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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.

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 harmonic active power sum expresses average electrical power as the sum of contributions from matching components of periodic terminal electric potential difference and electric current. With a common fundamental frequency and root mean square magnitudes for nonzero harmonic components, writeHere $V_0,I_0$ are the signed DC components. The passive sign convention makes the average product absorbed active power. Averaging over a complete fundamental period givesFor finite sums this follows from the cosine-product identity: products of different integer harmonic components average to zero, and matching components give half the cosine of their relative phase. For square-integrable periodic waveforms, the corresponding Fourier series identity follows by continuity of the inner product in the mean-square norm. The orthogonality is a property of the averaging interval and basis, not an assumption that the physical sources are independent.

For an original example, set $V_0=10\mathrm{V}$, $I_0=2\mathrm{A}$, $V_1=100\mathrm{V}$, $I_1=3\mathrm{A}$, and fundamental relative phase $\pi /3$. Add third-harmonic magnitudes $V_3=20\mathrm{V}$, $I_3=1\mathrm{A}$ with relative phase $\pi$. ThenThe third-harmonic contribution is $-20\mathrm{W}$, even though the total absorbed active power is positive. The root mean square values satisfy $V_{\mathrm{rms}}^2=10500{\mathrm{V}}^2$ and $I_{\mathrm{rms}}^2=14{\mathrm{A}}^2$, so apparent power is about $383.41\mathrm{VA}$ and total power factor about $0.3912$.

For a purely sinusoidal supply with no DC component, only $V_1$ survives, giving $P=V_1{I}_1\cos ({\alpha }_1-{\varphi }_1)$ even if the electric current is distorted. Current in other harmonic components can still increase conductor heating. The usual one-frequency complex electrical power construction is derived in the primary MIT treatment; the sum above applies that average-product reasoning separately to orthogonal components.^[ref]

A proposed measurement records the two waveforms synchronously and compares their directly averaged product with the resolved sum. A non-integer-period window, unresolved harmonic components, channel timing error or changing waveform can spoil the simple finite-record equality. The formula assumes a shared periodic model; multiplying separately measured RMS magnitudes gives apparent power instead.

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]