As before, consider expanding the transfer function as the ratio of two polynomials
If are the roots of and are the roots of we can write
where A is a real constant, are zeros of and are poles (infinities) of . Knowledge of and determines everywhere.
Lets now look at our two filter circuits. For a low-pass filter
and the filter has one pole at -1/(RC). For a high-pass filter
and it has one pole at -1/(RC) and one zero at 0. We refer to these two types of filters as single-pole filters.
There is a general rule that there must be at least as many reactive elements as poles. Based on the location of the poles we are able to deduce the general response properties of the filter. We will not do this here.
Example: If a transfer function has poles at and and a zero at (0,0), as shown in figure 3.5,
Figure 3.5: Poles and zeros in the complex plane.
- sketch on the interval .
The transfer function is given by
Plugging in values for gives the table 3.1.
Table 3.1: Numerical values of the transfer function.
Figure 3.6: The transfer function from the table above.
- If , what is the approximate value of at its highest point?
If then at is . Therefore