# RC Time Constant Calculator

> Computes the time constant τ = R·C of a resistor–capacitor circuit, the times to reach 63.2 %, 95 % and 99 % of the final voltage (and the 5τ 'full charge' convention), the −3 dB cutoff frequency 1/(2πRC), and optionally the charge level after a given time.

- Calculator id: `rc-time-constant` · Category: Engineering & Automotive (`engineering`) · Tool name: `calculate_rc_time_constant`
- Canonical page: https://tttkmbb.com/engineering/rc-time-constant · This document: https://tttkmbb.com/engineering/rc-time-constant.md · JSON definition: https://tttkmbb.com/engineering/rc-time-constant.json

## Purpose

Computes the time constant τ = R·C of a resistor–capacitor circuit, the times to reach 63.2 %, 95 % and 99 % of the final voltage (and the 5τ 'full charge' convention), the −3 dB cutoff frequency 1/(2πRC), and optionally the charge level after a given time.

**Use when:** You need how fast a capacitor charges or discharges through a resistor, or the corner frequency of a simple RC low-pass/high-pass filter.

**Do not use when:** The circuit contains an inductor (use lc-resonance), the capacitor is driven by a constant-current source, or you only need the combined value of several capacitors (use capacitors-series-parallel).

## Input

| Parameter | Type | Unit | Required | Description |
| --- | --- | --- | --- | --- |
| `resistance_ohm` | number | Ω | required | Series resistance in ohms (1 kΩ = 1000, 1 MΩ = 1e6). (> 0) |
| `capacitance_uf` | number | µF | required | Capacitance in microfarads (1 nF = 0.001 µF, 1 pF = 1e-6 µF, 1 F = 1e6 µF). (> 0) |
| `time_s` | number | s | optional | Optional time after the step; enables the charge-percentage output. (min 0) |

## Output

| Field | Type | Unit | Description |
| --- | --- | --- | --- |
| `time_constant_s` | number | s | τ = R × C. |
| `time_constant_formatted` | string |  | τ with an SI prefix, e.g. 100 µs. |
| `time_63_percent_s` | number | s | 1 τ: the capacitor reaches 1 − e⁻¹ = 63.2 % of the final voltage. |
| `time_95_percent_s` | number | s | −ln(0.05) τ ≈ 2.996 τ. |
| `time_99_percent_s` | number | s | −ln(0.01) τ ≈ 4.605 τ. |
| `time_5_tau_s` | number | s | Conventional full-charge time 5 τ, at which 99.33 % is reached. |
| `cutoff_frequency_hz` | number | Hz | f_c = 1 / (2π R C), the −3 dB corner of an RC filter. |
| `charge_percent_after_time` | number | % | 100 × (1 − e^(−t/τ)): voltage reached when charging (only when time_s is given). |
| `remaining_percent_after_time` | number | % | 100 × e^(−t/τ): voltage left when discharging (only when time_s is given). |

## Formula

`τ = resistance_ohm × capacitance_uf × 1e-6; t(p) = −τ × ln(1 − p); cutoff_frequency_hz = 1 / (2π τ); charging V(t)/V₀ = 1 − e^(−t/τ); discharging V(t)/V₀ = e^(−t/τ)`

## Data Sources

- Wikipedia – RC time constant — https://en.wikipedia.org/wiki/RC_time_constant (reference, retrieved 2026-09-24)
- HyperPhysics – Capacitor charging — http://hyperphysics.phy-astr.gsu.edu/hbase/electric/capchg.html (reference, retrieved 2026-09-24)

Data freshness: `static`. Deterministic formula with fixed constants; results never go stale. Inputs supplied by the caller determine the output.

## API

- `GET https://tttkmbb.com/api/v1/calculate/rc-time-constant?resistance_ohm=…&capacitance_uf=…`
- `POST https://tttkmbb.com/api/v1/calculate/rc-time-constant` with JSON body `{"inputs": {…}}`
- Response: unified envelope (`success`, `request`, `result.values`, `result.units`, `sources`, `freshness`, `timestamp`, `next_actions`, `links`); see https://tttkmbb.com/docs/response-format.md
- Schema: https://tttkmbb.com/api/v1/calculators/rc-time-constant · OpenAPI operationId `calculate_rc_time_constant` in https://tttkmbb.com/openapi.json
- Authentication: none. Rate limit: fair use, see https://tttkmbb.com/docs/rate-limits.md.

## MCP

- Server: `https://tttkmbb.com/mcp` (Streamable HTTP, JSON-RPC 2.0, no auth)
- Tool:  `run_calculator` with `{"calculator_id": "rc-time-constant", "inputs": {…}}`

## Example

- 10 kΩ and 100 µF, after 2 s: inputs `{"resistance_ohm":10000,"capacitance_uf":100,"time_s":2}` → `{"time_constant_s":1,"time_constant_formatted":"1 s","time_63_percent_s":1,"time_95_percent_s":2.995732,"time_99_percent_s":4.60517,"time_5_tau_s":5,"cutoff_frequency_hz":0.1592,"charge_percent_after_time":86.47,"remaining_percent_after_time":13.53}`
- 1 kΩ and 100 nF filter: inputs `{"resistance_ohm":1000,"capacitance_uf":0.1}` → `{"time_constant_s":0.0001,"time_constant_formatted":"100 µs","cutoff_frequency_hz":1591.5494}`

```
GET https://tttkmbb.com/api/v1/calculate/rc-time-constant?resistance_ohm=10000&capacitance_uf=100&time_s=2
```

## Limitations

The circuit contains an inductor (use lc-resonance), the capacitor is driven by a constant-current source, or you only need the combined value of several capacitors (use capacitors-series-parallel). All values are computed from the formula above; no measurement or live data is involved.

## FAQ

**Why is 5 τ called fully charged?**

Charging is exponential and never strictly completes; at 5 τ the capacitor holds 99.3 % of the final voltage, which is treated as complete in practice.

**Does the cutoff frequency apply to both low-pass and high-pass?**

Yes. The same RC pair gives a −3 dB point at 1/(2πRC) whether the output is taken across the capacitor (low-pass) or the resistor (high-pass).

## Related

- [LC Resonance Calculator](https://tttkmbb.com/engineering/lc-resonance.md) — Resonant frequency of an inductor–capacitor pair.
- [Capacitors in Series and Parallel Calculator](https://tttkmbb.com/engineering/capacitors-series-parallel.md) — Total capacitance to use as C.
- [Wavelength & Frequency Calculator](https://tttkmbb.com/physics/wavelength-frequency.md) — Wavelength of the cutoff frequency.