Table of Contents
RC Charging Explained
When a capacitor charges through a resistor from a constant voltage source, it follows an exponential curve. Initially, the full voltage appears across the resistor, driving maximum current into the capacitor. As the capacitor charges, the voltage across it increases, reducing the voltage across the resistor and therefore the charging current. The result is an exponentially decaying current and an exponentially rising capacitor voltage.
The rate of charging is determined by the time constant tau = RC. A larger resistance limits the charging current, slowing the process. A larger capacitance requires more charge to reach the same voltage, also slowing the process. The time constant is the single most important parameter in RC circuit analysis and appears in countless electronic circuits from timing circuits to filters to signal processing.
Charging Equations
The charging voltage asymptotically approaches Vs but theoretically never quite reaches it. In practice, the capacitor is considered fully charged after 5 time constants, when it reaches 99.33% of the supply voltage.
Charging Milestones
| Time Constants | % of Vs | % Remaining |
|---|---|---|
| 1 RC | 63.21% | 36.79% |
| 2 RC | 86.47% | 13.53% |
| 3 RC | 95.02% | 4.98% |
| 4 RC | 98.17% | 1.83% |
| 5 RC | 99.33% | 0.67% |
Frequently Asked Questions
Can a capacitor ever be fully charged?
Mathematically, the exponential charging curve approaches but never reaches the supply voltage. In practice, after about 5 time constants, the capacitor is within 0.67% of the supply voltage, which is close enough for virtually all applications. After 7 time constants, it reaches 99.91%, and after 10 time constants, 99.995%. The remaining voltage difference is below the noise floor of most circuits and is insignificant.
How does the discharge curve differ from charging?
The discharge curve is a mirror image: V(t) = V0 times e^(-t/RC). The voltage decays exponentially from its initial value toward zero with the same time constant. After 1 RC, the voltage drops to 36.8% of its initial value. After 5 RC, only 0.67% remains. The discharge current flows in the opposite direction to the charging current and also decays exponentially.
What is the 63.2% rule?
The 63.2% value comes from 1 - 1/e, where e is Euler's number (2.718). After exactly one time constant, the capacitor has charged to 63.2% of the supply voltage. This is a useful approximation for quick calculations: if you know the time constant, you immediately know the voltage at t = RC. Similarly, at t = 0.693RC (the natural log of 2 times RC), the capacitor reaches exactly 50% of the supply voltage, which is important for timing circuits.