Practical Electronics/Series LC

Circuit response

At equilibrium

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

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

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

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

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

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

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

$T=LC$

At resonance

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

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

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

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

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

$T=LC$

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

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

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