On the Choice of the Frequency Response of a Neutron Flux Fluctuation Measurement Channel for Obtaining Information on the Physical Parameters of the IRT-2000 Reactor
Abstract Generated abstract
The paper addresses how to choose the frequency response of a neutron flux fluctuation measurement channel for extracting physical information about the IRT-2000 reactor when detector efficiency is limited. It models reactor neutron noise as white noise transformed by the linear equations of reactor kinetics, adds frequency independent ionization chamber noise, and formulates the measurement problem as a Wiener optimal filtering problem. Using the Bode and Shannon method, the authors derive the shaping filter, analyze the poles and zeros of the combined spectral density, and obtain a physically realizable optimal filter transfer function for the reactor parameter range considered.
Full Text
Proceedings of the Tomsk Order of the Red Banner of Labor Polytechnic Institute named after S. M. Kirov
Vol. 162 1967
On the Selection of the Frequency Response of the Neutron Flux Fluctuation Measurement Channel for Obtaining Information on the Physical Parameters of the IRT-2000 Reactor
E. M. Belov, A. K. Martynichev
(Presented by the scientific seminar of the Faculty of Physics and Technology)
Recently, much attention has been devoted to statistical methods for measuring the dynamic characteristics of objects, in particular, to obtaining information about a reactor by the method of analyzing neutron-flux fluctuations. Since a nuclear reactor is a statistical system, fluctuations are inherent in the neutron density. These fluctuations, or reactor “noises,” are not of interest in themselves; however, since the statistical properties of reactor noise depend strongly on important reactor parameters, the study of noise in a number of cases makes it possible to determine these parameters in an easier way.
At present there exist a number of mathematical models for describing the process of the occurrence and transformation of neutron noise.
In the present work, the theory [1, 2, 3] is used to describe the noises, according to which reactor noises are considered as the result of the passage of a random process of the “white-noise” type through a linear system described by the well-known differential equations of reactor kinetics:
\[ \left. \begin{aligned} \frac{dn}{dt} &= \frac{\delta k-\beta}{l^*}\,n+\lambda c+s,\\ \frac{dc}{dt} &= \frac{\beta}{l^*}\,n-\lambda c \end{aligned} \right\} \tag{1} \]
As the primary source of reactor neutron noise, an equivalent source is taken, whose power spectral density \(G_s(\omega)\) does not depend on frequency and is constant [1]:
\[ G_s(\omega)=K_s^2. \]
The power spectral density of neutron fluctuations repeats, to within a constant factor, the form of the square of the modulus of the frequency response \(T(j\omega)\), which characterizes the relation of the neutron flux \(n\) to the source intensity \(s\) according to equation (1).
From equation (1) we find the frequency response:
\[ T(j\omega)=\frac{b_0}{a_0p^2+a_1p+1}, \]
where \(p=j\omega\), and
\[ b_0=-\frac{l^*}{\delta k}, \]
\[ a_0=-\frac{l^*}{\delta k\cdot\lambda}, \]
\[ a_1=-\frac{\beta+\lambda l^*-\delta k}{\delta k\lambda}. \tag{3} \]
In the case of a subcritical regime, the reactivity \(\delta k\) is a negative quantity, and all the coefficients of equation (2) are positive. The spectral density of neutron fluctuations is determined by the expression:
\[ G_n(\omega)=|T(j\omega)|^2\cdot \kappa_s^2 \]
or
\[ G_n(\omega)=\frac{b_0^2\kappa_s^2}{a_0^2\omega^4+(a_1^2-2a_0)\omega^2+1}. \tag{4} \]
In order to measure the statistical properties of noise, it is necessary to carry out the process of measuring neutron-flux fluctuations by means of a detector, such as, for example, an ionization chamber, which absorbs part of the neutrons and produces an electrical signal. However, the process of measuring the flux by an ionization chamber is also a random process; moreover, the lower the efficiency of the chamber, the more the readings of the ionization chamber fail to correspond to the actual values of the flux fluctuation.
If the chamber is placed in a region of constant flux, then, owing to the statistical nature of registrations, a signal will appear at the chamber output consisting of a constant component and fluctuations whose power spectral density will not depend on frequency, i.e., the fluctuation will have the character of “white” noise.
If the spectral density of the noise produced by the chamber is denoted by
\[ G_a(\omega)=m^2, \tag{5} \]
then the model of the measurement of neutron noise may be represented by the scheme in Fig. 1, where \(K(j\omega)\) is the frequency response of the measuring channel, including the time constant of the chamber.
When the noise-analysis method is used, it is necessary to measure neutron fluctuations having power spectral density \(G_n(\omega)\), under conditions of limited chamber efficiency.
Thus, the random process \(n(t)\), having spectral density \(G_n(\omega)\), may be regarded as the useful signal, while the incompleteness of the reproduction of this process by the chamber, owing to its limited efficiency, may be characterized in the form of an interference \(a(t)\) having power spectral density \(G_a(\omega)\). Then the problem of finding the frequency response \(K(j\omega)\), optimal in the sense of obtaining a minimum mean-square deviation of the output signal from the useful signal, is a Wiener problem of optimal filtering.
To find the frequency response we use the Bode and Shannon method [4], according to which it is necessary to find the frequency response of the so-called shaping filter \(Y_1(j\omega)\), satisfying the equation:
\[ |Y_1(j\omega)|^2=G_\Sigma(\omega)=G_n(\omega)+G_a(\omega), \tag{6} \]
where \(G_{\Sigma}(\omega)\) is the power spectral density of the total signal \(n(t)+a(t)\); since the detection process is statistically independent of the process of neutron-noise formation, the corresponding cross-spectral densities are equal to zero and do not contribute to the sum.
It is known [5] that the modulus of the frequency response uniquely determines the entire frequency response only in the case of so-called minimum-phase systems, when the frequency response has no zeros or poles in the lower half-plane of the complex variable. It is also necessary, for the existence of a physically realizable shaping filter, that the rational function \(G_{\Sigma}(\omega)\) have no real poles.
Fig. 1.
According to relations (4), (5), and (6), for \(G_n(\omega)\), \(G_a(\omega)\), and \(G_{\Sigma}(\omega)\) we obtain
\[ G_{\Sigma}(\omega)= \frac{a_0^2 m^2 \omega^4 + m^2(a_1^2-2a_0)\omega^2 + m^2+\kappa^2} {a_0^2\omega^4+(a_1^2-2a_0)\omega^2+1}, \tag{7} \]
where \(\kappa^2=b_0^2\kappa_s^2\).
The zeros of the function \(G_{\Sigma}(\omega)\) are
\[ \beta_{1,2}=\pm \sqrt{ \frac{ -m^2(a_1^2-2a_0)+ \sqrt{m^4(a_1^2-2a_0)^2-4a_0^2m^2(m^2+\kappa^2)} } {2a_0^2m^2} }, \]
\[ \beta_{3,4}=\pm \sqrt{ \frac{ -m^2(a_1^2-2a_0)- \sqrt{m^4(a_1^2-2a_0)^2-4a_0^2m^2(m^2+\kappa^2)} } {2a_0^2m^2} }; \tag{8} \]
the poles of the function \(G_{\Sigma}(\omega)\) are
\[ \gamma_{1,2}=\pm \sqrt{ \frac{ -(a_1^2-2a_0)+ \sqrt{(a_1^2-2a_0)^2-4a_0^2} } {2a_0^2} }, \]
\[ \gamma_{3,4}=\pm \sqrt{ \frac{ -(a_1^2-2a_0)- \sqrt{(a_1^2-2a_0)^2-4a_0^2} } {2a_0^2} }. \tag{9} \]
Since \((a_1^2-2a_0)>0\) (this follows upon substituting the values of \(a_1\) and \(a_0\) from (3) into the given inequality), it is clear from (8) and (9) that the rational function \(G_{\Sigma}(\omega)\) has no real zeros or poles, since
\[ (a_1^2-2a_0)> \sqrt{(a_1^2-2a_0)^2-4a_0^2}. \]
Since the zeros and poles of an even function with real coefficients are symmetric with respect to the real and imaginary axes [6], \(G_{\Sigma}(\omega)\) can be represented as the product of conjugate functions
\[ G_{\Sigma}(\omega)=\chi(\omega)\cdot\chi^*(\omega), \]
where the functions \(x(\omega)\), \(x^{*}(\omega)\) are constructed in such a way that \(x(\omega)\) has zeros and poles located in the upper half-plane, and \(x^{*}(\omega)\) in the lower half-plane. The frequency characteristic of the shaping filter is determined according to the expression
\[ Y_{1}(j\omega)=x(\omega). \]
Then, taking into account (8) and (9), the transfer function of the shaping filter can be written as:
\[ Y_{1}(p)=m\,\frac{(p-j\beta_{1})(p-j\beta_{3})}{(p-j\gamma_{1})(p-j\gamma_{3})}. \tag{10} \]
We determine the frequency characteristic of the “theoretically” optimal filter according to [4]:
\[ Y_{0}(j\omega)=\frac{G_{n}(\omega)}{G_{n}(\omega)+G_{a}(\omega)} =\frac{\kappa^{2}}{a_{0}^{2}m^{2}\omega^{4}+m^{2}(a_{1}^{2}-2a_{0})\omega^{2}+m^{2}+\kappa^{2}} \tag{11} \]
or, taking into account (8) and (9),
\[ Y_{0}(j\omega)= \frac{\kappa}{(p-j\beta_{1})(p-j\beta_{3})}\cdot \frac{\kappa}{(p-j\beta_{2})(p-j\beta_{4})}. \]
Then the transfer function of the theoretically optimal filter is determined as:
\[ Y_{0}(p)= \frac{\kappa}{(p-j\beta_{1})(p-j\beta_{3})}\cdot \frac{\kappa}{(p-j\beta_{2})(p-j\beta_{4})}. \tag{12} \]
Next, the transfer function \(Y_{2}(p)\) is determined:
\[ Y_{2}(p)=Y_{0}(p)Y_{1}(p)= \frac{\kappa^{2}m}{(p-j\gamma_{1})(p-j\gamma_{3})(p-j\beta_{2})(p-j\beta_{4})}, \tag{13} \]
where \(j\gamma_{1}\) and \(j\gamma_{3}\) are poles lying in the left half-plane,
\(j\beta_{2}\) and \(j\beta_{4}\) are poles lying in the right half-plane.
For the values of the parameters characterizing the IRT-2000 reactor, the poles \(j\gamma_{1}\) and \(j\gamma_{3}\) are real and distinct. The nature of the poles \(j\beta_{2}\) and \(j\beta_{4}\) depends on the ratio \(\kappa^{2}/m^{2}\), and for \(\kappa^{2}/m^{2}<100\) the poles are real and distinct. For real values of the chamber efficiency, the inequality \(\kappa^{2}/m^{2}<100\) is always satisfied. Thus, taking all the poles of equation (13) to be real and distinct, we decompose (13) into simple factors:
\[ Y_{2}(p)= \frac{A}{p-j\gamma_{1}}+ \frac{B}{p-j\gamma_{3}}+ \frac{C}{p-j\beta_{2}}+ \frac{D}{p-j\beta_{4}} \tag{14} \]
and take as \(Y_{3}(p)\) the sum of the simple fractions that do not contain poles in the right half-plane, i.e.,
\[ Y_{3}(p)= \frac{A}{p-j\gamma_{1}}+ \frac{B}{p-j\gamma_{3}}, \tag{15} \]
where the coefficients \(A\) and \(B\) are determined as follows:
\[ A= \frac{\kappa^{2}m}{(j\gamma_{1}-j\gamma_{3})(j\gamma_{1}-j\beta_{2})(j\gamma_{1}-j\beta_{4})}, \]
\[ B=\frac{\kappa^{2}m}{(j\gamma_{3}-j\gamma_{1})(j\gamma_{3}-j\beta_{2})(j\gamma_{3}-j\beta_{4})}. \]
\(Y_{3}(p)\) is a physically realizable part of \(Y_{2}(p)\). The transfer function of the optimal physically realizable filter is determined as
\[ K(p)=Y_{3}(p)\cdot Y_{1}^{-1}(p) = \left[ \frac{A}{(p-j\gamma_{1})} + \frac{B}{(p-j\gamma_{3})} \right] \frac{(p-j\gamma_{1})(p-j\gamma_{3})}{m(p-j\beta_{1})(p-j\beta_{3})} = \]
\[ = \frac{A(p-j\gamma_{3})}{m(p-j\beta_{1})(p-j\beta_{3})} + \frac{B(p-j\gamma_{1})}{m(p-j\beta_{1})(p-j\beta_{3})}. \]
A filter having this transfer function is physically realizable, since the degree of the numerator is less than the degree of the denominator and it has poles located in the left half-plane.
Thus, the problem of measuring fluctuations of the neutron flux under conditions of limited effectiveness of the measuring chamber reduces to the Wiener problem of optimal filtering, which is solved by the Bode–Shannon method.
The technical implementation of the filter will be considered in a separate paper.
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