Electronics/Electronics Formulas/Series Circuits/Series LC

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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

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

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

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

With Natural Response of a Wave function

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

With Resonance Response of a Standing Wave function

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