Problem 25-2 Source:
Continued Exercise 8.7 and Exercise 8.8 - page 190 - NILSSON, James W. & RIEDEL, Susan A. - Book: Circuitos Elétricos - Editora LTC - 5ª edição - 1999.
In the circuit shown in the Figura 25-02.1, the switch S has been in position 1 for a long time.
At t = 0 the key is shifted to the position 2.
Determine:
a)i(t) to t ≥ 0
b)vC(t) to t ≥ 0
Figura 25-02.1
Attention -In the source book
(Nilsson & Riedel), on page 189, there is a typo error in equation 8.60, where
the minus sign of α was omitted in the first part of the equation .
Solution of the Problem 25-2
Item a
From the previous problem, it is known that this circuit has a under-damped response. Thus, the answer is given by eq. 25-13. To find the values of B1 and B2, the values found in prob 25-1 will be used. See below:
i (0+) = I0 = 0
And from this information, we already found the value of B1, or:
B1 = i (0+) = 0
The other values are:
d i (0+)/ dt = 8 000 , α = ωd = 5 000 rad/s
As we have already calculated the value of B1, to find the value of B2 we use the relation below:
d i (0
+)/ dt / dt = - α B
1 + ω
d B
2
As B1 = 0, so the value of B2 is:
B2 = (d i (0+)/ dt / dt ) / ωd = 1.6
Now we can write the equation that governs the behavior of i (t), or
i (t) = 1.6 e-5 000t sin(5 000 t)
Item b
To solve item b, it is known that vC(0+) = 20 volts and by the
circuit we conclude that vC(∞) = 100 volts, for surely after a long time, the
capacitor will have the same voltage as the source. So based on eq. 25-15, we can write the equation
solution for the capacitor voltage, or:
vC (0)
= 100 + e-5 000t (B1 cos 5 000 t + B2 sin 5 000 t)
But to t = 0 we have:
20 = 100 + B1 ⇒ B1 = -80
To find the value of B2 it is necessary to know the value of the derivative of
vC (t) when t = 0. Now, as a capacitor cannot vary its voltage abruptly, so the derivative is null. Soon:
0 = α B1 + B2 ωd ⇒
B2 = -80
Replacing the calculated values in eq. 25-15 we get:
vC
(t) = 100 - 80 e-5 000t (sin 5 000 t + cos 5 000 t)
Note that for t = 0 resulting vC = 20 volts and when
t → ∞ resulting vC = 100 volts, these values agree with the problem.
