Root Locus

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Root Locus

• Introduction

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Closed Loop Transfer Function
Recall the closed loop system where
Let
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Characteristic Equation
The characteristic equation is
The roots of the characteristic equation form the
denominators of the terms of the solution and we
can determine stability by looking at the roots.
The roots of the characteristic equation change
as Kc is varied. Consider the following simple
CE
Root is a function of Kc
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• Consider the characteristic equation introduced
  earlier where tp1 12, tp2 6, tm 3 , tr 1
  and K2 2

Note that if Kc 0, roots are the denominator
roots of the Loop equation and if Kc is infinity,
roots are numerator roots
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Open Loop Transfer Function
For the CE, we can write
Where the OLTF is known as the Open Loop Transfer
Function
The Roots of P(s) are known as poles and roots of
Z(s) are known as zeros
An empirical method is available to allow the
locations of the roots of the CE to be plotted on
a Re vs Im diagram as K varies. This technique
uses the OLTF and is known as Root Locus Analysis.
Why do this? Roots falling in the right half
plane indicate instablity
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Table of Laplace Transforms
Factor
Solution
Root
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Location of roots