Battery Pack Formulas: Series and Parallel

Battery pack formulas: cells in series for voltage, strings in parallel for capacity, pack energy in Wh and internal resistance, with a 13S3P worked example.

The formulas

Vcell is the nominal cell voltage and Ccell the cell capacity in ampere-hours. A pack written 13S3P has 13 cells in series and 3 in parallel. The nominal cell voltage is 2 V for lead-acid, 3.2 V for LiFePO4 and 3.7 V for NMC lithium-ion. Use the voltage range from the cell data sheet for the charger and the cut-off, and a battery management system for lithium cells.

Worked example: 48 V, 10 Ah from 3.7 V, 3.5 Ah cells

1. s = 48 / 3.7 = 12.97, so use 13 cells in series, giving 48.1 V nominal
2. p = 10 / 3.5 = 2.857, rounded up to 3 in parallel, giving 10.5 Ah
3. Total cells: 13 × 3 = 39
4. Energy = 48.1 × 10.5 = 505.1 Wh
5. Fully charged at 4.2 V per cell: 13 × 4.2 = 54.6 V
6. With 25 mΩ cells, Rpack = 0.025 × 13 / 3 = 108.3 mΩ

Cells in series for a 48 V pack

Frequently asked questions

What does a series connection do?

It adds the cell voltages and keeps the capacity the same. The current through every cell in the string is the same.

What does a parallel connection do?

It adds the capacities and keeps the voltage the same. The load current divides between the strings.

Why does the capacity not come out exactly?

The cell count is a whole number, so the pack is rounded up to the next whole string. Check that the pack is not oversized for the charger or the space.

Do I need a battery management system?

For lithium cells, yes. It balances the cells, and it protects against overcharge, over-discharge, over-current and temperature.

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