Section 7–8 / Closed-Loop Frequency Response of Unity-Feedback Systems 477
Figure 7–79. The resonant peak is obtained as 5.2388 dB. The resonant frequency is 0.7906 radsec.
The bandwidth is 1.2649 radsec. These values can be verified from Figure 7–78.
7–8 CLOSED-LOOP FREQUENCY RESPONSE OF UNITY-
FEEDBACK SYSTEMS
Closed-Loop Frequency Response. For a stable, unity-feedback closed-loop sys-
tem, the closed-loop frequency response can be obtained easily from that of the open-
loop frequency response. Consider the unity-feedback system shown in Figure 7–80(a).
The closed-loop transfer function is
In the Nyquist or polar plot shown in Figure 7–80(b), the vector represents GAjv 1 B,
wherev 1 is the frequency at point A. The length of the vector is @GAjv 1 B@and the
angle of the vector is The vector the vector from the –1+j0point
to the Nyquist locus, represents 1+GAjv 1 B. Therefore, the ratio of to repre-
sents the closed-loop frequency response, or
OA
!
PA
!=
GAjv 1 B
1 +GAjv 1 B
=
CAjv 1 B
RAjv 1 B
PA
!
OA
!
,
PA
!
OA /GAjv 1 B. ,
!
OA
OA!
!
C(s)
R(s)
=
G(s)
1 +G(s)
1
+– s(0.5s+ 1) (s+ 1)
R(s) C(s)
Figure 7–78
Closed-loop system.
Frequency (rad/sec)
Bode Diagram
− 300
− 50
− 100
− 150
− 200
− 250
0
− 60
− 40
− 20
Phase (deg); Magnitude (dB)
20
0
10 −^1100101
Figure 7–79
Bode diagram of the
closed-loop transfer
function of the
system shown in
Figure 7–78.
