TUNABLE POLARIZERS FOR X-BAND RADAR AND TELECOMMUNICATION SYSTEMS

Background. Nowadays processing of signal polarizations is widely applied in modern information and telecommunication radio engineering systems for different purposes. Commonly polarization processing is carried out in polarization adaptive antenna systems. The essential elements of such systems are transformation devices for polarization processing. They perform the transformation of the types of polarization and separate the different types to isolated channels. The most simple, effective, technological and actual for analysis are polarizers based on square waveguides with irises and posts. Objective. The purpose of this work is to improve the electromagnetic characteristics of an adjustable polarizer by creating a mathematical model of such device. The device must provide optimized polarization and matching characteristics. Methods. The article presents a mathematical model of a waveguide polarizer with irises and posts by the decomposition method using wave transmission and scattering matrices. The developed model takes into account the influence of the polarizer design parameters on its characteristics. Results. The article contains the results of calculations based on the developed mathematical model of the polarizer. In addition, the results of modelling of the device using the finite element method are presented for comparison. For the developed waveguide polarizer we have compared the polarization characteristics and the matching. Conclusions. The created mathematical model allows us to effectively analyse the characteristics when the design parameters change. These parameters include the size of the wall of the square waveguide, the heights of the irises and posts, the distance between them, the thickness of the irises and posts. The developed polarizer is recommended for the application in modern telecommunication and radar systems.


Introduction
Fast progress in modern telecommunication satellite systems and radars encourages an increase in the data volumes transmitted in their wireless channels. In turn, this requires improvement of existing methods of signal processing and creation of new ones. Polarization signal processing is one of the leading methods for this purpose. Modern adaptive antenna systems, which perform polarization processing of signals, contain waveguide polarizers. The electromagnetic characteristics of such devices determine the overall performance of the communication or radar system. Polarization characteristics are very sensitive to the accuracy of manufacture. Therefore, accurate mathematical modelling and optimization of phase and polarization characteristics is an important problem for the development of modern microwave waveguide polarizers and antenna systems based on them.
But all listed structures have disadvantages, such as the complexity of the design and limited bandwidth. Therefore, a polarizer design containing two types of inhomogeneities was proposed. These are irises and posts. The presence of irises in the design allows you to provide a wide operating frequency band. The presence of posts in the design provides the adjustment of the polarizer.

Problem statement
The purpose of the presented article is to optimize the electromagnetic characteristics of a polarizer based on a square waveguide with diaphragms and post by changing the size of its structure. The problem is solved by creating an appropriate mathematical model of the square waveguide polarizer with irises and posts using wave matrices techniques.

Mathematical model of a square waveguide polarizer with irises and posts
The design of the waveguide polarizer is shown in Fig. 1. The structure contains two irises of height h and thickness w, two posts of height h p and diameter d, the distance between the iris and the post is l. The given design provides the basic polarization characteristics. The cylindrical pin provides adjustment and adjustment of characteristics due to change of length of a post in a waveguide.
According to the theory of microwave circuits [41][42][43], we present the scheme in the form of separate structural schemes, which are divided into elementary quadrupoles (Fig. 2). Fig. 2, a shows a block diagram of a waveguide polarizer with a post and inductive irises connected in parallel. Fig. 2, b shows a general block diagram of a waveguide polarizer with a post and capacitive irises connected in parallel.
Let us define the general wave matrix of scattering through elements of the general wave matrix of transfer [44]: 11 12 21 , T are matrix describing the iris; [ ] 2 T is a matrix describing a segment of a regular transmission line.
The wave transmission matrices are determined: where q 2 is electric length of a regular transmission line.
The electric length of a regular transmission line , T are matrices describing a segment of a regular transmission line. , where q 1 is electric length of a regular transmission line.
The electric length of a regular transmission line 1 2 , where l w is wavelength in the waveguide. The wavelength in the waveguide The wave transmission matrix for the post [ ] 11 12 where Y p is the conductivity of the post in the waveguide.
The conductivity of the post in the waveguide is determined by the formula [45] where a is the length of the wall of a square wave-guide; h p is the height of the post in the waveguide; k is wave number in vacuum; r is post radius.
To take into account the thickness of the iris used equivalent substitution schemes (Fig. 3).
For an inductive iris, the reactive supports of an equivalent circuit (Fig. 3, a) are determined by the expressions [46]: 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 [47] For a capacitive iris, the reactive conducti vities of an equivalent circuit (Fig. 3, b) are determined by the expressions [46]: 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 [47]

. t t t t + =
The characteristics of the polarizer are as follows: phase, matching and polarization. Phase and matching are the differential phase shift and the voltage state wave ratio (VSWR). The polarization characteristics of the polarizer are the axial ratio and the crosspolar discrimination (СPD).
Differential phase shift is determined by the expression 21 The axial ratio is determined in dB [48]:

Analysis of the developed mathematical model
Let us investigate the electromagnetic characteristics of the mathematical model of a waveguide polarizer in the X-frequency range from 7.7 GHz to 8.5 GHz.
Using our model, changing the height of the apertures h and pin h p , we achieve the required differential phase shift. To ensure a given match, adjust the distance between the diaphragms l. These changes must be made at the optimal diaphragm thickness. At this frequency we achieve optimal coordination with a small deviation of the differential phase shift from 90°.   Fig. 4 demonstrates that the maximum deviation of the differential phase shift from 90° is 2.3°. Fig. 5 shows that the maximum value of VSWR for both polarizations is 1.41.
Figs. 6 and 7 present the polarization characteristics of the developed mathematical model of the polarizer based on a square waveguide with two irises and two posts  Fig. 6 contains the dependence of the axial ratio on the frequency, and Fig. 7 contains the dependence of the VSWR on the frequency. In Fig.  5 we see that at a frequency of 8.5 GHz the axial ratio acquires its maximum value of 0.45 dB. Also, the СPD acquires a maximum value of 30 dB at this frequency. Thus, the proposed mathematical model in the X-band 7.7-8.5 GHz for a polarizer based on a square waveguide with two irises and two post provides the following characteristics: VSWR for horizontal and vertical polarization is less than 1.41, differential phase shift is within 90° ± 2.3°, axial ratio is less than 0.45 dB, crosspolar discrimination is higher than 30 dB.

Analysis of optimization results
Let us investigate the electromagnetic characteristics of a numerical model based on the finite element method in frequency domain [49][50][51] of a waveguide polarizer in the X-frequency range from 7.7 GHz to 8.5 GHz. Fig. 8 shows the phase and matching characteristics of the polarizer. Fig. 8 contains the dependence of the differential phase shift on the frequency, and Fig. 9 contains the dependence of VSWR on the frequency in the operating frequency range from 7.7 GHz to 8.5 GHz of the studied prototype.  Fig. 8 shows that the maximum deviation of the differential phase shift from 90° is 2.3°. Fig. 9 shows that the maximum value of VSWR for both polarizations is 1.29.
Figs. 10 and 11 show the polarization characteristics of the device in the operating frequency range from 7.7 GHz to 8.5 GHz. Fig. 10 contains the dependence of the axial ratio on the frequency, and Fig. 11 contains the dependence of the CPD on the frequency. In Fig. 10 we see that at the frequency of 8.45 GHz the axial ratio acquires its maximum value of 0.4 dB. Also the СPD acquires a maximum value of 29 dB at this frequency.  Such characteristics provide the optimal design of the polarizer, which are presented in Table 1. As we can see, the matching and polarization characteristics of the mathematical model and the numerical model by the finite element method coincide with the corresponding accuracy.
Therefore, the developed tunable waveguide polarizer simultaneously provides good matching and polarization characteristics. The range of change of the differential phase shift is 90° ± 2.3°. The polarizer provides VSWR 1.29, axial ratio 0.45 dB, СPD is 30 dB.

Conclusions
A new tunable waveguide polarizer has been suggested and developed. In the article we have created mathematical model of this type of polarizer based on a square waveguide with two irises and two posts. Theoretical model takes into account the influence of all design parameters on the polarization and matching characteristics of the waveguide polarizer. The developed model allows performing accurate optimization of the device. Optimal matching and polarization characteristics were achieved in the frequency range from 7.7 GHz to 8.5 GHz by changing the geometric dimensions of the structure. In addition, created mathematical model allows taking into account the influence of the height of irises and posts, the distances between them and their thickness on the electromagnetic characteristics of the polarizer. Therefore, the model can be recommended for the analysis and optimization of new tunable microwave polarizers based on waveguides with different numbers of irises and posts in the structure. Future research will focus on the development of the analytical model, which takes into account more reactive elements and more higher order modes.