162 ELECTRICAL APPARATUS

is the sum of the products of the horizontal components, and
of the vertical components, that is, of the real components and
of the imaginary components of these two quantities (as a
horizontal component of one does not represent any power with
a vertical component of the other quantity, being in quadrature
therewith).

PI I TTi 1 T ll

2 = |j&2 12 \)

where the brackets denote that the sum of the product of the
corresponding parts of the two quantities is taken.

As discussed in the preceding, the torque of an induction motor,
in synchronous watts, equals the power consumed by the primary
counter e.m.f.; that is:

and substituting (10) and (11) this gives:

D21 = seo2 {COST (g COST ± 61 sin r) + sin r (b cos r + 0| sin r) }

(12)

and herefrom follows the motor output or power, by multiplying
with (1 - s).
The sum of the torques of both motors, or the total torque, is :

2 Dt = Di + D2 = se02 {(ffi + 0) - (0i - flO cos 2r}. (13)
The difference of the torque of both motors, or the synchroniz-
ing torque, is:

2 D8 = se02 (61 - &) sin 2 r, (14)

where, by (7),

0i = ^J

m

SXi

mi m

= ri2 + s2£i2, m = (n + 2r)2 +

(15)

In these equations primary exciting current and primary
impedance are neglected. The primary impedance can be intro-
duced in the equations, by substituting (rx + $ro) for ri, and
(xi + xQ) for xi, in the expression of mi and m, and in this case
only the exciting current is neglected, and the results are suffi-
ciently accurate for most purposes, except for values of speed