Problem 24-10    Source:    Problem 8.14 - page 329 - NILSSON, James W. & RIEDEL, Susan A. - Book: Electric Circuits - Publisher Pearson Education do Brasil - 10th edition - 2016.

The switch in the circuit shown in Figure 24-10.1 has been in position a for a long time. At t = 0 it instantly moves to position b. Determine V_o (t) for t ≥ 0.

Solution of the Problem 24-10

Initially, we will find the equivalent resistance value in the circuit highlighted by the dashed rectangle represented in Figure 24-10.1. To do this we must calculate the Thévenin resistance, as learned in chapter 2. Therefore, we will apply a voltage source of 20 V to this circuit. Follow the circuit on Figure 24-10.2. Note that if through the resistor of 60 Ω circulates i₁ then through the 150 Ω resistor circulates 60 i₁/ 150 = 0.4 i₁. Therefore, the total current I_T circulating in the circuit is I_T = - 1.4 i₁. And we easily determine the value of i₁ by looping the dependent source and the 60 Ω resistor.

Solving this equation, we find:

Knowing the value of i₁ we can calculate the value of I_T, or

And now we can calculate the value of the Thévenin resistance, because:

Now let's calculate the value of α of a parallel RLC circuit. Using eq. 24-05, we have:

And the value of ω_o, using eq. 24-06 is:

Carrying out the calculation we have:

Note that in this case, α < ω_o, confirming an underdamped response. Therefore, the two roots of the characteristic equation are complex and the solution equation will be in the form of eq. 24-14 shown below for clarity.

We need to find the value of ω_d. Using eq. 24-13, we have:

Then the system's response to V_o (t) will be given by:

Observing the circuit shown in Figure 24-10.1, we easily determine that V_o (∞) = 0, as in this time the inductor will behave like a short circuit. Now it is necessary to find the values of B₁ and B₂. Let's determine what the voltage on the capacitor is when the switch is in position a, i.e. V_C(0⁺). Using a resistive voltage divider we find

And as we studied in the theoretical part, we know that:

And to find the value of B₂ it is necessary to know the value of the current in the capacitor at the instant (0⁺). We know that when we apply a voltage like jump in an inductor, the current flowing through it is zero. Therefore, the current that will flow through the capacitor is the current that will flow through the Thévenin resistance. Então:

Note the negative sign in i_C (0⁺), as the current is leaving the positive terminal of the capacitor . And to determine the value of B₂ we use the relationship studied in the theoretical part and reproduced below.

Performing numerical substitution and carrying out the calculation we find the value of B₂, or:

Then the complete answer can be written as: