# Electronics/Electronics Formulas/Series Circuits/Series RLC

## §Formula

### §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

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