GATE 2017 EE (EE2) – Question 18
Two parallel resistors have nominal values $R_1,R_2$ and additive uncertainties $\Delta R_1,\Delta R_2$. For equivalent resistance R, write $R_{,i}=\partial R/\partial R_i$. The standard deviation of its error is
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Correct answer: (A) $\pm\sqrt{(R_{,1}\Delta R_1)^2+(R_{,2}\Delta R_2)^2}$
Explanation
Here $R_{,1}=\partial R/\partial R_1$ and $R_{,2}=\partial R/\partial R_2$. For small independent errors, linearize $dR=R_{,1}dR_1+R_{,2}dR_2$. Independence makes the covariance term zero.
The variance is $\sigma_R^2=R_{,1}^2(\Delta R_1)^2+R_{,2}^2(\Delta R_2)^2$, giving **A**. For parallel resistance the derivatives are $R_2^2/(R_1+R_2)^2$ and $R_1^2/(R_1+R_2)^2$.
The source choices display ±; strictly the standard deviation is the nonnegative square root. The independence assumption is the conventional one underlying the key.