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Find the step response of the circuit shown in the Figure,

Find the step response of the circuit shown in the Figure,

enter image description here

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Solution:

Applying KVL to the circuit shown in Fig,

$$ x(t)=R i(t)+L \frac{d i(t)}{d t} $$

Applying Laplace transform,

$ \begin{gathered} X(S)=R I(S)+L S I(S) \\\\ X(S)=[R+L S] I(S)\\ \end{gathered} $

$ \begin{gathered} I(S)=\frac{X(S)}{[R+L S]} \\\\ Y(S)=L S \frac{X(S)}{[R+L S]}\\ \end{gathered} $

$ \text { For Step response } x(t)=u(t) \Rightarrow \quad X(S)=\frac{1}{S}\\ $

$ Y(S)=L S \frac{\frac{1}{S}}{R+L S}=\frac{L}{L S+R}=\frac{1}{S+\frac{R}{L}}\\ $

Applying Inverse Laplace transform,

$ y(t)=e^{-\frac{R}{L} t} u(t)\\ $

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