# Relation between poles, zeros and frequency response of filter?

#### AnalogDigitalDesigner

Joined Jan 22, 2018
121
Hello my dear and lovely friends,

I am a student of electronics and I am studying filters right now.

I would like to know what is the relationship between the poles, zeroes, and the actual frequency response of a filter? How does the s-place graph help us understand the response, and how do these poles and zeroes change the shape of the magnitude response bode plots?

Please I would really appreciate some help here. It's a mystery to me!

Best wishes and have a great day my friends!

Paulo

#### drc_567

Joined Dec 29, 2008
1,156
Basically, you are dealing with graphical representations of the circuit transfer function. This is the behavior of an output variable with respect to a stimulus or input variable. Generally, this takes the form of a numerator ... zero factors, and a denominator ... pole factors.

The Bode plot yields the circuit amplitude and phase as a function of frequency, giving a convenient view of these quantities with respect to frequency and usable bandwidth. The system stability as a function of frequency can also be ascertained.

The Root Locus, or S-plane plot, is oriented towards viewing transient performance ... how fast or slow a circuit output variable will take to arrive at a steady state. Long term circuit behavior is readily observed on the Root Locus plot ... left side indicates system stability, with long term decreasing amplitude, while right side indicates increasing, unstable performance. A typical Root Locus path segment originates at a pole root factor, and terminates at a zero root value, with the actual path location point being determined by the circuit transfer function parameters.
Example: A quadratic factor in the transfer function denominator will yield either real value, steady state results, or else complex number oscillating values, depending on the specific quadratic factor coefficients.
Another interesting aspect of the Root Locus plot is that the overall circuit performance can be altered, or designed, shifting the locus path by the addition and judicious placement of an extra pole or zero, the resulting effect being somewhat analogous to that of static electric charges lying in a plane.

Last edited:

Joined Mar 10, 2018
4,057
Here is an excellent Java app that allows you to add/remove poles and zeros
to S plane, move them around, and show effect of response.

Note Java may have to be updated on your machine and security exception
added in java for site url. In windows startmenu, search for "java", pick java console,