RC Circuits
A capacitor charging through a resistor: fast at first, then slower and slower. Find the voltage after a given time, how long it takes to reach a voltage, or the resistor or capacitor you need for a delay.
How a Capacitor Charges
A capacitor stores charge. When you connect an empty one through a resistor to a battery, current flows at first as if the capacitor were a wire. As it fills up its own voltage pushes back, the current drops and charging slows down: the voltage creeps towards the supply voltage but never quite reaches it.
The pace is set by the time constant τ = R·C. After one τ the capacitor is 63% charged, after five τ more than 99% — in practice full. Discharging follows the same curve the other way. This is how simple timers, delays and blinking circuits work.
Voltage and Current Over Time
τCharge per Time Constant
exact| Time | Charging: share of V | Discharging: share of V₀ left |
|---|---|---|
| 1 τ | 63.2% | 36.8% |
| 2 τ | 86.5% | 13.5% |
| 3 τ | 95.0% | 5.0% |
| 4 τ | 98.2% | 1.8% |
| 5 τ | 99.3% | 0.7% |
Worked Techniques
The charging formula turned around. Useful to design a delay: the time until a switching threshold is reached.
Every time constant the voltage drops to 37% of what it was.
Pick a convenient capacitor and calculate the resistor that gives the delay you need, or the other way round.
Sources & Verification
- A. R. Hambley, Electrical Engineering: Principles and Applications, 7th ed., Pearson, 2018 — Ch. 4 (Transients), Sec. 4.1 (First-Order RC Circuits).