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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,At fixed $P>0$, maximizing the magnitude of the power factor chooses the available integer $k$ minimizing $|Q_L-k{q}_s|$. Away from the end limits, nearest-stage rounding leaves $|Q_{\mathrm{net}}|\le q_s/2$. This capacitor-stage quantization error can be positive or negative. A policy forbidding leading operation instead restricts $k{q}_s\le Q_L$, 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.^[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.

Capacitor-stage quantization error is the residual reactive power caused by choosing among discrete settings of a capacitor bank instead of a continuously adjustable correction. In a fixed sinusoidal operating model, let the load absorb $Q_L$ and let the target residual be $Q_t$. The desired capacitive correction is $q_{\ast }=Q_L-Q_t$. Selecting available correction $q$ leavesThe error concerns the selected setting, separately from sensing error, changing load and component tolerance.

For equal stages of magnitude $q_s$ and an available range covering the desired correction, nearest-stage selection chooses an integer $k$ nearest $q_{\ast }/q_s$. HenceAt a half-stage tie either neighbor attains the bound. A policy forbidding a negative residual relative to zero instead rounds downward and gives $0\le Q_L-k{q}_s<q_s$, away from the bank's capacity limits. The two policies solve different constrained problems.

For an original zero-target example, let $Q_L=640\mathrm{var}$ and $q_s=250\mathrm{var}$. Three stages give $r=-110\mathrm{var}$; two give $r=+140\mathrm{var}$. With active power $P=1000\mathrm{W}$, nearest-stage selection guaranteesprovided the zero-reactive target is within range and the sinusoidal fixed-load assumptions hold. At only $P=200\mathrm{W}$, the same worst-case bound falls to about $0.8480$. A stage size suitable at high demand need not give the same numerical power factor at low demand.

Unequal stages require the actual attainable set. Stages of 100 and 400 vars permit $\{0,100,400,500\}$ vars. At a desired 250 vars, the nearest setting leaves 150 vars error, despite the smallest stage being only 100 vars. The relevant bound is half the largest gap between neighboring attainable settings in the tested range, not half the smallest component. Outside the bank's range, clipping to an endpoint can leave a larger error.

At fixed frequency, a fixed capacitance gives $q_s=\omega C_s{V}^2$, so stage magnitudes also change with RMS electric potential difference. Eaton's paper describes discrete stage sizes, combinations and automatic control; the error bounds above are original consequences of that ideal setting model.^[ref] A proposed test records the actual stage states and terminal quantities while slowly varying a known load across thresholds. Compare the observed residual with the attainable-setting prediction before assigning every deviation to quantization.

Reactive power overcompensation occurs when capacitive correction exceeds the inductive reactive power that it is intended to cancel. Under the passive sign convention and a positive-frequency sinusoidal supply, the source then sees negative net reactive power, producing a leading power factor. This can happen when a fixed capacitor bank remains connected as the inductive load falls. Eaton's application paper discusses excessive leading operation and the use of switched correction stages.^[ref]

Consider an ideal fixed RMS electric potential difference $V$ at angular frequency $\omega$, supplying a load that absorbs constant $P>0$ and $Q_L>0$. Let $Q_{\mathrm{cap}}=\omega C{V}^2\ge 0$ be the magnitude of the correction's reactive power; its signed contribution is $-Q_{\mathrm{cap}}$. Adding ideal shunt capacitance givesThe model holds the load and supply fixed while changing $C$. It omits losses and additional network dynamics. Optimal shunt capacitance for sinusoidal power factor correction follows by minimizing the squared root mean square electric current:For $C<C_{\ast }$ the remaining reactive power is positive and the power factor is lagging. For $C>C_{\ast }$ it is negative and the power factor is leading. The overcompensation threshold is $C_{\ast }$, but the current reduction interval for shunt reactive compensation is wider. Comparing the square of the corrected and uncorrected electric current givesThus source RMS current is reduced for $0<C<2C_{\ast }$, equals the uncorrected value at $C=2C_{\ast }$, and is larger for $C>2C_{\ast }$. An overcompensated operating point between $C_{\ast }$ and $2C_{\ast }$ still draws less source electric current than the uncorrected load.

For an original example, keep the parent article's chosen $P=1000\mathrm{W}$, $Q_L=750\mathrm{var}$, $V=200\mathrm{V}$ and 50 hertz. The uncorrected source draws 6.25 amperes at power factor0.8. Full cancellation needs $C_{\ast }\approx 59.68\mu \mathrm{F}$ and reduces source electric current to 5 amperes. A 100 microfarad correction gives $Q_{\mathrm{net}}\approx -506.64\mathrm{var}$ and 5.605 amperes, with a leading factor about0.8920. A 150 microfarad correction instead gives about $-1134.96\mathrm{var}$ and7.563 amperes, worse than no correction. The crossing is $2C_{\ast }\approx 119.37\mu \mathrm{F}$.

The optimum changes with the load. If the example load's $P$ and $Q_L$ both halve while its original optimal correction remains connected, then $P=500\mathrm{W}$ and $Q_{\mathrm{net}}=-375\mathrm{var}$. Source electric current is 3.125 amperes and the factor is 0.8 leading. This dependence motivates automatic power factor correction. The cited controller description measures system electric potential difference and electric current, using external current transformers, and switches discrete capacitor bank stages to follow a target.^[ref]

Retain the phase or sign of $Q_{\mathrm{net}}$ when reporting the result. A bare power factor magnitude does not distinguish leading from lagging; the manufacturer FAQ also illustrates how a meter's signed display depends on its convention.^[ref] The calculation assumes sinusoidal steady state; current distortion factor is a separate consideration for nonsinusoidal current.

Thomas M. Blooming is an electrical engineer whose coauthored Capacitor Application Issues by Thomas M. Blooming and Daniel J. Carnovale addresses practical use of capacitor banks in electric power systems. Its author biography describes work on power factor correction, harmonic filtering and related application measurements.^[ref]

A useful technical question is why improving fundamental displacement power factor can alter the response to other harmonic components. In an original lossless parallel LC circuit, the admittance isIts reactive terms cancel at the resonant frequencyFor stipulated $L=10\mathrm{mH}$ and $C=100\mu \mathrm{F}$, $f_r\approx 159.15\mathrm{Hz}$. Doubling $C$ moves it to about $112.54\mathrm{Hz}$. A correction component chosen for one operating frequency has therefore changed the network at other frequencies as well.

In this ideal parallel model the impedance diverges at resonance; a finite imposed harmonic electric current would demand an unbounded electric potential difference, signaling the ideal model's limit. Real losses and other network elements bound and shift the response. A proposed bench test uses a small known drive and sweeps frequency with and without the extra capacitor, recording both current and potential difference. It tests the frequency-dependent circuit rather than inferring behavior from the fundamental cosine alone.

The calculation is an original illustration of the network question, not an experimental result attributed to Blooming. His joint paper with Daniel J. Carnovale supplies the application context: component choice and operating conditions must be considered together. A correction that looks favorable in a single-frequency power factor calculation is not, by that fact alone, characterized across the full waveform.