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• Notes

$\text { The voltage pulse applied to the } 100 \mathrm{mH} \text { inductor in the shown Fig. }$

$\begin{array}{l}{\text { is } 0 \text { for } t<0 \text { and is given by the expression }} \\ {\qquad v(t)=20 t e^{-10 t} \mathrm{V}} \\ {\text { for } t>0 . \text { Also assunte } i=0 \text { for } t \leq 0 \text { . }} \\ {\text { Find the inductor current as a function of time. }}\end{array}$

$v*1=\left\{\begin{array}{cc}{20 t e^{-10 t}} & {t>0} \\ {0} & {t \leq 0}\end{array}\right.$

$i(t)=\frac{1}{L} \int_{t_{0}} V(t) d t+i\left(t_{0}\right)$

$i(0)=0 \quad \Rightarrow t_{0}=0$

$i(t)=\frac{1}{100*10^{-3}} \int_{0}^{t} V(t) dt+i(0)$

$i(t)=\frac{1}{100 * 10^{-3}} \int_{0}^{t} 20te^{-10 \mathrm{t}} \mathrm{dt}+0$

$i(t)=\frac{1}{0.1} \int_{0}^{t} 20 t e^{-10 t} d t$

$i(t)=200^{t}_{0}|\left[\frac{-1}{10} t e^{-10t}-\frac{1}{100} e^{-10t}\right]$

$i(t)=2\left[1-10 t e^{-{10} t}-e^{-10 t}\right]$                            (A)