Specialized conductor models for finite element eddy current simulation

Transformer and machine windings with complicated shapes and winding schemes can not be considered in full detail within an overall machine model. In this paper, besides the standard solid and stranded conductor models, specialized conductor models are developed for foil and machine windings which may exhibit particular skin and proximity effects. The conductor models globally behave as the true windings without requiring a full geometrical discretization or detailed winding scheme. They refine adaptively according to local error estimates in the global model, offer smaller computational times and are more reliable than if solid and stranded conductor models would be applied. The conductor models appear as magnetically coupled elements in an external circuit model. The field-circuit coupled model is conveniently represented by a coupled system of equations. Introduction Windings applied in transformers and rotating machine can have very complicated geometries and sophisticated winding schemes. Not only intentional eddy current effects aiming at forces or heat generation but also unintentional eddy current effects giving rise to local stress accumulation and hot spots inside the windings have to be simulated accurately. Two model scales are distinguished: the macro scale, i.e. at the global model and considering the fundamental machine behaviour, and the micro scale, i.e. inside the winding and dealing with skin and proximity effects with smaller wave lengths. The magnitude of induced phenomena does not only depend on the properties of the winding itself but also on the vicinity of highly permeable materials or moving parts and the spectrum of the applied excitation. Therefore, in general, phenomena at macroscale can not be decoupled from phenomena at micro-scale. The straightforward approach accounting for micro-scale effects requires a discretization of the entire machine up to micro-scale dimensions. This, however, yields huge models and unacceptable simulation times. Several model reduction techniques are developed in order to introduce micro-scale effects in global models: e.g. analytical macro-elements [7] and inner node elimination techniques [11]. They constitute a-priori model reductions, which are troublesome in case of non-linear materials and may hinder adaptive error control during finite element (FE) simulation. In this paper, we propose to approximate the detailed geometries and winding schemes by additional discretizations for the voltage and to insert these into the magnetic FE model. An error estimator automatically updates the multi-conductor model during the simulation.