Battery Pack Builder

A battery pack built from small round cells is described by two letters and two numbers. A laptop battery might be 4s2p: four cells in series, two in parallel. A larger pack might be 10S4P or 13S5P. The notation is compact because the idea is compact. One number buys voltage. The other buys capacity.

That is the whole of the arithmetic, and it takes a minute to learn. What takes longer is everything it does not say: how far the voltage drops under load, and why forty cells that look identical are not.

Series for volts, parallel for amp-hours

Connect cells end to end and each one adds its voltage to the total. An encyclopedia article on circuits puts it in a line: if the cells are connected in series, the voltage of the battery is the sum of the cell voltages. Connect cells side by side and the voltage stays that of one cell while the capacity, counted in ampere-hours, adds up. The same article notes that lithium-ion cells are often wired in parallel to raise the ampere-hour rating, and that each of four identical parallel cells supplies a quarter of the current.

So a pack of S series groups, each of P cells in parallel, has S times the cell voltage and P times the cell capacity. Take a cell rated 3.6 V and 3,000 mAh. Ten groups in series give 36 V. Four cells per group give 12 Ah. Multiply those and the pack holds 432 Wh, because energy is power over time and power is volts times amperes.

The energy does not care how the cells are arranged. Forty of them hold 432 Wh as 10S4P, as 20S2P or as 4S10P. The arrangement only decides how it is delivered: high voltage at low current, or the reverse.

Three voltages, and the one on the label

A lithium-ion cell does not have a voltage. It has a range. One data sheet for a 21700 cell lists three figures under cell voltage: 4.2 V for charge, 3.6 V nominal and 2.5 V for discharge. A student guide to battery specifications defines the last as the cut-off voltage, the minimum allowable, the one that generally defines the empty state. Ten such cells in series therefore swing from 42 V when full to 25 V when empty.

The label says 36 V anyway. Nominal voltage is the reference figure, sometimes thought of as the normal voltage of the battery. It sits between the limits, and the limits move with the chemistry. An encyclopedia entry on lithium-ion batteries gives the safe range as 2.5 V at the bottom and 3.65, 4.1, 4.2 or 4.35 V at the top, depending on the components of the cell. Both data sheets used here print the same three figures: 4.2, 3.6 and 2.5 V.

Where the middle figure falls depends on the chemistry too. The same entry gives the average voltage of a lithium cobalt oxide cell as 3.6 V when its anode is hard carbon and 3.7 V when it is graphite, and a space agency document puts the nominal voltage of lithium-ion cells at about 3.6 V. The choice is not neutral. The same ampere-hours multiplied by 3.7 instead of 3.6 come to 2.8 percent more watt-hours on paper.

This is why packs are named by nominal voltage and why the name is only a name. A device fed by a 36 V pack must tolerate 42 V on a full charge and must still work, or shut down gracefully, well below 36 V. Energy computed from the nominal voltage is an approximation too. That 21700 data sheet prints 4,200 mAh and 15.5 Wh for one cell, while 3.6 V times 4.2 Ah is 15.12 Wh.

What the multiplication leaves out

Every cell has internal resistance, and the pack inherits it. Series resistances add and parallel ones divide, so the pack’s resistance is the cell’s resistance times S and divided by P. The terminal voltage under load is the open voltage minus current times that resistance. The missing volts are the sag. The power lost inside the cells is the current squared times the resistance, and it leaves as heat.

A worked case makes the scale clear. The 21700 data sheet above gives a DC resistance of 16 milliohm and a continuous discharge current of 45 A. Build 10S4P from it and the pack’s resistance is ten times 16 divided by four, which is 40 milliohm. Draw 20 A and the sag is 0.8 V, a little over two percent of 36 V. The heat is 20 squared times 0.04, or 16 W, spread through forty cells.

The same pack holds 16.8 Ah, so 20 A empties it in 0.84 hours on paper, about 50 minutes. Each cell carries 5 A, which is a rate of roughly 1.19C: the C-rate is the current divided by the capacity, and 1C would empty the cell in one hour. Four cells rated 45 A each suggest a ceiling of 180 A for the group. Typed into the builder with a 1 mm gap between cells, the cells alone form a block about 225 by 89 by 70 mm, and at 70 g each they weigh 2.8 kg.

Each of those results carries a condition. The run time assumes the full rated capacity is delivered at the nominal voltage with nothing lost, and the student guide says plainly that capacity decreases with increasing C-rate. A space agency guideline lists higher discharge rates among the things that cut capacity, and adds that a cell run down to its cut-off at a high current still has some left to give at a lower one.

The 180 A ceiling assumes four cells share the current equally, which is true only when they are matched and joined by equal paths. The sag counts the cells and ignores every strip, joint, wire and switch between them and the load, so the real figure is larger.

There is a quieter trap in the resistance figure itself. Data sheets often give an impedance measured with a 1 kHz alternating signal, and it is not the number that sets sag under a steady load. The 21700 sheet lists 10 and 16 milliohm for AC and DC on the same cell. An encyclopedia entry on internal resistance warns that a measurement at 1 kHz may underestimate the resistance, because the frequency can be too high to register the slower electrochemical processes. A published comparison of methods measured 2.3 milliohm at 1 kHz against 4.5 to 6.6 with current steps.

None of those readings is wrong. The authors of that comparison conclude that a fixed-frequency AC figure is fast to take but cannot be used in general to compare different types of cell. A sag worked out from an AC figure should be read as a low estimate. The honest input is a DC resistance, measured the way the load will actually pull. Size matters as well. A space agency safety document puts the internal resistance of lithium-ion cells at 80 to 120 milliohm for small cells of 1 to 3 Ah and at about 0.8 milliohm for a cell of 190 Ah.

No two cells are identical

The arithmetic treats every cell as a copy of the data sheet. An encyclopedia entry on battery balancing says otherwise: the individual cells in a pack naturally have somewhat different capacities. It blames variation in manufacture and assembly, aging, impurities and uneven heating, and says the cells drift to different states of charge as the pack is cycled.

In a series string those differences compound. The weak cell fills first on charge and empties first on discharge. A space agency’s safety requirements describe the spiral: cycling a string of unbalanced cells widens the differences, until a weak cell can be over-discharged and driven into voltage reversal. The balancing entry adds thermal runaway to the list of what follows when a pack is not stopped in time.

Hence matching. The same requirements say the cells of a custom battery that could fail catastrophically shall be matched on charge retention, on internal resistance or impedance, and on ampere-hour capacity. One of its flow charts, for packs built from a small number of large cells, sets the limits at 0.1 percent for open-circuit voltage and mass and 5 percent for capacity. It adds a warning: matching and balancing circuits are no reason to accept a lot of cells that is not uniform to begin with.

Cell makers say the same thing as a rule. One safety data sheet tells users not to mix cells of different types and brands, or new cells with used ones, and never to short circuit, crush, puncture or solder a cell.

Even matched cells need electronics. A battery management system watches each series group and acts on what it sees. An encyclopedia entry lists what such a system may monitor: the voltages of individual cells, temperature, current and the state of balance. Balancing bleeds or shuttles charge so that the groups arrive at full together. The balancing entry states the rule the electronics enforce: discharge must stop when any cell first runs out of charge, and charging must stop when any cell reaches its maximum safe voltage.

The space agency guideline is as flat. Charge control is required for lithium-ion batteries, it says, to prevent an overcharge that can cause venting or rupture. The safety requirements ask for fusing as well, to limit the most a battery can deliver.

The stakes are not abstract. A federal consumer safety agency warned in 2021 that loose 18650 cells are often misused as stand-alone batteries although they have no protection circuits, and that a shorted cell can overheat, go into thermal runaway and expel burning contents. A workplace safety bulletin explains the chain reaction: heat from one failing cell damages its neighbors, which release more heat. It cites a count of more than 25,000 overheating or fire incidents over five years.

Names, sizes and the limits of a sum

The cells themselves are named by size. In the common scheme the first two digits are the nominal diameter in millimeters and the next two are generally the height, so an 18650 is about 18 by 65 mm, a 21700 about 21 by 70 and a 26650 about 26 by 65. An international standard for portable lithium cells covers designations, markings and dimensions.

Nominal is the word to watch. The two data sheets used here list maximums of 18.6 by 65.2 mm for an 18650 and 21.55 mm by 70.15 mm for a 21700. Half a millimeter per cell is nothing until there are ten in a row. Spacing matters as well. Cells packed in straight rows sit one pitch apart in both directions. Offset every other row by half a pitch and the rows close up to 0.866 of a pitch, because three touching circles form an equilateral triangle.

A calculator can do all of this in a blink, and that is exactly its limit. It multiplies, divides and places circles. It does not know how hot the middle cell of a block runs, whether a joint is sound, what happens when one cell fails, or how the pack behaves in the cold. It cannot choose a protection circuit or a fuse. The numbers describe a pack that would be reasonable to consider. They do not describe a pack that is safe to build, and no sum can.