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

The transfer function of the circuit in Figure 16–44 is:
A(s) 1 2R 1 R 2 Cc·s R 1 R 2 C 2 c 2 ·s 2
1 2R 1 Cc·s R 1 R 2 C 2 c 2 ·s 2
The coefficient comparison with Equation 16–23 yields:
a 1 4fcR 1 C
b 1 a 1 fcR 2 C
a 1 2
b 1
R
R 3
Active Filter Design Techniques
All-Pass Filter Design
To design the circuit, specify f C, C, and R, and then solve for the resistor values:
R 1 a 1
4fcC
R 2 b 1
a 1 fcC
R 3 R
(16–34)
(16–35)
(16–36)
(16–37)
(16–38)
(16–39)
Inserting Equation 16–34 into Equation16–30 and substituting ωC with Equation 16–27
gives the maximum group delay of a second-order all-pass filter:
t (16–40)
gr0 4R1C 16.7.3 Higher-Order All-Pass Filter
Higher-order all-pass filters consist of cascaded first-order and second-order filter stages.
Example 16–7. 2-ms Delay All-Pass Filter
A signal with the frequency spectrum, 0 < f < 1 kHz, needs to be delayed by 2 ms. To keep
the phase distortions at a minimum, the corner frequency of the all-pass filter must be
f C ≥ 1 kHz.
Equation 16–27 determines the normalized group delay for frequencies below 1 kHz:
Tgro tgr0 2ms·1 kHz2.0 TC Figure 16–42 confirms that a seventh-order all-pass is needed to accomplish the desired
delay. The exact value, however, is T gr0 = 2.1737. To set the group delay to precisely 2 ms,
solve Equation 16–27 for f C and obtain the corner frequency:
16-45