GATE 2022 EE Q.39: solution and answer

GATE 2022 · EEQ.39 · 2 marks · MCQ

The damping ratio and undamped natural frequency of a closed loop system as shown in the figure, are denoted as $ζ$ and $ω_{n}$ , respectively. The values of $ζ$ and $ω_{n}$ are

GATE 2022 EE question 39: Block diagram: R(s) enters summing junction 1 (R with +, feedback with −). — Figure from the GATE 2022 EE paper. It is part of the question.

(A)
$ζ = 0.5$ and $ω_{n} = 10$ rad/s
(B)
$ζ = 0.1$ and $ω_{n} = 10$ rad/s
(C)
$ζ = 0.707$ and $ω_{n} = 10$ rad/s
(D)
$ζ = 0.707$ and $ω_{n} = 100$ rad/s

Answer and solution

ANSWER

$ζ = 0.5$ and $ω_{n} = 10 rad/s$ , which is option (A).

Official answer: ASource: Official answer key, GATE 2022 EE (IIT Kharagpur)

Reduce the inner loop first. It is the first $\frac{10}{s}$ block with unity negative feedback, so by $\frac{G}{1 + G}$ : $\frac{10/ s}{1 + 10/ s} = \frac{10}{s + 10}$
This reduced block is in cascade with the second $\frac{10}{s}$ block, which gives the forward path of the outer loop: $G (s) = \frac{10}{s + 10} \cdot \frac{10}{s} = \frac{100}{s ( s + 10 )}$
Close the outer unity negative feedback loop: $\frac{Y ( s )}{R ( s )} = \frac{G ( s )}{1 + G ( s )} = \frac{100}{s ^{2} + 10 s + 100}$
Compare with the standard second-order form $\frac{ω _{n}^{2}}{s ^{2} + 2 ζ ω _{n} s + ω _{n}^{2}}$ . The constant term gives $ω_{n}^{2} = 100$ , so $ω_{n} = 10 rad/s$ .
The $s$ term gives $2 ζ ω_{n} = 10$ , so $ζ = \frac{10}{2 \times 10} = 0.5$ .
So $ζ = 0.5$ and $ω_{n} = 10 rad/s$ , which is option (A).

KEY IDEA

Reduce nested loops from the inside out. The inner unity-feedback loop around an integrator $\frac{K}{s}$ becomes a first-order lag $\frac{K}{s + K}$ . Closing the outer loop then gives a standard second-order denominator, and you read $ω_{n}$ from the constant term and $ζ$ from the $s$ coefficient.

WATCH OUT FOR

Ignoring the inner loop and treating the forward path as $\frac{100}{s ^{2}}$ . That gives $\frac{100}{s ^{2} + 100}$ , so $ζ = 0$ , which is not an option. Another slip is taking $ω_{n}$ as 100 (the value of $ω_{n}^{2}$ ) or setting $ζ ω_{n} = 10$ instead of $2 ζ ω_{n} = 10$ , which gives $ζ = 1$ .

From the GATE 2022 EE paper. Every question of this paper · Download the full paper · All 22 Time response questions, with notes

GATE 2022 EE Q.39: solution and answer

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