Thevenin's Theorem and Maximum Power Transfer

In electrical circuits, understanding how to deliver the maximum possible power to a load is crucial. This concept is addressed by the Maximum Power Transfer Theorem. This theorem is often used in conjunction with Thevenin's theorem, which simplifies a complex circuit into a simpler equivalent circuit consisting of a single voltage source and a single series resistance.

Maximum Power Transfer Condition

The Maximum Power Transfer Theorem states that for a given source, maximum power will be delivered to the load when the load resistance ($R_L$) is exactly equal to the Thevenin's equivalent resistance ($R_{Th}$) of the source network looking from the load terminals.

• Condition for maximum power transfer: $R_L = R_{Th}$

Calculating Maximum Power Delivered

Once the condition for maximum power transfer is met, the maximum power ($P_{max}$) delivered to the load can be calculated using the Thevenin's voltage ($V_{Th}$) and Thevenin's resistance ($R_{Th}$). The formula for maximum power is:

$$P_{max} = \frac{V_{Th}^2}{4R_{Th}}$$

Let's use the given values to calculate the maximum power delivered to the load in the network.

• Thevenin's voltage ($V_{Th}$) = 89.3 volts
• Thevenin's resistance ($R_{Th}$) = 46.98 ohms

Step-by-Step Power Calculation

Substitute the given values into the maximum power transfer formula:

$$P_{max} = \frac{(89.3 \, \text{V})^2}{4 \times 46.98 \, \Omega}$$

First, calculate the square of the Thevenin's voltage:

$$(89.3)^2 = 7974.49$$

Next, calculate the denominator:

$$4 \times 46.98 = 187.92$$

Now, divide the squared voltage by the calculated denominator:

$$P_{max} = \frac{7974.49}{187.92}$$

$$P_{max} \approx 42.433 \, \text{W}$$

Conclusion on Maximum Power

The calculated maximum power delivered to the load present in the network is approximately 42.43 Watts. This aligns with one of the provided options.