{title2=$I_1/I_{\rm rms}$}

The \b[<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> $\omega/(2\pi)$:
$$
i(t)=I_0+\sqrt2\sum_{n\ge1}I_n\cos(n\omega t-\phi_n).
$$
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> give
$$
I_{\mathrm{rms}}^2=I_0^2+\sum_{n\ge1}I_n^2.
$$
For purely sinusoidal <electric potential difference> $v(t)=\sqrt2V\cos\omega t$, only the fundamental <electric current> contributes to <active power>:
$$
P=\langle vi\rangle=VI_1\cos\phi_1,
\qquad
\boxed{\mathrm{pf}=\frac{P}{VI_{\mathrm{rms}}}=D_I\cos\phi_1.}
$$
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>, $\mathrm{THD}_I=\sqrt{\sum_{n\ge2}I_n^2}/I_1$. The <relation between current distortion factor and THD with DC current> is
$$
D_I=\frac1{\sqrt{1+\mathrm{THD}_I^2+(I_0/I_1)^2}}.
$$
With 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 $D_I=10/\sqrt{109}\approx0.9578$.

For an original zero-mean waveform, take a 230 volt RMS sinusoidal supply, $I_1=10\,\mathrm A$, $\cos\phi_1=0.8$, and RMS third/fifth <harmonic components> of 6 and 3 amperes. Then
$$
I_{\mathrm{rms}}=\sqrt{10^2+6^2+3^2}=\sqrt{145}\,\mathrm A,
\quad D_I\approx0.8305,
\quad\mathrm{THD}_I\approx0.6708.
$$
The <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.

\Image[/current-distortion-factor.png]
{title=Harmonic current lowers power factor beyond the fundamental phase displacement}
{description=Original sinusoidal-supply example. A 10 ampere RMS fundamental with displacement factor 0.8 is combined with 6 ampere third and 3 ampere fifth harmonics. The current distortion factor is about 0.8305 and the total power factor about 0.6644. Harmonic phases in the drawn waveform are chosen for illustration.}

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>:
$$
P=V_0I_0+\sum_{n\ge1}V_nI_n\cos(\alpha_n-\phi_n),
$$
with matching voltage/current phase conventions. The fundamental-only factorization above assumes a purely sinusoidal supply.


 Current distortion factor (source code)