The capacitor discharge equation is V(t) = V0 × e-t/RC. It predicts the exact voltage remaining across a capacitor as it bleeds off stored energy through a resistive path over time. Whether you are sizing a bleeder resistor for a 400V switch-mode power supply or calculating the timing interval for a 555 timer monostable circuit, this exponential decay formula is the foundational math governing RC (resistor-capacitor) networks.
On the bench, abstract theory only gets you so far. To use this formula safely and accurately, you need strict unit tracking, an understanding of parasitic effects like dielectric absorption, and a clear picture of what realistic time constants look like in actual hardware. Below, we break down the math, rearrange it for practical design work, and walk through real-world scenarios where the math meets physical components.
The Core Capacitor Discharge Equation and Symbol Definitions
The standard voltage decay formula for an isolated RC circuit is:
To use this equation without making catastrophic calculation errors, every variable must be converted to its base SI unit before plugging it in. Here is the definitive spec-sheet table for the formula's symbols.
| Symbol | Definition | Base SI Unit | Common Bench Units to Convert |
|---|---|---|---|
| V(t) | Voltage across the capacitor at time t | Volts (V) | Millivolts (mV) |
| V0 | Initial voltage across the capacitor at t = 0 | Volts (V) | Kilovolts (kV) |
| e | Euler's number (mathematical constant) | ~2.71828 | N/A |
| t | Elapsed discharge time | Seconds (s) | Milliseconds (ms), Microseconds (μs) |
| R | Resistance of the discharge path | Ohms (Ω) | Kilo-ohms (kΩ), Mega-ohms (MΩ) |
| C | Capacitance of the component | Farads (F) | Microfarads (μF), Nanofarads (nF), Picofarads (pF) |
| τ (tau) | Time constant (where τ = R × C) | Seconds (s) | N/A |
Reference: For a deeper physics-level derivation of the exponential decay constant, consult the HyperPhysics RC circuit module hosted by Georgia State University.
Rearranging the Formula: Solving for Time, Resistance, and Capacitance
In practical circuit design, you rarely just solve for V(t). Usually, you have a target voltage and a known capacitor, and you need to size the bleeder resistor or calculate the exact delay time. By applying natural logarithms (ln), we can rearrange the capacitor discharge equation to isolate any variable.
- Solve for time (t):
t = -R × C × ln(V(t) / V0)
Use case: Calculating how long a power supply takes to reach a safe-to-touch voltage after being unplugged. - Solve for resistance (R):
R = -t / (C × ln(V(t) / V0))
Use case: Sizing a bleeder resistor to ensure a 400V bus drops below 60V within 5 seconds. - Solve for capacitance (C):
C = -t / (R × ln(V(t) / V0))
Use case: Selecting a timing capacitor for a microcontroller reset delay circuit. - Solve for initial voltage (V0):
V0 = V(t) / e(-t/RC)
Use case: Forensic troubleshooting to determine what voltage a capacitor was charged to based on its current decay state.
Bench Work: Two Solved Problems with Strict Unit Tracking
The most common reason hobbyists and junior engineers get burned—sometimes literally—by RC math is failing to convert microfarads and milliseconds into base Farads and seconds. Here are two worked examples demonstrating strict unit tracking.
Problem 1: Finding the Voltage at a Specific Time
Setup: You have a 470 μF electrolytic capacitor (Nichicon UFW series) charged to 400V DC. It has a 10 kΩ bleeder resistor soldered directly across its terminals. What is the voltage across the capacitor 5 seconds after the power is cut?
- Convert to base units:
C = 470 μF = 470 × 10-6 F = 0.00047 F
R = 10 kΩ = 10 × 103 Ω = 10,000 Ω
V0 = 400 V
t = 5 s - Calculate the time constant (τ):
τ = R × C = 10,000 × 0.00047 = 4.7 seconds - Apply the discharge equation:
V(5) = 400 × e(-5 / 4.7)
V(5) = 400 × e(-1.0638)
V(5) = 400 × 0.3451 - Final Answer: 138.04 V
Bench takeaway: After 5 seconds, the capacitor still holds a lethal 138V. One time constant (4.7s) only drops the voltage to ~36.8% of its initial value.
Problem 2: Finding the Time to Reach a Safe Voltage
Setup: Using the exact same circuit from Problem 1, how many seconds must you wait for the voltage to drop to 50V (a generally accepted safe-touch threshold for dry skin)?
- Identify knowns (base units):
τ (RC) = 4.7 s
V(t) = 50 V
V0 = 400 V - Use the rearranged time formula:
t = -RC × ln(V(t) / V0) - Execute the math:
t = -4.7 × ln(50 / 400)
t = -4.7 × ln(0.125)
t = -4.7 × (-2.0794) - Final Answer: 9.77 seconds
Bench takeaway: It takes roughly 2.08 time constants to drop to 12.5% of the initial voltage. For a complete discharge (below 1% of V0), the industry rule of thumb is to wait 5τ (5 × 4.7 = 23.5 seconds).
Real-World Scenario: When the Math Meets a Camera Flash Circuit
Math on paper assumes ideal components. Physical capacitors on a workbench have parasitic properties that can bite you if you only rely on the basic capacitor discharge equation. Let us walk through a real-world failure scenario involving a vintage DSLR camera flash repair.
The Setup
An engineer is troubleshooting the IGBT gate driver on a flash board featuring a 330 μF, 350V Cornell Dubilier photoflash capacitor. The board has a 100 kΩ surface-mount bleeder resistor. The engineer unplugs the unit, waits exactly 60 seconds, and assumes the board is safe to probe because "a minute is plenty of time for a small cap to drain."
The Numbers
Let us run the math the engineer should have done:
τ = 100,000 Ω × 0.00033 F = 33 seconds.
At t = 60 seconds:
V(60) = 350 × e(-60 / 33) = 350 × e(-1.818) = 350 × 0.162 = 56.7V.
The Outcome and What Went Wrong
At 60 seconds, the capacitor still held nearly 57V. While 57V might not be lethal to a healthy adult, it was more than enough to forward-bias the parasitic body diode of the engineer's multimeter probe and short out the sensitive IGBT gate when the probe slipped, destroying a $15 surface-mount transistor.
But there is a second, deeper failure mode: Dielectric absorption (often called "soakage"). In aluminum electrolytic capacitors, the dielectric oxide layer traps charge dipoles. If the engineer had used a discharge stick to short the capacitor to 0V at 60 seconds, and then removed the stick, the capacitor would have spontaneously "recharged" itself to 10V or 20V over the next few minutes as the trapped dipoles relaxed. The basic discharge equation does not model dielectric absorption. For deep technical data on this phenomenon, refer to the RC Discharge Circuit tutorials and manufacturer application notes on electrolytic soakage.
Assumptions, Unit Traps, and Realistic Magnitudes
To use the capacitor discharge equation effectively, you must understand its boundaries. Here is a breakdown of the assumptions baked into the math, the unit traps that ruin calculations, and what realistic numbers look like on the bench.
When the Formula Applies (and Its Assumptions)
The equation V(t) = V0e-t/RC assumes an ideal, isolated RC loop. Specifically, it assumes:
- Constant Resistance: The resistor value does not change with voltage or temperature. (If your discharge path includes a voltage-dependent resistor like an MOV, or a thermistor that heats up and changes value, the simple exponential decay model breaks down).
- No Parasitic Inductance (ESL): It ignores Equivalent Series Inductance. At very high speeds (nanoseconds), the ESL of the capacitor and the PCB traces will cause ringing, meaning the voltage does not decay in a smooth exponential curve.
- No Dielectric Absorption: As noted in the camera flash scenario, it assumes the dielectric material releases all charge instantly, which is false for electrolytic and tantalum capacitors.
- No External Leakage: It assumes the only path for current is the intentional resistor R, ignoring the capacitor's internal leakage current and PCB surface contamination.
The Unit Mistakes That Break the Math
If your calculated discharge time is off by a factor of a million, you fell into the microfarad trap. Inline Data Highlight: 1 Farad is a massive amount of capacitance. Most bench components are rated in μF (10-6), nF (10-9), or pF (10-12). If you plug "470" into the C variable instead of "0.00047", your time constant τ will be calculated as 4,700,000 seconds instead of 4.7 seconds. Always write out the scientific notation (e.g., 470e-6) when using calculator software or SPICE simulators to force the unit conversion.
What a Realistic Answer Magnitude Looks Like
Developing an intuition for realistic time constants prevents sanity-check failures. Here is what you should expect based on the application:
- Power Supply Bleeder Networks: τ typically ranges from 0.5 to 10 seconds. The goal is to balance rapid discharge for safety against continuous power loss (I2R heating) while the unit is running.
- 555 Timer / Microcontroller Delays: τ typically ranges from 1 millisecond to 500 milliseconds. Resistors are usually in the 10kΩ to 1MΩ range, and capacitors are in the 1nF to 10μF range.
- Audio Coupling / DC Blocking: τ is usually designed to be greater than 5 seconds (often much higher) to prevent low-frequency signal attenuation (sag) during bass transients.
By anchoring your math to base SI units, respecting the physical limitations of real-world dielectrics, and using the rearranged forms of the equation to solve for your specific unknown, the capacitor discharge equation transitions from a textbook abstraction to a vital, daily bench tool.
