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2010-09-23
PIER B
Vol. 25, 349-367, 2010
download: 340
Radiophysical and Dielectric Properties of Ore Minerals in 12--145 GHz Frequency Range
Vasiliy V. Tikhonov , Dmitriy A. Boyarskii , Olga N. Polyakova , Andrey L. Dzardanov and Grigory N. Goltsman
The paper discusses a retrieval technique of complex permittivity of ore minerals in frequency ranges of 12--38 GHz and 77--145 GHz. The method is based on measuring frequency dependencies of transmissivity and reflectivity of plate-parallel mineral samples. In the 12--38 GHz range, the measurements were conducted using a panoramic standing wave ratio and attenuation meter. In the 77--145 GHz range, frequency dependencies of transmissivity and reflectivity were obtained using millimeter-band spectrometer with backward-wave oscillators. The real and imaginary parts of complex permittivity of a mineral were determined solving an equation system for frequency dependencies of transmissivity and reflectivity of an absorbing layer located between two dielectric media. In the course of the work, minerals that are primary ores in iron, zinc, copper and titanium mining were investigated: magnetite, hematite, sphalerite, chalcopyrite, pyrite, and ilmenite.
RADIOPHYSICAL AND DIELECTRIC PROPERTIES OF ORE MINERALS IN 12--145 GHZ FREQUENCY RANGE
2010-09-17
PIER B
Vol. 25, 331-348, 2010
download: 349
Comparative Performance of Gravitational Search Algorithm and Modified Particle Swarm Optimization Algorithm for Synthesis of Thinned Scanned Concentric Ring Array Antenna
Anirban Chatterjee , Gautam Mahanti and Narendra Nath Pathak
Scanning a planar array in the x-z plane directs the beam peak to any direction off the broadside along the same plane. Reduction of sidelobe level in concentric ring array of isotropic antennas scanned in the x-z plane result in a wide first null beamwidth (FNBW). In this paper, the authors propose pattern synthesis methods to reduce the sidelobe levels with fixed FNBW by making the scanned array thinned based on two different global optimization algorithms, namely Gravitational Search Algorithm (GSA) and modified Particle Swarm Optimization (PSO) algorithm. The thinning percentage of the array is kept more than 45 percent and the first null beamwidth (FNBW) is kept equal to or less than that of a fully populated, uniformly excited and 0.5 λ spaced concentric circular ring array of same scanning angle and same number of elements and rings.
COMPARATIVE PERFORMANCE OF GRAVITATIONAL SEARCH ALGORITHM AND MODIFIED PARTICLE SWARM OPTIMIZATION ALGORITHM FOR SYNTHESIS OF THINNED SCANNED CONCENTRIC RING ARRAY ANTENNA
2010-09-16
PIER B
Vol. 25, 315-330, 2010
download: 314
Synthesis of Optimal Narrow Beam Low Sidelobe Linear Array with Constrained Length
Jean-Jacques Fuchs and Benjamin Fuchs
The synthesis of optimal narrow beam low sidelobe linear array is addressed. Only the length of the array is constrained. The number, the positions and the weightings of the elements are left free. It is proven, that the optimal design is always an array with a small number of elements. One first demonstrates that among equally spaced linear arrays of given length, the sparsest Dolph-Chebyshev design, i.e., the one with the largest admissible inter-element distance, is the optimal one. Then, the restriction to equally spaced elements is removed, and the general problem is solved and discussed. It is shown that the sparsest Dolph-Chebyshev designs are optimal for array lengths in given specified intervals and close to optimal for all other lengths.
SYNTHESIS OF OPTIMAL NARROW BEAM LOW SIDELOBE LINEAR ARRAY WITH CONSTRAINED LENGTH
2010-09-15
PIER B
Vol. 25, 293-314, 2010
download: 263
Improved Analytical Model for Surface-Mounted PM Motors Considering Slotting Effects and Armature Reaction
Thierry Lubin , Smail Mezani and Abderrezak Rezzoug
This paper presents an exact analytical method for the computation of the magnetic field distribution in surface-mounted permanent-magnet (PM) motors for any pole and slot combinations including fractional slot machines. The proposed model takes into account the slotting effect and the armature reaction magnetic field. The analytical method is based on the resolution of the two-dimensional Laplace's and Poisson's equations in polar coordinates (by the separation of variables technique) for each subdomain, i.e., magnet, airgap and slots. Magnetic field distributions, back electromotive force and electromagnetic torque (cogging torque and load torque) computed with the proposed analytical method are verified with those obtained from finite element analyses.
IMPROVED ANALYTICAL MODEL FOR SURFACE-MOUNTED PM MOTORS CONSIDERING SLOTTING EFFECTS AND ARMATURE REACTION
2010-09-14
PIER B
Vol. 25, 277-292, 2010
download: 304
Planar Monopole UWB Antenna with Unii1/Unii2 WLAN-Band Notched Characteristics
Leonardo Lizzi , Giacomo Oliveri , Paolo Rocca and Andrea Massa
In this paper, a planar monopole antenna for Ultra-Wideband (UWB) communications with a notched behavior in the two sub-bands UNII1 and UNII2 of the Wireless Local Area Network (WLAN) band is presented. The antenna geometry is described by means of a spline curve and a rectangular slot. Numerical and experimental results are reported to assess the effectiveness of the proposed design in terms of impedance matching and radiation characteristics.
PLANAR MONOPOLE UWB ANTENNA WITH UNII1/UNII2 WLAN-BAND NOTCHED CHARACTERISTICS
2010-09-14
PIER B
Vol. 25, 261-275, 2010
download: 345
Design, Fabrication and Characterization of a Dielectric Resonator Antenna Reflectarray in Ka-Band
Mohd Haizal Jamaluddin , Ronan Sauleau , Xavier Castel , Ratiba Benzerga , Laurent Le Coq , Raphael Gillard and Thierry Koleck
A new reflectarray configuration is proposed for low-loss applications at millimeter waves. It is based on the use of dielectric resonator antennas (DRA) as radiating unit-cells. The phase response of the elementary cell is controlled by adjusting the length of a parasitic narrow metal strip printed on the top of each DRA. A 330° phase dynamic range is obtained for DRAs made in rigid thermosetting plastic (εr=10). As the antenna radiating aperture is non flat, an original low-cost fabrication process is also introduced in order to fabricate the parasitic strips on the DRA surface. A 24×24-element array radiating at broadside has been designed at 30 GHz and characterized between 29 and 31 GHz. The antenna gain reaches 28.3 dBi at 31 GHz, and the measured -1 dB-gain radiation bandwidth is 5.2%. The 3.2 dB loss observed between the measured gain and theoretical directivity is mainly due to the spillover loss (2.3 dB). The total dielectric and conductor loss is less than 0.9 dB.
DESIGN, FABRICATION AND CHARACTERIZATION OF A DIELECTRIC RESONATOR ANTENNA REFLECTARRAY IN KA-BAND
2010-09-14
PIER B
Vol. 25, 241-259, 2010
download: 347
Multi-Floor Indoor Positioning System Using Bayesian Graphical Models
Abdullah Al-Ahmadi , Abdusamea. I. A. Omer , Muhammad Ramlee Kamarudin and Tharek Bin Abdul Rahman
In recent years, location determination systems have gained a high importance due to its rule in the context aware systems. In this paper, we will design a multi-floor indoor positioning system based on Bayesian Graphical Models (BGM). Graphical models have a great flexibility on visualizing the relationships between random variables. Rather of using one sampling technique, we are going to use multiple sets each set contains a collection of sampling techniques, the accuracy of each set will be compared with each other.
MULTI-FLOOR INDOOR POSITIONING SYSTEM USING BAYESIAN GRAPHICAL MODELS
2010-09-10
PIER B
Vol. 25, 225-239, 2010
download: 274
Novel, Dual Band, Single and Double Negative Metamaterials: Nonconcentric Delta Loop Resonators
Cumali Sabah
Novel, dual band, single and double negative metamaterials composed of nonconcentric and different sized delta loop resonators are presented. The proposed structures provide two distinct resonant frequencies in the microwave region. Effective medium parameters of these metamaterial structures are extracted using retrieval method to demonstrate the presence of the mentioned frequencies. In addition, equivalent circuit model for the individual magnetic resonator and wire strip is presented to give a clear explanation for the resonance behavior of the structures and to validate the proposed designs. The results show that the proposed metamaterials can be used as an alternative to the known counterparts especially when a dual band operation is needed at the frequency region of interest.
NOVEL, DUAL BAND, SINGLE AND DOUBLE NEGATIVE METAMATERIALS: NONCONCENTRIC DELTA LOOP RESONATORS
2010-09-09
PIER B
Vol. 25, 211-224, 2010
download: 352
Rain Attenuation Modeling in the 10-100 GHz Frequency Using DROP Size Distributions for Different Climatic Zones in Tropical India
Saurabh Das , Animesh Maitra and Ashish Kumar Shukla
Rain drop size distributions (DSD) are measured with disdrometers at five different climatic locations in the Indian tropical region. The distribution of drop size is assumed to be lognormal to model the rain attenuation in the frequency range of 10-100 GHz. The rain attenuation is estimated assuming single scattering of spherical rain drops. Different attenuation characteristics are observed for different regions due to the dependency of DSD on climatic conditions. A comparison shows that significant differences between ITU-R model and DSD derived values occur at high frequency and at high rain rates for different regions. At frequencies below 30 GHz, the ITU-R model matches well with the DSD generated values up to 30 mm/h rain rate but differ above that. The results will be helpful in understanding the pattern of rain attenuation variation and designing the systems at EHF bands in the tropical region.
RAIN ATTENUATION MODELING IN THE 10-100 GHz FREQUENCY USING DROP SIZE DISTRIBUTIONS FOR DIFFERENT CLIMATIC ZONES IN TROPICAL INDIA
2010-09-07
PIER B
Vol. 25, 191-209, 2010
download: 468
Shape Slope Parameter Distribution Modelling of Electromagnetic Scattering by Rain Drops
Lakshmi Sutha Kumar , Yee Hui Lee and Jin Teong Ong
Gamma model parameters using 2nd, 3rd and 4th moments are calculated from the drop size data of Singapore. The gamma model is simplified into two parameter model by finding a relation between the shape and slope parameters, μ and Λ. Due to the poor correlation found between μ and Λ, the drop size data is filtered based on their rain rates before a good correlation between the two parameters can be found. The μ-Λ relations are then fitted for the different ranges of rain rate filtering. Scatter plots of μ and Λ are plotted with constant median volume diameter (D0) lines. The μ-Λ relations for the different rain types for the tropical island of Singapore are proposed and compared with the μ-Λ relations from three other countries of different climatic zones. T-Matrix calculations are performed to find the polarimetric variables at S-band by using the gamma DSD calculated from the Singapore's drop size data. The calculated differential reflectivity and horizontal reflectivity are used along with the best μ-Λ relations to find the gamma model parameters. The retrieved rain rate using polarimetric variables is compared with the distrometer's measured rain rate. Results show a good agreement between the retrieved rain rate and the measured rain rate. Therefore, the proposed shape slope relationship is found to be suitable for rain rate retrieval.
SHAPE SLOPE PARAMETER DISTRIBUTION MODELLING OF ELECTROMAGNETIC SCATTERING BY RAIN DROPS
2010-08-27
PIER B
Vol. 25, 171-189, 2010
download: 232
E-Field Extraction from h -Near-Field in Time-Domain by Using Pws Method
Blaise Ravelo
A novel technique of the electric- or E- field extraction from the magnetic- or H- near-field in timedomain is reported. This technique is based on the use of the Maxwell-Ampere relation associated to the plane wave spectrum (PWS) transform. It is useful for the E-near-field computations and measurements which are practically complicated in time-domain in particular, for the EMC applications. The considered EM-field radiation is generated by a set of electric dipoles excited by an ultra-short duration current having frequency bandwidth of about 10-GHz. The presented EM-field calculation technique is carried out by taking into account the evanescent wave effects. In the first step, the time-dependent H-field data mapped in 2-D plan placed at the height z0 above the radiating devices are transposed in frequency-dependent data through the fast Fourier transform. In order to respect the near-field approach, the arbitrary distance z0 between the EM-field mapping plan and radiating source plan should be below one-sixth of the excitation signal minimal wavelength. In the second step, one applies the PWS transform to the obtained frequency-data. Then, through the Maxwell-Ampere relation, one can extract the E-field from the calculated PWS of the H-field. In the last step, the inverse fast Fourier transform of the obtained E-field gives the expected time-dependent results. The relevance of the proposed technique was confirmed by considering a set of five dipole sources placed arbitrarily in the horizontal plan equated by z = 0 and excited by a pulse current having amplitude of 50 mA and half-width of about 0.6 ns. As expected, by using the Hx, Hy and Hz 2-D data calculated with Matlab in the rectangular plan placed at z0 = 3 mm and z0 = 5 mm above the radiating source, it was demonstrated that with the proposed technique, one can determine the three components of the E-field Ex, Ey and Ez.
E-FIELD EXTRACTION FROM H-NEAR-FIELD IN TIME-DOMAIN BY USING PWS METHOD
2010-08-27
PIER B
Vol. 25, 155-169, 2010
download: 260
How Lightning Tortuosity Affects the Electromagnetic Fields by Augmenting Their Effective Distance
Scott L. Meredith , Susan K. Earles , Ivica N. Kostanic , Niescja E. Turner and Carlos E. Otero
A novel approach for developing the electromagnetic fields from a lightning return stroke which follows a tortuous path will be presented. The proposed model is unique in that it recognizes that the symmetrical tortuosity of lightning directly impacts the observable distance r, which in turn, alters the resulting electromagnetic fields. In the literature, lightning return stroke models typically employ the assumption that the cloud-to-ground path is straight. Although this assumption yields fairly consistent results across an array of varying approaches, it does not account for lightning's natural physical appearance. Furthermore, straight-line models only account for the cloud-to-ground discharges and do not address branching and/or cloud-to-cloud discharges which are far more common. In reality, the ``steps'' which make up the lightning channel's initial descent are staggered or tortuous with respect to each other. Given this fact, the upward traveling current wavefront which follows this prescribed path will exhibit the same characteristics. In doing so, each current segment, which forms along its respective step, induces electromagnetic fields with angular aggregates that propagate outward from their origin. This, in turn, will generate spatial points where there are fields of higher and lower intensities. The results presented in this paper will show how the effective observable distance due to symmetrical tortuosity alters the resulting electromagnetic fields. Furthermore, it will be shown that as the observable distance r is increased, results from the proposed model closely resemble the straightline model which strongly suggests that symmetrical tortuosity is only influential at relatively close distances.
HOW LIGHTNING TORTUOSITY AFFECTS THE ELECTROMAGNETIC FIELDS BY AUGMENTING THEIR EFFECTIVE DISTANCE
2010-08-25
PIER B
Vol. 25, 131-154, 2010
download: 261
The Level Set Shape Reconstruction Algorithm Applied to 2D PEC Targets Hidden Behind a Wall
Mohammad Reza Hajihashemi and Magda El-Shenawee
The level set algorithm is extended to handle the reconstruction of the shape and location of objects hidden behind a dielectric wall. The Green's function of stratified media is used to modify the method of moments and the surface integral equation forward solver. Due to the oscillatory nature of the Sommerfeld integrals, the stationary phase approximation is implemented here to achieve fast and accurate reconstruction results, especially when the targets are located adequately far from the wall. Transverse Magnetic (TM) plane waves are employed for excitation with limited view for transmitting and receiving the waves in the far field at one side of the wall. The results show the capability of the level set method for retrieving the shape and location of multiple 2D PEC objects of arbitrary shapes even when there are located at a small distance from the wall. To reduce the computational expenses of the algorithm in the case of multiple hidden objects, the MPI parallelization technique is implemented leading to a reduction in the CPU time from hours on a single processor to few minutes using 128 processors on the NCSA Supercomputer Center.
THE LEVEL SET SHAPE RECONSTRUCTION ALGORITHM APPLIED TO 2D PEC TARGETS HIDDEN BEHIND A WALL
2010-08-25
PIER B
Vol. 25, 113-130, 2010
download: 232
Performance Evaluation of Phase-Angle Gradient Method for Phase Retrieval Based on Low-Frequency Amplitude-Only Near-Field Data
Markus Johansson , Hoi-Shun Lui and Mikael Persson
The Phase Angle Gradient Method (PAGM) is a recent technique developed for phase retrieval based on amplitude-only measurement data. Preliminary results have shown that the PAGM is able to perform phase retrieval at 100 MHz with accurate phase information based on measured field components on three planar surfaces. In this paper, a performance evaluation of the PAGM under different configurations is conducted. Phase retrieval based on field measurements for different plane sizes and separations between the planes are studied rigorously. In addition, the PAGM is tested for different initial phase distributions. The results show that the PAGM is capable of retrieving phase information even if the separation between the measurement planes is small in terms of wavelengths.
PERFORMANCE EVALUATION OF PHASE-ANGLE GRADIENT METHOD FOR PHASE RETRIEVAL BASED ON LOW-FREQUENCY AMPLITUDE-ONLY NEAR-FIELD DATA
2010-08-25
PIER B
Vol. 25, 93-111, 2010
download: 123
Data-Adaptive Resolution Method for the Parametric Three-Dimensional Inversion of Triaxial Borehole Electromagnetic Measurements
Faruk Omer Alpak and Carlos Torres-Verdin
We develop a new adaptive inversion procedure: Data-adaptive Resolution Inversion (DRI) method, which eliminates the need of selecting a parameterization prior to inversion. Instead, one performs a hierarchical search for the correct parameterization while solving a sequence of inverse problems with an increasing dimension of parameterization. A parsimonious approach to inverse problems usually involves the application of the same refinement consistently over the complete spatial domain. Such an approach may lead to over-parameterization, subsequently, to unrealistic conductivity estimates and excessive computational work. With DRI, the new parameterization at an arbitrary stage of inversion sequence is allocated such that new degrees of freedom are not necessarily introduced all over the spatial domain of the problem. The aim is to allocate new degrees of freedom only where it is warranted by the available data. Inversion results confirm that DRI is robust and efficient for multiparameter inversion of multicomponent borehole electromagnetic measurements.
DATA-ADAPTIVE RESOLUTION METHOD FOR THE PARAMETRIC THREE-DIMENSIONAL INVERSION OF TRIAXIAL BOREHOLE ELECTROMAGNETIC MEASUREMENTS
2010-08-25
PIER B
Vol. 25, 75-92, 2010
download: 291
Weather Radar Network with Pulse Compression of Arbitrary Nonlinear Waveforms: Ka-Band Test-Bed and Initial Observations
Hoon Lee and Yong-Hoon Kim
Short-wavelength radar networks are expected to complement current long-range weather radar systems. Accordingly, we proposed a configuration for such a network constituting pulse compression radars in order to use frequency resources efficiently and obtain multi-static information. We developed high resolution Ka-band pulse compression weather radar system as a test-bed. Using a commercial direct digital synthesizer (DDS) and field programmable gate array (FPGA) control, we generated linear and arbitrary nonlinear frequency modulated waveforms for low range sidelobes. Further, we completed a high duty factor system with a solid-state power amplifier. In a vertical-pointing mode, we were able to employ the developed radar to detect moderate rainfall up to 15 km. Details of the system design, hardware structure, data acquisition and processing algorithms were described. To validate the performance of the proposed radar system, we conducted several experiments by measuring cloud, snow and rain.
WEATHER RADAR NETWORK WITH PULSE COMPRESSION OF ARBITRARY NONLINEAR WAVEFORMS: KA-BAND TEST-BED AND INITIAL OBSERVATIONS
2010-08-20
PIER B
Vol. 25, 53-74, 2010
download: 328
Miniature and Higher-Order Mode Ferrite MIMO Ring Patch Antenna for Mobile Communication System
Seok Bae , Yang-Ki Hong , Jae-Jin Lee , Ji-Hoon Park , Jeevan Jalli , Gavin Abo , Hyuck M. Kwon and Chandana K. K. Jayasooriya
Miniaturized ferrite ring patch antennas (RPAs) were designed and fabricated for multiple-input multiple-out (MIMO) applications. Design parameters of higher-order mode ferrite RPAs, 1-RPA and 2-RPA, were optimized, and antenna performance of the ferrite 1-RPA was evaluated. The Z-type hexaferrite and 2%-weight borosilicate glass composite was used for the ferrite antenna disk. The measured permeability (μr) and permittivity (εr) of the hexaferrite were 2.59 and 5.7, respectively, at 2.5 GHz. Threemode orthogonal radiation of the ferrite 1-RPA was experimentally confirmed. With regard to the ferrite 2-RPA, excellent isolation (-40 dB) between ports 1 and 2 was achieved at 2.5 GHz. This excellent isolation property is attributed to both mode 3 orthogonal radiations of the bottom and top RPAs. The volumes of the 1- and 2-RPA were reduced to 14.5% and 34.5%, respectively, from 95 cm3 of a dielectric 2-circular patch antenna (2-CPA) volume.
MINIATURE AND HIGHER-ORDER MODE FERRITE MIMO RING PATCH ANTENNA FOR MOBILE COMMUNICATION SYSTEM
2010-08-20
PIER B
Vol. 25, 39-52, 2010
download: 442
Calculation of the Nonlinear Absorption Coefficient of a Strong Electromagnetic Wave by Confined Electrons in Doping Superlatices
Nguyen Quang Bau and Do Manh Hung
Analytic expressions for the nonlinear absorption coefficient (nonlinear absorption coefficient=NAC) of a strong electromagnetic wave (laser radiation) caused by confined electrons for the case of electron-optical phonon scattering in doping superlattices (doping superlattices=DSLs) are calculated by using the quantum kinetic equation for electrons. The problem is also considered for both the absence and the presence of an external magnetic field. The dependence of the NAC on the intensity E0 and the energy hΩ of the external strong electromagnetic wave (electromagnetic wave=EMW), the temperature T of the system, the doping concentration nD and the cyclotron frequency ΩB for case of an external magnetic field is obtained. Two cases for the absorption: Close to the absorption threshold ∣khΩ-0∣≤ε and far away from the absorption threshold ∣khΩ-0∣≥ε (k=0, ±1, ±2..., 0 and ε are the frequency of optical phonon and the average energy of electrons, respectively) are considered. The analytic expressions are numerically evaluated, plotted, and discussed for a specific DSLs n-GaAs/p-GaAs. The computations show that the NAC in DSLs in case presence of an external magnetic field is much more greater than to it is absence of an external magnetic field. The appearance of an external magnetic field causes surprising changes in the nonlinear absorption. All the results for the presence of an external magnetic field are compared with those for the absence of an external magnetic field to show the difference.
CALCULATION OF THE NONLINEAR ABSORPTION COEFFICIENT OF A STRONG ELECTROMAGNETIC WAVE BY CONFINED ELECTRONS IN DOPING SUPERLATICES
2010-08-20
PIER B
Vol. 25, 23-37, 2010
download: 267
FA-ScanSAR : Full Aperture Scanning Pulse by Pulse for the Nearspace Slow-Moving Platform Borne SAR
Bing Sun , Jie Chen , Chun-Sheng Li and Yin-Qing Zhou
Because the nearspace slow-moving platform borne synthetic aperture radar (SAR) can realize high resolution imaging using low pulse repetition frequency (PRF), a full-aperture ScanSAR (FA-ScanSAR) operation, which switches the range beam pulse by pulse, was proposed for wide swath imaging. This operation separates the wide swath into several sub-swaths, and each of which can be illuminated by a narrow range beam. The SAR antenna switches the range beam to point at each of the sub-swaths in turn, transmits pulses and receives echoes pulse by pulse. The design method of main system parameters and the calculating expressions of the performance indexes are addressed in the paper. A design example is given to compare the performance of the conventional strip operation, ScanSAR and FA-ScanSAR operation. The results show that FA-ScanSAR operation can obtain high resolution by full aperture accumulation in wide swath and improve the signal-to-noise ratio of SAR images for the nearspace slow-moving platform borne SAR.
FA-SCANSAR: FULL APERTURE SCANNING PULSE BY PULSE FOR THE NEARSPACE SLOW-MOVING PLATFORM BORNE SAR
2010-08-20
PIER B
Vol. 25, 1-21, 2010
download: 254
Factorization Method for Finite Fine Structures
Seil S. Sautbekov
This paper deals with the development of the Wiener-Hopf method for solving the diffraction of waves at fine strip-slotted structures. The classical problem for diffraction of plane wave at a strip is an important canonical problem. The boundary value problem is consecutively solved by a reduction to a system of singular boundary integral equations, and then to a system of Fredholm integral equations of the second kind, which effiectively is solved by one of three presented methods: A reduction to a system of the linear algebraic equations with the help of the etalon integral and the saddle point method numerical discretization based on Gauss quadrature formulas the method of successive approximations. The solution to the problem in the first method contains a denominator that takes into account the resonance process. Moreover, the precision of the main contribution of the short-wave asymptotic solution is ensured down to the quasi-stationary limit. The paper presents also comparisons of with earlier known results.
FACTORIZATION METHOD FOR FINITE FINE STRUCTURES