Parallel Cells Connection Overview

When two or more cells are connected in parallel, their positive terminals are joined together, and their negative terminals are joined together. This type of connection is commonly used to increase the current capacity of the power source while maintaining a consistent voltage (EMF) across the combination. However, if the cells have different electromotive forces (EMFs) and internal resistances, the equivalent EMF and equivalent internal resistance of the combination need to be calculated carefully.

Understanding Equivalent EMF

The concept of equivalent EMF (electromotive force) and equivalent internal resistance allows us to replace a complex arrangement of cells with a single, simplified cell that behaves identically to the original combination when connected to an external circuit. This simplifies circuit analysis significantly.

Derivation of Equivalent EMF for Parallel Cells

Consider two cells, Cell 1 and Cell 2, connected in parallel. Let their EMFs be \(\epsilon_1\) and \(\epsilon_2\), and their internal resistances be \(r_1\) and \(r_2\), respectively.

Let \(A\) and \(B\) be the two common points where the cells are connected in parallel. The potential difference across these two points, \(V_{AB}\), will be the same for both cells.

Current Distribution in Parallel Cells

For Cell 1, the potential difference \(V_{AB}\) can be expressed as:

$$V_{AB} = \epsilon_1 - I_1 r_1$$

Where \(I_1\) is the current flowing through Cell 1. From this, we can express \(I_1\) as:

$$I_1 = \frac{\epsilon_1 - V_{AB}}{r_1} \quad \text{(Equation 1)}$$

Similarly, for Cell 2, the potential difference \(V_{AB}\) is:

$$V_{AB} = \epsilon_2 - I_2 r_2$$

Where \(I_2\) is the current flowing through Cell 2. From this, we can express \(I_2\) as:

$$I_2 = \frac{\epsilon_2 - V_{AB}}{r_2} \quad \text{(Equation 2)}$$

Total Current and Equivalent Circuit

The total current \(I\) supplied by the parallel combination is the sum of the currents from individual cells:

$$I = I_1 + I_2 \quad \text{(Equation 3)}$$

Substitute Equation 1 and Equation 2 into Equation 3:

$$I = \left(\frac{\epsilon_1 - V_{AB}}{r_1}\right) + \left(\frac{\epsilon_2 - V_{AB}}{r_2}\right)$$

$$I = \frac{\epsilon_1}{r_1} - \frac{V_{AB}}{r_1} + \frac{\epsilon_2}{r_2} - \frac{V_{AB}}{r_2}$$

Rearrange the terms to group EMFs and potential differences:

$$I = \left(\frac{\epsilon_1}{r_1} + \frac{\epsilon_2}{r_2}\right) - V_{AB}\left(\frac{1}{r_1} + \frac{1}{r_2}\right)$$

Now, we want to express \(V_{AB}\) in the form of an equivalent cell: \(V_{AB} = \epsilon_{eq} - I r_{eq}\). Let's rearrange the equation to solve for \(V_{AB}\):

$$V_{AB}\left(\frac{1}{r_1} + \frac{1}{r_2}\right) = \left(\frac{\epsilon_1}{r_1} + \frac{\epsilon_2}{r_2}\right) - I$$

Combine the fractions on the left and right sides:

$$V_{AB}\left(\frac{r_1 + r_2}{r_1 r_2}\right) = \left(\frac{\epsilon_1 r_2 + \epsilon_2 r_1}{r_1 r_2}\right) - I$$

Now, isolate \(V_{AB}\):

$$V_{AB} = \frac{\left(\frac{\epsilon_1 r_2 + \epsilon_2 r_1}{r_1 r_2}\right)}{\left(\frac{r_1 + r_2}{r_1 r_2}\right)} - I \frac{1}{\left(\frac{r_1 + r_2}{r_1 r_2}\right)}$$

Simplify the expression:

$$V_{AB} = \frac{\epsilon_1 r_2 + \epsilon_2 r_1}{r_1 + r_2} - I \left(\frac{r_1 r_2}{r_1 + r_2}\right)$$

Comparing this equation with the standard form \(V_{AB} = \epsilon_{eq} - I r_{eq}\), we can identify the equivalent EMF (\(\epsilon_{eq}\)) and the equivalent internal resistance (\(r_{eq}\)).

The equivalent EMF of the parallel combination is:

$$\epsilon_{eq} = \frac{\epsilon_1 r_2 + \epsilon_2 r_1}{r_1 + r_2}$$

And the equivalent internal resistance is:

$$r_{eq} = \frac{r_1 r_2}{r_1 + r_2}$$

This formula for \(r_{eq}\) is consistent with two resistors connected in parallel: \(\frac{1}{r_{eq}} = \frac{1}{r_1} + \frac{1}{r_2}\).

Summary of Parallel Cell Characteristics

For two cells connected in parallel:

• Equivalent EMF: The equivalent EMF is a weighted average of the individual EMFs, weighted by the reciprocal of their internal resistances.
• Equivalent Internal Resistance: The equivalent internal resistance is calculated just like parallel resistors, i.e., the reciprocal of the sum of the reciprocals of individual internal resistances.

Therefore, based on our derivation, the equivalent EMF of the parallel combination is \(\frac{\epsilon_1 r_2 + \epsilon_2 r_1}{r_1 + r_2}\).