filters - university of colorado boulder
TRANSCRIPT
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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.