Oscillator-Phase-Noise-A-Tutorial

326IEEEJOURNALOFSOLID-STATECIRCUITS,VOL.35,NO.3,MARCH2000OscillatorPhaseNoise:ATutorialThomasH.Lee,Member,IEEE,andAliHajimiri,Member,IEEEAbstract—Lineartime-invariant(LTI)phasenoisetheoriesprovideimportantqualitativedesigninsightsbutarelimitedintheirquantitativepredictivepower.Partofthedifficultyisthatdevicenoiseundergoesmultiplefrequencytranslationstobecomeoscillatorphasenoise.Aquantitativeunderstandingofthisprocessrequiresabandoningtheprincipleoftimeinvarianceassumedinmostoldertheoriesofphasenoise.Fortunately,thenoise-to-phasetransferfunctionofoscillatorsisstilllinear,despitetheexistenceofthenonlinearitiesnecessaryforamplitudestabilization.Inaddi-tiontoprovidingaquantitativereconciliationbetweentheoryandmeasurement,thetime-varyingphase-noisemodelpresentedinthistutorialidentifiestheimportanceofsymmetryinsuppressingtheupconversionof1noiseintoclose-inphasenoise,andprovidesanexplicitappreciationofcyclostationaryeffectsandAM–PMconversion.TheseinsightsallowareinterpretationofwhytheColpittsoscillatorexhibitsgoodperformance,andsuggestnewoscillatortopologies.TunedLCandringoscillatorcircuitexamplesarepresentedtoreinforcethetheoreticalconsiderationsdeveloped.Simulationissuesandtheaccommodationofamplitudenoiseareconsideredinappendixes.IndexTerms—Jitter,low-noiseoscillators,noise,noisemeasure-ment,noisesimulation,oscillators,oscillatornoise,oscillatorsta-bility,phasejitter,phase-lockedloops,phasenoise,phase-noisesimulation,voltage-controlledoscillators.I.INTRODUCTIONINGENERAL,circuitanddevicenoisecanperturbboththeamplitudeandphaseofanoscillator’soutput.Ofnecessity,however,allpracticaloscillatorsinherentlypossessanampli-tude-limitingmechanismofsomekind.Becauseamplitudefluc-tuationsareusuallygreatlyattenuatedasaresult,phasenoisegenerallydominates.Thus,eventhoughitispossibletodesignoscillatorsinwhichamplitudenoiseissignificant(particularlyatfrequencieswellremovedfromthatofthecarrier),wewillfocusprimarilyonphasenoiseinthebodyofthistutorial.TheissueofamplitudenoiseisconsideredindetailintheAppendix,asarepracticalissuesrelatedtohowoneperformssimulationsofphasenoise.Webeginbyidentifyingsomeverygeneraltradeoffsamongkeyparameters,suchaspowerdissipation,oscillationfre-quency,resonator,andcircuitnoisepower.Thesetradeoffsarefirststudiedqualitativelyinahypotheticalidealoscillatorinwhichlinearityofthenoise-to-phasetransferfunctionisassumed,allowingcharacterizationbytheimpulseresponse.Althoughlinearityisdefensible,timeinvariancefailstoholdeveninthissimplecase.Thatis,oscillatorsarelineartime-varying(LTV)systems.Bystudyingtheimpulseresponse,ManuscriptreceivedAugust16,1999;revisedOctober29,1999.T.H.LeeiswiththeCenterforIntegratedSystems,StanfordUniversity,Stan-ford,CA94305-4070USA.A.HajimiriiswiththeCaliforniaInstituteofTechnology,Pasadena,CA91125USA.PublisherItemIdentifierS0018-9200(00)00746-0.Fig.1.“Perfectlyefficient”RLCoscillator.wediscoverthatperiodictimevariationleadstofrequencytranslationofdevicenoisetoproducethephase-noisespectraexhibitedbyrealoscillators.Inparticular,theupconversionof1noiseintoclose-inphasenoiseisseentodependonsymmetrypropertiesthatarepotentiallycontrollablebythedesigner.Additionally,thesametreatmenteasilysubsumesthecyclostationarityofnoisegenerators,andhelpsexplainwhyclass-Coperationofactiveelementswithinanoscillatormaybebeneficial.IllustrativecircuitexamplesreinforcekeyinsightsoftheLTVmodel.II.GENERALCONSIDERATIONSPerhapsthesimplestabstractionofanoscillatorthatstillre-tainssomeconnectiontotherealworldisacombinationofalossyresonatorandanenergyrestorationelement.Thelatterpreciselycompensatesforthetanklosstoenableaconstant-am-plitudeoscillation.Tosimplifymatters,assumethattheenergyrestorerisnoiseless(seeFig.1).Thetankresistanceisthereforetheonlynoisyelementinthismodel.Togainsomeusefuldesigninsight,firstcomputethesignalenergystoredinthetank(1)sothatthemean-squaresignal(carrier)voltageis(2)wherewehaveassumedasinusoidalwaveform.Thetotalmean-squarenoisevoltageisfoundbyintegratingtheresistor’sthermalnoisedensityoverthenoisebandwidthoftheRLCresonator(3)Combining(2)and(3),weobtainanoise-to-signalratio(thereasonforthis“upside-down”ratioisoneofconvention,aswillbeseenshortly)(4Sensiblyenough,onethereforeneedstomaximizethesignallevelstominimizethenoise-to-carrierratio.0018–9200/00$10.00©2000IEEELEEANDHAJIMIRI:OSCILLATORPHASENOISE327Wemaybringpowerconsumptionandresonatorexplicitlyintoconsiderationbynotingthatcanbedefinedgenerallyasproportionaltotheenergystored,dividedbytheenergydissi-pated:(5)Therefore(6)Thepowerconsumedbythismodeloscillatorissimplyequalto,theamountdissipatedbythetankloss.Thenoise-to-carrierratioishereinverselyproportionaltotheproductofres-onatorandthepowerconsumed,anddirectlyproportionaltotheoscillationfrequency.Thissetofrelationshipsstillholdsap-proximatelyforrealoscillators,andexplainsthenearobsessionofengineerswithmaximizingresonator,forexample.III.DETAILEDCONSIDERATIONS:PHASENOISEToaugmentthequalitativeinsightsoftheforegoinganalysis,letusnowdeterminetheactualoutputspectrumoftheidealoscillator.A.PhaseNo