An alternating current is given by the equation, i = i1 sin wt + i2 cos wt.
The irms will be;
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( \displaystyle\frac{1}{\sqrt{2}} \) (i12 + i22)1/2
The rms value is the square root of the mean of \( i^{2} \) over one cycle. (The paper writes w for the angular frequency \( \omega \).)
\( \displaystyle i^{2} = i_1^{2}\sin^{2}\omega t + i_2^{2}\cos^{2}\omega t + 2i_1 i_2 \sin\omega t\cos\omega t \)
Over a complete cycle the average of \( \sin^{2}\omega t \) and of \( \cos^{2}\omega t \) is \( \frac{1}{2} \), and the average of \( \sin\omega t\cos\omega t = \frac{1}{2}\sin 2\omega t \) is zero.
\( \displaystyle \langle i^{2}\rangle = \frac{i_1^{2}}{2} + \frac{i_2^{2}}{2} = \frac{i_1^{2} + i_2^{2}}{2} \)
\( \displaystyle i_{\mathrm{rms}} = \sqrt{\frac{i_1^{2} + i_2^{2}}{2}} = \frac{1}{\sqrt{2}}\left(i_1^{2} + i_2^{2}\right)^{1/2} \)
Shortcut: A sine and a cosine of the same frequency combine into a single sinusoid: \( i_1\sin\omega t + i_2\cos\omega t = I_0\sin(\omega t + \phi) \) with \( I_0 = \sqrt{i_1^{2} + i_2^{2}} \). Its rms value is \( I_0/\sqrt{2} \) directly.
Note: A similar question was also asked in CUET (UG) 2023 on 09-06-2023 (Slot 3).