Question

The current in a discharging LR circuit is given by i = i₀ e^−t/τ , where τ is the time constant of the circuit. Calculate the rms current for the period t = 0 to t = τ.

Answer 1

As per the question,
$i=i_0{e}^{\frac{-t}{\tau }}$
We need to find the rms current. So, taking the average of i within the limits 0 to τ and then dividing by the given time period τ, we get:
$$\begin{gathered} {i_{\mathrm{rms}}}^2=\frac{1}{\tau }{\int }_{\mathit{0}}^{\tau }{i}_0^2\mathit{}{\mathrm{e}}^{\mathit{-}\mathit{2}t\mathit{/}\tau }\mathit{}dt \\ \Rightarrow {i_{rms}}^2=\frac{i_0^2}{\tau }{\int }_0^{\mathrm{\tau }}{\mathrm{e}}^{\mathit{-}\mathit{2}t\mathit{/}\tau }\mathit{}dt \\ =\frac{i_0^2}{\tau }\times {\left[\frac{-\tau }{2}\mathit{}{\mathrm{e}}^{\mathit{-}\mathit{2}t\mathit{/}\tau }\right]}_0^{\tau } \\ =-\frac{i_0^2}{\tau }\times \frac{\tau }{2}\times \left[{\mathrm{e}}^{-2}-1\right] \\ =\frac{{i_0}^2}{2}(1-\frac{1}{{\mathrm{e}}^2}) \\ {i}_{\mathrm{rms}}=\frac{i_0}{e}\sqrt{\frac{e^2-1}{2}} \end{gathered}$$