Digital Signal Processing Reference
In-Depth Information
1
(V
gs
−
C
gs
∝
W
∝
V
T
)
2
,
fixed
I
bias
(A.9)
thus:
C
gs
if
V
gs
−
V
T
The answer to the first question is twofold. For the circuit depicted in Figure
A.1, the low frequency current gain (
g
m
r
ds
) of an intermediate stage is in-
versely proportional to the overdrive voltage. A reduced overdrive voltage re-
sults in an improved small-signal current gain. At the high frequency side of
the spectrum, the opposite option is more advantageous. From the definition of
f
T
and from Equations (A.8), (A.9), it follows that the current gain-bandwidth
product
f
t
increases proportionally with an increased overdrive voltage
V
gst
(A.10):
g
m
2
πC
gs
∝
f
T
=
V
gs
−
V
t
,
for fixed
I
bias
(A.10)
The reader should keep in mind that the simplified strong inversion model of
A.2 is not entirely accurate in a deep-submicron cmos technology. The
I
ds
versus
V
gs
characteristic becomes less quadratic for smaller transistor lengths.
Ultimately, beyond the velocity saturation point where the characteristic be-
comes linear, both
g
m
and
f
T
both become independent of the overdrive volt-
age. The general idea of the above findings, however, remains valid for the
complete spectrum of models in between the quadratic and a more linear tran-
sistor model.
The second experiment involves the case where the overdrive voltage
V
gs
−
V
T
stays constant, but this time only the dc bias current
I
bias
through the transistor
is increased. Once again using Equation (A.3), it follows that
g
m
is proportional
to the bias current. If the bias current through each section of the multistage
amplifier is scaled evenly, it is interesting to notice that the low frequency
current gain
g
m
r
ds
of the intermediate stage remains more or less constant: the
width
W
of the transistor scales proportional to the bias current and results in
an inverse proportional decrease of the output resistance of each stage:
g
m
∝
I
bias
1
W
∝
1
I
bias
r
ds
∝
g
m
r
ds
≈
constant
,
for fixed
V
gs
−
V
T
(A.11)
The same conclusion can be made for the cut-off frequency of an intermediate
transistor stage. Both the transconductance and the parasitic oxide capacitance
are in first-order approximation proportional to the bias current, which results
in a bias current independent gain-bandwidth product
f
T
:
