PIER B | PIER Journals

2009-03-11

PIER B

Vol. 13, 433-451, 2009

download: 121

Synthesis of Triple-Band and Quad-Band Bandpass Filters Using Lumped-Element Coplanar Waveguide Resonators

Min-Sou Wu , Yu-Zhi Chueh , Jen-Chun Yeh and Shau-Gang Mao

This paper develops a novel design method for synthesizing the multi-passband filter with high flexibility in various passband location and fractional bandwidth. Using the proposed compensation technology in the equivalent circuit of multi-passband resonator, the cutoff frequencies and matching property in passband regions can be improved. Triple- and quad-band bandpass filters operating in both wireless local area network (WLAN) 802.11 a/b/g and worldwide interoperability for microwave access (WiMAX) systems are presented to verify the design method. The lumped-element coplanar waveguide stub fabricated by the split-ring resonator is established to realize filter with compact size. All the measured, full-wave simulated and equivalent-circuit modeled results illustrate a good agreement among them, which validates the multi-passband design methodology and shows the advantages of DC elimination and deep rejection between each passband.

SYNTHESIS OF TRIPLE-BAND AND QUAD-BAND BANDPASS FILTERS USING LUMPED-ELEMENT COPLANAR WAVEGUIDE RESONATORS

2009-03-11

PIER B

Vol. 13, 409-431, 2009

doi:10.2528/PIERB09020802

download: 111

Simulating Wide Band Radar Response from PEC Targets Using Phase Extracted Basis Functions

Su Yan , Shiquan He , Zai-Ping Nie and Jun Hu

In this paper, we first review the Phase Extracted (PE) basis functions by recalling the derivation which shows that the induced current on a PEC surface has the propagating phase factor the same as the incident wave in a scattering problem. The wide band characteristic of this PE basis functions has been investigated by demonstrating that very wide band radar response from PEC objects can be simulated accurately by using PE bases, only based on a single coarse mesh grid. Besides, the resulted current coefficients are shown to vary slowly and smoothly with frequency changing and can be interpolated and extrapolated in very wide band easily. The piecewise cubic Hermite interpolation/extrapolation method with respect to the current coefficients is used to obtain the coefficients in the frequency band of interest. Numerical examples demonstrate very good accuracy and high efficiency in wide band radar response prediction in terms of the amplitude of the scattering field as well as its phase distribution.

SIMULATING WIDE BAND RADAR RESPONSE FROM PEC TARGETS USING PHASE EXTRACTED BASIS FUNCTIONS

2009-03-11

PIER B

Vol. 13, 385-407, 2009

doi:10.2528/PIERB09020601

download: 105

Mechanical Properties of a Ferrofluid Seal: Three-Dimensional Analytical Study Based on the Coulombian Model

Romain Ravaud and Guy Lemarquand

This paper presents a general method for studying the mechanical properties of a ferrofluid seal by using a three-dimensional analytical approach based on the coulombian model of a magnet. The fundamental Maxwell's equations lead us to define the concept of magnetic energy of the ferrofluid seal by using only the threedimensional equations of the magnetic field created by ring permanent magnets radially magnetized. Our study corresponds to the specific case when the ferrofluid is submitted to a very high magnetic field. Under these conditions, we assume that the mechanical properties of the ferrofluid depend only on the magnetic field created by the permanent magnets. Throughout this paper, the remanence polarization J of the magnets used is higher than 1T. The magnetic field we use in order to align the magnetic particles is very intense, greater than 400 kA/m. Consequently, the magnetic particles are assumed to be saturated and the magnetic field they create can be omitted. In this paper, a cylindrical structure consisting of two outer ring permanent magnets radially magnetized and an inner nonmagnetic cylinder is considered. In addition, a ferrofluid seal is placed between them. The calculation of the magnetic pressure of the ferrofluid seal has been analytically established in three dimensions in order to determine its shape. Moreover, the geometrical evolution of the ferrofluid seal shape is presented when the inner non-magnetic cylinder crushes the ferrofluid seal. The radial stiffness of the ferrofluid seal is determined in three dimensions when the inner cylinder is decentered. Furthermore, a way of obtaining the ferrofluid seal static capacity is discussed.

MECHANICAL PROPERTIES OF A FERROFLUID SEAL: THREE-DIMENSIONAL ANALYTICAL STUDY BASED ON THE COULOMBIAN MODEL

2009-03-05

PIER B

Vol. 13, 357-383, 2009

doi:10.2528/PIERB09012904

download: 122

A Projective Approach to Electromagnetic Propagation in Complex Environments

Emidio Di Giampaolo and Fernando Bardati

High frequency methods resort to numerical ray tracing for application to complex environments. A new method based on the geometrical projection performed by a ray-congruence has been developed as a preconditioning of the ray tracing procedure. It builds a visibility tree, i.e., a database, storing information on all possible ray paths inside a scenario. The method gives a solution to a class of open problems of ray tracing techniques: ray missing, double (multiple) counting, termination criterion, calculation upgrade. Other features of the method are the multipath map and the multipath classification that allow the user to know the relevance of multipath at any point of the scenario in advance, before ray-tracing calculation. The method can be systematically applied to scenarios pertaining to different applications provided that the objects belong to the class of polyhedrons. Reflected and diffracted contributions in a scene are modelled as secondary sources which are handled with an off-line electromagnetic field calculation. Numerical analysis is provided showing the efficiency of the method.

A PROJECTIVE APPROACH TO ELECTROMAGNETIC PROPAGATION IN COMPLEX ENVIRONMENTS

2009-03-05

PIER B

Vol. 13, 309-328, 2009

doi:10.2528/PIERB09012704

download: 127

Analytical Expression of the Magnetic Field Created by Tile Permanent Magnets Tangentially Magnetized and Radials Current in Massive Disks

Romain Ravaud and Guy Lemarquand

In this paper, we present new expressions for calculating the magnetic field produced by either tile permanent magnets tangentially magnetized or by radial currents in massive disks. These expressions are fully analytical, that is, we do not use any special functions for calculating them. In addition, they are three-dimensional and can be used for calculating the magnetic field for all regular points in space. The expressions commonly used for calculating the magnetic field produced by radial currents in massive disks are often based on elliptic integrals or semi-analytical forms. We propose in this paper an alternative analytical method that can also be used for tile permanent magnets. Indeed, by using the analogy between the coulombian model and the amperian current model, radial currents in massive disks can be represented by using the fictitious magnetic pole densities that are located on two faces of a tile permanent magnet tangentially magnetized. The two representations are equivalent and thus, the shape of magnetic field produced is the same for all points in space, with a smaller value in the case of it is produced by radial currents in massive disks. Such expressions can be used for realizing easily parametric studies.

ANALYTICAL EXPRESSION OF THE MAGNETIC FIELD CREATED BY TILE PERMANENT MAGNETS TANGENTIALLY MAGNETIZED AND RADIALS CURRENT IN MASSIVE DISKS

2009-03-04

PIER B

Vol. 13, 329-356, 2009

doi:10.2528/PIERB08122210

download: 110

Relativistic Laguerre Polynomials and Splash Pulses

Amalia Torre

New solutions of the homogeneous wave equation of the type usually referred to as relatively undistorted waves are presented. Such solutions relate to the so-called "splash modes", from which indeed they can be generated by applying the Laguerre polynomial operator. Accordingly, the solutions here presented resort to the relativistic Laguerre polynomials --- introduced about one decade ago within a purely mathematical context --- which in fact appear as modulating factor of the basic "splash mode" waveform. Similar solutions of the homogeneous spinor wave equation are also suggested.

RELATIVISTIC LAGUERRE POLYNOMIALS AND SPLASH PULSES

2009-03-03

PIER B

Vol. 13, 275-308, 2009

doi:10.2528/PIERB09012004

download: 115

Discussion About the Magnetic Field Produced by Cylindrical Halbach Structures

Romain Ravaud and Guy Lemarquand

This paper uses a three-dimensional analytical approach based on the Coulombian model for studying the magnetic field produced by cylindrical Halbach structures. Such structures, commonly used in magnetic couplings or in electrical machines, are composed of tile permanent magnets with rotating magnetizations. Such assemblies of tile permanent magnets allow one to easily optimize the radial field shape in the air gap of electrical machines. In addition, Halbach structures can be used in magnetic couplings for improving the torque transmitted between the two rotors. Analytical studies dealing with the optimization of such structures generally use a twodimensional analytical approach for calculating either the magnetic field produced by tile permanent magnets or the forces exerted between them. These two-dimensional expressions are useful because they have a very low computational cost. However, their accuracy depends greatly on the structure dimensions. We propose in this paper to use a three-dimensional analytical model based on the Coulombian model for determining the exact shape of the magnetic field produced by a Halbach structure. Such an approach also allows one to determine the demagnetizing magnetic field inside the tile permanent magnets. This element of information is important for the design of tile permanent magnets. In addition, we show that some effects cannot be predicted with the linearized analytical model. This implies that a linearized dimensional optimization is not accurate. This study has been carried out without any simplifying assumptions. Therefore, the calculations of the three magnetic field components are exact for all points in space, whatever the magnet dimensions. We can say that such a three-dimensional analytical approach is a good alternative to a finite element one because it has a lower computational cost and is more accurate.

DISCUSSION ABOUT THE MAGNETIC FIELD PRODUCED BY CYLINDRICAL HALBACH STRUCTURES

2009-03-03

PIER B

Vol. 13, 257-273, 2009

doi:10.2528/PIERB09010301

download: 195

Shape Recognition of Shallow Buried Metallic Objects at X-Band Using ANN and Image Analysis Techniques

Dharmendra Singh , N. K. Choudhary , Kailash Chandra Tiwari and Rajendra Prasad

A robust algorithm has been developed for improving the backscattered signal and recognizing the shape of the shallow buried metallic object using Artificial Neural Network (ANN) and image analysis techniques for remote sensing at X-band. An ANN with image analysis technique based on tangent analysis is proposed to recognize the shape of metallic buried objects and minimize the orientation effect of buried object. The experimental setup has been assembled for detecting the buried metallic objects of any size at different depths in the sand pit. The system uses only one pyramidal horn antenna for transmitting and receiving microwave signals at X-band (10.0 GHz). All the data to be processed by this algorithm has been received by moving the transmitter/receiver to different locations at a single frequency in X-band in the far field region. ANN technique has been found to be very efficient. An effective training technique has been used to improve the effectiveness of the algorithm. The retrieved result of shape is in good agreement with original shape.

SHAPE RECOGNITION OF SHALLOW BURIED METALLIC OBJECTS AT X-BAND USING ANN AND IMAGE ANALYSIS TECHNIQUES

2009-02-20

PIER B

Vol. 13, 237-256, 2009

doi:10.2528/PIERB09012302

download: 177

Waveguide Simulation Using the High-Order Symplectic Finite-Difference Time-Domain Scheme

Wei E. I. Sha , Xian-Liang Wu , Zhi-Xiang Huang and Ming-Sheng Chen

The high-order symplectic finite-difference time-domain scheme is applied to modeling and simulation of waveguide structures. First, the perfect electric conductor boundary is treated by the image theory. Second, to excite all possible modes, an efficient source excitation method is proposed. Third, the modified perfectly matched layer is extended to its high-order form for absorbing the evanescent waves. Finally, a high-order scattering parameter extraction technique is developed. The cases of waveguide resonator, waveguide discontinuities, and periodic waveguide structure demonstrate that the high-order symplectic finite-difference time-domain scheme can obtain better numerical results than the traditional finite-difference timedomain method and save computer resources.

WAVEGUIDE SIMULATION USING THE HIGH-ORDER SYMPLECTIC FINITE-DIFFERENCE TIME-DOMAIN SCHEME

2009-02-19

PIER B

Vol. 13, 203-236, 2009

doi:10.2528/PIERB09021102

download: 120

Combined Wiener-Hopf and Perturbation Analysis of the h -Polarized Plane Wave Diffraction by a Semi-Infinite Parallel-Plate Waveguide with Sinusoidal Wall Corrugation

Jian-Ping Zheng and Kazuya Kobayashi

The diffraction by a semi-infinite parallel-plate waveguide with sinusoidal wall corrugation is analyzed for the H-polarized plane wave incidence using the Wiener-Hopf technique combined with the perturbation method. Introducing the Fourier transform for the unknown scattered field and applying an approximate boundary condition together with a perturbation series expansion of the scattered field, the problem is formulated in terms of the zero-order and first-order simultaneous Wiener-Hopf equations. The Wiener-Hopf equations are solved via the factorization and decomposition procedure leading to the exact solutions. Explicit expressions of the scattered field inside and outside the waveguide are derived analytically by taking the inverse Fourier transform and applying the saddle point method. Far field scattering characteristics of the waveguide are discussed in detail via representative numerical examples.

COMBINED WIENER-HOPF AND PERTURBATION ANALYSIS OF THE H-POLARIZED PLANE WAVE DIFFRACTION BY A SEMI-INFINITE PARALLEL-PLATE WAVEGUIDE WITH SINUSOIDAL WALL CORRUGATION

2009-02-16

PIER B

Vol. 13, 187-201, 2009

doi:10.2528/PIERB09011603

download: 353

Switched Beam Antenna Array with Parasitic Elements

Muhammad Ramlee Kamarudin and Peter S. Hall

This paper describes the design of the disk-loaded monopole with a parasitic array for beam switching. Usually the radiation pattern of a single element such as a λ/4 monopole and the disk-loaded monopole provide low values of gain. The beamwidth is normally large and the coverage is wide. This may be appropriate in an on-body channel where the antenna orientation may not be easily controlled, such as when the users put the terminal in their pocket. In some non-body applications such as WLAN, it is necessary to design antennas with high gain to meet other demands such as high capacity or long range. Also, in the on-body environment it is essential to have such gain in order to minimize the path loss between the antennas, and hence increase the battery life. The antenna was excited using coaxial cable produced more gain and pattern compared to the single element top disk-loaded antenna. The reduced-size antenna namely a sector antenna array also has been discussed in detail in this paper. Such design has allowed at least 50% of the size reduction. The simulation results have shown very good agreement with the measurement for both antennas.

SWITCHED BEAM ANTENNA ARRAY WITH PARASITIC ELEMENTS

2009-02-16

PIER B

Vol. 13, 171-186, 2009

doi:10.2528/PIERB09011308

download: 200

A Comparison of Genetic Algorithms, Particle Swarm Optimization and the Differential Evolution Method for the Design of Scannable Circular Antenna Arrays

Marco A. Panduro , Carlos A. Brizuela , Luz I. Balderas and Diana A. Acosta

A comparison between different modern population based optimization methods applied to the design of scannable circular antenna arrays is presented in this paper. This design of scannable circular arrays considers the optimization of the amplitude and phase excitations across the antenna elements to operate with optimal performance in the whole azimuth plane (360^ο). Simulation results for scannable circular arrays with the amplitude and phase excitation optimized by genetic algorithms, particle swarm optimization and the differential evolution method are provided. Furthermore, in order to set which design case could provide a better performance in terms of the side lobe level and the directivity, a comparative analysis of the performance of the optimized designs with the case of conventional progressive phase excitation is achieved. Simulation results show that differential evolution and particle swarm optimization have similar performances and both of them had better performance compared to genetic algorithms when all algorithms are allowed equal computation time.

A COMPARISON OF GENETIC ALGORITHMS, PARTICLE SWARM OPTIMIZATION AND THE DIFFERENTIAL EVOLUTION METHOD FOR THE DESIGN OF SCANNABLE CIRCULAR ANTENNA ARRAYS

2009-02-13

PIER B

Vol. 13, 151-169, 2009

doi:10.2528/PIERB09011002

download: 175

Derivation and General Properties of Artificial Lossless Balanced Composite Right/Left-Handed Transmission Lines of Arbitrary Order

Carlos Camacho-Penalosa , Teresa M. Martin-Guerrero , Jaime Esteban and Juan E. Page

A circuit theory-based approach for systematically deriving all possible lossless balanced composite right/left-handed transmission lines is described. To illustrate the usefulness of the proposed approach, novel artificial transmission line unit-cells with tri- and quad-band behaviour are proposed. It is shown that the number of right-handed or left-handed frequency bands exhibited by such transmission lines is determined by the order of its unit-cell. It is explained why artificial lossless balanced transmission lines exhibit a stop-band around each pole of their associated continuous transmission line that can not be closed up. Since this approach allows for the systematic derivation of such transmission line unit-cells of arbitrary order, multi-band components based on metamaterial transmission lines are envisaged.

DERIVATION AND GENERAL PROPERTIES OF ARTIFICIAL LOSSLESS BALANCED COMPOSITE RIGHT/LEFT-HANDED TRANSMISSION LINES OF ARBITRARY ORDER

2009-02-12

PIER B

Vol. 13, 133-150, 2009

doi:10.2528/PIERB09011103

download: 137

A Dielectric Resonator-Based Route to Left-Handed Metamaterials

Jiafu Wang , Shaobo Qu , Hua Ma , Junhua Hu , Yiming Yang , Xiang Wu , Zhuo Xu and Meijuan Hao

In this paper, a new route for the realization of left-handed metamaterials (LHMs) is suggested. It is based on commercially available dielectric resonators with low loss and high temperature stability. By etching simple metallic strips on surface of dielectric resonators, the desired resonance modes can be enhanced while the undesired suppressed. In this way, resonance frequency of desired resonance modes can be tuned to the frequency range of interest. As a typical example, a wide-angle polarization-independent planar LHM based on disk-like dielectric resonators is proposed. Negative permeability and permittivity are realized by etching metallic strips along the electric field orientations of TE01_δ and HEM11_δ modes, respectively.

A DIELECTRIC RESONATOR-BASED ROUTE TO LEFT-HANDED METAMATERIALS

2009-02-11

PIER B

Vol. 13, 111-131, 2009

doi:10.2528/PIERB08122901

download: 194

Design and Analysis Considerations of 4-GHz Integrated Antenna with Negative Resistance Oscillator

Said H. Ibrahim

The purpose of this paper is to present a complete design and analysis of a new integrated balanced transmitter operated at 4 GHz using microstrip technology. It comprises a 4-GHz two-port negative resistance oscillator and a microstrip-patch antenna resonated at 4 GHz. Three different modules are designed and analyzed. The first one used a Lange coupler as a power splitter while the second and third used a one (two)-section branch couplers. The components of the three modules are designed using full-scale computer simulation program named with MSDES; performed by the author; which takes fully into account all the discontinuities included in the microstrip lines, while the modules are analyzed and optimized using APLAC V7.61 software. The design methodologies of the two-port negative resistance oscillator and microstrip patch antenna are introduced and explained. The analyses of the designed modules show a better efficiency and good performance. The modules give a transmit-antenna gain of 15.7 dB with antenna beam widths 91.7^o and 18.2^o in E-and H-planes, respectively. The complete schematic diagrams of the transmitter modules are drawn. These modules can be used in satellite communication, Doppler and other radars, active and semi-active seekers, radio altimeters, missile technology, weapon fuzing, manpack equipment, remote sensing, feed elements in complex antennas, satellite navigation receivers, and biomedical radiators.

DESIGN AND ANALYSIS CONSIDERATIONS OF 4-GHZ INTEGRATED ANTENNA WITH NEGATIVE RESISTANCE OSCILLATOR

2009-02-09

PIER B

Vol. 13, 75-110, 2009

doi:10.2528/PIERB08120704

download: 217

Diffraction by a Semi-Infinite Parallel-Plate Waveguide with Sinusoidal Wall Corrugation: Combined Perturbation and Wiener-Hopf Analysis

Jian-Ping Zheng and Kazuya Kobayashi

The diffraction by a semi-infinite parallel-plate waveguide with sinusoidal wall corrugation is analyzed for the E-polarized plane wave incidence using the Wiener-Hopf technique together with the perturbation method. The problem is formulated in terms of the simultaneous Wiener-Hopf equations by introducing the Fourier transform for the unknown scattered field and applying approximate boundary conditions in the transform domain. Employing the factorization and decomposition procedure together with a perturbation series expansion, the zero- and first-order solutions of the Wiener-Hopf equations are obtained. Explicit expressions of the scattered field inside and outside the waveguide are derived analytically by taking the inverse Fourier transform and applying the saddle point method. Far field scattering characteristics of the waveguide are discussed in detail via representative numerical examples.

DIFFRACTION BY A SEMI-INFINITE PARALLEL-PLATE WAVEGUIDE WITH SINUSOIDAL WALL CORRUGATION: COMBINED PERTURBATION AND WIENER-HOPF ANALYSIS

2009-02-09

PIER B

Vol. 13, 59-74, 2009

doi:10.2528/PIERB08122213

download: 202

TEM Horn Antenna for Ultra-Wide Band Microwave Breast Imaging

Reza K. Amineh , Aastha Trehan and Natalia K. Nikolova

A novel TEM horn antenna placed in a solid dielectric medium is proposed for microwave imaging of the breast. The major design requirement is that the antenna couples the microwave energy into the tissue without being immersed itself in a coupling medium. The antenna achieves this requirement by: 1) directing all radiated power through its front aperture,and 2) blocking external electromagnetic interference by a carefully designed enclosure consisting of copper sheets and power absorbing sheets. In the whole ultra-wide band the antenna features: 1) good impedance match, 2) uniform field distribution at the antenna aperture, and 3) good coupling efficiency.

TEM HORN ANTENNA FOR ULTRA-WIDE BAND MICROWAVE BREAST IMAGING

2009-02-09

PIER B

Vol. 13, 41-58, 2009

doi:10.2528/PIERB08112407

download: 1293

Modified Incomplete Cholesky Factorization for Solving Electromagnetic Scattering Problems

Tingzhu Huang , Yong Zhang , Liang Li , Wei Shao and Sheng-Jian Lai

In this paper, we study a class of modified incomplete Cholesky factorization preconditioners LL^T with two control parameters including dropping rules. Before computing preconditioners, the modified incomplete Cholesky factorization algorithm allows to decide the sparsity of incomplete factorization preconditioners by two fillin control parameters: (1) p, the number of the largest number p of nonzero entries in each row; (2) dropping tolerance. With RCM reordering scheme as a crucial operation for incomplete factorization preconditioners, our numerical results show that both the number of PCOCG and PCG iterations and the total computing time are reduced evidently for appropriate fill-in control parameters. Numerical tests on harmonic analysis for 2D and 3D scattering problems show the efficiency of our method.

MODIFIED INCOMPLETE CHOLESKY FACTORIZATION FOR SOLVING ELECTROMAGNETIC SCATTERING PROBLEMS

2009-02-09

PIER B

Vol. 13, 21-40, 2009

doi:10.2528/PIERB08110202

download: 289

Dispersion Flattened Optical Fiber Design for Large Bandwidth and High-Speed Optical Communications Using Optimization Technique

Somaye Makouei , Morteza Savadi-Oskouei , Ali Rostami and Zia Koozekanani

In this paper, design of the RII triple-clad structure as a dispersion flattened optical fiber including small pulse broadening factor as well as small dispersion and its slope applicable in broadband and fast communication is considered. The proposed optimization technique is based on the Genetic Algorithms (GA) consisting suitable fitness function for each application. The putting forward design method introduces the pulse broadening factor (?/?₀) about 1.0016 after 200 Km propagation at the zero dispersion wavelength that is so excellent compared to the structure (1.2794) reported in [2] recently. Meanwhile, the proposed structure provides high bit rate (more than 197.8 Gb/Sec at 100 km), large dispersion length (larger than 17400 km), uniform dispersion slope ([0.04,-0.08] ps/km/nm²) and broad bandwidths as well as small and uniform dispersion (smaller than 2.02 ps/km/nm) at [1.55-1.7] μm wavelength interval even for core diameter larger than 4.62 μm. Another important thing discussed in this paper is a proposal for optimization of the broadening factor on large wavelength duration, which is necessary for large bandwidth applications. The suggested technique is capable to minimize the pulse broadening factor over duration of wavelengths that is necessary for large bandwidth applications such as dense wavelength division multiplexing (DWDM) and optical time division multiplexing (OTDM). Our calculation for extracting optical properties of the proposed structure is evaluated analytically. For this purpose modal analysis of these fibers for obtaining possible wave vectors for given system parameters are done using Transfer Matrix Method (TMM) in cylindrical coordinate.

DISPERSION FLATTENED OPTICAL FIBER DESIGN FOR LARGE BANDWIDTH AND HIGH-SPEED OPTICAL COMMUNICATIONS USING OPTIMIZATION TECHNIQUE

2009-02-09

PIER B

Vol. 13, 1-20, 2009

doi:10.2528/PIERB08121901

download: 114

Magnetic Field Produced by a Tile Permanent Magnet Whose Polarization Is Both Uniform and Tangential

Romain Ravaud , Guy Lemarquand , Valerie Lemarquand and Claude Depollier

This paper presents the exact 3D calculation of the magnetic field produced by a tile permanent magnet whose polarization is both tangential and uniform. Such a calculation is useful for optimizing magnetic couplings or for calculating the magnetic field produced by alternate magnet structures. For example, our 3D expressions can be used for calculating the magnetic field produced by a Halbach structure. All our expressions are determined by using the coulombian model. This exact analytical approach has always proved its accuracy and its usefulness. As a consequence, the tile permanent magnet considered is represented by using the fictitious magnetic pole densities that are located on the faces of the magnet. In addition, no simplifying assumptions are taken into account for calculating the three magnetic field components. Moreover, it is emphasized that the magnetic field expressions are fully three-dimensional. Consequently, the expressions obtained are valid inside and outside of the tile permanent magnet, whatever its dimensions. Such an approach allows us to realize easily parametric studies.