Library / Electrical / ELE-17
Electrical
RC circuit response
Capacitor voltage at any time after a switch is thrown.
- Ref
- ELE-17
- Rev
- v1
- Field
- Electrical
- Result
- vC [V]
Vf the final voltage the capacitor heads towards, Vi the initial voltage, Rth the resistance the capacitor sees after the switching and C the capacitance. The time constant τ = Rth C.
After a switch, a capacitor’s voltage moves smoothly from where it started towards where the circuit wants it, covering about 63% of the gap in one time constant.
Calculator
- Time constant τ = RC100 ms
- Fraction of the step completed63.21 %
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- After 1τ the capacitor has covered 63.2% of the step, after 3τ 95%, and after 5τ over 99%.
- The same equation describes charging (Vi below Vf) and discharging (Vf = 0).
- A low-pass filter or debounce circuit is just an RC response watched at its output.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (V) | Local step | Change in result |
|---|---|---|---|---|---|---|
| Vf Final voltage | 1 to 50 V | 0.6321 | 31.61 | 30.97 | +10% | +10 % |
| C Capacitance | 1 to 1,000 µF | 0.04975 | 5 | 4.95 | +10% | −5.54 % |
| t Time since switching | 1 to 1,000 ms | 0.04975 | 5 | 4.95 | +10% | +5.54 % |
| R Resistance | 1 to 100 kΩ | 0.4758 | 5 | 4.524 | +10% | −5.54 % |
| Vi Initial voltage | 0 to 3 V | 3.161 | 4.264 | 1.104 | +10% | 0 % |
Sweep of Time since switching
| Time since switching (ms) | Capacitor voltage (V) |
|---|---|
| 1 | 0.04975 |
| 2.371 | 0.1172 |
| 5.623 | 0.2734 |
| 13.34 | 0.6242 |
| 31.62 | 1.356 |
| 74.99 | 2.638 |
| 177.8 | 4.155 |
| 421.7 | 4.926 |
| 1,000 | 5 |
Sweep one variable
Variables and how each one contributes
Capacitor voltage
What the equation gives you. Everything else feeds it.
Final voltage
Voltage the capacitor settles at after a long time. Raises the answer in proportion to the share of the step completed.
Initial voltage
Capacitor voltage at the instant of switching. Its influence fades as e^(−t/τ).
Resistance
Resistance seen by the capacitor after switching. Larger R slows the response.
Capacitance
Capacitor value. Larger C slows the response.
Time since switching
Elapsed time. Moves the voltage towards Vf.
Worked example with the values above
- Time constant: τ = RC = 10 kΩ × 10 µF = 100 ms.
- Decay factor: e−t/τ = e−1 = 0.3679.
- vC = 5 + (0 − 5) × 0.3679 = 3.161 V.
Assumptions and limits
- One switching event, constant sources afterwards.
- Ideal linear capacitor and resistor.
- Rth is the Thévenin resistance seen by the capacitor.
Common mistakes
- Using the supply resistor instead of the Thévenin resistance.
- Mixing ms with s or kΩ·µF units.
- Forgetting the initial charge.
Statistics and uncertainty
R and C are typically ±1–20% parts, so τ can be off by ±20% or more. Error in the voltage depends on where you are on the curve: the response is most sensitive to τ at around t = τ.
Where it is used
- Timing, filtering and debounce circuits.
- Charge time for firing circuits and flashes.
- Sensor settling time.
Maths you need
- Exponentials: e^(−x) decays to zero
- Time constant: τ = RC