PRINCIPLES OF ACTIVE NETWORK SYNTHESIS AND DESIGN GOBINDDARYANANI BellTelephoneLaboratories JOHN WILEY &SONS,NewYork Chichester Brisbane Toronto Singapore Copyright1976,byBenLaboratories,Inc. An rightsreserved.Published simultaneously inCanada. Reproductionor translationof anypartof thisworkbeyondthat permittedbySections 107or108of the1976UnitedStates Copy-rightActwithout thepermission of thecopyright owner isunlaw-ful.Requestsforpermissionorfunherinformationshouldbe addressed to the' Permissions Depanment, John Wiley & Sons. Inc. Li/WfU'yof CongressCatalogillg i"Publicatio"Data Daryanani, Gobind. Principles of activenetwork synthesis anddesign. Includes bibliographies andindex. I.Electricnetworks.2.Electronic circuit design. 3.Electric filters.1.Title. TK454.2.D27621.319'276-20659 Printed in Singapore 20191817 ToCarol PREFACE Integrated circuit technology profoundly influencesthe design of networks for voiceanddata communicationsystems.Integrationallowstherealizationof thesenetworkswithsmall-sizeandlow-costresistors,capacitors,andactive elements; thereby eliminating the need for inductors, which are relatively bulky andexpensive.Furthermore,activeRCnetworksprovideadvantagestheir passive counterparts do not, such as standardization and modularity of design, switchability, and ease of manufacture. These featureshave revolutionizedthe designof modern voice and data communication systems.More and more,the engineer is being facedwith the challenges and problems of active-RC network design.Thepurposeof thisbookistoprovidetheknowledgetomeetthese challenges. The approach usedinthebookisto developthe fundamentalprinciples of active andpassive networksynthesisinthe light of practical design considera-tions.ActiveNetworkSynthesis isaparticularly goodvehicleforintroducing manygeneraldesignconcepts,suchasperformanceversuscosttrade-offs, technologicallimitations,andcomputeraids.Theseideasarepresentedina simple way to allow assimilationbythe undergraduate electrical engineer, and are closelyrelatedtothe practical worldof engineering. Thebookissuitableforabasiccourseonnetworksynthesisoraninter-mediate course on circuits. The firsttwo chapters describe some simple analysis toolsandbasicpropertiesof activeandpassivenetworks.InChapter Three thestudentisintroducedtotheworldof filters:active,passive,electrome-chanical,anddigital.Examplesfromvoicecommunicationsystemsareused toillustratetheapplicationsofthebasicfiltertypes.InChapterFour,the filter approximation problem is discussed, with stress on the use of the standard approximation functionsrather than on their theoretical development. Animportant criterion ina practical designisthe sensitivity of the resulting circuittodeviationsinelementscausedbymanufacturingtolerancesand environmentalchanges(i.e.,temperature,humidity,andaging).Inkeeping withthepracticalorientationof thebook,sensitivityistreatedinChapter Five,priortodiscussingthesynthesisof circuits.Thispermitsthesynthesis stepstobecloselylinkedwiththisall-importantfigureofmeritandalso allowsalternatecircuitrealizationstobecomparedonthebasisoftheir sensitivities. Thesynthesisof passiveRLCnetworksisconsideredinChapterSixwith emphasisonthesynthesisof thedouble-terminatedladderfilter,astructure v viPREFACE most oftenusedinthe designof passivefilters . This structure alsoservesas a starting point for the synthesis of the coupled active filters described in Chapter Eleven. ChaptersSeventoTendealwithoperationalamplifierrealizationsof the biquadraticfunction, .whichisthefundamentalbuildingblockusedinboth cascadedandcoupledactivefilters.Thesingleamplifiercircuitrealizations are the subject of Chapters Eight and Nine, and the three amplifier realizations are coveredinChapter Ten.These circuits are compared on the basis of sensi-tivitiestopassiveandactiveelements,spreadsincomponentvalues,easeof tuning, andtypesof filterfunctionsthat canberealized.Abrief introduction tothedesignofcoupledfiltersandofgyratorandfrequency-dependent-negative-resistor realizations ispresented in Chapter Eleven. InChapter Twelvethenonidealproperties of theoperational amplifier and theireffectsonfilterperformanceisexploredingreaterdetailthaninthe precedingchapters.Finally,ChapterThirteendescribesthecompletedesign sequence,emphasizingcomputeraids,costminimization,anddesignopti-mization. The last part of this chapter briefly describes the discrete,thick-film, thin-film, andintegratedcircuittechnologiesusedinmanufacturingthefilter, concentratingontheprinciplesof designinsteadof onspecificdetails,which are expectedto change withtechnologicaladvances. Twocomputer programsareincludedinthetextdiscussion(AppendixD) asaidsindesign.TheMAGprogramcomputesthemagnitude,phase,and delayof functions;andthe CHEB program evaluates the Chebyshev approxi-mationfunctionforagivensetof filterrequirements.Theseprogramsare writteninANSIFORTRANIV,whichshouldbe ' compatiblewithmost computers. Copiesof theprogramon cardscanbeobtainedfromme.Equa-tionsandsampletables(ChapterFour)canbeusedinlieuof thecomputer programs. Althoughthebookisprimarilyaimedattheundergraduatelevel,itcan certainlybeusedforafirst-yeargraduatecourse,andbyengineersentering thefieldof activefilters . Theprincipalprerequisiteisabasiccircuitscourse. Thebookisdesignedtobecoveredinaone-semester coursebut,if needbe, severalof thesectionsinChaptersSix,Eleven,andTwelvemaybeomitted without loss of continuity.At a minimum, Chapters One toFive and Sevento Ten shouldbe coveredinthe course. In a graduate course thetechnicalpubli-cationsreferencedattheendof eachchaptermaybeusedassupplementary material. The material for this book evolved from my six years of work in the Network AnalysisandSynthesisDepartmentatBellLaboratories.Thisdepartment hasbeendeeplyinvolvedinthe area of active filterssincethe inception of this field. Thepresentformof thebookoriginatedfromanundergraduate course I taught at Southern University (Baton Rouge), where I was a Visiting Professor PREFACEvii on a program sponsored byBellLaboratories and froma similar one-semester coursegiventoBellLaboratoriesengineersaspartofthecompany'scon-tinuing education program. Gobind Daryanani ACKNOWLEDGMENTS It iswithgreatpleasurethatItakethisopportunitytothankmycolleagues atBellLaboratoriesfortheirhelpandencouragement.Inparticularlowe muchtoPaulFleischerforhiscompleteandthoroughreviewof theentire book. He made significant improvements in the choice of material for the book, the manner of presentation, and clarification of many ideas. I amverygratefultomycolleaguesDanHilberman,JosephFriend,James Tow,RenatoGadenz,DouglasMarsh,T a - ~ . f uChien,GeorgeThomas,and GeorgeSzentirmai fortheirthoroughreviewsandmany excellent suggestions onthesectionsrelatedtotheirareasof expertise.Iamespeciallygratefulto my wife,Carol, forher help withthe computer programs. I thank BellLaboratories forthe support provided meinthe writing of this book. Specifically,Ithankmydepartment headCarl Simone forhisconstant encouragement.MyappreciationalsogoestoDarleneKurotschkinand Patricia Cottman forthe typing of the manuscript. G.D. ix CONrENrs 1.Network Analysis 1.1RLCPassiveCircuits 1.2RLC Circuits with ActiveElements 1.2.1Dependent Current Sources 1.2.2DependentVoltageSources 1.3SimplifiedAnalysisof Operational-Amplifier Circuits 1.4ConcludingRemarks 2.NetworkFunctions and TheirRealizability 3 8 8 11 15 19 2.1NetworkFunctions31 2.2Propertiesof AllNetworkFunctions34 2.3Properties of DrivingPointFunctions37 2.3.1PassiveRLC Driving Point Impedances38 2.3.2PassiveRCDriving Point Impedances39 2.3.3PassiveLC Driving Point Impedances42 2.4Propertiesof TransferFunctions44 2.5Magnitude andPhasePlots of NetworkFunctions45 2.6TheBiquadraticFunction56 2.7ComputerProgramforMagnitude andPhase59 2.8ConcludingRemarks61 3.Introductory Filter Concepts 3.1CategorizationofFilters 3.1.1Low-Pass Filters 3.1.2High-Pass Filters 3.1.3Band-Pass Filters 3.1.4Band-Reject Filters 3.1.5GainEqualizers 3.1.6Delay Equalizers 3.2Passive,Active,andOtherFilters 3.3ConcludingRemarks 73 73 75 76 81 84 85 88 92 xi xiiCONTENTS 4.The ApproximationProblem 4.1BodePlot ApproximationTechnique 4.2Butterworth Approximation 4.3Chebyshev Approximation 4.4Elliptic Approximation 4.5BesselApproximation 4.6DelayEqualizers 4.7Frequency Transformations 4.7.1High-Pass Filters 4.7.2Band-Pass Filters 4.7.3Band-Reject Filters 4.8Chebyshev ApproximationComputerProgram 4.9ConcludingRemarks 5.Sensitivity 6. 5.1()) andQSensitivity 5.2Multi-ElementDeviations 5.3GainSensitivity 5.4Factors AffectingGainSensitivity 5.4.1Contributionof the Approximation Function 5.4.2Choiceof theCircuit 5.4.3Choiceof ComponentTypes 5.5Computer Aids 5.6ConcludingRemarks PassiveNetwork Synthesis 6.1SynthesisbyInspection 6.2DrivingPoint Synthesis 6.2.1Synthesis Using Partial FractionExpansion 6.2.2SynthesisUsingContinued FractionExpansion 6.3Low Sensitivityof PassiveNetworks 6.4TransferFunctionSynthesis 6.4. 1SinglyTerminated Ladder Networks 6.4.2ZeroShiftingTechnique 6.4.3Doubly Terminated Ladder Networks 6.5ConcludingRemarks 97 100 107 114 117 123 126 127 129 133 135 137 147 151 156 159 159 164 165 172 174 183 186 187 191 196 198 198 202 211 224 CONTENTS 7.Basics of Active Filter Synthesis 7.1FactoredFormsof theApproximationFunction 7.2TheCascadeApproach 7.3RealPolesandZeros 7.4BiquadTopologies 7.4.1Negative FeedbackTopology 7.4.2PositiveFeedbackTopology 7.5CoefficientMatching Technique forObtainingElement Values 7.6Adjusting theGainConstant 7.7ImpedanceScaling 7.8Frequency Scaling 7.9ConcludingRemarks 8.Positive FeedbackBiquadCircuits 9. 8.1PassiveRCCircuits UsedinthePositiveFeedback Topology 8.2SallenandKeyLow-PassCircuit 8.3High-Pass CircuitUsing RC-+CRTransformation 8.4SallenandKeyBand-PassCircuit 8.5Twin- T Networks forRealizingComplex Zeros 8.6ConcludingRemarks Negative FeedbackBiquadCircuits 9.1PassiveRCCircuitsUsedintheNegativeFeedback Topology 9.2ABand-PassCircuit 9.3Formationof Zeros 9.4TheUseofPositiveFeedbackinNegativeFeedback Topologies 9.5TheFriendBiquad 9.6Comparisonof Sensitivities of NegativeandPositive FeedbackCircuits 9.7ConcludingRemarks xiii 235 236 239 241 241 243 246 250 254 255 256 267 269 281 285 288 290 299 301 308 313 320 327 330 xivCONTENTS 10.The Three AmplifierBiquad 10.1TheBasicLow-Pass andBand-PassCircuit339 10.2Realizationof theGeneralBiquadraticFunction344 10.2.1TheSumming Four Amplifier Biquad345 10.2.2TheFeedforwardThreeAmplifier Biquad349 10.3Sensitivity350 10.4Comparisonof Sensitivitiesof ThreeAmplifier and SingleAmplifierBiquads353 10.5Tuning354 10.6SpecialApplications356 10.7ConcludingRemarks358 11.ActiveNetworksBasedonPassiveLadder Structures 11.1PassiveLadderStructures 11.2Inductor SubstitutionUsingGyrators 11.3TransformationofElementsUsingthe FDNR 11.4ACoupledTopologyUsingBlockSubstitution 11.5ConcludingRemarks 12.Effects ofRealOperational Amplifiers on ActiveFilters 367 370 375 380 387 12.1ReviewofFeedback Theory andStability397 12.2OperationalAmplifierFrequencyCharacteristics and CompensationTechniques402 12.3EffectsofOpAmpFrequencyCharacteristicson FilterPerformance407 12.4OtherOperationalAmplifierCharacteristics420 12.4.1Dynamic Range420 12.4.2Slew-Rate Limiting421 12.4.3OffsetVoltage423 12.4.4Input-Bias and Input-Offset Currents424 12.4.5Common-Mode Signals426 12.4.6Noise428 12.5ConcludingRemarks429 CONTENTS 13.DesignOptimization andManufacture of ActiveFilters 13.1Reviewof theNominalDesign 13.2DesignofPracticalFilters 13.2.1Overdesign 13.2.2Choiceof Components 13.3Technologies 13.3.1Integrated CircuitOperational Amplifiers 13.3.2DiscreteCircuits 13.3.3Thick-FilmCircuits 13.3.4Thin-FilmCircuits 13.4ConcludingRemarks APPENDIXES APARTIALFRACTIONEXPANSION BCHARACTERIZATIONOFTWO-PORTNETWORKS CMEANANDSTANDARDDEVIATIONOFA RANDOMVARIABLE DCOMPUTERPROGRAMS ANSWERSTOSELECTEDPROBLEMS INDEX xv 439 442 444 4 4 ~ 457 458 458 459 459 460 463 466 469 476 484 491 l, NETWORK ANALYSIS Inorder to develop the designprocedures foractive andpassive networks, itis firstnecessary to have good analysis techniques. While there are several different methods for analyzing these networks, nodal analysis willbe used in thisbook. Thismethodof analysisissimple,quitegeneral,andverysuitableforactive andpassivefiltercircuits.Althoughitisassumedthatthestudentisfamiliar withtheprinciplesanduseof nodalanalysis,abriefreviewisgiveninthis chapter.Inparticular, the analysis of circuits containing operational amplifiers, resistors, and capacitors, which are the elements constituting most active filters, iscoveredindetail.Theexampleschosennotonlyreviewnodalanalysisbut alsoservetointroducesomeelementaryprinciplesofsynthesis.Computer aidsthat canbeusedfortheanalysisof networksarereferencedattheendof the chapter. 1.1RLCPASSIVECIRCUITS Inthisintroductory sectionwereviewthesdomainnodalanalysisof passive networksbyconsideringthefollowingsimple exampleof acircuitcontaining resistors, capacitors, and anindependent current source. Example 1.1 FindthesdomainfunctionJ.j(s)/I1(s)forthecircuitshowninFigurel.la (knownasabridged-T network). Solution The firststep inthe analysis isto express the admittance of each element inthe s domain, as shown in Figure l.lb. In this circuit the voltages at nodes1,2, and 3 withrespect to groundare designatedVt(s),V2(s),andV3(s),respectively. Thenode equations are obtainedbyusingKirchhoff's currentlawatnodes 1,2,and 3,asfollows :t node1: or (1.1) Someothermethods[4]foranalyzingnetworksusemeshanalysis,theindefinitematrix approach, signal-flow graphtechniques, andthe state-spaceapproach. tHereafter. IandVareusedtomeanI(s) andV(s). 3 4NETWORKANALYSIS 1.1RLCPASSIVECIRCUITS5 c , or ( 1.3) v, CD The nodalmatrixrepresentation of the above equationsis 1 I,t R; + SCI RI - SCIVIII 11 RI RI+ R2+ sC2 R2 V2 0(1.4) (al 1 -SCI R2 R2+ SCI V30 sC, Using Cramer's rule[4],V3/ I I isgivenby 1 - + SCI RIRI V2 (s) V,(s) V3(s) CD 1.. CD .!. 0 R,R2 11 - + - + sC2 0 RIRIR2 I, (s) + SC2 V3 - SCI0 R2 (1.5) II1 RI+ SCI RI -SCI RI 11 - + - + sC2 RIR2R2 (bl" Figure 1.1(a)CircuitforExample1.1. 1 - SCI R2 R2+ SCI (b)Circuit showingadminancesins domain. node 2: (1.6) or (1.2) node3: 6NETWORKANALYSIS whichsimplifiesto Obserpations (1.7) 1.Since nodalanalysisusesKirchhoff's current law,it ismost convenientif theindependentsourcesare currentsources.Most circuits,however,are drivenbyvoltagesources.Theformulationofthenodalequationsof circuits containing voltage sourcesrequiresthe useof Norton'stheorem, which states that avoltage sourceVin series withan impedance Zcan be replacedby acurrentsourceI=VIZ,inparallelwiththeimpedanceZ. Figure1.2illustratesthisequivalence.Typically,Zwouldbetheinternal impedance of the voltage source. z v+ I=tt z Figure 1.2Norton'sequivalent circuit ofa voltagesourceinseries with animpedance. 2.Thenodaldeterminant,whichisthedenominatorofEquationI.S,is symmetricalaboutthediagonal.Thusthetermintheithrowandjth column [i.e., the (i,j) term], is equal to the U, i) term. This is a characteristic property of the nodal (and mesh) determinants of RLC networks. * 3.Supposewehadtheproblemof findinganetworkthatwillrealizethe function: VO S2+ es+ d -=2 l'Nns+ as (l.8) where the coefficients e, d, n, and a are positive numbers. Since this function has the same form asEquation1.7,itshould be possible to realize it using Allreciprocalnetworkshavethisproperty [4]. 1.1RLCPASSIVECIRCUITS7 the circuit of Figurel.la. The element valuesof the desired circuit can be obtained by equating the coefficients of equal powers of sin (1.7) and (1.8): II n= - + - (1.9) RlR2 (LIO) (UI) d==a(LI2) R1R2C1C2 C2 From (LII) and (LI2) it isseenthat the given coefficients must satisfy the relationship na -ed and the capacitor C 2isgivenby na C2 =- =-ed One choice of resistorsthat satisfiesEquations1.9is Substituting in(LIO),the remaining unknown,C 1> is n2 C1=-4a (LI3) Thus we see that if the given coefficients satisfy the relationship nle=aid, the transfer function of Equation 1.8 can be rea!ized by the circuit of Figure l.la with (LI4) The above discussionindicatesthat whenever acircuitisanalyzedthe resultsof theanalysiscanbeusedforthesynthesisof arelatedclassof functions. 4.Let us next consider the synthesis of aslightly different function: VO S2+ es+ d liNns2 + as+ b (LIS) 8NETWORKANALYSIS AcomparisonwithEquation1.7tellsusthatthecircuitof Figurel.1a willnotworkif bisnonzero. The questions the reader may now askare: (a)Canthefunctionberealizedwithreal(nonnegative)elementsusing a different circuit? (b)Supposethefunctionisdeterminedtoberealizable;howdoesone proceedtofindtherequiredcircuitandtheelementvaluesforthe circuit? Thefirstquestionrelatestotherealizabilityof functions-wewillhave more to say about thisinChapter 2.The answerstothe secondquestion constitutethesynthesisof networks,andthemajorpartof thistextis devoted to thisproblem. 1.2RLCCIRCUITSWITH ACTIVEELEMENTS ThusfarwehavediscussedtheanalysisofRLCcircuitswithindependent current andvoltage sources.Next wewillconsider the analysis of RLC circuits containingactivedevices.Themodelof anactivedevicewillalwaysinclude a voltage or current source whose value depends on a voltage across, or a current through,someotherpartof thecircuit.Thus,tobeabletoanalyzecircuits containing active devices,itbecomes necessaryto study the nodal formulation of circuitswith dependentcurrent and voltage sources. 1.2.1DEPENDENTCURRENTSOURCES Thetwotypesof dependentcurrentsourcesencounteredinactivenetworks are thevoltage controlledcurrent source (VCCS,Figure1.3a)andthecurrent controlled current source (CCCS,Figure1.3 b). Examplesof thesetypesof sourcesaretheVCCSmodelof thetransistor (Figure1.4b),andthe CCCSmodel of thetransistor (Figurel.4c). The models shown are commonly referredto as hybrid-n models. The firststep inthe nodal equation formulationof these circuits isto express a0 + v o0 (al(bl Figure 1.3(a)Voltagecontrolled currentsource(VCCS). (b)Currentcontrolled current source(CCCS) . 1.2RLCCIRCUITSWITHACTIVEELEMENTS9 ~ C o " ~ ' "(C) Base(8) (a) Emitter(F.) " C,. 8 1 '" v c"f g,.v ~ t (b) E ;8 '. c,. B E (e) Figure 1.4Hybrid-n modelsfor a transistor : (a)Symbol; (b)VCCSmodel ;(c)CCCSmodel. c '0 C thedependent current sourcesinterms of the circuitnodevoltages.The node equations can thenbe written asbefore, treating the dependent current sources asiftheywereindependentsources.Finally,theequationsarerearrangedso that only the independent sources occur on therighthand side of the equality. The following example illustrates theprocedure. Example 1.2 Find the nodal matrix equation of the transistor circuit of Figure 1.5a,using the hybrid-nmodelof Figurel.4b.Notethatforthepurposeof acanalysis,the collector of the transistor iseffectively at ground potential. 10NETWORKANALYSIS +5 V C Y'N -::-(a) V, " V2 CI' CD 8 0 C '"c"t I '0

tR;: Rs v30 E RL (b) Figure 1.5(a)Circuit forExample1.2.(b)Equivalentcircuit. Solution The circuit, withthe transistor modeled,isshown inFigure1.5b.In thisfigure the current source Iisgivenby The node equations are node1: (1.16) 1.2RLCCIRCUITSWITHACTIVEELEMENTS11 node 2: - ++ + sC"+ SCI')-+ Sc,,)=0(1.17) rxrxr"r" node 3: -+ SCn)++ + + Sc,,)=g",(V2- V3)(1.18) r"RLror" Thethirdequationisrearrangedsothat allof thenodevoltagesareonthe leftside of the equality: -+ sC"+ gm)+ V3(R1 + + + sC"+ g",)=0 '"Lror" The nodalmatrix equation is,therefore: II -+- 0 Rsrxrx 11 + Sc,,) - + - + sC"+ sC" rxr xr" 0 111 - + - + - + sCn + gm RLror" which can be solvedforVbV2,andV3using Cramer'srule. Observation VI V3

Rs 0 0 (1.19) Thepresenceof thedependentcurrentsource gm(V2- V3) makesthenodal determinantnon symmetrical[the(2,3)termisnotthesameasthe(3,2) term].Suchasymmetryusuallyoccursincircuitscontainingdependent sources. 1.2.2DEPENDENTVOLTAGESOURCES Thetwotypesof dependentvoltagesourcesencounteredinactivenetworks arethevoltagecontrolledvoltagesource(VCVS)andthecurrentcontrolled voltage source (CCVS), shown inFigure1.6aand1.6b,respectively. Examplesof theV C V S ,arethetriode (Figuru. 7)and-the differentialoper-ational amplifier (Figure(8). One example of the use ofa CCVS isin modeling the gyrator (seeProblem 1.8). A circuit containing dependent voltage sources can be analyzed by converting thevoltage sources to current sources using Norton's theorem, as explainedin Example 1'.1.This circuit, with dependent current sources, can then be analyzed just as inthe last section. The following example illustrates the procedure. o ~ - - - - - o + )J Vo ~ - - - - - o (a) Figure 1.6(a)Voltagecontrolledvoltage source(VCVS) . (b)Currentcontrolledvoltagesource(CCVS) . G Plate(PI 00 + Grid(GI v Cathode(CI C (al Figure 1.7VCVSmodelfora triode:(a)Symbol.(b) V' (a)(b) Model. Figure 1.8 (a)Symbol. VCVSmodelfor a differentialoperationalamplifier : (b)Model. 12 (bl )J V(bl + 1.2RLCCIRCUITSWITHACTIVEELEMENTS13 Example 1.3 FindthefunctionVO/ VINfortheoperationalamplifier(hereafterabbreviated as op amp) circuit showninFigure1.9a. Solution Theequivalentcircuit,obtainedbyusingtheopampmodelof Figure1.8b, isshown inFigure1.9b. The nodal equationsforthe current source equivalent of this circuit (Figure1.9c),are : node1: ( 111 )(1)VIN VI- +-+- - Vo- =-RsriRFRFRs node 2: R, 0 Rs Vor . I -=- -=- J. -=-fa) (Dv, Rf 0 + VIN R.m)or amultiplezero(m>n)on theimaginary axis,ats=joo. 2.3.1PASSIVERLCDRIVINGPOINTFUNCTIONS PassiveRLCnetworkscontainnoenergysourcesandassuchtheycanonly dissipate- but not deliver-energy. This dissipative nature of passive networks imposesafurtherrestrictiononthedpfunction.Therestrictionisthatifthe functionisevaluatedatanypointonthejwaxis,therealpartwillbenon-negative.Mathematically ReZ(jw)~0 forallw (2.12) A heuristic argument[7]to justify this statement follows . Suppose the network functionhasanegativerealpartatfrequencys= jwl Thenthefunctionat oS = jWIcan be written as ZUwd =-R(wl)+ jX(wd Wecanplaceacapacitance(orinductance)whosereactanceis- jX(wdin series withthe network (Figure 2.5),to cancel the reactance term inthe original circuit. This newnetwork, whichisalsopassive, hasanimpedance ZIUwd=-R(wl) whichisanegativeresistance.If weappliedavoltagesourceatthisfrequency tothisnetwork,thenetworkwoulddelivercurrenttothesource,thereby violatingthepassivenatureofthecircuit.Thusthenetworkcannothavea negativerealpart forany frequency on the jw axis. Another property of passive dp functionsisthat the residues of jw axispoles (definedinAppendixA)mustberealandpositive.Wereferthereader to[12] fortheproof. - jX(w, ) or v Z,(jw,)Z (jw,) Figure 2.5Cancellationofthereactanceof the RLCdp impedance Z(jw,) . RLC network 2.3PROPERTIESOFDRIVINGPOINTFUNCTIONS39 Summarizing, apassive driving pointfunction:* Must be arationalfunctionin s withreal coefficients. Maynothavepoles or zeros intherighthalf splane. May nothavemultiple poles or zeros on the jw axis. May not have the degrees of the numerator and denominator differing by more than one. Must have anonnegative realpart foralls= jw. Must have positive'and real residues forpoles on thejw axis. 2.3.2PASSIVERCDRIVINGPOINTFUNCTIONS Inthissectionweconsidernetworksconsistingof resistorsandcapacitors. Such networks willbeneededin the synthesis of active filtersusing operational amplifiers,resistors,andcapacitors.It isexpectedthatbyrestrictingtheclass ofusablecomponentsinanetwork,furtherlimitationsareimposedonthe correspondingnetworkfunction.RC drivingpointfunctionsmustnecessarily satisfy the properties of passive dp functions.Additional properties that must be satisfiedby RC networks are describedinthe following. It wasshown that the poles and zeros mustbe inthe lefthalf plane foralldp functions.Thefollowingisaheuristicargumenttoshowthatthepolesand zerosforRC networksmustlieonthenegativerealaxisof thesplane.If any poles are off the axis, the corresponding term in the impulse is given by Equation 2.9, 11(t)=2K lealCOSbe which,fornegativea,isanexponentiallydecayingsinusoid(Figure2.6). However,suchasinusoidalresponserequirescapacitors andinductorsinthe circuittostoreandreleaseenergyinalternatehalf cyclesof theresponse.A networkconsistingofonlyresistorsandcapacitorscannotproducesucha response;therefore,thecorresponding networkfunctioncannot have poles off the axis. Since the reciprocalof a dp function is also a dp function,the zeros also have thissamerestriction. Letusnextconsiderthebehaviorof RCimpedancefunctionsatdcandat infinity.Atdcallthecapacitorsbecome open circuitssotheRC networkwill reducetoaresistor,ortoanopencircuit.If itisresistive,thefunctionisa positiveconstantatde;if itisanopencircuit,thefunctionhasapoleatde. * The properties arc equivalenttothe positivereal (p.r.)property of passive driving pointfunctions whichstates that: I.H( s)isrealforrea l s 2.Re[H(sJJ2:0forRes2:0 Thep,r.propertyismentionedhereforcompleteness; adetaileddiscussionof thistopic canbe foundinanystandardtextbookonpassivenetworksynthesis[12]. 40NETWORKFUNCTIONSANDTHEIRREALIZABILITY \ jw \ \ \ xt h(t) --------r-------O x '\ '\ '\ I '\ '\ / / / / ;' ;' / Figure 2.6Impulseresponsedue tocomplexpolesinthelefthalf s plane. Again at infinite frequencythe capacitors are short circuits sotheRC network willreducetoaresistor,or toashortcircuit.If itisresistivethefunctionisa positive constant ; if a short circuit the functionhas a zero at infinite frequency. Usingtheaboveproperties,theRCdpimpedancefunctioncanbewritten inthe form. ZRC2

./ I--" 30 20 +-

- I-10 5678910 1

V n=5 '/ /

4V /v "" 3V I--' ./ V 40 1--'1--'1..-'./ 30 20 I-- I-10 . 1 .2.3.4 .5 .6.81.1.52345678910 0-Figure 4.10Lossof LP Chebyshevapproximationfor Am .. = 0.50 dB. 110 4.3CHEBYSHEVAPPROXIMATION111 / /V /V t

::: .8 0..J .6 V/ 1/ V/ V J VV .L. V 1III\ I-'"' -r----g.4 .. .2 .. II n=51ol V / /

IL /1/' V3vl--I-'" V ./ I--' V

120 110 100 90t 80en'0 70 ..J 601?2l 5015-V540 30 20 10 .... 1.2 , .3.4.5 .6.81. 1.52345678910 n Figure 4.11Lossof LP Chebyshevapproximationfor Am =1 dB. One of the objectives in considering an equiripple passband was to improve on the stopband attenuationprovidedby theButterworthapproximation. Letus comparethesetwoapproximationsforWWp.FromEquation4.7,the Butterworth attenuation for WWpisapproximately 20iog1oe(:J"(4.31) The Chebyshev attenuation isobtained fromEquation 4.28, where for WWp (i .e.,Op1)theterm eC"(O)1.Thus A(O) 101 20 loglo eC"(O) From (4.27), for 01 C"(O)2"-10" Usingthis expression, (4.32)reducesto 1 ==202"-IJ WpW/Wp1Wp (4.32) (4.33) Comparing(4.31)and(4.33),itisseenthattheChebyshevapproximation provides 20 iog(2)" - 1 =6(n- 1)dB(4.34) 112THEAPPROXIMATIONPROBLEM moreattenuationthanaButterworthofthesameorder.Therefore,forthe samelossrequirementstheChebyshevapproximationwillusuallyrequirea lower order than theButterworth. Let us next findthe roots ofthe function H(s).As in the Butterworth case these roots are foundby firstevaluating the roots of 1 H(sW,where 1H(s) 12 =1 + e2C;(Q)i{l~sli The rootsof theabove functioncanbe showntobe[3] k=0,1,2, ... , 2n- 1 where .1t(1+ 2k).(1.1) (Jk=sm"2--n- smh~smh-I s 1t(1+ 2k)(1.1) Wk=cos"2--n- cosh~smh-I s Asinthe Butterworth approximation the n lefthalf plane roots, corresponding tonegative (J,areassociatedwithH(s).Furthermore, from4.37a andb,itcan easily be seenthat whichisthe equation of an ellipse. Thus the roots of the Chebyshev approxima-tionlieonanellipseinthesplane,whoserealandimaginaryinterceptsare indicated inFigure 4.12. TheChebyshevapproximationfunctioncannowbeexpressedinfactored form,as 1 H(s)=n K (s- s) J where s jare the lefthalf plane roots of 1H(s) 12.The denominator constant Kis adjusted to provide a loss of 0 dB at the passband minima. The factored form of thelossfunctionforAmax=0.25dB,0.5dB,and1 dB,forordersupton=5 are given in Table 4.2. The polynomials given inthe tables apply to the normal-izedfilterrequirements,forwhichthepassbandedgefrequencyW=Wp=1. For the general LP filter, with the passband edge at W=Wp,these polynomials needtobe denormalizedbyreplacing s sby-Wp 4.3CHEBYSHEVAPPROXIMATION113 Table4.2Chebyshev Approximation Functions (a)Am".=0.25 dB IINumerator of H(s) s+ 4.10811 2S2+1.79668s+ 2.11403 3(S2+ 0.76722s+1.33863)(s+ 0.76722) 4(S2+ 0.42504s+1.16195)(s2+1.02613s+ 0.45485) 5(S2+ 0.27005s+1.09543)(S2+ 0.70700s+ 0.53642)(s+ 0.43695) (b)Ama.=0.5 dB nNumerator of H(s) s+ 2.86278 2S2+1.425625+1.51620 3(52+ 0.62646s+I. 14245)(s+ 0.62646) 4(S2+ 0.35071 s+1.06352)(S2+ 0.84668s+ 0.356412) 5(S2+ 0.22393s+1.03578)(S2+ 0.58625s+ 0.47677)(s+ 0.362332) (c)Amax=1 dB IINumerator of H(s) 1s+1.96523 2S2+1.09773s+1.10251 3(S2+ 0.494175+ 0.99420)(s+ 0.49417) 4(S2+ 0.27907s+ 0.98650)(S2+ 0.67374s+ 0.27940) 5(S2+ 0. 178915+ 0.98831) (S2+ 0.46841.1+ 0.42930)(s+ 0.28949) Theuseof thesetables isillustratedbythefollowingexample. Example 4.6 Denominator Constant K 4.10811 2.05405 1.02702 0.51352 0.25676 Denominator Constant K 2.86278 1.43138 0.71570 0.35785 0.17892 Denominator ConstantK 1.96523 0.98261 0.49130 0.24565 0.12283 Findthelow-pass approximation functionforthefilterrequirements Wp=200Ws=600Amax=0.5 dBAmin=20dB SOlution ThenormalizedstopbandedgefrequencyisQs =600/200=3.FromFigure 4.10,the required order is3.ThenormalizedLP function,H N(S),forn =3 and 114THEAPPROXIMATIONPROBLEM jw /COSh (tsinh-'t) sinh(-I;- sinh-1 1-) - - - - - + - - - - ~ - - ~ ~ - - - - - a Figure 4.12Locusof rootsfor Chebyshev approximation. A m a ~= 0.5dBisobtained fromTable 4.2b: (S2+0.62646s+1.14245)(s+0.62646) HN(S)=0.71570 The desired third-order LP filter function H(s), is obtained by denormalizing this function,byreplacing s by s/200: H() =(S2+125.3s+45698)(s+125.3) s5725600 4.4ELLIPTICAPPROXIMATION WehaveseenthattheChebyshevapproximation,whichhasanequiripple passband,yieldsagreaterstopbandlossthanthemaximallyflatButterworth approximation.Inboth approximationsthe stopbandlosskeepsincreasing at the maximum possible rate of 6n dB/octave for an nth order function. Therefore theseapproximationsprovideincreasinglymorelossthanthe fiatAminneeded abovetheedgeof thestopband.Thissource of inefficiencyisremediedbythe elliptic approximation (alsoknown astheeauer approximation). The elliptic approximation isthe most commonly used function in the design of filters.A typicalellipticapproximationfunctionissketchedinFigure4.13. The distinguishing feature of this approximation functionisthat ithas poles of attenuationinthestopband.Thustheellipticapproximationisarational t 4.4ELLIPTICAPPROXIMATION115 tv I I I r I I I I I I I I w_____ Figure 4.13Typicallosscharacteristicof a LP ellipticapproximation. functionwithfinitepoles and zeros,whiletheButterworth and Chebyshev are polynomials and as such have alltheir loss poles at infinity. In particular, in the elliptic approximationthelocation of thepolesmustbechosentoprovide the equiripple stopband characteristic shown. The pole closest to the stopband edge (wp,)significantlyincreasestheslopeinthetransitionband.The furtherpoles (wp2and infinity) are needed to maintain the required level of stopband attenua-tion.Byusingfinitepoles,theellipticapproximationisabletoprovidea considerablyhigherfiatlevelofstopbandlossthantheButterworthand Chebyshev approximations. Thus foragivenrequirementtheelliptic approxi-mationwill,ingeneral,requirealowerorderthantheButterworthorthe Chebyshev.Sincealowerorder correspondstolesscomponentsinthefilter circuit,theellipticapproximationwillleadtotheleastexpensivefilter realization. The mathematical development of the elliptic approximation isbased on the rathercomplextheoryof ellipticfunctions,whichisbeyondthescopeof this book. The interested reader isreferred to R.W.Daniel's text on approximation methods[3]fordetails.Thepolesandzerosof theLP elliptic approximation functionhavebeentabulatedforalargenumberofcasesbyChristianand Eisenmann[2].Asampleof somenormalizedellipticLPfilterfunctions,in factoredform,isgiveninTable4.3.Inthesetablesthefrequenciesn are normalized to the passband edge frequency Wp(i.e., n =w/wp).The denomina-torconstantK,showninthesecondcolumn,isdeterminedbythedeloss. Observethat adifferenttable isneededforeachvalueof Os, where _Ws_stopband edgefrequency s- Wp- -p-a-='ss-b-a-n-d-e-d-g-e-fr-e-q-u-e-n-cy AlsodifferenttablesareneededforeachA m a ~ ' 116 IIcf ~ '0 ... o 'iii c . ~ c .., o :..: 0:1 ~ CC .- 0:1EV;oc c0 OU ;;):::-000 - \0 .-..-. ' '"0; Cl 1.0 .8 .6 .4 .2 o .1 -r---......... .2.4 -r--. i'- ....., '" \. "'-'\.\ ..... r-,. '\\ I'r\ \ \. l'\: =1 \? "-\ " .6.81 Q 2 Figure 4.17Delayof LP Besselapproximations. Example 4.8 1/'='5 4 i?3 t V2 r 1

",.

',f/ " ./ 46810 '" r\1\ \\ , r\\1\ 'y\4 5 \.\ \ 1"-"\ i\. \ '-.. ....... , " ..............---i" .....::: .....46810 Find the LP Bessel approximation function forthe following filter requirements: (a)The delay mustbeflatwithin1 percent of the dc value up to 2 kHz. (b)The attenuation at 6 kHz must exceed25dB. Solution As a first attempt we try a fourth-order filter.From Figure 4.17, the fourth-order Besselapproximationisseento have adelay thatisflatto within1 percentup 122THEAPPROXIMATIONPROBLEM to "C '" '"o ...J 4 8 81216tsec (c) (b) Figure 4.18Characteristics of a fourth-orderBesselapproximation (Amax=3 dB);(a)Loss.(b)Delay.(c)Stepresponse. n to approximately 0=1.9. To satisfy the given delay requirements this normal-izedfrequencymust correspond to w =2 kHz. Thus, the normalized frequency corresponding to 6 kHz mustbe 6 Os= 2 (1.9)= 5.7 Atthisfrequencytheattenuation,fromFigure4.16,isonly22dB.Sincethe fourth-orderapproximationdoesnotmeetthelossrequirements,wewill nexttry afifth-orderfunction. FromFigure4.17thedelaystayswithin1percentuptoapproximately o =2.5,for11 =5.If this frequency corresponds to w=2 kHz, the normalized frequency corresponding to w= 6 kHz is 6 Os=2 (2.5)=7.5 From Figure 4.16, the fifth-order functionisseen to provide 29.5 dB of attenua-tionatthisfrequency.Thefifth-orderBesselapproximationthereforesatisfies both the delay andthelossrequirements. 4.6DELAYEQUALIZERS123 Thenormalizedfifth-orderfunctionislistedinTable4.4.Thisfunctionis denormalizedbyreplacing s by sTowhere, fromEquation 4.47, o2.5-4 To=w=2n(2000)=1.989(10)sec The resulting denormalizedgain functionis Vo3.608(108) ~ NS2+ 3.370(104)s+ 3.608(108) 1.8335(104) S+ 1.8335(104) Observation Incomparison,afifth-orderButterworthwithacutoff frequencyof 2kHz wouldprovideapproximately55dBof attenuationat6kHz.Thus,wesee thatwhile theBesselapproximation doesprovide aflatdelay characteristic, itsfilteringaction inthe stopband ismuch worse than the Butterworth! The poorstopbandcharacteristicsoftheBesselapproximationmakesitan impracticalapproximationformostfilteringapplications.Analternate solution to the problem of attaining aflatdelay characteristic isby the use of delay equalizers, whichisthe subject of thenextsection. 4.6DELAYEQUALIZERS Intheprecedingsectionwediscussedthetimeresponsedistortionresulting fromthenon flatdelaycharacteristicsof filters.TheBesselapproximation did yieldaflatdelayinthepassband; however, itsstopband attenuation proves to be inadequate formost filter applications. In this section we present an alternate way of obtaining flatdelay characteristics without sacrificing attenuation in the stopband. The approach used isto firstapproximate the required lossusing the Butterworth, Chebyshev, or elliptic functions.The delay of thisapproximation functionwill, 'of course,haveripplesandbumpsandwillcertainlynotbeflat (Figure4.14and4.15).Therefore,somemeansisneededtocompensatefor thedelaydistortionintroduced.AsmentionedinChapter3(page87),this compensation can be achieved by following the filter-circuitby delay equalizers. The purpose of the delay equalizer isto introduce the necessary delay shape to makethetotal delay (of thefilterandequalizer) asflataspossible inthepass-band.Furthermore,theequalizermustnotperturbthelosscharacteristic and thereforethelossof the equalizermustbeflatforallfrequencies. InChapter3itwasshownthatasecond-orderdelayequalizercouldbe realizedbythe all-pass function : VOS2- as+ b -= 2 ~ NS+ as+ b (4.48) 124THEAPPROXIMATIONPROBLEM jw X0 ------------r-----------a x o. Figure 4.19Polesandzerosofa second-order delayequalizer. forwhichthepole-zerodiagramisasshowninFigure 4.19.Thedelayof this functioncanbeevaluatedusingEquation3.13orfromtheMAGcomputer program(AppendixD).Theresultingdelaycharacteristicsfordifferentvalues of pole Q( =jbla) are sketchedinFigure 4.20.Thenormalizingfrequencyfor these curvesisthe pole frequency(w=jb), that is jb (4.49) FromthesecurvesitcanbeseenthatthedelaybumpsOCCuratorcloseto Q= 1 (i.e., w= jb) forQp>1.The parameter b in (4.49) therefore determines the location of the delaybump. The sharpness of thebump isafunctionof the parameter a.The curves showninFigure 4.20 are asample of the large variety of delay shapesthatcanbegeneratedbythe second-order equalizerfunction. Thegeneraldelayapproximationproblemconsistsof findingtheminimum numberof second-orderdelayfunctionsof theformof Equation4.48,whose delaysaddtogethertoyieldthedesireddelaycharacteristic.Theformof the desireddelay equalizer functionistherefore T(s)=fr S2- QjS+ bjj= IS2+ ajs+ bj(4.50) ThenumberofdelaysectionsNandtheirdefiningparameters(aj,b;)for approxi I11ating a given delay shape are usually obtained by computer optimiza-tion. Asanexample,considertheequalizationof thefourth-orderChebyshev delayfunctionsketchedinFigure4.21.Byusingacomputeroptimization program,theparametershavebeenfoundforthetwosecond-orderdelay sectionssketchedinthefigurewhichwillequalizethedelay.Theparameters 4.6DELAYEQUALIZERS125 ..; > '"OJCl '0 ! 9 J/i 7 / , 1-2 ....... I J!1\ 5 k:-1.5..;'1 IQp=0.5

,> ,,\ 3r-vV' '\ /I"--- -.....""- ,8 6 o_1 IIII 0.'0.3O. 0.70.9,. ,1.31.51.71. 9 .11 __ Figure 4.20DelayClfsecond-order delayequalizersfor different poleOs. definingthesetwo _Wp +-ws dex a::; ->dc Ingeneral,theHPpassband,fromWpto00,transformstotheLP passband 0 to1;andtheH Pstopband,0tows,transformstotheLPstopbandwp/ws to00.ThereforeEquation4.51transformsahigh-passfunction,THP(S),toa low-passfunction,1!.p(S), definedinthe Splane: (4.53) Fromthisequationitisseenthatthe attenuationof theHPfilterat S=SIis the same asfortheLP filterat S=wp/sl Torealizeagivensetof HPfilterrequirements characterizedbyAma.,Amin, Wp ,Ws(Figure4.23),wefirsttransformthese,usingEquation4.53,tothe 4.7FREQUENCYTRANSFORMATIONS129 equivalentnormalizedLPrequirements characterizedbyAma.,Amin,1,wp/ws (Figure4.24).TheseLPrequirementsarethenapproximatedusingthe Butterworth, Chebyshev, elliptic, Bessel, or other functions,the choice depend-ingonthefilterapplication.Finally,thenormalizedlow-passfilterfunction obtained,TLP(S),istransformedtothedesiredhigh-passfilterfunctionby usingEquation 4.53. Example 4.9 FindaButterworthapproximationforthehigh-passfilterrequirements characterizedby Amin=15dBAmax=3 dBWp=1000Ws=500 Solution The equivalentnormalizedLP filterrequirements are A.nin=15dBAmax=3 dB From Figure 4.5,itis seen that a third-order filter willmeet these requirements. Thenormalizedthird-orderLPButterworthfilterfunction,fromTable 4.1,is Vo(S)1 TLP(S)=V1N(S)=(S2+S+I)(S+1) ThecorrespondingH PfilterfunctionTHP(S)isobtainedbyreplacingSby 1000/s: THP(S)= (S2+1000s+106)(s+10(0) This functionwillhaveaButterworth characteristic (thatis,willbemaximally flatatinfinity) and willmeetthe prescribed high-pass requirements. 4.7.2BAND-PASSFILTERS Inthissectionweconsidertheapproximationofband-passfiltershaving requirementsasshowninFigure4.25.Again,usingfrequencytransformation techniques,thebandpassfunctionTBP(S)canbetransformedtoanormalized low-passfunctionTLP(S).Thefrequencytransformationthataccomplishes thisis (4.54) where B=W2 - WIisthepassband width of BP filter Wo=vi WI W2 isthe center (geometric mean) of thepassband 130THEAPPROXIMATIONPROBLEM t OJ "C ::lo -' Figure 4.25Atypicalband-pass function. Toshowthis,letusapplythetransformationtotheBPfunctionshownin Figure 4.25.For frequencieson theimaginary axis, s=jw and S=jil, so _W2 + w2 il=- 0 (w2 - wl)w (4.55) Using this equation, the center of the BP passband Wo(Figure 4.26) translates to 22 A__- Wo+ Wo_0 ~ ' o- -(W2- wdwo (4.56) The passband edge WIistranslated tothe low-pass frequency 22 ill=- -WI+ Wo=-1 (W2- wdwl (4.57) and thepassband edge W2 istranslated toil2=+ 1.Theband-pass passband, WIto w2,can be seen tobe transformed tothe frequencyband-1 to+ 1. Next, considering the stopband edge frequencies, we see that W3is transformed to and W4 istransformed to - w ~+ W6 (w2 - WI)W3 -wi + W6 (w2 - WI)W4 (4.58) (4.59) /\ I\ I\ I\ ,,-'/\ -_.....\ -1 t OJ "C tr o -' 4.7FREQUENCYTRANSFORMATIONS131 Figure 4.26Normalizedlow-pass function. If thestopbandattenuationexhibitsgeometricalsymmetryaboutthecenter frequency,thatis,if (4.60) then,from(4.58),W3andW4 willtranslatetothetwoedgesofthelow-pass passband: (4.61) Thus the stop bands of theband-pass functionaretransformedtothelow-pass stopbands fromils to00and-ils to- 00,where ils =(W4- (3)/(W2 - WI)' From the above discussion we see that the band-pass approximation function isrelatedtothelow-pass functionby (4.62) TorealizethesymmetricalBPrequirementsshowninFigure4.25,firstthe normalizedLP requirement characterized by (Figure 4.26) Amin isapproximated. The requiredband-pass functionTBP(S)isthenobtained from TLP(S)byusing Equation 4.62. 132THEAPPROXIMATIONPROBLEM 1-------1 i--II II II I I I I I I I ..,I I . I I I Itl I I - , W3 W, tw, W. W__ Figure 4.27NonsymmetricalBP requirements. Theabove methodcanbeadaptedtononsymmetricalBPrequirements,as follows.ConsidertheBPrequirementsshowninFigure4.27,whereAmin,# Amin>andW3W4 #wlw2. AnewBPrequirement(dottedlinesinFigure 4.27) whichdoeshavegeometricalsymmetrycanbegeneratedbyincreasingthe lower stopband attenuation to Amin> andbydecreasing W4 to sothat (4.63) The approximationfunctionforthis new, and more stringent, requirementwill certainly meet the original nonsymmetrical requirements. However, the resulting approximationfunctionwillbemorecomplex,andthecorrespondingcircuit realizationmoreexpensivethanisreallynecessarytomeettheoriginalBP requirements.MoreeconoJIlical,directmethodsofapproximatingnonsym-metricalrequirements do exist[3], but their discussionisbeyondthescope of thisbook. Example 4.10 Find the elliptic approximation forthe followingband-pass requirements: Amax=0.5dBAmin=20 passband=500Hz to1000Hz stopbands=dc to 275Hz and2000Hz to00 Solution From the giveninformation WI=2n500w2 =2nl000W3=2n275W4 =2n2000 4.7FREQUENCYTRANSFORMATIONS133 Observe thatthe stopband requirements are not symmetrical (WIW2#W3W4). To obtain geometrical symmetry the frequency W4can be decreased to the new frequency (Equation 4.63) =WIW2 =2nl818 W3 The equivalent normalizedLP requirements are then characterized by Amax=0.5 dBAmin=20Up=1 U= 1818- 275= 308 s1000- 500. From Table 4.3e, forAmax=0.5dBand Us=3.0,thisLP requirement canbe approximatedbythe second-order elliptic function Vo(S)0.083947(S2+ 17.48528) TLP(S)=JIf",(S)=(S2 +1.35715S+1.55532) Using Equation 4.62, theBP functionisobtainedbyreplacing S by S2+WI w2 S2+4n2(500000) (W2- Wl)S2n500s Making this substitution, the desiredBP approximation functionis 0.084(S4+ 2.12(1O)8s2 + 3.89(10)14) TBP(S)=S4+ 4.26(1O)3s3 + 5.48(lOfs2+ 8.42(1O)IOs+ 3.89(10)14 Observation FromEquation4.62weseethat eachLP pole (zero)transformstotwoBP poles (zeros),thustheorder of theBPfilterfunctionistwicethat of theLP filterfunction. 4.7.3BAND-REJECTFILTERS Inthissectionweconsidertheapproximationof thesymmetricalband-reject requirementsshowninFigure4.28.ThestopbandextendsfromW3toW4 , andthepassbandsextendfromdetoWIandW2toinfinity.Thegeometrical symmetry of therequirements imply that (4.64) Thefrequencytransformationusedtoobtaintheband-rejectfilterfunction TBR(S)fromthe equivalent normalizedlow-passfunctionTLP(S)is (4.65) where B=W2- WIisthepassband width 134THEAPPROXIMATIONPROBLEM al -0 ...J Figure 4.28Atypicalband-reject function. and Wo=J W 1 w2 isthe center of the stopband Considertheapplicationof thetransformationof Equation 4.65totheBR functionshowninFigure4.28.For frequenciesontheimaginaryaxis,s = jw and S=j!l, so 0= (w2- wdw _w2 + w6 (4.66) Proceedingasintheband-passcase,itcanbeshownthatthefrequencies wo, w 1, w2 ,w3,and W4trahsform as follows: BR(w)LP(n) Wo->XJ WI->+1 w2 ->-1 W3-> + W2- WI W4 - Wj W4 -> W2- WI W4 - Wj 4.8CHEBYSHEVAPPROXIMATIONCOMPUTERPROGRAM135 \ \ \ \ \ \ \ \ \ \ \ \ -1o Figure 4.29Normalizedlow-pass function. Thustheband-rejectstopbandW3toW4 isseentotransformtothelow-pass stopbandsOsto00and-Osto- 00,whereOs=(w2 - W1)f(W4 - w3); whiletheband-rejectpassbandstransformtothelow-passband- 1to+ 1. The band-reject filter functionTBR(S)is obtained from the low-pass filter function TLP(S),by using thetransformation (4.67) TorealizetheBRrequirementsshowninFigure4.28,wefirstapproximate theLP requirements characterized by (Figure 4.29): ThisLPrequirementisapproximatedusingtheButterworth,Chebyshev, elliptic, or Bessel approximation. Finally, the low-pass approximation function TLP(S)istransformedtothedesiredband-rejectfunctionusingEquation4.67. 4.8CHEBYSHEVAPPROXIMATION COMPUTERPROGRAM Theuseofstandardtablesgeatlyfacilitatesthecomputationsinobtaining approximationfunctions.However,eventhemostextensivetablescancover only alimitednumber of cases. If, forexample, thepassbandrequirementwas Am .. =0.6dB,thetabularapproachwouldconstrainthedesignertousethe closestlistedAmax(usually0.5dB).Acomputerprogramwhichsimulatesthe equations describingthe approximation steps doesnothavesuchalimjtation. 136THEAPPROXIMATIONPROBLEM Inthissectionwewilldescribeonesuchprogram(calledCHEB)fortheap-. proximationofChebyshevlow-pass,high-pass,band-pass,andband-reject filters.Similarprogramscanofcoursebewrittenfortheotherstandard approximations. The program listing and the input formatis giveninAppendix D. The inputs required for 'the program are: 1.The filtertype LP,HP,BP,BR 2.The filterattenuation requirements maximum passband attenuation minimum stopband attenuation 3.The filterpassband AMAXdB AMINdB For LP and HP:the passband edgefrequencyinHz For BP andBR: 4.The filterstopband the lower andupperpassbandfrequenciesinHz For LP and HP:thestopband edgefrequencyinHz For BP and BR:thelower andupper stopband frequenciesinHz 5.Frequencies forcomputation of gain FSstart frequencyHz F IfrequencyincrementHz F FfinalfrequencyHz Theoutputoftheprogramistheapproximationfunctioninthefollowing factoredform T(s)=Yo=fIM(J)S2+ C(J)s+ D(J) YINJ= 1N(J)s2+ A(J)s+ B(J) Theprogramalsocomputesandprintsthegainof thefilteratthespecified frequencies. Example 4.11 Find the Chebyshev approximation foraband-reject filterwhose are Amax=0.3dB,Amin=50 dB lowerpassband edge=200Hz upper passband edge=1000Hz lower stopband edge=400Hz upper stopband edge=500Hz Solution Using theCHEB program the filterfunctionisfoundtobe Ts_S2+ 7895683S2+ 7895683 ( ) - s+ 2297.72s+ 32279472S2+ 6892.5s+ 7895682 S2+ 7895683 S2+ 562s+ 1931312 FURTHERREADING137 From the program the calculated loss at the pass and stopband edges are Freq (Hz)Loss (dB) 2000.3 10000.3 40054.7 50054.7 The approximationfunctionisseentomeettheprescribedrequirementswith 4.7dB of attenuation to spare at the stopband edges. 4.9CONCLUDINGREMARKS In this chapter wehave discussed the approximation of filter requirements using . theButterworth, Chebyshev, elliptic,andBesselapproximations.Of these,the ellipticisthemostcommonlyusedbecauseitrequiresthelowestorderfora givenfilterrequirement.Thesynthesisof thesefunctionsusingpassiveRLC circuits is discussed in Chapter 6; and the synthesis using active filters is covered inChapters 8 to11 . A fewsampletables of approximationsforthe normalizedlow-pass require-mentswerepresentedinthischapter.The readerisreferredtoChristianand Eisenmann [2] for a more extensive set of tables for the Butterworth, Chebyshev, and elliptic approximations. The roots forhigher-order Besselapproximations have been tabulated by Orchard [10]. The synthesis of delay equalizers is not so straightforwardandwillusuallyrequireacomputer optimizationprogram. Thetransformationtechniquesdescribedwereapplicabletothedesignof H Pandsymme:ricalBPandBRfilters.Theapproximationofarbitrary stopbands(Figure3.2)andnonsymmetricalrequirementsismuchmore difficultandthe isreferredtoDaniels[3]formethodstoapproximate suchrequiremenls. FURTHERREADING I.N.Balbanian, r.A.Bickart,andS.Seshu,ElectricalNetll'orkTheory,Wiley,New York,1969, Alpendix A2.8. 2.E. ChristianandE.Eisenmann,FilterDesignTablesand Graphs,Wiley,NewYork, 1966. 3.R.W.Daniels.ApproximationMethods forElectronicFilterDesign,McGraw-Hill . . NewYork.1974. 138THEAPPROXIMATIONPROBLEM 4.A.J.Grossman,"Synthesisof Tchebycheffparametersymmetricalfilters,"Proc. IRE, 45,No.4, April1957,pp.454-473. 5.E.A.Guillemin,Synthesis of PassiveNetworks,Wiley,London,1957,Chapter14. 6.J.L.HerreroandG.Willoner,Synthesisof Filters,Prentice-Hall,EnglewoodCliffs, N.J.,1966,Chapter 6. 7.S.Karni,NetworkTheory :Analysis and Synthesis,Allynand Bacon,Boston,Mass., 1966, Chapter 13. 8.Y. J.Lubkin, Filters, Systems and Design: Electrical, Microwave, and Digital, Addison-Wesley,Reading,Mass.,1970. 9.S.K.MitraandG.C.Ternes,Eds.,ModernFilterTheoryandDesign,Wiley,New York,1973,Chapter 2. 10.H.J.Orchard, "The roots of maximally fiatdelaypolynomials,"IEEE Trans.Circuit Theory,CT-12,No. 3,September1965,pp. 452-454. II .L.Weinberg,Nelll"orkAnalysisandSynthesis,McGraw-Hili,NewYork,1962, ChapterII. 12.M.E.Van Valkenburg,IntroductiontoModernNetworkSynthesis, Wiley,New York, 1960,Chapter13. 13.A.I. Zverev,Handbookof FilterSynthesis,Wiley,NewYork,1967. PROBLEMS 4.1Bodeplotapproximation.Ahigh-passfilterrequirementisspecifiedby theparametersAmax=1 dB,Amin=28dB,fp =3500Hz,fs=1000Hz. ApproximatethisrequirementusingtheBode plotsof first- and second-orderhigh-passfunctions.Computethelossachievedbytheapproxi-mationfunctionat fpand Is . 4.2Butterworthloss.A fourth-orderLP Butterworth approximation function hasalossof 2 dBat100rad/ sec.Compute the lossat350rad/sec.Verify your answerusingFigure 4.5. 4.3Thepassbandlossofafourth-orderLPButterworthfilterfunctionis 1 dBat500Hz.Beyondwhatfrequencyisthelossgreaterthan40dB'? Verifyyour answerusing Figure 4.5. 4.4Sketch the loss characteristic of a seventh-order LP Butterworth approxi-mation forI:=0.1, Wp=1. 4.5Butterworthapproximation.Alow-passfilterrequirementisspecifiedby Amax=1 dB,Amin=35dB,fp=1000 Hz,fs=3500 Hz. (a)Find' theButterworth approximation functionneeded. (b)Determine the lossat9000 Hz. (c)Determine thepole Q'sof the gain function. PROBLEMS139 4.6Butterworth,poleQ.ShowthatthenormalizedLPButterworthloss functioncan be expressed as H(s)=jJI {S2- [2cosCk +: -1 ) ~ }+I} forneven.Findtheexpressionfornodd.Deriveanexpressionforthe maximumpoleQ(ofthegainfunction)foragivenorder.Verifyyour answer forn=4 using Table 4.1. 4.7Butterworth,slope.ProvethattheslopeoftheLPButterworthloss functionIHUw) I atthepassband edge frequencyis nl:2 wp(l+ 1:2)1 / 2 4.8Butterworthfilter. The network shown can be usedto realize a third-order function.FindvaluesforL1,C I,andC2 torealizetheButterworthfilter function VoK VIN =S3+ 2S2+ 2s+ 1 EvaluateKforthe solution obtained. L, c, FigureP4.8 4.9Chebyshevapproximation.FindtheChebyshevapproximationfunction neededto satisfy the followingLP requirements: Amax= 0.25dBAmin= 40dB Wp=1200rad/ secWs=4000rad/sec Compute the loss obtained at the stopband edge frequency.Byhowmuch maythewidthof thepassbandbeincreased,the other requirementsand the order remaining unchanged. 4.10ChebyshevvsButterworth.Afifth-orderLPChebyshevfilterfunction has a loss of 72 dB at 4000 Hz.Find the approximate frequency at which a fifth-orderButterworthapproximationexhibitsthesameloss,giventhat both approximations satisfy the same passbandrequirement. 140THEAPPROXIMATIONPROBLEM 4.11Chebyshev, order. Prove that the order n for the normalized LP Chebyshev approximationisgivenby cosh-I[(100.IAmn_l)/(IOO.IAma._1)]112 n= coshI(ws /wp) 4.12Chebyshevapproximation.Estimatetheorder of theChebyshevapproxi-mation functionneededto meetthe filterrequirement sketched inFigure P4.12.Obtainanexpressionforthetransferfunction.Howwouldyou changethecoefficientstoget10dBofgaininthepassband,without changing the shape of the filtercharacteristic? FigureP4.12 4.13Chebyshev polynomial properties. Show that ,1 C;(Oj="2[I+ C 2n(0)] whereCn(O)isthe nth-order Chebyshevpolynomial. 4.14Chebyshev,passbandmaxandmin.Findexpressionsfor(a)thenumber; (b) the magnitudes; and (c)the locations of the maxima and minima inthe passbandof anormalizedLPChebyshevapproximationlossfunction, intermsof nand 1:.Usetheresultstosketchthepassbandandstopband loss characteristic of afifth-ordernormalizedLP Chebyshev approxima-tion, given I:= 0.3. 4.15Chebyshev,slope.Showthattheslopeofannth-ordernormalizedLP Chebyshevapproximationfunctionatthepassbandedgefrequencyisn times that of the Butterworth approximation of the same order, assuming both approximations satisfy the samepassband requirement. 4.16Chebyshev, pole Q.Findan expressionforthepoleQof anormalizedLP Chebyshev approximation (gain) function, in terms of (1kand Wk (Equation 4.37). Hence, determine the maximum pole Q forthe case n=4 and I:=1. PROBLEMS141 4.17CHEBprogram.FindtheChebyshevapproximationfunctionneededto meet the following requirements, byusing the CHEB computer program: Amax= 0.2dB,Amin=30 dB,fp=1 kHz,fs=2.5kHz. 4.18Elliptic approximation.UseTable 4.3tofindan elliptic functionapproxi-mation thatwillmeetthe low-passrequirements describedbyAmax=0.5 dB,Amin= 40 dB,Wp=1000rad/sec,Ws=3200rad/sec.Computethe loss attained a ~the stopband edgefrequency. 4.19EllipticvsChebyshev. (a)Find the order of the Chebyshev approximation satisfying the requirements stated inProblem 4.18. (b)Determine the loss atWs=3200rad/seciftheorderusedisthatcomputedfortheelliptic approximation. 4.20Elliptic vsButterworth.RepeatProblem 4.19forthe Butterworth approxi-mationfunction. 4.21Maximally fiat function . Prove that forthe function 1 + blw + b2w2 + b3w3 tobemaximally flatatthe origin,the coefficientsmustsatisfytheequal-itiesal=bl,a2=b2.(Hint:Dividedenominatorintonumeratorto obtain aMaclaurin series expansion.) 4.22Maximally fiatdelay. Show that if the delay of the function ismaximally flatat the origin, and the dc delay isone second, then al=1, a2=2/ 5,a3=1/ 15.(Hint:usethe result stated inProblem 4.21.) 4.23Besselloss.Afifth-orderBesselapproximationfunctionhasadelay characteristicthatisflattowithin5percentofthededelayupto 100rad/ sec.Determinethelossat(a)250rad/ secand(b)600rad/ sec, usingFigures 4.16and 4.17. 4.24Bessel approximation.A low-passfilterisrequiredtohave3 dBof lossat thepassband edge frequency of 1000rad/sec, andat least30 dB of loss at 4000rad/sec.(a)FindtheBesselapproximationfunctionsatisfyingthe givenrequirements.(b)Uptowhatfrequencywillthedelaybeflatto within10percent of the dcdelay? 4.25The delay of aLP Besselapproximationisrequired tobeflatto within 5 percent of the dc delay of 1 msecupto the passband edge frequency of 1800 rad/sec, andthe loss must exceed 40 dB at14,000 rad/sec. Determine: (a)The Besselapproximation function. (b)The loss and delay at1800rad/sec. (c)The lossand delay at14,000rad/ sec. 142THEAPPROXIMATIONPROBLEM 4.26BesselvsBlltterworth.FindtheButterworthapproximationfunction neededtomeetthelossrequirementsspecifiedinProblem4.25,for Amax= I dB.Determine the frequencyup to which the delay stays within 5 percent of the dcdelay,byusing either theMAG computer program or Equation3.13. 4.27BesselvsChebyshev. RepeatProblem 4.26using aChebyshev approxima-tion. Once again assume Amax=IdB. 4.28Delayequalizer.Provethatthetotalarea (fromw =0 tow =(0) under any second-order delay curve is2 7tradians. Use this result to estimate the minimumnumberof second-orderdelaysectionsneededtorenderflat the delay characteristic shown. ~ > '" 0.4 ~0.1 w(rad/sec)FigureP4.28 4.29Use theMAG computer program (orFigure 4.20)to determine the aand bvaluesofthedelayequalizersection(s)neededtorenderthedelay characteristic of FigureP4.28flattowithin 3 percent. 4.30Chebyshevhigh-pass, order.Estimate the order of the Chebyshev approxi-mationfunctionneededtorealizethehigh-passrequirementofFigure P4.30. 1000 FigureP4.30 4.31Butterworthband-pass,order.EstimatetheorderoftheButterworth approximationfunctionneededtorealizethesymmetricalband-pass requirement of FigureP4.31, 3000 FigureP4.31 PROBLEMS143 w(logscale) rad/sec 432Ellipticband-pass,order. RepeatProblem4.31usingthe elliptic approxi-mation. 4.33High-Pass approximation. Find a Chebyshev approximation tosatisfy the followinghigh-passrequirements:Amax= 0.5dB,Amill= 20dB,Wp= 3000 rad/sec, Ws= 1000rad/sec. 4.34High-Passpoles. ShowthattheLP toH Ptransformationleadstoaset ofhigh-passpolesthatmaybeobtainedbyreflectingthenormalized low-passpoles about acircleof radius Wp . 4.35High-Passapproximation.Ahigh-passButterworthfiltermusthaveat least 45dBof attenuationbelow300Hz, andtheattenuationmustbeno more than0.5dBabove3000Hz.Findtheapproximationfunction. 4.36Band-Pass approximation.Find aChebyshev approximation functionfor the following band-pass requirements: Amax= 0.5 dBAmin= 15dB pass bands:200Hz to 400Hz stopbands:below100Hz and above1000Hz. 4.37C H EBprogram,band-pass.Expresstheapproximationfunctionfor Problem 4.36asaproduct of biquadratics, byusing the CHEB program. 4.38Band-Rejectapproximation.Transformthefollowingnormalizedlow-passfunctiontoasymmetricalband-rejectfunctionthathasitscenter frequency at1000 Hz and itslow frequencypassband edge at100Hz: 144THEAPPROXIMATIONPROBLEM 4.39CHEBprogram,band-reject.UsetheCHEBprogramtoobtaina Chebyshevapproximationfunctionthatmeetsthefollowingband-reject requirements: Amax=0.2 dBAmin=40 dB passbands:below1000Hz and above 6000Hz stopband:2000 Hz to3000 Hz. 4.40LP toBRtransformation.Show that the low-pass toband-reject transfor-mation of Equation 4.65 can be effectedby firsttransforming the low-pass to ahigh-passfunctionandthen applying aLP toBP transformation on the high-pass function. 4.41Narrow-bandLP toBP transformation. (a)ShowthattheLP toBPtransformationof Equation4.54transforms alow-passpole ontherealaxisto apair of complex conjugate poles. Findthe approximate locationof the complexpolesforthe so-called narrow-band case when Wo~B. (b)Forthenarrow-bandcase,showthatalow-passpoleat- ~- jQ transformstoan upper s plane pole at - ~~+ j( Wo- ~0. ) 4.42Usingtheresultsof Problem4.41,determinetheband-passfunctionob-t a i n ~ dby transfOl:;ming a third-order normalized LP Butterworth approxi-mation,givenWo=1000rad/sec,B=100rad/sec,andAmax=3dBfor theBP function.Plot thelow-passandband-pass pole-zeropatterns and determine the pole Q'sforthe low-pass andband-pass functions. 4.43Narrow-bandLPtoBRtransformation.ApplytheLP toBRtransforma-tion of Equation 4.65toplottheband-reject pole-zero locations obtained bytransforming: (a)A low-pass pole on thereal axis. (b)Apairof complexlow-passpolesat- ~ jQforthenarrow-band case Wo~B. 4.44Usetheresultsof Problem4.43tofindaChebyshevapproximationfor the symmetricalband-reject requirements: Amax=0.25dBAmin=35dB passband width=100 rad/sec stopband width= 25rad/sec center frequency Wo=1000 rad/sec. Checktheaccuracyof theanswerbycomparingitwiththeexactband-rejectfunction,obtainedbyusing the CHEB program. PROBLEMS145 4.45TransitionalButterworth-Chebyshev.TheTransitionalButterworth-Chebyshev (TBC) approximation representedby IHUQ)12=1 + S2(Q)2kC;_k(Q)0 ~k~n realizes a characteristic that isinbetween that of the Butterworth and the Chebyshev.ThelossfunctionreducestotheButterworthfork=n,and to the Chebyshev fork= o. (a)Showthat the TBCfunctionprovides 6(n- k- 1)dBmore attenua-tionthan theButterworth for0.~1. (b)ComparetheslopesoftheButtecworth,Chebyshev,andTBC(for k=2)at the passband edge frequency (0.=1). (c)Determinethenumberofderivativesof1HUQ) 12 thatarezeroat 0.=0forafourth-orderButterworth,Chebyshev,andtheTBC (k=2)approximations. 4.46InverseChebyshev.ThelossfunctionfortheInverseChebyshev approxi-mation isgi venby where Cn(w)isthe nth-order Chebyshev polynomial.Show that: (a)The lossfunctionismaximally flatat the origin (b)Thelossfunctionhasanequiripplecharacteristicinthestopband. Find the frequenciesof maxima andminima inthe stopband. (c)The minimum stopband lossisgivenby 2010g10C;2 S2) dB 5, SENSITIVITY Inthe preceding chapter weshowhowtoobtaintransfer functionsthatsatisfy givenfilterrequirements. The nextstepinthedesignprocessisthe choice of a circuit andthedeterminationof itselementvalues-the synthesis step.Aswill be seeninthe nextfewchapters onsynthesis,one has achoice of many circuit realizationsforaprescribedfilterfunqion.Givenperfectcomponentsthere wouldbelittledifferenceamongthevariousrealizations.Inpractice,real components willdeviate fromtheir nominalvalues due to the initialtolerances associatedwiththeirmanufacture;theenvironmentaleffectsof temperature humidity; andchemical changes duetothe aging of the components. Asa uence,theperformanceof thebuiltfilterswilldifferfromthenominal One way to minimize this difference is to choose components with small ringtolerances,andwithlowtemperature,aging,andhumidity IOelmC:lerlts.However, this approach willusuallyresultinacircuitthatismore thanisnecessary.A more practicalsolutionistoselect acircuitthat arowsensitivity to these changes. The lower the sensitivity of the circuit, the willitsperformance deviat.ebecause of element changes.Stated differently, lowerthesensitivitythelessstringentwilltherequirementsonthecom-beand, accordingly,the circuitbecomes cheaper to manufacture.For reason, sensitivity isone of the more important criteria usedfor comparing circuit realizations. Since a good understanding of sensitivity is essential the designof practicalcircuits,thissubjectistreatedinthisearlychapter. specifically, wewillstudy the definitions of differentkinds of sensitivities, describe ways of evaluating the sensitivity of a circuit. (J)ANDQSENSITIVITY a qualitative sense,the sensitivity of anetworkisameasure of the degreeof ,... .... UUIlofitsperformancefromnominal,duetochangesintheelements ngthenetwork.AsmentionedinChapter2,abiquadraticfilter can be expressedinterms of the parameters OJ p' OJ"Qp,Qz,andK, as (5.1) 147 148SENSITIVITY Inthissection,westudythesensitivity of thesebiquadratic parameterstothe elements andillustrate the evaluation of sensitivityby examples. Letusfirstconsider thesensitivity of thepole frequency wptoachange ina resistor R.Pole sensitivity is defined as the per-unit change in the pole frequency, !!.wplwp,causedby aper-unit change inthe resistor, !!.RI R.Mathematically !!.wp (5.2) (5.3) This isequivalent to (5.4) Note that the cost of manufacturing a component is a function of the percentage change (100x!!.RIR)rather thanthe absolute change (!!.R)of the component. Forthisreasonitisdesirabletomeasuresensitivityintermsof therelative changes in components, as isdone inEquation 5.2. Thesensitivitiesof theparameters w" Qp,Qz,andKtoany elementof the network are definedinasimilar way: Cow S"'p---p c- WoC p KRoK SR= -- etc KoR. (5.5) Equation 5.4canbe usedto develop some usefulrulesthatsimplify sensitivity calculations.The sensitivity of aparameter ptoanelement xis (5.6) If p isnot afunctionof x (e.g.,p=a constant), then =0(5.7) If p=cx,where cisaconstant sex=o(1nex)=o(1nc)+ o(lnx)=1 xo(1nx)a(1nx)a(lnx) (5.8) Another usefulrelationship is (5.9) 5.1wANDQSENSITIVITY149 This followsfromEquation 5.4,since _Sxl/P=_a(lnlip)__0 (-(1n p - - SP o(1nx)o(1nx)- x =-Sf/x Other usefulrelationships that can easily be provedare: SPIP2= SPI+ SP2 X X XSr = =+PI+ P2 (5.10) (5.11a) (5.11 b) (5. 11 c) (5.11d) (5.11 e) seJ(x)- SJ(X) x- x(5.1 If) here eisof x, andf(x) isafunctionof x. TheevaluatIOnof sensitivityusingtheserelationshipsisillustratedinthe examples. I/ztunple 5./ transfer functionVa/IIN for the passive circuit of Figure 5.1can be shown to Vas -=-------lINC2S1 s+-+_ RCLC the sensitivites of K,wP'andQpto the passive elements. + R c L PassivecircuitforExample5.1. (5.12) 150SENSITIVITY Solution. . From Equation 5. I, thebiquadratic parameters are. fieUsing thesensitivityrelationships developedinthe above section: ==-=- 1 - I Sfp =sL1Jl=- Sl=-2L=- "2=_st1e =-1 ==I SQp- JSR2CI L= -2t c- 2C Sp= -!SfCI L =- -!StI R2C =-1 Alltheother sensitivities are zero. ObservationshA'tivityof IThemagnitudesof thesensitivitiesarealllesstan sensl .one imliesthataIpercentchangeintheelementwillcausea1chanthe parameter (Equation 5.2).is considered a low In6 weseethat, in passiveladder structures canalways be designedtohave lowsenslt\Vltles.fh 2.Notethatthesensitivitiesevaluatedareequaltotheexponent0te element.For instance, inthe expression forpole Q Qp=RIC1I2L -1 / 2 thesensitivities are =I Ingeneral,if theparameter p isgivenby abc p =XI X2X-,d '11beequaltotheir respective thenthe sensitivityof p toXI'X2,anX3WI. exponents, that is Sp=aSP=b XIX2 SP=C x) (5.13) 3.The sensitivity of wp toRis zero. Thus,. anyinR not affect polefrequency.ThisisausefulconsiderationIfwewishtotune(1.. adjust) Qp without affecting wp ' 5.2MULTI-ELEMENTDEVIATIONS151 5.2MULTI-ELEMENTDEVIATIONS Inthelastsection,weobtainedanexpressionforthechangeinabiquadratic parameterduetoachangeinaparticularcircuitelement.Forinstance,the change inaresistance causes the pole frequencyto change by(Equation 5.2) (5.14) For small deviations inR (5.15) This isthechangeduetooneelement.Inwewillbeinterestedinthe change duetothesimultaneousvariationof alltheelementsinthecircuit,as discussedinthis section. Consider,forexample,thechangeinwpduetodeviationsof allthecircuit elementsXj(wheretheelements canberesistors,capacitors,inductors,orthe parameters describing the active device).The change .1wpmaybe obtainedby expanding itinaTaylor series, as awpaWpA cWp .1wp= - .1XI+ - ilX2+ .. . - .1Xm ax IaX2aXm + second- andhigher-order terms where m isthe totalnumber of elements inthe circuit.Sincethe changes inthe components .1Xj areassumedtobesmall,thesecond- andhigher-orderterms can beignored. Thus To bring thesensitivitytermintoevidence, (5.16)maybewrittenas m L j= I (5. I 6) (5.17) VXj =.1x/Xj istheper-unit change inthe element Xj' and isknown as the _.oII_I ..:...of x.From (5. I 7)the per-unit change inwpis (5.18) 152SENSITIVITY Similarly the per-unit changes in pole Q, w., Qz, and K, due to the simultaneous deviations of allthe components, are givenby m L j= I (S.19) -= K llQz=SQ. V Q XjXj Zj= I Example 5.2. TheactiveRCcircuitshowninFigureS.2realizesasecond-orderhIgh-pass function.FinditstransferfunctionVO/ JtfNandderiveexpressionsforthe sensitivity of wp and Qpto elements R" R2 ,CI ,C2 ,and the amplifier gainA. Solution The nodal equations forthe circuit are: node1: V- --+ - - Vo- =0 (k-l1)(1) RARARA node 2: V+(SCI++11) - vo(1 _1_) =V/NsCI R2+ - R2+C sC2 S2 The positive and negative terminal voltages of the op amp are relatedby Vo=(V+- V-)A c, + 1 Figure 5.2 ActiveRC circuit forExample5.3. 5.2MULTI-ELEMENTDEVIATIONS153 Solving these equations, the transfer functionisfoundtobe: VoS(s+I/R1C1)k(l+ k/ A)-' VIN Sl+S(RI1CI +R1IC1 +R1ICI (1- 1 +\/A)) +R,R21C,C2 (S.20) ComparingthisequationwithEquationS.1,thebiquadraticparametersare seen to be: (S.21) (S.22a) (S.22b) The sensitivityof wptothe components RI,R2,CI,andC2 are equaltotheir exponents, that is, ====-! The sensitivity of Qptothe components are evaluated as follows: SQp =SWp!(bw)p=SWp_S(bw)p R,R,R,R, From (S.21)and (S.22),usingtherelationship (S.11e): we get: _ + _1[1+1(1k)] 2(bw)pR2C2 R1CI - 1 + k/ A = + _1 [_1 +1(1k)] 2(bw)pR,C,R1CI - 1 + k/ A 111 sg:=- - + -- . --2R1C2 (bw)p 154SENSITIVITY Next, considering the sensitivities to the amplifier gain:

1IkAk =(bw) pR 2 C 1 . 1 + k/ A. ( - I). ""'(1-+---=-k/'-A-:-)R-2-=C::-t/""'k I1k2 1 = X R2 C1(1+.(bw)p (S.23) Observations 1.The expression for suggests that this term can be reduced by increasing the amplifiergain.Inparticular, foran idealop amp this sensitivityterm becomes zero,thatis, the pole Q becomes insensitivetothe op amp gain. 2.NotethatkdependsontheresistorsRAandRB(k=1 + R,l/ Rn),and isnota constant. The sensitivities of wp and Qpto RA /Rn can be computed by firstevaluating the sensitivities to k (seeProblem S.6). 3.The sensitivities of wpand Qp are related to their respective dimensions, as shown inthe following.From the example, weseethat (S.24a) and S/::+=-I+ =-I(S.24b) Now the pole frequency wp,which is of the formI / RC, has the dimensions dim(wp)=[Rr 1[C]-1 whilethepoleQ,givenbywp/(bw)p,isadimensionlessquantity.In other words Therefore,inthisexampleitisseenthatthesummedsensitivityofa parametertoalltheresistors(orcapacitors)isequaltothedimension ofresistance(orcapacitance)fortheparameter.Thispropertycanbe shown to hold for all active RC networks. The general form of this so-called 5.2MULTI-ELEMENTDEVIATIONS155 dimensionalhomogeneity property*is IS/:r= I=- I where the summation isover allthe resistors (capacitors). Ufllllple 5.3 Usingtheresultsof ExampleS.2: (S.2Sa) (S.2Sb) (a)Find the expressions forthe per-unit change in wpand Qp,giventhat the variabilityof everypassivecomponentis0.01,andthatof theamplifier gainisO.S." (b)Evaluate the per-unit change in wpand Qpforthe special case: Sollltion C1 =C2 =C wp=2n104 A=1000 (at10kHz) (a)The variabilities of the circuit elementt are giventobe Vc=0.01 Substituting in (S.18) Using EquationS.2Sb,andrecalling that=0,this reducesto

-p =0.01( -2) + O.S(O)=-0.02 wp Similarly, fromEquation S.19,S.23,and S.2Sa: = =O.Sk2 2_1_(S.26) QpAR2C1 (I (bw)p This property followsfromEuler's formulaforhomogeneous functions, [3J (page222). tin praclice thevariabilities are randomnumbers describedbyameanandastandard deviation, 'as willbe explainedinSection5.4.3. 156SENSITIVITY (b)For the specialcase, fromEquation 5.20 (bw)p=RIC(3- kk) 1 + 1000 14 W= - = 27110 pRC Dividing (5.28)by (5.27) whichyieldsk=2.96. From (5.26) Qp= ----:-k- = 20 3- ---=-k-1 + 1000 dQp0.5k2 wp Qp=It (1+ (bw)p Substituting fork,A, and Qpin(5.29) dQp=(2.96)220= 0.087 Qp1000 (1+ 0.00296)2 whichcorresponds to an 8.7percentincreaseinthepole Q. Observation (5.27) (5.28) (5.29) Thewsensitivitiesareaslow(= 1/ 2)asthoseobservedforthepassive network inExample 5.1; also, the contribution of the passive components to thepole Q iszero.However, the giveninthe gain the pole Q to change by 8.7percent, which ISfarfromneglIgible. Wewillsee later that one of the major problems associatedwithactiveRC filtersisthe sensitivity to amplifier gain, and it will therefore be necessary to study methods forreducing this effect. 5.3GAINSENSITIVITY Thusfartheeffectof elementdeviationsonthebiquadraticparameterswP' Qp,wz,Qz,andKhavebeenconsidered. are usually stated in terms of the maximum allowable devIatIOn In gam over specified bands of frequencies.In this section weshow how this gain deviation isrelated 5.3GAINSENSITIVITY157 tothebiquadratic parameter sensitivities;*furthermore, wealso suggestways of adapting the designprocess to minimize the gain deviation. Letusassumethat the filterfunctionhas beenfactoredinto biquadratics, as 2W z 2 s+-'s+w T(s)=nK;Qz,Zi ;-12wp2 - s+-'s+w Q Pi p, (5.30) The gain in dB is givenby G(w)= 20 loglol TUw)1 (5.31) Gain sensitivityisdefinedasthechangeingainindBt duetoaper-unit change inan element (or parameter) x: From this equation and forsmall changes inx 9'G(-----L--------L---oFigureP6.17 PROBLEMS229 6.18Synthesize anLC ladder network whichhas the z parameters (S2+ 1)(s2+ 3)(S2+ 4)(S2+ 5) Zll=S(S2+ 2)Z12=S(S2+ 2) using the topology shown.Sketchthepole-zeropattern at eachstep, and indicate the zero shifting and zeroproducing elements. 6.19LCladdertopologyprediction ..Sketch(butdonotsynthesize)anLC laddertopologyfortherealizationof the zparameters plottedinFigure P6.19. Zl10IE 0J( Z'200III( 0.Ji .j'j w____ FigureP6.19 6.20RepeatProblem 6.19forthe z parameters of FigureP6.20. Zl1)(0)(0)( Z'2'( 00uJ( 0v'f.j'jv'3J4v'5w____ FigureP6.20 6.21RepeatProblem 6.19forthe z parameters of FigureP6.21. Z110 )( 0)10 Z,2 00UIE 0 0v'1 .jj J3J4 w_ FigureP6.21 6.22RepeatProblem 6.19forthe z parameters of Figure P6.22. Zl1)(0H0J( Z'2 )(0H0 0 Jf .j'j J3J4 .j5 w_ Figure P6.22 230PASSIVENETWORKSYNTHESIS 6.23Singly-terminatedLPButterworthfilter.ConsiderthenormalizedLP Butterworth approximation function Va_K V1N - S3+ 2S2+ 2s+ 1 Synthesizethis functionusing anLC networkterminated at one end with a1 n resistor.Determine the constantKrealizedbythe network. 6.24Frequencyscaling.The networkshowninFigureP6.24realizesafourth-orderlow-passButterworthfunctionforwhichAmax= 3dB,Wp=1 rad/ sec.Scalethe elements of the networksothat: (a)The passband edge frequencyisat1000rad/sec. (b)The passband lossis1 dB at1000 rad/ sec (Hint:useEquation 4.21.) 1.53 H1.08H + 0.38 F FigureP6.24 6.25Impedancescaling.Showthatiftheimpedanceof eachelementinthe networkof FigureP6.24isincreasedbythesamefactorrx,thetransfer functiondoesnot change.Usethis resulttoimpedance scale the network tochangetheloadterminationto50nwithoutaffectingthetransfer function. 6.26Singly-terminated LP elliptic filter.Find an elliptic function approximation forthelow-passrequirementscharacterizedby:Wp=1 rad/sec,Ws=2 rad/ sec,Amax=0.5dB,Amin=30dB.UseTable4.3.Synthesizethe functionusing anLC network witha1 n termination.Scale the elements sothatthepassbandedgeisat10,000rad/secandtheloadtermination is600 n. 6.27High-passfiltersynthesisusingLPtoH Ptransformation.Thelow-pass filterofFigureP6.24realizesafourth-orderButterworthfunctionfor whichAmax=3dBandWp=1rad/ sec.Transformtheelementsto realize a fourth-order high-pass filter for which Amax=3 dB and Wp=100 rad/sec. (Hint:useEquation 4.51.) 6.28Band-passfiltersynthesisusingLP toBPtransformation.Transformthe elementsinthenormalizedlow-passfilterof FigureP6.24(describedin Problem6.27)torealizeaneighth-orderband-passfilterforwhichthe PROBLEMS231 passband width isB=100 rad/sec, center frequency is Wo=1000 rad/sec, and Amax=3 dB.(Hint:use Equation 4.62.) 6.29Singly-terminatedhigh-passfilter.FindanLCnetworkterminatedbya loadresistorof 600n satisfyingthehigh-passChebyshevfilterrequire-ments Wp=3000 rad/sec Amax=0.5dB Ws=1000rad/sec Amin=25dB (Hints:firstsynthesize the normalizedLP functionasgivenby Table 4.2, witha1 n termination. Then impedance scalethe network as inProblem 6.25.Finally use theLP to HP transformationof Equation 4.51.) 6.30Transducerandcharacteristicfunctions.Identifythetransducerfunction H(s)and the characteristic functionK(s)for: (a)Afourth-orderlow-passButterworthapproximationfunctionfor whichAmax =3 dB, Wp=1 rad/sec. (b)Afourth-orderlow-passButterworthapproxiinationfunctionfor whichAmax=1 dB, Wp=2 rad/sec. (c)A third-orderlow-pass Chebyshev approximationfunctionforwhich Amax=0.25dB, Wp=1 rad/sec. 6.31DoubleterminatedLPfilter.Synthesizeathird-orderLPButterworth approximationforwhichAmax=3 dBandWp=1rad/sec,usinganLC ladder networkterminated atboth ends with1 n resistors. 6.32A low-pass Chebyshev filter is required to meet the following specifications: fp=1200Hz Amax=0.5dB fs= 2400Hz Ami"=25dB SynthesizethefunctionusinganLCladdernetworkterminatedatboth endswith600n resistors.(Hint:firstsynthesizethenormalizedfunction with1 n terminations.) 6.33DoubleterminatedBPfilter.Aband-passfilterisrequiredtomeetthe followingspecifications: Am.x=3 dBAmi"=20 dB passband:1000rad/secto 2000 rad/sec stop bands :below500rad/sec and above 4000rad/sec Synthesize the Butterworth approximation function for these requirements using anLC ladder network terminated at both ends with50 n resistors. 232PASSIVENETWORKSYNTHESIS 6.34Doubleterminatedellipticfilter.Theellipticapproximationforthe low-passrequirementsshownischaracterizedby(see[1],page49, Os=2.5593): 4.886(s+ 1.05443)(S2+ 0.84976s+ 1.66129) H(s)=(S2+ 8.5589) ( ) _4.886s(S2+ 0.76528) Ks- (S2+ 8.5589) (a)RealizetheseLP requirements using thetopology shown. (b)Transformtheelements,usingEquation4.65,torealizetheband-rejectrequirements showninFigureP6.34c.Use 600 0terminations. 30111 2.5w(rad/sec) (a) 0.9484F 1n (b) PROBLEMS233 80rad/sec E w(rad/sec) (c) FigureP6.34 7, BASICS OF ACTIVE FILTER SYN1HESIS Inthe preceding chapterwediscussedthesynthesisof transferfunctionsusing passive Icomponents.Inthischapterwediscusssomebasicprinciplesrelated tothesynthesisof activenetworks.Broadlyspeaking,thetopologiesusedin active network synthesis can be classified into two groups, namely, cascaded and coupled.Thefundamentalblockusedinthesetopologiesisthebiquadratic function,which was introduced inSection 2.6.Because there exist several active realizationsof thebiquadratic function,itisusefultofurthercategorizethese realizationsthemselves.Inthischapter wediscussthe classification, some fun- properties,andtheprinciplesofsynthesisofthecommonlyused realizationsof thebiquadratic. The detaileddesignconsidera-tionsof cascadedandcoupledactivefiltersarethe subjectof theremainder of thisbook. 7.1FACTOREDFORMSOFTHE APPROXIMATIONFUNCTION approximation stepyields atransfer functionof theform: (s- zd(s'- Z2)'"(s- zn) T( s)=K (s- PI)(S- P2)' .. (s- Pm) (7.1) mentionedinChapter 2,this functioncan be factoredintobiquadratics, as N2+ d ( ) fl mjS+ CjSj TS=K j-'--'2--'----=-' j=1njS+ ajS+ bj (mj=lor 0) nj=lor 0 (7.2) form of the approximation function is quite general in that it can represent real or complex roots. In the case of a realpole, forinstance, we set n=0 a=1,*sothatthe denominator reducesto S+ b y,for a real zero, m=0 and C=1.For complex poles we set n=1, and complex zeros m=1.Thus, a complexpole-zeropair isrepresentedby T(s)=KS2+ CS+ d(7.3) S2+ as+ b subscripts are droppedwhenconsidering only onebiquadratic. 235 236BASICSOFACTIVEFILTERSYNTHESIS Table7.1Second-OrderFilter Functions Filter functionTransfer function I Low-passK S2+ as+ b S2High-passK S2+ as+ b s Band-passK S2+ as+ b S2 + d Band-rejectK S2+ as+ b S2- as+ b Delay equalizer S2+ as+ b Recall from Chapter 2 that this biquadratic can also be expressed in terms of the parameters K, WZ,wp, Qz,andQp,as (Equation 2.39): S2+ Wz S + w: T(s)= KQz-(7.4) W S2+ ---Es+ w2 QpP The second-order filterfunctionsdiscussedinChapter 3 are easily obtained as special cases of Equation 7.2, as shown in Table 7.1.For example, the low-pass functionisobtained fromEquation 7.2with m=Oc=Od=1n=1 AsmentionedinSection3.1.4,thefunctionrepresentingtheband-reject filteralsorealizesalow-passfilterwithazerointhestopband,knownasa low-pass-notchfilter.For this case,the zero frequencymustbehigherthanthe polefrequency,whichmeansthatd >b.Similarly,ahigh-pass-notchfilter isrealizedwhend R2, CIandC2 Now,thesolutionof threeequationsrequiresthreeindependentparameters. Sincewehavesixindependentparameters,itispossibleto fixthreepriorto solving the equations. One simple choice forthe fixedparameters is The remaining elements are then obtainedas follows.Dividing (7.30)by (7.32), weget so Substituting the values of C hC2,rl>and r2inEquation 7.31 1 RI=-a Finally, fromEquation 7.32,theremaining elementR2isgivenby Thus, one set of element values that will synthesize the given transfer function is: Observations 1.The element values are in ohms andfaradsand as such are definitely not practicaltoimplement.However,thesevaluescaneasilybescaledto yieldpracticalvalues asexplainedinSection 7.7. 2.Intheabovesynthesis,the choiceof thefixedparameters wasarbitrary. Inpractice,thischoiceisdictatedbyotherdesignconsiderations,such as the sensitivity of the network and the spread in the values of the elements. These and other practical designmatters willbe discussedinthe next few chapters. 3.Thesynthesiswasbasedontheopampbeingideal,thatis,A=00. Inmostsimplefilterdesignstheeffectof opampimperfectionsisquite * This choice is not completely arbitrary in that some choices do not lead to a solution. For example, thechoiceC1 = 1,C2 = 1,'2= 2"makesEquation7.30and7.32inconsistentand,therefory, does not allowa solution. 7.5COEFFICIENTMATCHINGTECHNIQUE249 smallandcanbeneglected.However,inapplicationswherethe require-mentsarestringent(highpoleQ's,highfrequencies,tighttoleranceson thepassbandperformance)itbecomesnecessarytoconsidertheeffects of the op amp. 4.The numeJator coefficient, 2b,waschosen so that the synthesis equations wouldyieldaconsistentsolution.Letusnowconsiderthesynthesisof the more general low-pass function d T(s)=-'s2'-+-a-s-+-b (7.33) where the term d isallowedtobe any positive constant. asintheexample,the oflikepowerinsin EquatIon 7.28and 7.33are equatedto obtain the followingrelationships: 1 + r2/rl =d(7.34) RIR2CIC2 11r2/rl --+-----=a RICI R2C1 R2C2 (7.35) (7.36) Asbefore,weletCI =C2 =1andrl=1.Thentheabovesynthesis equationscanbesolvedtoyieldthefollowingexpressionsforRI>R2, andr2: 2(d/b- 2) R2=-----;=.===== -a Ja2 + 4b(d/b- 2) 1 RI=--R2b d r2= b - 1 Itcanbeseenthatr2isnegativeford/bR2, RJ , CI,C2,andk)tosatisfythetwoconstraintsimposedbywpandQp(assuming thatKisanarbitraryconstant). Therefore,fourof theelementscanbefixed. One simple solutionisachievedby letting C 1=C2 =1RI=R2=R3=R Then the remaining elements are givenby fi R=R 1=R2=R3= -wp r2fi k=I+ - =4--rlQp Note thattheratior2/ rlispositive forQp>fi/3 (8.50a) (8.50b) 8.4SALLENANDKEYBAND-PASSCIRCUIT287 The above solutionresultsina gain constant of (8.5Oc) This gain constant can be changed byusing the techniques of input attenuation or gain enhancement, as mentionedinSection7.6. Example 8.5 Synthesizeasecond-orderBPfilterwithacenterfrequencyat1000rad/sec andapole Q of 10.The gain at thecenter frequencyisrequiredtobe 0 dB. $omtion The desiredtransfer functionis T(s)=l00s S2+l00s+ (1000)2 From Equation8.50, therequired elementvalues are C1=C2=IF fi RI=R2=RJ =1000= 1.414(1O)-Jn r2=3_J2 =2.858 rl10 To obtain practicalelementvalues,the elementsareimpedance scaledby107, to yield C1 = C2 =0.1 ,uF RI=R2=RJ =14.14 kn The ratio r 2/r 1 = 2.858 can be realized by making r 1 = 1 kfl and r 2 = 2.858kn. Withtheabovechoiceofelementsthegainconstantthatisrealized,from .Equation 8.5Oc,is K=2728 , thedesiredscalefactoris100, itisnecessarytoattenuate theinputbya of 27.28.UsingtheinputattenuationschemedescribedinSection7.6 inputresistanceRisreplacedbytheresistorsR4andRs,asshownin 8.7,where: Rs1 R4+ Rs27.28 we getR4=386 kn and Rs=14.7 kn. The complete circuit isshown Figure 8.7. 288POSITIVEFEEDBACKBIQUADCIRCUITS 14. 1kn R4= 386kn O.l,..F + Yo+14.1kn VIN -l'" ~ R,T k n ~ -=- -=-1 Figure 8.7Circuit forExample8.5. 8.5TWIN-TNETWORKSFOR REALIZATIONOFCOMPLEXZEROS In this section we present a qualitative description of a circuit for the realization of a second-order functionwithcomplex zeros, namely, (8.51) AsmentionedinChapter3,thisfunctionisusedintherealizationof band-reject, low-pass-notch, andhigh-pass-notchfilters. Fromthebasicpropertiesofthepositivefeedbacktopology,weknow that the zeros of thetransfer functionare the zeros of the feed forwardfunction of theRC network.Thus,weneedanRC structurethatexhibits zeros onthe jw axis. This requirement rules outRC ladder networks because they can only realize zeros on the real axis [10]. This fundamental property of RC ladders is a consequenceof thefactthatinaladdercircuitthezerosof transmissioncan only berealizedwhen aseries branch isanopen circuit or ashuntbranchisa short circuit.Put differently,the zeros of transmission are the impedance poles of the series branches and the impedance zeros of the shunt branches. However, itisknownthatRC impedances havealltheirpolesandzeros on the negative realaxis(Chapter2,page 42);hence,thetransmissionzerosof anRC ladder are constrained to lie on the negative real axis. Therefore, Equation 8.51cannot be implemented using the class of ladder networks shown in Figure 8.2, and we must consider alternate topologies.One such topology, isthe so-called Twin-T [6]showninFigure8.8.Inthistopologytherearetwopathsfrominputto output. Complex zeros of transmission are formedby choosing the component 8.5TWIN-TNETWORKSFORREALIZATIONOFCOMPLEXZEROS289 pR P;'c 3 CR 2' Figure 8.8Twin-T RC network. valuessothattheelectricalsignalsarriving attheoutputviathesetwopaths exactly cancel.Notice thatthis network has three capacitors and assuch it will realize athird-order function.However, if the elements are chosen as shownin the figure,a pole-zero cancellation occurs, resulting inasecond-order function with zeros onthe ;w axis,as desired. Inthenetwork' of Figure8.8,thedcgainof thefeed forwardfunctionTFFis seen tobeunity. This isverifiedby replacing the capacitors withopen circuits. Also,thegainatinfinitefrequency,obtainedbyshortingthecapacitors,can Figure 8.9Positive feedbackcircuitusingthe Twin- T, with a loadingnetwork. 290POSITIVEFEEDBACKBIQUADCIRCUITS be seen to be unity. Therefore, for the circuit of Figure 8.8,TFFwillhave the form (8.52) wherethepolefrequencyisequaltothezerofrequency.Ingeneral,however, the pole and zero frequencies are required to be different. One way of separating the pole and zero frequencies,as can easily beverified,istousethe RC loading network showninthecomplete activenetwork of Figure 8.9. 8.6CONCLUDINGREMARKS Inthischapterwediscussedthesynthesis,sensitivity,andsomepractical designaspectsof positivefeedbackbiquadstructures.Thesebiquadcircuits can be cascaded to synthesize amore complex filterfunction, as was mentioned inChapter 7.We only covered a sampling of the numerous RC circuits that are commonly used-a more complete selection can be foundin[6], [7], and[9]. We showed that the choice of the fixed elements in the solution of the synthesis equations greatly influenced the sensitivity of the resulting circuit. The analytical techniquespresentedinthischapterprovideasimpleyetusefulapproachto thisproblem.Amore detailedanalysistodeterminetheleastsensitivedesign isbestperformedusingMonte Carlo techniques describedinSection 5.5. Theimportanceof consideringthefinitegainof theopampwasillustrated by an example.A more complete treatment of the effectsof the op amp willbe covered inChapter12. FURTHERREADING I.A.Budak,PassiveandActiveNetworkAnalysisandSynthesis,HoughtonMifflin, Boston,Mass.,1974, Chapter10. 2.FairchildSemiconductor,TheLinearIntegratedCircuitsDataCatalog,1973, Fairchild Semiconductor, 464 EllisStreet,Mountain View,Calif. 3.S.S.Haykin,SYl1lhesisof RC ActiveFilterNetworks,McGraw-Hill,London,1969, Chapter 4. 4.L.P.Huelsman,Theory and D e s ~ q nof ActiveRC Circuits,McGraw-Hill,NewYork, 1968, Chapters 3 and 6. 5.W. J.Kerwin, "Active RC network synthesis using voltage amplifiers,"Active Filters, L.P.Huelsman, ed ..McGraw-Hill,New York,1970, Chapter 2. 6.G. S.Moschytz,LillearIntegrated Ne/lforks Design,VanNostrand, New York,1975, Chapter 3. 7.R.P.SallenandE.L.Key,"Apracticalmethodof designingRCactivefilters," IRE Trans. CircuitTheory,CT-2,May1955,pp. 74-85. PROBLEMS291 8.W.Saraga,.. Sensitivityof 2nd-orderSallen-Key-typeactiveRCfilters,"Electronics Lellers,3,10, October1967,pp.442-444. 9.A.S.Sedra,"Generation and classification of singleamplifier filters,"IntI. J. Circuit Theory and Appl. 2,I,March1974,pp. 51 - 67. 10.H.H.Sun.SyllIhesisof RC Networks,Hayden,N.Y.,1967,Chapter 3. PROBLEMS Low-pass synthesis.Synthesize thelow-pass transfer function K S2+ 100s+ 25(10)4 usingtheSaragadesign,withpracticalelementvalues.Showhowthe circuit canbe adaptedto: (a)Change the cutoff frequencyto 200rad/sec. (b)Achieveagain of dB at dc. Synthesizethe low-passfunction 20,000 (S2+ 2s+ 100)(s2+ 5s+ 200) Low-passChebyshevjilter.Alow-passfilterisrequiredtomeetthe following specifications: fp=1000Hzfs=3000Hz Ama,=0.5dB Amin=25dB degain=dB FindtheChebyshevapproximationfunctionfortheserequirementsand realizeitusingtheSa ragadesign.Youshouldnotneedextraelements foradjustingthegainconstant.(H illt:firstsynthesizethenormalized LP filter.) Low-passButterworthjilter.RepeatProblem8.4usingaButterworth approximation.Youmayuseoneextraelementforadjustingthegain constant. Low-passdesign.TheRCcircuitshownistobeusedinthepositive feedbacktopology torealize alow-passtransfer function. (a)Sketchthe activeRC circuit. (b)Obtainthetransferfunctionandonesetof designequations. (e)Computethesensitivitiesof (VpandQptothepassiveelementsRI andC1 forQp=.j3. 292POSITIVEFEEDBACKBIQUADCIRCUITS FigurePS.5 8.6Alternate designforSallen alldKey LP circuit. Three designs for the Sallen andKeyLP circuitwereevaluatedinSection8.2.Consideryetanother design (Design 4)basedon the followingchoice forthe fixedelements: k=2 (a)Find the synthesis equations for this design. Show that the resistor and capacitor ratios are equal toQp. (b)Compute the sensitivities of wpandQpto the passive elements andto thegain-bandwidthproductAocx.Assumethe op amp gainisA(s)= Aocx/s. (c)Determinethestatistics (Jiand 0')of thegaindeviationattheupper 3 dB passband edge frequency forthe filterdescribedinExamples 8. 1, 8.2,and 8.3. 8.7Generalexpressionsfor/!G.Forpurposesofcomparingdesignsitis convenienttodevelop expressionsforthestandard deviation of thegain change O'(/!G)atthe3 dB passband edge frequencies,as was done forthe Saraga designinEquation 8.34.Deriveasimilar expression forDesign1 (k=1) of the Sallen andKey low-pass circuit. AssumeQp1,and make reasonable approximations. Answer:[30'(/!Gm.sivecxp{8.686Qp[30'( YR. dJ}2 where CXp =1,CXA = 8.8RepeatProblem8.7forDesign2(k=3- I/Qp)of theSallenandKey low-pass circuit. Answer:cxp=5.5,CXA = 20.25 8.9RepeatProblem8.7forDesign4(Problem8.6)oftheSallenandKey low-pass circuit. Answer:cxp=2, CXA =4. 8.10OptimumkforSallenandKeyLPcircuit.Forthefilterdescribedin Examples8.2and8.3,determinethevalueof kthatyieldsthesmallest PROBLEMS293 gain deviation at the 3 dB passband edge frequency by sketching 30'(/!G), versus k forDesigns1,2,3, and 4.OIal Comparisonof Sal/enandKeyLPdesigns.Inproblems8.11to8.15 assumethecomponentstatisticstobethosegiveninExamples8.2and8.3, unless otherwise stated. thes.tatisticsof thegaindeviationduetothechangesinthe passive and actIve components forDesign1 (k=1)andDesign 3 (Saraga k= 4/3),at theupper3 dB passband edgefrequency,forasecond-order filterwithacutoff frequencyat2000HzandapoleQof 40. (Hint:usethegeneral expressions derivedinEquation 8.34andProblem 8.7.) 8.12Repeat Problem 8.11for alow-pass filter with a cutoff frequency at10,000 Hz and apole Q of 10. 8.13Show that of the three designs considered in Examples 8.2 and 8.3, Design 1(k=I)hasthesmallestgaindeviationatthecutofffrequency wp=27t2000rad/sec. 8.14The Sa raga design (k= 4/3) was foundto be superior toDesign1 (k= 1) forthe component tolerances specifiedinExamples 8.2 an