Summary : Procedure to plot root locus

Procedure   to plot Root locus

Step: 1  Determine the branch  number of loci.

Step: 2  Plot the poles and zeros on the S plane

Step: 3 Find real axis Loci .  

Step: 4   Find the number of Asymptotes and their angles, if any.

Step: 5  Determine the centre of Asymptotes  and draw.

Step: 6    Determine the break in/ away point , if any.

Step: 7   If the locus crosses the jw axis , determine the jw  cross over.

Step: 8  Calculate the angle of departure / angle of arrival  due to complex poles or zeros if any.

Sketch the root locus for the  system . (Brief discussion)

G(s)H(s) = K(s+4)(s+5)/[(s+3)(s+1)]

Steps

step 1:   m=2 (number of zeroes) , n=2 ( number of poles)

step 2:    n-m=0    ; No asymptotes //  no step required

Step 3:   Real axis loci      [ -3  -1]   and  [ -5   -4]

Step 4:                  1+G(s)H(s)=0

                          1+ K(s+4)(s+5)/[(s+3)(s+1)]=0

                           K=  -(s+1)(s+3)/(s+4)(s+5)

                                =  - s^2+4s+3/s^2+9s+20

                  dk/ds=  -   5s^2+34s+53/[(s^2+9s+20)]^2

              dk/ds=0    

               s=-2.42  and -4.38

    Plot

Note :   Root loci originate on the poles of G(s)H(s) (for K=0)

               and terminates on the zeros of G(s)H(s) (as K=∞).

Practice Question : Root Locus

Explain the term centroid and  asymptotes?

What is break away point?

Calculate the angle of asymptotes  and centroid for the system   having                                 G(s)= K(s+2)/[s(s+1)(s+4)

Find the  centroid and breakaway point for the system having   G(s) = K/[s(s+2)(s^2+6s+25)]

For G(s)H(s) = K/[s(s+1)(s+3)], determine the coordinates of valid break-away/break in points(s)

Sketch the root locus for  (i) G(s)H(s)= K/s(s+3)(s+5) and (ii) G(s)H(s) = K/[s(s+6)(s^2+4s+13)]

Explain the effect of addition of poles and zeros to a  second order control system. (root Locus)

For G(s) H(s)= K/s(s+4) Draw loot locus and determine the  value of K if the damping ratio is 0.707 .

How will you obtain the angle of departure and angle of arrival of root locii?

State the rule for obtaining the breakaway point  in the root locii.

Root Locus : Revision some points asked by students_5_fast exercise

Root Locus : Revision some points asked by students_4: intersection of root locus on Jw axis

Root Locus : Revision some points asked by students_3: break away/in point

Steps to determine the break away point:

1. Find the Characteristics Equation 1+G(s)H(s)=0 of the system.
2. Write  K in terms of  s.
3. Derive dK/ds  and Put dK/ds =0.
4. The roots of the equation dK/ds =0 are the break away points.
5. Note: If the value of K is positive for any roots of dK/ds=0, the roots   (are) valid break away /break in points.

In this example the value of K is positive  for S=-0.785. Hence this the break away point.In the video there is error in the last minutes. so correct by this.

Remaining points are ok.

Take Care

Jitender

Root Locus : Revision some points asked by students_2: Asimptotes and Centroid

Root Locus : Revision some points asked by students_1: real axis loci

There is no real root loci in between -2 and -1 since the sum of poles and zeroes is  even.