Handbook for Sound Engineers

712 Chapter 20

importance in modern audio applications and more
particularly those that are readily implemented by
means of active circuitry. Even though the emphasis

will be on active circuitry, a cursory examination of

some simple passive structures is of value.

Fig. 20-14 displays a few passive filter structures

along with their associated transfer functions. Note that

in each instance the filter transfer functions involve a

polynomial with S appearing in the denominator. The

characteristics of the various filters are associated with

the structure of these polynomials. With the possible

exception of antialiasing use, the filters employed in

audio work are restricted to pole orders of three or less

in order to maintain good transient response. A pole

order of three corresponds to an asymptotic slope rate of

18 dB per octave in the filter stop band. The most

popular polynomials for audio applications are the

Butterworth, with maximally flat amplitude response,

and the Bessel, with linear phase response (maximally

flat group delay). The Butterworth polynomials through

third order are:

1.

2.

Figure 20-13. Bode diagram for a dominant pole amplifier. 3.

120

100

80

60

40

20

0

0

30 o

60 o

90 o

0 10 10^2     103    104     105    106    107

Frequency—Hz

Open loop

Closed loop

20 dB log

Vo

Vi

S+Z 0

S^2 ++ 2 SZ 0 Z 02

S^3 +++ 2 S^2 Z 0 2 SZ 02 Z 03

Figure 20-14. Passive filter structures.

R

L

C

R

L

C R

C L R

R

L

C R

C

L R

R C

L

A = RC

S +

W 0

= S + W 0

1

RC

1

A =

W 0

S + W 0

L

S +

=

R

L

R

A =

LC

S^2 + RCS + LC^1

=

W 02

S^2 + RSWC^0 + W 02

1

L

A =

S^2 + S^2 + W 02

RW 0 S

LC

1

L

RS

L

RS

+

= RW

0 S+

L

C

L

C

A =

S^2

S^2

S^2

S^2 W 02

RW 0 S W 02

LC

1

LC

1

L

RS = + +

+ +

+ +

L

C

A = S

2

S^2 + S 2

S^2

LC SW^0 W 02

1

R

RC

S + =

+ C+

L

A = SS +

W 0

S

S +

=

RC

1

A =

W 0

S + W 0

S

S +

=

L

R

A. Single-pole low-pass.

C. Two-pole low-pass.

E. Two-pole bandpass.

B. Single-pole high-pass.

D. Two-pole high-pass.

F. Two-pole band reject.