Page 291 - Dynamics and Control of Nuclear Reactors

APPENDIX E Frequency response analysis of linear systems 293

A linear fit to the phase plot is shown in Fig. E.9. The slope of this straight line
corresponds to the transport time between the two detector locations. The knowledge
of the distance between the two detectors (3ft ( 0.91m)) can then be used to cal-
culate an average two-phase flow velocity in this BWR at the location of the detector
string. The estimated transport time (delay time) is 0.17s and the average flow veloc-
ity is 18ft/s (5.48m/s).

E.5 Frequency response of distributed systems

As shown in Appendix D, Section D.8, Eq. (D.34) gives the transfer function for a
fluid channel for temperature at position, x, per change in the channel inlet temper-
ature. The transfer function is
ð
Ux, sÞ ¼ U 0 e s + bÞx (E.31)
ð
With b¼1, Eq. (E.31) is rewritten as
ð
Ux, sÞ s +1Þx
¼ e ð (E.32)
U 0
The frequency response for this transfer function is determined by substituting s¼jω.
Ux, ωÞ x jωx
ð
¼ e e (E.33)
U 0
The magnitude and phase angle of the frequency response function at (x, ω) are
given by
Magnitude of Ux, ωÞ ¼ e x (E.34)
ð
Phase angle of Ux, ωÞ ¼ ω xÞ (E.35)
ð
ð
Note that the magnitude is independent of frequency and decreases as the position,
x, increases. The phase angle decreases as the frequency and position increase. This
illustrates that distributed parameter frequency responses are not minimum phase
systems as are lumped parameter frequency responses.

E.6 Frequency response measurements
Frequency response measurements are sometimes used to aid in the understanding of
a dynamic process or to check the validity of a theoretical dynamic model. The most
obvious approach is to introduce an input sinusoid. But many systems (including
nuclear reactors) lack hardware that can produce a sinusoidal input. They usually
are better able to introduce step changes. Therefore, methods have been devised
and implemented that can employ binary inputs (an input that takes two values
and steps up and down in a prescribed sequence).
The simplest binary signal is the square wave. Fourier analysis shows that the
square wave can be represented as a sum of sinusoids. The square wave has most