# Contents Introduction to Filter Concepts All-Pole Approximations

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1 Contents 1 Introduction to Filter Concepts Gain and Attenuation Functions Ideal Transmission Ideal Filters Real Electronic Filters Realizable Lowpass Filters Realizable Highpass (HP) Filters Realizable Bandpass (BP) Filters Realizable Band-Reject (BR) Filters Filter Technologies Designing a Filter ScalingandNormalization:SmartSimplification Approximation: The Heart of Filter Design ScalingandNormalization Impedance Scaling Frequency Scaling FullNormalization Prototype Filters CircuitOrder Problems References All-Pole Approximations Filter Specifications and Approximations All-Pole Transfer Functions and Approximations TheButterworthApproximation Optimization Using β asadesignparameter The 3 db Frequency of Butterworth Filters TheCut-offRate The Normalized Butterworth Lowpass Transfer Function Table of Prototype Butterworth Filters The Chebyshev Approximation Chebyshev Polynomials The All-Pole Chebyshev Approximation The 3-dB Frequency and The Cut-off Rate The Transfer Function ThePascalApproximation OptimizingthePascalApproximation OrderCalculation ix

2 x Contents The Transfer Function DesignExamples Comparison to Other Polynomial Approximations Chebyshev and Pascal Design Tables Table: Chebyshev Approximation Problems References Rational Approximations Introduction Rational Approximations The Inverse Chebyshev Approximation GainAnalysis RippleFactorandOrderCalculation Optimization TheInversePascalApproximation The Pascal Rational Function F P (Ω) Definition of the Inverse Pascal Approximation GainAnalysis Filter Design Using the Inverse Pascal Approximation The Order Inequality and the Order Nomograph The Transfer Function of the Inverse Pascal Filters InversePascalTables Problems References The Elliptic (Cauer) Approximation Introduction Calculation of K(x) and sn(u, x) The AGM and the Complete Elliptic Integral K(x) The Jacobi Nome q (Modular Constant) Theta Functions and the Jacobi Elliptic Sine sn(u, x) The Elliptic Approximation and the Elliptic Rational Function Properties of the Elliptic Rational Function R N (Ω S,Ω) Calculations Specifications and the Order of the Elliptic Approximation Elliptic Filter Design Optimization Stopband Optimization Ω S max The Transfer Function Elliptic Filter Design Aids TheOrderNomograph Transfer Function Tables Computer Aided Elliptic Filter Design Problems References Frequency Transformations The Lowpass to Highpass (LP-HP) Frequency Transformation The LP-HP Frequency Transformation at Transfer Function Level The LP-HP Frequency Transformation at Component Level

3 Contents xi 5.2 The Lowpass to Bandpass (LP-BP) Frequency Transformation The LP-BP Frequency Transformation at Transfer Function Level The LP-BP Frequency Transformation at Component Level The Lowpass to Band-Reject (LP-BR) Frequency Transformation The LP-BR Frequency Transformation at Transfer Function Level The LP-BR Frequency Transformation at Component Level Problems References Passive Filters: Basic Theory and Concepts PowerandMaximumPowerTransfer Insertion(TransmissionorEffective)Parameters The Effective Attenuation The Reflected Power and the Reflection Coefficient The Transmission Function T(s)and the Characteristic Function K(s) Relationship Between ρ(s) and T(s)or H(s) The Passive Filters Design Procedure Determination of Z 1 (s) of a Lowpass Filter Determination of Z 1 (s) via ρ(s) Determination of Z 1 (s) via ABCD Two-PortParameters Problems References Synthesis and Design of Passive Filters Preliminaries The Butterworth Approximation in Passive Filter Design Butterworth Filter Design by Analysis The Chebyshev Approximation in Passive Filter Design Even Order Passive Chebyshev Filters The Modified Chebyshev Approximation The Synthesis of Passive Chebyshev Filters The Pascal Approximation in Passive Filter Design Even Order Passive Pascal Filters TheModifiedPascalApproximation The Elliptic Approximation in Passive Filter Design Odd Order Elliptic Passive Filters Even Order Elliptic Passive Filters Problems References Active Simulation of Passive Ladder Filters Inductance Simulation Riordan s Simulated Inductor Antoniou s Simulated Inductor Floating Simulated Inductors A General Approach A Single Op Amp Simulated Inductor Frequency Dependent Impedance Scaling Linear Transformation Active Filters Connection of Linearly Transformed Two-Ports Input Termination

4 xii Contents OutputTermination Wave Active Filters Leap-Frog Filters Problems References Operational Amplifiers Introduction Operational Amplifier Models Voltage-Mode Operational Amplifiers Current-Mode Operational Amplifiers Basic Operational Amplifier Circuits VoltageFollowerorBuffer InvertingAmplifierCircuits CircuitswithNon-invertingVoltageAmplifier IntegratorsRevisited InvertingIntegrators Non-invertingIntegrators Operational Amplifier Imperfections The Frequency Dependent Finite Open-Loop Gain OtherImperfections Linear and Non-linear Operation Problems References Second Order Functions and Circuits Introduction Second Order Functions Second Order Lowpass (LP) Transfer Functions Second Order Highpass (HP) Transfer Functions Second Order Bandpass (BP) Transfer Functions Second Order Band-Reject (BR) Transfer Functions Second Order Allpass (AP) Transfer Functions Second Order Active-RC Circuits Impedance and Frequency Scaling TheRC-CRorLP-HPTransformation Sallen-KeyCircuits Sallen and Key Second Order Lowpass Filter Sallen and Key Second Order Highpass Filter Sallen and Key Second Order Bandpass Filter Deliyannis Circuits The Generalized Deliyannis Circuit Friend s Biquad Multiple Feedback (MF) Circuits The Lowpass Multiple Feedback Circuit The Highpass Multiple Feedback Circuit The Bandpass Multiple Feedback Circuit The Band-Reject Multiple Feedback Circuit TheBoctorCircuits Current Generalized Immittance Converter (CGIC) Circuits The Basic CGIC Biquad The Generalized (or 2-OA) CGIC Biquad

5 Contents xiii 10.8 Biquads with 3 Operational Amplifiers The Tow-Thomas Biquad TheStateVariableorKHNCircuit TheUniversalCircuit The 3-OA CGIC Biquad TheBainter3-OACircuit CreationofZeros Problems References Some Filter Design Mathematics Polynomials Even and Odd Part of Polynomials The Polynomial P( s) Hurwitz Polynomials The Continued-Fraction Property Strictly Hurwitz Polynomials Routh s Criterion Rational Functions Sturm s Theorem Retrieving Polynomial P(s)from P(jω) Partial Fractions Expansion Pole Residues Positive Real (PR) Functions Positive Real Functions: Definition I Positive Real Functions: Definition II Positive Real Functions: Definition III Problems References Synthesis of RLCM One-Port Circuits PoleRemovals Removal of Pole at s = Removal of Pole at s = Removal of a Conjugate Pole Pair at s =±jω PartialPoleRemoval RemovalofaConstant Minimum Positive Real Functions Synthesis of Minimum Positive Real Functions Brune s Method with X 1 < Brune s Method with X 1 > Synthesis of LC One-Ports Properties of the LC Functions FosterRealizations Cauer Realizations Problems References Index

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