The principle of maximum average power transfer states that the load resistance should be equal to the Thevenin equivalent resistance of the source network (looking from the load terminals) to achieve maximum power absorption by the load.

🪵Rough Schematic

Steps to Solve

1. Find the Thevenin Equivalent Resistance ($R_{th}$)

   To find $R_{th}$, we need to:

   • Remove the load resistor ($R_L$) from the circuit.

   • Zero the independent voltage source: Replace the 60∠53.2° V source with a short circuit.

   • Calculate the equivalent resistance looking into the open terminals where RL was connected. In this case, it's simply the given impedance:

     $R_{th}$ = 3 + j4 Ω

   • Important Note: For maximum power transfer, the load impedance should be the conjugate of the Thevenin impedance. However, since we're asked for a purely resistive load, we take the real part of the Thevenin impedance.

2. Determine the Load Resistance ($R_L$)

   For maximum average power transfer, the load resistance RL should be equal to the real part of the Thevenin impedance:

   $R_L$ = Re($R_{th}$) = 3 Ω

Therefore, the resistance of the purely resistive load must be 3 ohms to absorb the maximum average power.

Additional Insights (Not Required for the Problem, but Useful)

• Maximum Power Calculation: Once you have RL, you can calculate the maximum average power delivered to the load using the formula:

  $$ P_{max} = (V_{th}^2) / (4 * R_L) $$

  Where $V_{th}$ is the Thevenin voltage across the load terminals (you'd calculate this with the original circuit before removing $R_L$).

🧠Verified by Python

https://gist.github.com/viadean/8e184421b447723a0bc3c57ff0cda286

Explanation and How to Use

1. Import cmath: This module is necessary for complex number operations.
2. Thevenin Impedance: The code defines r_th (real part) and x_th (imaginary part/reactance) and combines them into a complex number z_th representing the Thevenin impedance.
3. Load Resistance: The load resistance r_load is set to the real part of the Thevenin impedance (r_th) as explained earlier.
4. (Optional) Maximum Power Calculation:
   • Crucially, you MUST replace the example Thevenin voltage (v_th) with your actual calculated Thevenin voltage. I've shown how to create a complex number from magnitude and angle (in degrees). You need to determine Vth from the original circuit using circuit analysis techniques (like mesh analysis, nodal analysis, or source transformations).
   • The code calculates p_max using the formula.
5. Output: The code prints the calculated load resistance and the maximum average power.

How to Run the Code:

1. Save the code as a .py file (e.g., max_power.py).
2. Run it from your terminal using python max_power.py.

Key Point: The most important part is to correctly calculate the Thevenin voltage (v_th) for your specific circuit. The rest of the code provides a framework for the calculation.