Determine i(t) for t 0 in the circuit of Fig. 8.96.

Question

Determine i(t) for t > 0 in the circuit of Fig. 8.96.

Accepted Answer

For [latex]\mathrm{t}=0^{-}, \mathrm{i}(0)=3+12 / 4=6[/latex] and [latex]\mathrm{v}(0)=0[/latex]
For [latex]\mathrm{t}>0,[/latex] we have a parallel R L C circuit with a step input.

[latex]\begin{array}{l}\alpha=1 /(2 \mathrm{RC})=(1) /(2 \times 5 \times 0.05)=2 \\\omega_{0}=1 / \sqrt{\mathrm{LC}}=1 / \sqrt{5 \times 0.05}=2\end{array}[/latex]

since [latex]\alpha=\omega_{0},[/latex] we have a critically damped response.

[latex]\mathrm{s}_{1,2}=-2[/latex]

Thus,

[latex]\mathrm{i}(\mathrm{t})=\mathrm{I}_{\mathrm{s}}+\left[(\mathrm{A}+\mathrm{Bt}) \mathrm{e}^{-2 \mathrm{t}}\right], \quad \mathrm{I}_{\mathrm{s}}=3[/latex]

[latex]\mathrm{i}(0)=6=3+\mathrm{A} \text { or } \mathrm{A}=3 \[/latex]
[latex]\mathrm{v}=\mathrm{Ldi} / \mathrm{dt} \text { or } \mathrm{v} / \mathrm{L}=\mathrm{di} / \mathrm{dt}=\left[\mathrm{Be}^{-2 \mathrm{t}}\right]+\left[-2(\mathrm{A}+\mathrm{Bt}) \mathrm{e}^{-2 \mathrm{t}}\right] \[/latex]
[latex]\mathrm{v}(0) / \mathrm{L}=0=\operatorname{di}(0) / \mathrm{dt}=\mathrm{B}-2 \times 3 \text { or } \mathrm{B}=6 \[/latex]
[latex]\text { Thus, } .\mathrm{i}(\mathrm{t})=3+[(3+6 \mathrm{t}) \mathrm{e}^{-2 \mathrm{t}}] \mathrm{A}[/latex]

Question 8.49: Determine i(t) for t > 0 in the circuit of Fig. 8.96.

Related Answered Questions