A battery consists of a collection of cells that are electrically connected with series and parallel combinations. The general nomenclature is (mS-nP), which means that m cells are connected in series and n of these series strings are connected in parallel. The total number of cells, N_c, is m × n. All of the cells together make up the battery (also referred to as a battery pack or simply pack). Sometimes, a subset of the cells is housed together in a module. The primary question we want to answer in this section is, for a given cell design, what is the best way to combine cells together to achieve the design objectives?

For a fixed number of cells, N_c, there are many layouts or different ways of electrically connecting the cells. A generic layout is shown in Figure 8.1. In the previous section, we saw how the voltage requirement determines the number of cells that must be placed in series. Here, we consider the resistance of the battery, which has a crucial impact on its peak power. In principle, any combination of n cells in parallel with m cells in series is acceptable so long as N_c = mn. Assuming that all cells are identical, each with an open-circuit potential of  and an internal resistance of R_int, the total resistance of the battery can be calculated:

(8.4)

Series connections (m) increase the resistance of the battery, and parallel connections (n) reduce the resistance of the battery. Similarly, series connections cause the battery voltage to increase, and parallel connections increase the capacity and current. Therefore, , and .

Assuming that the battery is ohmically limited,

(8.5)

Power from the battery is

(8.6)

Thus, we see the power from the battery is just the power from an individual cell times the number of cells, and is independent of how the cells are connected together. However, we must have a sufficient number of cells in series to meet the voltage requirements of the application as we have seen previously. Also, there is typically a maximum current specified for the system, since wires, connections, and electrical devices must be sized to handle the system current without, for example, excessive heating. A key advantage of higher voltages is that they result in lower currents and therefore smaller wires and smaller motors for a given power.

The above analysis only included the internal resistance of the cells in the battery pack. How does the situation change if, in addition to the internal resistance, the resistance of the connections between cells is incorporated? There is some resistance in the wires between cells and a contact resistance associated with each electrical connection. For this situation, we need to add the resistance external to the cells to the combined internal resistance calculated from Equation 8.4:

(8.7)

where  is the combined wire and connection resistance between each cell and on each end of the string. Clearly, the lower the external resistance, the higher the maximum power and the greater the energy obtained from the battery. As indicated in Equation 8.7, the external resistance can be calculated from the resistance of the connecting wires and the individual contact resistances. These connections, and therefore the external resistance, vary according to the battery layout. The effect of the battery layout is explored in Illustration 8.2.

ILLUSTRATION 8.2

A 48 kW·h battery is needed for 4 hours of energy storage. The cells available have an open-circuit potential of 2.0 V, a nominal capacity of 1 kW·h (C/4 rate), and an internal resistance of 2 mΩ. R_w is equal to 0.75 mΩ. Compare the nominal voltage, nominal current, and maximum power for four configurations: (4S-12P), (8S-6P), (12S-4P), and (48S-1P).

SOLUTION:

Note that each of the possible configurations has 48 cells in order to meet the total energy requirement with 1 kW·h cells.

Based on the discharge time of 4 hours, 12 kW of power is needed on average. Therefore, the nominal battery current and voltage are calculated as

where

For each layout, the total resistance is calculated. The battery current can be varied to find the maximum power. These values are tabulated on the right. The maximum power goes up slightly as the number of cells in series increases. More importantly, notice that the battery current becomes very large as cells are arranged in parallel. This dramatic increase is strong incentive to arrange the cells in series with higher battery voltages.

m	n	Vnom [V]	Inom [A]	Pmax kW
4	12	6.1	1980	16.3
8	6	12.3	976	16.9
12	4	18.4	647	17.1
48	1	74.7	161	17.3

There is one other term that comes up frequently in describing batteries: the module. In Section 8.1, we considered a 24 kWh device where 192 cells were combined to form the battery, with two strings in parallel. However, in practice, a battery design such as this would not likely be implemented as simply a 96S-2P arrangement. Rather, cells would be grouped into modules. For example, in this instance, four cells could be combined to form a module, where each module is 2S-2P and housed together in a single case with one pair of terminals. Forty-eight of these modules are then strung together in series to form the battery (Figure 8.2).

In this section, we have assumed that the cell design was fixed as a constraint, and we examined how cells of that type might be connected to meet the desired specifications. In the next sections, rather than use a fixed design, the design of the individual cell is considered. In Illustration 8.2, if we had a choice, we may have decided to select a smaller capacity cell and increase the voltage to, for example, 200 or 300 V, in order to further reduce the current.