6
4
2
0
2
4
6
6
Solution to ECE Test #6 S07 #1
1.
A feedback system has a forward-path transfer function H
1 feedback-path transfer function H
2
( )
= s 2 s
+
10
+
2 s
+
4
.
Is there a finite, positive value of K that makes this system unstable?
( )
=
( ) s
+
6
and a
Explain how you know T
( )
has a zero at s
=
3 . Therefore as K
Yes
is increased from zero, one branch of the root locus approaches this point which is in the right half-plane.
Therefore, before K reaches infinity, a pole of H
( )
is in the right half-plane.
2.
The pole-zero plots below are for the loop transfer function of a feedback system.
Sketch, directly on the pole-zero plots the root locus for each.
ω ω ω
4 2 0 2 4 6
[s]
σ
6
4
2
0
2
4
6
6 4 2 0 2 4 6
[s]
σ
6
4
2
0
2
4
6
6 4 2 0 2 4
2
6
[s]
σ

Solution to ECE Test #6 S07 #2
1.
A feedback system has a forward-path transfer function H
1 feedback-path transfer function H
2
( )
= s
2 s
+
10
+
2 s
+
4
.
Is there a finite, positive value of K that makes this system unstable?
( )
=
( ) s
+
6
and a
No
Explain how you know T
( )
has zeros at
s
=
3, 10 . All three poles are in the left half-plane. Two root locus branches terminate on the two finite zeros without going into the right half-plane. The other root locus branch terminates on a zero at infinity and the asymptote is radians so it approaches negative infinity and never goes into the right halfplane.
6
4
2
0
2
4
6
6
2.
The pole-zero plots below are for the loop transfer function of a feedback system.
Sketch, directly on the pole-zero plots the root locus for each.
ω [s] ω [s] ω [s]
4 2 0 2 4 6
σ
6
4
2
0
2
4
6
6 4 2 0 2 4 6
σ
6
4
2
0
2
4
6
6 4
2
2 0 2 4 6
σ

Solution to ECE Test #6 S07 #3
1.
A feedback system has a forward-path transfer function H
1 feedback-path transfer function H
Is there a finite, positive value of K that makes this system unstable?
( )
2
( )
= s
2 s 10
+
2 s
+
has a zero at
4
.
( )
=
( ) s
+
6
Yes
and a
Explain how you know T s s
=
10 . Therefore as K is increased from zero, one branch of the root locus approaches this point which is in the right half-plane.
Therefore, before K reaches infinity, a pole of H
( )
is in the right half-plane.
4
2
0
2
4
6
6
2.
The pole-zero plots below are for the loop transfer function of a feedback system.
Sketch, directly on the pole-zero plots the root locus for each.
ω
6
[s]
6
ω [s]
6
ω [s]
4 2 0 2 4 6
σ
4
2
0
2
4
6
6 4 2 0 2 4 6
σ
4
2
0
2
4
6
6 4 2 0 2 4 6
σ