# CST – Computer Simulation Technology

## Calculation of Cogging Torque in a Surface Permanent Magnet Motor

CST EM STUDIO® (CST EMS) is used to calculate the cogging torque in a 4-pole Surface Permanent Magnet Motor (PMSM). Cogging torque is an inherent drawback in permanent magnet machines and results from the interaction of the permanent magnet magnetomotive force (mmf) harmonics and the air-gap harmonics due to the stator slotting. An unexcited rotor will tend to attempt to align itself to a stable position. Its magnitude depends on parameters such as the rotor position, number of poles and stator teeth. Since this "pulsating" torque does not contribute to the net torque of the machine, it leads to speed ripple and induces vibrations and is a major design criterion in such machines.

For such a calculation a parametric sweep is performed to efficiently obtain the torque on the rotor as a function of the mechanical angle under a no-load condition. For this condition, only the permanent magnets are required for this type of simulation hence the stator winding excitation is not required....

A common technqiue for reducing the cogging torque, skewing of the stator stack or magnets, can be investigated via the simulation of a series of 2D slices or a full 3D simulation. This technique, although not addressed in this article, can be carried out with either the CST EMS 2D and 3D solvers.

Figure 1: Definition of one of the 4 radially magnetized permanent magnets

The geometry of the 4-pole surface permanent magnet motor can be easily constructed in the CST EMS modeler. The capability to import DXF is also available. In either case, parameters can be assigned to key geometrical features. Since the magnetostatic solver is used for the simulations, non-linear material properties are attribued to both the stator and rotor components.

The permanent magnets are firstly created as solid components. The magnetization is then attributed to each component according to its type, constant or radial, as shown in Figure 1. The magnetization vector is alternately inverted to create the required 4-pole field.

The simulations in this article were carried out using the 2D solver. To perform a 2D analyses, a 3D model is initially created. The user simply specifies the cut-plane or slice on which the 2D simulation is carried out. In the case where only a 2D simulation is required and the user does not wish to model the typically complicated winding geometry, special 2D excitation coils are available.

Figure 2: Mesh in the air gap of the motor

For the accurate calculation of torque, especially in the case of cogging torques which are generally quite small, 2nd order mesh elements in combination with a suitably defined mesh should be used as shown in Figure 2. The permanent magnets are located on the surface of the rotor meaning that the rotor torque calculation should be extended to the permanent magnets as well. The torque or force calculation is an integration of the fields on a surface around an object. When objects touch, the integration surface would normally follow a path at the interface between the two objects. This leads to an error and hence the need to extend this integration surface to exterior of both objects. The force calculation in CST EMS allows the user the option to include touching objects. CST EMS also warns the user of any touching objects during the force calculation.

Figure 3: Flux lines in the PMSM

The vector potential or flux lines can, in addition to other fields quantities, be visualized in the motor as shown in Figure 3.

Figure 4: Angular variation (mechanical degrees) of Cogging Torque (Nm) on rotor over a 15 degree span

A parametric analysis can be performed to investigate the effect of geometric parameters on the cogging torque. The dependence of torque on the tooth (or slot) width is shown in Figure 4. Due to the periodic nature of the torque, a span of 15 degrees is sufficient. As expected, the cogging torque increases as the slot width increases.

This article has demonstrated the possibility to obtain the cogging torque of a motor. This is a common and attractive alternative to transient simulations and allows the the user to efficiently establish the effect of several geometrical features on the cogging torque. The windings of the motor can also be assigned to determine other motor design parameters.