Question

An AC current is given by $I=I_0+I_1\sin \omega t$ then the rms value will be.

• A. $\sqrt{I_0^2+0.5I_1^2}$
• B. $\sqrt{I_0^2+0.5I_0^2}$
• C. $0$
• D. $\frac{I_0}{\sqrt{2}}$

Answer 1

The correct option is A $\sqrt{I_0^2+0.5I_1^2}$

Given current is $I=I_o+I_1\sin wt$
Rms current is given by $I_{rms}=\sqrt{\frac{{\int }_o^T{I}^{2}dt}{T}}$
$\therefore$ $I_{rms}=\sqrt{\frac{{\int }_o^T(I_o^2+I_1^2{\sin }^{2}wt+2I_o{I}_1\sin wt)dt}{T}}$
Or $I_{rms}=\sqrt{I_o^2\frac{{\int }_o^{T}dt}{T}+I_1^2\frac{{\int }_o^T{\sin }^{2}wtdt}{T}+2I_o{I}_1\frac{{\int }_o^T\sin wtdt}{T}}$
Using $\frac{{\int }_o^T{\sin }^{2}wt}{T}=0.5$ and $\frac{{\int }_o^T\sin wt}{T}=0$
We get $I_{rms}=\sqrt{I_o^2\frac{T}{T}+I_1^2\times 0.5+2I_o{I}_1\times 0}$
$⟹$ $I_{rms}=\sqrt{I_o^2+0.5I_1^2}$