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_0e (-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:
`i_{rms}^2 =1/τ  ∫_0^τ  i_0^2   e^-2t//τ  dt`

`⇒ i_{rms}^2 = i_0^2/τ ∫_0^tau e^-"2t"// τ dt ` 

=`t_0^2/τ xx [-r/2 e^-2t//tau ]_0^tau`

= `-i_0^2/2 xx r/2xx[e^-2 - 1]`

= `t_0^2/2 (1 - 1/e^2)`

`i_rms=i_0/e sqrt((e^2 - 1)/2)`