Handbook for Sound Engineers

Amplifier Design 715

(20-20)

Upon choosing R 1 =R 2 , Eq. 20-20 dictates that
C 1 =4C 2. If C 1 is chosen to be 0.02μF, then C 2
becomes 0.005μF and Eq. 20-19 then requires that R 1
be 10 k:. The reasonableness of these values allows the
design to be concluded with the circuit of Fig. 20-17.

Example 2. Second-order Bessel low pass with a zero
frequency group delay of 500μs.

The required transfer function is

(20-21)

with Z 0 =1/500μs. Taking Fig. 20-16C along with its
transfer function leads to the identification

(20-22)

from which it is found that

(20-23)

and

(20-24)

Upon choosing R 1 = R 2 , Eq. 20-24 requires that

Figure 20-17. Third-octave unity gain Butterworth low-pass

filter with fo = 500 Hz.

1

R 1 R 2 C 1 C 2

------------------------------

R 1 C 2 +R 2 C 2

R 1 R 2 C 1 C 2

--------------------------------

©¹

=§·

k 7

MF

k 7 k 7

MF

MF

Vin

Vout

+

• 
  
  
  
  • 
    
    
    
    • 
      
    
    
    
    
  
  
  
  

V 0

Vin

-------

3 Z 02

S

2

3 Z 0 S 3 Z 0

2

++

=-------------------------------------------

3 Z 02

S

2

3 Z 0 S 3 Z 0

2

++

-------------------------------------------

1

R 1 R 2 C 1 C 2

--------------------------

S^2

R 1 C 2 +R 2 C 2

R 1 R 2 C 1 C 2

--------------------------------

©¹

§·S^1

R 1 R 2 C 1 C 2

++--------------------------

-------------------------------------------------------------------------------------

=

Z 0 1

3 R 1 R 2 C 1 C 2

=---------------------------------

3

3 R 1 R 2 C 1 C 2

---------------------------------

R 1 C 2 +R 2 C 2

R 1 R 2 C 1 C 2

=--------------------------------

Table 20-1. Transfer Functions of Various Circuits of

Figure 20-16

Figure K Transfer function

20-16A = 1

20-16B = 1

20-16C = 1

20-16D = 1

20-16At 1

20-16Bt 1

20-16Ct 1

20-16Dt 1

20-16Ft 1

Vo

----- -Vi

1

RC

--------

S RC-------^1 -+

=-----------------

Vo

Vi

----- - S

S^1

RC

--------+

=-----------------

Vo

Vi

----- -

1

R 1 R 2 C 1 C 2

--------------------------

S^2

R 1 C 2 +R 2 C 2

R 1 R 2 C 1 C 2

©¹§·--------------------------------S^1

R 1 R 2 C 1 C 2

++--------------------------

=--------------------------------------------------------------------------------------

Vo

Vi

----- - S

2

S^2

R 1 C 2 +R 1 C 1

©¹--------------------------------R 1 R 2 C 1 C 2

§·S^1

R 1 R 2 C 1 C 2

++--------------------------

=--------------------------------------------------------------------------------------

Vo

Vi

----- -

K^1

RC

©¹§·--------

S RC-------^1 -+

------------------=

Vo

Vi

----- - KS

S^1

RC

--------+

=-----------------

Vo

Vi

----- -

KR^1

1 R 2 C 1 C 2

©¹§·--------------------------

S^2

R 1 C 2 ++R 2 C 2 1 – KR 1 C 1

R 1 R 2 C 1 C 2

©¹§·----------------------------------------------------------------------S^1

++R-------------------------- 1 R 2 C 1 C 2

----------------------------------------------------------------------------------------------------------------------------

=

Vo

Vi

----- -

KS^2

S^2

R 1 C 2 ++R 1 C 1 1 – KR 2 C 2

©¹---------------------------------------------------------------------R 1 R 2 C 1 C 2 -

§·S^1

R 1 R 2 C 1 C 2

++--------------------------

----------------------------------------------------------------------------------------------------------------------------

=

Vo

Vi

----- -

KS

R 1 C

----------

S^2

3 R 1 R 2 C+ 1 – KR 12 C

R 12 R 2 C^2

--------------------------------------------------------

©¹

̈ ̧

§·

S

R 1 +R 2

R 12 R 2 C^2

++-------------------

=-------------------------------------------------------------------------------------------------------

or

Vo

Vi

----- -

Ao

Q

©¹§·----- -ZoS

S^2

Zo

©¹------Q

§·S Z

o

++^2

=----------------------------------------