The discharge capacitor equation defines how voltage and current decay over time when a charged capacitor releases its energy through a resistive path. Whether you are sizing a bleeder resistor for a 400V power supply filter, calculating the low-time of a 555 timer astable circuit, or designing a supercapacitor backup for an RTC module, this single exponential formula is your primary design tool.
The foundational equation for capacitor voltage during discharge is:
V(t) = V0 × e-t / RC
The Core Discharge Capacitor Equation & Symbol Definitions
To use the formula correctly on the bench, you must map every variable to its strict SI unit before plugging in numbers. The most common calculation errors occur when mixing practical bench units (like microfarads and kilohms) without proper conversion.
| Symbol | Parameter | Strict SI Unit | Common Bench Unit | Conversion to SI |
|---|---|---|---|---|
| V(t) | Voltage across the capacitor at time t | Volts (V) | Volts (V) | None required |
| V0 | Initial voltage at the start of discharge (t=0) | Volts (V) | Volts (V) | None required |
| e | Euler's number (mathematical constant) | Dimensionless | ~2.71828 | None required |
| t | Elapsed discharge time | Seconds (s) | Milliseconds (ms) | Multiply ms by 10-3 |
| R | Discharge resistance (including ESR and load) | Ohms (Ω) | Kilohms (kΩ) | Multiply kΩ by 103 |
| C | Capacitance | Farads (F) | Microfarads (μF) | Multiply μF by 10-6 |
The product of R × C is known as the time constant, denoted by the Greek letter tau (τ). After one time constant (1τ), the capacitor voltage will always drop to exactly 36.8% of its initial value. After five time constants (5τ), it drops below 1%, which is the standard engineering threshold for considering a capacitor 'fully discharged'.
Real-World Component Values & Time Constants
Theoretical formulas assume ideal components, but bench work requires dealing with real-world capacitance ranges and practical resistor values. The table below maps common capacitor types to typical discharge scenarios, showing the resulting time constants and the time required to reach the 99% discharged state (5τ).
| Capacitor Type / Application | Typical C Value | Typical Discharge R | Time Constant (τ = RC) | Time to 99% Discharge (5τ) |
|---|---|---|---|---|
| Ceramic (High-frequency decoupling) | 100 nF (0.1 μF) | 10 kΩ | 1.0 ms | 5.0 ms |
| Aluminum Electrolytic (PSU Filter) | 4,700 μF | 220 Ω | 1.034 s | 5.17 s |
| Film (Motor Run / Snubber) | 50 μF | 100 kΩ | 5.0 s | 25.0 s |
| High-Voltage PSU (Bleeder network) | 1,000 μF | 47 kΩ | 47.0 s | 235.0 s (3.9 mins) |
| Supercapacitor (Memory backup) | 10 F | 100 Ω | 1,000 s | 5,000 s (83.3 mins) |
Note: In high-voltage power supplies, the bleed resistor (R) is deliberately chosen to balance standby power dissipation against the 5τ safety discharge time. A 47kΩ resistor across a 400V bus dissipates about 3.4W continuously, requiring a 5W or 10W rated wirewound resistor.
Rearranged Forms for Bench Troubleshooting
On the workbench, you rarely just solve for V(t). More often, you need to size a resistor to meet a safety deadline, or determine how long a backup circuit will hold up. Here are the algebraic rearrangements of the discharge capacitor equation, solving for every variable:
- Solve for time (t):
t = -R × C × ln(V(t) / V0)
Use when calculating how long it takes to drop to a specific logic threshold or safe-touch voltage. - Solve for resistance (R):
R = -t / (C × ln(V(t) / V0))
Use when sizing a bleeder resistor to ensure a PSU discharges within a mandated safety window (e.g., < 60 seconds). - Solve for capacitance (C):
C = -t / (R × ln(V(t) / V0))
Use when designing a hold-up circuit to keep a microcontroller alive during a brownout for a specific number of milliseconds. - Solve for initial voltage (V0):
V0 = V(t) / e-t / RC
Use in forensics or debugging: if you measure V(t) at a known time t, this tells you what the rail voltage was when the discharge began.
Worked Examples with Strict Unit Tracking
The fastest way to burn out a component or fail a safety inspection is by dropping a decimal during unit conversion. These two examples enforce strict SI unit tracking at every step.
Problem 1: Finding Voltage at a Specific Time
Scenario: You have a 555 timer discharge circuit. The timing capacitor is 470 μF, and it discharges through an internal transistor and external resistor totaling 2.2 kΩ. The capacitor is initially charged to 24V. What is the voltage across the capacitor after 0.5 seconds of discharge?
- Convert to strict SI units:
C = 470 μF = 470 × 10-6 F = 0.00047 F
R = 2.2 kΩ = 2.2 × 103 Ω = 2200 Ω - Calculate the time constant (τ):
τ = R × C = 2200 × 0.00047 = 1.034 seconds. - Calculate the exponent (-t / τ):
Exponent = -0.5 s / 1.034 s = -0.48356 (dimensionless). - Apply Euler's number:
e-0.48356 = 0.61659. - Multiply by initial voltage:
V(t) = 24V × 0.61659 = 14.8 V.
Problem 2: Sizing a Bleeder Resistor for Safety Compliance
Scenario: A tube amplifier power supply uses a 10,000 μF filter capacitor charged to 400V DC. Safety standards require the voltage to drop below 50V (a generally safe touch threshold) within 15 seconds of unplugging. What is the maximum value for the bleeder resistor?
- Convert to strict SI units:
C = 10,000 μF = 0.01 F
t = 15 s
V0 = 400 V
V(t) = 50 V - Select the rearranged formula:
R = -t / (C × ln(V(t) / V0)) - Calculate the natural log ratio:
Ratio = 50 / 400 = 0.125
ln(0.125) = -2.07944 - Solve for R:
R = -15 / (0.01 × -2.07944)
R = -15 / -0.0207944 = 721.3 Ω
Bench Decision: You must use a resistor of 721 Ω or lower to meet the 15-second requirement. A standard 680 Ω 5W wirewound resistor would be the correct choice here, yielding a discharge time of roughly 14.1 seconds.
Assumptions, Edge Cases, and Unit Traps
The discharge capacitor equation is an idealized mathematical model. To use it effectively in real-world circuit design, you must understand its boundaries.
When the Formula Applies (and When it Doesn't)
This equation assumes a purely resistive discharge path and constant capacitance. It applies perfectly to DC bench circuits, timing networks, and power supply bleeder designs.
However, it fails if the discharge path contains significant inductance (such as long wires or relay coils). In those cases, the circuit becomes an RLC network, and the discharge will exhibit underdamped ringing rather than a smooth exponential decay. It also assumes the capacitor is discharging into a fixed resistance; if the load is an active constant-current sink (like an LED driver or a switching regulator), the voltage decay will be linear, not exponential.
The 'Micro-Kilo' Cancellation Trap
A common shortcut taught in engineering schools is that multiplying microfarads (μF) by kilohms (kΩ) directly yields milliseconds (ms). While mathematically true (10-6 × 103 = 10-3), this shortcut causes catastrophic errors when mixed with other prefixes. If you multiply 100 nF by 10 kΩ using the 'micro-kilo' mental shortcut, you will get the wrong answer. Always convert to base Farads and base Ohms first, calculate the time constant in seconds, and then convert the final time constant to milliseconds or microseconds.
Realistic Answer Magnitudes & Sanity Checks
Develop an intuition for what a realistic answer looks like. If you calculate a discharge time of 4 seconds for a 100nF ceramic cap through a 10kΩ resistor, stop and check your math. The real answer is 1 millisecond. You likely forgot to convert nanofarads to farads. Conversely, if your calculated voltage V(t) is higher than your initial voltage V0, you have dropped a negative sign in the exponent.
The Dielectric Absorption 'Ghost Voltage'
One edge case the formula cannot predict is dielectric absorption. In large aluminum electrolytic and film capacitors, some charge gets trapped deep within the dielectric material during charging. If you rapidly short the terminals to 0V and then remove the short, the capacitor will spontaneously 'recover' a voltage over the next few minutes. For a 400V PSU capacitor, this ghost voltage can rebound to 20V or 30V—enough to deliver a painful shock or damage sensitive MOSFET gates. This physical reality is exactly why high-energy capacitors require permanent, hard-wired bleed resistors rather than relying on a one-time manual discharge with a shorting stick.
For deeper theoretical modeling of RC decay networks and non-ideal parasitic effects, the Georgia State University HyperPhysics database provides excellent interactive calculators and derivations to verify your bench math.
