The discharge of capacitor equation is V(t) = V0 · e-t/RC. This exponential decay formula dictates exactly how quickly a stored electrical charge dissipates through a resistive path. Whether you are designing a safe high-voltage DC bus for a variable frequency drive (VFD) or sizing a bleed resistor for a tube amplifier power supply, this equation is your primary tool for preventing lethal shocks and component damage. Below is the complete derivation framework, unit-tracked worked examples, and a concrete decision path for selecting physical components.
The Core Discharge of Capacitor Equation & Symbol Definitions
The standard formula for the voltage across a discharging capacitor at any given time is:
V(t) = V0 · e-t / (R · C)
| Symbol | Parameter | Standard Unit | Practical Notes |
|---|---|---|---|
| V(t) | Voltage at time t | Volts (V) | The target voltage you are solving for (e.g., safe touch voltage). |
| V0 | Initial voltage | Volts (V) | The voltage across the capacitor at t = 0 (the moment discharge begins). |
| e | Euler's number | Dimensionless | Mathematical constant ≈ 2.71828. The base of the natural logarithm. |
| t | Time elapsed | Seconds (s) | Must be converted from milliseconds or minutes before calculating. |
| R | Discharge resistance | Ohms (Ω) | The total resistance in the discharge path, including ESR and bleed resistors. |
| C | Capacitance | Farads (F) | Must be converted from μF or mF. 1 μF = 1 × 10-6 F. |
| τ (tau) | RC Time Constant | Seconds (s) | τ = R · C. Represents the time to discharge to 36.8% of V0. |
Rearranged Forms: Solving for Any Variable
In bench and jobsite scenarios, you rarely solve for V(t). You usually know your target safety voltage and need to size the resistor or calculate the wait time. Here are the algebraically rearranged forms:
- Solve for Time (t):
t = -R · C · 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 / (R · C)
Note: "ln" represents the natural logarithm (base e). Ensure your calculator is not set to base-10 log.
When the Formula Applies (and When It Fails)
This equation models an ideal RC circuit. It assumes constant resistance and capacitance, a purely resistive discharge path (no parasitic inductance), and ignores dielectric absorption.
At 1τ, the voltage drops to 36.8% of V0.
At 3τ, it drops to 5.0%.
At 5τ, it drops to 0.67%. In industrial safety standards like IEC 62368-1, 5τ is universally accepted as "fully discharged" for hazard analysis. If your 5τ time is 10 seconds, the capacitor is considered safe at 10 seconds, even if mathematically it never reaches absolute zero volts.
When it fails: If you are discharging a massive aluminum electrolytic capacitor bank at high speeds, the Equivalent Series Resistance (ESR) and the physical inductance of the busbars will alter the curve. Furthermore, electrolytic capacitance drops significantly at low temperatures; a 1000μF cap rated at 20°C might act like a 600μF cap at -20°C, discharging faster than your math predicts.
Worked Example 1: Calculating Safe Touch Time
Scenario: You have a 400V DC motor drive bus with a 1000μF filter capacitor. The manufacturer installed a 100kΩ bleed resistor. How long must a technician wait after power-off before the bus reaches the 60V safe-touch threshold?
- Identify and convert variables:
V0 = 400 V
V(t) = 60 V
C = 1000 μF = 0.001 F
R = 100 kΩ = 100,000 Ω - Calculate the time constant (τ):
τ = R · C = 100,000 Ω · 0.001 F = 100 seconds - Apply the rearranged time formula:
t = -τ · ln(V(t) / V0)
t = -100 s · ln(60 V / 400 V) - Execute the math:
t = -100 s · ln(0.15)
t = -100 s · (-1.8971)
t = 189.7 seconds (approx. 3 minutes, 10 seconds)
Verification: 189.7 seconds is roughly 1.9τ. At 2τ, voltage should be ~13.5% of 400V (54V). Our answer of 60V at 1.9τ aligns perfectly with the exponential curve.
Worked Example 2: Sizing a Bleed Resistor for a 400V Bus
Scenario: You are designing a 400V power supply with a 100μF output capacitor. IEC 62368-1 safety guidelines require the voltage to drop below 50V within 2 seconds of unplugging. What resistance value do you need, and what physical part should you buy?
- Identify variables:
V0 = 400 V
V(t) = 50 V
t = 2 s
C = 100 μF = 0.0001 F - Apply the rearranged resistance formula:
R = -t / [C · ln(V(t) / V0)]
R = -2 s / [0.0001 F · ln(50 / 400)] - Execute the math:
R = -2 / [0.0001 · ln(0.125)]
R = -2 / [0.0001 · (-2.0794)]
R = -2 / -0.00020794 = 9,618 Ω - Select standard value:
To guarantee discharge in under 2 seconds, we must round down to the nearest E24 standard value: 9.1 kΩ. - Calculate continuous power dissipation:
P = V02 / R = (400 V)2 / 9,100 Ω = 160,000 / 9,100 = 17.58 Watts.
Because the resistor is constantly connected across a 400V source during operation, it must dissipate 17.58W continuously. Applying a standard 50% safety derating for enclosed chassis environments, we need a resistor rated for at least 35W.
Unit Traps That Break Your Math
The most common reason RC circuit calculations fail on the bench is unit mismatch. The product of R and C must yield seconds.
Ohms (Ω) × Farads (F) = Seconds (s).
If you multiply kΩ by μF, you get milliseconds (ms).
If you multiply MΩ by μF, you get seconds (s).
Always convert to base SI units (Ω, F, s) before plugging values into the natural log equations to avoid magnitude errors.
Bleed Resistor Decision Tree: Picking the Exact Part
Calculating the ohms and watts is only half the job. High-voltage bleed resistors suffer from voltage coefficient drift, internal arcing, and thermal runaway if the wrong physical package is chosen. Use this decision matrix to select your exact component.
| Condition / Constraint | If True → Action | Concrete Part Recommendation |
|---|---|---|
| Bus voltage < 50V DC, Power < 2W | Use standard 1/4W or 1/2W axial metal film resistors. No special high-voltage rating needed. | Vishay PR02 series (2W metal film) or standard Yageo CFR-25. |
| Bus voltage 50V - 250V, Power 2W - 10W | Use metal oxide film resistors. Ensure the working voltage rating exceeds V0 to prevent internal arcing. | Vishay PR03 (3W, 750V max working voltage) or Xicor MOX series. |
| Bus voltage > 250V, Power > 10W (Continuous) | Must use chassis-mount wirewound resistors bolted to a heatsink. Derate by 50% for enclosed spaces. | Ohmite 160 Series (e.g., 160-50W wirewound, rated for 460V continuous). Bolt to chassis with thermal paste. |
| Discharge time required is < 1 second for high-energy banks (>500μF at >400V) | Passive resistors will overheat. Use an active discharge circuit (MOSFET switched across a lower-value resistor only when power is lost). | Active Bleed IC: Power Integrations CAPZero-3 (automatically disconnects bleed path during normal operation, connects on power loss). |
Final Default Recommendation: For the vast majority of 120V/240V AC rectified DC bus applications (approx 340V DC peak) requiring 2W to 10W of continuous dissipation, the Vishay PR03 metal oxide series is the optimal default pick. It handles the high continuous voltage without the parasitic inductance issues of wirewound resistors, and it costs under $1.00 in volume.
