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Introduction:

Thepurposeofthisexperimentwastomeasuretheoutputimpedanceofaninvertingoperational

amplifier.Theoperationalamplifier(orop-amp)isawidelyusedcomponentinelectroniccircuits;oneof

themainreasonsforthisisthatithasextremelylowoutputimpedancewhenusedinanegative

feedbackconfiguration.Havinglowoutputimpedanceallowstheop-amptoachieveextremelyhigh

gains.However,becausetheoutputimpedanceissosmall,itisalsoverydifficulttomeasure.

Figure

1:

generic

inverting

op-

amp

circuit.

Therearetwomainruleswhichgovernthebehaviorofop-amps:theop-ampcurrentrule(OACR)and

theop-ampvoltagerule(OAVR).TheOACRsaysthatthecurrentintoeachop-ampinputterminalis

approximatelyzero(1 =2),andtheOAVRsaysthatthevoltagedifferencebetweenthetwoop-ampinputterminalsisapproximatelyzero(+ = 0).Usingtheserulesonthecaseoftheinvertingamplifierandassumingthat1 (1+ 2)resultsinthefollowingexpression:

= 1 + 11+ 21 + 21

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whereistheopenloopgain,isthevoltagegain,isthecharacteristicoutputimpedanceofthebareop-amp,isthequantitywearelookingfor(theoutputimpedance),and1and2areasdefinedaboveinFigure1.Thisexpressioncanbefoundinmanytextbooksand generallyacceptedas

theequationfortheoutputimpedanceofaninvertingop-amp.However,thisrelationshipcantbeused

tomeasuresinceandarebothunknownsanddifferentforeachop-amp.

Themethodchosenfordeterminingtheoutputimpedanceofanop-ampinvolvedtakingadvantage

ofthefactthatrealop-ampsarenotideal(seeFig.2).Theideaisthatbecausetheop-ampisnon-ideal

andhasanoutputimpedance,Kirchhoffsvoltagelawrequiresthatiftheloadresistanceischanged,the

voltagedropacrossmustalsochange,therebychangingtheoutputvoltage.Thenbymeasuringtheoutputvoltagesformultiplevaluesof,theoutputimpedancecanbedetermined.

Figure2:theoreticalrepresentationofarealop-ampwithidealoutputvoltageandrealoutputvoltage.

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Procedure:

Thefirstpartoftheprojectinvolvedthedesignandconstructionofanop-ampcircuitfromwhichthe

outputimpedanceoftheop-ampcouldthenbedetermined.Thecircuit(showninFig.3)usedtwoop-

amps:thefirstonewithagainofapproximately1

1000andthesecondhavingagainofabout -1000.The

purposeofthefirstop-ampwasmerelytodecreasetheinputvoltageforthesecondop-amptokeep

theoutputofthesecondop-ampwithinits12vrange.A100 resistorinserieswitha1M variable

resistorwasusedfortheloadonthesecondop-amp.Theinputvoltageforthefirstop-ampwassetto

5VDCandmeasurementsweremadeoftheloadresistanceandtheoutputvoltagefromthesecondop-

ampacrosstherangeofthe1M variableresistor.Thisinitialtestingcircuitwas alsousedwithagainof

-100, -20,and -10.However,thevariationsofthesmallergainop-ampsoutputvoltageswerenearor

beyondthelimitofthedigitalmulti-meterbeingused,soonlytheresultsfromthe -100and -1000gain

configurationswerekept.

Theequationsfortheinitialtestingcircuitweremanipulatedwiththegoalofmakingtheslope

of

the

equation

for

a

line.

This

resulted

in

the

equation = +

wherethevariablesareasinFig.2withastheidealoutputvoltageandastherealoutputvoltage.Usingthisequationasaguide,aplotwasmadeoftheoutputvoltageversusnegativeonetimes

theoutputvoltage/loadresistanceforthetwogains.Thenbyusingtheleastsquaresmethod,the

equationforatrendlinewasobtainedalongwiththevalueoftheoutputimpedance.

Whileusingtheinitialtestingcircuit(Fig.3),itwasnoticedthatthevirtualgroundofthesecondop-

ampvarieddependingonthesetupofthecircuitandwasinfactnot0V.Dr.Braunsteinsuggestedtrying

tousethisfacttofindanewmethodtomeasure.Byconsideringtherealop-amptobea

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combinationofanidealop-ampwithabuiltinresistorasdescribedpreviouslyandsolvingtheresultingequationsgave

= (+ ( )

21)

whereisthevoltageoftheinvertinginputoftheop-amp.Tomakeuseofthisequation,anewcircuitwasbuilt(Fig.4).Unlikethecircuitforthepreviousattempt,theinputvoltagewasa44.7kHzsinewave

fromafunctiongeneratorattenuatedbyavoltagedividerandonlyoneop-ampwasused.Also,this

timetheloadresistancewaskeptataconstant366k andthegainoftheop-ampcircuitwas -2.Forthis

circuit,thefrequencyoftheinputsignal,invertinginputvoltage(

),outputvoltage(

),input

voltage(intheequationand inthediagram),andtheloadresistance()weremeasured.Thevariableforthesemeasurementswastheinputvoltage.

Basedonthelackofclearresultsfromthepreviousattemptsandthehighnoise-to-signalratio

involved,Dr.Braunsteinsuggestedusingalock-inamplifier.Dr.Braunsteinhadpreviouslydesignedand

constructedacircuitwithadataanalysisLabVIEWprogramwhichheprovidedtoreducesetuptime.The

diagramofthecircuitisFig.5andthelayoutoftheLabVIEWprogramcanbefoundinfigures6 -8.The

circuitusedtwoop-amps:oneusedasthecontrolop-amp(ofwhichtheoutputimpedancewastobe

measured),andanothertoproduceasinewaveofcontrolledamplitudeandDCoffsettouseasinput

forthecontrolop-amp.Thecircuitalsousedaresistornetworkfortheloadofthecontrolop-ampwhich

wasusedtocalibratetheoutputsignalfromthelock-inamplifier.TheinputdatafortheLabVIEW

programwastheoutputsignalfromthelock-inamplifierfortwodifferentconfigurations:one

configurationformeasuringtheoutputvoltageofthecontrolop-amp(),andtheotherformeasuringthecalibrationvoltagefromtheresistornetwork().Theseconfigurationshadtobedoneseparatelyforeachofthedifferentgains.TheLabVIEWprogramalsorequiredthevaluesofthe

calibrationnetworkresistorsandtheiruncertaintiesasinput.TheoutputfromtheLabVIEWprogram

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wasthevalueoftheoutputimpedanceofthecontrolop-ampandtheuncertaintyoftheoutput

impedance.Theprogramwasrunformultiplecases,witheachcasehavingadifferentgainforthe

controlop-ampcircuit.Thegainofthecontrolop-ampcircuitwasadjustedbychangingoutthe

feedbackresistor().

Foreachcase,morethan100datapointswerecollectedintheLabVIEWprogramforboththe

op-ampsignalandthecalibrationsignal.Uponcollectingseveralsetsofdatawiththismethod,Dr.

Braunsteinnoticedthatbycollecting100datapointstheerrorofthemeasurementwasincreaseddue

totemperaturedrift.Hesuggestedtakingonly10datapointspersetalongwithadjustingtheoffset

voltagetozerotheoutputbeforeeachrun.Thismethodwasadoptedandeachcasewasredone.

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Figure3:Circuitdiagramforthefirstattempt:

1=100,2=100k,3=100,=1M (variable),=5VDC.

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Figure4:circuitforsecondattempt

1=22k,2=22,3=10,1and2=1M (variable),=.5VAC,=100Hz.=100,=218

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Figure5:diagramoffinalmeasurementcircuit.

Resistorvalues:

R1=100

,Ra=1M

,Rb=1k

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Figure6:LabVIEWblockdiagram

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Figure7:LabVIEWblockdiagramalternatecases

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Figure8:LabVIEWfrontpanel

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Results:

Figure9:plotsfromthefirstattempt(DCinput)

= -2.10.1 withagainof -994and= -0.740.07 withagainof -99.9

y= -2.1356x+4.5286

R=0.9553

4.51

4.52

4.53

4.54

4.55

4.56

4.57

4.58

-0.021 -0.018 -0.015 -0.012 -0.009 -0.006 -0.003 0

Vout

-Vout/RL

OutputImpedanceofanInvertingOperationalAmplifier

with

a

Gain

of -

994

y= -0.7514x+0.4546

R=0.9526

0.4542

0.4545

0.4548

0.4551

0.4554

0.4557

0.456

0.4563

0.4566

-0.0021 -0.0018 -0.0015 -0.0012 -0.0009 -0.0006 -0.0003 0

Vout

-Vout/RL

OutputImpedanceofanInvertingOperationalAmplifier

withaGainof -99.9

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Theuncertaintiesfortheresultsofthefirstattemptarenottrueuncertaintiesbutmerelythe

uncertaintybasedontheR2value.Thisisbecauseonlyonemeasurementwasrecordedateachpoint.

Theresultsfromthesecondattemptwerenotkeptsincetheypredictedanegativevaluefor

on

theorderof105.

Figure10:plotfromthethirdattempt,inputsignalfrequency=13.10.2Hz

Theuncertaintiesinthisplotwerefoundusingthestandardmethodforpropagationofuncertainty

andtheequationsforthegainandoutputimpedanceofanop-amp.

y=3E-05x+0.0067

R=0.9992

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0 500 1000 1500 2000 2500

Zout()

Gain

OutputImpedanceofanInvertingOperationalAmplifier

vs.Gain

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Analysis:

Measuringtheoutputimpedanceofaninvertingop-ampwasmoredifficultthanoriginally

anticipated.Thefirsttwoattemptstogetavalueforoutputimpedanceresultedinvaluesthatweretoo

largeinmagnitudeandofthewrongsigncomparedtothetheoreticalvalue,howeverthefirstattempt

wasmuchclosertotheexpectedvalue.Theuseofthelock-inamplifiergreatlyimprovedtheaccuracyof

themeasurementsandtheuseoftheLabVIEWdatacollectionandanalysisprogrammademaking

measurementsquickandpainlessaftertheoffsetvoltageontheop-ampwasadjusted.Also,thelow

uncertaintiesintheresultsfromthethirdattemptarereassuringintheaccuracyofthemeasurements.

Theproblemwiththesecondattemptislikelytobeaninvalidassumptionorerrorinthederivation

oftheequationusedtocalculatethevaluefortheoutputimpedance.Theresultsfromthiscaseare

likelynotmeaningfulforthepurposeofthisexperiment.However,theresultsfromthefirstcasearea

bitmoreconcerning,sincethemethodusedtodeterminetheoutputimpedanceinthiscasewasvery

similartothemethodusedinthethirdcase.Iftheequationusedtofindthevalueofoutputimpedance

issolvedfor

,theresultingequationis

= 1 .

Notethattheonlytimethatisnegativeiswhentheidealoutputvoltage()islessthantherealoutputvoltage(),andthisiswhattheplotsshow.Theidealoutputvoltageisthelasttermintheequationforthetrendline,andtakingalookatthevalueforeachofthefirstattemptplotsshowsthat

themeasuredvoltageswerehigherthantheideal.Thisdoesnothelpresolvetheproblem,butitdoes

appeartheequationisworking.Unfortunately,theinputvoltagetothesecondop-ampcircuitofthe

firstattemptwasnotrecorded,sotheactualidealoutputvoltagecannotbecalculatedtoverifythe

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valuesfromtheplots.Itwouldbeworthrepeatingtheexperimenttoseeiftheactualidealoutput

voltageandthevaluefromtheplotsagree.

Anothermethodofverifyingtheresultswouldbetocheckthecharacteristicoutputimpedanceofthe

bareop-amparecomparethistothevaluearrivedatbyusingtheequation

=

ThisequationwasusedonthedataforthethirdattemptandtheresultingplotcanisFig.11.

Figure11:bare(nofeedback)op-ampcharacteristicoutputimpedance

Overall,thegoaloftheexperiment,tomeasuretheoutputimpedanceofaninvertingop-amp,was

accomplished.However,therearestillmanyvariables,suchasthefrequencyoftheinputsignal,whose

connectiontotheoutputimpedancewasnotinvestigated.Muchexperimentationremainstobedone

onthistopic.

0

2

4

6

8

10

12

14

0 500 1000 1500 2000 2500BareOp-AmpImpedance()

Gain

CharacteristicOutputImpedanceofanInverting

OperationalAmplifiervs.GainAssuminganOpen-LoopGainof210^5

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References

Floyd, Thomas L. "Ch. 12: Operational Amplifiers." Electronic Devices. 5th ed.

Upper Saddle River, NJ: Prentice Hall, 1999. N. pag. Print.

Simpson, Robert E. Introductory Electronics for Scientists and Engineers. 2nd

ed. Boston: Allyn and Bacon, 1987. Print.

Tocci, Ronald J. "Ch. 5: The Operational Amplifier." Fundamentals of Pulse and

Digital Circuits. 3rd ed. Columbus, OH: Merrill, 1983. N. pag. Print.