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FrequencyResponseandActiveFiltersThisdocumentisanintroductiontofrequencyresponse,andanintroductiontoactivefilters(filtersusingactiveamplifiers,likeopamps).YoumightalsowanttoreadasimilardocumentfromNationalSemiconductor,ABasicIntroductiontoFiltersActive,Passive,andSwitchedCapacitor.

FrequencyResponseBackground

Uptonowwehavelookedatthetimedomainresponseofcircuits.However it isoftenuseful to lookat the responseofcircuits in thefrequencydomain.Inotherwords,youwanttolookathowcircuitsbehaveinresponsetosinusoidalinputs.Thisisimportantandusefulfor several reasons: 1) if the input to a linear circuit is a sinusoid,thentheoutputwillbeasinusoidat thesamefrequency, thoughitsamplitudeandphasemayhavechanged,2)any timedomainsignalcan be decomposed via Fourier analysis into a series of sinusoids.Thereforeifthereisaneasywaytoanalyzecircuitswithsinusoidalinputs, the results can be generalized to study the response to anyinput.

To determine the response of a circuit to a sinusoidal signal as afunction of frequency it is possible to generalize the concept ofimpedancetoincludecapacitorsandinductors.Considerasinusoidalsignalrepresentedbyacomplexexponential:

where j=1)1/2 (engineers use j instead of i, because i is used forcurrent), is frequencyand t is time. It is a commonshorthand touse"s"insteadof"j".

Now let us look at the voltagecurrent relationships for resistorscapacitorsandinductors.

Foraresistorohmslawstates:

wherewe define the impedance, "Z", of a resistor as its resistance"R".

For a capacitor we can also calculate the impedance assumingsinusoidalexcitationstartingfromthecurrentvoltagerelationship:

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Notethatforacapacitorthemagnitudeoftheimpedance,1/C,goesdown with increasing frequency. This means that at very highfrequencies the capacitor acts as an short circuit, and at lowfrequencies itactsasanopencircuit.What isdefinedasahigh,orlow,frequencydependsonthespecificcircuitinquestion.

Likewise,foraninductoryoucanshowthatZ=sL.

Foraninductor,impedancegoesupwithfrequency.Itbehavesasashort circuit at low frequencies, and an open circuit at highfrequencies theoppositeofacapacitor.However inductorsarenotusedofteninelectroniccircuitsduetotheirsize,theirsusceptibilityto parisitic effects (esp.magnetic fields), and because they do notbehave as near to their ideal circuit elements as resistors andcapacitors..

ASimpleLowPassCircuit

To see how complex impedances are used in practice consider thesimplecaseofavoltagedivider.

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IfZ1isaresistorandZ2isacapacitorthen

Generally we will be interested only in the magnitude of theresponse:

Recallthatthemagnitudeofacomplexnumberisthesquarerootofthesumofthesquaresoftherealandimaginaryparts.Therearealsophase shifts associatedwith the transfer function (or gain, Vo/Vi),thoughtwewillgenerallyignorethese.

This is obviously a low pass filter (i.e., low frequency signals arepassed and high frequency signals are blocked).. If1 ) then the gaingoes to zero, asdoes the output. At =1/RC, called the breakfrequency (or cutoff frequency, or 3dB frequency, or halfpowerfrequency, or bandwidth), the magnitude of the gain is1/sqrt(20.71. In this case (and all first order RC circuits) highfrequencyisdefinedas>>1/RCthecapacitoractsasashortcircuitand all the voltage is across the resistance. At low frequencies,

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IfZ1isacapacitorandZ2isaresistorwecanrepeatthecalculation:

and

Athighfrequencies,>>1/RC,thecapacitoractsasashortandthegainis1(thesignalispassed).Atlowfrequencies,

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Ifyouderivethetransferfunctionforthecircuitaboveyouwillfindthatitisoftheform:

whichisthegeneralformforfirstorder(onereactiveelement)lowpass filters. At high frequencies (>>o) the capacitor acts as ashort, so the gain of the amplifier goes to zero. At very lowfrequencies (

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short,sothegainoftheamplifiergoestoH0=R1/R2.Atvery lowfrequencies (

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frequencies,theotherquantitiesareinHertz(i.e.fo=o/2,B=/2).Lookingattheequationabove,orthefigure,youcanseethatas0and>infinitythat|H(s=j)|0.Youcanalsoeasilyshowthatat=o that |H(s=jo)|=H0.Often youwill see the equation abovewritten in terms of the quality factor, Q, which can be defined interms of the bandwidth, , and center frequency, o, as Q=o/.ThustheQ,orquality,ofafiltergoesupasitbecomesnarroweranditsbandwidthdecreases.

Ifyouderivethetransferfunctionofthecircuitshownbelow:

HighQBandpassFilterwithOpAmp

youwillfindthatitactsasabandpassfilterwith:

andthecenterfrequencyandbandwidthgivenby:

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Radianfrequency Hertz

ThenotationR1||R2denotestheparallelcombinationofR1andR2,

.

SwitchedCapacitorFilters

Thereisaspecialtypeofactivefilter,theswitchedcapacitorfilter,thattakesadvantageofintegrationtoachieveveryaccuratefiltercharacteristicsthatareelectronicallytuneable.ThepageSwitchedCapacitorFiltersdescribestheseinmoredetail.