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=∞).