### 7.2.4.1 Derivation of the quadratic model

The quadratic model uses the same assumptions as the linear model except that the inversion layer charge density is allowed to vary in the channel between the source and the drain.

The derivation is based on the fact that the current at each point in the channel is constant and can be written as a function of the local charge density which in turn depends on the local channel voltage and the gate-to-source voltage.

Considering a small section within the device with width dx and channel voltage VC + VS one can still use the linear model, yielding:

(mf9)
where the drain-source voltage is replaced by the change in channel voltage over a distance dx, namely dVC. Both sides of the equation can be integrated from the source to the drain, so that x varies from 0 to the gate length, L, and the channel voltage VC varies from 0 to the drain-source voltage, VDS.
(mf10)
Using the fact that the DC drain current is constant throughout the device one obtains the following expression:
(mf11)
The drain current first increases linearly with the applied drain-to-source voltage, but then reaches a maximum value. According to the above equation the current would even decrease and eventually become negative. The charge density at the drain end of the channel is zero at that maximum and changes sign as the drain current decreases. As explained in the section on MOS capacitor, the change in the inversion layer does go to zero and reverses its sign as holes are accumulated at the interface. However these holes can not contribute to the drain current since the reversed-biased p-n diode between the drain and the substrate blocks any flow of holes into the drain. Instead the current reaches its maximum value and maintains that value, also for higher drain-to-source voltages. A depletion layer located at the drain end of the gate accomodates the additonal drain-to-source voltage. This behavior is refered to as drain current saturation.

Drain current saturation therefore occurs when the drain-to-source voltage equals the gate-to-source voltage minus the threshold voltage. The value of the drain current is then given by the following equation:

(mf12)
The quadratic model explains the typical current-voltage characteristics of a MOSFET which are normally plotted for different gate-to-source voltages. An example is shown in the figure below. The saturation occurs to the right of the dotted line which is given by ID = m Cox W/L VDS2.

mosfetiv.xls - mosfetiv.gif
Fig. 7.2.1 Current-Voltage characteristics of a MOSFET as obtained with the quadratic model. The dotted line separates the quadratic region of operation on the left from the saturation region on the right.
The drain current is again zero if the gate voltage is less than the threshold voltage.

The quadratic model can be used to calculate some of the small signal parameters, namely the transconductance, gm and the output conductance, gd.

### 7.2.4.2 Transconductance of a MOSFET

The transconductance quantifies the drain current variation with a gate-source voltage variation while keeping the drain-source voltage constant, or:
(mf14)

The transconductance in the quadratic region is given by:

(mf16)
which is proportional to the drain-source voltage for VDS < VGS - VT. In saturation the transconductance is constant and equals:
(mf17)

### 7.2.4.3 Output conductance of a MOSFET

The output conductance quantifies the drain current variation with a drain-source voltage variation while keeping the gate-source voltage constant, or:
(mf15)

The output conductance decreases with increasing drain-source voltage:

(mf18)
and becomes zero as the device is operated in the saturated region:
(mf19)

7.2 ¬ ­ ® 7.2.5

© Bart J. Van Zeghbroeck, 1996, 1997