Why do the oscillations in this simulator never die out?

The simulator models an ideal LC circuit with zero resistance (R=0). In reality, all wires and components have some resistance, which dissipates energy as heat, causing the oscillations to dampen over time. This idealization highlights the perfect, lossless energy exchange between the capacitor and inductor, which is a useful theoretical foundation before studying real-world RLC circuits.

What is the physical meaning of the phase difference between charge and current?

The current I(t) is the derivative of charge q(t). In sinusoidal oscillation, this results in a 90-degree (π/2) phase shift. When charge on the capacitor is at its maximum (and current is zero), all energy is stored electrically. A quarter cycle later, the charge is zero but the current is maximum, meaning all energy is stored magnetically in the inductor. This phase difference is a direct visualization of the energy transfer process.

How is this LC oscillator related to radio tuners or musical instruments?

The resonant frequency f₀ = ω₀/(2π) = 1/(2π√(LC)) is a key principle in tuning. In a radio, varying the capacitance (e.g., with a tuning knob) changes this resonant frequency, allowing the circuit to selectively amplify signals from a specific broadcasting station. Similarly, the concept applies to the acoustic resonance in wind instruments, where the geometry sets a natural frequency for sound waves.

Why does increasing the capacitance slow down the oscillations?

A larger capacitor can store more charge for a given voltage (C = Q/V). This means it takes longer to charge and discharge during each oscillation cycle. Mathematically, the angular frequency ω₀ = 1/√(LC) decreases as C increases, resulting in a longer period T = 2π√(LC). The inductor also plays a role: a larger inductance opposes changes in current more strongly, also slowing the oscillation.

LC Oscillator (Undamped)

Ideal series LC: q(t), I(t), ω₀ = 1/√(LC); U_C + U_L constant; vs RLC AC.

Live graphs

About this model

An ideal LC oscillator consists of an inductor (L) and a capacitor (C) connected in series, forming a resonant circuit with zero electrical resistance. This simulator visualizes the fundamental energy exchange that occurs in such a system. Initially, the capacitor is charged, storing energy in its electric field. When the circuit is closed, this charge flows through the inductor, creating a magnetic field and thus transferring energy from the capacitor to the inductor. Once the capacitor is fully discharged, the inductor's collapsing magnetic field drives current back to recharge the capacitor in the opposite polarity. This process repeats indefinitely, resulting in continuous, undamped sinusoidal oscillations of charge q(t) and current I(t). The dynamics are governed by Kirchhoff's voltage law, which leads to the differential equation d²q/dt² + (1/LC)q = 0. The solution is simple harmonic motion: q(t) = Q₀ cos(ω₀t) and I(t) = -ω₀Q₀ sin(ω₀t), where ω₀ = 1/√(LC) is the natural angular frequency of oscillation. A core principle demonstrated is the conservation of energy: the total electromagnetic energy U_total = U_C + U_L = (1/2C)q² + (1/2L)I² remains constant over time, oscillating perfectly between electric (capacitor) and magnetic (inductor) forms. This model is a significant simplification, assuming zero resistance, radiation losses, and internal impedance. By interacting with the simulator, students can explore how changing L and C affects the oscillation frequency and amplitude, visualize the 90-degree phase difference between charge and current, and verify energy conservation. It also provides a clean baseline for comparison with the damped oscillations of a real RLC circuit.

Who it's for: Undergraduate physics and electrical engineering students studying AC circuit theory, electromagnetic oscillations, and resonance phenomena.

Key terms

LC Circuit
Angular Frequency (ω₀)
Electromagnetic Oscillation
Energy Conservation
Simple Harmonic Motion
Resonant Frequency
Inductor
Capacitor

How it works

An ideal series LC loop obeys q'' + ω₀² q = 0 with ω₀ = 1/√(LC) and i = dq/dt. Electrostatic energy q²/(2C) and magnetic energy Li²/2 sum to a constant. For damping and AC resonance, see RLC Series (AC) in Electricity.

Key equations

Lq'' + q/C = 0 ⇒ ω₀ = 1/√(LC)

q(t) = Q₀ cos ω₀t + (I₀/ω₀) sin ω₀t, i = dq/dt

Frequently asked questions

Why do the oscillations in this simulator never die out?: The simulator models an ideal LC circuit with zero resistance (R=0). In reality, all wires and components have some resistance, which dissipates energy as heat, causing the oscillations to dampen over time. This idealization highlights the perfect, lossless energy exchange between the capacitor and inductor, which is a useful theoretical foundation before studying real-world RLC circuits.
What is the physical meaning of the phase difference between charge and current?: The current I(t) is the derivative of charge q(t). In sinusoidal oscillation, this results in a 90-degree (π/2) phase shift. When charge on the capacitor is at its maximum (and current is zero), all energy is stored electrically. A quarter cycle later, the charge is zero but the current is maximum, meaning all energy is stored magnetically in the inductor. This phase difference is a direct visualization of the energy transfer process.
How is this LC oscillator related to radio tuners or musical instruments?: The resonant frequency f₀ = ω₀/(2π) = 1/(2π√(LC)) is a key principle in tuning. In a radio, varying the capacitance (e.g., with a tuning knob) changes this resonant frequency, allowing the circuit to selectively amplify signals from a specific broadcasting station. Similarly, the concept applies to the acoustic resonance in wind instruments, where the geometry sets a natural frequency for sound waves.
Why does increasing the capacitance slow down the oscillations?: A larger capacitor can store more charge for a given voltage (C = Q/V). This means it takes longer to charge and discharge during each oscillation cycle. Mathematically, the angular frequency ω₀ = 1/√(LC) decreases as C increases, resulting in a longer period T = 2π√(LC). The inductor also plays a role: a larger inductance opposes changes in current more strongly, also slowing the oscillation.

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