Root Locus Plots
By: Oxygenδ
Ji Sun Sunny Choi
Sang Lee
Transfer Function: F(s)
F(s) = U(s)/Y(s)
• The roots of F(s) are called Poles when
F(s) = ∞
• The roots of F(s) are called zeros when
F(s) = 0
Root Locus Plots
• Graphical representations of the stability
of a control
– Stable conditions if the real part of the
poles are negative and unstable if they are
positive
• Poles are plotted on a complex
coordinate system with the imaginary
portion of the root plotted as a function
of the real portion of the root
Plotting a Root Locus Plot
• Given a function in P-only Control:
Kc
F ( s) 
13s 3  11s 2  11s  1  6 K c
Graph the Root Locus Plot
Solution
• First evaluate the poles of the transfer
function F(s) by setting F(s)=∞ and
varying Kc
• To do this, the denominator in F(s) is
set equal to 0 and the roots are solved
using the Solve[] function in Wolfram
Mathematica6
• The solutions were organized into a
table and plotted in Microsoft Excel
Solution – Table of Poles
Kc
Real
Imaginary
0 -0.09967
0
0 -0.37324 -0.795262
0 -0.37324
0.795262
0.004 -0.10229
0
0.004 -0.37193 -0.794815
0.004 -0.37193
0.794815
0.05 -0.13311
0
0.05 -0.35652
-0.79002
0.05 -0.35652
0.79002
0.2 -0.24174
0
0.2 -0.30221 -0.780201
0.2 -0.30221
0.780201
0.5 -0.46024
0
0.5 -0.19296
-0.79455
0.5 -0.19296
0.79455
Kc
Real
Imaginary
1 -0.71544
0
1 -0.06536 -0.865079
1 -0.06536
0.865079
1.33 -0.82973
0
1.33 -0.00821 -0.912391
1.33 -0.00821
0.932391
1.78 -0.95046
0
1.78 0.052153 -0.970861
1.78 0.052153
0.970861
1.9 -0.9781
0
1.9 0.065971 -0.985319
1.9 0.065971
0.985319
2
-1
0
2 0.076923 -0.997037
2 0.076923
0.997037
Solution – Root Locus Plot
Root Locus Plot of F(s)
1.5
Imaginary Roots
1
0.5
0
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
-0.5
-1
Real Roots
-1.5
0.2