Root Locus
Q
UESTION
7
For the system shown in Figure 7 with
(
29
2
1
C
G
s
s
=
and
(
29
1
s
G s
s
α
+
=
+
, sketch the Root Locus
Diagram by hand and using Matlab.
Note:
the free parameter is
α
. How many zeros does the
system contain and what are their values? How many poles does the system contain and what are
their values? What are the total number of branches and how many branches will go to infinity?
Where is the Root Locus on the real axis? If necessary, calculate the asymptote angles and real
axis intercept, imaginary axis crossing, and the value of
α
at the imaginary axis crossing.
G(s)
G
C
(s)
R(s)
+

Y(s)
Figure 7
The closed–loop transfer function is
(
29
(
29
(
29
(
29
(
29
(
29
(
29
(
29
3
2
3
2
3
1
1
1
1
1
1
1
C
C
s
s
s
G
s G s
s
T
s
s
G
s G s
s
s
s
s
s
α
α
+
+
+
+
=
=
=
+
+
+
+
+
+
+
.
The effective open–loop transfer function with
α
as the free variable is
(
29
(
29
2
3
1
eff
s
G
s
s
s
=
+
+
There are
m
= 2 zeros located at 0 and 0.
There are
n
= 3 poles located at –0.6823 and
0.3412
1.162
j
±
.
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Root Locus
There are
n
= 3 branches and
n
–
m
= 1 branch goes to infinity. Since one branch goes to infinity,
the asymptotic angle is 180
0
and the real axis intercept is not required.
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 Spring '11
 LANDERS
 Root Locus, Complex number, 1 g, 3 j, Geff, 1.162 j

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