(a) If the switch in Fig. 7.109 has been open for a long time and is closed at t = 0, find $v_{ o}$ (t).

(b) Suppose that the switch has been closed for a long time and is opened at t = 0. Find $v_{ o}$ (t).

Question

(a) If the switch in Fig. 7.109 has been open for a long time and is closed at t = 0, find v_{ o} (t).

(b) Suppose that the switch has been closed for a long time and is opened at t = 0. Find v_{ o} (t).

Accepted Answer

(a) [latex]\mathrm{v}_{\mathrm{o}}(\mathrm{t})=\mathrm{v}_{\mathrm{o}}(\infty)+\left[\mathrm{v}_{\mathrm{o}}(0)-\mathrm{v}_{\mathrm{o}}(\infty) ight] \mathrm{e}^{-\mathrm{t} / au}[/latex]

[latex]\mathrm{v}_{\mathrm{o}}(0)=0, \quad \mathrm{v}_{\mathrm{o}}(\infty)=\frac{4}{4+2}(12)=8[/latex]

[latex] au=\mathrm{R}_{\mathrm{eq}} \mathrm{C}_{\mathrm{eq}}, \quad \mathrm{R}_{\mathrm{eq}}=2 \| 4=\frac{4}{3}[/latex]

[latex] au=\frac{4}{3}(3)=4[/latex]

[latex]\mathrm{v}_{\mathrm{o}}(\mathrm{t})=8-8 \mathrm{e}^{-\mathrm{t} / 4}[/latex]

[latex]\mathrm{v}_{\mathrm{o}}(\mathrm{t})={8\left(1-\mathrm{e}^{-0.25 \mathrm{t}} ight) \mathrm{V}}[/latex]

(b) For this case, [latex]\mathrm{v}_{\mathrm{o}}(\infty)=0[/latex] so that

[latex]\mathrm{v}_{\mathrm{o}}(\mathrm{t})=\mathrm{v}_{\mathrm{o}}(0) \mathrm{e}^{-\mathrm{t} / \mathrm{t}}[/latex]

[latex]\mathrm{v}_{\mathrm{o}}(0)=\frac{4}{4+2}(12)=8, \quad au=\mathrm{RC}=(4)(3)=12[/latex]

[latex]\mathrm{v}_{\mathrm{o}}(\mathrm{t})=\mathbf{8} \mathrm{e}^{-\mathrm{t} / 12} \mathrm{V}[/latex]

Question : (a) If the switch in Fig. 7.109 has been open for a long tim...

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