# Why should each row of cells in battery grid have equal number of cells for maximum power transfer?

How a battery grid can be made using $Z$ cells of $r \space \Omega$ internal resistance which can deliver a maximum power to a load of $R \space \Omega$ resistance?

In a battery grid with $N$ cells in series and $M$ rows in parallel $E_{eq}=nE$ and $r_{eq}=\frac{Nr}{M}$.

Total cells $NM=Z$ and according to maximum power transfer theorem $\frac{N(r)}{M}=R$

My teacher said that for maximum power transfer, each row of cells should have equal number i.e. $N$ number of cells. Why is it so? Why can't each row have different number of cells?

IMO what your teacher actually meant to add was that all the cells are identical, so the internal resistance values are all the same and in parallel in any given row.
Under these condition if you removed (or added) a single cell in any given row, you change the matrix of internal resistances and move the maximum power point current flow.

If you don't have an equal number of cells you'll have an imbalance in the flow of electrons which results in some cells not being used to their full (electromotive) potential.

Obviously the cell numbers must match or there would be high circulation currents discharging the cells

The resistance matching maximizes power transfer but not efficiency and certainly causes a serious temperature rise but in theory is the maximal power transfer. With 10mohm typical ESR and 3.7V, this implies a 37 Amp rate with Pd=13.7W=I²R heating up the battery.

simulate this circuit – Schematic created using CircuitLab

Further analysis would show voltage inbalance per cell if the internal resistance, Ri is not balanced which results in thermal inbalance under high currents and unbalanced aging effects, so ESR is often matched <<1% when new and mismatch >>1% causes rapid aging of the hottest cell with highest Ri.

The total cell voltage *N and Ri ought to be balanced for ideal charging, which becomes critical at high current rates.

My teacher said that for maximum power transfer, each row of cells should have equal number i.e. N number of cells. Why is it so?

If you have a row of (say) ten perfect cells and each cell has a terminal voltage of (say) 1 volt, then the row voltage is 10 volts.

If you put that in parallel with a row with a different number of cells then you will create infinite current circulating around the cells.

However, it's still a bad situation if each cell has 0.1 ohm resistance because there is still a voltage mismatch when you connect rows up and you will get a large circulating current that may damage or massively reduce the energy storage in each row.