Problem 23-7    Source:   Problem 3.9 - page 11 -     Prof. Dr. João Costa Freire - Instituto Superior Técnico - Portugal     Book: Analysis of Circuits - 2006. Available at: http://web.ist.utl.pt/pedro.m.s.oliveira/ProbAC.pdf

Determine the expression for V_o in the circuit shown in Figure 23-07.1, when key S is in position 2, knowing that it has been in position 1 for a long time.

Solution of the Problem 23-7

Note that this problem can be solved by finding the expression for i_L. To find the final result, we use the fact that

When the S switch is in the 1 position, we can easily see that i_A = 3 A. Thus, the dependent source assumes the value of 2 i_A = 6 volts . As the inductor behaves like a short circuit, we quickly find that

To calculate the value of i_L for time t → ∞, we should consider the S key in position 2. In this case, we must find the equivalent Thévenin the circuit. So, we must work with the circuit shown in Figure 23-7.2.

Note that we indicate the inductor as a short circuit. So, we must find the value of i_L (∞). To do this, we will calculate the values of i_A and i₁. Then, making the mesh indicated by the green arrow, we have the following relation:

On the other hand, by making the mesh indicated by the arrow blue, we obtain the following relation:

Algebraically working the equation eq. 23-07.3 and along with the equation eq. 23-07.2 we get a system of two equations with two unknowns that are easy to solve and the values found are:

So, we have to

To find the value of R_th we must put in place of the inductor a convenient current source, such as I = 3 A, as shown in Figure 23-7.3.

Note that we use the fact that if the i_A circulates in the 4 ohm resistor, then in the 2 ohm resistor should circulate twice the current or, 2 i_A. So, we easily find the value of i_A, because of the circuit we have to

To find the value of V_ab just make the mesh, or

But as i_A = 1 A , then performing the calculation we find

At this point, we can calculate the value of R_th, because

And now we are going to calculate the time constant of the circuit, that is

With all the calculated data, we can write the equation of i_L (t), or

Now using the equation  eq. 23-07.1   developed at the beginning of the solution of the problem, we have the solution requested in the statement of the problem, or