Lecture Notes in Electrical Engineering 620 Mauro Parodi Marco Storace Linear and Nonlinear Circuits: Basic and Advanced Concepts Lecture Notes in Electrical Engineering Volume 620 Series Editors Leopoldo Angrisani, Department of Electrical and Information Technologies Engineering, University of Napoli Federico II, Naples, Italy Marco Arteaga, Departament de Control y Robótica, Universidad Nacional Autónoma de México, Coyoacán, Mexico Bijaya Ketan Panigrahi, Electrical Engineering, Indian Institute of Technology Delhi, New Delhi, Delhi, India Samarjit Chakraborty, Fakultät für Elektrotechnik und Informationstechnik, TU München, Munich, Germany Jiming Chen, Zhejiang University, Hangzhou, Zhejiang, China Shanben Chen, Materials Science and Engineering, Shanghai Jiao Tong University, Shanghai, China Tan Kay Chen, Department of Electrical and Computer Engineering, National University of Singapore, Singapore, Singapore Rüdiger Dillmann, Humanoids and Intelligent Systems Lab, Karlsruhe Institute for Technology, Karlsruhe, Baden-Württemberg, Germany Haibin Duan, Beijing University of Aeronautics and Astronautics, Beijing, China Gianluigi Ferrari, Università di Parma, Parma, Italy Manuel Ferre, Centre for Automation and Robotics CAR (UPM-CSIC), Universidad Politécnica de Madrid, Madrid, Spain Sandra Hirche, Department of Electrical Engineering and Information Science, Technische Universität München, Munich, Germany Faryar Jabbari, Department of Mechanical and Aerospace Engineering, University of California, Irvine, CA, USA Limin Jia, State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing, China Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland Alaa Khamis, German University in Egypt El Tagamoa El Khames, New Cairo City, Egypt Torsten Kroeger, Stanford University, Stanford, CA, USA 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 Möller, 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, Università degli Studi “Roma Tre”, Rome, Italy Yong Qin, State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing, China Gan Woon Seng, School of Electrical & Electronic Engineering, Nanyang Technological University, Singapore, Singapore Joachim Speidel, Institute of Telecommunications, Universität Stuttgart, Stuttgart, Baden-Württemberg, Germany Germano Veiga, Campus da FEUP, INESC Porto, Porto, Portugal Haitao Wu, Academy of Opto-electronics, Chinese Academy of Sciences, Beijing, China Junjie James Zhang, Charlotte, NC, USA The book series Lecture Notes in Electrical Engineering (LNEE) publishes the latest developments in Electrical Engineering—quickly, informally and in high quality While original research reported in proceedings and monographs has traditionally formed the core of LNEE, we also encourage authors to submit books devoted to supporting student education and professional training in the various fields and applications areas of electrical engineering The series cover classical and emerging topics concerning: • • • • • • • • • • • • Communication Engineering, Information Theory and Networks Electronics Engineering and Microelectronics Signal, Image and Speech Processing Wireless and Mobile Communication Circuits and Systems Energy Systems, Power Electronics and Electrical Machines Electro-optical Engineering Instrumentation Engineering Avionics Engineering Control Systems Internet-of-Things and Cybersecurity Biomedical Devices, MEMS and NEMS For general information about this book series, comments or suggestions, please contact leontina dicecco@springer.com To submit a proposal or request further information, please contact the Publishing Editor in your country: China Jasmine Dou, Associate Editor (jasmine.dou@springer.com) India Aninda Bose, Senior Editor (aninda.bose@springer.com) Japan Takeyuki Yonezawa, Editorial Director (takeyuki.yonezawa@springer.com) South Korea Smith (Ahram) Chae, Editor (smith.chae@springer.com) Southeast Asia Ramesh Nath Premnath, Editor (ramesh.premnath@springer.com) USA, Canada: Michael Luby, Senior Editor (michael.luby@springer.com) All other Countries: Leontina Di Cecco, Senior Editor (leontina.dicecco@springer.com) ** Indexing: The books of this series are submitted to ISI Proceedings, EI-Compendex, SCOPUS, MetaPress, Web of Science and Springerlink ** More information about this series at http://www.springer.com/series/7818 Mauro Parodi Marco Storace • Linear and Nonlinear Circuits: Basic and Advanced Concepts Volume 123 Mauro Parodi Department of Electric, Electronic, Telecommunications Engineering and Naval Architecture (DITEN) University of Genoa Genoa, Italy Marco Storace Department of Electric, Electronic, Telecommunications Engineering and Naval Architecture (DITEN) University of Genoa Genoa, 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 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remains neutral with regard to jurisdictional claims in published maps and institutional affiliations This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Preface In Volume of Linear and Nonlinear Circuits: Basic & Advanced Concepts,1 after introducing basic concepts, we considered only circuits containing memoryless components, whose equations (both topological and descriptive) are algebraic This volume is focused on components with memory and on circuits characterized by time evolution, that is, by dynamics The volume is articulated in three parts, and its structure follows the guidelines described in the preface to the whole book (Volume 1), with each part articulated in two independent lecture levels: basic and advanced The basic chapters are devoted to linear components and circuits with memory, whereas the advanced chapters focus on nonlinear circuits, whose nonlinearity is provided by memoryless components The analysis of dynamical circuits is carried out by making reference to the so-called state variables, and the concept of state runs through the whole volume, innervating with dendritic structure the language used to describe the concepts and even the equations describing a given circuit Moreover, owing to their proximity to the concept of energy, the state variables are an effective tool for understanding the behavior of a circuit not only from a mathematical standpoint, but also from a physical perspective The formalism adopted in this book to describe the state equations is based on system theory and on its general results, so that each circuit (together with its properties) can be viewed as a particular physical system To this end, the proposed theoretical results are intermingled with case studies that instantiate general ideas and point out methodological and applicative consequences Whenever possible, the circuits are compared to physical systems of different natures (mechanical or biological, for instance) that are governed by equations and properties completely similar and therefore exhibit the same dynamical behaviors To this end, we note the importance of normalizations, which make it possible to analyze, with the same conceptual tools, models of physical systems of any nature Published by Springer 978-3-319-61234-8) in 2018 (Print ISBN: 978-3-319-61233-1; eBook ISBN: v vi Preface The reader’s comprehension of the proposed concepts can be checked by solving the problems appearing at the end of each chapter (mainly the basic chapters) and comparing the obtained results with the solutions provided at the end of this volume Part V shows how to analyze circuits with one state variable, that is, first-order circuits In the linear case (treated in Chap 9, basic), the complete analytical solution can be obtained based on well-assessed approaches In the nonlinear case (Chap 10, advanced), we usually cannot easily find a closed-form analytical solution, but we can discover some important information about the general properties of the solution(s) To this end, we employ the tools of nonlinear dynamics 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 circuits exhibiting persistent oscillations In the linear case (Chap 13, basic), we describe how 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 to analyze circuits (called nonlinear oscillators) by working in the periodic steady state also in the absence of a forcing periodic input (autonomous oscillators) The analysis can also be carried out when some circuit parameters are varied, thus inducing qualitative changes (bifurcations) in the circuit dynamics and switching the 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 often abstract concepts provided by system theory to the physics of the considered examples In our opinion, this is a great help in understanding not only the specific case 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), circuits represent an excellent environment for better understanding the relationships between physics, mathematics, and system theory We are indebted to our friend Lorenzo Repetto, who carefully revised the preliminary version of this volume, reporting bugs and providing detailed comments We also acknowledge our colleagues and friends Giovanni Battista Denegri for his helpful comments on Sect 13.9.2, Matteo Lodi for his invaluable help with the simulations described in Sect 14.9, and Alberto Oliveri for constructive discussions and comments Genoa, Italy September 2019 Mauro Parodi Marco Storace Contents Part V Components with Memory and First-Order Dynamical Circuits Basic Concepts: Two-Terminal Linear Elements with Memory and First-Order Linear Circuits 9.1 Two-Terminal Linear Elements with Memory 9.1.1 Capacitor 9.1.2 Inductor 9.2 Capacitor and Inductor Properties 9.2.1 Energetic Behavior 9.2.2 Gyrator and Two-Terminals with Memory 9.2.3 Series and Parallel Connections 9.3 State and State Variables 9.3.1 Wide-Sense and Strict-Sense State Variables 9.3.2 Circuit Models of Algebraic Constraints 9.3.3 State Variables Method 9.4 Solution of First-Order Linear Circuits with One WSV 9.4.1 General Solution of the State Equation 9.4.2 Free Response and Forced Response 9.4.3 Circuit Stability 9.5 Forced Response to Sinusoidal Inputs 9.5.1 Sinusoids and Phasors 9.5.2 Phasor-Based Method for Finding a Particular Integral 9.5.3 Multiple Periodic Inputs: Periodic and Quasiperiodic Waveforms 9.6 Generalized Functions (Basic Elements) 9.7 Discontinuity Balance 3 7 10 11 13 16 17 20 22 30 33 33 36 38 41 44 vii viii Contents 9.8 Response of Linear Circuits with One WSV and One to Discontinuous Inputs 9.8.1 Step Response 9.8.2 Impulse Response 9.9 Convolution Integral 9.10 Circuit Response to More Complex Inputs 9.10.1 Multi-input Example 9.11 Normalizations 9.12 Solution for Nonstate Output Variables 9.13 Thévenin and Norton Equivalent Representations of a Charged Capacitor/Inductor 9.13.1 Charged Capacitor 9.13.2 Charged Inductor 9.14 First-Order Linear Circuits with Several WSVs 9.15 Problems References SSV 45 46 48 52 57 57 62 65 66 67 69 70 79 89 91 91 98 99 10 Advanced Concepts: First-Order Nonlinear Circuits 10.1 Asymptotic Solution of a Particular Class of First-Order Nonlinear Circuits 10.1.1 First-Order Circuits with More Than One WSV 10.1.2 Impossibility of Oscillations 10.1.3 Equilibrium Stability Analysis Through Linearization 10.2 Equilibrium Points and Potential Functions for First-Order Circuits 10.3 Analysis of First-Order Circuits with PWL Memoryless Components 10.3.1 Clamper 10.3.2 Half-Wave Rectifier 10.3.3 Hysteretic Circuit 10.3.4 Circuit Containing an Operational Amplifier 10.3.5 Circuit Containing a BJT 10.4 Bifurcations 10.4.1 Linear Case 10.4.2 Nonlinear Case 10.5 A Summarizing Example 10.5.1 Inverting Schmitt Trigger 10.5.2 Dimensionless Formulation 10.5.3 Analysis with Constant Input 10.5.4 Potential Functions 10.6 Problems References 100 101 107 107 109 110 113 117 121 121 122 126 127 128 129 133 136 139 Contents Part VI ix Second- and Higher-Order Dynamical Circuits 11 Basic Concepts: Linear Two-Ports with Memory and Higher-Order Linear Circuits 11.1 Coupled Inductors 11.2 Properties of Coupled Inductors 11.2.1 Series Connection 11.2.2 Passivity 11.2.3 Coupling Coefficient and Closely Coupled Inductors 11.2.4 Energy Conservation 11.2.5 Equivalent Models 11.2.6 Thévenin and Norton Equivalent Representations of Charged Coupled Inductors 11.3 Higher-Order Linear Circuits 11.3.1 General Method 11.3.2 Complementary Component Method and State Equations 11.3.3 State Equations in Canonical Form and I/O Relationships 11.4 Discontinuity Balance 11.5 Solution of the State Equations: Free Response and Forced Response 11.5.1 Free Response 11.5.2 Forced Response 11.6 Circuit Stability 11.7 Normalizations and Comparisons with Mechanical Systems 11.7.1 A Double Mass–Spring Chain and Its Circuit Model 11.8 Solution for Nonstate Output Variables 11.9 Response of LTI Dynamical Circuits to Discontinuous Inputs 11.10 Generic Periodic Inputs 11.10.1 Fourier Series 11.10.2 Some Supplementary Notes About Fourier Series 11.10.3 Mean Value of Circuit Variables 11.10.4 Root Mean Square Value 11.11 Multi-input Example 11.12 Problems References 143 143 147 147 148 149 150 154 158 159 160 161 163 168 172 173 180 193 202 205 207 208 211 212 214 218 221 230 234 247 Solutions Fig B.6 Qualitative graphical solutions to questions (a) and (b) of problem 11.11 501 (a) (b) di + 2Ri = e − 2Ra2 + 2Ra1 It is also a state equation dt d v L da1 dv LC + + 2RC + 2v = e − 2Ra2 − L dt R dt dt There are two WSVs, which are also SSVs, because there are no algebraic constraints 2R and λ2 = − The two natural frequencies are λ1 = − L RC λ1,2 < 0; thus the circuit is absolutely stable i(t) = i DC (t) + i AC1 (t) + i AC2 (t), with 11.13 L i DC (t) = A0 , 2R A1 [2R cos(ωt) + ωL sin(ωt)], (2R)2 + (ωL)2 R A2 i AC2 (t) = [ωL cos(2ωt) − R sin(2ωt)], R + (ωL)2 4R A1 ωL R A2 + i(0− ) = A0 + (2R)2 + (ωL)2 R + (ωL)2 i¯ = A0 ; i AC1 (t) = 502 Solutions (2R A1 ) (R A2 ) A20 + 21 (2R) +(ωL)2 + R +(ωL)2 Φ0 + i(0− ) i(0+ ) = L E E i(t) = i(0+ ) − A0 − e λ1 t + A + 2R 2R 10 From the discontinuity balance applied to the I/O relationship for v(t), we obtain v(0+ ) = v(0− ) By integrating the same equation between 0− and 0+ , we also dv Φ0 + L A E dv obtain = For t > 0, v(t) = K eλ1 t + K eλ2 t + , + dt 0+ LC dt 0− where K and K are solutions of the linear system ⎧ E ⎪ ⎨ v(0+ ) = K + K + dv ⎪ ⎩ = K λ1 + K λ2 dt 0+ ie f f = 2 di da1 d 2i de 11.14 LC + 2RC + i = −a2 + 2RC + C It is also a state equadt dt dt dt tion There are three WSVs and two SSVs, due to the presence of one algebraic constraint (loop involving only e(t) and C) −2RC ± (2RC)2 − 4LC The two natural frequencies are λ± = 2LC R C < L The circuit is absolutely stable Indeed, if the natural frequencies are complex conR jugate, then {λ± } = − < If they are real, they are also negative, because L (2RC)2 − 4LC < 2RC i(t) = i DC (t) + i AC1 (t) + i AC2 (t), with i DC (t) = −A2 , 2ω RC A [(1 − ω2 LC) cos(ωt) + 2ω RC sin(ωt)], (1 − ω2 LC)2 + (2ω RC)2 2ωC E i AC2 (t)= [4ω RC cos(2ωt) − (1 − 4ω2 LC) sin(2ωt)], (1−4ω LC)2 +(4ω RC)2 2ω RC A(1 − ω2 LC) 8(ωC)2 R E + , i(0− ) = −A2 + (1 − ω2 LC)2 + (2ω RC)2 (1 − 4ω2 LC)2 + (4ω RC)2 (2ω RC)2 ω A 4ω2 C(1 − 4ω2 LC)E di = − 2 dt 0− (1 − ω LC) + (2ω RC) (1 − 4ω2 LC)2 + (4ω RC)2 From the discontinuity balance applied to the I/O relationship for i(t), we obtain i(0+ ) = i(0− ) By integrating the same equation between 0− and 0+ , we also 2R A1 di E0 − E1 di = obtain + + dt 0+ L L dt 0− i AC1 (t) = Solutions 503 There is only a transient response By assuming λ+ = λ and λ− = λ∗ , we have ∗ i(t) = K eλt + K ∗ eλ t , where K is a solution of the linear system ⎧ + ⎨ i(0 ) = K + K ∗ di = K λ + K ∗ λ∗ ⎩ dt 0+ 11.15 LC d 2v da + αv = −Ra + 2L dt dt The two natural frequencies are λ± = ± − α LC α > If α > 0, the circuit is simply stable (two purely imaginary natural frequencies); if α = 0, the circuit is weakly unstable (λ = with multiplicity 2); if α < 0, the circuit is strongly unstable (two real natural frequencies, one of which is positive) From the discontinuity balance applied to the I/O relationship for v(t), we obtain v(0+ ) = v(0− ) By integrating the same equation between 0− and 0+ , we also A0 dv dv = obtain + dt 0+ C dt 0− di + R2 i = e(t) It is also a state equation dt d v dv R1 LC + (L + R1 R2 C) + R2 v = R1 e dt dt R2 The two natural frequencies are λ1 = − and λ2 = − L R1 C Circuit absolutely stable (two real negative natural frequencies) E0 E0 ; i(0− ) = i(t) = R2 R2 R1 E0 v(t) = R2 From the discontinuity balance applied to the I/O relationship for i(t), we obtain i(0+ ) = i(0− ) By integrating the same equation between 0− and 0+ , we also Φ0 + i(0− ) obtain i(0+ ) = L i(t) = K eλ1 t + i AC1 (t) + i AC2 (t), where 11.16 L E1 [R2 cos(ωt) + ωL sin(ωt)], + (ωL)2 E2 i AC2 (t) = − [2ωL cos(2ωt) − R2 sin(2ωt)], R2 + (2ωL)2 R2 E 2ωL E K = i(0+ ) − + R2 + (ωL)2 R2 + (2ωL)2 R1 E p¯ = − [1 + (ω R1 C)2 ][R22 + (ωL)2 ] i AC1 (t) = R22 504 Solutions d 2i di da de 11.17 R LC − RrC = −ra − RrC + (R + r )C It is also a state dt dt dt dt equation r The two natural frequencies are λ1 = and λ2 = L Circuit simply stable if r < 0; weakly unstable if r = 0; strongly unstable if r > Therefore, the stability condition is r < [−(r A + ωLb) cos(ωt) + r (ωL A − b) sin(ωt)], i(t) = ω RC[(ωL)2 + r ] where b = ωC [(R + r )E − Rr A] dv + v = Ra(t) It is also a state equation dt d i di da 2R LC + L = −R C dt dt dt and λ2 = The two natural frequencies are λ1 = − 2RC Circuit simply stable 2ω R C A1 R A1 − [2ω RC cos(ωt) − sin(ωt)]; v(0 ) = − v(t) = − + (2ω RC)2 + (2ω RC)2 2 R C A1 2ω R A1 (RC) i(t) = [2ω RC cos(ωt) − sin(ωt)]; i(0− ) = L[1 + (2ω RC) ] L[1 + (2ω RC)2 ] From the discontinuity balance applied to the I/O relationship for v(t), we obtain v(0+ ) = v(0− ) 11.18 2RC From the discontinuity balance applied to the I/O relationship for i(t), we obtain di di di i(0+ ) = i(0− ) and = Since we not have , instead of intedt 0+ dt 0− dt 0− − + grating the I/O relationship for i(t) between and , we can use the state R A0 di v(0+ ) di =− = v − Ra, thus obtaining + equation 2L dt dt 0+ 2L 2L v(t) = K eλ1 t + R A0 , where K = v(0+ ) − R A0 i(t) is sensitive to both natural frequencies, one of which is null, and therefore we cannot separate free response and forced response The solution is i(t) = K eλ1 t + K , where K and K are solutions of the linear system ⎧ + ⎨ i(0 ) = K + K di = K λ1 ⎩ dt 0+ −RC ± (RC)2 − 3LC 11.19 The two natural frequencies are λ± = Under LC the usual assumptions (R, L , C > 0), the circuit is absolutely stable for R C = 3L E + 2R A0 i¯∞ = 3R Solutions 2 p¯ = Ri ∞e f f , with i ∞e f f = 505 E + 2R A0 3R + E1 3R + A1 d 2v dv de da L1 11.20 L C + τ + v = ατ − M , where τ = It is also a state dt dt dt dt R equation The two natural frequencies are λ± = L 21 − 4L R C 2L RC Circuit absolutely stable (under the usual assumptions: R, L , C > 0) αωτ E v(t) = [β cos(ωt) + ωτ sin(ωt)], where β = − ω2 L C; β + (ωτ )2 αβωτ E v(0− ) = ; β + (ωτ )2 αω3 τ E dv = dt 0− β + (ωτ )2 From the discontinuity balance applied to the I/O relationship for v(t), we obtain dv dv v(0+ ) = v(0− ) and = By integrating the same equation between dt 0+ dt 0− dv ατ E M A0 dv =− − + 0− and 0+ , we also obtain dt 0+ L 1C L 1C dt 0− v(t) = K + eλ+ t + K − eλ− t , where K + and K − are solutions of the linear system ⎧ ⎨ v(0+ ) = K + + K − dv = K + λ+ + K − λ− ⎩ dt 0+ d v2 dv2 +τ + βv2 = ra(t), where T = n LC2 , τ = RC2 , β = dt dt n(α + n), r = (1 − n)R −τ ± τ − 4βT The three natural frequencies are λ± = (from the I/O rela2T tionship at point 1) and λ3 = (due to the presence of a cut-set involving only the independent current source and C1 ) The natural frequencies λ± are complex conjugates, provided that τ < 4βT , that is (in terms of the original circuit parameters), R C2 < 4n (α + n)L If this condition is satisfied, the circuit is simply stable (under the usual assumptions −τ < R, L , C > 0), since {λ± } = 2T dv2 rA v2 (t) = = = v2 (0− ); β dt 0− From the discontinuity balance applied to the I/O relationship for v(t), we dv2 dv2 obtain v2 (0+ ) = v2 (0− ) and = By integrating the same equadt 0+ dt 0− dv2 r Q0 = tion between 0− and 0+ , we also obtain dt 0+ T 11.21 T 2 −L ± 506 Solutions v2 (t) = K + eλ+ t + K − eλ− t , where K + and K − are solutions of the linear system ⎧ ⎨ v2 (0+ ) = K + + K − , dv2 = K + λ+ + K − λ− ⎩ dt 0+ A 11.22 i¯ = − , v¯ = R A Notice that under the prescribed assumptions, the circuit contains (in addition to the independent source) only passive elements, and some of them are dissipative You can check that it is absolutely stable The required mean power is | I˙1 |2 2 , with I0 = and p¯ = Ri Re f f = R I0 + (ωL A1 )2 | I˙1 |2 = R (1 − 2ω2 LC)2 + ω2 (L + R C)2 11.23 Notice that the circuit contains (in addition to the independent current sources) only passive elements, and some of them are dissipative You can check that it is absolutely stable 15 E R A, i¯e = 8R | I˙1 |2 (ωC M E )2 i 1e f f = i¯12 + Recall that the closely , with i¯1 = A and | I˙1 |2 = L 22 coupled inductors assumption implies L L = M v¯a = E0 E − 2R A0 A0 + (1 + Rg) , v¯ = + 2Rg R + 2Rg | I˙1 |2 | I˙2 |2 + , with p¯ = 2Ri e2f f = 2R i¯2 + 2 A0 E0 i¯ computed at point 1, but with g = (i.e., i¯ = + ) 3R 11.24 i¯ = | I˙1 |2 = | I˙2 |2 = E1 R and + 4(ω RC)2 A21 + 16(ω RC)2 R E R + R2 | I˙1 |2 | I˙2 |2 = R2 i¯2 + + , 2 11.25 v¯ a = p¯ = R2 i e2f f E 12 and | I˙2 |2 = (R + R2 )2 + (ωL )2 with i¯ = − E0 , R + R2 | I˙1 |2 = Solutions 507 11.26 We have to find the absolute stability condition The circuit’s natural R − R1 frequency is λ = ; therefore λ < if and only if R < R1 Notice that R R1 R2 C is the gain of a CCVS, thus in general, there are no restrictions on its sign E1 E2 v¯ = − 11.27 The circuits of Fig 11.77a, c are absolutely stable For instance, the state equations for the circuit of Fig 11.77a are ⎛ ⎧ v di ⎪ ⎨ = dt L v i a dv ⎪ ⎩ =− − + dt RC C C d ⇒ dt i v L ⎜ ⎜ =⎜ ⎝ 1 − − C RC ⎞ ⎟ ⎟ i ⎟ ⎠ v + a The eigenvalues λ1 , λ2 of the state matrix are the solutions of the characteristic equation λ λ2 + + = 0, RC LC and we have λ j < ( j = 1, 2) A completely analogous result holds for the circuit of Fig 11.77c Therefore, for instance, a sinusoidal input causes, for t → ∞, a sinusoidal steady state in both circuits Consider now the circuit of Fig 11.77b In this case, since v = e, the only SSV is the inductor current i The state equation and the corresponding eigenvalue λ are e di = ; dt L λ = Therefore, this circuit is simply stable For instance, when e(t) = E cos (ωt) · u(t) and for i(0) = I0 = 0, we immediately have, for all t ≥ 0, t i(t) = I0 + E cos (ωτ ) dτ = I0 + E sin (ωt) ω Inasmuch as i(t) contains the constant term I0 , this is not a sinusoidal steady state Analogous considerations hold for the circuit of Fig 11.77d, where the only SSV is the capacitor voltage v, and we still have a single eigenvalue λ = (simply stable circuit) The remaining part of the discussion is identical, mutatis mutandis, to that developed in the previous case 508 Solutions Problems of Chap 13 Solutions of problems of Chap 13 10α R0 V˙ = R+ + jω10R0 C I˙ By setting R0 = R/10 we obtain 13.1 Z ( jω) = Z ( jω) = R(ω) + j X (ω); R(ω) = R + αR −α R ωC ; X (ω) = + (ω RC) + (ω RC)2 For α > −1, we have R(ω) > for all ω In particular: • for −1 < α < 0, X (ω) > and Z ( jω) is resistive–inductive; • for α > 0, Z ( jω) is resistive–capacitive; • for α = 0, we have X (ω) = (purely resistive impedance) R2 C + jω C1 + R1 + R2 R1 + R2 R1 E E ω R1 R2 C 2 P = ; Q = − (R1 + R2 )2 (R1 + R2 )2 Inasmuch as the composite two-terminal is resistive–capacitive, the element to be connected in series is an inductor L with impedance jωL such that = jωL + Y ( jω) (R1 + R2 ) C1 + R2 C2 The result is L = (R1 + R2 ) + ω2 ((R1 + R2 ) C1 + R2 C2 )2 13.2 Y ( jω) = 13.3 Z ( jω) = We have j2ω R L (1 − α) (3 − 2α) R + j2ωL ⎧ ⎨ {Z ( jω)} > ⎩ {Z ( jω)} > for α < Therefore, Z ( jω) is resistive–inductive for α < 1 3 For α = 0, the admittance of the composite two-terminal is Y = − j R 2ωL Therefore, the condition { jωC + Y } = is satisfied for C = 2ω2 L R − ω2 LC + jωL a + jb =R c + jd R 2− + jω L + (ac + bd) + j (bc − ad) (a + jb) (c − jd) =R , the Inasmuch as Z ( jω) = R c2 + d c2 + d 2 R C condition {Z ( jω)} =0 is met for (bc − ad) =0, which implies L = + (ω RC)2 13.4 Z ( jω) = R ω2 LC R2C Solutions 509 13.5 The impedance matrix is ⎡ ⎢ ⎢ [Z ( jω)] = ⎢ ⎣ Z T H = R; R Rg + g (R + jωL) ˙ E˙ T H = Rg E jω (M − L ) (R + jωL) ˙ A; 13.6 E˙ T H = R + jω (L + L ) 13.7 A˙ N R = − ⎤ R + jωL + g (R + jωL) ⎥ ⎥ ⎥ ⎦ R + jωL Z T H ( jω) = jωL (R + jωL) R + jω (L + L ) jωCn E˙ (nω)2 LC − + jωn Lg ( jω) = ; Z N R n (g − jωnC) (nω)2 LC − + jωn Lg j E˙ ; 13.8 A˙ N R = ωL Z N R ( jω) = jωL − ω2 LC n + Rg n + jω RC ( jω) = − ; Z N R n + jω RC ng + R (ωC)2 − jωC n + ⎡ ⎤ R ⎦ 13.10 [T ( jω)] = ⎣ jωC + jω RC A˙ E˙ T H = − j ; Z T H ( jω) = R + ωC jωC R A2 + (ωC E)2 ωC (R A − E)2 P = ; Q = − + (ω RC)2 + (ω RC)2 Notice that P is the active power absorbed by the resistor R inside the two1 port, P = R I˙2 , whereas Q is the reactive power absorbed by the capacitor: 2 Q = − ωC V˙1 ⎡ ⎤ jωL R + jωL ⎥ ⎢2 + R ⎢ ⎥ 13.11 [T ( jω)] = ⎢ ⎥ ⎣ ⎦ 1 R RA E˙ T H = −R A; Z T H ( jω) = 2R + jωL; Z N R = Z T H ; A˙ N R = − 2R + jωL ⎡ ⎤ n (2R + jωL) n R ⎦ Notice that [Z ( jω)] is symmetric, and [Z ( jω)] = ⎣ nR R therefore the two-port is reciprocal 13.9 A˙ N R = A˙ 510 Solutions ⎡ ⎢ 13.12 [T A ( jω)] = ⎣ − n ⎤ ⎥ ⎦ −n ⎤ L1 − ⎥ ⎢ M ⎥ ⎢ [TB ( jω)] = ⎢ ⎥ ⎣ L2 ⎦ − − jωM⎡ M ⎤ L1 ⎥ ⎢ nM ⎥ ⎢ [T ( jω)] = T A · TB = ⎢ ⎥ ⎣ n nL2 ⎦ jωM M V˙2 I˙2∗ V˙1 j A E I˙2∗ V˙1 I˙1∗ + = + Therefore: P + j Q = 2 2 ⎧ P=0 ⎪ ⎪ ⎨ ⎡ M AE E2 L1 E A ⎪ ⎪ ⎩Q = − + 2n M 2ωL 2n L ⎡ + jω RC ⎢ − jωrC ⎢ 13.13 [T ( jω)] = ⎢ ⎣ jωC − jωrC ⎤ R ⎥ ⎥ ⎥ ⎦ 13.14 Denoting by E˙ the phasor representing the sinusoidal source e(t), we have ⎡ 1−β + − jωCαβ ⎢ R2 [Y ( jω)] = ⎣ R1 − jωCβ ⎤ jωCα jωC ⎥ ⎦; E˙ A˙ = ; R2 A˙ = The reciprocity condition Y12 = Y21 holds, provided that α = −β 13.15 ⎡ [Z ( jω)] = ⎣ 0 −2R ⎤ ⎦; E˙ = E˙ −1 ; ω2 LC E˙ = E˙ − 2R + jωL ω2 LC Solutions 511 13.16 ⎡ ⎢ jωC ⎢ [Z ( jω)] = ⎢ ⎣ jωC r+ ⎤ jωC r + jωL + R + jωC ⎥ ⎥ ⎥; ⎦ E˙ = A˙ ; jωC − ω LC E˙ = j A˙ ωC 13.17 The complex power delivered by the two-port is P + jQ = − V˙ A˙ ∗ E˙ I˙∗ + 2 = − E˙ A˙ ∗ − g E 6 Now, setting E˙ = E and A˙ = − j A, we obtain ⎧ gE2 ⎪ ⎪ , P = − ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ Q = − E A E − j2R A 13.18 We first obtain I˙∞ = and V˙∞ = − jωL I˙∞ Therefore, the com3R jωL I˙∞ , and plex power absorbed by the nullor is P + j Q = − ⎧ P = 0, ⎪ ⎪ ⎪ ⎨ ωL ⎪ ⎪ ⎪ ⎩Q =− 13.19 Z ( jω) = α R + jωL + L = R2C + (ω RC)2 Z h = Z ∗ = R α + E 3R 1 + (ω RC)2 2R ωL 13.21 The circuit equations are + 2A R + jω RC − jω L − 13.20 As a first step, we find Z = R − j R 1+ j R2C + (ω RC)2 2R Therefore, we have Z = Z 2∗ = ωL 512 Solutions ⎧ E˙ − V˙1 ⎪ ⎪ ⎨ = jωC2 V˙1 − V˙ + jωC1 V˙1 R1 V˙ ⎪ ⎪ ⎩ jωC1 V˙1 = − R2 From these equations we obtain, after few manipulations, H2 ( jω) = V˙ =− R1 (C1 + C2 ) E˙ + jω R1 C2 − j R2 C ω R2 C H2 ( jω) is a purely real value at the angular frequency ω0 such that ω0 R1 C2 = , ω0 R2 C1 that is, ω0 = √ R1 R2 C C H2 ( jω) can be recast as follows: H2 ( jω) = − 1+ j =− R2 C R1 (C1 + C2 ) ω R1 C R2 C R2 C 1 · − R1 (C1 + C2 ) ω R2 C1 R1 (C1 + C2 ) R2 C R1 (C1 + C2 ) ⎞= ⎛ = ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ω R C C ω 2⎟ ⎜ 1+ j − ⎟ ⎜ ω (C1 + C2 ) ⎜ R1 ω0 ⎟ ⎟ ⎜ ⎠ ⎝ = ω R1 R2 C R2 C R1 (C1 + C2 ) R1 (C1 + C2 ) =− =− ω0 ω ω0 ω 1+ j − + jQ − ω0 R1 (C1 + C2 ) ω0 ω ω0 ω Q Summing up, by defining ω0 = √ ; R1 R2 C C Q= ω0 R C C = , ω0 R1 (C1 + C2 ) (C1 + C2 ) H2 ( jω) assumes the same structure as H1 ( jω), apart from the constant coefficient R2 C 1 − , as expected R1 (C1 + C2 ) R Index A Active power, 348 Admittance, 328 resistive-capacitive, 330 resistive-inductive, 330 Apparent power, 351 Arnold tongue, 428 Attractor, 414 Autonomous, 93 B Band-pass filter, 387 Basin of attraction, 98, 414 Bifurcation, 121, 283 flip, 423 fold, 283 fold of cycles, 421 fold of equilibria, 283 global, 421 homoclinic, 434 Hopf, 284 local, 420 Neimark–Sacker, 427 Boucherot’s theorem, 356 C Canonical form first order, 71 first-order, linear, 20 first-order, nonlinear, 93 higher-order, linear, 163 higher-order, nonlinear, 274 Capacitance, Complementary component, 17, 161 Complex power, 350 Conductance, 329 Control space, 124 Convolution integral, 52 Coupling coefficient coupled inductors, 149 Cycle, 402 D Descriptive equations capacitor, coupled inductors, 143 inductor, Dirac δ-function, 42 Discontinuity balance, 44 Dissipative system, 302 E Energetic behavior conservative, 7, 150 Equilibrium center, 270 condition, 94, 276 point, 94, 275 saddle, 268 stable focus, 269 stable node, 267 unstable focus, 270 unstable node, 267 © 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/10.1007/978-3-030-35044-4 513 514 F Floquet multipliers, 410 Flow, 275 Fourier series, 212 convergence theorem, 213 Free response first order, 22 higher-order, 173 Frequency locking, 454 Frequency response, 373 Fundamental matrix, 408 G Global stability, 98 H Harmonic oscillator, 207 Hyperbolic equilibrium point, 100, 281 I Impedance, 326 resistive-capacitive, 328 resistive-inductive, 328 Inductance, Inner product, 213 Input matrix, 163 Input-output relationship, 18 first order, 20, 70 higher-order, linear, 164 Instability, 31, 194 Invariant set, 279 J Jacobian matrix, 281 L Leading eigenvalue, 262 Limit cycle, 402 Linear time-invariant circuit, 17 Liouville formula, 410 Liouville’s theorem, 408 Local stability, 98 Low-pass filter, 380 Lyapunov exponent, 417 M Maximum-power-transfer theorem, 370 Mean value Index periodic function, 212 Monodromy matrix, 409 Mutual inductance coupled inductors, 144 N Natural frequency, 22, 173 Nullcline, 275 O Order of a circuit, 16 Oscillator Colpitts, 297, 423, 438 simple harmonic oscillator, 196 van der Pol, 276, 282, 287, 406, 411, 419, 428, 450, 457, 462, 465 Wien bridge, 289, 316 P Parallel connection, 331 Parseval’s theorem, 223 Particular integral, 23 Period-doubling cascade, 425 Phase locking, 428 Phasor, 33 Poincaré section, 403 choice, theorem, 404 Power factor, 351 Primary inductance coupled inductors, 144 Problem solutions Ch 9, 483 Ch 10, 488 Ch 11, 494 Ch 13, 508 Pulse current, 43 voltage, 43 Q Quality factor, 385 R Reactance, 327 Reactive power, 348 Repellor, 282 Resistance, 327 Resonance, 385 Root mean square value, 221 Index Root mean square value theorem, 222 Route to chaos, 425, 440 S Secondary inductance coupled inductors, 144 Series connection, 331 Small signal, 306 Stability, 31, 194 absolute, 31, 194 simple, 31 Stable equilibrium point, 97 State, 10 State equation first-order, linear, 20 first-order, nonlinear, 93 higher-order, linear, 163 higher-order, nonlinear, 252, 274 State matrix, 163 State space, 93, 275 State variables, 11 method, 16 strict-sense, 12 wide-sense, 11 515 Steady state, 31 Structural stability, 446 Susceptance, 329 T Time constant, 23, 175 Torus, 414 Trajectory, 94 Transient response, see free response U Unit step, 41 V Variational equation, 408 Vector field, 93, 261, 275 Z Zero-input response (ZIR), 21 Zero-state response (ZSR), 21 ... 1 72 173 180 193 20 2 20 5 20 7 20 8 21 1 21 2 21 4 21 8 22 1 23 0 23 4 24 7 x Contents 12 Advanced Concepts: Higher-Order Nonlinear Circuits? ??State Equations and Equilibrium Points... Part VII 13 Basic State 13.1 13 .2 13.3 24 9 24 9 25 2 26 1 26 2 27 2 27 4 28 0 28 3 28 3 28 4 28 9 29 3 29 4 29 7 29 9 301 304 ... Technical Committee on Nonlinear Circuits and Systems (TC-NCAS) Mauro Parodi and Marco Storace are also the authors of Linear and Nonlinear Circuits: Basic & Advanced Concepts? ??Volume published