# Step-Skewed modeling of electrical machines

## Overview

Step-skewing of permanent magnets is a design procedure conceived to modify the spatial distribution of magnetomotive force in the air gap along the machine's axial length. The objective of skewing is to improve certain performance aspects of the machine by reducing undesired spatial harmonics of flux. In practice, a skewed machine design will generally exhibit
• a reduced cogging torque,
• diminished vibration and noise and
• reduced harmonic distortion in the induced electromotive forces of its windings
when compared to the corresponding unskewed design. Consequently, this class of machines cannot be properly represented in a Flux 2D project. On the other hand, representing a machine in Flux 3D may be time consuming, and the resulting project may require additional computer resources to be solved.
Step-skewed modeling of electrical machines may be easily achieved using the Flux Skew module to circumvent these simulation issues. Its main features and utilization cases for a simple and automatic extrusion of the geometry are provided below:
• A Simple (homogenous layers) approach allowing the easy parametrization of the layers length, the rotation angle between adjacent layers and the number of layers.
• An Advanced (layer by layer) method allowing the description of more complex topologies (V, W or zig-zag skewing, for instance) with a parametrization of the layers length and of the layer rotation angle on each layer.

## Example of application

This example considers the modeling of a step-skewed permanent magnet synchronous machine (PMSM), both in Flux Skew and in Flux 3D.

To compare these modules and the results yielded by them, a three-phase, eight-pole PMSM is considered. Its three-phase winding is distributed between several stator slots, with one phase per slot. Moreover, its rotor (which contains the magnets) is step-skewed: its permanent magnets are distributed into three skewed layers along the axial direction. Each layer has an axial length equal to 125 mm, and the rotation angle between them is 10°.

A first modeling approach for this device based on a transient magnetic simulation performed in Flux Skew is displayed in the Figure 2 below. It is worth remembering that, in Flux Skew, the whole project description is performed in 2D.

A simulation representing the same PMSM with step-skewed rotor has been performed in Flux 3D as well.

A first comparison between the results obtained with Flux 3D and Flux Skew is highlighted in Figure 3, which compares the machine torque and the flux linked to a phase winding yielded by both approaches. It may be remarked that the results computed with Flux Skew are higher and overestimate their Flux 3D counterparts.

This behavior is not really surprising, since the Flux 3D project takes into account the flux leakage at the extremities of the machine. This effect is not taken into account in Flux Skew, since it solves a series of linked 2D finite element problems instead.

On the other hand, those differences are compensated by a simpler and more straightforward description of the project in Flux Skew. Furthermore, the meshing and solving times are dramatically reduced in Flux Skew when compared to Flux 3D, as shown in the table below:
Table 1. Table comparing the Flux Skew and Flux 3D approaches.
Flux Skew Flux 3D
Mesh generation time 30 seconds 1 hour
Solving time 25 minutes 11 hours
Another comparison between the results computed by Flux 3D and Flux Skew for the machine considered in this example is provided in Figure 4. In this figure, the graph displays the magnetic flux density along a path going from the front of the machine (z = 0 mm) to its end (z = 375 mm) and passing through the center of a stator tooth (the red line shown in part (a) of Figure 4).Figure 4 shows that Flux Skew evaluates a constant magnetic flux density in each layer of the step-skewed machine along that specific path in the stator. This result follows once again from the aforementioned multi-2D approach implemented in Flux Skew, that describes the step-skewed machine problem in terms of a series of linked 2D finite element simulations. On the other hand, Flux 3D evaluates a more realistic magnetic flux density that varies continuously along the machine's axial length. Both solutions are in good agreement, especially at axial positions corresponding to the center of a layer of the Flux Skew project.

It follows from this example that Flux Skew provides results that are in overall good agreement with Flux 3D. This is achieved with a simplified project description and shortened computation times. Even so, the user should always keep in mind the underlying approximations in a Flux Skew simulation, notably while post-processing the results or when comparing them with Flux 3D or with the real device.