(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 tim...
(a) \mathrm{v}_{\mathrm{o}}(\mathrm{t})=\mathrm{v}_{\mathrm{o}}(\infty)+\left[\mathrm{v}_{\mathrm{o}}(0)-\mathrm{v}_{\mathrm{o}}(\infty)\right] \mathrm{e}^{-\mathrm{t} / \tau}
\mathrm{v}_{\mathrm{o}}(0)=0, \quad \mathrm{v}_{\mathrm{o}}(\infty)=\frac{4}{4+2}(12)=8\tau=\mathrm{R}_{\mathrm{eq}} \mathrm{C}_{\mathrm{eq}}, \quad \mathrm{R}_{\mathrm{eq}}=2 \| 4=\frac{4}{3}
\tau=\frac{4}{3}(3)=4
\mathrm{v}_{\mathrm{o}}(\mathrm{t})=8-8 \mathrm{e}^{-\mathrm{t} / 4}
\mathrm{v}_{\mathrm{o}}(\mathrm{t})={8\left(1-\mathrm{e}^{-0.25 \mathrm{t}}\right) \mathrm{V}}
(b) For this case, \mathrm{v}_{\mathrm{o}}(\infty)=0 so that
\mathrm{v}_{\mathrm{o}}(\mathrm{t})=\mathrm{v}_{\mathrm{o}}(0) \mathrm{e}^{-\mathrm{t} / \mathrm{t}}\mathrm{v}_{\mathrm{o}}(0)=\frac{4}{4+2}(12)=8, \quad \tau=\mathrm{RC}=(4)(3)=12
\mathrm{v}_{\mathrm{o}}(\mathrm{t})=\mathbf{8} \mathrm{e}^{-\mathrm{t} / 12} \mathrm{V}