0195136047.pdf

528 ELECTROMECHANICS

dW=Tedθm (12.4.7)

whereθmis expressed in electrical radians. Neglecting the winding resistances, the terminal

voltages are equal to the induced voltages given by

v 1 =e 1 =

dλ 1

dt

(12.4.8)

v 2 =e 2 =

dλ 2

dt

(12.4.9)

These are the volt-ampere equations, or the equations of motion, for the electrical side. Two

volt-ampere equations result because of the two sets of electrical terminals. Substituting these

into Equation (12.4.6), one gets

dWe=i 1 dλ 1 +i 2 dλ 2 (12.4.10)

Substituting Equations (12.4.7) and (12.4.10) into the differential form of Equation (12.4.5), we

have

dWm=dWe−dW=i 1 dλ 1 +i 2 dλ 2 −Tedθm (12.4.11)

By specifying one independent variable for each of the terminal pairs, i.e., two electrical variables

(flux linkages or currents) and one mechanical variableθmfor rotary motion, we shall attempt to

expressTein terms of the energy or the coenergy of the system.

Based on Equation (12.4.4) and the concepts of energy and coenergy, one has

Wm+Wm′=λ 1 i 1 +λ 2 i 2 (12.4.12)

Expressing Equation (12.4.12) in differential form, the expression for the differential coenergy

functiondWm′is obtained as

dWm′ =d(λ 1 i 1 )+d(λ 2 i 2 )−dWm

=λ 1 di 1 +i 1 dλ 1 +λ 2 di 2 +i 2 dλ 2 −dWm (12.4.13)

Substituting Equation (12.4.11) into Equation (12.4.13) and simplifying, one gets

dWm′=λ 1 di 1 +λ 2 di 2 +Tedθm (12.4.14)

Expressing the coenergy as a function of the independent variablesi 1 ,i 2 , andθm,

Wm′ =Wm′(i 1 ,i 2 ,θm) (12.4.15)

the total differential of the coenergy function can be written as

dWm′=

∂Wm′

∂i 1

di 1 +

∂Wm′

∂i 2

di 2 +

∂Wm′

∂θm

dθm (12.4.16)

On comparing Equations (12.4.14) and (12.4.16) term by term, the expression for the electromag-

netic torqueTeis obtained as

Te=

∂Wm′(i 1 ,i 2 ,θm)

∂θm

(12.4.17)

On the other hand, choosing the independent variablesλ 1 ,λ 2 , andθm, it can be shown that

Te=−

∂Wm(λ 1 ,λ 2 ,θm)

∂θm

(12.4.18)