0195136047.pdf

550 ELECTROMECHANICS

i 1

L 11

L 22

θ

Movable coil i^2

Fixed coil

Figure P12.4.11

Stator b-phase axis

θ

ia

ib

if

−

+

Stator a-phase axis

Rotor field winding

magnetic axis

+−

Figure P12.4.12

12.4.13Consider the machine configuration of Problem
12.4.12. Let the rotor be stationary and constant
direct currentsIa,Ib,andIfbe supplied to the
windings. Further, letIaandIbbe equal. If the
rotor is now allowed to move, will it rotate con-
tinuously or will it tend to come to rest? If the
latter, find the value ofθfor stable equilibrium.

12.4.14Let us consider an elementary salient-pole, two-
phase, synchronous machine with nonuniform air
gap. The schematic representation is the same
as for Problem 12.4.12. Structurally the stator
is similar to that of Figure E12.4.1 except that
this machine has two identical stator windings
in quadrature instead of one. The salient-pole
rotor with two poles carries the field winding con-
nected to slip rings. The inductances are given as

Laa=L 0 +L 2 cos 2θ; Laf=Lcosθ

Lbb=L 0 −L 2 cos 2θ; Lbf=Lsinθ;

Lab=L 2 sin 2θ

Lffis a constant, independent ofθ;L 0 ,L 2 , andL

are positive constants; andθis the angle between

the magnetic axes of the statora-phase winding

and the rotor field winding.

(a) Let the stator windings carry balanced two-

phase currents given byia=Iacosωtand

ib=Iasinωt, and let the rotor winding be

excited by a constant direct currentIf. Let the

rotor revolve at synchronous speed so that its

instantaneous angular positionθis given by

θ=ωt+δ. Derive an expression for the