Question

An alternating current is given by $i=i_1\cos \omega t+i_2\sin \omega t$. The rms current is given by

• A. $\frac{i_1+i_2}{\sqrt{2}}$
• B. $\frac{|i_1+i_2|}{\sqrt{2}}$
• C. $\sqrt{\frac{i_1^2-i_2^2}{2}}$
• D. $\sqrt{\frac{i_1^2+i_2^2}{2}}$

Answer 1

The correct option is D $\sqrt{\frac{i_1^2+i_2^2}{2}}$

We know, general form of current in AC circuitis
$i=i_0\sin (\omega t+\varphi )$ and
$i_{rms}=\frac{i_0}{\sqrt{2}}$.
Given,
$i=i_1\cos \omega t+i_2\sin \omega t$
we can rewritre it as
$=\sqrt{i_1^2+i_2^2}[\frac{i_1}{\sqrt{i_1^2+i_2^2}}\cos \omega t+\frac{i_2}{\sqrt{i_1^2+i_2^2}}\sin \omega t]$
Let us consider a traingle as shown
Therefore the equation becomes
$i=\sqrt{i_1^2+i_2^2}(\sin \varphi \cdot \cos \omega t+\cos \varphi \cdot \sin \omega t)$
$i=\sqrt{i_1^2+i_2^2}\left(\sin \right(\omega t+\varphi \left)\right)$

Therefore
$i_{rms}=\frac{\sqrt{i_1^2+i_2^2}}{\sqrt{2}}$