The LC circuit and the oscillation equation

A circuit containing only a capacitor (capacitance $C$) and an inductor (inductance $L$), with no resistors, behaves surprisingly: the energy oscillates indefinitely between the electrical form (charged capacitor) and the magnetic form (current in the solenoid). It is the electrical analogue of a pendulum, which exchanges gravitational potential energy and kinetic energy.

LC circuit: the capacitor $C$ and the inductor $L$ closed in a single loop, carrying the current $i(t)$.

Let us apply Kirchhoff’s loop law: going round the circuit in the direction of the current, the voltage drops must sum to zero. If $Q(t)$ is the charge on the positive plate of the capacitor:

$\frac{Q}{C} - L\,\frac{di}{dt} = 0$

By conservation of charge, $i = -dQ/dt$ (the current leaves the capacitor); differentiating again, $di/dt = -d^2Q/dt^2$. Substituting:

$\frac{Q}{C} + L\,\frac{d^2Q}{dt^2} = 0 \quad\Longleftrightarrow\quad \frac{d^2Q}{dt^2} = -\frac{1}{LC}\,Q$

Free oscillations of an LC circuit

The charge $Q(t)$ obeys the simple harmonic oscillation equation: $\ev{\frac{d^2Q}{dt^2} = -\omega^2\,Q} \quad\text{with}\quad \ev{\omega = \frac{1}{\sqrt{LC}}}$ The solution is $Q(t) = Q_0\,\cos(\omega t + \varphi)$: harmonic oscillation with frequency $f = \omega/(2\pi)$, exactly like a mass attached to a spring.

Key formula

$\omega_\text{LC} = \frac{1}{\sqrt{LC}}, \qquad T = 2\pi\sqrt{LC}$

The angular frequency depends only on the components $L$ and $C$, never on the amplitude $Q_0$ or on the initial conditions: it is a property of the circuit, not of how it is driven.

Collegamenti

Argomenti: Electromagnetic induction Concetti: LC and RLC oscillations · Simple harmonic motion

Esercizi collegati: Worked exercise — LC circuit, find the natural frequency · Problem — Natural frequency of an LC circuit · Problem — Capacitor of an AM radio receiver