Ok im working through question p6.109 of sedra smith part b. And im using the fact that Gv=10/10=vo/vsig to try and solve for Rd. My lecturer says the formular for vo= -gm.vsg.Rd. but i just cannot find it using kvl. Its negative because its 180 dregrees out of phase but i cannot kvl in my pie modle of the circuit to get the result. Also im confused on the effect the cappacitors have in regards to Rs when in ac analysis.

The last image shows my pie model but with all the earths connected. The the kvl entering at vo leaving through the earth.

Any help would be great. Cheers

 

Your Rs calculation looks good. But I do not see Rd value?

As for this 180 degrees out of phase. Notice that your source terminal is at AC ground thanks to Cs capacitor.
If Id = Vsg*gm and Vs = 0V we have
Id = (0 - Vin )*gm = - Vin*gm

We can explain this fact is a simple way. Any increase in an input voltage will cause Vsg to drop so, the Id current will also drop. And Vo will also drope becouse Vo = Id*Rd.
As you can see any increases in input voltage will cause the decrease in the output voltage. Hence we say the output voltage is 180 degrees out of phase. And this "minus sign" in the formula only informs us that the output voltage is the 180-degree phase shift with respect to the input voltage.