Opamp supplementary

Opamp supplementary Author(s): Fred-Johan Pettersen

Oslo University Hospital HF Last saved: 21/02/2020 08:24:00 OpampSupplementary-v5.docx

Page 1 of 11

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.



Page 2 of 11

2 LawsandverybasicintroductorystuffInmuchthesamewayaswehavetoobeylaws,electronshavetoobeylawstoo.Themostimportantonesarepresentedhere.

2.1 Ohm'slaw 𝑈 = 𝑅𝐼 (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: 𝑈 = !

" ∫ 𝐼𝑑𝑡 (2)

𝐼 = 𝐶 #$#% (3)

Equations2and3arereallythesameequationsaftersomefiddling.Thereshouldofcoursebeaconstantinequation2,butit'sleftoutforsimplicity.TheequationsshowthattherelationbetweenUandIissomewhatdifferentfromtherelationforresistorsasdescribedbyOhm'slaw.Asforresistors,thepositivesideofthevoltageacrossthecapacitorisonthesidethecurrententers.Ifweconsideracapacitorinthefrequencydomain,ithassomefunnyproperties.Itturnsoutthatthecapacitorcanbeseenasadevicewithfrequencydependentimpedance.Impedanceisanexpansionoftheresistanceconcept,andanimpedancevalueisactuallyavectorintheimaginaryplane.Weareonlygoingtoconsiderthevectorlength,orimpedancemagnitudehere.Thefrequencydependenceofthecapacitorimpedanceisgivenby



Page 3 of 11

|𝑍"| =

!&'("

. (4)Theessenceofequation4isthatthecapacitorishavinghighimpedanceatlowfrequenciesandlowimpedanceathighfrequencies.Thisissummarizedintable1alongwiththeimpedanceoftheresistor.

Device DC(f=0Hz) Lowfrequency Highfrequency Infinitefrequency|𝑍𝑅| R R R R|𝑍𝐶| ∞ 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 𝑈)$* = 𝐴+(𝑈,-. − 𝑈,-/) (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),itisdefinedas𝐴#0 = 20𝑙𝑜𝑔!1(𝐴).



Page 4 of 11

𝑈)$* = 𝐴+(𝑈,-. − 𝑈,-/)+𝐴"2 6$#$%.$#$&

&7 (6)

whereACMisthecommonmodegain.ACMisusuallyverylow,andcaneasilybelessthan0.01.Sinceweareinterestedinknowinghowmuchthecommonmodevoltageisrejected,weareusuallytalkingabouttheCommonModeRejectionRate(CMRR)whichusuallyisexpressedindecibels,andgivenby 𝐶𝑀𝑅𝑅 = 3'

|3()|= 20𝑙𝑜𝑔!1 6

3'|3()|

7 𝑑𝐵. (7)Asignalonthepowersupplies2mayalsobetransferredtotheoutput.Onceagain,wehavetoexpandtheequationdescribingtheopamp,andget 𝑈)$* = 𝐴+(𝑈,-. − 𝑈,-/)+𝐴"2 6

$#$%.$#$&&

7 + 𝐴56𝑈56 (8)whereAPSisthegainofasignalonthepowersupplyandUPSistheunwantedsignalthatliesontopofthepowersupplyvoltage.Theratioofrejectionisofinterest,andsimilarlytoCMRR,PowerSupplyRejectionRate(PSRR)isgivenby 𝑃𝑆𝑅𝑅 = 3'

|3*+|= 20𝑙𝑜𝑔!1 6

3'|3*+|

7 𝑑𝐵. (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:

• Thevoltagesonbothinputsareidenticalifsomesortoffeedbackisused.• Therewillnotflowanycurrentintotheinputs.

Theserulesofthumbapplyforallcircuitshere.Thetwomostcommonwaytodescribehowacircuitisprocessingasignalareanoutputfunctionorbyatransferfunction.Anoutputfunctionsimplystateswhattheoutputis,anda

2ThesignalistypicallynoiseorinterferenceaddedtothepureandsomewhatidealDCvoltageonthepowersupply.3Lecturersandauthorsjustlovethissentence.Firstofall,it'sfun,andthenit'saquetotheseriousreaderthatthisissmartifshe/he/itwanttolearnthestuff.



Page 5 of 11

transferfunctionmustbemultipliedbyaninputsignal.Forexample,equation10isanoutputfunction,andequation11isantransferfunctionofthesamecircuit. 𝑈)$* = 𝑈,- 61 +

7,7-7 (10)

𝐻 = $./0

$#$= 1 + 7,

7- (11)

4.2 Non-invertingamplifier

Figure2:Non-invertingamplifier.

Parameter Value/Description

Transferfunction 𝑈123𝑈45

= 1 +𝑅6𝑅7

Inputimpedance Veryhigh.Outputimpedance Verylow.

Pros Notinverting.Cons Gainunder1impossible.

Table3:Opampcircuitpropertiesfornon-invertingamplifier.

4.3 Unitygainbuffer

Figure3:Unity-gainbuffer.



= 1 +0∞ = 1

Inputimpedance Veryhigh.Outputimpedance Verylow.

Pros Buffer.Cons Mayaddnoise.

Table4:Opampcircuitpropertiesforunity-gainbufferamplifier.

4.4 Invertingamplifier



Page 6 of 11

Figure4:Invertingamplifier.



= −𝑅7𝑅6

Inputimpedance =R1Outputimpedance Verylow.

Pros Largerangeofgainpossible,evenbelow1.Lowoutputimpedance.

Cons Inverting.Lowinputimpedance.

Table5:Opampcircuitpropertiesforinvertingamplifier.

4.5 Integrator/low-passfilter

Figure5:Integrator/lowpassfilter.


Outputfunction 𝑈123 = −1𝐶6+𝐼45 𝑑𝑡 = −

1𝐶6+𝑈45𝑅6

𝑑𝑡 = −1

𝑅6𝐶6+𝑈45 𝑑𝑡

Inputimpedance =R1Outputimpedance Verylow.

Pros Largerangeofgainpossible.Lowoutputimpedance.


NeedR2orsomeotherformofdischargingofC1topreventbuild-upofcharge.Table6:Opampcircuitpropertiesforanintegrator/low-passfilter.

Asanexercise,lookatthetransferfunctionfortheinvertingamplifier,andthinkofthecapacitorasanimpedancewithvaluesthatvariesaccordingtotable1.Whichfrequenciesareletthrough,andwhicharestopped?

4.6 Derivator/high-passfilter



Page 7 of 11

Figure6:Derivator/high-passfilter.


Outputfunction 𝑈123 = −𝑅6𝐼45 = −𝑅6𝐶6𝑑𝑈𝑑𝑡

Inputimpedance Capacitive.Outputimpedance Verylow.


Cons Inverting.Capacitive/lowinputimpedance.

Table7:Opampcircuitpropertiesforaderivator/high-passfilter.Asanexercise,lookatthetransferfunctionfortheinvertingamplifier,andthinkofthecapacitorasanimpedancewithvaluesthatvariesaccordingtotable1.Whichfrequenciesareletthrough,andwhicharestopped?

4.7 Logarithmicandexponentialamplifier

Figure7:Exponentialandlogarithmicamplifiers.


Outputfunction 𝑈123 = −𝑅6𝐾 exp(𝑈45)

𝑈123 = −𝐾 ln7𝑈45𝑅68

Inputimpedance Low=R1

Outputimpedance Verylow.Pros Coolcircuits.

Lowoutputimpedance.Non-lineareffectsmaybeuseful.

Cons Inverting.Inaccurateandtemperaturedependent.

Lowinputimpedance.Table8:Opampcircuitpropertiesfortworelatednon-linearcircuits.

Sinceitmaybethatareaderdoesnotknoweverythingthereistoknowaboutdiodes,equation12describingthevoltage-currentrelationofadiode:

𝐼+ = 𝐼6 >𝑒/'890 − 1@. (12)



Page 8 of 11

Ifwesimplify,andacceptthattheconstantsarejustoffsetsandannoyingnumbers,wecanreduceequation12tosomethingmoreuseful,asshowninequations13andequation14. 𝐼+ = 𝐾𝑒$' (13) 𝑈+ = 𝐾𝑙𝑛(𝐼+) (14)Thismeansthatwecanimplementthefunctionsln 𝑋and𝑒8 .Andsincewecandoaddition(sesummingamplifierbelow),wecanimplementmultiplicationsince𝑒9: 3.9:0 = 𝐴𝐵.Howcoolisthat?

4.8 Summingamplifier

Figure8:Summingamplifier.


Outputfunction 𝑈123 = −𝑅:9𝑈;𝑅;

Inputimpedance =RnOutputimpedance Verylow.



Table9:Opampcircuitpropertiesforsummingamplifier.Itispossibletoadjustgainofeachinputbychangingthevalueofthecorrespondinginputresistor.

4.9 Differentialamplifier

Figure9:Differentialamplifier.



Page 9 of 11

Parameter Value/DescriptionOutputfunction 𝑈123 =

𝑅7/=𝑅6/>

(𝑈? −𝑈@)

Inputimpedance = 𝑅6/> + 𝑅7/=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 𝐼! =

$&/$%7A

(15)Sincenocurrentsareflowingintothe-inputsofthetwoopamps,weknowthatI1ispassingthroughtheresistorchain,andwecanuseOhm'slawtocalculatethevoltageacrosstheresistorchain 𝑈/& − 𝑈.& = 𝐼!(2𝑅3 + 𝑅;) =

$&/$%7A

(2𝑅3 + 𝑅;) (16)SincethevoltageacrosstheresistorchainisthesameastheoutputofstageI,wecandosomefiddlingonequation16,andweseethatthedifferentialgainofstageIisgivenby 𝐴+(6%=>?,) = 1 + &7B

7A (17)

SincethecommonmodevoltageontheoutputofstageIisthesameastheinput,thestageIcommonmodegainis1.NowthatwehaveanicelyamplifiedinputsignalavailableafterstageI,wewanttogetridofthecommonmodevoltage.ThisisdoneinstageIIbyusingadifferentialamplifierwheregainissetto1(orcloseto1inordertoreducenoiseanderrorandsuch).Notethatthereisapincalled



Page 10 of 11

UREF.Thisistypicallyplacedatsomereferencelevellikeground,butitcanbeanyreferencelevelyouchoose.ThepointisthattheoutputvoltageUOUTisrelativetothis.Theoutputfunctionisthen 𝑈)$* = (𝑈. − 𝑈/) 61 +

&7B7A7 (18)

relativetothepotentialonUREF.Bothstagescombinedwillthen:

• haveverylowinputimpedance.• haveadifferentialgainwhichcanbesetbychangingRG.• haveaverylowcommonmodegain.• 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/Sallen–Key_topology.

5 AmoreaccurateanalysisoftheinvertingopampcircuitTherulesofthumbcanbeabitfrustratingforsomeofyou,sohereisamorethoroughanalysisforonecircuitjusttoshowhowitcanbedone.Similaranalysescanbedoneforallcircuits.Theequationsthatdescribethecircuitare 𝑈8 = 𝑈)$* + 𝐼,B𝑅& = 𝑈)$* +

$#$/$.CD7,.7-

𝑅& (19)and 𝑈)$* = −𝑈8𝐴+ . (20)Byinserting20into19,weget



Page 11 of 11

− $./0

3'= 𝑈)$* + 𝑈,-

7-7,.7-

− 𝑈)$*7-

7,.7- (21)

whichcanbemanipulatedinto $./0

$#$= − 3'7-

3'7,.7,.7-. (22)

IfweassumethatADisverylarge,say∞,thisreducestothefamiliarform $./0

$#$= − 7-

7,. (23)

IfweassumethatADisverylarge,andthattheoutputiswithinreasonablelimits,equation5tellusthattheinputsmustbeveryclosetoeachother,andifADapproaches∞,thepotentialdiferencebetweenUIN+andUIN-approaches0.

Documents

Opamp supplementary