Poles and zeros of network functions:The network function or
system function is the ratio of Laplace transform of output to
transform of the input, neglecting initial conditions.It can be
expressed as the ratio of two poIynomials namely N(s) the numerator
polynomial and D(s) the denominator polynomial as
In general form it can be expressed as,
Where K =an/bm is a positive constant known as scale factor or
system gain factor.
The coefficients a,b,c and d are real and positive for passive
network and no dependent sources.
The numerator N(s) = 0 has 'n' roots, they are called as Zeros
of the T(s) The denominator D(s) = 0 has 'm' roots, they are called
as poles of the T(S)T(S) can be expressed in the factorised form
as,
Where Z1,Z2,Z3 ...... Zn are the roots of the polynomial N(s) =
0
P1,P2,P3 ...... Pm are the roots of the polynomial D(s) =
0Poles:
The values of s i.e. complex frequencies at which the system
function or network 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 value twice or more than that or
repeated value then it is called repeated pole.
A pair of poles conjugate values is called a pair of complex
conjugate poles.
The polynomial equation D(s) of the network function is called
characteristic equation of a system its roots are, known as
poles.Zeros:
The value of s i.e. complex frequencies at which the system
function or network function is zero are called zeros of the system
function.
There can be simple zeros, repeated zeros or complex conjugate
zeros. Singularities:
If the complex network function T(s) and all its derivatives
exist in a region in s-plane then T(s) is said to be analytic in
the region. The points in the s-plane at which the fun analytic are
called ordinary points.
There are some points in the s-plane at which the function T(s)
is not analytic i.e.T(s) and its derivatives are not existing, then
such points. are called singular points .or singularities of the
nerwork function.
As 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 value of the network function
at s = 0 is called d.c. gain of the network function. If we put s =
0 in the network function then the result is a constant value.
Pole-Zero plot:
The s-plane is a complex plane with x-axis indicating real axis
denoted as (-axis(or real axis) while y-axis indicating imaginary
axis denoted as jw-axis (or Im-axis). Such a plane indicate values
of the variables s and hence called s-plane. All the poles and
zeros are values of s that be indicated in s-plane. The plot
obtainecl by locating poles and zeros of the system function in the
s-plane is called as pole-zero plot of the system function. The
poles are located by cross(x) and zeros are located by zero (0) in
the s-plane, In case of repeated poles or zeros,the number of marks
should be equal to tbe number of repeated poles or zeros.
Fig. s-plane
Time domain reponse from pole-zero plot
It is possible to determine the time-domain repsonse from the
pole-zero plot of a network function. This is because the zeros of
network function are useful to obtain the magnitude response using
partial fractions and the poles are the complex frequencies that
help to determine time-domain behaviour of the response.Consider a
network function is given by,
Where Z1,Z2.Zn are the zeros and P1, P2 Pm are the poles of the
function H(s).
Using partial fraction, we can write as,
Where K1, K2 Km are the residues.
A particular residue Ki can be found as follows:
The each term (Pi - Zi) represents a vector drawn from Zi to
pole Pi and each term(Pi-Pk) represent a vector drawn from Pk to Pi
as shown in Fig. Each line is expressed in its polar form in
magnitude and phase.Let,
Therefore,
This equation shows that a particular residue Ki can be obtained
from
All the residuces K1, K2. Kn are calcultated using this
method.The time-domain response can be obtained by taking the
inverse Laplace transform.
