ON THE THEORY OF THE STATIC MODE OF AUTOMATIC DISTURBANCE CONTROL IN A SPECIAL TYPE OF CONTROLLER

V. M. Razin

(Presented by the scientific seminar of the Faculty of Automation and Computer Engineering)

In our work [1] an analysis was given of the operating conditions of a system for automatic compensation of the phase of electron injection into a betatron in the nonrelativistic case.

At present, the effort to increase the charge of the electrons accelerated in a betatron makes it necessary to use high injection voltages, reaching 400 kV and more. At such values of the electron injection voltage it is necessary to take relativistic effects into account when determining the operating conditions of the previously considered [1] system of automatic compensation of the injection phase. In the present work, results are presented for taking relativistic corrections into account and for solving certain other problems.

From the standpoint of modern terminology [2], the aforementioned system of automatic compensation may be regarded as a certain computing device that automatically changes the phase of injection of electrons into the betatron in the presence of external disturbances in the form of random fluctuations of the line voltage.

The problem solved by the computing device can be formulated mathematically as follows.

In the relativistic region of the electron, its energy, the magnetic-field strength in which the electron moves, and the radius of curvature of the electron trajectory are related by the relation [3]:

\[ Hr = \frac{10^4}{3}\sqrt{V(V+2V_0)}, \tag{1} \]

where \(H\) is the magnetic-field strength in oersteds;
\(r\) is the radius of curvature of the trajectory in cm;
\(V\) is the electron energy in megaelectronvolts (MeV);
\(V_0 = 0.511\) MeV is the rest energy of the electron.

When the betatron electromagnet is supplied with current of industrial frequency, the magnetic-field strength in the working zone varies as a function of time according to a sinusoidal law,

\[ H = H_m \sin \omega t, \tag{2} \]

where \(H_m\) is the amplitude of the magnetic-field strength at the radius of the equilibrium orbit, in oersteds;
\(\omega\) is the line frequency in rad/sec;
\(t\) is time in sec.

In writing relation (2), it is assumed that the origin of time is the instant at which the magnetic-field intensity at the radius of the equilibrium orbit passes through zero.

Combining (1) and (2), we obtain

\[ rH_m \sin \omega t=\frac{10^4}{3}\sqrt{V(V+2V_0)}, \tag{3} \]

or, solving equation (3) with respect to the variable \(t\), we have

\[ t=\frac{1}{\omega}\arcsin\left[\frac{10^4}{3rH_m}\sqrt{V(V+2V_0)}\right]. \tag{4} \]

Consequently, in accordance with equation (4), the computing device must solve the following problem.

Special high-accuracy sensors measure, in the preceding acceleration cycle or in several cycles with subsequent averaging, the amplitude of the magnetic-field intensity \(H_m\), the injection voltage \(V\), and the frequency of the supply network \(\omega\), and generate the corresponding signals supplied to the input of the computing device in Fig. 1. The computing device performs calculations according to equation (4) and, in accordance with the calculated value \(t\), generates a command synchronizing pulse at the required instant after the magnetic-field intensity has passed through zero at the beginning of the acceleration cycle.

Fig. 1. Block diagram of the computing device

For relatively small (5–10%) deviations of the network voltage from the nominal value, it appears expedient to use an approximate model of equation (4), technically implemented by the same means as was considered earlier [1].

Let us compile a more complete mathematical description of the controlling computing device than was done in [1], using the same notation as in the cited work [1].

As before, we assume that, at the place where the permalloy sensor is installed, the instant of transition of the magnetic field in the working zone changes according to the law

\[ H_g=H_{m_1}\sin(\omega t-\varphi), \tag{5} \]

where \(H_{m_1}\) is the amplitude of the magnetic-field intensity at the place where the sensor is installed, in oersteds;

\(\varphi\) is the phase shift caused by the demagnetizing action of the copper of the magnetizing coils or by other factors, in rad/sec.

Due to the biasing currents, a magnetic field is created at the same place:

\[ H_n=-\left[H_0-k_gH_{m_1}\cos(\omega t-\varphi_1)\right], \tag{6} \]

where \(H_0\) is the field due to the constant component of biasing, in oersteds;

\(k_gH_{m_1}\) is the field from the variable component;

\(k_g\) is a constant coefficient, with \(|k_g|\ll 1\);

\(\varphi_1\) is a phase shift small in magnitude, in rad/sec, caused by the fact that switching on a large activ-

... resistance does not reduce to zero the phase shift between the current and the voltage in the circuit generating the alternating component of the bias current.

The instant \(t_0\) of generation of the control pulse is determined from the condition \(H_g + H_n = 0\), that is,

\[ H_{m_1}\sin(\omega t_0-\varphi)=H_0-\kappa_g H_{m_1}\cos(\omega t_0-\varphi_1). \tag{7} \]

Let us rewrite equation (7) in the form

\[ t_0=\frac{1}{\omega}\arcsin\left[\frac{H_0}{H_{m_1}}-\kappa_g\cos(\omega t_0-\varphi_1)+\varphi\right]. \tag{8} \]

The solution of the transcendental—in the present case, trigonometric—equation obtained in implicit form with respect to the variable \(t_0\) will determine the instant of actual generation of the control pulse after the magnetic field on the equilibrium orbit passes through zero.

Thus, equation (8) may be regarded as an approximate mathematical model of the principal initial equation (4). Despite the apparent complexity of expression (8), the technical implementation for solving it proves to be much simpler in comparison with the construction of a computing device for realizing function (4).

Let us make permissible simplifications of the basic relations (4) and (8). Taking into account that electron capture in the process of betatron acceleration occurs within \(1\)–\(2\) electrical degrees after the instant at which the magnetic field on the orbit passes through zero, and that the angle \(\varphi_1\) can be made very small—in particular, the equality \(\varphi_1 \cong \omega t_0\) can be achieved—we may, with an error of the order of \(0.03\)–\(0.05\%\), rewrite equations (4) and (8) in the approximate form:

\[ t \cong \frac{1}{\omega}\frac{10^4}{3rH_m}\sqrt{V(V+2V_0)}, \tag{9} \]

\[ t_0 \cong \frac{1}{\omega}\left(\frac{H_0}{H_{m_1}}-\kappa_g+\varphi\right). \tag{10} \]

Let us note that equation (10) coincides with equation (13) in [1].

The conditions for choosing the design parameters \(H_0\) and \(\kappa_g\) were considered by us in [1] for the nonrelativistic case, for several possible methods of disturbance control.

Below we consider several possible methods for determining the parameters \(H_0\) and \(\kappa_g\) in the general case, which also includes the relativistic range of velocities of the electrons injected into the accelerator.

As an example, a method of disturbance control in an injection system is considered, with the charging transformer supplied from a winding located on the betatron magnetic core.

Such a method of supplying the charging transformer is associated with certain complications in the design of the electromagnet; however, it achieves automatic tracking of the transient processes in the electromagnet that arise as a result of random fluctuations of the mains voltage. The presence of automatic tracking leads to the fact that the quantities \(H_m\), \(H_{m_1}\), and \(V\) in equations (9) and (10) vary to some extent consistently, and, in certain transient modes, this ensures sufficiently accurate approximate mathematical modeling of equation (9) by equation (10) in these modes.

In the subsequent exposition, established static modes are meant, when the supply circuits of the charging transformer and the elec-

of the betatron electromagnet are independent, but are connected to one and the same source, which is the simplest to implement in practice.

In the established static regime, for the disturbance-control method under consideration, the following relations hold:

\[ H_{m_1}=\alpha U_c, \tag{11} \]

\[ H_m=\beta U_c, \tag{12} \]

\[ V=\gamma U_c, \tag{13} \]

where \(U_c\) is the nominal effective voltage of the supply network at the corresponding nominal values of \(H_{m_1}\), \(H_m\), and \(V\); \(\alpha,\beta,\gamma\) are proportionality coefficients, that is, constants.

Taking into account the relations (11–13), we write equations (9) and (10) in the form

\[ t \simeq \frac{1}{\omega}\,\frac{k'}{\beta U_c}\sqrt{2\gamma U_c V_0}\, \sqrt{1+\frac{\gamma U_c}{2V_0}}, \tag{14} \]

\[ t_0 \simeq \frac{1}{\omega}\left(\frac{H_0}{\alpha U_c}-\kappa_g+\varphi\right), \tag{15} \]

where the constant \(k'=\dfrac{10^4}{3r}\) depends on the design parameter of the betatron, \(r\), the radius of the equilibrium orbit.

To determine the parameters \(H_0\) and \(\kappa_g\), one may specify, for example, conditions of the following form: require that, at some nominal network voltage \(U_c=U_{c0}\), the equalities

\[ t(U_{c0})=t_0(U_{c0}), \tag{16} \]

\[ \frac{\partial t}{\partial U_c}(U_c=U_{c0}) = \frac{\partial t_0}{\partial U_c}(U_c=U_{c0}) \tag{17} \]

be satisfied.

Solving the system of two equations (16) and (17) with two unknowns \(H_0\) and \(\kappa_g\) gives, as a result,

\[ H_0=\frac{1}{2}\,\frac{k'\alpha}{\beta}\, \frac{\sqrt{2V_0\gamma U_{c0}}} {\sqrt{1+\dfrac{\gamma U_{c0}}{2V_0}}}, \tag{18} \]

\[ \kappa_g=\varphi-\frac{k'}{2\beta U_c}\, \frac{\sqrt{2V_0\gamma U_{c0}}} {\sqrt{1+\dfrac{\gamma U_{c0}}{2V_0}}} \left(1+2\frac{\gamma U_{c0}}{2V_0}\right). \tag{19} \]

In the notation adopted in [1], these parameters have the form:

\[ H_0=\frac{1}{2}\,\omega t_0 H_{m_1} \left(\frac{1}{1+\dfrac{\gamma U_{c0}}{2V_0}}\right), \tag{20} \]

\[ \kappa_g=\varphi-\frac{\omega t_0}{2} \left( \frac{1+2\dfrac{\gamma U_{c0}}{2V_0}} {1+\dfrac{\gamma U_{c0}}{2V_0}} \right). \tag{21} \]

Comparison of the obtained relations (20) and (21) with the corresponding formulas in [1] shows that the expres-

terms in the parentheses of formulas (20) and (21) are, in a certain sense, relativistic corrections to the conditions, obtained in the cited paper [1], for determining the parameters \(H_0\) and \(\kappa_g\).

It should be especially emphasized that, in this case, the values of the parameters \(H_0\) and \(\kappa_g\) are invariant with respect to the frequency \(\omega\) of the supply network, which is a very valuable property of the disturbance-control method under consideration. In practice this means that frequency fluctuations will not affect the adjustment of the controlling computing device.

Fig. 2. Mutual position of the modeled and modeling dependences

The fulfillment of conditions (16) and (17) means that the curve of the required injection phases \(t=t(U_c)\) and the curve of the actual injection phases \(t_0=t_0(U_c)\) produced by the computing control device have the relative position approximately shown in Fig. 2.

To determine the permissible deviations \(U_c\) from the nominal value \(U_{c0}\), one should determine the dependence of the absolute error of the injection phase on fluctuations of the line voltage.

Substituting (18) and (19) into (15), we have

\[ t_0=\frac{1}{\omega}\frac{k'}{2\beta}\, \frac{\sqrt{2V_0\gamma U_{c0}}}{\sqrt{1+\dfrac{\gamma U_{c0}}{2V_0}}} \left[ \frac{1}{U_c} + \frac{1}{U_{c0}} \left(1+2\frac{\gamma U_{c0}}{2V_0}\right) \right]. \tag{22} \]

Now let us find the expression for the absolute error \(\Delta t=t-t_0\), after simple algebraic transformations, in the following form:

\[ \Delta t(U_c)= \frac{k'\sqrt{2\gamma V_0}} {2\omega\beta\sqrt{1+\dfrac{\gamma U_{c0}}{2V_0}}} \left[ \frac{1}{\sqrt{U_c}} \left( 2\sqrt{1+\frac{\gamma U_c}{2V_0}} \sqrt{1+\frac{\gamma U_{c0}}{2V_0}} - \sqrt{\frac{U_{c0}}{U_c}} \right) - \frac{1}{\sqrt{U_{c0}}} \left(1+2\frac{\gamma U_{c0}}{2V_0}\right) \right]. \tag{23} \]

It is not difficult to see that for \(U_c=U_{c0}\) the right-hand side of expression (23) becomes zero, as it should in accordance with condition (16). By specifying a certain permissible value of \(\Delta t\), determined experimentally for each betatron, and solving the quadratic equation (23) with respect to the variable \(U_c\), one can find the permissible limits of variation:

\[ U_{c\min}\leqslant U_c\leqslant U_{c\max}. \tag{24} \]

It should be borne in mind that the exact determination of all quantities in relations (18) and (19) is a very difficult procedure, associated with simultaneous precision measurements of variable and pulsed signals. Furthermore, the exact setting of the calculated values of the parameters \(H_0\) and \(\kappa_g\) is also associated with major technical difficulties. In view of the above, it seems expedient to develop some practical methods of experimental

settings of the computing device after setting the reference values \(H_0\) and \(\kappa_g\), calculated from formulas (18) and (19).

Analysis of equations (14) and (22) and consideration of Fig. 2 show that, for experimental adjustment of the computing device, it is necessary in some way to produce systematic oscillations of the effective variable line voltage \(U_c\) in the neighborhood of the nominal value \(U_{c0}\) and to select the mutual ratio of the parameters \(H_0\) and \(\kappa_g\) in such a way that radiation appears only when \(U_c = U_{c0}\).

In order to widen the permissible limits of oscillation of the voltage \(U_c\) in the neighborhood of the nominal value \(U_{c0}\), one may somewhat modify the mutual arrangement of the dependences \(t(U_{c0})\) and \(t_0(U_c)\) in accordance with Fig. 3.

Such an arrangement of the dependences in the object and in the controlling model ensures a uniform approach of the modeled and modeling dependences to one another; moreover, in the case \(U_c = U_{c0}\), the maximum deviation does not exceed a certain absolute value of the admissible error \(\delta\).

In this case the conditions for determining the parameters \(H_0\) and \(\kappa_g\) must be specified by the equalities:

\[ t(U_{c0}) = t_0(U_{c0}) + \delta, \tag{25} \]

\[ \frac{\partial t}{\partial U_c}(U_c = U_{c0}) = \frac{\partial t_0}{\partial U_c}(U_c = U_{c0}). \tag{26} \]

Substituting (14) and (15) into (25) and (26), differentiating, and solving the resulting equations with respect to \(H'_0\) and \(\kappa'_g\), we obtain for \(H'_0\) exactly the same expression as in formula (18), while for the parameter \(\kappa'_g\) we obtain

\[ \kappa'_g = \kappa_g + \omega\delta, \]

\[ H'_0 = H_0. \tag{27} \]

It is now necessary to determine those values of \(U_c\) for which the error \(\Delta t(U_c)=t(U_c)-t_0(U_c)\) becomes zero. For this it is necessary to solve, with respect to \(U_c\), a quadratic equation analogous to expression (23)

\[ \frac{\kappa'}{2\omega\beta} \frac{\sqrt{2\gamma V_0}}{\sqrt{1+\dfrac{\gamma U_{c0}}{2V_0}}} \left[ \frac{1}{\sqrt{U_c}} \left( 2\left[ \sqrt{1+\frac{\gamma U_c}{2V_0}} \sqrt{1+\frac{\gamma U_{c0}}{2V_0}} \sqrt{\frac{U_{c0}}{U_c}} \right] \right) - \frac{1}{\sqrt{U_{c0}}} \left( 1+2\frac{\gamma U_{c0}}{2V_0} \right) \right] = -\delta . \tag{28} \]

In general form, the expressions for the roots of equation (28) are very cumbersome; therefore it should be recommended that equation (28) be solved with the aid of electronic computers performing calculations to an accuracy of up to 6 decimal places. Such accuracy is necessary because the very small right-hand side is obtained as a result of subtracting large quantities on the left-hand side of equation (28). Approximate values of the roots of equation (28) in general form can be obtained as follows. Let

\[ U_c = U_{c0} + \Delta U_c \]

and expand in a Taylor series the first two terms of the sum in the square brackets on the left-hand side of equation (28):

\[ \frac{1}{\sqrt{U_c}}\left(2\sqrt{1+\frac{\gamma U_c}{2V_0}}\sqrt{1+\frac{\gamma U_{c0}}{2V_0}}-\sqrt{\frac{U_{c0}}{U_c}}\right) = f(U_c)=f(U_{c0})+ \]

\[ + f'(U_{c0})\Delta U_c+f''(U_{c0})\frac{(\Delta U_c)^2}{2!}+\ldots \]

Retaining the first three terms in the expansion and substituting them into the left-hand side of equation (28), we find the approximate values:

\[ \frac{\Delta U_c}{U_{c0}} = \pm \sqrt{ \frac{\delta}{A} \left(1+\frac{\gamma U_{c0}}{2V_0}\right) \sqrt{U_{c0}} }, \tag{29} \]

where

\[ A= \frac{ \kappa'\sqrt{2\gamma V_0} }{ 2\omega\beta\sqrt{1+\frac{\gamma U_{c0}}{2V_0}} }. \tag{30} \]

Taking into account the notation (11–14), expression (29) can be rewritten in the form

\[ \frac{\Delta U_c}{U_{c0}} = \pm 2 \sqrt{ \frac{6\delta\omega H_m r}{10^4\sqrt{2VV_0}} \sqrt{\left(1+\frac{V}{2V_0}\right)^3} }. \tag{30'} \]

The found values of \(\Delta U_c\) can be used in the experimental adjustment of the computing device. In this case, under slow variations of \(U_c\) in the neighborhood of \(U_{c0}\), the ratios of the parameters \(H_0\) and \(k_g\) should be selected so that two radiation maxima are observed at \(U'_{c\min}=U_{c0}-\Delta U_c\) and \(U'_{c\max}=U_{c0}+\Delta U_c\).

The permissible deviations of the voltage \(U_c\) beyond \(U'_{c\max}\) and \(U'_{c\min}\) can evidently be found from the relation

\[ \frac{\Delta U'_c}{V_{c0}} = \pm 2 \times \]

\[ \times \sqrt{ \frac{2\delta}{A} \left(1+\frac{\gamma U_{c0}}{2V_0}\right) \sqrt{U_{c0}} }. \tag{31} \]

Fig. 3. Another variant of the relative arrangement of the modeled and modeling dependences.

Another possible approach to determining the conditions for selecting the parameters \(H_0\) and \(K_g\) is when we predetermine in advance the magnitude of the deviation \(\pm\Delta U_c\) from the nominal value at which the error \(\Delta t\) becomes zero.

To simplify the notation of the subsequent mathematical calculations, we introduce the following designations:

\[ a=\frac{1}{\omega}\frac{\kappa'}{\beta}\sqrt{2\gamma V_0}, \]

\[ b=\frac{\gamma}{2V_0}, \]

\[ U_{c0}=x_0, \]

\[ \begin{gathered} U_c=x,\\ \Delta U_c=\Delta x,\\ c=\frac{H_0}{a\omega},\\ d=\frac{\varphi_0-\kappa_g}{\omega}. \end{gathered} \tag{32} \]

Then the function being modeled will have the form

\[ t(x)=a\sqrt{\,b+\frac{1}{x}\,}, \tag{33} \]

and the modeling function will be

\[ t_0(x)=\frac{c}{x}+d. \tag{34} \]

The condition formulated above for determining the parameters \(c\) and \(d\) will be written as follows:

\[ \begin{gathered} a\sqrt{\,b+\frac{1}{x_0+\Delta x}\,}=\frac{c}{x_0+\Delta x}+d,\\ a\sqrt{\,b+\frac{1}{x_0-\Delta x}\,}=\frac{c}{x_0-\Delta x}+d. \end{gathered} \tag{35} \]

Solving the system of equations (35) gives the values of \(c\) and \(d\) in the following form:

\[ c=\frac{x_0^2-(\Delta x)^2}{2\Delta x}\,a\left(\sqrt{\,b+\frac{1}{x_0-\Delta x}\,}-\sqrt{\,b+\frac{1}{x_0+\Delta x}\,}\right), \tag{36} \]

\[ d=\frac{a}{2\Delta x}\left[(x_0+\Delta x)\sqrt{\,b+\frac{1}{x_0+\Delta x}\,}-(x_0-\Delta x)\sqrt{\,b+\frac{1}{x_0-\Delta x_0}\,}\right]. \tag{37} \]

Application of conditions (16) and (17) to equations (33) and (34) in the new notation gives

\[ c_1=\frac{1}{2}\frac{a}{\sqrt{\,b+\frac{1}{x_0}\,}}, \tag{38} \]

\[ d_1=\frac{a}{2x_0}\frac{1+2bx_0}{\sqrt{\,b+\frac{1}{x_0}\,}}. \tag{39} \]

By means of simple calculations it is not difficult to verify that

\[ \begin{gathered} \lim_{\Delta x\to 0} c=c_1,\\ \lim_{\Delta x\to 0} d=d_1, \end{gathered} \]

that is, conditions (16) and (17) are the limiting case of conditions (35) as \(\Delta x\to 0\). Recall that, when conditions (35) are prescribed, the modeled and modeling functions intersect at the points \(x_{\min}=x_0-\Delta x\) and \(x_{\max}=x_0+\Delta x\), whereas when conditions (16) and (17) are fulfilled, these functions are tangent at the point \(x=x_0\).

If it is required to ensure the best approximation of the modeled and modeling functions over a prescribed sufficiently wide range of variation of \(x\), from \(x=x_1\) to \(x=x_2\), then one should use the method of integral quadratic approximation of functions on an interval, i.e., the method of least squares [4]. For this it is necessary to determine the values of the quantities \(c\) and \(d\) that ensure the minimum of the integral:

\[ I=\int_{x_1}^{x_2} [t(x)-t_0(x)]^2\,dx = \int_{x_1}^{x_2} \left[ a\sqrt{b+\frac{1}{x}}-\left(\frac{c}{x}+d\right) \right]^2 dx. \tag{40} \]

The conditions for ensuring the minimum of the integral \(I\) are written in the form of a system of equations:

\[ \frac{1}{2}\frac{\partial I}{\partial c} = \int_{x_1}^{x_2} \left[ a\sqrt{b+\frac{1}{x}}-\left(\frac{c}{x}+d\right) \right]\frac{1}{x}\,dx =0, \]

\[ \frac{1}{2}\frac{\partial I}{\partial d} = \int_{x_1}^{x_2} \left[ a\sqrt{b+\frac{1}{x}}-\left(\frac{c}{x}+d\right) \right]dx =0. \tag{41} \]

Thus, in the present case the solution of the problem reduces to calculating

\[ I_1=\int_{x_1}^{x_2}\frac{a}{x}\sqrt{b+\frac{1}{x}}\,dx = -a\left. \left[ 2\sqrt{b+\frac{1}{x}} -\sqrt{b}\ln \frac{ \sqrt{b+\frac{1}{x}}-\sqrt{b} }{ \sqrt{b+\frac{1}{x}}+\sqrt{b} } \right]\right|_{x_1}^{x_2}, \]

\[ I_2=\int_{x_1}^{x_2} a\sqrt{b+\frac{1}{x}}\,dx = a\sqrt{b}\left. \left[ \sqrt{\left(x+\frac{1}{b}\right)x} +\frac{1}{b}\ln \left( \sqrt{x+\frac{1}{b}}+\sqrt{x} \right) \right]\right|_{x_1}^{x_2}. \]

and other tabulated integrals.

It is expedient to solve the system of equations (41), in view of the cumbersomeness of the computations, on electronic calculating machines.

After calculating the values of \(c\) and \(d\) from the system of equations (41), it is not difficult to obtain the required values of the quantities \(H_0\) and \(\kappa_g\) on the basis of relations (32), and to adjust the controlling computing device accordingly.

It should be borne in mind that, when determining the parameters \(H_0\) and \(\kappa_g\) in the last two cases, at \(U_c=U_{co}\) and in its neighborhood the admissible minimum phase error \(\Delta t\leq \delta\) is not ensured; that is, the actual value of the error may be greater than \(\delta\). In practice this will manifest itself in the fact that the radiation maximum will occur at the points \(U_{c1}=U_{co}-\Delta U_c\) and \(U_{c2}=U_{co}+\Delta U_c\), while at \(U_c=U_{co}\) and in the nearest neighborhood there may be observed a substantial decrease in the number of electrons captured into acceleration and, as a consequence, a sharp decrease in the dose power of betatron radiation. In these cases it is necessary, along with the system of automatic disturbance control, to use systems of automatic control by deviation, that is,

to employ combined systems of automatic regulation and control. In a combined control system for the electron-injection phase, the disturbance controller will provide high speed of response and create more favorable operating conditions for the deviation controller over a wide range of variation in the line voltage. In this case, the deviation controller will ensure the corresponding accuracy in maintaining the prescribed level of radiation-dose power, that is, it will stabilize this parameter independently of the nature of the external disturbing actions.

In conclusion, we briefly formulate the results obtained.

1. A relatively accurate mathematical description has been compiled of the operating conditions of an automatic disturbance-control system for the electron-injection phase in a betatron.

2. Relativistic corrections have been found for determining the parameters of the disturbance controller, making it possible to tune the automatic computing device at any injection voltage.

3. Several methods have been considered for calculating the principal parameters of the disturbance controller, including uniform approximation of the approximating function and the method of least squares.

4. It has been established that the most convenient method for calculations and practical application is the method of uniform approximation.

REFERENCES

1. V. M. Razin. Stabilization of betatron radiation. Izvestiya TPI, vol. 87, 1957.

2. A. A. Feldbaum. Computing Devices in Automatic Systems. Fizmatgiz, 1959.

3. A. P. Grinberg. Methods for Accelerating Charged Particles. GITTL, 1950.

4. B. P. Demidovich, I. A. Maron, A. Z. Shuvalova. Numerical Methods of Analysis. Fizmatgiz, 1962.