RL Circuit

Circuit	Configuration	Formual
RL series		${\displaystyle V_{L}+V_{R}=0}$ ${\displaystyle L{\frac {di(t)}{dt}}+i(t)R=0}$ ${\displaystyle {\frac {di(t)}{dt}}=-{\frac {1}{T}}i(t)}$ ${\displaystyle T={\frac {L}{R}}}$ ${\displaystyle \int {\frac {di(t)}{i(t)}}=-{\frac {1}{T}}\int dt}$ ${\displaystyle Lni(t)=-{\frac {1}{T}}+c}$ ${\displaystyle i(t)=Ae^{-{\frac {t}{T}}+c}=Ae^{-{\frac {t}{T}}}}$ ${\displaystyle A=e^{c}}$
Low pass filter		${\displaystyle {\frac {v_{o}}{v_{i}}}={\frac {R}{R+j\omega L}}={\frac {1}{1+j\omega {\frac {L}{R}}}}={\frac {1}{1+j\omega T}}}$ ${\displaystyle T={\frac {L}{R}}}$ ${\displaystyle \omega _{o}={\frac {1}{T}}={\frac {R}{L}}=2\pi f_{o}}$ ${\displaystyle v_{o}(\omega =0)=v_{i}}$ ${\displaystyle v_{o}(\omega =\omega _{o})={\frac {v_{i}}{2}}}$ ${\displaystyle v_{o}(\omega =00)=0}$
High pass filter		${\displaystyle {\frac {v_{o}}{v_{i}}}={\frac {j\omega L}{R+j\omega L}}={\frac {j\omega {\frac {L}{R}}}{1+j\omega {\frac {L}{R}}}}={\frac {j\omega T}{1+j\omega T}}}$ ${\displaystyle T={\frac {L}{R}}}$ ${\displaystyle \omega _{o}={\frac {1}{T}}={\frac {R}{L}}=2\pi f}$ ${\displaystyle v_{o}(\omega =0)=0}$ ${\displaystyle v_{o}(\omega =\omega _{o})={\frac {v_{i}}{2}}}$ ${\displaystyle v_{o}(\omega =0)=v_{i}}$