= Current-transformer phase displacement
{title2=$\alpha$}
<Current-transformer phase displacement> is the phase difference between the secondary-current waveform and the corresponding primary-current waveform after accounting for the chosen terminal polarity. Define $\alpha>0$ when the secondary reproduction leads the primary. NIST uses this sign convention and reports the angle directly; a positive amplitude correction factor does not specify the angle.<A Calibration Service for Current Transformers by John D. Ramboz and Oskars Petersons>[§1.3, printed p.5, Phase angle; p.6, phasor relationships]{ref}
For sinusoidal signals, choose primary <RMS phasor> $\underline I_p=I_pe^{-i\varphi}$ relative to a real supply phasor $V$. Let the nominally scaled secondary reproduce
$$
K_n\underline I_s=(1+\epsilon_I)e^{i\alpha}\underline I_p,
$$
where $\epsilon_I$ is the <current-transformer ratio error>. A meter using this current and an otherwise exact supply measurement infers
$$
\widehat P=VI_p(1+\epsilon_I)\cos(\varphi-\alpha).
$$
For $\cos\varphi\ne0$,
$$
\frac{\widehat P}{P}
=(1+\epsilon_I)\frac{\cos(\varphi-\alpha)}{\cos\varphi}
\simeq1+\epsilon_I+\alpha\tan\varphi.
$$
A lag-defined phase error would have the opposite sign in the last term, so the convention must accompany the number.
For an original example, set $V=100\,\mathrm V$, $I_p=5\,\mathrm A$, $\varphi=\pi/3$, $\epsilon_I=0$ and $\alpha=0.01$ radians. True <active power> is $250\,\mathrm W$, while the current-shifted indication is about $254.32\,\mathrm W$, an error of $+1.727\%$. At unity <displacement power factor>, the same phase displacement instead gives $\widehat P/P=\cos0.01\approx0.999950$, a much smaller second-order effect. Low <power factor> makes phase characterization especially important for relative power accuracy.
At zero true <active power>, a relative-error ratio is undefined. For $\varphi=\pi/2$, the shifted indication is $VI_p\sin\alpha$, which can be nonzero despite zero true average transfer. Report absolute error there rather than dividing by zero.
A proposed phase check compares the two current waveforms at the stated <frequency>, current level and <current-transformer burden>, with terminal direction and channel delay recorded. A time skew $\Delta t$ contributes a phase shift of magnitude $2\pi f|\Delta t|$; at 50 Hz, 10 microseconds is about 3.142 milliradians. The measurement chain must distinguish that delay from the transformer itself. The NIST publication's historical calibration apparatus measured ratio and phase separately; the examples here are derived illustrations, not reported NIST performance.
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