Handbook for Sound Engineers

706 Chapter 20

A second example will serve to further explore the
properties of transfer functions. Consider that the ampli-
fier of the previous example had an input resistance of
amount R. The input circuit is now to be modified by
connecting a capacitor of size C in series with this input
resistance to form a simple ac-coupled amplifier. Upon
denoting Z 0 c= 1 /RC the transfer function for this new
amplifier is given by

(20-6)

Eq. 20-6 indicates that the amplifier now has two
poles, the original one at S=Z 0 and a new one located
at S=Z 0 c. In addition, an examination of the numer-
ator of the transfer function indicates that there is now a
value of S for which the numerator becomes zero
namely at S=0. Values of S that make the numerator
zero are called the zeros of the transfer function. The
present amplifier has a single zero and a pair of poles
that can be displayed in a pole-zero diagram. A
pole-zero diagram is a drawing of the complex
frequency plane in which the pole locations are denoted
by X and the zero locations by 0. The pole-zero diagram
for the ac-coupled amplifier appears as Fig. 20-6.

Fig. 20-6 is a relatively simple pole-zero diagram as
the amplifier upon which it is based is simple. A few
conclusions based on more general amplifiers are worth
noting. Real poles may be singular while complex poles
always appear as conjugate pairs. The poles for ampli-
fiers that exhibit stable steady-state behavior may be
real or complex but must have negative real parts. The
zeros may appear anywhere in the S plane, but any zeros
with positive real parts are associated with nonmin-
imum phase behavior.

The Bode diagram for this example is arrived at by

the following steps. First, reform Eq. 20-6.

Substitute for Z 0 c in terms of Z 0 by examining the

pole-zero diagram, Z 0 c=1/4Z 0 therefore

Substitute S=jZ and find the absolute magnitude of the

resulting expression to obtain the gain function as indi-

cated here

Make a graph of versus. This

graph appears in Fig. 20-7. Next determine the phase

function I.

Figure 20-5. Nyquist diagram for the example amplifier.

Real part

(^510)
0
–5
–10
W
WWO
W ¾
Imaginary part
A
10 SZ 0
S+Z 0 c S+Z 0
=-----------------------------------------
Figure 20-6. Pole-zero diagram.
Pole
W¾ S Plane
O
Pole
WO Zero at origin
Imaginary
Real
A
S
S+Z 0 c

©¹
̈ ̧
§· 10 Z 0
S+Z 0
= ©¹§·---------------
A S
S
Z 0
4
+------

©¹
̈ ̧
̈ ̧
̈ ̧
§·
10 Z 0
S+Z 0
= ©¹§·---------------
GA=
Z
Z
2 Z 0
2
16
+--------

10 Z 0
Z
2
Z 0
2

• 
  

  = u--------------------------
  Z
  Z 0

  
  

  Z^2
  Z 02
  --------^1
  16
  ----- -+
  --------------------------^10
  Z^2
  Z 02
  --------+ 1
  = u----------------------
  20 dBlogG log ZZe 0