Solved: The initial value of the voltage in the circuit in Fig. 8.1 is zero, and the

Chapter 8, Problem 8.23

(choose chapter or problem)

The initial value of the voltage v in the circuit in Fig. 8.1 is zero, and the initial value of the capacitor current, \(i_{c}\left(0^{+}\right)\), is 45 mA. The expression for the capacitor current is known to be

\(i_{c}(t)=A_{1} e^{-200 t}+A_{2} e^{-800 t}, \quad t \geq 0^{+}\),

when R is \(250 \Omega\). Find

a) the values of \(\alpha, \omega_{0}, L, C, A_{1}\), and \(A_{2}\)

\(\left(\text { Hint: } \frac{d i_{C}\left(0^{+}\right)}{d t}=-\frac{d i_{L}\left(0^{+}\right)}{d t}-\frac{d i_{R}\left(0^{+}\right)}{d t}=\frac{-v(0)}{L}-\frac{1}{R} \frac{i_{C}\left(0^{+}\right)}{C}\right)\)

b) the expression for \(v(t), t \geq 0\),

c) the expression for \(i_{R}(t) \geq 0\),

d) the expression for \(i_{L}(t) \geq 0\).