LC circuit

Circuit at equilibrium

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

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

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

${\displaystyle i=Ae^{\pm j{\sqrt {\frac {1}{T}}}t}=Ae^{\pm j\omega t}=ASin\omega t}$

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

${\displaystyle T=LC}$

Circuit at resonance

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

${\displaystyle j\omega L+{\frac {1}{j\omega C}}=0}$

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

${\displaystyle T=LC}$

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

${\displaystyle v(\theta )=ASin(\omega +2\pi )-ASin(\omega -2\pi )}$