PROBLEMS 551torque under these conditions and describe
its nature.
(b) Compare the torque with that of Problem
12.4.12(b).
(c) Can the machine be operated as a motor? As
a generator? Explain.
(d) Suppose that the field currentIfis brought to
zero. Will the machine continue to run?12.4.15Consider the analysis leading up to Equation
(12.4.32) for the torque of an elementary cylin-
drical machine with uniform air gap.
(a) Express the torque in terms ofF,Fs, andδs,
whereδsis the angle betweenF ̄andF ̄s.
(b) Express the torque in terms ofF,Fr, andδr,
whereδris the angle betweenF ̄andF ̄r.
(c) Neglecting magnetic saturation, obtain the
torque in terms ofB, Fr,andδr,whereBis
the peak value of the resultant flux-density
wave.
(d) Letφbe the resultant flux per pole given by
the product of the average value of the flux
density over a pole and the pole area. Express
the torque in terms ofφ,Fr,andδr, whereφis
the resultant flux produced by the combined
effect of the stator and rotor mmfs.
12.4.16An electromagnetic structure is characterized by
the inductancesL 11 =L 22 = 4 +2 cos 2θ
andL 12 =L 21 = 2 +cosθ. Neglecting the
resistances of the windings, find the torque as a
function ofθwhen both windings are connected
to the same ac voltage source such that
v 1 =v 2 = 220
√
2 sin 314t12.4.17By using the concept of interaction between mag-
netic fields, show that the electromagnetic torque
cannot be obtained by using a four-pole rotor in
a two-pole stator.
12.4.18For each of the following devices, is a reluctance
torque produced when their coils carry direct
current?
(a) Salient-pole stator carrying a coil, and a
salient-pole rotor.
(b) Salient-pole rotor carrying a coil, and a
salient-pole stator.
(c) Salient-pole stator carrying a coil, and a
cylindrical rotor.
(d) Salient-pole rotor carrying a coil, and a cylin-
drical stator.
(e) Cylindrical stator carrying a coil, and a cylin-
drical rotor.
(f) Cylindrical rotor carrying a coil, and a cylin-
drical stator.
(g) Cylindrical stator carrying a coil, and a
salient-pole rotor.
(h) Cylindrical rotor carrying a coil, and a
salient-pole stator.
*12.4.19An elementary two-pole rotating machine with
uniform air gap, as shown in Figure E12.4.1, has
a stator-winding self-inductanceLssof 50 mH, a
rotor-winding self-inductanceLrrof 50 mH, and
a maximum mutual inductanceLof 45 mH. If the
stator were excited from a 60-Hz source, and the
rotor were excited from a 25-Hz source, at what
speed or speeds would the machine be capable of
converting energy?
12.4.20A rotating electric machine with uniform air gap
has a cylindrical rotor winding with inductance
L 2 =1 H and a stator winding with inductanceL 1
=3 H. The mutual inductance varies sinusoidally
with the angleθbetween the winding axes, with a
maximum of 2 H. Resistances of the windings are
negligible. Compute the mean torque if the stator
current is 10 A (rms), the rotor is short-circuited,
and the angle between the winding axes is 45°.
12.4.21The self and mutual inductances of a machine
with two windings are given byL 11 =( 1 +
sinθ),L 22 = 2 ( 1 +sinθ), andL 12 =L 21 =
M=( 1 −sinθ). Assumingθ=45°, and letting
coils 1 and 2 be supplied by constant currentsI 1
=15 A andI 2 =−4 A, respectively, find the
following:
(a) Magnitude and direction of the developed
torque.
(b) Amount of energy supplied by each source.
(c) Rms value of the current in coil 2, if the
current of coil 1 is changed to a sinusoidal
current of 10 A (rms) at 60 Hz and coil 2 is
short-circuited.
(d) Instantaneous torque produced in part (c).
(e) Average torque in part (c).
12.4.22Consider the elementary two-pole rotating ma-
chine with uniform air gap shown in Figure
E12.4.1. LetNsbe the number of turns on the
stator,Nrthe number of turns on the rotor,lthe
axial length of the machine,rthe radius of the
rotor, andgthe length of the air gap.