Series and Parallel Circuits - Combined Resistance

Objective

To calculate the combined resistance of two resistors connected in parallel.

Background

In a series circuit, components are connected end-to-end, so the current is the same through each component, but the total resistance increases. In a parallel circuit, components are connected side-by-side, so the voltage across each component is the same, and the total resistance decreases.

Parallel Circuits

When resistors are connected in parallel, they provide multiple paths for the current to flow. This results in a lower overall resistance compared to having the same resistors in series.

Formula for Combined Resistance (Parallel Resistors)

The combined resistance ($R_{eq}$) of two resistors in parallel is calculated using the following formula:

$ \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} $

Where:

• $R_{eq}$ is the equivalent resistance of the two resistors in parallel.
• $R_1$ is the resistance of the first resistor.
• $R_2$ is the resistance of the second resistor.

To find the equivalent resistance ($R_{eq}$), you can follow these steps:

1. Calculate the reciprocal of each individual resistance ($ \frac{1}{R_1}$ and $ \frac{1}{R_2}$).
2. Add the reciprocals together.
3. Take the reciprocal of the sum to find the equivalent resistance ($R_{eq}$).

Example Calculation

Consider two resistors connected in parallel with resistances $R_1 = 10 \, \Omega$ and $R_2 = 20 \, \Omega$. Calculate the combined resistance ($R_{eq}$).

Step	Calculation	Result
Calculate reciprocal of $R_1$	$ \frac{1}{R_1} = \frac{1}{10 \, \Omega} = 0.1 \, \Omega^{-1} $
Calculate reciprocal of $R_2$	$ \frac{1}{R_2} = \frac{1}{20 \, \Omega} = 0.05 \, \Omega^{-1} $
Add the reciprocals	$ \frac{1}{R_{eq}} = \frac{1}{10 \, \Omega} + \frac{1}{20 \, \Omega} = 0.1 \, \Omega^{-1} + 0.05 \, \Omega^{-1} = 0.15 \, \Omega^{-1} $
Take the reciprocal of the sum	$ R_{eq} = \frac{1}{0.15 \, \Omega^{-1}} = \frac{1}{0.15} \, \Omega = \frac{100}{15} \, \Omega = \frac{20}{3} \, \Omega \approx 6.67 \, \Omega $

Therefore, the combined resistance of the two resistors in parallel is approximately $6.67 \, \Omega$.

Key Points

• The combined resistance of resistors in parallel is always less than the value of the smallest individual resistance.
• The more resistors added in parallel, the lower the combined resistance will be.