Question

The electric current in circuit is given by $I=I_o\left(\frac{t}{T}\right)$ some time. What is the rms current for the period $t=0$ to $t=T$?

• A. $\frac{I_0}{\sqrt{3}}$
• B. $\frac{I_0}{2}$
• C. $\frac{I_0}{\sqrt{2}}$
• D. $\frac{I_0}{3}$

Answer 1

The correct option is A $\frac{I_0}{\sqrt{3}}$

RMS of a function $f\left(t\right)$ is given by
$f_{rms}=\sqrt{\frac{{\int }_0^T{f}^2\left(t\right)dt}{{\int }_0^{T}dt}}\therefore i_{rms}=\sqrt{\frac{{\int }_0^T{i}^2\left(t\right)dt}{{\int }_0^{T}dt}}==\sqrt{\frac{{\int }_0^T\frac{{I_0}^2{t}^2}{T^2}dt}{{\int }_0^{T}dt}}\text{Here},{\int }_0^T\frac{{I_0}^2{t}^2}{T^2}dt=\frac{{I_0}^{2}T}{3}\text{and}{\int }_0^{T}dt=T\therefore i_{rms}=\sqrt{\frac{{I_0}^2}{3}}=\frac{I_0}{\sqrt{3}}$