Electronics Handbook/Components/LC Network

LC Series
Circuit of L and C connected in series

Circuit Impedance

 * $$Z = Z_C + Z_L $$
 * $$Z = \frac{1}{j\omega C} + j \omega L$$
 * $$Z = \frac{1}{j\omega C} (j \omega^2 + \frac{1}{T})$$
 * T = LC

Circuit Natural Response
At equilibrium the total voltage of the circuit is zero
 * $$ V_L + V_C = 0$$
 * $$L \frac{dI}{dt} + \frac{1}{C}\int I dt = 0$$
 * $$\frac{dI}{dt} + \frac{1}{LC}\int I dt = 0$$
 * $$\frac{d^2I}{dt^2} + \frac{1}{LC} = 0$$
 * $$s^2 = - \frac{1}{LC}$$
 * s = ± $$\sqrt{-\frac{1}{LC}}$$ = ± j $$\sqrt{\frac{1}{LC}}$$


 * $$I = e^ (j \sqrt{\frac{1}{LC}} t) + e^(-j \sqrt{\frac{1}{LC}} t)$$
 * $$I = A Sin \sqrt{\frac{1}{LC}} t $$
 * $$I = A Sin \omega t $$

At equilibrium or The natural response of the LC series is the Sinusoidal Wave Oscillation 

Resonance
Resonance occurs when the frequency components cancel each other out. Therefore, at resonance
 * $$Z_L = Z_C$$
 * $$\omega L = \frac{1}{\omega C}$$
 * $$\omega = \sqrt{\frac{1}{LC}}$$


 * $$V_C = -V_L$$

At resonance the response of the LC series is the Standing Wave Oscillation 

Summary

 * The natural frequency reponse of the circuit is a Sinusoidal Wave
 * At resonance, the frequency reponse of the LC series is the oscillation of a Standing Wave