chapter 7 me 332 lecture notes

8/3/2019 Chapter 7 ME 332 Lecture Notes

1/39

Chapter 7

ROOT LOCUS METHOD


2/39

ROOT LOCUS

Root Loci: Gives graphical picture of the effect of selectedparameters on system poles and suggest what values

should be chosen to meet system design specificationon time constant and damping ratio to improve speedof the response

The root locus method is two stages process

1) Construction of the root loci which is the loci of closedloop poles position in s-plane as K or any other designparameter changesRoot loci are constructed from angle condition alone asthe loci of all points s for which the sum of vectorangles I from all open loop zeros to s minus the sumof the vector angles j from all open loop poles to sequal to odd number multiple of 180


3/39

ROOT LOCUS

2) The magnitude condition shows that the

value of k for which closed loop poles willbe located at a given points s along alocus equal the product of vector length B

jfrom all open loop poles to s divided by theproduct of vector length Ai from all open

loop zeros to s


4/39


5/39


6/39


7/39


8/39


9/39


10/39


11/39


12/39


13/39


14/39


15/39


16/39


17/39


18/39


19/39


20/39

n=[1];

den=[1 6 11 6];

%ors1=[1 1];s2=[1 2];s3=[1 3];d=conv(s1,conv(s2,s3))

sys=tf(n,d);

sym2=tf(n,den)

rlocus(sys)

figure

rlocus(sym2)


21/39

ROOT LOCUS

EXAMPLES


22/39

Example 1


23/39

Example Solution


24/39

Example Solution Contd


25/39



26/39



27/39



28/39



29/39



30/39


A line of constant damping is found by finding the angle anddrawing a line that will cross the root locus at at least one point as

shown in the previous root locus plot

Trial point is picked and the angle criteria is applied

Hence the point lie on the root locus


31/39


Value of open Loop gain constant K: Applying the magnitude criterion to

the above point gives

Closed loop poles for K=11.35: Since the closed loop system is a third order

system, there are three closed loop poles. Two of them are found at the point

(complex conjugate poles). The third lies on the real locus that extends from -5 to - . The value is calculated using the magnitude criterion as shownBelow

Solving for x gives x=0.73 which implies that s1= -5.73. hence the closed

loop poles at K=11.35 are


32/39



33/39

Example 2


34/39

Example 2 Solution


35/39

Example 2 Solution Contd


36/39



37/39



38/39



39/39


Plot of constant damping ratio on the root locus plot and test

trial points a long it using angle criterion as shown in theprevious figure at s=-0.8+j2.9 gives

Hence the point lie on the root locus

chapter 7 me 332 lecture notes

Documents