Electronics/Electronics Formulas/Series Circuits/Series RLC

Circuit Configuration

 * RLC_series_circuit.png

Circuit's Impedance
The total Impedance of the circuit
 * $$Z = Z_R + Z_L$$
 * $$Z = R + j\omega L$$
 * $$Z = \frac{1}{R} (1 + j\omega T)$$
 * $$T = \frac{L}{R}$$

Differential Equation
The Differential equation of the circuit at equilibrium
 * $$L \frac{di}{dt} + \frac{1}{C} \int i dt + iR= 0$$
 * $$\frac{d^2i}{dt^2} + \frac{R}{L} \frac{di}{dt} + \frac{1}{LC} = 0$$
 * $$s^2 + \frac{R}{L} s + \frac{1}{LC} = 0$$
 * $$s = (-\alpha \pm \lambda) t$$
 * $$\lambda = \sqrt{\alpha^2 - \beta^2}$$
 * $$\alpha = \frac{R}{2L}$$
 * $$\beta = \frac{1}{LC}$$

The Natural Response of the circuit

 * $$\lambda = 0 $$ . $$ \alpha^2 = \beta^2$$
 * $$i = e^(-\alpha t)$$


 * $$\lambda = 0 $$ . $$ \alpha^2 = \beta^2$$
 * $$i = e^(-\alpha t)[e^(\lambda t) + e^(-\lambda t)]$$


 * $$\lambda = 0 $$ . $$ \alpha^2 = \beta^2$$
 * $$i = e^(-\alpha t)[e^(j \lambda t) + e^(-j \lambda t)]$$

The Resonance Response of the circuit

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


 * $$ \omega = 0$$ . $$ \omega = 0$$