Electronics Handbook/Circuits/LC Series

LC Series Configuration
Picture

Circuit's Impedance

 * $$ Z = Z_L + Z_C$$
 * Z = ω L /_90 + $$\frac{1}{\omega C}$$/_-90


 * $$Z_L > Z_C$$
 * Z = Z_L - Z_C/_90


 * $$Z_L < Z_C$$
 * Z = Z_C - Z_L/_-90


 * $$Z_L = Z_C$$
 * Z = 0 /_ 0

Natural Response

 * $$L \frac{dI}{dt} + \frac{1}{C}\int I dt = 0$$
 * $$\frac{dI}{dt} + \frac{1}{LC}\int I dt = 0$$
 * $$\frac{d^2I}{dt^2} + \frac{1}{LC} = 0$$
 * $$s^2 = - \frac{1}{LC}$$
 * s = ± $$\sqrt{-\frac{1}{LC}} t$$ = ± j $$\sqrt{\frac{1}{LC}} t = \pm j\omega t$$

The Second Ordered Equation has two imaginary roots
 * $$I = e^ j \omega t + e^-j \omega t$$
 * $$I = A Sin\omega t$$

The natural response of the LC series is a Sinusoidal Wave. Therefore, LC series can be used as a Sin Wave Oscillator

Resonance Response
Resonance occurs in the circuit when $$Z_L - Z_C = 0. V_C + V_L = 0$$
 * $$Z_C = Z_L$$
 * $$\omega L = \frac{1}{\omega C}$$
 * $$\omega = \sqrt{\frac{1}{LC}}$$


 * $$V_L + V_C = 0 $$
 * $$V_L = - V_C $$

At Resonance, the LC series has the capability to generate Standing Wave Oscillation

Summary
LC series has a second order equation of current with two imaginaries roots
 * $$I(t) = e^(j\omega t) + e^(-j\omega t)$$

Which generates Sin Wave Oscillation
 * $$I(t) = A Sin \omega t$$
 * $$\omega = \sqrt{\frac{1}{T}}$$
 * T = LC

At resonance, Impedance of Inductor and Capacitor cancel out
 * ZL = ZC
 * $$\omega = \sqrt{\frac{1}{LC}}$$

Which generates Standing Wave Oscillation