When you power down a switch-mode power supply or a variable frequency drive (VFD), the energy stored in the DC bus capacitors does not vanish instantly. Calculating exactly how long that lethal voltage lingers requires the capacitor discharge formula. This equation governs the exponential decay of voltage across a capacitor discharging through a resistive path. Below, we break down the math, track the units through real bench problems, and examine a real-world scenario where ignoring the formula's boundary conditions resulted in a severe shock hazard.
The Core Capacitor Discharge Formula and Symbol Definitions
The voltage across a discharging capacitor at any given time is modeled by an exponential decay function. According to Georgia State University's HyperPhysics, the foundational equation is:
V(t) = V0 × e-t / (R × C)
Because current is directly proportional to voltage in a purely resistive circuit (Ohm's Law), the current decay follows the exact same exponential curve: I(t) = I0 × e-t / (R × C).
| Symbol | Definition | Standard SI Unit |
|---|---|---|
| V(t) | Voltage across the capacitor at time t | Volts (V) |
| V0 | Initial voltage across the capacitor at t = 0 | Volts (V) |
| t | Elapsed time since discharge began | Seconds (s) |
| R | Resistance of the discharge path (bleed resistor or load) | Ohms (Ω) |
| C | Capacitance of the component | Farads (F) |
| e | Euler's number (mathematical constant ≈ 2.71828) | Dimensionless |
| τ (tau) | Time constant (R × C), the time to reach 36.8% of V0 | Seconds (s) |
Rearranged Forms for Bench Calculations
On the workbench, you rarely just solve for V(t). You usually need to size a bleed resistor or calculate a safe wait time. Here are the algebraically rearranged forms:
- Solving for time (t): t = -R × C × ln(V(t) / V0)
- Solving for resistance (R): R = -t / (C × ln(V(t) / V0))
- Solving for capacitance (C): C = -t / (R × ln(V(t) / V0))
- Solving for initial voltage (V0): V0 = V(t) / e-t / (R × C)
Boundary Conditions: When the Formula Applies (And When It Fails)
The capacitor discharge formula assumes an ideal RC circuit. Before you trust the math to keep you from getting shocked, you must understand its assumptions and where real-world physics breaks the model.
Assumption 1: Constant Resistance and Capacitance. The formula assumes R and C do not change during discharge. In reality, the resistance of a wire-wound bleed resistor increases as it heats up from I²R losses, slightly slowing the discharge at the tail end.
Assumption 2: No Parasitic Inductance (ESL). At high frequencies or during rapid, low-resistance discharges (like a spot welder or camera flash), the Equivalent Series Inductance (ESL) of the capacitor creates an underdamped RLC ringing circuit. The simple exponential decay formula fails completely here; you must use differential equations for RLC circuits.
Assumption 3: No Dielectric Absorption (Soakage). This is the most dangerous assumption in high-voltage work. In electrolytic and film capacitors, the dielectric material absorbs some charge during prolonged high-voltage exposure. When you short the terminals to discharge it, the surface charge drains, but the absorbed charge slowly migrates back to the plates over minutes or hours. A capacitor can 'recharge' itself to a hazardous voltage after being fully discharged, a phenomenon well-documented in the All About Circuits DC textbook.
Unit Tracking: Two Solved Bench Problems
The most common reason engineers and technicians get the wrong answer is failing to convert microfarads (μF) to base SI Farads (F). Let's track the units explicitly.
Problem 1: Calculating Time to a Safe Touch Voltage
Scenario: A 400V DC bus uses a 470 μF electrolytic capacitor. A 10 kΩ bleed resistor is placed in parallel. How long does it take to discharge to the SELV (Safety Extra-Low Voltage) threshold of 50V?
- Identify and convert variables to base SI units:
V0 = 400 V
V(t) = 50 V
R = 10,000 Ω
C = 470 μF = 470 × 10-6 F = 0.00047 F - Calculate the time constant (τ):
τ = R × C = 10,000 Ω × 0.00047 F = 4.7 seconds - Apply the rearranged time formula:
t = -τ × ln(V(t) / V0)
t = -4.7 × ln(50 / 400)
t = -4.7 × ln(0.125)
t = -4.7 × (-2.0794) - Final Result:
t = 9.77 seconds
Problem 2: Sizing a Bleed Resistor for a Target Discharge Time
Scenario: You are designing a 230V AC motor drive. The rectified DC bus is 325V. You have a 1000 μF capacitor. Safety standards require the bus to drop below 60V within 3 seconds of power removal. What is the maximum bleed resistor value you can use?
- Identify variables:
V0 = 325 V, V(t) = 60 V, t = 3 s, C = 1000 μF = 0.001 F. - Apply the rearranged resistance formula:
R = -t / (C × ln(V(t) / V0))
R = -3 / (0.001 × ln(60 / 325))
R = -3 / (0.001 × ln(0.1846))
R = -3 / (0.001 × -1.689) - Final Result:
R = -3 / -0.001689 = 1,776 Ω
Bench Note: You would select the next standard lower resistor value (e.g., 1.5 kΩ) to ensure a safety margin, and verify its continuous power dissipation (P = V²/R = 325²/1500 = 70W) requires a heavy-duty chassis-mount power resistor.
Real-World Scenario: The VFD Bus Capacitor Shock Hazard
Math on a whiteboard is clean; the jobsite is not. Here is a documented failure mode involving a 480V AC Variable Frequency Drive (VFD) where the theoretical capacitor discharge formula clashed with physical reality.
The Setup:
A maintenance technician was tasked with replacing a cooling fan on a 50 HP VFD. The VFD's internal DC bus operates at approximately 680V DC, stored in a bank of four 2200 μF, 450V snap-in capacitors (configured in series/parallel for voltage and capacitance requirements). The manufacturer's schematic showed 47 kΩ bleed resistors across each capacitor.
The Numbers:
Using the formula, the effective capacitance of the bank was 2200 μF (0.0022 F) and the effective bleed resistance was 47,000 Ω. The time constant τ = 103.4 seconds. To reach a safe 50V from 680V, the math dictates: t = -103.4 × ln(50/680) = 269 seconds (about 4.5 minutes). The technician locked out the disconnect, waited 10 minutes (more than double the calculated time), and opened the chassis.
The Outcome:
While using a Fluke 87V multimeter to verify the bus was dead, the technician accidentally brushed the bus bar with their bare forearm. They received a severe, muscle-contracting shock. The meter, when properly connected, read 145V DC.
What Went Wrong:
Two physical realities broke the theoretical formula. First, one of the 47 kΩ surface-mount bleed resistors had failed open-circuit due to a prior voltage spike. This removed half the discharge path, doubling the actual time constant. Second, and more critically, dielectric absorption in the large aluminum electrolytic capacitors caused a voltage rebound. When the technician had briefly shorted the bus with a grounding stick 5 minutes prior, they only drained the surface charge. The deep dielectric charge soaked back to the plates, pushing the voltage back up to 145V. The lesson: always verify dead with a meter immediately before touching, and never trust the formula blindly on large electrolytic banks without a continuous shorting stick in place.
Common Unit Mistakes and Realistic Magnitudes
When plugging numbers into the capacitor discharge formula, magnitude errors are the most frequent culprit for blown components or safety hazards. Keep these rules of thumb on your bench:
- The Microfarad Trap: Capacitors are almost always labeled in μF, nF, or pF. The formula demands Farads. 1 μF = 10-6 F. 1 nF = 10-9 F. Forgetting this multiplier will result in a calculated time that is a million times too short.
- The Millisecond Mix-up: Oscilloscope readings and microcontroller timers often output in milliseconds (ms). You must convert ms to seconds (divide by 1000) before using 't' in the formula, or your calculated resistance will be off by three orders of magnitude.
- Realistic Magnitudes (The 5τ Rule): In practical engineering, a capacitor is considered 'fully discharged' for most logic-level applications at 5τ (five time constants), which equates to 0.67% of the initial voltage. If your calculated τ is 2 seconds, expect the discharge to take 10 seconds. If your math tells you a 10 μF cap and a 100 Ω resistor takes 50 seconds to discharge, you have a unit error (actual τ is 0.001s; 5τ is 5ms).
Understanding the capacitor discharge formula is not just about passing an electronics exam; it is about sizing bleed resistors correctly, designing safe power supplies, and knowing exactly when it is safe to put your hands inside a chassis. Track your units, respect the parasitics, and always verify with a meter.
