Page 86 - Analog and Digital Filter Design

3
CHAPTER

POLES AND ZEROES

This chapter contains material that is essential for those involved with analog
active filter design. Active filters can be designed from a set of numbers known
as the pole and zero locations. The pole and zero locations are obtained from a
filter’s transfer function, and typical pole and zero location patterns will be illus-
trated. This should give the reader a feel for how a filter will behave if it has a
certain pole-zero pattern, or a certain transfer function.

Tables of pole and zero locations are given in this chapter and can be used in
formulae (given in the next four chapters) to find resistor and capacitor values.
These normalized lowpass pole and zero values can be used to design lowpass,
highpass, bandpass, or bandstop filters. Scaling either the poles and zeroes, or
scaling the component values obtained from them, allows the frequency
response to be changed from the normalized lrads cutoff frequency. Pole
and zero placing formulae are given in the Appendix and include “natural” and
3 dB attenuation limited passbands. These formulae allow tables of pole and
zero values to be produced.

Knowledge of the origin or theory of poles and zeroes is not essential. However,
this information is provided for those readers who would like to understand the
ideas behind them. Poles and zeroes are located on a two-dimensional plane,
known as the S-plane. In the S-plane, one axis is “real” and is related to signal
decay. The other axis is “imaginary” and is related to frequency. The S-plane
will be explained further in a later section, as will an introduction to the Laplace
Transform.

This book is about filters, not poles and zeroes, so the theoretical coverage here
will be kept to a minimum. However, the excellent book by Robin Maddock
called Poles and Zeros in Electrical and Control Engineering would be useful
reading for anyone wishing to pursue this subject further.’ As Maddock’s title
indicates, he prefers the plural “zeros” rather than “zeroes,” which is my prefer-
ence; it’s just personal choice.