In a MOSFET, the transconductance in linear region can be expressed as:

Explanation:

For a MOSFET operating in the linear region, the current is given by:

\({I_D} = \frac{{W{\mu _0}{C_{ox}}}}{{2L}}\left[ {2\left( {{V_{GS}} - {V_T}} \right)\left( {{V_{DS}}} \right) - {{\left( {{V_{DS}}} \right)}^2}} \right]\)  ---(1)

W = Width of the Gate

Cox = Oxide Capacitance

μ = Mobility of the carrier

L = Channel Length

Vth = Threshold voltage

The transconductance is defined as the change in drain current for a given change in Gate-to-source voltage, i.e.

\({g_m} = \frac{{\partial {I_D}}}{{\partial {V_{GS}}}}\)

Differentiation equation (1) with VGS we get:

\({g_m} = \frac{{\partial {I_D}}}{{\partial {V_{GS}}}} = \frac{{W{\mu _0}{C_{ox}}}}{{2L}}\;\left[ {2{V_{DS}}} \right]\)

    \(=\frac{{W{\mu _0}{C_{ox}}}}{{L}}\;\left[ {{V_{DS}}} \right]\)Important Points

For a MOSFET in saturation, the current is given by:

\({I_{D\left( {sat} \right)}} = \frac{{W{μ _x}{C_{ox}}}}{{2L}}{\left( {{V_{GS}} - {V_{th}}} \right)^2}\)

The transconductance of a MOSFET is defined as the change in drain current(ID) with respect to the corresponding change in gate voltage (VGS), i.e. 

\({g_m} = \frac{{\partial {I_D}}}{{\partial {V_{GS}}}}\)

\(g_m = \frac{{W{μ _x}{C_{ox}}}}{{L}}{\left( {{V_{GS}} - {V_{th}}} \right)}\)

Hence in saturation mode, transconductance does not depend on VDS