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

II.1.c Bandpass Topologies
II.1.c.i Low-Pass to Bandpass Transformation
Regardless of the order, a bandpass filter [II.1] can be obtained from a low-pass filter
by means of the following transformation:
2 2
s + ω0
s → ,
s.
BWω
where BWω is the -3dB bandwidth
(II.6)
BW = ω −ω
, (II.7)
ω 2 1
expressed in pulsation, and ω0 is the resonant pulsation:
2
0 ω1ω2
ω = , (II.8)
In these formulas, ω1 and ω2 are pulsations which enable to build the bandpass filter
characterizing its passband and stopbands.
The low-pass to band-pass transform consists in replacing inductors in the circuit by a
series combination of inductor and capacitor, and capacitors by a parallel combination of
inductor and capacitor [II.3], as it may be seen in Figure 45.
Figure 45. Low-pass to Bandpass Transformation
Hence, the transfer function of a second order bandpass filter in Laplace domain has
the following form [II.1]:
1
H BPF ( s)
=
.
⎛ s ω0
⎞
(II.9)
1 + Q ⎜ − ⎟
⎝ ω0
s ⎠
This filter presents a resonance at a frequency, called central frequency, determined by the
reactive elements of the circuit:
1
f
2π
LC
0 = . (II.10)
Q is a term called quality factor, studied in more details later on, given by the formula:
f 0
Q =
BW
(II.11)
f
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