The LC Oscillator

Two components. No resistance. Pure oscillation. The simplest harmonic oscillator in electronics.

The Setup

An inductor and a capacitor, connected in a loop. No resistor. No source. Just $L$ and $C$, locked in a closed circuit, exchanging energy back and forth like dancers who cannot stop.

Begin with the capacitor charged to $V_0$ and no current flowing. The inductor sees an open circuit—nothing moves. But the capacitor is not content. The voltage difference creates an electric field, and that field pushes electrons. Current begins to flow.

The inductor resists. Not the way a resistor does—resistance dissipates, the inductor remembers. An inductor opposes any change in current. The changing current induces an electromotive force that opposes the change that caused it. Lenz's law. The inductor's voltage is $V_L = L\frac{dI}{dt}$. It is a reaction, not a dissipation.

The Dance

The capacitor discharges. Current builds through the inductor. Energy flows from the electric field (stored in the capacitor's plates) into the magnetic field (stored in the inductor's coil). When the capacitor is fully discharged, the current is maximum. All energy is in the inductor: $E = \frac{1}{2}LI_{\text{max}}^2$.

Now the inductor refuses to let current stop. The collapsing magnetic field induces a voltage that keeps pushing electrons. They pile up on the opposite capacitor plate. The capacitor recharges—opposite polarity this time. The inductor has handed the energy back.

Then the capacitor discharges again, current flowing the opposite direction. The inductor stores the energy once more. Then it pushes back. And so it goes, forever.

The Mathematics

Charge on the capacitor obeys:

$L\frac{d^2q}{dt^2} + \frac{q}{C} = 0$

Or equivalently:

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

This is the equation of a simple harmonic oscillator. The angular frequency is:

$\omega_0 = \frac{1}{\sqrt{LC}}$

The frequency in hertz is $f_0 = \frac{1}{2\pi\sqrt{LC}}$. This is the resonant frequency—determined entirely by the values of $L$ and $C$. No resistance, no damping, no decay. The oscillation continues forever.

The charge oscillates as $q(t) = Q_0\cos(\omega_0 t)$, and the current as $I(t) = -Q_0\omega_0\sin(\omega_0 t)$. The current lags the charge by 90 degrees. When charge is maximum, current is zero. When current is maximum, charge is zero. Energy sloshes between capacitor and inductor like water between two connected tanks.

Real World

No real LC circuit is lossless. Every wire has resistance. Every capacitor has leakage. Every inductor has core losses. The oscillation dampens, eventually stopping. The circuit becomes an RLC circuit, and the story changes from perpetual oscillation to damped oscillation—or, if resistance is too high, no oscillation at all.

But the ideal LC oscillator reveals something essential: energy can exist in two different forms, and the exchange between those forms produces oscillation naturally. This is the principle behind every radio tuner, every clock oscillator, every resonant circuit. The LC circuit is not a toy. It is the foundation.