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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.

A current transformer is a transformer used to reproduce a scaled version of an alternating primary electric current in a secondary measurement circuit. Many secondary turns around a primary with few turns make large primary electric currents measurable by an instrument. The DOE handbook describes this instrument arrangement and its magnetic operating conditions.^[ref]

For ideal magnitudes, primary and secondary turns $N_p,N_s$ giveThe polarity depends on the chosen terminal directions. The current-transformer burden is the connected secondary load, including the instrument and connection impedance. With a resistive secondary load $R_b$, $V_s\simeq I_s{R}_b$ and its dissipated power is $I_s^2{R}_b$.

For an original example, take $N_p=1$, $N_s=200$, primary current $100\mathrm{A}$ RMS, and total secondary electrical resistance $R_b=2\Omega$. The ideal secondary delivers $0.5\mathrm{A}$, develops $1\mathrm{V}$ and supplies $0.5\mathrm{VA}$ to this resistive burden. The reflected primary electrical resistance isThe primary is measured through magnetic coupling, while the secondary still needs a suitable connected circuit.

Real behavior includes current-transformer ratio error and current-transformer phase displacement. To see why the second matters, let the inferred current magnitude have fractional error $e$ and an extra lag $\delta$ relative to the true current. For a sinusoidal load with lag $\phi$, the estimated-to-true active power ratio isWith $e=0$, $\phi =\pi /3$ and $\delta =0.01$ radians, the first-order error is about $-1.73\%$. Correct amplitude scaling alone does not establish correct power measurement.

Opening an energized conventional current-transformer secondary removes its normal balancing current and can produce a large induced electric potential difference. The DOE handbook specifically explains this open-secondary behavior and its requirement for a closed secondary circuit when the instrument is absent.^[ref] A proposed measurement check therefore uses the intended connected burden and compares ratio and phase at relevant frequencies and current levels. Core response and saturation limit the ideal scaling model; a conventional inductive current transformer does not measure a steady DC component by the same transformer action.

The current-transformer burden is the load impedance connected to a current transformer's secondary, including the instrument and secondary connection leads. It determines the electric potential difference needed to drive the secondary electric current. Changing it can change both current-transformer ratio error and current-transformer phase displacement. ABB's application guide explicitly connects these errors to the secondary burden and distinguishes standard burdens from actual connected devices.^[ref]

For sinusoidal RMS secondary current $I_s$ and external burden $Z_b=R_b+i{X}_b$, the secondary terminal relations areHere $S_b$ is complex electrical power; its real part is dissipated active power. A volt-ampere burden magnitude alone does not determine its power factor or its complex complex impedance.

For an original resistive example, an instrument contributes $0.10\Omega$ and the complete outgoing-plus-return leads contribute $0.20\Omega$. At $5\mathrm{A}$ RMS, total burden is $0.30\Omega$, terminal magnitude is $1.5\mathrm{V}$, and burden apparent power is $7.5\mathrm{VA}$. The instrument dissipates $2.5\mathrm{W}$ and the leads dissipate $5\mathrm{W}$. Ignoring the leads would understate the required terminal magnitude by a factor of three. A stipulated rating of 5 VA at 5 A corresponds to only $0.20\Omega$ magnitude, so this setup would not satisfy that stated burden limit.

Equal VA values also need not mean equal complex loading. At 5 A, $Z_b=0.20\Omega$ and $Z_b=0.12+0.16i\Omega$ both have 5 VA magnitude. The second instead absorbs $3\mathrm{W}$ and $4\mathrm{var}$, with power factor0.6. The resulting phase and ratio effects depend on the transformer's own response.

The secondary winding's internal impedance is not part of the external burden, but both enter the induced-voltage and excitation calculation. NIST's equivalent circuit separates these terms explicitly.^[ref] A proposed calibration comparison records the complete connected impedance at the relevant frequency, including leads, and repeats ratio and phase measurements at selected burdens. It does not infer unchanged accuracy merely because the nominal turns remain the same.

A current-transformer excitation current is the part of the secondary-referred input current needed for the magnetic core's response and losses rather than delivered to the external secondary circuit. In an ideal current transformer this contribution is neglected. In a real equivalent circuit it helps explain both current-transformer ratio error and current-transformer phase displacement.

Choose terminal polarities so ideal current reproduction is in phase. Let $K_n$ be the nominal primary-to-secondary ratio, $I_0=I_p/K_n$ the secondary-referred input phasor, and $I_s$ the measured secondary phasor. The current balance isRepresent the excitation branch locally by current-transformer excitation admittance $Y_e$ and induced secondary phasor $E_s$. With total secondary winding-plus-burden impedance $Z_t$,The NIST treatment gives the corresponding magnetizing, leakage and burden equivalent circuit and shows how those impedances enter the complex ratio.^[ref]

For an original local model, write $Y_e=G-iB$ with $G,B>0$, and take $Z_t=R_t>0$. When $|Y_e{R}_t|\ll 1$,The amplitude deficit is approximately $G{R}_t$, while the secondary lead is approximately $B{R}_t$ radians. For stipulated $G=0.001\mathrm{S}$, $B=0.005\mathrm{S}$ and $R_t=1\Omega$, these are about $0.1\%$ and 5 milliradians. At an induced $1\mathrm{V}$, the excitation components are 1 milliampere in phase and 5 milliamperes in quadrature; the real branch power is $G|E_s{|}^2=1\mathrm{mW}$.

An increased current-transformer burden raises the induced potential required for a given secondary current. The actual excitation response need not remain linear: core behavior, frequency, waveform and prior magnetic state can change the local parameters. The NIST equivalent-circuit discussion explicitly notes the nonlinear core and current/burden dependence.^[ref]

A proposed characterization varies a known connected burden and compares measured current ratio and phase, keeping the primary-current and frequency conditions recorded. A model that fits one operating point is not thereby validated through saturation or for arbitrary harmonic components. Separating excitation from load delivery explains why a turns ratio alone cannot specify practical measurement accuracy.

John D. Ramboz is the first author, with Oskars Petersons, of NIST's 1991 account of a laboratory current transformer calibration service. The report describes ratio and phase measurements, their uncertainties and quality-control procedures for electrical measurement work.^[ref] It is a primary technical record of the service and its measurement conventions.

The separation of magnitude and phase has a direct consequence for electric power measurements. Consider an original sinusoidal model with root-mean-square electric potential difference $V$, secondary current magnitude $I_s$, true current ratio $K$ and primary-current lag $\varphi$. True active power isIf the secondary leads the primary by $\beta$, simply multiplying the secondary signal by nominal ratio $K_n$ givesprovided $\cos \varphi \ne 0$. This uses the lead-positive current-transformer phase displacement convention stated in the report.^[ref]

With exact magnitude conversion $K_n=K$ and small $\beta$ in radians, the relative power error is approximately $\beta \tan \varphi$. For stipulated $\beta =0.001$ rad and $\varphi =60^{\circ }$, that term is about $0.173\%$. At $\varphi =0$, its first-order value vanishes. Thus the same phase error can have different effects at different power factors.

This derivation is an explanatory example, not a reported performance test or an attribution of the calculation to Ramboz. It explains why a calibration report needs operating conditions, a sign convention and measurement uncertainty, in addition to a nominal transformer ratio. Precision in the current magnitude does not remove phase-dependent power error.

Oskars Petersons worked on electrical metrology at the National Institute of Standards and Technology. With John D. Ramboz, Petersons coauthored the 1991 report describing NIST's laboratory current transformer calibration service. The report treats current ratio and current-transformer phase displacement as distinct measurement results.^[ref]

A current transformer allows a secondary electric current to represent a larger or smaller primary current. Its marked ratio is a nominal conversion, whereas the measured ratio can differ. The report expresses the measured ratio as the marked ratio multiplied by the current-transformer ratio correction factor, and uses a secondary-leading sign convention for the reported phase displacement.^[ref]

An original calculation illustrates the distinction. A marked ratio of $200$ A:$5$ A gives $K_n=40$. If the current-transformer ratio correction factor is $1.0002$, the measured magnitude ratio is $K=40.008$. A secondary reading of 5 A then represents 200.04 A under those stated conditions, rather than exactly 200 A.

Magnitude alone is not a full alternating-current description. If the secondary phasor leads the primary by $\beta$ and their magnitude ratio is $K$, a consistent convention isThe magnitude correction and phase rotation address different errors. A measurement system can reproduce one accurately while requiring correction of the other. The joint report's calibration circuits, including compensated current comparators, give these distinctions an apparatus-level setting rather than leaving them as ideal-transformer identities.^[ref]

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.