GATE 2018 EE Q.9: solution and answer

GATE 2018 · EEQ.9 · 1 mark · MCQ

Match the transfer functions of the second-order systems with the nature of the systems given below.

Transfer functions

P: $\frac{15}{s ^{2} + 5 s + 15}$

Q: $\frac{25}{s ^{2} + 10 s + 25}$

R: $\frac{35}{s ^{2} + 18 s + 35}$

Nature of system

I: Overdamped

II: Critically damped

III: Underdamped

(A)
P-I, Q-II, R-III
(B)
P-II, Q-I, R-III
(C)
P-III, Q-II, R-I
(D)
P-III, Q-I, R-II

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Answer and solution

ANSWER

P is underdamped ( $ζ \approx 0.645$ ), Q is critically damped ( $ζ = 1$ ) and R is overdamped ( $ζ \approx 1.52$ ). The matching is P-III, Q-II, R-I, which is option (C).

Official answer: CSource: Official answer key, GATE 2018 EE (IIT Guwahati)

Compare each denominator with the standard form $s^{2} + 2 ζ ω_{n} s + ω_{n}^{2}$ . This gives $ω_{n} = constant term$ and $ζ = \frac{coefficient of s}{2 ω _{n}}$ .
Classification rule: $ζ < 1$ is underdamped, $ζ = 1$ is critically damped and $ζ > 1$ is overdamped. For a denominator $s^{2} + b s + c$ the same test can be run on the discriminant $b^{2} - 4 c$ : negative, zero or positive respectively.
P: $ω_{n} = 15 \approx 3.873$ , so $ζ = \frac{5}{2 15} \approx 0.645 < 1$ . Check: the discriminant is $25 - 60 = - 35 < 0$ , so the poles are complex, $s = - 2.5 \pm j \frac{35}{2} \approx - 2.5 \pm j 2.958$ . Underdamped (III).
Q: $s^{2} + 10 s + 25 = (s + 5)^{2}$ , so $ω_{n} = 5$ and $ζ = \frac{10}{2 \times 5} = 1$ . There is a repeated real pole at $s = - 5$ . Critically damped (II).
R: $ω_{n} = 35 \approx 5.916$ , so $ζ = \frac{18}{2 35} \approx 1.52 > 1$ . Check: the discriminant is $324 - 140 = 184 > 0$ , giving two distinct real poles $s = - 9 \pm 46 \approx - 2.22$ and $- 15.78$ . Overdamped (I).
So the matching is P-III, Q-II, R-I, which is option (C).

KEY IDEA

The damping ratio of a second-order system comes from its denominator alone: for $s^{2} + b s + c$ , $ζ = \frac{b}{2 c}$ . Equivalently, compare $b^{2}$ with $4 c$ . If $b^{2} < 4 c$ the poles are complex (underdamped), if $b^{2} = 4 c$ they are repeated (critically damped), and if $b^{2} > 4 c$ they are real and distinct (overdamped). The numerator plays no part.

WATCH OUT FOR

Taking $ζ$ to be the coefficient of $s$ divided by $2$ , forgetting the division by $ω_{n}$ . That gives $ζ = 2.5$ , $5$ and $9$ , which would wrongly call all three systems overdamped. A second mistake is to rank the systems by the size of the $s$ coefficient alone. R has the largest $s$ coefficient, but you only know it is overdamped once you check $1 8^{2} = 324 > 4 \times 35 = 140$ .

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GATE 2018 EE Q.9: solution and answer

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