Electromagnetic Simulation of New Tunable Guide Polarizers with Diaphragms and Pins

In this article we present the results of electromagnetic simulation, development and optimization of a guide polarizer with a diaphragm and pins. An original mathematical model was obtained using the wave matrix approach for a guide polarizer with one diaphragm and two pairs of pins. The discontinuity elements were modeled as inductive or capacitive conductivities for two kinds of linear perpendicular polarizations of the electromagnetic waves. The theoretical model is based on wave scattering and transfer matrices. The total matrix of a polarizer was developed using wave matrices of transmission of separate elements of a polarizer’s structure. Using elements of the general S-parameters the electromagnetic characteristics of a considered polarizer were obtained. In order to test the performance of a suggested mathematical model, it was simulated in a software based on the finite elements method in the frequency domain. The presented design of a polarizer is adjustable due to possibility of mechanical tuning of the heights of applied pins. Considered guide polarizer with one diaphragm and two pairs of pins provides a reflection coefficient of less than 0.36 and a transmission coefficient of more than 0.93 for both linear perpendicularly polarized modes. Therefore, a new theoretical technique was developed in the research for fast electromagnetic simulation of wave matrix elements of a guide polarizers with diaphragms and pins. Developed simulation approach can also be widely used for the development of new tunable guide filters, polarizers, rotators and other microwave components with diaphragms and pins. INDEX TERMS electromagnetic simulation; microwave engineering; waveguide components; electromagnetics; waveguide polarizer; microwave passive devices; wave matrix; polarization.


I. INTRODUCTION
HE modernization of state-of-the-art radio engineering systems has contributed to the emergence of new radar and satellite telecommunication systems, metrological equipment, navigation systems and mobile communication systems. Ku-band is widely used in satellite television systems. One of the main reasons is the shorter electromagnetic wavelengths, which allow the reception of signals by parabolic antennas of small sizes [1]. Such systems are applied in radio engineering systems with polarization signal processing [2][3][4][5][6][7][8][9][10][11]. The development of these systems expanded information capacity of communication channels by the use of modern microwave filters, orthomode converters and waveguide polarizers. In particular, modern telecommunication systems apply various adjustable waveguide filters and other components [12][13][14][15][16][17][18][19]. The operating frequency range is regulated by means of pins. Orthomode transducers support separation of the transmitted and received orthogonal signals for the same antenna system [20][21][22][23][24][25]. They also direct the transmitted wave's energy to an antenna and prevent the transmission to the sensitive receivers in radars.
Waveguide septum polarizers are also frequently used [57][58][59][60][61][62][63]. They represent a specially formed phase-shifting structure created from a conductive metal septum with steps. The advantages of septum polarizers are broadband operation, good ellipticity coefficient of about 1 dB, ease of installation in a waveguide, versatility due to the possibility of choosing the type of polarization by installing a septum in the waveguide and an acceptable cost for the consumer. It is especially important to provide resistance to fading and polarization mismatch for 5G wireless systems [64][65][66][67][68][69][70].
The main advantage of a waveguide polarizer with diaphragms over polarizers of other types is the ability to provide the most broadband mode of operation with good electromagnetic characteristics. A positive feature of using pin polarizers is the ability to tune them [79][80][81][82][83]. On the contrary, waveguide polarizers with diaphragms are not adjustable devices. Consequently, it is proposed to combine wideband diaphragm structures with adjustable pins. Thus, in this research we will introduce several pins into the design of a guide iris polarizer to enable its tuning.
Therefore, in this research we will solve an actual problem of the development of new adjustable square waveguide polarizers applying diaphragm and pins, which provides the effective electromagnetic characteristics in the operating frequency band.

II. THEORETICAL MODELING OF WAVEGUIDE POLARIZERS WITH DIAPHRAGM AND PINS
The mathematical calculation and initial estimation of electromagnetic characteristics of a waveguide polarizers with diaphragms is usually carried out by means of the theory of equivalent microwave circuits [84]. It was implemented before using wave scattering and transmission matrices in [85][86][87]. This approach makes it possible to express the basic values of the polarizer through the elements of the generalized transmission or scattering matrix. Let's use it to create our mathematical model.
The inner geometrical configuration of a developed and analyzed polarization converter containing a diaphragm and four pins is presented in Fig. 1. The design of this polarizer contains four identical pins and one diaphragm. Each pair of pins is located symmetrically with respect to the diaphragm, which in turn is located in the center of the waveguide section.
For the theoretical analysis we will apply a single-wave approach using the wave matrix technique [85][86][87]. The equivalent networks of a considered waveguide polarizer for the cases of capacitive and inductive diaphragms are given in Fig. 2a and 2b, respectively.  The equivalent conductivities of diaphragm in a square waveguide are determined by the following formulas [85]: where a is the width of walls of a square waveguide; d is the width of spacing or diaphragm window; λg is the length of fundamental wave in a considered square guide. Two equivalent circuits have uniform sections of the transmission line with electrical length θ, which is calculated by the formula: . (3) The wavelength in a square waveguide is determined as follows [7,8] where λ0 is the length of electromagnetic wave in vacuum, 2a is the cutoff length of the fundamental TE10 wave in a square guide. The equivalent conductivity of the pin in a rectangular waveguide is defined by the following expression [88]: , is pin's height, k stands for the wave number, r designates radius of a cylindrical pin.
For the vertically polarized fundamental mode TE10 the equivalent circuit of a square waveguide can be divided into five simple two-port circuits (Fig. 2, a). They include a twoport circuit that is equivalent to a capacitive diaphragm, two circuits that are equivalent to a capacitive pin, and two circuits of regular transmission line segments. These circuits are described by the transfer matrices [10]: cosec ln , (8) where and denote the conductivity of the pin and diaphragm, respectively; θ1 is an electric length of the equivalent transmission line.
According to the theory of wave matrices, general wave T-matrix of the waveguide polarizer is given as the product of partial transfer matrices [7,8]: . (9) For propagation of the main electromagnetic wave polarized in horizontal direction the equivalent circuit of the polarizer can be divided into three simple two-port circuits (Fig. 2b). They include a two-port inductive diaphragm equivalent circuit and two two-port transmission line section circuits. In this case the influence of an inductive pins is neglected. The equivalent circuits are described by the following transfer matrices: ; (10) , (11) where θ2 stands for an electric length of the equivalent transmission line. Then the general wave transmission matrix for the fundamental electromagnetic mode with horizontal linear polarization is determined as follows: .
The mathematical relation between wave T-matrix and S-matrix is determined by the following formula [84][85][86]: , (13) where denotes the determinant of transfer matrix calculated using (9) and (12). From the formula (13) we obtain expressions for the reflection factor and transfer factor of the analyzed waveguide polarizer [7,8] as follows: ; . (14) Therefore, the mathematical model of a guide polarizer with diaphragm and pins was developed based on the wave matrix method. In the following section we will apply created mathematical model for analysis of electromagnetic characteristics and development of a guide polarizer.

III. RESULTS OF MATHEMATICAL AND NUMERICAL SIMULATION OF ELECTROMAGNETIC PERFORMANCE
This section presents the results of calculations of the reflection and transfer factors of the waveguide polarizer obtained using the developed mathematical model for the satellite operating frequency range from 11.7 to 12.5 GHz. Fig. 3 indicates that module of the reflection coefficient of the developed mathematical model of a waveguide polarizer for both fundamental modes TE01 and TE10 is less than 0.37 in the frequency range from 11.7 GHz to 12.5 GHz. The lowest magnitude of the absolute value of the reflection factor of the polarizer is 0.06 at a frequency of 11.7 GHz.      5 shows the dependences of conductivity on frequency for the pin of developed polarizer for different heights. Due to the inaccuracy of manufacturing, the characteristics of a real polarizer would differ from those calculated using a mathematical model. They can be adjusted by changing the height of the pins. In Fig. 5 it is seen that with an increase in the height of the pin, the maximum value of the conductivity of the pin increases. Below we present the results of numerical modeling applying the finite element technique in the frequency domain [89,90]. As a result, graphical dependences of the reflection and transfer factors of the polarizer in the operating frequency range were obtained. Fig. 6 presents the dependences of module of reflection coefficient on frequency for both polarizations.     6 demonstrates that module of the reflection coefficient of developed waveguide polarizer for both fundamental modes TE01 and TE10 is less than 0.36 in the DBS-band from 11.7 GHz to 12.5 GHz. The lowest magnitude of the modulus of the reflection coefficient of the polarizer is 0 at a frequency of 12.06 GHz. Fig. 8 and Fig. 9 present the dependences absolute value of the transmission coefficient and its argument on frequency for the developed polarizer.  In Fig. 8 it can be seen that in the operating frequency range of 11.7-12.5 GHz the modulus of the transmission coefficient reaches a maximum value of 1.0 at a frequency of 12.06 GHz. Fig. 9 shows that the phase for horizontal polarization decreases from 165° to 78°, and for vertical polarization decreases from 78° to -20°. The minimum value of the modulus of the transmission coefficient of the waveguide polarizer is 0.931 at a frequency of 11.7 GHz.  In Fig. 10 it can be seen that in the operating frequency range of 11.7-12.5 GHz, the modulus of the cross polarization transformation reaches a maximum value of −86.5 dB at a frequency of 12.02 GHz. The minimum value of the cross polarization transformation by vertical polarization is −88.3 dB at 12.47 GHz. The minimum value of the cross polarization transformation by vertical polarization is −91.1 dB at 11.7 GHz and the maximum value of the cross polarization transformation by vertical polarization is −87.5 dB at 12.5 GHz. Table 1 presents main electromagnetic parameters of the optimized guide polarizer with a diaphragm and 4 pins in the operating band 11.7-12.5 GHz. These results were obtained by the mathematical and numerical models. Therefore, we can conclude from Table 1 that all the maximal levels of the reflection coefficient and minimal levels of the transmission coefficient almost coincided for both theoretical methods. This proves the correctness of the developed electromagnetic model of a waveguide polarizer with a diaphragm and pins.

IV. CONCLUSIONS
In this paper a new mathematical model of a polarizer based on a square waveguide with a diaphragm and four pins was developed. The proposed design of a polarizer provides the possibility to adjust the electromagnetic characteristics of the polarizer by changing the length of the pins. Using the developed mathematical model, the total scattering matrix elements were obtained and optimized in the operating satellite DBS-band. The waveguide polarizer maintains modulus of the reflection coefficient less than 0.37. The modulus of the transmission coefficient is higher than 0.93 for both perpendicular linear polarizations. The cross polarization transformation reaches a maximum value of −86.5 dB at a frequency of 12.02 GHz and reaches a minimum value of −91.1 dB at the frequency of 11.7 GHz. The developed theoretical model can be used to develop and optimize scattering matrix elements of waveguide polarizers and other components with diaphragms and pins. Next researches need to focus on the creation of mathematical models with a larger number of diaphragms and pins to ensure better performance of the polarizer. The proposed waveguide structure can also be suggested to create new adjustable waveguide filters and phase shifters for various radio engineering applications.