RC / LR Transient Plotter

Time-domain charge / discharge for a first-order RC circuit (τ = R · C) or rise / fall for a first-order LR circuit (τ = L / R). Pick Capacitor (RC) or Inductor (LR), flip between charging and discharging, and hit Play to watch the curve sweep over 5τ. Drag the sliders to retune; drag the scrub cursor to read instantaneous values. The legend below the controls maps blue to the primary trace and green-dashed to its mirror.

Mode

Phase

Animate

v_C — capacitor voltage

i_C — capacitor current

Resistance R

10.0 kΩ

Capacitance C

100 µF

Source voltage V_s

12.0 V

τ = 1.00 s

At t = τ: v_C ≈ 7.59 V (63%)

At t = 5τ: 11.9 V ≈ fully charged

Cursor

t = 1.00 s

v_C = 7.59 V

i_C = 441 µA

Quick sanity checks

The 63% rule. At t = τ a charging cap reaches $V_s(1 - 1/e) \approx 0.632\,V_s$ . Park the cursor at the τ tick and read the capacitor-voltage pill — it always lands at ~63 % of the source voltage no matter what R or C you pick.
The 5τ rule. At t = 5τ a charging cap is at $\approx 0.993\,V_s$ — call it fully charged. Park the cursor at the right edge of the plot and the readout confirms it.
Inductors are dual. Flip to LR mode. The current now plays the role voltage played in RC: it rises with the same shape, on the same time-constant rule (L/R). Switch phase to de-energising and the dashed back-EMF marker at t = 0 shows the polarity reversal that keeps current flowing through the inductor's stored field.

RC and RL transient response

When a voltage step is applied to an RC or RL circuit, the energy-storage element (capacitor or inductor) prevents an instantaneous change — the response follows an exponential curve governed by the time constant τ.

RC time constant

τ = R × C (seconds, with R in Ω and C in F)
At t = τ: capacitor charges to 63.2 % of the final voltage.
At t = 5τ: 99.3 % — considered fully charged for practical purposes.
Discharge: V(t) = V₀ × e^−t/τ

RL time constant

τ = L / R. Current rises to 63.2 % of I_final = V/R in one τ, and reaches steady state after 5τ. Inductor current cannot change instantaneously — switching off without a freewheeling diode generates a large voltage spike (V = L × dI/dt).

Learn more → Capacitors — Learn · Inductors — Learn

Formula reference

RC time constant: $\tau = R C$
10 kΩ with 100 µF gives τ = 1 s. More resistance means slower.
RL time constant: $\tau = \frac{L}{R}$
Note the inversion — more resistance makes an RL circuit faster, not slower.
Capacitor charging: $v(t) = V_f\left(1 - e^{-t/\tau}\right)$
63.2 % after one τ, 86.5 % after two, 99.3 % after five.
Capacitor discharging: $v(t) = V_0 \, e^{-t/\tau}$
Falls to 36.8 % of its starting value after one τ.

Symbol	Meaning	Unit
$\tau$	Time constant	s
$V_f$	Final (steady-state) value	V
$V_0$	Initial value at t = 0	V

Common mistakes

Using τ = RC for an RL circuit.
RL uses τ = L/R. The two behave oppositely: raising R slows an RC circuit but speeds up an RL one. Swapping the formulas inverts the trend entirely.
Expecting the capacitor to reach the supply voltage.
The approach is exponential and never technically completes. One τ reaches 63.2 %; five τ gets within 1 %, which is the practical definition of settled.
Ignoring source resistance.
The source's own output resistance adds to R and lengthens τ. A signal generator with 50 Ω output feeding a 100 Ω resistor makes the effective R 150 Ω.
Trusting the marked value of an electrolytic.
Electrolytics commonly run ±20 % or worse and drift as they age, so measured τ can differ substantially. Use film or ceramic where timing accuracy matters.
Switching an inductor without a flyback path.
Inductor current cannot stop instantly, so opening the circuit drives the voltage up until something breaks down. A reverse-biased diode across the coil gives that current somewhere to go.

Frequently asked questions

What is the time constant of an RC circuit?

Tau equals R times C. After one time constant the capacitor has charged to 63.2 percent of the final value, and after five it is within 1 percent, which is the usual definition of settled.

How long does a capacitor take to fully charge?

Strictly it never does, because the approach is exponential. In practice five time constants gets within 1 percent and is treated as complete. With R = 10 kilohm and C = 100 uF, tau is 1 second and settling takes about 5 seconds.

What is the time constant of an RL circuit?

Tau equals L divided by R. Note that increasing resistance speeds an RL circuit up, whereas it slows an RC circuit down — the two behave in opposite ways.

Why does my circuit settle slower than calculated?

The source resistance adds to R, and any load across the capacitor changes the effective resistance. Electrolytic capacitors also vary widely from their marked value, often 20 percent or more.

Why does switching off an inductor produce a voltage spike?

Inductor current cannot change instantly, so interrupting it forces the voltage to rise until something conducts. This is why a flyback diode across a relay coil is essential — without it the spike can reach hundreds of volts.