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

$Z_{t}=R$

$Z_{L}=-Z_{C}$

$i(\omega =0)=0$

$i(\omega =\omega _{o})={\frac {v}{R}}$

$i(\omega =0)=0$