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The current distortion factor is the ratio $D_I=I_1/I_{\mathrm{rms}}$, where $I_1$ is the root mean square magnitude of the fundamental electric current component and $I_{\mathrm{rms}}>0$ is the root mean square magnitude of the complete electric current. Its square is the fraction of the current's mean square supplied by the fundamental component. This ratio is distinct from displacement power factor, which measures the relative phase of the fundamental potential difference and current.

Use a Fourier series at fundamental frequency $\omega /(2\pi )$:Here $I_0$ is the DC component and the $I_n$ are RMS magnitudes of harmonic components. Averaging over a complete period, orthogonality and the Parseval identity giveFor purely sinusoidal electric potential difference $v(t)=\sqrt{2}V\cos \omega t$, only the fundamental electric current contributes to active power:Thus a displacement power factor of one can coexist with a lower total power factor.

For $I_1>0$, use fundamental-normalized current total harmonic distortion, ${\mathrm{THD}}_I=\sqrt{{\sum }_{n\ge 2}{I}_n^2}/I_1$. The relation between current distortion factor and THD with DC current isWith zero DC component, this reduces to $D_I=(1+{\mathrm{THD}}_I^2{)}^{-1/2}$. The definition of RMS here includes the DC component. For example, a 10 ampere fundamental and 3 ampere DC component, with no higher harmonic components, have zero total harmonic distortion but $D_I=10/\sqrt{109}\approx 0.9578$.

For an original zero-mean waveform, take a 230 volt RMS sinusoidal supply, $I_1=10\mathrm{A}$, $\cos {\varphi }_1=0.8$, and RMS third/fifth harmonic components of 6 and 3 amperes. ThenThe active power is 1840 watts, apparent power is about 2769.57 volt-amperes and total power factor is about 0.6644. This is an illustrative waveform model, not a measurement of a particular rectifier.

Terminology requires care. Overview of Functions: Harmonic Measurements by Schneider Electric uses distortion factor for the ratio of power factor to the fundamental cosine. Power Quality Considerations by Schneider Electric uses the label for harmonic content relative to the fundamental, or total harmonic distortion. The defining ratio identifies the intended quantity.

If the supply also contains harmonic components, use the harmonic active power sum:with matching voltage/current phase conventions. The fundamental-only factorization above assumes a purely sinusoidal supply.

The displacement power factor is $\cos {\phi }_1$, where ${\phi }_1$ is the phase difference between the fundamental components of terminal electric potential difference and electric current. It describes phase displacement at that frequency. For purely sinusoidal signals it equals total power factor; with distorted waveforms those quantities can differ. The one-frequency phase relation follows from the RMS average-power expression.^[ref]

For a purely sinusoidal potential difference of RMS magnitude $V$ and zero-mean periodic current, orthogonality of harmonics givesThe ratio $I_1/I_{\mathrm{rms}}$ is the current distortion factor under these assumptions. A displacement factor near one does not establish that the whole current waveform closely follows the supply sinusoid.

For an original example, let the supply be100V RMS. Let the fundamental current be4A RMS at a lagging phase with cosine0.8, and add a3A RMS third harmonic. Total RMS current is5A, active power320W and apparent power500VA. The displacement factor is0.8 while total power factor is0.64.

Now cancel only the fundamental quadrature current, whose magnitude is $4\sqrt{1-0.8^2}=2.4\mathrm{A}$. The remaining in-phase fundamental is3.2A and the third harmonic is still3A. RMS source current is $\sqrt{3.2^2+3^2}\approx 4.39\mathrm{A}$, so total factor is about0.73 even though the displacement factor has become one. This illustrates a limitation of phase-only power factor correction, not a universal model of a correcting device.

The cosine also loses lead/lag sign: phases $+{\phi }_1$ and $-{\phi }_1$ have the same displacement factor but opposite sinusoidal reactive power. Retain the phase or lead/lag convention when that distinction matters. If the potential difference itself contains harmonics, total active power needs the contributions of the corresponding harmonic pairs; this simple fundamental-only reduction is no longer sufficient.

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.

The relation between current distortion factor and THD with DC current accounts for the mean component omitted from the usual fundamental-normalized total harmonic distortion. For a periodic electric current, let $I_0$ be its signed DC component, $I_1>0$ the fundamental RMS magnitude, and $I_n$ the RMS magnitudes of higher harmonic components. The Parseval identity givesDefine ${\mathrm{THD}}_I=\sqrt{{\sum }_{n\ge 2}{I}_n^2}/I_1$ and current distortion factor $D_I=I_1/I_{\mathrm{rms}}$. Division by $I_1^2$ provesThe root mean square in this expression includes the mean, while the stated total harmonic distortion excludes it. An instrument reporting AC-only RMS or a differently normalized distortion quantity requires a correspondingly different conversion.

For an original example, take $I_1=4\mathrm{A}$, $I_0=3\mathrm{A}$ and combined higher-harmonic RMS magnitude $2\mathrm{A}$. Then ${\mathrm{THD}}_I=0.5$, $I_{\mathrm{rms}}=\sqrt{29}\mathrm{A}$ and $D_I=4/\sqrt{29}\approx 0.7428$. Neglecting the DC component would instead predict $1/\sqrt{1.25}\approx 0.8944$. In the same conductor electrical resistance $R$, the actual heating is $29R$, while that omission would predict $20R$.

For a zero-mean sinusoidal supply of $100\mathrm{V}$ RMS and an in-phase fundamental current, the example absorbs $400\mathrm{W}$. Its apparent power is approximately $538.52\mathrm{VA}$, so total power factor is $0.7428$ despite a displacement power factor of one. The DC component carries no average power from that particular zero-mean supply, but it still increases mean-square electric current. If the supply itself has a DC component, its contribution $V_0{I}_0$ must be included through the harmonic active power sum.

As $I_1$ tends to zero the defining current distortion factor can tend to zero, but fundamental-normalized total harmonic distortion is undefined at $I_1=0$. Do not evaluate the boxed relation there by silently dividing by zero. The manufacturer's documentation also uses differing labels for distortion ratios, reinforcing the need to state the defining normalization.^[ref][ref]