Handbook for Sound Engineers

796 Chapter 23

The values of R 8 , R 9 , R 10 , and R 11 are not critical and
should be chosen for minimum dc offset at each op-amp
stage.

23.3.1.3 All-Pass Filter

The circuit shown in Fig. 23-19 is an all-pass amplifier
with unity gain at all frequencies and having a phase
shift proportional to frequency according to

(23-51)

where,

T is the phase shift from input to output,

f 0 is 1 / (2SRC).

The phase shift is approximately proportional to the

frequency over a range of frequencies below and above

f 0. These circuits can be cascaded to induce more

phase-shift over the same frequency range or each

designed with a different f 0 to extend the range over

which phase is proportional to frequency.

(23-52)

Since phase is proportional to frequency, and from Eq.

23-50 phase is an expression of time, these circuits may

be used to introduce a small amount of delay.

23.3.2 Pole-Zero Analysis

A pole-zero plot, Fig. 23-20 is graphical way of repre-

senting the complex transfer function of a filter. The

pole-zero plot describes a surface that has peaks of infi-

nite magnitude that stretch the surface upward and zeros

that do the same downward. The height of the surface

along the Z axis, where V= 0, is the normal magnitude

response.

If the expression for the function is reduced to a

factored form in the s-plane where s is the Laplace

domain variable Eq. 23-35, then the transfer function of

a system can be represented as

Figure 23-18. Second-order state variable filter.

R 7

R 6

Vm R 12

C 1 C 2

R 1 R 2

R 8

R 4

R 11

R 3

R 9 R 10

Vo

Vo

10 k (^7) 10 k 7
2.4 k 7
10 k 7
10 k 7
10 k 7
10 k 7
5 k 7
MF MF
10 k 7
10 k 7
G
10 k 7
R 5
Low
pass

• 10 k 7
  Bandpass


• 
  

  (^)

  
  


• 
  

  Vo
  High pass
  Figure 23-19. All pass unity gain amplifier.
  T 2tan1–
  f 0
  f
  = ©¹§·--- -

  
  


• 
  

  R
  R
  C
  R
  TZ= t
  = 2 Sft