GATE 2017 EE – Question 16

Control Systems · Transient and Steady-state analysis of linear time invariant systems · 1 mark · Multiple choice

The transfer function of a system is given by $\dfrac{V_o(s)}{V_i(s)} = \dfrac{1 - s}{1 + s}$. Let the output of the system be $v_o(t) = V_m\sin(\omega t + \varphi)$ for the input, $v_i(t) = V_m\sin(\omega t)$. Then the minimum and maximum values of $\varphi$ (in radians) are respectively

$-\frac{\pi}{2}$ and $\frac{\pi}{2}$
$-\frac{\pi}{2}$ and 0
0 and $\frac{\pi}{2}$
$-\pi$ and 0

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Correct answer: (D) $-\pi$ and 0

Explanation

This is an all-pass system with magnitude 1 at every frequency. Its phase is $\varphi = -2\tan^{-1}(\omega)$, which is 0 at $\omega = 0$ and tends to $-\pi$ as $\omega \to \infty$. So the minimum is $-\pi$ and the maximum is 0.