Problem 15-17    Source:    Problem developed by the site author.

In the circuit shown in Figure 15-17.1, using the theorem of Thévenin , determine the value R so that it dissipates the maximum power.

Solution of the Problem 15-17

To use the Thevenin theorem, we must remove the resistor R from the circuit. Thus, the pair of terminals a and b appears. In these terminals we must determine the voltage of Thévenin in open circuit. For this, and based on the circuit shown in Figure 15-17.2 , using a current divider we can determine the values of I₁ and I₂. So

Performing the calculation we find the value of I₁, or

We use the same method to find the value of I₂.

Performing the calculation we find the value of I₂, or

Naturally, knowing the relation 10 = I₁ + I₂, we also find I₂ = 10 - I₁ = 6.75 A.

Now, knowing the values of I₁ and I₂ we can calculate the potential at terminals a and b, remembering that by resistors 4 Ω and 2.25 Ω current does not circulate. Then, making the mesh highlighted in orange, as shown in Figure 15-17.2, we have:

Then, we have:

To determine the value of R_th , we must eliminate the current source and after some arrangement in the circuit, the Figure 15-17.3 shows the new topology.

Through the circuit, it is verified that the resistors of 100 Ω and 150 Ω are in series, as well as the resistors of 120 Ω and 30 Ω . Therefore, adding the values we find 250 Ω e 150 Ω. After the calculation, we verify that these two resistors are in parallel. Then the equivalent resistance of the parallel will be:

Now, this equivalent resistance is in series with the resistors of 4 Ω and 2.25 Ω. Therefore, the resistance of Thévenin is

In the Figure 15-17.4 we see the final result. Notice that point b has a greater potential than the point a . To determine the value of R that dissipates the greatest power, we will use the maximum power transfer theorem. This theorem says that for maximum transfer of power for the load, it must have the same value as the resistance of Thévenin . Therefore