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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.
Solution
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)`
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