GATE 2023 CH – Question 62
The transfer function of a measuring instrument is $G_m(s) = \frac{1.05}{2s + 1}\exp(-s)$.
At time $t = 0$, a step change of +1 unit is introduced in the input of this instrument. The time taken by the instrument to show an increase of 1 unit in its output is _____ (rounded off to two decimal places).
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Show answer and explanation
Correct answer: 7.05 to 7.13
Explanation
For a unit step the output is zero until the delay of 1 min and then $y = 1.05\left(1 - e^{-(t - 1)/2}\right)$. Setting $y = 1$: $e^{-(t - 1)/2} = 1 - \frac{1}{1.05} = 0.04762$, so $t - 1 = 2\ln 21 = 6.089$ and $t = 7.09$ min.