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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For both Rauch and Sallen-Key filters, it is worth noticing that Q is in<strong>de</strong>pen<strong>de</strong>nt from<br />
C. Hence, while K remains constant, Q also stays constant. This enables to get a constant Qfactor<br />
frequency sweep, that is to say a constant harmonics rejection.<br />
As may be seen in Equation (IV.10), for RC filters the filter central frequency f0 is<br />
proportional to 1/C while a passive LC circuit, ma<strong>de</strong> of an inductor and a variable<br />
capacitance, resonates at a central frequency proportional to 1 / C . This is an interesting<br />
property for a frequency-tunable filter since the frequency tuning range is then enhanced:<br />
f<br />
f<br />
max max<br />
∝ (IV.15)<br />
min<br />
C<br />
C<br />
min<br />
For these two structures, it is also noticeable to say that gain is required to realize the<br />
selectivity. Now, it has to be kept in mind that the RF filter is located after a high gain LNA.<br />
Thus, consi<strong>de</strong>ring Friis formula, the filter amplification has to be as limited as possible to take<br />
advantage of the low NF of the LNA, since the NF of the receiver is required to be very low<br />
(about 4dB nowadays). This is also explained in [IV.3]. The structure offering the best<br />
selectivity versus gain tra<strong>de</strong>-off is looked for.<br />
From formulas in Table 17, trials have been performed to find this best tra<strong>de</strong>-off.<br />
Usual values used for the Sallen-Key filter are R1=R3=2R2. Thus,<br />
K<br />
Gain SK = (IV.16)<br />
3−<br />
K<br />
and<br />
1<br />
QSK = (IV.17)<br />
3−<br />
K<br />
Combining both equations leads to:<br />
1+ Gain SK<br />
QSK<br />
= (IV.18)<br />
3<br />
As far as the Rauch filter is concerned, it is usually used when R1=2R2 and R3=3R1.<br />
Hence, one gets:<br />
3K<br />
Gain R = (IV.19)<br />
11−<br />
2K<br />
So,<br />
K<br />
11Gain<br />
R<br />
= (IV.20)<br />
3+<br />
2Gain<br />
R<br />
Thus, one can replace K in the following last equation:<br />
3(<br />
1 − K)<br />
QR = (IV.21)<br />
11 − 2K<br />
QSK and QR versus the filter gain (GainSK and GainR) are plotted together in Figure<br />
143. It can be noticed that Rauch filters are more selective for a same filter amplification.<br />
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