Series and Parallel Circuits - Combined Resistance

Objective

To calculate the combined resistance of two or more resistors in series.

Series Circuits

In a series circuit, components are connected end-to-end, forming a single path for the current. The current is the same through all components in a series circuit. The total resistance in a series circuit is the sum of the individual resistances.

The formula for calculating the combined resistance (R_total) of resistors in series is:

$$R_{total} = R_1 + R_2 + R_3 + ... + R_n$$

where:

• R₁, R₂, R₃, ... , R_n are the individual resistances of the resistors in the series circuit.

Example: Series Circuit

Consider three resistors connected in series with resistances R₁ = 10Ω, R₂ = 20Ω, and R₃ = 30Ω. The total resistance is:

$$R_{total} = 10Ω + 20Ω + 30Ω = 60Ω$$

Parallel Circuits

In a parallel circuit, components are connected across each other, providing multiple paths for the current. The voltage across each component in a parallel circuit is the same. The reciprocal of the total resistance in a parallel circuit is the sum of the reciprocals of the individual resistances.

The formula for calculating the combined resistance (R_total) of resistors in parallel is:

$$\frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + ... + \frac{1}{R_n}$$

This can be rearranged to find the total resistance:

$$R_{total} = \frac{1}{\frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + ... + \frac{1}{R_n}}$$

Example: Parallel Circuit

Consider three resistors connected in parallel with resistances R₁ = 10Ω, R₂ = 20Ω, and R₃ = 30Ω. The total resistance is:

$$R_{total} = \frac{1}{\frac{1}{10Ω} + \frac{1}{20Ω} + \frac{1}{30Ω}}$$

$$R_{total} = \frac{1}{\frac{6}{60} + \frac{3}{60} + \frac{2}{60}}$$

$$R_{total} = \frac{1}{\frac{11}{60}}$$

$$R_{total} = \frac{60}{11}Ω \approx 5.45Ω$$

Summary

Understanding the difference between series and parallel circuits is crucial for calculating combined resistances. In series circuits, resistances add up. In parallel circuits, the reciprocal of the sum of the reciprocals of the resistances gives the total resistance.

Circuit Type	Formula for Combined Resistance
Series	$R_{total} = R_1 + R_2 + R_3 + ... + R_n$
Parallel	$R_{total} = \frac{1}{\frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + ... + \frac{1}{R_n}}$