Substituting this value of AD in equation (2) gives
true power = AC × AB cos φ (4)
Now the power factor may be defined as: that quantity by which the apparent watts must be multiplied in order to give the true power. That is
true power = apparent watts × power factor (5)
Comparing equations (4) and (5), AC × AB in (4) is equal to the apparent watts, hence, the power factor in (5) is equal to cos φ. That is, the power factor is numerically equal to the cosine of the angle of phase difference between current and pressure.
EXAMPLE I.—An alternator supplies a current of 200 amperes at a pressure of 1,000 volts. If the phase difference between the current and pressure be 30°, what is the true power developed?
In fig. 1,359, draw AB to scale, equal to 200 amperes, and draw AC of indefinite length making an angle of 30° with AB. From B, draw BD perpendicular to AC which gives AD, the active component, and which measured with the same scale as was used in laying off AB, measures 173.2 amperes. The true power developed then is
true watts = 173.2 × 1,000 = 173.2 kw.
The true power may be calculated thus: