Electricity/Electric circuit/LC circuit

LC circuit

Circuit at equilibrium

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

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

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

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

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

$T=LC$

Circuit at resonance

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

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

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

$T=LC$

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

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