RLC
Lq″ + Rq′ + q/C = 0 — the damped oscillator’s own code, running in copper.
Use the simulation above to change the variables and play through the guided stages. The explanation below describes the default starting values; the simulation updates its explanation as you experiment.
Setup
A charged capacitor, an inductor, a resistor — close the loop and the site’s most familiar mathematics returns in copper. This is not an analogy dressed up as a module; the charge on this page runs the damped oscillator’s own closed forms.
Isomorphism
Lq″ + Rq′ + q/C = 0 — letter for letter the damped spring, with inductance playing mass (current resists change), resistance playing the dashpot, and 1/C playing the spring constant. Physics reuses its best equations, and so does this site: the mapping here is executable code, not metaphor.
Solve
The same three regimes wear new clothes: ω₀ = 1/√(LC) sets the ring, γ = R/2L sets the drain, and the boundary sits at R_crit = 2√(L/C) — reachable on the slider. At the defaults the quality factor is exactly the spring module’s Q = 5, on purpose.
Discharge
A forty-millisecond event, honestly slowed: charge and current trade places hundreds of times a second while the energy bars slosh between plate and coil, the resistor skimming every pass. Set R to zero and the skimming stops — the lossless LC clock, which is every radio tuner ever built.
Audit
The audit makes the isomorphism literal: ω₀, γ, and the regime match the damped module’s derivation — regime and γ bitwise, ω₀ and ω_d to one part in 10¹⁵ — when fed (L, R, 1/C) as (m, c, k), and the equation is re-verified in electrical variables at machine precision. With R = 0 the energy ledger seals exactly — shared code, shared truth.