Electronics/LC transient

For a series LC consists of one capacitor connected with one coil in series in a closed loop

Circuit Impedance

In Polar Form

$Z = \omega L \angle 90 + \frac{1}{\omega C} \angle -90$

In Compled Form

$Z = j\omega L + \frac{1}{j\omega C}$

Circuit at equilibrium

$L\frac{dI}{dt} + \frac{1}{C} \int I dt = 0$

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

Time Constant

T = LC