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

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

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

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

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

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

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

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

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

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

$V_{L}+V_{C}=0$ .

${\displaystyle }$

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