# GROUPING OF CELLS

Grouping of cells

Series:

Eeq = ∑E1

And req = ∑ri; $$i=\frac{\sum{{{E}_{i}}}}{R+\sum{{{r}_{i}}}}$$.

If S number of identical cells each of marking (E, r) are connected in series across an external resistor R.

Eeq = SE

And req = Sr ⇒ $$I=\frac{SE}{R+Sr}$$.

Out of these S cells if x number of cells are reversely connected then Eeq = (S – 2x)E, $${{r}_{eq}}=Sr\Rightarrow I=\frac{(S-2x)E}{Sr+R}$$.

Parallel: If P number of identical cells each of marking (E, r) are connected in parallel across an external resistor R.

$${{E}_{eq}}=E\,{{r}_{eq}}=\frac{r}{P}\Rightarrow I=\frac{E}{\frac{r}{P}+R}$$.

For un-identical cells in parallel

$${{E}_{eq}}=\left( \sum{\frac{{{E}_{i}}}{{{r}_{i}}}} \right){{\left( \sum{\frac{1}{{{r}_{i}}}} \right)}^{-1}}$$.

And

$${{r}_{eq}}={{\left( \sum{\frac{1}{{{r}_{i}}}} \right)}^{-1}}\Rightarrow I=\frac{\left( \sum{\frac{{{E}_{i}}}{{{r}_{i}}}} \right){{\left( \sum{\frac{1}{{{r}_{i}}}} \right)}^{-1}}}{{{\left( \sum{\frac{1}{{{r}_{i}}}} \right)}^{-1}}+R}$$.

Mixed Grouping: Let S number of identical cells having marking (E, r) are connected in series in a row and there areP such rows are connected in parallel across an external resistor R. Then

Eeq = SE

$${{r}_{eq}}=\frac{Sr}{P}\Rightarrow I=\frac{SE}{\frac{Sr}{P}+R}=\frac{E}{\frac{r}{P}+\frac{R}{S}}$$.

This current I is maximum when $$\frac{S}{P}=\frac{R}{r}$$.

So, $${{I}_{\max }}=\frac{PE}{2r}$$ or $$\frac{SE}{2R}$$.