# SPQ and Power Factor in AC Systems

### SPQ

Apparent power, $$S$$, as a product of measured RMS voltage and current, can be split into the real and complex power conterparts.

Where $$\phi$$ is the phase angle, $$S$$ is the apparent power, $$Q$$ is the reactive power, $$P$$ is the active power.

Since $$Q$$ is purely complex, the direction of the vector is important and is defined as leading or lagging. Sign conventions are such that a leading reactive power is capacitive and generates power; lagging reactive power is inductive and absorbs power.

Calculations involving reactive power substitute resistance with reactance.

$$Q = \frac{V^2}{X}$$
$$Q = I^2 X$$

Where inductors are positive and capacitors and negative.

### Power Factor

Defined as

$$p.f. = \cos \phi$$