IEEE Circuits and Systems Magazine - Q2 2023 - 33
Similarly to (4), HsX() includes s +ω2 term in the
numerator and thus it exhibits a lossy behavior at its
high-pass shape as shown in Fig. 5(c) where HX 01
/ = Q . As in (5), by subtract()
=
and its peak gain is ωω
Hs Hs Hs
s
() = () −
= 2
ss
++
ωωω
1
21 2
The subtraction can be implemented either within
the feedback loop (VY in Fig. 5(a)) or out of the loop
( VBPF in Fig. 5(b)). Note that 1/Q term is not multiplied
with Hs
LPF() in (7) because Q is now dependent on ω1
and ω2 while the feedback path within the lossy integrator
in Fig. 5 has unity gain.
4) Biquad LPF With Cascaded Lossy Integrators: A second-order
filter can alternatively be implemented
by a cascade of two first-order lossy integrators
with two different poles at (, )ωω
12 as shown in
Fig. 6. The transfer function of this topology can
be calculated as below.
Hs
Cascade () =
1
⋅
1+
= 2
Q =
ss
1+
ωω
ωω
ss
ωω
++
ωω
12
+
The Q-factor of this topology has a maximum value
of 0.5, which one can derive by using xy xy+≥2. In
12
12
() +ωω ωω
12 ≤ 050.
12 12
(8)
Figure 6. Second-order LPF composed of cascaded lossy
integrators.
1
III. Notation Declaration and Assumptions
Throughout this article, transconductors ()gm will be
depicted using a 4-port drawing as shown in Fig. 7(a)
[24], instead of the conventional 3-port drawing style
that uses a single output port (rightside of Fig. 7(b)).
This is particularly needed to analyze the sourcefollower-based
filters, in which a single transistor acts
as a transconductor and multiple transconductors are
placed in a single bias current branch. We will discuss
2
12
ing VX and VLPF, one gets an ideal band-pass response.
YX LPF ()
ω
order to realize a wide Q tunability, a second-order filter
based on a two-integrator-loop topology is preferred
over a cascaded lossy integrators.
In the following sections, we will analyze the sec=Hs
BPF
()
(7)
ond-order filter circuits by interpreting them either
as lossy-first or as lossless-first two-integrator-loop
topologies. In addition, we will only deal with the
small-signal models excluding large signal behaviors
of the filter.
Figure 5. Two-integrator-loop representations with a lossless-first
configuration using two different poles.
SECOND QUARTER 2023
Figure 7. (a) The 4-port notation of a transconductor, (b)
implementation of a lossy integrator with a transconductor,
and block diagrams of a lossy integrator using (c) voltagemode
subtraction and (d) current-mode subtraction.
IEEE CIRCUITS AND SYSTEMS MAGAZINE
33
IEEE Circuits and Systems Magazine - Q2 2023
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