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GORESET
TheOp-ampIntegratorAmplifier
Intheprevioustutorialswehaveseencircuitswhichshowhowanoperationalamplifiercanbeusedaspartofapositive
ornegativefeedbackamplifierorasanadderorsubtractortypecircuitusingjustpureresistancesinboththeinputand
thefeedbackloop.Butwhatifweweretochangethepurelyresistive(R?)feedbackelementofaninvertingamplifierto
thatofafrequencydependantimpedance,(Z)typecomplexelement,suchasaCapacitor,C.Whatwouldbethe
effectontheoutputvoltage.ByreplacingthisfeedbackresistancewithacapacitorwenowhaveanRCNetworkacross
theoperationalamplifierproducinganOp-ampIntegratorcircuitasshownbelow.
Op-ampIntegratorCircuit
Asitsnameimplies,theOp-ampIntegratorisanoperationalamplifiercircuitthatperformsthemathematicaloperation
ofIntegration,thatiswecancausetheoutputtorespondtochangesintheinputvoltageovertimeastheop-amp
integratorproducesanoutputvoltagewhichisproportionaltotheintegraloftheinputvoltage.
Inotherwordsthemagnitudeoftheoutputsignalisdeterminedbythelengthoftimeavoltageispresentatitsinputas
thecurrentthroughthefeedbackloopchargesordischargesthecapacitorastherequirednegativefeedbackoccurs
throughthecapacitor.
Whenastepvoltage,Vinisfirstlyappliedtotheinputofanintegratingamplifier,theunchargedcapacitorChasverylittle
resistanceandactsabitlikeashortcircuitallowingmaximumcurrenttoflowviatheinputresistor,Rinaspotential
differenceexistsbetweenthetwoplates.NocurrentflowsintotheamplifiersinputandpointXisavirtualearthresulting
inzerooutput.Astheimpedanceofthecapacitoratthispointisverylow,thegainratioofXc/Rinisalsoverysmallgiving
anoverallvoltagegainoflessthanone,(voltagefollowercircuit).
AsthefeedbackcapacitorCbeginstochargeupduetotheinfluenceoftheinputvoltage,itsimpedanceXcslowly
increaseinproportiontoitsrateofcharge.ThecapacitorchargesupataratedeterminedbytheRCtimeconstant,(τ)of
theseriesRCnetwork.Negativefeedbackforcestheop-amptoproduceanoutputvoltagethatmaintainsavirtualearth
attheop-amp’sinvertinginput.
Sincethecapacitorisconnectedbetweentheop-amp’sinvertinginput(whichisatearthpotential)andtheop-amp’s
output(whichisnegative),thepotentialvoltage,Vcdevelopedacrossthecapacitorslowlyincreasescausingthe
chargingcurrenttodecreaseastheimpedanceofthecapacitorincreases.ThisresultsintheratioofXc/Rinincreasing
producingalinearlyincreasingrampoutputvoltagethatcontinuestoincreaseuntilthecapacitorisfullycharged.
Atthispointthecapacitoractsasanopencircuit,blockinganymoreflowofDCcurrent.Theratiooffeedbackcapacitorto
inputresistor(Xc/Rin)isnowinfiniteresultingininfinitegain.Theresultofthishighgain(similartotheop-ampsopen-
loopgain),isthattheoutputoftheamplifiergoesintosaturationasshownbelow.(Saturationoccurswhentheoutput
voltageoftheamplifierswingsheavilytoonevoltagesupplyrailortheotherwithlittleornocontrolinbetween).
Therateatwhichtheoutputvoltageincreases(therateofchange)isdeterminedbythevalueoftheresistorandthe
capacitor,"RCtimeconstant".BychangingthisRCtimeconstantvalue,eitherbychangingthevalueofthe
Capacitor,CortheResistor,R,thetimeinwhichittakestheoutputvoltagetoreachsaturationcanalsobechangedfor
example.
IfweapplyaconstantlychanginginputsignalsuchasasquarewavetotheinputofanIntegratorAmplifierthenthe
capacitorwillchargeanddischargeinresponsetochangesintheinputsignal.Thisresultsintheoutputsignalbeing
thatofasawtoothwaveformwhosefrequencyisdependantupontheRCtimeconstantoftheresistor/capacitor
combination.ThistypeofcircuitisalsoknownasaRampGeneratorandthetransferfunctionisgivenbelow.
RampGenerator
Weknowfromfirstprincipalsthatthevoltageontheplatesofacapacitorisequaltothechargeonthecapacitordivided
byitscapacitancegivingQ/C.ThenthevoltageacrossthecapacitorisoutputVouttherefore:-Vout=Q/C.Ifthe
capacitorischarginganddischarging,therateofchargeofvoltageacrossthecapacitorisgivenas:
ButdQ/dtiselectriccurrentandsincethenodevoltageoftheintegratingop-ampatitsinvertinginputterminaliszero,X
=0,theinputcurrentI(in)flowingthroughtheinputresistor,Rinisgivenas:
ThecurrentflowingthroughthefeedbackcapacitorCisgivenas:
Assumingthattheinputimpedanceoftheop-ampisinfinite(idealop-amp),nocurrentflowsintotheop-ampterminal.
Therefore,thenodalequationattheinvertinginputterminalisgivenas:
FromwhichwederiveanidealvoltageoutputfortheOP-ampIntegratoras:
Tosimplifythemath''salittle,thiscanalsobere-writtenas:
Whereω=2π?andtheoutputvoltageVoutisaconstant1/RCtimestheintegraloftheinputvoltageVinwithrespectto
time.Theminussign(-)indicatesa180phaseshiftbecausetheinputsignalisconnecteddirectlytotheinverting
inputterminaloftheop-amp.
TheACorContinuousOp-ampIntegrator
IfwechangedtheabovesquarewaveinputsignaltothatofasinewaveofvaryingfrequencytheOp-amp
Integratorperformslesslikeanintegratorandbeginstobehavemorelikeanactive"LowPassFilter",passinglow
frequencysignalswhileattenuatingthehighfrequencies.
At0HzorDC,thecapacitoractslikeanopencircuitblockinganyfeedbackvoltageresultinginverylittlenegativefeedback
fromtheoutputbacktotheinputoftheamplifier.Thenwithjustthefeedbackcapacitor,C,theamplifiereffectivelyis
connectedasanormalopen-loopamplifierwhichhasveryhighopen-loopgainresultingintheoutputvoltagesaturating.
Thiscircuitconnectsahighvalueresistanceinparallelwithacontinuouslycharginganddischargingcapacitor.The
additionofthisfeedbackresistor,Racrossthecapacitor,Cgivesthecircuitthecharacteristicsofaninvertingamplifier
withfiniteclosed-loopgainofR/R.Theresultisatverylowfrequenciesthecircuitactsasanstandardintegrator,while
athigherfrequenciesthecapacitorshortsoutthefeedbackresistor,Rduetotheeffectsofcapacitivereactance
reducingtheamplifiersgain.
TheACOp-ampIntegratorwithDCGainControl
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UnliketheDCintegratoramplifierabovewhoseoutputvoltageatanyinstantwillbetheintegralofawaveformsothat
whentheinputisasquarewave,theoutputwaveformwillbetriangular.ForanACintegrator,asinusoidalinput
waveformwillproduceanothersinewaveasitsoutputwhichwillbe90out-of-phasewiththeinputproducingacosine
wave.Furthermore,whentheinputistriangular,theoutputwaveformisalsosinusoidal.Thisthenformsthebasisof
aActiveLowPassFilterasseenbeforeinthefilterssectiontutorialswithacornerfrequencygivenas.
InthenexttutorialaboutOperationalAmplifiers,wewillanalysethecomplementoftheOp-ampIntegratoroperational
amplifiercircuitcalledtheDifferentiatorAmplifierthatproducesanoutputsignalwhichisthemathematical
operationofdifferentiation,thatisitproducesavoltageoutputwhichisproportionaltotheinputvoltage''srate-of-change
andthecurrentflowingthroughtheinputcapacitor.
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