268 ELECTRICAL APPARATUS

In the first case, if the reluctance is higher for rising, lower for
decreasing, current, the magnetism, which is proportional to
current divided by reluctance, is higher for decreasing than for
rising current; that is, its equivalent sine wave lags behind the
sine wave of current, and the e.m.f. or self-induction will lag
more than 90° behind the current; that is, it will consume
electrical power, and thereby deliver mechanical power, and do
work as a synchronous motor.

In the second case, if the reluctance is lower for rising, and
higher for decreasing, current, the magnetism is higher for rising
than for decreasing current, or the equivalent sine wave of
magnetism leads the sine wave of the current, and the counter
e.m.f. of self-induction lags less than 90° behind the current;
that is, yields electric power as generator, and thereby consumes
mechanical power.

In the first case the reactance will be represented by X = h +
jx, as in the case of hysteresis; while in the second case the
reactance will be represented by X = — h + fa*

153. The influence of the periodical variation of reactance
will obviously depend upon the nature of the variation, that is,
upon the shape of the reactance curve. Since, however, no
matter what shape the wave has, it can always be resolved in a
series of sine waves of double frequency, and its higher har-
monics, in first approximation the assumption can be made
that the reactance or the reluctance varies with double freauency
of the main current; that is, is represented in the form:

x — a + b cos 2 /3.

Let the inductance be represented by:
L = I + lr cos 2 0,

where y = amplitude of variation of inductance.
Let:
6 = angle of lag of zero value of current behind maximum value
of the inductance, L.
Then, assuming the current as sine wave, or replacing it by
the equivalent sine wave of effective intensity, I, current :
i = I V2sm(/3 - 0).