Solution: 10.8.5.2
The Bode magnitude and phase plots are shown in Figure 1, for K =100.
The Nyquist plot for the standard contour is shown in Figure 2 The Nyuist
-275
-250
-225
-200
-175
Phase in Degrees
-40
-20
0
20
40
60
80
100
Magnitude in Decibels
0.01 0.1 1 10 100
Frequency in Radians/Sec.
Phase in Degrees
Magnitude in Decibels
Figure 1: Bode Plots of GH =
K(s+2)
s
2
(s+1)(s+20)
For K =100
plot for the nonstandard contour of this problem is shown i Figure 3 Note
that the point ;1isalways encircled twice in the counterclockwise direc-
tion. Since travel around the contour in the s-plane in the counterclockwise
1
8
Im(GH)
Re(GH)
Figure 2: Nyquist plot for standard contour
8
Im(GH)
Re(GH)
Figure 3: Nyquist plot for nonstandard contour
2
direction, N =2,and the Nyquist equation is
Z = N +P
= 2+0
= 2:
There there are always two closed loop poles on the righthalfofthes plane
,and the system is unstable for 0 <K<1.
3