Combination of Resistances (Series and Parallel)
Introduction
In electrical circuits, resistors can be connected in different configurations to achieve desired resistance values. The two most common configurations are series and parallel. Understanding how resistances combine in these configurations is essential for circuit design and analysis. In this guide, we will explore the concepts of series and parallel resistances and provide numerical examples to demonstrate their calculation.
Series Combination of Resistances
When resistors are connected end-to-end, they are said to be in series. In a series circuit, the total resistance is the sum of the individual resistances. The current flowing through each resistor is the same, but the voltage drop across each resistor can vary.
The formula for the total resistance in a series circuit is:
Rtotal = R1 + R2 + R3 + … + Rn
Parallel Combination of Resistances
When resistors are connected side-by-side, they are said to be in parallel. In a parallel circuit, the total resistance is less than the smallest individual resistance. The voltage across each resistor is the same, but the current can vary.
The formula for the total resistance in a parallel circuit is:
1/Rtotal = 1/R1 + 1/R2 + 1/R3 + … + 1/Rn
Numerical Problems
Example 1: Series Combination
Problem: Three resistors of 5 Ω, 10 Ω, and 15 Ω are connected in series. Find the total resistance.
Solution:
Using the series resistance formula:
Rtotal = R1 + R2 + R3
Rtotal = 5 Ω + 10 Ω + 15 Ω = 30 Ω
Answer: The total resistance is 30 Ω.
Example 2: Parallel Combination
Problem: Two resistors of 6 Ω and 12 Ω are connected in parallel. Find the total resistance.
Solution:
Using the parallel resistance formula:
1/Rtotal = 1/R1 + 1/R2
1/Rtotal = 1/6 Ω + 1/12 Ω
1/Rtotal = (2 + 1) / 12 = 3 / 12 = 1/4
Rtotal = 4 Ω
Answer: The total resistance is 4 Ω.
Example 3: Combination of Series and Parallel
Problem: A circuit consists of three resistors: 4 Ω and 6 Ω in series, and this combination is in parallel with a 12 Ω resistor. Find the total resistance.
Solution:
First, find the series combination of 4 Ω and 6 Ω:
Rseries = 4 Ω + 6 Ω = 10 Ω
Now, calculate the parallel combination with 12 Ω:
1/Rtotal = 1/Rseries + 1/12 Ω
1/Rtotal = 1/10 Ω + 1/12 Ω = (6 + 5) / 60 = 11/60
Rtotal = 60/11 ≈ 5.45 Ω
Answer: The total resistance is approximately 5.45 Ω.
Important Note:
In series combinations, the total resistance increases as more resistors are added. In parallel combinations, the total resistance decreases as more resistors are added.
Frequently Asked Questions (FAQs)
1. How do resistors in series affect current and voltage?
In a series circuit, the current remains the same through all resistors, while the voltage divides across each resistor.
2. How do resistors in parallel affect current and voltage?
In a parallel circuit, the voltage remains the same across all resistors, while the current divides according to the resistance of each resistor.
3. What happens to the total resistance if more resistors are added in series?
The total resistance increases when more resistors are added in series.
4. What happens to the total resistance if more resistors are added in parallel?
The total resistance decreases when more resistors are added in parallel.
5. Can a combination of series and parallel resistors be simplified?
Yes, complex resistor networks can often be simplified by reducing series and parallel groups step by step.
