"Chapter 1 - The Op Amp's Place in the World" - HTL Wien 10

Band-Pass Filter Design
16-32
out affecting bandwidth, B, or gain, A m. For low values of Q, the filter can work without R3,
however, Q then depends on A m via:
Am 2Q 2
Example 16–5. Second-Order MFB Band-Pass Filter with f m = 1 kHz
To design a second-order MFB band-pass filter with a mid frequency of f m = 1 kHz, a quality
factor of Q = 10, and a gain of A m = –2, assume a capacitor value of C = 100 nF, and
solve the previous equations for R 1 through R 3 in the following sequence:
R2 Q
fmC
10
31.8 k
·1 kHz·100 nF
R1 R2
31.8 k
7.96 k
2Am 4
R3 AmR1 2Q2
2·7.96 k
80.4
Am 200 2
16.5.2 Fourth-Order Band-Pass Filter (Staggered Tuning)
Figure 16–32 shows that the frequency response of second-order band-pass filters gets
steeper with rising Q. However, there are band-pass applications that require a flat gain
response close to the mid frequency as well as a sharp passband-to-stopband transition.
These tasks can be accomplished by higher-order band-pass filters.
Of particular interest is the application of the low-pass to band-pass transformation onto
a second-order low-pass filter, since it leads to a fourth-order band-pass filter.
Replacing the S term in Equation 16–2 with Equation 16–7 gives the general transfer function
of a fourth-order band-pass:
A(s)
s 2 ·A 0 () 2
b 1
1 a1 ·s 2 b1 ()2·s
b1 2 a1 ·s b1 3 s4 (16–11)
Similar to the low-pass filters, the fourth-order transfer function is split into two second-order
band-pass terms. Further mathematical modifications yield:
A(s)
1 s
Q 1
Ami ·s
Qi (s) 2 ·
Ami ·
Qi s
1 1
Qi s s 2 (16–12)
Equation 16–12 represents the connection of two second-order band-pass filters in series,
where