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

H HPF
s
( s)
=
s
1 +
ω
This corresponds to inverting capacitors and inductors in the circuit (see
Figure 50), and to modify their value according to the transformation.
Figure 50. Low-pass to High-pass Transformation
1
- 43 -
(II.18)
Combining low-pass (with ω2 as cut-off frequency) and high-pass topologies, it leads
to the following bandpass filter transfer function:
s
H BPF ( s)
= H LPF ( s).
H HPF ( s)
=
.
2
⎛ 1 1 ⎞ s
(II.19)
1+
s ⎜ + +
ω1
ω ⎟
⎝
2 ⎠ ω1ω2
Assuming
ω1ω2
Q = and ω 0 = ω1ω2
, (II.20)
ω1
+ ω2
this gives:
1
H BPF ( s)
=
.
⎛ s ω0
⎞
(II.21)
1+
jQ
⎜ −
⎟
⎝ ω0
s ⎠
1
Thus, Q is maximum when ω1=ω2 and this gives max
2
= Q . The combination of a
low-pass and a high-pass filter gives a bandpass filter, but this filter has a selectivity limited to
1
. Figure 67, shown further, underlines that such selectivity is not enough to reach the RF
2
selectivity specifications.
II.1.c.iv Assessment of the Topology
The main advantage of the bandpass filter topology is its ability to reach a smaller
bandwidth than a combination of a low-pass and a high-pass filters. Based on only two
reactive components for a second order filter, the selectivity can be adjusted to whatever -3dB