# 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
Apr 29 '19 at 6:28

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:

$$-\text{v}\left(\text{s}\right)+\text{v}_\text{d}\left(\text{s}\right)+\text{R}\cdot\text{i}\left(\text{s}\right)=-\text{L}\cdot\left(\text{s}\cdot\text{i}\left(\text{s}\right)-\text{I}\left(0\right)\right)\tag2$$

• Can you explain to me the significance of the I(0), the initial current for the inductor ? Because in some other examples I've seen they don't include that even though the Laplace of the derivative has it. Apr 29 '19 at 7:44
• @MaizHalyym Well that is the initial condition in this circuit. So when there is an initial current trough the inductor it has to be placed there.
– Jan
Apr 29 '19 at 8:00