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: $$ 0 \leq P_f \leq 1 $$

## For a purely resistive load:

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

## For a purely reactive load:

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".

For a leading power factor, current leads voltage and a capacitive load is implied.

For a lagging power factor, current lags voltage and an inductive load is implied.

### Some points to consider:

When the power factor is less than unity (1) we know that voltage and current are not in phase and real power will not equal apparent power. A load with a low power factor draws more current than a load with a higher power factor for the same amount of power transferred. In such a case there is an energy loss in the system. When such a situation is encountered, steps may be taken to provide power factor correction where the power factor of a given load is increased (thus increasing efficiency).

Now that we have a decent understanding of Apparent Power and the Power Factor, lets continue on to the topic of Complex Power:

Continue on to Complex Power...