Lecture 11

Question 1

What is the root locus plot

Answer 1

Taking a parameter and plotting how it affects the roots of the characteristic equation as it is changed from 0 to Infinity

Question 2

What initial piece of math do we do before being able to plot the roots of the characteristic equation

Answer 2

Consider the characteristic equation as D(s) + K(N(s)) = 0
Where N(s) and D(s) are polynomials, and K is the parameter you want to vary

Question 3

Where does the plot of the root locus start

Answer 3

K is zero therefore D(s) + KN(s) = 0 -> D(s) = 0
So start at the roots of D(s)

Question 4

What will the root locus always beq

Answer 4

Symmetrical about the real axis as roots will either be a complex conjugate pair or real numbers

Question 5

Where does the root locus end

Answer 5

K is infinite, divide through by KD(s)
1/K + N(s)/D(s) = 0
N(s) = 0
So ends at the roots of N(s)

Question 6

For the characteristic equation s^2 + K*s + 1 = 0
What are D(s), N(s) and the start and end points

Answer 6

D(s) = s^2 + 1 N(s) = s
start = +-j
End s = 0 (at the origin)
But more starts than ends, therefore locus never ends - continue to infinite

Question 7

Where the locus never ends what happens

Answer 7

It tends towards an asymptote

Question 8

What is the widely used from of the characteristic equation

Answer 8

1 + K* N(s)/D(s) = 0

Question 9

What is the magnitude condition

Answer 9

MAG( K*N(s)/D(s) ) = 1

Question 10

What is the angle condition

Answer 10

angle ( K*N(s)/D(s) ) = -180

Question 11

What does factorising N(s) give

Answer 11

The end points of the root locus
-z1 , -z2, -z3

Question 12

What does factorising D(s) give

Answer 12

The end points of the root locus
-p1, -p2, -p3,

Question 13

What is (s+p1)

Answer 13

Vector subtraction, equal to the distance between the root and the start point p1

Question 14

What is (s+z1)

Answer 14

Vector subtraction, equal to the distance between the root and the end point z1