Corresponding author: Anatoliy G. Yuferov (anatoliy.yuferov@mail.ru)

Academic editor: Yury Kazansky

A number of issues pertaining to comparative analysis of possible options for algorithmic and circuit embodiment of reactimeters were examined from the standpoint of the general theory of measuring instruments and the theory of digital filters. Structural diagrams of the linear part of the reactimeter, as well as the functional algorithms and their numerical implementation are described in terms of transient characteristics and transfer functions. Parallel, straight, canonical, symmetrized, lattice and ladder block structural diagrams are examined. The corresponding difference equations are given. The obtained results allow comparing possible circuit design solutions from the viewpoint of a number of criteria: the complexity of the elemental composition (the number of integrators, summation units, multipliers, delay elements), the number of necessary computing operations, the identifiability of the hardware function of the reactimeter, the coherence between the calculated and the measured values, the sensitivity to parameter uncertainties, etc. The possibility of considering the equations of the reactimeter as autoregressive is demonstrated, which ensures adaptability of the reactimeter under operating conditions. Certain algorithms for identification of the transient response characteristic and transfer function of the reactimeter are indicated. The possibility is shown of using identical algorithms in the main computing unit for solving the direct and inverse problems of nuclear reactor kinetics for ensuring consistency between the calculated and the measured reactivity values. Upper and lower estimated reactivity values are suggested for the moment of switching on the reactimeter. Implementation of such estimations in the reactimeter design allows minimizing the time needed for reaching by the reactimeter of the operating mode. Certain methodological simplifications were used in the development of ladder and lattice circuit design solutions. The database containing parameters of the instrumental functions of the circuit design solutions of the reactimeters is available on a public website. A number of tasks and directions for further research are identified.

The purpose of the present study is the examination of certain issues pertaining to the comparative analysis of options of algorithmic and circuit design implementation of reactimeters from the viewpoint of criteria of the general theory of measurement instruments (

Practically all reactimeter designs (see, in particular, (

– Minimized the number of structural elements and operations

– Applied simple reactivity assessment algorithms correlated with reactivity calculation procedures accepted in the codes simulating the

– Ensured short time for reaching by the reactimeter the intended operation mode.

Certain aspects pertaining to the above issues are addressed in the present paper. The main attention is paid to the circuit design solutions for the reactimeter as the digital filter. Corresponding analogue circuits can be constructed following well known relations (

The set of differential equations of

Integration of the equation by parts reduces the latter to the form containing only power readings:

Reactimeter equations straightforwardly follow from (1), (2). Thus, following Eq. (1),

where ^{*} is the reactivity expressed in units of effective fraction of delayed neutrons β_{ef}: ρ^{*} = ρ/β_{ef}.

Reactivity according to Λ-scale is the relative rate of reproduction of prompt neutrons, i.e. the algebraic sum of relative rates of generation (probability 1/Λ) and loss (probability 1/

has the following form:

where _{j}_{j}_{ef}/Λ is the probability of generation of delayed neutrons (fraction of neutrons "spent” for the generation of precursors of delayed neutrons). _{ef}/Λ simplifies the comparison of relative rates of processes in the

Equations (1) – (3) ensure the required accordance between the solutions of the direct (calculation of reactor power dynamics) and the inverse (calculation or measurement of reactivity) problems if these problems use identical algorithms in their discrete realization

_{k}_{k}n_{k}_{k}_{ef}/Λ + _{k}

In the case of constant values of the source and the reactivity (1), (2) and (5) can be considered as the autoregression equations relative to the constants ^{*} and sampling values of hardware functions h, g. If sufficient number of measurements of the rate or reactor power is available the above equations are efficiently solved relative to the indicated regression factors using appropriate variant of the least square method (

In order to analyze algorithmic and circuit design options of the reactimeter let us write down its equation as the convolution equation by separating the linear part and converting it to the structure of linear filter

with input signal _{ef}/Λ)_{0} = β_{ef}/Λ explicitly as the amplification coefficient for the linear part of the reactimeter. This value can be the result of independent calculation or experimental estimation and it provides in this capacity additional possibility for testing adequacy of reactimeter adaptation. Thus, for comparing the circuit design solutions of the reactimeter it is sufficient to examine the structural transformations of the transfer function of the delayed neutron integral _{0}.

The direct discretization of the delayed neutron integral in the case when certain quadrature formula is applied in the calculations without accounting for the exponential representation of the kernel (4),

is characterized by the transfer function written in the following

where _{l}_{k,k}_{l}h_{l}_{k,k}_{l}

Analogue transfer function in

corresponds to the standard exponential form of the

Conventional circuit design solutions for analogue reactimeters are based on this form. Evident transformations of the

and the

structural forms.

In particular, the parameters of form (8) are equal to:

Sensitivity of these parameters to the delayed neutron constants is described by the following relations:

Complexity of element compositions of the hardware implementation of the reactimeter is characterized by the number of parameters and operators

where

Different options of such single-step discretization (quadrature formula) are used in the equation of digital reactimeter – the inverted solution of the kinetics equation (

^{–1}

and the input signal is initially processed by the block

Replacing the

This allows decreasing the number of operations in the calculations of reactivity _{k}_{k}_{k}

Change of the order of operations for processing the input signal by permutation of blocks

correspond.

In the given case only the input signal _{k}

Here, each calculation block (expression in the brackets) uses separate integrator but, however, the potential gain is associated with the fact that these blocks can operate in parallel. It is appropriate to call such layout

or

It is appropriate to call such form

Cascade structural forms are of interest since standard bilinear or biquadratic blocks can be used in the hardware implementation. However, in this case group parameters of delayed neutrons must be known.

_{j}_{J–j}

From the characteristic property of the all-pass filter

follow the equations of constraint for the lattice cascades

and the algorithm for calculating the cascade coefficients ^{j}

^{j}_{j}^{j}^{j}^{j}B^{j}_{j}^{j}_{J}_{j}^{j}

^{j}^{–1}(^{j}^{j}C^{j}^{j}q^{j}

Here the superscript index is the number of the cascade and the subscript index is the number of the coefficient in the polynomial.

According to the characteristic property (15) inputs of adjacent cascades are coupled as follows: ^{j}^{–1}(^{j}^{–1}(^{j}^{j}^{–1}(^{J}^{j}^{j}^{j}^{j}^{j}^{j}^{j}^{j}^{j}_{j}

Weights _{j}

by solving the set of linear equations linking the coefficients of polynomials ^{j}

_{1} and conductivity ^{*}_{1} + ^{*}^{*}_{2} and ^{*} so that _{1} + _{1} + 1/^{*}^{*} and so on. Such method of building up the structure corresponds to the procedure for calculating _{i}

_{1}(_{2} + ^{*}) + _{2}^{*}]/(_{2} + ^{*}) = _{1} + 1/(1/_{2} + 1/^{*}).

Such constructions are not unequivocal. Different options of equivalent structures are possible resulting in the decrease of the number of segments in case of correct selection of resistances _{i}_{i}_{2}_{i}

Elements of this matrix _{ij}_{1} = _{12}/(_{22} – 1), _{2} = _{12}, _{3} = _{12}/(_{11} – 1).

Following the interpretation of the transfer function _{1}(_{2} + _{3})/(_{1} + _{2} + _{3}).

Similar procedures are applicable to discrete TF of the straight-line structural form numerator and denominator of which are represented by polynomials (11). For them the first step of expansion of the TF into the continued fraction produces _{1}^{–1} + 1/_{1}(_{1}(_{1}(_{1}(_{1} is the real coefficient. In such case the delayed neutron integral _{k}_{1}(_{k}_{1}_{k}_{–1}). This transformation separates within the structure the negative feedback segment – the summand _{1}_{k}_{–1}. Since the power of polynomial _{1}(_{1}(_{2} + _{2}(_{2}. Following this we repeat the first step in relation to the transfer function _{2}(

Parameters of the above described circuit design solutions can be identified according to experimental data which solves the problem of reactimeter adaptation. Here the reduction of the number of elements of the circuit structure is possible if it is discovered that the identification produces zero values for respective parameters.

Identification of hardware function of the reactimeter is implemented in the most straightforward way (from the viewpoint of calculation) for the recursive structural form

in the situation when the leaving by the reactor of the steady-state operational mode is provoked by prompt pulsed or stepwise disturbance of the reactivity or the source. In such case the expression for direct estimation of the hardware function follows from Equation (5):

As applicable for the method of instantaneously removed source this formula takes the form

if the

Results of noise identification of the non-recursive reactimeter hardware function using Burg’s method (

Parameters of the straight-line structural form (11) can be found using Pade approximations (

Equation (1) was obtained in the assumption that the reactor was operated before the time moment

Estimation of the accumulated

Here, the lower estimation of reactivity (according to Λ-scale) equal to α(

where _{del} is the lifetime of delayed neutrons. This estimate majorizes the estimates corresponding to any other path for transition to the current state. It can only overestimate the real value of reactivity and, therefore, satisfies the nuclear safety requirements.

Approximate expression in the last formula is applicable for any circuit design solutions for presetting the value of reactivity at the time moment of turning on the reactimeter. It corresponds to the standard settings according to the doubling period _{2} > 10 s (in this case the value α in the denominator of the presented formulas can be neglected) and to the characteristic value of the generation time (allowing neglecting the first summand). Availability in the design of digital reactimeter of the possibility to perform such estimations allows reducing to zero the time needed for reaching by the reactimeter of its operation mode.

1. Possible options of circuit design implementation of reactimeter are described in terms of structural forms of transfer functions for the reactimeter linear part. Discrete TF were obtained under the condition of coincidence of transfer characteristics of the analogue and discrete implementations of the reactimeter linear part. Respective difference equations are given the structure of which determines the number of required multiplier units, memory elements and summator units in the hardware implementation of the reactimeter.

2. Represented difference equations can be used both in the calculations of reactivity and for calculating the reactor power dynamics. This unifies the direct and the inverse problems of the

3. Algorithms are described for identification of parameters of transfer functions ensuring reactimeter adaptation in operational conditions. From the viewpoint of simplicity of calculations, the non-recursive structural form appears to be the most attractive.

4. Relations are given that relate the coefficients of various structural forms of TF. Calculation of coefficients of transfer functions was performed. The current array of parameters of the reactimeter transfer functions is posted on public website.

5. Due to the linearity of the main computational block the suggested form of the reactimeter equation does not require the transition to small disturbance equations traditionally applied (

For further work under the considerate subject, it is appropriate to specify the following tasks:

– Construction of transfer functions of the digital reactimeter on the basis of different discretization methods applying, for instance, the bilinear transformation or

– Calculation of parameters for possible circuit design solutions for the known systems of delayed neutron data;

– Comparative analysis of suggested algorithms and circuit design solutions from the viewpoint of quality of noise suppression;

– Derivation of dispersion equations for the reactimeter (

– Generalization of the presented difference equations for multipoint models of

– Comparative analysis of the described circuit design solutions as applied to specific inventory of hardware components.

* Russian text published: Izvestiya vuzov. Yadernaya Energetika (ISSN 0204-3327), 2019, n. 4, pp. 95–108.