Modeling on an Analog Computer of the Dependence of the Intensity of $\gamma$-Radiation of a High-Current 25 MeV Betatron with External High-Voltage Injection on Its Principal Input Parameters

M. S. Aleinikov, V. A. Kochegurov

(Presented by the scientific seminar of NIIYaFEA)

Experience in operating the betatron has shown that its output parameter—the intensity of $\gamma$-radiation during operation—does not remain constant, but fluctuates within limits of up to 50%. These fluctuations are random in character and are determined by fluctuations of the input adjustable parameters (the phase and voltage of injection, the electromagnet voltage, etc.) and by external uncontrolled disturbances. In order to increase the average intensity of the betatron’s $\gamma$-radiation during operation, the problem arises of continuously selecting and maintaining such values of the input parameters at which the maximum intensity of bremsstrahlung $\gamma$-radiation is ensured. For this purpose, works [1, 2, 3, 4, 5] propose using automatic regulators of the extremal type. However, in these works no detailed investigations were carried out of the betatron as a controlled object, and the regulators were constructed without taking into account the stochastic dependence between the various parameters of the betatron. This, in our view, did not make it possible to improve the quality of operation of extremal systems by improving their structure.

It is known that in most cases the synthesis of regulators that ensure the search for and maintenance of the maximum of the output quantity under fluctuations of the input parameters and in the presence of certain disturbances requires knowledge of the mathematical dependence of the output parameter on the inputs.

In this respect, for the betatron there is at present still no unified theory that would give a quantitative estimate of the dependence of the intensity of $\gamma$-radiation on the principal input parameters.

In this connection, the synthesis of betatron regulators is most expediently carried out with the aid of adequate mathematical models constructed on the basis of qualitative analysis of certain dependences, with subsequent correction according to experimental investigations. Since the input adjustable parameters of the betatron do not remain constant over time, but undergo continuous changes, the tuning of regulators on an adequate mathematical model of the betatron must be carried out under conditions of continuous deviations of the input parameters. These deviations are purely random in character, and their properties can be studied by using statistical methods.

We conducted a passive experiment on a high-current betatron with external high-voltage injection in order to obtain statistical data.

By applying D. D. Brandon’s method [6], an equation of multiple regression was obtained describing the dependence of the intensity of the braking \(\gamma\)-radiation of the betatron on the injection phase, the injection voltage, the inflection voltage, and the voltage on the electromagnet winding.

The equation has the form:

\[ y = 11.021(1.047 - 0.052\,\Delta x_1 - 0.746\,\Delta x_1^2)(1.029 + 0.119\,\Delta x_2 - \tag{1} \]

\[ - 0.032\,\Delta x_2^2)\, \frac{1}{1.055 + 0.162\,\Delta x_3}\, (1.027 - 0.021\,\Delta x_4 - 0.025\,\Delta x_4^2), \]

where
\(y\) is the intensity of the \(\gamma\)-radiation of the betatron;
\(\Delta x_1, \Delta x_2, \Delta x_3, \Delta x_4\) are, respectively, the deviations of the injection phase, the voltage on the electromagnet winding, the inflection voltage, and the injection voltage from their mean values.

The adequacy of equation (1) was checked by calculating the ratio [6]:

\[ K = \frac{1}{N}\sum_{u=1}^{N} \frac{y_u}{y f_1(x_{1u}) f_2(x_{2u}) \ldots f_k(x_{ku})} = 0.995, \tag{2} \]

where \(y_u\) is the observed value of the intensity of the \(\gamma\)-radiation of the betatron, and
\(y f_1(x_{1u}) f_2(x_{2u}) \ldots f_k(x_{ku})\) is equation (1).

The quality of the chosen approximation may be considered satisfactory, since \(K = 0.995\) differs only slightly from 1.

On the basis of equation (1), using the blocks of the MNB-1 modeling machine, we constructed a model of the dependence of the intensity of the betatron \(\gamma\)-radiation on the injection phase, the voltage on the electromagnet winding, the inflection voltage, and the injection voltage (Fig. 1).

Since the input parameters of the betatron vary continuously, in order to model them it is necessary to know their spectral densities. The calculation of the spectral densities was carried out according to the procedure set forth in [7]. Figure 2 presents the spectral densities of oscillations of the injection phase (curve 1), the voltage on the electromagnet winding (curve 2), the inflector voltage (curve 3), and the injector voltage (curve 4).

It is known [6] that if “white” noise is applied to the input of a filter having the transfer function \(\gamma(\omega)\), then at the output of this filter one can obtain a signal with the required spectral density:

\[ S(\omega)=|\gamma(\omega)|^2. \tag{3} \]

Analysis of the spectral densities of the oscillations of the input parameters of the betatron under study showed that the transfer function of the filter in the model can be represented in the form:

\[ |\gamma(\omega)| = \frac{K}{\sqrt{(1 - T_1T_2\omega^2)^2 + [\omega(T_1 + T_2)]^2}}. \tag{4} \]

Taking expression (3) into account, we obtain

\[ T_1^2(T_2^2\omega^4 + \omega^2) + T_2^2\omega^2 + 1 - \frac{S(0)}{S(\omega)} = 0, \tag{5} \]

where \(S(0)=K^2\).

Using expression (5) and knowing the spectral densities of the oscillations of the input variables, one can calculate the filter parameters by successive approximation.

Figure 2 (curves \(1^0, 2^0, 3^0, 4^0\)) shows the calculated spectral densities at the output of the model filters.

Fig. 1. Block diagram of the model. Legend: \(NG\) — noise generator, \(U_{1-8}\) — amplifiers, \(MP\) — multiplication unit, \(NU\) — nonlinearity unit.

Fig. 2. Spectral densities.

A noise generator assembled according to the circuit described in [5] was used as the source of “white” noise.

Figure 3 presents, for comparison, curves of the time variation of the intensity of the betatron $\gamma$-radiation and of the voltage from the output of model 6.

In order to judge the accuracy with which the model reproduces the variations in the betatron $\gamma$-radiation intensity, the variances of the processes shown in Fig. 3 were calculated, and the spectral densities of their oscillations were computed (Fig. 4). The variance of process $a$ is equal to 6.51, and the variance for process $b$ is 4.

Using Cochran’s criterion [8] and the tables of [9], we find that

\[ G_{\max}=\frac{S_{\max}^{2}}{\sum_{i=1}^{N} S_i^{2}}=0.616, \tag{6} \]

where $S_{\max}^{2}$ is the maximum variance among the variances $S_1^{2}, S_2^{2}, \ldots, S_N^{2}$, is less than the tabulated value, equal to 0.621 at the 5% significance level. Consequently, the variances of the processes shown in Fig. 3 do not differ significantly from one another.

Fig. 3. Curves of variation of the intensity of the betatron $\gamma$-radiation, $a$, and of the voltage from the output of model 6, $b$.

Fig. 4. Spectral densities of oscillations of the intensity of the betatron $\gamma$-radiation, $a$, and of the voltage from the output of model 6, $b$.

Thus, the model ensures reproduction of the real process.

The model shown in Fig. 1 is used for developing extremal-type regulators and systems for stabilizing the intensity of a betatron’s $\gamma$-radiation.

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