IEEE TRANSACTIONS ON MAGNETICS, VOL. 47, NO. 4, APRIL 2011 805 Armature Reaction Magnetic Field of Tubular Linear Surface-Inset Permanent-Magnet Machines Yacine Amara, Georges Barakat, and Pascal Reghem GREAH, EA 3220, Université du Havre, 76063 Le Havre, France This paper presents an exact 2-D analytical model for predicting the armature reaction magnetic field in idealized structures of per- manent-magnet tubular linear machines with surface-inset mounted magnets. The armature reaction magnetic field distributions is analytically derived and compared to finite-element analyses. The analytical solution allows the prediction of the machine inductance and reluctance force in closed forms, and facilitates the evaluation of any possible partial irreversible demagnetization of the magnets. It can also be used to estimate resistance limited eddy-current losses in armature windings and permanent magnets. The developed an- alytical model can be advantageously used for the analysis and design of a class of linear tubular machines. Index Terms—Armature reaction field, electromagnetic analysis, linear machines, magnetic fields, permanent-magnet machines. NOMENCLATURE Pole pitch. Stator slot pitch. Magnet length. Magnet thickness. Stator slot height. Moving armature slot height. Air-gap thickness. Stator slot width. Number of stator slots. Number of pole pairs. Magnetic vector potential. , Flux density components in the polar coordinates system. I. INTRODUCTION N OWADAYS, the market of linear drives covers a wide range of industrial applications [1]. The need of fast and accurate analysis and design tools becomes increasingly pressing. As for radial flux rotary machines, the use of per- manent magnets allows improving performance of PM linear machines over other linear machine configurations. If well de- signed, permanent-magnet linear machines allow for high force density, high efficiency and excellent servo characteristics. This paper attempts to provide analytical tools to facilitate the analysis and design of a class of tubular linear PM machines Manuscript received September 20, 2010; revised December 20, 2010; ac- cepted December 22, 2010. Date of publication January 10, 2011; date of cur- rent version March 23, 2011. Corresponding author: Y. Amara (e-mail: yacine. amara@univ-lehavre.fr). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMAG.2011.2104971 Fig. 1. Tubular linear machines structures: (a) inner moving armature and (b) outer moving armature. (Fig. 1). The developed model gives exact field distribution due to armature reaction with taking into account stator slotting and moving armature saliency. Compared to the finite-element method, the analytical model is less time consuming and is more convenient to use in a design optimization process [2]. Compared to previous works [3]–[6], the proposed model takes into account both stator slotting and moving armature saliency in an explicit manner. The developed model embraces both internal and external moving armature topologies. The slotted stator has a classical configuration with straight teeth. The slots and teeth can be equally distributed or not [7], [8]. The slots and teeth can be arranged to accommodate any winding configuration [9], [10]. Results from this analytical model are compared to corresponding finite-element analyses. The analytical solution allows the prediction of the machines self and mutual inductances in closed forms, and facilitates the evaluation of any possible partial, irreversible demagnetization of the magnets [11]. It can also be used to estimate resistance limited eddy-current losses in armature windings [12]. Along with an open circuit analytical model, the proposed model can be used for prediction of on load field distribution [13], [14]. II. ARMATURE REACTION FIELD ANALYTICAL SOLUTION The model is formulated in two-dimensional polar coordi- nates (Fig. 2). The analytical solution for the magnetic field dis- tribution is set to cover only low permeability regions (stator slots (region I), air gap (region II), permanent magnets 0018-9464/$26.00 © 2011 IEEE