Electronics/Electronics Formulas/Series Circuits/Series LC

Circuit Configuration

 * RL_Series_Open-Closed.svg

Formula
The total Impedance of the circuit
 * $$Z = Z_R + Z_L$$
 * $$Z = R + j\omega L$$
 * $$Z = \frac{1}{R} (1 + j\omega T)$$
 * $$T = \frac{L}{R}$$

The Differential equation of the circuit at equilibrium
 * $$L \frac{di}{dt} + \frac{1}{C} \int i dt = 0$$
 * $$\frac{d^2i}{dt^2} + \frac{1}{LC} = 0$$
 * $$s^2 + \frac{1}{LC} = 0$$
 * $$s = \pm j \sqrt{\frac{1}{LC}} t$$
 * $$s = \pm j \omega t$$

The Natural Response of the circuit
 * $$i = A Sin \omega t$$

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

Summary
Series LC can be characterised by

2nd order Differential Equation
 * $$\frac{d^2i}{dt^2} + \frac{1}{T} = 0$$
 * $$T = LC$$

With Natural Response of a Wave function
 * $$i = A Sin \omega t$$

With Resonance Response of a Standing Wave function
 * $$i = A Sin \omega t$$