Electronics/Electronics Formulas/Series Circuits/Series LC

Circuit Configuration

Formula

The total Impedance of the circuit

$Z=Z_R+Z_L$

$Z=R+j\omega L$

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

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

The Differential equation of the circuit at equilibrium

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

$\frac{d^{2}i}{d{t}^2}+\frac{1}{LC}=0$

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

$s=\pm j\sqrt{\frac{1}{LC}}t$

$s=\pm j\omega t$

The Natural Response of the circuit

$i=ASin\omega t$

The Resonance Response of the circuit

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

$V_L+V_C=0$ . $V_C=-V_L$

Summary

Series LC can be characterised by

2nd order Differential Equation

$\frac{d^{2}i}{d{t}^2}+\frac{1}{T}=0$

$T=LC$

With Natural Response of a Wave function

$i=ASin\omega t$

With Resonance Response of a Standing Wave function

$i=ASin\omega t$