Problem 24-9    Source:    Question 31 - List of Electrical Circuits II - University of Algarve - Instituto Superior de Engenharia - Jorge Semião - 2009.

The switch in the circuit shown in Figure 24-09.1 has been open for a long time. At t = 0 it is closed. Calculate i_L (t) for t ≥ 0.

Solution of the Problem 24-9

As the problem statement says that the switch remained open for a long time, we can easily see that the inductor is behaving like a short circuit. In this case, we can determine the current flowing through it at t = 0^-. Observing Figure 24-09.1, we see that i_L (0^-) = 9 / 3 kΩ = 3 mA. When the switch is closed, we know that the current in the inductor cannot change suddenly, so we can say that i_L (0⁺) = 3 mA. This is the initial condition of the circuit. To facilitate the solution of the problem, we will perform a source transformation on the voltage source and the resistor R₂. In this way, we will obtain the circuit shown in Figure 24-09.2.

Note that we have two current sources in parallel. Then we can add their values, resulting in a current source of 9 mA. And the two resistors are also in parallel resulting in R_P = R₁ . R₂ / R₁ + R₂ = 2 kΩ.

See the final circuit in Figure 24-09.3. Now we can determine the parameters of the circuit.

We easily conclude that:

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

Note that in this case, α > ω_o, confirming an overdamped response. Therefore, the two roots of the characteristic equation are real and the solution equation will be in the form of eq. 24-04. Adapting to this case, we can write:

The values of r₁ and r₂ are given by the equations eq. 24-07 and eq. 24-08.

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

Observing the circuit shown in Figure 24-09.3, we easily determine that i_L (∞) = 9 mA, because in this time the inductor will behave like a short circuit and the current from the current source will pass entirely through it. And we have already determined that i_L (0⁺) = 3 mA. So, substituting these values into the equation above and considering t = 0, we obtain:

Solving this equation we obtain a first relationship between A₁ and A₂, or:

To find the solution we need a second relationship between A₁ and A₂. We know that for an inductor, we have:

From the circuit, we deduce that V_L (0⁺) = 0. Thus, substituting this value in the equation above and making t = 0, we obtain:

Carrying out the calculation we find the second relationship, or:

Substituting the value of A₁ above in the first relation, we find the values of A₁ and A ₂.

Therefore, the equation that governs the current in the inductor is given by: