Electronics/Electronics Formulas/Series Circuits/Series RLC

Circuit Configuration

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

$L\frac{di}{dt}+\frac{1}{C}\int idt+iR=0$

$\frac{d^{2}i}{d{t}^2}+\frac{R}{L}\frac{di}{dt}+\frac{1}{LC}=0$

$s^2+\frac{R}{L}s+\frac{1}{LC}=0$

$s=(-\alpha \pm \lambda )t$

$\lambda =\sqrt{{\alpha }^2-{\beta }^2}$

$\alpha =\frac{R}{2L}$

$\beta =\frac{1}{LC}$

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

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

$V_L+V_C=0$ .

${\displaystyle }$

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