What is the capacitor discharge formula?
V(t) = V0 × e^(−t/RC). The voltage falls to 36.8 % of its starting value after one time constant (τ = RC), to 13.5 % after two and below 1 % after five. The current follows the same curve, starting at V0 / R.
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Capacitor discharge calculator with charge time
Voltage after a time, or time to a voltage: this capacitor discharge calculator solves both, charging or discharging, through a resistor or at a constant current. Size a bleeder resistor that brings a bus or X capacitor below a safe voltage in time, and see what it dissipates.
Circuit
Examples
Find
V
Ω
F
s
Voltage after 1 s
4.415 V
- Time constant τ = RC
- 1 s
- Elapsed time in time constants
- 1 τ
- Voltage left
- 36.79 % of the starting voltage
- Current at the start
- 1.2 mA 12 V ÷ 10 kΩ
- Current at this point
- 441.5 µA
- Resistor power at the start
- 14.4 mW
- Energy stored at the start
- 7.2 mJ
- Energy stored at this point
- 974.4 µJ
- Heat in the resistor so far
- 6.226 mJ
Ideal capacitor: no leakage current and no ESR. Leakage speeds a discharge slightly; ESR adds to R.
Electrolytic and film capacitors recover part of their voltage after a quick discharge (dielectric absorption). Leave the discharge path connected and measure before touching the terminals.
How to calculate capacitor discharge time
Use t = RC × ln(V0 / V). A 100 µF capacitor discharging through 10 kΩ has τ = RC = 1 s, so going from 12 V to 1 V takes 1 s × ln 12 = 2.48 s. The voltage at any time is V0 × e^(−t/RC): after one time constant it is down to 36.8 % (4.415 V here), and after five it is under 1 %.
Capacitor charge and discharge formulas
Through a resistor, the capacitor voltage moves exponentially from where it starts toward where the circuit is pushing it: 0 V for a discharge, the supply voltage for a charge.
V(t) = Vf + (V0 − Vf) × e^(−t/τ)
- V0
- voltage on the capacitor at t = 0, in V
- Vf
- final voltage: 0 V for a discharge, the supply Vs for a charge
- τ
- time constant R × C, in s
- t
- time since the switch closed, in s
The capacitor discharge equation (to 0 V) and the capacitor charging equation (from 0 V) are the two common cases:
V(t) = V0 × e^(−t/RC) V(t) = Vs × (1 − e^(−t/RC))
Solved for the time to reach a target voltage V:
t = RC × ln((V0 − Vf) ÷ (V − Vf)) discharge: t = RC × ln(V0 ÷ V)
At a constant current I the voltage changes in a straight line, because i = C dV/dt. A bleeder resistor is the largest R that still reaches the safe voltage in time:
V(t) = V0 ± I × t ÷ C t = C × ΔV ÷ I R ≤ t ÷ (C × ln(V0 ÷ Vsafe))
The energy left on the capacitor at any point is ½CV²; during a discharge through a resistor, the rest has become heat in R.
Worked examples
Example 1: 100 µF discharging from 12 V through 10 kΩ
τ = 10 kΩ × 100 µF = 1 s. After 1 s the voltage is 12 × e^(−1) = 4.415 V. To reach 1 V takes t = 1 s × ln(12 ÷ 1) = 2.48 s. The discharge starts at 12 V ÷ 10 kΩ = 1.2 mA and turns 7.15 mJ into heat in the resistor on the way down to 1 V.
Example 2: bleeder for a 400 V, 220 µF bus (50 V within 1 minute)
R ≤ 60 s ÷ (220 µF × ln(400 ÷ 50)) = 131 kΩ. The nearest E24 value below is 130 kΩ, which reaches 50 V in 59.5 s. The catch is that it dissipates 400² ÷ 130 kΩ = 1.23 W for as long as the bus is up, and it has the full 400 V across it. Two 62 kΩ resistors in series (124 kΩ) reach 50 V in 56.7 s, with 200 V and 645 mW on each. The capacitor holds 17.6 J at 400 V.
Example 3: 470 nF X capacitor on 230 V AC (under 34 V within 1 s)
Unplugged at the crest, the capacitor starts at √2 × 230 V = 325.3 V. R ≤ 1 s ÷ (470 nF × ln(325.3 ÷ 34)) = 942 kΩ, so 910 kΩ (E24) gets there in 966 ms and dissipates 230² ÷ 910 kΩ = 58.1 mW while plugged in. That standby loss is why X capacitor discharge ICs exist: Power Integrations puts a 1 MΩ discharge resistor at 53 mW on 230 V AC (52.9 mW by the same formula). Check the test conditions in the standard: the highest rated line voltage and the component tolerances can push R lower.
Capacitor voltage after each time constant
Every RC charge and discharge has the same shape; only τ = RC stretches it. Multiply τ by the first column to get the time.
| Time | Discharge: voltage left | Charge: share of the step |
|---|---|---|
| 0.5 τ | 60.7 % | 39.3 % |
| 0.693 τ (half) | 50 % | 50 % |
| 1 τ | 36.8 % | 63.2 % |
| 2 τ | 13.5 % | 86.5 % |
| 3 τ | 4.98 % | 95 % |
| 4 τ | 1.83 % | 98.2 % |
| 4.61 τ | 1 % | 99 % |
| 5 τ | 0.674 % | 99.3 % |
| 7 τ | 0.0912 % | 99.9 % |
Bleeder resistors for a 400 V bus, 50 V within 1 minute
The largest resistor that meets the 50 V in 60 s limit for each capacitor, the nearest E24 value below it, and the power it burns at 400 V. Bigger capacitors need smaller resistors and more watts.
| Capacitor | Largest R | E24 value | Time to 50 V | Dissipation at 400 V |
|---|---|---|---|---|
| 47 µF | 614 kΩ | 560 kΩ | 54.7 s | 286 mW |
| 100 µF | 289 kΩ | 270 kΩ | 56.1 s | 593 mW |
| 220 µF | 131 kΩ | 130 kΩ | 59.5 s | 1.23 W |
| 470 µF | 61.4 kΩ | 56 kΩ | 54.7 s | 2.86 W |
| 1 mF | 28.9 kΩ | 27 kΩ | 56.1 s | 5.93 W |
What changes the curve in a real circuit
The calculator treats the capacitor as ideal. Leakage current adds a parallel path, so a real discharge is a little faster; ESR and wiring add to R, so a low-value discharge resistor is slower than the formula and its peak current is lower. Aluminum electrolytics have a wide capacitance tolerance (check the datasheet), and a bleeder has to meet its limit with the capacitor at the top of its tolerance and the line at its highest voltage.
Size the resistor for three things, not one: the continuous power V²/R while the circuit is on, the working voltage across it (several resistors in series share both), and, for a fast discharge, the pulse energy ½CV² it absorbs at once. Electrolytic and film capacitors also show dielectric absorption: shorted briefly, they recover part of their voltage, so leave the bleeder connected and measure before touching the terminals. A constant-current sink stops regulating when its own headroom runs out, so its straight line bends toward zero near the end.
Glossary
- Time constant (τ)
- R × C in seconds: the time for a discharge to fall to 36.8 % of its start, or a charge to reach 63.2 % of the step.
- Bleeder resistor
- A resistor permanently across a capacitor that drains its charge after the power is removed.
- X capacitor
- A safety-rated capacitor connected across the AC line (line to neutral) for EMI filtering. It stays charged at the plug pins after unplugging unless it is discharged.
- Dielectric absorption
- Charge held in the dielectric that returns after a short discharge, so the terminal voltage creeps back up.
- ESR
- Equivalent series resistance: the capacitor’s internal resistance, which adds to R and limits the peak current.
- Constant-current load
- A load that draws the same current regardless of voltage, which discharges a capacitor in a straight line.
Related guides
- RC time constant calculator: τ = RC and the charging curve on its own.
- Capacitor energy calculator: stored energy and hold-up time at constant power.
- Supercapacitor backup time calculator: hold-up with ESR included.
Frequently asked questions
What is the capacitor charging formula?
V(t) = Vs × (1 − e^(−t/RC)) when charging from 0 V through a resistor. The capacitor reaches 63.2 % of the supply after one time constant (τ = RC) and 99.3 % after five.
How do I calculate capacitor charge time?
Through a resistor, rearrange the capacitor charge formula to t = RC × ln((V0 − Vf) ÷ (V − Vf)), where V0 is the starting voltage, V the target and Vf the supply voltage. Charging 1 mF from 0 V toward 24 V through 100 Ω reaches 90 % in 230 ms.
How do I calculate the time for a capacitor to discharge to a given voltage?
t = RC × ln(V0 / V). For 100 µF and 10 kΩ (τ = 1 s), going from 12 V to 1 V takes 1 s × ln 12 = 2.48 s.
How long does a capacitor take to discharge fully?
Through a resistor, never exactly: the voltage approaches zero exponentially. Engineers use 5 τ as "discharged" (0.67 % left) or work out the time to a specific safe voltage with t = RC ln(V0 / V).
How do I size a bleeder resistor?
Use R = t ÷ (C × ln(V0 / Vsafe)) and round down to a standard value. A 220 µF capacitor on a 400 V bus needs at most 131 kΩ to reach 50 V within 60 s; 130 kΩ does it in 59.5 s but dissipates 1.23 W the whole time the bus is up.
How fast does a capacitor have to discharge to be safe?
It depends on the rule that applies to your product. Electrical Construction & Maintenance (2014) quotes US National Electrical Code section 460.6(A) as 50 V, nominal, or less within 1 minute of disconnection, for capacitors rated not over 600 V installed on electrical circuits, and California’s Low-Voltage Electrical Safety Orders (Title 8 §2534.6) set the same limit; check the code edition adopted where you are, and note that a capacitor built into other equipment may fall under that equipment’s standard instead. For X capacitors in appliances, Power Integrations quotes IEC 60335 as under 34 V within 1 s after unplugging.
What is the difference between discharging through a resistor and at constant current?
Through a resistor the current falls with the voltage, so the curve is exponential. A constant-current load (an electronic load or a current-regulated LED driver) pulls the voltage down in a straight line at I / C volts per second, and t = C × ΔV / I. Charging 100 µF to 5 V at 10 mA takes 50 ms.
How much energy is lost charging a capacitor through a resistor?
Charging from 0 V, the resistor turns as much energy into heat as the capacitor stores, whatever its value: half of what the supply delivers. Charging 1 mF to 24 V through 100 Ω heats the resistor by 288 mJ, the same as the 288 mJ stored.
More free PCB tools
See all free electronics tools →
- Capacitor energy calculatorEnergy and charge stored in a capacitor, the usable energy between two voltages, and hold-up time at a constant power.
- RC time constant calculatorTau = RC, the time to charge to any percentage, and the charging curve for your resistor and capacitor.
- LC resonant frequency calculatorResonant frequency of an inductor and capacitor, or the L or C for a frequency, with characteristic impedance.
- Inductive reactance calculatorReactance of an inductor at any frequency, or the inductance for a reactance, with winding resistance and Q.
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- Capacitive reactance calculatorReactance of a capacitor at any frequency, or the capacitance for a reactance, with ESR and a reactance-versus-frequency plot.
Sources
- OpenStax, University Physics Volume 2, §10.5 RC Circuits (charging, discharging and τ = RC)
- HyperPhysics (Georgia State University), Energy stored on a capacitor (half the supply energy is lost in R)
- Electrical Construction & Maintenance, Capacitor Discharge Time Requirements (quotes NEC 460.6(A), 2014)
- California Code of Regulations, Title 8 §2534.6, Drainage of Stored Charge
- Power Integrations CAPZero-3 (CAP300DG) datasheet: IEC 60335 X capacitor discharge, under 34 V in under 1 s
- Power Integrations, Reducing Idle Losses While Meeting IEC 62368-1 (X capacitor discharge resistor losses)
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