60 ELECTRICAL APPARATUS

42. Let, in a synchronous motor:

EQ = impressed voltage,

E = counter e.m.f., or nominal induced

voltage,

Z = r + jx = synchronous impedance,
/ = ij — ^"2 = current,

it is then:

?0 = E + ZJ

= E + (rii + xi2)

ii - ri2),

or:

(4)

(5)

or, reduced to absolute values, and choosing:

E = e = real axis in equation (4),
EQ = 60 = real axis in equation (5),

6o2 = (e + rii + a^*2)2 +'(xii — n'2)2 [e = real axis], (6)
4" = (e<> — n\ + ^)2 + OB£I — r^2)2 [eo = real axis]. (7)

Equations (6) and (7) are the two forms of the fundamental
equation of the synchronous motor, in the form most convenient
for the calculation of load and speed curves.

In (7), ii is the energy component, and £2 the reactive com-
ponent of the current with respect to the impressed voltage, but
not with respect to the induced voltage; in (6), i\ is the energy
component and iz the reactive component of the current with
respect to the induced voltage, but not with respect to the
impressed voltage.

The condition of motor operation at unity power-factor is :

Thus:

i2 = 0 in equation (7).

e =

(8)

at no-load, for ii = 0, this gives: e = e0, as was to be expected.

Equation (8) gives the variation of the induced voltage and
thus of the field excitation, required to maintain unity power-
factor at all loads, that is, currents, i\.

From (8) follows:

^l =

re<> ±

(9)