SIMULATION OF COMPACT POLARIZERS FOR SATELLITE TELECOMMUNICATION SYSTEMS WITH THE ACCOUNT OF THICKNESS OF IRISES

Background. One of the main problems in modern satellite telecommunication systems is to increase the volume of information transmission with simultaneous preservation of its quality. Key element of such systems is antenna systems with polarization processing, which is carried out using polarizers. Therefore, development of new polarizers and simple techniques for their analysis and optimization are important problems. The most simple, effective, technological and actual for analysis are polarizers based on waveguides with irises. Objective. The purpose of the paper is to create a mathematical model of the polarizer based on a square waveguide with irises, which allows analyzing the influence of polarizer’s design parameters on its electromagnetic characteristics. Methods. A mathematical model of the waveguide polarizer with irises is created by decomposition technique using transfer and scattering wave matrices. To take into account the irises’ thickness their equivalent Tand Π-shaped circuits were used. Results. We have developed mathematical model of the waveguide polarizer with irises, which takes into account their thickness and is based on the complete scattering wave matrix of the waveguide polarizer. The matrix has been obtained using the microwave circuit theory. The main characteristics of the waveguide polarizer were defined using matrix elements. The optimization of characteristics of a polarizer was carried out in the operating Ku-band 10.7–12.8 GHz. Conclusions. Suggested mathematical model of a waveguide polarizer with irises provides the account of heights of irises, distances between them and their thickness. The results obtained show that this model is simpler and faster for the calculation of electromagnetic characteristics compared to finite elements method, which is often used for analysis of microwave devices for various applications.


Introduction
Nowadays there is a rapid development of satellite telecommunication systems and the expansion of modern branches of science and technology that actively apply them. Often such systems require an increase of the volumes of information they are able to process and transmit. A key element of most modern satellite telecommunications systems are antenna systems with processing of signal polarization. Such systems use electromagnetic waves with orthogonal circular or linear polarizations. They improve information characteristics of telecommunication systems and increase the level of the received signal under adverse conditions of wave propagation. Polarization-spatial separation of channels provides the necessary characteristics of telecommunication systems. Application of the advantages of antennas with orthogonal polarizations in satellite telecommunication systems allows increasing their efficiency and information capacity [1].
Polarization processing units and devices for separation of signals with orthogonal polarizations are main elements of antenna systems with orthogonal polarizations. Such devices are used to solve problems of theory of detection and recognition of objects and investigate many phenomena of nature [2]. Such problems include prediction of the intensity of rainfalls, measuring the parameters of ice and snow cover, estimation of the parameters of icebergs, assessing the conditions of crops cultivation and many others.
The main types of signal polarization processing devices are based on structures with posts [3][4][5], ridged structures [6] and structures with irises [7,8]. Ridged structures and structures with irises are used to create broadband devices for micro wave engineering systems. The analysis of such structures is carried out using various analytical methods. Such methods include mode matching technique [9], transverse field-matching technique [10][11][12], magnetic field integral equation technique [13], and integral equations technique [14,15], in which it is possible to take into account the singularity fields on the edges, which excludes the relativity of convergence of series in the transverse field matching technique [16]. The structures of this type are also analyzed using the wave matrix method [17][18][19][20].
All listed above methods have one major drawback, which is the difficulty of calculating the complete structure of electromagnetic fields. Therefore, there is a need to focus on simpler methods based on matrix techniques of microwave circuit analysis. They use scattering and transmission wave matrices. Various microwave filters [21][22][23][24] and phase shifters [25][26][27][28] are often analyzed using such methods. They take into account the interaction of higher order modes without the application of a numerical optimization process using specialized computer programs.
Polarizers based on square waveguides with irises provide the best characteristics in wide and ultrawide operating frequency bands. Therefore, such a polarizer design was chosen to develop a mathematical model in our research.
Therefore, it is important to develop a new method for analyzing the characteristics of waveguide polarizers with irises. The new method makes it possible to take into account the thickness of the irises of a polarizer. Developed technique allows to determine all electromagnetic characteristics of the polarizer and does not require much time to perform calculations.

Problem statement
The purpose of the presented article is to improve the electromagnetic characteristics of a square waveguide polarizer with irises by optimizing its design for the operating frequency band. The problem is solved by creating an appropriate mathematical model of the waveguide iris polarizer using wave matrices techniques.

Mathematical model of a waveguide polarizer with irises
To create a mathematical model we consider a simple design of a polarizer based on a square waveguide with two irises. It is presented in Fig. 1. The transverse dimensions of the square waveguide of the polarizer are a×a. The design contains two identical irises with equal heights h, thickness w and distance between them l. We used a square waveguide because it provides better performance in a wide operating frequency band than a circular waveguide. Using the theory of microwave circuits [34], we present a waveguide polarizer with irises in a general equivalent scheme (Fig. 2). Let us divide the equivalent circuit into twoport circuits in the form of 1 section of a regular transmission line and 2 two-port circuits in the form of connected in parallel reactive elements. Each two-port circuit is described by the wave transfer matrix as follows: where q is the electric length of the equivalent regular transmission line.
The electric length of a regular transmission line is determined by the formula where l g is wavelength in the waveguide. The wavelength in the waveguide is determined by a known formula [34]: where | | T is determinant of the wave matrix of transmission.
From the scattering matrix we determine its elements through the T-matrix For the main wave of horizontal polarization, a simplified equivalent of the polarizer circuit contains inductors that are turned on in parallel (Fig. 3, a). For the main wave of vertical polarization of the equivalent circuit contains capacitors that are turned on in parallel (Fig. 3, b). Fig. 3. Equivalent circuit of a waveguide with two reactive elements To take into account the thickness of the diaphragms, we use more complicated T-and П-shaped equivalent circuits for each capacitive (Fig. 4, a) and inductive irises (Fig. 4, b).
For an inductive iris, the reactive resistances of an equivalent circuit (Fig. 4, a) are determined by the expressions [42]: where a is the size of the large wall of the waveguide; w is iris thickness; h is iris height.
To calculate the parameters of the wave matrix transmission of such a scheme using formulas For a capacitive iris, the reactive conductivities of an equivalent circuit (Fig. 4, b) are determined by the expressions [42] where a is the size of the large wall of the waveguide; w is iris thickness; h is iris height.
To calculate the parameters of the wave matrix transmission of such a scheme using formulas  Thus, elements of the general scattering matrix of our mathematical model were formed. Through these elements we determine the main electromagnetic characteristics of a waveguide polarizer with irises.
The differential phase shift at the output of the polarizer is determined by the expression 21 21 where 21 21 and 21 21 are elements of the general scattering matrix in the case of inductive and capacitive irises.
VSWR is calculated by the following formula The axial ratio can be determined by the following expression [43] in dB as follows using logarithmic scale

Analysis of the developed mathematical model
Let us consider the results of the calculation of the mathematical model of the waveguide polarizer in the Ku-band 10.7-12.8 GHz.
To ensure the required differential phase shift we have changed the height of the irises h. And to achieve a given matching we have adjusted the distance between the irises. This was performed for the optimal thickness of the irises. In the operating frequency band the optimal matching has been achieved with a small deviation of the differential phase shift from 90°. Fig. 5 we see that the maximum deviation of the differential phase shift from 90° is 7°.  From the figures we see that the maximum value of the axial ratio is 1.5 dB, and the crosspolar determination is greater than 21.5 dB.

Figs. 5-8 show the main electromagnetic characteristics of the polarizer. From
Thus, the proposed mathematical model in the Ku-band 10.7-12.8 GHz for a polarizer based on a square waveguide with 2 irises provides the following characteristics: VSWR for horizontal and vertical polarization is less than 2.15, differential phase shift is within 90° ± 7.0°, axial ratio is less than 1.5 dB, crosspolar discrimination is higher than 21.5 dB.

Analysis of optimization results
The developed mathematical model of a waveguide polarizer does not take into account some higher order modes. This can result in inaccuracies of calculation of the differential phase shift and polarization characteristics. Consequently, numerical techniques are applied for more accurate estimation of the polarizer's characteristics. Further optimization and modeling of a polarizer based on a square waveguide with two irises were performed by the finite integration technique in the operating Ku-band 10.7-12.8 GHz. Fig. 9 shows the dependence of the differential phase shift on the frequency. From the figure we see that the maximum deviation of the differential phase shift from 90° is 4.2° at 11.6 GHz.  Figs. 11 and 12 present dependences of the axial ratio and XPD on the frequency. From the figures we see that the maximum value of the axial ratio is 1.43 dB, and the crosspolar discrimination is higher than 21.7 dB. Therefore, within the operating frequency range 10.7-12.8 GHz the optimized polarizer based on the square waveguide with 2 irises provides the following characteristics: VSWR for horizontal and vertical polarization is less than 3.26, differential phase shift is within 90° ± 4.2°, axial ratio is less than 1.43 dB, crosspolar discrimination is higher than 21.7 dB.
The optimized parameters of the waveguide polarizer with two irises in the Ku-band 10.7-12.8 GHz are summarized in Table 1. Table 2 compares the optimized characteristics of the polarizer for the analytical method based on the developed mathematical model and the finite integration technique.
The small difference in sizes and characteristics given in the tables can be explained as follows. The analytical method and the finite integration technique used different numerical methods. In addition, the mathematical model of the analytical method does not take into account all the higher types of waves in the waveguide. Optimized by the created mathematical model structure of the polarizer has improved matching characteristics due to a slight increase in the deviation of the differential phase shift from the required 90°.

Conclusions
In this article we have developed a mathematical model of the polarizer based on a square waveguide with two irises. The mathematical model takes into account the influence of design parameters on the electromagnetic characteristics of the polarizer. It allows achieving better matching performance in the operating frequency band by changing all geometric dimensions of the irises. The novelty of the created model is its account of the influence of the iris thickness on the main characteristics of a waveguide polarizer. Developed mathematical model can be widely applied to create new waveguide polarizers and other devices based on different numbers of irises with different heights.
The proposed mathematical model of the polarizer allows determining the general wave scattering matrix. The main electromagnetic characteristics were determined using the elements of this matrix. Compared with the finite integration technique, the created mathematical model provides an opportunity to quickly analyze and optimize the electromagnetic characteristics by changing the inner sizes of the device. This approach makes it possible to achieve better matching characteristics simultaneously with an acceptable differential phase shift.
In further research it is advisable to focus on the development of more accurate mathematical model that will take into account more irises in the polarizer design and more higher order modes.