Page 90 - Dynamics and Control of Nuclear Reactors

7.9 Destabilizing negative feedback: A physical explanation 83

Note that the low frequency response (below around 1 rad/s) is determined by the
reactivity feedback (1/H). At higher frequencies, the feedback is small and the
response is identical to the response of a zero-power reactor (see Chapter 4 for
the zero-power frequency response). Ref. [2] describes the dynamic modeling and
experimental analysis of a commercial PWR. The experimental data, generated by
a pseudo-random binary sequence (PRBS) perturbation of the control rod reactivity,
were used to calculate the power-to-reactivity transfer function. The frequency domain
plots shown in Fig. 7.6 match very well with the reactor operational data analysis,
albeit using a point kinetics model with a simple feedback transfer function.

7.9 Destabilizing negative feedback: A physical explanation
Since destabilizing negative feedback can occur in power reactors, it is important to
understand the physical basis for this phenomenon. Destabilizing negative feedback
is an important issue for BWRs (see Chapter 13).
Feedback in a system can either augment or diminish the effect of input distur-
bances. One might think that negative feedback is always stabilizing, but this is not
true. Negative feedback can be stabilizing or destabilizing. In this section, we will
show the physical basis for destabilizing negative feedback.
The timing of negative feedback is the crucial issue. For example, if the process
that causes negative feedback is shifted in time, it can be experiencing a negative part
of an oscillation when the process being affected by the feedback is experiencing a
positive part of an oscillation. The result of negative feedback when the feedback
variable is negative is positive. That is, for instability to be caused by negative feed-
back, the feedback phase shift must be such that it changes the sign of the quantity
being fed back. This occurs when the feedback causes the phase shift to lie between
o
o
90 and 270 .
If the feedback is a single first order lag (H ¼ K/(s + a)), the phase shift varies
between 0° and 90°, the feedback cannot alter the sign of the feedback. So negative
feedback always is stabilizing in a system with a first order lag as feedback.
So, the feedback must be second order or higher for negative feedback to desta-
bilize a system. An example of a system with second order feedback will help in
understanding the phenomenon. Consider a simple third order system defined by
the following equations:
dx
¼ x Kz + f (7.7)
dt
dy
¼ 2x 2y (7.8)
dt

dz
¼ 3y 3z (7.9)
dt
The magnitude of the feedback from variable, z, in Eq. (7.7) is given by the coeffi-
cient, K. Eqs. (7.8) and (7.9) define the feedback. So, the feedback is second order
and the phase shift lies between 0° and 180°.