152 ELECTRICAL APPARATUS

frequency, but higher number of pairs of poles than the funda-
mental, and thus lower synchronous speeds, due to the deviation
of the space distribution of the motor winding from sine.

The fundamental motor torque, Ti, of Figs. 55 and 56, is given
by a sine wave of voltage and thus of flux, if the winding of each
phase is distributed around the circumference of the motor air
gap in a sinusoidal manner, as shown as F under "Sine," in Fig.
58, and the flux distribution of each phase around the circum-
ference of the air gap is sinusoidal also, as shown as <i> under
"Sine," in Fig. 58.«

This, however, is never the case, but the winding is always
distributed in a non-sinusoidal manner.

The space distribution of magnetizing force and thus of flux
of each phase, along the circumference of the motor air gap,
thus can in the general case be represented by a trigonometric
series, with co as space angle, in electrical degrees, that is, counting
a pair of poles as 2ir or 360°. It is then:

The distribution of the conductors of one phase, in the motor ;

air gap:

F — FQ {cos co + 0,3 cos 3 co + as cos 5 co + &? cos 7 co

+ a9 cos 9 co + ...}; (8) |

here the assumption is made, that all the harmonics are in phase, . I

that is, the magnetic distribution symmetrical. This is prac-
tically always the case, and if it were not, it would simply add
phase angle, am, to the harmonics, the same as in paragraphs 88 \

and 89, but would make no change in the result, as the component
torque harmonics are independent of the phase relations between I

the harmonic and the fundamental, as seen below. |

In a quarter-phase motor, the second phase is located 90° [

or a) = 5 displaced in space, from the first phase, and thus
represented by the expression:

7/ = Fo cos f co — ~1 + a3 cos (3 co —™] + a5 cos (5 co —™)
-f- 0,7 cos f 7 co —j2j 4- a9 cos (9 co —~0 + . . . |
f / 7r\ / 7T\ / 7T\
= Fo COS ( CO — s) + a3COS(3 CO + o +«5COS 5 CO — ™)
I \ ^/ \ Lt \ 2iJ
(9)