How to Calculate Poles & Zeros

Factor the Transfer Function

• 1

  Write out your transfer function. This should take the form of a polynomial with a number of terms on top and on the bottom. Either by hand or using a factoring program, find the factored form of this polynomial equation. This should give you something of the form H(s) = (s-z)/(s-p).

• 2

  List all terms in the denominator. These will correspond to your poles. All of your terms should be of the form (s-p). If it's of the form (s+p), rewrite it as (s-(-p)). If you remember that you're solving for zero, this means that s has to be equal to p. So, if the term is (s-3), s will equal 3. If the term is (s + 1/2), rewrite it as (s - (-1/2)) and s will equal -1/2. Do the same thing for zeros.

• 3

  Look for any terms that gave you a value that was 'plus or minus', or gave a complex conjugate, when you factored them. These are 'imaginary' values to your terms, and describe the imaginary part of the waveform. They lead to sinusoidal frequency responses. 'Real' values lead to exponential frequency responses.

• 4

  Draw all your poles and zeros on your chart. The 'real' axis is the x axis and the 'imaginary' axis is the Y axis. If there is no imaginary part to a pole or zero, just write an X for pole or O for zero on the graph at the corresponding value of s. If there is an imaginary part, write the X or O at both the positive and negative value of the imaginary component, with the line going through the real component. In other words, if a pole had a real component of 3 and an imaginary component of plus or minus 4, there would be poles at (3,4) and (3,-4).