GATE 2023 EC Q.18: solution and answer

GATE 2023 · ECQ.18 · 1 mark · MCQ

A series $R L C$ circuit has a quality factor $Q$ of 1000 at a center frequency of $1 0^{6}$ rad/s. The possible values of $R$ , $L$ and $C$ are

(A)
$R = 1 Ω$ , $L = 1 μ H$ and $C = 1 μ F$
(B)
$R = 0.1 Ω$ , $L = 1 μ H$ and $C = 1 μ F$
(C)
$R = 0.01 Ω$ , $L = 1 μ H$ and $C = 1 μ F$
(D)
$R = 0.001 Ω$ , $L = 1 μ H$ and $C = 1 μ F$

See the question in the paper

Tap to open full size

Answer and solution

ANSWER

(D) $R = 0.001 Ω$ , $L = 1 μ H$ and $C = 1 μ F$ . These give $ω_{0} = 1 0^{6} rad/s$ and $Q = ω_{0} L / R = 1000$ .

Official answer: DSource: Official answer key, GATE 2023 EC (IIT Kanpur)

All four options use the same $L = 1 μ H$ and $C = 1 μ F$ , so only $R$ has to be found.
Resonant (center) frequency of a series RLC circuit: $ω_{0} = \frac{1}{L C} = \frac{1}{1 0 ^{- 6} \times 1 0 ^{- 6}} = 1 0^{6} rad/s$ . This matches the given center frequency, so the $L$ and $C$ in the options are consistent with the question.
Quality factor of a series RLC circuit: $Q = \frac{ω _{0} L}{R} = \frac{1}{ω _{0} C R} = \frac{1}{R} \frac{L}{C}$
Here $ω_{0} L = 1 0^{6} \times 1 0^{- 6} = 1 Ω$ (equivalently $L / C = 1 0^{- 6} /1 0^{- 6} = 1 Ω$ ), so $Q = \frac{1 Ω}{R}$ .
Setting $Q = 1000$ gives $R = \frac{1 Ω}{1000} = 0.001 Ω$ . Check against the other options: $R = 1, 0.1, 0.01 Ω$ give $Q = 1, 10, 100$ respectively. Only option (D) gives $Q = 1000$ .

KEY IDEA

For a series RLC circuit, $ω_{0} = 1/ L C$ and $Q = ω_{0} L / R = \frac{1}{R} L / C$ . With $L = 1 μ H$ and $C = 1 μ F$ , the characteristic impedance $L / C$ is $1 Ω$ , so $Q = 1/ R$ and $Q = 1000$ needs $R = 1 m Ω$ .

WATCH OUT FOR

Using the parallel RLC formula $Q = R C / L$ (or $Q = ω_{0} R C$ ). In a series circuit a smaller $R$ gives a larger $Q$ . The parallel formula flips that and points to $R = 1000 Ω$ , which is not an option, or pushes a student to guess (A).

From the GATE 2023 EC paper. Every question of this paper · Download the full paper · All 34 AC, power & resonance questions, with notes

GATE 2023 EC Q.18: solution and answer

Answer and solution

More AC, power & resonance questions