Transformation: Transfer Function ↔ Pole Zero

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The pole-zero and transfer function representations of a system are tightly linked. For example consider the transfer function:

If we rewrite this in a standard form such that the highest order term of the numerator and denominator are unity (the reason for this is explained below).

This is just a constant term (b₀/a₀) multiplied by a ratio of polynomials which can be factored.

In this equation the constant k=b₀/a₀.  The z_i terms are the zeros of the transfer function; as s→z_i the numerator polynomial goes to zero, so the transfer function also goes to zero.  The p_i terms are the poles of the transfer function; as s→p_i the denominator polynomial is zero, so the transfer function goes to infinity.

In the general case of a transfer function with an mth order numerator and an nth order denominator, the transfer function can be represented as:

The pole-zero representation consists of the poles (p_i), the zeros (z_i) and the gain term (k). 

Note: now the step of pulling out the constant term becomes obvious.  With the constant term out of the polynomials they can be written as a product of simple terms of the form (s-zi).  This would not be possible if the highest order term of the polynomials was not equal to one.

Often the gain term is not given as part of the representation.  The nature of the behavior of the system is given by the poles and zeros (e.g., does it oscillate? decay quickly? ...), the gain term only determines the magnitude of the response.  In many cases a plot is made of the s-plane that shows the locations of the poles and zeros, and the gain term (k) is not shown.  See the example below.

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