LABORATORY MODULE
Control Principle
(DNT 354/3)
Semester 1 (2008/2009)
Experiment 6:
Root Locus
Name :_________________________________________
Matric No. :________________
Name :_________________________________________
Matric No. :________________
School of Electrical System Engineering
Universiti Malaysia Perlis (UniMAP)
EXPERIMENT 6
ROOT LOCUS
1.
OBJECTIVE:
1.1.
To illustrate root locus responses using transfer function.
1.2.
To study the effect of loop gain upon the system’s poles.
2.
2.1.
INTRODUCTION
Root locus of an open loop transfer function G(s) is the locus (plot) of all possible
locations of the closed loop poles with unity feedback and proportional gain K that is
varied from zero to infinity.
Figure 1:
Closed loop system with gain K and unity feedback
The closed loop transfer function is
T( s ) 
KGs 
1  KGs 
and thus the poles of the closed loop system are values of s such that
1 +kG(s)= 0
If we write
G( s ) 
bs 
as 
a(s)+ kb(s)= 0
Then this equation has the form of:
a(s)+ kb(s)= 0
Or
G( s ) 
as 
 bs   0
k
2
Let n = order of a(s) and m = order of b(s).
We will consider all positive values of k. In the limit as k goes to 0, the poles of the
closed loop system are a(s) = 0 or the poles of the G(s). In the limit as k goes to infinity,
the poles ofthe closed loop system are b(s) = 0 or the zero ofG(s). No matter what we
pick k to be, the closed loop system must always have n poles,
where n is the number of poles of G(s). The root locus have n branches, each branch
starts at a pole of G(s) and goes to a zero of G(s). If G(s) has more poles that zeros, then
there are n-m zeros at infinity
2.2. Matlab commands
Matlab commands can be used to generate root locus plot:
»rlocus(num,den) - calculates and plot the locus of the roots of 1 +kG(s)= 0
» [K,poles] = rlocfind(num,den) - find the gain K and its corresponding poles
for a given roots locus.
»sgrid(zeta, Wn) - draw lines (boundaries) of zeta and Wn.
3.
PROCEDURE
3.1. Given the following transfer functions, plot the root locus. For each G(s) indicate the
how many of poles, zeros and infinite zeros (if exist).
3.1.1.
Gs  
3.1.2.
Gs  
3.1.3.
G s  
1
3s  5s  1
2
5
s  3s  5s  1
3
2
s 2  6s  1
s 3  3s 2  5s  1
3.2. Using command ‘[K,poles] = rlocfind(num,den)’ find gain and its poles at these
respective conditions.
3.2.1.
At K → 0 when Gs  
1
3s 2  5s  1
3.2.2.
When
move
Gs  
both
poles
towards
each
other
and
meet
for
1
.
3s  5s  1
2
3.2.3.
When poles break away into the complex plane for Gs  
3.2.4.
When
poles
for Gs  
6
.
s  2s  8s  3
3
break
away
and
enter
3
.
3s  4s  1
the
2
left
plane
2
3
4.
Name
:
______________________________
Matrix No
:
______________________________
Name
:
______________________________
Matrix No
:
______________________________
RESULT
Teaching Engineer :
______________________________
4
Name
:
______________________________
Matrix No
:
______________________________
Name
:
______________________________
Matrix No
:
______________________________
Teaching Engineer :
______________________________
5
5.
DISCUSSION
6.
CONCLUSION
6