Lowest possible Vdsat. Could it negative?

vdsat equation

plotting the gm/id vs Vgs when sweeping Vgs from 0 to VDD says that vgs=0 is the best point of operation (the project i'm working on has no freq spec...so freq response isnt an issue at all) maybe even a negative vgs would give a higher value for gm/id. so having a vgs less than the threshold voltage means a negative vdsat.
Is that possible? How would I calculate the swing anyway when using hand calc just for intuition. is a max voltage of vdd-vdsat turn into vdd+vdsat ? I'm sure not

vdsat negatif

As Vgs-Vt approaches zero, the device is no longer working in strong inversion, and a different set of rules occurs because the conduction is no longer dominated by drift. Using vdsat = vgs-vt is only valid in the strong inversion region.

The device goes thru a transition region (moderate inversion) where operation can be difficult to describe using a single simple equation suitable for hand calculations because the current is a combination of drift current and diffusion current.

Beyond the moderate inversion region, the device will enter weak inversion, where the dominant current is due to diffusion. In this region the equations become much like the equations for a bipolar transistor (do a search on Vittoz or EKV to find equations suitable for hand calculations for the operation in this region). In this region, the vds.sat can be on the order of 100mV or so. Below this voltage, the exponential equation for current vs vds that can be found in the above stated models will be most useful, above this voltage, you can use the channel length modulation as you would in strong inversion to describe the current vs drain voltage.

vdsat subthreshold

with a vgs=0 I guarantee your circuit will not have much gain!

You should be wary of the accuracy of the mosfet models in this operating region. Because hand calculations are level 1 models, you will not be able to calculate by hand how the circuit will operate in this region.

You can search for "subthreshold" to see how a mosfet works when vgs<vth. The gm can be quite high because of the similarities to bipolar behavior.

mosfet vdsat equation

snafflekid, the subthreshold region is the region of operation that offers the highest possible gain because gain is directly propotional to gm and inversly propotional to the square root of the current. And the subthreshold region offers the highest gm/Id.

Anyway that wasn't the scope of my question.But thanks anyway. I understanded what confused me from JPR