PROCEEDINGS
OF THE TOMSK ORDER OF THE RED BANNER OF LABOR POLYTECHNIC
INSTITUTE named after S. M. KIROV

Volume 138　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　1965

ON THE ISSUE OF EXTREMAL CONTROL OF THE INTENSITY OF BETATRON RADIATION

E. M. Belov, N. Ya. Makarov, I. E. Naats

(Presented by the scientific seminar of the Research Institute of Nuclear Electronics)

For the purposes of industrial flaw detection it is necessary to obtain the maximum radiation intensity, whereas in carrying out various kinds of physical experiments it is sometimes necessary to operate at the required level of radiation intensity.

The authors set themselves the task of developing an electronic automatic tracking system that would make it possible to operate the betatron in both of the indicated modes. However, ensuring the operating mode of a betatron with maximum radiation-intensity output is technically difficult to accomplish because of the sharp peaked static characteristic of the controlled object.

Problems of extremal control differ substantially from problems of ordinary control in that the quantity sought is unknown and is not specified in advance; with the aid of the automatic-control system this quantity is determined and maintained with the prescribed accuracy.

The intensity of betatron radiation is affected by a number of factors, some of which are difficult, and sometimes practically impossible, to take into account [3]. The main parameters characterizing the accelerating installation are: the induction in the gap between the poles — \(B\), the injection voltage — \(u_i\), the current from the layer — \(I_{\mathrm{sl}}\), the position of the chamber in the gap between the poles of the magnet, the vacuum, the location of the peak transformer, and, finally, the phase of generation of the injection pulse — \(\varphi\).

Analysis of the experimental dependences shows that the principal parameters determining the radiation intensity are the voltage and the phase of the injection pulse. The general expression for the radiation intensity, taking into account only these two parameters, has the form

\[ I = f(u_i,\varphi). \tag{1} \]

Let us investigate the possibility of extremal control of the intensity of betatron radiation, i.e., such control in which the values of the parameters \(\varphi_0\) and \(u_0\) are automatically determined, at which the radiation intensity reaches the maximum possible value, all other conditions being equal. Before solving such a problem, it is necessary to show the existence of extrema, either complete or at least par-

practically within the operating range of variation of the parameters \(\varphi\) and \(u_i\). The condition for the existence of an extremum of the function \(I(\varphi,u_i)\) is written in the form

\[ \frac{\partial I(\varphi,u_i)}{\partial u_i}=0; \tag{2} \]

\[ \frac{\partial I(\varphi,u_i)}{\partial \varphi}=0. \tag{3} \]

Let us consider the possibility of extremal regulation by the injection voltage.

We assume that

\[ I \cong \kappa_1\sqrt{u_i}. \tag{4} \]

Then

\[ \frac{\partial I}{\partial u_i}=\kappa_2\frac{1}{\sqrt{u_i}}, \tag{5} \]

where \(\kappa_1\) and \(\kappa_2\) are certain constants.

It follows from expression (5) that \(\dfrac{\partial I}{\partial u_i}\ne 0\) in the operating range and, consequently, it is impossible to formulate an optimal regulation problem with respect to this parameter, since an increase in the radiation intensity is limited by the breakdown voltage at the injector.

Another parameter by which extremal regulation can be carried out is the phase of electron injection into the accelerating chamber [4].

For the selected parameters, \(\dfrac{\partial I}{\partial \varphi}\) becomes zero at least at one point in the operating range of phase variation from 0 to \(\varphi_{\max}\). Depending on the relations among the various parameters of the accelerator installation, two or more maxima of the radiation intensity are possible; therefore, it is more expedient to search for the principal extremum by the memorization method, since the use of the gradient method under these conditions is difficult.

Fig. 1. Block diagram of an extremal regulator.

The block diagram of a regulator with extremum memorization is shown in Fig. 1, where:

I — radiation sensor: 1 — ionization chamber with a “Cactus” roentgenometer; 2 — amplifying device for converting the direct-current voltage of the signal into a high-frequency voltage.

II — a two-channel information-conversion device; 3 — an alternating-current amplifier with peak detectors; 4 — integrators; 5 — comparison circuit; 6 — search and tracking circuit; 7—8 — operating-memory device.

III — synchronization channel: 9 — modulator of the ignition pulses; 10 — injection circuit.

IV — the controlled object—the betatron.

The radiation sensor—I—serves to convert radiation pulses into a high-frequency voltage, the amplitude of which is proportional to the radiation intensity.

The two-channel information-conversion device—II—is intended for processing the information arriving through the channels and for extracting the error signal that controls the synchronization circuits.

The synchronization channel serves to synchronize the electronic control circuit with the magnetic field of the betatron. The operating principle of the extremum regulator is as follows.

When the regulator operates in the mode of searching for the maximum value of the radiation intensity, the voltages at the outputs of the peak detectors accurately follow the change in radiation intensity; moreover, the rate of increase of the voltage of the current coordinate (peak detector 4) is higher than that of the reference coordinate (peak detector 4). Owing to this, the polarity of the mismatch signal, for the adopted phasing of the differential amplifier—5, remains negative. Thus, the motion of the injection pulse along the phase axis toward the maximum of the radiation intensity slows as the negative mismatch signal increases.

When the extremum is reached, the operating memory is switched off (the operating memory responds only to a change in the amplitude of the radiation-intensity pulse), and the magnitude of the mismatch signal decreases, thereby changing the scanning speed of the injection pulse along the phase axis.

The scanning stage (search) is switched to the mode of a direct-current amplifier with a slowly varying output voltage, as a result of which the phase of the injection pulse oscillates around the extremum \((t_1)\).

Fig. 2. Graph of the change in voltages on the integrators of the current and reference coordinates.

In the case of a sharp decrease in the radiation intensity, for example, as a result of a sharp increase in the voltage on the magnet, the voltage of the reference-coordinate integrator remains practically constant,

whereas on the current-coordinate integrator it decreases sharply \((t_2)\). The mismatch signal thereby becomes positive, and the injection pulse is shifted with increased speed toward a decrease in the phase value.

An analogous process is observed when the supply-line voltage is decreased. The voltage waveforms on the current- and reference-coordinate integrators are illustrated by the plots in Fig. 2, where: 1 is the voltage on the reference-coordinate integrator; 2 is the voltage on the current-coordinate integrator.

The changes in the voltages on the current- and reference-coordinate integrators at the time \(t^2\) correspond to such an increase in the magnet voltage at which the radiation intensity falls to zero.

The time \(t_3\) corresponds to the value of the radiation intensity at the newly established magnet voltage. The voltage waveforms at the time \(t_4\) correspond to the reduced voltage on the betatron magnet.

On the question of selecting the scanning rate

When designing an extremum regulator, it is necessary to determine the initial scanning rate of the injection pulse as the latter moves along the phase axis.

\[ c=\frac{I(\varphi_{k+1})-I(\varphi_k)}{t_{k+1}-t_k}, \tag{6} \]

where \(c\) is the scanning rate; \(\varphi_k,\ \varphi_{k+1},\ t_k,\ t_{k+1}\) are, respectively, the values of the phase of the injection pulse and the time interval between injection pulses through a period of the supply voltage.

Let us denote by \(\Theta\) the magnitude of the phase shift per one injection pulse:

\[ \Theta=\varphi_{k+1}-\varphi_k, \tag{7} \]

then the increment of the radiation intensity is

\[ \Delta I(\Theta)=I(\varphi_{k+1})-I(\varphi_k). \tag{8} \]

We choose a linear sweep with a constant scanning rate, i.e., a continuous forced change in the phase of the injection pulse. It follows from expression (8) that the increment of the radiation intensity for each injection pulse of constant amplitude is a function of three variables

\[ \Delta I=f(\varphi_k,I,\Theta), \tag{9} \]

or

\[ \Delta I_{\max}=\max f(\varphi_k,I,\Theta), \tag{10} \]

where \(\Delta I_{\max}\) is the maximum value of the increment of the radiation intensity between two injection pulses.

From the experimentally obtained characteristics \(I=f(\varphi_k)\) for the given accelerator, we select, in the region where the principal extremum is located, the points \(\varphi_k\) with the greatest derivative.

In this region

\[ \Delta I_{\max}>e, \tag{11} \]

where \(e\) is a quantity equal to the error of the measuring instrument.

In the particular implementation of the regulator, the measuring device used was a “Cactus” roentgenometer, whose error ...

10%. Thus, in the region of the principal maximum on the curve in Fig. 3 we mark off a segment \(0.1 I_{\max}\) along the ordinate axis and, projecting the points onto the phase axis, obtain the maximum value \(\Theta_{\max}\).

Fig. 3. Dependence of the betatron radiation intensity on the injection phase.

If the possible attainable voltages on the injector are taken into account, then for the given accelerator we have the value \(-\varphi_{\max}\); in this case the time for a complete traversal of all possible phase values from 0 to \(\varphi_{\max}\) is

\[ t=\frac{\varphi_{\max}}{\Theta_{\max}}\cdot T, \tag{12} \]

where \(T\) is the repetition period of the injection pulses.

An extremum regulator with continuous forced variation of the injection-pulse phase, with storage of the maximum value of the radiation intensity, was tested on betatrons with maximum accelerated-electron energies of 15 and 30 MeV. The results of the experimental test proved somewhat unexpected: the regulator with “operational memory” maintained a radiation intensity more than twice the value attainable by manual adjustment of the injection pulse to the optimum value, all other conditions being equal. The magnetic-field decay coefficient in the experiment was 0.68.

The electronic circuits of two versions of the extremum regulators, their discussion, and the results of experimental studies will be presented in subsequent papers.

REFERENCES

1. E. M. Belov, V. N. Titov. Device for regulating the gamma-radiation intensity of a betatron. Inventor’s certificate No. 140505.

2. E. M. Belov, I. E. Naats. Extremal regulator of gamma-radiation intensity. Inventor’s certificate No. 141 912.

3. E. M. Belov, V. N. Titov. Stabilizer of the gamma-radiation intensity of a betatron. Izvestiya VUZov, Radio Engineering section, No. 1, 1960.

4. V. M. Razin. Stabilization of betatron radiation. Izvestiya Tomsk Polytechnic Institute, vol. 87, 1957.