Analog Filters using MATLAB

Preface

Contents

Note to Instructors

Introduction to Analog Filters

1.1 Introduction

1.2 Signals and Signal Carriers

1.2.1 Analog Signals

1.2.2 Continuous-Time Signals

1.2.3 Signal Carriers

1.2.4 Discrete-Time and Digital Signals

1.3 Filter Terminology

1.3.1 Filter Synthesis

1.3.2 Filter Realizations

1.3.3 Implementation

1.4 Examples of Applications

1.4.1 Carrier Frequency Systems

1.4.2 Anti-aliasing Filters

1.4.3 Hard Disk Drives

1.5 Analog Filter Technologies

1.5.1 Passive Filters

1.5.2 Active Filters

1.5.3 Integrated Analog Filters

1.5.4 Technologies for Very High Frequencies

1.5.5 Frequency Ranges for Analog Filters

1.6 Discrete-Time Filters

1.6.1 Switched Capacitor Filters

1.6.2 Digital Filters

1.7 Analog Filters

1.7.1 Frequency Response

1.7.2 Magnitude Function

1.7.3 Attenuation Function

1.7.4 Phase Function

1.7.5 LP, HP, BP, BS, and AP Filters

1.7.6 Phase Delay

1.7.7 Group Delay

1.8 Transfer Function

1.8.1 Poles and Zeros

1.8.2 Minimum-Phase and Maximum-Phase Filters

1.9 Impulse Response

1.9.1 Impulse Response of an Ideal LP Filter

1.10 Step Response

1.11 Problems

Synthesis of Analog Filters

2.1 Introduction

2.2 Filter Specification

2.2.1 Magnitude Function Specification

2.2.2 Attenuation Specification

2.2.3 Group Delay Specification

2.3 Composite Requirements

2.4 Standard LP Approximations

2.4.1 Butterworth Filters

2.4.2 Poles and Zeros of Butterworth Filters

2.4.3 Impulse and Step Response of Butterworth Filters

2.4.4 Chebyshev I Filters

2.4.5 Poles and Zeros of Chebyshev I Filters

2.4.6 Reflection Zeros of Chebyshev I Filters

2.4.7 Impulse and Step Response of Chebyshev I Filters

2.4.8 Chebyshev II Filters

2.4.9 Poles and Zeros of Chebyshev II Filters

2.4.10 Impulse and Step Response of Chebyshev II Filters

2.4.11 Cauer Filters

2.4.12 Poles and Zeros of Cauer Filters

2.4.13 Impulse and Step Response of Cauer Filters

2.4.13.1 Cauer Filters with Minimum Q Factors

2.4.13.2 Cauer and Chebyshev II Filters of Type B

2.4.13.3 Cauer and Chebyshev I Filters of Type C

2.4.14 Comparison of Standard Filters

2.4.15 Design Margin

2.4.16 Lowpass Filters with Piecewise-Constant Stopband Specification

2.5 Miscellaneous Filters

2.5.1 Filters with Diminishing Ripple

2.5.2 Multiple Critical Poles

2.5.3 Papoulis Monotonic L Filters

2.5.4 Halpern Filters

2.5.5 Parabolic Filters

2.5.6 Linkwitz-Riley Crossover Filters

2.5.7 Hilbert Filters

2.6 Delay Approximations

2.6.1 Gauss Filters

2.6.2 Lerner Filters

2.6.3 Bessel Filters

2.6.4 Lowpass Filters with Equiripple Group Delay

2.6.5 Equiripple Group Delay Allpass Filters

2.7 Frequency Transformations

2.8 LP-to-HP Transformation

2.8.1 LP-to-HP Transformation of the Group Delay

2.9 LP-to-BP Transformation

2.10 LP-to-BS Transformation

2.11 Piecewise-Constant Stopband Requirement

2.12 Equalizing the Group Delay

2.13 Problems

Passive Filters

3.1 Introduction

3.2 Resonance Circuits

3.2.1 Q Factor of Coils

3.2.2 Q Factor for Capacitors

3.3 Doubly Terminated LC Filters

3.3.1 Maximum Power Transfer

3.3.2 Insertion Loss

3.3.3 Doubly Resistively Terminated Lossless Networks

3.3.4 Broadband Matching

3.3.5 Reflection Function

3.3.6 Characteristic Function

3.3.7 Feldtkeller’s Equation

3.3.8 Sensitivity

3.3.8.1 Passband Sensitivity

3.3.8.2 Stopband Sensitivity

3.3.9 Element Errors in Doubly Terminated Filters

3.3.9.1 Errors in the Reactive Elements

3.3.9.2 Errors in the Terminating Resistors

3.3.9.3 Effects of Lossy Elements

3.3.10 Design of Doubly Terminated Filters

3.3.10.1 LC Filters with Diminishing Ripple

3.4 Lowpass Ladder Structures

3.4.1 RCLM One-Ports

3.4.2 Generic Sections

3.4.2.1 Coupled Inductors

3.4.2.2 C Section with Tapped Coils

3.4.3 Lowpass Ladder Structures without Finite Zeros

3.4.4 Lowpass Ladder Structures with Finite Zeros

3.4.5 Design of Lowpass LC Ladder Filters

3.4.5.1 Element Values in Butterworth LC Ladders

3.4.5.2 Element Values in Chebyshev I LC Ladders

3.4.5.3 Element Values in Chebyshev II LC Ladders

3.4.5.4 Element Values in Cauer LC Ladders

3.5 Frequency Transformations

3.5.1 Changing the Impedance Level

3.5.2 Changing the Frequency Range

3.5.3 LP-to-HP Transformation

3.5.4 Multiplexers

3.5.5 LP-BP Transformation

3.5.6 LP-BS Transformation

3.6 Network Transformations

3.6.1 Dual Networks

3.6.2 Symmetrical and Antimetrical Networks

3.6.3 Reciprocity

3.6.4 Bartlett’s Bisection Theorem

3.6.5 Delta-Star Transformations

3.6.6 Norton Transformations

3.6.7 Impedance Transformations

3.6.8 Transformations to Absorb Parasitic Capacitance

3.6.9 Minimum-Inductor Filters

3.7 Lattice Filters

3.7.1 Symmetrical Lattice Structures

3.7.2 Synthesis of Lattice Reactances

3.7.3 Element Sensitivity

3.7.4 Bartlett and Brune’s Theorem

3.7.5 Bridged-T Networks

3.7.6 Half-Lattices

3.7.7 Reactance One-Ports

3.7.7.1 Foster I and II

3.7.7.2 Cauer I and II

3.8 Allpass Filters

3.8.1 Constant-R Lattice Filters

3.8.2 Constant-R Bridged-T Sections

3.8.3 Constant-R Right-L and Left-L Sections

3.8.4 Equalizing the Group Delay

3.8.5 Attenuation Equalizing

3.9 Electromechanical Filters

3.9.1 Mechanical Filters

3.9.1.1 Integrated Microelectromechanical Filters

3.9.2 Crystal Filters

3.9.3 Ceramic Filters

3.9.4 Surface Acoustic Wave Filters

3.9.5 Bulk Acoustic Wave Filters

3.10 Problems

Filters with Distributed Elements

4.1 Introduction

4.2 Transmission Lines

4.2.1 Wave Description

4.2.2 Chain Matrix for Transmission Lines

4.2.3 Lossless Transmission Lines

4.2.4 Richards’ Variable

4.2.5 Unit Elements

4.2.5.1 Matched Termination (Z2 = Z0)

4.2.5.2 Open-Ended (Z2 = ¥)

4.2.5.3 Short-Circuited (Z2 = 0)

4.3 Microstrip and Striplines

4.3.1 Stripline

4.3.2 Microstrip

4.3.3 MIC and MMIC Microstrip Filters

4.3.3.1 Superconducting Circuits

4.4 Commensurate-Length Transmission Line Filters

4.4.1 Richards’ Structures

4.5 Synthesis of Richards’ Filters

4.5.1 Richards’ Filters with Maximally Flat Passband

4.5.2 Richards’ Filters with Equiripple Passband

4.5.3 Implementation of Richards’ Structures

4.5.3.1 Stepped Impedance Filters

4.6 Ladder Filters

4.7 Ladder Filters with Inserted Unit Elements

4.7.1 Kuroda-Levy Identities

4.8 Coupled Resonators Filters

4.8.1 Immitance Inverters

4.8.1.1 General Immitance Inverter

4.8.1.2 Positive Impedance Inverters

4.8.1.3 Realization of GII

4.8.2 BP Filters Using Capacitively Coupled Resonators

4.9 Coupled Line Filters

4.9.1 Parallel-Coupled Line Filters

4.9.2 Hairpin-Line Bandpass Filters

4.9.3 Interdigital Bandpass Filters

4.9.4 Combline Filters

4.10 Problems

Basic Circuit Elements

5.1 Introduction

5.2 Passive and Active n-Ports

5.3 Passive and Active One-Ports

5.3.1 Passive One-Ports

5.3.2 Active One-Ports

5.3.2.1 Frequency-Dependent Negative Resistors (FDNRs)

5.4 Two-Ports

5.4.1 Chain Matrix

5.4.2 Impedance and Admittance Matrices

5.4.3 Passive Two-Ports

5.4.3.1 Transformer

5.4.3.2 Gyrator

5.4.4 Active Two-Ports

5.4.4.1 Controlled Signal Sources

5.4.4.2 Generalized Immitance Converters (GICs)

5.4.4.3 Generalized Immitance Inverters (GIIs)

5.5 Three-Ports

5.5.1 Passive Three-Ports

5.5.2 Active Three-Ports

5.6 Operational Amplifiers

5.6.1 Small-Signal Model of Operational Amplifiers

5.6.2 Implementation of an Operational Amplifier

5.7 Transconductors

5.7.1 Transconductance Feedback Amplifiers

5.7.2 Small-Signal Model for Transconductors

5.7.3 Implementation of a Transconductor

5.8 Current Conveyors

5.8.1 Current Conveyor I (CCI)

5.8.2 Current Conveyor II (CCII)

5.8.3 Current Conveyor III (CCIII)

5.8.4 Small-Signal Model for Current Conveyor II

5.8.5 CMOS Implementation of a CCII±

5.9 Realization of Two-Ports

5.9.1 Realization of Controlled Sources: Amplifiers

5.9.1.1 Controlled Sources with Operational Amplifiers

5.9.1.2 Controlled Sources with Transconductors

5.9.1.3 Controlled Sources with Current Conveyors

5.9.2 Realization of Integrators

5.9.2.1 Miller Integrator

5.9.2.2 Negative Integrator with Passive Compensation

5.9.2.3 Positive Integrators

5.9.2.4 Noninverting Integrator with Passive Compensation

5.9.2.5 Noninverting Integrators with Active Compensation

5.9.2.6 Transconductor-Based Integrators

5.9.2.7 Current Conveyor-Based Integrators

5.9.3 Realization of Immitance Inverters and Converters

5.9.3.1 Antoniou’s GIC

5.9.3.2 Transconductor-Based Gyrator

5.9.3.3 Current Conveyor-Based Gyrator

5.10 Realization of One-Ports

5.10.1 Integrated Resistors

5.10.1.1 Polycrystalline Resistors

5.10.1.2 Active Realization of Integrated Resistors

5.10.1.3 MOSFET Circuits

5.10.1.4 MOSFET Resistors

5.10.2 Differential Miller Integrators

5.10.3 Integrated Capacitors

5.10.4 Inductors

5.10.4.1 Integrated Inductors with Passive Components

5.10.4.2 Integrated Inductors with Active Components

5.10.4.3 Antoniou’s GIC of Type A

5.10.4.4 Antoniou’s GIC of Type B

5.10.4.5 Transconductor-Based Inductors

5.10.4.6 Current Conveyor-Based Inductors

5.10.5 FDNRs

5.11 Problems

First- and Second-Order Sections

6.1 Introduction

6.2 First-Order Sections

6.2.1 First-Order LP Section

6.2.2 First-Order HP Section

6.2.3 First-Order AP Section

6.3 Realization of First-Order Sections

6.4 Second-Order Sections

6.4.1 Second-Order LP Section

6.4.1.1 Q Factor for a Pole Pair

6.4.2 Second-Order HP Section

6.4.3 Second-Order LP-Notch Section

6.4.4 Second-Order HP-Notch Section

6.4.5 Second-Order BP Section

6.4.5.1 Second-Order AP Section

6.4.6 Element Sensitivity

6.4.7 Gain-Sensitivity Product

6.4.8 Amplifiers with Finite Bandwidth

6.4.9 Comparison of Sections

6.5 Single-Amplifier Sections

6.5.1 RC Networks

6.5.2 Gain-Sensitivity Product for SAB

6.5.3 Sections with Negative Feedback

6.5.3.1 NF1 Sections

6.5.3.2 NF1 LP Section

6.5.3.3 NF1 HP Section

6.5.3.4 NF1 HP-Notch Section

6.5.3.5 NF1 BP Section

6.5.3.6 NF1 AP Section

6.5.3.7 NF2 Sections

6.5.3.8 NF2 Generic Section

6.5.4 NF2 AP Section

6.5.5 Sections with Positive Feedback

6.5.5.1 PF1 Sections

6.5.5.2 PF1 LP Section

6.5.5.3 PF1 HP Section

6.5.5.4 PF1 BP Section

6.5.5.5 PF2 Sections

6.5.5.6 PF2 LP Section

6.5.5.7 PF2 HP Section

6.5.5.8 PF2 BP Section

6.5.5.9 PF3 Sections

6.5.5.10 PF4 Sections

6.5.5.11 PF4 LP Section

6.5.5.12 PF4 LP-Notch and HP-Notch Sections

6.5.5.13 PF4 HP Section

6.5.5.14 PF4 BP Section

6.5.6 ENF Sections

6.5.6.1 ENF Section: Deliyannis-Friend Section

6.5.6.2 ENF BP Section

6.5.7 Complementary Sections

6.6 Transconductor-Based Sections

6.7 GIC-Based Sections

6.7.1 GIC LP Section

6.7.2 GIC LP-Notch Section

6.7.3 GIC HP Section

6.7.4 GIC HP-Notch Section

6.7.5 GIC BP Section

6.7.6 GIC AP Section

6.8 Two-Integrator Loops

6.8.1 Two-Integrator Loops with Lossless Integrators

6.8.2 Kerwin-Huelsman-Newcomb Section

6.8.3 Transposed Two-Integrator Loop

6.8.4 Two-Integrator Loops with Lossy Integrators

6.8.5 Tow-Thomas SectionTow-Thomas Section

6.8.5.1 Integrators with Several Operational Amplifiers

6.8.6 Åkerberg-Mossberg Section

6.9 Amplifiers with Low GB Sensitivity

6.9.1 Differential Two-Integrator Loops

6.9.2 Transconductor Based on Two-Integrator Loops

6.9.2.1 LP Section

6.9.2.2 HP Section

6.9.3 Current Conveyors-Based Sections

6.9.3.1 LP Section

6.9.3.2 General Section

6.10 Sections with Finite Zeros

6.10.1 Summing of Node Signals

6.10.2 Injection of the Input Signal

6.10.2.1 Signal Injection Through Grounded Elements

6.11 Problems

Coupled Forms

7.1 Introduction

7.2 Taxonomy for Analog Filters

7.2.1 Coupled Forms

7.2.2 Simulation of Ladder Structures

7.3 Cascade Form

7.3.1 Optimization of Dynamic Range

7.3.2 Thermal Noise

7.3.2.1 Flicker Noise

7.3.3 Noise in Amplifiers

7.3.4 Noise in Passive and Active Filters

7.3.5 Distortion

7.3.6 Pairing of Poles and Zeros

7.3.7 Ordering of Sections

7.3.8 Optimizing the Section Gain

7.3.9 Scaling of Internal Nodes in Sections

7.3.9.1 General Tow-Thomas Section

7.3.9.2 General Åkerberg-Mossberg Section

7.3.10 LTC1562 and LTC1560

7.4 Parallel Form

7.5 Multiple-Feedback Forms

7.5.1 Follow-the-Leader-Feedback Form(FLF)

7.5.1.1 Companion Form

7.5.1.2 Shifted Companion Form (SCF)

7.5.1.3 Primary Resonator Block Structures

7.5.2 Inverse Follow-the-Leader-Feedback Form

7.5.3 Minimum Sensitivity Form

7.6 Transconductor-Based Coupled Forms

7.6.1 Inverse Follow-the-Leader-Feedback Form

7.6.2 Finite Transmission Zeros

7.7 Problems

Immitance Simulation

8.1 Introduction

8.2 PIC-Based Simulation

8.3 Gyrator-Based Simulation

8.3.1 Transconductor-Based Gyrator-C Filters

8.3.2 CCII-Based Gyrator-C Filters

8.4 Gorski-Popiel’s Method

8.5 Bruton’s Method

8.6 Problems

Wave Active Filters

9.1 Introduction

9.2 Generalized Wave Variables

9.2.1 Wave Transmission Matrix

9.2.2 Chain Scattering Matrix

9.2.3 Generalized Scattering Matrix

9.2.4 Voltage Scattering Matrix

9.3 Interconnection of Wave Two-Ports

9.4 Elementary Wave Two-Ports

9.5 Higher-Order Wave One-Ports

9.6 Circulator-Tree Wave Active Filters

9.7 Realization of Wave Two-Ports

9.7.1 Realization of a Generic Wave Two-Port

9.7.2 Differential Wave Two-Port

9.8 Realization Of Wave Active Filters

9.9 Power Complementarity

9.10 Alternative Approach

9.11 Problems

Topological Simulation

10.1 Introduction

10.2 LP Filters Without Finite Zeros

10.2.1 Lowpass Leapfrog Filters

10.2.2 Realization of the Signal-Flow Graph

10.2.3 Scaling of Signal Levels

10.3 Geometrically Symmetric BP Leapfrog Filters

10.4 Lowpass Filters Realized with Transconductors

10.5 LP Filters with Finite Zeros

10.5.1 Odd-Order Lowpass Filters with Finite Zeros

10.5.2 Even-Order Lowpass Filters with Finite Zeros

10.6 Problems

Tuning Techniques

11.1 Introduction

11.2 Component Errors

11.2.1 Absolute Component Errors

11.2.2 Ratio Errors

11.2.3 Dummy Components

11.3 Trimming

11.3.1 Trimming of Second-Order Sections

11.3.1.1 Lowpass Section

11.3.1.2 Highpass Section

11.3.1.3 Bandpass Section

11.3.2 LC Filters

11.4 On-Line Tuning

11.4.1 Pseudo-on-Line Tuning

11.4.2 Master-Slave Frequency Tuning

11.4.2.1 Integrator-Based Tuning

11.4.2.2 Phase-Locked Filter

11.4.2.3 Phase-Locked Oscillators

11.4.3 Master-Slave Q Factor Tuning

11.4.3.1 Phase-Locked Integrator

11.4.3.2 Passband Gain-Based Q Tuning

11.4.3.3 LMS-Based Q Tuning

11.4.3.4 Combined Frequency and Q Factor Tuning

11.5 Off-Line Tuning

11.5.1 Tuning of Composite Structures

11.5.1.1 Tuning of High-Order High-Q Filters

11.5.2 Parasitic Effects

11.6 Problems

References

Toolbox for Analog Filters

Index