RLC Circuit

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

When the circuit is at equilibrium

${\displaystyle V_{C}+V_{L}+V_{R}=0}$

${\displaystyle L{\frac {d}{dt}}i(t)+{\frac {1}{C}}\int i(t)dt+Ri(t)=0}$

${\displaystyle {\frac {d^{2}}{dt^{2}}}i(t)+{\frac {R}{L}}{\frac {d}{dt}}i(t)+{\frac {1}{LC}}i(t)=0}$

${\displaystyle {\frac {d^{2}}{dt^{2}}}i(t)+2\alpha {\frac {d}{dt}}i(t)+\beta i(t)=0}$

${\displaystyle {\frac {d^{2}}{dt^{2}}}i(t)=-2\alpha {\frac {d}{dt}}i(t)-\beta i(t)}$

Solution to the above equation

• 1 real root

${\displaystyle s=-\alpha }$

${\displaystyle i(t)=Ae^{-\alpha t}=A(\alpha )}$

• 2 real roots

${\displaystyle s=-\alpha \pm {\sqrt {\alpha -\beta }}}$

${\displaystyle i(t)=Ae^{(-\alpha \pm {\sqrt {\alpha -\beta }})t}}$

• 2 complex roots

${\displaystyle s=-\alpha \pm j{\sqrt {\beta -\alpha }}}$

${\displaystyle i(t)=Ae^{(-\alpha \pm j{\sqrt {\beta -\alpha }})t}=A(\alpha )Sin\omega t}$

${\displaystyle A(\alpha )=Ae^{-\alpha t}}$

${\displaystyle \omega ={\sqrt {\beta -\alpha }}}$

${\displaystyle \beta ={\frac {1}{T}}={\frac {1}{LC}}}$

${\displaystyle \alpha =\beta \gamma ={\frac {R}{2L}}}$

${\displaystyle T=LC}$

${\displaystyle \gamma =RC}$

When the circuit is at resonant

${\displaystyle Z_{t}=R}$

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

${\displaystyle i(\omega =0)=0}$

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

${\displaystyle i(\omega =0)=0}$