Odd Mathieu Functions Application to Synthesize a Multi-element Radiator Flat-topped Radiation Pattern

The paper studies synthesizing capabilities of a flat-topped radiation pattern when using the expansion of the target radiation pattern into a series in terms of odd Mathieu functions. As parameters for comparing the target and synthesized radiation patterns, we used a main-lobe width at a level of -1 dB and an irregularity of the top of the main-lobe of the radiation pattern. The sector-shaped radiation pattern has been synthesized for linear radiators of various lengths. The convergence of the coefficients of the Mathieu series in the synthesis of the sector-shaped radiation pattern has been estimated. It is shown that the use of piecewise-linear approximation of the target radiation pattern in the synthesis using a series expansion into odd Mathieu functions allows us to improve the quality of the radiation pattern formed.The task that involved finding the amplitude-phase distribution for a linear emitter with a length of 3λ, 4λ and 5λ (λ is operation wavelength) for a target radiation pattern was solved. The target amplitude distribution has the following electrical characteristics: the main-lobe width is 37.5° at a level of -1 dB and the side lobe level (SLL) is -20 dB. The synthesis procedure was performed for two cases. In the first case, the target radiation pattern is represented by a piecewise constant function with a given width. In the second case, the target pattern was specified using piecewise linear approximation of the top and slopes of the main lobe.Comparison of the radiation patterns obtained shows that in the first case, the main-lobe width of the radiation pattern at a level of -1 dB is 34°, the SLL varies from -15.6 to -17 dB, and the irregularity of the main-lobe top of the radiation pattern lies within 0.9 ... 1.2 dB. In the second case, the main-lobe width of the antenna radiation pattern at a level of -1 dB is 36.5°, the SLL is -17.5 dB, and the irregularity of the main-lobe top is 0.4 dB at most. When used, the considered under consideration enables us to obtain both the synthesized patterns for linear radiators of various lengths, and the corresponding amplitude-phase distributions and coefficients of the Mathieu series. An estimate of the convergence of the Mathieu series shows that the use of linear approximation of the target radiation pattern in some cases allows up to 2.7-fold increase in acceleration of the convergence of the Mathieu series. The accuracy of reproducing the sector-shaped pattern by the synthesis method using the expansion into odd Mathieu functions gives good results when synthesizing the amplitude-phase distribution for the linear radiators with an electric length of 5λ or more.

Solution of the Schrödinger torsion equation in the basis set of Mathieu functions: verification by numerical experiment

The efficiency of the algorithm for the numerical solution of the Schrödinger torsion equation in the basis of Mathieu functions has been considered. The computational stability of the proposed algorithm is shown. The energies of torsion transitions determined in the basis sets of plane waves and Mathieu functions have been compared with the results of spectroscopy. A conclusion about the applicability of the algorithm using the basis set of Mathieu functions to the solution of the Schrödinger equation with a periodic potential has been derived.

Application Of Mathieu functions And The Point Matching Method To Elliptic Conductors

Analysis of elliptic conductors, carrying a known total current, using the point matching method (PMM) in circular-cylinder coordinates [l] failed for the values of axes ratio b / a < 05, where a and b are the major and minor axes of the ellipse, respectively. This work shows that the use of elliptic-cylinder coordinates in conjunction with the point matching method overcomes this problem. This paper is a part of the work investigating the difficulties encountered in the point matching method, and is motivated by the fact that, despite its limitations, the method is still attractive as suggested by its use in recent papers [2]-141.

Application Of Mathieu functions And The Point Matching Method To Elliptic Conductors

Analysis of elliptic conductors, carrying a known total current, using the point matching method (PMM) in circular-cylinder coordinates [l] failed for the values of axes ratio b / a < 05, where a and b are the major and minor axes of the ellipse, respectively. This work shows that the use of elliptic-cylinder coordinates in conjunction with the point matching method overcomes this problem. This paper is a part of the work investigating the difficulties encountered in the point matching method, and is motivated by the fact that, despite its limitations, the method is still attractive as suggested by its use in recent papers [2]-141.

Transient Pressure-Driven Electroosmotic Flow through Elliptic Cross-Sectional Microchannels with Various Eccentricities

The semi-analytical solution for transient electroosmotic flow through elliptic cylindrical microchannels is derived from the Navier-Stokes equations using the Laplace transform. The electroosmotic force expressed by the linearized Poisson-Boltzmann equation is considered the external force in the Navier-Stokes equations. The velocity field solution is obtained in the form of the Mathieu and modified Mathieu functions and it is capable of describing the flow behavior in the system when the boundary condition is either constant or varied. The fluid velocity is calculated numerically using the inverse Laplace transform in order to describe the transient behavior. Moreover, the flow rates and the relative errors on the flow rates are presented to investigate the effect of eccentricity of the elliptic cross-section. The investigation shows that, when the area of the channel cross-sections is fixed, the relative errors are less than 1% if the eccentricity is not greater than 0.5. As a result, an elliptic channel with the eccentricity not greater than 0.5 can be assumed to be circular when the solution is written in the form of trigonometric functions in order to avoid the difficulty in computing the Mathieu and modified Mathieu functions.