Abstract
The development of new mathematical model for guide polarization converter with diaphragms was carried out in the research by the method of wave matrices. In addition, numerical modeling of the performance of a polarizer with diaphragms is made by simulating the propagation of the fundamental modes with perpendicular linear polarizations. The wave matrix model was obtained by splitting the polarizer into separate structural elements. Each element was described by its own wave transmission matrices. As a result, a general wave scattering matrix was formed. Based on the elements this matrix the electromagnetic characteristics of the considered polarizer were obtained theoretically. In particular, complex reflection and transmission coefficients were calculated. Their modules and phases were analyzed in the frequency interval 10.712.8 GHz. To check the correctness of the obtained results an independent numerical simulation was carried out applying the finite element methodology in the frequency interval. The results of both approaches are in good agreement. The engineered converter of polarization with four diaphragms provides a reflection coefficient modulus of less than 0.14 and a transfer coefficient modulus of more than 0.99 for two orthogonal types of polarizations. As a result, a rigorous mathematical method was developed to analyze the elements of the scattering matrix of a waveguide polarizer with diaphragms. It can be used for the development of new broadband waveguide polarizers and waveguide filters based on diaphragm elements.
Index Terms
Electromagnetic simulation; microwave passive devices; waveguide components; polarization; waveguide polarizer; diaphragm polarizer; wave matrix; reflection coefficient; transmission coefficient
I. INTRODUCTION
The fast evolution of modern communication systems in the millimeter range has contributed to the emergence and development of adaptive antenna systems with double polarization signal processing [^{1}1 D. M. Pozar, Microwave Engineering. Hoboken, New Jersey, USA: John Wiley and Sons, 2012.], [^{2}2 R. Sorrentino, and G. Bianchi, Microwave and RF Engineering. New Jersey, USA: John Wiley and Sons, 2010.]. The key elements of such systems are polarization transforming passive devices including orthomode converters, guide polarizers and microwave duplexers.
The polarizer of an antenna system converts linearly polarized input electromagnetic waves intocircularly polarized and performs the reverse transformation. Application of this waveguidecomponent allows to enhance the information volumes and channel's capacity in wirelesstelecommunication systems. Polarization processing devices have various designs based on the circular or rectangular/square waveguides. The idea of constructing of polarizers is based on the introduction of a quasiperiodic or inhomogeneous structure into the waveguide that will implement a phase shift of 90° between the modes with two perpendicular polarizations [^{3}3 G. Virone, R. Tascone, O. A. Peverini, G. Addamo, and R. Orta, “Combinedphaseshift waveguide polarizer”, IEEE Microwave and Wireless Components Letters, vol. 18, no. 8, pp. 509–511, August 2008. DOI: 10.1109/LMWC.2008.2001005.
https://doi.org/10.1109/LMWC.2008.200100...
][^{8}8 F. Dubrovka, et al., “Compact Xband steppedthickness septum polarizer”, IEEE Ukrainian Microwave Week (UkrMW), Kharkiv, Ukraine, September 2020, pp. 135–138. DOI: 10.1109/UkrMW49653.2020.9252583.
https://doi.org/10.1109/UkrMW49653.2020....
].
Historically, the first microwave devices for transformation of electromagnetic waves’ polarization type were based on waveguides with reactive pins [^{9}9 S. Piltyay, “Circular waveguide polarizer for weather radars and satellite information systems,” Journal of Microwaves, Optoelectronics and Electromagnetic Applications, vol. 20, no. 3, September 2021, pp. 475–489. DOI: 10.1590/217910742021v20i31183.
https://doi.org/10.1590/217910742021v20...
][^{13}13 A. V. Bulashenko, S. I. Piltyay, H. S. Kushnir, and O. V. Bulashenko, “Compact waveguide polarizer with three antiphase posts,” Visnyk of Vinnytsia Polytechnical Institute, no. 5, pp. 97–104, November 2020. (in Ukrainian). DOI: 10.31649/199792662020152597104.
https://doi.org/10.31649/199792662020...
]. Guide structures with posts provide developers with the possibility of adjustment of charactristics. On the other hand, the main disadvantage of the polarizer designs with pins is their relatively narrow operating frequency band. Operating range of waveguide devices can be extended using ridged structures and diaphragms. Recently, new kinds of polarizers, which are based on coaxial and other waveguides with mentioned discontinuities, began to appear [^{14}14 B. Subbarao and V. F. Fusco, “Compact coaxialfed polarizer,” IEEE Antennas and Wireless Propagation Letters, vol. 3, pp. 145–147, April 2004. DOI: 10.1109/LAWP.2004.831084.
https://doi.org/10.1109/LAWP.2004.831084...
][^{20}20 O. Sushko, et al., “Symmetrically fed 1‒10 GHz logperiodic dipole antenna array feed for reflector antennas”, IEEE Ukrainian Microwave Week (UkrMW), Kharkiv, Ukraine, September 2020, pp. 222–225. DOI: 10.1109/UkrMW49653.2020.9252778.
https://doi.org/10.1109/UkrMW49653.2020....
]. They provide broad frequency bands, but contain a significant number of reactive elements in the form of diaphragms. This leads to the larger overall sizes of their designs, which is their main disadvantage. Besides, several novel designs of polarizers based on structures with sectoral coaxial ridged waveguides are reported in [^{21}21 F. F. Dubrovka et al., “Eigenmodes of sectoral coaxial ridged waveguides”, Radioelectronics and Communications Systems, vol. 55, no. 6, pp. 239–247. June 2012. DOI: 10.3103/S0735272712060015.
https://doi.org/10.3103/S073527271206001...
][^{27}27 S. I. Piltyay, “Numerically effective basis functions in integral equation technique for sectoral coaxial ridged waveguides”, 14th International Conference on Mathematical Methods in Electromagnetic Theory (MMET), Kyiv, Ukraine, August 2012, pp. 492–495. DOI: 10.1109/MMET.2012.6331195.
https://doi.org/10.1109/MMET.2012.633119...
]. The application of coaxial guides allows to obtain dualband operation of antenna systems based on them.
All existent types of waveguide polarizers require the utilization of complex mathematical methods for their analysis. Among them we can highlight fieldmatching technique [^{28}28 J.M. Rebollar and J. Esteban, “CAD of corrugated circularrectangular waveguide polarizers,” Proceedings of Eighth Int. Conf. on Antennas and Propagation, Edinburgh, U.K., April 1993.][^{31}31 M. Warecka, R. Lech, and P. Kowalczyk, “Scattering and propagation analysis for the multilayered structures based on field matching technique,” Proceedings of 13th European Conference on Antennas and Propagation, Krakow, Poland, April 2019.] and integral equations technique [^{32}32 H. Kushnir et al., “Multiple reflections method for diaphragms polarizers development”, IEEE International Conference on Problems of Infocommunications. Science and Technology, Kharkiv, Ukraine, October 2021.][^{34}34 F. F. Dubrovka and S. I. Piltyay, “Eigenmodes analysis of sectoral coaxial ridged waveguides by transverse fieldmatching technique. Part 2. Numerical results”, Visnyk NTUU KPI Seriia – Radiotekhnika, Radioaparatobuduvannia, vol. 55, pp. 13–23. 2013. DOI: 10.20535/RADAP.2013.55.1323
https://doi.org/10.20535/RADAP.2013.55.1...
]. Consequently, creation of new easier approaches for the analysis of characteristics of waveguide polarizers is a stateoftheart scientific and engineering problem.
In dualband polarization processing units the polarizers are used in conjunction with orthomode transducers. An orthogonal modes converter is a guide device that discriminates two orthogonal signals with linear polarization within their common frequency range [^{35}35 F. F. Dubrovka et al., “A high performance ultrawideband orthomode transducer and a dualpolarized quadridged horn antenna based on it”, VIII International Conference on Antenna Theory and Techniques (ICATT), Kyiv, Ukraine, September 2011, pp. 176–178. DOI: 10.1109/ICATT.2011.6170737.
https://doi.org/10.1109/ICATT.2011.61707...
][^{39}39 E. Menargues et al., “Fourport broadband orthomode transducer enabling arbitrary interelement spacing,” IEEE Transactions on Microwave Theory and Techniques, vol. 66, no. 12, pp. 5521–5530, November 2018. DOI: 10.1109/TMTT.2018.2878208.
https://doi.org/10.1109/TMTT.2018.287820...
]. In addition, the application of waveguide filters [^{40}40 M. Omelianenko and T. Romanenko, “Eplane stepped–impedance bandpass filter with wide stopband,” Proceedings of IEEE 40th International Conference on Electronics and Nano technology, Kyiv, Ukraine, April 2020. DOI: 10.1109/ELNANO50318.2020.9088888.
https://doi.org/10.1109/ELNANO50318.2020...
][^{46}46 O. Myronchuk, O. Shpylka, and S. Zhuk, “Algorithm of channel frequency response estimation in orthogonal frequency division multiplexing systems based on Kalman filter,” Proceedings of IEEE 15th Int. Conf. on Advanced Trends in Radioelectronics, Telecommunications and Computer Engineering, LvivSlavske, Ukraine, Feb. 2020. DOI: 10.1109/TCSET49122.2020.235385.
https://doi.org/10.1109/TCSET49122.2020....
] in conjunction with polarizers improves the efficiency of polarization processing of signals in antenna systems.
Diaphragm polarizers are broadly utilized in stateoftheart satellite telecommunication systems [^{47}47 F. F. Dubrovka, et al., “Analytical and numerical method of constructive synthesis of optimal polarizers based on three irises in square waveguide,” Radioelectronics and Communications Systems, vol. 64, no 4, pp. 204–215, April 2021. DOI: 10.3103/S073527272104004X.
https://doi.org/10.3103/S073527272104004...
][^{52}52 S. I. Piltyay, A. V. Bulashenko, and I. V. Demchenko, “Waveguide iris polarizers for Kuband satellite antenna feeds”, Journal of Nano and Electronic Physics, vol. 12, no. 5, pp. 050241–050245, October 2020. DOI: 10.21272/jnep.12(5).05024.
https://doi.org/10.21272/jnep.12(5).0502...
]. Besides, there are effective guide polarizer designs with diaphragms and pins [^{53}53 L. PoloLopez, J. L. MasaCampos, and J. A. RuizCruz, “Design of a reconfigurable rectangular waveguide phase shifter with metallic posts,” Proceedings of European Microwave Conf., Nuremberg, Germany, October 2017. DOI: 10.23919/EuMC.2017.8231036.
https://doi.org/10.23919/EuMC.2017.82310...
][^{58}58 I. I. Dikhtyaruk et al., “Technique of mathematical synthesis of waveguide iris polarizers”, Journal of Nano and Electronic Physics, vol. 13, no. 5, pp. 050211–050215, 2021.]. The main advantage of a polarizer with diaphragms over guide polarizers of other types is the ability to provide the most broadband functioning with efficient parameters, which can be upgraded by adding additional diaphragms to the design. The disadvantage is the increase in the length of the polarizer, which occurs in this case.
Diaphragm polarizers can be realized in the form of separate plates, which are placed at a certain distance from each other. They can also be realized with one specially shaped plate. Such devices are called septum polarizers [^{59}59 A. V. Bulashenko, S. I. Piltyay, Y. I. Kalinichenko, and I. V. Zabegalov, “Waveguide polarizer for radar and satellite systems”, Visnyk NTUU KPI Seriia – Radiotekhnika Radioaparatobuduvannia, vol. 86, September 2021, pp. 5–13. DOI: 10.20535/RADAP.2021.86.513.
https://doi.org/10.20535/RADAP.2021.86.5...
][^{66}66 Z. Ding, R. Jin, S. Yang, and X. Zhu, “Design of septum waveguide feed manufactured by splitblock technique,” Proceedings of IEEE International Sumposium on Antennas and Propagation and North American Radio Science Meeting, Montreal, Canada, July 2020. DOI: 10.1109/IEEECONF35879.2020.9330045.
https://doi.org/10.1109/IEEECONF35879.20...
]. The septum divides the waveguide into two rectangular waveguides. These waveguides form ports that are used to transmit or receive linearly polarized or circularly polarized waves. Such polarizers are broadly applied in stateoftheart 5G telecommunication systems of the millimeter wavelength range [^{67}67 C. Shu, J. Wang, and X. Chen, “A wideband dualcircularpolarization horn antenna for mmWave wireless communications,” IEEE Antennas and Wireless Propagation Letters, vol. 18, no. 9, pp. 1726–1730, September 2019. DOI: 10.1109/LAWP.2019.2927933.
https://doi.org/10.1109/LAWP.2019.292793...
][^{75}75 C. Shu, S. Hu, X. Chen and Y. Yao, “Deeplearningbased inverse modeling with CMAES as applied to the design of a wideband highisolation septum polarizer,” Proceedings of IEEE International Sumposium on Antennas and Propagation and North American Radio Science Meeting, Montreal, Canada, July 2020. DOI: 10.1109/IEEECONF35879.2020.9330082.
https://doi.org/10.1109/IEEECONF35879.20...
]. The disadvantage of such guide polarizers is complex mathematical methods of their analysis.
Therefore, the development of new mathematical model of a waveguide polarizer with diaphragms, which will allow to carry out fast analysis and optimization of its characteristics, including magnitudes and phases of the transmission and reflection coefficients, is a relevant scientific problem.
II. THEORETICAL ANALYSIS OF WAVEGUIDE POLARIZER WITH FOUR DIAPHRAGMS
The inner structure and overall dimensions of an investigated polarizer based on a waveguide with diaphragms are demonstrated in Fig. 1.
The design of shown square guide polarizer contains four conducting diaphragms. The outer diaphragms are located symmetrically with respect to the inner diaphragms, which are located in the center of the waveguide.
For the theoretical analysis of the device we will use a singlewave approximation and the techniques of wave matrix theory [^{76}76 A. Bulashenko, S. Piltyay, and I. Demchenko, “Analytical technique for iris polarizers development”, IEEE International Conference on Problems of Infocommunications. Science and Technology (PIC S&T), Kharkiv, Ukraine, October 2020, pp. 593–598. DOI: 10.1109/PICST51311.2020.9467981.
https://doi.org/10.1109/PICST51311.2020....
][^{80}80 S. Piltyay, A. Bulashenko, I. Fesyuk, and O. Bulashenko, “Comparative analysis of compact satellite polarizers based on a guide with diaphragms”, Advanced Electromagnetics, vol. 10, no. 2, pp. 44–55, July 2021. DOI: 10.7716/aem.v10i2.1713.
https://doi.org/10.7716/aem.v10i2.1713...
]. The equivalent circuits of a waveguide polarizer for the case of inductive and capacitive diaphragms are given in Fig. 2.
The conductivity of inductance diaphragm in the square waveguide is determined by the formula [^{81}81 L. Lewin, Theory of Waveguides: Techniques for the Solution of Waveguide Problems. London, U.K.: NewnesButterworth, 1975.]
The conductivity of capacity diaphragm in the square waveguide is determined by the formula [^{81}81 L. Lewin, Theory of Waveguides: Techniques for the Solution of Waveguide Problems. London, U.K.: NewnesButterworth, 1975.]
Where a is the size of waveguide's cross section; d is the width of the gap or diaphragm window;
λ_{w} is the wavelength in a considered square guide.
Two circuits have regulatory sections of the transmission line with electrical length θ, which is calculated by the formula [^{82}82 J. Dobrowolski, Scattering parameters in RF and microwave circuit analysis and design. Norwood, USA: Artech House, 2016.]
where l is the length of transmission line segment.
The guide wavelength is determined as follows [^{83}83 W. L. Stutzman, Polarization in Electromagnetic Systems. Norwood, MA, USA: Artech House, 2018.]
where λ_{0} is the wavelength in vacuum, λ_{c} is critical wavelength in the rectangular waveguide.
The equivalent circuit of a square waveguide can be divided into seven simple twoport circuits (Fig. 2). They include two circuits equivalent to outer diaphragms, two circuits equivalent to central diaphragms, two circuits of regular transmission line segments of length l_{1} and circuit of a regular transmission line of length l_{2}. These circuits are described by transfer matrices [^{84}84 P.J. Clarricoats and A.D. Olver, Corrugate Horn Antennas. London, U.K.: Peter Peregrinus, 1984, 484 p.]:
Where Y_{1} and Y_{2} are conductivity of outer and middle diaphragms; θ_{1} and θ_{2} are electric length of the transmission lines.
The wave transfer matrix of the waveguide polarizer is calculated as follows [^{85}85 K. Kurokawa, An introduction to the theory of microwave circuits. New Jersey, USA: Murray Hill, 1969.]
Relations between the total wave transfer and scattering matrices are determined by the known expressions [^{86}86 I. Bahl, Lumped elements for RF and microwave circuits. Norwood, USA: Artech House, 2003.], [^{87}87 M. Golio, J. Golio, RF and microwave circuits, measurments, and modeling. New York, USA: Taylor & Francis Group, 2007, 774p.]
From (10) we obtain expressions for the reflection coefficient [^{88}88 Y.B. Siddik, Design of ultra wideband power transfer networks. New York, USA: John Wiley & Sons, 2010, 774p.]
From (10) we obtain expressions for the transmission coefficient [^{89}89 R. Ludwig, P. Bretchko, RF Circuit design: theory and applications. New Jersey, USA: Prentice Hall, 2000, 653p.]
Therefore, we will further analyze the obtained coefficients.
III. RESULTS OF MATHEMATICAL MODELING
The section presents the electromagnetic characteristics of the polarizer calculated using the developed model of the polarizer. The developed guide device with four diaphragms was designed for the operating frequency range from 10.7 to 12.8 GHz.
The dimensions of the optimal waveguide convertor of polarization calculated using the proposed matrix technique and the method of finite integration are as follows. The wall size of the square waveguide is 21.6 mm. The height of the outer diaphragm is 3.91 mm, height of the central diaphragm is 2.38 mm. The distance between the central diaphragm is 7.38 mm, the distance between the outer diaphragms is 7.38 mm. The thickness of the optimal diaphragms is 1 mm.
Frequency dependences of the module and phase of the reflection coefficients for both polarizations are presented in Fig 3a, 3b, respectively.
Dependences of reflection coefficient on frequency for both polarizations calculated by mathematical model: (a) module; (b) phase.
In Fig. 3a a typical behavior of the reflection of electromagnetic waves propagating inside a polarizer's structure versus frequency is observed for both polarizations. Namely, the module of the reflection coefficient for the horizontal polarization decreases with the increment of the frequency. On the contrary, the module of the reflection coefficient of the mode with vertical polarization creases with the increase of frequency of the fundamental electromagnetic mode.
In Fig. 3b it is possible to see that the phase of reflection coefficient for the vertical polarization is less −65° at a frequency of 10.7 GHz and for the horizontal polarization it is less than −150° at the lowest frequency of 10.7 GHz.
Fig. 4 presents the dependence the module and argument of transmission coefficient on the frequency for vertical and horizontal polarization the frequency interval 10.712.8 GHz.
Dependences of transmission coefficient on frequency for both polarizations calculated by mathematical model: (a) module; (b) phase.
In Fig. 4a it is observed that the module of transmission coefficient for the horizontal polarization is higher than 0.990 at a frequency of 10.7 GHz and for the vertical polarization is less 0.991 at a frequency of 12.8 GHz. In Fig. 4b it can be seen that the phase of transmission coefficient for the vertical polarization is less −50° at a frequency of 10.7 GHz and for the horizontal polarization is less 120° at a frequency of 10.7 GHz.
Consequently, in the frequency range 10.712.8 GHz the optimal design of a guide polarizer with four diaphragms provides the following matching characteristics. The reflection coefficient for both polarizations is less than 0.14. The transmission coefficient for both polarizations is greater than 0.99.
IV. NUMERICAL SIMULATION OF THE DEVELOPED WAVEGUIDE POLARIZER
This section contains the results of modeling a waveguide polarizer with four diaphragm applying the wellknown electrodynamic method. As a proposed method we used the finite element technique in the frequency domain [^{90}90 S. Piltyay, A. Bulashenko, Ye. Herhil, and O. Bulashenko, “FDTD and FEM simulation of microwave waveguide polarizers”, IEEE 2nd International Conference on Advanced Trends in Information Theory (IEEE ATIT), Kyiv, Ukraine, November 2020, pp. 357–363.DOI: 10.1109/ATIT50783.2020.9349339.
https://doi.org/10.1109/ATIT50783.2020.9...
][^{92}92 S. Piltyay, A. Bulashenko, et al., “Analytical modeling and optimization of new Kuband tunable square waveguide irispost polarizer”, International Journal of Numerical Modelling: Electronic Network, Devices and Fields, vol. 34, no. 5, pp. 1–27, 2021. DOI: 10.1002/JNM.2890.
https://doi.org/10.1002/JNM.2890...
]. The electromagnetic characteristics of the polarizer were simulated for the frequency interval from 10.7 to 12.8 GHz using the numerical method of finite elements in the frequency domain. This technique proved its computational speed and reliability for the calculation of the electromagnetic performance of waveguide polarizers in several recent investigations [^{93}93 I. Fesyuk et al., “Waveguide polarizer for radar systems of 2 cm wavelength range”, IEEE 3rd Ukraine Conference on Electrical and Computer Engineering, Lviv, Ukraine, August 2021, pp. 15–20.][^{96}96 A. Polishchuk et al., “Compact postsbased waveguide polarizer for satellite communications and radar systems”, IEEE 3rd Ukraine Conference on Electrical and Computer Engineering, Lviv, Ukraine, August 2021, pp. 78–83.].
In Fig. 5a it is possible to see that the module of the reflection coefficient of developed waveguide polarizer with diaphragms for both fundamental modes TE_{01} and TE_{10} is less than 0.11 in the frequency range 10.712.8 GHz. This result is in good agreement with the obtained before using wave matrix method. The minimum value of the modulus of the reflection coefficient of the polarizer is 0 at a frequency of 12.42 GHz. In Fig. 5b it can be seen that the phases of reflection coefficients of both polarizations decrease with the increment of frequency. Their difference is close to 50° in the whole operating frequency band 10.712.8 GHz.
Dependences of reflection coefficient on frequency for both polarizations calculated by numerical model: (a) module; (b) phase.
In Fig. 6a it can be seen that the transmission coefficient for the vertical polarization is less than 1, which corresponds to its physical sense. The peak value is reached at the frequency of 12.44 GHz. For the horizontal polarization the transmission coefficient is less than 0.997. Its maximal value is obtained at the frequency of 11.3 GHz.
Dependences of transmission coefficient on frequency for both polarizations calculated by numerical model: (a) module; (b) phase.
In Fig. 6b it is possible to see that the phase of transmission coefficient for the vertical polarization is less than 10° at a frequency of 10.7 GHz and for the horizontal polarization it is less than 100° at the frequency of 10.7 GHz.
Table I presents the characteristics of the developed polarizer, which were optimized for the Ku band for mathematical and numerical model. It demonstrates that the electromagnetic characteristics obtained using both the developed mathematical model and a numerical model of a waveguide polarizer with four diaphragms are in good agreement.
Consequently, it can be seen from Table 1 that all the parameters, which were calculated using the mathematical model and the numerical method, correlate with each other. This verifies the correctness of the created mathematical model. It can be widely applied for fast estimation of modules and arguments of the scattering parameters of polarizers, rotators, filters and other microwave devices based on discontinuities located in waveguides.
V. CONCLUSIONS
In this article we have developed a model of a microwave square polarizer with four diaphragms. Using the proposed model the electromagnetic parameters of a polarizer were optimized in the frequency range 10.712.8 GHz. The developed device ensures a reflection coefficient modulus of less than 0.14 for both linear polarization. The modulus of the transmission coefficient of a polarizer is more than 0.99 for both polarizations.
The results, which were obtained by the wave matrix model, can be used to develop and optimize waveguide polarizers with diaphragms. Further theoretical researches should focus on the creation of mathematical models for the devices with a larger number of diaphragms, which will provide better characteristics of the reflection and transmission coefficients. In addition, the presented structures can be effectively used for designing of new waveguide filters for various purposes.
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Publication Dates

Publication in this collection
15 Nov 2021 
Date of issue
Dec 2021
History

Received
23 May 2021 
Reviewed
03 June 2021 
Accepted
23 Sept 2021