GATE 2022 EE Q.27: solution and answer

GATE 2022 · EEQ.27 · 1 mark · NAT

An inductor having a $Q$ -factor of 60 is connected in series with a capacitor having a $Q$ -factor of 240. The overall $Q$ -factor of the circuit is ________. (round off to nearest integer)

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Answer and solution

ANSWER

The overall Q-factor of the series combination is $Q = \frac{Q _{L} Q _{C}}{Q _{L} + Q _{C}} = \frac{60 \times 240}{300} = 48$ .

Official answer: 48Source: Official answer key, GATE 2022 EE (IIT Kharagpur)

Model each lossy element with a series loss resistance. For the inductor, $Q_{L} = \frac{X _{L}}{R _{L}} = 60$ . For the capacitor, with equivalent series resistance $R_{C}$ , $Q_{C} = \frac{X _{C}}{R _{C}} = 240$ .
The Q-factor of a series RLC circuit is defined at resonance, where $X_{L} = X_{C} = X$ . Taking the given Q-factors at that frequency, $R_{L} = \frac{X}{60}$ and $R_{C} = \frac{X}{240}$ .
The same current flows through both loss resistances because they are in series, so the total loss resistance is $R = R_{L} + R_{C}$ .
Overall Q at resonance: $Q = \frac{X}{R _{L} + R _{C}} = \frac{1}{\frac{R _{L}}{X} + \frac{R _{C}}{X}} = \frac{1}{\frac{1}{Q _{L}} + \frac{1}{Q _{C}}}$
Substitute the values: $\frac{1}{Q} = \frac{1}{60} + \frac{1}{240} = \frac{4 + 1}{240} = \frac{5}{240} \Rightarrow Q = \frac{Q _{L} Q _{C}}{Q _{L} + Q _{C}} = \frac{60 \times 240}{300} = 48$
Check: the overall Q (48) is lower than the Q of either element, which is what we expect because the two losses add. Rounded to the nearest integer, $Q = 48$ .

KEY IDEA

In a series circuit at resonance both elements carry the same current and have the same reactance $X$ . Each loss resistance is $X / Q$ , and these resistances add, so the reciprocals of the Q-factors add: $\frac{1}{Q} = \frac{1}{Q _{L}} + \frac{1}{Q _{C}}$ . The rule works like resistors in parallel.

WATCH OUT FOR

Adding the Q-factors (60 + 240 = 300), averaging them (150), or multiplying them. The losses add, not the Q-factors, so the overall Q has to come out smaller than the smaller of the two (60). Another mistake is to take the lossless capacitor's Q as infinite and answer 60, even though the capacitor's Q is given as 240.

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GATE 2022 EE Q.27: solution and answer

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