Electricity/Electric circuit/RLC circuit

/RLC circuit/

RLC series

$v_L+v_C+v_R=0$

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

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

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

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

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

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

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

$T=LC$

$\gamma =RC$