Page 36 - Dynamics and Control of Nuclear Reactors

3.6 Transfer functions 27

This equation is the perturbation form of the kinetics equations. It is totally equiv-
alent to the standard form. In the perturbation form, all initial conditions are zero.
Note that the perturbation form of Eqs. (3.28) and (3.29) are also true for any
non-steady state nominal power P(0).
All of the terms in the perturbation equations are linear, constant coefficient terms
except (δρδP). If the use of the perturbation form is restricted to small perturbations,
then (δρ.δP) is small compared to {δρ.P(0)} and can be ignored. The final result for
the “small perturbation” form of the point kinetics equations is:
6
dδP δρ β X
¼ P 0ðÞ δP + λ i δC i (3.30)
dt Λ Λ
i¼1
β
dC i i
¼ δP λ i δC i i ¼ 1,2,…,6 (3.31)
dt Λ
Eqs. (3.30) and (3.31) may be rewritten by dividing them by P(0), thus using frac-
tional changes in the power. This representation will come in handy in the discussion
of fuel-to-coolant heat transfer dynamics.

3.6 Transfer functions
A transfer function is defined as the Laplace transform of a system’s output deviation
divided by the Laplace transform of the system’s input deviation. Appendix D
addresses Laplace transform theory. Laplace transform is a convenient tool for trans-
forming from a differential equation to an algebraic equation for expressing the rela-
tionship between two variables. Generally, one variable is considered as a system
output and the second variable as a system input. The analysis can be easily extended
to multiple-input multiple-output systems.
For a reactor, the transfer function that relates power to reactivity is δP(s)/δρ(s).
We derive this transfer function by Laplace transforming the small perturbation
equations. Initial conditions are zero because there is no deviation from steady state
initially. The result is
6
δρ β X
sδP ¼ P 0ðÞ δP + λ i δC i (3.32)
Λ Λ
i¼1
β i
sδC i ¼ δP λ i δC i (3.33)
Λ
where all variables are now functions of the Laplace transform parameter, s.
Solving for δP(s)/δρ(s) gives the desired transfer function:
δP 1
¼ (3.34)
δρ 0 β i 1
X 6 Λ
Λs 1+ C
B
@ A
i¼1 s + λ i Þ
ð