STABILIZATION OF BETATRON RADIATION

A. P. GRIGOR'EV, V. I. GORBUNOV

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

Owing to a number of its advantages over other accelerators for low energies, the betatron is finding ever wider application. In this connection, the tasks arise of achieving reliable operation of the betatron, increasing the radiation intensity, and stabilizing the intensity and energy of the radiation, which fluctuate within considerable limits as a result of fluctuations in the parameters of the supply network.

Stabilization of the radiation can be achieved either by stabilizing the entire power supply of the installation (Leonard scheme), or by stabilizing the parameters of individual units of the betatron [1], i.e., it requires the use of bulky and expensive devices and circuits. Recently, electronic circuits have also been developed which make it possible to stabilize the intensity [2, 3] and the energy [4, 5] at a specified level.

Circuits for stabilizing the intensity by deviation have a number of disadvantages:

1) they operate stably only at low level values, thereby reducing the effectiveness of the installation;

2) the integral value of the intensity is stabilized over a time interval determined by the inertia of the system, which can lead to the recording of false defects when using the scintillation method of flaw detection.

Known energy-stabilization circuits are very complex in their construction and adjustment. For the purposes of flaw detection and X-ray radiology they can be made substantially simpler by reducing the accuracy of stabilization and, consequently, more reliable.

Many requirements can be satisfied by the radiation-stabilization circuit described below. Its operation is based on the principle of automatically changing the phases of the injection and displacement pulses as a function of fluctuations in the basic parameters of the betatron installation, in accordance with the prescribed laws for the optimal injection phase (1) and the energy (2) of the γ-radiation:

\[ t_i=\frac{\sqrt{U_i}}{0.297r_0^{\omega}B_{0m}}, \tag{1} \]

\[ W=300r_0B_{0m}\sin\alpha_c, \tag{2} \]

where \(r_0\) is the radius of the equilibrium orbit,
\(B_{0m}\) is the maximum induction at the radius of the equilibrium orbit,
\(U_i\) is the injection voltage,
\(\alpha_c = \omega t_c\) is the phase of the current pulse in the bias winding.

Tuning the injection phase to the optimal value (1) makes it possible to obtain optimal capture conditions and thereby an intensity close to the maximum and relatively stable, provided that the radiation energy is stabilized (the lamp and injector filaments are stabilized by ordinary circuits). Energy stabilization can be achieved by changing the phase \(\alpha_c\) in accordance with the change in induction \(B_{0m}\) (2). As is clear from the foregoing, in this case, in contrast to previously known methods, stabilization of the radiation by excitation is used.

The task of automatically changing the injection and bias phases is performed by stages of “amplitude–time” converters [6]. They are gated multivibrators on pentodes (Fig. 1, tubes \(L_3, L_4\) and \(L_8, L_9\)). The duration of the rectangular pulse formed by the converter can be regulated not only by changing the time constant of the circuit \(\tau_g = C_g R_g\), but also by changing the potential on the screen grids. In this case, as the voltage on the screen grid of the left pentode \(L_8\) increases, the pulse duration increases, while as the voltage on the grid of the right pentode \(L_9\) increases, it decreases. This property of theirs was used in constructing the circuit. Information about oscillations of the induction, the injection voltage, and the frequency, which is contained in the amplitudes of the signals from the sensors of these quantities, is fed to the corresponding grids, thereby obtaining a delay of the injection or bias phase pulse that varies according to a definite law.

The circuit is adjusted on the basis of the following theoretical considerations.

The necessary change in the injection phase for obtaining its optimal value when the quantities entering into equation (1) oscillate can be obtained by the method of linear approximation. A monotonic analytic function \(t_i\) can be expanded in a Taylor series in the neighborhood of the selected operating point, which is determined by the choice of the voltage \(U_i\); as a result we obtain the linear model of equation (1):

\[ \Delta t_i = \frac{t_{i0}}{2U_i}\Delta U_i - \frac{t_{i0}}{B_{0m}}\Delta B_{0m} - \frac{t_{i0}}{\omega}\Delta \omega . \tag{3} \]

Thus, in order to obtain optimal capture conditions it is necessary to take into account the increments of three parameters, depending on which the injection phase must change. This change must occur along the line of optimal phases (Fig. 2).

Let us choose, as sensors of the induction \(B_{0m}\) and of the frequency \(\omega\), coils located on the yoke of the electromagnet, and as the sensor of the injection voltage, a resistance divider. Then the sensor equations can be written:

\[ U_{g1} = C_1 \omega B_{0m}, \tag{4} \]

\[ U_{g2} = C_2 U_i, \tag{5} \]

where \(C_1 = \mathrm{const}\) is determined by the number of turns of the sensor and by the magnetic flux encompassed by them;
\(C_2 = \mathrm{const}\) is determined by the parameters of the divider.

Fig. 1. Schematic circuit diagram for radiation stabilization.

Using the known method for calculating block diagrams of automatic-control systems [7] and equations (3–5), one can determine the coefficients for converting amplitudes into durations for the converter \((\mathrm{Л}_8\mathrm{Л}_9)\):

\[ K_B=-\frac{t_{i0}}{U_{g1}};\qquad K_\omega=-\frac{t_{i0}}{U_{g1}};\qquad K_{ui}=\frac{t_{i0}}{2U_{g2}}. \tag{6} \]

It turns out that \(K_B=K_\omega\), i.e., if the converter is adjusted to take into account only induction oscillations, then it also takes frequency oscillations into account, and the adjustment of the injection phase is carried out in accordance with equation (1) for both parameters. The converter coefficients (6) indicate that the injection phase must decrease as the quantities \(B_{0m}\) and \(\omega\) increase, and must increase as \(U_i\) increases.

Fig. 2. Plots of the dependence of the injection voltage on the optimum injection phase:
1 — at \(U_{c\,\mathrm{nom}}\); 2 — at \(U_{c\,\mathrm{nom}}+10\%\); 3 — at \(U_{c\,\mathrm{nom}}-10\%\).

Using the basic features of the converter \((\mathrm{Л}_8\mathrm{Л}_9)\) and taking equations (6) into account, we feed information about the injection voltage to the control grid of the left-hand tube \(\mathrm{Л}_8\), and information about \(B_{0m}, \omega\) to the grid of the right-hand tube \(\mathrm{Л}_9\). The action of the signals from the sensors must be such as to satisfy equations (1) and (3).

For the energy-stabilization circuit, one can give arguments analogous to those set forth above. As is seen from equation (2), the radiation energy at a constant ejection phase is a function only of the induction, since \(r_0\) is practically constant. The requirement of energy stability at a specified level may be represented in the form

\[ W^* = 300\,r_0 B_{0m}\sin\alpha_c=a_2=\mathrm{const}, \tag{7} \]

whence

\[ \sin\alpha_c=\frac{a_2}{B_{0m}}, \tag{8} \]

where

\[ a_2=\frac{W^*}{300r_0} \]

is a constant for the chosen energy level.

It follows from equations (7) and (8) that, in order to obtain stable energy, it is necessary that, during induction oscillations, the displacement phase change in accordance with equation (8). In practice, the displacement phase is changed by altering the delay of the cas-

cade of the converter and is measured in sec; therefore it is expedient to isolate the time \(t_c\) explicitly in expression (8). This will also make it possible to estimate the influence of the frequency on the stability of the energy and the possibility of correcting frequency oscillations. Then

\[ t_c=\frac{1}{\omega}\arcsin\frac{d_2}{B_{0m}} . \tag{9} \]

From this one can obtain a linear model for the displacement phase:

\[ \Delta t_c=-\frac{\sin^2\omega t_c}{\omega d_2\cdot\cos\omega t_c}\Delta B_{0m} -\frac{t_c}{\omega}\Delta\omega . \tag{10} \]

Using the sensor equation

\[ U_{q3}=C_3\omega B_{0m} \tag{11} \]

and the linear model (10), by the known method [7] we can obtain the conversion coefficients, required by the conditions for energy stability, from changes in the magnitude of the sensor signal into the duration of the delay:

\[ K'_B=-\frac{\tg\omega t_c}{\omega\cdot U_{q3}};\qquad K'_\omega=-\frac{t_c}{U_{q3}} . \tag{12} \]

As is seen from (12), in the present case the coefficients \(K'_B\ne K'_\omega\); and since the information on \(B_{0m}\) and \(\omega\) is obtained from a single sensor and is transmitted through one channel, it is obvious that a converter adjusted to take into account oscillations in the magnitude \(B_{0m}\) will not be able to correct frequency oscillations completely. They are corrected only when \(K'_B=K'_\omega\), when \(\tg\omega t_c=\omega t_c\), i.e., when the angle \(\alpha_c=\omega t_c\) is sufficiently small. However, the error caused by frequency oscillations is sufficiently small (does not exceed 0.14%).

From equations (10) and (12) it is seen that, as the quantities \(B_{0m}\) and \(\omega\) increase, the delay of the displacement pulse must decrease, as indicated by the minus signs; therefore it is necessary to feed the information on these quantities to the right grid of the converter. In the circuit (Fig. 1) this information is fed to the left grid, but through an amplifier stage that serves as a phase inverter and as a structural nonlinearity, making it possible to obtain the nonlinear coefficients \(K'_B\) and \(K'_\omega\) with greater accuracy.

The basic circuit (Fig. 1) was constructed and tested on the basis of the considerations set forth above. The circuit operates as follows. A negative pulse from the magnetic-field zero sensor is amplified (\(Л_1\)) and triggers the converter stages (\(Л_3Л_4\) and \(Л_8Л_9\)). The rectangular pulse formed is differentiated, and by its trailing edge, through an amplifier (\(Л_5\) and \(Л_{10}\)), triggers the thyratron stage (\(Л_6\) and \(Л_{11}\)), which forms the firing pulses for the injection circuits (lower channel) and displacement circuits (upper channel).

In the injection channel, information on the magnitude of the injection voltage is fed to the grid of the tube (\(Л_8\)) through a cathode follower (\(Л_7\)), while information on the quantities \(B_{0m}\) and \(\omega\) is fed to the grid of \(Л_9\). The circuits connected between the sensors and the screen grids, as well as the cathode follower (\(Л_7\)), serve to obtain the necessary steepness of conversion of amplitudes into duration, i.e., the coefficients \(K_B\), \(K_\omega\), \(K_U\).

Information on the induction and frequency is fed analogously in the displacement channel, where the amplifier \(Л_2\) is used, the purpose of which was described above.

The experimental data obtained in tuning the circuit coincide, with sufficient accuracy (which can be improved), with the necessary calculated data. The radiation energy can be stabilized with an accuracy of \(\le 1\%\) when the mains voltage fluctuates within \(\pm 10\%\) (Table 1).

Table 1

| | \multicolumn{16}{c}{Energy stabilization level} |
|---|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|
| \(W\), MeV | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 25 | 25 | 25 | 25 | 30 | 30 | 30 | 30 |
| \(\Delta W\), % | \(+10\) | 0 | \(-10\) | \(-15\) | \(+10\) | 0 | \(-10\) | \(-15\) | \(+10\) | 0 | \(-10\) | \(-15\) | \(+10\) | 0 | \(-10\) | \(-15\) |
| \(t_c \cdot 10^2\), \(\mu\)sec | 14,1 | 15,65 | 17,74 | 18,9 | 19,2 | 21,7 | 24,8 | 26,7 | 25,8 | 28,9 | 34,1 | 38,2 | 32,8 | 40,0 | — | — |
| \(U_{\mathrm{d3}}\) (V), calculated | 5,5 | 5 | 4,5 | 4,25 | 6,05 | 5,5 | 4,95 | 4,68 | 7,7 | 7 | 6,3 | 5,95 | 7,92 | 7,2 | 6,5 | — |
| \(U_{\mathrm{d3}}\) (V), experiment | 5,5 | 5 | 4,5 | 4,25 | 6,05 | 5,5 | 4,97 | 4,65 | 7,7 | 7 | 6,33 | 5,95 | 7,90 | 7,2 | 6,5 | — |

The injection channel also makes it possible to obtain values of the changes in injection phase under mains fluctuations that agree with the calculated values with high accuracy (Fig. 2).

The proposed circuit satisfies the basic requirements imposed on a betatron flaw detector; it may also be useful for betatrons of other purposes by virtue of one of its principal advantages: stabilization of the radiation is carried out from pulse to pulse.

At present, on the basis of this circuit, the development is being completed of a combined circuit for radiation stabilization, which will make it possible to obtain a highly stable intensity at high levels.

REFERENCES

1. V. M. Razin. Izvestiya TPI, vol. 87, pp. 206–215.

2. Yu. M. Akimov et al. Proceedings of the IV Interuniversity Conference on Electron Accelerators, pp. 324–328, Higher School, Moscow, 1964.

3. E. M. Belov, V. N. Titov. MEI, “Radiotekhnika,” 1960, No. 1.

4. N. N. Balamatov, B. A. Yur’ev. PTE, No. 4, Publishing House of the Academy of Sciences of the USSR, Moscow, 1963.

5. K. N. Geller. Nucl. Instrs methods 17, No. 2, 161–168, 1962.

6. Scientific report on topic 1004/63, “Flaw detection of plastics by the scintillation method.” Tomsk, TPI, 1963.

7. J. Taylor, M. Pestel. Analysis and Calculation of Nonlinear Automatic-Control Systems, “Energiya,” Moscow, 1964.