arXiv subjects
Publications and source records attributed to M. I. Ayzatsky.
Comparison of approximate solutions that were obtained by using different asymptotic methods of solutions of difference equations with the exact solution is presented. Results show that for the studied equation the method of transformation of the difference equation into the Riccati equation gives more correct solutions than the method of direct asymptotic expansion
A generalization of the already studied transformations of the linear differential equation into a system of the first order equations is given. The proposed transformation gives possibility to get new forms of the N-dimensional system of first order equations that can be useful for analysis of the solutions of the N-th order differential equations. In particular, for the third-order linear equation the nonlinear second-order equation that plays the same role as the Riccati equation for second-order linear equation is obtained.
The model of coupled oscillators plays an important role in modern physics. It is used for description of various processes: from vibrations atoms in solid states to electromagnetic oscillations in slow-wave structures. The model with short-range coupling is the most widely used, for which a separate oscillator is coupled with two adjacent ones only. There are two main types of oscillators coupling: capacitive (electric, power) and inductive (magnetic, inertial). In the first case, the coupling is proportional to the amplitudes of oscillations in the adjacent cells, in the second one - to the second derivative of these amplitudes. For numerical study of dynamics of a system that can be described by a model of coupled oscillators with an inductive coupling, it is necessary to find explicit expressions for the second derivatives of the amplitudes. To find these expressions, we propose to use the method that is based on the solution of difference equations. The results of the analysis of this method are given in the paper.
A novel approach to the synthesis of the electromagnetic field distribution in a chain of coupled resonators has been developed. This approach is based on the new matrix form of the solutions of the second-order difference equations. If a chain of coupled resonators can be described by the second-order difference equation for amplitudes of expansion of the electromagnetic field, two linearly independent solutions can be constructed on the basis of the solutions of nonlinear Riccaty equation. Setting the structure of one solution, from the Riccaty equation we can find the electrodynamical characteristics of resonators and coupling holes, at which the desired distribution of amplitudes is realized. On the base of this approach we considered the problem of separation of the electromagnetic field into forward and backward components in the inhomogeneous chain of resonators. It was shown that in the frame of considered model such separation is not defined uniquely.
Results of research of possibility of transformation of a difference equation into a system of the first-order difference equation are presented. In contrast to the method used previously, an unknown grid function is split into two new auxiliary functions, which have definite properties. Several examples show that proposed approach can be useful in solving different physical problems.
We developed the general approach that gives possibility to calculate the coupling coefficients for arbitrary chain of resonators without using the great number of eigen functions. For understanding this method and having possibility to control the accuracy of obtained results, we applied this procedure for simplest structure that has the analytical solutions - a circular cylindrical waveguide with uniform cross section.
We present the short description of the methods for calculation of the coupling coefficients in the Coupling Cavity Model of arbitrary chain of resonators. In the first part the procedure that is based on the Mode Matching Method is given. Then we present the new method that used only one eigen vector.
On the base of more general model we assessed accuracy of method that is widely used for post- tuning accelerator sections. Our consideration has shown that for wide range of disk loaded waveguide (DLW) parameters using simple tuning coefficients gives appropriate characteristics of DLWs. For injector sections, in which phase velocity and amplitude change, there are some deviation from the desired characteristics. It is necessary to evaluate the influence of additional phase shift on beam characteristics at the injector exit.
On the base of modified mode matching method we obtain some results that can be useful in the process of tuning of nonunifrom disk-loaded structures. Our consideration has shown that there are some parameters that depend only on the geometric sizes of sells and equals known values for the needed phase distribution of the amplitudes of field expansion. Changing the geometric sized in such way that these parameters will tend to a given values, we can numerically design the structure that have the given phase distribution of these amplitudes. We consider these tuning parameters as a good instrument under simulation of inhomogeneous structures with sizes that usually used in accelerators.
The coupled cavity model for calculation characteristics of inhomogeneous disc-loaded waveguides with taking into account the rounding of the disk hole edges was developed. On the base of this model simulation of section tuning process with the field measurement method was conducted. Accuracy of this method under tuning of disc-loaded waveguides was evaluated. As it follows from our investigation one can use simple coefficients in tuning process, but the needed values of these coefficients must be obtained by calculation of DLW parameters.
We have developed the mode matching technique that is based on the using the eigenmodes of circular cavities and the eigenwaves of circular waveguides as the basic functions for calculation the properties of nonuniform disc-loaded waveguides. We have obtained exact infinite systems of coupled equations which can be reduced by making some assumptions. Under such procedure we can receive more exact parameters of nonuniform equivalent circuits by solving the appropriative algebraic systems. These parameters of equivalent circuits are functions both geometric sizes and frequency. Moreover, under such approach all used values have interpretation. We called this approach as coupled cavity model.
On the base of general approach we obtain some results that can be useful in the process of tuning of nonunifrom disc-loaded waveguides. Our consideration has shown that simple values that characterize the detuning of the cells can be introduced only for the disc-loaded waveguide with parameters that change very slow. In general case it is needed to conduct full numerical simulation of specific disc-loaded waveguide and obtain all necessary coupling coefficients. After that one can start the tuning process on the base of bead-pull field distribution measurements.
This paper presents the results of studies in elaboration of a mathematical model of the cavity-chain slow wave structures. Considered is the problem of coupling of an infinitely long cylindrical cavity chain coupled through centerholes in the dividing walls of finite thickness without the assumption about smallness of any parameters. On the basis of a rigorous electrodynamic approach it is shown that the cavity chain can be described by a set of equations that describe the coupling of an infinite number of similar-type modes of short-circuit cavities. At the same time the coupling coefficients are determined by solving the basic sets of linear algebraic equations which describe the coupling the tangential electric fields on the right and left cylindrical hole cross-sections of the disks. The results are given of numerical simulations of the dependence of electrodynamic characteristics on the number of connections between the modes taken into account. Based on these calculations for different cavity dimensions, the number of coupling connections that give the sufficient accuracy of the description was determined.
This work presents research results on a novel analytical model of electromagnetic systems coupling through small size holes. The key problem regarding coupling of two cavities through an aperture in separating screen of finite thickness without making assumption on smallness of any parameters is considered. We are the first to calculate on the base of rigorous electromagnetic approach the coupling coefficients of the cylindrical cavities within the limit of small aperture and infinitely thin separating screen. The numeric results of electromagnetic characteristic dependencies that have been impossible to perform on the base of previous models are given.
Analytical solution of precise equations that describe the rf-coupling of two cavities through a co-axial cylindrical hole are given for various limited cases. For their derivation we have used the method of solution of an infinite set of linear algebraic equations, based on its transformation into dual integral equations.