Electronic Model of the Particle Motion Process in the Injection Line of Cyclic Accelerators
Abstract Generated abstract
This paper describes the use of analog electronic modeling to calculate and tune particle trajectories in injection channels of cyclic accelerators, where beam-shaping systems can require repeated solution of differential equations across multiple optical sections. Using the MNV-1 or MNB-1 analog machine with a specially developed programming device, the authors model channels containing quadrupole lenses, correctors, compensating capacitors, and external magnetic fields, including calculation both along the beam direction and in reverse from specified output beam parameters. The approach is presented as a practical means of selecting lens and corrector parameters, estimating required input beam dimensions and divergence, and assisting accelerator magnetic-field adjustment, including harmonic analysis of azimuthal field distributions and trajectory calculation on initial turns.
Full Text
ELECTRONIC MODEL OF THE PROCESS OF PARTICLE MOTION IN THE INJECTION CHANNEL OF CYCLIC ACCELERATORS
V. A. KOCHEGUROV, N. V. TRIKHANOVA, P. M. SHCHANIN
(Presented by the scientific seminar of the Faculty of Automation and Computer Engineering)
In high-energy cyclic accelerators, as a rule, external injection of particles is used. In this case the particles are accelerated to the injection energy by an additional injector accelerator, and then are introduced along the channel into the accelerating chamber. Certain requirements are usually imposed on the parameters of the injected particle beam; these are not always satisfied by the particles at the exit of the injector accelerator. Therefore, on the path of the particles from the injector to the accelerator, various shaping systems are installed: quadrupole lenses, magnetic correctors, deflecting magnetic and electrostatic devices. An analytical calculation of such systems is possible in the general case, but requires cumbersome computations.
For the calculation and selection of the optimal scheme for shaping and adjustment of the electron beam, we used an analog mathematical machine, MNV-1.
As an example, let us consider a method for calculating an injection channel intended for transporting an electron beam, consisting of two pairs of quadrupole lenses (sections 1, 3, 5, 7, Fig. 1), two correcting lenses (sections 9, 11), and three electric capacitors compensating for the influence of the external magnetic field (section 14). In the intervals between the lenses, the electrons are deflected by the external magnetic field \(B_z(y)\). The dependence of the magnetic-field intensity \(B_z\) on the coordinate \(y\) is shown in Fig. 2.
Fig. 1. Injection channel of 14 sections.
The initial data for solving the problem are the particle energy, the transverse dimensions, and the angular divergence of the electron beam at the entrance to the injection channel. It is required to determine the principal parameters of the shaping system that will provide, at the channel exit, a beam with the specified transverse cross section and angular divergence.
Let us choose a coordinate system in which the \(y\) axis coincides with the direction of motion of the electrons. The equations of the electron trajectory in li-
linear approximation in the vertical plane \(xy\), for sections \(1 \div 14\) have the form:
Fig. 2. Dependence of \(B_z/B_{z\max}\) on \(y\)
| Section | Equation |
|---|---|
| \(1,3,5,7\) | \[\frac{d^2x}{dy^2}+kx=0,\tag{1}\] where \(k=k_1,-k_2,+k_3,-k_4\) in sections \(1,3,5,7\), respectively |
| \(2,4,6,8,10,12\) | \[\frac{d^2x}{dy^2}-a=0.\tag{2}\] |
| \(9,11\) | \[\frac{d^2x}{dy^2}+b=0,\] where \(b=b_1\) in section 9, \(b=b_2\) in section 11. \(\tag{3}\) |
| \(13\) | \[\frac{d^2x}{dy^2}+Aa(y)=0.\tag{4}\] |
| \(14\) | \[\frac{d^2x}{dy^2}-[Bb(y)-Aa(y)]=0,\tag{5}\] where \(A\) and \(B\) are const |
The choice of the coefficients \(k_1 \div k_4\), \(b_1\), \(b_2\) from the conditions imposed on the beam can be made by solving successively the differential equations for sections \(1 \div 14\) for some set of coefficients. If unsatisfactory values of the coordinate \(x\) and of the derivative \(\frac{dx}{dy}\) are obtained at the exit of the channel, it is necessary to change the coefficients \(k_1 \div k_4\), \(b_1\), \(b_2\) and solve the equations for the new set of coefficients; such a calculation process must be continued until the required
solutions. Equations (1)–(5) have an analytical solution, but since for each set of coefficients it is necessary to solve 14 differential equations with different initial conditions, the process of selecting the coefficients is laborious.
To solve equations (1)–(5) we used the MNB-1 electronic modeling installation. Since, when using the modeling installation to solve the equation on each section, it is necessary to assemble its own structural circuit and set the initial conditions, a special programming device was developed for the MNB-1 model. The structural diagram of the model is shown in Fig. 3. The voltage \(U_2\) at point \(B\) is proportional to the displacement of the electron along the axis; the voltage at point \(A\), to the change in the angle of inclination of the electron trajectory. With contacts \(4P_1\) and \(3P_1\) of relay \(P_1\) closed, a circuit is created for solving equation (1); with contact \(3P_2\) of relay \(P_2\) closed, a circuit is formed for solving equation (2); with contacts \(4P_3\) and \(3P_3\) closed, we have a circuit for solving equation (1), and so on.
Fig. 3. Structural diagram for an injection channel consisting of 14 sections.
Control of the relay operation is carried out by a step selector. The step selector of the variable-coefficient block of type SB-3 of the IPT-5 model is used. A pulsed voltage of \(\pm 26\) V with a frequency of 1 Hz is applied to the winding of the step selector. A voltage of \(\pm 26\) V is applied to the selector wiper.
To construct a programming device designed for 400 sec of operation, four SB-3 blocks were used.
The schematic diagram of the programming device is shown in Fig. 4. Starting the step selector ShI1 is accomplished by contact \(1P\) of the relay located in integrating amplifier 1; this relay operates at the moment the integration process begins. On the first lamella of the step selector, relay \(P_1\) operates and prepares the circuit for solving equation (1). On the tenth lamella relay \(P_2\) operates, preparing the circuit for solving equation (2); at the same time, the normally closed contact of relay \(P_2\)—\(2P_2\)—opens the circuit of relay \(P_1\), etc. On the hundredth step of the first step selector, relay \(P_6\) operates, which by its contact \(3P_6\) starts the second step selector. The starting of the third and fourth step selectors is carried out analogously.
The error introduced into the calculation due to the inertia of the relay and the step selector is no more than 0.5%.
Fig. 4. Schematic diagram of the programming device.
Sometimes the design of the injection channel, as well as the type of the forming system, are specified, and it is possible, within certain limits, to regulate the parameters of the particles at the entrance to the injection channel. In such cases, in order to determine the characteristics of the elements of the forming system that ensure the specified beam parameters at the channel exit, it is expedient to calculate the particle trajectory “from the end,” making use of the reversibility property of the electron-optical system. The use of an analog machine with the corresponding program device considerably facilitates the solution of this problem.
As the initial conditions in the solving amplifiers, the values of the coordinate and the angle of inclination of the trajectory required at the exit of the forming system are set, while the direction of the magnetic fields is changed to the opposite. As a result of the solution, the required input parameters of the beam (dimensions and divergence angle) are determined. Thus, the solution answers the question: what parameters the beam must have at the entrance of a given forming system in order to obtain, at the exit, a beam with the specified parameters.
By the method indicated, in particular, the necessary input parameters of the electron beam were established and the characteristics were selected for a forming system consisting of two quadrupole lenses (sections 4, 6 in Fig. 5) and a corrector (section 2, Fig. 5) for the specified output parameters of the electron beam.
Fig. 5. Injection channel consisting of 6 sections.
In the intervals between the quadrupole lenses and the corrector, the electrons are acted upon by a uniform magnetic field. Since the distances between the quadrupole lenses and the corrector are specified, correction of the angular divergence and of the beam dimensions must be achieved by regulating the input parameters. In this case it is easy to determine the coefficients \(k\) of the quadrupole lenses and the current in the corrector, as well as what parameters the beam must have at the entrance of the forming system. The structural diagram of the model for calculating the electron trajectories in the horizontal plane of the channel under consideration is shown in Fig. 6, where 1, 2 are operational amplifiers in the integration mode; 3 is an operational amplifier
in inversion mode; \(\kappa_{21}\), \(\kappa'_{21}\), \(l_2\), \(l_4\) are constant-coefficient units; 4, 5 are variable-coefficient units, on which the dependence of the magnetic field on the coordinate \(y\) is set. The voltage \(U_0\) represents the action of the corrector magnetic field. When solving the equation on the first section, contacts \(3P_1\) of relay \(P_1\) and \(2P_4\) of relay \(P_4\) are closed; on the
Fig. 6. Block diagram for an injection channel of 6 sections.
second section, contacts \(3P_2\) of relay \(P_2\), \(2P_4\) and \(3P_4\) of relay \(P_4\) are closed; on the third section, contacts \(4P_4\), \(5P_4\) of relay \(P_4\) and \(3P_5\), \(4P_5\) of relay \(P_5\) are closed, and so on. The operation of the relays is controlled by the step selector of the program device.
The use of analog computers is of great assistance in adjusting the control magnetic field in cyclic accelerators. On an electronic model it is quite simple to calculate the components of the Fourier series of the azimuthal distribution of the magnetic field along the closed orbit of the accelerated particles, as well as the actual trajectory of the particles on the first turns.
Fig. 7. Block diagram for determining the Fourier coefficients.
For harmonic analysis of the azimuthal field distribution, the block diagram shown in Fig. 7 is used. It is known [2] that if a function \(f(t)\) with period \(T\) is applied to the input of the circuit, then at time \(t = T\) the voltages \(U_1, U_2\) are determined by the formulas:
\[ U_1 = -\frac{T}{2R_1C_1}\,a_n; \]
\[ U_2 = \frac{T^2}{4\pi n R_1R_2C_1C_2}\,b_n, \]
where \(a_n\) and \(b_n\) are the Fourier coefficients:
\[ a_n = \frac{2}{T} \int_0^T f(t) \cos \left( \frac{2\pi n}{T} \cdot t \right) dt, \]
\[ b_n = \frac{2}{T} \int_0^T f(t) \sin \left( \frac{2\pi n}{T} \cdot t \right) dt. \]
In this case the period of the natural oscillations of the circuit must satisfy the condition
\[ \omega_n = \frac{2\pi n}{T}. \]
To reproduce the function \(f(t)\), the electromechanical nonlinear unit NB-5 can be used in the “exact” mode.
REFERENCES
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