Problem 23-6    Source:   Problem 3.5 - 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 i_L in the circuit shown in the Figure 23-06.1.

Solution of the Problem 23-6

Note that this problem uses an independent source through step function. In this way, we know that it will act only after t = 0. Before t = 0, its value is null. Then, the dependent font 1.5 i_A will also be null . So, we already know that

To find the current in the inductor for time t → ∞ , we must calculate the Thévenin equivalent to the left of the inductor. Removing the inductor from the circuit, we are left with the circuit shown in the Figure 23-6.2.

We need to find a relation between i_A and i₁. Then, using the law of nodes for node a, we get:

Resolving this relation, we have

On the other hand, applying KVL to the circuit formed by the two resistors and the independent source, we have:

Replacing  eq. 23-06.1  in  eq. 23-06.2  and performing the calculation, we get

With the value of i_A we can calculate the value of V_ab (open circuit voltage), which is the value of V_th. So

To find the value of R_th , we must short-circuit the ab terminals on the circuit in Figure 23-6.2 and calculate the short-circuit current I_sc . We noticed that when making the short-circuit, the dependent source and the resistance of 20 ohms come out of the circuit. In this way, we have

With these data and using the eq. 15-01 we can calculate the value of R_th, or

In the Figure 23-6.3 we can see what the Thévenin equivalent circuit looked like.

At this point, we can calculate the current in the inductor for time t → ∞ , remembering that for this time the inductor behaves like a short circuit. Soon:

And now, based on the circuit of Figure 23-6.3 , we are going to calculate the time constant of the circuit, that is

With all the calculated data in hand, we can write the equation that expresses the circuit behavior shown in Figure 23-6.1. So