Electronics/Electronics Formulas/Series Circuits/Series RLC

The total Impedance of the circuit

${\displaystyle Z=Z_{R}+Z_{L}}$

${\displaystyle Z=R+j\omega L}$

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

${\displaystyle T={\frac {L}{R}}}$

The Differential equation of the circuit at equilibrium

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

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

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

${\displaystyle s=(-\alpha \pm \lambda )t}$

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

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

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

• ${\displaystyle \lambda =0}$ . ${\displaystyle \alpha ^{2}=\beta ^{2}}$

${\displaystyle i=e^{(}-\alpha t)}$

• ${\displaystyle \lambda =0}$ . ${\displaystyle \alpha ^{2}=\beta ^{2}}$

${\displaystyle i=e^{(}-\alpha t)[e^{(}\lambda t)+e^{(}-\lambda t)]}$

• ${\displaystyle \lambda =0}$ . ${\displaystyle \alpha ^{2}=\beta ^{2}}$

${\displaystyle i=e^{(}-\alpha t)[e^{(}j\lambda t)+e^{(}-j\lambda t)]}$

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

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

${\displaystyle }$

${\displaystyle \omega =0}$ . ${\displaystyle \omega =0}$