Electronics/Electronics Formulas/Series Circuits/Series RLC

Formula

Circuit's Impedance

The total Impedance of the circuit

${\displaystyle Z=Z_{R}+Z_{L}}$
${\displaystyle Z=R+j\omega L}$
${\displaystyle Z={\frac {1}{R}}(1+j\omega T)}$
${\displaystyle T={\frac {L}{R}}}$

Differential Equation

The Differential equation of the circuit at equilibrium

${\displaystyle L{\frac {di}{dt}}+{\frac {1}{C}}\int idt+iR=0}$
${\displaystyle {\frac {d^{2}i}{dt^{2}}}+{\frac {R}{L}}{\frac {di}{dt}}+{\frac {1}{LC}}=0}$
${\displaystyle s^{2}+{\frac {R}{L}}s+{\frac {1}{LC}}=0}$
${\displaystyle s=(-\alpha \pm \lambda )t}$
${\displaystyle \lambda ={\sqrt {\alpha ^{2}-\beta ^{2}}}}$
${\displaystyle \alpha ={\frac {R}{2L}}}$
${\displaystyle \beta ={\frac {1}{LC}}}$

The Natural Response of the circuit

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

The Resonance Response of the circuit

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