Current-transformer ratio error (${ϵ}_I$)

Current-transformer ratio error quantifies the amplitude mismatch between primary electric current and the value inferred by multiplying secondary current by a nominal ratio. For positive RMS magnitudes $I_p,I_s$ and marked primary-to-secondary ratio $K_n$, define the inferred-primary fractional error asA negative value means that nominal scaling underestimates the primary current. This explicitly states the error convention; a reported correction can have the opposite sign.

The true ratio is $K=I_p/I_s$. The current-transformer ratio correction factor used in NIST's report convention isNIST distinguishes marked ratio, true ratio, ratio correction and ratio correction factor in its terminology section.^[ref] Error and correction are approximately negatives for small errors, but their exact fractional relationship is nonlinear.

For an original example, take $K_n=200$, a reference primary current of $500\mathrm{A}$, and secondary current $2.49\mathrm{A}$. Nominal scaling gives $498\mathrm{A}$, so ${ϵ}_I=-0.004=-0.4\%$. The correction factor is $500/498\approx 1.004016$. Multiplying the nominal indication by that factor recovers 500 A; simply adding 0.4% would leave a small second-order residual.

A simplified secondary-referred equivalent circuit makes burden dependence visible. Let $I_p/K_n$ divide between the measured secondary current and a current-transformer excitation current. With excitation admittance $Y_e$ and total secondary series impedance $Z_t$,This original linearized model gives amplitude error approximately $-\mathrm{Re}(Y_e{Z}_t)$ and phase shift approximately $-\mathrm{Im}(Y_e{Z}_t)$ when $|Y_e{Z}_t|\ll 1$. The core response can be nonlinear; the formula is not a universal calibration curve. NIST's equivalent circuit shows the separate magnetizing, leakage and burden terms that motivate such a model.^[ref]

A proposed ratio test compares primary and secondary magnitudes against a reference at stated current, frequency and current-transformer burden. Record polarity and phase separately: a correct magnitude ratio does not establish a correct active power reading. The pair of ratio and current-transformer phase displacement measurements determines the relevant complex transfer at that operating point.

The current-transformer ratio correction factor multiplies a nominally scaled secondary-current indication to recover primary-current magnitude under stated calibration conditions. If $K_n$ is the marked primary-to-secondary ratio and $K=I_p/I_s$ is the true magnitude ratio, defineNIST distinguishes this factor from ratio correction, which is $\mathrm{RCF}-1$, and from the separately measured phase displacement.^[ref] Correcting amplitude does not remove current-transformer phase displacement.

With the inferred-primary error convention ${ϵ}_I=(K_n{I}_s-I_p)/I_p$, the exact relationship isFor an original example, $K_n=200$, $I_s=2.49\mathrm{A}$ and $\mathrm{RCF}=500/498\approx 1.004016$ give $I_p=500\mathrm{A}$. Using nominal scaling alone gives 498 A. The approximate correction $1-{ϵ}_I=1.004$ would instead give 499.992 A, leaving a small second-order difference.

A calibration correction has its own measurement uncertainty. Treating $K_n$ as an exact label and retaining correlation, first-order propagation givesFor a correction inferred from uncertain ${ϵ}_I$, its local sensitivity is $d\mathrm{RCF}/d{ϵ}_I=-(1+{ϵ}_I{)}^{-2}$. The GUM supplies the general sensitivity-and-covariance propagation rule used here.^[ref]

Nonlinearity also affects averaging. Suppose a stipulated error takes values $+0.01$ and $-0.01$ with equal probabilities. Its mean is zero, but the mean correction factor isThe correction at the mean error is instead one. Convexity of the reciprocal on $1+{ϵ}_I>0$ explains this Jensen's inequality effect. It is an illustrative probability model, not a claim about random drift of a particular transformer.

A correction factor should therefore travel with its current level, frequency, current-transformer burden and uncertainty information. A proposed check repeats the ratio comparison at selected operating conditions and tests whether one retained correction remains adequate. Applying more displayed digits cannot replace that validation.