Analog Circuit Theory and Filter Design in the Digital World: With an Introduction to the Morphological Method for Creative Solutions and Design [1st ed.] 978-3-030-00095-0;978-3-030-00096-7

Table of contents :
Front Matter ....Pages i-xii
Front Matter ....Pages 1-1
Mixed-Mode Signal Processing (George S. Moschytz)....Pages 3-18
Some Key Points from Network Theory (George S. Moschytz)....Pages 19-47
Filter Specifications and Approximation Theory (The Mathematical Approach to the Approximation Problem) (George S. Moschytz)....Pages 49-64
Filter Tables and Computer Programs (The Physical Approach to the Approximation Problem) (George S. Moschytz)....Pages 65-84
An Introduction to Signal-Flow Graph Theory (George S. Moschytz)....Pages 85-100
Controlled Sources, Nullors, Active Gain Devices, Impedance Converters and Inverters (Gyrators, NICs, Current Conveyors) (George S. Moschytz)....Pages 101-147
Passive LCR and Active-RC Filters (George S. Moschytz)....Pages 149-166
A Classification of Single-Amplifier Biquads (George S. Moschytz)....Pages 167-228
A Morphological Approach to the Design of Active Network Elements (George S. Moschytz)....Pages 229-267
Active Filter Design Techniques (George S. Moschytz)....Pages 269-298
Elements of Sensitivity Theory (George S. Moschytz)....Pages 299-333
Random Signals and Noise (George S. Moschytz)....Pages 335-356
Deriving Current-Mode from Voltage-Mode Circuits and Filters (George S. Moschytz)....Pages 357-366
Front Matter ....Pages 367-367
From Continuous Time to Discrete Time (George S. Moschytz)....Pages 369-380
The Sampling Theorem and Aliasing (George S. Moschytz)....Pages 381-397
The Laplace Transform of Sampled Signals: The Z-Transform (George S. Moschytz)....Pages 399-406
Switched-Capacitor Filters (George S. Moschytz)....Pages 407-415
The Four-Port Analysis of Switched-Capacitor Networks (George S. Moschytz)....Pages 417-452
Design of Switched-Capacitor Filters (George S. Moschytz)....Pages 453-482
The Transmission Matrix of SC Networks and Its Signal-Flow Graph (George S. Moschytz)....Pages 483-509
Back Matter ....Pages 511-550

Citation preview

George S. Moschytz

Analog Circuit Theory and Filter Design in the Digital World With an Introduction to the Morphological Method for Creative Solutions and Design

Analog Circuit Theory and Filter Design in the Digital World

George S. Moschytz

Analog Circuit Theory and Filter Design in the Digital World With an Introduction to the Morphological Method for Creative Solutions and Design

George S. Moschytz Bar-Ilan University Jerusalem, Israel

ISBN 978-3-030-00095-0    ISBN 978-3-030-00096-7 (eBook) https://doi.org/10.1007/978-3-030-00096-7 Library of Congress Control Number: 2018961406 © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To my wife, Brenda, who reinvigorated my life.

Preface

In the 1960s and 1970s, students in electrical engineering were studying mainly analog circuit theory and design, and only a small elite group had ventured into the digital circuit and system world. Today, the opposite is true; only a small number of electrical engineers are knowledgeable in analog circuit design. The digital revolution and with it the corresponding theory and design basics in an EE curriculum have taken over, leaving only a relatively small group of electrical engineers capable of analog circuit and system design. Since the “front end,” i.e., the interface between man and machine of most electronic systems, has remained largely analog – and presumably always will – this trend, i.e., the neglect of analog circuit theory and design in the EE curriculum, has resulted in a shortage of analog circuit and filter designers. On the one hand, to obtain the necessary but missing number of analog circuit specialists, a basic body of analog circuit theory and design courses should be included in the EE curriculum. On the other hand, because of the predominance of digital circuits and systems and the requirement of maintaining a 4-year time span for the first degree, the academic EE curriculum has of necessity dropped the traditional basic courses in circuit and system design that all earlier EE students were required to take. As a result of this development, there is a decided “shortfall” in current textbooks for courses on analog circuit and filter design. Whereas there is no need any more for the intense circuit and filter theory courses of yesteryear, there is a general realization that the neglect of analog know-how has been detrimental to the required, and expected, general knowledge of the average EE engineer. This book attempts to bring together a “general knowledge” of one aspect of analog engineering, e.g., that of network theory, filter design, system theory, and sampled-data signal processing. A single book cannot be comprehensive in this respect, but the intention is for it to be adequate for a start, to be followed, as required, by the specific needs of the practicing electrical engineer. It is hoped that this book will help to fill the existing gap in analog-circuit know-how by extracting many of the most important and useful features of analog circuit theory and design which the author found most important during his own activity in industrial and academic research and development (a decade at AT&T Bell Labs in the USA and three decades at the Swiss Federal Institute – ETH – in Zurich). The book is intended as a basis for one or more graduate courses – and for self-study – in analog and sampled-data circuit theory and design for ongoing or active electrical engineers wishing, or ­needing, to become proficient in analog circuit design on a system, rather than on a device, level. The material can be considered as a basic educational staple in the EE curriculum which has been neglected in recent decades and is now found to be missing in the general know-how of an academically educated electrical engineer. Another goal of this book is to emphasize methodology and creative analysis and design techniques that can be applied to areas beyond those specifically addressed in the book. This is reflected vii

viii

Preface

in the subtitle which refers to the “morphological method for creative solutions and design.” This is a generic method of creative design which was developed by a prolific Swiss/American scientist and inventor, Professor Fritz Zwicky. Zwicky claimed to have used this method for research and development topics ranging from astrophysics (discovery of super novae) to the design of airplane jet engines for takeoff on extremely short runways (the so-called jato system for aircraft carriers). The author has adapted and used this morphological method for the creative and successful design of analog circuits, devices, and systems. It is shown that much of the design of known as well as new circuit devices, e.g., gyrators, impedance converters, can be surprisingly easily accomplished using the morphological method of design. Numerous examples from different fields are presented, e.g., circuit devices, switched capacitor circuits and filters. However, there is a strong emphasis also on other analysis and design methodology throughout the book, besides that of the morphological method. The course material has been carefully prepared for audiences including both professional electrical engineers and graduate EE students. As such it was presented by PowerPoint slides with accompanying commentary and explanations. The book has a similar format. It consists of slides with accompanying commentary and explanations in the form of text. A CD with the slides of the book material is available. Jerusalem, Israel  George S. Moschytz

Acknowledgments

I am deeply grateful to Dr. Drazen Jurisic, Professor in the EE Department of the University of Zagreb, who is a former Ph.D. student of mine. Prof. Jurisic transferred onto PowerPoint most of the slides used in this book from my hand-written notes and illustrative material. He was also of immeasurable help throughout the writing of the book, patiently helping me with the quirks and intricacies of the Word and other computer programs used. It is also a pleasure to acknowledge with gratitude the direct and indirect contributions, as well as the inspiring enthusiasm of well over 50 Ph.D. students, and a far larger number of undergraduate and graduate students, during over more than three decades of joint study and acquisition of insight into the material covered in this book. I also thank my colleagues at the ETH for valuable interaction through those three decades, and especially to the late Professor James L. Massey and to Professor Qiuting Huang, together with whom I taught some of the materials in the book. Prof. Massey’s contributions to the mathematical formulation of sampling theory and discrete-time signal processing and Prof. Huang’s contributions to the presentation of noise in analog and discrete-time circuits were immensely helpful. Finally, I wish to acknowledge the fundamental contribution made by the late Carl C. Kurth, while working together at AT&T Bell Labs, on the four-port analysis of switched capacitor circuits. Finally, my wholehearted gratitude goes to my wife Brenda. Without her encouragement, patience, and devotion, I could not have written this book. Bar-Ilan University  George S. Moschytz Jerusalem, Israel

ix

Contents

Part I Continuous-Time Signal Processing 1 Mixed-Mode Signal Processing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   3 2 Some Key Points from Network Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  19 3 Filter Specifications and Approximation Theory (The Mathematical Approach to the Approximation Problem). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  49 4 Filter Tables and Computer Programs (The Physical Approach to the Approximation Problem). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  65 5 An Introduction to Signal-Flow Graph Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  85 6 Controlled Sources, Nullors, Active Gain Devices, Impedance Converters and Inverters (Gyrators, NICs, Current Conveyors). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 7 Passive LCR and Active-RC Filters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 8 A Classification of Single-Amplifier Biquads. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 9 A Morphological Approach to the Design of Active Network Elements . . . . . . . . . . . . 229 10 Active Filter Design Techniques. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 11 Elements of Sensitivity Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 12 Random Signals and Noise. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 13 Deriving Current-Mode from Voltage-Mode Circuits and Filters. . . . . . . . . . . . . . . . . 357 Part II Discrete-Time Signal Processing 14 From Continuous Time to Discrete Time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 15 The Sampling Theorem and Aliasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 16 The Laplace Transform of Sampled Signals: The Z-Transform. . . . . . . . . . . . . . . . . . . 399

xi

xii

Contents

17 Switched-Capacitor Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 18 The Four-Port Analysis of Switched-Capacitor Networks. . . . . . . . . . . . . . . . . . . . . . . 417 19 Design of Switched-Capacitor Filters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 20 The Transmission Matrix of SC Networks and Its Signal-­Flow Graph. . . . . . . . . . . . . 483 Appendix: Problems and Solutions for SC Analysis Using Four-Port Theory. . . . . . . . . . . 511 References and Additional Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537 Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539

Part I Continuous-Time Signal Processing

Chapter 1

Mixed-Mode Signal Processing

Slide 1.1

Chapter 1 Mixed-Mode Signal Processing

1

Introduction

This introductory chapter provides the motivation for this book. It gives examples of modern electronic systems that consist mainly of digital circuits. However, for the interface with the outside world, there must be a so-called analog front end, which is characterized by analog input and output signals. The resulting system becomes a “mixed-mode” system in which digital and analog signals must interact in an optimum fashion.

© Springer Nature Switzerland AG 2019 G. S. Moschytz, Analog Circuit Theory and Filter Design in the Digital World, https://doi.org/10.1007/978-3-030-00096-7_1

3

1  Mixed-Mode Signal Processing

4

The main reasons why analog circuits, and with them analog circuit design, remains of prime importance in modern digital systems can be summarized as follows: • The fact that “real-world” signals are almost exclusively analog makes a familiarity with this branch of circuit design essential; it will continue to be required in the future, both in the electrical engineering (EE) curriculum and in the EE work place. Thus, for example, the design of the “analog front end” (AFE), which provides the interface between a digital system and the real world, will, if anything, increase in importance as the sophistication of new technology makes higher demands on performance. • Depending on the technology used, analog signal processing (ASP) can have certain important advantages over digital signal processing (DSP), e.g., low power, no A/D conversion, and speed. • “Mixed-mode” signal processing, i.e., combining ASP with DSP on a VLSI chip, has become routine – depending on the application and the technology used. • Within the last decade or so, continuous-time analog circuits and filters – just as digital and discrete-­ time, e.g., “switched-capacitor” filters before them – have proved themselves in VLSI technology.

Sampled-Data & Digital Filters TIME QUANTIZATION

AMPLITUDE QUANTIZATION

IN

DIGITAL FILTER or

A D

SAMPLE

x(t) Sampling Period π Ts = ω

x*

s

A

DIGITAL SIGNAL PROCESSOR

xD

Data Bit Length

RECONSTRUCTION FILTER HOLD / LOW-PASS

D

OUT y (t)

y* (t)

yD

Filter Function Filter Algorithm Coefficient Bit Length

* without amplitude quantization

SAMPLED DATA FILTER Digital or switched capacitor ANALOG (continuous time) FILTER Conventional LC or Active RC

Signal form examples: Filter Function x(t)

x* [k]

xD

IV

IV

5V

t

k

5ms 10ms

1 2 3 4 5

yD 5V 1 2 3 4 5

k

1 2 3 4 5

12-bit picture of sample 3 5V 10

2

Slide 1.2

0101 1 010 0 01 1 + 28 + 27 + 25 + 21 + 20 = 144310 ∆ = 1,44V e.g.

t

y* [k]

y (t)

IV

IV

k

1 2 3 4 5 k

5ms 10ms

t

1 Introduction

5

This slide provides clarity between the designations used for analog, sampled-data, and digital circuits and filters. It emphasizes the distinction between them by relating the system parts of a processing chain to the signal types associated with them. An analog signal at the input to a sampler is quantized in time but remains analog in amplitude. This is fed into an analog-to-digital (A/D) converter whose output is now purely “digital.” The digital signal is processed by a DSP (digital signal processor) or digital filter, whose output is, in turn, fed into a digital-to-analog (D/A) converter. The output time-quantized analog signals are then generally passed through a reconstruction filter that smooths out the newly produced analog time-quantized signals and produces an analog output signal. Referring to the input and output signals of the corresponding parts of the processing chain or filter, we refer to an analog, sampled-data, and digital circuit or filter.

Typical Data Acquisition & Process Control System

ANALOG DATA

Analog ‘Outside World‛

COMPARISON ISOLATION GAINS, SCALING FILTERING LINEARIZATION COMPENSATION ANALOG SIGNAL CONDITIONG SETPOINTS

Analog

GAINS

AFE

Digital

ANALOG MULTIPLEXED A/D PROGRAMMED CONVERSION GAIN SAMPLE-HOLD CONVERSION CHANNELS INTERRUPT CONTROLS GAIN

INPUT TRANSDUCERS

S/H

MEMORY

DATA

DIGITAL PROCESSOR

OUTPUT PHYSICAL TRANSDUCERS PROCESS

EFFECTIVE VALVE SETTINGS

SWITCH SETTINGS

Data Adress Busses

CONTROL

COMMUNICATIONS LINKS HOST PROCESSOR

PROGRAMS

DISPLAYS

Slide 1.3 The analog front end (AFE) is the bottleneck in many mixed-mode IC system chips (even though it may take up only a small part of the chip area), and the filters very often represent one of the main problems. The AFE is the bridge to the outside world, as shown for the typical data acquisition and process control system shown in this slide. The boundary is shown between the analog circuitry, which provides the interface to the analog outside world, and the digital circuitry, which does the actual signal processing. Note, however, that there is a considerable amount of signal conditioning that the analog circuitry has to accomplish before the digital processing can take place. The conditioning will include such tasks as amplification, anti-alias filtering, comparing, linearizing, and various other functions, as shown in the slide.

1  Mixed-Mode Signal Processing

6

Key Elements of a Sampled Data System SIGNAL CONDITIONING A

D

DIGITAL

SIGNAL A CONDITIONING

D

AFE ANTIALIAS FILTER

ANALOG WORLD

S&H

TIMING PARAMETERS

GAINS

GAIN & COEFFICIENTS

Front end

AFE

A/D

DIGITAL SIGNAL PROCESSOR

L A T C H

DIGITAL PARAMETER & TIMING CONTROL

DIGITAL CONTROL OF SIGNAL CONDITIONING

DEGLITCH

D/A

FILTER

ANALOG WORLD

TIMING PARAMETERS

SCALE

CUTOFF COEFFICIENTS

Front end

Slide 1.4 Another example is this diagram of key elements in a typical sampled-data system with the signal conditioning by analog circuitry of the incoming analog signals. Here again, the analog circuitry constitutes the analog front end, which is inevitable in most systems that communicate with the outside, generally analog, world. As shown, the analog front end most often includes some kind of filtering, which is by no means trivial when realized on-chip. This is why, among other things, we are concentrating on filtering for the AFE in a number of chapters in this book.

1 Introduction

7

X.CLOCK X.SYNC X.MASTER XMIT FILTER

A.IN

(A/B SEL (X)) (A.SIG - IN) (B.SIG - IN)

ANTIALIASING FILTER

X.VREF 2 R2 X.VREF 1 R1

LOWPASS FILTER

Analog Front End

FIL OUT COD IN DEC OUT FIL IN A.OUT

(A/B SEL(R)) (A/B SIG OUT)

HIGHPASS FILTER

A.GND D.OUT

ENCODER

CAZ C1

AUTO-OFFSET

XMIT FILTER

LOWPASS FILTER

A.GND D.IN

DECODER

R.VREF 2 R2 R.VREF 1 R1

INTERPOLATION FILTER

R.CLOCK R.SYNC R.MASTER

Single chip filter/codec block diagram Slide 1.5 In this slide we see a filter/codec (coder-decoder) block diagram, again with the inevitable signal conditioning at the analog front end. Here the signal conditioning consists almost entirely of analog filtering on the mixed-mode, single chip.

Slide 1.6 To get some idea of the area taken up by the analog front end of a mixed-mode circuit chip, here is a photomicrograph of the codec chip shown in the previous slide and of the approximate area taken up by the AFE filters. This ratio of analog-to-digital chip area will, of course, change with the times and the technology. But, for a variety of reasons, it is to be expected that the required analog chip area will never be negligible.

1  Mixed-Mode Signal Processing

8

Digital Audio Studio System Analog MICROPHONES AMPLIFIERS, AND ADCs

DIGITAL OUTPUTS ANALOG OUTPUT

DAC

Digital

DIGITAL SIGNAL PROCESSOR

CONTROL CONSOLE

Slide 1.7 In this slide we see the block diagram of a digital audio studio system. Obviously, microphones, amplifiers, and analog-todigital converters (ADCs), among other circuits, require analog circuit-­ design knowhow.

DIRECT DIGITAL AUDIO INPUTS

Slide 1.8 This slide shows the block diagram of a typical receiver and Slide 1.9 the OSC CLK oscilloscope traces of the CONTROL GEN T band-pass filters measured O LO on the receiver chip. V AGC N E A DUAL AA, BP These and the previous E N L THRESH. TONE FILTER slides exemplify the many HEX HI C I AGC D O D O reasons for the processing DV E D A U of real-world signals. They AFE T ED T E T E include the extraction of DT/DP DEGLITCH DETECT information, e.g., ampliSUPERDIAL VISION tude, frequency, spectral PULSE content, and timing relaENHEX tionships; the reformatting of signals, e.g., in freBlock diagram of a receiver quency-division multiplex (fdm) and ­time-­division multiplex (tdm) systems; the compression of data, e.g., in modems, digital mobile radio, adaptive differential pulse-code modulation (adpcm) systems, and high-definition TV (hdtv); the generation of feedback control signals, e.g., in industrial process control; the extraction of signals from noise by FEATURES, TESTING

1 Introduction

9

filtering, autocorrelation, and convolution methods; and the storage of signal data in digital format for recovery and/or analysis using digital signal processing (dsp) techniques, e.g., fast Fourier analysis techniques.

Slide 1.10 Finally, this slide shows a typical multiple-system, mixed-mode chip and indicates the many different functions, analog and digital, included on it. Among other things, there are four phase-locked loop (PLL) circuits, a digital-to-analog converter (DAC) together with a switched-capacitor (SC) filter, several analogto-digital converters (ADCs), gain stages, voltage reference circuits, and several active-RC filter circuits.

1  Mixed-Mode Signal Processing

10

2

Analog Versus Digital Signal Processing

In this section, we introduce some general comparisons and considerations with regard to the benefits, or lack thereof, of analog vs. digital signal processing.

Analog Signal Conditioning and Processing Amplification (Gain) Impedance Transformation Removing Common-Mode Noise Isolation Cable Driving and Receiving Multiplication of Signals Dynamic Range Compression Programmable Amplification Filtering (Passive-LC and Active-RC)

Bit Sizes for 10V Full-Scale Converters Examples

Telephone(A, m-law) TV SC filters Compact Disk

Resolution 4-bit 6-bit 8-bit 10-bit 12-bit 14-bit 16-bit 18-bit 20-bit 22-bit 24-bit

LSB: Least Significant Bit FS: Full Scale

1 LSB 625mV 156mV 39mV 9.76mV 2.44mV 610mV 153mV 38mV 9.5mV 2.4mV 0.6mV

%FS 0.5 LSB 6.25 313mV 1.56 78mV 0.39 19.5mV 0.098 4.88mV 0.024 1.22mV 0.0061 305mV 0.0015 76mV 0.0004 19mV 0.0001 4.8mV 1.2mV 0.000024 0.3mV 0.000006

ppm FS dB FS -24 62500 -36 15625 -48 3906 -60 977 -72 244 -84 61 -96 15 4 -108 1 -120 0.24 -132 0.06 -144

Slide 1.11 In the question of “how much analog?” vs. “how much digital ?” that mixedmode signal processing of necessity implies, it helps to consider some of the many functions, listed in this slide, for which it is often advantageous to use analog rather than digital signal processing.

Slide 1.12 Continuing with these considerations, it is useful to compare the analog voltage range in a 10  V fullscale converter vs. the corresponding digital bit length necessary to cover the same voltage, or dynamic, range.

2  Analog Versus Digital Signal Processing

11

Trade-off between Resolution and Sampling Rate in ADCs

Down

Resolution 22 bits 20 bits 18 bits 16 bits 14 bits 12 bits 10 bits 8 bits

Sampling Rate 1 kSPS 4 kSPS 50 kSPS 500 kSPS Down 10 MSPS 25 MSPS 75 MSPS 500 MSPS

SPS: Samples per second

Filter Order vs. Sampling Frequency Filter Order N

maximum chip area & minimum dynamic range

SC maximum sampling frequency DSP Sampling frequency fs

Slide 1.13 It is important to be aware of the fact that there is a trade-off between the resolution and sampling rate in a typical analog-to-digital converter. Clearly this trade-off changes as technology progresses and develops; however the available channel bandwidth, no matter how large, will always set an upper bound on the sampling rate and on the corresponding possible resolution in bits.

Slide 1.14 Turning to digital vs. sampled-data circuits (e.g., switched-capacitor filters) demonstrates how in a digital filter or signal processor a high-order filter is limited to a low sampling rate and vice versa. By comparison, with a sampled-data filter, such as the switched-capacitor filter, the sampling rate is not limited by the filter order but rather by such factors as the bandwidth and cutoff frequency of the amplifiers used in the filter.

1  Mixed-Mode Signal Processing

12

Power Consumption vs. Dynamic Range power per pole edge frequency

[J] Digital: Reducing Power Consumption does not reduce dynamic range

10–6 digital limit

10–8

4mm(10pJ) 1mm(1pJ)

10–10 Analog: reducing power consumption reduces dynamic range

analog limit

10–12 Dynamic range in dB

40

60

80

100

Slide 1.15 This slide shows the interrelationship between power consumption, dynamic range, and downscaling chip dimensions for digital and analog circuitry. Using filter design as an example, the important message given is that reducing the power consumption in analog circuits (measured by the power required per filter pole, referred to the edge frequency of the amplifier and other devices) by downscaling the chip dimensions drastically reduces the available dynamic range in analog, but not in digital circuits. This, and most of the consecutive slides, was developed and published by Prof. Eric Vittoz, a renowned expert in general and low-power CMOS circuit design at Centre Electronique Horloger in  Neuchâtel, Switzerland. To the authors knowledge, the term “perceptive signal processing” (see below) is also due to E. Vittoz.

Why?

Combining Analog and Digital

Bridge to the outside world

Low power consumption No A/D conversion Speed Where?

The Analog Front End Applications requiring small size and battery operation hearing aids aids of all kind for the handicapped diagnostics tools mobile radio hand-held measurement instrumentation portable communication systems

Slide 1.16 Finally, this slide briefly summarizes some of the self-explanatory why’s and where’s of combining analog with digital circuitry.

3  Traditional Versus Perceptive Signal Processing

3

13

Traditional Versus Perceptive Signal Processing

The final theme of this chapter is to compare traditional signal processing with so-called perceptive signal processing. Traditional signal processing requires a certain level of signal-to-noise ratio (SNR) and precision which, with decreasing chip dimensions, is beyond the capability of analog, and ultimately even digital, circuits. Perceptive signal processing, on the other hand, does not require the high precision and high SNR, which is generally required in traditional signal processing. Being biologically (e.g., “brain”) inspired, it is based on collective, massively parallel analog or digital processors and systems. Thus, perceptive signal processing relies not so much on high SNR and high precision but on general impression, statistical intuition, and estimation. This resembles the decision-making process and perception found in biological systems and humans (perceptive signal processing), rather than in machines, robots, and computers (traditional or machine signal processing).

Slide 1.17 This slide shows the tradeoff between dissipated (also qualitatively: chip area) power versus precision Pmin/f [J] per pole [J] and dynamic range in a typical digital and analog Etr [J] –13 device. As mentioned 10 –9 10 process Digital –14 above, it is clear that the 10 evolution –15 cost of achieving low 10 power, in terms of –12 10 decreased dynamic range, is much more severe for Analog: Pmin= 8fkT SNR Digital: Pmin= mf Etr analog than for digital cir10–15 cuits. The expressions f = signal bandwidth used to calculate Pmin, i.e., m = number of gate-trans./period Etr = energy per transition 10–18 the minimum power for the analog and digital cases in terms of signal 90 0 30 60 120 SNR [dB] bandwidth, number of gate (also qualitatively: precision, linearity) transitions per period, and energy per transition, are also given in this slide. On the face of it, low-power, analog devices with reasonable dynamic range seem almost impossible to produce. An

al

og

Minimum Power for Signal Processing

1  Mixed-Mode Signal Processing

14

Traditional Processing Requires SNR and precision beyond capability of analog All digital (except very simple, limited SNR) Etr reduced by: scaled down processes reduced voltage reduced and controlled VT parallelism

Pmin [J ] f 10

–9

Digital

10–12

10

–15

og

al

10

An

–18

SNR [dB] 0

30

60

90

120

m reduced by: improved algorithms improved architectures optimum coding clock management

Slide 1.18 The conclusion above is emphasized in this slide, where the darkened, conventional working field in a typical “Pmin vs. dynamic range” signal-processing plane is shown. Reducing the dissipated power means shifting the useful, darkened, area closer to the coordinate origin (as shown in the next slide). As a result the dynamic range of an analog device rapidly decreases below any acceptable value. This makes traditional signal processing (TSP) with low-power analog devices virtually impossible; TSP requires a SNR and precision well beyond the capability of a low-power analog device. This is not the case for digital signal-processing (DSP) devices, at least within a limited degree of simplicity with limited SNR.

Perceptive Processing

(Perceptive: impressionist, intuitive vs. Restitutive: precision, detail.)

Fundamentally different from traditional tasks Pmin /f [J]

Goal: Perceive environment

10

–9

l

Digita

restitution

Collective processing in massively parallel systems

–15

al o

g

–18

analog for perception

An

10

Target: decision, action, cognition. Massive flux of analog information

10–12

10

(vision, audition, olfaction, tactile)

0

SNR [dB] 30

60

90

120

No need for large SNR or precision Low power. Compact Continuous time and amplitude Enabler: biological inspiration

Slide 1.19 The concept of perceptive signal processing (PSP) implies doing precisely what was considered not possible with low-power analog devices in the previous slide. It means shifting the working range in the “Pmin vs. dynamic range” plane closer to the origin  – where the dynamic range for TSP is generally too low – at least with analog devices.

4  Examples of Human Perceptive Signal Processing

15

However, with the biologically inspired concept of PSP, this is no longer a barrier. The biologically inspired concept of PSP implies collective processing in massively parallel systems as, for example, in the human brain. The human brain has an average of 100 billion “analog” neurons collectively active “in parallel.” In PSP the goal of processing is no longer precision and detail but, as in the human brain, intuitive, impressionist, and, indeed, perceptive. Examples of this are given in the slides that follow.

4

Examples of Human Perceptive Signal Processing

In the following three slides, the reader is asked to carry out a visual processing task that demonstrates the power of so-called perceptive signal processing. Each slide shows a picture of a crowd of people, and the reader is asked to find a certain object in the image as quickly as possible. This task would take considerable computing power and time (depending on the computer) in any optical scanning system, and yet it can be carried out “perceptively” in a few seconds by a human being, from the age of approximately 6 years old upward. The pictures are shown again at the end of this chapter with the sought after objects encircled in color.

Slide 1.20 In this example of visual perceptive processing, find: (i) A man with a straw hat (ii) A white plastic bag

16

1  Mixed-Mode Signal Processing

Slide 1.21 In this example of visual perceptive processing, find:



(i)  A man with black close-cut hair and glasses, looking down to the ground (ii)  A lady with a pink hat

Slide 1.22 In this example of visual perceptive processing, find: (i) A blond man with a beard and mustache (ii)  A baby carriage

4  Examples of Human Perceptive Signal Processing

17

Slide 1.23 (i) A man with a straw hat (ii) A white plastic bag

Slide 1.24 (i) A man with black close-cut hair and glasses, looking down to the ground (ii) A lady with a pink hat

18

1  Mixed-Mode Signal Processing

Slide 1.25 (i) A blond man with a beard and mustache (ii)  A baby carriage

Chapter 2

Some Key Points from Network Theory

Slide 2.1

Chapter 2 12 Key Points from Network Theory

1

Introduction

In the “Digital World” of today, many new university courses, related to digital systems and design, have had to be included in the EE curriculum, while maintaining the 4-year time span of a first EE degree (BSc). Due to pragmatism and sheer lack of frontal lecture time, this has necessitated the elimination of many previously required courses of the old “Analog World.” However, as discussed in “Chap. 1,” analog circuits are still vital, e.g., for the analog front end, as well as for many other electronic circuits and systems. Thus, in order to enable the study, e.g., the analysis and design, of modern analog and mixed-mode circuits and systems, a minimum of basic analog circuit-and-system theory and analysis is first required. This chapter is intended to meet this demand by introducing those fundamental “Key Points” that summarize the main body of what used to be a first-year Circuit Theory and Analysis (CTA) course. Furthermore, it is material necessary to understand the remaining chapters of this book. Thus, this chapter contains a concise and stripped-down presentation of material that was previously taught in a © Springer Nature Switzerland AG 2019 G. S. Moschytz, Analog Circuit Theory and Filter Design in the Digital World, https://doi.org/10.1007/978-3-030-00096-7_2

19

2  Some Key Points from Network Theory

20

CTA course in most first-degree electrical engineering programs. The Key Points are stated briefly, without long derivations, theorems, and proofs, but with short explanations and physical insights using the mathematics and physics taught in today’s first and second year EE curriculum. Thus, with this chapter, the students are prepared for an advanced course on the material in the remainder of the book which, typically, will be given in the fourth, and possibly already in the third, year of a 4-year EE first-degree curriculum.

2

A Brief Summary of Two-Port Relationships

Before we go to our key points from network theory, it is useful to recall some basic concepts and relationships of two-port theory. This theory looks at a network as a two-port or “black box” with a pair of input and output terminals, which make up the input and output port, respectively. Applying a voltage or current source to the input, or “driving,” port, and analyzing the voltage and current response at the output port, with or without a load, permits deductions to be made about the behavior of the “black box.” It does this in terms of the input to output voltages and currents which are expressed in terms of two-port matrix relationships. These relationships are summarized in the following slides.

Two-Port Network Parameters For passive and reciprocal circuits Impedance matrix [zij]

∆z = z11z22 – z12z21

Admittance matrix [yij]

∆y = y11 y22 – y12 y21

Hybrid matrix [hij]

∆h = h11h22 – h12h21

Hybrid matrix [gij]

∆g = g11g22 – g12g21

z V1 = z11 V2 21 y11 I1 I2 = y21

z12 z22 y12 y22

I1 I2 V1 V2

h11 h12 I1 V1 I2 = h21 h22 V2 g g I1 V1 = g11 g12 V2 21 22 I2

z12 = z21 y12 = y21 h12 = – h21 g12 = – g21

Transmission matrix [ABCD]

V1 I1

= A C

B D

V2 – I2

∆A = 1

Reverse transmission matrix [ABCD] ∆ A = AD – BC

V2 I2

= A C

B D

V1 – I1

∆A = 1

∆A = AD – BC

In Slide 2.2 the six most common two-port matrices are defined. Emphasis is on the [zij], the [yij], and the [ABCD] matrices, which for the purposes of this book are the most important.

2  A Brief Summary of Two-Port Relationships

21

Common Two-Port Configurations Circuit type

[zij]

z1

¥

- 1 Z1

- 1 Z1

1 Z1

2Z2

z1 2Z2 2 Z3

Z2

1 Z1 0 1 1

¥

Z2 Z2

z1 2

[ABCD]

1 Z1

Z2 Z2

z2

Z1

[yij]

Z1 + 2Z2 2

2Z2

2 Z1

- 2 Z1

2Z2

2Z2

-2 Z1

2 + 1 Z1 2Z2

2Z2

2Z2

2Z2

Z1 + 2Z2 2

Z1 + Z2

Z2

Z2

Z2 + Z3

- 2 Z1

- 2 Z1

2 Z1

(

1 Z1 Z3

Z1 2

1 2Z2

1 Z1 2

1

) (

1 1 + 1 Z1 Z2 Z3 -

Z1 4Z2

1+

2 + 1 Z1 2Z2

1 1 + 1 + 1 Z1 Z2 Z3

0

1 1 Z2

1 2Z2 -

1 Z1 Z3

1 1+ 1 Z3 Z1 Z2

1+

Z1 +1 Z2

)

Z1 4Z2 Z1Z3 Z2

Z1 + Z3 +

1 Z2

Z3 Z2

1+

Continued

Common Two-Port Configurations Z1 2

Z1

Z1 2

Z3

Z1 2Z2

2Z2

Z1 Z1 R

Z2

Z2

Z1 + Z2 2

R Z2

Z1(Z2 + Z3)

1

Z 1 + Z2 + Z 3

Z1Z3

1 1 + 2Z2 Z1 2Z2 1 2 + 1 2Z2 Z1 Z1

Z1

Z2

Z1 + Z2 2

Z2 Z2

[yij]

Z2

(

)

1 1 + Z1 2Z2

-

1 1 2Z1 2Z2

(

)

Z12 + 2Z1R + 2R2 Z1R(Z1 + 2R) 2(Z1 + R) Z1(Z1 + 2R)

1

-Z 1

-

(2Z1

1

Z1

)

1 2Z2

1 1 + 2Z1 2Z2

-

Z1 Z Z + 2Z2 1 1 + 2Z2 2 2 2

(

)

Z1 + Z2 2

1

Z2 Z2 +1 Z3 Z2 1 + 1 + Z2 1+ Z 1 Z 3 Z 1Z 3 Z1

- 1 Z2

1 1 + Z1 2Z2

1 Z1

Z2 - Z1 Z1 + Z2 2 2

Z1(Z1 + 2R)

Z1 + Z2 2

1 1 + Z1 2Z2

1 Z1

RZ12 + 2R2Z1 + 2R3

- Z2

1 Z2

1 + 1 Z 2 Z3

1 1 + 2Z1 2Z2

2R (Z1 + R) Z1(Z1 + 2R)

- Z2

- 1 Z2

Z3(Z1 + Z2)

2R2(Z1 + R) Z1(Z1 + 2R)

Z1 + Z2 2

1 + 1 Z1 Z2

Z1Z3

Z1 + Z2 Z2 - Z1 2 2

RZ12 + 2R2Z1 + 2R3 Z1(Z1 + 2R) 2

2 Z Z1 1 + 2Z2 2

[ABCD]

2(Z1 + R) Z1(Z1 + 2R)

Z12 + 2Z1R + 2R2 Z1R(Z1 + 2R)

1 1 + Z1 2Z2

(

1

)

1 1 2 + 2Z2 2Z2 Z1

1 1 + Z1 2Z2

Z1 + Z2 1 Z2 - Z1 2

2Z1Z2

2

2

Z1 + 2Z1R + 2R 2R(Z1 + R) Z1(Z1 + 2R) 2R2(Z1 + R)

FUNDAMENTALS OF LINEAR ACTIVE NETWORKS

[zij]

Circuit type

(Cont’d)

Z1 + Z2

Z1(Z1 + 2R) 2(Z1 + R) Z12 + 2Z1R + 2R2 2R(Z1 + R)

If Z1Z2 = R2 then Zimage = R

In Slides 2.3 and 2.4, the [zij], the [yij], and the [ABCD] matrices are given for some typical passive two-port network configurations. By “passive” we mean here that the networks contain no active devices.

2  Some Key Points from Network Theory

22

In Slides 2.5 and 2.6 the interrelationships between the six basic two-port matrices are given. These tables serve here mainly as general and useful reference material.

Matrix Interrelationships For Two-Port Networks V1

I1 Linear I2 network

V2

Two-Port Linear Network I1 V1 = [zij] V2 I2 ∆z = z11z22 – z12z21 I1 V = [yij] 1 V2 I2 ∆y = y11y22 – y12y21

I1 V1 = [hij] V2 I2

V1 A B = C D I1

∆h = h11h22 – h12h21 V I1 = [gij] 1 V2 V2

V2 –I2

∆A = AD – BC V2 A B = C D I2

∆g = g11g22 – g12g21

V1 –I1

∆A = AD – BC

Parameters Matrix

[zij]

[zij]

z11 z12 z21 z22

[yij]

[hij]

y22 ∆y

– y12 ∆y

– y21 ∆y

y11 ∆y

∆h h22

h12 h22

– h21 1 h22 h22

[gij]

1 –g12 g11 g11 g21 g11

∆g g11

A B C D

A B C D

A ∆A C C

D 1 C C

1 D C C

∆A A C C

Continued

Matrix Interrelationships For Two-Port Networks (cont’d) Matrix

[yij]

[hij]

[gij]

A B C D

A B C D

[zij] z22 – z12 ∆z ∆z – z21 z11 ∆z ∆z z12 ∆z z22 z22 – z21 1 z22 z22 1 – z12 z11 z11 z21 ∆z z11 z11 z11 ∆z z21 z21 1 z22 z21 z21 z22 ∆z z12 z12 1 z11 z12 z12

[yij]

[hij]

y11 y12 y21 y22 1 y11 y21 y11 ∆y y22

y12 y11 ∆y y11 y12 y22

– y21 1 y22 y22 – y22 y21 – ∆y y21

–1 y21 – y11 y21

– y11 y12 – ∆y y12

–1 y12 – y22 y12

1 – h12 h11 h11 h21 ∆h h11 h11 h11 h12 h21 h22 h22 –h12 ∆h ∆h –h21 h11 ∆h ∆h – ∆h – h11 h21 h21 – h22 – 1 h21 h21 1 h11 h12 h12 h22 ∆h h12 h12

A B C D

[gij] ∆g g22 – g21 g22 g22 ∆g –g21 ∆g

g12 g22 1 g22 – g12 ∆g g11 ∆g

g11 g12 g21 g22 1 g22 g21 g21 g11 ∆g g21 g21 – ∆g – g22 g12 g12 – g11 – 1 g12 g12

D B –1 B B D –1 D C A 1 A

–∆A B A B ∆A D C D –∆A A B A

A B C D D B ∆A ∆A C A ∆A ∆A

A B C D A B –∆A B

–1 B D B

1 B A A –∆A C A A C D ∆A D D ∆A

–1 D B D B ∆A

C A ∆A ∆A A B C D

Slide 2.6

2  A Brief Summary of Two-Port Relationships

23

Impedance and Gain Relationships for a 2-Port Network 1 Zs = Y s Es

I1 V1

+ –

I2 V2

Linear network

ZL = 1 YL

Zout

Zin

Terminated Linear Two-Port Network Parameters A C

B D

A C

B D

[zij]

[yij]

[hij]

[gij]

Zin

∆z + z11 ZL z22 + ZL

y22 + YL ∆y + y11 YL

∆h + h11 YL h22 + YL

g22 + ZL ∆g + g11 ZL

AZL + B CZL + D

B + DZL A + C ZL

Zout

∆z + z22 Zs z11 + Zs

y11 + Ys ∆y + y22 Ys

h11 + ZS ∆h + h22 ZS

∆g + g22 YS g11 + YS

B + DZS A + CZS

A ZS + B C ZS + D

z21 z22 + ZL

– y21 YL ∆y + y11 YL

– h21 YL h22 +YL

g21 ∆g + g11 ZL

1 CZL + D

A + C ZL

z21 ZL ∆z + z11 ZL

– y21 y22 + YL

– h21 ∆h + h11 YL

g21ZL g22 + ZL

1 A + BYL

B YL + B

–I2 * Ai – I1 Av =

V2 V1

In Slide 2.7 the impedance and gain relationships for two-ports, terminated with a load ZL = 1/YL, are given. In what follows we give an example of the calculation of the transfer function of a simple RC network from its [z] parameters.

∆A ∆A

* The reason for the negative sign in the expression for Ai is that Ai is the ratio of the load current (–I2) to the input current (I1).

Transfer Function in Terms of 2-Port Parameters

Two-port parameters: z  I  V   z [ z ] :  1  =  11 12   1 ; V2   z 21 z 22   I 2 

I   y [ y ] :  1  =  11  I 2   y21

V   A B   V2  [ ABCD ] :  1  =     I1  C D  − I 2 

T(s):

[z] z 21Z L ∆z + z11Z L

[y] − y21 y22 + YL

y12  V1  ; y22  V2 

Z L = ∞, YL = 0 [ABCD] 1 A + BYL

[z] [y] [A] z 21 − y21 1 z11 y11 A

In Slide 2.8 we first have the transfer function of a two-port expressed in terms of the [z], [y], and [ABCD] parameters.

2  Some Key Points from Network Theory

24

Poles and Zeros of the Transfer Function T(s)

Poles and Zeros: m

m

N (s) = T (s) = D( s)

∑ bi s i i =0 n

∑a s j =0

j

j

=K

∏ (s − z ) i =1 n

i

∏ (s − p ) i =1

j

s-plane:

A Reminder: Calculating 2-port parameters Example: z-parameters of a simple 2-port:

V1 = I1Z1 + ( I1 + I 2 )Z 2 V2 = ( I1 + I 2 )Z 2

V1   Z1 + Z 2 V  =  Z  2  2

∴ V1 = I1 (Z1 + Z 2 ) + I 2 Z 2 V2 = I1Z 2 + I 2 Z 2

Z 2   I 1   Z1 + Z 2 ⇒ Z 2   I 2   Z 2

Z 2   z11 =ˆ Z 2   z 21

z12  z 22 

Note: Because network is passive, it is reciprocal, i.e. z12=z21

In Slide 2.9 the general form of the numerator and denominator polynomials of the transfer function T(s) of a network is given, and the general location of its poles and zeros in the s-plane is illustrated. More to this representation is given in the Key Points below.

In Slide 2.10, we demonstrate, as a “reminder,” how to obtain the [z] parameters of the transfer function T(s) of a simple network in terms of the network components.

2  A Brief Summary of Two-Port Relationships

25

The Poles and Zeros of Passive RLC Networks

Two-Port Parameters:

V1   z11 z12   I1    ; = V2   z 21 z 22   I 2 

[z ]=  [A

V   A B  V2  B C D] =  1  =     I1  C D   I 2 

param:

[z ]

[ y]

T (s ) :

z21 Z L ∆ z + z11Z L

− y21 y22 + YL

I1

I1   y11 y12  V1  =    I 2   y21 y22  V2 

[ y ] = 

Z1

V1

V1   Z1 + Z 2 V  =  Z  2  2 T ( s) =

[ A B C D] [z ] [ y ] [ A] 1 A + BYL

I2 Z2

V2

ZL = ∞ YL = 0

In Slides 2.11 and 2.12, returning to the two-port parameters of Slide 2.8, we use the expression for T(s), in terms of the [z] parameters, to obtain the transfer function T(s). Note that the parallel connection of the impedance Z2 and the load ZL is designated Zp2L.

ZL

z21 y − 21 1 z11 y22 A

Slide 2.12

z21 Z L ∆ z + z11Z L

Z 2   I 1   Z1 + Z 2 ⇒ Z 2   I 2   Z 2

Z 2   z11 =ˆ Z 2   z 21

z12  z 22 

z 21Z L Z2ZL ⇒ ∆z + z11Z L Z 2 ( Z1 + Z 2 ) − Z 22 + ( Z1 + Z 2 ) Z L

Z2ZL Z p2L Z2ZL Z2 + ZL = = = Z1 + Z p 2 L Z1 ( Z 2 + Z L ) + Z 2 Z L Z + Z 2 Z L 1 Z2 + ZL

2  Some Key Points from Network Theory

26

3

Key Points from Network Theory

Slide 2.13 Key Point 1 states that the transfer function T(s) of a I1 I2 jω linear system or network, V1 V2 ZL Two-Port N jωz LHP 1 when seen as a two-port Az 1 with input voltage V1 and jω0 Ap output voltage V2, or input The transfer function T(s) of a stable i σ current I1 and output curnetwork (two-port) : Az 2 rent I2, must be a rational Θp j function with numerator Θ zi V2 / V1 = T(s)= N(s)/D(s) and denominator polynomi-jωz 1 als N(s) and D(s) with real must have all poles ( real and complex-conjugate) in the coefficients, thus T(s) = V2/ ω left-half s-plane (LHP), excluding the imaginary (j )-axis; V1  =  N(s)/D(s). Note that all complex poles and zeros must be complex conjugate. solving for N(s)  =  0 pro[Note: N(s) = 0 provides the zeros; D(s) = 0 provides the poles] vides the zeros, and D(s) = 0 the poles of the network. For stability, T(s) must have all poles (real and complex conjugate) in the left-half s-plane (LHP), excluding the imaginary jω-axis. This is fundamental for stable networks and systems alike. Furthermore, all complex poles and zeros must be complex conjugate. Theoretically, the zeros can be in the LHP and right-half s-plane (RHP). However, as we shall see later, there are some important constraints on the location of the zeros in the RHP. Furthermore, reference and discussion of the pole-­ zero display on the right of the slide is discussed later in this chapter (e.g., see Slide 2.27).

Key Point 1: Passive and Stable Network

Slide 2.14 Key Point 2 is a follow-up The transfer function of typical frequency- shaping and frequency to the previous Key Point selective LCR filters T(s)= N(s)/D(s) will have complexand describes the location conjugate poles in the left-half s-plane, excluding the imaginary of the poles and zeros of a (jω) –axis. Thus LCR filters are absolutely stable! passive RLC network consisting only of resistors The zeros may be real, complex conjugate, and located at the (R’s), inductors (L’s), and origin, or at infinity. No poles are permitted at the origin or at capacitors (C’s). infinity: (‘Hurwitz conditions’) s 3 + b2 s 2 + b1s + b0 Because such a network Example: General 3rd Order Transfer Function: T (s ) = K 3 s + a2 s 2 + a1s + a0 is passive, i.e., contains no jω (Imag) s – plane: s gain, e.g., active devices pole jω Where is the location of ω such as amplifiers (“active” the poles and zeros of ω=0 and “passive” from a netpassive RC (Real) σ ( i.e., inductorless) work theoretical point of networks in the zero view will be defined later), s-plane? s all poles, real and complex conjugate, must lie in the LHP, excluding the jω-axis. Furthermore, no poles are permitted at the origin or at infinity. These constraints are a result of the so-called Hurwitz conditions for D(s) [named after Adolf Hurwitz (1859–1919), a German mathematician who worked on algebra, analysis, geometry, and number theory at the ETH in Zurich].

Key Point 2: Poles and Zeros of an RLC Network

3  Key Points from Network Theory

27

It follows that because D(s) is “Hurwitz” for all passive RLC networks, such networks are absolutely stable. The zeros may be real, complex conjugate, and located at the origin, or at infinity. To summarize the contents of the two preceding slides, the main points are: (i) T(s) = N(s)/D(s) is a rational function in which both the numerator N(s) and the denominator D(s) are polynomials. The coefficients of the polynomials are rational numbers. (ii) Because an RLC network is passive, D(s) is a Hurwitz polynomial, which means that its roots, the poles of T(s), must lie in the LHP of the s = σ + jω complex-frequency coordinate system. (iii) Because the poles of T(s) lie in the left-half s-plane, RLC networks are always unconditionally stable. We now turn to the question of where the poles and zeros of passive RC, i.e., inductorless RC networks, lie in the s-plane? First, however, it is worth noting why, in fact, networks should be without inductors, i.e., “inductorless,” in the first place? Briefly, the reason is that inductors are three-dimensional components, whose Q (quality factor) rapidly decreases with miniaturization, i.e., with a reduction in size. This is because the Q factor of an inductor depends on the impedance of an inductance coil, relative to its resistive loss. With chip integration, this resistive loss increases, and therefore the Q of the inductance decreases accordingly. Passive RLC filters depend, for their frequency selectivity, on the resonance effects between the filter inductors and capacitors. These, in turn, depend on a sufficiently high-Q (sufficiently low loss) of the inductors and capacitors. Because of their high loss, miniaturized, i.e., low-Q, inductors are practically useless for high-selectivity LCR filters in miniaturized or integrated-circuit form. The losses in miniaturized capacitors are not effected in this way; their Q factor does not change with size reduction.

Key Point 3: Poles and Zeros of Passive RC Networks The poles of the transfer function of a passive RC network are single and negative real, excluding the origin and infinity.

T (s ) = K

s 3 + b2 s 2 + b1s + b0 s 3 + a2 s 2 + a1s + a0

s 3 + b2 s 2 + b1s + b0 Tˆ (s ) = Kˆ 3 s + aˆ2 s 2 + aˆ1s + aˆ0 What about the zeros ?

Poles and Zeros of 3rd-order RLC Network P1

Z1

jω

Z3 P3 Z2=Z1* P2=P1*

σ

P2

Poles and Zeros of 3rd-order RC Network

Z2

Z1

Z3 P3 Z2=Z1*

P2

jω

σ

P1 Z2

The zeros may be anywhere in the s-plane, if the topology allows it!

Slide 2.15 We have discussed the location of the poles and zeros of the transfer function T(s) of RLC filter networks and have further briefly discussed why inductors are difficult, if not impossible, to integrate into micro-miniaturized integrated circuits. The conclusion to these considerations is that, when limited to passive filter components, we are left with only passive RC filter networks to build our filters.

The following obvious question then follows: Where may the poles and zeros of a passive RC filter lie in the s = σ + jω complex-frequency plane, or similarly, what penalty, in terms of pole-zero location, are we subjected to when disposing only of passive RC components to build a frequency-selective filter? This brings us to Key Point 3, which states that the poles of the transfer function of a passive RC network are single, and negative real, excluding the origin and infinity. The exclusion of the origin and infinity is a result of D(s) being a Hurwitz polynomial since, as such, the jω axis is thereby excluded.

2  Some Key Points from Network Theory

28

Key Point 3 is demonstrated in the slide by showing two third-order transfer functions T(s), one of which has a tilde (^) over T(s) and over the coefficients of D(s), in order to indicate that it is the transfer function of a passive RC network. Without the ^ it is not possible to tell, other than by solving for the roots of D(s), whether T(s) refers to an RC or an RLC network. The above statement refers only to the poles of the transfer function T(s); it says nothing about the location of the zeros. Theoretically the zeros can lie anywhere within the s-plane. [Remember, the location of the zeros does not affect the stability of the network.] However, there are limits on the location of the zeros of an RLC and an RC network. They depend on the topology of the network and whether it is grounded or balanced with respect to ground. This will be discussed further on.

Poles and Zeros of RC and RLC Networks RC: s 3 + b2 s 2 + b1s + b0 ˆ ˆ T ( s) = K 3 s + aˆ 2 s 2 + aˆ1s + aˆ0 RLC: s 3 + b2 s 2 + b1s + b0 T ( s) = K 3 s + a2 s 2 + a1s + a0 RC:

Tˆ ( s ) = Kˆ

s 2 + ω z2 s2 +

RLC: T (s) = K

ωp qˆ p

s + ω p2

s + ω z2 2

s2 +

ωp qp

s + ω p2

1: Topology Dependence of Zero Location of Grounded RC Networks RC Ladder Network: I1

I2

V1

V2

jω Poles & Zeros multiple

single, not at origin not at

Slide 2.16 This slide provides more examples of the pole location of RLC and RC networks. Notice that the zeros are the same in each of the example pairs.

Slide 2.17 introduces the topology dependence on the location of the zeros of grounded RC networks. [Note that the pole location of RC networks – see Key Point 3  – is “iron clad.” There are no exceptions.] This slide shows that the zeros of grounded RC ladder networks are limited to the negative-­real axis but may occur in multiples.

3  Key Points from Network Theory

29

2: Topology Dependence of Zero Location of Grounded RC Networks Bridged-T Network: I

I V

V

Poles & Zeros

complex conjugate

Slide 2.18 shows that the zeros of so-called bridgedT networks may be complex conjugate, rather than only negative-real. However, since the bridged-T is an RC network, the poles must be single and on the negative real axis.

jω

σ single

Finally, Slide 2.19 shows that also two ladder networks, connected in parallel, may be designed to Parallel Ladder: produce complex-conjugate zero pairs. The socalled twin-T network, I I which we shall encounter V V later, is an example of two third-order ladder networks in parallel that produce complex-conjugate complex conjugate jω zero pairs. As always, Poles & Zeros LHP being an RC network, the RHP three poles of the twin-T are single and on the negative real axis, excluding σ the origin and infinity. Note that an intuitive reamultiple son for these RC networks producing complex zeros is that each network has two paths leading directly from the input to the output, whose phase at a certain predetermined frequency is 180° apart, thereby cancelling out the signal at that particular frequency.

3: Topology Dependence of Zero Location of Grounded RC Networks

2  Some Key Points from Network Theory

30

Slide 2.20 Key Point 4 summarizes Passive RC Networks and formulates the zero The zeros of passive RC ladder networks are negative real but location of grounded pasmay occur in multiples, including at the origin and at infinity. sive RC networks. I I It states, for grounded ladder V passive RC networks, (i) V that the zeros of passive RC ladder networks are negaThe zeros of bridged-T, and parallel-ladder (e.g. twin-T) tive real, but may occur in networks may be complex conjugate. They may even lie in the multiples, including at the right-half plane origin and at infinity, (ii) I I that the zeros of bridged-T and parallel-ladder (e.g., V V twin-T) networks may be complex conjugate, and bridged-T parallel-ladder (iii) that they may even lie in the right-half plane. Point (iii) is conditional and still quite vague. It leads to an important addition to Key Point 4 which is formulated in the next slide.

Key Point 4: Zero Location and Topology of Grounded

Limits on the Zero Location and Topology of Grounded Passive RC Networks

‘Fialkow-Gerst Conditions’*. The zeros of bridged-T, and parallel-ladder (e.g. twin-T) networks may be complex conjugate, however, only up to, but not including, a sector of π/m. m is the degree of the numerator of T(s) Corollary: A Linear Lumped-Parameter Finite (LLF) grounded RC network cannot have positive real zeros. Why not? ∗Α.D. Fialkow and I. Gerst, The transfer function of networks without mutual reactance, Quart. Appl. Math.,Vol. 12, pp. 117-131, 1954

Slide 2.21 This slide formulates the so-called Fialkow-Gerst conditions, which specify the boundary in the RHP on the zero location of grounded passive RC networks possessing two direct paths from input to output. [These conditions were published by A.D. Fialkow and I. Gerst: ‘The Transfer Functions Of General Twin TerminalPair RC Networks’ in the Quarterly Applied Math,

vol. 10. July 1952.] The Fialkow-Gerst conditions state the following: (i) The zeros of bridged-T and parallel-ladder (e.g., twin-T) networks may be complex conjugate and in the right-half plane, but only up to, and not including, a sector of π/m where m is the degree of the numerator of T(s). (ii) Corollary: A linear lumped-parameter finite (LLF) grounded RC network cannot have positive real zeros. The reason for this is that from (i) for zeros to be on the positive real axis, the numerator degree m would have to be infinite, which contradicts the definition of an LLF network. It also implies the fact that in so-called distributed RC networks, which do not consist of individual “lumped” RC components, but rather of distributed RC layers, real positive zeros are permitted.

3  Key Points from Network Theory

31

Example for Fialkow-Gerst Conditions Unbalanced (grounded) RC Network N.

1)

„degree of N(s)”

T (s) = K

s 3 + b2 s 2 + b1s + b0 N ( s ) = s 3 + a2 s 2 + a1s + a0 D ( s )

0

N =m=3 D0 = n = 3

„degree of D(s)”

Possible Pole/Zero Locations (provided the topology permits it): no zeros in this sector, including boundary!

Slides 2.22 and 2.23 present two illustrative examples of second- and third-order transfer functions of RC networks and their potential zero locations. Slide 2.23, in particular, is an illustrative example of the Fialkow-Gerst condi2) Passive RC Notch Network: tions. The example on the left is the pole-zero diagram of a second-order RC network; the denominator has two (negative-­ real) poles. The numerator has two zeros, it is second order, i.e., m = 2. The zeros Possible! Not possible! lie on the jω-axis. But the 2 2 2 2 Fialkow-Gerst conditions ( s + ω z )( s + Z 3 ) s + ωz T (s) = K T (s) = K state that the zeros may be ( s + P1 )( s + P2 )( s + P3 ) ( s + P1 )( s + P2 ) “complex conjugate and in the RHP, but in the RHP only up to, and not including, a sector of π/m, where m is the degree of the numerator of T(s).” Thus, for this second-order network, the zeros may lie close to, but not on, the boundary, which here is the jω-axis. On the right side of the slide, the pole–zero diagram of a third-order RC network has three (negative real) poles and three zeros. Thus, m = 3 and the zeros may lie in the RHP up to a boundary of π/3 or 60°; this, of course, includes the jω-axis.

Example for Fialkow-Gerst Conditions

(cont’d)

2  Some Key Points from Network Theory

32

More on Two-Port Parameters: [y] matrix (Admittance Matrix)

 I1   y '11 + y ' '11  I  =  y' + y' '  2   21 21

y '12 + y ' '12  V1  ; y '22 + y ' '22  V2 

[ y ] = [ y ]'+[ y ]' ' [ABCD] matrix (Chain, Transmission Matrix) V1   A B   V2   I  = C D  − I   2   1  A' B '   A' ' B ' '   A' A' '+ B ' C ' ' A' B ' '+ B ' D' '   =  C ' D ' C ' ' D ' ' C ' A' '+ D' C ' ' C ' B ' '+ D' D' '

[ ABCD ] = 

[ ABCD] = [ ABCD]'⋅[ ABCD]' ' Slide 2.24 adds some matrix relations of two-ports connected in parallel and cascade. These relations are useful in calculating parallel-connected RC networks (e.g., in order to provide ­complex-­conjugate zeros) in which case the [y] parameters are added. For cascaded RC two-ports, e.g., biquads, the [ABCD] parameters are multiplied.

More on Two-Port Parameters Loaded Two-port

∆y = y11 y22 − y12 y21 ∆[ A] = AD − BC

Z in =

AZ L + B V2 1 ; ; = CZ L + D V1 A + BYL

Z in =

y22 + YL V2 − y21 ; = ∆y + y ' ' YL V1 y22 + YL

For passive and reciprocal networks: y12 = y21

∆[ A] = 1

Slide 2.25 continues with useful two-port expressions, namely the input impedance of a loaded twoport, in terms of the [y] and the [ABCD] matrix.

3  Key Points from Network Theory

33

Slide 2.26 Key Point 5 is a consequence of the previous ones. It states that a highselectivity, passive, inducTypical filter characteristics ( e.g. Butterworth, Chebyshev, torless, i.e., RC, filter, in Elliptic, etc) cannot be realized with passive RC ( e.g. spite of all the benefits that inductorless) networks. it would provide, such as IC reliability, passivity, Why not? and unconditional stability, is not realizable. The Because they require complex-conjugate poles, reason for this is that to obtain any kind of filter selectivity, we require complex-conjugate poles. Passive RC networks, however, are restricted to poles that are single and on the negative real axis, excluding at the origin and at infinity. The selectivity limitation implied by this restriction on the pole location of RC networks can be further understood by returning to Slide 2.13. The graphical relationship between the poles and zeros of a filter transfer function T(s = jω) – when evaluated along the jω axis – and the amplitude response is shown there. The amplitude response is obtained at any frequency ω0 by dividing the product of the distances Azi (with i = 1…m) from each zero to ω0, by the product of the distances Apj (with j = 1…n) from each pole to ω0, and so on, as ω0 is moved along the frequency axis. Thus, the amplitude at the frequency ω0 is given by: Key Point 5: Practical Limits of Passive RC Networks

T ( ω0 ) =

Az1 ⋅ Az2 ⋅…⋅ Azm

(2.1) This expression helps to understand the fact that, traveling along the jω axis from the origin in the s-plane upwards, past the dominant (closest to the jω axis) pole, the closer to the jω-axis the poles are, the larger the ripple, and the steeper the amplitude slope will be. In the limit, if a pole were actually on the jω axis, the amplitude would have an extreme “infinite spike,” meaning that the system would oscillate at that frequency. This, of course, is the reason why poles must remain to the left of the jω axis, i.e., in the LHP. A similar graphical relationship exists for the phase angle at any frequency ω0. This is obtained from the sum of the angles Θzi from each zero to ω0 on the jω-axis, minus the sum of the angles Θpj from each pole to ω0 on the jω-axis. Thus, the phase at the frequency ω0 is given by:



Ap1 × Ap2 ⋅…⋅ Apn

m

n

i =1

j =1

φ (ω0 ) = ∑Θ zi − ∑Θ p j

(2.2)

These interrelationships between the pole-zero plot and the amplitude and phase response will be discussed in more detail in Chap. 3 (e.g., see Slide 3.5).

2  Some Key Points from Network Theory

34

Slide 2.27 This slide elaborates further on the relationship jω Elliptic ω between the pole-zero (Chebyshev-Cauer) Butterworth ω location of T(s) relative to (maximally flat) Chebyshev the jω-axis, and the ampli(equi-ripple) tude response. Using a fifth-order low-pass filter Most practical, frequencyas a practical example, it is selective, filters require ω ω shown that the “maximally complex-conjugate poles! flat,” i.e., ripple-free, Log. Amplitude Response: Butterworth filter has its poles on a semi-circle centered at the coordinate origin. By contrast, the poles of the equi-ripple Chebyshev and -30dB/oct (-100dB/dec) Chebyshev-­Cauer low-6dB/oct (-20dB/dec) pass filters lie on an ellipse, and in the ChebyshevCauer case, the zeros lie on the jω axis. With the order of the numerator polynomial of T(s) equal to m, and that of the denominator equal to n, the roll-off of the filter is −(n−m) times 6 dB/octave, or −(n−m) 20 dB/decade. Example of Typical Filter Characteristics: 5th-order Low-pass Filter

Some Key Points from Network Theory

Minimum-Phase and Non-Minimum-Phase Networks Key Point 6: A network with zeros in the left-half plane is called a minimum-phase network, with zeros in the right half plane it is called a non-minimum-phase network. If the poles are the same for both networks, but the zeros are symmetrical with respect to the imaginary axis, the difference between the two networks will be in the phase. The amplitude for the two networks will be identical.

Slide 2.28 Key Point 6 defines the concept of minimumphase and non-minimumphase networks. It follows that:

• A network with zeros in the left-half plane is called a minimum-phase network; with zeros in the right-half plane, it is called a non-minimumphase network. • If the poles are the same for both networks, but the zeros are symmetrical with respect to the imaginary axis, the difference between the two networks will be in the phase only, i.e., the amplitude for the two networks will be identical. • It follows that an all-pass filter, i.e., a delay network, is a two-port that has a flat amplitude response and affects only the phase of an incoming signal. It is made up of non-minimum phase networks whose poles and zeros are mirror images with respect to the jω-axis.

3  Key Points from Network Theory

35

Slide 2.29 Key Point 7 defines the The Degree/ Order of a Passive Network degree, or order, of a passive RLC or RC network. Key Point 7 : The order or degree of The order, degree, or number of poles, of a passive the network refers to the network is, in general, and for most practical order, or degree, of the purposes, equal to the number of reactive denominator polynomial elements minus the number of independent of the network transfer inductor or capacitor cut-sets and loops in the function, which is also the network.* number of its poles. In general, and for most *A cut set is a set of the smallest number of branches of a network which, when practical purposes, the cut, will divide the network into two parts order, degree, or number A Loop constitutes one or more branches forming a closed path of poles of a passive network is equal to the number of reactive elements, minus the number of independent inductor or capacitor cut sets and loops, in the network. Note that a cut set is a set of the smallest number of branches of a network which, when cut, will divide the network into two parts; a loop constitutes one or more branches forming a closed path. Examples are given in the next slide. Some Key Points from Network Theory

L’2

L’4

(1.106)

(0.679) C’4

C’2 1.0

C’1 (1.275)

C’5 (0.9788)

C’3 (0.786) (1.752)

(0.228)

K2 = 1

(a) L’1

L’3

L’5

(1.275)

(1.752)

(0.9788)

1.0

L’2 (0.228) C’2 (1.106)

L’4 (0.786) C’4 (0.679)

1 = 1

K

2

(b)

A Loop constitutes one or more branches forming a closed path A cut set is a set of the smallest number of branches of a network which, when cut, will divide the network into two parts L2 L4 R1

(58.7 mH)

(5 kΩ)

C2

V0

(41.6 mH) C4

C1 (0.48 nF) (2.7 nF)

C3 (1.44 nF) (3.7 nF)

C5 (2.07 nF)

R2 (5 kΩ)

(c) CC-05-25-49 low -pass filter: a, normalized minimum L version; b, normalized minimum C version; c, denormalized minimum L version.

T(s) = K

(s2 + 3.9525) (s2 + 1.8750) 2

(s + 0.5401) (s + 0.1612s + 1.0627) (s2 + 0.6482s + 0.68523)

Slide 2.30 This slide gives an example for Key Point 7. It shows an RLC low-pass filter network at the top, and its dual filter below. The upper filter is a so-called minimum-L network, referring to the fact that it has

2  Some Key Points from Network Theory

36

the minimum possible number of inductors; the filter in the middle of the slide is its dual filter, a minimum-C filter. Both are normalized (see Key Points 10 and 11). The filter at the bottom of the slide is the de-normalized minimum-L version of the filter. Each filter has seven reactive components, i.e., L’s and C’s. However, the denominator of the filter transfer function shown at the bottom of the slide is fifth order, meaning the low-pass filter has five rather than seven poles. This is because the minimum-­L filter has two capacitive loops; the minimum-C filter has two inductive cut sets. Being duals of each other, both filters have the same transfer function. Incidentally, because inductors are far more demanding in terms of cost, size, and lossiness, minimum-­L filters are generally preferred over their equivalent minimum-C dual filters. [Note that “duality” here is a network-theoretical concept that, to do it justice, would take up more space than the objectives of this book allow. It is amply covered in classical network analysis books, including those given in the bibliography listed at the end of this book.] Referring to the two dual RLC filters in this slide, note that LC series resonance is replaced by LC parallel resonance, that voltage drive is replaced by current drive, and so on. As we shall see in the next Key Point, by making these replacements in a “topologically correct” way, parallel resonance in a series branch generates transfer-function zeros in a voltage-driven filter; series resonance in a parallel branch does the same in the dual current-driven filter.

Slide 2.31 Key Point 8 states an Imaginary Zeros of RLC Ladder Networks important feature of LC resonance in RLC filters which is necessary for the Key Point 8 : network-theoretical overThe imaginary zeros of an RLC ladder filter are obtained by view of this chapter (withLC resonators in the branches of the ladder filter. out mentioning duality as The zeros are obtained by parallel resonators in the series was done in the previous branches, and series resonators in the parallel branches. slide). Thus: The imaginary zeros in the s-plane of an RLC ladder filter are obtained by LC resonators in the branches of a ladder RLC filter. They are obtained by parallel resonators in the series branches and series resonators in the parallel branches. Some Key Points from Network Theory

3  Key Points from Network Theory

1.0

C’1 (1.275)

37

L’2

L’4

(1.106)

(0.679)

C’2

C’4 C’5 (0.9788)

C’3 (0.786) (1.752)

(0.228)

K2 = 1

(a) L’1

L’3

L’5

(1.275)

(1.752)

(0.9788)

1.0

L’2 (0.228) C’2 (1.106)

L’4 (0.786) C’4 (0.679)

1 = 1

K

2

(b) L2

L4

R1

(58.7 mH)

(41.6 mH)

(5 kΩ)

C2

C4

V0

C1 (0.48 nF) (2.7 nF)

C3 (1.44 nF) (3.7 nF)

C5 (2.07 nF)

R2 (5 kΩ)

(c) CC-05-25-49 low-pass filter: a, normalized minimum L version; b, normalized minimum C version; c, denormalized minimum L version.

T(s) = K

(s2 + 3.9525) (s2 + 1.8750) 2

(s + 0.5401) (s + 0.1612s + 1.0627) (s2 + 0.6482s + 0.68523)

Slide 2.32 demonstrates Key Point 8 using the same RLC filters as was shown in Slide 2.31.

2  Some Key Points from Network Theory

38

Negative-Real Zeros of RC Ladder Networks C loop → 1st order:

1)

Rp =

R1 R2 ; R1 + R2

ωz = C p = C1 + C2

1 1 ; ωp = R pC p R1C1

T ( s → 0) = K

s + ωz T ( s) = K s +ωp

ωz R2 = ω p R1 + R2

T ( s → ∞) = K = R p < R1 C p > C1

C1 C1 + C2

>
>1

Slides 6.43, 6.44, 6.45, 6.46, 6.47, and 6.48 deal with the so-called current conveyor (CC). A current conveyor [introduced by the Canadian professors Adel S. Sedra and Kenneth C. Smith in 1968] is a three-terminal form of current amplifier with unity gain. There are three generations of the idealized device, CCI, CCII, and CCIII. The CCIII is the same as CCI but with reversed output current, the CCI has zero, and the CCII has infinite, input impedance. When combined with other circuit elements, current conveyors can perform some analog signal processing functions in a manner similar to that of opamps but with a better gain-bandwidth product.

ACTIVE GAIN DEVICES, cont’d 4. Current Conveyors (CCs) The General Current Conveyor Vy Vx

Iy Ix

y CC

z

Iz

x

Defining Equations: I y =α I x Vx = V y Iz = Ix

 I y  0 α 0 V y       Vx  = 1 0 0   I x   I z  0 1 0  Vz 

Vz

In Slide 6.43 the general current conveyor and its defining equations are introduced.

6  Controlled Sources, Nullors, Active Gain Devices, and Impedance Converters and Inverters…

120

In Slide 6.44, the CCI and, in Slide 6.45, the CCII are defined.

ACTIVE GAIN DEVICES, cont’d 1st Generation Current Conveyor (CCI) α Iy

Vy

y

Vx

z

CCI

Ix

Iz

x

Iy = Ix

 I y  0 1    Vx  = 1 0  I z  0 1

Vx = V y Iz = Ix

=1

Vz

0 V y    0  I x   0 Vz  Slide 6.45

ACTIVE GAIN DEVICES, cont’d 2nd Generation Current Conveyor (CCII) y

Vy Vx

Ix

Iy = 0 Vx = V y Iz = ±Ix

CCII+

z

CCII-

z

y

Vy Vx

x

Ix

x

Iz

Iz

 I y  0 0 0  V y       Vx  = 1 0 0   I x   I z  0 ± 1 0 Vz 

Vz

Vz

α =0

5  Active Gain Devices

121

ACTIVE GAIN DEVICES, cont’d

Nullor Realization of CCII-Type Current Conveyors Equivalent Circuit

CCII+

Realization with Nullors

Iy=0

Vy

I

z

Iy=0

y

I

x

Iy=0

Vy

I

x

Vx=Vy

CCII-

y

I

z

y

R

R

I

I

B I

y

x I

x

I

E

C z

I

“Diamond Transistor”

Vx=Vy

Slide 6.47

ACTIVE GAIN DEVICES, cont’d CCII Realization of Nullors y

CCII-

Vz=Vy I I

y’

I

x=x’

I

z z’

y’

I

z I y’ CCII-

I

CCII-

I

x’ z’

y

z

y’

z’

≡

CCII-

y

In Slides 6.46, 6.47, and 6.48, equivalent nullor circuits for the CCII+ and CCII− are given. The two differ in the direction of the output current. It is also interesting to recognize that the CCII- is nothing but a “perfect” transistor, sometimes referred to as a “diamond transistor.”

≡

x

x’

y’

z’

6  Controlled Sources, Nullors, Active Gain Devices, Impedance Converters and Inverters…

122

Slide 6.48

ACTIVE GAIN DEVICES, cont’d CCII Realization with Differential OpAmp and Current Sources Equivalent Circuit

OpAmp Circuit I

y

z

I=0

x y x

6

I

I=0

I

I I

Nullor-Based Active Devices with Finite Frequency Response

As has been pointed out above, idealized device concepts are represented by the nullor and are very useful for first-order feasibility studies in circuit design. However, often the idealization goes too far, in that it masks an important feature of a physical device, such as its finite bandwidth. Fortunately, the nullor concept can readily be made to include a finite frequency response by adding simple RC ­circuitry in order to introduce at least a single-pole roll-off to its frequency response. This is shown in the following slides.

Nullor-Based Active Devices with Finite Frequency Response One-Pole Frequency Response

Ex. 1: OpAmp

Slide 6.49 shows the onepole frequency response of an opamp.

6  Nullor-Based Active Devices with Finite Frequency Response

123

Nullor-Based Active Devices with Finite Frequency Response, cont’d

Slide 6.50 shows the corresponding mathematical model.

One-Pole Frequency Response

Ex. 1: OpAmp

A( s ) = A0

ωc ωt = s + ωc s + ωc

where ωt = A0 ⋅ ωc : Gain Bandwidth Product e.g. A( ω) s = jω = t A( ω) s = jωc =

A0 ≈ − j → ∠A( ω) = 270° jA0 + 1 A0 A A = 0 (1 − j ) = 0 e − j 45° 1+ j 2 2

In Slide 6.51 the frequency-dependent nullor equivalent of the nullor opamp (see Slide 6.36) is modified by replacing R2 by Z2, i.e., adding a capacitor in parallel with R2. This produces a one-pole rolloff for the nullor version of the opamp.

Nullor-Based Active Devices with Finite Frequency Response, cont’d Nullor Realization V1

Z1

Z2

Let: Z1 = 1[Ω] A0 Z 2 (s ) = A(s ) = 1 + s ωc ∴Y2 ( s ) =

1 1 s = + Z 2 (s ) A0 ωt

V2

Z1:

V2 Z =− 2 V1 Z1 Z2:

1[Ω]

A0[Ω]

C=

1 [F] ωt

6  Controlled Sources, Nullors, Active Gain Devices, Impedance Converters and Inverters…

124

Nullor-Based Active Devices with Finite Frequency Response, cont’d

Slide 6.52 demonstrates a similar procedure applied to the CVS of Slide 6.27.

One-Pole Frequency Response

Ex. 2: Current-controlled Voltage Source (CVS) I1

Ideal Nullor CVS

Ideal CVS

I2

R

~

V1= 0

V1

V2

V2

V1 = 0

V1 = 0

V2 = r0 I1

V2 = − RI1

∴ r0 = R

Nullor-Based Active Devices with Finite Frequency Response, cont’d CVS with One-Pole Frequency Response r ( s ) = r0 r(jω)

ωc 1 1 = + s s + ωc r0 r0ωc

G (s ) =

1 1 s = + r (s ) r0 r0ωc

= G0 + sC0

r0

where

G0 =

1 r0

C0 =

1 r0ωc

-6dB/oct

ωc

ωt

ω

Slides 6.53 and 6.54 demonstrate that this creates a one-pole roll-off to the frequency response of the CVS.

7  Stabilizing the Closed-Loop Gain of an Opamp (Basics from Control Theory)

125

Slide 6.54

Nullor-Based Active Devices with Finite Frequency Response, cont’d CVS with One-Pole Frequency Response Nullor Circuit R=r 0

V1

7

1 C = 0 r ω 0 c

V2

 tabilizing the Closed-Loop Gain of an Opamp S (Basics from Control Theory)

One of the most aggravating characteristics of active, as opposed to passive, networks is that the former can slip into a state of instability, if not actual oscillation. This is because active networks invariably contain gain elements within them and the resulting circuit functions can have poles that drift into the right half plane, thereby becoming unstable. This, by nature, passive networks cannot do (see Chap. 2). Thus, one of the most important non-ideal features that must be taken care of with active networks is to make sure that they are unconditionally stable. We emphasize “unconditionally,” because some active circuits get “hung up” in an unstable state, even if only while the power is turning on. Thus, stability must be guaranteed not only in the powered-on state but during the “turn-on period” as well. This important issue of stability, which is often the result of improper frequency compensation, is dealt with in this section. In what follows, we first summarize the basic concepts of control theory, as they apply to stability in feedback systems and networks. These concepts were formulated by H. Nyquist [Harry Theodor Nyquist (1889–1976), Swedish-born American electronic engineer, who made important contributions to communication and control theory] and H. Bode [Hendrik Wade Bode (1905–1982), American engineer, researcher, inventor, author, and scientist, of Dutch ancestry]. Among many other accomplishments, these two outstanding engineers (both of whom worked at the renowned Bell Telephone Labs, NJ) left their mark in control and circuit theory in important concepts bearing their name. Among them are the Nyquist diagram and the Bode plot, both of which are prominent in the following two slides. They are included here as an introductory example to demonstrate the stabilization of opamp closed-loop gain.

6  Controlled Sources, Nullors, Active Gain Devices, Impedance Converters and Inverters…

126

Stabilizing the Closed-Loop Gain of an Opamp Basics from Control Theory

T (s ) =

µ Out = In 1 + µβ

In Slide 6.55 a basic feedback circuit is shown as a block diagram and its equivalent signal-flow graph (SFG). This slide introduces the concepts of open-loop gain μ and loop gain μβ.

T(s): Closed-loop Gain µβ: Loop Gain

Stability conditions for Loop Gain [µβ] For

µβ = 1 ⇒ ∠µβ < 180° LG : 20 log µβ [dB] = 0

Polar diagram of LG (Nyquist Diagram)

(Bode Plot)

In Slide 6.56 a Bode plot of the loop gain of the circuit in the previous slide and a Nyquist diagram (polar diagram) of the loop gain are shown.

7  Stabilizing the Closed-Loop Gain of an Opamp (Basics from Control Theory)

Example: Non-Inverting Opamp

G=

127

In Slide 6.57 and what follows, these concepts are used as an example to describe the frequency stabilization of a non-inverting opamp. In this slide we have the circuit diagram, the corresponding SFG, and the expression for the closed-loop voltage gain of a non-inverting opamp.

Vout A = Vin 1 + { A/β LG

Non-Inverting Opamp (cont’d) G=

A ⇒ log G = log A − log(1 + LG ) LG >>1 ≈ 1 + A/β ≈ log A − log LG LG

log G ≈ log A − log

A

β

= log β

LG

rate of closure (r.o.c)

degree of stability inherent in a feedback system, as explained below.

In Slide 6.58 the logarithmic version of the closedloop gain expression for the non-inverting opamp is developed in order to obtain the gain in dB. This enables a better separation, both analytical and graphic, of the closed-loop gain (G), the loop gain (LG), and the open-loop gain (A). It also enables the introduction of the concept of “rate of closure” (r.o.c.; not to be confused with the commonly used “region of convergence” or ROC) which defines the angle between the open-loop and closed-loop gain. This quantity, first introduced by Bode, is critical for the

128

6  Controlled Sources, Nullors, Active Gain Devices, Impedance Converters and Inverters…

In Slide 6.59 the example of a non-inverting opamp is shown, whose open-loop Case 1 : Open-Loop Gain Frequency Compensated frequency response is ω0 for A s = A ( ) compensated such as to 0 One-Pole -6dB/oct Roll-Off s + ω0 obtain a one-pole -6  dB/ oct roll-off. The relationship between the slope of the open-loop gain of the opamp and the corresponding phase characteristic is shown. The phase of the open-loop gain decreases gradually by 90° as the pole of the openloop opamp is reached and continues on the frequency axis. At the same time, as mentioned above, the slope of the open-loop gain decreases by −6 dB per octave (−6 dB/oct) or, equivalently, −20 dB per decade (−20 dB/dec).

Non-Inverting Opamp (cont’d)

In Slide 6.60 the example is of a non-inverting opamp whose open-loop frequency response is not Case 2 : Open-Loop Gain NOT Frequency Compensated frequency compensated. Here the phase of the ω1ω2ω3 A( s ) = A0 open-loop gain decreases ( s + ω1 )( s + ω2 )( s + ω3 ) gradually by 90° as each pole of the open-loop opamp is reached and passed on the frequency axis. At the same time, the roll-off of the open-loop frequency response increases by another −6  dB/oct. Whereas the r.o.c. in the frequency-­ compensated case in the previous slide was −6 dB/oct, here in the case of three open-loop poles, the r.o.c. has increased to −18 dB/oct. As we shall see when the question of frequency compensation and stability is treated for the general case, this crossover angle between the closed-loop and open-loop gain, the “rate of closure” defined by Bode, is critical for the stability of the opamp.

Non-Inverting Opamp (cont’d)

8  Frequency Compensation and Stability

8

129

Frequency Compensation and Stability

Frequency Compensation and Stability In general, open loop gain (e.g. of Opamp) has more than one pole, e.g.:

A( s ) =

In Slide 6.61 we start once again with the mathematical formulation of the open-loop response of a typical opamp with three poles and DC gain A0.

A0ω1ω 2ω 3 ( s + ω1 )( s + ω 2 )( s + ω 3 )

In Slide 6.62 we have the Bode plot of A(s), as well Frequency Compensation and Stability, cont’d as that of the correspondWith the corresponding Bode plot: ing one-pole frequency-­ compensated open-loop |A(ω)| [dB] gain response. The r.o.c. open-loop for each case is also shown. gain − 12 dB oct − 6 dB oct According to Bode’s formulation, the slope of the closed-loop rate of gain amplitude response Loop Gain of frequency closure rate of compensated amplifier =12dB/oct closure decreases at a rate of 0 at the frequency of unity gain (where ϕ 0 : phase shift around feedback loop at unity gain frequency ω 0 ).

Phase Margin, Stability, and Frequency Compensation, cont’d

Slide 6.66

loop gain [dB]

0

ϕ

log ω

ω0

0° ϕ0

°

180

9

Phase margin

ω0

}

ϕm log ω

Methods of Frequency Compensation

In this section, several methods of frequency compensation are illustrated.

6  Controlled Sources, Nullors, Active Gain Devices, Impedance Converters and Inverters…

132

Various Methods of Frequency Compensation for r.o.c. ω2 ; ω2 < ωa , ωb ε 0 = 1 + 2  ω2 < ωa , ωb loop gain in the frequency-­ compensated opamp is decreased and with it the stabilizing effect of the decreased loop gain. Thus, frequency compensation involves a typical tradeoff – as so often in engineering – namely, in this case, the trade-off between frequency stability and stabilized closed-loop gain. Slides 6.71 and 6.72 show two more methods of frequency compensation, both of which guarantee an r.o.c. of 6 dB/oct.

Where have the open-loop poles gone?

(cont’d)

Various Methods of Frequency Compensation for r.o.c. = I1V1 1 − 2  ≠ 0 active or  g1  < lossy passive! Ideal Gyrator: y11 = y22 = 0 Active (ideal) Gyrator: g1 ≠ g 2

11

 atrix Parameters of Common Network Elements M and Controlled Sources

In Slides 6.90, 6.91, 6.92, 6.93, 6.94, 6.95, 6.96, and 6.97, the matrix parameters of impedance converters, inverters, and controlled sources are given. The tables given here are very useful and will be referred to several times in other chapters of this book.

Slide 6.90

IMPEDANCE CONVERTERS

Matrix Parameters of Some Common Network Elements

Element

[zij]

[yij]

Voltage Inversion General Impedance Converter (VGIC)

-

-

Current Inversion General Impedance Converter (CGIC)

[hij]

0 −1

-

−

[gij]

Z a ( s) 0 Z b ( s) − Z b ( s) Z a ( s) 0

0 1 Z a ( s) 0 Z b ( s)

-

0 1

[ABCD]

−1 − 0

Z a ( s) Z b ( s)

0 O

0

Z b ( s) Z a ( s) 0

1

1

0 O −

0

Z b ( s) Z a ( s)

Slide 6.91

IMPEDANCE CONVERTERS, cont’d

Matrix Parameters of Some Common Network Elements

Element Voltage Inversion Negative Impedance Converter (VNIC) Current Inversion Negative Impedance Converter (CNIC)

[zij]

-

-

[yij]

-

-

[hij]

[gij]

0 − k1 − k2 0

[ABCD]

0 − 1/ k2

− k1 O

− 1 / k1 0

1/ k2

0

k1

0

k2

0

1 / k1 0

0 1 / k2

0 k1

0 O − 1 / k2

0

Slide 6.92

IMPEDANCE CONVERTERS, cont’d

Matrix Parameters of Some Common Network Elements

Element

[zij]

Ideal Voltage Converter (IVC)

-

Ideal Current Converter (ICC)

-

[yij]

[hij]

[gij]

-

0 k −1 0

0 −1 1 0 k

-

0 −

1

β

[ABCD]

1

0 −β

0

1

k

0

0

1

1

0

0

β

0

6  Controlled Sources, Nullors, Active Gain Devices, Impedance Converters and Inverters…

144

Slide 6.93

CONVERTERS, INVERTERS:

Matrix Parameters of Some Common Network Elements

Element Converters Inverters

[hij]

-

1 − 0 m

0 ±

1 g

1 g

0 ±g

0

g

±

Ideal Gyrator IG

[yij]

0 m

±

Ideal Power Converter IPC

[zij]

[gij]

[ABCD]

0 −m

m 0 0 m

1 m

0

-

0

1 g

0

±

±g

0

-

Slide 6.94

INVERTERS, cont’d Matrix Parameters of Some Common Network Elements [zij]

Negative Impedance Inverter (NII)

0 m

1 g

1 g

0

Inverters

Element

Generalized Negative Impedance Inverter (GNII)

m

m

[yij] 0 m

[hij] 1 g

0 m 1 g (s )

1 g (s ) 0

[ABCD]

-

0

-

N mg 0

m g (s )

0

0 ±

N m g (s )

-

0

1 g

±

0

N mg

[gij]

-

1 g (s)

N m g (s)

0

Slide 6.95

Controlled Sources, Converters Matrix Parameters of Some Common Network Elements [zij]

Converters

Element Ideal Transformer (IT)

Controlled Sources

Current Controlled Voltage Source (CVS)

[yij]

-

-

0

0

r

0

-

[hij]

[gij]

0

-

1 n

±n

0

0

±

0

0

1 r

0

0 m

0 ±n mn

[ABCD]

m

1 n

-

0 O 1 n

11  Matrix Parameters of Common Network Elements and Controlled Sources

145

Slide 6.96

Controlled Sources Converters, cont’d Matrix Parameters of Some Common Network Elements Element

[zij]

-

Current Controlled Current Source (CCS)

-

Controlled Sources

Voltage Controlled Current Source (VCS)

[yij] 0 g

[hij]

[gij]

[ABCD]

0

0 0

0

-

-

0

0

α

0

-

0

0

1 g 0 0 1

α

Slide 6.97

Controlled Sources Converters, cont’d Matrix Parameters of Some Common Network Elements

Voltage Controlled Voltage Source (VVS)

Controlled Sources

Element

[zij]

[yij]

[hij]

-

-

-

[gij]

0

0

µ

0

[ABCD]

1

µ

0

0

0

In Slides 6.98, 6.99, 6.100, 6.101, and 6.102, nullator-norator versions of the four basic controlled sources are given. These will also be referred to and used in connection with various topics dealt with in the remainder of this book. Slide 6.98

Nullator-Norator Representation of Controlled Sources Controlled Sources with Single NullatorNorator

Controlled Source VVS

Controlled Sources with Two NullatorNorator Pairs

R2

V1

~ I1 = 0 V2 = µV1

V1

R1

Controlled Sources with Two NullatorNorator pairs

R2

V2 V1

R1

V2 V1

R1

I1 = 0

I1 = 0

I1 = 0

R  V2 =  2 + 1V1  R1 

R V2 = 2 V1 R1

V2 = −

R2

R2 V1 R1

V2

6  Controlled Sources, Nullors, Active Gain Devices, Impedance Converters and Inverters…

146

Slide 6.99

Nullator-Norator Representation of Controlled Sources, cont’d Controlled Sources with Two NullatorNorator Pairs

Controlled Sources with Two NullatorNorator Pairs

Controlled Sources with Single NullatorNorator

Controlled Source VCS

R V1

V1

V2 V1

R

V2 V1

V2

R

I1 = 0

I1 = 0

I1 = 0

I1 = 0

I 2 = gV1

1 I 2 = V1 R

1 I 2 = − V1 R

I2 =

1 V1 R

Slide 6.100

Nullator-Norator Representation of Controlled Sources, cont’d Controlled Sources with Single NullatorNorator

Controlled Source CVS

Controlled Sources with Two NullatorNorator Pairs

Controlled Sources with Two NullatorNorator Pairs

R

R V1

I1

~

V1

V1 = 0

V2

V1

V1 = 0

V2 = rI1

V2 = − RI1

V2 V1

R

V2

V1 = 0

V1 = 0

V2 = RI1

V2 = − RI1

Slide 6.101

Nullator-Norator Representation of Controlled Sources, cont’d Controlled Sources with Two NullatorNorator Pairs

Controlled Sources with Single NullatorNorator

Controlled Source CCS

R1 I1

V1 = 0 I 2 = α I1

V1

Controlled Sources with Two NullatorNorator Pairs

R1

R2

V1 = 0 R  I 2 = −  1 + 1 I1  R2 

R2

V1 = 0 I2 = −

R2

R1

V2 V1

R1 I1 R2

V2 V1

V2

V1 = 0 I2 =

R1 I1 R2

11  Matrix Parameters of Common Network Elements and Controlled Sources

Slide 6.102

Nullator-Norator Representation of Controlled Sources, cont’d Controlled Sources with Single Nullator-Norator

1) VVS

2) CCS

R1 = ∞; R2 = 0

V1

R1 = 0; R2 = ∞

V2

V1

V2

I1 = 0

V1 = 0

V2 = V1

I 2 = − I1

147

Chapter 7

Passive LCR and Active-RC Filters

Slide 7.1

Chapter 7

Passive LCR and Active- RC Filters

1

Introduction

This chapter forms the bridge between “passive” LCR (inductor-capacitor-resistor) filters and active-­RC (resistor-capacitance-gain) filters. Furthermore, it provides the motivation for replacing conventional passive LCR filters by inductorless, equivalent active-RC filter circuits that can be realized as IC (integrated circuit) chips. This means replacing “passive” frequency-filter circuits based on the electromagnetic resonance effect, by “active” frequency-filter circuits based on gain elements, e.g., amplifiers, connected with passive-RC circuits in positive- and/or negative-feedback circuits. “Active” here refers to the gain elements in the circuit that make up for the losses in the passive-RC circuit. (For a circuit theoretical definition of active and passive circuits, see Slide 6.86 in Chap. 6.) Unlike the rest of the book, the material presented here is in the form of a brief overview. This is because on the one hand, in today’s digital world, analog circuit theory and the concepts involved © Springer Nature Switzerland AG 2019 G. S. Moschytz, Analog Circuit Theory and Filter Design in the Digital World, https://doi.org/10.1007/978-3-030-00096-7_7

149

7  Passive LCR and Active-RC Filters

150

therein need no longer be dealt with in the same detail as in the past. On the other hand, many of these concepts are important, at least in the form of an overview, because analog circuits and filters are still used in practice. They are, in fact, often vital in those cases in which the ubiquitous digital signal processor cannot be used. In the remainder of this book, filters become one of the vehicles by which the analysis and design of analog circuits are described. In order to do this, conventional LCR filters and inductorless filters – in particular active-RC filters  – are discussed in this chapter. They will be encountered again for analog circuit design further on in this book.

2

Passive LC Filters

Slide 7.2 introduces the two generally assumed inventors of the LCR filter and the most prominent features of these filters. George 1915: first electromagnetic wave filter Ashley Campbell [1870– (G.A. Campbell, K.W.Wagner). 1954, American engineer, American Telegraph and Non-ideal inductors due to series Telephone (AT&T)] and resistance (“lossy” coils). Karl Willy Wagner [1883– 1953, German professor of Filter network unconditionally stable electrical communications] are credited with the invendue to finite component losses and tion of the first LCR filter termination resistors. for communication systems in 1915. “Classical” LCR filters were basic for telephone and telegraph communications for over half a century, but their design and manufacture (e.g., accurate filter tuning for a telecommunication channel) always remained both challenging and costly due to the lossy nature of the inductors. On the other hand, being passive, i.e., containing no gain elements, they were unconditionally stable and robust physically. Furthermore, they were remarkably insensitive to ambient disturbances and noise.

Passive LC Filters

2  Passive LC Filters

151

Typical LC Network L7 Rs

L1

RL1

C2

L2

RL2

L5

C4

RL5

RL7 C10

C8

∼

VIN

C1

L4 C3

C5

RL4

RL3

INSERTION LOSS [dB] 60

C7

C9

L6

AMPLITUDE SHAPING

C14 L8

RL6

C6

RL9

C13

C11 L3

L9

C12

RL8

L10

C15

VOUT

RL

RL10

ALL-PASS SECTIONS [DELAY COMPENSATION]

50 40 30 20 10 0 500

1000

1500

2000

2500

3000

f

Slide 7.3 shows a typical band-pass filter used in the bygone days of analog multichannel telephone communication systems. It combines an amplitude shaping and a delay equalization section. The resistors in series with the inductors are not physical resistors; they are the inevitable and omnipresent losses in the filter inductors. One of the main reasons for inductor lossiness is the resistivity of the wires making up the inductor. Capacitors can be realized – even on chip – with far fewer losses, meaning with a much higher component “quality factor,” Q. The purpose of showing this filter is to demonstrate the degree of complexity that a relatively sophisticated communication filter can reach. Hand in hand with this complexity comes the difficulty of tuning the filter to given specifications (fine-tuning individually, very often by hand) and, with that, the cost of manufacture. The filter contains several LC resonators or “tank” circuits that either “trap” or transfer the signal frequencies of resonance, thereby producing the desired band-pass-filter characteristic. If the LC components of the resonators are lossy, the “frequency-trapping” effect of the resonator will be accordingly jeopardized, i.e., the frequency selectivity of the “frequency-trapping” resonators will be less narrowband and the filtering effect less evident.

7  Passive LCR and Active-RC Filters

152

Slide 7.4 shows an LC parallel circuit, also referred to Passive Filters Cont’d as a resonator or “tank” circuit. The slide explains anaThe disadvantages of LC filters are lytically why the lossiness of the inductor decreases the directly related to the properties of the “frequency-trapping” capainductors. bility of a resonant tank circuit. The capacitor in the Example: Q of single LC resonator circuit is considered “ideal,” (“tank circuit”). i.e., lossless, which is a safe assumption in the frequency L bands in which filters of the ω0 L ω = 1 Cideal Q≈ ; 0 kind shown in the previous LCid RL RL slide are used. The inductor will be lossy, mainly due to the series resistance RL of the inductor wire. The resonance frequency of this LC tank circuit is ω0. The quality factor Q is given by ω0L/RL. This, as we shall see below, is given by the center frequency divided by the 3 dB bandwidth – in other words, the frequency selectivity or “frequency-trapping capability” of the resonance circuit. Thus, the smaller the inductor loss RL, the higher the selectivity, or Q, of the circuit. It is useful at this point to remember that the quality factor or Q of anything in engineering is generally defined as the desired quantity divided by an undesired quantity. Thus, in this case, the desired quantity is the impedance of the inductor at the resonance frequency, namely, ω0L; the undesired quantity is the inductor loss RL. Therefore, the Q of a resonant circuit is given by Q = ω0L/RL. It follows that the larger the inductance value L, the more wire is necessary for its realization, the larger will be the inductor loss RL, and the smaller its Q. This is demonstrated in the next slide.

LC

Slide 7.5 shows how, as the inductance value L of a resonant circuit increases at low frequencies, its losses, e.g., RL, will, of necessity, increase as well, thereby decreasing its Q. On the other hand, as the value of the inductance decreases, which is the case for high frequencies, the Q also decreases. This is related to the physical three-dimensional nature of the inductive device. As shown in the slide, somewhere in between, depending on the technology used to realize the inductor, there will be a maximum Q. Furthermore, with increasing L values, i.e., at low frequencies, there comes an increase in size and bulkiness of the inductor.

2  Passive LC Filters

153

Slide 7.6 shows a simple RLC circuit resulting in a second-order band-pass filter whose ratio of resonance frequency to passband, i.e., selectivity, is directly related to the Q of the tank circuit. Physically the phenomenon of resonance in an LC tank circuit is based on the periodic conversion of magnetic into electric energy and vice versa. In spite of the problems encountered with LCR filter circuits, it is important to remember that they have formidable advantages. Briefly, they can be listed as follows: • • • • •

They dissipate negligible power. As passive networks, they are unconditionally stable. They have extraordinarily low sensitivity to component tolerances and other ambient influences. They generate practically no noise. They provide either a DC path without offset or total isolation.

These advantages are significant and only partially, if at all, replaceable by inductorless filters and circuits. On the other hand, the list of disadvantages, although shorter, is decisive in today’s modern IC technology. Recapitulating briefly they are: • Bulkiness of inductors (weight and volume) • Low coil Q (particularly at low frequencies) • Fabrication methods incompatible with IC batch processing techniques In particular the last item of the list permits no mitigation. No matter what the considerable advantages provided by inductors (and transformers), if they cannot be manufactured with state-of-the-art technology, then they are, bluntly put, useless.

154

7  Passive LCR and Active-RC Filters

To summarize, all of the above indicates that inductors are problematic at low and high frequencies. Added to this comes the insurmountable difficulty – if not impossibility (depending on the frequency range) – of even realizing an inductor in any established integrated circuit (IC) technology, at any frequency. It is therefore not surprising that in analog circuitry, the necessity of eliminating inductors has become imperative. First among the circuits to which this imperative applies are LCR filters, and this opens the door to inductorless filters and, more specifically, to active-RC filters.

3

Active-RC Filters: (a) Basic Network Elements

The first impulse of most engineers faced with the problem of replacing one technology by another is to try and replace item by item, and feature by feature, with the new technology. This is the most direct – and “comfortable” – way of dealing with technology replacement but, as is recognized with time, often not the most successful. (The failure of directly replacing vacuum-tube circuits by transistorized circuits is a well-known example.) Generally, the most lasting and successful designs are the ones that are newly adapted to the features of the new technology. In Slide 7.7 we have two methods of accomplishing From Passive LC to Active RC technology replacement by & Digital Filters direct design. It shows the replacement of a passive LCR ladder filter by an inductorless digital filter and analog filter, respectively. At the center left of the slide is a simple LCR (elliptic) low-pass filter. Above it is an elegant way of replacing it by a digital circuit based on the theory of wave digital filters FDNR (WDFs). This theory was Opamps, Resistors developed by the reputed and Capacitors network theorist, Prof. Alfred Fettweis (1926– 2015, University of Bochum, Germany). The WDF replacement possesses not only the filtering capability of the original LCR filter but also other features normally associated exclusively with passive RLC filters; one of these is reciprocity, meaning, broadly speaking, operability from both terminal ends. Since this book is devoted to analog circuitry, the WDF method will not be pursued further. Judging by publications and application reports, it has been a successful method of going from LCR to (digital) inductorless filters. At the bottom of the slide is another elegant – and successful – method of inductorless filter design, for technology replacement by analog means. It is called the FDNR (frequency-dependent negative resistance) method. This method was developed by another reputed electrical engineer and scientist, Prof. Len Bruton (University of Calgary, Canada), and is based on an impedance transformation of the original LCR ladder filter, resulting in a ladder structure consisting of resistors and so-called FDNRs. The latter, being derived from capacitors, are sometimes called “super capacitors”; they consist of active-RC circuits suitable for IC realization. This method, along with less direct inductorless filtering techniques for technology replacement, will be dealt with in this and the following chapters.

3  Active-RC Filters: (a) Basic Network Elements

155

Slide 7.8 demonstrates the use of some of the network elements introduced in Impedance converter Chap. 6  – and briefly Ex: NIC: reviewed here  – for the A 0  [A B C D] =  design of inductorless,   0 D AZ L + B active-RC filters. As in Z in = = −k1k 2 Z L Chap. 6, we use the transA = m k1 ; D = ±1/k2 CZ L + D mission or [ABCD] matrix, Impedance inverter which is the most universal of the two-port matrices, to Ex: Gyrator: 0 B define the characteristics [ A B C D ] =   of these network elements. C 0  1 Furthermore, as in Chap. 6, Z in = B = 1 / g1 ; C = g 2 we consider these network g1 g 2 Z L elements as two-ports with g i = 1 / Ri [Ω −1 ] a load impedance ZL at the output terminals. It follows that the impedance converter is defined as a two-port for which B = C = 0 and the impedance inverter as a two-port with A = D = 0.

Active RC Filters a) Basic Network Elements

Basic network elements cont‛d: Negative Impedance Converter

Ex: A = m k1 = 1; D = ±1/k 2 = 1

ZL

NIC

ZL=R; ZIN=-R.

ZIN=-ZL

Gyrator R

ZL 2

ZIN=R /ZL

Ex: ZL=1/jωC; ZIN= jωR2C=jωLeq

In Slide 7.9 the symbols and definitions of two of the main representatives of converters and inverters are shown. For impedance converters, it is the negative impedance converter (NIC); for the impedance inverter, it is the gyrator. The transformer, ideal or not, is of course an even more conspicuous and well-known impedance converter, but it is not generally referred to in these terms.

7  Passive LCR and Active-RC Filters

156

Basic network elements cont‛d: Nullators & Norators 4mA

4mA

1mA 8V

4V

1k

8k

8V

4k 1mA

VBE≈0 4mA Vout 1k 4V

V1

V0

4k

RE

V2

RC

IC

I0

?

C

4V

E

RC=1k

R1=8k

4V

⇒

1k 4mA 4V

1mA

I2

I=0

Rp=R1IIR2

B

4V

V=0

I1

1k

8k

⇒

4V

Vin 4V

VB

12V

12V

R2=4k

C N E

B n

RE=1k

⇓

V1 = R p I1 V0 = V1 = I 0 RE I = IC + I0 = 0 V2 = I C RC 2 I C = − I 0 V2 = V2 = − RC V1 V0 RE

Basic network elements cont‛d:

nullator Nullator-Norator (Nullor):

v=0 i=0

norator

v,i arbitrary Ideal Transistor: C B E

⇒

Ideal Opamp:

B

C

E

1 2

3

⇒

1

3

2

Slides 7.10 and 7.11 summarize the derivation and definition of the nullator-norator (nullor) concept that was introduced in Chap. 6.

3  Active-RC Filters: (a) Basic Network Elements

157

Slide 7.12 shows a nullor realization of a negative Nullor Realization of Negative Impedance Converter impedance converter (NIC). 0  1 As we shall see further on, [ A B C D ] = 0 − Z 2  V=0 the nullor-based circuit I=0  Z1  shown here can also be I1 I2 adapted to realize other V = V 1 2 Z1 Z2 useful network elements V1 V2 ZL I0 Z2 that will be useful for I1 = − ( − I 2 ) inductorless active-RC filZ1 ter design. V1 = V2 ; I1 + I 2 = I 0 B C C This slide is useful in B ⇒ I 1 ⋅ Z1 = I 2 ⋅ Z 2 E drawing attention, once E Realization with opamp or again, to one of the prime AZ L + B Z transistor? benefits of the nullator-­ Z IN = = − 1 ⋅ ZL norator concept. We have CZ L + D B =C =0 Z2 ⇒ already seen in Chap. 6 that many, if not most, devices, e.g., transistors and opamps, can be traced back to the derivation from a controlled source. We have also shown in Chap. 6 how the four basic controlled sources can readily be realized as nullorbased circuits. Furthermore, we have seen how a nullor-based circuit can be realized by active devices such as transistors, opamps, current conveyors, and the like. As a result, the design of many devices and circuit elements can be carried out according to the following general procedure: Basic network elements cont‛d:

1. Derive the desired device or circuit from a combination of passive components and controlled sources. 2. Replace the controlled sources by their nullator-norator (nullor) equivalents. 3. Replace the nullors within the circuit by their equivalent active devices, e.g., transistors or opamps. 4. Complete the resulting idealized, and in its present form non-realizable, active circuit by adding the necessary biasing and other peripheral circuitry. This converts the circuit into a workable device while taking the envisaged technology into account. As we shall show in Chap. 9, many previously unknown circuits and devices can be developed by following this strategy. Before continuing with our overview of inductorless active-RC filter design, we need to recall the process of impedance scaling which was our Key Point 10, in Chap. 2; it is shown in the next slide.

7  Passive LCR and Active-RC Filters

158

Slide 7.13 serves to remind us of the process of impedance scaling, by which the impedance level and impedance nature of a network can be changed, without affecting its frequency response. This is F (rLi , C j / r , rRν , s ) = aF ( Li , C j , Rν , s ) accomplished by scaling each impedance of the network by a factor whose dimension depends on the dimension of the network function in question, i.e., impedance, admittance, or transfer function. In the case of a transfer function with given frequency response (which is to remain unchanged), each impedance of the network must be scaled by a dimensionless factor.

Basic network elements cont‛d:

Impedance Scaling an LCR Network by ω0/s L1 R1

ω0 ⋅ sL1 s

L5

L3 L2

L4

C2

C4

R2

ω0 ⋅ sL3 s

ω0 ⋅ sL2 s ω0 1 ⋅ s sC2

ω0 ⋅ s R1

ω0 ⋅ sL5 s

ω0 ⋅ sL4 s ω0 1 ⋅ s sC4

ω0 ⋅ s R2

FDNR! C1’=

1 [F] ω0R1

1 2 s ⋅D 2 D=C2 ω0 [ sec ] Ω Z’C2 =

R4’=ω0L4[Ω]

Frequency-Dependent Negative Resistor (FDNR):

Z ( s) =

1 1 1  → s = jω : Z ( ω) = − 2 [Ω] → [D ] =   ΩHz 2  ωD s D 2

Slide 7.14 uses the impedance scaling property to demonstrate how an LCR filter circuit (in this case an elliptic low-pass filter) can be impedance scaled by a dimensionless factor ω0/s such that every inductor becomes a resistor, every resistor a capacitor, and every capacitor a so-called frequency-­ dependent negative resistor (FDNR). This is the FDNR attributed to its inventor Prof. Len Bruton and referred to in Slide 7.10. More details of how the FDNR and other devices, e.g., the gyrator and NIC, can be designed will be discussed in Chaps. 9 and 10. However, in the overview of this chapter, the next two slides indicate in advance how the NIC nullor circuit of Slide 7.12 can be used for this purpose.

3  Active-RC Filters: (a) Basic Network Elements

159

Basic network elements cont‛d: Nullor Realization of NIC: (Negative Impedance Converter). From above:

Z IN = −

Z1 ZL Z2

Z1

Z2

ZL

Nullor Realization of General Impedance Converter (GIC), FDNR, Gyrator n1

Z IN =

Z1 Z 3 ZL Z2Z4

Z1

n2

Z2 N1

Z3

Z4 N2

Basic network elements cont‛d:

Ex: FDNR: 1 1 1 Z1 = , Z3 = , Z 2 = Z 4 = Z L = R, Z IN = 2 sC1 sC3 s C1C3 R Ex: Gyrator: 1 Z4 = , Z1 = Z 2 = Z 3 = Z L = R, Z IN = sC4 R 2 sC4 Ex: GIC: Z1 ( s ) ⋅ Z 3 ( s ) = k ( s ) : Z IN = k ( s ) Z L Z 2 ( s) ⋅ Z 4 ( s)

ZL

The top of Slide 7.15 shows the nullor-based NIC circuit that was shown in Slide 7.12. Cascading two of these nullor circuits, we have the nullor-based circuit shown in the lower half of this slide. As shown, the input impedance of this cascaded circuit is given by (Z1 Z3/Z2 Z4)/ZL. Note that the cascade of the two circuits eliminates the negative sign of the input impedance of the single nullor-based circuit. Slide 7.16 shows that depending on the choice of the components Z1 to Z4 and ZL in the nullor cascade of the previous slide, we obtain either an FDNR, a gyrator, or a general impedance converter (GIC). The latter multiplies the load ZL by a dimensionless, but frequency-dependent, factor k(s). Only resistors and capacitors are permitted for the impedances since the goal here is to design inductorless filters.

7  Passive LCR and Active-RC Filters

160

Basic network elements cont‛d:

Gyrator Realization with OP Amps Z1

Z2

Z3

Z4

−

−

A + 1

+

A2

ZIN Z5 ZIN =

FOR: Z4

=

Z1

Z3 Z2

Z5

Z4

1 jwC

Z1 = Z2 = Z3 = Z5 = R

ZIN = j w CR2

Slide 7.17 Using the nullor-opamp equivalence of Slide 7.11 and the component selection prescribed at the center of Slide 7.16, we obtain the opamp realization of a gyrator shown in this slide. It is obtained by using a capacitor for Z4 and resistors for the other four impedances. The equivalent “inductor” obtained with this gyrator has the value CR2.

Slide 7.18 shows the gyrator realization of a Gyrator-Capacitor Filters grounded inductor. It also makes the point that if the ωLeq  =  ⇒ QC = QL ! QL = loading capacitor of the Req  gyrator making up the INDUCTOR SIMULATION = ωRLC L  “inductor” is itself lossy,  RL R R2 RL with a leakage resistor of = QC = 1 RL, then its quality factor ωC L  Z R C R2 CL Z Q C = ωRLCL and the Q fac = ωRLC L tor of the resulting “induc GYRATOR LOAD z tor,” QL, are the same, i.e., QL = QC = ωRLCL. Lossy Inductor: [Incidentally, this was a factor in the problems R2 Z IN = + jωC L R 2 = Req + jωLeq related to the introduction RL of inductorless tantalum thin-film hybrid-integrated circuits which were propagated in the 1960s and 1970s. The thin-film capacitors had a reasonably high dielectric constant, but a dissipation factor made up, among other things, of the lead-in resistance and capacitive leakage. This was quite high – particularly when compared to that of discrete-component capacitors. Thus, because the Q factor QL of a simulated inductor could be no better than the Q factor QC of the capacitor on which the inductor depended, high-Q filter circuits were difficult to achieve.] Basic network elements cont‛d:

IN

L

L

L

IN

4  Active-RC Filters: (b) Cascading Biquads

161

Basic network elements cont‛d:

Frequency Rejection Network C2 2

1 C1

LC-FILTER

L1 1' 1

g2

g1 1'

L3

C2

2'

g2

2

CL2

C1 GYRATORCAPACITOR FILTER

C3

L2

CL3 CL1

C3

Slide 7.19 shows the realization of a floating inductor. In contrast to the gyrator simulation of a grounded inductor (see previous slide), the floating inductor requires two gyrators. They are connected in cascade as shown in the inductorless frequency-rejection network displayed in the slide.

g3 2'

2 Gyrators required for floating inductor!

4

Active-RC Filters: (b) Cascading Biquads

This section contains a brief overview of inductorless filter design based on the cascade of so-called biquads. As implied by its name, the biquad is a self-contained cascadable filter of which the denominator is quadratic, and the numerator has a maximum degree of two. Any realizable, even-order, filter transfer function can be broken up into the product of biquadratic transfer functions, each of which is realized by a corresponding number of biquads in cascade. In the case of an odd-order transfer function, a first-order term must be factored into the denominator of one of the biquads. Thus, a biquad is a cascadable filter of second or maximum third order, whereby the linear term making up the additional order of one is obtainable with a single resistor and capacitor, connected appropriately in a biquad circuit. Biquads can be cascaded without interaction between them, because the output of each is a voltage source with a low-output impedance, such as the output of an operational amplifier. Theoretically, this makes a cascade of biquads very versatile in that they can be cascaded in any sequence. However, there are practical considerations related to component sensitivity, dynamic range, and noise output that dictate guidelines on the cascading sequence. The “optimum” sequence differs according to the application and system environment.

7  Passive LCR and Active-RC Filters

162

Active RC Filters

b) Cascading Biquads m

N ( s ) bm s m + bm −1 s m −1 + = T ( s) = D ( s ) a n s n + a n −1 s n −1 +

+ b0 =K + a0

n even: = ∏ Ti ( s ) where: Ti ( s ) =

ni ( s ) = Ki di ( s)

ω s 2 + q zi s + ω2zi zi

ω s 2 + q pi s + ω2pi pi

i

i =1 n

∏ (s − p ) j

j =1

n/2 i =1

∏ (s − z )

Biquad Transfer Function

Slide 7.21

Cascading Biquads cont’d

n odd:

=

( n −1) / 2

k

∏ T ( s) s + α i

i =1

Slides 7.20 and 7.21 demonstrate how a given nthorder filter transfer function can be broken up into the product of biquadratic terms, each of which is to be realized by an active-RC biquad.

(Combined with biquad for 3rd-order fctn.)

Thus: cascade of either 2nd-order biquads or biquads with added RC pole: s2 + Ti ( s ) = K i

s2 +

ω zi

q zi ω pi q pi

s + ω2zi s + ω2pi

s2 +

or Ti ( s ) = K i

ω zi q zi

s + ω2zi

 2 ω pi  s + s + ω2pi (s + α )  q pi  

Slide 7.22 illustrates the equivalence between an nth-order filter and the corresponding cascade of filter biquads. (The overall order N of the filter is considered here to be even.)

Network Decomposition IN

N TH ORDER FILTER

OUT

EQUIVALENT TO IN

2 ND ORDER SECT.

2 ND ORDER SECT.

2 ND ORDER SECT.

1

2

N/2

OUT

4  Active-RC Filters: (b) Cascading Biquads

163

Slide 7.23 shows how a simple, i.e., second-order, filter transfer function can SINGLE FEEDBACK LOOP be obtained by interconnecting an amplifier with a passive-RC input network RC NETWORK t1(s) and an RC feedback t2 (S) network t2(s). This figure depicts the basic configu± ration, discussed in detail VOUT VIN RC NETWORK b in the next chapter, for all t1 (S) active-RC, single-­ amplifier, filter building VOUT b . t1 (S) blocks, or, as they are gen= erally known, “biquads.” VIN 1 + b t2 (S) (As we shall see, “multi-­ amplifier” biquads also exist and have their important place in the “biquad pantheon.”)

Low-Q Single OP-Amp Filters

Slide 7.24 is a follow-up on the previous slide in that it gives an example of how a sixth-order bandpass filter, designed with a cascade of three biquads, would appear. Each biquad consists of an opamp and passive-RC circuitry forming an input and a feedback network. More “real” examples are given next.

Single OP-Amp Filters STAGGER TUNED BAND-PASS FILTER db

w p1

w p2

b

w p3

w

b

b

V IN

V qp

L

1

,w p

1

qp

L2

,w p

2

qp

L3

,w

p3

OUT

7  Passive LCR and Active-RC Filters

164

5

Examples of Cascaded Biquad Active-RC Filters

Slides 7.25, 7.26, 7.27, 7.28, 7.29, and 7.30 show three examples of filters realized by biquads. The purpose of these slides is not to show the biquad circuits but to show the frequency response of each biquad and how, together, these responses make up the final desired overall filter response. Being ­plotted in dB, the individual response of each frequency-filter biquad must be added to the others rather than multiplied, in order to obtain the resulting overall filter response. The final example is of a delay equalizer which is realized by a cascade of all-pass filters or delay equalizers. In this case it is the delay, e.g., in microseconds, of each biquad that is added for the final response. The amplitude response of this circuit is flat.

Example of Cascaded Active RC Band-Pass Filter

In Slide 7.25 we see the five building blocks (biquads) schematically, and the specifications, of a high-precision tenth-order band-pass filter.

10th-order band-pass filter. Amplitude ripple 0.1 dB. Total tolerance (incl. delay equalizer) ±0.5dB. In Slide 7.26 we have the frequency response of each of the five biquads. Together they produce the overall response T(s) of a tenthorder band-pass filter.

5  Examples of Cascaded Biquad Active-RC Filters

Example of High-Precision Cascaded Active RC Delay Equalizer 16th-order delay equalizer: cascade of 8 2nd-order all-pass biquads. Overall tolerance (incl. transmit and receive filter: ±3µsec). Note: active RC delay networks always realized as cascaded biquads (rather than LC ladder simulation).

165

In Slide 7.27 we have the specifications of a sixteenth-order delay equalizer. With the active-RC biquad approach to filter design, this can be realized by eight all-pass biquads. Incidentally, as pointed out in the slide, inductorless delay lines and phase equalizers are more easily realized by cascaded allpass biquads than by other methods such as by LC ladder simulation.

In Slide 7.28 we have the individual and overall delay response of the delay-equalizer circuit. Note that for this circuit, the amplitude response is flat. It is the time delay of each of the eight biquads in microseconds, and the overall delay, that is plotted here. The overall delay is obtained by adding the contribution of each delay section or second-order all-pass filter  – which is what, in this case, each biquad really is.

166

7  Passive LCR and Active-RC Filters

Example of High-Precision Cascaded Active RC Frequency Rejection Filter

Finally in Slide 7.29, we have the specifications of a sixth-order frequencyrejection or band-stop filter.

6th-order frequency rejection filter. Realized with 6 cascaded biquads (12thorder!): 3 medium-Q frequency rejection biquads and 3 high-Q frequency emphasizing biquads. In Slide 7.30 the individual biquad amplitude response and the overall response of the frequency or band-rejection filter is shown. Note that the filter is sixth order which is normally realizable with three frequency-rejection biquads. However, for a high-Q circuit, it takes three additional “frequency-­emphasizing” biquads to “pull up” the slopes of the three medium-Q frequencyselective biquads. This is because it is difficult to maintain high accuracy and stability with single high-Q frequency-­ rejection biquads. Note that each of the filters in these examples represents a different group of filters. The first is a regular band-limiting bandpass filter. The second consists of all-pass or delay sections with flat amplitude all-pass characteristics. The last is a frequency-rejection or band-stop filter whose frequency-­rejection response becomes increasingly selective as it passes through the next biquad.

Chapter 8

A Classification of Single-Amplifier Biquads

Slide 8.1

Chapter 8

A Classification of Single-Amplifier Biquads (SABs) [Based on the Realization of Complex-Conjugate Poles]

1

Introduction

A plethora of single-amplifier second-order active filter circuits that can be cascaded into filters of higher order have been published in the literature and have been put into practice. These circuits are filter building blocks whose numerator and denominator are of second order, i.e., biquadratic; they are therefore referred to as “biquads” (see Chap. 7). Actually, the numerator of a biquad can be equal to or lower than second order. Furthermore, for odd-order filters, the extra pole needed to make a third-­ order cascadable building block can readily be obtained by adding a single resistor and capacitor to the input of the circuit – which is still referred to as a “biquad.” In this case the degree of the numerator can then also be up to three. The large variety of biquad filter building blocks is daunting and seems to make any overview or qualitative comparison of individual circuits close to impossible. However, in this chapter we demonstrate that of this large number and variety of biquads – published and in use – one very important © Springer Nature Switzerland AG 2019 G. S. Moschytz, Analog Circuit Theory and Filter Design in the Digital World, https://doi.org/10.1007/978-3-030-00096-7_8

167

8  A Classification of Single-Amplifier Biquads

168

group, namely, that of “single-amplifier biquads,” can actually be grouped into no more than four basic classes and two subclasses. This is achieved with the help of fundamental circuit-theoretical, analytical, and feedback concepts. At the same time, this classification permits a relevant evaluation, and performance comparison, to be made of the resulting circuits. The method of classification is general enough to serve as an example for other similar analog circuit design situations, in which the individual “circuit trees” initially detract from a recognition of the overall “circuit forest.” At the same time, since methodology in analog circuit design, and the important practical consequences resulting from it, is one of the targeted features of this book, the method of classification presented here fits neatly into the overall objectives of the book. Before beginning our classification, it is useful to keep in mind Slides 7.20, 7.21, 7.22, 7.23, 7.24, 7.25, 7.26, 7.27, 7.28, 7.29 and 7.30 of Chap. 7. There the analytical method of breaking up a given nth-order filter transfer function T(s) into a product of second order, i.e., biquadratic, or third order, transfer functions Ti(s), is dealt with. Each of these corresponds to the transfer function of a second- or third-order “biquad” filter building block. A cascade of these second- and/or third-order biquads results in the original nth-order filter transfer function and the corresponding nth-order filter circuit.

2

Design of General Filter Functions by Cascading Biquads

Slide 8.2 begins the classification of single-amplifier biquads. The classification is based on the method by which complex-conjugate pole pairs are realized by a particular biquad. [Remember from Chap. 2 that with passive RC circuits, only negative-real poles can be realized. On the other hand, practical LCR filters have transfer functions that invariably require complex-conjugate pole pairs.] Whereas Slides 7.20, 7.21, 7.22, 7.23, 7.24, 7.25, 7.26, 7.27, 7.28, 7.29 and 7.30 went through the general steps necessary for filter design based on the cascade of biquads, Slides 8.2, 8.3, 8.4 and 8.5 now go through a practical example, i.e., that of a seventh-order elliptic low-pass filter. This will set the stage for the classification that follows.

Biquad Cascade Design of General Filter Functions

Example:

Given typical 7th – order elliptic filter function:

T (s) = k ⋅

s 6 + b5 s 5 + s 7 + a6 s 6 +

+ b0 + a0

(1)

Break up into quadratic and simple terms: T (s) =

N (s) = D( s)

k ⋅ (s 2 + ω2z1 )(s 2 + ω2z2 )(s 2 + ω2z3 )

(2)

    ωp ωp ωp ( s + α) s 2 + 1 s + ω2p1   s 2 + 2 s + ω2p2   s 2 + 3 s + ω2p3  q q q p1 p2 p3    

Slide 8.2 shows a given seventh-order filter transfer function, followed by the same function broken up into quadratic and simple terms, such as to group complex and single roots into expressions with only real coefficients.

2  Design of General Filter Functions by Cascading Biquads

Biquad Cascade Design of General Filter Functions, cont’d

From Filter Table or Computer Program

Biquad Cascade Design of General Filter Functions, cont’d

Elliptic (CC) low-pass

169

Slide 8.3 shows the corresponding pole-zero location of the transfer function in the s-plane. This is readily found either from filter tables or computer programs.

Slide 8.4 shows the amplitude response of what is seen to be a seventh-order elliptic low-pass filter (see Chaps. 3 and 4).

8  A Classification of Single-Amplifier Biquads

170

Biquad Cascade Design of General Filter Functions, cont’d

T ( 0) = k

ω2z1 ω2z2 ω2z3

ω2p1 ω2p2 ω2p3 α

>0

Biquad Cascade Network 1 Rα

RC

1 ωα = Rα Cα

Network 2 ±β1

RC

Network 3 ±β1

RC

±β1

Slide 8.5 shows a conceptual arrangement of three biquads for the seventhorder elliptic low-pass filter given in the three preceding slides. The first biquad is third order, thanks to an added RC section connected to its input.

n −1 2

T ( s ) = T0 ( s ) ⋅ ∏ Ti ( s ) i =1

Slide 8.6 shows a possible Design Steps for Biquad Cascade Design pole-zero assignment for the biquads of the seventhorder filter in the s-plane. This pole-zero assignment reveals the first of the numerous degrees of freedom that the biquad designer is confronted with when designing inductorless filters with biquads. Thus, given the overall pole-zero distribution in the s-plane, the first question to be asked is “How do you combine the poles of the given filter into pairs with the zeros, such that each pair will constitute the poles and zeros of a biquad?” (For third and zeros order biquads, this will include negative-real poles. However, they have no influence on the pole-zero pairing assignment problem; they are obtained by pre- or post-connecting an RC combination to the biquad.)

2  Design of General Filter Functions by Cascading Biquads

Design Steps for Biquad Cascade Design, cont’d 1. Combine Pole/Zero pairs for maximum dynamic range and minimum sensitivity to component tolerance, e.g. : T1(s), T2(s), T3(s) [Rule of thumb: highest pole Q with nearest zero] 2. Distribute the factors among the n/2 ( n −1) k = k0 ⋅

2

∏k i =1

i

(or (n-1)/2 ) biquads for maximum dynamic range, e.g. k=k1· k2 · k3 3. Select sequence of biquads for maximum dynamic range, e.g. T2(s)

T1(s)

T3(s)

171

Slide 8.7 formulates the pole-zero pairing problem as the first of three initial steps that must be taken by the designer for biquad design. The problem is generally solved by combining the n poles and m zeros of the nth-order transfer function T(s) into pole-zero pairs, each of which will be realized by a biquad such as to minimize noise, maximize dynamic range, and minimize the sensitivity to component tolerances. A “rule of thumb” to fulfill these requirements is to

combine the pole pair with the highest pole Q with the nearest zero(s). [Referring to Slide 7.26 of the previous chapter, the five biquad filter functions Ti(s) with i = 1..5 which make up the final function T(s) are the result of five possible pole-zero pairings. There are many other possibilities of pairing the poles and zeros of the given filter. Some of the combinations may result in biquad responses that sharply attenuate the signal in the passband, only to be amplified by others in order to reach the specified final output signal level. Such a combination may introduce unnecessary noise and may also result in non-optimum sensitivity to component variations. This question of pole-zero pairing and the two following points are all interconnected; they have been dealt with thoroughly in the literature but go beyond the subject matter of this book.] The second and third design steps entail the distribution of the coefficients of the individual biquads, and the sequence in which the biquads are cascaded, both with the aim of maximizing the dynamic range of the resulting interconnected cascaded filter. [Here, and from here on, it is assumed without being mentioned that in the case of an odd-order filter, at least one of the biquads will be of third order.] Having determined the coefficients and pole-zero pairs, i.e., the individual filter transfer functions Ti(s) of the biquads to be cascaded in order to obtain a given nth-order filter function T(s), the design of the individual biquads begins. In this chapter we restrict ourselves to single-amplifier biquads, meaning biquads that contain a single-feedback amplifier. At least one amplifier (or other active device) is needed to produce the complex-conjugate poles that cannot be realized by a passive RC network alone, i.e., without including inductors in the design of the filter.

8  A Classification of Single-Amplifier Biquads

172

Design Steps for Biquad Cascade Design, cont’d Single - Amplifier Biquads are based on t ypical Active Feedback Amplifier RC Network Negative Feedback

µ(s)

Slide 8.8 shows the structure on which most singleamplifier biquads are based, namely, a positiveor negative-feedback amplifier 𝛃, a passive RC feedback network whose transfer function is:

n (s) , (8.1) dˆ ( s ) and a passive RC input β n( s ) T (s) = K µ( s ) = tˆ( s ) = network that provides the 1 + µ( s )β dˆ ( s ) multiplicative filter coeffiˆ cient K and the numerator β β d (s) N ( s) T (s) = K =K =K of the desired filter transfer n( s ) D( s) dˆ ( s ) + β n( s ) 1+ β function T(s). dˆ ( s ) Loop Gain As we know from Chap. 2, Key Point 3, the roots of d̂(s) are negative real, since they are the poles of a passive RC network. This is indicated by the ^ over d(s). On the other hand, the roots of the denominator D(s) of the biquad transfer function T(s) are complex conjugate, because they are the poles of the required biquad, and these, by definition, are complex conjugate for any practical filter applications. The overall biquad transfer function for the case of negative feedback is given by: Vin

K

T (s) = K

β

Vout

µ ( s ) = tˆ ( s ) =

N (s) β dˆ ( s ) β =K =K ˆd ( s ) + β n ( s ) n (s) D (s) 1+ β dˆ ( s )

(8.2)

The poles of T(s), which are the roots of the polynomial D(s) =  dˆ ( s ) + β n ( s ) = 0, are the complex-­ conjugate poles of the biquad that is to be designed. The expression for D(s) holds the key to obtaining complex-conjugate poles from the combination of the polynomial d̂(s), which has negative-real roots, β which is the gain of the feedback amplifier, and the numerator n(s) of μ(s). The polynomial n(s) is of maximum second order, i.e., it is either a constant, a linear term, or a second-order polynomial. Being the numerator of a passive RC feedback network, the location of its roots in the s-plane is restricted as described in Chap. 2.

3  Problem Formulation

173

In Slide 8.9 we see that for the case of negative feedback, the location of all possible roots of n( s ) ˆ ( s ) + β n ( s )  = 0 in = −1 180° Root Locus D ( s ) = dˆ ( s ) + β n( s ) = 0 ⇒ β D(s) =  d Characteristic dˆ ( s ) the s-plane can be obtained Equation by deriving the 180° root jω locus characteristic equation from it. [For reasons of space, we cannot review σ basic root locus theory P1 P2 here. There are many excellent publications and text books for this. dˆ ( s ) = ( s − P1 )( s − P2 ) : neg. real poles because they are However, as we proceed in poles of a passive RC network this chapter, we will advance those key points of root locus theory that are necessary for the understanding of the material at hand.] Because μ(s) is the transfer function of the passive RC feedback network, we know that the roots of its denominator d̂(s), which define the starting points of the root locus, are single, finite, and on the negative-real axis of the s-plane. Furthermore, the plus sign in the expression for D(s) indicates that the signal being fed back to the input by the amplifier β is negative, hence the reference to 180°, by which the inverting feedback amplifier “inverts” the feedback signal. [As we shall see later, in the case of positive feedback, we will have a minus sign in the expression for D(s) due to the “non-­inverting” feedback amplifier; we then refer to a 0° root locus characteristic equation.]

Design Steps for Biquad Cascade Design, cont’d

3

Problem Formulation

Problem Formulation Design a network with one active element (amplifier) such that we obtain the following 2nd-order transfer function:

T (s) =

N (s) = D( s)

N (s)

2

s +

qp =

ωp qp

ωp 2σ p

s +ω

2 p

ωp σp

Im Re σ

where the poles must be complex conjugate, i.e. q p ≥ 0.5

In Slide 8.10 we formulate the biquad classification problem, which is to design a filter network, or biquad, with one active element (e.g., amplifier) such as to obtain a secondorder transfer function with complex-conjugate poles, i.e., a pole Q, qp, larger than 0.5.

8  A Classification of Single-Amplifier Biquads

174

In Slide 8.11 we see that the problem formulation (“Initial Approach”): leads to a basic signal-flow General SFG with one active element ±β graph (SFG) representation of a feedback ampliˆt 32 fier circuit. The plus/minus β path represents the non2 3 inverting and inverting β tˆ12 feedback amplifier, respectˆ34 tively, and the t̂ij, with 4 1 i,j  =  1..4, represents pastˆ14 =‘ Leakage Path’ is assumed to be zero. sive RC networks where the subscripts refer to the ˆ ˆ N ( s) t t = tˆ14 ± β 12 34 T ( s) = nodes of the SFG and the ^ See next D( s) 1 β tˆ32 ( s ) slide indicates the passive RC nature of the networks. In given unknown particular, t̂32 represents the feedback network which is responsible for the method of creating complex-conjugate poles; t̂14 is the so-called leakage path. The latter is unaffected by the amplifier and will therefore be ignored in the present context, as will t̂34 for the same reason. The general transfer function T(s) for this SFG is given at the bottom of the slide.

±

In Slide 8.12 we see the simplified SFG due to the omission of the paths from nodes 1 to 4 and 3 to 4. The corresponding simplified transfer function T(s) now contains the denominator D(s) that was already discussed under Slide 8.8 and given in Eq. 8.2. This expression for D(s) represents the starting point for our biquad classification; it leads to two new ways of formulating our design problem.

3  Problem Formulation

175

General SFG with one active element, cont’d Analytical formulation of problem: (Polynomial Decomposition)

Decompose D(s) as follows:

D( s) = s 2 +

ωp ω s + ω2p = s 2 + 0 s + ω02 ± β n32 ( s ) qp qˆ dˆ32 ( s ) = ( s − P1 ) ( s − P2 )

qˆ < 0.5

In Slide 8.13 we have the analytical formulation of the design problem: “Decompose a quadratic polynomial whose roots are complex conjugate, into the sum of a quadratic polynomial with negative-­ real roots, plus or minus a factor β times a polynomial n(s) of maximum second order.” We will return to this formulation later.

In Slide 8.14 we have the second formulation of our design problem. This is a system-theoretical formulation and leads back to ˆ 1 ± β t32 ( s ) = 0 root locus theory. It does Because: this by converting D(s) of (s − Z ) (s − Z ' ) tˆ32 ( s ) = Eq. 8.2 into the character( s − P1 ) ( s − P2 ) neg. real poles istic equation that will therefore: result in the root locus of negative feedback all possible complex-con(s − Z ) (s − Z ' ) jugate roots available with = m1 β ( s − P1 ) ( s − P2 ) D(s) when β increases positive feedback from zero to infinity. Characteristic Equation The two roots of d̂(s), the starting points of the root locus, are single and on the negative-real axis, since they are the poles of the passive RC network constituting the feedback path t̂32(s). Thus, there are only two degrees of freedom at our disposal for the root locus. The first is the factor β, which is a positive or negative number, depending on the polarity of the feedback amplifier gain. The second is the location of the roots of n(s), i.e., the zeros of the numerator of t̂ij, which is the transfer function of a passive RC feedback network of second order. Theoretically, β can vary from zero to infinity. From root locus theory, we know that the roots of n(s) are the endpoints of the root locus, when β reaches infinity. System-theoretical Approach: (Characteristic Equation, Root Locus)

8  A Classification of Single-Amplifier Biquads

176

In Slide 8.15, the problem is now reformulated as follows: “Given the polarity P1, P2 are given as negative-real poles. of the feedback amplifier, find the roots of n(s) such Where may Z and Z’ be located such that the root as to obtain complex-conlocus with respect to β produces conjugate complex jugate roots, i.e., poles, in the left-half s-plane.” poles ? Starting with negative (s − Z ) (s − Z ' ) feedback, the correspond1. Negative feedback: β = −1 ing characteristic equation ( s − P1 ) ( s − P2 ) that defines the root locus is shown in the lower half of this slide. P1 and P2 are the given open-loop poles of the feedback loop transmission β t̂32(s); they are the starting points of the root locus. We now look for the possible location of all open-loop zero pairs, Z,Z’, i.e., roots of n(s), such that, with increasing value of β, they can become the endpoints of a useable root locus with respect to β. By useable we mean root loci that can provide a pair of closed-loop poles with the feedback loop βt̂32(s) that will, for a given value of β, serve as the poles for a specified biquad. In other words, for any given root locus, and for a given corresponding value of negative gain β, we will obtain the complex-conjugate closed-loop pole pair that we require for the feedback loop of our desired biquad. System Theoretical Formulation of the Problem:

In Slide 8.16 an assortment of root loci are shown that result from the problem formulation in the previous slide. At first glance, it would seem that there are an unlimited number of root loci that can fulfill our requirement. However, as we shall see in what folWe shall see in what lows, there are exactly follows that the root three basic groups of root locus 1 , 2 and 3 loci, each of which has its form the basis for the 3 own individual type of fundamental negativeendpoints, i.e., open-loop feedback-based singlezeros, corresponding to its amplifier Biquads. own kind of feedback network. Each of these will provide a complex-conjugate closed-loop pole pair p and p* in its own way; we shall refer to each as belonging to its own specific class of filter biquads. For this we need to relate the theoretical considerations discussed so far to a basic active-RC feedback structure. This brings us to the actual design of single-amplifier, single-feedback biquads and is dealt with in the next section.

4  Design of Single-Amplifier, Single-Feedback Biquads

4

177

Design of Single-Amplifier, Single-Feedback Biquads

Design of Single-Amplifier, Single-Feedback Biquads Basic Single-Amplifier, Single-Feedback (S.F) Biquad Structures [Classification: Type I, Single-Feedback Loop] 1

V1

∼

2

RC

3

β

3

Equivalent signal-flow graph (SFG) tˆ32

In Slide 8.17 a basic single-amplifier, single-feedback (SF) biquad structure is related to its equivalent signal-flow graph (SFG). The node numbers in the circuit correspond to the node numbers in the SFG.  In our classification this will be referred to as a Type I circuit because it has a single-feedback loop. The two polarities of β refer to positive and negative feedback, respectively.

tˆ12 1

2

β

3

Design of Single-Amplifier, Single-Feedback Biquads, cont’d

V2 = V1 ×t12 + V3 ×t32

t12 ( s ) =

V2 ; V1 V3 =0

t32 ( s ) =

V2 V3 V1 =0

t12 ( s ) , t 32 ( s ) : 2nd –order, passive RC networks tˆij (s ) =

nij (s ) = (s − P1 )(s − P2 )

nij (s ) ω0 s 2 + ij s + ω02ij qˆij

“^” indicates passive RC: qˆ < 0.5

In Slide 8.18 the individual branch transmission functions tij are deduced from the basic active-RC feedback structure shown in the previous slide. Quantities referring to passive RC circuits are indicated by a “^.” At this point we must specify whether we are referring to positive or negative feedback.

8  A Classification of Single-Amplifier Biquads

178

4.1

Negative Feedback In Slide 8.19 we begin with negative feedback and provide the general transmission terms for the corresponding SFG.

Realization of Complex-Conjugate Poles with Passive-RC Network and single amplifier ± β

a) negative feedback. tˆ32 (s) tˆ12 (s) 1

2

T ( s ) = −β ⋅

β

3

tˆ12 ( s ) =

n12 ( s ) dˆ12 ( s )

tˆ32 ( s ) =

n32 ( s ) dˆ32 ( s )

tˆ12 ( s ) N (s) = 1 + β tˆ32 ( s ) D( s )

Realization of Complex-Conjugate Poles with Passive-RC Network and single amplifier ±β, cont’d T(s) is to have complex - conjugate poles, i.e.: n12 n12 ( s ) dˆ12 = −β = T (s) = − β ⋅ ˆ n32 d ( s ) + β n32 ( s ) 1+ β dˆ32 dˆ12 =dˆ32 =dˆ ( s ) t (s) and t (s) 12

32

are obtained from N (s) N (s) the same network, = = ωp therefore poles are D( s) 2 2 s + s + ωp the same! qp ω Thus the polynomial D( s ) = s 2 + p s + ω2p = dˆ ( s ) + β n32 ( s ) decomposition has qp the form: q p > 0.5

In Slide 8.20 we take into account the fact that the individual branch transmission functions t̂ij refer to the same RC network but to different nodes. As a consequence, the denominator of each branch transmission function t̂ij, i.e., the transfer function poles of each, is the same.

4  Design of Single-Amplifier, Single-Feedback Biquads

Realization of Complex-Conjugate Poles with Passive-RC Network and single amplifier ±β, cont’d

Summary of Problem: ω Given: dˆ ( s ) = ( s − P1 )( s − P2 ) = s 2 + 0 + ω02 where qˆ < 0.5

qˆ

Find β, n32, and dˆ ( s ) such that

ωp D( s ) = dˆ ( s ) + β n32 ( s ) = ( s − p1 )( s − p 2 ) = s 2 + s + ω p2 qp

where q p > 0.5 .

179

In Slide 8.21, based on the conclusions reached so far, the problem of complexconjugate pole realization with passive RC networks and a single-feedback amplifier is reformulated. The solution leads us straight to the characteristic equation and the corresponding root locus, derived from the denominator D(s) of the overall transfer function T(s).

In Slide 8.22, we demonstrate the problem of realizing complex-conjugate poles with the means at How can we move hand, graphically. The root the neg. real poles locus of D(s) examines into the complex show the roots of the equaplane? tion D(s)  =  0 move in the s-plane as the variable β, i.e., the gain of the feedback amplifier, is increased from zero to infinity. For β = 0 these roots are simω ply the open-loop roots of qˆ < 0.5 0.5 ≤ q p = p < ∞ 2σ p d̂(s) = 0, i.e., P1 and P2 as shown in this slide. Having decided to start with negative, rather than positive, feedback, the only remaining available degree of freedom is now the location of the roots of the numerator n(s), i.e., the open-loop zeros. These must be placed such as to guarantee that the root locus of the closed-loop poles, the roots of D(s,β) = 0, will converge from their initial location P1 and P2 on the negative-real axis of the s-plane and meet at the so-called coalescence or breakaway point. When they meet, i.e., reach the coalescence point, they will split and symmetrically branch off the negative axis in the s-plane, thereby becoming complex conjugate. This requirement of meeting at the coalescence point puts an important constraint on the location of the open-loop zeros, i.e., the roots of n(s) = 0.

Realization of Complex-Conjugate Poles with Passive-RC Network and single amplifier ±β, cont’d

180

8  A Classification of Single-Amplifier Biquads

In Slide 8.23, we formulate the constraint for coalescence on the negative-real axis with a Key Point I shown at the botˆ Open loop zeros: end d ( s ) + β n32 ( s ) = 0 tom of this slide. It says points of root locus Characteristic Equation that for the 180° root locus (i.e., negative feedback), 180° n32 ( s ) n32 ( s ) β tˆ32 ( s ) = β =β = −1 the root locus exists on the ˆ Root-Locus ( s − P1 )( s − P2 ) d ( s) real axis only left of an Open loop poles: starting odd number of critical frepoints of root locus quencies, whereby “‘critical’ frequencies” mean Key Point I. For 180° root locus: root locus exists both poles and zeros. on real axis left of odd number of critical frequencies. Since for coalescence, the “Critical frequencies”: poles and zeros. root locus must exist between the two openloop poles, P1 and P2, Key Point I puts limits on the possible location of the open-loop zeros, i.e., on the nature of n(s).

Characteristic Equation for Root Locus of p, p* with respect to -β

Slide 8.24 takes us directly to the limits imposed by Key Point 1. This is a more detailed version of Slide 8.16 in that it graphically groups into three basic classes those open-loop zero pairs in the root locus that permit the closed-loop poles p and p* to lie in between the open-loop poles P1 and P2. Each of these classes permits the existence of a coalescence point on the negative-real axis of the root locus. Each of these classes therefore permits the generation of complex-conjugate closed-loop poles in the left-half s-plane (LHP). The following slides will now discuss each of these three classes, which we shall designate as the three Classes 1, 2, and 3.

Characteristic Equation for Root Locus of p, p* with respect to –β, cont’d

4  Design of Single-Amplifier, Single-Feedback Biquads

181

 lass 1 Biquads C We begin with Class 1 biquads. Like the other classes, this one will be additionally categorized as Type I because there is only one forward transmission path from the driving signal source to the amplifier input terminal. Later we shall see that there is a second group of biquads that have two forward transmission paths, one positive and one negative. Each one leads from the driving signal source to one of the two input terminals of the amplifier, which is typically a differential input opamp. This second group of biquads will be designated Type II. Furthermore these three negative-feedback biquad classes have only one feedback path, making them “single feedback” (SF). We shall see further on that there is yet another group of biquads among these classes that has dual feedback, i.e., a feedback transmission path from the amplifier output terminal to each of the differential input terminals. These biquads will be referred to as “dual-feedback” (DF) biquads. Consequently, the first class of biquads which is dealt with below is designated “Type I-Single Feedback-Class 1” or I-SF-1.

Characteristic Equation for Root Locus of p, p* with respect to –β, cont’d

Class 1: I-SF-1

Slide 8.25 shows the root locus of Class 1 biquads. It runs parallel to the s-plane imaginary axis ending at the open-loop zeros at “infinity.”

8  A Classification of Single-Amplifier Biquads

182

Characteristic Equation for Root Locus of p, p* with respect to –β, cont’d Class 1: (I-SF-1)

Zeros of tˆ32 (s ) , i.e. roots of n32 ( s ) lie at infinity

t32 ( s ) =

Slide 8.26 shows the corresponding feedback transmission function t̂32(s) whose numerator polynomial n32(s) must, in this case, be a constant, i.e., k32 ω02.

k32 ⋅ ω02 k32 ⋅ ω02 = ω s 2 + 0 + ω02 ( s − P1 )( s − P2 ) qˆ

tˆ32 ( s ) : low-pass function

tˆ32(s)

low-pass function

tˆ12 (s) 1

β

2

3

Characteristic Equation for Root Locus of p, p* with respect to –β, cont’d Class 1: (I-SF-1) 1

V1

2

RC

∼

3

2

3

β

low-pass function R2

R1

C2

C1

β

3

Slide 8.27 shows that for a Class 1 biquad, t̂32(s), which is the transmission function of an RC feedback network with a constant numerator n32(s) = k32 ω02, must be a secondorder passive RC low-­pass network.

4  Design of Single-Amplifier, Single-Feedback Biquads

Characteristic Equation for Root Locus of p, p* with respect to –β, cont’d Remember:

T (s) =

and

D( s) = s 2 +

N (s) = D( s)

N (s) ω s 2 + p s + ω2p qp

183

In Slide 8.28 we return to the analytical formulation, or polynomial decomposition, of Slide 8.13 for the Class 1 biquad.

ωp s + ω2p = dˆ ( s ) + β n32 ( s ) qp

2 For Class 1 networks: n32 ( s ) = k32 ( s ) ⋅ ω0

Thus:

s2 +

ωp ω s + ω2p = s 2 + 0 s + ω02 + β k32 ω02 qp qˆ

In Slide 8.29 we solve for ωp, and qp from the analytiCharacteristic Equation for Root Locus cal problem formulation of of p, p* with respect to –β, cont’d the previous slide, obtaining expressions from Solving for ωp, qp: which we can deduce certain characteristics of the ω p = ω0 1 + β k32 > ω0 Class 1 biquad  – without actually having seen the q p = qˆ 1 + β k32 > qˆ circuit. ω p ω0 For example, for a sec= = const. (independent of β) ond-order band-pass filter, qp qˆ the term ωp/qp is equal to its 3 dB bandwidth. As we Class-1 Complex-Conjugate Pole Generation see in this slide, ωp/qp is a ωp constant for the Class 1 = const. , for band-pass filter, 3dB Because biquad; therefore for a qp Class 1 band-pass biquad, bandwidth constant and independent of gain β. the 3 dB bandwidth will be constant, i.e., independent of the feedback amplifier gain β. Although with a little insight, this also becomes evident from a glance at the root locus itself; nevertheless, the analytical results obtained here speak for the benefit of interpreting analytical results independently of any other considerations.

8  A Classification of Single-Amplifier Biquads

184

Slide 8.30 shows a basic second-order negativefeedback network, designated as Class 1 due to the method in which it realizes R2 R1 its complex-conjugate pole 2 3 pair. It does so by connecting a second-order passive C2 C1 RC low-pass circuit in the feedback loop of an invert1 ing feedback amplifier. This says nothing yet V1 about how to realize the zeros of the resulting biquad, nor how a driving β signal source will make the circuit into a biquad with input and output terminals. Note however that we have already introduced a driving input voltage source at terminal 1 of the circuit. This is to help explain in advance the general method of converting the Class 1 feedback loop – and those of the other classes – into a useable biquad. The method is described in the next slide.

Realization of zeros [numerator of T(s)] for Class-1 Biquads

∼

In Slide 8.31, Key Point II of this chapter is formulated. It says that we can drive our Class 1 (i = 1…4) feedback network from an ideal signal Key Point II. Drive biquad from ideal voltage source such that the basic source (i.e. output of ideal opamp) such that nature of t̂32(s) will not be changed. This condition is basic nature of t32(s) (i.e. for Class-1 network: satisfied with the (ideal) low-pass characteristic) does not change. voltage source connected to terminal 1  in the previous Ex. I-SF-1 Band-pass (BP) Filter slide (naturally, after disconC1 R2 1 2 3 necting it from ground). β This is because an “ideal” V1 R1 C2 voltage source has zero source impedance, meaning that terminal 1 of the new circuit is at AC ground. In practice, of course, there is no such thing as an “ideal” voltage source; every physical signal source will inherently have some residual source impedance. Depending on how much this impedance deviates from zero, it will have to be corrected for as a second-order non-ideality. Note that the output terminal of an “ideal” opamp has the same feature of “zero” source impedance and may well require the same corrective treatment to take its actual finite output impedance into account. This is particularly the case for a high-frequency, and therefore low-impedance, biquad. In the lower part of Slide 8.31, the biquad circuit of Slide 8.30 is unraveled and redrawn with an ideal driving voltage source to expose what can be recognized to be a Type I-single feedback-Class 1 (I-SF-1) band-pass filter or biquad.

Realization of zeros [numerator of T(s)] for class-1 biquads

∼

4  Design of Single-Amplifier, Single-Feedback Biquads

Realization of zeros [numerator of T(s)] for class-1 biquads, cont’d

185

In Slide 8.32, the amplifier β of the previous slide is replaced by an operational amplifier.

Opamp Realization:

In Slide 8.33 the transmission functions t̂12(s) and Realization of zeros [numerator of T(s)] for t̂32(s), as well as the resultclass-1 biquads, cont’d ing expressions for ωp and qp, for the Class 1 bandpass filter of Slide 8.31 are For the above example: given. ω0 s k32 ω02 Note that terminal 1 tˆ12 ( s ) = k12 tˆ32 ( s ) = ω was chosen for the driving ω s 2 + 0 s + ω02 s 2 + 0 s + ω02 voltage source of the qˆ qˆ biquad in Slide 8.30 N (s ) ω0 s because it is a ground terT (s) = = k12 minal from which connectω ω s 2 + p s + ω2p s 2 + p s + ω2p ing an “ideal” voltage qp qp source would not affect the feedback circuit t̂32. This points to the important ω = ω 1 + k β ; q = q 1 + k β ˆ where: p 0 32 p 32 realization that Key Point 2 has its limits, namely, that there may not be many terminals available from which a driving voltage source can be introduced “without significantly altering the nature of the feedback network t32(s).”

8  A Classification of Single-Amplifier Biquads

186

In Slide 8.34 we see that in the case of our Class 1 circuit, there is only one other suitable terminal for the Ex. I-SF-1-Low-pass (LP) Filter introduction of a voltage R2 R12 3 2 source, namely, referring to Slide 8.30, at the interC2 C1 R11 connection between R1, R2, 1 and C1. Without changing the “basic nature” of the V1 feedback network, this can be achieved by breaking β up R1 into a voltage divider. This results in the loop R R2 11 1 2 3 gain and the forward gain β being reduced by the voltV1 R12 C1 C2 age dividing action of the two resistors. Both can readily be compensated for by a c­ orresponding increase in the gain β and in the multiplicand K. The result is the I-SF-1 low-pass filter biquad shown in this slide.

Examples:

∼

∼

Examples, cont’d

Ex: Application of Single-Feedback Class-1 Band-pass (I-SF-1 BP) Filter ωp tuning Rβ 1

C1 R1

R2

2

R0

3

C2

Slide 8.35 demonstrates one of the characteristic features of Class 1 band-pass biquads, namely, that the 3 dB bandwidth is constant irrespective of the center frequency, due to the nature of its root locus.

4  Design of Single-Amplifier, Single-Feedback Biquads

Examples, cont’d Voltage-Controlled band-pass filter with variable center frequency and constant bandwidth

187

Slide 8.36 shows how this can be used in practice, e.g., as a scanning secondorder band-pass filter with constant bandwidth.

Slide 8.37 lists this feature as an advantage of Class 1 biquads and also lists Advantages of Class 1 Networks: ­several disadvantages. However, it should be noted Constant bandwidth (e.g. interesting for constant that neither these advanbandwidth voltage-controlled band-pass filter) tages nor disadvantages are Disadvantages of Class 1 Networks: particularly significant. ωp depends on gain β (increases component Much rather they are listed sensitivity, i.e. to β); here to demonstrate that additional insight into the q p ~ β → β ≈ q 2p → high gain required for high nature of the biquad classes qp values  amplitude sensitivity to gain; discussed can be achieved Only LP and BP filters feasible. by a critical and discerning interpretation of the design method used and what it has produced. This follow-up interpretation is most important for any design activity; it is therefore repeated wherever possible or appropriate in this book.

Examples, cont’d

8  A Classification of Single-Amplifier Biquads

188

Characteristic Equation for Root Locus of p, p* with respect to –β, cont’d

In Slide 8.38 we now return to our set of root loci for complex-conjugate pole generation and discuss what we refer to as Class 2 biquads.

Class 2 Biquads

Class 2: I-SF-2

Zeros of tˆ32 (s ) , i.e. roots of n32(s) located at origin, i.e.,

tˆ32 ( s ) = k32

s2 s2 = k32 ω ( s − P1 )( s − P2 ) s 2 + 0 s + ω02 qˆ

sion function of a passive RC high-pass network.

Slide 8.39 shows the set of root loci corresponding to Class 2 (I-SF-2) networks. They correspond to passive RC feedback networks whose transmission functions tˆ32 ( s ) have pairs of negative-real zeros to the right of the openloop poles P1 and P2. This includes the special but most common case of two zeros at the origin of the s-plane. This case results in the transmission function at the bottom of the slide. This is the transmis-

Class 2 Filters

tˆ12 n12 ( s ) = −β ˆ 1 + β ⋅ tˆ32 d ( s ) + β ⋅ n32 ( s ) n32 ( s ) tˆ32 ( s ) = dˆ ( s ) T (s) = − β

s2

A : tˆ32 ( s ) = k32

s2 +

ω0 qˆ

s + ω02 |tˆ32 (jω)|

Slide 8.40 shows the root locus, the SFG, the transfer function and its frequency response, and the RC network, for the Class 2 circuit whose two zeros are at the origin of the s-plane.

Class 2 Filters B : tˆ32 ( s ) = k32

( s − Z1 )( s − Z 2 ) s2 +

ω0 qˆ

s + ω02 Z1Z 2 < P1 P2

|tˆ32 (jω)|

k32 = 1

Z1 =

1 ; R '1 C1

Z2 =

1 R '2 C2

t32 (0) = Z1 Z 2

ω

2 0

=

Z1 Z 2 >1

β>>1 (open-loop gain)

Examples, cont’d

Ex. I-SF-3 HP, LP, BP Filters: C1 R1

C2

R2

BP (R input) A→∞

C1 C0 R1

C2

R2

HP A→∞

(3 capacitors !)

qp =

ωp ≈ qz 2σ p

Slides 8.61 and 8.62 demonstrate the versatility of this class due to the use of the dual bridged-T circuits, thus providing low-pass (LP), high-pass (HP), and band-pass (BP) biquads.

4  Design of Single-Amplifier, Single-Feedback Biquads

Slide 8.62

Examples, cont’d

R1 C1

R2

199

C2

BP (C input)

A→∞

R1 R0

R2

C2

LP

A→∞

C1

Examples, cont’d

n12 ( s ) t12 ( s ) = ω s 2 + 0 s + ω02 qˆ T (s) = −

ωz s + ω2z qz t32 ( s ) = ω s 2 + 0 s + ω02 qˆ s2 +

n n12 β n12 ≈ − 12 = − ω n32 dˆ + β n32 β→ A→∞ s 2 + 0 s + ω02 qz q p ≈ qz

Slide 8.63 shows that, as long as the open-loop zeros of the feedback network tˆ32 ( s ) lie in the left-half s-plane [which they must for a bridged-T network (Fialkow-Gerst!)], the feedback amplifier can be operated with full, i.e., open-loop, gain. This is because the circuit is guaranteed absolute stability, since the closed-loop poles cannot “slide” into the RHP.  Theoretically, with “infinite” gain, the closedloop poles are then identical with the open-loop zeros, and the circuit is still stable.

8  A Classification of Single-Amplifier Biquads

200

Examples, cont’d

Advantages of I-SF-3 Networks p= 0: independent of gain (root locii with respect to gain: circular). qp≈qz or (with twin-T and qz=∞) q p = qˆ (1 + β). With bridged-T: open-loop gain opamp mode. LP, HP, BP filters feasible.

Finally, in Slides 8.64 (advantages) and 8.65 (disadvantages), the novel features of Class 3 biquads are listed. The advantages are noteworthy and more important in practice than those of the first two classes. The disadvantages are also significant, but as we shall see further on, they can readily be overcome. Slide 8.65

Examples, cont’d

Disadvantages: With bridged-T (which is most common) limit on max pole Q: qp. The higher qp the higher spread of RC components (i.e. qz  : R’s and/or C’s  ). Frequency reject or band-stop filters not possible (thus, e.g., elliptic filters impossible). 4.2

Positive Feedback

Realization of Complex-Conjugate Poles with Passive-RC Network and single amplifier: +β

b) positive feedback: tˆ32 (s) tˆ12 (s) 1

T (s) = β

2

β

3

tˆ12 ( s ) n12 ( s ) =β ˆ 1 − β tˆ32 ( s ) dˆ12 ( s )=dˆ32 ( s )=dˆ ( s ) d ( s ) − β n32 ( s )

Slide 8.66 brings us to single-amplifier biquads with positive feedback. The slide shows the basic SFG with positive gain β and the corresponding transfer function T(s).

4  Design of Single-Amplifier, Single-Feedback Biquads

201

In Slide 8.67 the characteristic equation for posiPositive Feedback, cont’d tive feedback is derived. This leads to the positive-­ Characteristic equation: feedback root locus, which is now a 0° root locus – in contrast to the 180° root 0° Rootn32 ( s ) n32 ( s ) β tˆ32 ( s ) = β =β = +1 locus of negative-­feedback ˆ ( s − P1 ) ( s − P2 ) d ( s) Locus circuits. Key Point IV formulates the critical feature  – Key Point IV: For 0° root locus: Root Locus exists on for our purposes – of the 0° real axis left of even number of critical frequencies root locus, namely, that it (“critical frequencies”: poles and zeros) and to right exists on the real axis left of the rightmost critical frequency. of an even number of critical frequencies (where “critical frequencies” are the poles and zeros) and to the right of the rightmost critical frequency.

Slide 8.68 shows that for the coalescence point to exist on the root locus in between the two open-­loop negative-real poles P1 and P2 the feedback network may have only one zero on the negative-real axis to the right of P1 and P2. This corresponds to a secondorder band-pass type of network with (i) a zero at the origin which is the most commonly used circuit or (ii) somewhere between the right of the rightmost open-loop pole P2 and the left of the s-plane origin. (Obviously, here again, the zero may not lie anywhere on the positive-real axis.) This results in only one possible type of root locus, namely, that consisting of circles with center on the real axis, positive or negative. Consequently, there is only one class of biquads with positive feedback; they are designated as “Class 4.”

Positive Feedback, cont’d

8  A Classification of Single-Amplifier Biquads

202

Class 4 Biquads Slide 8.69 shows the most common case of a Class 4 root locus with an openloop zero at the origin.

Class 4: I-SF-4

ω0 s ω s 2 + 0 s + ω02 qˆ tˆ32 ( s ) : band-pass function

Zeros of tˆ32 ( s ) are on either side of P1, P2, e.g. one at origin and one at infinity:

tˆ32 ( s ) = k32

Slide 8.70 shows the corresponding SFG and block diagram.

Class 4: I-SF-4, cont’d tˆ32 (s)

band-pass function

tˆ12 (s) 1

1

V1

∼

β

2

RC 3

2

3

β

band-pass function

3

4  Design of Single-Amplifier, Single-Feedback Biquads

203

Slide 8.71 shows three versions of a Class 4 feedback circuit, each of C2 R1 R2 C1 which consists of a simple 2 3 2 3 second-order passive RC R2 C1 R1 C2 band-pass network. We now briefly digress from our classification in β β order to demonstrate a feature of analog circuit C1 R 1 design that is sometimes 2 3 referred to as the art of R2 C2 analog design. It involves moving away from the strict rules and thought β processes of analog design and going into “musing” about the possibility of going beyond them. Obviously, these musings must follow the lines of rational thought and theory, i.e., “Ohm’s law must be obeyed.”

Examples

Examples, cont’d

Q: Are there other possible band-pass methods to be used for t32(s) in I-SF-4 biquad? Key Point V. Given a grounded 3-Pole network: 1

V1

∼

Passive Network 3

2

V2

t12 =

V2 V1 V3 =0

Thus, in Slide 8.72, although we already have several simple passive RC band-pass circuits for our feedback network providing t̂32(s) in the previous slide, we ask if and how we could find some more? To do so it is useful to reach back to a network theoretical relationship that the reader may be familiar with. It relates the transfer function of a passive grounded 3-pole network to that of the same 3-pole when the input and ground terminals are exchanged.

8  A Classification of Single-Amplifier Biquads

204

Slide 8.73 formulates the relationship between the transfer function t12(s)  = V2/V1 of the original 3-pole with that of the 3-pole in Exchanging the input terminal and ground which input and ground terminals are exchanged, we have: namely, t32(s)  =  V2/V3. It states, in Key Point V, that 3 2 Passive t12(s) + t32(s) = 1. V2 Network t32 = [It should be noted that V2 V3 V3 V1 =0 1 this is the special case of a theorem in network theory that applies to an n-port as It can be shown that : t12(s)+t32(s)=1 formulated several decades ago by various illustrious or t32 ( s ) = 1 − t12 ( s ) network theorists. Alas, this belongs to the age of sophisticated and elegant analog network theory for which today, in the “digital age,” there is neither time, space, nor even interest.]

Examples, cont’d

∼

Examples, cont’d

Ex. 1: 1

C

C

R

R

2C

R/2

2

3

t12 ( s ) =

s 2 + ω02 ω s 2 + 0 s + ω02 qˆ

: Band Reject

In Slide 8.74 the relationship of Key Point V is applied to the transfer function t12(s) of the well-­ known twin-T frequencyrejection network. [Note that t12(s), the transfer function of the conventional twin-T circuit, used as a frequency-rejection network in a Class 3 biquad, can provide complex-­conjugate openloop zeros in the right-half s-plane, but only as far into the RHP as “FialkowGerst permits”].

4  Design of Single-Amplifier, Single-Feedback Biquads

205

Examples, cont’d

ω0 s s +ω qˆ t32 ( s ) = 1 − = : Band-pass ω0 ω0 2 2 2 2 s + s + ω0 s + s + ω0 qˆ qˆ 2

2 0

Examples, cont’d

Ex. 2: Bridged-T

1

Z1

3 Z2

2 Z4

1

Ladder Topology Z4

Z3

interchange input and ground

Complex Conjugate Zeros: frequency reject circuit used in class-3 networks as t32(s).

2 Z2

Z3

Z1

3

Ladder Network: Zeros must lie on negative real axis no frequency reject network possible with class-3 networks!

In Slide 8.75, exchanging the input and ground terminals of the twin-T, the corresponding transfer function t32(s) = 1−t12(s) is shown to result in a bandpass network which may now be included in our set of Class 4 band-pass feedback circuits for t̂32(s).

In Slide 8.76 and Slide 8.77, this input-to-groundexchange relationship is applied to a bridged-T circuit. It demonstrates a duality between a bridged-T and a ladder network, transforming a frequency-­ rejection bridged-T network (with complex-conjugate zeros) into a band-pass ladder network (with zeros on the negative-real axis).

8  A Classification of Single-Amplifier Biquads

206

Slide 8.77

Examples, cont’d 3

2 Z1

Z2

Z4

1

Z3

Z4

Z2

Z1

1

Z3

3

interchange input and ground Ladder Network: Band-pass possible as t32 in Class-4 networks

2

Bridged-T Network: Complex conjugate zeros  frequency rejection network possible with Class-4 networks

Examples, cont’d

Ex. I-SF-4 Frequency Rejection Biquad:

1

C1

C2

R1 C3

R2 R3 3

2

β

3

In Slide 8.78, we use the input-to-ground-exchange relationship of Slide 8.73 to produce a Class 4 frequency-rejection biquad, which according to our classification is a I-SF-4 (“Type I-single feedback-­ Class 4”) frequency-rejection biquad.

4  Design of Single-Amplifier, Single-Feedback Biquads

207

Examples, cont’d

s 2 + ω02 tˆ12 ( s ) = k12 ω s 2 + 0 s + ω02 qˆ

k12 = 1

Slide 8.79 lists the individual t̂ij(s) functions and the overall transfer function T(s) of the resulting frequency-rejection biquad.

ω0 s q ˆ tˆ32 ( s ) = 1 − tˆ12 ( s ) = ω s 2 + 0 s + ω02 qˆ

n12 ( s ) T (s) = β =β dˆ ( s ) − β n32 ( s )

s 2 + ω02 ω s 2 + p s + ω2p qp

In Slide 8.80 we return to the basic SFG and transmission functions to obtain the characteristic features of Class 4 biquads.

Examples, cont’d

General: I-SF-4 Network: tˆ32 (s)

tˆ32 ( s ) = k32

tˆ12 (s) 1

β

2

3

ω0 s ω s 2 + p s + ω2p qˆ

Remember: D( s) = s 2 +

ωp ω s + ω2p = s 2 + 0 s + ω02 − β k32ω0 s qp qˆ

8  A Classification of Single-Amplifier Biquads

208

In Slide 8.81 we solve for the closed-loop expressions ωp and qp in terms of the open-loop quantities.  Solving for ωp, qp: Prominent in the resulting expressions is the fact that the closed-loop frequency ωp is independent of the Class-4 Complex Conjugate Pole Realization: gain β and that qp which must, by definition, be larger than 0.5 is made to ω p = ω0 be so by dividing the passive RC q̂ by the difference qˆ 1 qp = ; β < βcritical = between unity and an k32 qˆ 1 − k32 β qˆ expression involving the gain β. [On the face of it, such an expression is considered with apprehension because, for a critical value of β, namely, βcritical, this expression can become zero, resulting in poles on the jω axis and instability. This apprehension may well already have been felt at the sight of the root locus circle which crosses the jω axis into the RHP – this, being a manifestation of an infinite qp. However, as we shall see in the chapter on sensitivity, Chap. 11, this apprehension is unfounded. This is due to additional features, not discussed here, that guarantee stability – in many cases more so than the negative-feedback-based Class 1, 2, and 3 biquads.]

Examples, cont’d

Summary of General SFG with one active element: tˆ32

Initial Approach ( Ansatz):

tˆ12 β

2

T ( s ) = ±β

3

tˆ12 1 β tˆ32

±

1

Only tˆ32 ( s ) and β are responsible for the poles. neg. feedback

D( s ) = 1 ± β tˆ32 ( s ) = dˆ32 ( s ) ± β n32 ( s ) = 0 pos. feedback

Slide 8.82 summarizes our classification of singleamplifier biquads based on our initial approach (or “Ansatz”) of a corresponding SFG with a positive- or negative-feedback amplifier.

5  Type I, Type II, Single-Feedback, and Dual-Feedback Biquads

209

Four Classes of Single -Amplifier Biquads Based on Feedback Function t 32 (s) t32(s)

Cl.

1

k s +

ω ω

low-pass

2

k

s+ω

qˆ

Root Locus of ClosedLoop Poles

qp

β

Possible t12(s) Functions

qˆ 1 + kβ

β>1 = −

q p = qz

1− α > qz qz 1− α qˆ

α < α crit =

qˆ qˆ

ω0 s ω0 qˆ

s + ω 02

Example: II-DF-4 All-pass Network, cont’d

T (s) =

− Aα + At '12 1 + A(1 − α ) − At '32

=− A>>1

 2 ω0   s + s + ω 02  qz  α− ˆ d (s) =− k' ω s (1 − α ) − 32 0 dˆ ( s )

Slide 8.110 shows the forward and feedback RC networks and their transmission functions t’12 and t’32, where the prime indicates that they feed into the positive input terminal of the opamp.

α − t '12 (1 − α ) − t '32

Slides 8.111 and 8.112 go through the calculation of  the overall transfer function T(s).

8  Summary of Classification of Single-Amplifier Biquads (SABs)

225

Example: II-DF-4 All-pass Network, cont’d α 1  (α − 1) s 2 + ω 0 s −  + (α − 1)ω 02  qˆ q z  T (s) = − 1− α  (1 − α ) s 2 + ω 0 s − k '32  + (1 − α )ω 02  qˆ 

α 1  (1 − α ) s 2 − ω 0 s −  + (1 − α )ω 02  qˆ q z  = 1− α  (1 − α ) s 2 + ω 0 s − k '32  + (1 − α )ω 02  qˆ 

Example: II-DF-4 All-pass Network, cont’d

s2 − T ( s) =



ω0 s  α 1   −  + ω 02 1 − α  qˆ q z 

 1 k'  s 2 + ω 0 s − 32  + ω 02  qˆ 1 − α 

All-pass:

1  α 1   1 k '32   − = −  1 − α  qˆ q z   qˆ 1 − α 

Slide 8.113 presents the comprehensive transfer function T(s) of the II-DF-4 biquad. It is capable of realizing any kind of non-minimum-phase networks (i.e., zeros in the RHP, see Slide 2.28 in Chap. 2). The slide contains the design conditions for an all-pass filter (zeros mirrored on the jω axis).

8  A Classification of Single-Amplifier Biquads

226

Slide 8.114 contains the conditions for a notch filter (zeros on the jω axis) and for a general biquad (zeros anywhere in the right-half s-plane).

Example: II-DF-4 Biquad Network

α 1 − =0 qˆ q z

Notch Filter: General Biquad: ±

ω α 1  ωz =± 0  −  1 − α  qˆ q z  qz

ωp

 1 k'  = ω 0  − 32  qp  qˆ 1 − α 



Note:

k '32 1 < 1 − α qˆ

Examples of Type II All-pass/Notch Biquads*: (1-k)g

kg C/m

nR

C/n

(1-k)g R

A

(a) (1-k)g

C/n

A

mR

C

mR R

C

kg

(b) (1-k)g

kg A

mR

R

kg mR

A

C/n C

R (c)

C

nR

C/m C/n (d)

*Paper by Norbert Fliege

Slides 8.115, 8.116, 8.117, 8.118 and 8.119 give further examples of Type II biquads, some SF, others DF. As already pointed out, the latter affects only the poles.

8  Summary of Classification of Single-Amplifier Biquads (SABs)

227

Examples of Type II All-pass/Notch Biquads, cont’d (1-k)g

kg

R

C R

C/n C/m

C/n mR

(e)

nR

A

C

C/n

R

kg

R

A

mR

C

(1-k)g

(f)

C

C/n

mR

mR

fg

fg

A

A

(1-f)g

(1-f)g (h)

(g)

Examples of Type II All-pass/Notch Biquads, cont’d C/n

R

mR

C fg

A (1-f)g (i)

(1-k)g

kg

mR C/n

(1-k)g

A

nR

Rb

C/m C

R (a)

Rc

Ra

Ra

Rb

Rc

kg C/n

A mR

C (b)

8  A Classification of Single-Amplifier Biquads

228

Examples of Type II All-pass/Notch Biquads, cont’d (1-k)g

kg mR

R

C

kg

Rb

A

mR C

C/m

Rc

Ra

(1-k)g

A

Rb

C/m Rc

C/n

(c)

Ra

(d) (1-k)g

kg A

R

Rb

Rc C/n

C

Ra

(e)

Examples of Type II All-pass/Notch Biquads, cont’d R

C C/n

mR

C/n

R

mR

C fg

A

fg

A

[1-(f+k)]g

kg

[1-(f+k)]g

kg

(f)

(g)

[Note: Some of the circuits only “approximately” fit into our classification]. Note that some of the examples shown in these slides do not seem to fit into our classification because the passive RC circuit is grounded and not part of a frequency-dependent feedback loop. The feedback in these biquads is negative and constant and used to stabilize the closed-loop gain of the inverting opamp. Examples are the biquads in Slide 8.115 and on the top of Slide 8. 116. The reason for the lack of frequency-dependent feedback is that these biquads do not have complex-conjugate poles; they lie on the negative-real s-plane axis. However, these biquads fit into our classification in as much as they are Type II and have symmetrical poles and zeros on the negative- and positive-real s-plane axis. This makes them biquads whose poles and zeros are real; they are used as all-pass, delay, or other phase-shaping networks. Because they do not have complex-conjugate poles, they cannot, by definition, belong to one of the network classes of our classification.

Chapter 9

A Morphological Approach to the Design of Active Network Elements

Slide 9.1

Chapter 9 A Morphological Approach to the Design of Active Network Elements

1

Introduction

This chapter continues with the methodology of circuit design. Here, however, we are dealing with a general method of design which is known as morphological analysis (Wikipedia: “A method for systematically structuring and investigating the total set of relationships contained in multidimensional, usually non-quantifiable, problem complexes”). This method was developed by a Swiss/American physicist and inventor, Prof. Fritz Zwicky (1898–1974). It was successfully used by him in a variety of scientific and engineering fields ranging from astrophysics (discovery of “supernovae”) to the invention of airplane jet engines for takeoff on extremely short runways (so-called jato, i.e., “jet-­ assisted takeoff” system for aircraft carriers). Among many other areas of invention and ­problem-­solving, the morphological method was also used by him after he was recruited as advisor to the US government in order to solve the myriad problems encountered in rebuilding Europe, physically and ideologically, after the destruction of WWII. © Springer Nature Switzerland AG 2019 G. S. Moschytz, Analog Circuit Theory and Filter Design in the Digital World, https://doi.org/10.1007/978-3-030-00096-7_9

229

230

9  A Morphological Approach to the Design of Active Network Elements

In this chapter we present an adaptation of the morphological method for the design of analog circuits, devices, and systems. It is shown, for example, that various groups of circuits and devices, including gyrators, impedance converters, and active filters, can be directly and simultaneously obtained using the morphological design method. Many of the morphologically derived circuits and devices are new; others were previously invented by conventional means and described in separate publications. When using the morphological method, these and new circuits were obtained in whole groups and, as we shall see, almost automatically. One of the key tools of the morphological design method is Zwicky’s so-called morphological box which determines the “basic elements” of a problem and then combines variations of these basic elements into resulting groups of potential solutions. One group may well consist of “solutions” that are already known. Another of “solutions” which may not yet be known but hold promise and deserve further investigation with respect to their feasibility and adherence to initially specified constraints. A third group may be found to be physically unrealizable or incompatible with previously defined requirements and therefore not qualified for further attention. Ultimately, it is up to the expertise, as well as the creativity and ingenuity of the “thoroughly unbiased and open-minded” designer (Zwicky), to be able to distinguish between these three categories. This is followed by a critical decision, namely, which of them is worth pursuing and reducing to practice and which of them deserves neither further time nor development effort.

2

Zwicky’s Method of General Morphological Analysis

Conventional Problem Solving FORMULATION OF PROBLEM

SEARCH FOR SOLUTION TO PROBLEM

DOES SOLUTION MEET PROBLEM REQUIREMENTS

YES SOLUTION

NO

In Slide 9.2 we begin by considering a flowchart describing the conventional flow of “problem-­ solving.” The chart speaks for itself. It shows that after formulating a problem, a solution is looked for, tested for acceptability, and, if the answer is positive, applied. If the answer is negative, it means starting again, i.e., “going back to the drawing board.”

2  Zwicky’s Method of General Morphological Analysis

231

Slide 9.3 demonstrates a flowchart for Zwicky’s method of “morphological STEP 1 FORMULATION problem-solving.” It starts OF PROBLEM by formulating the prob1 & 2 are lem in step 1 while simulthe taneously (step 2) Most Critical identifying what we call Steps ! the main or basic elements STEP 3 of the problem. What MORPHOLOGICAL BOX (M-dimensional) exactly is meant by this will best be understood p solutions S 1 S 2 S3 Sµ S p from the examples that folsµ low below. STEP 4 REALIZABILITY In many cases it is posCONSTRAINTS AND PROBLEM SPECFICATIONS sible to limit the number of basic elements to three, S1 S2 Sυ Sq q solutions sν each of which has soq≤p called element alternaSTEP 5 PROGRESSIVE ELIMINATION tives. For example, if one FOR OPTIMUM SOLUTION(S) of the basic elements is a r optimum solutions sk “liquid,” then it is easy to S1 Sk Sr r ≤q see that within the subject matter at hand, e.g., food (water, milk, oil, juice, etc.), there is a large number of element alternatives. With three basic elements, the so-called morphological box (Zwicky’s designation) can be constructed (step 3), meaning that the element alternatives are listed along each axis of a three-­dimensional box. This leads to a number of “solutions” (step 4), each of which must be examined for its conformity to the realizability constraints and to the specifications of the problem. Step 5 then produces a final number of acceptable solutions, each of which must be examined individually for its feasibility within the framework of the problem at hand. Note that, although the morphological box is the most common, there are other methods of structuring the investigation, e.g., when there are more than three basic elements. These include the morphological tree and the morphological matrix, as will be shown in examples below. It must be pointed out that the morphological method is neither a shortcut around nor a substitute for creativity, basic knowledge, and expertise. However, it provides a framework and procedure for problem-solving that may be helpful in channeling and guiding ideas that are supportive of additional, often unexpected, and unpredictable solutions to a problem. It is also important to realize that because the problem formulation, and the choice of basic elements, is largely responsible for the outcome of a design, and the solution(s), to a problem, it is these two steps that are the most critical and important and deserving of the most initial attention and thought. It is possible that a problem may benefit from having more than one solution, as supplied by the morphological method. However, more often than not, the designer or problem-solver must use his/ her judgment and expertise to decide on the optimum solution from the number of final acceptable solutions presenting themselves.

Morphological Problem Solving

9  A Morphological Approach to the Design of Active Network Elements

232

P33 P32 P31

3 Basic Elements:

P11

P12

P13

P14

P15

P21

E3

E1, E2, E3, P22 and their Alternatives P23

Finally, before going to some illustrative examples, Slide 9.4 shows an example of a general morphological box made up of three basic elements E1, E2, and E3 and their element alternatives.

P24 P25

E2 E1

3

General Examples of Morphological Design

In this section we digress briefly from circuit theory and design to demonstrate the universality of the morphological method of problem-solving. We have deliberately selected examples that are very far from the subject matter of this book, in order to drive home the universality of the method. However, this section will also explain why it is very natural and useful to use this method for the novel solution of the design problems that are within the context of this book – as we shall show in the sections following this one.

Morphological Problem Solving STEP 1 FORMULATION OF PROBLEM

Example:

Methods of Transportation

STEP 3 MORPHOLOGICAL BOX (M-dimensional)

p solutions sµ

q solutions sν

S

S

S

S

S

S

S

S

S

q≤p r optimum solutions sk r ≤q

S

S

S

Slide 9.5 shows the flowchart for the problem of morphologically designing transportation systems, defined as power vehicles used to transport objects from one place to another.

3  General Examples of Morphological Design

233

Slide 9.6 shows the corresponding morphological box and the three basic elements of the problem. z 3 2 They are E1, or x, the type of vehicle (e.g., car, chair, sling, bed, etc.); E2, or y, the type 3 Basic of motor power (e.g., elements: x,y,z compressed air, chemy 1 4x7x8= 224 Solutions ical energy, electric, x x:Type of Vehicle: Car, Chair ,Sling, Bed... steam, magnetic force, Y:Type of Motive Power: Compressed air, Chemical energy, Electric, Steam, cable, belt, etc.); and Magnetic force, Cable, Belt,…. E3, or z, the type of z:Type of Medium: Air, Water, Oil, Hard surface, Friction surface, Rollers, Rails,… medium along which 1:Bed-type vehicle, driven by compressed air, operating on rails the vehicle is to be 2: Sling-type vehicle, powered electrically, operating in an air medium moved (e.g., air, 3:Bed-type vehicle, powered by a cable, operating in a water medium water, oil, hard surface, friction surface, rollers, rails, etc.). The total number of cells, or potential solutions, for this example is 4 × 7 × 8 = 224. Each cell can initially be considered as a basis for exploration. It may turn out that by doing so, some new transportation system will come to mind. In our example, three random “solutions” are shown: (1) a bed-type vehicle, driven by compressed air, operating on rails; (2) a sling-type vehicle, powered electrically, operating in an air medium; and (3) a bed-type vehicle, powered by a cable, operating in a water medium. Whether these “solutions” are worthy of further investigation is of no consequence here. They are merely intended to show how the method of morphological problem-solving – and in this case the morphological box – can stimulate creativity and the thought process. The method can lead to new solutions that simply never occurred to anyone before or that conventional thinking or rigor had not permitted consideration before. Thus, the creative aspect of invention is not eliminated by use of the morphological box, but it may afford a systematic way of causing new and feasible systems to be considered. We summarize the method of the morphological box with the following points: Methods of Transportation, i.e., getting something from one place to another via a powered vehicle..

• Determine the major dimensions, i.e., the basic elements Ei, of the problem. This is one of the most far-reaching and intricate steps because it determines the steps that follow and the outcome of the resulting solutions. • These dimensions become the basis for the space in which the box, or other means of graphic visualization, is constructed. The most common box is three dimensional. However, one can also work with a two-dimensional matrix or with a higher-dimensional box or tree. Later examples will demonstrate what this means. • The three-dimensional morphological box is a convenient communication tool and a good approach to organizing ideas. However, beyond three dimensions, the morphological box can often better be replaced and visualized by a relevance tree. • Once the major dimensions, or basic elements, Ei, are identified, each dimension can be partitioned into a set of descriptive coordinate entries which are the basic element alternatives. Further examples will illustrate this point.

9  A Morphological Approach to the Design of Active Network Elements

234

Morphological Problem Solving STEP 1 FORMULATION OF PROBLEM

Example:

Design of Beer Bottles

STEP 3 MORPHOLOGICAL BOX (M-dimensional)

p solutions sµ

S1

S2

S3

Sµ

Slide 9.7 presents our next rather far-flung example of morphological problemsolving. It deals with the prosaic problem of designing beer bottles.

or Matrix, Tree….etc

Sp

STEP 4 REALIZABILITY CONSTRAINTS AND PROBLEM SPECFICATIONS

q solutions sν

S1

S2

Sυ

Sq

q≤p r optimum solutions sk r ≤q

STEP 5 PROGRESSIVE ELIMINATION FOR OPTIMUM SOLUTION(S) S1

Sk

Sr

Slide 9.8 demonstrates that even when considering a familiar object such as a beer bottle, it is possible to use the morphological method to produce many of the forms in existence and to produce some completely new ones. The method generates a large number of alternatives; however, it still remains for the designer to sort out which alternatives to pursue.

3  General Examples of Morphological Design

235

Slide 9.9 shows that rather than using a three-dimenMorphological Chart for Beer Bottle sional box for this design problem, it is more convenient to derive a two(SOME) dimensional chart or TALL SHORT TRIANGULAR SQUARE CURVED matrix. Selecting one 12oz 20oz 8oz 6oz 90oz KEG. alternative from each attriMANY bute, we can define both BLUE CLEAR BROWN YELLOW CRYSTAL COLORED old and new configuraSMOOTH PEARLE ETCH SANDBLAST ROUGH STRIPED tions. Thus, the standard “stubby” bottle is short, 12 CAP TWIST PULL PLASTIC CORK GLASS oz., brown, smooth, and with a cap. A completely new configuration would be curved, 6 oz., crystal, etched, with pull top. Obviously this chart holds the key to many new and unusual beer bottle configurations.

Morphological Problem Solving STEP 1 FORMULATION OF PROBLEM

Example:

Methods of Energy Conversion

STEP 3 MORPHOLOGICAL BOX (M-dimensional)

p solutions sµ

S

S

S

S

S

STEP 4 REALIZABILITY CONSTRAINTS AND PROBLEM SPECFICATIONS

q solutions sν

S

S

S

S

q≤p r optimum solutions sk r ≤q

STEP 5 PROGRESSIVE ELIMINATION FOR OPTIMUM SOLUTION(S) S

S

S

or Matrix, Tree….etc

Slide 9.10 presents our next example, which is somewhat closer to home. It is that of finding methods of energy conversion.

9  A Morphological Approach to the Design of Active Network Elements

236

Slide 9.11 demonstrates that with the rather limited number of basic elements at our disposal (as far as this example goes), it again suffices here to draw a two-dimensional matrix. The basic elements are E1, initial energy form; E2, transmission form; and E3, final (storage) form. The same five basic element alternatives are valid for each. They are kinetic, elecEnergy Conversion Matrix : 1. K-E-C hydroelectric generation stored in trical, chemical, thermal, battery, 2. C-T-K internal combustion engine leading to energy being stored in a fly wheel; E-C-T refrigerator and nuclear. Three solutions that readily emerge from the energy conversion matrix are K-E-C, i.e., hydroelectric generation stored in a battery; C-T-K, i.e., an internal combustion engine leading to energy being stored in a fly wheel; and E-C-T, i.e., a refrigerator. Again, this is neither the place, nor is it our intention, to plunge deeper into the evaluation of these energy conversion systems. 3 Basic Elements

Morphological Problem Solving Design of a Forklift Truck

STEP 1 FORMULATION OF PROBLEM

Example:

STEP 3 MORPHOLOGICAL BOX (M-dimensional)

p solutions sµ

S

S

S

S

S

STEP 4 REALIZABILITY CONSTRAINTS AND PROBLEM SPECFICATIONS

q solutions sν

S

S

S

S

q≤p r optimum solutions sk r ≤q

STEP 5 PROGRESSIVE ELIMINATION FOR OPTIMUM SOLUTION(S)

S

S

S

Finally, in Slide 9.12, we round off the gamut of illustrative examples, by considering a rather more exotic and complex problem, namely, that of designing a forklift truck.

3  General Examples of Morphological Design

237

Slide 9.13 shows a variety of the so-called forklift trucks – needless to say, there are many more.

Slide 9.14 demonstrates that due to the many critiElements cal parameters, or basic Support Wheels Air Tracks Slides Spheres cushion elements, that determine the functioning of a forkSteering Turning Rails Air wheels thrust lift truck, here again a Stopping Reverse Brakes Block under Drag a chart or matrix of its basic power wheels weight on features is a useful structhe floor ture with which to morMoving Air thrust Power to Hauling along Linear phologically examine wheels a cable induction motor element combinations Power Electric Bottled Petrol Diesel Steam (here referred to as paramgas eters). As the pictures in TransHydraulic Gears and Belts or Flexible Slide 9.13 show, and the mission shafts chains cable many basic elements and Lifting Screw Hydraulic Rack and Chain or element alternatives makram pinion rope hoist ing up the final product Operator Seated Seated at Standing Walking Remote imply, the matrix will proat front rear control duce a very large number Morphological Chart for the Main Requirements of a Forklift Truck of possible “solutions.” Two, indicated by the blue and green dots, are shown in the matrix of this slide. We have digressed above, in order to show a variety of general design problems and how to use the morphological method to solve them. In what follows, we shall now apply this method to some of the real, and by no means trivial, problems encountered in the subject matter of this book. Foremost Basic

Parameters

Possible solutions

Element Alternatives

9  A Morphological Approach to the Design of Active Network Elements

238

among them is the design of devices which were introduced in theory, e.g., in matrix form or other network-theoretical terms, in Chap. 6. No indication was given there if, and how, they could actually be realized in circuit or device form. As we shall now show, such problems are no less suitable for their solution with the morphological method than the general problems given above. Indeed, the morphological method is not only an ideal starting point, but, as we shall see, it is hard to imagine any other methods that will provide not one but many realistic and feasible solutions for the type of design problems encountered in circuit and device design.

4

A Morphological Approach to the Design of Gyrators

Because the gyrator was invented and postulated as a theoretical, hypothetical, and even pathological network element by B.D.H. Tellegen in 1948, without any real thought of its ever seeing the light of day as a physical device, we deliberately begin with the morphological approach to the design of the gyrator. [It is considered “pathological,” because it contradicts a fundamental principle of network theory, being at the same time passive but nonreciprocal.] In fact, it took 17 years before B.A. Shenoi1 published a paper in which he described “a practical realization of a gyrator circuit.” Small wonder that the gyrator was not taken seriously for several decades; it was considered physically barely realizable. This makes it an ideal device to test the efficacy of design by the morphological method. To begin the design of the gyrator, morphological or otherwise, we do well to review some of its basic characteristics. For this we need to recall some fundamental principles of two-port theory.

A general two-port given by its [ABCD]matrix (designated by [A]) and terminated with a load impedance ZL has the input impedance: I2

I1

 A B C D   

V1

ZIN

V1 = AV2 − BI 2 I1 = CV2 − DI 2 Z IN =

ZL =

V2 − I2

V2

A B  C  D

[ A] = 

ZL

As shown in Slide 9.15, from Chap. 6, we know that the input impedance of a general two-port, terminated with a load impedance ZL, can be expressed in terms of its transmission or [ABCD] matrix, as follows: V AZ L + B Z IN = 1 = (9.1) I1 CZ L + D

V1 AZ L + B = I1 CZ L + D

B.A.Shenoi: ‘Practical Realization of a Gyrator Circuit and RC-Gyrator Filters’, IEEE Transactions on Circuit Theory, 12(3):374–380, October 1965. 1 

4  A Morphological Approach to the Design of Gyrators

239

A Gyrator is a two-port whose transmission or [ABCD] matrix is characterized by A=D=0:

A [ A] =  C

B  0 1 g   = D   g 0   

Slide 9.16 repeats the basic feature of the theoretically defined gyrator, namely, that in terms of the [ABCD] matrix of its twoport representation, A  =  D  =  0, or in the [y] matrix, y11 = y22 = 0.

and whose [y] matrix is characterized by y11=y22=0:

 0 g1    2 0

[ y ] = − g

For the loaded Gyrator (with the symbol shown):

I1

Z IN =

With A=D=0

B 1 C ZL

The Gyrator is an Impedance Inverter With :

0 1  g  g 0 

[ A] = 

Z IN =

1 g 2ZL

•Invented in 1948 by B.D.H.Tellegen: ‘ The Gyrator, a New Network Element‛ Philips Res. Report. •First Realization: B.A.Shenoi 1965 [17 years later!]: ‘A Practical Realization of a Gyrator Circuit…‛ IEEE Trans. On Circuit Theory,CT 12, pp374-380

Inductor Simulation with a Gyrator: Ideal Gyrator: Inverter (A=D=0) I1

0 1  g  g 0 

[ A] = 

Z IN =

1

1

≡ Zin

2

B 1 = 2 C g ZL

Z in = sLeq = s

CL g2

2

Simulation of an inductor by an equivalent gyrator-C combination

Slide 9.17 shows the symbolic representation of the gyrator and the fact that, as a two-port in terms of its [ABCD] matrix, and seen from its input terminals, its input impedance inverts the load at its output terminals. This makes it an impedance inverter, i.e., ZIN is proportional to ZL−1. More specifically, ZIN  =  (g2ZL)−1, where g is the gyrator conductance with dimension Ω−1 and ZL is the load impedance.

Slide 9.18 shows that if the gyrator is loaded by a capacitor, it plays the role of an equivalent inductor Leq. In the realm of integrated circuit design for analog inductorless circuits and filters, this role is essentially the “raison d’être” of the gyrator.

9  A Morphological Approach to the Design of Active Network Elements

240

Example of Gyrator Simulation of an Inductor R5 C1 (500Ω) (1.54nF)

LCR High-pass Filter

Equivalent Gyrator-RC Circuit

V1

Vin

C2

~

L2

500Ω 1.54nF

C3 (1.54nF)

(6.1nF) (618.4µH)

RL (500Ω)

Slide 9.19 shows an LCR high-pass filter and, below this, the equivalent highpass filter, with a gyratorcapacitor combination replacing the inductor.

1.54nF

6.1nF

~

500Ω 618.4⋅g2F

4.1

Transistor Design of Gyrators

In Slide 9.20 we are now ready, after this brief Morphological Problem Solving review of gyrator basics, to STEP 1 Transistor Design of FORMULATION Example: OF PROBLEM Gyrators approach the problem of designing a transistorized gyrator, using the morphological method. As has been emphasized above, STEP 3 MORPHOLOGICAL BOX the formulation of the (M-dimensional) problem with the morphop solutions S S S S S sµ logical method is critical – STEP 4 as, in fact, it is with most REALIZABILITY CONSTRAINTS AND PROBLEM SPECFICATIONS other design methods; it S S S S q solutions sν has a fundamental influence on the outcome of the q≤p STEP 5 solution. PROGRESSIVE ELIMINATION FOR OPTIMUM SOLUTION(S) Before we formulate r optimum solutions sk S S S the problem, it is useful to r ≤q recapitulate our design sequence for analog devices; it generally leads to a feasible solution. The design sequence is as follows: 1 . Describe the device in terms of one of the known two-port matrix parameters. 2. Rearrange the two-port parameters as the interconnection of the two-port parameters of controlled sources. [In Chap. 6 we pointed out that most devices can be represented in terms of controlled sources; these are often considered the basic elements making up an analog device.] 3. Replace the controlled sources by their nullator-norator equivalent representations.

4  A Morphological Approach to the Design of Gyrators

241

4. In the case of a transistorized device, replace individual nullator-norator pairs by transistors, taking care that only pairs are considered that have a common nullator-norator node. 5. In the case of a realization with opamps, replace nullator-norator pairs with opamps. Here there should be no common nullator-norator nodes. 6. Complete the biasing and other functions of the circuit according to standard principles and circuit-­ design rules. In Slide 9.21, starting with point 1 above, the gyrator is defined by its [y] matrix. The [y] matrix was found to be the most suitable for the derivation and design of a transistorized version of a gyrator.

Morphological Problem Solving STEP 1 FORMULATION OF PROBLEM

STEP 3 MORPHOLOGICAL BOX (M-dimensional)

p solutions sµ

S

S

S

S

S

STEP 4 REALIZABILITY CONSTRAINTS AND PROBLEM SPECFICATIONS

q solutions sν

S1

S2

Sυ

Sq

q p ≤

r optimum solutions sk r ≤q

STEP 5 PROGRESSIVE ELIMINATION FOR OPTIMUM SOLUTION(S)

S1

Sk

Sr

Slide 9.22 indicates that we must prepare to define the basic elements of the gyrator.

9  A Morphological Approach to the Design of Active Network Elements

242

EXAMPLE: Transistor Design of Gyrators, cont’d

Slide 9.23 shows a twoport representation of the gyrator in terms of its [y]-matrix.

TWO-PORT REALIZATION OF (1) I1

I2

V1

V2

As shown in Slide 9.24, and following point 2 EXAMPLE: Transistor Design of above, the two-port repreGyrators, cont’d sentation of the gyrator in the previous slide can now CONTROLLED-SOURCE REALIZATION OF (1) NON-INVERTING VOLTAGE-CONTROLLED CURRENT SOURCE E1 be broken up into the parallel connection of an inverting and a non-­ V inverting voltage-controlled current source V V (VCS). These become the two basic elements, E1 and V E2, for the transistorized design of a gyrator. In terms of the flowchart, this INVERTING VOLTAGE-CONTROLLED CURRENT SOURCE E2 means that m  =  2. [Note Two basic elements required for Gyrator design, namely ∴ m = 2 that, with a different probtwo VCS’s of opposite polarity. lem formulation, the basic elements are different. We shall see this later in the context of this chapter.] 2

1

2

1

4  A Morphological Approach to the Design of Gyrators

Voltage-Controlled Current Source

VCS

Non-inverting VCS with Single Nullator-Norator

243

Inverting VCS with Two NullatorNorator Pairs

with Two NullatorNorator pairs Non-inverting

Inverting

Non-inverting

R

V1

V1

Non-inverting VCS

R

I1 = 0

I1 = 0

I 2 = gV1

I2 =

V2 V

V

1 V1 R

V

V

R

I1 = 0

I1 = 0

1 I 2 = − V1 R

I2 =

1 V1 R

In Slide 9.25 we have a list of VCSs in terms of nullators and norators (see Slide 6.99 in Chap. 6) both inverting and non-inverting.

Equivalent Nullor Circuits for Voltage Controlled Current Sources VCS with noninverting transconductance (Element E1) E11 V1

E21 V2

R

1 g1 = R

R

V1

E12 V1

VCS with inverting transconductance (Element E2) g2 = −

V2

1 R

E22 V2

R

n1 ALTERNATIVES FOR ELEMENT E1

.. . E1 j .. .

E1n1

g1 =

1 R

V1

R1

R0

R2

V2

g2 = −

.. . n2 ALTERNATIVES FOR E.2 j .. ELEMENT E2

R0 R1R2

E2n2

Slide 9.26 shows that these two VCSs make up the available element alternatives, E1i and E2j, i.e., m = 2.

9  A Morphological Approach to the Design of Active Network Elements

244

Morphological Problem Solving STEP 1 FORMULATION OF PROBLEM

STEP 3 MORPHOLOGICAL BOX (M-dimensional)

p solutions sµ

S1

S2

S3

Sµ

We are now ready for a morphological representation, as implied in Slide 9.27.

Matrix

Sp

STEP 4 REALIZABILITY CONSTRAINTS AND PROBLEM SPECFICATIONS

q solutions sν

S1

S2

Sq

Sυ

q≤p STEP 5 PROGRESSIVE ELIMINATION FOR OPTIMUM SOLUTION(S)

r optimum solutions sk r ≤q

S1

Sk

Sr

The Morphological Box for Transistorized Gyrator Design n2 ELEMENT ALTERNATIVES E2 n1

A L T E R N A T I V E S E1

E22

E11

s11

s21

E12

s21

s22

. . . . .

E1n1

E2n2

. . . . .

E21

. . . . .

E L E M E N T

sµ

sn1n2

•TWO-DIMENSIONAL BOX BECAUSE m=2, i.e., TWO BASIC ELEMENTS (VOLTAGE CONTROLLED CURRENT SOURCES) REQUIRED PER GYRATOR •NUMBER OF POSSIBLE GYRATOR REALIZATIONS ACCORDING TO MORPHOLOGICAL BOX:

n1 ⋅ n2 ⋅ n3 (!)

Slide 9.28 shows the resulting morphological matrix which, as we shall see below, is actually a three-dimensional box, because for each of the solutions Sμ, there is more than one sub-solution.

4  A Morphological Approach to the Design of Gyrators

245

EXAMPLE 1: Gyrator Solutions S11 S11 CONSISTS OF VCS’s E11 AND E21 IN PARALLEL N1

N1 R1

V1

R2

N3

N3

N2

1 R1   − 1 R2 0  

[y] = 

V2

N2

0

GYRATOR ADMITTANCE MATRIX

EXAMPLE 1: Gyrator Solutions S11, cont’d GYRATOR REQUIRES 3 NULLORS EACH NULLOR MUST HAVE COMMON TERMINAL (TRANSISTORIZATION!) THE 3 NULLORS OF A SOLUTION (GYRATOR) MAY HAVE NEITHER A NULLATOR NOR A NORATOR IN COMMON, I.E., NEITHER A ROW NOR A COLUMN IN COMMON N1

N2

N3

N1

N1N1

N1N2

N1N3

N2

N2N1

N2N2

N2N3

N3

N3N1

N3N2

N3N3

: Solution (S11)1 : Solution (S11)2

Slide 9.29 shows the first solution to our gyratordesign problem, namely, S11 and the corresponding admittance matrix of the resulting gyrator. Now, when pairing nullators and norators for a transistorized circuit solution, more than one solution becomes evident because there is more than one way of combining the nullors.

Slide 9.30 illustrates the guidelines for pairing the nullators and norators (nullors) and carries out the actual pairing. The guidelines are: (i) A transistorized gyrator requires (at least) three nullors. (ii) For a transistorized gyrator, each nullor must have a common terminal. (iii) The nullator-norator pairing step is not unique. This creates a sub-matrix of nullators and norators which actually produces a

third dimension of solutions, i.e., a morphological box. (iv) The three nullors of a solution (gyrator) may have neither a nullator nor a norator in common, because the same nullator or norator cannot be used for more than one transistor. This means that a solution may have neither a row nor a column of the matrix in common. We now see in this slide that the solution S11 contains two valid solutions, namely, (S11)1 and (S11)2.

9  A Morphological Approach to the Design of Active Network Elements

246

Transistorized Gyrators (S11)1 and (S11)2, cont’d NULLOR

TRANSISTOR

C

C

In Slide 9.31 we recall the equivalence between a nullor and a transistor, in order to covert from a nullor circuit to a transistorized circuit.

N N

B

B E

E

Transistorized Gyrators (S11)1 and (S11)2 NULLOR GYRATOR S11 N1

N1 R1

V1

N3

R2 N3

N2 N2

V2

In Slide 9.32, we once again show the nullor gyrator solution S11 that is to be transistorized.

4  A Morphological Approach to the Design of Gyrators

247

Transistorized Gyrators (S11)1 and (S11)2, cont’d N1

N1

NULLOR GYRATOR S11

R1

V1

N3

GYRATOR (S11 )1 = (N1 N1 ,N2 N 2 ,N3 N 3 )

R2 N3

N2

V2

N2

(N1,N1)

R2 (N3,N3)

(N2,N2)

R1

GYRATOR (S11 )2 = (N1 N 2 ,N2 N 3 ,N3 N1 ) R2

(N3,N1)

R1

(N2,N3)

In Slide 9.33 we have the two transistorized circuit solutions contained in the nullor gyrator of the previous slide. Clearly, in their present form, these two circuits provide only skeletons of the final two transistorized circuit solutions.

Morphological Problem Solving STEP 1 FORMULATION OF PROBLEM

STEP 3 MORPHOLOGICAL BOX (M-dimensional)

p solutions sµ

S1

S2

S3

Sµ

Sp

STEP 4 REALIZABILITY CONSTRAINTS AND PROBLEM SPECFICATIONS

q solutions sν

S1

S2

Sυ

Sq

q p ≤

r optimum solutions sk r ≤q

STEP 5 PROGRESSIVE ELIMINATION FOR OPTIMUM SOLUTION(S) S1

Sk

Sr

Slide 9.34 now reminds us of step 4 above, namely, that having obtained these two, and the many additional solutions hidden in the morphological matrix, or box, of Slide 9.28, we must now apply our knowhow and expertise in circuit design to select the solution that best satisfies the required specifications and technology constraints imposed on the design.

248

9  A Morphological Approach to the Design of Active Network Elements

Evaluation of Solutions Si,j REALIZABILITY CONSTRAINTS

(E.G. - TRANSISTOR BIASING - PNP-NPN COMBINATIONS - IC - TECHNOLOGY)

Slide 9.35 summarizes some of these constraints and succinctly formulates some of Zwicky’s suggestions and advice to the designer/inventor as follows:

• The realizability and originality of the solutions (E.G. - y11=y22=0 depend on the experience - MINIMUM POWER) and ingenuity of the designer/inventor. It folREALIZABILITY AND ORIGINALITY OF lows that the morphologiSOLUTIONS DEPENDS ON EXPERIENCE AND cal method does not INGENUITY OF DESIGNER (INVENTOR). provide a shortcut around serious technical educaIMPORTANT: ‘UNBIASED APPROACH TO tion and experience. PROBLEM SOLVING’ (Zwicky)! • Most important is an “unbiased and unprejudiced approach to probAlways ask: WHY NOT?! lem-solving” (one of Zwicky’s key principles). • The designer/inventor should not take for granted what has been done in the past but go beyond the stage where others have stopped. • Always ask: Why not?!

PROBLEM SPECIFICATIONS

We have pointed out above that gyrator design remained elusive and that whenever a design was accomplished, it was proudly published. Thus, for example, our solution (S11)1 was a circuit published in Electronic Letters, in 1967.

4  A Morphological Approach to the Design of Gyrators

249

Realization of Gyrator Solution (S11)1 W.H. HOLMES, S. GRUETZMANN, W.E. HEINLEIN, Electron. Letters, Vol. 3, No. 2, 1967

+V +V

I1

Vout

(N3,n3)

CURRENT SOURCES +V

+V

I2

R2

≡

I1

Re

-V

+V

(N1,n1)

Vin

R1

-V

(N2,n2) I2

-V -V

-V

In Slide 9.36 the basics of the practical design are outlined. Nevertheless, since it is not the purpose of this chapter to go into the details of designing viable gyrators, we shall leave the further details of final design for other sources of design information (note NiN̅ j and Ninj are interchangeable). Note that a considerable number of solutions will result from the morphological design method. Each is deserving of serious consideration; some will be disqualified for one reason or another. Yet, given the previous dearth of practical gyrator circuits, the results achievable with the morphological approach are remarkable. To demonstrate this a little more, we shall examine another one of the many solutions available from our morphological matrix/box.

Example 2: Gyrator Solution S12 S12 CONSISTS OF VCS’s E11 AND E22 IN PARALLEL N3

N3 R3

V1

N1 R1

GYRATOR ADMITTANCE MATRIX

 0  [y] =    R0   −    R1 R2 

N1

N2

V2

N2 R2

R0

1 R3   0  =  − 1 R3 0  RR1==RR0 2

3

1 R3  0 

In Slide 9.37 we consider solution S12 which consists of VCSs E11 and E22 in parallel. The corresponding gyrator admittance matrix is given at the bottom of the slide.

9  A Morphological Approach to the Design of Active Network Elements

250

Slide 9.38 shows that solution S12 contains the two solutions (S12)1 and (S12)2 based on the reasoning given in Slide 9.30.

Example 2: Gyrator Solution S12, cont’d N1

N2

N3

N1

N1N1

N1N2

N1N3

N2

N2N1

N2N2

N2N3

: Solution (S12)1

N3

N3N1

N3N2

N3N3

: Solution (S12)2

Transistorized Gyrators (S12)1 and (S12)2 NULLOR GYRATOR S12  0  [y] =    R  −    RR 

1 R   0  =  − 1 R 0  RR ==RR

1R 0 

(N ,N ) (N N ) (N ,N )

GYRATOR (S12 )1 = (N1 N1 ,N2 N 2 ,N3 N 3 )

R

(N ,N )

R

R

R

(N ,N )

GYRATOR (S12 )2 = (N1 N 3 ,N2 N1 ,N3 N 2 )

R

(N ,N ) (N ,N ) R R

R

Slide 9.39 shows the two circuits corresponding to solutions (S12)1 and (S12)2.

5  A Morphological Approach to the Design of Opamp-Based General Impedance…

N2,N2’ N1,N1’ 1

R1

R0

2

T1

1 1’

1’

R2

R0

R2

N3,N3’

T8 ,T10 (p-n-pn-p-n)

N2,N2’ T7,T9 (p-n-pn-p-n)

N1,N1’

T4

2

T2

2’

251

T5

2’

T3 (n-p-n)

R1

N3,N3’

T6

R3

R3 V R0 1

T7

T1

T9

Input

Integrated Circuit Realization of Gyrator (S12)1 ( Bell Labs, 1970’s)

1’

R2

RDC T8

(N2,N2)

(N1,N1) T2

R1

T3 D1 D2

T4 2 2’

C7

T5

T6

(N3,N3)

D3

D1=D2=D3=D4=D5 =

T10

D4 D5

R3 =

V C7=1300pF

Slide 9.40 shows the gyrator circuit corresponding to solution (S12)1, completed for practical operation, including biasing and choice of transistors. This is a gyrator circuit that was realized as an integrated circuit and described in an unpublished memorandum at Bell Telephone Laboratories in the 1970’s.

5

 Morphological Approach to the Design of Opamp-Based A General Impedance Converters (GICs), Gyrators, and FDNRs

As was pointed out above, the formulation of a problem, and the way the solution is approached (“ansatz,” as explained in Chap. 3, Slide 3.13), critically determines the outcome of the solution. In this section, the emphasis is on designing gyrators and other devices with opamps rather than with transistors. As we shall see, this suggests a different formulation of the design.

9  A Morphological Approach to the Design of Active Network Elements

252

Remember:

Impedance Converters (B=C=0),

General Impedance Converter (GIC)

I1 k(s) V1 A( s ) 0    0 D( s)

[ A] = 

ZL

Z IN = k ( s ) Z L

k ( s) =

A( s ) D( s)

In Slide 9.41, it is useful, before we begin, to recall the definition of the general impedance converter (GIC). Basically, it is a two-port for which B=C  =  0  in the [ABCD] matrix and for which the remaining terms, A(s) and D(s), are frequency dependent. Thus, the input impedance to the GIC, loaded with the impedance ZL, is given by Zin = k(s) ZL and k(s)  =  A(s)/D(s). The multiplying factor k(s) must be dimensionless, and in most cases it will be frequency dependent (see Slide 6.81, Chap. 6).

In Slide 9.42 the new problem formulation is given. As with every problem formulation, and in particular OPAMP Design of GIC’s (incl. ‘Gyrators’ & within the framework of FDNR’s) Using Opamps the morphological method in which the basic eleProblem Formulation: ments of the problem must be identified, this formulaDesign a two-port whose input impedance is tion does not come without Z1 ⋅ Z 3 insight and background Z IN = ⋅ ZL knowledge of the subject Z2 ⋅ Z4 at hand. Thus, based on a familiarity with negative Constraint: minimize number of OPAMP’s (i.e. impedance converters, NULLORS) and even with some knowledge of their realization (see, e.g., Slide 7.15 in Chap. 7), the formulation of our problem leads to the approach shown in the next slide.

Problem Formulation:

5  A Morphological Approach to the Design of Opamp-Based General Impedance…

253

Slide 9.43 shows that the problem formulated in the previous slide can be GIC Realisation of (1): solved with two negative NIC2 NIC1 impedance converters (NICs) in cascade. These two NICs become our two z z Zl k1 ( s ) = − 1 k2 ( s) = − 3 basic elements E1 and E2. z2 z4 As mentioned earlier, and demonstrated above Z Z IN = k1 ( s ) ⋅ Z IN Z IN = k 2 ( s ) ⋅ Z L = − 3 ⋅ Z L with the design of transisZ4 Z1 ⋅ Z 3 torized versions of the ⋅ ZL = gyrator, a very promising Z2 ⋅ Z4 way to progress is, if posTwo basic elements required for GIC design, namely: two NIC’s or PIC’s sible, to start out with the m=2 device to be designed being reduced to the interconnection of controlled sources, whose nullor representation we are familiar with. Having obtained such an interconnection, a “skeleton” version of the circuit is readily obtainable by replacing the nullors by transistors or, in the present, case, by opamps. This is followed by “fleshing out” the circuit for operability, correct biasing, choice of transistors or opamps, etc. Thus, the question before us now is how to obtain an NIC from a controlled source? OPAMP Design of GIC’s (incl. Gyrators & FDNR’s), cont’d

1

2

2

Transmission Matrix of Two-Ports Exchanging Input and Ground II

I

1

[A

B C D]

3

2

[A

B C D]

2

A B  C D   

Exchanging Output and Ground III

3

1

[A

B C D]

1

A  A −1 C  1

B

 A + D − A − 1

2

3

1 D −1

 A + D − A −1  C 

B  D

We shall use these relationships to transform Controlled Sources into (Negative) Impedance Converters:

[A

A B C D ]NIC =  0

0 D  NIC

Zin=(A/D) ZL =-(Z1/Z2)ZL

For this we consider the [ABCD] matrix of a grounded two-port as given on the left side of Slide 9.44. Exchanging first the input terminal and ground, and then the output terminal and ground, gives us, for each case, a new [ABCD] matrix in terms of the original. This is shown in the middle- and righthand side of the slide.

9  A Morphological Approach to the Design of Active Network Elements

254

The 4 Basic Controlled Sources.

Exchanging Input and Ground

II

1. Voltage-Controlled Voltage Source (VVS).

+

+ -

V1

~

mV1 V2

-

I1 = 0 V2 = mV1

1 [ABCD] = m 0

I2

+

gV1 V2

V1

-

-

I1 = 0 I2 = gV1

[ABCD] =

éA ê A - 1 ëC 1

0 0

1 g 0

I2

V1 I1

-

~

rI1

+ V2

-

V1 = 0 V2 = rI1

0 [ABCD] = 1 r

4. Current-Controlled Current Source (CCS). I2 + + V1 = 0 0 V1 I1 aI1 V2 [ABCD] = 0 I2 = aI1

-

3

B

ù A + D - A - 1úû 1

Exchanging Output and Ground III

3. Current-Controlled Voltage Source (CVS).

+

B C D]

0

2. Voltage-Controlled Current Source (VCS).

+

[A

2

0

-

0

[A

1

0

1

0 1 a

D -1

B C D]

3

é A + D - A -1 ê C ë

2

Bù ú Dû

In Slide 9.45 we now carry out this terminal exchange on the [ABCD] matrix of two of the four controlled sources shown.

Transmission matrix of Two-Ports, cont’d Example VVS: Voltage Gain μ where μ > 1 1 [A B C D] =  µ 0

 1  0 → [A B C D ] = 1 − µ   0 0

VVS

Exchanging Input and Ground (II)

VNIC

1 A −1

A C 

B

 A + D − A − 1

 0  1

In Slide 9.46 we find that exchanging input and ground of a voltage-controlled voltage source (VVS), we obtain a voltage-based NIC (VNIC).

5  A Morphological Approach to the Design of Opamp-Based General Impedance…

Similarly, in Slide 9.47, we find that exchanging output and ground of a current-controlled current source (CCS), we obtain a current-based NIC (CNIC).

Transmission matrix of Two-Ports, cont’d Example CCS: Current Gain α where α > 1 0 0  1 0     [A B C D] = 1 → [A B C D] = 1  0  0   α  1 − α CNIC CCS

Exchanging Output and Ground (III)

 A + D − A −1  D −1 C  1

255

B  D

Nullator-Norator Representation of Controlled Sources Controlled Sources with Single NullatorNorator

Controlled Source VVS

Controlled Sources with Two NullatorNorator Pairs

R2

V1

~ I1 = 0 V2 = µV1

V1

R1

Controlled Sources with Two NullatorNorator pairs

R2

V2 V1

R1

I1 = 0

I1 = 0

R  V2 =  2 + 1V1 R  1 

R V2 = 2 V1 R1

V2 V1

R1

R2

V2

I1 = 0 V2 = −

R2 V1 R1

Only one Nullor

In Slide 9.48, we now go to our table of nullor versions of the VVS (see Chap. 6, Slide 6.98) and, if possible, find one with only one nullor. [There are several with two nullors, but in the problem formulation we specified “circuits with a minimum number of opamps,” i.e., nullors.]

9  A Morphological Approach to the Design of Active Network Elements

256

Slide 9.49 shows that this is readily found and carried out, resulting in our newly found nullor version of a VNIC.

Exchanging Input and Ground Nodes: VVS:

Problem: Floating Norator

R2 V1

R1

V2

1



Exchanging Input and Ground (II)

[ABCD] = 1 + Z 2

Z1

0

 0  0 

Z IN = −

VNIC! N

Z1

Z1 ⋅ ZL Z2

[A

N

Z2

 − Z1 B C D] =  Z 2  0

 0  1 

Nullator-Norator Representation of Controlled Sources

Controlled Source CCS

I1

V1 = 0 I 2 = αI1

Controlled Sources with Two NullatorNorator Pairs

Controlled Sources with Single NullatorNorator

R

R1 V1

Controlled Sources with Two NullatorNorator pairs

R2

V1 = 0 R  I 2 = − 1 + 1 I1 R  2 

V2

V1 = 0 I2 = −

R

R

R

R1 I1 R2

V V

V

V1 = 0 I2 =

R1 I1 R2

Only one Nullor

In Slide 9.50 we repeat the same steps for a CCS by first finding a one-nullor version of a CCS in the table of nullor-based CCSs (see Chap. 6, Slide 6.101).

5  A Morphological Approach to the Design of Opamp-Based General Impedance…

257

Exchanging Output and Ground Nodes: CNIC:

CCS:

n N

R1 V1

n

R2

Exchanging Output and Ground (III)

V2

0 0    [ A B C D ] = 0 1   1 + R1   R2 

I1 V1

N

R2

V2

0  1  [A B C D] =  − R2  0 R1  

Exchanging Output and Ground (III)

R1,2

I2 R1

Z IN = −

Z 1,2

Z1 ⋅ ZL Z2

In Slide 9.51 we exchange the output and ground terminals of the one-nullor CCS and obtain a nullor version of a CNIC.

Morphological Problem Solving STEP 1 FORMULATION OF PROBLEM

STEP 3 MORPHOLOGICAL BOX (M-dimensional)

p solutions sµ

S1

S2

S3

Sµ

Two basic elements required for GIC design, namely: two NIC‛s or PIC‛s m=2

Sp

STEP 4 REALIZABILITY CONSTRAINTS AND PROBLEM SPECFICATIONS

q solutions sν

S1

S2

Sυ

Sq

q p ≤

r optimum solutions sk r ≤q

STEP 5 PROGRESSIVE ELIMINATION FOR OPTIMUM SOLUTION(S) S1

Sk

Sr

Slide 9.52 indicates that we have now reached step 2 of our morphological design and that we have two basic elements to solve our problem, namely, a VNIC and a CNIC.

9  A Morphological Approach to the Design of Active Network Elements

258

TYPE

CNIC

VNIC

Slide 9.53 shows the VNIC and CNIC, their [ABCD] matrices, and their input impedance when loaded with impedance ZL. Other basic elements, including positive impedance converters or PICs, could be found, but are not needed.

NIC Design With One Nullor [ A B C D] NULLOR CIRCUIT ZIN 1  0 

 Z2 − Z  1 0

N

0  Z2  −  Z1 

Z1

N

Z2

Z IN = −

Z1 ⋅ ZL Z2

Z IN = −

Z1 ⋅ ZL Z2

N

 0  1

Z1

N

Z2

Table of Elements for Two-Nullor GIC Design : m=2 Element E1 Element E2 (One-Nullor NIC) (One-Nullor NIC) N

N

E11: CNIC

E21: CNIC Z1

N

Z1

Z2

N

E12: VNIC Z1

Z1

Z2

.. . n1 Alternatives for Element E1

E1 j

.. .

E1n1

Z2

N

E22: VNIC N

N

N

Z2

.. . n2 Alternatives for Element E2

E2 j

.. .

E2n2

Slide 9.54 shows the two basic elements E1 and E2 and the fact that m = 2. Note that for our design problem, it does not matter whether the NIC is voltage or current inverting. Thus a VNIC can be used with a CNIC, or each with itself, as element E1 and E2.

5  A Morphological Approach to the Design of Opamp-Based General Impedance…

Morphological Problem Solving STEP 1 FORMULATION OF PROBLEM

259

With Slide 9.55 we have reached step 3, namely, the assembly of the morphological box.

STEP 3 MORPHOLOGICAL BOX (M-dimensional)

p solutions sµ

S1

S2

S3

Sµ

Sp

STEP 4 REALIZABILITY CONSTRAINTS AND PROBLEM SPECFICATIONS

q solutions sν

S1

S2

Sυ

Sq

q≤p STEP 5 PROGRESSIVE ELIMINATION FOR OPTIMUM SOLUTION(S)

r optimum solutions sk r ≤q

S1

Sk

Sr

The ‘Morphological Box’ For Opamp GIC Design Two basic elements required for GIC design, namely: two NIC‛s or PIC‛s : m= 2

E21

E22

E11

S11

S12

E12

S21

S22

Note: Each solution Sij has more than one realization so that this ‘matrix‛ is really a 3-dimensional box

Slide 9.56 shows the morphological box for our problem in the form of a small matrix. However, here again we shall see that the matrix is actually three-dimensional because each solution cell Sij has at least one sub-solution.

9  A Morphological Approach to the Design of Active Network Elements

260

Solution S11 for GIC Design N2

N1

Solution (S11)1 Z1

Z3

Z N1 2

N2

Z4

N1

N2

N1

N1N1

N1N2

: (S11)11

N2

N2N1

N2N2

: (S11)12

Slide 9.57 shows solution S11. It has several sub-solutions within the bounds set by the nullors themselves. One such bound is that only one nullator and one norator can be assigned to the same opamp (the opamp being the intended active element for this problem). Another bound is that the nullator-norator pair may not have a common terminal. Within these bounds, this solution contains the two sub-solutions indicated in the slide.

TWO-NULLOR SOLUTIONS WITH NEITHER NULLATOR NOR NORATOR IN COMMON.

Solution (S11)2 for GIC Design N1

Z1

N2

Z3

Z N1 2

N2

Z4

N1

≡ N1

Solution (S11)2

N2 Z1

Z N1 2

Z3

N2

N2

N1

N1N1

N1N2

: (S11)21

N2

N2N1

N2N2

: (S11)22

Z4

S11 has four solutions! e.g.: (S11)12: ‘Antoniou GIC’ (S11)22: ‘Riordan GIC’

Slide 9.58 contains two more solutions. They result from the nature of the nullator for which two in tandem have the same effect, no matter in which order they are connected. It should be noted that most of these solutions result in perfectly feasible and useful circuits. Two of them were separately published in the literature, namely, (S11)12 (A. Antoniou in the Proc. IEEE, vol. 116 in 1969) and (S11)22 (R.H.S Riordan in Electronic Letters, vol. 3 in 1967).

5  A Morphological Approach to the Design of Opamp-Based General Impedance…

In Slides 9.59 we return to the cascade of two nullorbased negative impedance converters or NICs.

Nullor Realization of GIC, Gyrator, & FDNR Nullor Realization of NIC: (Negative Impedance Converter). From above:

Z IN = −

Z1 ZL Z2

Z1

Z2

261

ZL

Nullor Realization of General Impedance Converter (GIC), FDNR, Gyrator n1

Z IN =

Z1 Z 3 ZL Z2Z4

Z1

n2

Z2 N1

Z3

Z4 N2

ZL

FDNR, ‘Inductor’ (Gyrator), & GIC

Z IN

ZZ = 1 3 ZL Z2Z4

Ex: GIC:

n1

Z1

n2

Z2 N1

Z3

Z4 N2

Z1 ( s ) ⋅ Z 3 ( s ) = k ( s ) : Z IN = k ( s ) Z L Z 2 ( s) ⋅ Z 4 ( s)

Ex: FDNR:

Z1 =

ZL

1 1 1 , Z3 = , Z 2 = Z 4 = Z L = R, Z IN = 2 sC1 sC3 s C1C3 R

Ex: ‘Inductor' with Gyrator:

Z4 =

1 , Z1 = Z 2 = Z 3 = Z L = R, Z IN = sC4 R 2 sC4

In Slide 9.60 we note that by placing resistors and capacitors appropriately, the nullor circuit at the top of the slide has within it the capability of realizing a general impedance converter (GIC), a frequency-­ dependent negative resistor (FDNR) as introduced in Chap. 7, and a gyrator.

9  A Morphological Approach to the Design of Active Network Elements

262

Morphological Problem Solving STEP 1 FORMULATION OF PROBLEM

STEP 3 MORPHOLOGICAL BOX (M-dimensional)

p solutions sµ

S1

S2

S3

Sµ

Slide 9.61 indicates that we have now reached step 4 of the morphological design process. This is the step that is most dependent on the know-how and insight of the designer. Here, to illustrate the method, we shall pick out several solutions that have become established in practice.

Sp

STEP 4 REALIZABILITY CONSTRAINTS AND PROBLEM SPECFICATIONS

q solutions sν

S1

S2

Sυ

Sq

q≤p STEP 5 PROGRESSIVE ELIMINATION FOR OPTIMUM SOLUTION(S)

r optimum solutions sk r ≤q

S1

Sk

Sr

Circuits for Gyrator and FDNR Design ,N

Z1

Z2

R1

R2 ,N

Z3 R3

Z4 C4

Z5

R5

Z in =

Z1Z 3 Z 5 Z2Z4

With a capacitor for Z4, and resistors for the other impedances, we obtain the equivalent inductor L where: With a capacitor for Z1, a second capacitor for Z5 , and resistors for the other impedances, we obtain the Frequency-Dependent where: Negative Resistor or

Slides 9.62, 9.63 and 9.64 show how virtually the same circuit can be used to obtain either a gyrator or an FDNR. The one used in the following examples is one that has established itself in practice, namely, solution (S11)12.

5  A Morphological Approach to the Design of Opamp-Based General Impedance…

263

Opamp-RC Gyrator Circuit Simulating an Inductance.

R1

R2

R3

C4

1

Leq =

C4 g2

Z in =

2

sC Z1Z 3 Z 5 = 24 = sLeq g Z2Z4

Slide 9.63

Opamp-RC Circuit Realizing an FDNR (Frequency-Dependent Negative Resistor)

R2

C1

R3

R4 C5

D=

C1C5 R2 R4 R3

Z in =

Z1Z 3 Z 5 1 = 2 Z2Z4 s D

Slide 9.64

R5

9  A Morphological Approach to the Design of Active Network Elements

264

Gyrator Ladder Filter Doubly terminated gyrator-C ladder network a) grounded R5 C1 C3

(500Ω) (1.54nF)

Vin

~

C2 L2

(1.54nF)

(6.1nF) (618.4µH)

500Ω 1.54nF

RL (500Ω)

1.54nF

6.1nF V1

~

500Ω 618.4⋅g2F

Gyrator Ladder Filter, cont’d 25k 30.8p

30.8p 122p

50k 50k –

12.37p vin

+

R4(100K) 12.37p –

+

–

50k

50k vout

+

Leq

–

50k +

50k 50k 122p

25k 30.8p

30.8p

Doubly terminated gyrator-C ladder network b) balanced Slides 9.65 and 9.66 show how an LCR grounded and balanced high-pass filter, respectively, can be replaced by an equivalent inductorless gyrator-C ladder network.

5  A Morphological Approach to the Design of Opamp-Based General Impedance…

Slide 9.67 shows the frequency response of a 5thorder low-pass filter (designated CC 05 25 49 in classical filter tables, see Chap. 4).

Example: FDNR realization of 5th-order low-pass filter Am [dB] TB PB

0.3dB

265

SB Amin=40dB

Amax

fp=1kHz

1.32kHz

1

f

Ωs

Ω=f/fp

ω ω Ω S = StopBand = SB ωPassBand ωPB

FDNR-Transformation of 5th-order low-pass filter Minimum C RLC ladder filter

L1

R1

L5

L3

L2

L4

C2

C4

R’1 R2

C’1

Zero frequencies: 4 758 Hz

∼

5 293 2 233

R’2

R’3

6 951 Hz 5 802

1 051

3 249 10.02 10.09

3 001

1 051

2 233

R in Ω C in nF

3 001

3 157

DCRFilter C’2

D4

D2

10.11

FDNR

R’4

R’5

3 001

FDNR

3 279 15.03

15.08

Slides 9.68, 9.69 and 9.70 show the steps necessary to transform a minimum-C RLC ladder filter into an equivalent inductorless FDNR-based filter.

9  A Morphological Approach to the Design of Active Network Elements

266

Transformed LC network consists of resistors, capacitors, and FDNRs. R’1 C’1

  

R’2

R’3

R’4

DCRFilter

R’5

C’2

D4

D2

Each grounded capacitor becomes 1 FDNR. Each FDNR requires 2 opamps. Instead of 8 opamps with gyrators the equivalent FDNR network requires only 4 opamps!

FDNR realization of CC 05 25 49 5th-order low-pass filter 5 293 2 233

∼

5 802

1 051

10.11

3 249 10.02 10.09

3 001

1 051

2 233 FDNR

v

R in Ω C in nF

v 3 001

3 157

Slide 9.70

6 951 Hz

Zero frequencies: 4 758 Hz

3 001

FDNR

3 279 15.03

6

Slide 9.69

15.08

Further Typical Problems Dealt with Morphologically

It should be clear by now that the list of topics amenable to the morphological method of problem solving – a method that explores all the possible solutions to a multidimensional, non-quantified complex problem – is endless. Typical examples of fields in which it has successfully been used are the following: In the field of electrical engineering: • GIC design (gyrators, FDNRs, NICs) using two opamps • Design of very-low-frequency active filters for hybrid-integrated technology (Rmax  =  100  kΩ, Cmax = 20’000 pF!) • Microprocessor structures for digital filtering applications • Switched-capacitor filter synthesis • Displays of data (e.g., biodata) in readily accessible and recognizable form

6  Further Typical Problems Dealt with Morphologically

267

In government and academia: • Management in industry and commerce; local and state government • All branches of science and engineering • City traffic; medical care and research; public transportation/energy problems

3 Basic Elements: Mass, Binder, Flavor Flavor ginger almond pepper

Why not?!

garlic cinnamon onion

Egg-Nut-Onion Cake!

herring chocolate strawberry vanilla

corn meal ground nuts potato flour ground vegetables mashed potatoes

Mass

flour

water eggs milk oil

butter margarine fruit juice yoghurt

(+ baking powder) Binder

and a solution, shown in Slide 9.71 (“always ask why not….?”)

In domestic (and personal) life: • Vacation plans: {e.g., “basic elements”: 1, geographical location; 2, activities; 3, budget; 4, duration; 5, transportation; 6, participants; 7, accommodation} • Transfer of the author’s five-member family (three daughters) from nine-room suburban house in New Jersey, USA, to five-room city apartment in Zurich, Switzerland. Constraint: minimal reduction in privacy and comfort • Cooking, e.g., the morphology of cake baking {e.g., “basic elements,” mass, binder, and flavor},

Chapter 10

Active Filter Design Techniques

Slide 10.1

Chapter 10 Active - Filter Design Techniques

1

Introduction

In this chapter we summarize the main types of analog inductorless filter design and discuss some of their important features as they pertain to the constraints imposed by integrated-circuit technology. It is shown that, by and large, most of the design techniques known today can be related to the methodologies outlined in the previous chapters of this book. Thus, this chapter serves to demonstrate the power of these methodologies and, at the same time, serves to summarize the most important active filter design techniques known today. Based on the “PACE” (problem formulation, approach, calculation, evaluation) strategy introduced in Chap. 3, the chapter is broken down into four sections. In the first, the approach to obtaining ­inductorless active-RC filters is to start out with the transfer function of the desired filter (generally obtained from either filter tables or a filter design computer program). The transfer function is then © Springer Nature Switzerland AG 2019 G. S. Moschytz, Analog Circuit Theory and Filter Design in the Digital World, https://doi.org/10.1007/978-3-030-00096-7_10

269

270

10  Active Filter Design Techniques

broken up into a product of second-order “biquadratic” transfer functions which become the transfer functions of a cascade of second-order filters, the so-called biquads. These are then related to the classification of biquads derived and discussed in Chap. 8. In the second section, the so-called direct form of active filter design is described. Starting, as in Sect. 1, with the desired transfer function, signal-flow graph theory is used to represent the transfer function as a signal-flow graph (SFG). Individual paths of the SFG are shown to have the transmission function of an integrator – others are either simple adders or multipliers. These, the integrators, adders, and multipliers, can all be realized by simple feedback amplifier circuits which, when interconnected appropriately, result in the desired inductorless filter. [Note that integrators, adders, and multipliers can also be realized by other technologies, such as by digital or switched-capacitor circuits. All result in inductorless filters defined by the technology in question.] In the third section, starting with the desired filter specifications, we consult filter tables or a computer program to obtain the corresponding passive LCR filter circuit. Each component of the LCR filter circuit is then individually transformed such as to conform to the intended IC technology. In one case the resistors and capacitors are maintained as such, and the inductor is realized by active-RC gyrator circuits. In another the LCR filter is impedance transformed such that the inductors become resistors, the resistors become capacitors, and the capacitors become active, so-called frequency-­ dependent negative resistors or FDNRs. In the final section, by way of an illustrative example, a third-order high-pass notch filter is designed using each one of the above-mentioned approaches. The resulting circuits are then computer simulated and evaluated from the point of view of sensitivity, noise, and compatibility with common IC technology.

2

From Transfer Function to Active Filter: Cascaded Biquad Design

Slide 10.2 shows the nthorder transfer function T(s) = N(s)/D(s) of a given filter. As already discussed in previous chapters, n is  Given: m m −1 the order, or degree, of the N ( s ) bm s + bm −1 s + + b1 s + b0 = = T ( s) = denominator polynomial n n −1 D ( s ) a n s + a n −1 s + + a1 s + a0 D(s), i.e., the number of its n −1 n 2 2 poles. Similarly, m is the 1 = ∏ Ti ( s ) = ∏ Ti ( s ) ⋅ or order, or degree, of the s+α i =1 i =1 numerator polynomial ω zi N(s), i.e., the number of its zeros 2 2 s + s + ω zi finite zeros  – “finite” q zi ( s − zi )( s − zi* ) T ( s ) = K = Ki meaning not including the where: i i * ωp ( s − pi )( s − pi ) number of its zeros at s 2 + i s + ω2pi q pi poles infinity. D(s) can be broThis term determines whether the poles ken up into a product of are complex-conjugate or neg. real second- and third-order polynomials. Each of these is the denominator of a biquad which, when factored out, results in its two, or three, poles in the complex-frequency or s-plane. There is more flexibility with respect to the numerator polynomial, N(s). It too must be factored out into a product of biquad numerators, but the degree of each can range from zero (i.e., a constant), first,

From Transfer Function to Active Filter: Cascaded Biquad Design

2  From Transfer Function to Active Filter: Cascaded Biquad Design

271

second, or, in the case of a third-order biquad, third order. Note that the order of the numerator for any transfer function, i.e., biquad or T(s), may never exceed that of its denominator, i.e., for T(s), m ≤ n. The second-order denominator of each biquad consists of two complex-conjugate poles p1 and p2. These are generally combined into a second-order polynomial of the form s2 + (ω/qp)s + ωp2 where qp ≥ 0.5. [Note that if the poles were negative-real, we would not need an active-RC biquad, i.e., the biquad would be passive RC.]

In general zi ,

z *,

s2 + Ti ( s ) = K i

pi ,

i

s2 +

ωz

i

qzi

ωp

i

q pi

p* i

are complex conjugate: ~ jω

s + ω z2i

q pi ≥ 0.5 σ

s + ω p2i

Slide 10.3 summarizes these properties and expressions. Note that quantities that we wish to characterize as passive RC are given a “hat,” or “caret,” i.e., T̂(s), q̂p.

However, if Ti (s) is realized by a passive RC network, then pi , pi* will be negative real! ~ jω

Tˆ ( s ) =

N (s)

s2 +

ωp qˆ

s + ω p2

σ

For passive RC networks, it can be readily shown that:

q pi < 0.5

Slide 10.4 indicates that the pole Q, qp, is the decisive characteristic that determines whether a biquad has q pi ≥ 0.5 complex-conjugate or negative-real poles, i.e., whether is the pole Q of a passive RC network, we  Since it is active RC or passive have by definition, RC.  It therefore translates qˆ pi < 0.5 the whole question of how to transform a passive RC biquad into an active-RC  Question: How can we add an active element (i.e. biquad into the question of pos. or neg. amplifier) to make how, with the help of a facq pi = f qˆ pi , ± β ≥ 0.5 tor ±β, to increase the passive q̂p, which is smaller (where β: amplifier gain) ? than 0.5, into a qp which is equal to or larger than 0.5. Note that the approach taken here to active-RC biquad design is entirely different from the approach taken in Chap. 8. Accordingly, the results will also be different, namely, in this case, not nearly as differentiated and detailed as in Chap. 8. Among other things, in contrast to this chapter, an added objective in Chap. 8 was to include and demonstrate the usefulness for our classification of general design tools such as the root locus and the signal-flow graph (SFG). 

Thus the condition for Ti (s) to have complexconjugate poles is:

(

)

10  Active Filter Design Techniques

272



Let us consider the biquadratic transfer function of a passive RC Biquad: Tˆ ( s) Tˆ ( s ) =

N ( s)

s2 +

ωp qˆ

s + ω p2

where: qˆ < 0.5

1st method: Q multiplication (negative feedback)

T (s) = Thus:

T ( s) =

2.1

N (s)

where µ = 1 + β and β > 0

ω s + p s + ω p2 qˆ µ 2

N (s)

ωp

2

s +

qˆ(1 + β )

s +ω

p1 ⋅ p2 = ω p2

2 p

p1 p2 ( p1 + p2 ) −1 = qˆ

In Slide 10.5 we start out with the biquadratic transfer function T̂(s) of a passive RC circuit, i.e., q̂p = {qp  0. Thus, qp = q̂p(1 + β) and is larger than 0.5. [Obviously, for β = 0, qp = q̂p, and we have a passive RC network.] Consequently T(s) becomes the transfer function of an active-RC, rather than of a passive RC biquad. This design method is called the Q multiplication method.

Thus: T ( s) =

N ( s) ωp s2 + s + ω2p qˆ (1 + β)

= (1 + β)

=−

Where:

N ( s) ⋅ ωp 2 2 s + s + ωp 1 + β ⋅ qˆ

1+ β ˆ −β ⋅T ( s ) β 1 + β ⋅ tˆ1 ( s )

q p = qˆ (1 + β )

1 s 2 + ω2p ω s 2 + p s + ω2p qˆ

In Slide 10.6 the terms in T(s) are rearranged in order to comply with the basic SFG of a feedback amplifier with a passive RC feedback network (such as we encountered in Chap. 8).

2  From Transfer Function to Active Filter: Cascaded Biquad Design

tˆ (s ) 1

+ − 1 β ⋅Tˆ (s) β 1

tˆ12

tˆ32 2

−β

3

DF

2.2

273

Slide 10.7 shows the corresponding SFG and a typically resulting biquad. This is what, in Chap. 8, is called a Type I, DF, Class 3 band-pass biquad. This example shows that negative-feedback-based single-amplifier biquads can be traced back to Q multiplication by a term 1 + β.

Positive Feedback

Slide 10.8 also starts out with the biquadratic transfer function T̂(s) of a pasGiven biquadratic transfer function Tˆ ( s ) of a sive RC circuit, i.e., passive RC Biquad: where σ ˆ > ωp q̂p = {qp   ω . Thus, for an Thus T (s) can be rewritten as follows: p ˆ σ 2 ˆ active-RC biquad, σ must N ( s) 1 q p = qˆ p T ( s) = 2 ⋅ 2 s ˆ −χ s + 2σ σ ˆ s + ωp 1 − χ 2 be smaller than ω . p As 2 s 2 + 2σ ˆ s + ωp can  be seen in the slide, 1 χ = Tˆ ( s ) ⋅ in order to decrease σˆ , χ 1 − χ tˆ2 ( s ) we  subtract a quantity χ from it which is smaller than 2 σˆ . This implies a positive-feedback amplifier circuit with amplifier gain χ. 2nd method: Sigma Reduction (Positive feedback)

~ jω

10  Active Filter Design Techniques

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Slide 10.9 shows that by rearranging the terms in the previous slide in order to comply with the basic SFG of a positive-feedback amplifier with a passive RC feedback network, we obtain a biquad of the type shown in the lower 4 part of the slide. Referring to our classification in Chap. 8, this circuit is a Type I, SF, Class 4 lowpass biquad [In class 4 biquads SF and DF is the same]. This example shows that single-amplifier biquads based on positive feedback can be traced back to what we refer to as sigma reduction, i.e., reducing the term 2 σˆ by a quantity such as to obtain complex-­conjugate poles. Note that χ is limited in value such as to prevent right-half plane poles, i.e., circuit instability.

Classification of Single-amplifier Biquads tˆ (s) 32

1

tˆ (s) 12

2

− +β

Negative Feedback: Class 1: tˆ32 ( s ) → Class 2: tˆ32 ( s ) → Class 3:

tˆ32 ( s ) →

3

Low-pass

T (s):

Low-pass Band-pass

High-pass

T (s):

High-pass Band-pass Low-pass High-pass Band-pass

Bandreject T (s):

Positive Feedback: Class 4: tˆ32 ( s ) →

Band-pass

T (s):

Low-pass High-pass Band-pass Band-stop

Slide 10.10 summarizes the four classes of singleamplifier biquads, as they were derived in Chap. 8.

3  From Transfer Function to Active Filter: Direct Form Using Integrators

3

275

 rom Transfer Function to Active Filter: Direct Form Using F Integrators

From Transfer Function to Active Filter: Direct Form Using Integrators Given:

T ( s) =

N ( s ) bm s m + bm −1s m −1 + K + b1s + b0 = D ( s ) an s n + an −1s n −1 + K + a1s + a0

Without lack of generality, we let m=n and an=1, furthermore we divide N (s) and D (s) by s n, thus:

( s) =

bn + bn −1 ( 1s ) + K + b0 ( 1s ) n 1 + an −1 ( 1s ) + K + a0 ( 1s ) n

Equating T (s) with Mason’s General Reduction Rule for Signal-flow Graphs, we obtain:

T ( s) =

Pk ∆ k bn + bn −1 ( 1s ) + + b0 ( 1s ) n ∑ k = ∆ 1 + an −1 ( 1s ) + + a0 ( 1s ) n

where: ∆=Graph determinant= 1 − ∑ Li + ∑ Li L j Κ and ∑ Li L j , ∑ Li L j Lν concerns only non-touching loops Pk=Transmission of the k-th path from input to output ∆k=Cofactor of the k-th path, i.e. determinant of part of SFG that does not touch k-th path. We now make the following assumptions: (i) every forward path Pk touches every loop (∆k=1) (ii) all loops touch each other (∑ Li L j = ∑ Li L j Lk ' = Κ = 0)

Slide 10.11 starts with the familiar nth-order filter transfer function T(s), where n is the degree of the denominator polynomial D(s) and m the degree of the numerator polynomial N(s). With no loss of generality, we let m equal n. If this is not the case, then the corresponding coefficients bi are equal to zero. Dividing N(s) and D(s) by sn results in T(s) as a function of (s)-j with j = 1…n.

Slide 10.12 instructs how to convert the transformed version of T(s) of the previous slide into a signalflow graph using Mason’s general reduction rule for signal-flow graphs (see Slide 5.20, Chap. 5). In doing so, we apply the simplifying assumptions given in the slide.

10  Active Filter Design Techniques

276

Equating T (s) with Mason’s Multipath Rule for Signal-flow Graphs, we obtain: Integrators

T ( s) =

bn + bn −1 ( 1s ) + K + b0 ( 1s ) n ∑ Pk = n 1 + a n −1 ( 1s ) + K + a0 ( 1s ) 1 − ∑ Li

This permits two direct representations of T (s) as a SFG in which each coefficient ai and bi appears as a transmission value: Direct Form I :

1

1/s

-an-1

Direct Form II : -a2

-an-2

-a0

1/s b2

bn

-a0

n integrators 1/s

1/s

1/s

1/s

-a1

b1

b0

b0

b1

-a1

1/s

1/s b2

n integrators

-a2

-a3

1/s

bn-2

b3

bn-1

-an-2

1/s

bn-1

1/s

-an-1

1/s

1

bn

Note: Number of Integrators is equal to order of T(s).

Slide 10.13 shows the two resulting SFGs. They each consist of a series connection of terms (s)-j in the forward path and the coefficients aj of D(s) as the transmission values of individual feedback branches. Similarly, the coefficients bj of N(s) become the transmission values of individual forward branches of the SFG. Depending on whether all the feedback branches lead into the source or originate at the sink of the SFG, (the forward branches do the opposite) we obtain the two different SFGs, namely, the so-called Direct Form I and the Direct Form II.

From Transfer Function to Active Filter Cont’d Example: Biquadratic transfer function and the ‘Multi-amplifier Biquad’

T (s) = K

s2 ±

2

s ± B1s + B0 =K s 2 + A1s + A0 2

ωz  1  21   + ωz   qz  s  s =K 2 ωp  1 2 1  1 +   + ω p  qp  s  s 1±

Direct Form II:

s2 +

K

ωz qz

ωp qp

s + ω z2 s + ω p2

=K

1± 1+

1 B1  s + A1  1 + s

-ωp ω 2z/ω p

ωp

1/s ±ω /q z z

1

-ωp /qp

2

1 B0  s  2 A0  1  s

1/s

Slide 10.14 bridges the gap between SFG, transfer function T(s), and actual filter circuit. The key to the conversion into an actual circuit is the fact that the term (s)-j is the Laplace transform of an integrator. This already informs us that there will be at least as many integrators in the circuit as the order of T(s), i.e., the degree n of D(s). This slide provides an example for this design approach using a secondorder transfer function

3  From Transfer Function to Active Filter: Direct Form Using Integrators

277

T(s), which, typically, we refer to as the transfer function of a biquad. The first step in the design is to modify T(s) for direct-form design. This means dividing N(s) and D(s) by sn and then drawing the corresponding SFG with all the forward and feedback branches. The resulting Direct Form II SFG is shown at the bottom of the slide.

From Transfer Function to Active Filter Cont’d.

The

‘Multi-amplifier Biquad‛ R3 =R ωp

R1 =R qp C1

R4

C2

r

R2

r

Vin

VLP1

VBP R7

ω2p =

R5

1 R2 R3C1C2

R8 Vout

1/ 2

 C1   q p = R1   R2 R3C2  T (s) =

R6

VLP2

Vout R =− 8⋅ Vin R7

ωz

ω z2

qz

ω p  R R  s 2 + s  1 − 7 ⋅ 1  + q p  R5 R4  s2 +

ωp qp

 2  R7 R3  ω p  R ⋅ R − 1   6 4 

s + ω p2

Slide 10.15 shows the corresponding circuit. An analog integrator can readily be realized by an opamp with a capacitor in the feedback loop and a resistor connected to the input terminal. Thus, although this is the transfer function of a biquad, we see that with this design approach, the biquad will not be a single-amplifier circuit. Due to the number of integrators required to realize T(s) – and, as can be seen, an additional amplifier as an inverter – it will in any case be a multi-amplifier biquad. [This biquad is sometimes referred to as the “ring of 3 circuit” and also the “Tow-Thomas biquad” after Lee Thomas and James Tow who published the design rules for it; see J.  Tow, IEEE Spectrum, Vol.6, pp64–68, Dec. 1969.]

10  Active Filter Design Techniques

278

‘Multi-amplifier Biquad‛, cont‛d R2 C3 R4 R1

– +

C6 R5

– +

R8 R7

LP2

– +

LP1 BP

TBP(s) = –KBP TLP1(s) = KLP1

KBP =

wp2 + (wp/qp)s + wp2

KLP1 =

R2 R7 R1 R8

KLP2 =

R2 R1

S2

TLP2(s) = –KLP2 wp2 =

(wp/qp)s S2 + (wp/qp)s + wp2

S2

wp2 + (wp/qp)s + wp2

R8 R2 R5 R7 C3 C8

R4 R1

qp = R4 C3 wp

Tuning: (1)ƒp with R5, (2)qp with R4, (3) K with R1

Slide 10.16 shows the design equations for the multi-amplifier biquad of Slide 10.15.

3  From Transfer Function to Active Filter: Direct Form Using Integrators

279

Slide 10.17 shows the design flow chart for the multi-amplifier biquad of Input: ƒp, qp, C, K Slide 10.15.1 Finally, the question 1 Ro = seems imminent as to 2πƒpC when to use a singleamplifier and when a Print: Ro. Input: Rd multi-­amplifier biquad. Obviously, the savings in opamps (e.g., power and R2 = R7 = R8 = Rd C3 = C6 = C chip area) in single-ampli2 R R5 = o R4 = qpRo fier biquads is immediately Rd evident. Nevertheless, there are many other facPrint: tors, such as tunability, No Yes constant for sensitivity to component low-pass? and design tolerances, noise, dynamic range, R R R R1 = 4 KBP = K R1 = 2 KBP = 4 R1 adaptability to technologiK K R2 cal constraints, and KLP1 = KLP2 = K KLP1 = KLP2 = R1 requirements dictated by the application in question, that will affect the answer to this decision. However, Print: R1, R2, C3, R4, R5, C6, R7, R8, KBP, KLP1, KLP2, ƒp, qp this is not the place to discuss these rather intricate and application-specific design details; they go well beyond the core objectives of this book. However, at the end of the next section, and at the end of this chapter, we shall attempt to draw some general conclusions and guidelines with regard to the pros and cons of the design methods discussed here. GP2

G. S. Moschytz and Petr Horn: ‘Active Filter Design Handbook’, John Wiley & Sons, Chichester, 1981, pp. 82, 83.

1 

10  Active Filter Design Techniques

280

4

 rom Passive LC to Active-RC Filter: Inductor Simulation F (Gyrator) and Impedance Transformation (FDNR)

Slide 10.18 shows the attenuation response of a low-pass filter that is to be designed as an active-RC Am [dB] filter by inductor simulation. The attenuation response is given in terms TB used in filter tables and PB SB computer programs for the design of conventional LCR filters. [Note that the A =40dB Amax min attenuation response of a 0.3dB filter is generally used for conventional passive LCR fp=1kHz 1.32kHz f filter tables and their specifications; its inverse, the 1 Ωs Ω=f/fp gain, or frequency ωStopBand ωSB response is generally used ΩS = = for the specifications of ωPassBand ωPB active-RC and other inductorless filters.] As we know from previous chapters (e.g., Chap. 7), inductor simulation implies starting out with an LCR filter and then either replacing each inductor by a gyrator-C combination or impedance scaling the components of the filter by the dimensionless quantity ω0/s. Impedance scaling by the factor ω0/s transforms the inductors of the LCR filter into resistors, the resistors into capacitors, and the capacitors into frequency-dependent negative resistors (FDNRs). Both these methods are discussed in this section.

Example: Filter specifications in terms of frequency response:

Slide 10.19 shows the two conventional LCR filters that satisfy the specifications in the previous slide. C1=1.27524 L2 L4 Minimum L They are presented as they C2=0.22878 would be in typical filter L2=1.10584 C =1.75220 tables or computer pro2 1.0 3 K C2 C4 C1 C3 C5 C4=0.67860 grams. The two filters are 1 2 3 4 5 L4=0.78589 duals of each other and C5=0.97888 have the same frequency Normalized Dual Networks (both have the same frequency response!) Component response (see Chap. 4, Values Slide 4.21). They are Minimum C L1=1.27524 L1 L3 L5 designed to be equally terL2=0.22878 minated at both ends and C2=1.10584 1 L2 L4 1.0 are normalized for an K2 L3=1.75220 C2 C4 impedance level of unity. L4=0.67860 2 3 4 5 1 This is why only normalC4=0.78589 ized (dimensionless) LC L5=0.97888 component values are given. The upper LC filter is so-called minimum-L, because it requires a minimum number of induc-

Typical Filter Network and its Dual Example: 5th-order low-pass filter

4  From Passive LC to Active-RC Filter: Inductor Simulation (Gyrator)…

281

tors; the lower filter is “minimum-­C.” Conventional LC filter design is generally based on the minimum-L version because inductors are more costly, larger, and more difficult to handle than capacitors.

1. Gyrator Simulation of a Grounded Inductor

1

≡ Zin

1

Leq =

Slide 10.20 shows the simulation of a grounded inductor (Leq) by a gyratorcapacitor combination.

CL g2

2

2

Simulation of a grounded inductor by an equivalent gyrator-C combination

Example of Gyrator Simulation of a Grounded Inductor C1 R5 (500Ω) (1.54nF)

LCR High-pass Filter

Equivalent Gyrator-RC Circuit

V1

V in

C2

~

L2

500Ω 1.54nF

C3 (1.54nF)

(6.1nF) (618.4µH)

RL (500Ω)

1.54nF

6.1nF

~

500Ω 618.4⋅g2F

Slide 10.21 illustrates inductor simulation with a third-order high-pass notch filter. Note that a grounded L-simulation requires only one gyrator. Thus, since the filter function is not affected by the physical sequence in which the L and C is connected in the grounded branch of the filter, it is important that for a gyrator-C simulation the L and not the C is the grounded component of the branch.

10  Active Filter Design Techniques

282

2. Simulation of Floating Inductor with Gyrators Example: LCR network, from Filter Tables: C2

[Minimum L Filter] R1

C1

C4

L2

L4

C3

C2 R1

C1

g2

C’2 C’2=g22 L2

R2

C5 C4

g2

C3

g4

C’4

g4

C5

R2

C’4=g42 L4

Each floating inductor requires 2 gyrators. Each gyrator requires 2 opamps. Thus: to replace L2 & L4 we require 8 opamps! Floating inductors are typical for low-pass filters. Therefore use FDNRs, resulting in half as many opamps! Slide 10.22 emphasizes the last point. Here, we have a fifth-order LCR low-pass filter. Typically, as in any LCR low-pass filter, there are floating (non-grounded) inductors in the series branches of the filter. A floating inductor requires two gyrators for its simulation (e.g., see Chap. 7, Slide 7.19). Furthermore, each gyrator requires two opamps for its realization (e.g., see Chap. 9). Thus, the LCR filter shown here requires four gyrators and, for opamp realization, eight opamps to replace its two inductors.

4  From Passive LC to Active-RC Filter: Inductor Simulation (Gyrator)…

283

3. Impedance Transformation using FDNRs: [Scale every component by a dimensionless - but frequency-dependent – factor]

(i) Use dual LC Network from Tables: L1

R1

L5

L3

L2

L4

C2

C4

[Minimum C Filter] R2

(ii) Scale each impedance by ω0/s where ω0 is selected as filter cut-off (3dB) frequency. ω0 ⋅ sL1 s

ω0 ⋅ sL3 s

ω0 ⋅ sL2 s ω0 1 ⋅ s sC2

ω0 ⋅ s R1

ω0 ⋅ sL5 s

ω0 ⋅ sL4 s ω0 1 ⋅ s sC4

ω0 ⋅ s R2

FDNR! C1’=

1 [F] ω0R1

1 2 s ⋅D 2 D=C2 ω0 [ sec ] Ω

Z’C2 =

R4’=ω0L4[Ω]

Slide 10.23 shows how to circumvent the problem of excessive opamp numbers for the simulation of floating inductors by transforming the filter for FDNR design. This transformation means scaling each component of the filter by the dimensionless factor ω0/s. For this (and in contrast to most other practical applications) the LCR dual, i.e., the minimum-C version of the filter, is required.

(iii) Transformed LC network consists of resistors, capacitors, and FDNRs. L1 R1

L3

L5

L2

L4

C2

C4

R2

R’1 C’1

R’2 D2

R’3

R’4

R’5

D4

DCRFilter C’2

Each grounded capacitor becomes 1 FDNR. Each FDNR requires 2 opamps. Instead of 8 opamps with gyrators the equivalent FDNR network requires only 4 opamps! Slide 10.24 shows that with the FDNR transformation, the five inductors become resistors, the two resistors become capacitors, and the two grounded capacitors become FDNRs. Each FDNR requires two opamps. Thus, instead of the eight opamps necessary for the inductor simulation of this filter, the equivalent FDNR network shown in this slide requires only four opamps.

10  Active Filter Design Techniques

284

From Slides 10.25, 10.26 and 10.27, we are reminded from Chap. 9 that an FDNR can be realized with a circuit very similar to that of a gyrator; each one requires only two opamps.

Circuits for Gyrator and FDNR Design

Z1 R1

Z4

Z2

Z3

R2

R3

C4

Z5

R5

Z in =

Z1Z 3 Z 5 Z2Z4

With a capacitor for Z4 , and resistors for the other impedances, we obtain the equivalent inductor L eq , where: L eq =(R 1 R 3 R 5 C 4 )/ R 2 With a capacitor for Z1, a second capacitor for Z5 , and resistors for the other impedances, we obtain the Frequency-Dependent Negative Resistor or FDNR: (s 2 D) -1 , where: D= (C 1 C 5 R 2 R 4 )/ R 3

Slide 10.26

Opamp-RC Gyrator Circuit Simulating an Inductance.

R2

R1

R3

C4

R5

1 Leq =

C4 g2

Z in =

2

sC Z1Z 3 Z 5 = 24 = sLeq g Z2Z4

Slide 10.27

Opamp-RC Circuit Realizing an FDNR (Frequency-Dependent Negative Resistor)

R2

C1

R3

R4 C5

D=

C1C5 R2 R4 R3

Z in =

Z1Z 3 Z 5 1 = 2 Z2Z4 s D

4  From Passive LC to Active-RC Filter: Inductor Simulation (Gyrator)…

285

FDNR-Transformation of 5th-order low-pass filter Minimum C RLC ladder filter

L1

R1

L5

L3

L2

L4

C2

C4

R’1 R2

C’1

R’3

5 802

DCRFilter C’2

D4

3 249

1 051

10.11

10.02 10.09

3 001

1 051

2 233 3 001

R in Ω C in nF

R’5

6 951 Hz

5 293 2 233

FDNR

R’4

D2

Zero frequencies: 4 758 Hz

∼

R’2

3 001

3 157

FDNR

3 279 15.03

15.08

In Slide 10.28 we show the transformation of the minimum-C version of a fifth-order low-pass filter into an FDNR equivalent filter.

FDNR-realization of 5th-order low-pass filter Zero frequencies: 4 758 Hz 5 802

5 293 2 233

∼

6 951 Hz

1 051

10.11

3 249 10.02 10.09

3 001

1 051

2 233 FDNR

v

R in Ω C in nF

v 3 001

3 157

3 001

3 279 15.03

15.08

FDNR

In Slide 10.29 we show the final FDNR filter that is equivalent to the original LCR filter shown in Slide 10.22. Compared to the gyrator-C version of this filter the FDNR version requires half the number of opamps.

10  Active Filter Design Techniques

286

Biquad-realization of 5th-order low-pass filter r1

r0

r2 c1

c0

r3

r5 β1

c2 c3

r4

c4

r in Ω c in nF

r1=19 081 r2=19 081 r3=9 540 r4=93 529

c1=1.2 c2=1.2 c3=2.4 c4=2.7

c5 r7

β2

c6 c7

r8

c8

2nd-order biquad

3rd-order biquad r0=23 068 c0=5.6

r6

β1=1.84163

r5=12 389 r6=27 875 r7=8 577 r8=72 462

c5=2.7 c6=1.2 c7=3.9 c8=1.5

β2=1.694

In Slide 10.30 we show the equivalent single-amplifier (I-SF-4) biquad version of the FDNR filter shown in the previous slide. The two filter versions, FDNR and biquad, can be fine-tuned to provide the same frequency response, but not necessarily the same performance in terms of dynamic range, noise, ease of tuning, and power consumption. Furthermore, the FDNR filter has the inherent problem of terminations; the resistive terminations of the original LCR filter become capacitors; this defies the nature of a low-pass filter. In the example shown in Slide 10.29, a one-sided terminated filter is used which solves part of the problem – if a one-side terminated filter can be tolerated. A high-impedance shunt resistor can alleviate the problem for the other termination and, if necessary, for both terminations. Furthermore, the two biquads require only half the number of opamps compared to the FDNR version – and a quarter of the number required for the gyrator version. However, the twin-T is generally more difficult to tune – if tuning is possible at all. This brings us to the question, broached briefly at the end of Sect. 2 above, as to what is to be gained by using the one or the other of the design methods described in this chapter. The answer depends on many factors, including on the application, the technology used for its implementation, and so on. To illustrate these and other factors in a broad sense, we now review the realization of a given filter by several of the methods discussed so far and list some of the criteria that are often used to compare and evaluate their performance.

5

 Brief Review of Design Techniques for Inductorless Filters A (Example: A Third-Order High-Pass Notch Filter)

For our review we return to the third-order high-pass notch LCR filter that we previously examined in this chapter (see Slide 10.21). We shall use this “sample filter” in order to compare some of the practical considerations that may influence a decision as to which of the equivalent inductorless filter design methods discussed in this chapter might be the appropriate choice to replace the LCR high-pass notch (HPN) filter for a given application.

5  A Brief Review of Design Techniques for Inductorless Filters…

287

EXAMPLE: A THIRD-ORDER HIGH-PASS-NOTCH FILTER Frequency- Response Specifications A[dB] Stop Band

Amplitude- Response Stop Band

25

Pass Band

Pass Band

0.5 92.4 1.73nF

Vin

~

151.8

f[kHz] R5 C1 (500Ω) (1.54nF)

3.66nF

6.18nF 500Ω 550.8µH

LC Ladder Filter: Singly Terminated

Vin

~

C2 L2

C3 (1.54nF)

(6.1nF) (618.4µH)

RL (500Ω)

LC Ladder Filter: Doubly Terminated

Slide 10.31 shows our sample LCR, HPN filter again, both with single and double termination. Slide 10.32 shows the transfer function of the Filter Transfer Function Satisfying Specifications sample filter and the numerical values of its Transfer Function: coefficients. s ( s 2 + ω z2 ) The five different T (s) = K  2 ωp  2 inductorless filter realizas + ω p  ( s + ω0 )  s + where:  q tions of the original highp   pass notch LCR filter that we shall briefly review are ω z = 516.69 kHz (i) the twin-T biquad, (ii) ω p = 889.584 kHz the bridged-T biquad, (iii) q p = 2.2 the gyrator ladder simulation, (iv) the signal-flow ω 0 = 1.29974 MHz graph ladder simulation, K =1 and (v) the multi-amplifier biquad. Although this is not a comprehensive list of design methods, it demonstrates some of the constraints and considerations that are relevant for the selection of a design method for a given technology. A realistic performance comparison must be based on the technology for which the designs are intended and the constraints that this technology implies. We assume that the analog inductorless technologies for which our designs are intended are either of the integrated-circuit (IC) type or for discrete components. There is, of course, a multitude of IC technologies, most of which are in a constant state of flux. This leaves room for a large variety of technology-based constraints. Nevertheless, within this variety, there are certain typical fix points that are more or less valid for most of these technologies (and yet subject to constant change).

10  Active Filter Design Techniques

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For IC filter implementation, we select the following example of mainstream constraints: 1. Total capacitance Ctot (e.g., 400 pF); maximum capacitor spread Smax (e.g., 20). 2 . If possible, use opamps in inverting mode with a virtual ground at each amplifier input terminal. This is beneficial for optimum dynamic range. 3. Either the filter cutoff frequency or the notch frequency should be tunable in steps, by a capacitor or resistor array. Although the RC components on an IC chip can vary by a considerable amount (e.g., up to 50%), a high degree of tracking by like components can be assumed. Nevertheless, the tunability of at least the cutoff frequency, and/or the notch frequency, is generally required to overcome the large component tolerances. [Note: the cutoff frequency is tuned by the pole frequency and the notch frequency by the zero frequency. Either one can be used to accurately place the 3 dB frequency – referred to as the cutoff frequency – of the rising amplitude-response slope of the high-­ pass notch filter.] 4. For many applications, especially in communication systems, the filter must be balanced, i.e., symmetrical with respect to ground. 5. The total sum of resistor values should be minimal in order to minimize filter noise. For discrete-component filter implementation, we select the following mainstream set of constraints: 1. Bounds on capacitor values (typically 100 pF ≤ C ≤ 10 nF) Maximum capacitor spread (typically ≤100). 2. 99% of the circuits should remain within specs for resistor tolerances of 1% and capacitor tolerances of 2%. This means that the sensitivity of the filter response to component variations should be low. The concept of sensitivity to component variations will be discussed in the next chapter. 3. Resistor values should be as low as possible in order to minimize thermal output noise. For each of the methods listed above, the design equations and circuits designed to fulfill the specifications given in Slide 10.31 are given in the following slides.

(i) The Twin-T Single-Amplifier Biquad (hpn filter) R9 (1k) R5 (4.99k)

R10 (.313k)

– +

Vin C1 R 7 (1n) (.68k)

R8 C2 (4.57k) C3 (0.4n) (0.3n)

0.68k 4.99k

R6 (8.74k)

C4 (.1n)

1nF

+

C0 (1nF) R0 (1.74k)

0.3nF 8.74k

1nF Vout

Vin

0.4nF 0.05nF

1nF

–

0.4nF

2k 9.14k –

1.5k Vout

0.31k 0.31k 1nF

+ 4.99k

Grounded

0.68k

8.74k 0.3nF

Balanced (i) Slide 10.33 shows the grounded and balanced-to-ground circuit version of the twin-T biquad (I-DF-4, see Chap. 8).

5  A Brief Review of Design Techniques for Inductorless Filters…

289

Slide 10.34 shows the design equations for the twin-T biquad.

Design Equations for Twin-T Filter

T (s ) = K

(

)

s s 2 + ω z2   2 ωp  s + s + ω p2  (s + ω0 )   qp  

ω z2 =

1 1 = R5 R6CsC3 Rs R7C1C2  1 + Rs R8    1 + C4 C s 

ω p2 = ω z2 

(1 + C4 Cs )(ω p ω z ) ; qˆ = 2 qˆ (1 R8Csω z + Rs C4ω z ) − R10 R9

q p = qˆ

K=

1 + R10 R9 ; 1 + C4 C s

Cs =

C1C2 C1 + C2

1

(1 + C2 C1 )(1 + C2 C3 )

; Rs = R5 + R6 ; ω0 =

1 R0C0

Slide 10.35 gives the design equations for a twin-T null.

Design Equations for Twin-T Filter,cont‛d R9 (1k) R5 (4.99k)

R10 (.313k)

R6 (8.74k)

C0 (1nF)

– +

Vin C1 R 7 (1n) (.68k)

R8 C2 (4.57k) C3 (0.3n) (0.4n)

Vout

R0 (1.74k)

C4 (.1n)

The two conditions for a twin-T null are:

(i)

1 R5 R6C s C3

where

=

1 Rs R7 C1C2

, (ii )

C p = C1 + C2 , R p =

C3 R7

=

Cp Rp

R1 R2 R1 + R2

,

10  Active Filter Design Techniques

290

(ii) The Bridged-T Single-Amplifier Biquad (hpn filter) 1n

R1 (.388k) Vin

C4 (.5n)

.5n

.388k

C3 (1n)

1.18k

R5 (6.6k)

R2 R6 (5.9k) (1.18k)

+

Vin

R0 (.770k)

R9 (10k)

Vout

+

.11k 11.8k

Grounded

Vout

10k 1.18k

.388k

1.5k

2k

.11k

R8 (1k)

R7 (.11k)

1n

–

10k

C0 (1n)

–

6.6k

+ –

.5n

1n

6.6k

1n

Balanced (ii) Slide 10.36 shows the grounded and balanced-to-ground version of the bridged-T biquad (II-DF-­3, see Chap. 8).

Design Equations for Bridged-T filter ω 2p = ω z2 = C4 R2 qp =

+

R1 + R2  1

 R7 R8 −  R1R2C3C4  R5 R6 R9 ( R7 + R8 )  R1 + R2  1

− R1R2C3C4  R5

C3 + C4 R5

=

R7 ( R8 + R9 )  R6 R8 R9

R7 ( R8 + R9 )  C4 R8 R9

 +  R1

 

C3 + C4  R6

 

ωp

 R7 R8 R7 R8 ( R1 + R2 ) − −   C3C4  R5 R6 R9 ( R7 + R8 )  C3 R1R2 R9 ( R7 + R8 ) R8 K= R7 + R8

C3 + C4  1

Slide 10.37 gives the ­corresponding design equations.

5  A Brief Review of Design Techniques for Inductorless Filters…

(iii) Gyrator Ladder Simulation

The transmission matrix of a gyrator is:

291

(iii) Slide 10.38 and Slide 10.39 show the equations on which the gyrator ladder simulation is based.

0 1  A B  g [ ABCD ]gyrator =  =    g 0  C D  gyrator When terminated by a load impedance ZL the input to the gyrator is:

 AZ + B  B 1 1 Z in =  L = ⋅ = 2   CZ L + D  A= D=0 C Z L g Z L Slide 10.39

Equations for Gyrator Design, cont‛d

With a capacitor as the load, the input impedance becomes: Z in =

sCL = sLeq g2

The input impedance of the the twonullor cascade, loaded by ZL is: Z in =

Z1Z 3 Z L Z2Z4

10  Active Filter Design Techniques

292

Gyrator Ladder Filter C1 RS (25k) (30.8p)

R5

C1

(500Ω)(1.54nF)

C2

~

RL(25k) Vout R21(50k)

(6.1nF)

R22(50k)

RL

+

(500Ω)

(618.4µH)

−

L2

(1.54nF)

−

Vin

C2 (122p)

Vin ~

C3

C3 (30.8p)

R23 (50k)

+

C24(12.37p) R25(50k)

Doubly terminated gyrator-C ladder network a) grounded Slide 10.40 shows the third-order high-pass notch filter and its gyrator-C equivalent.

RS C1 RS

C1

C3 n1

C2

C3

R21 N1

C2 Vin

2RL Vout Vin

2L2

C1

Balanced LCR hpn

+ – + –

– +

R23 N2

C24 2R25

C2

C3

R22

N3 R21

C2 RS

n2

R22 n3

RS C1

N4 R23

Vout

C24 n4

C3

Balanced nullor version of hpn

+ – + –

– +

Balanced simulated inductor using multiple input-output amplifiers

Slide 10.41 shows the balanced LCR HPN filter, the balanced nullor version of the HPN filter, and the balanced simulated inductor using multiple input-output amplifiers.

5  A Brief Review of Design Techniques for Inductorless Filters…

V1 I1

V3 I3

Z1 Y2

VS

293

–1

–1

Z3

V2

Y4

I2

V4

VS

1

I4

V1 1 I1 Z1

1

1 V 1 Y2 2

I2

V3 1 Z3

I4 1 V 4 Y4

I3 1

–1 1

1

V S 1 V 1 Z1 I 1 1

I1 1 Z1

I2 Y2 V2 1 V3 Z3 I3 1 I4 Y4 V4

VS

1

1

I1

1

Z1 VS

Y2

V1

1

I2

I2 Z3

V2

Y4 V3

1

1

1

Z1

V4

+1

Y2

V4

V3 +1 I4

1

Z3

Y4 V3 –1 V4

V2

1

1

–1 Y4

I3

I2

V1 –1

VS

V2

I4

+1

+1 Z3

1

I1

I3

+1

–1 Y2 V1 +1

1

I4

1

I2

1

1

SFG Manipulation, cont‛d 1

I1

1

Y2

Z1 Vs

I3

I2

V1

-1

V2

I

V L

-1

V3

Rs

V4

1 sC

1

I

C

R

V

I

VR

V

-1

1 sL

I –1 sC

1

R0I1

1

1

1 sC2R0 1 sC1R0

R0I2

–

1 1

1 R0I1

R0I1 -1

Vs

–

V1 1

1

R0 sL2 –

1

V2

Vs

R0I3

R0I2

– 1 1 sC2R0 sC1R0

1 sL2

V2

1

I3 1

I2

1 - 1 sC1 sC2

V1

Vs

1

I4 1 RL

- 1 sC2

V3

-1 1

V4

1

V4

sL

C

V

1 sC

1

–1

V L

I

I1

C

Y4

Z3

1

I

I4

1

V1 1

R0 sL2

V2

1 sC3R0

V3

R0I4 1

-1

V4

1

1

1 sC3R0

1

-1

R0I3 1

V4

(iv)  Slide 10.42 and Slide 10.43 show the signal-flow graph ladder simulation and how to manipulate the initial ladder SFG into one that transforms all frequency- dependent transmission terms into terms s−1. These terms correspond to integrators in the Laplace domain.

10  Active Filter Design Techniques

294

Opamp realization of SFG HPN filter 1k

1

V4

–

–

+ –

+

1k

25k 1k

30.8p –

V2

1k

+

1

1k

Vout

30.8p

+

1

V1

–

Vs

25k

1 1k

1 sC3R0

+

-1

R0 sL2 –

– 1 – 1 sC2R0 sC1R0

25k

CL2 (49.5p) –

R0I1

R0I3

R0I2

+

R0I1

1

–

1

+

C2 (122p)

1 1

25k 25k

1k

1k

Vs ~

1k

+

1k

1k

1k

1k

–

–

1k

+

1k

–

+

1k

(v) Signal-Flow Graph Derivation for Multi-Amplifier Biquad Realization

T (s ) = K

s2 ± s2 +

ωz qz

ωp qp

ωz  1  21   + ωz   qz  s  s T (s ) = K 2 ωp 1 s + ω 2p   1+ qp s 1±

s + ω z2 s + ω p2

2

−ωp

Vin

K

ωz2 ωp

1 s

ωp

ωp −q p

1 s

1

ω ± qz z 1

Signal-flow graph corresponding to general Biquad function

V¢out

Slide 10.44 shows how each of the integrators in the Laplace domain is replaced by an analog integrator using opamps. Some integrators are inverting, which coincides with the simple opamp integrator which is inverting; others that are not require an additional opamp inverter to change their polarity.

(v) Slide 10.45 shows the multi-amplifier biquad design method and how the original Laplace-­t ransformed second-order transfer function T(s) of a biquad is transformed into an equivalent SFG. As in the previous method, the aim is to isolate branches with s−1 terms.

5  A Brief Review of Design Techniques for Inductorless Filters…

295

From Signal-Flow Graph to Multi-Amplifier Biquad −ωp ωz2 Vin

K

ωp

1 s

ωp

Integrators ωp −q p

1 s

1

ω ± qz z

V out

R3(=Rw ) p (22.5k)

1

R1(=Rq ) p (49.4k)

R4 (24.7k) Vin

C1 (50p) – +

R5 (2k)

Integrators R2 (22.5k)

C2 (50p)

R0

2

(1k)

R0

1

(1k)

– +

R8

R6 (1.37k)

R7 (1k)

– +

R (1k) – +

100p V out

17.4k

Vout

First-order high-pass circuit added on to obtain third-order hpn filter.

Slide 10.46 shows how the s-1 branches are realized by opamp-based integrators. As a biquad, the denominator of T(s) is of second order, so that two opamp integrators are expected in the forward path of the SFG. However the opamp integrator being inverting, and the feedback required to be negative, an additional simple opamp-based inverter is necessary in the forward path. This, with the feedback path, constitutes the realization of the denominator. For the general second-order numerator, an additional adder/subtractor opamp circuit is required to provide the three terms of the numerator for the biquad transfer function. Furthermore, a first-order high-pass circuit is added to the output of the biquad, in order to obtain the third pole for the third-order high-pass notch filter.

Slide 10.47 shows the general overall transfer function of the multi-amplifier ωz qz ω z2 biquad. The general sec  ω ond-order numerator term     R R R R s 2 + s  p 1 − 7 ⋅ 1  + ω p2  7 3 − 1 can provide the expresV R  R6 R4   q p  R5 R4  T (s ) = out = − sions required for every ω Vin R7 s 2 + p s + ω p2 possible biquadratic transqp fer function, ranging from high-pass/low-pass to 1 ωp = notch, band-pass, and R2 R3C1C2 all-pass. Note that the frequency C1 response of each one of q p = R1 R2 R3C2 these filter designs is identical and fulfills the prescribed specifications of the high-pass notch filter given in Slide 10.31. As pointed out above, which of these circuits is most appropriate for a given application depends on many individual, p­ roject- related factors. Briefly,

Design Equations for Multi-Amplifier Biquad

10  Active Filter Design Techniques

296

s­ urveying the five circuits for general usage, a performance evaluation related to the constraints given above results in the following observations: • The twin-T, bridged-T, gyrator, and FDNR circuits do not fulfill the requirement “opamps in inverting mode,” i.e., they require opamps in common-mode operation. • This leaves the signal-flow graph (SFG) ladder simulation and the multi-amplifier biquad (MAB) circuits to select from (given the technology constraints assumed in this study). • The SFG circuit has better thermal noise properties; the MAB circuit has better tuning properties. • The ultimate choice of circuits will depend on the application; both the SFG circuit and the MAB circuit have been successfully applied in analog front-end IC system chips. Finally, to round off this overview of active filter design techniques, it is important to point out that other design techniques exist that have not been explicitly dealt with here. They are based mainly on current-mode circuits, i.e., circuits that use current amplifiers or transconductance amplifiers, instead of opamps. Typical circuits of this kind are based on so-called gmC circuits, current-conveyor circuits, and current-controlled current-source circuits. Although the active devices are current based, the basic building blocks of the circuits are generally integrators and summing circuits, just as the circuits described in this chapter. Thus, these other design techniques are based on the same theoretical concepts that are dealt with here.

6

Increasing Dynamic Range by Using Balanced (Differential) Filters

With the increasing trend to lower-power IC electronic circuits and systems and the concurrent low battery voltage, the dynamic range of an analog circuit and of an inductorless filter becomes an important issue. (This is another one of those issues that was less critical with discrete-component circuits and LCR filters.) One way to improve dynamic range in low-power IC filters is to convert the grounded circuits into balanced ones. Moreover, in many telecommunication systems, balanced-to-ground circuits and filters are in any case the rule. The conversion from a grounded to a balanced circuit is relatively simple to carry out, as will be demonstrated in the final section of this chapter. Slide 10.48 shows a grounded Type I Class 4 single-amplifier (I-SF-4) third-order low-pass filter biquad. Being a Class 4 biquad, it is based on positive feedback to realize its complex-conjugate poles (the third pole is negative-real).

Increasing Dynamic Range by using Balanced (Differential) Filters Example: Class 4 (Pos. Feedback) C2 R1 Vin

C1

R2

R R3 C3

(β-1)R A→∞ Vout

Common Mode Operation

6 Increasing Dynamic Range by Using Balanced (Differential) Filters

297

Slide 10.49 shows the equivalent balanced filter. Note that the value of the components in series remains the same; the value of the parallel ones is such that their impedance is doubled. Thus, an imaginary ground plane halving the circuit horizontally would result in the original grounded circuit on either side of the ground plane. Furthermore, for this particular Class 4 circuit, which relies on common-mode operation, the balanced-circuit version has the advantage of operating in the virtual-­ ground opamp input mode which in itself also increases the dynamic range of the entire filter.

Slide 10.50 shows an actual grounded singleended third-order low-pass filter designed as an anti-­ aliasing filter for a data communication system. The component values are given in the diagram.

Single-ended 3rd-order Low-pass Filter: RG(0.935k) R1 (11.3k)

V1 ~

C1 (440 pf)

R2 (11k) C2 (180 pf)

R3 (12.1k) C3 (112 pf)

RF (1k)

– + V2

10  Active Filter Design Techniques

298

C2

IN_P

R1

R2

R3

11.3K

12.7K

13.0K

3 + 2 –

7

VCC

56P

VCC U1

R9 6 LT 1227 205 48

VEE

3 + 2 – R7 1780 R16

C3 56P

C1 120P

U3

81

7 LT 1206

EN

OUT_P

6 4 R11 VEE R13

562 A R17

150

562

A C5 33P

6190 C4

IN_N

R4

R5

R6

11.3K

12.7K

13.0K

VCC R19 4.12K 7 3 + 2 –

U2 6

8 1 U4 3 + 7 2 – EN LT 1206

R10

LT 1227 205 48

VEE

VCC

56P

R8 1780 R15 562 A R18

OUT_N

6 4 R12 VEE R14 562

150

A C6 33P

6190

Slide 10.51 shows the balanced version of the grounded filter in the previous slide. As with the latter, this circuit, coming from the “real world,” was not manufactured in the form of an integrated-­circuit chip, but rather with discrete components. As such, the opamps were realized by the cascade of two separate discrete amplifiers, and the passive components were also in discrete form. The capacitors in the input ladder network are tapered such as to decrease in value in a prescribed way from the filter input, to the opamp output, terminals. As we shall see in the next chapter, this strategy is referred to as “tapering” the input ladder circuit in order to decrease the sensitivity of the filter to ambient changes and component tolerances.

Chapter 11

Elements of Sensitivity Theory

Slide 11.1

Chapter 11

Elements of Sensitivity Theory

1

Introduction

The topic of network and filter sensitivity became relevant with the advent of inductorless, and in particular active-RC, filters. This is because the previously used passive LCR ladder filters have an extraordinary quality of being relatively insensitive to variations in component value caused by temperature and other ambient influences, as well as to initial tolerance of component values. Thus, neither sensitivity to ambient change nor filter stability in the sense of stray circuit oscillations was ever a serious – if any – issue with passive LCR filters. The situation is quite different with active-RC filters and networks. To obtain the necessary filter characteristics, these circuits depend on active feedback to obtain the frequency-selective features implicit in filter design. Feedback circuits are potentially unstable. Thus, even while ignoring the

© Springer Nature Switzerland AG 2019 G. S. Moschytz, Analog Circuit Theory and Filter Design in the Digital World, https://doi.org/10.1007/978-3-030-00096-7_11

299

11  Elements of Sensitivity Theory

300

extreme case of stray oscillations, such circuits are prone to unacceptable variations in overall circuit behavior. Consequently, the sensitivity of circuit performance to ambient variations and component tolerances has become a key issue for active-RC filter circuits; in fact, it has become an integral part of every discussion and evaluation of analog inductorless filters and other active circuits. This is why sensitivity theory is necessarily an integral part of the material in this book. In this chapter, we deal mainly with the mathematics and theoretical aspects of circuit sensitivity. The resulting sensitivity theory has become a useful and important tool for the performance evaluation of active-RC and other active inductorless circuits and filters. The practical implementation of this theory is demonstrated by the inclusion of illustrative examples within the chapter.

2

The Definition of Sensitivity

In Slide 11.2, the sensitivity of a linear system is defined as follows: if F(x1,x2,…,xn) is a Given a linear System: characterizing function of a Input F(x1,x2,...,xn) Output linear system and xi ≡ x is a parameter of the system, We wish to express the relative change then the sensitivity of F to small changes in xi is ∆F/F as a function of a relative change defined as the relative ∆xi/xi, where xi≡x is some parameter of the change ΔF/F as a function function F(x1,x2,…,xn). of a relative change Δxi/xi. ≈0 The sensitivity function is Taylor Series: derived from the Taylor 2 dF 1 2 d F series of F with respect to + ∆ x ⋅ 2 + ..... F ( x0 + ∆ x) = F ( x0 ) + ∆ x ⋅ small changes Δxi. The dx 2 dx emphasis here is on small, because the Taylor series is calculated from the mathematical derivatives of the function at a given point.

The Definition of Sensitivity

The Definition of Sensitivity cont’d ∆ F = F ( x0 + ∆ x ) − F ( x0 ) = ∆ x ⋅

dF dx

∆ F  ∆ x dF  x  dF / F  ∆ x ∆x = ⋅  = = S xF ⋅ ⋅ F  F dx  x  dx / x  x x dy 1 dx = → dy = d [ln x ] = Note: For y ( x ) = ln x → dx x x dF dx  Thus: = d [ln F ]; = d [ln x ]; F x Finally: ∆F dF / F d [ln F ] dF x F “Relative S xF = = = ⋅ ≈ x ∆ dx / x d [ln x ] dx F Sensitivity” x 

For: Computation Measurement

In Slide 11.3, we show that neglecting terms higher than the first, and rearranging, results in the so-called relative-sensitivity function of F with respect to x, designated S xF .

3  Some Useful Sensitivity Relations

301

For F(x1, x2,...xi...xn): n ∆ xi ∆F ∆x ∆x = S xF1 1 + S xF2 2 + .... = ∑ S xFi F x1 x2 xi i =1 ∆ x F i = VxFi : “variation of F with respect to xi” Let S xi xi ∆F n F Then: F = ∑Vxi i =1

∆F ≈ V XFµ F

F F If V X µ >> Vxi all i (i ≠ µ)

Sensitivity Definitions:

S Fx =

semi-relative sensitivity:

S Fx

semi-relative sensitivity: o F Sx (less common)

variation:

3

dF / F d [ln F ] = dx / x d [ln x ] dF dF = = dx / x d [ln x ] dF / F d [ln F ] = = dx dx

relative sensitivity:

absolute sensitivity: (gradient)

Slide 11.4 shows how the sensitivity function can be extended from differential to difference terms, using measurable function and parameter changes. This results in the so-called variation function of F with respect to xi, i.e., VxFi .

F '=

dF dx

VxF = S xF

Slide 11.5 summarizes the various definitions of sensitivity, ranging from relative sensitivity to absolute sensitivity (gradient) and variation. The most common in this context are relative and semi-­ relative sensitivities, as we shall see in the remainder of this chapter.

∆x x

Some Useful Sensitivity Relations

The expressions for the relative-sensitivity function given in Slide 11.3 permit a compendium of simple sensitivity relations for frequently occurring functions in linear circuit theory to be derived. These are listed in the following two slides.

11  Elements of Sensitivity Theory

302

Some Useful Sensitivity Relations dF dy dF x  Ex: F = y ( x) + c ⋅ = dx dx dx F dy x dy/ y y (x) y ( x) S xF = ⋅ = ⋅ = ⋅ S xy ( x ) dx y ( x) + c dx / x y ( x) + c y ( x) + c #12 S xF =

7) S xy = Suy1 ⋅ Suu21 ⋅ ... ⋅ S xun

1) S xx = S xc⋅x = S cx⋅x = 1

where y = y{u1 [u2 (...un ( x) ⋅ ⋅⋅)]}

2) S xc⋅ y = S xy

3) S = (S y x

4) S

yn x

)

∗* 8) S xu⋅v = S xu + S xv ∗* 9) S xu / v = S xu − S xv

x −1 y

= n ⋅ S xy

1 5) S xyn = ⋅ S xy n 6) S xy = Suy1 ⋅ S xu1 + Suy2 ⋅ S xu2 + ... where y = y (u1 , u2 ,..., un )

10) S 1x / y = − S xy 11) S1y/ x = − S xy

∗* 12) S xy +c = 13) S xu +v +...

1 ⋅ S xy ln y

dF du dv = ⋅v + ⋅u dx dx dx dF x ⋅ = S xF = S xu⋅v = dx u ⋅ v x du x dv = ⋅v⋅ + ⋅u ⋅ u ⋅v dx u ⋅ v dx

ϕ

16) S xy = S xy + jϕ y ⋅ S x y where y = y e

jϕ y

17) S xy = Re{S xy } 1

ϕ

18) S x y =

ϕy

Im{S xy }

19) S xy* = (S xy )

*

where y * = y e

S xu⋅v = S xu + S xv

− jϕ y

S

21) S xcos( y ) = y ⋅ tan ( y ) ⋅ S xy 22) S 23) S

cosh( y ) x

= y ⋅ coth ( y ) ⋅ S

#8 ;

Similarly:

20) S xsin( y ) = y ⋅ cot ( y ) ⋅ S xy sinh( y ) x

y ⋅ S xy * see example y+c 1 (u ⋅ S xu + v ⋅ S xv + ...) = u + v + ...

Ex: F = u ( x) ⋅ v( x)

y

14) S xe = y ⋅ S xe 15) S xln y =

In Slide 11.6 some of the expressions for the sensitivity function are listed. In order to understand how they were obtained, the derivation of the relation no. 12 is given at the top of the slide.

y x

= y ⋅ tanh ( y ) ⋅ S xy

u/v x

= S xu − S xv

#9.

Slide 11.7 lists some more useful relations and illustrates the derivation of sensitivity relation nos. 8 and 9 on the right-hand side of the slide. These relations will be used repeatedly in the course of this chapter. They will be seen to simplify much of what is included in sensitivity theory for active-RC networks.

3  Some Useful Sensitivity Relations

R0

Example:

303

(β-1)R0

C2

G1

V2

G1 C4 V s where: ω2p = G4 (G1 + G3 ) T (s) = 2 = K ⋅ 2 2 C2 C4 V1 s + (ω p / q p ) s + ω p G4 (G1 + G3 )C2C4 A(Gi , C j ) qp = = (G1 + G3 )(C2 + C4 ) + C2G4 − βC2G3 B (Gi , C j ) V1

G3

C4

∆q p q = ∑VGi p qp

G4

K = β⋅

∆ω p ω ω ω = ∑VGi p + ∑VCi p + ∑Vβ p ωp q q ∆K + ∑VCi p + ∑Vβ p = ∑VGKi + ∑VCKi + ∑VβK K

With Table of Sensitivity Relations: ω2

ω

ω

SGi p = 2 SGi p = − 2 S Ri p ex:

ω2

SG1p =

G1G4 SGG1G4 G1G4 + G4G3 1 1

qp = ex:

A B

1 ω2 ω S Ri p = − SGi p 2 G1 1 G1 ω = S R1p = − G1 + G3 2 G1 + G3

1 q SGip = SGAi − SGBi 2

1 G1 G (C + C4 ) q + 1 2 S R1p = − ⋅ 2 G1 + G3 B(Gi , C j )

1 q S Rip = − SGAi + SGBi 2 ∆F ∆R ( R1 ) = S RF1 1 R1 F F = ωp , qp.

Slide 11.8 presents an example of a single-amplifier band-pass filter biquad (Type I – DF-4; see Chap. 8) and its transfer function T(s). Furthermore, the overall variation of the pole Q, qp, the pole frequency ωp, and the constant K, expressed as the sum of their individual variations with respect to all the components of the circuit, is also given.

In Slide 11.9, examples of the sensitivity calculations which are part of the variations of the circuit given in the previous slide are readily calculated using the sensitivity relations of Slides 11.6 and 11.7.

11  Elements of Sensitivity Theory

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Example: Single-stage High-Frequency CMOS Common-source Amplifier Consider the following enhancement CMOS inverter with active load (Q2), common source driver (Q1), and all parasitic capacitances. Note: Vbs = 0 for this configuration.

VDD

Cgs2 Vb

Q2

Cbd2

Cgd2 Cgd1 RS Vin

a. Inverting CMOS Amplifier

Vo CL

Q1 Cbd1

~

Cgs1 VSS

Slide 11.11

b. Simplified MACROMODEL

Rin

g

d gmVgs

Vin

Rout Cout Vo

Cin s

By inspection Av= -gmRout = -gm1 (rds1//rds2) = -gm/(go1 + go2) Rin = Rs

In Slides 11.10, 11.11, 11.12, and 11.13, in preparation for an example of sensitivity analysis, a single-stage CMOS common-source amplifier is shown with the corresponding design expressions and typical process and component values.

Cin = Cgs1 + Cgd1(1 + gm1/(go1 + go2))

Cout=CL + Cbd1 + Cbd2 + Cgd2 + (1 + (go+go2)/gm1)Cgd1 Rout = 1/(go1 + go2)

3  Some Useful Sensitivity Relations

Typical C values are:-

305

Slide 11.12 Cgd1 = Cgd2 = 0.02pF Cbd1 = 0.05pF, Cbd2 = 0.01pF Let’s vary CL = 0 to 1pF Assume Rs = 100Ω

Process values Assume

(W/L)1 = 10, ID = 50 µA λ: modulation parameter

λ1= λ2 =0.01V–1 L=10 µm K1= 20µ A/v2, K2= 10µ A/v2 Cox = 3.5 × 10–4 pF/µ2.

Slide 11.13 Equations

KW ID = (Vgs-VT)2 2L gm = 2 KW ID , go = λI D 2L [gm/go] = 2 KW 2L

λ ID

A = –gm1/(go1+go2) = -2 K1W1 = –100 2L1 (λ1 + λ2) ID Cgs = (2/3) W L Cox = 0.23pF

Slide 11.14 shows the sensitivity analysis of the CMOS common-source amplifier voltage gain A to variations of the drain current ID and of the channelk1W1 / 2 L1 g m1 length modulation (CLM) A=− kW = −2 −2 1 1 parameter λ – again using (λ1 + λ 2 ) I D g o1 + g o 2 2 L1 the sensitivity relations of C1 =  ∆ I D ∆λ1  ∆A A ∆I D A ∆ λ1 λ1 + λ 2 Slides 11.6 and 11.7. [The = f , + Sλ1  = SID A ID λ1 CLM is the shortening of  I D λ1  kW the length of the inverted −2 1 1 -1/2 1 C 2 L1 channel region with S IAD = S I D1 ( I D ) = − ; C2 = 2 ID increase in drain bias -1 λ1 (drain to source voltage) C SλA1 = Sλ12 ( λ1 +λ 2 ) = − ; for large drain biases. λ1 + λ 2 Channel-length modulation is important because it decides the MOSFET output resistance, an important parameter in circuit design of current mirrors and amplifiers.]

Find Variation of gain A as a function of variations of Drain Current ID and of Channellenght modulation parameter λ.

11  Elements of Sensitivity Theory

306

Single-stage CMOS Common-source Amplifier (contd.) 1 ∆ ID λ1 ∆λ ∆A =− − ⋅ 1 A 2 I D λ1 + λ 2 λ1 ∆ A 1 ∆ I D ∆λ 1 ∆β + =− − ∆A A 2 ID λ 2 β ID λ1 A Special case: λ1 = λ 2 = λ; β = KW / 2L ∆I ∆λ ∆β 2 β ∆A = S IAD D + SλA + SβA ; A = gm / g0 = λ β ID λ ID A 1/ 2 −1 / 2 − 1 1 1 S IAD = S ICD1I D = − ; SλA = SλC2λ = −1; SβA = SβC3β = ; 2 2

4

In Slide 11.15, we continue the example of the previous slide by calculating the total variation of the amplifier voltage gain A to variations of the drain current ID and of the channel-length modulation parameter λ in terms of their respective sensitivities.

Some Important Sensitivity Expressions

In Slides 11.16, 11.7, and 11.18, we use the relations of Slides 11.6 and 11.7 to derive some important sensitivity expressions related to the sensitivity of the transfer function T(s) and of the characteristics it defines; these include the amplitude, phase, pole-frequency, and pole-Q sensitivity. These characteristics and their sensitivity to component tolerances, tuning errors, and ambient changes such as temperature and humidity, as well as aging, are important for the characterization of the performance of analog circuit functions.

Some Important Sensitivity Expressions 1. Magnitude and Phase Sensitivity

Given complex transfer function: T ( s ) s = jω = T ( jω) = T ( jω) ⋅ e jϕ( ω)

ln T (ω) = ln T ( jω) + jϕ(ω) = α(ω) + jϕ(ω)

α(ω) : amplitude response;ϕ(ω) : phase response. S xT ( ω) =

d [ln T (ω)] d [α(ω)] d [ϕ(ω)] = +j = S x α ( ω) + jS x ϕ( ω) d [ln x ] d [ln x ] d [ln x ]

S x α (ω) : amplitude sensitivity; S x ϕ(ω) : phase sensitivity.

Starting with Slide 11.16, the amplitude and phase sensitivity is shown to be readily derived as the real and imaginary part of the transfer function sensitivity to changes in a component x. [Note the difference in the letter S used for the relative and semi-relative sensitivity (see Slide 11.5).]

4  Some Important Sensitivity Expressions

307

In Slide 11.17, we continue with important sensiT ( ω) α ( ω) T ( ω) T ( ω) tivity expressions by Sx =Sx = Re S x = Ev S x s = jω showing that the amplitude 1 1 S xϕ( ω) = Im S xT ( ω) → S x ϕ( ω) = Im S xT ( ω) = Od S xT ( s ) s = jω and phase variation can be ϕ j obtained directly from the ∆x ∆x T ( ω) T ( ω) ∴ ∆α(ω) = Re S x ⋅ ; ∆ϕ(ω) = Im S x ⋅ . real and imaginative part x x 2. Pole and Zero Sensitivity of the transfer sensitivity to a component x. Given T (s) in terms of poles pj and zeros zi : m In the lower part of the Then: ( s − zi ) slide, the sensitivity of a p N (s) n m S x zi Sx j zero zi (zero sensitivity) = T ( s ) = K ⋅ in=1 T (s) K S = S − + D( s) x x and of a pole pj (pole sensis − z s − p (s − p j ) j =1 i =1 j i tivity) to change in a comj =1 dzi zi T (s) = lim ( s − zi ) S xT ( s ) ponent x is defined as the Zero Sensitivity: S x = dx / x = Res S x s → zi s = zi semi-relative sensitivity of dp j Pole Sensitivity: S x p j = = Res S xT ( s ) = lim ( s − p j ) S xT ( s ) zi and of pj, respectively, to s→ p j dx / x s= p j the change in the component or ambient factor x. Having defined the root, i.e., pole and zero semi-relative sensitivities to changes in x, these are shown to be the residues of the sensitivity to changes of x of the transfer function T(s), expressed as a partial fraction expansion. In this partial fraction expansion, all critical network frequencies, i.e., zeros zi and poles pj of T(s), become the poles.

Thus:

{

}

{

{

}

{

}

{

}

∏

∏

} {

{

}

}

∑

{

}

{

}

∑

In Slide 11.18, we continue with the discussion Pole, Frequency, and Q Variation of the sensitivity of T(s) by ∆p ∆x 1 ∆x p p Pole variation: p = S x x = p S x x expressing the variation of the poles of T(s) to changes ∆p 1 p p ∆  ∆  T ( s ) ∆x = {Res S x } = Re  + j Im   Thus: in a component or ambient p p x p    p factor x. The variation, i.e., It can readily be shown that: the relative change Δpi/pi ∆q p / q p ∆p  ∆p  ∆ω p Re   = ; Im   = − of a pole pi, is first 2 4q p − 1  p  p  ωp expressed in terms of the ∆ ω ∆ q / q ∆p p p p = −j Thus: residue of the sensitivity of 2 p ωp 4q p − 1 T(s) to x (see previous c r Furthermore: ∆ω p ω p ∆Ri ω p ∆C j ω p ∆β slide). The real part of this = ∑ S Ri + ∑ SC j + Sβ . Frequency expression is then shown R ωp C β j =1 i =1 j i variation: to be equal to the variation r c ∆q p q p ∆Ri q p ∆C j q p ∆β of the pole frequency = S S S . + + ∑ ∑ Ri Cj β Q Variation: q Ri Cj β Δωp/ωp and the imaginary i =1 j =1 p part proportional to the variation of qp, i.e., Δqp/qp. These two expressions, i.e., Δωp/ωp and Δqp/qp, are then broken up into the summation of their sensitivities to the components of the analog RC-active circuit, i.e., to the sensitivities with regard to its resistors, capacitors, and gain elements.

11  Elements of Sensitivity Theory

308

5

Summary of Sensitivity Expressions

In Slide 11.19, we summarize the sensitivity expressions discussed above by Given a transfer functionT (s)=N (s)/D (s). showing that most of them can be derived from the ∆T ( s ) ∆x = S xT ( s ) ⋅ ; S xT ( s ) : Transfer Sensitivity Then: transfer sensitivity of T(s) T ( s) x to the relative change in a Calculated S xT ( s ) component value (or an s = jω (computed) Quantity ambient change) desigResidue for s=p S xT ( jω ) = Re {S xT ( jω ) } + j Im {S xT ( jω ) } nated by x. S x p = Res {S xT ( jω) } Starting out with the s= p dα(ω) dϕ(ω) ; S x ϕ ( ω) = S x α ( ω) = transfer sensitivity, which dx / x dx / x is an analytic quantity at ∆ ∆ 1 p x T ( s ) the top center of the slide, also ∆x = {Res S x } ∆α(ω) = S x α ( ω) = f α (ω) possible p p s = jω x we see on the left that for x in this ϕ ( ω) ∆ x p p ∆  ∆  s = jω, the real part of the = f ϕ (ω) ∆ϕ(ω) = S x direction = Re  + j Im   x p p transfer sensitivity is the     measurable quantities semi-relative amplitude sensitivity and the imaginary part is the semi-relative phase sensitivity. Both these quantities are ­measurable, i.e., a deviation from the specified amplitude or phase can be readily determined by amplitude or phase measurement, respectively. On the right of the slide, the residue of the transfer sensitivity is shown to be equal to the semi-­ relative pole sensitivity. From this we obtain the relative pole variation, i.e., Δp/p resulting from a small relative change in x. Note that having Δp/p, we can also work our way back to obtain the amplitude and phase sensitivity with respect to a component or ambient change x.

Summary

In Slide 11.20, we continue with the chart of the previous slide, i.e., summarizing all the sensitivity ∆ω p ∆q p / q p ∆q p / q p ∆p ∆ω p expressions that can be − = −j derived from the transfer ωp p ωp 4q 2p − 1 4q 2p − 1 function sensitivity. The measurable quantities chart shows that the real jω ∆ωp ∆qp part of Δp/p is equal to the ωp qp  ∆Ri ∆C j ∆G  ∆ω p ∆f p pole-frequency variation,  = = Fω  , ,  Δωp/ωp, and the imaginary ωp ωp fp R C G j  i  part, divided by the term σ σp  ∆R ∆C j ∆G  ∆q p (4qp2–1)1/2, is equal to the  = Fq  i , , negative qp variation,  qp  Ri C j G  Δqp/qp. These two terms are measurable quantities. The diagram on the lower left of this slide shows that the term Δωp/ωp corresponds to a radial shift of the corresponding pole in the s-plane, and the term Δqp/qp corresponds to a circular shift about the origin of the s-plane. These two slides, i.e., 11.19 and 11.20, demonstrate how the sensitivities of all the relevant quantities included in the relative sensitivity of T(s = jω) to x are interconnected and can be derived from each other.

6  Some Practical Examples

6

309

Some Practical Examples

Example 1: Given a notch filter: 10k

10k

R0

(β-1)R0

T ( s) =

R 1k

V1

L C

jω

V2

6nF

V2 s 2 + ω02 =K⋅ ω V1 s 2 + 0 s + ω02 q

1 K = β; ω0 = LC 1 L q= R C

618µF

T(jω) σ

K

In Slide 11.21, we present the example of a simple active LCR notch filter. In it, we examine which component to adjust, and by how much, in order to shift the filter notch frequency in either direction by 5%.

ω

ω0

If the notch frequency is to be adjusted to within 5%, which component shall we adjust, and by how much?

Answer: ∆ω0 ∆L = S Lω0 or ω0 L ∆ω0 ∆L ω0 = S Lω0 + SC or ω0 L

S Lω0 = SCω0 = SC( LC )

with or

−1/2

∆ω0 ∆C = SCω0 ω0 C ∆C C

= S L( LC )

∆ω0 = ±5% → ω0

∆C = ±10% C

or

−1/2

=−

1 2

∆L = ±10% L

∆L ∆C + = ±10% L C

In Slide 11.22, we show that by using the design equations given here and the sensitivity formulas (1) and (4) in Slide 11.6, the L or C of the series resonant circuit must be varied by 10% in the direction opposite (because of the negative sensitivity) to the desired direction of the notch frequency.

11  Elements of Sensitivity Theory

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Slide 11.23 presents another practical example, Example 2: Given the following 2ndalso involving the tuning of a frequency-sensitive order active RC band-pass filter (biquad) circuit. It discusses the 2 R2 tuning of a single-ampliR5 1 R4 3 C2 R1 1 2 fier band-pass filter biquad q (I-SF/DF-4, see Slide p T( jω) Kω C1 R3 1 8.87). The three parame1 1 p ters K, ωp, and qp of T(s) 2 are to be tuned by circuit The transfer function is s resistors. [Depending on given by T ( s ) = K ⋅ ωp ωp the technology, tuning or 2 2 s + s + ωp adjusting resistors is often qp preferable to tuning The band-pass is to be tuned with resistors for K, capacitors.] ωp , and qp . There are five resistors in the circuit and three parameters to be tuned. The questions are: (i) Which resistors should be used such that only one resistor per parameter need be tuned and (ii) in which order should the resistors be tuned, so that each parameter, once tuned, needs no further corrective adjustment? In Slide 11.24, we show how to answer these two questions. First, the so-called sensitivity matrix is set up in which the sensitivity of each of the three parameters to each of the five resistors is assembled p numerically in a matrix. and qp , respectively, and give the tuning The higher the sensitivity sequence with respect to the 3 resistors. of a parameter to a resis dK / K   −1 0 1   dRK / RK  tor, the faster – or coarser –        dR / R  = − − d / 1/ 4 1/ 2 0 ω ω the tuning process will be. p p ω ω      dq p / q p   1/ 4 For speed or coarse tuning, 1.5 1.5  dR / R  q   q high-sensitivity values are desirable. Second, in order for one tuning resistor per parameter to exist, a three by three matrix must be extracted from the original matrix such that the upper triangle is as close to zero as possible. This will determine which resistor to tune for which parameter and in which order. The lower matrix gives the resulting optimum solution for this filter circuit. According to this matrix, the tuning process can proceed as follows:

The sensitivities of K, ωp and qp with respect to the resistors are given in the matrix form (sensitivity matrix).  dR1 / R1  RK − 1  dR2 / R2  0 0 1  dK / K   − 1        dR / R  ω = − − − d ω / 0 . 25 0 . 25 0 . 5 0 0 Rω p p p 3 3      dq p / q p   0.25 − 1.75 1.5 1.5 − 1.5 dR4 / R4  Rq   p  dR5 / R5  Select 3 resistors with which to tune K, ω

p

p

p

p

1. Tune K with R1. This will affect ωp and qp as well, but these will both undergo final tuning in the following steps. 2. Tune ωp with R3. This will not affect K but will affect qp. However, qp will undergo final tuning in the next step.

6  Some Practical Examples

311

3. Tune qp with R4. This will not affect ωp but will affect K. However, K, which determines the overall signal level of the final filter, can be corrected either with R5 or, in most cases, somewhere else in the cascade of circuits. Slide 11.25 presents a final example, starting with the specifications for the thirdorder high-pass notch filter discussed in Chap. 10 (see Slides 10.31 and 10.32). The inductorless activeRC biquad chosen to meet the specifications for this notch filter is the multiamplifier biquad of Chap. 10 (Slide 10.46).

Example 3: High-pass notch filter with adjustable notch frequency.

s (s 2 + ω2z ) Transfer function: T ( s ) = K ⋅  2 ωp  2 ωz = 516.69 kHz s + ( s + ω0 ) where s + ω p   qp ω p = 889.584 kHz q p = 2.2   ω0 = 1.29974 kHz K = 1 20log [dB] jω

T( jω)

s

jωz

σ -jωz

ωz

∆ωz ωz =±5%

ω

Which resistor of the following biquad shall we use to adjust the notch frequency? By how much must it be adjusted to obtain a ±°5% notch frequency tunability?

In Slide 11.26, we again show the multi-amplifier biquad. The question is now with which resistor to tune the notch frequency of the filter to within ±5% accuracy.

High-pass notch filter with adjustable notch frequency cont’d R3(=Rω ) p

(22.5K) R1(=Rq ) p

(49.4K) R4 (24.7K) Vin

C1 (50p) − +

R5 (2k) R7 (1k)

R2 (22.5K)

C2 (50p) R0 1 (1k)

− +

R8

R6 (1.37K)

R0 2 (1k) − +

R (1k) 100p

− +

V out

Opamp circuit derived from sfg with first-order high-pass circuit added on to obtain third-order hpn filter.

17.4K

Vout

11  Elements of Sensitivity Theory

312

Design Equations for Multi-Amplifier Biquad ωz s + ω2z qz T ( s) = K ⋅ ω s 2 + p s + ω2p qp s2 ±

ωz  1  21   + ωz   qz  s  s T ( s) = K ⋅ 2 ω 1 1 1 + p   + ω2p   qp  s  s 2

1±

ωz / qz

ω2z

 ω  R R   R R s 2 +  p 1 − 7 ⋅ 1  + ω2p  7 ⋅ 3 − 1 V R  R6 R4   q p  R5 R4  T ( s ) = out = − ⋅ ω Vin R7 s 2 + p s + ω2p qp 1 C1 ωp = R2 R3C1C2 q p = R1 R R C 2 3 2

Design Eqs. for Multi-Amplifier Biquad cont’d R7 R3

∆ωz ∆R = S Rω6z 6 ; R6 ωz

1 2 S Rω6z = S Rω6z 2

1 R7 R3 ∆ωz ∆R =− ⋅ 6 2 R7 R3 − R6 R4 R6 ωz

≈1

R7 R3 R6 R4 ⋅ S R6 1 R6 R4 ; = 2 R7 R3 − 1 R6 R4

∆R6 ∆ωz ≈ R6 ωz

!

Slide 11.27 presents the design equations for the multi-amplifier biquad. From the transfer function T(s), it is clear from the expression for ωz2 that the resistor R6 can vary the notch frequency, i.e., ωz, independently, without affecting any other coefficients.

In Slide 11.28, the sensitivity of ωz with respect to R6 is calculated, and the variation of ωz as a function of the variation of R6 is obtained. With the given component values of the circuit, we see that R6 is a suitable resistor with which to tune the notch frequency; it must be adjusted by approximately the same percentage as the desired shift in the notch frequency.

7  Relationships Between Amplitude, Phase, Pole-Frequency, and Pole-Qp Variation

7

313

 elationships Between Amplitude, Phase, Pole-Frequency, R and Pole-Qp Variation

In Slide 11.29, we show the sensitivity of the transfer function T(s) to changes of x, expressed as a partial fraction expansion in We found: which the residues are the dp j semi-relative pole and zero pj Sx = pj zi m n sensitivities and the critidx / x S S S xT ( s ) = S xK − ∑ x + ∑ x cal network frequencies, dz i i =1 s − zi j =1 s − p j S x zi = i.e., zeros zi and poles pj of dx / x T(s), become the poles (see Slide 11.17). p dx z dx ≈ ∆zi ∴ dp j = S x j ≈ ∆p j ; dzi = S x i At the bottom of this x x slide, we note that the differential terms dpj and dzi can be approximated by, and interchanged with, the difference (measureable) terms Δpj and Δzi, respectively. This is useful in order to relate measurable shifts in pole placement and the resulting changes in amplitude and phase, with their differential counterparts.

Relationship Between Amplitude, Phase, Pole-frequency, and Pole-Qp Variation

In Slide 11.30, we consider the variation of the transfer sensitivity ΔT(s)/T(s) in the vicinity of the dominant pole pd, resulting in the term Δpd/ (s-pd). This term, being the dominant pole, will dominate in the partial fraction expansion of ΔT(s)/T(s).

For a dominant pole pair: ∆pd ∆T ( s ) ∆x S x pd ∆x jω ≈ = S xT ( s ) ≈ pd T ( s) x s − pd x s − pd σ p*d

= dα (ω ) + jdφ (ω ) ≈

jω

ωp

σp

For s=jω: dT ( jω) ∆T ( jω) = d [ln T ( jω)] = ≈ T ( jω) ∆ 1 : ω p ≈ ωc

dα (ω p ) ≈

dq p qp

dφ (ω p ) ≈ 2q p

dpd jω − pd −

dω p

ωp

;

dω p

ωp

It is shown in this slide that for s = jω, where

dT ( jω ) / T ( jω ) ≈ dα ( ω ) + jdϕ ( ω ) dα ( ωp ) ≈ dq p / q p − dωp / ωp and dϕ ( ωp ) ≈ 2q p ( dωp / ωp )





(1)

(2)

(3) Thus, the amplitude response α(ω) is equally and directly dependent on frequency ωp and qp variations at the pole frequency ωp. On the other hand, the phase φ(ω) depends on frequency variations at ωp multiplied by 2qp.

11  Elements of Sensitivity Theory

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In Slide 11.31, we give a practical example of a sec20logT(jω) ond-order band-pass filter, ωp B=2σp qp= which is a good indicator  d ω p dq p  d ω 1 p 2σp − dα (ω p ± σ p ) ≈ ±   + q p for the behavior of a  2  ωp qp  ωp higher-order filter in the dω p 1 dq p ω ωp vicinity of the dominant + qp dϕ (ω p ± σ p ) ≈ ± ωp 2 qp pole. We show that at the 3 dB slopes on either side ∆ω p ∆ω p ; ∆ϕ(ω p ± σ p ) ≈ q p ∴ ∆α(ω p ± σ p ) ≈ q p of the pole frequency, the ωp ωp amplitude and phase variation caused by drift in Consequences: 1) Frequency stability determines either direction by the pole amplitude stability. frequency ωp is also qp 2) Tune pole frequency using the times larger than the drift phase characteristic response. of the pole Q, qp. This illustrates the seemingly counterintuitive fact that for high-performance and high-frequency selectivity, the stability of the pole frequencies, especially close to the dominant poles, is of more importance than that of the pole Qs. Because of the strong dependence on phase φ(ω) at the pole frequency, in the vicinity of the dominant pole frequency ωp, φ(ω) is also a far better indicator for the tuning of the pole frequency ωp than the amplitude response α(ω) (see previous slide). The important consequences of these observations given at the bottom of this slide are that (1) primarily the frequency stability determines the amplitude stability of a filter and (2) that the pole frequency ωp is best tuned using the phase as an accurate indicator. Example: 2nd Order Band-pass Filter

8

Homogenous Functions and Sensitivity Invariants

In Slide 11.32, we introduce the subject of homogenous functions. This is a subject that is very useful for the computation of the Given function of n variables F ( x1 , x2 , , xn ) sensitivity to component m If F ( λ x1 , λ x2 , , λ xn ) = λ F ( x1 , x2 , , xn ) variations within circuits λ>0 integrated on a chip. The then F ( x1 , x2 , , xn ) is a homogeneous reason for this is that the function, and m is its order of homogeneity. small-sized and “integrated” nature of the comDifferentiating both sides with respect to λ, ponents on an IC chip setting λ=1, and dividing by F, we obtain: causes the variations of the ∂F x1 ∂F x2 ∂F xn components on a chip to ⋅ + ⋅ + + ⋅ =m ∂x1 F ∂x2 F ∂xn F track very closely. Thus, Euler’s formula n all component values tend F for homogeneous Thus S xi = m to change by the same functions i =1 amount and in the same direction. This has direct and simplifying consequences on the analysis of the overall sensitivity of the IC circuit to component changes, as well as on the performance of the chip.

Homogeneous functions and Sensitivity Invariants

∑

8  Homogenous Functions and Sensitivity Invariants

315

As mentioned above, analyzing these tracking changes and their effect on the circuit performance, the so-called homogenous functions come into play very naturally, as will be discussed in what follows. The short and simple derivation of homogenous functions given on this slide produces Euler’s formula for homogenous functions. The formula is adapted here to the terminology of sensitivity theory, where m is the so-called degree, or order, of homogeneity.



All physical functions must have dimensional homogeneity. Example (from above): λ G4 (λ G1 + λ G3 ) 0 G4 (G1 + G3 ) ωp = =λ m=0 λ C2 λ C4 C2C4 ∴

g

∑S i =1

ωp Gi

m =1 

c

g

+ ∑ SC jp = 0

∑S

ω

i =1

j =1

ωp Gi

c

= −∑ SC jp ω

j =1

m = −1

Network function of circuit with tracking components n ∆ x Tracking components: ∆F ∆x ∆x n = ∑ S xFi i = ∑ S xFi = m ∆ xi ∆ x x ≡ F xi x i =1 i =1

∆ω p ∆G ∆C Ex : = − ωp G C ∆ R ∆ R ∆R S RG = S R1/ R =− R R R

xi

∴

∆ω p ωp

x

 ∆R ∆C  = − + = C   R

= −(TCR + TCC )∆Temp.

In Slide 11.33, it is stated that “all physical functions must have dimensional homogeneity.” This means that no separate comprehensive part of a physical quantity can have a dimension other than the overall dimension of the expression. Thus, for example, no comprehensive term in the expression for an impedance Z can have a dimension other than Ohm, and its degree of homogeneity, or m, will be unity. Similarly a dimensionless quantity, e.g., the quality factor Q,

will have the order of homogeneity equal to zero. The example given on this slide is for the pole frequency ωp of an active-RC network. It is shown that the degree of homogeneity m for the sum of conductances Gi in the expression for ωp is unity, and with respect to the sum of all capacitors, it is minus 1. Thus, the degree of homogeneity for the sum of all the components that determine ωp is zero. If all the components xi of a physical quantity track in the same direction, i.e., Δxi/xi ≡ Δx/x, then the variation of that quantity is m times the variation of its tracking components. Thus, as is well known, the frequency of an RC network can be stabilized with respect to temperature variations by using resistors and capacitors with equal but opposite temperature coefficients, as indicated at the bottom of the slide. We have seen above that the stability and robustness of the frequency response (amplitude and phase) of an active-RC filter with respect to component tolerances and ambient changes depend more on the stability of the pole frequency than on the pole Q. In what follows, we shall therefore examine how the pole frequency is affected by component tolerances and ambient changes.

11  Elements of Sensitivity Theory

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9

Pole-Frequency (ωp) Stability

In Slide 11.34, we set the stage for the question of G: Gain Device. pole-frequency (ωp) stabilr c ∆ω p ity by formulating the variω p ∆Ri ω p ∆C j ω p ∆G = S Ri + SC j + SG ation of ωp with regard to R C ωp G i =1 j =1 i j variations in the resistor, With tracking resistors and capacitors: capacitor, and gain eleIn general ments on which the pole ∆Ri ∆R ∆C j ∆C ≡ ; ≡ (e.g. Class 3,4) frequency ωp depends. We Ri R Cj C ≈0 do this in terms of the sum ∆ω p ∆R r ω p ∆C c ω p of the sensitivities of ωp to ω ∆G ∴ = S Ri + SC j + SG p variations in these circuit ωp R i =1 C j =1 G components. Because of the imporhomogeneity: homogeneity: Temp Coefficient in ppm/°C tance of frequency stabil=-1 =-1 ity, it follows that for ∆ω p ∆f p ∆R ∆C  = = − + Thus:  = −(TCR + TCC)∆°C active-RC filters, it is ωp fp C   R greatly advantageous to use circuits for which the pole frequencies depend only on the passive, i.e., RC components, and not on the gain elements. The gain, i.e., of an amplifier, in an active-RC network is inevitable in order to achieve Q (qp) values larger than 0.5, i.e., to obtain complex-conjugate poles. This is not the case for the pole frequency, and it is present in the pole frequency expression of only some biquads. In any case, it is not coincidental that the most useful and most commonly used single-­amplifier biquads are those of Classes 3 and 4; the pole frequency of both is independent of the feedback amplifier gain. Since the basis for active-RC networks and filters is to eliminate inductors so that they can be realized on IC chips, it is reasonable to assume tracking of the on-chip components with respect to tolerances and ambient changes. It follows that in order to examine the variation of the pole frequency with regard to ambient changes and component tolerances, we can make use of the homogeneity of the expression for ωp with regard to its defining components. Considering only the temperature changes, this results in the expression for the frequency variation Δωp/ωp at the bottom of this slide. As pointed out above, this shows the effect of the component temperature coefficients TCR and TCC and their polarity, and these, in turn, depend on the manufacturing process and the materials used for the final IC devices.

ωp Stability :

∑

∑

∑

∑

3 Main Causes for ∆f/f: (i) Temperature Coefficient: TCR, TCC (ii) Aging & Humidity: A/H (iii) Initial Error, Component Tolerance: initial  ∆f  ∆f  ∆f   ∆f   ∆f   ∴ = F   ,   ,   ,    f  f  TCR  f  TCC  f A/H  f  initial  Ex: ∆Temp=50 °C

 ∆f   f  = ( 0 ± 50ppm/ °C ) 50°C = ±0.25%   TCR  ∆f   f  = ( 0 ± 30ppm/ °C ) 50°C = ±0.15%   TCC

Ex: (cont’d)  ∆f   ∆f   ∆f  +  = ±0.1% ± 0.1% = ±0.2%  f  = f   A/H  A/H R  f  A/H C

 ∆f  ≤ ±0.2%  f    initial

 ∆f   f  ∆f   f

 = ±(0.25 + 0.15 + 0.2 + 0.2)% = ±0.8% Which is  worst correct??  case 1  = ± 0.252 + 0.152 + 0.22 + 0.22 = ±0.2%  rms 2  ("Schoeffler")

What are the consequences of worst case vs. RMS? It can be shown that for a biquad:

∆α(ω = ω3dB ) ≈ q p ⋅

∆ω p ωp

Ex: qp=20 ∴ ∆α(ω3dB )wc = 20 ⋅ 0.8% ≈ ±1.5dB;

∆α (ω3dB )rms = 20 ⋅ 0.2% ≈ ±0.4dB;

Drastic Consequences!

Ex: 2nd-Order Band-pass Filter: 1. worst case: ∆f ∆f p = 0.8% ⋅ f p = 8Hz BW=100Hz → q p = 10 8Hz 2. Rms: ∆f p = 0.2% ⋅ f p = 2Hz -3dB or wc 8% 2Hz?? ∆f p ∆f p f p ∆f = ⋅ = qp ⋅ p = ? f p rms 2% f p BW ω BW qp 950Hz f =1kHz 1050Hz p

p

In Slide 11.35, we consider the main causes for frequency variations  – in terms of f rather than radian frequency ω – in an active-RC circuit. They are ambient temperature (TCR/TCC), aging and humidity (A/H), and initial tuning errors and component tolerances (initial). Typical tolerances resulting from changes in ambient temperature are given at the bottom of the slide.

In Slide 11.36, we continue with assumptions for aging and humidity (A/H) and initial tuning errors and component tolerances (initial). We conclude by demonstrating the significant difference in result between a worst-case and an rms (root-mean-square) estimation of accuracy.

In Slide 11.37, we continue with the question of worst-case versus rms estimation by taking into account the crucial fact that the amplitude and phase response in the vicinity of a (dominant) pole frequency will vary by qp times the frequency variation. An example of a relatively high-Q bandpass filter is given in the slide. It shows clearly how the decision to use worstcase or rms values to characterize the frequency variation will play a critical role in the decision which technology to select for a given application.

11  Elements of Sensitivity Theory

318

Obviously, a general answer to this question cannot be given here, since it depends, among other things, on the application and on the resources available for manufacture. Suffice it to say that when the application is a critical one such as the design of a space vehicle [where a so-called success schedule “must be maintained” (~100% yield; no errors)] versus a commercial product, where a given yield and percentage of errors are taken for granted, working with worst case or rms may make all the difference on the technology and on the manufacturing methods decided upon for a given project.

10

Pole-Q (qp) Stability

As mentioned above, the gain in an active-RC filter is responsible for the creation of complex-­ conjugate poles by enlarging the pole Q, qp of a passive RC network from less than to greater than 0.5. Thus, in spite of the dominant role of frequency stability, the stability of the pole Q is very critical – especially in high-Q, highly selective, high-performance active-RC filters. In Slide 11.38, we deal with qp stability by first qp Stability formulating the variation of qp, namely, Δqp/qp, with r c ∆q p q p ∆Ri q p ∆C j q p ∆G respect to all the compo= ∑ S Ri + ∑ SC j + SG qp Ri Cj G i =1 j =1 nents of an active-RC circuit, i.e., resistors Ri,  With tracking resistors and capacitors: capacitors Cj, and gain G. ∆Ri ∆R ∆C j ∆C ≡ ; ≡ For the same reason as Ri R Cj C given with respect to the ∆q p ∆R r q p ∆C c q p q ∆G S Ri + SC j + SG p ∴ = frequency variation, we ∑ ∑ qp R i =1 C j =1 G are assuming tracking A[dB] ∆A ± resistors and capacitors on homogeneity: homogeneity: A A 0 Open Loop an IC chip. However, in =0 =0 Closed ± ∆G Loop this case, since qp is dimenG ∆q p q ∆G = SG p ∴ G sionless, its order of homoqp G ω geneity m with regard to all resistors, and with regard to all capacitors, is zero. This leaves only the variation of qp caused by the variation of the (amplifier) gain G. Note that the variation of the closed-­loop gain ΔG/G will in the case of a feedback amplifier be a function of the variation of the open-loop gain ΔA/A, as shown at the bottom of the slide.

10  Pole-Q (qp) Stability

S

qp G

319

=?

Class 1,2: q p = qˆ 1 + k32G Class 3: Class 4:

q

SG p =

1 k32G 1 ≈ 2 1 + k32G G ≈ A 2

k 32 G ≈1 1 + k 32 G G ≈ A q χ1 χ q qp = ; qˆ = q p (G = 0) = 1 SGp = p − 1 χ 2 − χ3G χ2 qˆ

q p = qˆ (1 + k32G )

Neg. feedback (Class 1,2,3)

∆q p 1 ∆G ≈ 2 G qp

∆q p qp

≈

∆G G

q

SGp =

Pos. feedback (Class 4)

∆q p qp

 q p  ∆G ≈  − 1  qˆ  G

Does this mean that Class-4 (pos. feedback) filters are much more tolerance-sensitive than Class 1,2,3 filters???

qp Stability (cont’d) 

Given single-amplifier biquads of the form: positive feedback In



RC

We have found that: Neg. feedback (Class 1,2,3)

 ∆q p   1  ∆G    q  ~  2  G ,  p  NF where i = 0 ,1

G

Out negative feedback

Pos. feedback (Class 4)

i

 ∆q p  q ∆G   ~  p − 1  q  q ˆ  G  p  PF 

In Slide 11.39, no assumptions are made on the nature of the gain G, i.e., feedback amplifier or otherwise, and the sensitivity of qp with respect to G for the four classes of singleamplifier biquads is listed. We note that the sensitivity of qp with respect to G is far smaller, namely, on the order of 0.5 to 1, for the negative-feedback classes, i.e., Classes 1, 2, and 3, than for Class 4 (positivefeedback) biquads.

As shown in Slide 11.40, the sensitivity of qp with respect to G for Class 4 biquads is proportional to qp/q̂p, which is larger than 2qp (since q̂p is smaller than 0.5). For high-Q biquads, this results in a very large sensitivity to gain variations. For a long time, this fact implied that Class 4 biquads could not be used for high-selectivity, i.e., high-Q, biquads.

11  Elements of Sensitivity Theory

320

qp Stability (cont’d) 

It would appear that, since : qˆ < 0.5 q q SGp PF >> SGp NF

( )

( )

and therefore that positive-feedback (Cl. 4) filters should not be used?! To find out if this is true, we must look at the term ∆G/G, and the opamp realization of G: R1

R2

SFG V3

V1

t32=

-A

A 1

2

R1 R1+R2

3 +A

qp Stability (cont’d) V3 ∆G ∆A A = S AG = ; G A V1 1 + t32 A t A 1 G ∆G ∆A =G⋅ 2 S AG = 1 − 32 = = G A 1 + t32 A 1 + t32 A A ∆G ∆A = G ⋅ 2 thus: We have found that A G ∆q p ∆A q ∆A ∆A = SGp ⋅ S AG ⋅ = G ⋅ SGq p ⋅ 2 = ΓGq p ⋅ 2 qp A A A

G=



In Slide 11.41, the conclusion above is examined more closely. We look at the most common case of an opamp with open-loop gain A that is used to obtain the closed-loop gain G with negative feedback. This circuit, with corresponding SFG, is shown at the bottom of the slide.

G A

Gain-Sensitivity Product (GSP)

(GSP)

In Slide 11.42, calculating the variation of G with respect to a variation of the open-loop gain A, we find that ΔG/G is equal to G·ΔA/A2. From this we find that Δqp/qp is equal to the closed-loop gain G, times the sensitivity of qp to G, multiplied by ΔA/A2. The closed-loop gain G times the sensitivity of qp to G is called the gain-sensitivity product, or GSP. Thus Δqp/qp is equal to the GSP times ΔA/A2.

10  Pole-Q (qp) Stability

321

In Slide 11.43, the GSP for the four biquad classes dis∆q p Filter qp qp cussed in Chap. 8 is listed. qp Γ SG A qp Class It is clear that taking into account the gain-sensitiv1 ∆G 1 k32G ≈ ity product changes the A 2 1 + k32G 2 G 1,2 qˆ 1 + k32G question of circuit stability 1 ∆ A 1 2 ≈ ≈ significantly. It is not the 2 2 A sensitivity but the gaink32G sensitivity product (GSP) ∆G ∆A 1 + k32G 3 qˆ (1 + k32G ) A ≈ ≈ that determines the stabilG A ≈ 1 G >> 1 ity of a circuit characteristic such as qp that depends q p χ1  q p  ∆G −1 −1   on opamp feedback. For χ 2 − χ 3G qˆ 4  qˆ  G χ1 q negative feedback (Classes χ1 p ∆A qˆ = qˆ = q p (G = 0) = 1, 2, 3), qp is proportional χ2 χ 2 ≈ q p A2 (G ≈ 1) to the opamp gain, and for high-qp (e.g., qp  >  5) this approaches the open-loop gain A. Thus, G ≈ A, and the GSP becomes very large. On the other hand, for positive feedback, where the closed-loop gain G is low, i.e., typically between one and two, the GSP will be correspondingly low, in fact generally lower than for negative feedback. In Slide 11.44, a numerical example for the GSP of a single-opamp biquad from Ex: A=100 each of the four classes ∆A Class 1,2 A ∆ A ≈ 50 ⋅ ⋅ discussed in Chap. 8 is i ∆q p q A2 2 A2 ≈ G ⋅ S G p ⋅ ∆ A = A 1  ⋅ ∆ A given. These examples   qp ∆A A2 q p >>0.5  2  A2 ∆A demonstrate the general A ⋅ 2 ≈ 100 ⋅ 2 i = 0,1 A importance of the GSP for Class 3 A the sensitivity characterFor positive feedback (Class 4) G is limited to small values ization of positive and neg(≤2÷3) irrespective of qp! a t iv e - f e e d b a c k - b a s e d Ex: G=1.5 ∆q p qp ∆ A  qp  ∆ A active-RC filters and q p = 10 ≈ G ⋅ S G ⋅ 2 ≈ G −1 ⋅ 2 A q >>1  qˆ circuits. qp  A qˆ = 0.45 G ≈1÷ 2 With the realization that ∆q p ∆A ≈ 30 ⋅ 2 it is the gain-sensitivity qp A product, and not the sensitivity, that determines the stability of feedback amplifier-based circuit parameters, any reluctance to use positive-feedback biquads because of their high-Q sensitivity to variations in gain is unfounded. In fact, for numerous reasons, their use in the field of IC chip design is often preferred. Accordingly, in the following section, we show that the gain-sensitivity product plays a fundamental role in the sensitivity optimization of active-RC analog circuits based on feedback amplifiers (opamps) – a role which goes well beyond that of active-RC filter design. Numerical Example for GSP of the Four Biquad Classes:

p

11  Elements of Sensitivity Theory

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11

The Gain-Sensitivity Product (GSP)

In Slide 11.45, we assume a circuit function F that The Gain-Sensitivity Product (GSP) depends on the open-loop gain A of an opamp. Due Assume a function F that depends on the to the negative feedback open-lop gain A of an opamp. In general (due which must always be to negative feedback which must always be present in an opamp-based present in an opamp-based circuit) the circuit, the function F can function F can be represented in the form: generally be represented in the form F(A) = fideal + fa/A, f where fid, fa are F ( A) = fideal + a where fideal and fa are indeindependent of A A pendent of A. If we now Let us now consider: consider the sensitivity of for large A this F(A) to A, we see (at the f /A f /F S AF ( A) = a ⋅ S Af a / A = − a ≈ 0 quantity cannot bottom of the slide) that F A A>>1 for large A, i.e., the openbe minimized! -1 loop gain of the opamp, the sensitivity of F to A approaches zero and therefore can be neither minimized nor even analyzed.

By contrast: GSP = ΓAF ( A) = A ⋅ S AF ( A) = −

fa F

≈ A>>1

fa f id

Independent of A, can readily be minimized!

Furthermore it can be shown that: q q ΓG p = ΓA p and in general: ΓGF ( G ( A)) = ΓAF ( G ( A))

Thus the GSP is valid for closed and open loop gain! Because gain characteristics are given in dB:

A[dB] A0 G

NOTE: OLG A LG ∆G ± G CLG

OLG: Open Loop Gain CLG: Closed Loop Gain LG: Loop Gain ± ∆A A < 12dB/oct! “rate of closure”

logω

A log ( A) − log (G ) = log   G  = log (1 + t32 A) ≈ log( t32 A) = LG[dB]. Thus: A / G ≈ At32 → G ≈ 1 / t32 LG

∆G G ∆A 1 ∆A ≈ ⋅ ≈ ⋅ A A LG A G

!

In Slide 11.46, we see, by contrast, that the GSP for F(A), which for large A becomes fa/fideal, can readily be analyzed and therefore also minimized. Furthermore, it is seen that the GSP is valid for closed and open-loop gain. Finally, in the lower part of the slide, the well-known fact is derived that the variation of the closed-loop gain, i.e., ΔG/G, is proportional to the variation of the open-loop gain ΔA/A, divided by the loop gain LG of the feedback amplifier. This, of course, is the reason for negative-­feedback amplifiers in the first place.

12  More Practical Examples

323

Example:R Non-inverting Opamp R 1

2

2 sfg

A V1

V3

-A 3

t32 V3

1 +A V1

1   1 1  1 1 V3 A t32 = = ≈ 1 −  =t −A 1 t32  V1 1 + t32 A 1 + 1 32 t32 A   At32 1 t32 A G 1 R +R = =0 A >> 1: G ≈ = 1 2 ; S A = 1 − 1 + 1 + t A t32 A A→∞ t32 R1 32 G=

A 1 R + R2 R = = 1 = 1+ 2 1 + t32 A A→∞ t32 R1 R1 R2 G minimum ΓA → R2 1 it follows that

SµTS < SµTM Single-loop is more stable than multi-loop!

Problem 4: Given band-pass filter specs for pilot-tone filter. Using a “Class 3” or “Class 4” single-opamp filter, assuming tracking components, and a temperature rise of 10˚C, a) what are the required temp. coefficients TCR and TCC?, b) the required GSP? T( jω)[dB] 0dB -3dB

2π⋅20Hz ±10% ±0.5dB

2π⋅800Hz

ω

Finally, in Slides 11.56, 11.57, 11.58, and 11.59, we derive the temperature coefficients and GSP that are necessary for a Class 3 and Class 4 pilot-tone band-pass filter such as to comply with given selectivity and stability requirements. For IC design we assume tracking components and require a biquad class in which the pole frequency ωp (which in the case of a second-order band-pass filter is the center frequency) is independent of gain; this is valid both for Class 3 and Class 4 biquads.

11  Elements of Sensitivity Theory

328

Solution: For a 2nd-order band-pass filter we have the transfer function: ωps T ( s) = where: ω p = 2π ⋅ 800Hz ωp 2 2 B3dB = 2σ p = 2π ⋅ 20Hz s + s + ωp qp ω q p = p = 40 a) Band-edge constraint: 2σ p

∆ω p ∆ω p 1  ∆q ∆ω p  dα(ω p ± σ p ) ≈  p + ± qp ⋅ ≈ qp ⋅  2  qp ωp  ωp ωp q p >>1

∆ω p 1 1 0.5dB → ≤ ∆α max = ⋅ = 0.144% ωp 40 8.68dB/Neper qp

Solution (cont’d) For tracking components we have (with β=closedloop gain) :

∆ω p ∆R r ω p ∆C c ω p ∆R ∆C  ω ∆β = − + S Ri + S C j + Sβ p =  ∑ ∑ C  R i =1 C j =1 ωp β  R = 0 (class 3, 4) = −1 = −1 ∴

− ∆Temp ⋅ (TCR + TCC ) ≤ 0.144%

∆Temp =10° C

⇒

< 144 ppm/ °C

ωp qp

∆B n ω p ∆ xi ∆ R r q p ∆C c q p q ∆β = ∑ S xi − S Rν − S C j − Sβ p ∑ ∑ β B x R C ν =1 i =1 j =1 i =0

∆ω p / ω p

∆B ∆ω p q ∆β = − Sβ p β B ωp q

p ± 10% 0.14% β ⋅ Sβ ⋅

GSP

=0

⇒ GSP ⋅

∆A A2

∆ A ∆B ∆ω p < − ≈ 0.09 ωp A2 B q

GSP = Γβ p ≤

Slide 11.58 shows that with tracking resistors and capacitors, the required temperature coefficient for resistors and capacitors results in the requirement that TCR  =  -TCC  ≤ 144 ppm/°C.

TCR + TCC
 ρo(f) or that the SNR always decreases with amplification – as already stated in Slide 12.32.

12  Random Signals and Noise

354

Since F0 is always greater than unity, it follows that ρs>ρ0, i.e. SNR ALWAYS DECREASES WITH AMPLIFICATION! F0 as defined above is function of operating frequency f and is referred to as SPOT NOISE FIGURE. In contrast: AVERAGE NOISE FIGURE F0: ∞

∫S

FO =

∞

NO

( f )df

For constant (white) noise (as for thermal noise) at device input, over ∆f:

−∞

∫ PG( f ) ⋅ S

NS

( f )df

F0 ≡ F0

−∞

10

Slide 12.40 designates the new interpretation of F0 as the “spot noise figure” since it is the SNR defined at the operating frequency f. Integrating the numerator and denominator of the expression for F0 given in Slide 12.37 over all frequencies, we obtain the “average noise figure” F0 . Note that for constant  – e.g., white or thermal  – noise, the average and regular noise figures are the same.

Equivalent Noise Temperature

Equivalent Noise Temperature For low-noise devices, F is close to unity. In such cases the equivalent noise temperature is easier to use. Linear 2-port device: Noise Figure F

Rs 4kTRs∆f 2 v nT

Rin=Rs (impedance matched)

Available noise power: N1

Available noise power into device: N1 [ For sinusoidal source, we saw earlier slide 12.38: PS ( f ) =

V02 ] 4 R( f )

Here: v2 4kTRS ∆f = kT ∆f N1 = nT = 4 RS 4 RS

Available noise power: N2

Slide 12.41 introduces the concept of equivalent noise temperature which is useful to describe the noise behavior of lownoise devices, in which case F is close to unity. Defining N1 as the available noise power at the input to a linear two-port device with noise figure F, it is shown in this slide that N1 = kTΔf.

11  Noise Figure of Composite Two-Port Networks

355

In Slide 12.42 the noise power Nd contributed by a two-port device to the total available output noise We define: N d = PG ⋅ k ⋅ Te ∆f power N2 is defined, namely, Nd = PG · k ·Te Δf, where PG where PG: available device power gain; Te: equivalent is the available device power noise temperature. gain and Te is the equivalent Total output Noise Power: N 2 = PG ⋅ N1 + N d = noise temperature. Then, as = PG ⋅ k ⋅ (T + Te )∆f shown in the slide, the total available output noise N2 PG ⋅ k ⋅ (T + Te )∆f T + Te Thus: F0 = power N2  =  PG · = = PG ⋅ N1 PG ⋅ k ⋅ T ∆f T N1 + Nd = PG · k ·(T + Te) Δf. From this expression the Thus: Te = T ⋅ ( F0 − 1) Equivalent Noise Temperature. output noise figure results, namely, F0 = N2/PG · N1. In terms of the noise temperature, it is then shown that F0 = (T + Te)/T and from this Te = T·(F0–1). This defines the equivalent noise temperature in terms of F0, where F0 is the noise figure of the device measured under matched input conditions with the noise source at temperature T.

Nd: Noise power contributed by 2-port device to total available output noise power N2.

11

Noise Figure of Composite Two-Port Networks

Having discussed the noise performance primarily in terms of the noise figure for single two-port networks and devices, it is now useful to examine the behavior of two-port networks in cascade. Again, we do so in terms of the resulting noise figure. In Slide 12.43 we consider two two-port networks in cascade, with power gain PG1 and PG2, noise figure Consider cascade of two noisy, impedance- matched, F1 and F2, and equivalent two-port networks: noise temperature Te1 and Nd1 Nd2 Te2, respectively. As in the available available PG1 PG2 power gain power gain previous section, N1 is the PG1 PG2 Noise Figure Noise Figure available noise power into F1 F2 the first device, generated N1 N2 N3 by the source resistance Rs. Te Te 1 2 Rin1=Rs Rin2=Rout1 From Slide 12.41 and With input impedance matching we found N1 = k ⋅ T ∆f assuming input impedance Nd1 matching, i.e., Rin1 = Rs, we = k ⋅ Te1∆f = k ∆f ( F1 − 1)T = N1 ( F1 − 1) From above: have N1  =  k·TΔf. PG1 Furthermore, Nd1 is the noise power contributed by the first two-port device to its total available output noise N2, and this leads into the second two port. Thus, from Slide 12.42, we have Nd1 = PG1·k· Te1Δf, and therefore Nd1/PG1 = k· Te1Δf = k· Δf T(F1–1). From this it is shown that we obtain Nd1/PG1 = N1(F1–1).

Noise Figure of Composite 2-port Networks

12  Random Signals and Noise

356

In Slide 12.44 we refer to Slide 12.42 where we have N2  =  PG1 · N1 + Nd1, and from Slide 12.43, we have Due to impedance matching Rin2=Rout1 we have: Nd1 = PG1 ·N1(F1–1). Thus, Nd 2 in this slide we obtain = k ⋅ Te 2 ∆f = k ∆f ( F2 − 1)T = N1 ( F2 − 1) PG2 N2  =  PG1 [N1 + N1 (F1– 1)] = PG1 ·F1 · N1. Thus: Throughout this analyN 3 = PG2 [ N 2 + N1 ( F2 − 1)] = PG2 [ PG1F1 N1 + N1 ( F2 − 1)] sis we assume that there is = PG1 ⋅ PG2 ⋅ F1 ⋅ N1 + PG2 ⋅ N1 ⋅ ( F2 − 1) impedance matching between the two ports, i.e., output noise power Rin1  =  Rs and Rin2  =  Rout1. N3 F −1 Thus, as in Slide 12.43, we = F1 + 2 Noise Figure: F0 = PG1 ⋅ PG2 ⋅ N1 PG1 have here Nd2/ PG2 = k·Te2Δf = N1(F2–1). output noise power for noiseless devices With N3 = PG2 [N2 + Nd2] it follows that the noise figure F0 = (output noise power/output noise power for noiseless devices) = N3/PG1 · PG2 · N1. From this we obtain the overall noise figure F0 of the two two-ports in cascade, in terms of the noise figures F1 and F2, respectively, and the power gain of the first two port, PG1, namely, F0 = F1 + (F2–1)/PG1.

From above: N 2 = PG1[ N1 + N1 ( F1 − 1)] = PG1 ⋅ F1 ⋅ N1

For the matched cascade of any number of two-port networks:

F0 = F1 +

F2 − 1 F3 − 1 F4 − 1 + + +K PG1 PG1 ⋅ PG2 PG1 ⋅ PG2 ⋅ PG3

If the first stage of the cascade has a high gain, the overall noise figure F0 is practically the same as that of the first stage!

In Slide 12.45 the noise figure for two two-ports in cascade obtained in the previous slide can be extrapolated into an expression for n two-ports in cascade. From this expression it becomes clear that if the first stage of the cascade has a high gain, PG1, the overall noise figure F0 is practically that of the first stage.

Chapter 13

Deriving Current-Mode from Voltage-Mode Circuits and Filters

Slide 13.1

Chapter 13 Deriving Current-Mode from Voltage-Mode Circuits and Filters

1

Introduction

Most existing active-RC filters were originally voltage-mode circuits, i.e., they used voltage-­controlled voltage sources (VVS) as the gain elements, and were part of voltage-in, voltage-out, circuits. More recently, due to the emergence of high-performance current-mode amplifying devices using MOSFET technology, current-mode circuits are increasingly used in analog circuits and filters. Current-mode circuits based on current-controlled current sources (CCCS) are realizable in IC form using the MOSFET-based transistor transconductance as a current gain device, combined with MOSFET passive components. In this chapter it is first shown that voltage-mode circuits can readily be transformed into current-­ mode circuits using a simple two-port transformation for voltage-mode to current-mode transfer functions. The resulting current-mode circuits are “inter-reciprocal” with respect to the original © Springer Nature Switzerland AG 2019 G. S. Moschytz, Analog Circuit Theory and Filter Design in the Digital World, https://doi.org/10.1007/978-3-030-00096-7_13

357

358

13  Deriving Current-Mode from Voltage-Mode Circuits and Filters

voltage-mode circuits, meaning they have identical properties, but with exchanged voltage and current quantities. With this transformation, the voltage-mode active-RC filters discussed in the previous chapters can be transformed into current-mode active-RC filters. Being inter-reciprocal, they have the same properties as the original voltage-mode active RC filters but permit realization with MOSFET-­ based current amplifying devices. [It should be pointed out that the concepts of reciprocity and inter-reciprocity conventionally belong to an extended branch of network theory which deals with so-called reciprocal and adjoint networks, also sometimes called transposed networks or dual networks. These terms distinguish between passive networks and passive and lossless networks (reciprocal), and active networks (inter-­ reciprocal, adjoint, transposed – and sometimes dual), etc. However, these are specialized and academically interesting subjects that have become victim to the condensed treatment of practice-oriented analog network theory considered essential in the present-day “digital world.”]

2

 wo-Port Transformation for Voltage-Mode to Current-Mode T Transfer Function

Slide 13.2 presents the two-port transformation for the voltage-mode to current-mode transfer function on which the N N I1 I2 I1 I2 material in this chapter is RC RC based. On the left is the V V 2 ∼ 1 [y] [y] definition of the voltage transfer function t12, i.e., − I1 y12 V2 y21 for a voltage source at the =− sˆ21 = =− tˆ12 = input, resulting in a voltI 2 V1 =0 y22 V1 I 2 =0 y22 age signal at the open-circuit output terminals, Voltage Transfer Function. Current Transfer Function. expressed in terms of the (open-circuit at output) (short-circuit at output) two-port [y] matrix. On the • Circuit N is the transpose of circuit N. right is the so-called reciprocal configuration in which the two port is driven by a current source at the two-port output terminals and the resulting current is measured at the short-circuited input terminals. The current transfer function s21 is also defined in terms of the two-port [y] matrix. The ^ over t12 and s21 points to the fact that, as shown in the slide, we are here considering passive RC networks. Similarly, when exchanging voltages and currents in active two-ports, in which all voltage sources, also within the circuit – become short circuits, and all currents sources, also within the circuit – are replaced by open circuits, then the two circuits are also called inter-reciprocal, adjoint, or transposed, meaning that V2/V1 is equal to −I1/I2 or t12 = s21 (without the ^).

Two-Port Transformation for Voltage-Mode to Current-Mode Transfer Function

2  Two-Port Transformation for Voltage-Mode to Current-Mode Transfer Function

Because the two-port is passive → y12=y21! Thus:

tˆ12 ( s ) = sˆ21 ( s ) = −

y12 y22

• Circuit N is the transpose of circuit N.

359

Slide 13.3 expresses the two-port transformation for the voltage-mode to current-mode transfer function of the corresponding passive RC networks. Some of the important consequences resulting from the voltage-mode to currentmode two-port transformation are as follows:

• When the two two-ports are passive RC, then they are reciprocal, meaning that y12 = y21. [A network is said to be reciprocal if the voltage appearing at the output due to a current applied at the input is the same as the voltage appearing at the input when the same current is applied to the output. An equivalent definition of reciprocity is obtained by exchanging voltage and current in the previous statement. A network that consists entirely of linear passive components, that is, resistors, capacitors, and inductors, is generally reciprocal. One exception was pointed out in Chap. 6, Slide 6.88 in which it was shown that the ideal gyrator is passive, lossless, but non-reciprocal]. • Circuit N is the transpose of circuit N • Using the transposition-based voltage-to-current transformation, the same categories of current-­based single-amplifier biquads (SABs) result, as are known for voltage-based SABs.  With this transformation, previously optimized voltage-based filters, e.g., with regard to sensitivity, can be directly transformed into optimized current-based filters. It follows that the classification for voltage-­based biquads can be extended to the transposed current-based biquads, as will be shown in what follows.

13  Deriving Current-Mode from Voltage-Mode Circuits and Filters

360

3

A Classification of Single-Amplifier Biquads

A Classification of Single-Amplifier Biquads t12 1

2

RC

VIN

3

±β

±α

VOUT

t32

3

s23

3 IIN 2

RC

3

IOUT 1

s21

Current-Based Biquad

Voltage-Based Biquad 1

2

RC

VIN

3

V2

~ ±βV

2

3

VOUT

3 IIN

2

I3

RC

3

IOUT

±αI3

1

Current-Controlled Current Source (CCS)

Voltage-Controlled Voltage Source (VVS)

In Slide 13.4 we apply the transposition-based voltage-mode to current-mode transformation to the voltage-mode SABs that were discussed in Chap. 8. We thereby obtain the equivalent classes of current-­mode single-amplifier biquads (SABs) that were obtained there. As mentioned above, the important feature of this transformation is that previously optimized voltage-mode filters are directly transformed into optimized current-mode filters or biquads. We see in this slide that a voltage-mode biquad designed with an opamp now requires a current amplifier instead. In the equivalent block diagram below, the voltage-controlled voltage source (VVS) becomes a current-controlled current source (CCS).

SFG of Voltage-Based Biquad t32 2

t12

α=β sˆ21 = tˆ12 sˆ23 = tˆ32

T (s) =

3

V2

V3

±β

3

I3

2 ±α

I2

1

s21

I sˆ21 ( s ) V3 tˆ ( s ) ! = ±β 12 ≡ I ( s) = 1 = ±α ˆ I3 1 αsˆ23 ( s ) V1 1 βt32 ( s ) ±

V1

s23

Transposition

±

1

SFG

SFG of Current-Based Biquad

I1

Slide 13.5 shows the transposition-based voltagemode to current-mode transformation in terms of the corresponding signalflow graphs (SFGs). Also shown is the corresponding transfer function T(s) which transforms into the corresponding transfer function I(s).

3  A Classification of Single-Amplifier Biquads

361

A Condensed Classification of Single Amplifier Second-Order Active Networks: SFGs of Voltage Mode and Current Mode Voltage Mode 1. Type I Single Feedback Class i I − SF − i 2. Type I Dual Feedback Class i I − DF − i

1

2

RC

3

±β

Current Mode s23

t32 (i=1...4)

V1 t12

2

1

3. Type II Single Feedback Class i II − SF − i 4. Type II Dual Feedback Class i II − DF − i

RC

3

2

RC

2'

i=4 2

RC

t12 V2

2'

RC

V3

−

A +

(i=3)

-A

t32

+A

V3

V1

+

V1

t12

V2

t'12 V' 2 (i=4)

+

s23

V3

t'32

−α

I3 +α

-A (i=1...4)

t'32

I1

I3

I'2 I2

+α s'23

s21 I1

−α

+A V'2

I2

s23 V3

t'12

β

s21

I1

I'2

s'23 1−β

s21

−α +α

I3

t32

V2

I2

3

−

2' Class 4

t12=1

(i=1...3) 3

3

2

I2

±α s23

2

−

A

1

I3

V3

t'32

t'12 V'

i=1,...,3

1

+A

Class 3 V'2 V1

1

-A

V2

+

2' −

±β

t32

t12 V1

V2

s'21

s21 I1

I'2

s'21

Slide 13.6 recalls Slide 8.84 showing the SFG representation of a condensed classification of voltagemode single-amplifier second-order active-RC networks. However, in this slide there is an additional column of the same four single-amplifier biquad (SAB) classes but in the equivalent current mode.

13  Deriving Current-Mode from Voltage-Mode Circuits and Filters

362

Four Classes of Single-Amplifier Biquads Based on Feedback Function t32(s) t32(s)

Cl.

1

2

k

k

ω2p ω s 2 + p s + ω2p qˆ

ωp

ω0 1 + kβ

s2 ω s 2 + p s + ω2p qˆ

ωz s + ω2z qz 3 k ω s 2 + p s + ω2p qˆ s2 +

4

Root Locus of ClosedLoop Poles

ωk

s

s2 +

ωp qˆ

s + ω2p

ω0 1 + kβ

qp

β

Possible t12(s) Functions

qˆ 1 + kβ

β  ωm. In this case the original frequency w w 3w –2ws –ws + 2s +ws + 2s +2ws spectrum, although occurring at multiples of ωs, is fold-over or aliasing individually unaffected by the sampling process. Below is an example of sampling when ωs ωm. important bridge between Then x (t ) is uniquely determined by its samples x (nT ), continuous-time (analog) n=0, ±1, ±2,... if 2π signals and their equivaωs = > 2ωm lent discrete-time (digital) T signals. In essence it states Given that ωs>2ωm, x (t ) can be exactly reconstructed from +∞ * the intuitively nonobvious its samples by passing the signal x (t ) = ∑n = −∞ x ( nT )δ(t − nT ) fact that, after a continuthrough an ideal low-pass filter with gain T and cutoff ous-time signal has been frequency satisfying: sampled, or quantized, in ωm ≤ ωcutoff ≤ ωs − ωm time, it can be completely recovered back in its original form, provided that the sampling frequency ωs is greater than or equal to twice the highest frequency component ωm of the original continuous-time signal. [The sampling theorem has a relatively long and diverse history, since many names are associated with its main statement. However, the names most intimately associated with it are Harry Nyquist, see Chap. 6, and Claude Elwood Shannon (1916–2001) who was an American mathematician, electrical engineer, and cryptographer known as “the father of information theory.”] Some Important Properties of the Sampling Process, cont’d:

15  The Sampling Theorem and Aliasing

384

This slide formulates the sampling theorem − an anchor in the foundation of discrete-time signal processing – in mathematical form. Below, the statement is added that, with ωs > ωm, x(t) can be perfectly reconstructed from its samples if the signal is passed through an ideal low-pass filter, the so-­ called reconstruction filter, whose cutoff frequency ωcutoff satisfies the condition ωm ≤ ωcutoff ≤ ωs−ωm. The reconstruction filter is necessary to filter out the frequency sidebands. These develop as a result of the sampling process with its inevitable periodicity of the spectrum of the original continuous-time signal, at multiples of the sampling frequency ωs.

Some Important Properties of the Sampling Process, cont’d:

This is schematically illustrated below: x(t)

x*(t) ws = 2p T

time domain:

x(t)

t |X (jw)|

|X*(jw)| 1 T

wm

x(t)

t

|X(jw)| 1

x(t)

x*(t)

t

frequency domain:

Reconstruction Low-pass (gain T)

1

w wm

ws 2

w ws wm ≤ wcutoff ≤ ws – wm

wm

w

Slide 15.6 illustrates the contents of the sampling theorem in graphical form. At the top the sampler is shown, followed by the reconstruction filter. Below this is an example in the time domain, of the original continuous-time signal, its discrete-time sampled version, followed by the reconstructed analog signal. Below that, the same sequence is shown but in the frequency domain.

2  Some Important Properties of the Sampling Process

2.3

385

Dependence on the Sampling Width

Slide 15.7 shows the third important property of the sampling process, namely, the dependence of the 3. Dependence on the Sampling Width: sampling process on the S x(t) 1 sampling width, or dura¿ Gain of 1/τ achieves x(t) x*(t) C tion, Ƭ, relative to the S normalization so that period T of the sampling the area under a pulse x* ~ x(t) x(t) frequency ωs. At the top of T equals unity. example: ¿ = 2 the slide, an example of an t ideal sampler is shown T 2T nT ¿ nT + ¿ which permits a simple S1 and S2 make closed description of the saminstantaneous contact S open t pling process and its critiat times nT and nT+τ, T nT 2T cal parameters. The ideal closed respectively. S switches S1 and S2, whose open t nT + ¿ T +¿ intermittent and regular 2T + ¿ ¿ switching pattern is shown at the bottom of the slide, make instantaneous contact at times nT and nT+ Ƭ. The gain of 1/Ƭ of the amplifier achieves normalization, so that the area under a pulse equals unity. Some Important Properties of the Sampling Process, cont’d:

1

2

1

2

Some Important Properties of the Sampling Process, cont’d: In terms of the unit step function u (t ) the output xˆ (t) can be expressed by: xˆ (t ) = ∑n =−∞x ( nT ){u (t − nT ) − u(t − nT − τ)} +∞

Since

L f (t − T ) → e − sT ⋅ F ( s )

1 s we can now calculate the Laplace Transform of 1 L x * (t ) → X * ( s ) = ⋅ Xˆ ( s ) τ

and

L u(t ) →

u(t), u(t-nT), and u(t-nT-nƬ), all of which are given in the slide.

Slide 15.8 contains the analytical expression for the output ̂x̂(t) of the sampler of the previous slide, at the input to the amplifier with gain 1/Ƭ, using the unit step function u(t) to do so. It also gives the relationships necessary to examine the frequency spectrum of x̂(t) resulting from sampling. This is done by calculating its Laplace transform, namely, Xˆ (s). In particular this entails the Laplace transform of the functions

15  The Sampling Theorem and Aliasing

386

Some Important Properties of the Sampling Process, cont’d xˆ (t ) = ∑ x ( nT ){u(t − nT ) − u(t − nT − τ)} +∞

n =−∞

↓L − sτ +∞ 1 e −   Xˆ ( s ) =  ⋅ ∑n =−∞x ( nT )e −snT   s 

[

]

Slide 15.9 continues with the calculation of Xˆ (s). With the Poisson summation formula (see Slide 14.22), we obtain the final expression for Xˆ (s) as a function of the sampling frequency ωs and the sample duration Ƭ.

Using the Poisson Summation Formula for the second term in brackets, we obtain: 1 − e − sτ   1 + ∞ Xˆ ( s ) =  ⋅ ∑n =−∞X ( s + jnωs )    s  T

Some Important Properties of the Sampling Process, cont’d Calculating the frequency spectrum, we let s =j ω and obtain: 1 − e − jω τ   1 + ∞  Xˆ ( jω) =   ⋅ T ∑n =−∞X ( j ( ω + nωs )) ω j   rewritten: e

− jω

τ 2

 jω 2τ − jω 2τ  τ − jω  sin ω τ  τ e −e ⋅ ⋅ ω τ = τ ⋅ e 2 ⋅  ω τ 2    2j  2  2  

In Slide 15.10 we let s = jω and rewrite the squarebracketed part of the expression for Xˆ (s) as shown. The resulting final expression now contains the familiar (sin x/x) term with which we can readily illustrate the effect of the sampling duration Ƭ on the frequency response of the sampled signal x(t). This we shall see in what follows.

2  Some Important Properties of the Sampling Process

387

Some Important Properties of the Sampling Process, cont’d 1 Thus: with X * ( jω) = Xˆ ( jω) : τ τ +∞ 1 − jω  sin ω τ  1 X * ( jω) = Xˆ ( jω) = ⋅ e 2 ⋅  ω τ 2  ⋅ ∑n =−∞X ( j ( ω + nωs )) τ T  2 

sin x x 1

–2

2

– ω= 2

In Slide 15.11 we first recall from the bottom of Slide 15.8 that X*(s) = Xˆ (s)/Ƭ. Thus, with s = jω, we obtain the final expression for the frequency spectrum X*(jω) of the sampled signal x*(t), as shown in this slide. Below is the graph of the (sin x/x) expression, which shows that the function is zero at multiples of x = ωƬ/2 = π.

x

ω= 4

Some Important Properties of the Sampling Process, cont’d Amplitude Response:

∗

X ( jω) =

τ→0

1) Ideal Sampler: ∗

X ( jω) =

Thus:

sin ω2τ ωτ 2

+∞ 1  sin ωτ2  ⋅  ⋅ ∑n =−∞X ( j ( ω + n ωs )) T  ωτ2 

→1

+∞ 1 ⋅ ∑n =−∞X ( j ( ω + n ωs )) T

1 T

3w − 2s

−

w − 2s

wm

ws 2

+ws

3ws 2

w

This is the purpose of the reconstruction low-pass filter shown in Slide 15.6.

In Slide 15.12 the effect of the sample duration Ƭ on the frequency spectrum of the sampled signal x(t) is illustrated. As Ƭ tends to zero, as it does for the ideal sampler, (sin x/x) tends to unity, and the sidebands of X*(jω) are all equally large. In general, this is not a desirable response, since we wish to make the frequency sidebands as small as possible so that we are left with only the main frequency lobe.

15  The Sampling Theorem and Aliasing

388

Some Important Properties of the Sampling Process, cont’d 2) Zero Order Hold: τ=T Thus:

–3ws 2

X ∗ ( j ω) =

–ws

–

+∞ 1  sin π ωωs  ⋅ ⋅ ∑−∞ X ( j (ω + nωs )) T  π ωωs 

ws 2

wm

ws 2

+ws

3ws 2

Some Important Properties of the Sampling Process, cont’d 3) Sample-and-temporary-hold: 0< τ