What is the RC time constant formula?

The time constant τ (tau) equals resistance times capacitance: τ = R × C. If R is in ohms and C is in farads, τ is in seconds. It describes how quickly a capacitor charges or discharges through a resistor — after one τ the capacitor reaches 63.2 % of the supply voltage when charging from zero.

How do I calculate the voltage on a capacitor at time t?

During charging from a discharged state: Vc(t) = V₀ × (1 − e^(−t/τ)). During discharging from a fully charged state: Vc(t) = V₀ × e^(−t/τ). Here V₀ is the supply or initial voltage and τ = RC. The calculator handles both modes and shows all intermediate steps.

What does the 5τ rule mean?

After five time constants (5τ) a capacitor is charged to about 99.3 % of the supply voltage, which is close enough to "fully charged" for most practical designs. Engineers use 5τ as the standard settling-time estimate for RC circuits, timers, and filter transients.

What is the cutoff frequency of an RC low-pass filter?

The −3 dB cutoff frequency is fc = 1 / (2π × τ) = 1 / (2π × R × C). At this frequency the output amplitude falls to 1/√2 ≈ 70.7 % of the input amplitude, and the phase shift is −45°. Below fc, the filter passes signals; above fc, it attenuates them.

How do I find R if I know the desired τ and C?

Rearrange τ = R × C to get R = τ / C. For example, to achieve τ = 1 ms with a 100 nF capacitor: R = 0.001 / 0.0000001 = 10 kΩ. Use the "Resistance (R)" solve mode and enter your target τ and capacitor value directly.

What is the RC half-life?

The half-life t½ is the time after which the capacitor voltage falls to exactly half its initial value during discharge (or rises to half V₀ during charging from zero). It equals τ × ln 2 ≈ 0.6931 × τ. This is directly analogous to radioactive decay — the same exponential law governs both.

RC Time Constant Calculator

An RC circuit pairs a resistor (R) and a capacitor (C) in series. It is one of the most fundamental building blocks in electronics: it appears in debounce filters, audio tone controls, power-supply smoothing, timing circuits, sensor conditioning, and countless embedded-system designs. The single most important number that characterises an RC circuit is the time constant τ (tau).

How it works

The time constant is defined by the product of resistance and capacitance:

τ = R × C

When R is in ohms and C is in farads, τ is in seconds. This single number governs three distinct behaviours:

Charging rate. A capacitor starting from 0 V and charging toward a supply V₀ follows the exponential curve Vc(t) = V₀ × (1 − e^(−t/τ)). After one τ it reaches 63.21 % of V₀; after five τ it is at 99.33 % — the standard “fully charged” threshold used in circuit design.
Discharging rate. A fully charged capacitor discharging through R decays as Vc(t) = V₀ × e^(−t/τ). The same τ controls how quickly it falls to zero.
Filter cutoff frequency. An RC network forms a first-order low-pass (or high-pass) filter. The −3 dB corner frequency is fc = 1 / (2π × τ), the point at which the output is attenuated to 70.7 % of the input and phase-shifted by 45°.

The calculator solves all three rearrangements of τ = R × C (find τ, find R, or find C), then derives fc, the half-life, the 5τ threshold, and optionally the exact capacitor voltage at any elapsed time you choose.

Worked example

A 10 kΩ resistor in series with a 100 µF capacitor:

τ = R × C = 10,000 Ω × 0.0001 F = 1 s
fc = 1 / (2π × 1) ≈ 0.159 Hz
t½ = 1 × ln 2 ≈ 0.693 s
5τ = 5 s  (fully charged)

If the supply voltage is 5 V, the capacitor voltage 2 seconds after connecting power is:

Vc(2 s) = 5 × (1 − e^(−2/1)) = 5 × (1 − e^(−2)) ≈ 5 × 0.8647 ≈ 4.32 V

R	C	τ	fc
1 kΩ	1 µF	1 ms	159 Hz
10 kΩ	100 µF	1 s	0.159 Hz
47 kΩ	10 nF	470 µs	338 Hz
1 MΩ	1 µF	1 s	0.159 Hz

Notice that the first and last rows give the same τ and fc — many R/C combinations yield identical time constants, which is why the solve-for-R and solve-for-C modes are so useful in real design work.

Formula note

The exponential shape of RC charging and discharging comes directly from solving the first-order linear differential equation R × C × dVc/dt + Vc = V₀. The solution is the decaying or rising exponential shown above. The time constant τ is the characteristic time of that solution — the inverse of the pole location in the Laplace domain at s = −1/τ. Every RC low-pass filter has exactly one pole, which is why it rolls off at −20 dB/decade above fc.

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