filters - university of colorado boulder

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FILTERSLAB3INTRO:MEASURINGTHEFREQUENCYDEPENDENCEOFLOWPASS,HIGHPASS,ANDBANDPASSFILTERS.

GOALS

Inthislab,youwillcharacterizethefrequencydependenceofthreepassivefilters.Youwillgainmoreexperiencemodelingboththeresponseofthefiltersandhowyourmeasurementtoolsaffectyourmeasurements.

Proficiencywithnewequipment:

o Oscilloscopeprobeo Capacitorsandinductors

§ Identifypolarizedcapacitorsanddeterminethecorrectinstallationorientation§ MeasurecapacitanceandinductancewithanLCRmeter.

Modelingthephysicalsystem:

o Developmathematicalmodelsoffrequencydependentvoltagedividerso Determinethelimitationsofthesemodelsandrangeofapplicability

Modelingmeasurementsystems:

o Refinethemodelofscopemeasurementtooltoincludecapacitanceofthecoaxcableo Refinethemeasurementsystemtoreducetheeffectofthecapacitanceofacoaxcable

DEFINITIONSScopeprobe–atestprobeusedtoincreasetheresistiveimpedanceandlowerthecapacitiveimpedancecomparedtoasimplecoaxcableprobe.Passband–therangeoffrequenciesthatcanpassthroughafilterwithoutbeingattenuated.Attenuationband-therangeoffrequenciesthatafilterattenuatesthesignal.Cutofffrequency(orcornerfrequencyor3dBfrequency),fc–thefrequencyboundarybetweenapassbandandanattenuationband.fcisthefrequencyatthehalf-powerpointor3dBpoint,wherethepowertransmittedishalfthemaximumpowertransmittedinthepassband.Theoutputvoltageamplitudeatf=fcis1 / 2 =70.7%ofthemaximumamplitude.

Lowpassfilter–afilterthatpasseslow-frequencysignalsandattenuates(reducestheamplitudeof)signalswithfrequencieshigherthanthecutofffrequency

Highpassfilter–afilterthatpasseshigh-frequencysignalsandattenuates(reducestheamplitudeof)signalswithfrequencieslowerthanthecutofffrequency

Bandpassfilter–adevicethatpassesfrequencieswithinacertainrangeandrejects(attenuates)frequenciesoutsidethatrange.

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BandpassfilterBandwidth–therangeoffrequenciesbetweentheupper(f+)andlower(f–)halfpower(3dB)points:

bandwidth ∆f=f+–f–.

APPLICATIONSOFFILTERS Afrequentprobleminphysicalexperimentsistodetectanelectronicsignalwhenitishiddeninabackgroundofnoiseandunwantedsignals.Thesignalofinterestmaybeataparticularfrequency,asinanNMRexperiment,oritmaybeanelectricalpulse,asfromanuclearparticledetector.Thebackgroundgenerallycontainsthermalnoisefromthetransducerandamplifier,60Hzpowerpickup,transientsfrommachinery,radiationfromradioandTVstations,cellphoneradiation,andsoforth.Thepurposeoffilteringistoenhancethesignalofinterestbyrecognizingitscharacteristictimedependenceandtoreducetheunwantedbackgroundtothelowestpossiblelevel.Aradiodoesthiswhenyoutunetoaparticularstation,usingaresonantcircuittorecognizethecharacteristicfrequency.Thesignalyouwantmaybelessthan10-6ofthetotalradiationpoweratyourantenna,yetyougetahighqualitysignalfromtheselectedstation. Manyexperimentsrequirespecificfiltersdesignedsothatthesignalfromthephenomenonofinterestliesinthepass-bandofthefilter,whiletheattenuationbandsarechosentosuppressthebackgroundandnoise. Thisexperimentintroducesyoutothefilteringpropertiesofsomewidelyusedbutsimplecircuits,employingonlyaresistorandcapacitorforhigh-andlow-passfiltersandanLCRcircuitforband-pass.

FILTERBASICSRCLow-andHigh-passfiltersTheresponseofRClow-passandhigh-passfilterstosinewavesisdiscussedinFCSections3.9&3.10.The3dBfrequencyis

fc =1

2πRC,

wherefcisthe3dBorhalf-powerpoint.

Theresponseofthefilterstoasquarewaveinthetimedomainisalsointeresting.

ParallelLCRBand-passfiltersSeeFCSection3.12(H&HSection1.22).TheresonantfrequencyandQaregivenby

f0 =1

2π LCQ =ω0RC =

f0Δf

whereω0=2πf0.Theresonantfrequency,f0,isthecenterfrequencyofthepassband,andtheQisequaltotheratioofthecenterfrequencytothebandwidth∆f.(ThesedefinitionsareexactlytrueonlyifQ>>1). ForaresonantLCRcircuitthecharacteristicimpedance,Z0,isthemagnitudeoftheimpedanceoftheinductororthecapacitorattheresonantfrequency:

CL

CLZ ===

000

1ω

ω

USEFULREADINGS1. FCSections3.4–3.18and10.1–10.62. H&HChapter1,especiallysections1.13-1.24.YouwillmakefrequentuseofthelasttopicinSection1.18,

“VoltageDividersGeneralized.”AppendixAonoscilloscopeprobes.

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LABPREPACTIVITIES

AnswerthefollowingquestionsusingMathematica.SavethecompletenotebookasapdfandturnitintoCanvasbymidnightthedaybeforeyourlabsectionmeets.Bringanelectroniccopyofyournotebooktolab,preferablyonyourownlaptop.Youwilluseittoplotyourdataduringthelabsession.

Question1

Low-andHigh-passfiltersa. DefinefunctionsinMathematicatocalculatethecut-offor3dBfrequency,fc,forthelow-andhigh-

passfiltersinFigure1(a)and(b).Theinputparameterstothisfunctionshouldbetheresistanceandcapacitanceofyourcircuit.Evaluatethefunctionsusingthenominalvaluesshownintheschematic.Duringthelab,youcaninputtheexactvaluesofyourcomponentsandthusquicklypredictthe3dByouexpectforyourcircuit.

b. CreatetwoBodeplots(oneforeachfilter)ofthefrequencyresponseofthelow-(1a)andhigh-pass(1b)filtersinFigure1.ABodeplotisalog-logplotof(Vout/Vin)versusfrequency.SeeH&HFig.4.31foranexample.Makesuretoincludealargeenoughrangeinfrequencytoseeboththepassandattenuationbands.HINT:DetailsaboutmakingplotsprettyareincludedinLabSkillActivity#2.

c. Duringthelabsection,youwillenteryourmeasurementsintoyourMathematicanotebookandplotthemwithyourmodelpredictions.Toprepareforthis,createalistof“fakedata”andplotitonyourBodeplots.Thiswillallowyoutocompareyourmodelandmeasurementsinrealtimeavoidinglosttimetakinglotsofdatawhensomethingiswrongwithyourcircuit.Thepointofthispartisjusttohaveyoucreateworkingcodetoenteralistofdataandplotitalongwiththefunction.Thenumericalvaluesofthefakedateareunimportant.HINT:ThereisahelpfulguideonourwebsiteundertheHINTSTabtitled“PlottingdataandtheorytogetherinMathematica.”

Question2

Band-passFiltersa. DefinefunctionsinMathematicatocalculatetheresonantfrequencyf0,thecharacteristic

impedanceZ0,andthequalityfactorQfortheband-passfilterinFigure1(c).Evaluatethefunctionsusingthenominalvaluesshownintheschematic.

b. CreateaBodeplotshowingthepredictedgain(|Vout/Vin|)versusfrequencyoftheband-passfilter.Makesuretoincludealargeenoughrangeinfrequencytoseeboththepassandattenuationbands.

c. Createalistof“fakedata”andplotitonyourBodeplots.Thepointofthispartisjusttohaveyoucreateworkingcodetoenteralistofdataandplotitalongwiththefunction.Thenumericalvaluesofthefakedateareunimportant.

Question3 Labactivitiesa. Readthroughallof the labstepsand identify thestep (or sub-step) thatyou thinkwillbe the

mostchallenging.b. Listatleastonequestionyouhaveaboutthelabactivity.

Figure1Filters.(a)low-pass,(b)high-pass,and(c)band-pass

(c)

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SETTINGUPTHECIRCUITSANDPREDICTINGTHEBEHAVIOR

Figure2.GeneralVoltageDividers.(a)resistivedivider,(b)low-passfilter,(c)high-passfilter,and(d)band-passfilter.Step1 BuildingtheCircuits

a. GatherallthecomponentstobeabletobuildthefourcircuitsshowninFig.2Ifyou

cannotfindcomponentsinstockwiththespecifiedvalues,takethenearestinvaluethatyoucanfind,within30%ifpossible.

o Resistivedivider:R1=10kΩ,R2=6.8kΩo Low-passfilter:R=10kΩ,C=1000pFo High-passfilter:R=10kΩ,C=1000pFo Band-passfilter:R=10kΩ,C=.01µF,L=10mH

b. Measureallcomponentsbeforeplacingthemintothecircuit.Recordthevaluesinyourlabbook.Drawdiagramsofallthecircuits.Makesuretousethesamelabelsonthediagramsandforthevaluesofthecomponents.

c. Buildallfourcircuitsonyourproto-board(makesuretheyareallseparate)Step2 UsetheMathematicamodelstopredictthebehaviorofthefilters.

a. Calculatetheexpectedtransferfunctionofthedivider.b. Calculatetheexpectedvaluesofthecut-offfrequenciesforthehigh-andlow-passfilters

usingtheactualcomponentvalues.c. Calculatetheexpectedresonantfrequencyf0andqualityfactorQfortheband-passfilter

usingtheactualcomponentvalues.HINT:Youshouldhavealreadydonethesecalculationsinyourlabprepnotebook.Justentertheexactvaluesofyourcomponents.

Step3 UsetheMathematicamodelstoplottheexpectedthebehaviorofthefilters.

a. Plotyourmathematicalmodelsofallthreefiltercircuits(threeindependentplots)usingyouractualcomponentvalues.Thefrequencyrangeshouldcoveratleastf=10-3fc(orf0)

tof=103fc(orf0)toshowthefullbehavior.b. Afterthe lab iscompletedandyouhaveyourmeasurementsontheseplotsaswell,you

willprintofftheplotsandtapethemintoyourlabbook.Makesuretoleaveroominyourlabbookfortheplots.

HINT:Youshouldhavealreadymade theseplots inyour labprepnotebook. Justenter theexactvaluesofyourcomponents.

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SETTINGUPTESTANDMEASUREMENTEQUIPMENT

Step4

Preparetotestthecircuits a. ConnectthecircuitboardtothefunctiongeneratorandtheoscilloscopeasshowninFig.

3.Itisalwayshelpfultodisplayboththeinputvoltageaswellastheoutputvoltageonthescopeatthesametime.

b. Testyoursetupbycreatinga1kHzsinewaveat1voltp-pusingthefunctiongeneratorandconfirmthewaveformfrequencyandamplitudebymeasuringthesignalonthescope.TriggerthescopeontheSync.outputofthefunctiongenerator.

Figure3.TestandMeasurementSet-up.Channel1will“pickoff”thefunctiongeneratorsignalonitswaytothecircuitboard.YoucandothisusingaBNC“T”connectormounteddirectlyontheoscilloscopeinput.

RESISTIVEVOLTAGEDIVIDER

Step5

a. Measurethefrequencydependenceofthevoltagedividera. Connectthesignalfromthefunctiongeneratortotheinputofthevoltagedivider.

Measurethetransferfunction(=Vout/Vin)overalargerangeinfrequency(100Hzto1MHzinapproximatelydecade(X10)steps).Recordyourmeasurementsinyourlabbook.

b. Atlowfrequencies(1kHz),compareyourmeasuredvalueofthetransferfunctiontowhatyourmodelpredictedusingyouractualcomponentvalues.Doesyourmeasurementagreewithyourprediction?Explicitlyrecordwhatcriteriayouusedtodeterminewhetherornotthemodelandmeasurementsagree.Ifthereisahighfrequencycut-off(3dBfrequency),measureitsvalue(wherethevoltageisreducedto0.7ofthelowfrequencyvalue).Recordthecut-offfrequency.

c. Holdontotheresistorsfromthisvoltagedivider.You’llbereturningtothiscircuitattheendofthelab.

LOWANDHIGHPASSFILTERS

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Step6

b. Measurethefrequencydependenceofthefilters.a. Connectthesignalfromthefunctiongeneratortotheinputofthelow-passfilter.

Measurethetransferfunction(=Vout/Vin)overalargerangeinfrequency(100Hzto1MHz)inatleastonestepperdecade,withseveralextrastepswithinthedecadearoundyourexpectedcutofffrequency.Recordyourmeasurementsinyourlabbook.Determineandrecordthecut-offfrequencyforthelow-passfilter.Compareyourmeasuredhalfpowerpoint(Vout/Vin=0.707)withthecut-offfrequencycomputedfromtheactualcomponentvaluesused.Includeyourcomparisoninyourlabbook.Thendothesameforthehigh-passfilter.

b. TestthepredictedfrequencyresponsebyplottingyourdatapointsdirectlyonyourtwoBodeplots.Doesthemodelagreewithyourdata?Explicitlyrecordwhatcriteriayouusedtodeterminewhetherornotthemodelandmeasurementsagree.

BANDPASSFILTER

Step7

a. Measurethefrequencydependenceoftheband-passfilter.a. Onresonance,Voutwillbeamaximumandthephaseshiftbetweentheinputandoutput

waveformswillbezero.Findtheresonantfrequencyfobothways.Adjustthefrequencysothat(1)theoutputhasmaximumamplitude(Vout/Vin=max),(2)thereiszerophasedifferencebetweenVoutandVin.Recordbothmeasurements.Whichmethodismoreprecise?

b. TheLCRmetermeasuredtheinductanceofyourinductorataparticularfrequency.Yourinductor’sinductancechangesslightlyatdifferentfrequencies.UseyourmeasurementsoffotogetamoreaccuratemeasureofLonresonancebydoingthefollowing.Comparethemeasuredfowiththeexpectedvalue .Refinethemodeloftheinductorby

calculatingacorrectedvalueofLfromthemeasuredvaluesoffoandC,andusethisrefinedvaluebelow.ComparethisvalueofLtothevalueyoumeasureusingtheLCRmeterinthelab.

c. DeterminethequalityfactorQbymeasuringthefrequenciesatthetwohalf-powerpointsf+andf–aboveandbelowtheresonanceatfo.Recordyourmeasurements.Recallthat

Q =Resonant frequency f 0

Bandwidth ΔfwhereΔf=f+–f–.

HINT:Thehalf-powerpointsarewhereVout=𝑽𝒐𝒖𝒕(𝒎𝒂𝒙)/ 𝟐not𝑽𝒊𝒏/ 𝟐.d. ComparethemeasuredvalueofQwiththatpredictedfrommeasurementsofcomponent

values.Dotheyagree?e. ItiscommoninallelectricalcircuitstofindQvaluesthataresomewhatlowerthanvalues

youpredictusingmeasuredcomponentvalues.Thisisduetoadditionallossesinthecircuit,inthiscaselossesareintheinductor.Measuretheinductor’s“equivalentseriesresistance”(ESR)usingaDMM.Youcanrefineyourmodelbyincludingthisresistanceinyourcircuit.Drawaschematicthatincludesthisresistor.WhatisthepredictedQwhenyouincludethisresistanceinyourmodel?HINT:Seehintssectionbelow.DoesthishavebetteragreementwithyourmeasuredQ?

f. Measurethetransferfunction(=Vout/Vin)asfunctionoffrequency.Useyourmodelpredictiontodecidewhatvaluesoffrequencytotakedata.Plotyourmeasurementsonthesamegraphasyourmodel.Note,yourtransferfunctiondidnotincludetherefinedvalueofQ.

Step8

b. Explorehigh-frequencybehaviorwiththescopeprobe.a. Now,returntothevoltagedividerfromStep5.Measurethetransferfunctionofthe

voltagedividerinafewstepsbetween1MHzandthemaximumfrequencyofyour

1 / 2π LC( )

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functiongenerator.NotewhetherVinchangesaswellasVout.b. Avoltagedividercontainingonlyresistorsshouldnothaveanyfrequencydependence.

However,acoaxcablehasacapacitanceof~25pF/foot.Youcouldrefineyourmodeltoincludethiscapacitance.However,inthiscase,refineyourphysicalsysteminsteadbyusingascopeprobe(seedefinitions)inplaceofthecoaxcabletoreducethecapacitanceofthemeasurementprobe.Repeatthemeasurements(andrecordtheminyourlabbook)ofQuestion5part(a)usingthe10xprobetomeasuretheoutputofthecircuit.Thisscopemodeldoesnotautomaticallydetectthepresenceofourscopeprobes,soyouwillhavetogointothescopesettingsandchangetheprobesettingto“10x.”Besuretoputitbackto“1x”whenyouremovethescopeprobe.

c. Doesyouoriginalmodelofjusttworesistorsbetterpredictthebehaviorofthecircuitwhenyouusea10Xprobe?

HINTS:REFINEDLCRBAND-PASSFILTERMODEL

Inductorsoftenhaveconsiderableresistanceastheyarejustwireswrappedaroundaferritecore.Onecanincludethisresistanceasaresistorinserieswiththeinductor.TherefinedmodeloftheQofthissystemis

𝑄0123415 =𝑅𝑅8

𝑅 𝐶𝐿 +

1𝑅8

𝐿𝐶

whereRListheequivalentseriesresistanceoftheinductor.Thisisnon-trivialtoderive.