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