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Find the Equivalent Resistance Between A and B

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When it comes to understanding electrical circuits, one of the fundamental concepts is finding the equivalent resistance between two points. Whether you are an electrical engineer, a student studying physics, or simply curious about how circuits work, this article will provide you with a comprehensive understanding of how to find the equivalent resistance between points A and B.

Understanding Resistance

Before diving into the specifics of finding the equivalent resistance, it is important to have a clear understanding of what resistance is. In simple terms, resistance is the measure of opposition to the flow of electric current in a circuit. It is denoted by the symbol R and is measured in ohms (Ω).

Resistance can be influenced by various factors, such as the material of the conductor, its length, cross-sectional area, and temperature. Different components in a circuit, such as resistors, capacitors, and inductors, contribute to the overall resistance of the circuit.

Series and Parallel Connections

When resistors are connected in a circuit, they can be arranged in two different ways: series and parallel connections. Understanding these connections is crucial for finding the equivalent resistance between two points.

Series Connection

In a series connection, resistors are connected end-to-end, forming a single path for the current to flow. The total resistance in a series connection is the sum of the individual resistances. Mathematically, it can be represented as:

RTotal = R1 + R2 + R3 + … + Rn

For example, consider a circuit with three resistors connected in series: R1 = 10Ω, R2 = 20Ω, and R3 = 30Ω. The equivalent resistance (RTotal) can be calculated as:

RTotal = 10Ω + 20Ω + 30Ω = 60Ω

Parallel Connection

In a parallel connection, resistors are connected side by side, providing multiple paths for the current to flow. The total resistance in a parallel connection can be calculated using the following formula:

1/RTotal = 1/R1 + 1/R2 + 1/R3 + … + 1/Rn

Using the same example as before, let’s assume the resistors are connected in parallel: R1 = 10Ω, R2 = 20Ω, and R3 = 30Ω. The equivalent resistance (RTotal) can be calculated as:

1/RTotal = 1/10Ω + 1/20Ω + 1/30Ω

Simplifying the equation gives:

1/RTotal = 0.1 + 0.05 + 0.0333

1/RTotal = 0.1833

Finally, by taking the reciprocal of both sides, we find:

RTotal = 1/0.1833 ≈ 5.45Ω

Combining Series and Parallel Connections

In real-world circuits, it is common to have a combination of series and parallel connections. To find the equivalent resistance in such cases, it is important to follow a systematic approach.

Step 1: Identify Series and Parallel Sections

Start by identifying the series and parallel sections in the circuit. This can be done by visually inspecting the circuit diagram or by analyzing the connections between the resistors.

Step 2: Simplify the Series Sections

For each series section, calculate the total resistance using the formula mentioned earlier. Replace the series section with a single resistor having the calculated resistance.

Step 3: Simplify the Parallel Sections

For each parallel section, calculate the total resistance using the formula mentioned earlier. Replace the parallel section with a single resistor having the calculated resistance.

Step 4: Repeat Steps 2 and 3

If there are still series or parallel sections remaining, repeat steps 2 and 3 until the entire circuit is simplified to a single equivalent resistor.

Example Circuit

Let’s consider an example circuit to illustrate the process of finding the equivalent resistance between points A and B.

Circuit Diagram

In this circuit, R1 = 10Ω, R2 = 20Ω, R3 = 30Ω, R4 = 40Ω, and R5 = 50Ω. We need to find the equivalent resistance between points A and B.

Step 1: Identify Series and Parallel Sections

By analyzing the circuit, we can identify the following series and parallel sections:

  • R1 and R2 are in series
  • R3 and R4 are in parallel
  • The combination of R3, R4, and R5 is in series with R2

Step 2: Simplify the Series Sections

For the series section R1 and R2, the total resistance is:

RTotal1 = R1 + R2 = 10Ω + 20Ω = 30Ω

Replace R1 and R2 with a single resistor having a resistance of 30Ω.

Simplified Circuit Diagram 1

Step 3: Simplify the Parallel Sections

For the parallel section R3 and R4, the total resistance is:

1/RTotal2 = 1/R3 + 1/R4 = 1/30Ω + 1/40Ω

Simplifying the equation gives:

1/RTotal2 = 0.0333 + 0.025

1/RTotal2 = 0.0583

Finally, by taking the reciprocal of both sides, we find:

RTotal2 = 1/0.0583 ≈ 17.14Ω

Replace R3 and R4 with a single resistor having a resistance of 17.14Ω.

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