# determine the Laplace domain of this RL circuit

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.

• You must show your attempt at constructing the differential equation. For the initial condition, assume the circuit has been connected for a long time, therefore the initial current is constant.
– Chu
Commented Apr 29, 2019 at 6:28

$$-\text{V}\left(t\right)+\text{V}_\text{d}\left(t\right)+\text{R}\cdot\text{I}\left(t\right)=-\text{L}\cdot\text{I}'\left(t\right)\tag1$$
Where $$\\text{I}\left(t\right)\$$ is the input current (which is the same trough all the components because it is a series circuit).
$$-\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$$