III. Gm-C Filtering - Epublications - Université de Limoges

and
As discussed in the previous chapter, in Figure 98, it is assumed that
1
r s = .
r g g
0
m1
m2
A first computation step gives:
g
V
⎛
⎞
⎜
1 1
= V
+ ⎟
out +
jC ω
⎜
b ⎟
⎝ r0
jLeff
ω + rs
⎠
- 84 -
Ca
L eff =
g g
m1
m2
m3
in
(III.1)
Hence, the filter transfer function can be deduced:
V
H ( jω)
=
V
out
in
=
r
r
s
0
g
ω
+ 1+
j
r
0
m3
( jL
Such a filter resonates at a central frequency
with a quality-factor set to
f
0
Q = r
0
1
=
2π
g
eff
ω + r )
2
( Leff
+ rs
r0C
b ) −ω
Leff
Cb
m1
a
1
gm
2 + 2
r0
C C
a
b
b
⎛ 1
CaCb
⎜ + g 2
⎝ r0
C + C
m1
g
m2
s
⎞
⎟
⎠
.
(III.2)
(III.3)
(III.4)
It is worth noticing that tunable capacitors Ca correspond to an equivalent tunable
inductor Leff. While passive inductor – varying capacitor resonator leads to a central frequency
proportional to1 / C , here it is possible to tune both Leff and C provided that Ca and Cb are
proportional. f0 then becomes proportional to1 / C . This is a very interesting property for a
tunable filter since it results in a larger tuning range for a same capacitance ratio:
f max Cmax
∝ . (III.5)
f C
min
min
That is why in the following, it is assumed Ca = Cb
= C . Moreover, when only tuning
Cb, it leads to a filter gain increasing with frequency, because Q increases as well, whereas
tuning both capacitors makes the gain remain constant.
This structure also permits to achieve a constant-Q frequency sweep because Q then
becomes constant, independent from C since:
r 1
= g . (III.6)
0 Q + 2
2 r0
m1g m2
As previously mentioned, this allows rejecting harmonic frequencies in a constant way.