Electricity/Electric circuit/RLC circuit

RLC series

 * $$v_L + v_C + v_R = 0$$
 * $$L \frac{d}{dt} i + \frac{1}{C} \int i dt + iR= 0$$
 * $$\frac{d^2}{dt^2} i = -2 \alpha \frac{d}{dt} i - \beta i$$
 * $$ i = A e^{(-\alpha \pm j \sqrt{\beta - \alpha})t} = A e^{-\alpha t} e^{\pm j \omega t} = A(\alpha) Sin \omega t$$
 * $$ \omega = \sqrt{\beta - \alpha}$$
 * $$ A(\alpha)= Ae^{-\alpha t}$$
 * $$ \beta = \frac{1}{T} = \frac{1}{LC}$$
 * $$ \alpha = \beta \gamma = \frac{R}{2L}$$
 * $$ T=LC$$
 * $$ \gamma=RC$$