I am currently learning control system design. I have a lot of confusion about the initial conditions of the inductor in the circuit. Could someone please correctly solve this by getting the differential equations and then take the Laplace transform.
Using Faraday's law of induction, we can write:
$$-\text{V}\left(t\right)+\text{V}_\text{d}\left(t\right)+\text{R}\cdot\text{I}\left(t\right)=-\text{L}\cdot\text{I}'\left(t\right)\tag1$$
Where \$\text{I}\left(t\right)\$ is the input current (which is the same trough all the components because it is a series circuit).
Using Laplace transform we can write:
$$-\text{v}\left(\text{s}\right)+\text{v}_\text{d}\left(\text{s}\right)+\text{R}\cdot\text{i}\left(\text{s}\right)=-\text{L}\cdot\left(\text{s}\cdot\text{i}\left(\text{s}\right)-\text{I}\left(0\right)\right)\tag2$$
