This simulator charges a capacitor through a series resistor after a voltage step, and it ties three views of the same system together: the physical circuit, the input-to-output block diagram and the oscilloscope-style exponential response. An energy ledger shows where the source energy goes.
• A 3D step-voltage supply, series resistor, cutaway capacitor, oscilloscope trace and first-order system block. • Three sliders: supply step (1-12 V), resistance (1000-10000 ohm) and capacitance (100-1000 microfarad). • Six live readouts: capacitor voltage, charging current, time constant RC, stored electrical energy, resistor heat and energy-balance error. • Two presets: Fast response (time constant 0.1 s) and Slow response (time constant 10 s, about 63.2% of the final step after 12 s).
The time constant is tau = R C and the step is applied at t = 2 s, so u = max(t - 2, 0). The capacitor voltage is Vc = Vs (1 - exp(-u / tau)) and the current is I = (Vs - Vc) / R. Stored energy is Ec = 0.5 C Vc2. The resistor dissipates Q = 0.5 C Vs2 (1 - exp(-2u / tau)), and the source work is C Vs2 (1 - exp(-u / tau)). At long times the source has delivered twice the energy stored in the capacitor, with half lost as heat in the resistor.
The resistor and capacitor are ideal and linear, with zero initial voltage, no leakage, no equivalent series resistance, no source resistance and no breakdown. Part geometry is representative. At one time constant the capacitor is at 63.2% of the final step, not 100%. Use the playback speed selector to slow the fast preset down, and compare the voltage trace with the stored-energy and heat readouts.
The response takes twice as long. RC sets the time scale of the exponential, not the final (DC) voltage, which is still the supply step.
Because the charging curve is exponential, Vc = Vs (1 - exp(-t/tau)). At t = tau that is 1 - exp(-1), about 63.2% of the final value.
For this charging process, half of the source work ends up stored in the capacitor and the other half is dissipated as heat in the resistor, regardless of the resistance value.
It is an ideal linear RC circuit with zero initial voltage and no leakage, ESR, source resistance or breakdown. The part geometry is representative only.