Rules for Constructing Bode Diagrams

This document will discuss how to actually draw Bode diagrams.  It consists mostly of examples.


Key Concept -
To draw Bode diagram there are four steps:
  1. Rewrite the transfer function in proper form.
  2. Separate the transfer function into its constituent parts.
  3. Draw the Bode diagram for each part.
  4. Draw the overall Bode diagram by adding up the results from part 3.

1. Rewrite the transfer function in proper form.

A transfer function is normally of the form:

A Transfer Function

As discussed in the previous document, we would like to rewrite this so the lowest order term in the numerator and denominator are both unity.

Some examples will clarify:

Example 1

Example 1

Note that the final result has the lowest (zero) order power of numerator and denominator polynomial equal to unity.

Example 2

Example 2

Note that in this example, the lowest power in the numerator was 1.

Example 3

Example 3

In this example the denominator was already factored.  In cases like this, each factored term needs to have unity as the lowest order power of s (zero in this case).


2.  Separate the transfer function into its constituent parts.

The next step is to split up the function into its constituent parts.  There are seven types of parts:

  1. A constant
  2. Poles at the origin
  3. Zeros at the origin
  4. Real Poles
  5. Real Zeros
  6. Complex conjugate poles
  7. Complex conjugate zeros

We can use the examples above to demonstrate again.

Example 1

Example 1

This function has 

The complex conjugate poles are at s=-1.5 ± j6.9 (where j=sqrt(-1)).  A more common (and useful for our purposes) way to express this is to use the standard notation for a second order polynomial

Underdamped

In this case

Omega_0, zeta

Example 2

Example 2

This function has 

omega_0, zeta  

Example 3

Example 3

This function has 


3. Draw the Bode diagram for each part.

The rules for drawing the Bode diagram for each part are summarized on a separate pageExamples of each are given later.


4. Draw the overall Bode diagram by adding up the results from step 3.

After the individual terms are drawn, it is a simple matter to add them together.  See examples, below.


Examples: Draw Bode Diagrams for the following transfer functions

These examples are compiled on the next page.

Example 1

A simple pole

 

Example 1

Full Solution

Example 2

Multiple poles and zeros

Example 2

Full Solution

Example 3

A pole at the origin and poles and zeros

Example 3

Full Solution

Example 4

Repeated poles, a zero at the origin, and a negative constant

Example 4

Full Solution

Example 5

Complex conjugate poles

Example 5

Full Solution

Example 6

A complicated function

Example 6

Full Solution


References

© Copyright 2005-2013 Erik Cheever    This page may be freely used for educational purposes.

Erik Cheever       Department of Engineering         Swarthmore College