Page 72 - Electric Machinery Fundamentals

48 ELECTRIC MACHINERY FUNDAMENTALS

I

+
v R

vir) z z ~ Z L e FIGURE 1-29
(a) A de voltage source supplying a
load with resistance R. (b) An ac
voltage source supplying a load with
(b) impedance Z = Z L 8 n.

The voltage applied to this load is
vCr) = V2V cos wt (1-56)
where V is the nns value of the voltage applied to the load, and the resulting cur-
rent flow is
;(1) = V21 cos(wt - 8) (1 -57)
where I is the rms value of the current flowing through the load.
The instantaneous power supplied to this load at any time t is
pet) = v(t);(t) = 2VI cos wt cos(wt - 8) (I- 58)
The angle 8 in this equation is the impedance angle of the load. For inductive
loads, the impedance angle is positive, and the current waveform lags the voltage
waveform by 8 degrees.
If we apply trigonometric identities to Equation (I-58), it can be manipu-
lated into an expression of the form
pet) = VI cos 8 (I + cos 2wt) + VI sin 8 sin 2wt (I-59)
The first term of this equation represents the power supplied to the load by the
component of current that is in phase with the voltage, while the second term rep-
resents the power supplied to the load by the component of current that is 90' out of
phase with the voltage. The components of this equation are plotted in Figure 1- 30.
Note that the first term of the instantaneous power expression is always pos-
itive, but it produces pulses of power instead of a constant value. The average
value of this term is
P = VI cos 8 (1- 60)
which is the average or real power (P) supplied to the load by term I of the Equa-
tion (1-59). The units of real power are watts (W), where 1 W = I V X 1 A.