Effect of Zeros on The Response of Circuit 7799

Department of Electrical Engineering, FAST-NU, Islamabad, Pakistan.28 April 2014

Abstract —In this document I am discussing the topic effect of zeroes on the response of circuits. Here I will describe how zeroes affect the response. I will explain how output change in presence of zeros or without zeroes means response with or without zeros in transform function. The whole discussion about the zeroes which take part in the amplitude of the steady and natural response. And how effect create in the time response which I will explain through the graphs and how the system response speeds up by adding a zero.

Keywords –Affect on magnitude, Transfer function, pole zero diagram, Time response, affect on pole.

I. INTRODUCTION

This document is describing about affect of zeros on the response of circuit. In this document I will explain what are zeros? How we can find zeros? Zeros of transfer functions. Graphical explanation of zeros on response. There affect on amplitude of response. How they affect the gaining factor. Iwill discuss the time response and frequency response explain the zeros behavior.This document is organized as follows: Section II describes the background. Section III explains the topic with examples. Section IV lists how would marks be allocated and/or deducted for the project. Section V describes the concept of peer-review.

II. Background

My whole topic is revolving about the zeros. In my topic I will use Laplace transform, partial fractions, pole zero diagram. Laplace transform is widely use in mathematics physics and also in circuit analysis. Here I will discuss its use in circuit analysis. We use Laplace transform in solving different circuit which is very tough to solve in time domain we convert a circuit in Laplace domain then after solving we again return in time domain by inverse Laplace. We use this definition to get s domain function :

F ( s )=L [ f ( t ) ]=∫0

∞

e−st f ( t )dt

In algebra, partial fraction of a rational fraction is the operation that expresses the fraction as a sum of a polynomial and one or several fractions with a simpler denominator. By partial fraction we can easily obtained several fractions which we can solve easily. We locate zeros graphically on pole zero diagram. By viewing pole zero diagram we can explain the behavior of the response. In pole zero diagram there are two axis one real and other one is imaginary axis where we can place pole and zeros of a function a simple pole zero diagram of a function is given from which we can take the idea of pole zero diagram.

Here -2 is zero and -2j ,+2j and -1 are poles.

I. Affect of zeros on response of circuit :

For a system, it is important to understand how the system is behaving which we are studying. The Laplace transform is a valuable technique that can be used to solve differential equations and obtain the dynamic response of a system. The Laplace transform make possible to understand the behavior of the system response without solving for the response. So by poles and zeros of a system tell us the behavior of the response without solving the whole equations to get response. Here I will only discuss about the behavior of zeros on the response of a circuit. Using a graphical method of pole zero diagram we can understand the affect more briefly. Now Conversion of time domain circuits to s-domain circuits and drawing the response of circuits on graph as a pole zero diagram which explains more briefly than the time domain equations and I will explain by this way how zeros affect the response of a circuit.First I want to explain what are zeros.A rational function may be expressed as the ratio of two factored polynomials. In other words we may write a Laplace function F(s) as

F ( s )=k ( s+z 1 ) ( s+z 2 )……… .. ( s+zn )(s+ p1 ) (s+ p2 )……… .. ( s+ pn )

Where k is a constant.For example:

F ( s )= 8 s2+120 s+4002 s4+20 s3+70 s2+100 s+48

F ( s )= 4(s¿¿2+15 s+50)s4+10 s3+35 s2+50 s+24

¿

F ( s )= 2 (s+5 ) ( s+10 )( s+1 ) ( s+2 ) ( s+3 ) (s+4 )

So the roots of numerator are –z1 –z2 ………-zn They are the values s at which F(s) becomes zero are called zeroes of F(s). in above example -5 and -10 are the zeros

Affect of zero on the response of a circuit can be explained best by its poles and zeros diagram. Zeros of a function may be complex as poles, so graphically we plot their locations on the complex s-plane whose axes represent the real and imaginary parts of the complex variable s. Such plots are known as pole zero plote. Zero location mark as circle (o) and a pole location a cross (X) as in given below figure. The location of the poles and zeros provide qualitative analyses of a system response.

First I want to explain what are zeroes of a transfer function ?. Let a transfer function H(s) which is as

H (s )= y (s)x (s)

here the roots y(s) are called zeros of H(s) or the values of y(s) which gives the H(s)=0. For example a

transfer function :

H (s )= s+5(s+3 )(s+6)

Here if we put s = -5 then the transfer function H(s) will become zero or we say that -5 is the root of nominator which are called zeros of a transfer function and the roots of denominator are called poles here I am explaining poles because it effects the stability of circuit which are also a ffected by the location of zero. I will first explain zeros in first order system. Here I will discuss about simple example to know about the effect of zero in a simple circuit response.[1]So we can say zeros take part in the generation of amplitude in steady and natural responses. Zero also affects the speed of theSo we can say zeros take part in the generation of amplitude in steady and natural responses. Zero also affects the speed of the response faster or slower which affect the location of the zero with respect to a pole. Zero does not affect the nature of the response.

II. Examples :

A transfer function H(s) of the circuit

H (s )= (s+2 )(s+5 )

Here -2 is a zero of a transfer function and -5 is pole. Now we let us see the unit step of a transfer function. Let Y(s) is the response or output.

Y (s )= ( s+2 )s (s+5 )

By partial fraction we solve it

Y (s )= As

+ Bs+5

A= s+2s+5

∨at s=0

A=2/5 and

B= s+2sat s=−5

B=3/5So the

Y (s )=

25s+

35s+5

Here in last one expression when we transferit in time domain as

y (t )=25+ 3

5e−5 t

Here 2/5 and 3/5 resulted of zeros and poles

So we can say zeros take part in the generation of amplitude in steady and natural responses. Zero also

affects the speed of the response faster or slower which affect the location of the zero with respect to a pole. Zero does not affect the nature of the response. Let us see by increasing number of zeros and taking

fix value of poles let us start poles of a transfer function (-1+2.82j) (-1-2.82j) and we will change the

values of zeros as -3 -5 and -10 we can see from figure that if a zero is closer to the pole a general

expression to see the behavior of zeros as

H (s )= s+a(s+b ) ( s+c )

where –a is zero of transfer function now by partial fraction

H (s )= As+b

+ Bs+c

A=−b+a−b+c

B=−c+a−c+b

H (s )=

−b+a−b+cs+b

+

−c+a−c+bs+c

If a will far away from both b and c then the equation could be written as

H (s )= a(s+b )(s+c)

So here zero is simple a gaining factor if a is near the poles and then it does not change the amplitude of the component relative to response.

this figure shows the affect of singal zeros one by one which are added to the response. Here in this example We can see that the closer the zero is to the poles, the greater its effect on the transient response. As the zero moves away from the dominant poles, the response approaches that of the two-pole system. This analysis can be understand via the partial-fraction expansion. The case in which zeros in LHP We see that the major impact of the zero is to increase the overshoot, with little impact on the settling time. But if the value of zero is positive then there will be a zero on the RHP then the response will be When the value of is negative, then there is a zero on the RHP, also called a nonminimum-phase zero. The transient response of the resulting system is quite different. In fact, the overshoot is suppressed to the point that the response first starts in the wrong direction and then changes sign.[3]Example 3 :

Circuit diagram;Let us consider a circuit in s domain

−is+ Va2+7 s

+ Va10s

=0

H (s )= 10 (2+7 s )10+s (2+7 s )

H (s )= 20+70 s

7 s2+2 s+10

Zero=−2070

=−0.28

Poles= -0.142+1.86i.

Here filled dots represents poles and empty dot represent zero, if we are calculating the phase angle w.r.t given point for example (s=jw) then we can calculate the phase angle just by subtracting the phase angle of pole to the angle of zero. And magnitude of H(s) is

III. REFERNCES

[1] http://aerostudents.com/files/aerospaceSystemsAndControlTheory/Book/ControlSystemsEngineeringCH4.pdf

[2] http://parlos.tamu.edu/MEEN364/Lecture12.pdf

[3] Control Systems Engineering, International Student Version, 6th Edition Norman S. Nise April 2011, ©2011