0195136047.pdf

12.4 FORCES AND TORQUES IN MAGNETIC-FIELD SYSTEMS 527

properties of the core material. Equations (12.4.1) and (12.4.2) may be interpreted graphically as
the area labeledenergyin Figure 12.4.1. The other area, labeledcoenergyin the figure, can be
expressed as

Wm′=vol

∫H 1

0

BdH=

∫i 1

0

λ(i) di=

∫F 1

0

φ(F)dF (12.4.3)

For a linear system in whichBandH,λandi,orφandFare proportional, it is easy to see that
the energy and the coenergy are numerically equal. For a nonlinear system, on the other hand,
the energy and the coenergy differ, as shown in Figure 12.4.1, but the sum of the energy and the
coenergy for a singly excited system is given by

Wm+Wm′ =vol·B 1 H 1 =λ 1 i 1 =φ 1 F 1 (12.4.4)
The energy stored in a singly excited system can be expressed in terms of self-inductance,
and that stored in adoubly excited systemin terms of self and mutual inductances, for the circuit-
analysis approach, as we pointed out earlier.
Let us now consider a model of an ideal (lossless) electromechanical energy converter that
is doubly excited, as shown in Figure 12.4.2, with two sets of electrical terminal pairs and one
mechanical terminal, schematically representing a motor. Note that all types of losses have been
excluded to form aconservativeenergy-conversion device that can be described by state functions
to yield the electromechanical coupling terms in electromechanics. A property of a conservative
system is that its energy is a function of its state only, and is described by the same independent
variables that describe the state. State functions at a given instant of time depend solely on the
state of the system at that instant and not on past history; they are independent of how the system
is brought to that particular state.
We shall now obtain an expression for the electromagnetic torqueTefrom the principle of
conservation of energy, which, for the case of asinkof electric energy (such as an electric motor),
may be expressed as

We=W+Wm

or in differential form as

dWe=dW+dWm (12.4.5)

whereWestands for electric energy input from electrical sources,Wmrepresents the energy stored
in the magnetic field of the two coils associated with the two electrical inputs, andWdenotes the
mechanical energy output.WeandWmay further be written in their differential forms,

dWe=v 1 i 1 dt+v 2 i 2 dt (12.4.6)

i 1

v 1

i 2

v 2

Doubly excited

lossless electromechanical

energy-conversion device

with conservative coupling fields

+

−

+−

Te, θm = ωmt + θm(0)

Figure 12.4.2Model of an ideal, doubly excited electromechanical energy converter.