RLC series

${\displaystyle v_{L}+v_{C}+v_{R}=0}$

${\displaystyle L{\frac {d}{dt}}i+{\frac {1}{C}}\int idt+iR=0}$

${\displaystyle {\frac {d^{2}}{dt^{2}}}i=-2\alpha {\frac {d}{dt}}i-\beta i}$

${\displaystyle i=Ae^{(-\alpha \pm j{\sqrt {\beta -\alpha }})t}=Ae^{-\alpha t}e^{\pm j\omega t}=A(\alpha )Sin\omega t}$

${\displaystyle \omega ={\sqrt {\beta -\alpha }}}$

${\displaystyle A(\alpha )=Ae^{-\alpha t}}$

${\displaystyle \beta ={\frac {1}{T}}={\frac {1}{LC}}}$

${\displaystyle \alpha =\beta \gamma ={\frac {R}{2L}}}$

${\displaystyle T=LC}$

${\displaystyle \gamma =RC}$

2 port LC series-R

2 port R-LC series