RC Circuit — Capacitor Charging & Discharging

The RC Circuit Simulator lets you explore how a resistor-capacitor circuit behaves during charging and discharging. The core relationship is the time constant τ = RC, which determines how quickly a capacitor charges or discharges through a resistor. During charging, the capacitor voltage follows V(t) = V₀(1 − e^−t/τ), reaching 63.2% of the supply voltage after one time constant and 99.3% after five time constants (5τ).

This simulator is built for electrical engineering students, electronics technicians, and physics learners studying transient analysis, exponential decay, cutoff frequency, and RC filter design. You can adjust resistance, capacitance, and supply voltage, then watch the capacitor charge or discharge in real time with animated current flow and a live voltage-time graph.

The simulator opens in Simulate mode with R = 1000 Ω, C = 100 μF, and V_supply = 12 V, giving a time constant of τ = 100 ms. To begin:

• Throw the Charge / Discharge switch. The schematic draws a real changeover (SPDT) switch: on Charge the supply feeds R and C, and on Discharge the battery is switched out of the loop entirely and the capacitor dumps its charge through R alone.
• Drag the R slider (1 Ω–10 MΩ) and C slider (1 nF–10 mF) to change the time constant and watch the curve genuinely stretch or compress against a real time axis. Both are logarithmic — seven decades will not fit on a linear track. Type an exact value into the box beside either slider if you need one.
• Adjust the V_supply slider (0.5–100 V) to change the voltage the capacitor charges toward.
• Pick a preset from the dropdown — standard bench, switch debounce, audio coupling, 555 timing, camera flash or a scope probe. The dropdown falls back to — Custom — as soon as you nudge anything away from a preset, and snaps back if you land on one again.
• Use the Calculations button for a full worked derivation at the cursor, and CSV / PNG to export the plotted data or the canvas.

Switch between the four modes — Simulate, Explore, Practice, Quiz — using the pill tabs at the top of the page, and between the two views — Charge / Discharge and Frequency Response — using the View tabs below them.

Simulate mode is the primary interactive workspace. The canvas shows the schematic with animated current dots on the left and an oscilloscope-style graph on the right.

The graph is a real instrument, not a normalised sketch. All three axes — volts, milliamps and seconds — carry their own full scale, chosen from the standard 1-2-5 sequence and re-ranged only when the signal overflows the screen or drops below 30 % of it. A small slider move therefore visibly grows or shrinks the trace; a large one re-ranges and announces the new value on the V/div, A/div and s/div readouts above the plot.

• Transport: Pause freezes the run so you can compare two settings; the t scrubber moves the cursor anywhere in the sweep; Restart returns to t = 0. Changing R, C or V does not throw the run away — the curve moves under the cursor, which is the whole lesson.
• Speed: real RC circuits are almost never watchable — τ spans nanoseconds to minutes over this slider range. The animation is compressed and the factor is printed on the canvas (“slow motion 1 : 2700” or “fast-forward 5 : 1”), so the numbers on screen are always the real ones. The Speed dropdown scales that further.
• Display Controls (top-left of the canvas): show or hide the equation, the labels and values, the current-flow dots, the τ markers, the I(t) trace and a background grid. Your choices are remembered on your next visit.
• Right-click the canvas for copy-value, CSV and PNG export, a grid toggle and reset.

The readout cards show capacitor voltage, the time constant, instantaneous current, charge percentage, and the full energy account: energy stored in C, heat dissipated in R, and energy drawn from the supply. Watch the current start at I₀ = V/R and decay exponentially, and watch the three energy figures settle at ½CV², ½CV² and CV².

Explore mode organises educational content into three categories:

• Fundamentals: Covers capacitor construction, capacitance definition (C = Q/V), electric field between plates, and the relationship between charge, voltage, and energy storage.
• Time Response: Explains the exponential charging and discharging curves, the meaning of τ = RC, the 5τ rule for full charge, and how to read time-domain waveforms.
• Applications: Describes real-world uses of RC circuits including low-pass and high-pass filters (cutoff frequency f_c = 1/(2πRC)), timing circuits, switch debouncing, and power supply smoothing.

Select any concept card to view a detailed explanation with formulas and an interactive canvas illustration.

Switch the View tabs to Frequency Response and the same two components become a filter. The canvas draws a full Bode plot — magnitude in decibels above, phase below, on a six-decade logarithmic frequency axis.

• Low-pass takes the output across the capacitor: flat at 0 dB, then falling at −20 dB per decade above the cutoff, with the phase sliding from 0° to −90°.
• High-pass takes the output across the resistor: the mirror image, rising at +20 dB per decade and leading by up to +90°.
• The cutoff f_c = 1/(2πRC) is marked, along with the −3 dB line and the straight-line asymptotes. At f_c the reactance X_C exactly equals R, the gain is 1/√2 = 0.7071, and the phase is ±45°.
• The Cursor f slider reads gain, gain in dB, phase and reactance at any frequency you like.

Like the time axis, the frequency window re-ranges only when the cutoff drifts out of the middle decades, so changing R or C visibly slides the knee along the plot instead of pinning it to the centre.

Practice mode generates unlimited calculation problems on RC circuits. Typical questions include: “Calculate the time constant for R = 2.2 kΩ and C = 47 μF”, “What is the capacitor voltage after 3τ when charging from 0 to 12 V?”, or “Find the cutoff frequency of an RC filter with R = 1 kΩ and C = 10 μF.” Enter your numeric answer, click Check, and see a step-by-step solution.

Quiz mode draws 5 questions — a mix of multiple choice and numeric — from a pool covering the time constant, exponential decay, energy storage, the 50 % charging efficiency result and filter behaviour. The multiple-choice options are shuffled on every draw, so the answer is never in a predictable position, and every question shows why its answer is right whether you got it right or wrong.

• Remember the 5τ rule: after 5 time constants, the capacitor is at 99.3% of its final value — effectively fully charged or discharged.
• Hit Pause and then move a slider. The curve redraws under a stationary cursor, so you can read the old and new values against the same time axis instead of losing the run.
• Use the Fast preset (10 Ω, 100 μF) to see rapid charging, then switch to Slow (10 kΩ, 1000 μF) to observe a much longer time constant — and note the s/div readout changing as the scope re-ranges.
• The capacitor is half charged at t = τ ln 2 = 0.693τ, not at 0.5τ. The curve is steepest at the start, so the first half of the journey takes less than half the time.
• Charging a capacitor through a resistor is exactly 50 % efficient no matter what R you choose — watch the “Heat in R” and “From Supply” cards settle at ½CV² and CV². Making R smaller only makes the loss happen faster.
• Watch the current readout during charging — it starts at I₀ = V/R and decays to near zero, confirming exponential behaviour.
• For filter design, remember that f_c = 1/(2πRC). A larger RC product gives a lower cutoff frequency.
• The energy stored in the capacitor (E = ½CV²) increases quadratically with voltage, so doubling voltage quadruples energy.
• Practice converting between time constant notation and actual seconds: τ = RC in seconds when R is in ohms and C is in farads.