Finding transconductace Gm=Isc/Vin for a common source MOS amplfier with source degeneration

Question

I am looking to find short circuit current over input voltage to find transconductance Gm for a common source amplifier with source degeneration.

To do this, I first shorted the output terminal and ignored resistor D because the voltage across it is zero. Then the small circuit model seems to be shown below. Let Vx = 0 and Isc = ix.

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

and

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

and

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.

Answer 1

Vs = Isc(ro//Rs) is not correct. You cannot club both the resistors here because you are interested in the current through Rs which is Isc. In other words, the current through the parallel combination of ro & Rs is not Isc.

Vs = Isc*Rs is correct.

Edit:

As you can see below, the current through ro||Rs is gmVgs