Directional Diagram Research Articles

Features computation diagram of annular direction antenna arrays what made on half-wave vibrators located above cylindrical surface.

The use of antenna-feeder devices, especially installed on moving objects, indicates the need to modernize and develop new type antenna devices to increase the efficiency of the radio communication system in conditions of active radio electronic suppression. One of the options for providing interference protection in communication channels with moving objects is the use of narrowly directed Smart antennas with a controlled directional pattern. Smart antennas, which are also called intelligent antennas, are one of the varieties of phased antenna arrays. Smart antennas use a set of radiating elements built in the form of grids. The most widespread are the ring antenna arrays, which are characterized by their compactness and great functionality. The relevance of the task of designing Smart antennas is to reduce the total number of emitters without degrading the antenna parameters. This task is particularly important for antennas with circular radiation patterns in communication systems with moving objects. The obtained results clearly show the possibilities of ring antenna arrays layout, depending on the radius of the antenna array and the radius of the cylindrical support, and under which conditions the geometric dimensions of the ring antenna arrays will preserve the coefficient of uniformity of the directional pattern. In order to reduce the number of vibrators in such antennas, the maximum distance between the vibrators is usually found, at which the uniformity of β directional diagrams does not decrease below a given level. To find the permissible distance, you can calculate the dependence of β on the radius of the grating directly when calculating the azimuthal directional diagrams of the antenna. To solve this problem, the external characteristics of the ring antenna arrays performed on half-wave symmetric vibrators in ideal conditions and vibrators placed over a cylindrical surface were considered.

Measurement of dynamic direction diagrams of phased antenna arrays in process of radar objects tracking

Measurement of dynamic direction diagrams of phased antenna arrays in process of radar objects tracking

Optimal Design of Sparse Matrix Phased Array Using Simulated Annealing for Volumetric Ultrasonic Imaging with Total Focusing Method.

The total focusing method (TFM) is often considered to be the 'gold standard' for ultrasonic imaging in the field of nondestructive testing. The use of matrix phased arrays as probes allows for high-resolution volumetric TFM imaging. Conventional TFM imaging involves the use of full matrix capture (FMC) for ultrasonic signals acquisition, but in the case of a matrix phased array, this approach is associated with a huge volume of data to be acquired and processed. This severely limits the frame rate of volumetric imaging with 2D probes and necessitates the use of high-end equipment. Thus, the aim of this research was to develop a novel design method for determining the optimal sparse 2D probe configuration for specific conditions of ultrasonic imaging. The developed approach is based on simulated annealing and involves implementing the solution of the sparse matrix phased array layout optimization problem. In order to implement simulated annealing for the aforementioned task, its parameters were set, the acceptance function was introduced, and the approaches were proposed to compute beam directivity diagrams of sparse matrix phased arrays in TFM imaging. Experimental studies have shown that the proposed approach provides high-quality volumetric imaging with a decrease in data volume of up to 84% compared to that obtained using the FMC data acquisition method.

Master formulas for N -photon tree level amplitudes in plane wave backgrounds

The presence of strong electromagnetic fields adds huge complexity to QED Feynman diagrams, such that new methods are required to calculate higher-loop and higher-multiplicity scattering amplitudes. Here we use the worldline formalism to present “master formulas” for all tree level amplitudes of two massive particles and an arbitrary number of photons, in a plane wave background, in both scalar and spinor QED. The plane wave is treated without approximation throughout, meaning in particular that our formulas are valid in the strong-field regime of current theoretical and experimental interest. We check our results against literature expressions obtainable at low multiplicity via direct Feynman diagram calculations. Published by the American Physical Society 2024

Studying the Influence of Moisture Content in Grain on the Efficiency of the Microwave Field Transfer into a Grain Layer with the Use of a Computer Model

The objective of these studies was to determine the dependence of the distribution of the microwave field into a grain layer and the efficiency of energy transfer from the magnetron into the grain layer, via a waveguide, on the moisture content. The subject of the research was a semicircular waveguide with slot radiators that supply a microwave field to the grain surrounding the waveguide. The change in the directional diagram of the microwave field in the longitudinal direction of the waveguide was studied using CST Microwave Studio 2019 software. It was found that the change in the moisture content of the processed grain leads to a significant change in both the radiation and total efficiency values of the waveguide. For instance, the radiation efficiency of the waveguide decreases by 15.1% (from 68.729 dB to 58.294 dB) when the moisture content of the processed grain increases from 14% to 26%. The total radiation efficiency also decreases (from 11.27 dB to 21.7 dB). In this case, not only the value of the radiation efficiency but also the shape of its dependence on the magnetron radiation frequency change. The results of this study enable the formulation of a requirement for a selective approach to the alignment of emitting waveguides depending on the moisture content of the processed grain. Data were obtained on the change in the Q-factor for resonators in which grain layers are processed, depending on the variation in grain moisture content. The research results suggest that when designing microwave convective zones for grain processing, the quality factor of the resonator should not be considered as the primary parameter. The main focus should be on maximizing the radiation efficiency of the waveguides.

Acoustic field reconstruction in the presence of masking objects

In acoustics, field reconstruction methods aim at retrieving acoustic fields (pressure, velocity and intensity) from acoustic measurements around a radiating source, which is often a vibrating structure of complex shape (pumps, engines…). If they are extensively used in laboratory conditions, their application to in situ characterization is not straightforward due to the presence of disturbing sources or masking objects, or to non-anechoic environments. The inverse Patch Transfer Function (iPTF) approach, thanks to the concept of virtual acoustic volume modelled by a finite element model, has already demonstrated its ability to deal with sources of complex shape and the presence of disturbing stationary sources in a non-anechoic acoustic environment. The objective of this article is to show how the presence of rigid masking objects can be easily and efficiently taken into account. A numerical experiment consisting of a thin, simply supported rectangular plate radiating noise in a semi-infinite acoustic field and partially masked by a rigid parallelepiped is presented. The acoustic fields identified and the directivity diagrams are compared with the reference and show that iPTF is able to cancel the presence of the masking object even if the latter completely covers the radiating plate. Finally, an industrial example consisting in an electric motor in operation is presented. Two configurations were tested: with and without the presence of a rigid object. Comparison of the results shows that the fields identified are in good agreement, demonstrating the ability of iPTF to cancel out the effect of masking objects.

Development of n-heptane/2-butyltetrahydrofuran mechanism based on decoupling approach

This study aims to investigate 2-butyltetrahydrofuran (2-BTHF) as a potential alternative to diesel engine fuel. In response to the fact that the chemical kinetic mechanism for the combustion of 2-BTHF blended with diesel has not yet been established, the present study simplified and optimized the mechanism of 2-BTHF and n-heptane by using the direct relationship diagram method and sensitivity analysis and constructed a combined 2-BTHF/n-heptane mechanism based on the decoupling method. The validation of experimental results shows that the combined mechanism can accurately predict the combustion process and agrees well with the experimental data, and thus a reliable 2-BTHF/n-heptane combined mechanism has been successfully established, which provides an effective tool for simulating the combustion of diesel and the mixture of 2-BTHF and diesel with a high degree of accuracy and applicability.

Application of Genetic Algorithm for Synthesizing Onedimensional Thinned Antenna Arrays to form Twodimensional Thinned MIMO Antenna Arrays

The possibility of using thinned classical linear antenna arrays to synthesize two-dimensional thinned MIMO antenna arrays on their basis is considered. The MIMO antenna array which receiving and transmitting elements are located on two mutually orthogonal lines is regarded. On the example of this array it is shown that the directivity diagram of its virtual aperture has the property of separability of spatial variables. It allows using the arrangement of elements of a thinned classical linear antenna array for the receiving and transmitting elements lines. It is demonstrated that a thinned MIMO antenna array can form a radiation pattern with a lower level of side lobes than a full array while keeping the width of the main lobe unchanged. The dependence allowing to determine the peak level of side lobes of a twodimensional thinned MIMO antenna array at a given thinning factor is given.

3‐D woven rectangular hollow structure conformal bearing microstrip antenna with different heights: Design, preparation, performance, and simulation

AbstractTraditional microstrip antennas have problems with heavy weight and poor gain. In order to reduce the weight of the antenna itself and improve its gain, the preforms of the 3‐D woven rectangular hollow structure conformal bearing microstrip antenna (3DWRHS‐CB‐MA)with different heights were woven from ultra‐high molecular weight polyethylene (UHMWPE) filament yarns and copper wires on a common loom through reasonable design. The preform was used as the reinforcement and epoxy resin was used as the matrix, four different heights (2, 4, 6, and 8 mm) of 3DWRHS‐CB‐MA were prepared by vacuum‐assisted resin transfer molding (VARTM) process. Finally, the electromagnetic performance was analyzed by high‐frequency structure simulator (HFSS) and experimental testing. The results showed that the frequency deviation of the 2 mm height microstrip antenna was minimal; while the 4 mm height microstrip antenna had better gain efficiency and directional diagram compared to others. The 4 mm height microstrip antenna has good radiation performance with a weight of only 35 g which shows that 3DWRHS‐CB‐MA are potential candidates for future intelligent textile materials and wireless communication fields.Highlights Weaving with UHMWPE filament yarns and copper wires on common looms. The preform was used as the reinforcement, and epoxy resin was used as the matrix. Simulation of electromagnetic characteristics of antennas using HFSS software. Proving the effectiveness of the electromagnetic simulation model. Revealing the electromagnetic radiation mechanisms.

Variable laser profiles and polarization states in the wave-mixing scheme for the generation of THz radiation in collisional-magnetized clustered-plasma

The four-wave mixing (FWM) mechanism for the generation of Terahertz (THz) radiation in magnetized-collisional clustered-plasma is studied. Clusters can trap the incident laser fields, leading to efficient energy transfer to plasma electrons and the creation of a strong nonlinear current. The size and density of clusters can be tuned to control the emission of THz radiation. The smaller clusters can bring about higher intensity THz radiation, while larger clusters can contribute to the broader THz frequency range. The THz radiation angular distribution patterns in the forward direction have been assessed, and the impact of plasma interaction length, cluster radius, and plasma electrons collision frequency on the generated THz wave patterns has been investigated. The FWM analysis highlights the importance of plasma clusters as well as magnetic fields in the efficient generation of THz emission. The model also takes into consideration how an induced electron current density initiated by DC external magnetic field can boost radiation power without changing the directivity diagram.

ЗНИЖЕННЯ ЗОВНІШНЬОГО ШУМУ БРОНЕАВТОМОБІЛЯ «НОВАТОР» ЗАСТОСУВАННЯМ ВИСОКОЕФЕКТИВНОГО ГЛУШНИКА ШУМУ

Armored vehicles are widely used for combat operations, for which it is extremely important to be as inconspicuous as possible. There are different types of masking, one of which is reducing the level of external noise. For the Novator armored car, a highly effective muffler of the exhaust gases of the internal combustion engine was devel-oped. The Novator armored car has a pickup-type body, a gross weight of 9,000 kg and a Ford F8 turbodiesel engine with a power of 300 hp. The uniqueness of the developed muffler is that it seeks to return all the sound waves that entered it back to the intake pipe. This effect is achieved due to the specific structure of the silencer housing. The case is an ellipsoid of rotation.At the same time, the section of the intake pipe (the pipe that goes into the muffler) is placed in the first focus of the ellipsoid of rotation, and the sound waves propagate to the inner wall of the elliptical body, reflect from it, con-verge at the second focus, and then continue to move to the opposite wall, reflect from it, converge in the first focus and spread further back into the intake pipe. In the intake pipe, there is partial interference of the reflected sound waves when they meet the incoming waves, and the final absorption by the sound-absorbing material. At the same time, exhaust gas-es, bouncing off the walls of the muffler, leave the latter through the exhaust pipe. At the same time, the exhaust pipe exits the ellipsoid of rotation coaxially with the intake pipe, which enters the pipe tel-escopically with a gap. Taking into account the directional diagram of sound energy radiation at the exit from the ex-haust pipe, a proposal was made to create a muffler not as a full ellipsoid of rotation, but in the form of a half ellipsoid of rotation. Such a solution will make the muffler significantly cheaper and much more compact, which is essential for a military vehicle. Keywords: armored car, “novator”, masking, camouflage, silencer, ellipsoid of rotation, directional diagram, acoustic efficiency, sound waves, sound absorbing material.

Combustion Characteristics of N-Butanol/N-Heptane Blend Using Reduced Chemical Kinetic Mechanism

The detailed mechanisms of n-heptane and n-butanol were reduced for the target condition of ignition delay time using the direct relationship diagram method based on error transfer, the direct relationship diagram method based on coupling error transfer and sensitivity analysis, and the total material sensitivity analysis method. The reduced n-heptane (132 species and 585 reactions) and n-butanol (82 species and 383 reactions) were used to verify the ignition delay time and concentrations of the major species, respectively. The results showed that the reduced mechanism has a good prediction ability for the ignition delay time. The predicted mole fraction results of the major species were in good agreement. These reduced mechanisms were combined to finally construct a reduced mechanism for the n-heptane/butanol fuel mixture, which included 166 species and 746 reactions. Finally, the reduced mechanism was used to simulate the HCCI combustion mode, and the results showed that the reduced mechanism can better predict the ignition and combustion timings of HCCI under different conditions and maintain the ignition and combustion characteristics of the detailed mechanism; this indicates that the mechanism model constructed in this study is reliable.

Three-dimensional analysis of THz radiation mechanisms in interaction of arbitrarily polarized multi-color laser pulses and obliquely magnetized plasma

Three-dimensional THz radiation generation via wave-mixing and photo-current mechanisms of multi-color femtosecond laser fields in obliquely magnetized plasma for various laser polarization states is considered. The main motive behind present study has been to investigate the development of more efficient and versatile THz radiation generation for a range of applications through a comprehensive mechanism describing the entire THz frequency band. Two and three-color laser waves interactions with plasma electrons results in a transverse nonlinear current density which causes THz radiation generation. Variation of THz electric field under different laser polarizations is examined analytically and numerically. The impacts of the applied oblique magnetic field and the field oblique angle on the THz electric field for both wave mixing and photo-current schemes are studied and result are compared. A method to implement THz radiation for special applications was developed showing that the external magnetic field is crucial in controlling plasma dynamics. The waveforms of THz angular distribution in forward direction are examined and the impact of different harmonics, various polarization states, external magnetic field, oblique angle of the applied external magnetic field, and the length of plasma on generated waveforms is considered. Based on the results of study, laser harmonics and applied external magnetic field increase electrons current density, leading to enhancement of radiation power without change in directionality diagram. These findings are in close agreement with experimental results.

Heavy metals in the bottom sediments of the Sasyk estuary

Purpose. The main aim of research is to study the bottom sediments of the Sasyk estuary, in particular, to establish their lithological characteristics and the peculiarities of the distribution of heavy metals in their layers. Method. The research was done with the results of the expedition work carried out in 2018-2019, 31 samples of the bottom sediments of the Sasyk estuary, channels and paleochannels of the Kogylnyk and Sarata rivers, and rocks of the shoreline of the studied reservoir were selected. We carried out sampling along the network of 31 stations from a motor boat using a bottom-grab and orientation by GPS navigation. The content of heavy metals was determined using geochemical spectral emission analysis. Statistical and cartographic processing of the research results was carried out using geographic information systems Mapinfo, QGIS, Global Mapper and Surfer. Also, to establish the possible routes of metal migration, we used the analysis of mesotopolineaments formed according to the NASA SRTM digital terrain model, with the construction of rose diagrams of the main directions of the lineaments. The results. We established that the bottom sediments were represented mainly by grey-black, black, viscous silt, often with an admixture of shell detritus. Less often, they were composed of sand, sometimes muddy with abundant fauna. We also determined the content of heavy metals in the system of bottom sediments, features of their distribution and possible ways of infusing. Thus, in the territory of the Sasyk estuary, heavy metals of the first (Pb), second (Ni, Cu, Cr, Mo) and third (V, Mn, Ba) hazard class are found in high concentrations. The average content of Pb for the bottom sediment system of the estuary is: for sand – 32 mg/kg, for silt – 41 mg/kg. The maximum content for both is 100 mg/kg. The average Ni content is: for sand – 27 mg/kg, for silt – 62 mg/kg. The maximum content for sand is 80 mg/kg, for silt – 100 mg/kg. The average Cu content in the silt of the Sasyk estuary is 81 mg/kg, and 26 mg/kg in the sand. An elevated content of Mn, Ba and Mo was also found in several samples. Оften areas with increased concentrations of metals have a point character. In almost all cases, they are connected to the river systems adjacent to the pond. The distribution of some metals is correlated with the main directions of estuary water circulation. Wastewater is a likely sources of metals entering the estuary geoecosystem. We also assume that the migration of these elements to the pond may occur through tectonic blocks. Scientific novelty. For the first time in recent years, we have carried out expeditionary work on research of the bottom sediments of the Sasyk estuary, along the complete network of the station. We obtained the data on the gross content of heavy metals, and determined their average, maximum and minimum concentrations. For the first time, maps of the spatial distribution of metals were constructed, which made it possible to establish the main regularities of their accumulation in the bottom sediment system of the pond. Practical significance. Pollution of the bottom sediments of the Sasyk estuary with heavy metals is one of the components of today's critical ecological state of the pond, which affects the well-being of the residents of the surrounding areas. The obtained results can be used at the state level to create effective programs for quality management of the resource potential of the estuary.

Stress simulation of 6-inch SiC single crystal

With the size of SiC boule increasing, the problem of crystal creaking becomes more and more serious. In this work, residual stress in SiC boule is calculated by conducting stress simulation based on COMSOL and STR-VR software. A direct diagram comparing on crystal stress distribution is given. Besides, we provide relevant indications for the complete realization of large size SiC bulk with low stress for growing mainstream 6-inch SiC bulk with competitive thickness (40–50 mm) and low residual stress. And a 6-inch stress-free SiC bulk with improved quality was successfully fabricated based on our theoretical investigations.

Vibration characteristics and elastic wave propagation properties of mirror-symmetric structures of trichiral ligaments

In this paper, two mirror-symmetric structures of three-chiral ligaments are proposed. The lattice theory and Bloch's theorem are combined to construct a theoretical model of the structures, and a finite element method is used to demonstrate that the structure has an ultra-wide bandgap coverage and the lowest bandgap of the new structure with a central entity can be reduced to 316.4 Hz. The rotational resonance and the offset resonance of the central entity of the trichiral ligament are closely related to the opening of the bandgap by vibration analysis. In addition, both the vibration transfer function and the stress cloud demonstrate that the structure is extremely capable of attenuating vibration and noise, with a maximum attenuation peak of − 380. The directional and regional properties of the elastic wave propagation are demonstrated by simulations of the phase constant plane, phase velocity, wave propagation direction diagram, group velocity and wave propagation path. This structure has the combined advantages of light weight, easy fabrication, broadband and low frequency vibration and noise control, while the integrated wave propagation analysis provides theoretical support for structural and bandgap optimization.

Forming an elliptical directional diagram of the sectoral horn antenna for flow irradiation of sugar beet seeds by electromagnetic field

The object of research is the process that forms an elliptical directional diagram of the H-sector horn antenna for flow irradiation of seeds with the electromagnetic field. The emitter of electromagnetic energy is presented as one of the main elements of installations for irradiating seeds with an electromagnetic field before sowing. This parameter was investigated by taking into account the values of the biotropic parameters of the low-energy electromagnetic field under the conditions of flow processing. This paper reports a study into the parameters of the H-sector horn emitter for irradiation of sugar beet seeds with a low-energy electromagnetic field at a frequency of 73...75 GHz in continuous flow. Thus, one should use the H-sectoral horn emitter with the following parameters: aperture width aa=20 mm; horn length RH=35 mm; b=1.8 mm. It is determined that in order to irradiate sugar beet seeds on the conveyor plane with a power flow density of P=100 μW/cm2, it is necessary to place two horns 1200 mm above the conveyor at a distance of 2540 mm from each other. It was checked that the treatment of sugar beet seeds with electromagnetic radiation in a continuous flow with a capacity of 300 kg/h is possible with a power of up to 2 W supplied to two horn antennas; the speed of the conveyor is 15 cm/s. The parameters of the sectoral horn for an elliptical directional diagram were studied by dividing the main task into internal and external. According to the results of the research, it is possible to build a base of geometric presets for adjusting installations for different types of seeds, the desired performance, the structural features of installations, as well as existing emitters

OPTIMIZATION OF INTELLIGENT TELECOMMUNICATION NETWORKS

The paper presents the results of research on the use of machine learning in telecommunication networks and describes the basics of the theory of artificial intelligence. The impact of dynamic Bayesian network (DBN) and DNN on the development of many technologies, including user activity detection, channel estimation, and mobility tracking, is determined. The indicators of the effectiveness of communications based on the theory of information bottlenecks, which is at the junction of machine learning and forecasting, statistics and information theory, are considered. A neural network model that is pretrained for high-level tasks and divided into transmitter-side and receiver-side uses is investigated. The process of learning the model, which is performed after its adjustment, taking ino account the existing transmission channels, is considered. New ANN learning techniques capable of predicting or adapting to sudden changes in a wireless network, such as federated learning and multiagent reinforcement learning (MARL), are reviewed. The DBN model, which describes a system that dynamically changes or develops over time, is studied. The considered model provides constant monitoring of work and updating of the system and prediction of its behavior. Distributed forecasting of channel states and user locations as a key component in the development of reliable wireless communication systems is studied. The possibility of increasing the number of degrees of freedom of the generalized wireless channel G(E) in terms of: the physical propagation channel, the directional diagram of the antenna array and mutual influence, electromagnetic physical characteristics is substantiated. The impact of ultra-highresolution theory on the development of many technologies, including localization algorithms, compressed sampling, and wireless imaging algorithms, is also identified. Mathematical expressions for optimizing the functional characteristics of 5G/6G radio networks are presented using new, sufficiently formal and at the same time universal mathematical tools with an emphasis on deep learning technologies, which allow systematic, reliable and interpretable analysis of large random networks and a wide range of their network models and practical networks.

Bandgap tuning and in-plane wave propagation of chiral and anti-chiral hybrid metamaterials with assembled six oscillators

In this paper, a hybrid acoustic metamaterial consisting of chiral and anti-chiral ligaments is proposed. The bandgap properties of the hybrid chiral and anti-chiral structure with and without oscillator intercalation are calculated using Bloch’s theorem and the finite element method, and the structure is verified and evaluated to have good vibration attenuation performance by means of transfer functions and stress clouds. By adjusting the resonant oscillator configuration and applying active strain, the structure is verified to have tunable capability, which provides a simple and easy scheme for bandgap tunability analysis. Finally, a comprehensive analysis of the structural vibration mode, phase constant plane, group velocity, phase velocity and wave propagation direction diagram lead to the conclusion that the directional and regional nature of wave propagation is responsible for the bandgap opening. The new structure in this paper provides a new way of thinking for vibration and noise reduction, and also provides a practical solution for bandgap adjustment. The comprehensive analysis of wave propagation also presents theoretical and technical support for the subsequent structure and bandgap optimization.

МАТЕМАТИЧЕСКИЙ АППАРАТ ДЛЯ АНАЛИЗА ПОМЕХОПОДАВЛЯЮЩИХ ПОЛОСКОВЫХ УСТРОЙСТВ С АСИММЕТРИЧНОЙ СТРУКТУРОЙ

A mathematical apparatus that is used to analyze protective devices based on a meander line (ML) turn. The apparatus takes into account the effect of asymmetry of cross-section on the propagation of pulse signal in a turn The developed mathematical apparatus includes: a model for calculating radiated emissions from a turn with an arbitrary cross-section; an analytical model for calculating the time response at the turn output; models for obtaining conditions that ensure the complete ultrashort pulse (USP) decomposition in multistage devices The proposed apparatus was evaluated on the example of structures with broad-side and side coupling (by the comparison with the results of computer simulation). As a result of comparison of frequency dependences of the maximum electric field strengths obtained using the model and numerically were produced ambiguous results. For the ML turn with broad-side coupling, the dependences were found to be in good agreement qualitatively in the frequency range up to 2.5 GHz. However, they can differ significantly in terms of quantity. More specifically, at some frequencies, there are pronounced maxima of the frequency dependences obtained numerically. For the turn with side coupling, the amplitudes of the maximum field strengths calculated by the model were found to be higher in the entire frequency range (with the exception of several maxima) than those calculated numerically. The constructed directional diagrams of the turn confirm the quantitative discrepancies. Meanwhile, the analysis of the obtained results allows us to claim that the real field strengths will be lower than those calculated by the model, so the model is applicable for preliminary estimates. As a result of evaluation of the analytical model for calculating the time response at the turn output revealed a complete coincidence of the time responses obtained on the basis of the model and numerically. Finally, the models to obtain the conditions for the complete pulse decomposition were evaluated on the example of 3-stage devices. The results demonstrated that the developed models were correct. The case was also considered when the conditions obtained using the models are not met, which leads to an increase in the signal amplitudes at the output of the devices. Thus, the entire presented mathematical apparatus can be used to analyze protective devices based on an ML turn with an asymmetric cross-section.