ELECTRONIC SIMULATION OF AN EXTREMUM REGULATOR FOR A BETATRON ON THE MN-7

V. M. Razin, N. Ya. Makarov, V. P. Sherstobitov

(Presented by the scientific seminar of the Scientific Research Institute of Electronic Introscopy)

The use of extremum regulators for betatrons makes it possible automatically to obtain from the accelerator the maximum intensity of gamma radiation, which is of great importance especially when a betatron is operated under the conditions of an industrial shop [1—3].

In [4] it is shown that the introduction into the regulator circuit of a variable step depending on the magnitude of the gamma-radiation intensity, and of a circuit for combining the comparison time of a pair of radiation pulses, improves the operation of the regulator. However, the choice of optimal parameters for the regulator circuits presents certain difficulties. This leads to the need to use computing devices for the calculations, which make it possible to investigate the problems of optimizing the system parameters more fully and, at the same time, in comparatively short periods already at the first stage of design [5]. Preliminary approximate calculations made it possible to find an initial step satisfying the condition of a short search time for the maximum radiation intensity. Further investigations, aimed at qualitatively observing the nature of the regulator’s operation, were carried out on the MN-7 analog electronic model.

The radiation of a betatron is pulsed in character; therefore, additional elements simulating the radiation processes were introduced into the model.

The basic equation of the extremum regulator has the following form:

\[ \Delta T = \kappa \int_{0}^{t} (I'' - I')\,dt, \tag{1} \]

where

\[ \begin{aligned} \Delta T &\text{ is the regulating action on the phase of electron injection into the accelerating chamber;}\\ I'' = f(T+\delta T) &\text{ is the intensity value at the instant of the trial phase shift } \delta T \text{ (step);}\\ I' = f(T) &\text{ is the intensity value in the absence of a step shift;}\\ \kappa &\text{ is the proportionality coefficient.} \end{aligned} \]

The magnitude of the trial step \(\delta T\) depends on the magnitude of \(I'\) and is expressed mathematically by the formula

\[ \delta T = \delta T_0 - a I'. \tag{2} \]

Here \(\delta T_0\) is the initial step at \(I' = 0\);

\(a\) is a coefficient whose value must be determined from the simulation results.

The dependence \(I = f(T)\) in the model is represented by individual straight-line segments (Fig. 1). To construct it, the averaged

Fig. 1. Dependence of the betatron radiation intensity on the injection phase.

Fig. 2. Electronic model of the extremum regulator of the betatron.

characteristic of the betatron was used. In this case the scale factors have the following values: the intensity scale

\[ m_I = 16.6 \frac{\text{rel. units}}{\text{V}}; \]

the scale for the phase (time) of injection

\[ m_T = 0.17 \frac{\mu\text{s}}{\text{V}}. \]

The circuit of the regulator model is shown in Fig. 2, where the designation of the blocks coincides with their designation on the MN-7 model.

The two channels being compared are modeled by two function generators 4 and 12 ($J'$ and $J''$, respectively), which, when a linearly increasing voltage is applied to the input, produce at the output two curves identical in absolute value and opposite in sign, corresponding to Fig. 1.

These curves do not require very high matching accuracy, since the operating points at each instant of time are separated along the abscissa axis by the value of the test step; that is, the ordinates always differ by an amount exceeding the adjustment error.

In this circuit the electrical pulses are obtained by means of relay RO at the output of amplifier 17, controlled by a voltage of 100 V through the “1 Hz” contact of the analog computer. The contact-closure time of 1RO and 2RO is 0.15 sec, i.e., the pulse duty factor is 6.7. In the real object, the duty factor of the pulse train of one channel has a value of $2 \cdot 10^4$. This difference is not fundamental, because in the present case it is sufficient to obtain the discharge time of the capacitors $C_1$ and $C_2$ of the comparison detectors in the same ratio to the pulse repetition period as in the real object. Integrating amplifier 5 directly models the optimizer integrator. Amplifiers 9, 11, 2, and 3 are needed for the normal operation of the nonlinearity blocks B-1 and B-3 of the BN-10 type. The initial step is produced by amplifier 1, and its change with increasing intensity $J'$ is carried out by means of amplifier 13, which has a limiting circuit in the feedback loop (valve $V_1$ and potentiometer $R_6$ with a source of $\pm 100$ V).

This makes it possible, as in the real optimizer, to obtain the minimum test step required for normal operation of the optimizer.

With the aid of switch “K” it is possible to switch the voltage $U_{\text{in}}$, simulating abrupt shifts of the characteristic $J = f(T)$.

Table 1 gives some experimental data recalculated for the real optimizer.

Table 1

The minimum step is everywhere equal to 0.5 $\mu$sec; at larger values of it, no “overshooting” of the maximum is observed.

The repetition period of a pair of radiation pulses is 40 msec (25 Hz). When a constant step is introduced (last line of the table), the time required to find the maximum of the function exceeds the permissible integration time for the MN-7 model. When “K” is switched on and $U_{\text{in}}$ is chosen corresponding to

...change of phase by 3 μsec, the movement of the operating point toward the maximum proceeds along the search curve (Fig. 3). The parameters were refined with the aid of the “Minsk-1” digital computer, and it was found that the optimal initial step lies in the range of 6–7 μsec.

Fig. 3. The process of searching for the maximum
(voltage from the output of amplifier 4).

A characteristic process of searching for the maximum is shown in Fig. 3 (voltage from the output of amplifier 4). This figure gives the curve for row 2 of the table.

Fig. 4. Change of the operating step in time
(voltage from the output of amplifier 1).

The interval between time marks on the real scale is 40 μsec. The change of the operating step for this case is shown in Fig. 4 (voltage from the output of amplifier 1).

Conclusions

1. The optimizer model makes it possible to qualitatively trace the operation of the optimizer and to select its parameters, close to optimal ones.

2. The introduction of a variable step considerably accelerates the process of searching for the maximum of the betatron gamma-radiation intensity.

3. The search time is substantially influenced by the initial step and by the transmission coefficient of the integrator.

With a transmission coefficient greater than 250 sec\(^{-1}\), in our case an oscillatory establishment of the maximum was observed and the accuracy of maintaining the maximum decreased.

References

1. L. M. Anan’ev, A. A. Vorob’ev, V. I. Gorbunov. Induction Electron Accelerator—Betatron. Atomizdat, 1961.

2. V. M. Razin. Stabilization of betatron radiation. Proceedings of the Tomsk Polytechnic Institute, vol. 87, 206–215, 1957.

3. A. P. Komar, G. F. Mikheev, N. N. Chernov. System of extremal regulation of the intensity of synchrotron γ-radiation. ZhTF, XXXI, 1, 109–115, 1961.

4. V. P. Sherstobitov, V. M. Razin, Ya. Ya. Makarchuk, A. S. Bobrovskii. Optimizer of the gamma-radiation intensity of a betatron. “Proceedings of the 5th Interuniversity Conference on Electron Accelerators,” Tomsk, 1964.

5. B. Ya. Kogan. Electronic Simulation Devices and Their Application to the Study of Automatic Control Systems. Fizmatgiz, 1959.