Page 42 - Servo Motors and Industrial Control Theory -

34 2 Feedback Control Theory Continued

Fig. 2.13 Nyquist plot for 6
a second order system with
ξ = 0.1. The frequency is
ω = 0 to
ω
changed from n
ω = ∞ 3
ω
n
imag
–6 –3 0 3 6

–3

–6
real

ω
2ζ ω
ϕ =− tan − 1 n (2.18)
1− ω 2
ω 2
n
The Nyquist plot for ζ = 0.1 is shown in Fig. 2.13. The locus starts at point 1 and it
ends at zero at high frequency. The intersection with the imaginary axis gives the
frequency of oscillation to a step input. The maximum phase angle is –180°. It can
be seen that as the locus becomes closer to the – 1.
Point the system becomes more oscillatory. The relative stability is defined by
Gain and Phase Margin. It should be noted that the variation of amplitude and phase
angle do not change linearly with the frequency. As the scattered points around the
– 90° show small variation of frequency produces large variation in Nyquist plot.
For third and higher order systems, the Nyquist plot will go in third or fourth
quadrant. It can be seen that for each power of highest order s in the denominator
of open loop transfer function there will be a 90° phase lag at high frequency. In
the above two examples the gain was taken as unity and, therefore, the Nyquist
plot at zero frequency starts at 1 on the real axis. If there is some integrator in the
denominator, for each integrator there will be a 90° phase lag in the Nyquist plot.
Therefore, system with two or more integrators in the denominator might become
unstable. As will be discussed later for this kind of system compensation network
must be designed.
The gain margin is defined as the amount of gain that can be increased without
the system becoming unstable. In Fig. 2.14, the distance of the intersection with
negative real axis to the origin is defined as d. Therefore,
1
GainM arg in = (2.19)
d