# TUTCRIS

## Air Gap Fields in Electrical Machines: Harmonics and Modeling of Movement

Tutkimustuotos

### Yksityiskohdat

Alkuperäiskieli Englanti Tampere University of Technology 78 978-952-15-3460-7 978-952-15-3455-3 Julkaistu - 13 helmikuuta 2015 G5 Artikkeliväitöskirja

### Julkaisusarja

Nimi Tampere University of Technology. Publication Tampere University of Technology 1281 1459-2045

### Tiivistelmä

The aim of this thesis is to develop mathematical tools for the efficient electromagnetic analysis of electromechanical devices. In the analysis of rotating electrical machines, the air gap fields play an important role. It is particularly useful to write the magnetic flux density in the air gap as a Fourier series expansion with respect to mechanical angle and/or time. Different terms in the series expansions are then called space and time harmonics and almost all relevant phenomena can be understood from the behaviour of the harmonics. The main topic of this thesis is the application of spectral Dirichlet-to-Neumann mappings to model rotating electrical machines in terms of air gap harmonics. A spectral Dirichlet-to-Neumann mapping for a rotor or stator subproblem in a time or frequency domain boundary value problem (BVP) completely characterizes the electromagnetic behaviour of the subproblem from the viewpoint of the remaining problem and can be used to replace the subproblem with an implicit Neumann boundary condition. This allows the developement of new efficient numerical and analytical methods for analysis of electrical machines. We discuss several analytical and numerical methods, where reformulation of one or both subproblems with Dirichlet-to-Neumann maps is used to speed up the solution or to obtain more accurate solutions. Moreover, properties of the spectral Dirichlet-to-Neumann maps yield information, which can be used to predict and understand the air gap field harmonics. The second topic of the thesis is the application of Riemannian geometry to construct BVPs for electromechanical devices in coordinate systems fixed to the moving bodies. Then, change of distances of the material points corresponds to change in the coordinate expression of the metric tensor in the air gap. In FE implementations, this allows modeling of movement without changes to the finite element mesh. We discuss the application of the approach to linear and rotating movement.

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