RL low-pass filter ($H(\omega )=R/(R+i\omega L)$)

An RL circuit with the output measured across its resistor is a first-order low-pass filter. For an ideal inductor, an ideal voltage source and a high-resistance output instrument, the electrical impedances $i\omega L$ and $R$ giveThe gain approaches one at low angular frequency and $R/(\omega L)$ at high angular frequency; at ${\omega }_c$ it is $1/\sqrt{2}$. The time-domain equation is $L\dot{I}+RI=V_{\mathrm{in}}$ and $V_{\mathrm{out}}=RI$, so the time constant is $L/R$. Instrument loading, source resistance and nonideal inductance can change the response.