EE214 standford ppt

T.H.LeeEE214FeedbackSystems©ThomasH.Lee,rev.October17,2004;AllrightsreservedPage1of34FeedbackSystems1.0IntroductionAsolidunderstandingoffeedbackiscriticaltogoodcircuitdesign,yetmanypracticingengineershaveatbestatenuousgraspofthisimportantsubject.WeagreewithMITPro-fessorJimRoberge’sfeelingthat“feedbackissofundamentallyimportantthatanalogengineerswhodon’tunderstanditshouldbelegallybarredfromcircuitdesign.”Thischapterisintendedasbutabriefoverviewofthefoundationsofclassicalcontroltheory,thatis,thestudyoffeedbackinsingle-input,single-output,time-invariant,linearcontinu-ous-timesystems.Asusual,we’llstartwithalittlehistorytoputthissubjectinitspropercontext.2.0ABriefHistoryofModernFeedbackAlthoughapplicationoffeedbackconceptsisveryancient(Ogannoytiger,tigereatOg),mathematicaltreatmentsofthesubjectarearecentdevelopment.Maxwellhimselfofferedthefirstdetailedstabilityanalyses,inapaperonthestabilityoftheringsofSaturn(forwhichhewonhisfirstmathematicalprize1),andalateroneonthestabilityofflyball-gov-ernorcontrolledsteamengines(“OnGovernors,”Proc.RoyalSoc.,no.100,1868).ThefirstconsciousapplicationoffeedbackprinciplesinelectronicswasapparentlybyrocketpioneerRobertGoddardin1912,inavacuumtubeoscillatorwhichemployedpos-itivefeedback.2Asfarasisknown,however,hispatentapplicationwashisonlywritingonthesubject(hewasmorethanalittlepreoccupiedwithrocketry,afterall),andhiscon-temporarieswerelargelyignorantofhisworkinthisfield.2.1ArmstrongandtheRegenerativeAmplifierEdwinHowardArmstrong’s1915paper3onvacuumtubescontainedthefirstpublishedexplanationofhowpositivefeedback(whichhecalledregeneration)couldbeusedtoincreasegreatlythevoltagegainofamplifiers.Althoughengineerstodayhaveaprejudiceagainstpositivefeedback,progressinelectronicsinthoseearlyyearswaslargelymadepossiblebyArmstrong’sregenerativeamplifier,sincetherewasnoothereconomicalwaytoobtainlargeamountsofgainfromtheprimitive(andexpensive)vacuumtubesoftheday.41.TheAdamsPrize,whichhewonin1857.Maxwellarguedthatstabilityoftheringswaspossibleonlyiftheyconsistedofsmallparticles.WenowknowfromdirectobservationbyVoyagerthatMaxwellwasright.2.U.S.Patent#1,159,209,filed1August1912,granted2November1915.3.“SomeRecentDevelopmentsintheAudionReceiver,”ProceedingsoftheIRE,v.3,pp.215-247,1915.4.Theeffectiveinternal“gmro”ofvacuumtubesbackthenwasonlyontheorderoffive,sothegainperstagewastypicallyquitelow,requiringmanyexpensivestagesifconventionaltopologieswereused.T.H.LeeEE214FeedbackSystems©ThomasH.Lee,rev.October17,2004;AllrightsreservedPage2of34WecanappreciatetheessentialfeaturesofArmstrong’samplifierbyexaminingthefol-lowingblockdiagram:FIGURE1.PositivefeedbackamplifierblockdiagramvINvOUT+afσHere,thequantityaisknownastheforwardgain,whilefisthefeedbackgain.Inourpar-ticularexample,arepresentsthegainofanordinary(i.e.,open-loop)singlevacuumtubeamplifier,whilefrepresentsthefractionoftheoutputvoltagethatisfedbacktotheampli-fierinput.Sincewehavetheblockdiagram,it’sstraightforwardtoderiveanexpressionfortheover-allgainofthisamplifier.First,recognizethat:σvINfvOUT⋅+=(1)Next,notethat:vOUTaσ⋅avINfvOUT⋅+()⋅==(2)Solvingfortheinput-outputtransferfunctionyields:Aa1af–--------------=(3)ItisevidentthatanypositivevalueofafsmallerthanunitygivesusanoverallgainAthatexceedsa,the“ordinary”(open-loop)gainofthevacuumtubeamplifier.Ifwemakeafequalto0.9,theoverallgainisincreasedtotentimestheopen-loopgain,whileanafprod-uctof0.99givesusafactorof100gainincrease,andsoon.Inthisway,Armstrongwasabletogetgainfromasinglestagethatotherscouldobtainonlybycascadingseveral.Thisachievementallowedtheconstructionofrelativelyinexpensive,high-gainreceiversandthereforealsoenableddramaticreductionsintransmitterpowerbecauseoftheenhancedsensitivityprovidedbythisincreasedgain.Inshortorder,thepositivefeedback(regenerative)amplifierbecameanearlyuniversalidiom,andWestinghouse(towhomArmstronghadassignedpatentrights)keptitslegalstaffquitebusytryingtomakesurethatonlylicenseeswereusingthisrevolutionarytechnology.T.H.LeeEE214FeedbackSystems©ThomasH.Lee,rev.October17,2004;AllrightsreservedPage3of342.2HaroldBlackandtheFeedforwardAmplifierWhileArmstrong’sregenerativeamplifierneatlymuchsolvedtheproblemofobtaininglargeamountsofgainfromvacuumtubeamplifiers,adifferentproblempreoccupiedthetelephoneindustry.Intryingtoextendcommunicationsdistances,amplifierswereneededtocompensatefortransmissionlineattenuation(ontheorderof20dB/mile).Usingampli-fiersavailableinthoseearlydays,distancesofafewhundredmileswereroutinelyachiev-ableand,withgreatcare,perhaps1,000-2,000mileswaspossible,butthequalitywaspoor.Afteratremendousamountofwork,acrudetranscontinentaltelephoneservicewasinauguratedin1915,witha68-year-oldAlexanderGrahamBellmakingthefirstcalltohisformerassistant,ThomasWatson,butthisfeatwasmoreofastuntthanapracticalachievement.Theproblemwasn’toneofinsufficientamplification;itwastrivialtomakethesignalattheendofthelinequiteloud,eventhoughtheamplifierswerecompensatingfortensofthousandsofdecibelsofattenuation.Rather,theproblemwasdistortion.Eachamplifiercontributedsomesmall(say,1%)distortio