# Electronics/Electronics Formulas/Series Circuits/Series LC

## 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

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

The Natural Response of the circuit

$i = A Sin \omega t$

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

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

With Natural Response of a Wave function

$i = A Sin \omega t$

With Resonance Response of a Standing Wave function

$i = A Sin \omega t$