Most domestic loads (such as washing machines, air conditioners, and refrigerators) and industrial loads (such as induction motors) are inductive and operate at a low lagging power factor.
An inductive load is modeled as a series combination of an inductor and a resistor.
The process of increasing the power factor without altering the voltage or current to the original load is known as power factor correction.
Consider the power triangle. If the original inductive load has apparent power \(S_1\).
\(\displaystyle P = S_1\cos(\theta_1)\)
\(\displaystyle Q_1 = S_1\sin(\theta_1) = P\tan(\theta_1)\)
To increase the power factor from \(\cos(\theta_1)\) to \(\cos(\theta_2)\) without altering the real power \(P\).
\(\displaystyle P = S_2\cos(\theta_2)\)
\(\displaystyle Q_2 = S_2\sin(\theta_2) = P\tan(\theta_2)\)
The reduction in the reactive power is caused by the shunt capacitor (in parallel with the inductive load).
\(\displaystyle Q_C = Q_1 - Q_2 = P\left[\tan(\theta_1) - \tan(\theta_2)\right]\)
But \(\displaystyle Q_C = V_\textrm{rms}^2 / X_C = \omega CV_\textrm{rms}^2\).
The value of the required shunt capacitance \(C\) is determined as
\(\boxed{C = \frac{Q_C}{\omega V_\textrm{rms}^2} = \frac{Q_1 - Q_2}{\omega V_\textrm{rms}^2} = \frac{P\left[\tan(\theta_1) - \tan(\theta_2)\right]}{\omega V_\textrm{rms}^2}}\)
where \(Q_C\) is the difference between the old and new reactive powers.
Note that the real power \(P\) dissipated by the load is not affected by the power factor correction because the average power due to the capacitance is zero.
It is also possible that the load is capacitive; that is, the load is operating at a leading power factor. In this case, an inductor should be connected across (in parallel with) the load for power factor correction. The value of the required shunt inductance \(L\) is determined as
\(\displaystyle Q_L = \frac{V_\textrm{rms}^2}{X_L} = \frac{V_\textrm{rms}^2}{\omega L}\)
\(\boxed{L = \frac{V_\textrm{rms}^2}{\omega Q_L} = \frac{V_\textrm{rms}^2}{\omega(Q_1 - Q_2)} = \frac{V_\textrm{rms}^2}{\omega P\left[\tan(\theta_1) - \tan(\theta_2)\right]}}\)
where \(Q_L\) is the difference between the old and new reactive powers.