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

RMS Current is $I_r_m_s = \frac {I_0}{e}(\sqrt (\frac {e^2 - 1}{2}))$

Explanation:

• The current in a discharging LR circuit is given by

I = $I_0 e^{-t/T}$

• Then, the rms current for the period t=0 to t= T can be obtained by:

$i_{rms}^{2} = \frac {1}{T}\int_{0}^{T}I_{0}^{2}e^{-2t/T} dt$

= $\frac {I_0^2}{T}\times [\frac {T}{2}e^{-2t/T}]_{0}^{T}$

= $\frac {I_0^2}{T} \times [\frac {T}{2}e^{-2t/T}-1]$

= $i_{rms}^{2} = \frac {I_0^2}{T} \times (1 - \frac {1}{e^2})$

So the rms current is  

$I_r_m_s = \frac {I_0}{e}(\sqrt (\frac {e^2 - 1}{2}))$