Electrical oscillation represents oscillation of electricity found in series LC and series RLC circuits

Series LC

When circuit is at equilibrium, the total voltages are zero

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

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

${\displaystyle {\frac {d^{2}}{dt^{2}}}i+{\frac {1}{LC}}i=0}$

${\displaystyle {\frac {d^{2}}{dt^{2}}}i=-{\frac {1}{T}}i}$

${\displaystyle i=A\sin \omega t}$

${\displaystyle \omega ={\sqrt {\frac {1}{T}}}}$

${\displaystyle T=LC}$

When circuit is at resonance, the total impedances are zero and the total voltages are zero

${\displaystyle Z_{L}+Z_{C}=0}$

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

${\displaystyle V(\theta )=A\sin(\omega _{o}+2\pi )-A\sin(\omega _{o}-2\pi )}$

${\displaystyle \omega _{o}=\pm j{\sqrt {\frac {1}{T}}}}$

${\displaystyle T=LC}$

Series RLC

When circuit is at equilibrium, the total voltages are zero

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

${\displaystyle L{\frac {d}{dt}}i+{\frac {1}{C}}\int i+iR=0}$

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

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

${\displaystyle i=A(\alpha )\sin \omega t}$

${\displaystyle \alpha =Ae^{-\alpha t}}$

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

${\displaystyle \alpha ={\sqrt {\frac {1}{T}}}={\frac {1}{LC}}}$

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

${\displaystyle T=LC}$

${\displaystyle \alpha =RC}$

When R is shorted, ${\displaystyle R=0}$ and ${\displaystyle \alpha =0}$

${\displaystyle {\frac {d^{2}}{dt^{2}}}i=-\beta i}$

${\displaystyle i=A\sin(\omega t)}$

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

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