# CHAPTER 2 OPERATIONAL AMPLIFIERS

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1 CHAPTER2OPERATIONALAMPLIFIERS ChapterOutline 2.1TheIdealOpAmp 2.2TheInvertingConfiguration 2.3TheNoninverting Configuration 2.4DifferenceAmplifiers 2.5IntegratorsandDifferentiators 2.6DCImperfections 2.7EffectofFiniteOpenLoopGainandBandwidthonCircuitPerformance 2.8LargeSignalOperationofOpAmp NTUEEElectronics L.H.Lu 21

2 2.1IdealOpAmp Introduction Theirapplicationswereinitiallyintheareaofanalogcomputationandinstrumentation Opampisverypopularbecauseofitsversatility Opampcircuitsworkatlevelsthatarequiteclosetotheirpredictedtheoreticalperformance Theopampistreatedabuildingblocktostudyitsterminalcharacteristicsanditsapplications Opampsymbolandterminals Twoinputterminals:invertinginputterminal()andnoninverting inputterminal(+) Oneoutputterminal TwodcpowersuppliesV+ andV Otherterminalsforfrequencycompensationandoffsetnulling Circuitsymbolforopamp Opampwithdcpowersupplies NTUEEElectronics L.H.Lu 22

3 Idealcharacteristicsofopamp Differentialinputsingleendedoutputamplifier Infiniteinputimpedance i1 =i2 =0(regardlessoftheinputvoltage) Zerooutputimpedance vO=A(v2v1)(regardlessoftheload) Infiniteopenloopdifferentialgain Infinitecommonmoderejection Infinitebandwidth Differentialandcommonmodesignals Twoindependentinputsignals:v1 andv2 Differentialmodeinputsignal(vId):vId =(v2v1) Commonmodeinputsignal(vIcm):vIcm =(v1+v2)/2 Alternativeexpressionofv1 andv2: v1 =vIcmvId/2 v2 =vIcm+vId/2 Exercise2.2(Textbook) Exercise2.3(Textbook) NTUEEElectronics L.H.Lu 23

4 2.2TheInvertingConfiguration Theinvertingcloseloopconfiguration ExternalcomponentsR1 andR2 formacloseloop Outputisfedbacktotheinvertinginputterminal Inputsignalisappliedfromtheinvertingterminal Invertingconfigurationusingidealopamp Therequiredconditionstoapplyvirtualshortforopampcircuit: Negativefeedbackconfiguration Infiniteopenloopgain Closedloopgain:G vO/vI = R2/R1 Infinitedifferentialgain:v2v1 =vO/A =0 Infiniteinputimpedance:i2 =i1 =0 Zerooutputimpedance:vO =v1i1R2 = vIR2/R1 Voltagegainisnegative Inputandoutputsignalsareoutofphase Closedloopgaindependsentirelyonexternalpassive components(independentofopampgain) Closeloopamplifiertradesgain(highopenloopgain) foraccuracy(finitebutaccurateclosedloopgain) NTUEEElectronics L.H.Lu 24

5 Equivalentcircuitmodelfortheinvertingconfiguration Inputimpedance:Ri vI/iI =vI /(vI/R1)=R1 Forhighinputclosedloopimpedance,R1 shouldbelarge,butislimitedtoprovidesufficientG Ingeneral,theinvertingconfigurationsuffersfromalowinputimpedance Outputimpedance:Ro =0 Voltagegain:Avo =R2/R1 Othercircuitexampleforinvertingconfiguration NTUEEElectronics L.H.Lu 25

6 Application:theweightedsummer Aweightedsummerusingtheinvertingconfiguration n Rf Rf Rf vO 0 R f ik ( v1 v2 ... vn ) k 1 R1 R2 Rn Aweightedsummerforcoefficientsofbothsigns R R R R R Rc vO v1 a c v2 a c v3 c v4 R1 Rb R2 Rb R3 R4 Exercise2.4(Textbook) Exercise2.6(Textbook) Exercise2.7(Textbook) NTUEEElectronics L.H.Lu 26

7 2.3Noninverting Configuration Thenoninvertingcloseloopconfiguration ExternalcomponentsR1 andR2 formacloseloop Outputisfedbacktotheinvertinginputterminal Inputsignalisappliedfromthenoninvertingterminal Noninverting configurationusingidealopamp Therequiredconditionstoapplyvirtualshortforopampcircuit: Negativefeedbackconfiguration Infiniteopenloopgain Closedloopgain:G vO/vI =1+R2/R1 Infinitedifferentialgain:v+v =vO/A =0 Infiniteinputimpedance:i2 =i1 =v /R1 Zerooutputimpedance:vO =v +i1R2 =vI(1+R2/R1) Closedloopgaindependsentirelyonexternalpassive components(independentofopampgain) Closeloopamplifiertradesgain(highopenloopgain) foraccuracy(finitebutaccurateclosedloopgain) Equivalentcircuitmodelforthenoninverting configuration Inputimpedance:Ri = (1+R2/R1)vi Outputimpedance:Ro =0 Voltagegain:Avo =1+R2/R1 NTUEEElectronics L.H.Lu 27

8 Thevoltagefollower Unitygainbufferbasedonnoninverting configuration Equivalentvoltageamplifiermodel: InputresistanceofthevoltagefollowerRi = OutputresistanceofthevoltagefollowerRo =0 VoltagegainofthevoltagefollowerAvo =1 Theclosedloopgainisunityregardlessofsourceandload Itistypicallyusedasabuffervoltageamplifiertoconnectasourcewithahighimpedancetoalow impedanceload Exercise2.9(Textbook) NTUEEElectronics L.H.Lu 28

9 Exercise1:Assumetheopampsareideal,findthevoltagegain(vo/vi) ofthefollowing circuits. (1)(2) (3)(4) NTUEEElectronics L.H.Lu 29

10 2.4DifferenceAmplifiers Differenceamplifier Idealdifferenceamplifier: RespondstodifferentialinputsignalvId RejectsthecommonmodeinputsignalvIcm Practicaldifferenceamplifier: vO =AdvId+AcmvIcm Ad isthedifferentialgain Acm isthecommonmodegain Commonmoderejectionratio(CMRR): | Ad | CMRR 20 log | Acm | Singleopampdifferenceamplifier R4 v v I 2 v R3 R4 v v R 1 R2 / R1 vO v iR2 v 1 R2 2 vI 1 vI 2 R 1 R1 1 R 3 / R4 1 R2 / R1 2 vIcm vId / 2 vIcm vId / 2 R R1 1 R3 / R4 1 R2 / R1 R2 1 1 R2 / R1 R2 1 1 R2 / R1 R2 1 R2 / R1 R2 vIcm vId Ad Acm 1 R3 / R4 R1 2 1 R3 / R4 R1 2 1 R3 / R4 R1 1 R3 / R4 R1 NTUEEElectronics L.H.Lu 210

11 Superpositiontechniqueforlineartimeinvariantcircuit SetvI2 =0 vO1 ( R2 / R1 )vI 1 vI1 R R4 SetvI1 =0 vO 2 1 2 vI 2 vO1 R 1 3 R R4 R 1 R2 / R1 vO vO1 vO 2 2 vI 1 vI 2 R1 1 R3 / R4 1 R2 / R1 R2 1 1 R2 / R1 R2 vIcm vId 1 R3 / R4 R1 2 1 R3 / R4 R1 1 1 R2 / R1 R2 1 R2 / R1 R2 CMRR 20 log / 2 1 R3 / R4 R1 1 R3 / R4 R1 vI2 vO2 1 1 R2 / R1 R2 1 R2 / R1 R2 Ad Acm 2 1 R3 / R4 R1 1 R3 / R4 R1 Theconditionfordifferenceamplifieroperation:R2/R1 =R4/R3 vO =(R2/R1)(v2v1) Forsimplicity,theresistancescanbechosenas:R3 =R1 andR4 =R2 DifferentialinputresistanceRid: Differentialinputresistance:Rid =2R1 LargeR1 canbeusedtoincreaseRid R2 becomesimpracticallylargetomaintainrequiredgain GaincanbeadjustedbychangingR1 andR2 simultaneously NTUEEElectronics L.H.Lu 211

12 Instrumentationamplifier vO R R Ad 4 1 2 vI 2 vI 1 R3 R1 DifferentialmodegaincanbeadjustedbytuningR1 Commonmodegainiszero Inputimpedanceisinfinite Outputimpedanceiszero Itspreferabletoobtainalltherequiredgaininthe1st stage,leavingthe2nd stagewithagainofone Exercise2.15(Textbook) Exercise2.17(Textbook) NTUEEElectronics L.H.Lu 212

13 2.5IntegratorsandDifferentiators Invertingconfigurationwithgeneralimpedance R1 andR2 ininvertingconfigurationcanbereplacedbyZ1(s)andZ2(s) Theclosedlooptransferfunction:Vo(s)/Vi(s)=Z2(s)/Z1(s) Thetransmissionmagnitudeandphaseforasinusoidinput canbeevaluatedbyreplacings withj Invertingintegrator Timedomainanalysis: t t 1 1 v (t ) vC (t ) VC i1 (t )dt VC I dt C0 C0 R t 1 RC 0 vO (t ) vC (t ) vI (t )dt VC Frequencydomainanalysis: Vo ( j ) Z 1 2 Vi ( j ) Z1 jRC Vo 1 f =90 Vi RC AlsoknownasMillerintegrator Integratorfrequency(int)istheinverseoftheintegratortimeconstant(RC) int =1/RC Thecapacitoractsasanopencircuitatdc( =0) openloopconfigurationatdc(infinitegain) Anytinydcintheinputcouldresultinoutputsaturation NTUEEElectronics L.H.Lu 213

14 TheMillerintegratorwithparallelfeedbackresistance Topreventintegratorsaturationduetoinfinitedcgain,parallelfeedbackresistanceisincluded G(dB) w(logscale) 1 1 RF C RC Vo ( j ) Z ( j ) 1 2 Vi ( j ) Z1 ( j ) R / RF jRC Closedloopgain=1/(jRF+R/RF) Closedloopgainatdc=RF/R Closedloopgainathighfrequency( >>1/RFC)1/jRC Cornerfrequency(3dBfrequency)=1/RFC Theintegratorcharacteristicsisnolongerideal LargeresistanceRF shouldbeusedforthefeedback NTUEEElectronics L.H.Lu 214

15 Theopampdifferentiator Timedomainanalysis: dvI (t ) iC dt dvI (t ) vO (t ) RC dt Frequencydomainanalysis: Vo ( j ) Z 2 jRC Vi ( j ) Z1 Vo RC f =90 Vi Differentiatoroperation: Differentiatortimeconstant:RC Gain(=RC)becomesinfiniteatveryhighfrequencies Highfrequencynoiseismagnified(generallyavoidedinpractice) NTUEEElectronics L.H.Lu 215

16 Thedifferentiatorwithseriesresistance Topreventmagnifyinghighfrequencynoise,seriesresistanceRF isincluded G(dB) w(logscale) 1 1 RC RF C Vo ( j ) jRC Vi ( j ) 1 jRF C Closedloopgain=jRC/(1+jRFC) Closedloopgainatinfinitefrequency=R/RF Closedloopgainatlowfrequency(

17 Exercise2: ForaMillerintegratorwithR =10k andC =10nF,ashuntresistanceRF is usedtosuppressthedcgain.FindtheminimumvalueofRF ifaperiodsignalwithaperiod of0.1sisappliedattheinput. Example2.4(Textbook) Example2.5(Textbook) Exercise2.18(Textbook) Exercise2.20(Textbook) NTUEEElectronics L.H.Lu 217

18 2.6DCImperfections* Offsetvoltage Inputoffsetvoltage(VOS)arisesasaresultoftheunavoidablemismatches Theoffsetvoltageanditspolarityvaryfromoneopamptoanother Theanalysiscanbesimplifiedbyusingthecircuitmodelwithanoffsetfree opampandavoltagesourceVOS atinputterminal TypicaloffsetvoltageisafewmV Effectofoffsetvoltageforaclosedloopamplifier VO VOS (1 R2 / R1 ) AdcvoltageVOS(1+R2/R1)existsattheoutputatzeroinputvoltage Inputoffsetvoltageiseffectivelyamplifiedbytheclosedloopgainastheerrorvoltageatoutput Someopampsareprovidedwithtwoadditionalterminalsforoffsetnulling NTUEEElectronics L.H.Lu 218

19 Inputbiasandoffsetcurrent DCbiascurrentsIB1 andIB2 arerequiredforcertaintypesofopamps InputbiascurrentisdefinedbyIB =(IB1+IB2)/2 InputoffsetcurrentisdefinedasIOS =|IB1IB2| TypicalvaluesforopampsthatusebipolartransistorsareIB =100nA andIOS =10nA Effectofinputbiascurrentforaclosedloopamplifiers Outputdcvoltageduetoinputbiascurrent:VO =IB1R2 IBR2 ThevalueofR2 andtheclosedloopgainarelimited. NTUEEElectronics L.H.Lu 219

20 Effectofinputoffsetvoltageonthethe invertingintegrator Theoutputvoltageisgivenby 1 t VOS VOS C 0 R vO VOS d t VOS t RC Theoutputvoltageincreaseswithtimeuntiltheopampsaturates Effectofinputbiascurrentontheinvertingintegrator Theoutputvoltageisgivenby 1 t I OS C 0 vO I B 2 R I OS d t I B 2 R t C Theoutputvoltagealsoincreaseswithtimeuntiltheopampsaturates NTUEEElectronics L.H.Lu 220

21 2.7EffectofFiniteOpenLoopGainandBandwidthonCircuitPerformance Practicalopampcharacteristics Opampwithfiniteopenloopgain:A(j)=A0 Opampwithfiniteopenloopgainandbandwidth:A(j)=A0/(1+j/b) Frequencyresponseofopamp: Openloopopamp ThefrequencyresponseofanopenloopopampisapproximatedbySTCform: A(j)=A0/(1+j/b) Atlowfrequencies( b),theopenloopopampisapproximatedby|A(jw)|A0/b Unitygainbandwidth (ft =t/2)isdefinedasthefrequencyatwhich|A(jt)|1 t =A0b NTUEEElectronics L.H.Lu 221

22 Invertingconfigurationusingopampwithfiniteopenloopgain Closedloopgain: vI (vO / A0 ) vI vO / A0 i1 R1 R1 v v v v / A vO O i1 R2 O I O 0 R2 A0 A0 R1 v R2 / R1 G O vI 1 (1 R2 / R1 ) / A0 ClosedloopgainapproachestheidealvalueofR2/R1 asA0 approachestoinfinite TominimizethedependenceofG onopenloopgain,weshouldhaveA0 >>1+R2/R1 vI vI vI R1 Inputimpedance: Ri i1 (vI vO / A0 ) / R1 (vI vI G / A0 ) / R1 1 G / A0 Outputimpedance: Ro 0 Invertingconfigurationusingopampwithfinitegainandbandwidth R2 / R1 R2 / R1 G 1 (1 R2 / R1 ) / A( j ) 1 (1 R2 / R1 ) / A0 /(1 j / b ) R2 / R1 1 (1 R2 / R1 ) / A0 j (1 R2 / R1 ) / b A0 ifA0 >>1+R2/R1 G G0 /(1+j/3dB) whereG0 =R2/R1 and3dB =A0b /(1+R2/R1)(A0 /|G0|)b NTUEEElectronics L.H.Lu 222

23 Exercise3: Consideraninvertingamplifierwheretheopenloopgainand3dBbandwidth oftheopampare10000and1rad/s,respectively.Findthegainandbandwidthoftheclose loopgain(exactandapproximatedvalues)forthefollowingcases:R2/R1 =1,100,200,and 2000. Exercise4:Anopamphasanopenloopgainof80dBanda3dBbandwidthof10rad/s. (1) TheopampisusedinaninvertingamplifierwithR2/R1 =100.Findthecloseloopgain atdcandat =1000rad/s. (2) TwoidenticalinvertingamplifierswithR2/R1 =100arecascaded.Findthecloseloop gainatdcandat =1000rad/s. (3) Forthecascadedamplifierin(2),findthefrequencyatwhichthegainis3dBlower thanthedcgain. Exercise2.12(Textbook) Exercise2.26(Textbook) Example2.6(Textbook) Exercise2.27(Textbook) Exercise2.28(Textbook) NTUEEElectronics L.H.Lu 223

24 2.8LargeSignalOperationofOpAmps Outputvoltagesaturation Ratedoutputvoltage(vO,max)specifiesthemaximumoutputvoltageswingofopamp Linearamplifieroperation(fortherequiredvO vO,max):vO =vO,max Themaximuminputswingallowedforoutputvoltagelimitedcase:vI,max =vO,max/(1+R2/R1) OutputistypicallylimitedbyvoltageincaseswhereRL islarge Outputcurrentlimits Maximumoutputcurrent(iO,max)specifiestheoutputcurrentlimitationofopamp Linearamplifieroperation(fortherequirediO iO,max):iL =iO,max iF Themaximuminputswingallowedforoutputcurrentlimitedcase: vI,max =iO,max[RL||(R1+R2)]/(1+R2/R1) OutputistypicallylimitedbycurrentincaseswhereRL issmall NTUEEElectronics L.H.Lu 224

25 Slewrate dv Slewrateisthemaximumrateofchangepossibleattheoutput:(V/sec) SR O max dt Slewratemaycausenonlineardistortionforlargesignaloperation Inputstepfunction Smallsignaldistortion(finiteBW) Largesignaldistortion(SR) vO (t ) V (1 e t t ) Fullpowerbandwidth DefinedasthehighestfrequencyallowedforaunitygainbufferwithasinusoidaloutputatvO,max vi (t ) Vo sin t vo (t ) Vo sin t vO dvo (t ) Vo cos t dt dv (t ) vO,max | o |max Vo SR distortionless dt SR dv (t ) | o |max Vo SR distortion dt w SR fM M wM 2 2vO ,max NTUEEElectronics L.H.Lu 225

26 Example2.7(Textbook) Exercise2.26(Textbook) Exercise2.30(Textbook) NTUEEElectronics L.H.Lu 226