Problem 24-4    Fonte:   Adapted from Question 3 of the Electrical Circuit Test II of PUCRS - 2019.

a) In the circuit shown in the Figure 24-04.1, determine the value of R_x so that the circuit has an underdamped response with oscillation frequency ω_d = 0.5 rad/s

b) After determining the value of R_x, determine the answer i_L (t) if V_i = 10 u_-1 (t).

Solution of the Problem 24-4

Item a

To solve this problem, first let's find the Thévenin equivalent of the circuit formed by the source and the two R_x resistors. We can see the result in the Figure 24-04.2.

Note that we now have a parallel RLC circuit that is perfectly solvable by applying the theory already studied. Then, with the values provided by the problem we can calculate the operating frequency of the circuit using the eq. 24 - 06, and we found:

We easily conclude that:

However, we know that there is a relation between ω_o, ω_d and α given by eq. 24-13 . How do we know the values of ω_o and ω_d, the value of α will be:

Performing the calculation, we find the value of α, or:

Note that in this case, α < ω_o, confirming a under-damped response. Thus, the two roots of the characteristic equation are complex and the equation for i_L(t) will be in the form of eq. 24-14 plus I_f. On the other hand, with the value of α we can calculate the value of R_x using eq. 24-05. However, we must be aware that the value we should use is R_x / 2 in the eq. 24-05. Then:

Performing the calculation, we find:

Item b

When the voltage source starts operating at t = 0 the inductor will behave like an open circuit. Therefore, we conclude that i_L (0) = 0  A. On the other hand, when t → ∞ The inductor behaves like a short circuit. In this case, we should look at the original circuit to calculate i_L (∞). Note that between the inductor and the source we only have the resistor R_x, then :