Cells in Series and Parallel

The study of cells in series and parallel configurations is fundamental to understanding how practical power sources, such as batteries, are constructed and how they behave in electrical circuits. An ideal cell is characterized solely by its electromotive force (EMF), denoted by $E$, which is the maximum potential difference it can provide when no current is drawn from it.

However, real cells possess an internal resistance, denoted by $r$, which causes a voltage drop within the cell itself when current flows. This internal resistance is responsible for the terminal voltage ($V$) being less than the EMF ($E$) when the cell is delivering current ($I$), given by the relation $V = E - Ir$.

Conceptual Foundation

Before delving into combinations, it's crucial to grasp the characteristics of a single cell:

Electromotive Force (EMF, $E$): — The work done by the cell per unit charge in moving the charge from the low potential terminal to the high potential terminal inside the cell. It's the maximum potential difference a cell can provide.
Internal Resistance ($r$): — The resistance offered by the electrolyte and electrodes of the cell to the flow of current. It causes some energy to be dissipated as heat within the cell.
Terminal Voltage ($V$): — The potential difference across the terminals of the cell when it is delivering current to an external circuit. $V = E - Ir$. When the cell is being charged, $V = E + Ir$. When no current flows, $V = E$.

Key Principles and Laws

Combinations of cells are analyzed using Kirchhoff's laws:

Kirchhoff's Current Law (KCL): — The algebraic sum of currents entering a junction (node) is equal to the algebraic sum of currents leaving the junction. This is a statement of conservation of charge.
Kirchhoff's Voltage Law (KVL): — The algebraic sum of changes in potential around any closed loop in a circuit is zero. This is a statement of conservation of energy.
Ohm's Law: — $V = IR$, where $V$ is the potential difference across a resistor $R$ through which current $I$ flows.

Cells in Series Connection

In a series connection, cells are connected end-to-end. The most common configuration is connecting the positive terminal of one cell to the negative terminal of the next. This arrangement is used to achieve a higher total EMF.

1. Series Connection (Aiding Polarity):

If $n$ cells, each with EMF $E$ and internal resistance $r$, are connected in series such that the positive terminal of one is connected to the negative terminal of the next (i.e., they are aiding each other), then:

Equivalent EMF ($E_{eq}$): — The EMFs add up. If all cells are identical, $E_{eq} = nE$.
Equivalent Internal Resistance ($r_{eq}$): — The internal resistances also add up. If all cells are identical, $r_{eq} = nr$.

If the cells are non-identical, with EMFs $E_1, E_2, dots, E_n$ and internal resistances $r_1, r_2, dots, r_n$, then:

Current in the External Circuit:

If this combination 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:

2. Series Connection (Opposing Polarity):

If one or more cells are connected with reversed polarity (e.g., positive to positive or negative to negative), their EMFs will subtract from the total. For example, if $n$ cells are connected in series, but $m$ of them are connected in reverse polarity, then:

Condition for Maximum Current in Series:

For $n$ identical cells in series, $I = \frac{nE}{R + nr}$.

If $R \gg nr$ (external resistance is much larger than total internal resistance), then $I \approx \frac{nE}{R}$. In this case, connecting cells in series is beneficial as the current increases proportionally with $n$.
If $R \ll nr$ (external resistance is much smaller than total internal resistance), then $I \approx \frac{nE}{nr} = \frac{E}{r}$. In this case, the current is limited by the internal resistance, and connecting more cells in series does not significantly increase the current. It might even be detrimental due to increased internal heating.

Cells in Parallel Connection

In a parallel connection, all positive terminals are connected to a common point, and all negative terminals are connected to another common point. This arrangement is primarily used to increase the current capacity and reduce the effective internal resistance.

1. Parallel Connection (Identical Cells):

If $n$ identical cells, each with EMF $E$ and internal resistance $r$, are connected in parallel:

Equivalent EMF ($E_{eq}$): — The EMF across the parallel combination remains the same as that of a single cell. $E_{eq} = E$. This is because all positive terminals are at the same potential, and all negative terminals are at the same potential, so the potential difference between the common positive and common negative points is simply $E$.
Equivalent Internal Resistance ($r_{eq}$): — The internal resistances combine like parallel resistors. $\frac{1}{r_{eq}} = \frac{1}{r} + \frac{1}{r} + \dots + \frac{1}{r}$ ($n$ times) $= \frac{n}{r}$. Therefore, $r_{eq} = \frac{r}{n}$.

Current in the External Circuit:

If this combination is connected to an external resistance $R$, the total current $I$ flowing through the circuit is:

2. Parallel Connection (Non-identical Cells):

If cells are non-identical (different EMFs $E_1, E_2, dots, E_n$ and internal resistances $r_1, r_2, dots, r_n$), the calculation is more complex. Using Kirchhoff's laws, it can be shown that:

Important Note for Non-identical Parallel Cells: If cells with different EMFs are connected in parallel, internal currents will flow between them even without an external load. This leads to energy dissipation and reduces the overall efficiency. Hence, parallel connection is most effective when cells are identical or very closely matched in EMF.

Condition for Maximum Current in Parallel:

For $n$ identical cells in parallel, $I = \frac{E}{R + \frac{r}{n}}$.

If $R \gg \frac{r}{n}$ (external resistance is much larger than total internal resistance), then $I \approx \frac{E}{R}$. In this case, connecting cells in parallel does not significantly increase the current beyond what a single cell provides.
If $R \ll \frac{r}{n}$ (external resistance is much smaller than total internal resistance), then $I \approx \frac{E}{r/n} = \frac{nE}{r}$. In this case, the current increases proportionally with $n$. This is the scenario where parallel connection is most beneficial, as it allows for a large current delivery by effectively reducing the internal resistance.

Mixed Grouping of Cells

Sometimes, cells are arranged in a combination of series and parallel connections to achieve a desired balance of voltage and current capacity. Consider a mixed grouping where there are $m$ rows, and each row contains $n$ identical cells connected in series. All $m$ rows are then connected in parallel.

EMF of one series row: — $E_{row} = nE$
Internal resistance of one series row: — $r_{row} = nr$

Now, these $m$ rows (each with $E_{row}$ and $r_{row}$) are connected in parallel. Since all rows are identical, the equivalent EMF of the entire combination will be the EMF of one row:

Equivalent EMF ($E_{eq}$): — $E_{eq} = nE$
Equivalent Internal Resistance ($r_{eq}$): — These $m$ identical rows in parallel will have an equivalent internal resistance of $r_{eq} = \frac{r_{row}}{m} = \frac{nr}{m}$.

Current in the External Circuit:

If this mixed grouping is connected to an external resistance $R$, the total current $I$ is:

Condition for Maximum Current in Mixed Grouping:

For maximum current to be drawn from a mixed grouping of cells, the external resistance $R$ should be equal to the equivalent internal resistance of the combination. This is a direct application of the maximum power transfer theorem. Thus, for maximum current:

Real-World Applications

Automotive Batteries: — Car batteries often consist of multiple lead-acid cells connected in series (typically 6 cells of 2V each to provide 12V). Some high-capacity batteries might also use parallel connections of these series strings.
Flashlights and Remotes: — Devices requiring higher voltage often use cells in series (e.g., two 1.5V AA cells in series for 3V).
Power Banks and Electric Vehicles: — These often use numerous lithium-ion cells arranged in complex series-parallel configurations to achieve the required voltage and high current capacity (e.g., '18650' cells in '3S2P' configuration means 3 cells in series and 2 such series strings in parallel).
Solar Panels: — Photovoltaic cells are connected in series to increase voltage and then multiple series strings are connected in parallel to increase current capacity, forming a solar panel.

Common Misconceptions

1
Ignoring Internal Resistance: — Many students initially overlook internal resistance, treating cells as ideal voltage sources. This leads to incorrect current and terminal voltage calculations, especially when the external resistance is comparable to or smaller than the internal resistance.
2
Incorrect Polarity in Series: — Forgetting that reversed polarity in series subtracts EMFs, not adds them. Internal resistances, however, always add up.
3
Parallel Connection of Non-identical Cells: — Assuming that non-identical cells in parallel will simply average their EMFs or that their equivalent EMF is the highest one. This is incorrect; internal currents will flow, and the equivalent EMF is a weighted average as derived earlier.
4
Confusing Voltage and Current Capacity: — Believing that series connections increase current capacity or parallel connections increase voltage. Series increases voltage, parallel increases current capacity (for identical cells) and reduces effective internal resistance.

NEET-Specific Angle

NEET questions on this topic typically involve:

Calculating equivalent EMF and internal resistance for series, parallel, or mixed groupings.
Calculating the current drawn from a combination of cells by an external resistor.
Identifying the conditions for maximum current or power transfer.
Conceptual questions about the advantages and disadvantages of series vs. parallel connections under different external load conditions.
Problems involving one or more cells connected in reverse polarity in a series circuit.

Mastering the derivations and the conditions for maximum current is crucial for NEET success.