[Ali-Hajimiri]Phase-Noise-in-Multi-Gigahertz-CMOS-

PhaseNoiseinMulti-GigahertzCMOSRingOscillatorsAliHajimiri,SotiriosLimotyrakis,ThomasH.LeeCenterforIntegratedSystems,Stanford,CA94305-4070AbstractAnanalysisofthephasenoiseindifferentialandsingle-endedringoscillatorsusingatime-variantmodelispre-sented.AnexpressionfortheRMSvalueoftheimpulsesensitivityfunction(ISF)isderived.Aclosed-formequa-tionforphasenoiseofringoscillatorsiscalculatedandalowerlimitonthephasenoiseofringoscillatorsisshown.Phasenoisemeasurementsofoscillatorsrunningupto5.5GHzareingoodagreementwiththetheory.IntroductionDuetotheirintegratednature,ringoscillatorshaverecentlybecomeanimportantbuildingblockinmanydigitalandcommunicationsystems[13.Theycanalsobeusedforsomelow-tierRFproducts.Recently,therehasbeensomeworkonmodelingthephasenoiseinringoscillators.[2]and[3]developmodelsforclockjitterbasedontimedomaintreatmentsofMOSandbipolardifferentialringoscillators,respectively.[4]proposesafre-quencydomainapproachtofindthephasenoisebasedonanLTImodelfordifferentialringoscillatorswithasmallnum-berofstages.Inthisworkwepresentageneralframeworktocalculatethephasenoiseofringoscillatorsbyapplyingatime-variantphasenoisemodel[5]toringoscillators.Basedonthisderivationweobtainalowerlimitonthephasenoiseofringoscillatorsinlongandshortchannelregimes.Goodagreementisobservedbetweenthepredictionsandmeasurementresultsofthephasenoiseofringoscillatorsrunningupto5.5GHz.BriefReviewoftheTime-VariantModelInanypracticaloscillator,thereisfluctuationsinamplitudeandphaseduetointernalandexternalnoise.Theamplitudefluctuationsaresignificantlyattenuatedbytheamplitudelimitingmechanismwhichispresentinanypracticalstableoscillatorandisverystronginringoscillators.Therefore,wewillfocusonphasevariation,whichisnotquenchedbysucharestoringmechanism.Theoutputofanoscillatorcanbewrittenas(1)Beinginterestedinitsphase,@(t),wecantreatanoscillatorasasystemthatconvertsvoltagesandcurrentstophase.Aswillbeseenshortly,forsmallperturbationsthisisalinearsystem.Itisalsoatime-variantsystemnomatterhowsmallwemaketheperturbations.Asanexample,considerthearbi-trarysingle-endedringoscillatorwithasinglecurrentsourceononeofthenodes,asshowninFig.1.AlsosupposethatthecurrentsourceconsistsofanimpulseofcurrentwithareaAq(incoulombs),occurringattimet=z.ThiswillcauseaninstantaneouschangeinthevoltageofthatnodewhichisgivenbyVo,t(t=A(t).f[w,t+@(U10-7803-4292-5/97/$10.0001998IEEEi(t)'+:=Fig.1.CMOSinverterchainringoscillator.rznsms4nsIFig.2.Timevarianceofthephaseresponse.(2)wherec,,&istheeffectivecapacitainceonthatnodeatthetimeofchargeinjection.Thiscorreslpondstoanequivalentshiftinthetransitiontimeforsmallchangesinvoltage.Thereforethechangeinthephase,@(1),isgivenbyA9AV=-Cnode(3)whereqmax=CnodeVswingandVswrngisthevoltageswingacrossthecapacitor.However,theproportionalityconstantistime-dependent.Thiscanbevisualizedbyconsideringtwoextremecases.Onecaseiswhentheimpulseisinjecteddur-inganoutputtransition.Thiswillresultinalargephaseshift.Astheotherextremecase,considerinjectinganimpulsewhiletheoutputissaturatedeithertosupplyorground.Thisimpulsewillhaveminimaleffectonthephaseoftheoscillator,asshowninFig.2.Unliketheamplituderesponse,oncethephaseshiftisintro-ducedintotheoscillatoritseffectpersistsindefinitely,sincesubsequenttransitionsareshiftedbythesameamount.Thus,thephaseresponseofanoscillatortoanimpulseisatimevaryingstep.Alsonotethataslongastheintroducedchangeinthevoltageduetothecurrentimpulseissmall,theresult-antphaseshiftislinearlyproportionaltotheinjectedcharge,andhencethetransferfunctionfromcurrenttophaseislin-ear.However,thetimevariantnatureofthesystemdoesnotdisappearevenforsmallperturbations.AV-AqA@=----Vswingqmax4.3.149IEEE1998CUSTOMINTEGRATEDCIRCUITSCONFERENCEWedefinetheunitimpulseresponseofthesystemastheamountofphaseshiftforaunitcurrentimpulse.Basedontheforegoingargument,oneobtainsthefollowingtimedependentimpulseresponse(4)Ymaxwhereu(t)istheunitstepandr(x)isperiodicunitlessfunc-tionwithperiod27c,whichgivesthetimevaryingpropor-tionalityconstantfor(3).Itislargewhenagivenperturbationcausesalargephaseshiftandsmallwhereithasasmalleffect[5].SinceT(x)representsthesensitivityofeverypointofthewaveformtoaperturbation,r(x)iscalledtheimpulsesensitivityfunction(ISF).Knowingtheimpulseresponse,onecancalculate$(t)usingthesuperpositionintegralwherei(t)representstheinputnoisecurrentinjectedintothenodeofinterest.Notethattheintegrationarisesfromtheclosedloopnatureoftheoscillator.Forawhitenoisecurrentsource,theargumentofthesecondintegralof(5),Ymaxhasthefollowingpowerspectrum(7)whereii/Afrepresentsthesingle-sidebandpowerspectrumofthenoisecurrentsourceandrrmsistherootmeansquare(RMS)valueoftheISF.$(t)isrelatedto~(t)throughanidealintegration;therefore,thesinglesidebandphasenoisespectrumforaringoscillatorwithNidenticalstagesiswherefrepresentsthefrequencyoffsetfromthecarrier.Inthecaseofmultiplenoisesources,i,,/Afrepresentsthetotalcurrentnoiseoneachnodeandisgivenbythepowersumofindividualsources[5].Inthepresenceofdevicel/fnoise,thedevicenoisepowerspectrum,(/Af,hasal/fregioninadditiontothewhitenoiseregion,wherefI