Parodi m linear and nonlinear circuits basic and advanced concepts vol 2 2020

In Volume 1 of Linear and Nonlinear Circuits: Basic & Advanced Concepts,1afterintroducing basic concepts, we considered only circuits containing memorylesscomponents, whose equations bot

Lecture Notes in Electrical Engineering 620

Shanben Chen, Materials Science and Engineering, Shanghai Jiao Tong University, Shanghai, China Tan Kay Chen, Department of Electrical and Computer Engineering, National University of Singapore, Singapore, Singapore

R üdiger Dillmann, Humanoids and Intelligent Systems Lab, Karlsruhe Institute for Technology, Karlsruhe, Baden-W ürttemberg, Germany

Haibin Duan, Beijing University of Aeronautics and Astronautics, Beijing, China

Gianluigi Ferrari, Universit à di Parma, Parma, Italy

Manuel Ferre, Centre for Automation and Robotics CAR (UPM-CSIC), Universidad Polit écnica de Madrid, Madrid, Spain

Sandra Hirche, Department of Electrical Engineering and Information Science, Technische Universit ät

M ünchen, Munich, Germany

Faryar Jabbari, Department of Mechanical and Aerospace Engineering, University of California, Irvine, CA, USA

Limin Jia, State Key Laboratory of Rail Traf fic Control and Safety, Beijing Jiaotong University, Beijing, China Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

Alaa Khamis, German University in Egypt El Tagamoa El Khames, New Cairo City, Egypt

Torsten Kroeger, Stanford University, Stanford, CA, USA

Qilian Liang, Department of Electrical Engineering, University of Texas at Arlington, Arlington, TX, USA Ferran Martin, Departament d ’Enginyeria Electrònica, Universitat Autònoma de Barcelona, Bellaterra, Barcelona, Spain

Tan Cher Ming, College of Engineering, Nanyang Technological University, Singapore, Singapore Wolfgang Minker, Institute of Information Technology, University of Ulm, Ulm, Germany

Pradeep Misra, Department of Electrical Engineering, Wright State University, Dayton, OH, USA

Sebastian M öller, Quality and Usability Lab, TU Berlin, Berlin, Germany

Subhas Mukhopadhyay, School of Engineering & Advanced Technology, Massey University, Palmerston North, Manawatu-Wanganui, New Zealand

Cun-Zheng Ning, Electrical Engineering, Arizona State University, Tempe, AZ, USA

Toyoaki Nishida, Graduate School of Informatics, Kyoto University, Kyoto, Japan

Federica Pascucci, Dipartimento di Ingegneria, Universit à degli Studi “Roma Tre”, Rome, Italy

Yong Qin, State Key Laboratory of Rail Traf fic Control and Safety, Beijing Jiaotong University, Beijing, China Gan Woon Seng, School of Electrical & Electronic Engineering, Nanyang Technological University, Singapore, Singapore

Joachim Speidel, Institute of Telecommunications, Universit ät Stuttgart, Stuttgart, Baden-Württemberg, Germany

Germano Veiga, Campus da FEUP, INESC Porto, Porto, Portugal

Haitao Wu, Academy of Opto-electronics, Chinese Academy of Sciences, Beijing, China

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Linear and Nonlinear

Circuits: Basic and Advanced Concepts

Volume 2

123

Mauro Parodi

Department of Electric, Electronic,

Telecommunications Engineering and Naval

Architecture (DITEN)

University of Genoa

Genoa, Italy

Marco StoraceDepartment of Electric, Electronic,Telecommunications Engineering and NavalArchitecture (DITEN)

University of GenoaGenoa, Italy

ISSN 1876-1100 ISSN 1876-1119 (electronic)

Lecture Notes in Electrical Engineering

ISBN 978-3-030-35043-7 ISBN 978-3-030-35044-4 (eBook)

https://doi.org/10.1007/978-3-030-35044-4

© Springer Nature Switzerland AG 2020

This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part

of the material is concerned, speci fically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on micro films or in any other physical way, and transmission

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The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made The publisher remains neutral with regard

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This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

In Volume 1 of Linear and Nonlinear Circuits: Basic & Advanced Concepts,1afterintroducing basic concepts, we considered only circuits containing memorylesscomponents, whose equations (both topological and descriptive) are algebraic.This volume is focused on components with memory and on circuits charac-terized by time evolution, that is, by dynamics The volume is articulated in threeparts, and its structure follows the guidelines described in the preface to the wholebook (Volume 1), with each part articulated in two independent lecture levels: basicand advanced The basic chapters are devoted to linear components and circuitswith memory, whereas the advanced chapters focus on nonlinear circuits, whosenonlinearity is provided by memoryless components.

The analysis of dynamical circuits is carried out by making reference to theso-called state variables, and the concept of state runs through the whole volume,innervating with dendritic structure the language used to describe the concepts andeven the equations describing a given circuit Moreover, owing to their proximity tothe concept of energy, the state variables are an effective tool for understanding thebehavior of a circuit not only from a mathematical standpoint, but also from aphysical perspective

The formalism adopted in this book to describe the state equations is based onsystem theory and on its general results, so that each circuit (together with itsproperties) can be viewed as a particular physical system To this end, the proposedtheoretical results are intermingled with case studies that instantiate general ideasand point out methodological and applicative consequences

Whenever possible, the circuits are compared to physical systems of differentnatures (mechanical or biological, for instance) that are governed by equations andproperties completely similar and therefore exhibit the same dynamical behaviors

To this end, we note the importance of normalizations, which make it possible toanalyze, with the same conceptual tools, models of physical systems of any nature

1 Published by Springer in 2018 (Print ISBN: 978-3-319-61233-1; eBook ISBN: 978-3-319-61234-8).

v

The reader’s comprehension of the proposed concepts can be checked by solvingthe problems appearing at the end of each chapter (mainly the basic chapters) andcomparing the obtained results with the solutions provided at the end of thisvolume.

Part V shows how to analyze circuits with one state variable, that is,first-ordercircuits In the linear case (treated in Chap.9, basic), the complete analyticalsolution can be obtained based on well-assessed approaches In the nonlinear case(Chap.10, advanced), we usually cannot easily find a closed-form analyticalsolution, but we can discover some important information about the generalproperties of the solution(s) To this end, we employ the tools of nonlineardynamics and bifurcation theory

Part VI generalizes the above concepts to higher-order circuits, both linear(Chap.11, basic) and nonlinear (Chap.12, advanced)

Part VII is focused on the analysis of periodic solutions, that is, on circuitsexhibiting persistent oscillations In the linear case (Chap.13, basic), we describehow to analyze circuits by working in the so-called sinusoidal steady state, induced

by a sinusoidal input In the nonlinear case (Chap.14, advanced), we show how toanalyze circuits (called nonlinear oscillators) by working in the periodic steadystate also in the absence of a forcing periodic input (autonomous oscillators) Theanalysis can also be carried out when some circuit parameters are varied, thusinducing qualitative changes (bifurcations) in the circuit dynamics and switchingthe circuit steady state from periodic to stationary or quasiperiodic or even chaotic.Some examples of oscillators (also of a noncircuit nature) are proposed and studied,isolated or coupled or even networked Also in these cases, the circuit analysis can

be easily related to energetic balances; this allows one to relate the general but oftenabstract concepts provided by system theory to the physics of the consideredexamples In our opinion, this is a great help in understanding not only the specificcase under consideration, but also the general laws it obeys With this perspective,

as already stated in the general preface to the whole book (see Volume 1), circuitsrepresent an excellent environment for better understanding the relationshipsbetween physics, mathematics, and system theory

We are indebted to our friend Lorenzo Repetto, who carefully revised the liminary version of this volume, reporting bugs and providing detailed comments

pre-We also acknowledge our colleagues and friends Giovanni Battista Denegri for hishelpful comments on Sect 13.9.2, Matteo Lodi for his invaluable help with thesimulations described in Sect.14.9, and Alberto Oliveri for constructive discussionsand comments

Genoa, Italy

September 2019

Mauro ParodiMarco Storace

Part V Components with Memory and First-Order Dynamical

Circuits

9 Basic Concepts: Two-Terminal Linear Elements with Memory

and First-Order Linear Circuits 3

9.1 Two-Terminal Linear Elements with Memory 3

9.1.1 Capacitor 3

9.1.2 Inductor 5

9.2 Capacitor and Inductor Properties 7

9.2.1 Energetic Behavior 7

9.2.2 Gyrator and Two-Terminals with Memory 8

9.2.3 Series and Parallel Connections 9

9.3 State and State Variables 10

9.3.1 Wide-Sense and Strict-Sense State Variables 11

9.3.2 Circuit Models of Algebraic Constraints 13

9.3.3 State Variables Method 16

9.4 Solution of First-Order Linear Circuits with One WSV 17

9.4.1 General Solution of the State Equation 20

9.4.2 Free Response and Forced Response 22

9.4.3 Circuit Stability 30

9.5 Forced Response to Sinusoidal Inputs 33

9.5.1 Sinusoids and Phasors 33

9.5.2 Phasor-Based Method for Finding a Particular Integral 36

9.5.3 Multiple Periodic Inputs: Periodic and Quasiperiodic Waveforms 38

9.6 Generalized Functions (Basic Elements) 41

9.7 Discontinuity Balance 44

vii

9.8 Response of Linear Circuits with One WSV and One SSV

to Discontinuous Inputs 45

9.8.1 Step Response 46

9.8.2 Impulse Response 48

9.9 Convolution Integral 52

9.10 Circuit Response to More Complex Inputs 57

9.10.1 Multi-input Example 57

9.11 Normalizations 62

9.12 Solution for Nonstate Output Variables 65

9.13 Thévenin and Norton Equivalent Representations of a Charged Capacitor/Inductor 66

9.13.1 Charged Capacitor 67

9.13.2 Charged Inductor 69

9.14 First-Order Linear Circuits with Several WSVs 70

9.15 Problems 79

References 89

10 Advanced Concepts: First-Order Nonlinear Circuits 91

10.1 Asymptotic Solution of a Particular Class of First-Order Nonlinear Circuits 91

10.1.1 First-Order Circuits with More Than One WSV 98

10.1.2 Impossibility of Oscillations 99

10.1.3 Equilibrium Stability Analysis Through Linearization 100

10.2 Equilibrium Points and Potential Functions for First-Order Circuits 101

10.3 Analysis of First-Order Circuits with PWL Memoryless Components 107

10.3.1 Clamper 107

10.3.2 Half-Wave Rectifier 109

10.3.3 Hysteretic Circuit 110

10.3.4 Circuit Containing an Operational Amplifier 113

10.3.5 Circuit Containing a BJT 117

10.4 Bifurcations 121

10.4.1 Linear Case 121

10.4.2 Nonlinear Case 122

10.5 A Summarizing Example 126

10.5.1 Inverting Schmitt Trigger 127

10.5.2 Dimensionless Formulation 128

10.5.3 Analysis with Constant Input 129

10.5.4 Potential Functions 133

10.6 Problems 136

References 139

Part VI Second- and Higher-Order Dynamical Circuits

11 Basic Concepts: Linear Two-Ports with Memory

and Higher-Order Linear Circuits 143

11.1 Coupled Inductors 143

11.2 Properties of Coupled Inductors 147

11.2.1 Series Connection 147

11.2.2 Passivity 148

11.2.3 Coupling Coefficient and Closely Coupled Inductors 149

11.2.4 Energy Conservation 150

11.2.5 Equivalent Models 154

11.2.6 Thévenin and Norton Equivalent Representations of Charged Coupled Inductors 158

11.3 Higher-Order Linear Circuits 159

11.3.1 General Method 160

11.3.2 Complementary Component Method and State Equations 161

11.3.3 State Equations in Canonical Form and I/O Relationships 163

11.4 Discontinuity Balance 168

11.5 Solution of the State Equations: Free Response and Forced Response 172

11.5.1 Free Response 173

11.5.2 Forced Response 180

11.6 Circuit Stability 193

11.7 Normalizations and Comparisons with Mechanical Systems 202

11.7.1 A Double Mass–Spring Chain and Its Circuit Model 205

11.8 Solution for Nonstate Output Variables 207

11.9 Response of LTI Dynamical Circuits to Discontinuous Inputs 208

11.10 Generic Periodic Inputs 211

11.10.1 Fourier Series 212

11.10.2 Some Supplementary Notes About Fourier Series 214

11.10.3 Mean Value of Circuit Variables 218

11.10.4 Root Mean Square Value 221

11.11 Multi-input Example 230

11.12 Problems 234

References 247

12 Advanced Concepts: Higher-Order Nonlinear Circuits—State

Equations and Equilibrium Points 249

12.1 Nonlinear State Equations 249

12.1.1 Existence of the State Equation 252

12.2 Linear Autonomous Circuits Revisited 261

12.2.1 Second-Order Circuits 262

12.2.2 nth-Order Circuits 272

12.3 Nonlinear Dynamical Circuits: Assumptions and General Properties 274

12.4 Equilibrium Stability Analysis Through Linearization 280

12.5 Bifurcations 283

12.5.1 Fold Bifurcation of Equilibria 283

12.5.2 Hopf Bifurcation 284

12.6 Wien Bridge Oscillator 289

12.6.1 Normalized State Equations 293

12.6.2 Hopf Bifurcation of the Equilibrium Point 294

12.7 Colpitts Oscillator 297

12.7.1 Normalized State Equations 299

12.7.2 Hopf Bifurcation of the Equilibrium Point 301

12.8 Small-Signal Analysis 304

12.8.1 Relationship with the Supercritical Hopf Bifurcation 316

References 317

Part VII Analysis of Periodic Solutions 13 Basic Concepts: Analysis of LTI Circuits in Sinusoidal Steady State 321

13.1 Sinusoidal Steady State 321

13.2 Circuit Equations in Terms of Phasors 322

13.2.1 Topological Equations 322

13.2.2 Descriptive Equations 323

13.3 Impedance and Admittance of Two-Terminal Elements 326

13.3.1 Impedance 326

13.3.2 Admittance 328

13.3.3 Relation Between Impedance and Admittance of a Two-Terminal Component 330

13.3.4 Series and Parallel Connections of Two-Terminal Elements 331

13.3.5 Reciprocity 334

13.4 Thévenin and Norton Equivalent Representations of Two-Terminal Elements 334

13.5 Two-Port Matrices 337

13.6 Thévenin and Norton Equivalent Representations

of Two-Port Elements 342

13.7 Sinusoidal Steady-State Power 347

13.7.1 Two-Terminal Components 347

13.7.2 Generic Components 352

13.8 Boucherot’s Theorem 356

13.9 Power Factor Correction 360

13.9.1 Advantages for the Consumer 367

13.9.2 Advantages for the Utility Company and Motivation for High-Voltage Transmission Lines 367

13.10 Theorem on the Maximum Power Transfer 370

13.11 Frequency Response 373

13.11.1 Network Functions of a Circuit 373

13.11.2 Some Properties of the Network Functions 374

13.12 Resonant Circuits and the Q Factor 383

13.13 Problems 390

14 Advanced Concepts: Analysis of Nonlinear Oscillators 401

14.1 Periodic Solutions and (Limit) Cycles 401

14.2 Poincaré Section 403

14.3 Floquet Multipliers 407

14.4 Poincaré Map 412

14.5 Stability of Generic Invariant Sets: Lyapunov Exponents 413

14.5.1 Particular Cases 418

14.6 Bifurcations 419

14.6.1 Fold Bifurcation of Limit Cycles 421

14.6.2 Flip Bifurcation 423

14.6.3 Neimark–Sacker Bifurcation 427

14.6.4 Homoclinic Bifurcations 434

14.7 Hindmarsh–Rose Neural Oscillator 441

14.7.1 Analysis of Equilibrium Points 442

14.7.2 Analysis of Limit Cycles 443

14.7.3 Equivalent Circuit 446

14.8 Forced Oscillators 450

14.9 Networks of Coupled Oscillators 455

14.9.1 Example 1: 3-Cell Network 457

14.9.2 Example 2: 4-Cell Network 462

14.9.3 Example 3: 30-Cell Network 465

14.10 Summarizing Comments 469

References 471

Appendix A: Complex Numbers 475

Appendix B: Synoptic Tables 479

Solutions 483

Index 513

About the Authors

Mauro Parodi was appointed full professor of Basic Circuit Theory by theEngineering Faculty at the University of Genoa, Italy, back in 1985 His scientific andteaching activity has been focusing on nonlinear circuits and systems theory, non-linear modeling, and mathematical methods for treatment of experimental data He iscurrently affiliated with the Department of Electrical, Electronic, Telecommuni-cations Engineering and Naval Architecture at the University of Genoa, where he hasbeen teaching Mathematical Methods for Engineers and Applied MathematicalModeling

Marco Storace received a Ph.D degree in Electrical Engineering from theUniversity of Genoa, Italy, in 1998 He was appointed full professor by the sameuniversity in 2011 and is currently affiliated with the Department of Electrical,Electronic, Telecommunications Engineering and Naval Architecture He was avisiting professor atÉcole Polytechnique Fédérale de Lausanne (EPFL), Lausanne,Switzerland, in 1998 and in 2002 The main focus of his research is on theory andapplications of nonlinear circuits, with a special emphasis on circuit models ofnonlinear systems, such as systems with hysteresis and biological neurons He isalso concerned with methods for piecewise linear approximation (and circuit syn-thesis) of nonlinear systems, and bifurcation analysis and nonlinear dynamics alike

He has been teaching Basic Circuit Theory, Analog and Digital Filters, andNonlinear Dynamics at the University of Genoa From 2008 to 2009 he served as anassociate editor of the IEEE Transactions on Circuits and Systems He is serving as

a Chair Elect (2019/2021) of the IEEE Technical Committee on Nonlinear Circuitsand Systems (TC-NCAS)

Mauro Parodi and Marco Storace are also the authors of Linear and NonlinearCircuits: Basic & Advanced Concepts—Volume 1 published by Springer in 2018,ISBN 978-3-319-61233-1 (Hardcover), ISBN 978-3-319-61234-8 (eBook)

xiii

AC Alternating current

CCCS Current-controlled current source

CCVS Current-controlled voltage source

DC Direct current

DP Driving point

KCL Kirchhoff’s current law

KVL Kirchhoff’s voltage law

l.h.s Left-hand side

LTI Linear and time-invariant

ODE Ordinary differential equation

PWL Piecewise-linear

r.h.s Right-hand side

RMS Root mean square

SI International system of units

SSV Strict-sense state variable

VCCS Voltage-controlled current source

VCVS Voltage-controlled voltage source

WSV Wide-sense state variable

ZIR Zero-input response

ZSR Zero-state response

xv

Part V

Components with Memory and First-Order Dynamical Circuits

Basic Concepts: Two-Terminal Linear

Elements with Memory and First-Order

Linear Circuits

Whenever humanity seems condemned to heaviness, I think I should fly like Perseus into a different space I don’t mean escaping into dreams or the irrational I mean that I have to change my approach, look at the world from a different perspective, with a different logic and with fresh methods of cognition and verification.

—Italo Calvino, Six Memos for the Next Millennium

Abstract In this chapter, we introduce two-terminal components with memory,

whose descriptive equations involve the time derivative of one descriptive variable.This implies generally speaking, that the circuit is described by a system of algebraicdifferential equations As a consequence, the generic circuit variable can no longer

be expressed as an algebraic function of the inputs, but has its own dynamics, which

will be analyzed for simple (first-order) circuits The concept of circuit stability isintroduced Moreover, discontinuous functions are defined, in particular the so-calledunit step and impulse functions, which are often used in circuit inputs

Here we introduce the descriptive equations for two widely used components withmemory

The capacitor (originally called a condenser) is a passive electrical device (some

examples are shown in Fig.9.1a), whose two-terminal model is shown in Fig.9.1b

© Springer Nature Switzerland AG 2020

M Parodi and M Storace, Linear and Nonlinear Circuits: Basic and Advanced Concepts,

Lecture Notes in Electrical Engineering 620,

https://doi.org/10.1007/978-3-030-35044-4_9

3

4 9 Basic Concepts: Two-Terminal Linear Elements with Memory …

Fig 9.1 Capacitor: a six

physical devices of different

It is apparent from the descriptive equation that knowing the specific value of v

at a given time t does not provide any information about the corresponding value

of i (t) Analogously, knowing the specific value of i at a given time t does not

provide any information about the corresponding value of v (t), since (by integrating

the descriptive equation and assuming that v (−∞) = 0)

v (t) = 1

C

 t

−∞i (τ)dτ. (9.2)

In both cases, to obtain one of the descriptive variables, we have to know the other

one’s history; in other words, this component keeps memory of the past Then it is

linear, time-invariant (assuming that the capacitance is constant), with memory.The shapes of real devices vary widely (see Fig.9.1a), but most of them contain

at least two electrical conductors (plates) separated by a dielectric; the model inFig.9.1b sketches this physical situation The conductors are typically thin films ormetal foils Materials commonly used as dielectrics include glass, ceramic, plasticfilm, paper, mica, and oxide layers

1 The unit is named after Michael Faraday (1791–1867), an English scientist who contributed to the study of electromagnetism and electrochemistry His main discoveries include the principles underlying electromagnetic induction, diamagnetism, and electrolysis.

The physical principle at the basis of the capacitor’s behavior is quite simple.

When a voltage v is applied across the conductive plates, a positive charge +q accumulates on one plate and a negative charge q on the other plate The ratio of the electric charge q (t) to the voltage v(t) is the capacitance C, namely q = Cv, where

the positive coefficient C depends on the physical (material) and geometric (shape)

characteristics of both dielectric and plates This could be chosen as the capacitor’sdescriptive equation, but in this book we have from the beginning chosen voltagesand currents as descriptive variables, since they can be easily measured (see Sect 1.3

in Vol 1) Then, by deriving this equation with respect to t, under the assumption that

C does not change with time, we obtain the capacitor’s descriptive equation (Eq.9.2)

in terms of the variables v and i

The term capacitor refers to the property of storing charge In any physical device,this can be done up to a given limit, after which the dielectric is no more able toinsulate the two plates, changes its nature and allows a flow of ohmic current, as if

it were a resistor This limit corresponds to the so-called breakdown voltage.

Other properties will be discussed in Sect.9.2

Particular case: when v (t) is constant, the descriptive equation reduces to

i (t) = 0, that is, the capacitor becomes equivalent to an open circuit, apart from

the aspects related to the stored energy (that will be treated in Sect.9.2.1) This is

the typical situation of the so-called DC steady state, as we will see in Sect.9.4

The inductor (also called coil or reactor) is a passive electrical device (some examples

are shown in Fig.9.2a), whose two-terminal model is shown in Fig.9.2b

Fig 9.2 Inductor: a three physical devices of different nature; b model

6 9 Basic Concepts: Two-Terminal Linear Elements with Memory …

The inductor descriptive equation is

v (t) = L di

L is a parameter called inductance The (derived) SI unit of measurement of

inductance is the henry,2whose symbol is H According to the descriptive equation,

it is evident that [H] = [V s A−1] Inductance values of typical inductors for use inelectronics range from microhenry to henry (see Table B.1 in Appendix B.1, Vol 1)

As well as the capacitor, the inductor is a component linear, time-invariant ing that the inductance is constant) and with memory

(assum-The shapes of real devices vary widely (see Fig.9.2a), but most of them tain at least a coil of conductive material, whose turns are often wound around aferromagnetic core (usually made of ferrite); the model in Fig.9.2b sketches a coilwinding

con-The physical principle at the basis of the inductor behavior is quite simple When

a current i flows through the coil, it induces a magnetic flux φ The ratio of the

magnetic fluxφ(t) to the current i(t) is the inductance L, namely, φ = Li, where

the positive coefficient L depends on the physical (material) and geometrical (e.g.,

shape, number of turns) characteristics of both coil and core Once more, this could

be chosen as the inductor descriptive equation, but we want voltages and currents

as descriptive variables Then, by deriving this equation with respect to t, under the assumption that L does not change with time, we obtain the inductor descriptive

equation (Eq.9.3) in terms of the variables v and i

In this book, we assume (unless otherwise stated) that the inductance is constantand then that the component is linear and time-invariant These are assumptions notalways satisfied by real inductors; for instance, inductors with ferromagnetic coresare nonlinear, since the inductance changes with the current Moreover, the modelneglects the presence of resistance (due to the resistance of the wire and energy losses

in core material) and capacitance in real devices

Remark: Every wire or other conductor generates a magnetic field when current

flows through it, so every portion of wire has some inductance

Particular case: when i (t) is constant, the descriptive equation reduces to

v (t) = 0, that is, the inductor becomes equivalent to a short circuit, apart from the

aspects related to the stored energy (see the next section) This is the typical situation

of the DC steady state (see Sect.9.4)

2 The unit is named after Joseph Henry (1797–1878), an American scientist who discovered the tromagnetic phenomenon of self-inductance He also discovered mutual inductance, independently

elec-of Michael Faraday.

9.2 Capacitor and Inductor Properties

It is common to assume that v (−∞) = 0 and i(−∞) = 0 In other words, we

assume that there exists a time in the past at which both the capacitor and the inductorwere uncharged Owing to this assumption, we have

w C (t) = 1

2Cv2(t) and w L (t) = 1

Capacitor and inductor are passive components, so we must have w C (t) ≥ 0 and

w L (t) ≥ 0 for all t (see Sect 3.3.5 in Vol 1) These inequalities are valid if and

only if we have, as anticipated, C > 0 and L > 0 respectively Moreover, passivity

also means that these components cannot deliver more energy than they absorbedpreviously However, in contrast to the resistor, which dissipates the absorbed power,

the capacitor and inductor store the absorbed energy; in particular, the capacitor

stores it through an electric field, the inductor through a magnetic field In otherwords, the resistor is passive and dissipative, whereas the capacitor and inductor are

passive and conservative This can be easily proved as follows (the proof is given for

the capacitor, but can be applied, mutatis mutandis, also to the inductor)

The energy variation between two generic times t A and t B > t Ais

Δw C = w C (t B ) − w C (t A ) = 1

2C (v2(t B ) − v2(t A )).

This means that if v (t B ) = v(t A ), the energy variation is 0, even if w C (t) changed in

the interval[t A , t B ] and independently of the shape of v(t) between v(t A ) and v(t B ),

that is, of the path followed Figure9.3shows two different voltage paths I and II,which connect the same starting and ending points, thus corresponding to the sameenergy variation

In mechanics, a completely analogous property holds when we consider a particlesubject to the action of a conservative3force: the work done by this force in movingthis particle from one point to another depends only on the initial and final positions

3 Two examples of conservative forces are gravitational forces and elastic spring forces.

8 9 Basic Concepts: Two-Terminal Linear Elements with Memory …

Fig 9.3 The capacitor as a conservative element

of the particle (with respect to some coordinate system), and it is independent of thepath followed

As stated in Vol 1, Sect 5.7.2, the most interesting application of the gyrator isthe possibility of converting an inductor into a capacitor and vice versa Indeed, if

a capacitor is connected to the second port of a gyrator, as shown in Fig.9.4a, we

Fig 9.4 a Inductor obtained by connecting a gyrator and a capacitor; b capacitor obtained by

connecting a gyrator and an inductor

This confirms that when a capacitor of capacitance C is connected to the second port of a gyrator with gyration conductance g m, the two-terminal “viewed” at the

first port is equivalent to an inductor of inductance C /g2

The parallel connection of two capacitors C A and C B (see Fig.9.5a) is equivalent

(macromodel) to a single capacitor with capacitance C A + C B (see Fig.9.5b)

This can be shown by considering that i A = C A

dt This is in line with the physical interpretation of the

capacitor as a container of charge: for any voltage v, the overall charge is the sum of the individual contributions C A v and C B v.

The series connection of two capacitors C A and C B (see Fig.9.6a) is equivalent(macromodel) to a single capacitor with capacitance C A C B

10 9 Basic Concepts: Two-Terminal Linear Elements with Memory …

Fig 9.7 a Series connection of inductors; b equivalent macromodel

Fig 9.8 a Parallel connection of inductors; b equivalent macromodel

Similarly, you can easily check that the series connection of two inductors L Aand

L B (see Fig.9.7a) is equivalent to a single inductor with inductance L A + L B (seeFig.9.7b), whereas the parallel connection of L A and L B(see Fig.9.8a) is equivalent

to a single inductor with inductance L A L B

L A + L B

(see Fig.9.8b)

9.3 State and State Variables

Systems described by a set of algebraic differential equations evolve with time, that

is, they are dynamical systems The presence of dynamics is related to memory, and

so modeling the evolution of a dynamical system requires some information about its

history The state of a dynamical system at a given time t0is the information necessary

and sufficient to summarize the circuit history before t0, that is, the information that,together with the knowledge of the system inputs4 for any t ≥ t0, allows one to

predict the state evolution for any t ≥ t0

Because the circuits are particular systems, we can define the state of a circuit asfollows

The state of a circuit at a given time t0 is a set of independent initial

con-ditions that are necessary and sufficient to determine the circuit evolution for

all t ≥ t0, provided that the system inputs are known for all t ≥ t0 The word

“independent” plays a key role; it means that these initial conditions might befixed arbitrarily, that is, there are no algebraic constraints involving exclusivelyone or more of these terms (and possibly one or more circuit inputs)

4 In a circuit, the inputs are voltages/currents impressed by voltage/current sources.

Fig 9.9 Example of a

circuit with state

The state variables are the circuit variables corresponding to the chosen set

of independent initial conditions

For instance, as shown in Vol 1, a memoryless circuit can be solved at any timewithout need of information about its past history, in which case it is stateless Incontrast, in the very simple circuit shown in Fig.9.9, the voltage v can be determined for some time t ≥ t0only if we know the circuit’s initial condition v (t0) The capacitor

descriptive equation and Kirchhoff’s current law (KCL) immediately give a (t) =

a (τ)dτ Then, given a(t) for t ≥ t0, knowing v for

t ≥ t0requires the knowledge of the initial condition v (t0) We remark that v is the

only state variable of this circuit

The above definition of state has two remarkable implications The first is that inthe presence of algebraic constraints, the choice of state variables can be nonunique.The second is that the circuit variables can be divided into two sets: the state variablesand the other circuit voltages and currents

On the one hand, the first implication requires an answer to the following question:

how can we choose the state variables? On the other hand, the second implication

provides a way to simplify the circuit analysis, which is called the state variables

method.

How can we choose the state variables? We can begin by considering the so-called

wide-sense state variables (WSVs) of the circuit, which are all the voltages across

capacitors and all the currents through inductors Indeed, the capacitor (inductor) is

a conservative component, and at time t0, its voltage (current) is all we need to know

about the component’s previous history to predict its evolution for t ≥ t0, owing tothe path-independence proved in Sect.9.2.1 The voltage of each capacitor and the

current of each inductor are candidates for being state variables, but they are not necessarily so, since they also have to be independent.

For instance, the circuit shown in Fig.9.10contains a capacitor, in which case v

is a WSV But in this circuit, KVL imposes that v (t) = e(t), which means that we

12 9 Basic Concepts: Two-Terminal Linear Elements with Memory …

Fig 9.10 Example of a circuit with one WSV and no SSV

Fig 9.11 Circuits with number of SSVs less than the number of WSVs

cannot impose arbitrarily an initial condition v (t0) This implies that v(t) cannot be a

state variable In fact, in this case we do not need an initial condition to find any circuit

variable, which can be found algebraically, because v (t) = e(t) and i(t) = C de

dt On

the other hand, in the “similar” example of Fig.9.9, there are no algebraic constraints

on v (t); thus we can have any initial condition v(t0) (which is needed to find v(t), as

shown above), and v (t) is a state variable.

The real state variables of a circuit are called strict-sense state variables (SSVs)

and are a subset of independent WSVs, that is, a set of WSVs none of which isrelated by algebraic constraints to other WSVs and/or to inputs, that is, independentsources In the example above (Fig.9.10), v was a WSV, but not an SSV, because of the algebraic constraint v (t) = e(t).

Two further examples are shown in Fig.9.11 In the circuit of Fig.9.11a we have

three capacitors and then three WSVs But KVL poses the algebraic constraint e (t) =

v1+ v2+ v3, meaning that the three WSVs are not independent, because a linearcombination of them (their sum) is determined by an input The same conclusionwould hold even in the more general case in which a WSV was a (linear or nonlinear)combination of other WSVs and/or inputs or input time derivatives In the consideredexample, the SSVs are two, and we can arbitrarily choose any pair of capacitorvoltages

In the circuit of Fig.9.11b we have two inductors and then two WSVs But KCL

poses the constraint a (t) = i1+ i2, meaning that the two WSVs are not independent,for the same reason as before In this case, there is only one SSV, and we can arbitrarilychoose any inductor current

In the considered examples, the algebraic relationships between WSVs are due

to specific topological structures, which are quite simple to detect by inspection:

(a) (b)

(c)

Fig 9.12 Circuits with number of SSVs less than the number of WSVs

loops containing only capacitors (as in Fig.9.5a) or capacitors and voltage sources(as in Figs.9.10and9.11a) and cut-sets involving only inductors (as in Fig.9.7a) orinductors and current sources (as in Fig.9.11b) In other cases, the constraints are due

to the presence of controlled sources or other memoryless two-ports Three examplesare shown in Fig.9.12 In the circuit of Fig.9.12a the algebraic constraint between

WSVs is v2+ e(t) = v1/n (KVL for the dashed loop) In the circuit of Fig.9.12b,KVL for the dashed loop (αv = v + e(t)) implies that v is not an SSV Finally, in the

circuit of Fig.9.12c, KCL for the dashed nodal cut-set (i1+ e(t)/R1 = 0) prevents

i1from being an SSV

Each independent algebraic constraint between a set of WSVs (and possibly inputs)implies the exclusion of one of them from the SSV set of the circuit After the variable

to be excluded is chosen (arbitrarily), the corresponding component (capacitor orinductor) can be represented in the circuit by an equivalent model, entirely formulated

in terms of nonstate variables and independent sources Let X denote the set of

WSVs subject to a given algebraic constraint inside a circuit and consider two rathergeneral forms of linear algebraic constraints: the first is concerned with capacitor

14 9 Basic Concepts: Two-Terminal Linear Elements with Memory …

voltages, and the second with inductor currents According to the examples discussed

in the previous section, these constraints on WSVs allow the presence of sources,

represented in compact form by a voltage source e (t) and by a current source a(t),

right-CCCS For all p = k, the driving current i p is that flowing through the pertinent

capacitor C p The last term of the right-hand side of Eq.9.9is an independentcurrent source

As an example, in the (previously discussed) circuit of Fig.9.12a, we can write the

algebraic constraint between WSVs as v2= v1

n − e(t) and replace the capacitor C2

with the model shown in Fig.9.14 After this replacement, the only state variable

in the circuit is v1 A completely analogous result holds, mutatis mutandis, by

choosing as SSV the voltage v2, that is, swapping the roles of the variables v1and

v2inside the algebraic constraint and using the model of the capacitor C1

• Similarly, a linear constraint between inductor currents (collected in the set X )

can be written as 

p :i ∈X

α p i p + a(t) = 0, α p = 0, (9.10)

Fig 9.13 Model for the

capacitor C k, whose voltage

v kis not an SSV

Fig 9.14 Model for the

capacitor C2 in the circuit of

Following the same line of reasoning as in the previous case, we can decide to

exclude i k ∈ X from the SSV set To this end, we write

The structure of this result is identical, mutatis mutandis, to that found in the

previous case Again, notice that the expression obtained does not involve state variables The corresponding circuit model for L kcontains a set of VCVSs, each

driven by a voltage v p ( p = k) and an independent voltage source, as shown in

Fig.9.15 As an example, the algebraic constraint between WSVs in the circuit

of Fig.9.11b can be written as i2 = −i1+ a(t) Therefore, in the circuit we can replace, for instance, the inductor L2(whose descriptive variables are i2and v2)with the model shown in Fig.9.16

16 9 Basic Concepts: Two-Terminal Linear Elements with Memory …

Fig 9.15 Model for the

inductor L k whose current i k

is not an SSV

Fig 9.16 Model for the

inductor L2 in the circuit of

Fig 9.11 b

Given a circuit containing N two-terminal linear components with memory (namely, N WSVs) and subject to M ( < N) independent algebraic constraints, these

models allow one to redraw the circuit so that it contains only the N − M

mem-ory components associated with the chosen SSVs and to write directly the circuitequations consistent with this setting

As will be shown later (see Sect.9.12), every other circuit variable can be expressedalgebraically in terms of state variables and inputs Owing to this general property,the original system of algebraic differential equations can be solved in two simplersteps:

• first, we solve a system of differential equations whose only unknowns are the

state variables (i.e., the SSVs);

• then, we solve a system of algebraic equations whose only unknowns are all the

other circuit variables

This is the core of the so-called state variables method, which will be detailed in

the rest of this chapter

The order of a circuit is the number of its SSVs.

Generally speaking, a linear circuit with N WSVs and M (< N) independent

algebraic constraints is said to be of order N − M In the next section, we start analyzing the simplest case, with N = 1 and M = 0, i.e., first-order circuits with

one WSV that is also an SSV

9.4 Solution of First-Order Linear Circuits with One WSV

A circuit containing only linear and time-invariant (LTI) components and

inde-pendent sources is called a linear time-invariant circuit.

In other words, an LTI circuit can be represented schematically as a linear

time-invariant P-port connected to P independent sources, as shown in Fig.9.17a

As stated above, in this section we consider LTI circuits with one WSV that isalso an SSV and analyze them by defining a common framework

The generic circuit belonging to this class is a first-order circuit It can be

rep-resented as the parallel connection of the component with memory (say, mc) and a memoryless two-terminal, called the complementary component, which contains the

rest of the circuit, as shown in Fig.9.17b, c

The complementary component can be represented through either its Théveninequivalent or its Norton equivalent, according to the admitted basis Generally speak-

ing, the equivalent source in both cases is a linear combination of the P independent

sources ˆu k (t) included in the complementary component; in other words, it can be

Fig 9.17 a General structure of an LTI circuit; b Particular case of a first-order LTI circuit

contain-ing only one two-terminal with memory (memory component mc), either an inductor or a capacitor;

c its equivalent representation: parallel connection between mc and a complementary component

18 9 Basic Concepts: Two-Terminal Linear Elements with Memory …

Fig 9.18 The four possible combinations of two-terminal with memory (L/C) and complementary

component (Thévenin/Norton equivalent)

There are four possible combinations, shown in Fig.9.18: panels a and b showtwo RC circuits, whereas panels c and d show two RL circuits

Before analyzing these circuits, we introduce the following concept

In a first-order circuit, an input–output (I/O) relationship is a differential

equation relating a given circuit variable (not necessarily the SSV) considered

as the circuit output to the circuit input(s) If the chosen output variable is also

the SSV, the I/O relationship is also called a state equation.

In the case of LTI circuits, the I/O relationship is a linear differential equation withconstant coefficients, as will become apparent from the analysis of the four circuitsshown in Fig.9.18

Case Study a

For the RC circuit shown in Fig.9.18a, we easily obtain (through KCL and

multiplying by R) the following differential equation for the state variable v:

This is both the state equation and the I/O relationship for v and can also be

For the RL circuit shown in Fig.9.18d, the state equation (I/O relationship)

for i can be easily obtained through KCL and multiplying by R:

L di

and it can also be recast as follows:

20 9 Basic Concepts: Two-Terminal Linear Elements with Memory …

The state equation for an LTI circuit with one WSV that is also an SSV (say,

x) can be always written either in the form (I/O relationship)

The left-hand side of Eq.9.20can be also written asL(x), to point out that we are

applying a linear operatorL = (a1

The homogeneous differential equation associated with Eq.9.20(9.21) is obtained

by setting ˆu k (t) = 0 for all k, thereby yielding L(x) = 0 ( ˆL(x) = 0).

In the following sections we provide two possible alternative ways for finding the

unique solution to Eqs.9.20and9.21with a given initial condition x (t0) = x0

In this section, we refer to the state equation in canonical form (Eq 9.21) Weintroduce a new variable ˜x, related to x as follows:

By deriving with respect to time, we obtain

Finally, we can express this solution in terms of the original variable x (recall that

x (t0) = x0and notice that owing to Eq.9.22, we have x (t0) = e a (t0−t0) ˜x(t0) = ˜x(t0)),

x (t) = x0e a (t−t0)

Z I R

+ e at P

In the above expression, ZIR is the zero-input response, that is, the response that

we would obtain in the absence of inputs (ˆu k (t) = 0 for all k) It depends only on the

circuit parameters (through a) and (linearly) on the initial condition The term ZSR

is the zero-state response, that is, the response that we would obtain with initial

condition x0= 0 It depends only on the circuit parameters (through the coefficients

a and b k) and (linearly) on the inputs

As an example, in the absence of inputs, the state variables of the two physicalsystems represented in Fig.9.19show a similar physical behavior In case (a), the

voltage v of the capacitor is governed by the (homogeneous) equation dv

by assuming that there are no inputs and v (t0) = V0, the solution is the Z I R term:

v (t) = V0e−(t−t0) RC Therefore, the energy initially stored by the capacitor is gradually

dissipated by the resistor R In case (b), a kayak of mass m has an initial velocity

v (t0) = V0 Inasmuch as nobody is paddling and there is no current, the only force

acting on the kayak is the viscous resistance K v oriented in the direction opposite

to v Newton’s law m dv

dt = −K v implies that v(t) = V0e−(t−t0)K m , which is a solutionsimilar to that for the capacitor voltage The role of the viscous force corresponds

to that played by the resistance R in the first case, from both a mathematical and a

physical (energy dissipation) standpoint

22 9 Basic Concepts: Two-Terminal Linear Elements with Memory …

Fig 9.19 Two physical systems showing a similar physical behavior in the absence of input: a an

RC circuit; b a kayak

A second way to find the (unique) solution x (t) to Eq.9.20(or9.21) with initial

condition x0 is based on the superposition principle In this case, the solution isfound by summing different contributions as follows:

where x f r (t) is called the free response (or transient response), x f o (t) is called

the forced response—since it is the part of the solution induced (forced) by the

corresponding inputs—and ˆx k (t) is a particular integral due to the kth input ˆu k (t).

The free response, or transient response, x f r (t) is the solution of the

homogeneous differential equation associated with Eq.9.20(L(x f r ) = 0) or

Eq.9.21( ˆL(x f r ) = 0) such that the initial condition holds, that is, such that

x f r (t0) + x f o (t0) = x0 It has the following structure:

x f r (t) = K e λ(t−t0) = ˜K e λt (9.25)

The termλ is called the natural frequency of the circuit In terms of Eq.9.20,λ

must be such thatL( ˜K e λt ) = 0, that is, (a1λ + a0)e λt= 0 This requires that

a1λ + a0= 0, (9.26)

which is called the characteristic equation (or auxiliary equation) of Eq. 9.20 Inpractice, the characteristic equation can be directly obtained from the homogeneousdifferential equation by replacing the term d x

dt withλ and x(= d0x

dt0) with λ0 = 1.From the characteristic equation, we obtainλ = − a0

a1

, provided that a1= 0.Similarly, the homogeneous differential equation associated with Eq.9.21is

˙x − ax = 0. (9.27)Then the corresponding characteristic equation is simplyλ = a.

In other words,λ is the eigenvalue of Eq.9.27 It contains information about theway in which a given circuit naturally reacts to any input change, namely, if it tends

to contrast this change (λ < 0), as in the kayak example, or if, on the contrary, it tends

to enhance the change (λ > 0) A deeper discussion about this point will be given

in Sect.9.4.3 For instance, in case studies a and b we find thatλ = − 1

RC, whereas

in case studies c and d, we haveλ = − R

L Notice that both natural frequencies are

negative

We remark that the natural frequency is an intrinsic property of the P-port shown

in Fig.9.17, and then it can be determined by turning off all of the P independent

sources That is whyλ is found by analyzing the homogeneous equation associated

with the I/O relationship

The reciprocal of the absolute value of a natural frequency has the physical

dimen-sion of time and is called the time constantτ of the circuit: τ = 1

|λ|.

A particular integral ˆx k (t) is a specific solution of the inhomogeneous

dif-ferential equation associated with Eq.9.20(resp.9.21) and related to the kth

input, that is,L( ˆx k (t)) = h k ˆu k (t) (resp ˆ L( ˆx k (t)) = b k ˆu k (t)).

At this point, we can show that solution9.24indeed solves Eq.9.20:

You can check that solution Eq.9.24also satisfies Eq.9.21

Remark: x (t) can be written as a sum of terms owing to the linearity of the

state equation (Eq.9.20or9.21), which makes it possible to apply the superpositionprinciple (see Sect 7.3 in Vol 1)

Once the particular integrals are found (see the examples below), the constant ˜K

in Eq.9.25can be determined by imposing the initial condition x (t0) = x0,

24 9 Basic Concepts: Two-Terminal Linear Elements with Memory …

Fig 9.20 Description of the relationship between solutions Eqs.9.23 and 9.24

We remark that solutions Eqs.9.23and9.24are equal However, their structures

are similar, but not identical: x f r (t) contains the ZIR and part of the ZSR, whereas

the remaining part of the ZSR is given by the forced response, as symbolicallysummarized in Fig.9.20

In the simplest cases, each particular integral can be found on the basis of a

similar-ity criterion, by assuming that ˆu k (t) is a function of the same form as the

correspond-ing input ˆu k (t) For instance, when λ = 0, the forced response term corresponding

to a constant (DC) input U is in turn a constant, say U0 Indeed,L(U0) = h k U ,

that is, a0U0= h k U , or equivalently, U0=h k U

a0 (recall that we assumedλ = 0, i.e.,

a0= 0)

Case Study 1: constant (DC) input

Solve the state equation a1

coef-ficients h, a0, and U are given by the problem and are expressed in terms of

circuit parameters The only unknown term is ˜K , which can be determined

by exploiting the initial condition, according to Eq.9.28: x (t0) = x f r (t0) +

Then the complete solution for t ≥ t0is

For instance, if we analyze the circuit of Fig.9.18a for t ≥ t0, with a (t) = A

and initial condition v (t0) = V0, the state equation (see Eq.9.12) is

The circuit natural frequency isλ = − 1

RC < 0 (under the standard

assump-tion R , C > 0) The free response is v f r (t) = K e− 1

RC (t−t0), and the (constant)

forced response isˆv = R A.

Therefore, the solution for t ≥ t0is

v (t) = K e− 1

RC (t−t0) + R A.

To determine K , we impose the initial condition v (t0) = K + R A = V0,

from which follows K = V0− R A.

Thus, the complete solution is

Similarly, the forced response term corresponding to an input ˆu k (t) = Ue σ t,

withσ = λ, is ˆx k (t) = U0e σ t Indeed,L(U0e σ t ) = h k U e σ t, and therefore,(a1σ +

a0)U0e σ t = h k U e σ t Then we obtain U0= h k U

a1σ + a0

; notice that the denominator

is nonzero, owing to the assumptionσ = λ.

As a final remarkable example, we consider a sinusoidal (AC) input ˆu k (t) =

angular frequency ˆx k (t) = K1cos(ωt) + K2sin(ωt), with K1 and K2 such that

L(K1cos(ωt) + K2sin(ωt)) = h k U cos (ωt) A detailed example is provided below.

26 9 Basic Concepts: Two-Terminal Linear Elements with Memory …

Case Study 2: sinusoidal (AC) input

Solve the state equation a1d x

dt + a0x = hU cos(ωt) for t ≥ t0, with initial

condition x (t0) = x0and λ = − a0

a1 < 0.

We already know that x f r (t) = K e λ(t−t0) Owing to the similarity criterion,

the particular integral is a sinusoidal term with the same frequency and, in eral, different amplitude and phase, that is, ˆx(t) = K1cos(ωt) + K2sin(ωt).

gen-Since ˆx(t) must be a solution of the state equation, we have

a1

d ˆx

dt + a0ˆx = hU cos(ωt)

that is,

−a1K1ω sin(ωt) + a1K2ω cos(ωt) + a0K1cos(ωt) + a0K2sin(ωt) = hU cos(ωt).

At this point, K1and K2can be easily identified by solving the following

a1K2ω + a0K1= hU,

−a1K1ω + a0K2= 0.

Now the only unknown term is K , which can be determined by

exploit-ing the initial condition, accordexploit-ing to Eq 9.28: x (t0) = x f r (t0) + ˆx(t0) =

K + K1cos(ωt0) + K2sin(ωt0) = x0, which leads K = x0− K1cos(ωt0) −

For instance, if we analyze the circuit of Fig.9.18c for t ≥ t0, with e (t) =

E cos (ωt) and initial condition i(t0) = I0, the state equation (see Eq.9.16) is

L di

dt + Ri = E cos(ωt).

You can check that each addend in the above equation has the physicaldimension of volts The circuit’s natural frequency isλ = − R

L < 0 (under the

standard assumption R , L > 0) The free response is i f r (t) = K e−R

L (t−t0), andthe forced response ˆi (t) is a sinusoid with the same angular frequency imposed

by the input: ˆi (t) = I ccos(ωt) + I ssin(ωt) By substituting ˆi(t) in the state

equation, we obtain

L [−I c ω sin(ωt) + I s ω cos(ωt)] + R[I ccos(ωt) + I ssin(ωt)] = E cos(ωt).

By solving the linear system

Therefore, the complete solution is

i (t) = [I0− I ccos(ωt0) − I ssin(ωt0)]e−R (t−t0)

In all cases, once the particular integral’s form is determined by similarity, ˆx k (t)

can be obtained by substitution in the state equation by applying the superpositionprinciple

Remark: From the complete solution expressed in terms of free response and

forced response, it is easy to obtain the specific initial condition that would ensurethat there is no transient response and the complete solution coincides with the

forced response In the first case study, it would be x0=hU

a0 (see Eq. 9.31); in

the second case study, it would be x0= K1cos(ωt0) + K2sin(ωt0) (see Eq.9.33).Similar conditions can be found for the circuit examples