Digital Signal Processing Reference
In-Depth Information
When the first order model (7.1) for a mos transistor in saturation is plugged
in, the relationship between the differential input voltage (
v
id
=
v
ninp
−
v
ninn
)and
the differential output current (
i
od
) is given by (7.3):
v
id
=
v
gs1
−
v
gs2
,
i
noutp
K
n
·
(W/L)
with
v
gs1
=
V
t
+
i
noutn
K
n
·
and
v
gs2
=
V
t
+
(7.3)
(W/L)
From Equation (7.2) an expression for both
i
noutp
and
i
noutn
as function of
I
bias
and
i
od
can be extracted. When substituted into the flattened relationship be-
tween the differential input voltage and the differential output current (7.3), the
following result may be obtained
2
(7.4):
2
K
n
(W/L)
I
B
1
1
i
od
I
B
−
i
od
I
B
v
id
·
=
+
−
(7.4)
This equation describes the differential input voltage as a function of the out-
put current, while the inverse function represents the transconductance of the
differential pair. The Taylor series expansion of the inverse function of (7.4)
can be obtained as follows (7.5):
2
K
n
(W/L)
I
bias
i
od
I
bias
suppose
x
=
v
id
·
and
y
=
1
1
then
x
=
+
y
−
−
y
1
x
2
4
y
=
x
·
−
1
8
x
3
y
≈
x
−
−···
2
K
n
(W/L)
I
bias
2
K
n
(W/L)
I
bias
v
id
·
3
i
od
I
bias
≈
1
8
v
id
·
−
−
...
(7.5)
As expected, no second order distortion components are present in Formula
(7.5). Note that this conclusion is valid only for a sufficiently high output
impedance of the tail current source of the differential pair. If for some rea-
son the parasitic capacitance at the common node is too high, or there is a
2
Under the assumption that no calculation errors were made by the author.
