GATE ECE 2019 | Question: 5

Question

Let $Y(s)$ be the unit-step response of a causal system having a transfer function

$$G(s)= \dfrac{3-s}{(s+1)(s+3)}$$

that is ,$Y(s)=\dfrac{G(s)}{s}.$ The forced response of the system is

• A. $u(t)-2e^{-t}u(t)+e^{-3t}u(t)$
• B. $2u(t)-2e^{-t}u(t)+e^{-3t}u(t)$
• C. $2u(t)$
• D. $u(t)$

Answer 1

The given transfer function is $G(s) = \frac{3 - s}{(s + 1)(s + 3)}$. The unit-step response of a causal system is given by $Y(s) = \frac{G(s)}{s}$.

To find the forced response of the system, we need to perform inverse Laplace transform on Y(s).

Let’s calculate:

$Y(s) = \frac{G(s)}{s} = \frac{3 - s}{s(s + 1)(s + 3)}$

This can be written as a sum of partial fractions:

$Y(s) = \frac{A}{s} + \frac{B}{s + 1} + \frac{C}{s + 3}$

Solving for A, B, and C, we get:

(A = 1, B = -2, C = 1)

So,

$Y(s) = \frac{1}{s} - \frac{2}{s + 1} + \frac{1}{s + 3}$

Taking the inverse Laplace transform, we get the forced response in the time domain:

$y(t) = u(t) - 2e^{-t}u(t) + e^{-3t}u(t)$

So, the correct option is A. $u(t) - 2e^{-t}u(t) + e^{-3t}u(t)$