IEEEJOURNALOFSOLID-STATECIRCUITS,VOL.33,NO.2,FEBRUARY1998179AGeneralTheoryofPhaseNoiseinElectricalOscillatorsAliHajimiri,StudentMember,IEEE,andThomasH.Lee,Member,IEEEAbstract—Ageneralmodelisintroducedwhichiscapableofmakingaccurate,quantitativepredictionsaboutthephasenoiseofdifferenttypesofelectricaloscillatorsbyacknowledgingthetrueperiodicallytime-varyingnatureofalloscillators.Thisnewapproachalsoelucidatesseveralpreviouslyunknowndesigncriteriaforreducingclose-inphasenoisebyidentifyingthemech-anismsbywhichintrinsicdevicenoiseandexternalnoisesourcescontributetothetotalphasenoise.Inparticular,itexplainsthedetailsofhow1=fnoiseinadeviceupconvertsintoclose-inphasenoiseandidentifiesmethodstosuppressthisupconversion.Thetheoryalsonaturallyaccommodatescyclostationarynoisesources,leadingtoadditionalimportantdesigninsights.Themodelreducestopreviouslyavailablephasenoisemodelsasspecialcases.Excellentagreementamongtheory,simulations,andmeasurementsisobserved.IndexTerms—Jitter,oscillatornoise,oscillators,oscillatorsta-bility,phasejitter,phaselockedloops,phasenoise,voltagecontrolledoscillators.I.INTRODUCTIONTHErecentexponentialgrowthinwirelesscommunicationhasincreasedthedemandformoreavailablechannelsinmobilecommunicationapplications.Inturn,thisdemandhasimposedmorestringentrequirementsonthephasenoiseoflocaloscillators.Eveninthedigitalworld,phasenoiseintheguiseofjitterisimportant.Clockjitterdirectlyaffectstimingmarginsandhencelimitssystemperformance.Phaseandfrequencyfluctuationshavethereforebeenthesubjectofnumerousstudies[1]–[9].Althoughmanymodelshavebeendevelopedfordifferenttypesofoscillators,eachofthesemodelsmakesrestrictiveassumptionsapplicableonlytoalimitedclassofoscillators.Mostofthesemodelsarebasedonalineartimeinvariant(LTI)systemassumptionandsufferfromnotconsideringthecompletemechanismbywhichelectricalnoisesources,suchasdevicenoise,becomephasenoise.Inparticular,theytakeanempiricalapproachindescribingtheupconversionoflowfrequencynoisesources,suchasnoise,intoclose-inphasenoise.Thesemodelsarealsoreduced-ordermodelsandarethereforeincapableofmakingaccuratepredictionsaboutphasenoiseinlongringoscillators,orinoscillatorsthatcontainessentialsingularities,suchasdelayelements.ManuscriptreceivedDecember17,1996;revisedJuly9,1997.TheauthorsarewiththeCenterforIntegratedSystems,StanfordUniversity,Stanford,CA94305-4070USA.PublisherItemIdentifierS0018-9200(98)00716-1.Sinceanyoscillatorisaperiodicallytime-varyingsystem,itstime-varyingnaturemustbetakenintoaccounttopermitaccuratemodelingofphasenoise.Unlikemodelsthatassumelinearityandtime-invariance,thetime-variantmodelpresentedhereiscapableofproperassessmentoftheeffectsonphasenoiseofbothstationaryandevenofcyclostationarynoisesources.Noisesourcesinthecircuitcanbedividedintotwogroups,namely,devicenoiseandinterference.Thermal,shot,andflickernoiseareexamplesoftheformer,whilesubstrateandsupplynoiseareinthelattergroup.Thismodelexplainstheexactmechanismbywhichspurioussources,randomordeterministic,areconvertedintophaseandamplitudevariations,andincludespreviousmodelsasspeciallimitingcases.Thistime-variantmodelmakesexplicitpredictionsoftherelationshipbetweenwaveformshapeandnoiseupcon-version.Contrarytowidelyheldbeliefs,itwillbeshownthatthecornerinthephasenoisespectrumissmallerthannoisecorneroftheoscillator’scomponentsbyafactordeterminedbythesymmetrypropertiesofthewaveform.ThisresultisparticularlyimportantinCMOSRFapplicationsbecauseitshowsthattheeffectofinferiordevicenoisecanbereducedbyproperdesign.SectionIIisabriefintroductiontosomeoftheexistingphasenoisemodels.SectionIIIintroducesthetime-variantmodelthroughanimpulseresponseapproachfortheexcessphaseofanoscillator.Italsoshowsthemechanismbywhichnoiseatdifferentfrequenciescanbecomephasenoiseandexpresseswithasimplerelationthesidebandpowerduetoanarbitrarysource(randomordeterministic).Itcontinueswithexplaininghowthisapproachnaturallylendsitselftotheanalysisofcyclostationarynoisesources.Italsointroducesageneralmethodtocalculatethetotalphasenoiseofanoscillatorwithmultiplenodesandmultiplenoisesources,andhowthismethodcanhelpdesignerstospotthedominantsourceofphasenoisedegradationinthecircuit.Itconcludeswithademonstrationofhowthepresentedmodelreducestoexistingmodelsasspecialcases.SectionIVgivesnewdesignimplicationsarisingfromthistheoryintheformofguidelinesforlowphasenoisedesign.SectionVconcludeswithexperimentalresultssupportingthetheory.II.BRIEFREVIEWOFEXISTINGMODELSANDDEFINITIONSTheoutputofanidealsinusoidaloscillatormaybeex-pressedas,whereistheamplitude,0018–9200/98$10.00 1998IEEE180IEEEJOURNALOFSOLID-STATECIRCUITS,VOL.33,NO.2,FEBRUARY1998Fig.1.Typicalplotofthephasenoiseofanoscillatorversusoffsetfromcarrier.isthefrequency,andisanarbitrary,fixedphaserefer-ence.Therefore,thespectrumofanidealoscillatorwithnorandomfluctuationsisapairofimpulsesat.Inapracticaloscillator,however,theoutputismoregenerallygivenby(1)whereandarenowfunctionsoftimeandisaperiodicfunctionwithperiod2.Asaconsequenceofthefluctuationsrepresentedbyand,thespectrumofapracticaloscillatorhassidebandsclosetot