Question

A current in circuit is given by $i=3+4\sin \omega t$. Then the effective value of current is?

• A. $5$
• B. $\sqrt{7}$
• C. $\sqrt{17}$
• D. $\sqrt{10}$

Answer 1

Answer: (C)

$\sqrt{17}$

Current is given as $i=I_o+I_1\sin wt$

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 $I_{rms}=\sqrt{I_o^2\frac{T}{T}+I_1^2\times 0.5+2I_o{I}_1\times 0}$
$⟹$ $I_{rms}=\sqrt{I_o^2+0.5I_1^2}$

Given : $I_o=3$ $I_1=4$

$⟹I_{rms}=\sqrt{3^2+0.5\times 4^2}=\sqrt{17}$

Correct answer is option C.