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ActiveBandPassFilter
AswesawpreviouslyinthePassiveBandPassFiltertutorial,theprincipalcharacteristicofaBandPassFilteror
anyfilterforthatmatter,isitsabilitytopassfrequenciesrelativelyunattenuatedoveraspecifiedbandorspreadof
frequenciescalledthe"PassBand".Foralowpassfilterthispassbandstartsfrom0HzorDCandcontinuesuptothe
specifiedcut-offfrequencypointat-3dBdownfromthemaximumpassbandgain.Equally,forahighpassfilterthepass
bandstartsfromthis-3dBcut-offfrequencyandcontinuesuptoinfinityorthemaximumopenloopgainforanactivefilter.
However,theActiveBandPassFilterisslightlydifferentinthatitisafrequencyselectivefiltercircuitusedinelectronic
systemstoseparateasignalatoneparticularfrequency,orarangeofsignalsthatliewithinacertain"band"of
frequenciesfromsignalsatallotherfrequencies.Thisbandorrangeoffrequenciesissetbetweentwocut-offorcorner
frequencypointslabelledthe"lowerfrequency"(?)andthe"higherfrequency"(?)whileattenuatinganysignals
outsideofthesetwopoints.
SimpleActiveBandPassFiltercanbeeasilymadebycascadingtogetherasingleLowPassFilterwitha
singleHighPassFilterasshown.
Thecut-offorcornerfrequencyofthelowpassfilter(LPF)ishigherthanthecut-offfrequencyofthehighpassfilter(HPF)
andthedifferencebetweenthefrequenciesatthe-3dBpointwilldeterminethe"bandwidth"ofthebandpassfilterwhile
attenuatinganysignalsoutsideofthesepoints.OnewayofmakingaverysimpleActiveBandPassFilteristoconnect
thebasicpassivehighandlowpassfilterswelookatpreviouslytoanamplifyingop-ampcircuitasshown.
ActiveBandPassFilter
Thiscascadingtogetheroftheindividuallowandhighpasspassivefiltersproducesalow"Q-factor"typefiltercircuit
whichhasawidepassband.Thefirststageofthefilterwillbethehighpassstagethatusesthecapacitortoblockany
DCbiasingfromthesource.Thisdesignhastheadvantageofproducingarelativelyflatasymmetricalpassband
frequencyresponsewithonehalfrepresentingthelowpassresponseandtheotherhalfrepresentinghighpass
responseasshown.
LH
Thehighercornerpoint(?)aswellasthelowercornerfrequencycut-offpoint(?)arecalculatedthesameasbefore
inthestandardfirst-orderlowandhighpassfiltercircuits.Obviously,areasonableseparationisrequiredbetweenthe
twocut-offpointstopreventanyinteractionbetweenthelowpassandhighpassstages.Theamplifieralsoprovides
isolationbetweenthetwostagesanddefinestheoverallvoltagegainofthecircuit.
Thebandwidthofthefilteristhereforethedifferencebetweentheseupperandlower-3dBpoints.Forexample,ifthe
-3dBcut-offpointsareat200Hzand600Hzthenthebandwidthofthefilterwouldbegivenas:Bandwidth(BW)=600-
200=400Hz.Thenormalisedfrequencyresponseandphaseshiftforanactivebandpassfilterwillbeasfollows.
ActiveBandPassFrequencyResponse
Whiletheabovepassivetunedfiltercircuitwillworkasabandpassfilter,thepassband(bandwidth)canbequitewide
andthismaybeaproblemifwewanttoisolateasmallbandoffrequencies.Activebandpassfiltercanalsobemade
usinginvertingoperationalamplifier.Sobyrearrangingthepositionsoftheresistorsandcapacitorswithinthefilterwe
canproduceamuchbetterfiltercircuitasshownbelow.Foranactivebandpassfilter,thelowercut-off-3dBpointis
givenby?whiletheuppercut-off-3dBpointisgivenby?.
InvertingBandPassFilterCircuit
HL
C2C1
Thistypeofbandpassfilterisdesignedtohaveamuchnarrowerpassband.Thecentrefrequencyandbandwidthofthe
filterisrelatedtothevaluesofR1,R2,C1andC2.Theoutputofthefilterisagaintakenfromtheoutputoftheop-amp.
MultipleFeedbackBandPassActiveFilter
Wecanimprovethebandpassresponseoftheabovecircuitbyrearrangingthecomponentsagaintoproducean
infinite-gainmultiple-feedback(IGMF)bandpassfilter.Thistypeofactivebandpassdesignproducesa"tuned"circuit
basedaroundanegativefeedbackactivefiltergivingitahigh"Q-factor"(upto25)amplituderesponseandsteeproll-off
oneithersideofitscentrefrequency.Becausethefrequencyresponseofthecircuitissimilartoaresonancecircuit,this
centrefrequencyisreferredtoastheresonantfrequency,(?r).Considerthecircuitbelow.
InfiniteGainMultipleFeedbackActiveFilter
Thisactivebandpassfiltercircuitusesthefullgainoftheoperationalamplifier,withmultiplenegativefeedbackapplied
viaresistor,RandcapacitorC.ThenwecandefinethecharacteristicsoftheIGMFfilterasfollows:22
Wecanseethenthattherelationshipbetweenresistors,RandRdeterminesthebandpass"Q-factor"andthe
frequencyatwhichthemaximumamplitudeoccurs,thegainofthecircuitwillbeequalto-2Q.Thenasthegain
increasessotodoestheselectivity.Inotherwords,highgain-highselectivity.
ExampleNo1
AnactivebandpassfilterthathasagainAvofoneandaresonantfrequency,?rof1kHzisconstructedusinganinfinite
gainmultiplefeedbackfiltercircuit.Calculatethevaluesofthecomponentsrequiredtoimplementthecircuit.
Firstly,wecandeterminethevaluesofthetworesistors,RandRrequiredfortheactivefilterusingthegainofthe
circuittofindQasfollows.
ThenwecanseethatavalueofQ=0.7071givesarelationshipofresistor,RbeingtwicethevalueofresistorR.
Thenwecanchooseanysuitablevalueofresistancestogivetherequiredratiooftwo.ThenresistorR=
10kΩandR=20kΩ.
Thecentreorresonantfrequencyisgivenas1kHz.Usingthenewresistorvaluesobtained,wecandeterminethevalue
ofthecapacitorsrequiredassumingthatC=C=C.
Thecloseststandardvalueis10nF.
ResonantFrequency
Theactualshapeofthefrequencyresponsecurveforanypassiveoractivebandpassfilterwilldependuponthe
characteristicsofthefiltercircuitwiththecurveabovebeingdefinedasan"ideal"bandpassresponse.Anactiveband
passfilterisa2ndOrdertypefilterbecauseithas"two"reactivecomponents(twocapacitors)withinitscircuitdesign
122
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21
1
2
12
andwillhaveapeakresponseorResonantFrequency(?r)atits"centrefrequency",?c.Thecentrefrequencyis
generallycalculatedasbeingthegeometricmeanofthetwo-3dBfrequenciesbetweentheupperandthelowercut-off
pointswiththeresonantfrequency(pointofoscillation)beinggivenas:
Where:
?istheresonantorCentreFrequency
?isthelower-3dBcut-offfrequencypoint
?istheupper-3dbcut-offfrequencypoint
andinoursimpleexampleabovetheresonantcentrefrequencyoftheactivebandpassfilterisgivenas:
The"Q"orQualityFactor
InaBandPassFiltercircuit,theoverallwidthoftheactualpassbandbetweentheupperandlower-3dBcornerpointsof
thefilterdeterminestheQualityFactororQ-pointofthecircuit.ThisQFactorisameasureofhow"Selective"or"Un-
selective"thebandpassfilteristowardsagivenspreadoffrequencies.ThelowerthevalueoftheQfactorthewideris
thebandwidthofthefilterandconsequentlythehighertheQfactorthenarrowerandmore"selective"isthefilter.
TheQualityFactor,QofthefilterissometimesgiventheGreeksymbolofAlpha,(α)andisknownasthealpha-peak
frequencywhere:
Asthequalityfactorofanactivebandpassfilter(Second-orderSystem)relatestothe"sharpness"ofthefiltersresponse
arounditscentreresonantfrequency(?r)itcanalsobethoughtofasthe"DampingFactor"or"DampingCoefficient"
becausethemoredampingthefilterhastheflatterisitsresponseandlikewise,thelessdampingthefilterhasthe
sharperisitsresponse.ThedampingratioisgiventheGreeksymbolofXi,(ξ)where:
The"Q"ofabandpassfilteristheratiooftheResonantFrequency,(?r)totheBandwidth,(BW)betweentheupper
andlower-3dBfrequenciesandisgivenas:
Thenforoursimpleexampleabovethequalityfactor"Q"ofthebandpassfilterisgivenas:
r
L
H
346Hz/400Hz=0.865.NotethatQisaratioandhasnounits.
Whenanalysingactivefilters,generallyanormalisedcircuitisconsideredwhichproducesan"ideal"frequency
responsehavingarectangularshape,andatransitionbetweenthepassbandandthestopbandthathasanabruptor
verysteeproll-offslope.However,theseidealresponsesarenotpossibleintherealworldsoweuseapproximationsto
giveusthebestfrequencyresponsepossibleforthetypeoffilterwearetryingtodesign.
ProbablythebestknownfilterapproximationfordoingthisistheButterworthormaximally-flatresponsefilter.Inthenext
tutorialwewilllookathigherorderfiltersanduseButterworthapproximationstoproducefiltersthathaveafrequency
responsewhichisasflatasmathematicallypossibleinthepassbandandasmoothtransitionorroll-offrate.
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