SINGLE-PHASE COMMUTATOR MOTORS 409

of the voltage ratio, t, which makes the commutation current, ig,
a minimum. This value is giyen by:

f=0, (120)

where ig is giTen by equation (106).

Since equation (106) contains t only under the square root,
the minimum. value of i0 is given also by :

where:

K = [1 - JV4 + Sfb (SX'i - coV'Ol2 -f [&X"4 - Sfe UfcV 4 + S\"4)]2.
Carrying out this differentiation, and expanding, gives:

(co2 + S2) Co2 4 S

This is the same value as the real component, f, of the complex
voltage ratio, T\, which caused the commutation current to
vanish entirely, and was given by equation (112).

It is, approximately:

Co* +- AS2'

Substituting (121) into (105) gives the value of the minimum
commutation current, igo.

Since the expression is somewhat complicated, it is preferable
to introduce trigonometric functions, that is, substitute:

tan 8 = ^> (123)

A 4

where 5 is the phase angle of X4, and therefore:

X"4 = X4 sin 5,

X,co35,J. (124)

and also to introduce, as "before, the speed angle (116):

hence:

tan
S }
c =— >
(125)

q = Vco2H-S2;j

/S = g sin a, j (126)
Co == 2 COS <T. I