Spectrum of the most powerful series–parallel combination of identical cells

Abstract

For a given total number of identical cells with internal resistance, we inquire the series–parallel combination of the cells that yields the maximum current through a load resistor linked to the combination. The aspect ratio or shape factor (number of parallel strings/number of serial cells in each string) of cell combination for the maximum load current depends on the resistance ratio (internal resistance/load resistance). Here we provide a systematic, algebraically simple and efficient way to find the (discrete) ‘spectrum’ of the shape factor for the maximum load current as a piecewise-defined function of resistance ratio. In particular, we can get the spectrum by computer programming even when the total number of series–parallel combinations is large as in the case of the electric eel. Furthermore, using the spectrum, we obtain the maximum load current as a piecewise-defined function of resistance ratio. All taken together, this article would be beneficial to university teachers and students because (i) it provides them with pedagogically useful information on the series–parallel combination of identical cells, (ii) its whole process is algebraically easy enough to understand, and (iii) it gives them a good example of using a computer programming in physics education. In addition, it is also of practical importance as the scheme introduced herein can be applied to electric vehicles whose batteries have the pack-module-cell structure.