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Grouping of Cells

Grouping of Cells :- It isn’t possible to make voltage sources and batteries for each specific voltage requirement. When a different voltage is necessary, two or more voltage sources(cells/batteries) are used in different combinations to produce the desired value of voltage and current. These cells can be connected in two basic types of combinations. These two combinations are :

Series Combination
Parallel Combination.

Series Combination / Series Grouping of Cells

When the voltage required by a load (such as a motor, appliance or electronic device) exceeds the voltage output of a single cell or battery, cells are connected in series to increase the total voltage. A series combination of cells refers to connecting multiple cells end-to-end so that the positive terminal of one cell is connected to the negative terminal of the next cell, and so on. This arrangement creates a single path for current flow through all the cells.

Let ε₁ , ε₁ be the emfs of two cells and r₁ , r₂ be their internal resistances respectively. Let the potentials of the three points A, B and C is denoted by V_A, V_B, and V_Cand the current flowing trough them is I.

Potential difference for the above cells :-

$\displaystyle {{V}_{AB}}={{V}_{A}}-{{V}_{B}}={{\varepsilon }_{1}}-I{{r}_{1}}$ …..(1)

$\displaystyle {{V}_{BC}}={{V}_{B}}-{{V}_{C}}={{\varepsilon }_{2}}-I{{r}_{2}}$ …..(2)

Potential difference for the series combination of cells :-

$\displaystyle {{V}_{AC}}={{V}_{A}}-{{V}_{C}}=({{V}_{A}}-{{V}_{B}})+({{V}_{B}}-{{V}_{C}})$

$\displaystyle \Rightarrow {{V}_{AC}}=({{\varepsilon }_{1}}-I{{r}_{1}})+({{\varepsilon }_{2}}-I{{r}_{2}})$

$\displaystyle \Rightarrow {{V}_{AC}}=({{\varepsilon }_{1}}+{{\varepsilon }_{2}})-I({{r}_{1}}+{{r}_{2}})$ …..(3)

Now if the series combination is replaced by a single cell of emf ε_eq and internal resistance r_eq as shown in figure, then

$\displaystyle {{V}_{AC}}={{\varepsilon }_{eq}}-I{{r}_{eq}}$ …..(4)

Comparing equations (3) and (4), we get

$\displaystyle {{\varepsilon }_{eq}}={{\varepsilon }_{1}}+{{\varepsilon }_{2}}$ …..(5)

$\displaystyle {{r}_{eq}}={{r}_{1}}+{{r}_{2}}$ …..(6)

In general if n cells are connected in series then,

Equivalent emf (ε_eq) = ε₁ + ε₂ + ε₃ + …..

Equivalent internal resistance (r_eq) = r₁ + r₂ + r₃ + …..

Note :-

In series combination if negative terminal of first cell is connected to the negative terminal of the second cell, then

$\displaystyle {{V}_{BC}}={{V}_{B}}-{{V}_{C}}=-{{\varepsilon }_{2}}-I{{r}_{2}}$

$\displaystyle \Rightarrow {{V}_{AC}}=({{\varepsilon }_{1}}-I{{r}_{1}})+(-{{\varepsilon }_{2}}-I{{r}_{2}})$

$\displaystyle \Rightarrow {{V}_{AC}}=({{\varepsilon }_{1}}-{{\varepsilon }_{2}})-I({{r}_{1}}+{{r}_{2}})$ …..(7)

Also for equivalent cell,

$\displaystyle {{V}_{AC}}={{\varepsilon }_{eq}}-I{{r}_{eq}}$ …..(8)

Comparing equations (7) and (8),

Equivalent emf (ε_eq) = ε₁ – ε₂

Equivalent internal resistance (r_eq) = r₁ + r₂

Parallel Combination / Parallel Grouping of Cells

The figure below shows a parallel combination of cells between points A and D in which positive terminal of each cell is connected to one point and negative terminal of each cell is connected to the other point.

Let ε₁ , ε₁ be the emfs of two cells and r₁ , r₂ be their internal resistances respectively and the cells are supplying currents I₁ and I₂ to the circuit.

Total current supplied by the cells,

I = I₁ + I₂ …..(9)

As the cells are connected in parallel so the potential difference (ΔV) across the cell is equal. For the first cell,

$\displaystyle \Delta V={{V}_{B}}-{{V}_{C}}={{\varepsilon }_{1}}-{{I}_{1}}{{r}_{1}}\Rightarrow {{I}_{1}}=\frac{{{\varepsilon }_{1}}-\Delta V}{{{r}_{1}}}$

For the second cell,

$\displaystyle \Delta V={{V}_{B}}-{{V}_{C}}={{\varepsilon }_{2}}-{{I}_{2}}{{r}_{2}}\Rightarrow {{I}_{2}}=\frac{{{\varepsilon }_{2}}-\Delta V}{{{r}_{2}}}$

Putting the values of I₁ and I₂ in equation (9), we get

$\displaystyle I=\frac{{{\varepsilon }_{1}}-\Delta V}{{{r}_{1}}}+\frac{{{\varepsilon }_{2}}-\Delta V}{{{r}_{2}}}$

$\displaystyle \Rightarrow I=\left( \frac{{{\varepsilon }_{1}}}{{{r}_{1}}}+\frac{{{\varepsilon }_{2}}}{{{r}_{2}}} \right)-\left( \frac{1}{{{r}_{1}}}+\frac{1}{{{r}_{2}}} \right)\Delta V$

$\displaystyle \Rightarrow I=\left( \frac{{{\varepsilon }_{1}}{{r}_{2}}+{{\varepsilon }_{2}}{{r}_{1}}}{{{r}_{1}}{{r}_{2}}} \right)-\left( \frac{{{r}_{1}}+{{r}_{2}}}{{{r}_{1}}{{r}_{2}}} \right)\Delta V$

$\displaystyle \Rightarrow \Delta V=\left( \frac{{{\varepsilon }_{1}}{{r}_{2}}+{{\varepsilon }_{2}}{{r}_{1}}}{{{r}_{1}}+{{r}_{2}}} \right)-I\left( \frac{{{r}_{1}}{{r}_{2}}}{{{r}_{1}}+{{r}_{2}}} \right)$ …..(10)

Now if the parallel combination of cells is replaced by a single equivalent cell of emf ε_eq and internal resistance r_eq as shown in figure above, then

$\displaystyle \Delta V={{\varepsilon }_{eq}}-I{{r}_{eq}}$ …..(11)

Comparing equations (10) and (11), we get

$\displaystyle {{\varepsilon }_{eq}}=\left( \frac{{{\varepsilon }_{1}}{{r}_{2}}+{{\varepsilon }_{2}}{{r}_{1}}}{{{r}_{1}}+{{r}_{2}}} \right)=\left( \frac{\frac{{{\varepsilon }_{1}}}{{{r}_{1}}}+\frac{{{\varepsilon }_{2}}}{{{r}_{2}}}}{\frac{1}{{{r}_{1}}}+\frac{1}{{{r}_{2}}}} \right)$ …..(12)

and

$\displaystyle {{r}_{eq}}=\left( \frac{{{r}_{1}}{{r}_{2}}}{{{r}_{1}}+{{r}_{2}}} \right)$ …..(13)

$\displaystyle \frac{1}{{{r}_{eq}}}=\left( \frac{{{r}_{1}}+{{r}_{2}}}{{{r}_{1}}{{r}_{2}}} \right)=\frac{1}{{{r}_{1}}}+\frac{1}{{{r}_{2}}}$ …..(14)

Dividing equation (12) by equation (13) we get,

$\displaystyle \frac{{{\varepsilon }_{eq}}}{{{r}_{eq}}}=\left( \frac{{{\varepsilon }_{1}}{{r}_{2}}+{{\varepsilon }_{2}}{{r}_{1}}}{{{r}_{1}}{{r}_{2}}} \right)=\frac{{{\varepsilon }_{1}}}{{{r}_{1}}}+\frac{{{\varepsilon }_{2}}}{{{r}_{2}}}$ …..(15)

In general if n cells are connected in parallel then, equivalent emf (ε_eq) and equivalent internal resistance (r_eq) are given by,

$\displaystyle \frac{{{\varepsilon }_{eq}}}{{{r}_{eq}}}=\frac{{{\varepsilon }_{1}}}{{{r}_{1}}}+\frac{{{\varepsilon }_{2}}}{{{r}_{2}}}+.......+\frac{{{\varepsilon }_{n}}}{{{r}_{n}}}$