A quasi-three-dimensional (3-D) analytical model of the magnetic field in an axial flux permanent-magnet synchronous machine is presented. This model is derived from an exact two-dimensional analytical solution of the magnetic field extended to the 3-D case by a simple and effective radial dependence modeling of the magnetic field. The obtained quasi-3-D solution allows rapid parametric studies of the air-gap magnetic field. Then, analytical modeling of the cogging torque is presented. It is based on the obtained quasi-3-D analytical solution. Results issued from the proposed model in the air gap are compared with those stemming from a 3-D finite-element method simulation as well as with prototype measured values.

Quasi-3-D analytical modeling of the magnetic field of an axial flux permanent-magnet synchronous machine

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

This paper presents a multi-slice analytical model for prediction of the open-circuit magnetic field in slotted semi-closed permanent-magnet axial flux synchronous machines. The technique is based on 2-D exact solution of the Maxwell's equations using the separation of variables method. The magnetic field expressions are developed in slots regions, slot opening regions, magnetic air gap regions, and permanent magnet regions leading to an exact calculation of the slot effects on the air gap magnetic field. The model is established considering that ferromagnetic parts are infinitely permeable, and end effects at inner and outer radii are neglected. The open circuit machine's global quantities (electromotive force and cogging torque) are then deduced from the local magnetic field expressions. Finally, the accuracy of the presented analytical model is validated by comparing its results to corresponding finite-element analyses for two different axial flux machines.

Analytical Modeling of the Open-Circuit Magnetic Field in Axial Flux Permanent-Magnet Machines With Semi-Closed Slots

The purpose of this study is to validate a Jacobian-based iterative method for real-time localization of magnetically controlled endoscopic capsules. The proposed approach applies finite-element solutions to the magnetic field problem and least-squares interpolations to obtain closed-form and fast estimates of the magnetic field. By defining a closed-form expression for the Jacobian of the magnetic field relative to changes in the capsule pose, we are able to obtain an iterative localization at a faster computational time when compared with prior works, without suffering from the inaccuracies stemming from dipole assumptions. This new algorithm can be used in conjunction with an absolute localization technique that provides initialization values at a slower refresh rate. The proposed approach was assessed via simulation and experimental trials, adopting a wireless capsule equipped with a permanent magnet, six magnetic field sensors, and an inertial measurement unit. The overall refresh rate, including sensor data acquisition and wireless communication was 7 ms, thus enabling closed-loop control strategies for magnetic manipulation running faster than 100 Hz. The average localization error, expressed in cylindrical coordinates was below 7 mm in both the radial and axial components and 5 $^\circ$ in the azimuthal component. The average error for the capsule orientation angles, obtained by fusing gyroscope and inclinometer measurements, was below 5 $^\circ$ .

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Jacobian-Based Iterative Method for Magnetic Localization in Robotic Capsule Endoscopy

Recent achievements in active capsule endoscopy have allowed controlled inspection of the bowel by magnetic guidance. Capsule localization represents an important enabling technology for such kinds of platforms. In this paper, the authors present a localization method, applied as first step in time-discrete capsule position detection, that is useful for establishing a magnetic link at the beginning of an endoscopic procedure or for re-linking the capsule in the case of loss due to locomotion. The novelty of this approach consists in using magnetic sensors on board the capsule whose output is combined with pre-calculated magnetic field analytical model solutions. A magnetic field triangulation algorithm is used for obtaining the position of the capsule inside the gastrointestinal tract. Experimental validation has demonstrated that the proposed procedure is stable, accurate and has a wide localization range in a volume of about 18 × 103 cm3. Position errors of 14 mm along the X direction, 11 mm along the Y direction and 19 mm along the Z direction were obtained in less than 27 s of elaboration time. The proposed approach, being compatible with magnetic fields used for locomotion, can be easily extended to other platforms for active capsule endoscopy.

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A discrete-time localization method for capsule endoscopy based on on-board magnetic sensing

A formula is presented for computing the field in axial-field permanent-magnet motors. The formula is based on a three-dimensional (3-D) analytical analysis and is expressed in terms of a finite sum of elementary functions. It is readily programmed and ideal for parametric studies of field strength. It is also well suited for parallel processing and could be developed into a motor model for real-time performance simulations. It is applied here to a practical motor geometry and verified by the use of 3-D finite element analysis (FEA).

M.A. Knewtson

Papers

A three-dimensional field solution for permanent-magnet axial-field motors