Gm-C Filter Synthesis for Modern RF Systems (Lecture Notes in Electrical Engineering, 807) 9811665605, 9789811665608

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Lecture Notes in Electrical Engineering 807

Vančo Litovski

Gm-C Filter Synthesis for Modern RF Systems

Lecture Notes in Electrical Engineering Volume 807

Series Editors Leopoldo Angrisani, Department of Electrical and Information Technologies Engineering, University of Napoli Federico II, Naples, Italy Marco Arteaga, Departament de Control y Robótica, Universidad Nacional Autónoma de México, Coyoacán, Mexico Bijaya Ketan Panigrahi, Electrical Engineering, Indian Institute of Technology Delhi, New Delhi, Delhi, India Samarjit Chakraborty, Fakultät für Elektrotechnik und Informationstechnik, TU München, Munich, Germany Jiming Chen, Zhejiang University, Hangzhou, Zhejiang, China Shanben Chen, Materials Science and Engineering, Shanghai Jiao Tong University, Shanghai, China Tan Kay Chen, Department of Electrical and Computer Engineering, National University of Singapore, Singapore, Singapore Rüdiger Dillmann, Humanoids and Intelligent Systems Laboratory, Karlsruhe Institute for Technology, Karlsruhe, Germany Haibin Duan, Beijing University of Aeronautics and Astronautics, Beijing, China Gianluigi Ferrari, Università di Parma, Parma, Italy Manuel Ferre, Centre for Automation and Robotics CAR (UPM-CSIC), Universidad Politécnica de Madrid, Madrid, Spain Sandra Hirche, Department of Electrical Engineering and Information Science, Technische Universität München, Munich, Germany Faryar Jabbari, Department of Mechanical and Aerospace Engineering, University of California, Irvine, CA, USA Limin Jia, State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing, China Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland Alaa Khamis, German University in Egypt El Tagamoa El Khames, New Cairo City, Egypt Torsten Kroeger, Stanford University, Stanford, CA, USA Yong Li, Hunan University, Changsha, Hunan, China Qilian Liang, Department of Electrical Engineering, University of Texas at Arlington, Arlington, TX, USA Ferran Martín, Departament d’Enginyeria Electrònica, Universitat Autònoma de Barcelona, Bellaterra, Barcelona, Spain Tan Cher Ming, College of Engineering, Nanyang Technological University, Singapore, Singapore Wolfgang Minker, Institute of Information Technology, University of Ulm, Ulm, Germany Pradeep Misra, Department of Electrical Engineering, Wright State University, Dayton, OH, USA Sebastian Möller, Quality and Usability Laboratory, TU Berlin, Berlin, Germany Subhas Mukhopadhyay, School of Engineering & Advanced Technology, Massey University, Palmerston North, Manawatu-Wanganui, New Zealand Cun-Zheng Ning, Electrical Engineering, Arizona State University, Tempe, AZ, USA Toyoaki Nishida, Graduate School of Informatics, Kyoto University, Kyoto, Japan Federica Pascucci, Dipartimento di Ingegneria, Università degli Studi “Roma Tre”, Rome, Italy Yong Qin, State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing, China Gan Woon Seng, School of Electrical & Electronic Engineering, Nanyang Technological University, Singapore, Singapore Joachim Speidel, Institute of Telecommunications, Universität Stuttgart, Stuttgart, Germany Germano Veiga, Campus da FEUP, INESC Porto, Porto, Portugal Haitao Wu, Academy of Opto-electronics, Chinese Academy of Sciences, Beijing, China Walter Zamboni, DIEM - Università degli studi di Salerno, Fisciano, Salerno, Italy Junjie James Zhang, Charlotte, NC, USA

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Vanˇco Litovski

Gm-C Filter Synthesis for Modern RF Systems

Vanˇco Litovski Faculty of Electronic Engineering University of Niš Niš, Serbia

ISSN 1876-1100 ISSN 1876-1119 (electronic) Lecture Notes in Electrical Engineering ISBN 978-981-16-6560-8 ISBN 978-981-16-6561-5 (eBook) https://doi.org/10.1007/978-981-16-6561-5 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 This work is subject to copyright. All rights are solely and exclusively licensed 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, expressed 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 Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

This is to express my appreciation to my son Ivan Litovski for his patience and unlimited support in the development of the RM software and the preparation of this book

Preface

There were, in our opinion, four major discoveries which fundamentally influenced everyone’s everyday life during history. First comes the needle. It allowed for pieces of fur to be connected together and to produce the first clothing. In that way, the son of the early tribal leader got serious chances to survive and to inherit the leadership, all that leading to continuation of the established social order for long. In other words, the health problem was fundamentally solved. Next comes the wheel. That discovery (Not given to the pre-Columbian American tribes by the visiting ancient aliens.) enabled ubiquitous transportation be it for personal, business, or military needs. So, the need for transportation was solved. The third most important discovery was the alternating current. In fact, Nikola Tesla brought unlimited amount of electrical energy to anyone who needed. It is fascinating how easy is now to do everything in household, in industry, in illumination, etc. Looking from the space at night, Earth looks like a shiny playing ball. Finally, the CMOS technology brought the last industrial revolution solving the problem of information storing, processing, and distribution. By dramatically reducing the amount of energy spent per logic state in a digital circuit, it allowed drastic reduction of the size of the fundamental pair of transistors (the inverter) and brought together billions of transistors in a single chip. As it usually happens, at the early days of CMOS, there were serious doubts as to what will be the largest size of the chip and the highest frequency in use. That was especially notable when the future of analog CMOS was to be predicted. Nowadays, by paramount reducing the size of the transistors (being difficult to imagine in the near past) and consequently the length of the interconnections within the chip, the parasitic capacitances were seriously diminished so enabling series production of analog functions working in the GHz part of the frequency spectrum. As for the analog filters, this progress brought a new component named CMOS Operational Transconductance Amplifier (OTA). Using CMOS OTAs, analog filter implemented in RF systems may be easily integrated nowadays. The aim of this book is to make it possible for every electronic engineer to design electronic filters based on OTAs and capacitors. These are known as Gm-C filters. vii

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Preface

We will elaborate method for circuit synthesis of three topologies: parallel, cascade, and the one being emanated from existing LC filters. After systematic comparisons based on several figures of merit, the cascaded will be selected and tables will be given containing circuit element values (transconductances and capacitances) for circuit synthesis of the best-known low-pass transfer functions. In addition, for all three topologies, methods and results will be given enabling synthesis of two-phase (poly-phase) filters in CMOS Gm-C technology no matter how complex the prototypes are. To our knowledge, most of the circuit synthesis part of the book (especially the parallel and the cascade synthesis) is fully original, i.e., here published for the first time. Niš, Serbia

Vanˇco Litovski

Contents

1 The Design of Gm-C Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Specifics of the RF Filter Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 5

2 A Glimpse to the Active and Lumped Passive Integrated High Frequency Electronic Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Passive Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Integrated Resistor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Integrated Capacitor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Integrated Planar Inductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Operational Amplifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Conventional CMOS Operational Amplifier . . . . . . . . . . . . . . 2.3.2 CMOS Operational Transconductance Amplifier . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 7 7 8 9 11 12 12 14 19

3 Parallel Realization of Gm-C Single- and Two-Phase Integrated Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Design Based on General Second-Order Cell . . . . . . . . . . . . . . . . . . . 3.3 Decomposition of the Transfer Function . . . . . . . . . . . . . . . . . . . . . . . 3.4 Physical Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 The Adder Cell (Summing Amplifier) . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Design Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Polyphase Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.1 Physical Implementation of the Polyphase Case . . . . . . . . . . 3.7.2 Example Two-Phase Linear-Phase Filter . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21 21 25 25 28 29 30 34 37 38 43

ix

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Contents

4 Cascade Realization of Active Gm-C Circuits . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Structure of the Cascaded Gm-C Filter . . . . . . . . . . . . . . . . . . . . . . . . 4.3 First-Order Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Low-Pass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 High-Pass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Zero on the Real Axis and All-Pass . . . . . . . . . . . . . . . . . . . . . 4.4 Second-Order Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Low-Pass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Band-Pass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 High-Pass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.4 Band-Stop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.5 Second-Order Low-Pass with a First-Order Zero at the σ-axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.6 Complex Zero and All-Pass . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 General Structure of a Gm-C Cell for Cascade Realization . . . . . . . 4.6 Design Example, a Band-Stop Cascaded Gm-C Filter . . . . . . . . . . . . 4.7 Design Example, a Band-Pass Cascaded Gm-C Filter . . . . . . . . . . . . 4.8 Two-Phase Cell Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 Design Example, a Two-Phase Cascaded Gm-C Filter . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45 45 46 48 48 49 50 51 51 53 54 56 57 58 60 61 63 66 70 74

5 Use of LC-to-Gm-C Transformation for Synthesis of Gm-C Single- and Two-Phase Integrated Filters . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.2 The Gyrator and the Simulated Inductor . . . . . . . . . . . . . . . . . . . . . . . 78 5.2.1 Floating Simulated Inductor . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.2.2 Simulated Ideal Grounded Transformer . . . . . . . . . . . . . . . . . 80 5.3 Circuit Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.4 Design Example No. 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.5 Design Example No. 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.6 Creation of the Two-Phase Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5.7 Design Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 6 Synthesis of Analog Gm-C Hilbert Transformer and Its Implementation for Band-Pass Filter Design . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Physical Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 The 6.5 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 On the Design of Arithmetically Symmetrical Wideband Selective Linear-Phase Band-Pass Gm-C Filters . . . . . . . . . . . . . . . . . 6.7 Design Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

111 111 115 116 120 120 124 125 128

Contents

7 Implementation Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Study of the Worst-Case Tolerance of Gm-C Filters . . . . . . . . . . . . . 7.3 Worst-Case Tolerance of Two-Phase Gm-C Filters Due to the Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 A Short Discussion on the Noise in Gm-C Filters . . . . . . . . . . . . . . . 7.5 On the Influence of the Electrical Characteristic of the Transconductor to the Filter Response . . . . . . . . . . . . . . . . . . . 7.6 Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 The Ultimate Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 How to Use the Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Polynomial Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 LSM Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Papoulis (Legendre or Optimal) Filters . . . . . . . . . . . . . . . . . . 8.3.3 Halpern Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.4 Butterworth (Maximally Flat) Filters . . . . . . . . . . . . . . . . . . . . 8.3.5 Chebyshev Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.6 Thomson (Bessel or Maximally Flat Group Delay) Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.7 Equi-ripple Group Delay Filters . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 LSM_Z Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2 Papoulis_Z Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.3 Halpern_Z Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.4 Butterworth_Z (Inverse Chebyshev) Filters . . . . . . . . . . . . . . 8.4.5 Modified Elliptic (Zolotarev) Filters . . . . . . . . . . . . . . . . . . . . 8.4.6 Thomson_Z Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.7 Equi-ripple-Td_Z Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

131 131 131 134 135 138 140 146 149 151 151 152 154 155 157 158 160 162 168 170 175 176 187 198 209 220 264 276 288

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

About the Author

Prof. Vanˇco Litovski was born in 1947 in Rakita, South Macedonia, Greece. He graduated from the Faculty of Electronic Engineering in Niš in 1970, and obtained his M.Sc. in 1974, and his Ph.D. in 1977. He was appointed as a teaching assistant at the Faculty of Electronic Engineering in 1970 and became a Full Professor at the same faculty in 1987. He was elected as Visiting Professor (honoris causa) at the University of Southampton in 1999. From 1987 until 1990, he was a consultant to the CEO of Ei, and was Head of the Chair of Electronics at the Faculty of Electronic Engineering in Niš for 12 years. From 2015 to 2017, he was a researcher at the University of Bath. He has taught courses related to analogue electronics, electronic circuit design, and artificial intelligence at the electro-technical faculties in Priština, Skopje, Sarajevo, Banja Luka, and Novi Sad. He received several awards including from the Faculty of Electronic Engineering (Charter in 1980, Charter in 1985, and a Special Recognition in 1995) and the University of Niš (Plaque 1985). Prof. Litovski has published 6 monographs, over 400 articles in international and national journals and at conferences, 25 textbooks, and more than 40 professional reports and studies. His research interests include electronic and electrical design and design for sustainability, and he led the design of the first custom commercial digital and research-oriented analogue CMOS circuit in Serbia. He has also headed 8 strategic projects financed by the Serbian and Yugoslav governments and the JNA, and has participated in several European projects funded by the governments of Germany, Austria, UK, and Spain, and the EC as well as the Black See Organization of Economic Cooperation (BSEC).

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Chapter 1

The Design of Gm-C Filters

1.1 Introduction By submitting the application for the US Patent No. 649621 (Fig. 1.1) on September 01, 1897, Nikola Tesla became the real inventor of the radio. It was patented as his first inventions in the field of wireless energy transmission [1]. It is our opinion that by this contribution, after solving the everlasting energy distribution problem by the invention of the alternating current, Tesla repeated himself and introduced the humanity into the era of unlimited information distribution. The first mass implementation of telecommunication systems was the telephony and its front-end, just after the microphone, was a filter (300–3200 Hz) defining the voice channel. It used transmission lines. To satisfy a need for reaching any point on earth be it on ships or on other continent, however, one was supposed to communicate wirelessly. Here comes Tesla. The hunger for information soon brought the trade to an explosion with ever rising need for frequency spectrum occupation which nowadays is reaching fantastic 60 GHz [2] and beyond. The traffic is becoming ubiquitous though the frequency spectrum is not a renewable natural resource. One needs filters to separate communication channels and to allow for signals to be extracted from a compound supporting simultaneously millions of different users. Our intentions in this part of the book are twofold. First, we want to continue our mission we started in [3]. Namely, we want to advocate a systematic approach to the filter design no matter the technology. What we may see in many publications now is many “homemade” one-shot solutions (e.g., bounded by the order of the filter or the type of the amplitude/phase characteristic) which are not based on systematic use of the theory of transfer function synthesis and are of no use for modified design requirements. Add to that ignorance, where the schematic, which is transfer function independent, is named by one of the possible approximation approaches to transfer function synthesis (e.g., elliptic for a schematic that may represent at least ten different approximation solutions). It is especially difficult to accept the perpetual promotion of Butterworth solutions which are known to © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 V. Litovski, Gm-C Filter Synthesis for Modern RF Systems, Lecture Notes in Electrical Engineering 807, https://doi.org/10.1007/978-981-16-6561-5_1

1

2

1 The Design of Gm-C Filters

Fig. 1.1 Part of the Nikola Tesla’s application for the US Patent No. 649621

be the worst among the selective filters. Further, one recommends polynomial solutions while use of transmission zeros on the ω-axis improves selectivity significantly without affecting the circuit complexity in most topologies and technologies. For example, a fourth-order selective low-pass filter with a transmission zero at the ωaxis is more selective and has controllable attenuation in the stopband while being realized with two biquads as opposed to a sixth-order polynomial filter which needs three biquads. We want to demonstrate a filter design process which starts from design requirements, goes through transfer function synthesis, and ends with system (or circuit) synthesis. By use of the RM software (or similar), any selective (and in the same time linear phase) transfer function may be synthesized and made available to the user so avoiding the restrictions imposed by the use of catalogs. Our second goal was to demonstrate the circuit synthesis procedures for Gm-C filters be it single (normal) or two phase. That also represents a continuation of the efforts reported in [3]. Namely, here we complete the Gm-C circuit synthesis by adding the cascade and the parallel solution. That is to be added to the synthesis based on LC prototypes so that a completion of the subject is done. Here, however, for the first time, complete two-phase Gm-C circuit synthesis is described for all three topologies. Comparisons are made to help the designer to come to hers/his own topology according to the technology available. So, we hope, based on the results reported, the reader will be capable to select his own structure of the filter (among the cascaded, parallel and emanated from passive LC) and to develop his own software for circuit synthesis. In that way, one will be

1.1 Introduction

3

capable to avoid the simplistic synthesis paradigms frequently encountered in the literature. Note, it was not our goal to solve a particular filter implementation problem, however. We simply supply the ideas and the tools. It is up to the designer to satisfy hers/his design requirements. The examples, given all through this part of the book, are here to illustrate both the transfer function synthesis and the circuit synthesis methods. Nevertheless, they are conceived to resemble some particular applications in software-defined radio [4], in wireless sensor networks [5], in hard disk drive read/write channels [6], video signal processors [7] and similar. We want also to reiterate that all transfer functions produced by the RM software, here transformed into Gm-C lumped circuit element filters, may be used as digital IIR (recursive) filters if bilinear transform is implemented [3].

1.2 Specifics of the RF Filter Synthesis We will here stress some features of the concepts of the modern radiotelecommunication systems in order to connect the contexts of transceiver systems and filtering within them. A general division of the system is to analog and digital part. The latter is in charge mainly for baseband processing and is not in the scope of this book. Analog processing may be viewed as a subsystem constituted of two parts, too. At the receiver side, the front-end of it constitutes most frequently of the antenna, the front-end band-select filter, and the low-noise amplifier as depicted in Fig. 1.2 [8]. Occasionally, it may contain a stage of band-reject filter to facilitate dealing with the image of the useful signal. The most important thing, from filtering point of view, at this stage, is the fact that the frequency spectrum under consideration is in the range of very high frequencies and, accordingly, the filtering circuits are normally realized using distributed parameters components such as surface acoustic waves (SAW) [9], bulk acoustic waves (BAW) [10], and microstrip filters [11]. On the opposite side, at the transmitter output, one meets also waveguides as the most convenient technology to deliver high power to the output antenna [12]. Again, all these are out of the scope of this book.

Fig. 1.2 Dual-IF heterodyne receiver (simplified from [8])

4

1 The Design of Gm-C Filters

We will be dealing with analog filtering based on lumped circuit elements and especially on two components: operational transconductance amplifiers (OTA) which are frequently nick-named transconductors and denoted as Gm or gm , and capacitors. Hence the filtering technology Gm-C. In the subsequent paragraph, we will try to make a case for this technology for implementation in the remaining part of the analog signal processing chain. The task of the remaining part of the analog subsystem is to perform channel selection. It “selects” the desired signal channel and “rejects” the interferers in the other channels. To do that mixing is to be performed and consequently part of the resulting spectrum is selected by a filter. There are several concepts of performing that activity. Three of them are mentioned in the sequel. Of course, our intention here is to locate the position and the type of filters necessary for the functionality of the system. Properties of these concepts and their mutual advantages and disadvantages are out of the scope of this chapter and may be found in [8]. Figure 1.2 depicts the structure of a dual intermediate frequency heterodyne receiver [8]. In this solution, the first mixing operation reduces the carrier frequency of the incoming signal to an intermediate frequency (IF) of relatively high value. The final IF is created after second mixing. The reason why this solution is mentioned here is the fact that band-pass filters are necessary for the baseband signal to be extracted. Figure 1.3 depicts the structure of a zero IF or a direct conversion receiver. Here, low-pass filters are needed only. Finally, Fig. 1.4 depicts the so-called low-IF receiver structure [13]. The main property of this structure is that the passband of the zero-IF solution is shifted toward higher frequency so that the lowest frequencies (DC and the pink noise region) are avoided. Here, two-phase (poly) filter is necessary to perform both the shifting and the selection. Throughout the following chapters, however, the opportunity will be exploited to demonstrate the synthesis methods of band-stop filters, Hilbert transformers and

Fig. 1.3 Direct conversion (zero IF) receiver (simplified from [8])

1.2 Specifics of the RF Filter Synthesis

5

Fig. 1.4 Low-IF receiver (simplified from [13])

arithmetically symmetrical band-pass filters most of them with linear phase responses in their passbands. In every solution, three variants will be offered: cascade, parallel, and a circuit obtained by transformation of an LC filter into a Gm-C filter. Comparisons will be given among the topologies in order to get the feeling on the effectiveness from circuit synthesis software development; physical feasibility; sensitivity to element value variations; noise susceptibility and similar point of view. The circuit synthesis methodology will be verified by statistical tolerance analysis.

References 1. Brenner P (2009) Tesla against Marconi: the dispute for the radio patent paternity. In: Proceedings of the IEEE EUROCON 2009. St.-Petersburg, Russia, pp 1035–1042 2. Kraemer M (2010) Design of a low-power 60 GHz transceiver front-end and behavioral modeling and implementation of its key building blocks in 65 nm CMOS. Thesis at L’Institut National des Sciences Appliquées de Toulouse, France 3. Litovski V (2019) Electronic filters, theory, numerical recipes, and design practice based on the RM software. Springer Science+Business Media 4. Collins TF, Getz R, Pu D, Wyglinski AM (2018) Software-defined radio for engineers illustrated edition. Artech House 5. D’Amico S, Ryckaert J, Baschirotto A (2006) A up-to-1 GHz low-power baseband chain for UWB receivers. In: 2006 Proceedings of the 32nd European solid-state circuits conference. Montreux, Switzerland, pp 263–265 6. Mehr I, Welland DR (1997) A CMOS continuous-time gm-C filter for PRML read channel applications at 150 Mb/s and beyond. IEEE J Solid-State Circuits 32(4):499–513 7. Gopinathan V, Tsividis YP, Tan K-S, Hester RK (1990) Design considerations for highfrequency continuous-time filters and implementation of an antialiasing filter for digital video. IEEE J Solid-State Circuits 25(6):1368–1378 8. Razavi B (2011) RF-microelectronics. Prentice Hall 9. Fischerauer G, Ebner T, Kruck P, Morozumi K, Thomas R, Pitschi M (2001) SAW filter solutions to the needs of 3G cellular phones. In: Digest of the 2001 IEEE MTT-S International microwave symposium. Phoenix, AZ, USA 10. Aigner R, Marksteiner S, Elbrecht L, Nessler W (2003) RF-filters in mobile phone applications. In: Digest of TRANSDUCERS ’03. 12th international conference on solid-state sensors, actuators and microsystems. Boston, MA, USA (2E124.P), pp 891–894

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1 The Design of Gm-C Filters

11. Al-Yasir YIA, Parchin NO, Abd-Alhameed RA, Abdulkhaleq AM, Noras JM (2019) Recent progress in the design of 4G/5G reconfigurable filters. Electronics 8(114):1–17 12. Miller M (2020) Waveguide makes a comeback in 5G—for antennas. Electronic Design. https:// www.electronicdesign.com/industrial-automation/article/21121326/waveguide-makes-a-com eback-in-5gfor-antennas 13. Behbahani F, Firouzkouhi Chokkalingam R, Delshadpour S, Kheirkhahi A, Nariman M, Conta M, Bhatia S (2002) A fully integrated low-IF CMOS GPS radio with on-chip analog image rejection. IEEE J Solid-State Circuits 37(12):1721–1727

Chapter 2

A Glimpse to the Active and Lumped Passive Integrated High Frequency Electronic Components

2.1 Introduction Since the beginning of electronic integration, producing high frequency active and lumped passive components in silicon was among the greatest challenges. With the advent of CMOS and the dramatic reduction of the dimensions, however, these “dreams” are becoming more and more true. Having a component (be it passive or active) being not dependent of the parasitics (which become aggressive at high frequencies) is becoming more and more affordable. That stands for the synthesis of active filters which went through many different technologies to arrive at integrated CMOS. In this chapter, we will try to give basic information on the potential components seen as building blocks of integrated active CMOS filters. By exposing the disadvantages of some, we will do elimination to reach the ultimate solution which is Gm-C technology. For that reason, the transconductor—the main active component in integrated CMOS filter—will be considered with comparably more attention.

2.2 Passive Components Three passive components intended to be used in high frequency integrated CMOS circuits will be considered in this paragraph. These are the integrated resistor, the MOS capacitor, and the planar spiral inductor.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 V. Litovski, Gm-C Filter Synthesis for Modern RF Systems, Lecture Notes in Electrical Engineering 807, https://doi.org/10.1007/978-981-16-6561-5_2

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2 A Glimpse to the Active and Lumped Passive Integrated …

2.2.1 Integrated Resistor There are several variants of realization of integrated resistors in MOS technology. Some of the simplest (smallest knowledge of the technology processes needed) are depicted in Fig. 2.1. Here, Fig. 2.1a depicts a “diffusion” resistor using the diffusion process intended to be used as a drain (or source) of the transistor for creation of the resistor. Lateral view is given with the two “metal” regions being the contacts. The distance between the contacts represents the length of the resistors, the depth of the diffusion is the resistor’s height, while the width is not shown. Considering the height of the resistor a technology constant, the resistance of this type of resistors may be expressed in the form R=ρ·

ρ L L = · = Rs · N S t W

(2.1)

where ρ is the specific resistance, L is the length, S is the cross-sectional area which is calculated as S = t · W, with t representing the thickness (depth) and W the width of the resistor. As can be seen, two new design variables are introduced: the sheet resistance Rs (being the technology parameter) and N “the number of squares” being the real design parameter.

metal

oxide

N+

P-

Nmetal

a

oxide

polysilic oksid -

N

b

metal P+ N-

oxide -

P

P+ c

Fig. 2.1 Integrated resistor in a p-well. a Diffusion of the source or the drain, b polysilicon resistor, and c p-well as a resistor

2.2 Passive Components

9

The above expression is valid for any kind of integrated resistors the main difference being in the sheet resistance. Two additional technology solutions (among, really, many others) are shown in Fig. 2.1. The main properties of the integrated resistor are [1]: low accuracy (in some cases worse than 20%), high temperature coefficient (e.g., 600 ppm/°C and more), and large N for reasonable values of resistance. In the last case, one has to lay out long resistors (many squares) which not only occupies large silicon area but collects large parasitic capacitance, too. For these reasons, one is trying to avoid resistors as much as possible. As we will see later on, no resistors will be needed for performing any filtering function in the solutions we are recommending in this book.

2.2.2 Integrated Capacitor The capacitance is by nature directly proportional to the area of the capacitor. That must be a serious consideration when integrated filters are designed. One is to attempt to minimize the maximum capacitance within the circuit. The other concern related to the capacitance value is related topics as accuracy, temperature coefficient, aging, voltage dependence, and similar. There are mainly two approaches to design a capacitor. The first one is based on the depletion capacitance of a p–n junction which must always be inversely polarized. Of course, to keep the capacitance value constant, the polarizing voltage must be also constant. Having many capacitors with different values and in the same time economizing the number of constant voltage sources, one has to be very careful with the design. One is not to forget that the polarizing voltage may disturb the operating conditions of the active components used within the filter. Alternatively, one may use MOS capacitor as depicted in Fig. 2.2. In this case the dielectric, being the thin oxide, is really thin which gives rise to the capacitance and allows minimization of area. The bottom plate of this capacitor is P+ diffusion (heavily doped P region) which has small but not negligible resistance which is the

Fig. 2.2 MOS capacitor in a CMOS integrated circuit

10

2 A Glimpse to the Active and Lumped Passive Integrated …

Fig. 2.3 A programmable capacitor

main parasitic of this capacitor. Of course, on a P-substrate, N+ diffusion will be used which is expected to have smaller resistance for the same doping concentration. There are other methods, within the general concept, to produce a MOS capacitor. Nevertheless, in general, the following figures are representing the properties of the produced integrated capacitor: oxide thickness (t ox ) larger than 15 nm, accuracy of the order of 10%, temperature coefficient always smaller than 100 ppm/o C, and voltage dependence about 50 ppm/V. Producing a capacitor with prescribed capacitance would need design of every capacitor individually which in the case of complex filters would be a laborious and error-prone task. Instead, one use common centroid structures [2] which are “programmed” to produce the desired value. The concept is based on the following formula C = C0 + C0

n 

2i · αi .

(2.2)

i=1

Here, C 0 is the minimum capacitance (primitive cell replicated within the structure) which is designed as a capacitor and for which a precise information on its value is available (including parasitics), α i is a single digit binary number, and n is defining the capacitance value. Equation (2.2) is illustrated in Fig. 2.3. So, if C 0 = 25 fF, n = 2, α 1 = 0, and α 2 = 1, one has C = 25 + 100 = 125 fF. Sophisticated algorithms were developed for programming the layout of a common centroid structure [3]. While the series resistance of the bottom plate of an integrated capacitor may be reduced by proper design tricks, one must also take into account the resistances of the switches which are in fact CMOS transmission gates (with short and wide transistors). Note, these switches are working in the linear region since the drain to source voltage will normally change sign during operation. Alternatives to the common centroid structure were suggested in [4, 5].

2.2 Passive Components

11

2.2.3 Integrated Planar Inductor One of the problems encountered in high frequency analog integrated filter synthesis is the area needed to produce a planar inductor. It is realized in a form of a flat spiral line the inductance of which is limited not only by the area but also by its huge parasitic capacitance. There are several realizations of such a component in CMOS as reported in [6–9]. The most important electrical parameters of an integrated inductor are the inductance (L), its resistance (R), its parasitic capacitance, and its Q-factor (Q) as a secondary parameter. The layout of one inductor of this kind is depicted in Fig. 2.4. One may see that the wires are twisted to reduce the parasitic capacitance. This inductor is specific in the sense that it has a tap terminal allowing specific uses. Figure 2.5 depicts the dependence of the reactance and the resistance of the integrated inductor on the signal frequency which is an additional problem when designing filters with this kind of components. The numerical values of the equivalent at f =

Fig. 2.4 Possible layout of a planar integrated inductor/transformer

Fig. 2.5 Typical frequency dependence of the impedance of a planar inductor

12

2 A Glimpse to the Active and Lumped Passive Integrated …

Fig. 2.6 Typical quality factor curve of a planar inductor

2.43 GHz are L = 6 nH, Rs = 9.3 , and C s = 220 fF (s stands for series). The value of the Q-factor is Q = 8.33 (Fig. 2.6). The frequency dependence (and the value) of the quality factor of this component makes it not desirable for filter design especially for filters with broad passband. Instead, its application is mostly limited to high frequency oscillators and mixers [9].

2.3 Operational Amplifiers Two versions of integrated CMOS operational amplifier will be considered with the goal to set the niches in which each of them belongs.

2.3.1 Conventional CMOS Operational Amplifier The main difference between the well-established bipolar operational amplifiers (e.g., µA741) is the difference between the very fundamental components. The bipolar transistor exhibits almost an order of magnitude higher transconductance than the MOS counterpart. That, as a consequence, is limiting the gain bandwidth product (GBW) of the CMOS operational amplifiers. In the next we will list some salient properties of the CMOS integrated OA (Operational Amplifier). The open-loop gain is the gain of the op-amp without positive or negative feedback, and for such an amplifier, the gain should be infinite, but typical real values range from about 20,000 to 200,000. The input impedance is assumed to be infinite to prevent any current flowing from the source excitation into the amplifier’s input circuitry. Real OAs have input leakage currents from a few pA to a few mA, and of course, input capacitance defined by the gate area of the input transistor and the Miller capacitance of the same (which altogether is less than a pF).

2.3 Operational Amplifiers

13

Fig. 2.7 Typical (approximated) amplitude characteristic of a quality CMOS operational amplifier

The output impedance of the ideal operational amplifier is assumed to be zero acting as an ideal voltage source with no internal resistance so that it can deliver as much current as necessary to the load. This (mainly) resistive feature is effectively in series with the load, thereby reducing the output voltage available to the load. Real OAs have output resistance in the 100 –20 k range. If no direct negative feedback is used, these values may be a serious obstacle to the implementation of OAs in filter design. Figure 2.7 depicts a typical frequency dependence of the gain of a high frequency CMOS operational amplifier. In this case, the open-loop gain is 49 dB (282 times), and the 3 dBcut-off frequency is 4 MHz. Finally, the GBW = 2 GHz [10]. Its performance is really impressive. However, one can read from Fig. 2.7 that for frequencies above 2 GHz there is no gain at all, which in fact makes questionable the application of the OA for higher frequencies. When claiming this, we have in mind that during every single design the gain was considered infinite at all frequencies. Note, as the CMOS transistor pairs are dimensioned for a large transconductances to achieve high GBW and high gain (at low frequencies), the accompanied parasitic capacitances severely erode the amplifier’s phase margin whereby reducing the GBW [11]. To summarize, for the implementation of the CMOS operational amplifier in high frequency active RC filters, first and most important obstacle is the need to be used in conjunction with capacitors and resistors. The latest, unfortunately, are the most undesirable components for integrated applications.

14

2 A Glimpse to the Active and Lumped Passive Integrated …

2.3.2 CMOS Operational Transconductance Amplifier So, if we eliminate the resistor, we are about to use capacitors and operational amplifiers only, which is of no use since there is no time constant. No time constant means no selectivity since there is no frequency dependence. The solution of this problem is that in place of using voltage-controlled voltage sources (VCVSs) one should use voltage-controlled current sources (VCCSs) in conjunction with capacitors. VCCS is named transconductor since its main parameter is transconductance and so one gets a new technology Gm- (coming from transconductance) C (coming from the capacitance), i.e., Gm-C circuits. Gm-C circuits allow for creation of filters (and not only filters) which may be fully integrated while not using inductors and resistors. The main trick is in the fact that the transconductor may be used as a resistor (R = 1/gm ) while avoiding all the negative properties of it. It is well known that the capacitors are components with the lowest tolerances, lowest temperature coefficients, small parasitics, and with no noise at all so what is remaining is to develop a quality transconductor which is a circuit with very high input impedance, with very high output impedance, and with controllable transconductance which is expected to be non-frequency-dependent. The new circuit element having these properties is named operational transconductance amplifier (OTA). Research in development of such circuits started early [12] to follow the shrinking of the dimensions within the CMOS technology [13].

Fig. 2.8 Transconductor. Schematic symbols a differential input differential output, b differential input single-ended output, c transconductor connected as a grounded resistor, and d transconductors connected as a floating resistor

2.3 Operational Amplifiers

15

Figure 2.8 depicts the schematic symbol used for the transconductor. Figure 2.8a represents a differential input differential output (DIDO) transconductor. Ideally, the following would be valid Iout = gm · (Vo+ − Vo− ).

(2.3)

Further, Fig. 2.8b represents the differential input single-ended output transconductor (DISEO) (being emanated from the previous one by ignoring the V o -terminal). Here, the following stands Iout = gm · Vo .

(2.4)

A grounded resistor is modelled by the circuit of Fig. 2.8c, while Fig. 2.8d represents a floating resistor where R = 2/gm . The OTA circuit is built of several stages as in the example circuit depicted in Fig. 2.9 [14]. The main stage is the differential transconductance amplifier here denoted as the “core”. It is supported by the “bias” circuit and by the “commonmode feedback” circuit. Here, a “start-up” circuit is added. It is a small miracle, and it shows how modern technology is creating unbelievable results. Namely, by this circuit, one may produce a resistance thousands of times larger than the resistance of a resistor occupying the same area designed in the same technology. The OTA itself is not perfect, too. Its main characteristics are the frequency dependence of its transconductance and output capacitance. Ideally, one would like to have a perfect OTA which means a component with zero-valued output capacitance and frequency-independent controllable transconductance. There is no such perfect component, however, despite the fact that improvements are reported almost on daily basis [16]. As an example, Fig. 2.10 depicts the frequency dependence of the transconductance of an OTA [15] obtained by simulation, while Fig. 2.11 depicts the corresponding output impedance. As can be seen high frequencies are reached (cut-off frequency is claimed to be 567 MHz), the transconductance is being much more frequency independent than the output impedance. Unfortunately, neither the transconductance nor the output impedances are given here in absolute values, so one is not capable to extract further conclusions. Risking to misinterpret the output impedance, we suppose that the value depicted is 20 · log (|Z out |/(1 )). If so, one may calculate that the modulus of the output impedance at low frequencies is of the order of 100 M. Having in mind the complexity of the circuit of a single transconductor and having the intention to verify designs of filters with many transconductors, a necessity arises to create a simplified model—macromodel—which may serve as a substitute during the design process so allowing multiple simulations (and optimization) within an acceptable time frame. We will here demonstrate some simple macromodels of OTA with gradually rising complexity.

16

2 A Glimpse to the Active and Lumped Passive Integrated …

Fig. 2.9 Example schematic of an OTA. Courtesy of the authors [14]

2.3 Operational Amplifiers

17

Fig. 2.10 Transconductance as a function of frequency [15]. Courtesy of the authors

Fig. 2.11 Output impedance (modulus) as a function of frequency [15]. Courtesy of the authors

The simplest of all macromodels of a transconductor is depicted in Fig. 2.12a. It consists of a simple VCCS. This is the model used during circuit synthesis of filters in subsequent chapters. First-order improvement is introduced in Fig. 2.12b. Here, the input and output capacitances are taken into account in conjunction with the output resistance. One may consider the Ro –C o combination as the model of the output impedance of the OTA. If so, Fig. 2.11 would represent an example for such output impedance. In that

18

2 A Glimpse to the Active and Lumped Passive Integrated …

Fig. 2.12 Simple macromodels of OTA. a Ideal transconductance b includes input and output imperfections [17], and c includes frequency dependence of the transconductance

case, one may use the diagram for extraction of the values of Ro and C o . The input capacitances C i are usually ignored due to their small values in comparison with the surrounding capacitances. By the circuit depicted in Fig. 2.12c, the simplest approximation of the frequency dependence of the transconductance is introduced. Here, the value of Ra is arbitrary, and the approximation goes for single pole roll-off, the cut-off frequency being. fc =

1 . 2 · π · Ra C a

(2.5)

The value of C a here may be extracted from a diagram equivalent to the one depicted in Fig. 2.10 and will depend on the choice of Ra . Various OTA designs were reported in literature in the last decades. We are interested here on the values of the macromodel parameters that were achieved for filter applications. One such result is given in [18] where the differential (open circuit) voltage gain (A0 ) of the transconductor was 122 dB, the GBW (with open output circuit) was 392 MHz, Ro = 28.5 G, and C o = 18 fF. An important additional advantage of the transconductor in comparison with the resistor is its property to be adjustable. One may control the transconductance of the differential pair (M1 and M2 in Fig. 2.9) by controlling the current of the main current source (M0 in Fig. 2.9). Such a dependence is depicted in Fig. 2.13. It follows the usual rule:  (2.6) gm ∼ k · Io where k is a proportionality constant, and I o is the drain current of the current source. Of course, this property is much of use in the case when the transconductor performs as an amplifier than as a resistor. It makes the filter design tractable since one

2.3 Operational Amplifiers

19

Fig. 2.13 Transconductance as a function of the quiescent current

may create several transconductors with different transconductances while changing the dimension of a single transistor in the layout. The last issue here is the dependence of the transconductance on the amplitude of the input signal. One would prefer a circuit allowing large swing of the input signal while keeping the transconductance constant. In other words, one would like to have linear behavior and no distortions. Much of research was devoted to this issue recently [19, 20]. It is shown that, depending on the value of I o , one may afford maximum amplitude of the input voltage between 0.2 and 0.4 V. Note, the discussion on the transconductors above was limited to CMOS technology only. While even in CMOS improvements may be expected, if one switches to SiGe BiCMOS, significant improvements may be reached [21]. A DC open-loop gain of 87 dB and unity gain frequency of 2.3 GHz was reported.

References 1. Gray PR, Hurst PJ, Lewis SH, Meyer RG (2009) Analysis and design of analog integrated circuits. Wiley 2. Maloberti F (2001) Analog design for CMOS VLSI systems. Springer 3. Tuinhout H, Wils N (2014) A cross-coupled common centroid test structures layout method for high precision MIM capacitor mismatch measurements. In: IEEE international conference on microelectronic test structures. Udine, Italy, pp 243–248 4. Saari V, Kaltiokallio M, Lindfors S, Ryynänen J, Halonen KAI (2009) A 240-MHz low-pass filter with variable gain in 65-nm CMOS for a UWB radio receiver. IEEE Trans Circ Syst I Reg Pap 56(7):1488–1499 5. Saari V (2011) Continuous-time low-pass filters for integrated wideband radio receivers. Aalto University Publication Series. Doctoral Dissertations 23/2011, Helsinki, Finland 6. Haobijam G, Palathinkal RP (2014) Design and analysis of spiral inductors. Springer India, New Delhi 7. Vashisht N (2008) RF modeling of passive components of an advanced submicron CMOS technology. MS Thesis. SJSU Scholar Works. https://scholarworks.sjsu.edu/etd_theses

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8. Kyranas A, Papananos Y (2002) Passive on-chip components for fully integrated silicon RF VCOs. Act Passive Electron Compon 25:83–95 9. Liu B, Gielen GCE, Zhao D (2011) Synthesis of integrated passive components for highfrequency RF ICs based on evolutionary computation and machine learning techniques. IEEE Trans Comput-Aid Des Integr Circ Syst 30(10):1458–1468 10. Kakoty P (2011) Design of a high frequency low voltage CMOS operational amplifier. Int J VLSI Des Commun Syst (VLSICS) 2(1):73–85 11. Lipka B, Kleine U (2007) Design of a cascaded operational amplifier with high gain. In: Proceedings of the 14th international conference on integrated circuits and systems. Ciechocinek, Poland, pp 260–261 12. Glozi´c D, Litovski V, Bayford R (1991) ASCOTA3—a new automatic hierarchical CMOS opamp synthesizer. Facta Univ Ser Electron Energ 4(1):81–105 13. Lo T-Y, Hung C-C (2009) 1V CMOS Gm-C filters design and applications. Springer 14. Mirkovi´c MD, Petkovi´c PM, Dimitrijevi´c I, Mirˇci´c I (2015) Operational transconductance amplifier in 350 nm CMOS technology. Electronics 19(1):32–37 15. Santos MM, Bertemes-Filho P, Vincence VC (2012) CMOS transconductance amplifier types for low power electrical impedance spectroscopy. In: XXIII congresso brasileiro em engenharia biomédica—XXIII CBEB, pp 1382–1386 16. Sanchez-Sinencio E, Silva-Martinez J (2000) CMOS transconductance amplifiers, architectures and active filters: a tutorial. IEE Proc Circuits Devices Syst 147(1):3–12 17. Bogason Ó, Werner KJ (2017) Modeling circuits with operational transconductance amplifiers using wave digital filters. In: Proceedings of the 20th international conference on digital audio effects (DAFx-17). DAFX-130-DAFX-137, Edinburgh, UK 18. Zeki A, Kuntman H (1999) High-output-impedance CMOS dual-output OTA suitable for widerange continuous time filtering applications. Electron Lett 35(16):1295–1296 19. Szczepa´nski S, Kozieł S (2004) Phase compensation scheme for feedforward linearized CMOS operational transconductance amplifier. Bull Pol Acad Sci Tech Sci 52(2):141–148 20. Ramachandran A (2005) Nonlinearity and noise modeling of operational transconductance amplifiers for continuous time analog filters. Master thesis at the Texas A&M University 21. Devarajan S, Gutmann RJ, Rose K (2004) A 87 dB, 2.3 GHz, SiGe BiCMOS operational transconductance amplifier. In: 2004 IEEE international symposium on circuits and systems. Vancouver, BC, Canada

Chapter 3

Parallel Realization of Gm-C Singleand Two-Phase Integrated Filters

In this chapter, a pair of Gm-C cells is developed allowing synthesis of circuits originating from any physical realizable transfer function (i.e., no restrictions on the position of the transmission zeros are obeyed). Then, the same is done for a two-phase Gm-C low-pass filter having shifted frequency response. Examples will be given for both the ordinary and the two-phase Gm-C filters demonstrating the unlimited power of the RM software in this domain.

3.1 Introduction When high frequency CMOS monolithic integrated filters are sought, the operational transconductance amplifier-C (OTA-C) or Gm-C solutions are becoming the successful alternative to both active RC filters [1] and filters using spiral inductors printed on the silicon surface [2] as discussed in [3, 4]. Gm-C solutions are found in the literature implemented in the frequency range up to several GHz. That trend, however, was not followed by proper design procedures. Namely, the exiting physical implementation procedures of the Gm-C technology may be grouped into two. The first, and the most frequently used, is the one which is based on substitution of the inductor in an existing LC solution obtained by cascade synthesis [5, 6], by a simulated one (which will be discussed in detail in Chap. 5). The floating simulated inductor, necessary to realize complex and transmission zeros at the imaginary axis by cells depicted in Fig. 3.1 [5], may be realized, according to [6], by the circuit of Fig. 3.2. As one can see excessive number of transconductors is necessary in some situations. The circuit of Fig. 3.1a needs four transconductors, while the cell depicted in Fig. 3.1b would need 12 transconductors since the transformer is obtained from Fig. 3.2 a by omitting the capacitor. That, in addition to the increased silicon area, gives rise to influence of the consequences of their imperfections such as finite (complex) output impedance and noise. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 V. Litovski, Gm-C Filter Synthesis for Modern RF Systems, Lecture Notes in Electrical Engineering 807, https://doi.org/10.1007/978-981-16-6561-5_3

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3 Parallel Realization of Gm-C Single- and Two-Phase …

Fig. 3.1 a LC cell realizing a zero at the ω-axis. b Cell realizing a complex zero

Fig. 3.2 Simulated floating inductor. Here, the value of the equivalent inductance L is C (Henries)

One is to be aware here that for now on in this book, unless otherwise stated, the transconducting amplifier will be modeled as an ideal voltage-controlled current source (VCCS) whose transconductance is denoted by gm . Note that a designer who created his own transfer function (not taken from a catalogue) faces two problems when synthesizing a passive LC filter: the choice of order of extraction and accumulation of numerical error during the extraction of the cells realizing the set of transmission zeros [5]. In the case of high order solutions both of these problems may lead to an unacceptable solution (e.g., a circuit with erroneous or even negative values of the passive elements). For the above reasons, this concept is usually implemented in low-pass realizations of Gm-C filters. In turn, one is usually profiting by the use of catalogued data on filters such as [7, 8], as discussed in [9]. Alternatively, to produce a band-pass solution, attempts were published for development of band-pass cells [10] which were supposed to be cascaded in order to improve selectivity. These are again, however, based on simulated inductances within passive resonant circuits separated by isolating transconductors. An attempt was reported in [11] to produce a fourth-order band-pass cell by cascading a low-pass and a high-pass one. These were derived from the corresponding Salen-and-Key active RC second-order cells [12]. Finally, following the same concept, a band-pass

3.1 Introduction

23

second-order cell was reported in [13] and pronounced “a band-pass filter” which is not wrong but is misleading since not a filter of higher order then two was synthesized. We are after an universal method which will be capable to produce schematics of the filtering circuits having transfer functions of any order with no restriction on the location of the passband (low-pass; band-pass; high-pass; band-stop; and all-pass) and with no restriction on the position of the transmission zeros (located at the origin; at infinity; at the positive and negative real axis; at the imaginary axis in conjugate pairs; and in conjugate pairs in the left or right half of the complex frequency plane). Of course, there are in the literature second-order (biquad) Gm-C cells allowing for the implementation of our concept. Having that in mind, we will address first the choice of the overall topology of the circuit. The transfer function obtained by the approximation process (or by reading a catalogue, e.g. [7]) may be written in factored form as m (s − z i )  . Ha (s) = A0 · H (s) = A0 · ni=1 − pi ) (s i=1

(3.1)

Here, A0 is properly chosen in order to get unity nominal gain. p and z are vectors of transfer function poles and zeros, respectively, while s is the complex frequency. n is the order of the filter, and m is the number of finite transmission zeros (order of the numerator). For the sake of simplicity, in the case of n- and m-even and m < n, (3.1) may be rewritten as H (s) = A0 · n/2

1

i=m/2+1

Di

·

m/2  Ni Di i=1

(3.2)

where Di = (s − pi )(s − pi ∗) represents a factor of the denominator related to a pair of conjugate poles, while Ni = (s − z i )(s − z i ∗) is the same for the numerator. Both Di and N i are polynomials with real coefficients which is crucial for circuit synthesis. It is intuitively natural for the schematic realizing a transfer function expressed as a product of simpler functions to resemble its structure and to take the form of cascaded network. That way of thinking is dominant especially in active RC synthesis. If so, the transfer function (3.2) may be transformed into a cascaded circuit in a large number of variants. That depends on the pairing of Ds and Ns into biquads and on the ordering of so obtained biquads into a cascade. According to [14], for example, in the case of n = m = 8, there are 18 possible combinations to create biquads, while for n = m = 12, one may create 1350 combinations. Of course, in the similar way rises the number of filter structures due to the ordering of the biquads in the cascade. Finally, one is not to forget that for almost every biquad cell there are variants which may be favored from this or that reason. Having that in mind, one may claim that there are some apparent advantages of parallel synthesis as compared with the cascaded one.

24

3 Parallel Realization of Gm-C Single- and Two-Phase …

First, as will be shown soon, when using parallel synthesis, the types of transfer functions of the cells (circuits) used to build the whole system are reduced to two: one for the first-order cell and another for the second-order cell. Then, there is no need for pole–zero pairing in order to reduce nonlinear distortions and noise since, as will be seen below, only poles and residues corresponding to them are used. In addition, there is no need for assigning specific gain values to the cells in order to keep the signals within prescribed limits. This issue is further elevated if the shape of the amplitude characteristics of the cells implemented is considered. When parallel synthesis is adopted, both the second-order and the first-order cell (as will be demonstrated) are never high-pass which means their noise bandwidth is limited even when high-pass filters as a whole are to be synthesized. Namely, in this case, at highest frequencies (as will be shown), the direct connection between the input and the output takes over which means no additional noise will be generated by the filter except for the summing amplifier noise. Furthermore, when parallel (in case of low-pass, band-stop, and all-pass filters) there is no amplification of the input DC offset. The price paid for all these advantages is the cost of the additional circuitry needed to perform the summing at the output of the filter. Having a solution for parallel synthesis of single-phase filter circuits in a form of a first- and second-order cell, we developed the corresponding two-phase cells so enabling synthesis of two-phase (polyphase) filters usually implemented in modern telecommunication systems. In addition, after studying the properties of the solutions so obtained, we went for arithmetically symmetrical band-pass solutions which (depending on the lowpass single-phase prototype) may have linear phase in the whole of the passband. No limitations on the relative passband width are seen so allowing for synthesis of linear-phase selective broad- and narrow-band band-pass filters being, as it is well known, the ultimate challenge in filter synthesis. In the sequel, we will first introduce the universal (generic) second-order Gm-C cell which will be in use later on to create a proper biquad. Then, the procedure of circuit synthesis will be explained. The transfer function will be decomposed into partial fractions which are realizable with first- and second-order cells. The schematics of the cells will be introduced together with proper set of design equations to allow completing the synthesis. Two serious examples will be presented fully confirming the power and versatility of the concept. A short analysis of the influence of the imperfections of the transconductor follows. Then, we will introduce the rationale and the concept of polyphase filtering. Proper two-phase cells will be developed, and element value evaluation advised. Example will be given showing an implementation convenient for software-defined radio. Then, the same procedure will be implemented in order to create arithmetically symmetrical frequency response band-pass filters no matter the size of the relative passband width.

3.2 Design Based on General Second-Order Cell

25

Fig. 3.3 A generic Gm-C biquad [16]

3.2 Design Based on General Second-Order Cell Several second-order cells were considered as candidates for implementation in our project all being capable to produce transmission zeros anywhere in the complex frequency plane [15–19]. Since there is no comparative study in the literature, we were supposed to select one based on some (subjective) criterion. More detailed explanations and developments given in [16] became the base for accepting the concept advised there. The schematic of the version of the biquad cell used in [16] is depicted in Fig. 3.3. This circuit is described by the following transfer function   s 2 bC1 C2 VV4i + s bC2 gm2 VV4i − aC1 gm3 VV3i + gm1 gm2 VV1i Vo = H (s) = Vi s 2 C1 C2 + sC2 gm2 + gm3 gm4

(3.3)

where a and b are design constants. V i and V o are the input and output terminal’s voltages, respectively. It is generic since by connecting the input V i to V 1 and by short circuiting V 3 or V 4 (or both) to V i one gets several variants of second-order cells one of which was adopted for our implementation here and will be elaborated later on.

3.3 Decomposition of the Transfer Function The continuous time transfer function (3.1) may be represented in a form of sum of partial fractions as follows [20],

Ha (s) = A0 ·

n  i=1

⎧ ⎪ ⎪ ⎨

n/2  He (s), n-even ri i=0 = . n/2  (s − pi ) ⎪ ⎪ ⎩ Ho (s) + He (s), n-odd i=1

(3.4)

26

3 Parallel Realization of Gm-C Single- and Two-Phase …

Fig. 3.4 Structure of the filter

In the above, index e is used to denote a second-order fraction constructed by a complex pair of poles, while o denotes a first-order fraction constructed by a simple real pole. r are the complex residues in the poles. Note that (3.4) is valid for n > m only. In the case n = m polynomial, long division must be carried out first (as explained in [20]) which leads to Ha (s) = A0 · [1 + H (s)].

(3.5)

The structure of the resulting filter is depicted in Fig. 3.4. As can be seen an auxiliary unity gain path is allowed for filters having n = m. The residues needed for the realization of the above computations are obtained (for the case of simple poles) as follows: ri =

lim{[(s − pi ) · H (s)]} = [(s − pi ) · H (s)]|s= pi s → pi

(3.6a)

or m

ri = A0 ·

j=1 pi n j=1 pi j=i

− zj − pj



(3.6b)

The summands in (3.4) in the case of a pair of conjugate poles may be collected into partial fractions as follows He (s) = G i

s + b0,i s 2 + a1,i s + a0,i

(3.7)

3.3 Decomposition of the Transfer Function

27

with G i = 2 · A0 · re{ri }, ⎧   ⎨ − re{ pi } + im{ri }·im{ pi } if im{ri } · im{ pi } > 0 re{ri }   , b0,i = ⎩ im{ri }·im{ pi } − re{ pi } if im{ri } · im{ pi } < 0 re{ri }

(3.8)

a1,i = −2re{ pi }, and ai0 = | pi |2 . In the case of a simple pole on the real axis, one has Ho (s) = G o

1 , s + ao

(3.9)

with G o = ro , and ao = − po . In the above, “re” stands for “real part” and “im” for “imaginary part”. We will denote pi = σ i + j · ωi , and zi = α i + j · β i , and r i = μi + j · ξ i . Accordingly (3.7), (3.8), and (3.9) may be rewritten as He (s) =

G i · s + G i · b0,i s 2 + a1,i s + a0,i

(3.10)

with G i = 2 · A0 · μi ,  G i · b0,i =

−2 · A0 · (μi σi + ξi · ωi ) if ξi · ωi > 0 , 2 · A0 · (ξi · ωi − σi · μi ) if ξi · ωi < 0

(3.11)

and a1,i = −2 · σi and ai0 = | pi |2 = σi2 + ωi2 , while Ho (s) = G o

1 , s + ao

(3.12)

with G o = A0 · ro = A0 · μ0 , and ao = − po = −σ0 . The developments expressed so far are (apart of the notation) equal to the ones used in [20] for active RC synthesis. The difference and, accordingly, the novelty we are introducing is that in the place of “multiple entry” active RC cells used in [20] we use standard and universally accepted circuits (cells) in Gm-C technology which are realizing (3.10) and (3.12). Since two types of cell transfer functions are in view, only two types of circuit cells will be involved. Note that the second-order cell has one zero at infinity and another on the real axis of the frequency plane being not restricted to any part of the real axis. That makes it a low-pass cell in any case.

28

3 Parallel Realization of Gm-C Single- and Two-Phase …

3.4 Physical Implementation In the case of parallel synthesis, the ability to produce a simple transmission zero located on the real axis is necessary. Both positive- and negative-valued zeros may be encountered during the synthesis process. According to the literature, e.g. [16], there are not many second-order cells in Gm-C technology which allow for a transmission zero on the real axis on either side of the origin. As already mentioned, we used the generic cell described in [16] and set the connections within it to produce the biquad depicted in Fig. 3.5. To get the design equations, we first simplify the notation. One may find easily by analogy that (3.10), for a given cell, may be rewritten as He (s) =

s2

g·s +q . +a·s+b

(3.13a)

where g = G and q = G · b0 (with the index i omitted). Now, after circuit analysis of the schematic depicted in Fig. 3.5, one gets He (s) =

gm3 ·gm1 C·C3 ·gm4 + gm3 C·C3

m3 − x·g ·s+ C3

s2 +

gm2 C

·s

.

(3.13b)

·gm1 ·gm4 m3 The design equations are now g = − x·g , q = gm3 , a = gCm2 and b = gm3 . C3 C·C3 C·C3 Note, by changing the sign of gm1 , we in fact transpose the transmission zero (being on the real axis) from one to the other side of the origin. Since there are more circuit elements than degrees of design freedom [expressed by the number of coefficients in (3.13a)], some of the element values are to be adopted as design constants. Speaking in normalized element values, we first adopt gm2 = gm3 = gm = 1 S. Then, we adopt x = 0.5. With that set, one may calculate C = 1/a, b·x , and gm1 = − q·x . C 3 = −x/g, gm4 = − a·g a·g Now,

Fig. 3.5 Second-order Gm-C cell used within the RM software. Note: 0 ≤ x ≤ 1. “ni” stands for “input node”, while “no” for the “output node”

3.4 Physical Implementation

29

Fig. 3.6 First-order Gm-C filter and node notation

if (g > 0 and q > 0), we choose a negative sign to the summing transconductor following the output of the proper cell. Set g = −g and q = −q; if (g > 0 and q < 0), we use negative sign to the summing transconductor following the output of the proper cell. Set g = −g and q = −q; if (g < 0 and q > 0), we proceed without intervention; if (g < 0 and q < 0) without intervention. The first-order cell is a bilinear circuit as depicted in Fig. 3.4. The transfer function of the circuit depicted in Fig. 3.6 may be expressed in the form Ho = g/(s + a),

(3.14a)

Here, related to (3.12), g = Go is the residue and a = ao the real pole. From circuit analysis, Ho =

gm1 /C . s + gm2 /C

(3.14b)

Now, by adopting C = 1F, for the normalized transconductances, we have gm1 = g and gm2 = a. Since there are no restrictions on the value of C, it may be set to be equal to the capacitances used within the second-order cells.

3.5 The Adder Cell (Summing Amplifier) To complete the schematic of the GM-C filter, the output signals of the cells have to be summed. The circuit used for this purpose within the RM software for filter design is depicted in Fig. 3.7. There are some issues to be addressed while analyzing this figure. First one is to note the auxiliary branch expressed by gmox . It will be present only if the order of the numerator and the order of the denominator of the transfer function are equal. We have such a situation in the cases of all-pass, high-pass and band-stop filters. Then, in cases when negative transconductance gmsi is to be realized, one needs to simply

30

3 Parallel Realization of Gm-C Single- and Two-Phase …

Fig. 3.7 GM adder cell used within the RM software for filter design

interchange the input terminals of the transconductor. Finally, the transconductor denoted gmo in fact represents the loading resistance. Direct connection of any impedance to the output of the filter will affect the transfer function of the overall circuit. If resistive loading is applied, it will change the overall gain, while reactive elements will affect the frequency response completely.

3.6 Design Example To illustrate the procedure, example will be given. The choice goes for encompassing as much of diversity of transmission zeros and frequency responses as possible. As the example, A transfer function specially synthesized for this purpose will be used. It is a sixteenth-order band-pass filter having four transmission zeros at the positive ω-axis and two zeros at the origin. The maximum passband attenuation was set to be amax = 3 dB, while the minimum stopband attenuation was set to be amin = 40 dB. Simultaneous amplitude and constant group delay (maximally flat) requirements were imposed. The central frequency was set to be 1 MHz. The required relative bandwidth was 0.4.

3.6 Design Example

31

The transfer function of the filter was produced by the program bptdam . The normalized zeros and poles are given in Table 3.1. For circuit synthesis, the program GM_C_par was used. Figure 3.8 depicts the frequency domain response of the filter obtained by SPICE simulation. Table 3.1 Zeros and poles (normalized) of the example band-pass pass filter Re{zero}

Im{zero}

Re{pole}

0

0

−0.2469279895

Im{pole} ±0.4237478118

0

0

−0.3664234507

±0.5936659440

0

±1.478557007

0.4277617776

±0.7439486870

0

±1.650920749

−0.4530516018

±0.8861414063

0

±0.52572423801

−0.4486080678

±1.024849052

0

±0.3878379628

−0.4146837227

±1.163047892

−0.3458693761

±1.304111671

−0.2235434043

±1.455583614

Fig. 3.8 Frequency response of the synthesized band-pass filter obtained by SPICE simulation (Note the linear scale for the frequency axis)

32

3 Parallel Realization of Gm-C Single- and Two-Phase …

Below is edited part of the .html report file for this example produced by the GM_C_par program of the RM software.

Welcome to The Electronic Filter Design Software Gm-C-Parallel, Program for synthesis of active Gm-C-circuits, in the form of parallel connection of Schaumann-Van Valkenburg biquads ---------------------------------------------------------Project name: VS_SIMUL_BP_10_16 -----------------------------------------Order of the numerator, n=10 Order of the denominator, m=16 Read in nominal transconductance is Gm0=1.00000e-005 ---------------------------------------------------------Partial fraction expansion Residues in the poles Re{res} Im{res} 8.367916454e-2 ±4.046974487e-1 7.132795768e+0 ±8.680662513e+0 -5.176849795e+1 ±3.6981418e+1 1.131165552e+2 ±4.315153155e+1 -1.084728338e+2 ±3.393442950e+1 4.8098250e+1 ±1.506116089e+1 -8.523340664e+0 ±2.984289716e+0 3.415597340e-1 ±1.076298443e-1 Residues after division with the nominal gain Re{res} Im{res} 5.368971467e-2 ±2.596595062e-1 4.5765873e+0 ±5.569633684e+0 -3.321538759e+1 ±1.929321059e+1 7.257715355e+1 ±2.768662222e+1 -6.959767734e+1 ±2.177280148e+1 3.085526513e+1 ±9.6634599e+0 -5.4686938e+0 ±1.914761748e+0 2.191494712e-1 ±6.905680360e-2 Ordered vector of transfer functions to be parallelized T(s)=(g·s+q)/(s2+a·s+b)

3.6 Design Example

33

g q a 1.0737943e-1 2.4657528e-1 4.9385598e-1 9.1530017e+0 -3.2591292e+0 7.3284690e-1 -6.6430775e+1 2.8977090e-1 8.5552356e-1 1.4515431e+2 1.6693867e+1 9.0610320e-1 -1.3919535e+2 -1.7816489e+1 8.9721614e-1 6.1710530e+1 3.1122419e+0 8.2936745e-1 -1.0937388e+1 1.2112189e+0 6.9173875e-1 4.3829894e-1 -1.0305707e-1 4.4708681e-1 ----------------------------------------------------------

b 2.4053564e-1 4.8670540e-1 7.3643979e-1 9.9050235e-1 1.2515648e+0 1.5246430e+0 1.8203329e+0 2.1686953e+0

SYNTHESIS OF THE PARALLEL ACTIVE GM_C FILTER Capacitances in [F] and transconductances in [S] Element values with reference to Fig. 3.5 and Fig. 3.7. Transconductances Cell No. gm1 gm2 gm3 gm4 1 -4.184760809e-5 1.e-5 1.e-5 4.082258824e-5 2 4.372877232e-6 1.e-5 1.e-5 6.530281030e-7 3 4.588766727e-8 1.e-5 1.e-5 1.166214566e-7 4 -1.142330720e-6 1.e-5 1.e-5 6.777826149e-8 5 -1.283934359e-6 1.e-5 1.e-5 9.019324754e-8 6 -5.472800156e-7 1.e-5 1.e-5 2.681046880e-7 7 1.440818994e-6 2.165397410e-6 8 4.733234308e-6 9.960445672e-5 Cell No. 1 2 3 4 5 6 7 8

C1 3.222699529e-013 2.171735226e-013 1.860322163e-013 1.756476994e-013 1.773875177e-013 1.918991925e-013 2.300795533e-013 3.559821941e-013

Capacitances C2 2.900429576e-012 1.954561703e-012 1.674289947e-012 1.580829294e-012 1.596487659e-012 1.727092733e-012 2.070715980e-012 3.203839747e-012

gms -1.e-5 -1.e-5 1.e-5 -1.e-5 1.e-5 -1.e-5 1.e-5 -1.e-5

C3 1.333956137e-011 1.564945061e-013 2.156221245e-014 9.868081193e-015 1.029053370e-014 2.321150834e-014 1.309631276e-013 3.268076532e-012

Finally, the transconductance of the output transconductor is Gmo=1.e-5 S. Here ends the report on the synthesis process.

34

3 Parallel Realization of Gm-C Single- and Two-Phase …

3.7 Polyphase Filters A special implementation of the parallel Gm-C synthesis procedure will be described here. The RM software is the first to implement parallel polyphase Gm-C filters in the way described below. Namely, in modern telecommunication systems (e.g., software-defined radio), a necessity arises for suppression of the received signal’s image at negative frequencies when the receiver runs in low intermediate frequencies (IF) mode (hence improving signal to noise ratio, i.e., noise figure) [21]. In fact, by using a low IF of only a few hundred kHz, it combines the integrability of the zero-IF receiver with the insensitivity of the IF receiver to parasitic baseband signals. An analog integrated asymmetric polyphase filter [22] is a key building block for the development of a high performance fully integrated low-IF receiver. The asymmetric polyphase filter makes it possible to suppress the mirror signal not at HF, but after quadrature demodulation at a low IF. The most important parameters for the polyphase filter are a high dynamic range and good mirror signal suppression [23]. The polyphase filter that is presented here has two inputs: in-phase (I) and quadrature (Q) and two quadrature outputs (I and Q). Therefore, two transfer functions characterize the filter. A low-IF receiver requires a polyphase filter with a passband from positive-to-positive frequencies, with an attenuation from negative-to-negative frequencies, and with no signal transfer from positive to negative frequencies and vice versa. Figure 3.9a depicts the baseband (low-pass) filter gain characteristic. As it can be seen, it is an even function meaning that positive and negative frequencies are equally treated. The goal here is to produce an amplitude characteristic favoring the positive frequencies and suppressing the negative ones. The transfer functions and the circuit synthesis of such a filter can be found by performing a linear frequency transformation on a low-pass filter characteristic. One example of the expected (desired) result of such a linear frequency transformation is depicted in Fig. 3.9b where the attenuation characteristic is shifted by 50 kHz. The following equation does that for a first-order low-pass filter. Hlp ( jω) =

1 1 ⇒ Hbp ( jω) = 1 + j · ω/ωc 1 + j · (ω − ω0 )/ωc

(3.15)

where ωc is the cut-off frequency of the low-pass prototype filter (here considered normalized to unity) while ω0 is the central frequency of the newly created band-pass filter. Implementation of this transform leads to a transfer function with complex coefficients. Also, by implementation of this transformation, the amplitude characteristic loses the property to be an even function. There are no circuit synthesis methods for this kind of functions. This is why the transformation is to be implemented directly to the circuit schematic by creating an equivalent circuit of the “polyphase” capacitor.

3.7 Polyphase Filters

35

Fig. 3.9 a Gain characteristic of an elliptic low-pass filter and b a shifted gain characteristic (of a polyphase filter)

Namely, implementation of the transformation to a grounded capacitor branch will lead to the circuit transformation depicted in Fig. 3.10a, where Ilp ( jω) = jωC · Vlp ⇒ Ibp ( jω) = j(ω − ω0 )C · Vbp .

(3.16a)

The last expression may be decomposed so that Ibp ( jω) = jωC · Vbp − jω0 C · Vbp = jωC · Vbp − jgm · Vbp ,

(3.16b)

hence gm = ω0 C. Here “lp” means low-pass, i.e., prototype, while “bp” means band-pass, i.e., transformed. Note that the 90° phase shift is added to V bp by the multiplication with j. It is established by an I-to-Q connection and vice versa. Figure 3.10b represents the complete implementation of the grounded capacitor in a two-phase system.

36

3 Parallel Realization of Gm-C Single- and Two-Phase …

Fig. 3.10 Implementation of the “polyphase” transformation a grounded capacitor, transformation b grounded capacitor complete c floating capacitor transformation, and d floating capacitor complete

3.7 Polyphase Filters

37

Figure 3.10c represents the “polyphase” transformation for the floating capacitor. Note that the value of gm is evaluated by the same formula gm = ω0 C. Finally, Fig. 3.10d represents the complete implementation of the floating capacitor in a two-phase system. In the sequel, we will first introduce the overall structure of a two-phase filter as unique offer by the RM software. Then, we will introduce the two-phase first- and second-order Gm-C cells which will be in use later on. Finally, the procedure for circuit synthesis will be explained. We will finish with a special example of design of a polyphase filter.

3.7.1 Physical Implementation of the Polyphase Case To create a schematic behaving as a two-phase filter, one is to create two channels (inphase and quadrature-phase). These are obtained by simple replicating the original signal path I and produce the Q. These channels are excited by signals being mutually phase-shifted by 90°. The circuit generating this kind of paired signals is named Hilbert transformer to which a special chapter will be devoted within this book. So, the if “I” input is sine, the “Q” input is cosine wave. Then, the capacitors are to be substituted by the models representing the transformation as depicted in Fig. 3.10 so making the coupling between the I and the Q channel of the filter. In the case of parallel synthesis, to create the whole, one needs polyphase firstand second-order cells and two identical summing circuits. Figure 3.11 depicts the overall structure of the system performing as a two-phase parallel filter. Note, only low-pass prototype filters are expected to be shifted. That means that the auxiliary

Fig. 3.11 Overall structure of a parallel two-phase Gm-C filter

38

3 Parallel Realization of Gm-C Single- and Two-Phase …

Fig. 3.12 First-order two-phase Gm-C cell

path which becomes active in band-stop and high-pass filters is not necessary. It was included, however, for completeness. One should notice the exceptional simplicity of synthesis of two-phase filters in a form of parallel structure. Namely, as is with ordinary parallel synthesis of active RC and active GM-C filters, one needs only two types of cells to create the filter. That simplifies the development of the synthesis software. Add to that the fact that no pole–zero pairing, gain accommodation, and order of extraction is necessary in this case, one may again conclude that parallel synthesis is preferable from software development point of view. The cells used in this structure are as follows. Figure 3.12 depicts the first-order two-phase cell. The second-order cell is depicted in Fig. 3.13. Both cells are created by combining the circuits depicted in Figs. 3.5, 3.6, and 3.10. Note that the summing is performed by two identical adder cells as depicted in Fig. 3.7.

3.7.2 Example Two-Phase Linear-Phase Filter For this example, a sixth-order modified elliptic filter was synthesized. Two transmission zeros at the positive ω-axis were introduced to produce maximum passband attenuation of amax = 0.5 dB and minimum stopband attenuation of amin = 35 dB.

3.7 Polyphase Filters

39

Fig. 3.13 Second-order two-phase Gm-C cell

The cut-off frequency was chosen to be f c = 200 kHz. The group delay characteristic was corrected by a fourth-order corrector to produce a passband approximation error of δ = 1%. For synthesis of the two-phase filter, the program POLY_Parallel was implemented. The nominal transconductance was chosen to be 10 µS. The characteristic was shifted for 210 kHz. In that way the resulting filter will be still low-pass while having large attenuation for the negative frequencies. The frequency response obtained by LTSpice simulation is depicted in Fig. 3.14. Note that since the version of LTSpice available to us does not allow for negative frequencies, we made a replica of the original filter and change the signs of the capacitances to obtain the “negative frequency” gain characteristic as shown in Fig. 3.14b. Since the frequency is negative (the frequency is in fact decreasing from zero to − 200 kHz) one should read this diagram from right to left. The negative sign of the group delay comes from the fact that by changing the sign of the frequency the phase starts rising (it is always falling for positive frequencies).

40

3 Parallel Realization of Gm-C Single- and Two-Phase …

Fig. 3.14 Frequency characteristics (gain and group delay) of the Gm-C example circuit realizing the two-phase selective filter with corrected group delay (Note the linear scale for the frequency). a Positive frequencies and b Negative frequencies obtained by changing the signs of the capacitances

3.7 Polyphase Filters

41

In the next the partly edited .html report of the POLY_Parallel program will be given.

Welcome to The Electronic Filter Design Software

POLY_parallel, Program for synthesis of active two-phase Gm-C-circuits , in the form of parallel connection of Schaumann-Van Valkenburg biquads ---------------------------------------------------------Project name: GMC_POLY_example_2 ---------------------------------------------------------Order of the numerator, n=8 Order of the denominator, m=10 ---------------------------------------------------------Normalized poles and zeros of the transfer function. Zeros Poles Re{} Im{} Re{} Im{} 2.976769139e-1 ±6.297736629e-1 -2.948161755e-2 ±1.001830981 3.483201475e-1 ±1.867755049e-1 -3.857505095e-1 ±4.098333315e-1 0. ±1.108674203 -1.491418204e-1 ±8.837808754e-1 0. ±1.326113003 -2.976769139e-1 ±6.297736629e-1 -3.483201475e-1 ±1.867755049e-1 -2.948161755e-2 ±1.001830981 Partial fraction expansion Residues in the poles Re{} Im{} -2.976439671e-1 ±9.788574828e-2 7.408724843e+1 ±9.245866010 2.686690750 ±1.931374531 -2.578985464e+1 ±4.331040717 -5.068644057e+1 ±4.843759632e+1 Residues after division with the gain Residues in the poles Re{} Im{} -3.322918681e-2 ±1.092803542e-2 8.271153764 ±1.032215139 2.999440900e-1 ±2.156200434e-1 -2.879197943 ±4.835205043e-1 -5.658670724 ±5.407608132

42

3 Parallel Realization of Gm-C Single- and Two-Phase …

Ordered vector of transfer functions to be parallelized T(s)=(g·s+q)/(s2+a·s+b) g q a b -6.6458374e-2 -2.3855389e-2 5.8963235e-2 1.0045345 1.6542308e+001 5.5351312 7.7150102e-1 3.1676682e-1 5.9988818e-1 4.7059016e-1 2.9828364e-1 8.0331192e-1 -5.7583959 -2.3231585 5.9535383e-1 4.8522641e-1 -1.1317341e+1 -1.9220406 6.9664029e-1 1.5621201e-1 ---------------------------------------------------------SYNTHESIS OF THE PARALLEL ACTIVE two-phase Gm-C FILTER Capacitances in [F] and transconductances in [S] Element values according to Fig. 3.13 and Fig. 3.7. Transconductances in the original and the summing cells gm2 gm3 gm1 -3.043865786e-5 1.e-5 1.e-5 -2.168529482e-6 1.e-5 1.e-5 -1.314961698e-5 1.e-5 1.e-5 -3.388224444e-6 1.e-5 1.e-5 -1.218932181e-6 1.e-5 1.e-5

Cell. No. 1 2 3 4 5

gm4 gms 1.281751517e-3 1.e-5 1.241015164e-7 -1.e-5 2.244680194e-5 -1.e-5 7.076813771e-7 1.e-5 9.906755095e-8 1.e-5

Transconductances in the coupling cells Cell. No. 1 2 3 4 5

gmc1 8.903853378e-5 6.804916481e-6 1.760069706e-5 8.818285454e-6 7.536170443e-6

gmc2 8.903853378e-5 6.804916481e-6 1.760069706e-5 8.818285454e-6 7.536170443e-6

gmc3 7.899681731e-5 3.173680571e-7 8.751631013e-6 9.117122379e-7 4.638898653e-7

Capacitances Cell. No. 1 2 3 4 5

C1 6.748058465e-11 5.157314740e-012 1.333922828e-11 6.683208189e-012 5.711517991e-012

C2 6.748058465e-11 5.157314740e-012 1.333922828e-11 6.683208189e-012 5.711517991e-012

C3 5.987016173e-11 2.405271194e-013 6.632692075e-012 6.909690921e-013 3.515731672e-013

The values of the load transconductance are gmo=1.e-005 S Here ends the synthesis process

3.7 Polyphase Filters

43

Fig. 3.15 Structure of the summing subsystem for both channels

References 1. Mohan PVA (2013) VLSI analog filters: active RC, OTA-C, and SC, modeling and simulation in science, engineering and technology. Springer, New York 2. Haobijam G, Palathinkal RP (2014) Design and analysis of spiral inductors. Springer India, New Delhi 3. Lo T-Y, Hung C-C (2009) 1V CMOS Gm-C filters design and applications. Springer Science + Business Media B.V., New York 4. Saari V (2011) Continuous-time low-pass filters for integrated wideband radio receivers. Aalto University publication series, Doctoral dissertations 23/2011 5. Scanlan JO, Rhodes JDI (1970) Unified theory of cascade synthesis. Proc IEE 117(4):665–669 6. Uzunov IS (2008) Theoretical model of ungrounded inductance realized with two gyrators. IEEE Trans Circ Syst II Exp Briefs 55(10):981–985 7. Zverev AI (2005) Handbook of filter synthesis. Wiley-Interscience, New York 8. Williams AB, Taylor FJ (2006) Electronic filter design handbook. McGraw-Hill, New York 9. Litovski VB (2019) Electronic filters, theory, numerical receipts and design practice using the RM software. Springer, New Delhi 10. Bhuiyan MAS, Omar MB, Reaz MBI, Minhad KN, Hashim FH (2014) A review on CMOS Gm-C band pass filters in RF application. J Theoret Appl Inf Technol 61(1):17–23 11. Ahmed RF, Awad IA, Ahmed M, Soliman AM (2006) New op-amp-RC to Gm-C transformation method. Analog Integr Circ Sig Process 49:79–86 12. Sallen RP, Key EL (1955) A practical method of designing RC active filters. IRE Trans Circuit Theory 2(1):74–85

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3 Parallel Realization of Gm-C Single- and Two-Phase …

13. Jl B et al (2017) Fully integrated high-quality factor GmC bandpass filter stage with highly linear operational transconductance amplifier. Adv Radio Sci 15:149–155 14. Hospodka J (2006) Optimization of Dynamic Range of Cascade Filter Realization. Radioengineering 15(3):31–34 15. Laker KR, Sansen WMC (1994) Design of analog integrated circuits and systems. McGraw-Hill College 16. Schaumann R, Van Valkenboug E (2001) Design of analog filters. Oxford University Press, New York 17. Temes GC (2020) CMOS active filters. https://slideplayer.com/slide/6192567/ 18. Geiger RL, Sanchez-Sinencio E (1985) Active filter design using operational transconductance amplifiers: a tutorial. IEEE Circuits Devices Mag 2(l):20–32 19. Sanchez-Sinencio E, Silva-Martinez J (2000) CMOS transconductance amplifiers, architectures and active filters: a tutorial. IEE Proc Circuits Devices Syst 147(1):3–12 20. Moran PL (1978) A low-cost parallel implementation for active filters. Electron Circuits Syst 2(1):21–25 21. Okanobu T, Tomiyama H, Arimoto H (1992) Advanced low-voltage single chip radio IC. IEEE Trans Consum Electron 38(3):465–475 22. Voorman JO (1988) Asymmetric polyphase filter. US Patent, US4696055A 23. Crols J, Steyaert M (1995) An analog integrated polyphase filter for a high performance lowIF receiver. In: Symposium on VLSI Circuits Digest of Technical Papers, June 1995, Kyoto, Japan, Paper no. 11–3, pp 87–88

Chapter 4

Cascade Realization of Active Gm-C Circuits

In this chapter, a set of Gm-C cells is developed allowing synthesys of circuits originating from any physical realizable transfer function (i.e., no restrictions on the position of the transmission zeros are obeyed). Then, the same is done for two-phase Gm-C low-pass filter having shifted frequency response. Examples will be given for both the ordinary and the two-phase Gm-C filters demonstrating the unlimited power of the RM software in this domain.

4.1 Introduction Cascaded Gm-C filters in general do not differ from the active RC solutions. Namely, all rules related to the pole–zero pairing and order of extraction remain the same. The difference is in the structure of the cells. In this chapter, we will go through a description of a set of Gm-C cells which are used in the GM_C_cascade program of the RM software for filter design. These are based on the theory described in [1]. As will be seen, the main difference between the active RC and the Gm-C cell is in the fact that the output impedance of the transconductor is large so that when the succeeding cell has finite input impedance it loads the output of the preceding cell so changing the overall transfer function. To avoid that within the program GM_C_cascade, all the Gm-C cells having finite input impedance are extended by a unity gain amplifier which is added to isolate them from the preceding cell. The unity gain amplifier has to have high frequency domain performances [2] to comply with the application in Gm-C technology. Having developed a complete set of Gm-C cells capable to realize any physically realizable transfer function, two examples are given representing a band-stop and a band-pass filter.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 V. Litovski, Gm-C Filter Synthesis for Modern RF Systems, Lecture Notes in Electrical Engineering 807, https://doi.org/10.1007/978-981-16-6561-5_4

45

46

4 Cascade Realization of Active Gm-C Circuits

The chapter then follows a procedure to develop two-phase Gm-C cells. Having in mind that one shifts only low-pass prototype functions, the set of cells produced is shorter that in the case of simple Gm-C filters. Here again, example is given demonstrating that the RM software is capable to manage any transfer function. Namely, in this example, the transfer function is taken from a publication (not created with the RM software). The POLY_Cascade program was used for synthesis.

4.2 Structure of the Cascaded Gm-C Filter In this document, the synthesis of active Gm-C filters in a form of a cascade of second-order cells (in the case of a filter of odd order, one additional first-order cell is added) will be described. The structure of the resulting schematic is shown in Fig. 4.1. During creation of the physical structure, however, there are several choices to be resolved. First, decomposition of the transfer function must be performed. That to be done decisions are to be made as to which pair of poles is to be paired with which zero (pair of zeros). Then, one has to decide which will be the order of extraction. As a result of that activity which will be described below in more details, a sequence of second-order transfer functions is obtained. For realization of every single second-order transfer function, usually referred to as a biquad, choices are to be made based on the type of the function (low-pass, high-pass, band-pass, band-stop, and all-pass), on the type of the complex zero if there is any, and on the value of the pole’s Q-factor (small and large). These choices are obvious, and proper circuit will be chosen based on the literature, e.g., [1]. The very cell structure and the element value calculation is based on the literature, e.g., [1]. To our knowledge, there are alternative solutions, e.g., [3] but we find that [1] gives a unified approach no matter the purpose of the cell and the location of its transmission zeros. Given the transfer function intended to be realized as a cascade of second-order cells (plus one of first order when odd order transfer functions are dealt with) beside

Fig. 4.1 Cascade of Gm-C cells

4.2 Structure of the Cascaded Gm-C Filter

47

the choice of the cell’s structure, two additional problems are to be solved. Namely, it is known that, by proper. • Pairing poles and zeros, and • Ordering the extraction of the cell one may improve several important properties of the final solution such as noise, linearity, and range of element values, i.e., total silicon area [4–7]. A transfer function of the form m (s − z i ) H (s) = A0 · ni=1 (4.1) i=1 (s − pi ) which, for the sake of simplicity, in the case of n- and m-even and m < n, may be rewritten as H (s) = A0 · n/2

1

i=m/2+1

Di

·

m/2  Ni D i i=1

(4.2)

where Di = (s − pi )(s − pi ∗) represent a factor of the denominator related to a pair of conjugate poles while Ni = (s − z i )(s − z i ∗) is the same for the numerator, may be realized in a large number of variants depending of the pairing of Ds and Ns into biquads and depending on the ordering of biquads so obtained. According to [4], for example, in the case of n = m = 8, there are 18 possible combinations to create biquads, while for n = m = 12, one may create 1350 combinations. Of course, in the similar way rises the number of filter structures due to the ordering of the biquads in the cascade. A procedure is implemented within the GM_C_cascade program enabling pairing in order to get the optimal biquads and ordering of the biquads in order to get optimum from linearity and noise point of view, which is based on the literature and will be not discussed here. First, pole–zero pairing is programmed. Then, the order of the cells (biquads) is established. After that, according to the properties of the cell. • Type of the function (low-, band-, high-, all-pass, or notch) • order (first or second) • sign of the gain (inverting or non-inverting), a choice is made as to which circuit should be the most appropriate for realization according to the following: • Pairing the transfer function poles having highest imaginary part with the attenuation poles having minimal frequency; • High-Q sections should be in the middle; • First sections should be low-pass or band-pass, to suppress incoming highfrequency noise; • All-pass sections should be near the input;

48

4 Cascade Realization of Active Gm-C Circuits

• Last stages should be high-pass or band-pass to avoid output DC offset. Furthermore, as will be discussed later one, an option is given to the designer to create a cascade in which every cell has a maximum gain (as a function of frequency) of unity. The designer is free to choose to keep the additional circuit or to simply short circuit its input and output. Finally, an option is on disposal for the overall gain at the central frequency to be adjusted to a desired value. In the next, an overview of the candidate cells will be given together with the design procedure.

4.3 First-Order Cell The transfer function of a general first-order cell may be expressed as Ho (s) = μ ·

s+α s+β

(4.3)

This function may be realized by the circuit depicted in Fig. 4.2. By circuit analysis, one gets Ho =

V2 a · s · C + gm1 s + gm1 /(a · C) = =a· V1 s · C + gm2 s + gm2 /C

4.3.1 Low-Pass To get a low-pass function of the form

Fig. 4.2 Generic first-order cell

(4.4)

4.3 First-Order Cell

49

Fig. 4.3 First-order Gm-C flow-pass filter and node notation

Ho (s) = μ · α ·

1 s+β

(4.5)

we will use the circuit of Fig. 4.3. The transfer function of the circuit depicted in Fig. 4.3 may be expressed in the form Ho = g/(s + β),

(4.6a)

gm1 /C . s + gm2 /C

(4.6b)

From circuit analysis, we get Ho =

Now, if only the pole coordinate is available, by adopting the value of gm1 = gm2 = gm0 , for the capacitance, we have C = gm2 /β.

(4.7)

4.3.2 High-Pass The transfer function of a general first-order cell may be expressed as Ho (s) = μ ·

s s+β

(4.8a)

This function may be realized by the circuit depicted in Fig. 4.4. Here, gm1 = 0 and a = 1 was chosen. By circuit analysis, we get Ho =

s . s + gm2 /C

(4.8b)

50

4 Cascade Realization of Active Gm-C Circuits

Fig. 4.4 First-order high-pass cell

If only the pole coordinate is available, after fixing gm2 = gm0 , one gets C = gm2 /β.

(4.9)

4.3.3 Zero on the Real Axis and All-Pass For the case when the pole and the zero have different modulus, with reference to (4.3), (4.4), and Fig. 4.2, one may write the following design equations α = gm1 /(a · C)

(4.10a)

β = gm2 /C

(4.10b)

Now, if only the zero and the pole are given, after adopting gm2 = gm0, we have C = β/gm2

(4.11a)

gm1 = a · C · α

(4.11b)

For the case of a first-order all-pass cell, the restriction α = −β is valid so that we choose a = 1 and gm1 = −gm2 .

(4.11c)

4.4 Second-Order Cell

51

4.4 Second-Order Cell The transfer function of a second-order cell in a general case may be expressed as He (s) = μ ·

s2 + g · s + q s2 + α · s + β

(4.12)

where α = −2 · σ0 and b = σ02 + ω02 where p = σ0 + j · ω0 is the pole being realized by the cell and g = −2 · γ0 and q = γ02 + δ02 where z = γ0 + j · δ0 is the zero being realized by the cell. Several second-order cells were considered as candidates for implementation in our project all being capable to produce transmission zeros anywhere in the complex frequency plane [1, 8, 9]. Since there is no comparative study in the literature, we were supposed to select one based on some (subjective) criterion. More detailed explanations and developments given in [1] became the base for accepting the concept advised there. The schematic of the version of the biquad cell used in [1] is depicted in Fig. 4.5. This circuit is described by the following transfer function

H (s) =

Vo = Vi

  s 2 bC1 C2 VV4i + s bC2 gm2 VV4i − aC1 gm3 VV3i + gm1 gm2 VV1i s 2 C1 C2 + sC2 gm2 + gm3 gm4

(4.13)

where a and b are design constants. V i and V o are the input and output terminal’s voltages, respectively. It is generic since by connecting the input V i to V 1 and by short circuiting V 3 or V 4 (or both) to V i , one gets several variants of second-order cells.

4.4.1 Low-Pass To create a low-pass second-order cell, we choose a = 0 and b = 0 and V 1 = V i so that (4.12) reduces itself into

Fig. 4.5 Generic Gm-C biquad [1]

52

4 Cascade Realization of Active Gm-C Circuits

Fig. 4.6 Second-order low-pass cell

Vo gm1 gm2 = 2 Vi s C1 C2 + sC2 gm2 + gm3 gm4 1 gm1 gm2 = 2 C1 C2 s + s · gm2 /C1 + gm3 gm4 /(C1 C2 )

H (s) =

(4.14)

The schematic of this cell is depicted in Fig. 4.6. One may find easily by analogy that (4.12), for this cell, may be rewritten as He (s) = μ ·

s2

q . +a·s+β

(4.15)

If only the coordinates of the poles p1,2 = σ0 ± j · ω0 are available, one may create only two design equations α = −2 · σ = gm2 /C1

(4.16a)

β = σ02 + ω02 = gm3 gm4 /(C1 C2 )

(4.16b)

Having two equations and six unknown parameters, we choose to adopt three gm2 = gm3 = gm4 = gm0, where gm0 is sort of normalization parameter to be supplied by the designer at the beginning of the design process. The value of gm1 may be fixed as gm1 = gm0 or left for adjustment of the gain (and its sign) of the cell during the redistribution of the overall gain of the filter. So, the rest will be calculated as C1 = gm2 /(−2 · σ )

(4.17a)

   C2 = gm3 gm4 / C1 · σ02 + ω02

(4.17b)

As an auxiliary equation allowing the gain at the origin to be equal to unity, one may have in mind gm1 = gm3 · gm4 /gm2

(4.17c)

4.4 Second-Order Cell

53

4.4.2 Band-Pass By putting b = 0, gm1 = 0 (the first transconductor is absent), and V 3 = V 1 = V i , one may transform (4.13) into Vo s(xC1 gm3 ) = 2 Vi s C1 C2 + sC2 gm2 + gm3 gm4 xC1 gm3 s = · 2 C1 C2 s + s · gm2 /C1 + gm3 gm4 /(C1 C2 )

H (s) =

(4.18)

The corresponding schematic is depicted in Fig. 4.7. If only the coordinates of the poles p1,2 = σ0 ± j · ω0 are available, one may create only two design equations α = −2 · σ = gm2 /C1

(4.19a)

β = σ02 + ω02 = gm3 gm4 /(C1 C2 )

(4.19b)

Having two equations and five unknown parameters, we choose to adopt three gm2 = gm3 = gm4 = gm0 , where gm0 is sort of normalization parameter to be supplied by the designer at the beginning of the design process. So, the rest will be calculated as C1 = gm2 /(−2 · σ )

(4.20a)

   C2 = gm3 gm4 / C1 · σ02 + ω02

(4.20b)

Note that according to (4.12) we have xC1 gm3 =μ·g C1 C2

Fig. 4.7 Band-pass biquad

(4.20c)

54

4 Cascade Realization of Active Gm-C Circuits

Fig. 4.8 Alternative band-pass biquad

which means that the gain of the cell may be controlled only by the parameter x since all other parameters are related to the locations of the poles. That may be a problem since the value of x must be always less than one. No negative gain may be produced too. As a convenient alternative, for the price of one additional transconductor, the circuit of Fig. 4.8 is recommended. Its transfer function obtained by circuit analysis is H (s) =

gm1 s Vo = · Vi C1 s 2 + s · gm2 /C1 + gm3 gm4 /(C1 C2 )

(4.21)

Now, we still use gm2 = gm3 = gm4 = gm0 and C1 = gm2 /(−2 · σ )

(4.22a)

   C2 = gm3 gm4 / C1 · σ02 + ω02

(4.22b)

while there is an additional freedom to control both the gain and the sign through gm1 = μ · g or gm1 = μ · g · C1 . C1

(4.22c)

4.4.3 High-Pass The target transfer function in this case is given by He (s) = μ ·

s2 . s2 + α · s + β

(4.23)

4.4 Second-Order Cell

55

Fig. 4.9 High-pass biquad

To create the high-pass function by the generic circuit of Fig. 4.1, one has to eliminate the first transconductance (gm1 = 0), to connect together the V 3 and V 4 terminals, and pronounce them as an input V i . The resulting schematic is depicted in Fig. 4.9. The corresponding transfer function emanated from (4.13) will be s 2 bC1 C2 Vo = 2 Vi s C1 C2 + sC2 gm2 + gm3 gm4 s2 =b· 2 s + s · gm2 /C1 + gm3 gm4 /(C1 C2 )

H (s) =

(4.24a)

To get that, however, the following should be satisfied bC2 gm2 = aC1 gm3 .

(4.24b)

Now, we still use gm2 = gm3 = gm4 = gm0 and C1 = gm2 /(−2 · σ )

(4.25a)

   C2 = gm3 gm4 / C1 · σ02 + ω02

(4.25b)

To complete, from (4.24b), we have b = a · C1 gm3 /(C2 gm2 ).

(4.25c)

The last expression is partly limiting the realizability of the cell since, after all, 1 > b > 0 must be satisfied. If not, the ratio gm3 /gm2 must be accommodated.

56

4 Cascade Realization of Active Gm-C Circuits

4.4.4 Band-Stop The target transfer function in this case is He (s) = μ ·

s2 + q s2 + α · s + β

(4.26)

To create the band-stop (notch) function by the generic circuit of Fig. 4.1, one has to connect together the V 1 , V 3 , and V 4 terminals and pronounce them as an input V i . The resulting schematic is depicted in Fig. 4.10. The corresponding transfer function emanated from (4.13) will be s 2 bC1 C2 + gm1 gm2 Vo = 2 Vi s C1 C2 + sC2 gm2 + gm3 gm4 s 2 + gm1 gm2 /(b · C1 C2 ) =b· 2 s + s · gm2 /C1 + gm3 gm4 /(C1 C2 )

H (s) =

(4.27a)

with the condition bC2 gm2 = aC1 gm3 .

(4.27b)

After adopting gm2 = gm3 = gm0, the following are the design equation for this case: C1 = gm2 /(−2 · σ )

(4.28a)

   C2 = gm3 gm4 / C1 · σ02 + ω02

(4.28b)

gm1 = q · (b · C1 C2 )/gm2

(4.28c)

To complete, from (4.27b), we have

Fig. 4.10 Band-stop (notch) biquad

4.4 Second-Order Cell

57

b = a · C1 gm3 /(C2 gm2 )

(4.28d)

which reduces itself into gm4 =

a σ02 + ω02 gm2 . · b (−2 · σ ) · (−2 · σ ) 1

(4.28e)

The last expression is partly limiting the realizability of the cell since, after all, 1 > b > 0 must be satisfied. If not, the ratio gm3 /gm2 must be accommodated.

4.4.5 Second-Order Low-Pass with a First-Order Zero at the σ -axis The target function is now He (s) = μ ·

s2

g·s +q +α·s+β

(4.29)

Both positive- and negative-valued zeros may be encountered during the synthesis process. To adopt the structure of the generic cell for this purpose, we choose b = 0 and V 3 = V 1 = V i . The schematic of the cell realizing this kind of transmission zero is depicted in Fig. 4.11. The transfer function emanated from (4.13) now becomes Vo s(−xC1 gm3 ) + gm1 gm2 = 2 Vi s C1 C2 + sC2 gm2 + gm3 gm4 −xgm3 s − gm1 gm2 /(xC1 gm3 ) = · 2 C2 s + sgm2 /C1 + gm3 gm4 /(C1 C2 )

H (s) =

m3 The design equations are g = − x·g ,q = C3

gm3 ·gm1 , C·C3

α=

gm2 C

and β =

(4.30) gm3 ·gm4 . C·C3

Fig. 4.11 Second-order Gm-C cell used within the RM software. Note: 0 ≤ x ≤ 1. “ni” stands for “input node”, while “no” for the “output node”

58

4 Cascade Realization of Active Gm-C Circuits

Note, by changing the sign of gm1 , we in fact transpose the transmission zero (being on the real axis) from one to the other side of the origin. Since there are more circuit elements than degrees of design freedom [expressed by the number of coefficients in (4.29)], some of the element values are to be adopted as design constants. We first adopt gm2 = gm3 = gm = gm0 . Then, we adopt the value of x (say x = 0.5). With that set, one may calculate C = gm2 /α, C3 = −x · gm3 /g, gm4 = −

β·x q·x , and gm1 = − . (4.31) α·g α·g

4.4.6 Complex Zero and All-Pass The target function is now He (s) = μ ·

s2 + g · s + q s2 + α · s + β

(4.32)

To create a circuit exhibiting complex zeros, one has to use V 1 = V 3 = V 4 = V i . The schematic is depicted in Fig. 4.10. The transfer function will be s 2 bC1 C2 + s(bC2 gm2 − aC1 gm3 ) + gm1 gm2 Vo = Vi s 2 C1 C2 + sC2 gm2 + gm3 gm4 2 s + s[gm2 /C1 − a · gm3 /(bC2 )] + gm1 gm2 /(b · C1 C2 ) =b· s 2 + s · gm2 /C1 + gm3 gm4 /(C1 C2 )

H (s) =

(4.33a)

Here, the condition bC2 gm2 = aC1 gm3 .

(4.33b)

must be satisfied. To produce complex zeros in the right-hand side of the complex frequency plane, one has to satisfy bC2 gm2 < aC1 gm3 .

(4.33c)

For an all-pass cell, one will need the following to be satisfied bC2 gm2 − aC1 gm3 = −C2 gm2

(4.34a)

gm1 gm2 /b = gm3 gm4 .

(4.34b)

4.4 Second-Order Cell

59

By comparison of (4.32) and (4.33), when only the zeros and poles are given, we get the following design equations g = gm2 /C1 − a · gm3 /(bC2 )

(4.35a)

q = gm1 gm2 /(b · C1 C2 )

(4.35b)

α = gm2 /C1

(4.35c)

β = gm3 gm4 /(C1 C2 ).

(4.35d)

In the case of complex zeros, there are four conditions and six unknowns. If we adopt the gm2 = gm3 = gm0 , the following may be produced C1 = gm2 /α

(4.36a)

C2 = a · gm3 /[b · (α − g)].

(4.36b)

gm1 = b · q · C1 · C2 /gm2

(4.36c)

gm4 = β · C1 · C2 /gm3 .

(4.36d)

In the case of an all-pass cell, there are two conditions only and six parameters. To satisfy (4.34a), we choose a = 0.5 and b = 0.5. Now, if gm2= gm3 = gm0 , we get C1 = gm2 /α

(4.37a)

C2 = agm3 /(2b · a),

(4.37b)

  gm4 = a · β · gm2 / 2b · α 2

(4.37c)

gm1 = b · gm3 · gm4 /gm2

(4.37d)

Of course, as in all previous cases, the choice of a and b is left to the designer. Note, in the case of all-pass filter, the gain of the sell becomes equal to b which mean one will need an additional amplifier to restore its value.

60

4 Cascade Realization of Active Gm-C Circuits

4.5 General Structure of a Gm-C Cell for Cascade Realization Figure 4.12 depicts the frequency dependence of the gain of a notch cell having relatively high Q-factor of the pole. One may observe a peak in the response which will be higher or lower depending on the value of the Q-factor. In a case of very high Q-factor, it may become very large. In addition, despite the care taken when order of extraction [4] was created it may happen that two (or more) consecutive cells have large peaks at frequencies which are not separated significantly. This situation will give rise to the local value of the gain and in some cases may create large signals within the cascade which may drive some OTAs into saturation so producing nonlinear distortions. To cope with this problem, the GM_C_Cascade program used for this kind of synthesis within the RM software for filter design provides for additional Gm single stage amplifier whose load transconductor (connected as a resistor) is automatically adjusted so that the maximum of the overall gain of the proper cell is unity. The structure of this amplifier is depicted in Fig. 4.13a. Having in mind the (occasional) necessity of a unity gain isolation amplifier as explained above, the very cell, and the additional controlling amplifier, the GM_C_Cascade program uses a complex cell as depicted in Fig. 4.13b. To reduce the overall complexity of the system, the designer of the filter is offered an additional opportunity to adjust the gain. Namely, if one finds that there will be no violation of the maximum amplitude within the whole circuit, the third part of the complex cell may be entirely or partially omitted. To adjust the final value of the overall gain, in such and in any case, additional amplifier is available connected in cascade to the overall output. Its gain is by default equal to unity and it may be adjusted at will by simply changing the value of the load resistor.

Fig. 4.12 Amplitude characteristic of a notch cell obtained by SPICE simulation (note the lin-lin scale)

4.6 Design Example, a Band-Stop Cascaded Gm-C Filter

61

Fig. 4.13 a Controlling amplifier and b the complex cascade cell

4.6 Design Example, a Band-Stop Cascaded Gm-C Filter As a first design example, a tenth-order band-stop LSM_Z filter [7] built of LSMbased fifth-order filter, with two transmission zeros at the positive ω-axis, was produced by the GM_C_cascade program of the RM software for filter design. The maximum passband attenuation was chosen to be amax = 3 dB, while the minimum stopband attenuation was set to amin = 55 dB. The cut-off frequency of the low-pass prototype was f norm = 100 kHz. The final band-stop filter transfer function was obtained by the program transformations with relative width of the stopband of BW r = 1. Below is the partly edited .html report followed by the simulation results produced by SPICE depicted in Fig. 4.14. Note that four types of second-order Gm-C cells were used in cascade to create the overall structure.

62

4 Cascade Realization of Active Gm-C Circuits

Welcome to The Electronic Filter Design Software Program GM_C_Cascade, SYNTHESIS OF THE CASCADE ACTIVE Gm-C FILTERS Project name: GMC_STOP Input data on the transfer function ---------------------------------------------------------Order of the numerator, n=10 Order of the denominator, m=10

Re{} 0. 0. 0. 0. 0.

Normalized transfer function poles and zeros Zeros Poles Im{} Re{} Im{} ±1.335788463e+000 -1.003824452e-001 ±5.017889198e-001 ±7.486215276e-001 -3.833311275e-001 ±1.916184766e+000 ±1.191926676e+000 -4.601415187e-001 ±8.878455850e-001 ±8.389777828e-001 -6.826556592e-001 ±1.114900134e+000 ±1. -3.994425089e-001 ±6.523618471e-001

EXTRACTION OF THE CELLS -----------------------------------------------------------------------Cell No. 1 Second order Gm-C band-stop cell C1=1.037972993e-011 C2=1.037972993e-011 gm1=1.517879224e-005 gm2=1.0e-005 gm3=1.0e-005 gm4=6.496929460e-005 (With reference to Fig. 4.10) -----------------------------------------------------------------------Cell No. 2 Second order Gm-C band-stop cell C1=5.828522072e-012 C2=5.828522072e-012 gm1=3.810708272e-006 gm2=1.0e-005 gm3=1.0e-005 gm4=9.168199036e-006 (With reference to Fig. 4.10). -----------------------------------------------------------------------Cell No. 3 Second order Gm-C band-stop cell C1=8.647064904e-012 C2=8.647064904e-012 gm1=5.903739278e-006 gm2=1.0e-005 gm3=1.0e-005 gm4=1.180747856e-005 (With reference to Fig. 4.10). -----------------------------------------------------------------------Cell No. 4 Second order Gm-C band-stop cell C1=9.961066959e-012 C2=9.961066959e-012 gm1=5.514452142e-006 gm2=1.0e-005 gm3=1.0e-005 gm4=9.168199036e-006 (With reference to Fig. 4.10). ------------------------------------------------------------------------

4.6 Design Example, a Band-Stop Cascaded Gm-C Filter

63

Type of the 5th cell=13 Cell No. 5 Second order Gm-C band-stop cell C1=3.963714539e-011 C2=3.963714539e-011 gm1=6.952149380e-005 gm2=1.0e-005 gm3=1.0e-005 gm4=6.496929460e-005 (With reference to Fig. 4.10). -----------------------------------------------------------------End of the synthesis procedure

Fig. 4.14 Frequency domain response of the Gm-C band-stop filter realized in a form of cascade of second-order notch cells

4.7 Design Example, a Band-Pass Cascaded Gm-C Filter As the second design example, a twelfth-order band-pass modified elliptic [7] filter with partial correction of the passband group delay will be shown. Its transfer function was obtained (first) by transformation of a modified elliptic fifth-order lowpass filter (produced by the Zolotarev program of the RM software for filter design), with two transmission zeros at the positive ω-axis. The maximum passband attenuation was chosen to be amax = 0.5 dB, while the minimum stopband attenuation was set to amin = 45 dB. The cut-off frequency of the low-pass prototype was f norm = 100 kHz. The final band-pass filter was obtained by the program transformations with relative width of the passband of BW r = 0.1. To that filter, a second-order corrector was added (using the program corrector_bp ) with imposed requirement of maximum relative group delay error of δ = 10%. The schematic of the filter/corrector cascade connection was synthesized by the

64

4 Cascade Realization of Active Gm-C Circuits

GM_C_cascade program. Below is the .html report followed by the simulation results produced by SPICE as depicted in Fig. 4.15. Note that four types of second-order Gm-C cells were used in cascade to create the overall structure.

Welcome to The Electronic Filter Design Software Program GM_C_Cascade, SYNTHESIS OF THE CASCADE ACTIVE Gm-C FILTERS Project name: gmc_CAS_elli_bp_corr Input data on the transfer function ---------------------------------------------------------Order of the numerator, n=11 Order of the denominator, m=12 Normalized zeros and poles of the transfer function Zeros Poles Re {} Im {} Re {} Im {} 3.393484080e-2 ±1.006421329e+0 -2.289601893e-2 ±9.997378518e-1 0. ±1.075370740e+0 -4.000190361e-3 ±1.053297076e+0 0. ±9.299118550e-1 -3.605559264e-3 ±9.493860756e-1 0. ±1.112279717e+0 -1.496367708e-2 ±1.037827590e+0 0. ±8.990544236e-1 -1.388985235e-2 ±9.633509136e-1 0. 0. -3.393484080e-2 ±1.006421329e+0 EXTRACTION OF THE CELLS ---------------------------------------------------------Cell No. 1 Second order Gm-C band-stop cell C1=9.946710578e-010 C2=9.946710578e-010 gm1=9.664440810e-002 gm2=1.000000000e-005 gm3=1.000000000e-005 gm4=1.733351782e-001 (With reference to Fig. 4.10). To this cell additional amplifier is connected in cascade (With reference to Fig. 4.13a). The value of gm5 is 5.57558e-006 S. -----------------------------------------------------------------------Cell No. 2 Second order Gm-C band-stop cell C1=2.659021280e-010 C2=2.659021280e-010 gm1=6.455795869e-003 gm2=1.000000000e-005 gm3=1.000000000e-005 gm4=1.202829448e-002 (With reference to Fig. 4.10). To this cell additional amplifier is connected in cascade (With reference to Fig. 4.13a). The value of gm5 is 5.36717e-006 S.

4.7 Design Example, a Band-Pass Cascaded Gm-C Filter

65

-----------------------------------------------------------------Cell No. 3 Schaumann-Van Valkenburg cell Second order ALL-pass cell gm1=5.503532426e-004 gm2=1.000000000e-005 gm3=1.000000000e-005 gm4=1.100706485e-003 xc1=1.172503976e-010 xc2=1.172503976e-010 xc3=5.862519882e-011 xc4=5.862519882e-011 (With reference to Fig. 4.10). To this cell additional amplifier is connected in cascade (With reference to Fig. 4.13a). The value of gm5 is 2.00000e-005 S. -----------------------------------------------------------------------Cell No. 4 Second order Gm-C band-stop cell C1=1.737801488e-010 C2=1.737801488e-010 gm1=2.061928638e-003 gm2=1.0e-005 gm3=1.0e-005 gm4=4.768920226e-003 (With reference to Fig. 4.10). To this cell additional amplifier is connected in cascade (With reference to Fig. 4.13a). The value of gm5 is 4.32368e-006 S. -----------------------------------------------------------------------Cell No. 5 Second order Gm-C band-stop cell C1=2.864590262e-010 C2=2.864590262e-010 gm1=5.237049987e-003 gm2=1.000000000e-005 gm3=1.000000000e-005 gm4=1.202829448e-002 (With reference to Fig. 4.10). To this cell additional amplifier is connected in cascade (With reference to Fig. 4.13a). The value of gm5 is 4.35394e-006 S. -----------------------------------------------------------------Cell No.=6 Second order band-pass GM-C cell gm2=1.000000000e-005 gm3=1.000000000e-005 gm4=1.000000000e-005 C1=1.103538532e-009 C2=1.103538532e-009 C3=1.273300138e-013 (With reference to Fig. 4.7). To this cell additional amplifier is connected in cascade (With reference to Fig. 4.13a). The value of gm5 is 1.95866e-001 S. -----------------------------------------------------------------End of the synthesis procedure

Fig. 4.15 Frequency domain response (obtained by SPICE) of the band-pass filter realized as cascade of Gm-C cells

66

4 Cascade Realization of Active Gm-C Circuits

4.8 Two-Phase Cell Synthesis The structure of the cascaded active two-phase Gm-C filter is depicted in Fig. 4.16. In fact, we have two signal paths as usual, and the cells are driving each other transferring the signal from the input to the output. The main goal here is to create a complete set of two-phase Gm-C cells necessary for creation of two-phase filters by shifting low-pass prototypes. For that reason, only the low-pass and the band-stop cells will be chosen from the set described above. That will include first- and second-order low-pass cells, second-order notch cell, first- and second-order cells having a transmission zero at the real axis, and a second-order cell having complex zeros including the all-pass cell. To that end, the normal Gm-C cells developed above will be combined with the transformed capacitors depicted in Fig. 3.10. We are starting with the first-order low-pass two-phase cell which is essentially the same as the one developed for parallel synthesis and depicted in Fig. 3.12. It is repeated here on Fig. 4.17 for completeness. The value of the coupling transconductance is, of course, gmc = ω0 · C. The rest of parameters are considered known from the ordinary Gm-C synthesis process. The first-order cell having a transmission zero at the real axis will be the next. By combining the Gm-C cell depicted in Fig. 2.4.2 and the couplings from Fig. 3.10, one may produce the new cell depicted in Fig. 4.18. In this cell, one uses gmc = (1 − a) · ω0 C

(4.38a)

gmcC = a · ω0 C.

(4.38b)

The second-order low-pass Gm-C cell from Fig. 4.6 in combination with the coupling depicted in Fig. 3.10a was used to produce the two-phase second-order low-pass Gm-C cell depicted in Fig. 4.19. Here, one has gmc1 = ω0 C1

Fig. 4.16 Cascaded two-phase filter structures

(4.39a)

4.8 Two-Phase Cell Synthesis

Fig. 4.17 First-order low-pass two-phase Gm-C cell

Fig. 4.18 First-order two-phase Gm-C cell exhibiting zero at the real axis

67

68

4 Cascade Realization of Active Gm-C Circuits

Fig. 4.19 Second-order low-pass two-phase Gm-C cell

gmc2 = ω0 C2 .

(4.39b)

Figure 4.20 depicts a second-order two-phase Gm-C cell behaving as band-stop, one with a simple zero at the real axis and one with complex pair of zeros. It is created by combination of the coupling cells depicted in Fig. 3.10 and the Gm-C cell

Fig. 4.20 Second-order two-phase Gm-C cell behaving as band-stop, as one with a simple zero at the real axis, and as one with complex pair of zeros

4.8 Two-Phase Cell Synthesis

69

depicted in Fig. 4.10. Here, we have gmc1 = (1 − a) · ω0 C1

(4.39c)

gmc2 = (1 − b) · ω0 C2

(4.39d)

gmcC1 = a · ω0 C1

(4.39e)

gmcC2 = b · ω0 C2 .

(4.39f)

Finally, starting with the Gm-C cell depicted in Fig. 4.2 and implementing the proper couplings, one gets the corresponding two-phase cell as depicted in Fig. 4.20. Here, the following is valid (Fig. 4.21). gmcC = a · ω0 C

(4.40a)

gmc = b · ω0 C.

(4.40b)

Fig. 4.21 First-order two-phase cell realizing zero at the real axis

70

4 Cascade Realization of Active Gm-C Circuits

4.9 Design Example, a Two-Phase Cascaded Gm-C Filter The example here will be based on a transfer function not synthesized by the RM software but taken from the literature [10]. The reason for that is twofold. First, we want to demonstrate that any transfer function found in the literature may be processed by the RM software system synthesis programs. Second, the transfer function selected is specific from the shape of the amplitude and group delay point of view which was achieved by use of transmission zeros at infinity, at the imaginary axis, and on the right half complex s plane. In addition, the function has a part which behaves as phase corrector. It is a thirteenth-order filter with 12 finite transmission zeros as depicted in Table 4.1 (for normalized data). The requirements were amax = 0.1 dB with decreasing amplitude with increased frequency in the passband; amin = 30 dB. The group delay was controlled to an error of δ = 2% in the whole passband and even beyond. The cut-off frequency used for denormalization was 1 MHz. Note, the reduction of the attenuation toward the edges of the passband may be related to the reduction of the derivative of the amplitude characteristic and consequently to the reduction of the overall sensitivity of the filter as discussed in [7]. To produce a feeling about the properties of the selected function, we used the LP-ANALYSIS program of the RM software. The results are depicted in Fig. 4.22. This filter may be stated as selective one since the width of the transition region is 25% of the passband width. Figure 4.22a depicts the overall gain characteristic, Fig. 4.22b depicts the passband gain of the filter, and Fig. 4.22c depicts the group delay of the filter obtained by running the transfer function analysis program. In order for this function to be synthesized in a form of a two-phase cascaded GmC filter, we choose gm0 = 10 μS and f shift = 1 MHz. The POLY_cascade program was implemented to produce the circuit. In the next, we will first demonstrate the simulation results obtained by SPICE, and then, we will add data (coming from the proper .html report) about the schematic. The results of SPICE simulation of the synthesized circuit are depicted in Fig. 4.23. Figure 4.23a depicts the overall response (amplitude and group delay) Table 4.1 Zeros and poles of the example transfer function Zeros

Poles

Re{}

Im{}

Re{}

Im{}

0.1686209

±0.9024604

−0.0870201

±1.0763062

0.1814326

±0.6527190

−0.3521763

±0.9347914

0.1819768

±0.3950624

−0.1895849

±0.4040078

0.1801890

±0.1323829

−0.2069522

±0.1429249

0

±1.2821873

−0.4464261

0

0

±1.7677102

−0.1686209

±0.9024604

−0.1814326

±0.6527190

4.9 Design Example, a Two-Phase Cascaded Gm-C Filter

71

Fig. 4.22 Frequency domain responses of the example function. a Overall gain; b passband gain; and c group delay

72

4 Cascade Realization of Active Gm-C Circuits

Fig. 4.23 Amplitude and group delay responses of the synthesized circuit obtained by SPICE simulation. a Complete characteristic and b passband characteristic

in the frequency domain obtained by SPICE simulation of the resulting two-phase circuit. As can be seen from Fig. 4.23b, where the passband response id depicted, the cut-off frequencies (0.1 dB) of the two-phase filter are f low = 0 Hz and f high = 2 MHz. Follows the edited .html report describing the circuit.

4.9 Design Example, a Two-Phase Cascaded Gm-C Filter

73

Welcome to The Electronic Filter Design Software Program POLY_Cascade, SYNTHESIS OF CASCADE TWO-PHASE ACTIVE Gm-C FILTERS Project name: GIVA_mod_elliptic ---------------------------------------------------------Order of the numerator, n=12 Order of the denominator, m=13 ---------------------------------------------------------START OF THE ELEMENT EXTRACTION PROCEDURE Capacitances in [F] and transconductances in [S] The shifting frequency is fshift=-1.00000e+006 [Hz] -----------------------------------------------------------------------Cell No. 1 Second order TWO-PHASE Gm-C band-stop cell (Fig. 4.20) C1=4.572361532e-012 C2=4.572361532e-012 gm1=5.158138424e-004 gm2=1.e-5 gm3=1.e-5 gm4=3.849481255e-004 gmc1=-2.872899480e-005 gmc2=-2.872899480e-005 gmcc1=-2.872899480e-005 gmcc2=-2.872899480e-005. To this cell additional amplifier is connected in cascade (Fig. 4.13a) The value of gm5 is 1.33996e-005 S. -----------------------------------------------------------------------Cell No. 2 Second order TWO-PHASE Gm-C band-stop cell (Fig. 4.20) C1=1.129795951e-012 C2=1.129795951e-012 gm1=1.656886281e-005 gm2=1.e-005 gm3=1.e-005 gm4=2.011364230e-005 gmc1=-7.098717319e-006 gmc2=-7.098717319e-006 gmcc1=-7.098717319e-006 gmcc2=-7.098717319e-006 To this cell additional amplifier is connected in cascade (Fig. 4.13a) The value of gm5 is 8.23762e-006 S. -----------------------------------------------------------------Cell No. 3 Second order TWO-PHASE ALL-pass cell (Fig. 4.20) gm1=1.852749329e-005 gm2=1.e-5 gm3=1.e-5 gm4=3.705498658e-005 C1=2.359656233e-012 C2=2.359656233e-012 C3=1.179828117e-012 C4=1.179828117e-012 gmc1=-1.482615737e-005 gmc2=-7.413078687e-006 gmcc1=-1.482615737e-005 gmcc2=-7.413078687e-006 To this cell additional amplifier is connected in cascade (Fig. 4.13a). The value of gm5 is 3.53553e-006 S -----------------------------------------------------------------Cell No. 4 Second order TWO-PHASE ALL-pass cell (Fig. 4.20) gm1=8.714131181e-006 gm2=1.000000000e-005 gm3=1.000000000e-005 gm4=1.742826236e-005 xc1=2.193031229e-012 xc2=2.193031229e-012 xc3=1.096515614e-012 xc4=1.096515614e-012

74

4 Cascade Realization of Active Gm-C Circuits

gmc1=-1.377922160e-005 gmc2=-6.889610798e-006 gmcc1=-1.377922160e-005 gmcc2=-6.889610798e-006 To this cell additional amplifier is connected in cascade (Fig. 4.13a). The value of gm5 is 3.53553e-006 S Input node=7 Output node=9 -----------------------------------------------------------------Cell No. 5 Second order TWO-PHASE GM-C cell having pair of complex zeros (Fig. 4.20) gm1=3.357168765e-006 gm2=1.e-5 gm3=1.e-5 gm4=7.068345510e-006 C1=2.098729159e-012 C2=2.098729159e-012 C3=1.070851376e-012 C4=1.070851376e-012 gmc1=-1.318670422e-005 gmc2=-6.728357632e-006 gmcc1=-1.318670422e-005 gmcc2=-6.728357632e-006 -----------------------------------------------------------------Cell No. 6 Second order TWO-PHASE GM-C cell having pair of complex zeros (Fig. 4.20) gm1=7.799781694e-007 gm2=1.e-5 gm3=1.e-5 gm4=1.973819233e-006 C1=1.922605112e-012 C2=1.922605112e-012 C3=1.027757722e-012 C4=1.027757722e-012 gmc1=-1.208008419e-005 gmc2=-6.457592217e-006 gmcc1=-1.208008419e-005 gmcc2=-6.457592217e-006 -----------------------------------------------------------------Cell No.=7 First order TWO-PHASE low-pass Gm-C cell (Fig. 4.17) c=3.565090462e-012 gm1=2.240012401e-005 gm2=1.e-5 gmc=-2.240012401e-005 -----------------------------------------------------------------To the system output an additional amplifier is connected in cascade The value of the loading transconductance gm5=3.212950000002e-007 [S] -----------------------------------------------------------------NOTE: The designer is advised to merge all additional two stage gain cells being introduced to adjust the gains of separate cells (if any). -----------------------------------------------------------------End of the synthesis procedure

References 1. Schaumann R, Van Valkenburg E (2001) Design of analog filters. Oxford University Press, New York 2. Monsurrò P, Pennisi S, Scotti G, Trifiletti A (2008) Unity-gain amplifier with theoretically zero gain error. IEEE Trans Instrum Meas 57(7):1431–1437 3. Ramachandran A (2005) Nonlinearity and noise modeling of operational transconductance amplifiers for continuous time analog filters. M.S. thesis. Texas A&M University 4. Hospodka J (2006) Optimization of dynamic range of cascade filter realization. Radioengineering 15(3):31–34

References

75

5. Xuexiang C, Sánchez-Sinencio CE, Geiger RL (1987) Pole-zero pairing strategies for cascaded switched-capacitor filters. IEE Proc G Electron Circ Syst 134(4):199–204 6. Chiou C-F, Schaumann R (1981) Refined procedure for optimizing signal-to-noise ratio in cascade active-RC filters. IEE Proc Pt G 128(4):181–191 7. Litovski V (2019) Electronic filters, theory, numerical recipes, and design practice based on the RM software. Springer Science+Business Media 8. Temes GC, CMOS active filters. https://slideplayer.com/slide/6192567/ 9. Laker KR, Sansen WMC (1994) Design of analog integrated circuits and systems. McGraw-Hill College 10. Litovski V, Milovanovi´c D (1983) New solution for the ideal filter approximation problem. IEE Proc Pt G 130(4):161–163

Chapter 5

Use of LC-to-Gm-C Transformation for Synthesis of Gm-C Singleand Two-Phase Integrated Filters

5.1 Introduction In the previous two chapters, we introduced concepts of Gm-C synthesis based on second (and first)-order cells being connected either in cascade or in parallel. Here, we will discuss (and give all necessary information) for the Gm-C synthesis that is based on transformation of existing LC filter solutions. Namely, the fundamental idea of implementation of simulated inductance is based on the availability of LC cascaded circuits which are synthesized by some other filter synthesis software system (such as [1]) or even extracted from an existing catalog such as [2]. In that way, the inductors, the elements being difficult to integrate in large numbers on a single silicon chip, are substituted by an equivalent circuit containing OTAs, while a capacitor and the rest of the filter elements (capacitors) remain the same. That, of course, is a very attractive method and even designer with extremely limited knowledge of filter design can produce successful designs. For cases, however, which cannot be found in catalogs, the designer (being in a possession of a proper transfer function) is expected to use (or develop) a computer program which is performing the passive LC synthesis. That was done for example, program in [1] and in this chapter circuit synthesis results produced by the software for filter design will be used. of the Solutions were found for circuits that simulate inductances [3, 4] of which we will later on elaborate the one based on gyrators [5] and described in [6]. Then, cells will be created to substitute the cells produced by a synthesis program for LCM filters. That means passive filters containing capacitors (C), inductors (L), and ideal transformers (M) are used as prototypes to create Gm-C filters. Since our intention is to create two-phase Gm-C solutions, only low-pass LCM synthesis will be taken into account in this chapter.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 V. Litovski, Gm-C Filter Synthesis for Modern RF Systems, Lecture Notes in Electrical Engineering 807, https://doi.org/10.1007/978-981-16-6561-5_5

77

78

5 Use of LC-to-Gm-C Transformation for Synthesis of Gm-C Single …

In the sequel, we will review very briefly the equivalent circuit to an inductor and systematically develop a synthesis process for Gm-C and two-phase Gm-C filters. The filter obtained will be tested using ideal OTAs (infinite input/output impedances) to check for the synthesis process and not for the properties of the practical realization. Synthesis examples will be given allowing the reader (who is developing his own software for circuit synthesis) to check for the correctness of the solution (as it was done in the previous chapters). That stands for parts of the synthesis process of passive LCM filters, too. Here, in addition, for some examples, the SPICE code will be given too, so allowing the reader to get the feeling on the behavior of this kind of solutions.

5.2 The Gyrator and the Simulated Inductor The fundamental building block which will be used to create the simulated inductor is the gyrator as depicted in Fig. 5.1. For this circuit, the nodal equations are I1 = gm V2 I2 = −gm · V1

(5.1)

When loaded by an impedance Z L , the output voltage will be V2 = −Z L · I2 .

(5.2)

After substitution in (5.1), one gets 2 · Z L · V1 , I1 = gm

(5.3)

For Z L = 1/( jωC), one has I1 /V1 =

Fig. 5.1 Gyrator realized by a pair of transconductors

2 1 gm = jωC jωL e

(5.4)

5.2 The Gyrator and the Simulated Inductor

79

with 2 . L e = C/gm

(5.5)

A gyrator loaded by capacitor will behave as an inductor. Since the capacitor has one terminal grounded, the resulting simulated inductor will be grounded, too. It is up to the designer to decide whether to use a fixed value for the transconductance or for the capacitor in order to create the desired value of the inductance.

5.2.1 Floating Simulated Inductor The schematic depicted in Fig. 5.2 represents a connection of two gyrators and a capacitor to produce a floating inductor. To show that we will write the nodal equations for the circuit as I1 = gm · VC I2 = −gm · VC jωC · VC − gm V1 + gm V2 = 0

(5.6)

After eliminating V C from the third equation and having in mind I 1 = − I 2 , one gets. ZL =

V1 − V2 jωC = 2 I1 gm

(5.7)

This means that the circuit of Fig. 5.2 behaves as a floating inductor of inductance 2 . L e = C/gm

Fig. 5.2 Simulated floating inductor. a Original version and b the GM-C version

(5.8)

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5 Use of LC-to-Gm-C Transformation for Synthesis of Gm-C Single …

Fig. 5.3 Simulated grounded transformer. a Original version and b the GM-C version

Note, at ω = 0, from (5.6), one gets V 1 = V 2 which corresponds to real inductor. The voltage V C is undefined and so are the currents I 1 and I 2 . The last two will be defined by the outer circuit as is the case with the inductor.

5.2.2 Simulated Ideal Grounded Transformer To produce a simulated grounded transformer based on gyrators, one may use the circuit of Fig. 5.3. The following are the nodal equations of this circuit I1 = gm1 · V0 I2 = −gm2 · V0 −gm1 V1 + gm2 V2 = 0.

(5.9)

The transformer’s equation is now gm1 V2 = . V1 gm2

(5.10a)

with n=

V2 V1

(5.10b)

Note the transformer depicted here has one terminal of both input and output, grounded. That is acceptable for the implementation of the cell realizing complex transmission zero (D-section depicted later in Fig. 5.8b). Negative “turn ratio” may be obtained by inverting (interchanging the input terminals of) both transconductance amplifiers in one of the gyrators.

5.3 Circuit Synthesis

81

5.3 Circuit Synthesis The circuit synthesis of this kind of filters is straightforward. One is to synthesize first an LC filter using a conventional synthesis procedure using a conventional synthesis of the software. The next step is to substitute the program, e.g., inductors and, if necessary, transformers with their models using OTAs. Here, we demonstrate the circuits equivalent to the ones described in [1] (Chap. 14). A limited set is given to save space. Nevertheless, this set is satisfactory for most physical realizations, especially when low-pass circuits are sought. Figure 5.4 depicts the equivalent circuit to the grounded inductor. As can be seen from now on the transconductance is considered a constant while the value of the capacitance is evaluated from (5.5) to be 2 . C = L · gm

(5.11)

At ω = 0, the equivalence is failing since no current flow toward the ground is possible. In the case of synthesis of Gm-C filters based on LC prototypes, this problem is usually mitigated by the fact that the inductor is either connected in series with a capacitor or there are two capacitors (to the left and to the right) which disconnect the inductor from DC signals. This will be demonstrated later on by the cell realizing a complex transmission zero without a transformer.

Fig. 5.4 Simulated grounded inductor. a Original version and b the GM-C version

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5 Use of LC-to-Gm-C Transformation for Synthesis of Gm-C Single …

This equation may be used as a reference when choosing the value of the transconductance. Namely, a small transconductance, e.g., gm = 10–6 S, would produce extremely small capacitances. If, for example L = 100 μH, one would produce C = 0.1 fF which is fairly small value and is in the range of the parasitic capacitances in any CMOS technology. In the opposite case, when large gm is chosen, the resulting capacitance may become very large. For example, if gm = 10–1 S and L = 100 μH, one gets C = 10 nF. It seems that for this inductance a value of gm = 10–3 S would be preferable. The question is, however, which is the output resistance of such an OTA. If satisfactory, the goal is reached. If not, one must go for a compromise. Figure 5.5b depicts the equivalent circuit to the one depicted in Fig. 5.5a. It is the equivalent to the Brune’s cell. No additional comments are necessary since the capacitance is calculated again from (5.11). Its alteration realizing transmission zero at the real axis with a cell using a transformer (as depicted in Fig. 5.6a) is depicted in Fig. 5.6b. Note, since for an ideal transformer the turn ratio is defined by V2 /V1 = 1/n,

(5.12)

having in mind the notation of Fig. 5.6, in comparison with (5.10), we have gmn = n · gm .

(5.13)

Fig. 5.5 PI-cell realizing a transmission zero at the ω-axis using simulated inductor. a Original LC version and b the GM-C version

5.3 Circuit Synthesis

83

Fig. 5.6 Equivalent circuit to the version of the Brune’s cell. a Original LC version and b the GM-C version

Figures 5.7b and 5.8b depict the equivalent circuits to the versions of the D-section depicted in Figs. 5.7a and 5.8a, respectively.

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5 Use of LC-to-Gm-C Transformation for Synthesis of Gm-C Single …

Fig. 5.7 Cell realizing a complex transmission zero without a transformer. a Original LC version and b the GM-C version

5.4 Design Example No. 1

85

Fig. 5.8 Cell realizing a complex transmission zero using a transformer. a Original LC version and b the GM-C version

5.4 Design Example No. 1

As an example, a ninth-order low-pass LSM filter with four zeros at the positive ω-axis exhibiting amax = 3 dB attenuation in the passband will be used. The stopband attenuation was set to amin = 50 dB. The cut-off frequency was set to 43 kHz. The and the transfer function was synthesized by sequential use of the programs of the software. As can be seen, gm = 10–6 S was used. Figure 5.9 depicts the SPICE simulation results. We will first present an edited version of the.html report containing data about the transfer function, the synthesis process, and finally the schematic of the resulting circuit. To make it possible for the reader to study the properties of the circuit and the influence of specific parameters (including realistic model of the transconductor), at the end of the example, we add the SPICE code (net-list) for the solution.

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5 Use of LC-to-Gm-C Transformation for Synthesis of Gm-C Single …

Welcome to The Electronic Filter Design Software Program: Gm_LC SYNTHESIS OF GM-C FILTERS BASED ON LOW-PASS PASIVE LC PROTOTYPES Project name: GMC_LC_tutorial_example ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Input data on the transfer function ---------------------------------------------------------Order of the numerator n=8; Order of the denominator m=9 You entered the transmission zeros and poles as follows: Re{} 0. 0. 0. 0.

Zeros Im{} ±1.207678823 ±1.347839032 ±1.774177189 ±3.290123935

Poles Re{} -7.170315101e-2 -2.565027588e-1 -1.451251999e+0 -5.858625045e-1 -1.112461676e+0

Im{} ±9.991239141e-1 ±9.818358904e-1 0. ±9.061210530e-1 ±6.430061028e-1

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ All polynomials in ascending order of s or ω2. The input transfer function ------------------------------------------------Poles-real part: -7.170315101e-002 -7.170315101e-002 -2.565027588e-001 -2.565027588e-001 -1.451251999e+000 -5.858625045e-001 -5.858625045e-001 -1.112461676e+000 -1.112461676e+000 Poles-imaginary part: 9.991239141e-001 -9.991239141e-001 9.818358904e-001 -9.818358904e-001 0.000000000e+000 9.061210530e-001 -9.061210530e-001 6.430061028e-001 -6.430061028e-001 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Ordered transmission zeros Zeros-real part: 0.000000000e+000 0.000000000e+000 0.000000000e+000 0.000000000e+000 9.999999990e+008 Zeros-imaginary part: 3.290123935e+000 1.774177189e+000 1.347839032e+000

5.4 Design Example No. 1

1.207678823e+000 9.999999990e+008 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Zeros of the reflection coefficient Real part -6.343710407e-002 -6.343710407e-002 -5.968329258e-002 -5.968329258e-002 -1.556952057e-001 -1.556952057e-001 -9.123290034e-001 -9.123290034e-001 0.000000000e+000 Imaginary part 2.213500246e-001 -2.213500246e-001 8.666427163e-001 -8.666427163e-001 -7.420305649e-001 7.420305649e-001 7.721385457e-002 -7.721385457e-002 0.000000000e+000 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Coefficients of the reflection coefficient (numerator): -0.000000000e+000 1.928115552e-002 1.016004006e-001 6.094896104e-001 1.484186221e+000 2.692741636e+000 3.657436880e+000 3.330116144e+000 2.382289212e+000 1.000000000e+000 Coefficients of the reflection coefficient (denominator): 2.882565345e+000 1.061974314e+001 2.430738234e+001 3.862231245e+001 4.634003434e+001 4.267802122e+001 3.006489385e+001 1.564068176e+001 5.504312179e+000 1.000000000e+000 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ INPUT IMPEDANCE Variant Zin=RS*(1-RO)/(1+RO) Numerator: -2.882565345e+000 -1.060046198e+001 -2.420578194e+001 -3.801282284e+001 -4.485584812e+001 -3.998527959e+001 -2.640745697e+001 -1.231056562e+001 -3.122022968e+000 Denominator: -2.882565345e+000 -1.063902430e+001 -2.440898274e+001 -3.923180206e+001 -4.782422056e+001 -4.537076286e+001 -3.372233073e+001 -1.897079791e+001 -7.886601391e+000 -2.000000000e+000 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ EXTRACTION OF THE CELLS All capacitances in F and all transconductances I S. ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ -----------------------------------------------------------------------k=1 (th) ZERO AT THE IMAGINARY AXIS=3.2901239e+000 -----------------------------------------------------------------------LC-version, variant with a resonant circuit (tank=L:Cm) in the series branch, PI_cell (Fig. 5.5a).

87

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5 Use of LC-to-Gm-C Transformation for Synthesis of Gm-C Single …

Simulated floating inductance: The capacitance (denoted L in Fig. 5.5b) within the simulated inductor is L*gm*gm=5.2452831e-013, Cp=2.1382177e009 Cm=2.4127365e-010 Cs=-2.1680919e-010. -----------------------------------------------------------------------k=2 (th) ZERO AT THE IMAGINARY AXIS=1.7741772e+000 -----------------------------------------------------------------------LC-version, variant with a resonant circuit (tank=L:Cm) in the series branch, PI_cell (Fig. 5.5a). Simulated floating inductance: The capacitance (denoted L in Fig. 5.5b) within the simulated inductor is L*gm*gm=5.8743338e-013, Cp=6.2896543e009 Cm=7.4088491e-010, Cs=-6.6280975e-010. -----------------------------------------------------------------------k=3 (th) ZERO AT THE IMAGINARY AXIS=1.3478390e+000 -----------------------------------------------------------------------LC-version, variant with a resonant circuit (tank=L:Cm) in the series branch, PI_cell (Fig. 5.5a). Simulated floating inductance: The capacitance (denoted L in Fig. 5.7b) within the simulated inductor is L*gm*gm=2.2310528e-013, Cp=6.8013276e009 Cm=3.3800058e-009, Cs=-2.2579092e-009. -----------------------------------------------------------------------k=4 (th) ZERO AT THE IMAGINARY AXIS=1.2076788e+000 -----------------------------------------------------------------------LC-version, variant with a resonant circuit (tank=L:Cm) in the series branch, PI_cell (Fig. 5.5a). Simulated floating inductance: The capacitance (denoted L in Fig. 5.7b) within the simulated inductor is L*gm*gm=2.6055821e-014, Cp=-4.4624834e009 Cm=3.6049205e-008 Cs=5.0929306e-009. -----------------------------------------------------------------------k=5 (th) ZERO AT INFINITY: parallel capacitance C=9.3862427e-010 -----------------------------------------------------------------------------------------Residual load impedance--------------------R=1.0000000e+003 ========================================================== Here ends the synthesis process The following is a copy of the corresponding SPICE net-list. +++++++++++++++++++++++++++++++ Welcome to the RM software for filter design *GM_Simulated inductance LC FILTERS. *PROJECT NAME: GMC_LC_tutorial_example

5.4 Design Example No. 1

v n0 0 ac 2 sin Rgen n0 1 1.00000e+003 * * Zero at the imaginary axis, w=3.2901239e+000 *Variant with no transformer-parallel LC in the series branch CP1 1 0 2.1382176851e-009 gleftt1 0 101 1 0 1.0000000000e-005 Rleftt1 0 101 1.0000000000e+012 gleftb1 1 0 101 0 1.0000000000e-005 Rleftb1 0 1 1.0000000000e+012 CsimL1 0 101 5.2452831369e-013 grigtht1 2 0 0 101 1.0000000000e-005 Rrigtht1 0 2 1.0000000000e+012 grigthb1 0 101 0 2 1.0000000000e-005 Rrigthb1 0 101 1.0000000000e+012 CM1 1 2 2.4127365418e-010 CS1 2 0 -2.1680919186e-010 * * Zero at the imaginary axis, w=1.7741772e+000 *Variant with no transformer-parallel LC in the series branch CP2 2 0 6.2896542945e-009 gleftt2 0 102 2 0 1.0000000000e-005 Rleftt2 0 102 1.0000000000e+012 gleftb2 2 0 102 0 1.0000000000e-005 Rleftb2 0 2 1.0000000000e+012 CsimL2 0 102 5.8743338264e-013 grigtht2 3 0 0 102 1.0000000000e-005 Rrigtht2 0 3 1.0000000000e+012 grigthb2 0 102 0 3 1.0000000000e-005 Rrigthb2 0 102 1.0000000000e+012 CM2 2 3 7.4088490786e-010 CS2 3 0 -6.6280975161e-010 * * Zero at the imaginary axis, w=1.3478390e+000 *Variant with no transformer-parallel LC in the series branch CP3 3 0 6.8013276156e-009 gleftt3 0 103 3 0 1.0000000000e-005 Rleftt3 0 103 1.0000000000e+012 gleftb3 3 0 103 0 1.0000000000e-005 Rleftb3 0 3 1.0000000000e+012 CsimL3 0 103 2.2310528099e-013 grigtht3 4 0 0 103 1.0000000000e-005 Rrigtht3 0 4 1.0000000000e+012 grigthb3 0 103 0 4 1.0000000000e-005

89

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5 Use of LC-to-Gm-C Transformation for Synthesis of Gm-C Single …

Rrigthb3 0 103 1.0000000000e+012 CM3 3 4 3.3800057945e-009 CS3 4 0 -2.2579092369e-009 * * Zero at the imaginary axis, w=1.2076788e+000 *Variant with no transformer-parallel LC in the series branch CP4 4 0 -4.4624834472e-009 gleftt4 0 104 4 0 1.0000000000e-005 Rleftt4 0 104 1.0000000000e+012 gleftb4 4 0 104 0 1.0000000000e-005 Rleftb4 0 4 1.0000000000e+012 CsimL4 0 104 2.6055821209e-014 grigtht4 5 0 0 104 1.0000000000e-005 Rrigtht4 0 5 1.0000000000e+012 grigthb4 0 104 0 5 1.0000000000e-005 Rrigthb4 0 104 1.0000000000e+012 CM4 4 5 3.6049204512e-008 CS4 5 0 5.0929306049e-009 * * ZERO AT INFINITY * parallel capacitance C015 5 0 9.3862427388e-010 RP5 5 0 9.9999999962e+002 *AP * Simulation settings---------------------------.ac dec 5000 5.0000e+002 5.0000e+005 .print ac v(5) .end +++++++++++++++++++++++++++++++++++++++++++++++++

5.4 Design Example No. 1

91

Fig. 5.9 SPICE simulation results for the first example. a Overall gain and group delay (logarithmic scale) and b passband gain and group delay (linear scale)

5.5 Design Example No. 2 As second example, a seventh-order low-pass LSM filter with two transmission zeros on the positive ω-axis exhibiting amax = 3 dB attenuation in the passband will be used. The stopband attenuation was set to amin = 40 dB. The cut-off frequency was set to 1 MHz, and as can be seen, gm = 10–5 S was used. The filter is extended in cascade with a second-order corrector producing a group delay error of 2%.

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5 Use of LC-to-Gm-C Transformation for Synthesis of Gm-C Single …

Fig. 5.10 SPICE simulation results for the second example (linear scale)

As can be seen, gm = 10–5 S was used. Figure 5.10 depicts the SPICE simulation results. We will first present an edited version of the .html report containing data about the transfer function, the synthesis process, and finally the schematic of the resulting circuit. To make it possible for the reader to study the properties of the circuit and the influence of specific parameters (including realistic model of the transconductor), at the end of the example, we add the SPICE code for the solution.

5.5 Design Example No. 2

93

Welcome to The Electronic Filter Design Software Program: Gm_LC SYNTHESIS OF GM-C FILTERS BASED ON LOW-PASS PASIVE L PROTOTYPES

Project name: LP_corr_LSM_3_40_7_4_2_1pc ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Input data on the transfer function ---------------------------------------------------------Order of the numerator n=6; Order of the denominator m=9 You entered normalized the zeros and poles as follows: Zeros Poles Re{} Im{} Re{} Im{} 5.393452892e-1 ±3.660571056e-1 -1.326870339e-1 ±9.951282463e-1 0.0 ±1.343133000 -1.476977197 0. 0.0 ±1.761057196 -4.981968802e-1 ±9.146829523e-1 -1.110790357 ±5.867176788e-1 -5.393452892e-1 ±3.660571056e-1 ----------------------------------------------------------------------Ordered transmission zeros Zeros-real part: 9.999999990e+008 9.999999990e+009 0.000000000e+000 0.00000000 0e+000 5.393452892e-001 9.999999990e+008 Zeros-imaginary part: 9.999999990e+008 9.999999990e+009 1.761057196e+000 1.34313300 0e+000 3.660571056e-001 9.999999990e+008 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Zeros of the reflection coefficient Real part -2.177140733e-001 -2.177140733e-001 -5.393452892e-001 -5.393452892e-001 -8.644932609e-002 -8.644932609e-002 -1.130145338e+000 -1.130145338e+000 0.000000000e+000 Imaginary part 5.466163662e-001 -5.466163662e-001 -3.660571056e-001 3.660571056e-001 -7.746380103e-001 7.746380103e-001 1.184003172e-001 -1.184003172e-001 0.000000000e+000 ----------------------------------------------------------------------All POLYNOMIALS in ascending order of s or w**2 -----------------------------------------------------------------------

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5 Use of LC-to-Gm-C Transformation for Synthesis of Gm-C Single …

Coefficients of the reflection coefficient (numerator): -0.000000000e+000 1.153913516e-001 6.729141612e-001 2.201674495e+000 4.805149585e+000 7.548458640e+000 8.640645179e+000 7.214495640e+000 3.947308052e+000 1.000000000e+000 ----------------------------------------------------------------------Coefficients of the reflection coefficient (denominator): 1.082837218e+000 6.286228865e+000 1.824900382e+001 3.399803525e+001 4.552810317e+001 4.524251238e+001 3.339820684e+001 1.765873425e+001 6.039016318e+000 1.000000000e+000 ++++++++++++++++++++++++++++++++++++++++++++++++++++ INPUT IMPEDANCE Variant Zin=RS*(1-RO)/(1+RO) Numerator: -1.082837218e+000 -6.170837513e+000 -1.757608966e+001 -3.179636076e+001 -4.072295359e+001 -3.769405374e+001 -2.475756166e+001 -1.044423861e+001 -2.091708265e+000 Denominator: -1.082837218e+000 -6.401620216e+000 -1.892191798e+001 -3.619970975e+001 -5.033325276e+001 -5.279097102e+001 -4.203885202e+001 -2.487322989e+001 -9.986324370e+000 -2.000000000e+000 ++++++++++++++++++++++++++++++++++++++++++++++++++++ EXTRACTION OF THE CELLS Capacitances in [F] and transconductances in [S] -----------------------------------------------------------------------k=1 (th) ZERO AT INFINITY: -----------------------------------------------------------------------Parallel capacitance C=1.5217700e-012 -----------------------------------------------------------------------k=2 (th) ZERO AT INFINITY: -----------------------------------------------------------------------Simulated series inductance (Fig. 5.2b). Capacitance C=2.7716003e-012 F. -----------------------------------------------------------------------k=3 (th) ZERO AT THE IMAGINARY AXIS=1.7610572e+000 ------------------------------------------------------------------------

5.5 Design Example No. 2

95

Simulated floating inductance: The capacitance (denoted L in Fig. 5.5b) within the simulated inductor is L*gm*gm=2.2192669e-012, Cp=2.8843404e-012, Cm=3.6803050e-013, Cs=-3.2638505e-013. -----------------------------------------------------------------------k=4 (th) ZERO AT THE IMAGINARY AXIS=1.3431330e+000 -----------------------------------------------------------------------Simulated floating inductance: The capacitance (denoted L in Fig. 5.5b) within the simulated inductor is L*gm*gm=3.8463685e-014, Cp=6.1332347e-012, Cm=3.6504914e-011, Cs=-5.2510067e-012 -----------------------------------------------------------------------k=5 (th) A PAIR OF COMPLEX ZEROS (all_pass) s=5.3934529e-001 w=3.6605711e-001 -----------------------------------------------------------------------Variant with one transformer (Fig. 5.8b) Simulated L1 by capacitance L1=8.0810786e-12 F, C4=7.3772288e-13, C3=2.0202697e-12. Simulated L2 by capacitance L2=2.9508915e-12 F, C1=-2.6211924e-13, C2=4.0658091e-13, gm0= 1.e-5, gmn=-1.e-5. -----------------------------------------------------------------------k=6 (th) ZERO AT INFINITY: -----------------------------------------------------------------------Parallel capacitance C=2.6211924e-013. ------------------Residual load impedance--------------------Residual is resistor R=1.e+5. Here ends the synthesis process

Figure 5.11 depicts the time domain response of the Gm-C filter obtained by SPICE simulation. As an excitation, in this case, a long pulse was used having a rise time of 0.1 ns. Since the transfer function of the circuit contains a pair of complex poles, the response has ringing at the beginning of the transient. Note that we are dealing with a passive LC prototype which is equally terminated; i.e., the source and the load resistances are equal. So, if the amplitude of the input pulse is 1 V, the output is halved. To compensate for that the output voltage of Fig. 5.10 is multiplied by 2 what is always done in frequency domain calculations due the definition of the attenuation.

Fig. 5.11 Response of the new Gm-C filter to a step function (approximated by a long pulse having rise time of 0.1 ns)

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5 Use of LC-to-Gm-C Transformation for Synthesis of Gm-C Single …

Follows the SPICE code for simulation in the frequency and time domain. Namely, the commands which may be used for transient simulation V1 …, and .tran … are presented as comments and may be restored if the ones related to frequency domain simulation (v1 …,.ac …, and .print ac …) are transformed into comments. Welcome to the RM software for filter design *GM_Simulated inductance LC FILTERS. *PROJECT NAME: LP_corr_LSM_3_40_7_4_2_1pc *V1 n0 0 PULSE(0 1 1n .1u 1u .1 6) v n0 0 ac 2 sin Rgen n0 1 1.00000e+005 * * ZERO AT INFINITY * parallel capacitance C011 1 0 1.5217699896e-012 * * ZERO AT INFINITY * Simulated serial inductance gleftt2 0 int2 1 0 1.000000000e-005 Rleftt2 0 int2 1.000000000e+012 gleftb2 1 0 int2 0 1.000000000e-005 Rleftb2 1 0 1.000000000e+012 Csim2 int2 0 2.771600253e-012 Rsavl12 int2 0 1.000000000e+012 grigtht2 2 0 0 int2 1.000000000e-005 RrigthT2 2 0 1.000000000e+012 grigthb2 0 int2 0 2 1.000000000e-005 Rrigthb2 0 int2 1.000000000e+012 * * Zero at the imaginary axis, w=1.7610572e+000 *Variant with no transformer-parallel LC in the series branch CP3 2 0 2.8843404122e-012 gleftt3 0 102 2 0 1.0000000000e-005 Rleftt3 0 102 1.0000000000e+012 gleftb3 2 0 102 0 1.0000000000e-005 Rleftb3 0 2 1.0000000000e+012 CsimL3 0 102 2.2192668783e-012 grigtht3 3 0 0 102 1.0000000000e-005 Rrigtht3 0 3 1.0000000000e+012 grigthb3 0 102 0 3 1.0000000000e-005 Rrigthb3 0 102 1.0000000000e+012 CM3 2 3 3.6803049868e-013 CS3 3 0 -3.2638504935e-013 *

5.5 Design Example No. 2

* Zero at the imaginary axis, w=1.3431330e+000 *Variant with no transformer-parallel LC in the series branch CP4 3 0 6.1332346504e-012 gleftt4 0 103 3 0 1.0000000000e-005 Rleftt4 0 103 1.0000000000e+012 gleftb4 3 0 103 0 1.0000000000e-005 Rleftb4 0 3 1.0000000000e+012 CsimL4 0 103 3.8463685430e-014 grigtht4 4 0 0 103 1.0000000000e-005 Rrigtht4 0 4 1.0000000000e+012 grigthb4 0 103 0 4 1.0000000000e-005 Rrigthb4 0 103 1.0000000000e+012 CM4 3 4 3.6504914132e-011 CS4 4 0 -5.2510066844e-012 * * Pair of complex zeros *Variant with one transformer *Simulated inductance 1 CsimL15 0 104 8.0810786301e-012 gleftta5 0 104 4 0 1.0000000000e-005 Rlefttb5 0 104 1.0000000000e+012 gleftbc5 4 0 104 0 1.0000000000e-005 Rleftbd5 0 4 1.0000000000e+012 grigthte5 8 0 0 104 1.0000000000e-005 Rrigthtf5 0 8 1.0000000000e+012 grigthbg5 0 104 0 8 1.0000000000e-005 Rrigthbh5 0 104 1.0000000000e+012 *Simulated inductance 2 CsimL25 0 204 2.9508915060e-012 gleftti5 0 204 5 0 1.0000000000e-005 Rlefttj5 0 204 1.0000000000e+012 gleftbk5 5 0 204 0 1.0000000000e-005 Rleftbl5 0 5 1.0000000000e+012 grigthtm5 7 0 0 204 1.0000000000e-005 Rrigthtn5 0 7 1.0000000000e+012 grigthbo5 0 204 0 7 1.0000000000e-005 Rrigthbp5 0 204 1.0000000000e+012 *Original capacitances C45 4 8 7.3772287650e-013 C35 4 5 2.0202696575e-012 C25 8 0 -2.6211924016e-013 C15 4 0 4.0658091122e-013 *Transformer glefttq5 0 304 7 0 1.0000000000e-005

97

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5 Use of LC-to-Gm-C Transformation for Synthesis of Gm-C Single …

Rlefttr5 0 304 1.0000000000e+012 gleftbs5 7 0 304 0 1.0000000000e-005 Rleftbt5 0 7 1.0000000000e+012 grigthtu5 8 0 0 304 -1.0000000000e-005 Rrigthtv5 0 8 1.0000000000e+012 grigthbw5 0 304 0 8 -1.0000000000e-005 Rrigthbx5 0 304 1.0000000000e+012 * * ZERO AT INFINITY * parallel capacitance C016 8 0 2.6211924016e-013 RP6 8 0 1.0000000000e+005 *In a case of nonmonotonic attenuation response you will probably *need a transformer to accommodate to equal terminations *AP * Simulation settings---------------------------*.tran .1n .005m UIC .ac dec 700 1.00000e+004 4.00000e+006 .print ac v(8) .end

5.6 Creation of the Two-Phase Cells Based on the set of Gm-C cells developed above and the circuit implementation of the two-phase transform depicted in Fig. 3.10, library of two-phase cells may be developed. Alike Chap. 3, in this case, the number of two-phase cells is larger so allowing to accommodate to the structure of the original Gm-C filter. To start with, Fig. 5.12 depicts the structure of the two-phase grounded inductor. The value of gmc in this case is given by 2 gmc = ω0 C = ω0 L · gm

(5.11)

Here, one is to have in mind that ω0 is the shifting frequency while gm is the normalizing transconductance supplied by the designer which expresses the technology in which the CMOS is implemented.

5.6 Creation of the Two-Phase Cells

99

Fig. 5.12 Two-phase grounded inductor

The same formula, (5.11), should be used for evaluation of the coupling transconductance in the case of the floating inductance. The corresponding two-phase cell is depicted in Fig. 5.13. Figure 5.14 depicts the two-phase cell realizing a PI cell used for implementation of a transmission zero at the ω-axis. In this case for the transconductances, we have: 2 gmcL = ω0 C = ω0 L · gm

(5.12a)

gmcM = ω0 CM

(5.12b)

gmcp = ω0

(5.12c)

gmcs = ω0 Cs

(5.12d)

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5 Use of LC-to-Gm-C Transformation for Synthesis of Gm-C Single …

Fig. 5.13 Two-phase floating inductor

Fig. 5.14 Two-phase Gm-C cell realizing the PI-cell of Fig. 5.7

5.6 Creation of the Two-Phase Cells

101

One has to be careful with the implementation of the schematic depicted in Fig. 5.14. Namely, the original LC PI-cell (of Fig. 5.7a) has the property that one of the two capacitances in the parallel branches (C P or C M ) is negative, which one will be negative depends on the order of extraction of the transmission zeros. If, for example, the previous cell was realizing a zero at infinity represented by a parallel capacitor, the value of C P would be negative. Otherwise, C M will be negative. As a consequence, one is first to create the complete schematic of the LC filter by summing the capacitances of the parallel branches of the succeeding cells. In other words, the gmc branch containing negative capacitance in Fig. 5.14 (gmcp or gmcs ) must be omitted while the remaining one must absorb the parallel negative capacitance before calculating the gmc value. The same is valid for the case of realization the complex zero which will be discussed later on. Next is the Brune cell (Fig. 5.15) which is used to realize a zero at the real axis of the complex frequency plane. In this case for the transconductances, we have:

Fig. 5.15 Two-phase Gm-C cell realizing the Brune cell of Fig. 5.8

102

5 Use of LC-to-Gm-C Transformation for Synthesis of Gm-C Single … 2 gmcL = ω0 C = ω0 L · gm

(5.12a)

gmC3 = ω0 C3

(5.12b)

gmn = ngm

(5.12c)

A two-phase cell realizing a complex transmission zero satisfying the condition of transformer-less realization is depicted in Fig. 5.16. In this case, for the transconductances, we have: 2 gmcL1 = ω0 C L1 = ω0 L 1 · gm

(5.12a)

2 gmcL2 = ω0 C L2 = ω0 L 2 · gm

(5.12b)

gmC1 = ω0 C1

(5.12c)

gmC2 = ω0 C2

(5.12d)

Fig. 5.16 Two-phase Gm-C cell realizing a complex transmission zero without a transformer as depicted in Fig. 5.9

5.6 Creation of the Two-Phase Cells

103

gmC3 = ω0 C3

(5.12e)

gmC4 = ω0 C4

(5.12f)

gmC5 = ω0 C5 .

(5.12g)

Figure 5.17 depicts the two-phase cell realizing a complex zero in an all-pass cell using a transformer. In this case, for the transconductances, we have: 2 gmcL1 = ω0 C L1 C = ω0 L 1 · gm

(5.12a)

2 gmcL2 = ω0 C L2 = ω0 L 2 · gm

(5.12b)

gmC1 = ω0 C1

(5.12c)

gmC2 = ω0 C2

(5.12d)

Fig. 5.17 Two-phase Gm-C cell realizing a complex transmission zero using a transformer as depicted in Fig. 5.6

104

5 Use of LC-to-Gm-C Transformation for Synthesis of Gm-C Single …

Fig. 5.18 Two-phase equivalent of a capacitor. a grounded and b floating

gmC3 = ω0 C3

(5.12e)

gmC4 = ω0 C4

(5.12f)

gmn = ngm .

(5.12g)

Finally, one is not to forget the capacitor as such, discussed in Fig. 3.10. Its twophase implementation is depicted in Fig. 5.18 (Fig. 3.10 repeated partly). Here, as usual, gmc = 2·π·f shift ·C, with f shift being the shifting frequency.

5.7 Design Example 3 Here, we will demonstrate the synthesis of a selective linear phase two-phase filter based on an LC prototype. It will be a ninth-order LSM filter with four transmission zeros at the positive imaginary axis the group delay of which is corrected by a fouthorder corrector. The maximum passband attenuation was chosen to be amax = 3 dB, the minimum stopband attenuation was amin = 40 dB, and the maximum passband group delay error was 1%. The cut-off frequency was set to f c = 1 MHz, while it was shifted for f shift = 2 MHz. To create this transfer function, the sequence of programs of the software for filter design was used.

5.7 Design Example 3

105

Fig. 5.19 SPICE simulation results of the second example

The so obtained transfer function was used for synthesis by the program. The following is the first ever presentation of the results obtained by this program. The frequency characteristic obtained by SPICE simulation is depicted in Fig. 5.19. Follows the edited .html report representing the transfer function and the structure of the filter.

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5 Use of LC-to-Gm-C Transformation for Synthesis of Gm-C Single …

Welcome to The Electronic Filter Design Software Program: POLY_GM_LC SYNTHESIS OF GM-C TWO-PHASE FILTERS BASED ON LOWPASS PASIVE LC PROTOTYPES Project name: GMC_poly_LSMZ_9_8_3_40_corr1 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Input data on the transfer function ---------------------------------------------------------Order of the numerator n=10; Order of the denominator m=13 Normalized zeros and poles of the transfer function. Zeros Poles Im{} Re{} Im{} Re{} 3.377829379e-1 ±6.262756554e-1 -6.613746484e-2 ±1.001314086e+0 3.617422307e-1 ±2.101758817e-1 -2.463787019e-1 ±1.002634972e+0 0.000000000e+0 0.00000000e+0 ±1.151553881e+0 -1.487930354e+0 0.00000000e+0 ±1.296819186e+0 -5.979308356e-1 ±9.556396147e-1 0.00000000e+0 ±1.797268335e+0 -1.162987338e+0 ±6.733625864e-1 -3.377829379e-1 ±6.262756554e-1 -3.617422307e-1 ±2.101758817e-1 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Ordered transmission zeros Zeros-real part: 9.999999990e+008 9.999999990e+009 0.000000000e+000 0.000000000e+000 0.000000000e+000 3.617422307e-001 3.377829379e-001 9.999999990e+008 Zeros-imaginary part: 9.999999990e+008 9.999999990e+009 1.797268335e+000 1.296819186e+000 1.151553881e+000 2.101758817e-001 6.262756554e-001 9.999999990e+008 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Zeros of the reflection coefficient Real part -6.343796533e-002 -6.343796533e-002 -3.617422307e-001 -3.617422307e-001 -5.968329237e-002 -5.968329237e-002 -1.556952137e-001 -1.556952137e-001 -9.123290083e-001 -9.123290083e-001 -3.377829379e-001 -3.377829379e-001

5.7 Design Example 3

0.000000000e+000 Imaginary part 2.213502439e-001 -2.213502439e-001 -2.101758817e-001 2.101758817e-001 8.666427193e-001 -8.666427193e-001 7.420305764e-001 -7.420305764e-001 7.721386299e-002 -7.721386299e-002 6.262756554e-001 -6.262756554e-001 0.000000000e+000 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++ All polynomials in ascending order of s or w**2 Coefficients of the reflection coefficient (numerator): 0.000000000e+000 1.708733759e-003 1.834695603e-002 1.258071139e-001 5.727261798e-001 1.832413042e+000 4.319902258e+000 7.904130329e+000 1.135585660e+001 1.294928699e+001 1.158855281e+001 7.833176545e+000 3.781341297e+000 1.000000000e+000 Coefficients of the reflection coefficient (denominator): 3.248448442e-001 2.911394127e+000 1.301853499e+001 3.781182331e+001 8.080792266e+001 1.333087934e+002 1.758972363e+002 1.877588614e+002 1.627670545e+002 1.134540915e+002 6.209546543e+001 2.542142404e+001 7.033849372e+000 1.000000000e+000 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++ INPUT IMPEDANCE Variant Zin=RS*(1-RO)/(1+RO) Numerator: -3.248448442e-001 -2.909685393e+000 -1.300018804e+001 -3.768601619e+001 -8.023519648e+001 -1.314763804e+002 -1.715773340e+002 -1.798547311e+002 -1.514111979e+002 -1.005048046e+002 -5.050691262e+001 -1.758824749e+001 -3.252508076e+000 Denominator: -3.248448442e-001 -2.913102861e+000 -1.303688195e+001 -3.793763042e+001 -8.138064884e+001 -1.351412065e+002 -1.802171385e+002 -1.956629918e+002 -1.741229111e+002 -1.264033785e+002 -7.368401824e+001 -3.325460058e+001 -1.081519067e+001 -2.000000000e+000 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++ EXTRACTION OF THE CELLS Capacitances in [F], inductances in [H], and transconductances in [S] Nominal read-in transconductance=1.00000e-005 S Read-in shifting frequency =2.00000e+006 Hz -----------------------------------------------------k=1 (th) ZERO AT INFINITY: parallel capacitance C=9.7865979e-011

107

108

5 Use of LC-to-Gm-C Transformation for Synthesis of Gm-C Single …

Coupling transconductance gmc=-1.229820159e-003 (According to Fig. 2.5.19). -----------------------------------------------------k=2 (th) Zero at infinity: SIMULATED series inductance. Original inductance =2.35576080e-004 Parallel capacitance after transformation C=2.3557608e-014 Coupling transconductance gmc=-2.960336326e-007 (According to Fig. 5.14). -----------------------------------------------------------------------k=3 (th) ZERO AT THE IMAGINARY AXIS=1.7972683e+000 Variant with a resonant circuit (tank=L:Cm) in the series branch, PI_cell Simulated floating inductance Cp=2.6411079e-010 Cm=3.1871095e-011 Cs=-2.8439241e-011 The capacitance (denoted L) within the simulated inductor is L*gm*gm=2.4604663e-014 gmcm=-4.005039968e-004 gmcs=3.573780432e-004 gmcp=-3.318914079e003 gmcL=-3.091913184e-007 (According to Fig. 5.15). -----------------------------------------------------------------------k=4 (th) ZERO AT THE IMAGINARY AXIS=1.2968192e+000 Variant with a resonant circuit (tank=L:Cm) in the series branch, PI_cell Simulated floating inductance Cp=3.3500933e-010 Cm=3.0902459e-010 Cs=-1.6074638e-010 The capacitance (denoted L) within the simulated inductor is L*gm*gm=4.8740320e-015 gmcm=-3.883317479e-003 gmcs=2.019998579e-003 gmcp=-4.209851458e003 gmcL=-6.124889236e-008 (According to Fig. 5.15). -----------------------------------------------------------------------k=5 (th) ZERO AT THE IMAGINARY AXIS=1.1515539e+000 Variant with a resonant circuit (tank=L:Cm) in the series branch, PI_cell Simulated floating inductance Cp=-8.0704042e-011 Cm=7.6417313e-010 Cs=9.0233577e-011 The capacitance (denoted L) within the simulated inductor is L*gm*gm=2.4996545e-015 gmcm=-9.602882716e-003 gmcs=-1.133908564e-003 gmcp=1.014156899e003 gmcL=-3.141158421e-008 (According to Fig. 5.15). -----------------------------------------------------k=6 (th) A PAIR OF COMPLEX ZEROS (all_pass) s=3.6174223e-001 w=2.1017588e-001 Variant with one transformer. Simulated L1 (=1.315720105e-003 H) by capacitance CL1=1.3157201e-013 F C4=1.0999196e-010 C3=3.2893003e-010 Simulated L2 (=4.399678273e-004 H) by capacitance CL2=4.3996783e-014 F C1=-2.9173226e-011 C2=3.9703918e-011 gm0=1.0000000e-005 gmn=-1.0000000e-005 Coupling transconductances: gmcc1=-4.989341518e-004, gmcc2=3.666015756e-004, gmcc3=-4.133456615e-003, gmcc4=-1.382199693e-003

5.7 Design Example 3

109

gmcl1=-1.653382645e-006, gmcl2=-5.528798770e-007 (According to Fig. 5.18). -----------------------------------------------------k=7 (th) A PAIR OF COMPLEX ZEROS (all_pass) s=3.3778294e-001 w=6.2627566e-001 Variant with one transformer. Simulated L1 (=4.247115068e-004 H) by capacitance CL1=4.2471151e-014 F C4=1.1779380e-010 C3=1.0617788e-010 Simulated L2 (=4.711751995e-004 H) by capacitance CL2=4.7117520e-014 F C1=-2.3382287e-011 C2=2.9173227e-011 gm0=1.0000000e-005 gmn=-1.0000000e-005 Coupling transconductances: gmcc1=-3.666015760e-004, gmcc2=2.938304853e-004, gmcc3=-1.334270550e-003, gmcc4=-1.480240546e-003, gmcl1=-5.337082192e-007, gmcl2=-5.920962175e-007 (According to Fig. 5.18). -----------------------------------------------------k=8 (th) ZERO AT INFINITY: parallel capacitance C=2.3382287e-011 Coupling transconductance gmc=-2.938304853e-004 (According to Fig. 5.19). ------------------Residual load impedance--------------------Residual is resistor R=1.0000000e+003 ========================================================= Here ends the synthesis process

We used the opportunity to demonstrate the versatility of the program and made another variant of this solution. It is shifted for f shift = 1 MHz. The resulting frequency domain characteristic as obtained by SPICE simulation is depicted in Fig. 5.20.

Fig. 5.20 Alternative shifting with respect to Fig. 5.19. Here, f shift = 1 MHz

110

5 Use of LC-to-Gm-C Transformation for Synthesis of Gm-C Single …

References 1. Litovski V (2019) Electronic filters, theory, numerical recipes, and design practice based on the RM Software. Springer Science+Business Media 2. Zverev AI (2005) Handbook of filter synthesis. Wiley-Interscience, New York 3. Tsividis YP, Voorman JO (1993) Integrated continuous-time filters: principles, design, and applications. IEEE, Piscataway, NJ 4. Lawanwisut S, Siripruchyanun M (2012) An electronically controllable active-only currentmode floating inductance simulator. In: Proceeding of the 35th international conference on telecommunications and signal processing (TSP), Prague, Czech Republic, pp 386–389 5. Tellegen BDH (1948) The gyrator, a new electric network element. Philips Research Rep 3:81– 101 6. Uzunov IS (2008) Theoretical model of ungrounded inductance realized with two gyrators. IEEE Trans Circ Syst-II: Express Briefs 55(10):981–985

Chapter 6

Synthesis of Analog Gm-C Hilbert Transformer and Its Implementation for Band-Pass Filter Design

6.1 Introduction The issue of design and implementation of linear-phase selective band-pass filters was considered for a long time now [1, 2]. Application of such filters is sought in several telecommunications and measurement applications where new generation broadband communication receivers require several to tens of MHz bandwidth in conjunction with very good noise and distortion performance. Filters with linear-phase responses, that is, constant group delay responses are needed in many applications for signal processing, image processing, and waveform transmission. Linear-phase selective band-pass filters are ideal for a wide range of applications such as antialiasing filters for high-resolution A/D converters, reconstruction filters for D/A converters in wireless communication receivers and transmitters, industrial and medical signal processing of optical and image processing filters, instrumentation and testing, RFID demodulation baseband filters, and many types of filtering in signal processing applications. The impulse radio system, for example, transmits modulated pulses having very short time duration. Information can be extracted in the receiver side based on the cross-correlation between received and reference pulses. Accordingly, the pulse distortion due to in-band group delay variation can cause serious degradation in system performance [3]. In other words, the overall system performance of an impulse radio can be degraded by pulse distortion caused by in-band group delay variation in antennas and filters. In the microwave communication systems, the band-pass filters help to avoid the inter-symbol interference caused by the signal distortion thanks to the flat magnitude response but also the linear phase. Designing selective linear-phase band-pass filters is a difficult task since the lowpass-to-band-pass transformation distorts the group delay [4]. To avoid that, one may proceed in several ways. The simplest one is described in [4] and consists of three steps. First one is to produce a selective low-pass filter. Then, a low-pass-to-band-pass transformation is © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 V. Litovski, Gm-C Filter Synthesis for Modern RF Systems, Lecture Notes in Electrical Engineering 807, https://doi.org/10.1007/978-981-16-6561-5_6

111

112

6 Synthesis of Analog Gm-C Hilbert Transformer and Its …

to be applied to get a selective band-pass filter [5]. As a consequence of the nature of the transformation, the resulting filter is geometrically symmetrical. Due to the arithmetical asymmetry of the original amplitude characteristic, the group delay suffers of asymmetry and strong distortions, too. Having such a selective band-pass filter, a phase corrector is synthesized to get the final result. Alternatively, one may use low-pass prototypes which are kind of compromise between selectivity and linear phase and transform them into band-pass [6–8]. That approach has limited scope since selectivity is sacrificed at the very beginning, and no zeros at the imaginary axis (to compensate for the loss of selectivity) are seen as possible. Finally, procedures were reported for direct (passive LC) synthesis of selective arithmetically symmetrical band-pass filters [9–13] to which correctors may be added. Successful solutions of arithmetically symmetrical band-pass filters with nonlinear-phase characteristics were offered [14, 15] based on implementation of polyphase (complex) paradigm. To the results of the implementation of these ideas for synthesis of a Gm-C arithmetically symmetrical linear-phase selective band-pass filters, this chapter is devoted. Real arithmetical symmetry in designing linear-phase band-pass filters was reported in [4]. These were produced by an algorithm optimizing simultaneously the amplitude and the phase. Results of such a synthesis are given as example in Chap. 3 in Fig. 3.8. The problem here is the limited selectivity due to the nature of the transfer function which in fact represents a linear-phase band-pass filter with partly corrected amplitude characteristic by introduction of zeros at the imaginary axis. Arithmetically symmetrical band-pass filters (highly selective and with linear phase) were already reported in this book. One should examine paragraph “4.7 Design example, A band-pass cascaded Gm-C filter” which, despite the fact it was intended to demonstrate the Gm-C synthesis method, illustrates the fundamental idea. This means that by using a two-phase technology one can shift a well-honed low-pass filter (e.g., [16–20]) and preserve all its properties which in this case are the linear phase and the selectivity. The only difference between the example given in Paragraph 4.7 (and potentially similar solutions) and the target filter to be designed here is in that the example is excited by two signals (I and Q) which are mutually shifted by 90°. What we need, however, is a filter which should be excited by a single signal. A circuit that is excited by a single signal at its input and produces two outputs which are mutually shifted by 90° is called Hilbert transformer and to its synthesis and application is mainly devoted this chapter. The analog Hilbert transformer is an electrical or electronic system that, given a sinusoidal input signal, produces two sinusoidal output signals mutually shifted in phase by π/2 radians. Such a system is depicted in Fig. 6.1. It is understood that analog solution to the Hilbert transform in which the phase shift is valid from the zeroth frequency is not possible. Consequently, band-pass solutions were sought [21, 22]. In these references, the physical realization was of main interest while living the transfer function synthesis methods hidden and

6.1 Introduction

113

Fig. 6.1 General structure of Hilbert transformer

making it very difficult or almost impossible to reproduce. We will partially follow the concept realized in [23] where synthesis of band-pass all-pass filters approximation of constant phase difference in equi-ripple manner was discussed. To start with one is to observe Fig. 6.2 where the shape of the “target” phase difference function is depicted. It is a phase difference of two all-pass filters of order 6 and 8, respectively. Do have in mind the maximally flat approach we are applying. To get this type of phase difference, one needs the phases to be linear (or, in general, the same shape) and properly shifted to each other. That is depicted in Fig. 6.3.

Fig. 6.2 Phase difference curve exhibiting constant value around a given frequency (here 100 kHz)

Fig. 6.3 Phase characteristics of two filters approximating linear phase around a given frequency (here 100 kHz)

114

6 Synthesis of Analog Gm-C Hilbert Transformer and Its …

Fig. 6.4 Group characteristics of two filters approximating linear phase having the same slope around a given frequency (here 100 kHz)

Now, Fig. 6.4 depicts the group delay characteristics of these two filters. Since the slopes of the phases are equal so are the values of the group delays. Difference of the group delay values would lead to different slopes, and consequently, no plateau in the curve of phase difference would be possible. That means that no two filters of the same order may produce a phase difference as depicted in Fig. 6.2. The larger the difference in the orders of the filters the larger the value of the phase difference may be obtained and vice-versa. Since, however, we are after π/2, in the example depicted above, the orders of the filters differ by two only. As for the interval of approximation of constant phase difference, it is defined by the filter of lower order. In the next, we will address the synthesis of band-pass all-pass filters approximating constant group delay in maximally flat manner [24] which is necessary for both branches of Fig. 6.1. This issue was fully elaborated in [4]. In the next, only the main equations will be given, and the reader is advised to consult [4]. The group delay of an nth order all-pass network whose RHP transmission zeros in the first quadrant are z k = σk + jωk , k = 1,2, …, n/2 may be calculated as τd (ω) =

n/2 

Tk (ω)

(6.1)

k=1

width Tk (ω) = r (ω) + q(ω) =

2 · σk 2 · σk + 2 . σk2 + (ω − ωk )2 σk + (ω + ωk )2

(6.2)

To get a maximally flat approximation of the group delay around the central frequency ω0 , with τd (ω0 ) = τ0 , where τ0 is a positive constant, one has to solve the following system of nonlinear equation with respect to the coordinates of zk F1 (σ, ω) = τd (ω0 ) − τ0 = 0

(6.3)

6.1 Introduction

115

Fi (σ, ω) = d i−1 τd /(dω)i−1 |(ω=ω0 ) = 0 for i = 2, . . . , n.

(6.4)

where σ and ω are vectors of the unknown coordinates of the zeros z k = σk + jωk . Implementing Newton–Raphson linearization leads to the following system of linear equations Fi +

n/2   ∂ Fi j=1

∂σ j

dσ j +

∂ Fi dω j ∂ω j

 = 0 for i = 1, . . . , n.

(6.5)

It should be solved iteratively. A strongly damped iterative process was implemented in order for the solution to converge for delays (τ0 ) of any value and for filters of any order [25]. Initial solutions for the iterative process are suggested in [4].

6.2 The Algorithm Having the procedures for synthesizing linear-phase band-pass all-pass filters, the algorithm depicted in Fig. 6.5 may be exercised to get the plateau of Fig. 6.2 at the level of π /2. In this procedure, the main loop is designed to converge toward d τ = π /2. One is supposed to set the final accuracy of the solution by giving ε a value (say ε = 10–5 ). Of course, to protect against divergence, the number of iterations in the main loop (mainiter) is limited by maxiter (say maxiter = 50). When the current value of d τ is Fig. 6.5 Algorithm’s pseudocode

116

6 Synthesis of Analog Gm-C Hilbert Transformer and Its …

Fig. 6.6 Result of implementation of the optimization algorithm

found, it is checked for its sign. If positive, the value of τ0 is incremented by properly chosen τ0 , and the loop cycle is completed. If d τ < 0, bisection is implemented meaning the τ 0 is halved and the value of τ 0 is reduced by τ 0 . Figure 6.6. represents a final result obtained after implementation of the above algorithm for n = 8. In this case, f 0 = ω0 /(2π ) = 100 kHz.

6.3 Physical Implementation Fact is that the Hilbert transform we are advertising here constitutes of two separate transfer functions. That is why, within the RM software for filter design, it may be synthesized in any of the following technologies: passive LC, active RC (cascaded and parallel), active SC, and active Gm-C (cascaded or parallel). Since, however, the parallel version of the Gm-C technology is the simplest to implement, in the next, we will suppose that the Hilbert transformer will be realized as such, and the theory of parallel Gm-C synthesis will be very shortly repeated below. The transfer function obtained by the approximation process (or by reading a catalogue, e.g., [26]) may be written in factored form as m (s − z i ) Ha (s) = A0 · H (s) = A0 · ni=1 i=1 (s − pi )

(6.6)

Here, A0 is properly chosen in order to get unity nominal gain. p and z are vectors of transfer function poles and zeros, respectively, while s is the complex frequency. n is the order of the filter, and m is the number of finite transmission zeros (order of the numerator). For the sake of simplicity, in the case of n- and m-even and m < n, (6.1) may be rewritten as

6.3 Physical Implementation

117

H (s) = A0 · n/2

1

i=m/2+1

Di

·

m/2  Ni D i i=1

(6.7)

where Di = (s − pi )(s − pi ∗) represents a factor of the denominator related to a pair of conjugate poles while Ni = (s − z i )(s − z i ∗) is the same for the numerator. Both Di and N i are polynomials with real coefficients which are crucial for circuit synthesis. Having a decomposed transfer function into partial fractions, we implement exactly the same procedure as described in Chap. 3. That stands for the Hilbert transformer and for the subsequent band-pass filter. To allow the reader to keep track on the physical synthesis of the Hilbert transformer and to associate the element values given in the succeeding example, we will here repeat the schematics of a parallel Gm-C filters and the cells which are constituting the whole. The overall structure of a GM-C filter realized as a parallel connection of secondorder cells is depicted in Fig. 6.7. One is to have in mind that all second-order cells have the same structure. After adopting the concept of cell synthesis advised in [27], the following set of cells was created. The schematic of the second-order cell is depicted in Fig. 6.8. Its general transfer function obtained after decomposition may be written as. He (s) =

g·s+q . s2 + a · s + b

(6.8a)

Now, after circuit analysis of the schematic depicted in Fig. 6.8, one gets He (s) =

Fig. 6.7 Structure of the filter

gm3 ·gm1 C·C3 ·gm4 + gm3 C·C3

m3 − x·g ·s+ C3

s2 +

gm2 C

·s

.

(6.8b)

118

6 Synthesis of Analog Gm-C Hilbert Transformer and Its …

Fig. 6.8 Second-order Gm-C cell used within the RM software. Note: 0 ≤ x ≤ 1. “ni” stands for “input node”, while “no” for the “output node” ·gm1 ·gm4 m3 The design equations are now g = − x·g , q = gm3 , a = gCm2 and b = gm3 . C3 C·C3 C·C3 Note, by changing the sign of gm1 , we in fact transpose the transmission zero (being on the real axis) from one to the other side of the origin. Since there are more circuit elements than degrees of design freedom [expressed by the number of coefficients in (6.8a)], some of the element values are to be adopted as design constants. Speaking in normalized element values, we first adopt gm2 = gm3 = gm = 1 S. Then, we adopt x = 0.5. With that set, one may calculate C = 1/a, b·x , and gm1 = − q·x . C 3 = -x/g, gm4 = − a·g a·g Now, if (g > 0 and q > 0), we choose a negative sign to the summing transconductor following the output of the proper cell. Set g = − g and q = − q; if (g > 0 and q < 0), we use negative sign to the summing transconductor following the output of the proper cell. Set g=− g and q=− q; if (g < 0 and q > 0), we proceed without intervention; if (g < 0 and q < 0) without intervention. The first-order cell is a bilinear circuit as depicted in Fig. 6.9. The transfer function of the circuit depicted in Fig. 6.6 may be expressed in the form

Ho = g/(s + a),

Fig. 6.9 First-order Gm-C filter and node notation

(6.9a)

6.3 Physical Implementation

119

Here, related to (12), g = Go is the residue and a = ao the real pole. From circuit analysis, Ho =

gm1 /C . s + gm2 /C

(6.9b)

Now, by adopting C = 1F, for the normalized transconductances, we have gm1 = g and gm2 = a. Since there are no restrictions on the value of C, it may be set to be equal to the capacitances used within the second-order cells. The outputs of all cells connected in parallel are driving the summing amplifier as depicted in Fig. 6.10.

Fig. 6.10 General structure of the summing subsystem

120

6 Synthesis of Analog Gm-C Hilbert Transformer and Its …

6.4 The Hilbert Program The Hilbert program of the RM software for filter design reads in the order of the larger filter, n, and the central frequency of the band-pass, f 0 . It performs the iterative procedure described above as “the algorithm”. This program produces two main results: the poles and zeros of the two all-pass filters and the final value of the group delay needed for the Hilbert transformer to be established. Accordingly, in order to get the schematic of the all-pass networks for a proper technology, the user is advised to perform the circuit synthesis procedure twice (for n and for n-2) with the same target group delay value using the program b0010 . If one needs a single SPICE description of the Hilbert transformer, one is expected to concatenate the two files obtained by system synthesis. Below, within the illustrative example that action was taken and the outputs obtained by simulation of the Hilbert transformer are denoted out_i and out_q.

6.5 Illustrative Example In this example, an eighth-order Hilbert transformer will be described. The central frequency selected was 100 kHz. The following is partly edited.txt report produced by the Hilbert program.

Welcome to The Electronic Filter Design Software HILBERT, Program for band-pass all-pass synthesis of two all-pass filters having output phases shifted by pi/2 using maximally flat approximation ======================================================== Project name: AP_0_6_1z5em5 ======================================================== Initial position of the RHP zeros: s[1]=1.780059089e-306 w[1]=1.012799782e-263 s[2]=1.403352208e-308 w[2]=7.991476891e-307 s[3]=4.142135784e-311 w[3]=1.113314951e-312 s[4]=9.242304572e-222 w[4]=1.781020606e-306 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++ SOLUTION

6.5 Illustrative Example

121

Order of the filters n_high=8 n_low=6 ======================================================== The results about the poles and zeros listed below are valid for frequency normalization with ωc=2·π·fc=6.28318530e+005 rad/s. Normalized poles (sign minus by the real part) and zeros (sign plus by the real part) of the Hilbert transformer Eighth order all-pass Sixth order all-pass Real part Imaginary part Real part Imaginary part ±.6081121269 ±.3898149221 ±.42308271 ±.6167146133 ±.7123040128 ±.8425676708 ±.511866084 ±1.010069716 ±.6725916124 ±1.237025311 ±.3972866106 ±1.387501112 ±.4732599575 ±1.629716642 ======================================================== FREQUENCY RESPONSE --------------------------------------------------------------Normalized group delay(w0) =9.75878e+000 --------------------------------------------------------------Frequency [Hz] group delay H [s] group delay L [s] Phase H [rad] Phase L

Phase_diff

3.12501e+003 1.43603e-005

8.59100e-006

-2.81842e-001 -1.68420e-001

-1.13423e-001

3.75002e+003 1.43645e-005

8.59991e-006

-3.38244e-001 -2.02174e-001

-1.36070e-001

--------------------------------------------------------2.50066e+006 2.52336e-008

1.36287e-008

-2.47376e+001 -1.86361e+001

-6.10146e+000

2.50129e+006 2.52209e-008

1.36219e-008

-2.47377e+001 -1.86362e+001

-6.10151e+000

2.50191e+006 2.52082e-008

1.36150e-008

-2.47378e+001 -1.86362e+001

-6.10155e+000

2.50254e+006 2.51956e-008

1.36082e-008

-2.47379e+001 -1.86363e+001

-6.10160e+000

======================================================== The program HILBERT successfully ends

We will give now a set of figures representing the frequency response of the newly synthesized Hilbert transformer. Figures 6.11 and 6.12 are obtained by processing the Hilbert’s program.csv output file (for n = 6 and n = 8) which contains only the responses of the filters. Figure 6.11 represents the phase characteristics obtained by merging the diagrams using MS Excel. The difference representing the “response” of the Hilbert transformer was created by subtraction within Excel. Figure 6.12 was obtained in similar manner. Here, in place of the phases, one was to depict the group delays on a single diagram. One may observe that in this case the approximation interval is about 50% of the central frequency which is acceptable even for ultra-wide-band systems. In other words, this solution will cover not only

122

6 Synthesis of Analog Gm-C Hilbert Transformer and Its …

Fig. 6.11 Phase responses of the two branches of the eighth-order Hilbert transformer

Fig. 6.12 Group delay responses of the two branches of the eighth-order Hilbert transformer

the passband of the potentially succeeding band-pass filter but also a large part of the skirt region, too. One is not to forget that the diagrams depicted in Figs. 6.11 and 6.12 are obtained from transfer function analysis with denormalization so that the (normalized) central frequency was shifted from f 0 = 1 Hz to f 0 = 100 kHz. Any shifting is allowed meaning that this solution may be used for very low and very high frequencies. The difference will be noticed when implementation technology comes in fore. If renormalization to a central frequency of f 0 = 1 MHz and Gm-C technology is implemented with parallel realization one gets circuits the phase responses of which are depicted in Fig. 6.13a. SPICE simulation was used. Figure 6.13b depicts the group delay for the same technology. Before proceeding, we will deliver the element values of the Hilbert filter just described (The 1 MHz case.). These are given in the following tables. Table 6.1 contains the element values of the cells (Fig. 6.8) of the sixth-order filter, while Table 6.2 does the same for the eighth-order filter. The tables contain a row with gsum which, according to Fig. 6.10, belongs to the summing amplifier.

6.5 Illustrative Example

123

Fig. 6.13 a Phase responses of the I and Q channel. b Group delay responses of the I and Q channel (I, the broader, is eighth order and Q, the narrower, is sixth order) Table 6.1 Element values of the sixth-order filter Cell No. 1

Cell No. 2

Cell No. 3

gm1

− 9.103828946e–006

3.067003563e–005

− 7.967382723e–008

gm2

1.000000000e–005

1.000000000e–005

1.000000000e–005

gm3

1.000000000e–005

1.000000000e–005

1.000000000e–005

gm4

7.639697416e–007

2.578688199e–006

3.491344038e–006

gsum

1.000000000e–005

1.000000000e–005

− 1.000000000e–005

C1

1.870593778e–013

1.546970218e–013

1.991743639e–013

C2

1.683534400e–012

1.392273196e–012

1.792569275e–012

C3

1.850352189e–013

3.286042808e–013

2.124334329e–013

124

6 Synthesis of Analog Gm-C Hilbert Transformer and Its …

Table 6.2 Element values of the eighth-order filter Cell No. 1

Cell No. 2

Cell No. 3

Cell No. 4

gm1

− 4.763555215e–6

6.659970742e–5

− 5.962001304e–7

− 1.571510701e–5

gm2

1.000000000e–5

1.000000000e–5

1.000000000e–5

1.000000000e–5

gm3

1.000000000e–5

1.000000000e–5

1.000000000e–5

1.000000000e–5

gm4

5.612103582e–8

7.489434850e–7

1.603217158e–7

2.024198281e–6

gsum

1.000000000e–5

1.000000000e–5

1.000000000e–5

1.000000000e–5

C1

1.30265592e–13

1.11366754e–13

1.17961906e–13

1.67614821e–13

C2

1.17239033e–12

1.00230078e–12

1.06165715e–12

1.50853339e–12

C3

2.08180037e–14

1.39590506e–13

1.73224119e–14

1.05926717e–13

Table 6.3 Group delay of the Hilbert transformer for different orders of the larger order filter Order (n)

4

6

8

10

12

Normalized group delay (s)

4.32715

7.04542

9.75878

12.4679

15.1735

As already mentioned, for a given order of the larger filter, there is only one value of the normalized group delay which will lead to a correct Hilbert transformer with maximally flat group delay response. These values are listed in Table 6.3 which contains the final values (obtained after iterations) of the group delay (or phase slope) for a set of filter orders. Note almost constant increments between columns. Again, these are normalized values, and the real value will be created by choosing the corresponding central frequency of the band-pass approximant.

6.6 On the Design of Arithmetically Symmetrical Wideband Selective Linear-Phase Band-Pass Gm-C Filters Having the Hilbert transformer and a two-phase filter synthesis program, one may proceed to a synthesis of a linear-phase arithmetically symmetrical selective bandpass filter. The structure of such a filter is depicted in Fig. 6.14. In the first stage, the Hilbert transformer is producing two signals shifted in phase for π /2. These are used to excite a two-phase (complex) filter performing shifting of the frequency response of a low-pass filter for a desired value of frequency. In that way, a band-pass filter with fully preserved shape of the frequency response of the low-pass prototype is obtained.

6.6 On the Design of Arithmetically Symmetrical Wideband Selective …

125

Fig. 6.14 General structure of the arithmetically symmetrical band-pass filter

No additional theoretical and practical information is needed. One is to implement the steps needed for synthesis of the Hilbert transformer and the steps for synthesis of the two-phase linear-phase arithmetically symmetrical band-pass filter as described in this book and in [4]. No limitations on the relative passband width are seen so allowing for synthesis of linear-phase selective broad- and narrow-band band-pass filters being, as it is well known, the ultimate challenge in filter synthesis. It is in the modern vocabulary to use the term UWB filters meaning ultra-wide band filters. These are part of the story here being not a special case of any kind.

6.7 Design Example The design process starts with the synthesis of the transfer function of the low-pass prototype. To that end, we use the LSM_Z filter [4] of fifth order with two transmission zeros on the ω-axis. The passband and stopband attenuations are amax = 3 dB and amin = 40 dB, respectively. Its cut-off frequency was set to 50 kHz. Its group delay was corrected with a second-order corrector [4] with maximal error of δ ± 1%. Table 6.4 contains the normalized values of the zeros and poles of the prototype filter. The complex filter was obtained by shifting its characteristic by 1 MHz. In that way, the relative passband width becomes Br = 10%. Table 6.4 Zeros and poles of the low-pass prototype Zeros Real part

Imaginary part

6.464310892e–001

± 3.982226727e–001

0.000000000e+000

± 1.862937662e+000

0.000000000e+000

± 3.160856024e+000

Poles − 2.975215539e–001

± 9.713175538e–001

− 1.786501773e+000

0.000000000e+000

− 1.167143041e+000

± 6.661311913e–001

− 6.464310892e–001

± 3.982226727e–001

126

6 Synthesis of Analog Gm-C Hilbert Transformer and Its …

Table 6.5 Element values of the complex filter (gm0 = 10 μS) CellNo gm1 (μS)

gm2 = gm3 (μS)

gm4 (μS)

C1 = C2 (pF)

C3 (pF)

gmc1 = gmc2 (μS)

38.5105

26.7468

70.6818

168.055 444.10708

− 10

42.8396

− 10

1

− 57.9336

0.1

2

− 6.35066

0.1

0.421067

3

− 2.26398

0.1

0.097964 12.31027

4

134.409 17.865

6.81814

31.83098

1.73240

0.699365 77.3478 200

gmc3 (μS)

10.885002

gms (μS)

4.3942412 10 10

Table 6.5 contains the values of the transconductances and the capacitances of the resulting complex filter realized using the POLY_Parallel program of the RM software for filter design. The values correspond to the cell depicted in Fig. 6.8. Figure 6.15 depicts the amplitude and group delay characteristic of the overall filter, Hilbert transformer (of n = 8) plus the complex filter as shown in Fig. 6.14, obtained by SPICE simulation (the SPICE file being produced automatically as part of the synthesis report). Having in mind the limitted frequancy interval in which the Hilbert transfromer exhibits phase difference of π/2 we investigated a “worst case” situation which woud be a quadrupled passband-width of the filter (new relative passband width of 40%) in conjunction with a minimal order of the Hilbert transformer, i.e., the order of the larger all-pass filter is n = 4. It came out that even in such a case (as can be seen from Fig. 6.16) the final result is only partly damaged in the deep stopband.

Fig. 6.15 Amplitude and group delay characteristic of the arithmetically symmetrical linear-phase selective band-pass filter synthesized as two-phase Gm-C structure

6.7 Design Example

127

Fig. 6.16 Solution with minimal order of the Hilbert transformer and quadrupled the passband width of the selective part of the system

To finalize, we will use the opportunity to study the properties of the resulting band-pass filter (with a Hilbert transformer with n = 8) in the time domain. Two situations will be considered. In the first one, we will bring to the input of the filter (as shown in Fig. 6.14) a sinusoidal signal of frequency f 0 = 1 MHz whose amplitude is modulated by a square pulse. The duration of the pulse was 60 μs. The resulting response obtained by SPICE simulation is depicted in Fig. 6.17. As one can see it corresponds to a response of a system having right-half plane zeros (nonminimum phase system). By studying this response, one may find modest values of both the overshoot is γ = 5% and the undershoot is β = 4%. Fast settling may be observed, too. The second situation is related to the effect of selection of a band-pass filter. Namely, a complex input signal consisting of a sum of three sinusoidal input signals of equal amplitudes will be brought to the input. The frequencies of the components will be the central frequency (f 0 = 1 MHz), the frequency of the first upper

Fig. 6.17 Time domain response of the band-pass filter to an amplitude modulated sinusoid

128

6 Synthesis of Analog Gm-C Hilbert Transformer and Its …

Fig. 6.18 Input and output waveforms of the band-pass filter

transmission zero (f u = 1.0931468831 MHz), and the frequency of the first bottom transmission zero (f b = 0.9068531169 MHz). The resulting input (v(1)) and output (v(5)) waveforms are depicted in Fig. 6.18. As one can see only the signal having the central frequency is transmitted, i.e., after steady state is reached, the output signal is monochromatic at the central frequency.

References 1. Lindalh CE (1961) A linear phase band-pass filter. Tech Memo No. 82 (3697-1-T). The University of Michigan Research Institute Ann Arbor 2. Wellekens CJ (1977) Filters d’amplitude et de phase spécifiées. L’onde Electrique 57(1):59–63 3. Myoung SS, Kwon BS, Kim YH, Yook JG (2007) Effect of group delay in RF BPF on impulse radio systems. IEICE Trans Commun 90(12):3514–3522 4. Litovski VB (2019) Electronic filters, theory, numerical receipts, and design practice based on the RM software. Springer 5. Temes GC, Mitra SK (eds) (1973) Modern filter theory and design. Wiley, New York 6. Sadughi S, Kim HK (1984) A new design for selective linear phase bandpass filters with arithmetical symmetry. I. J. Circuit Theory and applications, Letters to the Editor 7. Skwirzynsky JK, Zdunek J (1960) Design of networks with prescribed delay and amplitude characteristics. The Marconi Rev 115–139 8. Kwan K, Bach GC (1969) Simultaneous approximation in filter design. IEEE Trans Circ Theor CT-15:117–121 9. Watanabe H (1961) Approximation theory for filter-networks. IRE Trans Circ Theor 8(3):341– 356 10. Watanabe H (1958) Synthesis of band-pass ladder network. IRE Trans Circ Theor 5(4):256–264 11. Yu KC (1966) A study of arithmetically symmetrical bandpass filters. Master’s Report, University of Kansas, Manhattan, Kansas 12. Szentirmai G (1963) The design of arithmetically symmetrical band-pass filters. IEEE Trans Circ Theor 10(3):367–375 13. Szentirmai G (1964) A group of arithmetically symmetrical band-pass filter functions. IEEE Trans Circ Theor 11(1):109–118

References

129

14. Chou C-Y, Wu C-Y (2000) The design of wideband and low-power CMOS active polyphase filter and its application in RF double-quadrature receivers. IEEE Trans Circ Syst I: Regular Papers 52(5):825–833 15. Cuypers C et al (2002) General synthesis of complex analogue filters. In: 9th International conference on electronics, circuits and systems, vol 1, pp 153–156, Dubrovnik, Croatia 16. Christian S (1973) Beiträge zur Netzwerksynthese mit Hilfe von Cauerschen und verallgemeinerten q-Funktionen 17. Gutsche H (1973) Approximation of transfer functions for filters with equalized group delay characteristics. Siemens Forschung-und Entwicklung-Berichte 2(5):288–292 18. Hibino M, Ishizaki Y, Watanabe H (1968) Design of Chebyshev filters with flat group-delay characteristics. IEEE Trans Circ Theor 15(4):316–325 19. Litovski V (1979) Synthesis of monotonic passband sharp cutoff filters with constant group delay response. IEEE Trans Circ Syst CAS-26(8):597–602 20. Litovski V, Milovanovi´c D (1983) New solution for the ideal filter approximation problem. IEE Proceedings, Pt. G. 130(4):161–163 21. Saul PH (2004) Low power analogue 90° phase shifter. In: DATE ’04: Proceeding of the Conference on design, automation and test in europe, Paris, France, vol 3, pp 28–33 22. Hutchins B (2020) The design of wideband analog 90° phase differencing networks without a large spread of capacitor values. http://electronotes.netfirms.com/EN168-90degreePDN.PDF 23. Wiebach W (1973) Design and analysis of 90° phase difference networks. Distributed by the technical Information Service, U. S. Department of Commerce. https://apps.dtic.mil/dtic/tr/ful ltext/u2/769585.pdf 24. Gregorian R, Temes GG (1978) Design techniques for digital and analog all-pass circuits. IEEE Trans Circ Syst CAS-25(12):981–988 25. Litovski V, Zwolinski M (1997) VLSI circuits simulation and optimization. Chapman and Hall, London, UK 26. Zverev AI (2005) Handbook of filter synthesis. Wiley-Interscience, New York 27. Schaumann R, Van Valkenboug E (2001) Design of analog filters. Oxford University Press, New York

Chapter 7

Implementation Issues

7.1 Introduction Given the synthesis results with ideal elements and with so many significant figures, a question arises as to what will happen if real elements are implemented with limited accuracy. In addition, we are interested in the influence of the variation of the element values for reasons of accuracy of implementation, aging, or influence of the environment, e.g., temperature. In our case here, we have additional questions to answer related to topologies in use. Since we have parallel, cascade, and LC originating synthesis of Gm-C and two-phase Gm-C filters, one should wander which one is the best and which are the advantages and disadvantages of each. In addition, since these filters enter the chain of signal transmission, the noise issues have to be addressed. In other words, we would like to know which of the discussed configuration will produce the lowest signal-to-noise ratio. So, we will try in this chapter to find answers to these questions based on theoretical analysis, synthesis, and simulation under different conditions. To get the answers, a benchmark transfer function is to be selected which will be capable to expose the weaknesses of any topology. One of the criteria used for price estimation of an electronic circuit is the spread of the element values with an intention for the element variation to be as small as possible. For the examples under consideration, we will give a table containing the average values and ratios of the maximum and minimum value for capacitors and transconductors.

7.2 Study of the Worst-Case Tolerance of Gm-C Filters Two example-transfer functions will be considered in the next. These where synthesoftwarefor filter design [1]. Table 7.1 depicts the main information sized by the used for synthesis. To perform worst-case analysis [2–4], Gm-C circuit synthesis © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 V. Litovski, Gm-C Filter Synthesis for Modern RF Systems, Lecture Notes in Electrical Engineering 807, https://doi.org/10.1007/978-981-16-6561-5_7

131

132

7 Implementation Issues

Table 7.1 Enumeration of the example filters Example Filter no

Type

n

m

amax [dB]

amin [dB]

1

LSM_Z

9

4

3

30

2

LSM_Z

8

4

3

60

Table 7.2 Variations of the gain characteristic for the filter no. 1 (gm0 = 10 μS) Tol = 1%

a0 [dB]

Configuration

Par

Cas

f c [kHz]

gm1 varied

N

0.437

0.175

N

−5

gm4 varied

N

− 0.363

− 0.26

− 3.5

N

C 1 varied

N

N

N

2

N

4

LC

Par

Cas

amax [dB] LC

Par

Cas

LC

−8

5.15

N

N

1

−2

0.65

N

4.4

N

N

was performed for all three topologies described in the previous chapters: parallel, softcascade, and LC originated. Having the SPICE descriptions created by the ware, MonteCarlo simulation (50 samples) was performed with a single parameter considered variable and having maximum tolerance of 1%. Then, the worst case was extracted by SPICE. The frequency responses so obtained were analyzed, and the results are depicted in Table 7.2. Before proceeding, we want to stress that this kind of tolerance analysis is expected to cover parameter variation of any origin. These may be design errors, temperature variations, aging, and similar. All filter functions produced in this paragraph are obtained by setting the nominal transconductance gm0 = 10 μS. In the following tables, “Par” stands for parallel, “Cas” for cascade, and LC for LC-to-Gm-C transformed filter. “N” stands for not noticeable. a0 is the value in dB of the gain at the origin. Negative increment of a0 means smaller attenuation. Negative increment of amin means smaller gain. In the case of parallel realization, the element values are corresponding to Fig. 3.5, while in the case of the cascaded filter, one relates Fig. 4.6 for the low-pass cell and Fig. 4.10 for the band-stop cell. In the case of LC-to-Gm-C filters, gm1 denotes the top-left transconductor, gm4 the bottom right transconductor, and C 1 the capacitance used within the simulated inductor (Fig. 5.7). To get some background about the numbers given in Table 7.2, we are presenting here some highlights exposing specific properties of some of the responses. Figure 7.1 represents part of the passband gain response of the cascaded solution for nominal and worst case. The transconductances of the first transconductor in all cascaded cells were varied. As a result, the overall gain (statistically in worst case) was reduced by approximately 0.45 dB. That has to be taken into account when considering the reduction of the cut-off frequency. Namely, if “predistortion” was implemented so that the gain variation is compensated in advance, the amplitude characteristic would go upwards and the reduction of the passband width would disappear.

7.2 Study of the Worst-Case Tolerance of Gm-C Filters

133

Fig. 7.1 Passband gain (top line) and the worst-case gain (bottom line) response of the LSM_Z cascaded circuit with n = 9, m = 4, amax = 3 dB, amin = 30 dB, tol = 1% with gm1 as a parameter (gm0 = 10 μS)

Figure 7.2 represents the stopband response of the parallel solution under the same condition. Here, the minimum stopband attenuation is reduced which is inherent property of the parallel solution as discussed in [1]. Namely, since all cells connected in parallel are low-pass, at higher frequencies one manipulates with small numbers. If subtraction (between the output voltage values of the cells) occurs, numerical instability is introduced which, in turn, makes the gain value erroneous. Table 7.3 represents the same results as the previous one but for filter no. 2. By analysis of these result, we may easily conclude that from the LC-to-Gm-C solution performs excellent. It is followed by the cascade solution (which has some

Fig. 7.2 Passband gain (bottom line) and the worst-case gain (top line) response of the LSM_Z parallel circuit with n = 9, m = 4, amax = 3 dB, amin = 30 dB, tol = 1% with gm1 as a parameter (gm0 = 10 μS)

134

7 Implementation Issues

Table 7.3 Variations of the gain characteristic for the filter no. 2 (gm0 = 10 μS) Tol = 1%

a0 [dB]

Configuration

Par

Cas

LC

f c [kHz]

gm1 varied

0.1

− 0.34

N

gm4 varied

0.06

− 0.35

N

C 1 varied

N

N

N

N

Par

amax [dB]

Cas

LC

Par

Cas

LC

−2

− 1.5

N

− 17.3

N

N

− 4.5

N

N

− 9.5

N

N

N

N

−4

N

N

Note, there was no need to highlight any part of the LC-to-Gm-C solution

Table 7.4 Number of circuit elements and spread of element values of filter no. 1 (gm0 = 10 μS) No. of C

No. of gm

C max /C min

gm_max /gm_min

Par

Cas

LC

Par

Cas

LC

Par

Cas

LC

Par

Cas

LC

13

15

12

24

18 + 8 16

174

8.8

237,898

945.7

61

1

reduction of the gain in the passband) and the parallel solution (which perform very poor in the stopband). Another criterion to make comparisons among solutions is the spread of the values of the capacitors and transconductors of the nominal solution. In both cases, if large intervals for the element values is required, the price of the solution will rise. Table 7.4 represents the number of circuit elements and the ratios between the maximum and the minimum value for the capacitors and the transconductors. What can be seen from this table is that the number of circuit elements for different realizations is almost the same and that may not be accepted as a decisive criterion for distinguishing among solutions. As for the C max /C min and gm_max /gm_min , one may notice first the advantage of cascaded over the parallel solution. The LC-to-Gm-C solution would be preferable if there was no C max /C min = 237,898. As will be shown later on, however, that may be repaired effectively and bring advantages to this architecture.

7.3 Worst-Case Tolerance of Two-Phase Gm-C Filters Due to the Coupling In this case, the values of transconductances in the coupling between the I- and Qchannel within every cell were varied. We denote as gmc1a the left gmc1, as gmc3a the left gmc3 of Fig. 3.13, and as gmc1a the gmc of Fig. 3.12 for the parallel realization. Similarly, we denote as gmc1a the left gmcC1, as gmc3a the left gmcC2 of Fig. 4.20, as gmc1a the left gmc1, as gmc3a the left gmc2 of Fig. 4.19, and as gmc1a the gmcC of Fig. 4.18 for the cascade realization. Finally, we denote as gmc1a the left gmcM, as gmc3a the left gmcs of Fig. 5.15, and as gmc1a the left gmc of Fig. 5.14 for the CL-to-Gm-C realization.

7.3 Worst-Case Tolerance of Two-Phase Gm-C Filters …

135

Not much of a difference in the behavior may be noticed from the last two tables when compared with Tables 7.3 and 7.4.

7.4 A Short Discussion on the Noise in Gm-C Filters The noise on a system output is a consequence of the noise of the source (e.g., antenna) and the noise generated within the system. We are here, of course, interested in the second one. To simplify, we will begin with the assumption that white noise is present only. That means that the power spectral density of the noise is frequency independent. The system’s noise, in such a case, may be represented as vn2 = Vn2 · B

(7.1)

where vn2 is the noise power (developed on 1  load), Vn2 is the noise spectral density (here frequency independent), and B is the noise bandwidth (Note, the noise bandwidth and the system bandwidth are not the same number). The overscore denotes statistical average. Following this simplified representation and having in mind that all filters we want to compare have the same bandwidth we may conclude that Vn2 is the quantity needed to be observed. Now, let assume that we are capable to map all noise sources of the filter to the input. In that case Vn2 = |A| · Vni2

(7.2)

where Vni2 is the input referred noise spectral density and A is the gain of the filter (in all our cases, being equal to unity). This brings us to a common quantity needed to be analyzed from noise point of view. During the mapping from the output, through the circuit to the source, however, some of the noise sources map themselves into currents. That is the reason why the model depicted in Fig. 7.3 is adopted for representing the noise of four terminal components and systems. Fig. 7.3 Noise model of a four-terminal system

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7 Implementation Issues

In this figure, E n is the spectral amplitude of the equivalent input noise voltage while Jn is the spectral amplitude of the equivalent input noise current. It is important to note that E n and Jn , in general, are not white. Namely, even if only white noise sources were present within the four-terminal system, due to the reactive elements within it, one will get frequency-dependent spectral amplitudes. In the sequel, however, for the sake of simplicity, we will consider them white so the following will be accepted en = E n · jn = Jn ·

√ √

B

(7.3a)

B

(7.3a)

where en and jn are the equivalent (input referred) noise voltage and noise current, respectively. Since the quantities depicted in Fig. 7.3 are representatives of power, there is no orientation of the noise sources. Consequently, the noise analysis should be performed so that the system containing such components is analyzed separately for every noise source, and at the end, the output noise of the system is obtained as sum of squares of the output noise voltages [5]. For example, for the system of Fig. 7.3, one would have 2 vn,out = a12 · en2 + a22 · jn2

(7.4)

where a1 and a2 are absolute values of the proper voltage gain and transimpedance, respectively. The total noise of an OTA of transconductance gm can be expressed as noise voltage given by [6] k·T ·B π · gm

(7.5a)

k · T · gm B . π

(7.5b)

2 =γ eng v

and noise current given by 2 =γ jng j

In the above expressions, γv and γj stand for constants depending on the circuit structure of the transconductor (considered fully resistive), k is the Boltzmann constant, and T is the absolute temperature. As reported in [7],√in an example design, for a single transconductor, one may expect E ng = 33nV/ Hz and Jng = 0. That for a system with √ noise bandwidth of √ B = 10 MHz would produce eng = E ng B = 33 × 10−9 · 107 = 104 μV. Lower values of E ng were reported elsewhere [8].

7.4 A Short Discussion on the Noise in Gm-C Filters

137

To get the noise figure of a filter, one needs to substitute every transconductor with the noise model of Fig. 7.3 and to perform as many circuit analyses as the number of noise sources is. That will allow implementation of a formula similar to (7.4) but with as many pairs of addends as the number of transconductors is. That is illustrated in Fig. 7.4 where for the band-pass cell of Fig. 4.8 the noise circuit is created. After implementation of the analysis method described above, the following result was reported in [6] vn2

   k·T k·T γ2 γ1 gm1 · + γ4 + =γ · = γ2 + 2 · C2 Q gm4 2 · C2

(7.6)

2 = 0 for all transconductors was used. Here, γ is introduced as a constant where jng representing the band-pass cell as such. In that sense, (7.6) may be considered as a noise macromodel of the band-pass cell (of limited value due the absence of the influence of the capacitances within the transconductors). After these considerations, one may easily conclude that for comparisons of the filter architectures from the noise point of view one needs the schematic of every single transconductor which we have not. Generally speaking, such an information is usually considered highly confidential by the design company. In addition, to get realistic representation of the noise, one must drop the assumption of resistivity of the transconductor as a whole. That means one will need not only the dimensions of the transistors within the transconductor but the capacitances and the mutually coupling parasitics, too. Such information is not available before layout design, so

Fig. 7.4 Band-pass cell depicted in Fig. 4.8 with local noise sources associated to the transconductors shown

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7 Implementation Issues

we may conclude here that noise characterization of the filter is a post-layout activity for which sophisticated simulation tools must be used. Nevertheless, as we will see in the sequel, there is some information which, not quantitatively, but qualitatively may serve for some general conclusions. In addition to the concept of noise voltage and noise current, the concept of noise figure is in use. It represents the quotient of the overall input referred noise power and the noise power of the source (resistance). For the noise model of Fig. 7.3, one will get NF =

E n2 + Rs2 · Jn2 + 4kT Rs 4kT Rs

(7.7)

where NF is the noise figure. It is frequency dependent according to the frequency dependence of the noise of the filter. Here, however, white noise is considered all the time. We will use this concept to try to make some comparisons among the architectures from noise point of view.

7.5 On the Influence of the Electrical Characteristic of the Transconductor to the Filter Response In Chap. 2, we presented the simple versions of the micromodel of the transconductor. These will be used here to create some notion on the influence of the transconductor’s imperfections to the amplitude characteristic of the filter. In that, the topology of the filter will be considered and judged. The analysis will be based on SPICE simulation of three versions of the example Filter no. 1 described in Paragraph 7.2. To create a picture of the influence of the imperfection, we will implement the macromodel given in Fig. 2.12b with C i = 0. Fixed value of Ro = 1012  will be used, and in fact, the influence of the output capacitance will be examined. For convenience, this capacitance instead of C o will be noted as C out . Six values were assigned to Cout [fF]: 0, 20, 40, 60, 80, and 100. Accordingly, repetitive simulations were performed by SPICE. The simulation results are reported in the following set of figures. Figure 7.5a depicts the amplitude characteristic of the cascade topology with Fig. 7.5b being enlarged part related to the passband. Figure 7.6 depicts the amplitude characteristic of the parallel topology, and Fig. 7.7 depicts the amplitude characteristic of the LC-to-Gm-C topology. It will be discussed in the following chapter but we will here simply note that, apart from the cascaded architecture, no acceptable variant is present even in cases with small output capacitance of 20 fF.

7.5 On the Influence of the Electrical Characteristic …

139

a

b

Fig. 7.5 Influence of the output capacitance of the micromodel to the amplitude characteristic of the cascaded solution (gm0 = 10−5 S). a Complete characteristic and b passband characteristic

Fig. 7.6 Influence of the output capacitance of the micromodel to the amplitude characteristic of the parallel solution (gm0 = 10−5 S)

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7 Implementation Issues

Fig. 7.7 Influence of the output capacitance of the micromodel to the amplitude characteristic of the LC-to-Gm-C solution (gm0 = 10−5 S)

7.6 Comparisons To start with we will consider the effort to create a system synthesis program for generation of the circuit schematic. As discussed in [7], the synthesis in a form of parallel circuit is the simplest since there are only three different blocks to be manipulated: the second order cell, the first order cell, and the summing circuit. Of cause, this activity is preceded by the partial fraction expansion of the filter’s transfer function and creation of the biquad transfer functions. To synthesize the cascade schematic [9], one is first to perform pole-zero pairing so creating the biquad transfer function; to find the proper order of the cells within the cascade; and (optionally) to calculate the maximum gain of the cell in order to prevent “overdrive”, Follows the synthesis of the cells which, in this case, are large in number of types. To synthesize a Gm-C filter originating from an LC prototype, one needs first to create a complete synthesis program for passive LC circuits. That is a challenging task in many respects which we will not elaborate here. For that reason, mostly catalogs with tables of element values [10] are in use [11, 12]. That, of cause, dramatically limits the scope of this architecture. Having the passive LC circuit, one proceeds with substitution of the inductors which is a straightforward procedure except for the creation of a SPICE file (due to the need for naming the elements and to insert new nodes in an existing netlist). One has to have in mind that not all transmission zeros in a given transfer function will lead to transformerless cells or even to realizable ones. We find the parallel procedure to be the simplest and the LC-to-Gm-C transformation the most complex if no catalog is used. As the next criterion for comparison, we will consider the circuit complexity. It is expressed in number of transconductors and number of capacitors. In the previous

7.6 Comparisons

141

Table 7.5 Number of transconductors and spread of the transconductance values within the coupling for the two-phase version of filter No. 2 (gm0 = 10 μS) No. of gm

gm_max /gm_min

Par

Cas

LC

Par

Cas

LC

24

24

16

222

25

65,348

Table 7.6 Variations of the gain characteristic for the two-phase version of filter no. 2 due to variations in the coupling (BW stands for the passband width) (gm0 = 10 μS) Tol = 1%

a0 [dB]

BW [kHz]

amax [dB]

Configuration

Par

Cas

LC

Par

Cas

LC

Par

Cas

LC

gmc1a varied

N

N

−0.24

N

N

2.25

−13.6

N

2

paragraph, we first analyzed two synthesis tasks, and for each of them, we created three Gm-C (single-phase) filters. For all of them, the complexity is expressed in Tab. 7.4. There is not a large spread for the number of capacitors in the three categories the cascade version being the most complex. Similar is the situation for the transconductors. Note “ + 8” of transconductors for the cascaded solution means that there is additional (optional) amplifier (built of two transconductors) adjusting the gain of every biquad. So, there is not a decisive data in this stage of comparison. Table 7.5 contains complexity data related to the coupling networks of the two-phase filters. According to this, looking for the overall complexity, the LC-to-Gm-C is preferable (Table 7.6). The spread of element values is a very important issue in integrated circuit design. For the capacitors, large spread means small minimal (discretizing) value and consequently large number of separate capacitors to be added in parallel. For the transconductors, large spread of transconductances may make the design very difficult since several different designs of transconductors may become necessary. By analysis of Table 7.4, one comes to a conclusion that the parallel solution asks for considerably larger spread of the transconductance value while the LC-to-Gm-C (single-phase) solution has all its transconductances equal which is ideal. Table 7.5, however, makes the cascaded solution advantageous for the two-phase cases. One is to be very careful while falling into conclusions based on this table. In fact, it is to be considered together with the information on the spread of the capacitances. Table 7.4 shows that the cascade solution has incomparably smallest C max /C min ratio, while the LC-to-Gm-C solution has unreasonably large spread. The last one is coming from the inductance simulation formula (rewritten here for convenience) 2 C = L · gm

(7.8)

Namely, in the designs above, the nominal transconductance used was gm0 = 10 μS which in this formula multiplies the inductance value by a factor of 10–10 . So, an inductor of 1 mH (which is a large value) will need an equivalent capacitor of

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7 Implementation Issues

Table 7.7 Number of circuit elements and spread of element values of filter no. 1 (gm0 = 1 mS) C max /C min

gm_max /gm_min

Par

Cas

LC

Par

Cas

LC

174

2.73

7.45

945.7

78

1

100 fF. These values are much smaller than the rest of the capacitances, hence the large spread. To improve this property of the LC-to-Gm-C solutions, one may use a larger nominal transconductance. If gm0 = 1 mS were to be used in this case, the ratios depicted in Table 7.4 change into values given in Table 7.7. Before proceeding to the analysis of the results presented in Table 7.7, we would like to stress the following consequences of the increase of the nominal transconductance. Larger transconductance means wider transistors in the output circuit of the transconductor. That, in addition to the area occupied, will increase the output capacitance and so to deteriorate the performance at higher frequencies. Further, larger nominal transconductance rises the absolute values of the capacitances (due to the circuit synthesis formulae) leading to really large values (in these examples up to 600 + pF). That will seriously increase the silicon area. Finally, having in mind (2.5) larger transconductances will increase the power dissipation of the transconductor. For example, in [13], a design was reported where a transconductance of gm = 10.2 μS was obtained by a bias current of I bias = 2 μA with power consumption P = 0.14 μW. To get gm = 69.6 μS (6.82 times larger), they needed I bias = 50 μA (25 times larger) which is in accordance to (2.5) and Fig. 2.13. The power consumption was risen to P = 1.14 μW (8.14 times). One is not to undermine the number of transconductors in a filter being counted above. Now, if all these drawbacks are ignored, it comes that the increase of the nominal transconductance used in the design process may decisively influence the spread of the component’s values. For example, the ratio C max /C min for the LC-to-Gm-C architecture is reduced 10,000 times which makes this solution the most attractive one from this point of view. To see how the value of the nominal transconductance is controlling the influence of the output capacitance to the amplitude characteristics, we repeated the simulations with gm0 = 1 mS, and, again, with C out [fF]: 0, 20, 40, 60, 80, and 100. The simulation results are depicted in the following set of figures. Figure 7.8 depicts all six amplitude characteristics for the case of the cascaded architecture. They overlap showing that due to the dramatic rise of the rest of the capacitances in the circuit the influence of the output capacitance of the transconductor becomes negligible. Looking together Figs. 7.5 and 7.8, we may conclude that not much of improvements were obtained for the price paid and that there is room for a search for compromise value of gm0 . Figure 7.9 depicts all six amplitude characteristics for the case of the parallel architecture. One may see that dramatic improvement was obtained as compared with

7.6 Comparisons

143

Fig. 7.8 Influence of the output capacitance of the micromodel to the amplitude characteristic of the cascade solution (gm0 = 1 mS)

Fig. 7.9 Influence of the output capacitance of the micromodel to the amplitude characteristic of the parallel solution (gm0 = 1 mS)

the results depicted in Fig. 7.6. Still, parallel realization even with such a favorable value of the nominal transconductance suffers for distortions in the stopband. That, one would expect to be even worst in the case of filter no. 2, having in mind the part of Table 7.3 repeated below (for gm1 toleranced). amax [dB] Par

Cas

LC

− 17.3

N

N

144

7 Implementation Issues

Fig. 7.10 Influence of the output capacitance of the micromodel to the amplitude characteristic of the LC-to-Gm-C solution (gm0 = 1 mS)

Finally, Fig. 7.10 depicts all six amplitude characteristics for the case of the LC-to-Gm-C architecture. They overlap showing that due to the dramatic rise of the rest of the capacitances in the circuit the influence of the output capacitance of the transconductor becomes negligible. Looking together Figs. 7.9 and 7.7, we may conclude that the increase of gm0 is crucial for making this topology acceptable. Small values of transconductance lead inevitably to unacceptable spread of the capacitance values. Having in mind the spread of transconductances which is gm_max /gm_min = 1, it is advisable to consider a search for acceptable maximum value of the nominal transconductance. Let us go to the noise now. Figure 7.11 represents a simplified noise model of the parallel architecture in which the noise of the summing resistor is ignored while the active part is associated to the corresponding cell (with reference to Fig. 3.7). Considering an even order filter, with cells having infinite input resistances, for the noise figure one will get (white noise approximation only) k  NF = 1 +

i=1

2 2 E n,i + Rs2 · Jn,i

4kT Rs

 (7.9a)

Here, k is the number of paralleled cells (half of the order of the filter). Supposing all cells have the same input referred equivalent voltage and current this will be reduced to   k · E n2 + Rs2 · Jn2 NF = 1 + (7.9b) 4kT Rs Finally, if the noise current is neglected, which is usually done for circuits having MOS gate as input, one gets

7.6 Comparisons

145

Fig. 7.11 Simplified noise model of a parallel architecture

NF = 1 +

k · E n2 4kT Rs

(7.9c)

To conclude, the equivalent noise voltage of a parallel architecture will be k times larger than for a single cell and the noise figure will be increased by an amount depending on both the source resistance and the equivalent noise voltage of a single cell. For the cascaded architecture we will use only a two-stage example as depicted in Fig. 7.12. In this architecture, it is difficult to say that the cells have equal noise since their structures (in general, may) differ. Nevertheless, we will introduce the following assumptions: The input resistance of every cell is infinite, and the equivalent noise

Fig. 7.12 Noise model of the cascaded architecture

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7 Implementation Issues

currents of all cells are negligible. To that we will add the fact that in the synthesis method adopted, the maximum value of any cell gain is unity. This allows to map all the equivalent noise sources to the input and leads to k  NF = 1 +

i=1

2 E n,k−i+1 /

k−i+1

4kT Rs

j=1

a 2j−1

 (7.10)

a 2j−1 being the squared modulus of the (j − 1)th cell gain. a02 = 1. In the hypothetical case, when all the gains are equal to unity and all the equivalent noise voltages are equal, (7.10) reduces itself into (7.9c). Finally, for the LC-to-Gm-C architecture, the above analysis is practically impossible. The main reason for that is the finite (frequency dependent) input impedance of the filter. The problem is so obscure that no hypotheses may be established on the mapping of the transconductor noise to the input, and on the calculation of the noise figure. Nevertheless, one is not to forget that the remaining capacitors does not generate noise and that the number of transconductors, being defined by the number of inductors, is seen in Table 7.4 to be the smallest of all. So, in the attempt to compare the noise figures of the three architectures we may say that the parallel and the cascade one may have distinct values but not very much different to each other. As for the LC-to-Gm-C architecture, one may expect smaller noise figure due to the reduced number of transconductors being surrounded by capacitors which may shape the local noise frequency response within the filter.

7.7 The Ultimate Example Having collected knowledge and experience related to the influence of imperfections of any kind, we may conclude that the most promising or less susceptible to parameter variation is the cascaded architecture. This conclusion is not absolute but seems to us justified. In the analyses above, one parameter of a given type was considered variable, say the first transconductance in all cells or the first capacitor in all cells, and similar. The influence of the output capacitance of the transconductor was also analyzed separately from other parameter variations. Here, we will demonstrate the filter responses for a case when all parameters are statistical variables and the analysis will be done for simultaneous variations of the output capacitance. A two-phase filter will be designed with central frequency of the passband f 0 = 5 MHz obtained from a low-pass prototype having cut-off frequency f c = 5 MHz. The following design parameters will be implemented: maximum passband attenuation amax = 3 dB, minimum stopband attenuation amin = 40 dB, order of the filter n = 7, order of the numerator m = 4. For the prototype low-pass the LSM_Z [1], amplitude characteristic will be used. The nominal transconductance used in the physical synthesis was gm0 = 100 μS.

7.7 The Ultimate Example

147

The amplitude characteristics for positive and negative frequencies of the solution obtained are depicted in Fig. 7.13. We have no further comments to that. The influence of the variation of the parameters was done similar as in the previous paragraph. The difference is in that all variations were active simultaneously. All 60 transconductances in the two-phase filter were associated maximum statistical variation of 1%, all 22 capacitances were associated maximum statistical variation of 0.5%, and the output capacitances of all 60 transconductors were varied from zero to 100 fF in steps of 20 fF. Before proceeding with the results, we consider all maximum values of the tolerances and of output capacitances highly pessimistic which rises the confidence in the feasibility of the resulting filter’s performance. The results obtained by SPICE simulation will be demonstrated by three figures. Figure 7.14a depicts the complete amplitude characteristic.

a

b

Fig. 7.13 Amplitude characteristic of the seventh-order LSMZ two-phase filter. a positive frequencies and b negative frequencies

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7 Implementation Issues

a

b

c

Fig. 7.14 Amplitude characteristics of the seventh-order LSM_Z two-phase filter. a Overall characteristics b enlarged cut-off region and c enlarged first lobe of the upper stopband gain (All comments stated on (b) are valid for whole figure)

7.7 The Ultimate Example

149

Looking to this figure, one may conclude that high improvements were obtained as compared with the results reported in the previous paragraphs. One is not to forget the fact that here “everything” is toleranced. To get a better feeling on the effect of parameter variations, Fig. 7.14b depicts the amplitude characteristic at the edge of the passband. The pairs of lines in this figure represent the nominal and the worst-case amplitude characteristic for a given value of C out . So, the rightmost pair correspond to C out = 0 fF, the next to the left to 20 fF and so on. We can see that the output capacitance (60 of them) of the transconductor has much more influence to the cut-off frequency than the tolerances of both all the transconductances and all the capacitance together. That leads to a very useful design criterion for the transconductor. Note, various values of C out were reported in the literature. In [14] C out = 100 fF was obtained, in [15] values lover than 27 fF were reported, while in [16], C out = 6.9 fF was advertised, so witnessing of the advance (in time) of the CMOS technology. Having that in mind, one should consider the 20 fF curves as the reference one. Similar conclusion may be drawn from Fig. 7.14c. Here, the first lobe of the stopband gain is depicted to show that the 20 fF case introduces (in the worst-case) loss of attenuation less than 2 dB.

References 1. Litovski V (2019) Electronic filters. Springer Science + Business Media B.V. 2. Suiter D (1977) Worst-Case- und statistische Toleranzanalyse elektrischer Netzwerke. AEÜ 31(12):513–517 3. Special issue on Statistical Circuit Design. The Bell Syst Tech J 50(4) April 1971 4. Special issue on statistical design of VLSI circuits. IEEE Trans CAD Integr Circ CAD-5(1) (Jan 1986) 5. Litovski V, Zwolinski M (1997) VLSI circuit simulation and optimization. Chapman and Hall, London 6. Ananda Mohan PV (2013) VLSI Analog Filters Active RC, OTA-C, and SC. Springer Science + Business Media New York 7. Ramachandran A (2005) Nonlinearity and noise modeling of operational transconductance amplifiers for continuous time analog filters. Master thesis at the Texas A&M University 8. Mobarak M, Onabajo M, Silva-Martinez J, Sánchez-Sinencio E (2010) Attenuationpredistortion linearization of CMOS OTAs with digital correction of process variations in OTA-C filter applications. IEEE J Solid-State Circ 45(2):351–367 9. Hospodka J (2006) Optimization of dynamic range of cascade filter realization. Radioengineering 15(3):31–34 10. Zverev AI (2005) Handbook of filter synthesis. Wiley-Interscience, New York 11. Behbahani F, Firouzkouhi H, Chokkalingam R, Delshadpour S, Kheirkhahi A, Nariman M, Conta M, Bhatia S (2002) A fully integrated low-IF CMOS GPS radio with on-chip analog image rejection. IEEE J Solid-State Circ 37(12):1721–1727 12. Lo T-Y, Hung C-C (2009) 1V CMOS Gm-C filters, design and applications. Springer Science+Business Media B.V. 13. Szczepa´nski S, Kozieł S (2004) Phase compensation scheme for feedforward linearized CMOS operational transconductance amplifier. Bull Pol Acad Sci Tech Sci 52(2):141–148

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14. Wu P (1993) The design of high-frequency continuous-time integrated analog signal processing circuits. Dissertations and Theses. Paper 1162, https://pdxscholar.library.pdx.edu/open_a ccess_etds 15. Ergün BS, Kuntman H (2005) On the design of new CMOS DO-OTA topologies providing high output impedance and extended linearity range. J Electr Electron Eng 5(2):1449–1461 16. Zazerin A, Orlov A, Bogdan O (2013) Operational transconductance amplifier macromodel optimization for active piezoelectric filter design. Boctoqno-Evpopecki ypnal pepedovyx texnologi 6/12(66):30–34

Chapter 8

Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

8.1 Introduction After the analysis performed in the previous chapter, we came to a conclusion that the most appropriate topology of the Gm-C filters is the cascade one. Following that conclusion, we decided to make available to the filter design community a kind of catalog enabling fast instantiation, i.e., fast physical design. To make the catalog usable, we tried to encompass as broad a set of types of transfer functions as possible. That would first include polynomial (LSM, Papoulis, Halpern, Butterworth, Chebyshev, Thomson, and equi-ripple group delay) filters. In this case, the tables will cover orders of the filters from n = 3 to n = 12. As for the transfer functions having transmission zeros on the imaginary axis, we chose the ones having maximum number of zeros, e.g., m = 4 for the case of n = 5 and m = 6 for the case of n = 8. In this case, n = 11 was chosen as the maximum value. That is due to the fact that when n is higher extremely high (not physically feasible), selectivity is obtained in many cases. Of course, minimum value of n = 3 was used. First the transfer functions of the critical monotonic amplitude passband characteristics (CMAC) were extended with transmission zeros to get the LSM_Z, Papoulis_Z, Halpern_Z, and Butterworth_Z (Inverse Chebyshev) filters. The stopband attenuation was varied from amin = 30 dB to amin = 60 dB with steps of 5 dB. Then, the so-called modified elliptic filters were synthesized with four values for maximum passband attenuation amax = {0.1; 0.25; 0.5; 1} dB. Finally, the filter function exhibiting constant group delay was extended with transmission zeros to get Thomson_Z and Equi-rip-td_Z filters. Since low-pass filters were sought only, the list of Gm-C cells needed for physical realization is reduced to three: first-order low-pass, second-order low-pass, and second-order band-stop (notch) cells. These are redrawn in Figs. 8.1, 8.2 and 8.3 and labeled α, β, and γ, respectively.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 V. Litovski, Gm-C Filter Synthesis for Modern RF Systems, Lecture Notes in Electrical Engineering 807, https://doi.org/10.1007/978-981-16-6561-5_8

151

152

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Fig. 8.1 Circuit α. First-order Gm-C low-pass filter and node notation (labeled by α in the tables)

Fig. 8.2 Circuit β. The second-order low-pass cell (labeled by β in the tables)

Fig. 8.3 Circuit γ. A band-stop (notch) biquad (labeled by γ in the tables). Note: a · C 1 ≡ c1 ; (1 − a) · C 1 ≡ c2 ; b · C 2 ≡ c3 ; (1 − b) · C 2 ≡ c4

8.2 How to Use the Tables The tables below contain normalized values of the transconductances and capacitances. For the normal Gm-C case to compute the real element values (C and gm ), one is to supply the cut-off (normalization) frequency of the prototype low-pass filter (f c ) (with reference to Fig. 8.4) and the nominal (normalization) value of the transconductance (gm0 ). For the two-phase case, one needs in addition the frequency representing the shift (f sh ). For the normal Gm-C, the following formulas are applied gm = gm0 · gm,norm

(8.1)

8.2 How to Use the Tables

153

Fig. 8.4 Meaning of the frequencies (f c ≡ cutoff and f sh ≡ shift frequency)

where gm,norm is the normalized value of the transconductance as given in the tables below, and C = Cnorm ·

gm0 . 2 · π · fC

(8.2)

Here C and gm are the needed element values, while C norm (the normalized capacitance) and gm,norm are read from the tables below for every instance of the filters. For example, if C norm = 1.695714491 and gm,norm = 1.695714491 are read from a table, and if f c = 2.3 MHz and gm0 = 0.1 mS, one will get C = 1.695714491 · 0.0001/(2 · π · 2,300,000) = 11.734 pF and gm = 1.695714491 · 0.0001 = 0.1696 mS. The reader should have in mind that the cells depicted below do not contain the corrective amplifier used to adjust the gain at the origin. It should be adjusted by adding such an amplifier (depicted in Fig. 4.13a) mainly based on a single SPICE simulation of the newly synthesized filter. To create a two-phase filter, one needs the value of the transconductance(s) in the coupling circuit. We are repeating here the schematic of the simplest two-phase cell only to facilitate the understanding of the procedure. It is copied from Fig. 3.12. Here the value of the coupling transconductance is obtained from gmc = 2 · π · f sh · C.

(8.3)

So, in the case of f sh = 1 MHz, a capacitance of 11.734 pF will be followed with a coupling transconductance of gmc = 2 · π · 11.734 × 10–12 × 106 = 73.73 μS (Fig. 8.5). Before proceeding, we will remind the reader that amax is used for maximum passband attenuation all the way to the cut-off frequency (f c ) and amin is used to

154

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Fig. 8.5 First-order two-phase Gm-C cell

denote the minimum stopband attenuation above f s , the frequency of the upper edge of the transition band (or the lower edge of the stopband). In the sequel tables of element values will be listed. To make the use of the tables easier, every paragraph will be started by a figure illustrating the normalized attenuation characteristic of the corresponding filter category. To unify, n = 7 will be used for all categories, while for the Chebyshev and modified elliptic in addition, n = 8 will be illustrated. Note the transfer functions of the filters needed for physical synthesis were software for filter design [1]. produced by the Note, within the tables, the symbol “÷” was used to replace the construct “to”.

8.3 Polynomial Filters The transfer function of a polynomial filter may be expressed as H (s) = H0 · Pn (0)/Pn (s)

(8.4)

where Pn (s) is a Hurwitz polynomial of order n. The quantity H 0 is used in the case of even order Chebyshev filters and is calculated as H0 = 10−amax /20 .

(8.5)

8.3 Polynomial Filters

155

Fig. 8.6 Passband (left) and the overall attenuation characteristic of a seventh-order LSM filter

8.3.1 LSM Filters The attenuation characteristic of the seventh-order LSM filter is depicted in Fig. 8.6. It is normal for a member of the CMAC family to be normalized so that it exhibits 3 dB at the cut-off frequency (here normalized to unity). Using a value different to f c in (8.2) will stretch or compress the attenuation of the physical realization as compared to Fig. 8.6. In that way, one can accommodate to a different value of amax (Table 8.1).

156

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.1 Element values of the cascaded Gm-C LSM filters n

Cell no., type

3

1, β

c1 = 1.483202486

c2 = 0.7149769576

gm1 ÷ gm4 = 1

2, α

c = 1.695714491

gm1 = 1.695714491

gm2 = 1

1, β

c1 = 2.278975120

c2 = 0.4524587064

gm1 ÷ gm4 = 1

2, β

c1 = 0.9824612822

c2 = 2.662579004

gm1 ÷ gm4 = 1

1, β

c1 = 1.424145145

c2 = 0.7188981907

gm1 ÷ gm4 = 1

2, β

c1 = 0.4225587709

c2 = 1.588604500

gm1 ÷ gm4 = 1

3, α

c = 0.6821842290

gm1 = 0.6821842290

gm2 = 1

1, β

c1 = 2.220055990

c2 = 0.4711336857

gm1 ÷ gm4 = 1

2, β

c1 = 0.7407748674

c2 = 1.925565420

gm1 ÷ gm4 = 1

3, β

c1 = 0.3646971198

c2 = 1.436117724

gm1 ÷ gm4 = 1

1, β

c1 = 2.815139339

c2 = 0.3666636398

gm1 ÷ gm4 = 1

2, β

c1 = 0.9835114164

c2 = 1.356359816

gm1 ÷ gm4 = 1

3, β

c1 = 0.5012801979

c2 = 2.000907681

gm1 ÷ gm4 = 1

4, α

c = 0.9408441281

gm1 = 0.9408441281

gm2 = 1

1, β

c1 = 3.933469758

c2 = 0.2631752470

gm1 ÷ gm4 = 1

2, β

c1 = 1.321157138

c2 = 1.060187040

gm1 ÷ gm4 = 1

3, β

c1 = 0.7857591921

c2 = 2.422312029

gm1 ÷ gm4 = 1

4, β

c1 = 0.4559658499

c2 = 1.807072990

gm1 ÷ gm4 = 1

1, β

c1 = 4.643356242

c2 = 0.2204161876

gm1 ÷ gm4 = 1

2, β

c1 = 1.616973453

c2 = 0.7836926881

gm1 ÷ gm4 = 1

3, β

c1 = 0.9907233846

c2 = 1.916512026

gm1 ÷ gm4 = 1

4, β

c1 = 0.6119197581

c2 = 2.409533403

gm1 ÷ gm4 = 1

5, α

c = 1.091753524

gm1 = 1.091753524

gm2 = 1

1, β

c1 = 6.072313643

c2 = 0.168711482

gm1 ÷ gm4 = 1

2, β

c1 = 2.036653580

c2 = 0.621608568

gm1 ÷ gm4 = 1

3, β

c1 = 1.219444707

c2 = 1.520916744

gm1 ÷ gm4 = 1

4, β

c1 = 0.860542332

c2 = 2.836935582

gm1 ÷ gm4 = 1

5, β

c1 = 0.553049849

c2 = 2.196980165

gm1 ÷ gm4 = 1

1, β

c1 = 6.911975195

c2 = 0.147119602

gm1 ÷ gm4 = 1

2, β

c1 = 2.394160369

c2 = 0.497525746

gm1 ÷ gm4 = 1

3, β

c1 = 1.449958546

c2 = 1.162242461

gm1 ÷ gm4 = 1

4, β

c1 = 1.026393726

c2 = 2.308004347

gm1 ÷ gm4 = 1

5, b

c1 = 0.721279393

c2 = 2.835981994

gm1 ÷ gm4 = 1

6, α

c = 1.294074519

gm1 = 1.294074519

gm2 = 1

1, β

c1 = 8.645692014

c2 = 0.117691553

gm1 ÷ gm4 = 1

2, β

c1 = 2.899458820

c2 = 0.408277401

gm1 ÷ gm4 = 1

3, β

c1 = 1.725340163

c2 = 0.928123152

gm1 ÷ gm4 = 1

4, β

c1 = 1.222505551

c2 = 1.915678261

gm1 ÷ gm4 = 1

5, β

c1 = 0.942079228

c2 = 3.189120917

gm1 ÷ gm4 = 1

6, β

c1 = 0.659510093

c2 = 2.621809953

gm1 ÷ gm4 = 1

4 5

6

7

8

9

10

11

12

8.3 Polynomial Filters

157

Fig. 8.7 Passband (left) and the overall attenuation characteristic of a seventh-order Papoulis filter

8.3.2 Papoulis (Legendre or Optimal) Filters The attenuation characteristic of the seventh-order Papoulis filter is depicted in Fig. 8.7 (Table 8.2).

Table 8.2 Element values of the cascaded Gm-C Papoulis filters n

Cell no., type

3

1, β

c1 = 1.447095202

c2 = 0.741708896

gm1 ÷ gm4 = 1

2, α

c = 1.609899686

gm1 = 1.609899686

gm2 = 1

1, β

c1 = 2.156236247

c2 = 0.489052298

gm1 ÷ gm4 = 1

2, β

c1 = 0.908622837

c2 = 2.550404764

gm1 ÷ gm4 = 1

1, β

c1 = 3.252912423

c2 = 0.319791052

gm1 ÷ gm4 = 1

2, β

c1 = 1.286998535

c2 = 1.561515884

gm1 ÷ gm4 = 1

4 5

(continued)

158

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.2 (continued) n 6

7

8

9

10

11

12

Cell no., type 3, α

c = 2.134128868

gm1 = 2.134128868

gm2 = 1

1, β

c1 = 4.337270773

c2 = 0.237713657

gm1 ÷ gm4 = 1

2, β

c1 = 1.616919809

c2 = 1.060368289

gm1 ÷ gm4 = 1

3, β

c1 = 1.138247758

c2 = 3.505912693

gm1 ÷ gm4 = 1

1, β

c1 = 5.795628157

c2 = 0.176670670

gm1 ÷ gm4 = 1

2, β

c1 = 2.104038003

c2 = 0.717514869

gm1 ÷ gm4 = 1

3, β

c1 = 1.430521216

c2 = 2.282518049

gm1 ÷ gm4 = 1

4, α

c = 2.614747918

gm1 = 2.614747918

gm2 = 1

1, β

c1 = 7.247238676

c2 = 0.140653306

gm1 ÷ gm4 = 1

2, β

c1 = 2.571570606

c2 = 0.541449386

gm1 ÷ gm4 = 1

3, β

c1 = 1.663765146

c2 = 1.568828692

gm1 ÷ gm4 = 1

4, β

c1 = 1.360706541

c2 = 4.380825717

gm1 ÷ gm4 = 1

1, β

c1 = 9.068444674

c2 = 0.111998795

gm1 ÷ gm4 = 1

2, β

c1 = 3.176440854

c2 = 0.410549472

gm1 ÷ gm4 = 1

3, β

c1 = 2.010088411

c2 = 1.072916459

gm1 ÷ gm4 = 1

4, β

c1 = 1.614866804

c2 = 2.960754632

gm1 ÷ gm4 = 1

5, α

c = 3.067930019

gm1 = 3.067930019

gm2 = 1

1, β

c1 = 10.88537618

c2 = 0.093071644

gm1 ÷ gm4 = 1

2, β

c1 = 3.770097020

c2 = 0.330980039

gm1 ÷ gm4 = 1

3, β

c1 = 2.332795512

c2 = 0.811256919

gm1 ÷ gm4 = 1

4, β

c1 = 1.801081361

c2 = 2.053384507

gm1 ÷ gm4 = 1

5, β

c1 = 1.575144123

c2 = 5.207203421

gm1 ÷ gm4 = 1

1, β

c1 = 13.06988250

c2 = 0.077353172

gm1 ÷ gm4 = 1

2, β

c1 = 4.494089921

c2 = 0.267609894

gm1 ÷ gm4 = 1

3, β

c1 = 2.745117828

c2 = 0.617783248

gm1 ÷ gm4 = 1

4, β

c1 = 2.084327716

c2 = 1.413239250

gm1 ÷ gm4 = 1

5, β

c1 = 1.808324308

c2 = 3.613191483

gm1 ÷ gm4 = 1

6, α

c = 3.501664801

gm1 = 3.501664801

gm2 = 1

1, β

c1 = 15.25143535

c2 = 0.066187672

gm1 ÷ gm4 = 1

2, β

c1 = 5.210678577

c2 = 0.224799020

gm1 ÷ gm4 = 1

3, β

c1 = 3.143933665

c2 = 0.499156658

gm1 ÷ gm4 = 1

4, β

c1 = 2.340599191

c2 = 1.068351471

gm1 ÷ gm4 = 1

5, β

c1 = 1.965639533

c2 = 2.525263852

gm1 ÷ gm4 = 1

6, β

c1 = 1.782751268

c2 = 5.999946396

gm1 ÷ gm4 = 1

8.3.3 Halpern Filters The attenuation characteristic of the seventh-order Halpern filter is depicted in Fig. 8.8 (Table 8.3).

8.3 Polynomial Filters

159

Fig. 8.8 Passband (left) and the overall attenuation characteristic of a seventh-order Halpern filter

Table 8.3 Element values of the cascaded Gm-C Halpern filters n

Cell no., type

3

1, β

c1 = 1.569251490

c2 = 0.598924345

gm1 ÷ gm4 = 1

2, α

c = 2.122923083

gm1 = 2.122923083

gm2 = 1

1, β

c1 = 2.414401541

c2 = 0.390495076

gm1 ÷ gm4 = 1

2, β

c1 = 1.116621347

c2 = 2.842883394

gm1 ÷ gm4 = 1

1, β

c1 = 3.720969363

c2 = 0.254155361

gm1 ÷ gm4 = 1

2, β

c1 = 1.589839133

c2 = 1.308877392

gm1 ÷ gm4 = 1

3, α

c = 3.041672931

gm1 = 3.041672931

gm2 = 1

1, β

c1 = 5.015421417

c2 = 0.189919146

gm1 ÷ gm4 = 1

2, β

c1 = 1.999407517

c2 = 0.842467289

gm1 ÷ gm4 = 1

3, β

c1 = 1.509653934

c2 = 4.118706359

gm1 ÷ gm4 = 1

4 5

6

(continued)

160

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.3 (continued) n

Cell no., type

7

1, β

c1 = 6.738121778

c2 = 0.142209572

gm1 ÷ gm4 = 1

2, β

c1 = 2.589983978

c2 = 0.555087179

gm1 ÷ gm4 = 1

3, β

c1 = 1.902166094

c2 = 1.963048836

gm1 ÷ gm4 = 1

4, α

c = 3.878763286

gm1 = 3.878763286

gm2 = 1

1, β

c1 = 8.448294323

c2 = 0.114022727

gm1 ÷ gm4 = 1

2, β

c1 = 3.161271613

c2 = 0.417885586

gm1 ÷ gm4 = 1

3, β

c1 = 2.201291116

c2 = 1.242533031

gm1 ÷ gm4 = 1

4, β

c1 = 1.883371245

c2 = 5.326424833

gm1 ÷ gm4 = 1

1, β

c1 = 10.56808874

c2 = 0.091559406

gm1 ÷ gm4 = 1

2, β

c1 = 3.885079452

c2 = 0.318242501

gm1 ÷ gm4 = 1

3, β

c1 = 2.635024210

c2 = 0.809730744

gm1 ÷ gm4 = 1

4, β

c1 = 2.247247693

c2 = 2.608146756

gm1 ÷ gm4 = 1

5, α

c = 4.667680418

gm1 = 4.667680418

gm2 = 1

1, β

c1 = 12.67524669

c2 = 0.076618768

gm1 ÷ gm4 = 1

2, β

c1 = 4.596498625

c2 = 0.258205828

gm1 ÷ gm4 = 1

3, β

c1 = 3.044875409

c2 = 0.604613374

gm1 ÷ gm4 = 1

4, β

c1 = 2.490037984

c2 = 1.634116541

gm1 ÷ gm4 = 1

5, b

c1 = 2.242575339

c2 = 6.491891472

gm1 ÷ gm4 = 1

1, β

c1 = 15.17898350

c2 = 0.064183839

gm1 ÷ gm4 = 1

2, β

c1 = 5.449153240

c2 = 0.210637098

gm1 ÷ gm4 = 1

3, β

c1 = 3.556557828

c2 = 0.458947404

gm1 ÷ gm4 = 1

4, β

c1 = 2.853175315

c2 = 1.054545749

gm1 ÷ gm4 = 1

5, b

c1 = 2.596068752

c2 = 3.251349976

gm1 ÷ gm4 = 1

6, α

c = 5.423385494

gm1 = 5.423385494

gm2 = 1

1, β

c1 = 17.67013637

c2 = 0.055278648

gm1 ÷ gm4 = 1

2, β

c1 = 6.292368438

c2 = 0.178328278

gm1 ÷ gm4 = 1

3, β

c1 = 4.054397705

c2 = 0.371682293

gm1 ÷ gm4 = 1

4, β

c1 = 3.187443443

c2 = 0.780645742

gm1 ÷ gm4 = 1

5, b

c1 = 2.802562136

c2 = 2.025204604

gm1 ÷ gm4 = 1

6, β

c1 = 2.590585289

c2 = 7.627150653

gm1 ÷ gm4 = 1

8

9

10

11

12

8.3.4 Butterworth (Maximally Flat) Filters The attenuation characteristic of the seventh-order Butterworth filter is depicted in Fig. 8.9 (Table 8.4).

8.3 Polynomial Filters

161

Fig. 8.9 Passband (left) and the overall attenuation characteristic of a seventh-order Butterworth filter

Table 8.4 Element values of the cascaded Gm-C Butterworth filters n

Cell no., type

3

1, β

c1 = 0.999208594

c2 = 0.999208594

gm1 ÷ gm4 = 1

2, α

c = 0.999208594

gm1 = 0.999208594

gm2 = 1

1, β

c1 = 1.305788214

c2 = 0.764913026

gm1 ÷ gm4 = 1

2, β

c1 = 0.540875188

c2 = 1.846663402

gm1 ÷ gm4 = 1

1, β

c1 = 1.617266686

c2 = 0.617740906

gm1 ÷ gm4 = 1

2, β

c1 = 0.617740905

c2 = 1.617266687

gm1 ÷ gm4 = 1

3, α

c = 0.999525781

gm1 = 0.999525781

gm2 = 1

1, β

c1 = 1.931086458

c2 = 0.517433057

gm1 ÷ gm4 = 1

2, β

c1 = 0.706826701

c2 = 1.413653401

gm1 ÷ gm4 = 1

3, β

c1 = 0.517433057

c2 = 1.931086458

gm1 ÷ gm4 = 1

4 5

6

(continued)

162

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.4 (continued) n

Cell no., type

7

1, β

c1 = 2.246217737

c2 = 0.444890971

gm1 ÷ gm4 = 1

2, β

c1 = 0.801665829

c2 = 1.246556800

gm1 ÷ gm4 = 1

3, β

c1 = 0.554769967

c2 = 1.801326766

gm1 ÷ gm4 = 1

4, α

c = 0.999660937

gm1 = 0.999660937

gm2 = 1

1, β

c1 = 2.562156584

c2 = 0.390065114

gm1 ÷ gm4 = 1

2, β

c1 = 0.899709746

c2 = 1.110811464

gm1 ÷ gm4 = 1

3, β

c1 = 0.601166832

c2 = 1.662446839

gm1 ÷ gm4 = 1

4, β

c1 = 0.509644632

c2 = 1.960989752

gm1 ÷ gm4 = 1

1, β

c1 = 2.878627154

c2 = 0.347204919

gm1 ÷ gm4 = 1

2, β

c1 = 0.999736719

c2 = 0.999736719

gm1 ÷ gm4 = 1

3, β

c1 = 0.652531800

c2 = 1.531685516

gm1 ÷ gm4 = 1

4, β

c1 = 0.531948797

c2 = 1.878890435

gm1 ÷ gm4 = 1

5, α

c = 0.999736718

gm1 = 0.999736718

gm2 = 1

1, β

c1 = 3.195470004

c2 = 0.312794868

gm1 ÷ gm4 = 1

2, β

c1 = 1.101083923

c2 = 0.907766063

gm1 ÷ gm4 = 1

3, β

c1 = 0.706939396

c2 = 1.413878791

gm1 ÷ gm4 = 1

4, β

c1 = 0.561030281

c2 = 1.781591213

gm1 ÷ gm4 = 1

5, β

c1 = 0.506112728

c2 = 1.974909072

gm1 ÷ gm4 = 1

1, β

c1 = 3.512576783

c2 = 0.284568081

gm1 ÷ gm4 = 1

2, β

c1 = 1.203355154

c2 = 0.830650229

gm1 ÷ gm4 = 1

3, β

c1 = 0.763355888

c2 = 1.309438036

gm1 ÷ gm4 = 1

4, β

c1 = 0.594222523

c2 = 1.682142961

gm1 ÷ gm4 = 1

5, β

c1 = 0.520995787

c2 = 1.918570667

gm1 ÷ gm4 = 1

6, α

c = 0.999783594

gm1 = 0.999783594

gm2 = 1

1, β

c1 = 3.829891637

c2 = 0.261000786

gm1 ÷ gm4 = 1

2, β

c1 = 1.306304715

c2 = 0.765215585

gm1 ÷ gm4 = 1

3, β

c1 = 0.821177473

c2 = 1.217282207

gm1 ÷ gm4 = 1

4, β

c1 = 0.630111637

c2 = 1.586393058

gm1 ÷ gm4 = 1

5, β

c1 = 0.541089129

c2 = 1.847393844

gm1 ÷ gm4 = 1

6, β

c1 = 0.504214799

c2 = 1.982497792

gm1 ÷ gm4 = 1

8

9

10

11

12

8.3.5 Chebyshev Filters The normalized passband attenuation characteristics of the seventh and the sixthorder Chebyshev filters are depicted in Fig. 8.10a and b, respectively. Figure 8.10c depicts the overall attenuation characteristic of the seventh-order Chebyshev filter. Due to limited space, only circuits exhibiting amax = {0.1; 0.25; 0.5; 1} dB will be synthesized (Tables 8.5, 8.6, 8.7 and 8.8).

8.3 Polynomial Filters

163

Fig. 8.10 a and b Passband attenuation of the seventh and the sixth-order Chebyshev filter, respectively, and c the overall attenuation characteristic of a seventh-order Chebyshev filter

164

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.5 Element values of the cascaded Gm-C Chebyshev (amax = 0.1 dB) filters n

Cell no., type

3

1, β

c1 = 1.031559842

c2 = 0.573698585

gm1 ÷ gm4 = 1

2, α

c = 1.031559842

gm1 = 1.031559842

gm2 = 1

1, β

c1 = 1.892818277

c2 = 0.397218247

gm1 ÷ gm4 = 1

2, β

c1 = 0.784031001

c2 = 2.047534751

gm1 ÷ gm4 = 1

1, β

c1 = 3.002395589

c2 = 0.278732124

gm1 ÷ gm4 = 1

2, β

c1 = 1.146813067

c2 = 1.371212551

gm1 ÷ gm4 = 1

3, α

c = 1.855582522

gm1 = 1.855582522

gm2 = 1

1, β

c1 = 4.359449927

c2 = 0.203107334

gm1 ÷ gm4 = 1

2, β

c1 = 1.595669420

c2 = 0.899941918

gm1 ÷ gm4 = 1

3, β

c1 = 1.168111088

c2 = 3.250601790

gm1 ÷ gm4 = 1

1, β

c1 = 5.963671795

c2 = 0.153492191

gm1 ÷ gm4 = 1

2, β

c1 = 2.128409821

c2 = 0.623766043

gm1 ÷ gm4 = 1

3, β

c1 = 1.472905297

c2 = 2.056014554

gm1 ÷ gm4 = 1

4, α

c = 2.654083635

gm1 = 2.654083635

gm2 = 1

1, β

c1 = 7.814926986

c2 = 0.119645841

gm1 ÷ gm4 = 1

2, β

c1 = 2.744237419

c2 = 0.456130679

gm1 ÷ gm4 = 1

3, β

c1 = 1.833640821

c2 = 1.31030641

gm1 ÷ gm4 = 1

4, β

c1 = 1.554485628

c2 = 4.417893725

gm1 ÷ gm4 = 1

1, β

c1 = 9.913149935

c2 = 0.095688455

gm1 ÷ gm4 = 1

2, β

c1 = 3.442800842

c2 = 0.348121316

gm1 ÷ gm4 = 1

3, β

c1 = 2.247128658

c2 = 0.894418780

gm1 ÷ gm4 = 1

4, β

c1 = 1.831876066

c2 = 2.711202990

gm1 ÷ gm4 = 1

5, α

c = 3.442800842

gm1 = 3.442800842

gm2 = 1

1, β

c1 = 12.258305607

c2 = 0.078175652

gm1 ÷ gm4 = 1

2, β

c1 = 4.223924247

c2 = 0.274687048

gm1 ÷ gm4 = 1

3, β

c1 = 2.711926304

c2 = 0.649209870

gm1 ÷ gm4 = 1

4, β

c1 = 2.152196901

c2 = 1.695199301

gm1 ÷ gm4 = 1

5, β

c1 = 1.941524873

c2 = 5.570800701

gm1 ÷ gm4 = 1

1, β

c1 = 14.850373924

c2 = 0.065016143

gm1 ÷ gm4 = 1

2, β

c1 = 5.087511277

c2 = 0.222503289

gm1 ÷ gm4 = 1

3, β

c1 = 3.227294681

c2 = 0.494088488

gm1 ÷ gm4 = 1

4, β

c1 = 2.512237371

c2 = 1.142960146

gm1 ÷ gm4 = 1

5, β

c1 = 2.202651423

c2 = 3.354391321

gm1 ÷ gm4 = 1

6, α

c = 4.226857127

gm1 = 4.226857127

gm2 = 1

1, β

c1 = 17.689342722

c2 = 0.054892275

gm1 ÷ gm4 = 1

2, β

c1 = 6.033505381

c2 = 0.184065224

gm1 ÷ gm4 = 1

3, β

c1 = 3.792820041

c2 = 0.389848472

gm1 ÷ gm4 = 1

4, β

c1 = 2.910333178

c2 = 0.823031596

gm1 ÷ gm4 = 1

5, β

c1 = 2.499159757

c2 = 2.069578613

gm1 ÷ gm4 = 1

6, β

c1 = 2.328846150

c2 = 6.716517950

gm1 ÷ gm4 = 1

4 5

6

7

8

9

10

11

12

8.3 Polynomial Filters

165

Table 8.6 Element values of the cascaded Gm-C Chebyshev (amax = 0.25 dB) filters n

Cell no., type

3

1, β

c1 = 1.303402577

c2 = 0.573139935

gm1 ÷ gm4 = 1

2, α

c = 1.303402577

gm1 = 1.303402577

gm2 = 1

1, β

c1 = 2.352740056

c2 = 0.365794999

gm1 ÷ gm4 = 1

2, β

c1 = 0.974536840

c2 = 2.255993505

gm1 ÷ gm4 = 1

1, β

c1 = 3.703010632

c2 = 0.246523643

gm1 ÷ gm4 = 1

2, β

c1 = 1.414424201

c2 = 1.318005671

gm1 ÷ gm4 = 1

3, α

c = 2.288586431

gm1 = 2.288586431

gm2 = 1

1, β

c1 = 5.353784083

c2 = 0.175677916

gm1 ÷ gm4 = 1

2, β

c1 = 1.959620981

c2 = 0.809741598

gm1 ÷ gm4 = 1

3, β

c1 = 1.434542121

c2 = 3.535069430

gm1 ÷ gm4 = 1

1, β

c1 = 7.304906622

c2 = 0.130986612

gm1 ÷ gm4 = 1

2, β

c1 = 2.607090989

c2 = 0.543393609

gm1 ÷ gm4 = 1

3, β

c1 = 1.804162943

c2 = 1.959450120

gm1 ÷ gm4 = 1

4, α

c = 3.250989288

gm1 = 3.250989288

gm2 = 1

1, β

c1 = 9.556312548

c2 = 0.101215113

gm1 ÷ gm4 = 1

2, β

c1 = 3.355730710

c2 = 0.390423382

gm1 ÷ gm4 = 1

3, β

c1 = 2.242227575

c2 = 1.171842084

gm1 ÷ gm4 = 1

4, β

c1 = 1.900868752

c2 = 4.783086744

gm1 ÷ gm4 = 1

1, β

c1 = 12.107970044

c2 = 0.080465969

gm1 ÷ gm4 = 1

2, β

c1 = 4.205053867

c2 = 0.294846135

gm1 ÷ gm4 = 1

3, β

c1 = 2.744653985

c2 = 0.775648505

gm1 ÷ gm4 = 1

4, β

c1 = 2.237462429

c2 = 2.575534279

gm1 ÷ gm4 = 1

5, α

c = 4.205053867

gm1 = 4.205053867

gm2 = 1

1, β

c1 = 14.959862206

c2 = 0.065459364

gm1 ÷ gm4 = 1

2, β

c1 = 5.154817210

c2 = 0.231070823

gm1 ÷ gm4 = 1

3, β

c1 = 3.309596377

c2 = 0.553748486

gm1 ÷ gm4 = 1

4, β

c1 = 2.626510556

c2 = 1.512315589

gm1 ÷ gm4 = 1

5, β

c1 = 2.369409403

c2 = 6.018950413

gm1 ÷ gm4 = 1

1, β

c1 = 18.111979382

c2 = 0.054269193

gm1 ÷ gm4 = 1

2, β

c1 = 6.204887489

c2 = 0.186303418

gm1 ÷ gm4 = 1

3, β

c1 = 3.936109288

c2 = 0.417319631

gm1 ÷ gm4 = 1

4, β

c1 = 3.064003081

c2 = 0.989240500

gm1 ÷ gm4 = 1

5, β

c1 = 2.686422400

c2 = 3.181532239

gm1 ÷ gm4 = 1

6, α

c = 5.155206834

gm1 = 5.155206834

gm2 = 1

1, β

c1 = 21.564315749

c2 = 0.045709286

gm1 ÷ gm4 = 1

2, β

c1 = 7.355186519

c2 = 0.153606527

gm1 ÷ gm4 = 1

3, β

c1 = 4.623663538

c2 = 0.327216642

gm1 ÷ gm4 = 1

4, β

c1 = 3.547861816

c2 = 0.700889756

gm1 ÷ gm4 = 1

5, β

c1 = 3.046618010

c2 = 1.843984040

gm1 ÷ gm4 = 1

6, β

c1 = 2.838996027

c2 = 7.248803683

gm1 ÷ gm4 = 1

4 5

6

7

8

9

10

11

12

166

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.7 Element values of the cascaded Gm-C Chebyshev (amax = 0.5 dB) filters n

Cell no., type

3

1, β

c1 = 1.596280064

c2 = 0.548345863

gm1 ÷ gm4 = 1

2, α

c = 1.596280064

gm1 = 1.596280064

gm2 = 1

1, β

c1 = 2.851390062

c2 = 0.329760218

gm1 ÷ gm4 = 1

2, β

c1 = 1.181084435

c2 = 2.375564938

gm1 ÷ gm4 = 1

1, β

c1 = 4.465764150

c2 = 0.216189708

gm1 ÷ gm4 = 1

2, β

c1 = 1.705770120

c2 = 1.229626738

gm1 ÷ gm4 = 1

3, α

c = 2.759994030

gm1 = 2.759994030

gm2 = 1

1, β

c1 = 6.439143780

c2 = 0.151805169

gm1 ÷ gm4 = 1

2, β

c1 = 2.356890202

c2 = 0.719119713

gm1 ÷ gm4 = 1

3, β

c1 = 1.725363376

c2 = 3.691704548

gm1 ÷ gm4 = 1

1, β

c1 = 8.771438925

c2 = 0.112199132

gm1 ÷ gm4 = 1

2, β

c1 = 3.130490308

c2 = 0.471925758

gm1 ÷ gm4 = 1

3, β

c1 = 2.166366510

c2 = 1.818204388

gm1 ÷ gm4 = 1

4, α

c = 3.903657563

gm1 = 3.903657563

gm2 = 1

1, β

c1 = 11.462611642

c2 = 0.086211489

gm1 ÷ gm4 = 1

2, β

c1 = 4.025133932

c2 = 0.335124248

gm1 ÷ gm4 = 1

3, β

c1 = 2.689508508

c2 = 1.036706389

gm1 ÷ gm4 = 1

4, β

c1 = 2.280055218

c2 = 4.980968014

gm1 ÷ gm4 = 1

1, β

c1 = 14.512643709

c2 = 0.068276540

gm1 ÷ gm4 = 1

2, β

c1 = 5.040188266

c2 = 0.251348075

gm1 ÷ gm4 = 1

3, β

c1 = 3.289749251

c2 = 0.671706707

gm1 ÷ gm4 = 1

4, β

c1 = 2.681828161

c2 = 2.385020837

gm1 ÷ gm4 = 1

5, α

c = 5.040188266

gm1 = 5.040188266

gm2 = 1

1, β

c1 = 17.921525493

c2 = 0.055392493

gm1 ÷ gm4 = 1

2, β

c1 = 6.175336829

c2 = 0.196117869

gm1 ÷ gm4 = 1

3, β

c1 = 3.964810306

c2 = 0.474267527

gm1 ÷ gm4 = 1

4, β

c1 = 3.146491274

c2 = 1.335833890

gm1 ÷ gm4 = 1

5, β

c1 = 2.838490785

c2 = 6.259890499

gm1 ÷ gm4 = 1

1, β

c1 = 21.689251517

c2 = 0.045831454

gm1 ÷ gm4 = 1

2, β

c1 = 7.430406281

c2 = 0.157651331

gm1 ÷ gm4 = 1

3, β

c1 = 4.713524818

c2 = 0.355133339

gm1 ÷ gm4 = 1

4, β

c1 = 3.669170115

c2 = 0.855617204

gm1 ÷ gm4 = 1

5, β

c1 = 3.217013992

c2 = 2.943282984

gm1 ÷ gm4 = 1

6, α

c = 6.173404644

gm1 = 6.173404644

gm2 = 1

1, β

c1 = 25.815818474

c2 = 0.038543966

gm1 ÷ gm4 = 1

2, β

c1 = 8.805294925

c2 = 0.129707371

gm1 ÷ gm4 = 1

3, β

c1 = 5.535239791

c2 = 0.277330494

gm1 ÷ gm4 = 1

4, β

c1 = 4.247338876

c2 = 0.599685702

gm1 ÷ gm4 = 1

5, β

c1 = 3.647272579

c2 = 1.627510323

gm1 ÷ gm4 = 1

6, β

c1 = 3.398716980

c2 = 7.533732679

gm1 ÷ gm4 = 1

4 5

6

7

8

9

10

11

12

8.3 Polynomial Filters

167

Table 8.8 Element values of the cascaded Gm-C Chebyshev (amax = 1 dB) filters n

Cell no., type

3

1, β

c1 = 2.023592642

c2 = 0.497051222

gm1 ÷ gm4 = 1

2, α

c = 2.023592642

gm1 = 2.023592642

gm2 = 1

1, β

c1 = 3.583304772

c2 = 0.282889623

gm1 ÷ gm4 = 1

2, β

c1 = 1.484253435

c2 = 2.411395789

gm1 ÷ gm4 = 1

1, β

c1 = 5.589192421

c2 = 0.181032104

gm1 ÷ gm4 = 1

2, β

c1 = 2.134881535

c2 = 1.091107290

gm1 ÷ gm4 = 1

3, α

c = 3.454310886

gm1 = 3.454310886

gm2 = 1

1, β

c1 = 5.589192421

c2 = 0.181032104

gm1 ÷ gm4 = 1

2, β

c1 = 2.134881535

c2 = 1.091107290

gm1 ÷ gm4 = 1

3, β

c = 3.454310886

gm1 = 3.454310886

gm1 ÷ gm4 = 1

1, β

c1 = 10.93876926

c2 = 0.092092126

gm1 ÷ gm4 = 1

2, β

c1 = 3.904001550

c2 = 0.391989109

gm1 ÷ gm4 = 1

3, β

c1 = 2.701652898

c2 = 1.606177310

gm1 ÷ gm4 = 1

4, α

c = 4.868210306

gm1 = 4.868210306

gm2 = 1

1, β

c1 = 14.28235454

c2 = 0.070429130

gm1 ÷ gm4 = 1

2, β

c1 = 5.015295964

c2 = 0.275574657

gm1 ÷ gm4 = 1

3, β

c1 = 3.351113626

c2 = 0.875458894

gm1 ÷ gm4 = 1

4, β

c1 = 2.840936954

c2 = 5.009828112

gm1 ÷ gm4 = 1

1, β

c1 = 18.071779736

c2 = 0.055599965

gm1 ÷ gm4 = 1

2, β

c1 = 6.276263237

c2 = 0.205485311

gm1 ÷ gm4 = 1

3, β

c1 = 4.096539889

c2 = 0.556610895

gm1 ÷ gm4 = 1

4, β

c1 = 3.339529915

c2 = 2.103364527

gm1 ÷ gm4 = 1

5, α

c = 6.276263237

gm1 = 6.276263237

gm2 = 1

1, β

c1 = 22.30703722

c2 = 0.045006301

gm1 ÷ gm4 = 1

2, β

c1 = 7.686481185

c2 = 0.159743259

gm1 ÷ gm4 = 1

3, β

c1 = 4.935024705

c2 = 0.389282437

gm1 ÷ gm4 = 1

4, β

c1 = 3.916457781

c2 = 1.126613121

gm1 ÷ gm4 = 1

5, b

c1 = 3.533087605

c2 = 6.289486339

gm1 ÷ gm4 = 1

1, β

c1 = 26.98812268

c2 = 0.037176258

gm1 ÷ gm4 = 1

2, β

c1 = 9.245718605

c2 = 0.128092209

gm1 ÷ gm4 = 1

3, β

c1 = 5.865079574

c2 = 0.289915614

gm1 ÷ gm4 = 1

4, β

c1 = 4.565580012

c2 = 0.708286572

gm1 ÷ gm4 = 1

5, β

c1 = 4.002958250

c2 = 2.593588999

gm1 ÷ gm4 = 1

6, α

c = 7.681620629

gm1 = 7.681620629

gm2 = 1

1, β

c1 = 32.11503352

c2 = 0.031225797

gm1 ÷ gm4 = 1

2, β

c1 = 10.95383987

c2 = 0.105201890

gm1 ÷ gm4 = 1

3, β

c1 = 6.885871606

c2 = 0.225631680

gm1 ÷ gm4 = 1

4, β

c1 = 5.283715119

c2 = 0.491818973

gm1 ÷ gm4 = 1

5, β

c1 = 4.537229034

c2 = 1.371713638

gm1 ÷ gm4 = 1

6, β

c1 = 4.228024374

c2 = 7.565020936

gm1 ÷ gm4 = 1

4 5

6

7

8

9

10

11

12

168

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Fig. 8.11 Passband (left) and the overall attenuation characteristic of a seventh-order Thomson filter

8.3.6 Thomson (Bessel or Maximally Flat Group Delay) Filters The attenuation characteristic of the seventh-order Thomson filter is depicted in Fig. 8.11 (Table 8.9).

8.3 Polynomial Filters

169

Table 8.9 Element values of the cascaded Gm-C Thomson filters n

Cell no., type

3

1, β

c1 = 0.476619725

c2 = 0.998061517

gm1 ÷ gm4 = 1

2, α

c = 0.754857489

gm1 = 0.754857489

gm2 = 1

1, β

c1 = 0.501619806

c2 = 0.773040606

gm1 ÷ gm4 = 1

2, β

c1 = 0.364373512

c2 = 1.337564252

gm1 ÷ gm4 = 1

1, β

c1 = 0.521260423

c2 = 0.620599266

gm1 ÷ gm4 = 1

2, β

c1 = 0.361508495

c2 = 1.138349867

gm1 ÷ gm4 = 1

3, α

c = 0.664572291

gm1 = 0.664572291

gm2 = 1

1, β

c1 = 0.536378467

c2 = 0.512216474

gm1 ÷ gm4 = 1

2, β

c1 = 0.361241230

c2 = 0.967026490

gm1 ÷ gm4 = 1

3, β

c1 = 0.317650122

c2 = 1.219740800

gm1 ÷ gm4 = 1

1, β

c1 = 0.548623442

c2 = 0.432512771

gm1 ÷ gm4 = 1

2, β

c1 = 0.362008579

c2 = 0.828992648

gm1 ÷ gm4 = 1

3, β

c1 = 0.309654336

c2 = 1.092630582

gm1 ÷ gm4 = 1

4, α

c = 0.592714585

gm1 = 0.592714585

gm2 = 1

1, β

c1 = 0.559061758

c2 = 0.372145765

gm1 ÷ gm4 = 1

2, β

c1 = 0.363338433

c2 = 0.719039698

gm1 ÷ gm4 = 1

3, β

c1 = 0.304940616

c2 = 0.973745888

gm1 ÷ gm4 = 1

4, β

c1 = 0.284037174

c2 = 1.109403366

gm1 ÷ gm4 = 1

1, β

c1 = 0.568267153

c2 = 0.325197903

gm1 ÷ gm4 = 1

2, β

c1 = 0.364996873

c2 = 0.630904949

gm1 ÷ gm4 = 1

3, β

c1 = 0.302085765

c2 = 0.869562913

gm1 ÷ gm4 = 1

4, β

c1 = 0.276213808

c2 = 1.022646506

gm1 ÷ gm4 = 1

5, α

c = 0.537719833

gm1 = 0.537719833

gm2 = 1

1, β

c1 = 0.576562180

c2 = 0.287835061

gm1 ÷ gm4 = 1

2, β

c1 = 0.366844653

c2 = 0.559417875

gm1 ÷ gm4 = 1

3, β

c1 = 0.300372855

c2 = 0.780223251

gm1 ÷ gm4 = 1

4, β

c1 = 0.270960647

c2 = 0.937702367

gm1 ÷ gm4 = 1

5, β

c1 = 0.258952888

c2 = 1.019788486

gm1 ÷ gm4 = 1

1, β

c1 = 0.584146312

c2 = 0.257507042

gm1 ÷ gm4 = 1

2, β

c1 = 0.368795927

c2 = 0.500672908

gm1 ÷ gm4 = 1

3, β

c1 = 0.299401558

c2 = 0.704021587

gm1 ÷ gm4 = 1

4, β

c1 = 0.267307880

c2 = 0.859252349

gm1 ÷ gm4 = 1

5, β

c1 = 0.252080746

c2 = 0.956780075

gm1 ÷ gm4 = 1

6, α

c = 0.495026537

gm1 = 0.495026537

gm2 = 1

1, β

c1 = 0.591155052

c2 = 0.232469977

gm1 ÷ gm4 = 1

2, β

c1 = 0.370797968

c2 = 0.451784847

gm1 ÷ gm4 = 1

3, β

c1 = 0.298932665

c2 = 0.638926535

gm1 ÷ gm4 = 1

4, β

c1 = 0.264714521

c2 = 0.788623835

gm1 ÷ gm4 = 1

5, β

c1 = 0.247114949

c2 = 0.893373132

gm1 ÷ gm4 = 1

6, β

c1 = 0.239445543

c2 = 0.947311909

gm1 ÷ gm4 = 1

4 5

6

7

8

9

10

11

12

170

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Fig. 8.12 Normalized group delay and the overall attenuation characteristic of a seventh-order Equi-rip-td filter

8.3.7 Equi-ripple Group Delay Filters The attenuation characteristic of the seventh-order Equi-rip-td filter is depicted in Fig. 8.12. Due to limited space, only four values of δ = {1; 2; 5; 10} % will be synthesized (Tables 8.10, 8.11, 8.12 and 8.13).

8.3 Polynomial Filters

171

Table 8.10 Element values of the cascaded Gm-C of Equi-rip-td filters (error δ = 1%) n

Cell no., type

3

1, β

c1 = 0.628726378

c2 = 0.862025003

gm1 ÷ gm4 = 1

2, α

c = 1.045919935

gm1 = 1.045919935

gm2 = 1

1, β

c1 = 0.733380592

c2 = 0.529476184

gm1 ÷ gm4 = 1

2, β

c1 = 0.585026346

c2 = 1.745856190

gm1 ÷ gm4 = 1

1, β

c1 = 0.819942821

c2 = 0.345338911

gm1 ÷ gm4 = 1

2, β

c1 = 0.644821363

c2 = 1.073167209

gm1 ÷ gm4 = 1

3, α

c = 1.236184337

gm1 = 1.236184337

gm2 = 1

1, β

c1 = 0.906087288

c2 = 0.243715044

gm1 ÷ gm4 = 1

2, β

c1 = 0.708164554

c2 = 0.664874872

gm1 ÷ gm4 = 1

3, β

c1 = 0.668855573

c2 = 2.007134093

gm1 ÷ gm4 = 1

1, β

c1 = 0.972367733

c2 = 0.179117757

gm1 ÷ gm4 = 1

2, β

c1 = 0.757574563

c2 = 0.435324835

gm1 ÷ gm4 = 1

3, β

c1 = 0.710827637

c2 = 1.207035271

gm1 ÷ gm4 = 1

4, α

c = 1.400638350

gm1 = 1.400638350

gm2 = 1

1, β

c1 = 1.056363094

c2 = 0.140553953

gm1 ÷ gm4 = 1

2, β

c1 = 0.821548078

c2 = 0.311846629

gm1 ÷ gm4 = 1

3, β

c1 = 0.768214312

c2 = 0.748854788

gm1 ÷ gm4 = 1

4, β

c1 = 0.751804297

c2 = 2.252774243

gm1 ÷ gm4 = 1

1, β

c1 = 1.102512695

c2 = 0.110413252

gm1 ÷ gm4 = 1

2, β

c1 = 0.856503796

c2 = 0.228196935

gm1 ÷ gm4 = 1

3, β

c1 = 0.799310395

c2 = 0.484183495

gm1 ÷ gm4 = 1

4, β

c1 = 0.779485825

c2 = 1.324777840

gm1 ÷ gm4 = 1

5, α

c = 1.548525281

gm1 = 1.548525281

gm2 = 1

1, β

c1 = 1.189737512

c2 = 0.092639863

gm1 ÷ gm4 = 1

2, β

c1 = 0.923615550

c2 = 0.181060305

gm1 ÷ gm4 = 1

3, β

c1 = 0.860870679

c2 = 0.348692907

gm1 ÷ gm4 = 1

4, β

c1 = 0.837792661

c2 = 0.821993234

gm1 ÷ gm4 = 1

5, b

c1 = 0.829231399

c2 = 2.480181362

gm1 ÷ gm4 = 1

1, β

c1 = 1.219439114

c2 = 0.075785392

gm1 ÷ gm4 = 1

2, β

c1 = 0.946219934

c2 = 0.141636373

gm1 ÷ gm4 = 1

3, β

c1 = 0.881212497

c2 = 0.252976984

gm1 ÷ gm4 = 1

4, β

c1 = 0.856475006

c2 = 0.525298678

gm1 ÷ gm4 = 1

5, b

c1 = 0.845959297

c2 = 1.434810373

gm1 ÷ gm4 = 1

6, α

c = 1.685915241

gm1 = 1.685915241

gm2 = 1

1, β

c1 = 1.307691831

c2 = 0.066269800

gm1 ÷ gm4 = 1

2, β

c1 = 1.014358613

c2 = 0.119416230

gm1 ÷ gm4 = 1

3, β

c1 = 0.944130092

c2 = 0.200997700

gm1 ÷ gm4 = 1

4, β

c1 = 0.916829192

c2 = 0.378006384

gm1 ÷ gm4 = 1

5, b

c1 = 0.904378551

c2 = 0.887410403

gm1 ÷ gm4 = 1

6, β

c1 = 0.899288154

c2 = 2.685614261

gm1 ÷ gm4 = 1

4 5

6

7

8

9

10

11

12

172

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.11 Element values of the cascaded Gm-C of Equi-rip-td filters (error δ = 2%) n

Cell no., type

3

1, β

c1 = 0.676213857

c2 = 0.814973588

gm1 ÷ gm4 = 1

2, α

c = 1.142280520

gm1 = 1.142280520

gm2 = 1

1, β

c1 = 0.791324975

c2 = 0.483033092

gm1 ÷ gm4 = 1

2, β

c1 = 0.645237807

c2 = 1.794346671

gm1 ÷ gm4 = 1

1, β

c1 = 0.886745597

c2 = 0.310739476

gm1 ÷ gm4 = 1

2, β

c1 = 0.714692274

c2 = 1.015598155

gm1 ÷ gm4 = 1

3, α

c = 1.378997802

gm1 = 1.378997802

gm2 = 1

1, β

c1 = 0.983060104

c2 = 0.218591104

gm1 ÷ gm4 = 1

2, β

c1 = 0.788381283

c2 = 0.604813625

gm1 ÷ gm4 = 1

3, β

c1 = 0.751379799

c2 = 2.090798506

gm1 ÷ gm4 = 1

1, β

c1 = 1.050436233

c2 = 0.159587609

gm1 ÷ gm4 = 1

2, β

c1 = 0.840298361

c2 = 0.386750042

gm1 ÷ gm4 = 1

3, β

c1 = 0.796561560

c2 = 1.145687249

gm1 ÷ gm4 = 1

4, α

c = 1.573660597

gm1 = 1.573660597

gm2 = 1

1, β

c1 = 1.154155373

c2 = 0.126587913

gm1 ÷ gm4 = 1

2, β

c1 = 0.921962681

c2 = 0.277682021

gm1 ÷ gm4 = 1

3, β

c1 = 0.871534805

c2 = 0.687224978

gm1 ÷ gm4 = 1

4, β

c1 = 0.856181302

c2 = 2.374651453

gm1 ÷ gm4 = 1

1, β

c1 = 1.186336329

c2 = 0.097969101

gm1 ÷ gm4 = 1

2, β

c1 = 0.946853683

c2 = 0.199252727

gm1 ÷ gm4 = 1

3, β

c1 = 0.893615497

c2 = 0.428424405

gm1 ÷ gm4 = 1

4, β

c1 = 0.875352340

c2 = 1.256567162

gm1 ÷ gm4 = 1

5, α

c = 1.741099489

gm1 = 1.741099489

gm2 = 1

1, β

c1 = 1.306546094

c2 = 0.083951873

gm1 ÷ gm4 = 1

2, β

c1 = 1.042222681

c2 = 0.161017342

gm1 ÷ gm4 = 1

3, β

c1 = 0.982626623

c2 = 0.311576217

gm1 ÷ gm4 = 1

4, β

c1 = 0.960927989

c2 = 0.759971751

gm1 ÷ gm4 = 1

5, b

c1 = 0.952892178

c2 = 2.635636344

gm1 ÷ gm4 = 1

1, β

c1 = 1.307256595

c2 = 0.067102074

gm1 ÷ gm4 = 1

2, β

c1 = 1.042399880

c2 = 0.122848828

gm1 ÷ gm4 = 1

3, β

c1 = 0.982132532

c2 = 0.219433873

gm1 ÷ gm4 = 1

4, β

c1 = 0.959425170

c2 = 0.463754655

gm1 ÷ gm4 = 1

5, b

c1 = 0.949786303

c2 = 1.358878941

gm1 ÷ gm4 = 1

6, α

c = 1.894069492

gm1 = 1.894069492

gm2 = 1

1, β

c1 = 1.437166927

c2 = 0.060212075

gm1 ÷ gm4 = 1

2, β

c1 = 1.145691902

c2 = 0.106158932

gm1 ÷ gm4 = 1

3, β

c1 = 1.078948816

c2 = 0.178190266

gm1 ÷ gm4 = 1

4, β

c1 = 1.053253905

c2 = 0.338274601

gm1 ÷ gm4 = 1

5, b

c1 = 1.041549484

c2 = 0.822562374

gm1 ÷ gm4 = 1

6, β

c1 = 1.036762506

c2 = 2.861744289

gm1 ÷ gm4 = 1

4 5

6

7

8

9

10

11

12

8.3 Polynomial Filters

173

Table 8.12 Element values of the cascaded Gm-C of Equi-rip-td filters (error δ = 5%) n

Cell no., type

3

1, β

c1 = 0.764798828

c2 = 0.722187036

gm1 ÷ gm4 = 1

2, α

c = 1.328027116

gm1 = 1.328027116

gm2 = 1

1, β

c1 = 0.904939309

c2 = 0.411292463

gm1 ÷ gm4 = 1

2, β

c1 = 0.765596559

c2 = 1.837548692

gm1 ÷ gm4 = 1

1, β

c1 = 1.015422761

c2 = 0.260247281

gm1 ÷ gm4 = 1

2, β

c1 = 0.852033124

c2 = 0.906039269

gm1 ÷ gm4 = 1

3, α

c = 1.659241135

gm1 = 1.659241135

gm2 = 1

1, β

c1 = 1.130196418

c2 = 0.182711600

gm1 ÷ gm4 = 1

2, β

c1 = 0.945051083

c2 = 0.511310056

gm1 ÷ gm4 = 1

3, β

c1 = 0.912147050

c2 = 2.174236022

gm1 ÷ gm4 = 1

1, β

c1 = 1.196623220

c2 = 0.131996879

gm1 ÷ gm4 = 1

2, β

c1 = 0.998860060

c2 = 0.316439608

gm1 ÷ gm4 = 1

3, β

c1 = 0.960376681

c2 = 1.022500294

gm1 ÷ gm4 = 1

4, α

c = 1.903804925

gm1 = 1.903804925

gm2 = 1

1, β

c1 = 1.355854964

c2 = 0.108031752

gm1 ÷ gm4 = 1

2, β

c1 = 1.130680265

c2 = 0.231913965

gm1 ÷ gm4 = 1

3, β

c1 = 1.084946427

c2 = 0.595223689

gm1 ÷ gm4 = 1

4, β

c1 = 1.071160271

c2 = 2.537920085

gm1 ÷ gm4 = 1

1, β

c1 = 1.335625758

c2 = 0.080231728

gm1 ÷ gm4 = 1

2, β

c1 = 1.113160519

c2 = 0.158755594

gm1 ÷ gm4 = 1

3, β

c1 = 1.066896850

c2 = 0.346454428

gm1 ÷ gm4 = 1

4, β

c1 = 1.051174616

c2 = 1.112981568

gm1 ÷ gm4 = 1

5, α

c = 2.094085068

gm1 = 2.094085068

gm2 = 1

1, β

c1 = 1.556217523

c2 = 0.072853548

gm1 ÷ gm4 = 1

2, β

c1 = 1.296532955

c2 = 0.135465216

gm1 ÷ gm4 = 1

3, β

c1 = 1.241745984

c2 = 0.263294600

gm1 ÷ gm4 = 1

4, β

c1 = 1.221974237

c2 = 0.669445879

gm1 ÷ gm4 = 1

5, β

c1 = 1.214651568

c2 = 2.865988667

gm1 ÷ gm4 = 1

1, β

c1 = 1.453598991

c2 = 0.054453126

gm1 ÷ gm4 = 1

2, β

c1 = 1.210738515

c2 = 0.096427038

gm1 ÷ gm4 = 1

3, β

c1 = 1.159016034

c2 = 0.172027979

gm1 ÷ gm4 = 1

4, β

c1 = 1.139691523

c2 = 0.371140693

gm1 ÷ gm4 = 1

5, β

c1 = 1.131483451

c2 = 1.192081740

gm1 ÷ gm4 = 1

6, α

c = 2.258275590

gm1 = 2.258275590

gm2 = 1

1, β

c1 = 1.706885031

c2 = 0.052275671

gm1 ÷ gm4 = 1

2, β

c1 = 1.421465593

c2 = 0.089007777

gm1 ÷ gm4 = 1

3, β

c1 = 1.360283437

c2 = 0.148716953

gm1 ÷ gm4 = 1

4, β

c1 = 1.336918453

c2 = 0.285334428

gm1 ÷ gm4 = 1

5, β

c1 = 1.326263694

c2 = 0.724203896

gm1 ÷ gm4 = 1

6, β

c1 = 1.321898742

c2 = 3.110027187

gm1 ÷ gm4 = 1

4 5

6

7

8

9

10

11

12

174

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.13 Element values of the cascaded Gm-C of Equi-rip-td filters (error δ = 10%) n

Cell no., type

3

1, β

c1 = 0.849406724

c2 = 0.611885322

gm1 ÷ gm4 = 1

2, α

c = 1.515803346

gm1 = 1.515803346

gm2 = 1

1, β

c1 = 1.044983246

c2 = 0.349667496

gm1 ÷ gm4 = 1

2, β

c1 = 0.915066249

c2 = 1.829312094

gm1 ÷ gm4 = 1

1, β

c1 = 1.167804230

c2 = 0.218126301

gm1 ÷ gm4 = 1

2, β

c1 = 1.016680404

c2 = 0.791727706

gm1 ÷ gm4 = 1

3, α

c = 1.994137053

gm1 = 1.994137053

gm2 = 1

1, β

c1 = 1.303833400

c2 = 0.153258532

gm1 ÷ gm4 = 1

2, β

c1 = 1.132344394

c2 = 0.429658132

gm1 ÷ gm4 = 1

3, β

c1 = 1.103512447

c2 = 2.176975620

gm1 ÷ gm4 = 1

1, β

c1 = 1.366968374

c2 = 0.109677122

gm1 ÷ gm4 = 1

2, β

c1 = 1.185740951

c2 = 0.259268391

gm1 ÷ gm4 = 1

3, β

c1 = 1.152359090

c2 = 0.891379486

gm1 ÷ gm4 = 1

4, α

c = 2.290078069

gm1 = 2.290078069

gm2 = 1

1, β

c1 = 1.643975366

c2 = 0.095402900

gm1 ÷ gm4 = 1

2, β

c1 = 1.425069874

c2 = 0.200433457

gm1 ÷ gm4 = 1

3, β

c1 = 1.382960204

c2 = 0.526848323

gm1 ÷ gm4 = 1

4, β

c1 = 1.370309292

c2 = 2.680643327

gm1 ÷ gm4 = 1

1, β

c1 = 1.503043567

c2 = 0.065867315

gm1 ÷ gm4 = 1

2, β

c1 = 1.302383849

c2 = 0.127024580

gm1 ÷ gm4 = 1

3, β

c1 = 1.262843003

c2 = 0.279698725

gm1 ÷ gm4 = 1

4, β

c1 = 1.249441116

c2 = 0.958729331

gm1 ÷ gm4 = 1

5, α

c = 2.491829216

gm1 = 2.491829216

gm2 = 1

1, β

c1 = 1.895923444

c2 = 0.064858347

gm1 ÷ gm4 = 1

2, β

c1 = 1.642406721

c2 = 0.117240273

gm1 ÷ gm4 = 1

3, β

c1 = 1.591707286

c2 = 0.228327112

gm1 ÷ gm4 = 1

4, β

c1 = 1.573448999

c2 = 0.596758494

gm1 ÷ gm4 = 1

5, b

c1 = 1.566673743

c2 = 3.048270402

gm1 ÷ gm4 = 1

1, β

c1 = 1.609640884

c2 = 0.044134035

gm1 ÷ gm4 = 1

2, β

c1 = 1.394174873

c2 = 0.075849169

gm1 ÷ gm4 = 1

3, β

c1 = 1.350664816

c2 = 0.135083716

gm1 ÷ gm4 = 1

4, β

c1 = 1.334435706

c2 = 0.295291172

gm1 ÷ gm4 = 1

5, β

c1 = 1.327525071

c2 = 1.012469731

gm1 ÷ gm4 = 1

6, α

c = 2.651094162

gm1 = 2.651094162

gm2 = 1

1, β

c1 = 2.048400577

c2 = 0.045987003

gm1 ÷ gm4 = 1

2, β

c1 = 1.774005073

c2 = 0.075900525

gm1 ÷ gm4 = 1

3, β

c1 = 1.718221826

c2 = 0.126319377

gm1 ÷ gm4 = 1

4, β

c1 = 1.696947036

c2 = 0.243948045

gm1 ÷ gm4 = 1

5, β

c1 = 1.687216734

c2 = 0.637024822

gm1 ÷ gm4 = 1

6, β

c1 = 1.683221917

c2 = 3.263041129

gm1 ÷ gm4 = 1

4 5

6

7

8

9

10

11

12

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

175

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis The transfer function of this type of filters may be expressed as follows: H (s) = H0 · Pn (0) · Q m (s)/.[Q m (0) · Pn (s)]

(8.6)

where Pn (s) is a Hurwitz polynomial and Qm (s) has real coefficients and its zeros are on the imaginary axis symmetrically distributed with respect to the origin of the complex frequency plane. Here m = 2 · (n − 1)/2 and . denotes the floor function. The quantity H 0 is used for the transfer functions which exhibit attenuation at the origin (even order elliptic) and is calculated from (8.5). In addition to the element values, the following tables contain a specific information about the selectivity of the filter under consideration. That is the normalized frequency at which the attenuation reaches amin for the first time (f s ), i.e., the upper end of the transition band or the beginning of the stopband. The meaning of this quantity is illustrated by Fig. 8.13. Note that all xxx_Z filters have 3 dB at the cut-off frequency as shown by Fig. 8.14 to unify and to allow for comparisons, for the purpose of calculating the value of f s , the transfer functions of the modified elliptic filters were renormalized so that they exhibit 3 dB at the cutoff (this will be illustrated graphically in the corresponding paragraph). The element values of the modified elliptic filters, of course, are calculated from the original (not renormalized) transfer functions. Alike the polynomial filters, higher-order filters having transmission zeros at the imaginary axis exhibit extreme selectivity. That was the reason to reduce the maximum order of the filter considered in this case to n = 11. Seven values of the minimum stopband attenuation were used: amin = {30; 35; 40; 45; 50; 55; 60}.

Fig. 8.13 Meaning of f s and amin

176

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Fig. 8.14 Meaning of amax for monotonic passband attenuation response

8.4.1 LSM_Z Filters See Tables 8.14, 8.15, 8.16, 8.17, 8.18, 8.19 and 8.20. Table 8.14 Element values of the cascaded Gm-C LSM_Z filters (amin = 30 dB) n

fs

Cell no., type

3

1.836

1, γ

c1 ÷ c4 = 0.786438932

gm1 = 5.413280616

gm2 = gm3 = 1

2, α

c= 1.378357460

gm1 = 1.378357460

gm2 = 1

1, γ

c1 ÷ c4 = 1.274278288

gm1 = 7.890398990

gm2 = gm3 = 1

2, β

c1 = 0.781268460

c2 = 2.052471052

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.941436842

gm1 = 12.47640856

gm2 = gm3 = 1

gm4 = 3.742616311

2, γ

c1 ÷ c4 = 0.217435278

gm1 = 0.223757510

gm2 = gm3 = 1

gm1 = 0.223757510

3, α

c= 0.500154989

gm1 = 0.500154989

gm2 = 1

1, γ

c1 ÷ c4 = 1.570210429

gm1 = 17.97339553

gm2 = gm3 = 1

gm4 = 10.09200755

2, γ

c1 ÷ c4 = 0.356014062

gm1 = 0.438062068

gm2 = gm3 = 1

gm4 = 0.681586126

3, β

c1 = 0.299373318

c2 = 1.097839738

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.587886385

gm1 = 4.964104865

gm2 = gm3 = 1

4

5

6

7

1.531

1.450

1.239

1.176

gm4 = 2.457959019

gm4 = 6.562143396

gm4 = 1.734104780 (continued)

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

177

Table 8.14 (continued) n

8

9

10

11

fs

1.107

1.075

1.055

1.042

Cell no., type 2, γ

c1 ÷ c4 = 2.400154122

gm1 = 24.97218400

gm2 = gm3 = 1

gm4 = 23.54422914

3, γ

c1 ÷ c4 = 0.214007284

gm1 = 0.129085627

gm2 = gm3 = 1

gm4 = 0.418807988

4, α

c= 0.512620488

gm1 = 0.512620488

gm2 = 1

1, γ

c1 ÷ c4 = 0.307722141

gm1 = 0.770581657

gm2 = gm3 = 1

gm4 = 0.664725561

2, γ

c1 ÷ c4 = 0.857636587

gm1 = 2.534259854

gm2 = gm3 = 1

gm4 = 3.383452251

3, γ

c1 ÷ c4 = 3.530589787

gm1 = 31.61529180

gm2 = gm3 = 1

gm4 = 50.50382792

4, β

c1 = 0.294945021

c2 = 1.054499267

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.482440103

gm1 = 3.917700120

gm2 = gm3 = 1

gm4 = 1.422704441

2, γ

c1 ÷ c4 = 0.197040536

gm1 = 0.191597882

gm2 = gm3 = 1

gm4 = 0.430938677

3, γ

c1 ÷ c4 = 1.267504531

gm1 = 4.661478970

gm2 = gm3 = 1

gm4 = 7.244316888

4, γ

c1 ÷ c4 = 4.857063319

gm1 = 55.79776671

gm2 = gm3 = 1

gm4 = 95.51746482

5, α

c= 0.482242493

gm1 = 0.482242493

gm2 = 1

1, γ

c1 ÷ c4 = 0.272745436

gm1 = 0.722918283

gm2 = gm3 = 1

gm4 = 0.658459289

2, γ

c1 ÷ c4 = 0.662840840

gm1 = 1.730281996

gm2 = gm3 = 1

gm4 = 2.411209866

3, γ

c1 ÷ c4 = 1.712583996

gm1 = 7.743716134

gm2 = gm3 = 1

gm4 = 12.78428177

4, γ

c1 ÷ c4 = 6.605164958

gm1 = 98.81290469

gm2 = gm3 = 1

gm4 = 175.9679259

5, β

c1 = 0.273081999

c2 = 0.966265385

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.412925241

gm1 = 3.468074042

gm2 = gm3 = 1

gm4 = 1.273974278

2, γ

c1 ÷ c4 = 0.177738612

gm1 = 0.184306568

gm2 = gm3 = 1

gm4 = 0.431553410

3, γ

c1 ÷ c4 = 0.959860798

gm1 = 2.999641679

gm2 = gm3 = 1

gm4 = 4.733676164

4, γ

c1 ÷ c4 = 2.325688102

gm1 = 13.30361039

gm2 = gm3 = 1

gm4 = 23.29339793

5, γ

c1 ÷ c4 = 8.502601125

gm1 = 158.9540737

gm2 = gm3 = 1

gm4 = 291.4011459

6, α

c= 0.439208473

gm1 = 0.439208473

gm2 = 1

178

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.15 Element values of the cascaded Gm-C LSM_Z filters (amin = 35 dB) n

fs

Cell no., type

3

2.151

1, γ

c1 ÷ c4 = 0.770582311

gm1 = 7.181399149

gm2 = gm3 = 1

gm4 = 2.323462420

2, α

c= 1.468170025

gm1 = 1.468170025

gm2 = 1

c= 1.468170025

1, γ

c1 ÷ c4 = 1.234216857

gm1 = 9.118281735

gm2 = gm3 = 1

gm4 = 6.096285225

2, β

c1 = 0.823594695

c2 = 2.181938582

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.882375722

gm1 = 12.98489719

gm2 = gm3 = 1

gm4 = 3.248820345

2, γ

c1 ÷ c4 = 0.215246427

gm1 = 0.264161534

gm2 = gm3 = 1

gm4 = 0.353537827

3, α

c= 0.533093360

gm1 = 0.533093360

gm2 = 1

1, γ

c1 ÷ c4 = 1.460342833

gm1 = 17.68647044

gm2 = gm3 = 1

gm4 = 8.651593590

2, γ

c1 ÷ c4 = 0.349112903

gm1 = 0.479782911

gm2 = gm3 = 1

gm4 = 0.603533111

3, β

c1 = 0.312995183

c2 = 1.170133827

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 2.188408352

gm1 = 75.20141524

gm2 = gm3 = 1

gm4 = 19.47818291

2, γ

c1 ÷ c4 = 0.554407786

gm1 = 1.457240367

gm2 = gm3 = 1

gm4 = 1.469162209

3, γ

c1 ÷ c4 = 0.217030245

gm1 = 0.144679655

gm2 = gm3 = 1

gm4 = 0.381370734

4, α

c= 0.562498538

gm1 = 0.562498538

gm2 = 1

1, γ

c1 ÷ c4 = 0.800259071

gm1 = 5.589930746

gm2 = gm3 = 1

gm4 = 2.835064419

2, γ

c1 ÷ c4 = 3.209305311

gm1 = 37.93788762

gm2 = gm3 = 1

gm4 = 41.57323892

3, γ

c1 ÷ c4 = 0.304202462

gm1 = 0.249639806

gm2 = gm3 = 1

gm4 = 0.585763743

4, β

c1 = 0.316678768

c2 = 1.156267588

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.457923038

gm1 = 3.726239412

gm2 = gm3 = 1

gm4 = 1.197645309

2, γ

c1 ÷ c4 = 1.159873196

gm1 = 6.994972160

gm2 = gm3 = 1

gm4 = 5.934262370

3, γ

c1 ÷ c4 = 4.371328382

gm1 = 58.14995638

gm2 = gm3 = 1

gm4 = 77.20491846

4, γ

c1 ÷ c4 = 0.202646603

gm1 = 0.101491278

gm2 = gm3 = 1

gm4 = 0.393602122

5, α

c= 0.539449606

gm1 = 0.539449606

gm2 = 1

4

5

6

7

8

9

1.596

1.593

1.349

1.210

1.142

1.099

(continued)

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

179

Table 8.15 (continued) n

fs

Cell no., type

10

1.072

1, γ

c1 ÷ c4 = 0.620457160

gm1 = 3.908706534

gm2 = gm3 = 1

gm4 = 2.005301623

2, γ

c1 ÷ c4 = 0.271485419

gm1 = 0.302399756

gm2 = gm3 = 1

gm4 = 0.581637024

3, γ

c1 ÷ c4 = 1.558923643

gm1 = 6.654558909

gm2 = gm3 = 1

gm4 = 10.41154214

4, γ

c1 ÷ c4 = 5.937039954

gm1 = 82.51931398

gm2 = gm3 = 1

gm4 = 141.9241497

5, β

c1 = 0.297498649

c2 = 1.073974626

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.393443018

gm1 = 3.264437874

gm2 = gm3 = 1

gm4 = 1.071757327

2, γ

c1 ÷ c4 = 0.881130960

gm1 = 4.688052013

gm2 = gm3 = 1

gm4 = 3.857298506

3, γ

c1 ÷ c4 = 0.184224981

gm1 = 0.113985937

gm2 = gm3 = 1

gm4 = 0.395082251

4, γ

c1 ÷ c4 = 2.097468635

gm1 = 11.12182800

gm2 = gm3 = 1

gm4 = 18.74319028

5, γ

c1 ÷ c4 = 7.600113633

gm1 = 130.2367086

gm2 = gm3 = 1

gm4 = 232.5841532

6, α

c= 0.496166717

gm1 = 0.496166717

gm2 = 1

11

1.055

Table 8.16 Element values of the cascaded Gm-C LSM_Z filters (amin = 40 dB) n

fs

Cell no., type

3

2.543

1, γ

c1 ÷ c4 = 0.760631857

gm1 = 9.837437227

gm2 = gm3 = 1

2, α

c= 1.534914470

gm1 = 1.534914470

gm2 = 1

1, γ

c1 ÷ c4 = 1.207171184

gm1 = 10.92465161

gm2 = gm3 = 1

2, β

c1 = 0.858374609

c2 = 2.287851654

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.840275256

gm1 = 14.10855626

gm2 = gm3 = 1

gm4 = 2.914560689

2, γ

c1 ÷ c4 = 0.214198253

gm1 = 0.318462638

gm2 = gm3 = 1

gm4 = 0.331435052

3, α

c= 0.559753153

gm1 = 0.559753153

gm2 = 1

1, γ

c1 ÷ c4 = 1.380658512

gm1 = 18.16569032

gm2 = gm3 = 1

gm4 = 7.669487599

2, γ

c1 ÷ c4 = 0.345962820

gm1 = 0.542618357

gm2 = gm3 = 1

gm4 = 0.549384942

3, β

c1 = 0.324106199

c2 = 1.228758972

gm1 ÷ gm4 = 1

4

5

6

1.781

1.759

1.447

gm4 = 2.238692434

gm4 = 5.788074930

(continued)

180

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.16 (continued) n

fs

Cell no., type

7

1.268

1, γ

c1 ÷ c4 = 2.030294107

gm1 = 71.37942662

gm2 = gm3 = 1

gm4 = 16.68437208

2, γ

c1 ÷ c4 = 0.531538499

gm1 = 1.478115841

gm2 = gm3 = 1

gm4 = 1.287647417

3, γ

c1 ÷ c4 = 0.221109274

gm1 = 0.165186850

gm2 = gm3 = 1

gm4 = 0.354969478

4, α

c= 0.606297787

gm1 = 0.606297787

gm2 = 1

1, γ

c1 ÷ c4 = 2.968204680

gm1 = 83.08700408

gm2 = gm3 = 1

gm4 = 35.42672049

2, γ

c1 ÷ c4 = 0.759581995

gm1 = 2.288945161

gm2 = gm3 = 1

gm4 = 2.458225265

3, γ

c1 ÷ c4 = 0.303915841

gm1 = 0.267041807

gm2 = gm3 = 1

gm4 = 2.458225265

4, β

c1 = 0.336610983

c2 = 1.249410382

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 1.080361980

gm1 = 22.02799875

gm2 = gm3 = 1

gm4 = 5.032868343

2, γ

c1 ÷ c4 = 0.442130697

gm1 = 1.077276613

gm2 = gm3 = 1

gm4 = 1.042279913

3, γ

c1 ÷ c4 = 4.002139183

gm1 = 51.41997213

gm2 = gm3 = 1

gm4 = 64.57248706

4, γ

c1 ÷ c4 = 0.209518018

gm1 = 0.114012214

gm2 = gm3 = 1

gm4 = 0.367398865

5, α

c= 0.593120287

gm1 = 0.593120287

gm2 = 1

1, γ

c1 ÷ c4 = 0.591118307

gm1 = 3.725416796

gm2 = gm3 = 1

gm4 = 1.725215831

2, γ

c1 ÷ c4 = 1.444290376

gm1 = 8.959929316

gm2 = gm3 = 1

gm4 = 8.776927750

3, γ

c1 ÷ c4 = 5.428185176

gm1 = 84.06822012

gm2 = gm3 = 1

gm4 = 118.4238215

4, γ

c1 ÷ c4 = 0.273191046

gm1 = 0.181418781

gm2 = gm3 = 1

gm4 = 0.527334570

5, β

c1 = 0.321195399

c2 = 1.177773468

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.381447972

gm1 = 3.194937769

gm2 = gm3 = 1

gm4 = 0.931962497

2, γ

c1 ÷ c4 = 0.823684518

gm1 = 4.257472127

gm2 = gm3 = 1

gm4 = 3.253393514

3, γ

c1 ÷ c4 = 1.924761286

gm1 = 12.88524993

gm2 = gm3 = 1

gm4 = 15.60402127

4, γ

c1 ÷ c4 = 6.908244119

gm1 = 124.4557245

gm2 = gm3 = 1

gm4 = 191.9537840

5, γ

c1 ÷ c4 = 0.192110868

gm1 = 0.085630115

gm2 = gm3 = 1

gm4 = 0.369699973

6, α

c= 0.551489462

gm1 = 0.551489462

gm2 = 1

8

9

10

11

1.181

1.126

1.093

1.070

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

181

Table 8.17 Element values of the cascaded Gm-C LSM_Z filters (amin = 45 dB) n

fs

Cell no., type

3

3.028

1, γ

c1 ÷ c4 = 0.754238948

gm1 = 13.77392052

gm2 = gm3 = 1

2, α

c= 1.583298225

gm1 = 1.583298225

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 1.188442273

gm1 = 13.44327286

gm2 = gm3 = 1

2, β

c1 = 0.886405123

c2 = 2.372914485

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.809621475

gm1 = 15.84077086

gm2 = gm3 = 1

gm4 = 2.680222134

2, γ

c1 ÷ c4 = 0.213704111

gm1 = 0.389050719

gm2 = gm3 = 1

gm4 = 0.315721655

3, α

c= 0.581386193

gm1 = 0.581386193

gm2 = 1

1, γ

c1 ÷ c4 = 1.321545878

gm1 = 19.28926709

gm2 = gm3 = 1

gm4 = 6.974532042

2, γ

c1 ÷ c4 = 0.345174615

gm1 = 0.627975410

gm2 = gm3 = 1

gm4 = 0.510548149

3, β

c1 = 0.332999682

c2 = 1.275354081

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 1.909502768

gm1 = 70.18261561

gm2 = gm3 = 1

gm4 = 14.68941064

2, γ

c1 ÷ c4 = 0.515842964

gm1 = 1.548754740

gm2 = gm3 = 1

gm4 = 1.158588990

3, γ

c1 ÷ c4 = 0.225578486

gm1 = 0.190766930

gm2 = gm3 = 1

gm4 = 0.335525434

4, α

c= 0.644359119

gm1 = 0.644359119

gm2 = 1

1, γ

c1 ÷ c4 = 2.782977456

gm1 = 79.45219646

gm2 = gm3 = 1

gm4 = 31.02745283

2, γ

c1 ÷ c4 = 0.730349885

gm1 = 2.295264931

gm2 = gm3 = 1

gm4 = 2.189226339

3, γ

c1 ÷ c4 = 0.305705737

gm1 = 0.291703633

gm2 = gm3 = 1

gm4 = 0.488708195

4, β

c1 = 0.354606526

c2 = 1.333464816

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 3.714676076

gm1 = 278.1724192

gm2 = gm3 = 1

gm4 = 55.50549053

2, γ

c1 ÷ c4 = 1.020447632

gm1 = 6.116937095

gm2 = gm3 = 1

gm4 = 4.388192111

3, γ

c1 ÷ c4 = 0.432319949

gm1 = 0.636570464

gm2 = gm3 = 1

gm4 = 0.930697032

4, γ

c1 ÷ c4 = 0.217016124

gm1 = 0.129271478

gm2 = gm3 = 1

gm4 = 0.348165777

5, α

c= 0.643036475

gm1 = 0.643036475

gm2 = 1

c= 0.643036475

4

5

6

7

8

9

2.002

1.951

1.560

1.336

1.227

1.157

gm4 = 2.183924375

gm4 = 5.577461540

(continued)

182

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.17 (continued) n

fs

Cell no., type

10

1.116

1, γ

c1 ÷ c4 = 5.031022756

gm1 = 284.7463870

gm2 = gm3 = 1

gm4 = 101.5400195

2, γ

c1 ÷ c4 = 1.356930246

gm1 = 8.319103535

gm2 = gm3 = 1

gm4 = 7.605863837

3, γ

c1 ÷ c4 = 0.570820911

gm1 = 0.973162655

gm2 = gm3 = 1

gm4 = 1.524245683

4, γ

c1 ÷ c4 = 0.276834502

gm1 = 0.194311780

gm2 = gm3 = 1

gm4 = 0.487297406

5, β

c1 = 0.343959242

c2 = 1.277030215

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.781149394

gm1 = 14.00882012

gm2 = gm3 = 1

gm4 = 2.820667299

2, γ

c1 ÷ c4 = 0.374629836

gm1 = 0.918955029

gm2 = gm3 = 1

gm4 = 0.831355077

3, γ

c1 ÷ c4 = 1.791026365

gm1 = 11.59815579

gm2 = gm3 = 1

gm4 = 13.35096311

4, γ

c1 ÷ c4 = 6.364468817

gm1 = 109.3649706

gm2 = gm3 = 1

gm4 = 162.7370386

5, γ

c1 ÷ c4 = 0.200836594

gm1 = 0.096641494

gm2 = gm3 = 1

gm4 = 0.351256718

6, α

c= 0.604970049

gm1 = 0.604970049

gm2 = 1

11

1.087

Table 8.18 Element values of the cascaded Gm-C LSM_Z filters (amin = 50 dB) n

fs

Cell no., type

3

3.625

1, γ

c1 ÷ c4 = 0.750060137

gm1 = 19.5763353

gm2 = gm3 = 1

2, α

c= 1.617733836

gm1 = 1.61773383

gm2 = 1

1, γ

c1 ÷ c4 = 1.175219525

gm1 = 16.8760308

gm2 = gm3 = 1

2, β

c1 = 0.908638470

c2 = 2.440205065

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.786930344

gm1 = 18.2391573

gm2 = gm3 = 1

gm4 = 2.511501618

2, γ

c1 ÷ c4 = 0.213456861

gm1 = 0.47933695

gm2 = gm3 = 1

gm4 = 0.304241726

3, α

c= 0.599033542

gm1 = 0.59903354

gm2 = 1

1, γ

c1 ÷ c4 = 1.276907074

gm1 = 21.0236970

gm2 = gm3 = 1

gm4 = 6.468427368

2, γ

c1 ÷ c4 = 0.345862801

gm1 = 0.73881688

gm2 = gm3 = 1

gm4 = 0.482015950

3, β

c1 = 0.340008204

c2 = 1.311786928

gm1 ÷ gm4 = 1

4

5

6

2.264

2.170

1.689

gm4 = 2.147938536

gm4 = 5.430088067

(continued)

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

183

Table 8.18 (continued) n

fs

Cell no., type

7

1.412

1, γ

c1 ÷ c4 = 1.815577315

gm1 = 71.0357161

gm2 = gm3 = 1

gm4 = 13.22134423

2, γ

c1 ÷ c4 = 0.505111091

gm1 = 1.66430686

gm2 = gm3 = 1

gm4 = 1.064166328

3, γ

c1 ÷ c4 = 0.230042021

gm1 = 0.22186859

gm2 = gm3 = 1

gm4 = 0.320707796

4, α

c= 0.677177719

gm1 = 0.67717771

gm2 = 1

1, γ

c1 ÷ c4 = 2.638043442

gm1 = 78.1470829

gm2 = gm3 = 1

gm4 = 27.78031320

2, γ

c1 ÷ c4 = 0.709186852

gm1 = 2.36280880

gm2 = gm3 = 1

gm4 = 1.991452234

3, γ

c1 ÷ c4 = 0.308848516

gm1 = 0.32367486

gm2 = gm3 = 1

gm4 = 0.457406988

4, β

c1 = 0.370607808

c2 = 1.408252311

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 3.486597123

gm1 = 263.183128

gm2 = gm3 = 1

gm4 = 48.79027519

2, γ

c1 ÷ c4 = 0.974648387

gm1 = 5.98025790

gm2 = gm3 = 1

gm4 = 3.912973144

3, γ

c1 ÷ c4 = 0.426721284

gm1 = 0.66159873

gm2 = gm3 = 1

gm4 = 0.848027345

4, γ

c1 ÷ c4 = 0.224726843

gm1 = 0.14731358

gm2 = gm3 = 1

gm4 = 0.333521845

5, α

c= 0.689060205

gm1 = 0.68906020

gm2 = 1

1, γ

c1 ÷ c4 = 4.715039423

gm1 = 265.124232

gm2 = gm3 = 1

gm4 = 89.01982804

2, γ

c1 ÷ c4 = 1.289291481

gm1 = 7.93632816

gm2 = gm3 = 1

gm4 = 6.740731729

3, γ

c1 ÷ c4 = 0.556991616

gm1 = 0.97437150

gm2 = gm3 = 1

gm4 = 1.375583974

4, γ

c1 ÷ c4 = 0.281757596

gm1 = 0.21090702

gm2 = gm3 = 1

gm4 = 0.456766002

5, β

c1 = 0.365607117

c2 = 1.37119540

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 1.685639665

gm1 = 68.4756651

gm2 = gm3 = 1

gm4 = 11.68268018

2, γ

c1 ÷ c4 = 0.749423143

gm1 = 3.85203940

gm2 = gm3 = 1

gm4 = 2.500953084

3, γ

c1 ÷ c4 = 5.928663522

gm1 = 132.610370

gm2 = gm3 = 1

gm4 = 141.0471660

4, γ

c1 ÷ c4 = 0.371497125

gm1 = 0.38718368

gm2 = gm3 = 1

gm4 = 0.756599964

5, γ

c1 ÷ c4 = 0.210029660

gm1 = 0.10953023

gm2 = gm3 = 1

gm4 = 0.337370807

6, α

c= 0.656406603

gm1 = 0.65640660

gm2 = 1

8

9

10

11

1.280

1.193

1.142

1.106

184

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.19 Element values of the cascaded Gm-C LSM_Z filters (amin = 55 dB) n

fs

Cell no., type

3

4.355

1, γ

c1 ÷ c4 = 0.747294418

gm1 = 28.10905977

gm2 = gm3 = 1

2, α

c= 1.641919482

gm1 = 1.641919482

gm2 = 1

1, γ

c1 ÷ c4 = 1.165745944

gm1 = 21.50513551

gm2 = gm3 = 1

gm4 = 5.325142930

2, β

c1 = 0.926048899

c2 = 2.492792209

gm1 ÷ gm4 = 1

c1 = 0.926048899

1, γ

c1 ÷ c4 = 0.769909632

gm1 = 21.41254902

gm2 = gm3 = 1

gm4 = 2.387496470

2, γ

c1 ÷ c4 = 0.213301367

gm1 = 0.593849792

gm2 = gm3 = 1

gm4 = 0.295677472

3, α

c= 0.613522025

gm1 = 0.613522025

gm2 = 1

1, γ

c1 ÷ c4 = 1.242708614

gm1 = 23.39118315

gm2 = gm3 = 1

gm4 = 6.091389200

2, γ

c1 ÷ c4 = 0.347434954

gm1 = 0.879348414

gm2 = gm3 = 1

gm4 = 0.460677214

3, β

c1 = 0.345467363

c2 = 1.339932758

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 1.741509167

gm1 = 73.64898232

gm2 = gm3 = 1

gm4 = 12.11478154

2, γ

c1 ÷ c4 = 0.497859551

gm1 = 1.824130869

gm2 = gm3 = 1

gm4 = 0.993514777

3, γ

c1 ÷ c4 = 0.234267979

gm1 = 0.259162001

gm2 = gm3 = 1

gm4 = 0.309106956

4, α

c= 0.705338579

gm1 = 0.705338579

gm2 = 1

1, γ

c1 ÷ c4 = 2.522958270

gm1 = 78.69137297

gm2 = gm3 = 1

gm4 = 25.32355540

2, γ

c1 ÷ c4 = 0.693826020

gm1 = 2.484201454

gm2 = gm3 = 1

gm4 = 1.842605271

3, γ

c1 ÷ c4 = 0.312872149

gm1 = 0.363472892

gm2 = gm3 = 1

gm4 = 0.433038131

4, β

c1 = 0.384629205

c2 = 1.473864566

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 3.302888291

gm1 = 254.9327386

gm2 = gm3 = 1

gm4 = 43.68911168

2, γ

c1 ÷ c4 = 0.939270557

gm1 = 5.982656867

gm2 = gm3 = 1

gm4 = 3.554114954

3, γ

c1 ÷ c4 = 0.424138378

gm1 = 0.700934401

gm2 = gm3 = 1

gm4 = 0.785234430

4, γ

c1 ÷ c4 = 0.232374579

gm1 = 0.168291729

gm2 = gm3 = 1

gm4 = 0.322027668

5, α

c= 0.731142830

gm1 = 0.731142830

gm2 = 1

4

5

6

7

8

9

2.573

2.419

1.834

1.498

1.337

1.232

gm4 = 2.124020731

(continued)

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

185

Table 8.19 (continued) n

fs

Cell no., type

10

1.171

1, γ

c1 ÷ c4 = 4.459746034

gm1 = 252.5501878

gm2 = gm3 = 1

gm4 = 79.49443796

2, γ

c1 ÷ c4 = 1.236292259

gm1 = 7.745128320

gm2 = gm3 = 1

gm4 = 6.085693945

3, γ

c1 ÷ c4 = 0.547896033

gm1 = 0.995865054

gm2 = gm3 = 1

gm4 = 1.262933066

4, γ

c1 ÷ c4 = 0.287518845

gm1 = 0.231156498

gm2 = gm3 = 1

gm4 = 0.432833023

5, β

c1 = 0.385992329

c2 = 1.459817039

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 5.573858937

gm1 = 788.9648649

gm2 = gm3 = 1

gm4 = 124.5221723

2, γ

c1 ÷ c4 = 1.601453866

gm1 = 18.49460262

gm2 = gm3 = 1

gm4 = 10.41599522

3, γ

c1 ÷ c4 = 0.725737503

gm1 = 2.081133143

gm2 = gm3 = 1

gm4 = 2.258883946

4, γ

c1 ÷ c4 = 0.371040616

gm1 = 0.402780512

gm2 = gm3 = 1

gm4 = 0.699596071

5, γ

c1 ÷ c4 = 0.219429844

gm1 = 0.124338515

gm2 = gm3 = 1

gm4 = 0.326595689

6, α

c= 0.705612239

gm1 = 0.705612239

gm2 = 1

11

1.128

Table 8.20 Element values of the cascaded Gm-C LSM_Z filters (amin = 60 dB) n

fs

Cell no., type

3

5.246

1, γ

c1 ÷ c4 = 0.745447648

gm1 = 40.64402455

gm2 = gm3 = 1

2, α

c= 1.658747700

gm1 = 1.658747700

gm2 = 1

1, γ

c1 ÷ c4 = 1.158882360

gm1 = 27.71448895

gm2 = gm3 = 1

2, β

c1 = 0.939545098

c2 = 2.533494440

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.757004209

gm1 = 25.51952067

gm2 = gm3 = 1

gm4 = 2.294875379

2, γ

c1 ÷ c4 = 0.213165306

gm1 = 0.738439043

gm2 = gm3 = 1

gm4 = 0.289184696

3, α

c= 0.625489852

gm1 = 0.625489852

gm2 = 1

1, γ

c1 ÷ c4 = 1.216189530

gm1 = 26.45308072

gm2 = gm3 = 1

gm4 = 5.805286892

2, γ

c1 ÷ c4 = 0.349485775

gm1 = 1.055048130

gm2 = gm3 = 1

gm4 = 0.444494480

3, β

c1 = 0.349688475

c2 = 1.361515644

gm1 ÷ gm4 = 1

4

5

6

2.935

2.701

1.998

gm4 = 2.107998448

gm4 = 5.249432243

(continued)

186

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.20 (continued) n

fs

Cell no., type

7

1.595

1, γ

c1 ÷ c4 = 1.682430847

gm1 = 77.90636003

gm2 = gm3 = 1

gm4 = 11.26434851

2, γ

c1 ÷ c4 = 0.493061737

gm1 = 2.030383368

gm2 = gm3 = 1

gm4 = 0.939689895

3, γ

c1 ÷ c4 = 0.238128338

gm1 = 0.303519038

gm2 = gm3 = 1

gm4 = 0.299828993

4, α

c= 0.729457838

gm1 = 0.729457838

gm2 = 1

1, γ

c1 ÷ c4 = 2.430464895

gm1 = 80.82127957

gm2 = gm3 = 1

gm4 = 23.42682476

2, γ

c1 ÷ c4 = 0.682691887

gm1 = 2.656694278

gm2 = gm3 = 1

gm4 = 1.728448348

3, γ

c1 ÷ c4 = 0.317457447

gm1 = 0.411967992

gm2 = gm3 = 1

gm4 = 0.413717520

4, β

c1 = 0.396746217

c2 = 1.530637807

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 3.153095385

gm1 = 251.9751812

gm2 = gm3 = 1

gm4 = 39.73265015

2, γ

c1 ÷ c4 = 0.911734040

gm1 = 6.101298523

gm2 = gm3 = 1

gm4 = 3.277783157

3, γ

c1 ÷ c4 = 0.423731352

gm1 = 0.754015558

gm2 = gm3 = 1

gm4 = 0.736568802

4, γ

c1 ÷ c4 = 0.239774390

gm1 = 0.192440281

gm2 = gm3 = 1

gm4 = 0.312774266

5, α

c= 0.769322295

gm1 = 0.769322295

gm2 = 1

1, γ

c1 ÷ c4 = 4.250861987

gm1 = 245.3267366

gm2 = gm3 = 1

gm4 = 72.09249608

2, γ

c1 ÷ c4 = 1.194382654

gm1 = 7.705005697

gm2 = gm3 = 1

gm4 = 5.579681610

3, γ

c1 ÷ c4 = 0.542318864

gm1 = 1.035073410

gm2 = gm3 = 1

gm4 = 1.175906346

4, γ

c1 ÷ c4 = 0.293812311

gm1 = 0.255192637

gm2 = gm3 = 1

gm4 = 0.413638388

5, β

c1 = 0.405006971

c2 = 1.542551848

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 5.281256647

gm1 = 749.1244689

gm2 = gm3 = 1

gm4 = 111.6593383

2, γ

c1 ÷ c4 = 1.533477897

gm1 = 17.89463239

gm2 = gm3 = 1

gm4 = 9.434273239

3, γ

c1 ÷ c4 = 0.708151016

gm1 = 2.082708068

gm2 = gm3 = 1

gm4 = 2.071931777

4, γ

c1 ÷ c4 = 0.372551277

gm1 = 0.424980641

gm2 = gm3 = 1

gm4 = 0.655191259

5, γ

c1 ÷ c4 = 0.228848894

gm1 = 0.141155604

gm2 = gm3 = 1

gm4 = 0.318012338

6, α

c= 0.752422292

gm1 = 0.752422292

gm2 = 1

8

9

10

11

1.403

1.277

1.204

1.152

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

187

8.4.2 Papoulis_Z Filters See Tables 8.21, 8.22, 8.23, 8.24, 8.25, 8.26 and 8.27. Table 8.21 Element values of the cascaded Gm-C of Papoulis_Z filters (amin = 30 dB) n

fs

Cell no., type

3

1.852

1, γ

c1 ÷ c4 = 0.767402069

gm1 = 5.257812165

gm2 = gm3 = 1

2, α

c= 1.320697135

gm1 = 1.320697135

gm2 = 1

1, γ

c1 ÷ c4 = 1.209961856

gm1 = 7.317816252

gm2 = gm3 = 1

2, β

c1 = 0.733753332

c2 = 1.988730676

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 2.141542368

gm1 = 31.39123351

gm2 = gm3 = 1

gm4 = 18.52283446

2, γ

c1 ÷ c4 = 0.505449930

gm1 = 0.803155607

gm2 = gm3 = 1

gm4 = 0.946858951

3, α

c= 1.074817471

gm1 = 1.074817471

gm2 = 1

1, γ

c1 ÷ c4 = 3.057395007

gm1 = 42.46260817

gm2 = gm3 = 1

gm4 = 37.72795801

2, γ

c1 ÷ c4 = 0.705893165

gm1 = 1.357778952

gm2 = gm3 = 1

gm4 = 1.950883809

3,β

c1 = 0.579765438

c2 = 1.789356926

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 4.683043777

gm1 = 183.9071736

gm2 = gm3 = 1

gm4 = 88.50766142

2, γ

c1 ÷ c4 = 1.090173453

gm1 = 3.808416640

gm2 = gm3 = 1

gm4 = 4.930842180

3, γ

c1 ÷ c4 = 0.364025611

gm1 = 0.320879930

gm2 = gm3 = 1

gm4 = 0.641836488

4, α

c= 0.833130897

gm1 = 0.833130897

gm2 = 1

1, γ

c1 ÷ c4 = 1.479099930

gm1 = 11.99229847

gm2 = gm3 = 1

gm4 = 9.092876206

2, γ

c1 ÷ c4 = 6.223165926

gm1 = 109.1962581

gm2 = gm3 = 1

gm4 = 156.0676593

3, γ

c1 ÷ c4 = 0.494290680

gm1 = 0.563034929

gm2 = gm3 = 1

gm4 = 1.169097180

4, β

c1 = 0.452772765

c2 = 1.489835627

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.734491380

gm1 = 5.942859002

gm2 = gm3 = 1

gm4 = 2.594756475

2, γ

c1 ÷ c4 = 2.132273483

gm1 = 17.08235945

gm2 = gm3 = 1

gm4 = 18.95305619

3, γ

c1 ÷ c4 = 8.640729302

gm1 = 188.5071709

gm2 = gm3 = 1

gm4 = 300.5223126

4

5

6

7

8

9

1.460

1.210

1.140

1.085

1.062

1.044

gm4 = 2.320195731

gm4 = 5.829370855

(continued)

188

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.21 (continued) n

10

11

fs

1.034

1.026

Cell no., type 4, γ

c1 ÷ c4 = 0.281556758

gm1 = 0.175104403

gm2 = gm3 = 1

gm4 = 0.533266701

5, α

c= 0.668424979

gm1 = 0.668424979

gm2 = 1

1, γ

c1 ÷ c4 = 0.380118540

gm1 = 1.004132408

gm2 = gm3 = 1

gm4 = 0.895742120

2, γ

c1 ÷ c4 = 0.979083772

gm1 = 3.090562486

gm2 = gm3 = 1

gm4 = 4.481455627

3, γ

c1 ÷ c4 = 2.752578929

gm1 = 18.14758053

gm2 = gm3 = 1

gm4 = 31.43668903

4, γ

c1 ÷ c4 = 10.92466707

gm1 = 257.9983450

gm2 = gm3 = 1

gm4 = 479.8972842

5, β

c1 = 0.367030909

c2 = 1.246483494

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.557527733

gm1 = 4.489765186

gm2 = gm3 = 1

gm4 = 1.820787803

2, γ

c1 ÷ c4 = 0.229208210

gm1 = 0.239620173

gm2 = gm3 = 1

gm4 = 0.483393032

3, γ

c1 ÷ c4 = 1.379052167

gm1 = 5.350554266

gm2 = gm3 = 1

gm4 = 8.689874201

4, γ

c1 ÷ c4 = 3.692623691

gm1 = 31.17723660

gm2 = gm3 = 1

gm4 = 56.35674587

5, γ

c1 ÷ c4 = 14.21848576

gm1 = 428.8239879

gm2 = gm3 = 1

gm4 = 812.1842275

6, α

c= 0.555478072

gm1 = 0.555478072

gm2 = 1

Table 8.22 Element values of the cascaded Gm-C of Papoulis_Z filters (amin = 35 dB) n

fs

Cell no., type

3

2.172

1, γ

c1 ÷ c4 = 0.751907932

gm1 = 6.987586416

gm2 = gm3 = 1

2, α

c= 1.403154658

gm1 = 1.403154658

gm2 = 1

1, γ

c1 ÷ c4 = 1.171061629

gm1 = 8.478452713

gm2 = gm3 = 1

2, β

c1 = 0.771088322

c2 = 2.109665369

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 2.007792870

gm1 = 31.84699180

gm2 = gm3 = 1

gm4 = 16.15760445

2, γ

c1 ÷ c4 = 0.518068172

gm1 = 0.950610854

gm2 = gm3 = 1

gm4 = 0.893085949

3, α

c= 1.209090678

gm1 = 1.209090678

gm2 = 1

4

5

1.619

1.282

gm4 = 2.190928242

gm4 = 5.397893643

(continued)

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

189

Table 8.22 (continued) n

fs

Cell no., type

6

1.186

1, γ

c1 ÷ c4 = 2.843997871

gm1 = 40.64672474

gm2 = gm3 = 1

gm4 = 32.47779976

2, γ

c1 ÷ c4 = 0.703043280

gm1 = 1.463015536

gm2 = gm3 = 1

gm4 = 1.796690210

3, β

c1 = 0.634962911

c2 = 1.978780000

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 4.288282524

gm1 = 166.8960198

gm2 = gm3 = 1

gm4 = 73.99154922

2, γ

c1 ÷ c4 = 1.040609331

gm1 = 3.697393497

gm2 = gm3 = 1

gm4 = 4.289739850

3, γ

c1 ÷ c4 = 0.380405634

gm1 = 0.369262080

gm2 = gm3 = 1

gm4 = 0.599777144

4, α

c= 0.951357810

gm1 = 0.951357810

gm2 = 1

1, γ

c1 ÷ c4 = 5.673291489

gm1 = 187.2166065

gm2 = gm3 = 1

gm4 = 129.4308824

2, γ

c1 ÷ c4 = 1.391841770

gm1 = 5.724163366

gm2 = gm3 = 1

gm4 = 7.793748486

3, γ

c1 ÷ c4 = 0.498177332

gm1 = 0.594604649

gm2 = gm3 = 1

gm4 = 1.064244433

4, β

c1 = 0.501353328

c2 = 1.672741228

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 1.967018536

gm1 = 44.76535298

gm2 = gm3 = 1

gm4 = 15.79927992

2, γ

c1 ÷ c4 = 7.816449603

gm1 = 239.1430827

gm2 = gm3 = 1

gm4 = 245.5897206

3, γ

c1 ÷ c4 = 0.705886852

gm1 = 1.299583631

gm2 = gm3 = 1

gm4 = 2.236697029

4, γ

c1 ÷ c4 = 0.295438852

gm1 = 0.198141772

gm2 = gm3 = 1

gm4 = 0.493802231

5, α

c= 0.765584828

gm1 = 0.765584828

gm2 = 1

1, γ

c1 ÷ c4 = 0.924974799

gm1 = 6.179863244

gm2 = gm3 = 1

gm4 = 3.805066946

2, γ

c1 ÷ c4 = 2.521703046

gm1 = 21.16162775

gm2 = gm3 = 1

gm4 = 25.98530979

3, γ

c1 ÷ c4 = 9.855451181

gm1 = 238.5823766

gm2 = gm3 = 1

gm4 = 390.1685325

4, γ

c1 ÷ c4 = 0.383918838

gm1 = 0.325509141

gm2 = gm3 = 1

gm4 = 0.806210474

5, β

c1 = 0.407298909

c2 = 1.406015641

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.536515510

gm1 = 4.296717807

gm2 = gm3 = 1

gm4 = 1.555378371

2, γ

c1 ÷ c4 = 1.274989484

gm1 = 7.629322365

gm2 = gm3 = 1

gm4 = 7.188748589

3, γ

c1 ÷ c4 = 3.351634405

gm1 = 32.35198361

gm2 = gm3 = 1

gm4 = 45.94843621

7

8

9

10

11

1.113

1.083

1.058

1.045

1.034

(continued)

190

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.22 (continued) n

fs

Cell no., type 4, γ

c1 ÷ c4 = 12.77213857

gm1 = 380.0628892

gm2 = gm3 = 1

gm4 = 654.9046411

5, γ

c1 ÷ c4 = 0.240599927

gm1 = 0.124788405

gm2 = gm3 = 1

gm4 = 0.445518400

6, α

c= 0.636272191

gm1 = 0.636272191

gm2 = 1

Table 8.23 Element values of the cascaded Gm-C of Papoulis_Z filters (amin = 40 dB) n

fs

Cell no., type

3

2.571

1, γ

c1 ÷ c4 = 0.742178285

gm1 = 9.584790348

gm2 = gm3 = 1

2, α

c= 1.464117618

gm1 = 1.464117618

gm2 = 1

1, γ

c1 ÷ c4 = 1.144692629

gm1 = 10.18109721

gm2 = gm3 = 1

2, β

c1 = 0.801548981

c2 = 2.207906927

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 1.912751943

gm1 = 33.81026117

gm2 = gm3 = 1

gm4 = 14.56374025

2, γ

c1 ÷ c4 = 0.532741413

gm1 = 1.149300137

gm2 = gm3 = 1

gm4 = 0.860709956

3, α

c= 1.334971418

gm1 = 1.334971418

gm2 = 1

1, γ

c1 ÷ c4 = 2.689486446

gm1 = 40.64526549

gm2 = gm3 = 1

gm4 = 28.90643602

2, γ

c1 ÷ c4 = 0.706405762

gm1 = 1.622404666

gm2 = gm3 = 1

gm4 = 1.697293667

3, β

c1 = 0.688492221

c2 = 2.157593891

gm1 ÷ gm4 = 1

4, α

c= 1.069287040

gm1 = 1.069287040

gm2 = 1

1, γ

c1 ÷ c4 = 3.996329024

gm1 = 158.2140215

gm2 = gm3 = 1

gm4 = 64.06818716

2, γ

c1 ÷ c4 = 1.009828183

gm1 = 3.740375158

gm2 = gm3 = 1

gm4 = 3.862841773

3, γ

c1 ÷ c4 = 0.399943355

gm1 = 0.433480286

gm2 = gm3 = 1

gm4 = 0.574028434

4, α

c= 1.069287040

gm1 = 1.069287040

gm2 = 1

1, γ

c1 ÷ c4 = 5.263358363

gm1 = 172.1349897

gm2 = gm3 = 1

gm4 = 111.1645283

2, γ

c1 ÷ c4 = 1.332397098

gm1 = 5.532208329

gm2 = gm3 = 1

gm4 = 6.916116331

3, γ

c1 ÷ c4 = 0.507747182

gm1 = 0.645943853

gm2 = gm3 = 1

gm4 = 0.996334979

4

5

6

7

8

1.811

1.369

1.242

1.146

1.107

gm4 = 2.109353520

gm4 = 5.111529256

(continued)

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

191

Table 8.23 (continued) n

9

10

11

fs

1.074

1.058

1.043

Cell no., type 4, β

c1 = 0.550815959

c2 = 1.854859060

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 7.195916258

gm1 = 632.8228278

gm2 = gm3 = 1

gm4 = 207.8538323

2, γ

c1 ÷ c4 = 1.847659793

gm1 = 13.99236723

gm2 = gm3 = 1

gm4 = 13.64740056

3, γ

c1 ÷ c4 = 0.690713349

gm1 = 1.291278672

gm2 = gm3 = 1

gm4 = 1.998073860

4, γ

c1 ÷ c4 = 0.312251632

gm1 = 0.228430483

gm2 = gm3 = 1

gm4 = 0.468091984

5, α

c= 0.863994454

gm1 = 0.863994454

gm2 = 1

1, γ

c1 ÷ c4 = 9.047378999

gm1 = 617.3859040

gm2 = gm3 = 1

gm4 = 328.4680040

2, γ

c1 ÷ c4 = 2.351967753

gm1 = 19.07898329

gm2 = gm3 = 1

gm4 = 22.25010319

3, γ

c1 ÷ c4 = 0.890074318

gm1 = 2.002818435

gm2 = gm3 = 1

gm4 = 3.348758458

4, γ

c1 ÷ c4 = 0.392466060

gm1 = 0.348659880

gm2 = gm3 = 1

gm4 = 0.746127690

5, β

c1 = 0.448810790

c2 = 1.567574681

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 1.201274181

gm1 = 22.34980082

gm2 = gm3 = 1

gm4 = 6.166405980

2, γ

c1 ÷ c4 = 0.525887259

gm1 = 1.340381108

gm2 = gm3 = 1

gm4 = 1.376006913

3, γ

c1 ÷ c4 = 3.096468997

gm1 = 28.35452933

gm2 = gm3 = 1

gm4 = 38.78868626

4, γ

c1 ÷ c4 = 11.67290041

gm1 = 324.3599001

gm2 = gm3 = 1

gm4 = 546.6300736

5, γ

c1 ÷ c4 = 0.254473685

gm1 = 0.142220137

gm2 = gm3 = 1

gm4 = 0.420108037

6, α

c= 0.718327715

gm1 = 0.718327715

gm2 = 1

Table 8.24 Element values of the cascaded Gm-C Papoulis_Z filters (amin = 45 dB) n

fs

Cell no., type

3

3.063

1, γ

c1 ÷ c4 = 0.735923456

gm1 = 13.43344899

gm2 = gm3 = 1

2, α

c= 1.508144414

gm1 = 1.508144414

gm2 = 1

1, γ

c1 ÷ c4 = 1.126366736

gm1 = 12.55240546

gm2 = gm3 = 1

2, β

c1 = 0.825955528

c2 = 2.286353597

gm1 ÷ gm4 = 1

4

2.039

gm4 = 2.05659819

gm4 = 4.915313523

(continued)

192

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.24 (continued) n

fs

Cell no., type

5

1.471

1, γ

c1 ÷ c4 = 1.843772938

gm1 = 37.18946922

gm2 = gm3 = 1

gm4 = 13.45112318

2, γ

c1 ÷ c4 = 0.547759055

gm1 = 1.408409225

gm2 = gm3 = 1

gm4 = 0.841297302

3, α

c= 1.450957661

gm1 = 1.450957661

gm2 = 1

1, γ

c1 ÷ c4 = 2.575145183

gm1 = 42.08829348

gm2 = gm3 = 1

gm4 = 26.38675124

2, γ

c1 ÷ c4 = 0.713255073

gm1 = 1.836329402

gm2 = gm3 = 1

gm4 = 1.632163026

3, β

c1 = 0.739613833

c2 = 2.324140573

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 3.775506635

gm1 = 155.3888422

gm2 = gm3 = 1

gm4 = 57.01930691

2, γ

c1 ÷ c4 = 0.991647875

gm1 = 3.905802654

gm2 = gm3 = 1

gm4 = 3.570659756

3, γ

c1 ÷ c4 = 0.421411905

gm1 = 0.515155248

gm2 = gm3 = 1

gm4 = 0.558724918

4, α

= 1.186258895 gm1 = 1.186258895

gm2 = 1

1, γ

c1 ÷ c4 = 4.950883360

gm1 = 163.7833167

gm2 = gm3 = 1

gm4 = 98.15154191

2, γ

c1 ÷ c4 = 1.292014919

gm1 = 5.520297896

gm2 = gm3 = 1

gm4 = 6.304539554

3, γ

c1 ÷ c4 = 0.521139375

gm1 = 0.715905867

gm2 = gm3 = 1

gm4 = 0.952266257

4, β

c1 = 0.600777545

c2 = 2.034928858

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 6.717697897

gm1 = 585.7921034

gm2 = gm3 = 1

gm4 = 180.8902357

2, γ

c1 ÷ c4 = 1.760290240

gm1 = 13.36716128

gm2 = gm3 = 1

gm4 = 12.12638301

3, γ

c1 ÷ c4 = 0.684962491

gm1 = 1.323943972

gm2 = gm3 = 1

gm4 = 1.835250242

4, γ

c1 ÷ c4 = 0.331174153

gm1 = 0.266357457

gm2 = gm3 = 1

gm4 = 0.451198775

5, α

c= 0.963565191

gm1 = 0.963565191

gm2 = 1

1, γ

c1 ÷ c4 = 8.421915765

gm1 = 561.1545952

gm2 = gm3 = 1

gm4 = 284.3206807

2, γ

c1 ÷ c4 = 2.224950601

gm1 = 17.76836120

gm2 = gm3 = 1

gm4 = 19.59452703

3, γ

c1 ÷ c4 = 0.868660407

gm1 = 1.970566107

gm2 = gm3 = 1

gm4 = 3.031731638

4, γ

c1 ÷ c4 = 0.404426320

gm1 = 0.380789613

gm2 = gm3 = 1

gm4 = 0.705020040

5, β

c1 = 0.491437100

c2 = 1.730765214

gm1 ÷ gm4 = 1

6

7

8

9

10

1.307

1.185

1.134

1.093

1.072

(continued)

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

193

Table 8.24 (continued) n

fs

Cell no., type

11

1.054

1, γ

c1 ÷ c4 = 10.81711637

gm1 = 1887.710393

gm2 = gm3 = 1

gm4 = 469.0645091

2, γ

c1 ÷ c4 = 2.901558793

gm1 = 42.28900265

gm2 = gm3 = 1

gm4 = 33.67241822

3, γ

c1 ÷ c4 = 1.148996401

gm1 = 4.021552390

gm2 = gm3 = 1

gm4 = 5.446143488

4, γ

c1 ÷ c4 = 0.522668566

gm1 = 0.666319102

gm2 = gm3 = 1

gm4 = 1.251140028

5, γ

c1 ÷ c4 = 0.270203245

gm1 = 0.163788547

gm2 = gm3 = 1

gm4 = 0.402622735

6, α

c= 0.801712827

gm1 = 0.801712827

gm2 = 1

Table 8.25 Element values of the cascaded Gm-C Papoulis_Z filters (amin = 50 dB) n

fs

Cell no., type

3

3.670

1, γ

c1 ÷ c4 = 0.731833120

gm1 = 19.10582490

gm2 = gm3 = 1

2, α

c= 1.539393403

gm1 = 1.539393403

gm2 = 1

1, γ

c1 ÷ c4 = 1.113389815

gm1 = 15.78266077

gm2 = gm3 = 1

2, β

c1 = 0.845223287

c2 = 2.348119950

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 1.792868706

gm1 = 42.06256044

gm2 = gm3 = 1

gm4 = 12.65323311

2, γ

c1 ÷ c4 = 0.562102813

gm1 = 1.740987715

gm2 = gm3 = 1

gm4 = 0.829909059

3, α

c= 1.555953059

gm1 = 1.555953059

gm2 = 1

1, γ

c1 ÷ c4 = 2.488971821

gm1 = 44.82093293

gm2 = gm3 = 1

gm4 = 24.55616381

2, γ

c1 ÷ c4 = 0.721886509

gm1 = 2.109857319

gm2 = gm3 = 1

gm4 = 1.589261840

3, β

c1 = 0.787721717

c2 = 2.477311536

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 3.605524431

gm1 = 157.1185360

gm2 = gm3 = 1

gm4 = 51.85968152

2, γ

c1 ÷ c4 = 0.982047460

gm1 = 4.180379060

gm2 = gm3 = 1

gm4 = 3.367218788

3, γ

c1 ÷ c4 = 0.443927817

gm1 = 0.616772997

gm2 = gm3 = 1

gm4 = 0.550425793

4, α

c= 1.301489972

gm1 = 1.301489972

gm2 = 1

4

5

6

7

2.309

1.591

1.382

1.230

gm4 = 2.021910969

gm4 = 4.777710195

(continued)

194

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.25 (continued) n

fs

Cell no., type

8

1.166

1, γ

c1 ÷ c4 = 4.708263223

gm1 = 160.2950194

gm2 = gm3 = 1

gm4 = 88.58906771

2, γ

c1 ÷ c4 = 1.264989634

gm1 = 5.650524534

gm2 = gm3 = 1

gm4 = 5.868618134

3, γ

c1 ÷ c4 = 0.537115527

gm1 = 0.804978366

gm2 = gm3 = 1

gm4 = 0.924357085

4, β

c1 = 0.650867240

c2 = 2.211777242

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 6.342317084

gm1 = 557.5686640

gm2 = gm3 = 1

gm4 = 161.0139381

2, γ

c1 ÷ c4 = 1.695948363

gm1 = 13.12498399

gm2 = gm3 = 1

gm4 = 11.02294247

3, γ

c1 ÷ c4 = 0.686016817

gm1 = 1.391119239

gm2 = gm3 = 1

gm4 = 1.723115443

4, γ

c1 ÷ c4 = 0.351644512

gm1 = 0.312704176

gm2 = gm3 = 1

gm4 = 0.440286197

5, α

c= 1.064115204

gm1 = 1.064115204

gm2 = 1

1, γ

c1 ÷ c4 = 7.928739333

gm1 = 524.0263891

gm2 = gm3 = 1

gm4 = 251.7289551

2, γ

c1 ÷ c4 = 2.128848466

gm1 = 16.99710394

gm2 = gm3 = 1

gm4 = 17.65374905

3, γ

c1 ÷ c4 = 0.857045623

gm1 = 1.988964029

gm2 = gm3 = 1

gm4 = 2.807651586

4, γ

c1 ÷ c4 = 0.418925352

gm1 = 0.421734426

gm2 = gm3 = 1

gm4 = 0.676845450

5, β

c1 = 0.535019960

c2 = 1.895072243

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 10.13813123

gm1 = 1733.874925

gm2 = gm3 = 1

gm4 = 411.7112053

2, γ

c1 ÷ c4 = 2.750490091

gm1 = 39.52216359

gm2 = gm3 = 1

gm4 = 29.907863850

3, γ

c1 ÷ c4 = 1.112417878

gm1 = 3.895269008

gm2 = gm3 = 1

gm4 = 4.926499772

4, γ

c1 ÷ c4 = 0.524940700

gm1 = 0.690658208

gm2 = gm3 = 1

gm4 = 1.162687957

5, γ

c1 ÷ c4 = 0.287380782

gm1 = 0.189771233

gm2 = gm3 = 1

gm4 = 0.390460877

6, α

c= 0.886452869

gm1 = 0.886452869

gm2 = 1

9

10

11

1.115

1.089

1.067

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

195

Table 8.26 Element values of the cascaded Gm-C Papoulis_Z filters (amin = 55 dB) n

fs

Cell no., type

3

4.409

1, γ

c1 ÷ c4 = 0.729126152

gm1 = 27.44173873

gm2 = gm3 = 1

2, α

c= 1.561289502

gm1 = 1.561289502

gm2 = 1

1, γ

c1 ÷ c4 = 1.104069622

gm1 = 20.13758405

gm2 = gm3 = 1

2, β

c1 = 0.860255326

c2 = 2.396211104

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 1.754795935

gm1 = 48.63266727

gm2 = gm3 = 1

gm4 = 12.06894319

2, γ

c1 ÷ c4 = 0.575207087

gm1 = 2.164298993

gm2 = gm3 = 1

gm4 = 0.823513131

3, α

c= 1.649360775

gm1 = 1.649360775

gm2 = 1

1, γ

c1 ÷ c4 = 2.423067064

gm1 = 48.82861361

gm2 = gm3 = 1

gm4 = 23.19523662

2, γ

c1 ÷ c4 = 0.731204187

gm1 = 2.451387642

gm2 = gm3 = 1

gm4 = 1.561119155

3, β

c1 = 0.832340308

c2 = 2.616459017

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 3.472802813

gm1 = 162.7417070

gm2 = gm3 = 1

gm4 = 47.99099054

2, γ

c1 ÷ c4 = 0.978293941

gm1 = 4.561370257

gm2 = gm3 = 1

gm4 = 3.224245765

3, γ

c1 ÷ c4 = 0.466814159

gm1 = 0.741562665

gm2 = gm3 = 1

gm4 = 0.546961341

4, α

c= 1.414125968

gm1 = 1.414125968

gm2 = 1

1, γ

c1 ÷ c4 = 4.517076542

gm1 = 160.6215788

gm2 = gm3 = 1

gm4 = 81.38570030

2, γ

c1 ÷ c4 = 1.247476639

gm1 = 5.903533180

gm2 = gm3 = 1

gm4 = 5.553179059

3, γ

c1 ÷ c4 = 0.554779642

gm1 = 0.914650638

gm2 = gm3 = 1

gm4 = 0.907814056

4, β

c1 = 0.700691832

c2 = 2.384204151

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 6.043233221

gm1 = 543.3117520

gm2 = gm3 = 1

gm4 = 145.9869496

2, γ

c1 ÷ c4 = 1.648567190

gm1 = 13.18336014

gm2 = gm3 = 1

gm4 = 10.20692269

3, γ

c1 ÷ c4 = 0.692065519

gm1 = 1.490079800

gm2 = gm3 = 1

gm4 = 1.646297877

4, γ

c1 ÷ c4 = 0.373246879

gm1 = 0.368559691

gm2 = gm3 = 1

gm4 = 0.433620180

5, α

c= 1.165390634

gm1 = 1.165390634

gm2 = 1

4

5

6

7

8

9

2.627

1.732

1.470

1.281

1.203

1.140

gm4 = 1.998854064

gm4 = 4.679546580

(continued)

196

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.26 (continued) n

fs

Cell no., type

10

1.108

1, γ

c1 ÷ c4 = 7.533855976

gm1 = 500.6384399

gm2 = gm3 = 1

gm4 = 227.0406346

2, γ

c1 ÷ c4 = 2.055667599

gm1 = 16.62705078

gm2 = gm3 = 1

gm4 = 16.20487287

3, γ

c1 ÷ c4 = 0.852718064

gm1 = 2.049349886

gm2 = gm3 = 1

gm4 = 2.648188168

4, γ

c1 ÷ c4 = 0.435358025

gm1 = 0.471867474

gm2 = gm3 = 1

gm4 = 0.657891481

5, β

c1 = 0.579401543

c2 = 2.059957396

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 9.591072155

gm1 = 1628.720332

gm2 = gm3 = 1

gm4 = 368.1940475

2, γ

c1 ÷ c4 = 2.632209279

gm1 = 37.77755776

gm2 = gm3 = 1

gm4 = 27.07404309

3, γ

c1 ÷ c4 = 1.087654910

gm1 = 3.860248658

gm2 = gm3 = 1

gm4 = 4.545719036

4, γ

c1 ÷ c4 = 0.531400102

gm1 = 0.729431199

gm2 = gm3 = 1

gm4 = 1.099654391

5, γ

c1 ÷ c4 = 0.305723579

gm1 = 0.220605999

gm2 = gm3 = 1

gm4 = 0.382043224

6, α

c= 0.972538062

gm1 = 0.972538062

gm2 = 1

11

1.081

Table 8.27 Element values of the cascaded Gm-C Papoulis_Z filters (amin = 60 dB) n

fs

Cell no., type

3

5.313

1, γ

c1 ÷ c4 = 0.727316171

gm1 = 39.70088674

gm2 = gm3 = 1

2, α

c= 1.576520971

gm1 = 1.576520971

gm2 = 1

1, γ

c1 ÷ c4 = 1.097304246

gm1 = 25.97815543

gm2 = gm3 = 1

2, β

c1 = 0.871873559

c2 = 2.433324172

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 1.726000792

gm1 = 57.22204061

gm2 = gm3 = 1

gm4 = 11.63389315

2, γ

c1 ÷ c4 = 0.586810817

gm1 = 2.700658753

gm2 = gm3 = 1

gm4 = 0.820195808

3, α

c= 1.731119511

gm1 = 1.731119511

gm2 = 1

1, γ

c1 ÷ c4 = 2.372045293

gm1 = 54.19166516

gm2 = gm3 = 1

gm4 = 22.16454916

2, γ

c1 ÷ c4 = 0.740510550

gm1 = 2.872482462

gm2 = gm3 = 1

gm4 = 1.542892668

3, β

c1 = 0.873152379

c2 = 2.741406648

gm1 ÷ gm4 = 1

4

5

6

2.999

1.894

1.569

gm4 = 1.983385881

gm4 = 4.608630300

(continued)

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

197

Table 8.27 (continued) n

fs

Cell no., type

7

1.340

1, γ

c1 ÷ c4 = 3.367950897

gm1 = 171.9877896

gm2 = gm3 = 1

gm4 = 45.03299899

2, γ

c1 ÷ c4 = 0.978473111

gm1 = 5.053090407

gm2 = gm3 = 1

gm4 = 3.123476077

3, γ

c1 ÷ c4 = 0.489533015

gm1 = 0.893464120

gm2 = gm3 = 1

gm4 = 0.546874440

4, α

c= 1.523289438

gm1 = 1.523289438

gm2 = 1

1, γ

c1 ÷ c4 = 4.364561690

gm1 = 164.1732942

gm2 = gm3 = 1

gm4 = 75.84808560

2, γ

c1 ÷ c4 = 1.236785435

gm1 = 6.271083075

gm2 = gm3 = 1

gm4 = 5.322769128

3, γ

c1 ÷ c4 = 0.573449089

gm1 = 1.047171712

gm2 = gm3 = 1

gm4 = 0.899463032

4, β

c1 = 0.749853787

c2 = 2.551052369

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 5.801979364

gm1 = 540.0927850

gm2 = gm3 = 1

gm4 = 134.3870070

2, γ

c1 ÷ c4 = 1.613881699

gm1 = 13.49525398

gm2 = gm3 = 1

gm4 = 9.595022372

3, γ

c1 ÷ c4 = 0.701794904

gm1 = 1.620384400

gm2 = gm3 = 1

gm4 = 1.594901038

4, γ

c1 ÷ c4 = 0.395649817

gm1 = 0.435286348

gm2 = gm3 = 1

gm4 = 0.430084404

5, α

c= 1.267064815

gm1 = 1.267064815

gm2 = 1

1, γ

c1 ÷ c4 = 7.213711838

gm1 = 487.7004928

gm2 = gm3 = 1

gm4 = 207.9420144

2, γ

c1 ÷ c4 = 1.999825761

gm1 = 16.57494649

gm2 = gm3 = 1

gm4 = 15.10589414

3, γ

c1 ÷ c4 = 0.853889792

gm1 = 2.147157402

gm2 = gm3 = 1

gm4 = 2.535213056

4, γ

c1 ÷ c4 = 0.453270088

gm1 = 0.531913252

gm2 = gm3 = 1

gm4 = 0.645761069

5, β

c1 = 0.624404078

c2 = 2.224794910

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 9.144657024

gm1 = 1559.662583

gm2 = gm3 = 1

gm4 = 334.4608047

2, γ

c1 ÷ c4 = 2.538987633

gm1 = 36.81089520

gm2 = gm3 = 1

gm4 = 24.90254307

3, γ

c1 ÷ c4 = 1.071971426

gm1 = 3.899616924

gm2 = gm3 = 1

gm4 = 4.264417410

4, γ

c1 ÷ c4 = 0.541120931

gm1 = 0.781881385

gm2 = gm3 = 1

gm4 = 1.055040032

5, γ

c1 ÷ c4 = 0.325021609

gm1 = 0.256863135

gm2 = gm3 = 1

gm4 = 0.376366636

6, α

c= 1.059916558

gm1 = 1.059916558

gm2 = 1

8

9

10

11

1.245

1.168

1.130

1.097

198

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

8.4.3 Halpern_Z Filters See Tables 8.28, 8.29, 8.30, 8.31, 8.32, 8.33 and 8.34. Table 8.28 Element values of the cascaded Gm-C Halpern_Z filters (amin = 30 dB) n

fs

Cell no., type

3

1.805

1, γ

c1 ÷ c4 = 0.834593264

gm1 = 5.821411036

gm2 = gm3 = 1

2, α

c= 1.665448430

gm1 = 1.665448430

gm2 = 1

1, γ

c1 ÷ c4 = 1.345003969

gm1 = 8.603842271

gm2 = gm3 = 1

2, β

c1 = 0.871904048

c2 = 2.155349506

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 2.372604258

gm1 = 35.04496802

gm2 = gm3 = 1

gm4 = 24.08419599

2, γ

c1 ÷ c4 = 0.611930294

gm1 = 1.142270244

gm2 = gm3 = 1

gm4 = 1.259487816

3, α

c= 1.258315004

gm1 = 1.258315004

gm2 = 1

1, γ

c1 ÷ c4 = 3.357375277

gm1 = 47.63042418

gm2 = gm3 = 1

gm4 = 47.56819108

2, γ

c1 ÷ c4 = 0.861621493

gm1 = 1.983194688

gm2 = gm3 = 1

gm4 = 2.748003455

3, β

c1 = 0.663257030

c2 = 1.943775562

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 4.983445465

gm1 = 187.0644863

gm2 = gm3 = 1

gm4 = 103.4065092

2, γ

c1 ÷ c4 = 1.355978224

gm1 = 5.613050250

gm2 = gm3 = 1

gm4 = 7.410932713

3, γ

c1 ÷ c4 = 0.415362159

gm1 = 0.413119932

gm2 = gm3 = 1

gm4 = 0.733834671

4, α

c= 0.928767306

gm1 = 0.928767306

gm2 = 1

1, γ

c1 ÷ c4 = 6.488597433

gm1 = 211.5175274

gm2 = gm3 = 1

gm4 = 173.9178502

2, γ

c1 ÷ c4 = 1.859875245

gm1 = 9.405504536

gm2 = gm3 = 1

gm4 = 14.11327409

3, γ

c1 ÷ c4 = 0.564729660

gm1 = 0.729628189

gm2 = gm3 = 1

gm4 = 1.394752775

4, β

c1 = 0.500289824

c2 = 1.609556083

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.843697942

gm1 = 7.045630703

gm2 = gm3 = 1

gm4 = 3.226543339

2, γ

c1 ÷ c4 = 8.688501904

gm1 = 265.8094289

gm2 = gm3 = 1

gm4 = 309.4207247

4

5

6

7

8

9

1.433

1.198

1.132

1.081

1.060

1.042

gm4 = 3.057253861

gm4 = 7.840858340

(continued)

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

199

Table 8.28 (continued) n

10

11

fs

1.034

1.026

Cell no., type 3, γ

c1 ÷ c4 = 2.728655606

gm1 = 18.33238398

gm2 = gm3 = 1

gm4 = 30.72184722

4, γ

c1 ÷ c4 = 0.310157780

gm1 = 0.211619298

gm2 = gm3 = 1

gm4 = 0.568965762

5, α

c= 0.727029384

gm1 = 0.727029384

gm2 = 1

1, γ

c1 ÷ c4 = 1.128765620

gm1 = 8.085249275

gm2 = gm3 = 1

gm4 = 5.696590553

2, γ

c1 ÷ c4 = 0.419013703

gm1 = 0.537345202

gm2 = gm3 = 1

gm4 = 0.985069753

3, γ

c1 ÷ c4 = 3.557728179

gm1 = 29.73993790

gm2 = gm3 = 1

gm4 = 52.18466736

4, γ

c1 ÷ c4 = 10.73551437

gm1 = 248.5077763

gm2 = gm3 = 1

gm4 = 470.2547448

5, b

c1 = 0.397310080

c2 = 1.334226568

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.616787036

gm1 = 4.963923586

gm2 = gm3 = 1

gm4 = 2.069293072

2, γ

c1 ÷ c4 = 1.599479718

gm1 = 10.85427917

gm2 = gm3 = 1

gm4 = 11.31602852

3, γ

c1 ÷ c4 = 4.834760756

gm1 = 63.20564212

gm2 = gm3 = 1

gm4 = 96.32857287

4, γ

c1 ÷ c4 = 0.247493663

gm1 = 0.138123276

gm2 = gm3 = 1

gm4 = 0.500885085

5, γ

c1 ÷ c4 = 13.53404029

gm1 = 388.0215170

gm2 = gm3 = 1

gm4 = 744.0407952

6, α

c= 0.595326419

gm1 = 0.595326419

gm2 = 1

Table 8.29 Element values of the cascaded Gm-C Halpern_Z filters (amin = 35 dB) n

fs

Cell no., type

3

2.104

1, γ

c1 ÷ c4 = 0.817046430

gm1 = 7.661845199

gm2 = gm3 = 1

2, α

c= 1.791607520

gm1 = 1.791607520

gm2 = 1

1, γ

c1 ÷ c4 = 1.304055706

gm1 = 9.908395415

gm2 = gm3 = 1

2, β

c1 = 0.922337570

c2 = 2.298008234

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 2.239956339

gm1 = 35.99128161

gm2 = gm3 = 1

gm4 = 21.45434602

2, γ

c1 ÷ c4 = 0.630041210

gm1 = 1.351567287

gm2 = gm3 = 1

gm4 = 1.202221306

3, α

c= 1.441669099

gm1 = 1.441669099

gm2 = 1

4

5

1.580

1.262

gm4 = 2.903705957

gm4 = 7.335133744

(continued)

200

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.29 (continued) n

fs

Cell no., type

6

1.174

1, γ

c1 ÷ c4 = 3.154413733

gm1 = 46.38602082

gm2 = gm3 = 1

gm4 = 42.01936930

2, γ

c1 ÷ c4 = 0.860411319

gm1 = 2.132655131

gm2 = gm3 = 1

gm4 = 2.553280883

3, β

c1 = 0.734287589

c2 = 2.159004831

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 4.630473279

gm1 = 174.9813879

gm2 = gm3 = 1

gm4 = 89.45698619

2, γ

c1 ÷ c4 = 1.293103783

gm1 = 5.421738730

gm2 = gm3 = 1

gm4 = 6.450097188

3, γ

c1 ÷ c4 = 0.437735193

gm1 = 0.481033297

gm2 = gm3 = 1

gm4 = 0.694946241

4, α

c= 1.070346723

gm1 = 1.070346723

gm2 = 1

1, γ

c1 ÷ c4 = 6.014912043

gm1 = 192.9117989

gm2 = gm3 = 1

gm4 = 149.7724023

2, γ

c1 ÷ c4 = 1.746132839

gm1 = 8.664630213

gm2 = gm3 = 1

gm4 = 12.06248465

3, γ

c1 ÷ c4 = 0.573054728

gm1 = 0.778078644

gm2 = gm3 = 1

gm4 = 1.287354279

4, β

c1 = 0.557443794

c2 = 1.813335431

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 8.003278105

gm1 = 666.8747625

gm2 = gm3 = 1

gm4 = 263.1512227

2, γ

c1 ÷ c4 = 2.508049403

gm1 = 23.06435660

gm2 = gm3 = 1

gm4 = 25.44774145

3, γ

c1 ÷ c4 = 0.814361256

gm1 = 1.683035392

gm2 = gm3 = 1

gm4 = 2.810263695

4, γ

c1 ÷ c4 = 0.326999513

gm1 = 0.241092459

gm2 = gm3 = 1

gm4 = 0.529719825

5, α

c= 0.836735852

gm1 = 0.836735852

gm2 = 1

1, γ

c1 ÷ c4 = 9.863275791

gm1 = 642.2612419

gm2 = gm3 = 1

gm4 = 397.8165771

2, γ

c1 ÷ c4 = 1.070065097

gm1 = 3.615310558

gm2 = gm3 = 1

gm4 = 4.879913879

3, γ

c1 ÷ c4 = 3.246693015

gm1 = 25.35351221

gm2 = gm3 = 1

gm4 = 42.82856584

4, γ

c1 ÷ c4 = 0.424854990

gm1 = 0.396793183

gm2 = gm3 = 1

gm4 = 0.892505890

5, β

c1 = 0.442480685

c2 = 1.508598425

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.595146461

gm1 = 4.782761184

gm2 = gm3 = 1

gm4 = 1.777638186

2, γ

c1 ÷ c4 = 1.481716033

gm1 = 9.588993443

gm2 = gm3 = 1

gm4 = 9.415637994

3, γ

c1 ÷ c4 = 12.35263102

gm1 = 422.1119808

gm2 = gm3 = 1

gm4 = 621.0938902

7

8

9

10

11

1.106

1.079

1.055

1.043

1.033

(continued)

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

201

Table 8.29 (continued) n

fs

Cell no., type 4, γ

c1 ÷ c4 = 4.375531750

gm1 = 43.92721751

gm2 = gm3 = 1

gm4 = 78.10532694

5, γ

c1 ÷ c4 = 0.260508849

gm1 = 0.145885514

gm2 = gm3 = 1

gm4 = 0.462712579

6, α

c= 0.683783680

gm1 = 0.683783680

gm2 = 1

Table 8.30 Element values of the cascaded Gm-C Halpern_Z filters (amin = 40 dB) n

fs

Cell no., type

3

2.479

1, γ

c1 ÷ c4 = 0.805976004

gm1 = 10.43212438

gm2 = gm3 = 1

2, α

c= 1.887035864

gm1 = 1.887035864

gm2 = 1

1, γ

c1 ÷ c4 = 1.276410709

gm1 = 11.83033149

gm2 = gm3 = 1

2, β

c1 = 0.964169002

c2 = 2.415952658

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 2.145819393

gm1 = 38.55728511

gm2 = gm3 = 1

gm4 = 19.66890982

2, γ

c1 ÷ c4 = 0.650432114

gm1 = 1.631404973

gm2 = gm3 = 1

gm4 = 1.172638500

3, α

c= 1.622903288

gm1 = 1.622903288

gm2 = 1

1, γ

c1 ÷ c4 = 3.007813319

gm1 = 47.01266423

gm2 = gm3 = 1

gm4 = 38.20927676

2, γ

c1 ÷ c4 = 0.866580626

gm1 = 2.357746927

gm2 = gm3 = 1

gm4 = 2.434163873

3, β

c1 = 0.805494838

c2 = 2.366120455

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 4.371488446

gm1 = 170.3289266

gm2 = gm3 = 1

gm4 = 79.84047774

2, γ

c1 ÷ c4 = 1.253749344

gm1 = 5.455175822

gm2 = gm3 = 1

gm4 = 5.815271447

3, γ

c1 ÷ c4 = 0.464687442

gm1 = 0.572387578

gm2 = gm3 = 1

gm4 = 0.675370045

4, α

c= 1.215924458

gm1 = 1.215924458

gm2 = 1

c= 1.215924458

1, γ

c1 ÷ c4 = 5.666922484

gm1 = 182.9001127

gm2 = gm3 = 1

gm4 = 133.1566589

2, γ

c1 ÷ c4 = 1.667663346

gm1 = 8.308527198

gm2 = gm3 = 1

gm4 = 10.67841842

3, γ

c1 ÷ c4 = 0.588660548

gm1 = 0.854785303

gm2 = gm3 = 1

gm4 = 1.224238201

4, β

c1 = 0.616863053

c2 = 2.018799150

gm1 ÷ gm4 = 1

4

5

6

7

8

1.758

1.341

1.224

1.136

1.100

gm4 = 2.807037622

gm4 = 7.000242814

(continued)

202

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.30 (continued) n

fs

Cell no., type

9

1.070

1, γ

c1 ÷ c4 = 7.504290628

gm1 = 620.0434311

gm2 = gm3 = 1

gm4 = 231.7996997

2, γ

c1 ÷ c4 = 2.345822725

gm1 = 21.11102824

gm2 = gm3 = 1

gm4 = 21.82218347

3, γ

c1 ÷ c4 = 0.801140772

gm1 = 1.685936379

gm2 = gm3 = 1

gm4 = 2.541590238

4, γ

c1 ÷ c4 = 0.347569435

gm1 = 0.280203759

gm2 = gm3 = 1

gm4 = 0.505411709

5, α

c= 0.949737902

gm1 = 0.949737902

gm2 = 1

1, γ

c1 ÷ c4 = 9.234770845

gm1 = 588.2933625

gm2 = gm3 = 1

gm4 = 349.3800710

2, γ

c1 ÷ c4 = 3.012884998

gm1 = 29.67941785

gm2 = gm3 = 1

gm4 = 36.33460283

3, γ

c1 ÷ c4 = 1.034219837

gm1 = 2.642198749

gm2 = gm3 = 1

gm4 = 4.341046938

4, γ

c1 ÷ c4 = 0.436317080

gm1 = 0.427870122

gm2 = gm3 = 1

gm4 = 0.832350181

5, b

c1 = 0.489653795

c2 = 1.686861273

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 11.51799108

gm1 = 1860.805548

gm2 = gm3 = 1

gm4 = 540.9990569

2, γ

c1 ÷ c4 = 1.400067257

gm1 = 8.842808783

gm2 = gm3 = 1

gm4 = 8.138308161

3, γ

c1 ÷ c4 = 4.020890898

gm1 = 45.88701120

gm2 = gm3 = 1

gm4 = 65.26157246

4, γ

c1 ÷ c4 = 0.585294400

gm1 = 0.801736800

gm2 = gm3 = 1

gm4 = 1.583206324

5, γ

c1 ÷ c4 = 0.276450367

gm1 = 0.167066178

gm2 = gm3 = 1

gm4 = 0.437540912

6, α

c= 0.774529146

gm1 = 0.774529146

gm2 = 1

10

11

1.055

1.042

Table 8.31 Element values of the cascaded Gm-C Halpern_Z filters (amin = 45 dB) n

fs

Cell no., type

3

2.944

1, γ

c1 ÷ c4 = 0.798833896

gm1 = 14.54091101

gm2 = gm3 = 1

gm4 = 2.744644007

2, α

c= 1.957112017

gm1 = 1.957112017

gm2 = 1

c= 1.957112017

1, γ

c1 ÷ c4 = 1.257264924

gm1 = 14.51180547

gm2 = gm3 = 1

gm4 = 6.771249663

2, β

c1 = 0.998154801

c2 = 2.511517131

gm1 ÷ gm4 = 1

4

1971

(continued)

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

203

Table 8.31 (continued) n

fs

Cell no., type

5

1.434

1, γ

c1 ÷ c4 = 2.077450614

gm1 = 42.68536700

gm2 = gm3 = 1

gm4 = 18.41310889

2, γ

c1 ÷ c4 = 0.670931177

gm1 = 1.993702850

gm2 = gm3 = 1

gm4 = 1.159501322

3, α

c= 1.798821285

gm1 = 1.798821285

gm2 = 1

1, γ

c1 ÷ c4 = 2.899156328

gm1 = 49.16563702

gm2 = gm3 = 1

gm4 = 35.49050634

2, γ

c1 ÷ c4 = 0.876796028

gm1 = 2.658479713

gm2 = gm3 = 1

gm4 = 2.362292134

3, β

c1 = 0.875831437

c2 = 2.563108388

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 4.175488486

gm1 = 171.0202109

gm2 = gm3 = 1

gm4 = 72.90539044

2, γ

c1 ÷ c4 = 1.230274315

gm1 = 5.665245745

gm2 = gm3 = 1

gm4 = 5.386922684

3, γ

c1 ÷ c4 = 0.494860598

gm1 = 0.690503567

gm2 = gm3 = 1

gm4 = 0.668652013

4, α

c= 1.365290279

gm1 = 1.365290279

gm2 = 1

1, γ

c1 ÷ c4 = 5.402084818

gm1 = 178.6229106

gm2 = gm3 = 1

gm4 = 121.1391891

2, γ

c1 ÷ c4 = 1.613617783

gm1 = 8.225653417

gm2 = gm3 = 1

gm4 = 9.718521267

3, γ

c1 ÷ c4 = 0.609498870

gm1 = 0.959344275

gm2 = gm3 = 1

gm4 = 1.190433756

4, β

c1 = 0.678319704

c2 = 2.224855376

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 7.124815468

gm1 = 594.1425856

gm2 = gm3 = 1

gm4 = 209.2579404

2, γ

c1 ÷ c4 = 2.225347734

gm1 = 19.97181614

gm2 = gm3 = 1

gm4 = 19.25626532

3, γ

c1 ÷ c4 = 0.799497621

gm1 = 1.745389194

gm2 = gm3 = 1

gm4 = 2.367665571

4, γ

c1 ÷ c4 = 0.371057428

gm1 = 0.329825406

gm2 = gm3 = 1

gm4 = 0.490919398

5, α

c= 1.066288440

gm1 = 1.066288440

gm2 = 1

1, γ

c1 ÷ c4 = 8.759552626

gm1 = 555.3918746

gm2 = gm3 = 1

gm4 = 314.8155882

2, γ

c1 ÷ c4 = 2.835052351

gm1 = 27.31527240

gm2 = gm3 = 1

gm4 = 31.69644887

3, γ

c1 ÷ c4 = 1.014678187

gm1 = 2.620865100

gm2 = gm3 = 1

gm4 = 3.979303274

4, γ

c1 ÷ c4 = 0.452009318

gm1 = 0.470924477

gm2 = gm3 = 1

gm4 = 0.793438125

5, β

c1 = 0.538797622

c2 = 1.868804942

gm1 ÷ gm4 = 1

6

7

8

9

10

1.283

1.170

1.125

1.087

1.068

(continued)

204

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.31 (continued) n

fs

Cell no., type

11

1.051

1, γ

c1 ÷ c4 = 10.89791726

gm1 = 1736.807143

gm2 = gm3 = 1

gm4 = 485.0709569

2, γ

c1 ÷ c4 = 3.743876008

gm1 = 65.52701910

gm2 = gm3 = 1

gm4 = 55.97228785

3, γ

c1 ÷ c4 = 1.344141737

gm1 = 5.276147122

gm2 = gm3 = 1

gm4 = 7.254947290

4, γ

c1 ÷ c4 = 0.584010307

gm1 = 0.816293040

gm2 = gm3 = 1

gm4 = 1.450835942

5, γ

c1 ÷ c4 = 0.294695359

gm1 = 0.193498484

gm2 = gm3 = 1

gm4 = 0.420721445

6, α

c= 0.867824730

gm1 = 0.867824730

gm2 = 1

Table 8.32 Element values of the cascaded Gm-C Halpern_Z filters (amin = 50 dB) n

fs

Cell no., type

3

3.518

1, γ

c1 ÷ c4 = 0.794150015

gm1 = 20.59972818

gm2 = gm3 = 1

2, α

c= 2.007458056

gm1 = 2.007458056

gm2 = 1

1, γ

c1 ÷ c4 = 1.243745156

gm1 = 18.16803278

gm2 = gm3 = 1

2, β

c1 = 1.025294701

c2 = 2.587666748

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 2.026914007

gm1 = 48.49538727

gm2 = gm3 = 1

gm4 = 17.50632570

2, γ

c1 ÷ c4 = 0.690245835

gm1 = 2.455641379

gm2 = gm3 = 1

gm4 = 1.156076003

3, α

c= 1.966087765

gm1 = 1.966087765

gm2 = 1

1, γ

c1 ÷ c4 = 2.817036104

gm1 = 52.73049378

gm2 = gm3 = 1

gm4 = 33.49414205

2, γ

c1 ÷ c4 = 0.888941001

gm1 = 3.041011136

gm2 = gm3 = 1

gm4 = 2.320781093

3, β

c1 = 0.944202478

c2 = 2.748185303

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 4.023801370

gm1 = 176.0432498

gm2 = gm3 = 1

gm4 = 67.73824430

2, γ

c1 ÷ c4 = 1.217634970

gm1 = 6.030357519

gm2 = gm3 = 1

gm4 = 5.095037048

3, γ

c1 ÷ c4 = 0.527202236

gm1 = 0.840001158

gm2 = gm3 = 1

gm4 = 0.670938504

4, α

c= 1.517952427

gm1 = 1.517952427

gm2 = 1

4

5

6

7

2.224

1.544

1.351

1.211

gm4 = 2.703673218

gm4 = 6.610930977

(continued)

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

205

Table 8.32 (continued) n

fs

Cell no., type

8

1.153

1, γ

c1 ÷ c4 = 5.195316479

gm1 = 178.6137819

gm2 = gm3 = 1

gm4 = 112.1295941

2, γ

c1 ÷ c4 = 1.576840740

gm1 = 8.355320026

gm2 = gm3 = 1

gm4 = 9.040504204

3, γ

c1 ÷ c4 = 0.634146824

gm1 = 1.093471671

gm2 = gm3 = 1

gm4 = 1.177139211

4, β

c1 = 0.741516807

c2 = 2.430249936

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 6.826649771

gm1 = 582.7385686

gm2 = gm3 = 1

gm4 = 192.3240372

2, γ

c1 ÷ c4 = 2.135450587

gm1 = 19.42403960

gm2 = gm3 = 1

gm4 = 17.39964682

3, γ

c1 ÷ c4 = 0.806449238

gm1 = 1.854196078

gm2 = gm3 = 1

gm4 = 2.258080173

4, γ

c1 ÷ c4 = 0.396929004

gm1 = 0.391419984

gm2 = gm3 = 1

gm4 = 0.483329066

5, α

c= 1.186537597

gm1 = 1.186537597

gm2 = 1

1, γ

c1 ÷ c4 = 8.386209753

gm1 = 536.4147497

gm2 = gm3 = 1

gm4 = 288.8869267

2, γ

c1 ÷ c4 = 2.698706921

gm1 = 25.82417384

gm2 = gm3 = 1

gm4 = 28.30706104

3, γ

c1 ÷ c4 = 1.007179037

gm1 = 2.670281772

gm2 = gm3 = 1

gm4 = 3.736707858

4, γ

c1 ÷ c4 = 0.471047387

gm1 = 0.526188005

gm2 = gm3 = 1

gm4 = 0.769339745

5, β

c1 = 0.589883880

c2 = 2.054196086

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 10.41510963

gm1 = 1659.865598

gm2 = gm3 = 1

gm4 = 443.6008618

2, γ

c1 ÷ c4 = 3.525943284

gm1 = 60.42754198

gm2 = gm3 = 1

gm4 = 49.11597491

3, γ

c1 ÷ c4 = 1.307230487

gm1 = 5.149622894

gm2 = gm3 = 1

gm4 = 6.634156035

4, γ

c1 ÷ c4 = 0.589233931

gm1 = 0.852019644

gm2 = gm3 = 1

gm4 = 1.360411533

5, γ

c1 ÷ c4 = 0.314857216

gm1 = 0.225679243

gm2 = gm3 = 1

gm4 = 0.409600927

6, α

c= 0.963900231

gm1 = 0.963900231

gm2 = 1

9

10

11

1.106

1.083

1.063

206

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.33 Element values of the cascaded Gm-C Halpern_Z filters (amin = 55 dB) n

fs

Cell no., type

3

4.220

1, γ

c1 ÷ c4 = 0.791042775

gm1 = 29.51059136

gm2 = gm3 = 1

2, α

c= 2.043051887

gm1 = 2.043051887

gm2 = 1

1, γ

c1 ÷ c4 = 1.234057973

gm1 = 23.09925358

gm2 = gm3 = 1

2, β

c1 = 1.046662003

c2 = 2.647518485

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 1.989037904

gm1 = 56.24297593

gm2 = gm3 = 1

gm4 = 16.83832536

2, γ

c1 ÷ c4 = 0.707681645

gm1 = 3.040224145

gm2 = gm3 = 1

gm4 = 1.158165246

3, α

c= 2.121708758

gm1 = 2.121708758

gm2 = 1

1, γ

c1 ÷ c4 = 2.754020021

gm1 = 57.73007854

gm2 = gm3 = 1

gm4 = 31.99575943

2, γ

c1 ÷ c4 = 0.901661484

gm1 = 3.516272817

gm2 = gm3 = 1

gm4 = 2.298968050

3, β

c1 = 1.009568204

c2 = 2.919914366

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 3.904463672

gm1 = 184.9663114

gm2 = gm3 = 1

gm4 = 63.79437382

2, γ

c1 ÷ c4 = 1.212372695

gm1 = 6.544322130

gm2 = gm3 = 1

gm4 = 4.896148201

3, γ

c1 ÷ c4 = 0.560813543

gm1 = 1.026613843

gm2 = gm3 = 1

gm4 = 0.679699721

4, α

c= 1.673150774

gm1 = 1.673150774

gm2 = 1

1, γ

c1 ÷ c4 = 5.030886339

gm1 = 182.1098869

gm2 = gm3 = 1

gm4 = 105.1961111

2, γ

c1 ÷ c4 = 1.552427630

gm1 = 8.665131871

gm2 = gm3 = 1

gm4 = 8.556317596

3, γ

c1 ÷ c4 = 0.661514911

gm1 = 1.260272317

gm2 = gm3 = 1

gm4 = 1.178757876

4, β

c1 = 0.806093852

c2 = 2.633624029

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 6.586892106

gm1 = 582.4032563

gm2 = gm3 = 1

gm4 = 179.1990218

2, γ

c1 ÷ c4 = 2.068310362

gm1 = 19.33512551

gm2 = gm3 = 1

gm4 = 16.03345874

3, γ

c1 ÷ c4 = 0.819900844

gm1 = 2.010072546

gm2 = gm3 = 1

gm4 = 2.194183219

4, γ

c1 ÷ c4 = 0.424800056

gm1 = 0.466966404

gm2 = gm3 = 1

gm4 = 0.480887150

5, α

c= 1.310540762

gm1 = 1.310540762

gm2 = 1

4

5

6

7

8

9

2.522

1.672

1.430

1.257

1.186

1.129

gm4 = 2.676456733

gm4 = 6.496736759

(continued)

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

207

Table 8.33 (continued) n

fs

Cell no., type

10

1.100

1, γ

c1 ÷ c4 = 8.084930223

gm1 = 527.4525301

gm2 = gm3 = 1

gm4 = 268.7415122

2, γ

c1 ÷ c4 = 2.593659059

gm1 = 24.98067537

gm2 = gm3 = 1

gm4 = 25.78515192

3, γ

c1 ÷ c4 = 1.008817082

gm1 = 2.780123037

gm2 = gm3 = 1

gm4 = 3.578118637

4, γ

c1 ÷ c4 = 0.492823735

gm1 = 0.594601488

gm2 = gm3 = 1

gm4 = 0.756187371

5, β

c1 = 0.642823338

c2 = 2.242520392

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 10.02569719

gm1 = 1615.038049

gm2 = gm3 = 1

gm4 = 411.4603699

2, γ

c1 ÷ c4 = 3.353449924

gm1 = 57.01497182

gm2 = gm3 = 1

gm4 = 43.96425074

3, γ

c1 ÷ c4 = 1.284780488

gm1 = 5.148157124

gm2 = gm3 = 1

gm4 = 6.196126600

4, γ

c1 ÷ c4 = 0.599581621

gm1 = 0.907033476

gm2 = gm3 = 1

gm4 = 1.299778854

5, γ

c1 ÷ c4 = 0.336682399

gm1 = 0.264336183

gm2 = gm3 = 1

gm4 = 0.402582796

6, α

c= 1.062945438

gm1 = 1.062945438

gm2 = 1

11

1.075

Table 8.34 Element values of the cascaded Gm-C Halpern_Z filters (amin = 60 dB) n

fs

Cell no., type

3

5.078

1, γ

c1 ÷ c4 = 0.788964427

gm1 = 42.60181620

gm2 = gm3 = 1

2, α

c= 2.067932770

gm1 = 2.067932770

gm2 = 1

1, γ

c1 ÷ c4 = 1.227038843

gm1 = 29.71460126

gm2 = gm3 = 1

2, β

c1 = 1.063297069

c2 = 2.694054456

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 1.960335119

gm1 = 66.31017977

gm2 = gm3 = 1

gm4 = 16.33854530

2, γ

c1 ÷ c4 = 0.722948451

gm1 = 3.777336273

gm2 = gm3 = 1

gm4 = 1.163124637

3, α

c= 2.263401659

gm1 = 2.263401659

gm2 = 1

1, γ

c1 ÷ c4 = 2.705063615

gm1 = 64.28751615

gm2 = gm3 = 1

gm4 = 30.85149107

2, γ

c1 ÷ c4 = 0.914110398

gm1 = 4.099668167

gm2 = gm3 = 1

gm4 = 2.289841630

3, β

c1 = 1.071044382

c2 = 3.077315802

gm1 ÷ gm4 = 1

4

5

6

2.873

1.822

1.522

gm4 = 2.658231215

gm4 = 6.414335642

(continued)

208

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.34 (continued) n

fs

Cell no., type

7

1.310

1, γ

c1 ÷ c4 = 3.809398022

gm1 = 197.6972778

gm2 = gm3 = 1

gm4 = 60.72835897

2, γ

c1 ÷ c4 = 1.212059226

gm1 = 7.211208392

gm2 = gm3 = 1

gm4 = 4.761958634

3, γ

c1 ÷ c4 = 0.594886820

gm1 = 1.257124846

gm2 = gm3 = 1

gm4 = 0.693120793

4, α

c= 1.829861386

gm1 = 1.829861386

gm2 = 1

1, γ

c1 ÷ c4 = 4.898326182

gm1 = 188.7376495

gm2 = gm3 = 1

gm4 = 99.75461399

2, γ

c1 ÷ c4 = 1.536920839

gm1 = 9.139816954

gm2 = gm3 = 1

gm4 = 8.208984985

3, γ

c1 ÷ c4 = 0.690702263

gm1 = 1.463897324

gm2 = gm3 = 1

gm4 = 1.191452759

4, β

c1 = 0.871632661

c2 = 2.833583619

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 6.390936039

gm1 = 591.2830442

gm2 = gm3 = 1

gm4 = 168.7951461

2, γ

c1 ÷ c4 = 2.018322887

gm1 = 19.62639364

gm2 = gm3 = 1

gm4 = 15.01593672

3, γ

c1 ÷ c4 = 0.838286830

gm1 = 2.213756769

gm2 = gm3 = 1

gm4 = 2.163872782

4, γ

c1 ÷ c4 = 0.454364740

gm1 = 0.558947041

gm2 = gm3 = 1

gm4 = 0.482484602

5, α

c= 1.438249643

gm1 = 1.438249643

gm2 = 1

1, γ

c1 ÷ c4 = 7.837191111

gm1 = 526.4133987

gm2 = gm3 = 1

gm4 = 252.6935826

2, γ

c1 ÷ c4 = 2.512478627

gm1 = 24.64229781

gm2 = gm3 = 1

gm4 = 23.88086763

3, γ

c1 ÷ c4 = 1.017475650

gm1 = 2.945488698

gm2 = gm3 = 1

gm4 = 3.480774330

4, γ

c1 ÷ c4 = 0.516900479

gm1 = 0.677677791

gm2 = gm3 = 1

gm4 = 0.751496346

5, β

c1 = 0.697538017

c2 = 2.433272643

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 9.703921718

gm1 = 1594.206851

gm2 = gm3 = 1

gm4 = 385.7756940

2, γ

c1 ÷ c4 = 3.216214163

gm1 = 54.88022622

gm2 = gm3 = 1

gm4 = 40.03252458

3, γ

c1 ÷ c4 = 1.273558176

gm1 = 5.251078420

gm2 = gm3 = 1

gm4 = 5.889947044

4, γ

c1 ÷ c4 = 0.614084500

gm1 = 0.980968779

gm2 = gm3 = 1

gm4 = 1.261260659

5, γ

c1 ÷ c4 = 0.359996162

gm1 = 0.310410691

gm2 = gm3 = 1

gm4 = 0.398666151

6, α

c= 1.165112452

gm1 = 1.165112452

gm2 = 1

8

9

10

11

1.223

1.154

1.119

1.089

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

209

8.4.4 Butterworth_Z (Inverse Chebyshev) Filters See Tables 8.35, 8.36, 8.37, 8.38, 8.39, 8.40 and 8.41. Table 8.35 Element values of the cascaded Gm-C Butterworth_Z (Inverse Chebyshev) filters (amin = 30 dB) n

fs

Cell no., type

3

2.119

1, γ

c1 ÷ c4 = 0.535319454

gm1 = 3.430042057

gm2 = gm3 = 1

2, α

c= 0.880970368

gm1 = 0.880970368

gm2 = 1

1, γ

c1 ÷ c4 = 0.751566182

gm1 = 3.794001572

gm2 = gm3 = 1

2, β

c1 = 0.478112979

c2 = 1.542727392

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 1.128172156

gm1 = 13.71665524

gm2 = gm3 = 1

gm4 = 5.369234006

2, γ

c1 ÷ c4 = 0.294458088

gm1 = 0.356918158

gm2 = gm3 = 1

gm4 = 0.535285082

3, α

c= 0.679955854

gm1 = 0.679955854

gm2 = 1

1, γ

c1 ÷ c4 = 0.385752186

gm1 = 1.025671881

gm2 = gm3 = 1

gm4 = 0.863405181

2, γ

c1 ÷ c4 = 1.476583358

gm1 = 7.514103707

gm2 = gm3 = 1

gm4 = 9.112327420

3, β

c1 = 0.361876848

c2 = 1.244732351

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.543826575

gm1 = 4.381035563

gm2 = gm3 = 1

gm4 = 1.640945951

2, γ

c1 ÷ c4 = 0.217971821

gm1 = 0.216758863

gm2 = gm3 = 1

gm4 = 0.455149143

3, γ

c1 ÷ c4 = 2.037985715

gm1 = 12.18595052

gm2 = gm3 = 1

gm4 = 17.23032646

4, α

c= 0.531541106

gm1 = 0.531541106

gm2 = 1

1, γ

c1 ÷ c4 = 0.283426101

gm1 = 0.745632583

gm2 = gm3 = 1

gm4 = 0.682824802

2, γ

c1 ÷ c4 = 0.693318490

gm1 = 1.848137465

gm2 = gm3 = 1

gm4 = 2.530704625

3, γ

c1 ÷ c4 = 2.531484911

gm1 = 17.42226184

gm2 = gm3 = 1

gm4 = 26.42591722

4, β

c1 = 0.285593612

c2 = 1.002238381

gm1 ÷ gm4 = 1

5, α

c= 0.430658595

gm1 = 0.430658595

gm2 = 1

4

5

6

7

8

1.668

1.364

1.265

1.181

1.142

gm4 = 1.215295029

gm4 = 2.378811797

(continued)

210

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.35 (continued) n

fs

Cell no., type

9

1.108

1, γ

c1 ÷ c4 = 0.395523198

gm1 = 3.284413488

gm2 = gm3 = 1

gm4 = 1.198905388

2, γ

c1 ÷ c4 = 0.173423100

gm1 = 0.178770496

gm2 = gm3 = 1

gm4 = 0.428536703

3, γ

c1 ÷ c4 = 0.924435458

gm1 = 2.798394194

gm2 = gm3 = 1

gm4 = 4.293124930

4, γ

c1 ÷ c4 = 3.260312687

gm1 = 26.91736017

gm2 = gm3 = 1

gm4 = 43.59693709

5, α

c= 0.430658595

gm1 = 0.430658595

gm2 = 1

1, γ

c1 ÷ c4 = 0.226546029

gm1 = 0.654266767

gm2 = gm3 = 1

gm4 = 0.621158562

2, γ

c1 ÷ c4 = 0.502387553

gm1 = 1.228979551

gm2 = gm3 = 1

gm4 = 1.759459293

3, γ

c1 ÷ c4 = 1.129179253

gm1 = 3.837112400

gm2 = gm3 = 1

gm4 = 6.193374928

4, γ

c1 ÷ c4 = 3.903166366

gm1 = 37.09117183

gm2 = gm3 = 1

gm4 = 62.25372551

5, β

c1 = 0.234170880

c2 = 0.829084444

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.143799376

gm1 = 0.598735000

gm2 = gm3 = 1

gm4 = 0.416237708

2, γ

c1 ÷ c4 = 0.316939930

gm1 = 0.789822374

gm2 = gm3 = 1

gm4 = 1.046348356

3, γ

c1 ÷ c4 = 0.664455008

gm1 = 1.776514537

gm2 = gm3 = 1

gm4 = 2.818011739

4, γ

c1 ÷ c4 = 1.420202120

gm1 = 5.602250055

gm2 = gm3 = 1

gm4 = 9.494980430

5, γ

c1 ÷ c4 = 4.792242518

gm1 = 53.87131038

gm2 = gm3 = 1

gm4 = 93.52219322

6, α

c= 0.360059129

gm1 = 0.360059129

gm2 = 1

10

11

1.089

1.072

Table 8.36 Element values of the cascaded Gm-C Butterworth_Z (Inverse Chebyshev) filters (amin = 35 dB) n

fs

Cell no., type

3

2.519

1, γ

c1 ÷ c4 = 0.522990939

gm1 = 4.627210249

gm2 = gm3 = 1

2, α

c= 0.917068733

gm1 = 0.917068733

gm2 = 1

1, γ

c1 ÷ c4 = 0.723400083

gm1 = 4.486601539

gm2 = gm3 = 1

2, β

c1 = 0.492884946

c2 = 1.614296120

gm1 ÷ gm4 = 1

4

1.884

gm4 = 1.140570866

gm4 = 2.177398303

(continued)

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

211

Table 8.36 (continued) n

fs

Cell no., type

5

1.481

1, γ

c1 ÷ c4 = 1.049117552

gm1 = 13.96940915

gm2 = gm3 = 1

gm4 = 4.606220602

2, γ

c1 ÷ c4 = 0.293828239

gm1 = 0.418543876

gm2 = gm3 = 1

gm4 = 0.492763826

3, α

c= 0.737050216

gm1 = 0.737050216

gm2 = 1

1, γ

c1 ÷ c4 = 1.360811862

gm1 = 14.47513038

gm2 = gm3 = 1

gm4 = 7.696736644

2, γ

c1 ÷ c4 = 0.374681949

gm1 = 0.548682921

gm2 = gm3 = 1

gm4 = 0.770492540

3, β

c1 = 0.384146603

c2 = 1.345322819

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.508370064

gm1 = 4.198404478

gm2 = gm3 = 1

gm4 = 1.386848396

2, γ

c1 ÷ c4 = 1.848350678

gm1 = 17.09280409

gm2 = gm3 = 1

gm4 = 14.12836473

3, γ

c1 ÷ c4 = 0.221241912

gm1 = 0.157492675

gm2 = gm3 = 1

gm4 = 0.417672578

4, α

c= 0.587924646

gm1 = 0.587924646

gm2 = 1

1, γ

c1 ÷ c4 = 0.278312083

gm1 = 0.774719617

gm2 = gm3 = 1

gm4 = 0.605986767

2, γ

c1 ÷ c4 = 0.639946693

gm1 = 1.696647181

gm2 = gm3 = 1

gm4 = 2.105340356

3, γ

c1 ÷ c4 = 2.285224851

gm1 = 15.29843351

gm2 = gm3 = 1

gm4 = 21.48613264

4, β

c1 = 0.307744938

c2 = 1.100488385

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.372272346

gm1 = 3.084723110

gm2 = gm3 = 1

gm4 = 1.009724208

2, γ

c1 ÷ c4 = 0.839137140

gm1 = 4.437399930

gm2 = gm3 = 1

gm4 = 3.487055919

3, γ

c1 ÷ c4 = 0.177712751

gm1 = 0.109641157

gm2 = gm3 = 1

gm4 = 0.392942240

4, γ

c1 ÷ c4 = 2.921345666

gm1 = 22.91187244

gm2 = gm3 = 1

gm4 = 34.95495142

5, α

c= 0.481286929

gm1 = 0.481286929

gm2 = 1

1, γ

c1 ÷ c4 = 0.223960170

gm1 = 0.671524044

gm2 = gm3 = 1

gm4 = 0.550274348

2, γ

c1 ÷ c4 = 0.465761691

gm1 = 1.109359942

gm2 = gm3 = 1

gm4 = 1.457667019

3, γ

c1 ÷ c4 = 1.019053686

gm1 = 3.282093179

gm2 = gm3 = 1

gm4 = 4.991092596

4, γ

c1 ÷ c4 = 3.488205076

gm1 = 31.11132338

gm2 = gm3 = 1

gm4 = 49.67001965

6

7

8

9

10

1.347

1.237

1.186

1.141

1.116

(continued)

212

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.36 (continued) n

11

fs

1.094

Cell no., type 5, β

c1 = 0.254346465

c2 = 0.917911439

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.299704352

gm1 = 2.707191258

gm2 = gm3 = 1

gm4 = 0.880250089

2, γ

c1 ÷ c4 = 0.148173346

gm1 = 0.179691646

gm2 = gm3 = 1

gm4 = 0.381564713

3, γ

c1 ÷ c4 = 0.604921159

gm1 = 1.532665397

gm2 = gm3 = 1

gm4 = 2.282389213

4, γ

c1 ÷ c4 = 1.272182785

gm1 = 4.679219297

gm2 = gm3 = 1

gm4 = 7.566711240

5, γ

c1 ÷ c4 = 4.266196326

gm1 = 44.43997767

gm2 = gm3 = 1

gm4 = 74.06797419

6, α

c= 0.404725076

gm1 = 0.404725076

gm2 = 1

Table 8.37 Element values of the cascaded Gm-C Butterworth_Z (Inverse Chebyshev) filters (amin = 40 dB) n

fs

Cell no., type

3

3.012

1, γ

c1 ÷ c4 = 0.515111979

gm1 = 6.417906830

gm2 = gm3 = 1

2, α

c= 0.942491022

gm1 = 0.942491022

gm2 = 1

1, γ

c1 ÷ c4 = 0.703943338

gm1 = 5.481333164

gm2 = gm3 = 1

2, β

c1 = 0.504350473

c2 = 1.669873476

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.991872777

gm1 = 14.8831151

gm2 = gm3 = 1

gm4 = 4.087594560

2, γ

c1 ÷ c4 = 0.294680946

gm1 = 0.501776370

gm2 = gm3 = 1

gm4 = 0.463861800

3, α

c= 0.785246136

gm1 = 0.785246136

gm2 = 1

1, γ

c1 ÷ c4 = 1.275409351

gm1 = 14.56715857

gm2 = gm3 = 1

gm4 = 6.726181557

2, γ

c1 ÷ c4 = 0.367538403

gm1 = 0.604854183

gm2 = gm3 = 1

gm4 = 0.706035720

3, β

c1 = 0.403896635

c2 = 1.433802972

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.482952870

gm1 = 4.192015099

gm2 = gm3 = 1

gm4 = 1.211990415

2, γ

c1 ÷ c4 = 1.705696399

gm1 = 16.10412133

gm2 = gm3 = 1

gm4 = 11.99374095

3, γ

c1 ÷ c4 = 0.225653671

gm1 = 0.181258907

gm2 = gm3 = 1

gm4 = 0.391882967

4

5

6

7

2.141

1.617

1.442

1.301

gm4 = 1.093086230

gm4 = 2.042665523

(continued)

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

213

Table 8.37 (continued) n

8

9

10

11

fs

1.236

1.179

1.147

1.118

Cell no., type 4, α

c= 0.639110923

gm1 = 0.639110923

gm2 = 1

1, γ

c1 ÷ c4 = 0.600603101

gm1 = 3.916732503

gm2 = gm3 = 1

gm4 = 1.811535148

2, γ

c1 ÷ c4 = 2.098651842

gm1 = 19.80856781

gm2 = gm3 = 1

gm4 = 18.07938099

3, γ

c1 ÷ c4 = 0.276042502

gm1 = 0.242331204

gm2 = gm3 = 1

gm4 = 0.552257163

4, β

c1 = 0.328540589

c2 = 1.191765123

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.356169940

gm1 = 3.012351999

gm2 = gm3 = 1

gm4 = 0.879104576

2, γ

c1 ÷ c4 = 0.774953473

gm1 = 4.037481070

gm2 = gm3 = 1

gm4 = 2.930507780

3, γ

c1 ÷ c4 = 2.662275314

gm1 = 26.25058083

gm2 = gm3 = 1

gm4 = 28.98811651

4, γ

c1 ÷ c4 = 0.183074681

gm1 = 0.095994824

gm2 = gm3 = 1

gm4 = 0.368366138

5, α

c= 0.528886857

gm1 = 0.528886857

gm2 = 1

1, γ

c1 ÷ c4 = 0.439151077

gm1 = 2.726206983

gm2 = gm3 = 1

gm4 = 1.248743564

2, γ

c1 ÷ c4 = 0.223729713

gm1 = 0.270273344

gm2 = gm3 = 1

gm4 = 0.500588637

3, γ

c1 ÷ c4 = 0.935501005

gm1 = 2.920493390

gm2 = gm3 = 1

gm4 = 4.160158128

4, γ

c1 ÷ c4 = 3.169944922

gm1 = 27.12869048

gm2 = gm3 = 1

gm4 = 40.97533635

5, β

c1 = 0.273834692

c2 = 1.002742568

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.288176881

gm1 = 2.617234693

gm2 = gm3 = 1

gm4 = 0.765435017

2, γ

c1 ÷ c4 = 0.560393195

gm1 = 2.687602114

gm2 = gm3 = 1

gm4 = 1.912141205

3, γ

c1 ÷ c4 = 0.153553484

gm1 = 0.103266796

gm2 = gm3 = 1

gm4 = 0.357597074

4, γ

c1 ÷ c4 = 1.158934643

gm1 = 4.060540221

gm2 = gm3 = 1

gm4 = 6.233798792

5, γ

c1 ÷ c4 = 3.860931760

gm1 = 38.05992826

gm2 = gm3 = 1

gm4 = 60.62028146

6, α

c= 0.447531836

gm1 = 0.447531836

gm2 = 1

214

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.38 Element values of the cascaded Gm-C Butterworth_Z (Inverse Chebyshev) filters (amin = 45 dB) n

fs

Cell no., type

3

3.616

1, γ

c1 ÷ c4 = 0.509976118

gm1 = 9.067388051

gm2 = gm3 = 1

2, α

c= 0.960210032

gm1 = 0.960210032

gm2 = 1

1, γ

c1 ÷ c4 = 0.690204360

gm1 = 6.854332327

gm2 = gm3 = 1

2, β

c1 = 0.513176768

c2 = 1.712647822

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.949550488

gm1 = 16.42368366

gm2 = gm3 = 1

gm4 = 3.722451115

2, γ

c1 ÷ c4 = 0.296190249

gm1 = 0.610379807

gm2 = gm3 = 1

gm4 = 0.443513060

3, α

c= 0.825480280

gm1 = 0.825480280

gm2 = 1

1, γ

c1 ÷ c4 = 0.362858708

gm1 = 0.681398250

gm2 = gm3 = 1

gm4 = 0.659647906

2, γ

c1 ÷ c4 = 0.362858708

gm1 = 0.681398250

gm2 = gm3 = 1

gm4 = 0.659647906

3, β

c1 = 0.421211289

c2 = 1.510831044

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 1.595877594

gm1 = 51.04662850

gm2 = gm3 = 1

gm4 = 10.46684428

2, γ

c1 ÷ c4 = 0.464341687

gm1 = 1.330955367

gm2 = gm3 = 1

gm4 = 1.086914464

3, γ

c1 ÷ c4 = 0.230535879

gm1 = 0.210982374

gm2 = gm3 = 1

gm4 = 0.373435650

4, α

c= 0.685193085

gm1 = 0.685193085

gm2 = 1

1, γ

c1 ÷ c4 = 0.570907670

gm1 = 3.868334172

gm2 = gm3 = 1

gm4 = 1.600406107

2, γ

c1 ÷ c4 = 1.953878528

gm1 = 18.76769172

gm2 = gm3 = 1

gm4 = 15.63548818

3, γ

c1 ÷ c4 = 0.275529371

gm1 = 0.263898357

gm2 = gm3 = 1

gm4 = 0.513202183

4, β

c1 = 0.347871144

c2 = 1.275868589

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.344907692

gm1 = 3.031426076

gm2 = gm3 = 1

gm4 = 0.785257196

2, γ

c1 ÷ c4 = 0.725499890

gm1 = 3.797379256

gm2 = gm3 = 1

gm4 = 2.530639990

3, γ

c1 ÷ c4 = 2.459365394

gm1 = 24.03970501

gm2 = gm3 = 1

gm4 = 24.70107498

4, γ

c1 ÷ c4 = 0.188959792

gm1 = 0.109743821

gm2 = gm3 = 1

gm4 = 0.350708739

5, α

c= 0.573372687

gm1 = 0.573372687

gm2 = 1

4

5

6

7

8

9

2.441

1.774

1.550

1.374

1.292

1.221

gm4 = 1.062217849

gm4 = 1.949752062

(continued)

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

215

Table 8.38 (continued) n

fs

Cell no., type

10

1.182

1, γ

c1 ÷ c4 = 0.419423799

gm1 = 2.639121266

gm2 = gm3 = 1

gm4 = 1.098160974

2, γ

c1 ÷ c4 = 0.870541975

gm1 = 4.342677712

gm2 = gm3 = 1

gm4 = 3.562369461

3, γ

c1 ÷ c4 = 0.224985503

gm1 = 0.179266470

gm2 = gm3 = 1

gm4 = 0.464360475

4, γ

c1 ÷ c4 = 2.919701172

gm1 = 24.42443307

gm2 = gm3 = 1

gm4 = 34.72226533

5, β

c1 = 0.292504549

c2 = 1.083243525

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.280533661

gm1 = 2.604858852

gm2 = gm3 = 1

gm4 = 0.682812911

2, γ

c1 ÷ c4 = 0.526329626

gm1 = 2.489916266

gm2 = gm3 = 1

gm4 = 1.645706896

3, γ

c1 ÷ c4 = 1.070113526

gm1 = 5.267334597

gm2 = gm3 = 1

gm4 = 5.274621051

4, γ

c1 ÷ c4 = 3.540728747

gm1 = 39.80531655

gm2 = gm3 = 1

gm4 = 50.94318077

5, γ

c1 ÷ c4 = 0.159486360

gm1 = 0.068205696

gm2 = gm3 = 1

gm4 = 0.340349713

6, α

c= 0.488378313

gm1 = 0.488378313

gm2 = 1

11

1.146

Table 8.39 Element values of the cascaded Gm-C Butterworth_Z (Inverse Chebyshev) filters (amin = 50 dB) n

fs

Cell no., type

3

4.353

1, γ

c1 ÷ c4 = 0.506581988

gm1 = 12.96993282

gm2 = gm3 = 1

2, α

c= 0.972471426

gm1 = 0.972471426

gm2 = 1

1, γ

c1 ÷ c4 = 0.680348028

gm1 = 8.716778820

gm2 = gm3 = 1

2, β

c1 = 0.519927937

c2 = 1.745349669

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.917768299

gm1 = 18.63012710

gm2 = gm3 = 1

gm4 = 3.458444572

2, γ

c1 ÷ c4 = 0.297908423

gm1 = 0.749789705

gm2 = gm3 = 1

gm4 = 0.428800483

3, α

c= 0.858756177

gm1 = 0.858756177

gm2 = 1

1, γ

c1 ÷ c4 = 1.161485896

gm1 = 16.26129371

gm2 = gm3 = 1

gm4 = 5.529145310

2, γ

c1 ÷ c4 = 0.359760693

gm1 = 0.780053202

gm2 = gm3 = 1

gm4 = 0.625319949

3, β

c1 = 0.436239376

c2 = 1.577299186

gm1 ÷ gm4 = 1

4

5

6

2.793

1.955

1.672

gm4 = 1.041844469

gm4 = 1.884256907

(continued)

216

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.39 (continued) n

fs

Cell no., type

7

1.456

1, γ

c1 ÷ c4 = 1.509822207

gm1 = 51.31997137

gm2 = gm3 = 1

gm4 = 9.341114005

2, γ

c1 ÷ c4 = 0.450482359

gm1 = 1.407053462

gm2 = gm3 = 1

gm4 = 0.994700105

3, γ

c1 ÷ c4 = 0.235488407

gm1 = 0.247272070

gm2 = gm3 = 1

gm4 = 0.359835052

4, α

c= 0.726367299

gm1 = 0.726367299

gm2 = 1

1, γ

c1 ÷ c4 = 1.839433945

gm1 = 44.16902958

gm2 = gm3 = 1

gm4 = 13.82706722

2, γ

c1 ÷ c4 = 0.548069981

gm1 = 1.624220574

gm2 = gm3 = 1

gm4 = 1.443887756

3, γ

c1 ÷ c4 = 0.276097364

gm1 = 0.291462470

gm2 = gm3 = 1

gm4 = 0.483947856

4, β

c1 = 0.365679186

c2 = 1.352787348

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.686682052

gm1 = 12.96537498

gm2 = gm3 = 1

gm4 = 2.234153439

2, γ

c1 ÷ c4 = 0.337008837

gm1 = 0.884146670

gm2 = gm3 = 1

gm4 = 0.715672980

3, γ

c1 ÷ c4 = 2.297384042

gm1 = 22.63511398

gm2 = gm3 = 1

gm4 = 21.52239905

4, γ

c1 ÷ c4 = 0.195021339

gm1 = 0.126135696

gm2 = gm3 = 1

gm4 = 0.337616458

5, α

c= 0.614713443

gm1 = 0.614713443

gm2 = 1

1, γ

c1 ÷ c4 = 0.404578012

gm1 = 2.618554937

gm2 = gm3 = 1

gm4 = 0.986101526

2, γ

c1 ÷ c4 = 0.819059526

gm1 = 4.099331435

gm2 = gm3 = 1

gm4 = 3.118396851

3, γ

c1 ÷ c4 = 2.719067903

gm1 = 27.92119711

gm2 = gm3 = 1

gm4 = 30.07993373

4, γ

c1 ÷ c4 = 0.227176502

gm1 = 0.157680662

gm2 = gm3 = 1

gm4 = 0.437117293

5, β

c1 = 0.310255473

c2 = 1.159185561

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.275602245

gm1 = 2.651423250

gm2 = gm3 = 1

gm4 = 0.621425695

2, γ

c1 ÷ c4 = 0.499814687

gm1 = 2.368025965

gm2 = gm3 = 1

gm4 = 1.447749434

3, γ

c1 ÷ c4 = 0.999080946

gm1 = 4.842079054

gm2 = gm3 = 1

gm4 = 4.561963547

4, γ

c1 ÷ c4 = 3.282666830

gm1 = 36.08350400

gm2 = gm3 = 1

gm4 = 43.75321137

5, γ

c1 ÷ c4 = 0.165678569

gm1 = 0.077625703

gm2 = gm3 = 1

gm4 = 0.327535132

6, α

c= 0.527189380

gm1 = 0.527189380

gm2 = 1

8

9

10

11

1.355

1.268

1.220

1.177

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

217

Table 8.40 Element values of the cascaded Gm-C Butterworth_Z (Inverse Chebyshev) filters (amin = 55 dB) n

fs

Cell no., type

3

5.252

1, γ

c1 ÷ c4 = 0.504317437

gm1 = 18.70705347

gm2 = gm3 = 1

2, α

c= 0.980914347

gm1 = 0.980914347

gm2 = 1

1, γ

c1 ÷ c4 = 0.673195131

gm1 = 11.22232647

gm2 = gm3 = 1

2, β

c1 = 0.525066639

c2 = 1.770227627

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.893613857

gm1 = 21.59655671

gm2 = gm3 = 1

gm4 = 3.263635329

2, γ

c1 ÷ c4 = 0.299599986

gm1 = 0.927240728

gm2 = gm3 = 1

gm4 = 0.417944138

3, α

c= 0.886065665

gm1 = 0.886065665

gm2 = 1

1, γ

c1 ÷ c4 = 1.122971371

gm1 = 17.81718504

gm2 = gm3 = 1

gm4 = 5.149742211

2, γ

c1 ÷ c4 = 0.357694389

gm1 = 0.903848341

gm2 = gm3 = 1

gm4 = 0.599362817

3, β

c1 = 0.449172195

c2 = 1.634228259

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 1.441439763

gm1 = 52.87708404

gm2 = gm3 = 1

gm4 = 8.490897587

2, γ

c1 ÷ c4 = 0.440016236

gm1 = 1.517512014

gm2 = gm3 = 1

gm4 = 0.925054500

3, γ

c1 ÷ c4 = 0.240274341

gm1 = 0.290998336

gm2 = gm3 = 1

gm4 = 0.349563096

4, α

c= 0.762907767

gm1 = 0.762907767

gm2 = 1

1, γ

c1 ÷ c4 = 1.747617911

gm1 = 44.10196698

gm2 = gm3 = 1

gm4 = 12.45507051

2, γ

c1 ÷ c4 = 0.530234366

gm1 = 1.681605976

gm2 = gm3 = 1

gm4 = 1.324915768

3, γ

c1 ÷ c4 = 0.277312930

gm1 = 0.325247435

gm2 = gm3 = 1

gm4 = 0.461506532

4, β

c1 = 0.381954627

c2 = 1.422668188

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.655753328

gm1 = 12.82319635

gm2 = gm3 = 1

gm4 = 2.008651008

2, γ

c1 ÷ c4 = 2.166095986

gm1 = 39.61309088

gm2 = gm3 = 1

gm4 = 19.10475437

3, γ

c1 ÷ c4 = 0.331488683

gm1 = 0.511086599

gm2 = gm3 = 1

gm4 = 0.662748454

4, γ

c1 ÷ c4 = 0.201035864

gm1 = 0.145365859

gm2 = gm3 = 1

gm4 = 0.327658700

5, α

c= 0.652929099

gm1 = 0.652929099

gm2 = 1

4

5

6

7

8

9

3.203

2.161

1.811

1.548

1.425

1.321

gm4 = 1.028259886

gm4 = 1.837344711

(continued)

218

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.40 (continued) n

fs

Cell no., type

10

1.263

1, γ

c1 ÷ c4 = 0.777619934

gm1 = 3.957943255

gm2 = gm3 = 1

gm4 = 2.780029997

2, γ

c1 ÷ c4 = 0.777619934

gm1 = 3.957943255

gm2 = gm3 = 1

gm4 = 2.780029997

3, γ

c1 ÷ c4 = 2.555678695

gm1 = 26.42160067

gm2 = gm3 = 1

gm4 = 26.54337616

4, γ

c1 ÷ c4 = 0.229938583

gm1 = 0.173032630

gm2 = gm3 = 1

gm4 = 0.416118497

5, β

c1 = 0.327018241

c2 = 1.230440169

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.478889699

gm1 = 8.476184431

gm2 = gm3 = 1

gm4 = 1.296794265

2, γ

c1 ÷ c4 = 0.272595439

gm1 = 0.745798409

gm2 = gm3 = 1

gm4 = 0.574614037

3, γ

c1 ÷ c4 = 0.941376716

gm1 = 4.551696295

gm2 = gm3 = 1

gm4 = 4.018516781

4, γ

c1 ÷ c4 = 3.071345592

gm1 = 33.44477767

gm2 = gm3 = 1

gm4 = 38.27040133

5, γ

c1 ÷ c4 = 0.171933637

gm1 = 0.088513686

gm2 = gm3 = 1

gm4 = 0.317763197

6, α

c= 0.563917353

gm1 = 0.563917353

gm2 = 1

11

1.211

Table 8.41 Element values of the cascaded Gm-C Butterworth_Z (Inverse Chebyshev) filters (amin = 60 dB) n

fs

Cell no., type

3

6.344

1, γ

c1 ÷ c4 = 0.502796564

gm1 = 27.13396034

gm2 = gm3 = 1

2, α

c= 0.986708319

gm1 = 0.986708319

gm2 = 1

1, γ

c1 ÷ c4 = 0.667960112

gm1 = 14.57908757

gm2 = gm3 = 1

2, β

c1 = 0.528963536

c2 = 1.789084364

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.875085072

gm1 = 25.46780227

gm2 = gm3 = 1

gm4 = 3.117603580

2, γ

c1 ÷ c4 = 0.301150127

gm1 = 1.152076230

gm2 = gm3 = 1

gm4 = 0.409806069

3, α

c= 0.908337035

gm1 = 0.908337035

gm2 = 1

1, γ

c1 ÷ c4 = 1.092665085

gm1 = 19.88039492

gm2 = gm3 = 1

gm4 = 4.860189308

2, γ

c1 ÷ c4 = 0.356308737

gm1 = 1.056996152

gm2 = gm3 = 1

gm4 = 0.579400741

3, β

c1 = 0.460222560

c2 = 1.682682504

gm1 ÷ gm4 = 1

4

5

6

3.679

2.397

1.966

gm4 = 1.019139202

gm4 = 1.803345564

(continued)

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

219

Table 8.41 (continued) n

fs

Cell no., type

7

1.650

1, γ

c1 ÷ c4 = 1.386490374

gm1 = 55.61626203

gm2 = gm3 = 1

gm4 = 7.836130475

2, γ

c1 ÷ c4 = 0.432016948

gm1 = 1.662990935

gm2 = gm3 = 1

gm4 = 0.871419151

3, γ

c1 ÷ c4 = 0.244759011

gm1 = 0.343278555

gm2 = gm3 = 1

gm4 = 0.341652475

4, α

c= 0.795141346

gm1 = 0.795141346

gm2 = 1

1, γ

c1 ÷ c4 = 1.673069980

gm1 = 44.93892257

gm2 = gm3 = 1

gm4 = 11.39275480

2, γ

c1 ÷ c4 = 0.516124646

gm1 = 1.771442871

gm2 = gm3 = 1

gm4 = 1.232621843

3, γ

c1 ÷ c4 = 0.278892102

gm1 = 0.365742586

gm2 = gm3 = 1

gm4 = 0.443957789

4, β

c1 = 0.396725858

c2 = 1.485780393

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 2.058365441

gm1 = 137.6700032

gm2 = gm3 = 1

gm4 = 17.22705519

2, γ

c1 ÷ c4 = 0.630804509

gm1 = 3.660597571

gm2 = gm3 = 1

gm4 = 1.833511236

3, γ

c1 ÷ c4 = 0.327670102

gm1 = 0.544138915

gm2 = gm3 = 1

gm4 = 0.621643843

4, γ

c1 ÷ c4 = 0.206859286

gm1 = 0.167704399

gm2 = gm3 = 1

gm4 = 0.319924860

5, α

c= 0.688083617

gm1 = 0.688083617

gm2 = 1

1, γ

c1 ÷ c4 = 0.743834652

gm1 = 10.19837625

gm2 = gm3 = 1

gm4 = 2.516589569

2, γ

c1 ÷ c4 = 2.420913461

gm1 = 41.26308656

gm2 = gm3 = 1

gm4 = 23.79123459

3, γ

c1 ÷ c4 = 0.384599270

gm1 = 0.643623537

gm2 = gm3 = 1

gm4 = 0.833752736

4, γ

c1 ÷ c4 = 0.233024223

gm1 = 0.191150234

gm2 = gm3 = 1

gm4 = 0.399603455

5, β

c1 = 0.342752006

c2 = 1.296967168

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.462191032

gm1 = 8.390924304

gm2 = gm3 = 1

gm4 = 1.179176481

2, γ

c1 ÷ c4 = 0.893892480

gm1 = 8.522994721

gm2 = gm3 = 1

gm4 = 3.595086383

3, γ

c1 ÷ c4 = 2.896028490

gm1 = 45.78143496

gm2 = gm3 = 1

gm4 = 33.99843093

4, γ

c1 ÷ c4 = 0.270966905

gm1 = 0.276656456

gm2 = gm3 = 1

gm4 = 0.538140406

5, γ

c1 ÷ c4 = 0.178117417

gm1 = 0.100957566

gm2 = gm3 = 1

gm4 = 0.310149326

6, α

c= 0.598540787

gm1 = 0.598540787

gm2 = 1

8

9

10

11

1.503

1.379

1.310

1.249

220

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Fig. 8.15 Attenuation characteristics of seventh (top) and sixth (bottom)-order modified elliptic filters

8.4.5 Modified Elliptic (Zolotarev) Filters The run of the passband attenuation characteristics of the modified elliptic filters synthesized in this paragraph is already depicted in Fig. 8.10a and b (for n = 7, and n = 6), respectively. Below, Fig. 8.15, we show the stopband characteristics of the seventh- and sixth-order modified elliptic filters. One has to note the difference in the asymptotic slope of the attenuation which is in favor of the even order filters. In both cases, amax = 1 dB and amin = 30 dB were imposed in order to expose the mutual relation of the passband and stopband attenuation. As already mentioned, the value of f s given in the tables below was evaluated from renormalized characteristics of the modified elliptic filter so that they, as does the xxx_Z filters, exhibit 3 dB at cutoff. The renormalized characteristics of the seventhand sixth-order modified elliptic filters are shown in Fig. 8.16. One may note that, due to the exceptional selectivity of the modified elliptic filters, not much of a change

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

221

Fig. 8.16 Attenuation characteristics of seventh (left) and sixth (right)-order renormalized modified elliptic filters

(as compared with Fig. 7.15) may be noticed. The difference will be more noticeable for lower-order filters. There is one more consequence of the extreme selectivity of the modified elliptic filters. Namely for large values of amax (e.g., 1 dB), small values of amin (e.g., 30 dB), and extremely large orders (e.g., n = 11) of the filters, the conversion of the characteristic function (which is used in the iterative process) into the transfer function by implementation of Feldkeller’s equation [1] becomes numerical instable. That is a consequence of the fact that the polynomial to be solved (for n = 11, its order is 2n = 22) has zeros which practically overlap. For that reason, in some cases, the tables below are reduced to n = 10 (Tables 8.42, 8.43, 8.44, 8.45, 8.46, 8.47, 8.48, 8.49, 8.50, 8.51, 8.52, 8.53, 8.54, 8.55, 8.56, 8.57, 8.58, 8.59, 8.60, 8.61, 8.62, 8.63, 8.64, 8.65, 8.66, 8.67, 8.68 and 8.69).

222

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.42 Element values of the cascaded Gm-C of modified elliptic (Zolotarev) filters (amax = 0.1 dB amin = 30 dB) n

fs

Cell no., type

3

1.840

1, γ

c1 ÷ c4 = 0.597638325

gm1 = 5.615072227

gm2 = gm3 = 1

2, α

c= 0.943909143

gm1 = 0.943909143

gm2 = 1

1, γ

c1 ÷ c4 = 1.275499141

gm1 = 9.826430980

gm2 = gm3 = 1

2, β

c1 = 0.733865131

c2 = 1.754972952

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 3.106621615

gm1 = 61.13796156

gm2 = gm3 = 1

gm4 = 43.11727712

2, γ

c1 ÷ c4 = 0.647723433

gm1 = 1.335664001

gm2 = gm3 = 1

gm4 = 1.494919146

2, α

c= 1.313719156

gm1 = 1.313719156

gm2 = 1

1, γ

c1 ÷ c4 = 5.942341225

gm1 = 132.5315676

gm2 = gm3 = 1

gm4 = 149.7645676

2, γ

c1 ÷ c4 = 1.170864215

gm1 = 3.524282041

gm2 = gm3 = 1

gm4 = 5.072843243

2, β

c1 = 0.869042281

c2 = 2.108106996

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 13.81813237

gm1 = 1020.943134

gm2 = gm3 = 1

gm4 = 784.6841395

2, γ

c1 ÷ c4 = 2.542830418

gm1 = 16.78385737

gm2 = gm3 = 1

gm4 = 25.32892044

3, γ

c1 ÷ c4 = 0.668073540

gm1 = 0.992172746

gm2 = gm3 = 1

gm4 = 1.421551992

4, α

c= 1.347454840

gm1 = 1.347454840

gm2 = 1

1, γ

c1 ÷ c4 = 18.21885213

gm1 = 1078.986438

gm2 = gm3 = 1

gm4 = 1354.857755

2, γ

c1 ÷ c4 = 3.726951600

gm1 = 34.41522456

gm2 = gm3 = 1

gm4 = 54.49698739

3, γ

c1 ÷ c4 = 1.075141188

gm1 = 2.527584443

gm2 = gm3 = 1

gm4 = 3.903015875

4, β

c1 = 0.857275906

c2 = 2.191340645

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 61.69623685

gm1 = 19,647.50996

gm2 = gm3 = 1

gm4 = 15,314.96994

2, γ

c1 ÷ c4 = 11.14037315

gm1 = 310.2955660

gm2 = gm3 = 1

gm4 = 493.0356314

3, γ

c1 ÷ c4 = 2.615028296

gm1 = 14.52550434

gm2 = gm3 = 1

gm4 = 25.63609767

4, γ

c1 ÷ c4 = 0.703963795

gm1 = 1.015823740

gm2 = gm3 = 1

gm4 = 1.462274031

5, α

c= 1.445269452

gm1 = 1.445269452

gm2 = 1

4

5

6

7

8

9

1.392

1.149

1.078

1.032

1.0265

1.007

gm4 = 2.361708657

gm4 = 8.319372740

(continued)

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

223

Table 8.42 (continued) n

fs

Cell no., type

10

1.004

1, γ

c1 ÷ c4 = 114.5620772

gm1 = 44,123.48759

gm2 = gm3 = 1

gm4 = 52,663.52159

2, γ

c1 ÷ c4 = 20.64436854

gm1 = 962.0447841

gm2 = gm3 = 1

gm4 = 1698.397334

3, γ

c1 ÷ c4 = 4.795175976

gm1 = 47.50795212

gm2 = gm3 = 1

gm4 = 88.75593549

4, γ

c1 ÷ c4 = 1.217101147

gm1 = 3.002272051

gm2 = gm3 = 1

gm4 = 4.953052562

5, b

c1 = 0.914528697

c2 = 2.220100001

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 241.2237958

gm1 = 297,835.5475

gm2 = gm3 = 1

gm4 = 233,076.7765

2, γ

c1 ÷ c4 = 48.28556313

gm1 = 5780.100711

gm2 = gm3 = 1

gm4 = 9311.180777

3, γ

c1 ÷ c4 = 11.05471242

gm1 = 257.2959851

gm2 = gm3 = 1

gm4 = 481.5616151

4, γ

c1 ÷ c4 = 2.624236610

gm1 = 13.96550296

gm2 = gm3 = 1

gm4 = 25.60381978

5, γ

c1 ÷ c4 = 0.706915214

gm1 = 1.005138040

gm2 = gm3 = 1

gm4 = 1.462252561

6, α

c= 1.451412515

gm1 = 1.451412515

gm2 = 1

11

1.002

Table 8.43 Element values of the cascaded Gm-C of modified elliptic (Zolotarev) filters (amax = 0.1 dB amin = 35 dB) n

fs

Cell no., type

3

2.172

1, γ

c1 ÷ c4 = 0.569546529

gm1 = 7.329829541

gm2 = gm3 = 1

2, α

c= 0.971185886

gm1 = 0.971185886

gm2 = 1

1, γ

c1 ÷ c4 = 1.180817664

gm1 = 10.70633973

gm2 = gm3 = 1

2, β

c1 = 0.746315998

c2 = 1.823561357

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 2.648949082

gm1 = 54.09289934

gm2 = gm3 = 1

gm4 = 31.74329473

2, γ

c1 ÷ c4 = 0.622534762

gm1 = 1.425563673

gm2 = gm3 = 1

gm4 = 1.303421473

2, α

c= 1.409249808

gm1 = 1.409249808

gm2 = 1

1, γ

c1 ÷ c4 = 4.891332207

gm1 = 102.7601022

gm2 = gm3 = 1

gm4 = 102.3949006

2, γ

c1 ÷ c4 = 1.074843503

gm1 = 3.251916610

gm2 = gm3 = 1

gm4 = 4.120769325

2, β

c1 = 0.907916395

c2 = 2.261522788

gm1 ÷ gm4 = 1

4

5

6

1.546

1.218

1.117

gm4 = 2.160897264

gm4 = 7.193719369

(continued)

224

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.43 (continued) n

fs

Cell no., type

7

1.109

1, γ

c1 ÷ c4 = 10.34168086

gm1 = 657.3796788

gm2 = gm3 = 1

gm4 = 443.1250476

2, γ

c1 ÷ c4 = 2.086792191

gm1 = 12.25121311

gm2 = gm3 = 1

gm4 = 16.84537233

3, γ

c1 ÷ c4 = 0.633525351

gm1 = 0.936660586

gm2 = gm3 = 1

gm4 = 1.194218702

4, α

c= 1.425273549

gm1 = 1.425273549

gm2 = 1

1, γ

c1 ÷ c4 = 18.22921413

gm1 = 1237.895538

gm2 = gm3 = 1

gm4 = 1354.844760

2, γ

c1 ÷ c4 = 3.708827930

gm1 = 33.81824651

gm2 = gm3 = 1

gm4 = 53.52674474

3, γ

c1 ÷ c4 = 1.088321656

gm1 = 2.599225345

gm2 = gm3 = 1

gm4 = 3.814225734

4, β

c1 = 0.936142885

c2 = 2.365801210

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 41.94896220

gm1 = 10,322.26630

gm2 = gm3 = 1

gm4 = 7096.680139

2, γ

c1 ÷ c4 = 8.271145349

gm1 = 182.1799793

gm2 = gm3 = 1

gm4 = 270.0978245

3, γ

c1 ÷ c4 = 2.203266298

gm1 = 10.60715510

gm2 = gm3 = 1

gm4 = 17.56372797

4, γ

c1 ÷ c4 = 0.693426648

gm1 = 1.001584923

gm2 = gm3 = 1

gm4 = 1.264644029

5, α

c= 1.597352229

gm1 = 1.597352229

gm2 = 1

1, γ

c1 ÷ c4 = 74.95890003

gm1 = 20,723.66398

gm2 = gm3 = 1

gm4 = 22,578.56162

2, γ

c1 ÷ c4 = 14.72900689

gm1 = 509.9387302

gm2 = gm3 = 1

gm4 = 861.3918812

3, γ

c1 ÷ c4 = 3.863297377

gm1 = 31.37211983

gm2 = gm3 = 1

gm4 = 56.38455443

4, γ

c1 ÷ c4 = 1.133114688

gm1 = 2.627153322

gm2 = gm3 = 1

gm4 = 3.988983980

5, b

c1 = 0.974932915

c2 = 2.429943678

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 161.3605925

gm1 = 150,664.4466

gm2 = gm3 = 1

gm4 = 104,369.9175

2, γ

c1 ÷ c4 = 31.84014109

gm1 = 2662.748062

gm2 = gm3 = 1

gm4 = 4041.479997

3, γ

c1 ÷ c4 = 8.190531418

gm1 = 144.4657547

gm2 = gm3 = 1

gm4 = 261.4393988

4, γ

c1 ÷ c4 = 2.215826854

gm1 = 10.04704430

gm2 = gm3 = 1

gm4 = 17.52803843

5, γ

c1 ÷ c4 = 0.698169567

gm1 = 0.985183753

gm2 = gm3 = 1

gm4 = 1.264656226

6, α

c= 1.608471886

gm1 = 1.608471886

gm2 = 1

8

9

10

11

1.030

1.013

1.007

1.003

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

225

Table 8.44 Element values of the cascaded Gm-C of modified elliptic (Zolotarev) filters (amax = 0.1 dB amin = 40 dB) n

fs

Cell no., type

3

2.584

1, γ

c1 ÷ c4 = 0.551526958

gm1 = 9.943786129

gm2 = gm3 = 1

2, α

c= 0.990120577

gm1 = 0.990120577

gm2 = 1

1, γ

c1 ÷ c4 = 1.116575383

gm1 = 12.30716430

gm2 = gm3 = 1

2, β

c1 = 0.756098730

c2 = 1.877652888

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 2.343158353

gm1 = 51.83036339

gm2 = gm3 = 1

gm4 = 25.09778651

2, γ

c1 ÷ c4 = 0.606776529

gm1 = 1.589115510

gm2 = gm3 = 1

gm4 = 1.177393505

2, α

c= 1.490744637

gm1 = 1.490744637

gm2 = 1

1, γ

c1 ÷ c4 = 4.196440411

gm1 = 87.31004614

gm2 = gm3 = 1

gm4 = 75.96733751

2, γ

c1 ÷ c4 = 1.009223495

gm1 = 3.183688537

gm2 = gm3 = 1

gm4 = 3.507950881

2, β

c1 = 0.945112408

c2 = 2.402839109

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 8.606672650

gm1 = 530.5699284

gm2 = gm3 = 1

gm4 = 307.7398946

2, γ

c1 ÷ c4 = 1.903140791

gm1 = 11.17376870

gm2 = gm3 = 1

gm4 = 13.49091154

3, γ

c1 ÷ c4 = 0.669213604

gm1 = 1.114774539

gm2 = gm3 = 1

gm4 = 1.133090625

4, α

c= 1.679256895

gm1 = 1.679256895

gm2 = 1

1, γ

c1 ÷ c4 = 14.99399640

gm1 = 951.1015549

gm2 = gm3 = 1

gm4 = 919.1117687

2, γ

c1 ÷ c4 = 3.254330766

gm1 = 27.27935868

gm2 = gm3 = 1

gm4 = 40.59298434

3, γ

c1 ÷ c4 = 1.058313049

gm1 = 2.543907285

gm2 = gm3 = 1

gm4 = 3.364941456

4, β

c1 = 1.013788743

c2 = 2.581711143

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 30.59097854

gm1 = 6229.450040

gm2 = gm3 = 1

gm4 = 3783.799654

2, γ

c1 ÷ c4 = 6.490101298

gm1 = 120.2186575

gm2 = gm3 = 1

gm4 = 165.0416633

3, γ

c1 ÷ c4 = 1.928885837

gm1 = 8.419730589

gm2 = gm3 = 1

gm4 = 12.94962436

4, γ

c1 ÷ c4 = 0.693108670

gm1 = 1.022737417

gm2 = gm3 = 1

gm4 = 1.131505051

5, α

c= 1.742928807

gm1 = 1.742928807

gm2 = 1

4

5

6

7

8

9

1.733

1.304

1.166

1.078

1.045

1.022

gm4 = 2.036134507

gm4 = 6.475364756

(continued)

226

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.44 (continued) n

fs

Cell no., type

10

1.013

1, γ

c1 ÷ c4 = 52.84491665

gm1 = 11,322.63735

gm2 = gm3 = 1

gm4 = 11,240.33529

2, γ

c1 ÷ c4 = 11.15511183

gm1 = 306.3410436

gm2 = gm3 = 1

gm4 = 491.7665913

3, γ

c1 ÷ c4 = 3.248898364

gm1 = 22.68424642

gm2 = gm3 = 1

gm4 = 38.90240954

4, γ

c1 ÷ c4 = 1.078849670

gm1 = 2.413881649

gm2 = gm3 = 1

gm4 = 3.356538157

5, b

c1 = 1.035480104

c2 = 2.636423767

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 107.5587561

gm1 = 75,448.25791

gm2 = gm3 = 1

gm4 = 46,418.06172

2, γ

c1 ÷ c4 = 22.57099933

gm1 = 1424.007721

gm2 = gm3 = 1

gm4 = 2025.797293

3, γ

c1 ÷ c4 = 6.413968289

gm1 = 91.05167636

gm2 = gm3 = 1

gm4 = 158.1148749

4, γ

c1 ÷ c4 = 1.945452621

gm1 = 7.841239525

gm2 = gm3 = 1

gm4 = 12.91087975

5, γ

c1 ÷ c4 = 0.700338375

gm1 = 0.998598117

gm2 = gm3 = 1

gm4 = 1.131672275

6, α

c= 1.761519191

gm1 = 1.761519191

gm2 = 1

11

1.006

Table 8.45 Element values of the cascaded Gm-C of modified elliptic (Zolotarev) filters (amax = 0.1 dB amin = 45 dB) n

fs

Cell no., type

3

3.091

1, γ

c1 ÷ c4 = 0.539737062

gm1 = 13.84002121

gm2 = gm3 = 1

2, α

c= 1.003184509

gm1 = 1.003184509

gm2 = 1

1, γ

c1 ÷ c4 = 1.071787008

gm1 = 14.70618077

gm2 = gm3 = 1

2, β

c1 = 0.763709035

c2 = 1.919796388

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 2.130495400

gm1 = 52.74747851

gm2 = gm3 = 1

gm4 = 20.92843701

2, γ

c1 ÷ c4 = 0.596545951

gm1 = 1.825701610

gm2 = gm3 = 1

gm4 = 1.090504474

2, α

c= 1.558901374

gm1 = 1.558901374

gm2 = 1

1, γ

c1 ÷ c4 = 3.714388486

gm1 = 79.55365861

gm2 = gm3 = 1

gm4 = 59.93261802

2, γ

c1 ÷ c4 = 0.962062416

gm1 = 3.252734854

gm2 = gm3 = 1

gm4 = 3.088863136

2, β

c1 = 0.978367512

c2 = 2.528543661

gm1 ÷ gm4 = 1

4

5

6

1.956

1.407

1.226

gm4 = 1.956227144

gm4 = 5.996421799

(continued)

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

227

Table 8.45 (continued) n

fs

Cell no., type

7

1.112

1, γ

c1 ÷ c4 = 7.207458624

gm1 = 431.4735829

gm2 = gm3 = 1

gm4 = 217.0738248

2, γ

c1 ÷ c4 = 1.709325739

gm1 = 9.961427185

gm2 = gm3 = 1

gm4 = 10.64844819

3, γ

c1 ÷ c4 = 0.668962188

gm1 = 1.197932570

gm2 = gm3 = 1

gm4 = 1.040686741

4, α

c= 1.795965722

gm1 = 1.795965722

gm2 = 1

1, γ

c1 ÷ c4 = 12.18692401

gm1 = 702.2344741

gm2 = gm3 = 1

gm4 = 609.7173836

2, γ

c1 ÷ c4 = 2.825796219

gm1 = 22.01564648

gm2 = gm3 = 1

gm4 = 30.17014559

3, γ

c1 ÷ c4 = 1.016897584

gm1 = 2.463030283

gm2 = gm3 = 1

gm4 = 2.926618728

4, β

c1 = 1.064247334

c2 = 2.758653743

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 23.54856090

gm1 = 4184.180079

gm2 = gm3 = 1

gm4 = 2248.416955

2, γ

c1 ÷ c4 = 5.314948996

gm1 = 86.90280143

gm2 = gm3 = 1

gm4 = 109.7305548

3, γ

c1 ÷ c4 = 1.737713669

gm1 = 7.126032190

gm2 = gm3 = 1

gm4 = 10.09358392

4, γ

c1 ÷ c4 = 0.698812996

gm1 = 1.069307884

gm2 = gm3 = 1

gm4 = 1.036883768

5, α

c= 1.881227923

gm1 = 1.881227923

gm2 = 1

1, γ

c1 ÷ c4 = 39.48513905

gm1 = 6960.590522

gm2 = gm3 = 1

gm4 = 6286.970817

2, γ

c1 ÷ c4 = 8.852525077

gm1 = 203.1750198

gm2 = gm3 = 1

gm4 = 307.9675683

3, γ

c1 ÷ c4 = 2.823906925

gm1 = 17.61460202

gm2 = gm3 = 1

gm4 = 28.60696697

4, γ

c1 ÷ c4 = 1.043668771

gm1 = 2.300081286

gm2 = gm3 = 1

gm4 = 2.917952156

5, b

c1 = 1.095588329

c2 = 2.838310296

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 76.15621894

gm1 = 42,505.93433

gm2 = gm3 = 1

gm4 = 23,296.91367

2, γ

c1 ÷ c4 = 16.93134879

gm1 = 856.0665901

gm2 = gm3 = 1

gm4 = 1136.241120

3, γ

c1 ÷ c4 = 5.243699904

gm1 = 62.84956612

gm2 = gm3 = 1

gm4 = 103.9715964

4, γ

c1 ÷ c4 = 1.759156055

gm1 = 6.515941276

gm2 = gm3 = 1

gm4 = 10.05433393

5, γ

c1 ÷ c4 = 0.709415073

gm1 = 1.035301260

gm2 = gm3 = 1

gm4 = 1.037505089

6, α

c= 1.910569308

gm1 = 1.910569308

gm2 = 1

8

9

10

11

1.0653

1.033

1.020

1.010

228

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.46 Element values of the cascaded Gm-C of modified elliptic (Zolotarev) filters (amax = 0.1 dB amin = 50 dB) n

fs

Cell no., type

3

3.713

1, γ

c1 ÷ c4 = 0.531921345

gm1 = 19.59667631

gm2 = gm3 = 1

2, α

c= 1.012161321

gm1 = 1.012161321

gm2 = 1

1, γ

c1 ÷ c4 = 1.039953344

gm1 = 18.07750998

gm2 = gm3 = 1

2, β

c1 = 0.769582409

c2 = 1.952327278

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 1.978134493

gm1 = 56.21934498

gm2 = gm3 = 1

gm4 = 18.16994791

2, γ

c1 ÷ c4 = 0.589750050

gm1 = 2.143963765

gm2 = gm3 = 1

gm4 = 1.028604897

2, α

c= 1.615376315

gm1 = 1.615376315

gm2 = 1

1, γ

c1 ÷ c4 = 3.368226816

gm1 = 76.57345334

gm2 = gm3 = 1

gm4 = 49.58079164

2, γ

c1 ÷ c4 = 0.927297296

gm1 = 3.435328891

gm2 = gm3 = 1

gm4 = 2.790880748

2, β

c1 = 1.007755891

c2 = 2.639112787

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 6.233130034

gm1 = 374.8473593

gm2 = gm3 = 1

gm4 = 163.2328211

2, γ

c1 ÷ c4 = 1.569264511

gm1 = 9.355195138

gm2 = gm3 = 1

gm4 = 8.787365151

3, γ

c1 ÷ c4 = 0.671845792

gm1 = 1.313094997

gm2 = gm3 = 1

gm4 = 0.973264886

4, α

c= 1.902087027

gm1 = 1.902087027

gm2 = 1

1, γ

c1 ÷ c4 = 10.25886817

gm1 = 558.1704060

gm2 = gm3 = 1

gm4 = 433.8165362

2, γ

c1 ÷ c4 = 2.516739590

gm1 = 18.83760201

gm2 = gm3 = 1

gm4 = 23.58777733

3, γ

c1 ÷ c4 = 0.987463558

gm1 = 2.456514614

gm2 = gm3 = 1

gm4 = 2.609338950

4, β

c1 = 1.111697346

c2 = 2.924432569

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 18.91536283

gm1 = 3057.281885

gm2 = gm3 = 1

gm4 = 1454.888777

2, γ

c1 ÷ c4 = 4.501060990

gm1 = 67.52697204

gm2 = gm3 = 1

gm4 = 77.96392711

3, γ

c1 ÷ c4 = 1.599594599

gm1 = 6.339923764

gm2 = gm3 = 1

gm4 = 8.211638318

4, γ

c1 ÷ c4 = 0.707932726

gm1 = 1.136254801

gm2 = gm3 = 1

gm4 = 0.966636216

5, α

c= 2.011402237

gm1 = 2.011402237

gm2 = 1

4

5

6

7

8

9

2.220

1.528

1.298

1.152

1.091

1/049

gm4 = 1.904011238

gm4 = 5.667008520

(continued)

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

229

Table 8.46 (continued) n

fs

Cell no., type

10

1.030

1, γ

c1 ÷ c4 = 30.89738361

gm1 = 4699.641916

gm2 = gm3 = 1

gm4 = 3857.216758

2, γ

c1 ÷ c4 = 7.291565496

gm1 = 145.9222031

gm2 = gm3 = 1

gm4 = 207.6135817

3, γ

c1 ÷ c4 = 2.518610771

gm1 = 14.48080863

gm2 = gm3 = 1

gm4 = 22.11812167

4, γ

c1 ÷ c4 = 1.021208457

gm1 = 2.253370812

gm2 = gm3 = 1

gm4 = 2.600459553

5, b

c1 = 1.154687744

c2 = 3.034441888

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 56.86153874

gm1 = 26,559.51775

gm2 = gm3 = 1

gm4 = 13,004.10607

2, γ

c1 ÷ c4 = 13.28342308

gm1 = 564.8384665

gm2 = gm3 = 1

gm4 = 696.6584717

3, γ

c1 ÷ c4 = 4.435389154

gm1 = 46.66120411

gm2 = gm3 = 1

gm4 = 73.04280949

4, γ

c1 ÷ c4 = 1.626869110

gm1 = 5.687135763

gm2 = gm3 = 1

gm4 = 8.174715542

5, γ

c1 ÷ c4 = 0.722923201

gm1 = 1.090089185

gm2 = gm3 = 1

gm4 = 0.968105909

6, α

c= 2.055385233

gm1 = 2.055385233

gm2 = 1

11

1.016

Table 8.47 Element values of the cascaded Gm-C of modified elliptic (Zolotarev) filters (amax = 0.1 dB amin = 55 dB) n

fs

Cell no., type

3

4.472

1, γ

c1 ÷ c4 = 0.526694899

gm1 = 28.07088555

gm2 = gm3 = 1

2, α

c= 1.018313555

gm1 = 1.018313555

gm2 = 1

1, γ

c1 ÷ c4 = 1.017009690

gm1 = 22.68953300

gm2 = gm3 = 1

2, β

c1 = 0.774085955

c2 = 1.977260877

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 1.866473947

gm1 = 62.09915657

gm2 = gm3 = 1

gm4 = 16.26962041

2, γ

c1 ÷ c4 = 0.585143122

gm1 = 2.558773418

gm2 = gm3 = 1

gm4 = 0.983410381

2, α

c= 1.661805884

gm1 = 1.661805884

gm2 = 1

1, γ

c1 ÷ c4 = 3.112838917

gm1 = 76.98020606

gm2 = gm3 = 1

gm4 = 42.56728114

2, γ

c1 ÷ c4 = 0.901146727

gm1 = 3.724176792

gm2 = gm3 = 1

gm4 = 2.572568494

2, β

c1 = 1.033446888

c2 = 2.735400459

gm1 ÷ gm4 = 1

4

5

6

2.530

1.671

1.382

gm4 = 1.869430875

gm4 = 5.435272194

(continued)

230

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.47 (continued) n

fs

Cell no., type

7

1.200

1, γ

c1 ÷ c4 = 5.530079096

gm1 = 343.2863149

gm2 = gm3 = 1

gm4 = 129.1247427

2, γ

c1 ÷ c4 = 1.465232380

gm1 = 9.155091934

gm2 = gm3 = 1

gm4 = 7.509852943

3, γ

c1 ÷ c4 = 0.676405096

gm1 = 1.460625310

gm2 = gm3 = 1

gm4 = 0.922601820

4, α

c= 1.997728184

gm1 = 1.997728184

gm2 = 1

1, γ

c1 ÷ c4 = 8.887902924

gm1 = 471.4724617

gm2 = gm3 = 1

gm4 = 326.8723897

2, γ

c1 ÷ c4 = 2.288612323

gm1 = 16.92612842

gm2 = gm3 = 1

gm4 = 19.22787491

3, γ

c1 ÷ c4 = 0.967009504

gm1 = 2.512713022

gm2 = gm3 = 1

gm4 = 2.374933571

4, β

c1 = 1.156969259

c2 = 3.079992903

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 15.72653443

gm1 = 2391.772040

gm2 = gm3 = 1

gm4 = 1008.621201

2, γ

c1 ÷ c4 = 3.916981918

gm1 = 55.66953828

gm2 = gm3 = 1

gm4 = 58.46770586

3, γ

c1 ÷ c4 = 1.497655271

gm1 = 5.874388829

gm2 = gm3 = 1

gm4 = 6.915050083

4, γ

c1 ÷ c4 = 0.719322617

gm1 = 1.222966254

gm2 = gm3 = 1

gm4 = 0.913267894

5, α

c= 2.134474907

gm1 = 2.134474907

gm2 = 1

1, γ

c1 ÷ c4 = 25.09372254

gm1 = 3422.386327

gm2 = gm3 = 1

gm4 = 2549.477440

2, γ

c1 ÷ c4 = 6.188828076

gm1 = 111.8336819

gm2 = gm3 = 1

gm4 = 148.5414651

3, γ

c1 ÷ c4 = 2.292452787

gm1 = 12.46751554

gm2 = gm3 = 1

gm4 = 17.79961773

4, γ

c1 ÷ c4 = 1.007405234

gm1 = 2.255776703

gm2 = gm3 = 1

gm4 = 2.362559717

5, b

c1 = 1.212218190

c2 = 3.223707259

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 44.34014853

gm1 = 18,060.50310

gm2 = gm3 = 1

gm4 = 7918.517551

2, γ

c1 ÷ c4 = 10.80546064

gm1 = 401.8497697

gm2 = gm3 = 1

gm4 = 458.9473316

3, γ

c1 ÷ c4 = 3.855188629

gm1 = 36.75137593

gm2 = gm3 = 1

gm4 = 54.10768427

4, γ

c1 ÷ c4 = 1.530665037

gm1 = 5.160217513

gm2 = gm3 = 1

gm4 = 6.875011103

5, γ

c1 ÷ c4 = 0.739192707

gm1 = 1.160107761

gm2 = gm3 = 1

gm4 = 0.915219922

6, α

c= 2.195453786

gm1 = 2.195453786

gm2 = 1

8

9

10

11

1.121

1.067

1.042

1.024

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

231

Table 8.48 Element values of the cascaded Gm-C of modified elliptic (Zolotarev) filters (amax = 0.1 dB amin = 60 dB) n

fs

Cell no., type

3

5.396

1, γ

c1 ÷ c4 = 0.523178262

gm1 = 40.52544136

gm2 = gm3 = 1

2, α

c= 1.022520816

gm1 = 1.022520816

gm2 = 1

1, γ

c1 ÷ c4 = 1.000302491

gm1 = 28.92045584

gm2 = gm3 = 1

2, β

c1 = 0.777520574

c2 = 1.996268011

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 1.783205061

gm1 = 70.52778656

gm2 = gm3 = 1

gm4 = 14.91939119

2, γ

c1 ÷ c4 = 0.581961611

gm1 = 3.091084518

gm2 = gm3 = 1

gm4 = 0.949791415

2, α

c= 1.699727789

gm1 = 1.699727789

gm2 = 1

1, γ

c1 ÷ c4 = 2.920345816

gm1 = 80.12520980

gm2 = gm3 = 1

gm4 = 37.63129336

2, γ

c1 ÷ c4 = 0.881151672

gm1 = 4.122085489

gm2 = gm3 = 1

gm4 = 2.408841368

2, β

c1 = 1.055696432

c2 = 2.818534959

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 5.008191398

gm1 = 328.0675804

gm2 = gm3 = 1

gm4 = 106.3779009

2, γ

c1 ÷ c4 = 1.386246549

gm1 = 9.258997497

gm2 = gm3 = 1

gm4 = 6.599463484

3, γ

c1 ÷ c4 = 0.681752300

gm1 = 1.642971659

gm2 = gm3 = 1

gm4 = 0.883657150

4, α

c= 2.083276911

gm1 = 2.083276911

gm2 = 1

1, γ

c1 ÷ c4 = 7.883284504

gm1 = 418.6290178

gm2 = gm3 = 1

gm4 = 258.0738645

2, γ

c1 ÷ c4 = 2.116471994

gm1 = 15.83565138

gm2 = gm3 = 1

gm4 = 16.21737895

3, γ

c1 ÷ c4 = 0.952990867

gm1 = 2.623918681

gm2 = gm3 = 1

gm4 = 2.197902703

4, β

c1 = 1.199904296

c2 = 3.225011628

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 13.45537900

gm1 = 1980.693879

gm2 = gm3 = 1

gm4 = 740.4293931

2, γ

c1 ÷ c4 = 3.487092735

gm1 = 48.23975248

gm2 = gm3 = 1

gm4 = 45.87617241

3, γ

c1 ÷ c4 = 1.421880465

gm1 = 5.633091096

gm2 = gm3 = 1

gm4 = 5.993987348

4, γ

c1 ÷ c4 = 0.732566385

gm1 = 1.331409542

gm2 = gm3 = 1

gm4 = 0.872538460

5, α

c= 2.252300443

gm1 = 2.252300443

gm2 = 1

4

5

6

7

8

9

2.893

1.836

1.479

1.257

1.159

1.090

gm4 = 1.846314820

gm4 = 5.269533335

(continued)

232

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.48 (continued) n

fs

Cell no., type

10

1.057

1, γ

c1 ÷ c4 = 21.00617917

gm1 = 2650.702867

gm2 = gm3 = 1

gm4 = 1790.284350

2, γ

c1 ÷ c4 = 5.382543851

gm1 = 90.40726949

gm2 = gm3 = 1

gm4 = 111.5540709

3, γ

c1 ÷ c4 = 2.120426664

gm1 = 11.14627912

gm2 = gm3 = 1

gm4 = 14.79320592

4, γ

c1 ÷ c4 = 0.999465486

gm1 = 2.296585633

gm2 = gm3 = 1

gm4 = 2.178906663

5, b

c1 = 1.267559823

c2 = 3.404958512

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 35.81198592

gm1 = 13,149.30772

gm2 = gm3 = 1

gm4 = 5173.106391

2, γ

c1 ÷ c4 = 9.052273986

gm1 = 304.0318603

gm2 = gm3 = 1

gm4 = 320.5426022

3, γ

c1 ÷ c4 = 3.424899623

gm1 = 30.37551421

gm2 = gm3 = 1

gm4 = 41.83317420

4, γ

c1 ÷ c4 = 1.459141698

gm1 = 4.828195163

gm2 = gm3 = 1

gm4 = 5.937842590

5, γ

c1 ÷ c4 = 0.756947950

gm1 = 1.243498167

gm2 = gm3 = 1

gm4 = 0.873611309

6, α

c= 2.329824385

gm1 = 2.329824385

gm2 = 1

11

1.033

Table 8.49 Element values of the cascaded Gm-C of modified elliptic (Zolotarev) filters (amax = 0.25 dB amin = 30 dB) n

fs

Cell no., type

3

1.762

1, γ

c1 ÷ c4 = 0.762149347

gm1 = 6.852411607

gm2 = gm3 = 1

2, α

c= 1.181073161

gm1 = 1.181073161

gm2 = 1

1, γ

c1 ÷ c4 = 1.624298465

gm1 = 13.25608852

gm2 = gm3 = 1

2, β

c1 = 0.903843195

c2 = 1.898992692

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 4.113721020

gm1 = 91.93667526

gm2 = gm3 = 1

gm4 = 71.54802085

2, γ

c1 ÷ c4 = 0.803752302

gm1 = 1.852112400

gm2 = gm3 = 1

gm4 = 2.039916727

2, α

c= 1.569739196

gm1 = 1.569739196

gm2 = 1

1, γ

c1 ÷ c4 = 8.044091920

gm1 = 219.2665463

gm2 = gm3 = 1

gm4 = 266.3198837

2, γ

c1 ÷ c4 = 1.477701193

gm1 = 5.267662287

gm2 = gm3 = 1

gm4 = 7.653200715

2, β

c1 = 1.036428170

c2 = 2.209120007

gm1 ÷ gm4 = 1

4

5

6

1.341

1.123

1.061

gm4 = 3.085910678

gm4 = 12.02409594

(continued)

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

233

Table 8.49 (continued) n

fs

Cell no., type

7

1.024

1, γ

c1 ÷ c4 = 20.36331410

gm1 = 1977.868367

gm2 = gm3 = 1

gm4 = 1677.707368

2, γ

c1 ÷ c4 = 3.473784460

gm1 = 29.59223803

gm2 = gm3 = 1

gm4 = 45.86829809

3, γ

c1 ÷ c4 = 0.846133357

gm1 = 1.542286103

gm2 = gm3 = 1

gm4 = 2.007014553

4, α

c= 1.667147361

gm1 = 1.667147361

gm2 = 1

1, γ

c1 ÷ c4 = 39.31330661

gm1 = 4895.218357

gm2 = gm3 = 1

gm4 = 6218.773949

2, γ

c1 ÷ c4 = 6.634335339

gm1 = 97.93213231

gm2 = gm3 = 1

gm4 = 171.3737717

3, γ

c1 ÷ c4 = 1.517344768

gm1 = 4.786330984

gm2 = gm3 = 1

gm4 = 7.570998369

4, β

c1 = 1.075315299

c2 = 2.286011237

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 99.20528629

gm1 = 45,750.66025

gm2 = gm3 = 1

gm4 = 39,459.41388

2, γ

c1 ÷ c4 = 16.49542219

gm1 = 649.6503143

gm2 = gm3 = 1

gm4 = 1076.943010

3, γ

c1 ÷ c4 = 3.490150446

gm1 = 25.40833519

gm2 = gm3 = 1

gm4 = 45.16684791

4, γ

c1 ÷ c4 = 0.857524970

gm1 = 1.493416542

gm2 = gm3 = 1

gm4 = 2.008315896

5, α

c= 1.690833754

gm1 = 1.690833754

gm2 = 1

1, γ

c1 ÷ c4 = 191.0028489

gm1 = 114,002.8157

gm2 = gm3 = 1

gm4 = 146,106.6901

2, γ

c1 ÷ c4 = 31.64647895

gm1 = 2197.406972

gm2 = gm3 = 1

gm4 = 3983.937418

3, γ

c1 ÷ c4 = 6.621323841

gm1 = 89.62714077

gm2 = gm3 = 1

gm4 = 168.5266253

4, γ

c1 ÷ c4 = 1.526982390

gm1 = 4.700727554

gm2 = gm3 = 1

gm4 = 7.567684968

5, b

c1 = 1.082443204

c2 = 2.301054281

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 264.2853359

gm1 = 357,505.2053

gm2 = gm3 = 1

gm4 = 279,764.5049

2, γ

c1 ÷ c4 = 48.28738177

gm1 = 5780.536125

gm2 = gm3 = 1

gm4 = 9311.874711

3, γ

c1 ÷ c4 = 11.05471731

gm1 = 257.2962127

gm2 = gm3 = 1

gm4 = 481.5620407

4, γ

c1 ÷ c4 = 2.624236610

gm1 = 13.96550296

gm2 = gm3 = 1

gm4 = 25.60381978

5, γ

c1 ÷ c4 = 0.706915214

gm1 = 1.005138041

gm2 = gm3 = 1

gm4 = 1.462252561

6, α

c= 1.451412515

gm1 = 1.451412515

gm2 = 1

8

9

10

11

1.012

1.005

1.003

1.002

234

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.50 Element values of the cascaded Gm-C of modified elliptic (Zolotarev) filters (amax = 0.25 dB amin = 35 dB) n

fs

Cell no., type

3

2.074

1, γ

c1 ÷ c4 = 0.724043336

gm1 = 8.822459948

gm2 = gm3 = 1

2, α

c= 1.218833216

gm1 = 1.218833216

gm2 = 1

1, γ

c1 ÷ c4 = 1.494257270

gm1 = 14.09958778

gm2 = gm3 = 1

2, β

c1 = 0.921145182

c2 = 1.980859939

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 3.445495229

gm1 = 78.01043519

gm2 = gm3 = 1

2, γ

c1 ÷ c4 = 0.770341721

gm1 = 1.938782252

gm2 = gm3 = 1

2, α

c= 1.693654200

gm1 = 1.693654200

gm2 = 1

1, γ

c1 ÷ c4 = 6.486649179

gm1 = 161.8836895

gm2 = gm3 = 1

gm4 = 173.9675802

2, γ

c1 ÷ c4 = 1.350410444

gm1 = 4.762237673

gm2 = gm3 = 1

gm4 = 6.119861715

2, β

c1 = 1.090442382

c2 = 2.384749232

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 14.95659894

gm1 = 1230.938833

gm2 = gm3 = 1

gm4 = 907.9827079

2, γ

c1 ÷ c4 = 2.857606279

gm1 = 21.52690580

gm2 = gm3 = 1

gm4 = 30.30894638

3, γ

c1 ÷ c4 = 0.823173452

gm1 = 1.521153665

gm2 = gm3 = 1

gm4 = 1.711793149

4, α

c= 1.832962158

gm1 = 1.832962158

gm2 = 1

1, γ

c1 ÷ c4 = 27.70387614

gm1 = 2682.838516

gm2 = gm3 = 1

gm4 = 3094.298842

2, γ

c1 ÷ c4 = 5.213592545

gm1 = 63.21753545

gm2 = gm3 = 1

gm4 = 104.3313702

3, γ

c1 ÷ c4 = 1.394812970

gm1 = 4.143413518

gm2 = gm3 = 1

gm4 = 5.987220388

4, β

c1 = 1.142099572

c2 = 2.496141306

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 63.88747626

gm1 = 21,587.10126

gm2 = gm3 = 1

gm4 = 16,382.57265

2, γ

c1 ÷ c4 = 2.879909944

gm1 = 17.70287262

gm2 = gm3 = 1

gm4 = 29.64598041

3, γ

c1 ÷ c4 = 2.879909944

gm1 = 17.70287262

gm2 = gm3 = 1

gm4 = 29.64598041

4, γ

c1 ÷ c4 = 0.840362405

gm1 = 1.451419294

gm2 = gm3 = 1

gm4 = 1.713835252

5, α

c= 1.873546830

gm1 = 1.873546830

gm2 = 1

4

5

6

7

8

9

1.483

1.185

1.096

1.041

1.022

1.009

gm4 = 2.793450343

gm4 = 10.21916520

gm4 = 50.53132436

(continued)

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

235

Table 8.50 (continued) n

fs

Cell no., type

10

1.005

1, γ

c1 ÷ c4 = 117.6865461

gm1 = 47,378.86369

gm2 = gm3 = 1

gm4 = 55,503.83538

2, γ

c1 ÷ c4 = 21.63550576

gm1 = 1064.288169

gm2 = gm3 = 1

gm4 = 1854.345071

3, γ

c1 ÷ c4 = 5.201904646

gm1 = 56.07126440

gm2 = gm3 = 1

gm4 = 101.7782255

4, γ

c1 = 1.154322534

c2 = 2.522675749

gm1 ÷ gm4 = 1

gm4 = 5.980902532

5, b

c1 = 1.154322534

c2 = 2.522675749

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 285.8724139

gm1 = 428,231.1311

gm2 = gm3 = 1

gm4 = 327,150.6941

2, γ

c1 ÷ c4 = 49.56707128

gm1 = 6155.657259

gm2 = gm3 = 1

gm4 = 9786.972324

3, γ

c1 ÷ c4 = 11.69067229

gm1 = 288.8785509

gm2 = gm3 = 1

gm4 = 532.3522891

4, γ

c1 ÷ c4 = 2.891133334

gm1 = 16.97565341

gm2 = gm3 = 1

gm4 = 29.60654795

5, γ

c1 ÷ c4 = 0.844263449

gm1 = 1.434744759

gm2 = gm3 = 1

gm4 = 1.713882367

6, α

c= 1.882360028

gm1 = 1.882360028

gm2 = 1

11

1.002

Table 8.51 Element values of the cascaded Gm-C of modified elliptic (Zolotarev) filters (amax = 0.25 dB amin = 40 dB) n

fs

Cell no., type

3

2.462

1, γ

c1 ÷ c4 = 0.699709173

gm1 = 11.85825166

gm2 = gm3 = 1

2, α

c= 1.245205957

gm1 = 1.245205957

gm2 = 1

1, γ

c1 ÷ c4 = 1.406690558

gm1 = 15.92107426

gm2 = gm3 = 1

2, β

c1 = 0.934847138

c2 = 2.046131087

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 3.008594471

gm1 = 72.47181214

gm2 = gm3 = 1

gm4 = 38.74549697

2, γ

c1 ÷ c4 = 0.749479112

gm1 = 2.127346852

gm2 = gm3 = 1

gm4 = 1.567470600

2, α

c= 1.799656158

gm1 = 1.799656158

gm2 = 1

1, γ

c1 ÷ c4 = 5.473867211

gm1 = 132.1532585

gm2 = gm3 = 1

gm4 = 124.3981112

2, γ

c1 ÷ c4 = 1.261561581

gm1 = 4.564943777

gm2 = gm3 = 1

gm4 = 5.132040606

2, β

c1 = 1.138565726

c2 = 2.543908536

gm1 ÷ gm4 = 1

4

5

6

1.657

1.263

1.140

gm4 = 2.613498549

gm4 = 9.084955374

(continued)

236

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.51 (continued) n

fs

Cell no., type

7

1.063

1, γ

c1 ÷ c4 = 11.69476841

gm1 = 869.8819162

gm2 = gm3 = 1

gm4 = 556.8930067

2, γ

c1 ÷ c4 = 2.455605115

gm1 = 17.26877471

gm2 = gm3 = 1

gm4 = 21.82529997

3, γ

c1 ÷ c4 = 0.813725532

gm1 = 1.566492939

gm2 = gm3 = 1

gm4 = 1.517121548

4, α

c= 1.988659471

gm1 = 1.988659471

gm2 = 1

1, γ

c1 ÷ c4 = 20.88280818

gm1 = 1690.736143

gm2 = gm3 = 1

gm4 = 1761.791951

2, γ

c1 ÷ c4 = 4.300851810

gm1 = 45.35936692

gm2 = gm3 = 1

gm4 = 69.87902569

3, γ

c1 ÷ c4 = 1.313085899

gm1 = 3.791948856

gm2 = gm3 = 1

gm4 = 4.968771967

4, β

c1 = 1.207578084

c2 = 2.699519220

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 44.70390191

gm1 = 12,007.65009

gm2 = gm3 = 1

gm4 = 8031.171486

2, γ

c1 ÷ c4 = 8.982111227

gm1 = 217.5803064

gm2 = gm3 = 1

gm4 = 314.2509695

3, γ

c1 ÷ c4 = 2.482749854

gm1 = 13.55133679

gm2 = gm3 = 1

gm4 = 21.16012914

4, γ

c1 ÷ c4 = 0.837400307

gm1 = 1.466035279

gm2 = gm3 = 1

gm4 = 1.518375657

5, α

c= 2.049988561

gm1 = 2.049988561

gm2 = 1

1, γ

c1 ÷ c4 = 79.30058155

gm1 = 23,609.62688

gm2 = gm3 = 1

gm4 = 25,220.62720

2, γ

c1 ÷ c4 = 15.85392070

gm1 = 595.7372960

gm2 = gm3 = 1

gm4 = 990.3784718

3, γ

c1 ÷ c4 = 4.291350423

gm1 = 38.85668342

gm2 = gm3 = 1

gm4 = 67.53687888

4, γ

c1 ÷ c4 = 1.332434384

gm1 = 3.640802933

gm2 = gm3 = 1

gm4 = 4.960172179

5, b

c1 = 1.226976100

c2 = 2.742392437

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 170.2394247

gm1 = 171,578.2331

gm2 = gm3 = 1

gm4 = 116,068.4975

2, γ

c1 ÷ c4 = 33.72583459

gm1 = 3021.883494

gm2 = gm3 = 1

gm4 = 4517.937775

3, γ

c1 ÷ c4 = 8.890716030

gm1 = 171.0420208

gm2 = gm3 = 1

gm4 = 303.6475964

4, γ

c1 ÷ c4 = 2.498183820

gm1 = 12.79717850

gm2 = gm3 = 1

gm4 = 21.11593075

5, γ

c1 ÷ c4 = 0.843660013

gm1 = 1.440263124

gm2 = gm3 = 1

gm4 = 1.518562608

6, α

c= 2.065575409

gm1 = 2.065575409

gm2 = 1

8

9

10

11

1.035

1.016

1.009

1.004

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

237

Table 8.52 Element values of the cascaded Gm-C of modified elliptic (Zolotarev) filters (amax = 0.25 dB amin = 45 dB) n

fs

Cell no., type

3

2.942

1, γ

c1 ÷ c4 = 0.683836058

gm1 = 16.40165235

gm2 = gm3 = 1

2, α

c= 1.263481346

gm1 = 1.263481346

gm2 = 1

1, γ

c1 ÷ c4 = 1.345974728

gm1 = 18.77324688

gm2 = gm3 = 1

2, β

c1 = 0.945569252

c2 = 2.097411725

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 2.709695181

gm1 = 72.06513405

gm2 = gm3 = 1

gm4 = 31.57440757

2, γ

c1 ÷ c4 = 0.736096463

gm1 = 2.414095880

gm2 = gm3 = 1

gm4 = 1.440101275

2, α

c= 1.889205646

gm1 = 1.889205646

gm2 = 1

1, γ

c1 ÷ c4 = 4.784397811

gm1 = 116.7137326

gm2 = gm3 = 1

gm4 = 95.38098955

2, γ

c1 ÷ c4 = 1.198038093

gm1 = 4.582716215

gm2 = gm3 = 1

gm4 = 4.464478469

2, β

c1 = 1.181809028

c2 = 2.686868521

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 9.592716289

gm1 = 677.7315740

gm2 = gm3 = 1

gm4 = 375.8221939

2, γ

c1 ÷ c4 = 2.180800581

gm1 = 14.94190312

gm2 = gm3 = 1

gm4 = 16.78236814

3, γ

c1 ÷ c4 = 0.812497644

gm1 = 1.664596466

gm2 = gm3 = 1

gm4 = 1.382551015

4, α

c= 2.134324571

gm1 = 2.134324571

gm2 = 1

1, γ

c1 ÷ c4 = 16.57290820

gm1 = 1186.298250

gm2 = gm3 = 1

gm4 = 1111.939396

2, γ

c1 ÷ c4 = 3.681062714

gm1 = 35.33257210

gm2 = gm3 = 1

gm4 = 50.34051319

3, γ

c1 ÷ c4 = 1.256768326

gm1 = 3.617423286

gm2 = gm3 = 1

gm4 = 4.272540890

4, β

c1 = 1.270573120

c2 = 2.893945965

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 33.33052468

gm1 = 7573.062163

gm2 = gm3 = 1

gm4 = 4470.513958

2, γ

c1 ÷ c4 = 7.191659746

gm1 = 149.6587713

gm2 = gm3 = 1

gm4 = 199.4341000

3, γ

c1 ÷ c4 = 2.211252445

gm1 = 11.14742871

gm2 = gm3 = 1

gm4 = 16.08638794

4, γ

c1 ÷ c4 = 0.842976990

gm1 = 1.520152120

gm2 = gm3 = 1

gm4 = 1.381182380

5, α

c= 2.219312457

gm1 = 2.219312457

gm2 = 1

4

5

6

7

8

9

1.865

1.358

1.194

1.093

1.053

1.026

gm4 = 2.498990361

gm4 = 8.337104487

(continued)

238

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.52 (continued) n

fs

Cell no., type

10

1.015

1, γ

c1 ÷ c4 = 57.19728280

gm1 = 13,506.75465

gm2 = gm3 = 1

gm4 = 13,132.05081

2, γ

c1 ÷ c4 = 12.25913256

gm1 = 373.4625107

gm2 = gm3 = 1

gm4 = 588.3789032

3, γ

c1 ÷ c4 = 3.674861215

gm1 = 29.16139782

gm2 = gm3 = 1

gm4 = 48.16442238

4, γ

c1 ÷ c4 = 1.282776215

gm1 = 3.423083670

gm2 = gm3 = 1

gm4 = 4.263110436

5, b

c1 = 1.299672846

c2 = 2.958731331

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 114.9921736

gm1 = 88,179.87117

gm2 = gm3 = 1

gm4 = 52,985.22937

2, γ

c1 ÷ c4 = 24.45805834

gm1 = 1692.628390

gm2 = gm3 = 1

gm4 = 2367.423129

3, γ

c1 ÷ c4 = 7.104622097

gm1 = 112.3572721

gm2 = gm3 = 1

gm4 = 190.7251263

4, γ

c1 ÷ c4 = 2.231830002

gm1 = 10.34814242

gm2 = gm3 = 1

gm4 = 16.03962214

5, γ

c1 ÷ c4 = 0.852517245

gm1 = 1.482255182

gm2 = gm3 = 1

gm4 = 1.381798507

6, α

c= 2.244973280

gm1 = 2.244973280

gm2 = 1

11

1.008

Table 8.53 Element values of the cascaded Gm-C of modified elliptic (Zolotarev) filters (amax = 0.25 dB amin = 50 dB) n

fs

Cell no., type

3

3.530

1, γ

c1 ÷ c4 = 0.673334836

gm1 = 23.12558698

gm2 = gm3 = 1

2, α

c= 1.276077277

gm1 = 1.276077277

gm2 = 1

1, γ

c1 ÷ c4 = 1.302991355

gm1 = 22.84867917

gm2 = gm3 = 1

2, β

c1 = 0.953879543

c2 = 2.137247708

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 2.498186317

gm1 = 75.46991893

gm2 = gm3 = 1

gm4 = 26.93832501

2, γ

c1 ÷ c4 = 0.727318641

gm1 = 2.807572366

gm2 = gm3 = 1

gm4 = 1.349944270

2, α

c= 1.964029584

gm1 = 1.964029584

gm2 = 1

1, γ

c1 ÷ c4 = 4.296761136

gm1 = 109.6213058

gm2 = gm3 = 1

gm4 = 77.17192382

2, γ

c1 ÷ c4 = 1.151478360

gm1 = 4.770135931

gm2 = gm3 = 1

gm4 = 3.994658924

2, β

c1 = 1.220298841

c2 = 2.813744956

gm1 ÷ gm4 = 1

4

5

6

2.112

1.471

1.260

gm4 = 2.424486640

gm4 = 7.826852608

(continued)

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

239

Table 8.53 (continued) n

fs

Cell no., type

7

1.129

1, γ

c1 ÷ c4 = 8.164848954

gm1 = 569.5447134

gm2 = gm3 = 1

gm4 = 273.0343027

2, γ

c1 ÷ c4 = 1.985285067

gm1 = 13.70684859

gm2 = gm3 = 1

gm4 = 13.56954222

3, γ

c1 ÷ c4 = 0.815842220

gm1 = 1.808549839

gm2 = gm3 = 1

gm4 = 1.285369297

4, α

c= 2.268420367

gm1 = 2.268420367

gm2 = 1

1, γ

c1 ÷ c4 = 13.68914149

gm1 = 905.2288513

gm2 = gm3 = 1

gm4 = 760.2107046

2, γ

c1 ÷ c4 = 3.240996900

gm1 = 29.36123959

gm2 = gm3 = 1

gm4 = 38.36969171

3, γ

c1 ÷ c4 = 1.216714938

gm1 = 3.561349776

gm2 = gm3 = 1

gm4 = 3.772658922

4, β

c1 = 1.329928716

c2 = 3.077532356

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 26.10897615

gm1 = 5266.235191

gm2 = gm3 = 1

gm4 = 2747.059140

2, γ

c1 ÷ c4 = 5.983526633

gm1 = 111.7802013

gm2 = gm3 = 1

gm4 = 136.5519580

3, γ

c1 ÷ c4 = 2.018067353

gm1 = 9.694807216

gm2 = gm3 = 1

gm4 = 12.83216807

4, γ

c1 ÷ c4 = 0.853497788

gm1 = 1.604786945

gm2 = gm3 = 1

gm4 = 1.280247631

5, α

c= 2.380135201

gm1 = 2.380135201

gm2 = 1

1, γ

c1 ÷ c4 = 43.51988101

gm1 = 8612.582566

gm2 = gm3 = 1

gm4 = 7609.723113

2, γ

c1 ÷ c4 = 9.890778115

gm1 = 256.2740625

gm2 = gm3 = 1

gm4 = 380.2191326

3, γ

c1 ÷ c4 = 3.239534554

gm1 = 23.31611966

gm2 = gm3 = 1

gm4 = 36.33812307

4, γ

c1 ÷ c4 = 1.250804893

gm1 = 3.316759357

gm2 = gm3 = 1

gm4 = 3.764718147

5, b

c1 = 1.371693191

c2 = 3.170315696

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 83.05380192

gm1 = 51,666.16419

gm2 = gm3 = 1

gm4 = 27,656.36749

2, γ

c1 ÷ c4 = 18.67758234

gm1 = 1055.271350

gm2 = gm3 = 1

gm4 = 1374.558632

3, γ

c1 ÷ c4 = 5.902163379

gm1 = 80.15457683

gm2 = gm3 = 1

gm4 = 129.1903361

4, γ

c1 ÷ c4 = 2.045053343

gm1 = 8.836975165

gm2 = gm3 = 1

gm4 = 12.78841195

5, γ

c1 ÷ c4 = 0.867516069

gm1 = 1.551797485

gm2 = gm3 = 1

gm4 = 1.281928742

6, α

c= 2.420262632

gm1 = 2.420262632

gm2 = 1

8

9

10

11

1.075

1.039

1.023

1.012

240

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.54 Element values of the cascaded Gm-C of modified elliptic (Zolotarev) filters (amax = 0.25 dB amin = 55 5 dB) n

fs

Cell no., type

3

4.250

1, γ

c1 ÷ c4 = 0.666321423

gm1 = 33.03062115

gm2 = gm3 = 1

2, α

c= 1.284726643

gm1 = 1.284726643

gm2 = 1

1, γ

c1 ÷ c4 = 1.272100197

gm1 = 28.46539198

gm2 = gm3 = 1

2, β

c1 = 0.960271826

c2 = 2.167927396

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 2.344637299

gm1 = 82.24749925

gm2 = gm3 = 1

gm4 = 23.80047717

2, γ

c1 ÷ c4 = 0.721447628

gm1 = 3.325241800

gm2 = gm3 = 1

gm4 = 1.284433496

2, α

c= 2.025967227

gm1 = 2.025967227

gm2 = 1

1, γ

c1 ÷ c4 = 3.941303585

gm1 = 108.0845938

gm2 = gm3 = 1

gm4 = 65.10737238

2, γ

c1 = 1.254174834

c2 = 2.925098714

gm1 ÷ gm4 = 1

gm4 = 3.653289972

2, β

c1 = 1.254174834

c2 = 2.925098714

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 7.154232372

gm1 = 508.0943741

gm2 = gm3 = 1

gm4 = 210.1681124

2, γ

c1 ÷ c4 = 1.841746801

gm1 = 13.16355262

gm2 = gm3 = 1

gm4 = 11.40972735

3, γ

c1 ÷ c4 = 0.821613294

gm1 = 1.997136785

gm2 = gm3 = 1

gm4 = 1.212800538

4, α

c= 2.390332730

gm1 = 2.390332730

gm2 = 1

1, γ

c1 ÷ c4 = 11.67520513

gm1 = 739.0744858

gm2 = gm3 = 1

gm4 = 554.0866518

2, γ

c1 ÷ c4 = 2.918554242

gm1 = 25.71966045

gm2 = gm3 = 1

gm4 = 30.60201315

3, γ

c1 ÷ c4 = 1.188026593

gm1 = 3.597056806

gm2 = gm3 = 1

gm4 = 3.402597831

4, β

c1 = 1.385720593

c2 = 3.249690497

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 21.26734671

gm1 = 3956.490868

gm2 = gm3 = 1

gm4 = 1825.351617

2, γ

c1 ÷ c4 = 5.131607252

gm1 = 89.16009427

gm2 = gm3 = 1

gm4 = 99.29330441

3, γ

c1 ÷ c4 = 1.876036041

gm1 = 8.807997417

gm2 = gm3 = 1

gm4 = 10.62615624

4, γ

c1 ÷ c4 = 0.866698931

gm1 = 1.715601657

gm2 = gm3 = 1

gm4 = 1.203089152

5, α

c= 2.531391790

gm1 = 2.531391790

gm2 = 1

4

5

6

7

8

9

2.402

1.604

1.338

1.173

1.103

1.056

gm4 = 2.375279916

gm4 = 7.469989514

(continued)

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

241

Table 8.54 (continued) n

fs

Cell no., type

10

1.034

1, γ

c1 ÷ c4 = 34.55807138

gm1 = 5989.695364

gm2 = gm3 = 1

gm4 = 4803.160357

2, γ

c1 ÷ c4 = 8.256288402

gm1 = 189.2214611

gm2 = gm3 = 1

gm4 = 262.8463631

3, γ

c1 ÷ c4 = 2.921597983

gm1 = 19.61936917

gm2 = gm3 = 1

gm4 = 28.66887683

4, γ

c1 ÷ c4 = 1.230881336

gm1 = 3.289871352

gm2 = gm3 = 1

gm4 = 3.395155945

5, b

c1 = 1.442347079

c2 = 3.375867008

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 62.99242755

gm1 = 33,297.27330

gm2 = gm3 = 1

gm4 = 15,919.86343

2, γ

c1 ÷ c4 = 14.86654599

gm1 = 717.0988953

gm2 = gm3 = 1

gm4 = 866.4581348

3, γ

c1 ÷ c4 = 5.056981051

gm1 = 61.11407615

gm2 = gm3 = 1

gm4 = 92.92825485

4, γ

c1 ÷ c4 = 1.910632825

gm1 = 7.878090766

gm2 = gm3 = 1

gm4 = 10.58926754

5, γ

c1 ÷ c4 = 0.886476412

gm1 = 1.644122508

gm2 = gm3 = 1

gm4 = 1.206462421

6, α

c= 2.590903803

gm1 = 2.590903803

gm2 = 1

11

1.019

Table 8.55 Element values of the cascaded Gm-C of modified elliptic (Zolotarev) filters (amax = 0.25 dB amin = 60 dB) n

fs

Cell no., type

3

5.125

1, γ

c1 ÷ c4 = 0.661607399

gm1 = 47.59266882

gm2 = gm3 = 1

2, α

c= 1.290651155

gm1 = 1.290651155

gm2 = 1

1, γ

c1 ÷ c4 = 1.249653846

gm1 = 36.08085389

gm2 = gm3 = 1

2, β

c1 = 0.965159294

c2 = 2.191400502

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 2.230961643

gm1 = 92.44411454

gm2 = gm3 = 1

gm4 = 21.60099982

2, γ

c1 ÷ c4 = 0.717449256

gm1 = 3.992774141

gm2 = gm3 = 1

gm4 = 1.235876298

2, α

c= 2.076838885

gm1 = 2.076838885

gm2 = 1

1, γ

c1 ÷ c4 = 3.675941361

gm1 = 110.7704172

gm2 = gm3 = 1

gm4 = 56.76554837

2, γ

c1 = 1.283686501

c2 = 3.021882154

gm1 ÷ gm4 = 1

gm4 = 3.398974786

2, β

c1 = 1.283686501

c2 = 3.021882154

gm1 ÷ gm4 = 1

4

5

6

2.743

1.758

1.428

gm4 = 2.342458080

gm4 = 7.215859180

(continued)

242

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.55 (continued) n

fs

Cell no., type

7

1.225

1, γ

c1 ÷ c4 = 6.415456220

gm1 = 475.5021457

gm2 = gm3 = 1

gm4 = 169.3981057

2, γ

c1 ÷ c4 = 1.733758497

gm1 = 13.11124004

gm2 = gm3 = 1

gm4 = 9.895738479

3, γ

c1 ÷ c4 = 0.828553683

gm1 = 2.232772391

gm2 = gm3 = 1

gm4 = 1.157253431

4, α

c= 2.500156885

gm1 = 2.500156885

gm2 = 1

1, γ

c1 ÷ c4 = 10.22674326

gm1 = 638.5188949

gm2 = gm3 = 1

gm4 = 425.9211719

2, γ

c1 ÷ c4 = 2.678242674

gm1 = 23.56499527

gm2 = gm3 = 1

gm4 = 25.35857037

3, γ

c1 ÷ c4 = 1.168405566

gm1 = 3.716720690

gm2 = gm3 = 1

gm4 = 3.126064983

4, β

c1 = 1.439199713

c2 = 3.411033494

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 17.88956129

gm1 = 3168.035258

gm2 = gm3 = 1

gm4 = 1293.440075

2, γ

c1 ÷ c4 = 4.512761639

gm1 = 75.12524064

gm2 = gm3 = 1

gm4 = 75.89861982

3, γ

c1 ÷ c4 = 1.770313820

gm1 = 8.300594569

gm2 = gm3 = 1

gm4 = 9.077355148

4, γ

c1 ÷ c4 = 0.881994125

gm1 = 1.854600136

gm2 = gm3 = 1

gm4 = 1.143686729

5, α

c= 2.675171572

gm1 = 2.675171572

gm2 = 1

1, γ

c1 ÷ c4 = 28.40625351

gm1 = 4468.960675

gm2 = gm3 = 1

gm4 = 3248.639891

2, γ

c1 ÷ c4 = 7.084503103

gm1 = 148.3485087

gm2 = gm3 = 1

gm4 = 191.9256093

3, γ

c1 ÷ c4 = 2.682800672

gm1 = 17.21014382

gm2 = gm3 = 1

gm4 = 23.44492693

4, γ

c1 ÷ c4 = 1.219236826

gm1 = 3.323925227

gm2 = gm3 = 1

gm4 = 3.112685936

5, b

c1 = 1.510939200

c2 = 3.574182153

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 49.75508556

gm1 = 23,221.42398

gm2 = gm3 = 1

gm4 = 9939.077417

2, γ

c1 ÷ c4 = 12.23679270

gm1 = 522.6305735

gm2 = gm3 = 1

gm4 = 583.7690290

3, γ

c1 ÷ c4 = 4.441388769

gm1 = 49.18825243

gm2 = gm3 = 1

gm4 = 70.15589894

4, γ

c1 ÷ c4 = 1.811770128

gm1 = 7.267070121

gm2 = gm3 = 1

gm4 = 9.033380814

5, γ

c1 ÷ c4 = 0.907817983

gm1 = 1.756462150

gm2 = gm3 = 1

gm4 = 1.147655057

6, α

c= 2.756005336

gm1 = 2.756005336

gm2 = 1

8

9

10

11

1.137

1.076

1.047

1.027

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

243

Table 8.56 Element values of the cascaded Gm-C of modified elliptic (Zolotarev) filters (amax = 0.5 dB amin = 30 dB) n

fs

Cell no., type

3

1.692

1, γ

c1 ÷ c4 = 0.943900946

gm1 = 8.464034171

gm2 = gm3 = 1

2, α

c= 1.431246838

gm1 = 1.431246838

gm2 = 1

1, γ

c1 ÷ c4 = 2.021507220

gm1 = 17.94439614

gm2 = gm3 = 1

2, β

c1 = 1.085823489

c2 = 1.964393449

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 5.339477577

gm1 = 138.0722080

gm2 = gm3 = 1

gm4 = 116.8229099

2, γ

c1 ÷ c4 = 0.977933899

gm1 = 2.550948372

gm2 = gm3 = 1

gm4 = 2.803467506

2, α

c= 1.837794426

gm1 = 1.837794426

gm2 = 1

1, γ

c1 ÷ c4 = 10.65906717

gm1 = 357.5852818

gm2 = gm3 = 1

gm4 = 459.9925961

2, γ

c1 ÷ c4 = 1.832093739

gm1 = 7.765855262

gm2 = gm3 = 1

gm4 = 11.41870352

2, β

c1 = 1.217942535

c2 = 2.237039362

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 28.77614039

gm1 = 3615.185212

gm2 = gm3 = 1

gm4 = 3327.211246

2, γ

c1 ÷ c4 = 4.554981081

gm1 = 48.89953743

gm2 = gm3 = 1

gm4 = 78.17571397

3, γ

c1 ÷ c4 = 1.019971033

gm1 = 2.196024996

gm2 = gm3 = 1

gm4 = 2.770249031

4, α

c= 1.930515347

gm1 = 1.930515347

gm2 = 1

1, γ

c1 ÷ c4 = 57.15597362

gm1 = 9754.994357

gm2 = gm3 = 1

gm4 = 13,096.78441

2, γ

c1 ÷ c4 = 8.949263404

gm1 = 174.0889321

gm2 = gm3 = 1

gm4 = 310.7737795

3, γ

c1 ÷ c4 = 1.877027659

gm1 = 7.241256864

gm2 = gm3 = 1

gm4 = 11.40072995

4, β

c1 = 1.257627511

c2 = 2.297271004

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 154.0004385

gm1 = 101,615.6995

gm2 = gm3 = 1

gm4 = 94,944.49329

2, γ

c1 ÷ c4 = 23.73993189

gm1 = 1302.758180

gm2 = gm3 = 1

gm4 = 2229.184519

3, γ

c1 ÷ c4 = 4.570194253

gm1 = 43.07702404

gm2 = gm3 = 1

gm4 = 77.29096818

4, γ

c1 ÷ c4 = 1.030030753

gm1 = 2.143480681

gm2 = gm3 = 1

gm4 = 2.772365789

5, α

c= 1.950341573

gm1 = 1.950341573

gm2 = 1

4

5

6

7

8

9

1.299

1.103

1.050

1.019

1.009

1.003

gm4 = 4.093964456

gm4 = 17.33760321

(continued)

244

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.56 (continued) n

fs

Cell no., type

10

1.002

1, γ

c1 ÷ c4 = 302.9460289

gm1 = 271,438.2804

gm2 = gm3 = 1

gm4 = 367,261.8392

2, γ

c1 ÷ c4 = 46.92638732

gm1 = 4739.765261

gm2 = gm3 = 1

gm4 = 8758.485405

3, γ

c1 ÷ c4 = 8.933970099

gm1 = 162.1332739

gm2 = gm3 = 1

gm4 = 306.9134517

4, γ

c1 ÷ c4 = 1.885645411

gm1 = 7.147733887

gm2 = gm3 = 1

gm4 = 11.39956959

5, b

c1 = 1.263513274

c2 = 2.307881142

gm1 ÷ gm4 = 1

Table 8.57 Element values of the cascaded Gm-C of modified elliptic (Zolotarev) filters (amax = 0.5 dB amin = 35 dB) n

fs

Cell no., type

3

1.986

1, γ

c1 ÷ c4 = 0.893283686

gm1 = 10.73341785

gm2 = gm3 = 1

2, α

c= 1.481757198

gm1 = 1.481757198

gm2 = 1

1, γ

c1 ÷ c4 = 1.846592483

gm1 = 18.62326260

gm2 = gm3 = 1

2, β

c1 = 1.108736690

c2 = 2.056782599

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 4.389130476

gm1 = 112.2489428

gm2 = gm3 = 1

gm4 = 79.13918767

2, γ

c1 ÷ c4 = 0.933663925

gm1 = 2.618366840

gm2 = gm3 = 1

gm4 = 2.375721164

2, α

c= 1.992557003

gm1 = 1.992557003

gm2 = 1

1, γ

c1 ÷ c4 = 6.486649179

gm1 = 161.8836895

gm2 = gm3 = 1

gm4 = 173.9675802

2, γ

c1 ÷ c4 = 1.350410444

gm1 = 4.762237673

gm2 = gm3 = 1

gm4 = 6.119861715

2, β

c1 = 1.090442382

c2 = 2.384749232

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 14.95659894

gm1 = 1230.938833

gm2 = gm3 = 1

gm4 = 907.9827079

2, γ

c1 ÷ c4 = 2.857606279

gm1 = 21.52690580

gm2 = gm3 = 1

gm4 = 30.30894638

3, γ

c1 ÷ c4 = 0.823173452

gm1 = 1.521153665

gm2 = gm3 = 1

gm4 = 1.711793149

4, α

c= 1.832962158

gm1 = 1.832962158

gm2 = 1

4

5

6

7

1.431

1.159

1.096

1.041

gm4 = 3.662503878

gm4 = 14.47317363

(continued)

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

245

Table 8.57 (continued) n

fs

Cell no., type

8

1.022

1, γ

c1 ÷ c4 = 27.70387614

gm1 = 2682.838516

gm2 = gm3 = 1

gm4 = 3094.298842

2, γ

c1 ÷ c4 = 5.213592545

gm1 = 63.21753545

gm2 = gm3 = 1

gm4 = 104.3313702

3, γ

c1 ÷ c4 = 1.394812970

gm1 = 4.143413518

gm2 = gm3 = 1

gm4 = 5.987220388

4, β

c1 = 1.142099572

c2 = 2.496141306

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 94.10368871

gm1 = 43,202.43118

gm2 = gm3 = 1

gm4 = 35,466.34561

2, γ

c1 ÷ c4 = 16.34015763

gm1 = 650.7672984

gm2 = gm3 = 1

gm4 = 1048.205653

3, γ

c1 ÷ c4 = 3.692458935

gm1 = 28.66338342

gm2 = gm3 = 1

gm4 = 48.62477398

4, γ

c1 ÷ c4 = 1.003646448

gm1 = 2.053705844

gm2 = gm3 = 1

gm4 = 2.333488629

5, α

c= 2.166434605

gm1 = 2.166434605

gm2 = 1

1, γ

c1 ÷ c4 = 117.6803415

gm1 = 47,373.86812

gm2 = gm3 = 1

gm4 = 55,497.98315

2, γ

c1 ÷ c4 = 21.63552449

gm1 = 1064.290011

gm2 = gm3 = 1

gm4 = 1854.347908

3, γ

c1 ÷ c4 = 5.201904646

gm1 = 56.07126440

gm2 = gm3 = 1

gm4 = 101.7782255

4, γ

c1 ÷ c4 = 1.408754678

gm1 = 4.028408685

gm2 = gm3 = 1

gm4 = 5.980902532

5, b

c1 = 1.154322534

c2 = 2.522675749

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 311.5345932

gm1 = 508,564.7970

gm2 = gm3 = 1

gm4 = 388,512.1142

2, γ

c1 ÷ c4 = 49.42357287

gm1 = 6120.067161

gm2 = gm3 = 1

gm4 = 9731.071079

3, γ

c1 ÷ c4 = 11.69055749

gm1 = 288.8728773

gm2 = gm3 = 1

gm4 = 532.3412992

4, γ

c1 ÷ c4 = 2.891133668

gm1 = 16.97565734

gm2 = gm3 = 1

gm4 = 29.60655474

5, γ

c1 ÷ c4 = 0.844263449

gm1 = 1.434744759

gm2 = gm3 = 1

gm4 = 1.713882367

6, α

c= 1.882360028

gm1 = 1.882360028

gm2 = 1

9

10

11

1.009

1.005

1.002

246

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.58 Element values of the cascaded Gm-C of modified elliptic (Zolotarev) filters (amax = 0.5 dB amin = 40 dB) n

fs

Cell no., type

3

2.353

1, γ

c1 ÷ c4 = 0.861138639

gm1 = 14.28098022

gm2 = gm3 = 1

2, α

c= 1.517263730

gm1 = 1.517263730

gm2 = 1

1, γ

c1 ÷ c4 = 1.729884727

gm1 = 20.65287209

gm2 = gm3 = 1

2, β

c1 = 1.127106760

c2 = 2.131159444

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 3.781413144

gm1 = 101.0622702

gm2 = gm3 = 1

gm4 = 58.85739281

2, γ

c1 ÷ c4 = 0.906293593

gm1 = 2.830063962

gm2 = gm3 = 1

gm4 = 2.100714224

2, α

c= 2.126305977

gm1 = 2.126305977

gm2 = 1

1, γ

c1 ÷ c4 = 6.986024876

gm1 = 197.3663255

gm2 = gm3 = 1

gm4 = 198.2225896

2, γ

c1 ÷ c4 = 1.545391550

gm1 = 6.448014430

gm2 = gm3 = 1

gm4 = 7.420858252

2, β

c1 = 1.345467658

c2 = 2.592594815

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 15.51849463

gm1 = 1398.357648

gm2 = gm3 = 1

gm4 = 970.0811115

2, γ

c1 ÷ c4 = 3.105279534

gm1 = 26.18563329

gm2 = gm3 = 1

gm4 = 34.44079080

3, γ

c1 ÷ c4 = 0.972899306

gm1 = 2.163913427

gm2 = gm3 = 1

gm4 = 2.042076295

4, α

c= 2.317047655

gm1 = 2.317047655

gm2 = 1

1, γ

c1 ÷ c4 = 28.31514896

gm1 = 2911.235040

gm2 = gm3 = 1

gm4 = 3219.252429

2, γ

c1 ÷ c4 = 5.556703941

gm1 = 73.16954450

gm2 = gm3 = 1

gm4 = 115.9929568

3, γ

c1 ÷ c4 = 1.600398435

gm1 = 5.515222574

gm2 = gm3 = 1

gm4 = 7.237684663

4, β

c1 = 1.415705898

c2 = 2.725851493

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 63.29387930

gm1 = 22,223.20898

gm2 = gm3 = 1

gm4 = 16,051.99586

2, γ

c1 ÷ c4 = 12.10644372

gm1 = 379.3635727

gm2 = gm3 = 1

gm4 = 570.0624586

3, γ

c1 ÷ c4 = 3.134348193

gm1 = 21.18470911

gm2 = gm3 = 1

gm4 = 33.62400309

4, γ

c1 ÷ c4 = 0.996533657

gm1 = 2.052707491

gm2 = gm3 = 1

gm4 = 2.046763400

5, α

c= 2.376466517

gm1 = 2.376466517

gm2 = 1

4

5

6

7

8

9

1.592

1.231

1.120

1.052

1.028

1.013

gm4 = 3.399908378

gm4 = 12.70366612

(continued)

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

247

Table 8.58 (continued) n

fs

Cell no., type

10

1.007

1, γ

c1 ÷ c4 = 114.8654764

gm1 = 46,714.74032

gm2 = gm3 = 1

gm4 = 52,826.18210

2, γ

c1 ÷ c4 = 21.86490114

gm1 = 1103.192380

gm2 = gm3 = 1

gm4 = 1882.943606

3, γ

c1 ÷ c4 = 5.543572815

gm1 = 64.07376377

gm2 = gm3 = 1

gm4 = 112.7822325

4, γ

c1 ÷ c4 = 1.619016842

gm1 = 5.338657958

gm2 = gm3 = 1

gm4 = 7.228782030

5, b

c1 = 1.433494458

c2 = 2.759647122

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 284.0634388

gm1 = 442,664.9336

gm2 = gm3 = 1

gm4 = 322,891.7084

2, γ

c1 ÷ c4 = 48.62754587

gm1 = 6052.033561

gm2 = gm3 = 1

gm4 = 9394.226425

3, γ

c1 ÷ c4 = 11.99846995

gm1 = 306.7913923

gm2 = gm3 = 1

gm4 = 554.1271932

4, γ

c1 ÷ c4 = 3.149073089

gm1 = 20.20456304

gm2 = gm3 = 1

gm4 = 33.57256773

5, γ

c1 ÷ c4 = 1.002124906

gm1 = 2.024881549

gm2 = gm3 = 1

gm4 = 2.046976779

6, α

c= 2.389985959

gm1 = 2.389985959

gm2 = 1

11

1.003

Table 8.59 Element values of the cascaded Gm-C of modified elliptic (Zolotarev) filters (amax = 0.5 dB amin = 45 dB) n

fs

Cell no., type

3

2.808

1, γ

c1 ÷ c4 = 0.840246602

gm1 = 19.61786963

gm2 = gm3 = 1

2, α

c= 1.541978673

gm1 = 1.541978673

gm2 = 1

1, γ

c1 ÷ c4 = 1.649458417

gm1 = 24.02808480

gm2 = gm3 = 1

2, β

c1 = 1.141570234

c2 = 2.190052391

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 3.372453342

gm1 = 98.17570699

gm2 = gm3 = 1

gm4 = 46.88609989

2, γ

c1 ÷ c4 = 0.888915438

gm1 = 3.174228664

gm2 = gm3 = 1

gm4 = 1.914573263

2, α

c= 2.240305470

gm1 = 2.240305470

gm2 = 1

1, γ

c1 ÷ c4 = 6.030398515

gm1 = 169.1405220

gm2 = gm3 = 1

gm4 = 147.8862430

2, γ

c1 ÷ c4 = 1.461841265

gm1 = 6.369660100

gm2 = gm3 = 1

gm4 = 6.382195126

2, β

c1 = 1.400057458

c2 = 2.747247589

gm1 ÷ gm4 = 1

4

5

6

1.787

1.319

1.169

gm4 = 3.234006456

gm4 = 11.55055116

(continued)

248

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.59 (continued) 7

8

9

10

11

1.079

1.044

1.021

1.012

1.006

1, γ

c1 ÷ c4 = 12.47205635

gm1 = 1044.712837

gm2 = gm3 = 1

gm4 = 627.3152049

2, γ

c1 ÷ c4 = 2.726033126

gm1 = 22.00480697

gm2 = gm3 = 1

gm4 = 25.81254967

3, γ

c1 ÷ c4 = 0.969580761

gm1 = 2.274520240

gm2 = gm3 = 1

gm4 = 1.845697924

4, α

c= 2.493699046

gm1 = 2.493699046

gm2 = 1

1, γ

c1 ÷ c4 = 21.96230565

gm1 = 1945.883081

gm2 = gm3 = 1

gm4 = 1938.267009

2, γ

c1 ÷ c4 = 4.689367834

gm1 = 55.11907099

gm2 = gm3 = 1

gm4 = 81.11899681

3, γ

c1 ÷ c4 = 1.525241217

gm1 = 5.188970918

gm2 = gm3 = 1

gm4 = 6.155190432

4, β

c1 = 1.492249554

c2 = 2.930168049

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 45.78116493

gm1 = 13,201.73730

gm2 = gm3 = 1

gm4 = 8402.280518

2, γ

c1 ÷ c4 = 9.485689699

gm1 = 248.9414080

gm2 = gm3 = 1

gm4 = 346.2119563

3, γ

c1 ÷ c4 = 2.759685890

gm1 = 16.95771601

gm2 = gm3 = 1

gm4 = 24.95133348

4, γ

c1 ÷ c4 = 1.001103216

gm1 = 2.111740902

gm2 = gm3 = 1

gm4 = 1.848068372

5, α

c= 2.579482896

gm1 = 2.579482896

gm2 = 1

1, γ

c1 ÷ c4 = 80.14366874

gm1 = 24,978.14416

gm2 = gm3 = 1

gm4 = 25,724.55988

2, γ

c1 ÷ c4 = 16.49825250

gm1 = 656.1876882

gm2 = gm3 = 1

gm4 = 1064.848229

3, γ

c1 ÷ c4 = 4.679194430

gm1 = 46.56937403

gm2 = gm3 = 1

gm4 = 78.12362171

4, γ

c1 ÷ c4 = 1.550938721

gm1 = 4.956647925

gm2 = gm3 = 1

gm4 = 6.144497895

5, b

c1 = 1.519843687

c2 = 2.983037957

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 165.8872179

gm1 = 170,364.6025

gm2 = gm3 = 1

gm4 = 110,141.9020

2, γ

c1 ÷ c4 = 34.16587834

gm1 = 3173.416503

gm2 = gm3 = 1

gm4 = 4619.803271

3, γ

c1 ÷ c4 = 9.382024791

gm1 = 192.3782444

gm2 = gm3 = 1

gm4 = 333.3290402

4, γ

c1 ÷ c4 = 2.779867620

gm1 = 15.91630908

gm2 = gm3 = 1

gm4 = 24.89495727

5, γ

c1 ÷ c4 = 1.009937788

gm1 = 2.069296103

gm2 = gm3 = 1

gm4 = 1.848697900

6, α

c= 2.602649810

gm1 = 2.602649810

gm2 = 1

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

249

Table 8.60 Element values of the cascaded Gm-C of modified elliptic (Zolotarev) filters (amax = 0.5 dB amin = 50 dB) n

fs

Cell no., type

3

3.366

1, γ

c1 ÷ c4 = 0.826458849

gm1 = 27.53252073

gm2 = gm3 = 1

2, α

c= 1.559065428

gm1 = 1.559065428

gm2 = 1

1, γ

c1 ÷ c4 = 1.592772635

gm1 = 28.95292428

gm2 = gm3 = 1

2, β

c1 = 1.152829979

c2 = 2.236076938

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 3.086629720

gm1 = 101.0160164

gm2 = gm3 = 1

gm4 = 39.32097904

2, γ

c1 ÷ c4 = 0.877639193

gm1 = 3.658034666

gm2 = gm3 = 1

gm4 = 1.783747980

2, α

c= 2.336271740

gm1 = 2.336271740

gm2 = 1

1, γ

c1 ÷ c4 = 5.364108152

gm1 = 155.1535752

gm2 = gm3 = 1

gm4 = 117.1343447

2, γ

c1 ÷ c4 = 1.400885135

gm1 = 6.542071494

gm2 = gm3 = 1

gm4 = 5.658070806

2, β

c1 = 1.448874717

c2 = 2.885588563

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 10.45249258

gm1 = 850.2427533

gm2 = gm3 = 1

gm4 = 441.0629336

2, γ

c1 ÷ c4 = 2.460642031

gm1 = 19.73672185

gm2 = gm3 = 1

gm4 = 20.46668874

3, γ

c1 ÷ c4 = 0.972935654

gm1 = 2.451208909

gm2 = gm3 = 1

gm4 = 1.705916352

4, α

c= 2.658530853

gm1 = 2.658530853

gm2 = 1

1, γ

c1 ÷ c4 = 17.82083926

gm1 = 1429.465540

gm2 = gm3 = 1

gm4 = 1277.162980

2, γ

c1 ÷ c4 = 4.084113794

gm1 = 44.59347962

gm2 = gm3 = 1

gm4 = 60.40051875

3, γ

c1 ÷ c4 = 1.472316909

gm1 = 5.051364478

gm2 = gm3 = 1

gm4 = 5.387976502

4, β

c1 = 1.565136766

c2 = 3.124947626

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 35.02570045

gm1 = 8763.059275

gm2 = gm3 = 1

gm4 = 4920.687495

2, γ

c1 ÷ c4 = 7.760627481

gm1 = 179.1695464

gm2 = gm3 = 1

gm4 = 229.0091586

3, γ

c1 ÷ c4 = 2.497123314

gm1 = 14.43519411

gm2 = gm3 = 1

gm4 = 19.53526340

4, γ

c1 ÷ c4 = 1.012652075

gm1 = 2.216131508

gm2 = gm3 = 1

gm4 = 1.703544530

5, α

c= 2.773941263

gm1 = 2.773941263

gm2 = 1

4

5

6

7

8

9

2.020

1.425

1.230

1.112

1.064

1.032

gm4 = 3.126576372

gm4 = 10.77043622

(continued)

250

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.60 (continued) n

fs

Cell no., type

10

1.019

1, γ

c1 ÷ c4 = 59.40597353

gm1 = 15,101.46217

gm2 = gm3 = 1

gm4 = 14,139.06733

2, γ

c1 ÷ c4 = 13.05536886

gm1 = 431.6919090

gm2 = gm3 = 1

gm4 = 661.6514684

3, γ

c1 ÷ c4 = 4.078741907

gm1 = 36.28357466

gm2 = gm3 = 1

gm4 = 57.59793359

4, γ

c1 ÷ c4 = 1.506809271

gm1 = 4.753885363

gm2 = gm3 = 1

gm4 = 5.378816856

5, b

c1 = 1.605914390

c2 = 3.202679316

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 116.9061215

gm1 = 95,201.67020

gm2 = gm3 = 1

gm4 = 54,713.14275

2, γ

c1 ÷ c4 = 25.44604111

gm1 = 1877.524824

gm2 = gm3 = 1

gm4 = 2550.916822

3, γ

c1 ÷ c4 = 7.662203484

gm1 = 132.2339250

gm2 = gm3 = 1

gm4 = 218.2080351

4, γ

c1 ÷ c4 = 2.524132983

gm1 = 13.31374086

gm2 = gm3 = 1

gm4 = 19.48025761

5, γ

c1 ÷ c4 = 1.025999231

gm1 = 2.154843175

gm2 = gm3 = 1

gm4 = 1.705289708

6, α

c= 2.811306513

gm1 = 2.811306513

gm2 = 1

11

1.010

Table 8.61 Element values of the cascaded Gm-C of modified elliptic (Zolotarev) filters (amax = 0.5 dB amin = 55 dB) n

fs

Cell no., type

3

4.050

1, γ

c1 ÷ c4 = 0.817265352

gm1 = 39.20180696

gm2 = gm3 = 1

2, α

c= 1.570823400

gm1 = 1.570823400

gm2 = 1

1, γ

c1 ÷ c4 = 1.552165488

gm1 = 35.80119720

gm2 = gm3 = 1

2, β

c1 = 1.161519239

c2 = 2.271685021

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 2.881082845

gm1 = 108.6144651

gm2 = gm3 = 1

gm4 = 34.28861199

2, γ

c1 ÷ c4 = 0.870182972

gm1 = 4.301521160

gm2 = gm3 = 1

gm4 = 1.689188296

2, α

c= 2.416200722

gm1 = 2.416200722

gm2 = 1

1, γ

c1 ÷ c4 = 4.884004884

gm1 = 150.1511173

gm2 = gm3 = 1

gm4 = 97.18847448

2, γ

c1 ÷ c4 = 1.355486685

gm1 = 6.930338421

gm2 = gm3 = 1

gm4 = 5.136174967

2, β

c1 = 1.492083453

c2 = 3.007841180

gm1 ÷ gm4 = 1

4

5

6

2.293

1.550

1.302

gm4 = 3.055860623

gm4 = 10.22823044

(continued)

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

251

Table 8.61 (continued) n

fs

Cell no., type

7

1.152

1, γ

c1 ÷ c4 = 9.049400314

gm1 = 739.7332391

gm2 = gm3 = 1

gm4 = 330.9049206

2, γ

c1 ÷ c4 = 2.268135330

gm1 = 18.62016193

gm2 = gm3 = 1

gm4 = 16.94670074

3, γ

c1 ÷ c4 = 0.979799282

gm1 = 2.689374541

gm2 = gm3 = 1

gm4 = 1.602433166

4, α

c= 2.809718817

gm1 = 2.809718817

gm2 = 1

1, γ

c1 ÷ c4 = 14.98051035

gm1 = 1131.288398

gm2 = gm3 = 1

gm4 = 903.1573647

2, γ

c1 ÷ c4 = 3.644867975

gm1 = 38.18463919

gm2 = gm3 = 1

gm4 = 47.23545049

3, γ

c1 ÷ c4 = 1.433946680

gm1 = 5.049040740

gm2 = gm3 = 1

gm4 = 4.821207788

4, β

c1 = 1.633242895

c2 = 3.308626144

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 28.00054701

gm1 = 6344.482814

gm2 = gm3 = 1

gm4 = 3146.428718

2, γ

c1 ÷ c4 = 6.567307680

gm1 = 138.7068415

gm2 = gm3 = 1

gm4 = 161.9617398

3, γ

c1 ÷ c4 = 2.306177327

gm1 = 12.88860629

gm2 = gm3 = 1

gm4 = 15.93622355

4, γ

c1 ÷ c4 = 1.027968135

gm1 = 2.357570031

gm2 = gm3 = 1

gm4 = 1.593680737

5, α

c= 2.957767808

gm1 = 2.957767808

gm2 = 1

1, γ

c1 ÷ c4 = 46.19841554

gm1 = 10,064.67864

gm2 = gm3 = 1

gm4 = 8554.062909

2, γ

c1 ÷ c4 = 10.72987622

gm1 = 308.0004921

gm2 = gm3 = 1

gm4 = 443.1519528

3, γ

c1 ÷ c4 = 3.645992238

gm1 = 29.89082451

gm2 = gm3 = 1

gm4 = 44.60760857

4, γ

c1 ÷ c4 = 1.478947770

gm1 = 4.675795645

gm2 = gm3 = 1

gm4 = 4.816820663

5, b

c1 = 1.690858341

c2 = 3.417153054

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 86.52740599

gm1 = 58,519.63902

gm2 = gm3 = 1

gm4 = 29,979.67157

2, γ

c1 ÷ c4 = 19.85072256

gm1 = 1223.088535

gm2 = gm3 = 1

gm4 = 1544.182722

3, γ

c1 ÷ c4 = 6.476631923

gm1 = 97.83522682

gm2 = gm3 = 1

gm4 = 152.7311475

4, γ

c1 ÷ c4 = 2.341841399

gm1 = 11.67368975

gm2 = gm3 = 1

gm4 = 15.89333370

5, γ

c1 ÷ c4 = 1.047469210

gm1 = 2.273660115

gm2 = gm3 = 1

gm4 = 1.597809602

6, α

c= 3.015343273

gm1 = 3.015343273

gm2 = 1

8

9

10

11

1.089

1.047

1.028

1.015

252

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.62 Element values of the cascaded Gm-C of modified elliptic (Zolotarev) filters (amax = 0.5 dB amin = 60 dB) n

fs

Cell no., type

3

4.880

1, γ

c1 ÷ c4 = 0.811092896

gm1 = 56.36418306

gm2 = gm3 = 1

2, α

c= 1.578888866

gm1 = 1.578888866

gm2 = 1

1, γ

c1 ÷ c4 = 1.522729169

gm1 = 45.12566486

gm2 = gm3 = 1

2, β

c1 = 1.168179537

c2 = 2.299023708

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 2.730009289

gm1 = 120.8195282

gm2 = gm3 = 1

gm4 = 30.80765108

2, γ

c1 ÷ c4 = 0.865166560

gm1 = 5.135875261

gm2 = gm3 = 1

gm4 = 1.619379935

2, α

c= 2.482179811

gm1 = 2.482179811

gm2 = 1

1, γ

c1 ÷ c4 = 4.528878028

gm1 = 151.6146523

gm2 = gm3 = 1

gm4 = 83.62791957

2, γ

c1 ÷ c4 = 1.321076561

gm1 = 7.525720216

gm2 = gm3 = 1

gm4 = 4.749936928

2, β

c1 = 1.529919878

c2 = 3.114723833

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 8.038466312

gm1 = 678.6273476

gm2 = gm3 = 1

gm4 = 261.3156557

2, γ

c1 ÷ c4 = 2.124603761

gm1 = 18.28235511

gm2 = gm3 = 1

gm4 = 14.51812602

3, γ

c1 ÷ c4 = 0.988373564

gm1 = 2.990682388

gm2 = gm3 = 1

gm4 = 1.523634117

4, α

c= 2.946785764

gm1 = 2.946785764

gm2 = 1

1, γ

c1 ÷ c4 = 12.96422289

gm1 = 952.1408369

gm2 = gm3 = 1

gm4 = 676.8626889

2, γ

c1 ÷ c4 = 3.319101909

gm1 = 34.29793040

gm2 = gm3 = 1

gm4 = 38.48248048

3, γ

c1 ÷ c4 = 1.406832253

gm1 = 5.164742307

gm2 = gm3 = 1

gm4 = 4.396558657

4, β

c1 = 1.697785951

c2 = 3.480825611

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 23.18728743

gm1 = 4925.133015

gm2 = gm3 = 1

gm4 = 2158.824396

2, γ

c1 ÷ c4 = 5.709997017

gm1 = 113.9150772

gm2 = gm3 = 1

gm4 = 120.8843909

3, γ

c1 ÷ c4 = 2.163694760

gm1 = 11.95706378

gm2 = gm3 = 1

gm4 = 13.43207153

4, γ

c1 ÷ c4 = 1.045383440

gm1 = 2.534118238

gm2 = gm3 = 1

gm4 = 1.507882081

5, α

c= 3.130709632

gm1 = 3.130709632

gm2 = 1

4

5

6

7

8

9

2.615

1.700

1.387

1.200

1.120

1.066

gm4 = 3.008785718

gm4 = 9.843878276

(continued)

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

253

Table 8.62 (continued) n

fs

Cell no., type

10

1.040

1, γ

c1 ÷ c4 = 37.34012266

gm1 = 7255.152936

gm2 = gm3 = 1

gm4 = 5590.235004

2, γ

c1 ÷ c4 = 9.092086103

gm1 = 234.7722225

gm2 = gm3 = 1

gm4 = 315.3421179

3, γ

c1 ÷ c4 = 3.324597621

gm1 = 25.76532738

gm2 = gm3 = 1

gm4 = 35.93480776

4, γ

c1 ÷ c4 = 1.462312407

gm1 = 4.691305386

gm2 = gm3 = 1

gm4 = 4.391108980

5, b

c1 = 1.773851653

c2 = 3.625163088

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 66.95250924

gm1 = 39,218.98915

gm2 = gm3 = 1

gm4 = 17,954.12127

2, γ

c1 ÷ c4 = 16.07558870

gm1 = 861.3183732

gm2 = gm3 = 1

gm4 = 1006.706697

3, γ

c1 ÷ c4 = 5.626978164

gm1 = 76.83464606

gm2 = gm3 = 1

gm4 = 112.7870072

4, γ

c1 ÷ c4 = 2.208915714

gm1 = 10.62497347

gm2 = gm3 = 1

gm4 = 13.39944654

5, γ

c1 ÷ c4 = 1.072357999

gm1 = 2.421401072

gm2 = gm3 = 1

gm4 = 1.514747494

6, α

c= 3.213838532

gm1 = 3.213838532

gm2 = 1

11

1.022

Table 8.63 Element values of the cascaded Gm-C of modified elliptic (Zolotarev) filters (amax = 1 dB amin = 30 dB) n

fs

Cell no., type

3

1.612

1, γ

c1 ÷ c4 = 1.217919988

gm1 = 11.32170400

gm2 = gm3 = 1

2, α

c= 1.787233754

gm1 = 1.787233754

gm2 = 1

1, γ

c1 ÷ c4 = 2.638715701

gm1 = 26.80177938

gm2 = gm3 = 1

2, β

c1 = 1.349663178

c2 = 1.948271823

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 7.385932399

gm1 = 235.3487590

gm2 = gm3 = 1

gm4 = 218.6024671

2, γ

c1 ÷ c4 = 1.242295285

gm1 = 3.851549084

gm2 = gm3 = 1

gm4 = 4.287028179

2, α

c= 2.214958635

gm1 = 2.214958635

gm2 = 1

1, γ

c1 ÷ c4 = 15.23200475

gm1 = 679.4422670

gm2 = gm3 = 1

gm4 = 928.7885274

2, γ

c1 ÷ c4 = 2.403155247

gm1 = 12.87187184

gm2 = gm3 = 1

gm4 = 19.29133032

2, β

c1 = 1.490262328

c2 = 2.175297201

gm1 ÷ gm4 = 1

4

5

6

1.254

1.082

1.038

gm4 = 6.029400528

gm4 = 27.91435274

(continued)

254

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.63 (continued) n

fs

Cell no., type

7

1.013

1, γ

c1 ÷ c4 = 44.24405231

gm1 = 7802.000497

gm2 = gm3 = 1

gm4 = 7832.430631

2, γ

c1 ÷ c4 = 6.382098265

gm1 = 92.55108172

gm2 = gm3 = 1

gm4 = 153.1574012

3, γ

c1 ÷ c4 = 1.284761800

gm1 = 3.425785656

gm2 = gm3 = 1

gm4 = 4.255855516

4, α

c= 2.303455367

gm1 = 2.303455367

gm2 = 1

1, γ

c1 ÷ c4 = 90.80933272

gm1 = 23,220.87932

gm2 = gm3 = 1

gm4 = 32,989.87787

2, γ

c1 ÷ c4 = 12.96487504

gm1 = 357.8730058

gm2 = gm3 = 1

gm4 = 652.3443986

3, γ

c1 ÷ c4 = 2.442730191

gm1 = 12.15171861

gm2 = gm3 = 1

gm4 = 19.21902952

4, β

c1 = 1.522235445

c2 = 2.215616724

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 264.8008305

gm1 = 275,885.1865

gm2 = gm3 = 1

gm4 = 280,491.6326

2, γ

c1 ÷ c4 = 37.23982581

gm1 = 3109.660054

gm2 = gm3 = 1

gm4 = 5490.170243

3, γ

c1 ÷ c4 = 6.396453192

gm1 = 83.59151744

gm2 = gm3 = 1

gm4 = 151.9531639

4, γ

c1 ÷ c4 = 1.293535891

gm1 = 3.367241684

gm2 = gm3 = 1

gm4 = 4.259154152

5, α

c= 2.319558096

gm1 = 2.319558096

gm2 = 1

1, γ

c1 ÷ c4 = 368.2612970

gm1 = 379,436.0499

gm2 = gm3 = 1

gm4 = 542,527.6243

2, γ

c1 ÷ c4 = 76.19721057

gm1 = 12,281.31241

gm2 = gm3 = 1

gm4 = 23,105.15618

3, γ

c1 ÷ c4 = 12.94761189

gm1 = 338.7941900

gm2 = gm3 = 1

gm4 = 646.6932752

4, γ

c1 ÷ c4 = 2.450499902

gm1 = 12.04626557

gm2 = gm3 = 1

gm4 = 19.22325698

5, β

c1 = 1.526951612

c2 = 2.222308288

gm1 ÷ gm4 = 1

8

9

10

1.006

1.002

1.001

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

255

Table 8.64 Element values of the cascaded Gm-C of modified elliptic (Zolotarev) filters (amax = 1 dB amin = 35 dB) n

fs

Cell no., type

3

1.885

1, γ

c1 ÷ c4 = 1.145617098

gm1 = 14.04520032

gm2 = gm3 = 1

2, α

c= 1.858754025

gm1 = 1.858754025

gm2 = 1

1, γ

c1 ÷ c4 = 2.385997060

gm1 = 26.90867792

gm2 = gm3 = 1

2, β

c1 = 1.381109025

c2 = 2.049923808

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 5.917085537

gm1 = 180.6094753

gm2 = gm3 = 1

gm4 = 140.1084925

2, γ

c1 ÷ c4 = 1.178923859

gm1 = 3.854510686

gm2 = gm3 = 1

gm4 = 3.562911609

2, α

c= 2.416661431

gm1 = 2.416661431

gm2 = 1

1, γ

c1 ÷ c4 = 11.56550168

gm1 = 438.2820800

gm2 = gm3 = 1

gm4 = 535.1496135

2, γ

c1 ÷ c4 = 2.138318409

gm1 = 10.81797466

gm2 = gm3 = 1

gm4 = 14.54834013

2, β

c1 = 1.561083804

c2 = 2.362655274

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 29.91104336

gm1 = 4101.490044

gm2 = gm3 = 1

gm4 = 3578.953531

2, γ

c1 ÷ c4 = 5.002176947

gm1 = 60.24884243

gm2 = gm3 = 1

gm4 = 91.54082395

3, γ

c1 ÷ c4 = 1.234924047

gm1 = 3.253888210

gm2 = gm3 = 1

gm4 = 3.511815680

4, α

c= 2.552948145

gm1 = 2.552948145

gm2 = 1

1, γ

c1 ÷ c4 = 58.40914061

gm1 = 10,521.88224

gm2 = gm3 = 1

gm4 = 13,646.89729

2, γ

c1 ÷ c4 = 9.624665014

gm1 = 203.8879914

gm2 = gm3 = 1

gm4 = 353.9260855

3, γ

c1 ÷ c4 = 2.199087818

gm1 = 9.999013247

gm2 = gm3 = 1

gm4 = 14.54289046

4, β

c1 = 1.619924816

c2 = 2.431155773

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 151.3278718

gm1 = 102,651.8319

gm2 = gm3 = 1

gm4 = 91,601.66515

2, γ

c1 ÷ c4 = 24.41947580

gm1 = 1402.775636

gm2 = gm3 = 1

gm4 = 2343.426218

3, γ

c1 ÷ c4 = 5.023921907

gm1 = 52.36670609

gm2 = gm3 = 1

gm4 = 90.38902435

4, γ

c1 ÷ c4 = 1.249851893

gm1 = 3.164514024

gm2 = gm3 = 1

gm4 = 3.517320340

5, α

c= 2.585029371

gm1 = 2.585029371

gm2 = 1

4

5

6

7

8

9

1.373

1.132

1.064

1.025

1.012

1.005

gm4 = 5.302185426

gm4 = 22.74082964

(continued)

256

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.64 (continued) n

fs

Cell no., type

10

1.002

1, γ

c1 ÷ c4 = 293.4390433

gm1 = 262,491.2691

gm2 = gm3 = 1

gm4 = 344,428.8941

2, γ

c1 ÷ c4 = 47.40813110

gm1 = 4887.758378

gm2 = gm3 = 1

gm4 = 8908.745991

3, γ

c1 ÷ c4 = 9.606099950

gm1 = 188.0731778

gm2 = gm3 = 1

gm4 = 348.7341432

4, γ

c1 ÷ c4 = 2.211199638

gm1 = 9.843367697

gm2 = gm3 = 1

gm4 = 14.53911053

5, β

c1 = 1.629185622

c2 = 2.444847459

gm1 ÷ gm4 = 1

Table 8.65 Element values of the cascaded Gm-C of modified elliptic (Zolotarev) filters (amax = 1 dB amin = 40 dB) n

fs

Cell no., type

3

2.227

1, γ

c1 ÷ c4 = 1.100095972

gm1 = 18.41519771

gm2 = gm3 = 1

2, α

c= 1.909464643

gm1 = 1.909464643

gm2 = 1

1, γ

c1 ÷ c4 = 2.219338069

gm1 = 29.12617411

gm2 = gm3 = 1

2, β

c1 = 1.406887672

c2 = 2.132585098

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 5.004891781

gm1 = 155.9222534

gm2 = gm3 = 1

gm4 = 100.0875071

2, γ

c1 ÷ c4 = 1.140139571

gm1 = 4.086618458

gm2 = gm3 = 1

gm4 = 3.105784532

2, α

c= 2.592837184

gm1 = 2.592837184

gm2 = 1

1, γ

c1 ÷ c4 = 9.421188555

gm1 = 329.1937181

gm2 = gm3 = 1

gm4 = 354.7773015

2, γ

c1 ÷ c4 = 1.977984244

gm1 = 9.977145289

gm2 = gm3 = 1

gm4 = 11.86865454

2, β

c1 = 1.644089482

c2 = 2.538353631

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 21.98565655

gm1 = 2554.727860

gm2 = gm3 = 1

gm4 = 1932.977034

2, γ

c1 ÷ c4 = 4.146388752

gm1 = 44.36416476

gm2 = gm3 = 1

gm4 = 61.07314357

3, γ

c1 ÷ c4 = 1.210909422

gm1 = 3.250151281

gm2 = gm3 = 1

gm4 = 3.034283359

4, α

c= 2.788120932

gm1 = 2.788120932

gm2 = 1

4

5

6

7

1.521

1.197

1.099

1.042

gm4 = 4.866660070

gm4 = 19.61554462

(continued)

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

257

Table 8.65 (continued) n

fs

Cell no., type

8

1.022

1, γ

c1 ÷ c4 = 41.11904324

gm1 = 5745.824650

gm2 = gm3 = 1

gm4 = 6762.032777

2, γ

c1 ÷ c4 = 7.608330452

gm1 = 132.8506484

gm2 = gm3 = 1

gm4 = 217.2470670

3, γ

c1 ÷ c4 = 2.039952053

gm1 = 8.796000163

gm2 = gm3 = 1

gm4 = 11.67515173

4, β

c1 = 1.717785088

c2 = 2.643730872

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 96.78615756

gm1 = 47,789.03047

gm2 = gm3 = 1

gm4 = 37,467.86082

2, γ

c1 ÷ c4 = 17.45013101

gm1 = 757.2219145

gm2 = gm3 = 1

gm4 = 1185.496239

3, γ

c1 ÷ c4 = 4.177578502

gm1 = 36.98715734

gm2 = gm3 = 1

gm4 = 59.97586648

4, γ

c1 ÷ c4 = 1.234366745

gm1 = 3.119503533

gm2 = gm3 = 1

gm4 = 3.043406209

5, α

c= 2.844713896

gm1 = 2.844713896

gm2 = 1

1, γ

c1 ÷ c4 = 180.2014796

gm1 = 108,266.7290

gm2 = gm3 = 1

gm4 = 129,885.7891

2, γ

c1 ÷ c4 = 32.35620192

gm1 = 2355.239165

gm2 = gm3 = 1

gm4 = 4127.244196

3, γ

c1 ÷ c4 = 7.590644440

gm1 = 118.8871669

gm2 = gm3 = 1

gm4 = 212.4467603

4, γ

c1 ÷ c4 = 2.058034939

gm1 = 8.578283756

gm2 = gm3 = 1

gm4 = 11.66598804

5, b

c1 = 1.734045911

c2 = 2.668349526

gm1 ÷ gm4 = 1

9

10

1.009

1.005

Table 8.66 Element values of the cascaded Gm-C of modified elliptic (Zolotarev) filters (amax = 1 dB amin = 45 dB) n

fs

Cell no., type

3

2.652

1, γ

c1 ÷ c4 = 1.070678026

gm1 = 25.04868544

gm2 = gm3 = 1

2, α

c= 1.944973207

gm1 = 1.944973207

gm2 = 1

1, γ

c1 ÷ c4 = 2.105543306

gm1 = 33.28379221

gm2 = gm3 = 1

2, β

c1 = 1.427382623

c2 = 2.198637089

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 4.404268123

gm1 = 146.8276883

gm2 = gm3 = 1

gm4 = 77.38886616

2, γ

c1 ÷ c4 = 1.115787501

gm1 = 4.516214694

gm2 = gm3 = 1

gm4 = 2.800609590

2, α

c= 2.744533096

gm1 = 2.744533096

gm2 = 1

4

5

1.701

1.278

gm4 = 4.594399528

gm4 = 17.61289502

(continued)

258

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.66 (continued) n

fs

Cell no., type

6

1.144

1, γ

c1 ÷ c4 = 8.002601806

gm1 = 271.5818658

gm2 = gm3 = 1

gm4 = 255.7385092

2, γ

c1 ÷ c4 = 1.861509640

gm1 = 9.667146283

gm2 = gm3 = 1

gm4 = 10.06575913

2, β

c1 = 1.716414068

c2 = 2.699857537

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 17.20647982

gm1 = 1807.647752

gm2 = gm3 = 1

gm4 = 1183.438335

2, γ

c1 ÷ c4 = 3.583530153

gm1 = 35.89391525

gm2 = gm3 = 1

gm4 = 44.26670997

3, γ

c1 ÷ c4 = 1.202748232

gm1 = 3.368416764

gm2 = gm3 = 1

gm4 = 2.711119368

4, α

c= 3.009928550

gm1 = 3.009928550

gm2 = 1

1, γ

c1 ÷ c4 = 30.97630114

gm1 = 3611.779907

gm2 = gm3 = 1

gm4 = 3836.537456

2, γ

c1 ÷ c4 = 6.304407033

gm1 = 95.94283739

gm2 = gm3 = 1

gm4 = 146.3175784

3, γ

c1 ÷ c4 = 1.932417934

gm1 = 8.131996524

gm2 = gm3 = 1

gm4 = 9.787133201

4, β

c1 = 1.814105688

c2 = 2.850750731

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 67.38095873

gm1 = 26,323.57610

gm2 = gm3 = 1

gm4 = 18,157.42148

2, γ

c1 ÷ c4 = 13.30123302

gm1 = 468.3030493

gm2 = gm3 = 1

gm4 = 681.1880602

3, γ

c1 ÷ c4 = 3.622510104

gm1 = 28.59155682

gm2 = gm3 = 1

gm4 = 43.14016717

4, γ

c1 ÷ c4 = 1.235975998

gm1 = 3.176642773

gm2 = gm3 = 1

gm4 = 2.720676688

5, α

c= 3.097461630

gm1 = 3.097461630

gm2 = 1

1, γ

c1 ÷ c4 = 120.6660379

gm1 = 53,243.17551

gm2 = gm3 = 1

gm4 = 58,235.09600

2, γ

c1 ÷ c4 = 23.67601364

gm1 = 1312.416540

gm2 = gm3 = 1

gm4 = 2194.914826

3, γ

c1 ÷ c4 = 6.289202169

gm1 = 82.99410311

gm2 = gm3 = 1

gm4 = 141.8250523

4, γ

c1 ÷ c4 = 1.958182619

gm1 = 7.837860043

gm2 = gm3 = 1

gm4 = 9.774497377

5, b

c1 = 1.840408600

c2 = 2.891007856

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 260.1415940

gm1 = 387,266.7504

gm2 = gm3 = 1

gm4 = 270,683.4767

2, γ

c1 ÷ c4 = 51.20233318

gm1 = 6848.195188

gm2 = gm3 = 1

gm4 = 10,384.52280

7

8

9

10

11

1.065

1.035

1.016

1.009

1.004

(continued)

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

259

Table 8.66 (continued) n

fs

Cell no., type 3, γ

c1 ÷ c4 = 13.17346716

gm1 = 372.8689573

gm2 = gm3 = 1

gm4 = 660.1363453

4, γ

c1 ÷ c4 = 3.642611444

gm1 = 27.12538691

gm2 = gm3 = 1

gm4 = 43.06757998

5, γ

c1 ÷ c4 = 1.244189015

gm1 = 3.127046420

gm2 = gm3 = 1

gm4 = 2.721363079

6, α

c= 3.118339422

gm1 = 3.118339422

gm2 = 1

Table 8.67 Element values of the cascaded Gm-C of modified elliptic (Zolotarev) filters (amax = 1 dB amin = 50 dB) n

fs

Cell no., type

3

3.175

1, γ

c1 ÷ c4 = 1.051336333

gm1 = 34.92074922

gm2 = gm3 = 1

2, α

c= 1.969622900

gm1 = 1.969622900

gm2 = 1

1, γ

c1 ÷ c4 = 2.025854769

gm1 = 39.57385135

gm2 = gm3 = 1

2, β

c1 = 1.443434265

c2 = 2.250639923

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 3.991295145

gm1 = 147.5805821

gm2 = gm3 = 1

gm4 = 63.46555738

2, γ

c1 ÷ c4 = 1.100169074

gm1 = 5.145094563

gm2 = gm3 = 1

gm4 = 2.588306023

2, α

c= 2.873351020

gm1 = 2.873351020

gm2 = 1

1, γ

c1 ÷ c4 = 7.029886861

gm1 = 241.7320846

gm2 = gm3 = 1

gm4 = 197.1620193

2, γ

c1 ÷ c4 = 1.776721537

gm1 = 9.767512874

gm2 = gm3 = 1

gm4 = 8.820717034

2, β

c1 = 1.781082200

c2 = 2.845808179

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 14.13394000

gm1 = 1411.663616

gm2 = gm3 = 1

gm4 = 798.1231229

2, γ

c1 ÷ c4 = 3.197777417

gm1 = 31.27238447

gm2 = gm3 = 1

gm4 = 34.22128550

3, γ

c1 ÷ c4 = 1.204993493

gm1 = 3.592244429

gm2 = gm3 = 1

gm4 = 2.485473948

4, α

c= 3.219823811

gm1 = 3.219823811

gm2 = 1

1, γ

c1 ÷ c4 = 24.57664275

gm1 = 2529.807839

gm2 = gm3 = 1

gm4 = 2414.305547

2, γ

c1 ÷ c4 = 5.414454862

gm1 = 75.07296974

gm2 = gm3 = 1

gm4 = 105.8013735

4

5

6

7

8

1.917

1.376

1.199

1.094

1.053

gm4 = 4.419334831

gm4 = 16.27407944

(continued)

260

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.67 (continued) n

9

10

11

fs

1.026

1.015

1.007

Cell no., type 3, γ

c1 ÷ c4 = 1.857629768

gm1 = 7.806990358

gm2 = gm3 = 1

gm4 = 8.472963784

4, β

c1 = 1.907019089

c2 = 3.050054087

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 50.05066127

gm1 = 16,483.99380

gm2 = gm3 = 1

gm4 = 10,016.80158

2, γ

c1 ÷ c4 = 10.65314881

gm1 = 321.7770533

gm2 = gm3 = 1

gm4 = 431.5769213

3, γ

c1 ÷ c4 = 3.240804056

gm1 = 23.68476312

gm2 = gm3 = 1

gm4 = 32.97798498

4, γ

c1 ÷ c4 = 1.248033723

gm1 = 3.308512292

gm2 = gm3 = 1

gm4 = 2.489474939

5, α

c= 3.341540156

gm1 = 3.341540156

gm2 = 1

1, γ

c1 ÷ c4 = 86.59430154

gm1 = 30,137.86236

gm2 = gm3 = 1

gm4 = 29,988.20177

2, γ

c1 ÷ c4 = 18.28824914

gm1 = 819.7017310

gm2 = gm3 = 1

gm4 = 1299.286736

3, γ

c1 ÷ c4 = 5.403780182

gm1 = 62.61376616

gm2 = gm3 = 1

gm4 = 101.5785691

4, γ

c1 ÷ c4 = 1.893060268

gm1 = 7.421756740

gm2 = gm3 = 1

gm4 = 8.460714651

5, b

c1 = 1.947139069

c2 = 3.111161515

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 176.7050446

gm1 = 201,437.7645

gm2 = gm3 = 1

gm4 = 124,886.5016

2, γ

c1 ÷ c4 = 36.99359285

gm1 = 3802.841589

gm2 = gm3 = 1

gm4 = 5396.315773

3, γ

c1 ÷ c4 = 10.53002235

gm1 = 244.7447063

gm2 = gm3 = 1

gm4 = 414.1347213

4, γ

c1 ÷ c4 = 3.268343414

gm1 = 22.10237617

gm2 = gm3 = 1

gm4 = 32.90319889

5, γ

c1 ÷ c4 = 1.260855970

gm1 = 3.234168834

gm2 = gm3 = 1

gm4 = 2.491320584

6, α

c= 3.376491650

gm1 = 3.376491650

gm2 = 1

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

261

Table 8.68 Element values of the cascaded Gm-C of modified elliptic (Zolotarev) filters (amax = 1 dB amin = 55 dB) n

fs

Cell no., type

3

3.816

1, γ

c1 ÷ c4 = 1.038472930

gm1 = 49.49740962

gm2 = gm3 = 1

2, α

c= 1.986632743

gm1 = 1.986632743

gm2 = 1

1, γ

c1 ÷ c4 = 1.969035732

gm1 = 48.44900634

gm2 = gm3 = 1

2, β

c1 = 1.455876178

c2 = 2.291102736

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 3.697965010

gm1 = 155.8798821

gm2 = gm3 = 1

gm4 = 54.40973482

2, γ

c1 ÷ c4 = 1.089969102

gm1 = 5.996074014

gm2 = gm3 = 1

gm4 = 2.436023262

2, α

c= 2.981423940

gm1 = 2.981423940

gm2 = 1

1, γ

c1 ÷ c4 = 6.339030093

gm1 = 228.4461695

gm2 = gm3 = 1

gm4 = 160.1710250

2, γ

c1 ÷ c4 = 1.713920600

gm1 = 10.20798054

gm2 = gm3 = 1

gm4 = 7.932428917

2, β

c1 = 1.838634792

c2 = 2.975860430

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 12.05020791

gm1 = 1189.704075

gm2 = gm3 = 1

gm4 = 579.8255474

2, γ

c1 ÷ c4 = 2.922688233

gm1 = 28.84652354

gm2 = gm3 = 1

gm4 = 27.79000122

3, γ

c1 ÷ c4 = 1.212956314

gm1 = 3.910431215

gm2 = gm3 = 1

gm4 = 2.321109736

4, α

c= 3.414892346

gm1 = 3.414892346

gm2 = 1

1, γ

c1 ÷ c4 = 20.29695253

gm1 = 1927.747395

gm2 = gm3 = 1

gm4 = 1646.123461

2, γ

c1 ÷ c4 = 4.779330657

gm1 = 62.56435807

gm2 = gm3 = 1

gm4 = 80.82045874

3, γ

c1 ÷ c4 = 1.803816877

gm1 = 7.713436387

gm2 = gm3 = 1

gm4 = 7.513743516

4, β

c1 = 1.994497580

c2 = 3.239949748

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 39.09681967

gm1 = 11,402.05845

gm2 = gm3 = 1

gm4 = 6110.932372

2, γ

c1 ÷ c4 = 8.866376961

gm1 = 240.1259806

gm2 = gm3 = 1

gm4 = 295.0165072

3, γ

c1 ÷ c4 = 2.967696153

gm1 = 20.69822526

gm2 = gm3 = 1

gm4 = 26.40712315

4, γ

c1 ÷ c4 = 1.266067024

gm1 = 3.499875529

gm2 = gm3 = 1

gm4 = 2.316095684

5, α

c= 3.574394042

gm1 = 3.574394042

gm2 = 1

4

5

6

7

8

9

2.173

1.492

1.265

1.131

1.075

1.039

gm4 = 4.304638690

gm4 = 15.35154936

(continued)

262

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.68 (continued) n

fs

Cell no., type

10

1.023

1, γ

c1 ÷ c4 = 65.63316312

gm1 = 19,061.41664

gm2 = gm3 = 1

gm4 = 17,225.18184

2, γ

c1 ÷ c4 = 14.74405210

gm1 = 560.8000928

gm2 = gm3 = 1

gm4 = 837.0753326

3, γ

c1 ÷ c4 = 4.776199700

gm1 = 50.25086846

gm2 = gm3 = 1

gm4 = 76.87410924

4, γ

c1 ÷ c4 = 1.851378722

gm1 = 7.224401905

gm2 = gm3 = 1

gm4 = 7.509248366

5, b

c1 = 2.053146097

c2 = 3.327288371

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 126.7902787

gm1 = 116,574.6543

gm2 = gm3 = 1

gm4 = 64,292.25089

2, γ

c1 ÷ c4 = 28.15160608

gm1 = 2351.953483

gm2 = gm3 = 1

gm4 = 3108.216374

3, γ

c1 ÷ c4 = 8.751013367

gm1 = 174.4900850

gm2 = gm3 = 1

gm4 = 280.2344423

4, γ

c1 ÷ c4 = 3.004815277

gm1 = 18.97953778

gm2 = gm3 = 1

gm4 = 26.34578651

5, γ

c1 ÷ c4 = 1.285345061

gm1 = 3.394941415

gm2 = gm3 = 1

gm4 = 2.320790817

6, α

c= 3.630010458

gm1 = 3.630010458

gm2 = 1

11

1.012

Table 8.69 Element values of the cascaded Gm-C of modified elliptic (Zolotarev) filters (amax = 1 dB amin = 60 dB) n

fs

Cell no., type

3

4.597

1, γ

c1 ÷ c4 = 1.029850845

gm1 = 70.94928932

gm2 = gm3 = 1

2, α

c= 1.998323007

gm1 = 1.998323007

gm2 = 1

1, γ

c1 ÷ c4 = 1.927988845

gm1 = 60.61414878

gm2 = gm3 = 1

2, β

c1 = 1.465444525

c2 = 2.322304185

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 3.484406514

gm1 = 171.0396745

gm2 = gm3 = 1

gm4 = 48.25243942

2, γ

c1 ÷ c4 = 1.083196417

gm1 = 7.108778601

gm2 = gm3 = 1

gm4 = 2.324247107

2, α

c= 3.071170067

gm1 = 3.071170067

gm2 = 1

1, γ

c1 ÷ c4 = 5.833934034

gm1 = 226.3645360

gm2 = gm3 = 1

gm4 = 135.5510595

2, γ

c1 ÷ c4 = 1.666577783

gm1 = 10.96174161

gm2 = gm3 = 1

gm4 = 7.280663936

2, β

c1 = 1.889327003

c2 = 3.090355847

gm1 ÷ gm4 = 1

4

5

6

2.474

1.628

1.343

gm4 = 4.228535474

gm4 = 14.70173100

(continued)

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

263

Table 8.69 (continued) n

fs

Cell no., type

7

1.175

1, γ

c1 ÷ c4 = 10.57612824

gm1 = 1064.358827

gm2 = gm3 = 1

gm4 = 446.3981893

2, γ

c1 ÷ c4 = 2.720052225

gm1 = 27.82006493

gm2 = gm3 = 1

gm4 = 23.44378683

3, γ

c1 ÷ c4 = 1.223702630

gm1 = 4.321224760

gm2 = gm3 = 1

gm4 = 2.197069727

4, α

c= 3.593199511

gm1 = 3.593199511

gm2 = 1

1, γ

c1 ÷ c4 = 17.30534943

gm1 = 1571.391006

gm2 = gm3 = 1

gm4 = 1196.214014

2, γ

c1 ÷ c4 = 4.310903015

gm1 = 54.86903679

gm2 = gm3 = 1

gm4 = 64.50227523

3, γ

c1 ÷ c4 = 1.764394104

gm1 = 7.799668850

gm2 = gm3 = 1

gm4 = 6.791155599

4, β

c1 = 2.075973144

c2 = 3.419114149

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 31.77379094

gm1 = 8528.902407

gm2 = gm3 = 1

gm4 = 4035.225559

2, γ

c1 ÷ c4 = 7.604828762

gm1 = 191.3106712

gm2 = gm3 = 1

gm4 = 214.0920410

3, γ

c1 ÷ c4 = 2.765335305

gm1 = 18.85784049

gm2 = gm3 = 1

gm4 = 21.91389249

4, γ

c1 ÷ c4 = 1.286952260

gm1 = 3.742011243

gm2 = gm3 = 1

gm4 = 2.180429763

5, α

c= 3.793289974

gm1 = 3.793289974

gm2 = 1

1, γ

c1 ÷ c4 = 51.96433500

gm1 = 13,173.73429

gm2 = gm3 = 1

gm4 = 10,796.00558

2, γ

c1 ÷ c4 = 12.30124267

gm1 = 413.0599345

gm2 = gm3 = 1

gm4 = 577.1939409

3, γ

c1 ÷ c4 = 4.316471985

gm1 = 42.39496056

gm2 = gm3 = 1

gm4 = 60.78469410

4, γ

c1 ÷ c4 = 1.825861058

gm1 = 7.186665805

gm2 = gm3 = 1

gm4 = 6.796592363

5, b

c1 = 2.157388386

c2 = 3.538006011

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 95.73896503

gm1 = 74,516.33137

gm2 = gm3 = 1

gm4 = 36,654.33977

2, γ

c1 ÷ c4 = 22.34908556

gm1 = 1588.525692

gm2 = gm3 = 1

gm4 = 1947.043727

3, γ

c1 ÷ c4 = 7.500824341

gm1 = 132.9685188

gm2 = gm3 = 1

gm4 = 201.3954808

4, γ

c1 ÷ c4 = 2.814563180

gm1 = 16.98720567

gm2 = gm3 = 1

gm4 = 21.88394599

5, γ

c1 ÷ c4 = 1.314940991

gm1 = 3.601384452

gm2 = gm3 = 1

gm4 = 2.190323247

6, α

c= 3.877865840

gm1 = 3.877865840

gm2 = 1

8

9

10

11

1.103

1.055

1.033

1.018

264

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Fig. 8.17 Passband (a) and stopband (b) attenuation of the seventh-order Thomson (Bessel) filter

8.4.6 Thomson_Z Filters The attenuation (passband and stopband) characteristics of the seventh Thomson (Bessel) filter is depicted in Fig. 8.17. Figure 8.18 depicts the group delay characteristic of the seventh-order Thomson (Bessel) filter. One has to note that after the optimization procedure is used to create the transmission zeros (by extending the transfer function), the attenuation characteristic was renormalized so that to exhibit 3 dB at the cutoff. Therefrom, the reduction of the approximation interval of constant group delay as compared to the polynomial Bessel filter. Nevertheless, as can be seen, the approximation interval of constant group delay is still much broader than the width of the passband (Tables 8.70, 8.71, 8.72, 8.73, 8.74, 8.75 and 8.76).

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

265

Fig. 8.18 Normalized group delay characteristic of the seventh-order Thomson (Bessel), respectively

Table 8.70 Element values of the cascaded Gm-C of Thomson_Z filters (amin = 30 dB) n

fs

Cell no., type

3

3.083

1, γ

c1 ÷ c4 = 0.211257679

gm1 = 1.164084889

gm2 = gm3 = 1

2, α

c= 0.669168448

gm1 = 0.669168448

gm2 = 1

1, γ

c1 ÷ c4 = 0.206821116

gm1 = 0.772743815

gm2 = gm3 = 1

2, β

c1 = 0.300467149

c2 = 1.102972920

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.177956476

gm1 = 1.439602396

gm2 = gm3 = 1

gm4 = 0.839930777

2, γ

c1 ÷ c4 = 0.123417729

gm1 = 0.226231526

gm2 = gm3 = 1

gm4 = 0.317572396

2, α

c= 0.453765288

gm1 = 0.453765288

gm2 = 1

1, γ

c1 ÷ c4 = 0.171769442

gm1 = 1.042619475

gm2 = gm3 = 1

gm4 = 1.047171448

2, γ

c1 ÷ c4 = 0.115683623

gm1 = 0.193193332

gm2 = gm3 = 1

gm4 = 0.373558774

2, β

c1 = 0.203448079

c2 = 0.781217777

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.154282596

gm1 = 2.499248860

gm2 = gm3 = 1

gm4 = 1.268456051

2, γ

c1 ÷ c4 = 0.101803202

gm1 = 0.310461089

gm2 = gm3 = 1

gm4 = 0.436684908

3, γ

c1 ÷ c4 = 0.087080265

gm1 = 0.108640792

gm2 = gm3 = 1

gm4 = 0.283402589

4, α

c= 0.333363607

gm1 = 0.333363607

gm2 = 1

4

5

6

7

2.687

2.525

2.505

2.502

gm4 = 0.477545439

gm4 = 0.648891923

(continued)

266

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.70 (continued) n

fs

Cell no., type

8

2.502

1, γ

c1 ÷ c4 = 0.150097026

gm1 = 1.759111547

gm2 = gm3 = 1

gm4 = 1.502265540

2, γ

c1 ÷ c4 = 0.097549184

gm1 = 0.276398595

gm2 = gm3 = 1

gm4 = 0.505310672

3, γ

c1 ÷ c4 = 0.081870525

gm1 = 0.095935121

gm2 = gm3 = 1

gm4 = 0.313162417

4, β

c1 = 0.152516728

c2 = 0.595705730

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.139080993

gm1 = 4.289288499

gm2 = gm3 = 1

gm4 = 1.747450238

2, γ

c1 ÷ c4 = 0.089331447

gm1 = 0.499762857

gm2 = gm3 = 1

gm4 = 0.578529100

3, γ

c1 ÷ c4 = 0.073934219

gm1 = 0.157059281

gm2 = gm3 = 1

gm4 = 0.347399550

4, γ

c1 ÷ c4 = 0.067602166

gm1 = 0.065423478

gm2 = gm3 = 1

gm4 = 0.270097054

5, α

c= 0.263209329

gm1 = 0.263209329

gm2 = 1

1, γ

c1 ÷ c4 = 0.135914399

gm1 = 2.898065979

gm2 = gm3 = 1

gm4 = 2.003099196

2, γ

c1 ÷ c4 = 0.086477178

gm1 = 0.443873390

gm2 = gm3 = 1

gm4 = 0.655761407

3, γ

c1 ÷ c4 = 0.070807621

gm1 = 0.143837180

gm2 = gm3 = 1

gm4 = 0.384983214

4, γ

c1 ÷ c4 = 0.063874210

gm1 = 0.058401563

gm2 = gm3 = 1

gm4 = 0.288962315

5, β

c1 = 0.122087183

c2 = 0.480794419

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.128295416

gm1 = 6.998165589

gm2 = gm3 = 1

gm4 = 2.268467328

2, γ

c1 ÷ c4 = 0.080998246

gm1 = 0.774897749

gm2 = gm3 = 1

gm4 = 0.736600525

3, γ

c1 ÷ c4 = 0.065757237

gm1 = 0.249962660

gm2 = gm3 = 1

gm4 = 0.425273263

4, γ

c1 ÷ c4 = 0.058708537

gm1 = 0.098875933

gm2 = gm3 = 1

gm4 = 0.311093569

5, γ

c1 ÷ c4 = 0.055364219

gm1 = 0.043880187

gm2 = gm3 = 1

gm4 = 0.263467805

6, α

c= 0.217444274

gm1 = 0.217444274

gm2 = 1

9

10

11

2.052

2.502

2.502

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

267

Table 8.71 Element values of the cascaded Gm-C of Thomson_Z filters (amin = 35 dB) n

fs

Cell no., type

3

3.632

1, γ

c1 ÷ c4 = 0.219232856

gm1 = 1.720906046

gm2 = gm3 = 1

2, α

c= 0.694430189

gm1 = 0.694430189

gm2 = 1

1, γ

c1 ÷ c4 = 0.216518568

gm1 = 1.045440737

gm2 = gm3 = 1

2, β

c1 = 0.314555487

c2 = 1.154689241

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.189924086

gm1 = 1.779648712

gm2 = gm3 = 1

gm4 = 0.839930777

2, γ

c1 ÷ c4 = 0.131717597

gm1 = 0.293274525

gm2 = gm3 = 1

gm4 = 0.317572396

2, α

c= 0.484281097

gm1 = 0.484281097

gm2 = 1

1, γ

c1 ÷ c4 = 0.183019509

gm1 = 1.238965887

gm2 = gm3 = 1

gm4 = 1.047171448

2, γ

c1 ÷ c4 = 0.123260341

gm1 = 0.244378803

gm2 = gm3 = 1

gm4 = 0.373558774

2, β

c1 = 0.216772943

c2 = 0.832383760

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.165035736

gm1 = 2.852391603

gm2 = gm3 = 1

gm4 = 1.268456051

2, γ

c1 ÷ c4 = 0.108898650

gm1 = 0.358357762

gm2 = gm3 = 1

gm4 = 0.436684909

3, γ

c1 ÷ c4 = 0.093149558

gm1 = 0.137083824

gm2 = gm3 = 1

gm4 = 0.283402589

4, α

c= 0.356598279

gm1 = 0.356598279

gm2 = 1

1, γ

c1 ÷ c4 = 0.160315327

gm1 = 1.957970995

gm2 = gm3 = 1

gm4 = 1.502265540

2, γ

c1 ÷ c4 = 0.104190134

gm1 = 0.314061116

gm2 = gm3 = 1

gm4 = 0.505310672

3, γ

c1 ÷ c4 = 0.087444104

gm1 = 0.120512968

gm2 = gm3 = 1

gm4 = 0.313162417

4, β

c1 = 0.162899757

c2 = 0.636260164

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.148780970

gm1 = 4.692591583

gm2 = gm3 = 1

gm4 = 1.747450239

2, γ

c1 ÷ c4 = 0.095561724

gm1 = 0.543943272

gm2 = gm3 = 1

gm4 = 0.578529100

3, γ

c1 ÷ c4 = 0.079090641

gm1 = 0.177713444

gm2 = gm3 = 1

gm4 = 0.347399550

4, γ

c1 ÷ c4 = 0.072316970

gm1 = 0.082417709

gm2 = gm3 = 1

gm4 = 0.270097054

5, α

c= 0.281566436

gm1 = 0.281566436

gm2 = 1

4

5

6

7

8

9

3.005

2.721

2.677

2.665

2.663

2.663

gm4 = 0.477545439

gm4 = 0.648891924

(continued)

268

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.71 (continued) n

fs

Cell no., type

10

2.663

1, γ

c1 ÷ c4 = 0.145277904

gm1 = 3.120633300

gm2 = gm3 = 1

gm4 = 2.003099195

2, γ

c1 ÷ c4 = 0.092434822

gm1 = 0.477091211

gm2 = gm3 = 1

gm4 = 0.655761407

3, γ

c1 ÷ c4 = 0.075685746

gm1 = 0.162292239

gm2 = gm3 = 1

gm4 = 0.384983214

4, γ

c1 ÷ c4 = 0.068274674

gm1 = 0.073456381

gm2 = gm3 = 1

gm4 = 0.288962315

5, β

c1 = 0.130498094

c2 = 0.513917627

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.137243008

gm1 = 7.467814951

gm2 = gm3 = 1

gm4 = 2.268467327

2, γ

c1 ÷ c4 = 0.086647234

gm1 = 0.821460056

gm2 = gm3 = 1

gm4 = 0.736600525

3, γ

c1 ÷ c4 = 0.070343285

gm1 = 0.266972488

gm2 = gm3 = 1

gm4 = 0.425273263

4, γ

c1 ÷ c4 = 0.062802994

gm1 = 0.111695801

gm2 = gm3 = 1

gm4 = 0.311093569

5, γ

c1 ÷ c4 = 0.059225436

gm1 = 0.055277237

gm2 = gm3 = 1

gm4 = 0.263467805

6, α

c= 0.232609296

gm1 = 0.232609296

gm2 = 1

11

2.663

Table 8.72 Element values of the cascaded Gm-C of Thomson_Z filters (amin = 40 dB) n

fs

Cell no., type

3

4.312

1, γ

c1 ÷ c4 = 0.225001514

gm1 = 2.537957139

gm2 = gm3 = 1

2, α

c= 0.712702679

gm1 = 0.712702679

gm2 = 1

1, γ

c1 ÷ c4 = 0.224332191

gm1 = 1.408577139

gm2 = gm3 = 1

2, β

c1 = 0.325907020

c2 = 1.196359136

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.200753348

gm1 = 2.215038570

gm2 = gm3 = 1

gm4 = 0.839930777

2, γ

c1 ÷ c4 = 0.139227989

gm1 = 0.376561902

gm2 = gm3 = 1

gm4 = 0.317572396

2, α

c= 0.511894271

gm1 = 0.511894271

gm2 = 1

1, γ

c1 ÷ c4 = 0.193376812

gm1 = 1.483106435

gm2 = gm3 = 1

gm4 = 1.047171448

2, γ

c1 ÷ c4 = 0.130235798

gm1 = 0.304892860

gm2 = gm3 = 1

gm4 = 0.373558774

2, β

c1 = 0.229040395

c2 = 0.879489399

gm1 ÷ gm4 = 1

4

5

6

3.880

2.934

2.851

gm4 = 0.477545439

gm4 = 0.648891923

(continued)

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

269

Table 8.72 (continued) n

fs

Cell no., type

7

2.820

1, γ

c1 ÷ c4 = 0.175128211

gm1 = 3.283868263

gm2 = gm3 = 1

gm4 = 1.268456052

2, γ

c1 ÷ c4 = 0.115558158

gm1 = 0.416481032

gm2 = gm3 = 1

gm4 = 0.436684909

3, γ

c1 ÷ c4 = 0.098845958

gm1 = 0.169665714

gm2 = gm3 = 1

gm4 = 0.283402589

4, α

c= 0.378405430

gm1 = 0.378405430

gm2 = 1

1, γ

c1 ÷ c4 = 0.169875406

gm1 = 2.199587071

gm2 = gm3 = 1

gm4 = 1.502265541

2, γ

c1 ÷ c4 = 0.110403301

gm1 = 0.358440175

gm2 = gm3 = 1

gm4 = 0.505310672

3, γ

c1 ÷ c4 = 0.092658656

gm1 = 0.148257312

gm2 = gm3 = 1

gm4 = 0.313162417

4, β

c1 = 0.172613953

c2 = 0.674202247

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.157901547

gm1 = 5.174603896

gm2 = gm3 = 1

gm4 = 1.747450238

2, γ

c1 ÷ c4 = 0.101419852

gm1 = 0.597179055

gm2 = gm3 = 1

gm4 = 0.578529100

3, γ

c1 ÷ c4 = 0.083939058

gm1 = 0.201375044

gm2 = gm3 = 1

gm4 = 0.347399550

4, γ

c1 ÷ c4 = 0.076750147

gm1 = 0.101622100

gm2 = gm3 = 1

gm4 = 0.270097054

5, α

c= 0.298827032

gm1 = 0.298827032

gm2 = 1

1, γ

c1 ÷ c4 = 0.157901547

gm1 = 5.174603896

gm2 = gm3 = 1

gm4 = 1.747450238

2, γ

c1 ÷ c4 = 0.101419852

gm1 = 0.597179055

gm2 = gm3 = 1

gm4 = 0.578529100

3, γ

c1 ÷ c4 = 0.083939058

gm1 = 0.201375044

gm2 = gm3 = 1

gm4 = 0.347399550

4, γ

c1 ÷ c4 = 0.076750147

gm1 = 0.101622100

gm2 = gm3 = 1

gm4 = 0.270097054

5, β

c= 0.298827032

gm1 = 0.298827032

gm2 = 1

1, γ

c1 ÷ c4 = 0.145656691

gm1 = 8.020365944

gm2 = gm3 = 1

gm4 = 2.268467328

2, γ

c1 ÷ c4 = 0.091959143

gm1 = 0.876838130

gm2 = gm3 = 1

gm4 = 0.736600525

3, γ

c1 ÷ c4 = 0.074655680

gm1 = 0.286619951

gm2 = gm3 = 1

gm4 = 0.425273263

4, γ

c1 ÷ c4 = 0.066653132

gm1 = 0.126185232

gm2 = gm3 = 1

gm4 = 0.311093569

5, γ

c1 ÷ c4 = 0.062856251

gm1 = 0.068151447

gm2 = gm3 = 1

gm4 = 0.263467805

6, α

c= 0.246869408

gm1 = 0.246869408

gm2 = 1

8

9

10

11

2.816

2.816

2.816

2.816

270

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.73 Element values of the cascaded Gm-C of Thomson_Z filters (amin = 45 dB) n

fs

Cell no., type

3

5.148

1, γ

c1 ÷ c4 = 0.229091646

gm1 = 3.736032772

gm2 = gm3 = 1

2, α

c= 0.725658363

gm1 = 0.725658363

gm2 = 1

1, γ

c1 ÷ c4 = 0.230511201

gm1 = 1.892015282

gm2 = gm3 = 1

2, β

c1 = 0.334883810

c2 = 1.229311676

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.210399720

gm1 = 2.767765293

gm2 = gm3 = 1

gm4 = 0.839930777

2, γ

c1 ÷ c4 = 0.145918014

gm1 = 0.480535081

gm2 = gm3 = 1

gm4 = 0.317572396

2, α

c= 0.536491233

gm1 = 0.536491233

gm2 = 1

1, γ

c1 ÷ c4 = 0.202879328

gm1 = 1.783622884

gm2 = gm3 = 1

gm4 = 1.047171448

2, γ

c1 ÷ c4 = 0.136635571

gm1 = 0.376908660

gm2 = gm3 = 1

gm4 = 0.373558774

2, β

c1 = 0.240295416

c2 = 0.922707412

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.184630004

gm1 = 3.805952973

gm2 = gm3 = 1

gm4 = 1.268456050

2, γ

c1 ÷ c4 = 0.121827906

gm1 = 0.486594403

gm2 = gm3 = 1

gm4 = 0.436684909

3, γ

c1 ÷ c4 = 0.104208965

gm1 = 0.206959918

gm2 = gm3 = 1

gm4 = 0.283402589

4, α

c= 0.398936276

gm1 = 0.398936276

gm2 = 1

1, γ

c1 ÷ c4 = 0.178870417

gm1 = 2.489269448

gm2 = gm3 = 1

gm4 = 1.502265540

2, γ

c1 ÷ c4 = 0.116249227

gm1 = 0.410625718

gm2 = gm3 = 1

gm4 = 0.505310672

3, γ

c1 ÷ c4 = 0.097564991

gm1 = 0.179345494

gm2 = gm3 = 1

gm4 = 0.313162417

4, β

c1 = 0.181753973

c2 = 0.709901688

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.166527526

gm1 = 5.743563685

gm2 = gm3 = 1

gm4 = 1.747450238

2, γ

c1 ÷ c4 = 0.106960302

gm1 = 0.660734281

gm2 = gm3 = 1

gm4 = 0.578529100

3, γ

c1 ÷ c4 = 0.088524552

gm1 = 0.228357465

gm2 = gm3 = 1

gm4 = 0.347399550

4, γ

c1 ÷ c4 = 0.080942919

gm1 = 0.123060297

gm2 = gm3 = 1

gm4 = 0.270097054

5, α

c= 0.315151608

gm1 = 0.315151608

gm2 = 1

4

5

6

7

8

9

3.821

3.173

3.035

2.973

2.962

2.960

gm4 = 0.477545439

gm4 = 0.648891924

(continued)

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

271

Table 8.73 (continued) n

fs

Cell no., type

10

2.960

1, γ

c1 ÷ c4 = 0.162351661

gm1 = 3.697219674

gm2 = gm3 = 1

gm4 = 2.003099195

2, γ

c1 ÷ c4 = 0.103298206

gm1 = 0.562751343

gm2 = gm3 = 1

gm4 = 0.655761407

3, γ

c1 ÷ c4 = 0.084580699

gm1 = 0.206621822

gm2 = gm3 = 1

gm4 = 0.384983214

4, γ

c1 ÷ c4 = 0.076298642

gm1 = 0.109288270

gm2 = gm3 = 1

gm4 = 0.288962315

5, β

c1 = 0.145834857

c2 = 0.574315696

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.153614365

gm1 = 8.661222396

gm2 = gm3 = 1

gm4 = 2.268467328

2, γ

c1 ÷ c4 = 0.096983154

gm1 = 0.942020828

gm2 = gm3 = 1

gm4 = 0.736600525

3, γ

c1 ÷ c4 = 0.078734350

gm1 = 0.309181822

gm2 = gm3 = 1

gm4 = 0.425273263

4, γ

c1 ÷ c4 = 0.070294598

gm1 = 0.142359958

gm2 = gm3 = 1

gm4 = 0.311093569

5, γ

c1 ÷ c4 = 0.066290282

gm1 = 0.082503942

gm2 = gm3 = 1

gm4 = 0.263467805

6, α

c= 0.260356645

gm1 = 0.260356645

gm2 = 1

11

2.960

Table 8.74 Element values of the cascaded Gm-C of Thomson_Z filters (amin = 50 dB) n

fs

Cell no., type

3

6.175

1, γ

c1 ÷ c4 = 0.231961721

gm1 = 5.496719067

gm2 = gm3 = 1

2, α

c= 0.734749457

gm1 = 0.734749457

gm2 = 1

1, γ

c1 ÷ c4 = 0.235333903

gm1 = 2.536175647

gm2 = gm3 = 1

2, β

c1 = 0.341890172

c2 = 1.255031052

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.218860657

gm1 = 3.466202485

gm2 = gm3 = 1

gm4 = 0.839930777

2, γ

c1 ÷ c4 = 0.151785908

gm1 = 0.610763415

gm2 = gm3 = 1

gm4 = 0.317572396

2, α

c= 0.558065495

gm1 = 0.558065495

gm2 = 1

1, γ

c1 ÷ c4 = 0.211539496

gm1 = 2.151463495

gm2 = gm3 = 1

gm4 = 1.047171449

2, γ

c1 ÷ c4 = 0.142468038

gm1 = 0.463139123

gm2 = gm3 = 1

gm4 = 0.373558774

2, β

c1 = 0.250552738

c2 = 0.962094379

gm1 ÷ gm4 = 1

4

5

6

4.340

3.446

3.236

gm4 = 0.477545439

gm4 = 0.648891924

(continued)

272

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.74 (continued) n

fs

Cell no., type

7

3.128

1, γ

c1 ÷ c4 = 0.193585652

gm1 = 4.431400538

gm2 = gm3 = 1

gm4 = 1.268456051

2, γ

c1 ÷ c4 = 0.127737281

gm1 = 0.570592557

gm2 = gm3 = 1

gm4 = 0.436684908

3, γ

c1 ÷ c4 = 0.109263717

gm1 = 0.249857922

gm2 = gm3 = 1

gm4 = 0.283402589

4, α

c= 0.418287047

gm1 = 0.418287047

gm2 = 1

1, γ

c1 ÷ c4 = 0.187369630

gm1 = 2.832952466

gm2 = gm3 = 1

gm4 = 1.502265542

2, γ

c1 ÷ c4 = 0.121772929

gm1 = 0.471859555

gm2 = gm3 = 1

gm4 = 0.505310672

3, γ

c1 ÷ c4 = 0.102200892

gm1 = 0.214143059

gm2 = gm3 = 1

gm4 = 0.313162417

4, β

c1 = 0.190390201

c2 = 0.743633401

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.174727582

gm1 = 6.409208949

gm2 = gm3 = 1

gm4 = 1.747450237

2, γ

c1 ÷ c4 = 0.112227182

gm1 = 0.735939664

gm2 = gm3 = 1

gm4 = 0.578529100

3, γ

c1 ÷ c4 = 0.092883629

gm1 = 0.259114364

gm2 = gm3 = 1

gm4 = 0.347399550

4, γ

c1 ÷ c4 = 0.084928665

gm1 = 0.146796461

gm2 = gm3 = 1

gm4 = 0.270097054

5, α

c= 0.330670129

gm1 = 0.330670129

gm2 = 1

1, γ

c1 ÷ c4 = 0.170210418

gm1 = 4.060885923

gm2 = gm3 = 1

gm4 = 2.003099195

2, γ

c1 ÷ c4 = 0.108298435

gm1 = 0.616993660

gm2 = gm3 = 1

gm4 = 0.655761407

3, γ

c1 ÷ c4 = 0.088674892

gm1 = 0.232800378

gm2 = gm3 = 1

gm4 = 0.384983214

4, γ

c1 ÷ c4 = 0.079991936

gm1 = 0.130068351

gm2 = gm3 = 1

gm4 = 0.288962315

5, β

c1 = 0.152894105

c2 = 0.602115887

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.161182347

gm1 = 9.403180829

gm2 = gm3 = 1

gm4 = 2.268467329

2, γ

c1 ÷ c4 = 0.101761137

gm1 = 1.018092534

gm2 = gm3 = 1

gm4 = 0.736600525

3, γ

c1 ÷ c4 = 0.082613285

gm1 = 0.335024398

gm2 = gm3 = 1

gm4 = 0.425273263

4, γ

c1 ÷ c4 = 0.073757739

gm1 = 0.160256127

gm2 = gm3 = 1

gm4 = 0.311093569

5, γ

c1 ÷ c4 = 0.069556146

gm1 = 0.098336890

gm2 = gm3 = 1

gm4 = 0.263467805

6, α

c= 0.273183404

gm1 = 0.273183404

gm2 = 1

8

9

10

11

3.105

3.098

3.097

3.097

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

273

Table 8.75 Element values of the cascaded Gm-C of Thomson_Z filters (amin = 55 dB) n

fs

Cell no., type

3

7.429

1, γ

c1 ÷ c4 = 0.233952097

gm1 = 8.080078683

gm2 = gm3 = 1

2, α

c= 0.741054065

gm1 = 0.741054065

gm2 = 1

1, γ

c1 ÷ c4 = 0.239059924

gm1 = 3.394611963

gm2 = gm3 = 1

2, β

c1 = 0.347303289

c2 = 1.274901848

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.226176247

gm1 = 4.349517476

gm2 = gm3 = 1

gm4 = 0.839930777

2, γ

c1 ÷ c4 = 0.156859471

gm1 = 0.774247183

gm2 = gm3 = 1

gm4 = 0.317572396

2, α

c= 0.576719274

gm1 = 0.576719274

gm2 = 1

1, γ

c1 ÷ c4 = 0.219358954

gm1 = 2.599538327

gm2 = gm3 = 1

gm4 = 1.047171448

2, γ

c1 ÷ c4 = 0.147734302

gm1 = 0.566840045

gm2 = gm3 = 1

gm4 = 0.373558774

2, β

c1 = 0.259814302

c2 = 0.997657745

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.201994150

gm1 = 5.176240676

gm2 = gm3 = 1

gm4 = 1.268456051

2, γ

c1 ÷ c4 = 0.133285619

gm1 = 0.670705490

gm2 = gm3 = 1

gm4 = 0.436684909

3, γ

c1 ÷ c4 = 0.114009646

gm1 = 0.299496959

gm2 = gm3 = 1

gm4 = 0.283402589

4, α

c= 0.436455572

gm1 = 0.436455572

gm2 = 1

1, γ

c1 ÷ c4 = 0.195415015

gm1 = 3.237497668

gm2 = gm3 = 1

gm4 = 1.502265541

2, γ

c1 ÷ c4 = 0.127001685

gm1 = 0.543485538

gm2 = gm3 = 1

gm4 = 0.505310672

3, γ

c1 ÷ c4 = 0.106589253

gm1 = 0.253240266

gm2 = gm3 = 1

gm4 = 0.313162417

4, β

c1 = 0.198565284

c2 = 0.775563958

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.182552355

gm1 = 7.183040227

gm2 = gm3 = 1

gm4 = 1.747450238

2, γ

c1 ÷ c4 = 0.117253018

gm1 = 0.824163672

gm2 = gm3 = 1

gm4 = 0.578529100

3, γ

c1 ÷ c4 = 0.097043208

gm1 = 0.294223602

gm2 = gm3 = 1

gm4 = 0.347399550

4, γ

c1 ÷ c4 = 0.088732000

gm1 = 0.172975205

gm2 = gm3 = 1

gm4 = 0.270097054

5, α

c= 0.345478429

gm1 = 0.345478429

gm2 = 1

4

5

6

7

8

9

4.948

3.760

3.458

3.291

3.249

3.232

gm4 = 0.477545439

gm4 = 0.648891924

(continued)

274

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.75 (continued) n

fs

Cell no., type

10

3.229

1, γ

c1 ÷ c4 = 0.177697562

gm1 = 4.481034027

gm2 = gm3 = 1

gm4 = 2.003099195

2, γ

c1 ÷ c4 = 0.113062221

gm1 = 0.680151657

gm2 = gm3 = 1

gm4 = 0.655761407

3, γ

c1 ÷ c4 = 0.092575486

gm1 = 0.262025289

gm2 = gm3 = 1

gm4 = 0.384983214

4, γ

c1 ÷ c4 = 0.083510587

gm1 = 0.152790947

gm2 = gm3 = 1

gm4 = 0.288962315

5, β

c1 = 0.159619547

c2 = 0.628601507

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.168410551

gm1 = 10.25132655

gm2 = gm3 = 1

gm4 = 2.268467327

2, γ

c1 ÷ c4 = 0.106324604

gm1 = 1.106145410

gm2 = gm3 = 1

gm4 = 0.736600525

3, γ

c1 ÷ c4 = 0.086318068

gm1 = 0.364587872

gm2 = gm3 = 1

gm4 = 0.425273263

4, γ

c1 ÷ c4 = 0.077065397

gm1 = 0.179943670

gm2 = gm3 = 1

gm4 = 0.311093569

5, γ

c1 ÷ c4 = 0.072675384

gm1 = 0.115653866

gm2 = gm3 = 1

gm4 = 0.263467805

6, α

c= 0.285434283

gm1 = 0.285434283

gm2 = 1

11

3.229

Table 8.76 Element values of the cascaded Gm-C of Thomson_Z filters (amin = 60 dB) n

fs

Cell no., type

3

8.956

1, γ

c1 ÷ c4 = 0.235325753

gm1 = 11.87168606

gm2 = gm3 = 1

2, α

c= 0.745405184

gm1 = 0.745405184

gm2 = 1

1, γ

c1 ÷ c4 = 0.241917663

gm1 = 4.539552202

gm2 = gm3 = 1

2, β

c1 = 0.351454977

c2 = 1.290142113

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.232398666

gm1 = 5.462595290

gm2 = gm3 = 1

gm4 = 0.839930777

2, γ

c1 ÷ c4 = 0.161174891

gm1 = 0.979696732

gm2 = gm3 = 1

gm4 = 0.317572396

2, α

c= 0.592585613

gm1 = 0.592585613

gm2 = 1

1, γ

c1 ÷ c4 = 0.226345558

gm1 = 3.143970538

gm2 = gm3 = 1

gm4 = 1.047171449

2, γ

c1 ÷ c4 = 0.152439654

gm1 = 0.691899540

gm2 = gm3 = 1

gm4 = 0.373558774

2, β

c1 = 0.268089413

c2 = 1.029433243

gm1 ÷ gm4 = 1

4

5

6

5.658

4.120

3.707

gm4 = 0.477545439

gm4 = 0.648891923

(continued)

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

275

Table 8.76 (continued) n

fs

Cell no., type

7

3.442

1, γ

c1 ÷ c4 = 0.209852485

gm1 = 6.060959934

gm2 = gm3 = 1

gm4 = 1.268456050

2, γ

c1 ÷ c4 = 0.138470933

gm1 = 0.789530105

gm2 = gm3 = 1

gm4 = 0.436684908

3, γ

c1 ÷ c4 = 0.118445051

gm1 = 0.357285307

gm2 = gm3 = 1

gm4 = 0.283402589

4, α

c= 0.453435340

gm1 = 0.453435340

gm2 = 1

1, γ

c1 ÷ c4 = 0.203026362

gm1 = 3.709957394

gm2 = gm3 = 1

gm4 = 1.502265541

2, γ

c1 ÷ c4 = 0.131948357

gm1 = 0.626957333

gm2 = gm3 = 1

gm4 = 0.505310672

3, γ

c1 ÷ c4 = 0.110740867

gm1 = 0.297434619

gm2 = gm3 = 1

gm4 = 0.313162417

4, β

c1 = 0.206299334

c2 = 0.805771907

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.190035834

gm1 = 8.074551797

gm2 = gm3 = 1

gm4 = 1.747450239

2, γ

c1 ÷ c4 = 0.122059641

gm1 = 0.926819597

gm2 = gm3 = 1

gm4 = 0.578529100

3, γ

c1 ÷ c4 = 0.101021359

gm1 = 0.334342410

gm2 = gm3 = 1

gm4 = 0.347399550

4, γ

c1 ÷ c4 = 0.092369445

gm1 = 0.201861789

gm2 = gm3 = 1

gm4 = 0.270097054

5, α

c= 0.359640835

gm1 = 0.359640835

gm2 = 1

1, γ

c1 ÷ c4 = 0.184859193

gm1 = 4.964546314

gm2 = gm3 = 1

gm4 = 2.003099194

2, γ

c1 ÷ c4 = 0.117618895

gm1 = 0.753237393

gm2 = gm3 = 1

gm4 = 0.655761407

3, γ

c1 ÷ c4 = 0.096306497

gm1 = 0.294721606

gm2 = gm3 = 1

gm4 = 0.384983214

4, γ

c1 ÷ c4 = 0.086876261

gm1 = 0.177544766

gm2 = gm3 = 1

gm4 = 0.288962315

5, β

c1 = 0.166052591

c2 = 0.653935632

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.175340515

gm1 = 11.21550762

gm2 = gm3 = 1

gm4 = 2.268467327

2, γ

c1 ÷ c4 = 0.110699779

gm1 = 1.207328839

gm2 = gm3 = 1

gm4 = 0.736600525

3, γ

c1 ÷ c4 = 0.089869990

gm1 = 0.398370772

gm2 = gm3 = 1

gm4 = 0.425273263

4, γ

c1 ÷ c4 = 0.080236578

gm1 = 0.201552192

gm2 = gm3 = 1

gm4 = 0.311093569

5, γ

c1 ÷ c4 = 0.075665919

gm1 = 0.134468520

gm2 = gm3 = 1

gm4 = 0.263467805

6, α

c= 0.297179682

gm1 = 0.297179682

gm2 = 1

8

9

10

11

3.396

3.364

3.357

3.356

276

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Fig. 8.19 Passband (a) and stopband (b) attenuation of the eleventh-order Equi-rip-td_Z filter

8.4.7 Equi-ripple-Td_Z Filters At the end of this “catalog,” we will change the order of the example xxx_Z filter to n = 11. That will allow for demonstration of the exceptional capabilities of the software. Figures 8.19 and 8.20 are depicting the attenuation and group delay characteristics of the eleventh-order Equi-rip-td_Z filter with amin = 30 dB, δ = 5%. To complete the demonstration, Fig. 8.21 depicts the same characteristics but produced (after circuit synthesis) by SPICE simulation (Tables 8.77, 8.78, 8.79, 8.80, 8.81, 8.82 and 8.83).

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

277

Fig. 8.20 Group delay of the eleventh-order Equi-rip-td_Z filter

Fig. 8.21 SPICE attenuation (top) and group delay (bottom) responses for the amax = 3 dB, amin = 30 dB, δ = 5%, f c = 1 kHz, n = 9 and m = 8

278

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.77 Element values of the cascaded Gm-C of Equi-rip-td_Z filters (amin = 30 dB, δ = 5%) n

fs

Cell no., type

3

2.766

1, γ

c1 ÷ c4 = 0.310658446

gm1 = 1.944212916

gm2 = gm3 = 1

2, α

c= 1.078879373

gm1 = 1.078879373

gm2 = 1

1, γ

c1 ÷ c4 = 0.356085664

gm1 = 1.786055418

gm2 = gm3 = 1

2, β

c1 = 0.602511034

c2 = 1.446118520

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.295226235

gm1 = 3.006929283

gm2 = gm3 = 1

gm4 = 3.901761264

2, γ

c1 ÷ c4 = 0.247721974

gm1 = 0.981005370

gm2 = gm3 = 1

gm4 = 0.940393152

2, α

c= 0.964822794

gm1 = 0.964822794

gm2 = 1

1, γ

c1 ÷ c4 = 0.347844437

gm1 = 3.165055740

gm2 = gm3 = 1

gm4 = 6.185685070

2, γ

c1 ÷ c4 = 0.290861620

gm1 = 1.115109373

gm2 = gm3 = 1

gm4 = 1.848293560

2, β

c1 = 0.561469266

c2 = 1.338344189

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.289116630

gm1 = 5.593502138

gm2 = gm3 = 1

gm4 = 9.065541757

2, γ

c1 ÷ c4 = 0.241334991

gm1 = 1.889403405

gm2 = gm3 = 1

gm4 = 3.156558267

3, γ

c1 ÷ c4 = 0.232037006

gm1 = 0.872478475

gm2 = gm3 = 1

gm4 = 0.939243428

4, α

c= 0.919958187

gm1 = 0.919958187

gm2 = 1

1, γ

c1 ÷ c4 = 0.343008797

gm1 = 6.068005255

gm2 = gm3 = 1

gm4 = 12.55052278

2, γ

c1 ÷ c4 = 0.286043336

gm1 = 2.132961646

gm2 = gm3 = 1

gm4 = 4.875429841

3, γ

c1 ÷ c4 = 0.274473434

gm1 = 0.982664090

gm2 = gm3 = 1

gm4 = 1.822754113

4, β

c1 = 0.541971531

c2 = 1.284103296

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.286169884

gm1 = 9.561300253

gm2 = gm3 = 1

gm4 = 16.64710205

2, γ

c1 ÷ c4 = 0.238504697

gm1 = 3.731621602

gm2 = gm3 = 1

gm4 = 7.011787707

3, γ

c1 ÷ c4 = 0.228592288

gm1 = 1.705588346

gm2 = gm3 = 1

gm4 = 3.079472407

4, γ

c1 ÷ c4 = 0.225223657

gm1 = 0.826625435

gm2 = gm3 = 1

gm4 = 0.944467229

5, α

c= 0.897353284

gm1 = 0.897353284

gm2 = 1

4

5

6

7

8

9

2.444

2.639

2.405

2.656

2.394

2.664

gm4 = 1.059003817

gm4 = 2.200233147

(continued)

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

279

Table 8.77 (continued) n

fs

Cell no., type

10

2.388

1, γ

c1 ÷ c4 = 0.339729675

gm1 = 10.33019873

gm2 = gm3 = 1

gm4 = 21.36090234

2, γ

c1 ÷ c4 = 0.283039301

gm1 = 4.137594562

gm2 = gm3 = 1

gm4 = 9.570965852

3, γ

c1 ÷ c4 = 0.271079045

gm1 = 1.909713996

gm2 = gm3 = 1

gm4 = 4.716184771

4, γ

c1 ÷ c4 = 0.266762778

gm1 = 0.923171101

gm2 = gm3 = 1

gm4 = 1.825351794

5, b

c1 = 0.530328410

c2 = 1.251317870

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.284450798

gm1 = 14.86216320

gm2 = gm3 = 1

gm4 = 26.69450015

2, γ

c1 ÷ c4 = 0.236926098

gm1 = 6.487109829

gm2 = gm3 = 1

gm4 = 12.55600654

3, γ

c1 ÷ c4 = 0.226804667

gm1 = 3.393287610

gm2 = gm3 = 1

gm4 = 6.737369372

4, γ

c1 ÷ c4 = 0.223023107

gm1 = 1.630659704

gm2 = gm3 = 1

gm4 = 3.070780280

5, γ

c1 ÷ c4 = 0.221416892

gm1 = 0.801405970

gm2 = gm3 = 1

gm4 = 0.949165995

6, α

c= 0.883831506

gm1 = 0.883831506

gm2 = 1

11

2.668

Table 8.78 Element values of the cascaded Gm-C of Equi-rip-td_Z group delay filters (amin = 35 dB, δ = 5%) n

fs

Cell no., type

3

3.141

1, γ

c1 ÷ c4 = 0.330036088

gm1 = 2.835061938

gm2 = gm3 = 1

2, α

c= 1.146175591

gm1 = 1.146175591

gm2 = 1

1, γ

c1 ÷ c4 = 0.372928450

gm1 = 2.319463125

gm2 = gm3 = 1

2, β

c1 = 0.631009693

c2 = 1.514519655

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.317765843

gm1 = 3.665455415

gm2 = gm3 = 1

gm4 = 3.901761265

2, γ

c1 ÷ c4 = 0.266634780

gm1 = 1.234155204

gm2 = gm3 = 1

gm4 = 0.940393152

2, α

c= 1.038484027

gm1 = 1.038484027

gm2 = 1

1, γ

c1 ÷ c4 = 0.366343608

gm1 = 3.539565256

gm2 = gm3 = 1

gm4 = 6.185685067

2, γ

c1 ÷ c4 = 0.306330314

gm1 = 1.381530361

gm2 = gm3 = 1

gm4 = 1.848293560

2, β

c1 = 0.591329500

c2 = 1.409520427

gm1 ÷ gm4 = 1

4

5

6

2.662

2.796

2.574

gm4 = 1.059003818

gm4 = 2.200233145

(continued)

280

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.78 (continued) n

fs

Cell no., type

7

2.822

1, γ

c1 ÷ c4 = 0.310604942

gm1 = 6.146153602

gm2 = gm3 = 1

gm4 = 9.065541757

2, γ

c1 ÷ c4 = 0.259271979

gm1 = 2.142294393

gm2 = gm3 = 1

gm4 = 3.156558264

3, γ

c1 ÷ c4 = 0.249282931

gm1 = 1.101301849

gm2 = gm3 = 1

gm4 = 0.939243428

4, α

c= 0.988333183

gm1 = 0.988333183

gm2 = 1

1, γ

c1 ÷ c4 = 0.362217845

gm1 = 6.383666956

gm2 = gm3 = 1

gm4 = 12.55052277

2, γ

c1 ÷ c4 = 0.302062227

gm1 = 2.369671362

gm2 = gm3 = 1

gm4 = 4.875429843

3, γ

c1 ÷ c4 = 0.289844392

gm1 = 1.220497299

gm2 = gm3 = 1

gm4 = 1.822754113

4, β

c1 = 0.572322816

c2 = 1.356015163

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.307141752

gm1 = 10.06427610

gm2 = gm3 = 1

gm4 = 16.64710205

2, γ

c1 ÷ c4 = 0.255983436

gm1 = 3.972271240

gm2 = gm3 = 1

gm4 = 7.011787703

3, γ

c1 ÷ c4 = 0.245344600

gm1 = 1.928975061

gm2 = gm3 = 1

gm4 = 3.079472401

4, γ

c1 ÷ c4 = 0.241729100

gm1 = 1.044555142

gm2 = gm3 = 1

gm4 = 0.944467229

5, α

c= 0.963115531

gm1 = 0.963115531

gm2 = 1

1, γ

c1 ÷ c4 = 0.359406081

gm1 = 10.61943773

gm2 = gm3 = 1

gm4 = 21.36090234

2, γ

c1 ÷ c4 = 0.299432324

gm1 = 4.360776417

gm2 = gm3 = 1

gm4 = 9.570965852

3, γ

c1 ÷ c4 = 0.286779355

gm1 = 2.128934901

gm2 = gm3 = 1

gm4 = 4.716184772

4, γ

c1 ÷ c4 = 0.282213100

gm1 = 1.148724008

gm2 = gm3 = 1

gm4 = 1.825351795

5, b

c1 = 0.561043882

c2 = 1.323791489

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.305118271

gm1 = 15.34641390

gm2 = gm3 = 1

gm4 = 26.69450015

2, γ

c1 ÷ c4 = 0.254140547

gm1 = 6.721349898

gm2 = gm3 = 1

gm4 = 12.55600653

3, γ

c1 ÷ c4 = 0.243283719

gm1 = 3.606045366

gm2 = gm3 = 1

gm4 = 6.737369368

4, γ

c1 ÷ c4 = 0.239227400

gm1 = 1.841506069

gm2 = gm3 = 1

gm4 = 3.070780280

5, γ

c1 ÷ c4 = 0.237504481

gm1 = 1.013140196

gm2 = gm3 = 1

gm4 = 0.949165995

6, α

c= 0.948048462

gm1 = 0.948048462

gm2 = 1

8

9

10

11

2.559

2.833

2.551

2.839

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

281

Table 8.79 Element values of the cascaded Gm-C of Equi-rip-td_Z group delay filters (amin = 40 dB, δ = 5%) n

fs

Cell no., type

3

3.627

1, γ

c1 ÷ c4 = 0.345100534

gm1 = 4.141584799

gm2 = gm3 = 1

2, α

c= 1.198492596

gm1 = 1.198492596

gm2 = 1

1, γ

c1 ÷ c4 = 0.387823715

gm1 = 3.028247465

gm2 = gm3 = 1

2, β

c1 = 0.656213072

c2 = 1.575011615

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.339826374

gm1 = 4.570914772

gm2 = gm3 = 1

gm4 = 3.901761263

2, γ

c1 ÷ c4 = 0.285145595

gm1 = 1.523059401

gm2 = gm3 = 1

gm4 = 0.940393152

2, α

c= 1.110579593

gm1 = 1.110579593

gm2 = 1

1, γ

c1 ÷ c4 = 0.383393474

gm1 = 4.031965537

gm2 = gm3 = 1

gm4 = 6.185685068

2, γ

c1 ÷ c4 = 0.320587122

gm1 = 1.678950191

gm2 = gm3 = 1

gm4 = 1.848293560

2, β

c1 = 0.618850353

c2 = 1.475120408

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.331579445

gm1 = 6.922161696

gm2 = gm3 = 1

gm4 = 9.065541756

2, γ

c1 ÷ c4 = 0.276780075

gm1 = 2.428967793

gm2 = gm3 = 1

gm4 = 3.156558264

3, γ

c1 ÷ c4 = 0.266116486

gm1 = 1.356282570

gm2 = gm3 = 1

gm4 = 0.939243427

4, α

c= 1.055073258

gm1 = 1.055073258

gm2 = 1

1, γ

c1 ÷ c4 = 0.379827243

gm1 = 6.769458020

gm2 = gm3 = 1

gm4 = 12.55052277

2, γ

c1 ÷ c4 = 0.316747130

gm1 = 2.640211929

gm2 = gm3 = 1

gm4 = 4.875429841

3, γ

c1 ÷ c4 = 0.303935319

gm1 = 1.489393480

gm2 = gm3 = 1

gm4 = 1.822754114

4, β

c1 = 0.600146570

c2 = 1.421938504

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.327570695

gm1 = 10.74535945

gm2 = gm3 = 1

gm4 = 16.64710205

2, γ

c1 ÷ c4 = 0.273009683

gm1 = 4.252775027

gm2 = gm3 = 1

gm4 = 7.011787703

3, γ

c1 ÷ c4 = 0.261663225

gm1 = 2.188012857

gm2 = gm3 = 1

gm4 = 3.079472404

4, γ

c1 ÷ c4 = 0.257807247

gm1 = 1.287694188

gm2 = gm3 = 1

gm4 = 0.944467228

5, α

c= 1.027175311

gm1 = 1.027175311

gm2 = 1

4

5

6

7

8

9

2.921

2.932

2.740

2.968

2.721

2.984

gm4 = 1.059003818

gm4 = 2.200233145

(continued)

282

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.79 (continued) n

fs

Cell no., type

10

2.711

1, γ

c1 ÷ c4 = 0.377397947

gm1 = 10.96430888

gm2 = gm3 = 1

gm4 = 21.36090234

2, γ

c1 ÷ c4 = 0.314421903

gm1 = 4.611563375

gm2 = gm3 = 1

gm4 = 9.570965855

3, γ

c1 ÷ c4 = 0.301135528

gm1 = 2.373821813

gm2 = gm3 = 1

gm4 = 4.716184771

4, γ

c1 ÷ c4 = 0.296340686

gm1 = 1.404485381

gm2 = gm3 = 1

gm4 = 1.825351794

5, b

c1 = 0.589129734

c2 = 1.390060480

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.325221481

gm1 = 15.98025454

gm2 = gm3 = 1

gm4 = 26.69450016

2, γ

c1 ÷ c4 = 0.270885007

gm1 = 6.993070658

gm2 = gm3 = 1

gm4 = 12.55600653

3, γ

c1 ÷ c4 = 0.259312860

gm1 = 3.853120296

gm2 = gm3 = 1

gm4 = 6.737369365

4, γ

c1 ÷ c4 = 0.254989284

gm1 = 2.087132391

gm2 = gm3 = 1

gm4 = 3.070780283

5, γ

c1 ÷ c4 = 0.253152848

gm1 = 1.249766377

gm2 = gm3 = 1

gm4 = 0.949165995

6, α

c= 1.010512166

gm1 = 1.010512166

gm2 = 1

11

2.993

Table 8.80 Element values of the cascaded Gm-C of Equi-rip-td_Z group delay filters (amin = 45 dB, δ = 5%) n

fs

Cell no., type

3

4.245

1, γ

c1 ÷ c4 = 0.356265364

gm1 = 6.059744031

gm2 = gm3 = 1

2, α

c= 1.237266708

gm1 = 1.237266708

gm2 = 1

1, γ

c1 ÷ c4 = 0.400673699

gm1 = 3.972172175

gm2 = gm3 = 1

2, β

c1 = 0.677955753

c2 = 1.627197370

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.361387374

gm1 = 5.764920958

gm2 = gm3 = 1

gm4 = 3.901761263

2, γ

c1 ÷ c4 = 0.303237257

gm1 = 1.867093667

gm2 = gm3 = 1

gm4 = 0.940393152

2, α

c= 1.181042655

gm1 = 1.181042655

gm2 = 1

1, γ

c1 ÷ c4 = 0.399337217

gm1 = 4.694303224

gm2 = gm3 = 1

gm4 = 6.185685070

2, γ

c1 ÷ c4 = 0.333919011

gm1 = 2.002746378

gm2 = gm3 = 1

gm4 = 1.848293561

2, β

c1 = 0.644585771

c2 = 1.536464545

gm1 ÷ gm4 = 1

4

5

6

3.231

3.063

2.897

gm4 = 1.059003818

gm4 = 2.200233146

(continued)

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

283

Table 8.80 (continued) n

fs

Cell no., type

7

3.096

1, γ

c1 ÷ c4 = 0.352197532

gm1 = 7.997757673

gm2 = gm3 = 1

gm4 = 9.065541757

2, γ

c1 ÷ c4 = 0.293990658

gm1 = 2.738292805

gm2 = gm3 = 1

gm4 = 3.156558264

3, γ

c1 ÷ c4 = 0.282663992

gm1 = 1.640153626

gm2 = gm3 = 1

gm4 = 0.939243428

4, α

c= 1.120679231

gm1 = 1.120679231

gm2 = 1

1, γ

c1 ÷ c4 = 0.396150632

gm1 = 7.260480416

gm2 = gm3 = 1

gm4 = 12.55052278

2, γ

c1 ÷ c4 = 0.330359598

gm1 = 2.952181629

gm2 = gm3 = 1

gm4 = 4.875429841

3, γ

c1 ÷ c4 = 0.316997189

gm1 = 1.785790548

gm2 = gm3 = 1

gm4 = 1.822754112

4, β

c1 = 0.625938362

c2 = 1.483047481

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.347640951

gm1 = 11.68054993

gm2 = gm3 = 1

gm4 = 16.64710205

2, γ

c1 ÷ c4 = 0.289736985

gm1 = 4.570192732

gm2 = gm3 = 1

gm4 = 7.011787704

3, γ

c1 ÷ c4 = 0.277695329

gm1 = 2.475930854

gm2 = gm3 = 1

gm4 = 3.079472404

4, γ

c1 ÷ c4 = 0.273603096

gm1 = 1.556584546

gm2 = gm3 = 1

gm4 = 0.944467228

5, α

c= 1.090110338

gm1 = 1.090110338

gm2 = 1

1, γ

c1 ÷ c4 = 0.394023942

gm1 = 11.38201585

gm2 = gm3 = 1

gm4 = 21.36090234

2, γ

c1 ÷ c4 = 0.328273535

gm1 = 4.894158709

gm2 = gm3 = 1

gm4 = 9.570965855

3, γ

c1 ÷ c4 = 0.314401837

gm1 = 2.651563953

gm2 = gm3 = 1

gm4 = 4.716184771

4, γ

c1 ÷ c4 = 0.309395762

gm1 = 1.688270928

gm2 = gm3 = 1

gm4 = 1.825351795

5, b

c1 = 0.615083422

c2 = 1.451298597

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.344952952

gm1 = 16.82526886

gm2 = gm3 = 1

gm4 = 26.69450016

2, γ

c1 ÷ c4 = 0.287319837

gm1 = 7.303997429

gm2 = gm3 = 1

gm4 = 12.55600653

3, γ

c1 ÷ c4 = 0.275045597

gm1 = 4.136587881

gm2 = gm3 = 1

gm4 = 6.737369368

4, γ

c1 ÷ c4 = 0.270459706

gm1 = 2.363564498

gm2 = gm3 = 1

gm4 = 3.070780283

5, γ

c1 ÷ c4 = 0.268511852

gm1 = 1.510871818

gm2 = gm3 = 1

gm4 = 0.949165995

6, α

c= 1.071820820

gm1 = 1.071820820

gm2 = 1

8

9

10

11

2.879

3.116

2.877

3.128

284

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.81 Element values of the cascaded Gm-C of Equi-rip-td_Z group delay filters (amin = 50 dB, δ = 5%) n

fs

Cell no., type

3

5.019

1, γ

c1 ÷ c4 = 0.364274653

gm1 = 8.874641104

gm2 = gm3 = 1

2, α

c= 1.265082003

gm1 = 1.265082003

gm2 = 1

1, γ

c1 ÷ c4 = 0.411488895

gm1 = 5.229787068

gm2 = gm3 = 1

2, β

c1 = 0.696255492

c2 = 1.671119539

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.382176767

gm1 = 7.300251678

gm2 = gm3 = 1

gm4 = 3.901761263

2, γ

c1 ÷ c4 = 0.320681471

gm1 = 2.290622258

gm2 = gm3 = 1

gm4 = 0.940393152

2, α

c= 1.248984042

gm1 = 1.248984042

gm2 = 1

1, γ

c1 ÷ c4 = 0.414464218

gm1 = 5.564569240

gm2 = gm3 = 1

gm4 = 6.185685070

2, γ

c1 ÷ c4 = 0.346567952

gm1 = 2.359693345

gm2 = gm3 = 1

gm4 = 1.848293559

2, β

c1 = 0.669002852

c2 = 1.594666232

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.372513979

gm1 = 9.439367221

gm2 = gm3 = 1

gm4 = 9.065541757

2, γ

c1 ÷ c4 = 0.310949453

gm1 = 3.070612298

gm2 = gm3 = 1

gm4 = 3.156558264

3, γ

c1 ÷ c4 = 0.298969411

gm1 = 1.957166507

gm2 = gm3 = 1

gm4 = 0.939243428

4, α

c= 1.185325400

gm1 = 1.185325400

gm2 = 1

1, γ

c1 ÷ c4 = 0.411482617

gm1 = 7.909301561

gm2 = gm3 = 1

gm4 = 12.55052278

2, γ

c1 ÷ c4 = 0.343145312

gm1 = 3.304325760

gm2 = gm3 = 1

gm4 = 4.875429841

3, γ

c1 ÷ c4 = 0.329265745

gm1 = 2.106388745

gm2 = gm3 = 1

gm4 = 1.822754113

4, β

c1 = 0.650163687

c2 = 1.540444997

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.367458493

gm1 = 12.96102405

gm2 = gm3 = 1

gm4 = 16.64710204

2, γ

c1 ÷ c4 = 0.306253668

gm1 = 4.914940520

gm2 = gm3 = 1

gm4 = 7.011787705

3, γ

c1 ÷ c4 = 0.293525568

gm1 = 2.784474812

gm2 = gm3 = 1

gm4 = 3.079472404

4, γ

c1 ÷ c4 = 0.289200053

gm1 = 1.855092341

gm2 = gm3 = 1

gm4 = 0.944467228

5, α

c= 1.152252926

gm1 = 1.152252926

gm2 = 1

4

5

6

7

8

9

3.603

3.208

3.046

3.213

3.030

3.236

gm4 = 1.059003817

gm4 = 2.200233145

(continued)

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

285

Table 8.81 (continued) n

fs

Cell no., type

10

3.019

1, γ

c1 ÷ c4 = 0.409559522

gm1 = 11.90388655

gm2 = gm3 = 1

gm4 = 21.36090234

2, γ

c1 ÷ c4 = 0.341216707

gm1 = 5.215386643

gm2 = gm3 = 1

gm4 = 9.570965854

3, γ

c1 ÷ c4 = 0.326798076

gm1 = 2.967390517

gm2 = gm3 = 1

gm4 = 4.716184771

4, γ

c1 ÷ c4 = 0.321594621

gm1 = 1.996246219

gm2 = gm3 = 1

gm4 = 1.825351795

5, b

c1 = 0.639334937

c2 = 1.508520412

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.364448426

gm1 = 17.96369278

gm2 = gm3 = 1

gm4 = 26.69450015

2, γ

c1 ÷ c4 = 0.303558099

gm1 = 7.651535590

gm2 = gm3 = 1

gm4 = 12.55600653

3, γ

c1 ÷ c4 = 0.290590164

gm1 = 4.453679304

gm2 = gm3 = 1

gm4 = 6.737369368

4, γ

c1 ÷ c4 = 0.285745095

gm1 = 2.662550918

gm2 = gm3 = 1

gm4 = 3.070780283

5, γ

c1 ÷ c4 = 0.283687155

gm1 = 1.799641951

gm2 = gm3 = 1

gm4 = 0.949165995

6, α

c= 1.132396196

gm1 = 1.132396196

gm2 = 1

11

3.249

Table 8.82 Element values of the cascaded Gm-C of Equi-rip-td_Z group delay filters (amin = 55 dB, δ = 5%) n

fs

Cell no., type

3

5.979

1, γ

c1 ÷ c4 = 0.369904214

gm1 = 13.00106836

gm2 = gm3 = 1

2, α

c= 1.284632792

gm1 = 1.284632792

gm2 = 1

1, γ

c1 ÷ c4 = 0.420405728

gm1 = 6.908129136

gm2 = gm3 = 1

2, β

c1 = 0.711343127

c2 = 1.707332168

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.401601437

gm1 = 9.250248760

gm2 = gm3 = 1

gm4 = 3.901761263

2, γ

c1 ÷ c4 = 0.336980557

gm1 = 2.819889295

gm2 = gm3 = 1

gm4 = 0.940393152

2, α

c= 1.312465409

gm1 = 1.312465409

gm2 = 1

1, γ

c1 ÷ c4 = 0.428984869

gm1 = 6.667099982

gm2 = gm3 = 1

gm4 = 6.185685070

2, γ

c1 = 0.692441201

c2 = 1.650534968

gm1 ÷ gm4 = 1

gm4 = 1.848293560

2, β

c1 = 0.692441201

c2 = 1.650534968

gm1 ÷ gm4 = 1

4

5

6

4.047

3.383

3.193

gm4 = 1.059003817

gm4 = 2.200233146

(continued)

286

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.82 (continued) n

fs

Cell no., type

7

3.326

1, γ

c1 ÷ c4 = 0.392406818

gm1 = 11.30194990

gm2 = gm3 = 1

gm4 = 9.065541757

2, γ

c1 ÷ c4 = 0.327554648

gm1 = 3.448639542

gm2 = gm3 = 1

gm4 = 3.156558263

3, γ

c1 ÷ c4 = 0.314934853

gm1 = 2.306134498

gm2 = gm3 = 1

gm4 = 0.939243428

4, α

c= 1.248623661

gm1 = 1.248623661

gm2 = 1

1, γ

c1 ÷ c4 = 0.426059051

gm1 = 8.782185201

gm2 = gm3 = 1

gm4 = 12.55052278

2, γ

c1 ÷ c4 = 0.355300953

gm1 = 3.683766699

gm2 = gm3 = 1

gm4 = 4.875429844

3, γ

c1 ÷ c4 = 0.340929714

gm1 = 2.453577857

gm2 = gm3 = 1

gm4 = 1.822754113

4, β

c1 = 0.673195202

c2 = 1.595013997

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.386990716

gm1 = 14.67803450

gm2 = gm3 = 1

gm4 = 16.64710205

2, γ

c1 ÷ c4 = 0.322532553

gm1 = 5.278382565

gm2 = gm3 = 1

gm4 = 7.011787703

3, γ

c1 ÷ c4 = 0.309127892

gm1 = 3.113675882

gm2 = gm3 = 1

gm4 = 3.079472405

4, γ

c1 ÷ c4 = 0.304572456

gm1 = 2.185370946

gm2 = gm3 = 1

gm4 = 0.944467229

5, α

c= 1.213500825

gm1 = 1.213500825

gm2 = 1

1, γ

c1 ÷ c4 = 0.424252119

gm1 = 12.57653335

gm2 = gm3 = 1

gm4 = 21.36090234

2, γ

c1 ÷ c4 = 0.353457563

gm1 = 5.580102838

gm2 = gm3 = 1

gm4 = 9.570965855

3, γ

c1 ÷ c4 = 0.338521677

gm1 = 3.317540954

gm2 = gm3 = 1

gm4 = 4.716184771

4, γ

c1 ÷ c4 = 0.333131552

gm1 = 2.327880125

gm2 = gm3 = 1

gm4 = 1.825351794

5, b

c1 = 0.662270530

c2 = 1.562637289

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.383715000

gm1 = 19.50242277

gm2 = gm3 = 1

gm4 = 26.69450016

2, γ

c1 ÷ c4 = 0.319605704

gm1 = 8.028360292

gm2 = gm3 = 1

gm4 = 12.55600653

3, γ

c1 ÷ c4 = 0.305952220

gm1 = 4.797037834

gm2 = gm3 = 1

gm4 = 6.737369368

4, γ

c1 ÷ c4 = 0.300851016

gm1 = 2.980154949

gm2 = gm3 = 1

gm4 = 3.070780282

5, γ

c1 ÷ c4 = 0.298684283

gm1 = 2.119077313

gm2 = gm3 = 1

gm4 = 0.949165996

6, α

c= 1.192260347

gm1 = 1.192260347

gm2 = 1

8

9

10

11

3.172

3.348

3.163

3.362

8.4 Filters with Maximum Number of Transmission Zeros at the ω-Axis

287

Table 8.83 Element values of the cascaded Gm-C of Equi-rip-td_Z group delay filters (amin = 60 dB, δ = 5%) n

fs

Cell no., type

3

7.161

1, γ

c1 ÷ c4 = 0.373825302

gm1 = 19.06845992

gm2 = gm3 = 1

2, α

c= 1.298250258

gm1 = 1.298250258

gm2 = 1

1, γ

c1 ÷ c4 = 0.427609231

gm1 = 9.145762870

gm2 = gm3 = 1

2, β

c1 = 0.723531738

c2 = 1.736586697

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.419189125

gm1 = 11.71388392

gm2 = gm3 = 1

gm4 = 3.901761264

2, γ

c1 ÷ c4 = 0.351738245

gm1 = 3.484861155

gm2 = gm3 = 1

gm4 = 0.940393151

2, α

c= 1.369943369

gm1 = 1.369943369

gm2 = 1

1, γ

c1 ÷ c4 = 0.442934351

gm1 = 8.033296856

gm2 = gm3 = 1

gm4 = 6.185685070

2, γ

c1 ÷ c4 = 0.370374195

gm1 = 3.256604012

gm2 = gm3 = 1

gm4 = 1.848293560

2, β

c1 = 0.714957604

c2 = 1.704206111

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.411724862

gm1 = 13.63156114

gm2 = gm3 = 1

gm4 = 9.065541757

2, γ

c1 ÷ c4 = 0.343680044

gm1 = 3.910389218

gm2 = gm3 = 1

gm4 = 3.156558264

3, γ

c1 ÷ c4 = 0.330438981

gm1 = 2.681096085

gm2 = gm3 = 1

gm4 = 0.939243428

4, α

c= 1.310092948

gm1 = 1.310092948

gm2 = 1

1, γ

c1 ÷ c4 = 0.440078545

gm1 = 9.942626385

gm2 = gm3 = 1

gm4 = 12.55052278

2, γ

c1 ÷ c4 = 0.366992148

gm1 = 4.081520181

gm2 = gm3 = 1

gm4 = 4.875429842

3, γ

c1 ÷ c4 = 0.352148023

gm1 = 2.832593788

gm2 = gm3 = 1

gm4 = 1.822754113

4, β

c1 = 0.695346723

c2 = 1.647498011

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.405696324

gm1 = 16.90432882

gm2 = gm3 = 1

gm4 = 16.64710205

2, γ

c1 ÷ c4 = 0.338122508

gm1 = 5.664330362

gm2 = gm3 = 1

gm4 = 7.011787704

3, γ

c1 ÷ c4 = 0.324069918

gm1 = 3.473364858

gm2 = gm3 = 1

gm4 = 3.079472403

4, γ

c1 ÷ c4 = 0.319294290

gm1 = 2.544587011

gm2 = gm3 = 1

gm4 = 0.944467229

5, α

c= 1.272156679

gm1 = 1.272156679

gm2 = 1

4

5

6

7

8

9

4.572

3.598

3.352

3.431

3.310

3.455

gm4 = 1.059003818

gm4 = 2.200233146

(continued)

288

8 Element Values of Cascaded Gm-C and Two-Phase Gm-C Filters

Table 8.83 (continued) n

fs

Cell no., type

10

3.299

1, γ

c1 ÷ c4 = 0.438305474

gm1 = 13.46824901

gm2 = gm3 = 1

gm4 = 21.36090234

2, γ

c1 ÷ c4 = 0.365165848

gm1 = 5.985714821

gm2 = gm3 = 1

gm4 = 9.570965851

3, γ

c1 ÷ c4 = 0.349735210

gm1 = 3.691382814

gm2 = gm3 = 1

gm4 = 4.716184771

4, γ

c1 ÷ c4 = 0.344166538

gm1 = 2.687271173

gm2 = gm3 = 1

gm4 = 1.825351795

5, b

c1 = 0.684208246

c2 = 1.614399661

gm1 ÷ gm4 = 1

1, γ

c1 ÷ c4 = 0.402660406

gm1 = 21.54640472

gm2 = gm3 = 1

gm4 = 26.69450016

2, γ

c1 ÷ c4 = 0.335385801

gm1 = 8.425527434

gm2 = gm3 = 1

gm4 = 12.55600653

3, γ

c1 ÷ c4 = 0.321058194

gm1 = 5.159465402

gm2 = gm3 = 1

gm4 = 6.737369366

4, γ

c1 ÷ c4 = 0.315705125

gm1 = 3.321549491

gm2 = gm3 = 1

gm4 = 3.070780283

5, γ

c1 ÷ c4 = 0.313431413

gm1 = 2.468006202

gm2 = gm3 = 1

gm4 = 0.949165995

6, α

c= 1.251126578

gm1 = 1.251126578

gm2 = 1

11

3.469

Reference 1. Litovski VB (2019) Electronic filters, theory, numerical receipts and design practice using the RM software. Springer, New Delhi

Index

A Active filters, 7 All-pass, 66 Antialiasing filter, 111 Approximation interval, 121 Arithmetically symmetrical band-pass filters, 5, 125 Attenuation characteristic, 170

B Band-pass, 53, 63 Band-pass biquad, 54 Band-pass filter, 4, 65 Band-stop, 132 Band-stop biquad, 56 Band-stop filter, 4, 29, 61 Baseband (low-pass) filter, 34 Base-band signal, 4 Bessel filters, 264 Bilinear transform, 3 Biquad, 28 Bisection, 116 Broadband, 111 Brune’s cell, 82, 83 Bulk Acoustic Waves (BAW), 3 Butterworth_Z filters, 151, 209 Butterworth filters, 161

C Carrier frequency, 4 Cascaded Gm-C filter, 46, 70, 112 Cascaded network, 23 Cascade realization, 60 Cascade synthesis, 21

Catalogue, 151 Central frequency, 48, 146 Chebyshev filters, 154, 162 Circuit synthesis, 81 Common centroid structure, 10 Complex cell, 60 Complex coefficients, 34 Complex frequency plane, 58 Complexity, 60 Complex transmission zero, 103 Complex zero, 58 Coupling transconductance, 66 Current source, 18 Cut-off frequency, 13

D Decomposition of the transfer function, 25, 46 Depletion capacitance, 9 Differential Input Differential Output (DIDO) transconductor, 15 Differential Input Single-Ended Output transconductor (DISEO), 15 Differential pair, 18 Diffusion, 10 Direct conversion, 4 Distortions, 24

E Equi ripple group delay filters, 151 Equi-rip-td_Z filters, 151, 276 Equivalent (input referred) noise, 136

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 V. Litovski, Gm-C Filter Synthesis for Modern RF Systems, Lecture Notes in Electrical Engineering 807, https://doi.org/10.1007/978-981-16-6561-5

289

290 F First order cell, 29, 66 First order two-phase Gm-C cell, 67 First order zero at the σ-axis, 57 Floating resistor, 15 Floating simulated inductor, 79

G Gain Bandwidth Product (GBW), 12 General second order cell, 25 Grounded resistor, 15 Gyrator, 80

H Halpern_Z filters, 151, 198 Halpern filters, 159 Heterodyne receiver, 4 High-pass, 49, 54 High-pass biquad, 55 High-pass filter, 38 Hilbert transformer, 4, 37, 127 Hurwitz polynomial, 175

I Ideal grounded transformer, 80 IIR filters, 3 Impulse radio, 111 Inductor, 77 In-phase, 37 Input referred noise, 138 Integrated capacitor, 9 Integrated circuit design, 141 Integrated planar inductor, 11 Integrated resistor, 8 Intermediate Frequency (IF), 4 Inter-symbol interference, 111 Inverse Chebyshev filters, 151, 210

L Legendre filters, 157 Limited accuracy, 131 Linearity, 47 Linear phase, 38 Low-IF receiver, 4 Low-Noise Amplifier (LNA), 3 Low-pass, 48, 51 Lowpass-to-bandpass transformation, 111 LSM_Z filter, 61, 151, 185 LSM filters, 155, 156

Index M Macro-model, 18 Maximally flat, 124 Maximally flat filters, 168 Maximally flat group delay filters, 168 Micromodel, 144 Microwave communication systems, 111 Miller capacitance, 12 Mixer, 12 Mixing, 4 Modified elliptic filter, 38, 151, 221 Monte Carlo simulation, 132 MOS capacitor, 9 MOS technology, 8

N Negative frequencies, 40 Net-list, 85 Noise, 34, 146 Noise band-width, 135 Noise figure, 137 Noise macro-model, 137 Noise model, 145 Nonlinear distortions, 60 Normalization, 152 Notch biquad, 56 Notch cell, 66 Numerical error, 22

O Open loop gain, 13 Operational Amplifier (OA), 13 Operational transconductance amplifier (OTA), 14, 15 Optimal filters, 157 Order of extraction, 22 Oscillator, 12 Oxide thickness, 10

P Papoulis_Z filters, 151, 187 Papoulis filters, 157 Parallel synthesis, 37 Partial fraction, 140 Phase margin, 13 PI-cell, 82, 101 Pink noise, 4 P-n junction, 9 Polynomial filter, 175 Polynomial long division, 26 Poly-phase filtering, 24

Index Polyphase filters, 34 Post-layout activity, 138 Power consumption, 142 Power spectral density, 135 Prototype low-pass filter, 152 Q Quadrature demodulation, 34 Quadrature-phase, 37 Q-factor, 12 R Receiver, 3 Relative bandwidth, 31 Residue, 29 RF filter, 3 RM software for filter design, 29, 45, 77, 131, 154 S Salen-and-Key, 22 Saturation, 60 Second order cell, 29, 117, 140 Second order low-pass two-phase Gm-C cell, 68 Selectivity, 175 Sheet resistance, 8 Signals, 1 Signal to noise ratio, 131 Simulated grounded inductor, 81 Simulated inductance, 77 Single-phase filter, 24 Software defined radio, 3 Spectral amplitude, 136 SPICE, 78 SPICE simulation, 31, 91, 105, 122, 126, 127, 153, 276 Statistical tolerance analysis, 5 Steady state, 128 Stopband, 61 Structure, 60 Summing amplifier, 29 Surface Acoustic Waves (SAW), 3 Switch, 10 System bandwidth, 135 T Telecommunication, 1 Telephony, 1 Temperature coefficient, 14 Tesla, 1

291 Thin oxide, 9 Thomson_Z filters, 151, 264, 265 Thomson filters, 168, 169 Time domain, 127 Time domain response, 95 Transceiver, 3 Transconductor, 7 Transfer function, 1, 61 Transformation, 77 Transient, 96 Transimpedance, 136 Transistor, 142 Transition band, 175 Transition region, 70 Transmission gate, 10 Transmission lines, 1 Transmission zero at the real axis, 82 Transmission zeros, 30, 38 Transmitter, 3 Turn ratio, 82 Two-phase cascaded Gm-C filter, 70 Two-phase cell, 69 Two phase (complex) filter, 124 Two-phase floating inductor, 100 Two-phase parallel filter, 37 Two-phase (polyphase) filters, 24

U Ultra-Wide-Band (UWB) systems, 121 Unity gain amplifier, 45

V Video signal processors, 3 Voltage Controlled Current Source (VCCS), 14 Voltage-Controlled Voltage Source (VCVS), 14

W Waveguide, 3 White noise, 144 Wideband, 124 Wireless, 1 Wireless sensor networks, 3 Worst-case tolerance, 131

Z Zero at the real axis, 66, 101 Zero-IF receiver, 34 Zolotarev filters, 262