Problem 14-6    Source:    Adapted from Problem 4.98 - page 155 -    NILSSON & RIEDEL -    Book: Electric Circuits - 10th edition - Ed. Pearson Education of Brazil - 2015.

In the circuit shown in Figure 14-06.1, calculate the value of V_x and i using the Superposition method.

Solution of the Problem 14-6   -    Superposition Method

We must pay attention to the fact that the circuit has a controlled or dependent voltage source. In this case, we cannot exclude this source from the circuit to use the superposition method. So, starting by excluding the 90 V source, we are left with the circuit shown in Figure 14-06.2. Note that we changed the circuit topology a little to make the calculations easier. To differentiate the calculation of currents with different sources, we will assume the index a for the calculation with the first source and the index b for the calculation with the second source.

Note from the circuit shown in the figure above that the resistor R₃ is between the dependent source 2.5 V_x and the voltage across the resistor R₁. With this in mind, we can easily conclude that across the resistor R₃ we have a voltage drop equal to 3.5 V_x, as indicated in the figure above. This way, we easily calculate the value of V_x by applying the Nodal method to node b.

Remembering that for this case we have V_b = - V_x. Using this relationship in the equation above, we find:

Applying LKC to node c we will find a relationship between V_x and i_a. Then:

So this is the contribution of the 40 V source to the current in the dependent source. Now let's calculate the source contribution of 90 V. To do this we "kill" the 40 V source, obtaining the circuit shown in Figure 14-06.3.

Applying the Nodal method to node b, we have:

On the other hand, making the external mesh of the circuit, we obtain:

Substituting this relation into the previous one we obtain the value of V_x, or:

Substituting this value in the previous relationship we obtain the value of V_b, or:

And with these values we can easily calculate the values of I₁ and I₂, or:

And from the circuit shown in Figure 14-06.3 we know that i_b = I₁ + I₂ . Then:

Now that we know the contribution of each source separately we can calculate the value of the current i through the dependent source, that is: i = i_a + i _b. Then:

And to find the value of V_x just subtract the values of the two independent voltage sources. That way: