LC Oscillations Explained Simply with Easy Examples

LC oscillations occur when a charged capacitor is connected to an inductor and the energy of the circuit begins to swing back and forth between them. The capacitor electric field energy converts into the inductor magnetic field energy and then back again, over and over, producing a current that alternates with a natural frequency. It is the electrical equivalent of a swinging pendulum or a mass on a spring.

This back-and-forth exchange matters because it is the beating heart of every radio transmitter and receiver. The same oscillation that a school circuit shows for a few cycles before dying out is sustained in real equipment to generate the carrier waves of radio, television and mobile phones. Understanding LC oscillations means understanding where radio frequencies come from.

Key Concepts

• LC circuit: a loop containing only an inductor L and a capacitor C, ideally with no resistance.
• Energy exchange: energy oscillates between the capacitor (electric field, ½CV²) and the inductor (magnetic field, ½LI²); total energy stays constant in the ideal case.
• Natural frequency: the fixed frequency of oscillation set only by L and C.
• Charge oscillation: the charge on the capacitor varies sinusoidally, and the current is its rate of change.
• Damping: in real circuits resistance slowly converts the oscillating energy into heat, so the amplitude decreases with time.
• Analogy with SHM: charge plays the role of displacement, inductance the role of mass, and 1/C the role of the spring constant.

Formulas

The angular frequency of LC oscillations is ω₀ = 1 / √(LC), so the frequency is f₀ = 1 / (2π√(LC)) hertz, with L in henrys and C in farads. The charge on the capacitor varies as q = q₀ cos(ω₀t) and the current as i = −ω₀ q₀ sin(ω₀t), so the maximum current is I₀ = q₀ / √(LC). The total energy remains constant: U = q₀² / (2C) = ½ L I₀², in joules. These results assume zero resistance; with resistance the oscillations decay exponentially.

Worked Example

A capacitor of 100 μF is charged to a maximum charge of 2 × 10⁻³ C and then connected to an inductor of 0.1 H. Find the frequency of oscillation and the maximum current.

1. f₀ = 1 / (2π√(LC)) = 1 / (2π√(0.1 × 100 × 10⁻⁶)) = 1 / (2π × 3.16 × 10⁻³) ≈ 50.3 Hz.
2. I₀ = q₀ / √(LC) = 2 × 10⁻³ / 3.16 × 10⁻³ ≈ 0.63 A.
3. Check with energy: initial energy = q₀²/(2C) = (4 × 10⁻⁶) / (2 × 10⁻⁴) = 0.02 J. Also ½LI₀² = 0.5 × 0.1 × 0.63² ≈ 0.02 J. The two agree.

The charge and current keep exchanging roles about fifty times per second, with a peak current of about 0.63 A. In a real circuit, wire resistance would slowly shrink both the charge and current amplitudes as the 0.02 J turns into heat.

Common Misconceptions

• The capacitor and inductor have maximum energy at the same instant. Wrong. When the capacitor is fully charged, the current is zero and all energy is electric. When the capacitor is fully discharged, the current is maximum and all energy is magnetic. They are a quarter cycle out of step.
• LC oscillations continue forever in any circuit. Wrong. Every real wire and coil has resistance, which drains the energy as heat. Sustained oscillations in transmitters need an energy source and an amplifier to replace the losses.
• Doubling both L and C doubles the frequency. Wrong. Frequency varies as 1/√(LC). Doubling both makes LC four times larger and halves the frequency.

Real-World Applications

Every radio and television tuner uses an LC circuit whose frequency is adjusted to match the wanted station. Radio transmitters generate their carrier waves with LC oscillators maintained by amplifier circuits. Metal detectors and induction heaters exploit changing LC oscillation when metal comes near a coil. Quartz watches replace the LC pair with a vibrating crystal for greater stability, but clock and communication circuits still begin from the LC idea, and tank circuits in amplifiers select the frequency band to be amplified.

LC oscillations are a continuous exchange of energy between capacitor and inductor at the natural frequency 1/(2π√(LC)). This simple loop, losing energy slowly through resistance in practice, is the seed from which all radio-frequency technology grows.