The discharge equation for a capacitor is V(t) = V₀e^(-t/RC). It defines exactly how voltage decays across a capacitor when it discharges through a resistive path. Whether you are sizing a bleeder resistor for a high-voltage power supply, calculating the safe-to-touch time for a motor drive, or debugging a camera flash circuit, this formula is the bedrock of RC timing and safety analysis. Below, we break down the derivation, track units through solved problems, and explore a real-world bench failure where the math didn't match reality.
The Core Discharge Equation and Its Assumptions
When a charged capacitor is connected across a resistor, the current flow is initially high and decays exponentially as the voltage drops. The governing equation for the voltage at any given time t is:
V(t) = V₀ × e^(-t / RC)
| Symbol | Parameter | Standard Unit | Description |
|---|---|---|---|
| V(t) | Voltage at time t | Volts (V) | The remaining voltage across the capacitor after time t has elapsed. |
| V₀ | Initial Voltage | Volts (V) | The starting voltage across the capacitor at t = 0. |
| e | Euler's Number | Dimensionless | Mathematical constant approximately equal to 2.71828. |
| t | Time | Seconds (s) | The elapsed time since the discharge path was established. |
| R | Resistance | Ohms (Ω) | The total resistance of the discharge path (including ESR and bleeder). |
| C | Capacitance | Farads (F) | The total capacitance of the bank or component. |
| τ (tau) | Time Constant | Seconds (s) | The product of R × C. At 1τ, the capacitor has discharged by 63.2%. |
When the Formula Applies (and When It Doesn't)
This equation assumes an ideal capacitor and a constant resistance. It applies perfectly to bench setups with film capacitors and standard carbon-film resistors. It begins to break down in high-voltage electrolytic banks due to dielectric absorption (where the dielectric material slowly releases trapped charge back into the circuit after the initial RC decay) and in circuits where the discharge path includes non-linear components like diodes or transistors. For safety-critical applications, the industry standard is to wait 5τ (five time constants), at which point the capacitor is considered 99.3% discharged.
Rearranged Forms and Critical Unit Mistakes
On the bench, you rarely just solve for V(t). You usually need to size a resistor (R) or calculate a wait time (t). Here are the algebraically rearranged forms of the discharge equation:
- Solve for time (t): t = -RC × ln(V(t) / V₀)
- Solve for resistance (R): R = -t / (C × ln(V(t) / V₀))
- Solve for capacitance (C): C = -t / (R × ln(V(t) / V₀))
- Solve for initial voltage (V₀): V₀ = V(t) / e^(-t/RC)
The Unit Trap: What Breaks the Math
The most common reason a calculated bleeder resistor fails on the bench is a unit conversion error. The natural logarithm (ln) and Euler's number (e) are dimensionless, meaning the exponent (-t / RC) must also be dimensionless. If you plug in milliseconds for time and microfarads for capacitance without converting to base SI units, your exponent will be off by orders of magnitude.
Worked Problem 1: Sizing a Bleeder Resistor for a 500V Supply
Scenario: You are designing a linear power supply with a 500V DC output and a 1000 µF main filter capacitor. Safety regulations require the output to drop below 50V within 60 seconds of power-off. What bleeder resistor value do you need, and what wattage rating should it have?
- Identify and convert variables:
V₀ = 500V
V(t) = 50V
t = 60s
C = 1000 µF = 0.001 F - Select the rearranged formula for R:
R = -t / (C × ln(V(t) / V₀)) - Plug in the numbers:
R = -60 / (0.001 × ln(50 / 500))
R = -60 / (0.001 × ln(0.1)) - Calculate the natural log:
ln(0.1) ≈ -2.30258 - Finish the calculation:
R = -60 / (0.001 × -2.30258)
R = -60 / -0.00230258
R ≈ 26,057 Ω
Component Selection & Power Check: The nearest standard E24 resistor value is 27 kΩ. But we must check the power dissipation. While the supply is ON, the resistor constantly dissipates heat: P = V² / R = 500² / 27,000 = 250,000 / 27,000 = 9.25 Watts. A standard 5W resistor will overheat and fail open-circuit, leaving the capacitor charged and creating a lethal shock hazard. You must use a 15W or 20W wirewound resistor, or put three 91kΩ 5W resistors in parallel to share the thermal load.
Worked Problem 2: Calculating Safe-to-Touch Time for a Motor Drive
Scenario: You are servicing a 400V DC bus on a servo drive. The bus uses two 2200 µF capacitors in series (yielding a total capacitance of 1100 µF). The factory-installed bleeder resistor is 150 kΩ. How long must you wait after killing the main breaker before the bus drops to a safe 50V?
- Identify and convert variables:
V₀ = 400V
V(t) = 50V
R = 150,000 Ω
C = 1100 µF = 0.0011 F - Select the rearranged formula for t:
t = -RC × ln(V(t) / V₀) - Calculate the time constant (τ = RC):
τ = 150,000 × 0.0011 = 165 seconds - Plug into the full equation:
t = -165 × ln(50 / 400)
t = -165 × ln(0.125) - Calculate the natural log and final time:
ln(0.125) ≈ -2.0794
t = -165 × -2.0794
t ≈ 343.1 seconds (approx. 5.7 minutes)
Practical Takeaway: The math says 5.7 minutes. However, as a best practice on the bench, you never trust the math alone for high-voltage DC buses. After waiting 6 minutes, you must verify the voltage with a Category III/IV rated multimeter before touching any busbars.
Real-World Bench Scenario: The VFD Shock Hazard
Formulas assume ideal conditions. Real-world components do not. Here is a scenario that highlights why understanding the limitations of the discharge equation is critical for your safety.
The Setup
A technician is troubleshooting a 480V AC input Variable Frequency Drive (VFD). The rectified DC bus sits at roughly 650V DC. The schematic shows a single 220 kΩ bleeder resistor wired directly across a 3300 µF electrolytic capacitor bank. The tech calculates the time to reach 50V using the standard formula:
t = -(220,000 × 0.0033) × ln(50 / 650) = -726 × -2.565 = 1,862 seconds (31 minutes).
The Outcome
The tech locks out the main disconnect, sets a timer for 35 minutes to give a safety margin, and then opens the chassis. Assuming the bus is dead, they reach in to check a gate drive ribbon cable. Their forearm brushes the DC busbar, resulting in a severe shock and a minor arc flash. The multimeter later reads 140V across the bus.
What Went Wrong?
Two distinct physical realities broke the ideal math:
- Dielectric Absorption: Large aluminum electrolytic capacitors suffer from dielectric absorption. The insulating oxide layer 'soaks' up charge during prolonged high-voltage operation. When the main voltage is removed and the capacitor discharges through the bleeder, the surface charge dissipates according to the RC time constant. However, the deep dielectric charge slowly migrates back to the plates over time, causing the terminal voltage to 'rebound' after the initial discharge. Standard RC tutorials often omit this, but it is a known hazard in high-voltage repair.
- Component Drift: The 220 kΩ bleeder resistor had been subjected to years of thermal cycling inside the hot VFD chassis. Carbon composition and thick-film resistors can drift significantly upward in value when stressed. The actual resistance had drifted to over 350 kΩ, pushing the real-world time constant far beyond the calculated 31 minutes.
Mastering the discharge equation for capacitors means knowing how to crunch the numbers flawlessly, but more importantly, knowing exactly when the physical components on your bench will defy those numbers. Treat the math as your baseline, and your multimeter as your final authority.
