H ∞ Engineering and Amplifier Optimization

271 Pages·2004·10.53 MB·English

by  Jeffery C. Allen

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
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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