RC circuit charge and discharge
τ = R × C
The transient regime of a resistor-capacitor circuit, both ways. The tool returns the time constant, the voltage and current at the requested instant, the fraction of voltage reached, the stored charge and energy, the settling time at five constants, and, on request, the time needed to reach a target voltage. The current carries its sign, because that is what tells a filling capacitor from an emptying one, and the charging voltage is computed so as to stay correct even on times a thousand billion times shorter than the time constant.
3.1606 V
- Calculation
- u(t) = E(1 − e^(−t/τ)) = 5 × (1 − e^(−1)) = 3.1606 V
- Time constant
- τ = R × C = 1000 × 1E-6 = 0.001 s
The voltage reaches 63.2121% of its final value, after 1 time constant(s).
The current falls in both regimes, charging included: it is not the capacitor that sets the current but the resistor, and the voltage across it drops as the capacitor voltage rises. At the first instant of a charge, the empty capacitor behaves like a wire: the whole source falls across the resistor, and the current is at its maximum.
The regime is taken as settled after five time constants, that is 0.005 s: 0.7% of the gap is left at that point. The convention is not fundamental, an exponential never reaching its limit; it only says that beyond it the gap drops below measurement noise.
Stored energy is ½Cu²: it grows as the square of the voltage. At half the final voltage, the capacitor holds only a quarter of its energy, not half.
The model assumes a perfect voltage source, a constant resistance and an ideal capacitor with no series resistance and no leakage current. On a real electrolytic capacitor, leakage makes the final voltage drift, and the rated capacitance often carries a 20% tolerance.
Scientific dossier
What the tool computes, what it assumes, where it stops being valid, and where its data comes from.
Method & formulasτ = R × C
τ = R × C
charging: u(t) = E(1 − e^(−t/τ))
discharging: u(t) = E e^(−t/τ)
i(t) = ±(E/R) e^(−t/τ)
W = ½ C u²
One differential equation governs both regimes, and one quantity sums up their speed: the time constant τ = RC, which has the dimension of a time. It depends neither on the source voltage nor on the instant considered. The landmarks that follow are the ones worth remembering: 63% after one constant, 95% after three, 99.3% after five.
- τ
- · time constant, R × C, in seconds. It sets the speed of the regime.
- E
- · source voltage when charging, initial capacitor voltage when discharging.
- u(t)
- · voltage across the capacitor: it is continuous, it never jumps.
- i(t)
- · current in the circuit. Maximal at t = 0 in both regimes, it always decays.
- q
- · stored charge, q = C u, in coulombs.
- W
- · stored energy, ½ C u², in joules. It grows as the square of the voltage.
Validity domainThe model assumes a perfect voltage source, a constant resistance and an ideal capacitor with no equivalent series resistance and no leakage current.
The model assumes a perfect voltage source, a constant resistance and an ideal capacitor with no equivalent series resistance and no leakage current. On a real electrolytic, leakage makes the final voltage drift and the rated capacitance often carries a 20% tolerance, far larger than the precision of this calculation. The target voltage must stay inside the genuinely reachable range: charging tends to E without reaching it, discharging tends to zero without reaching it, and the tool refuses the open bound rather than returning an infinite time. On times very short against the constant, the charging voltage is computed with expm1 rather than E(1 − e^(−t/τ)), whose subtraction would return exactly zero.
Reading the resultThe number of time constants elapsed is read before the voltage: it says where you are in the regime, independently of R, C and E.
The number of time constants elapsed is read before the voltage: it says where you are in the regime, independently of R, C and E. The current often surprises when charging, where one expects it to grow with the voltage: it is the opposite. The current is set by the resistor, and the voltage across it is E minus that of the capacitor; it therefore falls as the capacitor fills. At the first instant, an empty capacitor behaves like a wire.