# Mathematical Models of Polarization Adaptive Antenna Arrays Based on First-Kind Fredholm Integral Equations

## Authors

• A. O. Marchenko The National Defence University of Ukraine, Kyiv, Ukraine
• Yu. A. Husak The National Defence University of Ukraine, Kyiv, Ukraine
• S. V. Khamula Yevgeny Berezniak Military Academy, Kyiv, Ukraine
• V. V. Voitko Yevgeny Berezniak Military Academy, Kyiv, Ukraine
• A. B. Steiskal Research Institute of the Ministry of Defense of Ukraine, Kyiv, Ukraine
• V. V. Kuzmenko Institute of special communication and information protection National Technical University of Ukraine «Ihor Sikorsky Kyiv Polytechnic Institute», Kyiv, Ukraine

## Keywords:

adaptive antenna array, polarization, inhomogeneous translucent structure, impedance body, resonance wavelength, mathematical model, Fredholm integral equation of the first kind

## Abstract

Formulation of the problem in general. During radio relay communication, electromagnetic waves propagate along the earth's surface, and due to refraction, the polarization of signals can change, which leads to loss of signal power. This problematic situation can be solved by polarization adaptation of antenna systems built on the basis of lattice structures that convert signals with any polarization into a circle. Such antenna arrays are polarization-holographic antennas.

Analysis of recent researches and publications. In the theory of antenna synthesis, closed translucent surfaces, methods of synthesis of directional properties of reflective antenna arrays, which take into account the presence of mutual communication between irradiators of any type, as well as models of electrodynamic structures for arrays of different shapes are considered. This approach has a generalized nature and requires the adaptation of mathematical methods and models to antenna systems of a specific design. The polarization-holographic antenna can be considered as a non-homogeneous translucent structure, the problem of diffraction of electromagnetic waves in which it is expedient to solve the Fredholm integral equations of the first kind.

Presenting the main material. The mathematical formalization of the electrodynamic model of a non-homogeneous translucent body, which in its properties corresponds to the polarization-adaptive antenna array, is considered as the formulation and solution of the inverse electrodynamic problem connecting the primary electromagnetic field, surface current, and surface impedance. This surface impedance is the holographic kernel of the integral equation, which makes it possible to synthesize the impedance surface for circularly polarized waves. Diffraction of electromagnetic waves on a multilayer impedance body is described by a system of Fredholm integral equations of the first kind for different resonant wavelengths.

Conclusion. The method of Fredholm integral equations of the first kind makes it possible to determine the parameters of the antenna array through the transformation of the primary electromagnetic field into a secondary one based on the principles of holography. The result of solving the integral equation is its holographic core, which corresponds to the transparent (reflector) of the antenna. A system of integral equations was obtained, which mathematically formalizes the electrodynamic model of a planar antenna array with several layers, taking into account the mutual influence of these layers.

The perspectives of future researches. Further studies should be considered the description of the propagation process of electromagnetic waves, taking into account their multiple reflections, in a planar impedance body and the improvement of the matrix method for determining the transmission and reflection coefficients in such bodies.

## Author Biographies

### A. O. Marchenko , The National Defence University of Ukraine, Kyiv, Ukraine

Candidate of Engineering Sciences

### Yu. A. Husak , The National Defence University of Ukraine, Kyiv, Ukraine

Doctor of Military Sciences, professor

### S. V. Khamula , Yevgeny Berezniak Military Academy, Kyiv, Ukraine

Candidate of Engineering Sciences, Docent

### V. V. Voitko , Yevgeny Berezniak Military Academy, Kyiv, Ukraine

Candidate of Engineering Sciences, Senior Researcher

### A. B. Steiskal , Research Institute of the Ministry of Defense of Ukraine, Kyiv, Ukraine

Candidate of Engineering Sciences, Senior Researcher

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Marchenko A. O., Husak Yu. A., Voytko V. V. Bahatosharova polyaryzatsiyno-holohrafichna antena. Patent na korysnu model' № 142499; zayavl. 06.12.2019; opubl. 10.06.2020 [Multilayer polarization-holographic antenna. Utility model patent No. 142499; statement 06.12.2019; published 10.06.2020].

2023-09-30

## How to Cite

Марченко , А. О., Гусак , Ю. А., Хамула , С. В., Войтко, В. В., Стейскал, . А. Б. and Кузьменко , В. В. (2023) “Mathematical Models of Polarization Adaptive Antenna Arrays Based on First-Kind Fredholm Integral Equations: ”, Visnyk NTUU KPI Seriia - Radiotekhnika Radioaparatobuduvannia, (93), pp. 52-57. doi: 10.20535/RADAP.2023.93.52-57.