### Finite element analysis is used to analyze the effects of different designs on the reduction of cogging torque.

Permanent magnet machines are used in many industrial applications because of their ability to produce high power densities. The market for such machines has been expanding due to the availability of affordable magnet materials, technological improvements, and advances in design and control. While still a relatively new phenomenon in wind turbines, permanent magnet generators are increasingly the focus of R&D in that field.

In any application, the interaction of the permanent magnets with the stator teeth or rotor poles in permanent magnet machines can give rise to cogging torque, which is unwanted pulsation in the shaft torque that causes structural vibrations and noise. Due to the absence of axisymmetry in rotor geometry, cogging torque will vary with the angular position of the rotor. The periodicity is determined by the number of stator slots and rotor poles, while the magnitude is determined by a number of geometric factors such as pole arc angle, magnet dimensions, geometry of the stator teeth, etc.The cogging variation in the torque may interfere with other components such as position sensors. Vibration and noise are amplified further when the frequency of the cogging torque matches the mechanical resonant frequency of the stator or rotor. It is therefore essential to evaluate the cogging torque produced by various design choices for a permanent magnet machine.

Finite element analysis (FEA) is used to analyze the effects of different designs on the reduction of cogging torque, and to enable faster prototyping of the final product. Abaqus FEA from SIMULIA, Dassault Systèmes, was used to compute cogging torque in a permanent magnet generator designed for a wind turbine. To reduce numerical noise, techniques were employed involving a sliding mesh, and then repeated meshes, using the advanced meshing functionality.

One of the challenging aspects of computing a cogging torque curve using the finite element method is reducing the numerical noise generated from the mesh. The noise arises due to the complex variation of the magnetic field in the air gap, often leading to numerical cancellation errors that are sensitive to the nature of the chosen mesh. The analyses were based on the geometry of the stator-internal permanent magnet generator proposed by Zhang et.al. for wind power generation applications (Figure 1).

The simulation consisted of a number of individual analyses, each considering a different angular position of the rotor. Cogging torque computations are sensitive to the nature of the mesh; noise may be introduced if the mesh topology varies in each angular position. The subsequent cogging torque curve can be noisy, and the periodicity of the curve may be lost.

To minimize the numerical noise, a sliding mesh technique was used. With this approach, the stator mesh remains fixed and the rotor mesh is circumferentially re-positioned for each individual analysis. The rotor and stator meshes have fixed topologies and their common interface (the center of the air gap) is divided into equally spaced segments with every angular degree of the interface spanned by seven equally spaced nodes. This allows the rotor mesh to be moved circumferentially in discrete angular increments for each analysis while maintaining spatially coincident nodes at the interface.The nature of the mesh in the stator teeth also influences the numerical noise. Abaqus/CAE was used to generate a repeated mesh in the stator teeth to help reduce the numerical noise. To maintain mesh quality, controls such as edge seeds and single/double biased edge seeds are used. The biased edge seeds can be used to generate smaller elements at the stator-air and the rotor-air interfaces in the air gap. Biased meshing helps resolve the field variation at these interfaces as the magnetic flux leaves the stator tooth and enters the rotor poles, and vice versa. The two-dimensional magnetostatic problem is modeled using an extruded three-dimensional mesh that has only one element along the thickness direction.

A 2D magnetostatic analysis was performed for various angular positions of the rotor. A 2D analysis ignores the end effects and assumes that the field is invariant along the length of the device. This is a reasonable assumption for many motor applications, and allows for fast prototyping of the device. A full 3D analysis of the model can be performed at the end of the design either to confirm or make minor modifications to the design. The angular positions of the rotor considered here range from 0° to 12° with an increment of one-sixth of a degree.

The magnetic field output was postprocessed for each angular position of the rotor to compute the torque on the rotor. The Maxwell stress-tensor-based approach was adopted to compute the torque. In this approach, the torque is computed as an integral on a surface that encompasses the rotor. For the current analysis, the integration surface is chosen at the center of the air gap to minimize numerical noise.

The contour plot of the magnetic flux density at zero angular rotation of the rotor is shown in Figure 2. Notice that the magnetic field is saturated (red regions) in the bridge regions. If the analysis did not account for nonlinearity, the magnetic flux would completely pass through the bridge regions and avoid the high-reluctance air gap, and hence the rotor, altogether.

The cogging torque as a function of angular position is extracted by postprocessing the field output. The torque curve is very smooth and does not exhibit any noise. For an electrical machine, the periodicity of the cogging torque in degrees is given by 360/LCM(M,N), where M is the number of stator slots, N is the of number rotor poles, and LCM signifies the least common multiple. The current model has 12 stator slots and 10 rotor poles and hence the periodicity is six degrees. The computed torque curve has the expected periodicity of six degrees.

*This work was done by Krishna Gundu, Engineering Specialist, at SIMULIA, Dassault Systèmes. For more information, visit http://info.hotims.com/45607-122.*