# Analysis of Permanent Magnet Motors in High Frequency--A Review
## 3 Analysis Tools
### Finite Element Methods (FEM)
#### 3.1.2 Analysis of Coil Conductors  
In low-frequency FEM analysis, generally considering the winding coils to be bulk coils might be accurate enough, as the main performance such as the joule losses or the torque capacity do not depend on the supplying frequency. However, when reaching higher frequencies, the detailed modeling of the winding and its fine mesh is essential as the current distribution is not uniform anymore due to skin and proximity effects. The mesh size must be smaller than the skin depth at that frequency to obtain accurate results [63]. Solving a detailed problem might increase the computational load notably, so it is essential to simplify the problem, trying to find a good trade-off between the computational load and the accuracy of the results. Different papers working on this purpose can be found in the literature. In [72; 84] four different FEM models for axial high-speed Permanent Magnet Synchronous Machine (PMSM) are analyzed. Bulk coils and individual conductors with fine mesh, both in 2-D and 3-D domains. In the case of the 3D FEM simulations, the end windings are considered. The most accurate model to calculate the AC losses in the coils is the 3-D complete model, as it considers end, skin and proximity effects. If bulk coils  
Figure 5: Iron sheet Effects. (**a**) Lamination effect [83]. (**b**) End effect [78]. are used, these effects are not considered. Considering 2D models, usually, even with fine mesh, these models underestimate AC copper losses, as it is stated in [66]. Three methods are proposed to obtain accurate results reducing the computational effort of a 3-D fine mesh:  
* [PERSON] et al. propose to use a bulk coil 3-D model with coarse mesh, arguing that it is enough to calculate the magnetic flux density. Then, applying an analytical formulation, the AC copper eddy current losses are calculated. Skin and proximity losses due to neighboring conductors are neglected in this method, as the analyzed conductors are smaller than the skin depth at the calculated frequency of 1.6 kHz. The authors mention that this method may be useful for coreless permanent magnet machines, where the major part of AC component of copper losses is caused by the permanent magnet flux passing over the conductors [72]. * In the case of [84], a 2-D model with detailed conductors and fine mesh is proposed. Then, based on the flux density variation along the active length of the machine in a coarse mesh 3-D model, some coefficients are calculated to account the variation of losses along the end winding and the active length. * [PERSON] et al. calculate the flux density for different conductor layers, and then the AC loss is calculated analytically for radial flux machines. If the skin depth is lower than the conductor radius, the flux density must be calculated in different points in each conductor [71]. In Table 1, an overview of the different electromagnetic simulations is shown. Most cases include the rotor in the FEM model, whereas the end winding is just included in 3-D models. The mesh, if mentioned in the articles, is usually lower than the skin depth in the analyzed frequency, to calculate skin and proximity effects accurately. In some cases bulk conductors are used (not each turn conductor), so these effects are not taken into account. Some cases just mention that iron core properties are considered, without further explanation. However, most cases consider the iron losses, and even the lamination effect, with 3-D models or equivalent 2-D models. Some models use complex permeabilities, and nonlinear BH curves, while others just keep the permeability constant. The lower part of the table shows some cases where the main objective is to obtain the AC copper losses for high-speed PMSM, with common characteristics with high-frequency models.