Table Of Content__ eor o
LINEAR
PHYSICAL
SYSTEMS
Theory of physical systems from the viewpoint of
classical dynamics, including Fourier methods
eor o
PHYSICAL
SYSTEMS
Theory of physical systems from the viewpoint of
classical dynamics, including Fourier methods
Ernst A. Guillemin
Massachusetts Institute of Technology
Dover Publications, Inc.
Mineola, New York
Bibliographical Note
This Dover edition, first published in 2013 is an unabridged
republication of the work originally published by John Wiley &
Sons, Inc., New York, in 1963.
International Standard Book Number
ISBN-I 3: 978-0-486-49774-7
ISBN-JO: 0-486-49774-7
Manufactured in the United States by Courier Corporation
49774701
www.doverpublications.com
To Mary Grace and Russ
Preface
This deals mainly with topics that lie between those treaded in
Introductory Circuits Theory and those in Synthesis ofP assive Networks.
These are copies in linear system theory with restriction to physical
systems but not confined to passive or bilateral systems.
Briefly, some of these topics have to do with the topological, algebraic
and matrix aspects of setting up equilibrium equations involving, besides
the usual two-terminal passive R's, l 's and C's, elements which are
symbolically represented by boxes with three or more terminals projecting
incidentally, the embedding of such elements tells us how to deal with
networks that have been tom apart into smaller sections and hence include
the analysis of networks by the method of tearing.
Matrix aspects touch upon synthesis as well as upon analysis questions,
and to establish the pertinent relationship between matrices and the related
impedance functions the concept of normal coordinate transformations in
essential. This is discussed not only for two-element-kind networks, where
its application has long been known, but for RLC networks as well.
Two different approaches are taken here: one depends upon expressing the
network equilibrium on a mixed-variable basis; the other upon applying a
suitable complex transformation directly to the matrices on loop or node
basis. The interesting aspect of the latter method is that elements of the
transformation matrix tum out to be the residues of the associated
impedances obtained by familiar methods of partial-fraction expansion.
The astounding result here is that (e xcept for a simply formable
normalization factor) the direction cosines of the normal coordinates are
the residues of the pertinent impedance functions.
vm
vm PREFACE
Another topic that belongs in this category is a construction of the
impulse response of a network in terms of its poles and zeroes in a form
that is usually gotten by Laplace methods, which shows that this result
does not require Laplace theory for its derivation.
This group of topics (expanding those in Introductory Circuit Theory)
is rounded out by discussing the synthesis problem starting with
parameters matrices for one-, two- and three-element-kind networks,
deriving the essential energy relations for the characterization of
impedances, and in terms of these deriving the properties of impedances
(for example, the positive-real property) and methods or testing a given
functions for these properties. These subjects are dealt with in a simplified
manner compared with that used in Synthesis of Passive Networks but
sufficiently detailed to be meaningful and useful.
The remaining part of the book (which is the larger part) is devoted to
Fourier and related theory, and its application to the solution of linear
systems. Most of this section of the book deals only with Fourier Theory;
the part dealing with its application to linear systems is restricted to a
single chapter (e xcept for the discussion of numerical methods of
evaluation).
Fourier theory is introduced as a method of creating a desired interference
pattern from steady sinusoids, by applying this theory first to periodic
functions and then extending the method to aperiodic ones.
The paramount issue here is the evaluation of error and its dependence
upon the spectral width, which result in the fact that one can make the
error arbitrarily small, although not zero, with a sufficiently large but still
finite spectral width. Through the simple use of Cauchy's integral law one
can then extend the procedure to constructing such interference patterns
with sinusoids involving frequencies: that is to say, through the use of
sinusoids having exponentially modulated amplitudes. This amount to
extending the Fourier integral to a double-ended Laplace transform, and
one thus recognizes that the Fourier transform is the Laplace transform
evaluate along }-axis. Here one must, of course, include impulse in the
spectral functions, which is a process that fits in with familiar time-domain
procedures and leads directly to the numerical evaluation of inverse
transforms by means of impulse-train methods.
PREFACE
IX
A topic that logically fits in at this point is the evaluation of time-and
frequency-domain errors, the mean-square type of error and the kind of
frequency-domain truncation causing it, the related Gibbs phenomenon,
the Cesaro and the other types of truncation or summation and their time
domain aspects.
All these thinks are presented before Fourier theory is applied to
network analysis. Thus, the limitations due to Fourier theory alone and
those that introduced by network properties are presented separately,
avoiding are confusion in the student's mind.
It is emphasized that in the use of Fourier methods for finding network
response everything in the analysis remains as it is in the familiar classic
differential-equations approach except that the excitation function is
represented as an interference pattern of sinusoidal (involving real or
complex frequencies). The solution still consists of a forced part
(particular integral) and a force-free part (complementary function).
However, if the real parts of corresponding complex frequencies whose
interferences pattern yields the desired excitation are longer or at least as
large as the largest of all the natural frequencies, then the forced part of the
solution dominates and the force-free parts is negligible. The solution is
formed from the particular integral alone. When the particular integral
does not dominate one must add another integral yielding the force-free
(transient) part. The use of Fourier methods is still possible now but no
longer simple. These matters are seldom if ever pointed out in other
textbooks on Fourier theory and its application.
Rather, in many or the modem textbooks, Fourier methods are dealt with
as a completely new approach which is based upon a set of postulates
having little if anything to do with classical dynamics. According to this
attitude the linear system has no natural frequencies, no normal
coordinates, not even an equilibrium equation. The total solution is given
by the forced part of the solution alone-according to dictum, not because
of any physical reason. In fact, one is supposed to use only mathematical
reasoning. There is a basic inconsistency between such an approach and
the one is based upon classical dynamics. This abstract point of view may
be acceptable, however, for certain types of application where the system
is a black box and has no normal physical properties.
PREFACE
X
Fourier or Laplace theory introduced in this way is useful for handling
certain problems, for instance, in the discussion of ideal bandpass filtering.
But let it not be implied that it is a general approach that this likewise
applicable to physical systems, for this is untrue. I believe that this
textbook is the first one to point out that such inconsistencies exist in the
teaching of Fourier theory. Let us give the mathematically abstract
approach a different name like "black-box" Fourier theory so that student
will be forewarned to expect discrepancies between this brand of Fourier
theory and the kind that is consistent with and based upon classical
dynamics, which has been the basic for all the circuit theory he has been
taught so far.
Next in the list or topic presented is that dealing with methods of
numerical evaluation which are based upon the impulse-train concept. The
sampling or functions, arbitrarily or uniformly, is closely related subject.
Important in this discussion is a related method of error evaluation and
error transformation from frequency to time domains. It is extremely
interesting to find here that this process is basically the same as the almost
forgotten C. L. Fortescue method of "symmetrical coordinates".
Finally, since the topic dealing with real-part sufficiency is closely
related to the exact and approximate methods of Fourier transformation it
seems logical to close with a discussion of Hilbert transforms. These lead
rather logically to Bode's resistance-integral theorem and its applications,
which are used to present practical application or illustrations.
In closing, I wish to express gratefulness to all my associates who,
through the years, have helped teach this subject and thereby have made
many new and useful contributions that are reflected in the present state of
this text material. I would like to single out for special mention Thomas G.
Stockham, Jr., who has contributed many interesting discussions and
useful points of view.
ERNEST A. GUILLEMIN
Wellesley Hills, Mass,
May 1965
Contents
I Solution of Algebraic Equations When Auxiliar Conditions Are
Specified
I
1. The Mixed Situation I
2. The Auxiliary Conditions are Linear Constraints 4
3. S mmetr Conditions 7
4. Abridgment 8
Problems-2
2
12
1. Direct Algebraic Approach 12
2. Introduction of Auxiliary Variables; The Topological
Approach 20
3. Introduction of Auxiliar Variables Throu h an Al ebraic
Approach 32
4. Parameter Matrices 39
5. Equilibrium Equations in Mixed Form 42
6. Accommodation of Multi-terminal E1ements 46
Problems 47
3 Calculation and Characterization of Response 54
1. Piecewise Anal sis of Networks 54
2. Determinant Evaluation in Terms of Link Sets or Tree Sets 55
3. Response Functions Directly Related to Branch Conductances 59
xi
