GATE 2022 EE Q.30: solution and answer

GATE 2022 · EEQ.30 · 1 mark · NAT

Q.30 If only 5% of the supplied power to a cable reaches the output terminal, the power loss in the cable, in decibels, is ________. (round off to nearest integer)

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Answer and solution

ANSWER

The power loss in the cable is $10 lo g_{10} (1/0.05) = 10 lo g_{10} 20 \approx 13 dB$ .

Official answer: 13Source: Official answer key, GATE 2022 EE (IIT Kharagpur)

Power ratio: only $5%$ of the input power reaches the output, so $\frac{P _{o u t}}{P _{in}} = 0.05$ and $\frac{P _{in}}{P _{o u t}} = \frac{1}{0.05} = 20$ .
Decibel definition for a power ratio: $Loss (dB) = 10 lo g_{10} \frac{P _{in}}{P _{o u t}}$ . The factor is $10 lo g_{10}$ , not $20 lo g_{10}$ , because this is a ratio of powers, not of voltages or currents.
$Loss = 10 lo g_{10} (20) = 10 (lo g_{10} 2 + lo g_{10} 10) = 10 (0.3010 + 1) = 13.01 dB$
Rounded to the nearest integer, the cable loss is $13 dB$ . Put the other way, the cable's gain is $10 lo g_{10} (0.05) = - 13 dB$ ; a loss is quoted as a positive number.

KEY IDEA

For a power ratio, decibels are $10 lo g_{10} (P_{in} / P_{o u t})$ . If $5%$ gets through, the ratio is $20$ , and $10 lo g_{10} 20 = 10 (1 + 0.301) \approx 13 dB$ .

WATCH OUT FOR

Using $20 lo g_{10}$ , which is the voltage or current formula, and getting $26 dB$ . The other common slip is taking the log of the lost fraction ( $95%$ ) instead of the ratio of input to output, which gives about $0.22 dB$ . A third is entering $- 13$ : the question asks for a loss, so the answer is positive.

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GATE 2022 EE Q.30: solution and answer

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