Charging a capacitor through a resistor from a DC source follows exponential RC dynamics; add an inductor and the same loop rings as an underdamped RLC oscillator:
RC mode: V_c(t) = V·(1 − e^(−t/τ)), τ = R·C
I(t) = (V/R)·e^(−t/τ)
RLC mode: V_c(t) ≈ V·(1 − e^(−ζωt)·cos(ω_d t)), ω_d = √(1/LC − (R/2L)²)
- Source voltage — the DC EMF driving the loop.
- Resistance R — limits current and, in RLC mode, controls damping of the ringing.
- Capacitance C — energy-storage element; larger C means slower charging (larger τ) but more stored charge at steady state.
- RC / RLC toggle — switches between a pure charging transient and a damped oscillation (adds a fixed 10 mH inductor).
- Phase-plane trace — plots capacitor voltage against loop current instead of time; an RC transient draws a single curve into the origin, an RLC transient spirals in.
The same physics drives camera-flash charging circuits, debounce filters, and the resonant tank circuits used in radio tuners and switch-mode power supplies.