运算放大器噪声关系1f噪声均方根(RMS)噪声与等效噪声带宽

MT-048TUTORIALOpAmpNoiseRelationships:1/fNoise,RMSNoise,andEquivalentNoiseBandwidth"1/f"NOISEThegeneralcharacteristicofopampcurrentorvoltagenoiseisshowninFigure1below.3dB/OctaveWHITENOISELOGfCORNER1fNOISEnV/√HzorμV/√Hzen,inkFCkFC1fen,in=Figure1:FrequencyCharacteristicofOpAmpNoiseAthighfrequenciesthenoiseiswhite(i.e.,itsspectraldensitydoesnotvarywithfrequency).Thisistrueovermostofanopamp'sfrequencyrange,butatlowfrequenciesthenoisespectraldensityrisesat3dB/octave,asshowninFigure1above.Thepowerspectraldensityinthisregionisinverselyproportionaltofrequency,andthereforethevoltagenoisespectraldensityisinverselyproportionaltothesquarerootofthefrequency.Forthisreason,thisnoiseiscommonlyreferredtoas1/fnoise.Notehowever,thatsometextbooksstillusetheoldertermflickernoise.Thefrequencyatwhichthisnoisestartstoriseisknownasthe1/fcornerfrequency(FC)andisafigureofmerit—theloweritis,thebetter.The1/fcornerfrequenciesarenotnecessarilythesameforthevoltagenoiseandthecurrentnoiseofaparticularamplifier,andacurrentfeedbackopampmayhavethree1/fcorners:foritsvoltagenoise,itsinvertinginputcurrentnoise,anditsnon-invertinginputcurrentnoise.Thegeneralequationwhichdescribesthevoltageorcurrentnoisespectraldensityinthe1/fregionisf1Fk,i,eCnn=,Eq.1wherekisthelevelofthe"white"currentorvoltagenoiselevel,andFCisthe1/fcornerfrequency.Rev.0,10/08,WKPage1of6MT-048Thebestlowfrequencylownoiseamplifiershavecornerfrequenciesintherange1Hzto10Hz,whileJFETdevicesandmoregeneralpurposeopampshavevaluesintherangeto100Hz.Veryfastamplifiers,however,maymakecompromisesinprocessingtoachievehighspeedwhichresultinquitepoor1/fcornersofseveralhundredHzoreven1kHzto2kHz.Thisisgenerallyunimportantinthewidebandapplicationsforwhichtheywereintended,butmayaffecttheiruseataudiofrequencies,particularlyforequalizedcircuits.RMSNOISECONSIDERATIONSAswasdiscussedabove,noisespectraldensityisafunctionoffrequency.Inordertoobtainthermsnoise,thenoisespectraldensitycurvemustbeintegratedoverthebandwidthofinterest.Inthe1/fregion,thermsnoiseinthebandwidthFLtoFCisgivenby⎥⎦⎤⎢⎣⎡==∫LCCnwFFCnwCLrms,nFFlnFvdff1Fv)F,F(vCLEq.2wherevnwisthevoltagenoisespectraldensityinthe"white"region,FListhelowestfrequencyofinterestinthe1/fregion,andFCisthe1/fcornerfrequency.Thenextregionofinterestisthe"white"noiseareawhichextendsfromFCtoFH.ThermsnoiseinthisbandwidthisgivenbyCHnwHCrms,nFFv)F,F(v−=Eq.3Eq.2and3canbecombinedtoyieldthetotalrmsnoisefromFLtoFH:)FF(FFlnFv)F,F(vCHLCCnwHLrms,n−+⎥⎦⎤⎢⎣⎡=Eq.4Inmanycases,thelowfrequencyp-pnoiseisspecifiedina0.1Hzto10Hzbandwidth,measuredwitha0.1to10Hzbandpassfilterbetweenopampandmeasuringdevice.Themeasurementisoftenpresentedasascopephotowithatimescaleof1s/div,asisshowninFigure2belowfortheOP213.Page2of6MT-048........................................10010090........................................20mV1s20nV/div.(RTI)+-100Ω900ΩACTIVEBPF0.1-10HzGAIN=1000G=100TOSCOPENOISEGAIN=10TOTALGAIN=1,000,000OP213120nV1s/div.Figure2:0.1Hzto10HzInputVoltageNoisefortheOP213vnw510152025300.1110100FREQUENCY(Hz)INPUTVOLTAGENOISE,nV/√Hz0.1Hzto10HzVOLTAGENOISEForFL=0.1Hz,FH=10Hz,vnw=10nV/√Hz,FC=0.7Hz:Vn,rms=33nVVn,pp=6.6×33nV=218nV1/FCORNERFC=0.7Hz(WHITE)200nVTIME-1sec/DIV.Figure3:InputVoltageNoisefortheOP177Itispossibletorelatethe1/fnoisemeasuredinthe0.1to10Hzbandwidthtothevoltagenoisespectraldensity.Figure4aboveshowstheOP177inputvoltagenoisespectraldensityontheleft-handsideofthediagram,andthe0.1to10Hzpeak-to-peaknoisescopephotoontheright-hand‹‹‹‹Vn,rms(FL,FH)=vnwFClnFCFL+(FH–FC)Page3of6MT-048side.Equation2canbeusedtocalculatethetotalrmsnoiseinthebandwidth0.1to10HzbylettingFL=0.1Hz,FH=10Hz,FC=0.7Hz,vnw=10nV/√Hz.Thevalueworksouttobeabout33nVrms,or218nVpeak-to-peak(obtainedbymultiplyingthermsvalueby6.6—seethefollowingdiscussion).Thiscompareswelltothevalueof200nVasmeasuredfromthescopephoto.Itshouldbenotedthatathigherfrequencies,thetermintheequationcontainingthenaturallogarithmbecomesinsignificant,andtheexpressionforthermsnoisebecomes:LHnwLHrms,nFFv)F,F(V−≈.Eq.5And,ifFH>>FL,HnwHrms,nFv)F(V≈.Eq.6However,someopamps(suchastheOP07andOP27)havevoltagenoisecharacteristicsthatincreaseslightlyathighfrequencies.Thevoltagenoiseversusfrequencycurveforopampsshouldthereforebeexaminedcarefullyforflatnesswhencalculatinghighfrequencynoiseusingthisapproximation.Atverylowfrequencieswhenoperatingexclusivelyinthe1/fregion,FC>>(FH–FL),andtheexpressionforthermsnoisereducesto:⎥⎦⎤⎢⎣⎡≈LHCnwLHrms,nFFlnFv)F,F(V.Eq.7Notethatthereisnowayofreducingthis1/fnoisebyfilteringifoperationextendstodc.MakingFH=0.1HzandFL=0.001Hzstillyieldsanrms1/fnoiseofabout18nVrms,or119nVpeak-to-peak.Thepointisthataveragingresultsofalargenumberofmeasurementsoveralongperiodoftimehaspracticallynoeffectonthermsvalueofthe1/fnoise.Amethodofreducingitfurtheristouseachopperstabilizedopamp,toremovethelowfrequencynoise.Inpractice,itisvirtuallyimpossibletomeasurenoisewithinspecificfrequencylimitswithnocontributionfromoutsidethoselimits,sincepracticalfiltershavefiniterolloffcharacteristics.Fortunately,measurementerrorintroducedbyasinglepolelowpassfilterisreadilycomputed.Thenoiseinthespectrumabovethesinglepolefiltercutofffrequency,fc,extendsthecornerfrequencyto1.57fc.Similarly,atwopolefilterhasanapparentcornerfrequencyofapproximately1.2fc.Theerrorcorrectionfactorisusuallynegligibleforfiltershavingmorethantwopoles.Thenetbandwidthafterthecorrectionisreferredtoasthefilterequivalentnoisebandwidth(seeFigure4below).Page4of6MT-048GAUSSIANNOISESOURCEGAUSSIANNOISESOURCESINGLEPOLELOWPASSFILTER,fCBRICKWALLLOWPASSFILTER,1.57fCIDENTICALLEVELSSAMERMSNOISELEVELEQUIVALENTNOISEBANDWIDTH=1.57×fCFigure4:EquivalentNoiseBandwidthItisoftendesirabletoconvertrmsnoisemeasurementsintopeak-to-peak.Inordertodothis,onemusthavesomeunderstandingofthestatisticalnatureofnoise.ForGaussiannoiseandagivenvalueofrmsnoise,statisticstellusthatthechanceofaparticularpeak-to-peakvaluebeingexceededdecreasessharplyasthatvalueincreases—butthisprobabilityneverbecomeszero.Thus,foragivenrmsnoise,itispossibletopredictthepercentageoftimethatagivenpeak-to-peakvaluewillbeexceeded,butitisnotpossibletogiveapeak-to-peakvaluewhichwillneverbeexceededasshowninFigure5below.NominalPeak-to-Peak2×rms3×rms4×rms5×rms6×rms6.6×rms**7×rms8×rms%oftheTimeNoisewillExceedNominalPeak-to-PeakValue32%13%4.6%1.2%0.27%0.10%0.046%0.006%**Mostoftenusedconversionfactoris6.6Figure5:RMStoPeak-to-PeakRatiosPage5of6MT-048Peak-to-peaknoisespecifications,therefore,mustalwaysbewrittenwithatimelimit.Asuitableoneis6.6timesthermsvalue,whichisexceededonly0.1%ofthetime.REFERENCES1.HankZumbahlen,BasicLinearDesign,AnalogDevices,2006,ISBN:0-915550-28-1.AlsoavailableasLinearCircuitDesignHandbook,Elsevier-Newnes,2008,ISBN-10:0750687037,ISBN-13:978-0750687034.Chapter1.2.WalterG.Jung,OpAmpApplications,AnalogDevices,2002,ISBN0-916550-26-5,AlsoavailableasOpAmpApplicationsHandbook,Elsevier/Newnes,2005,ISBN0-7506-7844-5.Chapter1.Copyright2009,AnalogDevices,Inc.Allrightsreserved.AnalogDevicesassumesnoresponsibilityforcustomerproductdesignortheuseorapplicationofcustomers’productsorforanyinfringementsofpatentsorrightsofotherswhichmayresultfromAnalogDevicesassistance.Alltrademarksandlogosarepropertyoftheirrespectiveholders.InformationfurnishedbyAnalogDevicesapplicationsanddevelopmenttoolsengineersisbelievedtobeaccurateandreliable,howevernoresponsibilityisassumedbyAnalogDevicesregardingtechnicalaccuracyandtopicalityofthecontentprovidedinAnalogDevicesTutorials.Page6of6