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

Figure 39. H3 and N+5 rejections of a first order low-pass filter
A higher order low-pass filter can be obtained increasing the number of poles in the
transfer function by the addition of reactive elements in the circuit. Figure 40 presents second
order low-pass filters built either cascading two RC first order filters or combining an
inductor to a capacitor.
Vin
R1
C1
R2
C2
- 37 -
Vout
Vin
Figure 40. Second order RC and LC low-pass filters
The transfer function of a second order filter is then given by the formulas:
α
1
H LPF ( s)
= =
, 2
2
(II.4)
( s − p1
)( s − p2
) 1+
β1s
+ β 2 s
α being a constant. Such filter allows getting a steeper slope in order to reject harmonics more
strongly as depicted in Figure 41 and Figure 42.
For second order (and more) filters, different topologies may be used, as illustrated in
Figure 43, corresponding to different sets of coefficients βis in the transfer function [II.2].
These topologies, known as Chebychev, elliptic or Butterworth filters, enable to get a flatter
passband or a steeper slope (see Figure 43). Their rejections are plotted in Figure 44. Though
elliptic filters show higher rejections, it is worth adding that this kind of filter uses one
reactive element more than the other two filters, which creates a transmission zero (notch).
For instance, the normalized transfer function of a second order Butterworth low-pass
filter is given by:
1
H Butterworth
( s)
= . 2
(II.5)
2
1+
2s
+ s
L
C
Vout