This paper presents a synthesis of analytical calculations of magnetic parameters (field, force, torque, stiffness) in cylindrical magnets and coils. By using the equivalence between the amperian current model and the coulombian model of a magnet, we show that a thin coil or a cylindrical magnet axially magnetized have the same mathematical model. Consequently, we present first the analytical expressions of the magnetic field produced by either a thin coil or a ring permanent magnet whose polarization is axial, thus completing similar calculations already published in the scientific literature. Then, this paper deals with the analytical calculation of the force and the stiffness between thin coils or ring permanent magnets axially magnetized. Such configurations can also be modeled with the same mathematical approach. Finally, this paper presents an analytical model of the mutual inductance between two thin coils in air. Throughout this paper, we emphasize why the equivalence between the coulombian and the amperian current models is useful for studying thin coils or ring permanent magnets. All our analytical expressions are based on elliptic integrals but do not require further numerical treatments. These expressions can be implemented in Mathematica or Matlab and are available online. All our models have been compared to previous analytical and semianalytical models. In addition, these models have been compared to the finite-element method. The computational cost of our analytical model is very low, and we find a very good agreement between our analytical model and the other approaches presented in this paper.

Cylindrical Magnets and Coils: Fields, Forces, and Inductances

Design and experiment of a high-static-low-dynamic stiffness isolator using a negative stiffness magnetic spring

In this paper, a theoretical analysis of an axial magnetic coupling is presented, leading to new closed-form expressions for the magnetic axial force and torque. These expressions are obtained by using a 2-D approximation of the magnetic coupling geometry (mean radius model). The analytical method is based on the solution of Laplace's and Poisson's equations by the separation of variables method. The influence of geometrical parameters such as number of pole pairs and air-gap length is studied. Magnetic field distribution, axial force, and torque computed with the proposed 2-D analytical model are compared with those obtained from 3-D finite elements simulations and experimental results.

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Simple Analytical Expressions for the Force and Torque of Axial Magnetic Couplings

For the first time, the power output of an electromagnetic magnetic levitation vibration energy harvester was studied when placed on 10 human participants while walking and running on a treadmill from 2 mph (3.2 km/h) to up to 7 mph (11.3 km/h). The power generated from the device when participants walked at 3 mph (4.8 km/h) averaged 71 μW. When running at 6 mph, the power increased to 342 μW. The testing on participants revealed that due to unique gaits and body structure, acceleration spectrum and damping can vary significantly between participants. Taller participants had a lower step frequency and therefore lower frequency acceleration content, signifying that a single design may not be optimal for all participants. Additionally, the estimated damping force varied largely between participants, from 3 to 8 mN. To minimize the effects of damping, the paper studies the effect of angle of attachment and damping reduction techniques using low friction materials and a guide rail system, which improve power output by over 50% when compared to the sub-optimal design.

Kinetic energy harvesting from human walking and running using a magnetic levitation energy harvester

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.

/pdf/mutual-inductance-and-force-exerted-between-thick-coils-a25558qsd2.pdf

Mutual inductance and force exerted between thick coils

This paper presents some improved analytical expressions of the magnetic fleld produced by arc-shaped permanent magnets whose polarization is radial with the amperian current model. First, we show that the radial component of the magnetic fleld produced by a ring permanent magnet whose polarization is radial can be expressed in terms of elliptic integrals. Such an expression is useful for optimization purposes. We also present a semi-analytical expression of the axial component produced by the same conflguration. For this component, we discuss the terms that are di-cult to integrate analytically and compare our expression with the one established by Furlani (1). In the second part of this paper, we use the amperian current model for calculating the magnetic fleld produced by a tile permanent magnet radially magnetized. This method was in fact still employed by Furlani for calculating the magnetic fleld produced by radially polarized cylinders. We show that it is possible to obtain a fully analytical expression of the radial component based on elliptic integrals. In addition, we show that the amperian current model allows us to obtain a fully analytical expression of the azimuthal component. All the expressions determined in this paper are compared with the ones established by Furlani (1) or in previous works carried out by the authors.

/pdf/comparison-of-the-coulombian-and-amperian-current-models-for-2ajcql640d.pdf

Comparison of the Coulombian and Amperian Current Models for Calculating the Magnetic Field Produced by Radially Magnetized Arc-Shaped Permanent Magnets

This paper deals with the modeling of parallelepipedic magnets of various polarization directions. For this purpose, we use the coulombian model of a magnet for calculating the magnetic potential in all points in space. Then, we determine the three components of the magnetic fleld created by a parallepiped magnet of various polarization direction. These three components and the scalar magnetic potential are also expressed in terms of fully analytical terms. It is to be noted that the formulas determined in this paper are more general that the ones established in the literature and can be used for optimization purposes. Moreover, our study is carried out without using any simplifying assumptions. Consequently, these expressions are accurate whatever the magnet dimensions. This analytical formulation is suitable for the design of unconventional magnetic couplings, electric machines and wigglers.

/pdf/magnetic-field-produced-by-a-parallelepipedic-magnet-of-pm6fwxar40.pdf

Magnetic Field Produced by a Parallelepipedic Magnet of Various and Uniform Polarization

This paper presents a comparison of cylindrical and plane air gap magnetic couplings in which the tile permanent magnet polarizations can be either radial or tangential or axial. The expressions of the torque transmitted between the two rotors of each coupling are determined by using the colombian approach. All the calculations are performed without any simplifying assumptions. Consequently, the expressions obtained are accurate and enable a fast comparison between the structures presented in this paper.

/pdf/magnetic-couplings-with-cylindrical-and-plane-air-gaps-371gv0er3b.pdf

Magnetic couplings with cylindrical and plane air gaps: influence of the magnet polarization direction

The aim of this paper is to show the interest of using equivalence models for calculating the electric fleld produced by cylindrical capacitors with dielectrics. To do so, we use an equivalent model, based on the dual Maxwell's Equations for calculating the two electric fleld components created inside the capacitor and outside it. This equivalent model uses flctitious currents generating a electric vector potential that allows us to determine the electric fleld components in all points in space. The electric fleld produced by charge distributions as capacitor with dielectrics is generally determined by using the coulombian model. Indeed, it is well known that the electric fleld derives from a scalar potential. By using the Maxwell's equations, this scalar potential is in fact linked to the existence of charge distributions that are generally located on the faces of the capacitors. However, this last model does not allow us to obtain reduced analytical expressions since it involves the calculation of charge volume density appearing in the dielectric material for arc- shaped cylindrical topologies. Consequently, it is interesting to look for another approach that gives analytical expressions with a lower computational cost. In this paper, we show that the use of flctitious currents instead of charges allow us to obtain 3D analytical reduced expressions with a lower computational cost. This analytical approach is compared to the coulombian model for showing the equivalence between the two approaches.

https://hal.archives-ouvertes.fr/hal-00430850/document

Romain Ravaud

Papers

Comparison of the Coulombian and Amperian Current Models for Calculating the Magnetic Field Produced by Radially Magnetized Arc-Shaped Permanent Magnets

Magnetic Field Produced by a Parallelepipedic Magnet of Various and Uniform Polarization

Magnetic couplings with cylindrical and plane air gaps: influence of the magnet polarization direction

Introducing Fictitious Currents for Calculating Analytically the Electric Field in Cylindrical Capacitors