Question

In the circuit below, use a source transformation to determine Vo.

Answer 1

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How to Use Source Transformation to Find Vo in a Circuit: A Step-by-Step Guide

In electrical engineering, simplifying complex circuits is a fundamental skill. One of the most powerful techniques for this is source transformation, which allows you to replace a voltage source with an equivalent current source, or vice versa. This guide will walk you through a classic example, demonstrating how to use source transformation to find the output voltage (Vo) in a given circuit.

The Problem

We are given the following electrical circuit and asked to determine the voltage Vo across the 8Ω resistor using the source transformation method.

Original Circuit:

The circuit contains:
A 3 Ampere (A) ideal current source.
A 4 Ohm (Ω) resistor in parallel with the current source.
A 2 Ω resistor.
An 8 Ω resistor, across which the output voltage Vo is measured.

The Solution: A Detailed Walkthrough

We will solve this problem in two main steps: first by applying source transformation to simplify the circuit, and then by analyzing the new, simpler circuit to find Vo.

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Step 1: Convert the Current Source to a Voltage Source

The core of the source transformation technique lies in converting between source types. The rule is:

A current source (I) in parallel with a resistor (R) can be replaced by a voltage source (V) in series with the same resistor (R). The voltage is calculated using Ohm's Law: V = I × R.

In our circuit, we have a 3A current source in parallel with a 4Ω resistor.

1. Calculate the equivalent voltage (V):
   V = I × R
   V = 3 A × 4 Ω
   * V = 12 V

2. Redraw the circuit:
   Replace the 3A current source with a new 12V voltage source.
   The 4Ω resistor, which was in parallel, is now placed in series with the 12V source.
   * The polarity of the voltage source is determined by the direction of the original current source. Since the arrow pointed up, the positive terminal of the voltage source is on top.

The transformed circuit now looks like this:

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Step 2: Find the Output Voltage (Vo) in the Simplified Circuit

After the transformation, we have a much simpler series circuit. All components (the 12V source, 4Ω resistor, 2Ω resistor, and 8Ω resistor) are now connected in a single loop.

1. Calculate the total resistance (R_total) of the series circuit:
   In a series circuit, resistances add up.
   R_total = 4 Ω + 2 Ω + 8 Ω
   * R_total = 14 Ω

2. Calculate the total current (I) flowing through the loop:
   Using Ohm's Law for the entire circuit (I = V / R_total):
   I = 12 V / 14 Ω
   * I = 6/7 A

3. Calculate the output voltage (Vo):
   The voltage Vo is the voltage drop across the 8Ω resistor. We can find this using Ohm's Law again (Vo = I × R).
   Vo = I × R_8Ω
   Vo = (6/7 A) × 8 Ω
   Vo = 48/7 V

Converting this fraction to a decimal gives us the final answer.

Final Answer

The output voltage Vo across the 8Ω resistor is:

Vo = 6.857 V

By applying source transformation, we converted a more complex parallel-series circuit into a simple series loop. This simplification made it easy to calculate the current and, ultimately, the desired output voltage Vo. This example highlights why source transformation is an essential tool in circuit analysis.