IEEE Circuits and Systems Magazine - Q2 2023 - 34
this type of filters in Section V. Fig. 7(b) shows an example
of a gCm lossy integrator implementation. The
block diagram of this circuit can be described either
with voltage-mode (Fig. 7(c)) subtraction or currentmode
(Fig. 7(d)) subtraction. We will use both representations
interchangeably.
We assume that the intrinsic gain of the transistor is
sufficiently large, i.e., gr
mo 1, where ro denotes the
output impedance of a transistor, therefore, the load impedance
of each transconductor can be approximated
as 1/ .sC The body effect of the transistor is also ignored,
i.e., gmb = 0.
IV. OTA-Based Filters
A. Second-Order LPF
Fig. 8 shows the OTA-based second-order LPF adopted
in the early silicon cochlea designs [1], [4], [25], [26]. The
circuit consists of 3 OTAs and 2 capacitors leading to
a gCm
mm
range of the filter [4] but at the expense of voltage headroom.
The transfer function of this circuit is described
below.
Hs
OTA-LPF () =
gg
CC
ss
2+− +mm mm m
C1
ω==
gg
CC
mm
12
12
Q
g 1
C1
g 1
g 3
C1
g 2
C2
+
gg
CC
−+
C1
g 3
mm
gg
C
12
12
mm mg 2
C2
2
mm
12
12
12
1C
(9)
The basic structure of Fig. 8 is equivalent to the cascaded
lossy integrators (or first-order LPFs) in Fig. 6.
However, the added positive feedback gm3 makes a
Hs
OTA-LPF1() =
gg
CC
ss
topology. Note that the diode-connected source
degeneration technique was used in the OTAs of the
feed-forward path (, ),gg
12 to extend the linear input
2 mm m
++
g 2
C2
ω0==
gg
CC
mm
12
12
Q
gg
CC
mm
12
12
12
12
gCg
gC
m
m
12
21
(10)
With this parameter setting, the positive feedback
path is removed and thus the feedback stability is easier
to be ensured.
In the original paper [1] that proposed the LPF in
Fig. 8 for the cochlea channel, the following choices
were made: CC C==12 and gg gmm m==12 leading to
the following equations for the transfer function, Q, and
ω0. The derived equation in (11) is the same as the one
introduced in [1].
Hs
OTA-LPF2 () =
ω0==
C
ss
gm
2
Q
g
C
g
C
2
m
2
+−
21
2 mm m
+
g 3
C
1
−
gm3
2gm
g
C
2
2
(11)
significant difference to the transfer function in (8) because
it cancels out the negative feedback within the
first lossy integrator when gg
from the transfer function that −gCm31/
13=
mm It can be seen
cancels out
13=
mm
.
gCm11/ when gg thereby the overall topology
reduces to the lossless-first two-integrator-loop
structure (see Fig. 5(a)). In other words, the positive
feedback converts a lossy integrator into a lossless
integrator. This in turn leads to complex poles in its
transfer function and its maximum Q value is no longer
limited to 0.5 in contrast to the case of cascaded
lossy integrators (Fig. 6 and (8)). The transfer function,
ω0, and Q factor of the OTA-based LPF when
gg
mm are given below.
13=
Figure 8. OTA-based second-order LPF with (a) gmC equivalent circuit and (b) small-signal diagram.
34
IEEE CIRCUITS AND SYSTEMS MAGAZINE
SECOND QUARTER 2023
IEEE Circuits and Systems Magazine - Q2 2023
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