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GATE BT 2025 | Question: 33

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​​​​​The output $y(t)$ of a first-order process is governed by the following differential equation

\[
\tau_{p} \frac{d y}{d t}+y=K_{p} f(t)
\]
where $\tau_{p}$ is a non-zero time constant, $K_{p}$ is the gain and $f(t)$ is the input with $f(0)=0$.

Assume $y(0)=0$. The transfer function for this process is (consider $s$ as the independent variable in the Laplace domain)

  1. $\frac{K_{p}}{\tau_{p} s+1}$
  2. $\frac{\tau_{p}}{K_{p} s+1}$
  3. $\frac{\tau_{p}}{K_{p}(s+1)}$
  4. $\frac{K_{p}}{\tau_{p}(s+1)}$

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