Problem 15-10    Source:    Problem elaborated by the author of the site.

Determine the Thévenin equivalent for the terminals a - b of the circuit shown in Figure 15-10.1. Subsequently, find the value of R for maximum power transfer and calculate this power.

Solution of the Problem 15-10   -    Thévenin/Norton Method

First let's do some source transformations to make the circuit simpler and easier to compute the Thévenin equivalent. So let's transform the current source of 6.75 A, which is in parallel with the 8 ohms resistor, into a voltage source of 8 x (6.75) = 54 volts. In this transformation we get the resistors of 8 and 10 ohms in series. This series results in a resistor of 18 ohms. In other words, we get a voltage source of 54 volts in series with an 18 ohms resistor. This enables a new source transformation, resulting in a current source of 3 A in parallel with the 18 ohms resistor.

In the same way we can proceed with the voltage source V / 3 in series with the 12 ohms resistor. In the transformation, we have a current source V / 36 in parallel with the 12 ohms resistor. See the Figure 15-10.2.

Note that in this circuit three resistors appeared in parallel. Therefore, by making the parallel we find an equivalent resistance of 6 ohms . So we were able to reduce the circuit to a more suitable configuration to find the Thevenin equivalent , as we can see on the right side of the Figure 15-10.2.

Working with the right circuit, we can find the value of Thévenin's Voltage by making the mesh. With this, we have:

Solving this equation we find the desired voltage, or:

To calculate the Thévenin resistance there are two possible methods to apply. Let's go study each of them separately.

Method 1

Let us take as a reference the circuit shown in the Figure 15-10.3. As a first step let's put the output, that is, the terminals a-b, in short circuit. Let's calculate the current of Norton (I_n) which passes between the terminal a and the b terminal through the short. Please note that by placing the output short, we have V = 0.

In this way, the dependent current source is canceled, leaving only the current source independent whose value is 3 A. However, to calculate I_n simply apply a current divider because the two resistors are in parallel. Then:

Now, to calculate the Thévenin resistance, simply obtain the quotient between the Thévenin voltage and the Norton current, or:

Method 2

Another way to calculate Thévenin's resistance is to use the traditional technique of eliminating all independent sources in the circuit, staying only dependent sources, if any. In this case, the independent source was deleted.

Since we have a dependent source, then we need to introduce an independent source on the a-b terminals. Thus, we have the circuit shown in the Figure 15-10.4. Note that for the source introduced in the circuit we choose a voltage of value equal to 1 volt. It could be any other value. To find the value of the Thévenin resistance, we must calculate the value of I. Notice that by setting a font with value 1 volt, we have already defined the value of V = 1 volt. So, making the mesh by the source and the resistors, we have:

Solving the equation we find for I the value of:

Therefore, the Thévenin resistance will have the value of:

Exactly the same value found by the other method. Therefore, the circuit was reduced as shown in the Figure 15-10.5.

By the theory we know that for R to dissipate the maximum power must have the same value of Thévenin resistance, that is:

Therefore the power dissipated by R will be equal to: