Bias Circuit

4.3 Transconductance Parameter
The transconductance parameter, gm, was introduced in Unit 2 in the treatment on the
rudimentary electronic amplifier; it is the proportionality constant of the linear relationship
between the output (responding) current and the input (control) voltage [(2.3)]. For the
MOSFET, NMOS, or PMOS, Id = gmVgs, where Id is into the drain for both transistor types. An
ideal transistor can be modeled with this alone. A simple model, which includes no other
components, would often be adequate for making estimates of circuit performance.
To obtain an expression for gm as a function of the general form iD = f(vGS, vDS, vSB) [e.g.,
(3.8)], we use the definition [from (4.1)]
Equation 4.2
Using (3.8) to express iD, the resulting relation for gm is
Equation 4.3
where [(3.7)] and Veffn = VGS – Vtno. Note that the use of VDS is consistent
with the partial derivative taken with respect to vGS, that is, Vds = 0. Also, the use of Vtno
implies that vSB = 0. In general, VSB could be nonzero, although in the definition of gm, Vsb
must be zero. For the case of nonzero VSB (bias), one substitutes for Vtno a constant Vtn(VSB) in
the gm expression.
Alternative forms for the gm expression can be obtained from (3.8), which is, solving for Veffn,
Equation 4.4
Using (3.8), (4.3), and (4.4), gm takes on altogether three forms:
Equation 4.5
Usually, in initial design, kn replaces to eliminate the VDS dependence without a serious
penalty in accuracy.