RC Circuit Charging Simulator
Enter values to simulate.
Simulation summary:
RC Pulse Pilot Mini-Game
Pulse the switch to steer the capacitor voltage toward each inspection target and feel how the RC time constant governs response speed.
Modifier: None — steady state.
Charging an RC Circuit in the Real World
The ubiquitous resistor–capacitor (RC) circuit hides in nearly every electronic device. From camera flashes to timing networks in microcontrollers, a lone capacitor connected through a resistor to a voltage source stores charge gradually rather than instantaneously. The simple arrangement of a battery, a resistor, and a capacitor exhibits rich transient behavior when the circuit is switched on. Although textbooks present the exponential rise of voltage across the capacitor as an abstract formula, watching the charge accumulate in real time illuminates the physics of electrical energy conversion. This simulator elevates the traditional calculator by pairing numeric predictions with a dynamic canvas rendering of the circuit. The animation shows a stylized battery, resistor, and capacitor. As time advances, a colored bar grows within the capacitor to indicate its voltage, while patterned energy bars display how energy partitions between the electric field of the capacitor and heat dissipated by the resistor. The interface updates instantly when you adjust parameters, inviting experimentation with resistance, capacitance, and time step while preserving the underlying purpose of computing RC transients.
Variables, Assumptions, and Governing Equations
In its most idealized form, the RC circuit consists of a resistor with resistance , a capacitor with capacitance , and a direct‑current supply of magnitude . At the moment the switch closes, the capacitor voltage is typically zero. Charge flows through the resistor, raising asymptotically toward . Ohm’s law and the capacitor relation combine to form the differential equation that governs the system: . Rearranging yields the standard first‑order form . Solving analytically gives the familiar exponential rise . The current follows as . These relations presume an ideal voltage source and neglect parasitic inductance, leakage currents, and temperature dependence. Yet even this stripped‑down model captures the essential physics of charging: the resistor limits current, enforcing a finite rate at which the capacitor’s electric field builds, while the integral of over time reveals the heat dissipated in the resistor.
Numerical Scheme and Stability
Although an analytic solution exists, the simulator integrates the differential equation numerically to illustrate the mechanics of time stepping and to enable extensions such as variable sources. We implement a classical fourth‑order Runge–Kutta (RK4) method. For a state variable , the update reads where each term evaluates the derivative at successively advanced intermediate states. RK4 offers excellent accuracy for smooth problems but still requires a stable time step. The user‑specified is clamped between and seconds. Smaller steps reduce numerical error and energy drift but require more iterations to reach a given time. Because the true solution is exponential, the method remains stable provided is small compared with the time constant . For pedagogical purposes, the simulator displays the accumulated energy in the capacitor and the energy dissipated in the resistor . Conservation dictates that the sum approaches . A bar chart beneath the canvas depicts this partition, and the numeric summary reports the instantaneous voltage and time.
Worked Example
Consider a flash circuit employing a 100 Ω resistor and a 1 mF capacitor charged from a 5 V supply. The time constant is s. Starting with an uncharged capacitor, the analytic solution predicts . Running the simulator with s reproduces this curve. After 0.1 s the capacitor reaches 63% of the supply voltage, 0.3 s yields 95%, and by 0.5 s the voltage is essentially at 99%. The CSV export provides the data needed to plot voltage versus time externally. Downloading and opening the CSV reveals rows of time, capacitor voltage, current, and cumulative energies, enabling deeper analysis. If you increase the resistance to 1 kΩ, the charging slows dramatically; at 0.1 s the voltage sits around 9%. In contrast, reducing the resistance to 10 Ω produces a swift 63% rise after only 0.01 s but dissipates energy ten times faster in the resistor.
Parameter Comparison
The table below compares the baseline example with two variants, illustrating how resistance and capacitance shape the transient. The quantity denotes the time to reach 63% of , equal to one time constant.
| R (Ω) | C (F) | τ = RC (s) | t63 (s) |
|---|---|---|---|
| 100 | 0.001 | 0.10 | 0.10 |
| 1000 | 0.001 | 1.00 | 1.00 |
| 100 | 0.0001 | 0.01 | 0.01 |
How to Read the Animation
The canvas renders a simplified circuit: the supply appears on the left, a resistor is drawn as a zigzag, and the capacitor consists of two vertical plates. A vertical orange bar inside the capacitor indicates its current voltage relative to the supply. Below, the blue bar labeled “Capacitor” grows as energy accumulates in the electric field, while the orange bar labeled “Resistor” tracks heat dissipated. The summary beneath the bars reports the simulated time and capacitor voltage, and the hidden text summary mirrors this information for screen readers. Pressing Play starts the RK4 integration; Pause freezes it; Reset clears the history and restarts. The time‑step field lets you experiment with stability: small values yield smooth curves, whereas overly large steps cause visible error and energy mismatch. The canvas is keyboard‑focusable, and pressing the space bar toggles play/pause. Tooltips on each input clarify units, reinforcing that all values are in SI. CSV downloads capture the time series at the chosen step, allowing offline plotting or verification of the conservation relation .
Limitations and Possible Extensions
The simulation assumes an ideal resistor and capacitor and ignores inductance, dielectric absorption, and voltage‑dependent capacitance. Real components exhibit parasitic series resistance and leakage currents that alter the transient. The heat bar tracks energy dissipated only within the resistor; in practice, battery internal resistance and wiring also convert energy to heat. Numerical integration may accumulate rounding errors, especially with coarse time steps or extremely large resistance‑capacitance products. Extending the model to include a time‑varying supply or to simulate discharging would require only minor modifications to the differential equation. A natural next step is modeling an RLC circuit, revealing oscillatory behavior akin to a mass‑spring system. Incorporating temperature effects, non‑linear capacitors, or switched circuits could further enrich the tool. Engineers may also explore control strategies by injecting a feedback term representing an active regulator.
References
- R. C. Dorf and J. A. Svoboda, Introduction to Electric Circuits, 9th ed., Wiley, 2013.
- J. D. Irwin and R. M. Nelms, Basic Engineering Circuit Analysis, 11th ed., Wiley, 2015.
Related calculators: RC Circuit Discharging Calculator, RL Circuit Current Calculator, RLC Circuit Damping Calculator.