HAPTE FILTES AND TUNED AMPLIFIES hapter Outline. Filter Traniion, Type and Specification. The Filter Tranfer Function. Butterworth and hebyhev Filter. Firt Order and Second Order Filter Function.5 The Second Order L eonator.6 Second Order Active Filter Baed on Inductor eplaceent.7 Second Order Active Filter Baed on the Two Integrator Loop Topology.8 Single Aplifier Biquadratic Active Filter.9 Senitivity. Tranconductance Filter. Switched apacitor Filter NTUEE Electronic L. H. Lu
. Filter Traniion, Type and Specification Filter Tranfer Function A filter i a linear two port network repreented by the ratio of the output to input voltage Tranfer function T o / i Traniion : evaluating T for phyical frequency = j Tj = Tj e j Gain: log Tj db Attenuation: log Tj db Output frequency pectru : o = T i Type of Filter NTUEE Electronic L. H. Lu
Filter Specification Paband edge : p Maxiu allowed variation in paband traniion : A ax Stopband edge : Miniu required topband attenuation : A in Low Pa Filter Band Pa Filter The firt tep of filter deign i to deterine the filter pecification Then find a tranfer function T whoe agnitude Tj eet the pecification The proce of obtaining a tranfer function that eet given pecification i called filter approxiation NTUEE Electronic L. H. Lu
Tranfer Function. Filter Tranfer Function The filter tranfer function i written a the ratio of two polynoial: M am a T N b M M N N... a... b The degree of the denoinator filter order To enure the tability of the filter N M The coefficient a i and b j are real nuber The tranfer function can be factored and expreed a: am z z... zm T p p... p N Zero: z, z,, z M and NM zero at infinity Pole: p, p,, p N Zero and pole can be either a real or a coplex nuber oplex zero and pole ut occur in conjugate pair The pole have to be on the LHP of plane NTUEE Electronic L. H. Lu
The Butterworth Filter. Butterworth and hebyhev Filter Butterworth filter exhibit onotonically decreaing traniion with all zero at = Maxially flat repone degree of paband flatne increae a the order N i increaed Higher order filter ha a harp cutoff in the tranition band The agnitude function of the Butterworth filter i: T j / p N equired tranfer function can be defined baed on filter pecification A ax, A in, p, T p NTUEE Electronic L. H. Lu 5
Natural Mode of the Butterworth Filter The natural ode pole locate on a circle / N The radiu of the circle i p / Equal angle pace T p p K p N... p N N K / N where p/ p... pn NTUEE Electronic L. H. Lu 6
Deign Procedure of the Butterworth Filter Deign Specification A ax, A in, p, Deign Procedure. Deterine fro A ax Aax / T j p Aax[ db] log. Deterine the required filter order N fro p,, A in Attenuation. Deterine the N natural ode pole with p, p,... A [ db] log[/ p N / p N ] log[ / / N p / p N ] A in. Deterine T K T p p N / N where p /... pn NTUEE Electronic L. H. Lu 7
acade Filter Deign. Firt Order and Second Order Filter Function Firt order and econd order filter can be cacaded to realize high order filter acade deign i one of the ot popular ethod for the deign of active filter acading doe not change the tranfer function of individual block if the output reitance i low Firt Order Filter Bilinear tranfer function a a T b a a NTUEE Electronic L. H. Lu 8
Firt Order Filter ont d NTUEE Electronic L. H. Lu 9
NTUEE Electronic L. H. Lu Second Order Filter Biquadratic tranfer function Pole frequency: Pole quality factor: Q Pole: Bandwidth: / Q a a a T, Q j Q p p Q BW
Second Order Filter ont d NTUEE Electronic L. H. Lu
Second Order Filter ont d NTUEE Electronic L. H. Lu
The eonator Natural Mode.5 The Second Order L eonator Parallel eonator urrent Excitation oltage Excitation urrent Excitation I o i Y / L / / / / L / L oltage Excitation o i / / L / L / / L Q The L reonator can be excited by either current or voltage ource The excitation hould be applied without change the natural tructure of the circuit The natural ode of the circuit are identical will not be changed by the excitation ethod The iilar characteritic alo applie to erie L reonator NTUEE Electronic L. H. Lu
ealization of Traniion Zero alue of at which Z = and Z Z behave a a hort alue of at which Z = and Z Z behave a an open ealization of Filter Function Low Pa Filter High Pa Filter Bandpa Filter o T i / L / / L o T i / / L o T i / / / L NTUEE Electronic L. H. Lu
Notch Filter Low Pa Notch Filter High Pa Notch Filter NTUEE Electronic L. H. Lu 5
.6 Second Order Active Filter Inductor eplaceent Second Order Active Filter by Op Ap ircuit Inductor are not uitable for I ipleentation Ue op ap circuit to replace the inductor Second order filter function baed on L reonator The Antoniou Inductance Siulation ircuit Inductor are realized by op ap circuit with negative feedback The equivalent inductance i given by Z L in eq I / / 5 5 L eq NTUEE Electronic L. H. Lu 6
The Op Ap eonator The inductor i replaced by the Antoniou circuit The pole frequency and the quality factor are given by / 6 5 / Q A iplified cae where = = = 5 = and = 6 = 6 6 6 6 5 / Q / 6 NTUEE Electronic L. H. Lu 7
Filter ealization Low Pa Filter High Pa Filter Bandpa Filter Notch Filter NTUEE Electronic L. H. Lu 8
LPN Filter HPN Filter All Pa Filter NTUEE Electronic L. H. Lu 9
NTUEE Electronic L. H. Lu.7 Second Order Active Filter Two Integrator Loop Derivation of the Two Integrator Loop Biquad High pa ipleentation: Band pa ipleentation: Low pa ipleentation: / Q K T Q K hp hp hp i hp / Q K T hp bp / Q K T hp lp
NTUEE Electronic L. H. Lu ircuit Ipleentation I High pa tranfer function: Band pa tranfer function: Low pa tranfer function: Notch and all pa tranfer function: Q K Q f hp f hp f i f hp ] / [ ] / [ T i hp hp ] / [ ] / [ T i bp bp ] / [ ] / [ T i bp bp ] / [ / / / T L F B F H F i o
ircuit Ipleentation II Tow Thoa Biquad Ue an additional inverter to ake all the coefficient of the uer the ae ign All op ap are in ingle ended ode The high pa function i no longer available It i known a the Tow Thoa biquad An econoical feedforward chee can be eployed with the Tow Thoa circuit T o i r Q NTUEE Electronic L. H. Lu
.8 Single Aplifier Biquadratic Active Filter haracteritic of the SAB ircuit Only one op ap i required to ipleent biquad circuit Exhibit greater dependence on the liited gain and bandwidth of the op ap More enitive to the unavoidable tolerance in the value of reitor and capacitor Liited to le tringent filter pecification with pole Q factor le than Synthei of the SAB ircuit Ue feedback to ove the pole of an circuit fro the negative real axi to the coplex conjugate location to provide elective filter repone Step of SAB ynthei: Synthei of a feedback loop that realize a pair of coplex conjugate pole characterized by and Q Injecting the input ignal in a way that realize the deired traniion zero Natural ode of the filter: t N / D L At AN / D The cloed loop characteritic equation: L t / A P The pole of the cloed loop yte are identical to the zero of the network NTUEE Electronic L. H. Lu
NTUEE Electronic L. H. Lu Network with coplex traniion zero haracteritic Equation of the Filter t Q t Q Let = =, =, = / Q / Q
NTUEE Electronic L. H. Lu Injection the Input Signal The ethod of injection the input ignal into the feedback loop through the grounded node A coponent with a ground node can be connected to the input ource The filter traniion zero depend on where the input ignal i injected / i o 5
Generation of Equivalent Feedback Loop a b t a c t Equivalent Loop haracteritic Equation: haracteritic Equation: L At A t At A NTUEE Electronic L. H. Lu 6
Generation of Equivalent Feedback Loop ont d NTUEE Electronic L. H. Lu 7
Filter Senitivity.9 Senitivity Deviation in filter repone due to the tolerance in coponent value Epecially for coponent value and aplifier gain laical Senitivity Function Definition: For all change: S S y x y x y / y li x x / x y x x y S y x y / y x / x NTUEE Electronic L. H. Lu 8
. Tranconductance Filter Liitation of Op Ap ircuit Suitable for audio frequency filter uing dicrete op ap, reitor and capacitor High frequency application liited by the relatively low bandwidth of general purpoe op ap Ipractical for I ipleentation due to: The need for large capacitor and reitor increae the I cot The need for very precie value of tie contant require expenive triing/tuning The need for op ap that can drive reitive and large capacitive load Method for I Filter Ipleentation Tranconductance filter: Utilize tranconductance aplifier or tranconductor together with capacitor for filter High quality and high frequency tranconductor can be eaily realized in MOS technology Ha been widely ued for ediu/high frequency application up to hundred of MHz MOSFET filter: eplace reitor with MOSFET in linear region Technique have been evolved to obtain linear operation with large input ignal Switched capacitor filter: eplace the required reitor by witching a capacitor at a relatively high frequency The reulting filter are dicrete tie circuit a oppoed to the continuou tie one I ideally uited for ipleentation low frequency filter in I for uing MOS technology NTUEE Electronic L. H. Lu 9
Tranconductor An ideal tranconductor ha infinite input reitance and infinite output reitance The output can be poitive or negative depending the current direction Tranconductor can be ingle ended or fully differential NTUEE Electronic L. H. Lu
Baic Building Block Negative tranconductor ued to realize a reitance Tranconductor loaded with a capacitor a an integrator Firt Order G Low Pa Filter o G G i G / G / G NTUEE Electronic L. H. Lu
NTUEE Electronic L. H. Lu Second Order G Low Pa Filter / / / / / / G G G G G G G G G G i i G G G Q G G LP dc gain BP center - freq gain G G G G
Siplified ircuit G = G = G = = G G Q G Fully Differential ircuit NTUEE Electronic L. H. Lu
Baic Principle. Switched apacitor Filter A capacitor witched between two node at a ufficiently high rate i equivalent to a reitor The reitor in the active integrator can be replaced by the capacitor and the witche Equivalent reitor: vi T iav c eq vi i av T c Equivalent tie contant for the integrator = T c / NTUEE Electronic L. H. Lu
Practical ircuit an realize both inverting and non inverting integrator Inenitive to tray capacitance Noninverting witched capacitor S integrator Inverting witched capacitor S integrator NTUEE Electronic L. H. Lu 5
NTUEE Electronic L. H. Lu Filter Ipleentation ircuit paraeter for the two integrator with the ae tie contant c T c K T K T T c c Q T Q c 5 5 5 6