Opamp supplementary Author(s): Fred-Johan Pettersen
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OpampsupplementaryFred-JohanPettersen1,21OsloUniversityHospitalHF,DepartmentofClinicalandBiomedicalEngineering,Norway2UniversityofOslo,DepartmentofPhysics,Norway
AbstractAverybriefsupplementarytotheofficialsyllabus-definingliterature.Thissupplementaryisdescribingoperationalamplifiersverybrieflyalongwithsomenicecircuits.
Keywords:opamp,circuits,FYS4250,FYS9250,FYS3240,FYS4240
1 Introduction
1.1 Whythissupplementary?Ademandforabriefsupplementarytoothersourcesofknowledgeaboutoperationalamplifiers(opamps)hasbeenvoiced,andthisisanattempttomakesuchasupplementary.Itisonlyintroductory,andisbyanymeannotaimingtobeareferencedocument.Thereareliterallytonsofbooksthatcoveropampsbetter,morethoroughly,moreelegant,andsoon;sogoaheadandstudythesubjectmoreifyoulike.Ifyouonlywantonebookonelectronics,andonethatcoversapracticalapproach,youshouldconsiderTheArtofElectronics,thirdedition,byPaulHorowitzandWinfieldHill(https://artofelectronics.net).Ifyoujustwanttolookatsomecoolopampapplications,trythis: https://en.wikipedia.org/wiki/Operational_amplifier_applications.
1.2 Whyopamps?Mostphysiologicalquantitieswewanttohavealookatstartasaphysicalquantity-utterlyuselessandinaccessibleforthephysician.Butdon'tdespair-therearewaysoftransformingthephysicalquantitiestosomethingthatcanbedisplayedonascreen.Thewayittypicallyhappensislikethis:
1. Physicalquantity.2. Transducerconvertphysicalquantitytoanalogueelectricalquantity=>Current,voltage,
charge,etc.3. Processingofelectricquantity:
a. Transformationbetweenelectricalrepresentations(current,voltage,charge,etc.).b. Amplification.c. Summation.d. Filtering.e. Othermathematicalstuff.
4. Conversionformanaloguerepresentationtodigitalrepresentation.5. Processingbycomputer.6. Displayofresult.
Itisintheprocessingoftheanaloguesignalsopampsplayanimportantrolesinceopampswithafewexternalconnectionsandcomponentscanperformmostrequiredmathematicaloperationsonthesignals.
Opamp supplementary Author(s): Fred-Johan Pettersen
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2 LawsandverybasicintroductorystuffInmuchthesamewayaswehavetoobeylaws,electronshavetoobeylawstoo.Themostimportantonesarepresentedhere.
2.1 Ohm'slaw π = π πΌ (1)Anyresistorwillbehaveaccordingtothislaw.Thepositivesideis,bydefinition,wherethecurrententers.Theimpedance(ZR)ofaresistorisalwaysthesameasit'sresistance,R.
2.2 Kirchhoff'sfirstlawThesumofallcurrentsintoanodeiszero.
Thissimplymeansthatchargeisconserved,andthatanelectricnodecannotstorechargeinanyway.
2.3 Murphy'slawWhatcangowrong,will.
Justforfun.Butthereisatouchofexperienceheresayingthatitisimportanttokeepcircuitsassmallandsimpleaspossible.
2.4 ThecapacitorIfyouknowanythingaboutcapacitors,pleasedon'treadthissinceitisaveeeeerysimplifiedviewofhowwecanuseacapacitor.OK,soyoudon'tknowmuchaboutcapacitors.That'sfine.Hereistheinformationyouneedtoenableyoutoreadthistext.Someusefulformulaswhenlookingatthecapacitorinthetimedomain: π = !
" β« πΌππ‘ (2)
πΌ = πΆ #$#% (3)
Equations2and3arereallythesameequationsaftersomefiddling.Thereshouldofcoursebeaconstantinequation2,butit'sleftoutforsimplicity.TheequationsshowthattherelationbetweenUandIissomewhatdifferentfromtherelationforresistorsasdescribedbyOhm'slaw.Asforresistors,thepositivesideofthevoltageacrossthecapacitorisonthesidethecurrententers.Ifweconsideracapacitorinthefrequencydomain,ithassomefunnyproperties.Itturnsoutthatthecapacitorcanbeseenasadevicewithfrequencydependentimpedance.Impedanceisanexpansionoftheresistanceconcept,andanimpedancevalueisactuallyavectorintheimaginaryplane.Weareonlygoingtoconsiderthevectorlength,orimpedancemagnitudehere.Thefrequencydependenceofthecapacitorimpedanceisgivenby
Opamp supplementary Author(s): Fred-Johan Pettersen
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|π"| =
!&'("
. (4)Theessenceofequation4isthatthecapacitorishavinghighimpedanceatlowfrequenciesandlowimpedanceathighfrequencies.Thisissummarizedintable1alongwiththeimpedanceoftheresistor.
Device DC(f=0Hz) Lowfrequency Highfrequency Infinitefrequency|ππ | R R R R|ππΆ| β High Low 0
Table1:Impedanceforresistorsandcapacitorforsomefrequencies.
3 Theopamp
3.1 TheidealopampTheopampisadevicemadeofanumberofothercomponentsliketransistors,resistors,andcapacitorsneatlypackagedintoasmallpackage.Fourourconvenience,suppliersofopampsmaketheminavarietyofsizesandwithavarietyofcapabilities.Thebasicopampisathree-pindeviceasshowninfigure1a),whileinrealityithasatleastfivepinsasshowninfigure1b).Wewillusethethree-pinsymbolforsimplicity.
a)Opamp,threepinsymbol. b)Opamp,fivepinsymbol.
Figure1:Opampsymbols.Thebasicfunctionisdescribedbyequation5 π)$* = π΄+(π,-. β π,-/) (5)whereUOUTistheoutputvoltage,UIN+isthevoltageatUIN+,UIN-isthevoltageattheUIN-,andADisthedifferentialgain.ADistypicallyverylarge,solargethatitispointlesstousetheopampwithoutanyformoffeedback.ADcaeasilybeintherange1000to1000000,andiscommonlyexpressedindecibels1.
3.2 CMRRandPSRRSinceopampsarenotideal,wehavetolookatsomemorefeatures.Fromequation5,wecanbeledtobelievethatopampsareignoringcommonmodevoltages.Well,theyusuallydoaprettygoodjobatit,buttheyarenotperfect.Acommonmodevoltageasavoltagethatispresentonbothinputs.Asaresult,wehavetoexpandequation6abit,andget:
1Decibelisalogarithmicscale,andforsignalvalues(notpowervalues),itisdefinedasπ΄#0 = 20πππ!1(π΄).
Opamp supplementary Author(s): Fred-Johan Pettersen
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π)$* = π΄+(π,-. β π,-/)+π΄"2 6$#$%.$#$&
&7 (6)
whereACMisthecommonmodegain.ACMisusuallyverylow,andcaneasilybelessthan0.01.Sinceweareinterestedinknowinghowmuchthecommonmodevoltageisrejected,weareusuallytalkingabouttheCommonModeRejectionRate(CMRR)whichusuallyisexpressedindecibels,andgivenby πΆππ π = 3'
|3()|= 20πππ!1 6
3'|3()|
7 ππ΅. (7)Asignalonthepowersupplies2mayalsobetransferredtotheoutput.Onceagain,wehavetoexpandtheequationdescribingtheopamp,andget π)$* = π΄+(π,-. β π,-/)+π΄"2 6
$#$%.$#$&&
7 + π΄56π56 (8)whereAPSisthegainofasignalonthepowersupplyandUPSistheunwantedsignalthatliesontopofthepowersupplyvoltage.Theratioofrejectionisofinterest,andsimilarlytoCMRR,PowerSupplyRejectionRate(PSRR)isgivenby πππ π = 3'
|3*+|= 20πππ!1 6
3'|3*+|
7 ππ΅. (9)
3.3 SomeotherpropertiesSomepropertiesoftheopamparepresentedinthetablebelow.
Property Idealopamp Real-worldopampInputimpedance Infinite. Veryhigh.
Outputimpedance(outputisseenasavoltagesource) Zero. From1Wto100W.Differentialgain Infinite. Veryhigh.
Commonmodegain Zero. Verylow.Powersupplygain Zero. Verylow.
Bandwidth Infinite. Finite.Noise Zero. Yes.
Table2:Opamppropertiesforidealandreal-worldopamps.
4 Somecoolopampcircuits
4.1 AnalysisAnalysisiflefttothereaders3.Tosimplifyanalysisofasensibledesignedopampcircuit,therearetworulesofthumbthatwillmakeiteasy:
β’ Thevoltagesonbothinputsareidenticalifsomesortoffeedbackisused.β’ Therewillnotflowanycurrentintotheinputs.
Theserulesofthumbapplyforallcircuitshere.Thetwomostcommonwaytodescribehowacircuitisprocessingasignalareanoutputfunctionorbyatransferfunction.Anoutputfunctionsimplystateswhattheoutputis,anda
2ThesignalistypicallynoiseorinterferenceaddedtothepureandsomewhatidealDCvoltageonthepowersupply.3Lecturersandauthorsjustlovethissentence.Firstofall,it'sfun,andthenit'saquetotheseriousreaderthatthisissmartifshe/he/itwanttolearnthestuff.
Opamp supplementary Author(s): Fred-Johan Pettersen
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transferfunctionmustbemultipliedbyaninputsignal.Forexample,equation10isanoutputfunction,andequation11isantransferfunctionofthesamecircuit. π)$* = π,- 61 +
7,7-7 (10)
π» = $./0
$#$= 1 + 7,
7- (11)
4.2 Non-invertingamplifier
Figure2:Non-invertingamplifier.
Parameter Value/Description
Transferfunction π123π45
= 1 +π 6π 7
Inputimpedance Veryhigh.Outputimpedance Verylow.
Pros Notinverting.Cons Gainunder1impossible.
Table3:Opampcircuitpropertiesfornon-invertingamplifier.
4.3 Unitygainbuffer
Figure3:Unity-gainbuffer.
Parameter Value/Description
Transferfunction π123π45
= 1 +0β = 1
Inputimpedance Veryhigh.Outputimpedance Verylow.
Pros Buffer.Cons Mayaddnoise.
Table4:Opampcircuitpropertiesforunity-gainbufferamplifier.
4.4 Invertingamplifier
Opamp supplementary Author(s): Fred-Johan Pettersen
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Figure4:Invertingamplifier.
Parameter Value/Description
Transferfunction π123π45
= βπ 7π 6
Inputimpedance =R1Outputimpedance Verylow.
Pros Largerangeofgainpossible,evenbelow1.Lowoutputimpedance.
Cons Inverting.Lowinputimpedance.
Table5:Opampcircuitpropertiesforinvertingamplifier.
4.5 Integrator/low-passfilter
Figure5:Integrator/lowpassfilter.
Parameter Value/Description
Outputfunction π123 = β1πΆ6+πΌ45 ππ‘ = β
1πΆ6+π45π 6
ππ‘ = β1
π 6πΆ6+π45 ππ‘
Inputimpedance =R1Outputimpedance Verylow.
Pros Largerangeofgainpossible.Lowoutputimpedance.
Cons Inverting.Lowinputimpedance.
NeedR2orsomeotherformofdischargingofC1topreventbuild-upofcharge.Table6:Opampcircuitpropertiesforanintegrator/low-passfilter.
Asanexercise,lookatthetransferfunctionfortheinvertingamplifier,andthinkofthecapacitorasanimpedancewithvaluesthatvariesaccordingtotable1.Whichfrequenciesareletthrough,andwhicharestopped?
4.6 Derivator/high-passfilter
Opamp supplementary Author(s): Fred-Johan Pettersen
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Figure6:Derivator/high-passfilter.
Parameter Value/Description
Outputfunction π123 = βπ 6πΌ45 = βπ 6πΆ6ππππ‘
Inputimpedance Capacitive.Outputimpedance Verylow.
Pros Largerangeofgainpossible.Lowoutputimpedance.
Cons Inverting.Capacitive/lowinputimpedance.
Table7:Opampcircuitpropertiesforaderivator/high-passfilter.Asanexercise,lookatthetransferfunctionfortheinvertingamplifier,andthinkofthecapacitorasanimpedancewithvaluesthatvariesaccordingtotable1.Whichfrequenciesareletthrough,andwhicharestopped?
4.7 Logarithmicandexponentialamplifier
Figure7:Exponentialandlogarithmicamplifiers.
Parameter Value/Description
Outputfunction π123 = βπ 6πΎ exp(π45)
π123 = βπΎ ln7π45π 68
Inputimpedance Low=R1
Outputimpedance Verylow.Pros Coolcircuits.
Lowoutputimpedance.Non-lineareffectsmaybeuseful.
Cons Inverting.Inaccurateandtemperaturedependent.
Lowinputimpedance.Table8:Opampcircuitpropertiesfortworelatednon-linearcircuits.
Sinceitmaybethatareaderdoesnotknoweverythingthereistoknowaboutdiodes,equation12describingthevoltage-currentrelationofadiode:
πΌ+ = πΌ6 >π/'890 β 1@. (12)
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Ifwesimplify,andacceptthattheconstantsarejustoffsetsandannoyingnumbers,wecanreduceequation12tosomethingmoreuseful,asshowninequations13andequation14. πΌ+ = πΎπ$' (13) π+ = πΎππ(πΌ+) (14)Thismeansthatwecanimplementthefunctionsln πandπ8 .Andsincewecandoaddition(sesummingamplifierbelow),wecanimplementmultiplicationsinceπ9: 3.9:0 = π΄π΅.Howcoolisthat?
4.8 Summingamplifier
Figure8:Summingamplifier.
Parameter Value/Description
Outputfunction π123 = βπ :9π;π ;
Inputimpedance =RnOutputimpedance Verylow.
Pros Largerangeofgainpossible.Lowoutputimpedance.
Cons Inverting.Lowinputimpedance.
Table9:Opampcircuitpropertiesforsummingamplifier.Itispossibletoadjustgainofeachinputbychangingthevalueofthecorrespondinginputresistor.
4.9 Differentialamplifier
Figure9:Differentialamplifier.
Opamp supplementary Author(s): Fred-Johan Pettersen
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Parameter Value/DescriptionOutputfunction π123 =
π 7/=π 6/>
(π? βπ@)
Inputimpedance = π 6/> + π 7/=Outputimpedance Verylow.
Pros Lowoutputimpedance.Accurateifgainislow.
Cons Lowinputimpedance.Largeerrorifgainishigh.
PoorCMRR,especiallyifgainishigh.Table10:Opampcircuitpropertiesfordifferentialamplifier.
4.10 Instrumentationamplifier
Figure10:Aninstrumentationamplifier.
Thisisabeauty,soitwillbeexplainedinabitmoredetail.Thecircuithastwostages,andtheinputstageisprovidingdifferentialgainandhighinputimpedance.Ourruleofthumbsaysthatthevoltagesontheopampinputsareidentical,whichmeansthatU+andU-arethevoltagesoneachsideofRG.I1cannowbecalculated πΌ! =
$&/$%7A
(15)Sincenocurrentsareflowingintothe-inputsofthetwoopamps,weknowthatI1ispassingthroughtheresistorchain,andwecanuseOhm'slawtocalculatethevoltageacrosstheresistorchain π/& β π.& = πΌ!(2π 3 + π ;) =
$&/$%7A
(2π 3 + π ;) (16)SincethevoltageacrosstheresistorchainisthesameastheoutputofstageI,wecandosomefiddlingonequation16,andweseethatthedifferentialgainofstageIisgivenby π΄+(6%=>?,) = 1 + &7B
7A (17)
SincethecommonmodevoltageontheoutputofstageIisthesameastheinput,thestageIcommonmodegainis1.NowthatwehaveanicelyamplifiedinputsignalavailableafterstageI,wewanttogetridofthecommonmodevoltage.ThisisdoneinstageIIbyusingadifferentialamplifierwheregainissetto1(orcloseto1inordertoreducenoiseanderrorandsuch).Notethatthereisapincalled
Opamp supplementary Author(s): Fred-Johan Pettersen
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UREF.Thisistypicallyplacedatsomereferencelevellikeground,butitcanbeanyreferencelevelyouchoose.ThepointisthattheoutputvoltageUOUTisrelativetothis.Theoutputfunctionisthen π)$* = (π. β π/) 61 +
&7B7A7 (18)
relativetothepotentialonUREF.Bothstagescombinedwillthen:
β’ haveverylowinputimpedance.β’ haveadifferentialgainwhichcanbesetbychangingRG.β’ haveaverylowcommonmodegain.β’ beadifferentialamplifieronsteroids.
Luckyforus,therearealoadofthesecommerciallyavailable.
4.11 Activefilterexamples(Sallen&Key)
Figure11:SallenandKeyfilters.Thetopcircuitsalow-passfilterandthebottomcircuitisahigh-passfilter.
Asanexampleoffiltersusingopamps,theSallen&Keyfiltersarenice.Thesearesecondorderfilterswithanopampamplifier.Theycanbebothlow-passandhigh-pass.Itisverycommontousethesamevalueonbothresistorsandbothcapacitors.Moreinformationhere:https://en.wikipedia.org/wiki/SallenβKey_topology.
5 AmoreaccurateanalysisoftheinvertingopampcircuitTherulesofthumbcanbeabitfrustratingforsomeofyou,sohereisamorethoroughanalysisforonecircuitjusttoshowhowitcanbedone.Similaranalysescanbedoneforallcircuits.Theequationsthatdescribethecircuitare π8 = π)$* + πΌ,Bπ & = π)$* +
$#$/$.CD7,.7-
π & (19)and π)$* = βπ8π΄+ . (20)Byinserting20into19,weget
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β $./0
3'= π)$* + π,-
7-7,.7-
β π)$*7-
7,.7- (21)
whichcanbemanipulatedinto $./0
$#$= β 3'7-
3'7,.7,.7-. (22)
IfweassumethatADisverylarge,sayβ,thisreducestothefamiliarform $./0
$#$= β 7-
7,. (23)
IfweassumethatADisverylarge,andthattheoutputiswithinreasonablelimits,equation5tellusthattheinputsmustbeveryclosetoeachother,andifADapproachesβ,thepotentialdiferencebetweenUIN+andUIN-approaches0.