Battery Series/Parallel Configuration
Works out the nominal voltage, capacity, energy and cell count of a battery pack built from identical cells, given how many are in series and how many strings are in parallel. Use it to plan an e-bike, tool, robot or solar storage pack, or to read a pack label written as, for example, 13S4P.
Equations
Vpack = Ns·Vcell Cpack = Np·Ccell Epack = Vpack·Cpack N = Ns·Np
Ns is the number of cells connected end to end in each string, and Np the number of those strings connected side by side. Cells in series add their voltages while the current through them is the same. Strings in parallel add their capacities at the same voltage [1]. Vcell is the nominal cell voltage in volts, and Ccell is the cell capacity in amp-hours. Epack is the pack energy in watt-hours, and N is the total number of cells to buy.
The Cell chemistry list fills in the standard nominal voltage for lead-acid, nickel and primary cells [2]–[5]. Lithium-ion cells have no single standard value, so choose Custom and enter the maker’s nominal voltage, usually 3.6 or 3.7 V.
Limits and common mistakes
- The results are nominal values. A charged Li-ion cell sits near 4.2 V and an empty one near 3.0 V, so a pack’s actual voltage swings well above and below Vpack. Size chargers, converters and capacitors for the full range.
- Only connect matched cells: the same chemistry, capacity, age and state of charge. Mismatched cells in series drift apart, and the weakest one gets over-discharged.
- Lithium packs need a battery management system to balance and protect each series group.
- Capacity in Ah adds only in parallel. To compare packs of different voltages, compare energy in Wh.
Worked example
An e-bike needs a pack of about 48 V nominal and at least 600 Wh. The builder has Li-ion cells rated 3.6 V and 3.5 Ah, and plans a 13S4P layout.
- Leave Cell chemistry at Custom (the default), since Li-ion has no standard value in the list.
- Enter Cell voltage (Vcell) 3.6 V and Cell capacity (Ah) 3.5.
- Enter Cells in series (Ns) 13 and Strings in parallel (Np) 4, and press Solve. The Solution shows:
Pack voltage: 46.8 V
Pack capacity: 14 Ah
Pack energy: 655.2 Wh
Total cells: 52
Check by hand: 13 × 3.6 = 46.8 V, 4 × 3.5 = 14 Ah, and 46.8 × 14 = 655.2 Wh, which meets the 600 Wh target. As another check, one cell holds 3.6 × 3.5 = 12.6 Wh, and 52 cells hold 52 × 12.6 = 655.2 Wh. Fully charged, the pack reaches 13 × 4.2 = 54.6 V, so the controller must accept that voltage.
