Apparent Power and the Power Factor

Before directly addressing the definition of complex power, we will talk about "Apparent power" and the "power factor"

Apparent Power and the Power Factor:

Consider the voltage and current at the terminals of circuit. (defined as): $$v(t) = V_m\cos(\omega t +\theta_v)$$ $$i(t) = I_m\cos(\omega t + \theta_i)$$ We know from our discussion on Average Power, that average power is defined as: $$P = \frac{1}{2} V_m I_m \cos(\theta_v-\theta_i)$$ ...and average power in terms of RMS values is: $$P = V_{rms} I_{rms} \cos(\theta_v-\theta_i)$$ let: $$S = V_{rms} I_{rms}$$ Average power now becomes: $$P = S \cos(\theta_v-\theta_i)$$ We now define "Apparent Power" as:

$$S = Apparent \; Power = V_{rms} I_{rms}$$

Additionally we define the "Power Factor" as:

$$P_f = Power \; Factor = \cos(\theta_v-\theta_i) \qquad,(Eqn\;1)$$

We now notice that average power is the product of apparent power and the power factor. $$P = S P_f \qquad,(Eqn\;2)$$ Apparent power gets its name because it seems "apparent" that power is the product of voltage and current. The units of measurement for apparent power are volts-amperes (VA) (not watts as with average or real power). The power factor is dimensionless and is the ratio of the average power absorbed by a circuit to the apparent power exchanged between the circuit and its source. By manipulating equations 1 and 2 we get: $$P_f = \frac{P}{S} = \cos(\theta_v-\theta_i)$$ The power factor angle is the factor by which the apparent power must be multiplied in order to obtain the real/average power.

Power Factor Values:

Power factor values fall within the following range: $$0 \leq P_f \leq 1$$

Voltage and current are in phase. $$\theta_v-\theta_i = 0$$ Therefore: $$P_f = 1$$ ...and real/average power equals apparent power: $$S = P$$

Voltage and current are 90 degrees out of phase. $$\theta_v-\theta_i = \pm 90^{\circ}$$ Therefore: $$P_f = 0$$ ...and $$P = 0$$

When the power factor falls in between 0 and 1:

In such cases, the power factor can either be "leading" or "lagging".