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Passive Network Synthesis: Advances With Inerter
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Passive Network Synthesis: Advances With Inerter
Michael Z. Q. Chen, Kai Wang, Guanrong Chen
After the invention of a new mechanical element called 'inerter' in 2002, research interest in passive network synthesis has been revived and this field has again become active and essential.
The unique compendium highlights the synthesis of passive electrical or mechanical networks, which is motivated by the vibration control based on a new type of mechanical elements named inerter. It introduces important fundamental concepts of passive network synthesis, and presents recent results on this topic.
These new results concern mainly the economical realizations of lowdegree functions as RLC networks (damperspringinerter networks), the synthesis of nport resistive networks, and the synthesis of lowcomplexity mechanical networks. They can be directly applied to the optimization and design of various inerterbased mechanical control systems, such as suspension systems, vibration absorbers, building vibration systems, etc.
This useful reference text provides important methodologies and results for researchers in the fields of circuit theory, vibration system control, passive systems, control theory, and electrical engineering.
The unique compendium highlights the synthesis of passive electrical or mechanical networks, which is motivated by the vibration control based on a new type of mechanical elements named inerter. It introduces important fundamental concepts of passive network synthesis, and presents recent results on this topic.
These new results concern mainly the economical realizations of lowdegree functions as RLC networks (damperspringinerter networks), the synthesis of nport resistive networks, and the synthesis of lowcomplexity mechanical networks. They can be directly applied to the optimization and design of various inerterbased mechanical control systems, such as suspension systems, vibration absorbers, building vibration systems, etc.
This useful reference text provides important methodologies and results for researchers in the fields of circuit theory, vibration system control, passive systems, control theory, and electrical engineering.
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Año:
2020
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World Scientific Publishing Co. Pte. Ltd.
Idioma:
english
Páginas:
254
ISBN 10:
9811210888
ISBN 13:
9789811210884
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Passive Network Synthesis Advances with Inerter 11567_9789811210877_TP.indd 1 27/8/19 5:09 PM b2530 International Strategic Relations and China’s National Security: World at the Crossroads This page intentionally left blank b2530_FM.indd 6 01Sep16 11:03:06 AM Passive Network Synthesis Advances with Inerter Michael Z Q Chen Nanjing University of Science and Technology, China Kai Wang Jiangnan University, China Guanrong Chen City University of Hong Kong, China World Scientific NEW JERSEY • LONDON 11567_9789811210877_TP.indd 2 • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI • TOKYO 27/8/19 5:09 PM Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library CataloguinginPublication Data A catalogue record for this book is available from the British Library. PASSIVE NETWORK SYNTHESIS Advances with Inerter Copyright © 2020 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN 9789811210877 For any available supplementary material, please visit https://www.worldscientific.com/worldscibooks/10.1142/11567#t=suppl Printed in Singapore Steven  11567  Passive Network Synthesis.indd 1 270819 4:00:47 PM August 15, 2019 11:32 wsbook9x6 Passive Network Synthesis: Advances with Inerter 11567main To the memory of Professor Rudolf E. Kalman (1930–2016) Professor Rudolf E.; Kalman and Dr. Michael Z. Q. Chen at Professor Kalman’s residence, Zurich, Switzerland, on July 12, 2014 v page v b2530 International Strategic Relations and China’s National Security: World at the Crossroads This page intentionally left blank b2530_FM.indd 6 01Sep16 11:03:06 AM August 15, 2019 11:32 wsbook9x6 Passive Network Synthesis: Advances with Inerter 11567main Preface This book is concerned with the synthesis of passive electrical or mechanical networks, which is motivated by the vibration control based on a new type of mechanical elements named inerter. The inerter was proposed by Professor Malcolm Smith from the University of Cambridge in 2002, which is a twoterminal passive element whose force applied at the terminals are proportional to the relative acceleration. Since any nport passive electrical network can be constructed with resistors, inductors, capacitors, and transformers, where transformers can be avoided for the oneport case, the “birth” of inerters completes the analogy between the passive mechanical networks and electrical networks under the “forcecurrent” framework, where dampers, springs, inerters, and levers are analogous to resistors, inductors, capacitors, and transformers, respectively. As a result, the analysis and synthesis of electrical networks can be transplanted into those of mechanical networks. For instance, any positivereal impedance (or admittance) can be realized as a oneport mechanical network consisting of dampers, springs, and inerters (damperspringinerter network), by properly following a network synthesis procedure, such as the BottDuffin synthesis. To date, many investigations have focused on applying damperspringinerter networks to a series of passive or semiactive vibration control systems, such as vehicle suspensions, train suspensions, building vibration systems, landing gears, wind turbines, etc. The passive mechanical networks are actually positivereal controllers of the vibration control systems, and system performances can certainly be enhanced by introducing inerters. For the design of inerterbased vibration control systems, the first step is to determine the transfer function of the controller according to the performance requirements, which is usually the mechanical impedance vii page vii August 15, 2019 11:32 viii wsbook9x6 Passive Network Synthesis: Advances with Inerter 11567main Passive Network Synthesis: Advances with Inerter (or admittance); the second step is to realize the resulting function as a damperspringinerter network based on the theory of passive network synthesis. Although the realization as a oneport damperspringinerter network is always available, the classical synthesis methods and results always generate many redundant elements, and the synthesis problem of damperspringinerter networks using the least number of elements is far from being completely solved. Unlike electrical systems, the number of elements is a critical index to be considered for mechanical systems due to the limitation of space, weight, cost, etc. As a consequence, the significance of investigating passive network synthesis has become appealing again, and many investigations have focused on this topic motivated by the inerterbased vibration control. Notably, Professor Rudolf E. Kalman has made an independent call for the renewed investigation on passive network synthesis. In addition, this topic can provide important impacts on electronic engineering, biometric image processing, control theory, etc. In this book, some important fundamental concepts of passive network synthesis are introduced, and some recent results by the authors on this topic are presented. These results are mainly concerned with the economical realizations of lowdegree functions as RLC networks (damperspringinerter networks), the synthesis of nport resistive networks, and the synthesis of lowcomplexity mechanical networks. Many of these results can be directly applied to the optimization and design of various inerterbased vibration control systems. In Chapter 1, the development of passive network synthesis and its application to inerterbased vibration control are introduced. In addition, an outline of this book is presented. In Chapter 2, some important fundamental results of passive network synthesis are introduced, including the properties of positiverealness, some classical synthesis procedures, and graph theory for passive networks. Some concepts and results will be utilized in the following chapters. In Chapter 3, the realization problem of biquadratic impedances as RLC networks is discussed. Since the biquadratic synthesis is a classical problem in network synthesis, electrical circuits are utilized to describe the networks. In Chapter 4, the synthesis of nport resistive networks is discussed, including a review of the investigations on this problem, and some recent results obtained by the authors. In Chapter 5, the synthesis problems of lowcomplexity mechanical networks are investigated, where the number of dampers and inerters is also an important index to be considered in addition to the total number of elements. Finally, Chapter 6 presents the summary of this book. page viii August 15, 2019 11:32 wsbook9x6 Passive Network Synthesis: Advances with Inerter Preface 11567main ix This book is readable by graduate students and researchers in related fields. Some basic knowledge of mathematics, circuit theory, and control theory is needed to follow the content in this book. The authors are very grateful to those who have contributed to the development of this book, and to the editorial office of the publishing company. Michael Z. Q. Chen, Kai Wang, and Guanrong Chen page ix b2530 International Strategic Relations and China’s National Security: World at the Crossroads This page intentionally left blank b2530_FM.indd 6 01Sep16 11:03:06 AM August 15, 2019 11:32 wsbook9x6 Passive Network Synthesis: Advances with Inerter 11567main Acknowledgments This work is supported by the National Natural Science Foundation of China under grants 61873129 and 61703184, and by the Hong Kong Research Grants Council under several GRF grants in the 1990s. xi page xi b2530 International Strategic Relations and China’s National Security: World at the Crossroads This page intentionally left blank b2530_FM.indd 6 01Sep16 11:03:06 AM August 15, 2019 11:32 wsbook9x6 Passive Network Synthesis: Advances with Inerter 11567main Contents Preface vii Acknowledgments xi 1. 2. Introduction 1 1.1 1.2 1.3 1 4 9 Preliminaries of Passive Network Synthesis 2.1 2.2 2.3 2.4 2.5 2.6 2.7 3. Synthesis of Passive Networks . . . . . . . . . . . . . . . . New Research Motivation: InerterBased Mechanical Control Outline of the Book . . . . . . . . . . . . . . . . . . . . . PositiveReal Function and Foster Preamble Synthesis of OnePort Lossless Networks . . The Brune Synthesis . . . . . . . . . . . . . The BottDuffin Synthesis . . . . . . . . . . The Darlington Synthesis . . . . . . . . . . Graph Theory for Passive Networks . . . . . Principle of Duality . . . . . . . . . . . . . . 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 15 16 19 25 27 33 Biquadratic Synthesis of OnePort RLC Networks 37 3.1 3.2 3.3 3.4 37 38 41 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . Basic Notations and Results . . . . . . . . . . . . . . . . . A Canonical Biquadratic Impedance . . . . . . . . . . . . Realizations of Biquadratic Impedances with No More than Four Elements . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Realizations with No More than Three Elements . 3.4.2 Realizations with Four Elements . . . . . . . . . . xiii 43 43 44 page xiii August 15, 2019 11:32 xiv Passive Network Synthesis: Advances with Inerter 11567main Passive Network Synthesis: Advances with Inerter 3.5 3.6 3.7 3.8 4. wsbook9x6 Realization of Biquadratic Impedances as FiveElement Bridge Networks . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Preliminary Lemmas . . . . . . . . . . . . . . . . 3.5.2 FiveElement Bridge Networks with Two Reactive Elements of the Same Type . . . . . . . . . . . . . 3.5.3 FiveElement Bridge Networks with One Inductor and One Capacitor . . . . . . . . . . . . . . . . . 3.5.4 Main Results . . . . . . . . . . . . . . . . . . . . . Generalized Synthesis without RealPart Minimization for Biquadratic Impedances . . . . . . . . . . . . . . . . . . . 3.6.1 Preliminary Lemmas . . . . . . . . . . . . . . . . 3.6.2 Biquadratic Impedances with Real Zeros and Arbitrary Poles . . . . . . . . . . . . . . . . . . . 3.6.3 Further Generalization to General Biquadratic Impedances . . . . . . . . . . . . . . . . . . . . . . A Generalized Theorem of Reichert for Biquadratic Minimum Functions . . . . . . . . . . . . . . . . . . . . . SevenElement SeriesParallel Realizations of a Specific Class of Biquadratic Impedances . . . . . . . . . . . . . . 3.8.1 Preliminary Lemmas . . . . . . . . . . . . . . . . 3.8.2 Realizations as ThreeReactive SevenElement SeriesParallel Networks . . . . . . . . . . . . . . . 3.8.3 Realizations as FourReactive SevenElement SeriesParallel Networks . . . . . . . . . . . . . . . 3.8.4 Realizations as FiveReactive SevenElement SeriesParallel Networks . . . . . . . . . . . . . . . 3.8.5 Main Results . . . . . . . . . . . . . . . . . . . . . 53 54 58 67 74 79 79 81 85 89 106 108 110 112 121 126 Synthesis of nPort Resistive Networks 127 4.1 4.2 127 128 128 130 4.3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . A Review of nPort Resistive Network Synthesis . . . . . 4.2.1 Realizations with n ≤ 3 . . . . . . . . . . . . . . . 4.2.2 General Properties of nPort Resistive Networks . 4.2.3 Realizations of Admittance Matrices with n + 1 Terminals . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Realizations of Admittance Matrices with More than n + 1 Terminals . . . . . . . . . . . . . . . . Synthesis of nPort Resistive Networks Containing 2n Terminals . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 138 141 page xiv August 15, 2019 11:32 wsbook9x6 Passive Network Synthesis: Advances with Inerter 11567main Contents 4.3.1 4.4 5. . . . . . . . . . . . . . . . . 142 145 147 148 150 151 157 165 Mechanical Synthesis of LowComplexity OnePort Networks 167 5.1 5.2 167 5.3 5.4 6. A Necessary and Sufficient Condition for Realization . . . . . . . . . . . . . . . . . . . . 4.3.2 Element Value Expressions . . . . . . . . . . . 4.3.3 Numerical Example . . . . . . . . . . . . . . . Minimal Realization of ThreePort Resistive Networks 4.4.1 Minimal Realization with Four Terminals . . . 4.4.2 Realization with at Most Four Elements . . . . 4.4.3 Realization with Five Elements . . . . . . . . . 4.4.4 Some Examples . . . . . . . . . . . . . . . . . xv Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . Realization of a Special Class of Admittances with One Damper, One Inerter, and Finite Springs . . . . . . . . . . 5.2.1 Realizability Conditions when the Impedance of Spring Network Exists . . . . . . . . . . . . . . . . 5.2.2 Final Realization Results . . . . . . . . . . . . . . Realizations of a Special Class of Admittances with Strictly Lower Complexity than Canonical Configurations . . . . . 5.3.1 Cases with Zero Coefficients . . . . . . . . . . . . 5.3.2 Preliminary Lemmas . . . . . . . . . . . . . . . . 5.3.3 Realizations with No More than Four Elements . . 5.3.4 Realizations of FiveElement DamperSpring Networks . . . . . . . . . . . . . . . . . . . . . . . 5.3.5 Realizations of FiveElement DamperSpringInerter Networks . . . . . . . . . . 5.3.6 Final Condition . . . . . . . . . . . . . . . . . . . Synthesis of a OneDamper OneInerter Network Containing No More than Three Springs . . . . . . . . . . 5.4.1 Realizability Conditions under a Particular Assumption . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Final Realization Results . . . . . . . . . . . . . . 5.4.3 Some Examples . . . . . . . . . . . . . . . . . . . Future Outlook Bibliography 172 173 182 187 188 190 192 197 200 208 209 209 219 223 229 231 page xv b2530 International Strategic Relations and China’s National Security: World at the Crossroads This page intentionally left blank b2530_FM.indd 6 01Sep16 11:03:06 AM August 15, 2019 11:32 wsbook9x6 Passive Network Synthesis: Advances with Inerter 11567main Chapter 1 Introduction 1.1 Synthesis of Passive Networks Passive network synthesis is to physically realize a given function describing the port behavior of a passive network using only passive elements, under the assumption that the network and each element must be linear, timeinvariant, and lumped. This assumption will be valid throughout this book. In contrast, network analysis is to determine the external or internal behavior of a given network. Therefore, it can be noted that network synthesis is the inverse process of network analysis. As an important branch of system theory, passive network synthesis was widely investigated from the 1930s to the 1970s. Brune [Brune (1931)] first solved the passive realizability problem of oneport passive networks by establishing a systematic realization procedure, which is named the Brune synthesis. It is shown in [Brune (1931)] that the impedance (resp., admittance) of a oneport passive network is positivereal, and any positivereal impedance (resp., admittance) is realizable as a oneport passive network consisting of a finite number of resistors, inductors, capacitors, and transformers. Darlington [Darlington (1939)] proposed an alternative synthesis procedure, named the Darlington synthesis, through which any positivereal impedance (resp., admittance) can be realized as the cascade connection of a twoport lossless network (containing inductors, capacitors, and transformers) and one resistor. However, it should be noted that transformers are not preferred in practice. Bott and Duffin [Bott and Duffin (1949)] first showed that only resistors, inductors, and capacitors are needed to construct a oneport passive network through establishing a transformerless synthesis procedure, which is named the BottDuffin synthesis. On the other hand, it should be noted that transformers are avoided at the 1 page 1 August 15, 2019 11:32 2 wsbook9x6 Passive Network Synthesis: Advances with Inerter 11567main Passive Network Synthesis: Advances with Inerter cost of increasing the number of redundant elements, although lately some modified approaches were further proposed [Pantell (1954); Reza (1954)]. To solve the synthesis problem of oneport RLC networks using the least number of elements, many investigations focused on the minimal realization problems of lowdegree impedances (resp., admittances), especially biquadratic impedances [Ladenheim (1964); Seshu (1959); Vasiliu (1970)], which have not been completely solved so far. In addition, the synthesis problems of multiport passive networks have also been widely investigated, which can be referred to [Anderson and Vongpanitlerd (1973); Newcomb (1966)]. Nevertheless, it should be noted that transformers cannot always be avoided in the multiport case, even for the simple multiport resistive networks (see Chapter 4). Notably, from the 1970s to the 1990s, the research interest in passive network synthesis declined in spite of some new developments on this topic. After the invention of a new mechanical element called “inerter ” in 2002 [Smith (2002)], research interest in passive network synthesis has been revived and this field has again become active and practically essential, motivated by the design of inerterbased mechanical control (see the next section for more details). Chen and Smith [Chen (2007); Chen and Smith (2009b)] first investigated the lowcomplexity mechanical network synthesis utilizing dampers, springs, and inerters. Then, there have been a series of new results in this field during the past decade (see [Chen et al. (2013a, 2015c); Hughes and Smith (2014); Jiang and Smith (2011)], for instance). Moreover, Kalman [Kalman (2010, 2014); Lin et al. (2011)]1 has made an independent call for the renewed investigation on this topic. The remaining part of this section will introduce some basic concepts and results. Definition 1.1. [Anderson and Vongpanitlerd (1973), pg. 21] Assuming that there is no energy stored at t0 , an nport network is defined to be passive, if Z T ε(T ) = v T (t)i(t)dt ≥ 0, t0 for any t0 , T , and port voltage vector v(·) ∈ Rn and current vector i(·) ∈ Rn satisfying the constraints of the network. 1 Reference [Lin et al. (2011)] is a document summarizing the main content of Professor Rudolf E. Kalman’s lecture for the Berkeley Algebraic Statistics Seminar on October 26, 2011. page 2 August 15, 2019 11:32 wsbook9x6 Passive Network Synthesis: Advances with Inerter Introduction 11567main 3 Moreover, the passivity of elements can be defined based on Definition 1.1, since an element can be regarded as a specific network. It can be checked that resistors, inductors, capacitors, and transformers with constant element values are linear, timeinvariant, lumped, and passive elements. An important result that can be proved by Tellegen’s Theorem [Anderson and Vongpanitlerd (1973), pg. 22] is stated as follows. Theorem 1.1. [Anderson and Vongpanitlerd (1973), pg. 22] Any network consisting of a finite number of passive elements must be a passive network. It should be noted that the converse of Theorem 1.1 is not always true, which means that a passive network can sometimes contain active elements. However, the task of passive network synthesis requires that the passive network to be realized must only contain passive elements. By taking the Laplace transforms, an nport linear, timeinvariant, and lumped network can be described by a transfer function matrix, whose entries are realrational functions. For instance, the impedance matrix Z(s) ∈ Rn×n (s) of an nport network satisfies ˆ V̂ (s) = Z(s)I(s), ˆ are the Laplace transforms where the ndimensional vectors V̂ (s) and I(s) of voltage vector v(t) and current vector i(t), respectively. Similarly, the admittance matrix Y (s) ∈ Rn×n (s) of an nport network satisfies ˆ = Y (s)V̂ (s). I(s) Moreover, some other transfer function matrices, such as scattering matrix and hybrid matrix, can also be utilized to describe the port behavior of a linear, timeinvariant, and lumped network. Remark 1.1. Although impedance and admittance matrices are transfer function matrices that are most commonly used, the impedance or admittance matrices may not exist for some special networks. If the impedance and admittance matrices of a network simultaneously exist, then Z(s) = Y −1 (s). Specifically, for the oneport case, these two matrices are scalars and can be called the impedance and admittance for brevity. Definition 1.2. [Anderson and Vongpanitlerd (1973), pg. 51] A realrational function matrix H(s) is positivereal if H(s) is analytic and H(s) + H ∗ (s) 0 for all s with <(s) > 0, that is, in the open righthalf plane. Here, H ∗ (s) denotes the complex conjugate transpose of H(s), and means nonnegative definite. page 3 August 15, 2019 11:32 4 wsbook9x6 Passive Network Synthesis: Advances with Inerter 11567main Passive Network Synthesis: Advances with Inerter Theorem 1.2. [Anderson and Vongpanitlerd (1973), Sections 2.7 and 2.8] The impedance matrix Z(s) (resp., admittance matrix Y (s)) of an nport passive network is positivereal. Moreover, if the network is reciprocal (see [Newcomb (1966)]), then Z(s) (resp., Y (s)) is symmetric, that is, Z(s) = Z T (s) (resp., Y (s) = Y T (s).) Theorem 1.3. [Anderson and Vongpanitlerd (1973), Chapters 9 and 10] Any positivereal impedance matrix Z(s) ∈ Rn×n (s) (resp., admittance matrix Y (s) ∈ Rn×n (s)) is realizable as an nport passive network consisting of resistors, inductors, capacitors, transformers, and gyrators. Moreover, if the positivereal impedance matrix Z(s) (resp., admittance matrix Y (s)) is symmetric, then it is realizable as an nport passive network consisting of resistors, inductors, capacitors, and transformers. 1.2 New Research Motivation: InerterBased Mechanical Control Since 2002, the invention of a new kind of mechanical elements named “inerter ” and its successful applications has renewed the research interest in passive network synthesis [Smith (2002); Chen et al. (2009)]. The inerter is a twoterminal mechanical element with its terminal dynamics satisfying F = b(v̇1 − v̇2 ), where b is called the inertance, F is the force applied to its two terminals, and v̇1 and v̇2 are the accelerations of the two terminals. The inerter was first proposed and constructed by Professor Malcolm C. Smith from the University of Cambridge. The mechanical model of a rackandpinion inerter is shown in Fig. 1.1. Moreover, there are some other methods of constructions for inerters, such as hydraulics and screw mechanisms [Smith (2008)]. rack terminal 2 Fig. 1.1 pinions gear ﬂywheel terminal 1 Schematic of the mechanical model of a rackandpinion inerter [Smith (2002)]. page 4 August 15, 2019 11:32 wsbook9x6 Passive Network Synthesis: Advances with Inerter Introduction 11567main 5 Based on the conventional forcecurrent analogy framework, which means that force is analogous to current and velocity is analogous to voltage, the passivity of a Rmechanical network can be defined according to DefiniT tion 1.1, that is, t0 v T (t)F (t)dt ≥ 0 for any t0 , T , and external velocity n vector v(·) ∈ R and force vector F (·) ∈ Rn satisfying the constraints of the network. In practice, passive mechanical networks are widely employed in many control systems, such as vehicle suspensions, train suspensions, machine vibration systems, etc. Compared with the active control approach, such a passive control method can have a lower cost and higher reliability. For instance, no power is needed and some serious practical problems such as measurement errors and actuator failures can be avoided. In order to physically realize the passive mechanical network utilizing the conventional passive elements such as springs, dampers, etc., engineers traditionally use the “trial and error” approach for design, which obviously lacks a theoretical foundation. Based on the forcecurrent analogy, if one can find passive mechanical elements that are analogous to the basic passive electrical elements: resistors, inductors, capacitors, and transformers (not necessarily in the oneport case), then the theory of passive electrical network synthesis can be completely transplanted into the design of passive mechanical networks. Conventionally, the damper, spring, and lever in the mechanical system can be analogous to the resistor, inductor, and transformer in the electrical system, respectively. However, for a long period of time, people used a mass to make a partial analogy to a capacitor, which is actually analogous to a grounded capacitor. A summary of the conventional incomplete analogy is presented in Table 1.1. Recalling that the force applied to its two terminals is proportional to the relative acceleration between them, the inerter is acctaully the “missing” mechanical element that is analogous to a capacitor (see Fig. 1.2). Therefore, the “birth” of such an element completes the analogy between passive, linear, timeinvariant, lumped, reciprocal mechanical networks and the electrical ones. As a result, the physical design of passive mechanical networks become much more convenient and systematic by using the theory of passive network synthesis. Specifically, based on the BottDuffin synthesis, any oneport mechanical network can be constructed by using at most three types of elements: dampers, springs, and inerters. To date, the passive mechanical networks with inerters have been applied to a series of passive or semiactive vibration control systems [Chen et al. (2012, 2015a); Hu et al. (2014); Hu and Chen (2015); Papageorgiou and Smith (2006); Smith and Wang (2004); Wang et al. (2009, 2012)], page 5 August 15, 2019 11:32 wsbook9x6 6 Passive Network Synthesis: Advances with Inerter 11567main Passive Network Synthesis: Advances with Inerter Table 1.1 The conventional incomplete forcecurrent analogy between the mechanical and electrical systems, which can be referred to [Smith (2002)]. Mechanical system Electrical system Force Velocity Zero velocity point Damper Spring Mass Lever Mechanical interconnection Current Voltage Zero potential point Resistor Inductor Grounded capacitor Transformer Electrical interconnection Electrical Mechanical F F v2 dF dt F v1 = k(v2 − v1 ) F v1 v2 F = b d(v2dt−v1 ) F F v2 v1 F = c(v2 − v1 ) Y(s) = k s i di dt spring Y(s) = bs i damper v2 i v1 i v2 Y(s) = 1 Ls = L1 (v2 − v1 ) inductor i v1 Y(s) = Cs i = C d(v2dt−v1 ) inerter Y(s) = c v2 i v1 i = R1 (v2 − v1 ) capacitor Y(s) = 1 R resistor Fig. 1.2 The analogy between springs, inerters, and dampers in mechanical networks and inductors, capacitors, and resistors in electrical networks [Smith (2002)]. where the mechanical networks are actually the passive control devices. The results show that introducing inerters can indeed enhance system performances. In [Hu et al. (2014)], a direct comparison idea is proposed to study the influence of adding one element at a specific position for vehicle suspensions, where the performance index for a complex configuration is decoupled as two parts: the part corresponding to the original configuration and the part corresponding to the added element. In [Chen et al. (2014a)], a fundamental property is presented that an inerter can reduce the natural frequencies of vibration systems based on a general multidegreeoffreedom system. In [Hu and Chen (2015)], the inerterbased dynamic vibration absorber (IDVA, also known as inerterbased tuned mass damper) is proposed, and page 6 August 15, 2019 11:32 wsbook9x6 Passive Network Synthesis: Advances with Inerter Introduction 11567main 7 the H∞ and H2 performances for IDVA are evaluated. In [Hu et al. (2015)], inerterbased isolators are proposed, and analytic solutions for the H∞ and H2 performances of several inerterbased isolators are derived. In [Chen et al. (2015a)], the idea of decoupling the inerterbased semiactive suspensions as a passive part and a semiactive part is proposed, and the semiactive suspensions with the passive part as several given inerterbased networks are evaluated. In [Chen et al. (2014c)], the idea of semiactive inerter is proposed. The performance of semiactive inerter for vehicle suspensions is studied in [Chen et al. (2014c, 2016a)]. In [Hu et al. (2017a)], the physical embodiment of semiactive inerter is proposed by using a controllableinertia flywheel. In [Hu et al. (2017b)], the skyhook inerter idea is proposed, and the semiactive realization of the skyhook inerter idea by using semiactive inerters is studied. In [Hu et al. (2018a)], a fundamental fact is revealed that masschain systems with inerters may have multiple natural frequencies, and a necessary and sufficient condition for natural frequency assignment problem of inerterbased masschain systems is derived. In [Dong et al. (2015)], the effect of introducing inerters to suppress the shimmy vibration of aircraft landing gear structures is investigated. In [Liu et al. (2015)], some nonlinearities in the landing gear model with inerters are analyzed. In [Hu and Chen (2017); Hu et al. (2018b)], the inerter is applied in offshore wind turbines for the first time, and its performance is evaluated by using the FAST code developed by the National Renewable Energy Laboratory. Some of the recent results on inerterbased mechanical control are presented in the book [Chen and Hu (2019)]. An illustrative example is shown in Fig. 1.3, which is the suspension control system based on a quartercar model. In this model, ms denotes the sprung mass, mu denotes the unsprung mass, kt denotes the spring stiffness of the tyre, and K(s) denotes the mechanical admittance of a oneport passive network containing inerters. The control diagram of this model is shown in Fig. 1.4, where w is the external input, z is the output to be controlled, and the admittance K(s) is a positivereal controller to be determined such that the control system can meet certain requirements. The complete process of designing an inerterbased vibration control system is summarized as follows. • Given a vibration control system, determine a suitable positivereal admittance or impedance (matrix) K(s) of a mechanical network, which is the system controller or part of the controller. The design process should consider the passivity of the mechanical network or page 7 August 15, 2019 11:32 8 wsbook9x6 %RG\ Passive Network Synthesis: Advances with Inerter 11567main Passive Network Synthesis: Advances with Inerter 6XVSHQVLRQ some further constraints, such as the total number of elements, the structure requirement, etc. network • Using the theory of passive synthesis, realize K(s) as a :KHHO passive mechanical network consisting of dampers, springs, inerters, levers (if necessary), etc. 7\UH As can be seen, the approaches and results of passive network synthesis are essential in the above design. D E Fs %RG\ %RG\ ms zs 6XVSHQVLRQ ks 6XVSHQVLRQ Ks :KHHO :KHHO mu zu 7\UH 7\UH kt zr Fig. 1.3 A quartercar vehicle suspension system model, where ms denotes the sprung mass, mu denotes the unsprung mass, kt denotes the spring stiffness of the tyre, and K(s) denotes the mechanical admittance of a passive network containing dampers, springs, inerters, and possibly levers. z w G(s) F v2v1 K(s) Fig. 1.4 Control diagram for the quartercar vehicle suspension system model, where the mechanical admittance K(s) is a positivereal controller, w is the external input, and z is the output to be controlled. page 8 August 15, 2019 11:32 wsbook9x6 Passive Network Synthesis: Advances with Inerter Introduction 11567main 9 Since it is difficult to implement levers with unrestricted ratios in practice, no lever (transformer) is preferred. Moreover, low complexity is often required for mechanical systems due to the limitation of cost, space and weight. When no transformer (lever) is contained, the synthesis of oneport RLC (damperspringinerter) networks following a classical synthesis procedure yields a large number of redundant elements in many cases, and its minimal realizability problem is unsolved; the synthesis of multiport RLC (damperspringinerter) networks is not solved, even for the multiport resistive networks. As a result, the research interest in passive network synthesis has been renewed, and a series of new results have appeared since the invention of inerters [Chen and Smith (2009b); Chen et al. (2013a); Hughes and Smith (2014); Jiang and Smith (2011)]. In addition to the mechanical control applications, passive network synthesis can also be applied to the field of electronic engineering [Lavaei et al. (2011); Mukhtar et al. (2011)], and can provide longterm impacts on many other areas, such as biometric image processing [Saeed (2014)], passivitypreserving model reduction [Reis and Stykel (2011)], open and interconnected systems [Willems (2007)], etc. 1.3 Outline of the Book This book introduces some recent advances on passive network synthesis, mainly including three important topics in this field: realizability of biquadratic impedances as oneport RLC networks, synthesis of nport resistive networks, and synthesis of lowcomplexity mechanical networks. In Chapter 2, some important preliminaries of passive network synthesis will be reviewed. For the synthesis of oneport passive networks, some basic properties of positivereal functions, and some classical oneport synthesis procedures will be presented. Moreover, some basic results of graph theory for network analysis and synthesis will be presented. Finally, the principle of duality will be explained. In Chapter 3, some recent results on the realizability of biquadratic impedances as oneport RLC networks will be introduced, which are mainly referred to [Chen et al. (2016b, 2017); Wang and Chen (2012); Wang et al. (2014, 2018)]. Biquadratic synthesis of oneport RLC networks has been an important topic in passive network synthesis, where its minimal realizability problem is still unsolved. In Chapter 4, some recent results on the synthesis of nport resistive networks will be introduced, which are mainly referred to [Chen et al. (2015b); Wang and Chen (2015)]. The synthesis of nport resistive networks is another important research topic. page 9 August 15, 2019 11:32 10 wsbook9x6 Passive Network Synthesis: Advances with Inerter 11567main Passive Network Synthesis: Advances with Inerter In Chapter 5, some recent results on the lowcomplexity synthesis of passive mechanical networks will be introduced, which are mainly referred to [Chen and Smith (2009b); Chen et al. (2013a,b, 2015c)]. In addition, examples for mechanical control will be presented. Chapter 6 will present some interesting problems to be further investigated in the field of passive network synthesis. page 10 August 15, 2019 11:32 wsbook9x6 Passive Network Synthesis: Advances with Inerter 11567main Chapter 2 Preliminaries of Passive Network Synthesis 2.1 PositiveReal Function and Foster Preamble This section will give a brief introduction of positivereal functions and the properties, which are referred to [Baher (1984); Brune (1931); Chen and Smith (2009a); Guillemin (1957); Van Valkenburg (1960)]. Definition 2.1. [Baher (1984), pg. 27], [Van Valkenburg (1960), pg. 72] A realrational function H(s) is said to be positivereal if H(s) is analytic and <(H(s)) ≥ 0 for all s with <(s) > 0, that is, in the open righthalf plane. It is obvious that Definition 2.1 is a special case of Definition 1.2. Based on Definition 2.1, the following theorems can be obtained. Theorem 2.1. [Brune (1931)] If H(s) and W (s) are both positivereal functions, then W (H(s)) is positivereal. Theorem 2.2. [Baher (1984), pg. 27] If H1 (s) and H2 (s) are two positivereal functions, then αH1 (s) + βH2 (s) is positivereal for any α > 0 and β > 0. Since 1/s, ks, and s + k are all positivereal functions for any k > 0 by Definition 2.1, the following corollaries of Theorem 2.1 can be directly obtained. Corollary 2.1. If H(s) is a positivereal function, then H −1 (s) and H(s−1 ) are both positivereal. Corollary 2.2. If H(s) is a positivereal function, then αH(βs) + γ is positivereal for any α > 0, β > 0, and γ ≥ 0. 11 page 11 August 15, 2019 11:32 12 wsbook9x6 Passive Network Synthesis: Advances with Inerter 11567main Passive Network Synthesis: Advances with Inerter A classical criterion for testing positiverealness is presented as follows. Theorem 2.3. [Baher (1984), pg. 33] A realrational function H(s) is positivereal if and only if 1. H(s) is analytic for any <(s) > 0; 2. <(H(jω)) ≥ 0 for all ω ∈ R with s = jω not being a pole of H(s); 3. any pole of H(s) on jR ∪ ∞ is simple and have a positive residue. Definition 2.2. [Baher (1984), pp. 29–30] A real polynomial P (s) is said to be a Hurwitz polynomial if all its zeros are in <(s) ≤ 0 (closed lefthalf plane) with the zeros on jR (imaginary axis) being simple. Specifically, P (s) is called a strictly Hurwitz polynomial if all its zeros are in <(s) < 0. Since checking the residue conditions may be a complex task especially for higherdegree functions, the following necessary and sufficient condition for positiverealness might be easier to use. Theorem 2.4. [Weinberg and Slepian (1958)] A realration function H(s) = p(s)/q(s) with p(s) and q(s) being coprime polynomials is positivereal, if and only if 1. p(s) + q(s) is a Hurwitz polynomial; 2. <(H(jω)) ≥ 0 for ω ∈ R with s = jω not being a pole of H(s). In some cases, it is desirable to allow p(s) and q(s) to contain common roots on jR, which means that the convenient test by Theorem 2.4 is no longer applicable. Therefore, the following new positivereal criterion is useful. Theorem 2.5. [Chen and Smith (2009a)] A realration function H(s) = p(s)/q(s) with p(s) and q(s) having no common root in <(s) > 0 is positivereal, if and only if 1. p(s) + q(s) has no root in <(s) > 0; 2. <(H(jω)) ≥ 0 for ω ∈ R with s = jω not being a pole of H(s). Note that any positivereal function H(s) can be written in the form of m h0 X 2hi s + + h∞ s + H1 (s), H(s) = s s2 + ωi2 i=1 (2.1) page 12 August 27, 2019 14:9 wsbook9x6 Passive Network Synthesis: Advances with Inerter Preliminaries of Passive Network Synthesis 11567main 13 where h0 ≥ 0, hi ≥ 0, and h∞ ≥ 0 are the residues of the poles of H(s) at s = 0, s = ±jωi , and s = ∞, respectively, in which the zero value means that there is no pole. Theorem 2.6. [Baher (1984), pg. 34] Consider a positivereal function H(s) as expressed in (2.1), where h0 ≥ 0, hi ≥ 0, i = 1, 2, . . . , m, and h∞ ≥ 0. Then, H1 (s) in (2.1) is positivereal. Theorem 2.6 shows that any positivereal function maintains positiverealness after extracting all its poles on jR ∪ ∞ with the McMillan degree (or called degree)1 of the function being reduced. It is noted that h0 /s, h∞ s, and 2hi s/(s2 + ωi2 ) are impedances (resp., admittances) of a capacitor (resp., an inductor), an inductor (resp., a capacitor), and the parallel (resp., series) connection of a capacitor and an inductor, respectively. Therefore, any positivereal impedance Z(s) can be preliminarily realized through removing all its poles (resp., zeros) on jR ∪ ∞ as extracting these lossless components in series (resp., in parallel). Theorem 2.7. [Guillemin (1957); Van Valkenburg (1960)] If H(s) is a positivereal function, then H(s) − χ is positivereal, where χ is no larger than the minimum value of <(H(jω)) for any ω ∈ R ∪ ∞. Theorem 2.7 shows that any positivereal function maintains the positiverealness after subtracting a constant that is equal to the minimum value of the real part of the function on jR∪∞. Therefore, any positivereal impedance Z(s) can be preliminarily realized through extracting a series (resp., parallel) resistor whose resistance (resp., conductance) is equal to the minimum value of the real part of Z(s) (resp., Z −1 (s)) on jR ∪ ∞. Definition 2.3. [Van Valkenburg (1960), pg. 161] A realrational function H(s) is said to be a minimum function if (i) H(s) is positivereal, (ii) H(s) contains no pole and zero on jR ∪ ∞, and (iii) there exists a finite real value ω1 6= 0 such that H(jω1 ) = jX1 with X1 6= 0. As a consequence, the Foster preamble is defined as follows. Definition 2.4. [Van Valkenburg (1960), pg. 161] Given a positivereal impedance Z(s), the removal of the poles and zeros on jR ∪ ∞ and the 1 For any realrational function H(s) = a(s)/b(s) with polynomials a(s) and b(s) being coprime, the McMillan degree of H(s) is equal to the maximum degree of a(s) and b(s), which is denoted as δ(H(s)) = max{deg(a(s)), deg(b(s))} [Anderson and Vongpanitlerd (1973), Chapter 3.6]. page 13 August 15, 2019 11:32 14 wsbook9x6 Passive Network Synthesis: Advances with Inerter 11567main Passive Network Synthesis: Advances with Inerter minimum constant of <(Z(jω)) or <(Z −1 (jω)) correspond to the extraction of resistors, capacitors or inductors (see Theorems 2.6 and 2.7). The Foster preamble is the successive removal of these poles, zeros, and minimum constant values, such that both the remaining impedance Z1 (s) and admittance Y1 (s) = Z1−1 (s) are minimum functions with lower degrees or one of Z1 (s) and Y1 (s) is zero. Remark 2.1. Since the admittance Y (s) of any oneport network is equal to Z −1 (s), which always exists provided that the impedance Z(s) is nonzero, the realization of a positivereal admittance Y (s) as a oneport passive network using the Foster preamble and any other synthesis approach can be converted into the realization of the impedance Z(s). Therefore, one only needs to discuss the the realization problem of either the impedance or the admittance as oneport networks. Remark 2.2. It can be verified that H −1 (s) is not always a minimum function if H(s) is a minimum function, e.g., H(s) = (s2 + s + 1/2)/(s + 1)2 . Therefore, the Foster preamble can only terminate when both the resulting impedance and admittance are minimum functions. Otherwise, if the resulting impedance Z1 (s) (resp., admittance Z1−1 (s)) is a minimum function with the admittance Z1−1 (s) (resp., impedance Z1 (s)) not being one, then the Foster preamble can still continue by extracting the minimum value of <(Z1−1 (s)) (resp., <(Z1 (s))), which further yields a pole (resp., zero) of the impedance at zero or infinity. The Foster preamble can complete the realization of a given positivereal impedance Z(s) if the resulting impedance or admittance is zero. Otherwise, other realization procedures need to be further utilized. An illustrative example is presented as follows. Example 2.1. Consider a positivereal impedance 12s3 + 6s2 + 7s + 2 . 4s3 + 4s2 + 3s + 2 Then, by the Foster preamble, it can be written as −1 !−1 1 s Z(s) = 1 + + +2 , 2s 2s2 + 1 Z(s) = which is realizable as the configuration in Fig. 2.1 with R1 = 1 Ω, R2 = 2 Ω, L1 = 2 H, L2 = 1 H, and C1 = 2 F. page 14 August 15, 2019 11:32 wsbook9x6 Passive Network Synthesis: Advances with Inerter Preliminaries of Passive Network Synthesis 11567main 15 L1 R1 L2 R2 C1 Fig. 2.1 2.2 A realization of Example 2.1. Synthesis of OnePort Lossless Networks Definition 2.5. [Baher (1984), pg. 48] A positivereal function H(s) is called a reactance function (or Foster function) if H(s) is an odd rational function, that is, Ev(H(s)) := (H(s) + H(−s))/2 = 0 for all s. Theorem 2.8. [Baher (1984), pg. 51] Any reactance function H(s) can be written in the form of ±1 2 2 (s + ω12 )(s2 + ω32 ) · · · (s2 + ω2n−1 ) , (2.2) H(s) = k 2 s(s2 + ω22 )(s2 + ω42 ) · · · (s2 + ω2n−2 ) where k > 0 and 0 ≤ ω1 < ω2 < ω3 < ω4 < . . .. A oneport lossless network is a special type of passive networks containing only reactive elements (inductors and capacitors) and transformers. Since no resistor is involved, the network is lossless and there is no dissipation. Therefore, as shown in [Baher (1984), pp. 47–48], the impedance Z(s) of any oneport lossless network must be a reactance function. Conversely, consider any reactance impedance function Z(s). By Theorem 2.8, all the poles and zeros of Z(s) must be on jR ∪ ∞ and are alternatingly interlaced with each other, in the form of (2.2). Through extracting all the poles (or zeros) of Z(s) based on Theorem 2.6, which is called the partial fraction expansion approach, Z(s) is realizable as a oneport lossless network consisting of only inductors and capacitors (LC network). Such a realization is called Foster’s form. Any reactance impedance function Z(s) is realizable as a oneport LC network, through the successive removal of poles at s = ∞ (or s = 0) from the function and the subsequently inverted remainders, which is called the continued fraction expansion approach. Such a realization is called Cauer’s form. It is noted that both of these two approaches belong to special cases of the Foster preamble. page 15 August 15, 2019 11:32 16 wsbook9x6 Passive Network Synthesis: Advances with Inerter 11567main Passive Network Synthesis: Advances with Inerter As a summary, the following result can be established. Theorem 2.9. [Baher (1984), Chapter 3] The impedance (resp., admittance) of a oneport lossless network must be a reactance function, and any reactance function is realizable as the impedance (resp., admittance) of a oneport lossless network consisting of only inductors and capacitors. Moreover, synthesis results of oneport RL and RC networks can be similarly derived, which can be referred to [Van Valkenburg (1960)] for details. 2.3 The Brune Synthesis As discussed in the previous section, any positivereal impedance (resp., admittance) can be converted into a minimum function after the Foster preamble. Consequently, considering a minimum impedance, Brune [Brune (1931)] first established a systematic approach to realize such a function using a finite number of passive elements. Assume that a given impedance Z1 (s) is a minimum function. Then, there must exist a finite ω1 > 0 such that <(Z1 (s)) is zero at s = ±jω1 with =(Z1 (s)) being nonzero, that is, Z1 (jω1 ) = jX1 with X1 6= 0. It is noted that the function Z1 (s) − sX1 /ω1 must contain a zero at s = ±jω1 . The case of X1 /ω1 < 0 is first discussed. Letting L1 = X1 , ω1 (2.3) a negative inductor L1 < 0 can be extracted in series based on W1 (s) = Z1 (s) − L1 s. Then, W1 (s) must be a positivereal function, and W1−1 (s) contains a pole at s = ±jω1 with the residue K1 > 0. Therefore, it follows that 2K1 s W2−1 (s) = W1−1 (s) − 2 , s + ω12 which implies that W2 (s) is still positivereal and the extracted inductor L2 and capacitor C1 satisfy L2 = 1 , 2K1 C1 = 2K1 . ω12 According to the above discussion, W2 (s) can be expressed as W2 (s) = −L1 s3 + Z(s)s2 − ω12 L1 s + ω12 Z(s) , (2K1 L1 + 1)s2 − 2K1 Z(s)s + ω12 (2.4) page 16 August 27, 2019 14:9 wsbook9x6 Passive Network Synthesis: Advances with Inerter Preliminaries of Passive Network Synthesis 11567main 17 which implies that W2 (s) must contain a pole at s = ∞, since Z1 (s) contains no pole and no zero on jR ∪ ∞. By the extraction of a series inductor L3 = −L1 , 2K1 L1 + 1 (2.5) the pole of W2 (s) at infinity can be removed, yielding Z2 (s) = W2 (s) − L3 s = Z1 (s)s2 − 2ω12 K1 L21 s + ω12 (2K1 L1 + 1)Z1 (s) , (2K1 L1 + 1)((2K1 L1 + 1)s2 − 2K1 Z1 (s)s + ω12 ) which implies that Z2 (s) is a positivereal function. It is noted that the McMillan degree of Z2 (s) satisfies δ(Z2 (s)) = δ(W2 (s))−1 = δ(W1 (s))−3 = δ(Z1 (s))−2. Therefore, the above realization yields a Brune cycle as shown in Fig. 2.2, where the element values L1 , L2 , L3 and C1 satisfy (2.3)–(2.5), and the McMillan degree of Z2 (s) is lower than that of Z1 (s). Based on the equivalence of the “T structure” of L1 , L2 , and L3 and a transformer, one can always obtain an equivalent Brune cycle as shown in Fig. 2.3, where Lp = L1 + L2 , Ls = L2 + L3 , M = L2 . (2.6) It follows that Lp Ls = 1 L1 + 2K1 1 L1 − 2K1 2K1 L1 + 1 = 1 > 0, 4K12 (2.7) which implies that Lp > 0 due to Ls = L2 + L3 > 0. Therefore, one concludes that the Brune cycle in Fig. 2.3 contains only passive elements when X1 /ω1 < 0. L1 L3 L2 Z1(s) Z2(s) C1 Fig. 2.2 A Brune cycle, where L2 > 0, C1 > 0, and L1 L3 < 0 [Van Valkenburg (1960)]. The other case of X1 /ω1 > 0 can be similarly discussed, which can be referred to [Van Valkenburg (1960), pp. 170–172] for details. For this case, Z1 (s) can be similarly realized as the Brune cycle in Fig. 2.3 according to (2.3)–(2.6), where Z2 (s) is a positivereal function with δ(Z2 (s)) = δ(Z1 (s)) − 2. It should be noted that W1 (s) = Z1 (s) − L1 s contains a zero page 17 August 27, 2019 14:9 18 wsbook9x6 Passive Network Synthesis: Advances with Inerter 11567main Passive Network Synthesis: Advances with Inerter M Lp Ls Z1(s) Z2(s) C1 Fig. 2.3 A Brune cycle that is equivalent to Fig. 2.2, where Lp = L1 + L2 > 0, Ls = L2 + L3 > 0, and M = L2 > 0 [Van Valkenburg (1960)]. at s = ±jω1 without preserving the positiverealness. However, after the Brune cycle, Z2 (s) becomes a positivereal function again by extracting a negative inductor L3 in series. For this case, it is clear that L1 > 0, L2 > 0, L3 < 0, and C1 > 0. Furthermore, the condition (2.7) also holds, which implies that Ls > 0 due to Lp = L1 + L2 > 0. Therefore, the Brune cycle in Fig. 2.3 also contains only passive elements when X1 /ω1 > 0. As a consequence, combining the Foster preamble and Brune’s work, the following theorem can be obtained. Theorem 2.10. [Brune (1931)] Any positivereal impedance (resp., admittance) is realizable as a oneport passive network containing a finite number of resistors, inductors, capacitors, and transformers. As a summary, the Brune synthesis procedure is stated as follows. Algorithm (the Brune synthesis) Step 1. Given a positivereal impedance Z(s), utilize the Foster preamble to obtain a resulting impedance Z1 (s). If Z1 (s) or Z1−1 (s) is zero, then the synthesis procedure is finished. Otherwise, if Z1 (s) is a minimum function, then turn to the next step. Step 2. Realize a given minimum impedance function Z1 (s) as a Brune cycle shown in Fig. 2.3 according to (2.3)–(2.6). Step 3. If the resulting positivereal impedance Z2 (s) is of degree zero, then the synthesis procedure is finished by realizing Z2 (s) as a resistor. Otherwise, let Z2 (s) → Z(s) and return to Step 1. page 18 August 15, 2019 11:32 wsbook9x6 Passive Network Synthesis: Advances with Inerter Preliminaries of Passive Network Synthesis 2.4 11567main 19 The BottDuffin Synthesis Given any positivereal impedance Z(s), a minimum impedance function Z1 (s) can be obtained using the Foster preamble. Then, there must exist a finite ω1 > 0 such that Z(jω1 ) = jX1 with X1 6= 0. Bott and Duffin [Bott and Duffin (1949)] first established a realization procedure for any positivereal impedance as a oneport passive network without the use of transformers, which is called the BottDuffin synthesis. The derivation of such a synthesis procedure is based on Richards’s Theorem [Richards (1947)], which is stated as follows. Theorem 2.11 (Richards’s Theorem). [Richards (1947)] Given a positivereal function H(s), the function kH(s) − H(k)s , R(s) = kH(k) − sH(s) is also positivereal for any k > 0, and the McMillan degree of R(s) does not exceed that of H(s), that is, δ(R(s)) ≤ δ(H(s)). Applying Richards’s Theorem to the minimum impedance function Z1 (s), for any k > 0, one obtains kZ1 (s) − Z1 (k)s , (2.8) R1 (s) = kZ1 (k) − sZ1 (s) which implies that −1 −1 1 s k R1 (s) Z1 (s) = + + + . (2.9) Z1 (k)R1 (s) kZ1 (k) Z1 (k)s Z1 (k) By (2.9), Z1 (s) is realizable as the configuration in Fig. 2.4, where Z1 (k) 1 L1 = , C1 = , (2.10) k kZ1 (k) and the two resulting impedances Z1 (k)R1 (s) and Z1 (k)/R1 (s) are positivereal with δ(Z1 (k)R1 (s)) ≤ δ(Z(s)) and δ(Z1 (k)/R1 (s)) ≤ δ(Z(s)). It is noted that the McMillan degrees of these two impedances can be further reduced while preserving the positiverealness, provided that each of them contains a pole or a zero on jR. Recalling that Z(jω1 ) = jX1 with X1 6= 0, the case of X1 > 0 is first discussed. Then, X1 /ω1 > 0. Since Z1 (s) is a minimum function, it is clear that Z1 (k)/k is a continuous function taking values from zero to infinity for k ∈ (0, +∞). Therefore, there always exists k > 0 such that X1 Z1 (k) = , (2.11) k ω1 page 19 August 15, 2019 11:32 20 wsbook9x6 Passive Network Synthesis: Advances with Inerter 11567main Passive Network Synthesis: Advances with Inerter C1 Z1(k)R1(s) Z1(s) Z1(k)/R1(s) L1 Fig. 2.4 A realization of the minimum impedance function Z1 (s) based on (2.9), where Z1 (k)R1 (s) and Z1 (k)/R1 (s) are both positivereal impedances whose McMillan degrees do not exceed that of Z1 (s). which means that 1/R1 (s) has a pole at s = ±jω1 , whose residue is assumed to be α > 0, that is, 1 2αs = 2 + P (s). R1 (s) s + ω12 (2.12) Then, letting k satisfy (2.11), one obtains 1 1 2αs P (s) = − , = 2 2 Z2 (s) Z1 (k)R1 (s) Z1 (k)(s + ω1 ) Z1 (k) Z1 (k) 2αZ1 (k)s Z3 (s) = − 2 = Z1 (k)P (s), R1 (s) s + ω12 (2.13) (2.14) where the resulting impedances Z2 (s) and Z3 (s) must be positivereal with δ(Z2 (s)) = δ(R1 (s)) − 2 ≤ δ(Z1 (s)) − 2 and δ(Z3 (s)) = δ(R1 (s)) − 2 ≤ δ(Z1 (s)) − 2. Therefore, the above realization yields a BottDuffin cycle as shown in Fig. 2.5, where the element values L1 and C1 satisfy (2.10), and L2 = Z1 (k) 2αZ1 (k) 2α 1 , L3 = , C3 = . , C2 = 2 2α ω12 ω1 Z1 (k) 2αZ1 (k) (2.15) For the case of X1 < 0, it is obvious that ω1 X1 < 0. Recalling that Z1 (s) is a minimum function, kZ(k) is continuous and takes values from zero to infinity for k ∈ (0, +∞). Therefore, there always exists k > 0 such that kZ(k) = −ω1 X1 , (2.16) page 20 August 15, 2019 11:32 wsbook9x6 Passive Network Synthesis: Advances with Inerter 11567main Preliminaries of Passive Network Synthesis 21 L2 C1 Z2(s) C2 Z1(s) C3 L3 L1 Z3(s) Fig. 2.5 The BottDuffin cycle for the case of X1 > 0, where δ(Z2 (s)) ≤ δ(Z1 (s)) − 2 and δ(Z3 (s)) ≤ δ(Z1 (s)) − 2. which means that R1 (s) has a pole at s = ±jω1 , whose residue is assumed to be β > 0, that is, R1 (s) = 2βs + Q(s). s2 + ω12 (2.17) Then, letting k satisfy (2.16), one obtains 2βZ1 (k)s = Z1 (k)Q(s), s2 + ω12 R1 (s) 2βs Q(s) 1 = − , = 2 2 Z3 (s) Z1 (k) Z1 (k)(s + ω1 ) Z1 (k) Z2 (s) = Z1 (k)R1 (s) − (2.18) (2.19) where the resulting impedances Z2 (s) and Z3 (s) are positivereal with δ(Z2 (s)) = δ(R1 (s)) − 2 ≤ δ(Z1 (s)) − 2 and δ(Z3 (s)) = δ(R1 (s)) − 2 ≤ δ(Z1 (s)) − 2. Therefore, the above realization yields a BottDuffin cycle as shown in Fig. 2.6, where the element values L1 and C1 satisfy (2.10), and L2 = 2βZ1 (k) Z1 (k) 1 2β , L3 = , C2 = , C3 = 2 . 2 ω1 2β 2βZ1 (k) ω1 Z1 (k) (2.20) As a consequence, by repeatedly utilizing the Foster preamble and the BottDuffin cycle, the following theorem can be obtained. Theorem 2.12. [Bott and Duffin (1949)] Any positivereal impedance (resp., admittance) is realizable as a oneport passive network containing a finite number of resistors, inductors, and capacitors, that is, a oneport RLC network. page 21 August 15, 2019 11:32 22 wsbook9x6 Passive Network Synthesis: Advances with Inerter 11567main Passive Network Synthesis: Advances with Inerter Z2(s) C1 C2 L2 Z1(s) L3 L1 Z3(s) C3 Fig. 2.6 The BottDuffin cycle for the case of X1 < 0, where δ(Z2 (s)) ≤ δ(Z1 (s)) − 2 and δ(Z3 (s)) ≤ δ(Z1 (s)) − 2. In summary, the BottDuffin synthesis procedure is stated as follows. Algorithm (the BottDuffin synthesis) Step 1. Given a positivereal impedance Z(s), utilize the Foster preamble to obtain a resulting impedance Z1 (s). If Z1 (s) or Z1−1 (s) is zero, then the synthesis procedure is finished. Otherwise, if Z1 (s) is a minimum function, then turn to the next step. Step 2. Consider a given minimum impedance function Z1 (s), where Z1 (jω1 ) = jX1 with ω1 > 0 and X1 6= 0. If X1 > 0, then realize Z1 (s) as the BottDuffin cycle in Fig. 2.5 according to (2.10)–(2.15). If X1 < 0, then realize Z1 (s) as the BottDuffin cycle in Fig. 2.6 according to (2.10) and (2.16)–(2.20). Step 3. If the remaining positivereal impedances Z2 (s) and Z3 (s) are of degree zero, then the synthesis procedure is finished by realizing Z2 (s) and Z3 (s) as resistors. Otherwise, continuously repeat Steps 1 and 2 for the remaining positivereal impedances, until the McMillan degrees of them are all zero. It is noted that the BottDuffin synthesis generates many more elements than the Brune synthesis. In order to further simplify the realizations, Pantell [Pantell (1954)], Reza [Reza (1954)], et al. established some modified BottDuffin synthesis approaches. The realizations can be regarded as extensions of the BottDuffin synthesis based on the principle of balanced bridges and Y –∆ or ∆–Y transformation. Two types of Pantell’s page 22 August 15, 2019 11:32 wsbook9x6 Passive Network Synthesis: Advances with Inerter 11567main page 23 Preliminaries of Passive Network Synthesis 23 modified BottDuffin cycles are shown in Figs. 2.7 and 2.8 (see [Chen (2007), pp. 28–32], [Balabanian (1958), pp. 109–113] for details). Considering the modified BottDuffin cycle in Fig. 2.7, when X1 > 0, any minimum impedance function Z1 (s) is realizable as the configuration in Fig. 2.7(a), where (C1 + C3 )C2 L2 (C1 + C3 )L3 , L6 = 2 , C1 C1 + C1 C2 + C2 C3 C 2 + C1 C2 + C2 C3 C1 C 3 , C6 = 1 , C5 = C1 + C3 C1 + C3 L5 = (2.21) and L1 , L2 , L3 , C1 , C2 , C3 , Z2 (s), and Z3 (s) are determined according to (2.10)–(2.15). When X1 < 0, any minimum impedance function Z1 (s) is realizable as the configuration in Fig. 2.7(b), where (L1 + L2 )L1 L3 L5 = L1 + L2 , L6 = 2 , L1 + L1 L2 + L2 L3 L2 C2 (L2 + L1 L2 + L2 L3 )C3 C5 = , C6 = 1 , L1 + L2 (L1 + L2 )L1 (2.22) and L1 , L2 , L3 , C1 , C2 , C3 , Z2 (s), and Z3 (s) are determined according to (2.10) and (2.16)–(2.20). L C C Z s C L Z s Z s C Z s Z s L (a) L Z s L C (b) Fig. 2.7 A Pantell’s modified BottDuffin cycle for the case of (a) X1 > 0 and (b) X1 < 0, where δ(Z2 (s)) ≤ δ(Z1 (s)) − 2 and δ(Z3 (s)) ≤ δ(Z1 (s)) − 2. Consider an alternative modified BottDuffin cycle as shown in Fig. 2.8. When X1 > 0, any minimum impedance function Z1 (s) is realizable as the August 15, 2019 11:32 wsbook9x6 Passive Network Synthesis: Advances with Inerter 11567main 24 Passive Network Synthesis: Advances with Inerter configuration in Fig. 2.8(a), where L2 C 2 (C12 + C1 C2 + C2 C3 )L3 , L8 = , C1 + C2 (C1 + C2 )C1 (C1 + C2 )C1 C3 C7 = C1 + C2 , C8 = 2 , C1 + C1 C2 + C2 C3 L7 = L1 L3 (L1 + L3 )C 3 C7 = , , L1 + L3 L1 (L L2 + L1 L2 + L2 L3 1 + L3 )L2C2 , C8 = 2 , L8 = 1 L1 + L3 L1 + L1 L2 + L2 L3 L7 = Z s C Z s C C C L (2.24) and L1 , L2 , L3 , C1 , C2 , C3 , Z2 (s), and Z3 (s) are determined according to (2.10) and (2.16)–(2.20). and L1 , L2 , L3 , C1 , C2 , C3 , Z2 (s), and Z3 (s) are determined according to (2.10)–(2.15). When X1 < 0, any minimum impedance function Z1 (s) is realizable as the configuration in Fig. 2.7(b), where (2.23) Z s L Z s C L L L Z s Z s (a) (b) Fig. 2.8 An alternative Pantell’s modified BottDuffin cycle for the case of (a) X1 > 0 and (b) X1 < 0, where δ(Z2 (s)) ≤ δ(Z1 (s)) − 2 and δ(Z3 (s)) ≤ δ(Z1 (s)) − 2. Z(s) = Example 2.2. Consider a positivereal impedance given by 3s2 + 2s + 3 . s2 + s + 2 It can be checked that min <(Z(jω)) occurs at ω1 = 1 such that <(Z(jω1 )) = 1. Therefore, through extracting a resistor R1 = 1 Ω in page 24 wsbook9x6 Passive Network Synthesis: Advances with Inerter 11567main 25 R L C L C R (a) R (b) An example for (a) the Brune synthesis and (b) the BottDuffin synthesis. The Darlington Synthesis Another wellknown synthesis procedure of oneport passive networks is the Darlington synthesis [Darlington (1939)]. This procedure is based on the resistive extraction approach as shown in Fig. 2.10, where a resistor R is extracted such that the realization is converted into that of a twoport lossless network. As a consequence, at most one resistor is needed for realizing a positivereal impedance Z(s) by following the Darlington synthesis procedure. For the general configuration in Fig. 2.10, with R = 1 Ω, its impedance is obtained as 2 (z11 z22 − z12 )/z11 + 1 Z(s) = z11 , (2.25) z22 + 1 where z11 , z22 , and z12 are the entries of the impedance matrix of a lossless network: z z Z2×2 (s) = 11 12 . (2.26) z12 z22 C C 2.5 R Fig. 2.9 L Ls M Lp series, one can function as obtain a minimum impedance 2 2s + s + 1 Z1 (s) = Z(s) − R1 = 2 . s +s+2 It is noted that Z1 (jω1 ) = jX1 = j1, which means that X1 = 1 > 0. Following the Brune synthesis procedure, Z(s) is realizable as the configu ration in Fig. 2.9(a), where R1 = 1 Ω, R2 = 1/2 Ω, C1 = 1 F, Lp = 2 H, Ls = 1/2 H, and M = 1 H. Following the BottDuffin synthesis procedure, it can be solved to obtain as the configura k = 1, and Z(s) is realizable tion in Fig. 2.9(b), where R1 = 1 Ω, R2 = 1/2 Ω, R3 = 2 Ω, L1 = 1 H, L2 = 1/2 H, L3 = 2 H, C1 = 1 F, C2 = 2 F, and C3 = 1/2 F. R Preliminaries of Passive Network Synthesis page 25 August 15, 2019 11:32 August 15, 2019 11:32 wsbook9x6 26 Passive Network Synthesis: Advances with Inerter 11567main page 26 Passive Network Synthesis: Advances with Inerter I I V V /RVVOHVV 1HWZRUN Zs R Fig. 2.10 The general realization configuration for the Darlington synthesis, which is based on the resistive extraction [Van Valkenburg (1960)]. Consider a given positivereal impedance in the form of m1 + n1 , m2 + n2 Z(s) = $ $ P $ where m1 and n1 are the even and odd parts of the numerator, and m2 and P n2 are the even and odd parts of the denominator. Supposing that Z(s) is multiplied in numerator and denominator by$aP common factor m0 + n0 , $ $ with m0 and n0 being even and odd parts,Prespectively, one obtains $ 1 + n1 m0 + n0 m $P $ N m2 + n2 m0 + n0 $ $N (m0 m1 + nN 0 n1 ) + (m m0 + n01 0 n1 + n0 m1 ) = =: 01 . (m0 m2 + n0 n2 ) + (n0 m2 + m0 n2 ) m2 + n02 $P $P $P Z(s) = Comparing (2.25) with (2.27), it follows that p 0 0 m1 m2 − n01 n02 m02 m01 z11 = 0 , z22 = 0 , z12 = n2 n2 n02 $ or $ $ n0 $ n0 = 10 , z22 = 20 , z12 = m2 m2 $ where $N (Case A) z11 (2.27) $N $ $Q $ p 0 $0 n1 n2Q− m01 m02 , m0 Q 2 (Case B) $Q $ $Q m01 m02 − n01 n02 = (m1 m2 − n1 n2 )(m20 − n20 ). As shown in [Van Valkenburg (1960), Section 14.2], one can always $L $ $L determine aL common factor m0 + n0 according to the zeros of m1 m2 − L L n1 n2 , such that (m1 m2 − n1 n2 )(m20 − n20 ) is a full square polynomial. As L $L Furthermore, it is $shown a consequence, $zL12 is a realrational function. in L [Van Valkenburg (1960), Section 14.2] that z11 , z22 , and z12 in both Case A $L $ $L $L $L $L $ $ $ $L $ $ $ $ $ $ August 15, 2019 11:32 wsbook9x6 Passive Network Synthesis: Advances with Inerter 11567main Preliminaries of Passive Network Synthesis page 27 27 and Case B constitute a positivereal impedance matrix Z2×2 (s) in the form of (2.26), such that Z2×2 (s) can be written as Z2×2 (s) = N X K (i) fi (s), (2.28) i=1 where fi (s) is a positivereal function in one of the three forms: s/(s2 +ωi2 ), s, and 1/s, and " # (i) (i) k11 k12 (i) K = (i) (i) k12 k22 (i) (i) (i) is a nonnegative definite matrix satisfying k11 k22 − (k12 )2 = 0. Therefore, it can be proved that K (i) fi (s) is realizable as the twoport lossless (i) (i) (i) (i) configuration in Fig. 2.11, where ni  = k22 /k12  = k12 /k11 , and (i) Z (i) = k12 fi (s) ni is realizable with at most two reactive elements. As a result, the given positivereal impedance Z(s) is realizable as the configuration in Fig. 2.12 by following the Darlington synthesis procedure. ni Zi Fig. 2.11 (1960)]. 2.6 /RVVOHVV 1HWZRUN The twoport lossless configuration realizing K (i) fi (s) [Van Valkenburg Graph Theory for Passive Networks For the analysis and synthesis of nport networks, some basic concepts and results from graph theory are presented in this section. Definition 2.6. [Seshu and Reed (1961),$pg. 9] A linear graph is the collection of edges and vertices, where an edge is a line segment together with its endpoints and a vertex is an endpoint of an edge. $ $P $ P $ Definition 2.7. [Boesch (1966)] Consider an nport RLC (resp., damperP springinerter) network containing e elements and v nodes. The augmented$ $ $P $ P N $N $N $ $N $ $ $ $ $Q Q $Q $ August 15, 2019 11:32 wsbook9x6 28 11567main Passive Network Synthesis: Advances with Inerter Passive Network Synthesis: Advances with Inerter n Z /RVVOHVV 1HWZRUN n Z R nN ZN Fig. 2.12 The configuration realizing Z(s) by the Darlington synthesis procedure [Van $ $P Valkenburg$P (1960)]. $ $ P graph G $isP formulated by letting each port or element correspond to an $ $P P and letting each node correspond to a graph vertex. The subgraph edge consisting of all the edges corresponding to the ports is called a port graph $P $ $P Gp . The subgraph consisting of all the edges corresponding to the elements is called a network graph Ge . Furthermore, an edge belonging to the port $N $N graph Gp is called a port edge, and an edge belonging to a network graph Ge is called a network edge. $ N $N $N By Definition 2.7, it is clear that G is the union of Gp and Ge . An example to illustrate the concepts in Definition 2.7 is shown in Fig. 2.13. $ $ A'2 $ $ $Q $ QA1 A'1 A2 $Q $Q A3 A threeport network $L Fig. 2.13 L L $L L $L $ $L Augmented graph $L $L Definition 2.8. [Seshu and$LReed (1961), pg. 15] A circuit $L (or called a loop) $L is a closed edge sequence [Seshu and Reed (1961), pg. 14] with the degree $L $L [Seshu and Reed (1961), pg. 14] of each vertex being two. $ $ Port graph An example illustrating the concepts of the port graph and network graph. $L $ $ $ A'3 $ $Q $ $ $ $ $ $ page 28 August 15, 2019 11:32 wsbook9x6 Passive Network Synthesis: Advances with Inerter Preliminaries of Passive Network Synthesis 11567main 29 Definition 2.9. [Seshu and Reed (1961), pg. 24, pg. 26] For any connected graph [Seshu and Reed (1961), pg. 15], a subgraph that contains all the vertices without any circuit is called a tree. An edge of a tree is called a branch; an edge of the complement of a tree (cotree) is called a chord (or called a link ). Theorem 2.13. [Seshu and Reed (1961), pg. 24, pg. 26] A graph is a tree if and only if there is one and only one path [Seshu and Reed (1961), pg. 14] between any two vertices. Theorem 2.14. [Seshu and Reed (1961), pp. 25–26] A connected graph with v vertices and ne ≥ v − 1 edges must contain a tree, where the number of branches is v − 1 and the number of chords is ne − v + 1. Theorem 2.15. [Seshu and Reed (1961), pp. 26–27] For a connected graph with v vertices, its subgraph Gs is made part of a tree,2 if and only if Gs does not contain any circuit. Moreover, if a subgraph Gs contains v − 1 edges with no circuit, then Gs constitutes a tree. Definition 2.10. [Seshu and Reed (1961), pg. 27] The f circuits of a connected graph (with v vertices and ne ≥ v − 1 edges) for a certain tree are ne + v − 1 circuits, each of which is formed by a chord and its unique tree path. Definition 2.11. [Seshu and Reed (1961), pg. 27] The rank of a graph with v vertices and p maximal connected subgraphs is defined to be v − p. Specifically, the rank of a connected graph is v − 1. Definition 2.12. [Seshu and Reed (1961), pg. 28] A cutset is a set of edges of a connected graph, such that the removal of these edges will reduce the rank of the graph by one and the removal of any proper subset of these edges cannot do so.3 Theorem 2.16. [Seshu and Reed (1961), pg. 34] For a connected graph, its subgraph Gs is made part of the complement of a tree (cotree),4 if and only if Gs does not contain any cutset. 2 The special case where Gs constitutes a tree is included. removing edges, any isolated vertex is assumed to be a maximal connected subgraph for the calculation of the rank. 4 The special case where G constitutes a cotree is included. s 3 After page 29 August 15, 2019 11:32 30 wsbook9x6 Passive Network Synthesis: Advances with Inerter 11567main Passive Network Synthesis: Advances with Inerter Definition 2.13. [Seshu and Reed (1961), pg. 31] For a connected graph with v vertices, the f cutsets with respect to a tree are v − 1 cutsets, each of which is formed by one branch of the tree and some chords. In the above definition, the chords corresponding to a certain branch can be uniquely determined based on the following theorem. Theorem 2.17. [Seshu and Reed (1961), pg. 31] An f cutset determined by a branch of a tree exactly contains the chords whose corresponding f circuits contain the branch. To better analyze nport passive networks, all the edges can be assigned with orientations to form a directed graph. Definition 2.14. [Seshu and Reed (1961), pg. 91] For a connected graph with v vertices and ne ≥ v − 1 edges, the f circuit matrix Bf = [bij ] is a (ne − v + 1) × ne matrix, whose rows correspond to ne − v + 1 f circuits and whose columns correspond to ne edges. If edge j belongs to the ith f circuit and has the same (resp., opposite) orientation as that of the f circuit, then bij = 1 (resp., bij = −1); if edge j does not belong to the ith f circuit, then bij = 0. Here, the f circuit orientation coincides with that of the defining chord. Definition 2.15. [Seshu and Reed (1961), pg. 97] For a connected graph with v vertices, the f cutset matrix Qf = [qij ] is a (v − 1) × ne matrix, whose rows correspond to v − 1 f cutsets and whose columns correspond to ne ≥ v − 1 edges. If the jth edge belongs to the ith f cutset and has the same (resp., opposite) orientation as that of the f cutset, then qij = 1 (resp., qij = −1); if the jth edge does not belong to the ith f cutset, then qij = 0. Here, the f cutset orientation coincides with that of the defining branch. Consider an nport RLC (resp., damperspringinerter) network containing e elements. Then, the augmented graph G, port graph Gp , and network graph Ge of the network can be formulated based on Definition 2.7. Without loss of generality, one can assume that G is connected with n + e edges and v vertices. Otherwise, one can obtain a connected and separable augmented graph [Seshu and Reed (1961), pg. 35] by letting one vertex of a component be common with that of another. Therefore, a tree T must exist by Theorem 2.14. For each edge of the network graph Ge , that is, network edge, its orientation is assigned with the same direction of the reference page 30 August 15, 2019 11:32 wsbook9x6 Passive Network Synthesis: Advances with Inerter Preliminaries of Passive Network Synthesis 11567main 31 element current.5 For each edge of the port graph Gp , that is, port edge, its orientation is assigned with the opposite direction to the port current.6 Furthermore, n+e−v +1 f circuits and v −1 f cutsets can be uniquely determined with respect to the tree T , which can be denoted as Bf = Bf 1 Bf 2 , and Qf = Qf 1 Qf 2 , respectively, where the columns of Bf 1 ∈ R(n+e−v+1)×e and Qf 1 ∈ R(v−1)×e correspond to the edges of the network graph Ge , and the columns of Bf 2 ∈ R(n+e−v+1)×n and Qf 2 ∈ R(v−1)×n correspond to the edges of the port graph Gp . Then, Kirchhoff’s laws for the network can be expressed as " # Û =0 (2.29) Bf 1 Bf 2 V̂ and " Qf 1 Qf 2 # Jˆ = 0, −Iˆ (2.30) where Jˆ and Û are the Laplace transforms of the element currents and voltages, and Iˆ and V̂ are the Laplace transforms of the port currents and voltages. It is known [Seshu and Reed (1961), pg. 123] that the following conditions hold: " # T Bf 1 ˆ Jˆ Tˆ = Bf Im = Im , (2.31) ˆ BfT2 −I and " Û V̂ # = QTf V̂n QTf1 = V̂n , QTf2 (2.32) where Iˆm are the Laplace transforms of the currents corresponding to the chords of the tree T , and V̂n are the Laplace transforms of the voltages corresponding to the branches of the tree T . 5 It is known that the direction of the actual current of an element may not be the same as that of the reference current, where they have the same direction when the value of the reference current is positive and they have opposite directions when the value of the reference current is negative. 6 The directions of port currents are chosen as the actual directions, which are determined by the polarities of the ports. page 31 August 15, 2019 11:32 32 wsbook9x6 Passive Network Synthesis: Advances with Inerter 11567main Passive Network Synthesis: Advances with Inerter By Ohm’s law, it is clear that Jˆ = Gd Û , ˆ Û = Dd J, (2.33) where Gd is a diagonal matrix whose diagonal entries are element admittances (1/Ri , 1/(Lj s), or Ck s) and Dd is a diagonal matrix whose diagonal entries are element impedances (Ri , Lj s, or 1/(Ck s)). Combining (2.29)–(2.32), one obtains ˆ Qf 1 Gd QTf1 V̂n = Qf 2 I, (2.34) Bf 1 Dd BfT1 Iˆm = −Bf 2 V̂ . (2.35) and As discussed in [Boesch (1966)], if the impedance (resp., admittance) of an nport RLC network exists, then any current vector Iˆ (resp., voltage vector V̂ ) is permitted, which means that the port graph Gp cannot contain any cutset (resp., circuit) of the augmented graph. By Theorem 2.16 (resp., Theorem 2.15), the port graph must be made part of a cotree (resp., tree). Conversely, if the port graph Gp is made part of a cotree (resp., tree), then any branch (resp., chord) must be a port edge. Therefore, Qf 1 (resp., Bf 1 ) can be written as Qf 1 = [Iv−1 , Qf 12 ] (resp., Bf 1 = [In+e−v+1 , Bf 12 ]), where Iv−1 (resp., In+e−v+1 ) is an identity matrix, which implies that Qf 1 GQTf1 (resp., Bf 1 DBfT1 ) must be nonsingular. Therefore, by (2.32) (resp., (2.31)), (2.34) (resp., (2.35)) is equivalent to V̂ = QTf2 (Qf 1 Gd QTf1 )−1 Qf 2 Iˆ (resp., Iˆ = BfT2 (Bf 1 Dd BfT1 )−1 Bf 2 V̂ ), which means that the impedance matrix (resp., admittance matrix) exists. The above discussion can be summarized as the following two theorems. Theorem 2.18. [Boesch (1966)] Consider an nport RLC (or damperspringinerter) network whose connected augmented graph G contains v vertices and n + e edges. The impedance matrix of the network exists, if and only if its port graph Gp can be made part of a cotree G −T of its augmented graph G. Moreover, the impedance matrix can be expressed as Z(s) = QTf2 (Qf 1 Gd QTf1 )−1 Qf 2 , (2.36) where Qf 1 = [Iv−1 , Qf 12 ] and Qf 2 constitute the f cutset matrix Qf = [Qf 1 , Qf 2 ] of G with respect to the tree T , the columns of Qf 1 correspond to network edges, the columns of Qf 2 correspond to port edges, and Gd is a diagonal matrix whose diagonal entries are element admittances (ai , bj /s, or ck s for ai , bj , ck > 0). page 32 August 15, 2019 11:32 wsbook9x6 Passive Network Synthesis: Advances with Inerter Preliminaries of Passive Network Synthesis 11567main 33 The admittance matrix of the network exists, if and only if its port graph Gp can be made part of a tree T of its augmented graph G. Moreover, the admittance matrix can be expressed as Y (s) = BfT2 (Bf 1 Dd BfT1 )−1 Bf 2 , (2.37) where Bf 1 = [In+e−v+1 , Bf 12 ] and Bf 2 constitute the f circuit matrix Bf = [Bf 1 , Bf 2 ] of G with respect to the tree T , the columns of Bf 1 correspond to network edges, the columns of Bf 2 correspond to port edges, and Dd is a diagonal matrix whose diagonal entries are element impedances (a0i , b0j s, or c0k /s for a0i , b0j , c0k > 0). Remark 2.3. After a proper rearrangement of rows and corresponding columns, Gd (resp., Dd ) can be written as Gd = Gd1 u s−1 Gd2 u sGd3 (resp., Dd = Dd1 u sDd2 u Dd3 ). Specifically, Gd (resp., Dd ) is a real diagonal matrix for an nport resistive network. Remark 2.4. If the port graph Gp is exactly a cotree, that is, n = n + e − v + 1, then the augmented graph G contains n f circuits, and by (2.36) the impedance matrix can be expressed as Z(s) = LDd LT , (2.38) where [In , L] is the f circuit matrix of G. If the port graph Gp is a tree, that is, n = v − 1, then the augmented graph G contains n vertices, and by (2.37) the admittance matrix can be expressed as Y (s) = W Gd W T , (2.39) where [In , W ] is the f cutset matrix of G. Remark 2.5. By (2.36) and (2.37), it can be seen that changing the orientation of any edge in the network graph Ge (network edge) does not affect Z(s) and Y (s), and changing the orientation of an edge in the port graph Gp (port edge) corresponds to a crosssign change [Brown and Reed (1962a)] (see Definition 4.2) of Z(s) and Y (s). Here, the orientation change of a port edge corresponds to switching the polarity of the port. 2.7 Principle of Duality In addition to linear graphs, any oneport (that is, twoterminal) RLC (damperspringinerter) network N can be described by a oneterminalpair labeled graph N with two distinguished terminal vertices (see [Seshu and page 33 August 15, 2019 11:32 34 wsbook9x6 Passive Network Synthesis: Advances with Inerter 11567main Passive Network Synthesis: Advances with Inerter Reed (1961), pg. 14]), in which the labels designate passive circuit elements regardless of the values of the elements, namely resistors, capacitors, and inductors, which are labeled as Ri , Ci , and Li , respectively. Two natural maps acting on the labeled graph are defined as follows: (1) GDu := Graph duality, which takes the oneterminalpair graph into its dual (see [Seshu and Reed (1961), Definition 312]) while preserving the labeling. (2) Inv := Inversion, which preserves the graph but interchanges the reactive elements, that is, capacitors to inductors and inductors to capacitors, with their labels Ci to Li and Li to Ci . Consequently, one defines7 Dual := network duality of oneterminalpair labeled graph := GDu ◦ Inv = Inv ◦ GDu. An example to illustrate GDu, Inv, and Dual can be referred to Fig. 3.5. Denoting the oneterminalpair labeled graphs of the configurations in Figs. 3.5(a), 3.5(b), 3.5(c), and 3.5(d) as N2a , N2b , N2c , and N2d , respectively, the following relations hold: N2b = GDual(N2a ), N2c = Inv(N2a ), and N2d = Dual(N2a ). Consider a network N whose oneterminalpair labeled graph is N . Let Inv(N ) denote the network whose oneterminalpair labeled graph is Inv(N ), resistors are of the same values as those of N , and inductors (resp., capacitors) are replaced by capacitors (resp., inductors) with reciprocal values, which is called the frequency inverse network of N . Let GDu(N ) denote the network whose oneterminalpair labeled graph is GDu(N ) and whose elements are of the reciprocal values to those of N , which is called the frequency inverse dual network of N . Let Dual(N ) denote the network whose oneterminalpair labeled graph is Dual(N ), resistors are of reciprocal values to those of N , and inductors (resp., capacitors) are replaced by capacitors (resp., inductors) with same values, which is called the dual network of N (see [Seshu and Reed (1961), Definition 65]). It can be proved that Z(s) (resp., Y (s)) is realizable as the impedance (resp., admittance) of a network N whose oneterminalpair labeled graph is N , if and only if Z(s−1 ) (resp., Y (s−1 )) is realizable as the impedance (resp., admittance) of Inv(N ) whose oneterminalpair labeled graph is 7 Such an approach of defining GDu, Inv, and Dual based on oneterminalpair labeled graphs was suggested by Professor Rudolf E. Kalman in his private communication with the first author on July 13, 2014. page 34 August 15, 2019 11:32 wsbook9x6 Passive Network Synthesis: Advances with Inerter Preliminaries of Passive Network Synthesis 11567main 35 Inv(N ), if and only if Z(s−1 ) (resp., Y (s−1 )) is realizable as the admittance (resp., impedance) of GDu(N ) whose oneterminalpair labeled graph is GDu(N ), and if and only if it is realizable as the admittance (resp., impedance) of Dual(N ) whose oneterminalpair labeled graph is Dual(N ). Therefore, if a necessary and sufficient condition is derived for H(s) = Pm Pm i j i=0 ai s / j=0 bj s to be realizable as the impedance (resp., admittance) of a oneport network whose oneterminalpair labeled graph is N , then the corresponding condition for Inv(N ) can be obtained from that for N through conversions ak ↔ am−k and bk ↔ bm−k for k = 0, 1, ..., bm/2c (the principle of frequency inversion). The corresponding condition for GDu(N ) can be obtained from that for N through conversions ak ↔ bm−k for k = 0, 1, ..., m (the principle of frequencyinverse duality). Furthermore, the corresponding condition for Dual(N ) can be obtained from that for N through conversions ak ↔ bk for k = 0, 1, ..., m (the principle of duality). page 35 b2530 International Strategic Relations and China’s National Security: World at the Crossroads This page intentionally left blank b2530_FM.indd 6 01Sep16 11:03:06 AM August 15, 2019 11:32 wsbook9x6 Passive Network Synthesis: Advances with Inerter 11567main Chapter 3 Biquadratic Synthesis of OnePort RLC Networks 3.1 Introduction The realization of biquadratic impedances as passive RLC networks has been an essential topic in passive network synthesis. Although a series of investigations have been made, this problem is still unsolved. It can be seen from the formulation of the BottDuffin synthesis procedure that the RLC realization of biquadratic impedances can provide important guidance on positivereal functions with higher degrees by induction. Practically, the impedances (or admittances) of many mechanical networks or electrical networks in mechatronic systems are in biquadratic forms (see [Papageorgiou and Smith (2006); Wang et al. (2009)], for instance). Therefore, it is important to investigate biquadratic synthesis of RLC networks, especially its minimal realization. Through the BottDuffin synthesis procedure (resp. Pantell’s modified BottDuffin synthesis procedure), nine (resp., eight) elements are needed to realize the entire class of positivereal biquadratic impedances as seriesparallel (nonseriesparallel) RLC networks, where the realizations contain fewer elements for special cases, such as biquadratic minimum impedances or the impedances directly realizable by the Foster preamble. However, the BottDuffin approach cannot guarantee the minimality of realizations, and it is necessary to discuss the realization problem of a biquadratic impedance as a kelement network for k = 1, 2, ..., 8. In [Ladenheim (1948)], Ladenheim first investigated the realization of biquadratic impedances by listing 108 configurations, which cover all the possible irreducible networks containing no more than two reactive elements and no more than three resistors. Furthermore, based on the method of enumeration, biquadratic synthesis of threereactive fiveelement networks and sixelement seriesparallel 37 page 37 August 15, 2019 11:32 38 wsbook9x6 Passive Network Synthesis: Advances with Inerter 11567main Passive Network Synthesis: Advances with Inerter networks have been investigated in [Ladenheim (1964); Vasiliu (1969)] and [Vasiliu (1970, 1971)], respectively. In addition, some other realization problems of biquadratic impedances have been investigated in [BarLev (1962); Chang (1969); Eswaran and Murti (1973); Reichert (1969); Steiglitz and Zemanian (1962); Tirtoprodjo (1972)]. During recent years, by defining a new concept called regularity [Jiang and Smith (2011)] and investigating its properties, Jiang and Smith reconsidered the the realization problems of biquadratic impedances as fiveelement networks and sixelement seriesparallel networks in [Jiang (2010); Jiang and Smith (2011, 2012)]. As a result, the investigations are more systematic and the realization results are better combined, where it is shown [Jiang and Smith (2011)] that the regularity of a biquadratic impedance is equivalent to the realization as a tworeactive fiveelement seriesparallel network. Following previous investigations on minimal realizations of biquadratic minimum impedances, Hughes and Smith continued to investigate such a problem in terms of the minimality of reactive elements for both seriesparallel [Hughes and Smith (2014)] and nonseriesparallel cases [Hughes (2017)]. This chapter presents some recent results of biquadratic synthesis in [Chen et al. (2016b, 2017); Wang and Chen (2012); Wang et al. (2014, 2018)]. In this chapter, networks are assumed to be oneport passive transformerless networks containing no more than three kinds of passive elements, which are resistors, capacitors, and inductors (RLC networks). Element values are assumed to be positive and finite if not specially mentioned. 3.2 Basic Notations and Results The general form of a biquadratic impedance is a2 s2 + a1 s + a0 , (3.1) Z(s) = b2 s2 + b1 s + b0 where ai ≥ 0, i = 0, 1, 2, and bj ≥ 0, j = 0, 1, 2. For brevity, the following notations are introduced:1 A = a0 b1 − a1 b0 , B = a0 b2 − a2 b0 , C = a1 b2 − a2 b1 , Da := a1 A − a0 B, Db := −b1 A + b0 B, Ea := a2 B − a1 C, Eb := −b2 B + b1 C, M := a0 b2 + a2 b0 , ∆a := a21 − 4a0 a2 , ∆b := b21 − 4b0 b2 ∆ab := a1 b1 − 2M, 1 These notations were suggested by Professor Rudolf E. Kalman in his private communication with the authors on July 16, 2014. page 38 August 15, 2019 11:32 wsbook9x6 Passive Network Synthesis: Advances with Inerter Biquadratic Synthesis of OnePort RLC Networks 11567main 39 R := AC − B 2 , Γa := R + b0 b2 ∆a , Γb := R + a0 a2 ∆b . Define R0 (a, b, s) as the resultant [Gantmacher (1980), Chapter XV] of a(s) := a2 s2 + a1 s + a0 and b(s) := b2 s2 + b1 s + b0 in s, that is, a2 0 R0 (a, b, s) = b2 0 a1 a2 b1 b2 a0 a1 b0 b1 0 a0 . 0 b0 Then, it is clear that R = −R0 (a, b, s). It is known from [Gantmacher (1980), Chapter XV] that there exists a common factor between a(s) and b(s) if and only if R = −R0 (a, b, s) = 0. Lemma 3.1. [Chen and Smith (2009a); Foster (1962)] A biquadratic function Z(s) in the form of (3.1) with ai ≥ 0, i = 0, 1, 2, and bj ≥ 0, j = 0, 1, 2, √ √ is positivereal, if and only if ( a2 b0 − a0 b2 )2 ≤ a1 b1 . Lemma 3.2. [Foster (1963), pg. 527] A biquadratic function Z(s) in the form of (3.1) with ai ≥ 0, i = 0, 1, 2, and bj ≥ 0, j = 0, 1, 2, is a minimum √ function, if and only if ai > 0, i = 0, 1, 2, bj > 0, j = 0, 1, 2, and ( a2 b0 − √ a0 b2 )2 = a1 b1 . Lemma 3.3. [Jiang and Smith (2011), Lemma 5] A biquadratic function Z(s) in the form of (3.1) with ai ≥ 0, i = 0, 1, 2, and bj ≥ 0, j = 0, 1, 2, is regular, if and only if at least one of the following four conditions holds: 1. B ≤ 0 and Db ≥ 0; 2. B ≤ 0 and Ea ≥ 0; 3. B ≥ 0 and Eb ≥ 0; 4. B ≥ 0 and Da ≥ 0. Lemma 3.4. [Jiang (2010), Lemma 8] Any positivereal biquadratic impedance (3.1), with any of its six parameters equal to zero, can be realized by a seriesparallel network with no more than two reactive elements and two resistive elements through the Foster preamble. Definition 3.1. Consider a oneport network N , containing no more than three kinds of elements (resistors, inductors, and capacitors). Letting each element correspond to an edge [Seshu and Reed (1961), pg. 9] and each voltage node correspond to a vertex [Seshu and Reed (1961), pg. 9] yields a linear graph, called the network graph of N . The subgraph [Seshu and Reed (1961), pg. 12] with edges corresponding to reactive elements (inductors and capacitors) is called the reactiveelement graph, whose edges are called reactiveelement edges. The subgraph with edges corresponding to resistors is called the resistor graph, whose edges are called resistor edges. page 39 August 15, 2019 11:32 40 wsbook9x6 Passive Network Synthesis: Advances with Inerter 11567main Passive Network Synthesis: Advances with Inerter Let P(i, j) denote a path whose two endvertices are i and j, and let C(i, j) denote a cutset that separates the network graph into two connected subgraphs containing vertices i and j, respectively. Then, denote a path P(i, j) (resp., cutset C(i, j)) whose edges only correspond to resistors, inductors, or capacitors as RP(i, j), LP(i, j), or CP(i, j) (resp., RC(i, j), LC(i, j), or CC(i, j)), respectively. Denote a path P(i, j) (resp., cutset C(i, j)) whose edges exactly correspond to two kinds of elements, which are inductorscapacitors, resistorsinductors, or resistorscapacitors, as LCP(i, j), RLP(i, j), or RCP(i, j) (resp., LCC(i, j), RLC(i, j), or RCC(i, j)), respectively. Let E(i, j) denote an edge incident with [Seshu and Reed (1961), pg. 12] vertices i and j. Then, vertex i is said to be adjacent to vertex j by E(i, j). Moreover, let RE(i, j) denote a resistor edge incident with vertices i and j. Also, for any oneport network N whose two terminals are denoted as a and a0 , P(a, a0 ) denotes the path whose terminal vertices (see [Seshu and Reed (1961), pg. 14]) are a and a0 , and C(a, a0 ) denotes the cutset that separates N into two connected subgraphs N1 and N2 containing two terminal vertices a and a0 , respectively. Lemma 3.5. The network graph of a network N with two terminals a and a0 realizing a biquadratic impedance Z(s) in the form of (3.1), where ai > 0, i = 0, 1, 2, and bj > 0, j = 0, 1, 2, can neither contain any path P(a, a0 ) nor contain any cutset C(a, a0 ) whose edges correspond to only one kind of reactive elements. Proof. Assume that there exists such a path P(a, a0 ) or cutset C(a, a0 ). Then, it is known from [Seshu (1959)] that the impedance of N must contain zeros or poles at s = 0 or s = ∞. This contradicts the assumption. More generally, the following lemma can be obtained. Lemma 3.6. Any biquadratic impedance Z(s) in the form of (3.1) with ai > 0, i = 0, 1, 2, and bj > 0, j = 0, 1, 2, is not realizable as the network shown in Fig. 3.1. Proof. In [Jiang (2010), Sec. 3.2], it is shown that there exist poles for the impedance of Fig. 3.1(a) and zeros for that of Fig. 3.1(b) at s = jω or s = ∞, which implies that some of the impedance’s coefficients must be zero. Thus, the lemma is proved. page 40 GXDO GXDO l l EE l l GXDO GXDO l l 11567main Biquadratic Synthesis of OnePort RLC Networks 41 EE 3DVVLYH 3DVVLYH 1HWZRUN 1HWZRUN 3DVVLYH 3DVVLYH 1HWZRUN 1HWZRUN /RVVOHVV /RVVOHVV GXDO GXDO 1HWZRUN 1HWZRUN EE DD Passive Network Synthesis: Advances with Inerter GXDO GXDO wsbook9x6 August 15, 2019 11:32 /RVVOHVV /RVVOHVV 1HWZRUN 1HWZRUN (a) DD E(b) E Fig. 3.1 The network structures of a lossless subnetwork and any passive subnetwork (a) in series or (b) in parallel. The following lemma provides the equivalence of two classes of networks. Lemma 3.7. [Lin (1965)] Any passive network as shown in Fig. 3.2(a) is equivalent to a passive network as shown in Fig. 3.2(b), where Zu and Zv are positivereal impedances, α = a(a+b)/b, β = a+b, and γ = c(a+b)2 /b2 . ȕȕZu bZu aZu cZv ĮĮZu (a) ȖȖZv (b) Fig. 3.2 Two equivalent networks, where Zu and Zv are the impedances of any two passive networks, α = a(a + b)/b, β = a + b, and γ = c(a + b)2 /b2 (see [Lin (1965), Fig. 3]). 3.3 A canonical form Zc (s) for biquadratic impedances (3.1), first considered in [Reichert (1969)], is expressed as √ s2 + 2U W s + W √ , (3.2) Zc (s) = s2 + (2V / W )s + 1/W where r A Canonical Biquadratic Impedance W = a 0 b2 , a 2 b0 a1 U= √ , 2 a0 a2 b1 V = √ . 2 b0 b2 3DVVLYH 3DVVLYH (3.3) Here, Zc (s) can be obtained from Z(s) through Zc (s) = αZ(βs), where 1HWZRUN 1HWZRUN p 4 3DVVLYH /RVVOHVV 3DVVLYH /RVVOHVV α = b2 /a2 and β = a0 b0 /(a 2 b2 ). If Z(s) is realizable as a network N , 1HWZRUN 1HWZRUN 1HWZRUN 1HWZRUN /RVVOHVV /RVVOHVV 1HWZRUN 1HWZRUN page 41 August 15, 2019 11:32 42 wsbook9x6 Passive Network Synthesis: Advances with Inerter 11567main Passive Network Synthesis: Advances with Inerter then the corresponding Zc (s) must be realizable as another one Nc , with the same oneterminalpair labeled graph by a proper transformation of the element values, and vice versa. Therefore, the realizability condition for Zc (s) in terms of U , V , W > 0, as a network whose oneterminalpair labeled graph is N , can be determined from that of Z(s) in terms of ai > 0, i = 0, 1, 2, and bj > 0, j = 0, 1, 2, via the transformation √ √ a2 = 1, a1 = 2U W , a0 = W, b2 = 1, b1 = 2V / W , b0 = 1/W. (3.4) Conversely, the realizability condition for Z(s) as a network with oneterminalpair labeled graph N in terms of ai > 0, i = 0, 1, 2, and bj > 0, j = 0, 1, 2, can be determined from that for Zc (s) in terms of U , V , W > 0, through the transformation (3.3). Furthermore, through (3.4), it is concluded that Zc (s) is positivereal if and only if σc := 4U V + 2 − (W + W −1 ) ≥ 0, as stated in [Jiang and Smith (2011)]. Notations ∆ab , R, Γa , and Γb , defined above are respectively converted into ∆abc := 4U V − 2(W + W −1 ), Rc := −4U 2 − 4V 2 + 4U V (W + W −1 ) − (W − W −1 )2 , Γac := −4V 2 + 4U V (W + W −1 ) − (W + W −1 )2 , and Γbc := −4U 2 + 4U V (W + W −1 ) − (W + W −1 )2 . Also, M R+2a0 a2 b0 b2 ∆ab is converted to −(W +W −1 )3 +4U V (W +W −1 )2 − 4(U 2 + V 2 )(W + W −1 ) + 8U V . Moreover, for brevity, denote λc := 4U V − 4V 2 W + (W − W −1 ). With ρ∗ (U, V, W ) = ρ(U, V, W −1 ) and ρ† (U, V, W ) = ρ(V, U, W ) for any † rational function ρ(U, V, W ), it can be verified that λ∗† c W , λc /W , λc , and λ∗c correspond to Da , Db , Ea , Eb , respectively, through (3.4). Besides, by denoting ηc := 4U 2 + 4V 2 + 4U V (3W − W −1 ) + (W − W −1 )(9W − W −1 ) and ζc := −4U 2 − 4V 2 + 4U V (W + W −1 ) − (W − W −1 )(3W − W −1 ), corresponding to −R + 4a0 b2 (a1 b1 + 2B) and R − 2a0 b2 B, respectively, one has ηc∗ = ηc∗† and ζc∗ = ζc∗† corresponding to −R + 4a2 b0 (a1 b1 − 2B) and R + 2a2 b0 B, respectively. page 42 September 11, 2019 8:51 wsbook9x6 Passive Network Synthesis: Advances with Inerter 11567main Biquadratic Synthesis of OnePort RLC Networks 3.4 43 Realizations of Biquadratic Impedances with No More than Four Elements This section will discuss the realization problem of a biquadratic impedance Z(s) in the form of (3.1) to be realizable as an RLC network containing no more than four elements. Based on Lemma 3.4, to investigate realizations with no more than four elements, it suffices to assume that ai > 0, i = 0, 1, 2, and bj > 0, j = 0, 1, 2. A necessary and sufficient condition for the realization of such an impedance with no more than three elements is presented in Lemma 3.8. Furthermore, the main result of this section is shown in Theorem 3.5, which presents a necessary and sufficient condition for the realization of such an impedance with no more than four elements. Figures 3.5–3.7 are the fourelement realization configurations, whose realizability conditions are summarized in Table 3.1. 3.4.1 Realizations with No More than Three Elements Lemma 3.8. A biquadratic impedance Z(s) in the form of (3.1), where ai > 0, i = 0, 1, 2, and bj > 0, j = 0, 1, 2, is realizable with no more than three elements, if and only if R = 0. Proof. Sufficiency. Since R = 0, there exists a common factor between the numerator and denominator of Z(s), which means that Z(s) can be written as Z(s) = (α1 s+α0 )/(β1 s+β0 ), where αi > 0, i = 0, 1, and βj > 0, j = 0, 1. Thus, Z(s) is realizable as a configuration shown in Fig. 3.3 by the Foster preamble when α0 β1 − α1 β0 6= 0, or as a single resistor when α0 β1 − α1 β0 = 0. Necessity. By the principle of duality, one only needs to discuss the network graphs shown in Fig. 3.4. For Fig. 3.4(a), the only one edge should correspond to a resistor, otherwise it will result in a path P(a, a0 ) or a cutset C(a, a0 ) corresponding to one kind of reactive elements, which is impossible by Lemma 3.5. For Figs. 3.4(b), 3.4(c), and 3.4(d), the networks that can be equivalent to one containing fewer elements are not considered to avoid repetition, as the discussion is in the order of the increasing numbers of elements from Fig. 3.4(a) to Fig. 3.4(d). Furthermore, by Lemmas 3.5 and 3.6, the network graphs in Figs. 3.4(b) and 3.4(c) are directly eliminated, and Edge 1 and only one of Edge 2 or Edge 3 of the graph in Fig. 3.4(d) correspond to resistors, yielding the networks shown in Figs. 3.3(a) and 3.3(c). By the page 43 VļVVļV VļVVļV GXDO GXDO August 15, 2019 11:32 GXDO GXDO Passive Network Synthesis: Advances with Inerter wsbook9x6 F F F 44 F GXDO GXDO l l GXDO GXDO E E E E Passive Network Synthesis: Advances with Inerter l l D E D 11567main page 44 3.4.2 GXDO GXDO l Realizations with Four Elements l GXDO GXDO D E D E Z(s) in the form of (3.1), where Lemma 3.10. If a biquadratic impedance ai > 0, i = 0, 1, 2, bj > 0, j = 0, 1, 2, and R 6= 0, is realizable with four elements, then the number of reactive elements is two or three. R (a) R R R L R C R R (b) L R C D D D GX l GX l GX GX D D D D (c) Proof. Assuming that such two elements exist, the network N can be equivalent to one containing no more than three elements, which implies R = 0 by Lemma 3.8. Thus, this lemma is proved. Lemma 3.9. If a biquadratic impedance Z(s) in the form of (3.1), where ai > 0, i = 0, 1, 2, bj > 0, j = 0, 1, 2, and R 6= 0, is realizable as a four not contain any two elements of the same element network N , then N does kind in series or in parallel. D l In the remaining part of this Dsection, onlyl needs to consider case of E D it E the R 6= 0. E principle of duality, one obtains Figs. 3.3(b) and 3.3(d). It can be verified D D E E by calculation that impedances of these networks satisfy R = 0. (d) Fig. 3.3 Threeelement configurations realizing Z(s) in the form of (3.1), where ai > 0, i = 0, 1, 2, and bj > 0, j = 0, 1, 2, discussed in Lemma 3.8, whose oneterminalpair labeled graphs are (a) N1a , (b) N1b , (c) N1c , and (d) N1d , satisfying N1b = GDual(N1a ), N1c = Inv(N1a ), and N1d = Dual(N1a ), respectively. 3DVVLYH 3DVVLYH /RVVOHV / 1HWZRUN 1HWZRUN 1HWZRU 1 3DVVLYH 3DVVLYH /RVVOHV / 1HWZRUN 1HWZRUN 1HWZRU 1 August 15, 2019 11:32 wsbook9x6 Passive Network Synthesis: Advances with Inerter 11567main Biquadratic Synthesis of OnePort RLC Networks 1 a 1 a' 45 2 a a' (a) (b) 3 1 2 a 1 a' (c) Fig. 3.4 3 a 2 a' (d) Network graphs of the networks with at most three elements (one half). Proof. Assume that there is no more than one reactive element. Then, it follows from [Anderson and Vongpanitlerd (1973), pg. 370] that Z(s) can be expressed as a bilinear function whose McMillan degree is at most one, implying that R = 0. Assume that the network contains four reactive elements. Then, all the poles of Z(s) must be at s = jω or s = ∞ [Guillemin (1957)], which contradicts the fact that all the coefficients are positive. Thus, this lemma is proved. Theorem 3.1. A biquadratic impedance Z(s) in the form of (3.1), where ai > 0, i = 0, 1, 2, bj > 0, j = 0, 1, 2, and R 6= 0, is realizable with four elements, if and only if Z(s) is realizable as one of the configurations shown in F