CHAPTER VII
HIGHER HARMONICS IN INDUCTION MOTORS
88. The usual theory and calculation of induction motors,
as discussed in " Theoretical Elements of Electrical Engineer-
ing" and in "Theory and Calculation of Alternating-current
Phenomena," is based on the assumption of the sine wave. That
is, it is assumed that the voltage impressed upon the motor
per phase, and therefore the magnetic flux and the current, are
sine waves, and it is further assumed, that the distribution of
the winding on the circumference of the armature or primary,
is sinusoidal in space. While in most cases this is sufficiently
the case, it is not always so, and especially the space or air-gap
distribution of the magnetic flux may sufficiently differ from sine
shape, to exert an appreciable effect on the torque at lower
speeds, and require consideration where motor action and
braking action with considerable power is required throughout
the entire range of speed.
JLet then:
e = ei cos 0 + £3 cos (3 <t> — a*) + £5 cos (5 <t> — as) + e7 cos
(7<£ - a7) + e* cos (9 0 - afl) + . . . (1)
be the voltage impressed upon one phase of the induction motor.
If the motor is a quarter-phase motor, the voltage of the
second motor phase, which lags 90° or -0 behind the first motor
z
phase, is:
a'= 61 cos (0 — ™j + e3 cos (3 <£—™ — a3) + 02 cos (50— -^ — <*
+ e? cos (7 cj>-----------a7} + 69 cos (9 <j> — ~ — agj + - -
= ei cos \<t> — |j + e3 cos (3 <t> - az + 0 + e5cos (d 4> — <*5 —
+ e7 cos (? <£ - a7 + !) + e9 cos (9 * - «9 - !) + . . • (2)
The magnetic flux produced by these two voltages thus con-
sists of $ series of component fluxes, corresponding respectively
144