Electricity/Electric circuit/RLC circuit

RLC circuit

RLC series

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

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

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

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

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

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

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

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

$T=LC$

$\gamma =RC$