Question

The current in a certain circuit varies with times as shown in the figure. The rms current in terms of $I_0$is , $\frac{I_0}{\sqrt{x}}$. Find $x$

Answer 1

Solution:-

We know,

RMS current $I_{rms}=\sqrt{\frac{{\int }_0^{\tau }[I\left(\tau \right){]}^{2}d\tau }{{\int }_0^{\tau }d\tau }}⟶e{q}^n\left(i\right)$

where \tau is time period of given waveform. Now, first we solve the integral as

${\int }_0^{\tau }[I\left(\tau \right){]}^{2}d\tau ⟶e{q}^n\left(ii\right)$

Now, as we can see from given waveform I varies proportionally with \tau as zero to $I_0$ in period zero to $\frac{\tau }{2}$ and $-I_0$ to zero in period $\frac{\tau }{2}$ to So, integral [$e{q}^n\left(ii\right)$] becomes

${\int }_0^{I_0}{I}^{2}dt+{\int }^0-I_0{I}^{2}dI$

$\frac{I^3}{3}{|}_0^{I_0}+\frac{I^3}{3}{|}_{-I_0}^0=\frac{2{I_0}^3}{3}⟶e{q}^n\left(iii\right)$

Similarly we get ${\int }_0^{\tau }d\tau =2I_0⟶e{q}^n\left(iv\right)$

Use $\left(iii\right)\&\left(iv\right)ine{q}^n\left(i\right)$

we get $I_{rms}=\sqrt{\frac{2{I_0}^3}{3\left(2I_0\right)}}=\frac{I_0}{\sqrt{3}}$

So, x=3