Supercapacitor Energy Formulas
Supercapacitor formulas: the usable energy between two voltages, the run time at constant power or constant current and the loss in the series resistance.
The formulas
| To find | Formula |
|---|---|
| Usable energy between V1 and V2 | E = ½ × C × (V1² − V2²) |
| Time at constant power | t = E / P |
| Time at constant current | t = C × (V1 − V2) / I |
| Voltage step and loss in the ESR | ΔV = I × ESR and P = I² × ESR |
C is the capacitance in farads, V1 the starting voltage, V2 the lowest voltage that the load accepts, P the load power in watts, I the load current in amperes and ESR the equivalent series resistance in ohms. A supercapacitor cannot give all its energy, because the voltage falls as it discharges and the load stops working below V2. The constant-power time uses the usable energy, so it ignores the ESR loss and the converter efficiency. The values are examples, and the data sheet gives the rated voltage, the ESR and the leakage.
Worked example: a 100 F cell from 2.7 V to 1.35 V
- E = 0.5 × 100 × (2.7² − 1.35²) = 273.4 J
- At a constant 10 W, t = 273.4 / 10 = 27.34 s
- At a constant 5 A, t = 100 × (2.7 − 1.35) / 5 = 27 s
- With an ESR of 10 mΩ, the voltage step is 5 × 0.01 = 0.05 V and the loss is 5² × 0.01 = 0.25 W
Usable energy of the 100 F cell by lowest voltage
| Lowest voltage | Usable energy | Share of the full energy |
|---|---|---|
| 0 V | 364.5 J | 100% |
| 0.9 V | 324 J | 88.89% |
| 1.35 V | 273.4 J | 75% |
| 1.8 V | 202.5 J | 55.56% |
Frequently asked questions
Why is only part of the energy usable?
The voltage of a capacitor falls in step with its charge. Below the lowest voltage the load works poorly or not at all, so the energy left in the cell stays there.
What does discharging to half the voltage give?
It gives 75% of the stored energy, because the energy goes with the square of the voltage.
Why do cells go in series?
One cell has a low rated voltage, often 2.7 V. A series string raises the voltage, and balancing keeps each cell within its rating. The capacitance of the string is the cell value divided by the number of cells.
How does this differ from a battery?
A supercapacitor has a falling voltage and a low energy per kilogram, but it gives a high current and many cycles. See the battery pack article for the cell arithmetic of a battery.
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