Periodically switched RL circuit ($I_{\max }=(V/R)/(1+e^{-R\Delta t/L})$)

An RL circuit alternating for equal intervals $\Delta t$ between zero drive and constant drive $V$ has a unique attracting periodic electric current when $R,L,\Delta t>0$. Write $s=e^{-R\Delta t/L}$ and $J=V/R$. Sampling just before each zero-drive interval gives the recurrence relation $I_{n+1}=s^2{I}_n+J(1-s)$. Subtracting its fixed point yields $I_n-J/(1+s)=s^{2n}[I_0-J/(1+s)]$. Thus the limiting upper and lower electric currents are $J/(1+s)$ and $sJ/(1+s)$; convergence is geometric. This gives a simple exact ripple estimate for a periodically driven inductor.