Practical Electronics/Series LC


 * RL_Series_Open-Closed.svg

At equilibrium
When circuit is at equilibrium, the total voltage of the circuit is zero
 * $$v_L + V_C = 0$$
 * $$L \frac{di}{dt} + \frac{1}{C} \int i dt = 0$$
 * $$\frac{d^2i}{dt^2} + \frac{1}{LC} = 0$$
 * $$\frac{d^2i}{dt^2} = - \frac{1}{T}$$
 * $$i = A e^{\pm j \omega t} = A \sin \omega t$$
 * $$ \omega =\sqrt{\frac{1}{T}}$$
 * $$T = LC$$

At resonance
When circuit is a resonance, the total impedance of the circuit is zero
 * $$Z_L + Z_C = 0 $$
 * $$Z_C = -Z_L $$
 * $$ \frac{1}{\omega C} = -\omega L$$
 * $$\omega_o = \pm j \sqrt{\frac{1}{T}}$$
 * $$T=LC$$


 * $$v_L + v_C = 0 $$
 * $$v_C = -v_L $$
 * $$v(\theta) = A \sin(\omega_ot+2 \pi) - A \sin(\omega_ot-2 \pi) $$