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An AC current is given by $i(t)=I_1\sin wt+I_2\cos wt$. Find the rms value of current.

(a) $i_{rms}={\sqrt{I_1^2+I_2^2}\over2}$

(b) $i_{rms}={\sqrt{I_1^2+I_2^2\over2}}$

(c) $i_{rms}={I_1^2+I_2^2\over2}$

(d) $i_{rms}=2\left(\sqrt{I_1^2+I_2^2}\right)$
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Answer: (b) $i_{rms}={\sqrt{I_1^2+I_2^2\over2}}$

Solution:

$i(t)=I_1\sin\omega t+I_2\cos\omega t$

So, we could write:

$\implies i(t)=\sqrt{I_1^2+I_2^2}\sin(\omega t+\phi)$

[Where $\phi = tan^{-1}\left({I_2\over I_1}\right)$]

$\implies i(t)=\sqrt{I_1^2+I_2^2}\sin(\omega t+\phi)$

So rms value would be

$i_{rms}={\sqrt{I_1^2+I_2^2\over2}}$

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