Mathematical models of spheroidal spiral-frame radiating structures
- Authors: Tabakov D.P.1, Valiullin R.M.1
-
Affiliations:
- Povolzhskiy State University of Telecommunications and Informatics
- Issue: Vol 26, No 1 (2023)
- Pages: 38-48
- Section: Articles
- URL: https://journals.ssau.ru/pwp/article/view/16926
- DOI: https://doi.org/10.18469/1810-3189.2023.26.1.38-48
- ID: 16926
Cite item
Abstract
The article considers mathematical models of two spheroidal spiral-frame emitters, built on the basis of a general approach involving the use of an integral representation electromagnetic field. The internal problem of electrodynamics is reduced to a system of Fredholm integral equations of the 1st kind. The resulting system was solved by the method of moments with piecewise constant basis functions and delta functions as test functions. In this case, the local linearization of the generating conductors of the structures under consideration was carried out. The dependences of current distributions, input resistance, and radiation characteristics of structures on frequency have been studied. It is shown that standing, traveling, and mixed current waves can exist in the structures under consideration. The current regime is determined by the wave sizes and the geometry of the structures and determines the behavior of the wave resistance in the frequency range. Despite the similar geometry, the characteristics of the considered structures have certain differences.
Full Text
Introduction
Helical antennas (HAs) represent a wide class of radiating structures, whose geometry and characteristics meet predetermined and diverse requirements. The main advantages of HAs include the ability to achieve a wide operating frequency band, good radiation directivity characteristics, the ability to electrically control the polarization characteristics of the radiation, and various shapes of emitting elements. Subject to the principle of angles and complementarity, frequency-independent variants of HA are implemented, the overlap coefficient of which reaches several tens of units [1]. HAs are used in antenna technology as self-sufficient radiating structures, as feeds for mirror antennas, in phased antenna arrays, to construct slow-wave systems, and in other elements of microwave devices [2].
Currently, interest in spiral elements is also associated with the development of the theory of metamaterials [3], while chiral structures can be considered a special case [4]. The introduction of conductive particles of various configurations (in this case, spiral particles) into the source material changes its electrodynamic properties. Such structures can be used as frequency-selective elements in polarization converters as low-reflective coatings and microwave energy concentrators. Naturally, the construction of the abovementioned structures requires the presence of mathematical models (MMs) of their basic elements, which in this case are spiral elements, generally described by sufficiently numerous parameters. Currently, phenomenological equations operating with the chirality parameter
The problem with the currently existing exact methods used in computer-aided design systems (CST MWS, FEKO, and HFSS) is the high requirements for computing resources, which are a consequence of their versatility, as well as the numerical nature of the results obtained, which cannot always be interpreted correctly. Therefore, the construction of rigorous and computationally efficient MMs of spiral elements and structures using these elements as their basis is a relevant task.
The most accurate MMs are constructed on the basis of integral equations (IEs) of various types [8–11]. The most widely used MMs are in the form of Fredholm IE systems of the first kind, obtained using the thin-wire (TW) approximation [12]. Herein, the complete MM of the structure under consideration must comprise the solution of exterior (determination of EMF [electromagnetic field] at any point in space) and interior (determination of current functions from boundary conditions on the elements of the structure) electrodynamic problems. The IE system represents a solution to only an internal problem. Therefore, the MM should be constructed on the basis of the corresponding integral representation of the EMF (EMF IR), which maintains a continuous relationship between the current functions and their EMF generated at any point in space.
In [13], based on TW EMF IR, the construction of MMs of cylindrical spiral elements of two types (conventional and combined) was performed. The problem of diffraction of these elements has been solved. It was revealed that the interior structure of the element considerably influences the scattered field characteristics. Thus, on a combined spiral element in a rather wide frequency range, the effect of orthogonal scattering occurs when the angle between the wave vectors of the primary and scattered electromagnetic waves is near 90°.
In [14], an ellipsoidal spiral particle was discussed, in which the MM was also constructed on the basis of the TW EMF IR. At the same time, a detailed analysis of the solution to the spectral problem was performed, which consisted of determining the behavior of the eigenfunctions and eigenvalues of the integral operator in the frequency band. It was demonstrated that the solution to the interior problem as a whole is determined by the eigenfunctions that have the smallest absolute value of the associated eigenvalues.
In this article, the MMs of two types of spiral-frame emitters are proposed, for which a numerical solution of the interior problem is studied in the frequency range. The radiation characteristics and input resistance are then determined. These emitters can be used as self-sufficient antennas or as a part of more complex antenna systems.
1. Physical models and geometry of the radiating structures
The geometry of the emitters is shown in Fig. 1. Both structures include a straight axial conductor A’A located along the axis Oz and a pair of spiral conductors AP and A’P connected to each other at a point P and at the corresponding ends A and A’ of the axial conductor. The structure presented in Fig. 1a contains a regular ellipsoidal helix (hereinafter referred to as A-helix). The structure presented in Fig. 1b comprises a spiral with a kink, and its lower part is a mirror image of the upper part relative to the plane XOY.
The structure conductors have the same radius, equal to
The generalized parametric equation of the spiral generatrix Ls has the form:
Here, a is the spheroid radius, and c is its semiaxis; j is a parameter on the generatrix (essentially the azimuth of the cylindrical coordinate system), and Nl is the number of helical turns. For A-helix,
and substitute it into Eq. (1). In this case, this problem can only be solved numerically using the inverse interpolation method [15]. The equation for the axial conductor Lv is written directly in the natural parameter l:
2. MMs of the radiating structures
To construct MMs of emitters in the previously considered formulation, an EMF IR is used [13]:
Here,
is the EMF IR from the current
are the kernels of the EMF IR;
G(R) is Green’s function defined for free space;
are kernel components, and
is the distance between the source and observation points, regularized by a small parameter ε, whose function is performed by the radius of the conductors. In our case, Eq. (4) involves a pair of conductors
The integral representation of the EMF of Eq. (4) contains current functions
The result of applying the presented boundary equation to the EMF IR is a system of IEs of the form:
The given IE system is classified as a Fredholm IE system of the first kind [16]. Let us discretize Eq. (4) by replacing the generators with kinked curves:
where
Here,
is the center of the segment, the unit tangent vector on the segment, and its length, respectively,
Use of Eq. (11) presupposes knowledge of unknown current amplitudes
If the condition is met that
for all j values, a stable solution of the SLAE is obtained [12]. The reliability of the results obtained on the basis of Eq. (11) and Eq. (12) is confirmed in [17].
3. Results of the numerical modeling
During numerical modeling, A- and B-helices were studied, the geometry of which was determined by the following parameters:
Fig. 2 presents the results of calculating the amplitude current distributions on the conductor AP of both structures for various ratios
Fig. 2. Comparison of the amplitude distributions of the current on the conductor of the A-helix (curve 1) and B-helix (curve 2) at different values
Figs. 3–6 show the results of calculating the input resistance of the structures under consideration for four ratios
Fig. 3. Dependence of the amplitude (a) and phase (b) of the input resistance on the ratio for the A-helix;
Fig. 4. Dependence of the amplitude (a) and phase (b) of the input resistance on the ratio for the B-helix;
In the HF region, the dependence of the input resistance on frequency at
Fig. 5. Dependence of the amplitude (a) and phase (b) of the input resistance on the ratio for the A-helix;
Fig. 6. Dependence of the amplitude (a) and phase (b) of the input resistance on the ratio for the B-helix;
Fig. 7 presents a comparison of the normalized amplitude directional patterns (DPs) of the considered emitters in the meridian plane, calculated at different ratios of
Fig. 7. Comparison of the normalized amplitude radiation patterns of the A-helix (curve 1) and B-helix (curve 2) at different values
Conclusion
This study considers two variants of spheroidal spiral-frame emitters (A-helix and B-helix). The emitters differ in the internal structure of the spiral elements. Note that HAs, which have a geometry near that of the emitters considered, are widely used in practice. MMs of emitters are proposed and constructed on the basis of an integral representation of the electromagnetic field recorded in the TW approximation. These MMs allow for a quantitative assessment of the electrodynamic parameters of the structures under consideration. The interior problem is described as a system of Fredholm IEs of the first kind. A method for reducing it to a SLAE presented with respect to unknown values of complex current amplitudes on segments is presented, and a condition is given for correctly implementing this procedure.
Based on the models presented, numerical solutions were obtained for the interior and exterior electrodynamic problems. For some ratios
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, RussiaRuslan M. Valiullin
Povolzhskiy State University of Telecommunications and Informatics
Author for correspondence.
Email: ruslanvaliullin1998@yandex.ru
post-graduate student of the Department of Physics, Povolzhskiy State University of Telecommunications and Informatics, Samara, Russia.
Research interests: electrodynamics, microwave devices and antennas.
Russian Federation, 23, L. Tolstoy Street, Samara, 443010, RussiaReferences
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