Electricity/Electric circuit/LC circuit

/LC circuit/

Circuit at equilibrium

$v_L+v_C=0$

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

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

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

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

$T=LC$

Circuit at resonance

$Z_L+Z_C=0$

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

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

$T=LC$

$V_L+V_C=0$

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