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

As it may be seen in Figure 64, an Rp-limited filter will give a constant gain frequency
sweep when tuning the capacitance. One can also notice that the Q-factor decreases with
frequency. Indeed, the quality factor of the parallel circuit is then defined as:
Q = R p
C
L
while higher frequencies are reached with smaller capacitances.
// (II.26)
However, most of the time, LC resonators are Rs-limited because of the poor Q-factors
of discrete or of on-chip inductors. Such a filter presents a constant bandwidth when sweeping
the frequency with the capacitance. The quality factor of this resonant circuit is given by:
1
Qs
=
Rs
L
C
(II.27)
Then, the bandwidth becomes independent from the tuning capacitance C:
f 0
BW =
Q
1
= Rs
2π
LC
C Rs
=
L 2πL
(II.28)
s
Another particularity of this Rs-limited circuit is that gain strongly increases with
frequency, due to the increase of the Q-factor, as illustrated in Figure 64.
Rp-limited circuit
Constant gain
Figure 64. Rp-limited (a) and Rs-limited LC filter when sweeping the capacitance
The large gain increase of an Rs-limited filter can be handled by the decrease of the
current I flowing through the resonator, by means of a reduced input transconductance gm, as
shown in Figure 65. This compensates for the gain increase but this will also result in a filter
with lower noise performances. Dividing the whole tuning range in sub-bands is another good
solution to limit the gain increase to a certain amount.
Figure 65. Current-fed resonating circuit
- 52 -
Rs-limited circuit
Constant BW = 20 MHz