LC Circuit Oscillator: Electromagnetic Resonance

simulator intermediate ~8 min
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f₀ ≈ 503 Hz — resonant frequency of a 10 mH, 10 μF LC circuit

An LC circuit with 10 mH inductance and 10 μF capacitance oscillates at approximately 503 Hz. The initial charge of 5 μC on the capacitor stores 1.25 μJ of energy, which sloshes back and forth between the electric field of the capacitor and the magnetic field of the inductor.

Formula

f₀ = 1/(2π√(LC)) (resonant frequency)
Q(t) = Q₀ cos(ω₀t) e^(-Rt/2L) (damped charge)
E_total = Q₀²/(2C) = ½LI_max² (energy conservation)

The LC Circuit: Nature's Electromagnetic Oscillator

An LC circuit is one of the simplest yet most important circuits in electronics. Connect an inductor and a capacitor in a loop, charge the capacitor, and release — the charge oscillates back and forth at a precise natural frequency. This electromagnetic oscillation is the electrical analog of a pendulum or a mass on a spring, and it underlies all radio communication.

Energy Exchange: Electric to Magnetic and Back

When the capacitor is fully charged, all energy is stored in its electric field (E = Q²/2C). As the capacitor discharges, current flows through the inductor, building up a magnetic field. When the capacitor is empty, all energy has transferred to the inductor's magnetic field (E = LI²/2). The inductor then drives current to recharge the capacitor in the opposite polarity, completing the cycle.

Damping and the Quality Factor

Real circuits have resistance, which dissipates energy as heat and causes the oscillations to decay exponentially. The quality factor Q = (1/R)√(L/C) quantifies this damping — high Q means many oscillation cycles before significant energy loss. Superconducting circuits can achieve Q factors in the billions, while typical electronic circuits range from 10 to 1000.

Applications: From Radio to Quantum Computing

LC circuits are everywhere in modern technology. Radio transmitters and receivers use them to generate and select specific frequencies. Crystal oscillators in every computer are essentially LC circuits where the crystal provides both L and C. At the quantum scale, superconducting LC circuits form the basis of transmon qubits — the building blocks of quantum computers.

FAQ

What is an LC circuit?

An LC circuit consists of an inductor (L) and capacitor (C) connected in a loop. Energy oscillates between the electric field of the capacitor and the magnetic field of the inductor at the resonant frequency f₀ = 1/(2π√(LC)). It is the electrical analog of a mass on a spring.

What is the resonant frequency?

The resonant frequency f₀ = 1/(2π√(LC)) is the natural oscillation frequency of the circuit. At this frequency, energy transfers completely between the capacitor and inductor each half-cycle. It depends only on L and C, not on the initial charge.

What is the quality factor Q?

The quality factor Q = (1/R)√(L/C) measures how many oscillation cycles occur before the energy is significantly damped. High Q means sharp resonance and low energy loss. A Q of 100 means the circuit rings for about 100 cycles.

How is an LC circuit used in radio?

Radio receivers use LC circuits to select a specific frequency — tuning the capacitance changes the resonant frequency to match the desired station. The circuit amplifies signals at its resonant frequency and rejects others, acting as a bandpass filter.

Sources

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