0195136047.pdf

PROBLEMS 549

Ns turns

vs

is

− +

ωm

Rotor axis

Rotor

Stator pole axis

Stator

(d-axis or

direct axis)

Interpole axis

(q-axis or

quadrature axis)

θ = ωmt − δ

θ

Figure P12.4.8

12.4.10A two-winding system has its inductances given

by

L 11 =

k 1

x

=L 22 ;L 12 =L 21 =

k 2
x
wherek 1 andk 2 are constants. Neglecting the
winding resistances, derive an expression for the
electric force when both windings are connected
to the same voltage sourcev =Vm sin ωt.
Comment on its dependence onx.
12.4.11Two mutually coupled coils are shown in Figure
P12.4.11. The inductances of the coils areL 11 =
A,L 22 =B, andL 12 =L 21 =Ccosθ. Find the
electric torque for:
(a)i 1 =I 0 ,i 2 =0.
(b)i 1 =i 2 =I 0.
(c)i 1 =Imsinωt,i 2 =I 0.
(d)i 1 =i 2 =Imsinωt.
(e) Coil 1 short-circuited andi 2 =I 0.
*12.4.12Consider an elementary cylindrical-rotor two-
phase synchronous machine with uniform air
gap, as illustrated in the schematic diagram in
Figure P12.4.12. It is similar to that of Figure
E12.4.1, except that Figure P12.4.12 has two
identical stator windings in quadrature instead
of one. The self-inductance of the rotor or field
winding is a constant given byLffH; the self-
inductance of each stator winding is a constant

given byLaa=Lbb. The mutual inductance be-

tween the stator windings is zero since they are in

space quadrature; the mutual inductance between

a stator winding and the rotor winding depends

on the angular position of the rotor,

Laf=Lcosθ; Lbf=Lsinθ

whereLis the maximum value of the mutual in-

ductance, andθis the angle between the magnetic

axes of the statora-phase winding and the rotor

field winding.

(a) Let the instantaneous currents beia,ib,andif

in the respective windings. Obtain a general

expression for the electromagnetic torqueTe

in terms of these currents, angleθ, andL.

(b) Let the stator windings carry balanced two-

phase currents given byia=Iacosωt, and

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 the torque expression un-

der these conditions and describe its nature.

(c) For conditions of part (b), neglect the re-

sistance of the stator windings. Obtain the

volt–ampere equations at the terminals of

stator phasesaandb, and identify the speed–

voltage terms.