Electrical Machines Calculation of the Electromagnetic Field by Coupling Finite Elements / Macro-Element

The development of any technology relies not only on experimentation, which is always necessary, but also on a simulation model of the behavior of the device being designed. The physical phenomena that govern electrical devices such as rotating machines are described by Maxwell's equations. To design, maintain or understand the operation of electrical devices, we need precise knowledge of the spatial and temporal distribution of the electromagnetic field within the device under study. To do this, the problem must first be formulated on the basis of Maxwell's equations and the physical characteristics of the materials making up the devices. The formulation of the problem leads to non-linear equations of the parabolic type. The analytical resolution of such equations calls on over-simplifying assumptions, which therefore do not reflect the reality of the physical phenomena. In order to take account of the saturation state of ferromagnetic materials and the complexity of the geometric shapes of devices such as electrical machines, numerical solving methods are generally used. From the 1970s, researchers became interested in adapting the Finite Element Method (FEM), previously used in mechanics and civil engineering, to solving electromagnetic problems. As a result, finite element analysis of two- and three- dimensional electromagnetic field problems has undergone significant development in recent decades. The choice of the degree of approximation in FEM makes it possible to improve the accuracy of the determination of the electromagnetic field. To better model the complex geometric shapes of electromagnetic devices, various finite elements, both rectilinear and curvilinear elements, have been developed. The FEM is also used to solve parabolic equations, thanks to the finite difference discretization of the term representing the variation of the field with respect to time. However, in the dynamic analysis of the electromagnetic field in electrical machines, such as synchronous machines linked to power electronics, the finite elements modelling the air gap become distorted when we try to simulate the movement of the rotor. In practice, this leads to poorly accurate numerical results. Therefore, instead of keeping the topology constant during movement, it is possible to use a variable topology or a mixed topology. However, the former gives rise to discontinuity in movement while the latter is characterized by complex implementation. In order to avoid such drawbacks and to more accurately determine the electromagnetic field in electrical machines, the air gap in rotating machines can be modelled using a special finite element known as a ‘Macro-Element’. This element will be coupled to the conventional finite elements that model the rest of the domain representing the machine. Then, the Finite Elements (FE) - Macro-Element (ME) coupling will be applied to the simulation of the machine operation while taking into account the rotor motion. The development of the various approaches described in the book is based on the example. These approaches range from the formulation of the problem to the determination of the machine parameters, via the calculation of the electromagnetic field using a process of coupling numerical elements and analytical elements. The techniques presented in the book constitute an algorithmic approach that allows the reader to create his own calculation code. The first chapter of the book will be devoted to presenting the finite element formulation of the electromagnetic field equations governing an electrical device. The second chapter will describe the process of calculating the 2D distribution of the vector potential in an electrical machine. The third and final chapter will present the technique for representing the air gap using the Macro-Element, the process for coupling the latter to the finite elements, and the application to the study of the rotating machine and the calculation of global quantities such as the electromagnetic torque. Finally, a selection of bibliographical references is provided to enable the reader to obtain more detailed information on a particular aspect of the book. This bibliography is made up of four categories of references: - The basics of the finite element method, - The foundations of the Finite Element/Macro-Element coupling technique, - Adaptations and applications of the Finite Element/Macro-Element coupling technique, - Other developments.