Table of Contents

Cover

Title

Preface

Introduction

1 Continuous-time Systems: General Properties, Feedback, Stability, Oscillators

1.1. Representation of continuous time signals
1.2. Representations of linear and stationary systems and circuits built with localized elements
1.3. Negative feedback
1.4. Study of system stability
1.5. State space form
1.6. Oscillators and unstable systems
1.7. Exercises

2 Continuous-time Linear Systems: Quadripoles, Filtering and Filter Synthesis

2.1. Quadripoles or two-port networks
2.2. Analog filters
2.3. Synthesis of analog active filters using operational amplifiers
2.4. Non-dissipative filters synthesis methods
2.5. Exercises

Appendix: Notions of Distribution and Operating Properties

A.1. Dirac distribution or Dirac impulse δ_a or δ(x − a)
A.2. Derivation of a distribution and derivation of discontinuous functions
A.3. Laplace transform of distributions
A.4. Distribution in principal value p.v. following Cauchy’s definition
A.5. Solving equations with discontinuous functions derivatives

Bibliography

Index

End User License Agreement

List of Illustrations

1 Continuous-time Systems: General Properties, Feedback, Stability, Oscillators

Figure 1.1. Representation of a sinusoidal signal on the complex plane
Figure 1.2. Spectrum of a sinusoidal signal (amplitude solid line, phase dotted)
Figure 1.3. Spectrum of a periodic signal of repetition frequency f₁ (modules in bold and arguments in dotted lines)
Figure 1.4. Triangular signals (left) and their spectrum (FT) (right). For a color version of this figure, see www.iste.co.uk/muret/electronics2.zip
Figure 1.5. Bode and Nyquist diagrams of a first-order low-pass filter [H₁(u)]^-1 in full lines and H₁(u) as a dotted line in the complex plane. For a color version of this figure, see www.iste.co.uk/muret/electronics2.zip
Figure 1.6. Bode and Nyquist diagrams of a second-order low-pass filter with transmittance [H₂(u)]^-1 (from the highest to the lowest curve, the ζ values, displayed as z inside the figure, are 0.05, 0.5, 0.707, 5 in the Bode diagrams at left and from the lowest to the highest 0.2, 0.5, 5 in the Nyquist diagram at right). H₂(u) is plotted in the case ζ = 0.5 as a dotted line in the complex plane. For a color version of this figure, see www.iste.co.uk/muret/electronics2.zip
Figure 1.7. Bode and Nyquist diagrams of a second-order band-pass filter (from the highest to the lowest curves, the ζ values are 0.06, 0.6, 0.707, 6 in the Bode diagram at left and 0.2, 0.6, 6 in the Nyquist diagram at right). For a color version of this figure, see www.iste.co.uk/muret/electronics2.zip
Figure 1.8. Bode and Nyquist diagrams of a second-order high-pass circuit (from the highest to the lowest curves, the ζ values are 0.06, 0.6, 0.707, 6 in the Bode diagram at left and 0.2, 0.6, 6 in the Nyquist diagram at right). For a color version of this figure, see www.iste.co.uk/muret/electronics2.zip
Figure 1.9. Map of transmittance poles
Figure 1.10. Block diagram of a negative feedback system
Figure 1.11. Inverting amplifier (left) and non-inverter (right) circuit
Figure 1.12. Divider (left) and square root function (right), with V₀ as a constant voltage
Figure 1.13. “No threshold” rectification
Figure 1.14. Asymptotic Bode diagram of operational amplifier circuit gains
Figure 1.15. Impedance converter circuit
Figure 1.16. Nyquist diagrams of the complex variable s= σ + jω and of the transfer function H(s)
Figure 1.17. Bode diagrams of the gain modulus and the reverse of the feedback coefficient, and loop gain argument, for a closed-loop system stable in open loop (full line, system stable in closed loop since |1/B| > |A| for f = f₂; dotted line, system unstable in closed loop since |1/B| < |A| for f = f₁; the frequencies corresponding to Arg{AB}=π in each case)
Figure 1.18. Block diagram of a system in state representation
Figure 1.19. Sinusoidal Wien bridge oscillator
Figure 1.20. Sinusoidal oscillator using an inverter circuit based on an operational amplifier
Figure 1.21. Colpitts oscillator
Figure 1.22. Symbol and equivalent circuit of quartz resonator
Figure 1.23. Hartley oscillator
Figure 1.24. Clapp oscillator
Figure 1.25. Quartz oscillator operating with an inverting logic gate
Figure 1.26. Amplitude oscillator stabilization mechanism
Figure 1.27. Dynamic circuit with negative conductance −G_a and circuit with series (at left) or parallel (at right) resonant network, damped by positive conductance G_p
Figure 1.28. Nonlinear dipole N characteristics to the left and S to the right, with negative dynamic resistances and conductance (1/R_d or G_d = - G_a), and load lines suited to relaxation oscillator operation
Figure 1.29. Relaxation oscillator circuits with N dipole to the left and S to the right
Figure 1.30. Start cycle and limiting cycle of a quasi-sinusoidal oscillator obtained from simulation results
Figure 1.31. Chua oscillator
Figure 1.32. Characteristic i = g(u) of dipole D used in the Chua oscillator and load lines corresponding to R equal to (i) R₁, (ii) R₂ and (iii) R₃
Figure 1.33. Eigenvalues of A in case (i) for the operating point at the origin. For a color version of this figure, see www.iste.co.uk/muret/electronics2.zip
Figure 1.34. Eigenvalues of A in case (ii) for operating point “0” around the origin. For a color version of this figure, see www.iste.co.uk/muret/electronics2.zip
Figure 1.35. Eigenvalues of A in case (ii) for operating points P_±1. For a color version of this figure, see www.iste.co.uk/muret/electronics2.zip
Figure 1.36. Cycles completed by the operating point u₂(i₃) in various conditions
Figure 1.37. FFT of the Chua oscillator signal for β =21, γ =0. For a color version of this figure, see www.iste.co.uk/muret/electronics2.zip
Figure 1.38. FFT of the Chua oscillator signal for β =24, γ =0. For a color version of this figure, see www.iste.co.uk/muret/electronics2.zip
Figure 1.39. FFT of Chua oscillator signals in deterministic chaotic conditions obtained with β =30, γ =0 in red (higher spectrum) and with β =24, γ =0.24 in blue (lower spectrum). For a color version of this figure, see www.iste.co.uk/muret/electronics2.zip

2 Continuous-time Linear Systems: Quadripoles, Filtering and Filter Synthesis

Figure 2.1. Input and output receptor conventions in a quadripole or two-port network
Figure 2.2. Association of quadripoles in series (on the left) and in parallel (on the right), using the appropriate models
Figure 2.3. Quadripole with receptor convention at left (input) and generator convention at right (output)
Figure 2.4. Current–voltage feedback on the left and voltage–current feedback on the right
Figure 2.5. Current–current feedback on the left and voltage–voltage feedback on the right
Figure 2.6. Representation of a type I quadripole with a single linked source
Figure 2.7. Series Foster′s synthesis
Figure 2.8. Parallel Foster’s synthesis
Figure 2.9. Ladder network obtained through Cauer’s synthesis, the order of the last two elements L₅ and C₆ being arbitrary
Figure 2.10. Ladder network obtained through Cauer synthesis up to L₃, then Foster synthesis in series for the element C₇ and finally using Cauer synthesis for the last two
Figure 2.11. Representations of a quadripole, including currents, voltages and complex parameters on the left and using incident and reflected waves, and s-parameters on the right
Figure 2.12. Chain of two quadripoles described by their chain parameters, deduced from s-parameters
Figure 2.13. Generator with current source and internal admittance Y_g loaded by an admittance Y_u
Figure 2.14. Quadripole matched to the input generator and the output load
Figure 2.15. Active quadripole described by its parameters Y, inserted between a generator and a load
Figure 2.16. Quadripole inserted between a generator and a load for the description by the s-parameters
Figure 2.17. Quadripole described by its s-parameters with terminations of any kind
Figure 2.18. Quadripole in a situation of image-matching, where Z_e and Z_s are also, respectively, the input and output impedance of the quadripole
Figure 2.19. Quadripole with hybrid parameters (type II) between a generator and a load
Figure 2.20. Block diagram representing the quadripole and termination elements
Figure 2.21. Block diagram giving V₂ from E_g for the quadripole and termination elements
Figure 2.22. Ladder admittance and impedance network
Figure 2.23. Block diagram of an unspecified portion of the ladder network of Figure 2.22
Figure 2.24. Sinc function
Figure 2.25. Integration contour for the application of the Cauchy theorem to the function H(jω) − H(∞) divided by s−jω₁
Figure 2.26. Transmittance and attenuation template of a low-pass filter
Figure 2.27. Transmittance modulus and group delay for fourth-order low-pass filters (Bessel in red broken line, Butterworth in blue dash-dot line, Chebyschev in full light green line, elliptical in dark green dash). For a color version of this figure, see www.iste.co.uk/muret/electronics2.zip
Figure 2.28. Active filters with a single operational amplifier
Figure 2.29. Second-order active filter using three operational amplifiers
Figure 2.30. Delyannis–Friend second-order active filter
Figure 2.31. Multiple feedback loop cell with forward transmittances T_n, T_n-1, … T₂, T₁ and feedback transmittances R₁, R₂, … R_n-1 R_n
Figure 2.32. Ladder sixth-order low-pass filter
Figure 2.33. Network of the normalized Chebyschev filter obtained after the transformation low-pass to band-pass with a normalized bandwidth Δω = 0.3 and a ripple of 0.1 dB within the bandwidth
Figure 2.34. Transmittance modulus of the eighth-order Chebyshev band-pass filter with normalized bandwidth equal to 0.3 and a ripple of 0.1 dB within the bandwidth
Figure 2.35. Sixth-order low-pass elliptic filter
Figure 2.36. Fourth-order elliptic filter synthesis based on its normalized admittance y₂₂
Figure 2.37. Low-pass filter synthesis based on its normalized impedance z₁₁ = Z₁₁/R₁
Figure 2.38. Low-pass filter synthesis based on its normalized admittance y₂₂ = Y₂₂ R₂
Figure 2.39. Full filter
Figure 2.40. Low-pass filter synthesized from its normalized impedance z₁₁
Figure 2.41. Z₁₁ computation network
Figure 2.42. Y₂₂ computation network
Figure 2.43. Final network of the Chebyschev sixth-order low-pass filter
Figure 2.44. Four-cell low-pass filter, two of each type, with the same image impedance at each port, given by the expression of Z_i1 in the previous table, provided that m₂ = m₁
Figure 2.45. Modulus of image impedances normalized by the characteristic impedance R₀ for the low-pass (a) and (b) cells (real if ω < 1, imaginary if ω > 1): Z_i1(b) in solid lines for m = 0.5; Z_i1(a) in dotted line and Z_i2(a) = Z_i2 (b) in dash-dotted line. For a color version of this figure, see www.iste.co.uk/muret/electronics2.zip
Figure 2.46. Modulus of the image impedances normalized by the characteristic impedance R₀ for high-pass cells (a′) and (c) (imaginary for ω < 1, real for ω > 1): Z_i2(c) in solid lines for m′ = 0.5; Z_i1(c) = Z_i1(a′) in dash-dotted line and Z_i2(a′) in dotted line. For a color version of this figure, see www.iste.co.uk/muret/electronics2.zip
Figure 2.47. Ripple envelope in the bandwidth of low-pass filters of parameter m₁ = 0.5 characterizing the end cells for three values of the parameter μ = 1.02; 1.07; 1.12 in ascending order of the values at ω = 0 (respectively, in dotted line, solid line and dash-dotted line). For a color version of this figure, see www.iste.co.uk/muret/electronics2.zip
Figure 2.48. Schematic of a low-pass filter, symmetrical and composed of two doubled cells of type (b)-(bi) and two simple terminal (b) and (bi) cells. Terminations are not illustrated
Figure 2.49. Image-transmittance in red and actual transmittance in blue for the low-pass filter synthesized in this section. For a color version of this figure, see www.iste.co.uk/muret/electronics2.zip
Figure 2.50. Deviation of the actual attenuation in the bandwidth with respect to 6 dB for the low-pass filter synthesized in this section
Figure 2.51. Elementary type-(d) band-pass cell, with an attenuation pole in the upper lateral stop band
Figure 2.52. Elementary type (e) band-pass cell, with an attenuation pole in the lower lateral stop band
Figure 2.53. Complementary cells of band-pass cells
Figure 2.54. Image-matching filter, composed of cascading cells represented by rectangles, with their termination resistances
Figure 2.55. Image attenuation in red and effective attenuation in blue, obtained by simulation, for the filter whose template is defined in the example above. For a color version of this figure, see www.iste.co.uk/muret/electronics2.zip
Figure 2.56. Equivalent circuit and symbol of the quartz resonator, where the damping resistance (Joule losses) is neglected
Figure 2.57. Another equivalent circuit of the quartz resonator
Figure 2.58. Lattice filter

First published 2018 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

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The rights of Pierre Muret to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Control Number: 2017961003

British Library Cataloguing-in-Publication Data
A CIP record for this book is available from the British Library
ISBN 978-1-78630-182-6

Preface

Today, we can consider electronics to be a subject derived from both the theoretical advances achieved during the 20th Century in areas comprising the modeling and conception of components, circuits, signals and systems, together with the tremendous development attained in integrated circuit technology. However, such development led to something of a knowledge diaspora that this work will attempt to contravene by collecting both the general principles at the center of all electronic systems and components, together with the synthesis and analysis methods required to describe and understand these components and subcomponents. The work is divided into three volumes. Each volume follows one guiding principle from which various concepts flow. Accordingly, Volume 1 addresses the physics of semiconductor components and the consequences thereof, that is, the relations between component properties and electrical models. Volume 2 addresses continuous time systems, initially adopting a general approach in Chapter 1, followed by a review of the highly involved subject of quadripoles in Chapter 2. Volume 3 is devoted to discrete-time and/or quantized level systems. The former, also known as sampled systems, which can either be analog or digital, are studied in Chapter 1, while the latter, conversion systems, we address in Chapter 2. The chapter headings are indicated in the following general outline.

Each chapter is paired with exercises and detailed corrections, with two objectives. First, these exercises help illustrate the general principles addressed in the course, proposing new application layouts and showing how theory can be implemented to assess their properties. Second, the exercises act as extensions of the course, illustrating circuits that may have been described briefly, but whose properties have not been studied in detail. The first volume should be accessible to students with a scientific literacy corresponding to the first 2 years of university education, allowing them to acquire the level of understanding required for the third year of their electronics degree. The level of comprehension required for the following two volumes is that of students on a master’s degree program or enrolled in engineering school.

In summary, electronics, as presented in this book, is an engineering science that concerns the modeling of components and systems from their physical properties to their established function, allowing for the transformation of electrical signals and information processing. Here, the various items are summarized along with their properties to help readers follow the broader direction of their organization and thereby avoid fragmentation and overlap. The representation of signals is treated in a balanced manner, which means that the spectral aspect is given its proper place; to do otherwise would have been outmoded and against the grain of modern electronics, since now a wide range of problems are initially addressed according to criteria concerning frequency response, bandwidth and signal spectrum modification. This should by no means overshadow the application of electrokinetic laws, which remains a necessary first step since electronics remains fundamentally concerned with electric circuits. Concepts related to radio-frequency circuits are not given special treatment here, but can be found in several chapters. Since the summary of logical circuits involves digital electronics and industrial computing, the part treated here is limited to logical functions that may be useful in binary numbers computing and elementary sequencing. The author hopes that this work contributes to a broad foundation for the analysis, modeling and synthesis of most active and passive circuits in electronics, giving readers a good start to begin the development and simulation of integrated circuits.

Outline

1) Volume 1: Electronic Components and Elementary Functions [MUR 17].
1. i) Diodes and Applications
2. ii) Bipolar Transistors and Applications
3. iii) Field Effect Transistor and Applications
4. iv) Amplifiers, Comparators and Other Analog Circuits
2)
Volume 2: Continuous-time Signals and Systems.
1. i) Continuous-time Stationary Systems: General Properties, Feedback, Stability, Oscillators
2. ii) Continuous-time Linear and Stationary Systems: Two-port Networks, Filtering and Analog Filter Synthesis
3) Volume 3: Discrete-time Signals and Systems and Conversion Systems [MUR 18].
1. i) Discrete-time Signals: Sampling, Filtering and Phase Control, Frequency control circuits
2. ii) Quantized Level Systems: Digital-to-analog and Analog-to-digital Conversions

Pierre MURET
November 2017

Introduction

This volume is dedicated to the study of linear and stationary systems in which time is considered as a continuous variable, as well as certain extensions in the case of nonlinear systems. It is mainly centered on single-input and single-output systems but a method capable of generalizing studies to linear or nonlinear multi-input and multi-output systems is also addressed. Generally, in order to highlight the properties of these systems, one must necessarily rely on the analysis of electrical signals that either characterize their response to an excitation signal or their natural (or proper) response. The former output signal is dependent on the input signal and is called forced response, whereas their natural response is independent of the excitation signal placed on their input. Therefore, it is essential to begin with the representations of signals by forming a close correlation between the time domain and the frequency domain, which are connected by the Fourier transform or decomposition into Fourier series. It is then natural to customize the study to the case of stationary systems, for which the forced response is invariant under time translation of the signal applied on input, and which, in addition, follow the principle of causality. The unilateral Laplace transform then proves to be useful and it leads us to the notion of transfer function or transmittance, together with the Fourier transform in the case of finite energy signals. The properties of these two types of transforms and their application to the case of electronic systems are covered in the first part of Chapter 1 while the consequences of causality are addressed in Chapter 2.

The second part of Chapter 1 is dedicated to the study of feedback and its applications, and then to the different methods for studying the stability of the systems, or to means able to control their instability, as is the case for oscillators. A system is stable if, after a finite life span excitation, it finally returns to its previous idle state, namely without any variation of electrical quantities, and it is unstable otherwise. In the early stages of electronics, feedback was paramount and it led to much progress and the development of a multitude of applications, which are reviewed here. The mathematical tools constituted by the time–frequency transforms mentioned earlier or representations in the complex plane are then used to address problems of system stability, including the case of those that incorporate a feedback loop, known as looped systems. The extension to state variables and state representation, which is based on the decomposition of the response of a system into a set of first-order differential equations, is then addressed. The previous concepts finally make it possible to detail the different ways for analyzing oscillators’ operation, which initially can be considered as linear systems at the limit of stability, but which in practice are always subject to a limitation of the amplitude that requires nonlinearity to be taken into account. The transition from predictable operation to a chaotic regime is presented in the case of a model system.

In Chapter 2, the properties of stable electronic systems are particularized to the case of networks and specially quadripoles. The different representations of networks in the form of quadripoles are discussed, as well as all notions of impedance or admittance deriving therefrom. Some are measurable, thus experimentally feasible, while others are fictional, such as image impedances, but open a highly fruitful scope of application, which is the subject of the last section of this chapter. The concepts of matching, whether power or impedance matching, are detailed, as well as their consequences and rules to apply in practice in order to optimize the operation of electronic assemblies and to best take advantage of the components that are included.

The last part of Chapter 2 is devoted to stable systems that can be analyzed as analog filters, namely satisfying the principle of causality, of which the general consequences are presented. There are either circuits incorporating one or more active devices such as operational amplifiers or passive circuits, limited here to non-dissipative cases. The synthesis of these analog filters is thorough, and can be used to determine the value of all the components of a filter based on imposed criteria, most often a template in the frequency domain. Two topics are presented; on the one hand for active filters and on the other hand for non-dissipative passive filters. In the second case, the method using effective parameters is an exact method, but not covering all the applications, while the method of image parameters is suitable to most requirements, with a deviation from the template that can be minimized. The ways to make adjustments and all circuits necessary for the practical implementation of the filters are detailed. Examples are given for each important case, based on the transfer functions calculated by means of software programs (here, MATLAB). The different possible choices for the computational functions are presented in relation to the criteria to be verified. In the case of the synthesis based on image parameters, formulas allowing the calculation of all elements are demonstrated. Although the case of systems with distributed (or scattered) elements, essential when the wavelength becomes comparable to the dimensions of the circuit, is not explicitly addressed, the description of the quadripoles using s-parameters, as detailed in Chapter 2, easily adapts.

Continuous-time Signals and Systems

Outline