LC circuit (${\omega }_0=1/\sqrt{LC}$)

An ideal electrical circuit exchanges energy between an inductor and a capacitor. With no electrical resistance or applied drive, the electric charge obeys $L{q}^{\prime \prime }+q/C=0$, giving angular frequency ${\omega }_0=1/\sqrt{LC}$. A series RLC circuit adds dissipation and can include an external drive.

An ideal inductor and capacitor in parallel have electrical admittance $1/Z=1/(j\omega L)+j\omega C$ for phasors with time dependence $e^{j\omega t}$. The branch currents cancel at angular frequency ${\omega }_0=1/\sqrt{LC}$, giving zero total electrical admittance and an infinite limiting electrical impedance. Below this resonance its electrical reactance is positive; above it the electrical reactance is negative. Losses make the peak finite.

An ideal voltage source drives a series resistor feeding a parallel inductor and capacitor. In the $e^{i\omega t}$ convention, the branch voltage ratio is the title expression. It is a band-pass response with unit peak at $1/\sqrt{LC}$, phase from $+\pi /2$ through zero to $-\pi /2$, and half-power bandwidth $1/(RC)$. At resonance the input current vanishes while equal and opposite inductor/capacitor currents remain finite. Reactive branch-current magnitudes must be combined as phasors, not added as positive scalars.