# Electricity/Electric circuit/RL circuit

## RL circuit

### RL series

${\displaystyle v_{L}+v_{R}=0}$
${\displaystyle L{\frac {d}{dt}}i+iR=0}$
${\displaystyle {\frac {d}{dt}}i=-{\frac {1}{T}}i}$
${\displaystyle i=Ae^{-{\frac {t}{T}}}}$
${\displaystyle T={\frac {L}{R}}}$

### 2 port LR

${\displaystyle {\frac {v_{o}}{v_{i}}}={\frac {R}{R+j\omega L}}={\frac {1}{1+j\omega T}}}$
${\displaystyle T={\frac {L}{R}}}$
${\displaystyle \omega _{o}={\frac {1}{T}}}$
${\displaystyle \omega =0}$  . ${\displaystyle v_{o}=v_{i}}$
${\displaystyle \omega _{o}=\omega _{o}}$  . ${\displaystyle v_{o}={\frac {v_{i}}{2}}}$
${\displaystyle \omega _{o}=00}$  . ${\displaystyle v_{o}=0}$

### 2 port RL

${\displaystyle {\frac {v_{o}}{v_{i}}}={\frac {j\omega L}{R+j\omega L}}={\frac {j\omega T}{R+j\omega T}}}$
${\displaystyle T={\frac {L}{R}}}$
${\displaystyle \omega _{o}={\frac {1}{T}}}$
${\displaystyle \omega =0}$  . ${\displaystyle v_{o}=0}$
${\displaystyle \omega _{o}=\omega _{o}}$  . ${\displaystyle v_{o}={\frac {v_{i}}{2}}}$
${\displaystyle \omega _{o}=00}$  . ${\displaystyle v_{o}=v_{i}}$