Presets
Playback
Circuit
The capacitor is charged through R from the source while the lamp discharges it. The steady voltage is the divider Vs R_L/(R + R_L); the time constant is C R R_L/(R + R_L).
Events and controls
Model checks
The science
What is modelled
A source with internal resistance R (a Thevenin equivalent) charges a capacitor C. A lamp of resistance R_L is connected across the capacitor and drains it. The lamp is treated as a fixed resistor, so its power is V²/R_L and it glows in proportion. The page counts the time the capacitor stays above 2.5 V as the service time of the lamp.
Equations
- Node equation:
C dV/dt = (Vs - V)/R - V/RL. - Exact solution for a constant source:
V(t) = V∞ + (V0 - V∞) e-t/τwithV∞ = Vs RL/(R + RL)andτ = C R RL/(R + RL). - Energy: source
∫Vs I dt= resistor heat∫I²R dt+ lamp∫V²/RL dt+½C(V² - V0²).
Checks run in the page
- RK4 against the exact exponential.
- The energy balance, integrated with the trapezoid rule on the stepped solution.
- The time to reach 63.2 % of the way against C R R_L/(R + R_L).
The checks use a constant source and no boost, dimming or backup; the graphs show the scripted case.
Honest simplifications
An ideal capacitor (no leakage or series resistance), a linear lamp (a real filament has a strongly temperature-dependent resistance), and an ideal source. The 3 A figure is a game-style limit, not a rating.
What to try
- Switch the scripted sags off and compare the blue curve with the grey exact solution: they coincide.
- Raise the capacitance: the time constant grows and the voltage rides through the sag.
- Tick Boost charger and watch the peak current approach the 3 A limit.