IEEEJOURNALOFSOLID-STATECIRCUITS,VOL.39,NO.9,SEPTEMBER2004AStudyofInje - PDF document

IEEEJOURNALOFSOLID-STATECIRCUITS,VOL.39,NO.9,SEPTEMBER2004AStudyofInjectionLockingandPullinginOscillatorsBehzadRazavi,Fellow,IEEEInjectionlockingcharacteristicsofoscillatorsarede-rivedandagraphicalanalysisispresentedthatdescribesinjectionpullingintimeandfrequencydomains.Anidentityobtainedfromphaseandenvelopeequationsisusedtoexpresstherequisiteos-cillatornonlinearityandinterpretphasenoisereduction.Thebe- Fig.1.Oscillatorpullingin(a)broadbandtransceiverand(b)RFtransceiver.Injectionlockingbecomesusefulinanumberofapplica-tions,includingfrequencydivision[8],[9],quadraturegenera-tion[10],[11],andoscillatorswithnerphaseseparations[12].Injectionpulling,ontheotherhand,typicallyprovesundesir-able.Forexample,inthebroadbandtransceiverofFig.1(a), ,islockedtoalocalcrystaloscillatorwhereasthereceiveVCO, ,islockedtotheincomingdataandhencepotentiallyaslightlydifferentfrequency.Thus,thetwooscillatorsmaypulleachotherasaresultofcouplingthroughthesubstrate.Similarly,thehigh-swingbroadbanddataattheoutputofthetransmittermaypull and asitcontainssubstantialenergyinthevicinityoftheiroscillationfrequencies.Fig.1(b)depictsanotherexampleofundesirablepulling.Thepoweramplier(PA)outputinanRFtransceivercontainslargespectralcomponentsinthevicinityof ,leakingthroughthepackageandthesubstratetotheVCOandcausingpulling.II.I (thuscontributingnophase0018-9200/04$20.00©2004IEEE IEEEJOURNALOFSOLID-STATECIRCUITS,VOL.39,NO.9,SEPTEMBER2004 Fig.2.(a)Conceptualoscillator.(b)Frequencyshiftduetoadditionalphaseshift.(c)Open-loopcharacteristics.(d)Frequencyshiftbyinjection.shift),andtheidealinvertingbufferfollowsthetanktocreateatotalphaseshiftof360 aroundthefeedbackloop.Whathap-pensifanadditionalphaseshiftisinsertedintheloop,e.g.,asdepictedinFig.2(b)?Thecircuitcannolongeroscillateat becausethetotalphaseshiftatthisfrequencydeviatesfrom360 by .Thus,asillustratedinFig.2(c),theoscillationfrequencymustchangetoanewvalue suchthatthetankcontributesenoughphaseshifttocanceltheeffectof .Notethat,ifthebufferand contributenophaseshift,thenthedraincurrent mustremaininphasewith underallcondi-Nowsupposeweattempttoproduce asinu-soidalcurrenttothedraincurrentof [Fig.2(d)].Iftheam-plitudeandfrequencyof arechosenproperly,thecircuitin-deedoscillatesat ratherthanat andinjectionlockingoccurs.Underthiscondition, and bearaphasedifference[Fig.3(a)]because:1)thetankcontributesphaseat ,rotating withrespecttotheresultantcurrent, ,and2) stillremainsinphasewith andhenceoutofphasewithrespectto ,requiringthat formanangle .(If and wereinphase,then wouldalsobeinphasewith andthuswith ).Theangleformedbe- and issuchthat becomesalignedwith (and )afterexperiencingthetankphaseshift, ,at Inordertodeterminethelockrange(therangeof whichinjectionlockingholds),weexaminethephasordiagramofFig.3(a)as departsfrom .Tomatchtheincreasinglygreaterphaseshiftintroducedbythetank,theanglebetween and mustalsoincrease,requiringthat rotatecoun-terclockwise[Fig.3(b)].Itcanbeshownthat (1) (2) Fig.3.Phasedifferencebetweeninputandoutputfordifferentvaluesof ! jandI whichreachesamaximumof (3)if DepictedinFig.3(c),theseconditionstranslatetoa90 betweentheresultantand ,implyingthatthephasediffer-encebetweenthe ,andtheoutput, ,reachesamaximumof .Tocomputethevalueof cor-respondingtothiscase,werstnotethatthephaseshiftofthetankinthevicinityofresonanceisgivenby(SectionIII-A) andrecognizefromFig.3(c)that and .Itfollowsthat (Thisresultisobtainedin[3]usingadifferentapproach.)Wedenotethismaximumdifferenceby ,withtheunderstandingthattheoveralllockrangeisinfact around Thedependenceofthelockrangeupontheinjectionlevel, ,istobeexpected:if decreases, mustformagreateranglewith soastomaintainthephasedifferencebetween and at [Fig.3(d)].Thus,thecircuitmovesclosertotheedgeofthelockrange.Asaspecialcase,if ,then(2)reducesto Wecall lockrange. RAZAVI:STUDYOFINJECTIONLOCKINGANDPULLINGINOSCILLATORS Fig.4.Phaseshiftinaninjection-lockedoscillator.implyingthat issmalland .Equations(5)and(7)thereforegive fortheinput-outputphasedifferenceacrossthelockrange.AsevidentfromFig.3(c),for thisdifferencereaches attheedgeofthelockrange,aplausibleresultbecauseifthezerocrossingsoftheinputfallonthepeaksoftheoutput,littlephasesynchronizationoccurs.Thelockrangeinthiscasecanbeobtainedfrom(6)or(8): Thesubtledifferencebetween(6)and(9)playsacriticalroleinquadratureoscillators(asexplainedbelow).Fig.4plotstheinput-outputphasedifferenceacrossthelockrange.Incontrasttophase-locking,injectionlockingto mandatesoperationawayfromthetankresonance.1)ApplicationtoQuadratureOscillators:Withtheaidofafeedbackmodel[13]oraone-portmodel[14],itcanbeshown(unilateral)couplingoftwoidenticaloscillatorsforcesthemtooperateinquadrature.Itcanalsobeshown[14]thatthistypeofcoupling(injectionlocking)shiftsthefrequencyfromresonancesothateachtankproducesaphaseshiftof (10)where denotesthecurrentinjectedbyoneoscillatorintotheotherand isthecurrentproducedbythecoreofeachoscillator.Useof(5)thereforegivestherequiredfrequencyshift Interestingly,(9)wouldimplythateachoscillatorispushedtotheedgeofthelockrange,but(6)suggeststhatfor,say, Fig.5.(a)Injection-lockeddivider.(b)Equivalentcircuit. ,thelockrangeexceeds(9)by3.3%.Inotherwords,fora90 phasedifferencebetweenitsinputandoutput,aninjection-lockedoscillatorneednotoperateattheedgeofthelockrange.2)ApplicationtoDividers:Fig.5(a)showsaninjec-tion-lockedoscillatoroperatingasa stage[15].Whilepreviousworkhastreatedthecircuitasanonlinearfunctiontoderivethelockrange[15],itispossibletoadoptaviewtosimplifytheanalysis.Switchingatarateequaltotheoscillationfrequency, and formamixerthattranslates to ,withthesumcomponentsuppressedbythetankselectivity.Thus,asdepictedinFig.5(b),injectionof at intonode isequivalenttoinjectionof (where isthemixerconversiongain)at directlyintotheoscillator.If and switchabruptlyandthecapacitanceat isneglected,then ,and(9)canbewrittenas Ifreferredtotheinput,thisrangemustbedoubled: rmedbysimulations,(13)representstheupperboundonthelockrangeofinjection-lockeddividers.III.IIftheinjectedsignalfrequencyliesoutof,butnotveryfarfromthelockrange,theoscillatorisWestudythisbehaviorbycomputingtheoutputphaseofanoscillatorunderlow-levelinjection.A.PhaseShiftThroughaTankForsubsequentderivations,weneedanexpressionforthephaseshiftintroducedbyatankinthevicinityofresonance.Asecond-orderparalleltankconsistingof , ,and exhibitsaphaseshiftof (14) IEEEJOURNALOFSOLID-STATECIRCUITS,VOL.39,NO.9,SEPTEMBER2004 ,and ,wehave Ifthecurrentowingthroughthetankcontainsphasemodula-tion,i.e., ,thenthephaseshiftcanbeobtainedbyreplacing in(15)withtheinstantaneousinputfre-quency, : Validfornarrow-bandphasemodulation(slowly-varying thisapproximationholdswellfortypicalinjectionphenomena.B.OscillatorUnderInjectionConsiderthefeedbackoscillatorysystemshowninFig.6,wheretheinjectionismodeledasanadditiveinput.Theoutputisrepresentedbyaphase-modulatedsignalhavingacarrierfre-quencyof (ratherthan ).Inotherwords,theoutputisassumedtotracktheinputexceptfora(possiblytime-varying)phasedifference.Thisrepresentationisjustiedlater.Theob-jectiveistocalculate ,subjectittothephaseshiftofthetank,andequatetheresultto Theoutputoftheadderisequalto (17) Thetwotermsin(18)cannotbeseparatelysubjectedtothetankphaseshiftbecausephasequantitiesdonotsatisfysuperpositionhere.Thus,theright-handsidemustbeconvertedtoasinglesinusoid.Factoring andde wewrite (20)Since and ,wehave andhence Upontravelingthroughthetank,thissignalexperiencesaphaseshiftgivenby(16): (22) Fig.6.oscillatorunderinjection.Equatingthisresultto ,weobtain Wealsonotefrom(19)that (24) (25) Itfollowsfrom(23),(24),and(26)that (27) OriginallyderivedbyAdler[1]usingasomewhatdifferentap-proach,thisequationservesasaversatileandpowerfulexpres-sionforthebehaviorofoscillatorsunderinjection.Underlockedcondition, ,yieldingthesameresultasin(9)forthelockrange.If ,theequationmustbesolvedtoobtainthedependenceof upontime.Notethat istypicallyquitesmallbecause,from(28),itreachesamaximumofonly .Thatis, variesslowlyunderpullingconditions.sequationcanberewrittenas Notingthat ,makingachangeofvariable ,andcarryingouttheinte-gration,wearriveat (30)where Thispaperintroducesagraphicalinterpretationofthisequationthatconfersinsightintothephenomenonofinjectionpulling.Interestingly, isequaltothegeometricmeanof +! ! differencebetween andtheupperendofthelockrange)and ! ! (thedifferencebetween andthelowerendofthelockrange). RAZAVI:STUDYOFINJECTIONLOCKINGANDPULLINGINOSCILLATORS Fig.7.Phasevariationofaninjection-pulledoscillator.C.Quasi-LockLetusrstexaminetheaboveresultforaninputfrequencyjustbelowthelockrange,i.e., but .Underthiscondition, isrelativelysmall,andtheright-handsideof(30)isdominatedbytherstterm solongas islessthanone,approachingalargemagnitudeonlyforashortduration[Fig.7(a)].Notingthatthecyclerepeatswithaperiodequalto ,weplot asshowninFig.7(b).Thekeyobservationhereisthat isnear90 mostofthetimeasiftheoscillatorwereinjection-lockedtotheinputattheedgeofthelockrange.Attheendofeachperiodandthebeginningofthenextperiod, undergoesarapid360 andreturnstothequasi-lockcondition[Fig.7(c)].Wenowstudythespectrumofthepulledoscillator.Thespec-trumhasbeenanalyticallyderivedusingdifferenttechniques[4],[5],butadditionalinsightcanbegainediftheresultsinFig.7areutilizedasthestartingpoint.Thefollowingobser-vationscanbemade.1)Theperiodicvariationof atarateof impliesthattheoutputbeatswiththeinput,exhibitingside-bandswithaspacingof .Notethat isafunctionofboth and (andhencetheinjectionlevel).2)Sincetheoscillatorisalmostinjection-lockedtotheinputforalargefrac-tionoftheperiod,weexpectthespectrumtocontainsignienergyat RedrawingFig.7(b)withthemodulo- transitionsattheendofeachperiodremoved[Fig.8(a)]andwritingtheinstantaneous Fig.8.Instantaneousfrequencyandspectrumofaninjection-pulledoscillator.frequencyoftheoutputas weobtaintheresultdepictedinFig.8(b).Theinterestingpointhereisthat,for thelockrange,theinstantaneousfrequencyoftheoscillatorgoesonlyabove ,exhibitingapeakvalueof asobtainedfrom(28).Thatis,theoutputspectrumcontainsmostlysidebandsabove Wenowinvokeausefulobservationthattheshapeofthespec-trumisgivenbytheprobabilitydensityfunction(PDF)oftheinstantaneousfrequency[16].ThePDFisqualitativelyplottedinFig.8(c),revealingthatmostoftheenergyisconnedtotherange andleadingtotheactualspectruminFig.8(d).Themagnitudeofthesidebandsdropsapproximatelylinearlyonalogarithmicscale[4],[5].Isitpossibleforoneofthesidebandstofallatthenaturalfrequency, ?Thefollowingmusthold: ,where isaninteger.Thus, .Since isoutofthelockrange,theleftsideofthisequationexceedsunityandnovalueof canplaceasidebandat .Wethereforesaytheoscillatorisfromitsnaturalfrequency.This IEEEJOURNALOFSOLID-STATECIRCUITS,VOL.39,NO.9,SEPTEMBER2004 Fig.9.Pullingbehaviorforinjectionsomewhatfarfromthelockrange.alsojustiestheuseof ratherthan forthecarrierfrequencyoftheoutputinFig.6.D.FastBeatItisinstructivetoexaminetheresultsobtainedaboveas deviatesfartherfromthelockrangewhileotherparametersre-mainconstant.Rewriting(30)as werecognizethattheverticaloffsetdecreaseswhereastheslopeofthesecondtermincreases.Theright-handsidethereforeap-pearsasdepictedinFig.9(a),yieldingthebehaviorshowninFig.9(b)for .Thus,comparedtothecaseillustratedinFig.7:1)thebeatfrequencyincreases,leadingtoawiderseparationofsidebands;2) staysrelativelyconstantforashorterpartoftheperiodandexhibitsafastervariationatthebeginningandend;and3)theinstantaneousfrequencyisnear forashorterdu-ration[Fig.9(c)],producingasmallerspectrallineatthisfre-quency.Infact,if issufcientlyfarfrom ,theenergyat Fig.10.One-portrepresentationofanoscillatorunderinjection. fallsbelowthatatthenextsideband [Fig.9(d)].Eventually,thecomponentsat and haveapprox-imatelyequallevels[4],[5].Interestingly,theanalysesin[4]and[5]onlyrevealthespec-truminFig.9(d).Ontheotherhand,theapproachpresentedhere,particularlytheuseofthePDFoftheinstantaneousfre-quency,correctlypredictsbothquasi-lockandfastbeatcondi-Inquadratureoscillators,pullingmayoccurifthefrequencymismatchbetweenthetwocoresexceedstheinjectionlockrange.Withinsufcientcoupling,theoscillatorsdisplayabehaviorsimilartothatdepictedinFigs.7and9.Notethattheresultingsidebandsareduetointermodulationbetweenthetwooscillatorsignals.Forexample,thespacingbetweenthesidebandsisafunctionofthecouplingfactor.IV.REQUISITESCILLATOROuranalysisofinjectionlockingandpullinghasthusfarig-norednonlinearitiesintheoscillator.Whilethismayimplythatcanbeinjectionpulledorlocked,weknowfromthesuperpositionprinciplethatthiscannothappen.Specif-ically,alinearoscillatorwouldsimplygenerateasinusoidat inresponsetoaninitialconditionandanotherat isresponsetotheinput.Toresolvethisparadox,wereexaminetheoscilla-torysystemunderinjection,seekingitsenvelopebehavior.Inthiscase,itissimplertomodeltheoscillatorasaone-portcircuitconsistingofaparalleltankandamildlynonlinearnegativeconductance(Fig.10),where representsthelossofthetank.Forexample, andtheinvertingbufferinFig.2(a)constituteanegative cell.Inthiscircuit Nowletusassume and ,where denotestheenvelopeoftheoutput.Substitutingtheexponentialtermsin(32)andseparatingtherealandimaginaryparts,wehave (33) Alinearoscillatorcanbedenedasoneinwhichtheloopgainisexactlyunityforallsignallevels. RAZAVI:STUDYOFINJECTIONLOCKINGANDPULLINGINOSCILLATORSTosimplifytheseequations,weassume:1)theenvelopevariesslowlyandbyasmallamount;2)themagnitudeoftheenvelopecanbeapproximatedasthetankpeakcurrentproducedbythe circuit, ,multipliedbythetankresistance ;3) ;4) whereapplicable;and5)thephaseanditsderivativesvaryslowly.Equations(33)and(34)thusreduceto (35) rstisAdlersequation,whereasthesecondexpressesthebehavioroftheenvelope.Todevelopmoreinsight,letusstudytheseresultswithinthelockrange,i.e.,if .Writing givesthefollowingusefulidentity: For thatis,thecircuitrespondsbyweakening circuit(i.e.,allowingmoresaturation)becausetheinjectionaddsin-phaseenergytotheoscillator.Ontheotherhand,for wehave ,asifthereisnoinjection.Fig.11illustratesthebehaviorof acrossthelockrange.Whilederivedforamildly-nonlinearoscillator,theabovere-sultdoessuggestageneraleffect:theoscillatormustspendlesstimeinthelinearregimeas movescloserto oscillatorthereforedoesnotinjectionlock.V.PThephasenoiseofoscillatorscanbereducedbyinjectionlockingtoalow-noisesource.Fromatime-domainperspective,effectofinjectionmanifestsitselfascor-rectionoftheoscillatorzerocrossingsineveryperiod,therebyloweringtheaccumulationofjitter.Thisviewpointalsorevealsthatthereductionofphasenoisedependsontheinjectionlevel,anditreachesamaximumfor [Fig.12(a)](wherethezerocrossingsof greatlyimpactthoseof )andamin-imumfor [Fig.12(b)](wherethezerocrossings coincidewiththezero-slopepointson Usingtheone-portmodelofFig.10andtheidentityex-pressedby(38),wecanestimatethephasenoisereductioninamildlynonlinearoscillatorthatisinjection-lockedtoanoiselesssource.AsdepictedinFig.13,thenoiseofthetankandthe cellcanberepresentedasacurrentsource .Intheabsenceofinjection,the(average)valueof cancels ,and experiencesthefollowingtransimpedance: (39) Fig.11.Variationof acrossthelockrange. Fig.12.Conceptualillustrationofeffectofinjectionlockingonjitter(a)inthemiddleand(b)attheedgeofthelockrange. Fig.13.Modelforstudyingphasenoise. isampliedbyanincreasinglyhighergainasthenoisefrequencyapproaches Nowsupposeaniteinjectionisappliedatthecenterofthelockrange, .Then,(38)predictsthattheoveralltankadmittancerisesto .Inotherwords,thetankimpedanceseenby at fallsfrominnity(withnoinjection)to underinjectionlocking.Asthefre-quencyof deviatesfrom continuestodom-inatethetankimpedanceuptothefrequencyoffsetatwhichthephasenoiseapproachesthatofthefree-runningoscillator(Fig.14).Todeterminethispoint,weequatethefree-runningnoiseshapingfunctionof(39)to andnotethat and Thus,thefree-runningandlockedphasenoiseprolesmeetattheedgesofthelockrange.Forverysmallfrequencyoffsets,thenoiseshapingfunctionassumesaLorentzianshapeandhenceanitevalue. IEEEJOURNALOFSOLID-STATECIRCUITS,VOL.39,NO.9,SEPTEMBER2004AsillustratedinFig.12(b),iftheinputfrequencydeviates ,theresultingphasenoisereductionbecomeslesspro-nounced.Infact,as approacheseitheredgeofthelock dropstozero,raisingtheimpedanceseenbythenoisecurrent.InCMOStechnology,itisdifculttorelyonthephasenoisereductionpropertyofinjectionlocking.Sincethelockrangeistypicallyquitenarrowandsincethenaturalfrequencyofoscil-latorsincurssignicanterrorduetoprocessvariationsandpoormodeling,thelockingmayoccurneartheedgeofthelockrange,therebyloweringthephasenoiseonlyslightly.Forexample,ifthetwo-sidedlockrangeisequalto %andthenaturalfre-quencyoftheoscillatorvariesby %withprocessandtem-perature,then,intheworstcase,theinjectionlockingoccursat .Itfollowsfrom(37)andtheaboveobser-vationsthattheimpedanceseenbythenoisefallsfromin ,yieldinga4.4-dBdegradationinthephasenoisecomparedtothecaseof VI.IULLINGINSCILLATORSTheanalysisinSectionIIIdealswithpullinginnominallyfree-runningoscillators,ararecaseofpracticalinterest.Sinceoscillatorsareusuallyphase-locked,theanalysismustaccountforthecorrectionpoducedbythePLL.Inthissection,weas-sumetheoscillatorispulledbyacomponentat whilephase-lockedsoastooperateat .Wealsoassumethattheoscillatorcontrolhasagainof andcontainsasmallperturbation, ,aroundadclevel.ExaminingthederivationsinSectionIIIforaVCO,weob-servethat(19)and(21)remainunchanged.In(22),ontheotherhand,wemustnowadd to .Forsmallpertur- canbeneglectedinthedenominatorof and Equatingthephaseof(41)to andnotingthat(24)and(26)canbeshowntostillhold,wehave Fig.15showsaPLLconsistingofaphase/frequencydetector(PFD),achargepump(CP),andalow-passlter( and andtheVCOunderinjection.Sincewithalowinjectionlevel,thePLLremainsphase-lockedto ,itismoremeaningfultoexpresstheoutputphaseas ratherthan .Thus, and (43) whereitisassumed radian.Thisapproximationisrea-sonableifpullingdoesnotexcessivelycorruptthePLLoutput. Fig.14.Reductionofphasenoiseduetoinjectionlocking. Fig.15.PLLunderinjectionpulling.TheaboveresultcannowbeusedinaPLLenvironment.InFig.15,thePFD,CP,andloopltercollectivelyprovidethefollowingtransferfunction: wherethenegativesignaccountsforphasesubtractionbythePFD.Wethereforehave Substitutingfor in(44)anddifferentiatingbothsideswithrespecttotime,weobtain (47)where .ThisrevealsthatthePLLbehavesasasecond-ordersysteminitsresponsetoinjectionpulling. (48) wehave (50)where denotesthephaseofthetransferfunctionatafrequency Adual-loopmodeldevelopedbyA.Mirzaeiarrivesatasimilarresultbutwithadifferentvalueforthepeakamplitudeofthecosine[17]. RAZAVI:STUDYOFINJECTIONLOCKINGANDPULLINGINOSCILLATORS Fig.16.(a)VCO.(b)Diephotograph. Fig.17.Measuredspectrumoffree-runningoscillatorunderinjection.(a)Quasi-lock.(b)Fastbeat. Fig.18.Measuredspectrumof(a)free-runningoscillatorunderinjection,and(b)phase-lockedoscillatorunderinjection.Equation(50)leadstoseveralinterestingandimportantobservations.First,theVCOoutputphaseismodulated,therebycreatingonlytwosymmetricsidebands(forlowinjection).Thesidebandsresideat ,i.e.,at and .Second,(46)suggeststhatthecontrolvoltagealsovariessinusoidallyatafrequencyof ,possiblyservingasapointformonitoringthestrengthofpulling.Third,thepeakvalueof in(50)andhencethesidebandmagnitudesvarywith ;infact,theyapproachzeroforboth and ,assumingapeakinbetween.ThisisbecausethePLLsuppressestheeffectofpullingif iswellwithintheloopbandwidthandtheoscillatorpullingbecomeslesssignicantif islarge.Thebandpassbehaviorofthepeakphasein(50)standsinsharpcontrasttotheresponseofPLLstophaseattheoutputoftheVCO.Forexample,theVCOphasenoiseexperi-encesahigh-passtransfertotheoutput.ThesymmetryofsidebandscanalsobeinterpretedwiththeaidoftherelationshipbetweentheshapeofthespectrumandthePDFoftheinstantaneousfrequency, .Ifthesidebandswere IEEEJOURNALOFSOLID-STATECIRCUITS,VOL.39,NO.9,SEPTEMBER2004 Fig.19.Measuredproleofsidebands.asymmteric,sowouldthePDFof be.Thatis, wouldspendmoretimeatoneofitsextremes.ThePLLwouldthenapplyagreatercorrectionatthatextreme,eventuallycreatingasymmetricspectrum.Theforegoinganalysisassumesarst-orderlooplterandcontinuous-timeloopoperation.Theadditionofasecondca-pacitorfromtheoscillatorcontrollinetogroundandthedis-crete-timenatureoftheloopleadtosidebandmagnitudesthataresomewhatdifferentfromthosepredictedby(50).Forthisreason,circuitsimulationsareoftennecessarytodeterminethesidebandlevelsaccurately.VII.EXPERIMENTALESULTSA1-GHzcharge-pumpphase-lockedloopincludinga cir-cuithasbeendesignedandfabricatedin0.35- mCMOStech-nology.Fig.16(a)showstheoscillatorandthemethodofin-jection,withthetransistorwidthsshowninmicrons(thelengthsareequalto0.35 m),andFig.16(b)depictsthediephoto.Thedistinctionbetweenquasi-lockandfastbeatcasesisdemonstratedinthespectraofFig.17forthefree-runningoscil-lator.Here,theinjectedlevelisapproximately38dBbelowtheoscillationlevelandthelockrangeisequalto 1.5MHz.In-deed,forinjection110kHzoutsidethelockrange,[Fig.17(a)],thesidebandat displaysthelargestmagnitude.As furtherdeviatesfromthelockrange(710kHzoutsidethelockrange),thecomponentat becomesdominant.Fig.18(a)and(b)comparestheoutputspectrumbeforeandafterphase-locking,respectively.Here,theinjectedlevelisap-proximately53dBbelowtheoscillationlevel.AstheanalysisinSectionIIIpredicts,thesidebandsbecomesymmetricaftertheloopisclosed.TheleftsidebandinFig.18(b)islocatedat andisslightlylargerthantherightone.Thisisbecause alsofeedsthroughtheoscillatortotheoutput.Inotherwords, ismovedtoabove ,thentherightsidebandbecomeslarger.Measurementsalsoconrmthatthespectrumremainssymmetricevenforaverysmallchargepumpcurrentbutthesidebandsriseinbothmagnitudeandnumber.Fig.19plotstheproleofthesidebandsas variesfrom tolargevalues,conrmingthebandpassbehaviorofpulling.Theseresultsagreewellwithsimulations.Thetheoreticalpre-dictionsoverestimatethepeakofthisprolebyabout7dB.Thisisderivedfromthemeasuredlockrange: =I (! =! [1]R.Adler,Astudyoflockingphenomenainoscillators,Proc.IEEEvol.61,pp.13801385,Oct.1973.[2]K.Kurokawa,Injectionlockingofmicrowavesolid-stateoscillators,Proc.IEEE,vol.61,pp.13361410,Oct.1973.[3]L.J.Paciorek,Injectionlockingofoscillators,Proc.IEEE,vol.53,pp.1727,Nov.1965.[4]H.L.Stover,Theoreticalexplanationoftheoutputspectraofunlockeddrivenoscillators,Proc.IEEE,vol.54,pp.310311,Feb.1966.[5]M.Armand,Ontheoutputspectrumofunlockeddrivenoscillators,Proc.IEEE,vol.59,pp.798799,May1969.[6]A.E.Siegman,Lasers.MillValley,CA:UniversityScienceBooks,[7]R.R.Ward,TheLivingClocks.NewYork:AlfredKnopf,1971.[8]V.Manassewitsch,FrequencySynthesizers,3rded.NewYork:Wiley,[9]E.Normann,Theinductance-capacitanceoscillatorasafrequencydi-vider,Proc.IRE,vol.24,Oct.1946,pp.799[10]C.J.M.Verhoeven,Ahigh-frequencyelectronicallytunablequadratureoscillator,IEEEJ.Solid-StateCircuits,vol.27,pp.10971100,July[11]A.Rofougaranetal.A900MHzCMOSLCoscillatorwithquadratureIEEEISSCCDig.Tech.Papers,Feb.1996,pp.392[12]J.KimandB.Kim,AlowphasenoiseCMOSLCoscillatorwitharingIEEEISSCCDig.Tech.Papers,Feb.2000,pp.430[13]T.P.Liu,A6.5GHzmonolithicCMOSvoltage-controlledoscillator,IEEEISSCCDig.Tech.Papers,Feb.1999,pp.404[14]B.Razavi,DesignofIntegratedCircuitsforOpticalCommunica-.NewYork:McGraw-Hill,2002.[15]H.R.RateghandT.H.Lee,Superharmonicinjection-lockedfrequencydividers,IEEEJ.Solid-StateCircuits,vol.34,pp.813821,June1999.[16]H.E.Rowe,SignalsandNoiseinCommunicationSystems.Princeton,NJ:VanNostrand,1965.[17]A.Mirzaei,privatecommunication. BehzadRazavi03)receivedtheB.Sc.degreeinelectricalengineeringfromSharifUniversityofTechnology,Tehran,Iran,in1985andtheM.Sc.andPh.D.degreesinelectricalengineeringfromStanfordUniversity,Stanford,CA,in1988and1992,respectively.HewasanAdjunctProfessoratPrincetonUniver-sity,Princeton,NJ,from1992to1994,andatStan-fordUniversity,Stanford,CA,in1995.HewaswithAT&TBellLaboratoriesandHewlett-PackardLabo-ratoriesuntil1996.Since1996,hehasbeenAssociateProfessorandsubsequentlyProfessorofelectricalengineeringattheUniversityofCalifornia,LosAngeles.HesitheauthorofPrinciplesofDataConversionSystemDesign(IEEEPress,1995),RFMicroelectronics(PrenticeHall,1998)(alsotranslatedintoJapanese),DesignofAnalogCMOSIntegratedCircuits(McGraw-Hill,2001)(alsotranslatedintoChineseandJapanese),andDesignofIntegratedCircuitsforOpticalCommunications(McGraw-Hill,2003),andtheeditorofMonolithicPhase-LockedLoopsandClockRecoveryCircuitsPress,1996),andPhase-LockinginHigh-PerformanceSystems(IEEEPress,2003).Hiscurrentresearchincludeswirelesstransceivers,frequencysynthe-sizers,phaselockingandclockrecoveryforhigh-speeddatacommunications,anddataconverters.Dr.RazavireceivedtheBeatriceWinnerAwardforEditorialExcellenceatthe1994IEEEInternationalSolid-StateCircuitsConference(ISSCC),thebestpaperawardatthe1994EuropeanSolid-StateCircuitsConference,theBestPanelAwardatthe1995and1997ISSCC,theTRWInnovativeTeachingAwardin1997,andtheBestPaperAwardattheIEEECustomIntegratedCircuitsCon-ferencein1998.Hewastheco-recipientofboththeJackKilbyOutstandingStudentPaperAwardandtheBeatriceWinnerAwardforEditorialExcellenceatthe2001ISSCC.Hehasbeenrecognizedasoneofthetoptenauthorsinthe50-yearhistoryofISSCC.HeservedontheTechnicalProgramCommitteesoftheISSCCfrom1993to2002andtheVLSICircuitsSymposiumfrom1998to2002.HehasalsoservedasGuestEditorandAssociateEditoroftheIEEEOURNALOFTATE,IEEETRANSACTIONSONIRCUITSAND,andtheInternationalJournalofHighSpeedElectronics.HeisanIEEEDistinguishedLecturer.