§§§
º
º
§§
- H º º



Analysis and Design
of
Engineering Systems

Class Notes for M.I.T. Course 2.751
by
Henry M. Paynter
Associate Professor of Mechanical Engineering
Massachusetts Institute of Technology
with the assistance of
Peter Briggs
The M.I.T. PreSS
Cambridge, Massachusetts
COPYRIGHT © 1960, 1961 BY
THE MASSACHUSETTS INSTITUTE OF TECHNOLOGY
All rights reserved
Library of Congress Catalog Card Number:
PRINTED IN THE UNITED STATES OF AMERICA


Original book
was bound
with text
OVCTSCW11.
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best capture
possible.










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TO
TO
TO
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~~~~
DEDICATION
BERN DIBNER in appreciation of his keen sense of scientific
history and magnificent standards of aesthetic quality;
GEORGE A. PHILBRICK in small token of more than a decade of
friendship and service as foil and mentor par excellence;
HAROLD A. WHEELER as yet another scholar and craftsman whose
seminal concepts were a prime catalyst for this effort.
PREFACE
These notes represent a partial evolution from a primeval jungle
of hand-duplicated class notes used over the last seven years in a highly
successful one-semester course for seniors and graduate students at the
Massachusetts Institute of Technology. More recently, a second semester
has been added; thus the reader should be forewarned that the rather abrupt
and preclimactic closure of these notes coincides with the end of the first
term (2.75l) syllabus.
The present complete course is intended to provide both a practical
and a theoretical background for engineers of all pedigree who find them-
selves dealing with complex systems which operate simultaneously in several
media and over a broad band of frequency.
The systems here of principal concern are real and material; as
such, they may be described in energetic terms. System behavior is so
viewed from the standpoint of energy continuity and power balance: a gen-
eralized Poynting vector is defined, valid both for continuous and for
reticular systems. The useful and significant subsystems are then class-
ified both with respect to the number of energy ports through which energy
is exchanged with the environment and also in terms of the particular intern-
all power transformations involved. Thereby the synthesis and design of
Systems involves a selection and interconnection of a set of standard multi-
port elements, appropriate to accomplish the required tasks.
It will be noted that both active and passive systems are treated;
thus automatic feedback control, together with power and signal amplifiers,
are assumed as natural means of providing such activation. Moreover, while
energy and power remain central throughout this treatment, signal flow in
real devices is considered consistently as a form of low-power-level com-
munication bond. Thus such recondite topics as channel-capacity, gain-band-
Width and indeterminacy, may be treated in a workmanlike fashion.
Electrical engineers should be particularly tolerant in criticism
of this approach to engineering systems; but lest they be tempted to condemn
this material as mere generalized (or perhaps, worse, debased) circuit theor
they should remark the central role played here by the philosophical concept
of relations, (as opposed to functions), by the logical concept of serial
order (as opposed to measure), and by the engineering concept of essential
multiports (as opposed to 1-port impedances). Such topics and others beside
while largely absent and perhaps superfluous in linear circuit theory, becam.
foundation stones upon which a general framework of practical theory for
engineering systems may be laid with confidence.
Certain of the material presented in these notes will also appear
in a forthcoming book Ergs and Bits: The Flow of Energy and Signals in
Engineering Systems to be published by the McGraw-Hill Book Company.
One should bear in mind that for the M.I.T. students this abbrevi-
ated text is richly supplemented with about eighty home and examination
problems, covering a broad range of applications, and serving to amplify
and to clarify many points only briefly mentioned here. Regrettably, it
has proved impractical to include the problem material herewithin.
The reader may thank the diligence of Mr. Peter Briggs for much of
the early part of the text; he put many hours into a valiant effort to cap-
ture on paper the flavor of previously undocumented lectures. Other partic-
ular thanks for comments and corrections are due to many former students
and present colleagues, and especially to Mr. David R. Waughan. Also ap-
preciated was the continuing advice and counsel of Prof. S. A. Coons,
Prof. J. L. Shearer, and Prof. T. B. Sheridan.
The manuscript is the work of Mrs. Amy Botelho, with the help of
Mrs. Addison Dahmen and Miss. Bonny Davis. The symbols and figures are
largely the effort of Miss. Alice Griffin of the M. I. T. Illustration
Service. Of course, all faults and blemishes must be laid to the author
who assures the reader that criticism will be respectfully welcomed.
Henry M. Paynter
Cambridge, Massachusetts
June, 1961
Part I.
Part II.
A.
B.
Part III.
A.
B.
Part IV.
A.
B.
C.
D.
E.
Part W.
A.
B.
C.
D.
E.
F.
G.
Part, WI.
A.
B.
C.
Part VII.
A.
B.
C.
TABLE OF CONTENTS
Introductory Remarks
Engineering Systems
Engineering and Pure Science
Specific Examples of Engineering Systems
Systems and Abstractions
The System Concept: Identity, Structure, and Properties
Description of a System
Reticulation
Variables and Parameters of Energetic Systems
Introduction and Historical Background
Description of Energy Transactions
The Continuity of Energy
Introduction and Historical Background
Properties of the Field
Generalized Continuity Equation
Continuity of Energy and the Generalized Poynting Vector
Continuity of Entropy
Energy Ports and Power Bonds
Introduction
Application of the Divergence Theorem
Reticulation of the Energy Influx
Reticulation of Energy Storage
Reticulation of Energy Dissipation
The Reticulated Equation of Energy Continuity
Power Bonds
Multiported Systems and Elements
Introduction , ºs
Multiports
Ideal Energy Junctions
Classes and Relations
Relations and Structure
The Concept of a Class
The Concept of a Relation
Part VIII.
Part IX.
A.
B.
C.
D.
E.
F.
Part X.
A.
B.
Part XI.
A.
B.
C.
D.
E.
Part XII.
A.
B.
C.
D.
E.
F.
G.
H.
Continuum Logic and Hyperpolyhedral Functions
Introduction
Classes
Order
Continuum Logic
Two-Valued or Binary Logic
Multivalued Logic (Post Logics)
Hyperpolyhedral Functions
The Steady-State of Energetic Systems
Tntroduction sº sº
The Static Case
The Stationary Case
Determination of the Steady-State
System Reticulation for Steady-State Behavior
Nonlinearity
Functional Transformations and Computing Functionals
Introduction
Computing Functionals
Diagrams and the Coding of System Structure
Signs
Communication of a Computing Structure
Combinatorial Topology - Incidence Matrix
Coded Representation of Graphs and Digraphs
Coding the Energetic Structure of Multiport Systems
State-Determined Systems
Introduction
Elements of a State-Determined System
The Mathematical Construct of a State-Determined System
The Wariables of State in Generalized Dynamics
The Tetrahedron of State
The Characteristic Static Relations
The Three State-Determined Elements (R, C, I)
The Concept of Circuits and Networks
77
77
77
8||
87
91
92
1 OO
1 OO
1 OO
1 O1
102
103
105
1 O8
1 O8
1 O8
113
113
11 lº
117
123
125
13O
130
132
133
135
136
137
1 l;3
1 lº&
Part XIV.
A.
B.
D.
E.
Part, XW.
A.
B.
C.
Part XVI.
A.
C.
D.
E.
F.
Part XVII.
B.
C.
D.
E.
Distribution of Energy Over Space, Time, and Frequency
Introduction
Energy Principles for State-Determined Systems
Fields, Potentials, and Transmittances
Field Form Factors
Rectangle Diagrams
The Temporal Response of Physical Systems
System Response in the Time and Frequency Domains
Linear System Response in Terms of Potential Functions
One-Port Elements and the Impedance Concept
The Flow of Power and Energy in Systems
Two-Port Elements and Energy Transport Processes
Generalized. Two-Port Elements
Primitive Energy Transport Processes
Linear Two-Port Elements
Some Standard Forms of Two-Port Nets
Description of Linear Two-Ports
Transformers and Transducers
The Concept of IdeaTTwo-Fort Elements
Energy Transformation Elements
Energy Transduction Elements
Energy Transmission Elements
The Two-Port Element: • TM*
The Canonical Transmitter Matrix
Generalized Transmitters and Wavelike Transmitters
Ideal Wavelike Transmitters
Modeling Diffusive Transmission
The Dynamics of Monotone Processes
Energy Modulation and Amplification
General Three–Fort Elements
Generalized Power Modulators as Three-Port Elements
Generalized Amplifiers
The Trinode as a Three-Port Element
Cascading and Feedback of Amplifiers
153
153
15l.
159
178
186
193
198
211
21 l;
219
223
223
225
227
233
235
21:0
2k1
250
256
256
259
267
277
28O
287
287
288
293
299
Analysis and Design
Of
Engineering Systems
Part I 2.751 CLASS NOTES 1
I. Introductory Remarks
A. Engineering Systems
Characteristics and Classifications
At the outset it may in general be stated that an engineering
system is conceived, designed, and constructed to perform a specific
task. The content of this course will be concerned with material or
physical systems--machines, structures, instruments--which are to be dis-
tinguished from the more abstract, nonphysical systems such as economic
or social complexes. However, this latter type is no less real than the
concrete, physical system; indeed, it is conceivable that the nation's
economy might be modeled by delineating the dynamic interaction among
elements analogous to the inertial, dissipative, and elastic elements of
physical systems.
The following are a few examples of material engineering sys-
tems of the type we wish to consider:
1. Services and utilities--water supply, electric power
generation, communication;
2. Structures--buildings, houses, bridges;
3. Instruments --clocks, computers;
li. Vehicles--submarines, aircraft, spacecraft, ships,
automobiles%.
There are, of course, transcendental systems whose boundaries encompass
two or more of the above. A large missile, for example, is necessarily
a complex structural system as well as a vehicle, and in addition, it re-
quires electronic computing elements to effect its guidance and control.
By the same token, it is often the case that one of the above types may
be viewed as an integration of well-defined and physically distinct sub-
systems; in many vehicles the power plant may be thought of as a propul-
Sion system, conceived apart from the system as a whole.
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * = an an im, sº ºne sºme sm sm; me as sºme me sºme amº an as ºn as as am ºn as ame me ºn as me as as as as as as sºme
* The latter two are examples of interfacial vehicles.
2 2.751 CLASS NOTES
Engineering systems are historically improved and proliferated.
Improvements may occur as a result of lighter, faster, more compact, or
more reliable components being substituted for older ones; or as a re-
sult of a reconception, redesign, or rearrangement of configuration
which utilizes the same or similar components but yields a more per-
fectly integrated whole. There is, in addition, an all-pervasive trend
towards greater sophistication which demands a higher level of ability
among those who are responsible for the conception and design of new sys-
tems.
B. Engineering and Pure Science
Engineering generally predates science. Historically, inven-
tion, the art of the utilization of natural phenomena to realize certain
practical objectives, has preceded science which subjects the phenomena
to rational analysis and seeks a thorough understanding thereof. Indeed,
it has often been the case that an inventor, who has employed incorrect
reasoning and analysis, has succeeded in contriving a workable system!
The conception and design of the first airplanes was certainly not
founded on a thorough understanding of aerodynamic lift. In fact, lift
was not fully understood until long after the first successful flight.
To the contrary, some early mathematical analyses predicted that air-
planes could not fly!
This perhaps dramatizes the fact that traditional modes of
analysis often fail to account for those crucial factors which limit the
performance of a system, or even those very agencies which enable the
system to operate at all. As mentioned, the first analyses of flight
overlooked the true nature of the fluid circulation which results in a
lift force or sidethrust. More generally, it is the failure to properly
and completely account for the flow of matter, energy, information, en-
tropy, etc. which is the downfall of most classical analyses. Biological
systems are elusive for this very reason. The well-heralded "second
industrial revolution" could conceivably occur once these flow phenomena
are more perfectly understood.

Analysis and Design
of
Engineering Systems

Class Notes for M.I.T. Course 2.751
by
Henry M. Paynter
Associate Professor of Mechanical Engineering
Massachusetts Institute of Technology
with the assistance of
Peter Briggs
The M.I.T. PreSS
Cambridge, Massachusetts
COPYRIGHT © 1960, 1961 BY
THE MASSACHUSETTS INSTITUTE OF TECHNOLOGY
All rights reserved
Library of Congress Catalog Card Number:
PRINTED IN THE UNITED STATES OF AMERICA
TO
TO
TO
DEDICATION
BERN DIBNER in appreciation of his keen sense of scientific
history and magnificent standards of aesthetic quality;
GEORGE A. PHILBRICK in small token of more than a decade of
friendship and service as foil and mentor par excellence;
HAROLD A. WHEELER as yet another scholar and craftsman whose
seminal concepts were a prime catalyst for this effort.
PREFACE
These notes represent a partial evolution from a primeval jungle
of hand-duplicated class notes used over the last seven years in a highly
successful one-semester course for seniors and graduate students at the
Massachusetts Institute of Technology. More recently, a second semester
has been added; thus the reader should be forewarned that the rather abrupt
and preclimactic closure of these notes coincides with the end of the first
term (2.75l) syllabus.
The present complete course is intended to provide both a practical
and a theoretical background for engineers of all pedigree who find them-
selves dealing with complex systems which operate simultaneously in several
media and over a broad band of frequency.
The systems here of principal concern are real and material; as
such, they may be described in energetic terms. System behavior is so
viewed from the standpoint of energy continuity and power balance: a gen-
eralized Poynting vector is defined, valid both for continuous and for
reticular systems. The useful and significant subsystems are then class-
ified both with respect to the number of energy ports through which energy
is exchanged with the environment and also in terms of the particular intern-
al power transformations involved. Thereby the synthesis and design of
systems involves a selection and interconnection of a set of standard multi-
port elements, appropriate to accomplish the required tasks.
It will be noted that both active and passive systems are treated;
thus automatic feedback control, together with power and signal amplifiers,
are assumed as natural means of providing such activation. Moreover, while
energy and power remain central throughout this treatment, signal flow in
real devices is considered consistently as a form of low-power-level com-
munication bond. Thus such recondite topics as channel-capacity, gain-band-
Width and indeterminacy, may be treated in a workmanlike fashion.
Electrical engineers should be particularly tolerant in criticism
of this approach to engineering systems; but lest they be tempted to condemn
this material as mere generalized (or perhaps, worse, debased) circuit theory
they should remark the central role played here by the philosophical concept
of relations, (as opposed to functions), by the logical concept of serial
order (as opposed to measure), and by the engineering concept of essential
multiports (as opposed to 1-port impedances). Such topics and others besides
while largely absent and perhaps superfluous in linear circuit theory, becams
foundation stones upon which a general framework of practical theory for
engineering systems may be laid with confidence.
Certain of the material presented in these notes will also appear
in a forthcoming book Ergs and Bits: The Flow of Energy and Signals in
Engineering Systems to be published by the McGraw-Hill Book Company.
One should bear in mind that for the M.I.T. students this abbrevi-
ated text is richly supplemented with about eighty home and examination
problems, covering a broad range of applications, and serving to amplify
and to clarify many points only briefly mentioned here. Regrettably, it
has proved impractical to include the problem material herewithin.
The reader may thank the diligence of Mr. Peter Briggs for much of
the early part of the text; he put many hours into a valiant effort to cap-
ture on paper the flavor of previously undocumented lectures. Other partic-
ular thanks for comments and corrections are due to many former students
and present colleagues, and especially to Mr. David R. Waughan. Also ap-
preciated was the continuing advice and counsel of Prof. S. A. Coons,
Prof. J. L. Shearer, and Prof. T. B. Sheridan.
The manuscript is the work of Mrs. Amy Botelho, with the help of
Mrs. Addison Dahmen and Miss. Bonny Davis. The symbols and figures are
largely the effort of Miss. Alice Griffin of the M. I. T. Illustration
Service. Of course, all faults and blemishes must be laid to the author
who assures the reader that criticism will be respectfully welcomed.
Henry M. Paynter
Cambridge, Massachusetts
June, 1961
Part I.
A.
B.
C.
D.
Part II.
A.
B.
Part III.
A.
B.
Part IV.
A.
B.
C.
D.
E.
Part W.
A.
B.
C.
D.
E.
F.
G.
Part, WI.
A.
B.
C.
Part VII.
A.
B.
C.
TABLE OF CONTENTS
Introductory Remarks
Engineering Systems
Engineering and Pure Science
Specific Examples of Engineering Systems
Systems and Abstractions
The System Concept: Identity, Structure, and Properties
Description of a System
Reticulation
Variables and Parameters of Energetic Systems
Introduction and Historical Background
Description of Energy Transactions
The Continuity of Energy
Introduction and Historical Background
Properties of the Field
Generalized Continuity Equation
Continuity of Energy and the Generalized Poynting Vector
Continuity of Entropy
Energy Ports and Power Bonds
Introduction
Application of the Divergence Theorem
Reticulation of the Energy Influx
Reticulation of Energy Storage
Reticulation of Energy Dissipation
The Reticulated Equation of Energy Continuity
Power Bonds
Multiported Systems and Elements
Introduction Atº
Multiports
Ideal Energy Junctions
Classes and Relations
Relations and Structure
The Concept of a Class
The Concept of a Relation
Part VIII.
Part X.
A.
B.
Part XI.
A.
B.
C.
D.
E.
Part XII.
A.
B.
C.
D.
E.
F.
G.
H.
Continuum Logic and Hyperpolyhedral Functions
Introduction
Classes
Order
Continuum Logic
Two-Walued or Binary Logic
Multivalued Logic (Post Logics)
Hyperpolyhedral Functions
The Steady-State of Energetic Systems
Introduction
The Static Case
The Stationary Case
Determination of the Steady-State
System Reticulation for Steady-State Behavior
Nonlinearity
Functional Transformations and Computing Functionals
Introduction
Computing Functionals
Diagrams and the Coding of System Structure
Signs
Communication of a Computing Structure
Combinatorial Topology - Incidence Matrix
Coded Representation of Graphs and Digraphs
Coding the Energetic Structure of Multiport Systems
State-Determined Systems
Introduction
Elements of a State-Determined System
The Mathematical Construct of a State-Determined System
The Wariables of State in Generalized Dynamics
The Tetrahedron of State
The Characteristic Static Relations
The Three State-Determined Elements (R, C, I)
The Concept of Circuits and Networks
77
77
77
8||
87
91
92
1 OO
1 OO
1 OO
1 O1
102
103
105
1O8
1 O8
1O8
113
113
11 ||
117
123
125
130
130
132
133
135
136
137
1 lº
1 l;8
Part XV.
Distribution of Energy Over Space, Time, and Frequency
Introduction
Energy Principles for State-Determined Systems
Fields, Potentials, and Transmittances
Field Form Factors
Rectangle Diagrams
The Temporal Response of Physical Systems
System Response in the Time and Frequency Domains
Linear System Response in Terms of Potential Functions
One-Port Elements and the Impedance Concept
The Flow of Power and Energy in Systems
Two-Port Elements and Energy Transport Processes
Generalized. Two-Port Elements
Primitive Energy Transport Processes
Linear Two-Port Elements
Some Standard Forms of Two-Port Nets
Description of Linear Two-Ports
Transformers and Transducers
The Concept of IdeaTTwo-Fort Elements
Energy Transformation Elements
Energy Transduction Elements
Energy Transmission Elements
The Two-Port Element: • TM*
The Canonical Transmitter Matrix
Generalized Transmitters and Wavelike Transmitters
Ideal Wavelike Transmitters
Modeling Diffusive Transmission
The Dynamics of Monotone Processes
Energy Modulation and Amplification
General Three-Port Elements
Generalized Power Modulators as Three-Port Elements
Generalized Amplifiers
The Trinode as a Three-Port Element,
Cascading and Feedback of Amplifiers
153
153
15k
159
178
186
193
198
211
21 l;
219
223
223
225
233
235
21:0
2k1
250
256
256
259
267
277
28O
287
287
288
293
299
Analysis and Design
Of
Engineering Systems
Part I 2.751 CLASS NOTES 1
I. Introductory Remarks
A. Engineering Systems
Characteristics and Classifications
At the outset it may in general be stated that an engineering
system is conceived, designed, and constructed to perform a specific
task. The content of this course will be concerned with material or
physical systems--machines, structures, instruments--which are to be dis-
tinguished from the more abstract, nonphysical systems such as economic
or social complexes. However, this latter type is no less real than the
concrete, physical system; indeed, it is conceivable that the nation's
economy might be modeled by delineating the dynamic interaction among
elements analogous to the inertial, dissipative, and elastic elements of
physical systems.
The following are a few examples of material engineering sys-
tems of the type we wish to consider:
1. Services and utilities--water supply, electric power
generation, communication;
2. Structures--buildings, houses, bridges;
3. Instruments --clocks, computers;
li. Vehicles--submarines, aircraft, spacecraft, ships,
automobiles%.
There are, of course, transcendental systems whose boundaries encompass
two or more of the above. A large missile, for example, is necessarily
a complex structural system as well as a vehicle, and in addition, it re-
quires electronic computing elements to effect its guidance and control.
By the same token, it is often the case that one of the above types may
be viewed as an integration of well-defined and physically distinct sub-
systems; in many vehicles the power plant may be thought of as a propul-
Sion system, conceived apart from the system as a whole.
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * = * * * * * * * * * * = ame as as as as as as me as
* The latter two are examples of interfacial vehicles.
2 2.751 CLASS NOTES
Engineering systems are historically improved and proliferated.
Improvements may occur as a result of lighter, faster, more compact, or
more reliable components being substituted for older ones; or as a re-
sult of a reconception, redesign, or rearrangement of configuration
which utilizes the same or similar components but yields a more per-
fectly integrated whole. There is, in addition, an all-pervasive trend
towards greater sophistication which demands a higher level of ability
among those who are responsible for the conception and design of new sys-
tems.
B. Engineering and Pure Science
Engineering generally predates science. Historically, inven-
tion, the art of the utilization of natural phenomena to realize certain
practical objectives, has preceded science which subjects the phenomena
to rational analysis and seeks a thorough understanding thereof. Indeed,
it has often been the case that an inventor, who has employed incorrect
reasoning and analysis, has succeeded in contriving a workable system!
The conception and design of the first airplanes was certainly not
founded on a thorough understanding of aerodynamic lift. In fact, lift
was not fully understood until long after the first successful flight.
To the contrary, some early mathematical analyses predicted that air-
planes could not fly!
This perhaps dramatizes the fact that traditional modes of
analysis often fail to account for those crucial factors which limit the
performance of a system, or even those very agencies which enable the
system to operate at all. As mentioned, the first analyses of flight
overlooked the true nature of the fluid circulation which results in a
lift force or sidethrust. More generally, it is the failure to properly
and completely account for the flow of matter, energy, information, en-
tropy, etc. which is the downfall of most classical analyses. Biological
systems are elusive for this very reason. The well-heralded "second
industrial revolution" could conceivably occur once these flow phenomena
are more perfectly understood.
Part, I 2.751 CLASS NOTES 3
C. Specific Examples of Engineering Systems
So as to clarify what is meant by the "analysis and design of
engineering systems" and to point out the objectives toward which such
analysis will be directed, two examples will now be considered.
Men have traditionally relied on the celestial clock as the
standard time keeper; however, within the past few years we have been
able to manufacture clocks which are sufficiently precise to enable a
measurement of the inaccuracies and variabilities of celestial time as
observed from the noisy and unstable platform of the earth's surface.
Thus, the ultimate time keeper will undoubtedly be an engineering system,
i.e., an instrument whose sole purpose is to indicate a running time co-
Ordinate. -
Let us consider the common pendulum clock, reducing it to its
basic elements and identifying the variables with which the interactions
among the elements may be delineated. It is worthwhile to point out
that by removing the veil of material embodiment given to a particular
type of clock--the pendulum clock--it will become evident that clocks in
general consist of three basic elements: (i) a source of energy; (ii) a
gate or regulator of the flow of energy from this source; (iii) an indi-
cator to "read out" or display the desired running time coordinate. The
concept "clock" is thus reduced to a schematic diagram showing the flow
of energy through its essential elements. Such a schematic simplifies
the analysis of the system and facilitates the incorporation of improved
elements into the structure as these become available.
Now, the pendulum, by itself, is a nonlinear damped oscillating
System; in particular, its period depends upon the amplitude. Thus by
merely initiating an oscillation and counting subsequent decrementing
SWings we cannot hope to achieve the type of time keeper we desire--one
that continually reads out a running time coordinate--due to the fact
that the free pendulum is a transient device inherently nonlinear and of
"irregular periodicity." Moreover, we are, of course, unable to resolve
the swings as the amplitude approaches zero. However, by providing a
2.751 CLASS NOTES

Time coordinate
display: indi-
cating pointer (I)
^:
./
§§ ###
*|†g
Mi úl º y
M2
Nº-LL’ êe
Energy flow regulator
or modulator: pendulum
and escapement (R)
Energy source:
falling Weight
(E)
Mi(t) Me(t)
ë, (t) ěa(t)
A PENDUIUM CIOCK
Figure 1.
Part I 2.751 CLASS NOTES 5
source of energy and a gate to regulate the in-flow of this energy, it
is possible to maintain the amplitude of the pendulum nearly uniform;
then with the incorporation of an indicating pointer intermittently
driven by the energy source under control of the oscillating pendulum a
true clock is achieved.
This flow of energy through the system is crucial. Thus, we
identify the variables MA(t), 31(t), and Me(t), 32(t)--the torques and
angular velocities at each of the two interfaces within the system. The
product Mé is the instantaneous power or time rate of energy flow.
Therefore
P1(t) = M1(t) - é1(t) = M1(t) • *1(t)
Pa(t) = Me(t) - 82(t) = Me(t) . oe/t)
Pi and Pe are thus the instantaneous rates of energy flow at the two
interfaces. Usually power is conceived as a time-and-space averaged in-
variant, but we shall see that such an interpretation leads to inconsist-
encies in any analysis predicated thereon.
Let us scrutinize the E-R interface; it is apparent that the
torque M1(t) is dependent upon the velocity, a 1(t). Thus we may regard
col(t) as an "input" to the energy source and M1(t) as an "output."
The roles are necessarily reversed from the viewpoint of the pendulum-
escapement. Thus, a more precise delineation of the interaction at this
interface might be sketched as follows:
R
-4-ol.(t)-6-
The ideas introduced in this example will be dealt with further
in the second example--an analysis of a typical vehicular propulsion sys-
tem.
* * * * * * * * * * * * * * * * * *
The purpose of a vehicle is to transfer itself and its cargo
from one point in space to another. The path along which the vehicle
may travel in executing a given mission is usually subject to many
6 2.751 CLASS NOTES
constraints; these arise, for example, as a result of the necessity to
maximize economy, avoid collision with other vehicles or stationary ob-
jects, etc. The generalized vehicle problem may be viewed as two compo-
nent problems, both of which involve the concept of control or regula-
tion. It is first necessary to control the attitude or orientation of
the vehicle; secondly, it is necessary to regulate the speed and path or
trajectory. The solution to the second problem is embodied in the ve-
hicular propulsion system. We shall now rather cursorily analyze the
propulsion system of a ship merely as a further illustration of the tech-
niques to be employed in this course.
Some basic assumptions must first be made. It will be supposed
that we are studying a ship whose hull shape and power plant have been
fixed. Presumably, these have been chosen so as to achieve optimum econ-
omy and performance in light of the service for which the ship is in-
tended. For example, let the power plant be an oil-fired steam boiler
With a direct turbo-drive.
A fundamental problem which confronts the propulsion system
analyst is this: how does the ship respond to a command to accelerate
from one speed to another? The duration and nature of the transient is,
needless to say, a function of the power plant and its associated con-
trols, the hull shape and surface preparation, and the state of the
fluid in which the ship is floating. Three possible transients are
sketched in Figure 2.
The propulsion system of the vessel may be represented sche-
matically as Figure 3—a. Isolating the elements of the system and de-
lineating the interactions among them by way of energy bonds is indi-
cated in Figure 3-b.
Again, it is well to point out that each of these energy bonds
refers to a local energy state in the system; for example, the bond be-
tween the screw and the reduction gear may be labeled (M,a) Where M is
the torque in the drive shaft and Co its angular velocity. Thus, the
power flowing to the screw is
Part I
2.751 CLASS NOTES
Velocity

I
• *
- -, * * Final
1--" ‘. :- Steady-State
Initial -->4.
—
* * * * * * * me m ms me
de ©
Steady-State •3,
-> Time
Figure 2
8 2.751 CLASS NOTES
Screw * Fuel in
Reduction
Gear
Condenser Feedwater Pump
Figure 3-a
--
{:}–{:}+(ºr) + (tº)—{*} {*}-
{ cona)— º
Figure 3-b

Part, I 2.751 CLASS NOTES 9
A bond is shown between the screw and the effective mass of
the ship (the mass of the ship plus the virtual mass of the water moving
with the ship); this bond is labeled F, W, so that the power flowing
from the screw is
P., H F v
Finally, the available energy imparted to the effective ship
mass is ultimately dissipated by way of an effective resistance which
accounts for the frictional resistance acting on the hull itself as well
as a virtual resistance resulting from eddy formation in the surrounding
fluid.
Consider the energy state on either side of the screw driven
at constant speed Co while the ship is moving at a constant velocity W.
The screw is a transducer or two-port device with P. = M - Co flowing in
and Po = F : V flowing out. Now, only a certain two of the four vari-
ables, F, W, M, Co, may be considered as independent or input variables,
While the remaining two are labeled dependent or outputs. Suppose, for
example, that we choose V, Co., as the independent variables. Then
there must exist two relations 1
F = *CW, Co)
M = {{V, co)
which are characteristics of the screw and describe completely its be-
havior. A causally directed signal flow graph of the screw might then
be sketched as follows:
Ship Screw
* * * * * * * * *º dº sº tº -l
I (Jet) &M Nº.
| W MM
| ! ~
F <–1 5 < ( ) `s Co
| Jot ) |
9F —l

lO 2,751 CIASS NOTES
Such a diagram both delineates the signal flow from and to the
screw and also states the constraints upon the variables. However, the
same information is embodied in the following simpler diagram:
}H (screw) —|{
This tells us that the two "flow" variables V, Co., are "inputs" to the
screw while the two "effort" variables, F, M, are "outputs" from the
screw and therefore "inputs" to the adjacent elements. The technique of
formulating and manipulating such bond diagrams will be a primary con-
cern of this course.
D. Systems and Abstractions
Some degree of abstraction is inevitable in the analysis of
any physical system. The second example considered above illustrates
this point nicely; in particular, we modeled the effect of the fluid
contiguous to the vessel in terms of a virtual mass and a virtual resist-
ance. In so doing the important interactions between the ship and the
fluid were isolated and shown as localized energy bonds in the concep-
tual schematic of the propulsion system.
The artful act of abstracting from the totality of interac-
tions between the elements of a physical system and the elements of its
environment, and from among the various parts of the system itself, only
those interactions which are relevant to the specific questions being
asked, and then expressing these mathematically, is certainly a crucial
step in the analysis of system behavior. Once this has been accomplished
we are no longer talking about the physical system which is the subject
of the analysis but rather about a conceived, abstract substitute system
or model which is embodied in the mathematical relationships connecting
its parts. Thus, an appreciation of the properties of abstract systems
is indeed a prerequisite to the incisive analysis of physical systems.
Part II 2.751 CLASS NOTES 11
II. The System Concept: Identity, Structure, and Properties
A. Description of a System
Although material systems are of primary interest in this
course, the inevitability of abstraction as a step in the analysis of a
material system was pointed out at the end of Part I. We have also al-
luded to systems which are purely abstract. Thus, in formulating an
approach to the description of a system, it behooves us to concentrate
on the properties and governing relationships possessed by all Systems --
both material and abstract. Once a sufficiently general systems theory
has been developed, the task of specializing to a particular type will
be correspondingly simpler.
The description of a system necessarily begins With the
identification of a universe (U)--or "universe of discourse" as it is
often called in logic --which is a domain or set of sufficient scope to
include all elements within the system, plus all exterior elements with
which the system may be interacting. The system (S) is then a well-
defined subset of U, the elements of S possessing properties and inter-
relationships which happen to be of particular interest to us. The com-
plementary set, U less S, we shall label the environment Et hence, E is
the set of elements which interact with the elements of S but are not in
S.
The rational process of endowing a system with structure we
call reticulation. The act of separating S from U, and thus defining
the interface between S and E, is the first step in this process. The
System is further reticulated by conceptually tearing it apart into its
essential elements. Since the structural attribute of a system which
interests us most is the functional connectedness of its elements, the

l2 2.751 CLASS NOTES
final step in the reticulation process is the sketching of the important
relations and bonds of interaction among the elements and between each
of the elements and the environment f
Further description of a properly reticulated system involves
a more careful delineation of the functional connectedness of its ele-
ments. For the purposes of our general treatment of material systems,
this may be accomplished by either of two techniques:
1. An energy bond may be conceived as an interaction;
associated with each bond are two variables, the first
pertaining to an effort and the second to a flow, their
product yielding the power or energy flow rate.
2. Alternatively, an interaction may be conceived as a
bilateral signal flow between two elements, thus
attributing a direction of causality to the inter-
action.
A reticulated energy bond diagram facilitates a general under-
standing of the functional connectedness of a material system and there-
fore might well be used as a tool in synthesis and preliminary analysis.
However, for the purposes of a more detailed analysis the noncausal
energy bond reticulation is usually not adequate and must be transformed
into the bilateral signal flow reticulation. The description of the

Part II 2.751 CLASS NOTES 13
system is then completed by conceptually substituting for each element a
black box for which the input-output functional dependency is specified.
ºxxºnº
&
C޺
O
QXXXXXXXC ©
&
V ECTOR XXXXXXXXX V E CTO R
INPUT \ J OUTPUT
FUNCTIONAL
DEPENDENCY
B. Reticulation
The Latin form, reti, meaning (fish) net, is the stem from
which the Word reticulation is derived. The verb, to reticulate, means
literally to make into or like a network. Hence, for our use, reticula-
tion is a peculiarly vivid term to impart the idea of a conscious act of
structuring in the form of a network. How better can a system be char-
acterized than as a group of elements tenaciously bonded to one another
as are the meshpoints of a net?
We have said that reticulation begins with the partitioning of
the universe (U) into two subsets, the system (S) and the environment (E).
It should be emphasized that the environment can only be defined relative
to the system; that is, a conjugate relationship exists between E and S
Such that E is defined once U has been identified and those elements be-
longing to S have been marked off.
The system communicates with its environment across the E-S
interface. Unfortunately, traditional modes of analysis have often
placed constraints upon the forms in which such communication can occur.
For example, classical thermodynamics deals entirely with systems whose
boundaries are permeable only to matter and energy; thus, all living
processes, which continuously import negative entropy and information,
are perforce excluded from conventional thermodynamical analysis unless
the existence of a class of truly open systems is admitted.

1 l; 2.751 CIASS NOTES
The interface between E and S is inevitably a function of the
observer and, in particular, the questions he intends to ask concerning
the behavior of the system. Generally it is not established by the sur-
face of a material embodiment as might be anticipated in the case of a
simple physical system. For example, consider a straight-backed chair.
A practical analysis of a chair would certainly not be concerned with
the isolated object "chair;" instead, it would consider the interaction
of this object with the seat of a human occupant, with the floor on
which it rests, etc. Thus the demarcation of the system boundary is far
more sophisticated than might superficially appear.
Reticulation has been defined as the process by which a system
is endowed with structure. This process is facilitated if we view the
system as the integration of a number of subsystems. Thus, the elements
of a system are simply systems of lower order, and therefore interac-
tions occurring between two elements are of the same class as those
which may occur between two systems. Likewise, it is convenient to view
the environment as being structured to the extent that all S-E interac -
tions may be viewed as occurring between two systems --one an element of
S and the other of E. Hence, all possible interactions occurring within
interactions.
We have implied that a system is, in truth, a hierarchy of
subsystems. It is interesting to note, in the case of engineering or
man-made material systems, that there is a level of decomposition such
that all systems of a lower order are natural systems (See Hall and
Fagen). In some instances this level is rather low, a case in point
|being that of a thermonuclear device employing tritium as the fusable
material. Since the "element" tritium is itself man-made, the reticula-
tion would have to be carried all the way to the nuclear particles--in
particular, a proton and two neutrons. The tritium nucleus is certainly
a system in every respect; it may be reticulated into three elements
With energy bonds linking these elements.
Part II 2.751 CLASS NOTES 15
e’
--&
Although we lack sufficient knowledge to further reticulate a "proton"
or a "neutron," there is no a priori reason for denying that such a re-
ticulation could be performed. Hence, we can only say that a further
reticulation makes little sense in the context of this course, since at
this base level the fundamental properties of the elementary constituents
must be assumed.

16 2.751 CLASS NOTES
Background Reading
1. General Systems. Theory Yearbook, Vol. I. Introduction by
Ludwig von BERTAIANFFY.
The requirements of a "general systems theory" are
discussed. Also, profound insights into the nature
of open and closed systems are given (see excerpts
from pages 3-5).
2. KRON, Gabriel. Tensors for Circuits.
Read the two Introductions, one by HOFFMAN and the
other by KRON himself, for some appreciation of
KRON's method of "tearing" (diakoptics).
3. HALL and FAGEN. Definition of a System.
This reference imparts a general understanding of
the ideas dealt with in Part II. Sections 1–10,
11.1, and 13, are particularly valuable.
Part III 2.751 CLASS NOTES 17
III. Variables and Parameters of Energetic Systems
A. Introduction and Historical Background
At about the turn of the century the analysis of the macro-
scopic behavior of systems on an energetic basis had gained considerable
stature. However, the advent of quantum and relativistic mechanics de-
flated the theoretical structure to such a degree that the study of Ener-
getics per se lay dormant until the World War II era, at which time it
received renewed attention. The more recent treatments re-evaluated the
theory as it had previously stood and, for that matter, criticized the
well-entrenched theories of classical thermodynamics. Quoting from the
foreword of BRÖNSTED's Principles and Problems in Energetics:
Brønsted's Energetics is not to be confused with the
energetics associated with the names of Helm, Mach, and
Ostwald of the decade 1890-1900. The original proposals
failed for many reasons; at least in part in not recog-
nizing the necessity of the coupling of processes and in
some cases an incorrect assignment of the intensity and
capacity factors of the energy which still persist in text-
books published during the past year. Brønsted recognized
these pitfalls and has been most careful to avoid them.
The analysis of material systems on an energy basis will prove
most fruitful and illuminating in the context of the present course.
B. Description of Energy Transactions
In the diagram of a reticulated system, the presence of inter-
action bonds is schematically indicated. No attempt is made to qualify
or describe the form of functional connectedness; there is only the bare
Statement that such exists.
Energy bonding is a particular type of functional connected-
ness. The presence of an energy bond will be indicated schematically by
a heavy bar between the bonded systems.
Sl S2
More than one energy bond can link two given systems. Thus, in general
We show one bar for each form of bonding which We Wish to consider.
z-S
Sl S2
18 2.751 CLASS NOTES
The ever-present possibility of a multiplicity of bonding is a fundamen-
tal property of energy which we may call coextensivity or interpenetra-
|bility--many forms of energy may occupy the same region of space.
We have established that the analysis of a complex System in
its (complex) environment is reliably based upon a general treatment of
simple system-to-system interactions. In particular, we shall now focus
our attention on a class of material systems called energetic systems
and attempt to describe the energy transactions which can occur across
the boundaries of such systems.
1. Noncausal Description
Energy transactions across the E-S interface may be defined by
a pair of variables which together are a measure of the power or flow
rate of energy.
P
One of the variables is an extensive factor in that its magnitude is
dependent upon the extent of the portion of the system entering into the
transaction. The other variable is an intensive factor, being a func-
tion only of the field in which the system resides. If the two varia-
'bles are properly chosen, their product will yield the instantaneous
power exchanged. The factoring of power into two components is funda–
mental in mechanics (power = force : velocity) and thermodynamics
(power = pressure , rate of volume change). LOTKA relates that attempts
have been made in the analysis of social behavior to employ intensities
and extensities which combine to yield a quantity analogous to power,
although he admits the analogy is a rather loose one.
For our purposes it is convenient to think of the intensive
variable as an effort (e) and the extensive variable as a flow (f) so that

Part III 2.751 CLASS NOTES 19
Power = Effort, , Flow
IP -: e º f
We then view the system, with which we associate the over-all energy
state E, , as entering into energy transactions with its environment at
a number of localized regions on its boundary surface.
e, j *\ſ. */ſ.
N

/
/
\
f f © O © fe.
n en 5
Thus, in the case of the noncausal energy bond reticulation, the quanti-
ties e4 and fi ( i = 1, 2, 3, ..., n) are the external variables of the
system.
A reticulated energy bond diagram facilitates an over-all
grasp of the behavior of a system and an appreciation of the functional
connectedness of its elements. More specifically, it imparts an under-
standing of the transformation and flow of energy within the system and
assists in the isolation of the essential energy interactions with the
environment.
The manifestation of elusive environmental energy interactions
in the behavior of a system has traditionally been the stimulus for inno-
Vating discoveries. We are told that NEWTON "discovered" gravity as a
result of an apple falling on his head! HUNT gives an account of the
discovery of motional impedance by KENNELLY and PIERCE in 1912 which
dramatically underscores the importance of environmental interaction in
the study of acoustical phenomena.
2O 2,751 CLASS NOTES
2. Causal Description
If we wish to attribute a direction of causality to the inter-
action between two systems, it is imperative that we provide for bilateral
communication. Employing for the moment the terminology of the previous
section, suppose that a flow of energy occurs between two systems, S1
and Sa. Now, if we wish to endow this interaction with causality, we
might be prone to say, for example, that the flow occurs from S1 to Se
loy virtue of an effort supplied by Sl. However, this statement is only
half true since the effort which S1 must supply to achieve a given flow
is inevitably a function of the impedance of Se to that flow; hence, a
means of "communicating" the nature of this impedance back to S1 must be
provided.
A noncausal energy bond can be converted into a causal bond in
the form of a bilateral signal flow. That is
e
Sl S2
f
is equivalent to
X
Sl—- S2
y
where the signals x and y are so chosen that x . y = IP . Hence, for
the general system with n energy transactions occurring across its bound-
ing surface, X
2 U2
9,
9m Kn © O © 95 X5 94
We may now imagine a segregation of the n-inputs Xi from the n-outputs
Wi and a conceptual deformation of the system such that all the inputs
enter at one face while all the outputs leave from the opposite face.

Part III 2.751 CLASS NOTES 21
It is convenient to combine the Xi to form an input vector
Y = [x1, x2, ... xn], and the y, to form an output vector Y = [yl,
Y22 - - - *al. Now, for each input-output pair (x ) there corresponds
i’ Wi
a power component P + = Xiyi, so that the total power exchanged is
P = Xo Y
*iyi
-
#.
It is interesting to note, in this connection, that power is
an invariant under a coordinate transformation. In particular, let the
matrix of the transformation be T. Then, LE CORBEILLER demonstrates
that either X or Y Will be contravariant, i.e., it will transform like
the coordinates according to the matrix T; while the other will be co-
Variant, i.e., it will transform according to the transpose of T-l o
Suppose it is X that is covariant. Then the power is given by the
matrix multiplication
P =X Y.
where Y t denotes the transpose of Y. In the new coordinate frame
P=[T]x] TY-x T- TY=x Y=P
Finally, the system S is conceptually replaced by a module
which performs the operation on X which is required to yield up Y.
x → p Y

22 2.751 CLASS NOTES
The functional dependency Y Up (X) is of a most general
form such that the entire history of X is scanned to yield a present
value of Y. Therefore, it is applicable to the analysis of all proc-
esses in which the system S might be involved, and in particular to
transition processes from one steady-state condition to another.
For a purely steady-state analysis, a simpler static function
(p is applicable, yielding the present value of Y corresponding to
a present value of X.
Thus, we see that in the case of a bilateral signal flow reti-
culation the variables of the system are the vectors X and Y ; Or,
more precisely, their respective components.
Contained in the function'ſ are the intrinsic properties of
the system in the form of a set of parameters. In the most general case
the parameters may vary with time and with the environment of the system.
Thus we may display the scheme:
System cºnsteristics
T |
Wariables Parameters
| |
ſ | ſ |
X 8 Y (B 8. f Geometrical Material
Input (exogenous) Effort, Diameter, Elastic and inertial
Output (endogenous) Flow Length, etc. Properties, etc.
As an example, consider the propeller shaft in the turbo-drive
propulsion system discussed in Part I.
}*
º
|*|
º

Part III 2.751 CLASS NOTES 23
First the system is suitably isolated from its environment; it
is then conceptually transfigured into a module possessing the properties
of the shaft.

Power Shaft Power
<= =\ {= <+,
Output 7 Parameters Input
A practical analysis of a system, such as the propeller shaft
in the above example, is directed either toward a determination of be-
havior for a given design, or the design which will achieve a given be-
havior, in both cases the environment being taken as fixed. Thus, it is
sometimes convenient to view the parameters of the system also as "inputs,"
redrawing the module as shown below:
System S
Input | | Output
Variables | | Variables
|

A&
| |
| |
Z Parameters
Hence, problems of the first type are involved with finding Y as 8,
function of X for a given Z , while those of the second type involve
finding Zas a function of Y for a given X º
2l;
2.751 CLASS NOTES
Background Reading
(1)
(2)
(3)
(l)
(5)
(6)
BRÖNSTED, J. N. Principles and Problems in Energetics, Chapters I
and II.
Chapter I imparts some feeling for the relationship between
Energetics and the classical theories of thermodynamics.
Chapter II demonstrates the inadequacy of the classical work
concept and delineates more carefully the requirements of a
generalized Work principle.
LOTKA, A. J. Elements of Mathematical Biology, pp. 260-266: 303-305.
One is led to the belief that there are characteristics Of
social systems which are, at least in an abstract sense,
describable energetically.
DE GROOT, S. R. Thermodynamics of Irreversible Processes, pp. 1-9.
The author points out, in connection with the effort-flow
concept, that there are many examples of "cross-effects" --
coupling between two processes, such as in the thermo-electric
effect, wherein an effort of one form produces, in addition to
its corresponding flow, a flow of another form. Thus, one's
attention is directed towards the problem of describing such a
coupling by way of energy bonds. A presentation of ONSAGER's
theory provides some understanding as to how the problem might
be handled.
HUNT, Frederick W. Electro-Acoustics, p. 96.
See particularly the interesting account of the discovery of
motional impedance by KENNELLY and PIERCE.
TRIMMER, J. D. Response of Physical Systems, pp.99-103.
The ideas of conjugate variables and impedances are briefly
introduced.
ZWTKKER, C. Physical Properties of Solid Materials, pp. 72-73.
An appreciation is imparted of the parametric description of
a system in the form of a set of compliances or rigidities,
for the Static case, and conductances or resistances in the
stationary case.
Part III 2.751 CLASS NOTES 25
(7) IE CORBETILER, P. Matrix Analysis of Electrical Networks, pp.59-61.
The invariance of power under a coordinate transformation is
demonstrated for a system residing in a potential force field.
The distinction between eontravariant quantities and covariant
quantities is made.
26 2.751 CLASS NOTES
TV. The Continuity of Energy
A. Introduction and Historical Background
Isaac NEWTON's view of the physical universe was that of a
system of mass points whose interactions and over-all behavior were de-
scribable in terms of certain basic laws of mechanics, these laws being
interpreted for a specific particle as the total differential equations
of motion. NEWTON and his followers attempted to extend this explanation
of reality to encompass all forms of interaction. However, their mecha-
nistic explanation of light, being, as it were, founded on particle in-
teraction, left much to be desired.
Man's appreciation of the universe was dramatically broadened
with the advent of James Clerk MAXWELL's use of field concepts for light
and electricity. Here was a theoretical structure of sufficient scope
to serve as a skeleton for the analysis of the electromagnetic radiation
phenomena which had eluded physicists of the Newtonian school. Although
he did not view fields specifically as "energy fields," MAXWELL was cer-
tainly aware of the energetic aspect of fields of all types.
Albert EINSTEIN sums up the effect which MAXWELL had on
physics in the following quotation:
Before Clerk Maxwell people conceived of physical reality--
in go far as it is supposed to represent events in nature--as
material points, whose changes consist exclusively of motions
which are subject to total differential equations. After
Maxwell they conceived physical reality as represented by con-
tinuous fields, not mechanically explicable, which are subject
to partial differential equations. This change in the concep-
tion of reality is the most profound and fruitful one that has
come to physics since Newton; but it has at the same time to
be admitted that the program has by no means been completely
carried out yet. The successful systems of physics which have
been evolved since rather represent compromises between these
two schemes, which for that very reason bear a provisional,
logically incomplete character, although they may have achieved
great advances in certain particulars.
John Henry POYNTING extended MAXWELL's formulation to the
consideration of energy transport in an electrical network. The scope
of POYNTING's analysis is best described in his own words:
A space containing electric currents may be regarded as
a field where energy is transformed at certain points into
Part IV 2.751 CLASS NOTES 27
the electric and magnetic kinds by means of batteries, dynamos,
thermoelectric actions, and so on, while in other parts of the
field this energy is again transformed into heat, work done by
electromagnetic forces, or any form of energy yielded by cur-
rents. Formerly a current was regarded as something travelling
along a conductor, attention being chiefly directed to the con-
ductor, and the energy which appeared at any part of the cir-
cuit, if considered at all, was supposed to be conveyed thither
through the conductor by the current. But the existence of
induced currents and of electromagnetic actions at a distance
from a primary circuit from which they draw their energy, has
led us, under the guidance of FARADAY and MAXWELL, to look
upon the medium surrounding the conductor as playing a very
important part in the development of the phenomena. If we
believe in the continuity of the motion of energy, that is, if
we believe that when it disappears at one point and reappears
at another it must have passed through the intervening space,
we are forced to conclude that the surrounding medium contains
at least a part of the energy, and that it is capable of trans-
ferring it from point to point.
On the basis of POYNTING's formulation, Charles Proteus
STEINMETZ developed a regimented theory for the practical analysis and
design of circuits and laid the foundations for electrical engineering.
Oliver HEAVISIDE continued the thread of POYNTING's work to a general-
ized statement of energy continuity, while Sir Oliver LODGE discussed
the identity and conservation of both matter and energy. Thus, the con-
cept of energy-matter conservation in space as well as time became
firmly implanted in the structure of the theory of Energetics.
Maxwell Early
|
Poynting Genesis of the
Energy Concept from
Maxwell
| | W
Steinmetz Heaviside Lodge
Practical Continuity Identity
Electrical Of Of
Engineering Energy Energy

In parallel with the line of development sketched above was
that of the classical theory of thermodynamics, which generated an energy
Concept of a rather divergent nature. From the fundamental experiments
28 2.751 CIASS NOTES
of Julius Robert MAYER, James Prescott JOULE, and William Thomson
( Lord KELVIN ) there emerged the principles of energy conservation,
energy transformation, and the equivalence of heat and work--ideas upon
which LODGE drew heavily. However, as has been previously stated, the
axiomatic structure of thermodynamics is applicable only to systems
whose boundaries are of limited permeability. Rudolf CLAUSIUS made
sweeping Statements concerning the energy and entropy of the entire uni-
verse, but only on the basis of the assumption that the universe could
be considered an isolated system: Regarding the conservation of energy,
HEAVISIDE had this to say:
The principle of the continuity of energy is a special
form of that of its conservation. In the ordinary understand-
ing of the conservation principle it is the integral amount of
energy that is conserved, and nothing is said about its dis-
tribution or its motion. This involves continuity of exist-
ence in time, but not necessarily in space also.
But if we can localise energy definitely in space, then we
are bound to ask how energy gets from place to place. If it
possessed continuity in time only, it might go out of exist-
ence at one place and come into existence simultaneously at
another. This is sufficient for its conservation. This view,
however, does not recommend itself. The alternative is to
assert continuity of existence in space also, and to enuncil-
ate the principle thus:--
When energy goes from place to place it traverses the
intermediate space.
This is so intelligible and practical a form of the
principle, that we should do our utmost to carry it out.
But one now has the right to inquire, as did MAXWELL, "If some-
thing is transmitted from one particle to another at a distance, what is
its condition after it has left the one particle and before it has
reached the other?" It was indeed necessary to endow the void with cer-
tain material properties in order to conceive of a transfer of energy
through it. For this purpose the ether was contrived--a substance to
permeate all space through which energy might pass. That MAXWELL was
unable to conceive of an energy transfer apart from an intervening me-
dium is clearly indicated by the following quotation:
In fact, whenever energy is transmitted from one body to
another in time there must be a medium or substance in which
Part IV 2.751 CLASS NOTES 29
the energy exists after it leaves one body and before it
reaches the other, for energy, as Torricelli remarked, "is
a quintessance of so subtle a nature that it cannot be con-
tained in any vessel except the inmost substance of mate-
rial things."
An ether was also essential in the eyes of POYNTING, LODGE, and HEAVI-
SIDE, for these men felt that the transformation of energy was always
accompanied by a transfer, and vice versa. This was the very essence of
the principles of continuity and identity which they propounded.
As man's understanding of the physical universe was extended
his conception of the ethereal substance was correspondingly altered. A
modern viewpoint is submitted by EINSTEIN by way of conclusion to his
essay, "Relativity and the Ether:"
We may sum up as follows: According to the general
theory of relativity space is endowed with physical quali-
ties; in this sense, therefore, an ether exists. In ac-
cordance with the general theory of relativity space with-
out an ether is inconceivable. For in such a space there
would not only be no propagation of light, but no possi-
bility of the existence of scales and clocks, and there-
fore no spatio-temporal distances in the physical sense.
But this ether must not be thought of as endowed with the
properties characteristic of ponderable media, as composed
of particles the motion of which can be followed; nor may
the concept of motion be applied to it.
For our present purposes the universe will be viewed as a
field--a heterogenous, anisotropic continuum, i.e., its properties are
Variable over space and polarized with respect to orientation. The
ether is then a conceptual artifice which we shall dispense with; indeed,
We shall take the even stronger position that energy may exist in and of
itself, requiring no material vessel in which to be stored or transported.
B. Properties of the Field
If we think of all the universe as constituted from a rela-
tively small number of basic particles (about thirty), then the only
difference between the air, a wall, a floor, a desk, etc., is the den-
sity and distribution of these particles. Then consider a system S
residing in a field F:
3O 2.751 CLASS NOTES
IF

Q&
QºS
* , (x,y,z,t, ...)
Č. (2), t, ...)
We describe the system in terms of variables and parameters which fall
into two categories: (i) Intensive quantities 4;(x,y,z,t, ...) Which
are characteristics of the field F; (ii) Extensive quantities
& 1.(?,t, ...) which depend also on the extent (2) of the system involved.
However, the existence of extensive quantities cannot be reconciled di-
rectly with the field description which we seek; it is thus convenient
to define specific extensities or field densities which are the densi-
ties of the extensive quantities & k. We define
e k = Jºo Čſ M = e k(x,y,z,t) . . . )
Thus, the e is are point quantities and are conformable to the field de-
scription. Perhaps the most familiar example of a specific extensity is
the mass density, p , where with mass, M, in a volume, ?/:
p = lim M/?/
T/—- O
C. Generalized Continuity Equation
The statement of the principle of mass conservation as a field
equation is familiar to us,
—º. div(pV) + (a|p/at) = 0
We know that div(p W) measures the convective transfer of mass densities
away from a point in the field while (9p/6t) is the time rate of
change in the local density. Setting the right-hand side equal to zero
assumes there are neither sources or sinks of matter in the field,
Part IV 2.751 CLASS NOTES 31
tantamount to common parlance "mass is neither created nor destroyed."
More generally, however, in the light of modern physics and mass-energy
equivalence, we might insert a source term or and write
div(p W) + (op/ 2 t) = a
Now, it is quite appropriate that we similarly demand the
continuity of any and all of the specific extensities é Therefore,
k"
we may state the generalized equation of continuity as follows:
div(eV) + (9e/ 9t) = a
where e V is the convective transport of € and W is the appropriate
transport velocity. Sometimes, in order to avoid the necessity of meas-
uring V it is convenient to define a € - flux vector, Ž, 8. S
3 = e V
so that We can Write:

div(?) + ( ae/a t) = a
D. Continuity of Energy and the Generalized Poynting Vector.
MAXWELL dealt specifically with the localized energies of a
field. For example, he attributed to an electrostatic field a distrib-
uted potential energy proportional to the product of the potential W
and the electrical displacement e. He likewise defined magnetic and
electrokinetic energies in terms of distributed quantities--energy den-
Sities, as it were.
The discussion of energy on a localized basis allows one to be
most specific, and, in general, uncompromising in the analysis of a par-
ticular system. Classical modes of analysis always require, as has been
repeatedly pointed out, a "cloaking" of the system with a quasi-impene-
trable veil, and generally demand that energy be integrated over the
domain of the system before any acceptable statements concerning its
conservation or transformation may be made. Thus, we might say that the
conception of energy as a localized or distributed quantity permits the
fullest possible exploitation of the available information concerning
the nature of the system, its bounding surface, and its conceivable reti-
culations. Hence, following HEAVISIDE's logic directly, we shall state
the equation of energy continuity for a field:
32 2.751 CLASS NOTES

- div F = (ae/at) + £2 a
where p is the generalized Poynting Vector, E is the local energy den-
sity, and ſp d is the rate of energy loss or dissipation. This statement
will be the foundation for our analyses of energetic systems.
E. Continuity of Entropy
It is appropriate at this point to introduce parenthetically
and without development a conjugate continuity equation for entropy.
- div S = (a gºat) + 42
Where 5 = entropy flux, % = entropy density, and 4 = entropy sink.
This equation is most important in that it permits the analysis of a sys-
tem which is exchanging entropy and information with its environment
rather than, or perhaps in addition to, energy. Biological systems are
notable examples of systems to which it is applicable. We shall discuss
the relation of this equation to energy continuity somewhat later on.
Part IV 2.751 CLASS NOTES 33
Background Reading
(1)
(2)
(l)
EINSTEIN, A. Essays in Science.
"Clerk Maxwell's Influence on the Evolution of the Idea of the
Physical Universe"
"The Problem of Space, Ether, and the Field of Physics"
"Relativity and the Ether"
This group of essays imparts a very broad appreciation of the
development of the physical concepts relevant to Part IV from
Newtonian mechanics, through MAXWELL's field theory, to the
modern theories of relativity and quantum mechanics.
POYNTING, J. H. On the Transfer of Energy in the Electromagnetic
Field, Philosophical Transactions of the Royal Society of
London for the Year 1881, Vol. 175, pp. 3]{3-361.
The manner in which energy is transported from point to point in a
space carrying electrical currents and is transformed at various
points by Way of batteries, motors, etc., is analyzed.
HEAVISIDE. O. Electro-magnetic. Theory, pp. 73-77.
The principle of energy continuity is stated by way of an extension
to POYNTING's conception of energy transport.
LODGE, O. On the Identity of Energy, The London, Edinburgh, and
Publin. Philosophical Magazine and Journal of Science, Vol. XIX,
January–June (1885) pp. 1182–187.
As another extension to POYNTING's energy transport concept, the
idea of the identification of energy, as it is transferred from one
point to another, is discussed. -
MAXWELL, J. C. Electricity and Magnetism, Vol. III, pp. 270-274; h93.
In the first reference MAXWELL develops expressions for the various
energies of a field. In the second he makes a concluding statement
concerning what he feels to be the nature of the ether.
31, 2.751 CLASS NOTES
(6) SOMMERFELD, A. Thermodynamics and Statistical Mechanics, pp. 152-153.
An equation for entropy continuity is derived from that for energy
continuity for the special case of a homogeneous, isotropic solid.
(7) PRIGOGINE, I. Thermodynamics of Irreversible Processes, pp. 32-34.
A general equation of entropy continuity is developed similar to
the one stated above.
Part W 2.751 CLASS NOTES 35
W. Energy Ports and Power. Bonds.
A. Introduction
We have come to view the universe as a field in which resides
the system S. We are now in a position to require that energy transac-
tions between S and E and within S be subject to the general equation of
energy continuity
. - 9 (£
div p +: 3t + ſ’d
The purpose of Part W is to reticulate the continuity equation
for application to the analysis of energetic systems. This amounts to
assuming: (1) that the S-E interface is permeable to the passage of
energy only over a relatively few areas of restricted extent--energy
ports as we shall call them; (2) that the energy storage function of the
system is not distributed continuously throughout its volume, but is
rather lumped in discrete localities or regions; (3) that the dissipa-
tive property of the system is also confined to discrete regions. Al-
though this is admittedly an idealization, it is a very practical and
necessary one. p
2 º
Discrete regions of energy
Storage
3
E Discrete power entry ports
Discrete regions of power dissipation
Reticulation of Energy Transactions,
Storages, and Dissipations

36 2.751 CLASS NOTES
B. Application of the Divergence Theorem
The Divergence Theorem will now be applied to the continuity
equation as stated above in order that it be converted to a form amenable
to reticulation. The Theorem states
J F. maa - Jaiv Faſº/
Cl 2P
where Q is the closed surface bounding the volume 2ſ and n is the unit
outward normal. This single theorem is perhaps the most profound in all
applied mathematics since from it every other theorem may be derived.
Substituting the generalized Poynting vector, p , for the
generic vector, F , and letting Z/ be the volume of S and & the area
of its bounding surface
J P. na a - Jay Fa ºr
CL
Performing the volumetric integration on all terms in the continuity
equation, and substituting from the above we thus obtain
-
| Fia ſº. £ºv
-
9 (£
# a 2ſ + ſpa 2-
3t d
The continuity equation now stands ready to be reticulated in the manner
We desire.
C. Reticulation of the Energy Influx
We consider first the convective transfer of energy across the
System boundary; the assumption is made that there are l areas (2. 1 over
Which the S-E interface is permeable to energy flow. It is further as-
sumed that each of the (2-4 is of sufficiently limited extent that jj
may be considered everywhere parallel to the associated unit normal ni,
and that nonuniformities in the magnitude of P may be absorbed by an
appropriate averaging Operation such that
J. P. Fa al - P, a ,
Part W 2.751 CLASS NOTES 37
Thus, the boundary of S is no longer viewed as a shapeless bag, but
rather a multifaceted surface, each facet corresponding to an energy
port.
Now, for each port the product p i Q. i is the power leaving
through that port, P 1. Hence, the total power flux through the
boundary of the system is
l l
2. P. a = X PA
i=1
D. Reticulation of Energy Storage
It would appear sensible to assume that energy storage is a
function of a relatively few (m) localized regions in S rather than
being uniformly distributed throughout its volume. Hence, when the inte-
gration
9° a 2-
º, at
is performed over the entire volume of S there will only be a discrete
number of regions º in which
9 @
~ + O
3 t #
Thus
IIl -
f24 av. - X. |ſ ge d ſly ,
3t j=1 2P. 3 t J
2- J

38 2.751 CLASS NOTES
Now, a reticulation of the energy storage in S will have mean-
ing only if the volumes 2, in which storage is supposed to occur are
fixed in size and disposition. Assuming this to be the case, we may
carry the time differentiation outside the integral and change it from
partial to total giving
96 . . m d
†: a* - : + ſea ºr,
2, 9t j=1 dt 2P -
j
But, we now have the right to define
E3 = J ea 2/3
2-3
as the instantaneous energy stored in each little storage region 22, *
Hence, we finally obtain
3 (£ Il a B.
— d ?/- = S —%
3 tº j=1 dt
2-
E. Reticulation of Energy Dissipation
Following the same logic, it is a plausible assumption that
energy dissipation will occur only in a discrete number (n) of restricted
regions º, , i.e., the resistive or dissipative property of S is lumped
in the regions 2 : rather than being distributed throughout 2- .
It is therefore evident that
Jr., & - ; J Pa *.
2~ * 2:
With the definition
! …
% Pad & k = (£a).
We finally obtain
Il
ſº - ? (4%).
Part W 2.751 CLASS NOTES 39
F. The Reticulated Equation of Energy Continuity
Combining the above results we may write the reticulated equa-
tion of energy continuity as follows:
l m dlf. Il
- X: P , = X → * > (£3),
i=1 j=1 dt k=1
This equation makes the irrefutable statement that a net flux
of energy into a system is either stored or dissipated, and on it we may
found the practical analysis of any energetic system. However, the iden-
tification and calculation of the P 1, the E 3, and the (ſ’), in
terms of the variables and parameters of an actual system is no trivial
task. A simple example will serve to reveal the first major pitfall
which must be avoided.
dashpot Spring

P —— H. Huº-T Iſla SB — P.
& = u + x
m = z – u
|D
|
—— dashpot — spring
~ energy
is dissipated
energy is stored
IP
2
Iº, SS

liO 2.751 CLASS NOTES
We intend to analyze a spring-mass-dashpot system in Which the
spring and dashpot are nonlinear, and we wish to account for the rela-
tivistic variation in the momentum of the mass. In particular, Suppose
we are given the following characteristic relations :
e,( ; )
e
S
ea = eaſ; )
-:
-:
Spring force
-:
Damping force
-:
Momentum of mass = p = p(2)
Since energy is stored in the mass and the spring, and is dissipated in
the dashpot, we may say with assurance
d
P. - P.-- (E. E. . (Pa
:
; : ſº.” + Jºãp + eač
In this case 0° a is self-evident; however, one might be prone to write
instead of IE, and IE, what we might label the complementary energies,
: – ſmae. E := ſpa,
IE: "de, ; IE * pdź
It is noteworthy that IE s ºf IE ; and IE Iºl
spring and constant mass. Thus, it is imperative that extreme caution
IE m only for a linear
=
be exercised in evaluating the "energies" of a system so that the inclu-
Sion of "incorrect" energy terms may be avoided. Needless to say, the
continuity equation is valid only if the energy terms are properly evalu-
ated.
A similar difficulty can arise in connection with the power
flows P,. Power is carried across the system boundary by transmission
links--shafts, ducts, electrical conductors, waveguides, etc. We have
previously stated that the power state of a transmission link may be in-
dicated by the product of an intensive variable (effort--e) and an exten-
sive variable (flow--f), which tacitly assumes that two such variables
may be identified for a given transmission link. For the description of
the power flow through a shaft, for example, we would be prone to pick
the torque as the effort and the angular velocity as the flow. But what
is the torque of a shaft? Indeed, a shaft possesses neither a single,
Part W 2.751 CLASS NOTES li
characteristic quantity "the torque," nor a single angular velocity
common to all its parts; Granted that we could integrate the moment of
the shear stress (r) over the cross-section of the shaft and get a
torque M = J r raaſr), and We could also calculate a mean angular
* /
A." (r) dA(r)
velocity w = , but who is to guarantee that the
dA (r
ſº
product M . J will yield the true power transmitted? Our only recourse
is to calculate (or measure) one of the variables, say the torque, and
then to assume an angular velocity a, (# ūj) such that M - (j does in
fact give the power. A course of action similar to this must be taken
in the case of any real (i.e., nonideal) transmission link.
We shall suppose that it will always be possible to find a
pair of effort and flow variables such that, for a given system,
P. = e, f, ; 1 = 1, 2, 3, ' ' ', 1
The energy coupling between a system and its environment is
often rather elusive. In particular, a single transmission link may
appear to be the medium of exchange for several, or perhaps even an in-
finite number of energies. For example, any small region of space is
energetically coupled to every other region by way of a spectrum of
electromagnetic radiations. It behooves us, therefore, to allot one
energy port to each transaction; hereafter, we shall speak of an energe-
tic interaction between two systems as a power bond. Thus, the power
bonding between a system and its environment, or between two systems Sl
and S2 is reticulated as sketched below into l separate bonds.
Sl LX S2
e-jºb-
Or, more simply, /~
2-->
l

l;2 2.751 CLASS NOTES
where one bond is allotted to each energetic interaction.
Once the power bonding between S1 and S2 has been fully reticu-
lated, and each bond has been described in terms of an effort X flow
product, we are in a position to define the vectors:
© it: ſel, €2, e3, . . . ; ell
f -: [fl, f2, f3, • * * * fil
Thus, the total power transacted is given by the matrix product
IP = e, , f
t Tel T [fl, f2, f2, ..., fil
€2
e3
G. Power Bonds
The schematic representation of a reticulated energetic system
which we have adopted shows the elements as linked by heavy bars, each
bar denoting a power bond. Thus, for a two-port system which is reticu-
lated into three multiported elements,
the Simple bond diagram would appear as follows
2">
dºº-ºº. Sl S2 -
We note immediately the similarity between such a diagram and
a chemical bond diagram. Indeed, the similarity is by no means superficial

Part W 2.751 CLASS NOTES l;3
the mechanism by which a chemical bond is created between two atoms is
closely analogous to the formation of a power bond linking two systems,
the atomic valency in the former case corresponding to the energetic
portality in the latter. Thus, we have the chemists to thank for some
of the essential ideas incorporated into Our Schematic representation.
In particular, KEKUIE and FRANKLAND were early proponents of the bond
diagram.
as ºne as ºn me an ame me tº ims ºn m = an am sº mº " ºr me = *- * *
Causality implies the existence of two variables, one inde-
pendent and the other dependent--such as in a mathematical relationship
y = f(x) wherein a y-value inevitably follows once a x-value is speci-
fied.
From }–{:
Independent Dependent
f(x) F: y
Suppose, for example, that
y = ax + b = f(x)
we can also write, in this case, the inverse relation
x = (y/a) - (b/a) = f'(y)
Thus, there is no indication of causality inherent in the sign of
equality; rather, by convention an equation is generally written so that
the dependent variable is on the left, thus implying a right to left
causality. No ambiguity results, however, if we were to indicate the
functional dependency of y on x by writing
x – f – y
Then, applying the inverse
x ––E H-y—[r]--—x
Such a representation emphasizes the signal sense and de-emphasizes the
nature of the functional dependence and the form of the "signals" x and
Y.
We are reminded of the care With which the chemist endows his
equations with causality, this being due to the fact that the direction
lil, 2.751 CLASS NOTES
in which a reaction proceeds is so intimately dependent upon the ambient
conditions. Thus, for example, the direction of causality in the equa-
tion
2H2 + O2 *Ts 2H2O + heat
is partly determined by the ambient temperature. The reverse (right to
left) reaction predominates at extremely high temperatures, whereas the
forward reaction predominates at lower temperatures.
For our purposes it is imperative that a direction of causality
be imparted to an energetic exchange since no quantitative analysis of
any form is possible until this is done. Once a power bond has been de-
scribed in terms of an effort-flow couple then it may be endowed with
causality. In the case of a system communicating energetically with its
environment either the effort or the flow may be viewed as determined by
the environment, i.e., as independent, so that the other variable is
looked upon as dependent. Consider, for example, an ideal fluid system
in the steady-state:
€2
Suppose that for this system the pressure is determined by the environ-
ment and is held constant (but not necessarily uniform) over the entire
boundary of the system. Assuming the fluid to be incompressible, it
would be natural to set
e
-
+ = pressure in lb. /ft2
f.
# = flow rate in ft”/sec
-
in Order to describe the four power bonds between S and E. Since the
pressure is environmentally determined

Part W 2.751 CLASS NOTES l:5
el = E1 = constant
e2 = E2 = constant
e3 = E3 = constant
-
e4 = E4 = constant
The continuity equation States, for this example
2IP = 2 e,f, = 2E, F, - o
iT i
since there cannot be internal energy dissipation or storage. In addi-
tion, each of the IP, is constant so that each of the fi is also con-
stant. Thus, the efforts and flows associated with the internal bonds
must be constant .
We have stated that an energetic interaction is endowed with
causality if it is conceived as a bilateral signal flow. Thus, a power
bond
e e
Ezº. , or simply,
f
becomes
e e
-º- —sº-
-º- Or —º-
f f
The assignment of causality to a bond is equivalent to adding
a single bit of information to the noncausal bond. Hence, it is theo-
retically possible to accomplish this addition with a single stroke,
thus:
e e
~ X is equivalent to H
f f
e e
{ : }* ~ * H-
f f
A useful mnemonic is the association of the flow variable, f, with a
direction parallel to or along the bond, and the effort variable, e,
With the transverse stroke.
F | Ow

l,6 2.751 CLASS NOTES
Thus, in the four-port fluid system we would indicate the fact that the
pressure is given on the boundary as follows:
e2
f4
Assigning causality arbitrarily to the internal bonds, we may represent
the system completely in the following succinct form:
*/
tº ſº º ºr nº wº, ºr mºr sºr smººr-
Thus far we have only spoken in general terms about the depic -
tion of power bonds, presumably in preparation for a detailed quantita-
tive analysis, and have not concerned ourselves specifically with what
happens when two systems are coupled together. Naturally the power
transferred across a bond is a function of the characteristics of the
two participating systems. It is generally possible to conceive of one
of the systems as the "supplier" and the other as the "recipient" of
power. Consider two coupled systems, Si and S2, operating in the steady-
State ;
Now, it is plausible to assume that S1 has a falling e-f characteristic
such that e decreases as f increases. It is also to be expected that S2
has a rising characteristic such that e and f increase together. If
this happens to be the case the steady-state equilibrium point Will cor-
respond to the intersection of the two characteristics.

Part, W 2.751 CLASS NOTES l;7
equilibrium point of operation
2'
2 *N
, -’ falling supply characteristic
rising demand characteristic
-º-
f
If the e-f characteristics of S1 and S2 are such that there is no defi-
nite intersection, then there exists no point of equilibrium operation.
On the other hand, there may be systems which possess characteristics as
sketched below:
e
supply
demand
\ unstable equilibrium
* f.
It is apparent that the intersection, though definite, generally corre-
Sponds to an unstable equilibrium point.
It is all very well to be able to determine the equilibrium
power transfer, but we often wish to do much more than this. In particu-
lar, how can one maximize the power transferred in the steady-state op-
eration of two coupled systems S1 and Sa?
Since IP = e . f, the curves of constant power are equilateral
hyperbolas on the e-f plane.
e f
i
sº
^
f




l;8 2.751 CLASS NOTES
Thus, plotting these curves on the same grid as the supply and demand
characteristics of S1 and S2
* N / Initial demand characteristic
2^ Final demand characteristic
it is apparent that the equilibrium point (1) is not optimal while (2)
is. We may say in general that a design problem of a rather complex na-
ture must be solved in order to match the characteristics of two coupled
systems so as to achieve an optimum power transfer. The modification in
Sa which resulted in the movement of the equilibrium point from (1) to
(2) is an example of "load matching." Presumably Si might have been
altered instead to achieve a different optimal operating condition--an
example of "source matching."


Part VI 2.751 CIASS NOTES l;9
VI. Multiported Systems and Elements
A. Introduction
At this point it is clear that, once reticulated, any material
system may be conceived as a multiported device with multiported ele-
ments. Thus, we shall consider briefly certain properties of multiports
in general, and discuss in greater detail a particular universally en-
countered multiport, namely, the ideal energy junction.
If the number and variety of multiport components of an
engineering system is sufficiently large, more than one
operable structure, circuit, or system could be assembled
from the same parts. Abstractly, this is equivalent to the
statement that for any given parts list (or molecular formu-
la) more than one possible bond diagram (or structural for-
mular) may exist. Systems which possess the same list of
components but have differing bond structures may be con-
sidered as structural isomers, and the situation may be
referred to as structural isomerism, borrowing a usage
from chemistry. The number of possible structural isomers
increases very rapidly as the number and variety of compo-
nents increases. In this connection the equivalent situa-
tion in Organic chemistry is instructive. For example,
Butane [C4H10 J has two isomers, Octane [Cah is J has eighteen,
while calculations indicate that the homologous polymer
CaoHa2 has 62, 191,178,805,831 theoretically possible
isomers: As to variety, more than one million diverse Or-
ganic compounds have already been identified involving just
the four atoms, C, H, O, and N, and this figure is growing
exponentially with time. In the engineering systems field
some aspects of Structural isomerism have already been treated
extensively in connection with circuit theory as we shall
learn presently.
We have emphasized the necessity of abstracting from the many
attributes of a system those properties that are essential to the delin-
eation of the functional connectedness of its elements. Indeed, a truly
incisive analysis is one which is detached from a specific material em-
bodiment and which focuses upon the functions of the elements and the
manner in which they are bonded together. The properties with which the
elements of a reticulated system are endowed are transcendent properties,
i.e., the artificial boundaries between hydraulics, electronics, and
thermodynamics are largely overlooked. For example, a transducer is a
two-port--an energy converter--and a concern as to whether the conversion
is electromechanical, hydromechanical, or thermoelectrical is often sec-
Ondary.
5O 2.751 CLASS NOTES
B. Multiports
Although the denumerably infinite universe of all possible
combinations of one-, two-, and three-port elements is not sufficiently
Broad to enclose all conceivable energetic systems, its extent is so
great as to include most systems of practical interest. Thus, we shall
confine our attention primarily to one-, two-, and three-port elements
and combinations thereof.
1. One-ports
A one-port may be thought of as a generalized impedance, some
specific examples being resistance elements, capacitance elements, and
inertance elements, together with all one-ported networks composed of
such elements. The one-port is schematically represented in an energe-
tic bond diagram simply as
A –
Thus, if we consider the universe of all one-port combinations, we note
that it has but a single additional member, namely
A — B
A two-port may be conceived as a generalized transport process,
i.e., a process by which energy is transformed, transmitted, or trans-
duced. Thus, a communication system may be looked upon as a String of
two-ports. The viewing of an ordinary triode amplifier as a two-port is
generally accepted and is subject only to the assumption of a constant
power supply, i.e., the power supply is located Within the conceptual
boundaries of the two-ported element. Hugh SKILLING in his text, Elec-
trical Engineering Circuits, discusses the two-port, or two-port net as
he calls it, from the standpoint of the electrical engineer. He sche-
matically depicts various internal reticulations of the two-port and con-
cerns himself primarily with the description of the transfer character-
istics of such an element. Contemporary books on transistor theory and
application, such as the text, Transistor Circuit Engineering, demonstrate
that linear two-ports may be mathematically represented by way of a 2 x 2
transformation matrix.
Part VI 2.751 CLASS NOTES 51
The universe of all possible combinations of two-ports has but
one member since the elements in the chain
— C — D — E — F —
may be coalesced to yield a single, equivalent two port
— G —
If we admit combinations of one- and two-ports, two new members
are added, namely
A – G –
A — G — B
It is immediately evident, however, that the universe of one-
and two-ports combinations is far too simple and restrictive.
With the admission of the three-port the universe expands from
the five members identified thus far to one which is denumerably infi-
nite. The richness of this universe obtains from the possibility of
branch structure which is attributable, of course, solely to the pres–
ence of the three-port. Thus, the three-port is a singular and most
assential element.
We may think of the three-port as a generalized modulator,
including (triportal) ideal energy junctions, power and signal modula-
tors, power and signal amplifiers, and power exchangers as specific ex-
amples. Classical mechanics recognizes but a single three-port, namely
the triportal energy junction; in this realm all systems are conceived
as interconnected sets of one-ports (generalized impedances) and ideal
energy junctions.
C. Ideal Energy Junctions
For a generic multiported element, A, the equation of energy
continuity states :
52 2.751 CLASS NOTES
/...N. i=1
Now, let us restrict A to be ideal, by which we shall mean that 2, is
identically zero, and if it further lacks the capacity for energy stor-
age, then IE vanishes as well, leaving simply the condition:
Il
X IP + = O
i–1
A large class of energetic elements approximately satisfy this fundamen-
tal condition and the continued discussion of such elements is by no
means trivial. Several of the most useful ideal elements are 1
a . Energy Junctions
b. Ideal Transformers and Gyrators
c. Ideal Transducers
f. Differentials
g. Ideal Structural Modulators
In particular, we shall presently concern ourselves with the class of
ideal energy junctions.
From this point onward a duality of characteristic relation-
ships must be emphasized, for there exist two conjugate energy junctions-
the effort junction and the flow junction. We may best depict this
duality by carrying the development of both types in parallel, thus:
Part WI 2.751 CIASS NOTES 53
º º e e º o e º o e º º º de º º & º º tº º tº º e º º & © e ve o tº © & ©
Effort Junction º |Flow Junction
Both junctions are characterized by the condition that one of
the two conjugate variables is common to all bonds, i.e., for a junction
With n bonds :
fi = f (i = 1, 2, 3, ..., n) . ei = e (i = 1, 2, 3, ..., n)
Then it immediately follows that
Il e Il
X e = 0 © X. f. = 0
# i te # Ti
hereafter we shall represent the two junctions respectively as
/ tº N /
N
/...N /...N
The conjugate relationships

f. = f' (i = 1, 2, 3, ..., n) e , = € (i = 1, 2, 3, ..., n)
(Loop Law) (Node Law)
play a dominant role in the idealized analysis of energetic systems.
Carlo FERRARI depicts this role for a variety of media, perhaps the most
familiar of which are the electrical network and the mechanical linkage.
The conjugate junctions law are simple generalizations of KIRCHHOFF's
Loop and Node Laws in the electrical case, and, borrowing FERRARI's ter-
minology, the Laws of Velocity and Equilibrium in the mechanical case.
The paper by J. C. SHOENFELD and the text by M. F. GARDNER and J. L.
BARNES develop various aspects of the electromechanical analogy. In
5|| 2.751 CLASS NOTES
particular, SCHOENFELD noted that the flow junction (Node Law) in an
electrical network and the effort junction (Equilibrium Law) in a me-
chanical system were isomorphic; thus, the importance of the duality in
the concept of the general energy junction is underlined.
The student, KIRCHHOFF, based upon a query in Neumann's
physics seminar at Koenigsburg, made the first comprehensive
study of the general electrical network problem by showing
the relation between coarse reticulations (macroreticulations)
and the field theorems (microreticulations). This was carried
out in terms of Stoke's Theorem (Loop or Effort Conservation)
and Gauss's Theorem (Node or Flow Conservation); the results
were published first as an appendix to a paper in 1815 and
then in more complete detail in 1847.
As a warning against offhand use of the terms "Kirchhoff's
First Law" and "Kirchhoff's Second Law," it is interesting to
note that the laws appeared as follows in the two papers 1
1815 18||7
1) Il + I2 + ... + Ipi = 0 I. w kilki + at k2lk2 + . . .
2) Il w 1 + Te . 0 2 . . . = Ekl + Ek2 + . . .
+ Iy w y = sum of the EMF II. IX1 + IX2 + . . . = 0
Thus the node and loop rules are transposed in the two papers.
Implied in the assertion e = e (i = 1, 2, 3, ..., n) and
f’. = f' (i = 1, 2, 3, ..., n) is the assumed uniformity of the ener-
i
getic medium by which the energy junction A is communicating with its
environment. In other words, the bonds are either all electrical con-
ductors or all mechanical links or all fluid-carrying ducts, etc.
Energy junctions are associative and dissociative with respect
to the triportal primitives; therefore, any multiported junction may be
conceived as the combination of several three-ports. Thus for example:
------ (-|--|--|-
Historically, the notion of the energetic junction, in all its
generality, has not been exploited effectively. In the analysis of elec-
trical networks the concept has been developed more extensively than in
Part VI 2.751 CLASS NOTES 55
other areas--notably by way of Kirchhoff's Laws. As a result, investi-
gators in heat transfer, hydraulics, and so forth, have often resorted
to the contrivance of electrical analogs. The sophistication of electri-
cal schemata undoubtedly contributed to the attractiveness of this ap-
proach. However, we see now that such artifices are unwarranted in the
light of the general formulation here presented. Just the same, it is
illuminating to depict energy junctions as series and parallel electri-
cal networks as sketched below:
eb fb eb fb
** *—ºr
º |
fa : fo tº fa : fo
** tº -º- * -** O= sº-ſº
“...ſº ... (1)... ..] ec * “...l. ... I (O) Me,
Let us next interpret the following interconnection of energy junctions
and one-ports
A B
| |
— O — 1 —
in terms of the equivalent electrical network. Recalling that A and B
are generalized impedances, we have for the above
Or, more simply

56 2.751 CLASS NOTES
It is noteworthy that the circuit dual-–the network resulting from a
O —- 1
transposition -- is immediately recognizable.

This, in fact, is an important attribute of a schematic representation
in the form of the energetic bond diagram.
Part WT 2.75l. CLASS NOTES 57
Background Reading
1.
SKILLING, H. H. Electrical Engineering Circuits, Chapter 18.
This reference gives some insight from the electrical engineering
viewpoint into the nature and function of the two-port; attention
is focused upon the problems of functionally or operationally de-
scribing such elements knowing their internal structure.
SHEA, R. F. (editor). Transistor circuit. Theory, pp. 1-3, 21-22,
Appendix.
The two-port is discussed relative to the description and analysis
of transistors. One-ports and multiports are also mentioned. The
matrix representation of linear elements is presented.
FERRARI, C. Relazione Generale sui "Modeli Analogie."
This paper presents the conjugate junction laws for a number of
important engineering media, thus lending breadth to the notion
of the ideal energy junction.
SCHöENFELD, J. C. Analogy of Hydraulic, Mechanical, Acoustic, and
Electrical Systems.
Recognition is given to the importance of the duality in the energy
junction concept in delineating the electromechanical isomorphism.
GARDNER, M. F., and J. L. BARNES. Transients in Linear Systems,
Chapter II.
Particular emphasis is given to the schematic representation of
electrical and mechanical systems so as to exploit the analogies
existing between them.
58 2.75l. CLASS NOTES
VII. Classes and Relations
A. Relations and Structure
Bertrand RUSSELL states: "To exhibit the structure of an
object is to mention its parts and the ways in which they are inter-
related." In the analysis of systems we are confronted with the task
of establishing an order, a conceptual structure, in an initially form-
less universe. First, the S-E dichotomy is depicted in U, and then
both S and E are further reticulated to the degree appropriate to the
objectives of the analysis. The essential step in the process, however,
is the recognition of the significant interrelationships among the re-
ticulated elements. A system is not described by a parts list alone,
but rather by the combination of a parts list and a delineation of the
interconnections and interactions among the parts.
In dealing with relationships among the elements of a system,
we must account for the existence of functional dependencies of the most
general character. Thus far we have made mention of the generalized
functional, "P , which scans the input vector X (t - T) for 0 < t < do
and yields up a value for the output vector Y(t).
A particular form, and one for which we shall find frequent use in the
sequel, is the vector-to-scalar transformation
y(t) = (P [X (t)]
For example, consider the correlation functional which yields the energy
stored in an ideal element, namely:
OO
/P (e-r) at
yº- • f (t — r ) d r
O
Wherein we might identify |X -kº, *} == {e,f).
Y (t) = IE (t)

Part VII 2.75l CLASS NOTES 59
Because of the fundamental role of relations in our analytical
constructs we shall now turn to their general characterization in the con-
text of the theory of classes, this being the most fundamental mathemat-
ical system available and therefore the most appropriate medium in which to
couch a generalized description of relations.
B. The Concept of a Class
Out of a chaotic universe of sensory impressions and mental
images, our reasoning mind struggles for Order and understanding. The
fundamental ordering principle upon which all this effort is based is
that of likeness, resemblance or similarity. All thought springs from
beginnings in comparative studies in which similar objects and phenomena
are brought together into classes.
Thus the concept of a class (or alternatively, a set, collection,
ensemble or aggregate) becomes the simplest component of mathematics and
logical thought itself. The first step in establishing a class is that of
determining the property of membership.
A class is determined (or established) the moment one arrives at
a property (or rule, test or condition) which any object (or entity) with-
in the universe under consideration must possess (or satisfy) in order to
be a member of (or belong to) the class. Thus the concept of the class it-
self and the required rules for membership are inextricably interwoven. We
Shall inquire further into the nature of these conditions and properties
below.
It is first worthwhile to introduce mathematical symbolism to make
these concepts more precise. We shall accordingly denote various classes
by Roman capitals:
CLASSES: A, B, C, etc.
The individual objects which comprise any of these classes we
Shall speak of as elements (or members or components) and denote by lower
Case Roman letters:
60 2.75l. CLASS NOTES
ELEMENTS: a, b, c, x, y, z, etc.
Properties possessed by the elements, including those properties
upon which membership is based, will be denoted by small Greek letters:
PROPERTIES: a, 9, Y, 3... etc.
2 3
Between these elements and classes we have possible membership
relations. The fact that a given element, a, is a member of a class, A, ,
we can conveniently express in the form
a € A
by employing the membership symbol
MEMBERSHIP: € read "-- is a member of --"
Schematic diagrams frequently are used to aid in the compre-
hension of relationships between classes and elements. One approach is to
depict the elements as geometric points and the classes as sets of points.
Clearly, however, many other portrayals are possible. All such representa-
tions have justification to the degree that they lead to a self-evident
or intuitive understanding of interrelationships.
It is frequently necessary to deny or negate the existence of a
relationship between two objects; in a common symbolism the "operation" of
denial is accomplished through the use of a vertical slash: " | ". Thus,
to deny the membership relation we write, for example
b # A
indicating that the element b does not satisfy the requirements for mem-
bership in the class A, that is, b is not a member of A.
No confusion should result if the relations of membership and non-
membership are stated in reverse fashion, thus:
A P a
A # b
Indeed, in the sequel, great significance will be attached to the so-
called converse relationships of which those just above are examples.
Part VII 2.75l CLASS NOTES 61
Referring to the sketch we see that the following statements
hold true:
a, b, c e A but a, b, c & B
d, e < B but d, e 4 A
a, b, c, d, e < C
In an allegory we may liken the establishment of a class to the
action of a small boy at the beach becoming interested in gathering white
pebbles. We observe him gathering pebbles one by one, looking at them
to see if they are white and either putting them into his pocket or throw-
ing them back onto the beach. The defining property involved here is that
of whiteness. All the pebbles in his pocket then are members of an evolv-
ing class of white pebbles and those thrown back belong to many other classes
but in particular to the class of non-white pebbles. Any given pebble in his
pocket can be considered as an element of the given class of white pebbles.
We may distinguish here at least two classes: the class A of white
pebbles and the complementary class B of non-white pebbles. Any given peb-
ble in the pocket we may distinguish by the lower case letter x and write
the fact of membership in the form
x 6 A.
Since most classification and gathering processes are not at any
Śiven time exhaustive, we must consider the existence of subclasses and the
Situation of containment within a class.

62 2.75l. CLASS NOTES
For example, in the allegory, the pebbles in the boy's pocket
form a subclass C of the class of white pebbles A. By this we mean that
C is part of but not all of A. We may symbolize this fact by the state-
ment
C C A
employing the symbol
PART SYMBOL: C , read as "--is a (proper) part of --".
We refer to the class C as a proper subclass or part of the class
Often, however, we do not wish, (or are not able), to establish the fact tha
C is only a part of A, but wish merely to express the fact that C is con-
tained in or included in A. We may still refer to C as a subclass of A but
we may also wish to cover the possibility that C and A may be coincident or
coextensive; that is, that C might include in some cases, each and every
element of A. Thus a subclass is either a part or the whole. In this more
general case we would write symbolically
C C A
contained
CONTAINMENT: C read as "--is Or in--"
included
It is interesting to note that the relations C and C ,
which can hold between two classes, are respectively analogous to the re-
lations 2 and 2 which may exist between two real variables. Indeed, this
analogy can be a useful mnemonic device for those who are unfamiliar with
the containment relations.
The Operation of denial or negation may be applied to each of the
containment relations, employing as before the vertical slash. Thus, for
example,
D 4: A
As in the case of membership, it may frequently be convenient to
Write the containment relations in converse form. For instance, in the
statement
A D C
Part VII 2.75l CLASS NOTES 63
the symbol, D , reads "--partly consists of--".
Two elements, a and b, of a given class are said to be equal
or identical,
a = b
if they can be regarded as interchangeable with respect to the class and
the associated class property. Thus, identity of elements only implies a
certain relative indifference or indistinguishability within the context
of a given class.
r Elements which are not identical are said to be distinct and are
indicated symbolically
a + b
Identity_ºf_Sleššeš
'In the pebble allegory, presume that through some quirk of
geology, in a rather short time the boy had gathered together all the white
bebbles on the beach. Then, if we continue to recognize the class C as
she pebbles in his pocket and the class A as white pebbles, We
might wish to express the fact that the class C had exhausted the class A;
;hat is, each and every member of A was included in C. We can simply ex-
ress the fact by the statement that the class C is identical to the class A,
Jut we can also put very simple conditions on the two classes for this to be
IUle.
Any two classes, X and Y, are said to be abstractly identical
ºr equivalent, Written:
X
Y
f, and only if, X CY and Y C X. The equivalence symbol is read as follows:
equivalent
QUIVALENCE: E , read as "-- is Or to --".
identical
Thus the identity between the class of pebbles in the boy's pocket
nd the class of white pebbles on the beach is established merely by de-
ermining simultaneously whether all the white pebbles are in the boy's pocket
nd whether all the pebbles in the boy's pocket are white pebbles.
6|| 2.75l. CLASS NOTES
Classes which have no elements in common are said to be
disjunct or disjoint. We should particularly note that this is not the
same as the fact that they are not equivalent, symbolized
x + Y
which merely means that X and Y do not consist of the identical set of
elements.
C. The Concept of a Relation
The purpose of the above discussion was to establish the context
in which we shall seek an understanding of relationships or relations in
their most general form. In doing this we inevitably encountered several
Specific relationships, namely those of membership, containment, and identity
or equivalence. Each of these is an example of a relation between two objects
or terms-- a so-called diadic relation, or simply a diad. The totality of
objects linked by a given relation we call its range and it is thus apparent
that a numbering of these objects affords a convenient approach to the class-
ification of relations. That is, we may usefully distinguish between monads,
diads, triads, tetrads, etc.
We shall here employ illuminated Roman capitals to denote relations,
thus:
RELATIONS: IR, T, W, X, etc.
If two relations, Rand $ are precisely the same, we may indicate
their equivalence by way of the familiar notation
The diad, "x bears the relation IR to y" could be written symbolic-
ally either
X IR y Or IR (x,y)
However, the nature of the terms or objects x and y is quite irrevelant :
hence, the existential graph:
with a specification of the realm of its applicability imparts the same in-
formation as the first form.
Part VII 2.75l. CLASS NOTES 65
The converse of the diadic relation TR , when it has meaning,
is written in our symbolism
tº JHI tº
a form which is most suggestive of the significance of the converse. A
particular form of the converse, namely the "inverse" of a mapping or trans-
formation is often written IR-l. It will shortly become evident that certain
theorems which apply to transformations and their inverses also hold for re-
lations when the inverse, IR-1, is replaced by the more general converse, HI.
A restricted set of relations, namely those that express some form
of identity, are symmetric in the terms such that IR E PI. In writing
such relations it is often convenient to employ a suggestive symbolism which
exploits those letters that are inherently symmetric: T 5 IHI, 0,etc.
With this introduction to abstract relations, it is now propitious
to focus our attention on certain specific types of relations of immediate
present value. The objective of our study will be the establishment of a
secure basis from which we may approach the relationships to be encountered
in the generalized analysis of systems with increased understanding and in-
Sight.
We may organize this treatment on the method of categorization
briefly introduced above, namely that founded on an enumeration of the ob-
jects linked by a given relation.
Monads
c The statement: "there exists the object x" is an example of a
monadic relation or monad--its range is the single entity x. In customary
mathematical symbolism it is written
Ex
The monad is so simple, and its statement and structure so
Buccinct, that one is hard put to elaborate upon it. However, a considera-
tion of the grammatical structure of the literal statement monad is perhaps
illuminating. Let us, therefore, examine in greater detail the exemplary
nonadic statement, "there exists the object x." The converse form, "the
bbject x exists," suggests the symbolism
x IE
66 2.75l. CLASS NOTES
\
wherein the existence relation is denoted by an illuminated capital in
conformance with our notation. We note that the "subject" and "kernel"
of the statement "x exists" have their counterparts in the symbolic state-
ment. That is,
Literal Statement Symbolic Statement
Subject the object x X
Kernel exists – IE
The denial of existence is accomplished with the application of the
IE or #
When applied to x, this new monad would be read:
vertical slash, thus:
"The object x does not exist."
The generic monad is written simply – IR , together with a specifi-
cation of its field of applicability. That is, when we say
x IR or IR (x)
we imply that - IR may be meaningfully applied throughout the class X whose
elements x have the certain common property characteristic of the class.
Diads
As a result of the utter simplicity of the monad its significance
as a relation tends to elude the intuitive grasp which one has for higher
order relations. The diad, then, is the simplest relation that has a im-
mediate intuitive significance. The range of the diad consists of two ob-
jects, a and b for example. Symbolically, the diadic relationship may be
expressed
a. IR b
An alternative form may at times be appropriate; it is written
IR (a, b)
The first form has many mnemonic advantages and has by far the widest use,
but we frequently employ both forms. It is noteworthy, in the second form,
that in general the commutation of the terms inside the parenthesis is not
valid. That is,
IR (a, b) # RG, a
Part VII 2.75l. CLASS NOTES 67
We may represent a diad by an existential graph if to the re-
lationship symbol IR we append two tails thus
sº IR tºº
which indicate its diadic nature. The relation - IR- is denied or nega-
ted by application of the vertical slash, - • e
Three fundamental properties which are either present or absent
in any diadic relation are the following:
1) Reflexivity; a IR a
Any relation satisfying this condition is said to be reflexive;
if a a, the relation is irreflexive.
2) symmetry: If a R b then b R a
Any relation satisfying this condition is said to be symmetric;
if a IR b but b a then the relation is asymmetric. A
third important possibility is that of antisymmetry: a IR b
and b R a if and only if a = b.
3) Transitivity: If a IR b and b IR c then a IRc
Any relation satisfying this condition is said to be transitive;
Otherwise it is said to be intransitive.
It is possible to regard any diadic relation as directed or polar-
. ized. That is in an existential graph:
a. *—IR º b
Corresponding, then, to any such polarized relation, IR , there will often
be a unique and well-defined converse, JHI , such that if a IR |b then
b JPII a. It is important to note, however, that not every IR possesses 8.
meaningful converse. Referring to the definition of symmetry, it becomes
Obvious that relations which are symmetric not only do possess converses,
but in addition satisfy the condition that
IR EJHI
In the light of the above discussion let us consider two all-
important classes of diads--the abstract equivalence and ordering relations.
These will play a dominant role in the development of the generalized func-
tions and transformations which are required to describe the behavior of
real systems.
68 2.75l. CLASS NOTES
Equivalence Relations. A diad, - I - , is said to be an
equivalence, relation if, and only if, I is: (i) reflexive; (ii) sym-
metric; and (iii) transitive.
So far we have encountered the two equivalence relations--
Identity of Elements (=) and the Equivalence of Classes ( = ) -- but there
are many other examples of equivalence relations in all branches of mathe-
matics and logic--in particular, Congruence ( = ) and Similarity ( ~ ) in
ordinary Euclidean geometry.
The fundamental property of any equivalence relation is that it
divides the range over which it applies into a k-fold set of mutually ex-
clusive equivalence classes (IK"). Thus,
k
a II b, if, and only if (a, b) = IK
The number, k, may be either finite or infinite, and in the latter case,
either demumerable (i.e., countable) or nondenumerable.
In the simplest possible case the range will be merely bisected
giving rise to a dichotomy or dichotomic categorization (e.g., up, down;
positive, negative; etc.). Thus, such classification systems will in
general give rise to polychotomies or manifold categories as indicated in
the following table:
Example of Equivalence Classes
Class: Transducers
|
| | T
Subclasses: Electro- Fluid- Electro-
Mechanical Mechanical Thermal
Transducers Transducers Transducers
Elements: Motors Turbines Heaters
Generators Pumps Thermopiles
Microphones Pistons Thermocouples
\ J
Y

Mutually Exclusive Equivalence Classes
In all cases, any element in a subclass is abstractly equivalent
Part VII 2.75l. CLASS NOTES 69
or identical, but only with respect to the defining property or condition
; )f the subclass. For example, it is obvious that the prime number 7 is
not "equal" to the prime ll in the sense of ordinary arithmetic, but only
"equivalent" in the sense that they are both prime numbers.
Ordering Relations. If we are given any asymmetric ordering
relation, - O) -, applicable over a range (a, b) in a class IK, we can
construct a corresponding antisymmetric ordering relation - ID) — by de-
fining ID) to be the same as either J) Or I where — I - is an equiv-
alence relation.
We may say then that J) is a strong or serial ordering relation,
it being: (i) irreflexive; (ii) asymmetric, and (iii) transitive. On
the other hand, ID is a Weak or partial ordering relation since it is:
(i) reflexive, (ii) anti-symmetric, and (iii) transitive.
The following table gives examples of these ordering relations
which are already familiar to us:
Range O) II ID)
|Re al > : >
Numbers
Classes D E 2
It is important that we distinguish between the various ways
diads may be combined. The three diad combinations which we shall briefly
consider are: (i) composition, (ii) alternation, and (111) conjunction.
Composition. Suppose X IR, y and y IR 2 Z. Then ×IR 3 Z where
IRA = R, 3 IR a
the symbol R. () IR 2 denoting the composition of the two relations
R and IRa. By way of an example, suppose IR, * = MI and IR, i: IF
Where MI and IF are respectively the relations of motherhood and fatherhood.
Then, if IR 3 = MOIF and X R 3% , then x is the paternal grandmother
Of z.
Ln connection with the composition of two relations it is note-
Worthy that if
IP () () = IR
7O 2.75l. CLASS NOTES
then
which is seen to be a generalization of the more familiar statement for trans-
formations:
1
o' P " - (Po)"
Furthermore, we note that in the case of any transitive relation T that
T & T = T
Alternation. The alternation of two relations R. and R2 5
Written
IRs – R, UIR,
is the result of applying either IR, or R, . Thus, if again R. = MI -:
motherhood and Re = IF = fatherhood, then IR 3 -: MI U IF is the re-
lation of parenthood.
Conjunction. The conjunction of two relations IR, and IR 2
Written
R - Rſ. R.
is the result of applying both Rand R, e Thus, IR 3- MI ſ] IF is, in
mammalian biology, impossible since there is no x and y such that x Mſ).IFy,
i.e., x is both the mother and the father of y. This fact may be expressed by
asserting that MI ſh F is true. -
Any relation IR whose range consists of three terms is a triadic
relation Or a triad, indicated in symbolic form
IR (a, b, c)
or existentially: º IR cº-
|
A specific triadic relationship is, "b is between a and c." More-
' a
over, any operation or rule of combination by which two terms "produce'
third term is a triad. A triadic relationship exists between a mother, a
father, and a child, for example. In algebra and arithmetic, instead of
saying two given mumbers a and b determine a third mumber, c, such as their
Part VII 2.75l CLASS NOTES 71
Sll II]
or their product
C = ab
we may say that the three terms satisfy a triadic relation IR among a, b,
and c.
The structural or topological properties of triadic relationships
are not so simple as those of diadic relations. However, several possi-
bilities varying from complete symmetry to various types of asymmetry may
be distinguished.
An example of complete symmetry occurs in the relationship be-
tween the three sides or the three vertices of an equilateral triangle.
Here clearly
IR (a, b, c) IR(b, c, a) = IR (c., a, b)
IR(c, b, a.) IR (b, a, c)
IR(a, c, b)
We may also speak of this as permutative symmetry since all permutations
are allowable and are equivalent. Such symmetry also occurs for example in
the algebraic equation
a + b + c = O
However, some triads are symmetric only over a part of their range,
such as the examples of sum and product mentioned above which are clearly
symmetric in a and b; that is
Sum: C = a + b = b + a.
Product: C = * : * ~ * * *
The "between" relation also possesses this limited symmetry since "b is be-
tween c and a "." It is appropriate that we designate such limited symmetry
Commutative symmetry by analogy to the commutative properties of the sum and
product operators.
The relations of sum and product, as well as many other familiar
triads, have an implicit polarization or directionality. That is, combining
à and b yields, respectively, the sum, c = a + b, or the product, c = a . b:
72 2.75l CLASS NOTES
the uniting of male and female results in the procreation of offspring so
that the father-mother-child triad is also polarized. It is convenient to
symbolize such directionality; for example,
IR (a, b, c)
Or IR(( a, b) -- c)
Or C IR(a , b)
to indicate that c was the result of the combination of a and b.
Corresponding to certain types of asymmetric triads we can
establish the existence of alternative relations. Consider, for example,
the triadic relationship between father, mother, and child in which we
distinguish at least the three polarized forms
R. (f, m; c)
IR,(n, c; f)
R, ºr, c; m)
Assuming normal wedlock these might be read as:
IR, ; c is the child of n and the child of f;
IR
f is the father of c and the husband of In;
IR : In is the mother of c and the wife of f.
We note that while R. is symmetric in m and f, clearly R, and R,
are asymmetric, but certainly in a broad sense the triadic relationship is
established by each of the alternative forms so that they are, to this degree,
equivalent.
A similar example to the above is the algebraic relation x/y - zº
Part VII 2.75l. CLASS NOTES 73
With the monad, diad, and the triad we can build up a rich
universe of polyadic relations--this because of the three-tailed
property of the triad. That is, the relations
may be linked so as to construct a polyad of any order. However, the
universe of polyadic relations so obtained is by no means exhaustive, al-
though it is sufficiently broad for most of our purposes.
The abstract treatment of relations in mathematical literature
has, for the most part, excluded the triad, concentrating instead of diads.
It is self-evident, however, that the triad is essential if we are to con-
sider even the simplest polyadic structures, since the compounding of diadic
relations can never produce anything but diads. It is therefore possible to
view the monad, diad, and the triad as the basic "building blocks" out of
which all more complex relations can be constructed.
The general polyadic or "n-adic" relation may be symbolized
R. (a1, a2, ..., a,)
or, graphically
We may thus depict the reticulation of a tetrad into two triad primitives:
| |
–F– = –IR,- IR;-
If IR is an algebraic relation then it may always be reticulated into a
System involving only the four triadic primitives:
-º- -º-
-º- -º-
7|| 2.75l. CLASS NOTES
However, for the generalized relations which are necessary to describe the
behavior of real systems such a simple reticulation is not possible. In-
deed, we shall see that there is a set of logical triadic primitives
which is sufficiently general to serve as the building blocks for the con-
struction of all such relations.
The relationships which bind together the characteristic variables
of a physical system are clearly polyadic. We may think of them as falling
into one or more of the following categories:
1) Correspondences
2) Functions
3) Transformations
l;) Operators
We have been using the generic term functional to include functions, trans-
formations and operators.
To properly describe the behavior of a system we must be ready to
admit to the "functional domain" multivalued and discontinuous functions,
unlike the traditional strategy of mathematicians, and indeed, many engineers.
For example,
. -l
x = sln y
is a permissible relation. Certainly "a function" is a relation so that
y = F(x) is equivalent to the statement y IF x; the converse, x = F"(y),
if it is meaningful, is then written x TI y .
A correspondence is merely the statement that there is an *1 which
corresponds to a Y12 symbolized perhaps by the following notation:
Multivaluedness may exist wherein the following correspondence could result;
for example:
×1
Part VII 2.75l. CLASS NOTES
75
The generalized functional Y =%P (X) is one with which
we are now familiar, and affords a sophisticated example of a polyadic
relation.
76
2.75l. CLASS NOTES
Background Reading
(l)
(2)
(3)
(l)
(5)
(6)
RUSSELL, Bertrand. Human Knowledge--Its Scope and Limits. Chapter III
This reference discusses aspects of structure as they pertain to the
meaning of words, and the connexity of sentences and complex but,
meaningful sounds. It affords a valuable insight into the importance
of relationships from a non-engineering viewpoint.
PEIRCE, C. S. Collected Fapers
See particularly
Vol. 3 The Logic of Relatives: 3. 156 - 3.191
Vol. 1 Trichotomic Mathematics: 11.309 - || .310
Vol. 5 The Valency of Concepts: 5. H69
Without question, Peirce, the founder of pragmatism, was first to
realize the singular character of the triadic relation. His use of
bond diagrams for logical thought is prophetic and revealing. His
philosophic concepts of Firstness (quality), Secondness (effect), and
Thirdness (meaning) are grounded in the properties of monads, diads,
triads, respectively. A word of caution -- Peirce's style runs the
(deliberate?) gamut from extreme lucidity to perverse obscurity! But
for those who like to climb mountains "just because they are there"
Charles Sanders Peirce is a man to know.
TARSKI, A. Introduction to Logic, Chapter W.
Tarski covers diadic relations in this text in a way which is easy to
follow. He introduces the idea of a polyad, but without development.
CHURCH, A. Introduction to Mathematical Logic, pp. lj-23.
The author discusses aspects of functions which are pertinent as
background reading for the consideration of generalized relations.
SUPPES, P. Introduction to Logic, Chapter 10.
The mathematical properties of (diadic) relations are discussed in
an understandable fashion. Particular note should be given to the
definition of anti-symmetry.
BELL, E. T. Development of Mathematics, pp. 553-5911.
This reference sketches the history of the development of mathe-
matical logic from Leibniz (1666) until Godel (1931).
Part VIII. 2.75l CTASS NOTES 77
VIII. Continuum Logic and Hyperpolyhedral Functions
A. Introduction
A class of extremely flexible, n-dimensional piecewise-
linear functions may be generated through the use of an extension of
the logical operations of union and intersection. These hyperpoly-
hedral functions, as we shall call them, will be employed in the
description and modelling of the behavior of physical systems. Such
functions were first described by George Arthur PHILBRICK.
The union and intersection operations on classes have their
basis in, and may be derived from, the diadic ordering relation, 2 wº
Thus, the process of comparison and subsequent establishment of Order
is fundamental to the development of the hyperpolyhedral functions.
B. Classes
Taking as fundamental the relations of membership and inclusion,
5 and 2 respectively, and their denial, denoted by the vertical slash, ,
the operations of logical union and intersection may be developed. If,
for a given class X, the element x 4. X, then we define the complement-
+ +
ary class X such that x 6 X.
Union--Outer Selection
We desire a "least outer bounding class" X which, for the
aggregate {X, |k = 1, 2, 3, . . . , n}, satisfies the conditions that :
(1){X} < X ; (ii) for every class ¥29,3,3Gy. We then define
the operator U such that
x - ſix. - U (X 1) - X, U X, U ' ' ' U X,
The class X, by definition, contains all the Xº, and, what is
more, it is absolutely the smallest, most restricted class which does
so. Thus, we refer to it as the least outer bound (l. o.b. ) by analogy
to real number theory. For two classes, X, and %2,
l
X = X, U X,
78 2.75l CLASS NOTES
wherein the symbol U is often read "cup". The union operator for this
case may be illustrated by depicting the classes as shaded areas; then,
the union is enclosed by the dotted envelope:
If a third disjunct class X3 is added, the union is as sketched
below:
The union operation is associative and commutative; that is:
Associative property: X, U (x20x3) = (X,\JXe)ux,
X;UX2U X3
:
E
Commutative property: AUža X-JX,
If the aggregate tºl. = 1, 2, 3, . . . , n} is extended without
limit it becomes convenient to speak in terms of the universal class S
such that, for any Y whatsoever,
*
X & S


part VIII 2.75l CTASS NOTES 79
We now seek a "greatest inner bounding class" X, which, for the
aggregate { %. | k = 1, 2, 3, . . 2 * satisfies the conditions: (i) 3&xi;
(ii) for any class YS (x, y Y G X. We then define the operator ſl such
that
X
ſh (X, ) = x, n X, n . . . n X
ń x
Il
1. I Tk
The class 3. is, by definition, contained by all the *k, and what
is more, it is the largest, most extensive class which is so contained.
Thus, we refer to it as the greatest inner bound (g. i. b. ) by analogy to
the greatest lower bound (g. l. b. ) of real number theory. For two classes
X1 and X2,
x = X, ſh;K,
wherein the symbol ſ] is often read "cap". The intersection operation for
this simple case is illustrated by the sketch below:
The intersection operation is associative and commutative; that
is:
Associative property: x,ſ (x-ſix.) = (X1 ſh X2) ſ] X3
B
x, ſh %2 ſh X3
Commutative property: : X ſix. = x ſ] x

8O 2.75l CLASS NOTES
In expressions which involve both union and intersection these operations
are mutually distributive. For example,
Uſ ſlºx1,x2, ), X3 J E X3 U [x, ſh X2]
IX: U x, ſl [x: U xe
[x, U x,] ſ (xe Ux,
E
E
obviously, if the aggregate {x, k = 1, 2, 3, ..., n} include:
one or more disjunctive classes then the class X is empty, i.e., there
are no elements xex. The concept of the empty or null class is an es-
sential one; we denote this class by the symbol G). Evidently then,
O C X
Indeed, the following succession of inclusion relations holds foº
the classes we have defined thus far:
o g3 c X, < x < S
Thus, if the aggregate { | k = 1, 2, 3, . . . , n} includes all possible
classes in the universe, § , then, and only then, Will
*N
Y
x - X. - fi(x,) - o
X = Xo - Ö (x,) - s
The concept of the complementary class is fundamental to the es-
tablishment of order. If, for a given class X the element x é X, then we
+ -X-
define the complementary class X such that x < X. As was done in the
case of the union and intersection of classes, the complementation of a
class X may be looked upon as an operation, the operator being denoted
*
( )".
part VIII 2.75l CTASS NOTES 8l.
C. Order
We have seen that the act of comparison among an aggregate of
classes {X. | k = 1, 2, 3, . . . , n , and the subsequent ordering by way
of the diadic relation S. , is basic to the establishment of the inner and
outer bounding classes 3. and 3. Indeed, ordering is perhaps the most im-
portant operation in the universe; certainly it is fundamental to the es-
tablishment of any scale of measurement.
tº sº ºn tº gº any
The concept of order is often confounded with the idea of a
scale of values or of numbers. It is important to demonstrate that order-
ing is independent of a number scale, and we shall do this by considering
lexicographic order—-the result of a weak. ordering followed by additional
weak ordering within the equivalence classes so produced, and continued
until simple order is achieved. The prototype series, from which the name
"lexicographic" is taken, may be thought of as an ordinary set of listings
in a dictionary, telephone directory, or other lexicon.
In establishing such a series we first recognize that the universe
$, which is to be ordered, comprises the totality of letter groupings
formed from the twenty-six letters and the blank space. A sample would be
the following:
ABSOLUTE
ABSOLUTE ZERO
HYDROGEN
HYDROGEN ATOM
HYDROGENATE
HYSTERESIS
ZERO
82 - 2.75l CLASS NOTES
Here, in establishing the series, the lexicographer first
groups all entries into mutually exclusive classes based upon the
initial letters (A, . . . , H . . . . Z). Within each such weak ordered class
there are in general several elements. Each of these elements is in-
different (or identical) with respect to any given equivalence class,
say A. These may then be sorted into the proper subclasses A , AA, . . . .
AH, ..., AZ. This sorting and arranging operation can be repeated until
every element has been placed into a unique class such as the class
ABSOLUTE. This ultimate class, consisting of but one element, constitutes
a term in a series and will generally have a unique predecessor and suc-
cessor, except in the case of the bounding classes A and ZZZ . . .
Continuum Order
It is propitious at this point to consider an ordering wherein
we suppose the universe $ to envelope a single continuum of classes x(k)
Selecting any two classes from Š we perform a test of comparison and es- *g
tablish which is greater in the abstract sense of the relation C ; to the
smaller of the two we assign k = l, for instance, and to the larger k = 2,
This process is repeated again and again over a large sample of the ag-
gregate {x(k)} . The result (after possible renumbering) is an ordering
such as
X(1) CX (2) Cx (3)C... Cx (n)
If the process is extended ad infinitum, we can imagine that the
continuum is completely ordered:
C C C C
G) S- X < x (k) C X, C. §
We wish now to define and to interpret the complementation of
the ordered continuum. It is evident that the classes X (k) have been
Strung out along the ordered coordinate k in the fashion

*
k
_-º’
X (k)
Now if we were to convert or reverse the rank ordering; that is to rank
from the "greatest" to "least", we would have
part VIII 2.75l CTASS NOTES 83
§ 2 & 2 x (k) 2 x 2 G
We may now establish a biunique, 1–l correspondence between the classes
in the original order and those in the reverse order as follows:
Original Order: O G x < x(1) © x (2) … x (k) “x (n) C3 CŞ
l—l Correspondence: | | | | | | | |
Converse Order: $2 X > X(n) > X(n-1)... x(k.) ... X(1) > 3 > O
\-/
In general as a result of the homomorphism or biunicity the BROUWER
theorem leads us to expect a fixed point, say ko, which may or may not
actually "exist" within the range of values. Moreover we can now
define the class complementary to k as the mate in the correspondence a-
loove; then:
X (k) = X (n + 1 - k)
Such converse order complementation may be conceived as a
"reflection" or "rotation" of the continuum about the fixed point ko.
In physical measuring processes it is usually convenient (but not necess-
ary) to take ko as O -- the physical zero or datum
It should be emphasized that neither a metric nor the concept of
number is required to establish the ordered continuum; rather, order is
founded on the simple act of comparison. Indeed, the establishment of
Order through comparison is a pre-requisite to the construction of a metric
Scale.
ºpe: 533 Lºwer Selection_in_the_Continº
Once order has been established in $ then the operation of
Outer selection on the aggregate { X (k)} will yield the class X which
\-f
corresponds, in the ordered scale, to the class X *ax” That is,
Ux (k)
3 = X *ax)
In the same manner
ſlx (k)
/*
X = * (*min)
F
8l. 2.75l CTASS NOTES
Thus, in a sense, the operators U and ſl may be conceived as upper and
lower selectors, respectively, in the ordered continuum. It is exactly
in this sense that we wish to consider these operators in the discussion
of continuum logic which follows.
D. Continuum Logic
The discussion up to this point has been completely general and
unrestricted. It is now necessary to specialize to the case wherein the
measure of the classes X(k) is some physical variable or value--perhaps,
something as abstract and qualitative as a utility (as in the theory of
games and decision-making), or something as concrete and quantitative as
a weight, length, or voltage. We still need not suppose a scale--of weight,
for example--to order the X(k) since the ordering could in this case be ac-
complished through the use of a balance. The establishment of a scale is a
result, not a pre-requisite, of the ordering process.
With this specialization, the ordering relation C. particularizes
to -- and the operators U and ſl, which now are, in reality, upper and
lower selectors, are to be interpreted as operators--special cases of
generalized functional "P. According to its definition the operation U
on an input bundle
X
yields a single output value:
Y = U (X) = 3.
==
{x (k)}
X(k__..)
the greatest of the X (k) Iſa, X
:
E
Similarly, in the case of the operator ſl,
e-
Y = ſl (X) = x = the least of the X (k) = x (kata)
There are many devices which can perform the operations U and ſl,
i.e., we can actually realize these operations in terms of computing
elements as schematically depicted below:

x LX U H-Y
\–

x TXſ n H-Y
Part VIII 2.75l CTASS NOTES
85
It is often the case that X is an ensemble of time-varying functions;
that is,
X = X(t) = {x, (t) | x = 1, 2, 3, ..., n}
Then, of course TJ {X (t)} will be the greatest of the X, at time t,
While ſl{X (t)} will be the least. These then yield the dotted en-
velopes:
Perhaps the most common method for realizing upper and lower
Selection is the electrical scheme employing diodes. An upper selector
is shown below:
Y NJ
e (t) LT
e.,(t) D-
e=U {*. (t)}
e.(t) P} ~
- E

86 2.75l CLASS NOTES
In electronics such a circuit has long been called an "auctioneer"
circuit; in the same tenor the ſll-circuit is called a "shopper".
Of course, upper and lower selection may also be realized
mechanically and hydraulically:
-Z zº. 2.
—Z- A × Zº Z LZ Z. 2. z Z. Z. Z Z. LZ ZT 7 Z ZT Z
z
t
J. T.
|
T r Jºrn.
X + #3, +x,
Check Walve

part VIII 2.75l CTASS NOTES 87
For the case where n = 2, i.e., where Y ={x1, *2), it is in-
formative to draw contours of the surfaces U (X) and ſl (X).


U- U. Uri U.2 2^
Thus, we see that there is no basic distinction between the operations
U (xi, *2) and ſl (x1, *2), and, for example, the more familiar
) = x + xe or p = (x, *2) = x1,x
algebraic functions such as dº, (x1, X l l' --2'
2
E. Two-Walue or Binary Logic
As a particularization of continuum logic, wherein the %: C8, Cl
assume any values whatever, we now consider two-value or binary logic
Wherein the %k may be equal either to zero or to one. These two values
are often taken as signifying, respectively, falsity (F) or truth (T).
The convention is thus established that
O K 1 Or F K T
O% = 1 Or Fºk = T
Note that the fixed point under complementary order reversal is non-
existent (despite the fact we might call it "1/2"):
When there are two independent variables, i.e., when n = 2 in the
aggregate {*. |k = 1, 2, 3, . . . , n} , the operations of union, inter-
Section, and complementation may be conveniently portrayed in functional
88 2.75l CLASS NOTES
matrix form as shown below:
X. X.
U(X, X2) O | 1 ſl(X, X2) O | 1
O | O O || O
X
2 1 || 1 || 1 X. 1 O || 1


X.
(X)"| O | 1
1 O
X2 1 || 1 || O

A second useful representation completely equivalent to the
above is by way of the truth table, wherein the symbols T and F are
substituted for 1 and 0.

X | Xa U(X, X.)|n(X,X2 (X, )* | (Xa)”
F F F F T T
F T T F T F
T | F T F F T
T | T T T F F
Part VIII 2.75l CTASS NOTES 89
The following interpretations of the union, intersection, and
complementation operations may now be made in classical binary logic:
Operation Symbolic notation Logical interpretation
Union TJ "( ) or ( )"
Intersection ſ]] "( ) and ( )"
Complementation ( )* "not ( y
There are a total of sixteen distinct binary operations L 2
considering polarization; a convenient coding system may be employed as
will be illustrated for the three operations introduced thus far. In
the matrix form the elements are labelled generically Ol } £8 T Y, 8 8. S
indicated below:
X,


O
Cl
9
Y
X2
1.
8
Y a
The entries for a given operation IL are written in the fixed Order Ol 3 Y 8,
and the number for which these are the binary digits is then taken as
the code number of the operation.
Code
Operation Binary Decimal
"or" of 11 7
"and" Ooo! 1
"not X. 11 1 of O 1 O
"not X2" 1 1 Oc 12
The sixteen binary operations are by no means independent. Indeed,
90 2.75l CTASS NOTES
they all may be established from the operations (1), (7), and (10). In
fact, even this set of basic operations is not minimal for we may, for
example, construct (1) from (7) and (10). The truth of this may be de-
monstrated by way of the signal flow graph shown below:

X 2* | |
Tº T ---------- –
X1 | X_2 | (IO) (IO) (7) | (IO)
O O 1 1. 1 O
1. O O 1. 1 O
O 1 1. O 1 O
1. 1 O O O 1

Indeed from the triplet (1, 7, 10) both the pair (7, 10) and the pair
(1, 10) suffice as logical primitives, but the remaining pair (1, 7) does
not.
The question now comes to mind: "Is there a single binary opera-
tion on the basis of which all sixteen operations may be established?" The
two Sheffer Stroke operations "nor" (8) and "nand" (1H) are each such com-
plete logical primitives. A suggestive symbolism will be used for these
operations, namely:
(8) = tº J = nor = not-or = "dagger" ( , )
(1}) = ſh = nand = not-and = "stroke" ( . )
It is easy to demonstrate that these are indeed logical primitives;
we shall simply verify that from (8) the operations (10) and (7)--which to-
gether are a set of logical primitives--may be constructed and we leave the
part VIII 2.75l CTASS NOTES 91
remainder of the proof to the interested reader. The following signal
flow graphs delineate the construction:


F. Multivalued Logic (Post Logics)
We may conceive of a spectrum of multi-valued or n-valued logics
with binary and continuum logic occupying the extreme positions. If we
think of the ends of the real line [o, 1] as corresponding, respectively,
to absolute falsity and absolute truth, then we might interpret all
intermediate positions as corresponding to partial truth, as it were.
It is easy to construct the matrix form for the logical operations
in such n-valued logics; an illustration is given below in the case of the
operation |U :
X,
U(X, X2)| O || 1 || 2 || 3 | . . . . n
O | O | 1 2 3 n
1 || 1 || 1 || 2 || 3 n
2 || 2 || 2 2 || 3 n
X 2 || 3 || 3 || 3 || 3 || 3 n
n n n n n tº ſº tº º n

The n-valued TJ and ſ] Oper 3. Lors then give rise to Post algebras.
92 2.75l CTASS NOTES
G. Hyperpolyhedral Functions
At the outset it was stated that a class of functions was sought
which could be used to describe the behavior of physical systems in general
By proper choice of parameters such functions must be capable of conform-
ance to not only continuous, well-behaved functional characteristics, but
also to characteristics which are inherently nonlinear or discontinuous.
Indeed, functions which play havoc with conventional mathematics must be
rendered into articulate and tractable form in order to describe the be-
havior of many commonplace elements. Take, for example, the simple diode;
plotting the voltage-current characteristic of a real diode generally yields
a curve similar to the one sketched below:
S.

A —-i
It is hardly necessary to point out the discontinuity in the slope of the
characteristic at the origin. Thus, we seek a construct in the context of
which all behavioral characteristics may be described. Such a construct is
founded upon the operations of upper and lower selection, U and ſl, i. e.
the operators:


x [X U H-Y x [ix] n H-Y
shall be utilized as the fundamental building blocks in the synthesis of
complex, multi-dimensional functions.
Consider, for example, the operation sketched below:

X —— U Ha-Y
A
Ž
Part VIII 2.75l CTASS NOTES 93
This yields a function Y = Y(X) which may be sketched as follows:
A Y

+ º-
X
If now We take as a unit cell or polygonal primitive
x –
–H
–M —— |

Cº-
U
Y
Ž O
A Y
Y = U (o, X-M)
- X
and then write
Y = X. a U (0, x - M.)
k=1

the result is a polygonal function which might appear as sketched below:

| Y
^^T\_-_2^
91; 2.75l CTASS NOTES
We can imagine that this is indeed a funicular polygon; i.e., a two
dimensional curve consisting entirely of line segments.
Next, we consider a more general operation which embodies
both U and ſl:

X. ——
x, —
ſl H- U Hº- – Y
ZZZZZZZZZZ O
%2, |z Y = U ſo, ſh(X, X,)]
/ Ys
2^ Y 2
/ Y
/ 1
Y = o -X,

An immediately evident generalization yields the polyhedral primitive, --

X4 ——L
* - La
* H | T T.

part VIII 2.75l CTASS NOTES 95
From which we construct the polyhedral function
Iºl t *
a 13 U [ O, ſ] (X, -M, %2 -º) J
1 i = 1
Y
:
:
J
A representation of this surface would reveal that it is composed en-
tirely of triangular, facets; the following sketch illustrates this:
Y

X2
Generalizing further, we arrive at the n-dimensional hyperpolyhedral
function
Y
— Kri. tº j
i, j, k, . . .u *…..." U [o, ſl(X, M1, X2 M : ,
X3- M. , . . . Xa-M. )]
|r = 1,2,3, ' ' 'n } = Ha ( X )
Hn
{
Xr
96 2.75l CTASS NOTES
It is now a relatively simple conceptual step to hyperpolyhedral
computing systems which utilize, as elements, hyperpolyhedral functions
Ha (X). We thus realize the fantastically variegated universe of functions
at our disposal for modelling system behavior. ſt is to be emphasized,
moreover, that such functions may be embodied in practical computing hardware.
Consider, for example, the polyhedral multiplier -ºx
z - % uſe, n(x-1,x)] + 3 Uſon (X,Y-9]
O j = 1
which is a first approximation to the product Z = XY.
When X and Y are positive, this does, in fact, give exact results for
integers.
Into the realm of hyperpolyhedral computing systems we certainly
must admit implicit functions wherein the output depends upon itself as
well as the inputs. That is,
Y = p (X,Y)
Schematically, the implicit feature appears as single or multiple feed-
back loops within the structure of the function which insert the output Y
at various stages in the forward computation process. A simple example is
the following:
Y = U [o.ſ. (K, X+Y)]

part VIII 2.75l CTASS NOTES 97
The open loop response characteristic is simply
AY
-->
X
With the addition of the feedback loop the response becomes:
+ A
K
A
—Cº-
Such a function might, for instance, be used to model the behavior of
a control valve as sketched below:

98 2.75l CTASS NOTES
Background Reading--Continuum Logic
1. Alonzo Church, Introduction to Mathematical Logic.
In the context of true-false or binary logic the important "con-
nectives" or operators are introduced.
2. Paul Rosenbloom, The Elements of Mathematical Logic, pp. 51-65.
A discussion of multi-valued logic is given. The reader is cautioned
to note that Rosenbloom interchanges the usage of U and ſl e
3. Hans Reichenbach, The Theory of Probability, pp. 387-389.
The author discusses some of the implications of a multi-valued logic ;
the reader should note that n-valued logic occupies a position on a continuus
at one end of which is binary logic and at the other end of which is continue
logic.
Background Reading--Hyperpolyhedral Functions
1. George A. Philbrick, Continuous Electronic Representation of Nonlinear
Functions of n Variables. (Palimpsest)
The author introduces the concept of piecewise linear functions built up
from U , ſl, and " + " for use in analog computing when it is desired to
fit an analog model to a body of empirical data.
2. Thomas E. Stern, Piecewise-Linear Network Analysis and Synthesis.
A formalism for dealing with piece-wise linear networks is developed
from the fundamentals, although a rather unfamiliar nomenclature is used.
Included are examples of polyhedral and pyramidal functions, as well as more
sophisticated surfaces.
3. S. A. Ginsburg, Logical Method for Synthesizing Function Generators.
The development and viewpoint in this paper are basically similar to
that in the Philbrick paper mentioned above. Here, however, somewhat more
attention is given to the background of the logical constructs which give
rise to the synthesis of piecewise linear functions.
part VIII 2.75l CTASS NOTES 99
4. G. A. Korn and T. M. Korn, Electronic Analog Computers. Chapter 6.
In the context of a general discussion of analog computer techniques
are included brief descriptions of polyhedral multipliers and diode function
generators.
5. H. J. Zimmerman and S. J. Mason, Electronic Circuit. Theory.
Use is made of networks of ideal diodes in the synthesis of models
of various essential circuit elements.
1 OO 2.75l CTASS NOTES
IX The Steady-State of Energetic Systems
A. Introduction
The analysis of the steady-state plays a dominant role in pro-
viding us with an overall understanding of the behavior of energetic sys-
tems. Although the steady-state case, per se, is rather sterile and un-
interesting (but by no means trivial!), an insight into the steady-state
behavior of a system forms the basis upon which the analysis of its stabil-
ity and transient behavior may be founded. Consider, for instance, the fact
that the stability of a system may be evaluated by observing the result of
small excursions about a steady operating point. Moreover, a transient
condition in a stable system is the means by which its operating state al-
ters from one steady condition to another.
We shall wish to distinguish between two types of steady-states:
(i) the static case, wherein the power flux is identically zero and all
that is required is a statement of the distribution of internally stored
energy; (ii) the stationary case, wherein the power flux is constant, at
least in the mean.
In rendering the description of the steady-state in terms of mathès
matical relationships we are faced with the problem of modelling all sorts
of nonlinear, as well as linear, behavior.
B. The Static Case
Consider a four-ported system S. We define S to be static if it
is in equilibrium with its environment E such that all the f" s, both in-

part IX. 2.75l CTASS NOTES 1 OT
ternal and external, are identically zero. If this is indeed the case,
then certainly the power flow IFP is everywhere zero, and the state of
the system is entirely described through a specification of the distribu-
tion of internally stored energy IE. In any real structure, which is in-
herently deformable, this is tantamount to a specification of the deform-
ation of the system, i.e., the displacement of every particle thereof.
Cases in point are electro- and hydro-static fields.
A static system, then, is one which has passed through some sort
of transient condition during which power flows, from the environment and
among the various parts of the system, were occurring. That is, we must
conceive of the attainment of static equilibrium as a process requiring
a finite interval of time. As the equilibrium state is approached, the
power flows all decrease, and of course, vanish utterly when that state is
reached. However, in any such process of practical interest there has been
a net influx of power, leaving the system with internally stored energy
which is distributed in a manner characteristic of the conditions on its
boundary. The fact that such an energy distribution is ultimately re-
ducible to a distribution of deformation leads us to consider displacement
quantities
t
- OO
In particular, at each of the ports we are concerned with the pair of
conjugate variables (e q+), similarly, a pair (e, q) may be identified
i”
at each internal bond.
C. The Stationary Case
Referring to the sketch at the beginning of the previous section,
if S is operating in a stationary state, then it is in dynamic equilibrium
With E. In the case of strict stationarity the time derivatives of the f"s
both internal and external, vanish identically. A weaker condition is that
Of quasi-stationarity wherein the time averages of the f's are Zero, i.e. ,
each of the flows is fluctuating about some steady mean value.
1 O2 2.75l CLASS NOTES
Strict Stationarity: f (t) = o or f (t) = F = constant
for all bonds
..". IP = constant
for all bonds
Quasi-Stationarity: f (t) = c or f (t ) = F = constant
for all bonds, e.g.
f (t ) = F * F2 sin (Ut
•". IPD = constant
for all bonds.
D. Determination of the Steady-State
By "determination of the steady-state" is meant an analysis
which leads to a computation of the equilibrium magnitudes of the signif-
icant dependent variables of a system corresponding to a given set of in-
dependent variables, i.e., the impressed conditions at the boundaries.
Such an analysis is by no means trivial in the case of a complex system.
It has been stated repeatedly that in order to perform any sort
of incisive quantitative analysis the non-causal energy bond reticulation
must be transformed into a causal bond reticulation. Thus, in the
case of the four-port, D,
f | e
e 2 D 2
|
f3
f e
the assignment of causality might result in the diagram
Now, if there be any determinant stationary condition we must
be able to write
part IX. 2.75l CTASS NOTES 103
f = @, (e1, f2, e3, fl.)
ea-ſp; (e., f
f3 –d5; (e1, f2, e3, fi.)
el, -d: (e1, f2, e3, fi.)
If we are searching for a static condition then, of course, we replace all
the f's by displacement quantities (q); hence, there must exist a set
. 1
q =(p. (e1, 92, e3, q.)
2
e2 = (p. (e1, 92, e3, q.)
— A3
q3 = (p. (e1, 92, e3, q.)
l,
el, - (p. (e1, q2, e3, q,]
In the above it is to be understood that the functions (p are Of
the most general type. However, an important special case is that of
linear functions, although in the real world true linearity is never
found. Still, linear or linearized analysis facilitates the computation-
al process and permits approximate answers to be obtained quickly. These
are Only in error to the degree that the system cannot be made to conform
to a linear characteristic within its operating range. The fact remains,
however, that one can quickly cite examples of elements which are essentially
nonlinear in character (i.e., linearization is not possible); indeed, such
nonlinearity is exploited by the designer and therefore cannot be overlooked
by the analyst. 1
Suppose that pr is indeed linear; then, for small changes in
the independent quantities,
This, evidently, has reduced the problem to the ultimate in Simplicity,
yet the ,---the influence coefficients -- are not always easy to evaluate.
E. System Reticulation for Steady-State Behavior
The most general case of a static functional transformation, i.e.,
One which yields up an output value Y (t) corresponding to an input value
X (t), is the operator (ſ)

|X dº IX Y.
1Ol; 2.75l CTASS NOTES
In the present context we wish to consider practical measures which will
facilitate the determination of the steady-state behavior, and to this
end it is necessary that we reticulate the function @ . Consider, for
example, the two port

E
Corresponding to each bond there is a pair of conjugate variables, say
e and f. Thus associated with this system are a total of fourteen
quantities, only two of which are environmentally determined, i.e., are
independent or input quantities. Hence there are twelve dependent or out-
put quantities, each of which must be evaluated in order to specify the
steady-state behavior. That is,
{ X , , X a
Y
Y { Y . , Ya, . . . , Y, )
The only practicable decomposition of the function {i} is One
which will yield up each of they, individually. That is, we shall have
to reticulate @ into twelve primitive operators of the form

X T d), HY, i = 1, 2, 3, . . . , 12
In the most general case, wherein implicit operators are employed, we
would thus arrive at the reticulation:
Part IX 2.75l. CLASS NOTES 105
Z% Żºl Y
1
Ž 77% dº Y2
F. Nonlinearity
It is propitious at this point to dwell upon the problem of non-
linearity, and how, in general terms, nonlinear behavioral characteristics
are rendered into tractable form for the purposes of quantitative analysis.
We must deal with two types of nonlinearity: (i) curvilinearity,
Which may be linearized for small excursions about a steady operating point;
(ii) essential nonlinearity which cannot be linearized. Examples are

1O6 2.75l CTASS NOTES
sketched below:

Y
A

|-
Curvilinear Essentially Nonlinear
Characteristic Characteristic
Obviously it is impossible to linearize the essentially nonlinear
characteristic in the vicinity of the point O without overlooking a most
significant aspect thereof.--namely, the discontinuity in slope at 0.
Curvilinear characteristics may be approximated to any degree
of precision by functions constructed from the basic connectives or opera-
tions
The commutivity of the two inputs in the case of "sum" and "multiply" is
most important. However, "minus" and "divide" are asymmetric, and there-
for non-commutative. These four operations are the basis for all
algebraic functions.
A class of "logical functions" —- the hyperpolyhedral function3-"
have been introduced which suffice to construct any linear, curvilinear,
or essentially nonlinear characteristic to an arbitrary degree of precision
Part IX. 2.75l CLASS NOTES 1O7
These are founded upon the operations
-->4)--- I U H-

Due to the piece-wise linear property of the hyperpolyhedral functions
a curvilinear characteristic is automatically linearized, while the dis-
continuity ºf an essentially nonlinear characteristic may be exactly
preserved.
108 2.75l CLASS NOTES
X. Functional Transformations and Computing Functionals
A. Introduction
We have said that a multi-ported system which is undergoing
some sort of generalized energetic process may be conceived as an element
that accepts an input vector X = {x, | i = 1, 2, 3, . . . , n} upon which
it operates according to the functional "P to yield an output vector
Y (Y, | = 1, 2, 3, …, n ; ; the functional "P is such that the
entire past state of TX is scanned to yield a single present value of
Y . Now, in order to compute the system state at any particular in-
stant it is essential that not only the external outputs Yi (t) but also the
States of each of the internal bonds be computed. We thus desire a retic-
ulation of "HP which permits each of these internal States, as well as the
external output variables to be displayed individually.
B. Computing Functionals
To permit any sort of computing or quantitative description,
whether by machine or by hand, the element
x IXL up Y
must be reticulated into a set of primitive scalar output functionals
x LXI tº H-Y
We are thus concerned about the specific way [P must be reticulated for
computing purposes to allow the state of the system to be completely
described. To emphasize that this concern stems from the desire to com-
pute we shall introduce a special symbolism for the primitive computing
functional:

Part X. 2.75l CTASS NOTES 109
The small circle indicates the output side of the functional. Deleting
the box, we have simply
Thus, in order to model a two-ported element two computing functionals
might be interconnected as follows:
X T
1 Y.
T. Y
2
As is the case in this illustration, it is generally necessary to employ
feedback loops in the synthesis of complex computing functionals. As before,
the presence of such loops results in implicit computing functionals. Now
the functional
represents all possible (i.e., conceivable) deterministic transformations
of inputs {x, | i = 1, 2, 3, . . . , n) into an output Y. It determines a
present value for Y from the present and all past values of the Xi. Under
no circumstances whatever can we demand, nor is there any use for a functional
which requires a future value of an input to compute a present Y. Indeed,
this may be regarded as the rule or law for the construction of computing
functionals.


11O 2.75l CLASS NOTES
Consider as two examples of commonly accepted functional
operators the ordinary time differentiator and time integrator, symbol-
ized
ID
Derivative :
Now, it is theoretically impossible to compute the "exact"
derivative of a variable without a knowledge of its immediate future. That
is, by definition,
Y (t) = ID [X (t) ) = d [X (t))
d:t

= lim X (t + At) - X (t - At)
At —- o 2. At
Thus, according to our conception of an "allowable" computing functional
differentiation is not physically realizable.
On the other hand, the operation of running integration is readily
constructible for
* (*) - ſ & 6), - I ſat . . ()
requires only a knowledge of the past history of X to compute the present
value of Y. Thus, we conclude that, as a rule of thumb, differentiations
should be avoided in computing -- indeed there is no way to accurately
differentiate -- while a very accurate integrating operation may be physic-
ally realized in analog or digital form.
A second important conjugate pair of functional elements are the
time advance and time delay:
IE
Advance:
A
Delay:
Part X 2.75l CTASS NOTES 1 11
We immediately note that IE is not an allowable computing functional Since
Y (t) = IE * [X (t)} = X (t + T)
On the other hand, A is realizable since
Y (t) = A + [X (t)) = x (t - T)
Thus, we see, by way of the above examples, that while the rela-
tion
T T's I ( II º' identity)
holds in theory there are many cases where the converse.or inverse of a
functional is not realizable, and hence the symbol TT has no physical
Significance. Moreover, although computing functionals are the basic
building blocks in any computing program, analog or digital, many function-
als may be realized only in one of the two media. For example, precise
integration is possible only on the analog machine while precise time delay
is a digital operation. However, both the physical model realizable on the
analog computer, or the "logical engine" resulting from a digital program
may be depicted as shown below:

112
2.75l CLASS NOTES
That is, in general a computing representation or model of a given system
may be depicted as a network of computing functionals, containing multiple
feedback loops and yielding as outputs all the variables of the system
necessary to fix its state.
Background Reading - Computing Systems
(l)
(2)
HARTREE, D. R. Calculating Instruments and Machines.
This classic work treats the principles underlying both analog and
digital machines and describes some of the instruments of historical
significance.
IVALL, T. E. Electronic Computers.
A collection of readable British essays on analog and digital
electronic devices originally appearing in "Wireless World".
SCOTT, N. R. Analog and Digital Computer Technology.
A contemporary work detailing the structure and applications of
modern high speed machines. ©
von NEUMANN, J. The Computer and the Brain.
A most provocative posthumous essay by the late great mathematician,
leaving unanswered the query as to how nature yields such accurate
and reliable signals from noisy and erratic components.
Part XI. 2.75l CTASS NOTES ll 3
XI. Diagrams and the Coding of System Structure
A. Signs
We are here concerned with a problem of communication -- specific-
ally, the transmittal of the description of a reticulated system from
one human mind to another. We seek a form of description which is complete
yet sufficiently succinct, and of such a nature as to permit a verbal trans-
mittal, over the telephone for example. Thus, an encoding of the schematic
description is indicated.
To provide a background for this discussion we consider briefly
the general theory of signs or semiotics. Charles Sanders PEIRCE states:
"A Sign, or Representamen, is a First which stands in such a genuine triadic
relation to a Second, called its Object, as to be capable of determining a
Third, called its Interpretant, to assume the same triadic relation to its
Object in which it stands itself to the same Object."
All sorts of human communication is accomplished by way of a sign-
activity. That is, an individual A employs a sign S to communicate an
idea of an object 0 to a second individual B in whose mind an interpre-
tation I (also a sign) is evolved as a result of perceiving S. The
Situation is not uncommon in engineering analysis wherein the individuals
A and B are the same person, and S is a sketch or diagram drawn as an aid
in problem-solving -- a form of self-communication.
Peirce is to be credited with the trichotomy of signs into the
classes: (i) Icons; (ii) Indices; (iii) Symbols. Quoting directly from
Peirce :
"A sign is either an icon, an index, or a symbol. An icon
is a sign which would possess the character which renders it
significant, even though its object had no existence; such as a
lead-pencil streak as representing a geometrical line. An index
is a sign which would, at once, lose the character which makes it
a sign if its object were removed, but would not lose that char-
acter if there were no interpretant. Such, for instance, is a
piece of mould with a bullet hole in it as a sign of a shot; for
without the shot there would have been no hole; but there is a
hole there, whether anybody has the sense to attribute it to a
shot or not. A symbol is a sign which would lose the character
which renders it a sign if there were no interpretant. Such is
any utterance of speech which signifies what it does only by
virtue of its being understood to have that signification."
11 l; 2.75l CTASS NOTES
Thus, an icon is a characterizing sign which exhibits in and by
itself the properties which an object must possess to be denoted by it.
Examples of icons are photographs, models, star charts, and chemical
diagrams.
An index is a directing sign which refers to its object by a
dynamical or spatial connection and otherwise bears no resemblance to the
object. Sub- and superscripts, index marks, clocks and meters, and any-
thing which focuses attention or startles may be considered an index.
A symbol is a characterizing sign which always involves a rule
or convention to establish the connection with the implied object. The
utility relies utterly upon the mind of the interpreter to conjure up its
meaning and significance. For example, names of people, things, stars,
and elements, as well as code marks and mathematical notations, are all
symbols.
A sign -- a schematic diagram, for example -- which refers to a
physical system as its object, embodies all three classes of sign-action.
The bare skeleton of the diagram is iconal, exhibiting directly certain
properties of the system. This skeleton, however, is endowed with various
labels, arrows, etc. which involve indicial and symbolic sign-action.
For example, in a block diagram a component might be labeled "W l',
which directs the reader's attention, or perhaps memory, to the previously
made definition of this functional -- as distinguished from the definitions
Of V, V, etc. -- and thus involves both indicial and symbolic sign
activity.
B. Communication of a Computing Structure
Schemata of various sorts -- block diagrams, signal flow graphs,
etc. -- are invaluable aids to the description of systems and to the
communication of their structure. We are specifically concerned with the
problem of describing and communicating the nature of a computing structure,
i.e., a network of computing functionals T. We desire a method which is
sufficiently flexible to describe the most general types of nonlinear
networks and which will lend itself to encoding for the purpose of verbal
Part XI. 2.75l CLASS NOTES 115
transmittal.
Two essential dichotomies may be discerned in the realm of
schematic representations of system structure. The first is now
familiar to us: the causal (bilateral signal flow) vs. the non-causal
(energy bond) representations. The second dichotomy subdivides the
large and variegated class of "branch-node" schemata into, on the one
hand, those representations which identify the functional operators with
the nodes and the signal variables with the branches (block diagrams);
and, on the other hand, those representations which identify the variables
with the nodes and the operators with the branches (Mason-Tustin signal
flow graph ).
y2
- | X3 €3 € 5
§§ %.N- 5 e, /ſ, Z N
y; X4 Xs `s fs €
—sº-
B → C T. B —Éa— C +
y4 Y6 fa fe
Causal Bilateral Signal Flow Non-Causal Energy Bond
Diagram Diagram
X
X 2
X 3
Functional Block Diagram Signal Flow Graph
Operators: Nodes Variables: Nodes
Variables: Branches Operators: Branches

116 2.75l CTASS NOTES
The non-causal representation, a generalized circuit diagram,
uncluttered and simple, enables the experienced analyst to visualize
quickly the behavior of a system, while the causal description is es-
sential for a detailed quantitative understanding of its performance.
The block diagram is especially suited to determining the transfer char-
acteristic of a structure of interconnected elements, provided the bound-
aries of the elements have been correctly chosen. In the case of a com-
puting structure, which is our present concern, these boundaries are gen-
erally self evident. The block diagram has the distinct advantage of be-
ing applicable to the case of nonlinear as well as linear systems. The
signal flow graph, on the other hand, may be used precisely only to de-
scribe linear networks since a summary action is implied at each of the
nodes; that is, for example
x + IF ~ x
1 - of To 21 T2
For all these cases, however, we seek a representation which is
capable of being encoded, and for this purpose the following branch-node
structure suggests itself;

Part XI. 2.751 CTASS NOTES 117
But this structure may be easily encoded by way of the following tabu-
lation:
Y IT X
l IT, 1, 2, 3
2 Ta 2, 3, 1
3 T 3, 1, 2
3

Corresponding to each node there is a single output y, that results from
the operation of the associated functional IT upon the input signals,
which in this case are simply the outputs of all three nodes. Thus, for
example, the first row of the table might be read, "the signal yı re-
aults from the operation of T, upon Wi 2 y2” and WA". In actuality, of
course, the entries in the T-column would indicate the nature of the
functionals, say by way of a numerical coding: 1 for an upper selector,
2 for a lower selector, 3 for an integrator, etc. It is thus possible
to communicate succinctly a complex structure in the form of a table or
sequence of numbers only. The task of transforming this number sequence
into a readable diagram and vice versa is almost trivial.
} What we have done here is to treat a specific application of the
broader theory of graphs, which in turn stems from the mathematical
discipline of combinatorial topology. This general study deals with the
ways in which the structural connexity of a space may be described and
communicated; we recognize this as precisely the problem with which we
have been concerned, wherein "the space" happens to include a computing
system and the connectedness of interest to us embraces the functional re-
lationships between the several computing components. In combinatorial
topology connexity is communicated by way of incidence matrices, a
condensed form of which are the coded tables here suggested for use in
Communicating system structure.

c. Combinatorial Topology - Incidence Matrix
A. W. TUCKER states: "Topology deals with the rudimentary
geometrical properties which depend on continuity rather than on size
and shape." The domain of discourse is a space in which the topologist
attempts to establish theorems related to connexity and structure.
118 2.75l CTASS NOTES
Henri Poincaré is generally cited as the originator of this branch of
mathematics, which he named analysis situs.
Connexity is depicted by way of linear graphs or, alternative-
ly, by incidence matrices. A linear graph is constituted from nodes and
branches. A digraph (directed graph) is a linear graph in which the
'branches have been endowed with a directional sense. An example of an
ordinary linear graph is given below:
In this graph there are nine branches and six nodes. The associated
incidence matrix may be easily written:
8, lo C d e f g h i
1 1 O 1 O O O O O O
2 1 1 O 1 1 O O O O
3 O 1 O O O 1 1 O O
l; O O 1 1 O O O 1 O
5 O O O O 1 1 O 1 1
6 O O O O O O 1 O 1
In this matrix an entry of "1" indicates a branch-node impingement, while
an entry of "O" indicates no impingement. The elements are therefore
labelled incidence numbers.
A topological space is a complex constituted from a number of
simplexes or cells; these are labelled, according to convention, as

Part XT 2.75l CLASS NOTES 119
follows:
O-cells : nodes
1-cells : branches
2-cells : loops
Hence, the incidence matrix discussed above, which depicted a node-branch
structure, is called the "O1" incidence matrix, or simply II c. Poincare
defined the numbers
*k = number of k-cells in a complex
ao = number of O-cells
a. = number of 1-cells
a2 = number of 2-cells
The rank of the incidence matrix Ik K+1 is denoted *k. Since no signific-
y
ance has been attributed to II for k=-1 it is necessary to restrict
k, k+1
this definition to hold only for k = 0, 1, 2, ... . Hence, we say that
rk = rank of "... (for k = 0,1,2,...) ; r_j = O
We also define the kth order Betti number
bk = *k - *k - *k-
so that, in particular, the zeroeth and first Betti numbers are given by
b = a - r 3 b. = a- - r. - r
O O O 1 1 1 O
Which requires that some significance be attached to bo. Accordingly, we
define
°o = number of separate connected parts in a complex.
With this it is now convenient to define the rank R of the linear graph as
R = r = a – b
O O O
which yields an alternative definition of the first Betti number for linear
&raphs, since r = 0, namely
1
12O 2.75l CLASS NOTES
It is also propitious to observe that some authors refer to the first
Betti number as the nullity, N.
The Euler characteristic is defined in terms of either the b
k or the *k
as follows:
E
k
K = 2. ". (-1)
k
2. al. (-1)
The Euler characteristic for a connected linear graph of W nodes
and B branches is simply
K = b - b. = a - a
O o
Or K = 1 - N = W - B
Since R = W - 1 we thus obtain the fundamental invariant relation for all
linear graphs

B = R + N
which is identical to the previous result b. : al - R.
By way of illustration of the significance of some of the above character-
istic numbers three theorems are stated.
Theorem 1. If we start with the O-cells of a linear graph and
add the 1-cells one by one, the number of 1-cells added joining nodes not
previously connected is *o and the number of 1-cells added joining vertices
already connected is b. .
In connection with this theorem it is well to point out that a
complex which contains no loops -- i.e., no closed paths within the structure-
but which would contain a loop with the addition of a single branch, is called
a tree. A forest is a complex consisting of a number of disconnected trees.
Theorem 2. The first Betti number of a forest is zero.
Theorem 3. If the first Betti number of a graph is bi, We C8 Il
remove b. 1-cells from it, but no fewer, which will reduce it to a forest.
These theorems are stated without proof for the purpose of illus-
tration only. From them we observe the importance of the rank R and nullity
N in the topological characterization of a space.
part XI. 2.75l CLASS NOTES T 21
Background Reading - Signs

(1)
(h)
PEIRCE, C. S. Philosophical Writings, (edited by J. Buchler), Logic
as Semiotic: Theory of Signs.
Peirce presents his form of the theory of signs-—the logic of semiotic.
Much of the point of view adopted in this course originates with Peirce,
although this subject has been taken up and colored by more recent
thinkers in this field (and occasionally presented in more readable
fashion).
GALLIE, W. B. Peirce and Pragmatism
Gallie presents a compact summary of Peirce's semiosis and theory of
signs.

YOUNG, J. W. Lectures on Fundamental Concepts of Algebra and Geometry,
pp. 226-239, (Growth of Algebraic Symbolism, by
U. G. Mitchell)
The history of the use of symbols in algebra and arithmetic is traced.
MORRIS, C. W. Foundations of the Theory of Signs
Morris presents (without adequate citation) much of Peirce's thought
on this subject.
CHERRY, C. On Human Communication, Chap. 3, pp. 219–226.
This is a modern text in which signs are discussed as a part of the
loroader subject of communication. Much of Peirce's thought is again
represented.
TRUXAL, J. G. Automatic Feedback Control System Synthesis, Chap. 2.
A discussion is given of the disadvantages of block diagrams and
the Mason signal flow graph is presented as a useful tool in systems
analysis.
1 22
2.75l CTASS NOTES
Background Reading - Topology
(1)
(2)
(3)
SYNGE, J. L. The Fundamental Theorem of Electrical Networks, Quarterly
of Applied Math., July 1951, p. 113.
In his development of the theorems and concepts leading up to the
"fundamental theorem" the author employs a very readable intuitive
approach. Much of this development is purely a discussion of
topology and digraphs which is direct support of the material on
this subject presented herein.

TUCKER, A. W. The Topological concept of Space. (A lecture given
at the Galois Institute of Mathematics).
Tucker discusses many of the essential concepts of topology without
resorting to formal mathematical proofs. Thus, his approach lends
itself to a deepening insight into this subject, beyond the super-
ficial statements made in these notes.

SINGER, James. One-Dimensional Analysis Situs. (A lecture given at
the Galois Institute of Mathematics). (1935)
This reference contains much of the material used in these notes.
The theorems merely stated herein are stated and proved by Singer, as
are several additional theorems which concern the structure and
connexity of linear graphs.

SINGER, James. Two-Dimensional Analysis Situs. (A lecture given at
the GalCTs Institute SFVEEHematics). (1936)
Many of the statements made by Tucker are discussed more thoroughly
in this reference which extends, along intuitive lines, into the
topology of two-dimensional spaces (surfaces).
Part XI 2.75l CLASS NOTES 123
D. Coded Representation of Graphs and Digraphs
The original branch-node incidence matrix of the previous
section may be encoded in a simple array merely by condensing or col-
lapsing either rows or columns in the following alternative fashions:
ROW CODE COLUMN CODE
1 a c a 1 2
|b 2 3
2 a b d e C 1 l;
3 b f g d 2 l;
l; c d h e 2 5
f 3 5
5 e f h i g 3 6
h l; 5
6 g i i 5, 6
It is readily apparent that an encoding by rows gains rapidly in
efficiency and simplicity as the connexity of the structure increases if
the specific node and branch tags are both to be transmitted. Nevertheless.
We shall have frequent occasion to use both forms of coding as required.
* * * * * * * * * * * * * * * * * sms - ºr *s sm - * * * * * * * * *
In addition to the first (node-branch) incidence matrix, the
matrix indicating the cyclic or closed-loop character of the system structure
is also of fundamental topological interest. This circuital or branch-loop
incidence may be determined for any reticulate system by indicating the
incidence of all branches upon N + 1 independent loops where N is the nullity
(i.e. the number of branches-out-of-tree) of the structure.
L O O P S
r _A_ —Y
I II III IV W
B r 8, 1 1 º º o
R b 1 ſº & 1 *
A 1 1 e • Q

12); 2.75l CLASS NOTES
Dual Graphs.
For the graph previously depicted and discussed, the rank
R = 5 and the nullity N = li. Therefore, for the dual graph, the rank
R* = l; and the nullity N* = 5. This dual graph may be constructed directly
from the transpose of the second incidence matrix, merely using the topo-
logical dual isomorphism:
(N + 1) Loops –- (R* + 1) Dual Nodes
(B) Branches <!—- (B+) Dual Branches
Thus the first incidence matrix of the dual graph is merely the
transpose of the second incidence matrix of the original graph. This gives,
in coded form:
I a” b c 1 gº h! if
II at c : dº
III dº e º hº
IV b e º f :
V f : gº i !
corresponding to the graphical form:
DUAL:
II dº III e i IV ft W
º
NHé
B% = 9
by contrast to the original figure:
oRIGINAL * ~ *
N = l;
B = 9
A well-known theorem of topology due to Hassler WHITNEY states that
a dual graph can exist only for a planar graph (i.e. a linear graph which ca"
be homeomorphically mapped onto a plane or a sphere).


Part XI 2.75l CLASS NOTES 125
E. Coding the Energetic Structure of Multiport Systems
The previous incidence matrices and equivalent codes may be
used for the topological structuring of multiport systems, provided that
the System is closed, and the following correspondence is employed:
MULTIPORT ELEMENTS <!—- NODES
POWER BONDS <!—e BRANCHES
First, it is possible to close all otherwise open multiport
Systems by a Simple artifice. Since an n-ported system S must necessar-
ily be bonded to an n-ported environment E , we can always annex the en-
vironment to the System itself to form a necessarily closed system, in
the fashion:
e.
s
To emphasize the complementary aspect of the environment in this circum-
stance we may denote the environment of S by the underscored symbol, S.
Thus the closed system becomes
,-S
* : * S
SL’
The duality between S and S is complete since
S
S
Which means in words that the environment of the environment of a system
is the system itself.
Thus if the system S is coded as:
S a b c . . . n
then we may designate the environment S of S as
S a b c . . . n
and the two subsystems as a single closed system becomes in coded form:
S a b c . . . n
S a b c . . . n
If the system is closed to begin with no complementary element, S ,
is required for closure.
126 2.75l CLASS NOTES
In the nontrivial case the internal energetic reticulation is
also given. For example the structure:
~" s.
f
might be encoded as
S a b
A a c f
B C d e
C b d e f
From the code itself we may infer the following facts, among others:
1) The overall system is a 2-port
2) It has been reticulated into 3 multiports, namely
a) two 3-ports
b) one li-port
The assignment of CAUSALITY may be accomplished using the code
alone as follows:
I ... EFFORT inputs are unmarked;
II . . . FLOW inputs are underscored.
Thus a typical causality would be as follows:
:
b:
:
e f
This means that S itself is of the form
S a b
since S a b = S a b is its complement.
Part XI 2.75l CLASS NOTES 127
The Mechanics of System Interconnection
Consider that we were assembling the original system from the
subsystems A, B, C, whose ports have been assigned consistent causality.
In this case we would have started with
A a b c 5 B a' bºc' 3 C a” c' d"
The primes would not generally appear in the separate listings and are
here indicated only to prevent confusion.
The initial step requires the unique labelling or, better,
numbering of all ordered ports, for example as follows:
A 1-2-3. B 4.5-6. C 7.8.9.10
This array corresponds to the (element-bond) incidence matrix

The particular interconnections given previously may now be
expressed by
i
:
.
O
These column identifications result in column additions in the
System matrix and reduce the matrix to:
;
!
:

!
1
* >
e
:
*
ſe
tº
tº
f#
128 2.75l CLASS NOTES
where the environmental S row may be added such that the column sum
vanishes identically for every column.
The corresponding coded system may now be written:
:
:
:
:
f
To render this in a coded form identical to that of the original,
only simple permutation of letters is required in the form:
$ f' 'b C
|b C f
This yields the equivalent code
:
b
f
f
J
|
.
:
b.
which is merely a permutation of the first system and is therefore topo-
logically or structurally identical.
Background Reading -- Graphs, Digraphs, and Networks
(1) CAYLEY, A. On the Analytical Forms called Trees, with Application to
the Theory of Chemical Combinations, Report of the British
Association for the Advancement of Science, pp. 257-305 (1875)
*mº ºmsºmº uºmºsºmºmºmºm sºme
(2) KEMPE, A. B. A Memoir on the Theory of Mathematical Form, Philosophical
Transactions, pp. 1-70 (1886).
A little known and truly remarkable anticipation of combinatorial topology
whose origin is usually credited to the papers of POINCARE.
(3) KOENIG, D. Theorie der Endlichen und.Unendlichen Graphen, Chelsea
Publishing Co., New York (1950).
This relatively recent book has now become a classic in this field.
Part XI 2.75l CLASS NOTES 129
Background Reading -- Graphs, Digraphs, and Networks (continued)
(l) WHTTNEY, H. Non-separable and Planar Graphs, Transactions of the American
- Mathematical Society, Vol. 3), pp. 339-362 (1932).
The author here proves for the first time that duals exist only for planar
graphs and therefore for planar logical and electrical networks.
(5) HOHN, F. E., S. SESHU, and D. D. AUFENKAMP. The Theory of Nets, Trans-
actions of the IRE, Vol. EC-6, No. 3, pp. 1511-161 (September,
1957).
The authors generalize the concept of a digraph into a net to include
certain higher order structural information. Many theorems and properties
of universal value may then be adduced.
(6) SHIMBEL, A. Structure in Communication Nets, Proceedings of the Symposium
on Information Networks, Polytechnic Institute of Brooklyn,
Brooklyn, pp. 199-2O3 (1954).
This paper propounds concepts and methods which enable the determination
of the minimum paths and resultant trees in any communication digraph.
(7) HARARY, F. Structural Duality, Behavioral Science, Vol. 2, No. 1,
pp. 255-265 (October, TT957).
A very readable treatment of various duality transformations applied
to graphs and digraphs.
(8) GRAYBEAL, T. D. Block Diagram Network Transformation, Electrical Engineer-
ing, pp. 985-990 (November, 1951).
(9) STOUT, T. M. A Block-Diagram Approach to Network Analysis, AIEE Trans-
actions, pp. 255-260 (November, 1952).
(10) MASON, S. J., Feedback Theory.--Some Properties of Signal Flow Graphs,
º Proc. Inst. Radio Engrs. H1, 114-1156 (September, 1953).
(ii) .......... Feedback Theory.--Further Properties of signal Flow Graphs,
Proc. Inst. Radio Engrs. Hº, 920–926 (July, 1956).
The four papers above deal with transformations and equivalencies of flow
graphs.
SESHU, S. and REED, M. B. Linear Graphs and Electrical Networks .
This excellent text has a summary of much of the above and an excellent
bibliography.

130 2.75l CLASS NOTES
XII. State-Determined Systems
A. Introduction
tº tº º ºs º ºs ºs º ºss tº A gian ºr ºr elements of great practical importance is
associated with the historical concept of state-determined systems. For
such idealized systems the specification of a certain delimited set of
parameters is sufficient to describe completely the behavior of the
system. These parameters are said to determine the state of the system
in the sense that when the values of such parameters are known at any
instant, the behavioral configuration is also known at that instant.
Any particular system is then specified merely by fixing a set
of static, generally nonlinear relationships between these state variables.
All problems in the generalized dynamics of such state-determined systems
are reduced to the purely kinetic or kinematic "motion" of a representa-
tive point in a multidimensional abstract space, called the phase space.
In such a phase space any single point represents a possible State of the
system, and a connected set of such points, or phase trajectory, represents
a "history" of the system.
The essential significance of the concept of a state-determined
system rests in the fact that the future behavior of such a system is
determined completely in terms of the complete specification of the instant-
aneous present state of the system, together with the temporal fluctua-
tions of all "external forces" during the future period. In many prac-
tically important cases where the external effects are small enough to
be neglected, so that the system may be regarded as effectively isolated
or closed, the future behavior is determined once and for all upon
specification of the initial state alone.
This fascinating notion utterly dominated the growth and evolu-
tion of classical mechanics until just before the outset of the present
century. Indeed, we may quote the following from G. D. BIRKHOFF
(Dynamical Systems--1927) :
In dynamics we deal with physical systems whose
state at time t is completely specified by the values
of n real variables
x1, x2 x3 . . . . ºn
part XII 2.75l. CLASS NOTES 131
Accordingly, the system is such that the rates
of change of these variables, namely .
dx1/āt, *2/4t, *3/4t, • * * * *n/it
merely depend upon the values of the variables them-
selves, so that the laws of motion cºn be expressed
'by means of n differential equations of the first
order,
dx+/it = X, (*, *2,43, ...,xn) (i = 1, 2, 3,...,n)
However, more recent physical treatments have acknowledged the fact that
no material system is ever isolated, nor is the instantaneous state ever
capable of complete determination; out of this realization have been
evolved modern statistical mechanics and other stochastic views of the
"knowable" physical world. Despite this, state-determined "models" of
material systems will continue to play an extremely useful role in en-
gineering analyses which are directed toward the practical prediction
of approximate performance.
Quoting from the paper by Arturo Rosenblueth and Norbert Wiener,
The Role of Models in science:
No substantial part of the universe is so simple
that it can be grasped and controlled without abstrac-
tion. Abstraction consists in replacing the part of
the universe under consideration by a model of a sim-
ilar but simpler structure. Models, formal or intel-
lectual on the one hand, or material on the other, are
thus a central necessity of scientific procedure.
Thus, whenever an engineering problem must be studied, other
than by direct manipulation or experimentation with the actual system
involved, it is necessary to have recourse to models of some type.
- Often these are real or actual models, in which physical counter-
Pºrts are involved. Sometimes they are earlier versions of the same type
* System or are extant and similar devices. Often they are simplified
o: Scaled down versions in the form of research models and pilot plants.
Again, physical models in the form of analogies are frequently used
"ith considerable effectiveness.
In many other circumstances, only conceptual models are employed,
132 2.75l CLASS NOTES
in which rational abstractions and idealizations are made to correspond,
with more or less validity, to real situations. These may be rendered
precise--but not necessarily accurate--in the form of mathematical
models, in which the component elements are mathematical variables
interrelated through various mathematical operations.
If these operations can be restricted to a small collection
of simple computing operations, we can thus construct a computing model
corresponding, at least approximately, to any given engineering situa-
tion. Lastly, if the operations upon and interconnections between vari-
ables are actually realized in a physical device, the corresponding com-
puting system does, in fact, constitute a physical model
In previous chapters we have discussed the nature and descrip-
tion of computing models in considerable detail. We now turn our at-
tention to the state-determined model of the physical universe, which
is intrinsically a mathematical model, and, although it has certain
shortcomings which will be pointed out, it does succeed in describing
a great variety of phenomena.
B. Elements of a State-Determined System
Any state-determined system may be reticulated into just two
kinds of multi-ported elements: (i) one-port impedances, which are
generically denoted -X, and which include ideal resistances (—R),
capacitances (-C), and inertances (-I) together with the ideal effort
source, (-E), and ideal flow source, (-F); (ii) multi-ported energy
junctions, which are generically denoted . J., and which include the flow
junction -o- and the effort junction 1. . Hence, we have at our dis-
posal a field of seven elements:
1 Port Impedances 3 Port Energy Junctions
-X -J-
-R
–C -Q-
-I
-E -1-
-F
part XII 2.751 CLASS NOTES 133
The dynamical models of Newton and Faraday and the field model of
Maxwell for electro-mechanical interaction may all be reconstructed
from this seven-element universe. Moreover, Lagrangian and Hamiltonian
mechanics treats systems which may also be reticulated into these same
elements. Hence, at first glance, we might be prone to say that all the
universe may be modeled as a network of these seven multi-ports:
A further reflection will reveal, however, that there are
certain two-and three-port elements which for example satisfy the con-
dition2Peo yet are not energy junctions, namely, ideal transducers,
transformers, levers, and differentials. Obviously, without such ele-
ments it is impossible to model or to construct a host of essential man-
made devices. Thus, we must conclude that the seven-element universe
of classical mechanics is by no means complete. To represent all con-
ceivable systems we must not only add other state-determined elements,
but also augment these with more general multi-port elements.
C. The Mathematical Construct of a State-Determined System
We shall now concern ourselves with the mathematical structure
of the state-determined model. The most general form consistent with
the underlying assumptions and concepts of state-determinism will be
compared with the classical form which was stated briefly by way of the
quotation above.
First of all, the state-determined system is one whose condi-
tion at any time (t) is precisely and completely specified by a finite
set of variables X (t) = {x1(t) || 1 = 1,2,...,n}. It is therefore
easy to see that an instantaneous condition or state of the system is
represented By a point in an n-dimensional phase-space. As the condi-
tion of the system alters, due to the action of disturbances at its
boundaries the state-vector Xtraces a path in the phase space which
* appropriately call the phase trajectory.
Before proceeding further it is well to point out that a system
whose condition is completely described only by specifying an infinite
"mber of variables cannot be "state-determined"-- indeed there is no
*y of computing its state in a finite time.
we have said that the state-vector X is a function of time, t,

13|| 2.75l CTASS NOTES
more generally, it varies according to some running parameter which
usually possesses the dimension of time. Actually there is no "clock"
capable of keeping a perfectly continuous running record of time; how-
ever, we may think of the parameter t as being either discrete or con-
tinuous for the purpose of computation. In particular, analog computa-
tion usually employs a continuous time parameter (either "real", or scaled),
while digital computation uses a discrete time parameter, i.e., the state-
vector is computed every hundredth of a second, for example.
With these notions understood we may now turn to the primary
assumption in the construction of the state-determined model. This
concerns the manner in which the instantaneous state is related to
earlier states. This result is equivalent to restricting the nature
of the phase trajectory which may pass through a certain state, say X (o).
In the most general form, this condition is expressed as follows, in
accordance with the functional notation we have employed up to this
point :
X = Tº Dº X
That is, in words, a unique present state X (t) is determined by the
trajectory X(t-rlt-o). Such a determination involves a static
(possibly implicit) function (p of all the variables {x, | i = 1, 2, .. ..,n}
and a time translation operator T of a peculiar type. Let us immediately
compare this purportedly general form with the classical form which we
have already seen, namely wherein the rate of change of the state-vector
X is a static function of the state variables {x, | i = 1,2,...,n ).
That is,
d x º
at T di (*, 1 = 1,2,...,n}}
This we may write in the equivalent compact form
di X
d_f
Or, in the integral form
x - I ſ” at . D (x) = Tº j k X
Upon comparison with the form first stated it is evident that the time
= f(X)
translation operator T corresponds to, and therefore must be of the same
Part XII 2.75l CLASS NOTES 135
general nature as the running integrator [ ſ dt J.
5. The Wariables of State in Generalized Dynamics
Newton founded his axiomatic dynamics upon the concepts of
force and momentum; thus, a system of mass points was viewed as con-
nected by way of force interactions, and the motion of each particle
was determined according to the fundamental law:
[ Vector summation of forces = time rate of change of momentum.
The forces active on a given particle could arise either from disturb-
ances of an origin external to the system, or from internal interactions.
Newton's second law is often loosely stated in the form
X. F = ma
rather than the original and correct form
d:t
' there ensued a
Following Newton's enuciation of his "laws of motion'
battle royal among the learned dynamicists of the next two centuries
over the notion of "force" and its true usefulness in the dynamical
description of a system. Hertz, in his introduction to The Principles
of Mechanics, summarizes these arguments in a most profound manner, and
discusses the various "images" of a generalized dynamical situation.
In our position, however, it is perfectly acceptable to define
a certain quantity, p , as the generalized momentum of an element, which
is related to the associated generalized force or effort, e , according
to the exact law
C = # or p = ſ edit
This form is in agreement with Newton's original statement, and is
acceptable also in the light of relativistic mechanics which tell us
that there is, in reality, a functional dependency between momentum and
Velocity such that as the speed of light is approached the momentum
becomes infinite.
There arose, as a result of the work of Lagrange and Hamilton,
136 2.75l CLASS NOTES
another image of a dynamical situation wherein it was the potential and
kinetic energies which defined the state of the system. We must realize
in connection with this that there is always a correspondence between
the total strain or displacement of a system and its potential energy,
which may arise as a result of internal deformations as well as gross
displacements in a force or potential field. Maxwell points out that the
correspondence is usually not a simple one, particularly in the case of
the energy of deformation. On the other hand, there is a simple relation,
ship which connects the kinetic energy of a particle and its velocity.
Thus, it might be said that the two state variables which arise out of
this energetic description are the generalized displacement, q , and the
generalized velocity or flow, f . These are, of course, connected by
the relationship
r – # Or" a - ſ rat
The purpose of the above discussion is simply to justify the
selection of the four variables e , p , f , and q as the variables of
state with which the dynamical or energetic condition of any physical
state-determined system may be described. Thus, for a multi-degree of
freedom system the following state vectors would be employed:
© : (e. | i = 1,2,3,...,n) f = (f1 | 1 = 1, 2, 3, ...,n)
p :: (Pi | 1 = 1,2,3,...,n) CL = (q; | 1 = 1,2,3,...,n)
==
E. The Tetrahedron of State
It is now possible to identify each of the four state variables
with a vertex of a "tetrahedron of state" and consider a given system
as characterized by the functional relationships between the variables,
these being associated with the edges of a tetrahedron;
(e

Part XII 2.75l CLASS NOTES 137
The correspondences between (? and Ip and between f and CI are,
of course, implicit in the construct itself; that is,
Ip - J eat or (ě = ºp
- CI
q = J f at Or f - —º
However, the other essential correspondences, which are but three in
number, are static vector functions peculiar to a given system, namely,
Gº = iſ a (f) Or f - if, (e)
q = ºc (e) or e = is (q)
p = p, (f) or f = Pr(p)
Thus, we see that the characterization of all dynamical or time-
dependent interactions is embodied in the Ip- e and ſ - f relations,
while the remaining relationships are of a simple static nature. These
latter will now be dealt with in detail.
F. The Characteristic Static Relations
Before discussing in any detail the static relationships
introduced in the previous section, it is well to emphasize that the
structure of these relations is peculiar to a given system; it is, in
fact, derived from the reticulation of that system into one-port impedance
elements and multi-ported energy junctions. Thus, for example, the form
of the static vector function be or its dual, ſº e is deterºined by
the disposition of resistive elements (-R) in the reticulated system.
Similarly, *. and gº 1 are determined according to the disposition of
capacitive and inertive elements, respectively.
Below we emphasize the expected (indeed inevitable:) non-
linearity of these relations. Also, in each case the dualism of relation-
ships is recognized, along with the resultant necessity to distinguish
between two complementary "energies" associated with each impedance
element. Failure to make this careful distinction in the presence of
nonlinear relationships between the state variables has, at times,
resulted in serious and substantial errors in the calculation of energy
storage and dissipation terms.
138 2.75l. CLASS NOTES
1. Resistance-Conductance Relations and Generalized Energy Dissipation
ſ * * *-* - sº mº sºm. T
P [X –––R tº a
f --~~~ d
| |
|- — — — — — — — –
Pure Energy Dissipation
IP (t) = (F,
TZZZZZZZZZZZZZZZZZZZZZZ
State Relations
Por any one-port element the generally nonlinear static relation-
ship between effort (e) and flow (f) can always be considered as a:
!
RESISTANCE Oſr CONDUCTANCE
Relationship T Relationship
R ! G
We will distinguish between these two converse modes, particularly when We
consider causal sense, as follows:
f —- IR —— e e —- G --— f
GENERALIZED
CONDUCTANCE
f = P, (e) = G (e)
GENERALIZED
RESISTANCE
e = p...(f) = IR(r)
* R




Part XII 2.751 CLASS NOTES 139
These clearly reduce to the ordinary linear relations in the
special case, namely:
LINEAR CONDUCTANCE
f = G - e
LINEAR RESISTANCE
e = R • f
Whenever steady-state resistance, (or conductance) is present in
any one-port, it is clear from energy continuity that available energy is
dissipated into heat in the amount:
(P. - r . R(t) ! (Pa = e : G (e)
!
f
since no energy is stored in a purely resistive element. For the lineal
C8 Se:
(P - Frº ! (P – Geº
1.
Which is well-known.
Let us now consider that we have a one-port system containing 17
Separate resistances. Then the total dissipation must be:
42 2
(P - * - 2 - . ,
d . . e
J * 3-1 3
1
(P - f. R. (P. - e. G
.
if G , f y IR, and (G are treated as £- coordinate vectors. Thus the
energy dissipation is a scalar additive function, summed over all available
energy sinks.
Moreover, if we define the functions:
!
f ſ’ Røar . cocontRNT. F. = ſ."G (e) de
Then it is clear that the dissipation (P. := Pr + Fe and we obtain a dualistic
CONTENT: P

11,0 2.75l. CLASS NOTES
generalization of the RAYLEIGH dissipation function, which we shall investiga
later. This particular choice of terminology follows from the work of CHERRY,
a ud MILLAR, both separately, and together. However, very similar notions sta
from the "power function" concept of WELIS and FJERTES.
2. Capacitance Relations and Generalized Potential Energy
Rºşeşººl
* , a n. * is *.* * * * * * * *
& S. s." " -, * R - º * *
S. * ~ * .*.*-* S.
-, *.. ºšš|
ºf s' s 4°,
* *
.*
tº sº.
*
e Tº: § 3.
f C. .*
Potential Energy Storage
IP (t) = alE/at
f z zº z
State Relations
For any one-port element, the generally nonlinear static relation-
ship between effort and displacement can always be considered as a capaci-
tance relationship. Again we distinguish between two converse relations,
namely:
e -- (C -- q q —- S --- *
GENERALIZED
EFFORTANCE
GENERALIZED
DISPLACEANCE
q = p. (e) - C (e); e - P. (a) - $ (q);


Part XII 2.751 CLASS NOTES 1 lºl
These clearly reduce to the ordinary linear relations in the
special case, namely:
!
LINEAR T LINEAR
CAPACITANCE ſº SUSCEPTANCE
!
1.
q = C - e ? e = S • q
T
Whenever capacitance is present in any one-port, it is clear from
energy continuity that available energy is stored as generalized potential
energy in the amount:
E = q . $ (q)
P
E = e - C (e)
p
E
==
E
e
Q1
since no energy is dissipated in a (purely) capacitive element. For the
linear case:
u = (1/2) ce” U = (1/2) Sq’
:
which is well-known.
For the general case we shall find it useful to define the dual
pair of energy functions, where 20 = Ue + Udi
POTENTIAL
ENERGY
COPOTENTIAL
ENERGY
v, -ſ co, . . v, -ſ so a
which will later permit us to keep proper energy books. The first concepts
of dualistic or complementary energy forms date back to the work of CLERK
MAXWELL and ENGESSER, but these ideas are fully developed in the work of
CHERRY and MILLAR previously cited.
1 l;2 2.75l. CLASS NOTES
3. Inertance Relations and Generalized Kinetic Energy
tº ºs ºs º ºs º º ºs
P(t) = alE./at
State Relations
For any one-port element the generally nonlinear static relation-
ship between flow and momentum can always be considered as an inertance re-
lationship. Once again, we distinguish the two converse aspects, namely:
f -- I -- p p —- T -- f
GENERALIZED
FLUANCE
GENERALIZED
MOMENTANCE
p-ºriſt) - I (r): f = Pr(P) - T (p),
These clearly reduce to the ordinary linear relations in the specia
case, namely:
t
LINEARIZED T LINEARIZED
INERTANCE ! FLUANCE
t
p = I. r ! f = T. p
?
!

Eart XII 2.75l CLASS NOTES 11.3
Whenever inertance is present in any one-port, it follows from
energy continuity that available energy is stored as generalized kinetic
energy in the amount:
* = 2.T = f' - p
E = f : I (r) F, - p - T (p)
since no energy is dissipated in a purely inertive element. For the linear
case:
T = (1/2) I F2 º T = (1/2) Tre
which is Well-known.
- As before, for the general nonlinear case, we shall find it useful
to define the dual pair of kinetic energy functions:
COKINETIC
ENERGY
f
* - ſ II (F) • dif'
KINETIC
ENERGY
P
º, -/ T (p) . dp
which are needed for the energy principles to be discussed. As before, it
follows that 2:I = T, + T .
f : "p
G. The Three State-Determined Elements (R, C, I)
We see from the above that all primitive state-determined relations
can be expressed by the three 1-port elements:
R — ; C — ; I —
The above properties of these elements may all be compactly Summarized in the
single grand tabulation attached. The linearized values of the parameters
(R, c, I) are indicated; in the general case the corresponding nonlinear re-
lations must be generated by algebraic or by hyperpolydral functions.
1 lºl,
Resistancs-Conductance
Relations
*sistascº metanosis a =$e (£)
Converse ºs -->
consoucuasicrºssanoes: f ** (e)
ºfs, ; S. º
* * -"
&_ i- * (2 `Tº ... -
Z º P(t)= Pd(e)
f d'E/&t=O
of Sºle
—ſ ©
<}
*ā-e exº~. :
Inertance Wºelations
womentariº Raanoºp-º,G)
Converse WR:
Fluatics welaroºf-º-(p)
f-E--): (*-E-f
- c-lºº
local
PoſHMTML ENERGY:Crºtºhar.
kNETIC ENERGY: Inertiveshwag
or 2P = P + Pe
cº-º y =ſed; - P(p
coeniº P. =ſide = P(e)
Lºlºr CASE: Constard Resistance
e =R}, f = #-e R=rssuance
P = ſº-ſºfa =##,
We =ſide- # ede-#e"
...Turlºnsarºeſhly Pr= Pe = W
–E–
zº’s P +P.+ ####e"= rſ = †
d(eſ)= ed'ſ #fde dºº-ed, tº . .Y d(ſp)= falo 4
Gesslyed ey General (ſp)=fa|p-H pdf
º }...ºft. Tºlºsſal ſºu-ex =ſº ſºle :* zr=ſp=ſ++ſºd;
or 2U = Uq. + Ue
(ºlonesi)
*B-syuºſºdy-uſº
Complementary P.E.
or Corolºntial Easyucaſ:gde=U(e)
LINEARCASE: Constant Capacitance
zºº. #4 ; G = canaenavºce
U.-ſedg-ſ’āqlū-à-1:
U.aſºle-ſcede - #- e'
...reli-oº-oºl Ug-Ue=U |
or zT= Tºp + Tº
uMEArcase:Consai Inertancel
p=1}, {=#-P I===srael
tº-ſºap-ſºrdp=#-p" |
T =ſºdf =ſºdf -4-f" |
zºr=tp:Tº-#p-ºf-f.
-RurlinearCaſe0mly :














Part XII 2.75l. CLASS NOTES 1 h9
Background Reading -- State-Determined Systems
(1)
(2)
(3)
(h)
(5)
))
MAXWELL, James Clerk. Matter and Motion
Starting from elementary concepts Maxwell demonstrates in a beautiful
fashion, and without recourse to any sophisticated mathematical form-
ulation, his various views on mechanics and dynamics.
HERTZ, Heinrich. The Principles of Mechanics
Much is to be gained merely by reading Hertz's introduction to this
short work. One is lead by his reasoning to a deeper insight into
the fundamentals of mechanics. He points out, in particular, the
ambiguity in Newton's definition of force, as it is implied by the
three laws of motion.
MAGIE, William Francis. A Source Book in Physics
Material of historical interest is presented on most of the contribu-
tors to physical science. In particular, chapters are devoted to the
work of Maxwell and Hertz.
LANCZOS, Cornelius. The Variational Principles of Mechanics
Again, the introduction serves as an excellent appreciation of the
basis of the variational approach to the description of dynamical
situations. In particular, the fundamental differences between Newton-
ian dynamics and the energetic method of Euler and Lagrange is pointed
out; namely, the former views a system as characterized by the momenta
of, and the force interactions among, its elements, while the latter
relies on constraints upon the potential and kinetic energies.
ROUTH, Edward John. cs of a System of Rigid Bodies (First
edition, 1
WHTTTAKER, E. T. Analytical Dynamics (First edition, 1904)
WEBSTER, Arthur Gordon. The Dynamics of Particles (First edition, 1904)
These are standard classical references on dynamics. They represent
specific excellent integrations of Newtonian and variational mechanics.
1 l;6
2.75l CLASS NOTES
(8)
(9)
However, the applications are largely restricted to problems of academic
interest.
BIRKHOFF, George D. Dynamical Systems (Published in 1927)
This is a classical modern treatise on the dynamics of state-determined
systems which developed out of a series of lectures given by Birkhoff
in 1920.
CHERRY, B. Colin. Duality, Partial Duality and Contact Transformations
The application of Hamiltonian mechanics to electro-magnetic systems
is indicated.
Background Reading--One-Port State Determined Elements
(1)
(2)
(3)
(l)
(5)
(6)
(7)
CHERRY, E. C. Generalized Concepts of Networks, Proceedings of the
Symposium on Information Networks, Polytechnic Institute of
Brooklyn, New York, 1951, pp. 176–177.
CHERRY, E. C. Some General Theorems for Non-linear Systems Possessing
Reactance, Phil. Mag. (7), v 1.2, p. 1161 (1951).
CHERRY, E. C. The duality between interlinked electric and magnetic
circuits and the formation of transformer equivalent circuits,
Proc. Phys. Soc. B., v. 62, p. 101 (1949).
CHERRY, E. C. and W. M.ILLAR, Some New Concepts and Theorems Concerning
Nonlinear Systems, in Automatic and Manual Control, Butterworths
London, 1952, pp. 263-27ſ. sººm
MILLAR, W. Some General Theorems for Non-linear Systems Possessing
Resistance, Phil. Mag. (7), v B2, p. 1150 (1951).
WELLS, D. A. J. App. Phys., v 16, p. 535 (1945).
FUERTES, F. A. On the Power Function, J. App. Phys., v 17, p. 712 (1916).
rt XII 2.75l CTASS NOTES 1 l;7
º ground Reading -- Models and Analogs
ROSENBLUETH and Norbert WIENER. The Role of Models in Science, Philosophy
of Science, Vol. 12, No. 1 (October, 1915) pp. 316-321.
ARBER, Agnes. Analogy in the History of Science in Studies and Essays
in the History of Science and Learning, Schuman, New York,
T5|TFETET235.
ZINSSER, Hans H., M.D., Pitfalls of Physiological Modelling, University of
Southern California Medical Bulletin, pp. 6-13 (July, 1953).
BRODBECK, May. Models, Meaning and Theories in Decisions, Values and
Groups, Vol. I, (1960).
DEUTSCH, Karl W. Mechanism, Organism, and Society: Some Models in Natural
and Social Science, Philosophy of Science, Vol. 18, No. 3,
July, 1951, pp. 230-252.
JONES, Richard W. Models, Analogues and Homologues in Regelungstechnik:
Moderne Theorien und ihre Verwendbarkeit, pp. 326–328.
SCHOENFELD, J. C. Analogy of Hydraulic, Mechanical, Acoustic and Electric
Systems, Applied Scientific Research, Section B, Vol. 3,
pp. 117-150.
1 |& 2.75l. CLASS NOTES
H. The Concept of Circuits and Networks
Classical dynamics has been primarily--perhaps nearly exclusively.--
concerned with reticular systems and processes which can be effectively .
conceived as composed of state-determined one-ports suitably interconnected,
These models generally assume storage and dissipation of energy at a finite
number of localized regions, "lumps", or "points"; e.g., "mass points" in :
mechanics; "lumped circuits" in electricity. Such substitutes for the actº
underlying field continuum have often been remarkably useful and dramatical
productive. The relations between the macroproperties of the one-port lump.
impedances and the microproperties of the continuous fields we shall treat
the next article. Here We shall be concerned with certain of the system prº
erties of lumped constant systems. ;
In terms of the previous relationships it is now possible to general?
the classical "mechanical system of many particles" or the traditional "eles
trical network" to deal with any engineering system in which the "m" prim-
itive parts are all state-determined one-port elements in any medium, each
containing generally nonlinear resistance, capacitance, and inertance prop-
erties. It is not the least necessary that the system be differentiated in
regard to heterogeneity of medium, since the basic relations above are valis
for all media.
It is first necessary to show that all possible sti uctural combina-
tions of one-ports may be obtained using only the two ideal junctions (0, 1.
Electrical symbolism offers the most efficient explanatory language, but We
can readily verify the result for mechanical systems.
Two electrical impedances in series may be shown thus:
1 X2 |
O—[THO. ––)-(-|--|- — 1 —
Xi X2
part XII 2.75l CLASS NOTES 11:9
hwhile two impedanege in parallel are Written X X
8, 8.
| |
ſ ] = -------
Xb Xb Xb

Any "grounded" impedance (i.e., where one of the efforts is the zero
potential) can always be written:
X X

| |
[ } T]— 3|| [ ; — 1 -: = [ ! — X |
O
Eince the one-junction can be absorbed directly into the impedance relation-
ship, itself. Similar results hold for the dual situation, which is partic–
:
Aularly significant for mechanical inertias, namely:
Zero *
-º-º: F. –
X
With these properties and conventions, the damped electrical oscillator:
L R
r:
O. *—?.
Could be written as the radical:
R
— 1 - O –
L C

15O 2.75l CLASS NOTES
The electrical dual structure may be drawn immediately by employing
the relations:
1 = O
This results in the system:
R
— O ~ 1 —
C I
c -
O- * -TööUN–O
Sº wº -O
On the other hand, both direct and dual systems may be diagramed for
mechanical systems, namely:
* * * * * *
* * *-*.

part XII 2.75l CLASS NOTES 151
Thus in practical systems, we are not usually confronted with a
single one-port element, but rather with a plurality of one-ports inter-
connected through ideal energy functions. In electrical Science Such
networks are customarily represented by a meshwork of lines, each one of
which represents a general one-port impedance element:
5

3
Moreover each such element may itself be a complex net and so on, ad
infinitum, but this is immaterial. The points are usually called term-
inals or nodes. The line joining any two nodes is called a leg or branch,
and any closed path made up of branches is called a loop or circuit or
mesh. The topological properties of such networks are then obvious from
Our previous treatment.
Frequently, it has been assumed that all networks can be constructed
from one-port elements. This is of course not true, and it is possible to
construct other kinds of networks which contain multiports of various kinds;
even since the earliest days of electromagnetism, following the work of
Joseph HENRY and Michael FARADAY, the role of mutual induction--a two-port
phenomenon-- has been of signal importance.
But this generalized network concept is not limited to electrical
Science alone. Largely through the pioneering Work of Gabriel KRON, the
true role played by reticular fields and generalized nets is now better
understood. References are given in the reading list to applications in
Such diverse fields as:
NUMERICAL ANALYSIS CONFORMAL MAPPING
DIRICHLET PROBLEM NONI,INEAR NETS
FIELD PROBLEMS ALGEBRAIC TOPOLOGY
SCHRODINGER EQUATION RADIATION ANALYSIS
ELASTICITY and PLASTICITY FTUID FLOW
152 2.75l CLASS NOTES
Background Reading - Networks
PHILLIPS, H. B., and N. WIENER, Nets and the Dirichlet Problem,
Journal of Mathematics and Physics, Vol. 2, pp. 105-124
(1923).
LYUSTERNIK, L. A. On Electrical Modelling of Symmetric Matrices,
Uspekhi Matem. Nauk (N. S.) Vol. 41, pp. 198-200 (1949).
WON MISES, R. On Network Methods in Conformal Mapping and in Related
Problems. Construction and Application of Conformal
Maps, U. S. Dept. of Commerce, Natl. Bur. of Standards,
Applied Math. Ser. Vol. 18, pp. 1-6 (1952).
BIRKHOFF, G. D., and J. B. DIAZ. Non-linear Network Problems, Quarterl
of Applied Mathematics, Vol. 13, No. 4, pp. 131III: (T556,
OPPENHEIM, A. K. Radiation Analysis by the Network Method, Transactions
of the ASME, Vol. 78, pp. 75-735 (1956).
KRON, G. Numerical Solution of Ordinary and Partial Differential Equation
by means of Equivalent Circuits, Journal of Applied Mechaº
ics, Vol. 16, pp. 176-186 (1915).
-----. Equivalent Circuit of the Field Equations of Maxwell, Proceedings
of the IRE, Vol. 32, pp. 289-299 (1914).
* * * * * • Electric Circuit Models of the Schroedinger Equation, Physics
Review, Vol. 67, pp. 39–43 (1915).
ºn ºn tº tº ſº • Equivalent Circuits of the Elastic Field, Journal of Applied
Mechanics, Vol. 11, pp. A119-161 (1977).
-----. Equivalent Circuits of Compressible and Incompressible Fluid Flow
Fields, Journal of Aeronautical Science, Vol. 12, pp. 221-
231 (1915). tºº.
* * * * * . A Set of Principles to Interconnect the Solutions of Physical
Systems, Journal of Applied Physics, Vol. 24, pp. 965-989
(1953).
* * * * * • Solution of Complex Non-linear Plastic Structures by the Method
of Tearing, Journal of Aeronautical Science, Vol. 23,
pp. 557-562 (1956). T
BRANIN, F. H. Kron's Method of Tearing and its Applications, Proceedings
Qf the Second Midwest Symposium.on Circuit. Theory, MHChīgas
State University, Ep. 3. TEŽ.75T(T556). :
ROTH, J. P. An Application of Algebraic Topology: Kron's Method of
Tearing, Quarterly of Applied Mathematics, Vol. XVII,
No. 1, p. 1 (April, 1959).
Part XIII 2.75l CLASS NOTES 153
XTTI. Distribution of Energy over Space, Time and Frequency
A. Introduction
We are concerned in this section with a quantitative description
of the distribution of power and energy over space and time. The spatial
distribution requires that we consider the properties of continuous and
reticular fields. In the general case, these fields are nonstationary or
unsteady. but for most engineering purposes we may consider the fields as
pseudo-static (or quasi-stationary). Such fields for energetic systems are
governed by scalar potential functions, together with their derived and
associated vector fields. If the fields are truly dynamic, we can preserve
the field language if we employ retarded potentials and a corresponding
integral formulation.
At any given point (or bond) in an energetic field, the local
power state is instantaneously related to the boundary or environmental
power states through dynamic transfer characteristics or operators. These
may be employed either to describe the local behavior in the time domain
or to interpret the response characteristics in terms of frequency or
spectral sensitivity. Thus the local energy distribution over time of
any linear (or linearizable) system may alternatively be viewed as a
distribution of the same energy over the frequency band.
These fundamental facts and concepts, sufficient to deal with
multiport systems, are few in number and are outlined in the paragraphs
below.
In Part IV we treated the continuity of energy in a generalized
field while Part V introduced the concept of field reticulation and the
corresponding reticulation of field energy and power. These reticular
energies were then evaluated for state-determined elements. We now pro-
ceed to the precise restatement of the reticular energy principles and
power balances for state-determined systems. This is the form in which the
generalized energy concepts were first obtained; but, consistent with the
assumptions of modern relativistic mechanics and quantum physics, it is
energy and not material structure which is the foundation point for rational
Science.
15l. 2.75l. CLASS NOTES
B. Energy Principles for State-Determined Systems
* * * * * * * * * * * * * * * * * * * * * dºs dº ſºme tº ſº º ºsº º .* ºne º ims tº º ºs ºs º º ºs tº
It is useful to introduce the classically important concepts of
pure sources of effort and flow as indicated below. These correspond,
for the continuous case, to the traditional DIRICHLET and NEUMANN boundary
conditions. Moreover, it is possible to represent nearly all boundary tra
ports of energy in terms of the generalized HEIMHOLTZ and THEVENIN equiv-
alent sources, the names corresponding to the traditional linear electrical
equivalents. Merely by extending the boundaries of the system to include
the source impedance functionals, we can represent all power transport a-
cross the boundaries of an n-port state-determined system in terms of the
ideal DIRICHLET or NEUMANN sources. In that which follows below, for sim-
plicity, let us assume all boundary ports as equivalent to a finite or in-
finite set of such ideal elements.
DIRICHLET PORT: | Element, : E – NEUMAN PORT: | Element: F — |
CONSTANT FLOW SOURCE:
f = F = constant
CONSTANT EFFORT SOURCE:
e = E = constant
de ôf
# = 0 3E = O
Rise e Rise e
Zero Impedance F Infinite
Impedance
|
f
Content Pº E f (Drop e) df =
f (–E)df
Or' Fr = - Ef
(i.e., Power GAIN = E • f)
Cocontent
Content
e
Cocontent Fe E ſ
f'de = f F(-iº,
Or P = - Fe
e
(i.e., Power GAIN = F • e)
:


Pure Sources of Effort, and Flow
* * * *-ºsmºs ººmmºns sºmeºmºmºsºme
part XIII 2.75l CLASS NOTES 155
* = * * * * * * * * * * * * * * * * *-* * * * * * * * * * * * * *
The classical LAGRANGIAN function was historically the first intro-
duced to deal with generalized state-determined systems. This may be de-
fined in terms of our relations above in the form:
L (f,q) = Tº - U a
X. Cokinetic Energies — X. Potential Energies
:
Lagrangian L
=
:
Total Free Energy
that is:
-
T,
IIl
2. Tr: (f1) = "F1 + Te2 + · · · + Ten
Im
9 iſ "… (4) :- 941 + "q2 + = • * + "qm
U
We may then write the Lagrange Equation in the form:

a 3D, 3L 9 P.
— (–) - — + = O
dt, of a q 0f
Since We can arrange that IP f includes all the energy sources as well as
Sinks.
However, it was not realized until comparatively recently that
there also exists a completely parallel and dual form of the Lagrange Equation
expressible entirely in terms of the effort vector, G , and momentum vector,
|) . Here, the dual Lagrangian would be a complementary free energy in terms
Of the total copotential energy less the total kinetic energy. Thus we may
eXpress the two dual energy formulations side-by-side in the form of Table I
below.
156
2.75l CLASS NOTES
TABLE I
LAGRANGE EQUATIONS
!
Classical Form : Dual Form
f
d 3T., 3 U 3 IP ; d. 3 U 3 T 3 IP
* Hi , + = ; :- #) + —£ 4. # = 0
dt, dt 0 8 0
f QI of (€ |) (e
COKINETIC ENERGY: T , -- U , coPOTENTIAL ENERGY
POTENTIAL ENERGY: U Cl sº-> T D : KINETIC ENERGY
CONTENT: IP , -- IP e : COCONTENT
Let us next see how we may obtain a generalized power balance in
any state-determined system.
The dynamics of a normal Lagrangian system
would be governed by the pair of equations:
a 3 T 3U 3|P
sºme —#) + —3 + # = O
dit, of, d qi of,
d (q.)
gºmºsºs = f'
dt, 9i i
If these expressions are multiplied together, and Summed, there
results:

3
dt,
(T, U.) + F = P,
* f all
where P, is the active content (or power supplied to the system) from
energy sources.
By considering these sources as energy ports, we obtain
Part XIII 2.75l CLASS NOTES 157
the previous energy expression:
(a E/dt) + (P - P
Alternatively, we may carry out a power balance in terms of the
system Hamiltonian function
H = f; p - L (f, I) - H (p, q)
In terms of the individual elements:
L X. (*p. º War)
i
This gives the general result:
IHI = 2 (T., 4 U.) = E = Total Stored Energy
i pi qi
Then a general Hamiltonian power balance gives:

d IHI 3 IP
+ X, f, —# = 0
dt, i 0 f.
l
*——” \– -v- _*
Change of Energy Sources
Stored Energy and Sinks
again merely a specialization of the reticulated energy continuity.
We may then summarize all the previous results in the form of the
generalized energy diagram depicted in the figure below. This result follows
by time-integrating the power flow for an isolated or closed system (i.e.,
X. IP = O). In this case we may write:
158 2.75l CLASS NOTES
IE,(t) = T, == 2, *pi
E,(:) E U. - 2, "qi
t t
E.6) - ſ 20t) dt = 3 ſ Pai dt
for Which
IE,(t) + IE,( ) + IE,(t) = COnst = E,
Representa-
tive Point,



Part XIII 2.75l CLASS NOTES 159
C. Fields, Potentials, and Transmittances
The finite macroreticulations discussed previously always repre-
sent merely approximate partitionings of the continuous field energies into
* finite number of abstract individual elements of state-determined or gen-
eral functional form. For the finite state case, these elements are marked
ty the relational symbols, IR 3. (C , and I , and therefore represent
finite coordinate systems which are the natural generalizations of rigid
body dynamics.
Electrical circuits and networks are but special cases in the
electromagnetic domain of the general concept of reticular fields wherein
the fluxes and potentials are assumed to conform to a prescribed meshwork.
The lumped circuit concept is thus to electrical science what the Newtonian
mass point is to mechanics and the Hookian linear spring to elasticity.
In particular, these approximations ignore more or less completely
the finite velocity of energy propagation and the consequent field retarda-
tion effects. Thus, while a more rigorous field theory would formulate
functional or operational relations between the field quantities, the
reticular field concept presumes that simple static functions relate local
Or total variables.
For these and other reasons any particular system representation
can at most hold only within a restricted amplitude and frequency domain.
Outside these limits, we must inevitably expect discrepancies between
analysis and experiment, between prediction and performance. While we can
partially reduce these divergences by making the reticulation and correspond-
ing model more complex, we cannot ever expect complete equivalence between
the multiply-infinite order of physical reality and the modest finite order
of our conceptual models.
16O 2.75l. CLASS NOTES
Quasi-Stationary Processes and Slowly Varying Fields
Most of the problems of engineering analysis and system design lis
within the domain of slowly varying fields. For most cases this condition
will hold whenever the dimensions of the system are small compared to the
characteristic wavelengths of all disturbances. Under these circumstances
the field retardation may be neglected and the field properties may be cal-
culated as for stationary processes. In this manner static linear or non-
linear relations may be established among the principal variables which may
be taken as integral forms of the corresponding field quantities . }
The resultant inertance, capacitance, and resistance relations
depend in addition to the material constants only upon the geometry of the
field and are the result of integrations or averaging processes performed
over the space coordinates. Thus only an integration with respect to time
remains.
While a precise treatment of rapidly varying processes demands
consideration of transient field effects, usually represented by systems of
partial differential equations, a description in terms of ordinary differ-
ential equations suffices for slowly varying fields. Moreover, for the
static and stationary states of equilibrium these relations reduce still
further to systems of purely algebraic equations.
By contrast, the introduction of the concept of a retarded potenti!
for rapidly varying unsteady motion represents an attempt to preserve the
field concept for the description of nonstationary phenomena in continuous
media. We shall encounter a particular instance of this technique in deal-
ing with wavelike transmission in Part XVI.
Part XIII 2.75l. CLASS NOTES 161
- as as me as sº sº, sº me -e ss sº as eas ºs ºs eme sm * = * * * * *
The concept of a stationary field occurs in many branches of
physics and engineering; for example: electrodynamics, aerodynamics,
hydrodynamics, elasticity, heat conduction, and gravitational theory.
insofar as the phenomena admit the definition of scalar potential functions,
their abstract form and treatment is essentially identical. This makes it
possible to establish formal analogies of strict equivalence and to trans-

late solutions and experimental results from any one field into all analogous
fields. Each one of these particular media is characterized by a fundamental
scalar which satisfies the Poisson or Laplace equation, by a derived field
vector which is defined as the gradient of the associated Scalar potential,
and by an associated field vector which is tensorially related to the grad-
*ient vector. There are additional analogous concepts for each applied field
which makes it only necessary to obtain the solution of a problem in one
branch in detail in order to be able to predict for every other branch the
similar solution merely by making the proper correspondence of terms. For
our purposes here we shall find it convenient to deal entirely in terms of
a generalized language as indicated in the first row of the appended table.
It is of particular interest to note that associated with each scalar poten-
tial field there is, in general, an associated vector field. In keeping with
our generalized symbolism, we may think of one set of fields as intrinsic or
effort fields and the other set as extrinsic or flow fields. In the elec-
trical case we have the electrostatic field and its relation with the cap-
acitance in the form of the associated field transmittance; in the magneto-
static case, the associated magnetic flux and field permeance. It is
frequently helpful and suggestive to interchange terminology and other imagery
from one medium to another, so that the maximum cross-fertilization of ideas
can occur. On the other hand, by manifesting that all particularizations
arise from a single, common generalization, all field analogies are rigorized
and a methodical approach to field concepts is made evident.
162
2.75l CLASS NOTES
SPECIALIZATIONS OF

Potential Potential Equipotential Potential
Function Difference Surface Gradient
GE ZED 2 -- —- * *
; Ul ul - up –ſ U - d.s u = const. U = (+)-7,
l
2 * * * *
ELECTROMAGNETIC 3 - £4 =f H ds | 3 = const. H = - V fº
Magnetostatic l (Ideal Iron Magnetic Field
Potential Surfaces) Strength
é, -é, -r E. W. & --> *
1 - C-2 = E • dis = const. E = - V &
ELECTROSTATIC Elect rostatic f l (Ideal Conductor Electric Field
Potential Surfaces) Strength
& &n – & ſ f : . 35 | E t # = -\7 &
l - 2 = E • 8 > COIlST, e. : -
*;, Electrostatic l (Electrode
Potential Surfaces) do
2 * ** º *
STEADY T * - tº -ſ U • dis T = const. U = - VT
TEMPERATURE Temperature l Isotherms Temperature
Temp. Differential Gradient
q, 2 -a- • * -º
FLUID $, -be =ſ W ds = const. W = -V (p
Velocity l Equipotential
VELOCITY Potential Surfaces Velocity
2 -- -º- * **
FLUID H * - He-ſ U - d.s H = const. U = - V H
SEEPAGE Pressure Head l *:::::::s .
Differential Head
2 * -lº -º- -*.
U U2 - U -ſ g ds U = const. g = + V U
GRAVITATIONAL Gravitational l Isogravimetric Gravitational
Potential Potential Difference Surfaces Acceleration
Part XIII
163
2.75l CLASS NOTES
ISOTROPIC POTENTIAL FIELDS

Material Associated Associated Field Remarks
Constant Field Wector Flux Transmittance
tº
-º- -à- -àe —he
K v = k U * - ſº . Î || 7-v/(a, - wº) ||7- Aſi,
Transmittance = k LA
- Al B = p H q = | B dA (P = d/(? - ?:)
Permeability Magnetic Induction Magnetic Flux Permeance
- * -* -*. —h-
8. D = & E a -ſ; . . C = Q/(€1 - & 2)
Dielectric
Constant Electric Induction | Electric Charge Capacitance
a J = or E 1 -ſi- iſ G = I/(81 - &2)
Electric
*Conductivity Current Density Current Conductance
*
k J = k[J a -ſi- iſ G = Q/(T1 - Ta)
Thermal
Conductivity Heat Flux Heat Flow Thermal Conductance
o p = 9 V W = | p - d. A G = W/(P - Pe)
Mass Density Momentum Flux Mass Flow
Or W = or U * - ſº . . F = Q/(H1 - Ha)
Permeability Seepage Velocity Seepage Flow Seepage Conductance
*-
16], 2.75l CLASS NOTES
tºp º ºr ſº ºm º ºs º ºs ºs ºs ºn tº º ºs ºf flºw tº 4º dº ºms ºf tº * * * * * * * * * * * * *
Consider a heat conducting medium with the configuration indicati

Ambient Temperature T
/ - mpera */ (e.g., Ground Surface)
/23///, // '7, 4–4 ‘’
/6% " [...] /
(e.g., Steam Pipe) /// (e.g., Utility Duct)
/ / / / / / / / / / / /
We might consider the situation where the temperatures were all considered
as functions of time, To(t), Ti(t), T2(t), and we desire information on
the resultant flow of heat, particularly, for example, the heat loss from
the steam pipe or the heat flow into the utility duct.
This system may be viewed as a 3. port thermal element:
T **
T Thermal T
—H System Hº-
If the element can be assumed linear and in the steady state then we know
that the following equivalent sets of linear equations hold:
T = IR Q
! º
-i. | 1.4-lie-i-º-||..i.
- ie. - |-ºl.i.e. i-º- |..º.
T3 *31 ; R32 : F23 83
Or’ Q : G e
ems ame mºs ºs º * * * * * * * * * * * - º dº sº * * ºn tº ºs
tº tº tºe tº º tº ºs ºs º ºs - * * * * - tº ºn ºss tº - tº tº gº. --
Q
2
:
2
1
G
2
2
2
3
2
Part, XIII 2.75l. CLASS NOTES 165
However, since we are presumably dealing with a potential field,
GREEN’S THEOREM (see below) results in MAXWELL'S RECIPROCAL RELATIONS,
namely
*i; E Fji 3 94; E °3;
which merely express the fact that the resistance and conductance matrices
are symmetric. But two additional nodicity conditions Will also hold,
namely:
I • * * & + 82 + Q2 = O
3
II - - - Q (T) Q ( T, II T) (T = Const)
The first relation expresses the continuity condition while the second
equation states the relativity condition (i.e. that the heat flow
depends only upon relative, and not upon absolute, temperatures). As
We discuss later in connection with trinode amplifiers, Condition I
then requires that the row sums of IR (and the column sums of G ) must
vanish identically while Condition II requires that the column sums of
R (and the row sums of G ) must also vanish.
. Finally, as a result of the reciprocity and nodicity conditions
above, this field problem may be conceived in terms of the following retic-
ulation:

O
R
2. ' \,.” *o/s. \\º
R
J’ e ST? Ti ^^^. T2
& 812
If the values of the three equivalent thermal resistances (Ra, Rp, R.) Were
known, the instantaneous heat flows could be calculated.
166 2.75l CLASS NOTES
This model derives its validity from the presumed superposibility
(or linearity) of the governing temperature fields. This means, for instanc.
that we can consider the total heat loss from the steam pipe as the sum of
810) and the loss to the duct (812). While this
is reasonable and obvious, it is not so obvious that these effects are ideal.
the loss to the atmosphere (
independent of each other, such that the loss may be assumed in the form:
Q = Qio + 812 = GaſT1 - To) + G. (T1 - T2)
Where Ga and Ge represent overall linear heat transmittances or thermal
conductances. As we shall see, each of these conductances may be derived or
estimated directly from the form (alone) of the temperature field, in the
fashion:

* * * : * : **
Thermal zºº. \º Form
Conductivity Length Factor
Thus it is that the macroscopic properties (e.g., G.) are related to the
microscopic properties (e.g., kºm) and the absolute size (e.g., L).
All the above concepts were first employed in a general and
consistent fashion by James Clerk MAXWELL for electromagnetic fields. We
demonstrate below that these tools have universal application and great
utility. In particular, all one-port linear properties may be related
very simply to the geometrical parameters (= size X shape) and material
properties (= transmissivity) in terms of the overall transmittance:

(Transmittance) = (Transmissivity) - (Size) . (Form)
\- L/ \. Af
Property Geometry
Part XIII 2.75l. CLASS NOTES 167
It has not been customary in applied mathematics to consider the
particular engineering significance of abstract concepts, nor in specific
physical or engineering sciences, to display clearly the logical necessity
and interrelationship of diverse, separate historical discoveries. Thus
we find valuable at this point a terse summary of fundamental concepts upon
which the field description of all material systems depend.
We are concerned here both with scalar quantities (u) which are
the analogs of geometrical points, and also vector quantities (U or U),
the analogs of directed line segments. Pressure, density, temperature are
typical scalars; while force, velocity, heat flux, current density are repre-
Sentative vectors. The treatment below is somewhat different than the usual
in the introduction and use of matrix notation for vectoral relations.
Of course, the ordinary time and space derivatives transform scalar
fields into new scalar fields, and vectors into new vectors. But in addition
to these familiar operations we must consider also three additional vector
derivative operations as follows:
GRADIENT
V
_--—s

Scalar Vector ROTATION
Fields Fields VX
DIWERGENCE
Vo
To complete the field description, we need also to account for the
material or phenomenological operators relating vectors to associated vectors
in the form:
V = IK U
In an isotropic field, the general matrix IK reduces to a Simple scalar
constant, K .
168 2.75l CLASS NOTES
Scalar Fields
gº º ºsmº ºm ºmº ºne ºm me time tº ºne ºmº wº
If any scalar quantity, u, such as temperature, pressure,
density, voltage, etc., is defined at every point in a region we have
a scalar field. The general scalar field varies with time as well as
position, which in Cartesian coordinates we would indicate by
u = u(x,y,z,t). But the field is clearly independent of the coordinate
system in which it is measured; to emphasize this invariance it is prefer-
able to Write u = u(R,+), where f measures the position vector in an
arbitrary frame.
If the field is static or stationary, it is time-invariant and
We have u = u(R), alone.
If we connect all the adjacent points in a scalar field having
equal values of u, we obtain level surfaces, isopleths, or isotimics.
(These only become equipotentials if the scalar u is a potential function).
In two dimensions the isotimics are curves, the analogs of the familiar
contour lines. In the ensuing discussion, we shall find it convenient to
depict our results, whenever appropriate, for this planar case since it
is best suited to the limitations of the printed page (or blackboard!).
The First_Derived Quantity: The GRADIENT of a SCALAR FIELD

- h.
This derived Trai u = j Generates a
operation ~ * 9. tºº N-> VECTOR FIELD

The GRADIENT is a DERIVED VECTOR QUANTITY equal to
the MAXIMUM SPATIAL RATE of CHANGE of the SCALAR FIELD FUNCTION.
º asº.
The vector U = Vu may be computed by matrix means; for
Cartesian coordinates, this may be written:
U V O ll
U [& / ex ; o /ö y : 3/2 z] u
pºrt XIII 2.75l. CLASS NOTES 169
The gradient is always normal to the isotimics and measures
the maximum spatial rate of change of the scalar field. In planar fields,
de gradient is thus directed at right angles to the contours and measures
de direction and magnitude of the steepest ascending slope.
This concept of a gradient is fundamental to the field descrip-
tion of physical systems. The gradient generates a vector field from the
even scalar field; we might perhaps best visualize this process in terms
* the gradient field derived from a set of elevation contours above a
wo-dimensional surface such as a plane or sphere.
In general, the gradient vector will be different at each and
avery point of the scalar field; we thus obtain a derived vector field
*ich may be represented by an equivalent system of fieldlines or gradient,
linese
These gradient lines may be approximated by the following simple
construction.
--~~ – - Approximation to the gradient line
- between u = 1 and u = 2
`. d 6 u = 2 through point a.
_c}-T
| * 2.
* / 2' 2
/ Af
€ ... ."
u = 1
Continuing in this manner until a summit is reached, an approximation to
a curvilinear gradient line may be obtained. If a system of such field-
lines are drawn, any scalar field may be represented in conjugate form
by the system of gradient lines as follows:


17O 2.75l CLASS NOTES
Original Representation Conjugate Representation
by
CONTOUR LINES
a’ ---T... --→" ~ |
/ … --- | |-- _-
º/ /2-
`-- / / ſ \
J. --T / | ~/
–
Thus we have, in fact, replaced the original scalar field by an
associated vector-field-–the gradient field. Clearly, we could recover
the original contour lines by the same process used to obtain the
gradient lines •
The line integral of a gradient field between any two points
is independent of the path and is equal to the difference between the
values of the associated scalar at the terminal points of the path; thus
2 -> -º-
Vu dR = ue - ul
1
If the path is closed, the line integral must necessarily vanish:
-*- -*.
§ V u • dB = O
The significance of this last result lies particularly in the converse
interpretation: namely, that
—A. **- *. *—º
If $ F - d.R = 0, then F = grad u
Where u is an associated scalar field.

Part XIII 2.75l. CLASS NOTES 171
The Second Derived Quantity:
* = * * * * * * * * * * * * * * * * * * * * * * * * *
(The Divergence of a Vector Field)

*- * Generates a
This derived º *
div - F = V - F SCALAR FIELD
operation

The DIVERGENCE is a DERIVED SCALAR QUANTITY equal
to the LOCAL PRODUCTION of FIELD LINES per unit volume.
** **
The scalar divergence or = V - F may be computed by matrix means;
for Cartesian coordinates this inner product would be written:
O = Wo - F
F
[3/3 xjö/øyjø/32] .
O-
-:
The Gauss Divergence Theorem
Let us consider any closed control surface A bounding a volume
in any vector field F.
Then the following fundamental theorem due originally to Karl
Friedrich GAUSS will always hold:

ſſ, F. J. = ſ.ſ. (v. 7) a v-
\—y—’ \–y-
Surface Wolume
Evaluation Evaluation
Control Element
SURFACE: A

VOLUME ºf
ºf 1
172 2.75l. CLASS NOTES
The GAUSS Theorem (together with its STOKES' conjugate below)
may properly be considered as the basic theorems of mathematical physics,
Since from them may be derived all conservation and continuity principles,
as well as GREEN'S theorem and the reciprocity principle.
Field Tubes and Solenoidal Fields
* * * * *=º ºr ºne ºs º ºsmº gº tº º tº ºsº ºn tº ºne ºf ºr tº tº iº sº º ºs º ºs ºs º gº tº tº
If in a general vector field, F. We restrict attention to regions
which are divergence-free, where
V . F = O
then we can readily see that there will be neither production nor destruc –
tion of field lines within this volume. Thus any internal field line will
|be conserved.
In such regions consider the field between two arbitrary surfaces
A and B. (These would become curves or points for two- and one-dimensional
fields, respectively).
Let us draw an arbitrary closed contour, a, on surface A. Through
each point, Cl , on a, a field line will pass which will intersect the Sur-
face B at point B. As the point O is moved around curve a, the point
will trace a corresponding closed curve, b, on B. Moreover, the field line
segments, C. B , serve as generators or directrices of a tube T, which we
shall define as a field tube.
FIELD TUBE T
Curve
Surface Surface
A B

2.75l CLASS NOTES 173
In electromagnetism such tubes were first named by Michael FARADAY
"sphondyloids" but James Clerk MAXWELL used the term "solenoid" from the
Greek a w A m ‘ v (solen) "a tube". The current usage in electrodynamics is
tube of induction or flux tube. In fluid mechanics such tubes are called
- Everywhere in such tubes the field is conserved, resulting in a
continuity relation. Moreover, by definition no FIELD WECTOR F can inter-
sect a field tube.
* = * * * * * = * * * * * * * * * * * * * * * *
(The ROTATION of a VECTOR FIELD)
This derived rot F = VX F Generates a
*Cº-
operation WECTOR FIELD
Nºxº~ \\
The CURL or ROTATION is a DERIVED VECTOR QUANTITY
equal to the LOCAL CIRCULATION per unit area.
We may compute the vector G = V X F very simply by matrix means;
in Cartesian coordinates this outer product becomes:
G
WX . IF
Gl O -o/öz;+0/0y F.
-ā--|-|1337; 5-3-373; ..]".
-āj--|-375;#1375;}~5-- ||--Fi-
3 ! ! 3
* * * * * * * * * ~ * - * * * * * * * * * * * * * * * * *
Let us consider any closed contour C bounding an area A in any
tector field F.

17 l; 2.75l. CLASS NOTES
Then the following fundamental theorem, due originally to Georg
Gabriel STOKES, will always hold:

2. F E f. F. ãR = Jſ, W x F. dA
| \–v-” \– Y A
Circulation Boundary Surface
Evaluation Evaluation
Irrotational Fields
* * * * * * * * * * * * * * * * * * *
If we know that, T is identically Zero in some region, then we
may always Write
F = + grad p = + V4,
since V X V% = 0 for any scalar field. The q, so associated with F is
called a scalar potential function, following the Original terminology Of
George GREEN. The corresponding field is known as a potential field.
Conversely all gradient fields are irrotational by identity.
Solenoidal Fields
ems ſº wº º ºss ºn tº tº ºs mº assº ºn eme * * * *
V - G = O
Since V : V X F vanishes identically for any
vector field F. Such divergence-free G fields are called solenoidal.
Conversely, the vector F associated with G is said to be a
vector potential function and generates a corresponding vector potential
field.
art. XIII 2.75l CLASS NOTES 175
, - - - - - - - - - - - - - - - - - - - - - - - - - * * * * * * *
We may now summarize the results above for the dual cases of
ſcalar and vector potential functions.
IRROTATIONAL FIELDS SOLENOIDAL FIELDS
(Conservative) i (Source-free)
-*. -> ſ —he ->
If? V × ºf = 0 ; If: V - f = 0
-*. --—A- | -> —- -
Then: f = + grad e Then: f = + rot F
-> → l -> *-* ->
since; V × Ve = 0 since; V - V × F = 0
;
y
hen e(s, y, z) is said to be a # Then F(x,y,z) is said to be a
!
;CALAR POTENTIAL FUNCTION which ! WECTOR POTENTIAL FUNCTION which
generates a (SCALAR) POTENTIAL ; generates a VECTOR POTENTIAL
|
FIELD. º FIELD.
—he -—h- : -*- —-
'aking f = + grad e we have for >{ Taking f = + rot F we have for the
She components # components
º
{
de/2x ; ( ò F/? y) $º ( * r ſ & 2)
be/?y ( ò F / o z) tº-º- ( ò F/ē x)
be/32 { ( ò F/ ex) º ( ò F./A y)
'or the two-dimensional or planar field we find F. E F, = O and F, may
le taken as the scalar, f. Then we obtain the results:
T - 2e/2x be/oy) = [ of/ oy - of/ ox]
This conjugate plane potential field is of singular importance
20 the analysis of all physical systems. Its relation to conformal
lapping we next discuss.
176 2.75l CLASS NOTES
Plane E2tentials and Conformal Maps
Many plane potential field problems may be solved readily em-
ploying functions of a complex variable. We outline here only a brief
account of the resultant conformal mappings.
If the argument of any algebraic or transcendental function is
a complex number z = x + Jy, then, in general the function w = f(z) will
also be a complex number W = u + jv. If this function is analytic then
>
(2 u/2 x) + j ( 2 v/ex) f : (z)
( ò u/? y) + j (2 v/2 y) = jf'(x)
Comparing terms gives the CAUCHY-RIEMANN equations:
ou/ ox = ov/öy ; Ou/ö y = - bv/öx
But these conditions have two interesting consequences. First they en-
sure that every transformation w = f(z) which satisfies these conditions
Will map portions of the Z-plane into the w-plane such that relative
directions and angles are preserved; such transformations are called
conformal mappings and would map any set of three z-points as follows:
Z-plane

Part XIII 2.75l CLASS NOTES 177
as a result of conformality a rectangular grid in the z-plane is mapped
into an orthonormal meshwork as indicated.
2-plane -
777-7777-77777.7/7/ -I
The second interpretation of the CAUCHY-RIEMANN equations results from
the fact that they are precisely equivalent to the results we obtained
above for plane potential fields, namely
* - gº e = [oe/ox be/ > y)
*
also = rot F = ſ of/ oy - a f/2 x]
Thus any conformal mapping produces a plane potential field.
We may always interpret One set of lines as equipotentials and the con-
jugate set as the corresponding fieldlines. However, and perhaps more
importantly, it becomes apparent that the field form factor A is a con-
formal invariant and represents nothing but the appropriate aspect ratio
Of the conformally equivalent rectangle: This fact we treat in more
detail below.
Background Reading -- Conformal Mapping
(1) BECKENBACH, E. F., Editor construction and Application of conformal
Maps, Proceedings of a Symposium, National Bureau of Standards
Applied Mathematics Series, Vol. 18 (1952).
Perhaps the single most useful reference in this subject.
(*) ROTHE, R, OLIENDORFF, F. and POHIEHAUSEN, K. : Theory of Functions, (1933)
A translation of a classical German text, which predated American
engineering application.

178 2.75l CLASS NOTES
Background Reading -- Conformal Mapping (Continued)
(3) HEWIEY, L. V. : Two-Dimensional Fields in Electrical Engineering (19.
An excellent treatment of the practical use of conformal maps to
several fields of engineering.
(h) KOBER, H: Dictionary of Conformal Representation (1952)
A useful table in applying mapping functions.
(5) NEHARI, Zeev. Conformal Mapping (1952)
(6) CARATHEODORY, C.: Conformal Representation, Number 28, Cambridge
Tracts in Mathematics and Mathematical Physics (1952
The two above monographs deal with the analytical and formal pro-
perties of conformal maps, including 3-dimensional transformation.
D. Field Form Factors
As indicated earlier in this section, the transmittance of a
homogeneous energetic field depends only upon the property constant of
the medium and upon the field geometry in terms of (size) times (shape).
This latter shape constant is measured by the field form factor, A ,
which then can depend only upon the (normalized) boundary conditions.
For a field tube, these boundary constraints consist of fieldlines along
the Walls and two terminal isotimic surfaces.
If the field is a potential field, the form factor is unique
and invariant under conformal mappings. These properties of 2\ yield a
number of Simple bounds and estiuating techniques as indicated below.
Part XIII 2.75l. CLASS NOTES 179
The general field relations above may now be applied specifically
to the problem of evaluation overall transmittances of a field tube. We
assume in each case the existence of the following primitive vectorial re-
lations:
f * * * * * * * * * * * * * * * * * * mass sm sº tº as mm as tº sº tº ess tº me as ame amº me me ºs ºse tº ºs ºms ºms ºne ºf ºn sm sº tºº sm; sº tº º smº 7
; A SCALAR POTENTIAL: u = u(R, t) ;
: The POTENTIAL GRADIENT: U = - grad u :
: An ASSOCTATED VECTOR: W = k U :
* — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — J
We may represent an arbitrary field tube in the following planar
fashion:
l, sº LA
Boundary
Boundary Equipotentials
Fieldlines
Then along the tube between the two bounding equipotentials we have
the relation: * -- ~~
POTENTIAL DIFFERENCE: u, - ue = ſ U - d.R
While across the tube between the bounding fieldlines we have the relation:
TOTAL FLUX: V = J V - d. A
A
We are now in position to define the overall field transmittance in
the fashion: f —he -º-
A
E
W .
FIELD TRANSMITTANCE: Jºe —#--— = —
ſ T. is u, - u2
1
In every field of physical origin, it is this quantity which is of greatest
*ignificance and importance at the macroscopic level. Historically, these
Phenomenological relations were discovered independently and generally are
*med after their discoverers (e.g. OHM's Law).


18O 2.75l. CLASS NOTES
However, it is our present purpose to demonstrate that given the
existence of an underlying relatively homogeneous and isotropic field, the
corresponding transmittance may be evaluated from the field geometry and
material transmissivity.
It may be readily demonstrated from dimensional reasoning that, since
==-ſº =º,
U = - grad u and W = k U , then:
-* =mº, A
U dA = V
Jº- * TFIF k –F–
*} - d.R. ll
Where Ay and Pu are an appropriate area and length, respectively, to absorb all
information about the field geometry. But we may further factor the geomet-
rical aspect into a (size) x (shape) product since
| 2
A/L, - (A/L,) N = (If/L) A = L A
Here An and I'm are arbitrary or nominal dimensions but L measures absolute siz:
and X measures the characteristic form.
Thus we obtain finally:
! l
; FIELD TRANSTITANCE: = k - I - A = (Property) - (Size). (Form):
\ _0 \ J. :
* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ---------------X------
MATERIAL GEOMETRICAL
Parameter Parameters
The value of X depends only upon the form of the field-–the boundar
configurations and conditions--and not in any way, upon the absolute scale or
the particular medium. This fact is not only the principal justification for
the practical use of field analogies; it also permits rapid estimates of trans-
mittance by simple inspection of field geometry.
For two-dimensional or planar fields the dimension L can be taken as
the transverse width of the medium, giving a particularly simple result for the
transmittance per unit width, namely:
This relation clearly implies that geometrically similar planar fielº
have identical form factors. But the same result is also true for all potentiæ
fields.
Part XIII 2.75l CLASS NOTES 181
,- - - - - - - - - - - - - - - -
Given the steady potential field e(x,y, z) and the associated
-> S.
vector field f(x,y, z) = - k grad e, the field energy contained in an
arbitrary closed region is given by:
- 2 ... — P. zTA º
IE = | f | * dºſ = k J. | grad elº d’
*f ºr
But using the divergence theorem these volume integrals may be directly
equated to the Surgace integral to obtain
- → al, —A- -*.
|E == - / efs dA = k /. e grad e º dA
In the discussion below, we shall restrict attention to the
special case of plane potential fields. However the methods and relations
derived are of general validity.
For this case, the above general principles reduce to the
following relations:
/-
2. §
IE /* = k #,G# * + (#)*. dxdy = off, (#): + (#): dixdy
^ 2 2 2-
* } • # as - ; }, e # de = k 2, e if
>
The form expressed at the lower right is most easily evaluated
Blone e-lines and f-lines as indicated below:
Boundary c —

182 2.75l CLASS NOTES
Thus we obtain for the energy per unit. Width:
|E /L
-
k # e dif = |k (es – ei) • (f), – fi)
= k • E • F
It is obvious now that this is merely the Simple summation of the equal
energies distributed in each Orthonormal cell. Moreover, Since F = A E
then we obtain finally
t
+
*-----------------------------------------------
Thus the form factor A not only serves as a measure of the field trans-
mittance; it also is a direct measure of the energy storage or dissipa-
tion in any potential field.
Returning to the volumetric evaluation of field energies We are
now in a position to state a pair of complementary extremum principles
which may be used to determine the field energy and therefore the form
factor. These we shall name after their promulgators, namely:
DIRICHLET’s Principle THOMSON's Principle
Of all arbitrary scalar fields Of all solenoidal vector fields
Y
|
in a region Satisfying a given ; in a region Satisfying a given
boundary condition, the POTENTIAL boundary condition, the IRROTATIONAL
field has the MINUMUM energy. field has the MINIMUM energy.
ſ
The two principles may then be used very effectively to
estimate upper and lower bounds for the field form factor to obtain re-
Sults in the form:
D T
part XIII 2.75l CLASS NOTES 183
For DIRICHIET's principle we make an arbitrary assignment of
isotimics compatible with the boundary conditions; for THOMSON's princi—
ple, the fieldlines are assumed. If, and only if, the assumptions corre-
spond to equipotentials and gradient lines, respectively, Will the
corresponding field energies be minimized.
For some of the interesting history behind these dual principles
we may quote POLYA from the work cited below:
An important special case of Thomson's principle • . .
Was already known to Gauss. . . That the two principles
can be used to obtain estimates of the capacity from
opposite sides, has been observed by Maxwell to whom
the method was suggested by an investigation Of Lord
Rayleigh...Maxwell, however, did not derive upper or
lower bounds from his method in concrete cases, Ob-
serving that the "operations. . . that are in general too
difficult for practical purposes".
That this is not quite the case, particularly in this era of
machine computation, is amply testified to in the book of POLYA and also
in CRANDALL*s text, which is referenced below:
It is perhaps easiest to visualize this application for the
case of dissipation in a planar field. However analogous developments
are possible for all energetic fields.
DIRICHLET Principle y THOMSON Principle
Let e(x,y) be an arbitrary Y Let f(x,y) be an arbitrary
Scalar function such that e = 1 source free vector function such
On Surface A of a fieldtube and ; that the fieldlines are normal to
e = 0 on surface B. Then E = 1. * A and B, coincide With the field-
tube boundaries, and enclose one
ſ
|
ſ
3.
unit of flow. Then F = 1 .
----------------------------------- *-
18|| 2.75l CLASS NOTES
--- —A- -
P. - e ſº. It ºv IP = 0 ||f| av
v. i v.
= Go - B = GD ; = Rr • Fº = Rr
But the above minimum principles tell us
? -
--> —-
Go = G i R = R
and the equality Signs hold only for:
-> -->
V*e = 0 ; Vx f = 0
º
Thus we may obtain upper and lower bounds in the form:
mº sºme sºme sºme emº ºme ºne sº ame ºn sºme ºf sº tº tº sº tº º smºs sº sº º Aº º º T
y
-> -> y
GD > G = GT i
i ~ R -- ;
; F = R = Rr
3 _j
If the bounds are close then
T---- Y
; g = G - G ;
; D T i
i VT. R.
; F = # - RF
† R
(Note in the above approximation if arithmetic means were used different
estimates would be obtained for conductance and resistance, which is not
particularly logical.)
Background Reading -- Dirichlet - Thomson Principles
(1) POLYA, G. and SZEGO, G. : Isoperimetric Inequalities in Mathematical
Physics. Annals of Mathematics Studies, Number 27 (1951).
(2) COURANT, R.; Dirichlet's Principle, Conformal Mapping, and Minimal
Surfaces. (1950).
Part XIII 2.75l. CLASS NOTES 185
Maxwell's Principle of Transmissivity
A universal field principle of great utility was first ennunciated
by James Clerk MAXWELL. It may be stated in general dualistic terms in the
following Words.
smº an arms ºne man ºn tº m ºms ame as ºs º ºs - ºn tº eme ºn tº mº m sºme - * * *s as sm sº am was amas m = m, sº tº mº me ºn as tº me ºn sºme mº me as ºw ame ºn tº me sm as º ºs
In any field, if the transmissivity is locally
i
f T
t f
; 1.
! }
; !
: DECREASED INCREASED ;
f
; the overall transmittance is correspondingly
! 1.
; TECREASED ; INCREASED ;
; ; !
f f
* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - I
It then follows that fixing any
fieldline equipotential
DECREASES INCREASES
the Overall field transmittance.
These last conditions amount to a loosening and generalization of
the
THOMSON Principle DIRICHLET Principle
:
respectively.
The transmissivity principle has practical application to uniform
and nonuniform, linear and nonlinear, continuous and reticular fields of all
sorts. As indicated above, it may provide bounded estimates of the form
factors and transmittances of fields; but, in addition, by judicious com-
pensatory or balanced increases and decreases of transmissivity (or local
transmittance) throughout the extent of a field a rational estimate may be
readily obtained.
186 2. Tjl CLASS NOTES
E. Rectangle Diagrams
In the earlier discussion of state-determined elements, the concept
of complementary energies was introduced. While these notions originated
with MAXWELL and HELMHOLTZ in the last century, only recently a striking inte:
pretation was given to the distribution of generalized energy over reticular
fields in terms of rectangle diagrams. These figures, introduced chiefly
through the efforts of CHERRY and MILLAR, represent the generalization of the
concept of form factor to arbitrary linear and nonlinear fields. They serve
to portray graphically the local distribution of stored or dissipated energy
over the extent of a reticular 1. port system or network of like-kind 1-port
elements (i.e. all-resistor, all-capacitor, or all-inertor reticular fields).
For each of the three species of nets, the rectangle diagrams are
mosaics as follows:
RESISTIVE FIELD CAPACITIVE FIELD INERTIAL FIELD
rºl Fºº Fiºr. M.
e Fºr 3. a ſujº. 4 ºriº
2.lk 4R, tº “º. 31. 4U, * | * Zik 24t.



F- d f
P = x. P. U = X U. T = x. T.
k k- p’
Since the appropriate generalized energies are all compatibly distributed, so
must also be the corresponding normal and complementary energies. For a unifori
continuous field, with unit property constant, the rectangle diagram simply be:
comes the conformal mapping of the field into an equivalent rectangle of iden-
tical form factor; as before, the form factor, \ , measures the aspect ratio
of this rectangle, all of whose elements are equal squares.
However, the importance of the rectangle diagram is rather for depicº
ing the case of nonuniform and nonlinear reticular fields, where the elements
themselves will also generally all be diverse but compatible and contingent
part XIII 2,75l. CLASS NOTES 187
rectangles. Clearly, the micro-reticulation can then extend indefinitely
to the field substructure if desirable, but this is not a necessary require-
ment.
Whenever we deal with continuous or reticular, linear or nonlinear
fields of like-kind elements operating under steady conditions, the dual
pair of Lagrange Equations cited previously yield a simple set of dual extremum
Principles for each type of element as follows:
APACTTANCE : = {
! 0U / 2d so
RESISTANCE : 3 IP iſ a f = O
INERTANCE: 3 T E/ 3 f = 0
oU./ O (€2 = O
a P.) 9 @ = 0
9T./ ope O
Hince all these pairs have identical topological structure, we shall center
httention on the resistance case. These dual extremum conditions give us
Bimple generalizations to arbitrary fields of the classical
THOMSON Principle DIRICHLET Principle
:
liscussed above.
For the resistance case, their significance is the following: in
liny reticular field embedded in a fixed environment, where all bounding parts
have been represented as equivalent:
Neumann (Helmholtz) Ports Dirichlet (Thevenin) Ports
:
!or any arbitrary internal assignment of
Flow f
Effort (€2
ſ
188 2.75l. CLASS NOTES
compatible with the boundary conditions, that set which minimizes total
Cocontent IP
Content IP, e
i
represents the equilibrium or steady-state configuration. Similar statements
hold for capacitive and inertial fields.
Not only do these principles offer a particularly attractive way tº
estimate the steady-state solution for such fields but they also lead directl
as before to bounded estimates of overall transmittance.
Again, for the resistance case an approach to the minimum for:
C O N T E N T : C O C O N T E N T
W i l l f u r n i s h a n a p p r o x i m a t e
L O W E R B O U N D : U P P E R B O U N D
;
for the overall conductance (= resistive transmittance).
These properties may be made self-evident through use of rectangle
diagrams as indicated below.
We may demonstrate the relation of the above principles to rectang!
diagrams in terms of a loaded resistive Wheatstone bridge. For simplicity,
us consider the case where the variables have been normalized as indicated.
loroader application of these ideas should be obvious from this example.
Part XIII 2.75l CLASS NOTES 189
f
GENERALIZED THOMSON PRINCIPLE ; GENERALIZED DIRICHLET PRINCIPLE
T
7.
!
Assignment of (*A. *E) : Assignment of (ea. ep)
t
; 6 = 4
! ~~~~
!
!
!
J| 1-f,
A 1.
;
: 6a €e
f
| | 1 – fe :
!
f
2 ºz º.º. P º zºº º º T
1.
!, : € = O
!
1.
Rectangle Diagram: ! Rectangle Diagram:
!
!
1.
º
+ !
A :
º :
°3% º ;
£Pf 34 (6.3% :
• S. §3. º y P ;
**
f= O g 3% F#2 f= 1 ;
§§ ºł1& ;
*** -arº, ;
• 3%. 26: ... !
:- - - ** * *** 'ſs: ;
+ f
f
B. ;
1.
f
1.
!
f
;
-> T >
f f, equil. ; e e, equil.
!
!
For the linear case: : For the linear case:
!
!
> ! *
Re wº- R ; *e -> G
tº gº - - tº gº tºº tºms as ºn tº ºn tº +-------------
! !
! !
f !
; R = R = R
! f - * e !
1. !
; i
<
: Gr -: G → *e t
t t
t t
! !
* ------------------------- 1.


190 2. Tjl CLASS NOTES
In the above relations, the equality signs hold only for the
equilibrium case, corresponding to a correct assignment of system variables.
The Coons Construction
tº ºr ºs º ºs ºs ºn tº me ims tº ºm º ºs º ºs tº mº me tºns tº sº
As another example of the application of rectangle diagrams, conside
the problem of determining the overall transmittance of the following bridge
structure :
E
=–
1
C
— 4
where the local linear transmittances are as indicated. The solution of all
such linear (and nonlinear !) bridges may be elegantly determined using a
simple rectangle diagram construction originating with Stephen A COONS as
follows:
Coon' s Construction Rectangle Diagram
<}
<!
4 º
2 4


Part XIII 2.75l. CLASS NOTES 191
his gives a final transmittance:
J = 5/7 = 0.7
This result may be check using the MAXWELL Transmissivity Principle
is follows:
LOWER BOUND UPPER BOUND
Zero Central Transmittance Infinite Central Transmittance

ZZ :
% : |
% S 3.C) i 2, O
2}\\ : § |
N : N
<!— 2, O —- ;
7 = 2/3 = c.67 ; J = 1.5/2.0 = 0.75

- * * * * * * - - - * * * wº-
jackground Reading -- Rectangle Diagrams
1) CHERRY, E. C. and MILLAR, W.; Some New Concepts and Theorems Concerning
Non-linear Systems, Automatic and Manual Control, pp. 263-274 (1952).
3) GARDNER, Martin: Mathematical Games, scientific American, Vol. 199,
Number 5, pp. 136–1 || 2 (November, 1958)
192 2.751 CLASS NOTES
Background Reading -- Potential Functions
(1) GREEN, George: An Essay on the Application of Mathematical Analysis
to the Theories of Electricity and Magnetism (1828).
* =s=" sº-ººººººmsºmº- ºnemes amº-wºmmemºmºmº-e wºmeºmº-me
(2) ------------- : Mathematical Investigations concerning the Laws of
Equilibrium of Fluids analogous to the Electric Fluid, with
other similar Researches. Phil. Trans. (1833).
These two papers by a self-taught genius (and miller!) are the
foundations of modern potential theory.
(3) KBILOGG, 0. D. : Foundations of Potential Theory (1929).
(h) MACMILLAN, W. D. : The Theory of the Potential, Second Edition (1958).
The above two books are the classic references.
Background Reading -- Fields
(1) MAXWELL, J. C. : A Treatise on Electricity and Magnetism, Third Revised
Edition, Vol. Tand IT (T85T).
Maxwell's contribution to field theory ranks with Newton's gift to
dynamics.
(2) MASON, Max and WEAVER, Warren: The Electro-Magnetic Field (1929)
A definitive work of an analytical nature.
(3) SKILLING, H. H. : Fundamentals of Electric Waves (1942)
An outstanding lucid introduction to field phenomena treated by
vectorial mechanics.
(H) WEBER, Ernst: Electromagnetic Fields, Vol. I - Mapping of Fields
Second Edition (1951).
Very instructive and readable introduction to field theory.
(5) ROGERS, W. E.: Introduction to Electric Fields (1954).
Especially elegant graphical and analog field plots.
(6) SCHNEIDER, P. J. : Conduction Heat Transfer (1955)
An excellent treatment of transient and steady field phenomena dis-
guised under the rubric of an engineering speciality.
(7) BITTER, Francis: Currents, Fields and Particles (1956)
A text. Which has been used in the basic undergraduate curriculum at
M.I.T.
Part XIII 2.75l. CLASS NOTES 193
F. The Temporal Response of Physical Systems
*
Let us use the closed, state-determined system as an example.
The conditions at the boundary between system and environment can always
be at least approximately represented in terms of set of (possibly mixed)
Thevenin and Helmholtz equivalent sources, where:
Thevenin Source Helmholtz Source
E – 1 – F – O –
Z Z.
We may then further idealize these sources by extending the prime retic-
ulation -- the fixing of the system boundary -- to include the source
impedances as parts of the System itself in the fashion:
Ideal ; ; ; ºriginal Ideal : Augmented
Source T : T System Source . System
! Z ©
For both continuous and reticular fields this strategem results in
the classical boundary conditions of Dirichlet or of Neumann, or else mix-
tures of these forms. The specifications of E = E(t) and F = F(t) of
course include constants and null values as important special cases.
Let us now consider a multiport state-determined system S which
has been rendered into such a form:
F3
º
F. B2
*
S
N
F
ll
N
…
JT1
In general each internal state variable will be determined adjacent
º an appropriate energy junction; this situation we might indicate in the
fashion:
19|| 2.75l CLASS NOTES
On the above diagram the (m = n) ideal boundary sources are all independent
variables or inputs while the internal energetic variables e; and fi. 3.I.'é
each and every one dependent or output variables. We may then indicate the
final explicit functional solutions in the form:
«»
Nº. I E1(t), ..., E,(t); F (t), ..., FA(t) .
:
e (t)
:
t; (b) - Y, ( , (), ..., B.G.), F(t), ..., F.()
tº
Q º
* <>
We are merely stating that the instantaneous value of any internal state
variable is a function only and entirely of the history of the ideal environ-
ment. We should note that the output variables conjugate to each external
input source are now determined from within the system and are therefore
included in the set ſe; y fi.l.
It is also interesting to note the singular result which arises for
the special case of a constant environment. In this case, for a state-de-
termined system, the response trajectory is uniquely determined by the initial
State, alone.
gº tº - º Aº- - * * * * * * * * * * * * * * * * * * *
A system is linear if all the elemental components of a system
are linear, in the sense that the governing relationships among the energy
variables are all linear. In the case of the state determined systems, the

Part XIII 2.75l CLASS NOTES 195
rimitive characteristics are linear static functions while for the more
eneral n-port linear elements, the reticulation may be carried only to the
evel of linear functional Operators.
If the functionals Y. and Y, are all linear operators then
the Superposition property implies the additional functional reticulation:
e; F • e a + IF, E, + . . . -- Z, F, + . . .
*k -: jº º ºs. + Ya, Rh + tº º ºr + IF, F, + & © -->
* º
--> © *
* tº º
The operators IF, and IFº
While the dimensioned operators Z, and Yº, are called transfer impedances
being dimensionless, are termed transfer ratios,
ind transfer admittances, respectively; all four classes of operators may be
tonsidered transfer functions.
- Using Scaling constants, the transfer impedances and admittances may
Blso be expressed in pure ratio form; for example:
Z. F F,
Y G IF,
-
:
(E./F.) IF ji
(F /E.) IF kh
-:
-
kh
By this means all the transfer operators reduce to dimensionless operators
F ab, or to such ratios multiplied or divided by a nominal resistance
Bonstant. We may then restrict attention to the single element:
X y
y = IF x Ol." ——FH-

ºfer Characteristics of Iineer state-Petermined Reticular Systems
Such systems are reticular complexes of the following seven linear
* linearized elements:
| E., F., R., C., I-, -0. , -1 - 1
196 2.75l CLASS NOTES
A simple example involving all seven might be the system:
f
'N
I R
The general functional relations for e and f would be written:
E F
i
f
:
Y, ( , ; )
e = W
a ( E, F )
which in turn linearize to the form:
f
:
(1/R). F, + F. • F
e : IF • E + Re IF • F
2l 22
using the resistance R as the scaling constant.
Any of a wide variety of linear reduction schemes will yield for
the above system:
RC) D 1
F, --H # TGayBe IF. - 1 + (RC) D + (IC) De
1 I/R) D + 1
IF, = ~TTOREETCHER2 IF, = --H # ſº

Since each operator is dimensionless, and, indeed, the numerator and
denominators are separately nondimensional, the coefficients of the powers
of D are accordingly time constants raised to the corresponding power.
Thus the three physical parameters (R, C, I) manifest their effects
through the three time constants (T1, T2, Ts) where
2 V/IC 3 T3 > I/R
T. = RC 3 T
Part XIII 2.75l. CLASS NOTES 197
In these terms the above operators become:
T1D 1
". . . . . . . 5 F. : T.I.
1 tºº (Tºp + 1)
IF, - 1 + T1D + Tgpº 3 IF, 2: 1 + T1D + TâD
The fact that IF IF is a consequence of the reciprocity principle hold-
2l 2
l
|ng for all such linear passive Systems.
F
From the above we may see that the general case of a linear system
having r external bonds to the environment and a total of s additional
itate variables, will entail ( r + s ) energy junctions whose ( r + s ) output
States Y may be related to the r input variables X by the linear matrix form:
Y T - X
IF º ºg & IF 1z | Fall
IF
Yl ſIF
IF
ll. l 2
Y2 & © tº IF2r X2
2]. 22
I” + S : := • : © e & . • | . r
& • e * * >
r"OWS I’OWS
X.
Wr IF r1 |Fre & © & IFrr r
& º ſº e º “w-d -
'rts IF rts, 1TFrts, 2 to e e IFrts, r
* |- *-
\. \ L/
Y-
r – columns
Background Reading -- Linear Systems
li) GENG, D. K. : Analysis of Linear systems (1959)
(2) TRIMMER, J. D. : Response of Physical Systems, Second Printing (1953)
198 2.75l CLASS NOTES
G. System Response in the Time and Frequency Domains
tº dº ſº tº ºne ºne º º ºs ºme tº ºn tº ſº tº º ſº tº ºn tº ºne º º ſº º &º gº tº tº fº tº ſº º ºs ºf tº
Under normal operating conditions the variables of most physical and
engineering systems will undergo arbitrary variations over time. The general
situation will involve stochastic signals, the word deriving from the Greek
a raj Xa a rv kos (stochastikos) meaning, surprisingly enough, both "to
aim" and "to guess". Such variable signals are those which have some prob-
abilistic element and are thus not completely deterministic. At the extremes
of the stochastic range, we find the purely deterministic (i.e. point–predict-
able) signals at the one end and purely random (i.e. distribution-predictable)
signals at the other end.
Since any deterministic functional operator, "P , applied to a
stochastic signal, X , will produce another stochastic signal Y , we are
necessarily concerned in all systems with an adequate description of arbitrary
stochastic Signals.
The detailed description of random signals and processes We leave to
be considered in 2.752; here we shall concern ourselves with purely determin-
is cic but otherwise arbitrary signals.
Consider first the unit step or jump function, U(t):
U(t),


1 \ 1
U(t) = 2 [1 + sgn t| !
y- 9 <—1/2
Time t
This discontinuous but nevertheless analytic function was first introduced
into system analysis by Oliver HEAVISIDE and is frequently called in his honor
the Heaviside function.
A completely arbitrary stepwise varying signal can be defined by the
Part XIII 2.75l CLASS NOTES 199
equation:

*rts

OO T
x(t) = 2 a U(t-T,) – r+l,
k=1 r--1
r-H2
Here the coefficients, *k, and jump times, T., are assumed at will (subject
to the implicit ordering; T & T. +1).
If we wish the jumps to be at synchronous clock intervals, T, we
= kT, to obtain: X
k
need merely to set "k

T-
CO H
X(t) = 2 a U(t-kT) JT- –
=- OO | |- H |

{-
Now the sequence [ ak ] is a precise specification of any such X(t), given a
Specified clock interval, T.
The significance of the above results lies in the fact that such
descriptions in the time domain are the precise equivalents of the conventional
Fourier series in the frequency domain. To demonstrate this let us for sim-
plicity consider an even periodic function. This would be sketched and ex-
2OO 2.75l. CLASS NOTES
panded as indicated
CO
* * * * *msº —º- X(t) = X. bk cos k a t
t k=O
* tº ſº tº
* * * * *m.
— m/o-
Now by analogy to the last expansion above we have
Clock Interval: T “—- CU : Fundamental Frequency
Jump Amplitude: *k <!--> bi: ; Harmonic Amplitude
Jump Time : kT *-* kaj : Harmonic Frequency
Of course the general expression for a Fourier series will involve
both odd and even terms since any function can be expressed as the sum of an
odd and an even function. Thus the general Fourier series is written in any
of the equivalent forms:
X(t)
ź. al, sin (kajit º,)
= 3. (*): Sin kwºb, cos kayf)
F. : ". C OS (katt fº)
The angles 6, and $. are called the harmonic phases. We shall generally
find the last form most convenient in use.
It is also not without interest to speculate on the frequency analº
of the asynchronous arbitrary stepwise time signal above. This may be post”
lated as the (generally) aperiodic signal:
X(t) = :*. cos (wit + $1.)

Part XIII 2.75l CLASS NOTES 2O1
This is clearly a generalization of the Fourier series which will frequently
give the appearance of a quasi-random Signal if the set of I (Uk | are rela-
tively incommensurable.
Thus with either the set I a..., T, 1 or the set [ b, w, b, .
we may describe to an arbitrary degree of precision any normally encountered
X(t) over a finite period of time. We are now in possession of a descriptive
mechanics sufficient to treat the dynamics of any linear system either in the
time or the frequency domain.
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
Any functional operator, + , is said to be a linear operator A y
if it satisfies the linearity condition:

E
A ( &n X + (5 Y.) (I. A X + b AY
for arbitrary constant matrices (CI) and |b. Of course the vector variables
Y and Y must be such as to lie within the domain of definition of A º
The above condition is both necessary and sufficient. Thus if the
operator is "known" to be linear, then the linearity condition is necessarily
satisfied. The above condition is customarily written for scalars and in the
form of two simpler conditions as follows:
T------------------------------------------------ T
: AG, x.) - A x, Ax, :
f f
; A ºx E B. Ax :
i------------------------------------------ * * * * * * i
It is perhaps easier to see now that linearity implies only that A is
distributive with respect to addition and commutative With respect to scalar
multiplication.
For the systems of interest to us, we are concerned in the scalar
Case With situations.
y - "HH [ x )
2O2 2.75l. CLASS NOTES
But from the above we know that:
If: Y = A x
X. °kºk
k
: & Ax
X
:
and:
Y
:=
then:
Thus the response of a linear System to a disturbance composed of a weighted
sºmºsºs ºsmºs ºmºmºsºmºmºs º ºsmºsºme
signal acting alone.
This superposition principle for linear systems is the basis for the
application of nearly all mathematical and scientific theory to the real world.
Up until its promulgation by Daniel BERNOULLI in connection with the vibrating
string problem in 1753 the practical application of mathematics and analysis
was severely restricted. The first dramatic results after its enunciation
were the masterful trigonometric series expansions of Joseph FOURIER in 1822.
The superposition property is obviously also valid for linear multi-
port systems in the form
A X
3. c. X,
3. c. A X,
-:
Y
|X
Y
:
In the paragraphs below We indicate the application of the Super-
position principle first to the representation of behavior in the time domain
and, secondly, to representation in the frequency domain.
* * * * ºne ºn tº º ºs ºn tº ſº ºne tº * * * * * * * * * * * * * +º º tº ſº ſº º º ºs º ºs
Consider the time-invariant linear transfer operator IF (D). We
define the step response, F(t), as
F(t) = IF (D) U(t)
Part XIII 2.75l CLASS NOTES 2O3
Thus the step response is merely the output Y(t) resulting from an input
X(t) = U(t). We may then determine the response to an arbitrary jump
function by using the superposition property as follows:
Given: Y(t) = IF x(t)
d: X(t) = U(t...T
8.Il ( : al, U(t-T.)
and: F(t) = IFu(t)
Then: F(t-T.) IF U(t-T.) [time invariance]
and: IF X(t) = X. *k Fu(t-t') [ superposition )
k
Therefore: Y(t) = X. *k F(t-T,)
k

This result implies that for linear operators a knowledge of the step response
alone is adequate to determine the behavior in the time domain to an arbitrary
degree of precision. It is this fact which has raised the step response to
the eminence which it has held for the last seventy years or more. Of course,
if the system is essentially nonlinear and the normal input is arbitrary then
the Step response has little if any value as a behavioral measure; it is worth
stressing this point in the light of contemporary proclivities for obtaining
Such meaningless data.
If we wish to pass to the limiting case of a smooth X(t) then the
Šums must go over into integrals. This transition is natural if Stieltjes
integration is employed. Thus we define X in the form:
t;
X(t) = J. dx(T)
if X is (purely) continuous in real time, t, (and • umbral time, T ) then
this reduces to the Riemanian identity:
t
x(t) = ſ. ax(t) = x(t)
2Ol. 2.75l. CLASS NOTES
But if X is (purely) discontinuous at a series of discrete times I "k |
then the integral is evaluated as the sum of jumps or salti in X up to
time t, namely:
<
T.4 t
X(t) = 3. a U(t-T.)
Having established this relation we may then state the integral form of supe
position as convolution integral:

t;
Y(t) = J. :(- T) dx(T)
Due to linearity, it is simple to demonstrate that a complementary form of
this convolution integral exists, namely:

t .
Y(t) = f X(t-T) d'E(T)
— OO
Furthermore, if F(t) has no discontinuities (including one at the origin!)
then we may usefully introduce the concept of the impulse response or
weighting function, f(t), where
t;
f(t) = d.R(t)/dt or F(t) = f f(t)dt
—- CO
Substituting f(t) dt = dF(t) into the second convolution integral -- a step
valid and useful only if f(t) is finite ( F(t) continuous) -- we obtain the
far more common -- but less useful -- form:

t
Y(t) = f X(t-T) f(T) dt
sº
These convolution integrals are all originally credited to DUHAMH
Except in electrical engineering they were little used until the advent of
Norbert WIENER. Now due largely to the central role they play in his Writi.
they have come into increasing use throughout engineering and physical scie
Part XIII 2.75l CLASS NOTES 2O5
* * * * * * * * * * * * * * * tº- * * * tº tº tº ºf ºw tº tº tº tº tºs tº ºn tº * * * * * *
Let us next consider that we disturb a system with an arbitrary
input which we characterize in the previous generalized Fourier form, namely:
X(t) = 2. by cos( a t + $ )
Then if the system is stationary and linear, the response, Y(t), must be that
iue to the superposition of responses, Y., due to each *k acting alone. But
k”
these individual responses may be derived by examining the behavior of a
linear operator excited by a pure sinusoid of frequency (U k'
A unit amplitude sinusoidal function of time may always be repre-
sented as the instantaneous average of two-counter-rotating vectors (or
sinors or phasors) in the form:
X(t) = 7,
This principle is actually used
.8 a common type of sinusoidal driver or
Vibrator.
In this way, we may avoid the artificial use of real and imaginary parts and
lay introduce the symmetrical occurrence of positive and negative frequencies.

2O6 2.75l CLASS NOTES
tº ºf tº tº ºn gº ºs ºng ºn tº sº tºns ºf tº tºº smºs º º
It is then simple to demonstrate that the n-th time derivative of
X(t) becomes:
n_+j w t
[ (+j (U )"e - w t )
D* x(t) = } + (-j w )"e
and therefore that an arbitrary linear operator, IF (D), acting upon X give
the result:
x(t) - F (B) zº) - , F (3 w Ye" " " . F(-; a ye- " " .
This may be visualized in terms of phasor diagrams.
Im
It is evident that IF (-j (U )
*
-: IF is the complex con-
jugate of IF just as XT and YT
Re are the instantaneous conjugates
Of X* and y", respectively.
Observe also that the magnitude |F | and the phase |IF have direct inter-
pretation both on the polar and the temporal plots.

\rt XIII 2.75l. CLASS NOTES 2O7
- Clearly, then, the response of any linear system IF to steady
nusoidal excitation is uniquely characterized by the behavior of IF (j a ),
particularly, | F (j (U) | and | IF (; a ). This frequency response may
indicated either by the polar locus or Nyquist Plot (in honor of Harry NYQUIST),
the magnitude vs. phase locus or Nichols Plot (following Nathaniel B. NICHOLS)
! by the pair of gain (= log magnitude) vs frequency and phase Vs frequency
fes or Bode Plots (after Henryk W. BODE). Certain of these we discuss fur-
r below.
Historically, the principal reason for the interest in sinusoidal
sponse lay in the characteristic variance of waveform under linear trans-
rmations. However, this property is not restricted to sine waves alone; in
ct, it also holds for a very natural generalization in the exponentially
mped or attenuated sinusoid of the form:
X(t) = e * * cos w t
it we may now readily generalize as well the previous concept of rotating
asors to include this complex form in the fashion:
+
x(t) = 1 ( e^* + 2*
where s = OT + j (U
s34 = OT - j (U
| Such conjugate phasors not only
Enter-rotate but spirally swell (or > 0) | X(t)
'decay (O & 0) in magnitude as well.
**, the complex frequency, s , repre- |
ts a domain which is simultaneously
*ting and changing scale, but one in
*h the relative phases and magnitudes of
W phasors remain invariant!

2O8 2.75l CLASS NOTES
As before we may examine the behavior of Dh X(t). The result is
readily obtained as
D” x(t) - 4. I (s)” e^* + (g)” e” )
:
which is equivalent to: | p"| ( a * + a 2)n/2
and: | Dº
It is particularly enlightening to interpret these results in
n tan" ( w/o )
=
phasor form for the three cases:
O & O O = G O > O
(DECAYING (CONSTANT (GROWING
AMPLITUDE) AMPLTTUDE) AMPLITUDE)
a > 90° a « 90°
This representation leads directly to a particularly elegant con-
ceptual picture of the characteristic roots of linear operators simply as
those complex values of s for which 1/ IF (s) = 0 and therefore which give
Vectorial equilibria of the complex phasor diagrams.
The complex frequency response of any linear operator IF (D) is


Part XIII 2.75l CLASS NOTES 209
then:
Y(t)
-:
4. | IF (s)e^* + IF (ex)e^**
>
4 F x , F x
Again, as with ordinary frequency response, the operator IF (D) is character-
ized by IF (s).
The above process can be very effectively interpreted as a conformal
mapping process of an s-plane into an IF -plane. In particular, the ordinary
frequency response becomes a mapping of the imaginary axis S = ja ” into a
corresponding curve on the IF -plane. This locus is simply the sº Plot
in the form:
S - plane —º- IF - plane
(COMPLEX FREQUENCY) (NYQUIST PLOT)
Im. A Im.
* *
2.
2-&
Z —º Re
J
Nyquist
Plot,

21 O 2.751 CLASS NOTES
But we now have a much more general context for this mapping, namely
the complete complex frequency response in the form:
5 - plane -> IF - plane
All the conventional results of frequency response may be obtained
from this field map, but useful additional properties may be inferred there-
from. Moreover, this characterization leads directly to the description next
following.

Part XIII 2.75l CLASS NOTES 211
H. Linear System Response in terms of Potential Functions
= ~ * * * * * tº tº tº dº tº ºmº tº º ºs ºs tº
Consider a general system transfer characteristic, IF (s) in
terms of the complex root-variable, S = OT + j (0. This may generally be
expressed as a ratio of polynomials, finite or infinite, in the form:
IF(s) = P(s)/Q(s) = ( *** / ***
However, these may in turn be written, at least implicitly, in the factored
forms, to give
3:... . ." .
cº-º- (s - p, ) (s - p.) . . . (s - p.)
P(S 1 2 k
F (-) - #: A H - A TH+
t f Poles —
The real constant, A, measures the infinite frequency gain of the system,
While the roots of the numerator and denominator polynomials give rise to
the ZEROES Of IF and the POLES of IF , respectively. Since IF is completely
determined by these terms, one may consider that in this sense, any transfer
characteristic may be considered as characterized completely by A and the
poles and zeroes.
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
If one now divides the transfer characteristic by the constant A
and takes the logarithm, there is obtained the normalized transmission charac-
teristic, G = in(IF/A). Thus
G(s) = ln(F/A) = º + 3 ºf (s)
log phase
magnitude angle
ratio
since IF - A . e? g e 3% - (Meg)xe (Phase)
By considering the characterization of IF in terms of its poles and zeroes,
212 2.75l. CLASS NOTES
the transmission characteristic (G can be Written:
G (s) = 2. ln(a - p.) - 2 in(s - d.)
"Sources" G) "Sinks" G)
The terms on the right may be considered as the potential function
due to a SUM of sources G) and sinks G) located at the zeroes and poles,
respectively. This may be visualized and, indeed, realized in an analog con-
sisting of electrical charges, fluid wells, or other situations, thus giving
rise to a FIELD in which the equipotentials are contours of constant log
magnitude and the fieldlines are contours of constant phase angle.
* * * * * * * * * * * * * * * * * * * * * * * * * * * * *
Consider the function:
w(s) = + in s
+ for ZERO; – for POLE
NOW:
RADIAL field lines are curves of CONSTANT PHASE
CIRCULAR equipotentials are curves of CONSTANT MAGNITUDE
90°
|
1 35° - * |- * 15° Phase: Z_G
N 2 N. / ZERO POLE
X | N SOurce Sink
/ /
/ \\ + / \ + º
/ N2-TSSJZ \ ! / G !
/ N / N \ - +

Part XIII 2.75l. CLASS NOTES 213
All system functions can be found from this one distribution by
locating sources at every pole, sinks at every zero, and then summing up
phase angles and log amplitudes for each distribution. This, of course, is
identical to vector multiplication of all pole vectors and zero vectors,
and is the basis of Walter D. EVANS" "Spirule" and other more complex devices.
Transmission Functions as Potentials
* = as and sºme ºne * * * * * * * * * * * * = * * * * * * * * * * * * * * * * *
However, in many practical applications of linear system response,
it is helpful to return to the basic conception that both IF (s) and (G (s)
are analytic functions of the complex frequency s = OT-F j(U, for which the
Cauchy–Riemann equations imply the existence of a set of orthogonal potential
functions.
In this light, (G ( s) can be viewed as a conformal mapping of
F (s) which is in turn another mapping of the s-plane in the following
fashion, for the particular case of a pure delay IF (s) = e-Ts,
IF – plane
Im
/
/
/ 2.
.”
2^
Re
º 1 - |
SS / |
,” N.
2 * Žss /
__* / \ /
Y ~ ||-- /
/ \ /
> *
— L-- T \
Clearly in this instance there is neither need for, nor value in,
determining the "poles" and "zeroes" of |F before constructing the map of (G y
Since this latter map is even simpler than that for IF itself. Indeed, per-
haps the greatest value of the transmission function lies in its generally
Simple form both for finite reticulations and for continuous systems.


21 l; 2.75l. CLASS NOTES
I. One-Port Elements and the Impedance Concept
Generalized One-Port Relations
* * ºne tºs ºs º ºsº tº tº sº tº º ºsº ºne tº iº gº ºn tº * * * * * * * * * * *
Consider the interaction between two open or closed systems con-
nected by a single energy bond:
The causality of this bond could be assigned in only two possible ways:
S. H S2 Or S. – S2
If we now replace each system by a functional operator, "HP, We find that
both cases may be represented by the unique causal scheme:

UHT I IIH
These two relations, UHT a.” and "H.,
ized impedance relations. If both Systems are otherwise isolated from the
We shall speak of as general
environment, the relations are generally deterministic in nature; if one
or both are nonisolated, the operators will necessarily take on stochastic
properties. In either event, we recall that the meaning of the functionals
H. and "HP is that the entire history of the input variables (f and e,
respectively) is required to establish merely the present value of the
output variables (e and f, respectively). Both"P functions could repre-
sent extremely complex fields, networks, processes, or other systems, but
We could still always Speak of them as impedance relations, so long as but
one port were involved. Some writers, such as KRON, have attempted to
generalize further the words impedance and admittance to cover n-port systems
where fand © vectors were used as inputs, respectively. This usage ber
COIſle S proportionately clumsier and more specialized as the portality of the
elements increases, and/or essential nonlinearities are present.
Part XIII 2.75l. CLASS NOTES 215
Therefore we now restrict attention to a deterministic system
or subsystem capable of exchanging power only at a single port as indicated
above. It is clear that for such elements the over-all behavior is defined
by specifying the functional relationship between effort and flow at the
Single port of entry.
- -, -, * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
The static characteristics or steady-state relationships for any
one-port element are generally nonlinear static functions of the form:
Y = qi) (X)
For the practical (nonideal) case this single curve is usually
presented in the form of a graph, defining the range of all possible opera-
ting points for the component. This static function may be approximated to
an arbitrary degree of precision by a polygonal function (particularly for
essentially nonlinear elements), or by an algebraic function (particularly
for curvilinear elements).
Dynamic Impedances
From a causal Standpoint, Since We have seen that the power trans-
fer must depend upon the product of one input variable, X(t), and one out-
put variable Y(t), two general possibilities exist for the nonequilibrium
or transient case, namely:
IMPEDANCE FUNCTIONALS:
t) = "P ... + r(t
X(t) = f(t); Y(t) = e(t) e(t) er * f(t)
3.
ADMITTANCE FUNCTIONALS:
f(t) = H. * e(t)
X(t) = e(t); Y(t) = f(t)
It is necessary to make a distinction between these two forms,
Since for the general nonlinear case, a well-defined converse of a given
functional relationship may not exist. The choice of the terms, of course,
has an historical |background.
216 2.75l CLASS NOTES
State-Determined Impedances
ams ºr ºms, sºme amº as smºs ºº em sºme ºs º ºs tº ims tº * * * * * * * * * * *
The classical state-determined elements [ E., F., R., C., I. J
may now all be interpreted as special instances of the generalized impedance
functional since
e = E UH
E f
e - RF : H. f
e - Sa - $ . . . f -"P.
f = F = H. e
f = Ge 2: H. e
f = Tº = T . . . e -"P. e
History of the Impedance Concept
is ºne º sº ame tº enm mºs º ºs ºs ºss tº tº assºs ºm ms tº sº tº gº ºne tº gº º ºs ºs º º sº ſº sº tº
In electricity the impedance concept grew out of the desire to
generalize Ohm's Law and the notion of resistance to make certain elementary
direct or constant current concepts applicable to problems involving period-
ically varying current. Historically, this need arose in the last few
decades of the nineteenth century, under pressure of the growing electric
power and telephone industries. In dynamic analysis, even before 1900,
German and British physical scientists, notable among them HEIMHOLTZ,
KELVIN, MAXWELL, and HEAVISIDE, saw the analogous structure of electro-
dynamics and classical dynamics. For these early writers the natural analog
of the electrical impedance, relating voltage to current, was the relation
of force to velocity. However, largely due to the historical precedence of
static elastic analysis in mechanical problems, the principal variables in
mechanics were taken to be force and displacement. In this sense, mechanical
impedance as a force to displacement ratio may be considered as the attempt
to generalize Hooke's Law and the notion of a spring constant to problems
in dynamics. In short, the principal justification for differing definitions
of impedance between the electrical and mechanical fields is the practical
requirement that the steady-state conditions should reduce, at least for
Small variations, to linear algebraic relations between the principal
variables in the respective domains.
Part XIII 2.75l CLASS NOTES 217
In our treatment below we shall loosely use the term impedance
to describe all general effort-flow dynamic functional relationships. This
renders unnecessary any such distinction between varying definitions.
Besides its roots in classical dynamics, the impedance concept
is also firmly founded in the field substructure of material systems, so
that the general properties of all energetic one-port elements are de-
rivable from the nature of their underlying fields as outlined elsewhere
in these notes. An excellent treatment of the dynamical aspects of the
field basis for impedances is given in classical papers by CARSON and by
SCHELKUNOFF. The term impedance is credited to BEDELL and CREHORE and its
use became widespread through the prolific writings of Charles Proteus
STEINMETZ.
Linear One-Port Impedances
* * * * * * * * * * * * * * * * * * * * * * * * * * *
If the general functional operators, "HP, and "P., are assumed
to be linear operators in the form:
Z = HH.,
Y = HFr.
LINEAR FORM
then we have the more customary definitions of impedance based upon linear
System behavior.
Conventionally, these linear operators, themselves, have been
associated with the concept of a linear impedance, namely Z, and its
reciprocal, the linear admittance, Y , since now:
e = "HP ... * f = Z. . f
ef
f = "HP, ≤ e - Y - e
fe
Thus Z - Y = 1; Or Y - 1/Z
However, for linear systems with constant (i.e., non-time-varying)
parameters, these operators may be advantageously expressed in the form:
Z= Z(D); Y_ Y(D); D = d/dt
218 2.75l CLASS NOTES
However, an ambiguity in nomenclature would still exist when we
attempted to interpret these expressions in causal form. We can preserve
our original meaning by interpretating causal impedances and causal
admittances in the sense:
IMPEDANCE (Relation) : f —- e :
ADMITTANCE (Relation) ; e –- f Y
as indicated previously.
In any case, all such linear systems become linear one-ports.
In the steady-state, a single straight line characteristic will always
relate effort to flow.
Background Reading -- Impedance Concept
(1) BEDELL, F. and CREHORE, A. : Derivation and Discussion of the General
Solution of the Current Flowing in a Circuit Contianing Resistance,
Self-Inductance and Capacity, with Any Impressed Electromotive
Force, Journal AIEE, Vol. IX, pp. 303-374 (1892)
(2) CARSON, J. R. : Electromagnetic Theory and the Foundation of Electric
Circuit. Theory, The Bell System Technical Journal, pp. 1-17.
(manuary, 1927).º
(3) SCHELKUNOFF, S. A. : The Impedance Concept and Its Application to
Problems of Reflection, Refraction, Shielding and Power Absorption,
The Bell System Technical Journal, Vol. 17, pp. 17-48 (January, 1938
(h) CHENEA, P. F. : On Application of Impedance Method to Continuous
Systems, Journal of Applied Mechanics, pp. 571-571 (December, 1953)
Part XIII 2.75l. CLASS NOTES 219
j. The Flow of Power and Energy in Systems
We are now in possession of all the basic tools needed both to answer
the questions raised in Part IX concerning the steady-state of energetic
systems and also to consider the transient flow of energy over the extent of
the system.
We may view this process either in the time domain or in the frequency
domain. In the first instance, we merely consider IP 1.(t) on each bond k
pf the system; in the second case P. is further spectrally decomposed into
P k' i
power State. Then by a simple generalization of the conventional description
(U 1) where the (U , are all the frequency components present in the
employed for a-c power systems, we consider the real power flow, P.( CU 1). and
reactive power flow, &Q CU 1), along each bond k at frequency 1 . It may
sometimes prove convenient to reticulate further each bond into its spectral
components, (P.1. &1).
At the beginning of this section we discussed the application of energy
principles to state-determined systems, ending with a triangle diagram depict-
ing the history of total system energy for a closed system. Now we may ask
how this power and energy is distributed over the extent of the system, recog-
hizing that the microstructure is either continuous or quantized depending
pon the ultimate level of energetic reticulation.
It is easiest to demonstrate this state of affairs for a reticular
Bystem. Let us take for example the simple R-C system driven by a unit step
in effort:
E — is
C
}learly, we may sensibly inquire as to the values of [ Pºſt), IP R(t), P. (c)]
for each of the three bonds, and therefore determine also the corresponding
mergies E,(t), E (t), E.(t) i.
22O 2.75l. CLASS NOTES
If the system is linear then the following results are readily obtain
in terms of T = RC:
Pºſt) - (pºſs) --" ;IE,(*) - (TE:/F) ( - e."
Pºſt) +: (E*/R) e-2t/T ;IE,(t) -: (TE*/2R) [1 - e-2t/T j
P. (e) -: (E*/R) fe-t/T - e-2t/T | , E.(t) - (TE*/2R) [1 - ce-t/T + e-et/ .
Ps tº: IP's + Pc 3|Er : IEF + IEc
IE | TE*/2R = c E%2///
IEE
• * * * * * * * * * * * * * * > * *
... • * * * * s sº 2 : : *
Similar results will obtain if IR and (C are nonlinear elements.
Moreover, diagrams such as these can be drawn for all bonds of any general
reticular system in contact with a particular environment.
If the system is not so reticulated we must return to the field de-
scription in terms of a local instantaneous Poynting vector, =º- (t), and energ
density (E (t). However, solutions for such cases are impossible to obtain
or yet to conceive except in the simplest classical linear cases.

ºrt XIII
2.75l CLASS NOTES 221
i.e. - - - - - * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ~ * * * * * * * * * ~ * * * * * * * * * * * * *
Based upon our remarks above and assuming a generalized Fourier
representation for an arbitrary but stationary power state [ e(t), f(t)
any single frequency component will have the form:
e
=
Vz E, cos ( w it)
~/2 F cos( (U , t) + v2. F, a sin( a , t)
1. ip 1. iQ i
f
:
Thus the corresponding power component P, : e, (t) g f;(t) |becomes:
IP
i
r:
:
2F1F1P cosº ( (U it) + *19 cos( (U it) sin( (U it)
FiFiP [1 + cos(2 (U it) | + **, 3 [sin (2 (U it) |
P; [1 + cos(2 w it) } + 84. [sin (2 w it) ]
P. + Ps
In electrical engineering, the first term has for many years been called the
à-phase or real power component and the second term the quadrature or reactive
power component. These components have the temporal form:
is
REAL POWER
REACTIVE POWER
|Pio.
tº Sgn P. =
i
leading.
, ſ\'ºſ) / Nº.,
V V
Sgn 8, then the reactive power is said to be lagging otherwise it


222 2.75l. CLASS NOTES
In a-c power nets a very useful convention has evolved for indicat-
ing the flow of real and reactive power. We may modify this for our purposes
as follows for any bond:
—H-
Q
It is then simple to demonstrate that for an ideal energy junction;
Pi P2 P, + P2 + P3 = O
—º- <}-
— J —
—Hº- <+- Q O
+ + :=
Q1 | t Q2 Q; 2 83
P3' 83
Similarly, we may demonstrate for linear elements the following
results:
P O O
—- –º- -->
R C I
O Q Q
Thus we may apply all the tools of a-c network analysis to the
determination of ſºld, &l for each power bond, k , and spectral line, i
In nonlinear devices, such as rectifiers and parametric amplifiers,
a number of significant relations exist amongst these doubly reticulated
spectral power bonds. Some of these constraints have been studied recently
by Paul PENFIELD as generalizations of the earlier MANLEY-ROWE formulas.
Since power may always flow out of a multiport element in a differ"
ent frequency band than it enters, the macroscopic irreversibility of micro-
scopically reversible systems is no paradox. The generalized forms of the
second law of thermodynamics represent attempts to express this intrinsic
"band-and-bond" scattering property of all physical systems.
Part XIV 2.75l CLASS NOTES 223
Two-Port Elements and Energy Transport Processes
A. Generalized Two-Port Elements
We have earlier suggested that the behavior of many standard
engineering components may be studied by considering them as two-port
elements. These two energy ports we shall arbitrarily designate generally
as the "upstream" and "downstream" ports, 1 and 2, respectively and are
the only ones for which the device is to be represented and investigated.
Thus we are led to the problem of specifying the necessary characteristic
quantities and relationships to define adequately system behavior, even
When the exact internal contents and construction of the two-port system
may be unknown or, indeed, "unknowable".
e e2
PORT 1: — TWO-PORT — ; PORT 2
f f2
In all cases, the behavior will be characterized in terms of the
two power flows:
P (t) = e, (t) . f, (t); Pºſt) = ea (t) . fe (t)
and therefore in terms of four variables:
e1 = f; , e2 - f2
Since at each port only one of these variables may be taken as an input,
We Will generally have the situation depicted below, which indicates that
the output variable at each port is functionally determined by the input
Variables at both ports, that is:
Y, (t) = *H, [x, (t), x, (t))
-


X. —- ſº- X º D-
1 - Pºr Vb —-Y. 1 Y. —y,
y - W
A - ſh-
X _º _º º
1 Ya -uſ -T— X2 X2 º l C- NA, Hº- ¥2
Causally Reticulated Two-Ports
22h 2.75l CLASS NOTES
Moreover, only four possible particularizations of this causal
sequence exist, namely:
I: IMPEDANCE: H 2-Port — III: ADPEDANCE: — 2-Port —
II: IMMITTANCE: H 2-Port H. IV: ADMITTANCE: — 2–Port H.
The names employed for the four configurations follow from the
partitioning of the words
im - pedance
ad - mittance
and associating the prefixes with the upstream port and the suffixes, with
the downstream port of the two-port element. The four possibilities then
follow simply from the mnemonic scheme:
im pedance
ad mittance
Thus we say that a given two-port is in the form of an impedance,
we merely imply that the causality of each port, taken alone, is in the
form of an impedance functional; by contrast, an adpedance two-port implies
that the upstream port has an admittance causality and the downstream port,
an impedance causality.
The static characteristics or steady-state relationships for any
two-port element are generally nonlinear static functions of the form:
Yi = PA(x, X2)
Y2 = q’s(x, X2)
Eor the general case, these curves are usually depicted in the form of two
graphs, each of which relates two of the quantities with the third as a
parameter to give a family of curves, as follows:

Y
A Increasing 2 A \
_º, \\ y
22 Plus N Increasing Xı
N
Y. d

—ſº-
P-
—P
X
X, 2
Part XIV 2.75l CTASS NOTES 225
However, particularly for operation over both signs of X1, X2,
the use of "contour" or "hill" characteristics is common; these have the
form:
Legend:
Y1;
-X * * * * * * *
1 Ye:
Such static characteristics have the important consequence that
they provide a complete specification of the range of "operating points"
at which the device or component may be maintained under steady operation,
and over which, it may course during transient operation.
B. Primitive Energy Transport Processes
If We consider the typical vehicle propulsion system as indicated
below, we can distinctly recognize a small number of basic processes in-
Volving the transport of energy from an upstream port to one downstream.
The elements used for these purposes may frequently be classified into three
primitive types, namely:
a) ENERGY TRANSFORMATION ELEMENTS: [ — Transformers — ) = [ —TF— j
Generalizations of the lever, gear, hydraulic jack,
and electrical transformer.
b) ENERGY TRANSDUCTION ELEMENTS: ſ — Transducers — j = [ —TD–– )
Generalizations of the motor – and - generator,
pump – and - turbine, magnetohydrodynamic devices,
heat pump, etc.
c) ENERGY TRANSMISSION ELEMENTS: I — Transmitters — j = i —TM– J
Generalizations of the rod, shaft, pipe, wire, conduit, etc.

226 2.75l CLASS NOTES
Thus the physical scheme:
- TURBINE - SHAFT - GEAR - SHAFT - PROPELLER -
Can be represented in general terms by the elements:
-TRANSDUCER—TRANSMITTER-TRANSFORMER-TRANSMITTER-TRANSDUCER-
Or TD. ITM TF ITM TD
The principal benefit of such a generalization is that it readily
permits the cataloguing of linear and nonlinear two-port relationships for
such components and devices once and for all, quite independently of the
media in which the devices operate.
WEHICLE PROPULSION SYSTEM

Part XIV 2.75l CLASS NOTES 227
C. Linear Two-Port Elements
If the general functional operators for the two-port,"P.,
and p, can be presumed linear over the practical range of operation
of a particular component, then a most significant and powerful simpli-
fication subsists. This is manifested in the reductions:
F, • X1 + IF, a . x=
Fe, • X1 + |Fea • X2
Yl = # a *[X1, X2]
-:
Y2 : | |b *[X1, X2]
which may then be summarized in the single causal statement, in matrix
form:
The matrix M\then has the four physically realizable particularizations;
these have evolved into a fairly standard symbolism over the last several
decades, at least as regards the electrical field. The four causal
matrices, in standard forms, are as follows:
º
Configuration I . . . . . . . . .Impedance Matrix: Z E *i-is
º !
r f
Configuration II . . . . . . . .Immittance Matrix: IHI E +++)
º !
r !
Configuration III . . . . . . . Adpedance Matrix : (G E ++º]
º t
!
Configuration IV ........Admittance Matrix: Y = ºriº:
Val y22
The relation of these causal matrices to the possible combinations of
2-ports will be indicated subsequently.
228 - 2,75l CLASS NOTES
Transmission Matrices:
In addition to these four causal matrices a most significant
standard form was long ago developed, which established a direct spatial
correspondence to the ports themselves. This relates the power states,
5, and 52, at each end of the linear two-port through a transmission
matrix in the form:
$1
MI
$
el A
!
!
sº tº #º hº := | * * * * * { *s sº sº º ºs © nº gº tº *
!
t
fi C
-1
Of course, the inverse transmission matrix,MI , relates $.
to 5, in the form:
- 1
$ 2 : MI $
-1
While the MI and MI matrices are clearly noncausal, they have
the peculiar advantage that the overall coupled transmission matrix for
two 2-ports in tandem or cascade may be obtained by direct matrix multi-
plication in the fashion:
el 2. Port 2. Port, e2
&E
gº
*
tºº
:
tº
tº
gº
sº
gºs
tº-
*º
tºº
i
º
gº
*º
*
º
gº
tºº
tº
©
tº-º
tºº
*
*
*
*
*
tº
i
º
iº
gº
*
tºº
gºe
tºº
tºº
º
º
fº
sº
*
gº
sº
tº
$º
:
sº
$º
gº
ſº
tº-
gº
tºº
sº
º
*
sº
sº
dº
tº sº.
*
*
º
wº
i
iº
$-º
*
tºº
gºs
*
iº
*
$º-
iº
º
tºº
º
*
gº
º
*
º
(s
$ºs
*
º
wº
Ca F ºr DaL) fa
Part XIV 2.75l CLASS NOTES 229
Historical Notes
A long history is associated with the development of linear two-
port concepts. In electrical engineering these systems have been known by
the various alternative names:
FOUR TERMINAL NETWORK
TWO-TERMINAL-PAIR NETWORK
QUADRUPOLE ( or QUADRIPOLE)
FOURPOLE ( or "VIERPOL" in German)
as well as a number of others. The designation fourpole was first used by
Breisig in 1921. While the first practical use of such concepts was in
the theory of long power transmission lines and associated apparatus in-
cluding transformers, Breisig apparently was the first to have used the
A y IB 3. (C 2 ID) , operators for communication lines and networks. The
application of the matrix notation is credited to Strecker and Feldtkeller.
Much additional valuable material on general linear two-ports can be found
in Works by Baerwald, Guillemin, Pipes, Le Corbeiller, and many others.
Origin of the Term Port
The word "port" used in this connection apparently originated with
Harold A. WHEELER to describe the coupling holes in waveguides. This
usage was presented by Wheeler to the IRE in the following words:
It has been customary to designate each entrance or exit
of a network as a pair of terminals, based upon the circuit
concept of wires and conduction. The result was cumbersome
terms such as "four-terminal pair" with the unobvious mean-
ing of a network with two pairs of terminals. Furthermore,
the terminal-pair concept becomes artificial in the case of
electromagnetic fields transmitting power within boundaries,
#hrough holes, and from one region to another in Space.
After considering many alternatives the writer has
adopted the term . . . "port" as the general designation of
an entrance or exit of a network. A Self-impedance be-
comes a "one-port". The usual transducer becomes a "two-
port" . . . The general network . . . a "multi-port". This
plan . . . is first put to use in this monograph.
230 2.75l. CLASS NOTES
* - - - * * * * * * * * sºns amas ºn east ºm ºms ºne ºm as imm.
Certain special relationships frequently exist for linear two-
ports which may be directly expressed in terms of the matrix elements,
namely:
Z12 = Z213 Y22 = Y21
det M = A-AD - IB C= 1
PASSIVE RECIPROCITY: - - . . {
SPACE SYMMETRY: - - - . . . . . A = |D)
Thus any symmetrical, reciprocal (i.e., PASSIVE) linear two-port
can be described completely in terms of only two linear operators, for
example, A and IB Since
; IB RECIPROCAL
* -- - 2 - - - - - - +------ f'Or and tº tº gº. LINEAR TWO PORTS
(A-1)/IB; A SYMMETRIC
Two-Ports Composed of 1 –Port Impedances and 3–Port Junctions
An extensive and useful subclass of two-port elements arises from
i
#
the interconnection of one-port elements, coupled through energy junctions,
in a polymerized chain of the form:
Where
I — I — I – — J – J = O or 1
Z. Z Z Z = 1 –Port Impedances
Since the individual one-ports could be either impedance or ad-
mittance functionals, both forms of energy junction are involved and two
different particularizations exist:
Part XIV 2.75l CLASS NOTES 231
We could call these structures impedance two–ports and admittance
two-ports in strict conformance with our general usage. However, we
should note that Since the conventional causality for each junction would
give the forms:
— 1 H H O —
T
y Z.
there results a "cross-over" between these causal usages and the conven-
tional noncausal usage immediately following. The best solution seems
to be to call the structure with the effort junction a SERIES IMPEDANCE
and that with the flow junction, a SHUNT ADMITTANCE, which agrees with
standard electrical terminology.
For linear 1 –ports the corresponding series impedance and shunt
admittance 2-port transmission matrices become:
SERIES IMPEDANCE SHUNT ADMITTANCE
— 1 — — O —
Z Y
! t
--1-i.Z. -4------
!
O 1 Y : 1
From our previous analysis it should now be amply clear that both
Z and Y might consist of any number of energy Storage and dissipating
elements, so long as only one energy port communicates with the rest of
the system. However, for each matrix, two simple cases are particularly
noteworthy; namely, where:
ID CD
Z = { or ; Y= 3 or
R G
232 2.75l CLASS NOTES
We may indicate these by bond graphs and corresponding matrices:
i
-
+º
wººd
-
3.
-
*
Aº
sº
i
-
-
-
sº
5
i
-
-
*sº
tºº.
$
-
*d
tºº
*ºr
i
-
-
*
*
Examples of Polymerized Chains
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
Using the above elements, we may now demonstrate their unusual
value for rapid determination of 2-port characteristics of typical polymers
For example, a commonly encountered form of damped oscillator in any medium
has the following properties:
— 1 1 O O —
I R C G
; ; ; !
-1-#P-I I-1_i_5_1 | -1_i_9_1 | _1_i_9_
O ; 1 O : 1 D ; 1 G : 1
! ! ? !
! !
; ID + R. 1 ; O
* * * * * * * * + - - - - - - - - --------T-------
O ; 1 CD + G :
!
---1+(IPfä)(CPig)-i----- +--t------
ſ
CD + G : 1
In a similar fashion we may depict a certain linearized fluid
system in the form:
— TANK– RESISTANCE TANK — WALWE —
O 1 O O
Cl R C2 G
; ; : ;
---4--------|--|--- i-----|---4--------|--- !--------
C1D : 0 1 C2D ; 1 G : 1
!
tº tº º 1 + RG + Tl2P i R
was nºt sº tº º me tº º ºs º ºn tº him sºme º ºn tº tºº sº ºn tº ºne ºn tº ºn ºn tº F----------------------------
G[1+(T11+Tal+T22)D+Til Teapºl i 1 + T11D
where: T11 = RCl; Ti2 = RC2; T21 = Ci/G; T22 = C2/G
Part XIV 2.75l. CLASS NOTES 233
D. Some Standard Forms of Two-Port Nets:
A number of recurring 2-port structures formed from impedance
functionals have been given names in electrical science. As mentioned
previously, the topology of such nets can be described simply in terms
of the junction structure alone, using the conventions:
-:
E
I – 1 — ( – 1 – [ — ZZ — )
Z.
[ — YY — )
:-
E
I – Q – I – Q – )
Y
The canonical structures may then be enumerated in a unique
simple order as follows:
1) The " L - Net " : [ — O — 1 — | = [ — EL — )
2) The " PI - Net " : ſ — O — 1 — O – || = | — PI — I
3) The "Tee - Net" X ſ — 1 — O — 1 — 1 = [ — TEE – )
O — *-
N /*
h) The "Latti Net" ſ |
e 8, C e - e : I — [ — LAT —
or "Bridge - Net"
i
X
©
->
E
These structures may in turn be cascaded or polymerized to form
the following ladder nets:
==
[
tºº
O
º
1
-
O
-
1
wº
-
O
tº-
1
-
J
5) The " L - Ladder " I — EL — in
6) The " Pi – Ladder " I — PI — I"
-
*
O
*
-
O
-
1
--
*
O
-
1
-
O
*
|
7) The "Tee-Ladder" ſ — TEE —]”
2-
º
1
wº-
O
--
sº
Q
*
-
1
-
O
-
1
*
]
23);
2.75l CLASS NOTES
and may be combined in various other ways to form other named structures
such as
8) The "Bridged – T"
9) The "Parallel - T'
I —
2%
O
N
TEE
O — )
E
*
|
$E
[
*E=-º-º-º:
— 1 — O — )
|
— O — 1
— O — 1
|
O — j
|
— O — 1
The above topologies may all be represented uniquely in coded
form as follows:
a)
b)
1)
2)
3)
lº)
8)
ZZ ab ; 1 ab.
YY ab ; O ab.
EL ab ; YY ac
PI ab ; YY ac
TEE ab ; ZZ ac
LAT ab 3 1 acd.
O fºrmn
‘Bridged TEE ab;
Parallel TEE ab;
}
3
ZZ be.
ZZ, cd
YY cd
1 bef
ZZ gk
O acd
O acd
3
YY bol.
ZZ bal.
O cgh
ZZ jn
O bef
O bef
5
3
O dij ;
ZZ him ;
ZZ ce ;
TEE ce ;
O ekl ; O fiſſ
ZZ il. .
TEE dif's
TEE dif'.
Part XIV 2.75l CLASS NOTES 235
E. Description of Linear Two-Ports
. Each of the six forms of matrix for the linear two-port (i.e. Z,
G , IHI, Y, MI, M-1) involves for the general case four independent
functional operators, which most Simply are the corresponding four matrix
elements, themselves. The specification of these four elements completely
determines the behavior of a linear two-port; in particular, at any given
frequency four suitable measurements will suffice to describe response.
If the network is passive reciprocal, only three of the functional
operators are independent and there is thus one constraining relation among
the set of four; now, at any single frequency three measurements will suffice
to define the system.
These various methods of specifying system behavior are inter-
related so that given one set we may find any of the others. We shall
here consider these relations.
The Interconnection of Two-Ports
tº as sº sº tº gº dº ſº tº º ºr gº ºn tº ſº * * ºr eme ºs ºne ºf wºme tº ºn tº me tº ºne tº tºº tº
There are five possible ways of interconnecting a pair of two-
port elements; these may be described as follows:
1) Cascade: — A tº
2) Series - Series: — 1 dº
B
3) Series – Shunt: ~— 1 ~~ *~ O —
& T- B _T
li) Shunt – Series: --- O ~ s-T 1 —
5) Shunt – Shunt: — O Pr"> o –
236 2.75l CLASS NOTES
In the cascade connection we have already seen that the
MI - matrices Simply multiply to obtain the interconnected behavior.
The remaining connections are governed by the Z , IHI, (G y Y matrices
respectively, as follows:
2) Series - Series:
Z Z Z.
a. b
3) Series – Shunt : IHI = H, + H.
li) Shunt – Series: G = G. + (G |b
5) Shunt – Shunt: Y Y, + Y.
The extension of these results to systems of two-ports cascaded
and coupled through energy junctions is Straightforward.
Transfer Characteristics of Two-Port Nets
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
The questions which arise in connection with systems involving
linear two-ports are commonly grouped in the following categories:
I. The TRANSFER Problem: wherein One seeks the effort or flow
at the downstream port in response to effort or flow at the
upstream port, with ideal terminations generally assumed at
the downstream port;
II. The TRANSMISSION Problem: wherein the power state at one port
is required in terms of the power state at the second port With:
a) Unrestricted terminal conditions, or
b) Terminal impedances specified;
III. The INSERTION Problem: wherein is sought the effect of in-
Serting a two-port into a system in place of a through bond.
Typically these problems are "filtering" and "protection"
situations, where performance is measured in terms of the
change in power, effort, or flow after insertion from that
occurring before insertion.
All these problems require consideration of the transfer character:
istics of a given two-port. These may be readily determined in terms of the
elements of the MI - matrix as indicated in the attached table.
Part XIV 2.751 CLASS NOTES 237
MATRIX FORM: CAUSAL FORM:
F-— — — — — — — — — —
f f2 * E | *2
-* -* 1 ––3–2–
O— | —O H |
A IB | |IF IF
. * * * * * * * * e | |-11 22 || |
1 C ID 2 | |
O– –O
| | IF |
12 Hº- <!-
y. | x
" ––––––––––– — 2
Determinant A = #| = [\ ID - IBC
= 1 for PASSIVE (RECIPROCAL) SYSTEMS
C W A R T A B L E S T R A N S F E R O P E R A T O R S
A
Inputs Output,
S pu utputs IF, IF2 F. IF as
E | * | *2 W1 | W2 dy,/0x, dy, /ö xe ôy2/3 x, dy2/032



Y e, es | f | f | ID/B |-A/IB | 1/IB |-A/IB
G e f f 'e C/A +A / A 1/A || –IB/A
|H| | f e e f IB / ID |+A / ID 1/ID | – C / ID
Z f f e e A/C —A / C 1 / C —ID / C
TRANSFER OPERATORS FOR LINEAR TWO-PORTS
238 2.75l. CLASS NOTES
Background Reading -- Early 2. Port Literature
(1) EVANS, R. D. and SELS, H. K. : Transmission Line Constants and
Resonance, Electrical Journal, Vol. 18, p. 306 (1921)
Introduced ABCD constants to power engineers.
(2) BREISIG, F. : Theoretische Telegraphie, Second Edition (1924)
(3) STRECKER, F. and FELDTKELLER, R.: Grundlagen der Theorie des
allgemeinen Vierpols, Elektrische Nachrichten Tecknik, Vol. 6,
p. 93 (1929)
(h) BAERWALD, H. G. : Die Eigenschaften Symmetrischer lin-Pol...,
Sitzb- d. Preuss Akad. d. Wiss, Phys-Math. Kl., Vol. 33, p. 784 (19.
The above three works carried over the use of linear 2. port concepts
into communications circuits.
(5) WAGNER, K. W. : Operatorenrechnung (1910)
Related the 2. port operators to transient response.
Background Reading -- Early Use of 2. port Matrices
(1) GUIII.EMIN, E. A. : Communication Networks, Vol. II (1935)
(2) PIPES, L. A. : The Matrix Theory of Four Terminal Networks, Phil. Mag.
Vol. 30, p. 370 (1910). -
These two authors are principally responsible for the present nearly
universal use of matrices for linear system analysis.
Part XIV 2.751 CLASS NOTES 239
Background Reading -- Current Works
(1)
(2)
(3)
(l)
Le COREEILIER, P. Matrix Analysis of Electrical Networks (1950)
A very readable account of the causal 2. port matrices.
WEBER, Ernst; Linear Transient Analysis, Vol. II (1956)
Nearly the entire volume is devoted to linear passive and active 2. ports
largely treated in terms of transmission matrices.
KUPFMULIRR, K. : Einfuhrung in die theoretische Electrotechnik,
Fifth Edition (TS57).
This classic German text has been kept up-to-date and matrix methods
are used throughout.
LAURENT, Torbern: Vierpoltheorie und Frequenztransformation (1956)
A book principally concerned With frequency domain behavior for
communication Systems.
Background Reading -- Gyrators
(1)
TELEGEN, B. D. H. : The Gyrator, a New Electric Network Element,
Philips Research Reports, p. 81, (April, 1948).
This epic-making paper has stirred up a revolution in microwave
techniques.
210 2.75l. CLASS NOTES
XV. Transformers and Transducers
A. The Concept of Ideal Two-Port Elements
We have earlier introduced the concept of a class of ideal multi-
port elements for which
X. IPD = O
Where the sum is carried over all ports. In the particular case of three
ports, the ideal effort and flow junctions satis ed this condition. We
shall now consider two-port elements of this class indicated as follows:
Ideal Element—J Or [ IE—]
[
Losses of effort and flow, as well as dynamic effects, such as in-
ertance and capacitance, can then be appended to these ideal elements to
model certain types of real elements, in the fashion:


LOSses and
Dynamics
[ — O . 1 Ideal 1 ... O – ||
N- Element,
- * - - - - - - - - v — — — — — — — —”
a model for
2— — — —-— — — —s
Real
Element
[ —
Particularly for energy transducers this need for "interdiffusion"
of media exists; it is not generally possible to represent the real process
of energy conversion without energy losses which involve state variables
from both media concerned.
Part XV 2.75l. CLASS NOTES 241
However, for many energy transformers, it is often possible to
embed the losses and dynamics in transmission elements at the two ports in
the fashion:
Transmission Ideal Transmission
Element Element Element,
or: — TM IE TM —
The use of the above reticulations for practical representations
involving the transformation and transduction of power we shall now dis-
CllSS •
B. Energy Transformation Elements [ — TF — )
These useful devices may always be considered in a lossless static
form, since dynamics and dissipation can be readily included immediately
upstream and downstream of the transformer ports. This may be seen
readily in terms of particular instances. Consider, for example, the
treatment of pivot friction in a lever:

F. V. y F. F. V.
C
= [ — 1 — TF — ) or [ — TF — 1 — )
Ö) R R
SCHEMATIC MOLECULES
Similarly, the inertia of a real set of spur gears could be ap-
pended to the terminals of the primitive transformer in the fashion:
CD 2
,722-sv [ — 1 — TF — —
*y- \\ Il I2
/ / //
- {{ o – ‘Z or ( – 1 — TF – ) or I — TF – 1 —
I. N--> * == I2 I I”

In this case, the inertial impedances can be appended to both sides
ºf an inertialess transformer or reflected entirely to either side.
2l;2 2.75l CLASS NOTES
This last example, then, gives us a prototype reticulation for
handling the general case of a lossy, reactive transformer, namely:
1 —
— 1 — TF
Zl Z2
where the element, TF, can be considered static and lossless.
This is usually considered as an ideal transformer for vihich
:
O
dE/dt = O ; 42. =

1 – 1-1 2" 2 2
ºs ºs tº as tº ºne tº imº ºm * * * * * * * * * mºs, sºme sº tº time tº ºs
We may then obtain the relations between efforts and flows for an
ideal TF by considering a two-port matrix whose A y IB y C 5 ID) , Operati
are all constant at the values a, b, c, d, respectively. Then we have:
2
*
sº
tº
{-º
sº
tºº
tº
tºmº
*
tºº
sº
The input power may be expressed:
elfi
(aea + bfa) (cea +- df2)
2 22
(ae)e: + (ad + be) eaf2 + (bd)r:
Since the power balance requires that IP, E IP, then the
following three conditions must hold:
ac = 0 5 ad + bo = 1 3 |bd E O
Only two possibilities exist: the first corresponds to the normal ideal
transformer; the second case to the gyrating transformer or gyrator
first discussed by TELLEGEN in l918. We can treat these two devices in
parallel fashion as follows:
Part XV 2.75l. CLASS NOTES 2113
GYRATING TRANSFORMER
(Ideal Gyrator)
NORMAL TRANSFORMER
(Idinal Transformer)
!
!
!
!
;
Conditions: ; Conditions:
!
b = c = 0 5 d = 1/a = a a = d = 0 $ c = 1/b = b
!
Matrix: ; Matrix:
t
! O ! O b
[ --à-i-º- ! [ --ºr?----
O ; at : b'; O
!
Relations: ; Relations:
;
T
!
e1 = aea ; e1 = |bf 2
!
fi = a' fa ; fi : b'e 2
;
- !
Or ; Or
f
1
!
e1 = aea ; e1 = bfa
!
fe : af1 ; e2 = bfl
.. :
Determinant: A = + 1 ; Determinant: A = - 1
The relations between these two types of ideal transformers are readily
conceived in terms of a primitive unit gyrator or gyrating matrix, y
playing a role analogous to the identity matrix, , defined as follows:
* * * * * * sº tºº ſº gº tº º sº tº gº º ſº º
-
I
:
;
i
;
3
:
-
;
i
*
gº
}
ſº
tºº
tº-
Thus II or [ — ) represents the normal or "undisturbed" bond while
G or [ GY
While det G = - 1.
In these terms, the two transformers are related as follows:
| represents a "gyrated" bond. Moreover det II = + 1
— NORMAL TRANSFORMER — —— GYRATING TRANSFORMER ——
– TF — O — TF O GY
a O 1 : O a ! O O : 1
---1-- * ! - - - - - - - — — — 1--- wº ---4---
-ā-j-ž, -º-;-ī- -ā-ā-ā, -j-i-º-
2ll 2.75l CLASS NOTES
This permits us to include the gyrator within the framework of ordinary
ideal transformers in all that follows.
The constant a measures the (flow) transformation ratio. For the
various common particularizations this is interpreted as:
MECHANICAL LEVER: lever ratio
8,
GEAR TRAIN: a = gear ratio
8,
8,
HYDRAULIC JACK! = area ratio
ELECTRICAL TRANSFORMER: turns ratio
-
Causality of Ideal Transformers
* * *s ºs º ºsmº tº sºme same * * * * * * * * * * * * * * * * * * * * * *
The two causal bond diagrams for an ideal transformer would have
the form:
ADPEDANCE IMMITTANCE
— TF — H TF H
which can both be expressed by the diagram
| "Can be reflected (1 = 2)
|
1 2
4.
Immittance |- – - – - a Adpedance
--> W <!-
f — He f
1 — 2
The gyrator is included within this scheme simply by crossing ové
or gyrating inputs and outputs as follows:
Impedance)
Form
Indicated |

Part XV 2.75l CLASS NOTES 21:5
- - - - - - - * * * * * * * * * *
. The gyrator has assumed great practical importance in electrical
engineering, particularly in the domain of microwaves. It has been en-
dowed with status as a standard circuit element having the symbol:
...] [.
Although realizations of the purely electrical gyrator have been
proposed using active elements such as amplifiers, it must be recalled
that the ideal gyrator is inherently a passive device, namely one which
neither stores nor dissipates energy.
From the standpoint of internal energy reticulation, the ideal
electric gyrator instantaneously converts electric to magnetic energy and
Vice-versa. This indicates at once its enormous practical importance
since a capacitor connected to one port looks like an inductor at the
other port. We may see this immediately using the unit gyrator GY in the
form:
GY C
el O ; e2
TÉ. -----------------| |#:
!
thus el = CD e2
fl = e2
Or el : CD a

Whence the capacitance acts as if it were an inertance. The importance
of this result for UHF communications cannot be overestimated!
This instantaneous conversion IE € == IE, suggests an electrical
realization in the form:
El Electrostatic F Mechanical to E2
to Mechanical Electromagnetic
Il Transducer V Transducer I2
Since, if the two transducers were themselves ideal, the conversion would
be instantaneous.
2k6 2.75l CLASS NOTES
More recently, the gyrator has been approximated in solid state
devices, using, for example, (a) the Hall effect, and (b) the Faraday
rotation of a ferrite.
These practical realizations then permit gyration in another media
by the Simple scheme:
Medium Medium
X X
el X - Electric Fa Electrical Fo Electric - X e2
fl Transducer *a Gyrator To Transducer fa
As long as the two transducers are themselves similar, near-ideal devices,
effective gyration will take place between ports 1 and 2.
In all such practical embodiments the self-impedances Z11 and Zea
will not be zero nor the transfer-impedances Z12 and Zal be perfectly
skew-symmetric, as is the case with the ideal gyrator.
The "Perfect" Electrical Transformer
* * * * me ims tº an ºn am sºm. * * * * * * * * * * * * * * * * * * * * * * * * *
In the problem material considerable attention is given to the
modeling of electrical transformers as two-port elements, It is par-
ticularly interesting to note that a so-called "perfect" transformer
(with unity coupling) may be represented in the form:
TF —
Q
Which in conventional electrical symbols becomes

a : 1
,-\–S
*—V-
Ideal TF
\– -Y. _/
Perfect, TF
Part XV 2.75l CLASS NOTES 21,7
The corresponding 2-port matrix would become:
! !
1 !
IVII -: ----4---- º ----1----
! !
! !
! º
-
º
tºº
sº
-
-
º
-
*
º-
tºº
wº-
ſº-
i
Furthermore, a possible causal representation of this structure
might have the form:
— O — TF —
-l
I
With the corresponding computer flow graph
f <!
If the transformer were linear as implied,—- P->transforms w-#7
{ /L.
!
This same structure may then also be indicated in linear block diagram
form as:
Finally, it is most important to realize that by no means is it
possible to reticulate this system into a purely I E, F, 0, 1, R, C, I
Structure; such a transformer is an essential two-port device.


218 2.75l CLASS NOTES
Geometrical Constraints as Space Transformers
* * * * *mºs º ºs ºs º ºs ºne tº tº mis tº º gº tº ºr tº dº sº ºms ºne tº ºs ºs ºs sº tº º ºs ºs ºss tº gº ºne º ºs ºº sm tº sº ºms tº
The introduction of the ideal transformer as a fundamental 2-port
element now allows us to represent many two-and three-dimensional geo-
metrical constraints as simple transformer couplings between the several
axes involved. Some examples can serve to elucidate this possibility.
For example, the Ordinary flyweight mechanism, originally in-
troduced by James WATT and still employed as a speed sensing element, has
the following form:
The important "gyroscopic transformer action" or "rotary-angular
transduction" involved can be expressed by the well established dynamical
relations:
cºrrugal. Torous: Ma - II (9) - w w = \{P w
GYROSCOPIC TORQUE: Mo = II (6) a 6 - H. §
However, it should be noted immediately that the existence of one of these
relations immediately implies the existence of the second. Moreover the
modulus "HP of any space transformer or gyrator will always be a functional
of (at least) the adjacent bond flows ( f). In a holonomic system the
functional gives way to an ordinary (static) function of the coordinates
or displacements ( [l ), while for the rheonomic systems the functional
becomes a function of the rates or flows (f). In the example above We
have a mixed situation with the modulus
UP = qi) ( 9, a j = qi ( (IL, f)

Part XV 2.75l. CLASS NOTES 21:9
It is also interesting to note that for small changes about the
equilibrium
di) = c OnSt. = b
and we have the previous case of an ideal gyrator. This implies that
the parts of the system each side of the gyrator are mutually dual; thus
for example it is 09 and not I 9 which adds to the *w to give the effec-
tive inertia when looking into the flyweights from the rotary bond.
Another enlightening example is the case of a suspended vehicle
or platform for which the rotary as well as linear energies must be taken
into account. Such a system might have the abstract form:
V T
V
\
TF— 1 — TF
2^ N
°-TF— —my -9
A. | A |
A I A
1 CU 2
| |
F F
There is now no need to "transfer" or "reflect" the linear and rotary
inertias to the two support points; the four transformers indicated take
completely into account both the efforts and the flows associated with
the two inertances.
Similar considerations will apply for all dynamical problems in
two-and three-dimensional space. By these means, physical space itself
becomes either a variable or a parameter, depending respectively upon
Whether an energy is, or is not, associated with a corresponding spatial
Variation. Moreover, the relation between system geometry and system
topology is thereby made directly evident.
The technique of using ideal transformers to relate space axes
derives from original work of Wannever BUSH, Gabriel KRON and many others.
It forms the basis of highly successive use of passive electrical analogs
by G. D. McCANN and his followers.

250 2.75l. CLASS NOTES
C. Energy Transduction Elements [ — TD — )
An energy transducer or energy converter is used to convert
available energy in one medium into available (or possibly unavailable)
energy in another. Some of the more common forms of transducer are mani-
fested on the transduction tetrahedron below.
Mechanical
TRANSDUCTION TETRAHEDRON
In recent times, some of the branches of this tetrahedron have
even become well-established domains of engineering science such as
thermoelectricity and magnetohydrodynamics (MHD). The most general relation
ships for such transducers are clearly no more than that for a general two-
port, but considerable insight is gained by considering some useful trans-
duction models and representations.
Below We indicate models of the form:
[ — TM — TF — TM —
I — TD — J
since a 100 per cent efficient transducer is always equivalent to an ideal
transformer.

Part XV 2.75l CLASS NOTES 251
- sm * * *
-
* = • *
*
- ::::::::
FLUID - MECHANICAL
MECHANO – ELECTRICAL
This we may see for the fluid-mechanical piston transducer Since
F(t) = A - P(t)
Q(t) = A • V(t)
Thus:
+|-|+iº-|-|3.
W O #7. Q
Similarly for a solenoid transducer:
F(t) = (Bl) . I(t)
E(t) = (Bl) • V(t)
3.]...g.i.a.l. |3.
f ſº O W
Note in the latter case that transduction is in the form of a gyrator.
In all such cases the coupling modulus serves merely as a transformation
*atio, with no loss of energy. All dynamic and dissipative actions are then
included in external generally nonlinear impedances, as we have noted has
been standard procedure for many years in electrical transformer practice.


252 2.75l CLASS NOTES
This gyrating model of electro-mechanical transduction is
considerably more general than would appear at first glance. For example,
a very refined representation of a 3-phase synchronous machine can be
obtained from the directly generalized relations
M = UPI - II
IE = [[J • N
where IE = [E1(t), E2(t), B3(b)], II : III.(t), I2(t), Ig(t)] are the
instantaneous phase voltages and currents of the three phases and the modulº
UHT = UPI ( II, N )
has the three phase components
UPI,
UHe
UHT
Ó 1 (II) - sin (kNt - O T )
qī) a(II) • sin (kNt - #
Ó 3(II) • sin (kMt. - #7)
-:
>
T )
3
Yet another generalization is possible following standard acousti).
practices, particularly where the transducer is approximately linear, namel
an impedance matrix representation, which in this application is originally
due to Henri POINCARE, and is conventionally written:
Z 1, Z 2 are the (self) impedances of Medium I and Medium II, respectives
While T 12 and Tal are the so-called transduction operators that describ;
the coupling between Medium I and Medium II. The dramatic history behind
Such developments is discussed delightfully by HUNT; some recent applica-
tions to electrical machinery are treated by RIDEOUT and SWIFT.
Another form of description has evolved out of the field of fluid
machinery in the form of relations based upon dynamic similarity, which are
used extensively for problems involving pumps, turbines, and aircraft and
marine propellers.

Part XV 2 75l. CLASS NOTES 253
The general turbomachine may be depicted in the causal form:
H.- Turbomachine +
Here it is only necessary to realize that a geometrically
similar flow field will exist for any operating point along a line of
Similitude defined by:

Q = r aw
This constraint imposes severe restrictions upon the possible form of the
resultant machine characteristics, giving as one of several possibilities
the pair of relations.

M = f(r) - w 2
P = g(r) - Q2
The corresponding computing structure may be realized in the form:

P — X He pe —º (pro X |o-M
Particularly simple models result upon taking
f(r)
g(r)
a + br- crº + . . .
(e/rº) + (f/r) + h + . . .
*
*
Where the series coefficients (a, b, c, ..., e, f, h, ...) depend upon the
£eometry of the device, including the possibly variable blading and other
modulating parameters.
25l. 2.75l CLASS NOTES
Lastly, we may obtain certain general results for the case of
constant efficiency reciprocal transducers operating over a Small range.
A per-unit notation reveals:
aIP, dIP a
= u1 + Vl $ = u2 + V2
Po IPo2
Where u = de/e = d(ln e) and v = dif/f = d(in f)
thus Ull a ; b lla
----| = |---4---| | | ----
Vl c ; d. V2
If efficiency is constant:
a + c = 1 3 b + d = 1
and if the transducer is reciprocal:
ad - bc = +1
These three conditions reduce the matrix to the two forms
f a - - 1 7
-----#---! Or P_-_1 -----
1 – a £2 — a 2 - b;1 – b
-
A value a = 1 or b = 1 then yields 100 per cent efficiency, and corresponds wo
the ideal transformer and gyrator, respectively.
Besides their use as power-level energy converters, transducers form
E
the essential elements of most continuous measuring instruments for physical
and engineering processes. A very complete list of instrument transducers
has been prepared by D. B. KRET of the Du Mont Laboratories, entitled Transdº
A Compilation. Useful Primarily in Oscillography.

Art XV 2.751 CLASS NOTES 255

#ikground Reading -- Electro-Acoustic Transducers
|) HUNT,...F.W.; Flectroacoustiº. The Analysis of Transduction and Its
. Historical Background (1951)T * sº
A superb account of the history of the supfect.
2) FISCHER, F. A. Fundamentals of Electroacoustics (1955)
Presents the physical priuciples underlying electro-acoustic transduction.

Ackground Reading -- Electro-Mechanical Transducers
1) RIDEOUT, W. C. : Analysis of D. C. Rotating Machines as Two-Port
Networks, AIEE Conference Paper, CP-57-760 (1957).
2) SWIFT, W. B. : Analysis of D. C. Rotating Machines as Two-Port
Networks, AIEE Conference Paper, CP-57–761 (1957)
These companion papers give an excellent introduction to the d. c.
machine as a transducer.
3) WHTTE, D. C. and WOODSON, H. H. : Electromechanical Energy Conversion (1959)
The current definitive work in this subject.
256 2.75l. CLASS NOTES
XVI. Energy Transmission Elements
A. The Two-Port Element: [ — TM — )
To convey or transmit energy from one place where it is available -.
the source -- to another place where it is to be used -- the load -- a 2-port
transmitting medium or transmission element [ – TM — 1 is required. Some
common realizations might be indicated as in the following table:
º
-
-
rº-
º
*
*
*
º
º
*-
&-
4-
-
*
tºº
iº
tºº
tºº
*
A-
B-
*-
*
Aº
*
**
*
*-
*
*
*-
T-----------------------------
FORM OF ENERGY
dºms º ºs-ºs ºne º ºs ºn tº emº ºme sºme ºs me tº º ºs º ºs ºmit tº ſºme tº * * * * * * tº dº º º 'º º ſº º jº 㺠dº
Fluid Pipe or Duct
Mechanical Rod or Shaft
Electrical Wire Or Conductor
º º ºs tº ºn
The absolutely ideal "TM" is the simple power bond:
[ — )
But all real transmission elements possess static and dynamic properties re-
Sulting in the dissipation, scattering, and storage of energy.
Thus, in the case of power transmission, nonideal behavior manifests
itself in power losses; while in devices for the transmission of signals
and information, all real transmission links Will delay, distort, attenuate,
scatter, and contaminate the desired Signals.
The paragraphs below deal with, and distinguish between both situa-
tions, and relate behavior to the limiting cases of pure wavelike and pure
diffusive transmission,
* Bºº, º ºn 4-
Within and around any •TM- element, part of the available energy bei
transported is continuously consumed and converted into heat. Under steady
operating conditions, along the entire transmission path, from the source to
the load, there will be a net convergence of the Poynting vector, p , re-
sulting in a power gradient approximately parallel to the transmitter and a
continuous decrease in power level along its length.
2.75l. CLASS NOTES 257
| No practical devices exist which can transmit power over space
thout Such corresponding parasitic guidance- or support-losses. However
}. effects of these power losses on the overall flow of power in engineer-
systems are usually restricted within fairly narrow limits.
On the One hand, since any resistance in the -TM- consumes useful
over and thus lowers the efficiency of transmission, it is generally un-
conomical to permit too high a value of transmission resistance. On the
#er hand, since lower resistance usually implies larger quantities of trans-
ission materials, there are also lower practical limits to power loss.
. As a result of the above considerations and other factors, the
teady rated percent power loss of most transmitting elements will be modest.
any case, all such losses in energy can be reduced to combinations of the
no forms:
Loss in Effort : RESISTANCE : I — 1 — )
R
Loss in Flow : LEAKAGE : ſ — O —
G
The resultant steady loss of any transmission system must therefore
e capable of representation in the form:
I — 1 – 0 — J"
R G
here the index n is taken sufficiently large. Under most circumstances, by
º ining resistance and conductance relations, the overall loss relations
sy be modeled:
I – 1 – 9 –
R
e e
mere Re and Ge are equivalent resistances and conductances.
*
3. * * * * * * tº ims tº dº º ºsº ims tº º ſºme º ºss ºne º sº ºr ims as ºn tº ºs ºs ºn tº º ºs dº
Under transient conditions of operation, the field effects of energy
torage, in the form of inertance and capacitance distributed along the trans-
ission path, will produce significant effects upon system behavior. These
ºil be manifested in local variations in the distribution of power and energy
wer the extent of the transmitter.











258 2.75l. CLASS NOTES
While a transmission system can be roughly characterized by a
reticular model of low order, whenever the wavelengths of the power state
variables ſe(t), f(t) l become small by comparison to the length of the trans-
mission path, the continuous nature of the TM- element must be taken into
account.
* ºr tº tº º tº º tº ºt * * * * *
An actual length of transmission line in any medium may be conceived
as having all effort and flow losses concentrated in reticular fashion at the
upstream and/or downstream ends in the following manner:
[ — Actual Transmitter with Losses }
/ N
[ Loss Lossless Loss In
Element Transmitter Element,
Clearly, if this situation is assumed to hold for a sufficiently large number
n, of appropriately small transmitter segments, any real transmission system
may be approximated to an arbitrary degree of practical accuracy. Frequently
only a few such elements are necessary. The particular sections or junctions
at which all energy losses are presumed to occur may be called loss junctions.
It is particularly important to realize that while certain forms of
linear loss may sometimes be handled in other analytical fashions, this use
of loss junctions becomes mandatory for the representation of essential non-
linear and/or reticular resistance and leakage (NONTINEAR: e.g.: electrical:
corona loss; fluid: turbulent loss; thermal: radiation loss) (RETICULAR: e.g
electrical: suspension insulators; fluid: bend losses; mechanical: bearing
resistance).
Under rapid transient conditions, many factors conspire to make actus'
dynamic losses somewhat more complex. The flow will usually vary from point
to point due to distributed capacitance; this will necessarily cause the local
losses to vary. If the flow has sufficiently high frequency components, add-
itional resistance and scattering phenomena are present which do not manifest
themselves at lower frequencies or in the steady state. Indeed, all experi-
mental evidence in every field medium indicates that the higher frequency com-
ponents attenuate more rapidly than lower frequency signals.
pºrt XVI 2.75l CLASS NOTES 259
The Canonical Transmitter Matrix ( T )
tiºns ºf Šºć-2:Pºst--Peticular Transmitters
Consider the general asymmetric 2-port:
[ — TP — )
A symmetric 2-port can always be produced by connecting two such two-ports
back-to-back in either of the alternative fashions:
— TP — PT — Or — PT — TP —
º- =º
But we know that for any linear reciprocal, symmetric 2' port only two
of the four operators can be taken as independent.
A canonical form for the transmission matrix of all such reversible
elements can be obtained through two new defined operators:
cosh' A
Characteristic Impedance Zo- ^/ IB / C E 1/Y.
which results in a final transmission matrix:
E
Propagation Operator T
Jºnsider now a chain or cascade of n such identical reversible elements;
I — TT — TT — ... — TT — l = I — TT — "
1 2 Il
* may consider any such system as a reticular transmitter.
Then:
M-T #ifºis - ºr- ºr
º




260 2.75l. CLASS NOTES
The value of the T matrix is now obvious since it is the only form for
MI which preserves its nature upon multiplication; these properties
result directly from the identities of hyperbolic trigonometry.
tº º º ſº tº º ºsmº º sº tº ºt
is ºn tº ºn tºne ºf ºsmº
Consider a uniform one-dimensional linear transmitter:
:
!
}*— Total Series Impedance z –-
|

Wire, Rod, Shaft, Pipe, Duct, etc.
*— Total Shunt, Admittance Y. —-
:!
Since the overall structure is symmetric, this system has a T —matrix:
T Cosh T : Z. Sinh T
- | * * * * * * * * * * * !. iºd
Y sinh T ; cosh T
!
Moreover, we may consider this system to be reticulated into n identical
symmetric micro-elements to obtain
If n is sufficiently large the structure of the micro-element is not critica”.
Either of the following two molecules would serve as micro-elements:
Part XVI 2.75l. CLASS NOTES 261
!
ſ TEE jn f [ PI In
!
ſ 1 O 1 in ! [ O 1 O jn
†: * «» ! tº & tºr
º
Z/2n Yºſn Z/2n f Yº/2n Z/n Yeſ2n
!
f
pºss ! 2 — Il ? — ; — Il
Z, Y Z, Z. Y ! Z, Y, i.
1 + → ; * + → 1 1 + → ; z/n
2 : 3 2 : t
#----i-B lin ? £----i
* * * * * * * * * * * -: * * * * * * * * * * * * ! ------------, * = * * * * * * * * * ºn
Z, Y ! Yi. Zºº Y." | Z, Y
YE/n 1 + + f +++ ; ; 1 + +
: 2n ! Il lin’ 2n
* * * f sº
?
2
1
For both cases, there follows directly, for cosh(#T) = A and Z. E IB/C,
the values:
!
cogn(#T) = 1 + Gº. ; cosh(; II") = 1 + Gº.
!
1.
2 1 t 2 Y 1
Z. = (Z/Y.)[1+ Gº". Z - (</x)/(1 + Gººd
$
We may determine the T matrix for the continuous transmitter merely by let-
ting the number of microelements, n, become infinite. In this case, for
either "TEE- or -PI- elements there results:
2
lim T = Z. . Y.
n —- CO 2
lim, Z = Z. / Y.
n —- CO
yielding the final results

Overall Propagation Operator T = . / Žt © ¥t
Characteristic Impedance Z.- /Z / Yt
262 2.75l CLASS NOTES
Substituting these results into the system T-matrix gives:

Note that specification of either of two pairs of operators, (Žt, YE) Or
(T 9 Z c) is sufficient to specify T in any instance. It is also obvio
that both direct and converse relations exist, between these two sets in the
T = x/Z, - Y, 3 Z = T - Z.
Z.- * / Y. 3 Yt IT / Z.
fashion:

:=
The T-matrix above is the canonical form for all linear uniform
transmission elements, regardless of the nature of Žt and Yt- We shall find
it convenient in the treatment below to deal with two limiting cases of trans
mitters, namely:
— WAWELIKE — — DIFFUSIVE —
TM TM
Žt B ItD Žt E Re
Yt E C.D Yt = CD
Part XVI 2.75l CLASS NOTES 263
C. Generalized Transmitters and Wavelike Transmitters
Consider the one-dimensional generalized transmission process
governed by the matrix differential equation:

(d. 5 /dx) + N S = o
Where S; B: [+] is the power state vector. This relation is simply a
formulation of the intuitive relation:
5 + d $ = I II - N dr] $
Let us assume a matrix Solution of the form:
S. = M. S.
Where S$ 2 is the downstream power State and M X. is the 2-port transmission
matrix for a length x of “TM- .
Then:
:
d5/dx = (MI/dx) - $
NS = N M S,
Which yield an equivalent differential equation for MI , namely
(d. M/dx) + NMI- 0
This relation could be used directly to determine differential equa-
tions for the A 9 IB 9 C y ID) elements, but a much more instructive pro-
cedure, at least for wavelike transmitters, is to first transform the power
State S to a characteristic state |Rin terms of characteristic (or scatter-
2
and
>
ing) variables (u,v) where:
IR
IHI
IE
$
26l; 2.75l. CLASS NOTES
with the converse relations
* * * iº tºº tº
wº
e
Jº
]
*
O
O
*
1.
*
1
-
ſ
}-
]
We can relate the characteristic state, IR. at any upstream
point, x, to the downstream state ,IR a , by the expression
JH E ME *H R,
IR
IX.
IR,
Q e R 2
Thus the characteristic matrix, Q, (which is directly related
to the scattering matrix) plays the same role for characteristic variables
as the transmission matrix MI plays for power variables.
If we define a matrix T analogous to N above, by the relation:
T = HIE N E *H
then the differential equation for the general characteristic matrix, Q 3.
(d Q/dx) + T Q = o
This last equation can then be solved for the case is a uniform lossless
is given by:
wavelike transmitter in a very direct fashion as follows:
For the Wavelike TM
N
:=
tºº
*
*º
*
gºe
i
*
*
*
*
*
Then T can be found to be
vºll & D : 0 Y : 0
T
!
o AT&T o i-y
if the scaling constants are taken as R. = 1/Go # // C .
Part XVI 2.75l CLASS NOTES 265
Writing:
o: [-º]
& 20°
we find: *: /
w - d /x - |#%
Q () & ‘i Zºr’
TQ = [+,- | [44] - [+º, |
o ;-y & & -Y & -y º
Adding matrices We obtain:
2: Y&#3.2/3 | -
Č' - y & Arº y A.
From the primitive statement Q a Q = ( I + T dx) Q , taking into
account the nature of T , then necessarily:
/3 F 62 E O
For (Z and Af , we have the parallel developments:
(Z + Y (Z = 0 2 : y & -
Aſ (Z'-- y & * + y &
## in (Z-- Y :* in 20-y
(Z 24%
d in (Z= - Y ax a la -/- Y ax
Q. O ! 20° O
Jala (Z. - Y dx ſ a la C. J. Ya.
O in & - Y - . in 47- - Y -


266 2.75l CLASS NOTES
This yields the final universal transmission relation:

o ſº-Hº-
|-Fr.
tº
i
which is in the form of a transformer.
This matrix is actually applicable to a broader class of transmitter
than merely the wavelike variety, but the latter instance is our present con-
CeIT le
For this wavelike case:

T = x/~! C : x D = v/ J , ) ( & ) = MILC, D = TD
where T is simply the Wave propagation time from upstream to downstream
port. The corresponding (ſ) —matrix is

tº ºs ºs ºne ºn tº tº
Q
:=
*
tºº
Yº.
tºº
*
wº
Aº
i
tº
As a last Step We can transform this result back to the transmission
matrix M , to obtain
M = IE *H Q HIE

Or
cosh TD
MI sº ºn tº time tº tº tº dºes hºt ºn tº am tº
G Sinh TD
O
tº tº tº dº sº tº ºne ºč ºn tº º ſºme
cosh TD
i
E
I
With
Fo E 1/Go E a /I./C. ; T t " 9t
Part XVI 2.75l CLASS NOTES 267
D. Ideal Wavelike Transmitters
We are here concerned with continuous elements of the form:
| WIRE, FIPE, ROD, SHAFT, DUCT, etc.]
TM
having the fundamental transmission matrix:
!
cosh TD ; R., sinh TD
------------------
!
G, sinh TD : cosh TD
t
T - Propagation Time 2: It Ct
R. = 1/G. = Surge Resistance = VI, / Ct
This same matrix can be used for general operational analyses,
describing system behavior in either the frequency or time domains. Contrary
to common opinion, transients and vibrations in such transmitters become ex-
tremely simple to investigate so long as the most effective description is
employed.
A significant simplification takes place if the variables are trans-
formed in such a way that both effort and flow are measured in the same units.
This normalization will then eliminate one of the parameters in the trans-
mission matrix, as indicated in the following tabulation:

*-
* * * * W_9_5_%–4–5–1–4–4. T-I-9–W--5–2.É.-3. M_E_3_--------------
ITEM x= * *
SCHEME A. SCHEME B SCHEME C
Normalized
State
Wector € € = e € = G e 6 = . I G - e = ſIF
S = O O e
q = RAf % = f p = F. : : - / P
*— f
! ! !
Normalized cosh TD'ſ sinh TD cosh TD; Sinh TD || _ cosh_TDHsinh_TD
Transmission >| = |-------4------- = as tº a sº sº tº ºt #::::::-t: = as as ess ame ºn sm tº sº sm as sº sºn ºn was tº
sinh TD; cosh TD sinh TD; cosh TD sinh TD; cosh TD
Matrix T
268 2.751 CLASS NOTES
Intrinsic Wavelike Transmission Matrix
We may consider the reduction to intrinsic variables as a factoring
of the transmission matrix in the form:
- cosh TD ; R. sinh TD
IMI = | --------------------- i---------------------
* Go Sinh TD ! Cosh TD *
Yºsi O cosh TD ; Sinh TD Yºe; O
|M|| : ----T---- || || -------- F-------- || || ---- F----
o MG...] | sinh TD cosh TD o MR.
The matrix IK is that for an ideal transformer of modulus w/º:
While the matrix T represents the ideal wavelike transmitter with transit
time T. We shall find it convenient to adopt the symbolism:
C S
T = |--------
S : C
where C = cosh TD and S = sinh TD. Note that the well-known identity
cº dº sº = 1
merely expresses the reciprocity condition for such transmitters.
* - - tº lº º mº tº * * * * * * * * * * * * * * *
Consider the transformation:
[*] - 4 +++)
(1/x/2) (e + f) [Sum]
(1/ w/2) (e – f.) [Diff.
Or:
li
-:
V.
:
Part XVI 2.75l CLASS NOTES 269
The inverse transformation is readily determined as:
-- r ll
-T- º lº
(1/ V 2) (u + v) [Sum]
(1/ V 2) (u - v) [Diff. J
-
*
tº-
*s
-
-
tº-
*
i
O
r
e
-
f
-
The instantaneous power, IP (t), can now be expressed:
- * * * *
IP = e - f = u−/2 - vº/2 = IP t - IP .
IP = const.
Maximum P always occurs
When uf – max , vº -- min - O
Zero IP corresponds
to V = + Ul
Here:
EX
TRANSMITTED POWER º IP = u−/2
(or COFLUENT Power) º t -
<H
REFLECTED POWER w IP = wº/2
(or COUNTERFLUENT Power) ' r T
Thus u(t) measures the instantaneous value of the downstream flowing
or cofluent power and v(t), the instantaneous value of upstream flowing or
Sounterfluent power. -
The variables (u,v) are known in electrical science as scattering
Sarameters and in fluid mechanics as characteristic variables. We may define
* Characteristic state vector, IR , by the definition:

27O 2.75l CLASS NOTES
which is analogous to the normal State vector:
S = ſ ----
- f
Then the relation between these two measures of state is given by
the converse pair of relations:
IR = IHI • $
[-,-] = + [-,----|-]
IHI R
# -----|-] . [+]
§
;
-
:
~
*
;
2
Note that since IHI = [ ! *] I , the scattering operator, HI, is
analogous to the gyrating Operator G , in being another of the (many)
square roots of I , and therefore representing a duality transformation.
=
Characteristic_Relâtions_for_WäVelike Tränämiššion:
Let us now apply the scattering matrix IHI fore and aft of an
ideal transmitter T, thus:
N
H
T
IHI
: +|-|-- }- [-º,-----|- + [-,-----
y 2 L 1 ; -1 s--T--a y 2 L 1 ; -1
*e +[ ___C +_S !----à-t-8------ |-|--|--|-]
º 2 C - 5 ; S – C i----
-*. -------------------+-----------*--------- ]
* O ; 3-----s
> [− !----------- 3--------- ]
O ; a -TD

Or N : [----- * * * * * * * * * * * * #------------- --]
A †
Thus we may write
R : N dº IR 2
-1
-i. A E} O u2
Part XVI 2.75l CLASS NOTES 271
Where A, - e.” is the time delay operator. We may then obtain the two
time domain relations:

:
u2(t)
v (t) : vaſt * T)
u, (t - T)
These characteristic relations were first obtained by Bernhard RIEMANN in
1860 for the nonlinear case of Sound waves of finite amplitude.
Background Reading -- Method of Characteristics
(1) RTEMANN, Bernhard: Ueber die Fortpflanzung ebener Luftwellen von
endlicher Schwingungsweite, (1860) Gesammelte Mathematische Werke,
pp. 156-175, Second Edition (1953).
(2) MASSAU, Janius: Unsteady Flow in Open Channels, Annales de l'Associa-
tion des Ingénieurs sortis des ecoles spéciales de Gand T. 23
FE-95-31T (T555).
(3) BERGERON. L. : . Du Coup de Bélier en Hydraulique au Coup de Foudre
en Électricite (Tg50)
Background Reading -- Scattering Matrices
(1) CARLIN, H. J. : The Scattering Matrix in Network Theory, IRE Trans-
actions on Circuit Theory, Vol. CT-3, Number 2, pp. 88-97
(TRE-1556).
(2) REDHEFFER, R. R.: Difference Equations and Functional Equations in
Transmission-line Theory, Modern Mathematics for the Engineer,
Second Series, pp. 282-337 (1961) —r-
272 2.75l CLASS NOTES
* * * me eme emº ºn as ºs ºn tº sm ºme as ºne sºme º saw sº em as * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
System: e e2
E –F– TM f R
Structure: 1 2
Disturbarace: e1(t) = E sin (Ut
Response: f(t) F sin (aſt - p)
TRANSFER CHARACTERISTIC:
Setting the ratio R2/R., = r, the intrinsic transmission matrix
yields:
*
*
*
t-
4-
iº
º
-
º-
i
Therefore the operational transfer characteristic becomes:
- ºf - * : * ~ *2 = 1 - = cann Tor
F11 - e c. 4 S r + 7. T r + tanh TD
FREQUENCY RESPONSE:
In terms of frequency response
IF, (ja) = ; # : # # ----------(Periodic Functions)
Since the arguments are periodic, the transfer function will itself be
periodic, Moreover, any transformation of the form w = (a + bz)/(c + dz)
with w and z complex numbers is a bilinear form which always transforms
CIRCLES into CIRCLES (note that [St. LINES] C [CIRCLES] and [INVERSE TF] C
[BILINEAR TF]).
In this case Z = j tan (Ut is a straight line, and F.G.) is
therefore a circle. To determine the circle, it is only necessary to fix
three points. Since all frequency responses of linear systems must give
polar plots symmetric about the horizontal axis, we know that the center of
the IF 11 circle must lie on the axis. The two additional points may be
Part XVI 2.75l CLASS NOTES 273
quickly found by noting:
(1) At: a t = 0, f TT, it 2 T , . . . , t 2n TT , ...
tan (Ut, E O
F11 = 1/r
T
(2) At : wt = + +, + +, • * * > + 2n + 1 Tr 2 & & ©
tan (U tº -> CO
F11 = r
Thus the final locus has the appearance:
CIRCUILAR
FREQUENCY
LOCUS:
traversed
at uniform
Speed.






27].
2.75l CLASS NOTES
* * * * * * * * *
A general variable parameter wavelike transmitter can always be
considered as a finite or infinite sequence of small elements, thus:
Variable Mater,
Parameters
Variable Tul v
CROSS-SECTION £5.5.£33 XSSSSS. =#-
º3. Ž Aº -* NJRs
% _* º sº-s
&- * * T TN
Ø __* Y- * | Nº
Q? º
ZZZZZZZZZZ, Sv ~ y – - º - -, - SS
f \ \ W * + _ -" |- -/ T-
! \ – \ !— H - * | A S
\ \ | R-2— — — i- /
\ N. 77.lºvº V &
\º £7. {& º 2ZZ ZZ
\s 9.
Upstream Match
Equivalent Pair
of Uniform TM §
Downstream Match
Of of t
Half Section Half Section :
CHARACTERISTIC CHARACTERISTIC
IMPEDANCE < * IMPEDANCE
Żol = Zu Zoº F *




2.75l CLASS NOTES
275
Each small section may be considered as two lengths of uniform trans-
mitter having equal transmission times Tu :
T d = T. The properties and
corresponding characteristic impedance of one half-section are taken as that
at the beginning of the physical element while the parameters of the other
half-section are those corresponding to the end of the physical element.
It is then apparent that as the lengths and corresponding transmission
times shrink to zero this model will become exact, both for abrupt and for
gradual non-uniformities.
The advantage of synchronous timing for all the elements can now be
made evident.
into the intrinsic forms:
We may factor the transmission matrix for a double element
Ru, T ! Rd., T
!
Upstream ! DOWnstream
Half-Section ! Half-Section
º
!
ºf-2. ||.g. j.g. |Yºj.g. |}|{#.g. | | g. i.e.||:/ºj.g.
0 i-ſgu S C O MRu ‘L 0 </Rd 8 : C 0 v Rd
!
*/Rà/Ru o
T *-ā- –4 F---------- T
O , VRu/Rd
r º O
T | - ! T
O 1/r
Q
T - IE • T
º
?
$
8
$
!
*— TM TF TM
Thus it is that the matrix for the general case may be expressed (except
for the terminal scaling) by the finite or infinite product:
M. - 'Iſ T. E.
276 2.75l. CLASS NOTES
In the special case where the transformer ratios, *k, remains every-
where close to unity, the matrix M. may be approximately refactored to
the forms:
M. – (T^*) (TE)
Il 2n
(TTE) (Te')
k=1
Under these conditions, behavior is comparable to that of an equiv-
alent uniform transmitter for which the local intrinsic variables
e (x)
q (x)
satisfy the wave relations. This result is known as Green's Law after
*/Go • e(x)
v R, . f(x)
George GREEN, who in 1837 derived these relations for gravity waves in
shallow Water channels.
A practical example of the use of Green's Law for engineering estimates
may be indicated in terms of the pressure surge following sudden shutoff in
flow at the small end of a tapered pipe. The Green's Law estimate would be
obtained from the formula:
P(s)/P. = x/ ZoCs)/Zoo -: ~/DeſD(s)
where D is the pipe diameter. The following tabulation gives typical result:
tº
º-
tº-
tº
º
-
º-
*
tºº
tº-
ſº
tºs
tºº
º-
tº-
-
*
º
tº
*
sº
tº
*-
t-
tºº
tº
tºº
tºº
sº
-
tº-
tº-
*
º
ſº
tºº
ºm
tº-
-
tº-
tº-
º
tº
tº
tºº
tº
º
tº-
º
tº
tºº
tº-º
º
tºº
tº-
-
tºº
º
F--------------- T ! f
; ; ; Calculated Pressure Rise ;
! 1. * --------------------------------------- 4.
! ! 1. f
; Station ; Diameter ; Exact Results ; Green's Law :
i-------------- i------------ ----------------------- +------------------- ;
; O : 1. OO : 1 - OO : 1 - OO :
: 1 ; 1 .. 20 ; O.82 ; O.83 :
f ! !
: 2 ; 1. HO ; O.69 ; O.71 :
f f f
; 3 ; 1.6O ; O.60 ; O.63 !
! f ! f !
; l; ; 1.8O ; O.53 ; O. 56 ;
! f ! ! !
; 5 ; 2. OO ; O. l;7 ; 0.50 ;
! 1. ! t !
1. 1. 1. 1. 1.
tº ims tº ºm º sºme m tº me sºme ºn tºº sma am ºn m a.ms sºme sm * m ms was em ms ºus ºw amº ºms am ºn am time ºm m ºn tº *
Part XVI 2.75l. CLASS NOTES 277
E. Modeling Diffusive Transmission
Wavelike Transmitters with Dispersion
sº a sm ºn tº * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
Consider the transmission micro-element:
:
*
where Y(d) # CD but is a general function of D. This situation is a much
closer approximation to reality than the pure capacitive case.
For example, consider Y(D) = C.D + ſc, D/(1 + T,D)] corresponding to
the 1 port element:
It is easy to demonstrate that this results in a dispersion or
scattering action. Such dispersion is a vitally important factor in all
communication and other transmission of information, since, for example it
determines the effective "channel capacity" in the fundamental Shannon
Formula:
C = W ln [ 1 + (S/N)
Let us consider a more general case in which °o = O and
Il
Y = X. [c, D/(1 + TD)]
k
Then Il
for low frequencies: Y -- ( X C.) • D = C,D
Il
for high frequencies: Y - X. Cºſ". = 2 G = G
This latter corresponds to a pure diffusive transmitter which was
considered by William Thomson, Lord KELVIN, in connection with SUBMARINE
CABLES.
278 2.75l. CLASS NOTES
tº dº ſºme tº º ºs ºs ſº tº 4 º żº º Aºss dº º ºf tº * *-* * * * * * *
Duals
O – 1 —
Consider the micro. 1 Q e tº
elements R C G I

~
This geometry first arose in J. B. J. FOURIER'S analysis of transient
heat conduction: Theorie Analytique de Chaleur. The FOURIER mathematics
was exhaustively applied by KELVIN to the RC cable with singular consequence
In our general TM parlance we now have:
A/TD
T = R. C
R.A/TD tº t

Z = R. wº T =v Žt Ye
t; t Žo
Note now that the characteristic impedance is an operator, no longer a
scalar, while the time constant T = R.C. is an R-C parameter. This is the
microscopic counterpart of the distinction between restoring times and wave
=
R
:
C
D
periods of macrosystems.
In order to interpret these universal properties let us consider the
special case
----------- Relative Length = 1 --------
el Diffusive *x Diffusive °o Infinite
F. Transmitter *x Transmitter fo- O Impedance
F---- ( 1 — x )---- F-----( x ) ----- ++
!
?
Then: -:- - Azi IB, eo
sºssº ---T-S-- e tº ºn tº tº
DX X; IIDX O
!
-j- - _All_i El...] . [...is
! * * *
This gives e A cosh X vſ TD

el A1 cosh–YTD
Part XVI 2.75l CLASS NOTES 279
Then for example at the "insulated" end (x = 0)
ex(t) -Gº e1(t) = F(D) - e, (t)
Here as in all such diffusive systems, the transfer operator IF (D) is a
monotone process which bears to diffusion the corresponding relation that
vibratory processes bear to Wavelike transmission. This we shall treat next.
However, first we may interpret our result by factoring the "cosh"
function into its characteristic roots. This yields a representation as an
infinite set of tapered first order lags.
Fo---- f ––– a re--4-
===TZm- K = 1 1 + a TD *k (H-T)2
$ºssº 1 te 1 * > —— tº º ve
* TT5 T5 TD TTO.O.TTE TT5:05-II
resulting in the responses:
STEP RESPONSE:
|
IMPULSE RESPONSE: f(t) _/>=

28O 2.75l CLASS NOTES
F. The Dynamics of Monotone Processes
Introduction
* * * * * * * * * * * *
A large number of fluid, thermal, chemical and other industrial
and organic processes are characterized by a step response which is mono-
tonic non-decreasing in time as indicated.

RESPONSE
F(t) --------------
*--- ſº-
Lºr
TIME
The corresponding frequency response, at least for most continuous processes,
would have a non-increasing amplitude and non-decreasing phase lag with in-
creasing frequency as follows:


| F| i- - Z F V-
FREQUENCY FREQUENCY
All linear Systems giving rise to such response can be called
monotone processes.
Monotone response is manifested through:
(a) Time delay or dead time.
(b) Dispersion or rise time.
In physical processes, time delay is usually associated with
propagation or transport phenomena as measured by the ratio of travel dis-
tance to propagation or transport velocity. The dispersion in any process
can ultimately be attributed to the law of increasing entropy, whereby the
Part XVI 2.75l CLASS NOTES 281
distributed resistances in any system cause an attenuation increasing with
frequency. Such scattering action is reduced by isolation and relaying
methods, but is always present to some degree.
Oscillatory processes, characterized by the presence of comple-
mentary energy storage elements, Will have monotonic response whenever the
energy dissipated per cycle becomes sufficiently large compared to the energy
stored in each mode.
ims ºr me as ºne ºs ºs ºs tº ºne nº ºn tº ame * * * * * * * *
One finds recurrent need to define quantities which are definite
integrals with respect to a given monotone distribution function F(t).
Thus we may write the expectation of a function IE(g), 8, S
+ e CO
Expectation: IE (g) = f g dF(t)
Thus one finds that the moments about the origin of a monotone
may be expressed compactly in the Stieltjes form by the relations:
k, * * *
k-th moment: a. * |E (t”) = J.' • dº(t)
- "OO
The infinite set of Such moments, of course, measures the dis-
tribution properties of a monotone function; in particular, the zero
moment:
+ e CO
*o - ſº
measures the total area of the distribution. One conventionally normalizes
the distribution, if possible, such that *o is identically unity to give
F(- o) = 0 and F(q) = 1. We shall henceforth assume this to be the case.
The first moment:
+ 2 CO
al - ſº ar(t)
-" OO
yields the mean effective position or centroid of the distribution if ac = I.
282 2.75l CLASS NOTES
In the same way the second moment:
+
2 -ſ tº . dF(t)
Measures the mean square position of the distribution, and so forth.
In addition, the (bilateral) Laplace transform F(s) of a mono-
Laplace transform:
+ OO
F (-) - E (") - ſcº. are
-"OO
while the corresponding Fourier transform, F (w), becomes Fourier trans-
form:
+ 90
IE (e-joº) -ſ e” . ar(t)
IF (a)
In general, for monotone functions, the above integral transform
functions all exist in the mathematical sense, and uniquely characterize
the given function. It is readily understood that all physically realizable
dynamic monotone responses must satisfy the condition.
F(t) = o for t < o
since otherwise an effect would occur in the absence of cause, which is a
Situation not normally encountered.
Moreover, certain relationships exist between the above transforms,
For example, if in the equation above, the exponential e-st is expanded in
an infinite power series,
OO k
1 2 2 - S k
- *- : - - - - - - -2. #– t
and the result is integrated term-by-term, there results the series expansion
for the general monotone operator, namely:
OO k
F (-) - ¥ ºf A
k=o •
part XVI 2.75l CLASS NOTES 283
Thus, the Laplace transform of any monotone distribution function is very
simply expressed as an alternating power series whose coefficients are
directly related to the moments about the origin of its original (or time)
distribution.
In the same way, we may also write the Fourier transform of the
distribution in terms of a power series which may be found simply by placing
s = j (U in the expansion above to obtain:
k
Q al.(-ja)
IF (0) = X →-
k=O
If we separate this series into its real and imaginary parts, there results
3. 8,
Re F(a) = a, - = w # , " - ...
a. 8.
- Im F (a) = e, w - #w #w *- e & P
Which means that the real and imaginary components of the frequency response
of any monotone process are given by simple alternating power series ex-
pansions of even and odd powers, whose coefficients are directly proportional
to the corresponding moments of the step response. This fact offers one
possibility for determining the moments, and therefore the transient response
characteristics directly from observed frequency response data.
28l. 2.75l. CLASS NOTES
* * * * * * * * *ms - -
The operational sum of a sequence of monotone responses is itself a
monotone response. Such a Situation arises whenever two or more monotone
processes are placed in parallel as indicated.

º F. ->
o—— : -º-O
-º- IF, Cº-
This situation is expressed by the operational equation:
Il
F(s) : F + F + ... + F = 2F ( )
Therefore the moments, *kp? of the resultant distribution are given by:
Il
* - 2 "a
m=1
ºne ºne tº ºms º ºs ºn tº amº &m ame tº ºne tº ºn eam tº emm sº ºn ºm º ºs º ºsº tº em ºn tº ºms
The operational product of a sequence of monotone responses
is itself a monotone response. Such a situation occurs whenever two or
more monotone processes are put in tandem or cascade as indicated.


O-º- F. - e Gº e o—— IF Hº-o
Part XVI 2.75l CLASS NOTES 285
This circumstance is represented by the operational equation:
FG) - F - F. .... F, - it. F(.)
m=I
The logarithms of the F. will, however, add, in the form:
Il
log F. > X. log IF
Iſl
m=1
The transmission operator, log IF , corresponding to any monotone IF , can
also be expanded in a power series of the form
CD c
log F := X. # (- s)*
k=o
where the coefficients c, are called (by statisticians) the cumulants
or semi-invariants of the distribution F(t).
Thus, for any cascade of n monotones, the cumulants are additive
in the form
The moments *k and corresponding cumulants c, are related by the
k
identity: -
k-1 -
*k = Cl, + X. (::) *mºk-m
m=I
Since c. has the dimensions of *k and therefore **, it is conven-
k
ient and significant to define the following set of constants:
Attenuation 8 = ln °o
Mean Delay T = C
Iſl 1 1/2
Dispersion Time T. = (c 2)
Skew Time Ta = (e)'ſ
Excess Time Te = (...)"
286 2.75l. CLASS NOTES
In terms of these new constants any monotone process may be
characterized by the transform:
1 2 2 1 3_3 1 l, l;
8 - T.s +-- T.’s - -ā- Taºs 4 == T, sº - ...
F(s) = e
In terms of the frequency response, this expansion demonstrates that
the amplitude depends only on the even powers of Cu5 and the phase only on
the odd powers since
Gain
1 2 - 2 1 l; l;
8- + T ºw: * + T "w" - ...
log |F|
Phase
1 3 tº- tº e º
- Tº a + -ā-Taº
Z IF
Thus, the description of a monotone process is unique only if an
infinite set of parameters is specified. However, any monotone may be
approximated with increasing accuracy by matching an increasing number of
the cumulants of the actual process by their counterparts in a model.
One may also profitably introduce into dynamic context, the conven-
tional statistical dimensionless coefficients:
Coefficient of Variance Al Tºſºn
T.3/T 3
8, ' S
l;
Coefficient of Skew O.
E
Coefficient of Excess B º,"/.
Part XVII 2.75l CLASS NOTES 287
XVII. Energy Modulation and Amplification
A. General Three-Port Elements
The causal relationships between the three inputs and the
three outputs of a general three-port element are of the form:
Y, (t) = Y, [x, (t) x_(t) x,(t)]
Yºſt) = Y, [x, (t) Xe(t) x,(t)]
Y,(t) = Y, [x, (t) Xe(t) X,(t)]
which would be diagrammed as follows:
For static behavior the NZ is reduce to Ó 's, while for
linearized dynamic response the N/ 's partition in the form:
Y, - Fº X, + IF2 . x2 + IFs x,
Ye = IF2, X, + IF a . x2 + IF, x,
x, - Fe, x + F = x2 + F = x,
Which is the particularization for 3-ports of the linear relation:
Y = A : X

288 2.75l CLASS NOTE:
B. Generalized Power Modulators as Three-Port Elements
Perhaps the most significant single development of the twentieth
century has been the sharpening of the concepts and practice concerned with
the modulation and amplification of power and signals. The primitive
elements required for all such transformations involve significant energy
transfer at a minimum of three ports. Thus the three-port element serves
as the prototype generalized modulator.
We are here concerned with modulators in all media (i.e., electrical
fluid, mechanical, thermal, and so forth), but it is important to emphasize
that for a three-port to be considered as a true power modulator it will
generally have the bond pattern indicated:
Medium B
M di
edium Medium
A M&D A
\ Same ____*
Medium
While it is often possible to have Medium B = Medium A, one would not usually
consider three-ports involving three different media as true modulators.
Some typical species and realizations are indicated in the morphological
matrix of Table I.
Thus the normal power flow for a modulator would appear as follows:
al. T
LOAD
MOD
POWER wº
SOURCE | 1:... £, “I H
LO POWER
CONTROL
Moreover, it is very convenient to have a canonical ordering of the ports
as follows:
1 — MOD — 2
3

Part XVII
2.75l CLASS NOTES 289
Table I
SPECIES OF MODULATORS
A Morphological Matrix of Certain Realizations
Principal Modulating Medium B
Medium
A Fluid Electrical Mechanical Thermal
Pneumatic Electrically WALWES
Or
Fluid Hydraulic Operated OF
Operated Valves ALL
Valves MHD TYPES
T)evices -
WACUUM TUBES | SWITCHES Bimetal
TRANSISTORS RELAYS Switches
Electrical Saturable SUPER-
Reactors CONDUCTIVE
RELAYS
Modulated Magnetic CIUTCHES
Mechanical FIUID Clutch DIFFERENTIALS
COUPLING Variable
Speed
Drives
BISURFACE
Thermal MODULATED
CONVECTOR

290 2.75l CLASS NOTES
for Which under normal steady conditions the flow of power would be
1 −-2 with the inequalities
P, e P, x > P.
Two very broad classes of modulators exist according to whether
the above weak inequality becomes a weak equality or strong inequality.
These may be considered as follows:

WEAK EQUALITY STRONG INEQUALITY
fºL/
STRUCTURAL or PARAMETRIC DISSIPATIVE or THROTTLING
MODULATORS MODULATORS
Modulate by Modulate by
TRANSFORMER ACTION CONVERTING !”-- Pd
HIGH EFFICIENCY LOW EFFICIENCY
Prototype Example: Prototype Example:
NEEDLE WALWE GATE WALWE
(T\ ſ _- E
©===X==> E @HHBX--> -
| rº-
Henceforth we shall find it convenient for visualization
purposes to take the fluid valve as a prototype modulator and the gate
valve as the most representative type.
C. Generalized Amplifiers
Any modulator can be used as an amplifier by the elementary
Part XVII
2.75l CLASS NOTES
291
transposition of configuration indicated:
/
1 ... 2
Hi Pá –- Moń – F MedIP
sº . . ."
ROTATING o, 2’
^
about ye Lo IP
the
13-Axis
N a'-
IP 1 *=3 ... ".3-2 IP
Lo -: ~A- F Med
PRODUCES: :* -
*::::
& HiP
2. Source
This is readily seen in terms of the gate valve:
! !
1 MOD —” "— AMP — ”
| 31
1 2 21
—-| =2,35E |—- ->
- * Y —
3
The modulator (MOD) and the amplifier (AMP) can thus be con-
Sidered as perturbations of the same primitive device, merely resulting
from interchange of supply and control ports.
Frequently an amplifier is viewed as an active (i.e., IlCIl-

292 2.75l CLASS NOTES
reciprocal) two-port in the fashion:
\
2-
TH F-––––.
M
| D | |
| | | | |
i is | Es
––––– –––––
Where "ES -" represents an energy source.
Thus a cascade or chain of amplifiers in any medium could be
represented by the bond diagram:
tº C, @ :AM TP :AM: ITP- 2AM. tº º &
where —TP– represents any two-port coupling system.
Similarly the use of EFFORT FEEDBACK or FLOW FEEDBACK around
FB– for the two-
any amplifier can be indicated as follows, using
port feedback elements:
EFFORT FEEDBACK FLOW FEEDBACK
(Current, Motion, Flow, etc.)
Hº-
— O — AM— O—
(Voltage, Force, Pressure, etc.)
Hin-
— 1 — AM — O —
These cases will be discussed further below.
Part XVII 2.75l. CLASS NOTES 293
D. The Trinode as a Three-Port Element
The following relations are true for any three-port:
e e2 f, - Y, (e., e, e.) • . . I
1 1 s -1 2” 3
f f2
THT fe = Yeſe, ea, eº) ... • II
e e2,
3
f f3 - Yºſe,
e.) . . . III
3 3
But frequently in addition there exist additional constraints upon the
effort and flow variables.
For example, if all ports connect the same medium, it is
reasonable to expect that the continuity equation will hold for the
three flows, namely, for positive inward flows:
f + f2 + fº = O . . . IV
Moreover, any three-port which depends only upon relative or differential
efforts will obey the condition for any constant effort, E:
N/ ((e. + E), (ea + E), (es + E)) = Y (e., e2, e3) . . . W
for all three flow functionals.
Such a three-port we shall find useful to recognize and denote
as a trinode. Particularly noteworthy are the particular instances of
Vacuum triodes and transistors.
If the trinode be linear, we could write a linear admittance
matrix in the form:
Tf — |- ! f — ſe. T
1 | | 11:12; 13 | | 1
|---
f Y ły #y e
23|_| 21:22; .23 |. .3
; !
is,
t
{
291. 2.751 CLASS NOTES
But the continuity condition (IV) further requires that the sum of each
X. Y: tº tº º º (I )
while the datum relativity condition (V) requires that each row sum vanish:
:* -o ... (Vt)
These results were first pointed out by SHEKEL and mean that if
we consider each of the constraints upon the general three-port matrix, we
would find the following:
GENERAL Three-port 9 Elements
Less Continuity –3
Less Relativity –2
ACTIVE Trinode l; Elements
Less Reciprocity -1
PASSIVE Trinode 3 Elements
Less Symmetry –2
UNIFORM Trinode (flow Jet.) 1 Element
Thus it is clear then that the principal difference between, on the one
hand, a linear active trinode, with internal energy sources, and, on the
other hand, passive elements, lies in the failure of the former element
to satisfy the reciprocity conditions.
Trinode Amplifiers
lº, º ºs º ºs ºs tº $º tº º tºº, º ºsmº tº º sº tº twº
The practical significance of the active or nonreciprocal trinode
modulating element lies in its value as an amplifier or relay. This will
generally be of the two-port form:
— AM —
Ol": — XX. —
Part XVII 2.75l. CLASS NOTES 295
which may be realized from any active trinode by inserting the trinode into
the junction structure:

This is, of course, merely a generalization of the ordinary electric circuit
configuration:


- O
O- TV-y *O T-O
We may then derive the various configurational permutations in an entirely
Systematic way as shown below.
* * * *m' was ºn tº an ºm m - was mº m men wºn tº me = * * * * * * * * * * * * *
The admittance matrix for a generalized linear trinode would have
the form:
8, To C
e
8, eb ; 8.
*S _^* *aa i *ab i *ac
}* * * * * * * * * * * * * * * * F------ *
TRI -: ; ; b
* * | *...*.i.as
gºes tº ºm º ºs tº f------- * was m tº ºne ºn tº
! !
Y : Y ! Y C
e f ca : "cb TCC

296 2.75l. CLASS NOTES
If we consider Port-a as the low power input, there exist three
amplifier configurations all of the general form:
Lo IP 91 9 * *- : =2 . Hi IP
Input Tf - f, Output
1
1 | º | 2
- O -
e
O
The effort, eo is common to both sides of the amplifier
( —X >
). The permutations are as indicated:
Common-a Common-b Common-g
Port X. : C 8,
Port y : b C
Port z : 8, To C
All three of the two-port amplifier configurations may now be
determined from the general results:
Two-Port, *xx ! Y
ADMITTANCE : Y = #-j-ž
:
Y
Y
Y
Y
Determinant : Axy
Two-Port 1 *yy: 1
Matrix Xy 3.A.: "º
The corresponding matrices and matrices for each particular
case are as follows:
CONFIGURATION: . Common-a ! Common-b ! Common-c
- * * ! !
STRUCTURE; & c *zrb ! a szc f aszz-b
§ 8. ! b ! C
* 1 ! ?
Y : . lºssíº.. *#23 ! *sejº
f;---| | | -s;--??--- I - I -;-- f;---
º *befºb *ca; CC *bajºub
& ! !
tº *bb; 1 ! -1 *ce; 1 f 1 *bb 1 1
f ! gººmsºmº -----|--|--
Part XVII 2.75l. CLASS NOTES 297
(Neglecting Interelectrode Capacitance)
This commonly used trinode element has the following standard
symbol and linear admittance:
PLATE 9. D k
p { =º as and sº tº as us ------f
ſ 1.
() : 0 0 :
GRIP / -- \ 1---------- 4---------- sº ... }
1.
t
g : º 1.
Y | Ém ; ëp (8, t ër);
ſ ſ * * :
k ; + g :
* -g g
CATHODE -- P * * *
1---------------------------------- !


Plate Conductance
Where ëp = 1/r
Ém *Hép Grid - Plate Transconductance
* * * *ms ºs ºm º ºs tº º ºs º ºs ºn tº sºme tº sºme ºs º ºs ºs tº ºne tº ºr tº * * * * * * * * * * * * * * * * * * * *
(Neglecting all but first order effects)
This commonly used element has the following symbol and approx-
imate admittance:
e C b
EMITTER COLLECTOR ; ------------,
ºi ; O -a, ;
€ C ; l ;
tº º ºs ºn tº ºne ºn tº º ºsmº erºs * --- - - - - - - - - - - - - - - - - - - - - - ?
! f
Y == - O. g ; 8 + O. g., - g ;
b i ; O i o
! º
BASE ! —- - - - - - - - . . . . . ;
! º
*( ºr - * ſº * * * !
(a - 1)s, E. : (1 -øerts,
!---------------------------------- ;
where a = a (D) = Current Amplification Factor
ëi. = Input Conductance ~ &b
go = output Conductance ~ go - 0
g


298 2.75l CLASS NOTES
sº sº º ºs * * * * * * * * * * * * * * * * * * * *
A useful measure of the "activeness" or activity of a non-
reciprocal two-port derives from the determinant of the governing trans-
mission matrix. From the previous results We have:
A = }</.
If the element or system is reciprocal
Y = Y
Xy yx
and A
Since, physically, *xy Iſled, Slli'e S of ſ de, and *yx : ofy d ex, an amplifie:
with near-infinite power gain would have
E
1
A —- O
This is true for the idealized triodes and transistors usually considered.
Then if we measure the activity of a two-port system as the unity
complement of the absolute value of the determinant we have, using the
Activity N. = 1 -|A|
Hebrew letter aleph:

2art XVII 2.75l. CLASS NOTES 299
E. Cascading and Feedback of Amplifiers
- -, * * * = * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
If we connect across an amplifier ( — » X— ) another two-
port element, using either O or 1 junctions (indicated by J), we arrive
at the commonly encountered feedback scheme:
FE
— J —X > — J —
The two-port " — FE — " indicates the subsystem of particular feedback
elements employed. The four particularizations of the energy junction
situation may be denoted as follows (with typical electrical usage also
indicated).
| FE | FE
— O X > - Q — — O - >> - 1
Flow-Flow Feedback Flow-Effort Feedback
(SHUNT-SHUNT) (SHUNT-SERIES)
Y - Y, Y, G= G, G,
H FE – H FE –
— 1 - > X • O — 1 - > X • 1 —
Effort-I'l OW Feedback Effort-Effort Feedback
(SERIES-SHUNT) (SERIES-SERIES)
H= H, HI, Z = Za + Z F
*** - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Consider a chain of amplifiers with the reticulation:
[ — AC — ) = [ AM CN ]" = ( — As — ) *
Where I — CN — 1 is the interstage coupling network. If the elements are
3OO
2.75l CLASS NOTES
all linear, each stage may be represented in the form:
amplifier as an example.
[ – AS — :- [ — AM – ) e [ — CN – )
As Hs. > Aa Ba. As Hs.
Cs IDs C a Da Ce D.
We might take the vacuum tube common-cathode reactively coupled
Here the single stage, – AS — , is given by:
CN
— AM — o

H
H
O-6 -O- O
f 2 + TD; R
-/*#/A ||†††
F------ • I - - - - - - f------
O O 1/R 1 .
! !
r r "
2 LP: - R —P.
I.i.5, HäIII: TD). Tº
!
! O
|- ; .
The voltage amplification ratio for a chain of "n" such
amplifiers is therefore:
E_(t) Il Il Il
# - (A)" - (-k) (HH-)
pu /(2 + f)
(1 + f)/(2 + f)
where K
k
f
riſk R
Part XVII 2.75l. CLASS NOTES 3O1
-, * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
A very useful and significant system configuration results when-
ever an ultrahigh gain, low-pass, inverting amplifier is employed with
additional input and feedback elements, according to the scheme:
FE
H.--
— IE — O — FA — O —
(x) (y)
Where: FA : Feedback Amplifier
FE : Additional Feedback Elements
IE : Input Elements
The admittance matrices of (– FE –) and (– FA –) will add,
since they are coupled "flow-to-flow."
! !
! y Y
Y_ = |--|--|--|- Y - |-º-º:
FA : FE y: yy
1.
*x : *xy
dº * = Y +- Y 2: #-F#-ºº--
amp FA FE yx ; Tyy
!
D
-
y
* =
*
tº
i
3O2 2.75l CLASS NOTES
the overall system matrix can be determined as
M = M : MI amp
- A : #IBI O ; O
tº º ºsº lº ºf fi--- -:-----5--
(Ci ; l Xy :
f
- ºx...i......g.....
º Y ! O
The overall voltage transfer ratio is then, simply
E2/E1 = 1/Bºx, * * zy/B,
If we take the very simple configuration:
then the resulting transfer ratio is

B2/E = - Ze/Z,
This structure, developed about twenty years ago in connection
With gun directors and fire control apparatus, is the primitive element
underlying the contemporary electronic analog computing machine or
"electronic differential analyzer". The amplifiers (— FA —) developed
for such use are referred to as d. c. computing amplifiers or operational
amplifiers. Standardized input and feedback circuits permit the realizatiº
of scaling, summing, integrating, and other operations, as indicated in the
Voluminous literature on modern electronic analog computers.

Part XVII 2.751 CLASS NOTES 3O3
asseround Reading -- Matrix Methods for Active Systems
(1) ABBOTT, W. R. : Analysis of Four-Terminal Networks Containing Vacuum
Tubes, Misc. Paper H6-2011, AIEE (September, 1946).
(2) PETERSON, L. C. : Equivalent Circuits of Linear Active Four-Terminal
Networks, The Bell Systems Technical Journal, Vol. XXVII, No. 4
pp. 593-622 (october, T58).
(3) BROWN, J. S. and BENNETT, F. D. : The Application of Matrices to
Vacuum-Tube Circuits, Proc. IRE, Vol. 36, pp. 811-852 (1918)
(l) EPSTEIN, H. : Solution of Transients in Active Four-Terminal Networks,
. J. Franklin Institute, Vol. 251, pp. 607-616 (1951)
(5) HSU, H. : On Transformations of Linear Active Networks with Applications
at Ultra-High Frequencies, Proc. IRE, Vol. 1:1, pp. 59-67 (1953)
The five papers above were principally responsible for the introduc-
tion of 2-port matrix techniques to the design of vacuum-tube and
transistor circuits.
(6) MIDDIEBROOK, R. D. : An Introduction to Junction Transistor Theory (1957)
(7) SHEA, R. F., Editor: Transistor Circuit Engineering (1957)
(8) ------------------- : Principles of Transistor Circuits (1953)
These three books amply testify to the value of linear and nonlinear
2-port concepts in the design and applications of solid state amplifiers.
(9) WEBER, Ernst: Linear Transient Analysis, Vol. II (1956)
An excellent Summary of much of the material in the above sources.
Iliſiii.
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