GATE 2024 IN Q.31: solution and answer

GATE 2024 · INQ.31 · 1 mark · NAT

The current $i (t)$ drawn by a circuit is given as

i (t) = 4 + 30 cos (t) - 20 sin (t) + 15 cos (3 t) - 10 sin (3 t) A

The root-mean-square value of $i (t)$ is ________ A (rounded off to one decimal place).

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Answer and solution

ANSWER

The rms value of $i (t)$ is $828.5 \approx 28.8 A$ .

Official answer: 27.0 to 30.0Source: Official answer key, GATE 2024 IN (IISc Bengaluru)

Orthogonality of sinusoids: every term of $i (t)$ repeats with the common period $T = 2 π$ . Averaged over one period, every cross term averages to zero: dc × sinusoid, sinusoids at different frequencies, and $cos$ × $sin$ at the same frequency. So the mean-square value of $i (t)$ is the sum of the mean-square values of its separate components: $I_{rms}^{2} = I_{0}^{2} + \frac{I _{m 1}^{2}}{2} + \frac{I _{m 3}^{2}}{2}$
DC term: the mean-square contribution is $4^{2} = 16$ .
Fundamental ( $ω = 1 rad/s$ ): $30 cos t - 20 sin t$ is a single sinusoid with peak $I_{m 1} = 3 0^{2} + 2 0^{2} = 1300$ , so its mean-square value is $\frac{1300}{2} = 650$ .
Third harmonic ( $ω = 3 rad/s$ ): $15 cos 3 t - 10 sin 3 t$ has peak $I_{m 3} = 1 5^{2} + 1 0^{2} = 325$ , so its mean-square value is $\frac{325}{2} = 162.5$ .
Total mean square: $I_{rms}^{2} = 16 + 650 + 162.5 = 828.5 A^{2}$
RMS value: $I_{rms} = 828.5 \approx 28.78 A \approx 28.8 A$

KEY IDEA

For a sum of a dc term and sinusoids at distinct frequencies, the mean squares add because the cross terms average to zero: $I_{rms} = I_{0}^{2} + \sum_{k} I_{mk}^{2} /2$ . First merge the $cos$ and $sin$ terms at each frequency into one sinusoid of peak $a^{2} + b^{2}$ . The dc term counts at full value, not halved.

WATCH OUT FOR

Common mistakes: (1) halving the dc term, i.e. using $4^{2} /2$ instead of $4^{2}$ ; (2) adding peak or rms amplitudes directly instead of their squares, which gives $4 + 30/ 2 + \dots$ ; (3) treating the $cos$ and $sin$ terms at the same frequency as if their peaks add linearly ( $30 + 20$ ) instead of in quadrature ( $3 0^{2} + 2 0^{2}$ ). The negative signs on the sine terms have no effect on the rms value.

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GATE 2024 IN Q.31: solution and answer

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