Page 286 - Dynamics and Control of Nuclear Reactors

288 APPENDIX E Frequency response analysis of linear systems

Certain other useful information can be obtained from a Bode plot. For instance,
we observe inFig. E.2that theamplitude of thefrequencyresponseof 1 is essentially
s + a
constant out to about a radians per second. This simply means that the
inertia of the system is sufficiently small for the output to keep up with the input
for these frequencies. At higher frequencies, the system output is unable to
keep up with the system input and the amplitude decreases. This is also observed in
the phase plot. The frequency range over which the amplitude is at least
1
p ffiffi ¼ 0:707as large as the steady state ω ¼ 0ð Þ amplitude is called the bandwidth
2
of the system.

E.3 Systems with oscillatory behavior
A system frequency response may also be used to provide information about
the oscillatory characteristics of the system. If the amplitude peaks at some
frequency, this means that the system will greatly amplify any component of the
input at that frequency. This phenomenon is called resonance. In general, a Bode
plot with a tall, narrow peak in the magnitude indicates that the system tends to be
highly oscillatory. Damped oscillatory behavior is also noticed in neutron power
response in a BWR, primarily caused by the void reactivity feedback.

Example E.3
As an example, consider the system with the following transfer function
ω 2 n
GsðÞ ¼ (E.25)
s +2ζω n s + ω 2 n
2
Rewrite G(s) in the form

1
GsðÞ ¼ (E.26)
s 2 2ζs
ω 2 + ω n +1
n
The frequency response function is given by
1
ð (E.27)
ω 2ζ
GjωÞ ¼ 2
1 + j ω
ω 2 n ω n
Fig. E.6 shows the frequency response for ω n ¼1.0 and ζ¼0.1.