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GATE 2016 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 $\frac{Y(s)}{R(s)} = \frac{s}{s + 2}$. The steady state output $y(t)$ is $A \cos(2t + \varphi)$ for the input $\cos(2t)$. The values of $A$ and $\varphi$, respectively are

  1. $\frac{1}{\sqrt{2}}, -45°$
  2. $\frac{1}{\sqrt{2}}, +45°$
  3. $\sqrt{2}, -45°$
  4. $\sqrt{2}, +45°$

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Correct answer: (B) $\frac{1}{\sqrt{2}}, +45°$

Explanation

For a sinusoidal input at $\omega = 2$, the steady-state output has amplitude $|G(j2)|$ and phase $\angle G(j2)$. Here $G(j2) = \frac{j2}{2 + j2}$, so $|G| = \frac{2}{\sqrt{8}} = \frac{1}{\sqrt{2}}$ and the angle is $90° - 45° = +45°$.