IEEE TRANSACTIONS ON MAGNETICS, VOL. 44, NO. 7, JULY 2008 1751 Analytical and Numerical Techniques for Solving Laplace and Poisson Equations in a Tubular Permanent-Magnet Actuator: Part I. Semi-Analytical Framework B. L. J. Gysen, E. A. Lomonova, J. J. H. Paulides, and A. J. A. Vandenput Electromechanics and Power Electronics Group, Department of Electrical Engineering, Eindhoven University of Technology, Eindhoven 5600 MB, The Netherlands We present analytical and numerical methods for determining the magnetic field distribution in a tubular permanent-magnet actuator (TPMA). In Part I, we present the semianalytical method. This method has the advantage of a relatively short computation time and it gives physical insight. We make an extension for skewed topologies, which offer the benefit of reducing the large force ripples of the TPMA. However, a lot of assumptions and simplifications with respect to the slotted structure have to be made in order to come to a relatively simple semianalytical description. To model the slotting effect and the related cogging force, we apply a Schwarz–Christoffel (SC) mapping for magnetic field and force calculations in Part II of the paper. Validation of the models is done with finite-element analysis. Index Terms—Analytical, permanent magnet, skewing, tubular actuator. I. INTRODUCTION T UBULAR permanent-magnet actuators are increasingly being employed in the industry due to their high force density, high bandwidth, virtually zero attraction force, and the absence of end windings. In general, the efficiency is therefore higher then conventional linear permanent-magnet machines and the latter can be replaced in many applications as for example in position engineering [1], pick and place machines, and free piston energy conversion. Many numerical and analytical methods exist for analysis, design, and optimization of electrical machines, each with their own strength and pitfalls. The magnetic equivalent circuit (MEC) is widely used because of its simplicity and relative small computation time [2]–[4]. These equivalent circuits have the analogy of an electrical circuit and are a simplification of the governing quasi-static Maxwell equations (1) (2) In general, the simplification (1) in electrical networks is valid since the currents, , are concentrated in wires and the volt- ages, , are clearly defined. However, in a magnetic circuit the simplification (2) is not always valid since the flux, , does not have predetermined paths due to the slot/tooth structure of the stator and/or translator. The MEC method therefore, needs preliminary knowledge about the flux paths for calculation of the different reluctances, , and the field distribution is only calculated at a few discrete points of the structure, which makes the force calculation very inaccurate. In the case of nonlinear ferromagnetic material, the calculation is performed iteratively due to the nonlinear reluctances. Finite-element analysis (FEA) Digital Object Identifier 10.1109/TMAG.2008.922416 Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. can be regarded as the most accurate one, because it takes satu- ration into account and almost no simplification of the actual ge- ometry is needed. But the long computation time makes it some- times unsuitable for design and optimization. However, only the initial – -curve of the materials are taken into account and, therefore, hysteresis is neglected. Schwarz–Christoffel (SC) conformal mapping is a technique which originates from the Riemann mapping theorem (1851) [5], and it is becoming useful nowadays, due to the compu- tational capabilities of the microprocessors. It states that any simply connected region in the complex plane can be confor- mally mapped onto any other, provided that neither of these re- gions spans the entire plane. This method can only handle ge- ometries with a limited number of points but it is able to calcu- late the field solution numerically at every point within a closed polygon. This method cannot take saturation into account which limits its ability to the linear case. However, the computational time is relatively short compared to the finite-element method (FEM). This method is already applied for the various electrical machine types with successful results [6]–[8]. The most elegant method for determination of the electro- magnetic fields is the analytical solution, which also gives in- sight into the dependency of the geometry and the material prop- erties on the performance characteristics. The disadvantage of this method is that it only applies to the linear case (no satura- tion), and a lot of assumptions and simplifications regarding the geometry have to be made. This paper considers the tubular permanent-magnet actuator (TPMA) shown in Fig. 1. It consists of a translator made of iron with surrounding magnets, which are magnetised in pos- itive and negative radial directions alternatively. The inner part of the translator can be made of a nonmagnetic material with a low mass density in order to reduce the total mass of the trans- lator. The stator consists of iron slots with a three phase con- centrated winding topology. The teeth have saliency in order to reduce the slotting effect. The translator moves in axial direc- tion with a force that consists of four components: • mean force caused by the magnetic field of the magnets and the current excitation with the same spatial frequencies; 0018-9464/$25.00 © 2008 IEEE