Time domain equation for RC circuit with AC input

[eyeweyew]

TL;DR Summary: How to find Time domain equation for RC circuit with AC input from inverse laplace transform

For a simple RC circuit with AC input such as this: https://www.electronics-tutorials.ws/wp-content/uploads/2013/08/rc12.gif?fit=310,151?fit=310,226. If the AC input is just a simple sinusoidal v_I(t)=V_icos(ωt) and the transfer function is V_o/V_i=1/(1+sRC). The inverse Laplace transform of V_o/V_i is e^-t/RC/RC. How exactly do I find the Time domain equation for v_O(t) from this inverse laplace transform? I cannot seem to find any reference to show how to do that.

 

[DaveE]

 

[DaveE]

Have you tried searching the web for this. I looked for "solve RC circuit with Laplace transforms" and got lots of links that looked good.

Is this a home work problem? At what point in the analysis are you getting confused? Can you show us what you've tried so far?

 

[DaveE]

If $\frac{V_o(s)}{V_i(s)} = \frac{1}{(1+sRC)}$ then $V_o(s) = \frac{1}{(1+sRC)} V_i(s)$

$v_o(t) = \mathcal L^{-1}(V_o(s)) = \mathcal L^{-1}(\frac{1}{(1+sRC)} V_i(s))$

What is $V_i(s)$?

 

[eyeweyew]

I think I got it. Thanks!