Input transconductance relationship ti the differential or common mode voltage

Dear friends,

I have read that the input transconductance of the differential input transistors is also depending on the level of the input common mode voltage and or the amount of the differential input voltage. But I haven't seen the formula that states this change. I usually use the gm = sqrt(2 I K) which is not involved with the common or differential voltage level.

Hope you can explain it to me with equations


Thank you very much

The formula you use is the simplest level2 square law model for the transistor. Actual models are not that simple, but for hand calculations your formula should be good. In anyway, you want to get in the ballpark correct schematic sizing when you do hand calculations and then adjust the rest by simulation.
Transconductance can vary with the input common mode if, for example you have your bulks connected to gnd or vdd and then the common mode will change the Vsb and the bulk effect can change the effective transconductance.
Differential voltage can also bring to a change in the transconductance because as you know transconductance changes with the current through the transistor. As you increase the differential voltage you also make the difference in the currents of the two diff pair transistors bigger and they have different transconductances.
These are best seen from simulation.

Common and differential mode input voltage affect more or less the drain current and are in so far already covered by the gm square law. Vds dependency has to be considered additionally.

Thank you friends for your help,

How can I mathematically prove that gm decreases when the differential input voltage increases ?

Thank you once again

I have personally never done this but I assume you start with the large signal equations of the differential current in the diff pair vs Vid. The you can plot the current vs Vid and even you may find derivatives at different points along the curve. Since the curve is compressive which eventually saturates, this means that gm is max around vid=0 and then decreases as vid increases.

Thank you Suta,

The way you explained it surely explain the behavioral from simulation result which is logically explained by you.

As you suggested. may be I will find the answer in large signal model not from the small signal expression of Gm

The first approximation is 1-tgh^2(vin/nVt), which is true for BJT and weak inversion. For general mosfet diff pair, it transconductance is described by derivative of non elementary Lambert W function (however the result is not far away from BJT approximation for most of the cases)

Dear friends,

Thank you again for your help,

Indeed I also have found the detailed mathematical derivation from the book of Analog CMOS Integrated Circuit by Behzad Rezavi, page 107-108