solving circuit using Laplace Transform

[andrewllewop]

having issues with solving this circuit using Laplace transform. firstly i applied KVL to the ciruit[V=V(L)+V(R)+Vo], then i substituted the voltage of the inductor for V(L) and the current through the capacitor to give V(R)=R*(i*dv/dt). resulting in the final equation:
L*di/dt + R*(i*dv/dt) + Vo. I then laplaced that equation to give: V(s) = sLI(s) + RCSVo(s) + Vo. But I'm basically stuck there, just need Vo/Vin maybe its my math but i don't see it.

If there is another way or if i am doinf something wrong I'd appreciate it if someone could look at it for me
Thnx.

click on the image above.

 

[albbg]

You can proceed in two different ways:

1. Using KVL:

VL+VR+Vo=Vi

we know that the current is the same for all the components (single mesh): VR=R*i, VL=L*di/dt and i=C*dVo/dt. Deriving this last di/dt=C*d²Vo/dt². Let's now substitute in the KVL

L*C*d²Vo/dt²+R*C*dVo/dt+Vo=Vi

Laplacing: (S²*L*C+S*R*C+1)*Vo=Vi

2. Using directly the laplace transform of each single component. We have a voltage divider in which the impedance to ground is Zc=1/(S*C) while the series is ZL+R=SL+R then simply:

Vo/Vi=[1/(S*C)]/[1/(S*C)+S*L+R]=1/[1+S²*L*C+S*R*C]

 

[andrewllewop]

hey thnx I see it now the first method is very clear I feel like I"m blind or something. I also get the 2nd part.
Appreciate it.