Valerie Lemarquand

Analytical Calculation of the Magnetic Field Created by Permanent-Magnet Rings

We present analytical formulations, based on a coulombian approach, of the magnetic field created by permanent-magnet rings. For axially magnetized magnets, we establish the expressions for the three components. We also give the analytical 3-D formulation of the created magnetic field for radially magnetized rings. We compare the results determined by a 2-D analytical approximation to those for the 3-D analytical formulation, in order to determine the range of validity of the 2-D approximation.

MUTUAL INDUCTANCE AND FORCE EXERTED BETWEEN THICK COILS

We present exact three-dimensional semi-analytical ex- pressions of the force exerted between two coaxial thick coils with rect- angular cross-sections. Then, we present a semi-analytical formulation of their mutual inductance. For this purpose, we have to calculate six and seven integrations for calculating the force and the mutual induc- tance respectively. After mathematical manipulations, we can obtain semi-analytical formulations based on only two integrations. It is to be noted that such integrals can be evaluated numerically as they are smooth and derivable. Then, we compare our results with the flla- ment and the flnite element methods. All the results are in excellent agreement.

The Three Exact Components of the Magnetic Field Created by a Radially Magnetized Tile Permanent Magnet

This paper presents the exact analytical formulation of the three components of the magnetic field created by a radially magnetized tile permanent magnet. These expressions take both the magnetic pole surface densities and the magnetic pole volume density into account. So, this means that the tile magnet curvature is completely taken into account. Moreover, the magnetic field can be calculated exactly in any point of the space, should it be outside the tile magnet or inside it. Consequently, we have obtained an accurate 3D magnetic field as no simplifying assumptions have been used for calculating these three magnetic components. Thus, this result is really interesting. Furthermore, the azimuthal component of the field can be determined without any special functions. In consequence, its computational cost is very low which is useful for optimization purposes. Besides, all the other expressions obtained are based on elliptic functions or special functions whose numerical calculation is fast and robust and this allows us to realize parametric studies easily. Eventually, we show the interest of this formulation by applying it to one example: the calculation and the optimization of alternate magnetization magnet devices. Such devices are commonly used in various application fields: sensors, motors, couplings, etc. The point is that the total field is calculated by using the superposition theorem and summing the contribution to the field of each tile magnet in any point of the space. This approach is a good alternative to a finite element method because the calculation of the magnetic field is done without any simplifying assumption.

Permanent Magnet Couplings: Field and Torque Three-Dimensional Expressions Based on the Coulombian Model

This paper presents three-dimensional expressions for the optimization of permanent-magnet couplings. First, we give a fully analytical expression of the azimuthal field created by one arc-shaped permanent magnet radially polarized which takes into account its magnetic pole volume density. Such an expression has a very low computational cost and is exact for all points in space. Then, we propose two semianalytical expressions of the azimuthal force and the torque exerted between two arc-shaped permanent magnets. These expressions are valid for thick or thin arc-shaped permanent magnets. Furthermore, this approach allows us to realize easily parametric studies and optimizations. The analytical approach taken in this paper, based on the Coulombian model, is a good alternative compared to the finite element method generally used to study such configurations.

New Formulas for Mutual Inductance and Axial Magnetic Force Between a Thin Wall Solenoid and a Thick Circular Coil of Rectangular Cross-Section

This paper presents new analytic formulas for determining the mutual inductance and the axial magnetic force between two coaxial coils in air, namely a thick circular coil with rectangular cross-section and a thin wall solenoid. The mutual inductance and the magnetic force are expressed as complete elliptical integrals of the first and second kind, Heumans Lambda function and one well-behaved integral that must be solved numerically. All possible singular cases are automatically handled by the proposed formulas. The results of the work presented here have been verified by the filament method and previously published data. The new formulas provide a substantially simple alternative over previously published approaches, which involve either numerical techniques (finite element method, boundary element method, method of moments) or other semianalytic or analytic approaches.

DISCUSSION ABOUT THE ANALYTICAL CALCULATION OF THE MAGNETIC FIELD CREATED BY PERMANENT MAGNETS

This paper presents an improvement of the calculation of the magnetic field components created by ring permanent magnets. The three-dimensional approach taken is based on the Coulombian Model. Moreover, the magnetic field components are calculated without using the vector potential or the scalar potential. It is noted that all the expressions given in this paper take into account the magnetic pole volume density for ring permanent magnets radially magnetized. We show that this volume density must be taken into account for calculating precisely the magnetic field components in the near-field or the far-field. Then, this paper presents the component switch theorem that can be used between infinite parallelepiped magnets whose cross-section is a square. This theorem implies that the magnetic field components created by an infinite parallelepiped magnet can be deducted from the ones created by the same parallelepiped magnet with a perpendicular magnetization. Then, we discuss the validity of this theorem for axisymmetric problems (ring permanent magnets). Indeed, axisymmetric problems dealing with ring permanent magnets are often treated with a 2D approach. The results presented in this paper clearly show that the two-dimensional studies dealing with the optimization of ring permanent magnet dimensions cannot be treated with the same precisions as 3D studies.

HALBACH STRUCTURES FOR PERMANENT MAGNETS BEARINGS

This paper is the third part of a series dealing with permanent magnet passive magnetic bearings. It presents analytical expressions of the axial force and stifiness in radial passive magnetic bearings made of ring permanent magnets with perpendicular polarizations: the inner ring polarization is perpendicular to the outer ring one. The main goal of this paper is to present a simple analytical model which can be easily implemented in Matlab or Mathematica so as to carry out parametric studies. This paper flrst compares the axial force and stifiness in bearings with axial, radial and perpendicular polarizations. Then, bearings made of stacked ring magnets with alternate polarizations are studied for the three kinds of polarizations, axial, radial and perpendicular. The latter correspond to Halbach structures. These calculations are useful for identifying the structures required for having great axial forces and the ones allowing to get great axial stifinesses.

Analytical Design of Permanent Magnet Radial Couplings

This paper presents a three-dimensional analytical approach for computing the torque transmitted by synchronous radial couplings with radially polarized tile permanent magnets. An expression is given as a finite sum of elliptic functions, which has no underlying simplificating assumptions. Such an expression allows parametric studies and optimizations of coupling structures. It is applied first to the study of conventional radial couplings. Then, structures with “short pitch” magnets are considered and compared with conventional ones. Eventually, special designs are presented, as the ones dedicated to screwing applications, for instance, which require creating an asymmetrical torque.

Magnetic Field Created by Tile Permanent Magnets

This paper presents an analytical calculation of the three components of the magnetic field created by tile permanent magnets whose magnetization is either radial or axial. The calculations are based on the Coulombian model of permanent magnets. The magnetic field is directly calculated, without the magnetic potential. Both axial and radial magnetization of the tiles are considered. The expressions obtained give the magnetic field in all the space. Such analytical expressions are very useful for the design and optimization of many industrial applications.

Calculation Method of Permanent-Magnet Pickups for Electric Guitars

This paper first presents the structures of permanent-magnet pickups for electric guitar and the considered device: string, magnet, coil. It then describes a method to calculate the induced electromotive force (EMF) in the pickup coil when the string moves. The method of calculation links the EMF in the pickup coil with the flux cut by the string when it moves. The EMF is of course nonlinear. The harmonics of the EMF can be calculated. This is a first step towards the final aim that will consist in studying the frequency spectrum of the electrical signal given by the pickup, which is quite complicated, in order to link the magnetic structure and the string movement with the musical effect. Analytical calculations using the Coulombian model of magnets are used to evaluate the magnetic field created by the magnet and the electromotive force in the pickup coil.