Practical Electronics/Series RL

A circuit of two components R and L connected in series

Natural Response of the cicuit can be obtained by setting the differential equation of the circuit to zero

$v_{L}+v_{R}=0$

$L{\frac {dI}{dt}}+IR=0$

${\frac {dI}{dt}}=-{\frac {1}{T}}I$

$\int {\frac {dI}{I}}=-{\frac {1}{T}}\int dt$

$LnI=-{\frac {1}{T}}+C$

$I=e^{(-{\frac {1}{T}}+C)}$

$I=Ae^{-{\frac {t}{T}}}$

$T={\frac {L}{R}}$

$e^{C}=IR$