Power Factor Correction⚠ unverified

Understanding the Parameters

• $Q_{old}$ — the lagging reactive power drawn by the (typically inductive) load, in volt‑amperes reactive. See Reactive Power.

• $Q_{new}$ — the smaller reactive power you want the source to see after adding the capacitor.

• $Q_c$ — the leading reactive power the capacitor supplies locally so the source no longer has to.

Derivation (Approaching a Proof)

For a load of real power $P$ at phase angle $\varphi$, the reactive power is $Q = P\tan\varphi$ (from the power triangle, see Apparent Power). To move from an initial angle $\varphi_{old}$ to a corrected $\varphi_{new}$ at constant real power $P$, the capacitor must supply the reactive‑power difference:

$$Q_c = Q_{old} - Q_{new} = P\big(\tan\varphi_{old} - \tan\varphi_{new}\big).$$

A capacitor produces leading reactive power that cancels part of the load's lagging reactive power, so the net $Q$ seen by the source drops from $Q_{old}$ to $Q_{new}$, raising the power factor $\cos\varphi$ toward unity. The capacitance needed follows from $Q_c = \omega C V^2$:

$$C = \frac{Q_c}{\omega V^2}.$$

Because $P$ is unchanged while $S=\sqrt{P^2+Q^2}$ falls, the line current $I = S/V$ falls in proportion — the whole point of correction.

History

Shunt capacitor banks for power‑factor correction spread through utility and industrial systems in the early‑to‑mid 20th century as inductive motor loads grew and utilities began billing on reactive demand. Modern installations add switched or automatic (thyristor‑controlled) banks and, increasingly, active power‑factor correction in switch‑mode supplies.

Related Concepts: Power Factor, Reactive Power, Apparent Power, Watt's Law

Notes: Registry calculator power-factor-correction (unverified; the live card mislabels $Q_{old}$, $Q_{new}$, and the result as dimensionless — they are reactive powers in var).