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• Notes

$\begin{array}{l}{\text { The switch in the circuit shown in Fig. has been in position } x \text { for a long time. At } t=0 \text { , }} \\ {\text { The switch moves instantaneously to position } y \text { . Find the following: }}\end{array}$

$\begin{array}{l}{\text { a) } v_{\mathrm{C}}(t) \text { for } t \geq 0} \\ {\text { b) } v_{\mathrm{O}}(t) \text { for } t \geq 0} \\ {\text { c) } i_{\mathrm{O}}(t) \text { for } r \geq 0^{+}}\end{array}$

$\text { at position } x$

DC

$\frac{d V}{d t}=0$                        $i_{c}(t)=C \frac{d v}{d t}=0 \quad \rightarrow 0 C$

KVL

$-100+v_{0}=0$

$V_0=100V$

$\text { at position y }$

$\frac{60 * 240}{60+240}=48$

$32+48=80 k \Omega$

$\tau =R C=80*0.5 \mu F=0.04 sec$

$V_0 (t)=V_{0} e^{\frac{-t}{2}}=100e^{\frac{-t}{0.0}}$

$V_{c}(t)=100 e^{-25 t} \quad (V)$

$V_{0}=V_ c * \frac{48}{32+48}=10 e^{-25 t} * \frac{48}{32+48}$

$V_0(t)=60 e^{-25t}$

$i_{0}=\frac{V_{0}}{R}$

$i_{0}=\frac{60 e^{-25 t}}{60}$

$i_{0} (t)=e^{-25 t} \quad(A)$