The core Capacitor Discharge Resistor (CDR) formula calculates the exact resistance required to safely bleed a capacitor down to a target voltage within a specific timeframe. The master equation is R = -t / [C × ln(Vt / V0)]. Whether you are designing an offline switch-mode power supply (SMPS) or a solar inverter, this formula is the mathematical bridge between lethal stored energy and IEC 62368-1 safety compliance.

Below, we break down the derivation, map out the rearranged forms, track units through two bench-ready problems, and examine a real-world failure where the math was right but the component selection was fatally wrong.

The Master CDR Formula and Symbol Definitions

The discharge of a capacitor through a fixed resistor follows an exponential decay curve. By rearranging the standard RC time-domain equation, we isolate the resistance needed to hit a specific voltage threshold at a specific time.

R = -t / [C × ln(Vt / V0)]

Symbol Parameter Standard Unit Notes
R Bleeder Resistance Ohms (Ω) Must handle steady-state power dissipation
t Discharge Time Seconds (s) IEC often mandates 1s for plug pins, 3-5s for internal
C Capacitance Farads (F) Requires conversion from µF or mF
Vt Target Voltage Volts (V) Safe touch threshold (typically 60V DC or 42.4V peak)
V0 Initial Voltage Volts (V) Maximum nominal bus voltage before power-off
ln Natural Logarithm Dimensionless Base-e log, NOT base-10

Assumptions and Realistic Magnitudes

This formula assumes an ideal, isolated RC network. It ignores Equivalent Series Resistance (ESR), capacitor leakage current, and any parallel PCB trace leakage. In practice, these parasitic elements only aid the discharge, meaning your calculated R value is a conservative maximum limit.

Realistic Magnitudes: For offline AC/DC power supplies (400V DC bus), expect R to fall between 100 kΩ and 1 MΩ, C between 10 µF and 100 µF, and t between 1 and 5 seconds.

Rearranged Forms: Solving for Every Variable

On the bench, you rarely solve for R alone. You might need to verify how long an existing prototype takes to discharge, or determine the maximum allowable bus capacitance for a given safety standard. Here are the algebraic rearrangements:

  • Solving for Time (t): t = -R × C × ln(Vt / V0)
  • Solving for Capacitance (C): C = -t / [R × ln(Vt / V0)]
  • Solving for Target Voltage (Vt): Vt = V0 × e(-t / RC)
  • Solving for Initial Voltage (V0): V0 = Vt × e(t / RC)

Unit Mistakes That Break the Math

The CDR formula is unforgiving of lazy unit tracking. If your calculator spits out a resistance of 0.0004 Ω or 4,000,000,000 Ω, you have fallen victim to one of these three traps:

  1. The Microfarad Trap: Datasheets list capacitance in µF. The formula demands Farads. You must multiply your µF value by 10-6. Entering "47" instead of "0.000047" will yield a resistor value one million times too small.
  2. The Logarithm Confusion: Your calculator's log button is base-10. The formula requires the natural logarithm, which is base-e. You must use the ln button. Using base-10 will skew your result by a factor of 2.302.
  3. Time in Milliseconds: Safety standards might state "discharge within 500 ms". You must convert this to 0.5 seconds before plugging it into the equation.

Worked Problem 1: Sizing a Bleeder for a 400V DC Bus

Scenario: You are designing the primary side of a 250W offline SMPS. The rectified 230V AC mains yields a nominal 400V DC bus. You are using a 10 µF bulk capacitor. IEC 62368-1 guidance requires the internal bus to discharge to 60V within 3 seconds of power removal.

Step 1: Convert and Define Variables

  • V0 = 400 V
  • Vt = 60 V
  • C = 10 µF = 0.00001 F
  • t = 3 s

Step 2: Calculate the Natural Log

  • ln(Vt / V0) = ln(60 / 400) = ln(0.15)
  • ln(0.15) ≈ -1.8971

Step 3: Solve for R

  • R = -3 / [0.00001 × -1.8971]
  • R = -3 / -0.000018971
  • R ≈ 158,136 Ω

Verification & Power Rating: Select the next standard E24 value up to ensure faster discharge: 160 kΩ. Now, calculate steady-state power dissipation while the unit is running: P = V² / R = 400² / 160,000 = 160,000 / 160,000 = 1.0 Watt. You must use a minimum 2W rated metal oxide or wirewound resistor to provide a safety margin.

Worked Problem 2: Calculating Discharge Time for an Existing Circuit

Scenario: You are auditing a legacy motor drive board. The DC bus runs at 325V (rectified 230V AC). The bulk capacitance is 22 µF, and the existing bleeder resistor is 220 kΩ. You need to know how long it takes to reach a safe 50V.

Step 1: Define Variables

  • V0 = 325 V
  • Vt = 50 V
  • C = 22 µF = 0.000022 F
  • R = 220 kΩ = 220,000 Ω

Step 2: Calculate the RC Time Constant (τ)

  • τ = R × C = 220,000 × 0.000022 = 4.84 seconds

Step 3: Solve for t

  • t = -τ × ln(Vt / V0)
  • t = -4.84 × ln(50 / 325)
  • t = -4.84 × ln(0.1538)
  • t = -4.84 × -1.8718
  • t ≈ 9.06 seconds

Outcome: A 9-second discharge time is generally unacceptable for user-accessible terminals under modern safety standards. The board requires a lower resistance value or an active bleed circuit.

Real-World Scenario: The Smoked Resistor

Bench War Story: When the Math is Right, but the Physics Bites Back

Setup: A junior engineer was tasked with adding a bleeder resistor to an 800V DC bus (a solar inverter PFC stage) with a massive 470 µF capacitor bank. The safety requirement was to discharge to 60V within 5 minutes (300 seconds) to allow for safe servicing.

Numbers: Using the CDR formula: R = -300 / [0.00047 × ln(60/800)]. The math yielded 246,548 Ω. The engineer selected a standard 250 kΩ through-hole carbon film resistor and soldered it across the bus.

Outcome: The moment the 800V bus was energized, the resistor violently popped, vented magic smoke, and failed open-circuit. The capacitor remained fully charged at 800V indefinitely, creating a lethal shock hazard for the next technician.

What Went Wrong: The engineer solved for Resistance but ignored Power Dissipation. At 800V, a 250 kΩ resistor dissipates P = 800² / 250,000 = 2.56 Watts continuously. A standard 1/4W or 1/2W carbon film resistor will thermally destruct within seconds at that load. The Fix: Use a 5W chassis-mount wirewound resistor, or better yet, a series string of three 82 kΩ 1W resistors to divide both the voltage stress and the thermal load across the PCB.

When the Formula Applies (and When It Doesn't)

The passive CDR formula is the gold standard for simple, isolated RC networks. However, you must abandon it under the following conditions:

  • Active Bleed Circuits: If your design uses a MOSFET or BJT to switch the bleeder resistor on only when AC is removed (to avoid steady-state power loss), the discharge curve is no longer a simple exponential from t=0. You must calculate the turn-on delay and adjust your time variable.
  • Parallel Loads: If the capacitor is feeding a microcontroller or a high-impedance voltage divider that remains active during power-down, that parallel resistance alters the equivalent discharge R. You must use the Thevenin equivalent resistance of the entire network.
  • Massive ESR / Dielectric Absorption: In older electrolytic capacitors or specific film chemistries, dielectric absorption can cause the voltage to "rebound" after the initial discharge. The formula assumes the voltage stays at Vt; in reality, a rebound might push it back above the 60V safety threshold. Always verify with a multimeter on the bench.

Safety Caveat: High-voltage DC bus capacitors store lethal energy. Never rely solely on calculated bleeder resistors for personal safety. Always de-energize the system, wait the calculated time, and verify the terminals are dead using a Category III or IV rated multimeter before touching the circuit. For formal product certification, consult the latest IEC safety standards and your local Authority Having Jurisdiction (AHJ).

For a deeper dive into the physics governing these curves, review the foundational RC time constant principles and application notes from major component manufacturers regarding power supply safety design.