IEEE Solid-States Circuits Magazine - Summer 2018 - 51

si, g m = G spec IC and fT = fT spec IC ,
which are well-known results.
All the aforementioned expressions
are relatively simple because they are
all versus the IC. To take the derivatives versus VGS , we need to convert
them versus VGS . This is why IC (v) is
given in Figure 5. The first derivative
of these expressions provides g m ^VGS h,
shown in Figure 6.
Again, the expressions have been
taken from [1]-[3], except for those
involving vsat. If vsat is involved, a
kind of parallel combination is taken
as given by

The EKV model also allows easy prediction
of the distortion.
The fitting model allows the calculation of the second derivative. This is
zero at a value of v denoted by v c . It
is given by

larger. The fitting model of Figure 6
can thus be used.
The g m values can now be plotted
versus v, as shown in Figure 8, together with the derivatives. The inflection
point, which gives zero IM 3, is clearly shown.

v c = 1 ln 1 = 1 ln L .
2
mc
2
L sat

(5)

wi + si

1

g m tot

=

1

g m wi

+

1

g m sat

(4)

for wi + vsat. This g m is denoted by
g m tot . Then, the resulting values of
g m tot are compared with the values
of g m obtained by numerical differentiation of IC ^VGS h . They are
all given in Figure 6. The IP3 values
are added in the pure regions wi, si,
and vsat. Also, parameter values are
added in the corner. Parameter theta
is also added as it used to represent
vsat, which is now characterized by
parameter lambdaC m c .
In (4), the parameters are linear.
They can also have exponents of <1 for
better fitting purposes. Values between
1 and 0.8 have been obtained. The
question is now how accurately these
crossover expressions are in predicting
IM 3 and IP3 for the wi + si + vsat and wi
+ vsat regions.

[. . .]2
si

e2v

v2
wi + si + vsat
[. . .]2

wi + vsat

θ = λc /4nUT

1 + λc /2 [. . .]

si + vsat
v2
1 + λc /2 v
v>0

In (1 + λc e2v)
λc

vsat
v
λc /2
v>0

Gsp = gmsat λc = Ispec/nUT
[. . .] = In (1 + ev )

FIGURE 5: The IC versus vGS for all regions.

wi + si
Gsp

Distortion in wi + si + vsat
For the full model of wi + si + vsat,
two different g m values are compared. They are:
■ g lm IC, which is obtained by taking
the derivative of IC (v) given in the
center block of Figure 5
■ g m tot, given by (4) with g m wi replaced by g m for wi + si in the top
rectangle of Figure 6
■ g m sat taken from the bottom circle of Figure 6.
The result is shown versus v in Figure 7. The matching is very good. For
small v, in wi, the expression is the
same for both. In vsat, g lm IC is slightly

EKV

wi

[. . .]
1 + e-v
EKV

wi
Gsp e2v

IP3 = √ 8 nUT

Gsp v

gmsat =

θ = λc /4nUT

λc [. . .]
λc [. . .] + 1 + e-v

si + vsat
λc v

gmsat

λc e2v
1 + λc e2v

IP3 = ∞

wi + si + vsat
gmsat

wi + vsat

si

vsat
gmsat

IP3 = ∞

1 + λc v
v>0

Gsp = gmsat λc = Ispec/nUT
[. . .] = In (1 + ev )

FIGURE 6: The value of gm (vGS) in all regions of operation.

IEEE SOLID-STATE CIRCUITS MAGAZINE

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51
IEEE Solid-States Circuits Magazine - Summer 2018

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