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RC circuit charge and discharge calculator

Time constant τ (tau), voltage and current at a given instant, charge, energy, and the time to reach a target voltage.

Circuit

The circuit

Switch closed onto the source: voltage starts at zero and tends to E.

VWhen discharging, this is the initial capacitor voltage.
The observed instant
VOptional: gives the time needed to reach it.

Transient

charging · 0 to 6 τ
0255075100% U01234560255075100time (τ)% i₀63.2% at ττ = 0.001 su = 3.16 Vi₀ = 0.005 A
t = 1 τ
Calculation
u(t) = E(1 − e^(−t/τ)) = 5 × (1 − e^(−1)) = 3.1606 V
Time constant
τ = R × C = 1000 × 1 × 10⁻⁶ = 0.001 s
Note

With every time constant, the voltage covers 63% of the remaining gap.

The law: u and iThe instant t

Result

continuous computation
At time t
u3.1606 Vat t = 1 τ
Fraction of E
63.2121 %
Current at that instant
0.001 839 4 A
Stored charge
3.1606 × 10⁻⁶ C
Stored energy
4.99471 × 10⁻⁶ J
Circuit landmarks
Time constant
0.001 s
Initial current
0.005 A
Settled to 99.3%
0.005 s
Energy once charged
1.25 × 10⁻⁵ J

Calculation, assumptions and sources

OverviewTransient regime of an RC circuit, charging or discharging: time constant τ, voltage and current at a given instant, fraction of E reached, stored charge and energy, settling time at 5 τ, and the time to reach a target voltage.

Transient regime of an RC circuit, charging or discharging: time constant τ, voltage and current at a given instant, fraction of E reached, stored charge and energy, settling time at 5 τ, and the time to reach a target voltage. The sign of the current distinguishes charging from discharging.

Method & formulasτ = R × C
τ = R × Ccharging: 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 describes both regimes; its time constant τ = RC does not depend on E. Landmarks: 63% of the gap covered after τ, 95% after 3 τ, 99.3% after 5 τ.

τ
time constant, R × C, in seconds.
E
source voltage when charging, initial capacitor voltage when discharging.
u(t)
voltage across the capacitor, a continuous function of time.
i(t)
current in the circuit, maximal at t = 0 in both regimes, then decaying.
q
stored charge, q = C u, in coulombs.
W
stored energy, ½ C u², in joules.
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. The target voltage excludes E when charging and zero when discharging, reached only after an infinite time.

Reading the resultThe number of time constants elapsed, t/τ, places the instant in the regime independently of R, C and E.

The number of time constants elapsed, t/τ, places the instant in the regime independently of R, C and E. When charging, the current falls while the capacitor voltage rises: it equals (E − u)/R. At t = 0, an empty capacitor behaves like a wire. After five time constants, 0.7% of the gap is left and the regime is taken as settled; beyond that, the gap drops below measurement noise. Energy grows as the square of the voltage: at half the final voltage, the capacitor holds a quarter of its energy.