Electricity/Electric circuit/RC circuit

RC circuit

RC series

$v_{C}+v_{R}=0$

$C{\frac {d}{dt}}v+{\frac {v}{R}}=0$

${\frac {d}{dt}}v=-{\frac {1}{T}}v$

$v=Ae^{-{\frac {t}{T}}}$

$T=RC$

2 port RC

${\frac {v_{o}}{v_{i}}}={\frac {\frac {1}{j\omega C}}{R+{\frac {1}{j\omega C}}}}={\frac {1}{1+j\omega T}}$

$T=RC$

$\omega _{o}={\frac {1}{T}}$

$\omega =0$ . $v_{o}=v_{i}$

$\omega _{o}=\omega _{o}$ . $v_{o}={\frac {v_{i}}{2}}$

$\omega _{o}=00$ . $v_{o}=0$

2 port CR

${\frac {v_{o}}{v_{i}}}={\frac {R}{R+{\frac {1}{j\omega C}}}}={\frac {j\omega T}{R+j\omega T}}$

$T=RC$

$\omega _{o}={\frac {1}{T}}$

$\omega =0$ . $v_{o}=0$

$\omega _{o}=\omega _{o}$ . $v_{o}={\frac {v_{i}}{2}}$

$\omega _{o}=00$ . $v_{o}=v_{i}$