# Fundamental Physics/Electricity/Electric circuits/RLC circuit

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