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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 current-transformer burden is the load impedance connected to a current transformer's secondary, including the instrument and secondary connection leads. It determines the electric potential difference needed to drive the secondary electric current. Changing it can change both current-transformer ratio error and current-transformer phase displacement. ABB's application guide explicitly connects these errors to the secondary burden and distinguishes standard burdens from actual connected devices.^[ref]

For sinusoidal RMS secondary current $I_s$ and external burden $Z_b=R_b+i{X}_b$, the secondary terminal relations areHere $S_b$ is complex electrical power; its real part is dissipated active power. A volt-ampere burden magnitude alone does not determine its power factor or its complex complex impedance.

For an original resistive example, an instrument contributes $0.10\Omega$ and the complete outgoing-plus-return leads contribute $0.20\Omega$. At $5\mathrm{A}$ RMS, total burden is $0.30\Omega$, terminal magnitude is $1.5\mathrm{V}$, and burden apparent power is $7.5\mathrm{VA}$. The instrument dissipates $2.5\mathrm{W}$ and the leads dissipate $5\mathrm{W}$. Ignoring the leads would understate the required terminal magnitude by a factor of three. A stipulated rating of 5 VA at 5 A corresponds to only $0.20\Omega$ magnitude, so this setup would not satisfy that stated burden limit.

Equal VA values also need not mean equal complex loading. At 5 A, $Z_b=0.20\Omega$ and $Z_b=0.12+0.16i\Omega$ both have 5 VA magnitude. The second instead absorbs $3\mathrm{W}$ and $4\mathrm{var}$, with power factor0.6. The resulting phase and ratio effects depend on the transformer's own response.

The secondary winding's internal impedance is not part of the external burden, but both enter the induced-voltage and excitation calculation. NIST's equivalent circuit separates these terms explicitly.^[ref] A proposed calibration comparison records the complete connected impedance at the relevant frequency, including leads, and repeats ratio and phase measurements at selected burdens. It does not infer unchanged accuracy merely because the nominal turns remain the same.

A current-transformer excitation current is the part of the secondary-referred input current needed for the magnetic core's response and losses rather than delivered to the external secondary circuit. In an ideal current transformer this contribution is neglected. In a real equivalent circuit it helps explain both current-transformer ratio error and current-transformer phase displacement.

Choose terminal polarities so ideal current reproduction is in phase. Let $K_n$ be the nominal primary-to-secondary ratio, $I_0=I_p/K_n$ the secondary-referred input phasor, and $I_s$ the measured secondary phasor. The current balance isRepresent the excitation branch locally by current-transformer excitation admittance $Y_e$ and induced secondary phasor $E_s$. With total secondary winding-plus-burden impedance $Z_t$,The NIST treatment gives the corresponding magnetizing, leakage and burden equivalent circuit and shows how those impedances enter the complex ratio.^[ref]

For an original local model, write $Y_e=G-iB$ with $G,B>0$, and take $Z_t=R_t>0$. When $|Y_e{R}_t|\ll 1$,The amplitude deficit is approximately $G{R}_t$, while the secondary lead is approximately $B{R}_t$ radians. For stipulated $G=0.001\mathrm{S}$, $B=0.005\mathrm{S}$ and $R_t=1\Omega$, these are about $0.1\%$ and 5 milliradians. At an induced $1\mathrm{V}$, the excitation components are 1 milliampere in phase and 5 milliamperes in quadrature; the real branch power is $G|E_s{|}^2=1\mathrm{mW}$.

An increased current-transformer burden raises the induced potential required for a given secondary current. The actual excitation response need not remain linear: core behavior, frequency, waveform and prior magnetic state can change the local parameters. The NIST equivalent-circuit discussion explicitly notes the nonlinear core and current/burden dependence.^[ref]

A proposed characterization varies a known connected burden and compares measured current ratio and phase, keeping the primary-current and frequency conditions recorded. A model that fits one operating point is not thereby validated through saturation or for arbitrary harmonic components. Separating excitation from load delivery explains why a turns ratio alone cannot specify practical measurement accuracy.

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.^[ref]

For sinusoidal signals, choose primary RMS phasor ${\underline{I}}_p=I_p{e}^{-i\phi }$ relative to a real supply phasor $V$. Let the nominally scaled secondary reproducewhere ${ϵ}_I$ is the current-transformer ratio error. A meter using this current and an otherwise exact supply measurement infersFor $\cos \phi \ne 0$,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 $V=100\mathrm{V}$, $I_p=5\mathrm{A}$, $\phi =\pi /3$, ${ϵ}_I=0$ 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 $\hat{P}/P=\cos 0.01\approx 0.999950$, 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 $\phi =\pi /2$, the shifted indication is $V{I}_p\sin \alpha$, 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.

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.

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.

John D. Ramboz is the first author, with Oskars Petersons, of NIST's 1991 account of a laboratory current transformer calibration service. The report describes ratio and phase measurements, their uncertainties and quality-control procedures for electrical measurement work.^[ref] It is a primary technical record of the service and its measurement conventions.

The separation of magnitude and phase has a direct consequence for electric power measurements. Consider an original sinusoidal model with root-mean-square electric potential difference $V$, secondary current magnitude $I_s$, true current ratio $K$ and primary-current lag $\varphi$. True active power isIf the secondary leads the primary by $\beta$, simply multiplying the secondary signal by nominal ratio $K_n$ givesprovided $\cos \varphi \ne 0$. This uses the lead-positive current-transformer phase displacement convention stated in the report.^[ref]

With exact magnitude conversion $K_n=K$ and small $\beta$ in radians, the relative power error is approximately $\beta \tan \varphi$. For stipulated $\beta =0.001$ rad and $\varphi =60^{\circ }$, that term is about $0.173\%$. At $\varphi =0$, its first-order value vanishes. Thus the same phase error can have different effects at different power factors.

This derivation is an explanatory example, not a reported performance test or an attribution of the calculation to Ramboz. It explains why a calibration report needs operating conditions, a sign convention and measurement uncertainty, in addition to a nominal transformer ratio. Precision in the current magnitude does not remove phase-dependent power error.

Oskars Petersons worked on electrical metrology at the National Institute of Standards and Technology. With John D. Ramboz, Petersons coauthored the 1991 report describing NIST's laboratory current transformer calibration service. The report treats current ratio and current-transformer phase displacement as distinct measurement results.^[ref]

A current transformer allows a secondary electric current to represent a larger or smaller primary current. Its marked ratio is a nominal conversion, whereas the measured ratio can differ. The report expresses the measured ratio as the marked ratio multiplied by the current-transformer ratio correction factor, and uses a secondary-leading sign convention for the reported phase displacement.^[ref]

An original calculation illustrates the distinction. A marked ratio of $200$ A:$5$ A gives $K_n=40$. If the current-transformer ratio correction factor is $1.0002$, the measured magnitude ratio is $K=40.008$. A secondary reading of 5 A then represents 200.04 A under those stated conditions, rather than exactly 200 A.

Magnitude alone is not a full alternating-current description. If the secondary phasor leads the primary by $\beta$ and their magnitude ratio is $K$, a consistent convention isThe magnitude correction and phase rotation address different errors. A measurement system can reproduce one accurately while requiring correction of the other. The joint report's calibration circuits, including compensated current comparators, give these distinctions an apparatus-level setting rather than leaving them as ideal-transformer identities.^[ref]