Current distortion factor ($I_1/I_{\mathrm{rms}}$)

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 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]