Maximum Current in Series-Parallel Cell Grouping

When we combine multiple cells together, we form a battery. Cells can be grouped in series, in parallel, or in a combination of both, known as series-parallel grouping. The way cells are grouped affects the total electromotive force (EMF) and the total internal resistance of the resulting battery.

Consider a series-parallel grouping where there are m parallel rows, and each row contains n cells connected in series. Let each individual cell have an EMF e and an internal resistance r.

• The total EMF of each series row is the sum of the EMFs of the cells in series, which is \( ne \). Since the rows are connected in parallel, the total EMF of the battery remains \( E_{total} = ne \).
• The total internal resistance of each series row is the sum of the internal resistances, which is \( nr \). Since there are m such rows in parallel, the total internal resistance \( r_{total} \) of the battery is given by \( r_{total} = \frac{nr}{m} \).

Now, if this battery is connected to an external resistance R, the total current I flowing through the circuit is given by Ohm's law for the entire circuit:

\( I = \frac{E_{total}}{R + r_{total}} \)

Substituting the expressions for \( E_{total} \) and \( r_{total} \):

\( I = \frac{ne}{R + \frac{nr}{m}} \)

To get the maximum current for a given external resistance R (especially when optimizing the grouping parameters m and n for a fixed total number of cells), the current I is maximized when the denominator \( R + \frac{nr}{m} \) is optimized. The condition for maximum current delivered to the external resistance R in such a grouping is achieved when the external resistance is equal to the total internal resistance of the battery.

That is, the condition for maximum current in the external circuit is:

\( R = r_{total} \)

Substituting the expression for \( r_{total} \):

\( R = \frac{nr}{m} \)

This condition ensures that the maximum possible current is delivered to the external load resistance R from the series-parallel arrangement of cells. This is also related to the maximum power transfer theorem, which states that maximum power is transferred to the load when the load resistance equals the source resistance.

Based on this analysis, the condition required to get the maximum current in a series-parallel grouping of cells is when the external resistance is equal to the total internal resistance of the battery.