The Core Equation Capacitor Sizing: Derivation and Worked Examples

The fundamental equation capacitor sizing formula for constant-current discharge is C = (I × Δt) / ΔV. If you need a capacitor to supply a known current ($I$) for a specific duration ($\Delta t$) while allowing a maximum voltage drop ($\Delta V$), this rearranged form of the capacitor current law gives you the exact required capacitance ($C$) in Farads. No guessing, no rule-of-thumb multipliers—just direct physics.

The Core Equation: Capacitor Sizing and Derivation

The relationship between current and voltage in a capacitor is defined by the differential equation $I = C \frac{dV}{dt}$. This states that the current ($I$) flowing through a capacitor is equal to its capacitance ($C$) multiplied by the rate of change of voltage across it over time ($\frac{dV}{dt}$).

In practical circuit design—such as sizing a hold-up capacitor for a microcontroller brownout or a ride-through capacitor for a relay coil—the load draws a relatively constant current over a short, defined window. By assuming $I$ is constant, we can integrate the differential equation over the time interval $\Delta t$, yielding the algebraic equation capacitor sizing formula:

$$C = \frac{I \cdot \Delta t}{\Delta V}$$

When the Formula Applies (and Its Assumptions)

This equation capacitor model assumes an ideal capacitor and a constant current load. It applies perfectly to digital logic hold-up, linear regulator input buffering, and steady-state analog loads. It explicitly ignores three real-world factors:

• Equivalent Series Resistance (ESR): The formula calculates the capacitive droop only. The instantaneous voltage drop at $t=0$ caused by ESR ($V_{ESR} = I \times ESR$) must be added to $\Delta V$ if the ESR is high.
• Dielectric Absorption: In high-precision sample-and-hold circuits, dielectric memory causes voltage rebound, which this formula ignores.
• Leakage Current: For long-duration hold-up (hours or days), the capacitor's internal leakage current acts as a parallel load and must be added to $I$.

Rearranged Forms, Unit Pitfalls, and Magnitudes

Depending on what you are solving for, the equation capacitor formula can be rearranged. Keep these forms handy for bench debugging:

• Solve for Capacitance: $C = \frac{I \cdot \Delta t}{\Delta V}$ (Used for sizing a new component)
• Solve for Hold-up Time: $\Delta t = \frac{C \cdot \Delta V}{I}$ (Used to calculate how long an existing capacitor will sustain a load)
• Solve for Voltage Droop: $\Delta V = \frac{I \cdot \Delta t}{C}$ (Used to verify if a chosen capacitor will cause a brownout)
• Solve for Peak Current: $I = \frac{C \cdot \Delta V}{\Delta t}$ (Used to calculate inrush or peak burst current capability)

Realistic Answer Magnitudes

If your final $C$ value falls outside these ranges, double-check your unit conversions:

• 1 pF to 1 µF: High-frequency decoupling, RF filtering, timing oscillators (Ceramics).
• 1 µF to 10,000 µF (10 mF): Power supply bulk filtering, audio coupling, short-term digital hold-up (Electrolytics, Tantalums, Large MLCCs).
• 0.1 F to 5,000 F: Memory backup, motor start-assist, energy harvesting ride-through (Supercapacitors / EDLCs).

Worked Example 1: Relay Coil Ride-Through

Scenario: You are designing a safety interlock circuit. A 12V DC relay (coil resistance 240Ω) needs to remain energized for at least 100 ms after the main power rail is cut to allow a mechanical latch to fully engage. The relay's minimum drop-out voltage is 8V.

Step 1: Identify the known variables and convert to SI base units.

• Supply Voltage ($V_{initial}$) = 12 V
• Drop-out Voltage ($V_{final}$) = 8 V
• Allowable Droop ($\Delta V$) = $12\text{ V} - 8\text{ V} = 4\text{ V}$
• Discharge Time ($\Delta t$) = $100\text{ ms} = 0.1\text{ s}$
• Coil Current ($I$) = $V / R = 12\text{ V} / 240\text{ }\Omega = 0.05\text{ A}$ (Assuming constant current draw for this short window)

Step 2: Apply the equation capacitor formula with explicit unit tracking.

$$C = \frac{0.05\text{ A} \cdot 0.1\text{ s}}{4\text{ V}}$$

$$C = \frac{0.005\text{ A}\cdot\text{s}}{4\text{ V}}$$

$$C = 0.00125\text{ F}$$

Step 3: Convert to practical units and select a part.

$0.00125\text{ F} = 1250\text{ }\mu\text{F}$.
Standard values do not include 1250 µF. We must round up to the next standard E12 series value, which is 1500 µF. Because the initial voltage is 12V, we select a 16V or 25V rated capacitor.
Concrete Pick: Panasonic EEV-FK1V152 (1500 µF, 35V SMD Aluminum Electrolytic, low ESR).

Worked Example 2: RTC Supercapacitor Hold-Up (With Leakage)

Scenario: An offline datalogger uses a DS3231 Real-Time Clock (RTC). When main power fails, the RTC switches to the VBAT pin, drawing 1.0 µA. The system might be unpowered for up to 14 days. The DS3231 minimum VBAT operating voltage is 2.0V, and the supercapacitor charges to 3.3V.

Step 1: Identify variables and account for leakage.

• $\Delta V = 3.3\text{ V} - 2.0\text{ V} = 1.3\text{ V}$
• $\Delta t = 14\text{ days} \times 24\text{ h} \times 3600\text{ s} = 1,209,600\text{ s}$
• Load Current ($I_{load}$) = $1.0\text{ }\mu\text{A} = 1 \times 10^{-6}\text{ A}$
• Edge Case Adjustment: A typical 1F supercapacitor has a leakage current of roughly 1.0 µA after 72 hours. Total discharge current $I_{total} = I_{load} + I_{leakage} = 2.0\text{ }\mu\text{A} = 2 \times 10^{-6}\text{ A}$.

Step 2: Apply the formula.

$$C = \frac{(2 \times 10^{-6}\text{ A}) \cdot (1,209,600\text{ s})}{1.3\text{ V}}$$

$$C = \frac{2.4192\text{ A}\cdot\text{s}}{1.3\text{ V}}$$

$$C = 1.86\text{ F}$$

Step 3: Select a part based on voltage derating.

We need at least 1.86 F. However, supercapacitors are typically rated for 2.5V or 2.7V. Our charge voltage is 3.3V. We must use two 2.7V supercapacitors in series to handle the 3.3V safely. Putting two identical capacitors in series halves the total capacitance ($C_{total} = C / 2$). Therefore, each individual capacitor must be $1.86\text{ F} \times 2 = 3.72\text{ F}$.
Concrete Pick: Two KEMET FC0V505ZFTBR24 (5.0 F, 2.7V) supercapacitors wired in series, yielding a 2.5 F bank rated for 5.4V maximum.

Decision Tree: Picking the Right Capacitor Chemistry

Do not default to a single chemistry for all applications. Use this decision path to terminate your design with the correct component class and a verified default part number.

For further reading on component selection and derating curves, consult the Analog Devices capacitor selection guide and the Cornell Dubilier application guides. Always verify your final $\Delta V$ on an oscilloscope under worst-case thermal and load conditions, as datasheet nominal values can shift by up to 20% at temperature extremes.