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GATE 2022 EE – Question 39

Control Systems · Transient and Steady-state analysis of linear time invariant systems · 2 marks · Multiple choice

The damping ratio and undamped natural frequency of a closed loop system as shown in the figure, are denoted as $\zeta$ and $\omega_n$, respectively. The values of $\zeta$ and $\omega_n$ are

Diagram for GATE 2022 EE question 39
  1. $\zeta=0.5$ and $\omega_n=10$ rad/s
  2. $\zeta=0.1$ and $\omega_n=10$ rad/s
  3. $\zeta=0.707$ and $\omega_n=10$ rad/s
  4. $\zeta=0.707$ and $\omega_n=100$ rad/s

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Correct answer: (A) $\zeta=0.5$ and $\omega_n=10$ rad/s

Explanation

The inner loop around the first integrator $\frac{10}{s}$ with unity feedback gives $\frac{10}{s+10}$. In series with the second integrator, the open loop is $\frac{100}{s(s+10)}$, and closing the outer unity feedback loop gives $\frac{100}{s^2+10s+100}$. So $\omega_n=10$ rad/s and $2\zeta\omega_n=10$, giving $\zeta=0.5$.