# Fundamental Physics/Electronics/RLC Circuit

## RLC Circuit A circuit of 3 component a resistor and a capacitor and an inductor connected in series

## When the circuit is at equilibrium

$V_{C}+V_{L}+V_{R}=0$ $L{\frac {d}{dt}}i(t)+{\frac {1}{C}}\int i(t)dt+Ri(t)=0$ ${\frac {d^{2}}{dt^{2}}}i(t)+{\frac {R}{L}}{\frac {d}{dt}}i(t)+{\frac {1}{LC}}i(t)=0$ ${\frac {d^{2}}{dt^{2}}}i(t)+2\alpha {\frac {d}{dt}}i(t)+\beta i(t)=0$ ${\frac {d^{2}}{dt^{2}}}i(t)=-2\alpha {\frac {d}{dt}}i(t)-\beta i(t)$ Solution to the above equation

• 1 real root
$s=-\alpha$ $i(t)=Ae^{-\alpha t}=A(\alpha )$ • 2 real roots
$s=-\alpha \pm {\sqrt {\alpha -\beta }}$ $i(t)=Ae^{(-\alpha \pm {\sqrt {\alpha -\beta }})t}$ • 2 complex roots
$s=-\alpha \pm j{\sqrt {\beta -\alpha }}$ $i(t)=Ae^{(-\alpha \pm j{\sqrt {\beta -\alpha }})t}=A(\alpha )Sin\omega t$ $A(\alpha )=Ae^{-\alpha t}$ $\omega ={\sqrt {\beta -\alpha }}$ $\beta ={\frac {1}{T}}={\frac {1}{LC}}$ $\alpha =\beta \gamma ={\frac {R}{2L}}$ $T=LC$ $\gamma =RC$ ## When the circuit is at resonant

$Z_{t}=R$ $Z_{L}=-Z_{C}$ $i(\omega =0)=0$ $i(\omega =\omega _{o})={\frac {v}{R}}$ $i(\omega =0)=0$ 