Problem 24-1    Source:   Example 8.1 - page 177 - NILSSON, James W. & RIEDEL, Susan A. - Book: Circuitos Elétricos - Editora LTC - 5ª edição - 1999.

In the circuit shown in the Figure 24-01.1, we have to R = 200 Ω, L = 50 mH and C = 0.2 µF. Determine:

a) the roots of the characteristic equation that describes the transient behavior of the circuit.

b) Is the circuit response over-damped, under-damped, or critically damped?

c) repeat the items (a) and (b) to R = 312.5 Ω.

d) what should be the value of R for the answer to be critically damped?

Solution of the Problem 24-1

Item a

Initially, we must calculate the values of α and ω_o, because we know the values of R, L and C. So:

And for the value of ω_o:

With this data we can write the roots of the characteristic equation using the eq. 24-06 and eq. 24-07. So:

Performing the calculation:

To find the other root, we must use eq. 24-07. After the numerical substitution and making the calculation, we obtain:

Item b

With the values calculated in item a, we find that α > ω_o and therefore the circuit response is over-damped.

Item c

Changing the value of R para 312.5 Ω, we get a new value for α but the value of ω_o remains the same as its value does not depend on R. The new value of α is:

As α < ω_o, so the circuit has a underdamped response. For this type of answer, it is known that the roots of the characteristic equation are complex. Then, substituting the numerical values and after performing the calculation, we obtain:

Item d

In order to achieve a critically damped response, we know that α = ω_o. In this case, α = ω_o = 8 000 rad/s. Then the value of R will be:

Doing the numerical substitution and making the calculation: