= Power factor correction
<Power factor correction> changes the current drawn from a source so that a specified <active power> is supplied with a more favorable <power factor>. For a sinusoidal inductive load, a parallel <capacitor> can supply an opposite <reactive power> component. It changes the source current's quadrature component while an ideal lossless correction branch leaves the load's <active power> unchanged.
Let the original load absorb $P+jQ_L$ at RMS potential difference $V$ and angular frequency $\omega$. A parallel ideal capacitance contributes $Q_C=-\omega CV^2$, so
$$
Q_{\rm source}=Q_L-\omega CV^2.
$$
To change a lagging phase angle from $\varphi_1$ to a smaller lagging angle $\varphi_2$ at fixed $P>0$,
$$
C=\frac{P(\tan\varphi_1-\tan\varphi_2)}{\omega V^2}.
$$
These follow from <complex electrical power> and the capacitive phase relationship.<AC Power Flow in Linear Networks by James L. Kirtley Jr.>[Energy and Power, equations61–65; capacitive/inductive terms in equation71]{ref}
For an original ideal example, take $P=1000\,\mathrm W$, $V=200\,\mathrm V$ and50Hz. An initial factor0.8 corresponds to750var. A target factor0.95 corresponds to about328.68var, so a capacitance about33.53µF supplies the difference. RMS source current changes from6.25A to about5.26A. Resistive feeder loss at fixed resistance changes by $(0.8/0.95)^2\approx0.709$, while the1000W load conversion remains the same in this model.
<Reactive power overcompensation> occurs if the capacitive contribution exceeds the intended inductive component. The net $Q$ then changes sign; continuing to add capacitance does not keep improving the magnitude of the factor. At a different load or frequency the same component can produce another result. Component ratings and nonideal circuit behavior require their own model rather than being determined by this one-frequency algebra.
Correction of <displacement power factor> is also distinct from correction of waveform distortion. Harmonic current can remain after the fundamental quadrature component is canceled. A useful measurement compares synchronized source potential-difference/current waveforms, RMS magnitudes and their average product before and after a change, retaining the operating condition. Separately measuring only RMS magnitudes cannot establish the resulting <active power>.
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