IEEE Circuits and Systems Magazine - Q2 2023 - 36
A simpler form of the BPF as shown in Fig. 10 uses
V. Source-Follower-Based Filters
the OTA as a core building block [13], [30]. This circuit
adopts a capacitively-coupled instrumentation amplifier
(CCIA) [31] associated with a buffer-based DC-servo
loop. Its transfer function can be derived as in (14)
−s
Hs
CCIA-BPF () =
ss
gm1
C2
2 mm m
++
g 2
C1
ω0==
gg
CC
mm
12
12
using a nodal analysis in (15).
iv vsC
vv sC gv v
ZZ BPF
=−
→=
vZ
→=
2
()
=−
F
() +−
()+
BPF
gs CC
m21
++
F
vBPF
−= +−mBPF12 ()
()+
− m
ZZ
Z
sC C2
sCs F g 1
F
gv sCvv vsC
v
BPFF
≈
(15)
Here, we assume CC C12, F as used in [31]. We
also assume the frequency range of this BPF is far
lower than gC
mF1 /
gg
12,
such that the filter parameters
12,
are solely determined by CC capacitors and
mm transconductors. From Fig. 10(b), we see
that this BPF circuit has a lossy-first two-integratorloop
topology shown in Fig. 3(c), with an added input
differentiation stage sC1. Therefore, although the
block diagram might look like a second-order LPF, the
transfer function shows a BPF response because of
the differentiator. A parallel filter array adopting the
BPF topology in Fig. 10 was implemented with an onchip
mixed-signal decision tree classifier in [13], and
demonstrated a 2-class VAD task.
sistor Mr =
Here, rgo111= / ds is the output impedance of trangdsb
is the output impedance of tail
L =() =+ dsb
1,/1ob
current source, and RsCr sC g11()// ob
11
is the load impedance seen at the VOUT node. The approximation
in (17) is valid assuming sufficiently large
intrinsic gain of the transistor gr ggmo m
=gCm /
ds 1
Since its cutoff frequency ω01 1 includes transconductance
and capacitance, the SF-LPF is a firstorder
gCm
filter or a lossy integrator as discussed
in Section II and presented in Fig. 7(b). As shown in
Fig. 11(b), the SF implements a local feedback around
the source node of the M1
transistor. Because the
closed-loop gain at DC is given in the first term of (16),
()= /.
1+ s
IN Zm Z
gsCvBPFmF + sC1s vIN
()
12()
gm1
=
mds
1
C1
gm1
⋅
gg g11 + s
++ dsb 1
gg g
1
C
m11ds
++ dsb
(17)
Q
gg
CC
12
12
gm1C1
gCm22
(14)
A. Source-Follower (First-Order LPF)
Fig.
11 shows a transistor-level
schematic, gCm
equivalent circuit, and block diagram of the SF-based
first-order LPF. The minus output port (sink) of gm1
transconductor, which is the M1 transistor, is tied to
AC GND (corresponding to VDD in the schematic) in
the gCm equivalent circuit. The transfer function of
the SF-LPF is given as below, denoting the equivalent
transconductance of the circuit as GI VmD
Chapter 3.2.5 in [32]).
=∂ ∂/
IN (see
Hs GR()==
SF
mL
1
gm11 1gr
1
++
RL 1
1
mo
1
(16)
⋅ RL
Figure 10. CCIA-based second-order BPF [13] with (a) gmC equivalent circuit and (b) small-signal diagram.
36
IEEE CIRCUITS AND SYSTEMS MAGAZINE
SECOND QUARTER 2023
IEEE Circuits and Systems Magazine - Q2 2023
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