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LowPassFilterIntroduction
Basically,anelectricalfilterisacircuitthatcanbedesignedtomodify,reshapeorrejectallunwantedfrequenciesofan
electricalsignalandacceptorpassonlythosesignalswantedbythecircuitsdesigner.Inotherwordsthey"filter-out"
unwantedsignalsandanidealfilterwillseparateandpasssinusoidalinputsignalsbasedupontheirfrequency.
Inlowfrequencyapplications(upto100kHz),passivefiltersaregenerallyconstructedusingsimpleRC(Resistor-
Capacitor)networks,whilehigherfrequencyfilters(above100kHz)areusuallymadefromRLC(Resistor-Inductor-
Capacitor)components.Passivefiltersaremadeupofpassivecomponentssuchasresistors,capacitorsandinductors
andhavenoamplifyingelements(transistors,op-amps,etc)sohavenosignalgain,thereforetheiroutputlevelisalways
lessthantheinput.
Filtersaresonamedaccordingtothefrequencyrangeofsignalsthattheyallowtopassthroughthem,whileblockingor
"attenuating"therest.Themostcommonlyusedfilterdesignsarethe:
1.TheLowPassFilter–thelowpassfilteronlyallowslowfrequencysignalsfrom0Hzto
itscut-offfrequency,?cpointtopasswhileblockingthoseanyhigher.
2.TheHighPassFilter–thehighpassfilteronlyallowshighfrequencysignalsfromitscut-
offfrequency,?cpointandhighertoinfinitytopassthroughwhileblockingthoseanylower.
3.TheBandPassFilter–thebandpassfilterallowssignalsfallingwithinacertain
frequencybandsetupbetweentwopointstopassthroughwhileblockingboththelowerand
higherfrequencieseithersideofthisfrequencyband.
SimpleFirst-orderpassivefilters(1storder)canbemadebyconnectingtogetherasingleresistorandasinglecapacitor
inseriesacrossaninputsignal,(Vin)withtheoutputofthefilter,(Vout)takenfromthejunctionofthesetwo
components.Dependingonwhichwayaroundweconnecttheresistorandthecapacitorwithregardstotheoutput
signaldeterminesthetypeoffilterconstructionresultingineitheraLowPassFilteroraHighPassFilter.
Asthefunctionofanyfilteristoallowsignalsofagivenbandoffrequenciestopassunalteredwhileattenuatingor
weakeningallothersthatarenotwanted,wecandefinetheamplituderesponsecharacteristicsofanidealfilterbyusing
anidealfrequencyresponsecurveofthefourbasicfiltertypesasshown.
IdealFilterResponseCurves
Filterscanbedividedintotwodistincttypes:activefiltersandpassivefilters.Activefilterscontainamplifyingdevicesto
increasesignalstrengthwhilepassivedonotcontainamplifyingdevicestostrengthenthesignal.Astherearetwo
passivecomponentswithinapassivefilterdesigntheoutputsignalhasasmalleramplitudethanitscorresponding
inputsignal,thereforepassiveRCfiltersattenuatethesignalandhaveagainoflessthanone,(unity).
ALowPassFiltercanbeacombinationofcapacitance,inductanceorresistanceintendedtoproducehighattenuation
aboveaspecifiedfrequencyandlittleornoattenuationbelowthatfrequency.Thefrequencyatwhichthetransitionoccurs
iscalledthe"cutoff"frequency.Thesimplestlowpassfiltersconsistofaresistorandcapacitorbutmoresophisticated
lowpassfiltershaveacombinationofseriesinductorsandparallelcapacitors.Inthistutorialwewilllookatthesimplest
type,apassivetwocomponentRClowpassfilter.
TheLowPassFilter
AsimplepassiveRCLowPassFilterorLPF,canbeeasilymadebyconnectingtogetherinseriesasingleResistorwith
asingleCapacitorasshownbelow.Inthistypeoffilterarrangementtheinputsignal(Vin)isappliedtotheseries
combination(boththeResistorandCapacitortogether)buttheoutputsignal(Vout)istakenacrossthecapacitoronly.
Thistypeoffilterisknowngenerallyasa"first-orderfilter"or"one-polefilter",whyfirst-orderorsingle-pole?,becauseit
hasonly"one"reactivecomponent,thecapacitor,inthecircuit.
RCLowPassFilterCircuit
AsmentionedpreviouslyintheCapacitiveReactancetutorial,thereactanceofacapacitorvariesinverselywith
frequency,whilethevalueoftheresistorremainsconstantasthefrequencychanges.Atlowfrequenciesthecapacitive
reactance,(Xc)ofthecapacitorwillbeverylargecomparedtotheresistivevalueoftheresistor,Randasaresultthe
voltageacrossthecapacitor,Vcwillalsobelargewhilethevoltagedropacrosstheresistor,Vrwillbemuchlower.At
highfrequenciesthereverseistruewithVcbeingsmallandVrbeinglarge.
WhilethecircuitaboveisthatofanRCLowPassFiltercircuit,itcanalsobeclassedasafrequencyvariablepotential
dividercircuitsimilartotheonewelookedatintheResistorstutorial.Inthattutorialweusedthefollowingequationto
calculatetheoutputvoltagefortwosingleresistorsconnectedinseries.
WealsoknowthatthecapacitivereactanceofacapacitorinanACcircuitisgivenas:
OppositiontocurrentflowinanACcircuitiscalledimpedance,symbolZandforaseriescircuitconsistingofasingle
resistorinserieswithasinglecapacitor,thecircuitimpedanceiscalculatedas:
Thenbysubstitutingourequationforimpedanceaboveintotheresistivepotentialdividerequationgivesus:
So,byusingthepotentialdividerequationoftworesistorsinseriesandsubstitutingforimpedancewecancalculatethe
outputvoltageofanRCFilterforanygivenfrequency.
ExampleNo1
ALowPassFiltercircuitconsistingofaresistorof4k7Ωinserieswithacapacitorof47nFisconnectedacross
a10vsinusoidalsupply.Calculatetheoutputvoltage(Vout)atafrequencyof100Hzandagainatfrequencyof10,000Hz
or10kHz.
Atafrequencyof100Hz.
Atafrequencyof10kHz.
FrequencyResponse
Wecanseeabove,thatasthefrequencyincreasesfrom100Hzto10kHz,theoutputvoltage(Vout)decreasesfrom9.9v
to0.718v.Byplottingtheoutputvoltageagainsttheinputfrequency,theFrequencyResponseCurveorBode
Plotfunctionofthelowpassfiltercanbefound,asshownbelow.
FrequencyResponseofa1st-orderLowPassFilter
TheBodePlotshowstheFrequencyResponseofthefiltertobenearlyflatforlowfrequenciesandalloftheinputsignal
ispasseddirectlytotheoutput,resultinginagainofnearly1,calledunity,untilitreachesitsCut-offFrequencypoint
(?c).Thisisbecausethereactanceofthecapacitorishighatlowfrequenciesandblocksanycurrentflowthroughthe
capacitor.
Afterthiscut-offfrequencypointtheresponseofthecircuitdecreasesgivingaslopeof-20dB/Decadeor(-6dB/Octave)
"roll-off"assignalsabovethisfrequencybecomegreatlyattenuated,untilatveryhighfrequenciesthereactanceofthe
capacitorbecomessolowthatitgivestheeffectofashortcircuitconditionontheoutputterminalsresultinginzero
output.
ForthistypeofLowPassFiltercircuit,allthefrequenciesbelowthiscut-off,?cpointthatareunalteredwithlittleorno
attenuationandaresaidtobeinthefiltersPassbandzone.ThispassbandzonealsorepresentstheBandwidthofthe
filter.Anysignalfrequenciesabovethispointcut-offpointaregenerallysaidtobeinthefiltersStopbandzoneandthey
willbegreatlyattenuated.
This"Cut-off","Corner"or"Breakpoint"frequencyisdefinedasbeingthefrequencypointwherethecapacitivereactance
andresistanceareequal,R=Xc=4k7Ω.Whenthisoccurstheoutputsignalisattenuatedto70.7%oftheinputsignal
valueor-3dB(20log(Vout/Vin))oftheinput.AlthoughR=Xc,theoutputisnothalfoftheinputsignal.Thisisbecauseit
isequaltothevectorsumofthetwoandistherefore0.707oftheinput.
Asthefiltercontainsacapacitor,thePhaseAngle(Φ)oftheoutputsignalLAGSbehindthatoftheinputandatthe-3dB
cut-offfrequency(?c)andis-45outofphase.Thisisduetothetimetakentochargetheplatesofthecapacitorasthe
inputvoltagechanges,resultingintheoutputvoltage(thevoltageacrossthecapacitor)"lagging"behindthatoftheinput
signal.Thehighertheinputfrequencyappliedtothefilterthemorethecapacitorlagsandthecircuitbecomesmoreand
more"outofphase".
Thecut-offfrequencypointandphaseshiftanglecanbefoundbyusingthefollowingequation:
Cut-offFrequencyandPhaseShift
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Thenforoursimpleexampleofa"LowPassFilter"circuitabove,thecut-offfrequency(?c)isgivenas720Hzwithan
outputvoltageof70.7%oftheinputvoltagevalueandaphaseshiftangleof-45.
Second-orderLowPassFilter
Thusfarwehaveseenthatsimplefirst-orderRClowpassfilterscanbemadebyconnectingasingleresistorinseries
withasinglecapacitor.Thissingle-polearrangementgivesusaroll-offslopeof-20dB/decadeattenuationof
frequenciesabovethecut-offpointat?.However,sometimesinfiltercircuitsthis-20dB/decade(-6dB/octave)angle
oftheslopemaynotbeenoughtoremoveanunwantedsignalthentwostagesoffilteringcanbeusedasshown.
Second-orderLowPassFilter
Theabovecircuitusestwopassivefirst-orderlowpassfiltersconnectedor"cascaded"togethertoformasecond-order
ortwo-polefilternetwork.Thereforewecanseethatafirst-orderlowpassfiltercanbeconvertedintoasecond-ordertype
bysimplyaddinganadditionalRCnetworktoitandthemoreRCstagesweaddthehigherbecomestheorderofthe
filter.Ifanumber(n)ofsuchRCstagesarecascadedtogether,theresultingRCfiltercircuitwouldbeknownasan"n-
order"filterwitharoll-offslopeof"nx-20dB/decade".
Soforexample,asecond-orderfilterwouldhaveaslopeof-40dB/decade(-12dB/octave),afourth-orderfilterwouldhave
aslopeof-80dB/decade(-24dB/octave)andsoon.Thismeansthat,astheorderofthefilterisincreased,theroll-off
slopebecomessteeperandtheactualstopbandresponseofthefilterapproachesitsidealstopbandcharacteristics.
Second-orderfiltersareimportantandwidelyusedinfilterdesignsbecausewhencombinedwithfirst-orderfiltersany
higher-ordern-valuefilterscanbedesignedusingthem.Forexample,athirdorderlow-passfilterisformedby
connectinginseriesorcascadingtogetherafirstandasecond-orderlowpassfilter.
ButthereisadownsidetoocascadingtogetherRCfilterstages.Althoughthereisnolimittotheorderofthefilterthatcan
beformed,astheorderincreases,thegainandaccuracyofthefinalfilterdeclines.WhenidenticalRCfilterstagesare
cascadedtogether,theoutputgainattherequiredcut-offfrequency(?c)isreduced(attenuated)byanamountin
relationtothenumberoffilterstagesusedastheroll-offslopeincreases.Wecandefinetheamountofattenuationatthe
selectedcut-offfrequencyusingthefollowingformula.
PassiveLowPassFilterGainat?c
where"n"isthenumberoffilterstages.
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3dB
th
th
Soforasecond-orderpassivelowpassfilterthegainatthecornerfrequency?cwillbeequalto0.7071x0.7071=0.5Vin
(-6dB),athird-orderpassivelowpassfilterwillbeequalto0.353Vin(-9dB),fourth-orderwillbe0.25Vin(-12dB)andso
on.Thecornerfrequency,?cforasecond-orderpassivelowpassfilterisdeterminedbytheresistor/capacitor(RC)
combinationandisgivenas.
2nd-OrderFilterCornerFrequency
Inrealityasthefilterstageandthereforeitsroll-offslopeincreases,thelowpassfilters-3dBcornerfrequencypointand
thereforeitspassbandfrequencychangesfromitsoriginalcalculatedvalueabovebyanamountdeterminedbythe
followingequation.
2nd-OrderLowPassFilter-3dBFrequency
where?cisthecalculatedcut-offfrequency,nisthefilterorderand?isthenew-3dBpassbandfrequencyasa
resultintheincreaseofthefiltersorder.
Thenthefrequencyresponse(bodeplot)forasecond-orderlowpassfilterassumingthesame-3dBcut-offpointwould
looklike:
FrequencyResponseofa2nd-orderLowPassFilter
Inpractice,cascadingpassivefilterstogethertoproducelarger-orderfiltersisdifficulttoimplementaccuratelyasthe
dynamicimpedanceofeachfilterorderaffectsitsneighbouringnetwork.However,toreducetheloadingeffectwecan
maketheimpedanceofeachfollowingstage10xthepreviousstage,soR2=10xR1andC2=1/10thC1.Second-
orderandabovefilternetworksaregenerallyusedinthefeedbackcircuitsofop-amps,makingwhatarecommonly
knownasActiveFiltersorasaphase-shiftnetworkinRCOscillatorcircuits.
LowPassFilterSummary
Sotosummarize,theLowPassFilterhasaconstantoutputvoltagefromD.C.(0Hz),uptoaspecifiedCut-offfrequency,
(?c)point.Thiscut-offfrequencypointis0.707or-3dB(dB=-20logVout/Vin)ofthevoltagegainallowedtopass.The
frequencyrange"below"thiscut-offpoint?cisgenerallyknownasthePassBandastheinputsignalisallowedtopass
throughthefilter.Thefrequencyrange"above"thiscut-offpointisgenerallyknownastheStopBandastheinputsignal
-3dB
isblockedorstoppedfrompassingthrough.
Asimple1storderlowpassfiltercanbemadeusingasingleresistorinserieswithasinglenon-polarizedcapacitor(or
anysinglereactivecomponent)acrossaninputsignalVin,whilsttheoutputsignalVoutistakenfromacrossthe
capacitor.Thecut-offfrequencyor-3dBpoint,canbefoundusingtheformula,?c=1/(2πRC).Thephaseangleofthe
outputsignalat?candis-45foraLowPassFilter.
Thegainofthefilteroranyfilterforthatmatter,isgenerallyexpressedinDecibelsandisafunctionoftheoutputvalue
dividedbyitscorrespondinginputvalueandisgivenas:
ApplicationsofpassiveLowPassFiltersareinaudioamplifiersandspeakersystemstodirectthelowerfrequencybass
signalstothelargerbassspeakersortoreduceanyhighfrequencynoiseor"hiss"typedistortion.Whenusedlikethisin
audioapplicationsthelowpassfilterissometimescalleda"high-cut",or"treblecut"filter.
Ifweweretoreversethepositionsoftheresistorandcapacitorinthecircuitsothattheoutputvoltageisnowtakenfrom
acrosstheresistor,wewouldhaveacircuitthatproducesanoutputfrequencyresponsecurvesimilartothatofaHigh
PassFilter,andthisisdiscussedinthenexttutorial.
TimeConstant
Untilnowwehavebeeninterestedinthefrequencyresponseofalowpassfilterandthatthefilterscut-offfrequency
(?c)istheproductoftheresistance(R)andthecapacitance(C)inthecircuitwithrespecttosomespecified
frequencypointandthatbyalteringanyoneofthetwocomponentsaltersthiscut-offfrequencypointbyeitherincreasing
itordecreasingit.
Wealsoknowthatthephaseshiftofthecircuitlagsbehindthatoftheinputsignalduetothetimerequiredtochargeand
thendischargethecapacitorasthesinewavechanges.ThiscombinationofRandCproducesachargingand
dischargingeffectonthecapacitorknownasitsTimeConstant(τ)ofthecircuitasseenintheRCCircuittutorials
givingthefilteraresponseinthetimedomain.
Thetimeconstant,tau(τ),isrelatedtothecut-offfrequency?cas.
orexpressedintermsofthecut-offfrequency,?cas.
Theoutputvoltage,Voutdependsuponthetimeconstantandthefrequencyoftheinputsignal.Withasinusoidalsignal
thatchangessmoothlyovertime,thecircuitbehavesasasimple1storderlowpassfilteraswehaveseenabove.But
whatifweweretochangetheinputsignaltothatofa"squarewave"shapedON/OFFtypesignalthathasanalmost
verticalstepinput,whatwouldhappentoourfiltercircuitnow.Theoutputresponseofthecircuitwouldchange
dramaticallyandproduceanothertypeofcircuitknowncommonlyasanIntegrator.
TheRCIntegrator
TheIntegratorisbasicallyalowpassfiltercircuitoperatinginthetimedomainthatconvertsasquarewave"step"
responseinputsignalintoatriangularshapedwaveformoutputasthecapacitorchargesanddischarges.
ATriangularwaveformconsistsofalternatebutequal,positiveandnegativeramps.Asseenbelow,iftheRCtime
constantislongcomparedtothetimeperiodoftheinputwaveformtheresultantoutputwaveformwillbetriangularin
shapeandthehighertheinputfrequencythelowerwillbetheoutputamplitudecomparedtothatoftheinput.
TheRCIntegratorCircuit
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Thisthenmakesthistypeofcircuitidealforconvertingonetypeofelectronicsignaltoanotherforuseinwave-generating
orwave-shapingcircuits.
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