Analog Filters

370 Pages·1996·6.79 MB·English

by  Kendall L. Su (auth.)

Analog Filters
Analog Filters
Kendall L. Su
Regents' Professor Emeritus, School of Electrical and Computer Engineering,
Georgia Institute of Technology, Atlanta, USA.
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SPRlNGER-SCIENCE+BUSINESS MEDIA, B.V.
First edition 1996
© 1996 Kendall L. Su
Originally published by Chapman and Hali in 1996
Softcover reprint of the hardcover 1s t edition 1996
ISBN 978-1-4613-5851-0 ISBN 978-1-4615-2051-1 (eBook)
DOI 10.1007/978-1-4615-2051-1
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Contents
Preface xi
1 Introduction 1
1.1 Preliminary remarks 1
1.2 The analog filter . . 3
1.3 Ideal and approximate filter characteristics 4
1.4 MAT LAB ... 7
1.5 Circuit analysis 9
1.6 Normalization and denormalization
- scaling ......... . 12
1.6.1 Frequency scaling. 12
1.6.2 Impedance scaling 14
1.6.3 Normalization and denormalization . 15
1. 7 Steps involved in the design of filters 15
Problems .......... . 19
2 The approximation 23
2.1 The Butterworth lowpass characteristic ....... 24
2.1.1 The normalized Butterworth lowpass charac-
teristic . . . . . . . . . . . . . . . . . . . . . . 24
2.1.2 Using a normalized Butterworth characteristic for
a filtering requirement . . . . . 27
2.2 The Chebyshev lowpass characteristic 28
2.2.1 The Chebyshev polynomial . . 29
2.2.2 The Chebyshev lowpass characteristic 32
2.3 Other Chebyshev-related characteristics 35
2.4 The elliptic-function filter characteristic 37
vi Contents
2.5 Comparison of standard lowpass characteristics 39
2.6 Summary 41
Problems 41
3 Network functions 47
3.1 General procedure ............ . 47
3.2 Network functions for Butterworth filters 51
3.3 Network functions for Chebyshev filters . 56
3.4 Network functions for elliptic-function filters. 60
3.5 Bessel-Thomson filter functions 64
3.6 Delay equalization 69
3.7 Summary 72
Problems 72
4 Frequency transformation 75
4.1 Lowpass-to-highpass transformation 76
4.2 Lowpass-to-bandpass transformation 78
4.3 Lowpass-to-bandreject transformation 84
4.4 Summary 86
Problems 86
5 Properties and synthesis of passive networks 91
5.1 The driving-point function of a passive
oneport - the positive real function . . . ...... . .. 91
5.2 The driving-point function of a lossless
oneport - the lossless function . . . . . 92
5.2.1 Properties of a lossless function . 92
5.2.2 Foster's expansion of a loss less function 94
5.2.3 Foster's realizations of a loss less function 97
5.2.4 Removal of poles at infinity .. 98
5.2.5 Removal of poles at the origin . 102
5.2.6 Removal of finite nonzero poles 104
5.2.7 Mixed canonic realization 105
5.2.8 Noncanonic realization . 107
5.3 Properties of loss less twop orts . 108
5.4 LC ladder twoport .. . ... 110
Contents vii
5.5 Foster's preamble 113
5.6 Summary 116
Problems 117
6 Singly-terminated LC ladders 121
6.1 LC ladder with a current source . 121
6.1.1 Transmission zeros at the origin and infinity. 123
6.1.2 Zero shifting .................. 129
6.1.3 LC ladder with finite nonzero transmission zeros 131
6.2 LC ladder with a voltage source. . . 134
6.3 Other singly-terminated orientations 136
6.4 Summary 137
Problems 138
7 Doubly-terminated LC ladders 143
7.1 Basic formulation ....... . 143
7.2 LC ladders with equal terminations. 147
7.3 LC ladders with unequal terminations 156
7.4 A doubly-terminated filter used in reverse 161
7.5 Summary 165
Problems 165
8 Sensitivity 169
8.1 Definition of sensitivity 171
8.2 Properties of first-order sensitivity 172
8.3 Sensitivities of network performance 173
8.4 Sensitivity calculation . . . . . 175
8.5 Unnormalized sensitivity .... 178
8.6 Multiparameter and statistical
sensitivities .... ..... 179
8.7 A case for low sensitivity of passive filters 181
8.8 Summary 182
Problems 182
9 Basics of active filters 185
9.1 Comparison of passive and active filters . 185
viii Contents
9.2 The operational amplifier (op amp) 188
9.3 Some simple op amp circuits 190
9.4 First-order sections ..... . 193
9.5 RC single-op amp circuit relationships 195
9.5.1 Finite-gain single-op amp configuration 195
9.5.2 Infinite-gain single-op amp configuration. 198
9.6 Gain adjustments ..... 198
9.6.1 Gain reduction .. 199
9.6.2 Gain enhancement 199
9.7 RC-CR transformation. 201
9.8 Types of biquads 204
9.9 Summary 205
Problems 205
10 Biquad circuits 215
10.1 Sallen-Key biquads 215
10.1.1 Lowpass biquad . 215
10.1.2 Highpass biquad 221
10.1.3 Bandpass biquad 223
10.1.4 Bandreject biquad 225
10.2 Infinite-gain multiple-feedback (MFB)
biquads .. . . . . . . . 228
10.2.1 Lowpass biquad . 228
10.2.2 Highpass biquad 230
10.2.3 Bandpass biquad 231
10.2.4 Bandreject and allpass biquads . 235
10.3 The two-integrator biquads . . . . . . 236
10.3.1 The Kerwin-Huelsman-Newcomb (KHN) biquad 237
10.3.2 The Tow-Thomas biquad 240
10.3.3 The Fleischer-Tow biquad 242
10.4 Summary 243
Problems 244
11 High-order active filters 249
11.1 The cascade realization 249
11.2 The state-variable method of realization 260
Contents ix
11.2.1 Realization of all-pole transfer functions 261
11.2.2 Realization of general transfer functions 262
11.2.3 Realization using lossy integrators 266
11.3 Lowpass-to-bandpass transformation .. 270
11.3.1 Coupled biquads with infinite Q 270
11.3.2 Bandpass filters using the primary resonator
blocks . . . . 272
11.4 Internal gain change 273
11.5 Summary 276
Problems 276
12 Active simulation of passive filters 279
12.1 Some active twoports ....... . 280
12.2 The generalized impedance converter
(GIC) . . . . . . . . . . . . . . . . . 282
12.3 Simulation of inductances in an LC ladder. 284
12.3.1 Simulation of grounded inductors. 285
12.3.2 Simulation of floating inductors . 286
12.3.3 Simulation of groups of inductors 288
12.4 Simulation using frequency-dependent
negative resistances (FDNR's) . . . . . 292
12.5 Functional simulation of passive filters 294
12.5.1 The leap-frog realization. . . . 294
12.5.2 Leap-frog realization of the lowpass LC ladder 297
12.5.3 Leap-frog realization of bandpass filters using bi-
quads . . . . . . . . . . . . . . . . . . . . . . .. 299
12.5.4 Leap-frog realization of bandpass filters using in-
tegrators 299
12.5.5 Simulation of a special bandpass filter 301
12.6 Summary 306
Problems 306
13 Switched-capacitor filters 313
13.1 An introduction .......... . 314
13.2 Simulation of resistors by switched
capacitors . . . . . . 315
13.3 Simple basic circuits ....... . 318
x Contents
13.3.1 All-capacitor op amp circuits 319
13.3.2 The inverting integrator . . . 320
13.3.3 The inverting lossy integrator 321
13.3.4 The inverting weighted summing integrator 321
13.3.5 A noninverting integrator . . . . . . . . . 322
13.3.6 Inverting summer-integrator combination 322
13.3.7 The differential integrator . . . 323
13.3.8 The differential lossy integrator 323
13.3.9 The differential amplifier 324
13.3.lOThe first-order section 325
13.4 Switched-capacitor biquads . 326
13.5 Functional simulation of LC ladders 329
13.6 Summary 330
Problems 333
Appendix A Tables of filter functions 337
Bibliography 355
Index 359
Preface
This book is intended as an intermediate-level introduction to the ba
sic theory of analog filters. It covers three major fundamental types of
analog filters - passive, active, and switched-capacitor. The only ba
sic knowledge required to follow the material in this book is some basic
circuit theory, signal analysis, Laplace transforms, and mathematics typ
ically required by most engineering curriculums at the sophomore level.
The emphasis of this book is on giving the student some fundamental
principles behind the various techniques of analog filter design. It is tar
geted toward students in communications, signal processing, electronics,
controls, etc. It is not meant to be an in-depth or comprehensive treat
ment of the entire area of filter theory as network theorists and filter
designers would like to present such a subject. Rather, it is meant to ex
pose the student to the elegant theory behind the development of analog
filters. It also introduces the student to the jargon used and techniques
practiced in analog filters.
The design of standard filters is now a fairly routine matter. In fact,
one can purchase software at a very modest cost and generate filters
without ever knowing anything about the underlying principles. The
main purpose of this book is to engender some understanding of the
mathematical basis of network synthesis and filter theory. Although
the mechanical steps for generating filters are covered, they are not the
major focus of this book. It stresses the mathematical bases and the
scholastic ingenuity of analog filter theory. In other words, the student
will learn why analog filters work as well as how they can be generated.
The book should help nonspecialist electrical engineers in gaining a back
ground perspective and some basic insight into the development of real
time filters. In many modern advances in signal handling, their concepts
and procedures have close links to analog filters, either conceptually or
mathematically. The material in this book will provide engineers with
a better perspective and more penetrating appreciation of these modern
techniques.