Page 204 - Mechanical Engineers' Handbook (Volume 2)

2 Basic Analog Filters 193

V output (s)
H(s) (6)
(s)
V input
is the output-to-input gain of a standard two-port system.

2.1 Butterworth
The Butterworth filter has a smooth passband region (frequencies less than ƒ hertz) and a
c
smooth stop band (frequencies greater than ƒ ) and a comparatively wide transition region
c
as shown in Figs. 3 and 4. Let S be the cutoff frequency; then a low-pass rational function
c
approximation is as follows:
1
2

H(s)
(7)
1 (s/s ) 2N
c
where 2 ƒ s . In factored form,
c
c
1
H(s) (8)
(s p )(s p ) (s p N 1 )
1
0
where p j and
i
i
i
2 ƒ cos (N 2i 1) i 0,..., N 1
c
i
2 ƒ sin 2N (N 2i 1) (9)

i
c
2N i 0,..., N 1

Figure 3 Frequency response to a third-order Butterworth filter.