Page 43 - Servo Motors and Industrial Control Theory -

2.6 Nyquist Plot 35

Fig. 2.14 Nyquist plot for 0.5
open loop transfer function of 0.5
third order with one integra-
tor y = 0.25 )
( e s (0.25s + 0.1s + 1) 0.25
2

image y –0.5 –0.25 0 0.25 0.5

–0.25

–0.5 –0.5
–0.5 x 0.5
real

Usually the gain margin is expressed in units of decibels, that is,
1
Gainm arg in = 20log =− 20log d (2.20)
d
For stable system d < 1 and, therefore, the gain margin for stable system is posi-
tive and it is meaningless for unstable system. The phase margin is defined as the
amount of degrees that the Nyquist plot must be rotated until the system becomes
unstable. The phase angle between the negative real axis and M = 1 is defined as
phase margin. This is shown in Fig. 2.14 as ψ. Figure 2.14 shows that the intersec-
tion of the locus with negative real axis is at d = 0.25. Converting this to decibels
gives

Gainmargin = 20log(0.25) = 12db

Phase margin = 90 °

With above gain margin and phase margin, the closed loop system will have an ac-
ceptable response to step input.
Usually for a stable system, to have a minimum number of oscillation the Gain
Margin > 6 db and the Phase Margin > 45° must be obtained in the system at the
design stage. For the system of Fig. 2.15, the gain can be increased by 6 db and the
response would still be acceptable.
The intersection between the Nyquist plot and the negative real axis gives the
Gain Margin. So this point must be calculated carefully. As Fig. 2.15 shows there
is a large variation in the Nyquist plot and frequency near the intersection point.
The crosses on the diagram show the exact point for a given frequency. It can be