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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g<br />
V<br />
m3<br />
in<br />
+ g′<br />
′ V<br />
3<br />
m3<br />
in<br />
Replacing:<br />
3<br />
g 1g<br />
2r<br />
⎛<br />
0<br />
3 g 1r0<br />
g 1r<br />
⎞<br />
m m<br />
m<br />
m 0<br />
3 3<br />
3 1 jr0C<br />
bω<br />
( αVin<br />
βV<br />
) ⎜ ′<br />
⎛ ⎞<br />
+<br />
=<br />
+ in + gm<br />
2 + g′<br />
′<br />
⎟<br />
m2<br />
( αVin<br />
+ βVin<br />
) + ( αVin<br />
+ βVin<br />
)<br />
1 jr0C<br />
aω<br />
⎜ 1 jr0C<br />
aω<br />
⎜<br />
1 jr0C<br />
aω<br />
⎟<br />
+<br />
+<br />
⎟<br />
r0<br />
⎝<br />
⎝ + ⎠ ⎠<br />
(<strong>III</strong>.15)<br />
Thus, this leads to:<br />
α<br />
m3<br />
= ,<br />
gm1g<br />
m2r0<br />
1 + jr0C<br />
bω<br />
+<br />
1 + jr C ω r<br />
which corresponds to the gain of the filter transfer function.<br />
0<br />
a<br />
g<br />
- 86 -<br />
0<br />
(<strong>III</strong>.16)<br />
The second equation gives:<br />
3<br />
g 1g<br />
2r<br />
⎛<br />
0 g 1r0<br />
g 1r<br />
⎞<br />
m m<br />
m<br />
m 0<br />
3 1 jr0C<br />
bω<br />
gm3<br />
β<br />
⎜ ′<br />
⎛ ⎞ +<br />
′′ =<br />
+ gm<br />
2 + g ′′<br />
⎟<br />
m2<br />
α + β<br />
1 jr0C<br />
aω<br />
⎜ 1 jr0C<br />
aω<br />
⎜<br />
1 jr0C<br />
aω<br />
⎟<br />
(<strong>III</strong>.17)<br />
+<br />
+<br />
⎟ r0<br />
⎝<br />
⎝ + ⎠ ⎠<br />
Hence,<br />
β<br />
g ′′<br />
This finally gives:<br />
⎛<br />
− ⎜ g<br />
g ′′<br />
m1<br />
0<br />
m1<br />
0<br />
3<br />
m3<br />
m2<br />
m2<br />
⎜ 1 + jr0C<br />
aω<br />
⎜1<br />
jr0C<br />
aω<br />
⎟ ⎟<br />
⎝<br />
⎝ + ⎠<br />
=<br />
⎠<br />
(<strong>III</strong>.18)<br />
gm1g<br />
m2r0<br />
1 + jr0C<br />
bω<br />
+<br />
1 + jr C ω r<br />
0<br />
r<br />
a<br />
⎛<br />
+ g′<br />
′ ⎜<br />
0<br />
g<br />
r<br />
3<br />
⎞ ⎞<br />
⎟ ⎟α<br />
3<br />
3<br />
⎛ g 1 2 0 1 ⎞ ⎛<br />
⎞<br />
0 ⎜ ′′ ⎛ ⎞<br />
m gm<br />
r + jr Cbω<br />
gm1r0<br />
gm1r0<br />
3<br />
g ′′<br />
′<br />
⎟<br />
m3<br />
⎜ +<br />
2<br />
2<br />
3<br />
1<br />
⎟ − gm<br />
+ gm<br />
⎜<br />
0<br />
0<br />
1<br />
⎜<br />
0 1 ⎟ gm<br />
⎝ + jr C<br />
⎠ +<br />
⎟<br />
aω<br />
r<br />
jr C<br />
⎝<br />
aω<br />
⎝ + jr0C<br />
aω<br />
⎠<br />
β =<br />
⎠<br />
4<br />
, (<strong>III</strong>.19)<br />
⎛ g 1g<br />
2r0<br />
1 + jr ⎞<br />
m m<br />
0Cbω<br />
⎜ +<br />
1<br />
⎟<br />
⎝ + jr0C<br />
aω<br />
r0<br />
⎠<br />
which corresponds to the third or<strong>de</strong>r non-linear term.<br />
Determining by <strong>de</strong>sign the achievable IIP3 of all transconductors leads to the various<br />
gmi3, thanks to the formula:<br />
4 IIP3<br />
g ′′ mi = . gmi<br />
. 10 . (<strong>III</strong>.20)<br />
3<br />
β and α can thus be computed and this gives the achievable IIP3 of the filter.