A4.1 Measurement error of powers

Assume the input power of a load with power factor (lagging) between 0.6 and 1.0 is measured with a current transformer (CT), an ammeter, a potential transformer (PT), and a voltmeter, as shown in Fig. A4.1. The ammeter and the voltmeter have an error of ɛ_A and ɛ_V, respectively. The CT has a ratio correction factor RCF_CT and a phase angle shift γ measured in minutes. The PT has a ratio correction factor RCF_PT and a phase angle shift β measured in minutes [97–100, Chapter 8].

The input power is expressed as

$P=VIcos\phi$

The measurement error of the input power is given by

$\frac{\mathrm{\Delta }P}{P}=\frac{\mathrm{\Delta }V}{V}+\frac{\mathrm{\Delta }I}{I}-\frac{sin\phi \cdot \mathrm{\Delta }\phi }{cos\phi }=\left(RC{F}_{PT}-1+\frac{{\varepsilon }_V}{V}\right)+\left(RC{F}_{CT}-1+\frac{{\varepsilon }_A}{I}\right)-\frac{sin\phi }{cos\phi }\frac{\pi \left(\beta -\gamma \right)}{180\cdot 60}\text{.}$

For power factors ranging from 0.6 to 1.0 (lagging, based on consumer notation), the worst case in power measurement error is when cosφ = 0.6, therefore,

$\frac{\mathrm{\Delta }P}{P}\approx \left(RC{F}_{PT}-1+\frac{{\varepsilon }_V}{V}\right)+\left(RC{F}_{CT}-1+\frac{{\varepsilon }_A}{I}\right)-\frac{\left(\beta -\gamma \right)}{2600}=\left(TC{F}_{PT}-1+\frac{{\varepsilon }_V}{V}\right)+\left(TC{F}_{CT}-1+\frac{{\varepsilon }_A}{I}\right)=\left(\frac{{\varepsilon }_{PT}+{\varepsilon }_V}{V}\right)+\left(\frac{{\varepsilon }_{CT}+{\varepsilon }_A}{I}\right)$

where

$TC{F}_{CT}=RC{F}_{CT}+\gamma /2600\text{,}$

$TC{F}_{PT}=RC{F}_{PT}-\beta /2600\text{,}$

${\varepsilon }_{CT}=\left(TC{F}_{CT}-1\right)\cdot I\text{,}$

${\varepsilon }_{PT}=\left(TC{F}_{PT}-1\right)\cdot V\text{.}$

According to American National Standard [97, Chapter 8], transformer correction factors TCF_CT and TCF_PT (ɛ_CT and ɛ_PT) can directly be obtained based on the accuracy class from the device nameplates. Normally, the maximum absolute errors for ammeters and voltmeters are based on the full-scale errors. For instrument transformers, the maximum absolute errors (ɛ_CT and ɛ_PT) are proportional to the actually measured values of currents and voltages, as derived above.

Note that the maximum errors of an instrument transformer derived above are systematic errors that can be reduced by ratio and angle corrections. If the measured results are not corrected, the overall uncertainty of measurement is derived from the systematic (s) uncertainty and random (r) uncertainty:

$u=\sqrt{u_s^2+u_r^2}$

The random uncertainty is normally much smaller than the systematic uncertainty and can be ignored; therefore,

$u=u_s={\varepsilon }_{CT}\mathrm{or}{\varepsilon }_{PT}\text{.}$

A4.2 Nameplate data of measured transformers