296

ELECTRICAL APPARATUS

i:

hence:

5 = e-be[A

and, substituting the trigonometric for the exponential functions,
gives ultimately:

6 = #e~6*cos (00 + 7). (25)

That is, the motor steadies down with an oscillation of period:

/o = Of

sin (a - /S) (c2 + pP0)2

4 TrzMo 64 x2M02

and decrement or attenuation constant:

(26)

(27)

170. It follows, however, that under the conditions considered,
a cumulative surging, or an oscillation with continuously increas-
ing amplitude, can not occur, but that a synchronous motor,
when displaced in phase from its mean position, returns thereto
either aperiodically, if Z>2 > a, or with an oscillation of vanishing
amplitude, if J>2 < a. At the worst, it may oscillate with constant
amplitude, if b = 0.

Cumulative surging can, therefore, occur only if in the differ-
ential equation (19):

de

(28)

the coefficient, b, is negative.

Since c2, representing the induction motor torque of the damp-
ing device, etc., is positive, and pP0 is also positive (p being
the exponent of power variation with speed), this presupposes

—h2
the existence of a third and negative term, 5 ,..,, , in 6:

O TTjlKtQ

, _ c2 + pP0 -A2. (M)

A

This negative term represents a power:

P2 = _fc2s. (30)

that is, a retarding torque during slow speed, or increasing /3, and
accelerating torque during high speed, or decreasing /?.

The source, of this torque may be found external to the motor,
or internal, in its magnetic circuit.

V,