Spectral characteristics of the integral operator of the internal problem of electrodynamics for elliptical frame structure
- Authors: Tabakov D.P.1, Mayorov A.G.1
-
Affiliations:
- Povolzhskiy State University of Telecommunications and Informatics
- Issue: Vol 26, No 1 (2023)
- Pages: 58-69
- Section: Articles
- URL: https://journals.ssau.ru/pwp/article/view/16928
- DOI: https://doi.org/10.18469/1810-3189.2023.26.1.58-69
- ID: 16928
Cite item
Abstract
The article is devoted to the analysis of electrodynamic properties elliptical frame structure. Taking into account double symmetry internal problem of electrodynamics for the structure under consideration in the framework of the thin-wire approximation is reduced to four integral Fredholm equations of the first kind, written with respect to independent current functions. A study of spectral characteristics of the integral operators of the corresponding integral equations for various values of the electrical length and ellipticity of the frame. It is shown that the eigenfunctions of integral operators for close values of these parameters have a high degree of correlation, with In this case, the eigenfunctions are close in form to trigonometric functions. Features of the frequency dependence of the eigenvalues integral operators. The conclusion is made about the resonant nature of these dependences, what makes an elliptical frame structure in many respects similar to the previously considered tubular vibrator and spherical spiral particle. The results presented in the article form an in-depth understanding of the processes occurring in the structure under consideration, and also serve as a guideline in the construction of approximation models for solving the internal tasks.
Full Text
Introduction
Loop antennas are one of the most common types of antennas, and they have several applications (television, cellular communications, radio communications, etc.). Theoretical research on loop antennas has been conducted for quite a long time; hence, a rather large number of scientific works on this subject now exist. At present, the characteristics of such structures can be calculated with a high degree of accuracy using computer-aided design systems, engineering equations, and developed models of other works that have varying degrees of complexity. In [1], a loop antenna is examined in the approximation of the uniform current distribution. In [2], the long-line theory is applied for the calculations. In [3; 4], in the cross section of a conductor with small wave sizes, a quasistatic approximation is introduced for the current distribution. In [5], a ring stripline antenna is considered, for which an infinite set of integral equation systems (IESs) are developed with respect to the Fourier harmonics of the components of the surface current density vector on the strip. The obtained results allow estimation of the relationship between the amplitudes of the longitudinal and transverse current components.
It is worthy to note that rigorous mathematical models have been designed mostly only for ring frames with the simplest geometry. The axial symmetry of the structure present in this case significantly simplifies the solution of the interior problem. Rigorous models of frames of more complex configurations (e.g., elliptical, polygonal) are not so common; therefore, developing such mathematical models is urgently needed. Even for rigorous models developed in the form of IE (including singular ones), authors, as a rule, limit themselves to the analysis of the quantitative characteristics of current distributions without investigating the reasons that result in the formation of these distributions. This aspect is critical to producing an adequate pattern of the interior physical processes in the structures being considered. This problem can be solved using the eigenfunction method (EFM) developed in [6]. Previously, this method was applied by the authors to the analysis and construction of an approximation of the interior problem solution for a tubular electric dipole [7; 8]. An alternative to EFM is the characteristic mode method [9–11]. Its advantages over EFM are the simplicity of the numerical implementation; however, a significant disadvantage of this method is the low stability of the computational procedure.
We discuss a mathematical model of an elliptical frame (EF) structure expressed in the form of four independent IEs. The solution to the interior electrodynamic problem is developed on the basis of EFM. The method of EF excitation was not specified to increase the generality of the presented results, i.e., the developed model can be employed to solve both antenna and diffraction problems. In a given frequency range for different variants of the EF geometry, the spectral characteristics of the integral operators of the corresponding IEs were analyzed.
1. Statement of the problem
Consider solving the interior problem of electrodynamics on an EF structure using the eigenfunction method. The geometry of the structure is shown in Fig. 1. An EF conductor, which has infinitely high conductivity, has a circular cross section with a diameter
Fig. 1. Thin-wire model of an elliptic frame antenna
Рис. 1. Тонкопроволочная модель эллиптической рамочной антенны
The parametric equation of the EF generatrix L has the form
Here, t is the azimuth of a cylindrical or spherical coordinate system and
Here,
Within the adopted model, the EFM structure is described by an integral representation (IR), described in detail in [13]:
Here,
The structure under consideration with
In this case,
Here,
on each generatrix, and thus, the following IE system is obtained
Here,
are the tangential components of the external electric field on the generatrices and kernels of the IE system, respectively. Due to the symmetry of the structure, we obtain the following equations:
For functions
where
Regarding functions
Physically, the planes XOZ and YOZ are an electric or magnetic wall for the structure being considered; thus, the following boundary conditions are valid for the functions
We approximate the generatrices
where
Here,
The complete eigenvalue problem (EVP) for a matrix
In this equation,
The index “T” is the transpose operation. The matrix
In our case, the length of the EF generatrix L should be selected as the main parameter normalized to the wavelength
It should also be noted here that with
2. Numerical modeling and analysis of the results
The problem of Eq. (10) was solved in a rectangular area:
At intervals
The number of segments N during linearization of the generatrix was assumed to be 100, and the ratio
Calculation of EVP X and EV
Let
that is, calculations must be conducted for matrices located in close points of region
Next, each line
In our case, the matrix
Numerical calculations have two aims. The first aim, which has predominantly practical significance, is associated with determining the possibilities of constructing an approximation model for solving an interior problem based on the EFM. To realize this goal, it becomes necessary to solve some tasks. Task 1 includes the analysis of the degree of correlation of EV calculated at various points in region
The second aim is mainly of theoretical significance and is associated with determining the nature of the frequency dependence of EVP. Earlier in [6; 7], for other structures, it has already been revealed that this dependence has a resonant nature; therefore, the main contribution to the formation of a solution to the interior problem is made by only a small part of EF. In this case, we need confirmation of this fact with some additional details for the structure being considered. To estimate the residual between vector or matrix arrays V calculated at a pair of points in region
Figure 2 illustrates the graphs of values
Fig. 2. Dependence
Рис. 2. Зависимость
Figure 3 exhibits the graphs of values
Fig. 3. dependence
Рис. 3. Зависимость
Figure 4 shows the graphs of values
Fig. 4. Dependence
Рис. 4. Зависимость
Figure 5 shows the graphs of the real and imaginary parts of the first four eigenfunctions calculated at
Fig. 5. View of the first four of its own functions
Рис. 5. Вид первых четырех собственных функций
The ratio of the intensities of the real and imaginary parts of the eigenfunctions can be estimated by the magnitude
Fig. 6. Dependence
Рис. 6. Зависимость
Figure 7 shows the graphs of the values
Fig. 7. Dependencies
Рис. 7. Зависимости
An important aspect in the analysis was considering the symmetry, since in the structure under consideration a degeneracy effect is noted, which consists of the coincidence of resonance points for the EVP of matrices of various SLAEs. This effect is noted for
Generally, we can conclude that the structure under study in terms of the behavior of eigenvalues and the shape of eigenfunctions is in many ways similar to the previously considered tubular dipole [7; 8] and a spherical spiral particle [6]. Thus, the previously proposed strategies regarding the construction of an approximation model to solve the interior electrodynamic problem are fully applicable to the structure under consideration.
Conclusion
This study considers a variant of a mathematical model to solve an interior electrodynamic problem for an elliptical spiral structure constructed by thin-wire approximation. The structure has double mirror symmetry, which allows the generation of a mathematical model in the form of four independent Fredholm IEs of the first kind, written relative to the corresponding current functions that meet the boundary conditions for the electric or magnetic wall at the points of intersection of the generatrix of the structure with the symmetry planes. Within the method of moments, the reulting IEs were reduced to SLAEs relative to the values of the current functions on the segments of the linearized generatrix. Solutions of the SLAE are expressed in terms of the eigenvectors and eigenvalues of the SLAE matrix. The eigenvectors of the SLAE approximate the eigenfunctions of the integral operator of the corresponding IE. For each IE, the behavior of the eigenfunctions and eigenvalues of the integral operator was examined depending on the electrical length of the generatrix of the structure and the ellipticity coefficient at a fixed electrically small radius of the conductor.
An estimate of the residual between the eigenfunctions calculated for different values of the specified parameters is given. It is shown that the discrepancy increases with increasing electrical length of the generatrix of the structure and with decreasing ellipticity coefficient; however, at the selected step of changing the parameters, it has rather small values. A more detailed analysis allowed us to conclude that the most significant contribution to the value of the residual is made by the eigenfunctions of the lower types. This information forms the primary guideline when constructing an approximation model to solve the interior problem of the structure under consideration.
Analysis of the forms of eigenfunctions showed their closeness to trigonometric functions. Therefore, they can be approximated by the corresponding series, which in this case exhibit rapid convergence. Inthe limiting case, when the ellipse degenerates into a circle, each eigenfunction can be analytically precisely determined by a pair of trigonometric functions.
An analysis of the dependence of the eigenvalues on the electrical length of the generatrix confirmed their resonant nature. Thus, the structure considered from this viewpoint is in many ways similar to the electric dipole and spherical spiral particles previously considered by the authors. That is, it can be argued that a rather limited set of eigenfunctions makes a significant contribution to the solution of the interior problem. It is also worth mentioning here that considering the symmetry of the structure significantly simplifies the numerical analysis when the degeneracy effect occurs, when one eigenvalue can correspond to more than one eigenfunction, which is noted in this case with values of the ellipticity coefficient tending to unity. At lower values of the ellipticity coefficient, only the effect of the degeneration of resonant frequencies is noted.
This work has both theoretical and practical significance. The theoretical significance is related to the development of methods for the electrodynamic analysis of frame emitting and re-emitting structures. The proposed approach offers an in-depth understanding of the behavior of the structures under consideration from the viewpoint of electrodynamics and significantly simplifies the interpretation of the obtained numerical results compared with direct solving of integral equations and their systems. The applied significance is related ot the fact that the obtained results can serve as good reference in constructing approximation models to solve the interior problem for the structure under consideration, as well as for structures with similar geometry.
About the authors
Dmitry P. Tabakov
Povolzhskiy State University of Telecommunications and Informatics
Email: illuminator84@yandex.ru
Doctor of Physical and Mathematical Sciences, professor of the Department of Physics, Povolzhskiy State University of Telecommunications and Informatics, Samara, Russia.
Research interests: electrodynamics, microwave devices and antennas, optics, numerical methods of mathematical modeling.
Russian Federation, 23, L. Tolstoy Street, Samara, 443010, RussiaAndrey G. Mayorov
Povolzhskiy State University of Telecommunications and Informatics
Author for correspondence.
Email: andrey.mayorov.92@yandex.ru
engineer of the Department of Radioelectronic Systems, Povolzhskiy State University of Telecommunications and Informatics, Samara, Russia.
Research interests: electrodynamics, microwave devices and antennas, numerical methods of mathematical modeling.
Russian Federation, 23, L. Tolstoy Street, Samara, 443010, RussiaReferences
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