Question

In an ac main, the rms voltage V_ac, rms current I_ac and power W_ac are measured as: V_ac = 100 V ± 1%, I_ac = 1 A ± 1% and W_ac = 50 W ± 2% (errors are with respect to readings). The percentage error in calculating the power factor using these readings is
1. 1%
2. 2%
3. 3%
4. 4%

Answer 1

Correct Answer - Option 4 : 4%

Concept:

W = V × I × cos ϕ   .............. (1)

Where

W = active power measured

V = voltage measured

I = current measured

ϕ  = power factor angle

cos ϕ = power factor

By differentiating the above equation (1)

We get

\(\frac{{dW}}{W}\times 100 = ± \left( {\frac{{dV}}{V}\times100 + \frac{{dI}}{I} \times100+ \frac{{dcos\ ϕ }}{{cos\ ϕ }}}\times100\right)\)

\(\frac{{dcos\ ϕ }}{cos\ ϕ }\times100 = ± \left( {\frac{{dV}}{V}\times100 + \frac{{dI}}{I}\times100+ \frac{{dW }}{{W }}}\times100 \right)\) ..................(2)

\(\frac{{d\cos \phi }}{{\cos \phi }} \times 100\) = Percentage relative error in cos ϕ

Calculation:

Given:

\(\frac{{dV}}{V}\) = 1%, \(\frac{{dI}}{I}\) = 1%, \(\frac{{dW}}{W}\) = 2%

By substituting in equation (2)

We get

\(\frac{{dcos\ \phi}}{cos\ \phi}\) = ± ( 1% + 1% + 2% )

= ± 4%

Percentage Relative errors can be added in multiplication and division format equations.

Note: In division relative errors can be added, numerator and denominator should be independent to each other.