On the theoretical and experimental determination of the
electromagnetic torque in electrical machines
Citation for published version (APA):
Kamerbeek, E. M. H. (1970). On the theoretical and experimental determination of the electromagnetic torque in electrical machines. Technische Hogeschool Eindhoven. https://doi.org/10.6100/IR109161
DOI:
10.6100/IR109161
Document status and date: Published: 01/01/1970
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ON THE THEORETICAL AND
EXPERIMENTAL DETERMINATION
OF THE ELECTROMAGNETIC
TORQUEIN ELECTRICALMACHINES
ON THE THEORETICAL AND
EXPERIMENTAL DETERMINATION
OF THE ELECTROMAGNETIC
TORQUE IN ELECTRICAL
MACHINES
PROEFSCHRIFT
TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL TE EINDHOVEN, OP GEZAG VAN . DE RECTOR MAGNIFICUS, PROF. DR. IR. A.A.TH.M. VANTRIER, HOOGLERAAR IN DE AFDELING DER ELECTROTECHNIEK, VOOR EEN COMMISSIE UIT DE SENAAT TE
VERDEDIGEN OP VRIJDAG 22 MEI 1970 DES NAMIDDAGS TE 4 UUR
DOOR
EVERT MARIUS HENDRIK
KAMERBEEK
PROF. DR. IR. J. G. NIESTEN PROF. DR. JR. J. P. SCHOUTEN
groep Electromechanica van de Technische Hogeschool te Eindhoven. Gaarne betuig ik de leden van deze groep - in het bizonder de heer H. J. van den Braken-, alsmede Ir. C. Kooy en Ir. T. Scharten uit de groep Theore-tische Electrotechniek mijn dank voor alle hulp die ik bij de totstandkoming van dit proefschrift van hen mocht ontvangen.
De Direktie van het Natuurkundig Laboratorium der N.V. Philips' Gloei-lampenfabrieken te Eindhoven ben ik zeer erkentelijk voor de gelegenheid welke mij werd geboden om dit proefschrift in deze vorm te voltooien na mijn indiensttreding bij het Natuurkundig Laboratorium op 1 oktober 1968.
Gaarne bedank ik mijn collega's voor de van hen ondervonden medewerking en voor de met hen gevoerde stimulerende gesprekken.
INTRODUCTION . . . .. . . . .
PART 1: ON THE THEORETICAL DETERMINATION OF
ELEC-TROMAGNETIC TORQUE. 3
INTRODUCTION . . . 3 1. METHODS OF CALCULA TION USING THE MACROSCOPIC
IMPULSE RELATION. . . 9 1.1. Derivation of Maxwell's equations from the electron theory of
Lorentz . . . 9 1.2. The macroscopie electromagnetic impulse relation . . . 12 1.3. Maxwell's field equations for systems with "slo~ly" moving
mat-ter . . . 14 2. THE POWER BALANCE OF AN ELECTROMECHANICAL
SYSTEM . . . . 17
2.1. The power balance for a system without· electrical or magnetic hysteresis phenomena . . . 18
2.2. Quasi-statie electromechanical systems . 20
2.2.1. Systems of the magnetic type . . 21
2.2.2. Systems of the electrostatic type 22
2.3. Network representation of electromechanical systems 24 2.3.1. Systems of the magnetic type . . . 24 2.3.2. Systems of the electrostatic type . . . 27 2.4. The electromagnetic-field energy of a system with permanently
magfietized and/or permanently polarized matter . 28
2.4. 1. Systems with permanent magnets 28
2.4.2. Systems with electrets . . . 31 3. THE PRINCIPLE OF VIRTUAL DISPLACEMENT AS APPLIED
TO SYSTEMS OF THE MAGNETIC TYPE . . . 32 3.1. The ponderomotive force density of electromagnetic origin 32 3.2. The electromagnetic torque of an electrical machine as a function
of the currents and the rotor position . . : . . . . 35 3.3. Choice of the coordinate system . . . 38 3.4. Physical interpretation of coordinate transformations 40 3.5. Commutator machines . . . 42 4. PRACTICAL APPLICATIONS IN ELECTRICAL MACHINES 46 4.1. Field calculations . . . 46
4.1.2. Machines with a single cylindrical structure . . . 49 4.2. A qualitative investigation into the equations of motion of a
machine with a single cyliru:lrical structure 50
4.3. Application of the Maxwell stress tensor . . . 53
PART II: ON THE EXPERIMENTAL DETERMINATION OF
ELECTROMAGNETIC TORQUE . 58
INTRODUCTION . . . 58
5. SURVEY OF CURRENT TORQUE-MEASURING METHODS 60
5.1. Measurements at the mechanica) gates . 5.1.1. The rotor gate
5.1.2. The stater gate . . . .
5.1.3. Conclusion . . . . 5.2. Measurements at the electrical gates ~
5.3. Conclusion 60 60 62 63 64 67
6. TORQUE MEASUREMENTS USING HALL GENERATORS 68 6.1. Theoretical premises . . . 69
6.2. Further description of the measuring method . 74
6.2.1. Application of Hall generators . . . . 74
6.2.2. Choice of the number of Hall generators . 76
6.2.3. Nature and significance of the (space) harmonies in the
air-gap field . . . .. . . . 79 6.3. Torque measurements in symmetrical three-phase machines . . 82 6.4. Torque measurements in commutator machines. . . 85 6.5. Measuring errors due to an inaccurate axial positioning of Hall
generators in symmetrical a.c. machines with skewed rotor
con-ductors . . . 88
6.5. l. Measuring errors in asynchronous machines 88
6.5.2. Measuring errors in synchronous machines . 90
6.5.3. Conclusion . . . 92
7. TORQUE MEASUREMENTS WITH THE AID OF
MEAS-URING WINDINGS . . . 93
7. l. Theoretica! pre mises . . . 93
7.2. Practical realization of the measuring windings 97
7.2.1. Slot windings . 98
8.1. Torque measurements in a tbree-phase induction machine with wound rotor . . . 102 8.1.1. Measurements with Hall generators . . . 104 8.1.1. l. Design and realization of the measurement circuit 104 8.1.1.2. Test of the filter effect of the measuring circuit . . 109 8.1.1.3. Measurements during steady-state operation. . . 110 8.1.1.4. A genera! investigation into the effect of stator iron
loss on the electromagnetic torque and the measur-. ing signa! obtained with Hall generators or
measur-ing windmeasur-ings . . . 112 8.1.1.5. Recording of the measuring signa! Uhg during
steady-state operation . . . 118 8.1.1.6. Recording of the measuring signa! Uhg during
dynamic operation . . . 122
8.1.2. Measurements with slot windings 125
8.1.2.1. The measuring circuit . 125
8.1.2.2. Test of the measuring circuit 125 8.1.2.3. Measurements during steady-state operation. 128 8.1.2.4. Recording of the measuring signa! Usw during
steady-state operation . . . 128 8.1.2.5. Recording of the measuring signa!
u.w
duringdynamic operation . . . 128 8.1.3. Comparison between measurements with Hall generators
and measurements with slot windings . . . 128 8.2. Torque rneasurements in a three-phase induction machine with
cage rotor . . . 131 8.2.1. Measurements with slot windings . . . 132 8'.2.1.1. Measurements during steady-state operation . 132 8.2.1.2. Measurements during dynamic operation . . 135 8.2.2. Measurements with foil windings . . . 137 8.2.3. Comparison between the measurements with slot windings
and the measurements with foil windings . . . 137 8.3. Torque measurements in a synchronous machine with salient poles
and starting/damping cage at the stator side . . . 139 8.3.1. Measurements during steady-state operation . . . 141 8.3.2. Measurements during dynamic operation. . . 142 8.4. Torque measurements in a d.c. commutator machine with Hall
generators . . . . 8.4.1. Meàsurements during steady-state operation 8.4.2. Measurements during dynamic operation. .
145 147 148
Integral formulae; vector statements . List of symbols References Summary Samenvatting . ft·
151
152
156
158
161
INTRODUCTION
In the theory of rotating electrical machines· the most outstanding quantity is the torque resulting from the electromagnetic field within the machine. The electromagnetic torque and the angular velocity of the shaft determine the magnitude of the power involved in the conversion of energy as a first approx-imation.
The designer of electrical machines will take the desired electrical or mechanica! power as a starting point for his calculations. His objective may be to obtain optima! efficiency during the electromechanical energy-conversion process. Calculating the electromagnetic torque, which under steady-state conditions equals the mechanica! torque applied to the shaft of the electrical machine if friction is ignored, is the main part of his design.
Many a research worker in the field of electromechanical energy conversion is confronted with the phenomenon of electromagnetic forces. Thus he might find himself devoting his efforts to devising new models in which high demands as to power density, efficiency, torque-speed characteristics, controllability of torque and/or angular velocity are met. In that case he will benefit not only from fundamental theoretica! insight into the electromagnetic forces but also from experimental methods of verifying his theoretica! expectations.
Most electrical machines can be compared both physically and mathemati-cally - if necessary by using coordinate transformations - with a definite working condition of a special generalized educational machine *) constructed · for this purpose. The usefulness of such a generalized machine will naturally increase considerably if a torque-measurement device operating under both statie and dynamic conditions is available.
In electrical-machine practice a method of measuring torques under widely different conditions is desirable. For instance, control of torque is only possible if measured values are available.
In connection with
a
fundamental study of new torque-measuring devices a theoretica! study of the fundamentals of electromagnetic forces appeared to be indispensable. Moreover, it was feit desirable to deal with machines using modern permanent-magnet materials, which are being increasingly employed, especially in machines of small dimensions. The use of modern permanent-magnet materials frequently results in a simplified construction and a higher power density.The extensive literature on the subject of forces arising in the presence of electromagnetic fields bas been consulted in abundance in the course of this study. This literature is usually not easily accessible to the investigators working
*) Some generalized machines are frequently used for practical work and lecture demon-strations in the Electromechanical Department of the Technologica! University of Eindhoven.
2
-in the field of applied electromechanics and consequently little known to them · in most instances.
Due to these considerations it appeared to be useful also to discuss the fun-damental principles of electromagnetic farces. In part l these principles are involved in an evaluation of the theoretica! background to methods of calculat-ing electromagnetic torque as employed in the theory of electrical machines. The possibilities and limitations of each method are stressed.
The theory of the first part also largely serves as a basis for part Il, in which the experimental determination of electromagnetic torques is considered. A short review of the possibilities offered by known measuring methods is followed by detailed investigations into the merits of other methods suitable for measure-ments under dynamic conditions.
PART 1: ON THE THEORETICAL DETERMINATION
OF ELECTROMAGNETIC TORQUE
INTRODUCTION
The understanding of electromagnetic force action can no doubt benefit from a comparison with gravitational forces. The law of gravity formulated by New-ton (1643-1727) describes the macroscopie forces resulting from the presence of matter ha ving a certain mass. This law leads to the definition of a gravitational field in which a certain mass experiences an acceleration due to gravitational forces. The magnitude and direction of this field - by definition the gravitational force per unit mass - follows from the spatial distribution of _mass. For this rea:son the mass of matter is indicated as the source of the gravitational field in field theory. Assuming that the law of gravity is also valid for the elementary particles within an atom, the direct gravitational force Fg,4 exerted on a body is found by summation of the above forces on the elementary particles of the body, so that
where m1 indicates the mass of the ith particle and g1 the gravitational field at the location of the ith particle. The variations in the value of g1 within matter are, of course, unknown. However, by taking the average of a large number of particles both in space and in time, the summation referred to can be rewritten as an integral over the volume filled by the particles:
Fg.d
=
fem
g dV; vf!m in this case indicates the macroscopie mass density and g the macroscopie gravitational field within the matter. As f!m is now a continuous quantity, the matter may be assumed to be a continuum. NaturaJJy, the macroscopie field of gravitation does not show the fluctuations due to the individual elementary particles. The direct gravitational force on a body may also be found by writing
Fg,d
=
J
f!mgo dV, vin which g0 represents the gravitational field due to all matter outside the body concerned. The quantity f!m g0 , however, is not a measure for the density of the direct gravitational force within the matter, since this assumption would lead to the conclusion that this density within the earth is related only to the local mass density and to the mass outside the earth.
Finally, it may be pointed out that the ponderomotive gravitational force which is exerted on a body in a liquid or gas is given by
in which
4
-Fa.h =
J
-ph n dA Ais the result of the mechanica! forces exerted on the surface A under the influence of the hydrostatic pressure, p1" on the body. The ponderomotive and direct forces will be equal only in a vacuum.
The forces of electromagnetic origin were first systematically investigated and formulated by Coulomb, Ampère and Oersted. To explain these phenomena Faraday (!"791-1867) introduced the concepts of electrical and magnetic fields. It was Maxwell (1831-1879) who finally succeeded in describing all macro-scopically observable eiectromagnetic phenomenà in his well-known field equations.
The definition of the electrical field E is based on Coulomb's law. This law describes the action of force between two statie point charges in a vacuum. The electrical field determines the force exerted on a statie point charge q 'in a vacuum and is given by
F=qK.
The magnetic field may be defined in the same way with Coulomb's magnetic law, which gives us the action of force between two hypothetical point charges in a vacuum. Historically, this hypothesis goes back to the time before Oersted, when electrical and magnetic phenomena were not thought to be related.
At present the magnetic field Bin a vacuum is preferably defined by the force exerted on a point charge q moving with velocity v, with respect to an observer in a magnetic field B. The formula derived by Lorentz is written as
F = qvxB.
In this way both the electrical and the magnetic field are defined by the force of electromagnetic origin Fem exerted on a moving point charge in a vacuum. This force, named the Lorentz force, is given by the expression
Fem = q(E
+
vxB).From the above it immediately follows that observers moving with respect to each other will in general find different values for both the electrical and the magnetic field. In the special case in which the observers move in relation to each other in straight lines and with constant velocity, the exact relation between the observed values can be derived from the well-known Lorentz transforma-tions 1 ). If the difference between these velocities is small with respect to the speed of light, the classica! Galilean transformation might give a good approxi-mation, though in fact the theory of the efoctromagnetic field is not invariant for Galilean transformations.
The actual electromagnetic field within the matter cannot be determined experimentally. The effect of matter on the electromagnetic phenomena outside the matter, however, can be defined effectively by using Maxwell's field equa-tions. As wilt be evident later, movement, if any, of matter should be duly taken into account.
In our considerations we assume t_hat the electromagnetic phenomena taking place within the matter on a microscopie scale can be described in principle with the electron theory of Lorentz. In this theory all electromagnetic phenom-ena within the matter are described by means of a microscopie electromagnetic ,_ field, which is again defined as the force exerted on a point charge:
F.m = q (e
+
v xb).In this formula e and b represent the microscopie electrical and magnetic fields. According to this theory all electromagnetic phenomena are ascribed to electri-cally charged particles either at rest or in motion. lt is found that this physical model can indeed be used as a basis for Maxwell's equations, for, after averaging the quantities and equations derived by means of the electron theory, equations are obtained which are unconditionally similar to Maxwell's (macroscopie) field equations.
At present increasing use is being made of a model with magnetic (polariza-tion) charges in the material in order to describe macroscopie magnetic phenom-ena. A review of the various formulations of Maxwell's laws together with their advantages and disadvautages has been published by Hofmann 2 ) and by Penfield and Haus 3). We prefer Boffi's formulation 4 ), since it can be based on the electron theory which, as such, is simple in conception and, from the physical point of view, extremely refined.
From the electron theory it fellows tbat the direct electromagnetic force on a body (the sum of all direct forces exerted on the charged particles of that body) IS
F.m =
L:
(q;e,
+
qt Vtx
b;), ,fin which q, and V; respectively represent the charge and th~ velocity of the ith particle, while e1 and b1 are the microscopie electrical and magnetic field at the
location of the ith particle. The average values of e and b are equated to the macroscopie-field quantities E and B, respectively. As in the procedure followed in determining the macroscopie mass density of matter, average values should be taken of q1 and q1 v1• However, complications now arise. These
complica-tions are all due to the fact that the distribution of sources within matter made up of positive and negative elementary charge carriers depends on the resulting electromagnetic field (in éietennining the mass density of solids any similar influe'!ce of the gravitational force can be entirely neglected).
6
-In the case of conductors and semiconductors the electrical field will cause a shift of the free electrons (comparable with the transport of gas molecules by gravitational forces within a closed system). In dielectric materials polarization will take place as a result of interatomic shifts of elementary positive and nega-tive charges. In magnetic materials magnetization will take place as a result of interatomic displacements of elementary circulating currents.
Hofmann points out that in the necessary process of averaging the charge and current density two fundamentally different procedures may be followed. In the first procedure the averaging process is carried out over elements of volume having an arbitrary spatial boundary and therefore not always containing an integral number of molecules (this is what he calls a "mathematica! boundary"). In contrast to this summation, the second procedure consists in averaging over elements of volume containing an integral number of molecules, so that the boundaries are not arbitrary (this is called a "physiqtl boundary") 5 ). In cal-culating physical force densities the physical boundary should in principle be employed 6 ). The mathematica! boundary leads to a macroscopie force density r.m M, the physical boundary to a macroscopie force density femPH. Both are functions .of quantities found in Maxwell's equations.
Calculation of electromagnetic forces on bodies surrounded by an electro-magnetically neutra! medium (e.g. vacuum) can be done either by integrating femPH over the volume of these bodies or by a similar integration of r.m M. In this case surface forces of electromagnetic origin are altogether absent and it is not important how the boundary is chosen 7 ). A further important conclusion is that the calculation of electromagnetic torques exerted on these bodies is simplest when the mathematica! boundary is used. In this case the simple expression
is valid, in which r is the radius vector drawn from an arbitrary point. When the physical boundary is used this calculation is more complicated, since in this case a macroscopie torque on the physically bounded element of volume has to be taken into account 8 ). '
The preceding genera! consideration leads to the conclusion that determina-tion of the electromagnetic torque necessitates a knowledge of the electro-magnetic-field distribution. However, such a field calculation is feasible only in exceptional cases. It means that solutions have to be found for the Maxwell equations in a space .in which, in the case of electromechanical energy conver-sion, moving matter is present. In spite of this, the method of calculation indicated gains in significance since the possibility of setting up an impulse relation for the electromagnetic field is present. Taking this impulse relation in to consideration, the conclusion is justified that in practical cases the neces-sary volume integration of the force density can be reduced to integration of an
electromagnetic stress along a surface enclosing the body in question. In this case success or failure depends on the possibility of calculating the electro-magnetic field outside the body in question.
Other methods of calculation employ the law of conservation of energy as applied to an electromechanical system. Applying this law in the case of a virtual displacement of matter within the system, it appears to be possible under certain conditions to find an expression for a force density of electromagnetic origin. According to the procedure followed this force density has a pondero-motive character which implies that, like fem M, it cannot be interpreted as a physical force density. The limiting conditions result from the necessary require-ments that the energies involved in the energy conversion have to be described with so-called "state functions" 9 ). Consequently, systems with mechanica!, electrical or magnetic hysteresis cannot be investigated by these methods. For-tunately, the influence of the above phenomena is usually negligible so far as the action of the electromagnetic forces is concerned. The "hysteresis motor'', of course, is an important exception.
A considerable simplification results if the electromagnetic-field energy of the system under consideration can be calculated with reasonable accuracy from the fields of zero order 10). In this case the electrical or magnetic field follows from the field equations of the electrostatic or magnetostatic field (by the magnetostatic field is meant the field due to permanent magnets and constant currents). Fields of this kind are termed quasi-statie fields. Under these con-ditions the electromagnetic-field energy within the system no longer depends explicitly on time and the velocity of moving matter in th~ system. Further simplifications are possible if the matter within the system is isotropic and rigid. Applications of the method under consideration to systems with anisotropic and elastic matter are found e.g. in the work of Stratton 11 ).
Finally, if it is possible to describe the sources of the electrical and magnetic fields with currents (i) in conductors of small cross-section, charges (q) on con-ductors and variables determining the position of the matter, then the behaviour of the electromechanical system can be described w.ith the aid of an electro-mechanical network. Variables and components of the network are usually accessible for practical measurements. The components can be identified with resistors, inductors and capacitors ("lumped constants") if the media within the system are electrically and magnetically linear, viz. s, = s,(x,y,z) and
µ,
=
µ,(x,y,z). Though the methods of calculation based on the law of con-servation of energy also require a field calculation, an approxi-mation of the field distribution is found in some cases to lead to very good and useful results. Chapter 1 describes the method of calculation based on the macroscopie impulse relation of the electromagneti_c field. In connection with this considera-tion a rough description is given in sec. 1 .1 of the averaging process applied to the equations from electron theory. In sec. 1.2 a derivation is given of the8
-macroscopie electromagnetic impulse relation, while sec. 1.3 gives a derivation
of the macroscopie field equations for slowly moving matter.
Chapter 2 presents considerations regarding the power balance of an elec tro-mechanical system ne~lecting relativistic inftuences. In sec. 2.1 the power balance is given for an electromechanical system without electrical or magnetic hysteresis phenomena. Section 2.2 considers the co;1ditions under which a quasi-statie treatment is possible. With regard to this approach a distinction is made between
electromechanical systems of the electrical and of the magnetic type. In sec. 2.3 attention is devoted to the network conception of these two types of systems. ·
Finally, in sec. 2.4 expressions for the electrical- and magnetic-field energy of a system with permanently magnetized and polarized matter are derived. ·
Chapter 3 describes the method of calculating electromagnetic forces and torques from the power balances examined in chapter 2 and the principle
of virtual displacement. The discussion is limited to systems of the magnetic type. In sec, 3.1 resulting expressions for the ponderomotive-force density and
surface-force density of electromagnetic origin are given, while in sec. 3.2 an expression is deri.ved for the electromagnetic torque of electrical machines with
thin current-carrying conductors and negligible iron loss. The latter expression has the various circuit currents and rotor position as variables. In sec. 3.3 it is
shown that the expression for the torque found in sec. 3.2 may in many cases
be simplified by suitable choice of the coordinate system (x,y,z) of the stationary
observer. In sec. 3.4 the physical meaning of simplifying coordinate trans-formations is investigated, while sec. 3.5 presents a treatment of commutator machines, illustrating the use of the transformations discussed.
In chapter 4 possibilities for applying the preceding theoretica! results are
further investigated. In this connection sec. 4.1 deals with the calculation of
the magnetic-field distribution within an electrical machine. Furthermore in
sec. 4.2 the usefulness of a qualitative treatment of the field distribution within an electrical machine with salient poles is illustrated. Finally, sec. 4.3 indicates how the surface-force density of electromagnetic origin can be used in
1. METHODS OF CALCULATION USING THE MACROSCOPIC IMPULSE RELATION
1.1. Derivation of Maxwell's equations from the electron theory of Lorentz In Lorentz' electron theory the microscopie electrical and magnetic fields e and b are given by the equations
ob rot e = - - ,
àt div b
=
0,1 be
rot b = µo Im1
+ - - ,
c2 bt . . (2 d1v e = - . eo (1.1) (1.2) (1.3) (1.4) In (I .3) Im1 represents the microscopie current density and in (1.4)e
is the microscopie charge density; c = (e0 µ0) - 112 is the velocity of light in a vacuum. Introduction of the microscopie charge densitye
ahd the microscopie current density Im1=
e
vm1 indicates that the elementary charges are considered as microscopically small clouds of charge. The law of conservation of charge is implied in the relations (1.3) and (1.4) and leads to(1.5) The microscopie force density of electromagnetic origin is given by
fem
=
(2 (e+
Vmi x b)=
(2 e+
Imi x b. (1.6) Ultimately, it is this microscopie force density on which the electromechanical energy conversion is based.The equations (1.1)-(1.6) describe the microscopie electromagnetic phenom-ena as these present thefuselves to a stationary observer in a coordinate system (x,y,z). The structure of the equations is independent of the movement of matter,. for all microscopie and macroscopie movements of matter are as-similated in the source functions e(x,y,z,t) and Irn1(x,y,z,t). Within matter the
microscopie electromagnetic field is not ac<;essible for calculation. However, by averaging the relations (1.1)-(1.6) over space and time and applying the mathematica! boundary to the elements of volume, equations are obtained containing macroscopie quantities which can in principle be determined with theoretica! methods. From (1.1 )-(1.6), after the above averaging process (in-dicated by the symbol
<
)
),
the following set of equations is found:- 10 -oB rotE = - - , àt div B
=
0, 1 oE rot B=
µ0 11+ - - ,
c2 àt d. 1vE=-, (!1 éo à(!, ..-+
d1vl1 =0, àt (1.7) (1.8) (l.9) (!.10) (1.11)in which E
=
<e), B=
<b), I,=
<Imi) and (!, "~ <e). If it is further sup-posed that the elements of volume are bounded mathematically and may be regarded as infinitely small (which in fact means that macroscopically an ideal continuum: is present), the macroscopie direct force density can be written as( 1.12)
The force density in (1.12) naturally has a mathematica! character (indicated by the superscript M) and should therefore not be interpreted as a physical
force density within the matter. Nevertheless, the simple relation ( 1.12) is used below because the object is to calculate and measure the total force F.m and
torque Tem of a body in air.
The macroscopie charge density can be given as
(1.13)
In relation (1.13) (!f represents the density of the free charge carriers (free to travel within the matter). The quantity (!p represents the charge density which manifests itself under the influence of the electrical polarization phenomena.
lt is worthwhile to introduce a polarization vector P and postulate
ev
=
-\J. P. (l'.l4)The macroscopie current density can be split up int9 three components: ( 1.15)
The current density If is attributed to free charge carriers, while the
polariza-tion-current density lv is related to polarization phenomena. From relations
(1.11) and (l.14) it follows that
oP
lp= -.
It will be seen that a macroscopie transport of charge is not necessarily
accom-panied by a macroscopie density of charge. In this connection reference may be made to currents in metal conductors. Therefore a macroscopie current
density
Im
(the third component) also has to be taken into account. Though this component does not play a part in the transport of charge over large distances it does have a macroscopie effect. These currents are termed ei reu lat-ing currents, clearly indicatlat-ing their divergence-free character.Ifthe matter is stationary the macroscopie circulating currents may be directly
connected with the average value of elementary atomie circulating currents, which in the case of magnetic materials can lead to macroscopically perceptible magnetization. It may be worthwhile to represent the· distribution of the macroscopie circulating currents by a magnetization vector M, as follows:
( 1.17) That only circulating currents are involved is clearly apparent from the relation
\l. I'" = \l. \l xM = 0. ( 1.18)
The averaging process roughly presented here and applied to eqs (1.1)~(1.4) can be effected completely mathematically 12 •13 •14). In this process it becomes
apparent that not only the dipole moments, but in principle also the quadru-poles etc., as well as the movement of matter, are involved in finding the vectors P and M. If
D
=
e0 E+
P ( 1.18)and
B = µ0 (H
+
M) ( 1.19)are introduced and the relations (1.13)-(1.17) used, the eq uations (!. 7)-(1.11) will assume the same form as the well-known Maxwell field equations
oB \l XE = - -,
ot
\l . B=
0, oD \l XH = lf+ - ,ot
\l . D=
r!J· (1.20) (1.21) (1.22) (1.23) The field equations (1.20)-(1.23) contain (!f and If as source functions. The influence of the matter is expressed by the relations (1.18) and (1.19). lf thematter is stationary,
e
1 and If describe the sources with respect to the stationary matter; this implies a considerable simplification in using these field equations. In the stationary condition, however, electromechanical energy conversion is-
12-Maxwell's field equations for systems with moving matter will be investigated further in sec. J .3.
1.2. The macroscopie electromagnetic impulse relation
The total electromagnetic force exerted on a body bounded by a vacuum (air) can always be written as
F em =
J
fem M d V, v(1.24)
where fem Mis obtained from (l.12). Using (1.9) and (l.10), expression (1.12) can
be written as
fem M = êo ('V . E) E + - -('V
x
B)
X B - êo EX ('V X E) - êo -à (EX B).µo àt
With vector statement V this may be written a5 fem M
=
êo ('\l . E) E+
êo (E. 'V) E--!
'\}
(e0 E2 )+
+
~
('V . B) B+
~
(B. \/) B- t 'V (82) - µ0 e0~(Ex_!_)·
(1.25)µo µo µo àt µo
The term (l/µ0 ) ('V . B) B, which equals zero, has been includyd in relation
( 1.25), in order to underline the symmetry between the electrical and the magnetic components.
The integral formula B enables the expression (1.24) to be written as
(1.26) with
Pem
=
êo (E . n)E
-
-!
n ( e0 E2 )+
(B . n)_!_
_
-! n
(~).
µo µo (1.27)
If we integrate over space in its entirety, the surface integral approaches zero
and the result wil\ be
(J .28)
If the mechanica! impulse is represented by Gmech and the force of other than
electromagnetic origin by F', it is found from elementary mechanics that · dGmech
F' +Fem =
-dt (1.29)
Using (1.28) and (l.29), integration over space in its entirety (F' = 0) results m:
dGmech
r (
B ) .- - +
s0µ0Ex- dV=O.
dt · · µo
v
(1.30)
On the basis of relation (1.30), one arrives at the following definition for an electromagnetic density of impulse:
ge=
2_
(E
x
_!_)
cl µo and at the following for an electromagnetic impu1se
Ge=
J
gemdV.
v
(1.31)
(1.32) According to (1.30), (1.31) and (1.32) it is valid to state that for space in its entirety
Ge
+
Gmech = constant. (1.33)For a finite volume V, surrounded by a surface A in air, it is found with the aid of (l.26) and (1.29) that
d
- (Gmech
+
Ge) =J
Pem dA+
F'. (J.34)dl A
The quantities fem M and Pem may also be written as
(l .35a) and
Pem = n.
2s,
(1.35b)with the symmetrical stress tensor:
B2 B B
2S
=
t
s0 E1 1 - so E E+
t
-
I - - - . (l .35c)µo µo
In electromechanical systems with statie electric and magnetic fields, the force of electromagnetic origin exerted on the arbitrarily polarized and magnetiz~d matter wil! be
Fem
=
J
Pem dA,A
with the surface A choseR as being in air outside the matter.
(1.36)
In electromechanical systems in which the electromagnetic field can be regarded as quasi-statie the derivation with respect to time of the electro-magnetic impulse - due, amongst other things, to the factor c2 = (s0 µ
0 ) -1
in the denominator of (1.31) - is always negligibly small with respect to the other terms in relation (1.34). Consequently, relation (1.36) can be used in these cases. Because of the symmetry of the tensor 2S it follows (see, for example,
Tem=
f
r XPem dA. (1.37)A
The important conclusion from the above is that the expressions (1.36) and (1.37) are valid for an arbitrary distribution of the polarization Pand magnetiza-tion M within the closed surface A.
A practical calculation of Fem and Tem naturally requires knowledge of the values of B and E at the surface A in air. Further investigations into systems with moving matter are therefore conducted in sec. 1.3 in order to obtain Maxwell's field equations for these cases in a more convenient formulation. 1.3. Maxwell's field equations for systems with "slowly" moving matter
With a view to the theory in chapter 2 it would also be useful to investigate whether the sources of field equations ( 1.7)-(1. IO) could be expressed in terms
of the sources observed by an observer moving along with the matter and in terms of velocity v of the matter (assumed rigid) with respect to the stationary observer.
An observer in a c.oordinate system (x',y',z') firmly connected with the matter will describe the sources of the fields E' and B' observed by him in. terms of the functions: e,'(x',y',z',t') and l,'(x',y',z',t') and specify these sources in greater detail with the quantities
e/,
f/,
P' and ~M'. Neglecting relativistic inftuences-( v
«
c, so that t=
t' and d V=
d V'), the stationary observer can describe,the dashed quantities in his own coordinate system (x,y,z) by using the Galilean transformations. Quantities transformed in this way will be indicated by *,so that t!J*(x,y,z,t)
=
e/(x',y',z',t'), P*(x,y,z,t)=
P'(x',y',z',t'), etc. From the definition of charge dènsity it follows thatf!J
=
f2J* (1.38)and
f!u=e/=-\J.P*. (1.39)
From (1.14) and (1.39) it is found that
p =P*. (1.40)
Furthermore it will be considered valid to state that
l:r =lf*
+
e/v
.
(1.41)As
v .
Im=
0, the conclusion may be drawn that as a result of the magnetiza-tion M*, the stamagnetiza-tionary observer will experience a current density (circulating-current distribution):I(M*) =lm*
=
v
xM*. (1.42)The current density which will be noted by the stationary observer as a result of the polarization P* is
in analogy with (1.41), with IP* = (oP*/ot)mat· From vector statement 1 (proof of which is given in ref. 16) and relations (1.39) and (1.43) it follows that
àP*
I(P*) = - + \J x(P* xv). àt
(1.44) According to the above theory, the sum of current densities l(M*) and l(P*)
sh~rnld equal the sum of Im and IP. Therefore
àP* àP
\J xM* + - + \J x(P*xv) = \J xM + - .
àt àt
As P
=
P* it follows thatM=M*+P*xv. ( 1.45)
Moving polarized matter can therefore contribute to the magnetization M. Substitution of the relations (1.38)-(1.41) and (1.45) in the original field equa-tions (1. 7)-(1.10) gives the field equaequa-tions
àB \l xE = - - , (1.46) àt \l . B
=
0, (1.47) B W* ~ \l x- = 11* +e
1*
v + \l x(M* + P* xv) + - - + Bo - , (1.48) µo àt àt Bo \l . E =e.
*.
(1.49)The equations (1.46)-(1.49) are known as Maxwell's equations for "slowly" moving matter.
The forrnula for the Lorentz force indicates that
E*
=
E+
vxB. ( l.50)At this stage it would also be interesting to know the relationship between B*, E, B and v when relativistic infiuences are neglected. From relations (1.7) and (1.50) it follows that
àB =-\! xE*
+
\J x(vxB) = (oB*) + \l x(vxB). (1.51)àt Of mat
Using (1.8) and vector statement 1 we can conclude that B =B*.
For the sake of completeness we rnay also write:
B
H =--M
=
H*+
vxP*, µo(1.52)
D = e0E+P =D*-e0vxB*. (1.54) lt is easy to see that relations ( 1.52) and (1.50) in principle conflict. The Jatter relation can therefore only be explained by the relativity principle. If we assume that y
=
{l - (v/c)2 } 112 ~Land that the occurring accelerations are small, thisprinciple leads to the following relations between dashed and non-dashed variables 1 7 ): v
ei
=er'
+ - . 1/, c2 (1.55) 1, = 1/ +ver', (1.56) v p =P' + -xM' (1.57) 2 ' C· M =M' + P'xv, (1.58) E = E' -vxB', (1.59) v B =B' +-xE' (1.60) 2 ' c v D =D'--xH' (1.61) 2 ' c H = H'+vxD'. (1.62)Comparison of the above relations (1.55)-(1.62) with relations (l.38)-(1.54) shows that care has sometimes to be exercised in applying the Galilean in-variance.
2. THE POWER BALANCE OF AN
ELECTROMECHANICAL SYSTEM
-lt can be shown that with a certain amount of calculation the following electromagnetic power balance may be derived from relations (1.7)-(1.11):
f
n. (Ex!_) dA+
~
j
(t
e0 E2+ -
1- B2 ) dV+
f
E. 1, dV = 0. (2.1)µ0 dt 2µo
A V V
lt should be noted that the terms in this balance are valid for macroscopie phenomena o~ly and that a physical interpretation of the individual terms should be approached with caution beçause of the mathematica! bounding used. In all the present investigations, it will be assumed that the enclosing surface A lies in air.
The first term in relation (2;1) can be interpreted as the electromagnetic power passing through surface A in an outward direction. The electromagnetic power delivered can therefore be represented by
Pe
=-J
n.(Ex!_)dv. (2.2). A µO
The second term leads toa definition of the energy density of the electromagnetic field:
B2
w
=
t
êo E2+ -- .
2µo
(2.3) The remaining term E. 1, sholild now constitute a measure for the power per unit volume converted into some other form of energy under the influence of moving charge carriers, e.g. chemica! energy (accumulator), heat (electrical dissipation within the material), energy stored inside the material (polarization and magnetization energy) and, of course, mechanica! energy. The mechanica! energy can either leave the system through a mechanica! "gate", e.g. a rotating shaft, or be stored partly or fully inside the system in kinetic or potential form. On the other hand it may be dissipated due to mechanica! friction.
If the macroscopie speed of the matter is indicated by the vector v, the power converted into mechanica! form per unit volume is
femM. v
=
(e,
E+
1, xB). v. (2.4) The total electromechanical power then equalsP.m
=
J
r.m M. v dv
=
J
(e,
E+
1,x
B). v dv.
(2.5)v v
This electromechanical power can be represented as a separate term in the power balance:
- 18
-d ( B1 )
Pe =
-!
!
e0 E1 + - dV + f (E + v xB). (1, - (h v) dV + Pem· (2.6)dt 2 µ0
v v
Relation (2.6) is the starting point for further investigation in this chapter. 2.1. The power balance for a system without electrical or magnetic hysteresis·
phenomena
If relation (I.50) is used and
(2.7) introduced, power balance (2.6) can be written as
d ( B1 )
Pe
=
-!
!
êo E1+ - -
d V X f E* . 1, * d V+
Pem·dt 2 µ0
v v
(2.8)
Let us now suppose that the relation between 1/ and E' can be represented by Ohm's law in the specific form
1/
=
a (E'+
Fq'), (2.9)where ais the specific conductivity and
F/
a force per unit charge which does not have its origin in the electrical field (e.g. the force of chemica! origin in an accumulator). For the sake of simplicity we shall assume that, inside V, Fq' is zero. Neglecting relativistic influences, the relationis also valid, so that
(1 *)1
f E*.1,*dV= f
_!~dV+
f E*.l/dV+f
E*.lm*dV. (2.11)v v v v
If the medium is free from electrical and magnetic hysteresis, the first integral in the right-hand term can be identified as the power Pd dissipated inside the matter. The second term can be identified as the rate of change of the polariza-tion energy Wp and the third as the rate of change of the magnetization energy
WM. The balance (2.8) can now be rewritten as
dW dWp dWM
P = - + - + - + P d + P ,
e dt dt dl em (2.12)
where
(2.13}
(2.14) lf the material inside volume V is at rest ( v
=
0) then naturally E*=
E and Ii* =
11 so that the power balance can be written in the form- f
n. (E xH) dA=
:t
f (
l
E. dD+
j
H. dB) d V+
f
E~
11 d V.A V O O V (2.15)
The vector S
=
Ex H in the left-hand term of relation (2.15) is known as the Poynting radiation vector 18). The density of the electromagnetic-field energycan now be represented by
(2.16) where D We = JE. dD (2.17) 0 and B Wm=JH.dB. (2.18) 0
The expressions (2.17) and (2.18) therefore respectively describe the density of the electrical- and magnetic-field energy in place~ in which no moving matter is present, provided that electrical and magnetic hysteresis phenomena are not present.
Using (2.14), we write (2.12) in the form
(2.19) where in accordance with (2.5) and with v
=
ds/dt(2.20)
The vector s in the above relation represents the displacement of an element of volume rigidly bound to the matter in a coordinate system (x,y,z). If the moving substance is rigid and rigidly connected toa coordinate system (x',y',z'), the displacement of a fixed point of this body can be represented by
s
=
s(a,b,c; a,{3,y; x',y',z'), (2.21) in which a, b and c correspond with the coordinates of the origin O', and a,f3
and y with the angles fixing the position of coordinate system (x',y',z') with respect to coordinate system (x,y,z). In the case of a free body the variables-
20-In order to apply the principle of the virtual displacement the electromagnetic-field energy Wem has to be a state function; in other words, it has to be possible to derive the field energy from the state of the above system at any moment without taking into account the history of the system.-For this reason electrical and magnetic hysteresis phenomena cannot be taken into consideration.
Suppose the electromagnetic-field energy is given by the state function (2.22) in which q 1 , ••• , q n are the independent varia bles of the system. A quasi-statie treatment is possible, providing Wem does not áplicitly depend on the velocity v and the time t, so that
(2.23) in which the independent variables of the system qi. ... , qm are independent of the velocity.
The conditions for treatment as a quasi-statie system and the form of the power balance under these conditions are investigated in sec. 2.2.
2.2. Quasi-statie electromechanical systems
If relativistic influences are neglected, the total electromagnetic-field energy can be written as
Wem =
j (
J'E
'.
dD'+
j'H'
.
dB')d
V.v 0 0
(2.24)
The dashed quantities can be observed in coordinate systems rigidly connected to the matter (v
=
0). Outside the matter, where P = M=
0, use can be made of an arbitrary coordinate system, e.g. the coordinate system (x,y,z) of the stationary observer. From relation (2.24) it can be concluded that Wem wilt be independent of v, if E' = E, B' = B, D' = D and H' = H. According to relations (1.58)-(1.61) this is only approximately the case iflv xB
'
I
« IE'I,
(2.25)lvxD
'
I
« IH'I,
(2.26)1 :2
xH
'
I
« ID'I,
(2.27)1 : 2
xE'I
« IB'I
.
(2.28)However, in the case of electromechanical energy transfer (v
=F
0), these con-ditions cannot be satisfied simultaneously.assume that v
<
200 m/s, E'<
107 V/m, H'<
105 A/m, D'<
10-5 C/m2and B'
<
2 Wb/m2 • These converters can be subdivided into 2 groups, the magnetiç and the electrostatic. The farmer group is characterized by high values of the field quantities B' and H', and low values of the field quantities E' and D', while in the second group the opposite is the case.2.2.1. Systems of the magnetic type
The magnetic fields are generated by conduction currents in metallic
con-ductors and by magnetizable materials. High current densities (5, 106 A/m 2)
can be obtained with the aid of low electrical-field strength (/1
=
a E withar::::; 5.107 Q-1 m-1), white magnetization M can magnify the magnetic in-duction B by a factor of 1000 or more
Cv
xB=
µ0 11+
µ0 Im)· Thefre-quencies of the conduction currents are low (
<
1000 Hz, in most cases 50or 60 Hz).
In the systems of the magnetic type conditions (2.26) and (2.28) are satisfied, which means that H' = H and B' = B. The magnetic-field energy is now given by
B
W m =
f ( f
H . dB) d V (2.29)v 0
and is no longer an explicit function of the velocity v. The electrical-field energy,
however, is still an explicit function of v, but can be neglected with respect to
the magnetic-field energy owing to the low values of E and D, so that
(2.30)
Neglecting the displacement current
oD
/
ot
(the value of D and that of the frequency are low) also has the consequence that Wem is no longer an explicit function of time.The field equations for an electromechanical system of the magnetic type can now be simplified to
oB
\1 XE = - - ,ot
\1 . B=
0, \l xH=
11 =1/, t:0 \1 . E= e,,
(2.31) (2.32) (2.33) (2.34)where B = ,u0 (H
+
M), with M = M*. The (uriivalent) relation between B,where
µ,
=
µr(x',y',z',H)and
M0 ~= M0(x' ,y' ,z').
The H field is now determined by the sources
\l xH
=
11*
and 1 ' / \l . H = - - (H . \J,u
,
+
\l . M0 ). µ, (2.35) (-2.36) (2.37) (2.38) (2.39)From relations (2.35), (2.38) and (2.39) it is evident that within an electro-mechanical system of thé; magnetic type the field quantities B and H are func-tions of the independent variables a, b, c, a, (3, y and the current distribution
11
=
1/.
Using relations (2.31), (2.38) and the vector statement I, it can now be shown that for the present system type
Pe- Pd=
jH.(àB) dV
v àt ma't
(2.40)
and that from the power balance (2.19) it follows that
f
H. (dB)mat d V=
dJ (
Î
H. dB) d V+
f
(fem • ds) d V.v v 0 v
(2.41)
Using the principle of virtual displacement an expression for the pondero-motive force density of magnetic origin will be derived from relation (2.41) in sec. 3.1.
2.2.2. Systems of the electrostatic type
In generating electrostatic fields, high concentrations of charge (high values of r}f) and media which can be well polarized (high values of !2v) are used.
The frequencies involved are of the same order of magnitude as in systems of the magnetic type (
< 1000
Hz). The present systems are less.suitable than the systems of the magnetic type for electromechanical energy conversion, due to the fact that the maximum electrostatic force densities and energy densities aresmall in comparison with the corresponding magnetic quantities:
!121 E;"' •. '
«
1
1
1 xBlmax( 1 B2)
_ (-} êo E2)max
« - -
·
2 µo max
These systems satisfy conditions (2.25) and (2.27), hence E'
=
E and D=
D'. The electrical-field energy W. is now given by(2.42)
and is no Jonger an explicit function of the velocity v. The magnetic-field energy is still an explicit function of v, but can be neglected with respect to
w
.,
so that (2.43) Neglecting the term àB/àt (thus \l xE "'' 0) results in Wem no longer being an explicit function of time.-~he field equations for an electromechanical system of the electrostatic type read: \l xE
=
0, \l . B = 0,B
àD \J-X-=
Q/
v+
\J x(Pxv)+
-
,
µoot
co \l . E=
Q,=
Q, *, where D = t:0 E+
P, with P = P*. (2.44) (2.45) (2.46) (2.47)In analogy with relation (2.35) the univaJent relation between D, E and P will be represented by D
=
c, co E+
Po, (2.48) where t:, = c, (x',y',z', E) (2.49) and Po=
Po (x' ,y',z'). (2.50)The E field is now determined by the two relations
v
xE = 0 (2.51)and
1
co V · E
=
-
-
(Bo E · \l êr+
\l • Po* - Q/). (2. 52)c,
From relations (2.48), (2.51) and (2.52) it appears that in an electromechanical system of the electrostatic type the fields E and D are functions of the
Using relations (2.21), (2.46) and vector statement I, we can write for the pres-ent type of system
P.-Pd=JE.(bD) dV,
i)f mai v
from which it follows that the power balance (2.19) can be deduced as
f
E. (dD)mal dV~
dJ (
f
E. dD) dV+
J
(fem. ds) dV.v v 0 v
(2.53)
(2.54)
Using the principle of virtual displacement, an expression for the pondero-motive force density of electrostatic origin can be derived from relation (2.54). 2.3. Network representation of electromechanical systems
2.3.1. Systems of the magnetic type
If, within the volume Vof an electromechanical system of the magnetic type, the distribution of the current density
1/
is fully determined by the position of thin conductors and the circuit currents flowing in lhese conductors (eddy currents and skin effect can be neglected in this case) and if, moreover, the magnetic induction at the surface A in air is negligibly small, a network repre-sentation of this type of system is possible. It is assumed that the exchange of electromagnetic power with the surroundings occurs by means of thin, current-carrying conductors at rest.Introducing B = V
x
A, -expression (2.40) for a system of the magnetic type can be deduced as(P.-Pd) dt
=
f
(dA)ma•. V xH dV-f
(dA)ma•. (n xH) dA. (2.55)v À
As the magnetic induction at surface A is negligibly small and V xH
=
lf*we find from (2.55):
(P. - Pd) dt
=
f
1/.
(dA)mat dv.
(2.56)v
lntegration need only be performed over the volume including the Nm
current-carrying circuits. Expression (2.56) can therefore be further reduced to
Nm
(P. - Pd) dt
=
~ ikf
(dA)mat. dC, (2.57) k= 1 ckwhere Ck represents the contour and ik the current of the kth circuit.
For the sake of sirnplicity we will suppose that within the considered volume V the current paths are not interrupted by capacitive elements, e.g. a capacitor
(capacitance). Current paths leaving the volume are reduced to a closed path by means of a connecting line on the surface A between connecting terminals of tha circuit under consideration (see fig. 2.1). The result of expression (2.57) is not effected by this, as the contribution to the contour integral at the surface is zero.
Fig. 2. l. Electrical circuit C k with terminals at a surface A. Circuit C k is transformed to a closed contour by means of a curve between terminals 1 and 2.
With Stokes' theorem relation (2.57) can now be written as Nm
(Pe - Pd) dt =
1:
ikf
(dB)mat. n dA, (2.58)k= l Ak
where Ak is a surface with C" as a boundary curve. If the linked flux of the kth electrical circuit is further defined by
• </>k
=
f
B . n dA, (2.59)·Ak
it is found from (2.58) and (2.59) that Nm
(Pe - Pd) dt =
1:
ik d<Pk· (2.60)k=l
1t should be noted that the change of flux dc/>k can be due to a change of B
with time, but also to a movement of contour Ck. These two contributions to
d</>k become quite clear if, using vector statement
I,
we writed</>k
=
f
dB . n dA+
f
\J x (B x ds) . n dA. (2.61) Using relation (I .51) the Faraday-Maxwell law of induction follows from (2.61):d</>k
ek
=
f
E*. dC= - - .
(2.62) ck dtIf the entire contour consists of a thin current-carrying conductor, then E* = 11
*
/a
and (2.62) can be reduced to- 26
-(2.63) where Rk is the resistance of the kth electrical circuit.
If the contour is partly situated on the surface A (see fig. 2.1), then
d</>k . 1
ek
= -
~=
Rk ik+
f
E* . dC.dt 2
(2.64)
The tangential component of E*, however, is rotation-free on the surface A (bB/bt ~ 0) and can consequently be derived from a scalar potential U, in view of the condition that E*
=
- '\l U. In this case (2.64) can be rewritten asd<f>k R . U <2J U <o ek
= - -
= k Ik+
k - k •dt (2.65)
The p9tential difference uk= uk<o - uk<2> is called the terminal voltage of
the kth electrical circuit and results in the well-known Kirchhoff relations for voltages:
(2.66)
For the powers P. and P4 we can now write: Nm P. = ~ uk i" (2.67) k= 1 and Nm Pd
=
~ i"2 Rk. (2.68) k= 1Anticipating the discussion of sec. 2.4 we assume that if all currents in V are zero the magnetic-field energy within the volume V is given by the state function W mo· (The energy W mo will be related to the permanent magnetiza-tion M0 in sec. 2.4.1). The total magnetic-field energy for arbitrary values of · the currents in V can then be represented by
,
Wm
=
w
mO+
f
(l
H . dB) dv,
v
(2.69)
where B0 is the magnetic induction in the zero-current condition. Introducing currents ik and linked fluxes cf>k allows relation (2.69) to be written as
Nm if>k
Wm
=
Wmo+
~J
ik d</>k> (2.70)where <Pko is the flux linked by the kth electical circuit if all currents ik are zero. The energy law (2.41) can now be written as
Nm Nm <bk
~ik d</>k
=
dWmo+
~ dJ
ik d<f>k+
Pem dt. (2. 71)k= 1 k= 1 <l>ko
The mechanica! variables and the currents ik (or the linked fluxes <Pk) can be treated as independent variables.
The energy balance lends itself without further difficulty to application of the principle of virtual displacement.
In
sec. 3.2 expression (2.71) will be the starting point for determiniog an expression for the electromagnetic torque of an elec-trical machine.2.3.2. Systems of the electrostatic type
If, within the volume V of an electromechanical system of the electrostatic type the charges (!f are present only as surface charges on conductors and if the electrical field E at surface A in air is negligibly small, the present system can be simplified to a network as follows:
·. As
E =-\JU and
\JU• (dD)mat
=
\J • [U (dD)ma1] - U \J • (dD)mat (2.53) can be written as(Pe - Pd) dt
=
f
U \J . (dD)mat d V -f
n. U (dD)mat dA.y A
As D at A is negligibly small and (!f
=
\J . D, the above expression can be simplified to(Pe- Pd) dt = JU (def)mat dV. (2.72)
v
Since, moreover, {!f is different from zero only at the surface of the conductors, the volume integration in (2.72) can be reduced to a surface integration cov-ering the Ne conductors, hence
Ne
(Pe - Pd) dt
=
~ UiJ
(daf)mat dA, (2.73)!=! At
where Ui is the potential of the /th conductor and af the surface-charge density. The charge on the /th conductor follows from
