ANALYSIS OF THE INJECTION SYSTEM IN A BETATRON USING MULTIVARIABLE CONTROL THEORY

V. M. Razin

(Presented at the scientific seminar of the Department of Computer Engineering)

In [1, 2] we considered the systems we developed for stabilizing the injection pulse voltage and the emission current of the injector cathode, or the mean value of the electron current to the conducting layer of the betatron vacuum chamber. As already noted, these two systems form a single system—in this case, a two-variable control system—since the injection voltage affects the emission current, and the emission current, being the load for the injection pulse voltage, in turn affects the amplitude of this voltage. An increase in the injector current leads to a corresponding decrease in the amplitude of the pulse voltage at the injector. Further, as can be seen from the diagrams in [1, 2], both stabilization systems are under the action of an external disturbance in the form of random changes in the mains voltage. Analysis of the physical processes in these circuits leads, if the corresponding nonlinear dependences are linearized, to the general diagram shown in Fig. 1.

In this diagram, the number 1 denotes the element controlling the magnitude of the emission current; block 3 represents the dynamic properties of the filter at the input of measuring element 1; block 4 corresponds to the magnetic amplifier; block 5 takes into account the thermal inertia of the injector cathode; block 6, with the corresponding weight, reproduces the external disturbance \(U_c\). The element controlling the injection voltage is denoted by the number 2; the dynamic properties of the filtering and amplification circuits are shown here by blocks 7 and 8; the dynamic properties of the magnetic amplifier correspond to block 9; the action of the external disturbance \(U_c\) with weight coefficient \(K_8\) is represented by block 10. The effect of the injection voltage \(U\) on the emission current \(I_\mathrm{e}\) in the linear approximation is represented by the relation

\[ I_\mathrm{e} = I_{\mathrm{e}0} + K_{10}\Delta U_i + K'\Delta T, \tag{1} \]

where

\[ \begin{aligned} I_{\mathrm{e}0} &\text{— initial value of the emission current;}\\ \Delta U_i &\text{— increment of the injection voltage;}\\ \Delta T &\text{— increment of the cathode temperature;}\\ K_{10}\ \text{and}\ K' &\text{— constant coefficients.} \end{aligned} \]

Equation (1) is modeled by summing element 11 and block 13.

We shall represent the reverse effect of the injector emission current on the injection voltage in the form of the equation

\[ U_i = U_{i0} + K''\Delta U - K_9\Delta I_\mathrm{e}, \tag{2} \]

Fig. 1. Diagram of two-loop control of the electron injection system.

where \(U_{i0}\) is the initial value of the injection voltage;
\(\Delta I_{\mathrm{e}}\) is the increment of the emission current;
\(\Delta U\) is the increment of the voltage at the output of the magnetic amplifier;
\(K''\) and \(K_{0}\) are constant coefficients.

Equation (2) is modeled by block 14 and summing element 12.

For the purpose of analyzing the properties of the circuit in Fig. 1 and optimizing its parameters, we use the theory of multivariable control systems [3].

Applying the widely known methods for composing the dynamic equations of automatic-control and regulation systems in operator form [3], taking into account the notation of the circuit in Fig. 1, we obtain:

\[ I_{\mathrm{e}}(p)\left[ \prod_{i=1}^{3} K_i + \prod_{i=1}^{3} (1+pT_i) \right] - U_i(p)K_{10}\prod_{i=1}^{3}(1+pT_i) = K_2K_3(1+pT_1)I_{\mathrm{e,ref}}(p) + (1+pT_1)(1+pT_2)K_3K_7U_c(p), \tag{3} \]

\[ U_i(p)\left[ \prod_{i=4}^{6} K_i+\prod_{i=4}^{6}(1+pT_i) \right] + I_{\mathrm{e}}(p)K_0\prod_{i=4}^{6}(1+pT_i) = K_6(1+pT_4)(1+pT_5)U_{i,\mathrm{ref}}(p) + K_8\prod_{i=4}^{6}(1+pT_i)U_c(p). \]

Using the notation adopted in [3], we write the system of equations (3) in matrix form

\[ A\gamma = K_{\mathrm{ob}}\gamma_{\mathrm{ref}} + DF, \tag{4} \]

where

\[ A= \left\| \begin{array}{cc} \displaystyle \prod_{i=1}^{3}K_i+\prod_{i=1}^{3}(1+pT_i),& \displaystyle -K_{10}\prod_{i=1}^{3}(1+pT_i) \\[1.2em] \displaystyle K_0\prod_{i=4}^{6}(1+pT_i),& \displaystyle \prod_{i=4}^{6}K_i+\prod_{i=4}^{6}(1+pT_i) \end{array} \right\|, \tag{5} \]

\[ \gamma= \left\| \begin{array}{c} I_{\mathrm{e}}(p)\\ U_i(p) \end{array} \right\|; \qquad K_{\mathrm{ob}}\gamma_{\mathrm{ref}}= \left\| \begin{array}{c} K_2K_3(1+pT_1)I_{\mathrm{e,ref}}(p)\\ K_6(1+pT_4)(1+pT_5)U_{i,\mathrm{ref}}(p) \end{array} \right\|; \]

\[ DF= \left\| \begin{array}{c} (1+pT_1)(1+pT_2)K_3K_7U_c(p)\\ \displaystyle K_8\prod_{i=4}^{6}(1+pT_i)U_c(p) \end{array} \right\|. \]

The system of equations with respect to the controlled quantities in matrix form has the form

\[ \gamma=A^{-1}\left[K_{\mathrm{ob}}\gamma_{\mathrm{ref}}+DF\right], \tag{6} \]

where the matrix inverse to the matrix \(A\), calculated by known methods [4], is

\[ A^{-1}=\frac{1}{\Delta} \left\| \begin{array}{cc} \displaystyle \prod_{i=4}^{6}K_i+\prod_{i=4}^{6}(1+pT_i),& \displaystyle K_{10}\prod_{i=1}^{3}(1+pT_i) \\[1.2em] \displaystyle K_0\prod_{i=4}^{6}(1+pT_i),& \displaystyle \prod_{i=1}^{3}K_i+\prod_{i=1}^{3}(1+pT_i) \end{array} \right\|. \]

Here the determinant of the system of equations (3) is

\[ \Delta = \begin{vmatrix} \displaystyle \prod_{i=1}^{3} K_i+\prod_{i=1}^{3}(1+pT_i),& \displaystyle K_{10}\prod_{i=1}^{3}(1+pT_i) \\[1.2em] \displaystyle K_9\prod_{i=4}^{6}(1+pT_i),& \displaystyle \prod_{i=4}^{6}K_i+\prod_{i=4}^{6}(1+pT_i) \end{vmatrix}. \]

Substituting (8) into (7), and (7) into (6), taking into account the notations for \(T\), \(K_{\mathrm{ob}}\), \(Y_{\mathrm{et}}\), and \(DF\), we obtain equations for the controlled quantities in the form

\[ I_{\mathrm{e}}(p)= \frac{1}{\Delta} \left\{ \left[ \prod_{i=4}^{6}K_i+\prod_{i=4}^{6}(1+pT_i) \right] \left[ K_2K_3(1+pT_1)I_{\mathrm{et}}(p)+ \right. \right. \]
\[ \left. +(1+pT_1)(1+pT_2)K_3K_7U_c(p) +K_{10}\prod_{i=1}^{3}(1+pT_i) \left[ K_6(1+pT_4)(1+ \right. \right. \]
\[ \left. \left. +pT_5)U_{i\mathrm{et}}(p) +K_8\prod_{i=4}^{6}(1+pT_i)U_c \right] \right\}, \tag{9} \]

\[ U_i(p)= \frac{1}{\Delta} \left\{ K_9\prod_{i=4}^{6}(1+pT_i) \left[ K_2K_3(1+pT_1)I_{\mathrm{et}}(p)+ \right. \right. \]
\[ \left. \left. +(1+pT_1)(1+pT_2)K_3K_7U_c \right] + \left[ \prod_{i=1}^{3}K_i+\prod_{i=1}^{3}(1+pT_i) \right] \left[ K_6(1+pT_4)(1+ \right. \right. \]
\[ \left. \left. +pT_5)U_{i\mathrm{et}}(p) +K_8\prod_{i=4}^{6}(1+pT_i)U_c(p) \right] \right\}. \tag{10} \]

The characteristic equation of the two-coupled control system under consideration will be

\[ \Delta= \left[ \prod_{i=1}^{3}K_i+\prod_{i=1}^{3}(1+pT_i) \right] \left[ \prod_{i=4}^{6}K_i+\prod_{i=4}^{6}(1+pT_i) \right] + \]
\[ +K_9K_{10}\prod_{i=4}^{6}(1+pT_i) = \prod_{i=1}^{6}K_i+ \prod_{i=1}^{3}K_i\prod_{i=4}^{6}(1+pT_i)+ \]
\[ +\prod_{i=4}^{6}K_i\prod_{i=1}^{3}(1+pT_i) + (1+K_9K_{10})\prod_{i=1}^{6}(1+pT_i). \tag{11} \]

The obtained relations make it possible: 1) in the steady-state regime, to find the generalized matrix of error coefficients and minimize their magnitude; 2) to investigate the system for stability and evaluate the quality of the control process; 3) to determine the basic properties of automatic-control systems from the \(D\)-partition curve and to solve a number of other practically important problems.

The formulas presented above were obtained under the assumption that all quantities vary continuously. In fact, because of the pulsed action of the injection circuit, the two-coupled control system under consideration should be regarded as pulsed; for this purpose it is necessary, in the circuit of Fig. 1, to provide pulse elements \(K_1\) and \(K_2\), closing synchronously with a frequency equal to the frequency of the supply network. However, taking into account that the time constants of the inertial elements in this control system

turn out to be considerably greater in magnitude than the duration of the switching period of the pulsed elements, the methods of investigation developed for continuous-action systems can be applied to the system [5]. In this case the input filters should be regarded, in a first approximation, as zero-order memory elements, i.e., in the relations obtained one should set \(T_1 = T_4 = 0\). As for the disturbing action of the line voltage \(U_c(p)\), the Laplace transform of this quantity should be formed with respect to the envelope of the amplitude of the alternating voltage.

The theoretical investigations carried out in [1] and their comparison with experimental results showed discrepancies of up to 30–50% in such parameters as the overall gain coefficient, the duration of the transient process, the magnitude of the overshoot, etc., which is explained by the presence of substantial nonlinearities in the system under consideration. On the basis of the foregoing, one should conclude that the relations obtained can be used and can yield useful practical results; however, a certain caution must be exercised, since, owing to the presence of substantial nonlinearities in the system, there is always a danger of obtaining qualitatively incorrect results. In conclusion, we note that the analysis of the betatron injection system that we have carried out also has independent significance, since systems of this kind are used in many types of charged-particle accelerators and also have independent applications.

REFERENCES

1. V. M. Razin. Izv. TPI, vol. 87, p. 201, 1957.
2. V. M. Razin. Izv. TPI, vol. 87, p. 216, 1957.
3. M. V. Meerov. Systems of Multivariable Control. Moscow, Nauka, 1965.
4. F. R. Gantmakher. Theory of Matrices. Moscow, Nauka, 1966.
5. A. A. Krasovskii, G. S. Pospelov. Fundamentals of Automation and Engineering Cybernetics. GEI, Moscow–Leningrad, 1962.