Find v(t) for t 0 in the circuit of Fi. 8.98.

Question

Find v(t) for t > 0 in the circuit of Fi. 8.98.

Accepted Answer

Let i= inductor current and v= capacitor voltage.

[latex] ext { At } t=0, v(0)=0 ext { and } i(0)=i_{0}[/latex]

For [latex]\mathrm{t}>0,[/latex] we have a parallel, source-free [latex]\mathrm{LC}[/latex] circuit [latex](\mathrm{R}=\infty)[/latex]
[latex]\alpha=1 /(2 \mathrm{RC})=0[/latex] and [latex]\omega_{\mathrm{o}}=1 / \sqrt{\mathrm{LC}}[/latex] which leads to [latex]\mathrm{s}_{1,2}=\pm \mathrm{j} \omega_{\mathrm{o}}[/latex]

[latex]\begin{array}{l}\mathrm{v}=\mathrm{Acos} \omega_{0} \mathrm{t}+\mathrm{B} \sin \omega_{\mathrm{o}} \mathrm{t}, \mathrm{v}(0)=0 \mathrm{A} \\\mathrm{i}_{\mathrm{C}}=\mathrm{Cdv} / \mathrm{dt}=-\mathrm{i} \\\mathrm{dv} / \mathrm{dt}=\omega_{0} \mathrm{Bsin} \omega_{0} \mathrm{t}=-\mathrm{i} / \mathrm{C} \\\mathrm{dv}(0) / \mathrm{dt}=\omega_{0} \mathrm{B}=-\mathrm{i}_{0} / \mathrm{C} ext { therefore } \mathrm{B}=\mathrm{i}_{0} /\left(\omega_{0} \mathrm{C} ight) \\\mathrm{v}(\mathrm{t})=-\left(\mathrm{i}_{0} /\left(\omega_{0} \mathrm{C} ight) ight) \sin \omega_{0} \mathrm{t} \mathrm{V} ext { where } \omega_{0}=1 / \sqrt{\mathrm{LC}}\end{array}[/latex]

Question 8.51: Find v(t) for t > 0 in the circuit of Fi. 8.98.

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