WAVEFORM GENERATORS外文文献.docx

WAVEFORM GENERATORS外文文献.docx

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WAVEFORMGENERATORS外文文献

WAVE-FORMGENERATORS

1TheBasicPricipleofSinusoidalOscillators

Manydifferentcircuitconfigurationsdeliveranessentiallysinusoidaloutputwaveformevenwithoutinput-signalexcitation.Thebasicprinciplesgoverningalltheseoscillatorsareinvestigated.Inadditiontodeterminingtheconditionsrequiredforoscillationtotakeplace,thefrequencyandamplitudestabilityarealsostudied.

Fig.1-1showsanamplifier,afeedbacknetwork,andaninputmixingcircuitnotyetconnectedtoformaclosedloop.TheamplifierprovidesanoutputsignalX0asaconsequenceofthesignalXiapplieddirectlytotheamplifierinputterminal.TheoutputofthefeedbacknetworkisXf=FX0=AFXi,andtheoutputofthemixingcircuit（whichisnowsimplyaninverter）is

Xf’＝-Xf=-AFXi

FromFig.1-1theloopgainis

Loopgain=Xf’/Xi=-Xf/Xi=-FA

SupposeitshouldhappenthatmattersareadjustedinsuchawaythatthesignalXf’isidenticallyequaltotheexternallyappliedinputsignalXi.Sincetheamplifierhasnomeansofdistinguishingthesourceoftheinputsignalappliedtoitatwouldappearthat,iftheexternalsourcewereremovedandifterminal2wereconnectedtoterminal1,theamplifierwouldcontinuetoprovidethesameoutputsignalXoasbefore.Note,ofcourse,thatthestatementXf’=XimeansthattheinstantaneousvaluesofXf’andXiareexactlyequalatalltimes.TheconditionXf’=Xiisequivalentto–AF=1,ortheloopgain,mustequalunity.

Fig-1-1AnamplifierwithtransfergainAandfeedbacknetworkFnotyetconnectedtoformaclosedloop.

TheBarkhausenCriterionWeassumeinthisdiscussionofoscillatorsthattheentirecircuitoperateslinearlyandthattheamplifierorfeedbacknetworkorbothcontainreactiveelements.Undersuchcircumstances,theonlyperiodicwaveformwhichwillpreserve,itsformisthesinusoid.ForasinusoidalwaveformtheconditionXi=Xf’isequivalenttotheconditionthattheamplitude,phase,andfrequencyofXiandXf’beidentical.Sincethephaseshiftintroducedinasignalinbeingtransmittedthroughareactivenetworkisinvariablyafunctionofthefrequency,wehavethefollowingimportantprinciple:

Thefrequencyatwhichasinusoidaloscillatorwilloperateisthefrequencyforwhichthetotalshiftintroducedasasignalproceedsfromtheinputterminals,throughtheamplifierandfeedbacknetwork,andbackagaintotheinput,ispreciselyzero（or,ofcourse,anintegralmultipleof2π）.Statedmoresimply,thefrequencyofasinusoidaloscillatorisdeterminedbytheconditionthattheloop-gainphaseshiftiszero.

Althoughotherprinciplesmaybeformulatedwhichmayserveequallytodeterminethefrequency,theseotherprinciplesmayalwaysbeshowntobeidenticalwiththatstatedabove.Itmightbenotedparentheticallythatitisnotinconceivablethattheaboveconditionmightbesatisfiedformorethanasinglefrequency.Insuchacontingencythereisthepossibilityofsimultaneousoscillationsatseveralfrequenciesoranoscillationatasingleoneoftheallowedfrequencies.

Theconditiongivenabovedeterminesthefrequency,providedthatthecircuitwilloscillateatall.AnotherconditionwhichmustclearlybemetisthatthemagnitudeofXiandXf’mustbeidentical.Thisconditionisthenembodiedinthefollowingprinciple:

Oscillationswillnotbesustainedif,attheoscillatorfrequency,themagnitudeoftheproductofthetransfergainoftheamplifierandthemagnitudeofthefeedbackfactorofthefeedbacknetwork（themagnitudeoftheloopgain）arelessthanunity.

Theconditionofunityloopgain-AF=1iscalledtheBarkhausencriterion.Thisconditionimplies,ofcourse,boththat|AF|=1andthatthephaseof-Aiszero.TheaboveprinciplesareconsistentwiththefeedbackformulaAf=A/（1+FA）.Forif–FA=1,thenAf→∞,whichmaybeinterpretedtomeanthatthereexistsanoutputvoltageevenintheabsenceofanexternallyappliedsignalvoltage.

PracticalConsiderationsReferringtoFig.1-2,itappearsthatif|FA|attheoscillatorfrequencyispreciselyunitytthen,withthefeedbacksignalconnectedtotheinputterminals,theremovaloftheexternalgeneratorwillmakenodifference*IfIFAIislessthanunity,theremovaloftheexternalgeneratorwillresultinacessationofoscillations.Butnowsupposethat|FA|isgreaterthanunity.Then,forexample,a1-Vsignalappearinginitiallyattheinputterminalswill,afteratriparoundtheloopandbacktotheinputterminals,appeartherewithanamplitudelargerthan1V.Thislargervoltagewillthenreappearasastilllargervoltage,andsoon,Itseemsjthen,thatif|FA|islargerthanunity,theamplitudeoftheoscillationswillcontinuetoincreasewithoutlimit,Butofcourse,suchanincreaseintheamplitudecancontinueonlyaslongasitisnotlimitedbytheonsetofnonlinearit