Question

Using source conversion, convert the given circuit into an equivalent circuit containing a single resistance and voltage source.

Answer 1

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Mastering Circuit Analysis: A Step-by-Step Guide to Source Transformation

Source transformation, also known as source conversion, is a fundamental technique in electronic circuit analysis used to simplify complex circuits. It allows you to convert a voltage source in series with a resistor into an equivalent current source in parallel with the same resistor, or vice versa. This method is an application of Thevenin's and Norton's theorems and is invaluable for making circuit problems easier to solve.

Let's walk through an example step-by-step, based on the provided image, to find an equivalent circuit with a single voltage source and a single resistor.

The Problem

Q. Using source conversion, convert the given circuit into an equivalent circuit containing a single resistance and voltage source.

The initial circuit consists of four parallel branches between terminals A and B:
1. An 8 Ω resistor.
2. A 6 A current source, with the current flowing towards terminal A.
3. A 3 V voltage source (positive terminal at A) in series with a 1 Ω resistor.
4. A 12 V voltage source (positive terminal at A) in series with a 4 Ω resistor.

Step-by-Step Solution with Corrections

We will simplify the circuit by converting the voltage sources into current sources, combining the parallel components, and then converting the result back into a final voltage source circuit.

Step 1: Convert the 3V Voltage Source to a Current Source

A voltage source (V) in series with a resistor (R) can be converted into a current source (I) in parallel with the same resistor.

• Rule: I = V / R
• The direction of the current source's arrow points towards the positive terminal of the voltage source.

For this branch:
V = 3 V
R = 1 Ω
* I = 3 V / 1 Ω = 3 A

Since the positive terminal of the 3V source is at the top (towards A), the 3A current source will also point up towards A. The 1Ω resistor is now in parallel with this new source.

Step 2: Convert the 12V Voltage Source to a Current Source

We apply the same rule to the fourth branch.

• V = 12 V
• R = 4 Ω
• I = 12 V / 4 Ω = 3 A

Correction Note: The positive terminal of the 12V source is also at the top (towards A). Therefore, the resulting 3A current source must also point upwards towards A. The provided image incorrectly shows this current source pointing down.

Step 3: Combine All Parallel Current Sources

After the transformations, our circuit has three parallel current sources and three parallel resistors. Let's combine the current sources. All three sources point in the same direction (upwards, towards A), so we can simply add their values.

• I_total = (Original 6A source) + (Converted 3A source) + (Converted 3A source)
• I_total = 6 A + 3 A + 3 A = 12 A

Correction Note: The calculation in the image, I = 6 + 3 - 3 = 6 A, is incorrect because it is based on the wrongly directed current source in Step 2. The correct total equivalent current is 12 A, flowing towards terminal A.

Step 4: Combine All Parallel Resistors

Now, let's find the equivalent resistance (Req) of the three parallel resistors (8Ω, 1Ω, and 4Ω).

• Formula: 1/Req = 1/R1 + 1/R2 + 1/R3
• 1/Req = 1/8 + 1/1 + 1/4
• To add these fractions, we find a common denominator, which is 8:
  1/Req = 1/8 + 8/8 + 2/8 = 11/8
• Therefore, the equivalent resistance is:
  Req = 8/11 Ω (approximately 0.727 Ω)

At this point, we have simplified the circuit to its Norton equivalent: a 12 A current source in parallel with an 8/11 Ω resistor.

Step 5: Convert the Final Current Source to a Voltage Source

The final step is to convert this equivalent current source circuit back into a voltage source circuit (Thevenin equivalent) as requested.

• Rule: V = I * R
• The positive terminal of the new voltage source will be at the end where the current arrow points (terminal A).

Using our corrected values:
V_eq = I_total Req
V_eq = 12 A (8/11) Ω = 96/11 V
* V_eq ≈ 8.73 V

Correction Note: The image's final calculation, V = 6 A * (8/11) Ω = 4.36 V, is incorrect as it uses the wrong current value from Step 3.

Conclusion: The Final Equivalent Circuit

By correctly applying the rules of source transformation, the original complex circuit is simplified to an equivalent circuit containing:

• A single voltage source of 96/11 V (approximately 8.73 V), with the positive terminal at A.
• A single series resistor of 8/11 Ω (approximately 0.727 Ω).

This example highlights the power of source transformation and underscores the importance of carefully checking the polarity of voltage sources and the direction of current sources at every step to ensure an accurate result.