Frequency Response and Filters
Objectives:
Thisexperimentprovidespracticalexperienceswithfrequencyresponsesofanalogfilters.Filterswillbe constructedandgraphsofgainmagnitudeandphaseversusfrequencywillbecreatedbasedondata
collectedinthelab.
Pre Lab:
1.ReadandunderstandHambleyChapter6 Sections6.1,6.2and6.5coveringlow passfiltersandhighpassfilters.
2.Thegain(alsocalledthetransferfunction)ofafilteristheratioofthephasoroutput voltagetothephasorinputvoltage.Usingphasoranalysis,thetransferfunctionofa first-orderRCLowPassfilterasafunctionoffrequencyisgivenbytheequations below.
3.1st
orderlowpasstransferfunction:
H(ω)=Vout =
Vin
1/jωC=
R+1/jωC1+
1
jωRC
4.Evaluatethetransferfunctioninitem3aboveusingthenominalvaluesofyour resistorandcapacitorcomponentsasshowninFigure1todeterminethe magnitudeandphaseat frequenciesof50Hz,200Hz,1kHz,and5kHz.These valueswillbeusedtodetermineifyourcircuitiscorrectinthelab.Showall calculationsinyourlabreport.
5.UsethevoltagedividerruletoperformphasoranalysisontheHighPassfilter circuitshowninFigure2.Noticethattheinputsignalismeasuredby Channel1 (Vin)andtheoutputvoltage(Ch2)ismeasuredacrossthe1KΩresistor. The transferfunctionwillbedifferentfromthatshowninitem3.
6.Evaluatethetransferfunctionyouderivedinitem5usingthenominalvaluesofthe resistorandcapacitortodeterminetheoreticalgainandphaseat frequenciesof
500Hz,1kHz,5kHz,and50kHz
7.MarkthefourpointscalculatedinthePrelabs(forthelowpassandhighpass filters)onthegraphsforPartsAandB.
PartA:LowPassFilterFrequencyResponse
Procedure:
1. Measurethe1kΩresistorusingtheFluke45asanOhmmeter.Measurethe1μFcapacitor usingtheuniversalbridgeontheinstructor’stable.Enterthesevaluesonthedatasheet.
2. ConstructthelowpassfiltercircuitillustratedinFigure1.
3. Setthefunctiongeneratoroutputtoa sinusoidwithamplitudeof1Vpeak-to-peakandan initialfrequencyof50Hz.
4. Theoscilloscopedisplaymayhavea naturalcoarseness,whichcanbereducedby selecting theAveragefunction.Trytouseasfewsamplesaspossibletoavoidlongdelayswhile movingfromonedisplayto another.
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NOTE: If thedisplayed wave is extremelyjumpy, the problem is generallyfaulty banana leads or anungroundedprobe. If you cannot remedythe problem, ask forassistance.
Figure1:TheLowPassFilter Circuit
5. PushtheAUTOSCALEbuttontodisplaybothchannels.Repositionsothatthe0Visat the midlineonthebothchannelsandthewaveformsappeartooverlap.Adjustthe
Volts/divisionto500mV/divisionforbothchannelsandthesec/divisiontodisplaymorethen onecompleteperiod.Athighfrequencies,theamplitudeofthefilteredsignalmaybetoo small tobedetectedby theAutoscalefeature.YoumayneedtosettheVolts/divisionscale manuallyinordertoseethesignal.
6. PresstheMEASUREbuttontodeterminetheVoltagepk-pkamplitudesforbothchannels, andrecordinthedatasheetforthefrequencyof50Hz.Atleast onefullperiodofthe waveformmustbeinviewtofortheoscilloscopemeasurefunctionstobeaccurate.
7. PresstheCURSORbuttonandadjustthecursorstomeasurethechangeintimebetween peaks.RecordtheΔtinthedatasheetforthefrequencyof50Hz.
8. CalculatetheGainmagnitudeandPhaseusingthefollowingformulasandenterinTable1of thedatasheet.
|Gain|=VOut
VIn
Phase=Frequencyx Δtx360
9. Comparewithwhatyouwouldexpectforthiscircuitbasedontheoreticalcalculationsfrom theprelab.
10.Repeatsteps6through8 to makemeasurementsforall frequenciesshowninTable1and recordthesevaluesinTable1.AtveryhighfrequenciesAUTOSCALEmaynotwork becausethefilteredsignalistoosmall.Usemanualadjustmentsofthevolts/divisionto revealthesignal.
11.GraphtheGainandPhaseonthesemilogpaperprovided.Doyourresultsagreewithwhat youwouldexpectfora lowpassfilter?Explainwhatis meantby a lowpassfilter.
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PartB:HighPassFilterFrequencyResponse
Procedure:
1. Measurethe1-kΩresistorusingtheFluke45asanOhmmeter.Measurethe0.1-μFcapacitorusing theuniversalbridgeontheinstructor’stable.Enterthesevaluesonthedatasheet.
2. ConstructthehighpassfiltercircuitillustratedinFigure2.
3. Usinganymethod,setthewaveformgeneratoroutputtoa sinusoidwithanamplitudeof1V
peak-to-peakandaninitialfrequencyof10kHz.
4. Theoscilloscopedisplaymayhavea naturalcoarseness,whichcanbereducedby selectingthe
Averagefunction.Trytouseasfewsamplesaspossibletoavoidlongdelayswhilemovingfrom
NOTE: If thedisplayed wave is extremelyjumpy, the problem is generallyfaulty banana leads or anungroundedprobe. If you cannot remedythe problem, ask forassistance.
onedisplayto another.
Figure2:TheHighPassFilter Circuit
5. PushtheAUTOSCALEbuttonto displaybothchannels.Repositionsothatthe0Visat the midlineonthebothchannelsandthewaveformsappeartooverlap.AdjusttheVolts/divisionto
500mV/divisionforbothchannelsandthesec/divisiontodisplaymorethanonecompleteperiod. Atlowfrequencies,theamplitudeofthefilteredsignalmaybetoosmalltobedetectedby the Autoscalefeature.YoumayneedtosettheVolts/divisionscalemanuallyinordertoseethe signal.
6.
7. PresstheMEASUREbuttontodeterminetheVoltagepk-pkamplitudesforbothchannels,and recordinthedatasheetforthefrequencyof10kHz.Atleast onefullperiodofthewaveform mustbeinviewtofortheoscilloscopemeasurefunctionstobeaccurate.
8. PresstheCURSORbuttonandadjustthecursorsto measurethechangeintimebetweenpeaks.
RecordtheΔtinthedatasheetforthefrequencyof10kHz.
9. CalculatetheGainmagnitudeandPhaseusingthefollowingformulasandenterinTable2ofthe datasheet.
Gain=VOut
VIn
Phase=Frequencyx Δtx360
10.Repeatsteps6through8 to makemeasurementsforall frequenciesshowninTable2andrecord thesevaluesin Table2.Atlowfrequencies,theamplitudeofthefilteredsignalmaybetoosmall
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tobedetectedby theAutoscalefeature.YoumayneedtosettheVolts/divisionscalemanuallyin ordertoseethesignal.
11. GraphtheGainandPhaseonthesemilogpaperprovided.Doyourresultsagreewithwhatyou wouldexpectfora highpassfilter?Explainwhatis meantby a highpassfilter.
PartC:UsingFiltersToIsolateFrequenciesInAnAMSignal
Procedure:
1. Forthispart,youwillusethefunctiongeneratortocreateanamplitudemodulated(AM)signalto simulatethesoundofanenginerunningat6000RPMwithanassociatedhigh-frequencyvibration.
TheAMsignalisgivenby
vAM(t)=A[1+μcos(ωmt)]cos(ωct)
Expandingtheequationandusingthetrigonometricidentityforthesumofcosines,wehave:
v(t)=Acos(ωt)+Aμcos[(ω
+ω)t]+Aμcos[(ω
−ω)t]
AMc2
mc2mc
inwhichAisthecarrieramplitudewhichwewillset at 1V,μ is themodulationindexwhichwe chooseas0.3,cos(ωmt)isthemodulatingwaveformwhichrepresentsa high-frequency(5kHz) vibration,andcos(ωct)isthecarrierwaveformthatresultsfromenginerotation.Foranengine speedof6000rpm,thecorrespondingcarrierfrequencyis 100Hz.
NoticethattheAMsignalhasthreecomponentswithfrequenciesof100Hz,4900Hzand5100
Hz.The100Hzcomponentresultsfromenginerotationwhilethe4900and5100Hz componentsresultfromvibration.Wewillusea lowpassfiltertopassthe100Hzcomponent andreject thevibrationcomponents.Thenwewillusea highpassfiltertopassthevibration componentsandrejectthe100Hzcomponent.
2. SetthewaveformgeneratortosinusoidalmodeasshowninFigure3(A).
SwitchtotheAMmodebypressingtheMODbutton(Fig3B)andselectingAMusingthesoft keysshownin(Fig3C).
Toreturntothemain menupresstheMODbutton(Fig3B).
SelectFREQusingthesoftkeysshownin(Fig3C)andentera frequencyof100Hzforthecarrier waveform.Thenumberscanbesetbyusingtheknoborbypressingthebuttons(Fig3D).
SelectAMPLusingthesoftkeysshownin(Fig3C)andenter1Vpk-pkfortheamplitudeofthe carrierwaveform.ThisstepisdonebypressingthenumberoneandselectingVpponthescreen PresstheMOD(Fig3B)thenAMPLbutton(Fig3C)to settheDEPTHto30%.Thiscorresponds toμ=0.3.
PressFREQbutton(Fig3C)tosetthefrequencyofthemodulatingwaveformandenter5kHz.
3. ConnectthelowpassfilterandoscilloscopeasillustratedinFigure1.TurnoffChannel2ofthe oscilloscopeandexaminetheAMsignalastheinputtothefilter.Adjustthevolts/divisionand seconds/divisionsothatoneperiodfillsthescreen.
4.TurnonChannel2sothatonecompletewaveformis viewedontheoscilloscope.Atleast onefull periodofthewaveformmustbeinviewtofortheoscilloscopemeasurefunctionstobeaccurate. Examinethephase,amplitude,andappearanceofthesignalthatistheoutputofthelowpass
filter.HaveyourTAcheckyourdisplayandsignthedatasheet.Explaininyourlabreport theeffectthatthelowpassfilterhasontheAMsignal.Couldthisfilteredsignalbeusedto determinetheengineRPM?
5.Disconnectthelowpassfilterandthenconnectthehighpassfilterandoscilloscopeasillustrated inFigure2.Examinetheresultingwaveformsforthefilterinputandoutput.Examinethe appearanceofthesignalthatistheoutputofthehighpassfilter.HaveyourTAcheckyour displayandsignthedatasheet.Explaininyourlabreporttheeffectthatthehighpassfilterhas ontheAMsignal.
Log-LinearPlotting
Inmanyapplications,theabilitytostronglyrejectsignalsinagivenfrequencyband isofprimary importance. Comparingthe performanceof various filters, itis helpfultoexpressthemagnitudesofthetransfer functionindecibels. Toconvertatransfer-functionmagnitudetodecibels,thecommonlogarithmofthe transfer-function needs to bemultiply by 20
|H(ƒ)|DB=20log|H(ƒ)|
Oneofthebestwaysofplottingtransferfunctionsisusingalogarithmicscale. Inthisscale,the variableismultipliedbyagivenfactor forequalincrementsoflengthalongtheaxis. Onthelinearscale,equal lengthson the scalecorrespond toaddinga givenamounttothevariable.
Theadvantageofalogarithmicfrequencyscalecomparedwithalinearscaleisthatthevariationsin themagnitude or phaseofatransfer functionforalowrange of frequency,aswell asthevariationsinahigh range,canbeclearlyshownonasingleplot.Withthelinearscale,eitherthelowrangewouldbeseverely
compressed orthe high rangewould beoff scale.
Figure4and5showsexamplesofLinear-Linearand Linear-Log graphsusing the following formula:
|H(ƒ)|DB=1/(1+j(f/fb),where fb is a fix constant(100) and the frequencyfgoesfrom1 to 10000, incrementingbyone.
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GraphsforPartA
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GraphsforPartB
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Part A: LowPass Filter Measurements
Measured1kΩresistorvalue
Measured1μFcapacitorvalue
Table 1
f(Hz)CH-1pk-pkCH-2pk-pkΔtgain
50
100
200
500
800
1k
2k
5k
10k
H(f)
dB
phase
"+"ifCH-2leads
Part B: HighPass Filter Measurements
Measured1kΩresistorvalue
Measured0.1μFcapacitorvalue
Table 2
f(Hz)CH-1pk-pkCH-2pk-pkΔtgain
50
100
200
500
800
1k
2k
5k
10k
H(f)
dB
phase
"+"ifCH-2leads
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PartC:UsingFilters ToIsolateFrequenciesInAnAMSignal
TALowPassfilterVerification TAHighPassfilterVerification
YourName
LabInstructor
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