25621

ELECTROMECHANICAL ENERGY CONVERSION

25

Electromagm:!tic and Electrostatic Forces

The energy balance relationships given by (1.3-21) may be arranged as
K

I~ fek dXk =
k=1

I:
j=1

J

efj ij

dt - dWf

(1.3-58)

In order to obtain an expression for feb it is necessary to first express W and then f take its total derivative. One is tempted to substitute the integrand of(1.3-22) into (1.3-58) for the infinitesimal change of field energy. This procedure is, of course, incorrect because the integrand of (1.3-22) was obtained with all mechanical displacements held fixed (dXk = 0), where the total differential of the field energy is required in (1.3--58). The force or torque in any electromechanical system may be evaluated by employing (1.3-58). In many respects, onegains a much better understanding of the energy conversion process of a particular system by starting the derivation of the force or torque expressions with (1.3-58) rather than selecting a relationship from a table. However, for the sake of completeness, derivation of the force equations will be set forth and tabulated for electromechanical systems with K mechanical inputs and J electrical inputs[2]. For an electromagnetic system, (1.3-58) may be written as

(1.3-59)

With ij and Xk selected as independent variables we have
Wf

= Wf (il, , ij ;Xl, ,XK) Aj = Aj(i 1, , ij ;Xl, ,XK)

(1.3-60) (1.3-61)

From (1.3-60) and (1.3-61) we obtain

(1.3-62)

(1.3-63)

In (1.3-62) and (1.3-63) and hereafter in this development the functional notation of (iI, ... , ij; Xl, ... , XK) isabbreviated as (ij, Xk). The index n is used so as to avoid confusion with the index j because each dAj must be evaluated for changes in all currents in order to account for mutual coupling between electrical systems.

26

BASIC PRINCIPLES FOR ELECTRIC MACHINE ANALYSIS

[Recall that we did this in (1.3-42) for J = 2.] Substituting (1.3-62) and (1.3-63) into (1.3-59) yields

(1.3-64) Bygathering terms, we obtain

(1.3-65) When we equate coefficients, we obtain (1.3-66)

(1.3-67) Although (1.3-67) is of little practical importance, (1.3-66) can be used to evaluate the force at the kth mechanical terminal of an electromechanical system with only magnetic coupling fields and with ij and Xk selected as independent variables. A second force equation with ij and Xk as independentvariables may be obtained from (1.3-66) by incorporating the expression for coenergy. For a multiexcited system the coenergy may be expressed as
J

We

= Li)j j=l

Wf

(1.3-68)

Because ij and Xk are independent variables, the partial derivative with respect to x is (1.3-69) Hence, substituting (1.3-69) into (1.3-66) yields (1.3-70)

ELECTROMECHANICAL ENERGY CONVERSION

27

Table1.3-1 Electromagnetic Force at kth Mechanical Inputa

aFor rotational systems replace fek with Tek and

Xk

with

rh.

It should be recalled that positive fek and positive dXk are in the same direction. Also, if the magnetic system is linear, then We = Wf.

By a procedure similar to that used above, force equations may be developed for magnetic coupling with Aj and Xk as independentvariables. These relations are given in Table 1.3-1 without proof. In Table 1.3-1 the independent variables to be used are designated in each equation by the abbreviated functional notation. Although only translational mechanical systems have been considered, all force relationships developed herein may be modified for the purpose of evaluating the torque in rotational systems. In particular, whenconsidering a rotational system, fek is replaced with the electromagnetic torque Teb and Xk is replaced with the angular displacement 8k . These substitutions are justified because the change of mechanical energy in a rotational system is expressed as (1.3-71) The force equation for an electromechanical system with electric coupling fields may be derived by following a procedure similar to that...