# Electricity/Electric circuit/RC circuit

## RC circuit

### RC series

${\displaystyle v_{C}+v_{R}=0}$
${\displaystyle C{\frac {d}{dt}}v+{\frac {v}{R}}=0}$
${\displaystyle {\frac {d}{dt}}v=-{\frac {1}{T}}v}$
${\displaystyle v=Ae^{-{\frac {t}{T}}}}$
${\displaystyle T=RC}$

### 2 port RC

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

### 2 port CR

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