Systems & Control: Foundations & Applications Series Editor Tamer Ba§ar, University of Illinois at Urbana-Champaign Editorial Board Karl Johan Äström, Lund Institute of Technology, Lund, Sweden Han-Fu Chen, Academia Sinica, Beijing William Helton, University of California, San Diego Alberto Isidori, University of Rome (Italy) and Washington University, St. Louis Petar V. Kokotovic, University of California, Santa Barbara Alexander Kurzhanski, Russian Academy of Sciences, Moscow and University of California, Berkeley H. Vincent Poor, Princeton University Mete Soner, K05 University, Istanbul Jeffery C. Allen H°° Engineering and Amplifie r Optimization Springer Science+Business Media, LL C Jeffery C. Allen Space and Naval Warfare Systems Center Naval Command Control and Ocean Surveillance San Diego, CA 92152 U.S.A. Library of Congress Cataloging-in-Publication Data Allen, Jeffery, C. H-[infinity] engineering and amplifier optimization / Jeffery C. Allen. p. cm. - (Systems & control) On t.p. "[infinity]" appears as the infinity symbol. Includes bibliographical references and index. ISBN 978-1-4612-6478-1 ISBN 978-0-8176-8182-1 (eBook) DOI 10.1007/978-0-8176-8182-1 1. Broadband amplifiers-Mathematical models. 2. H [oo] control. 3. Hardy spaces. 4. Multidisciplinary design optimization. I. Title: Hardy spaces engineering and amplifier optimization. II. Title. III. Series. TK7871.58.B74A43 2004 621.3815'35-dc22 2004047749 CIP AMS Subject Classifications: 94C65,47N70,46N10,47A48,47A56,47A20 ISBN 0-8176-3780-X Printed on acid-free paper. ©2004 Springer Science+Business Media New York Originally published by Birkhäuser Boston in 2004 Softcover reprint of the hardcover 1st edition 2004 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to property rights. Copyright is not claimed to any part herein that is an original work prepared by a U.S. government officer or employee as part of that person's official duties. 98765432 1 SPIN 10954560 www. birkhasuer-science. com To Martin and Esther for friendship and family, to Elaine for insight and faith, and to the memory of Kathryn. Preface Amplifying a weak, noisy, wide band signal is a canonical problem in electrical engineering. Figure 0.1 illustrates the classical solution where an amplifier is connected to the input and output matching circuits. The amplifier designer is given the amplifier and must design matching circuits that boost the input signal. The design problem is that the amplifier does increase the power of the weak input signal-but also amplifies the input noise, adds its own noise, and can be unstable for certain matching circuits. Input signal Input Output signal Amplifier Output -'\r+ matching (Given) matching -'\r+ circuit circuit Fig. 0.1. The amplifier and matching circuits. For example, anyone fiddling with a car radio learns that turning up the volume of a noisy signal simply produces a louder version of the same noisy signal. In addition, the amplifier must be stable. The characteristic "howl" or feedback of a public address system is an ear-splitting example of an unstable amplifier. The amplifier designer tinkers with the matching circuits to simul taneously trade gain for noise while maintaining stability. Thus, the Amplifier Matching Problem is really a multiobjective optimization problem [22]. The bulk of the amplifier literature is devoted to narrow-band solutions using the circuit of Figure 0.1 [106], [53], [140]. These solutions become sub optimal as the bandwidth of the amplifier increases, stability margins become more stringent, and as multiple amplifiers are added to get greater gain. viii Preface This monograph shows how recent developments in Hoc engineering equip the amplifier designer with new tools that • compute the best possible performance available from any matching cir cuits, • benchmark existing matching solutions, and • generalize to multiple amplifier configurations. This monograph is aimed at two groups: applied mathematicians with an interest in electrical engineering and electrical engineers with an interest in Hoc theory. For mathematicians, there are few engineering subjects where an advanced topic like Hoc theory has such an immediate connection to actual physical devices. For engineers, many Hoc methods are still at the first level of development. The flexibility of the Hoc methods coupled with the plethora of electrical devices offer splendid research opportunities for electrical engi neers. Finally, both groups can attack the fundamental questions connecting circuits and HOC functions. We hope our readers realize a rich harvest from the research opportunities explicitly called out in this monograph. Chapter 1 reviews the necessary circuit theory. The intended audience is the mathematician with a modicum of electrical engineering background. The scattering formalism is the framework for this discussion. In the scattering formalism, we review the lumped elements, how the lumped elements may be connected to construct matching circuits, how Belevitch's Theorem gives the structure of these matching circuits, the associated scattering matrices and linear-fractional forms, the power and gain functions, the orbits generated by the matching circuits, and a basic model of an amplifier. The foundation of these circuit-scattering techniques is the Existence Theorem for the Scattering Matrix. Roughly speaking, any passive circuit admits a scattering matrix that belongs to the unit ball of HOC. We will see that the matching circuits corre spond to the inner functions. In this sense, circuit design really corresponds to optimization over specified classes of Hoc functions. Chapter 2 makes explicit that the gain, noise, and stability functions of an amplifier are functions of the matching circuits. The Amplifier Matching Problem is to find input and output matching circuits that simultaneously trade off the competing objectives: AMP-l maximize the gain, AMP-2 minimize the noise, AMP-3 guarantee stability. Because the matching circuits correspond to HOC functions, the Amplifier Matching Problem is really the multiobjective optimization of the gain, noise, and stability functions over specified classes of Hoc functions. Chapter 3 sets out the necessary tools of HOC engineering. The intended audience is the electrical engineer with some background in control theory. Nehari's Theorem is the fundamental result that essentially defines HOC engi neering. J. W. Helton and his colleagues have made enormous progress adapt- f>reface Ix ing Nehari's Theorem to electrical engineering [65], [64], [9], [10], [11], [66], [15]; control theory [62], [63], [68], [69], [71], [741, [75]; and signal processing [85]. This chapter collects the technical results aiming at the interplay between Hoo theory, which supports the computations for amplifier optimization, and the disk algebra, which corresponds to lumped matching circuits. Chapter 4 presents the classes of matching circuits. At this point, the dis tinction between mathematicians and electrical engineers is starting to blur. The intellectual orientation is provided by the Circuit-Scattering Correspon dence and the key question: What elements in Hoo correspond to a circuit and conversely? The State-Space Representation Theorem characterizes this correspondence for the lumped circuits. There are nontrivial open questions when a corre spondence between distributed circuits and Hoo is attempted. These research questions are made explicit with a warning on some of the pitfalls. Chapter 5 gathers the preceding material into a high-level presentation of the Amplifier Matching Problem. The amplifier designer optimizes over se lected classes of matching circuits. Each collection of matching circuits pushes the input and output loads around to generate the input and output re ftectances. These collections of reftectances are called the orbits of the input and the output loads generated by the particular class of matching circuits. Thus, the amplifier designer can either optimize over the matching circuits directly or optimize over the corresponding orbits. Likewise, the gain, noise, and stability functions are viewed as either functions on the matching circuits or functions on the associated orbits. Both the matching circuits and the or bits are Hoo functions. Darlington's Theorem specifies when these orbits are dense in the unit ball of the real disk algebra, which is a strict subset of Hoo. Nehari's Theorem solves the optimization problems on HOO. The challenge is to link the computable Nehari bound to the optimization problem on the orbits. Chapter 6 presents the multiobjective optimization of several amplifiers. These examples acquaint us with the amplifier data and the numerical multi objective optimizers. As such, these examples are the "raw material" for the subsequent analysis. The Amplifier Matching Problem then generalizes to an Hoo multiobjective optimization problem that leads to the Stable Amplifier Conjecture and the Hoo Multidisk Method. Chapter 7 presents the Hoo Multidisk Method, which is a direct gener alization of the single-frequency disk method found in every amplifier text [82], [106], [140]. The following example of this single-frequency disk method illustrates the gain and noise tradeoff's that an amplifier designer typically encounters. For this particular design, the gain GT and noise figure Fare functions of the input matching circuit only. At a single frequency, the input matching circuit corresponds to a point SG in the unit disk of the complex plane: ISGI :::; 1. Consequently, the gain GT(SG) and the noise figure F(SG) are functions on the unit disk. A key result is that the level sets of the gain and x Preface noise figure are circles. That is, the set of complex numbers where the gain is constant is a circle in the unit disk. Likewise, the noise figure is constant on circles in the unit disk. Now an amplifier designer might want the gain to exceed 8 dB and the noise figure to fall below 2 dB. These inequalities result in the gain and noise disks plotted in Figure 0.2. The real and imaginary axes 0.8 ~6 ~4 ~2 F(SG)<2dB (i~ SG » 8 dB ~ ~2 -OA ~~ ~.8 -I -I -0.5 o 0.5 9{ Fig. 0.2. Gain and noise disks in the unit disk at 4 GHz. are labeled "~" and "~," respectively. Each point in the gain disk corresponds to a input matching circuit that delivers gain in excess of 8 dB. Each point in the noise disk corresponds to a input matching circuit that keeps the noise below 2 dB. Because the gain and noise disks intersect, the constraints are feasible and an input matching circuit may be extracted from the intersection. If the disks do not intersect, no matching circuit exists and the constraints are not feasible. In this case, the amplifier designer must relax the constraints and plot anew. However, these gain and noise performance bounds hold only at a single frequency. The Hoo Multidisk Method follows this engineering approach for wide band amplifier design. The designer specifies wideband gain, noise, and stability Preface xi constraints. At each frequency, the corresponding gain, noise, and stability disks are computed. As a functions of frequency, these disks turn into gain, noise, and stability tubes. Figure 0.3 presents such a gain tube. The vertical slices contain a gain disk similar to Figure 0.2. As the frequency sweeps from o to 20 GHz, the gain disks sweep out the gain tube. The Amplifier Matching Problem admits a solution if and only if the intersection of the gain, noise, and stability tubes with Hoo is nonempty. NIl32A8ofA: O.~dB <l'oCiain1\lbe .... r ....... o.~ ...... .. ~ n 0 ....... j.. ...•••..... .o.s ·1 1 XI ·1 0 ft O~ Fig. 0.3. Gain tube. So the first issue the amplifier designer must settle is whether the gain, noise, and stability tubes have a nonempty intersection at all. If so, the con straints are said to be LOO feasible. If the constraints are LOO feasible, the second issue to settle is whether a matching circuit lies in this intersection. Because the matching circuits are Hoo functions, the problem to determine if the nonempty intersection of the gain, noise, and stability tubes also intersect Hoo. If so, the constraints are said to be HOO feasible. Continuity then lets us assert that lumped matching circuits exist that can force the amplifier to meet the design constraints to arbitrary precision. Thus, the Hoo Multidisk Method computes the best possible gain, noise, and stability trade-offs for the amplifier circuit of Figure 0.1. The problem is that the circuit topology