Analog Electronic Filters: Theory, Design and Synthesis

585 Pages·2012·15.64 MB·English

by  Hercules G. Dimopoulos (auth.)

Analog Electronic Filters
ANALOGCIRCUITSANDSIGNALPROCESSING
SeriesEditors:
Mohammed Ismail.TheOhioStateUniversity
Mohamad Sawan.ÉcolePolytechniquedeMontréal
Forfurthervolumes:
www.springer.com/series/7381
Hercules G. Dimopoulos
Analog Electronic
Filters
Theory, Design and Synthesis
Dr.HerculesG.Dimopoulos
DepartmentElectronics
Technol.Educ.Inst.ofPiraeus(T.E.I.)
P.Ralli&Thivon250
Egaleo12244
Greece
[email protected]
SeriesEditors:
MohammedIsmail MohamadSawan
205DreeseLaboratory ElectricalEngineeringDepartment
DepartmentofElectricalEngineering ÉcolePolytechniquedeMontréal
TheOhioStateUniversity Montréal,QC,Canada
2015NeilAvenue
Columbus,OH43210,USA
ISBN978-94-007-2189-0 e-ISBN978-94-007-2190-6
DOI10.1007/978-94-007-2190-6
SpringerDordrechtHeidelbergLondonNewYork
LibraryofCongressControlNumber:2011938013
©SpringerScience+BusinessMediaB.V.2012
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patienceandwisdom
Preface
Filters are essential subsystems in a huge variety of electronic systems. Filter ap-
plicationsareinnumerable;theyareusedfornoisereduction,demodulation,signal
detection, multiplexing, sampling, sound and speech processing, transmission line
equalizationandimageprocessing,tonamejustafew.Inpractice,noelectronicsys-
temcanexistwithoutfilters.Theycanbefoundineverythingfrompowersupplies
to mobile phones and hard disk drives and from loudspeakers and MP3 players to
homecinemasystemsandbroadbandInternetconnections.
Thistextbookintroducesbasicconceptsandmethodsandtheassociatedmathe-
maticalandcomputationaltoolsemployedinelectronicfiltertheory,synthesisand
design.Sincetheapproximationproblemmustbesolvedbeforeafiltercanbede-
signed,asignificantpartofthebookisconcernedwithapproximationsinamanner
thatallowsdeepunderstandingandeasygeneralization.Theapproximationproce-
dureusesthefilterspecificationstoderiveacircuitfunction,fromwhichtheelec-
tronicfiltercircuitcanthenbesynthesized.Thesynthesismethodtobeusedinorder
toderiveanelectroniccircuitfromamathematicalcircuitfunctiondependsonthe
technologywhichistobeemployedforthefinalimplementationofthefilter.Syn-
thesismethodsarethereforedependentonthetechnologiescurrentlyavailableand
changeastechnologyevolves.Inthisbook,synthesismethodshavebeenrestricted
totheclassicalpassiveandactive-RCcases,whichareexcellentrobustparadigms
withhighdidacticvalue.MorerecenttechnologieslikeOTA-C,log-domain,current
conveyorsbasedcircuitsandsoonarenotpresentedastheyareoverlyspecialized
forageneralfiltertextbookwhichisaimedmainlyatsupportingstudentsofunder-
graduatefiltercourses.
Mostofthematerialofthebookcanbetaughtaspartofundergraduatecourses
onfilters.Chapter1isageneralintroductiontofilterconceptsanddesignmethods
andprovidesabasicbackgroundtotheseaswellasthecommonlanguageforfilter
synthesisanddesign.Chapter2dealswiththeclassicalpolynomialapproximations
(ButterworthandChebyshev)thatleadtoall-polelowpasstransferfunctionswithout
zeros.ItalsointroducestherecentlydocumentedPascalapproximation.InChap.3,
rationalapproximationsarepresented.Rationalapproximations,suchastheinverse
Chebyshev,leadtolowpasstransferfunctionswithjω-axiszerosandgainoratten-
uationrippleinthestopband.Chapter4isexclusivelydevotedtoaspecialrational
approximation,theellipticapproximation,notonlyforitsimportance,butalsobe-
causeithasnotbeengiventhefullattentionitdeservesintheliteratureduetothe
vii
viii Preface
complicated mathematics involved. In this chapter, all mathematical formulae are
providedandthecorrespondingcomputationaltoolsareexplicitlypresented.
Chapters2–4applytolowpassfilters.Chapter5presentsthetransformationsthat
canbeusedforothertypesoffilters,i.e.highpass,bandpassandband-rejectfilters.
Chapter6providesanintroductiontopassivefilters,whiletheactualdesignofsuch
filtersisdealtwithinChap.7.Theactive-RCsimulationofpassiveladderfiltersis
presentedinChap.8.
Operational amplifiers and first and second order circuits are presented in
Chaps. 9 and 10, while in Chap. 11 some specific filter synthesis mathematics is
presented.Finally,Chap.12coversthesynthesisofRLCMone-portcircuits,used
extensivelyinpassivefilterdesign.
Iwouldliketoexpressmygratitudetolaboratorystaff,Dr.ElenaSarriandMr.
MakisZigirisfortheirhelpinpreparingthemanuscriptandfordrawingmostofthe
figures.SpecialthanksareduetoMr.MiltosPoulizosforhisassistanceindrawing
partofthefigures.Finally,Iwouldliketothankallmyfriendsandcolleaguesfor
encouragingmetowritethistextbookandforallofthestimulatingdiscussionswe
havehadduringtheprocess.
Athens,Greece HerculesG.Dimopoulos
Contents
1 IntroductiontoFilterConcepts . . . . . . . . . . . . . . . . . . . . . 1
1.1 GainandAttenuationFunctions . . . . . . . . . . . . . . . . . . . 1
1.2 IdealTransmission . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 IdealFilters . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 RealElectronicFilters . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.1 RealizableLowpassFilters . . . . . . . . . . . . . . . . . 8
1.3.2 RealizableHighpass(HP)Filters . . . . . . . . . . . . . . 10
1.3.3 RealizableBandpass(BP)Filters . . . . . . . . . . . . . . 11
1.3.4 RealizableBand-Reject(BR)Filters . . . . . . . . . . . . 11
1.4 FilterTechnologies. . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.5 DesigningaFilter . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.5.1 ScalingandNormalization:SmartSimplification . . . . . . 17
1.5.2 Approximation:TheHeartofFilterDesign . . . . . . . . . 18
1.6 ScalingandNormalization. . . . . . . . . . . . . . . . . . . . . . 22
1.6.1 ImpedanceScaling . . . . . . . . . . . . . . . . . . . . . . 23
1.6.2 FrequencyScaling . . . . . . . . . . . . . . . . . . . . . . 24
1.6.3 FullNormalization . . . . . . . . . . . . . . . . . . . . . . 25
1.6.4 PrototypeFilters . . . . . . . . . . . . . . . . . . . . . . . 29
1.7 CircuitOrder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2 All-PoleApproximations . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.1 FilterSpecificationsandApproximations . . . . . . . . . . . . . . 37
2.1.1 All-PoleTransferFunctionsandApproximations . . . . . . 39
2.2 TheButterworthApproximation. . . . . . . . . . . . . . . . . . . 41
2.2.1 OptimizationUsingβ asaDesignParameter . . . . . . . . 44
2.2.2 The3dBFrequencyofButterworthFilters . . . . . . . . . 46
2.2.3 TheCut-offRate . . . . . . . . . . . . . . . . . . . . . . . 47
2.2.4 TheNormalizedButterworthLowpassTransferFunction . 49
2.2.5 TableofPrototypeButterworthFilters . . . . . . . . . . . 53
2.3 TheChebyshevApproximation . . . . . . . . . . . . . . . . . . . 59
2.3.1 ChebyshevPolynomials . . . . . . . . . . . . . . . . . . . 60
ix
x Contents
2.3.2 TheAll-PoleChebyshevApproximation . . . . . . . . . . 61
2.3.3 The3-dBFrequencyandTheCut-offRate . . . . . . . . . 70
2.3.4 TheTransferFunction . . . . . . . . . . . . . . . . . . . . 71
2.4 ThePascalApproximation. . . . . . . . . . . . . . . . . . . . . . 77
2.4.1 OptimizingthePascalApproximation. . . . . . . . . . . . 84
2.4.2 OrderCalculation . . . . . . . . . . . . . . . . . . . . . . 86
2.4.3 TheTransferFunction . . . . . . . . . . . . . . . . . . . . 86
2.4.4 DesignExamples . . . . . . . . . . . . . . . . . . . . . . 88
2.4.5 ComparisontoOtherPolynomialApproximations . . . . . 91
2.5 ChebyshevandPascalDesignTables . . . . . . . . . . . . . . . . 94
2.5.1 Table:ChebyshevApproximation . . . . . . . . . . . . . . 94
2.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
3 RationalApproximations . . . . . . . . . . . . . . . . . . . . . . . . 103
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.2 RationalApproximations . . . . . . . . . . . . . . . . . . . . . . 104
3.3 TheInverseChebyshevApproximation . . . . . . . . . . . . . . . 110
3.3.1 GainAnalysis . . . . . . . . . . . . . . . . . . . . . . . . 113
3.3.2 RippleFactorandOrderCalculation . . . . . . . . . . . . 116
3.3.3 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . 118
3.4 TheInversePascalApproximation . . . . . . . . . . . . . . . . . 130
3.4.1 ThePascalRationalFunctionF (Ω) . . . . . . . . . . . . 134
P
3.4.2 DefinitionoftheInversePascalApproximation. . . . . . . 137
3.4.3 GainAnalysis . . . . . . . . . . . . . . . . . . . . . . . . 139
3.4.4 FilterDesignUsingtheInversePascalApproximation . . . 139
3.4.5 TheOrderInequalityandtheOrderNomograph . . . . . . 141
3.4.6 TheTransferFunctionoftheInversePascalFilters . . . . . 146
3.5 InversePascalTables . . . . . . . . . . . . . . . . . . . . . . . . 153
3.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
4 TheElliptic(Cauer)Approximation . . . . . . . . . . . . . . . . . . 165
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
4.2 CalculationofK(x)andsn(u,x) . . . . . . . . . . . . . . . . . . 169
4.2.1 TheAGMandtheCompleteEllipticIntegralK(x) . . . . . 170
4.2.2 TheJacobiNomeq (ModularConstant) . . . . . . . . . . 170
4.2.3 ThetaFunctionsandtheJacobiEllipticSinesn(u,x) . . . . 171
4.3 TheEllipticApproximationandtheEllipticRationalFunction . . . 172
4.4 PropertiesoftheEllipticRationalFunctionR (Ω ,Ω) . . . . . . 173
N S
4.5 Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
4.6 SpecificationsandtheOrderoftheEllipticApproximation. . . . . 180
4.7 EllipticFilterDesignOptimization . . . . . . . . . . . . . . . . . 184
4.7.1 StopbandOptimizationΩ . . . . . . . . . . . . . . . 187
Smax
4.8 TheTransferFunction . . . . . . . . . . . . . . . . . . . . . . . . 190
4.9 EllipticFilterDesignAids . . . . . . . . . . . . . . . . . . . . . . 204