Solved: Assume that the capacitor voltage in the circuit of Fig. 8.15 is underdamped
Chapter 8, Problem 8.57(choose chapter or problem)
Assume that the capacitor voltage in the circuit of Fig. 8.15 is underdamped. Also assume that no energy is stored in the circuit elements when the switch is closed.
a) Show that \(d v_{C} / d t=\left(\omega_{0}^{2} / \omega_{d}\right) V e^{-\alpha t} \sin \omega_{d} t\).
b) Show that \(d v_{C} / d t=0\) when \(t=n \pi / \omega_{d}\), where n = 0, 1, 2, . . . .
c) Let \(t_{n}=n \pi / \omega_{d}\), and show that \(v_{C}\left(t_{n}\right)=V-V(-1)^{n} e^{-\alpha n \pi / \omega_{d}}\).
d) Show that
\(\alpha=\frac{1}{T_{d}} \ln \frac{v_{C}\left(t_{1}\right)-V}{v_{C}\left(t_{3}\right)-V}\),
Where \(T_{d}=t_{3}-t_{1}\).
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