RC Circuit Calculator

The RC time constant τ = R × C determines how fast a capacitor charges.

At t=1τ, the capacitor reaches 63.2% of supply voltage. At t=5τ, it reaches 99.3% and is considered fully charged.

For R=10kΩ and C=10μF, τ = 0.1 seconds.

The cutoff frequency of an RC filter is fc = 1/(2π × R × C). This is the frequency at which the filter's output drops to −3 dB (70.7% of input).

Below fc, a low-pass RC filter passes the signal; above fc, it attenuates it.

During charging: Vc(t) = Vs × (1 − e^(−t/τ)).

During discharging: Vc(t) = V0 × e^(−t/τ).

Enter your supply voltage, initial voltage, resistance, capacitance, and time to find the exact capacitor voltage at any point.

To calculate the RC time constant, multiply the resistance by the capacitance: τ = R × C. If R is in ohms (Ω) and C is in farads (F), the result τ is in seconds.

For example, R = 10 kΩ (10,000 Ω) and C = 10 μF (0.00001 F) give τ = 10,000 × 0.00001 = 0.1 s. This single product governs charging speed, discharge speed, and filter response.

As HyperPhysics (Georgia State University) explains, the capacitor reaches 63.2% of its final voltage after one τ and is treated as fully charged after five τ (99.3%). Larger R or C slows the circuit and lowers its cutoff frequency.

The RC time constant is measured in seconds (s), the SI unit of time defined by BIPM via the caesium-133 transition. It arises because resistance in ohms (Ω) multiplied by capacitance in farads (F) yields seconds: 1 Ω × 1 F = 1 s, since 1 F = 1 A·s/V and 1 Ω = 1 V/A.

Cutoff frequency fc is measured in hertz (Hz), meaning cycles per second, defined by NIST as the reciprocal of the period. Angular frequency ω = 2πfc is expressed in radians per second (rad/s).

Always convert prefixes carefully:

• 1 kΩ = 1,000 Ω
• 1 μF = 10⁻⁶ F
• 1 nF = 10⁻⁹ F

The RC cutoff frequency is calculated with fc = 1 / (2π × R × C), which equals 1 / (2πτ). For R = 10 kΩ and C = 10 μF, τ = 0.1 s, so fc = 1 / (2π × 0.1) ≈ 1.592 Hz. This is the −3 dB point where output amplitude falls to about 70.7% of input, per Electronics Tutorials and IEEE filter conventions.

The corresponding angular cutoff frequency is ω = 2πfc = 1/(RC) = 1/τ = 10 rad/s in this example.

Because fc is inversely proportional to R and C, doubling either component halves the cutoff frequency, shifting the filter's transition band lower.

A capacitor's charging progress depends only on multiples of τ:

• After 1τ it reaches 63.2%
• after 2τ 86.5%
• after 3τ 95.0%
• after 4τ 98.2%
• and after 5τ 99.3%, at which point engineers treat it as fully charged

Discharging follows the mirror curve: after 1τ the voltage drops to 36.8% of its start, and after 5τ to about 0.7%.

For τ = 0.1 s, full charge takes roughly 5 × 0.1 = 0.5 s. Khan Academy notes this exponential behavior means the capacitor never mathematically reaches 100%; it approaches it asymptotically, so the five-τ rule is a practical engineering convention, not an exact limit.

RC circuits appear throughout electronics.

• Timing circuits, such as the classic 555 timer, use τ = RC to set precise delays and oscillation periods.
• Low-pass RC filters smooth power-supply ripple and remove high-frequency noise from audio and sensor signals, while high-pass RC filters block DC offset and pass audio treble.
• Encoders and debounce circuits use RC delays to filter contact bounce from mechanical switches.
• In signal processing, RC networks shape pulse edges and form integrator or differentiator stages.

Encyclopaedia Britannica notes that capacitor charging behavior also underpins camera flash charging and defibrillator energy storage, where controlled charge and discharge times are critical to safe operation.

Both RC filter types share the same cutoff formula fc = 1/(2πRC), but differ in where the output is taken.

In a low-pass filter, output is measured across the capacitor: it passes frequencies below fc and attenuates those above at −20 dB per decade. In a high-pass filter, output is taken across the resistor: it passes frequencies above fc and blocks lower ones.

Per IEEE and IEC filter definitions, the roll-off of a single RC stage is first-order (−20 dB/decade).

To design a filter, pick fc first, choose a standard capacitor value, then solve for R = 1/(2π × fc × C). Cascading stages increases roll-off steepness.

The most frequent error is mixing unit prefixes. Capacitance is usually given in microfarads (μF), nanofarads (nF), or picofarads (pF), so forgetting to convert to farads produces τ values off by factors of thousands.

• Confusing fc = 1/(2πRC) with the time constant 1/RC = ω is another common slip; fc is in hertz, while ω is in radians per second.
• Some users assume a capacitor charges linearly, but the curve is exponential, per NIST and HyperPhysics.
• Others treat 5τ as truly 100% charged when it is 99.3%.
• Finally, remember low-pass and high-pass outputs are measured across different components even though the cutoff formula is identical.

Suppose Vs = 5 V, R = 10 kΩ, and C = 10 μF, giving τ = 0.1 s.

To find the charging voltage at t = 0.05 s, use Vc(t) = Vs × (1 − e^(−t/τ)). Here t/τ = 0.05/0.1 = 0.5, so e^(−0.5) ≈ 0.6065, and Vc = 5 × (1 − 0.6065) ≈ 1.97 V, about 39% charged.

For discharging from V0 = 5 V, use Vc(t) = V0 × e^(−t/τ); at t = 0.1 s (one τ), Vc = 5 × e^(−1) ≈ 5 × 0.3679 ≈ 1.84 V, matching the 36.8% rule cited by Electronics Tutorials. The instantaneous current through the resistor follows Ohm's law — divide the resistor voltage (Vs − Vc) by R, or drop the numbers into our ohms law calculator to verify. Our calculator automates these exponential computations.