To calculate capacitor discharge time to a safe voltage, use the rearranged time formula: t = -R × C × ln(Vt / V0). For a standard 400V DC bus discharging to the 50V NFPA 70E safe-touch limit, a 220kΩ bleeder resistor across a 470µF capacitor takes roughly 214 seconds. If you are designing a bleeder network and need the resistor value to hit a specific time target, use R = -t / (C × ln(Vt / V0)).
The Core Capacitor Discharge Formula & Symbol Definitions
The foundational equation governing the voltage decay of a capacitor discharging through a fixed resistor is derived from Kirchhoff’s Voltage Law and the definition of capacitance (I = C × dV/dt). As the capacitor releases its stored energy, the voltage drops exponentially, not linearly. This is the engine behind any reliable capacitor discharge calculator.
Vt = V0 × e(-t / (R × C))
| Symbol | Parameter | Standard Unit | Description |
|---|---|---|---|
| Vt | Voltage at time t | Volts (V) | The target voltage remaining on the capacitor terminals. |
| V0 | Initial Voltage | Volts (V) | The starting voltage across the capacitor at t=0. |
| e | Euler’s Number | Constant | Mathematical constant approximately equal to 2.71828. |
| t | Time | Seconds (s) | The elapsed discharge time. |
| R | Resistance | Ohms (Ω) | The resistance of the discharge path (bleeder resistor). |
| C | Capacitance | Farads (F) | The nominal capacitance value. |
Rearranged Forms for Circuit Design
On the bench, you rarely know all variables and just need to find ‘Vt’. Usually, you are designing a bleeder circuit and need to solve for R or t. Here are the algebraically rearranged forms of the core equation:
- Solve for Time (t):
t = -R × C × ln(Vt / V0) - Solve for Resistance (R):
R = -t / (C × ln(Vt / V0)) - Solve for Capacitance (C):
C = -t / (R × ln(Vt / V0)) - Solve for Initial Voltage (V0):
V0 = Vt / e(-t / (R × C))
Note: ‘ln’ represents the natural logarithm (base e). Ensure your calculator is set to natural log, not log base 10.
Unit Traps That Break Your Calculations
The most common reason a capacitor discharge calculator yields a wildly incorrect answer is a unit mismatch. The formula demands strict SI base units. If you feed it the units printed on the component’s jacket, your answer will be off by orders of magnitude.
| Component Marking | Wrong Input (Breaks Math) | Correct SI Input | Multiplier |
|---|---|---|---|
| 470 µF | 470 | 0.00047 | × 10-6 |
| 220 kΩ | 220 | 220,000 | × 103 |
| 50 ms (milliseconds) | 50 | 0.050 | × 10-3 |
The Reality Check: If your calculated time ‘t’ comes out to 0.00004 seconds for a massive motor-run capacitor, you forgot to convert microfarads to farads. If your calculated resistor ‘R’ is 12 Ω for a 400V bus, you forgot to convert microfarads, and you are about to specify a resistor that will instantly vaporize.
Worked Problem 1: Sizing a Bleeder Resistor for a 400V DC Bus
Scenario: You are designing a power supply with a 400V DC bus. The main filter capacitor is 470µF. Safety standards require the bus to discharge to 50V or less within 60 seconds of power removal. What standard resistor value do you need, and what is its minimum power rating?
Step 1: Identify knowns and convert to SI units.
- V0 = 400 V
- Vt = 50 V
- t = 60 s
- C = 470 µF = 0.00047 F
Step 2: Select the rearranged formula for R.
R = -t / (C × ln(Vt / V0))
Step 3: Execute the math with intermediate tracking.
- Vt / V0 = 50 / 400 = 0.125
- ln(0.125) = -2.07944
- Denominator: C × ln(Vt / V0) = 0.00047 × -2.07944 = -0.0009773
- R = -60 / -0.0009773 = 61,393 Ω
Step 4: Select the standard E24 resistor value and calculate power.
The closest standard E24 value higher than 61.3kΩ is 62kΩ (or 68kΩ for a wider safety margin). Let’s use 68kΩ to ensure we easily beat the 60-second mark.
Peak power dissipation occurs at t=0: P = V02 / R = 4002 / 68,000 = 160,000 / 68,000 = 2.35 Watts.
Result: Specify a 68kΩ resistor rated for at least 5W (derating by 50% for reliability).
Worked Problem 2: Calculating Safe-Touch Time for a Motor Run Capacitor
Scenario: An HVAC control board uses a 50µF motor run capacitor charged to 240V AC (peak DC ~340V, but we will calculate based on a nominal 240V RMS equivalent DC charge for conservative baseline, or assume it was charged to 240V DC in a test jig). The board has a fixed 220kΩ bleeder resistor. How long until it reaches the 50V safe-touch threshold?
Step 1: Identify knowns and convert to SI units.
- V0 = 240 V
- Vt = 50 V
- R = 220 kΩ = 220,000 Ω
- C = 50 µF = 0.00005 F
Step 2: Select the rearranged formula for t.
t = -R × C × ln(Vt / V0)
Step 3: Execute the math.
- Calculate the RC time constant (τ): 220,000 × 0.00005 = 11 seconds.
- Calculate the voltage ratio: 50 / 240 = 0.20833.
- Calculate the natural log: ln(0.20833) = -1.5686.
- Multiply: t = -11 × -1.5686 = 17.25 seconds.
Result: The capacitor reaches 50V in 17.25 seconds. According to standard RC discharging circuit theory, it takes roughly 5 time constants (55 seconds) to reach <1% voltage (2.4V), but the safe-touch threshold is crossed much earlier.
Decision Tree: Picking Your Exact Bleeder Resistor
Calculating the ohms is only half the battle. High-voltage capacitor discharge generates a massive instantaneous thermal pulse. Standard thin-film SMD or 1/4W carbon film resistors will crack, arc over, or drift permanently out of spec. Use this decision path to select the correct physical component.
| Condition / Circuit Parameter | Action / Requirement | Concrete Component Pick |
|---|---|---|
| IF V0 < 50V AND Energy < 0.5 Joules | Standard metal film is sufficient. Focus on standard 1% tolerance. | Vishay MRS25 (1/4W Metal Film) |
| IF V0 is 120V-240V AND continuous dissipation < 0.5W | Require high-voltage rating to prevent internal arcing. Standard 1/2W is fine. | Vishay HVR37 (1/2W, 3500V peak Metal Glaze) |
| IF V0 > 300V AND Energy > 5 Joules (e.g., 400V bus, >100µF) | Must handle high surge energy without cracking. Carbon composition or thick film pulse resistors are mandatory. | Stackpole CB220KJT (220kΩ, 2W Carbon Comp) |
| IF continuous power dissipation > 2W | Use chassis-mount aluminum housed resistors bolted to the enclosure for heatsinking. | Ohmite RHA Series (Wirewound, Chassis Mount) |
Realistic Magnitudes and Bench Realities
When plugging numbers into a capacitor discharge calculator, you should have a mental model of what a ‘normal’ answer looks like. If your results fall outside these realistic magnitudes, re-check your inputs.
- Bleeder Resistance: Typically falls between 100kΩ and 1MΩ. Values below 10kΩ will waste too much continuous power as heat during normal operation. Values above 2MΩ take too long to discharge and are highly susceptible to PCB flux leakage currents overriding the resistor.
- Discharge Time: For mains-connected equipment, safety standards (like IEC 62368-1) often require the plug pins to drop below 60V within 1 to 2 seconds. For internal DC bus capacitors, a 30-second to 300-second discharge to 50V is standard practice.
- Dielectric Absorption (Soakage): If you short a large electrolytic capacitor with a screwdriver, spark it to zero, and remove the screwdriver, the voltage will slowly creep back up to 10V-20V over the next hour. This is dielectric absorption. A bleeder resistor prevents this rebound, which is why a resistor is always superior to a manual discharge tool for leaving a circuit in a safe state.
Always verify your calculated discharge time on the bench. Charge the capacitor to nominal voltage, kill the power, and use a high-impedance digital multimeter to log the voltage at t=1s, t=10s, and your calculated target time. If the measured voltage is more than 10% higher than your calculated Vt, your capacitor’s actual leakage or ESR is altering the RC curve, and you must step down to the next lower standard resistor value.
