RLC circuit

A circuit of 3 component a resistor and a capacitor and an inductor are connected in series

Circuit	Configuration	Formula
RLC series		At equilibrium ${\displaystyle V_{L}+V_{C}+V_{R}=0}$ ${\displaystyle L{\frac {d^{2}i}{dt^{2}}}+{\frac {1}{C}}\int idt+iR=0}$ ${\displaystyle {\frac {d^{2}i}{dt^{2}}}+{\frac {R}{L}}{\frac {di}{dt}}+{\frac {1}{LC}}i=0}$ ${\displaystyle {\frac {d^{2}i}{dt^{2}}}=-{\frac {R}{2L}}{\frac {di}{dt}}-{\frac {1}{LC}}i}$ ${\displaystyle {\frac {d^{2}i}{dt^{2}}}=-2\alpha {\frac {di}{dt}}-\beta i}$ 1 real root . ${\displaystyle \alpha =\beta }$ . ${\displaystyle i=Ae^{-\alpha t}=A(\alpha )}$ 2 real roots . ${\displaystyle \alpha >\beta }$ . ${\displaystyle i=Ae^{(-\alpha \pm {\sqrt {\alpha -\beta }})t}}$ 2 complex roots . ${\displaystyle \alpha <\beta }$ . ${\displaystyle i=Ae^{(-\alpha \pm j{\sqrt {\beta -\alpha }})t}=A(\alpha )Sin\omega t}$ ${\displaystyle \beta ={\frac {1}{T}}={\frac {1}{LC}}}$ ${\displaystyle \alpha =\beta \gamma ={\frac {R}{2L}}}$ ${\displaystyle T=LC}$ ${\displaystyle \gamma =RC}$ ${\displaystyle A(\alpha )=Ae^{-\alpha t}}$ ${\displaystyle \omega ={\sqrt {\beta -\alpha }}}$ At resonance ${\displaystyle Z_{L}=-Z_{C}}$ ${\displaystyle \omega _{o}={\sqrt {\frac {1}{T}}}}$ ${\displaystyle T=LC}$ ${\displaystyle Z_{t}=Z_{L}+Z_{C}+Z_{R}=Z_{R}=R}$ ${\displaystyle i={\frac {v}{R}}}$ ${\displaystyle i(\omega =0)=0}$ ${\displaystyle i(\omega =\omega _{o})={\frac {v}{R}}}$ ${\displaystyle i(\omega =00)=0}$