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ActiveHighPassFilters
ThebasicelectricaloperationofanActiveHighPassFilter(HPF)isexactlythesameaswesawforitsequivalentRC
passivehighpassfiltercircuit,exceptthistimethecircuithasanoperationalamplifierorop-ampincludedwithinitsfilter
designprovidingamplificationandgaincontrol.Likethepreviousactivelowpassfiltercircuit,thesimplestformof
anactivehighpassfilteristoconnectastandardinvertingornon-invertingoperationalamplifiertothebasicRChigh
passpassivefiltercircuitasshown.
FirstOrderActiveHighPassFilter
Technically,thereisnosuchthingasanactivehighpassfilter.UnlikePassiveHighPassFilterswhichhavean
"infinite"frequencyresponse,themaximumpassbandfrequencyresponseofanactivehighpassfilterislimitedbythe
open-loopcharacteristicsorbandwidthoftheoperationalamplifierbeingused,makingthemappearasiftheyareband
passfilterswithahighfrequencycut-offdeterminedbytheselectionofop-ampandgain.
IntheOperationalAmplifiertutorialwesawthatthemaximumfrequencyresponseofanop-ampislimitedtothe
Gain/Bandwidthproductoropenloopvoltagegain(A)oftheoperationalamplifierbeingusedgivingitabandwidth
limitation,wheretheclosedloopresponseoftheopampintersectstheopenloopresponse.
AcommonlyavailableoperationalamplifiersuchastheuA741hasatypical"open-loop"(withoutanyfeedback)DC
voltagegainofabout100dBmaximumreducingatarolloffrateof-20dB/Decade(-6db/Octave)astheinputfrequency
increases.ThegainoftheuA741reducesuntilitreachesunitygain,(0dB)orits"transitionfrequency"(?t)whichis
about1MHz.Thiscausestheop-amptohaveafrequencyresponsecurveverysimilartothatofafirst-orderlowpass
filterandthisisshownbelow.
FrequencyresponsecurveofatypicalOperationalAmplifier.
V
Thentheperformanceofa"highpassfilter"athighfrequenciesislimitedbythisunitygaincrossoverfrequencywhich
determinestheoverallbandwidthoftheopen-loopamplifier.Thegain-bandwidthproductoftheop-ampstartsfrom
around100kHzforsmallsignalamplifiersuptoabout1GHzforhigh-speeddigitalvideoamplifiersandop-ampbased
activefilterscanachieveverygoodaccuracyandperformanceprovidedthatlowtoleranceresistorsandcapacitorsare
used.
Undernormalcircumstancesthemaximumpassbandrequiredforaclosedloopactivehighpassorbandpassfilteris
wellbelowthatofthemaximumopen-looptransitionfrequency.However,whendesigningactivefiltercircuitsitis
importanttochoosethecorrectop-ampforthecircuitasthelossofhighfrequencysignalsmayresultinsignal
distortion.
ActiveHighPassFilter
Afirst-order(single-pole)ActiveHighPassFilterasitsnameimplies,attenuateslowfrequenciesandpasseshigh
frequencysignals.Itconsistssimplyofapassivefiltersectionfollowedbyanon-invertingoperationalamplifier.The
frequencyresponseofthecircuitisthesameasthatofthepassivefilter,exceptthattheamplitudeofthesignalis
increasedbythegainoftheamplifierandforanon-invertingamplifierthevalueofthepassbandvoltagegainisgiven
as1+R2/R1,thesameasforthelowpassfiltercircuit.
ActiveHighPassFilterwithAmplification
Thisfirst-orderhighpassfilter,consistssimplyofapassivefilterfollowedbyanon-invertingamplifier.Thefrequency
responseofthecircuitisthesameasthatofthepassivefilter,exceptthattheamplitudeofthesignalisincreasedbythe
gainoftheamplifier.
Foranon-invertingamplifiercircuit,themagnitudeofthevoltagegainforthefilterisgivenasafunctionofthefeedback
resistor(R2)dividedbyitscorrespondinginputresistor(R1)valueandisgivenas:
GainforanActiveHighPassFilter
Where:
A=thePassbandGainofthefilter,(1+R2/R1)
?=theFrequencyoftheInputSignalinHertz,(Hz)
?c=theCut-offFrequencyinHertz,(Hz)
Justlikethelowpassfilter,theoperationofahighpassactivefiltercanbeverifiedfromthefrequencygainequation
aboveas:
1.Atverylowfrequencies,?
2.Atthecut-offfrequency,?=?c,
3.Atveryhighfrequencies,?>?c,
Then,theActiveHighPassFilterhasagainAthatincreasesfrom0Hztothelowfrequencycut-offpoint,?at
20dB/decadeasthefrequencyincreases.At?thegainis0.707Aandafter?allfrequenciesarepassband
frequenciessothefilterhasaconstantgainAwiththehighestfrequencybeingdeterminedbytheclosedloop
bandwidthoftheop-amp.
Whendealingwithfiltercircuitsthemagnitudeofthepassbandgainofthecircuitisgenerallyexpressed
indecibelsordBasafunctionofthevoltagegain,andthisisdefinedas:
MagnitudeofVoltageGainin(dB)
Forafirst-orderfilterthefrequencyresponsecurveofthefilterincreasesby20dB/decadeor6dB/octaveuptothe
determinedcut-offfrequencypointwhichisalwaysat-3dBbelowthemaximumgainvalue.Aswiththepreviousfilter
circuits,thelowercut-offorcornerfrequency(?c)canbefoundbyusingthesameformula:
F
FC
CF,C
F
ThecorrespondingphaseangleorphaseshiftoftheoutputsignalisthesameasthatgivenforthepassiveRCfilter
andleadsthatoftheinputsignal.Itisequalto+45atthecut-offfrequency?cvalueandisgivenas:
Asimplefirst-orderactivehighpassfiltercanalsobemadeusinganinvertingoperationalamplifierconfigurationas
well,andanexampleofthiscircuitdesignisgivenalongwithitscorrespondingfrequencyresponsecurve.Againof
40dBhasbeenassumedforthecircuit.
InvertingOperationalAmplifierCircuit
FrequencyResponseCurve
ExampleNo1
Afirstorderactivehighpassfilterhasapassbandgainoftwoandacut-offcornerfrequencyof1kHz.Iftheinput
capacitorhasavalueof10nF,calculatethevalueofthecut-offfrequencydeterminingresistorandthegainresistorsin
thefeedbacknetwork.Also,plottheexpectedfrequencyresponseofthefilter.
Withacut-offcornerfrequencygivenas1kHzandacapacitorof10nF,thevalueofRwillthereforebe:
o
Thepassbandgainofthefilter,Aisgivenasbeing,2.
Asthevalueofresistor,Rdividedbyresistor,Rgivesavalueofone.Then,resistorRmustbeequaltoresistorR,
sincethepassbandgain,A=2.Wecanthereforeselectasuitablevalueforthetworesistorsofsay,10kΩ''seachfor
bothfeedbackresistors.
Soforahighpassfilterwithacut-offcornerfrequencyof1kHz,thevaluesofRandCwill
be,10kΩ''sand10nFrespectively.Thevaluesofthetwofeedbackresistorstoproduceapassbandgainoftwoaregiven
as:R=R=10kΩ''s
Thedataforthefrequencyresponsebodeplotcanbeobtainedbysubstitutingthevaluesobtainedaboveovera
frequencyrangefrom100Hzto100kHzintotheequationforvoltagegain:
Thisthenwillgiveusthefollowingtableofdata.
Frequency,?(Hz)VoltageGain(Vo/Vin)Gain,(dB)20log(Vo/Vin)
1000.20-14.02
2000.39-8.13
5000.89-0.97
8001.251.93
1,0001.413.01
3,0001.905.56
5,0001.965.85
10,0001.995.98
50,0002.006.02
100,0002.006.02
Thefrequencyresponsedatafromthetableabovecannowbeplottedasshownbelow.Inthestopband(from100Hzto
1kHz),thegainincreasesatarateof20dB/decade.However,inthepassbandafterthecut-offfrequency,?=1kHz,the
gainremainsconstantat6.02dB.Theupper-frequencylimitofthepassbandisdeterminedbytheopenloopbandwidth
oftheoperationalamplifierusedaswediscussedearlier.Thenthebodeplotofthefiltercircuitwilllooklikethis.
TheFrequencyResponseBode-plotforourexample.
F
2112
F
12
C
ApplicationsofActiveHighPassFiltersareinaudioamplifiers,equalizersorspeakersystemstodirectthehigh
frequencysignalstothesmallertweeterspeakersortoreduceanylowfrequencynoiseor"rumble"typedistortion.When
usedlikethisinaudioapplicationstheactivehighpassfilterissometimescalleda"TrebleBoost"filter.
Second-orderHighPassActiveFilter
Aswiththepassivefilter,afirst-orderhighpassactivefiltercanbeconvertedintoasecond-orderhighpassfiltersimply
byusinganadditionalRCnetworkintheinputpath.Thefrequencyresponseofthesecond-orderhighpassfilteris
identicaltothatofthefirst-ordertypeexceptthatthestopbandroll-offwillbetwicethefirst-orderfiltersat40dB/decade
(12dB/octave).Therefore,thedesignstepsrequiredofthesecond-orderactivehighpassfilterarethesame.
Second-orderActiveHighPassFilterCircuit
Higher-orderhighpassfilters,suchasthird,fourth,fifth,etcareformedsimplybycascadingtogetherfirstandsecond-
orderfilters.Forexample,athirdorderhighpassfilterisformedbycascadinginseriesfirstandsecondorderfilters,a
fourth-orderhighpassfilterbycascadingtwosecond-orderfilterstogetherandsoon.ThenanActiveHighPass
Filterwithanevenordernumberwillconsistofonlysecond-orderfilters,whileanoddordernumberwillstartwithafirst-
orderfilteratthebeginningasshown.
CascadingActiveHighPassFilters
Althoughthereisnolimittotheorderofafilterthatcanbeformed,astheorderofthefilterincreasessotodoesitssize.
Also,itsaccuracydeclines,thatisthedifferencebetweentheactualstopbandresponseandthetheoreticalstopband
responsealsoincreases.
Ifthefrequencydeterminingresistorsareallequal,R1=R2=R3etc,andthefrequencydeterminingcapacitorsareall
equal,C1=C2=C3etc,thenthecut-offfrequencyforanyorderoffilterwillbeexactlythesame.However,theoverall
gainofthehigher-orderfilterisfixedbecauseallthefrequencydeterminingcomponentsareequal.
Inthenexttutorialaboutfilters,wewillseethatActiveBandPassFilters,canbeconstructedbycascadingtogether
ahighpassandalowpassfilter.
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