Laplace transform for RC integrator

integratorlaplace transform

This is my RC integrator:

rc integrator

From my two different analysises:

$$ V_{out}(s) = V_{in}\dfrac1{sRC} $$

and

$$ V_{out}(t) = \dfrac1{RC}\int V_{in} dt $$

These are of course different as one is in the frequency domain and one in the time domain.

If I understand correctly, \$ \mathcal{L}\{V_{out}(t)\} = V_{out}(s) \$

However, Wolfram Alpha tells me \$ \mathcal{L}\{V_{out}(t)\} = V_{in}\dfrac1{s^2RC} \neq V_{out}(s) \$

What am I doing wrong?

Best Answer

Your equations are correct, assuming initial conditions are zero, but your Vi should be Vi(s) in the first equation. I would guess that the Vin in Wolfram is a step of magnitude Vi, with Laplace transform Vi/s, which gives the Laplace output, Vout(s), in your final equation.