I am looking to find short circuit current over input voltage to find transconductance Gm for a common source amplifier with source degeneration.
Looking at sources online, it seems like the next step is to do use KVL such following the path through ro and Rs. Something like:
(Isc - gm(vgs))ro + (Rs)Isc = 0
Vgs = Vin - Vs where Vs = Isc(Rs)
Solving for this gives an expression for Isc/Vin:
gmro / ro + Rs + gm(ro)(Rs)
However, I'm confused as to why I can't seem to get the correct answer when making the assumption that ro and Rs is in parallel. From my understanding, both resistors are connected at the same node and to ground, and thus they should paralle. When solving it this way, I set
gmVgs = Isc
Vgs = Vin - Vs where Vs = Isc(ro//Rs)
Using these steps gives an expression for Gm where
Gm = (1/gm + Rs//ro)^-1
I'm not quite sure why these two methods give different answers. Am I making an assumption that cannot be made? I feel like I'm misunderstanding the conditions for parallel resistance, but I'm not quite sure what it is.