Electronics/Electronics Formulas/Series Circuits/Series LC

Formula

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

${\displaystyle L{\frac {di}{dt}}+{\frac {1}{C}}\int idt=0}$
${\displaystyle {\frac {d^{2}i}{dt^{2}}}+{\frac {1}{LC}}=0}$
${\displaystyle s^{2}+{\frac {1}{LC}}=0}$
${\displaystyle s=\pm j{\sqrt {\frac {1}{LC}}}t}$
${\displaystyle s=\pm j\omega t}$

The Natural Response of the circuit

${\displaystyle i=ASin\omega t}$

The Resonance Response of the circuit

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

Summary

Series LC can be characterised by

2nd order Differential Equation

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

With Natural Response of a Wave function

${\displaystyle i=ASin\omega t}$

With Resonance Response of a Standing Wave function

${\displaystyle i=ASin\omega t}$