Electrical – Derivation of transcondutance gm using Taylor expansion

circuit analysiscircuit-designintegrated-circuitmosfettransistors

From the fist page of the lecture, there is a derivation of transconductance of MOSFET using Taylor expansion.

As seen from the picture bellow, drain current is a function of vGS, vDS and vBS.
iD is a function of three variables here. However, in Taylor expansion around DC operating point (VGS, VDS, VBS) the author didn't take vDS, vBS as variables.
So iD is just like a function of vGS only.

I am wondering why they can do that?
This seems to be wrong to me.
Could anyone confirm?

enter image description here

Best Answer

As stated in the introduction, all other voltages are held constant:

Derivation: consider for example the relationship of the increment in drain current due to an increment in gate-source voltage when the MOSFET is saturated-- with all other voltages held constant.

An for this reason only the change of the gate-source voltage is considered.

Of course to extend this model a similar derivation will be done for vds and vbs resulting in gds and gmbs.

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