STUDY OF A CLOSED-ORBIT ANALYZER BASED ON ELEMENTS OF UNIVERSAL ANALOG COMPUTERS

V. A. Kocheturov, E. E. Ivanchenkova

(Presented by the seminar of the Department of PM)

Control of closed orbits is of great importance in high-energy accelerators, since various perturbations of the magnetic field cause distortions of closed orbits and lead to a decrease in the intensity of the beam of accelerated particles. The difficulties of analytical methods for analyzing closed orbits consist in the fact that small errors in coordinate measurements can lead to divergent solutions when determining perturbations of the magnetic field. Such a problem belongs to the class of “ill-posed problems,” whose numerical solution is based on the application of various smoothing methods. This article considers issues in the development and investigation of an analog computing device which, from the results of measurements of the parameters of a closed orbit, makes it possible to localize dipole perturbations of the magnetic field, which may arise, in particular, due to displacement of periodic elements of the magnetic system from the ideal orbit. According to the theory of “smoothed” closed orbits, the set of true particle trajectories is replaced by the averaged trajectory of the center of the beam along a certain equivalent circle. The differential equation of the “smoothed” motion has the form:

\[ \frac{d^{2}x}{d\vartheta^{2}} + Q^{2}x(\vartheta)=\delta(\vartheta), \tag{1} \]

where \(x\) is the coordinate of the transverse (radial or vertical) oscillation;

\[ \begin{aligned} \vartheta &\text{ is the azimuthal angle;}\\ Q &\text{ is the number of free oscillations per 1 revolution;}\\ \delta(\vartheta) &\text{ is the perturbation of the magnetic field.} \end{aligned} \]

For

\[ \delta(\vartheta)= \begin{cases} k, & \text{for } \vartheta_{1}<\vartheta_{2},\\ 0, & \text{everywhere,} \end{cases} \]

the solution of differential equation (1) will be:

\[ x(\vartheta)=c_{1}\cos Q\vartheta+c_{2}\sin Q\vartheta+\frac{1}{Q^{2}}\delta(\vartheta)(1-\cos Q\vartheta). \tag{2} \]

Thus, the coordinates of the closed orbit are harmonic oscillations with superimposed perturbations proportional to \(\delta(\nu)\).

Measurement of the closed orbit in accelerators with strong focusing is carried out by means of discrete sensors arranged in a definite manner along the equilibrium orbit. Therefore, a computing device for closed-orbit analysis must contain some interpolating filter, whose input receives information in discrete form on the parameters of the smoothed closed orbit, and a computational filter that removes from the signal proportional to the coordinates of the beam orbit the oscillations determined by the number \(Q\).

Fig. 1. Block diagram of a closed system for analyzing and compensating magnetic-field perturbations.

\(A\) — closed-orbit analyzer;
\(IF\) — interpolating filter;
\(RU\) — recording device (indicator);
\(K\) — device correcting the perturbation of the magnetic field

The block diagram of the closed system for analyzing and compensating the perturbation is given in Fig. 1.

1. Interpolating Filter

In solving interpolation problems, the proper choice of the sampling rate of the closed-orbit coordinates (the placement of sensors along the orbit) is of great importance. The discretization interval is established by Kotelnikov’s theorem, according to which any function \(f(t)\), consisting of frequencies from 0 to \(f_c\) (the highest harmonic of the signal), can be represented with sufficient accuracy by means of numbers taken at intervals

\[ \Delta t=\frac{1}{2f_c}\ \text{sec}. \]

In accordance with the theorem, the ideal reconstructing filter is one having the impulse transient response

\[ g(t)=\frac{\sin \omega_c(t-k\Delta t)}{\omega_c(t-k\Delta t)}. \]

Because of the physical unrealizability of such a filter, in practice a number of circuits are used that have approximate impulse transient responses. For our study we selected a filter with the transfer function [1]:

\[ W(p)=\frac{1}{\Delta t}\cdot\frac{1-e^{-p\Delta t}}{p}. \tag{3} \]

Its logarithmic amplitude-frequency and phase-frequency characteristics have the form:

\[ L(\omega)=20\lg \Delta t \left( \frac{\sin \frac{\omega \Delta t}{2}} {\frac{\omega \Delta t}{2}} \right),\ \text{dB}, \]

\[ \varphi=-\operatorname{arctg}\,2 \frac{\sin^2 \frac{\omega \Delta t}{2}} {\sin \omega \Delta t},\ \text{deg}. \tag{4} \]

The filter has a sufficiently linear phase-frequency characteristic, but introduces a delay by the time \(\Delta t\).

The accuracy of reconstruction of the sinusoid is \(1, 2, 5\%\) for the number of samples per period \(N=22;\ 16;\ 10\).

2. Computational Filter

Its transfer function can be obtained from equation (1) and has the form:

\[ W(p)=\frac{p^2+\omega_0^2}{(p+\omega_1)^2}. \tag{5} \]

The filter is characterized by the presence of two zeros on the imaginary axis \((p_{1,2}=\pm \omega_0)\) and a double damping pole located on the real axis \((P_{1,2}=\pm \omega_1)\). The logarithmic frequency characteristics of the computer have the form

\[ L(\omega)=20\lg \frac{ \frac{\omega_0^2}{\omega_1^2} \left|1-\frac{\omega^2}{\omega_0^2}\right| }{ 1+\frac{\omega^2}{\omega_1^2} },\ \text{dB}. \]

\[ \varphi=-\operatorname{arctg} \frac{\frac{2\omega}{\omega_1}} {1-\frac{\omega^2}{\omega_1^2}},\ \text{deg}. \tag{6} \]

The block diagram of the analyzer, containing the interpolating filter IF and the computer CF, is given in Fig. 2. The investigations were carried out for different values of the damping poles \(\omega_1\) of the transfer function of the computational filter tuned to the frequency of free oscillations \(\omega_0\). In this case the closed orbit was implemented on analog-computer blocks in accordance with equation (1). The magnetic-field perturbations \(\delta\) were represented on the variable-coefficient block with stepwise approximation. The independent variable \(v\) is represented in the model by machine time \(\tau_m\), while the perturbation \(\delta_{(\omega)}\) and the dependent variable \(x\) are represented by voltage.

The relation between the computer variables and the closed orbit is described by the equations:

\[ \tau_{\mathrm{M}} = m_{\nu}\cdot \nu,\qquad Q^{2} = \omega_{0}^{2} m_{\tau}^{2}, \text{ where} \]

\[ U_{\delta} = m_{\delta}\cdot \delta(\nu),\qquad \omega_{0}^{2} = \alpha_{5} a_{7} a_{10} k_{41} k_{51} k_{62}\,\frac{m_{x}}{m_{\delta}}, \]

\[ U_{x} = m_{x}\cdot x(\nu). \]

Fig. 2.

Fig. 3.

It was found that as the pole \(\omega_1\) is increased, the sensitivity of the computer decreases. At the same time, the phase shifts also decrease; these make it possible to judge the accuracy of localization of magnetic-field perturbations. The choice of the optimal value of the damping pole \(\omega_1\) is determined by the requirements on the accuracy of localization and on the sensitivity of the system. In particular, when the parameters of the magnetic system of the Yerevan synchrotron [2] were used as the initial parameters—\(Q=5.4\); \(n=48\), the number of pairs of periodic elements; \(\Delta x=0.5\) mm, the permissible transverse displacement of a block; \(A=20\) mm, the permissible amplitude of betatron oscillations; and \(\tau_m=50\) sec—it turned out that at the output of the computer filter it is possible to determine the displacement of a magnet sector to within 0.02 mm, which is 4% of the permissible value. The phase shift was equal to 0.013 sec for a perturbation duration \(\Delta \tau=0.5\) sec, and the magnitude of the output signal depended linearly on the magnitude of the perturbation. Figure 3 shows oscillograms of the unperturbed orbit (a), the perturbed orbit (b), the real perturbation (c), and the perturbation at the computer output (d).

Thus, the investigations carried out on the analog computer of the closed-orbit analyzer showed that, with a simple hardware implementation, it can be successfully used for continuous monitoring of closed orbits and for automatic compensation of perturbations, provided that there is a corresponding number of sensors measuring, at discrete points, the coordinates of the closed orbit.

REFERENCES

1. Hulsman. Active Filters. Moscow, 1972.

2. Proceedings of the International Conference on Accelerators. Edited by A. A. Kolomenskii, A. B. Kuznetsov, and A. N. Lebedev. Moscow, Atomizdat, 1964.