RC Time Constant Guide: τ = RC Formula & Circuit Design
What Is the RC Time Constant?
The RC time constant (τ, tau) describes how quickly a capacitor charges or discharges through a resistor. It's the product of resistance and capacitance:
τ = R × C
- τ = time constant (seconds)
- R = resistance (ohms)
- C = capacitance (farads)
Charging Curve
A capacitor charges exponentially through a resistor:
V(t) = Vs × (1 − e−t/τ)
| Time | % Charged | % Remaining to Charge |
|---|---|---|
| 1τ | 63.2% | 36.8% |
| 2τ | 86.5% | 13.5% |
| 3τ | 95.0% | 5.0% |
| 4τ | 98.2% | 1.8% |
| 5τ | 99.3% | 0.7% |
5τ rule: After 5 time constants, the capacitor is considered fully charged/discharged (99.3%).
Common τ Values
| R | C | τ | Settling (5τ) |
|---|---|---|---|
| 1 kΩ | 1 μF | 1 ms | 5 ms |
| 10 kΩ | 10 μF | 100 ms | 500 ms |
| 100 kΩ | 10 μF | 1.0 s | 5.0 s |
| 1 MΩ | 1 μF | 1.0 s | 5.0 s |
| 10 kΩ | 100 μF | 1.0 s | 5.0 s |
| 47 kΩ | 47 μF | 2.2 s | 11.0 s |
| 1 MΩ | 100 μF | 100 s | 500 s |
RC Low-Pass Filter
An RC circuit acts as a low-pass filter (passes low frequencies, attenuates high):
Cutoff frequency: fc = 1 / (2π × R × C)
At fc, the output is −3 dB (70.7%) of the input. Frequencies above fc are attenuated at −20 dB/decade.
Cutoff Frequency Examples
| R | C | fc | Application |
|---|---|---|---|
| 10 kΩ | 100 nF | 159 Hz | Audio low-pass filter |
| 10 kΩ | 10 nF | 1.59 kHz | Audio crossover |
| 1 kΩ | 100 nF | 1.59 kHz | Signal filtering |
| 10 kΩ | 1 nF | 15.9 kHz | Anti-aliasing filter |
| 100 kΩ | 10 μF | 0.159 Hz | DC smoothing |
Energy Stored in a Capacitor
E = ½ × C × V²
A 100 μF capacitor charged to 12V stores: E = 0.5 × 0.0001 × 144 = 0.0072 J = 7.2 mJ.
A 1 F capacitor at 12V stores: E = 0.5 × 1 × 144 = 72 J — enough to power an LED for several seconds.
Common Mistakes
- Using μF directly: The formula requires farads. Convert: 100 μF = 0.0001 F. Using 100 instead of 0.0001 gives τ = 1,000,000× too large.
- Ignoring ESR: Real capacitors have Equivalent Series Resistance (ESR). A 100 μF cap with 0.1Ω ESR in a 10Ω circuit has τ = 10.1 × 0.0001, not 10 × 0.0001.
- Confusing charge and discharge: Charging: V(t) = Vs(1 − e−t/τ). Discharging: V(t) = V0 × e−t/τ. Different formulas.
Standards Reference
- IEC 60062 — Capacitor Marking Codes
- IEEE Standard 100 — Electrical Definitions
- NFPA 70 — NEC (capacitor applications)
Frequently Asked Questions
What is the RC time constant formula?
τ = R × C, where τ is in seconds, R is in ohms, and C is in farads. It's the time for a capacitor to charge to 63.2% of the applied voltage (or discharge to 36.8%). After 5τ (99.3%), the capacitor is considered fully charged or discharged.
How do you calculate the time to charge a capacitor?
Use V(t) = V_s × (1 − e^(−t/τ)) where τ = R × C. To find time for a specific voltage: t = −τ × ln(1 − V/V_s). Example: time to reach 10V on a 12V supply with τ = 1s: t = −1 × ln(1 − 10/12) = −1 × ln(0.167) = 1.79 seconds.
What is the 5 tau rule?
After 5 time constants (5τ), a capacitor is 99.3% charged or discharged. Engineers use this as the 'settling time' — the time to consider the circuit at steady state. For τ = 0.1s, settling time = 0.5s. For τ = 1s, settling time = 5s.
What is an RC filter used for?
RC filters pass or block specific frequencies. A low-pass RC filter (capacitor to ground) passes DC and low frequencies while attenuating high frequencies — used for smoothing power supplies and removing noise. A high-pass RC filter (capacitor in series) blocks DC and passes AC signals.
How do I choose R and C for a specific time constant?
For τ = 1 second: use 10 kΩ × 100 μF, or 100 kΩ × 10 μF, or 1 MΩ × 1 μF. Higher R = lower current draw but more susceptible to noise. Lower R = faster response but more current. Choose based on your circuit's impedance requirements.