Practical Electronics/Series LC

Circuit response

At equilibrium

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

$v_L+V_C=0$

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

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

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

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

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

$T=LC$

At resonance

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

$Z_L+Z_C=0$

$Z_C=-Z_L$

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

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

$T=LC$

$v_L+v_C=0$

$v_C=-v_L$

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

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