Handbook for Sound Engineers

Amplifier Design 713

These are often written in a normalized form such as

1.

2.

3.

In the Butterworth polynomials, Z 0 =2Sf0, where f 0

is the frequency at which the response is 3 dB down.

The Butterworth polynomials yield excellent amplitude

response characteristics while their phase and group

delay characteristics are far from being ideal. Their use

in constant resistance crossover networks is almost

universal.

The Bessel polynomials in normalized form are

1.

2.

3.

Here, Z 0 has the significance that the group delay at
zero frequency is just the reciprocal of Z 0. The group
delay for any filter at any frequency is given by the
negative of the first derivative of phase response with
respect to Z:

(20-12)

For a system not to introduce any phase distortion, it
is necessary that I be either independent of frequency or
of the form

(20-13)

In the first instance tg= 0 and for Eq. 20-13 tg=k,
where k is a constant. Bessel filters are nearly ideal in
this respect as their group delays are constant or nearly
so throughout their passbands. Unfortunately, the ampli-
tude response of Bessel filters for orders higher than
one, though without ripples, is not as flat as the corre-
sponding Butterworth filter. The first-order Bessel and
Butterworth filters are identical.

Operational amplifiers make significant contribu-

tions in the area of active filter implementation. The

following examples, though by no means exhaustive,

will serve as an introduction to this important subject.

The circuit of Fig. 20-15 simulates a physical

inductor. A physical inductor at low frequencies, where

interturn capacitance is not of importance, can be

thought of as a pure resistance in series with a pure self-

inductance. As such, a physical inductor has an imped-

ance Z that has both a real and an imaginary part.

A physical inductor also has a quality factor or Q.

These properties are summarized by the following equa-

tions:

(20-14)

(20-15)

Third-order or higher filters are readily obtained by

cascading two or more sections of the examples

displayed in Fig. 20-16. The transfer functions of the

various filters appear in Table 20-1.

This discussion of active filters employing opera-

tional amplifiers will now be concluded by exploring

two design examples.

Example 1. Third-order Butterworth low pass with a

corner frequency f 0 of 500 Hz and unity gain.

This filter can be implemented by cascading a first-

order section followed by a second-order section. The

required overall transfer function is

(20-16)

Taking Fig. 20-16A for the first-order section along

with its transfer function leads to the identification

S

Z 0

------+ 1

S

2

Z 0

------^2 S

Z 0

++---------- 1

S^3

Z 03

--------^2 S

2

Z 02

--------^2 S

Z 0

+++------ 1

S

Z 0

------+ 1

S^2

3 Z 02

------------ S

Z 0

++------ 1

S^3

15 Z 03

---------------^2 S

2

5 Z 02

------------^2 S

Z 0

+++------ 1

tg –dI

dZ

=---------

I –kZ+= constant

Figure 20-15. The resistor, capacitor, operational amplifier

combination presents the signal source with the same

impedance at all frequencies as does the physical inductor

in the dotted enclosure. The two circuits are equivalent.

+

R

L

Vi Vi

R/2

R/2

4 L

R^2

C =

• 
  

ZRj+= ZL

Q ZL

R

=-------

V 0

Vin

-------

Z 0

S+Z 0

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

Z 02

S^2 ++SZ 0 Z 02

= u-------------------------------------