Circuit Theory/LC Tuned Circuits

Series LC
A circuit of one Capacitor and one inductor connected in series

Circuit Impedance

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

Natural Response
At equilibrium, the total voltage of the two components are equal to zero
 * $$L\frac{dI}{dt} + IR = 0$$
 * $$\frac{dI}{dt} = - I \frac{R}{L}$$
 * $$\int \frac{dI}{I} = - \frac{R}{L} \int dt $$
 * $$ln I = - \frac{t}{T} + C$$
 * $$I = e^(- \frac{t}{T} + C)$$
 * $$I = A e^(- \frac{t}{T})$$

The Natural Response of the circuit is a Exponential Decrease in time

Resonance Response

 * $$ Z_L - Z_C = 0$$ . $$ V_L + V_C = 0$$
 * $$\omega L = \frac{1}{\omega C}$$
 * $$\omega = \sqrt{\frac{1}{LC}}$$
 * $$ V_C = - V_L $$

In Resonance, Impedance of Inductor and Capacitance is equal and the sum of the Capacitor and Inductor's voltage are equal result in Standing Wave Oscillation. Therefore, Lossless LC series can generate Standing Wave Oscillation

LC in Parallel
A circuit of one Capacitor and one inductor connected in parallel

Circuit Impedance

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