Root Locus

1. BAE 3023 Root Locus Introduction

2. Closed Loop Transfer Function BAE 3023 Recall the closed loop system where: Let:

3. Characteristic Equation BAE 3023 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

4. BAE 3023 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

5. Open Loop Transfer Function BAE 3023 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

6. Table of Laplace Transforms BAE 3023 Factor Solution Root

7. Location of roots BAE 3023