Note: from the constancy of the drain current throughout the device, it can be seen that as and the electric field F(L) diverges.
The differential drain conductance

tends to zero when and the I-V characteristics may be extrapolated in the voltage region assuming a constant (independent of drain current
may be found by substituting from Eq.(5.8) into Eq.(5.7), which results in a highly complicated expression, however, it can be simplified for gate voltages close to the threshold voltage
Note: this approach is only valid when the channel electrons do not suffer any velocity saturation due to high electric fields.
Note: modern day MOSFETs have extremely small gate lengths, and the channel has high electric fields (more than the critical electric field required for velocity saturation), which creates the velocity saturation effects for the channel electrons.

Fig.5.7 The I-V characteristics of an n-channel MOSFET for different values of gate voltage . The dashed line represents the drain-to-source saturation voltage.

Fig.5.8 The variation of the drain saturation current with gate voltage for three different values of substrate doping.
For very small the terms under the curly brackets in Eq.(5.15) can be expanded in Taylor series, leading to the following simplified expression for the I-V characteristics in the linear region:
A physical justification of Eq.(5.16) can be given as follows:
At very small the charge induced in the channel is, to the first order, independent of the channel potential, thus,
(5.17)
Now, for small the electric field F in the channel is nearly constant, and is
given by
The drain current is entirely due to drift, and is given by the electrons in transit model:

since

5.2.3 The Charge Control Model
A simplified description of the I-V characteristics of a MOSFET can be obtained by using the charge control model.
In this model, it is assumed that the concentration of free carriers induced in the channel is given by
Compare Eq.(5.19) with Eq.(5.2): in Eq.(5.19), the variation of the depletion charge density with the channel potential has been neglected.
The drain current can now be given by
Compare Eq.(5.20) with Eq.(5.5).
Equation (5.20) can be rewritten as
Integrating Eq.(5.21) from x = 0 (source side) to x = L (drain side), which corresponds to a change in from the following expressions for the I-V characteristics are obtained:

Fig.5.9 The I-V characteristics of an n-channel MOSFET calculated using the charge control model (solid curve) and the Shockley model (dashed curve).
The differential transconductance is defined as
From Eqs.(5.22) and (5.23),

where is referred to as the device transconductance parameter, with is referred to as the process transconductance parameter.
Thus, in order to achieve a high value for the transconductance gm, the following steps may be taken.
Higher value of low field electron mobility
Thinner gate dielectric layers, which in turn gives large values for the insulator capacitance per unit area
Large widths (W) and short lengths (L).
Note: for short channel devices, where velocity saturation effects are important, the dependence of transconductance on the low-field electron mobility and the gate length gets strongly affected.

EXAMPLE 5.1: An n-channel MOSFET with the process transconductance parameter the threshold voltage is biased at Determine the drain current ID, the transconductance and the drain conductance

SOLUTION:

Hence, the device is under linear mode of operation
.

Note the huge change in transconductance in saturation as compared to the linear region: this is due to the square law dependence of current on the gate voltage in the saturation region (as against the linear variation in the linear region).

Drain Conductance

This is due to the independence of the saturation drain current on the drain voltage. In reality, channel length modulation creates a change in drain current with respect to the drain voltage in saturation, and finite drain conductance

Effect of Source and Drain Series Resistance

The analysis so far neglects the effects of the source/drain series resistance, and the entire voltage is assumed to drop along the channel.
However, for modern day MOSFETs, this effect cannot be ignored, due to smaller diffusion cross-sections and smaller drain currents.
The extrinsic (measured) voltages can be related to the intrinsic (device) voltages by the following equations:

where are the source and drain resistances respectively.
The extrinsic transconductance is related to the intrinsic transconductance

where is the intrinsic drain conductance.
Similarly, the extrinsic drain conductance is related to the intrinsic drain conductance