0195136047.pdf

12.4 FORCES AND TORQUES IN MAGNETIC-FIELD SYSTEMS 529

Depending on the convenience in a given situation, either Equation (12.4.17) or (12.4.18) can be
used. For the case of a translational electromechanical system consisting of only one-dimensional
motion, say, in the direction of the coordinatex, the torqueTeand the angular displacementdθm
are to be replaced by the forceFeand the linear displacementdx, respectively. Thus,

Fe=

∂Wm′(i 1 ,i 2 ,x)

∂x

(12.4.19)

and

Fe=−

∂Wm(λ 1 ,λ 2 ,x)

∂x

(12.4.20)

For a linear magnetic system, however, the magnetic energy and the coenergy are always
equal in magnitude. Thus,

Wm=Wm′ =

1

2

λ 1 i 1 +

1

2

λ 2 i 2 (12.4.21)

In linear electromagnetic systems, the relationships between flux linkage and currents (in a doubly
excited system) are given by

λ 1 =L 11 i 1 +L 12 i 2 (12.4.22)

λ 2 =L 21 i 1 +L 22 i 2 (12.4.23)

whereL 11 is the self-inductance of winding 1,L 22 is the self-inductance of winding 2, and
L 12 = L 21 = Mis the mutual inductance between windings 1 and 2. All of these in-
ductances are generally functions of the angleθm(mechanical or spatial variable) between
the magnetic axes of windings 1 and 2. Neglecting the iron-circuit reluctances, the electro-
magnetic torque can be found from either the energy or the coenergy stored in the mag-
netic field of the air-gap region by applying either Equation (12.4.17) or (12.4.18). For a
linear system, the energy or the coenergy stored in a pair of mutually coupled inductors is
given by

Wm′(i 1 ,i 2 ,θm)=

1

2

L 11 i 12 +L 12 i 1 i 2 +

1

2

L 22 i 22 (12.4.24)

The instantaneous electromagnetic torque is then given by

Te=

i^21

2

dL 11

dθm

+i 1 i 2

dL 12

dθm

+

i 22

2

dL 22

dθm

(12.4.25)

The first and third terms on the right-hand side of Equation (12.4.25), involving the angular
rate of change of self-inductance, are thereluctance-torqueterms; the middle term, involving the
angular rate of change of mutual inductance, is theexcitation torquecaused by the interaction of
fields produced by the stator and rotor currents in an electric machine. It is this mutual inductance
torque (or excitation torque) that is most commonly exploited in practical rotating machines.
Multiply excited systems with more than two sets of electrical terminals can be handled in
a manner similar to that for two pairs by assigning additional independent variables to the
terminals.
If none of the inductances is a function of the mechanical variableθm, no electromagnetic
torque is developed. If, on the other hand, the self-inductancesL 11 andL 22 are independent of
the angleθm, the reluctance torque is zero and the torque is produced only by the mutual term
L 12 (θm), as seen from Equation (12.4.25). Let us consider such a case in the following example
on the basis of thecoupled-circuit viewpoint(orcoupled-coils approach).