Question: why does transconductance gm of a MOSFET not vary with changing Vgs, ideally?

[u/Own_Ability280]

The circuit in question

First-time poster here, Idk if I'm breaking any rules or something
So this question made me realize that I really don't get gm and small-signal analysis. The question says to verify that gm = K(Vin - Vout - Vt) (Vin and Vout being the operating point).

This result is obtainable by taking the partial derivative of Id with respect to Vgs as per the formula, but looking at the way we obtain that derivative in the first place, I don't see why we didn't consider v_OUT as another independent variable contributing to v_gs (since v_gs = v_in - v_out)? Basically if v_in changes, so will v_gs, which will induce a current change, and that would affect v_out, which would change v_gs again?

So I expected that I'd have to differentiate both v_in and v_out with respect to v_in in the current expression k(v_in - v_out - v_t)²/2 instead of taking the partial derivative, which expanded into some crazy quadratic in terms of v_out which I solved and differentiated, but obviously it was entirely wrong. And I have gone through a bunch of explanations online but still haven't the foggiest as to this entire thing. Can someone here explain how we can seemingly ignore v_gs's dependence on v_out? Thanks in advance

[u/kthompska]

Actually gm does change with different dc Vgs voltages. You will see this when you take the d(Id)/d(Vgs) derivative about many different operating point Vgs.

In your particular circuit you will see that how much gm changes for a given VIN change depends on the value of R. If R >> 1/gm then it won’t change much. As R approaches 1/gm then gm will change much more rapidly. You can solve for a few points and plot it.

[u/Own_Ability280 → u/kthompska]

Yeah, I got the fact that gm will have different values for different operating point values. Where I got confused was the gm = K(Vin - Vout - Vt) expression that I had to verify. Basically, I get this if I differentiate Id (K(Vin -Vout - Vt)²/2) with respect to Vin while keeping Vout constant. But I didn't understand why we wouldn't also treat Vout as a function of Vin and differentiate it too? Because the output voltage depends on the input voltage indirectly, right? That, incidentally, is how I got the quadratic I alluded to in my post, by substituting for Id in the expression Vout = Id*R (I can share my working but it's extremely messy). Why do we treat Vout as a constant here when its value changes with any change in Vin?

[u/UplevelEngineering → u/Own_Ability280]

Your total-derivative approach is valid, but it finds the circuit's dId/dVin, not the transistor's gm = ∂Id/∂Vgs (with Vds and Vbs held fixed). Using the ideal model in your question, dId = gm(dVin - dVout) and dVout = R dId, so dId/dVin = gm/(1 + gm R). The output feedback is therefore included when you solve the circuit; it isn't part of the device's gm definition. Changing the DC bias still changes gm.

[u/Own_Ability280 → u/UplevelEngineering]

Uh, what is Vbs? Also, yes, to your point, I had got gm = partial derivative of Id and Vgs, but then Vgs itself is Vin - Vout (not just a single independent expression), so I don't know how it's any different in this case from computing dId/dVin, since Vgs should simplify into Vin and Vout?

[u/Own_Ability280]

In case it helps, I got K(Vin - Vout - Vt){1 - 1/√(KR(Vin-Vt))} (The formatting is probably horrible but I'm on my phone)

[u/positivefb]

It does, and the partial derivative and non-linear terms are quite different depending on where you operate the transistor.

In general, transient analysis is the only real analysis. Everything else is a tool.