RC circuits and time constants: how capacitors charge and discharge

The time constant τ = R × C

The time constant is the product of resistance and capacitance: τ = R·C, with R in ohms and C in farads. The unit works out to seconds, which is a good sanity check: 10 kΩ (10,000 Ω) multiplied by 100 µF (0.0001 F) gives 1.0 s. Reading τ in seconds is also a quick way to compare designs, because doubling either component doubles the response time.

A larger resistance slows the current that charges the capacitor, and a larger capacitance needs more charge for the same voltage, so both increase the time constant. The supply voltage does not appear in τ at all: it changes the final voltage but not the shape or timing of the curve.

Charging and discharging curves

When a capacitor charges from zero through a resistor, its voltage follows V(t) = V₀·(1 − e^(−t/τ)). After one time constant it has reached 1 − e^(−1), which is about 63.2% of the final voltage; after two, about 86.5%; after five, about 99.3%.

Discharging through the same resistor reverses the shape: V(t) = V₀·e^(−t/τ). The voltage falls to about 36.8% after one time constant and to about 0.7% after five, which is why circuits are usually treated as settled after 5τ in either direction.

These percentages are properties of the exponential, not of the particular components. They are the reason 63.2% and 5τ are quoted so often in electronics even for very different resistor and capacitor values.

Worked example: 10 kΩ and 100 µF

With R = 10 kΩ and C = 100 µF, the time constant is 10,000 × 0.0001 = 1.0 s. Charging from 0 V toward a 5 V supply gives about 3.16 V after 1 s, about 4.32 V after 2 s and about 4.97 V after 5 s.

The cutoff frequency for the same pair is f_c = 1 ÷ (2π × 1.0 s) ≈ 0.159 Hz. A 10 kΩ resistor with a 10 µF capacitor, by contrast, has τ = 0.1 s and f_c ≈ 1.59 Hz: ten times smaller capacitance gives a ten times shorter time constant and a ten times higher cutoff.

Notice that a one-second time constant corresponds to a fairly low cutoff frequency. Timing and filtering are two views of the same RC product, and the calculator reports both so the connection stays visible.

Cutoff frequency and first-order filters

For a first-order RC low-pass filter, the cutoff frequency is f_c = 1 ÷ (2πRC). Below f_c the filter passes most of the signal; above it the output rolls off at about 20 dB per decade. The same RC pair can also be arranged as a high-pass filter with the same cutoff.

Since τ = RC, the cutoff frequency can also be written f_c = 1 ÷ (2πτ). That reciprocal relationship is why a circuit with a long time constant responds slowly and filters only low frequencies, while a short time constant responds quickly and passes higher frequencies.

The cutoff is the point where the output falls to about 70.7% of its low-frequency value, or half the power. It is a useful single number for comparing filter designs, but it does not by itself describe the full response.

Where RC timing matters and where the model stops

RC networks debounce mechanical switches, set timers, shape audio signals, couple amplifier stages and filter sensor outputs. In each case the time constant tells you how fast the circuit responds and how much of a changing signal it passes.

The simple model assumes ideal, constant components driven by a step voltage. Real capacitors have leakage and equivalent series resistance, resistors drift with temperature, and whatever is connected to the output loads the circuit. Those effects can shift the measured curve, which is why a calculation is a starting point for measurement. A useful check is to measure the voltage at a known time and compare it with the curve: a large disagreement usually means component tolerance, output loading or a supply that is not a clean step.