Question

For the circuit shown below, find the value of the load resistance \(R_L\) that will absorb the maximum amount of power from the source circuit.

• A. 5 \(\Omega\)
• B. 1.818 \(\Omega\)
• C. 7.333 \(\Omega\)
• D. 2.857 \(\Omega\)

Hint:

To apply the Maximum Power Transfer theorem, always focus on finding the Thevenin equivalent circuit (\(V_{TH}\) and \(R_{TH}\)) as seen by the load. For purely resistive circuits, maximum power is achieved when \(R_L = R_{TH}\).

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The question asks for the value of the load resistor \(R_L\) that ensures maximum power transfer (MPT).
Step 2: Key Formula or Approach:
The Maximum Power Transfer Theorem states that maximum power is delivered from a source network to a load resistor \(R_L\) when the value of \(R_L\) is equal to the Thevenin equivalent resistance (\(R_{TH}\)) of the source network as seen from the terminals of the load.
\[ R_L = R_{TH} \] To find \(R_{TH}\), we must deactivate all independent sources in the circuit (voltage sources are replaced by short circuits, and current sources by open circuits) and then calculate the equivalent resistance looking back into the load terminals.
Step 3: Detailed Explanation:
The calculation shown in the image is \(`R_TH\) = ... = 40/14 = 20/7 = 2.857 \(\Omega\)`. This numerical result corresponds to the parallel combination of the 10 \(\Omega\) and 4 \(\Omega\) resistors: \[ R_{eq} = 10 \Omega \parallel 4 \Omega = \frac{10 \times 4}{10 + 4} = \frac{40}{14} = \frac{20}{7} \Omega \approx 2.857 \, \Omega \] This suggests that when finding the Thevenin resistance, the 5 \(\Omega\) resistor is not part of the calculation. This would occur in a circuit topology where deactivating the 10V source also shorts out the 5 \(\Omega\) resistor. A possible circuit configuration for this is where the 5 \(\Omega\) resistor is directly in parallel with the 10V source. When the source is shorted, the 5 \(\Omega\) resistor is also shorted and can be ignored. Then, if the 10 \(\Omega\) and 4 \(\Omega\) resistors are in parallel with respect to the load terminals, we get the calculated \(R_{TH}\).
Assuming this interpretation is correct based on the provided answer: 1. Find the Thevenin Resistance (\(R_{TH}\)): We follow the result from the image's calculation. \[ R_{TH} = \frac{20}{7} \, \Omega \approx 2.857 \, \Omega \] 2. Apply the Maximum Power Transfer Theorem: For maximum power transfer, the load resistance must equal the Thevenin resistance. \[ R_L = R_{TH} = 2.857 \, \Omega \] Step 4: Final Answer:
To achieve maximum power transfer, the load resistance \(R_L\) must be equal to the Thevenin resistance of the circuit, which is calculated to be 2.857 \(\Omega\).