OK, I see now.
The relations between Vgs, Vth and Vds that you pointed out are mostly valid in stron inversion.
To put some order in all this, let me say the following:
1. Let's call the regions of operation of the transistor active and ohmic. In active region the MOSFET behaves more or less as a current source. In the ohmic region it behaves as an ohmic resistor - somewhat nonlinear, especially farther away you move from the origin (I'm talking about Vds-Id coordinate system).
2. In this same coordinate system you usually look at family of curves and the parameter is Vgs. When you increase Vgs you jump to a curve from that family which corresponds to a higher Id in its flat part - this I hope should be quite familiar to you since it is the basics.
3. When you decrease Vgs, you go to a lower curve. Imagine you decrease Vgs so much that it becomes equal or lower than the threshold voltage. You move to a very low current curves, ideally current should be zero, but in reality there is still some conduction - this is weak inversion. But you're still on a curve from the family of curves that more or less preserves the ohmic and the active regions of the curve. One can define boundaries in this set-up - until about Vgs-Vth~70-80mV you are in weak inversion. For Vgs-Vth>120-140mV you're in strong inversion. In-between your are in moderate inversion. But important thing to remember is that going from one mode of operation into the other you are still on that family of curves.
4. Operation in weak inversion is different than that in strong inversion. In strong inversion there is a channel under the gate and the electric field in this channel moves the carriers. In weak inversion there's no channel as such, but you have processes similar to a bipolar transistor i.e. diffusion. In a way the MOSFET in weak inversion is like a crappy bipolar transistor - it can be either in saturation (your ohmic region) or in active mode when it behaves like a current source.
In moderate inversion you have a mixture of diffusion transport and drift.
5. So far it has to be clear that all throughout the different modes of operation - weak inversion, moderate inversion and strong inversion those Id-Vds curves have a flat part where the transistor is like a current source with some gds in parallel. The gds is an incremental quantity i.e. valid for small signals or small increments. You put your transistor in a certain operating point and around that point you give some small increment of Vds and see what increment of Id results. The ratio between those increments is gds, not the ratio of Id/Vds at the operating point.
6. gds is a very difficult to define quantity. In older technologies, may be even in 0.5u you can still probably get the Id-Vds curves cross at one single point on the x-axis - something like an Early voltage. Then defining gds is not difficult. But in modern-day technologies, 0.18u and below this is no longer the case. There are many effects that contribute to the gds and even you can not find a crisp point where you can say the MOSFET enters into active region. It's all very gradual. Then, it is practically useless to try to come with a formula for gds. Transistor models are too complicated for those technology nodes and you'll be better off to just simulate and extract gds.
Waddy said:
I am still pretty new to all of this, still trying to wrap my head around the idea of weak inversion (all of my previous classes never even mentioned it). I think the technology is 0.5um, I will check tomorrow to see if that is correct. Also, thanks for pointing out that gds=Id/Vds is not valid with DC values, that will be very helpful.
As I remember from my classes a MOSFET is in the active region if Vgs>Vth and Vds>(Vgs-Vth) and in the triode region if Vgs>Vth and Vds<(Vgs-Vth). Does this carry over to the weak inversion except with Vgs<Vth? Thanks.