Digital Signal Processing Reference
In-Depth Information
β
load
/β
load
represent the transconductance mismatch of the input (gain) and
the output (load) transistor pair. The first part of this expression can be vali-
dated quite easily: offset in the input pair threshold voltage (
V
t,gain
)canbe
modelled as a fixed offset voltage applied to the differential input of the am-
plifier. Finding the corresponding offset at the output is then just a matter of
scaling the virtual offset voltage at the input with the voltage gain of the ampli-
fier. Offset between the threshold voltages in the output pair is directly visible
at the output. The second part of (7.12) is somewhat counter-intuitive, in this
respect that a larger overdrive voltage (
V
gs2
−
V
t
) seems to result in a larger
sensitivity towards errors in the transconductance factor
β
. However, this result
is quite logical: for a fixed bias current, a larger overdrive voltage (
V
gs2
−
V
t
)
of the diode-connected output pair corresponds to a increased load impedance
(1
/g
m, load
). A current-mismatch between the branches of the input (or output)
pair is thus translated in a larger offset voltage at the output.
The second-order harmonic distortion ratio of the nonlinear loaded amplifier
can be approximated by (7.13):
2
3
16
v
idp
V
gs1
−
I
I
V
t
V
gs1
−
·
β
1
β
1
=
V
t
·
·
V
t
+
hd
2
(7.13)
From Formula (7.13), it is clear that the hd
2
is the product of two small val-
ues: (1) current mismatch and (2) offset of the threshold voltage. When the tail
currents of the gain- and load pair are chosen to be equally, second-order distor-
tion should not be of any significance. However, in the real implementation of
the amplifier, a small portion of the output current will flow through the finite
output impedance of the transistors connected to the output node. This is even
more true for increasing frequencies, at which the impedance of the parasitic
output capacitances becomes a significant portion of the output impedance.
As a consequence, a more realistic distortion model would take into account
that a non-negligible portion of the total differential output current is injected
into a linear impedance. This fraction of the output signal should then be char-
acterized by the mismatch expressions of the resistively loaded pair (7.10).
Continuing in the same vein, the reader must keep in mind that third-order
distortion partly originates from the fact that a fraction of the output current
does not flow through the loading pair. Nevertheless, for the part of the sig-
nal that does flow through the diode-connected pair, the following expression
holds (7.14):
2
1
32
v
idp
V
gs1
−
hd
3
=
·
V
t
2
I
I
2
3
2
β
gain
β
gain
V
t
V
gs1
−
3
4
+
+
(7.14)
V
t
