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

Table of Contents for the Digital Edition of IEEE Circuits and Systems Magazine - Q2 2023
