Engineering Find the output resistance in a amplifier (MOSFET) small-signal model

[Martel]

Homework Statement

Provided as pictures

Relevant Equations

Provided as pictures

I have to solve a problem related with an 'invented' (non-real) MOSFET working in its saturation region (amplifier). I have solved all the questions, but I'm unable to get the last one. Basically, I need to determine the output resistance of the amplifier as a function of other given 4 variables. I have read the book and I know that I'm supposed to apply a voltage test to the the amplifier's output, measure the obtained current and divide them to get the output resistance, which in the typical amplifier schema with a load resistor is exactly that load resistor (this is actually the situation in my problem).

The problem is that I need to express this answer as a function of other given 4 variables, and I have tried to do it like 1000 times and there is no way for getting what I need to do. The problem and the question is below:

The main problem's description:

The provided small signal model:

The question:

I have been able to obtain gm, which is K*VDS^2

Still, I just can't get the right result for the output resistance (ro). It should be R, but how can I express it as a function of the mentioned variables (notice I can't use VS).

 

[gneill]

I don't see any way to avoid referencing the resistance R in your output resistance calculation. It obviously represents a path to (signal) ground via V_s.

 

[alan123hk]

In order to know the exact values of gm and ro in the small signal model, it seems unavoidable that the supply voltage (VS) and bias resistance (R) are involved.
According to the obtained equation gm = K * VDS ^ 2, gm is related to the bias voltage (VDS), so the values of the supply voltage (VS) and the bias resistance (R) should be provided to determine the bias voltage.

Note: Obviously, according to the given equation, the value of ro is also forced to depend on the bias voltage (VDS)