# 7.2.4 The Quadratic Model

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## 7.2.4 The quadratic model

### 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 *V*_{C} + *V*_{S}
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 *dV*_{C}.
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
*V*_{C} varies from 0 to the drain-source voltage,
*V*_{DS}.
(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
*I*_{D} = m *C*_{ox}
*W/L V*_{DS}^{2}.

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, *g*_{m}
and the output conductance, *g*_{d}.

### 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
*V*_{DS} < *V*_{GS} -
*V*_{T}. 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
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© Bart J. Van Zeghbroeck, 1996, 1997