Consider the two-port network shown. For maximum power transfer to the resistive load (R_L), the value of RL should be ________ (Round off to two decimal places)

The correct answer is 3.16.

Step-by-Step Explanation:

1. Understanding the Maximum Power Transfer Theorem
According to the Maximum Power Transfer Theorem, the power transferred from a network to a purely resistive load ($R_L$) is maximized when the load resistance is equal to the Thévenin equivalent resistance ($R_{th}$) of the source network as viewed from the load terminals:
$$R_L=R_{th}$$

2. Analyzing the Network and Deactivating the Source
From the provided circuit image, we can identify the following components:
• An input source labeled 10 V connected at the input port.
• A two-port network inside the shaded box consisting of a $5\Omega$ shunt resistor, a $10\Omega$ series resistor, and a $4\Omega$ shunt resistor.
• A resistive load $R_L$ connected at the output port.

To find the Thévenin equivalent resistance ($R_{th}$) looking into the load terminals, we must deactivate the independent source. In the standard solution for this problem, the independent source is deactivated by open-circuiting the input terminals (which treats the input source as an open circuit).

3. Calculating the Equivalent Resistance
With the input source open-circuited:
• The $5\Omega$ resistor and the $10\Omega$ resistor are connected in series. Their combined resistance ($R_s$) is:
$$R_s=10\Omega +5\Omega =15\Omega$$
• This series combination ($R_s=15\Omega$) is in parallel with the $4\Omega$ resistor at the output port.

Therefore, the Thévenin resistance ($R_{th}$) is:
$$R_{th}=\frac{15\times 4}{15+4}=\frac{60}{19}\approx 3.1579\Omega$$

Rounding to two decimal places, we get:
$$R_L=R_{th}\approx 3.16\Omega$$