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

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