ISRO EC 2023 Q.67: solution and answer

ISRO 2023 · ECQ.67 · 1 mark · MCQ

The characteristic equation of a closed loop system is $s^{2} + 2 s + 2 = 0$ . Then the system is

(A)
Over damped
(B)
Centrally damped
(C)
Under damped
(D)
Un damped

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Answer and solution

ANSWER

The system is under damped, option (C), since $ζ \approx 0.707$ lies between 0 and 1 and the roots are $- 1 \pm j$ .

Official answer: CSource: ISRO's revised and final answer key of 16 February 2024

Compare the characteristic equation with the standard second-order form $s^{2} + 2 ζ ω_{n} s + ω_{n}^{2} = 0$ .
Matching the constant term: $ω_{n}^{2} = 2$ , so $ω_{n} = 2 rad/s$ .
Matching the coefficient of $s$ : $2 ζ ω_{n} = 2$ , so $ζ = \frac{1}{ω _{n}} = \frac{1}{2} \approx 0.707$ .
Check with the roots: $s = \frac{- 2 \pm 4 - 8}{2} = - 1 \pm j$ . These are complex conjugates with negative real part, consistent with $0 < ζ < 1$ .
Complex-conjugate roots in the left half-plane give a decaying oscillation, so the system is under damped.
The other options do not fit. Over damped needs $ζ > 1$ (real distinct roots). Un damped needs $ζ = 0$ (roots on the imaginary axis). 'Centrally damped' is not a standard term; critical damping needs $ζ = 1$ (equal real roots).

KEY IDEA

Read the damping ratio off the characteristic polynomial by comparing it with $s^{2} + 2 ζ ω_{n} s + ω_{n}^{2}$ . A value $0 < ζ < 1$ means complex-conjugate roots in the left half-plane and an under damped response.

WATCH OUT FOR

Taking the coefficient 2 of $s$ as $ζ$ itself (giving $ζ = 2$ , over damped), instead of $2 ζ ω_{n}$ with $ω_{n} = 2$ .

From the ISRO EC 2023 paper. Every question of this paper · The full paper (PDF, ISRO) · All 2 ISRO Time response questions, with notes