-
l J' i ...~' :n~it'f. l,:qttf:~AX.Je r:~L ~'(l. ~IJ I' , .. .11.
C. i .: 'd . I, . '.' I r lJ. , IiIBUOH CA ..
, I,,)
jt/ ) ELECTROMECHA~CAL
;'.' ....., ENERGY. Inll" FREDDY PUn. .. ( ; CIV. 8620 Aviam
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AniSlant Profeuor of Electrical Engineering
Profeuor of Electrical Engineering V"
The Monachuse"s Institute of Technology
Deportment of Electricol Engineering
HERBERT H. WOODSON
/...
DAVID C. WHITE
~
,/ ;,',j
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-
, h.,;;.. ...... ....., t~"I.c.u~-'" .....-- rl',I'lIAI'.
B1BLlOTECA
Oonado port F 0 R E W 0 R D
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"
This book is one of several resulting from a recent reVISIon of
the 1Electrical Engineering Course at The Massachusetts Institute
of Tech::/l'"
nology. Thc books have the general format of texts and are being
used as such. However, they might well be described as reports on a
research program aimed at the evolution of an undergraduate core
curriculum in Electrical Engineering that will form a basis for a
continuing career
.. in a field that is ever-changing. The development of an
educational program in Electrical Engineering ,
to keep pace with the changes in technology is not a new
endeavor at ,
" The Massachusetts Institute of Technology. In the early
1930's, the Faculty of the Department undertook a major review and
reassessment of its program. By 1940, a series of new courses had
been evolved,
("
.... and resulted in the publication of four rebted hooks,
The new technology that appeared during World War II brought
great change to the field of Electrical Engincering. In recognition
of this ... "\ fact, the Faculty of the Department undertook
another reassessment of its program. By about 1952, a pattern for a
curriculum had been evolved and its implementation was initiated
with a high degree ofTS; ", ;~! J"4 -".,'~
enthusiasm and vigor. ~'f)."- ~;;j t.t'
The new curriculum subordinates option structures built around
areas of industrial practice in favor of a common core that
provides a broad
t "('tiM!" base for the engineering applications of the
sciences. This core structure
11, includes a newly developed laboratory program which stresses
thl: role of experimentation and its relation to theoretical
model-making in the
.))'
solution of engineering problems. Faced with the time limitation
of a four-year program for the Bachelor's degree, the entire core
curriculum gives priority to basic principles and methods of
analysis rather than ~~'"
to the presentation of current technology. J. A. STRATTON
yii
-
i
j.
~"
" Ii P R E F A C E" ';;;
~
-
'Jill'
PREFACEx J' sound base on which to build an understanding of
electromechanical interactions. Although such an approach is useful
for solving some pre~cnt-day problems. it does not give adequate
accent to the more general problem of electromechanical
interactions.
This book has been strongly influenced by three sources. The
initial impetus and major driving force was the formulation of a
required, onesemester. senior course in electromechanical energy
conversion that would treat in some depth fundamental concepts and
at the same time treat specific transducers in sufficient
generality that physical insight into transducer dynamics could be
obtained. It was felt that with the everincreasing use of more
accurate feedback control systems, the engineer must be aware of
the dynamic properties of the transducers that he is substituting
for human muscle. The second source of information was research
sponsored by the U. S. Air Force in which the dynamic behavior of
aircraft generating systems was studied. The third source was
graduate teaching and thesis research.
Since electromechanical energy conversion can occur only by the
interaction of electromagnetic fields and material bodies in
motion, it is reasonable to begin with a treatment of macroscopic
electrodynamics. We have experimented with this starting point in
our senior course and find that a student who masters this topic
has excellent preparation for proceeding to more advanced work. But
we also have found that for the average student the time required
for an adequate coverage of electrodynamics leaves little time for
a treatment of lumped-parameter systems to the degree necessary if
dynamic behavior is to be studied. Furthermore, it is in general
impractical to describe the general dynamic behavior of energy
conversion systems only in terms of fields. These.. considerations,
plus the presence in our curriculum of a junior course in fields,
energy, and forces which introduces electrodynamics, prompted us to
start with a lumped-parameter approach with some references to
field theory as the basis for determining when lumped parameters
can be used and in defining the system parameters.
Having chosen the lumped-parameter approach, we chose to make
the derivation of equations of motion in Chapter 1 quite general,
the principal limitation being that the coupling system (Le.,
energy storage in fields) be conservative. Two approaches have been
used: the method of arbitrary displacement and conservation of
energy, and Hamilton's principle leading to Lagrange's equations.
The first approach yields only electromechanical coupling terms;
consequently, the equations of motion of a system must be written
using force laws (Kirchhoff's laws and d'Alembert's principle), and
much bookkeeping must be done in
PREFACE xi
complicated problems. The Lagrangian formulation, on the other
hand, can be generalized to include the nonconservative parts or
the system, so all bookkeeping is done automatically. Some
educators may feel that the use of the Lagrangian formulation may
tend to make crank-turners out of the students; however, we feel
that the Lagrangian has further significance than just being a tool
for solving problems. It introduces students to variational
principles which are in one sense as fundamental as the
conservation principles. The Lagrangian formulation also introduces
generalized coordinates and lays a firm foundation for the meaning
of independent variables. The Lagrangian state function and the
Legendre transformation provide an entry into state functions in
general and a tie with state functions commonly used in
thermodynamics. We feel that these advantages ot" the Lagrangian
formulation make its introduction well worthwhile. This is in line
with the desire to broaden the scope of the present treatment and
to lay a foundation which in the future can be expanded to include
systems other than those with only electrical and mechanical in
teractions.
Chapter 2 treats mathematical techniques for solving equations
of motion, starting with the straightforward classical solution of
linear differential equations with constant coefficients and ending
with the use of analog computers to solve nonlinear equations. As
is obvious from the content, the purpose of this chapter is not to
teach mathematics but to illustrate the use of known techniques in
analyzing typical transducers. No attempt has been made to include
all the known methods for analyzing linear equations. Notable
omissions are electrical analogs, flow graphs, and operational
tcchniques other than Laplace transforms. We feel that the
techniques used are representative. No examples of the use of
digital computers are included. However, we recognize the
increasing importance of digital computation in system analysis and
the general equations derived for electromechanical energy
convertors are directly applicable to digital computation.
Chapters 3 and 4 represent an errort to integrate the earlier
work t,,: of Kron, Gibbs, and others into a unified treatment of
the dynamics of
I' !;, ~ '~.' rotating machines. By defining a general physical
model consisting
~ 7, 'l)' of concentric magnetic cylinders with current sheets
on their surfaces, {,\ (. a field solution is obtained, parameters
are defined, and equations of
motion are derived. Starting from a single model, the complete
dynamic I~ . I '. equations of most conventional machines and many
unconventional ones
;~i; are obtained by selected constraints to the general model.
Transforma '... tions are introduced that mathematically describe
the physical change of variables made by a commutator and put the
equations of motion in
I~ ~ ; ~ ~ \,~, ; I",; ;, ., ..
_-----oLL
-
xiii xii PREFACE
PREFACE
IJ.'. ,ii,
'" ' ,,':!i.
'. ~
.. .~.;",I},~,;:,
'i
".
more recognizahle and solvable forms. Thus one physical model
with one set of equations dcsnibes all of conventional steady-state
machine theory, the dynamic behavior of conventional machines, and
the steadystate and dynamic characteristics of many unconventional
machines.
The usc uf the generalized approach with the aid uf a Iahoratory
machine having the same generality reduces steady-state machine
theory to the solution of steady-state a-c and d-c circuit prohlems
with which our students arc fully familiar. Furthermore, the
general field solution allows interpret:1tion of energy conversion
properties in terms of fields. Since so little time is required for
this aspect of machine theory, much time remains for study of the
dynamic properties which are so important in the engineering of
today and which will become more important in the future.
There may he some question about the choice of a two-phase
machine as the deviL'e for teaching machine theory. It is shown in
Chapter 10 that the energy conversion characteristics of any
polyphase machine with symmetrical impedances can be obtained from
an equivalent two[,hase machine for which the equations of motion
arc ohtained by a str:llghtforward change of variables. We feel
that the mathematical cUll1pkxily of L'ven a two phase systcm tends
to ohscurc many of the concepts contained in the treatment. The use
of a three-phase o;ystem as the analytical vehicle only introduces
additional mathematical complexities and further detracts from the
understanding of the nature of energy conversion in rotating
machines.
Chapter 5 gives a simple introduction to feedback control system
theory. We feel that such an inclusion is desirable because those
students who do not take such a course as an elective should
nevertheless be exposed to the essential ideas involved. In
addition, this chapter illustrates through an example how the
dynamic behavior of an interconnected system of machines is
affected by the characteristics of the machines.
Chapters I through 5 contain the material prepared for a
first-term senior course. More recently, it is being taught as a
second-term junior,; course. The classroom work is augmented by a
laboratory in which transducers and the generalized machine are
used to get the fundamental concepts across. In addition,
commercial machines and transducers are used in experiments that
stress dynamics, interconnected systems, and feedback control
theory.
Chapters 6 through 9 are detailed and specialized treatments of
specific devices. Their purpose is to illustrate techniques and to
provide
t
t
j I !
information on applications of the general approach of the
earlier char: ters. This material is primarily intended for use in
graduate courses and for the use of practicing engineers. It has
also been used as part of a one-semester senior elective subject on
electric machine systems.
Chapters 1() and 1 I are necessary for generality in the
treatment. Chapter IO.provides the analytical methods for reducing
any n-m phase machine to an equivalent two-phase machine for
considerations of dynamic energy c(lnversion properties and thus
justifies the lise of the two-phase model in the carlier treatment.
Chapter 11 shows how the results of the analysis of Chapter 3 with
sinusoidal currcn t shcL:ts Can be extended to analyze a machine
with current sheets (windings) of any arbitrary distrihution. It
also contains a rigorous justification for the concept of analyzing
a commutator by a simple transformation of variables. This one
concept is of course vital to the establishment of the unity among
aIt machine types.
The topics in this book were first presented to a group of
seniors at MIT in the fall of 1954, and the material prepared at
that time was a group efTort. It h:1S sincL: gonc through several
modifications, but lllany of the basic ideas around which this text
is formed were estahlished by the initial group, which consisted of
the authors plus Professors MahmOUd Riaz, Robert M. Saunders,
I-Jerman Koenig, and Richard H. Fnlzier. In addition to the initial
group effort, special recognition is due to Professor Mahmoud Riaz
for his contribution to Chapters 4, 6, 8, and 10; to Professors
Leonard A. Gould and Robert M. Saunders for their contribution to
Chapter 5; and to David Bobroff for his contribution to Chapter 1.
Others who have made many contributions to the effort have been
Professors Richard B. Adler, Lan J. Chu, David J. Epstein, _ Robert
M. Fano, Charles Kingsley, Jr., Alexander Kusko, Osman K. Mawardi,
Norman H. Meyers, and Karl L. Wildes. Special mention should also
be made of the very excellent works of Drs. Gabriel Kron and W. J.
Gibbs, whose very fine publications in this field were an
invaluable aid to the development of many of the concepts presented
here.
The authors are also greatly indebted to Bernard Lovell, whose
diligence in editing and checking derivations was invaluable. We
wish to. thank also the Misses Lucia Hunt, Ruth Coughlan, Lydia
Bonazzoli, and Evelyn Fraccastoro for their perseverance in typing
the drafts of the manuscript.
Special mention is due our wives Glorianna G. White and Blanche
S. Woodson for the invaluable assistance, both tangible and
intangible, given during the writing of this text.
Above all, the authors wish to express their appreciation for
the
-
.. , ifV"\1'
xiv PREFACE
dynamic leadership and stimulation provided hy Dr. Gordon S.
Brown, c o N T E N T s
without whme foresight, perseverance, and courage this book
would not
have been possible. DAVID C. WHITE HERBERT H. WOODSON
Cambridge, Mass. November, 1958
n;
l.;
11~
The Dynamic Equations of Motion of Electromechanical,riU Chapter
1 Systems
2 Analytical Techniques for Treating Electromechanical Equa
tions of Motion Including Typical Transducers as Examples 87
. 3 The Generalized, Magnetic Field Type, Rotating, Electro.n I'
../ . mechanical Energy Converter 170tt
4 Two-Phase Transformations and the Generalized Machine 254
';of +t 5 Fundamentals of System Dynamics 360
f" 6 Dynamics of Transducers 395
''l
7 Dynamics of Commutator Machines 421,~~l:, ) ~~
"",,) . OU1; ~i III .~1~:k}~~i 8 Dynamics of Induction Machines
477,.~~ "'I', ,r, (1 ~';, 1t1 .llj'I, .... ! 9 Dynamics of
Synchronous Machines 508
'-:1:! ~.~')".1: -'
.~.~
);f 10 Generalized Analysis of the nm Winding Machine 545/
'~:;:'
\rf!~r"'! .j. 2; t Space Harmonic Analysis in Machines,Ii; 11
603 ~i ~. '~ Index 639
;,;;1 I:l '1'1 i
~. .~tt4 !:'.m ' ..
'JO"IIJ.b
Xy
~
"'-- -
-
pc H A T E R o N E
,
~
The Dynamic Equations of Motion
of Electromechanical Systems
,I.:~:;
1.0 Introduction
'{-; '. Electromechanical energy conversion is the result of
electrouynamic interactions and a rigorous and thorough treatment
requires a st1ldy of moving, charge-carrying material bodies in
electromagnetic fields. However, for quasi-static (low-frequency)
and low-velocity electromechanical systems the dynamic equations of
motion can be formulated to a high degree of accuracy in terms of
lumped electric circuit parameters that arerf. evaluated from
static field solutions.'"
1 The subject of this book is the characterization of
electromechanicalI systems by lumped parameters. This involves
three major problems: (I)
a physical description of the system, (2) derivation of the
differential equa,W tions of motion for the system, and (3)
solution of the equations subject\i
to the specified operating conditions of interest. The physical
description . ~UJ
1,;; of the system entails the establishment of an idealized
model whose
character is determined by the physics of the problem and whose
,+ ~i:.',~ .. ~ completeness depends on the application. The
equations of motion
\ .~'
J are obtained from the idealized model in several ways, some of
which ,:; ~ are covered in the present chapter. The solution of the
equations of
~ '-,'; /, motion is in general difficult because the equations
for any energy confit version system are nonlinear. for some
applications the equations can
I, Ii be linearized (e.g., for small signals); in others a
change of variables
l~' J,,' 1 leads to simplification. In a few cases it is
necessary to solve the * For a discussion of the relationship
between circuits and fields for systems in ~......... relative
motion see R. M. Fano and L. J. Chu, Fields, Enel'!:';'. alld
Forces. John Wiley.
' .. '.. New York, 1959.
\ ~ t ...........j;:~.fIo... _~_ )- -_.
-
ELECTROMECHANICAL ENERGY CONVERSION 2
complete, nonlinear equations, and machine computation
techniques may bc requi red. The types of nonlinearities
encountered and the simplification possible depend upon the
electromechanical systems and their applications.
1.1 The Various Approaches to the Study of the Dynamics of
Electromechanical Systems
The dynamic equations of motion of electromechanical systems can
be . determined from physical laws using either force density from
electromagnetic field theory or an arbitrary displacement and
conservation of energy to obtain the mechanical forces of
electrical origin. Alternatively, the equations can be obtained
from variational principles applied to selected energy functions.
It is difficult to state if either of these approaches is more
basic, particularly since both will lead to the correct results if
properly applied. There is, however, a great difference between
them as to the formality of the analytical techniques employed.
The application of force laws to obtain the equations of motion
is probably the least formal analytically and requires a good deal
of insight and judgment when dealing with complicated systems
containing many variables. The electric coupling terms resulting
from mechanical motion and the equations of motion for the
electrical and mechanical parts of the system are obtained from
known force laws, such as Faraday's law, Coulomb's law, Kirchhoff's
law, and d'Alembert's principle. The method of arbitrary
displacement and conservation of energy is then used to obtain the
mechanical forces of interaction between systems, because it lends
itself easily to the derivation of mechanical force equations
expressed in terms of electric circuit quantities. Alternatively,
these coupling terms can be obtained by integrating force densities
obtained from electro
magnetic field theory. The derivation of the equations of motion
from variational principles
is significantly different from the previous method. First, one
establishes a common terminology for all types of systems, whether
electrical, mechanical, thermal, acoustical, etc., by defining
state functions (energy functions) in terms of sets of generalized
variables. Then by the use of a single fundamental postulate, e.g.,
Hamilton's principle, the equations of motion for all systems
including any coupling terms are determined. The variational
approach is quite formal analytically; and, as a result, insight
into physical processes can be lost in the mathematical procedures.
However, if the method is properly understood, and if adequate
attention is given to the selection of the state function, physical
insight can be gained because of the generality of the method. The
variational method is one
EQUATIONS OF MOTION OF ELECTROMECHANICAL SYSTEMS 3
of the most pnwcrful techniq lies of dynamics; and, .tltllout!h
beginners may tend to become "crank turners," the long-term value
of variational techniques precludes their dismissal merely on the
ground of conceptual difficulties or lack of physical insight when
employed by the novice.
+
1.2 Fundamental Relationships in Electromechanics
j
The fundamental force relationship of statics is that for
equilibrium the summation of all forces acting on a body is zero.
This basic concept was used by d'Alembert who postulated that the
sum of all forces equals zero for dynamic equilibrium of mechanical
systems, * For a multiloop dynamic system d'Alembert's principle
requires that at the kth mechanical nodet
2: r
(A, - fk) = 0 (1-10) I-I
-
,. ELECTROMECHANICAL ENERGY CONVERSION the force equation is
expressed as follows: the sum of all voltage drops around a loop
(kth loop) equals zero,
" ek, =0 (I-Ie)~
i--l
where ek, = the ith voltage in the kth loop. The
continuity-of-charge relationship or continuity of current is
expressed by stating that the sum of all currents into a node
(kth node) must equal zero. $'
1
Lr
ikl = 0 (1- Id) I-I
where iAj = the ith current Howing into the kth node.
D'Alembert's principle, Eq. I-la, the continuity-of-space
relationship, Eg. I-Ih, and Kirchhoff's laws, Egs. I-Ie and d,
express the complete equations of motion for electromechanical
systems providing the mechanical forces of electrical origin are
included in d'Alembert's principle, Eq. I-la, and the electric
voltages and currents used in Kirchhotf's laws, Eqs. I-Ie and d,
include the effects of mechanical motion.
The fundamental laws needed to study the dynamics of
electron'~~hanical
devices from a circuit viewpoint are now complete.
Unfortunately, the mechanical forces of electromechanical coupling,
when expressed in a form easily used for electromagnetic field
problems, are not readily adapted to an equivalent circuit
treatment of connected electrical and mechanical systems. The
direct extension of the macroscopic force equations in terrr.s of
electromagnetic field quantities to force equations in terms of
electric circuit quantities is difficult by integration methods
unless the physical device is quite simple. It proves easier to
derive these coupling terms, using an arbitrary displacement and
the conservation of
energy.
1.1.1 Mechanical Forces of Electromechanical Coupling Derived by
an Arbitrary Displacement and Conservation of Energy
~
The first step in analyzing a complicated electromechanical
system by ail arbitrary displacement* and conservation of energy is
to reduce the system containing electromechanical coupling terms to
a minimum. To do this, separate out all purely electrical parts and
all purely mechanical parts of
~' if,i,'",',,",
I, 'I the system including losses, as shown in Fig. 1-1. This
separation procedure is carried out to the extent that each
electrical terminal pair is For treatments of the principle of
virtual displacement, see Goldstein, loc. cit., or E. T. Whittaker.
Analytical Dynamics, Dover Publications, New York, 1944.
1
EQUATIONS OF MOTION OF ELECTROMECHANICAL SYSl EMS 5
coupled to one energy storage, either electrical or magnetic.
Any internal interconnections between circuits that are coupkd to
different energy storages are included in the external electrical
network. The Illechani,cal variables represented by the mechanical
terminal pairs arc those whi~h
affect energy storage in the ekctric and magnetic fields. Any
purely mechanical couplings between mechanical variables such as
gear trains, springs, etc.. are included in the external mechanical
network.
Compte-t~ fiectromechamCOtI System . _ _ __
r------------------------. I
'.1_"--++ tt flectucal Network' (Ieetflcal ElectrICal
lQualllJlls of motton fromcnput 10 " Input toKirchhoff's
lawssY5tem I coupling fields -
Electromechancal
L-t:
Network (repfesentlng coupling
fre!dc,-electrlC .Illd IT1.I~rlr.ll()
!:.ljUatlOfls 01 Fllohon from conSerVatIon 01 t'nt"I~Y and ,.HI
arblltMy ulspl.lC\.'rlIent
(f,),+ '-
Mechanical Network: + t - I Mechanical f MechanicalEQuatlolis of
moflon fromi:t Impul fo Input 10d'AleflltHHI's prillclple
system coupling fields! II and Conllnl,Jlty of ipace
ji I - I
I IL J
Fig, I-I. Simplification of electromechanical system for
analysis by an arbitrary displacement and conservation of
energy.t
1< (
The separation procedure results in the general conservative
electromechanical coupling network depicted in Fig. 1-2 in which
there are n electrical terminal pairs and m mechanical terminal
pairs. By virtue of the lumped-parameter approach, each electrical
terminal pair will be
il.' coupled to either a magnetic field energy storage or an
electric field
iJ energy storage. To fix ideas, assume that the electrical
terminal pairs I ~ i ~ I are coupled to electric field storage and
terminal pairs, I + I ~ i ~ n are coupled to magnetic field
storage. The total stored' energy W in the coupling network is
given by
W = We + Wm " (1-2)
~ where W, is energy stored in electric fields and W m is energy
stored il) magnetic fields.
t The lumped-parameter representation of the coupling network of
Fig. 1-2 requires the system variables to be functionally related.
For example,
, ..=eM.... . __, .
~
-
--
7 6
r .
ELECTROMECHANICAL ENERGY CONVERSION EQUATIONS OF MOTION OF
ELECTROMECHANICAL SYSTEMS
--" ... +
("1
il; +-...:....-~---
Electrical Inplll~ to U. coupling
fields 0-=--- _
in
+ Un
fj ~
+ ~({')I
XI
fk -...::~----
0-:; ~--- ({.IkMechanical
mputs to X. coupling
fields 0-------
~
+ -- ({.Im %"1
,1
Fig. 1-2. Dcfinition of coupling system for and conservation of
energy.
Electromechanical Network: (representing couplmg fields
-electric and magnetic)
All dissipative elements removed to external circuits; therefore
system is conservative.
.:tR g,~ ~~ r ~q
f10rJ ,,'
'1'''
, "'
1"
analysis by an arbitrary displacement
.~
if the electromcchanical coupling network can be characterized
electrically lincar inductanccs. the kth nux linkage I~ givcn
hy
by.
.:\k = " 2: i =,11 I
Lk;ii (1-30)
where tL.~ inductances L Ai Xb ,xm '
are functions of the mechanical coordinates
LAi == Lki(x" ... ,xm ) (1-3b)
Similarly, for a system of electrically linear capacitances
there would occur
qk = 1
2: i,~ I
Cki/li (I-Jc)
Where the capacitances Cki Xl' .'" Xm ,
are functions of the mechanical coordinates.
Ckl = Cki(X" ... , X",) (1-3d)
In general. the system may be nonlinear; thus a set of
parameters Land C cannot be defined. In such cases only general
functional relationships among the variables can be established,
such as
flux linkage and current of kth inductor: I~
or .:\k = ,\(i/+" ... , i,,; xI> ... , x",) ( 1-4a)
ik = ik("I+I, ... , '\,,; xI> ... , XIII) ( 1-4h)
voltage and charge of kth capacitor:,~!
" ,.!~
or Uk = Uk(qlo ... , ql; X Io .. , X",) (l-4c)
\:! ~ qk = qk(l'lo .. , VI; XIo , X",) ( 1-4d)
"",,'
r"J.
't~
;\.-""
~;
In any case, regardless of whether the relationships between the
varia hies are linear or nonlinear, the relations are restricted to
be single-valucd functions because it is assumed that the energy
stored in the electromechanical coupling fields can be described by
state functions.
The assumption that the stored energy in the electromechanical
coupling network is a state function forces it to bc a
single-valued function of the system variables. independent of the
derivatives and integrals of the variables. Thus. the stored energy
W may bc a function of the instantaneous configuf
-
8 ELECTROMECHANICAL ENERGY CONVERSION
function is a stronger one than the requirement that the
expressions of Eqs. 1-4a-d be si ngle-valued functions. Stored
energy that is a state function always yields singh~-valued
internal constraints; hut single-valued inl\:rnal constraints alone
do not guarantee that the stored energy is a state functillll (see
Sec. 1.4.6).
In sUllllllary. the constraint of requiring the energy functions
to be state functions places the following limits on the
electromechanical system:
I. The lumped parameters evaluated from electromagnetic fields
must be derivable from static (zero-order) fields. *
::. The functional relationships between variables (Eqs. 1-4a-d)
must be single-valued.
3. Hysteresis cannot be included in expressing the functional
dependence of the variables in Eqs. 1-4a-d. This does not mean that
the loss due to hysteresis cannot be represented by a resistive
element located outside the coupling sy.,tem.
Restricting the energy functions which describe the
electromechanical coupling lields to b . , An; i/+1> . , i,,; x"
... , xm;fJ, .. ,fm) (1-5a) j
l W = iq'q,:AJ! 1.,\n;Xtxm '!. '!! 1 ,L . L [v;(ql"'" o, ...
;0;0 0;0... 0 i~ I j~l " ql; Xl' , " Xm) dq, ..t.~.
..~~ .,
+ i;(>\;r\> , A;,; xi, ... , x;,,) dA; ~ ~i~ + f;(q;, ,
q;; 1.;+10' .. , A;,; X;, . .. , x;") dXiJ (1-5b)
where primes denote variables of integration. The energy
function of 'jr
tj Eqs. I-Sa and h is evaluat!:d by performing the line integral
over all the ~
system variables. The constraint of the proper single-valuedness
of the
, functional relationship indicated in Eq. J-5h to yield a state
function allows the evaluation of the line integral by choosing any
convenient path of j
I.;.."i ;~/
integration. Also, because of the interdependencies of the
variables in
Eq. 1-5b. it is clear that of the six variables 'Ii' Vi' ii' Ai'
fj, and xj there
See Fano and Chu, loc. cit.
-
EQUATIONS OF MOTION OF ELECTROMECHANICAL SYSTEMS . 9
can be only three indepcndent variables, the other lhn:!:
variah1l.:s heing required to satisfy the interna~constraint
equations of the form
Vi - 11;(lJ" ... ,q,; Xb ... , x",) (1-611)
il = i,(A/1 1, , An; Xh ... X,.,) ( I -flh)
jj = jj(qh ... , 'II; A'+h' , An; Xh ... , X",) ( 1-6(')
In view of these constraint equations, it follows that the
energy function W can be expressed as a function of three
independent variables. which lor the constraint equations I-oa-('
are 'Ii' ,'1/, and x j yielding :In energy function
W = W(ql"'" ql; 1.'+10 An; Xh ... ,Xm ) (1-6d)
Even though qi' \, and x j have been considered as the
independent variables above, the single-valued relations of Eqs.
l-6a-c can be solved to allow any three of the six variables to be
treated as independent.
In gener,l!, in electromechanical systems it is possible to
evaluate constraint equations 1-6a and h by solving for the
electric and magnetic static field solutions determined by the
physical configuration. The evaluation of the constraint equation
for the force fj is usually much more difficult. Thus it is common
practice to find the constraint relationships for the electrical
variables and to use these to evaluate the stored energy,
utilizin'g the fact that since the energy is a state function the
correct energy function can be obtained by keeping all electrical
variables Ai and qi zero when assembling the mechanical system and
then holding all the displacements x j constant when establishing
the final values of the electrical variables. For this path of
assembly of the system the energy function becomes
W(qh ... ,ql; 1.1+ 10 An; x .. ... xm)
i qlt .. lq,; ),,+l ... An;xll ...xm n = L [v;(q~, ... , q;; Xh
x ) dq;m0.....0;0.....0;,l'1.......... /_1 + i;(A;+1... , A~;
XI> x ) d,\;J (1-7)m
The energy functions given by Eq. 1-7 can then be used to find
the force constraints fj and hence the force of electromechanical
coupling.
The forms of the stored energies and the nature of the network
in Fig. 1-2 indicate that n electrical variables and m mechanical
variables can be specified independently, This can be interpreted
as meaning that the coupling network has (n + m) degrees of
freedom. With the usual interpretation that a degree of freedom
represents one independent energy storage element in a lossless
system, the coupling network appears to
-
10 ELECTROMECHANICAL ENERGY CONVERSION
have (n + m) independent energy storages in spite of the fact
that in Eq. 1-7 energy storage was assigned only to each of the n
electrical terminal pairs. This apparent ambiguity is resolved by
realizing that mechanical energy supplied at the mechanical
terminals goes into either electric or magnetic field storage.
Consequently, in a coupled electromechanical network the electric
and magnetic fields can be interpreted as storing both electrical
and mechanical energy and hence can represent both electrical and
mechanical degrees of freedom.
With the electromechanical coupling network of Fig. 1-2
specified, the terminal characteristics of the network can be
found. According to the discussion above, one variable at each of
the (n + m) terminal pairs can be specified independently. First,
consider the I electrical terminal pairs that are coupled to
electric field storage. When the qj and Xj are specifie.
independently, the current in the ith terminal pair is t
. dqjI = (l-8)
I dt
and the Yoltage Vi at the ith terminal pair is given by the
internal constraint of Eq. 1-6a. Next, consider the (n - I)
electrical terminal pairs that are coupled to magnetic field
storage. When the Aj and xj are specified independently, the
voltage at the ith terminal pair is given by Faraday's law
d\ V, = di (1-9)
and the current il is given by the internal constraint of Eq.
1-6b. Thus, the Yolt-ampere characteristics at the electrical
terminals have been determined so the effects of the electrical
part of the coupling network can be ineluded in the equations for
the external electrical network described in Fig. 1-1. It should be
mentioned that instead of specifying qj for the I electric field
storages and A; for the (n - I) magnetic field storages, the
voltage Vi at the electric field storages and i j at the magnetic
ficld storages could have been considered as independent, in which
case the internal constraints of Eqs. 1-6a and b would be used to
obtain qj and AI for inclusion in Eqs. 1-8 and 1-9.
The next problem is to find the force due to the
electromechanical coupling. Since the m mechanical terminal pairs
are characterized by m independent variables, it is possible to
consider each mechanical terJl1inal pair individually to find the
electromechanical force. Defining the force (le)k shown in in Fig.
1-2 as the force applied to the kth mechanical coordinate (node) by
the electromechanical coupling network, the force (le)k can be
found by considering that an arbitrary displacement
EQUATIONS OF MOTION OF ELECTROMECHANICAL SYSTEMS II
dXk occurs in the kth mechanical coordinate in time dt with all
other mechanical coordinates held fixed, i.e..
dXJ = 0 for j 'I- k
The electrical variables change during the time ill. and these
changes are completely arbitrary except that the internal
con~traints of Eqs, 1-6a and b must be satisfied. This means that
only one electrical variable at each electrical terminal pair can
be changed arbitrarily. During the arbitrary displacement the
conservation of energy must hold. The various energies involved in
the arbitrary displacement are:
energy supplied at electrical terminals = n2: v,i; dt
1=1 (I-lOa)
~ .. ;
energy supplied at mechanical terminals = - (I.h-':k dt = -
(le)k dXk (I-lOb)
change in stored electrical and magnetic
energy of coupling field = dW (I-JOe)
where W is the total stored electrical and magnetic energy of
the coupling fields (Eq. 1-5b)
energy lost in dissipation = 0 (I-IOd)
i
f.!,
(1-11)-(i')k dXk + i V,il dt = dW ;=1
All lossy elements are removed from the network of Fig. 1-2. The
conservation of energy requires that the sum of the input
energy
must equal the change in stored energy, thus, from Eqs. 1-10:
j
.,it.
From this expression the force applied to the kth mechanical
node by the electromechanical coupling field is:
(f,)k = d.i.... ( i viii cit X k ;=1
- dW) ( 1-12)
When the n independent electrical variables and the fit
independent mechanical coordinates are specified, Eq. 1-12 gives
the force applied to the kth mechanical node and the velocity of
the kth mechanical node is dxk/dt = Xk' Thus the force-velocity
characteristics at the mechanical terminal pairs of the
electromechanical network of Fig. 1-2 have been specified. The
force of Eq. 1-12 can be included with d'Alemhert's
it
.. I'
-
12 ELECTROMECHANICAL ENERGY CONVERSION EQUATIONS OF MOTION OF
ELECTROMECHANICAL SYS rEMS 13
principle to write the equilibrium equation for the kth
mechanical node aN energy estahlishes that the energy input from
all sources j, 'l(lrct! as magnetic field energy W",(A" ... , An;
XI> .. , X n,).
ilk - Uk + Umh + (j,)d = 0 (1-13) Wm = input electrical energy +
input mechanical energy (1-14a)
where Pk is the inertia force, fk the externally applied
mechanical force,
and (t;"h the force applied hy mechanical springs. All three of
the or
terms arc considered to be included in the mechanical network of
Fig. 1-l.
J. A, .....Ann
W",(AI>' .. A,,; x., ... , x",) = .~ i;(A;, . .. A:,; x;, .
.. ," ~~ , '.' I.'Eqs. 1-6a and h. Consequently, this force is the
true force regardless of T .,,:- f/Al, ... , A", XI' .. , x m )
d'\j (1-14b)0 .....0 J .. I
t!le external lerminal colIslraints that may he imposed on the
coupling
This is tonetwork hy the external electrical and mechanical
system. for the general case of
emphasize the point that only the inlernal constraints had to be
satisfied II A graphical plot of the total energy Wm'
A;(x;, ... x;"; i;, ... , i;'), where A; is a nonlinear.
single-valued function in the arbitrary displacement.
The mechanical force (f,h of electromechanical coupling contains
II
terms due to two types of energy storage fields, electrical (W,)
and
magnetic (Wm ). At low electrical frequencies and low.
mechanical .
velocities such that the electrical system can be represented by
lumped:'" .~
parameters, all electric energy storage (W,) will be in
capacitances and.
all magnetic energy storage (Wm ) will be in inductances. Thus,
,he two: ~rlca,
J sy~tem
problems of electric and magnetic field coupling can be treated
separately
---
and the results can be combined, with the proper external or
terminal
constraints, to describe a system having both electric and
magnetic field
coupling with a mechanical system. The forces caused by each of
these
two coupling fields will be treated in the next two
sections.f..
1.2.2 Mechanical Force Due to MagnetiC Field Coupling.J:'\'.
'':.'A system of current-carrying coils is represented
schematically in
Fig. 1-30. The only important, or first-order, electromagnetic
coupling
field between various elements of this network for quasi
statics' and for f'
To evaluate thelow-velocity mechanical motion is the magnetic
field.
stored energy in the magnetic field, assume that the final flux
linkage AI
Fig. 1-3a. Coupled current-carrying coils.
in each of the coils in Fig. 1-30 and the final positions of the
coils Xl are
For any givenobtained by any arbitrary paths compatible with the
internal constrain
ts of the displacements and currents, is shown in Fig. l-3b.
( of both the electrical and mechanical variables between zero
and their system, where A; is a single-valued function of the x j
and ii. the total
final values. Assuming all mechanical storage elements and all
non stored magnetic energy Wm is determined uniquely by the
paramete
rs
coupled electrical storage clements plus all dissipative
elements to be XI>' .. , x", and AI,' .. ,A" and only depends
upon the final values of these
The total stored magnetic energy is a state function. Theif ;1
external to the electromechanical coupling field, the conservation
of J parameters.~' amount of the storeL! magnetic energy which is
supplied by electrical,:,f:}:~
.
For alternate treatments see E. Fitzgerald and C. Kingsley,
Electric Mach
ines, sources and that which is supplied by mechanical sources
depend upo
n McGraw-Hili. New York, 1952; R. E. Doherty and R. H. Park,
"Mech
anical Force how the system is assembled in reaching its final
state. The energ
y j- between Electric Circuits," Trans. AlEE, Vol. 45, 1926, pp.
240-252.
...~)
.1.....,...;;
j...I.."
~.
-
14 ELECTROMECHANICAL ENERGY CONVERSION EQUATIONS OF MOTION OF
ELECTROMECHANICAL SYSTEMS 15
'I
i-z ii
(Stored magnetic energy~i"" i~d)"~
, 0
~~~~
Magnetic coenergy.
I i" A'di'o n A
i~t 71
Fig. I-lb. Path of operation to reach final energy when
mechanical coordinates are held at xi = XI' xi = x 2 , , X~ = x m
while the electrical variables are simultaneously brought to their
final values.
",I
&1 ij
X2(2:i..:c,;;;i{. .i~)
i2 ii
-.. ,. f.
ill i~
Fig. I-le. Path of operation to reach final energy when
mechanical coordinates are held at xi = XI' xi = xi, x~ = x;. while
the electrical linkages >'1 are brought to their final values,
then the mechanical coordinates are brought to their final
values.
j
supplied from ekctrical sources and mechanical sources 1',)1' a
particular path of asst:J1lbly or Ihc system shown in Fig. 1-.111
i, drawn in hg. I-.~c. From these ligures it should be clear thal
it is possible to arbltral'lly supply any portion of the total
energy from either source by the prllper choice of path used to
assemble the system. For example, hold all !lux linkages at zero
and mechanically assemble the system, and then establish the flux
linkages with the mechanical coordinates held at their final
values. The energy stored in the magnetic field for the given
values of flux linkage will have LV be supplied by electrical
sources. For this case, the stored magnetic energy is
Wm(AI> ... , An; XI . , xm)
"I .... '''" n=2: i;(/,;, ... , A~; XI, ... , x ) dA; (1-15)
mJ0, ....0 ,-I 1 where Wm is evaluated as the integral of idA for
any fixed spacing, i.e.,
,~! all Xl are constants.
..1t2 EXAMPLE lEI
As an example of the method of evaluation of the integral of Eq.
1-14b. consider the case in which
n=3 '$' " and the system is electrically linear '1
il = TllAl + T 12AZ + T13A3 i2 = TZ1A I + Tn)o.z + T 23 A3
(lEI-I)
-"I i3 = T 31 A1 + TnAz + T33 A3 ~'" /. ~ \1~)with the T's
functions of the xj only and
T IZ = T2b T13 = r3l> T Z3 = Tn f)J,
The meaning of Eq. 1-14b :1
J"I'''2'''3 3 I.'
Wm = 2: i;(A;, A;, Ai; Xl> , X ) dA; (IEI-2)m0,0.0 i=1
is that with the xj held constant each flux linkage is brought
to its final value holding all other flux linkages fixed. The order
in which the flux '" linkages are brought to their final values is
immaterial because the energy is a state function. For the purposes
of illustration assume that the flux
--~
-
16 ELECTROMECHANICAL ENERGY CONVERSION
linkages are brought to their final values in the sequence
AI> .12, .13 ; then Eq. IEI-2 can be written out as
rA1 0 0 0fA' l'A2'Wm = Jl 1;(,\;,0,0) dA; + I~(AJ> .1;.0) dA;
0,0.0 Al'O,O
JAPAz.A l+ ii(AI> Az, A;) dA; (IEI-3) A1.AZ.0 . Wm = t l
TIlA; dA; + f: 2 (T21 A1 + TZ2A;) dA;
+ foAl (T3 \A 1 + T 32A2 + Tn'\;) dA; (11-4)
Recognizing that unprimed variables in the above integrations
are held fixed, there results
W m = !TllA~ + r 21 A1A2 + !f'22A~+ 1'31 A1A3 + T32A2A3 + tr33Ai
which can be written as
J 3
W m, = L L !TijA/A] /~I j~ I
which is the conventional expression for energy storage in a
linear magnetic field device.
" The interchange between electrical and mechanical energy via
the stored energy in the magnetic field is a direct manifestation
of the energy con version process. The fact that the stored energy
can be determined for any configuration of the system, and that
this stored energy is a state function defined solely by the
functional relationships between variables" and by the final values
of these variables, provides a powerful tool for determining the
coupling forces of electromechanics.
Now that the stored magnetic energy has been determined, the
arbitrary displacement and conservation of energy as expressed by
Eg. 1-11 can be used to evaluate the mechanical forces on the
system of Fig. 1-3. Assuming that the stored electric field energy
W~ is zero, it follows from Eq. 1-12 that
" n
C('.). dXk = 2: II dA, - dWm (1-16) /=1
where dA; = v/ dt. To obtain Eq. 1-12 and hence Eq. 1-16, an
arbitrary displacement dx, of the kth mechanical node was assumed
to take place. At the samc time no explicit restrictions were
placed on the changes in Aj and I ; consequently, it may appear
that the force (J;)k will depend on
EQUATIONS OF MOTION OF ELECTROMECHANICAL SYSTEMS 17
how).., and i; vary during the arbitrary displacement. It will
be shown, however, that th~/ force (.f..}k is independent 0/ the
mriatiollS (If ,\, lind i; tlurinK the arhitl'w')' displacel1lent
pl'Ol'ided the changcs in '\; al/l/ i, ji,f!oll' the /unctional
relationships (the internal constraints) oj' thc I.I'\ICIII hcc
Fig. 1-3b and Eg. 1-7). In other words, the fact that only one of
the two variables A; and ii can be treated as indepcndent must be
recogni1SJ as an internal system constraint.
The arbitrary displacement is taken at the point defined (see
Fig, 1-3b) by
Ai = >';(11 ... ill; xI> ... , X",) (1-l7a)
This expression establishes the dependence of Aon i and X which
must be maintained during the arbitrary displacement defined in
obtaining Eq. 1-16. This single-valued dependence results from the
stored magnetic j
,energy Wm= ~f/",(il> ... , in; XI> . X",) (1-17b)
being a state function. The second term on the right of Eq.
1-16, dWm, is a total differential (see Eq. 1- 11) and becomes
- 8Wmd ~: 8W"'d'dWm - -
-
18 ELECTROMECHANICAL ENERGY CONVERSION EQUA"I,JNS OF MOTION OF
ELECTROMECHANICAL SYSTEMS 19
Substituting Eqs. 1-18 and 1-21 into Eq. 1-16 yields It is
possible to put Eq. 1-27 in a simpler form by writing it in terms
of the magnetic coenergy W:n which is defi ned as:
- - 0Wmd ~, . all; d (/.),k dX" - -')-- x" + L. I; ~ x" , l'l
.... ,/n n.. ,., 0'of(;"" 1=1 uX" W m = .~ A;(lp ... , I,,; xl>
... , Xm ) dl, ( 1-28)
~.(~fJA;d) ~fJWmd'+ .L. I, L. Y I, -.L. y- II (1-22) 1 __:1 r=l
I, Iz~1 I,
Rearranging terms and interchanging indices in the last term
change Eq. 1-22 to the form
- oW " aA') " (- oW "aA)(Ie)" dx" = ( a m + ,2 i; ~ dx" +2: ~ +
2: i, ~ di;x" I-I uX" 1=1 uI; ,=1 ull ( 1-23)
In order that the force (f,)" be independent of the change in i;
and AI during the arbitrary displacement, the coefficient of dil in
Eq. 1-23 must
be zero. - oWm ~. OA, _ 0-,,-.- + L I, -:;;:- - (1-24)
u1i r= luI;
It can be seen graphically how this condition makes (Ie)"
independent of the changes in i j and II; by referring to Fig. 1-4.
For any di; there is a corresponding dll; (two examples are shown
in Fig. 1-4); thus, if (I,)" is independent of di j , it is also
independent of dA;.
To show that Eq. 1-24 always holds, the definition of stored
magnetic energy given by Eq. 1-15 will be used and integrated by
parts (f idA = iA - J Adi) to obtain the alternative
definition:
n nlill.... i n Wm = ~ i,A, - ~ A;U;, ... i~; xl>' .. , x m)
di; (1-25)
,-1 0... ,0 , 1
Substitution of Eq. 1-25 into 1-24 and evaluation of the
indicated derivatives (keeping in mind the functional relationships
of Eq. 1-17a)
yield
- () Wm ~. oA, ~ . oA, \ \ ~. OA, 0 ( 26)-.,.- + L. 1,-:;;:- = -
L. I,~ - "; + II; + L l,~ = I01; r-I uti r= I uti r= 1 uli
which is always satisfied. Consequently, the force (Ie)" is
always given by:
(/.) = - oWm(i1> ... , in; XI> , Xm) " ax"
+ i i; oA;(;1> ... , in; xl> .. , X m) (1-27) ;-1 aX"
This is a perfectly general expression for (I,h which holds
regardless of how A; and i; are changing with time in the
system.
0.. ,,0 .-1
~: di di ~r 'T 'i_
~j f- I i dX; I
dX, Change dUring K"1 arbitrary displacement
Ai (i;, ,i~iXl',,,,XRt,,,xm)
I Change dunng arbitrary displacement I
r
I I I I I
;"_~,1i\jt..~.f
,-,. (," '''i' ,xk +dxk ... .x )'\ I' , n' m I I
I t o ii i , ~
Fig. 1-4. Illustrating how Ai and ii can change during an
arbitrary displacement dXk.
The relation between energy and coenergy has already been
established as a consequence of writing Wm as in Eq. 1-25 thus:
n Wm = L i;A; - W,:, ( 1-29)
;=1
This relation is illustrated graphically for the ith CircUit In
Fig. 1-5. Substitu';')n of Eq. 1-29 into Eq. 1-27 and subsequent
simplification lead to
aw'(' .(I.)" = m I" , In; XI> . , xm ) (1-30) aXk
If:..o..L
-
20 ELECTROMECHANICAL ENERGY CONVERSION
Equations 1-27 and 1-30 give the force U.)k when the
displacements and currents are used as the independent variables.
If it is desired to express the energy with the flux linkages ('\)
and displacements (x) as independent variables, Eqs. 1-27 and 1-30
must be modifted. For this new functional dependence the stored
magnetic energy is expressed as
Wm = Wm(>'lo .. , , A,,; xI> ... , xm) (1-31)
and i j = ij(AI> ... , A,,; xl> .. , x m) (1-32)
AI
AI
c -,Iiu
Fig. 1-5. Graphical relation between magnetic energy and
coenergy.
Since \ and Xj are the independent variables the
differentiations of
Eq. 1-16 yield:
- aWm(AJ, ... , An; Xl> .. , x m ) d AU;')k dXk = Xk OXk
~ aWm(A" ... , An; Xl>"" X m ) dA/ i-I OA;
n + L i;(AI> ... , An; Xl> , X m ) dA/ (1-33)
;=,
EQUATIONS OF MOTION OF ELECTROMECHANICAL SYSTEMS 21
With the stored magnetic energy given by Eq. 1-15, the last two
terms of Eq. 1-33 subtract to zero, giving for the
electromechanical coupling force U,oh applied to the kth mechanical
node:
(n = - ()Wm(,\\ 0 An; XI> ... xm ) (1-34), , , I. OXk
This force can also bc evaluated in terms of the coenergy When
Eq. 1-29 is substituted into Eq. 1-34. remembering that the'" and
,Ij arc the independent variables. there results:
aw,;,p'l> . An; Xl> . , , X",)U.h = OXk
_ )- A, ai;(Al> . , . An; Xl> .. xm ) (1-35) ;::;'1 '
aXk
Equation 1-35 is an equivalent way of expressing Eq. 1-34. The
several forms of the electromechanical coupling force ef..)k
applied
to the kth mechanical node by a magnetic coupling field as found
by an
TABLE I-I. Mechanical Force Caused by Magnetic Coupling
Field
= t,,A. .-1i i; d)..;Stored magnetic energy W m (1-15) 0"".0 ro
Magnetic coenergy W~ = .,,';. ~ A; di; (1-28) 0...0 I-I
Relation between energy and coenergy W.. + W~ = L i,A,
(1-29)
'-I
Conservation of energy during arbi L i, dA, = dW.. + (f.)t dXt
(1-16)trary displacement dXt (Iossless) 1-1
Independent Force Evaluated Force Evaluated Variables from
Stored Energy from Coenergy
Currents i, ([.) _ -oW.. i. OA I ([.) = iJW~t - -- +
1/Coordinates x, } iJXk I-I OXt k OXk Flux linkages A, 1 ([.) =
-oWm (f.)k = 0 W,~ _ I AI .oil Coordinates x J J k OXk OXk I_I
OXk
arbitrary displacement of the kth mechanical node XI. are
summarized in Table 1-1. The four expressions for U.)k given in
Table I-I are equivalent and will yield identically the same force,
which is the true force, for a given
-
22 .
EQUATIONS OF MOTION OF ELECTROMECHANICAL SYSTEMS
23ELECTROMECHANICAL ENERGY CONVERSION
state of the system, i.e., for a given set of ii' Ai> and Xj'
In order to find the dynamic path of the system the force of Table
(-I must be used with d"Aklllbl:rt's principle and Kirchhoff's laws
to establish the cquations of dynamic equilibrium.
The results of Table I-I are complctely general and independent
of electrical source variations (assuming low electrical
frequcncies and low mechanical velocities such that a quasi-static
solution is valid). It is worthwhile to examine some of the results
more closely.
For instance, the force obtained from the coenergy with i l and
x) as independent coordinates was given by Eq. 1-30
(f.)k = oW~(il> ... , in; Xl> ... , xm) (1-30)oXk
It has already been shown that this force is independent of the
changes in A; and ii which take place during the arbitrary
displacement; consequently, this expression is valid regardless of
how>"; and ii vary, if the variation is compatible with the
internal constraints, and therefore it is a general expression for
the force. On the other hand, considering Eq. 1-30 from a
mathematical point of view, since il and XI are independent
variables the partial derivative is taken with respect to Xk,
holding all other x's and all i's constant. The holding of the i's
constant is a mathematical restriction imposed by the selection of
independent coordinates and has nothing to do with electrical
terminal constraints. The mathematical restrictions are often
misinterpreted as electrical terminal constraints, and some
confusion about the generality of the force expressions
results.
Statements similar to those just given about Eq. 1-30 can be
made about all the force equations in Table 1-1. These force
equations are general; the mathematical restrictions placed on the
derivatives by the choice of independent coordinates have nothing
to do with electrical terminal constraints in general.
On the other hand, the general expressions of Table 1-1 can be
used to interpret specialized electrical terminal constraints. For
instance, if all changes in flux linkages d>"; are constrained
to zero, there can be no energy flow between electrical sources and
magnetic fields; conseq uently, energy conversion must take place
solely between the magnetic field and the mechanical system. This
is illustrated by noting that the electromechanical coupling force,
when evaluated from stored magnetic energy with>"; and x) as the
independent variables, is simply the negative rate of change of
stored magnetic energy with respect to mechanical displacement with
the flux linkages held constant. In this case the elcctrical
tcrminal constraints of the special case coincide with the
mathematical restrictions of the general case.
Another interpretation of this type can bc made by considering a
system excited by electrical constant-current sources and by
inquiring about the energy conversillll. From Tahle I-I the force
is evaluatn! from the coenergy lIsing i, and x, as independent
coordinates; the force is mathematically given by the ratc of
change of magnetic coenergy with the currents held constant. In
this case the mathematical restrictions in the general case
coincide with the electrical terminal constraints in the special
case. This leads to an interpretation of the coenergy W;n as a
measure of the converti bility of electrical energy from
constant-current sources.
When a system is electrically linear some general statelT1l:nts
can be made ahnut energy conversion and about the relation between
energy and coenergy. By electrically linear it is meant that the
flux linkages are linear functions of the currents, thus
where
"
A; = n
L [i,i; (1-36) r= I
[I, =: [1,(Xh .. Xn,) = I" (1-37)
is a general single-valued function of the displacements. The
use of Eq. 1-36 in the definition of Wm (Eq. 1-15) yields for the
stored magnetic energy
;l .... IO n (n )WIN = L i; L Ii' di; z:
n
L n
-tli,i;i, (1-38)J0 .....0 ;=1 r=1 i-=d r=,-" J From Eq. (I -29)
the magnetic coenergy is
n
W~ = L i;Ai - WIN (1-39) i=1
Substitution of Eqs. 1-36 and 1-38 into Eq. 1-39 yields the
result:
fI n
W;" = Wm = L L 1!,J,i, ( 1-40) ;=1 r=1
Thus in the electrically linear case the stored magnetic energy
is equal to the magnetic coenergy. This can be seen geometrically
by considering Fig. 1-5 with a linear relation between A; and i;.
"
The fact that the energy and coenergy are equal in the
electrically linear case has led to the use of the two state
functions interchangeably. Investigation of Table 1-1 shows that
energy and coenergy must be distinguished; otherwise in the
electrically linear case the sign of the mechanical force will be
in error if the wrong state function is used.
-
24 ELECTROMECHANICAL ENERGY CONVERSION
Equation 1-16 can be used to write the conservation of energy
for an arbitrary displacem'~nt c/.>;J.. as:
" 2: i; d>.; = dWm + U"h dXk (1-41) i ,
' __ ....... -.J'--.,--' ~
cl,,'rtrli.:.a1 Morcd Ill",,,; 1\;ll11Gl I Iliput liclt.1
ouqHlt
l'lll"rM)' energ)' Cliel ~y
Next, the individual terms on the right-hand side of Eq. 1-41
can be cV.lluated, assuming an electrically linear system. Using
the force from Table I-I and the coenergy with i j and x j as
independent variables, the energy converted from electrical to
mechanical form in an electrically linear system is
" " 1 of. (1-42)U,)k dXk = ;~I '~I 2" o_~: iii, dXk
The change in stored field energy is found from Eq. 1-40 as:
_ ~. ,. .. ~. ~~ ~ 01;, ..dWm - ... ~ 1;,l j dl, + ... ~ ').
Ijl, dXk (1-43)
; " , ; - - I , I - (IXk
According to Eq. 1-41 the sum of Eqs. 1-42 and 1-43 equals the
electrical input power. When all the electrical sources are
constrained to be constant-current sources,
di, = 0
and the electrical energy converted to mechanical form becomes
equal to the change in stored magnetic energy. Thus, when an
electrically linear system is excited by constant-current sources,
the electrical input energy is divided equally between stored field
energy and converted energy.
1.2.3 Mechanical Force Due to Electric Field Coupling-
The mechanical forces produced by magnetic coupling fields in an
electromechanical system have been determined. A similar
development can be made for finding mechanical forces due to
electric field coupling in an electromechanical system. Consider
the case where the only sign iticant stored energy is electric
flcld energy (sec Fig. l-6a). The electrical stored energy W, can
be found in a manner like that used to find the magnetic stored
energy in the previous section. Assume that all purely electrical
or mechanical energy storage elements plus all dissipative elements
have been removed from the system (see Sec. 1.2.1); then, from
For an alternate treatment SL"C Fitzgerald and Kingsley, [(/('.
cit.
.,
EQUATIONS OF MOTION OF ELECTROMECHANICAL SYSTEMS .25
the conservation of ener.gy the stored electrical energy W" must
t:qu,il the input energy from all sources,
W, = input electrical energy + input mechanical energy
(1-.44)
l"' .. ,,' ' W.(ql> ... , q,; XI' ... , xm ) =2: I';(q;, _.
_, 'I;; x;, ... , X;,,) !/l/ n.....o ,. I l
'!'\'m,~ " I. ,I.'I I+ . ..:. Ij(q" - . - , q" XI' ... \",) d\j
(1-45) n... :0 j I .
" ... ;;:.
f.".i"l
., ~
Fig. l-'a. C41ed charge-earrying conductors.
The expression for stored elee.trical energy as given by Eq.
1-45 is plotted in Fig. l-6b and c. The amount of the total stored
electrical energy supplied by electrical sources and that supplied
by mechanical sources arc dependent upon the manner in which the
system is ,assemhled; however, since the electrical stored energy
is a statc function, YV. can be expressed in the simple form:
W,(ql> ... ql; Xl> . , x )m
1"\ .....'11 I = 2: v;(q;, .. " q;; x", .. XIII) tlq; (1-46)
n....n j I
-
26 ELECTROMECHANICAL ENERGY CONVERSION EQUATIONS OF MOTION OF
ELECTROMECHANICAL SYSTEMS 27
. . jq,II; t Stored electncal energy = (} 1/; dq; qj f,\,
-
:w I:LEC IKOMECIIANICAL ENERGY CONVERSION
Now Eqs. 1-4l\ and \-49 lIsed with Eq. \-47 give the force U;.h
as:
- i'IV ' ?q,) , (- i)W ' (1q)
U~). d.\. = ( -----;-,-,-" + ~ P, ~ tis. + ~ -i:!'+ 2 /',
-i:!--~ tlv, (I-50)e\. I~I (','. j-I Vi '-1 Vj It can be shown
quite readily that
- oW, !.; oq, 0 (I-51).-.-,- + .:... V, 0:;- = (,Vi ,-1 vV;
Therefore the forcc (f,). is independent of how Vi and q, arc
externally constrained (providing Eg. 1-48 is satisfied) during the
arbitrary displacement oecause the coef1icient of tlv, is zero in
Eg. \-50. The resulting expression for the force is:
(f..) = -OW,,(PI""; 1',: x, ... , x",) k cJx.
~ V oll'(I'. I' . X X )+ ') "',1'-1,",," (1-52)I ':-1 OX.I
It is possible to put Eq. I-52 in a simpler form by writing it
in terms of the electrical coenergy W;.. which is defmcd as:
ul ..... , ' W~ =
v
2 q;dv; (I-53)i0.... ,0 j.=, The relation between the electrical
energy and coenergy is (see Fig. 1-6b)
, W, + W~ = L vjql (1-54)
'-I Substitution of Eq. 1-54 into Eq. I-52, making certain to
keep Vj and xJ as the independent variables, leads to
(/')1( = oW~(VI< ... , v,; XI< , , x",) (l-55) cXk
Equations I-52 and I-55 give the force (f,). when the
displacements xJ and the voltages /'i are independent
variables.
Expressions can be obtained for the force (fe)' that are
equivalent to.. Eqs. I-52 and I-55 but expressed in terms of the
displacements X j and the charges qj as independent variables.
Assume that the voltages are
expressible as:
VI = V;(ql' ... ,q,; XI> .. x",) (I-56)
Then the stored electrical energy can be written as:
W, = W~(qh"" q,; Xh , x m) (I-57)
EQUATIONS OF MOTION OF ELECTROMECHANICAL SYSTEMS 29
The use of Eqs. I-56 and I-57 with Eq. 1-47 yields the
result
(f.) = _ oWr(q" .. , q,: Xl, ' ,X",) ( I-58) ,. OXk
which is the correct force regardless of how Vi and qj vary
during the arbitrary displacement.
Substituting from Eq. ]-54 for W, into Eq. I-58 and recognizing
that the x j and lfi are the independent variables yield the force
in terms of the electrical coenergy as
()W~(ql< . , .. q,: Xl' .. Xm)Cf.h = (lx. ~ Ol';(q,. ' ..
,q,: XI< .. Xm) - .;.. qi " (I-59)
i-.I ex.
The various forms of the force U,)' which have been found for
electric field coupling are summarized in Table 1-2. The four
expressions for
_'J' TABLE 1-2. Mechanical Force Caused by Electric Field
Coupling
I'I" .. '" ~ 'd'Stored electrical energy W, = .. V, q, (1-46)
0.....0 1-1 Iv' .....r l "d'Electrical coenergy W' = L q, v, (1-53)
, 0.....0 '-1
r.t ',t . ";,:~ , I
Relation between energy and coenergy W, + W; = L v,q, (I-54)'-I
,
Conservation of energy during arbi 2: VI dq, = dW, + ([.)_ dx_ (
\-47) trary displacement dx_ (loss less) ;~ '-I
.;1'; '\I~
Independent Force Evaluated from Force Evaluated Variables
Stored Energy from Coenergy
Voltages v, } , -oW' oW:(f.h = --' + ') v oq, (1.)_ =
oXCoordinates XI ax_ I~ I ox_ 1 Charges q, } -oW aw' 'ou,(1.)_ =
--' (f.) , - L q-Coordinates XI ax_ , k = eXt I_I I ox_
~;
force (f,h are equivalent and will yield exactly the same force,
which is the true force, for a given state of the system, i.e ..
for a given set of qj,
i Ilj. and Xj' In order to find the dynamic behavior of a system
the force of
of Table 1-2 must be used with d'Alembert's principle and
Kirchhoff's laws to establish the equations of dynamic
equilibrium.
-
38 ELECTROMECHANICAL ENERGY CONVERSION
The force expressions of Table 1-2 ~re;i completely general and
the mathematical restrictions imposed by the differentiations
indicated must not be confused with external electrical constraints
imposed on the system. Equation I-58, which yields the force for
any general case, shows that in the special case of a terminal
constraint of constant charge, the energy is converted between
stored electrical energy and mechanical energy with no input from
the electrical source. The general expression for force of Eq. I-55
also shows that when a special external electrical constraint of a
constant-voltage source is applied, the energy converted from
electrical to mechanical form is equal to the change in electrical
coenergy. The electrical coepergy can be considered as a measure of
the convertibility of energy from constant-voltage sources through
electric field coupling, In addition, it can be shown that when an
electrically linear electric field system is excited by
constant-voltage sources, the electrical input energy divides
equally, half going to electric field storage and the other half to
mechanical energy. These results are clearly analogous to those
derived in the previous section for magnetic field coupling. They
also indicate that for electric field coupling a constant-voltage
constraint and a constant charge constraint are analogous
respectively to a constant-current constraint and a constant-flux
linkage constraint for magnetic field coupling.
1.3 Hamilton's Principle-
In the previous sections the equations of motion of
electromechanical systems were obtained from force laws, These
force relations were determined from basic experiments or
postulates of physics, and they form a set of "differential
principles," i.e., principles concerned with incremental changes in
the system.
It is also possible to develop an alternate approach to the
problem of describing the path of a dynamic system by postulating
that the dynamic path of the system is determined by finding the
extremum of certain integral functions. This alternate approach is
called a variational method and is based on a set of "integral
principles," i.e., principles relating to gross motion of the
system.
When a system is completely described by one of the principles,
the other principle can be derived. This is just another way of
saying that one physical system is described by one dynamic path of
operation regardless of how the equations of motion are derived.
The integral principle
For a derivation of Hamilton's principle from the principle of
virtual work, see Goldstein, loc. elf. or Whittaker, loc. cit.
EQUATIONS OF MOTION OF ELECTROMECHANICAL SYSTEMS jJ
usually considered to be the most fundamental is Hamilton's
principle. Hamilton's principle can be derived for mechanical
systcms from d'Alembert's principle and the principle of virtual
work. However. Hamilton's principle proves to be significant for
other than just mechanical systems, and as such overshadows
d'Alembert's principle and is at least a more general relationship
if not a more fundamental one.
The usual statement of Hamilton's principle is that the
variation of the time integral of the Lagrangian L between fixed
end points qi(tt) and
-
'''").~
32 ELECTROMECHANICAL ENERGY CONVERSION
representation of the meaning of an extremum is shown in two
dimensions in Fig. 1-7. ,>,
The condition for an extremum of the function 1 is that the
variation of the function I equal zero,
M = S f.1 2L(ql(/), ... , qN(/); (Ml), " . ,QN(I); I) dt .. 0
(1-61) 'I
subject to the end conditions that.
oqj(t\) = 0 and oqj(l2) = a for i = 1, 2, 3, ... , N
In the above equation the symbol 8 means the time-independent
variation as used in the calculus of variations and is analogous to
a differential in ordinary ditTen.:ntial calculus. (Sec Sec.
1.3.2.)
Equation 1-61 is Hamilton's principle. It is now necessary to
see what set of relationships must exist among the N coordinates,
the qj, and their N derivatives, the q" in order to satisfy Eq.
1-61. The reduction of Hamilton's principle to a set of
differential equations is best done using :f the calculus of
variations. Therefore, a short digression into the calculus of
variations is a worthwhile next step. ,}
1.3.1 Calculus of Variations
The calculus of variations is concerned chiefly with the
determination of maxima and minima (more exactly extrema) of
expressions involving unknown functions. It differs from
differential calculus in that the variables are known in ordinary
calculus and a minimum or maximum of a . function of these known
variables is desired. In the calculus of variations,'1 the
variables are unknown and it is desired that the relationships
among { the variables be found which will form an extremum (e.g., a
maximum. or minimum) of some integral containing these variables.
I
The essential techniques of the calculus of variations can be
determined by procedures analogous to those of finding maxima and
minima of j differential calculus. As an example, consider the
function I defined as~
'";1
the integral between 11 and t2 of another function L, where L is
a function1i of the unknown variables q(t) and !J(t) and of the
independent variable t:$
' ,1= 2 L(q(t); ,jet); 1) dt (1-62)J
'I
A typical problem is to find the function q(t) and also (j(l)
which will
h,r " Illor.: delaile,1 treatmcnt of Ihe I:alculus of
v;lriations see, e.g., F. B. Hildebrand, Methods oj Applied
Mathematics, Prentice-Hall. New York, 1954, Chltp. 2.
EQUATIONS OF MOTION OF ELECTROMECHANICAL SYSTEMS 33
make the integral I an extremum" and which will satisfy the end
conditions
that q(ll) - q\ and q(12) - q2 (1-63)
where q\ and q2 are fixed end points. As a first step in finding
the extremum of I assume that qo(t) is the
actual function which makes I an extremum, and choose any
continuously J differentiable function 7)(t) which vanishes at the
end points q(t\) = ql
~l
and q(lz) = qz. Then for any constant a, the function
q(t) = qo(l) + a 7)(1) (1-64)
II will satisfy the end conditions of Eq. 1-63. The
integral/:
!
~ \, " I(a) == ". L(qo(t) + a 1)(t); 40(1) + IX >j(t); t) dl
( 1-65) rI
J
11
" ~ .', is obtained from Eq. 1-62 by setting
~ ~ q(t) = qo(t) + IX 7)(t) (1-64) '1\'
and 4(1) = 40(1) + IX 7j(/) (1-66)
The function l(a), Eq. 1-65, is only a function of lX, once
qo(t) and 1](t) are assigned, and furthermore, the function I(a) is
an extremum when a = a because qo(l) was chosen to make it so.
However, this is only possible if
.1
dljda = 0 when ex = 0 (1-67) . J" \ (If>: Equation 1-67 is,
therefore, the condition for an extremum and is a
.rJ' defining relation which relates differential calculus to
the calculus of
'I;f.!,, variations. , . , The condition for an extremum, Eq.
1-67, can now be applied to Eq.
;': :}" 1-6'5 to find the differential equation which will
result. Since a, in'".. ~, Eq. 1-65, is a constant in the
integration. the differentiation wi.th respect ~.
to a can be taken under the integral sign to yield it
dl(a) II2 [OL () 8L,( )] d 0--= -1]1+---;;-7)11= (I -68) da 'I
oq oq
Before investigating Eq. I-68 further, it is desirable to
introduce the variational notatilln of the calculus of variations.
The only variation to be studied here is the time-independent
variation.
To tix idcas
-
II
I
:
ELECTR.OMECHANICAL ENERGY CONVERSION EQUATIONS OF MOTION OF
ELECTROMECHANICAL SYSTEMS 3S34
Now return to the condition for an extremum, Eq. I-M!, and note
that1.3.2 Time-Independent Variations (0 Variations) from the
definition of oL, Eq. 1-74, ,
Consider the case in which times 'I and 12 of the end points are
held I fixed and in which the integrand (L in Eq. 1-65) is an
explicit function of j a dl(a) = J" 8L dt ( 1-75) the variables q,
q, and'. Definc a variation of the function L in terms of cia '1
variations of q and q and not of the time. This is a
time-independent Since u. is a constant in the integration with
respect to I, the expression of (8) variation. Eq. 1-75 mu~t equal
zero if Eq. 1-68 equals zero. Thus, the condition
The function qo(t) has been defined as the true function which
produces for an extremum is an extremum, and any other function
with the same end points was \ I' defined by Eq. 1-64 as 2 oLdt = 0
(1-76) (1-64) '1q(t) = qo(t) + a 7)(/)
Since the vanatlon is time-independent, it can be taken outside
theNow define the variation oq as the variation of the function
q(/) from the .~: integral sign to yield
true function qo(t) and obtain: ~,~
""!~,
8q = q(t) - qo(t) = a 7)(t) (1-69) Sf = S J'2L dt = J/2 SL dt =
0 (1-77)
'1 '1
Similarly, the variation oq of the velocity q from the true
velocity 40 is Expanding SL by Eq. 1-74, the condition for an
extremum in variational
(1-70)=oq = q(t) - 40(1) a 1j(t) notation can be expressed
as
The next step is to define the variation of a function, e.g.,
the variation ' 2 (8L f:JL.)81 = a Sq + 8"'" 8q dt = 0 (1-78)of the
function L(q(/); .:j(/); I). To do this find the difference which
results J'I q q from a small variation in q(/) from the true
function qo(/). This difference
Several forms of the condition for an extremum expressed in
variationalis notation are given by Eqs. 1-76, 1-77, and 1-78. An
extremum of theoL = L(q(t, a); q(t, a); t) - L(qo(t); qo(t); t)
(1-71) function f in the calculus of variations is found by setting
the variation of
where q(l, a) and 4(/,
-
36 ELECTROMECHANICAL ENERGY CONVERSION
(1-80)
for i = I, 2, 3, ... ,N. Since the variation 0 has been defined
to be independent of the time f, it can be taken under the integral
sign as in
Eq. 1-77 to yield
oj = J'2 SL(qt(f), ... ,qN(/); eMf), ., ., tiNCt); I) dl 'I
.,1'\
IlJ J/2!1: (8L Il 8L 8') 1 = .L. a qi + Y lJj C f = II ,-I q/
q,
Expanding the oL in Eq. 1-80 gives
0 (1-81) if
To simplify this expression, integrate the second term by parts.
ovariation is independent of time; consequently,
The
oq = ~ oq (1-82)
and the second term of Eq. 1-81 can be integrated by parts to
obtain
.~, 5 ' 2 ~ 8L 0'. d.:- ',' q, t 111
-
38 ELECTROMECHANICAL ENERGY CONYERSION t EQUATIONS OF MOTION OF
ELECTROMECHANICAL SYSTEMS 39
IA State Functions II For example, it means that a knowlcdge of
onc type of physical system may prove helpful in gaining an insight
into a physical system of
anII entirely ditTerent nature. It also means that an engineer,
whose work
In the previous sections the Euler-Lagrange equation of motion
was
takes him outside his field of specialization, nced not fecI in
completely
developed from Hamilton's principle; and, in so doing, the
Lagrangian
strange territory.state function was introduced. The Lagrangian
and the other sta
te
functions are of central importance in the characterization of
physical Unfortunately, the class of systems exactly describable by
state fun
ctions
docs not include all electromechanical systems. Dissipation must
be
systems (electrical, mechanical, electromechanical, chemical,
thermo-
excluded from systems if they are to be described by state
functions. A
dynamic, etc.). The state functions include the total energy of
the system
form of dissipation which proves particularly troublesome is
hysteresis.
and other closely associated functions such as the Lagrangian.
These
At first, this appears to be a severe limitation; however, the
main use of
functions are called state functions because, at a given instant
of time,
they depend solely on the state of the system at that instant of
time, and
i.' i. (,not on past history. Their importance has been
recognized for a lo
ng -'-- -, -+--+ Lossy + Loss lesstime in thermodynamics and in
the statistical and quantum mechanical v' +
I electrical Vi electromechanical x,treatment of atomic systems,
although their first noteworthy use was
in system - system
advanced classical dynamics. These state functions, and the
variables
- :,~". ," For example, to describe a thermodynamic system, the
I:
,'
"\.
Fig, 1-8. Lossy electromechanical system divided into simpler
component parts.describing them, are in many cases used by
engineers
without explicitly ..;""~,..'." realizing it. heating engineer
will use such variables as temperature and entropy; th
e ,,.. state functions will be to obtain a general formulation
of the equations of , -,;1
control system engineer will use force and displacement to
describe his
I
connected mechanical system; the aeronautical engineer will talk
about -
~.~
...~ motion of a system. The quantities which are most difficult
to obtain
are the coupling terms between different types of systems, e.g.,
the
the roll torque and roll angle in discussing the stability of an
aircraft; the
electromechanical coupling terms in eJectromcchanics.
Fortunately,
electrical engineer will use voltage and charge for describing
electric l..,~
these terms arc determined by the conservative part of the
system and
circuit behavior; and the chemical engineer will employ such
terms as ,'.r.
Each of these engineers works in [ ,. are derivable from state
functions. Thus, it becomes practical to separate
chemical potential and mole number. the problem into two parts,
consisting of (I) an energy conversion p
art his field of specialization, talking about these widely
different physic
al For an
systems in terms which seem equally unrelated. In actuality,
these that is dissipationless and (2) other parts with
dissipation.
>'loR;' electromechanical system, this takes the form of a
lossy electrical system
physical systems have much in common. (including hysteresis
losses), a lossless electromechanical system, and
a For example, in each of the above cases denote the first
variable by
It is possible in this way
11 and the second variable by qj; then, irrespective of the
nature of the .,11r' lossy mechanical system as shown in Fig. 1-8.
to study the lossless electromechanical system and bring in
dissipationsystem, it is always possible to write l' ,i. when
considering the over-all system performance. dW =/; dql (1-86)
,f;L}I;',; , 1.4.. 1 The Characterization of Physical Systems
(Without Hysteresis where dW represents a differential change in
energy produced by a t'~l' .},'. ~,',:,::,-;.. and Dissipation)
The variables f; and q, aredifferential change dqj in the
variable q;. '.'
generalized vatiables and their product describes an energy
relation in The constitution of a physical system from a dynamic
point of view can
each of the above systems. The energy functio,ns may be, and
usually be regarded as consisting of a number of particles, subject
to inte
r- 'I
are, state functions and contain much valuable information about
the connection and constraints of one kind or another. The
configurati
on
system described by them. Actually, Eq. 1-86 is a manifestation
of the of a given system at any time can be specified in terms of
quantities call
ed
fact that in spite of their vastly different natures all
physical systems have the coordinates of the system. The choice of
the set of coordinates f
or
a fundamental similarity and lend themselves to a common
mathematical a system is usually somewhat arbitrary, but in general
each individu
al
This fact has far-reaching consequences for the engineer. energy
storage element of the system can have a set of coordinates.
For
description.
-
-41
40 ELECTROMECHANICAL ENERGY CONVERSION
example, every discrete element of mass can have its position
specified in terms of three space coordinates, each inductance
element can have its nux linkage specified, or each capacitor can
have its total charge specified., Examrk's of possible coordinates
for several systems are shown in Fig. 1-9. When dealing with static
systems (systems in static equilibrium) the values of the
coordinates completely specify the system. For a dynamic system,
however, the coordinates do not completely specify the system and.
an additional set of dynamic variables equal in number to the
coordinates must be used. These dynamic variables can be the first
derivatives of the coordinates, the velocities, or they can be a
second set of variables, e.g., the momenta. The velocities and the
momenta are associated variables and either set can be chosen as
the dynamic variables.
So far, only the number of variables that can be ascribed to a
particular system have been discussed; however, in any given system
all of these variables may not be independent and hence they cannot
all be specified independently. The question of how many variables
;ire independent is determined by the constraints of the system.
The problem of handling the constraints is one of the most
diflicult single questions of dynamics. Constraints are of two
essential types-holonomic and nonholonomi~
constraints. The holonomic constraints are represented by sets
of relations among the coordinates or, if expressed as
differentials, they can be integrated to yield these relations. For
example, if 11 coordinates can be ascribed to a system and then 111
equations of the form
.~(q\> ... ,q,,: t) = 0 j = I, ... , m (1-87)
can be written, it is possible to reduce the number of
coordinates from n to (n - 111) by using these rn constraint
equations to eliminate m variables. Holonomic constraints are
always expressible in the form of Eq. 1-87; furthermore. for a
system which has only holonomic constraints it is always possible
to select a set of independent coordinates which does not contain
the constraint equations. Thus, if n is the number of coordinates
determined from all energy storage elements and m is the number of
holonomic constraints, then there are (n - m) independent
cr~rdinates
.. and (n - Ill) velocities, or a total of 2(n - m) variables
which can be used to describe uniquely the dynamic motion of the
system. The minimum value of (n - m) that can be found is also the
number of degrees of freedom of a holonomic system. When a system
is described, using a selected set of coordinates which eliminates
the various system con-straints, it is accepted practice to caIl
these coordinates the generalized coordinah:s of the system. For a
system of N = (n - m) degrees of freedom there will always be 2N
generalized variables needed to describe the dynamic path of the
system (i.e., N coordinates and N velocities).
EQUATIONS OF MOTION OF ELECTROMECHANICAL SYSTEMS
m ~
Adiabatic vwalls L e
s T
GasJiJ q, '" II = charge on C
q I Xl =displacement 01 mass q2 -J;' dt. !Otegral .Ofq, .. S ..
entropy q l %2 = displacement of sprlOg Curren! In I.
qz =V =volume OR ({II (b) '1 X=I!ux I1nkmg L
q2 Z.,JF;dt =. Integral of voltage on C
(e) ,
c'OD~
q,. Q, Chafge on (.', :1'1 q2:= Q2:=: Charge on C
2 qJ fit dt :Il integral of current in L 1 q. ji2dt =integral of
current in L 2
OR q 1=X -= dIsplacement
q\ '" A, tOl.1 flux linking L \ q2" Q:o Charge on C
q2 - .\.2 total flux linking L 2
q3f;, dt Integral of current in Lq, ~ fe, dl -_,ntegral of
voltage .cross C\ (t)
q4 - fi2 dt integral of voltage across C2 (d)
,.~
I'r.:
~! Vr .~"l ?
~~
q 1 - :c 1 displacement of M C L q2 '" "'2 ~ displacement of
spri",
Q3'" A" flux linking L q .. j'.dl .. integral volt.ge on C
Iron
(f)
Fig. 1-9. Examples of physical systems and variables ql
associated with each energy storage, sources and dissipation
omitted for simplicity.
~I"'
"
L
-
42 ELECTROMECHANICAL ENERGY CONVERSION EQUATIONS OF MOTION OF
ELECTROMECHANICAL SYSTEMS 43
The questions of independence of coordinates which arise when
there required and can be expressed as ql(/), qz(/) . ... , qN(I)
where the q;(t)
ex.ist nonholonomic constraints are much morc dil1icult to
resolve. A are the genl'l"u/i::cd coordinafes. Systems so described
are calkd dynami
c
For dynamic systems a second set of N quantities can
benonholonomic constraint is one in which equations of constraint
do not systems.
ex.ist among the coordinates, i.e., constraints not satisfying
Eq. 1-87. introduced such as the PI(/). P2(t), ... , PN(I). called
the genera/b,
d
The slale of a dynamic systcm at a Ricen inSlanl of lillie isFor
example, a nonholonomic constraint is obtained when the gcncral
momenta.
determined by the particular values or the N generalized
coordinates and
form of thc constraint equation is Thus the statc or athe N
generalized momenta at that instant or time. i((/I> ... , 4n; t)
= 0 (l-88) dynamic system may be represented as a point in a
2N-dimensional
where the resulting differcntia1 cquation is not intcgrab1e. An
example
of such a nonholonomic constraint is the commutator in electric
machines
por the constraint imposcd on a rolling ball by a perfcctly
rough surface
.
Another type of nonholonomic constraint is the inequality type
of
constraint such as that imposed by the wall of a container upon
a gas
", r
I,
particle contained therein. In this case the nonholonomic
constraint is At'..1,of the form
(ql)2 - b2 < 0 (1-89) " ",
where ql is the coordinate of the gas particle and b is the
coordinate of ".;(""the wall of the container.
"\ .When dealing with nonho10nomic constraints it is not
possible to find " ... % 116I Xoa set of generalized independent
coordinates equal in number to the ~.',
number of degrees of freedom. Instead, it is necessary to choose
a
number of coordinates equal to the number of degrecs of freedom
plus
the number of nonho10nomic constraints. For problems of this
type the
Euler-Lagrange equation derived in Sec. 1.3.3 from Hamilton's
principle
cannot be used since the coordinates are not independent. In
general,
\,'>"i J any problem with nonholonomic constraints is very
difficult to handle
unless some trick can be devised to reduce it to an cquivalent
holonomic ~i.l 1
problem. A mcthod for doing this will be discussed latcr in Sec.
1.6. ~
":.Fig. 1-10. Path in phase space for system of Fig. 1-9a
when p = 0 and x = ."0 at I = O. 1.4.2 Generalized Coordinates
for Holonomic Systems
There are two types or states of physical systems: static and
dynamic. space, the 2N dimensions being the N coordinates ql and
the N momenta
For the static state, only the description of the system in
static equilibrium
PI' Thic ~pace is called phase space. Once the state of a
conservative
with its environment is given. The state of a static system is
completely f
system is established at one time its path in phase space is
completely
specified by the values of its N generalizcd coordinates. For a
system in determined. This means that oncc a given set of ql(ll)
and 1'1(11) is estab- .,static equilibrium there can be no
dissipation. In discussing stati
c lished, then ql(l) and