforthedraincurrentgiventheterminalvoltagesoftheMOSdevice.Anexamplewillhelptoillustratethisaswellasshowhowthemodelisappliedtothep-channeldevice.APPLICATIONOFTHESIMPLEMOSLARGE-SIGNALMODELAssumethatthetransistorsinFig.3.1-1haveaW/Lratioof5jLm/l!LffiandthatthelargesignalmodelparametersarethosegiveninTable3.1-2.Ifthedrain,gate,source,andbulkvoltagesofthen-channeltransistorare3V,2V,0V,and0V,respectively,findthedraincurrentRepeatforthep-channeltransistorifthedrain.gate,source,andbulkvoltagesare-3V,-2V,0V.and0V.respectively.Wemustfirstdetermineinwhichregionthetransistorisoperating.Equation(3.1-15)givesv05(sat}as2V-0.7V==1.3V.SincevD.5is3V,then-channeltransistorisinthesaturationregion.UsingEq.(3.1-18)andthevaluesfromTable3.1-2,wehave.Jn==K/.W2L(vas1102VTN){1xw-6(5jLm)l(ljLm)+XNvos)2(2-0.7)(I+0.04X3)=520!LAEvaluationofEq.(3.1-15)forthep-channeltransistorisgivenasllso{sat)""Vsa-IVrPI=2V-0.7V=1.3V3.2OtherMOSLarge-SignalModelParameters79Sincevsois3V,thep-channeltransistorisalsointhesaturationregion,andEq.(3.1-17)isapplicable.ThedraincurrentofFig.3.[-l(b)canbefoundusingthevaluesfromTable3.1-2as.K;.wlo=-u-D2(1+)...,vso)x~~:~IJ.m)(2-0.7)2(1+0.05X3)=24311-AItisoftenusefultodescribevasintermsofi/)insaturationasshownbelow:Vus=Vr+vu;;ifJ(3.1-19)ThisexpressionillustratesthattherearetwocomponentstovGs--anamounttoinvertthechannelplusanadditionalamounttosupponthedesireddraincurrent.ThissecondcomponentisoftenreferredtointheliteratureasVoN·ThusVoNcanbedefinedas(3.1-20)ThetermVoNshouldberecognizedasthetermforsaturationvoltageVns(sat).Theycanbeusedinterchangeably.3.2OTHERMOSLHRGE-SIGNHLMODELPHRHMETERSThelarge-signalmodelalsoincludesseveralothercharacteristicssuchasthesource/drainbulkjunctions,source/drainohmicresistances,variouscapacitors,andnoise.Thecompleteversionofthelarge-signalmodelisgiveninFig.3.2-1.ThediodesofFig.3.2-1representthepnjunctionsbetweenthesourceandsubstrateandthedrainandsubstrate.Forpropertransistoroperation,th~-ediodesmustalwaysbereversebiased.Theirpurposeinthedemodelisprimarilytomodelleakagecurrents.Thesecurrentsareexpressedas.180[(qvso)kT-1]=1,exp(3.2-1)and(3.2-2)whereI,isthereversesaturationcUI'!'entofapnjunction,qisthechargeofanelectron,kisBoltzmann'sconstant.andTistemperatureinkelvinunits.Theresistorsr0andrsrepresenttheohmicresistanceofthedrainandsource,respectively.'JYpically,theseresistorsmaybe50-100fi*andcanoftenbeignoredatlowdraincurrents.•Forsilicilkprocess,lheseresislallceswillbemuchleiiS-ontheorderof5-l0n.80CMOSDEVICEMODELINGFigure3.2-1Completelarge-signalmodelfortheMOSII'lll1sistor.BGsThecapacitorsofFig.3.2-1canbeseparatedintothreetypes.ThefirsttypejncludescapacitorsC80andC85,whicharea.'!sociatedwiththeback-bia.'!eddepletionregionbetweenthedrainandsubstrateandthesourceandsubstrate.ThesecondtypeincludescapacitorsCc;0,Cas.andCa~~owhichareallcommontothegateandaredependentontheoperatingconditionofthetransistor.Thethirdtypeincludesparasiticcapacitors,whichareindependentoftheoperatingconditions.Thedc;pletioncapacitorsareafunctionofthevoltageacrossthepnjunction.Theexpressionofthisjunction-depletioncapacitanceisdividedintotworegionstoaccountforthehighinjectioneffects.ThefirstisgivenasVsxCax=(CJ)(AX)[I-PB]-MJ•vexs(FC)(PB)whereX=DforC80orX=SforCBSAX=areaofthesource(X=S)ordrain(X=D)CJ=zero-bias(v8x=0)junctioncapacitance(perunitarea)qes;Nsua2PBPB=bulkjunctionpotential[sameasandC08asafunctionofVa.swithVosconstantandV63=0.3.2OtherMOSLarge-SignalModelParameters8SandC08isapproximatelyequaltoC1+2C5•Asv0sapproachesVrfromtheoffregion.athindepletionlayerisformed,creatingalargevalueofC4•SinceC4isinserieswithC2,Httleeffectisobserved.Asvasincreases,thisdepletionregionwidens,causingC4todecreaseandreducingCell·WhenVc;s=VT,aninversionlayerisformedthatpreventsfurtherdecreasesofc4(andthusC08).C.,C2,andC3constituteCasandCoD·TheproblemishowtoallocateC2toCasandCav·Theapproachusedistoassumeinsaturationthatapproximatelytwo-thirdsofC2belongstoCasandnonetoC00.Thisi~;,ofcourse,anapproximation.However,itha.~beenfoundtogivereasonablygoodresults.Figure3.2-7showshowCasandC00changevaluesingoingfromtheofftothesaturationregion.Finally,whenv08isgreaterthanv08+Vr;theMOSdeviceentersthenonsaturatedregion.[nthiscase,thechannelextendsfromthedraintothesourceandC2issimplydividedevenlybetweenCavandCcsasshowninFig.3.2-7.Asaconsequenceoftheaboveconsiderations,weshallusethefollowingformulasforthecharge-storagecapacitancesoftheMOSdeviceintheindicatedregions.Off(3.2-9a)(3.2-9b}(3.2-9aredefinedas.)g,.=iHo(evaluatedatthe.qutescentpomt(3.3-3)iJio.•=-(evaluatedatthequtescentpomt)iJvss(3.3-4)-a\Ius8mb.randa~..8dl=-,.-(evaluatedattheqwescentpomt)""ns(3.3-5)3.3.Small-SignalModelfortheMOSTransistor89Thevaluesofthesesmall-signalparametersdependonwhichregionthequiescentpointoccursin.Forexample,inthesaturatedregiong.,canbefoundfromEq.(3.1-18)as(3.3-6)whichemphasizesthedependenceofthesmall-signalparametersonthelarge-signaloperatingconditions.Thesmall-signalchanneltransconductanceduetov58isfoundbyrewritingEq.{3.3-4)as(3.3-7)UsingEq.(3.1-2)andnotingthatatD/iJVr=-aiofavas.weget*1'(3.3-8)Thistransconductancewillbecomeimportantinoursmall-signalanalysisoftheMOStransistorwhentheacvalueofthesource-bulkpotentialv..,isnotzero.Thesmall-signalchannelconductance,gd,{g0),isgivenas(3.3-9)ThechannelconductancewillbedependentonLthrough).,whichisinverselyproportionaltoL.WehaveassumedtheMOStransistorisinsaturationfortheresultsgivenbyEqs.(3.3-6),(3.3-8),and(3.3-9).Theimportantdependenceofthesmall-signalparametersonthelarge-signalmodelparametersanddevoltagesandcurrent~isillustratedinThble3.3-1.Inthistableweseethatthethreesmall-signalmodelparametersofg,.Kmm.andKd•haveseveralalternatefonns.Anexampleofthetypicalvaluesofthesmall-signalmodelparametersfollows.TYPICALVALUESOFSMALL-SIGNALMODELPARAMETERSFindthevaluesofKm•Km~n.andR.uusingthelarge-signalmodelparametersinTable3.1-2forbothann-channelandap-channeldeviceifthedevalueofthemagnitudeofthedraincurrentis50f.LAandthemagnitudeofthedevalueofthesource-bulkvoltageis2V.AssumethattbeWILratiois1j.~.m/1tJ.rn.•NotethatabsolutesignsareusedforV88inordertopreventg-frombecominginfinite.However,inafewrarecasesthesoiii'CHulkjunctionisforwardbiasedandinthiscasetheabsolutesignsmustberemovedandVs8becomesnegative(forn-channeltransistor).90CMOSDEVICEMODELINGTABLE3.3-1DependenceoftheSmall-SignalModelParametersonthedeValuesofVoltageandCurrentintheSaturationRegionSmall-SignalModelParametersB..deCurrent..(2K'(DWfl-)112IdeVoltageK'WL')'[fj(VGJ-V7)]112""-(Vos-Vr)"'t(2Jo(J)lflg..,lr.t.deCurrentandVoltage2;\+IVsailills>.loUsingthevaluesofTable3.1-2andEqs.(3.3-6),(3.3-8),and(3.3-9)givesg,.,=105fJ.A/V,70.7fJ.AIV,8m~nBm~n~12.811-A/V,and8.ts=2.0ILA/Vforthen-channeldeviceandBm=12.0~/V,andB.ts=2.5JJ.A/Vforthep-channeldevice.=AlthoughMOSdevicesarenotoftenusedinthenonsaturationregioninanalogcircuitdesign,therelationshipsofthesmall-signalmodelparametersinthenonsarurationregionaregivenas(3.3-10)(3.3-11)and8rbefJ(Vas-Vr-V~)(3.3-12)Table3.3-2summarizesthedependenceofthesmall-signalmodelparametersonthelargesignalmodelparametersanddevoltagesandcurrentsforthenonsaturatedregion.Thetypicalvaluesofthesmall-signalmodelparametersforthenonsaturatedregionareillustratedintbefollowingexample.TYPICALVALUESOFTHESMALL-SIGNALMODELPARAMETERSINTHENONSATURATEDREGIONFindthevaluesofthesmall-signalmodelparametersinthenonsaturationregionforann-channelandap-channeltransistorifVas=5V,Vos='lV,andIV851""'2V.AssumethattheWILratioforbothtransistorsis1p.m/1p.m.AlsoassumethatthevalueforK'inthenonsaturationregionisthesameasthatforthesaturation(generallyapoorassumption}.3.3Small-SignalModelfortheMOSTransistor91TABLE3.3-2DependenceoftheSmall-SignalModelParametersonthedeValuesofVoltageandCurrentintheNonsaturationRegionSmall-SignalModelParameter:sdeVoltageand/orcurrentDependence2(21~,1+1Vsall112•I!(Vn.,-Vr-Vlls)SolutionFirst,itisnecessarytocalculatethethresholdvoltageofeachtransistorusingEq.(3.1-2).TheresultsareaVrof1.02Vforthen-channeland-1.14Vforthep-channel.Thisgivesadecurrentof383j.LAand168j.LA,respectively.UsingEqs.(3.3-10),(3.3-11).and(3.3-12),wegetKm=110~J.A/V,Cmru=13.4~J.A/V,andr.u=3.05kilforthen-channeltransistorand8m=50IJ..A/V,8mru=8.52j.LA/V,andrtb=6.99kOforthep-channeltransistor.Thevaluesofrdandr:.areassumedtobethesameasr0andrsofFig.3.2-1.Likewise,forsmall-signalconditionsC8.,Csd•C~~·Cbd,andC""areevaluatedforCv,CBd•andC8,byknowingtheregionofoperation(cutoff,saturationornonsaturation)andforCbdandCbsbyknowingthevalueofV11oandVns·Withthisinfonnation,C8,,C11d•C8b•Cbd,andCb,canbefoundfromCus.Cuo·Ca8•Cso•andCss.respectively.IfthenoiseoftheMOStransistoristobemodeled,thenthreeadditionalcurrentsourcesareaddedtoFig.3.3-1asindicatedbythedashedlines.Thevaluesofthemean-squarenoisecurrentsourcesaregivenasi~rD=(4::)Afi~rS=(~;)A/(A2)(3.3-13)(Al)(3.3-14)and(3.3-15)Thevariousparametersfortheseequationshavepreviouslybeendefined.Withthenoisemodelingcapability,thesmall-signalmodelofFig.3.3-1isaverygeneralmodel.Itwillbeimportanttobefamiliarwiththesmall-signalmodelforthesaturationregiondevelopedinthissection.Thismodel.alongwiththecircuitsimplificationtechniquesgiveninAppendixA,willbethekeyelementinanalyzingthecircuitsinthefollowingchapters.923.4CMOSDEVICEMODELINGCOMPUTERSIMULATIONMODELSThelarge-signalmodeloftheMOSdevicepreviouslydiscussedissimpletouseforbandcal·culationsbutneglect~manyimportantsecond-ordereffect~.Wbileasimplemodelforhandcalculationanddesignintuitioniscritical,amoreaccuratemodelisrequiredforcomputersimulation.Therearemanymodelchoicesavailableforthedesignerwhenchoosingadevicemodeltouseforcomputersimulation.Atonetime,HSPICE"'supported43differentMOSFETmodels[2](manyofwhichwerecompanyproprietary)whileSmartSpicepublishessupportfor14[9).Whlcbmndelistherightonetouse?Inthefablesssemiconductorenvironment,theusermustusethemodelprovidedbythewaferfoundry.Jncompanieswherethefoundryiscaptive(i.e.,thecompanyownsitsownwaferfabricationfacility)amodelinggroupprovidesthemodeltocircuitdesigners.ItisseldomthatadesignercbOO$ellamodelandperformsparameterextractiontogelthetermsforthemodelchosen.TheSPICELEVEL3demodelwillbecoveredinsomedetailbecauseitisarelativelystraightforwardextensionoftheLEVEL2model.TheBSJM3v3modelwillbeintroducedbutthedetailedequationswillnotbepresentedbecauseofthevolumeofequationsrequiredtodescribeit-thereareothergoodtextsthatdealwiththesubjectofmodelingexclusively{10,11],andthereislittleadditionaldesignintuitionderivedfromcoveringthedetails.Modelsdevelopedforcomputersimulationhaveimprovedovertheyearsbutnomodelhasyetbeendevelopedthat,withasinglesetofparameters,coveisdeviceoperationforallpossiblegeometries.Therefore,manySPICEsimulatorsofferafeaturecalled•·modelbinning:•Parametersarederivedfortransistorsofdifferentgeometry(W'sandl.'s)andthesimulatordetennineswhichsetofparameterstousebasedontheparticularWandLcalledoutinthedeviceinstantiationlineinthecircuitdescription.Thecircuitdesignerneedonlybeawareofthissincethebinningisdonebythemodelprovider.SPICELEVEL3ModelThelarge-signalmodeloftheMOSdevicepreviouslydiscussedissimpletouseforhandcalculationsbutneglectsmanyimportantsecond-ordereffects.Mostofthesesecond-ordereffectsaredue10narroworshortchanneldimensions(le.-;sthanabout3p.m).lnthissection,wewillconsideramorecomplexmodelthatissuitableforcomputer-basedanalysis(circuitsimulation,i.e.,SPICEsimulation).Inparticular,theSPICELEVEL3modelwillbecovered(seeTable3.4-1}.ThismodelistypicallygoodforMOStechnologiesdowntoabout0.8~~om.WewillalsoconsidertheeffectsoftemperatureontheparametersoftheMOSlarge-signalmodelWefirstconsidersecond-ordereffectsduetosmallgeometries(Fig.3.4-1).WhenVasisgreaterthanV11thedraincurrentforasmalldevicecanbegivena.~{2]follows:DrainCurrent.['os=BETAVas-w.lf(l+ft,)]Vr--2VoEllo£w.rrBETA=KP.."cox4Ir=....L..rr•HSP(CEisnowownedbyAvant!Inc:.andhasbeenn:namedStar-H.spice.(3.4-1)(3.4-2)3.4ComputerSimulationModels93TABLE3.4-1TypicalModelParametersSuitableforSPICESimulationsUsingLEVEL-3Model(ExtendedModel)*TypicalParameterValueParameterSym6otParameterOescripCI'onvroThresholdMobilityNarrow-widththresholdadjustmentfactorStatic-feedbackthresholdadju.•tmentfactorSaturationfieldfactorInchannellengthm~JdulalionMobilitydegradationfactorSubs!ratedopingOxidelhickne~;sMelllllurgicaljunctiondeplhDeltawidthLareraldiffusionPsrameterforweakinversionmodelinguoDELTAEl'AKAPPATHETANSUBTOXXJWDLDNFScosorr-ChtmnelCGDO-0.7:!:0.15v6602.4210t:m1N-s1.250.10.10.152.50.13X10160.16x1016140140em-'A0.20.2J.Ull0.0167X10110.0156XIOH:220X10-la:220X10-11:220X10-12220xININJ.Ull100110CJCJSWMJMJSWl.hriU0.7:!:0.15xw-•2xw-•Jsoxw-12CGBOp-Ch«mree0.50.38w-••700X10-IZ560X10"6350X10-lt)1.111cm-2F/mFlmFlmP/m2F/m0.50.35"The-o;evaluesatebasedona0.8J.Ullsilicon-gatebulkCMOSn-wellprocessandincludecapacitanceparame1en1fromTable3.2-1.I...,=L-2{LD)(3.4-3)w.lf=w-2(WD)(3.4-4)min(vru..,vos(sat))(3.4-5)GAMMA·f.fn+4(PHI+vss)112(3.4-6}lfoe=1"=GateBulkFigure3.4-1IllustrationoftbeshortchanneleffectsinrbeMOStransistor.94CMOSDEVICEMODELINGNotethatPHIistheSPICEmodeltermforthequantity2(/)p.AlsobeawarethatPHIisalwayspositiveinSPICEregardlessofthetransistortype(p-orn-cbannel).Inthistellt,thetermPHIwillalwaysbepositivewhiletheterm2(/)pwillhaveapolaritydeterminedbythetr.msistortypeasshowninTable2.3-1.fn=DELTAw.aJ;•=(3.4-7)l-4rrXJ{LDXJ+we[l-(XJ+wpwpwp=xd(Plflxd=TBs;2·C.,.2112)]_LD}XJ(3.4-8)+"ss)112(3.4-9)2.e.)11251(q•NSUB(3.4-10)(3.4-11)kt=0.0631353,kl=0.08013292,k3=0.01110777ThresholdVoltageVr=V~n-ETA-8.14X10-22)(c....3~.Vns+GAMMA·/.(PHI+Vss)112+/n{PHI+(3.4-12)llss)(3.4-13)orvbl=vro-GAMMA·VPHl(3.4-14)SaturationVoltageVwVDs(sat)v,s-Vr=1+fb=vc=v..,+Vc-(3.4-15)(v;..+~~~)'12-·VMAX·l.,ffp;.lfVMAXisnotgiven,thenv00(sat)(3.4-16)(3.4-17)~v001•3.4ComputerSimulationModels95EffectiveMobilityII"-I~UOwhenVMAX=0,..,-1+THETA(vGI-V7)'{3.4-18)p,v•whenVMAX>0;otherwisel'ctt=p..P..ff=(3.4-19)1+~VcChannelLengthModulationti.L=xd[KAPPA(vDs-vDS(sat))]'12,tiL=ep~xJl+[ep~Jy+whenVMAX=0{3.4-20)J12KAPPA·xtf(vDS-VDs(sat)),(3,4-21)whenVMAX>OwhereVc(vcep=+Vos(sat))(3.4-22}L.ttVDS(sat}.iDslos=1(3.4-23)_tJ.Thetemperature-dependentvariablesinthemodelsdevelopedsofarincludetheFermipotential,PHI,EO,bulkjunctionpotentialofthesourc~ulkanddrain-bulkjunctions,PB,tbereversecurrentsofthepnj'unctions,/8,andthedependenceofmobilityonremperacure.Thetemperaturedependenceofmostcfthesevariablesisfoundintheequationsgivenpreviouslyorfromwell-knownexpressions.ThedependenceofmobilityontemperatureisgivenasUO(T}=UO(T0)T)BBX(To(3.4-24).whereBEXisthetemperatureexponentformobilityandistypically-1.5.llthormkTq(3.4-25)(T)""-EG(T)PHI(T)=.'T}_vbl\[r+~:os.o]T)[3In(ToT)+PHI(To).(=1.16-1.02•10-•To.,.,.-VIR\•o}+-vlhernlo(T}PHJ(T)-PIU(T0)2+(3.4-26)EG(T0)EG(T)]vobonD(To)-vtherm(T)EG(T0)-EG(T)2(3.4-27)(3.4-28)96CMOSDEVICEMODfLINGVTO(T)=vbl{T)+GAMMA[VPHJ(T)](3.4-29)(NSUB)(3.4-30)PHI(7)=2v~~>ormInni.,T)n/..T)=1.45·1016(!:._)T312exp[EG·0(.!.1)(I)To2•vlheml(To)(3.4-31)Fordrainandsourcejunctiondiodes,dlefollowingrelationshipsapply:(T)-PB(T)=PB·To((T)EG(T0)EG(T)]v~~~mn(T)3InTo+vlllonn(To)-v.......,(T)(3.4-32)andls(T)]2(PHT_+vs8)(3'52)NFSisaparameterusedintheevaluationofVoNandcanbeextractedfrommeasurements.Thedraincurrentintheweakinversionregion,VasCribingweakinversionoperationisgivenas•Wloa;Llooexp"'(Vo.r)n(kT/q)(3.5-5)wherethetermnisthesubthresholdslopefactor,and100isaprocess-dependentpar-.nneterthatisdependentalsoonvs11andV7•Thesetwotermsarebestextractedfromexperimentaldata.'!Ypicallynisgreatertbaa18lldlessthan3(1-Vn(5.5-13)222CMOSAMPUFIERSFigure5.5-3(a)SourcefoUowerwithaMOSdiodeload.louTioUT-OUT-OUT(a)(b)Sourcefollowerwithacurrent-sinkload.(b)assumingthat1ltNcanbetakentoV00andnooutputcurrentisflowing.However,V71isafunctionofVoursothatwemustsubstituteEq.(3.1-2}intoEq.(5.5-13)andsolveforvOUT.Tosimplifythemathematics,weapproximateEq.(3.1-2)as(5.5-14)SubstitutingEq.(5.5-14)intoEq.(5.5-13)andsolvingforv0urgives2Vour(max)aVoo'YI+2-Vr01-2'Y1y'r12+4(VJ>o-Vss-Vnn)=(5.5-15)=UsingthenominalvaluesofTable3.1-2andassumingthatVoD-Wssl2.5V,welindthatvmrrllp5.2-1.ProblemsvourVssFigurePS.l-14n-channelinputdifferentialamplifierwhen/.'iS=IOOJ!AandW11L1=W,YLz=Wf~W;JL.t=1,assumingthatallchannellengthsarecqll8..1andhaveavalueoflJ.LID.RepeatifWlL,=WJ~10W,IL3=IOWJL....I.RepeatProblem5.2-1forrhep-chanuelinputdif·ferentialamplifier.DeveloptheexpressioasforV1c(max)andV1clmin)forthep-channelinputdifferentialamplifieroffig.5.2-7.Fmdthemaximuminputcommon-modevoltage,~/C(max),andtheminimuminpurcommonmodevoltage,v1c(min).ofthen-imilattoFig.().2-7(b).Onlyasecond-order(two-pole)systemhasbeenconsideredthusfar.Inpractice,therearemorethantwopolesinthetransferfunctionofaCMOSopamp.Therestofthistreatmentwillconcentrateonthetwomostdominant(smaller)polesandtheRHPzero.Figure6.2-8Figure6.2-8Atwo-stageopampwithvariousparnsiticandcircuitcapacitancesshown.6.2CompensationofOpAmps259Figure6.2-9lllustrationoftheimplementationofthedominantpolethroughtheMillereffectonCc.M6istreatedasanNMOSforthisillustration.,.illustratesatypicalCMOSopampwithvariousparasiticandcircuitcapacitancesshown.TheapproximatepoleIUldzerolocationsresultingfromthesecapacitancesaregivenbelow:(6.2-14);(6.2-15)and(6.2-16)Theunity-gainbandwidthasdefinedinSection6.1iseasilyderived(seeProblem6.2-3)andisshowntobeapprollimately(6.2-17)Theabovethreerootsareveryimportanttothedynamicperformanceofthetwo-stageopamp.Thedominantleft-halfplanepole,p1,iscalledtheMillerpoleandaccomplishesthedesiredcompensation.Intuitively,itiscreatedbytheMillereffectonthecapacitance,Cc,asillustratedinFig.6.2-9,whereM6isassumedtobeanNMOStransistor.ThecapacitorC,ismultipliedbyapproximatelythegainofthesecondstage,g11R11,togiveacapacitorinparallelwithR1ofg1~11C...MultiplyingthiscapacitancetimesR1andinvertinggivesEq.(6.2-14).Thesecondrootofimportanceisp2•ThemagnitudeofthisrootmustbeatleastequaltoGBandisduerothecapacitanceattheoutputoftheopamp.ltisoftencalledtheourpurpole.Generally,C11isequaltotheloadcapacitance,CL•whichmakestheoutputpolestronglydependentontheloadcapacitance.Figure6.2-10showsintuitivelyhowthisrootdevelops.Figure6.2-10Illustrationofthebowtheoutputpoleinatwo-stageopampiscreated.M6i.streatedasanNMOSforthisillustration.260CMOSOPERATIONALAMPLIFIERSFigure6.2·11lllu~lrationofhowtheRHPzeroisdeveloped.M6istreatedasanNMOSforthisillustration.v"Since~~isnearorgreaterthanGB.thereactanceofC,isapproximatelyli(GB•C,)andisverysmall.ForallpracticalpurposesthedrainofM6isconnectedtothegateofM6,formingaMOSdiode.Weknowthatthesmall-signalresistanceofaMOSdiodeis1/g.,.MultiplyingJlg,.,ubyC11(orCvandinvertinggivesEq.(6.2-15).ThethirdrootistheRHPrero.Thisisaveryundesirablerootbecauseitbooststheloopgainmagnitudewhilecausingtheloopphaseshifttobecomemorenegative.Bothoftheseresultsworsenthestabilityoftheopamp.InBITopamps,theRHPzerowasnotseriousbecauseofthelargevaluesoftransconductance.However,inCMOSopamps,theRHPzerocannotbeignored.ThiszerocomesfromthefactthattherearetwosignalpathsfromtheinputtotheoutputasillustratedinFig.6.2~11.OnepathisfromthegateofM6throughthecompensationcapacitor,C.,totheoutput(V"toV.,...).Theotherpathisthroughthetransistor,M6,totheoutput(V'toV.,.J.Atsomecomplexfrequency,thesignalsthroughthesetwopathswillbeequalandoppositeandcancel,creatingthezero.TheRHPzeroisdevelopedbyusingsuperpositiononthesetwopc~thsasshownbelow;II.(s)=(-g016R11(llsC,))R11out+1/sC.,V'+(R11R11+llsC06K6V,....(sat)mdd!enusingthepR"viousR"lationshlptofind1..Ofcourse.thepropermirrorbetweenM3andM4isnolongerguaranteed.10.llesigaS,toachievetbcdesiredcum:ntratinabetweenljand1,..tIJ.CJeckgainandpowerdissipationspecificatiOM,=A•2/l.aKmt.1,(~,+~.)(~+~,),·12.Ifthegainspecificationi~notmet.lhelllhecurren~/~8lldt.c1111bedecrea~~edortheWILratiosorM2and/orM6increa.'ied.Thepre.viou5calculationsmustberecheckedtoensurelhatIIIeyhavebeen!lllti~lied.Ifthepowerdissipationistoohigh.thenonecanonlyrrduccthecurrcniS1,andt•.RcducllonofcurrentswillprobablynecessitateanincreaseofsomeoftheWILratio.tosatisfyinputandoutputswings._;••·13.SimulatetbecircuittochecktoseelbatallspecificationsBR'mer.l________________________________________________________________________________276CMOSOPERATIONALAMPUFIERS-'thattheseadjustmentstoimprovenoiseperfonnancedonotadverselyaffectsomeOlberimportantperfonnanceparameteroftheopamp.Thepower-supplyrejectionratioistoalargedegreedeterminedbylheconfigurationused.SomeimprovementinnegativePSRRcanbeachievedbyincreasingtheoutputresistanceofMS.ThisisusuallyaccomplishedbyincreasingbothWsand4proportionatelywilhoutseriouslyaffectinganyotherperfonnance.TransistorM7shouldbeadjustedaccordinglyforpropermatching.AmoredetailedanalysisofthePSRRofthetwo-stageopampwillbeconsideredinthenextsection.Thefollowingexampleillustratesthestepsindesigningtheopampdescribed.DESIGNOFATWO-STAG£OPAMPUsingthematerialanddeviceparametersgiveninTables3.1-1and3.1-2,designanamplifiersimilartothatshowninFig.6.3-1thatmeetsthefollowingspecificationswithaphaseIIlW'ginof60°.Assumethechannellengthistobe1IJ.m.A,>SOOOVIVV00=2.5VGB=5MHzV.,.trange=±2VCL=IOpFVss=-2.5VSR>10V/IJ.SPdi,.:S2mWICMR=-1to2VljtiMu.J,IThefirststepistocalculatetheminimumvalueofthecompensationcapacitorCc,whichisCc>(2.2/10)(10pP)=2.2pFChooseCcas3pF.Usingtheslew-ratespecificationandC,.calculate15,Nextcalculate(WILhusingICMRrequirements.UsingEq.(6.3-12)wehave(w.ZL)3-(50X30X10-610-)[2.5-2-0.856-15+0.55]2-Therefore,(W/Lh=(W/L)4=15Nowwecancheckthevalueofthemirrorpole,p3,tomakesurethatitisinfactgreaterthan10GB.AssumetheCoa=2.47fF/IJ.m2•Themirrorpolecanbefoundas-g.,3-Y2K;Si39P3..,2C,.3=2(0.667)W~Co,=2.81X10radlsor448MHz.Thus,p3andz3arenotofconcerninthisdesignbecausep3>>I0GB.t6.3DesignoftheTwo-StageOpAmp277ThenextstepinthedesignistocalculateKmlusingEq.(6.3-13).Therefore,(W/L)1is(W/L)1=(WILng!l=2KNJ=12(94.25)2.IIO.=2.79""3.015Next,calculateVI>SSusingEq.(6.3-15).VDss=(-1)-(-2.5)-~30XJ061106xto-.-0.85•3=0.35VUsingVDSScalculate(W/L)sfromEq.(6.3-16).2(30xw-6>(W/L)s=(110X10-6)(0.35l=4.49.,.4.5FromEq.(6.2-20),weknowthat8m62:IOgmlo;:942.5JLSAssumingthatg..,=942.5....Sandcalculatingg""'as150.,..s,weuseEq.(6.3-18)toget8m6s6=s4~=8m4942.5150t5·--=94.25=94Calculate/6usingEq.(6.3-19).l6=(9425.X10-6)2(2)(50XlO6)(94}=94.5.,.A...95AILDesigningS6byusingEq.(6.3-20)givesS6""15.SincetheWILratioof94fromaboveisgreater,themaximumoutputvoltagespecificationwillbemet.Finally,calculate(W/L},usingEq.(6.3-21).(W/L},:=4.5G~!~=:)=14.25=14LetuschecktheVou1(min}specificationalthoughtheW/LofM7islargeenoughthatthisisprobablynotnecessary.ThevalueofVnu1(min}isVm;n(out){"'2-95=V057(sat)=\1'~=0.351Vwhichislessthanrequired.Atthispoint.thefirst-cutdesigniscomplete.278CMOSOPERATIONALAMPLIFIERSThepowerdissipationcanbecalculatedasPm..=5V•(30}LA+95fLA)=0.625mWNowchecktoseethatthegainspecificationhasbeenmet.A="30(2)(92.45X610(0.04X101)(942.5X10-6)+0.05}95X10-6(0.04+0.05)=7696V/Vwhichmeetsspecifications.Ifmoregainweredesired,aneasywaytoachieveitwouldbetoincreasetheWandLvaluesbyafaclOrof2,whichbecauseofthedecreasedvalueof>..wouldincreasethegainbyafactorof20.Figure6.3-3showstheresultsofthefirst-cutdesign.Thenextphaserequiressimulation.Vss=-2.5VFigure63-3ResultofExample6.3-l.NullingResistor,MillerCompensationItmaylikelyoccurthattheundesiredRHPzeromaynotbenegligibleintheabovedesignJlfO"cedure.ThiswouldoccuriftheGBspecificationwaslargeoriftheoutputstagetransconductance(gm6)wasnotlarge.Inthiscase,itbecomesnecessarytoemploythenullingresistorcompensationmethod.WeshallusetheresultsofSection6.2toillustratehowtoapplythissolution.Section6.2describedatechniquewherebytheRHPzerocanbemovedtothelefthalfplaneandplacedonthehighestnondominantpole.Toaccomplishthis.aresistorisplacedinserieswiththecompensationcapacitor.Figure6.3-4showsacompensationschemeusingtransistorMSasaresistor.ThistransistoriscontrolledbyacontrolvoltageVrthatadjuststheresistorsothatitmaintainsthepropervalueoverprocessvariatioliS[6].Withtheadditionoftheresistorinthecompensationscheme,theresultingpolesandzerosare[seeEqs.(6.2~37)to(6.2-40)]Cm2g..,[A.CcA.C.PI~---=---(6.3-23)6.3DesignoftheTwo-StageOpAmp279IVssFigure6.3-4CMOStwo-stageopampusingnullingresistorcompensation.(6.3-24)(6.3-25)(6.3-26)whereA.=8mt8mtJR,R11•Inordertoplacethezeroontopofthesecondpole(Pi),thefollowingrelationshipmuslhold:(6.3-27)TheresistorR,isrealizedbythetransistorMS.whichisoperatingintheactiveregionbecausethedecurrentthroughitiszero.Therefore,R,canbewrittenasR-_av_rus_l-----~-.,....--•-iiiDBv.,...-o-KJ.Ss(Vsos-fVrpi)(6.3-28)ThebiascircuitisdesignedsothatvoltageVAisequaltoV8•Asaresult.(6.3-29)Inthesaturationregion(6.3-30)280CMOSOPERATIONALAMPLIFIERSSubstitutingintoEq.(6.3-28)yieldsR.I'=0.01V-•.Also,assumethattheminimumdevicedimensionis2f)omandchoosethesmallestdevicespossible.DesignC,andR,togiveGB=1MHzandtoeliminatetheinfluenceoftheRHPzero.Howmuchloadcapacitanceshouldthisopampbecapableofdrivingwithoutsufferingadegradationinthephasemargin?Whatistheslewrateofthisopamp?AssumeVoo=-V~"'2.SVandR8=IOOkil.6.3-9.UsetheelectricalmodelparametersofthepreviousproblemtodesignW3,L3,W4•Z...W5,~-C..,andR,ofPig.P6.3-8ifthedecurrentsarein·creasedbyafactorof2andifW1=~=W2=Lz=2~toobtainalow-frequency.differentialVvv=2.5vS/1rVss=-2.5VFigureH-3-7344CMOSOPERATIONALAMPUFIERSK'N=24p.A/V2,0.75v,'low=0.01voltagegainof5000ll!ldaGBof1MHz.AlldevicesshouldbeinsarucationundernormaloperatingconditionsandtheeffectoftheRHPshouldbecanceled.Howmuchloadcapacitanceshouldthisopampbeabletodrivebeforesufferingadegradationinthephasemargin?Whatistheslewrateofthisopamp?6.3-10.FortheopampshowninFig.P6.3-10.as~umealltranSistorsareoperatinginthesaturationregionandfind{a)thedevalueof1~.11,and/8,(b)thelow-frequencydifferentialvoltagegain,A.,..(O),(c)theGBinHz.(d)thepositiveandnegativeslewrates,(c)thepowerdissipation,and{0thephasemarginassumingtllattheopen-loopunitysaiDis1MHz.-2.0VFigureP6.3-106.3-11.AsimpleCMOSopampisshownFig.P6.3-ll.Usethefollowinsmodelpatametersandfindthenumericalvalueofthesmall-signaldifferentialvoltagegain~•.,/~;..outputresistanceR.,...thedominantpolep1,theunity-gainbandwidthGB,theslewrateSR,andthedepowerdissipation.K;.=8p.AIV2•Vm=-Vll'"'v-•.and'Ap=0.02v-1•20J.LA6.3-12.Onalog-logplotwiththeverticalaxishavingarangeofIo·Jto10+3andthehorizontalaxishavingarangeof1toLOOp.A,plotthelow-freqi1CncygainA,.(O).theunity-gainbandwidthGB.thopowerdissipationPm...•theslewrateSR,theoutputresistanceR08,themagnitudeofthedominlllltpoleIPd·andthemagnitudeoftheRHPzeroz.allnormalizedtotheirrespectivevaluesatI11=If.LAasafunctionofI8fromItoI00p.Aforthestandardtwo-stageCMOSopamp,AssumethecurrentinMSisk118andtheoulpulcurrent(inM6}is~,.6.3-13.DeveloptheexpressionsimilartoEq.(6.3-32}fortheWILratioofM6AinFig.P6.3-l3thatwillcausetheRHPzerotocanceltheoutputpole.RepeatExample6.3-2usingthecircuitofFig.P6.3-l3andthevaluesofthetransistorsinExample6.3-J.6.3-14.UsetheintuitiveapproachpresentedinSection5.2tocalculatethesmall-signaldifferentialvoltagegainofthetwo-stageopampofFig.6.3-1.Problems345V..,(min)=0.75YSlewrate=:!:IOVI..,sV.,(min)=1VVk(max)=2VGB=10MHzPhasemargin=60"whentbcoulputpolt"'2GBandlheRHPzero=JOGB.Keepthemirrorpolel!!:10GB(C.,.=O.SfF/1whYourdesignshouldmeetorexceedthesespecifical.ions.IgnorebulkeffectsinIbisproblemandsummarizeyourWvaluestothenearestmicron,thevalueofC,(pF),andI(J.A.A)inthefollowingtable.Usethefollowingmodelparameters:K'N=24J.I.A/V2,K',.=8J.I.A/V2,Vm-VTP=0.75V,~0.01v-1,andll,.=0.02V-1•!!preN..J.UNuUingresistorimplementedbyaMOSdiode.63-15.ACMOSopampcapableofopemti.ngfrom1.5VpowersupplyisshowninFig.P6.3-15.Alldevicelengthsarelfl-mandaretooperateinlhesaruralionregion.DesignalloftheWvaluesofeveryttan.s.isrorofthisopwnptomeetthefollowingspeciticati0118:==,,{WI•W2W3~W4W5•WdW6W7W9~WIOWll•WI2,..,.+1.5v6.3-16.TheCMOSopampshowninPig.P6.3-16wasdesigned.fabricated.andtested.Itworkedsatisfactorilyexceptthatwhentheopampwasusedinaunity-gainconfiguration,thepositivepeakofa:!:1.5Vsinusoidoscillatedasillustrated.Whatcausedthisoscillationandhowcanitbefixed?Assumethattheelectricalparametersofthis=-VTP=0.7V,K'N"'28p..AfVz.K'p=8ll-AIV~.)w=A,.=O.OJV-1,'YN0.35ylfz,"'N=0.9V112,and2141f1=0.5V.opampareVmFigureP6.3-15=~VnutFigurePfi.J-16C:::::...J346CMOSOPERATIONALAMPLIFIERSFigurePM-46.4-1.Sketchtheasymptoticfl'tquency~nscofPSRR+andPSRR-ofthetwo-stageopampdesignedinExample6.3-1.6.4-2.Findthelow-frequencyPSRRandallrootsofthepositiveandnegativepower-supplyrejectionratioperfonnanccforthetwo-slageopampofFig.P6.3-10.6.4-3.RepeattheanalysisofthepositivePSRRofFig.6.4-2iftheMillercompensationcircuittyofFig.6.2-16(a)isused.Comparethelow-frequencymagnitudeandroo!llwiththoseofthepositivePSRRforFig.6.4-2..,.,..tv.,.,-6.4-4.lnFig.P6.4-4.findandidentifythelow-frequencygainandtheroots.Thisrepresentsthecasewhereanoisyacgroundcaninfluencethenoiseperformanceofthetwo-stageopamp.6.4-S.RepelltlbeanalysisofFigs.6.4-2and6.4-4forthep-channelinput,two-stageopampshowninFig.P6.4-5.Voo+--~--------~--------~VBIAS6.5-2.6.5-3.6..5-4.6.5-5.6.U.VssFigureP6.4-56.s-t.AssumethatinFig.6.5-l(a)thecurrentsinMlandM2are50)LAandtheWILvaluesoftheNMOSlrallSistorsare10andofthePMOSmmsis-6.s-7.totsareS.WhatisthevalueofV81AStbatwillcausethedrain--!rourcevoltageofMIandM2tobeequaltoVa.,(sal)?DesignthevalueofRtokeepthesource--dminvoltageofM3andM4equal10V.a(sat).Findanexpressionforthesmall-signalvoltagegainofv.,1/~1,forFig.6.5-l(a).lftheWIC.valuesofMI.M2,MCt,andMC2inFig.6.5-1(b)are10andthecurrentsinMIandM2are50...,A,findtheWILvaluesofMBIlhrougbMBSthatwillcausethedraln-$0urcevoltageofMIandM2tobeequaltoV.,.,(sat).AssumethatMB3=MB4111ldthecurrentthroughMB5is5jlAWhatwillbetilecurrentflowingthroughMS?InFig.6.5-1(a),findthesmall-signalimpedance10acgroundlookingintothesourcesofMC2IUidMC4,assumingthereisnocapacitanceattachedtotheoutput.Assumethecapacitancetogroundatthesenodesis0.2pF.WhatisthevalueofthepolesatthesourcesofMC3andMC4?Repeatifacapacitorof10pFisattachedtotheoutput.RepeatExample6.5-ltofindnewvaluesofW1andW1thatwillgiveavoltagegainof10.000.FindthedifferentialvoltagegainofFig.6.5-1(a)wheretheoutputistakenatthedrainsofMC2andMC4,W1/L1=Wi~=10)l.rnlltJ.ID,Wc1/l.c1=Wdl.c=Wdl.o=WcJLo!=l)I.IIIII11m.W-jy=WJ~=IJLm/1Jl.ffi,andI,,.100!JA.UsethemodelparametersofTable3.1-2.IgnorethebulkeffecJS.DiscusstheinlluencethatthepoleatthegateofM6inFig.6.5-2willhaveontheMillercompensationofthisopamp.SketchanapproximaterootlocusplotasC,.isvariedfrom:zerotothevalueusedforcompensation.VerifyEqs.(6.5-4)through(6.5-8)forthetw~stageopampofFig.6.5-3havingaca~secondstage.IfthesecondstagebiascurrentisSOtJ.Aa.ndW£)4,=Wcr/Lct,=Wc.7/.l.o=W.,lProblems347=L,1JIJTI{lJl.m,whatistheautputresistanceoflhisamplifieru.~ingtheparametersofTable3.1-27U.S.VerifyEqs.(6.5-9)through(6.5-1.1)assumingthatM3=M4=M6=M8andM9=MIO=Mil""M12andgiveanexpressionfortheo"eralldiffereulialvoltagegainofFig.6.5-4.6.5-9.Aninternallycompensated,ca5-..---...--o..l'()lllUnbufferedOutputopampstage:•Ct-J-·....-.RL~:.·-..-··:....-:~poles,p;andp~.Thebufferwillintroduceanotherpole,p;.Withnocompensation,theopenloopvoltagegainoftheopampisgivenas(7.1-3)wherepfandpiaretheuncompensatedpolesoftheunbufferedopampandp]isthepoleduetotheoutputstage.WeshallassumethatJp{ltorMP3sem;estheoutputcurrentthroughtransistorM6,andintheeventofexcessivelylargeoutputcurrents,thebiasedinvenerformedbytransistorsMP3andMN3trips,thusenablingtransistorMP5.OncetransistorMP5isenabled,thegateoftransistorM6ispuJleduptowardthepositivesupplyVvo·Therefore,thecurrentinM6islimitedtoapproximately60mA.Inasimilarmanner,transistorsMN3A,MP3A.MP4A.MN4A,andMNSAprovideshort-circuitprotectionforcurrentsinking.TheopampofFig.7.1·8iscompensatedbymethodsstudiedinChapter6.Eachamplifier,AIandA2,isindividuallycompensatedbytheMillermethodS-coupleddifferential-inputstage.390HIGH-PERFORMANCECMOSOPAMPSand_Km3Vid2(7.3-6)Wenotethatthesmall-signalperformanceofFig.7.3-9isequivalenttoasoun:e-coupleddifferential-inputpair.Infact,thecross-coupleddifferential-inputstageissimplytwoc:nllisconnectedsource-coupleddifferentialpairs.whereonetransistorisNMOSandtheotherPMOS.ThecompleteopampisshowninFig.7.3-10.Weseethatthecurrentsgeneratedbythecross-coupleddifferential-inputstagearesimplysteeredtoapush-pullcascodeloadtogivetheequivalentgainofatwo-stageopamp.Theinputcommon-moderangeofFig.7.3-10ismuchlessthantheotheramplifierspesentedabove.Thenegativeinputcommon-modevoltageiswhereVGs=Vr+VoN·2V7+3V0...,couldbeasmuchas2V.Thegainisequivalenttoatwostageopampandcompensationisaccomplishedbythecapacitancetogroundateachou1putCommon-ModeOutputVoltageStabilizationThepurposeofcommon-modestabilizationistokeepthecommon-modevoltagemidwaybetweenthelimitsofthesignalswing{normallypower-supplyvoltage~~),Thi$canbedonewitbinternalorexternalcommon-modefeedback.Theconceptofaninternalcommon-modefeed·backschemeisillustratedinFig.7.3-11.Thiscircuitmodelstheoutputofeachofthemdifferential-in,differential-outopampsconsideredabove.Fortheopampsthatarenotpush-Figure7.3-10Cla.~sAB,differential-outputopampusingacross-coupleddifferential-inputstage.7.3Differential-OutputOpAmps--Figure7.3-11Modeloftheoutputofadifferential-outputopampandtheimplememationofacommonmodefeedbackscheme.,IJiI.391pull(ClassAB),thelowerdependentcurrentsourcesarereplacedwithindependentcurrentsources.Thisisthecaseforthetwo-stageMilleropampofFig.7.3-3,thefolded-cascadeopampofFig.7.3-5,andthetwo-stagedifferential-output,folded-cascodeopampofFig.7.3-7.Thecommon-modefeedbackcircuitofFig.7.3-11worksbysensingthecommon-modevoltageonlhegatesofMIandM2.Ifthecommon-modevoltageshoulddecrease,thesource-gatevoltagesofMlandM2areincreased.Thiscausesmorecurrenttobeinjectedintothetopconnectionofthesourcingcurrentsources,R,1andR02•Assumingbalancedconditions,thiscurrentsplitsandflowsthroughR.,1andR03andthroughR02andRo4.ThevoltagedropsacrossR.,3andRoo~increasewithrespecttoVssandopposetheoriginalcommon-modevoltagedecrease.Figure7.3-12showsbowthismethodofcommon-modefeedbackstabilizationisappliedtothetwo-stageMilleropampofFig.7.3-3.Itisobservedthatthepowersupplyvaluemustbeincreasedtoaccommodatethecommon-modefeedbacktransistors,MI0andM11.Thiswillalsoresultinadecreaseinthepositiveoutputvoltageswing.ThestabilizationtechniqueofFig.7.3-11isprobablysufficientformostcasesbutithasthedisadvantageofnotbeingreferenced.Itsimplyopposeschanges.Aschemethatwilldrivethecommon-modeoutputvoltagetoadesiredvalueisshowninFig.7.3-13.lnthiscase,M3andM4havetheirgatestakentothedesiredcommon-modeoutputvoltage,V,..,,,.ThiswillgenerateacurrentofIasshown.NowifthegatesofM1andM2aretakentotheactualdifferentialoutputvoltages,v.,1andv.,2,theywillcreateacurrentcalled/5•Thiscurrentismirroredtobecome16•IfMIthroughM4arematched,thenwhenthecommon·modevalueoh..1QC1JJFigure7..3-12Two-stageMillerdifferential-in,differential-outopampwithcommon-modestabilization.t392HIGH-PERFORMANCECMOSOPAMPSFigure7.3-13Acommon-modestabiliza.tionschemethatreferencestheconectiontoa.desiredconunon-modevoltage,V..,.,.Tocorrectioncin:uitryandvu7.isequaltoV""""lot3flshouldequall6•Ifthecommon-modevalueofv.1andv.2isgreaterthanV"'.,../6willbegreaterthanl""mandvoltagetothecorrectioncircuitrywilldecrease.Ifthecommon-modevalueofV01andVo2islessthanV""""16w111belessthanI,.,,..andvoltagetothecorrectioncircuitrywillincrease.Withproperselectionofthecorrectioncircuitry,common-modestabilizationcanbereferencedtothedesiredvalueofcommon-modeoutputvoltage(seeProblem7.3-6).Sincethecommon-modestabilizationtechniquesare11.formofnegativefeedback,onemustbecarefultoensurethathigh-gainloopsarestable.Iftheoutputofthedifferential-outputopampusesthecascodeconfiguration,thecommonmodeloopgaincanea.~ilybeequivalenttoatwo-stageopamp.lbeabovecommon-modeoutputvoltagestabilizationschemeswereinternaltotheopamp.Inmanyapplicationsitispossibletostabilizethecommon-modeoutputvoltageusingexternalcircuitry.Thisisparticularlyattractiveforswitched-capacitorcircuits.Figure7.3-14showshowadifferential-outputopampcanbestabilizedbyexternalmeans[12].Inthistypeofcircuit,theopampisusedonlyduringthe~pha.~.TheterminaloftheopampdesignatedasCMbias,isaninputthatdeterminesthecommon-modeoutputvoltage.Duringthe1/11phase,thetwocapacitors,Ccmoarechargedtothedesiredvalueoftheoutputcommon-modevoltage,V,_.NotethatVacmisconnectedtoCMbiasduringthisphase.Duringthe~phase,theCcmcapacitorsthathavebeenchargedtVoanareconnectedbetweenthedifferentialoutputsandtheCMbiasnode.Eventhoughadifferentialoutputvoltagemayexist,theaveragevoltageappliedtotheCMbiasnodewillbeV_,.ItisassumedthatthephaseperiodsaresmallenoughthatthevoltageacrossCcmdoesnotchange.Manyothertypesofcommon-modeoutputvoltagestabilizationtechniquesarepossiblebuttheyallfollowthegeneralprinciplesillustratedabove.MostpracticalrealizationsofintegratedcircuiL~us.ingopampswillusedifferential-outputopampstoincreasethesignalswingandremoveoddharmonics.Inaddition,commonnoisesuchaschargeinjectionfromFigure7.3-14Ane~ttemaloutput,common-modevollagestabilization5Cbeme.7.4MicropowerOpAmps393swilchesisgreatlydiminished.Whilewewillcontinuetodevelopvarioustypesofopampsusingsingle-endedoutputs,thereadermustkeepinmindthisisprimarilyforpurposesofsimplificationandnotnecessarilythepracticalfonnofaparticularopampimplementation.UHICROPDWERUPRHPSIInthissection,opampsthatrequireminimumpowerareconsidered.Thistypeofopampprimarilyoperatesintheweak.inversionregion.Opampsoperatingintheweak.inversionregionhavebecomeveryuseful[13-17]becausetheyoperatenotonlyatlowpower-supplycurrentsbutalsoatverylowpower-supplyvoltages.Ourfullttaskinthissectionistodevelopthesmall-signalequationsfortransistorsoperatinginweakinversion.ThesewillbeappliedtounderstandingafewbasicamplifierarchitecturesthatworkwellU.'iingmicropowertechniques.ThetechniquesthatareintroducedinthissectiontocreatehighcurrentoverdrivecanbeusedinstronginversioncircuitsaswcU.Two-StageMillerOpAmpOperatinginWeakInversionFirst,considertheequationsthatmodelthelarge-signalbehavioroftransistorsoperatingatverylowcurrentdensities.Assumingoperationintbesaturationregion,subthresholddraincurrentwasgiveninEq.(3.5-5)as~'I(7.4-1)Fromthisequation,thetransconductancecaneasilybederivedas(7.4-·2)Thisresultisveryinterestinginthatitshowsalinearrelationshipbetweentransconductanceanddraincurrent.Furthermore,thetransconductanceisindependentofdevicegeometry.Thesetwocharacteristicssetthesubthresholdregionapartfromthestronginversionregion,wheretherelationshipbetween8mandInisasquare-lawoneandalsoafunctionofdevicegeometry.Infact,thetransconductanceoftheMOSdeviceoperatingintheweakinversionregionlooksverymuchlikethatofabipolartransistor.Equation(7.4-1)showsnodependenceofdraincurrentondrain-sourcevoltage.Ifsuchwerethecase,thenthedeviceoutputimpedancewouldbeinfinite(whichisobviouslynotcorrect).Thedependenceofi1>onVoscanbeapproximatedinthesamewayasitwasforthesimplestronginversionmodel,wherethedraincurrentismodulatedbytheterm1+hVos·Notethattheweakinversion"-maynotnecessarilybethesameasthatextractedfromstronginversionmeasurements.Theexpressionforoutputresistanceinweakinversionislr""'"-Mo(7.4-3)394HIGH-PERFORMANCECMOSOPAMPSFi1111re1.4-11\VOstageMilleropampoperatinginwellkinven;ion.Likethetransconductance,theoutputresistanceisalsoindependentofdeviceaspectratio.WIL(atconstantcurrent).Since).isafunctionofchannellength,itistheonlycontrolthedesignerhasonthegain(g.,ru)ofasinglestageoperatinginweakinversion.Withthesethingsinmind,considerthesimpleopampshowninflig.7.4-t.Thedegainofthisamplifieris(7.4-4)Intermsofdeviceparametersthisgaincanbeexpressedas(7.4-5)Thegainbandwidthg.,11CisGB=Im(n1kT!q)C(7.4-6)ItisinterestingtonotethatwhilethedegainoftheopampisindependentofI,.theGBisnoLThisbecomesalimitingfactorinthedynamicperfonnanceoftheopampoperatinginweakinversionbecausethedecurrentissmallandthustheGBissmall.Theslewrateofthisamplifieris(7.4-7)GAINANDCiBCALCULATIONSFORSUBTHRESHOLDOPAMPCalculatethegain,GB.andSRoftheopampshowninFig.7.4-1.Thecurrent,lm.is200nA.Thedevicelengthsare1~m.Valuesfornare1.5and2.5forp-channelandn-channeltran·sistors,respectively.Thecompensationcapacitoris5pF.UseTable3.1-2asrequired.Assumethatthetemperatureis27oc.7.4MicropowerOpAmps395SolutionUsingEq.(7.4-5),thegainis1A""'2•(1.5)(2.5)(0.026)(0.04+0.05)(0.04+0.05)""'48701V/V'ThegainbandwidthisGB·""'SR100w-9x2.5(0.026)(5xw-12>=307,690rps=="49.0kHz=(2)(307,690)(2.5)(0.026)=0.04V/~OtherOpAmpsOperatingIntheWeakInversionRegionConsiderthecircuitshowninFig.7.4-2asanalternativetothetwo-stageopampdescribedabove.ThedifferentialgainofthefirststageisIntennsofdeviceparameters,thisgaincanbeexpressedasA..,=IUJ!I-..V,=[Dlfl4.,.l11)4112Yt1[)411a(7.4-9)Verylittleifanygainisavailablefromthefirststage.Thesecondstagehowever,does,provideareasonableamountot'gain.ThetotalgainofthecircuitisbestcalculatedassumingthatthedevicepairsM3-M8,M4-M6,andM9-M7actascurrentmirrors.Therefore,=A""Kmt(SJS4)(gtb6+K=(SJS4)(J\,+:ll.7)n1V,(7.4-10}Figure7.4-2Push-pulloutputopampforoperationinweakinvenion,396HIGH-PERFORMANCECMOSOPAMPSAtroomtemperature(V,=0.0259V)andfortypicaldevicelengths,gainsontheorderof60dBcanbeobtained.Thegainbandwidthcanbeexpressed(7.4-1l)wherethecoefficientbistheratioofWJLr,toWiL4(Ws/LstoW3/L3}.lfhighergainsuerequiredusingthisba~iccircuit,twothingscanbedone.ThefirstistobleedoffsomeofthecurrentflowingindevicesM3andM4sothattheirtransconductanceswillbelowerthantheinputdevices,thusgivingthefirststageagaingreaterthanone.Onlyasmallimprovementcanbeobtainedwiththistechnique.Theotherwaytoincreasethegainistoreplacetheoutputstagewithacii8CodeasillustratedinFig.7.4-3.Thegainofthisamplifieris(7.4-12)Asimplecalculationshowsthatthiscascodeamplifiercanachievegainsgreaterth1111SOdBandallofthegainisachievedattheoutput.IncreasingtheOutputCurrentforWeakInversionOperationThedisadvantageoftheamplifierspresentedthusfaristheirinabilitytoprovidelargeoutputcurrentswhilestillmaintainingmicropowerconsumptionduringquiescentconditions.Aninterestingsolutiontothisproblemhas.beenpresentedintheliterature1131.Thebasicidea,showninFig.7.4-4,istoprovideaboostoftailcurrent(whichisultimatelyavailableattheoutputthroughmirroring)wheneverthereisadifferentialinputvoltage.Figure7.7-4consistsofthecircuitshownwithinthedottedboxofFig.7.4-3plusthecircuitrynecessarytoboostthetailcurrent,I,,whenadifferentialinputsignalisapplied.---------------v~Figure7.4-3OpampofFig.7.4-2w;ingcascadeoutputsforgainimpmvcmenlinweakinversionoperation.7.4MicropowerOpAmps397Figure7.4-4Dynamicallybiaseddifferentialamplifierinputstage.AssumethattransistorsM18throughM21areequaltoM3andM4andthattransistorsM22,M23,M24,M25.M26,andM27areallequal.FurtherassumethatM28throughM29arerelatedinthefollowingway:(7.4-13)(7.4-14)Duringquiescentconditions,thecurrentsitandi2areequal;therefore,thecurrentinM24isequaltothecurrentsuppliedbyMl9.AsaresultnocurrentflowsinM26,andthusnoneinM28.Similarly,nocurrentflowsinM29.Therefore,noadditionalcurrentillprovidedforthedifferentialstage.However,ifv11>v12,theni2>itandthetailcurrentsuppliedtothedifferentialstageisincreasedbytheamountofA(i2-i1).Ifv11i1andwecanwritethat(7.4-15)FromtherelationshipforthedraincurrentinweakinversionandthedefinitionofvJN,wecanexpressMitasVJN)-;-i,_It=exp(-nV(7.4-16)7Ifwedefinetheoutputcurrentasbtimesi2ioiJT=-i1,wemaywritetheoutputcurrent,iour.asbl{exp(:~.)-1)___;:....___.:.,_:;:___:..,-.....,....(1+A)-(A-I)exp(-vc_:-~)nV,(7.4-17)398HIGH-PERFORMANCECMOSOPAMPSInFig.7.4-3,bistheratioofM5toM4(andalsoM8toM3whereM7equalsM6).Figure7.4-5showsparametriccurvesofnonnalizedoutputcurrentasafunctionofinputvoltageforvariousvaluesofA.Thisconfigurationcanbeveryusefulwhenverylowquiescentcurrentisrequiredandconsiderabletransientcurrentsmaybeneededtodrivecapacitorsinsampleddatafilterapplications.TheenhancedperfonnanceofFig.7.4-4isaresultofbothnegativeandpositivefeedback.ThiscanbeseenbytracingthesignalpathfromthegateofM28throughM2tothegaleofM19andbacktothegateofM28.Thenegative-feedbackpathstartsatthegateofM28throughMItothegateofM20andthroughM24backtothegateofM28.Forthissystemtobestableduringlinearoperation,theinfluenceofthenegative-feedbackpathmustoverwhelmthepositive-feedbackpath.Itiseasytoseethatthegainofthepositive-feedbackloopis...,kloopgam.Positive-.eedbac(8m19)8m!9=A=(8m2$)---=A-g...gm4(7.4-18)8m26Thegainofthenegative-feedbackpathisNegative-feedbackloopgain=(8"')gm328(R.ao)(gm24)=ASm22Km26(7.4-19)Asavoltagedifferenceisappliedattheinput.thecurrentsinthetwopathschangeandthegainofthepositive-feedbackpathincreaseswhilethegainofthenegative-feedbackpathdecreases.ThisleadstotheoverdriveseeninFig.7.4-5.IfAbecomestoolarge,thenthesystemisindeedunstableandthecurrentwillrendtoinfinity.Thecurrent.however,willnotreachinfinity.Theinput11ansistorswillleavetheweakinversionregionandEq.(7.4-17)isnolongervalid.Ithasbeenshownthatfromalarge-signalviewpoint,thissystemisstable[13].ThemaximumpossibleoutputcurrentwillbedeterminedbytheproductofK'andWILandthesupplyvoltage.Theaboveanalysisassumesidealmatchingofthecurrentmirrors.Ifbettercurrentmirrorsarenotused,thismismatchmuslbeconsidered.Figure7.4-5Parametriccurvesshowingnonnalizedoutputcurrentversusinputvoltagefll£differentvaluesofA.7.4MicropowerOpAmps399Figure7.4-6(a)CurrentmirrorwithM2operatingintheactiveregion.(b)lllusttationofthellraincurrentsofMlandM2.(a)Anotheradvantageofoperatingthetransistorsinthesubthresholdregionisthatthegate-sourcevoltageappliedforpropercircuitoperationcaneasilybebelowthethresholdvoltageby100mYormore.Therefore,theVnssaturationvoltageistypicallybelow100mV.Asaresultofthe.o;esmallvoltagedrops.theopampoperatinginweakinversioncaneasilyfunctionwitha1.5Vsupply.providedthatsignalswingsarekeptsmall.Becauseofthislowvoltageoperation,circuitsoperatinginweakinversionareveryamenabletoimplantable,biomedicalapplications,wherebatterysizeandcapacityarelimited.IncreasingtheOutputCurrentforStrongInversionOperationThetechniqueusedaboveforboostingtheoutputcurrentcanalsobeusedforamplifiersworkinginstronginversion.Inthiscao;e,Ashouldbelessthanonesincenomechanismexiststolimitthecurrentasitincrea.'IC:seKceptthepowersupplies.Othertechniquesexistthatallowboostingoftheoutputcurrentabovethequiescentvalue.OneoftheseisshowninFig.7.4-6.Thisisasimplecurrentmirrorwiththeoutputtransistoroperatingintheactiveregion.IfthemirrorisdesignedsothatMlandM2haveequalcurrentswhenM2isintheactiveregion,thenifM2canbemovedfromtheactiveregiontothesaturationregion,thecurrentmirrorwillexperienceacurrentgain.Letusconsideranexampletoillustratethisconcept.CURRENTMIRRORWITHM2OPERATINGINTHEACTIVEREGIONAssumethatM2hasavoltageacrossthedrain-sourceof0.1V"'(sat).DesigntheW2f.L,.ratiosothat/1=/2=100J.LAifW.JL110.Findthevalueof/2ifM2issaturated.=lftJMU.I,IUsingtheparametersofThble3.1-2,wefindthatthesaturationvoltageofM2isV.,..(sat)21.[2(ii:J=~.Kiv(Wz/L,)=\f~=0.4264VNow,usingtheactiveequationofM2,weset/2=100..-.AandsolveforWi.L,..lOOj.LA=K:V(Wi~)[V.u1(sat)·V&a-0.5V~.z]=llOj.LA/V2(WiLl)£0.426•0.0426-0.5·0.04262)y2.=1.883XL0-6(W2fL;I}400HIGH-PERFORMANCECMOSOPAMPSThus,IOO=1.883(W,jL,)-+r:w=53.12NowifM2shouldbecomesaturated,thevalueoftheoutputcurrentofthemirrorwith100J.LAinputwouldbe53111-Aoraboostingof5.31times/1•Theaboveexampleillustratestheconcept.TheimplementationofthisconceptisshowninFig.7.4-7andiseasytodousingtbedifferential-outputopampsconsideredinSection7.3.ThecurrentmirrorsthathavetheboostingappliedtothemconsistofM7andM9,M81111dMIO.M5andMil,andM6andM12.Normally,M9,Ml0,Mll,andMI2areoperatingintheactiveregionbecausethegate-sourcedropsofMl3andM21,Ml4andM22,MISandM23,andMl6andM24arede.~ignedtoputthesetransistorsintheactiveregionwhenthequiescentvalueofi1ori2isflowing.Assumethatwhenadifferentialinputisapplied,i1willincreaseandi2decrease.Theincreasedvalueofi1flowingthroughM21andM24willbringM9andMl2backintosaturation,allowingthecurrentmirrorsM7andM9andM6andM12toachieveacurrentgainofk.wherekistheboostingfactorofthemirror(k=5.31inExample7.4-2).Thecurrentmirrorsinthei2pathwillhaveagainlessthanone,whichalsoincreasesthecurrentavailableatthedifferentialoutput~.Unfonunately,asM9,MlO,M11,andM12movetowardthesaturationregionasthegate-sourcevoltagesofM21,M22,M23,andM24increase,thegate-sourcevoltageofMI3,Ml4,MIS,andM16alsoincrease,keepingM9throughMl2frombecomingsaturated.Thecurrentmirrorboostingconceptappliedtotheregulated-cascadecurrentmirroravoidsthisproblemandgivesbetterperformance.Agoodexampleofalow-poweropampwiththeabilitytosinkandsourcelargeoutputcurrentsisfoundinacommerciallyavailableCMOSopamp[18j.Figure7.4-8showsthedetailsofthisopamp.Thegainstagesareincludedwithintheopampsymbolandthetransistorsshownareforthepurposesofprovidingthelargesinkandsourceoutputcurrents.Thekeytothisdesignisthefloatingbattery(whichofcourseisimplementedbyMOSFETs)Figure7.4-7Implementationofthecurrent-mirrorboostingconceptusingthedifferentialoutputopampofFig.7.3-10.Vss7.4MicropowerOpAmps401Figure7.4-8Outputcircuitofalow-power,higb·output-currentopamp.designedtogivethethresholdvoltageplus0.1Vacrossthegate-sourcesofM2andM3.Jnotherwords,thevalueofthisbatteryvoltageisV~~~~t=IVr~·l+OJv+Vm+0.1v=IVTPI+v.,..,.+0.2v(7.4-20)Also,thequiescentvoltage,V1(Q),oftheoutputofthegainstageisassumedtobeequaltoNotethatthegattHourcevoltageofMl,whichisVm+0.1V.istranslatedtothegate-sourceofMlviathepathM4-M5-M6.AlsonotethatthepathM4-M5-M6-M7ispositivefeedbackbutthepresenceofV8ntmakesthegainclosetozero(theimpedanceoftheVantimplementationwillbeofimportance).Assumethattheoutputvoltageofthegainstage,Jr1,increasesbyanamountAv.Inthiscase,thegate-sourcevoltageofMlisVm+0.1V+Avandthesource-gatevoltageofM2isIVTP\+O.lV-Av.Therefore,MlisonandM2isoff.Ifv1decreasesbyanamountAv,MlwillbeoffandM2on.Thisparticularopampisspecifiedashavingaquiescentcurrentof1.2f.I.Aandtypicalsink/sourceoutputcurrentsof300tt.A/600tt.Aforpowersuppliesthatvaryfrom2.5tol0V.Thefollowingexampleillustratesthedegreeofoverdrivecurrentavailableinthisopamp.OVERDRIVECURRENTAVAILABLEINFIG.7.4-8Assumethatv1variesaboutV1(Q)by~0.3VandthattheW/LvaluesofMlandM2are50and150,respectively.Calculatethesink/sourceoutputcurrents.SolutionIfv1=±0.1V.then.effectively,:±:0.2Vaboutthequiescentvalueofgate-sourcevoltageisappliedtoMlandM2.ThesinkingcurrentwilloccurwhenMJisonandM2isoffbecausev1>0.Whenv1=V1(Q)+0.3V,thecurrentinMlis402HIGH-PERFORMANCECMOSOPAMPSWhenv1=V1{Q)-0.3V,thecurrentinM2isim""K'·W22PLz(0.4V)2=SO·2150(0.16}j.LA=60011-AAsthevariationaroundV1(Q)increases,theoutputsink/sourcecurrentswillalsoincrease.Illthisparticularopamp,theoutputsink/sourcecurrentscanbeaslargeas2mA.Thissectionhasexaminedthesubjectoflow-poweropamps.Inmostlow-poweropamps,thetransistorsareworkinginthesubthresholdorweakinversionregion.Thisalsoallowsthepowersuppliestobesmall.whichisanadvantageforbattery-poweredcircuits.Thedisadvantageofweak.inversionoperationisthatthesmallcurrentsimplylowbandwidths.Underdynamicconditions,itispossibletoachievelargeoutputcurrentsusingtechniquesthatboosttheoutputsink/sourcecurrentcapability.7.5LOW-NOISEOPRMPSLow-noiseopampsareimportantinapplicationswherealargedynamicrangeisrequired.Thedynamicrangecanbeexpressedasthesignal-to-noiseratio(SNR).Thesignificanceofdynamicrangecanbeappreciatedwhenconsidering!hedynamicrangenecessaryforsignalresolutionintermsofdigitalbits.ItwillbeshowninChapterI0thatadynamicrangeof6dBisrequiredforeverybitofresolution.Thus,ifanopampisprocessingasignalthatwillbeconvertedintoa14bitdigitalsignal,adynamicrangeofover84dBor16,400isrequired.AsCMOStechnologycontinuestoimprove,thepowersuppliesaredecreasingandmaximumsignallevelsof1Varenotunusual.ThermsvalueofasinusoidwithalVpeak-to-peakvalueis0.354Vrms.Ifthisisdividedby16,400,thentheopampmusthaveanoiseHoorthatislessthan21.6f.LVrms.Inadditiontonoiseasalowerlimit,onemustalsoconsiderthelinearityoftheopamp.Iftheopampisnonlinear,thenapuresinusoidalsignalwillgenerateharmonics.Ifthetotalharmonicdistortion(THD)oftheseharmonicsexceedsthenoise,thennonlinearitybecomesthelimitingfactor.Sometimesthenotationofsignal-to-noiseplusdistortion(SNDR)isusedtoincludebothnoiseanddistortioninthedynamicrangeconsideration.Oneshouldalsoconsidertheinftuenceofunwantedsignalssuchaspower-supplyinjectionorchargeinjec·tion(duetoswitches).Low-noiseopampsmusthaveasufficientlyhighPSRRinordertoachievethedesiredlowerlimitofthedynamicrange.Inthissection,wewillmaketheimportantassumptionthattheopampsarelinearandhaveahighPSRRandfocusonlyonthenoise.InSection.3.2,thenoiseperformanceofaMOSFETwasdescribed.ThenoiseinaMOSFETwasmodeledasamean-squarecurrentgeneratorinthechannelgiveninEq.(3.2-12)repeatedhereforconvenience.(7.5-1)7.5Low-NoiseOpAmpsI111,403Itiscustomarytoreflectthisnoiseasamean-squarevoltagegeneratorinserieswiththegatebydividingEq.(7.5-1)byg~togetEq.(3.2-13)alsorepeatedhereforconvenience.2eeq=(8k7tl+q)3gmKF]+2fC""WL/('~~(7.5-2)ItisimportanttorememberthatEq.(7.5-2)isonlyvalidforthecasewherethesourceoftheMOSFETisonacground.Forexample,Eq_.(7.5-2)wouldnotbevalidifthetransistorwasinlhecascadeconfiguration.Inthatcase,onewouldeitherusetheeffectivetransconductance[seeEq.(5.2-35)1orsimplyuseEq.(7.5·1).ThenoiseoftheMOSFETconsistsoftwopartsasshowninEq.(7.5-l)or(7.5-2).ThefirstpartisthethermalnoiseandthesecondiscaUedtheIifnoise.Inmanyrespects,thethermalnoiseofaMOSFETdeviceisequivalenttothethennalnoiseofaBJT.Unfortunately,the1/fnoiseofaMOSFETismuchlargerthanthe1/.fnoiseofaBJT.Onemightfeelthatthe1/fnoiseisonlyimportantatlowfrequenciesbecauseitvariesinverselywithfrequency.However.thellfnoiseisaliasedaroundclockfrequenciesandthereforebecomessignificantevenathigherfrequencies.Forexample,whenMOSFETsareusedtobuildahigh-frequencyVCO,thespectralpurityofthesinusoidisllmitedamongotherthingsbythelifooise[19].MinimizingthethennaJnoiseofopampsisreasonablystraightforward.FromthefirstterminEq.(7.5-2),weseethatwewantthesmall-signaltransconductance,g,.,tobelargetominimizetheequivalentinput-mean-squarenoisevoltage.ThiscanbedonebylargedecurrentsorlargeWILratios.FortheIifnoise,thereareatlea.c;tthreeapproachestominimizingthelifnoiseofCMOSopamps.ThelintistominimizethenoisecontributionoftheMOSFETsthroughcircuittopologyandtransistorselection(NMOSversusPMOS),decurrents,andWILmtios.ThesecondistoreplacetheMOSPETsbyBJTstoavoidtheIifnoise.Thethirdistouseexternalmeans,suchac;chopperstabilization,tominimizethe1/fnoise.SincetheII[noiseisgenerallymorecrucial,wewillfocusonreducingthisnoiseintheremainderofthissection.Low~NoiseOpAmpsUsingMOSFETsThefirstapproachtominimizingthe1/fnoiseusescircuittopologyandtransistorselection.Thetransistorselectioniseasy.Empirically,PMOStransistorshaveabouttwotofivetimeslessl(fnoL..ethanNMOStransistors.*Therefore,PMOStransistorsshouldbeusedwhereitisimportanttoreducethelifnoise.Thecircuittopologyisalsostraightforward.Theoneprinciplethatiskeyinminimizingnoiseistomolcethefirstslagegainashighaspossible.Thismeansthatiftheinputisadifferentialamplifier,thesource-coupledtransistorsshouldbePMOSandthegainofthedifferentialamplifiermustbeaslargeaspossible.Thiswas~howntobetrueinSection5.2forthedifferentialamplifier,whereitwasfoundthatthelengthsoftheloadtransis·torsshouldbegreaterthanthelengthsoftheinputtransistorstominimizetheI({noise.Figure7.5-lshowsaCMOSopampthatachieveslownoisebycarefulselectionoftheW/l.ratiosf20].Thisopampissimilartothetwo-stageopampofSection6.3exceptforthecascodedevices,M8andM9,whichareusedtoimprovethePSRRasdiscussedinSection6.4.Also,returningthecompensationcapacitortothesourceofM9allowstheoutputpoletobe•valuesforKFfera0.611-mTSMCCMOStechnalogyareKF=1X10-23F·Aforthen-channelandlandtheminimumchannellength,L.ntn(,....m)[241.ThemotivationfordecreasingthechannellengthistoincreasetheMOSFETfrandtoallowmorecircuitstobeimplementedinthesamephysicalarea,whichsustainsthemovefromverylarge-scaleintegrated(VLSDcircuitstoultralarge-scaleintegrated(ULSI}circuits.Thereductioninvoltagecomesaboutbecausethereduceddimensionsreducethevoltagebreakdownsofthetechnology.Inaddition,thepowerdissipationindigitalcircuitsinproportionaltothesquareofthepowersupply.lnordertoreducethepowerdissipationinULSlcircuits,itisnecessarytoeithercoolthechiportoreducethepowersupplyorboth.However,fromtheviewpointoftheanalogdesigner,thetrendsindicatedonFig.7.6-1becomeaproblem.Asthepower-supplyvoltagesaredecreased,thedynamicrangeoftheanalogsignalsisdecreasedIfthethresholdvoltageisscaledwiththepower-supplyvoltage,thedynamicrangewouldnotbestronglyltffected.Thereasonthatthethresholddoesnotsignificantlydecreaseisbecauseofthedigitallogic.WerecallthatcurrentflowsintheMOSFETevenbelowthethresholdvoltage,whichistheweakinversionregion.Ifthethresholdvoltageistooclosetozero,appreciablecurrentwouldflowintheMOSFETlogicdevicesevenwhentheyareoff.Ifthissmallcurrentismultipliedbythousandsofdevices,itbecomesasignificantamountofpowerdissipation.CMOStechnologycapbemodifiedtohavelowerthresholdvoltllgesandhavenoleabgecurrentbutthiswillprobablynotbecomeastandardfortheCMOStechnologybecauseitisnotneededfromthedigitaldesigner'sviewpoint.Thebestthresholdforlogictogivethehighestnoisemarginisapproximatelyhalfwaybetweenpowersupply.Ifa1.5Vpowersupplyisused,then0.75Vthresholdsgivemorenoisemargin.Consequently,theanalogdesignerwillhavetofacetheissueofdecretofonnthechannelsinMlandM2.IfthevalueofilNisgreaterthanIr>ss.thenV85isgreaterthanzeroandthecurrentmirrorfunctionsasanormalcurrentmirror.Withthevalue7.6Low-VoltageOpAmps423Figure7.6-11(a)Simplebulk-drivencurrentmirror.(b)Cascodebulk-drivencurrentmirror.":'"(a)(b)ofV8slllightlygreaterthanzero,thevalueofVMINattheinputacrossthecurrentsourcecanbesmall.ExperimentalresultsshowthatV85islessthan0.4Vforcurrentsupto500j.i.A.Thesmall-signalinputresistanceis1/gm/J,andthesmall-signaloutputresistanceisrbthesameasthatofagate-drivencurrentmirror.Figure7.6-ll(b)showsacascadecurrentmirrorusingbulk-drivenMOSFETs.Theminimuminputvoltageofthecurrentmirrorisequaltothesumofthebulk-sourcevoltagesforMlandM3.Again,thevalueofimmustbegreaterthanlossofthetransistors.Theminimuminputvoltageislessthan0.5Vformostcurrentsunder100JJ.A..Theadvantageofthecascadecurrentmirrorisbettermatchingandhigheroutputresistance.Figure7.6-12showstheexperimentaloutputcurrentsfora2IJ.ID.,p-wellCMOStechnology.Anotherapproachtolow-voltagecurrentmirrorsistolevelshiftthedrainbelowthegate.ThisapproachisillustratedinFig.7.6-13,wherethebase-emittervoltageofaBITisusedtoaccomplishthevoltageshift.WeknowthatthedraincanbeVrlessthanthegateandstillbeinsaturation.Therefore,a"longasthebase-emitterdropoftheBITislessthanVnthismethodwillallowtheinputvoltageofthesimplecurrentmirrortoapproachVDS1(sat).Unfonunately,CascadeCurrentMirrorAllWIL's200,.uni41J.m=,I~4xl0-5:;3xl0-5.!2lin=SOIJ.Ahn=40!!AIIlin=3011Avv_::lin=20!!Alin=10IJ.A000.20.40.6Vout(V)0.81Figure7.6-12E~tperimentalres11ltsof•bulk-drivencasc:odec11rrentmirror(foursamples).424HIGH-PEFIFORMANCECMOSOPAMPSFigure1.6-13SimpleCllJ'I'elltmirrorwithlevelshiftingofthedrainofMI.thismethodonlyworksforeitherann-channelorap-channelmirrorbutnotbothbecausetheBJTrequiresafloatingwellandonlyonetypeofwellisavailableinmostprocesses.NotethattheBITcanbelateralorvertical.TheclassicalbandgapvoltagegeneratordiscussedinSection4.6usesthesummationoravoltagethatisproportionaltoabsolutetemperature(PTAT)andavoltagewhosetemperaturecoefficientisbasedonthatofapnjunction.Whenthesevoltagesareinseries,thebandgaptopologyiscalledthevoltagemode.ThismodeofoperationisshowninFig.7.6-14(a).Unfonunately.itrequiresatleastthebandgapvoltageofsilicon(1.205Vat27"C)plusaMOSFETsaturationvoltage.Thismeansthattheminimumpower-supplyvoltagewouldbearound1.5V.IfbettercurrentmirrorsareusedinthebandgapgeneratoraswasthecaseforFig.6.3-9,theminimumpowersupplyvoltagewouldbethebandgapvoltageplusadiodedropplusfiveMOSFBTsaturationvoltages.Thiscouldeasilybeaminimumof2.5V.Consequently,thecurrentmodeofvoltage-i:urrent-modearchitecnnesofFigs.7.6-14(b)and7.6-14(c)aremoresuitableforlow-power-supplyvoltageoperation.(/NLinthesefiguresisanonlinearcurrentusedtocorrectforthebandgapcurvatureproblem.)ItwasshowninSection4.5howtocreateavoltagethatisPTAT.Whenthisvoltageisplacedacrossaresistor,thecurrentisPTAT.EventhoughthetemperaturecoefficientoftheresistormaymakethiscurrentdifferentfromaPTATcurrent,whenthiscurrentisreplicatedlnanotherresistorwiththesametemperaturecoefficient,thevoltageacrossthatresistorwillbePTAT.ThisapproachisillustratedinFig.7.6-15forgeneratingcurrentswithtemperaturecoefficientsofV8EandVPTAT·(a)(b)Figure7.6-14(a)Voltage-modebandgaptopology.{b)Current-modebandgaptopology.(c)Voltage-current-modebandgaptopology.7.6Low-VoltageOpAmps425Figure7.6-15MethodofgeneratingcurrentswithtemperaturecoefficientsequaltothaiofVaEandVP'f/ll'•ThePTATcurrentisgeneratedbythedifferenceofthebase-emitterdropsofQlandQ2superimposedacrossR1•Thiscurrentflowsthroughdiode-connectedM4.Anyp-channeltransistorwhosesourceandgateareconnectedtoM4willcreateaPTATcurrent.Forexample,thevoltageVouucanbeexpressedas(7.6-15)HthetemperaturecoefficientsofR,.andR1areidentical,thenVowlwillbePTAT.Thecurrent,whichbasatemperaturecoefficientofthebase-emitter,IVBaisgeneratedbyputtingR'1acrossthebase-emitterjunctionofQ1.NotethatPTATcurrentflowsthroughQI.ThecurrentfvaEisgeneratedbythenegative-feedbackloopconsistingofQl,M6,M7,MS,andR3•ThisfeedbackloopcausesthecurrentinQltobePTATandcausesthecurrentinM8tobe/VB£·Anyp-cbanneltransistorwhosesourceandgateareconnectedtothesourceandgateofMSwillhaveacurrentwhosetemperaturecoefficientisthatofabase-emitterjunction.ThiscurrentcanbeusedtocreateavoltagewithatemperaturecoefficientofV8e.Thisisseeninthefollowingequation:(7.6-16)Again,ifthetemperaturecoefficientsofR3andR4areidentical,thenVouahasthetemperarurecoefficientofabase-emitterjunction.Unfortunately,thetemperaturedependenceoftheba·Thisgiveslhemaximuminputcommon-modevoltageandallowsthepowersupplytovarywithoutlimitingthepositiveinputcommon-modevoltage.Thesignalcurrentsofthedifferentialoutputarefoldedthroughthesetransistors(M6andM7)andconvertedtosingle-endedsignals7.6Low-VoltageOpAmpsFigure7.6-17Alow-voltage,two-stageopamphavingV00i!!:427v,...withthen-channelcurrentmirror(M8andM9).Finally,asimpleClassAoutputstageusingMillercompensationisusedforthesecond-stagegain.Thisopampwillhavethesameperformanceasatheclassicaltwo-stageopampbutcanbeusedatalowerpower-supplyvoltage.Ithastheadvantageovertheclassicaltwo-stageopampthattheloadsoftheinputdifferentialstagearebalanced.DESIGNOFALOW-VOLTAGEOPAMPUSINGTHETOPOLOGYOFFIG.7.6-17UsetheparametersofTable3.1-2todesigntheopampofFig.7.6-17tomeetthespecificationsgivenbelow.VoD=2VV~cm(max.)=2.5Vv.,..,(max)=1.75vVout(min)=0.5VSlewrate=±10V/fJ.SPhasemarginV1,..(min)=IVGB=lOMHz=60°forCL=10pFSolutionAssumingtheconditionsforatwo-stageop:ampnecessarytoachieve60°phasemarginandthattheRHPzeroisatleast10GBgivesCc=0.2C,_=2pFTheslewrateisdirectlyrelatedtotbecurrentinMSandgivesIs=Cc·SR=2X10-11•107=2011-AWealsoknowtheinputtransconductancesfromGBandC,.Theyaregivenas8m1=8m2=GB•C.,=20?rXIif·2X10-12=125.67,..S428HIGH-PERFORMANCECMOSOPAMPSKnowingthecurrentflowinMIandM2givestheW/Lratiosas(125.67X10-~=?.IS2·llOXl0-6•10X10-62Next,wefindtheWILofM5thatwillsatisfyV~mt(min)specification.Thisgives2.10110.7.18-0.75=1-0.159-0.75Vvss(sat)=1-=0.0909vTherefore,V055sat()=0.0909=J2/sKiv(Ws/Ls)Ws~-=Ls2·20==441l0•(0.0909)2ThedesignofM3andM4isaccomplishedfromtheupperinputcommon-modevoltageandisVkm(max)=V00-Vsm(sat)+VTN=2-Vsm(sat)+0.75=2.5VSolvingforV503(sat)gives0.25V.Lr:tusassumethatthecurrent.~inM6andM'7are20j.LA.Thisgivesacurrentof30j.LAinM3andM4.KnowingthecurrentinM3(M4)givesVsoJ(sat)s~2·30wlw450.(Wj~)-+~=L..·16so·12.s•20xw=(J)•t2.8.I•2.47X10-LS2(j).w6•L6.Co.69=7.59X10rad/sWehaveassumedthatthechannellengthis1~mintheabovecalculation.Thisvalueisabout100timesgreaterthanGBsowesbouldbeokayeventhoughwehaveneglectedthedraingroundcapacitancesofMlandM3.Finally,theWILratiosofthesecondstagemustbedesigned.Wecaneitherusetherelationshipfor60°phasemarginofg.,14=10g,1=1256.7,...SorconsiderpropermirroringbetweenM9andMl4.Combiningtheequationsforsaturationregion,wegetw-==Lg,..KA-V0s(sat)Substituting1256.7p.SforCmland0.5VforV0Sl4givesW1JL14=22.85.Thecurrentcorrespondingtothisg,andWILis/14=314~A.TheWILofMl3isdesignedbythenecessarycurrentratiodesiredbetweenthetwotransistorsandisWuluL13/1231420-=-Ill=-·12.8=201Now,wemustchecktomakesurethattheV""'(max)issatisfied.ThesaturationvoltageofMl3is2.31450.2010.25Vwhichexactlymeetsthespecification.F01:propermirroring,theWILratioofM9shouldbew9=19~/14w,4=1.42Lr4SinceWr/~wasselectedas1,thisiscloseenough.430HIGH-PERfORMANCECMOSOPAMPSLetuschecktoseewhatgainisachievedatlowfrequencies.Thesmall-signalvoltagegaincanbewrittenas=1j.IS,8ths=0.8j.LS,8.u13=15.7j.LS,and81hl4aboveequationgives8th?Ynutvia=(125612567·)(·)1.828.26="'=12.56j.J.S.Puttingthesevaluesinthe69.78·44.47=3103VNThepowerdissipation,includinglaJAsof201.1-A,is7081.1-W.Theminimumpowersupplypossiblewithnoregardtotheinputcommon-modevoltagerangeisVr+3V0MWithVr=0.1VandVoN...0.25V,thisopampshouldbecapableofoperatingwithapowersupplyof1.5V.TheopampofFig.7.6-17isagoodexampleofalow-voltageopampusingstandardtechniquesofdesigningopamps.Ifthevoltagedecreasesbelow2Vrothenproblemsstarttoarise.Theseproblemsincludeadecreasedinputcommon-modevoltagerange.ThiscanbesomewhatalleviatedbyusingthepamllelinputstageofFig.7.6-4.Theothermajorproblemisthatmostofthetransistorswillhaveadrain-sourcevoltagethatisclosetothesaturationvoltage.ThismeansthatRdswillbelarger(orrthsmaller).Thiswillcausethegaintodecreasesignificantly.Intheaboveexample,ifthetransistorlambdasareincreasedto0.12fortheNMOSand0.15forthePMOSthegaindecreasesbynine,resultinginagainof345VN.Ofcourse,onecanuselongerchannellengthstocompensatethisreductionbutthepenaltyismuchlargerareasandmorecapacitance.Typically,whatmusthappenisthatmorestagesofgainarerequired.ItmaybenecessarytofollowtheoutputoftheopampofFig.7.6-17witbtwoormorestages.Thiscausesthecompensationtobecomemorecomplexasthenumberofgainstagesincrease.Them11ltipathnestedMillercompensationmethoddiscussedbrieflyattheendofSection7.2isfoundinmoredetailintheliterature[7,311.TobuildCMOSopampsthatoperateatpowersupplieslessthan1.5-2Vrequiresareductioninthethresholdvoltageortheuseofdifferentapproaches.ManyCMOStechnologieshavewhatiscalled11naturaltransistor.ThistransistoristhenormalNMOStransistorwithouttheimplanttoshiftthethresholdtoalargervoltage.ThethresholdofthenaturalMOSFETisaround0.1-0.2V.Suchatransistorcanbeusedwhereneededintheopamp·toprovidethenecessaryinputcommon-modevoltagerangeandthegain.Onemustrememberthatthenaturaltransistorsdonotgotozerocurrentwhenthegate-sourcevoltageiszero,whichisnotaproblemwithmostanalogapplications.Thealternativetousingamodifiedtechnologyistousethebulk-driventechniquediscussedearlierinthissection.Thistechniquewillbeusedtoillustrateanopampthatcanworkwithapower-supplyvoltageofl.25Vr.WewillusethisapproachtoimplementaCMOSopampworkingfromaIVpowersupplyhavinganinputcommon-modevoltagerangeof25mVoftherail,anoutputswingwithin25mVoftherail,andagainof275[29].Thegaincouldbeincreasedbyaddingmoregainstagesassuggestedabove.7.6low-VoltageOpAmps431Figure7.6-18AIV,two-stageoperationalamplifier.Figure7.6-18showsa1VCMOSopampusingbulk-drivenPMOSdevicesandtheinputtransistorsforthedifferentialinputstage.ThecurrentmirrorconsistingofM3,M4,andQ5istheoneshowninFig.7.6-13.Q6isabufferandmaintainsthesymmetryofthemirror.TheoutputstageisasimpleClassAoutput.Longchannellengthswereusedtotrytomaintainthegain.TheperformanceofthisissummarizedinTable7.6-1.ThegaincouldbeincreasedbyTABLE7.6-1PerformanceResultsfortheCMOSOpAmpofFig.7.6--18Specification(VDD=0.5V,V55=-0.5V)deOpen-loopgainPower-supplyCIJITeft!Unity-gainbandwidlh(GBJPhasemarginInput-offsetvoltageInputcommon-modevoltagerangeOutputswingPositiveslewmreNegativeslewrateTHO,closed-loopgainof~IVNTHD,closed-loopgainof+IVNSpectralnoisevoltagedcrulityPositivepowerS11pplyrejectionNegativepow«supplyrejectionMeasuredPerformance(Ct=22pF)49dB(Vk:mmidrange)300,..A1.3MH2(V~cmmidrange)57"(V,..midrange)l:lmV-0.475to0.450V-0.475to0.491V+0.71'-V/s-1.6jJ.V/s-60dB(0.75Vpp,IkHzsinewave)-59dB(0.75Vpp,10kHz~inewave)-59dB(0.7SVpp,IkHzsinewave}-51dB(0.75Vpp,10kHzsinewave)367nv/\11{;@1kHz181nVIv'ib@tokHz81nV/\.o'lh@100kHz444nVI\.o'lh@IMH261dBatlOkHz55dBat100kHz22dBatlMHz45dBatlOkHz27dBat100kHzSdBatlMHz432HIGH-PERFORMANCECMOSOPAMPSusingthecascadeofseveralmoreClassAinvertingstagescompensatedwiththemultipathnestedMillercompensationapproach.Thetechniqueofforward-bia.~ingsomeofthebulk-sourcejunctionsofaCMOSopamphasbeenusedtoimplementafolded-cascadeopamphavingagainofalmost70dBfora1Vpowersupply[32).Unfortunately,asthepower-supplyvoltageisdecreased,thepowerdissipatedinlowvoltageopampsdoesnotdecreaseproportionally.Thisisduetothefactthatthegainisproportionaltothe8mrlhproduct.Asr~~sdecreasesbecauseoftheincreasingvalueoflambda(channelmodulationparameter).thevalueofg,.,mustgoup.However,thetransconductanceisincreasedbylargerWILvaluesandmorecurrent.AslargeWILvaluesareapproached,moreareaisusedandlargerparasiticcapacitancesresultTherefore,thedecurrentstendtobelarge,resultinginhigherpowerdissipation.Thisisanareawherebipolarjunctiontransistorshaveasignificantadvantagebecausetheycanachievemuchlargervaluesoftransconduc•tancesatlowercurrents.7.7SUMMARYThischapterbasdiscussedCMOSopampswithperformancecapabilitiesthatexceedthoseoftheunbufferedCMOSopampsofthepreviouschapter.Theprimarydifferencebetweenthetwotypesistheadditionofanoutputstagetothehigh-performanceopamps.OutputstagespresentedincludethosethatuseonlyMOSdevicesandthosethatusebothMOS1111dbipolardevices.Theoutputresistanceofthesource-oremitter-followeroutputscouldbereducedevenfurtherbyusingnegativefeedback.Addingtheoutputstageusuallyintroduce..~morepolesintheopen-loopgainoftheoperationalamplifier,makingitmoredifficulttocompen·sate.Theprimaryfunctionofthebufferedopampsistodrivealow-resistanceandalargecapacitanceload.Anotherareaofimprovementwa.'>inthefrequencyresponse.Understandingwhatlimitsthefrequencyresponseoftheopampallowedtheunity-gainbandwidthtobeoptimized.Theuseofcurrentfeedbackactuallyallowstheabovefrequencylimittobeexceeded.resultinginvery-high-frequencyopamps.Thischapteralsoincludedthedifferentialoutputopamp.Ex·amplesofconvertingsingle-endedoutputopampstodifferentialoutputopampsweregiven.Thissectionwasimportantbecausemostanalogsignalprocessingdonetodayusesadifferentialsignaltorejectnoiseandincreasethedynamicrange.Astheanalogcontentoflargemixed-signalintegratedcircuit.-.increases,itisimponanttominimizethepowerdissipation.Opampswithminimumpowerdissipationwereintroducedalongwiththemeansofincreasingtheoutputcurrentwhendrivinglargecapacitiveloads.ltwasseenthatalow-poweropampcouldbeobtainedattheexpenseoffrequencyresponseandotherde...irablecharacteristics.Mostlow-poweropampsworkintheweakinversionmodeinordertoreducedissipationandthereforeperformlikeBJTopampcircuits.Asthepower-supplyvoltagesdecreasebecauseoftechnologyimprovementsandthedesiretominimizepowerdissipation.manychallengesfacetheanalogdesigner.Oneistokeepthenoiselevelaslowaspossible.Methodsofdesigninglow-noiseopampswerepresentedalongwithseveralexamples.Inaddition,theopampsmustbedesignedtoworkwiththeeverdecreasingpower-supplyvoltages.Aspower-supplyvoltagesbegintoapproach2Vr-newtechniquesmustbeused.TwosolutionsaretousetechnologiesthathavespecialtransistorssuchasthenaturalMOSFETortousetheMOSFETsinanunconventionalwaylikethebulk·drivenMOSFETs.Problems433TheCMOSopampdesignsofthischapterareagoodexampleoftheprincipleoftradingdecreasedperfonnanceinoneareaforincreasedperformanceinanother.Dependingontheapplicationoftheopamp,thistrade-offleadstoincreasedsystemperformance.ThecircuitsandtechniquespresentedinthischaptershouldfindapplicationinmanypracticalareasofCMOSanalogcircuitdesign.~LEHS.AssumethatV00"'-Vssand/17andlminPig.7.1-1are100j.iA.DesignW11/L18andWt9f'Lt9togetV.rot•=Vsat9=1.5V.DesignWltll.ztandWv/Lnso!hatthequiescentcurrentinM21andM22isalso100f.!-A.!.CalculatethevalueofVAandV8infig.7.1-2andthereforethevalueofVc·I.Ass11methatK:V=47jLA!V1,K'p=17J.IA!Vz,Vm=0,7V,VTP=-0.9V,'YN=0.85,V112,'Yr=0.25V1a.2lrf>,l=0.62V.)."'=o.osv-•.and)..,.=0.04V-•.UseSPICEtosimulateFig.7.1-2andobtainthesimulatedequivalentofPig.7.1-3.~.modelparametersofThble3.1-2.Whatisthe-3dBfrequencyofthisbufferifCL=10pF?7.1-7.ACMOScircuitusedasanoutputbufferforanOTAisshowninFig.P7.1-7.Findthevalueofthesmallsignaloutputresistance,R-andfromthisvalueestimatethe-3dBbandwidthifaSOpFCllpiK:itorisattachedtotheoutpUtWhatisthemlll(imumandminimumoutputvoltageifalkfire~istorisattachedtotheo11tput?Whatisthequiescentpowerdissipationofthiscircuit?UscSPICEtoplotthelotalharmonicdistortion(THO)oftheoutputstageofFig.7.1-2asafunc-tionofthennsoutputvoltageatIkHzforaninput-stagebiascurrentof20f.l-A.UsetheSPICEmodelparametersgiveninProblem7.l-3.i,AnMOSoutputstageisshowninfig.P7.1-5.Drawasmall-signalmodelandcalculatetheac++lowfrequency.Assumethatbulkeffects~beneglected.voltagegainatWILvaluesinmicronsVss=-3VFlgureP7.1·7Vout7.1-8.WhattypeofBITisavailablewithabulkCMOSVin0--1VssFigureP7.1-Si.Fmdthevalw:ofthesmall-signaloutp11tresistanceofFig.7.1-9iftheWvaluesofMIandM2an:increasedfrom10f.!-mto100f.!-ID.Usethep-welltechnology'lAbulkCMOSn-weUtechnology?7.1-9.AssumethatQIOofFig.7.1-11isconnecteddirectlytothedrainsofM6andM7andthatM8andM9arenotpresent.Giveanexpressionforthesmall-signaloutputresistanceandcomparethiswithEq.(7.1-9).JfthecurrentinQIO-Mllis500j.iA.thecurrentinM6andM7is100!JA,and{JF=100,usetheparametersofThble3.1-2assumingI...,mchannellengthsandcalculatethisresistanceatroomtemperature.434HIGH-PERFORMANCECMOSOPAMPSVout7.1-10.FlRdthedominantrootsoftheMOSfoUowerandlheBITfollowerfor11M:buffered.ClassAopampofExample7.1-1.UsethecapacitancesofTable3.2-1.Comparetheserootlocationswiththefac1thatGB=5MHz.7.1-11.GiventheopampinFig.P7.1-ll,findthequiescentcurrentstlowingintheopampandthesmallsignalvoltagegain,ignoringanyloadingproducedbytheoutputstage.AssumeKfv=25IJ.A/V2andx;,'=10IJ.A/V2•flindthesmall-signaloutputresistanceassumingthar"=0.04V-•.7.2-1.FmdtheGBofatwo-stageopampusingMillercompensationusinganullingresistorthatbas60"phasemarginwherethesecondpoleis-10XIIfrad/8andtwohigherpolesboth.ar-100X1Ifrad/s.AssumethattbeRHPzeroisusedtoCIIIICCItbesecondpoleandthattheloadcapacitancestaysconstantIftheinputtransconductanceis500!J.AIV,whatisthevalueofC,?7.2-2.ForanoparnpwherethesecondpoleissmallerthananylargerpolesbyafactorofI0,wecansetthesecondpoleat2.2GBtoget60°phasemargin.Use11M:polelocationsdetenninedinExample7.2-2andfindtbeconstantmultiplyingGBthatdesignsP6for60"phasemargin.7.2-3.WhatwillbethephasemarginofExample7.2-2ifCt=lpF'?7.2-4.UsetbetechniqueofExample7.2-2toextendtheGBoflbecascadeopampofExample6.5-2asmuchaspossiblewhilemaintaininga60"phasemargin.WhuistheminimumvalueofCLforthemaximumGB'?7.2-S.ForthevoltageamplifierusingacurrentmirrorshowninFig.7.2-11,designthecunentsinMl,M2,M5,andM6andtheWILratiostogiveanoutputresistancethatisatleastIMOandaninputresistancethatislessthanIk.O.{Thiswouldallowavoltagegainof-10tobeachievedtJsingR1=10k.OandR2=IMO.7.2-6.InExample7.2-3.calculatethevalueoftheiDJIIIIpoleoftbecurrentamplifierandcomparewithlhemagnitudeoftheoutputpole.7.2-7.AddasecondinputtothevoltageamplifierofPig.7.2-12usinganotherR1resistorconnectedfromthisinputtotheinputofthecurrentamplifier.UsingtheconfigurationofFig.P7.2-7.calcula!etheinputresistance,outpUtresistance,and-3dBfrequencyofthiscircllit.AssumethevaluesforFig.7.2-12asdevelopedinExample7.2-3butleithetwoR1resistorseachbe1000!l.FigureP7.2-77.2-8.ReplaceR1inFig.7.2-.12withadifferentialamplifierusingacurrent-mirrorload.Designthedifferentialtransconductance,g..,sothatitisequaltoIIR1•7.3-1.ComparetbedifferentialoutputopampsofFigs.7.3-3,7.3-5,7.3-6,7.3-7,7.3-8,and1.3-10fromtheviewpointof(a)noise,(b)PSRR,(c)ICMR[V~max)andV1,(min)],(d)OCMR[VJmax)andV0(min)].(e)SRassumingallinputdifferentialcurrentsareidentical,and(f)powerdissipationifProblems435+----~~~--~~--~VnrASFiguftP7.3-3currentoftheinputdifferentialamplifimareidenticalandpowersuppliesareequal7.3-5,7.3-6,7.3-7,7.3·8,and7.3-10.Includethedifferential-in,differential-outvoltagegain,thenoise,andthePSRR.all7.3-2.ProvethattheloadseenbythedifferentialoutputsoftheopampsinFig.7.3-4areidentical.Whatwouldbethesingle-endedequivalentloadsifCLwasreplacedwitharesistor,RL?7.3-1.1\vodifferentialoutputopampsareshowninPig.P1.3-3.(a)Showhowtocompensatelheseopamps.(b)Ifalldecurrentsthroughalltransistorsare50p.AandallWILvaluesareIO~~om/1j.l.m,usetheparametersofTable3.1-2andfindthedifferential-in,differential-outsmall-signalvoltagegain.7.3-4.ComparativelyevaluatetheperfonnanceofthetwodifferentialoutputopampsofFig.P1.3-3withthedifferentialoutputopampsofFigs.7.3-3,7.3-5.FigUICP7.3-5showsadifferential-in,differentialoutopamp.Developanexpressionforthesmallsignal,differential-in,differential-outvoltagegainandthesmall-signaloutputresistance.7.3-6.Usethecommon·modeoutputstabilizationcircuitofFig.7.3-13tostabilizethedifferentialoutputopampofFig.7.3-3togroundassumingthatthepowers11ppliesaresplitaroundground{VDIJ=IVssl>·Designacorrectioncircuitthatwillfunctionproperly.7.3-7.(a)rralltransistorsinFig.7.3-12haveadecurrentof50ji.AandaWILof10f.Ull/1j.UD.findthegainofVssFigureP7.3·5436HIGH-PERFORMANCECMOSOPAMPStbecommon-modefeedbackloop.(b)IftheoutpU!ofthisamplilicriscascoded,thenrepealpart{a).of20f,LA,WIL..ISOj.Lm/10j.l.m,Kj.,"'25f,LA/V2•andR0isI00kfl.7.!-8.Showhowtouscthecommonfeedbackcirc11itof7.5·2.RepeatExample7.5-1withW1=W2=.500jiJilandL1=~=0.5j.LMtodecreasethenoisebyafactorof10.Fig.5.2-15tostabilizethecommon-modeoutpUtvoltageofFig.7.3-5.Whatwo11ldbetheapproximategainofthecommon-modefeedbackloop(intermsofg..andrd,landhowwouldyoucompensatethecommon-modefeedbackloop?7.4-1.Calculatethegain,GB,SR.andP-forlhefoldcd-cascodeopampofFig.6.5-7(b)if¥00=-Vss=1.5V,tbec11rrentintbedifferentialamplifierpairis.50nAeachandthec11rrentinthesources,M4andM5,is150nA.Ass11mcthetransi~ton;areall10f.Lm/lf.LMandtheloadcapacitoris2pF.7.4-2.Calculatethegain,GB,SR,andP-fortheopampofFig.7.4-3,whereI,100nA.alltransistorwidthsare10J.Lmandlengthsare1j.Lm,andV00""-Vss=1.5V.lfthesaturationvoltageis0.1V,de-=signthecorrectbiasvoltageforMI0andMIIthatachievesmaximumandminimumoutputswingassumingthetransiston>M12andMI.5haveSOnA.Assumethat/00=2nA,n,""1.5andn.=2.5.7.4-!,DeriveEq.(7.4-17).IfA..2.atwhatvalueofviN/nV,williour=Sf,?7.4-4.Designthecum:ntboostingmirrorofFig.7.4-6(a)10achieveI00f,LAoutputwhenM2issaturated.Assumethat11=toj.LAandW1/L1=10.FindW;zl~andthevalueofV0~wherei2=10f,LA.7.4-5.IntheopampofFig.7.4-7,thecurrentboostingideaillustratedinFig.7.4-6suffersfromtheproblemthatasthegateofMISorM16isincreasedtoachievecum:ntboosting,thegate-sourcedropofthesetransistorsincreasesandpreventsthevruoftheboostingtransistor(MllandMl2}fromreachingsat11ration.Showhowtosolvethisproblemandconfirmyoursolutionwithsimulation.7.5-1.ForthetransistoramplifierinFig.P7..5-l,whatistheequivalentinput-noisevoltageduetothermalnoise?AssumethetransistorhasadedraincurrentFigureP7.5-17.5-!.Interchangealln-channelandp-channeltransistorsinFig.7.5-1andusingtheWILvaluesdesignedinExample7..5-1,findtheinputequivalent1/fnoise,theinputequivalenttbermalnoise,thenoisecornerfrequencyBDdthermsnoiseina1~to100kHzbandwidth.7.5-4.FmdtheinputequivalentnnsnoisevoltageoftheopampdesignedinExample6.3-1overabandwidthofIHzto100kHz.7.5-5.FindtheinputequivalentnnsnoisevoltageoftheopampdesignedinExample6.5-2ofabandwidthof1Hz;toI00kHz.7.6-1.IftheWandLofalltransistorsinFig.7.6-3are100j.LMand1j.LM.respectively,findthelowestsupplyvoltagethatgivesazerovalueofICMRifthedecum:ntinM.5is100jJA.7.6-2.RepeatProblem7.6-1ifMtandM2arenaturalMOSFETswithaVr=0.1VandtheotherMOSFETparametersaregiveninTable3.1-2.7.6-3.RepeatProblem7.6-1ifMIandM2aredepletionMOSFETswithaVr=-IVandtheotherMOSFETpanuneten>aregiveninTable3.1-2.7.6-4.FindthevaluesofV..,.andV...,ofFig.7.6-4iflheWandLvaluesofalltransistorsareI011-mand1!J.m,respectively,andthebiascu=ntsinMN5andMPSareI00j.t.Aeach.7.6-5.Twon-channelsource-coupledpairs,oneusingregulartransistorsandtheotherwithdepletiontransistorshavingaVr=-JV.areconnectedwiththeirgatescommonandthesourcestakentoindividualcurrentsinks.Thetransistorsaremod·eledbyTable3.1-2eJtceptthethresholdis-1Vforthedepeletiontransistors.Designthecombinedsource-coupledpairstoachieverail-to-railfora0-2Vpowersupply.TrytokeeptheequivalentinputtransconductanceconstantovertheICMR.Showbowtorecombinethedraincurrentsfromeachsource-coupledpairinordertodriveasecond-stagesingle-ended.7.6-6.ShowhowtocreatecurrentmirrorsbyappropriatelymodifyingthecircuitsinSection4.4thatwillhaveexcellentmatchingandaVMIN(in)=VONandVwiN(out)=VON.References437U-7.ShowhowtomodifyFig.7.6-16tocompensateforthetemperaturerangetotheleftofwherethetwocharacteristicscross.ordertocalculatethebulk-source/draindepletioncapacitors(assumezerovoltagebias).WhatisthenewvalueofGBandthevalueofCc?7.6-8.FortheopampofEllample7.6-1.findtheoutputandhigher-orderpoleswhileincreasingtheGBasmuchaspossibleandstillmaintaininga60°phasemargin.AssumethatLl+L2+L3=2IJ.min7.6-9.ReplaceMSandM9orFig.7.6-17withahighswingcascadecurrentmirrorofFig.4.3•7andrepealEurnple7.6-1.RHERENCESl.M.Milkovic,"CurrentGainHigh-FrequencyCMOSOperationalAmplifiers,"IEEEJ.Sefid.StateCitf:uils,Vol.SC-20,No.4.W·845-l!S1.Aug.1985.2.D.G.Maeding."ACMOSOperationalAmplifierwithLowImpedanceDriveCapability,"IEEEJ.Solid-StateCitf:uits,jVoi.SC-18.No.2,pp.227-229,Apr.1983.3.K.E.BrehmerandI.B.Wieser,"LargeSwingCMOSPowerAmplifier,"IEEEJ.Solid-SrateCircuit$,Vol.SC-18.No.6.pp.624-629.Dec.1983.4.R.B.Blackman."EffectorFeedbackonImpedance,"BellSyst.Tech.1.,Vol.22.pp.269-277.1943.5.S.Ma.~uda,Y.Kitalllura.S.Ohya,andM.Kikuchi,"CMOSSampledDifferential.PushPullCascodeOperationalAmplifier,"Proceedingsofthe/984/ntenwlionalSymposiumonCircuitsandSystems,Monlreal.Canada,May1984,pp.1211-1214.I6.P.E.Allen,andM.B.Terry,''TheUseofCurrelllAmplifiersforHighPerformanceVoltageAmplification;•IEEE1.,Solid-StateCin:uirs,Vol.SC-15,No.:Z..pp.155-162,Apr.1980.'1.R.G.H.EschauzierandJ.H.Huijsing.FrequenqCompensationTechniquesforLow-PowerOperationalAmplifiers,'Norwell,MA:KluwerAcademicPublishers,1995,Chap.6.8.S.RabiiandB.A.Wooley,"A1.8VDigital-AudioSigma-DeltaModulatorin0.8IJ.MCMOS,"IEEEJ.Solid-StateCircuits,Vol.32,No.6,pp.783-796,June1997.9.J.Grilo,B.MacRobbie,R.Halim,andG.Ternes,"A1.8V94dBDynamicRange4I:ModulatorforVoice1IApplications."/SSCCDig.Tech.Papers.pp.230-231.Feb.1996.·1.0.Y.Huang,G.C.Temes.andH.Hoshizawa."AHigh-Linearity,Low-Voltage,AU-MOSFETDelta-SigmaModulator,"~1'roc.CJCC'97,pp.13.4.1-13.4.4,May1997.'11.P.W.Li.M.I.Chin.P.R.Gray,andR.Castello,"ARatio-IndependentAlgorithmicAnalog-to-DigitalConversionTechnique,"IEEE.J.Solid-StaleCircuits,Vol.SC-19,No.6.pp.828-836,Dec.1984.·12.S.LewisandP.Gray,"APipelined5-Msample/s9-bitAnalog-to-DigitalConverter,"IEEEJ.Sclid-StateCircuits,Vol.SC-22,No.6,pp.954-961,Dec.1987.13.M.G.Degrauwe,l.Rijrnenants,E.A.Vittoz,andH.J.DeMan,"AdaptiveBiasingCMOSAmplifiers,"IEEE1.Solid-StateCircuits.Vol.SC-17,No.3,pp.522-528,June1982.14.M.Degrauwe,E.Vittoz.andI.Verbauwhede."AMicropowerCMOS-InstrumentationAmplifier,"IEEEJ.Solid.StateCircuit~·.Vol.SC-20.No.3,pp.805-807,June1985.[5.P.VanPeteghem,1.Verbauwbede,andW.Sansen,"MicropowerHigh-PerformanceSCBuildingBlockforIntegratedLow-LevelSignalProcessing.''IEEE1.Solid-StateCircuits.Vol.SC-2,No.4,pp.837-844,Aug.1985.i16.D.C.Stone,J.E.Schroeder,R.H.Kaplan.andA.R.Smith,''ArutlogCMOSBuildingBlocksfurCustomandScmicustomApplications,"IEEEJ.Solid-StateCirr:uits,Vol.SC-19,No.I,pp.55-61,Feb.1984.17.P.Krummenacher,"MicropowerSwitchedCapaci[O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onthefigure.TheidealaspectofthismodelisthewayinwhichtheoutputmakesatransitionbetweenVoLandVon·Theoutputchangesstatesforaninputchangeof/iV.where/iVapproacheszero.Thisimpliesagainofinfinity,asshownbelow.Gain=A=lim•AV--fllv:oH-v:OLfiV(8.1-1)Figure8.1-4showsthedetransfercurveofafirst-ordermodelthatisanapproximationtoarealizablecomparatorcircuit.Thedifferencebetweenthismodelandthepreviousoneis!hegain,whichcanbeexpressedasA.-_Von-VoLVm-VJL(8.1-2)v,.-whereVmandV,Lrepresenttheinput-voltagedifferencevNneededtojustsaturatetheoutputatitsupperandlowerlimit,respectively.Thisinputchangeiscalledtheresolulionofthecomparator.Gainisaveryimportantcharacteristicdescribingcomparatoroperation,foritdefinestheminimumamountofinputchange(resolution)necessarytomaketheoutputswingbetweenthetwobinarystates.Thesetwooutputstatesareusuallydefinedbytheinputrequirementsofthedigitalcircuitrydrivenbythecomparatoroutput.ThevoltagesV08andVoLmustbeadequatetomeettheVIHandV1Lrequirement.~ofthefollowingdigitalstage.ForCMOStechnology,thesevaluesareusually70%and30%,respectively,oftheraiJ-to-railsupplyvoltage.ThetransfercurveofFig.8.1-4ismodeledbythecircuitofFig.8.1-5.ThismodellookssimilartothemodelofFig.8.1-3.theonlydifferencebeingthefunctionsf1andfoFigure8.1-lIdealtransfercurveofacomparator.~--VoH--------+-------•vp-VNVoL--...8.1CharacterizationofaComparator441Figure8.1-3Modelforanidealcomparator.Vpfor(vp-VN)>0VoLfor(vp-VN)<0Thesecondnonidealeffectseenincomparatorcircuit~isinput-offsetvoltage,V09•InFig.8.1-2theoutputchangesastheinputdifferencecrosseszero.Iftheoutputdidnotchangeuntiltheinputdifferencereachedavalue+V0s.thenthisdifferencewouldbedefinedastheoffsetvoltage.Thiswouldnotbeaproblemiftheoffsetcouldbepredicted,butitvariesrandomlyfromcircuittocircuit[11foragivendesign.Figure8.1-6illustratesoffsetinthetransfercurveforacomparator,withthecircuitmodelincludinganoffsetgeneratorshowninFig.8.1-7.The:tsignoftheoffsetvoltageaccountsforthefactthatV03isunknowninpolarity.Inadditiontotheabovecharacteristics,thecomparatorcanhaveadifferentialinputresistanceandcapacitanceandanoutputresistance.Inaddition,therewillalsobeaninputcommon-moderesistance,R1""'.AlltheseaspectscanbemodeledinthesamemanneraswasdonefortheopampinSection6.1.Becausetheinputtothecomparatorisusuallydifferential.theinputcommon-moderangeisalsoimportant.TheICMRforacomparatorwouldbethatrangeofinputcommon-modevoltageoverwhichthecomparatorfunctionsnormally.Thisinputcommon-moderangeisgenerallytherangewherealltransistorsofthecomparatorremaininsaturation.Eventhoughthecomparatorisnotdesignedtooperateinthetransitionregionbetweenthetwobinaryoutputstates,noiseisstillimportanttothecomparator.Thenoiseofacomparatorismodeledasifthecomparatorwerebiasedinthetransitionregionofthevoltage-transfercharacteristics.ThenoisewillleadtoanuncertaintyinthetransitionregionasshowninFig.8.1-8.Theuncertaintyinthetransitionregionwillleadtojitterorphasenoiseinthecircuitswherethecomparatorisemployed.DynamicCharacteristicsThedynamiccharacteristicsofthecomparatorincludebothsmall-signalandlarge-signalbehavior.Wedonotknow,atthispoint,howlongittakesforthecomparatortorespondtothegivendifferentialinput.Thecharacteristicdelaybetweeninputexcitationandoutputtransitionisthetimeresponseofthecomparator.Figure8.1-9illustratestheresponseofacomparatortoaninputasafunctionoftime.Notethatthereisadelaybetweentheinputexcitationandtheoutputresponse.Thistimediffereneeiscalledthepropagationdelaytimeofthecomparator.ItisaveryimportantparametersinceitisoftenthespeedlimitationintheconversionFigure8.1-4Transfercurveof11comparatorwithfinitegain.VoL-.-442COMPARATORSFigure8.1-5Modelforacomparatorwithfinitegain.VpVNOHfor(vp-VN)fr(vp-VN)=>0A,..(vp-VN)forVJL6+8th7)(8.2-3)UsingEqs.(8.2-1)through(8.2-3)andEq.(8.1-4)givestheresolutionofthecomparator,whichisV1.(min).Figure8.2-1Two-stagecomparator.446COMPARATORSThecomparatorofFig.8.2-lha.'l.twopolesofinlefest.Oneistheoutputpoleofthefirststage,p.,andthesecondistheoutputpoleofthesecondstage,p2•Thesepolesareexpressedas-1PI=----Ct(g.ta+BtLM)(8.2-4a)-1P2=-----(8.2-4b)andC~t(KdiJ.+8ds4)whereC1isthesumofthecapacitancesconnectedtotheoutputofthefirststageandC11isthesumofthecapacitancesconnectedtotheoutputofthesecondstage.C11willgenerallybedominatedbyCL.Thefrequencyresponseofthetwo-stagecomparatorcanbeexpressedu.~ingtheaboveresultsas(8.2-5)Thefollowingexampleillustratesthepracticalvaluesforthetwo-stage,open-loopcomparator.PERFORMANCEOFATWO-STAGECOMPARATOREvaluateVoH•VouAv(O),V~n(min),p1•andp2forthetwo-stagecomparatorshowninFig.8.2-1.AssumethatthiscomparatoristhecircuitofExample6.3-1withnocompensationcapacitor,Cc,andtheminimumvalueofV06""0V.Also,a.~sumethatC1=0.2pFandC11=5pP.lji1Mit.j.iUsingEq.(8.2-1),wefindthatVoH=2.5-(2.5-0-0.7)[1-~1-62•95X10]=2.42V50X10-6•14(2.5-0-0.7)2ThevalueofVOL.isfoundfromEq.(8.2-2)andis2.5V.Equation(8.2-3)wasevaluatedinExample6.3-1asAv(O)=7696.Thus,fromEq.(8.1-2)wefindtheinputresolutionas.Vla(mm)=Von-VoL4.92VA.(O)=7696=0.640mV"8.2Two-Stage,Open-loopComparators447Next.wefindthepolesofthecomparator,p1andp2•FromExample6.3-1wefindthatp1=Ktb2+Kti•4Cr610-(0.04+0.05)_=15X0.211X106.75X106(1.074MHz)andPz=Kt~.E+KtU7=95)(10-6(0.04Cu+0.05)=1.71Xlif(0.670MHz}5X10-12Theresponseofatwo-stage,open-loopcomparatorhavingtwopolestoastepinputofv,nisgivenas(8.2-6)wherep1"-h~.....::..::'--......::..::.._::-'-......:.......:..:.,_,.:_______:_c11(8.2-8)IftherateofriseorfallofEq.(8.2-6)exceedsthepositiveornegativeslewrate,thentheoutputresponseissimplyarampwhoseslopeisgivenbyEq.(8.2-7}or(8.2-8).Assumingthatslewdoesnotoccur,thestepresponseofthetwo-polecomparatorofEq.(8.2-6)canbeplottedbynormalizingtheamplitudeandtime.Theresultisv:,.,..(t.)=Vour(l.)A,(O)V1nm_,=I---e•m-1+--1em-1-lilt•(8.2-9)where(8.2-10)andr,=tp,t=.,.1(8.2-11)4-48COMPARATORSIfm=1,thenEq.(8.2-9)becomesv'(t)=I-p1e-••""'n~-•.PI=1-e-•.-te-t,"(8.2-12)wherep1isassumedtobeunity.Equations(8.2-9)and(8.2-12)areshowninFig.8.2-2forvaluesofmfrom0.25to4.Iftheinputstepislargerthanv~o(min),thentheamplitudesofthecurvesinFig.8.2-2becomelimitedatVoH·Wenotewithinterestthattheslopeatt=0iszero.ThiscanbeseenbydifferentiatingEq.(8.2-9)andlettingt0.ThesteepestslopeofEq.(8.2-9)occursat=ln(m)m-I(8.2-B)r.(max)=--whichisfoundbydifferentiatingEq.(8.2-9)twiceandsettingtheresultequaltozero.Theslopeatr.(max)canbewrittenasdv:.O,,(tnCmax))dz.m-[exp(-ln(m))=m-1m-1(ln(m))]exp-m--m-1{8.2-14)Iftheslopeofthelinearresponseexceedstheslewrate,thenthestepresponsebecomes!>.lewlimited.IftheslewrateisclosetothevalueofEq.(8.2-14)itisnotclearhowtomodelthestepresponse.Onecouldassumeaslewlimitedresponseuntiltheslopeofthelinearresponsebecomeslessthantheslewratebutthispointisnoteasytofind.1fthecomparatorisoverdriven,Vm>Vm(min),theniftheslewrateissmallerthanEq.(8.2-14),theslewratecanbeusedtopredictthestepresponse.Figure8.2-2Linearstepresponseofacomparatorwithtworealaxispoles,p,and1'7.·ItilJ0.8~~0.6a!0.4m=P2~0.2z0Pl0246=NormalizedTime(r11"'tp1tlrl)8108.2Two-Stage,Open-LoopComparators449STEPRESPONSEOFEXAMPLE8.2-1FindthemaximumslopeofExample8.2-1andthetimeatwhichitoccursifthemagnilUdeoftheinputstepisV1n(min).IfthedebiascurrentinM7ofFig.8.2-1is100/LA.atwhatvalueofloadcapacitance,Ct.wouldthetransientresponsebecomeslewlimited?IfthemagnilUdeoftheinputstepisIOOVm(min)andVoH-VoL=IV,whatwouldbethenewvalueofCLatwhichslewingwouldoccur?SolutionThepolesofthecomparatorweregiveninE:x.ample8.2-1asp1=~6.75X106rad/sandp2=-1.71X106rad/s.Thisgivesavalueofm=0.253.FromEq.(8.2-13),themaximumslopeoccursattn{max)=1.84s.Dividrngby/Pdgivest(max)=0.273IJ.S.TheslopeofthetransientresponseatthistimeisfoundfromEq.{8.2-14)asdv:,...{r.(max))dIn=-0.338[exp(-1.84)-exp(-0.253•1.84)]=0.159VIsMultiplyingtheaboveby/PI!givesdv:n.,(t(max))=1072V/1-LSdt.Therefore,iftheslewrateofthecomparatorislessthan1.072V/!J.S,therransientresponsewillexperienceslewing.Iftheloadcapacitance,CL,becomeslargerthan(100p.A)I(1.072VI~LS)or93.3pF,thecomparatorwillexperienceslewing.Ifthecomparatorisoverdrivenbyafactorof100V1.,(min),thenwemustunnonnalizetheoutputslopeasfollows:dv:,1(1(max))dtV1n==V;.,(min)dv:.,Jt(max))dt·.=100·1.072V/11-s=107.2V/sTherefore,thecomparatorwillnowslewwithaloadcapacitanceof0.933pF.ForJargeoverdrives,thecomparatorwillgenerallyexperienceslewing.Itisofinteresttopredictthepropagationdelaytimeforthetwo-polecomparatorwhenslewingdoesnotoccur.Todothis,wesetEq.(8.2-9)equaltoO.S(VoH+V0Jandsolveforthepropagationdelaytime,tp.Unfortunately,thisequationisnoteasilysolved.AnalternateapproachistoreplacetheexponentialtermsinEq.(8.2-9)withtheirpowerseriesrepresentations.Thisresultsin(8.2-15)··-)]450COMPARATORSEquation(8.2·15)canbesimplifiedto1)Vout\tn""'Settingvout(t11)equalto0.5(V011delaytime,lpn•ast1"1...mt!4,.(0)V10(8.2-16)2+V01.)andsolvingfort11givesthenormalizedpropagationVoH+VoL=JV~n(rrrln)mA,..(O)V~nmV1n=_1_v;;;k(8.2-17)wherekwadefinedinEq.(8.1-8).ThisresultapproximatestheresponseofFig.8.2-2asaparabola.Forvaluesofr.thatareles.sthan1.thisisareasonableapproximation.Ifonecon-siderstheinfluenceofoverdrivingtheinput,thenEq.(8.2-17)isevenabetterapproximation.Theinfluenceofoverdrivingistoonlyusetheinitialpartoftheresponse(i.e.,itislikeV0uisbeingloweredandbroughtclosertozero)ThefollowingexampleexplorestheapplicationofEq.(8.2-17)topredictthepropagationtimedelayforatwo-polecomparator.PROPAGATIONDELAYTIMEOFATWO-POLECOMPARATORTHATISNOTSLEWINGFindthepropagationdelaytimeofthecomparatorofExample8.2-1ifV10=10mV,100mV,andIV.FromExample8.2-1weknowthatV;.(rnin)=0.642mVandm=0.253.ForV1•=10mV,k=15.576,whichgives11"'=0.504.ThiscorrespondswellwithFig.8.2-2,wherethenor·malizedpropagationtimedelayisthetimeatwhichtheamplitudeis1/{2k)or0.032.DividingbyIP11givesthepropagationtimedelayas72.7ns.Similarly,forV1n=100mVand1Vwegelpropagationdelaystimeof23.6nsand7.5ns,respectively.InitialOperatingStatesfortheTwo-Stage,Open-LoopComparatorInordertoanalyzethepropagationtimedelayofaslewing,two-stage,open-loopcomparator,itisnecessarytofirstbeabletofindtheinitialoperatingstatesoftheoutputvoltagesofthefirstandsecondstages.Considerthetwo-stage,open-loopcomparatorofFig.8.2·3.ThecapacitancesattheoutputsofthefirstandsecondstagesareC1andC11,respectively.Ourapproachwillbetochooseoneoftheinputgatesequaltoadevoltageandfindtheoutputvoltageofthefirstandsecondstageswhentheotherinputgateisaboveandbelowthedevoltageonthefirl>tgate.Actually,weneedtoconsidertwocasesforeachoftheprevi•ouspossibilities.ThesecasesarewhenthecurrentsinMIandM2aredifferentbutneitheriszeroandwhenoneoftheinputtransistorsha"acurrentofIssandtheothercurrentiszero.Webeginbyfirsta~sumingthatvmisequaltoadevoltage,V02,andthatv01>Vmwithi10.lnthiscase,aslonga."M4remainsinsaturation,i4=i3=i1,whichisgreaterthani2•Consequently,v,.1increasesbecauseofthedifferencecurrentll.owingintoC1•8.2Two-Stage,Open-LoopComparators451Figure8.2-3Two-stage,open-loopcomparatorusedt~;~findinitialstates.AsV01continuestoincrease,M4willbecomeactiveandi4Vm.itUndertheconditionsofEq.(8.2-18),thevalueofY.sG'6outputvoltagewillbe>Vm.theni1=Issandi2=0andv.,1willbeatVonandYou,isstillatVss.Next,assumethatv02isstillequaltoV02,butnowVGJ0andi2>YuhthepreviousresultsarestillvaliduntilthesourcevoltageofMlorM2causesM5toleavethesaturatedregion.Whenthathappens,IssdecreasesandV01canapproachVssandVw1willbedeterminedaswasillustratedinExample5.1-2.Next.IISSumethatvc;1isstillequaltoVel•butnowv02i2,whichgivesi4>i2,causingV01toincrease.AslongasM41ssaturated,i4>iz.WhenM4enterstheactiveregion,4willdecreaseuntili4=i2atwhichpointv.,1stabilizesandisgivenas{8.2-24}UndertheconditionsofEq.(8.2-24),thevalueofVsG6:J}=.(8.4-19)11(8.4-20)VssVss(a)(b)Figure8.4-JZ(a)ComparatorofFig.8.4-llwhereVinisverynegativeandincreasingtowardV.;:,,.(b)Comparatoroffig.8.4-11where""'isverypositiveanddecrcusinstowardVn,,..8.4ImprovingthePerformanceofOpen-LoopComparators473KnowingthecurrentsinbothMlandM2,itiseasytocalculatetheirrespectivevasvoltages.SincethegateofMlisatground,thedifferenceintheirgate-sourcevoltageswillyieldthepositivetrippointasgivenbelow:VaSJ2"=(;:Vasz2"=(;:)112+Vr1(8.4-21))'12+Vn(8.4-22)(8.4-23)Oncethethresholdisreached.thecomparatorchangesstatesothatthemajorityofthetailcurrentnowftowslhroughM2andM4.Asaresult,M7isalsoturnedon,thusturningoffM3,M6,andMt.Asinthepreviouscase,astheinputdecreasesthecircuitreachesapointatwhichthecurrentinMJincreasesuntilitequalsthecurrentinM7.TheinputvoltageatthispointisthenegativetrippointViRP·TheequivalentcircuitinthisstateisshowninFig.8.4-12(b).Tocalculatethetrippoint,thefollowingequationsapply:(W/L),.;,=(W/L)4(8.4-24)'4i,(8.4-25);,=i'l+it(8.4-26)ilr::Therefore,.14is.=1+[(W/Lhi(WIL).J='2it=is-i2(8.4-27)(8.4-28}UsingEqs.(8.4-21)and(8.4-22)tocalculateVaS>thetrippointis(8.4-29)Theseequationsdonottakeintoaccounttheeffectofchannellengthmodulation.Theiruseisillustratedinthefollowingexample.CALCULATIONOFTRIPVOLTAGESFORACOMPARATORWITHHYSTERESISConsiderthecircuitshowninFig.8.4-11.UsingthetransistordeviceparametersgiveninTable3.1-2calculatethepositiveandnegativelhresholdpointsifthedevicelengthsareallI,...mandthewidthsaregivenas:w.=W:t=WI()=w••=10!LIDandw3=w4=2,...m.474COMPARATORSThegateofMlistiedtogroundandtheinputisappliedtothegateofM2.Thecurrenti5=20J,LA.Simulatetheresultsusingsimulation.Tocalculatethepositivenippoint,assumethattheinputpositive.,(W/L)6.16.ha.~beennegativeandisheading.=(WILh13=(5/1)(13)i5•1120,u.A13=1+[(WIL)J(WILh]==1+5=3·33p.Ai2=i5-v0s1=i1(2i.)=20-3.33=16.67p.A112{i;2i:z)Van=({3+Vn1122=(2.(5)3.33)1101!2(2.1667)+Vn=(S)~O+0.71!2+0.7=1=0.946-0.810=ViRP~VaS2-Vest=0.81V0.946V0.136VDetenniningthenegativetrippoint,similaranalysisyields;4=3.33p.Ait=16.67f.l.AVGS2=0.81VVast=0.946VVftu.aVGSl-VcsJ=0.81-0.946=-0.136VPSPICEsimulationresultsofthiscircuitareshowninFig.8.4-13.2.62.4-Figure8.4-13SimulationofthecomparatorofExample8.4-2......2.22.....21.8(volts)1.61.41.2'1'---.I-O.S-0.4-0.3-0.2-0.10.00.10.20.30.4'1"111(volta)o.s8.5Discrete-TimeComparatorsFigure8.4-14Completecomparatoroutputstage.475withinternalhysteresisincludinganThedifferentia]stagedescribedthusfarisgenerallynotusefula1oneandthusrequiresanoutputstagetoachievereasonablevoltageswingsandoutputresistance.Thereareanumberofwaystoimplementanoutputstageforthistypeofinputstage.OneoftheseisgiveninFig.8.4-14.Differential-to-single-endedconversionisaccomplishedattheoutputandthusprovidesaClassABtypeofdrivingcapability.~.5DISCRHE-TIMECOMPRRRTORSInmanyapplicationsthecomparatoronlyfunctionsoveraportionofatimeperiod.Suchcircuitsaredrivenbyaclockandwillhaveaportionoftimeorphasewhenthecomparatorisfunctioningasacomparatorandaphasewhenthecomparatorisnotbeingused.Inthiscircumstance,otherformsofcomparatorscanbeusedthatareefficientandhaveasma11propagationdelaytime.Wewillexaminetwosuchcomparatorsinthissection.Theyaretheswitchedcapacitorcomparatorandtheregenerativecomparator.SwitchedCapacitorComparatorsTheswitchedcapacitorcomparatorusesacombinationofswitchedcapacitorsandopen-loopcomparators.Theadvantagesofthes.witcbedcapacitorcomparatorarethatdifferentialsigna1scanbecomparedusingsingle-endedcircuitsandtheswitchedcapacitorcomparatornaturallylendsitselftoautozeroingthedeoffsetvoltageoftheopen-loopcomparator.Figure8.5-1(a)showsatypicalswitchedcapacitorcomparator.Thevoltagesappliedtothecircuitarenormallysampledandheldsothatcapitalvariablesareused.Whenthe1/11switchesareclosedinFig.8.5~1,thecapacitorCautozerostheoffsetvoltageofthecomparator,Vo.~·ThecapacitorCPrepresentstheparasiticcapacitancefromtheinputofthecomparatortoground.Werecallthatthecomparatormustbestableinunity-gain476COMPARATORSFigure8.5-1(a)Aswitchedcapacitorcomparator.(b)Equivalentcircuitof{a)whenthe4J2switchesareclo:;ed.operationforthiscircuittowork.properly.ItcanbeshownthatthevoltageacrossC1andCPattheendofther/11phaseperiodis(8.5-1)and(8.5-2)NowwhenthetP2switchisclosed,theequivalentcircuitatthebeginningofthef/11phaseperiodisshowninFig.8.5-l(b}.Inthiscircuit,thevoltageacrosseachcapacitorhasbeenremovedandrepresentedbyastepvoltagesource.Thisallowsustousetheprincipleofsuperpositiontoeasilysolvefortheoutputasillustratedbelow.V2CVcw(lil2)=-A[--C+CVt-Vos)CVosCp]+-C+Cp+AVosc,C+Cpc)](8.5-3)=-A[cv1-v.>_c_-v~(-c-+__P_+Alf.s=-A(Vl-v.>_c_C+~C+~C+~C+~IfCpissmallerthanC.thenEq.(8.5-3)canbesimplifedtoVoutCC'/);0""'A(Vl-V2)(8.5-4)Therefore,thedifferencebetweenthevoltagesV1andV2isamplifiedbythegainoftbecomparator.Thegainofthecomparatorusedfortheswitchedcapacitorcomparatormustbelargeenoughtosatisfytheresolutionrequirements.Inmanycac;es,theresolutionislarge(i.e.,100mV),sothataverysimplesingle-stageamplifierissufficientforthecomparator.Thespeedofthecomparatordependsonhowlongittakesthecircuittosettletoitssteadystageaftertheswitcheshavebeenclosedforagivenperiod.Duringthe=VoH-VoLe''"~"LAv1VaH-Voc.(8.5-16)ItisimportanttoremembertbataV1willalwaysbelessthanVoH-VovThepropagationtimedelayofalatchcanbefoundbysettingEq.(8.5-16)equalto0.5.Theresultis(8.5-17)480COMPARATORSFigun8.S·SNormalizedtime-domainresponseofalatch.SinceAV1isalwayslessthan0.52200,andoutputvoltageswingwithin1.5Vofeitherrail.UseTables3.1-2and3.3-1.UseI11-mchannellengthsforalltransistors.1.3-1.AssumethatthedecurrentinM5ofFig.8.3-1is10011-A.IfWr/4=5(WJ~)andWw'L.-.!='S(W3/L3).whatisthepropagariondelayrjmeofthiscomparatorifCL=10pFandVnp=1~-Vss=2V?a.:nIfthefolded-cascodeopampofExample6.5-3isusedasacomparator,findthedominantpoleifCl5pF.If!beinputstepisI0mV.detenninewhethertheresponseislinearorslewingandfindthepropagationdelaytime.=4898.4-1.IfthecomparatorusedinFig.8.4-1hasadominantpoleatHl"radlsandagainofIfrl,howlongdoesitlakeC..utocharge1099%ofitstina/value,V051Whatisthefinalvaluethatthecapacitor,CA2>willchargetoifleftintheconfigurationofFig.8.4-l(b)foralongtime?8.4-2.UsethecircuirofFig.8.4-9anddesignahystef1:Sischaracteristicthatha.~Vii,.=0VandVfRP=IVifVnH=2VandVoL=OV.LetR1=lOOkll.8.4-3.RepeatProblem8.4-2forFig.8.4-10.8.4-4.AssumethatalltransistorsinFig.8.4-11areope·ratinginthesaturationmode.Whatisthegainofthepositive-feedbackloop,M6-M7.usingtheWILvaluesandcurrentsofEll&mple8.4-2?8.4-S.RepeatExample8.4-1todesigno.sv.8.4-6.RepeatExample8.4-2ifi5usingasimulator.v:lll'=-ViR,.==5011-A.Confirm490COMPARATORS8.5-1.Listtheadvantagesanddisadvantagesoftheswill::hedcapacitorcomparatorofFig.S.5-loveranopen-loopcomparatorhavingthesamegainandliequeocyresponse.S.S-2.IfthecurrentandWILvaluesofthetwolatchesinFig.8.5-3arcidentical,whichlatchwillbefaster?Why?8.6-2.Whatisthegainand-3dBbandwidth{inHz)ofFig.PS.6-2ifCL=1pf?Ignorereverse-billlivoltageeffectsonthepnjunctionsandassumethebulk-sourceandbulk~nareasarcgivenbyWX51J.ID.8.5-J.RepeatExample8.5·1ifa1V0111=(O.S)(VoH-VuJ.8.5-4.RepeatExample8.5-lifthedelatchcurrenti&50fJ.A.8.5-5.Redeveloptheexpressionfora1V..,1.1V,forthecirl:uitofFig.PS.5-5,where6v0111=v42-v.1and~v{=v;1-va.·'FigurePU-28.6-J.RepeatProblem8.6-2forFig.P8.6-3.TheWILratiosforMIandM2an:10fJ.mllfJ.IIIandforlheremainingPMOStransistorstheWILratiosareall2fJ.IIIIJf.LIII.FigurePS.S-58.5-6.ComparelhedynamiclatchofFtg.8.5-SwiththeNMOSandPMOSlatchesofFig.8.5-3.Whatarctheadvantagesanddisadvantagesofthetwolatches?8.5-7.Uselheworst-casevaluesofthetransistorparametersinThble3.1-2andcalculatetheworstcasevoltageoffsetfortheNMOSlatchofFig.8.5-J(a).8.6-1.Assumeanopamphasalow-frequencygainof1000VNandadominantpoleat-104rad/s.Compan:the-3dBbandwidthsoftheconfigura.lionsinPig.P8.6-Jusingthisopamp.FigureP8.6-3R(a)R(b)FigureP8.6-15RReferences8.6-4.Assumethatacomparatorconsistsofanamplifiercascadedwithalatch.AssumetheamplifierhasagainofSVNanda-3dBbandwidthofIITL.>whereTListhelatchtimeconstant.Findthenormalizedpropagationdelaytimefortheoverallconfigurationiftheappliedinputvoltageis0.05(V08-VoL)andthevoltageappliedtothelatchis(a)4V1=0.05(V08-V0.J,(b)4V1=0.1(Von-Vo.J,(c)4V,=O.lS(VoH-Vo.J.and(d)4V,=0.2CVoH-Vot.).Fromyourresults,whatvalueof.1V1wouldglveminimumpropagationdelaytime?8.C..5.Asswnethatacomparatorconsistsoftwoidenticalamplifierscascadedwithalatch.AssumetheamplifierhasthecharacteristicsgiveninProblem8.6-4.Whatwouldhethenormaliz.edpropagationdelaytimeiftheappliedinputvoltageis0.05(V08-V0andthevoltageappliedtothelatchis4V1=O.l(VoH-V0,J1v,......1/in=O.OS~H-VoL):~I=0r4918.6-6.RepealProblem8.6-5iftherearethreeidenticalamplifierscascadedwithalatch.WhatwouldbethenormalizedpropagationdelaytimeiftheappliedinputvoltageisO.OS(V08-V01)andthevoltageappliedtothelatchis.1V1=0.2(VoN-Vov?8.6-7.AcompararorconsistsofanamplifiercascadedwithalatchasshowninFigureP8.6-7.TheamplifierhasvoltagegainofI0VNandf-3dB=I00MHzandthelatchha.qatimeconstantof10ns.ThemaximumandminimumvoltageswingsoftheamplifierandlatchareY08andV01••Whenshouldthelatchbeenabledaftertheapplicationofastepinputtotheamplifierof0.05(VoH-VoL)togetminimumoverallpropagationtimedelay?Whatisthevalueoftheminimumpropagationtimedelay?ItmayUSefultorecallthatthepropagatingtimedelayofthelatchisgivenaslp='TLln(VoH-Vot)2v11wherev11isthelatchinput(aV,ofthetext).M·-----·~-----------11Amplifier""Latch:A,(O)=lOVIV~-(-::-lolTL=IOns•-JdB:o:IOOMHzVi•~---~-----~~~~~~---:VoutIII-----·•lefftgweP8.6-7''.IEFERENCES1.D.J.Allslot,"APrecisionVariable-supplyCMOSCompararor,"IEEEJ.Solid-StateCircuits,Vol.SC-17,No.6,pp.1080-1087,Dec.1982.2.M.Bazcs."'fwoNovelFullComplementarySelf-BiasedCMOSDifferentialAmplifiers,"IEEEJ.Solid-StateCircuit.t,Vol.26,No.2,pp.165-168.Feb.1991.3.J.MillmanandC.C.Halkias,lntegratedElectronics:At~a/ogandDigitalCircuitsandSystems.NewYork:McGrawHill,1972.4.A.S.SedraandK.C.Smith,MicroelectronicCircuit.•,4thed.NewYork:OxfordUniversityPress.1998.S.J.MillmanandH.Taub,Pulse.Digital,andSwitchingWa~·eforms.NewYork:McGraw-Hill,1965.6.T.B.ChoandP.R.Gray,"AlOb,20Msamples/s,35mWPipelineAIDConverter,"IEEEJ.Solid-StareCircuits,Vol.30,No.3,pp.166-172,Mar.1995.1.A.L.CohanandP.E.Allen."A1.5V.ImWAudio4'!,Modulatorwith98dBDynamicRange,"Proc.Int.Solid·SwreCircuitCmif..pp.50-51.Feb.1999.8.K.Klltani,T.Shibata,andT.Ohmi,"CMOSCharge-TransferPreamplifierforOffset-FluctuationCancellationinLow-PowerAIDConverters,"IEEEJ.Solid-StateCircuits,Vol.33,No.5,pp.762-769,May1998.9.J.Doemberg,P.R.Gray,andD.A.Hodges,"A!O-bit5-Msample/sCMOSTwo-StepFlasbADC,wIEEEJ.Solid-StareCircuits,Vol.24,No.2,pp.241-249,Apr.1989.10.R.J.Baker,H.W.Li.andD.E.Boyce,CMOSCircuitDesign,l.ayoat,andSimulation.Piscataway,NJ:IEEEPress,1998,Chap,26.Chapter9SwitchedCapacitorCircunsUntiltheearly1970s,analogsignal-processingcircuitsusedcontinuoustimecircuitsconsistingofresistors,capacitors,andopamps.Unfonunately,theabsolutetolerancesofresistorsandcapacitorsavailableinstandardCMOStechnologiesarenotgoodenoughtoper·formmostanalogsignal-processingfunctions.Intheearly1970s,analogsampled-datatechniqueswereusedtoreplacetheresistor,resultingincircuitsconsistingofonlyMOSFETswitches,capacitors,andopamps[1,2].Thesecircuitsarecalledswitchedcapacitorcircuitsandhavebecomeapopularmethodofimplementinganalogsignal-processingcircuitsinstandardCMOStechnologies.Oneoftheimponantrea.2clockhalf-periodsorphases.Therefore,werewriteEq.(9.1-1)asi1(average)=fITi1(t)dr=~Ji1(t)dt+(7'120ITi1(l)dr)(9.L-13)7'120UsingtheresultofEq,(9.1-4)wecanexpresstheaveragevalueof/1asrm.(}_1r1average--Tfdq(t)+IJdq1()t-_qt(T/2)1-T0qt(O)+qr(T)-T7'12qt(T/2)T(9.1-14)Therefore,/1(average)canbewrittenintennsofC1,C2,Vet,andve2as,C2[vo(TI2)-vo(O)]r1(average)=T+Ct[Vct(T)-Vet(T/2))T(9.1-15)Atr=0,T/2,andT,thecapacitorsinthecircuithavethevoltagethatwaslastacrossthembeforeS1andS2opened.Thus,thesequenceofswitchesinFig.9.1-3(b)causeve2(0)=V.,vdT/2)V~ovc.(T/2)0,andVct(T)=V1-V::~;.ApplyingtheseresultstoEq.(9.1-15)gives=='EquatingEqs.(9.1-11)and(9.1-16)givesthedesiredrelationship.whichisR=TCt+Ca(9.1-11)9.1SwitchedCapacitorCircuitsr.497DESIGNOFASERIES-PARALLELSWITCHEDCAPACITORRESISTOREMULATIONIfC1=C2=C.findthevalueofCthatwiUemu1atea1Mfiresistoriftheclockfrequencyis250kHz.SolutionTheperiodoftheclockwavefonnis4fLS.UsingEq.(9.1-17)wefindthatCisgivenasT2C=-=RT4Xl0-6=4pFlOr.Therefore,C1=C2=C""2pF.Thble9.1-1summarizestheequivalentresistanceofeacbofthefourswitchedcapacitorresistoremulationcircuitsthatwehaveconsidered.Itissignificanttonotethat,ineachcase,theemulatedresistanceisproportionaltothereciprocalofthecapacitance.ThisisthecharacteristicofswitchedcapacitorcircuitsimplementedinCMOStechnologythatyieldsmuchmoreaccuratetimeconstantsthancontinuoustimecircuits.AccuracyofSwitchedCapacitorCircuitsThefrequencyortimeprecisionofananalogsignal-processingcircuitisdetenninedbytheaccuracyofthecircuittimeconstants.Tomustratethis,considerthesimplefirst-order,low-passTABLE9.1-1SummaryoftheEmulatedResistanceofFourSwitchedCapacitorResistorCircuitsSwitchedCapacitorResistorEmulationCln::ultEquivalentReslstan~~Parallel,SeriesSeri~elBili!lelll'Tctc.+c,.T4C498SWITCHEDCAPACITORCIRCUITSfiltershowninFig.9.1-4.Thevoltage-transferfunctionofthiscircuitinthefrequencydomainis(9.1-18)where(9.1-19)iscalledthetimeconstantofthecircuit.Inordertocomparetheaccuracyofacontinuoustimecircuitwithadiscretetime,orswitchedcapacitor,circuit,letusdesignateT1as1c;.Theaccuracyof1ccanbeexpressedas1'1(9.1-20)Weseethattheaccuracyisequaltothesumoftheaccuracyoftheresistor.R1,andtheaccuracyofthecapaci1or,C2•InstandardCMOSlechnology,theaccuracyof1ccanvarybetweeo5%and20%dependingonthetypeofcomponentsandtheirphysicalsizes.Thisaccuracyisinsufficientformostsignal-processingapplications.Nowletusconsiderthecasewheretheresistor,R1,ofFig.9.1-4isreplacedbyoneoftheswitchedcapacitorcircuitsofTable9.1-1.Forexample,letusselecttheparallelswitchedcapacitoremulationofR1•Ifwedesignatethetimeconstantforthiscaseas7'"'thentheequivalenttimeconstantcanbewrittenas(9.1-21)where.f..isthefrequencyoftheclock.TheaccuracyofTocanbeexpressedas(9.1-22)Thisisanextremelysignificantresult.Itstatesthattheaccuracyofthediscretelimeconstant,To,isequaltotherelativeaccuracyofC1andC2andtheaccuracyoftheclockfrequency.Assumingthattheclockfrequencyisperfectlyaccurate.thentheaccuracyofT9canbeassmallas0.1%instandardCMOStechnology.Thisaccuracyismorethansufficientformostsignal-processingapplicationsandistheprimaryreasonforthewidespreaduseofswitchedcapacitorcircuitsinCMOStechnology.R1~T-0-0Figure9.1-4Continuoustime.,fu:st-Qrder,low-passcircuit9.'SwitchedCapacitorCircuits499AnalysisMethodsforSwitchedCapacitorCircuitsUsingTwo-Phase,NonoverlappingClocksSwitchedcapacitorcircuitsareoftencalledanalogsampled-datacircuitsbecausethesignalsarecontinuousinamplitudeanddiscreteintime.Anarbitrarycontinuoustimevoltagewaveform,v(t),isshownonFig.9.1-5bythedashedline.Atthetimest=0,T/2,T.3TI2,..•tbisvoltagehasbeensampledandheldforahalf~period(T/2).Thesampled-datawaveform,v*(t),ofFig.9.1-S(a)istypicalofaswitchedcapacitorwaveformassumingthattheinputsignaltotheswitchedcapacitorcircuithasbeensampledandheld.Thedarkershadedandlighteronshadedrectanglescorrespondtotheq,1phaseandthe~phase,respectively,ofrhetwo-phasenonoverlappingclockofFig.9.1-2.ItisclearfromFig.9.1-5thatthewaveforminFig.9.1-5(a)is~ualtotheswnofthewaveformsinFigs.9.1-5{b)and9.l-5(c).Thisrelationshipcanbeexpressedas(9.1-23)wherethesuperscriptodenotestheoddphase(q,1)andthesuperscriptedenotestheevenphase(t/1~.Foranygivensamplepoint,t=nT/2,Eq.(9.1-23)maybeexpressedasv*(nT/2)I=V'((nn=l.2.3.4....-1)~)I=~((n-no=l,l....t>i)\(9.1-24)~t=2,4....Figure9.1·5(a)A!IIIIIlpled-d.atavoltagewavefonnforatwo-phaseclock.(b)Wavefonnfortheoddclock(11>1).(c)Waveformfortheevenclock(r/12.).v"(l}~(I)--i:!;::Ji...:;;;:·'h-~;.;i.n'+-_.~~~~--~_.--~~~~~--.,0TIZT3TI22TST/23T7TI24T9TI2ST(c)500SWITCHEDCAPACITORCIRCUITSToexamineswitchedcapacitorcircuitsinthefrequencydomain,itisnecessarytotrans·formthesequenceinthetimedomaintoa;:-domainequivalentexpression.Toillustrate,considertheone-sided;:-transformofasequence,v(nT),definedas[5)V(z)""=:Lv(nT)z-"=v(O)+v(T)t-1+v(2T)z-2+...(9.1-2S)n=OforallzforwhichtheseriesV(z)converges.Now,Eq.(9.1-23)canbeexpre.o;sedin!be;:-domainasV"'(z)=V"(z)+~(z)(9.1·26)Thet-domainformatforswitchedcapacitorcircuitsallowsonetoanalyzetransferfunctions.Aswitchedcapacitorcircuitviewedfroma;:-domainviewpointisshowninFig.9.1-6.Boththeinputvoltage,V,(z),andoutputvollage,V0(Z),canbedecomposedintoitsoddandevencomponentvoltages.Dependingonwhethertheoddorevenvoltagesareselected,therearefourpos~ibletransferfunctions.Ingeneral,theyareexpressedas(9.1·27)whereiandjcanbeeithereoro.Forexample,H""(z)representsv:(z)IVr(z).Also,atransferfunction.H('l.),canbedefinedasH(z)=V~(z)=v:(z)+v:(z)V,{z)V{(z)+V/'(z)(9.1-28)Theanalysisapproachforswitchedcapacitorcircuitsusingatwo-phase,nonoverlappinsclockconsistsofanalyzingthecircuitinthetimedomainduringaselectedphaseperiod.Becausethecircuitconsistsofonlycapacitors(chargedanduncharged}andvoltagesources,theequationsareeasytoderiveusingsimplealgebraicmethods.Oncetheselectedpbaseperiodbasbeenanalyzed.thenthefollowingphaseperiodisanalyzedcarryingovertheinilialconditionsfromthepreviousanalysis.Atthispoint,atime-domainequationcanbefoundthatrelatestheoutputvoltageduringthesecondperiodtotheinputsduringeitherofthephaseperiods.Next,thetime-domainequationisconvertedtothez-domainusingEq.(9.1-25).Thedesired~-domaintransferfunctioncanbefoundfromthisexpression.Thefollowingexamplewillillustratethisapproach.SwitchedCapacitorCircuit"'•r/>2Figure9.1-6Input-outputvoltage5ofageoeralswitchedcapac:imrcircuitinthe~-domain.9.1SwitchedCapacitorCircuits,.501Flgure9.1-7{a)Switchedcapacitor,low-passfilter.(b)Clockphasing.,,~¢2f~'~¢2~oiJJ~oiJ2~•n~}n~Jn-~nn+!n+lf(b)(a)Itisconvenienttoassociateapointintimewitheachclockphase.Theobviouschoicesareatthebeginningoftheclockphaseortheendoftheclockphase.Wewillarbitrarilychooselhebeginningoftheclockphase.However,onecouldequallywellchoosetheendoftheclockphase.Thekeyistobeconsistentthroughoutagivenanalysis.Inlhefollowingexample,thetimepointisselectedasthebeginningofthephaseperiodasindicatedbythesingleparenthesisinFig.9.1-7(b)associatingthebeginningofthephaseperiodwiththatphaseperiod.ANALYSISOFASWITCHEDCAPACITOR,FIRST-ORDER.LOW-PASSFILTERUsetheaboveapproachtofindthet-domaintransferfunctionofthefirst-order,low-passswitchedcapacitorcircuitshowninFig.9.1-7(a).Thiscircuitwasdevelopedbyreplacingtheresistor,R"ofFig.9.1-4withtheparallelswitchedcapacitorresistorcircuitofTable9.1·1.Figure9.1-7(b)givesthetimingoftheclocks.Thistimingisarbitr~andisusedtoassisttheanalysisanddoesnotchangetheresult.SolutionLetusbeginwiththe4>1pha~eduringthetimeintervalfrom(n-I)Tto(n-~)T.Figure9.1-8(a)istheequivalentofFig.9.1-7(a)duringthistimeperiod.Duringthistimeperiod,C1ischargedtovj(n-l)T.However,C2remainsatthevoltageofthepreviousperiod.v~(nFigure9.1-8(b)showsausefulsimplificationtoFig.9.1-8(a)byreplacingC2,whichhasbeenchargedto~(nbyanunchargedcapacitor,C2,inserieswithavoltagesourceofvi(n-~)T.ThisvoltagesourceisastepfunctionthatstartT.v2(n-l)T=vi(n-~)T{9.1-29)v~n-1)1(b)Figure9.1·8(a)EquivalentcircuitofFig.9.1-7(a)duringtheperiodfrom1~(n-l)Tto{b)Simplifiedequivalentof(a).1~(n-i>T.502SWITCHEDCAPACITORCIRCUITSNow.letusconsiderthenextclockperiod,tP2_,duringthetimefromt=(n-!)Ttot=nT.TheequivalentcircuitofFig.9.1-7(a)duringthisperiodisshowninFig.9.1-9.WeseethatC1withitspreviouschargeofvl(n-l)TisconnectedinparallelwithC2,whichhasthevoltagegivenbyEq.(9.1-29).Thus.theoutputofFig.9.1-9canbeexpressedasthesuperpositionoftwovoltagesources,vt(n-l)Tandv;(n-I)Tgivenas(9.1-30)Figure9.1-9EquivalentcircuitofFig.9.1-7(a)duringthetimefromt=(n=!>Tto1=nT.IfweadvanceEq.(9.1-29)byonefullperiod,T,itcanberewrittenasv2(n)T=l1(n-Or(9.1-31)SubstitutingEq.(9.1-30)intoEq.(9.1-31)yieldsthedesiredresultgivenas(9.1-32)Thenextstepistowritethe.:-domainequivalentexpressionforEq.(9.1-32).Ifweexpressthe.:-domainequivalenceasv(nT)++V(,""11-111-0.S11rt+0.511+1n+l.5FigureP9.2-59.2-6.FindH..lzJI=v;t.zJ/VTWloftheswitchedcapacitorcircuitshowninFig.P9.2-6.Replace0).AssumingthatClisuncharged.findanexpressionfortheoutputvoltage,V...,,afterthet/11clockisFigure1'9.2-8602SWITCHEDCAPACITORCIRCUITSapplied.Assumethatri~andfalltimesofthe411dockareslowenoughsothatthechanneloftheNMOSlnlllsistorswitchtracksthegatevoltage.Theonandoffvolragesoff/11areJOVandOV.respectively.EvaluatethedeoffsetattheoutputifthevariousparameterSforthisproblemareVr'""1V,C1,'""Cg;~"'100IF,C1'""5pF,andC~=IpF.9.2-9.AswitchedcapacitoramplifierisshowuinFig.P9.2-9.Whati~themaximumclockfrequencythatwouldpermittheidealoutputvoltagetobetead1edtowithin1%iftheopamphasadeg.ainofI0,000andasingledominantpoleat-100radls'lA55umeidealswitches.ingIP2·Useopamps,capacitors.andswit1orofJzindicatingthephaseduringwhichtheswit/td.,..J·IfC2=C4=10pF0010kHz.findthevalueofclandcltoimplementthefollowingtransferfunction:and/,Vout(s)=-10Yu.f5)(-;-~:.:.._+_1)610+1!1.6-1.CombineFigs.9.6-2(a)and9.6-2(b)tofonnacontinuoustimebiquadcircuit.Replacethenegativeresistorwithaninvertingopampandfindlhes-domainfrequencyrespotJSe.CompareYOIUanswerwithEq.(9.6-1).',6.2.(a)Uselhelow-QswitchedcapacitorbiquadcircuitshowninFig.P9.6-2todesignthecapacitorratiosofalow-pass,second-orderfilterwithapolefrequencyofIkHz,Q=S,andagainatdeof-10iftheclockfre..quencyis100kHz.Whatisthetotalcapaci·9.6-4.F'mdthez-domaintransferfunctionH(z.)V0111{z)IV18(z)inthefonnofFigure1'9.6-2=fortheswitchedcapacitorc:ircuitshoWPinFig.P9.6-4.Evaluatethea1'sandb,'sintennsofthecapacitors.Next,assumethatwT<<1andfmdH{s).Whattypeofsc:cond-ordercircuitisthis?606SWITCHEDCAPACITORCIRCUITSFigure1'9.6-49.6-5.Findthez-domaintransferfunctionH(z)""Y001(follows:Weightingfactoroftheith-bit=~~:,'(~:)=2N-I-tLSBs(10.3-2)and..±O.SLSBt100Accuracyofthe1th-btt=N_1_1LSB=N-I=N-I%222(10.3-3)Forexample,theMSB-bitofFlg.10.3-1(i=0}musthavetheaccuracyof±t12M~K.AttheM-bit,theaccuracymustbe:!:112/f.IftheK-bitsareperfectlyaccurate,thenthescalingfactorof112Mwouldhavetohavetheaccuracyof±112K.However,theseconsiderationsonlyholdforasingle-bit.Ifmultiple-bitsareconsidered,aworst-caseapproachmustbetaken.Letusillustratethisinthefollowingexample.ILLUSTRATIONOFTHEINFLUENCEOFTHESCALINGFACTORAssumethatM=2andK=2inFig:.10.3-1andfmdthetransfercharacteristicofthisDACifthescalingfactorfortheLSBDACis3/8insteadof1/4.AssumethatVREP=IV.Whatisthe±INLand±DNLforthisDAC'?lsthisDACmonotonicornot7IJ!1MU.I,ITheidealDACoutputisgivenasTbeactualDACoutputcanbewrittenasvooT(actual)=b0~2b3b2416+-1+-3b3+-3216b08b6b,.3b=-+-1+-+-332323232TheresultsaretabulatedinTable10.3-1forthisexample.Table10.3-lcontainsalltheinformationweareseeking.AnLSBforthisexampleis1116or2132.Thefourthcolumngivesthe+INLas1.5LSBandthe-JNLasOLSB.Thefifthcolumngivesthe+DNLas0.5LSBandthe-DNLas-1.SLSB.Becausethe-DNLisgreaterthan-ILSB,thisDACisnotmonotonic.10.3ExtendingtheResolutionofParallelDigital-AnalogConverters637TABL£10.3-1IdealandActualAnalogOutputfortheDACinExample10.3-1lnpl.ltDigitalWordChangeinVour(IKlual)-2/'3.2VOUT(actual)-VOUTvOUT(adual)VOUT00000132013200013/3261322/329/3261322/323/32010001018/3211/32!1/321013201321/320110011114/3217/3212/3214/322/323132100010011613219/3216/321813201321132101022132201322132l/32lOll25/323/3211001132-3/32110124/3227/3222/3224/3211103013211113313200100011013211324/32113211321132-3/3211321/321132-3/32113201321/32l/32213226/3228/3230132113211323/32Example10.3-1illustratestheinfluenceofthescalingfactorofFig.10.3-1.ThefollowingexampleshowshowtofmdthetoleranceofthescalingfactorthatwillpreventanerrorfromoccurringintheDACusingthearchitectureofFig.l0.3-1.FINDINGTHETOLERANCEOFTHESCALINGFACTORTOPREVENTCONVERSIONERRORSFmdtheworst-casetoleranceofthescalingfactor(x=112M=114)intheaboveexamplethatwillnotcauseaconversionerrorintheDAC.SolutionBecausethescalingfactoronlyaffectsthelSBDAC,weneedonlyconsiderthetwoLSB-bits.Theworst-caserequirementfortheidealscalingfactorof114isgivenas~-(x±Ax)2+-b3(x±4Ax)xb2s-2xb3+-4±-I32or1:s32638DIGITAL-ANALOGANDANALOG-DIGITALCONVERTERSTheworst-c8llevalueofaxoccurswhenbothb2andb3are1.Therefore,wegetThescalingfactor,x.canbeexpresseda~116Ix+l:J.x=-+-=-+4-2424-24Therefore,thetolerancerequiredforthescalingfactor;cis5/24to7124.Thiscorrespondstoanaccuracyof::!:16.7%.whichislessthanthe::!:25%(::!:100%/2K)becauseoftheinfluenceoftheLSB-bits.ItcanbeshownthatthelNLandDNLwillbeequalto±O.SLSBorJess(seeProbleml0.3-6).AnothermethodofcombiningtwoormoreDACsistoincreasetheresolutionisshowninFig.10.3-2.InsteadofscalingtheoutputofthesubDACs,thereferencevoltagetoeachsu!J.DACisscaled.ThismethodofcombiningsubDACsiscalledsubranging.TheanalogoutputofthesubrangingOACofFig.10.3-2canbeexpressedas(l0.3-4a)or(l0.3-4b)WenotethatEq.(10.3-4b)isidenticaltoEq.{l0.3-1b)forthemethodthatdividestheanalogoutputofthesubOACs.TheaccuracyconsiderationsofthismethodaresimilartothemethoddiscussedaboveforFig.10.3-1.ItcanbeshownthattherequirementforthetoleranceonthescalingfactorforthereferencevoltageisidenticaltothatofthescalingconstantfortheoutputvoltageoftheLSBDAC..Figure10.3-3showsacurrentsc.zdingDACusingthecombinationoftwo,4-bitcurrentscalingsubDACs.ThescalingoftheLSBsubDACisaccomplishedthroughthecurrentFigure10..3-lCombininganM-bitandK-bitsubOACtoformanM+K-bitDACbydividingtheVRilftotheK-LSBDAC(subranging).M-MSBlbitsJK-l.SB}bits10.3ExtendingtheResolutionofParallelDigital-AnalogConverters639I.SBsubDACMSBsubDACFigure10.3-3CombinationofcurrentscalingsubDACsusingacw1ntdivider.dividerconsistingofRandISR.WeseethatthecurrentbeingsunkbytheLSBDAC,i1,is16i2•Thesumofi2plusthecurrentssunkbytheMSBDACflowsthroughthefeedbackresistor,Rp.tocreatetheoutputvoltage,VoUT•Theoutputvoltagecanbeexpressedas(10.3-5)TheindividualDACsorsubDACscanberealizedusinganyofthemethodsconsideredintheprevioussectionforcurrentscalingDACs.ThevoltagescalingDACdoesnotadaptwelltothemethodofincrea.~ingtheresolutionshowninFig.10.3-1orl0.3-2becauseofthehighoutputresistance.ThehighoutputresistancemakesitdifficulttosumtheoutputsofsubDACsthatusevoltagescaling.Tosolvethisproblem,bufferamplifiersshouldbeaddedtoeachsubDACoutput.Figure10.3-4showsachargescalingDACusingthecombinationoftwo,4-bitchargescalingsubDACs.ThescalingoftheLSBsubDACisaccomplishedthroughthecapacitorc..Thevalueofthescalingcapacitor.c..canbefoundasfollows.Theseriescombinationofc.andtheLSBarraymustterminatetheMSBarrayorequalC/8.Therefore,wecanwritec1-=---81I-+-c.2CVReFo----L--~--~~--~-----L--~--~~~Fl~10.3-4Combinationoftwo,4-bitchargescalingsubDAC~toforman8-bitcharge(10.3-6)640DIGITAL-ANALOGANDANAlOG-DIGITALCONVERTERSor18]16115-=---=---=c.C2C2C2C2C(10.3-7)Thus,thevalueofthescalingcapacitor,C.,shouldbe2CI15.LetusnowshowthattheoutputvoltageofFigure10.3-4isequivalenttothatofan8-bitchargescalingDAC.First,wemustfindtheTheveninequivalentvoltageoftheMSBarray,V~oandtheLSBarrayplusthetenninatingcapacitor,V2•Thesevoltagescanbewrittenas(10.3-8)and(10.3-9)UsingthetwoequivalentvoltagesgiveninEqs.(10.3-8)and(10.3-9),Fig.10.3-4canbesimplifiedtothecircuitshowninFig.10.3-5.Fromthisfigure,wecanwritetheoutputvoltageas~_1_+~2C2C_OUT-(1152C+2C+)8lSC+(VI_8_)V:15C211582C+2C+15C_(1s+15•15)+(-15+15•IS+16VI15_(16•15-16·ts+162C2v2-~~lSC/82-=Vt"':""+16(10.3-10})v2)v+(16.1516+16)v-l6v_15+1l6v1t>r_nt,.r'""'C,•201S+1615•IS212Figure10.3-5Simplifiedequivalenlcin:uitofFig.10.34.10.3ExtendingtheResolutionofParallelDigital-AnalogConverters641+1.....LSBArrayMSBArrayFigure10.3-6Combinationoftwo.4-bit,binary-weighted,chatgeamplifiersubDACstoforman8-bit,binary-weighted,chargeamplifierDAC.SubstitutingEqs.(10.3-8)and(10.3-9)intoEq.(10.3-10)givestheanalogoutputvoltageofFig.10.3-4.(10.3-11)InthecaseofFig.10.3-5,theaccuracyofthescalingcapacitor,C:..influencestheMSBarrayaswellastheLSBarraybecauseitformspartoftheterminatingcapacitorfortheMSBarray.Figure10.3-6showsacombinationoftwochargescalingDACsthatusethechargeamplifierapproach.ThisMSBsubDACisnotdependentontheaccuracyofthescalingfactorfortheLSBsubDAC.TheaboveideasarerepresentativeofmethodsusedtoincreasetheresolutionofaDACbycombiningtwoormoresubDACsusingthesametypeofscaling.Manydifferentcombinationsoftheseideaswillbefoundintheliteratureconcerningintegratedcircuitdigitalanalogconverters.CombinationofDifferentlyScaledDACsThesecondapproachtoextendingthere."WlutionofparallelDACsusessubDACshavingdifferentscalingmethods.OneoftheadvantagesofthisapproachisthatthedesignercanchoseascalingmethodthatoptimizestheMSBsandadifferentscalingmethodtooptimizetheLSBs.Themostpopularexampleofthisapproachisthecombinationofthevoltagescalingandchargescalingmethods[6).Figure10.3-7illustratesaDACthatusesvoltagescalingfortheMSBsubDACandchargescalingfortheLSBsubDAC.TheMSBsubDACisM-bitsandtheLSBsubDACisK-bits,givingaDACwiththeresolutionofM+K-bits.TheM-bitvoltagesubDACoffig.10.3-7scalesthereferencevoltagetoVR!lP/2M,whichisusedasthereferencevoltageforthechargescalingsubDAC.ThissubDACconsistsofaresistorstringof2MequalresistorsconnectedbetweenVIIEFandground.TherearetwoM-to-2Mdecodersconnectedtotheresistortapsasshown.642DIGITAL-ANALOGANDANALOG-DIGITALCONVERTERSFigure10.3-7M+K-bitDACusinganM-bitvoltagescalingsubDACfortheMSBsandaK-bitcharge!iCalingsubDACfortheLSBs.ThesedecoderscanbeimplementedbyFig.10.2-7(a)or10.2-8.TheM-bitscauseoutputofdecoderAtobeconnectedtothetop(higherpotential)andtheoutputofdecoderBtothebottom(lowerpotential)ofoneofthe2"'resistors.TheoutputofdecoderAisbusAandtheoutputofdecoder8isbusB.Thevoltagescaling,chargescalingDACofFig.10.3-7worksasfollows.First,thetwoswitchesSFandswitchesS18throughSK,Bareclosed,dischargingallcapacitors,IftheoutputoftheDACisappliedtoacomparato.rthatcanalsofunctionasaunity-gainbuffer.autozeroingcouldbeaccomplishedduringthisstep.Next,theMMSBsareappliedtothevoltagesca1ingsubDACinthemannerdescribedabovetoconnectbusAatthetopoftheproperresistorandbus8tothebottomofthisresistor.Inreality,thedesiredanalogoutputvoltagewillbebetweenthevoltagesatthetopandbottomofthisresistor,whichisdeterminedbytheMMSBs.TheequivalentcircuitoftheDACofFig.I0.3-7atthispointisgivenbyFig.l0.3-8(a).Figure10.3-8(a)EquivalentcircuitofFig.10.3-7forthevoltagescalingsubDAC.(b)EquivalentcircuitoftheentireDACofFig.10.3-7.~!!...........JL--.......1--·-'--L___J{a)..,..+z-.wvREl'+VQIJTVQUT..,..(b)t1..,..10.3ExtendingtheResolutionofParallelDigital-AnalogConverters643ThelowervoltagesourceofFig.10.3-S(a),VkF.n:presentsthevoltagefromgroundtothebottomterminaloftheselectedresistor,wherebusBisconnected.Itcanbewrittenas,~(bobrhu-2bM-1)=VIU!J'-21+-+22.+...+1!f-12M.(10.3-12)wherethebitsb0throughbM-ldeterminethevalueofVREF·TheuppervoltagesourceofFig.I0.3-S(a)representsthevoltagedropacrosstheresistorconnectedbetweenbusesAandB.ItisequaltotheLSBofthevoltagesubDACofFig.10.3-7.ThefinalstepintheconversionistoconnectthecapacitorsinthechargescalingDACtobusAiftheirbitis1andtobusBiftheirbitis0.IfweletthecapacitorsthatareconnectedtobusAbedesignatedasC,q,thenFig.l0.3-8(b)becomesamodeloftheentireDACoperation.TheoutputvoltageofthechargescalingsubDAC,VoUT•canbeexpressedas(10.3-13)AddingVREFofEq.(10.3-12)to'I'oUTofEq.(10.3-13)givestheDACoutputvoltageas(10.3-14)whichisequivalenttoanM+K-bitDAC.TheadvantagesoftheDACofFig.10.3-7isthattheMSBsareguaranteedtobemonotonic.TheaccuracyoftheLSBsshouldbegreaterthantheMSBsbecausetheyaredeterminedbycapacitors.Thecomponentspreadisdeterminedbythebinary-weightedcapacitorsandiszK-I.Unfortunately,whiletheMSBsaremonotonic,theaccuracyoftheresistorsisnotasgoodasthecapacitorsandnonmonotonicityisli.kelytooccurwhentheMSBsandLSBsarecombined.Figure10.3-9showsaDACthatu..~esachargescalingsubDACfortheMSBsandavoltagescalingsubDACfortheLSBs.ThisDACwillhavetheadvantageofbetteraccuracyintheMSBsandmonotonicLSBs.Becausetherequiredtoler14bitsAlgorilhmicMultiple-bitpipelineFoldinga.DdinterpolatingLow~'IOiution>6bits654DIGITAL-ANALOGANDANALOG-DIGITALCONVERTERSStaticCharacterizationofADCsTheinputofanADCisananalogsignal,typicallyananalogvoltage,andtheoutputisadigitalcode.Theanaloginputcanhaveanyvaluebetween0andVRI!Fwhilethedigitalcodeisrestrictedtofixedordiscreteamplitudes.PopulardigitalcodesusedforADCsareshowninTable10.5-2andincludebinary,thermometer,Gray,andtwo'scomplement.Themostwidelyuseddigitalcodeisthebinarycode.Somecodeshaveadvantagesoverothersthatmakethemattractive.Forexample,!:heGrayandthermometercodesonlychange1-bitfromonecodetothenext.ThestaticcharacterizationofADCsisbasedontheinput-outputcharacteristicshowninFig.I0.5-3fora3-bitADC.InthiKparticularcharacteristic,theinputhasbeenshiftedsothattheidealstepchangesoccuratanaloginputvaluesof0.5LSB(2i-1),wherejvariesfrom1toNforanN-bitADC.Beneaththeinput-outputcharacteristicofFig.10.5-3isaplotofthequantizationnoiseasafunctionoftheinput.Thequantizationnoiseisaplotofthedifferencebetweentheinfiniteresolutioncharacteristicandtheideal3-bitcharacteristicasafunctionoftheinputvoltage.TheidealADCcharacteristicwillhaveaquantizationnoisethatliesbetween:t0.5l.SB.Thedefinitionsfordynamicrange,thesignal-to-noiseratio(SNR),andtheeffectivenumberofbits(ENOB)oftheADCare!:hesameasthosegiveninSection10.1fortheDAC.ThesequantitieswerereferencedtotheanalogvariableandinthecaseoftheADCarereferencedtothedigitaloutputword.Theresolutionof!:heADCisthesmallestanalogchangethatcanbedistinguishedbyanADC.ResolutionmaybeexpressedinpercentoffuUscale(FS)butistypicallygiveninthenumberofbits,N,wheretheconverte-rhas2Npossibleoutputstates.Theprimarycharacteristicsthatdefinethestaticperformanceofconvertersareoffseterror,gainerror.integralnonlinearity(/NL),anddifferentialnonlinearity(DNL).ForanADCwithoffset,letusshifttheinfiniteresolutioncharacteristiclinehorizontallyuntilthequantizationnoiseissymmetricalwhenreferencedtothisline(hereweareassumingthatothererrorssuchasgainandnonlinearityarenotdominantorhavebeenremovedfromthecharacteristic).Thehorizontaldifferencebetweenthislineandtheinfiniteresolutioncharacteristicthatpassesthroughtheoriginisoffseterror:OffseterrorisillustratedinFig.10.5-4(a).Gainerrorisadifferencebetweentheactualcharacteristic,andtheinfiniteresolutioncharacteristic,whichisproportionaltothemagnitudeoftheinputvoltage.ThegainerrorcanbethoughtofasachangeintheslopeoftheinfiniteresolutionlineaboveorbelowavalueofI.TABLE10.5-2DigitalOutputCodesUsedforADCsDecimalBinaryThermometerOlO0i5001001z.8(a)Vin.li881lYR.EF88(b)Figure10.54(a)ElUilllpleofoffsetenorfera3-bitAOC.(b)Exampleofgaillerrorfora3-bitADC.656DIGITAL-ANALOGANDANALOG-DIGITALCONVERTERSFigure10.5-5ExampleofJNLandDNT..fora3-bitADC.Differentialnonlinearity(DNL)oftheADCisdefinedasameasureoftheseparationbetweenadjacentcodesmeasuredateachverticalstepinpercentorLSBs.ThedifferentialnonlinearityofanADCcanbewrittenasDNL=(Dcr-1)LSBs(10.5-l)whereDaisthesizeoftheactualverticalstepinLSBs.Figure10.5-5showstheintegralanddifferentialnonlinearityfora3-bitADCreferencedtothedigi!aloutputcode.WeseethatthelargestandsmallestvaluesofINLare+1LSBand-lLSB,respectively.ThelargestandsmallestvaluesofDNLare+llSBandOLSB,respectively.AscomparedtotheDAC,aDNLof-LSB,whichresultedfromthecasewhereastepshouldhaveoccurred,doesnothappenfortheADC.Forexample,ataninputvoltageoffi;onFig.10.5-5,thefactthataverticaljumpdoesnotoccurcannotbeconsideredasaDNLof-ILSB.NonmonotonicityinanADCoccurswhenaverticaljumpisnegative.NonmonotonicitycanonlybedetectedbyDNLBecauseoutputislimitedtodigitalcodes,alljumpsareintegers.Nonnally,theverticaljumpisILSB.Ifthejumpis2LSBsorgreater,missingoutpUtcodesmayoccur.IftheverticaljumpislessthanOLSB,thentheADCisnotmonotonic.Figure10.5-6showsa3-bitADCcharacteristicthatisnotmonotonic.NonmonotonicitygenerallyoccurswhentheMSBdoesnothavesufficientaccuracy.1bechangefrom0Jill....to10000....isthemostdifficultbecausetheMSBmusthavetheaccuracyof±0.5LSBorexcessiveDNLwilloccur.JIJ'§t.;lJO101s100gOils~010001Figure10.5-6Exampleofnonmonotonie3-bitADC.10.5IntroductionandCharacterizationofAnalog-DigitalConverters657INLANDDNLOFA3-BITADCFmdtheTNLandDNLforthe3-bitADCinFig.10.5-6.SolutionThelargestvalueofINLforthis3-bitADCoccursbetweenfi;andft;orf;;and..\:andislLSB.ThesmallestvalueofINLoccursbetween{iand~andis-2/.SBs.ThelargestvalueofDNLforthisexampleoccursat-hor~andis+lLSB.Thesmalle.~tvalueofDNl..occursat/6andis-2LSBs,whichiswheretheconverterbecomesnonmonotonic.DynamicCharacteristicsofADCsThedynamiccharacteristicsofADCshavethesamedependenceasfoundinDACs,namely,parasiticcapacitancesandtheopamps.Inaddition,inallADCsatleastonecomparatorisused.Thecomparatorisusedtodeterminewhethertheanaloginputisaboveorbelowaparticularvoltage.ThestaticanddynamicperformancesofthecomparatorwasstudiedindetailinSection8.1.ThisinformationwillbeusedtodeterminethedynamicbehaviorofADCsandshouldbereviewedattheappropriatepointinthischapter.Insomecases,theADCmayuseanopampthatwillinfluenceboththestaticanddynamicperformances.ThematerialnecessarytounderstandtheinfluenceoftheopamphasbeenpresentedinSection10.1inregardtothestaticanddynamicperformancesofDACs.Sample-and-HoldCircuitsBecausethesample-and-hold(S/H}circuitisakeyaspectoftheADC.i.tisworthwhiletodetermineit.~influenceontheADC.Figure10.5-7showsthewaveformsofapracticalsampleand-holdcircuit.Theacquisitiontime,indicatedbytl1'isthetimeduringwhichthesampleand-holdcircuitmustremaininthesamplemodetoensurethatthesubsequenthold-modeoutputwillbewithinaspecifiederrorbandoftheinputlevelthatexistedattheinstantofthesample-and-holdconversion.Theacquisitiontimeassumesthatthegainandoffseteffectshavebeenremoved.Thesettlingtime,indicatedby~..isthetimeintervalbetweenthesample-andholdtransitioncommandandthetimewhentheoutputtransientandsubsequentringinghavesettledtowithinaspecifiederrorband.Thus,theminimumsample-and-holdtimewouldbe(10.5-2)HoldSampleSIHCommandHold/:OutputofSJHvalidforADC-+_,s_..:::IIIIIIconversionllii\'(t)II___,..••IVin(l)I',.....,...,.___._Figure10.5-7Wavefonn.sforasample-and·holdcircuit.658DIGITAL-ANALOGANDANALOG-DIGITALCONVERTERSTheminimumconversiontimeforanADCwouldbeequaltoT....,p~.andthemaximumsamplerateis(10.5-3)InadditiontotheabovecharacteristicsofaS/Hcircuit,thereisanaperturetime,whichisthetimerequiredforthesamplingswitchtoopenaftertheS/Hcommandhasswitchedfromsampletohold.Anotherconsiderationoftheaperturetimeisapenurejitter.whichisavariationintheaperturetimeduetoclockvariationsandnoise.DuringtheholdperiodoftheS/HakTICnoiseexisL~becauseoftheswitchandholdcapacitor.Sample-and-holdcircuitscanbedividedintotwocategories.ThesecategoriesareS/HcircuitswithnofeedbackandS/Hwilhfeedback.Ingeneral,theuseoffeedbackenhancestheaccuracyoftheS/Hatthesacrificeofspeed.TheminimumrequirementforaS/Hcircuitisaswitchandastorageelement.Typically,thecapacitorisuseda.~thestorageelement.Asimpleopen-loopb11fferedS/HcircuitisshowninFig.10.5-B(a).Theunity-gainopampisusedtobufferthevoltageacrosstheholdcapacitor.TheidealperfonnanceofthisS/HcircuitisshowninFig.10.5-B(b).ThesamplemodeoccurswhentheswitchisclosedandtheanalogsignalissampledonacapacitorCH.Duringtheswitch-opencycleortheholdmode,thevoltageisavailableattheoutput.TheS/HcircuitofFig.10.5-B(a)issimpleandfast.Thecapacitor,CH,ischargedwiththeRCtimeconstantoftheswitchonresistanceplusthesourceresistanceofvin(t).Onedisadvantageisthatthesource,v;.(t).mustsupplythecurrentnecessarytochargeCH.Theunitygainopamppreventsthevoltagefromleakingoffthecapacitorandprovidesalow-resistancereplicaoftheheldvoltage.Thedeoffsetoftheopampandchargefeedthroughoftheswitchwillcausethisreplicatobeslightlydifferent.AnimportantdynamiclimitationoftheS/HcircuitisthesettlingtimeoftheopampsuchastheoneusedinFig.10.5-B(a).Whenanopampwithadominantpoleat·~rn>v;...vlh0gratesnegativelywithaconstantslope,becauseYaEFisconstant.WhenV;01(/)becomeslessthanthevalueofV111,thecounterisstoppedandthebinarycountcanbeconvertedintothedigitalword.Thisisdemonstratedbyconsideringthetimeatwhichv;...(t)equalsVlh.Theintegratorvoltageatt1+t2isgivenas(10.6-2)SubstitutingEq.(10.6-1)intoEq.(10.6-2)gives[KNaEFTvf-0,thanVRI>v~.ThegoaloftheM-MSB-bitl'istofindthelargestvoltageoftheladderOncethisisfound,thenbusBisconnectedtothispointandbusAisstringwhereVR1aspossible.Thesequenceofcomparatoroutputsisadigitalcodecorrespondingtotheunknownanaloginputsignal.BydrivingthecapacitorarraydirectlythroughtheMOSswitches,therearenooffseterrorsifenoughtimeisaJlowedfortheswitchingtransientstosettle.Also,theparasiticcapacitorsofallswitchesexceptSFdonotcauseerrorsbecauseeverynodeisdriventoafinalvoltagethatisindependentofthecapacitorparasiticsaftertheswitchtransientshavesettled.TheADCinFig.10.7-4iscapableof12-bitmonotonicconversionwithadifferentialnonlinearityoflessthan±0.5LSBandaconversiontimeof50p.s[6).672DIGITAl-ANALOGANDANALOG-DIGITALCONVERTERSFigure10.7·7Asuccessive-approximationADCusingtheserialDACofFig.10.4-1.VR£FSequenceand~onlrollogicAsuccessive-approximationADCusingtheDACofFig.10.4-1isshowninFig.10.7-7.ThisconverterworksbyconvertingtheMSBaN-Ifirst.(Theith-bitisdenotedasd1fordigitalanalogconversionanda1foranalog-digitalconversion.)Thecontrollogictakesaverysimplefonnbecausethedigital-analoginputstringatanygivenpointintheconversionisjustthepreviouslyencodedwordtakenl..SBfirst.Forexample,considerthepointduringtheanaJog-digilalconversionwherethefirstKMSBshavebeendecided.Todecidethe(K+I)MSB,a(K+1)-bitwordisfannedinthedigital-analogcontrolregisterbyaddingaIastheLSBtotheK-bitwordalreadyencodedinthedatastorageregi.~ter.A(K+l)-bitdigital-analogconversionthenestablishesthevalueofaN-K-tbycomparisonwiththeunknownvoltageV~.Thebitisthenstoredinthedatastorageregisterandthenextserialdigital-analogconversionisinitiated.TheconversionsequenceisshownindetailinTh.bleI0.7-1.Figure10.7-8illustratesa4-bitanalog-digitalconversionforV~=VRI!f'·Altogether,N(N+1)clockcyclesarerequiredforanN-bitADCusingtheconfi~tionofFig.10.7-7.i\PipelineAlgorithmicADCAnalgorithmicADCpatternedafterthealgorithmicDACofSection10.4isshowninFig.10.7-9.ThisN-bitADCconsistsofNstagesandNcomparatorsfordeterminingthesignsofTABLE10.7-1ConversionSequencefortheSerialDACofFig.10.7-7Digital-AnalogConversionNumber23Digitai-AJ1alogInputWorddod,dzd,.zd,._,NumberofChargingStepsliN-~2">~-llir;-tu,....zComparatcwOutputUN-Ia,._,46.'v·2NNTotalnumberofchwgingr.leps=NCN+I)10.7Medium-SpeedAnalog-DigitalConvertersluxlluFigure10.7-8lliustrationoflheoperationofthesuccessive·approxi.mationADCofFig.10.7-7fortheconversionofthesampledanaloginputvoltageofVRE'F.Thedigitalwordoutisbo=1,b1=1.11,.=0,andb3=l.1101lllx673+-+~·-.:13116n-1.O.OOfo__._..__-.____....._____tlf120123401234S6012345678theNoutputs.EachstagetaJcesitsinput.multipliesitby2,andaddsorsubtractsthereferencevoltagedependingonthesignofthepreviousoutput.ThecomparatoroutputsformanN-bitdigitalrepresentationofthebipolaranaloginputtothefirststage.EachofthestagesofthepipelinealgorithmicADCareidentical.Theithstagetakestheoutputofthepreviousstage,V1-handduringthenextclockcycleitcomparesthisvoltagewithgroundandoutputstheith-biLInaddition,thevoltageV1-1ismultipliedby2andthereferencevoltage,VREF,isaddedorsubtracteddependingonwhetherthecomparatoroutputis]oworhigh,respectively.Thisismathematicallydescribedasv,=2V,_1-(10.7-6)bl-1VREFwhereb1-1isgivenasb,_,=MSB+1{-1if¥1-1>0ifYt-11'I'1/J6~6116iS/161--1r7rm!'-';.,...1'..··I...~-sfV":....~i!i:.-·I/.....Jdbl0()000000Ib20000IIII0b300II00II0b4oI0I0I0I0IlIIII000III0II00II010I0DigimllnpurCQ(jcFigurePIO.I-410.1-5.AIVpeak-to-peaksinusoidalsignalisappliedtoanideal10-bitDAC.whichhasaVRill'of5V.FindthemaJt-c!llleDNLinunitsofLSBsandatwhattru.nsitiondoesitoccur?FigurePlO.:Z-610.2-14.Abinary-weightedDACusingachargeamplifierisshowninF'J.g.PI0.2-14.Atthebeginningofthedigital-to-analogconversion,allcapacito111aredischarged.IfabitiRI,lhecapacitoris718DIGrTAL~ANALOGANDANALOG-DIGITALCONVERTERSFigureP10.2·llFigureP10.1·13....,_..,1-~--~--....0'-'0UTcapncitormatchingis0.2%(regardlessofcapac·itorsiu:s)whatisthemaximumvalueofNforidealoperation?10.2-15.ThecircuitsbowninFig.Pl0.2-15isanequiva·lentfortheoperationofaDAC.Theopampdif·ferentialvoltagegain,A,.J,.s).ismodeledasFigureP10.2·14(a)If"'•goestoinfinitysothatA.,(s)=A"lO),whatistheminimumvalueofAw~(OlthatwiUcauseconnectedtoVREPandifthebitis0thecapacitorisconnected10ground.(a)DesignCx10getVorn-bob24b,.._,)=(-+-1+...+--v.l'l'2"(b)Identifylheswitchesbyb1,wherei=0istheMSBandi=N-1istheC.SB.(c)Whatisthemaximumcomponentspreadforlhecapa·cito111?(d)IsthisDACfa~torslow?Why?(e)CanthisDACbenonmonotonic?(f)IftheVi!EFFigurePlO.l-lSProblems719FigurePlO.l-16a:!:0.5l..SBerrorforan8-bitDAC?(b)IfA,.t(O)ismuchlargerthanthevaluefoundin(a).whatistheminimumconversiontimeforan8-bitDACthatdesignatedasx.(b)FindthelargestvalueofxthatcausesaILSBDNL(c)1-"indthesmallestvalueof.tthatcau~esa2LSBDNLgivesa±O.Sl.SBcrrorifGB=IMlli?10.2-16.AchargescalingDACisshowninFig.P10.2-16thatusesaC-2Cladder.Allcapacitorsaredischargedduringthe4>1pha~e.(a)WhatvalueofC1-isrequiredtomakethisDACworkcorrectly'!(b)WriteanexpressionforvoLTduringrPlintermsofthebits,b,.andthereferencevoltage,VIWP(c)DiscussatleasttwoadvantagesmdtwodisadvantagesofthisDACcomparedloothertypesofDACs.10.3-1.TheDACofFig.10.3-1hasM=2andK=2.lfthedivisorha~anincorrectvalueof2,expressthe±INLandthe:!:DNLintermsofLSB.randdeterminewhetherornottheDACismonotonic.Repealifthedivisoris6.10.3-2.RepealProblem10.3-lifthedivisoris3and6.10.3-3.RepealProblem10.3-1ifthedivisoriscorrect(4)andtheVRI!FfortheMSBsubDACis0.75VRHFandtheVIW'furtheLSBsubDACis10.3-6.ShowfortheresultsofExampleI0.3-2thattheresultinglNLan.dDNLwillbeequalto±O.SLSBotJess.10.3-7.A4-bitDACisshowninFig.P10.3-7.WhenabitisI,theswitchpertainingtothatbitisconnected10theopampnegativeinputterminal;otherwise.itisconnectedtogrou.nd.Identifytheswitchesbythenotationb"'b1•b2,orbl•whereb1correspondstothelth-bitandhoistheMSBandb3istheLSB.SolveforthevalueofR,thatwillgiveproperDACperfonnance.10.3-8.AssumeR1=BR.R4=R8=2R.R2-=R6=4R.R3=R7=inFig.Pl0.3-8.(a)FindthevaluesofR9andR1ointermsofRthatgiveanideal8-bitDAC.(b)Findtherangeofvalue5ofR.,intennsofRthatkeeptheINLs±0.5LSB.A:~sume!hatR10hasitsidealvalue.Clearlystaleanyassumptionyoumakeinworkingthispart.(C)FindtherangeofR111intermsofRthatkeepstheconvenermonotonic.AssumethatR9hasitsidealvalue.Clearlystateanyassu.mptionsyoumakeinworkingthispart.l.25VREP•)10.3-4.Findtheworst-casetolerance(±%)ofthedivi~~Dr,x.fortheDACofFig.I0.3-1thatha~M=3andK=3.Assume!hatthesubDACsareidealinallrespectsexceptforthedivisor.10.3-5.TheDACofFig.10.3-2hasM=3andK=3.(a)FindtheidealvalueofthedivisorofVJWllR~=16R,andthattheopampisideal10.3-9.Designa10-bittwo-stagechargescalingDACsimilartoFig.10.3-4usingtwo5-bitsectionswithacapacitiveattenuatorbetweenthestages.FigurePlO.J-7720DIGITAL-ANALOGANDANALOG-DIGITALCONVERTERSFigureP10.3-8GiveallcapacitancesintermsofC,whichisthesmallestcapacitorofthedesign.10.3-10.Atwo-stagechargescalingDACisshowninFig.PIO.J-10.(a)DesignC,intermsofC.theunitcapacitor,toachievea6-bittwo-stagechargescalingDAC.(b)Ifc.isinerrorbyIJ.C.,findanexpressionfor"OUTinlermsofC.,IJ.C.,bl-i•andVREP.(c)Iftheexpressionforllot-Tinpart(b)isgiven115whatistheaccuracyofc.necessaryto11voidanerrorusingworst-l:asecoosideralions.10.3-11.IftheopwnpsinthecircuitofFig.PI0.3-11haveadegainof10"andadominantpoleat100Hz.atwhatclockfrequencywilltheeffectivemnnherofbits(ENOB)equal7-bits,IISSUmingthatthecapacitonandswitchesareideal?UseawotlitcaseapproachtothisproblemandassumethatlimeresponsesoftheLSBandMSBstagesaddtogivetheoverallconversiontime.10.3-12.TheDACinFig.PIO.J-12usestwoidentical2-bitDACstoachievea4-bitDAC.Giveanexpressionfor"OUTa.~afunctionofVRF.Fandthebitsb~~ob1•b2,andb3duringthe1aodtharenonoverlappiogclocks.Thenoise,n.,.ofageneralL-loopllmodulatorisexpressedliSReferences729Y(z)/B«foand.fs"'2/o/t=lo-0.5/s..h=!u+0.5Jsil'fsFigun!P10.9-7tbtJwhereM=f.J(2f8).WhatisthevalueofIsfora14-bitADCusingthismodulatorifthesamplingfrequency,[.,is10MHz?PulseCodeModulation(PCM)¢2Flgun!Pl0.9-8(a)AssumetbatthequantizationlevelforeachquantizerisA=0.5VR1!Fandfindthedynamicrangeindecibelsthatwouldresultiftheclockfrequencyis100MHzandthebandwidthoftheresultingADCis1MHz.(b)WhatwouldthedynamicrangebeindecibelsifthequantizCfllare2-bit?10.9-9.Afirst-order,1-bit,bandpassAImodulatorisshowninFig.P10.9-9.Findthemodulationnoisespectraldensity.N(/),andintegratethesquareofthemagnitudeofN(/)overthebandwidthofinterest(/1tofuandfindanexpressionforthenoisepower,11.(/).inthebandwidthofinterestintermsofAandtheoversamplingfactorM,10.9-10.Afirst-order,1-bit,bandpw;s,AImodulatori&showninFig.PI0.9-10.Findthemodulationnoisespectraldensity,N(f).andintegratethesquareofthemagnitudeofN(/)overthebandwidthofinterest(Jjtoj2)andlindanell:pressionforthenoisepower,II,(/},inthebandwidthofinterestinte!liiSufAIIRdtheovel"!tamplingfactorM,whereM=/)(2/8).Whatisthevalueof/11fora12-bitADClll\ingthismodulatorifthesamplingfrequency./,.isIOOMHz?Assumethatf,=4.f.,andfsFI•X.IlL,andAW.Wehaveassumedthat211/>FIcouldbedeterminedbyiterationifN5u8wereknown.(IfN5u8isnotknown,thenNsusmustbemea.~uredbyothermeans,forexamp1e,bulkresistance.)Also,onemustrememberthatthesemodelparameters,withtheexceptionofX,IlL.andAW.aredependentontemperature.8.21/FNOISEInmanyapplications,goodnoiseperformanceisaveryimportantrequirementforananalogdesign.Consequently,thenoiseperformanceoftransistorsmustbecharacterized.Theequationdefiningthemean-squarenoisecurrentinanMOStransistorgiveninEq.(3.2-12)isrepeatedhere.•2_f.-[SkTg,.(13+'II)+(KF)lu].,,JC""L2UJ1(A)(B.2-l)754CMOSDEVICECHARACTERIZATION.AllnotationisconsistentwiththatinSection3.2.AthighfrequenciesthefirstterminEq.(8.2-1)dominates,whereasatlowfrequenciesthesecondtenndominates.Sincethesecondtermistheonlyonewithmodelparameters,itistheonlyponionoftheexpressionthatmustbeconsideredforcharacterization.Equation(B.2-2)describesthemean-squarenoisecurrentatlowfrequenciesas(8.2-2)Thismean-squarenoisecurrentisrepresentedasacurrentsourceacrossthedrainandsourcenodesinthesmall-signalmodelforthetransistor.ThisisillustratedinFig.8.2-1.Sincenoiseismoregenerallyconsideredattheinputratherthantheoutput.theinput-referrednoisecurrentisgivenbymultiplyingEq.(8.2-2)byg;/toget(8.2-3)Substitutingtherelation.'lhipfor8minthesaturationregion8m=V2Ks(WIL)TD(B.2-4)intoEq.(B.2-3)givesaconvenientfonnfortheinput-noisevoltage2_[e,-KF]2K$WLfCoxllf(v2)(8.2-5)Forcharacterizationpurposes.assumethatthenoisevoltageismeasuredata1HzbandwidthsothattheMtermisunity.Equation(8.2-5)canberewrittenaslogle~J=log[2K'~CS]-log[/]oxByplottinglog[f]versuslog[e!Jandmeasuringtheintercept.whichisIog[2K5~cJonecanextracttheparameterKF.(8.2-6)8.3.CharactefizationofOthefActiveComponents7558.3CHRRBCTERIZRTIONOFOTH£0ACTIVECOMPONENTSIntheprevioussections,characterizationofmostofthemoreimportantparametersofthelarge-signalMOSmodelhasbeeneovered.ThissectionwiUbedevotedtocharacterizationofothercomponentsfoundinatypicalCMOSprocess.OneoftheimportantactivecomponentsavailabletotheCMOSdesignerisasubstratebipolarjunctiontransistor(BIT)(seeSection2.5).ThecollectorofthisBITisalwayscommonwiththesubstrateoftheCMOSpttK:ess.Forexample,iftheCMOSprocessisap-wellprocess,thensubstrateistheeollector.thepwellisthebase,andthen+diffusionsinthepwellaretheemitter.ThesubstrateBlTisusedprimarilyfortwoapplications.Thefirstisasanoutputdriver.Becausethe8mofaBJTisgreaterthanthe8mofanMOSdevice,theoutputimpedance,whichistypicallyllg,..islowerforaBJT.Thesecondapplicationisinbandgapvoltage-referencecircuits.Forthesetwoapplications,theparametersofinterestarethedebeta.fJrJkT/q,(8.3-1)and(8.3-2}A,..isthecross-sectionalareaoftheemitter-basejunctionoftheBIT.InordertodeterminetheparameterfJr~c.Eq.(B.3-2)canberearrangedtotheformgiveninEq.(B.3-3),whichgivesiEasafunctionofi8inalinearequation:(8.3-3)Thecurrenti8canbeplottedasafunctionofi&andtheslopemeasuredtodeterminef3rJd•88g,..88g..,,88-91Geometriccenterfrequency,577g.,,88-91g,.},..88-91Gradingcoefficient,32.SeealsoMJ.MJSWGuurdbars,5IHarmonicdistortion,221High-speedopamps,368High-swingcascode.Seecascode,highswingHistogramtest,664HSPICE.72,92Hysteresis.Seecomparator,hysteresisICMR(inputcommon-moderange),181,416Impactioni1.ation,27Inductorcutsets,592Initialoperatingstatesofatwo-stagecompllllltor450INL.Seelinearity,integralInputoffsetvoltagetV~).181,441Iotegrallinearity.Seelinearity,integralIntegratorcancellationofphaseerror,527continoou~time,520frequency,521switchedcapacitor,523timeconstant,521Intrinsiccarrierconcentration(n1).'74Inverteractiveloadinvener,168-172currentsourceloadinvener,172--175noiseanalysis,1711-180push-pull,176-178loversion.Seemoderateinversion,stronginversion,weakinversionIonimplantation.19,21Is(IS),36,SO,53.Seealsopnjunction,saturationcurrentJunctioncapacitance.Seecapacitor.junctionJunctiondepth.21K'.75KAPPA.9lKf..745Kf.86.403K$.745Ladderfilter.580.581t,5821LAMBDA.'Y·75Large-signalmodel.Seealsomodel,large-signalactiveluadinverter,168cascadeamplifier,199curre111sourceloadinverter,172differeDtialamplifier.182push-pullinverter,176Latch,477LatchtimeCODSillllt,479Latch-up,50LateralBJT.SeeBIT,lateralLateraldiffu.~ion(LD),83,93.Seea.booverlapLayout.SS,2113Layoutrules.SeedesignrulesLD.Seelateraldift"usionLDD.SeelightlydupeddrainLength,37,73Length,effective.73.84LEVELImodel.72.78.Seeal$0simpleMOSmodelLEVEL3model,72,92Lightlydopeddrain(LDD),27IndexLinearregression,746Linearitydifferential(DNL),618,629,632,644.654--{;.57integralONL),618,629,632,644,654-657LOCOS,26,29Lownoiseamplifier.Seeamplifier,lownoiseLowvoltageamplifier,426bias,422Macromodel.323-341Mask,22.24MASH.SeeAID.delta-sigma.MASHMatching,56Meshanalysis,735Micropoweramplifier.Seeamplifier,micropowerMillercapacitance,199.207compensation.256,355effect,206simplification.742Minoritycarrierconcentration,34,35MJ.81MJSW,82.103Moat,26.SeealsoactiveareaMobility,43,93degradaliun,93,746surface,73temperaturedependence,52,95Model,modelinglarsesignal,79parameters,75second-ordereffectssimpleMOS(SeealsoSah),43smallsignal.87nonsaturatedregion,90saturatedregion.89SPICE.MODELstatement.102weakinversion,98,99Moderateinversion.99Modulators,sigma-della(ordella-sigma)ca.•cade,706distributedfeedback.706distributedfeedforward,706MOMcapacitor.Seecapacitor.MOMMonotonic,monotonicity,619,656NFS.93.98Negativephotoresist.Seephotoresist,negativeNitride.silicon.26Nodalanalysis,734Noisealiasing,599differentialslagenoise,193effectivenoisebandwidlh.532flicker,1/j.55,86,40378tinverterstagenoise,178kT/C,414.532.599,658MOSnoisecurrent,86quantization,615,616~haping,698shot.54switchedcapacitorcircuits,532,599thennal,55,86Noninvertingamplifier,507Nonmonotonicity,618Noni>Vetlappingclocks.494Nonsarurationregion.75.90Normalizedlow-pa~stonormalizedbandpasstransformation.577Normalizedlow-passtounnormalizedbandpassrransformation,577Normal.Wttion,bandpass,577Not.ation,6NRD(equivalentnumberofsquaresforthedrain),100NRS(equivalentnumberofsquaresforthesource),100NSUB{substrnteconcentrationordoping).93Nyquistrate,652.698Nullingresistor,278n-wellresistor.Seeresistor,n-weUOffresistance(switch),114Offset,114,181,246,251,442,465Offseterror,617,655,676Ohmspersquare,62Onresistance(switch),I14Operationalamplifiers.Seeamplifier.operationalOperationaltransconductanceamplifiers(OTA).Seeamplifier.operationaltransconductanceOutdi:tfusion.Seelateraldiffu~ionOverlap.83,84.SeealsolateraldiffusionOverlapcapacitance.Seecapacitance,overlapOversamplingAID.SeeAID,oversamplingandAID,delta-sigmaOversamplinga~sumption,505Oversamplingratio,698Overshoot,317,771Oxidation,19Oxide.19growth,20thin,27.SeealsoTOXParallelAID.SeeAID,parallelParallelswiu:hedcapacitorequivalentresistor.493PardSiticcapliCitance,44,46Passivationlayer.28Passivecomponents,43performancesummary,45PB,80,95PD.IOOPennilivity,74Pbasemargin.254.260782INDEXPHl(2t/>jJ,95.103Photolithographicinvariance,59Photolithography,22.23Photomask,22-25Photoresistoogative,24positive,24__PipelineAID.SeeND,pipelinePipelineD/A.SeeDfA.pipelinePlasmaetching.Seeetching.plasmaPLI.Seephotolithographicinvariancepnjunction.21,29,33capacitance,32charge.32depletionregion.32.35saturationcurrent,36step.30Polycide.29Polysilicide,47Polysilicon,27Polysilicondeposition,27Polysiliconresistance,45,47Positivefeedback,477Power-supplyrejectionratio,305Processingsteps,25Projectionprinting,25Propagationdelay,441,443,444.450,452,454Proximityprinting,25PS(perimeterofthesource),82,100PSRR,248,28~293,305simulation,313P'IAT.153,425Pushpull,176,224,226Quan!Uationnoi!le,615Quantixersmultibit.704singlebit,703RD.103Referencebandgapvoltagereference,153-159,424BIT.144bootstrap,148,I52current,143lowvoltage,424vl'threshold,148.149,151VRF.•144voltage.143n:net,l47Regenerativecomparator.Seecomparator,regenerativeRegionsofoperation.75cutoff,75nonsaturated,76saturated.76Resistance.Seealsoresistorcalculation,62contact,760,762Resistordiffused.47layout,60n-well.47polysilicon,47Resistoremulation.493,497Resolution,614,654Reversesaturationcurrent,79RHPzero,257,262,270,280,366RS,103RSH,103SID.SeemoatSahequation,43,73Salicide.29Sallen-Keyfilter.596Sampleandhold.657,661SAR.Seesuccessiveapprmlim.ationregisterSaturationcwrent.Seepnjunction,saturationcurrentSaturationregion,76Saturationvoltage,76SCR.SeesiliconcontrolledrectifierSecond-ordersystem,768Self-calibratingAID.SeeAID,self-calibratingSelectivity.22Sensitivily,144SerialD/A.SeeDlA.serialSeriesswitchedcapacitorresistor,495Series-parallelswitchedcapacitorresistor.495Settlingtime,248.249,657simulation,317Sheetresistance,62ShichmanandHodgesmodel.43,73Sidewall,81Signalanalog,2analogsampled-data,2continuoustime.492digital,2discretetime,492Signaltonoiseratio(SNR).616.617,654Signal-dependentinputoffset.528Silicide.29Siliconconstants,74dioxide,19nitride,23,26,27teChnology,19Silicon-controlledrectifier(SCR),50Silicondioxide,19Siliconnitride,22.23Simulation,99-109Simulationofopamps,310-323IndexSimulationofswitchcd-capacitotcircuits,541Single-slopeAID.SeeAID,singleslopeSlewrateorslewingrate,248,270,442,447simulation,317Small-signalmodel,87.Seealsomodel,small-signalactiveloadinverte~170cascodeamplifier,202cascodecurrentsink,140currentsourceloadinverter,173differentialamplifier,188push-pullinverter,177Source,7,37Sourcefollower.221Sourcerearrangement,738Sourcereduction,740Somcesubstitution,738Spaceroxide.27SPICE,72.78Spun-onglass,28Sputtering,21Stateequations,584Statevariables,583Stepjunction,30Stepresponse,3!7,771Storistors,541Stronginven;ion,39surfacepotential,39Subthreshold,97,174,393.SeealsoweakinversionSuccessiveapproximationAJD.SeeAID,successiveapproximationSurfacemobility,73potential.74,75Switch,MOS.113Switchedcapacitoramplifierpara.~iticinsensitiveinverting,515parasiticinsensitivenoninverting,5JSbiquad,551low-Q,551bigh-Q.555Fleischer-Laker,558filter,561first-order,544integrator,523inverting,526nonim·erting,523second-order,550Switchfeedlhrough.Seeclockfeedlbrough.andchargeinjectionSymbology,6Symbolscurrentsource,8operationalamplifier,8transistor,8voltageso!IICe,8783Thmperaturemobilitydependence.52.95MOSdevicedependence,52pnjunctiondependence,53thresholddependence,52,96Temperaturecoefficientfractional.52,ISOpas.~ivecomponents,45ThstingA/0,662-66.5THD.SeetotalhammnicdistortionThennalnoise.Seenoise,tl!ern!alTt!ETA.93Thinoxide,27Thresholdvoltage,39,41,74,75Temperaturedependence,52.96limeconstant,498Tuneconstantofalatch.Seelatchtimeconstantlime-interleavedA/0.SetND,timeinterleavedTtrnevarianceandinvariance,537TotaJhannon.icdistortion,221,622TOX93Transconductancecharacteristic,77bulk-channel(g.-).Seelll!lbsgate-channel(g.J.Setg.,parameter(13),75Transferfunctions.Seefiltersnoisetr.msferfunction(delta-sigmamodulators),702signallr'ansferfunction(delra-sigmBmodulators),702Transresistance,switchedcapacitor,513Transientanalysis(SPICE).lOBTransistorBJT,48,408,409depletion,42MOS,36regionsofoperation,75Transistorsymbol.Seesymbol,transistorTripvoltageoruippoint.Seecomparator,trippointUnitmatchingprinciple,56,283Unity-gainbandwidth.SeegainbandwidthUnity-gainstability,265UO,93.Seea/$omobilityV0s(sat).76,270VerticalBJT,48VIH,440v,~,.440VMAX,94,95VMIN•127-129Vo11•440,445Vot,440Voltagecoefficient,46Voltage-controUedcurrentsource(VCCS),7.8,230Voltage-controlledvoltagesource(VCVS),7,8,230Voltagereference.See~eference784INDEXVoN•79,l30Vro.74,75XJ,93XQC,L04VTO,IOJYiannoulospath,59Wafer,19WD.93Weakinversion,173,395,396Weakinversionmodel,98Well.25Wetetching.Seeetching,wetWidth,37,73Width,effective,73,84Wilson.Seecurrentmirror,WilsonWorkfunction,39Z-domain,500analysis,537models,532Zenerbreakdown.Seebreakdown.zcnerZenerdiode,34
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