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$$
\begin{array}{l}{\text { The initial energy stored in the circuit shown is zero. At } t=0, \text { a de current source of } 24 \mathrm{mA} \text { is }} \\ {\text { applied to the circuit. }}\end{array}
$$

$$
\begin{array}{l}{\text { a) What is the initial value of } i_{l}(t) ?} \\ {\text { b) What is the initial value of } \frac{d i}{d t} ?} \\ {\text { c) What are the roots of the characteristic equation? }} \\ {\text { d) What is the numerical expression for } i_{l}(t) ?}\end{array}
$$

$$
V_{l} (t)=l \frac{d i (t)}{d t}
$$

$$
V_{l}(0)=l \frac{di(0)}{d t}
$$

$$
0=x \times 0
$$

$$
\frac{d i(0)}{d t}=0
$$

$$
S^{2}+\frac{1}{R _C} S+\frac{1}{L_ C}=0
$$

$$
S^{2}+\frac{1}{625 * 25*10^{-9}}S+\frac{1}{95 * 10^{-3}*25 *10^{-9}}=0
$$

$$
S_{1}=-32000+J24000
$$

$$
S_{2}=-32000-J24000
$$

$$
\text { complex } \Rightarrow \text {under damped}
$$

$$
i_{l}(t)=I_{f}+\beta_1 e^{- \alpha t}cos(wdt)+\beta_2e^{-\alpha t}sin (wdt)
$$

DC

$$
\frac{d i}{d t}=0 , 
V_l=0 \Rightarrow s \cdot c
$$

$$
I_f=24 \mathrm{mA}
$$

$$
i_l(t)=24*\beta_1 e^{-32000t}cos (24000t)+\beta_{2} e^{-32000 t} \sin (24000 t)
$$

$$
i_{l}(0)=0 \ ,\quad  \frac{d i (0)}{d t}=0
$$

$$
i_l (0)=0=24+\beta_1
$$

$$
\beta_{1}=-24 \mathrm{mA}
$$

$$
\frac{d i(t)}{d t}=-\Delta \cos (0)*32000+\beta_2 \ cos (0)*24000=0
$$

$$
\beta_{2}=-32 \mathrm{mA}
$$

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