LC Oscillator Circuit

Simulate the continuous exchange of energy between an inductor and a capacitor in an ideal LC circuit.

LC Oscillator Circuit

Concept Overview

An LC circuit, also called a resonant circuit, tank circuit, or tuned circuit, is an idealized electrical circuit consisting of an inductor (L) and a capacitor (C) connected together. When charged, the circuit acts as an electrical resonator, storing energy oscillating at the circuit's resonant frequency. The capacitor stores energy in an electric field, while the inductor stores energy in a magnetic field. As the capacitor discharges, its current builds a magnetic field in the inductor. When the capacitor is fully discharged, the inductor's magnetic field collapses, inducing a current that recharges the capacitor with the opposite polarity, perpetuating the cycle. In a purely ideal LC circuit with no resistance, this oscillation would continue infinitely.

Mathematical Definition

The behavior of an ideal LC circuit can be modeled using Kirchhoff's voltage law. The sum of the voltage across the inductor and the voltage across the capacitor must be zero. This leads to a second-order linear differential equation:

L(d²q/dt²) + q/C = 0

where q is the charge on the capacitor. The solution to this equation reveals that the charge, and consequently the voltage and current, oscillate sinusoidally over time. Assuming an initial voltage V₀ on the capacitor and zero initial current, the voltage V(t) and current I(t) are given by:

V(t) = V₀ cos(ωt)

I(t) = V₀ √(C/L) sin(ωt)

where the resonant angular frequency is ω = 1 / √(LC).

Key Concepts

Resonant Frequency: The natural frequency at which the LC circuit oscillates, determined by the values of inductance and capacitance. It is given by f = 1 / (2π√(LC)).
Energy Conservation: In an ideal LC circuit, the total energy remains constant. It continuously shifts back and forth between the electrical energy stored in the capacitor (E_C = 1/2 CV²) and the magnetic energy stored in the inductor (E_L = 1/2 LI²).
Phase Difference: The voltage across the capacitor and the current through the inductor are exactly 90 degrees (π/2 radians) out of phase. When voltage is at a maximum, current is zero, and vice versa.

Historical Context

The foundational principles of LC circuits were established in the mid-19th century. Lord Kelvin published mathematical models describing electrical oscillations in 1853, noting that the discharge of a Leyden jar (an early capacitor) through a coil could be oscillatory. In 1887, Heinrich Hertz famously utilized LC circuits in his spark-gap transmitter to generate and detect radio waves, providing the first experimental proof of James Clerk Maxwell's electromagnetic theory.

Real-world Applications

Radio and Television: LC circuits form the core of tuners used to select a specific frequency from a mix of ambient signals. Adjusting the variable capacitor changes the resonant frequency to match the desired station.
Oscillators: They are essential components in signal generators and oscillators, converting DC power into AC signals at specific frequencies for use in transmitters and electronic clocks.
Filters: LC networks are used to construct band-pass, band-stop, low-pass, and high-pass filters to isolate or reject specific frequency ranges in audio and radio frequency applications.

Related Concepts

RC & RL Circuits — the foundational first-order circuits analyzing transient exponential behaviors without oscillation.
Harmonic Oscillator — the mechanical analogue to the LC circuit, demonstrating identical differential equations for mass-spring systems.
Electromagnetic Waves — demonstrating the relationship between changing electric and magnetic fields, fundamentally tied to high-frequency LC oscillations.

LC Oscillator Circuit