Pulse Width Modulation in Power Electronics 9811234574, 9789811234576

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Other World Scientific Titles by Mayergoyz

Plasmon Resonances in Nanoparticles ISBN: 978-981-4350-65-5 Fundamentals of Electric Power Engineering ISBN: 978-981-4616-58-4 Quantum Mechanics: For Electrical Engineers ISBN: 978-981-3146-90-7 ISBN: 978-981-3148-01-7 (pbk) Hysteresis and Neural Memory ISBN: 978-981-120-950-5

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

Library of Congress Control Number: 2021931305 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

PULSE WIDTH MODULATION IN POWER ELECTRONICS Copyright © 2021 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 978-981-123-457-6 (hardcover) ISBN 978-981-123-458-3 (ebook for institutions) ISBN 978-981-123-459-0 (ebook for individuals)

For any available supplementary material, please visit https://www.worldscientific.com/worldscibooks/10.1142/12217#t=suppl Typeset by Stallion Press Email: [email protected] Printed in Singapore

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TO OUR FAMILIES

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Preface

You have in your hands a book on pulse width modulation (PWM) in power electronics. This modulation is used in the design of inverters which convert dc input voltages into ac output voltages of desired and controllable frequencies and peak values. These inverters are widely used in such applications as the frequency control of speed of ac motors, the design of uninterruptible power supplies (UPS) as well as the integration of renewable energy sources into existing power grid systems. Pulse width modulation techniques are used in inverters to approximate sinusoidal waveforms by sequences (trains) of rectangular pulses of the same peak value. This approximation can only be achieved if the pulse widths are properly modulated. A common feature of PWM techniques is the suppression of low order harmonics in the Fourier spectra of PWM voltages. This suppression is achieved at the expense of some amplification of high order harmonics in the Fourier spectra. However, these high order harmonics are efficiently suppressed in output voltages by energy storage elements (inductors and capacitors) of load circuits of the inverters. As a result, almost sinusoidal output voltages are achieved. In the book, the main emphasis is on the discussion of conceptual foundations of PWM techniques. For this reason, the book has a very strong theoretical flavor. The discussion covers various PWM techniques. However, special efforts are made to present in detail the optimal time-domain PWM techniques advanced by the authors. The book has four chapters. The first chapter presents the review of some basic and selective facts of electric circuit theory. This is done with the purpose to engage a wider audience of readers. The first three sections of this chapter cover terminal relations for basic circuit elements, Kirchhoff laws, the phasor technique and phasor diagrams for the ac steady state ana-

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lysis of electric circuits as well as the discussion of three-phase circuits. The next three sections of the first chapter deal with the steady-state analysis of electric circuits subject to time-periodic non-sinusoidal voltage sources. Such sources typically appear as a result of periodic switchings in power converters. There are two analytical techniques that are used for the analysis of electric circuits with the above voltage sources. The first one is the frequency-domain technique, which is based on Fourier series expansion of time-periodic voltages sources and subsequent use of the phasor technique. The second one is the time-domain technique, which is based on the formulation of the steady-state circuit analysis as a boundary value problem for ordinary differential equations with periodic boundary conditions. In the subsequent chapters of the book, both of these techniques are extensively used for the analysis of electric circuits subject to pulse width modulated voltage sources. The second chapter of the book deals with the pulse width modulation in single-phase inverters. It starts with the discussion of the circuit structure and performance of single-phase bridge inverters. This discussion leads to the realization that pulse width modulation techniques are needed for dc-to-ac conversion. The second section of this chapter covers the version of PWM when the pulse widths are sinusoidally modulated. In this PWM technique, each half-cycle is subdivided into equal number of time intervals, and the centers of the rectangular pulses coincide with the midpoints of these time intervals. By using the midpoint approximation for the evaluation of Fourier coefficients, the analytical study of this sinusoidal PWM technique is carried out for large number of rectangular pulses. This study results in the “sparse-twin” Fourier spectrum which reveals the suppression of low order harmonics. The latter is achieved at the expense of amplification of high order harmonics which form periodically repeated sidebands. It turns out that these sidebands have two dominant “twin” harmonics. The third section of chapter 2 deals with selective harmonic elimination technique. The realization of this technique requires the solution of coupled nonlinear transcendental equations. It turns out that the solution of these equations can be reduced to the computing of the roots of a single univariate polynomial. This is accomplished by using the Chudnovsky method, which is based on the mathematical machinery of Chebyshev polynomials as well as modified Newton’s Identities for elementary symmetric polynomials. The fourth section of the chapter deals with the time-domain analysis of electric circuits subject to PWM voltages. This analysis is performed for different load circuits. It starts with the dis-

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cussion of the traditional LR load circuit, and it is demonstrated that this analysis is reduced to the solution of simultaneous linear algebraic equations with two-diagonal matrices. It turns out that analytical solutions of such equations can be found. These solutions lead to the explicit expression for output voltages in terms of switching-times, which determine the widths of rectangular pulses. In the case of L-RC load circuits, the timedomain technique leads to simultaneous, linear algebraic equations with four-diagonal matrices. It is remarkable that explicit analytical solutions to these equations can be found. This is because these equations have a special mathematical structure which allows for their reduction to two sets of decoupled simultaneous equations with two-diagonal matrices. Similarly, in the case of L-C-LR load circuits, the time-domain analysis leads to the solution of simultaneous algebraic equations with six-diagonal matrices. These equations can be reduced to three sets of decoupled simultaneous equations with two-diagonal matrices. In this way, the explicit analytical expressions for output voltages in terms of switching times can be obtained. The fifth and the sixth sections of the second chapter deal with the optimal pulse width modulation technique for LR and L-CR load circuits. These techniques are based on explicit analytical expressions for output voltages in terms of switching times. Consequently, these switching times can be chosen to fulfill specific optimization criteria. One of such criteria that is extensively discussed in the book is the minimization of the total harmonic distortion (THD) of the desired sinusoidal output voltages. The explicit analytical expressions for THD in terms of switching times are derived for LR and L-RC load circuits. Furthermore, it is demonstrated that the above optimization technique can be generalized to simultaneously perform the selective elimination of specific harmonics as well as the minimization of THD due to remaining high order harmonics. This is done by treating harmonic elimination equations as constraints and by using the method of Lagrange multipliers for the solution of the constrained minimization problem. Finally, a brief description of the use of such optimization techniques as Sequential Quadratic Programming (SQP) and Interior-Point method for optimal pulse width modulation is presented. The third chapter deals with pulse width modulation in three-phase inverters. The switching used to realize the three-phase PWM is much more sophisticated than the switching used in single-phase inverters. This switching is discussed in detail in the first section of the chapter. The ultimate goal of this switching is to impose the proper symmetry conditions as well as certain constraints on PWM line voltages. The symmetry con-

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ditions which are imposed are: T3 -translational symmetry which results in the elimination of all harmonics with orders divisible by three; half-wave symmetry which results in the elimination of all even order harmonics; and quarter-wave symmetry which is mathematically equivalent to odd symmetry and results in the elimination of all cosine-type harmonics. There are two constraints: the single-leg switching constraint which means that the state of only two switches in the same leg of the inverter are simultaneously changed, and the Kirchhoff Voltage Law (KVL) constraint on PWM line voltages. The effect of above symmetry conditions and constraints on the switching can be clearly elucidated by using the pqr -technique, which is based on the division of all rectangular pulses during one half-cycle into three distinct groups labeled as p, q and r pulse-groups. By using this subdivision, it is established that T3 -translational and half-wave symmetries as well as KVL constraint imply that each pulse in the q group is the sum of two specific pulses from the p and r groups. Furthermore, it is demonstrated that the single-leg switching constraint leads to two specific and alternating in time patterns of switching for the realization of the KVL constraint. Finally, by using the above pqr -technique, the three-phase PWM with sinusoidal modulation of pulse widths is developed. This technique leads to the same results as the space-vector pulse width modulation (SVPWM). The second section of the chapter deals with the Fourier analysis of the three-phase sinusoidal pulse width modulation of line voltages. Conceptually, this analysis is similar to the discussion of the Fourier spectra for single-phase inverters. The main distinction of the three-phase inverters is that only the pulses in the q-group are midpoint centered, while the pulses in the p and r groups are off-centered. The latter leads to the appearance of intermediate bands of harmonics in the spectra of three-phase line voltages. These intermediate bands exist in addition to the presence of main bands of harmonics which have the same structure as in the case of single-phase inverters. By using the midpoint approximation for the evaluation of the Fourier coefficients, the analytical study of intermediate bands of harmonics is carried out, and it is demonstrated that the first intermediate band has four dominant harmonics. It is also shown that the obtained analytical results are well-compared with numerical results based on accurate evaluation of Fourier coefficients. It is worthwhile to mention here that in our spectral analysis of PWM voltages the conventional (i.e. single-variable) Fourier series are used. In literature, the approach based on double Fourier series has been advanced for the same purpose, and the

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Fourier coefficients have been obtained in terms of Bessel and trigonometric functions. It seems to us that the double Fourier series approach is more relevant when the exact periodicity of PWM line voltages is violated, and these voltages are quasi-periodic functions of time. Whereas, the case of strict periodicity can be properly treated by using single-variable Fourier series. The third section of chapter 3 deals with the time-domain analysis of three-phase circuits subject to PWM line voltages. It is typical that loads of the three-phase inverters are star-connections of identical linear circuits. This arrangement can be viewed as a balanced load of three-phase inverters. For this reason, the per-phase time-domain analysis is developed for such load circuits. It leads to the analysis of load electric circuits subject to multi-level voltage sources. The multi-level structure of these voltage sources is determined by using the pqr -technique. Subsequently, the efficient analytical technique is suggested which leads to the exact formulas for currents and voltages in load circuits as explicit functions of switching timeinstants for PWM line voltages. This technique is illustrated for the LR, L-CR and L-C-LR load circuits. The fourth section of the chapter deals with the optimal pulse width modulation technique for three-phase inverters. This technique is based on the explicit analytical expressions found in the previous section for output voltages in load circuit in terms of switching times for line voltages. These switching times can be chosen to minimize the total harmonic distortion of the desired sinusoidal output voltage. The explicit analytical expressions for such THD in terms of switching times are derived in the case of LR and L-CR load circuits. It is also demonstrated by using the constrained optimization that the above minimization technique can be extended to simultaneously eliminate selected harmonics as well as minimize THD due to remaining harmonics. Finally, it is shown that the total number of independent switching times for line voltages involved in the minimization can be substantially reduced. This reduction is based on the symmetry considerations as well as KVL and single-leg switching constraints, which lead to certain relations between switching times. The last section in chapter 3 deals with AC-to-AC converters. Such converters are often constructed by cascading rectifiers with three-phase inverters. To completely describe the structure of AC-to-AC converters, the design of three-phase bridge rectifier and the analysis of its operation are presented in the section. As a result, a very simple analytical formula for the output voltage of three-phase bridge rectifier is derived and carefully analyzed. In the conclusion of this section, the discussion is presented of how AC-to-AC converters are used in semiconductor drives for frequency control of speed

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of induction and synchronous motors. In the case of induction motors, the discussion is based on the electromagnetic torque vs speed characteristics of such motors. By using these characteristics, it is established that the operational speeds of induction motors are very close to the synchronous speeds of rotating magnetic fields created by three-phase stator windings of the motors. This fact is utilized for the frequency control of speed. The section is concluded with the discussion of the “constant volts per hertz” criterion which is used to maintain constant electromagnetic torque in the process of frequency control of motor speed. The fourth chapter of the book deals with magnetic issues which can be useful in the design and applications of inverters as well as other power converters. It is known that inductors are widely used in power electronic circuits. Inductors are designed by using high magnetic permeability ferromagnetic cores to enhance the values of their inductances. The ferromagnetic cores exhibit hysteresis. Furthermore, they also have a finite electric conductivity which results in the induction of eddy currents when inductor cores are subject to time-varying magnetic fields. This leads to the energy losses due to hysteresis phenomena and eddy currents. Such losses are usually called core losses. The core losses also occur in ferromagnetic parts of ac motors, and they are frequency dependent. For this reason, these losses may be quite pronounced in the case of frequency control of speed of these motors. The fourth chapter deals with the analytical study of hysteresis and eddy current losses. The first section of this chapter mostly covers the classical results on core losses. The novel parts of this section are related to the analysis of hysteresis losses for non-periodic (non-cyclic) variations of magnetic fields as well as the analysis of eddy current losses in the case of circular and elliptic polarizations of magnetic fields. Such polarizations occur in the magnetic cores of ac machines whose principle of operation is based on rotating magnetic fields. In the first section, the analysis of eddy currents is performed by neglecting the magnetic fields created by induced eddy currents. This is tantamount to the assumption of uniform distribution of magnetic flux density across the thickness of the core laminations. This assumption is removed in the subsequent sections of the chapter. In the second section, the rigorous analysis of eddy currents is performed in the case of circular polarization of magnetic fields and magnetically isotropic media. In this case, due to the rotational symmetry of the problem, there is no generation of high order harmonics despite nonlinear magnetic properties of ferromagnetic media. Remarkably, this allows for the derivation of exact analytical solutions to nonlinear

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Maxwell partial differential equations in the case of power law approximations of magnetization curves. These solutions exhibit almost rectangular profile of magnetic flux density and a finite depth z0 of penetration of electromagnetic fields into magnetically nonlinear conducting media. By using the above solutions, simple analytical formulas for eddy current losses are derived. When the thickness of lamination is less than or equal to 2z0 , these losses are the same as discussed in the first section of the chapter. In the third section of chapter 4, eddy currents are analyzed in the case of elliptical polarization of magnetic fields. This polarization is treated as a perturbation of circular polarization, and the efficient perturbation technique is developed to account for generation of high order harmonics and for the calculation of eddy currents. A similar perturbation technique can be used to account for anisotropic magnetic properties of ferromagnetic cores. The fourth section of the chapter deals with the calculation of eddy currents in the case of linear polarization of magnetic fields. By using the mathematical machinery of self-similar solutions, it is demonstrated that nonlinear diffusion of electromagnetic fields into ferromagnetic laminations occurs as an inward progress of almost rectangular profiles of magnetic flux density of variable in time height. As a consequence, magnetic flux density is not uniform across the lamination thickness even for relatively low frequencies. This results in the increase of eddy current losses usually known as “excess losses”. The validity of the above explanation of excess eddy current losses has been experimentally verified. The last section of the chapter is quite different in nature from the previous sections of the same chapter. Its purpose is to discuss the latest developments in spintronics which may be very promising for power conversion at nanoscale and at very low values of voltages and power. Furthermore, it is remarkable that spintronics based power converters may be designed to operate without repetitive switching but due to the unique physical phenomena occurring in nanomagnetic devices. Nevertheless, it must be remarked here that the discussion presented in this section is only suggestive in nature, and there are many issues that must be investigated and resolved before spintronics based power converters become a reality. In the book, the presented analytical results are illustrated by numerous computations. These computations have been performed by using standard MATLAB software. Similar software can be used in modern digital power electronics controllers for the realization of various pulse width modulation techniques in real-time. In writing this book, we endeavored to make it accessible and appealing

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to a broad audience of readers. To achieve this goal, a special emphasis has been placed on the clarity of exposition of various concepts and facts related to pulse width modulation in power electronics. It is for the readers to judge to what extent our efforts have been successful. No attempts have been made to refer to all relevant publications. For this reason, the reference list is not exhaustive but rather suggestive. Our thanks to Yu Shan Tay from World Scientific for her valuable assistance and patience. Finally, the first author of the book gratefully acknowledges the financial support derived from the Alford L. Ward professorship that made this project possible.

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Contents

Preface 1.

Review of Some Basic Facts of Electric Circuit Theory 1.1 1.2 1.3 1.4 1.5 1.6

2.

Review of Electric Circuit Theory Phasor Analysis . . . . . . . . . . Three-Phase Circuits . . . . . . . Fourier Series . . . . . . . . . . . Frequency-Domain Technique . . Time-Domain Technique . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

1 . . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

Pulse Width Modulation in Single-Phase Inverters 2.1 2.2 2.3 2.4 2.5 2.6

3.

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Single-Phase Bridge Inverter . . . . . . . . . . . . . Sinusoidal Pulse Width Modulation and Its Fourier Analysis . . . . . . . . . . . . . . . . . . . . . . . . Selective Harmonic Elimination and Chudnovsky Technique . . . . . . . . . . . . . . . . . . . . . . . Time Domain Analysis of Pulse Width Modulation in Single-Phase Inverters . . . . . . . . . . . . . . . Optimal Pulse Width Modulation . . . . . . . . . . Optimal Pulse Width Modulation for L-RC circuit

Pulse Width Modulation in Three-Phase Inverters 3.1 3.2 3.3

1 12 28 37 46 59 71

. . . .

71

. . . .

83

. . . . 101 . . . . 114 . . . . 129 . . . . 161 183

Three-Phase Bridge Inverter . . . . . . . . . . . . . . . . . 183 Fourier Analysis of Three-Phase Sinusoidal PWM . . . . . 204 Per-phase Time-Domain Analysis of Three-Phase Inverters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 xv

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3.4 3.5 4.

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Optimal Pulse Width Modulation in Three-Phase Inverters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 AC-to-AC Converters . . . . . . . . . . . . . . . . . . . . . 279

Magnetic Aspects 4.1 4.2 4.3 4.4 4.5

Hysteresis and Eddy Current Losses . . . . . . . . . Advanced Analysis of Eddy Current Losses for Circular Polarizations of Magnetic Fields . . . . . . . Perturbation Technique for Non-circular Polarization of Magnetic Fields . . . . . . . . . . . . . . . . . . . Eddy Currents in the Case of Linear Polarization of Magnetic Fields . . . . . . . . . . . . . . . . . . . On the Promising Use of Spintronics in Nanoscale Power Electronics . . . . . . . . . . . . . . . . . . . .

293 . . . 293 . . . 306 . . . 332 . . . 354 . . . 382

Bibliography

391

Index

399

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Chapter 1

Review of Some Basic Facts of Electric Circuit Theory

1.1

Review of Electric Circuit Theory

In circuit theory, there are two distinct types of basic relations [1]. In the first type, these relations are based on the physical nature of circuit elements, and they are called terminal relations. The second type of relations reflect the connectivity of the electric circuit, namely, how the circuit elements are interconnected. For this reason, they are sometimes called topological relations. These relations are expressed by the Kirchhoff Current Law (KCL) and Kirchhoff Voltage Law (KVL). We start with terminal relations and consider five basic two-terminal elements: resistor, inductor, capacitor and ideal (independent) voltage and current sources. These circuit elements are ubiquitous in power engineering. However, in power electronics, multi-terminal circuit elements are used as well. These multi-terminal circuit elements are models of power semiconductor devices and they are not discussed in this brief review. A two-terminal element is schematically represented in Figure 1.1. Each two-terminal element is characterized by the voltage v(t) across the terminals and by the current i(t) through the element. In order to write the meaningful equations relating the voltages and currents in electric circuits, it is necessary to assign a polarity to the voltage and a direction to the current. These assigned (not actual) directions and polarities are called reference directions and reference polarities. These reference directions and polarities are assigned arbitrarily. The actual current directions and voltage polarities are not known beforehand and they may change with time. The reference direction for a current is indicated by an arrow, while the reference polarity is specified by placing plus and minus signs next to element terminals (see Figure 1.1). The reference directions and polarities are

1

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usually coordinated by choosing the reference direction of the current from the positive reference terminal to the negative reference terminal. i(t) + v(t) − Fig. 1.1

As discussed below, the reference directions and polarities are used in writing KCL and KVL equations. These equations are then solved and the signs of currents and voltages are found at any instant of time. If at time t a current is positive, i(t) > 0,

(1.1)

then the actual direction of this current coincides with its reference direction. If, on the other hand, the found current is negative at time t, i(t) < 0,

(1.2)

then the actual direction of this current at time t is opposite to its reference direction. Similarly, if a voltage is found to be positive, v(t) > 0,

(1.3)

then the actual polarity coincides with its reference polarity. On the other hand, if the voltage is found to be negative, v(t) < 0,

(1.4)

then the actual voltage polarity is opposite to its reference polarity. It is clear from the above discussion that the reference directions and polarities allow one to write KCL and KVL equations, then solve them and eventually find actual directions and polarities of circuit variables. Now, we consider a resistor. Its circuit notation is shown in Figure 1.2. The terminal relation for the resistor is given by Ohm’s law,

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Review of Some Basic Facts of Electric Circuit Theory

i(t)

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R −

+ v(t) Fig. 1.2

v(t) = Ri(t),

(1.5)

where R is the resistance of the resistor. By using the terminal relation (1.5), we find the expression for instantaneous power p(t) for the resistor, p(t) = v(t)i(t) = Ri2 (t),

(1.6)

or, p(t) =

v 2 (t) . R

(1.7)

It is clear from equations (1.6) and (1.7) that the instantaneous power for the a resistor is always positive. This implies that the resistor is an energy consuming element of electric circuit. For this reason, resistors are used in circuit models for actual power devices to describe their electric energy consumption. This is (for instance) typical for circuit models of electric motors. Next, we discuss an inductor. Its circuit notation is shown in Figure 1.3. The inductor is characterized by inductance L and its terminal relation is given by the formula: v(t) = L

di(t) . dt

(1.8)

The last equation implies that the current through an inductor is differeni(t) +

L v(t)

−

Fig. 1.3 tiable and, consequently, a continuous function of time. The latter means that for any instant of time t0 the value of the current i(t0− ) immediately

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before t0 is equal to the value of the current i(t0+ ) immediately after t0 : i(t0− ) = i(t0+ ).

(1.9)

It is clear from equation (1.8) that in the case of dc currents (i(t) = const) the voltage across the inductor is equal to zero and the inductor can be replaced by a “short-circuit” branch. On the other hand, if the inductor (as a part of a circuit) is connected by some switch to a source and the initial current through the inductor before switching is zero, then according to the continuity condition (1.9) the current through the inductor will remain zero immediately after switching. This means that immediately after switching the inductor is an “open-circuit” branch. It is apparent from equation (1.8) that the instantaneous power p(t) for the inductor is given by the formula
L d i2 (t) di(t) = . (1.10) p(t) = v(t)i(t) = Li(t) dt 2 dt It follows from the last equation that power is positive if the absolute value of inductor current increases with time and it is negative if the absolute value of the current decreases with time. This implies that when the absolute value of the inductor current is increasing with time, the electric power is being consumed and stored in the inductor’s magnetic field. On the other hand, when the absolute value of the inductor current is decreasing with time, then the electric power is “given back” at the expense of the energy previously stored in the magnetic field. This clearly suggests that the inductor is an energy storage element and it finds many applications as such in electric power engineering [2–5]. Another important application of the inductor is for “ripple suppression” in power electronics [2, 3, 5]. Power electronics converters are switching-mode converters in which semiconductor devices are used as switches that are repeatedly (periodically) switched “on” and “off” to achieve the desired performance of the converters. This periodic switching results in periodic components of electric currents which often manifest themselves as ripples in output converter voltages. These ripples can be suppressed by using inductors. Indeed, from equation (1.8) we find Z 1 t v(τ )dτ. (1.11) i(t) = i(0) + L 0

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If the current is a periodic function of time with period T , then i(0) = i(T )

(1.12)

and Z

T

v(τ )dτ = 0.

(1.13)

0

It is apparent that the second term in the right-hand side of equation (1.11) can be construed as a ripple in electric current. It is clear from formula (1.11) that this ripple can be suppressed by increasing the value of inductance L. It is also clear from formulas (1.11) and (1.13) that the same ripple can be suppressed by decreasing T , i.e., by increasing the frequency of switching. This leads to the “trade-off” that is practiced in power electronics: the higher the frequency of switching of power semiconductor devices, the smaller the value of inductance that is needed for the ripple suppression. Inductance is also used in power electronics in order to realize the performance of a diode as a free-wheeling diode. This performance is achieved by using a series connection of inductor and diode. As a result of this connectivity, the diode is turned on to maintain the continuity of the current through the inductor. It is important to stress that the continuity condition (1.9) is always valid for a single inductor. The situation can be very different in the case of two (or more) coupled inductors. In the latter case, the continuity of the current (1.9) through an individual inductor may not be valid and it is replaced by the continuity of total magnetic energy created by coupled inductors. This fact is utilized (for instance) in the design of power electronic flyback converters [5]. Now, we proceed to the discussion of a capacitor. Its circuit notation is shown in Figure 1.4. The capacitor is characterized by capacitance C and its terminal relation is given by the formula i(t) = C

dv(t) . dt

(1.14)

The last equation implies that the voltage across a capacitor is differentiable and, consequently, a continuous function of time. The latter means that for any instant of time t0 the value of the voltage v(t0− ) immediately

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C i(t) −

+ v(t)

Fig. 1.4 before t0 is equal to the value of the voltage v(t0+ ) immediately after t0 : v(t0− ) = v(t0+ ).

(1.15)

It is apparent from equation (1.14) that in the case of dc voltages (v(t) = const) the current through the capacitor is equal to zero and the capacitor is an “open-circuit” branch. This is consistent with the fact that the current through the capacitor is a displacement current, which exists only when the electric field in the capacitor varies with time. In the case when an uncharged capacitor with zero voltage is connected through some switch to a source, then according to the continuity condition (1.15) the voltage across the capacitor will remain zero immediately after switching. This means that immediately after switching the capacitor is a “short-circuit” branch. This implies that capacitors connected in parallel with power equipment may protect this equipment from large initial transient currents, which mostly flow through these capacitors. It is apparent from equation (1.14) that the instantaneous power p(t) for the capacitor is given by the formula
C d v 2 (t) dv(t) = . (1.16) p(t) = v(t)i(t) = Cv(t) dt 2 dt It follows from the last equation that power is positive if the absolute value of capacitor voltage increases with time and it is negative if the absolute value of capacitor voltage decreases with time. This implies that, when the absolute value of the capacitor voltage is increasing with time, the electric power is being consumed and stored in the electric field within the capacitor. On the other hand, when the absolute value of the capacitor voltage is decreasing with time, then the electric power is “given back” at the expense of energy stored in the electric field. This clearly reveals that the capacitor is an energy storage element and it is used as such in many power-related applications [2–5]. Another important application of the capacitor is for “ripple suppres-

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sion” in power electronics. Indeed, from equation (1.14) we derive Z 1 t i(τ )dτ. v(t) = v(0) + C 0

page 7

7

(1.17)

If the capacitor voltage is a periodic function of time with period T , then v(0) = v(T )

(1.18)

and T

Z

i(τ )dτ = 0.

(1.19)

0

It is clear that the second term in the right-hand side of formula (1.17) can be construed as a ripple in capacitor voltage. It is apparent from formula (1.17) that this ripple can be suppressed by increasing the value of capacitance C. It is also apparent from formulas (1.17) and (1.19) that the same ripple can be suppressed by decreasing T , i.e., by increasing the frequency of switching. Hence, the trade-off: the higher the frequency of switching of semiconductor devices in power converters, the smaller the value of the capacitance that is needed for ripple suppression. Finally, we shall discuss ideal (independent) voltage and current sources, whose circuit notations are shown in Figures 1.5 and 1.6, respectively. An

− +

i(t) vs (t) Fig. 1.5

is (t) −

+ v(t)

Fig. 1.6 ideal voltage source is a two-terminal element with the property that the

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voltage across its terminals is specified at every instant of time, v(t) = vs (t),

(1.20)

where vs (t) is a given (known) function of time. It is apparent from the given definition that the terminal voltage does not depend on the current through the voltage source, which is reflected in the terminology independent voltage source. An ideal current source is by definition a two-terminal element with the property that the current through this element is specified at every instant of time, i(t) = is (t),

(1.21)

where is (t) is a given (known) function of time. It is clear from the given definition that the terminal current does not depend on the voltage across the current source; in this sense, this is an ideal (or independent) current source. Previously we discussed the terminal relations, which are determined by the physical nature of the circuit elements. Now, we proceed to the brief discussion of the relations which are due to the connectivity of elements in an electric circuit. There are two types of such relations. We begin with the Kirchhoff Current Law (KCL). KCL equations are written for nodes of electric circuits. A node of an electric circuit is a “point” where three or more elements are connected together. KCL states that the algebraic sum of electric currents at any node of an electric circuit is equal to zero at every instant of time. This is mathematically expressed as follows: X

ik = 0.

(1.22)

k

The term “algebraic sum” implies that some currents are taken with positive signs while others are taken with negative signs. Two equivalent rules can be used for sign assignments. One rule is that positive signs are assigned to currents with reference directions toward the node, while negative signs are assigned to currents with reference directions from the node. KCL equations can be written for any node. However, only (n − 1) equations will be linearly independent, where n is the number of nodes in a given circuit. The (n − 1) nodes for which KCL equations are written can be chosen arbitrarily. The KCL equation for the last (n-th) node can be obtained by summing up the previously written KCL equations. This clearly sug-

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gests that the equation for the last (n-th) node is not linearly independent. A “point” in an electric circuit where only two elements are connected is not qualified as a node because of the triviality of the KCL equation in this case, which simply suggests that the same current flows through both circuit elements, i.e., these two circuit elements are connected in series. Next, we discuss equations written by using the Kirchhoff Voltage Law (KVL). These equations are written for loops. A loop is defined as a set of branches that form a closed path with the property that each node is encountered only once as the loop is traced. A branch is defined as a single two-terminal element or several two-terminal elements connected in series. KVL states that the algebraic sum of branch voltages around any loop of an electric circuit is equal to zero at every instant of time. This is mathematically expressed as follows: X

vk = 0.

(1.23)

k

The term “algebraic sum” implies that some branch voltages are taken with positive signs while others are taken with negative signs. The following rule can be used for sign assignment. If the tracing direction of the branch coincides with the reference direction of the branch current then the positive sign is assigned to the branch voltage, otherwise the negative sign is assigned to the branch voltage. Since there may be many possible loops for any given circuit, determining which loops must be traced in order to write linearly independent KVL equations is not entirely obvious. One method which will always produce the correct number of linearly independent KVL equations is based on the use of a graph tree. By definition, a graph tree of an electric circuit is a subset of branches with the property that all nodes of the circuit are connected together, but there are no closed loops formed by these branches. It is clear that any graph tree of an electric circuit with n nodes contains (n − 1) branches. By adding a new branch to the graph tree we create a new loop and a new KVL equation can be written for the created loop. This KVL equation will contain a new variable − branch voltage for the added branch. KVL equations written in this way will be linearly independent because each new equation contains a new variable. It is easy to see that the total number of linearly independent KVL equations written in the described way is equal to b − (n − 1), where b is the total number of branches in the circuit. This is so because b − (n − 1) branches should be

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removed to form a graph tree and, consequently, b − (n − 1) loops will be formed by adding one by one the removed branches. It is clear that the total number of linearly independent equations written by using KCL and KVL (i.e., the total number of equations (1.22) and (1.23)) is equal to the number b of branches. An additional b equations are obtained by using terminal relations (1.5), (1.8), (1.14), (1.20) and (1.21). Thus, the total number of equations is 2b, which is the total number of circuit variables, i.e., the total number of branch currents and branch voltages. These are ordinary differential equations for which the initial conditions can be found by using the continuity conditions (1.9) and (1.15). Thus, the framed equations (1.5), (1.8), (1.9), (1.14), (1.15), (1.20), (1.21), (1.22) and (1.23) form the foundation of electric circuit theory. In this theory, various analysis techniques have been developed which exploit the connectivity of electric circuits as well as the particular nature of excitation of these circuits. Some of these techniques will be discussed in this book due to their wide use in electric power engineering and particularly in power electronics. It is worthwhile to note that in electric circuit theory the basic (framed) relations (1.5), (1.8), (1.9), (1.14), (1.15), (1.20), (1.21), (1.22) and (1.23) are treated as axioms (postulates) which are fully consistent with experimental facts. However, within the framework of electromagnetic field theory, all these fundamental circuit relations can be derived by using approximations relevant to the notion of electric circuits with lumped parameters. It is worthwhile to stress in the conclusion of this section that electric circuits are models for actual devices − models which are based on some simplifications and approximations. This may lead in some cases to intrinsic (logical) contradictions between the basic circuit relations. We shall illustrate such possible contradictions for two cases of very simple electric circuits shown in Figures 1.7 and 1.8. SW i(t) + is (t)=I0

L v(t)

− R

Fig. 1.7

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SW + − +

vs (t)=V0

C

v(t)

R

− Fig. 1.8

In Figure 1.7, at time t0 = 0 a dc current source is connected by switch (SW ) to an inductor and a resistor connected in series. It is clear that prior to the switching the current through the inductor is equal to zero, i(0− ) = 0.

(1.24)

According to the continuity condition (1.9), the current i(0+ ) immediately after switching must be equal to zero, i(0+ ) = 0.

(1.25)

However, according to KCL the same current must be equal to the current I0 of the current source, i(0+ ) = I0 6= 0

(1.26)

which is a contradiction. Similarly, for the circuit shown in Figure 1.8, according to the continuity condition (1.15) the voltage v(0+ ) across the capacitor is equal to zero: v(0+ ) = 0,

(1.27)

if the capacitor was not charged before switching. However, according to KVL the same voltage must be equal to the voltage V0 of the dc voltage source: v(0+ ) = V0 6= 0,

(1.28)

which is a contradiction. The above contradictions appear because in the circuits shown in Figures 1.7 and 1.8 some small parameters have been neglected. For instance, for an actual (real) inductor there is always small parallel to L capacitance between the turns of the inductor. The presence of this capacitance will remove the contradiction between equations (1.25) and (1.26) because im-

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mediately after switching the current I0 will flow through this capacitance. Similarly, the presence of a small resistance of connecting wires in the circuit shown in Fig. 1.8 will remove the contradiction between equations (1.27) and (1.28), because immediately after switching the voltage V0 will be applied across such resistance. The presented discussion suggests that in the actual devices represented by the circuits shown in Figures 1.7 and 1.8 the initial stages of transients will be controlled by neglected small parameters. This implies that small parameters can be important for proper modeling of the performance of actual devices. In power engineering, there are many devices whose performance is controlled by small parameters. For instance, in the case of transformers and induction machines, these small parameters are leakage inductances. Whereas, in the case of boost and buck-boost power converters these small parameters are the small resistances of inductors used in their circuits [5]. 1.2

Phasor Analysis

In many electric power applications, electric circuits are excited by ac (sinusoidal) sources. Under steady-state conditions, all voltages and currents in such circuits will be sinusoidal. A special and very useful circuit analysis technique exists which exploits this fact. It is known as the phasor technique and it uses complex numbers to represent time-harmonic sinusoidal quantities. The main advantage of the phasor technique is that it reduces calculus operations on time-harmonic sinusoidal quantities to algebraic operations on complex numbers (phasors). As a result, the basic differential equations of electric circuits discussed in the previous section are reduced to linear algebraic equations with respect to the phasors. This significantly simplifies the analysis of electric circuits under ac steady-state conditions. This technique was originally proposed for the analysis of electric circuits by C. P. Steinmetz [6]. With time, this technique found applications in many different areas of electrical engineering. For instance, time-harmonic electromagnetic fields are usually studied by using the Maxwell equations written in the phasor form [7]. The central idea of the phasor technique can be described as follows. Every time-harmonic sinusoidal quantity is fully characterized by three numbers: its frequency, peak value and initial phase. In ac steady-state analysis, the frequency of sinusoidal quantities is fixed and known. Consequently, every time-harmonic quantity of known frequency is fully characterized by

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two numbers: its peak value and initial phase. The same is true for complex numbers. In the polar form, each complex number is characterized by its magnitude (absolute value) and polar angle. This suggests to represent any time-harmonic quantity by a phasor, which is a complex number whose magnitude is equal to the peak value of the time-harmonic quantity and whose polar angle is equal to the initial phase of this time-harmonic quantity. This definition of the phasor is illustrated by the following two formulas for time-harmonic (sinusoidal) voltage and current, respectively, v(t) = Vm cos (ωt + ϕV ) ↔ Vˆ = Vm ejϕV ,

(1.29)

i(t) = Im cos (ωt + ϕI ) ↔ Iˆ = Im ejϕI ,

(1.30)

where Vˆ and Iˆ are the corresponding phasors of sinusoidal voltage and sinusoidal current. It is clear from the above formulas that if sinusoidal quantities are given then it is easy to write their phasors. On the other hand, if phasors are known and represented in the polar form, then it is easy to write the corresponding time-harmonic quantities. Now, it is easy to see that in the case of ac steady state all terminal relations can be represented in the phasor form. Indeed, consider first a resistor. Then, by substituting sinusoidal voltage and current into formula (1.5), we find: Vm cos (ωt + ϕV ) = RIm cos (ωt + ϕI ) .

(1.31)

The last equality implies that Vm = RIm ,

(1.32)

ϕV = ϕI .

(1.33)

Formula (1.33) reveals that in the case of resistors, their time-harmonic voltages and currents are in phase. From the last two formulas, we also derive ˆ Vˆ = Vm ejϕV = RIm ejϕI = RI.

(1.34)

Thus, it is established that the terminal relation for the resistor can be

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written in the phasor form as follows: ˆ Vˆ = RI.

(1.35)

Next, we consider an inductor. Then, by substituting sinusoidal voltage and current into formula (1.8), we find Vm cos (ωt + ϕV ) = −ωLIm sin(ωt + ϕI ),

(1.36)

 π . Vm cos (ωt + ϕV ) = ωLIm cos ωt + ϕI + 2

(1.37)

or

The last equality implies that Vm = ωLIm , π ϕV = ϕI + . 2

(1.38) (1.39)

Formula (1.39) reveals that in the case of an inductor, its time-harmonic voltage leads its time-harmonic current by π2 . Equivalently, the current lags behind the voltage by π2 . From the last two formulas we also derive ˆ Vˆ = Vm ejϕV = ωLIm ej(ϕI +π/2) = jωLIm ejϕI = jωLI.

(1.40)

Thus, it is established that the terminal relation for the inductor can be written in the phasor form as follows: ˆ Vˆ = jωLI.

(1.41)

Finally, we consider a capacitor. By substituting sinusoidal voltage and current into formula (1.14), we find Im cos (ωt + ϕI ) = −ωCVm sin (ωt + ϕV )

(1.42)

 π . Im cos (ωt + ϕI ) = ωCVm cos ωt + ϕV + 2

(1.43)

or

The last equality implies that Im = ωCVm , π ϕI = ϕV + . 2

(1.44) (1.45)

Formula (1.45) reveals that in the case of a capacitor, its time-harmonic

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current leads its time-harmonic voltage by π2 . Equivalently, the voltage lags behind the current by π2 . From the last two formulas we also derive Iˆ = Im ejϕI = ωCVm ej(ϕV +π/2) = jωCVm ejϕV = jωC Vˆ .

(1.46)

Thus, it is established that the terminal relation for the capacitor can be written in the phasor form as follows: j ˆ Vˆ = − I. ωC

(1.47)

It is also clear that given sinusoidal voltage and current sources can also be written in phasor forms: vs (t) = Vms cos (ωt + ϕVs ) ↔ Vˆs = Vms ejϕVs ,

(1.48)

is (t) = Ims cos (ωt + ϕIs ) ↔ Iˆs = Ims ejϕIs .

(1.49)

Thus, it can be concluded that in the case of ac steady state all terminal relations can be written in algebraic phasor forms. Now, we turn to KCL and KVL equations and write them in the phasor form as well. To do this we shall use the fact that the phasor of the sum of sinusoidal quantities is equal to the sum of the phasors of sinusoidal quantities being summed. The proof of this fact is based on the following mathematical relation between sinusoidal quantities and phasors: h i h i v(t) = Vm cos(ωt + ϕV ) = Re Vm ej(ωt+ϕV ) = Re Vˆ ejωt . (1.50) Next, consider a sum of sinusoidal quantities of arbitrary physical nature: X g(t) = Gmk cos(ωt + ϕk ) (1.51) k

and we want to prove that ˆ= G

X k

ˆk. G

(1.52)

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By using the Euler formula and formula (1.50), we find i i X h h X Re Gmk ejϕk ejωt g(t) = Gmk Re ej(ωt+ϕk ) = k

k

=

X

"

i

h

! X

ˆ k ejωt = Re Re G

ˆk G

# e

jωt

i h ˆ jωt , = Re Ge

(1.53)

k

k

which proves the equality (1.52). Having established the last fact, we can write KCL equations (1.22) and KVL equations (1.23) in the phasor form as follows: X

Iˆk = 0,

[(n − 1) lin. ind. eqs.],

(1.54)

Vˆk = 0,

[b − (n − 1) lin. ind. eqs.].

(1.55)

k

X k

Iˆ +

R

L

VˆR

VˆL VˆC

Vˆ

C

− Fig. 1.9

Next, we shall discuss the very important concept of impedance. To this end, consider a branch where a resistor, an inductor and a capacitor are connected in series (see Figure 1.9). According to KVL, we have Vˆ = VˆR + VˆL + VˆC .

(1.56)

By using phasor relations (1.35), (1.41) and (1.47), we transform the last equation as follows:    j ˆ ˆ 1 Vˆ = RIˆ + jωLIˆ − I = I R + j ωL − . (1.57) ωC ωC Now, the impedance Z of the RLC branch can be naturally introduced by

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the formula   1 = R + jX, Z = R + j ωL − ωC

(1.58)

where X=

  1 ωL − ωC

(1.59)

is called the reactance of the branch and it is determined by the energy storage elements in the branch. By using the definition (1.58) of impedance, we shall write the terminal relation for the RLC branch in the form: ˆ Vˆ = IZ.

(1.60)

In the polar form, the impedance can be written as Z = |Z|ejϕ .

(1.61)

By substituting the last formula into equation (1.60), we find Vm ejϕV = Im |Z|ej(ϕI +ϕ) ,

(1.62)

Vm = Im |Z|,

(1.63)

ϕV − ϕI = ϕ.

(1.64)

which implies that

Thus, the magnitude of branch impedance relates peak values of branch voltage and current, while the polar angle of the impedance is equal to the phase shift in time between branch voltage and current. From equations (1.58) and (1.61) the following useful expressions for |Z| and ϕ can be obtained: s 2  p 1 2 2 2 , (1.65) |Z| = R + X = R + ωL − ωC

tan ϕ =

ωL − X = R R

1 ωC

.

(1.66)

The branch shown in Figure 1.9 has the most general composition when

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three physically distinct two-terminal elements are connected in series. The expression (1.58) for the impedance of such a branch is naturally simplified when only one or two distinct two-terminal elements are present in the branch. These simplifications are given by the following formulas: Z = R,

(1.67)

if a branch contains only a resistor; Z = jωL,

(1.68)

if a branch contains only an inductor; Z=−

j ωC

(1.69)

if a branch contains only a capacitor; Z = R + jωL,

(1.70)

j , ωC

(1.71)

for an RL branch; Z =R− for an RC branch and 

1 Z = j ωL − ωC

 (1.72)

for an LC branch. The last formula suggests that the impedance is equal to zero if: ω=√

1 . LC

(1.73)

It is also clear that under the condition (1.73), the impedance of the RLC branch is given by the formula Z = R.

(1.74)

In other words, under condition (1.73) an RLC branch acts as a pure resistor and branch voltage and current are in phase. This phenomenon is called resonance and it has important implications. The presented discussion can be summarized as follows. At ac steady state, each branch can be characterized by impedance and the expression for the branch impedance is determined by the composition of the branch.

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Let us consider a branch number k and let Zk be its impedance. Then, the phasor branch voltage Vˆk and the phasor branch current Iˆk are related by the formula (see (1.60)) Vˆk = Iˆk Zk .

(1.75)

Formula (1.75) can be used in the phasor form of KVL equations (1.55). In these equations (as well as in equations (1.54)) we can also identify known phasors of voltage and current sources and move them with appropriate signs to the right-hand sides. The described transformations eventually result in the following ac steady-state equations for the phasors of branch currents: X X (1.76) Iˆk = − Iˆsk , k

X k

Iˆk Zk = −

k

X

Vˆsk .

(1.77)

k

These are simultaneous linear algebraic equations, and it is apparent that the total number of these equations is equal to the total number of passive (i.e., without sources) branches. By solving these equations, Iˆk can be found and then by using formula (1.75) the phasors of branch voltages Vˆk can be determined. As soon as this is done, instantaneous voltages and branch currents can be computed by using formulas similar to (1.50). In the circuit theory, various analysis techniques have been developed. They usually exploit the specific connectivity of electric circuits. One example of these techniques is the method of equivalent transformations which exploits series and parallel connections of various branches to achieve the overall simplification of electric circuits. It is interesting to mention that from the mathematical point of view the phasor technique allows one to find particular periodic solutions of ordinary differential equations (ODEs) which describe the performance of electric circuits in the case of their time-harmonic excitation. This phasor technique is more efficient and more powerful than the technique of undetermined coefficients used in courses on ordinary differential equations [8]. That is why, the phasor technique is frequently used in power electronics to derive simple expressions for particular periodic solutions to ODEs when actual regimes of power converters are not ac steady states. As has been emphasized in our previous discussion, the main idea of the phasor technique is to reduce the operations of calculus on sinusoidal

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quantities to algebraic operations on their phasors. It turns out that this idea can be extended to a broader class of voltages and currents. Consider the following voltage: v(t) = Vm eσt cos (ωt + ϕV ) .

(1.78)

By using the Euler formula, the last equation can be transformed as follows: i h i h v(t) = Vm eσt Re ej(ωt+ϕV ) = Re Vm ejϕV e(σ+jω)t . (1.79) Now, we introduce the phasor Vˆ , Vˆ = Vm ejϕV ,

(1.80)

as well as the complex frequency s = σ + jω.

(1.81)

By using the last two formulas in equation (1.79), we find h i v(t) = Re Vˆ est .

(1.82)

The last formula extends the notion of phasors to voltages (1.78) which are characterized by complex frequency s. Similarly, for a current of complex frequency s we have h i ˆ st , i(t) = Im eσt cos (ωt + ϕI ) = Re Ie (1.83) where as before Iˆ = Im ejϕI . Now, it can be shown that the phasor terminal relations for resistors, inductors and capacitors in the case of voltages and currents of the same complex frequency s are similar to formulas (1.35), (1.41) and (1.47). Indeed, in the case of a resistor we have Vm eσt cos (ωt + ϕV ) = RIm eσt cos (ωt + ϕI ) . By using formulas (1.79)-(1.83), we derive h i h i ˆ st Re Vˆ est = Re RIe

(1.84)

(1.85)

which implies that ˆ Vˆ = RI.

(1.86)

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In the case of an inductor, we find Vm eσt cos (ωt + ϕV ) =σLIm eσt cos (ωt + ϕI )  π . + ωLIm eσt cos ωt + ϕI + 2 The last equality can be transformed as follows: h i h i ˆ st , Re Vˆ est = Re (σ + jω)LIe

(1.87)

(1.88)

or h i h i ˆ st , Re Vˆ est = Re sLIe

(1.89)

ˆ Vˆ = sLI.

(1.90)

which implies that

By the same line of reasoning, it can be shown that in the case of a capacitor we have 1 ˆ I. Vˆ = sC

(1.91)

By using this algebraization of terminal relations, it can be demonstrated that in the case of an RLC branch subject to a voltage (1.78) of complex frequency s we have the branch current of the same complex frequency and their phasors are related by the impedance which is a function of the same complex frequency. Namely, ˆ Vˆ = IZ(s),

(1.92)

where Z(s) = R + sL +

1 . sC

(1.93)

In applications, it is quite rare that electric circuits are excited by sources of complex frequency. However, voltages and currents of complex frequency regularly appear in the case of transients in electric circuits. For this reason, the notion of complex frequency as well as the notion of impedance Z(s) as a function of complex frequency s can be useful in the analysis of transients in electric circuits. Indeed, it can be shown that the complex frequencies of transient response in RLC circuits are zeros of impedance Z(s) = 0. This is true because the transient responses are solutions to homogeneous differential equations. The detailed discussion of this matter is beyond the

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scope of this section. It turns out that the phasor technique admits very simple geometrical interpretation which is given by phasor diagrams. Phasor diagrams are ubiquitous in electric power engineering [4,5]. They provide geometric visualization for the phase shifts in time between different sinusoidal quantities and for their peak values. The starting point in the discussion of phasor diagrams is the representation of a sinusoidal quantity by a uniformly rotating vector. Consider a time-harmonic voltage v(t) = Vm cos (ωt + ϕV ) .

(1.94)

Fig. 1.10

We can represent this voltage by a vector Vˆ (called a phasor) whose length is equal to the peak value Vm of v(t) and whose initial angle with the x-axis of the Cartesian coordinate system is equal to the initial phase ϕV of v(t). This angle is called “initial” because the vector Vˆ is uniformly rotating in the counterclockwise direction with constant angular velocity equal to the angular frequency ω of v(t). Thus, at time t the angle ϕV (t) between the vector Vˆ and the x-axis (see Figure 1.10) is equal to ϕV (t) = ωt + ϕV ,

(1.95)

i.e., angle ϕV (t) is the sum of the initial angle ϕV and the angle ωt through which the vector has rotated over the time t. If at any instant of time we consider the projection of this vector onto the x-axis, it is clear that this projection is equal to the instantaneous value of voltage v(t). In this sense, v(t) is represented by the uniformly rotating vector Vˆ . Now consider a

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time-harmonic current of the same angular frequency ω, i(t) = Im cos (ωt + ϕI ) .

(1.96)

It can also be represented by a uniformly rotating vector (phasor) Iˆ whose length is equal to the peak value Im of i(t) and whose initial angle with the x-axis is equal to the initial phase ϕI of i(t). Since this vector is uniformly rotating in the counterclockwise direction with angular velocity ω, the angle ϕI (t) between Iˆ and the x-axis at time t (see Figure 1.10) is equal to ϕI (t) = ωt + ϕI .

(1.97)

Again, it is apparent that at any instant of time t the projection of vector Iˆ onto the x-axis is equal to the instantaneous value of current i(t). Now, the important observation emerges. If we consider the angle ϕ between the two ˆ we find according to formulas (1.95) and (1.97) rotating vectors Vˆ and I, that ϕ = ϕV (t) − ϕI (t) = ϕV − ϕI = const.

(1.98)

Thus, this angle ϕ does not change with time and it is equal to the phase shift in time between sinusoidal voltage v(t) and sinusoidal current i(t). In this sense, one may say that the rotation of the vectors does not matter because it does not change the lengths of the vectors and the angle between them. These lengths and the angle are the most important because they represent the peak values of the sinusoidal quantities and their phase shift in time. For this reason, the rotation of vectors (phasors) can be completely ignored and we represent sinusoidal quantities by vectors (phasors) whose lengths are equal to the peak values of the sinusoidal quantities and the angles between the vectors are equal to the phase shifts in time between the sinusoidal quantities. This is the central idea of the phasor diagrams, i.e., to represent the phase shifts in time by geometric angles between the vectors (phasors). This idea helps to visualize different relations between sinusoidal quantities and to use geometry in calculations. Phasor diagrams can be constructed for complicated electric circuits. These constructions are based on generic phasor diagrams for the three basic two-terminal elements (resistor, inductor and capacitor). These generic phasor diagrams are shown in Figure 1.11. Consider first the resistor. According to formula (1.33), time-harmonic voltage across the resistor and time-harmonic current through the resistor are in phase, i.e., the phase shift in time between the voltage and current is equal to zero. That is

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Fig. 1.11

why a generic phasor diagram for the resistor has the form shown in Figure 1.11a. This diagram is called generic because it reflects the main feature, zero phase shift, while the lengths of phasors VˆR and IˆR may vary from problem to problem. Now consider the inductor. According to formula (1.39), time-harmonic voltage across the inductor leads the time-harmonic current through the inductor by π2 . For this reason, in the generic phasor diagram the geometric angle between vectors VˆL and IˆL is equal to π ˆ ˆ 2 (as shown in Figure 1.11b) and vector VL leads vector IL in the sense of counterclockwise rotation. Finally, consider the capacitor. According to formula (1.45), time-harmonic current through the capacitor leads the time-harmonic voltage across the capacitor by π2 . This means that in the generic phasor diagram the geometric angle between vectors VˆC and IˆC is equal to π2 (as shown in Figure 1.11c) and vector IˆC leads vector VˆC in the sense of counterclockwise rotation. Now, through several examples we shall demonstrate how the phasor diagrams can be constructed for actual circuits Iˆ +

R

L

VˆR

VˆL VˆC

Vˆ

C

− Fig. 1.12 Example 1. Consider the RLC circuit shown in Figure 1.12 excited by time-harmonic voltage. First, we shall write the KVL equation for this circuit, Vˆ = VˆR + VˆL + VˆC .

(1.99)

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This equation implies that in the phasor diagram vector Vˆ is the vectorial sum of vectors VˆR , VˆL and VˆC . As a general rule, we start the construction of the phasor diagram by identifying the quantity which is common to the resistor, inductor and capacitor. This quantity is the current, so we shall ˆ Then, according to the generic diagram 1.11a, the vector first draw vector I. VˆR has the same direction as vector Iˆ (see Figure 1.13), while according to the generic diagrams 1.11b and 1.11c vectors VˆL and VˆC are shifted from vector Iˆ by π2 in counterclockwise and clockwise directions, respectively. By performing the vector addition of the three vectors VˆR , VˆL and VˆC , we arrive at the final form of the phasor diagram shown in Figure 1.13. In this diagram, the geometric angle ϕ between vectors Vˆ and Iˆ represents the phase shift in time between the input voltage and input current.

Fig. 1.13 It is of interest to consider a particular form of this phasor diagram corresponding to the case of resonance. The resonance occurs under the condition specified by formula (1.73). At resonance, the input voltage and current are in phase. This leads to the phasor diagram shown in Figure 1.14. It is apparent from this figure that the voltages across the inductor and capacitor have the same peak values but opposite phases (i.e., shifted in phase by π). For this reason, they compensate one another at any instant of time. It is also apparent from Figure 1.14 that the peak values of voltages across the inductor and capacitor may be much larger than the peak value of the input voltage. This may never happen in dc circuits. Example 2. Consider the RL circuit shown in Figure 1.15. Suppose that by using a voltmeter the peak values of the input voltage and the voltage across the resistor have been measured and found to be 50 V and 30 V, respectively. The question is what the peak value of the voltage across the inductor and the phase shift in time between the input voltage and input current are.

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Fig. 1.14

Iˆ +

R VˆR VˆL

Vˆ

L

− Fig. 1.15

Fig. 1.16

The visualization of the phasor diagram for the above circuit appreciably facilitates and simplifies the solution of this problem. A well-versed person will visualize this phasor diagram in mind (without actually drawing it) to come up with the immediate answers that the peak value of the voltage across the inductor is 40 V, while the phase shift in time between the input voltage and input current is arctan 4/3. For pedagogical reasons, we draw this phasor diagram shown in Figure 1.16, which is a particular case (VˆC = 0) of the phasor diagram shown in Figure 1.13. From the right triangle in Figure 1.16, the above-stated solution of the problem becomes immediately apparent. Example 3. Consider a circuit shown in Figure 1.17. First, we write

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L1

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IˆR

1

VˆL1

mainbook

IˆC

VˆR

R

VˆL2

L2

C

Vˆ − 2

Fig. 1.17

Fig. 1.18 relevant KCL and KVL equations: Vˆ12 = VˆR + VˆL2 , Iˆ = IˆR + IˆC

(1.100)

Vˆ = VˆL1 + Vˆ12

(1.102)

(1.101)

where Vˆ12 is the phasor of the voltage across the nodes 1 and 2. Second, as a general rule, we start the construction of the phasor diagram from the

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“end” of the circuit, i.e., from the branch RL2 and we identify the current IˆR as the quantity common to R and L2 . So, we draw vector IˆR . According to the generic phasor diagrams shown in Figure 1.11a and 1.11b, vector VˆR has the same direction as the vector IˆR , while vector VˆL2 is perpendicular to the vector IˆR and its orientation reflects that the voltage across the inductor leads the current by π2 . According to equation (1.100), the vectorial sum of VˆR and VˆL2 results in Vˆ12 (see Figure 1.18). According to the generic diagram shown in Figure 1.11c, vector IˆC is perpendicular to vector Vˆ12 and “leads” it in the sense of counterclockwise rotation. The vectorial sum ˆ Finally, according of IˆC and IˆR results according to equation (1.102) in I. to the generic diagram shown in Figure 1.11b, vector VˆL1 is perpendicular to vector Iˆ and “leads” it. According to equation (1.102), the vectorial sum of VˆL1 and Vˆ12 results in Vˆ , and this completes the construction of the phasor diagram. 1.3

Three-Phase Circuits

Three-phase circuits are ubiquitous in electric power engineering because ac electric power is generated, transmitted, distributed and often consumed as three-phase power [4, 5]. In this section, we discuss the very basic facts related to three-phase circuits and their analysis. a − +

Vˆa

Za neutral

O0

Vˆc

Zb

−

O

+

+

−

c

Zc

Vˆb

b

Fig. 1.19 In three-phase circuits, there are three voltage sources va (t), vb (t) and vc (t). These are very special voltage sources because they have the same peak value, the same frequency and are phase shifted with respect to one

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another by 2π/3: va (t) = Vm cos ωt,   2π vb (t) = Vm cos ωt − , 3   4π vc (t) = Vm cos ωt − . 3

(1.103) (1.104) (1.105)

The last two equations can be written in a different form to clearly reveal the shift in time between the voltages va (t), vb (t) and vc (t). This is done by defining the time-period T = 2π ω of these voltages. Next, by using this definition we can write:      T T vb (t) = Vm cos ω t − = va t − , (1.106) 3 3      2T 2T = va t − . (1.107) vc (t) = Vm cos ω t − 3 3 Thus, it is clear that these voltage sources are shifted in time by T /3 with respect to one another. This fact would be instrumental in the discussion of three-phase inverters. These voltage sources are usually connected into “star” (Y) or “delta” (∆). In this brief review, we shall only discuss the star connection of voltage sources and star connection of loads. This configuration of threephase circuits is illustrated by Figure 1.19. It is clear from this figure that there are four lines that connect the voltage sources with the loads. These lines are marked as a, b and c and the neutral. Impedances Za , Zb and Zc represent different phase loads that are connected between the neutral and lines a, b and c, respectively. Vˆa , Vˆb and Vˆc are correspondingly the phasors of voltages va (t), vb (t) and vc (t). It follows from formulas (1.103)-(1.105) that these phasors can be written as: Vˆa = Vm ,

2π Vˆb = Vm e−j 3 ,

4π Vˆc = Vm e−j 3 .

(1.108)

It is customary in the theory of three-phase systems to use the notation α = e−j

2π 3

.

(1.109)

It is obvious that α2 = e−j

4π 3

,

α3 = 1,

α4 = α.

(1.110)

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From formulas (1.108)-(1.110) we find Vˆa = Vm ,

Vˆb = αVˆa ,

Vˆc = α2 Vˆa .

(1.111)

It is clear from the presented discussion that multiplication of a phasor by α is equivalent to the phase shift by 2π/3 for the corresponding sinusoidal quantity. It can be easily proved that 1 + α + α2 = 0.

(1.112)

Indeed, by treating the left-hand side of formula (1.112) as a geometric series, we find that 1 + α + α2 =

1 − α3 = 0, 1−α

(1.113)

because α3 = 1 (see 1.110). From formulas (1.111)-(1.112), we derive Vˆa + Vˆb + Vˆb = 0.

(1.114)

The last equation implies that va (t) + vb (t) + vc (t) = 0.

(1.115)

The last formula expresses a simple mathematical fact that the sum of three sinusoidal quantities of the same peak value, the same frequency and phaseshifted with respect to one another by 2π/3 is equal to zero at any instant of time. This mathematical fact will be used in our subsequent discussions.

Fig. 1.20

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It is customary in the case of three-phase circuits to speak about phase and line voltages. Phase voltages are measured between the neutral and one of the lines a, b and c. It is clear that Vˆa , Vˆb and Vˆc are the phasors of the phase voltages and their peak values are the same: |Vˆa | = |Vˆb | = |Vˆc | = Vm = Vph ,

(1.116)

where Vph is the notation for the common peak value of the phase voltages. Line voltages are measured between different pairs of lines. This means that there are three line voltages Vˆab = Vˆa − Vˆb , Vˆbc = Vˆb − Vˆc , Vˆca = Vˆc − Vˆa .

(1.117) (1.118) (1.119)

It is clear from Figure 1.19 and the last three formulas that Vˆab represents the phasor of the line voltage measured between lines a and b, Vˆbc is the phasor of the line voltage measured between lines b and c, while Vˆca means the phasor of the line voltage measured between lines c and a. On the basis of symmetry, it is clear that the peak values of line voltages are the same: |Vˆab | = |Vˆbc | = |Vˆca | = Vl ,

(1.120)

where Vl is the common peak value of the line voltages. The last equality can also be made geometrically transparent by constructing the phasor diagram shown in Figure 1.20a. In this diagram, the lengths of phasors Vˆa , Vˆb and Vˆc and geometric angles between them are consistent with formulas (1.103), (1.104) and (1.105), while the diagram representations of phasors Vˆab , Vˆbc and Vˆca are consistent with formulas (1.117)-(1.119). It is clear from Figure 1.20a that isosceles triangles aO0 b, bO0 c and cO0 a are identical and can be obtained from one another through rotation by 2π/3. This implies the validity of formula (1.120). By using Figure 1.20b, it is easy to find the relation between the peak values of line and phase voltages. Indeed, from this figure follows that √ π 3 ˆ ˆ = 2Vph , (1.121) Vl = |Vab | = 2|Va | cos 6 2 which leads to Vl =

√

3Vph .

(1.122)

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In the US, the typical rms (root mean square) values of phase and line voltages are 120V and 208V, 208V and 360V, 360V and 620V, etc. Furthermore, it is clear from Figure 1.20a that the line voltages are phase-shifted with respect to one another by 2π/3. Consequently, Vˆbc = αVˆab ,

Vˆca = α2 Vˆab .

(1.123)

From the last formula and equation (1.112) follows that: Vˆab + Vˆbc + Vˆca = 0.

(1.124)

The last formula implies that at any instant of time vab (t) + vbc (t) + vca (t) = 0.

(1.125)

It is clear that the above formula (1.125) is also consistent with (and follows from) KVL. It is interesting to find explicit expressions for vab (t), vbc (t) and vca (t). It is clear from Figure 1.20b that  √ π . (1.126) vab (t) = 3Vm cos ωt + 6 Taking into account that line voltages are phase-shifted by 2π/3, we find   √ π 2π vbc (t) = 3Vm cos ωt + − , (1.127) 6 3

vca (t) =

√

  π 4π 3Vm cos ωt + − . 6 3

(1.128)

Now, it is easy to write expressions for the phasors of the line voltages: √ √ √ π π 7π Vˆab = 3Vm ej 6 , Vˆbc = 3Vm e−j 2 , Vˆca = 3Vm e−j 6 . (1.129) The question can be asked what function the neutral plays in the threephase circuit shown in Figure 1.19. It is clear from this figure that as a result of the presence of the neutral the same peak (or rms) value of voltages can be maintained across the loads represented by impedances Za , Zb and Zc . This is accomplished despite the possible variations in time of these impedances due to changing loads. Furthermore, the neutral can also be utilized for the detection of faults in three-phase power systems. This is because the occurrence of faults may result in non-zero currents through the neutral. The latter is due to the transition from the balanced load to the unbalanced

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load situation in the case of fault occurrence. Finally, it is also interesting to mention that the name neutral is used with a different connotation for final distribution of ac power to residential loads in the US. In this residential distribution, transformer’s secondary winding is tapped at the center, which makes available a center-tap connection in addition to the two terminals of the secondary winding. This center-tapped connection is quite often also called the neutral. Typically, the voltage measured between either of the two secondary terminals and this neutral is about 120 V rms. In such case the voltage between the two secondary terminals is 240 V rms. This is because the two voltages between the neutral and the two secondary terminals are in phase. This is in contrast with the three-phase system, where the voltages between the neutral and line terminals are shifted in phase by 2π 3 . The above conclusion that the neutral in three-phase systems allows to maintain the same peak (or rms) value of voltages across the load impedances was achieved by neglecting in the three-phase circuit shown in Figure 1.19 the impedances of connecting wires a, b and c and the neutral. To analyze the effect of these impedances on the ability to maintain more or less constant peak value of voltages across the loads, we shall analyze the electric circuit shown in Figure 1.21. The intrinsic simplicity of the circuit shown in this figure is revealed by the observation that there are only two nodes (O and O0 ) in this circuit. To take advantage of this simplicity, we choose node O0 as a reference node with zero potential, VˆO0 = 0.

(1.130)

Then, the analysis of the above circuit can be performed by using the following two steps. Step 1. It is apparent from Figure 1.21 and equation (1.130) that: Vˆa − VˆO , Iˆa = Za + Za0

Vˆb − VˆO Iˆb = , Zb + Zb0

Vˆc − VˆO Iˆc = , Zc + Zc0

VˆO Iˆn = − . (1.131) Zn

Next, we introduce admittances Ya =

1 , Za + Za0

Yb =

1 , Zb + Zb0

Yc =

1 , Zc + Zc0

Yn =

1 . (1.132) Zn

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Za0

Iˆa

a − +

Vˆa

Za Zn

O

Vˆc

Iˆn

0

Zc

O

Zb

+

+

−

−

Vˆb

c

Zb0

Iˆb

Zc0

Iˆc

b

Fig. 1.21

and rewrite the equations (1.131) as follows:       Iˆa = Ya Vˆa − VˆO , Iˆb = Yb Vˆb − VˆO , Iˆc = Yc Vˆc − VˆO , Iˆn = −Yn VˆO .

(1.133)

It is apparent that if VˆO is found then all currents can be found by using the last formulas in (1.133) or the formulas in (1.131). Step 2. From the Kirchhoff Current Law for node O, we find Iˆa + Iˆb + Iˆc + Iˆn = 0.

(1.134)

By substituting the expressions from (1.133) into the last equation, we arrive at       Ya Vˆa − VˆO + Yb Vˆb − VˆO + Yc Vˆc − VˆO − Yn VˆO = 0, (1.135) which can be further transformed as Ya Vˆa + Yb Vˆb + Yc Vˆc = (Ya + Yb + Yc + Yn )VˆO ,

(1.136)

Ya Vˆa + Yb Vˆb + Yc Vˆc . VˆO = Ya + Yb + Yc + Yn

(1.137)

which leads to

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This completes the analysis of the electric circuit shown in Figure 1.21. Indeed, by computing VO from formula (1.137) and then using the computed value of VO in formulas from (1.131), we can find all the currents. Formula (1.137) is of special interest because the deviations of VO from zero reflect deviations of voltages across the phase loads. Indeed, if |Zn | is very small and, consequently, |Yn | is very large, then according to (1.137) we find VO ≈ 0.

(1.138)

Furthermore, if line impedances Za0 , Zb0 and Zc0 are small in comparison with load impedances Za , Zb and Zc , then the peak values of the voltages across the load impedances can be maintained approximately the same. Next, we consider the important case of balanced load described by the equalities: Za + Za0 = Zb + Zb0 = Zc + Zc0 .

(1.139)

Then, according to (1.132), we have Ya = Yb = Yc = Y.

(1.140)

By using the last formula as well as formula (1.114) in equation (1.137), we derive   Y Vˆa + Vˆb + Vˆc = 0. (1.141) VˆO = 3Y + Yn From equations (1.131) and (1.141), we conclude that Iˆa =

Vˆa , Za + Za0

Iˆb =

Vˆb , Zb + Zb0

Iˆc =

Vˆc , Zc + Zc0

Iˆn = 0. (1.142)

Now, taking into account formulas (1.111) and (1.139), we find Iˆb = αIˆa ,

Iˆc = α2 Iˆa .

(1.143)

Thus, in the case of balanced load, the current through the neutral is equal to zero, while Iˆa , Iˆb and Iˆc have the same peak values and are phaseshifted with respect to one another by 2π/3. This is a very important fact. The reason is that such currents through stationary (but distributed) windings may create uniformly rotating magnetic fields. Such fields are at the very foundation of the principles of operation of ac electric machines

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(i.e., generators and motors). The ability to create uniformly rotating magnetic fields by stationary windings was historically one of the main reasons why three-phase circuits (and three-phase power systems) were introduced. Furthermore, unbalanced loads are very detrimental to the operation of synchronous generators. The reason is that they result in induction of appreciable eddy currents in the solid rotors of these generators, leading to their heating. Thus, the conclusion can be reached that the balanced load is the preferable mode of operation of power systems. For this reason utility companies usually take special measures to achieve the mode of operation which is sufficiently close to the balanced load.

Za0 − +

Vˆa

Iˆa

Za

Fig. 1.22

In the case of balanced load, the analysis of three-phase circuits can be substantially simplified by using the concept of “per-phase analysis.” Indeed, consider a very simple single-phase circuit shown in Figure 1.22 instead of the three-phase circuit shown in Figure 1.21. According to Figure 1.22, we have Iˆa =

Vˆa , Za + Za0

(1.144)

which is identical to the first formula in (1.142). As soon as Iˆa is found, formulas in (1.143) can be used to find Iˆb and Iˆc . Thus, in the case of balanced load, per-phase analysis leads to the same result as the analysis of the three-phase circuit shown in Figure 1.21. In Chapter 3, the described per-phase analysis will be further generalized. Namely, it will be performed in the time-domain in the case when three-phase currents are not sinusiodal. This generalization will be instrumental in the analysis of pulse width modulation in three-phase inverters.

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37

Fourier Series

In power electronics, analysis of steady-state performance of switchingmode power converters is reduced to the analysis of electric circuits excited by time-periodic non-sinusoidal sources. There are two analytical techniques that will be extensively used in this book for the analysis of such circuits. The first one is the frequency-domain technique, which is based on the Fourier series expansions of time-periodic functions (sources) and subsequent use of the phasor technique. The second one is the time-domain technique, which is based on the formulation of the steady-state circuit analysis as a boundary value problem for ordinary differential equations with periodic boundary conditions. We begin with the frequency-domain technique and we first review in this section the basic facts related to the Fourier series [9]. These series are used for periodic functions. Below, we consider periodic functions of time, which are relevant to circuit analysis. However, the Fourier series are also very instrumental in the design of windings of synchronous and induction machines used in and beyond power engineering [5, 10]. In these cases, Fourier series are used for the expansion of periodic functions in space rather than in time. f (t)

0

t

T /2

T t+T

3T /2

2T

t

Fig. 1.23 Function f (t) is said to be periodic with period T (see Figure 1.23) if f (t + T ) = f (t).

(1.145)

It is apparent that a multiple of T by any natural number is a period of f (t) as well. It will be assumed in the following that T is the smallest period of

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f (t). The fundamental angular frequency ω=

2π T

(1.146)

will be associated with this period. It turns out that under some general conditions periodic functions can be expanded into trigonometric Fourier series f (t) = c0 +

∞ X

[an cos nωt + bn sin nωt],

(1.147)

n=1

where numbers c0 , an and bn are called the Fourier expansion coefficients. The next step is to find the expression for these expansion coefficients in terms of function f (t). This can be accomplished by using the following “orthogonality” relations for trigonometric functions: Z T cos nωtdt = 0, (1.148) 0

Z

T

sin nωtdt = 0,

(1.149)

cos nωt sin mωtdt = 0,

(1.150)

0 T

Z

0 Z T

cos nωt cos mωtdt =

T δnm , 2

(1.151)

sin nωt sin mωtdt =

T δnm , 2

(1.152)

0

Z 0

T

where the symbol δnm is the so-called Kronecker delta defined by the formula: ( 1, if n = m, δnm = (1.153) 0, if n 6= m. These orthogonality conditions are easy to prove. Formulas (1.148) and (1.149) are immediately obvious because functions cos nωt and sin nωt are periodic with period Tn , and the integrals of sine and cosine functions over any number of periods are equal to zero. The proof of formulas (1.150), (1.151) and (1.152) is only slightly more complicated. This proof is based on the well-known trigonometric identities which reduce products of sine and cosine functions to sums of these functions with modified arguments.

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Afterwards, formulas similar to (1.148) and (1.149) can be used. This line of reasoning can be followed when n 6= m. In the case when n = m, the trigonometric identities relating squares of cosine and sine functions to cosine functions of double argument can be used to arrive at formulas (1.151) and (1.152). We omit the details of the outlined derivations and encourage the reader to perform these derivations, which will be beneficial for proper understanding of the material. In calling formulas (1.148)-(1.152) orthogonality conditions, we use geometric language which implies the analogy between formulas (1.148)-(1.152) and orthogonality of vectors. In this sense, the functional set consisting of a constant function and trigonometric functions is an orthogonal functional set, and formulas (1.148) and (1.149) can be understood as orthogonality relations between a constant function and cosine and sine functions. It is known that orthogonal vectors are linearly independent and they may be used as bases for expansions of an arbitrary vector. Similarly, the functional set consisting of a constant function and cosine and sine functions can be used as the functional basis for the expansion of an arbitrary function f (t), and formula (1.147) can be understood as such an expansion. The Fourier expansion coefficients can be interpreted as “projections” of f (t) on the “axes” identified with a constant function and cosine and sine functions. This interpretation is consistent with the following formulas for the expansion coefficients: 1 T

Z

2 T

Z

2 bn = T

Z

c0 = an =

T

f (t)dt,

(1.154)

f (t) cos nωtdt,

(1.155)

f (t) sin nωtdt.

(1.156)

0 T

0 T

0

These formulas can be derived by using the orthogonality relations. Indeed, by integrating both sides of formula (1.147) over T and by using formulas (1.148) and (1.149) we arrive at equation (1.154). In geometric language, it can be said that c0 is the projection of f (t) on the constant (unity) function. Similarly, by multiplying both sides of formula (1.147) by cos nωt, subsequently integrating both sides over T and using the orthogonality relations (1.148), (1.150) and (1.151), we arrive at the formula (1.155). In geometric language, it can be said that an is the projection of f (t) on cos nωt. Finally, by multiplying both sides of formula (1.147) by sin nωt,

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subsequently integrating both sides over T and using orthogonality relations (1.149), (1.150) and (1.152), we arrive at the formula (1.156). In geometric language, it can be said that bn is the projection of f (t) on sin nωt. Next, the following two remarks are in order. Remark 1. Integrands in formulas (1.154), (1.155) and (1.156) are periodic functions of period T . It is apparent from the geometric point of view that integrals of such functions over period T do not depend on the location of the time interval of integration. Consequently, the last three formulas can also be written in the following form: 1 T

Z

1 T

Z

1 bn = T

Z

c0 = an =

T 2

f (t)dt,

(1.157)

f (t) cos nωtdt,

(1.158)

f (t) sin nωtdt.

(1.159)

− T2 T 2

− T2 T 2

− T2

Remark 2. Under some general conditions, expansion coefficients an and bn tend to zero with the increase of n. Namely: lim an = 0,

n→∞

lim bn = 0.

n→∞

(1.160)

It is interesting to note that formulas (1.160) are valid despite the fact that the integrands in formulas (1.155) and (1.156) do not tend to zero. This raises the question of the generic intuitive explanation for the validity of relations (1.160). This explanation proceeds as follows. The functions cos nωt and sin nωt become very fast oscillating as n is increased. In other words, the period Tn of these functions becomes smaller and smaller. A sufficiently regular (normal) function f (t) can be accurately approximated by a piecewise constant function with constant values in each time interval of Tn -duration. According to (1.148) and (1.149), for such piecewise constant functions, the integrals in (1.155) and (1.156) are equal to zero. A simple and rigorous proof of the validity of relations (1.160) can be given by assuming that f (t) is differentiable and using integration by parts in formulas (1.155) and (1.156). The reader is encouraged to carry out this proof. A periodic function f (t) may have some symmetry properties. These symmetry properties may result in substantial simplifications of Fourier

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f (t)

−T

−T /2

−t 0

t

T /2

T

t

Fig. 1.24 series (1.147). Below, we shall discuss three types of symmetries: even symmetry, odd symmetry and half-wave symmetry. a) Even symmetry A function f (t) is called even (or even symmetric) if for any t we have f (t) = f (−t).

(1.161)

An example of the graph of such function is shown in Figure 1.24, and it is clear that this graph exhibits mirror symmetry with respect to the vertical axis. It is apparent from formula (1.161) as well as from Figure 1.24 that for any even function we have Z T2 Z T2 f (t)dt = 2 f (t)dt. (1.162) − T2

0

b) Odd symmetry A function f (t) is called odd (or odd symmetric) if for any t we have f (t) = −f (−t).

(1.163)

An example of the graph of such function is shown in Figure 1.25, and it is clear that this graph exhibits rotational symmetry with respect to the origin. It is apparent from formula (1.163) as well as from Figure 1.25 that for any odd function we have Z T2 f (t)dt = 0. (1.164) − T2

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Remark 3. It follows from the definitions (1.161) and (1.163) that the product of two even or two odd functions is an even function, while the product of an even function and an odd function is an odd function. c) Half-wave symmetry A function f (t) is called a half-wave symmetric function if for any t we have   T f t+ = −f (t). (1.165) 2 where T is the period of f (t). An example of the graph of such function is shown in Figure 1.26. It is apparent from the definition (1.165) and Figure 1.26 that a half-wave symmetric function has two identical but of opposite sign half-cycles. For this reason, Z T f (t)dt = 0. (1.166) 0

Remark 4. Half-wave symmetric functions are related to functions of period T2 . Indeed, it is easy to see from formula (1.165) that the product of two half-wave symmetric functions is a function of period T 2 , while the product of a half-wave symmetric function and a function of period T2 is a half-wave symmetric function. f (t)

−T

−T /2

−t

0

t

T /2

T

t

Fig. 1.25 Now, we demonstrate symmetry-related simplifications of Fourier series (1.147). We start with the case when f (t) is an even function. We recall that cos nωt are even functions, while sin nωt are odd functions. From the

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f (t)

−T

t

0

−T /2

T /2

t + T /2 T

t

Fig. 1.26

last observation and Remark 3, we find that the integrands in formulas (1.157) and (1.158) are even functions, while integrands in formula (1.159) are odd functions. From these facts and formulas (1.162) and (1.164), we conclude that

bn = 0,

f (t) = c0 +

∞ X

an cos nωt,

(1.167)

(1.168)

n=1

2 c0 = T

Z

4 T

Z

an =

T 2

f (t)dt,

(1.169)

f (t) cos nωtdt.

(1.170)

0 T 2

0

The achieved simplification is twofold. First, only a constant term and cosine terms are present in the Fourier series expansion (1.168). Second, the calculation of c0 and an requires the evaluation of integrals over T2 rather than over T . This simplification is especially useful when the above integrals are evaluated numerically. Next, we consider the case when f (t) is an odd function. In this case, according to Remark 3 the integrands in formulas (1.157) and (1.158) are odd functions, while the integrands in formulas (1.159) are even functions. From these facts and formulas (1.162) and (1.164), we conclude respectively

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that:

f (t) =

c0 = 0,

(1.171)

an = 0,

(1.172)

∞ X

bn sin nωt,

(1.173)

n=1

bn =

4 T

Z

T 2

f (t) sin nωtdt.

(1.174)

0

As before, the achieved simplification is twofold. First, only sine terms are present in the Fourier series expansion (1.173). Second, the calculation of bn requires the evaluation of integrals over T2 rather than over T . Finally, we consider the case when f (t) is a half-wave symmetric function. According to formulas (1.166) and (1.154), we immediately find that: c0 = 0.

(1.175)

To achieve further simplification in the Fourier series expansion, we observe that for odd n functions cos nωt and sin nωt are half-wave symmetric functions, while for even n functions cos nωt and sin nωt are periodic functions with period T2 . From the last observation and Remark 4, we find that the integrands in formulas (1.155) and (1.156) are of half-wave symmetry for even n, while these integrands are periodic functions of period T2 for odd n. Thus, according to formula (1.166) all even Fourier coefficients are equal to zero, while odd Fourier coefficients are not equal to zero and integration over T can be reduced to twice the integration over T2 . Thus, we arrive at the following simplification of Fourier series: f (t) =

∞ X

[a2n+1 cos(2n + 1)ωt + b2n+1 sin(2n + 1)ωt] ,

(1.176)

n=0

4 T

Z

4 = T

Z

a2n+1 = b2n+1

T 2

f (t) cos(2n + 1)ωtdt,

(1.177)

f (t) sin(2n + 1)ωtdt.

(1.178)

0 T 2

0

Once again, the achieved simplification is twofold. First, all even order harmonics in the Fourier series expansion (1.176) are eliminated. Second,

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the calculation of the coefficients a2n+1 and b2n+1 requires the evaluation of integrals over T2 rather than over T . It is worthwhile to mention that the case of half-wave symmetric functions is encountered quite often in electric power engineering in general, and power electronics in particular. This is because half-wave symmetry results in complete elimination (filtering out) of all even harmonics in the Fourier series expansion. We conclude this section by the discussion of an alternative form of Fourier series. This form is very convenient for the coupling of the Fourier series expansion with the phasor technique. This coupling is the foundation of the frequency-domain technique for the analysis of electric circuits with periodic non-sinusoidal sources. Consider one term of the infinite sum in Fourier series expansion (1.147) and perform the following equivalent transformation. First we find that: an cos nωt + bn sin nωt =

p

a2n + b2n p

an cos nωt + b2n

a2n

! bn sin nωt . +p a2n + b2n

(1.179)

Next, we introduce ϕn by the formulas an cos ϕn = p , 2 an + b2n bn , sin ϕn = − p 2 an + b2n

(1.180) (1.181)

which means that tan ϕn = −

bn . an

(1.182)

It is easy to see that the above definition of ϕn is consistent with trigonometric identity cos2 ϕn + sin2 ϕn = 1. We shall also introduce the notation p cn = a2n + b2n .

(1.183)

(1.184)

By substituting formulas (1.180), (1.181) and (1.184) into equation (1.179),

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we obtain an cos nωt + bn sin nωt = cn (cos ϕn cos nωt − sin ϕn sin nωt) = cn cos(nωt + ϕn ).

(1.185)

By using the last relation in formula (1.147), we arrive at the following alternative form of the Fourier series: f (t) = c0 +

∞ X

cn cos(nωt + ϕn ),

(1.186)

n=1

where c0 , cn and ϕn can be computed by using the following formulas: 1 c0 = T

Z

2 T

Z

an =

T

f (t)dt,

(1.187)

f (t) cos nωtdt,

(1.188)

0 T

0

Z 2 T f (t) sin nωtdt, T 0 p cn = a2n + b2n , bn tan ϕn = − . an bn =

(1.189) (1.190) (1.191)

It is remarkable that the form (1.186) contains only cosine terms. In the particular case when f (t) is an even symmetric function, all initial phases ϕn are equal to zero and the Fourier series (1.190) is reduced to the Fourier series expansion (1.168). This concludes the review of the Fourier series. 1.5

Frequency-Domain Technique

In this section, we consider the frequency-domain technique for the analysis of steady-state regimes of electric circuits excited by periodic nonsinusoidal sources. This technique is based on the combination of Fourier series expansions and phasors. We first present the general description and justification of the frequency domain technique and then we illustrate this technique by two examples. Consider an electric circuit shown in Figure 1.27. Here, vs (t) is a given

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periodic non-sinusoidal voltage source vs (t + T ) = vs (t),

(1.192)

while LEC is the abbreviation for a generic linear electric circuit with given lumped parameters. It is required to find the input electric current i(t) in the steady-state regime, i.e., i(t + T ) = i(t).

(1.193)

i(t) − +

vs (t)

LEC

Fig. 1.27 The frequency-domain technique for the solution of the stated problem consists of the following three steps. Step 1. The given periodic function vs (t) is expanded into Fourier series by using formulas (1.186)-(1.191). These formulas in the notation relevant to our problem can be written as follows:

vs (t) = Vs0 +

∞ X

Vsn cos(nωt + ϕsn ),

(1.194)

n=1

Vs0 = an =

1 T

Z

2 T

Z

T

vs (t)dt,

(1.195)

vs (t) cos nωtdt,

(1.196)

0 T

0

Z 2 T bn = vs (t) sin nωtdt, T p 0 Vsn = a2n + b2n , bn tan ϕsn = − , an

(1.197) (1.198) (1.199)

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where ω=

2π . T

(1.200)

Each term in the expansion (1.194) can be interpreted as a voltage source: vsn (t) = Vsn cos(nωt + ϕsn ),

(1.201)

and the given voltage source vs (t) can be interpreted as the series connection of these voltage sources (see Figure 1.28): vs (t) = Vs0 +

∞ X

vsn (t).

(1.202)

n=1

i(t)

.. . LEC

LEC vsn (t)

− +

− +

vs (t)

− +

Vs0

i(t)

.. .

Fig. 1.28

Step 2. Next, we shall use the superposition principle illustrated in Figure 1.29 and consider the current i(t) as the sum of currents I0 and in (t) excited in LEC when only one of the voltage sources Vs0 or vsn (t), respectively, is active while all others are set to zero. Thus, i(t) = I0 +

∞ X

in (t).

(1.203)

n=1

The calculation of I0 requires dc analysis of LEC subject to dc voltage Vs0 . In this analysis, inductors in LEC are replaced by short-circuit

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i(t)

49

I0

=

LEC

Vso

− +

− +

vs (t)

page 49

LEC

in (t) − +

+ · · · + vsn (t)

LEC

+···

Fig. 1.29 branches, while capacitors are replaced by open-circuit branches. Thus, the determination of I0 is reduced to the dc analysis of the resistive electric circuit corresponding to LEC (see Figure 1.30). I0 resistive version of LEC

− +

Vs0

Fig. 1.30

Iˆn − +

Vˆsn

impedance version of LEC

Fig. 1.31 The calculation of in (t) can be carried out by using the phasor tech-

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nique. Namely, the time-harmonic voltage source vsn (t) is represented by the phasor vsn (t) = Vsn cos(nωt + ϕsn ) → Vˆsn = Vsn ejϕsn ,

(1.204)

each branch of LEC is represented by its impedance evaluated at the angular frequency nω and the phasor analysis technique is used to find the phasor Iˆn in the impedance version of LEC (see Figure 1.31). Having found the phasor Iˆn = Imn ejϕIn , the current in (t) can be written as follows: Iˆn = Imn ejϕIn → in (t) = Imn cos (nωt + ϕIn ) .

(1.205)

Step 3. By using the superposition principle as well as formulas (1.203) and (1.205), the final expression for i(t) can be represented in the form i(t) = I0 +

∞ X

Imn cos (nωt + ϕIn ) .

(1.206)

n=1

We conclude the general description of the frequency-domain technique with the following two remarks. Remark 1. In many power electronics-related applications, the first term in the right-hand side of formula (1.206) can be interpreted as the main desired signal, while the infinite sum in (1.206) can be interpreted as undesirable “ripple.” Thus, the frequency-domain technique leads to the clear separation between the main desired component of the signal and its ripple. Remark 2. In the presented general description of the frequency-domain technique, the calculation of the input current i(t) was discussed. However, it is easy to see that the same three steps can be applied to the calculation of any branch current or any branch voltage of LEC. Next, we illustrate the described general frequency-domain technique with the following two examples. Example 1. Consider the electric circuit shown in Figure 1.32, where the voltage source vs (t) is a periodic sequence (train) of identical rectangular pulses (see Figure 1.33). Thus, V0 , T , t0 , L and R are given, and it is required to find i(t) and vR (t) in the steady-state regime, i.e., when i(t + T ) = i(t),

vR (t + T ) = vR (t).

(1.207)

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51

L

i(t) − +

vs (t)

page 51

R

Fig. 1.32

vs (t) V0

0

t0

T

2T

3T

t

Fig. 1.33

Step 1. We represent vs (t) as the following Fourier series: vs (t) = Vs0 +

∞ X

Vsn cos (nωt + ϕsn ) ,

(1.208)

n=1

where ω=

2π . T

(1.209)

To find Vs0 , Vsn and ϕsn , we sequentially use the formulas (1.195)-(1.199). Consequently, Z Z 1 t0 t0 1 T Vs0 = vs (t)dt = V0 dt = V0 . (1.210) T 0 T 0 T By introducing the notation D for so-called duty factor, D=

t0 , T

(1.211)

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we find (1.212)

Vs0 = DV0 . Next, Z 2V0 t0 vs (t) cos nωtdt = cos nωtdt T 0 0   t0 2V0 2V0 1 = sin nωt = sin nωt0 . T nω nωT

2 an = T

Z

T

(1.213)

0

Taking into account in the last formula the relation (1.209), we find an =

V0 sin nωt0 . πn

(1.214)

Similarly, Z 2V0 t0 sin nωtdt vs (t) sin nωtdt = T 0 0   t0 1 2V0 2V0 − cos nωt = (1 − cos nωt0 ) . = T nω nωT

2 bn = T

Z

T

(1.215)

0

Invoking again the relation (1.209), the last formula can be written as follows: bn =

V0 (1 − cos nωt0 ) . πn

(1.216)

Next, p

V0 πn

q

2

sin2 nωt0 + (1 − cos nωt0 ) r V0 p V0 nωt0 = 2 (1 − cos nωt0 ) = 4 sin2 , πn πn 2

Vsn =

a2n + b2n =

(1.217)

which implies that Vsn

2V0 nωt0 = sin . πn 2

(1.218)

0 In the last formula we use sin nωt because Vsn is always positive while 2

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0 may be negative for some value of n. Finally, sin nωt 4

tan ϕsn =

cos nωt0 − 1 . sin nωt0

(1.219)

Thus, the explicit analytical expressions are found for Vs0 , Vsn and ϕsn . This concludes this first step. Step 2. First consider the dc analysis of the resistive version of the electric circuit shown in Figure 1.32. This resistive version is illustrated by Figure 1.34. Now, the analysis is trivial and results in I0 =

Vs0 DV0 = , R R

VR0 = DV0 .

(1.220)

Second, consider the phasor analysis of ac steady state in the circuit shown

I0 − +

Vs0

R

Fig. 1.34

Iˆn L − +

Vˆsn

R

Fig. 1.35

in Figure 1.35 at the frequency nω. Here, we find Vˆsn = Vsn ejϕsn ,

(1.221)

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and Vsn and ϕsn are given by formulas (1.218) and (1.219). Furthermore, Vˆsn Iˆn = , Zn

(1.222)

where Zn = R + jnωL =

p R2 + n2 ω 2 L2 ejϕn

(1.223)

and tan ϕn =

nωL . R

(1.224)

By substituting formulas (1.221) and (1.223) into equation (1.222), we end up with Vsn ej(ϕsn −ϕn ) . + n2 ω 2 L2

(1.225)

Vsn cos (nωt + ϕsn − ϕn ) . R2 + n2 ω 2 L2

(1.226)

Iˆn = √

R2

From the last formula, we find in (t) = √

Step 3. The input current i(t) is the sum of currents I0 and in (t) for all ac steady-state regimes: i(t) = I0 +

∞ X

in (t),

(1.227)

n=1

or ∞ sin nωt0 DV0 2V0 X 2 √ + i(t) = cos (nωt + ϕsn − ϕn ) , R π n=1 n R2 + n2 ω 2 L2

(1.228)

where we have used formula (1.218) for Vsn in the equation (1.226). By multiplying the last formula by R we find the voltage across the resistor ∞ sin nωt0 2V0 R X 2 √ vR (t) = DV0 + cos (nωt + ϕsn − ϕn ) . π n=1 n R2 + n2 ω 2 L2 (1.229)

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In some applications, ωL  R

(1.230)

p R2 + n2 ω 2 L2 ≈ nωL.

(1.231)

and, consequently,

This leads to the following simplification of formula (1.229): ∞ 0 2V0 R X sin nωt 2 cos (nωt + ϕsn − ϕn ) . vR (t) ≈ DV0 + πωL n=1 n2

(1.232)

It is clear that due to the inequality (1.230) the second term (the ripple) in the right-hand side of formula (1.232) is small. It is also clear from the last formula that the ripple suppression is controlled by the product ωL. This implies the trade-off for ripple suppression between the value of L and the frequency of switching used to produce the train of rectangular pulses shown in Figure 1.33. We point out that this trade-off has already been discussed in general terms in section 1. Example 2. Consider the electric circuit shown in Figure 1.36, where the voltage source vs (t) is a periodic function of time shown in Figure 1.33. It is assumed that V0 , T , t0 , R, L and C are given, and it is required to find voltage vR (t) across the resistor in the steady-state regime, i.e., vR (t + T ) = vR (t).

(1.233)

i(t) L − +

vsn (t)

C

R

Fig. 1.36 Step 1. As before, we represent vs (t) by the Fourier series: vs (t) = Vs0 +

∞ X n=1

Vsn cos (nωt + ϕsn ) ,

(1.234)

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Iˆn L − +

Vˆsn

IˆRn

IˆCn C

R

Fig. 1.37 where ω = 2π T and Vs0 , Vsn and ϕsn can be computed by using formulas (1.212), (1.218) and (1.219). In other words, the first step in this example is identical to the first step in the first example, because the electric circuits in these examples are excited by identical voltage sources. Step 2. The dc analysis of the electric circuit shown in Figure 1.36 and subject to dc voltage source Vs0 (instead of voltage source vs (t)) leads to the circuit shown in Figure 1.33. The result of this analysis is obvious: VR0 = Vs0 = DV0 .

(1.235)

Next, we consider the phasor analysis of ac steady state in the electric circuit shown in Figure 1.37 at the frequency nω. Here, as before, Vˆsn = Vsn ejϕsn ,

(1.236)

where Vsn and ϕsn are found in the first step (see formulas (1.218) and (1.219)). It is apparent that the phasor Iˆn of the input current is equal to Vˆsn , Iˆn = Zn

(1.237)

where Zn is the input impedance of the electric circuit shown in Figure 1.37 evaluated at the frequency nω. It is clear that this input impedance is found as follows: Zn = jnωL +

j R − nωC j − nωC +R

.

(1.238)

By using simple algebraic transformations, we find Zn = jnωL +

R R − n2 ω 2 LCR + jnωL = . 1 + jnωCR 1 + jnωCR

(1.239)

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By substituting the last formula into (1.237), we arrive at Iˆn = Vˆsn

1 + jnωCR . R − n2 ω 2 LCR + jnωL

(1.240)

Now, by using the current divider rule, we obtain IˆRn = Iˆn

j − nωC j − nωC

+R

= Iˆn

1 . R + jnωCR

(1.241)

By substituting (1.240) into the last equation, we find IˆRn =

Vˆsn , R − n2 ω 2 LCR + jnωL

(1.242)

which leads to VˆRn = Vˆsn

R . R − n2 ω 2 LCR + jnωL

(1.243)

By using formula (1.236) and simple transformations, the last equation can be written as follows: VˆRn = q

Vsn R (n2 ω 2 LCR

ej(ϕsn −ϕn ) , 2

− R) +

(1.244)

n2 ω 2 L2

where tan ϕn =

nωL . R − n2 ω 2 LCR

(1.245)

From formula (1.244) we obtain Vsn R vRn (t) = q cos (nωt + ϕsn − ϕn ) . 2 (n2 ω 2 LCR − R) + n2 ω 2 L2

(1.246)

Step 3 Now, by using superposition principle, we arrive at vR (t) = VR0 +

∞ X

vRn (t).

(1.247)

n=1

By using formula (1.235) for VR0 and formula (1.246) for vRn (t) as well

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as formula (1.218) for Vsn , we obtain the final expression for vR (t): vR (t) = DV0 ∞ sin nωt0 2V0 R X 2 q cos (nωt + ϕsn − ϕn ) . + π n=1 2 n (n2 ω 2 LCR − R) + n2 ω 2 L2

(1.248) In certain applications, the second term in the right-hand side of the last equation can be construed as a ripple. This ripple will be effectively suppressed if the lumped parameters of the circuit shown in Figure 1.36 are chosen in such a way that ω 2 LC  1 and ω 2 C 2 R2  1.

(1.249)

The last inequalities imply, respectively, that n2 ω 2 LCR  R

(1.250)

n4 ω 4 L2 C 2 R2  n2 ω 2 L2 .

(1.251)

and

Formulas (1.250) and (1.251) mean that q 2 (n2 ω 2 LCR − R) + n2 ω 2 L2 ≈ n2 ω 2 LCR.

(1.252)

This leads to the following simplification of equation (1.248): ∞ 0 2V0 X sin nωt 2 vR (t) ≈ DV0 + cos (nωt + ϕsn − ϕn ) . πω 2 LC n=1 n3

(1.253)

Now, it is apparent that more efficient suppression of ripple can be achieved in the circuit shown in Figure 1.36 in comparison with the circuit shown in Figure 1.32. Indeed, each term in the infinite sum of formula (1.253) decays as 1/n3 rather than 1/n2 as in formula (1.232). In addition, the suppression of the ripple in formula (1.253) is controlled by the product ω 2 LC rather than by the product ωL. This means that the increase in switching frequency suppresses the ripple more efficiently in the circuit

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shown in Figure 1.36 as compared with the circuit shown in Figure 1.32. Finally, the dependence of ripple suppression on ω 2 LC reveals, as before, the trade-off between the values of energy storage elements L and C and the frequency of switching. 1.6

Time-Domain Technique

Now, we proceed to the discussion of the time-domain technique for the analysis of steady-state regimes of linear electric circuits excited by periodic non-sinusoidal sources. This technique is based on the formulation of steady-state analysis as a boundary value problem for ordinary differential equations with periodic boundary conditions. We illustrate this technique by the following two examples. i(t) − +

vs (t)

L

R

Fig. 1.38

vs (t) V0

0

t0

T

2T

3T

t

Fig. 1.39 Example 1. Consider the electric circuit shown in Figure 1.38 excited by the voltage source vs (t), where vs (t) is the periodic function of time shown in Figure 1.39. Here, V0 , T , t0 , L and R are given, and it is required to

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find i(t) and vR (t) which are periodic with period T : i(t + T ) = i(t),

vR (t + T ) = vR (t).

(1.254)

It is apparent that this example is identical to Example 1 from the previous section. This is done on purpose in order for the reader to compare advantages and disadvantages of the time-domain and frequency-domain techniques. The time-domain technique can be presented as a sequence of the following three distinct steps. Step 1. The intent of this step is to formulate the problem of steady-state analysis as a boundary value problem with periodic boundary conditions. To this end, we first write the KVL for the circuit shown in Figure 1.38: L

di(t) + Ri(t) = vs (t). dt

(1.255)

It is apparent from Figure 1.39 that the last equation can be written as two distinct equations for two time intervals: di(t) + Ri(t) = V0 , dt di(t) + Ri(t) = 0, L dt L

if 0 < t < t0 , if t0 < t < T.

(1.256) (1.257)

These two equations can be complemented by two conditions i(0) = i(T ),

(1.258)

i(t0− ) = i(t0+ )

(1.259)

The equation (1.258) is the periodic boundary condition which follows from (1.254) for t = 0, while the equation (1.259) is the interface condition which expresses the continuity of electric current through the inductor. The last four equations constitute the boundary value problem for differential equations (1.256) and (1.257) with periodic boundary and interface conditions (1.258) and (1.259), respectively. As soon as the solution of this boundary value problem is found, the value of i(t) can be found at any time (i.e., not only in time interval [0, T ]) by using the first equation in (1.254). Indeed, by using this equation, we can extend the solution from the time interval [0, T ] to the time interval [T, 2T ], and then from the time interval [T, 2T ] to the time interval [2T, 3T ] and so on. In other words, by using the first equation in (1.254), the solution of the boundary value prob-

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lem (1.256)-(1.259) with periodic boundary condition can be periodically extended to the infinite time interval. It is clear that this periodically extended solution has the physical meaning of the steady state in the electric circuit shown in Figure 1.38. Step 2. The intent of this step is to find general solutions of differential equations (1.256) and (1.257). We start with equation (1.256). This is a linear inhomogeneous equation of first order with constant coefficients. Its general solution has two distinct components: a particular solution ip (t) of inhomogeneous equation (1.256) and a general solution ih (t) of the corresponding homogeneous equation. Namely, i(t) = ip (t) + ih (t),

(1.260)

dip (t) + Rip (t) = V0 , dt

(1.261)

dih (t) + Rih (t) = 0. dt

(1.262)

where L while L

In mathematics, the particular solution of inhomogeneous differential equation (1.261) is sought in the same form as the right-hand side of the equation: ip (t) = G = const.

(1.263)

By substituting formula (1.263) into equation (1.261) we find RG = V0 ,

G=

V0 R

(1.264)

and ip (t) =

V0 . R

(1.265)

It is apparent that ip (t) is identical to the dc steady state in the electric circuit shown in Figure 1.38 excited by the dc voltage V0 . This observation is very helpful and can be used for the calculation of particular solutions of differential equations for more complicated circuits excited by dc or ac voltage sources. In the latter case, the phasor technique can be used for the calculation of particular solutions.

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Now, we proceed to the calculation of ih (t) by using equation (1.262). We look for a solution of this equation in the form ih (t) = A1 est ,

(1.266)

where A1 and s are some constants. By substituting the last formula into equation (1.262), we arrive at sLA1 est + RA1 est = 0,

(1.267)

sL + R = 0

(1.268)

which leads to

and s=−

R . L

(1.269)

Thus, the general solution ih (t) has the form R

ih (t) = A1 e− L t .

(1.270)

By substituting formulas (1.265) and (1.270) into equation (1.260), we find the general solution of equation (1.256): i(t) =

R V0 + A1 e − L t , R

if 0 < t < t0 .

(1.271)

Next, we consider the general solution of equation (1.257). This equation is identical in structure to equation (1.262). Consequently, the general solution has the form R

i(t) = A2 e− L t ,

if t0 < t < T,

(1.272)

where A2 is some constant. Step 3. The intent of this step is to find constants A1 and A2 from the periodic boundary condition (1.258) and interface boundary conditions (1.259). These conditions lead to the following equations: RT V0 + A1 = A2 e − L , R Rt0 Rt0 V0 + A1 e − L = A2 e − L . R

(1.273) (1.274)

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Indeed, by using formula (1.271) for the evaluation of i(0) and formula (1.272) for the evaluation of i(T ), we end up with equation (1.273). Similarly, by using equation (1.271) for the evaluation of i(t0− ) and formula (1.272) for the evaluation of i(t0+ ), we arrive at equation (1.274). The last two equations can be easily solved. Indeed, these equations can be written as follows: V0 , R V0 Rt0 A1 − A2 = − e L . R

A1 − A2 e −

RT L

=−

Then, by subtracting the second equation from the first, we find   V  Rt0  RT 0 A2 1 − e − L = e L −1 R

(1.275) (1.276)

(1.277)

and hence Rt0

V0 e L − 1 A2 = . R 1 − e− RT L

(1.278)

Now, by substituting the last formula into equation (1.276), after simple transformations we obtain R(t0 −T )

V0 e L −1 . A1 = RT − R 1−e L

(1.279)

By using the last two formulas in equations (1.271) and (1.272) we arrive at the final expression    R(t0 −T ) V0 Rt L e −1  − L  1+ e , if 0 < t < t0 , − RT i(t) = R Rt0 1−e L (1.280)  − Rt  V0 e L −1 L , if t0 < t < T. RT e R 1−e− L By taking into account that vR (t) = i(t)R,

(1.281)

we find

vR (t) =

   V0 1 +  V0

e

R(t0 −T ) L −1 RT 1−e− L

Rt0 e L

−1

1−e−

RT L

Rt

e− L ,

e

− Rt L

 ,

if 0 < t < t0 , (1.282) if t0 < t < T.

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Formulas (1.270) and (1.272) are the final results of our analysis. It is interesting to deduce from the last formula the expression for vR (t) in the case when RT  1. L

(1.283)

By using the inequality (1.283), we conclude that Rt

e− L ≈ 1,

if 0 < t < T.

(1.284)

Then, by using equation (1.283) and the first two terms of the Taylor expansions, we find: R

e− L (T −t0 ) − 1 ≈ 1 −

R R (T − t0 ) − 1 = − (T − t0 ), L L

R

R R T = T, L L

(1.286)

R R t0 − 1 = t0 . L L

(1.287)

1 − e− L T ≈ 1 − 1 +

R

e L t0 − 1 ≈ 1 +

(1.285)

Consequently,    R(t0 −T )  V0 1 + − RTL = V0 tT0 = DV0 , L vR (t) ≈ R t   V0 LR T0 = V0 tT0 = DV0 ,

if 0 < t < t0 ,

(1.288)

if t0 < t < T.

L

Thus, as expected, the ripple was suppressed by the large inductance and we find: vR (t) ≈ DV0 .

(1.289)

Example 2. Consider the electric circuit shown in Figure 1.40 excited by voltage source vs (t), where vs (t) is a periodic function of time given by the formula vs (t) = Vm | sin ωt|

(1.290)

and shown in Figure 1.41. Here, Vm , ω = 2π T , L, C and R are given and it is required to find vR (t) at the steady state: vR (t + T /2) = vR (t).

(1.291)

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The stated problem is encountered in the analysis of rectifiers in power electronics. i(t) L − +

vs (t)

iR (t)

iC (t) C

R

Fig. 1.40

vs (t) Vm

0

T

T 2

3T 2

t

2T

Fig. 1.41

Step 1. First, it is apparent that vR (t) = vC (t).

(1.292)

Then, by using KVL and KCL, we respectively find L

di(t) + vC (t) = vs (t). dt

i(t) = iC (t) + iR (t) = C

dvC (t) vC (t) + . dt R

(1.293)

(1.294)

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By substituting the last formula into equation (1.293), we obtain LC

L dvC (t) d2 vC (t) + + vC (t) = vs (t). 2 dt R dt

(1.295)

Now, we consider the last equation for the time interval between 0 and T /2, which is the period of vs (t). It is clear from formula (1.290) as well as Figure 1.41 that for this time interval vs (t) = Vm sin ωt.

(1.296)

By substituting this formula into equation (1.295) we end up with LC

L dvC (t) d2 vC (t) + + vC (t) = Vm sin ωt. dt2 R dt

(1.297)

This is a second-order differential equation, which is consistent with the fact that in the circuit being discussed there are two energy storage elements L and C. At the steady state, current i(t) and voltage vC (t) satisfy the periodic boundary conditions i(0) = i(T /2),

(1.298)

vC (0) = vC (T /2).

(1.299)

Form the last two formulas and formula (1.294), we find dvC dvC (0) = (T /2). dt dt

(1.300)

Thus, the problem of the analysis of the steady state is reduced to the boundary value problem for the second-order differential equation (1.297) with two periodic boundary conditions (1.299) and (1.300). This concludes the first step. Step 2. A general solution of differential equation (1.297) is the sum of a (p) (h) particular solution vC (t) of this equation and a general solution vC (t) of the corresponding homogeneous equation (h)

LC

(h)

d2 vC (t) L dvC (t) (h) + + vC (t) = 0. dt2 R dt

(1.301)

We consider the specific particular solution of equation (1.297) which corre-

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sponds to the ac steady state of the circuit shown in Figure 1.40 excited by the voltage source Vm sin ωt. This particular solution can be found by using the phasor technique. Actually, this has been already done in the previous section when we used the phasor technique to analyze the circuit shown in Figure 1.37. There are only two minor differences. First, the phasor of the voltage source Vˆs in our case is given not by formula (1.236) but by the following formula : π Vˆs = −Vm ej 2 .

(1.302)

Second, in formulas (1.246) and (1.245) n must be set to 1. Thus,   Vm R π (p) vC (t) = − q cos ωt + − ϕ , (1.303) 2 2 (ω 2 LCR − R) + ω 2 L2 tan ϕ =

ωL . R − ω 2 LCR

(1.304)

Next, we look for a solution of equation (1.301) in the form (h)

vC (t) = Aest .

(1.305)

By substituting the last formula into equation (1.301), we find after simple transformations that LCs2 +

L s + 1 = 0. R

(1.306)

This quadratic equation has two roots which are given by the formulas: r 1 1 1 s1 = − + − , (1.307) 2 2 2RC 4R C LC r 1 1 1 s2 = − − − . (1.308) 2RC 4R2 C 2 LC For the sake of simplicity consider the typical case when s1 and s2 are real and distinct. Other cases can be treated in a similar way. Then, a general solution of homogeneous equation (1.301) is (h)

vC (t) = A1 es1 t + A2 es2 t ,

(1.309)

and a general solution of equation (1.297) is given by the formula
Vm R cos ωt + π2 − ϕ vC (t) = − q + A1 es1 t + A2 es2 t . (1.310) 2 2 2 2 (ω LCR − R) + ω L

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This concludes the second step. Step 3. To find constants A1 and A2 we use periodic boundary conditions (1.299) and (1.300). This leads, after simple transformations, to the following simultaneous equations for A1 and A2 :     s2 T s1 T 2RVm sin ϕ (1.311) A1 1 − e 2 + A2 1 − e 2 = q 2 (ω 2 LCR − R) + ω 2 L2     s1 T s2 T 2ωRVm cos ϕ A1 s1 1 − e 2 + A2 s2 1 − e 2 = − q . 2 (ω 2 LCR − R) + ω 2 L2 (1.312) The solution to these equations is given by the formulas A1 =

2RVm (s2 sin ϕ + ω cos ϕ) ,  q s1 T 2 (s2 − s1 ) 1 − e 2 (ω 2 LCR − R) + ω 2 L2

(1.313)

A2 =

2RVm (s1 sin ϕ + ω cos ϕ) . q  s2 T 2 (s1 − s2 ) 1 − e 2 (ω 2 LCR − R) + ω 2 L2

(1.314)

By substituting the last two formulas into equation (1.310), we obtain the analytical solution for the steady state of the electric circuitshown  in Figure 1.40. This analytical solution is valid for the time interval 0, T2 . By using formula (1.291), it can be periodically extended for any time interval. This ends the discussion of example 2. We conclude this section with the following remark. In power electronics problems related to pulse width modulation, the switching is performed in such way that the corresponding voltage source vs (t) is half-wave symmetric. Namely,   T vs (t) = −vs t + . (1.315) 2 This type of switching is done in order to eliminate all even harmonics (see section 4). At the steady state, all currents and voltages will also be halfwave symmetric. This means that for any branch current i(t) in this circuit we have   T i(t) = −i t + . (1.316) 2

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In such problems, it is sufficient to find the solution only for the time T interval 0, 2 . Indeed,  when the solution is found, it can be extended to  the time-interval T2 , T using formula (1.316), and subsequently it can be extended to all periods by using the periodicity of i(t). It is clear that halfwave symmetry of i(t) can be imposed by using the anti-periodic boundary condition i(0) = −i(T /2).

(1.317)

The last formula follows from equation (1.316) by setting t = 0. This type of anti-periodic boundary condition will be extensively used in the analysis of pulse width modulation in the subsequent chapters.

B1948

Governing Asia

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Chapter 2

Pulse Width Modulation in Single-Phase Inverters

2.1

Single-Phase Bridge Inverter

In this chapter, some basic principles of dc-to-ac energy conversion are presented. The power electronics circuits that accomplish this conversion are called inverters [2, 5, 11]. The voltage-source single-phase inverters are discussed below. These inverters convert energy from fixed dc voltage sources into ac energy with voltages of desired and controllable frequencies and peak values. This is achieved by using pulse width modulation. We begin with the discussion of the single-phase bridge inverter. The electric circuit of such inverter is shown in Figure 2.1. This circuit contains a dc voltage source V0 as an input, four switches SW1 , SW2 , SW3 and SW4 on the four shoulders of the bridge and an LR branch with an output voltage vout (t) designated as the voltage across the terminals of the resistor R. The main challenge is to develop a proper strategy of switching that results in sinusoidal voltage vout (t) of desired frequency and peak value across the resistor R. It is clear that such a switching strategy should periodically invert the polarity of the output voltage. This polarity inversion can be achieved by periodically repeating the following two steps of switching: step #1 of simultaneously turning SW1 and SW3 “on” and SW2 and SW4 “off”; and step #2 of simultaneously turning SW1 and SW3 “off” and SW2 and SW4 “on”. It is apparent from Figure 2.1 that during the first step voltage V0 of positive polarity appears across the nodes 1 and 2, while during the second step voltage V0 of inverted (opposite) polarity appears across the same nodes 1 and 2. This implies that for the described switching the electric circuit shown in Figure 2.1 can be replaced by the equivalent circuit shown in Figure 2.2. In this figure, the equivalent voltage source veq (t) is a periodic

71

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3 SW1 L

− +

V0

SW2

1

R 2

i(t) + vout (t) −

SW4

SW3

4 Fig. 2.1 train (sequence) of rectangular voltage pulses of alternating polarity. The plot of veq (t) is shown in Figure 2.3. It is apparent that veq (t) is a function of half-wave symmetry. Namely, the following relation is valid:   T . (2.1) veq (t) = −veq t + 2 The latter means that at steady state the current i(t) in the electric circuit shown in Figure 2.2 (as well as in the electric circuit shown in Figure 2.1) is a function of half-wave symmetry,   T . (2.2) i(t) = −i t + 2 By using this fact, we can formulate the steady-state analysis for the above electric circuit as the following boundary value problem with “antiperiodic” boundary condition: L

di(t) T + Ri(t) = V0 , if 0 < t < , dt 2   T i(0) = −i . 2

(2.3)

(2.4)

It is clear that the “antiperiodic” boundary condition (2.4) follows from formula (2.2) by setting t equal to zero. It is also clear that after  Tsolving  the boundary value problem (2.3)-(2.4) for the time 0, 2 , the   T interval current i(t) can be extended to the time interval 2 , T by using the half-

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wave symmetry expressed by formula (2.2). After that, current i(t) can be extended for subsequent time intervals using the periodicity of i(t), that is i(t) = i(t + kT ),

(2.5)

where k is a positive integer. L 2

i(t) − +

veq (t)

+ R

vout (t) −

1 Fig. 2.2

Fig. 2.3 It is evident that a general solution to the differential equation (2.3) can be written in the form   R T V0 0 0. i 2 From the last two inequalities and formula (2.11) follows that   T vout (0) < 0, while vout = −vout (0) > 0. 2

Fig. 2.4

(2.17)

(2.18)

(2.19)

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By taking into account inequalities (2.17), (2.18) and (2.19) and using formulas (2.10) and (2.12), the plots of i(t) and vout (t) can be constructed. These plots are shown in Figures 2.4a and 2.4b, respectively. It is apparent from Figure 2.4a that the current i(t) changes its sign and, consequently, its direction of flow within each half-cycle. This fact has important implications concerning the design of the switches SW1 , SW2 , SW3 and SW4 shown in Figure 2.1. Indeed, single transistors are unidirectional (unilateral) switches. For instance, a BJT conducts an electric current from emitter to collector and, similarly, a power MOSFET (or IGBT) is designed to conduct an electric current from drain to source [5, 12]. This implies that a single transistor being in the “on” state cannot accommodate the flow of electric current in two opposite directions as needed in order to realize the current flow depicted in Figure 2.4a. It turns out that due to the presence of the inductor in the circuit shown in Figure 2.1, bidirectional (bilateral) switches SW1 , SW2 , SW3 and SW4 can be designed as parallel connections of power MOSFETs (or IGBTs) with freewheeling diodes [2, 5, 13]. This is shown in Figure 2.5. 3

T r1

T r2 L

− +

V0

D1 1

R 2

i(t)

T r4

D2

+ vout (t) −

D4

T r3

D3

4 Fig. 2.5 The operation of the electric circuit shown in this figure can be elucidated as follows. Immediately prior to the time instant t = 0, transistors T r2 and T r4 are in conducting (“on”) states, while transistors T r1 and T r3 are in “off” states. This results in negative polarity voltage V0 across the nodes 1 and 2 and in the current flow in the direction opposite to the one shown in Figure 2.5, which is consistent with Figure 2.4a. At time t = 0, transistors T r2 and T r4 are turned off, while transistors T r1 and T r3 are

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turned on, producing inversion of the polarity of the voltage V0 across the nodes 1 and 2. However, transistors T r1 and T r3 cannot conduct the current i(t) due to its direction at t = 0. To maintain the continuity of the current through the inductor L, diodes D1 and D3 are turned on and they form the closed path for freewheeling i(t). Physically, diodes D1 and D3 are turned on because any discontinuity of i(t) through L results in the induction of large voltage L di(t) dt of such polarity that will force these diodes into the conduction state. As soon as the current i(t) is  reduced to zero and its direction is reversed during the half-cycle 0, T2 , transistors T r1 and T r3 start to conduct this current. At time t = T2 , transistors T r1 and T r3 are turned off, while transistors T r2 and T r4 are turned on. However, T r2 and T r4 cannot conduct the positive current i(t) (with the direction shown in Figure 2.5). To maintain the continuity of the current through the inductor L, diodes D2 and D4 are turned on and form the closed path for the freewheeling current i(t) until its direction is reversed. The described conduction pattern is periodically repeated.

Fig. 2.6 As mentioned above, power MOSFETs or IGBTs are often used for the design of switches. For this reason, we now briefly discuss the design of these transistors starting with power MOSFETs [14, 15]. As with most power semiconductor devices, the structure of this device is vertical. A schematic depiction of this structure is presented in Figure 2.6. By using a vertical structure and lightly doped (drift) n− region, it is possible for power MOSFETs to sustain high blocking voltages and high currents. The

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principle of operation of the power MOSFET is essentially the same as the principle of operation of the lateral MOSFET commonly used in microelectronics. Namely, when a positive gate voltage is applied, an n-channel is formed in the p regions through the inversion process. This n-channel connects the n+ regions of the source contacts with the n− region. As the voltage between the drain and source is applied, the electron flow from source to drain is established resulting in drain current ID . The curves ID versus VDS are qualitatively similar to those for lateral MOSFET (see Figure 2.7). Power MOSFETs are fabricated by using a vertical double diffusion process to create n+ and p regions. For this reason, these power MOSFETs are sometimes called VDMOSFETs or DMOSFETs. Power MOSFETs are fabricated as multicell devices and Figure 2.6 represents the schematics of one cell. A large number of such cells are closely packed in a single silicon chip and all these cells are connected in parallel. The number of parallelconnected cells varies (depending on the geometric dimensions of the chip) from several thousand to more than twenty thousand. As a result of parallel connectivity of the cells, the overall on-state resistance (i.e., resistance between drain and source terminals in the conducting state) is substantially reduced in comparison with the on-state resistance of an individual cell [16, 17].

Fig. 2.7

Power MOSFETs can be fabricated as Si-based or SiC-based devices. For Si the energy gap is Eg = 1.1 eV,

(2.20)

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while for SiC the energy gap is appreciably larger (see [18, 19]): Eg = 3.2 eV.

(2.21)

For this reason, SiC belongs to the class of wide-band semiconductors which are very attractive and promising in the area of power electronics. The reason is that high energy gaps lead to appreciably higher breakdown electric fields, which is beneficial for the operation of semiconductor devices at high voltages. Furthermore, high energy gaps also result in much higher operating temperatures [20] and higher radiation hardness [21]. The former is important for the operation of semiconductor devices at high currents and voltages. Currently, a relatively new wide-band gallium nitride (GaN) semiconductor is a focus of extensive research [22–25]. For this material the band gap (see [18, 19]) is: Eg = 3.4 eV.

(2.22)

Furthermore, GaN has a direct energy gap, while the energy gap for SiC is indirect. It is clear from Figure 2.6 that the sides of each power MOSFET cell can operate as a pn− n+ diodes which are called “body-diodes”. This suggests that power MOSFET devices can operate as desired parallel connections of MOSFETs and diodes (see Figure 2.5) removing the necessity of using separate diode devices. This idea of using body-diodes in Si and SiC power MOSFETs is of interest and being currently investigated [26–28]. It turns out that the structure of the power MOSFET can be modified to create another power semiconductor device called the IGBT or COMFET [29,30]. The first abbreviation stands for “insulated-gate bipolar transistor”, while the second abbreviation stands for “conductivity modulated field-effect transistor”. A schematic depiction of one cell of the IGBT is shown in Figure 2.8. In actual IGBT devices a very large number of such cells are connected in parallel. It is evident from this figure that the main structural difference between the power MOSFET and IGBT is the replacement of the n+ region of the power MOSFET by a p+ region in the IGBT. This replacement has important consequences. Indeed, on two sides of each cell of the IGBT two p+ n− p bipolar transistors are formed as a result of the above design modification. When a positive voltage is applied to the gate resulting in the formation of two n channels in the p regions, the flow of electrons will be

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Fig. 2.8 caused by the application of drain-to-source voltage. This electron current entering the n− region (which is the base region for the bipolar transistors on the sides) will trigger these bipolar transistors, resulting in the side flow of holes from drain to source. Thus, the current in the IGBT has two distinct components, the electron current due to the MOSFET action and the hole current due to the side BJT action: IIGBT = IMOSFET + IBJT .

(2.23)

This leads to the increase in IGBT current for the same value of VDS in comparison with the power MOSFET, where only the electron component of the drain current is present. This increase in the drain current results in the reduction of on-state resistance and on-state losses [31]. It must be remarked that the introduction of the p+ region in the IGBT creates a four-layer vertical structure p+ n− pn+ similar to the one used in the design of thyristors. This may lead to parasitic thyristor action in the IGBT which is usually called thyristor latch-up [29, 32]. This parasitic thyristor action may compromise the gate control over the drain current. Special techniques have been developed to achieve non-latch-up operation of the IGBT [33–35]. The discussion of these techniques is outside the scope of this book. It turns out that the bidirectional switches shown in Figure 2.5 can be used to control the width of rectangular pulses at each half-cycle, which is very important for the device realization of pulse width modulation. Particularly, the pattern of veq (t) shown in Figure 2.9 can be produced

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through the appropriate switching. Indeed, immediately prior to the time instant t = t1 , transistors T r1 and T r3 are in “on” states, transistors T r2 and T r4 are in “off” states, and the current i(t) > 0, i.e., its direction coincides with the one shown in Figure 2.5. At time t1 , transistor T r3 is turned off. Due to the presence of the inductor, the continuity of i(t) must be maintained. This is only possible if diode D2 is turned on and freewheels the current i(t) through the closed path formed by the LR branch, diode D2 and transistor T r1 . For this closed path, the voltage across the nodes 1 and 2 is equal to zero as shown in Figure 2.9. At time T2 +t0 , transistor T r1 is turned off, while transistors T r2 and T r4 are turned on. This switching action causes the voltage between nodes 1 and 2 to change to −V0 as shown in Figure 2.9. Then, at time T2 +t1 , when i(t) < 0 and i(t) has the direction opposite to the one shown in Figure 2.5, transistor T r4 is turned off, forcing diode D1 to turn on, forming in this way the closed path for i(t) through D1 , T r2 and the LR branch. This results in zero voltage across the nodes 1 and 2 as consistent with the plot of veq (t) in Figure 2.9. The described switching pattern is periodically repeated.

Fig. 2.9 The steady-state analysis of the electric circuit shown in Figure 2.2 with veq (t) shown in Figure 2.9 can be carried out in a similar way as before. Namely, it is easy to see that electric current i(t) during the half-cycle

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 T 0, 2 can be represented as follows: R

i(t) = A1 e− L t , R V0 + A2 e − L t , i(t) = R R

i(t) = A3 e− L t ,

0 ≤ t ≤ t0 ,

(2.24)

t0 ≤ t ≤ t1 ,

(2.25)

T , 2

(2.26)

t1 ≤ t ≤

where A1 , A2 and A3 are some constants. These constants are determined from the boundary condition (2.4) as well as from the following continuity conditions for electric current i(t) at t0 and t1 : i(t0− ) = i(t0+ ),

(2.27)

i(t1− ) = i(t1+ ).

(2.28)

Having determined A1 , A2 and A3 from conditions (2.4), (2.27) and (2.28), we arrive at the final expressions for i(t): Rt1

Rt0

V0 e L − e L − RL (t+ T2 ) , 0 ≤ t ≤ t0 e R 1 + e− RT 2L " # Rt0 T R V0 e L (t1 − 2 ) + e L − R t L i(t) = 1− e , t0 ≤ t ≤ t1 . RL R 1 + e− 2T i(t) = −

Rt1

i(t) =

Rt0

V0 e L − e L − R t e L , R 1 + e− RT 2L

t1 ≤ t ≤

T . 2

(2.29) (2.30)

(2.31)

A plot of i(t) based on formulas (2.29), (2.30) and (2.31) is shown in Figure 2.10. The plot of vout (t) is a scaled (by R) version of the plot for i(t). It is apparent from the above plot that i(t) is a half-wave symmetric periodic function of t with period T . This period and, consequently, the fundamental frequency of i(t) is controlled by the pattern of switching of SW1 , SW2 , SW3 and SW4 that can be chosen appropriately. It is also clear from the above plot as well as formulas (2.29), (2.30) and (2.31) that i(t) is a piecewise exponential function of time t. This is true for the excitation of the electric circuit in Figure 2.2 by any sequence of rectangular pulses. It is also evident that the waveform of i(t) shown in Figure 2.10 has more resemblance to a sinusoidal function than the waveform shown in Figure 2.4. This suggests that by choosing a more elaborate sequence of rectangular pulses for veq (t) much better resemblance of i(t) (and vout (t)) with a sinusoidal function can be achieved. The latter can indeed be accomplished

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Fig. 2.10

by using the pulse width modulation (PWM) technique discussed in the subsequent sections. 2.2

Sinusoidal Pulse Width Modulation and Its Fourier Analysis

Pulse width modulation is used in power electronic inverters to approximate low frequency waveforms by a sequences (trains) of rectangular pulses whose widths are properly modulated. A large number of PWM techniques have been developed over the years and discussed in literature (see, for instance, [11, 36–42]). A common feature of these techniques is the suppression of low order harmonics in the Fourier spectra of PWM voltages. This suppression is achieved at the expense of some amplification of high order harmonics in the Fourier spectra. However, these high order harmonics are suppressed in the output voltage vout (t) by an inductor in the LR branches of the inverters. Next, we discuss one simple version of PWM when the pulse widths are sinusoidally modulated [5, 43]. In this version, each half-cycle T2 is subdivided into N equal time intervals and the centers τi of these time intervals are specified by the formula   T 1 τi = i+ , (i = 0, 1, 2, . . . , N − 1). (2.32) 2N 2

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The pulse width modulated voltage consists of a sequence of rectangular pulses centered at τi . The widths of these pulses are sinusoidally modulated: ∆ti =

mT sin ωτi , 2N

(2.33)

where as before ω=

2π , T

(2.34)

while m is called the modulation index (or depth of modulation) and usually 0 < m < 1.

(2.35)

It is shown below that by varying m the peak value of the sinusoidal output voltage can be controlled. An example of such pulse width modulated voltage is presented in Figure 2.11. For the sake of drawing simplicity, in this example N = 5. However, for PWM to be effective, N is usually quite large, N  1.

(2.36)

Such sequences of width modulated rectangular pulses can be obtained by using specific switching patterns of transistors in the circuit shown in Figure 2.5. Some details of the realization of these switching patterns are discussed later in this section.

Fig. 2.11

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Now, we will be concerned with the study of the Fourier spectra of the above PWM voltages. It is clear from Figure 2.11 that veq (t) has two types of symmetry: odd symmetry and half-wave symmetry. This implies (see Chapter 1 section 4) that in the Fourier series for veq (t) there are only odd sine-type harmonics. Namely, veq (t) =

∞ X

V2n−1 sin(2n − 1)ωt,

(2.37)

n=1

where V2n−1

4 = T

Z

T 2

veq (t) sin(2n − 1)ωtdt.

(2.38)

0

The immediate purpose of the subsequent discussion is the evaluation of the Fourier coefficients V2n−1 . To this end and by taking into account that veq (t) is a sequence of N rectangular pulses of the same peak value V0 , the last formula can be transformed as follows: N −1 Z 4V0 X V2n−1 = sin(2n − 1)ωtdt. (2.39) T i=0 ∆ti According to formulas (2.33) and (2.36), the widths of pulses ∆ti are quite small. For this reason, we shall use the following midpoint approximation: Z sin(2n − 1)ωtdt ≈ ∆ti sin(2n − 1)ωτi . (2.40) ∆ti

This is the only approximation that is used in our subsequent derivation. It is evident that this approximation is quite accurate for small n when the period T2n−1 of sin(2n − 1)ωt is quite large in comparison with ∆ti : T2n−1 =

T mT  ∆ti = sin ωτi . 2n − 1 2N

(2.41)

This approximation may not be accurate when T2n−1 and ∆ti are comparable. This suggests that our derivation will lead to accurate results for loworder sinusoidal harmonics in the Fourier series expansion (2.37). Namely, it will be demonstrated that the low order harmonics will be suppressed at the expense of high order harmonics. However, this approximation will also lead to some deviations from the actual Fourier spectrum. This matter will be discussed in detail later.

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Returning to our derivation and by substituting formula (2.40) into equation (2.39), we find V2n−1 ≈

N −1 4V0 X ∆ti sin(2n − 1)ωτi . T i=0

(2.42)

The last equation is further transformed by using expression (2.33) for ∆ti : V2n−1 =

N −1 2mV0 X sin(2n − 1)ωτi sin ωτi . N i=0

(2.43)

By recalling the trigonometric identity sin α sin β =

1 [cos(α − β) − cos(α + β)] , 2

the last formula can be transformed as follows: "N −1 # N −1 X mV0 X V2n−1 = cos(2n − 2)ωτi − cos 2nωτi . N i=0 i=0

(2.44)

(2.45)

From formulas (2.32) and (2.34) it follows that ωτi =

π (2i + 1) 2N

(2.46)

and, consequently, the relation (2.45) can be written as "N −1 # N −1 X mV0 X (n − 1)π nπ V2n−1 = cos (2i + 1) − cos (2i + 1) . N N N i=0 i=0

(2.47)

Thus, the subsequent derivation is based on the evaluation of the following two sums: S1 =

N −1 X

cos

(n − 1)π (2i + 1), N

(2.48)

cos

nπ (2i + 1). N

(2.49)

i=0

S2 =

N −1 X i=0

It is apparent that the expression for S1 can be transformed as follows: "N −1 # " # N −1 X (n−1)π (n−1)π X (n−1)2π j N (2i+1) j N j i N S1 = Re e = Re e e . (2.50) i=0

i=0

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By introducing the notation q = ej

(n−1)2π N

,

(2.51)

we find that N −1 X

ej

(n−1)2π i N

=

i=0

N −1 X

qi .

(2.52)

i=0

The last sum is a geometric series and, consequently, N −1 X

qi =

i=0

1 − qN . 1−q

(2.53)

Now, by combining formulas (2.51), (2.52) and (2.53), we derive: N −1 X

ej

(n−1)2π i N

i=0

=

1 − ej(n−1)2π 1 − ej

(n−1)2π N

.

By using the last expression in formula (2.50), we arrive at   j(n−1)2π (n−1)π 1 − e S1 = Re ej N . (n−1)2π 1 − ej N

(2.54)

(2.55)

To conclude the evaluation of S1 , we consider two distinct cases. Case a) when n − 1 is not divisible by N , that is, for any natural number r we have the inequality n − 1 6= rN.

(2.56)

The latter means that q = ej

(n−1)2π N

6= 1,

(2.57)

while on the other hand ej(n−1)2π = 1.

(2.58)

Thus, according to equation (2.55) we find S1 = 0.

(2.59)

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Case b) deals with those “rare” instances when n − 1 is divisible by N . This means that such a natural number r can be found that n − 1 = rN.

(2.60)

The latter implies that q = ej

(n−1)2π N

= ej2πr = 1.

(2.61)

In this case, the fraction in formula (2.55) is not defined. It turns out that in this case the sum S1 can be evaluated differently. Namely, from formulas (2.61) and (2.52) we find N −1 X

ej

(n−1)2π i N

= N.

(2.62)

i=0

Furthermore, according to equality (2.60), we have ej

(n−1)π N

= ejrπ = (−1)r .

(2.63)

By using the last two formulas in the expression (2.50), we obtain S1 = (−1)r N.

(2.64)

Thus, we have concluded the evaluation of S1 . The evaluation of sum S2 can be carried out in a similar way. However, it is easy to perform this evaluation based on the observation that S1 and S2 have identical mathematical structures. Indeed, it is clear that S1 can be transformed into S2 by replacing n − 1 by n. This implies that as far as the value of S2 is concerned, there are two distinct cases: Case a) when n 6= rN

(2.65)

S2 = 0,

(2.66)

and

as well as

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Case b) when n = rN

(2.67)

S2 = (−1)r N.

(2.68)

and

It is easy to see that the condition (2.60) is equivalent to 2n − 1 = 2rN + 1.

(2.69)

Consequently, we conclude that if

2n − 1 = 2rN + 1,

then

S1 = (−1)r N.

(2.70)

Similarly, the condition (2.67) is equivalent to 2n − 1 = 2rN − 1.

(2.71)

Consequently, if

2n − 1 = 2rN − 1,

then

S2 = (−1)r N.

(2.72)

It is also easy to see from (2.56), (2.59), (2.65) and (2.66) that S1 = S2 = 0,

if

2n − 1 6= 2rN ± 1.

(2.73)

Now, from formulas (2.47), (2.48), (2.49), (2.70), (2.72) and (2.73) we conclude that V2rN ±1 = (±)(−1)r mV0 ,

(2.74)

while V2n−1 = 0,

if

2n − 1 6= 2rN ± 1.

(2.75)

In particular, for r = 0 from equation (2.74) we obtain V1 = mV0 .

(2.76)

By using the last three formulas in equation (2.37), we find veq (t) = mV0 sin ωt + mV0

∞ X (±)(−1)r sin(2rN ± 1)ωt. r=1

(2.77)

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Fig. 2.12 This means that under approximation (2.40) the pulse width modulated voltage veq (t) has a “sparse-twin” spectrum as illustrated in Figure 2.12. This spectrum is “sparse-twin” because “most” of the terms in the Fourier series expansion (2.37) are equal to zero, and those terms which are not equal to zero appear as pairs (i.e., twins) with equal peak values. Next, we shall use formula (2.77) in the analysis of the electric circuit shown in Figure 2.2. We shall treat each term in (2.77) as an ac voltage source of frequency (2rN ± 1)ω. Then, by using the superposition principle and ac steady-state analysis, we derive the following expression for the current i(t): mV0 sin(ωt − ϕ) i(t) = p 2 R + (ωL)2 ∞ X   (±)(−1)r p + mV0 sin (2rN ± 1)ωt − ϕ± r , 2 2 2 2 R + (2rN ± 1) ω L r=1 (2.78) where ωL , R (2rN ± 1)ωL tan ϕ± . r = R tan ϕ =

(2.79) (2.80)

For sufficiently large number N of pulses (see (2.36)), we have (2rN ± 1)ωL  ωL

for all

r ≥ 1,

(2.81)

and all terms in the sum in the formula (2.78) are small and can be neglected. This leads to mV0 i(t) ≈ p sin(ωt − ϕ) 2 R + (ωL)2

(2.82)

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and mV0 R sin(ωt − ϕ). vout (t) ≈ p R2 + (ωL)2

(2.83)

Thus, PWM leads to practically sinusoidal output voltage vout (t), and its peak value can be controlled by varying the modulation index m. It is also clear from the last formula and formula (2.34) that the frequency of ac voltage can be controlled by varying T . Similar analysis can be performed when the RL branch in Figure 2.1 is replaced by the L-RC branch as shown in Figure 2.13, which leads to the equivalent circuit 2.14. This branch is typical for single-phase inverters used in the design of uninterruptible power supplies (UPS). The analysis of the circuit shown in Figure 2.14 with veq (t) given by equation (2.77) can be performed using the same line of reasoning as discussed in example 2 in section 5 of Chapter 1. Again, we shall treat each term in (2.77) as an ac voltage source of frequency (2rN ± 1)ω. Then, by using the superposition principle and ac steady-state analysis, we can derive the following expression for the output voltage: mV0 R sin(ωt − ϕ) vR (t) = q 2 (R − ω 2 LCR) + ω 2 L2 + mV0 R

∞ X r=1

±(−1)r

  1 ± ± sin (2rN ± 1)ωt − ϕr . Zr

(2.84)

where ωL , R − ω 2 LCR (2rN ± 1)ωL tan ϕ± , r = R − (2rN ± 1)2 ω 2 LCR q 2 Zr± = [R − (2rN ± 1)2 ω 2 LCR] + (2rN ± 1)2 ω 2 L2 , tan ϕ =

(2.85) (2.86) (2.87)

For a sufficiently large number of pulses N we find that each term in the sum in formula (2.84) is small and can be neglected. This leads to: mV0 R vR (t) ≈ p sin(ωt − ϕ) 2 (R − ω LCR)2 + ω 2 L2

(2.88)

which is a sinusoidal output voltage. It is worthwhile to mention again that the derivation of formula (2.77)

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3 SW1 − +

V0

SW2

C L

2

1 R + vR (t) −

SW4

SW3

4 Fig. 2.13 +

L − +

veq (t)

C

R

vR (t) −

Fig. 2.14 has been based on approximation (2.40). For this reason, it is of interest to compare the “sparse-twin” spectrum shown in Figure 2.12 with the actual spectrum numerically computed without using this approximation. These computations can be performed by using the following formula V2n−1 =

N −1 X 2V0 [cos(2n − 1)ωt2i+1 − cos(2n − 1)ωt2i+2 ] , (2.89) (2n − 1)π i=0

where ∆ti , 2 ∆ti t2i+2 = τi + , 2 (i = 0, ..., N − 1). t2i+1 = τi −

(2.90) (2.91)

Equation (2.89) is derived from equation (2.39) by explicit integration. The time-instants t2i+1 and t2i+2 arethe switching time-instants for the ith pulse of veq (t) in the interval 0, T2 . By using the last three formulas, the actual Fourier spectra for the

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sequence of rectangular pulses with sinusoidally modulated widths have been numerically computed. These computations have been performed for N = 21, N = 51, N = 101 and N = 201 as well as for the following values of modulation index: m = 0.3, m = 0.5, m = 0.7 and m = 0.9. The computational results for the amplitudes of the first sideband harmonics are presented in Figures 2.15, 2.16, 2.17 and 2.18. The amplitudes of the six most significant first sideband harmonics are presented in Table 2.1. It is clear from the presented numerical results that the formula (2.77) based on the approximation (2.40) properly captures the main features of the Fourier spectrum of sinusoidally modulated PWM voltage. Namely, it reveals the suppression of low order harmonics. Furthermore, this formula also demonstrates that the latter is achieved at the expense of amplification of high order harmonics which form periodically repeated sidebands. It also captures that these sidebands have two dominant “twin” harmonics for small or moderate values of the index of modulation m. Finally, formula (2.77) reveals that the magnitude of the twin harmonics does not depend on N . However, the approximation (2.40) leads to some deviations from the actual spectra. Namely, according to formula (2.77) the magnitude of these twins remains constant for all r while it should diminish to zero as r increases (see Remark 2 in Chapter 1 section 4, formula (1.160)). Finally, formula (2.77) does not capture other harmonics except the sideband “twins”. It is apparent from the Table 2.1 that the magnitude of significant sideband harmonics remains more or less the same for a wide range of values of N above 50. In this sense, these values are universal. Once they have been computed for a relatively small value of N , they can be readily used for larger number of pulses N as well. Similar universality was observed in our computation of the magnitudes of the significant second sideband harmonics.

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|V2n−1 /V1 |

(a) m = 0.3 1.00 0.80 0.60 0.40 0.20 0.00 36

38

40

42 44 2n − 1

46

48

46

48

46

48

46

48

|V2n−1 /V1 |

(a) m = 0.5 1.00 0.80 0.60 0.40 0.20 0.00 36

38

40

42 44 2n − 1

|V2n−1 /V1 |

(a) m = 0.7 1.00 0.80 0.60 0.40 0.20 0.00 36

38

40

42 44 2n − 1

|V2n−1 /V1 |

(a) m = 0.9 1.00 0.80 0.60 0.40 0.20 0.00 36

38

40

42

44

2n − 1

Fig. 2.15: Sidebands centered around 2N for N = 21.

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|V2n−1 /V1 |

(a) m = 0.3 1.00 0.80 0.60 0.40 0.20 0.00 96

98

100

102 104 2n − 1

106

108

106

108

106

108

106

108

|V2n−1 /V1 |

(a) m = 0.5 1.00 0.80 0.60 0.40 0.20 0.00 96

98

100

102 104 2n − 1

|V2n−1 /V1 |

(a) m = 0.7 1.00 0.80 0.60 0.40 0.20 0.00 96

98

100

102 104 2n − 1

(a) m = 0.9

|V2n−1 /V1 |

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1.00 0.80 0.60 0.40 0.20 0.00 96

98

100

102

104

2n − 1

Fig. 2.16: Sidebands centered around 2N for N = 51.

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|V2n−1 /V1 |

(a) m = 0.3 1.00 0.80 0.60 0.40 0.20 0.00 196

198

200

202

204

206

208

206

208

206

208

206

208

2n − 1

|V2n−1 /V1 |

(a) m = 0.5 1.00 0.80 0.60 0.40 0.20 0.00 196

198

200

202 204 2n − 1

|V2n−1 /V1 |

(a) m = 0.7 1.00 0.80 0.60 0.40 0.20 0.00 196

198

200

202 204 2n − 1

|V2n−1 /V1 |

(a) m = 0.9 1.00 0.80 0.60 0.40 0.20 0.00 196

198

200

202

204

2n − 1

Fig. 2.17: Sidebands around 2N for N = 101.

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|V2n−1 /V1 |

(a) m = 0.3 1.00 0.80 0.60 0.40 0.20 0.00 396

398

400

402

404

406

408

406

408

406

408

406

408

2n − 1

|V2n−1 /V1 |

(a) m = 0.5 1.00 0.80 0.60 0.40 0.20 0.00 396

398

400

402 404 2n − 1

|V2n−1 /V1 |

(a) m = 0.7 1.00 0.80 0.60 0.40 0.20 0.00 396

398

400

402 404 2n − 1

(a) m = 0.9

|V2n−1 /V1 |

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398

400

402

404

2n − 1

Fig. 2.18: Sidebands around 2N for N = 201.

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Table 2.1: Most significant sidebands centered around 2N for different values of m and N . (a) N = 21

2n − 1 37 39 41 43 45 47

m = 0.3 0.000240 0.030414 0.897860 0.888050 0.039847 0.000615

V2n−1 /V1 m = 0.5 m = 0.7 0.001763 0.006263 0.077468 0.132920 0.733510 0.524520 0.709730 0.486920 0.098550 0.161380 0.004367 0.014772

m = 0.9 0.015404 0.182880 0.306320 0.260670 0.207460 0.033939

(b) N = 51

2n − 1 97 99 101 103 105 107

m = 0.3 0.000325 0.033080 0.895010 0.890970 0.036966 0.000478

|V2n−1 /V1 | m = 0.5 m = 0.7 0.002362 0.008283 0.083572 0.141560 0.726590 0.513480 0.716790 0.497990 0.092264 0.153300 0.003426 0.011774

m = 0.9 0.020006 0.191180 0.292750 0.273950 0.201350 0.027654

(c) N = 101

2n − 1 197 199 201 203 205 207

m = 0.3 0.000359 0.034026 0.894020 0.891980 0.035988 0.000436

|V2n−1 /V1 | m = 0.5 m = 0.7 0.002599 0.009073 0.085709 0.144510 0.724170 0.509650 0.719230 0.501830 0.090099 0.150440 0.003136 0.010835

m = 0.9 0.021769 0.193860 0.288070 0.278580 0.199000 0.025633

(d) N = 201

2n − 1 397 399 401 403 405 407

m = 0.3 0.000377 0.034510 0.893510 0.892490 0.035496 0.000416

|V2n−1 /V1 | m = 0.5 m = 0.7 0.002726 0.009492 0.086797 0.145990 0.722940 0.507700 0.720460 0.503770 0.089003 0.148980 0.002996 0.010377

m = 0.9 0.022697 0.195180 0.285700 0.280930 0.197760 0.024639

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Next, we shall briefly discuss how PWM voltages can be generated. Both analog and digital techniques can be used for this purpose. One analog technique (see [11]) is to generate a triangular (low-voltage) waveform v1 (t) with the frequency which is 2N times the inverter output frequency as well as to generate a low-voltage sinusoidal (modulating) function v2 (t) (see Figure 2.19): v1 (t) = vtr (t),

(2.92)

v2 (t) = Vm sin ωt.

(2.93)

These low-voltage (and low-power) waveforms can be generated by using operational amplifiers, for instance [44, 45]. The difference of these waveforms vG (t) = v2 (t) − v1 (t)

(2.94)

can be used as a voltage controlling the switching of the transistors in the inverter circuit. For instance, this voltage vG (t) can be applied as a gate voltage for the power MOSFETs or IGBTs. These transistors will be in “on” states and produce rectangular pulses with time durations equal to the time intervals for which vG > 0.

(2.95)

According to Figure 2.19, these time intervals ∆ti can be computed as ∆ti ≈ αVm sin ωti .

(2.96)

On the other hand, α≈

T /(2N ) . Vtr

(2.97)

By substituting the last formula in (2.96), we find ∆ti ≈

Vm T sin ωti Vtr 2N

(2.98)

It is apparent that the last formula coincides with formula (2.33) when the modulation index is defined as m=

Vm . Vtr

(2.99)

Thus, by controlling the ratio of peak values of the sinusoidal and triangular

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waveforms, the modulation index m and, consequently, the peak value of the sinusoidal output voltage vout (t) (see equation (2.83)) can be controlled.

Fig. 2.19 In practice, digital controllers are often used to control the switches of inverters to generate PWM line-voltages [46, 47]. These controllers are equipped with digital microprocessors which can be properly programmed to achieve the desired PWM as well as desired variation of frequency and peak value. Digital signal processors (DSPs) and Field Programmable Gate Arrays (FPGAs) used for the implementation of PWM have specialized PWM modules encoded for its easy implementation.

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101

Selective Harmonic Elimination and Chudnovsky Technique

As mentioned at the beginning of the previous section, the main goal of pulse width modulation is to suppress low order harmonics in modulated voltage sources. In the case of sinusoidal PWM, the latter is accomplished by using a large number of rectangular pulses. It is natural to raise the question of how to exactly eliminate low order harmonics by using a small number of rectangular pulses. It turns out that this elimination can be achieved by employing the technique which is known as selective harmonic elimination (SHE) [48–51]. The central idea of this technique is to use the exact formula (2.89) for the Fourier coefficients and to find such switching time-instants t2i−1 and t2i that specific Fourier coefficients V2n−1 have desired values. As a result, one obtains a set of simultaneous (i.e., coupled) nonlinear equations. It turns out that the solution of these coupled nonlinear equations can be found by computing the roots of a single univariate polynomial. The latter is accomplished by using the Chudnovsky method [52, 53]. In this method, the reduction of coupled nonlinear transcendental equations to a single polynomial equation is achieved by using the mathematical machinery of Chebyshev polynomials and modified Newton’s Identities. We start the discussion by rewriting formula (2.89) in the following form: N

V2n−1 =

X 2V0 [cos(2n − 1)α2i−1 − cos(2n − 1)α2i ] , π(2n − 1) i=1

(2.100)

where αi = ωti .

(2.101)

It can be shown that imposing half-wave symmetry along with oddsymmetry implies quarter-wave Hence all the switching time symmetry.  instants in the time interval T4 , T2  can be expressed in terms of switching time-instants in the interval 0, T4 . Using this fact, equation (2.100) can be rewritten as:

V2n−1 =

4V0 [cos(2n − 1)α1 − cos(2n − 1)α2 + . . . + cos(2n − 1)αN ] . (2n − 1)π (2.102)

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According to the principle of selective harmonic elimination, the following conditions are imposed: V1 = Vm , V2n−1 = 0,

(2.103) n = 2, 3, . . . , N

(2.104)

where Vm is a desired peak value of the first fundamental harmonic. By using formulas (2.102), (2.103) and (2.104) we arrive at the following set of coupled nonlinear equations: cos α1 − cos α2 + · · · + cos αN =

πVm , 4V0

(2.105)

and cos 3α1 − cos 3α2 + . . . + cos 3αN = 0, cos 5α1 − cos 5α2 + . . . + cos 5αN = 0, .. .

(2.106)

cos(2N − 1)α1 − cos(2N − 1)α2 + . . . + cos(2N − 1)αN = 0. Now, we introduce new variables βi = αi ,

if i is odd,

(2.107)

βi = π − αi , if i is even,

(2.108)

as well as the following notation A=

πVm . 4V0

(2.109)

By using the last three formulas, the equations (2.105) and (2.106) can be represented in the form: cos β1 + cos β2 + . . . + cos βN = A, cos 3β1 + cos 3β2 + . . . + cos 3βN = 0, .. .

(2.110)

cos(2N − 1)β1 + cos(2N − 1)β2 + . . . + cos(2N − 1)βN = 0. Equations (2.110) are coupled, nonlinear transcendental equations. We next reduce them to algebraic equations by using Chebyshev polynomials Tn (x) [54]. This can be done by using the following property of these

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polynomials: cos nβ = Tn (cos β),

(2.111)

cos β = x

(2.112)

cos βi = xi , for n = 1, 2, . . . , N.

(2.113)

where

and, consequently,

Now, the coupled equations (2.110) can be written in terms of Chebyshev polynomials as follows: T1 (x1 ) + T1 (x2 ) + . . . + T1 (xN ) = A, T3 (x1 ) + T3 (x2 ) + . . . + T3 (xN ) = 0, .. .

(2.114)

T2N −1 (x1 ) + T2N −1 (x2 ) + . . . + T2N −1 (xN ) = 0, The above equations can be written in the concise form as: N X

T1 (xi ) = A,

(2.115)

i=1 N X

T2n−1 (xi ) = 0, for n = 2, 3, . . . , N.

(2.116)

i=1

Next, we shall briefly summarize the basic properties of the Chebyshev polynomials Tn (x) which will be helpful in our subsequent discussion. These polynomials are orthogonal on the interval [−1, 1] with the following weight function 1

(1 − x2 )− 2 . The latter means that Z 1 Tm (x)Tn (x) √ dx = 0, 1 − x2 −1

(2.117)

if

m 6= n.

(2.118)

Furthermore, Chebyshev polynomials satisfy the following recurrent relations Tn+1 (x) = 2xTn (x) − Tn−1 (x).

(2.119)

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By setting: T0 (x) = 1

and

T1 (x) = x,

(2.120)

from the formula (2.119) we can find the expression for Chebyshev polynomial for any n. Namely, T2 (x) = 2x2 − 1, T3 (x) = 4x3 − 3x,

(2.121)

T4 (x) = 8x4 − 8x2 + 1, T5 (x) = 16x5 − 20x3 + 5x,

and so on. It is apparent from formula (2.121) (and more generally, from formulas (2.119)-(2.120)) that the Chebyshev polynomials T2n−1 (x) are odd functions of x. This implies that: T2n−1 (x) =

n X

cn,m x2m−1 .

(2.122)

m=1

The above coefficients cn,m can be found from the following general formula for the Chebyshev polynomials,   bn/2c n X (−1)r n − r (2x)n−2r , Tn (x) = 2 r=0 n − r r where bn/2c is the largest integer less than or equal to n/2, and binomial coefficient defined as:   p p! = . q q!(p − q)!

(2.123) p q




is the

(2.124)

Another very interesting property of the Chebyshev polynomials related to their historical discovery is that T˜n (x) defined by the formula T˜n (x) = 21−n Tn (x)

(2.125)

provides the best polynomial approximation for (i.e., the least deviation from) zero in the interval [−1, 1] among all polynomials of degree n. Now, we return to equations (2.115)-(2.116). By substituting expressions (2.122) into formulas (2.115)-(2.116) and interchanging summation with respect to i and m, we will get the following equations for the odd

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powers of xi N X

xi = A.

(2.126)

i=1 2n−1 X

cn,m

m=1

N X

x2m−1 = 0, i

(n = 2, 3, . . . , N ).

(2.127)

i=1

Next, by introducing the notation N X

x2m−1 = s2m−1 , i

(m = 1, 2, . . . , N )

(2.128)

i=1

we arrive at the following linear equations: s1 = A, 2n−1 X

cn,m s2m−1 = 0,

(2.129) (n = 2, 3, . . . N ).

(2.130)

m=1

Expressions (2.129) and (2.130) form a set of linear simultaneous equations for s2m−1 with known coefficients and a triangular matrix. They can be easily solved to obtain s2m−1 , for all m = 1, 2, . . . , N . The following analytical expression for s2m−1 is reported in [52]:   2m − 1 A , where m = 1, 2, . . . , N. (2.131) s2m−1 = m−1 4 m−1 Thus, nonlinear coupled transcendental equations (2.110) have been reduced to algebraic equations for the odd power sums of xi . The next step is to reduce equations (2.128) to a single univariate polynomial whose roots are xi . It is a well-established fact that in the case when all (odd and even) power sums are known, this can be accomplished by using Newton’s Identities [55,56]. Namely, the coefficients of the above polynomial coincide (up to a sign) with elementary symmetric polynomials. These symmetric polynomials can be found by using Newton’s identities if all power sums of the roots xi are known. However, the problem we face here is that we do not know all power sums, but only odd power sums. Remarkably, Chudnovsky brothers demonstrated [52,53] that the above difficulty can be circumvented and the desired polynomial can be reconstructed by knowing only odd power sums. Their approach is of general mathematical interest and, for this reason, it is discussed below in some generality.

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We start with the following expression for the sought monic polynomial P (x) of degree n in terms of its roots xi : P (x) =

n Y

(x − xi ) .

(2.132)

i=1

Next, we compute the logarithmic derivative of this polynomial: n

P 0 (x) X 1 d [ln P (x)] = = . dx P (x) x − xi i=1

(2.133)

Each term in the sum of (2.133) can be expanded into power series in terms of x1 . This can be done by replacing x by y1 , and then using Taylor series in terms of y. This eventually leads to the following series expansion: ∞

X xk 1 i = . x − xi xk+1

(2.134)

k=0

By substituting (2.134) into (2.133) and interchanging summations with respect to i and k, we end up with ∞ n X d 1 X k [ln P (x)] = x . dx xk+1 i=1 i

(2.135)

k=0

Now, by recalling the definition of sk , the last formula can be written as ∞

X sk d [ln P (x)] = . dx xk+1

(2.136)

s0 = n.

(2.137)

k=0

It is clear that

Consequently, formula (2.136) can be reduced to ∞

d n X sk [ln P (x)] = + . dx x xk+1

(2.138)

k=1

By integrating both sides of the last expression we arrive at: ln P (x) = n ln x −

∞ X sk kxk

k=1

(2.139)

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which can be further transformed as: ∞ X sk P (x) = x exp − kxk n

! .

(2.140)

k=1

The last formula relates polynomial P (x) to all (odd and even) power sums sk . The most crucial next step is to remove even power sums s2k in formula (2.140). This can be achieved by considering the polynomial P (−x) which, according to (2.140), is given by the expression ! ∞ X (−1)k sk n n . (2.141) P (−x) = (−1) x exp − kxk k=1

Now, by dividing P (x) by P (−x), we end up with:  ! ∞  X 1 − (−1)k sk P (x) n = (−1) exp − P (−x) kxk

(2.142)

k=1

It is clear that the even terms in the sum in the last formula are equal to zero. Consequently,  ∞  ∞ X X 1 − (−1)k sk s2k+1 = 2 . (2.143) kxk (2k + 1)x2k+1 k=1

k=0

This leads to the following simplification of equation (2.142): ! ∞ X P (x) s2k+1 n = (−1) exp −2 P (−x) (2k + 1)x2k+1

(2.144)

k=0

Next, by introducing the notation P˜ (x): P˜ (x) = (−1)n P (−x),

(2.145)

formula (2.144) can be written in the concise form as: P (x) = P˜ (x)G(x),

(2.146)

G(x) = eV (x)

(2.147)

where

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and V (x) = −2

∞ X k=0

s2k+1 (2k + 1)x2k+1

(2.148)

or, V (x) = −2

s

1

x

+

 s5 s3 + + · · · . 3x3 5x5

(2.149)

Now it is clear from formulas (2.146)-(2.149) that polynomial P (x) is related to only the odd power sums of its roots. This is exactly what was intended to be accomplished. The last step is to demonstrate how the coefficients of the polynomial P (x) can be found from equation (2.146) in terms of s2k+1 . To this end, we introduce the variable y=

1 x

(2.150)

and write equation (2.147) as G(y) = eV (y) =

∞ X

gj y j ,

(2.151)

j=0

where the last term is the power series expansion of G(y). Formula (2.149) can also be written as a power series expansion for V (y): V (y) =

∞ X

vj y j ,

(2.152)

j=1

where according to (2.149) we have v2j−1 = −

2s2j−1 and 2j − 1

v2j = 0.

(2.153)

Our purpose is to find gj in terms of vj . To this end, we shall differentiate equation (2.151) and arrive at dV (y) dV (y) dG(y) = eV (y) = G(y) . dy dy dy Now, by using power series expansions for

dG(y) dy ,

G(y) and

(2.154) dV (y) dy ,

equation

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(2.154) can be written as g1 + 2g2 y + 3g3 y 2 + . . . = g0 + g1 y + g2 y 2 + . . .


v1 + 2v2 y + 3v3 y 2 + . . . ,




(2.155)

which can be further transformed to arrive at g1 + 2g2 y + 3g3 y 2 + . . . = v1 g0 + (2v2 g0 + v1 g1 )y + (3v3 g0 + 2v2 g1 + v1 g2 ) y 2 + . . . .

(2.156)

By equating the terms of like powers in the last formula, we end up with the following set of equations for gi : g1 = v1 g0 , 2g2 = 2v2 g0 + v1 g1 , 3g3 = 3v3 g0 + 2v2 g1 + v1 g2 , .. .

(2.157)

which can be written in the concise form as jgj =

j X

kvk gj−k .

(2.158)

k=1

Since g0 = 1, equation (2.157) can be sequentially used to find gj in terms of vk , and consequently, according to formula (2.153), in terms of sk . As soon as gj are found, G(x) can be written in the form G(x) =

∞ X gj . j x j=0

(2.159)

Now, by using this formula in equation (2.146), coefficients of polynomial P (x) can be determined. For the sake of simplicity, this will be demonstrated for the example when P (x) is a third-degree polynomial: P (x) = x3 + p1 x2 + p2 x + p3 .

(2.160)

By using formulas (2.160), (2.159) and (2.146), we end up with (x3 + p1 x2 + p2 x + p3 )   g2 g1 + 2 + ··· . = −(−x3 + p1 x2 − p2 x + p3 ) 1 + x x

(2.161)

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Now, by matching the like powers of

1 x,

we find

1 → 0 = −g1 p3 + g2 p2 − g3 p1 + g4 , x 1 → 0 = −g2 p3 + g3 p2 − g4 p1 + g5 , x2 1 → 0 = −g3 p3 + g4 p2 − g5 p1 + g6 . x3 The last equations can be written in the matrix form as      g3 −g2 g1 p1 g4 g4 −g3 g2  p2  = g5  . g5 −g4 g3 p3 g6

(2.162)

(2.163)

By using the last equation, the polynomial coefficients p1 , p2 , p3 can be found. It is worthwhile to write the last equation in another form by introducing another variables p˜k related to the polynomial coefficients pk as follows: p˜k = (−1)k pk .

(2.164)

Then, the linear equations (2.163) can written as:      g3 g2 g1 p˜1 g4 g4 g3 g2  p˜2  = − g5  . g5 g4 g3 p˜3 g6

(2.165)

It is clear that in case of polynomials of higher degrees, we obtain linear equations for p˜k with the matrix whose pattern is similar the one in equation (2.165):      gn gn−1 · · · g1 p˜1 gn+1  gn+1 gn · · · g2   p˜2  gn+2       = − (2.166)  .   . .   .. .. .. .. .  ..     . . . . .  g2n−1 g2n−2 · · · gn

p˜n

g2n

Now, we arrive at the point that the algorithm of computing the switching time-instants ti resulting in selective harmonic elimination can be stated as a sequence of the following steps. Step 1 is to find the odd power sums s2m−1 by using formulas (2.109) and (2.131). Step 2: By using formula (2.153), find gj by solving linear equation (2.157) (or (2.158)).

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Step 3: By solving equation (2.166) and using formula (2.164), find the coefficients pk for polynomial P (x) with roots equal to xi . Step 4 is to find the roots xi of the above polynomial P (x). Step 5: By using the roots xi and formulas (2.113), (2.107), (2.108), and (2.101), find the switching time-instants ti which result in selective harmonic elimination. (a) PWM voltage for SHE

veq (t) V0

0

ωt T 4

T 2

3T 4

T

−V0 veq (t) V0

0

(b) PWM voltage for Sinusoidal PWM

ωt T 4

T 2

3T 4

T

−V0 Fig. 2.20: Comparison of PWM voltage for SHE-PWM and Sinusoidal PWM, when N = 7.

By using the above stated algorithm, αi and the corresponding switching times were calculated for N = 7, 9, 11, 15, 17 and N = 19. The results of calculations are compared with the sinusoidal pulse width modulation (SPWM). This comparison is presented in Table 2.2 as well as in Figures 2.20 and 2.21. It is apparent from Table 2.2 and these figures that the values of αi (and consequently, switching-times ti ) are very close for selective harmonic elimination (SHE) and sinusoidal PWM for sufficiently large

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(a) PWM voltage for SHE

veq (t) V0

0

ωt T 4

T 2

3T 4

T

−V0 veq (t) V0

0

(b) PWM voltage for Sinusoidal PWM

ωt T 4

T 2

3T 4

T

−V0 Fig. 2.21: Comparison of PWM voltage for SHE-PWM and Sinusoidal PWM, when N = 11.

values of i. This suggests that the analytical expression for αi known for sinusoidal PWM can be used to compute the initial guesses for finding the roots of the corresponding polynomial equation. It has been observed in our calculations that the polynomials we deal with have real coefficients and real roots. Such polynomials are called hyperbolic (or real-rooted) polynomials and they are encountered in many different areas of mathematics [57]. It would be interesting to prove that the polynomials of the above technique are indeed hyperbolic for any N . It is worthwhile to mention that the equation (2.166) has the structure of Toeplitz system. It is known that the matrix of Toeplitz equations becomes ill-conditioned with the increase of its dimension [58]. Indeed, we observe in our calculations numerical instability for the case N = 21. In section 5, another approach to selective harmonic elimination will be discussed within the framework of time-domain optimization technique.

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Table 2.2: Values of αi for Selective harmonic elimination (SHE) and Sinusoidal PWM (SPWM), when Vm /V0 = 0.9 for different values of N . (b) N = 9

(a) N = 7 α1 α2 α3 α4 α5 α6 α7

SHE 0.3080 0.4239 0.6255 0.8525 0.9659 1.3026 1.3600

SPWM 0.1794 0.2693 0.5472 0.7991 0.9400 1.3040 1.368

α1 α2 α3 α4 α5 α6 α7 α8 α9

SHE 0.2572 0.3337 0.5189 0.6689 0.7906 1.0092 1.0807 1.3653 1.4069

SPWM 0.1472 0.2018 0.4450 0.6021 0.7523 0.9929 1.0741 1.3693 1.4137

(c) N = 11 α1 α2 α3 α4 α5 α6 α7 α8 α9 α10 α11

(d) N = 15 α1 α2 α3 α4 α5 α6 α7 α8 α9 α10 α11 α12 α13 α14 α15

SHE 0.1723 0.2037 0.3455 0.4077 0.5205 0.6120 0.6984 0.8173 0.8804 1.0243 1.0684 1.2348 1.2650 1.4507 1.4734

SPWM 0.0948 0.1145 0.2850 0.3432 0.4764 0.5707 0.6699 0.7961 0.8662 1.0187 1.0658 1.2380 1.2692 1.4535 1.4765

α1 α2 α3 α4 α5 α6 α7 α8 α9 α10 α11 α12 α13 α14 α15 α16 α17

SPWM 0.084727 0.1000 0.2544 0.2999 0.4249 0.4990 0.5966 0.6969 0.7701 0.8930 0.9456 1.0871 1.1237 1.2787 1.3043 1.4677 1.4876

SPWM 0.1245 0.1610 0.3750 0.4817 0.6298 0.7981 0.8914 1.1077 1.1619 1.4085 1.4423

(f) N = 19

(e) N = 17 SHE 0.15526 0.1803 0.3111 0.3608 0.4681 0.5415 0.6269 0.7227 0.7884 0.9047 0.9534 1.0885 1.1234 1.2750 1.3001 1.4653 1.4852

SHE 0.2209 0.2752 0.4442 0.5510 0.6726 0.8286 0.9097 1.1110 1.1612 1.4045 1.4371

α1 α2 α3 α4 α5 α6 α7 α8 α9 α10 α11 α12 α13 α14 α15 α16 α17 α18 α19

SHE 0.14129 0.1617 0.2829 0.3236 0.4254 0.4856 0.5692 0.6479 0.7146 0.8106 0.8624 0.9742 1.0132 1.1393 1.1680 1.3067 1.3282 1.4767 1.4944

SPWM 0.0765 0.0888 0.2297 0.2662 0.3834 0.4432 0.5380 0.6194 0.6936 0.7944 0.8506 0.9681 1.0093 1.1402 1.1697 1.3105 1.3321 1.4788 1.4964

In this approach, the selective harmonic elimination will be achieved by using constrained optimization where the constraints appear as nonlinear equations for selective harmonic elimination. In this way, selective harmonic

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elimination is achieved simultaneously with the minimization of the total harmonic distortion. Furthermore, by using the constrained optimization, selective harmonics can be eliminated which are not necessarily the group of the first sequential low order harmonics. The latter may (or may not) be useful in the development of advanced pulse width modulation techniques. (a) PWM voltage for SHE PWM

v(t) V0

ωt

0

T 4

T 2

3T 4

T

−V0 v(t)

(b) PWM voltage for Sinusoidal PWM

V0

0

ωt T 4

T 2

3T 4

T

−V0 Fig. 2.22: Comparison of PWM voltage for SHE-PWM and Sinusoidal PWM, when N = 15.

2.4

Time Domain Analysis of Pulse Width Modulation in Single-Phase Inverters

In this section, we are concerned with the exact analytical time-domain analysis of PWM for different load circuits used in applications. This analysis is of interest in its own right, and it will be used in the next section as a foundation for the development of optimal PWM techniques. These techniques result in minimization of the total harmonic distortions along

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with possible selective harmonic eliminations. Here, we discuss the analytical time-domain analysis of PWM for singlephase inverters. This analysis can be performed when veq (t) is any periodic sequence (train) of rectangular voltage pulses with half-wave symmetry. The developed technique will be extended to the three-phase inverters in the next chapter. This will be done by using per-phase time-domain analysis of these inverters. The loads discussed below are traditional LR circuits as well as more complex L-RC and L-C-LR circuits. These circuits are used in applications, for instance in uninterruptible power supplies (UPS) [59–63]. It turns out that these higher-order load circuits more efficiently suppress high order harmonics. It is worthwhile to point out that the presented analytical technique can be in principle used for the analysis of any higherorder linear load circuit subject to PWM voltages. We start the discussion with the case of the LR load circuit (see Figures 2.1 and 2.2). This analysis can be stated as the boundary value problem of finding the solution to the following differential equation: ( 0, if t2j < t < t2j+1 , di(t) + Ri(t) = (2.167) L dt V0 , if t2j+1 < t < t2j+2 , where t2j−1 and t2j are the switching “on” and switching “off” timeinstants, respectively, j = 0, 1, 2 . . ., N while t0 = 0,

t2N +1 =

T . 2

(2.168)

The solution of equation (2.167) must satisfy the following continuity conditions at the switching time-instants: + i(t− 2j ) = i(t2j ),

i(t− 2j+1 )

=

i(t+ 2j+1 ),

(2.169) (2.170)

where j = 0, 1, . . . , N , as well as the anti-periodic boundary condition stated below:   T . (2.171) i(0) = −i 2 The conditions (2.169)-(2.170) reflect the continuity of electric current through the inductor, while the anti-periodic boundary condition (2.171) is the result of half-wave symmetry of i(t) caused by the half-wave symmetry of veq (t).

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It is clear from equation (2.167) that the current i(t) is given by the formula: ( R if t2j < t < t2j+1 , A2j+1 e− L t , (2.172) i(t) = V0 t −R L + R , if t2j+1 < t < t2j+2 , A2j+2 e where j = 0, 1, . . . , N , while coefficients A2j+1 and A2j+2 must be determined from the continuity of electric current i(t) at times t2j and t2j+1 as well as from the “antiperiodic” boundary condition (2.171). This leads, respectively, to the following simultaneous equations for the above coefficients: A2 − A1 = − A3 − A2 =

V0 Rt1 e L , R

(2.173)

V0 Rt2 e L , R

(2.174)

.. . A2j − A2j−1 = − A2j+1 − A2j =

V0 Rt2j−1 e L , R

(2.175)

V0 Rt2j e L , R

(2.176)

V0 Rt2N e L , R

(2.177)

.. . A2N +1 − A2N = and RT

A1 + A2N +1 e− 2L = 0.

(2.178)

Formulas (2.173)-(2.177) can be viewed as linear simultaneous equations with a two-diagonal matrix. These equations can be solved analytically as follows. By summing up all equations from (2.173) to (2.177), we find A2N +1 − A1 =

2N Rtj V0 X (−1)j e L . R j=1

(2.179)

Now, we need to solve the two simultaneous equations (2.178) and (2.179). This leads to the following expressions: A2N +1

V0 = R

P2N

j=1

R

(−1)j e L tj RT

1 + e− 2L

,

(2.180)

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V0 A1 = − R

P2N

j=1

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R

(−1)j e L tj

1+

mainbook

RT e− 2L

RT

e− 2L .

(2.181)

Having found A1 , all other A-coefficients can be computed by using the formula Aj = A1 +

j−1 R V0 X (−1)n e L tn for j = 2, 3, . . . , 2N. R n=1

(2.182)

which can be obtained by summing up the first (j − 1) equations in (2.173)(2.177). Thus, by using the last three formulas along with equation (2.172), we can find explicit analytical expression for the current i(t) as well as the output voltage vout (t). In this way, the voltage becomes a known function of V0 , R, L as well as the switching time-instants t1 , t2 , . . . , t2N : vout (t) = F (t, V0 , R, L, t1 , t2 , . . . , t2N ).

(2.183)

Examples of vout (t) computed by using the above formulas are shown in Figure 2.23a for N = 11 and Figure 2.23b for N = 21. Sinusoidal PWM has been used to calculate the switching time-instants (see section 2.2), while the computations have been performed for the values of modulation index m = 0.8, frequency f = 60 Hz, R = 10Ω and L = 100 µH. Next, we proceed to the analysis of the output voltage vR (t) for the L-RC load circuit shown in Figure 2.24. By using KVL and KCL for the above circuit, the following differential equations can be written: di(t) + vR (t) = veq (t), dt dvR (t) vR (t) C + = i(t). dt R L

(2.184) (2.185)

By differentiating equation (2.185) and then substituting it in (2.184), we arrive at: ( 0, if t2j < t < t2j+1 , d2 vR (t) L dvR (t) LC + + vR (t) = . (2.186) 2 dt R dt V0 , if t2j+1 < t < t2j+2 , Now, the analysis can be stated as the following boundary value problem of finding the solution of equation (2.186) subject to the following conditions:

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(a) N = 11

vout (t) V0

t

0

T

T 2

−V0 (b) N = 21

vout (t) V0

0

t 3T /2 2T

−V0 Fig. 2.23

L 2

i(t)

+

− +

veq (t)

C

R

vR (t) −

1 Fig. 2.24

(a) continuity of voltage vC (t) = vR (t) at the switching time-instants tk : + vR (t− k ) = vR (tk ),

k = 1, 2, . . . , 2N ;

(2.187)

(b) continuity of the derivative of the voltage vC (t) = vR (t) at the same

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time-instants: dvR − dvR + (tk ) = (t ), dt dt k

k = 1, 2, . . . , 2N ;

(2.188)

  T , 2

(2.189)

(c) anti-periodic boundary conditions: vR (0) = −vR

dvR dvR (0) = − dt dt

  T . 2

(2.190)

The validity of formula (2.188) follows from equation (2.185) and continuity of vC (t) = vR (t) as well as i(t). Similarly, formula (2.190) follows from equations (2.185), (2.189) and the half-wave periodicity of i(t). Next, we proceed to the solution of the equation (2.186). The general (h) solution vR (t) to the homogeneous equation corresponding to equation (2.186) can be written as follows: (h)

vR (t) = Aes1 t + Bes2 t ,

(2.191)

where s1 and s2 are roots of the following quadratic equation: LCs2 +

L s + 1 = 0. R

(2.192)

Here, for the sake of simplicity we consider the generic case when the above roots are distinct. (p) On the other hand, the particular solution vR (t) to equation (2.186) when veq (t) = V0 is given by the formula: (p)

vR (t) = V0 . Thus, by using equations (2.191) and (2.193), we arrive at: ( A2j+1 es1 t + B2j+1 es2 t , if t2j < t < t2j+1 , vR (t) = A2j+2 es1 t + B2j+2 es2 t + V0 , if t2j+1 < t < t2j+2 ,

(2.193)

(2.194)

where j = 0, 1, 2, . . . , N . The last formula can be rewritten in the concise form as: vR (t) = Ak es1 t + Bk es2 t + χk ,

if tk−1 < t < tk ,

(2.195)

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where χk =


V0 · 1 + (−1)k , 2

for k = 1, 2, . . . , 2N + 1.

(2.196)

It is clear that χk takes two values: 0 or V0 . Furthermore, from formula (2.195) we find: dvR (t) = s1 Ak es1 t + s2 Bk es2 t , dt

if tk−1 < t < tk .

(2.197)

We now proceed to find the expressions for the coefficients Ak and Bk . By using equations (2.195), (2.197), (2.187) and (2.188), we obtain: Ak es1 tk + Bk es2 tk + χk = Ak+1 es1 tk + Bk+1 es2 tk + χk+1 , s1 Ak e

s1 tk

+ s2 Bk e

s2 tk

= s1 Ak+1 e

s1 tk

s2 tk

+ s2 Bk+1 e

,

(2.198) (2.199)

where k = 1, 2, . . . , 2N . Formulas (2.198) and (2.199) define a set of simultaneous linear algebraic equations with a four-diagonal matrix. It is remarkable that explicit analytical solutions to these equations can be found. This is because these equations have a special mathematical structure which allows for their reduction to two sets of decoupled simultaneous equations with two-diagonal matrices for coefficients Ak and Bk , respectively. Indeed, by multiplying equation (2.198) by s2 and subtracting it from (2.199) we find: (s1 − s2 )Ak es1 tk − s2 χk = (s1 − s2 )Ak+1 es1 tk − s2 χk+1 ,

(2.200)

which can then be rearranged as follows: Ak − Ak+1 =

s2 [χk − χk+1 ] e−s1 tk . s1 − s2

(2.201)

It is apparent from equation (2.196) that χk − χk+1 = (−1)k V0 .

(2.202)

Thus, from the last two equations, we obtain: Ak − Ak+1 = (−1)k V0

s2 e−s1 tk . s1 − s2

(2.203)

It is evident that equations (2.203) are simultaneous linear equations with two-diagonal matrix which are similar to the equations in the case of LR load circuits. For this reason, the same technique can be used for their solution. Namely, if we add equations (2.203) for k = 1, 2, . . . , 2N , we

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arrive at: A1 − A2N +1 = V0

2N s2 X (−1)n e−s1 tn . s1 − s2 n=1

(2.204)

Next, from equations (2.195), (2.197) and the boundary conditions (2.189)(2.190), we find: T

T

A1 + B1 = −A2N +1 e 2 s1 − B2N +1 e 2 s2 , s1 A1 + s2 B1 = −s1 A2N +1 e

T 2

s1

− s2 B2N +1 e

(2.205) T 2

s2

.

(2.206)

Again, by multiplying equation (2.205) by s2 and then subtracting it from (2.206), we obtain: T

A1 + A2N +1 e 2 s1 = 0.

(2.207)

By solving simultaneous equations (2.204) and (2.207) we find: A1 =

A2N +1

T 2N X V0 s2 e 2 s1   (−1)n e−s1 tn , T (s1 − s2 ) 1 + e 2 s1 n=1

2N X V0 s2   =− (−1)n e−s1 tn . T (s1 − s2 ) 1 + e 2 s1 n=1

(2.208)

(2.209)

Having found A1 , the other coefficients Ak can be computed as follows: Ak = A1 −

k−1 s2 V0 X (−1)n e−s1 tn . s1 − s2 n=1

(2.210)

The last expression is derived by adding the first (k − 1) equations defined by formula (2.203). Now, we proceed to derive the expressions for the Bk -coefficients. On the one hand, this can be accomplished by multiplying equation (2.198) by s1 and subtracting it from equation (2.199) and then proceeding in the same way as in the derivation of formulas (2.208)-(2.210). On the other hand, the expression for the coefficient Bk can be immediately found by using the permutational symmetry of equations (2.198) and (2.199) with respect to interchanging s1 and s2 as well as Ak and Bk . By using this symmetry and replacing in formulas (2.208)-(2.210) Ak by Bk and interchanging s1 and

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s2 we arrive at the following expressions: T 2N X V0 s1 e 2 s2   (−1)n e−s2 tn , B1 = T (s2 − s1 ) 1 + e 2 s2 n=1

B2N +1 = −

2N X V0 s1   (−1)n e−s2 tn . T s 2 (s2 − s1 ) 1 + e 2 n=1

Bk = B1 −

k−1 s1 V0 X (−1)n e−s2 tn . s2 − s1 n=1

(2.211)

(2.212)

(2.213)

Finally, by using these analytical expressions for Ak and Bk along with equation (2.194), we arrive at the analytical formula for vR (t) which is valid for any choice of switching time-instants tk . As an example, the numerically computed output voltage waveforms are shown in Figure 2.25a for N = 11 and in Figure 2.25b for N = 21. Sinusoidal pulse width modulation has been used to calculate the switching time-instants (see section 2.2), and the computations have been performed for the values of modulation index m = 0.8, frequency f = 60 Hz, R = 10Ω, L = 100 µH and C = 100 µF. Next, we extend the previous analysis to the case of the L-C-LR circuit shown in Figure 2.26. We intend to derive the analytical expression for the output current i1 (t) in this figure. As before, the voltage veq (t) is described by equation: ( 0, if t2j < t < t2j+1 , veq (t) = (2.214) V0 , if t2j+1 < t < t2j+2 , and it satisfies the half-wave symmetry condition. By using KVL and KCL the following differential equations can be written: 1 1 di(t) = − vC (t) + veq (t), dt L L di1 (t) R 1 = − i1 (t) + vC (t), dt L1 L1 dvC (t) 1 1 = i(t) − i1 (t). dt C C

(2.215) (2.216) (2.217)

Next, we represent the above equations in the state-variable form. To

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(a) N = 11

vout (t) V0

t

0

T 2

T

−V0 vout (t)

(b) N = 21

V0

t

0

T 2

T

−V0 Fig. 2.25 this end, we define the state-vector 

 i(t) x(t) =  i1 (t)  , vC (t) and represent the equations (2.215)-(2.217) as follows:   1 0 0 − L1 L ˙ x(t) =  0 − LR1 L11  x(t) +  0  veq (t). 1 1 0 0 C −C

(2.218)

(2.219)

Now the analysis can be stated as the following boundary value problem of finding i1 (t) from the equations (2.218) and (2.219) subject to the following conditions:

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L 2

i1 (t)

i(t) R + vC (t) −

− +

veq (t)

C L1

1 Fig. 2.26

(a) continuity of current i1 (t) at the switching time-instants tk : + i1 (t− k ) = i1 (tk ),

k = 1, 2, . . . , 2N ;

(2.220)

(b) continuity of the derivative of the current i1 (t) at the same time-instants: di1 + di1 − (t ) = (t ), dt k dt k

k = 1, 2, . . . , 2N ;

(2.221)

(c) continuity of the second derivative of i1 (t) at the same time-instants: d2 i1 − d2 i1 + (t ) = (t ), k dt2 dt2 k

k = 1, 2, . . . , 2N ;

(2.222)

(d) anti-periodic boundary conditions:   T i1 (0) = −i1 , 2   di1 T di1 (0) = − , dt dt 2   d2 i1 T d2 i1 (0) = − . dt2 dt2 2

(2.223) (2.224) (2.225)

The continuity conditions can be explained as follows. It is clear that the state-variables i(t), i1 (t) and vC (t) are continuous. Moreover, it follows from equation (2.216) and (2.217), the derivatives of i1 (t) and vC (t) are also continuous. Furthermore, by differentiating equation (2.216), we can conclude that the second-order derivative of i1 (t) is continuous as well. Similarly, equations (2.223)-(2.225) follow from the half-wave symmetry of veq (t) and consequently, the half-wave symmetry of state variables i(t), i1 (t)

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and vC (t) and their derivatives. Next, we proceed to the solution of the equations (2.218)-(2.219) and we use some facts from the theory of state-variable equations [64]. According to this theory, we shall first consider the characteristic equation for the matrix in formula (2.219):   1 1 1 R R 2 3 s + + s+ = 0. (2.226) s + L1 C L L1 LL1 C For the sake of simplicity, we shall deal with the generic case when the three roots s1 , s2 , and s3 of the above equation are distinct. This implies that (h) the general solution i1 (t) to the homogeneous equation corresponding to equation (2.219) has the form: (h)

i1 (t) = Aes1 t + Bes2 t + Ces3 t .

(2.227) (p)

On the other hand, it is easy to see that the particular solution i1 (t) to the state-equation (2.219), for the time-intervals when veq (t) = V0 can be written as follows: (p)

i1 (t) =

V0 . R

(2.228)

Using formulas (2.227) and (2.228), the general solution for i1 (t) can be expressed as: ( A2j+1 es1 t + B2j+1 es2 t + C2j+1 es3 t , if t2j < t < t2j+1 , i1 (t) = V A2j+2 es1 t + B2j+2 es2 t + C2j+2 es3 t + R0 , if t2j+1 < t < t2j+2 , (2.229) where j = 0, 1, . . . , N . The above equation can be represented in the concise form: i1 (t) = Ak es1 t + Bk es2 t + Ck es3 t + χ ˜k ,

if tk−1 < t < tk .

(2.230)

Here, χ ˜k =


V0 · 1 + (−1)k , 2R

(2.231)

for k = 1, 2, . . . , 2N + 1. It is apparent that χ ˜k takes two values: 0 or VR0 . Furthermore, it is clear that the following formulas can be written for the derivatives of i1 (t): di1 (t) = s1 Ak es1 t + s2 Bk es2 t + s3 Ck es3 t , dt

if tk−1 < t < tk ,

(2.232)

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d2 i1 (t) = s21 Ak es1 t + s22 Bk es2 t + s23 Ck es3 t , dt2

if tk−1 < t < tk .

(2.233)

We now proceed to derive the expressions for the unknown coefficients Ak , Bk and Ck . This is done by using the continuity conditions for i1 (t) as well as for its first and second derivatives as specified in equations (2.220)-(2.222). By applying these continuity conditions at tk for all k = 1, 2, . . . , 2N , the following equations can be written in accordance with formulas (2.230), (2.232) and (2.233): Ak es1 tk + Bk es2 tk + Ck es3 tk + χ ˜k =Ak+1 es1 tk + Bk+1 es2 tk + Ck+1 es3 tk + χ ˜k+1 ,

(2.234)

s1 Ak es1 tk + s2 Bk es2 tk + s3 Ck es3 tk = s1 Ak+1 es1 tk + s2 Bk+1 es2 tk + s3 Ck+1 es3 tk ,

(2.235)

s21 Ak es1 tk + s22 Bk es2 tk + s23 Ck es3 tk = s21 Ak+1 es1 tk + s22 Bk+1 es2 tk + s23 Ck+1 es3 tk .

(2.236)

These are linear coupled equations for the coefficients Ak , Bk and Ck with the six diagonal matrix. It is remarkable that these equations can be decoupled and reduced to three separate sets of simultaneous linear equations with two-diagonal matrices. This is accomplished as follows. Multiplying equation (2.234) by s2 and subtracting it from equation (2.235), we obtain: (s1 − s2 )Ak es1 tk + (s3 − s2 )Ck es3 tk − s2 χ ˜k = (s1 − s2 )Ak+1 es1 tk + (s3 − s2 )Ck+1 es3 tk − s2 χ ˜k+1 .

(2.237)

Similarly, multiplying equation (2.234) by s22 and subtracting it from equation (2.236), we get: (s21 − s22 )Ak es1 tk + (s23 − s22 )Ck es3 tk − s22 χ ˜k =(s21 − s22 )Ak+1 es1 tk + (s23 − s22 )Ck+1 es3 tk − s22 χ ˜k+1 .

(2.238)

It is clear that coefficients Bk have been eliminated. We next eliminate coefficients Ck by multiplying (2.237) by (s3 + s2 ) and subtracting it from (2.238). After simple algebraic transformations, the following equations for Ak emerge: Ak − Ak+1 =

s2 s3 [χ ˜k+1 − χ ˜k ] e−s1 tk (s1 − s2 )(s1 − s3 )

(2.239)

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for k = 1, 2, . . . , 2N . It is clear from (2.231) that: χ ˜k+1 − χ ˜k = (−1)k+1

V0 . R

(2.240)

Thus, from the last two formulas, we conclude : Ak − Ak+1 =

V0 s2 s3 (−1)k+1 e−s1 tk . R (s1 − s2 )(s1 − s3 )

(2.241)

By adding the last equations for k = 1, 2, . . . , 2N , we arrive at the following formula: A1 − A2N +1 =

2N X s2 s3 V0 (−1)n+1 e−s1 tn . R (s1 − s2 )(s1 − s3 ) n=1

(2.242)

Next, by using the half-wave anti-periodic boundary conditions (2.223)(2.225) for i1 (t) as well as its first and second derivatives along with (2.168), (2.230), (2.232) and (2.233) we obtain:

A1 + Bk + Ck T

T

T

= −A2N +1 es1 2 − B2N +1 es2 2 − C2N +1 es3 2 ,

(2.243)

s1 A1 + s2 Bk + s3 Ck T

T

T

T

T

= −s1 A2N +1 es1 2 − s2 B2N +1 es2 2 − s3 C2N +1 es3 2 ,

(2.244)

s21 A1 + s22 Bk + s23 Ck T

= −s21 A2N +1 es1 2 − s22 B2N +1 es2 2 − s23 C2N +1 es3 2 .

(2.245)

Again, using the same steps as before, from the above three formulas we can derive the following equation: T

A1 + A2N +1 es1 2 = 0.

(2.246)

Finally, we can solve the simultaneous equations (2.242) and (2.246). This yields the following result: A1 =

2N X

V0 s2 s3 

 T

R(s1 − s2 )(s1 − s3 ) 1 + e−s1 2

(−1)n+1 e−s1 tn .

n=1

(2.247)

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Having obtained A1 , all other coefficients Ak can be computed according to the formula: Ak = A1 −

k−1 X V0 s2 s3 (−1)n+1 e−s1 tn . R(s1 − s2 )(s1 − s3 ) n=1

(2.248)

This formula is obtained by adding the first (k − 1) equations in equation (2.241). By using as before the permutational symmetry argument, similar expressions for coefficients Bk and Ck can be immediately written by the appropriate interchanging (permutation) of s1 , s2 and s3 . This leads to B1 =

2N X

V0 s1 s3 

R(s2 − s1 )(s2 − s3 ) 1 + e−s2 2

Bk = B1 −

(−1)n+1 e−s2 tn .

 T

(2.249)

n=1

k−1 X V0 s1 s3 (−1)n+1 e−s2 tn , R(s2 − s1 )(s2 − s3 ) n=1

(2.250)

and C1 =

2N X

V0 s1 s2 

 T

R(s3 − s1 )(s3 − s2 ) 1 + e−s3 2

Ck = C1 −

(−1)n+1 e−s3 tn .

(2.251)

n=1

k−1 X V0 s1 s2 (−1)n+1 e−s3 tn . R(s3 − s1 )(s3 − s2 ) n=1

(2.252)

In this way, the explicit analytical expression for the current i1 (t) is obtained. Hence, the output voltage vout (t) across the resistor R can also be computed. As an example, numerically computed output voltage waveforms are shown in Figure 2.27a for N = 11 and Figure 2.27b for N = 21. As before, the switching time-instants have been calculated using sinusoidal PWM (see section 2.2). The computations have been performed for the values of modulation index m = 0.8, frequency f = 60 Hz, R = 1Ω, L = 500 µH, C = 100 µF and L1 = 100 µH. It is clear from the comparison of the last figure with Figure 2.23 and Figure 2.25 that the ripple in the output voltage has been appreciably reduced due to the presence of additional energy storage elements in the circuit shown in Figure 2.26. Our discussion of time-domain analysis was based on the analytical solution of many simultaneous equations corresponding to the entire number N of rectangular pulses of PWM voltages. It turns out that there is another

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vout (t)

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(a) N = 11

V0

t

0

T 2

T

−V0 vout (t)

(b) N = 21

V0

0

t T 2

T

−V0 Fig. 2.27 equivalent approach which is based on the superposition principle. In this approach, the analytical solution for a single rectangular pulse is first found by using the anti-periodic boundary condition. Afterwards, the solution for the entire number N of rectangular pulses can be obtained by the proper superposition of solutions for single rectangular pulses. It is suggested to the interested reader to pursue further this approach, which will be left here unattended. 2.5

Optimal Pulse Width Modulation

In the previous section, we discussed the analysis of electric circuits subject to PWM voltages. Particularly, we derived the analytical expressions for the output voltages vout (t) for single phase inverters and different load circuits. These expressions are explicit functions of the switching timeinstants t1 , t2 , . . . , t2N which describe the PWM voltage veq (t). Hence, for given specific values of V0 , T and load circuit parameters, we can express

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the output voltage vout (t) as a known function of switching-time instants: vout (t) = vout (t, t1 , . . . , t2N ).

(2.253)

Using this fact, various optimal PWM techniques can be developed to choose the switching time-instants to fulfill specific criteria for optimization. We start our discussion with the selection of switching time-instants to minimize the following function: Z T2 2 2 [vout (t) − Vm sin(ωt − ϕ)] dt. (2.254) E2 (t1 , . . . , t2N ) = T 0 Here, Vm is a desired peak value of the first harmonic of the output voltage vout (t) and ϕ is its initial phase. The minimization of E2 (t1 , . . . , t2N ) must be performed under the constraint that the switching time-instants form a strictly monotonically increasing sequence: 0 < t1 < t2 < · · · < t2N