Question

An alternating current is given by the equation, $i=i_1\sin \omega t+i_2\cos \omega t$. The RMS value of current will be :

• A. $\frac{1}{2}(i_1^2+i_2^2{)}^{\frac{1}{2}}$
• B. $\frac{1}{\sqrt{2}}(i_1^2+i_2^2{)}^{\frac{1}{2}}$
• C. $\frac{1}{\sqrt{2}}(i_1+i_2{)}^2$
• D. $\frac{1}{\sqrt{2}}(i_1+i_2)$

Answer 1

The correct option is B $\frac{1}{\sqrt{2}}(i_1^2+i_2^2{)}^{\frac{1}{2}}$

Given:
$i=i_1\sin \omega t+i_2\cos \omega t$
$\Rightarrow i=i_1\sin \omega t+i_2\sin ({90}^{\circ }+\omega t)$

So, net current,
$I_o=\sqrt{i_1^2+i_2^2+2i_1{i}_2\cos \theta }$

$\Rightarrow I_o=\sqrt{i_1^2+i_2^2+2i_1{i}_2\cos {90}^{\circ }}$

$\Rightarrow I_o=\sqrt{i_1^2+i_2^2}$

We know that,
$I_{RMS}=\frac{I_o}{\sqrt{2}}$

$\Rightarrow I_{RMS}=\frac{\sqrt{i_1^2+i_2^2}}{\sqrt{2}}=\frac{1}{\sqrt{2}}{(i_1^2+i_2^2)}^{\frac{1}{2}}$