= Automatic power factor correction
<Automatic power factor correction> adjusts a correction device as the measured load changes, rather than leaving one fixed compensating setting connected. In a staged <capacitor bank>, a controller uses terminal <electric potential difference> and <electric current> information to choose which stages are connected. The controlled quantity and target must distinguish <active power>, signed <reactive power>, waveform distortion and the chosen lead/lag convention.
For an original single-phase sinusoidal model, available stages each supply $q_s$ of capacitive <reactive power> at the operating <electric potential difference>. With $k$ connected stages,
$$
Q_{\rm net}=Q_L-kq_s,\qquad
\mathrm{pf}(k)=\frac{P}{\sqrt{P^2+(Q_L-kq_s)^2}}.
$$
At fixed $P>0$, maximizing the magnitude of the <power factor> chooses the available integer $k$ minimizing $|Q_L-kq_s|$. Away from the end limits, nearest-stage rounding leaves $|Q_{\rm net}|\le q_s/2$. This <capacitor-stage quantization error> can be positive or negative. A policy forbidding leading operation instead restricts $kq_s\le Q_L$, so it can choose a different setting.
Suppose $P=1000\,\mathrm W$, $Q_L=640\,\mathrm{var}$ and $q_s=250\,\mathrm{var}$. Three stages leave $-110\,\mathrm{var}$ and factor about $0.9940$ leading; two stages leave $+140\,\mathrm{var}$ and factor about $0.9903$ lagging. Both outcomes are consistent with their respective objectives. If the load changes to $P=200\,\mathrm W$, $Q_L=100\,\mathrm{var}$, leaving all three stages connected produces $-650\,\mathrm{var}$; zero stages is closer to cancellation than one. A stored stage decision must therefore be updated when the load changes.
Actual feedback must measure the corrected network. The Eaton paper places the <current transformer> so it senses the relevant load and capacitor contribution, and explains how a wrong phase, polarity or location produces misleading feedback.<Capacitor Application Issues by Thomas M. Blooming and Daniel J. Carnovale>[Section VI.C, printed p.8, CT Location for Automatic Banks]{ref} Measuring only the correction branch would not establish the combined source <power factor>.
A proposed controller test varies a known resistive-inductive load through several operating points, recording stage state, signed <reactive power>, RMS <electric current> and synchronized terminal waveforms. Check both increasing and decreasing demand, including small excursions around a stage threshold. A dead band and timing rules can prevent repeated switching in response to small fluctuations; they must be specified along with the target. Stage size, switching transients and the load's <harmonic components> require additional checks beyond the ideal rounding model.
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