Poles and zeros of network functions:
The network function or system system function is the ratio ratio of Laplace transform of output to transform of the input, input, neglecting neglecting initial conditions.It conditions.It can be expressed as the ratio of two poIynomials namely N(s) the numerator polynomial and (s) the d enominator polynomial as
In general form it can be expressed as,
!here " #an$bm is a positi%e constant known as scale factor or system gain factor.
The coefficients a,b,c and d are real and positi%e for passi%e& network and no dependent sources.
The numerator N(s) # ' has n roots, they are called as eros of the T(s)
The denominator (s) # ' has m roots, they are called as poles of the T(*)
T(*) can be expressed in the factorised form as,
!here +,,- ...... n are the roots of the polynomial N(s) # ' +,,- ...... m are the roots of the polynomial (s) # ' Poles:
The %alues %alues of /s0 /s0 i.e. i.e. comple complex x fre1uen fre1uencie ciess at which which the system system functi function on or networ network k function is infinite are called poles of the system function .
If such a poles are real and non repeated, these are called simple poles.
If a particular pole has same %alue twice or more than that or repeated %alue then it is called repeated pole.
2 pair of poles con3ugate %alues is called a pair of complex con3ugate con3ug ate poles.
The polynomial e1uation (s) of the network function is called characteristic e1uation of a system its roots are, known as poles.
Zeros:
The %alue of
‘s’ i.e.
complex fre1uencies at which the system function or network
function is 4ero are called 4eros of the system function. There can be simple 4eros, repeated 4eros or complex con3ugate 4eros. Singularities:
If the complex network function T(s) and all its deri%ati%es exist in a region in s5plane then
T(s) is said to be analytic in the region. The points in the s5plane at which the fun analytic
are called ordinary points. There are some points in the s5plane at which the function T(s) is not analytic i.e.T(s) and its deri%ati%es are not existing, then such points. are called singular points .or singularities of the nerwork function. 2s a network function is infinity at all poles i.e. a network function does not exist. Therefore poles are the singularities of the network function. & D.C. Gain: & The %alue of the network function at s # ' is called d.c. gain of the network function. If
we put s # ' in the network function then the result is a constant %alue. Pole-Zero plot 6 The s5plane is a complex plane with x5axis indicating real axis denoted as σ5axis(or real
axis) while y5axis indicating imaginary axis denoted as 3w5axis (or Im5axis). *uch a plane indicate %alues of the %ariables /s0 and hence called s5plane. 2ll the poles and 4eros are %alues of /s0 that be indicated in s5plane. The plot obtainecl by locating poles and 4eros of the system function in the s5plane is called as pole54ero plot of the system function. The poles are located by /cross0(x) and 4eros are located by /4ero0 (') in the s5plane, In case of repeated poles or 4eros,the number of marks should be e1ual to tbe number of repeated poles or 4eros.
Fig. s-plane
Time domain reponse from pole-zero plot
It is possible to determine the time5domain repsonse from the pole54ero plot of a network function. This is because the 4eros of network function are useful to obtain the magnitude response using partial fractions and the poles are the complex fre1uencies that help to determine time5domain beha%iour of the response. 7onsider a network function is gi%en by,
!here +,88.n are the 4eros and +, 88 m are the poles of the function 9(s). :sing partial fraction, we can write as,
!here " +, " 88 " m are the residues. 2 particular residue K i can be found as follows6
The each term (i 5 i) represents a %ector drawn from Z i to pole i and each term(i5k ) represent a %ector drawn from k to i as shown in ;ig.
Therefore,
This e1uation shows that a particular residue " i can be obtained from
2ll the residuces " +, " 8. " n are calcultated using this method. The time5domain response can be obtained by taking the in%erse Laplace transform.
