At equilibrium

When circuit is at equilibrium, the total voltage of the circuit is zero

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

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

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

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

${\displaystyle i=Ae^{\pm j\omega t}=A\sin \omega t}$

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

${\displaystyle T=LC}$

At resonance

When circuit is a resonance , the total impedance of the circuit is zero

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

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

${\displaystyle {\frac {1}{\omega C}}=-\omega L}$

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

${\displaystyle T=LC}$

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

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

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