Jerry P. Selvaggi

Calculating the external magnetic field from permanent magnets in permanent-magnet motors-an alternative method

We present an alternative method for calculating the magnetic field from a set of permanent magnets in a permanent-magnet motor. The method uses a cylindrical coordinate system to model the geometry of the structure enclosing the magnets. A Fourier series expansion yields an alternative to the more familiar multipole expansion given in spherical coordinates. The expansion is developed by using Greens function in cylindrical coordinates. A technique called charge simulation allows computation of an equivalent point charge distribution. Finally, Coulombs law is applied to express the magnetic scalar potential in a mathematically tractable form.

Computation of the Three-Dimensional Magnetic Field From Solid Permanent-Magnet Bipolar Cylinders By Employing Toroidal Harmonics

We present an analytical method, employing toroidal harmonics, for computing the three-dimensional (3-D) magnetic field from a circular cylindrical bipolar permanent magnet. Bipolar magnets are those which are polarized perpendicular to the axis of the cylinder. We take a completely analytical approach in order to facilitate parametric studies of the external 3-D magnetic field produced by bipolar magnets. The results of our analysis are verified by comparing them to previously published results. The application of toroidal harmonics are ultimately shown to be well-suited for both parametric studies as well as numerical computation.

Computation of the External Magnetic Field, Near-Field or Far-Field, from a Circular Cylindrical Magnetic Source using Toroidal Functions

A method is developed for computing the magnetic field from a circular or noncircular cylindrical magnetic source. A Fourier series expansion is introduced which yields an alternative to the more familiar spherical harmonic solution, Elliptic integral solution, or Bessel function solution. This alternate formulation coupled with a method called charge simulation allows one to compute the external magnetic field from an arbitrary magnetic source in terms of a toroidal expansion which is valid on any finite hypothetical external observation cylinder. In other words, the magnetic scalar potential or the magnetic field intensity is computed on a exterior cylinder which encloses the magnetic source. Also, one can compute an equivalent multipole distribution of the real magnetic source valid for points close to the circular cylindrical boundary where the more familiar spherical multipole distribution is not valid. This method can also be used to accurately compute the far field where a finite-element formulation is known to be inaccurate

Coupling Factor Between the Magnetic and Mechanical Energy Domains in Electromagnetic Power Harvesting Applications

Micro-power generation is an area developing to support autonomous and battery-free wireless sensor networks and miniature electronic devices. Electromagnetic power harvesting is one of the main techniques for micro-power generation and it uses the relative motion between wire coils and miniature magnets to convert mechanical energy to electricity according to Faradays law of induction. Crucial for the design and analysis of these power systems is the electromechanical coupling factor K , which describes the coupling between the mechanical and electromagnetic energy domains. In current literature K is defined as NBl : the product between the number of turns in the coil (N), the average magnetic induction field (B), and the length of a single coil turn (l) . This paper examines the validity of the current K definition and presents two case studies involving cylindrical permanent magnets and circular coil geometries to demonstrate its limitations. The case studies employ a numerical method for calculating K which uses the toroidal harmonics technique to determine the magnetic induction field in the vicinity of the cylindrical magnet.

The Newtonian force experienced by a point mass near a finite cylindrical source

The Newtonian gravitational force experienced by a point mass located at some external point from a thick-walled, hollow and uniform finite circular cylindrical body was recently solved by Lockerbie, Veryaskin and Xu (1993 Class. Quantum Grav. 10 2419). Their method of attack relied on the introduction of the circular cylindrical free-space Green function representation for the inverse distance which appears in the formulation of the Newtonian potential function. This ultimately leads Lockerbie et al to a final expression for the Newtonian potential function which is expressed as a double summation of even-ordered Legendre polynomials. However, the kernel of the cylindrical free-space Green function which is represented by an infinite integral of the product of two Bessel functions and a decaying exponential can be analytically evaluated in terms of a toroidal function. This leads to a simplification in the mathematical analysis developed by Lockerbie et al. Also, each term in the infinite series solution for the Newtonian potential function can be expressed in closed form in terms of elementary functions. The authors develop the Newtonian potential function by employing toroidal functions of zeroth order or Legendre functions of half-integral degree, (Bouwkamp and de Bruijn 1947 J. Appl. Phys.18 562, Cohl et al 2001 Phys. Rev.A 64 052509-1, Selvaggi et al 2004 IEEE Trans. Magn.40 3278). These functions are monotonically decreasing and converge rapidly (Moon and Spencer 1961 Field Theory for Engineers (New Jersey: Van Nostrand Company) pp 368?76, Cohl and Tohline 1999 Astrophys. J.527 86). The introduction of the toroidal harmonic expansion leads to an infinite series solution for which each term can be expressed as an elementary function. This enables one to easily compute the axial and radial forces experienced by an internal or an external point mass.

An application of toroidal functions in electrostatics

A method employing the use of toroidal functions is introduced for calculating the scalar potential and the electric field from a charged conducting ring. This method is an alternative to the well-known elliptic integral formulation and is usually easier to formulate than the elliptic integral solution.

Employing Toroidal Harmonics for Computing the Magnetic Field From Axially Magnetized Multipole Cylinders

We employ a toroidal harmonic expansion in order to develop a three-dimensional solution for the magnetic field due to a permanent-magnet multipole cylinder. The equations derived in this paper can be used for the optimization and design of various devices that employ cylindrical multipole magnets. The analytical equations employ hypergeometric functions derived from the analytical integration of zeroth-order toroidal functions. Hypergeometric functions are quite general and are very useful for parametric studies.

The vector potential of a circular cylindrical antenna in terms of a toroidal harmonic expansion

A toroidal harmonic expansion is developed which is used to represent the vector potential due to a circular cylindrical antenna with a rectangular cross section at any arbitrary point in space. The singular part of the antenna kernel is represented by an associated toroidal harmonic expansion and the analytic part of the kernel is represented by a binomial expansion. A simple example is given to illustrate the application of the toroidal expansion.

Simulation of broken rotor bars in an induction motor

This paper presents the results of running both a transient, finite element model and an equivalent circuit model of an induction motor with broken rotor bars and/or broken end rings. The frequency analysis of the terminal current shows a component that can be identified as being caused by the broken rotor bar.

Computational Electromagnetics and the Search for Quiet Motors

The paper reviews the progress in understanding and modeling the noise and vibration and the stray fields produced by electric machines. In machines that are already quiet, numerical methods are necessary to find the smaller sources of vibrations or emitted electrical signals. Several examples are given for permanent magnet machines and induction motors.