IEEE Solid-States Circuits Magazine - Summer 2018 - 49

g m . It corresponds with the value of IC
where g m shows an inflection point.
This point can easily be calculated by
use of these curves.
These calculations are easier if
the second derivative of g m is taken
rather than the third derivative of
the current I DS . The curves are thus
needed of g m versus VGS rather than
versus IC. This conversion is carried
out first. Figure 1 shows this inflection
point, for linear scales of I DS and VGS ,
leading to zero IM 3 .
These derivatives are simple provided the MOST operates in one region
only. For example, in wi, the MOST
I DS -VGS characteristic is an exponential, which is easily approximated by
a power series around a fixed biasing
point, as given by [5], [6]

gm
vs
si
wi

Inflection Point
VGS

gm′

VT

gm″

IM3 = 0 at VGS ≈ VT

VGS
FIGURE 1: The inflection and zero-IM3 points of gm .

i DS = a 1 VGS + a 2 V

2
GS

+ a3 V

3
GS

+ g. (1)

The first coefficient a 1 is the g m of
the MOST. It is the first derivative of
the i DS -VGS curve. Coefficient a 2 is
given by the second derivative and
coefficient a 3 by the third derivative.
However, higher-order derivatives are difficult. It is easier to find
closed-order expressions for g m versus v GS. In this case, only the second
derivative is required. As a result, the
third-order intercept point (in volts),
which is always inversely proportional to IM 3, is given by [5]
IP3 V =

4 a1 =
3 a3

8g m
,
m
gm

VGS - VT
nU T

,

Gsp √ IC

Gsp

1/2

where the I spec is defined in [1]-[3],
and U T = kT/q ≈ 26 mV at room temperature; n is the subthreshold parameter. It depends on the voltages used
but is taken to be about 1.35 such that
nU T ≈ 35 mV. The coefficient is a 1 = 1
and a 3 = 1/6 such as IP3 V = 8 nU T or
8 # 35 mV or 100 mV.

Gsp

1

= gmsat
= WCoxvsat
IC

gm
IDS

-1/2
λc
nUT

1
nUT

fTsp
1/2
fTspIC
wi

1
nUTλc IC
IC

-1

(2)

(3)

λc

λc

GspIC

fT

m is the second derivative
where g m
g
of m versus VGS . It is clear that
g m (v GS )m is a lot easier to calculate
than i DS (v GS )n.
For a MOST in wi [6], the current is
given by
I DSwi = I spec exp

gm

fTsp
λc

=

vsat
= fTsat
2πL

1
1

si

1/λc2

vs

IC

FIGURE 2: The transconductance gm, ratio gm/IDS, and fT versus IC.

A MOST has three regions of operation. At larger currents, it enters the
si region in which the i DS ^v GS h has a
square-law characteristic. At even larger
currents, vsat is reached in which the
i DS ^v GS h curve becomes linear [1]-[3]
and the transconductance is constant.
In both regions, the higher-order derivatives are zero. No IM 3 is thus generated.
Unfortunately, a MOST never operates in pure wi, si, or vsat. It always operates somewhere in between. These

IEEE SOLID-STATE CIRCUITS MAGAZINE

su m m e r 2 0 18

49



IEEE Solid-States Circuits Magazine - Summer 2018

Table of Contents for the Digital Edition of IEEE Solid-States Circuits Magazine - Summer 2018

Contents
IEEE Solid-States Circuits Magazine - Summer 2018 - Cover1
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IEEE Solid-States Circuits Magazine - Summer 2018 - Contents
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IEEE Solid-States Circuits Magazine - Summer 2018 - Cover3
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