GATE 2020 EC – Question 59
A system with transfer function $G(s)=\dfrac{1}{(s+1)(s+a)}$, $a>0$ is subjected to an input $5\cos3t$. The steady state output of the system is $\dfrac{1}{\sqrt{10}}\cos(3t-1.892)$. The value of $a$ is ________.
Practise this question in The GATE Grind →
Show answer and explanation
Correct answer: 4
Explanation
At $\omega=3$, $|G|=\frac{1}{\sqrt{1+9}\sqrt{a^2+9}}$, so the output amplitude is $\frac{5}{\sqrt{10}\sqrt{a^2+9}}$. Setting this equal to $\frac{1}{\sqrt{10}}$ gives $\sqrt{a^2+9}=5$, so $a=4$. Check the phase: $-\tan^{-1}3-\tan^{-1}\frac34=-71.57^\circ-36.87^\circ=-108.4^\circ=-1.892$ rad.