Electricity/Electric circuit/RC circuit

/RC circuit/

RC series

$v_C+v_R=0$

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

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

$v=A{e}^{-\frac{t}{T}}$

$T=RC$

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

$\frac{v_o}{v_i}=\frac{R}{R+\frac{1}{j\omega C}}=\frac{j\omega T}{R+j\omega T}$

$T=RC$

${\omega }_o=\frac{1}{T}$

$\omega =0$ . $v_o=0$

${\omega }_o={\omega }_o$ . $v_o=\frac{v_i}{2}$

${\omega }_o=00$ . $v_o=v_i$