Bias Circuit

The output-characteristic equation in the linear region corresponds to VDS ranging from the
condition of Fig. 3.1(b) to that of Fig. 3.1(c). As VDS increases from zero, the channel begins to
close off at the drain end (i.e., the channel becomes progressively more wedge shaped). The
result is an increase in the resistance of the channel as a function of VDS, and therefore a
sublinear current – voltage relation develops.
When VGS > Vtno, the electron charge in the channel can be related to the gate voltage by Qchan
= Cox(VGS – Vtno) (per unit area of MOSFET looking down at the gate), where Cox is the parallelplate
capacitance (per unit area) formed by the MOS structure. This provides a simple linear
relation between the gate voltage and the charge in the channel.
The conductivity in the channel is σchan = µnQchan/tchan, where µn is the mobility of the electrons
in the channel and tchan is the thickness of the channel into the semiconductor. Thus, in the
case of a uniform channel (i.e., for VDS 0), the channel conductance is
Equation 3.2
where
Equation 3.3
and where KPn = µnCox is the SPICE transconductance parameter (the n subscript is the
equation symbolic notation for the NMOS; the parameter in the device model is just KP), W is
the physical gate width, and L, again, is the channel length. Parameter KPn is related to the
electron mobility in the channel and the oxide thickness. Therefore, it is very specific to a
given MOSFET device.
As VDS increases, but is less than Veffn [transition from Fig. 3.1(b) to 3.1(c)], the wedge-shaped
effect on the channel is reflected functionally in the channel conductance relation as
Equation 3.4
This leads to an output characteristic equation for the linear region, which is
Equation 3.5
The derivation leading to (3.4) and (3.5) is given in Unit 3.4. The linear-region relation, (3.5),
is applicable for VDS up to VDS = Veffn, which is the boundary of the linear and active regions.
The active-region equation is then obtained by substituting into (3.5), VDS = Veffn, giving
Equation 3.6