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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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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
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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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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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
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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
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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
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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.
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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.
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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
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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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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)
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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)
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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.
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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]
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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)
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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.
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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
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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
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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).
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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
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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
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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
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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
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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
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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
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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
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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)
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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
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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)
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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
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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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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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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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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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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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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