The average power (in terms of the rms values) is a product of two terms. The product \(V_\textrm{rms}I_\textrm{rms}\) is known as the apparent power \(S\). The factor \(\cos(\theta_v - \theta_i)\) is called the power factor \(\mathrm{pf}\).
\(\displaystyle P = \frac{1}{2} V_m I_m \cos(\theta_v - \theta_i) = V_\textrm{rms}I_\textrm{rms}\cos(\theta_v - \theta_i)\)
The apparent power (in \(\textrm{VA}\)) is the product of the rms values of voltage and current.
\(\boxed{S = V_\textrm{rms}I_\textrm{rms}}\)
The apparent power is measured in volt-amperes or \(\textrm{VA}\) to distinguish it from the average (real) power, which is measured in watts.
The power factor is the cosine of the phase difference between voltage and current. It is also the cosine of the angle of the load impedance.
\(\boxed{\mathrm{pf} = \frac{P}{S} = \cos(\theta_v - \theta_i)}\)
The power factor is dimensionless, since it is the ratio of the average (real) power \(P\) to the apparent power \(S\). The angle \(\theta_v - \theta_i\) is called the power factor angle, since it is the angle whose cosine is the power factor.
The power factor angle is equal to the angle of the load impedance if \(\mathbf{V}\) is the voltage across the load and \(\mathbf{I}\) is the current through it.
\(\displaystyle \mathbf{Z} = \frac{\mathbf{V}}{\mathbf{I}} = \frac{V_m\angle\theta_v}{I_m\angle\theta_i} = \frac{V_m}{I_m}\angle(\theta_v - \theta_i)\)
\(\displaystyle \mathbf{Z} = \frac{\mathbf{V}_\textrm{rms}}{\mathbf{I}_\textrm{rms}} = \frac{V_\textrm{rms}\angle\theta_v}{I_\textrm{rms}\angle\theta_i} = \frac{V_\textrm{rms}}{I_\textrm{rms}}\angle(\theta_v - \theta_i)\)
The value of power factor (\(\mathrm{pf}\)) ranges between zero and unity. For a purely resistive load, the voltage and current are in phase, so that \(\theta_v - \theta_i = 0\) and \(\mathrm{pf} = 1\). This implies that the apparent power \(S\) is equal to the average power \(P\). For a purely reactive load, \(\theta_v - \theta_i = \pm 90^\circ\) and \(\mathrm{pf} = 0\). In this case the average power is zero. In between these two extreme cases, \(\mathrm{pf}\) is said to be leading or lagging. Leading power factor means that current leads voltage, which implies a capacitive load. Lagging power factor means that current lags voltage, implying an inductive load.