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Deliyannis, Theodore L. et al "Frontmatter" Continuous-Time Active Filter Design Boca Raton: CRC Press LLC,1999 Theodore L. Deliyannis University of Patras, Greece Yichuang Sun University of Hertfordshire, U.K. J. Kel Fidler University of York, U.K. Continuous-Time Active Filter Design ©1999 CRC Press LLC Preface “In this digital age, who needs continuous-time filters?” Such an obvious question, and one which deserves an immediate response. True, we do live in a digital age—digital com- puters, digital communications, digital broadcasting. But, much though digital technology may bring us advantages over analog systems, at the end of the day a digital system must interface with the real world—the analog world. For example, to gain the advantages that digital signal processing can offer, that processing must take place on bandlimited signals, if unwanted aliasing effects are not to be introduced. After the processing, the signals are returned to the real analog world after passing through a reconstruction filter. Both bandlimiting and reconstruction filters are analog filters, operating in continuous time. This is but one example—but any system that interfaces with the real world will find use for continuous-time filters. The term continuous-time perhaps needs some explanation. There was the time when analog filters were just that—they processed analog signals in real time, in contrast to digital filters which performed filtering operations on digital representations of samples of signals, often not in real time. Then in the 1970s, along came sampled data filters. Sampled data filters did not work with digital representations of the sampled signal, but operated on the samples themselves. Perhaps the best known example of such an approach is that of switched-capacitor filters, which as the name suggests, use switches (usually transistor switches) together with capacitors and active devices to provide filter functions. Note that these filters are discontinuous in time as a result of the switching which takes place within the circuits; indeed continuous bandlimiting and reconstruction filters are needed as a result. Much research took place in the 1970s and 1980s on switched capacitor filters as a result of the advantages for integrated circuit realization that they promised. There was so much stress on research in this area that development of the more conventional analog filters received relatively little attention. However, when switched capacitor filters failed to provide all the solutions, attention once again turned to the more traditional approaches, and the name continuous-time filters was coined to differentiate them from their digital and sampled data counterparts. This book is about continuous-time filters. The classic LCR filters built with inductors, capacitors, and resistors are such filters, of course, and indeed are still much in use. How- ever, these filters are unsuitable for implementation in the ubiquitous integrated circuit, since no satisfactory way of making inductors on chip has been found. That is why so much attention has been paid to active continuous-time filters over the years. Active filters offer the opportunity to integrate complex filters on-chip, and do not have the problems that the relatively bulky, lossy, and expensive inductors bring—in particular their stray magnetic fields that can provide unwanted coupling in a circuit or system. It is therefore active continuous-time filters on which we shall concentrate here. As just mentioned, active filters have been around for some time as a means of overcom- ing the disadvantages associated with passive filters (of which the use of inductors is one). It is a sobering realization that the Sallen and Key circuit (which uses a voltage amplifier, resistors, and capacitors, and is one of the most popular and enduring active-RC filter “architectures”) has been around for about 40 years, yet research into active filters still pro- ceeds apace after all that time. Tens of thousands of journal articles and conference papers must have been published and presented over the years. The reasons are manifold, but two ©1999 CRC Press LLC particular ones are of note. First, the changes in technology have required new approaches. Thus as cheap, readily available integrated circuit opamps replaced their discrete circuit counterparts (early versions of which used vacuum tube technology, mounted in 19” racks), it became feasible to consider filter circuits using large numbers of opamps, and new improved architectures emerged. Similarly the development of integrated transconductance amplifiers (the so-called OTA, or operational transconductance amplifiers) led to new filter configurations which reduced the number of resistive components, and allowed with advantage filter solutions to problems using currents as the variables of interest, rather than voltages. In the limit this gives rise to OTA-C filters, using only active devices and capacitors, eminently suitable for integration, but not reducing the significance of active-RC filters which maintain their importance in hybrid realizations. Second, the demands on filter circuits have become ever more stringent as the world of electronics and communications has advanced. For example, greater demands on bandwidth utilization have required much higher performance in filters in terms of their attenuation characteris- tics, and particularly in the transition region between passband and stopband. This in turn has required filters capable of exhibiting high “Q,” but having low sensitivity to compo- nent changes, and offering dynamically stable performance – filters are not meant to oscil- late! In addition, the continuing increase in the operating frequencies of modern circuits and systems reflects on the need for active filters that can perform at these higher frequencies; an area where the OTA active filter outshines its active-RC counterpart. What then is the justification for this new book on continuous-time active filters? For the newcomer to the field, the literature can be daunting, in both its volume and complexity, and this book picks a path through the developed field of active filters which highlights the important developments, and concentrates on those architectures that are of practical significance. For the reader who wants to be taken to the frontiers of continuous-time active filter design, these too are to be found here, with a comprehensive treatment of transconductance amplifier-based architectures that will take active filter design into the next mil- lennium. All this material is presented in a context that will enable those readers new to filter design (let alone continuous-time active filter design) to get up to speed quickly. This book will be found interesting by practising engineers and students of electronics, communications or cognate subjects at postgraduate or advanced undergraduate level of study. It is simply structured. Chapters 1 through 3 cover the basic topics required in intro- ducing filter design; Chapters 4 through 7 then focus on opamp-based active-RC filters; finally, Chapters 8 through 12 concentrate on OTA-Capacitor filters (and introduce some other approaches), taking the reader up to the frontiers of modern active continuous-time filter design. A book such as this requires much work on the part of the authors. In this case it is an achievement of which the authors are particularly proud because it represents the success- ful collaboration of three engineering academics from quite different cultural back- ground—Greece, China, and the United Kingdom. The catalyst to this collaboration has been Nora Konopka from CRC Press in the U.S., to whom all of us are grateful. In addition, we have many to thank as individuals. Theodore Deliyannis particularly thanks his colleagues I. Haritantis, G. Alexiou, and S. Fotopoulos in Patras, Prof. A. G. Constantinides of Imperial College, London, and the IEE for allowing him to reproduce parts of their com- mon papers published in the Proceedings of the IEE. He also expresses his gratitude to Mrs. V. Boile and his postgraduate student K. Giannakopoulos for their help in preparing the manuscript. Finally he thanks his wife Myriam for her encouragement and understanding. Yichuang Sun acknowledges Prof. Barry Jefferies of the University of Hertfordshire, U.K., for his support, and helpful comments on his work; he is also grateful to Tony Crook for his help in preparing the manuscript. He also expresses his thanks to his wife, Xiaohui, and son, Bo, for their support and patience. ©1999 CRC Press LLC Kel Fidler is particularly grateful to his co-authors Theodore and Yichuang for their incredibly hard work, and their patience and civility at times when things became a little quiet! He also thanks all his friends and colleagues in York for their forbearance and understanding when they observed that, once again, he had taken on too much! In particular he thanks Navin Sullivan, without whom, in many complex ways, these authors would never have come together to write this book. TLD, Patras YS, Hatfield JKF, York ©1999 CRC Press LLC Authors Professor Theodore L. Deliyannis is Professor of Electronics in the University of Patras, Greece. Dr. Yichuang Sun is Reader in Electronics in the University of Hertfordshire, U.K. Professor J. Kel Fidler is Professor of Electronics in the University of York, U.K. ©1999 CRC Press LLC Contents Chapter 1 Filter Fundamentals 1.1 Introduction 1.2 Filter Characterization 1.2.1 Lumped 1.2.2 Linear 1.2.3 Continuous-Time and Discrete-Time 1.2.4 Time-Invariant 1.2.5 Finite 1.2.6 Passive and Active 1.3 Types of Filters 1.4 Steps in Filter Design 1.5 Analysis 1.5.1 Nodal Analysis 1.5.2 Network Parameters 1.5.2.1 One-Port Network 1.5.2.2 Two-Port Network 1.5.3 Two-Port Interconnections 1.5.3.1 Series–Series Connection 1.5.3.2 Parallel–Parallel Connection 1.5.3.3 Series Input–Parallel Output Connection 1.5.3.4 Parallel Input–Series Output Connection 1.5.3.5 Cascade Connection 1.5.4 Network Transfer Functions 1.6 Continuous-Time Filter Functions 1.6.1 Pole-Zero Locations 1.6.2 Frequency Response 1.6.3 Transient Response 1.6.3.1 Impulse Response 1.6.3.2 Step Response 1.6.4 Step and Frequency Response 1.7 Stability 1.7.1 Short-Circuit and Open-Circuit Stability 1.7.2 Absolute Stability and Potential Instability 1.8 Passivity Criteria for One- and Two-Port Networks 1.8.1 One-Ports 1.8.2 Two-Ports 1.8.3 Activity 1.8.4 Passivity and Stability 1.9 Reciprocity 1.10 Summary References and Further Reading Chapter 2 The Approximation Problem 2.1 Introduction ©1999 CRC Press LLC 2.2 Filter Specifications and Permitted Functions 2.2.1 Causality 2.2.2 Rational Functions 2.2.3 Stability 2.3 Formulation of the Approximation Problem 2.4 Approximation of the Ideal Lowpass Filter 2.4.1 Butterworth or Maximally Flat Approximation 2.4.2 Chebyshev or Equiripple Approximation 2.4.3 Inverse Chebyshev Approximation 2.4.4 Papoulis Approximation 2.4.5 Elliptic Function or Cauer Approximation 2.4.6 Selecting the Filter from Its Specifications 2.4.7 Amplitude Equalization 2.5 Filters with Linear Phase: Delays 2.5.1 Bessel-Thomson Delay Approximation 2.5.2 Other Delay Functions 2.5.3 Delay Equalization 2.6 Frequency Transformations 2.6.1 Lowpass-to-Lowpass Transformation 2.6.2 Lowpass-to-Highpass Transformation 2.6.3 Lowpass-to-Bandpass Transformation 2.6.4 Lowpass-to-Bandstop Transformation 2.6.5 Delay Denormalization 2.7 Design Tables for Passive LC Ladder Filters 2.7.1 Transformation of Elements 2.7.1.1 LC Filters 2.7.1.2 Active RC Filters 2.8 Impedance Scaling 2.9 Predistortion 2.10 Summary References Chapter 3 Active Elements 3.1 Introduction 3.2 Ideal Controlled Sources 3.3 Impedance Transformation (Generalized Impedance Converters and Inverters) 3.3.1 Generalized Impedance Converters 3.3.1.1 The Ideal Active Transformer 3.3.1.2 The Ideal Negative Impedance Converter 3.3.1.3 The Positive Impedance Converter 3.3.1.4 The Frequency-Dependent Negative Resistor 3.3.2 Generalized Impedance Inverters 3.3.2.1 The Gyrator 3.3.2.2 Negative Impedance Inverter 3.4 Negative Resistance 3.5 Ideal Operational Amplifier 3.5.1 Operations Using the Ideal Opamp 3.5.1.1 Summation of Voltages 3.5.1.2 Integration 3.5.2 Realization of Some Active Elements Using Opamps ©1999 CRC Press LLC