Principal Passive Network Synthesis: Advances With Inerter
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Passive Network Synthesis: Advances With Inerter

, ,
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 low-degree functions as RLC networks (damper-spring-inerter networks), the synthesis of n-port resistive networks, and the synthesis of low-complexity mechanical networks. They can be directly applied to the optimization and design of various inerter-based 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.
Año:
2020
Editorial:
World Scientific Publishing Co. Pte. Ltd.
Idioma:
english
Páginas:
254
ISBN 10:
9811210888
ISBN 13:
9789811210884
File:
PDF, 6.44 MB
Descarga (pdf, 6.44 MB)

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Passive
Network Synthesis
Advances with Inerter

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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

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Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication 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
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ISBN 978-981-121-087-7

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Printed in Singapore

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Passive Network Synthesis: Advances with Inerter

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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

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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 two-terminal passive element whose force applied at the terminals are
proportional to the relative acceleration. Since any n-port passive electrical network can be constructed with resistors, inductors, capacitors, and
transformers, where transformers can be avoided for the one-port case, the
“birth” of inerters completes the analogy between the passive mechanical networks and electrical networks under the “force-current” 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 positive-real impedance (or admittance) can be realized as a one-port mechanical network consisting of
dampers, springs, and inerters (damper-spring-inerter network), by properly following a network synthesis procedure, such as the Bott-Duffin synthesis.
To date, many investigations have focused on applying damper-springinerter networks to a series of passive or semi-active vibration control systems, such as vehicle suspensions, train suspensions, building vibration
systems, landing gears, wind turbines, etc. The passive mechanical networks are actually positive-real controllers of the vibration control systems,
and system performances can certainly be enhanced by introducing inerters. For the design of inerter-based 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

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(or admittance); the second step is to realize the resulting function as a
damper-spring-inerter network based on the theory of passive network synthesis. Although the realization as a one-port damper-spring-inerter network is always available, the classical synthesis methods and results always
generate many redundant elements, and the synthesis problem of damperspring-inerter 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 inerter-based 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 low-degree functions as RLC networks (damperspring-inerter networks), the synthesis of n-port resistive networks, and the
synthesis of low-complexity mechanical networks. Many of these results can
be directly applied to the optimization and design of various inerter-based
vibration control systems.
In Chapter 1, the development of passive network synthesis and its application to inerter-based 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 positive-realness, 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 n-port 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 low-complexity 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.

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Preface

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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

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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.

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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: Inerter-Based Mechanical Control
Outline of the Book . . . . . . . . . . . . . . . . . . . . .

Positive-Real Function and Foster Preamble
Synthesis of One-Port Lossless Networks . .
The Brune Synthesis . . . . . . . . . . . . .
The Bott-Duffin Synthesis . . . . . . . . . .
The Darlington Synthesis . . . . . . . . . .
Graph Theory for Passive Networks . . . . .
Principle of Duality . . . . . . . . . . . . . .

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11
15
16
19
25
27
33

Biquadratic Synthesis of One-Port 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
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Passive Network Synthesis: Advances with Inerter

3.5

3.6

3.7
3.8

4.

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Realization of Biquadratic Impedances as Five-Element
Bridge Networks . . . . . . . . . . . . . . . . . . . . . . .
3.5.1 Preliminary Lemmas . . . . . . . . . . . . . . . .
3.5.2 Five-Element Bridge Networks with Two Reactive
Elements of the Same Type . . . . . . . . . . . . .
3.5.3 Five-Element Bridge Networks with One Inductor
and One Capacitor . . . . . . . . . . . . . . . . .
3.5.4 Main Results . . . . . . . . . . . . . . . . . . . . .
Generalized Synthesis without Real-Part 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 . . . . . . . . . . . . . . . . . . . . .
Seven-Element Series-Parallel Realizations of a Specific
Class of Biquadratic Impedances . . . . . . . . . . . . . .
3.8.1 Preliminary Lemmas . . . . . . . . . . . . . . . .
3.8.2 Realizations as Three-Reactive Seven-Element
Series-Parallel Networks . . . . . . . . . . . . . . .
3.8.3 Realizations as Four-Reactive Seven-Element
Series-Parallel Networks . . . . . . . . . . . . . . .
3.8.4 Realizations as Five-Reactive Seven-Element
Series-Parallel Networks . . . . . . . . . . . . . . .
3.8.5 Main Results . . . . . . . . . . . . . . . . . . . . .

53
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79
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81
85
89
106
108
110
112
121
126

Synthesis of n-Port Resistive Networks

127

4.1
4.2

127
128
128
130

4.3

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
A Review of n-Port Resistive Network Synthesis . . . . .
4.2.1 Realizations with n ≤ 3 . . . . . . . . . . . . . . .
4.2.2 General Properties of n-Port 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 n-Port Resistive Networks Containing 2n
Terminals . . . . . . . . . . . . . . . . . . . . . . . . . . .

133
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Contents

4.3.1

4.4

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165

Mechanical Synthesis of Low-Complexity One-Port 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 Three-Port 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 Five-Element Damper-Spring
Networks . . . . . . . . . . . . . . . . . . . . . . .
5.3.5 Realizations of Five-Element
Damper-Spring-Inerter Networks . . . . . . . . . .
5.3.6 Final Condition . . . . . . . . . . . . . . . . . . .
Synthesis of a One-Damper One-Inerter 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

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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,
time-invariant, 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 one-port 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 one-port passive network is positive-real, and any positive-real
impedance (resp., admittance) is realizable as a one-port 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 positive-real
impedance (resp., admittance) can be realized as the cascade connection of
a two-port 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 one-port passive network through establishing a transformerless synthesis procedure, which is named the Bott-Duffin synthesis. On
the other hand, it should be noted that transformers are avoided at the
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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 one-port RLC networks using the least
number of elements, many investigations focused on the minimal realization problems of low-degree 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 multi-port 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 multi-port case, even for the simple multi-port 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 inerter-based mechanical control (see the next
section for more details). Chen and Smith [Chen (2007); Chen and Smith
(2009b)] first investigated the low-complexity 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 n-port 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.

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Introduction

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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, time-invariant, 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 n-port linear, time-invariant, and
lumped network can be described by a transfer function matrix, whose
entries are real-rational functions. For instance, the impedance matrix
Z(s) ∈ Rn×n (s) of an n-port network satisfies
ˆ
V̂ (s) = Z(s)I(s),
ˆ are the Laplace transforms
where the n-dimensional 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 n-port 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, time-invariant, 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 one-port 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 positive-real 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 non-negative definite.

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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 n-port
passive network is positive-real. 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 positive-real impedance matrix Z(s) ∈ Rn×n (s) (resp., admittance matrix Y (s) ∈ Rn×n (s)) is realizable as an n-port passive network consisting
of resistors, inductors, capacitors, transformers, and gyrators. Moreover,
if the positive-real impedance matrix Z(s) (resp., admittance matrix Y (s))
is symmetric, then it is realizable as an n-port passive network consisting
of resistors, inductors, capacitors, and transformers.

1.2

New Research Motivation: Inerter-Based 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 two-terminal 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 rack-and-pinion 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 rack-and-pinion inerter [Smith (2002)].

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Introduction

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5

Based on the conventional force-current 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 force-current 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
one-port 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, time-invariant, 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 Bott-Duffin synthesis, any one-port 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 semi-active 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)],

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Table 1.1 The conventional incomplete force-current
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 multi-degree-of-freedom system.
In [Hu and Chen (2015)], the inerter-based dynamic vibration absorber
(IDVA, also known as inerter-based tuned mass damper) is proposed, and

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Introduction

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7

the H∞ and H2 performances for IDVA are evaluated. In [Hu et al. (2015)],
inerter-based isolators are proposed, and analytic solutions for the H∞ and
H2 performances of several inerter-based isolators are derived. In [Chen
et al. (2015a)], the idea of decoupling the inerter-based semi-active suspensions as a passive part and a semi-active part is proposed, and the
semi-active suspensions with the passive part as several given inerter-based
networks are evaluated. In [Chen et al. (2014c)], the idea of semi-active
inerter is proposed. The performance of semi-active inerter for vehicle suspensions is studied in [Chen et al. (2014c, 2016a)]. In [Hu et al. (2017a)],
the physical embodiment of semi-active inerter is proposed by using a
controllable-inertia flywheel. In [Hu et al. (2017b)], the skyhook inerter
idea is proposed, and the semi-active realization of the skyhook inerter
idea by using semi-active inerters is studied. In [Hu et al. (2018a)], a fundamental fact is revealed that mass-chain systems with inerters may have
multiple natural frequencies, and a necessary and sufficient condition for
natural frequency assignment problem of inerter-based mass-chain 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 inerter-based
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 quarter-car 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
one-port 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 positive-real controller to
be determined such that the control system can meet certain requirements.
The complete process of designing an inerter-based vibration control system
is summarized as follows.
• Given a vibration control system, determine a suitable positive-real
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

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%RG\

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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 quarter-car 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

v2-v1
K(s)

Fig. 1.4 Control diagram for the quarter-car vehicle suspension system model, where
the mechanical admittance K(s) is a positive-real controller, w is the external input, and
z is the output to be controlled.

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Introduction

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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 (damper-spring-inerter) 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 multi-port
RLC (damper-spring-inerter) networks is not solved, even for the multi-port
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 long-term impacts on many other areas, such as biometric
image processing [Saeed (2014)], passivity-preserving 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 one-port RLC networks, synthesis of n-port resistive networks, and synthesis of low-complexity mechanical networks.
In Chapter 2, some important preliminaries of passive network synthesis
will be reviewed. For the synthesis of one-port passive networks, some basic
properties of positive-real functions, and some classical one-port 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 one-port 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 one-port 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 n-port resistive networks will be introduced, which are mainly referred to [Chen et al. (2015b);
Wang and Chen (2015)]. The synthesis of n-port resistive networks is another important research topic.

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In Chapter 5, some recent results on the low-complexity 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.

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Chapter 2

Preliminaries of Passive Network
Synthesis

2.1

Positive-Real Function and Foster Preamble

This section will give a brief introduction of positive-real 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
real-rational function H(s) is said to be positive-real if H(s) is analytic and
<(H(s)) ≥ 0 for all s with <(s) > 0, that is, in the open right-half 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 positive-real
functions, then W (H(s)) is positive-real.
Theorem 2.2. [Baher (1984), pg. 27] If H1 (s) and H2 (s) are two positivereal functions, then αH1 (s) + βH2 (s) is positive-real for any α > 0 and
β > 0.
Since 1/s, ks, and s + k are all positive-real 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 positive-real function, then H −1 (s) and
H(s−1 ) are both positive-real.
Corollary 2.2. If H(s) is a positive-real function, then αH(βs) + γ is
positive-real for any α > 0, β > 0, and γ ≥ 0.
11

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A classical criterion for testing positive-realness is presented as follows.
Theorem 2.3. [Baher (1984), pg. 33] A real-rational function H(s) is
positive-real 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 left-half
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 higher-degree functions, the following necessary and sufficient condition
for positive-realness might be easier to use.
Theorem 2.4. [Weinberg and Slepian (1958)] A real-ration 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 positive-real criterion is
useful.
Theorem 2.5. [Chen and Smith (2009a)] A real-ration 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 positive-real 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)

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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 positive-real 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 positive-real.
Theorem 2.6 shows that any positive-real 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 positive-real 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
positive-real function, then H(s) − χ is positive-real, where χ is no larger
than the minimum value of <(H(jω)) for any ω ∈ R ∪ ∞.
Theorem 2.7 shows that any positive-real function maintains the
positive-realness after subtracting a constant that is equal to the minimum
value of the real part of the function on jR∪∞. Therefore, any positive-real
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 real-rational function
H(s) is said to be a minimum function if (i) H(s) is positive-real, (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 positive-real
impedance Z(s), the removal of the poles and zeros on jR ∪ ∞ and the
1 For any real-rational 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].

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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 one-port network is equal
to Z −1 (s), which always exists provided that the impedance Z(s) is nonzero,
the realization of a positive-real admittance Y (s) as a one-port 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 one-port 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 positive-real
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 positive-real 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.

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L1
R1
L2
R2
C1

Fig. 2.1

2.2

A realization of Example 2.1.

Synthesis of One-Port Lossless Networks

Definition 2.5. [Baher (1984), pg. 48] A positive-real 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 one-port 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 one-port 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 one-port 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
one-port 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.

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As a summary, the following result can be established.
Theorem 2.9. [Baher (1984), Chapter 3] The impedance (resp., admittance) of a one-port lossless network must be a reactance function, and any
reactance function is realizable as the impedance (resp., admittance) of a
one-port lossless network consisting of only inductors and capacitors.
Moreover, synthesis results of one-port 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 positive-real 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 positive-real 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 positive-real 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)

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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 positive-real 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 positive-real function with δ(Z2 (s)) =
δ(Z1 (s)) − 2. It should be noted that W1 (s) = Z1 (s) − L1 s contains a zero

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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 positive-realness. However, after the
Brune cycle, Z2 (s) becomes a positive-real 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 positive-real impedance (resp., admittance) is realizable as a one-port 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 positive-real 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 positive-real 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.

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19

The Bott-Duffin Synthesis

Given any positive-real 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
positive-real impedance as a one-port passive network without the use of
transformers, which is called the Bott-Duffin 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
positive-real function H(s), the function
kH(s) − H(k)s
,
R(s) =
kH(k) − sH(s)
is also positive-real 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 positive-realness, 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

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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 positive-real 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 positive-real 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 Bott-Duffin 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)

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21

L2
C1

Z2(s)
C2

Z1(s)
C3

L3

L1
Z3(s)

Fig. 2.5 The Bott-Duffin 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 positive-real 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 Bott-Duffin 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
Bott-Duffin cycle, the following theorem can be obtained.
Theorem 2.12. [Bott and Duffin (1949)] Any positive-real impedance
(resp., admittance) is realizable as a one-port passive network containing
a finite number of resistors, inductors, and capacitors, that is, a one-port
RLC network.

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Z2(s)
C1
C2

L2

Z1(s)
L3
L1

Z3(s)
C3

Fig. 2.6 The Bott-Duffin 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 Bott-Duffin synthesis procedure is stated as follows.
Algorithm (the Bott-Duffin synthesis)
Step 1. Given a positive-real 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 Bott-Duffin cycle in Fig. 2.5 according to (2.10)–(2.15). If X1 < 0,
then realize Z1 (s) as the Bott-Duffin cycle in Fig. 2.6 according to (2.10)
and (2.16)–(2.20).
Step 3. If the remaining positive-real 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 positive-real impedances, until the McMillan degrees of them are
all zero.
It is noted that the Bott-Duffin 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 Bott-Duffin synthesis approaches. The realizations can be regarded
as extensions of the Bott-Duffin synthesis based on the principle of balanced bridges and Y –∆ or ∆–Y transformation. Two types of Pantell’s

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Preliminaries of Passive Network Synthesis



23







modified Bott-Duffin 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 Bott-Duffin 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 Bott-Duffin 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 Bott-Duffin cycle as shown
in Fig. 2.8.


When X1 > 0, any minimum impedance function Z1 (s) is realizable as the

































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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
Bott-Duffin 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 positive-real 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

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25


R







L


























C



L



C





R




(a)



R



(b)











An example for (a) the Brune synthesis and (b) the Bott-Duffin synthesis.


The Darlington Synthesis

Another well-known synthesis procedure of one-port 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 positive-real 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 Bott-Duffin
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





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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 positive-real 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 real-rational function.
in


L
[Van Valkenburg (1960), Section 14.2] that z11 , z22 , and z12 in both Case A
$L

$

$L

$L
$L

$L

$
$

$

$L

$

$

$

$

$

$

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page 27

27

and Case B constitute a positive-real 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 positive-real 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 non-negative definite matrix satisfying k11 k22 − (k12 )2 = 0. Therefore, it can be proved that K (i) fi (s) is realizable as the two-port 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
positive-real 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 two-port lossless configuration realizing K (i) fi (s) [Van Valkenburg

Graph Theory for Passive Networks

For the analysis and synthesis of n-port 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 n-port
RLC (resp., damper-P
spring-inerter) network containing e elements and
v nodes. The augmented$
$


$P

$

P

N
$N

$N

$

$N

$


$

$



$

$Q
Q

$Q

$

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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 three-port 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

$

$

$

$

$

$

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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 (co-tree) 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 cut-set 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 (co-tree),4 if and
only if Gs does not contain any cut-set.
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 co-tree is included.
s
3 After

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Definition 2.13. [Seshu and Reed (1961), pg. 31] For a connected graph
with v vertices, the f -cut-sets with respect to a tree are v − 1 cut-sets, 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 -cut-set determined
by a branch of a tree exactly contains the chords whose corresponding f circuits contain the branch.
To better analyze n-port 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 -cut-set matrix Qf = [qij ] is a (v − 1) × ne matrix,
whose rows correspond to v − 1 f -cut-sets and whose columns correspond
to ne ≥ v − 1 edges. If the jth edge belongs to the ith f -cut-set and has
the same (resp., opposite) orientation as that of the f -cut-set, then qij = 1
(resp., qij = −1); if the jth edge does not belong to the ith f -cut-set, then
qij = 0. Here, the f -cut-set orientation coincides with that of the defining
branch.
Consider an n-port RLC (resp., damper-spring-inerter) 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

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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 -cut-sets 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.

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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 n-port RLC network exists, then any current vector Iˆ (resp., voltage
vector V̂ ) is permitted, which means that the port graph Gp cannot contain
any cut-set (resp., circuit) of the augmented graph. By Theorem 2.16 (resp.,
Theorem 2.15), the port graph must be made part of a co-tree (resp., tree).
Conversely, if the port graph Gp is made part of a co-tree (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 n-port RLC (or damperspring-inerter) 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 co-tree 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 -cut-set 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).

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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 n-port resistive network.
Remark 2.4. If the port graph Gp is exactly a co-tree, 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 -cut-set 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 cross-sign 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 one-port (that is, two-terminal) RLC
(damper-spring-inerter) network N can be described by a one-terminal-pair
labeled graph N with two distinguished terminal vertices (see [Seshu and

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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 one-terminal-pair graph into
its dual (see [Seshu and Reed (1961), Definition 3-12]) 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 one-terminal-pair labeled graph
:= GDu ◦ Inv = Inv ◦ GDu.
An example to illustrate GDu, Inv, and Dual can be referred to Fig. 3.5.
Denoting the one-terminal-pair 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 one-terminal-pair labeled graph is N .
Let Inv(N ) denote the network whose one-terminal-pair 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 one-terminal-pair 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 one-terminal-pair 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 6-5]).
It can be proved that Z(s) (resp., Y (s)) is realizable as the impedance
(resp., admittance) of a network N whose one-terminal-pair 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 one-terminal-pair labeled graph is
7 Such an approach of defining GDu, Inv, and Dual based on one-terminal-pair labeled
graphs was suggested by Professor Rudolf E. Kalman in his private communication with
the first author on July 13, 2014.

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Inv(N ), if and only if Z(s−1 ) (resp., Y (s−1 )) is realizable as the admittance (resp., impedance) of GDu(N ) whose one-terminal-pair labeled graph
is GDu(N ), and if and only if it is realizable as the admittance (resp.,
impedance) of Dual(N ) whose one-terminal-pair 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 one-port network whose one-terminal-pair 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 frequency-inverse 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).

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Chapter 3

Biquadratic Synthesis of One-Port
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 Bott-Duffin synthesis procedure that the RLC
realization of biquadratic impedances can provide important guidance on
positive-real 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 Bott-Duffin synthesis procedure (resp. Pantell’s modified
Bott-Duffin synthesis procedure), nine (resp., eight) elements are needed
to realize the entire class of positive-real biquadratic impedances as seriesparallel (non-series-parallel) 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
Bott-Duffin approach cannot guarantee the minimality of realizations, and
it is necessary to discuss the realization problem of a biquadratic impedance
as a k-element 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 three-reactive five-element networks and six-element series-parallel
37

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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 [Bar-Lev
(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 six-element series-parallel 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 two-reactive five-element series-parallel
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 series-parallel [Hughes and Smith (2014)] and non-series-parallel 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 one-port 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.

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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 positive-real, 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 positive-real biquadratic
impedance (3.1), with any of its six parameters equal to zero, can be realized by a series-parallel network with no more than two reactive elements
and two resistive elements through the Foster preamble.
Definition 3.1. Consider a one-port 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 reactive-element graph, whose edges are called
reactive-element edges. The subgraph with edges corresponding to resistors
is called the resistor graph, whose edges are called resistor edges.

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Let P(i, j) denote a path whose two end-vertices are i and j, and let
C(i, j) denote a cut-set that separates the network graph into two connected subgraphs containing vertices i and j, respectively. Then, denote
a path P(i, j) (resp., cut-set C(i, j)) whose edges only correspond to resistors, inductors, or capacitors as R-P(i, j), L-P(i, j), or C-P(i, j) (resp.,
R-C(i, j), L-C(i, j), or C-C(i, j)), respectively. Denote a path P(i, j) (resp.,
cut-set C(i, j)) whose edges exactly correspond to two kinds of elements,
which are inductors-capacitors, resistors-inductors, or resistors-capacitors,
as LC-P(i, j), RL-P(i, j), or RC-P(i, j) (resp., LC-C(i, j), RL-C(i, j), or
RC-C(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 R-E(i, j) denote a resistor edge
incident with vertices i and j.
Also, for any one-port 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 cut-set
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 cut-set 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 cut-set 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.

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GXDO
GXDO





l
l



EE













l
l


















GXDO
GXDO
l
l

























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Biquadratic Synthesis of One-Port 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

ws-book9x6









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 positive-real 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

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Passive Network Synthesis: Advances with Inerter

then the corresponding Zc (s) must be realizable as another one Nc , with
the same one-terminal-pair 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 one-terminal-pair
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 oneterminal-pair 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 positive-real 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.

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Biquadratic Synthesis of One-Port 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 cut-set 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

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GXDO GXDO

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GXDO
GXDO
Passive Network Synthesis:
Advances
with Inerter

ws-book9x6

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






11567-main


















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 Three-element 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 one-terminal-pair
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

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Biquadratic Synthesis of One-Port 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