# poles and zeroes

In network theory, there is a theorem which states that all the network functions in S domain can be expressed as the ratio of two polynomials A(S)/B(S). The books define the poles to be the roots of the polynomial B(s) and zeroes to be the roots of the polynomial A(s). However, going through the book I found that these definitions are not just sufficient, they don't cover the whole thing abt poles and zeroes. Can anyone tell me what exactly does pole, zero and residue mean in context of network analysis

• Possibly relevant reading: Design of RLC-Band Pass Filters www.euitt.upm.es/uploaded/519/bandfilter_script.pdf – DarenW Nov 6 '12 at 22:16

## 2 Answers

The zeros of a network function are the values of $s$ for which the function is zero (the numerator is zero).

The poles of a network function are the values of $s$ for which the function goes to infinity (the denominator is zero).

The numerator and denominator of the network function are polynomials in the complex variable $s$.

Using partial fraction expansion, the network function can be expressed as the sum of terms of the form:

$\dfrac{r_i}{s - p_i}$

where $r_i$ is the residue associated with the pole $p_i$

For example:

$\dfrac{s + 1}{s^2 + 5s + 6} = \dfrac{2}{s+3} + \dfrac{-1}{s+2}$

Clearly, this network function has:

1 zero: $s_z = -1$

2 poles: $s_{p1} = -3, s_{p2} = -2$

2 residues: $r_1 = 2, r_2 = -1$

@Alfred has it right. For a historical context and further reading this comes from complex analysis. This is the same field has contour integration in it and in particular that is where the residue concept comes from. Same math is used in fluid flow and other boundary values problems.