# Optimization of Torque in a Permanent Magnet Synchronous Motor (PMSM) for Traction Applications

This article summarizes the simulation and optimization of a 200 Hz, 8-Pole Permanent Magnet Synchronous Motor (PMSM) typically used for traction applications in automotive and transportation systems. The aim is to apply CST EM STUDIO® (CST EMS) to the optimization of two conflicting goals - maximum average torque and minimum torque ripple.

Figure 1 shows the CST EMS model of the principal features of the motor. Generally, electrical machines are, for efficiency and accuracy reasons, preferably simulated in 2D. However, to facilitate easy switching between 2D and 3D, the core model is constructed in 3D. The user may then determine whether the appropriate 2D or 3D solver is used. A clear advantage is that the geometry, materials and other model parameters are consistent between the models - critical when switching between 2D and 3D simulations....

This explains the 3D construction of the model even though the simulations are performed with the 2D transient motion solver in CST EMS. The user can simply specify the cut-plane for use in this solver. It should also be added that the conductivity in the permanent magnets has been set to zero since the focus of this article is on the optimization of the torque in the motor. The inclusion of the eddy currents in the permanent magnets would adversely affect the simulation time without any appreciable effect on the accuracy of the results especially due to the fact that the magnets are embedded in the rotor as opposed to being surface mounted.

Figure 1: General PMSM model description

In CST EMS a rotation gap can be defined to which a constant speed can be attributed. Alternatively, it's also possible to apply the calculation of the equation of motion instead. The rotation gap facilitates a moving mesh technique for the transient time-step simulation. A clear advantage with the moving mesh technique, as opposed to re-meshing at each time-step, is that mesh noise is virtually eradicated - an issue which can critically affect the calculation of cogging torque.

A parameter, speed, has been attributed to this rotation gap to facilitate parametric data extraction as a function of speed e.g. Back-EMF versus speed.

A non-linear M19 BH characteristic is applied to both the stator and rotor components. The barriers have been modeled as air.

Figure 2: Definition of permanent magnet and barrier shapes and positions

Figure 2 focuses on the parametric definition of the barriers. A polygon curve definition is applied in which separate parameters have been attributed to the local u,v coordinates of the individual polygon points. These parameters are to be used in the optimization. In this model, the initial and final points in the definition are locked to the end points of the adjacent permanent magnet.

The distances and angles shown in the figure also indicate the parametric set-up of the permanent magnets.

Figure 3: Extraction of torque and torque ripple used in the optimization

Figure 3 shows the torque in the motor as function of time for the initial model geometry. Also shown is the definition of the average torque and ripple. These single values are used for the optimization. These definitions may be arbitrarily defined by the user to allow more complex goals.

Figure 4: Sum of all goals versus optimizer step (Nelder Mead Simplex Algorithm)

The CST® Optimizer is a multidimensional one which can simultaneously optimize several parameters. Local optimization techniques include, amongst others, the Trust Region Framework and the Nelder Mead Simplex methods. Global optimizers such as CMA Evolutionary Strategy, a Genetic and a Particle Swarm are also available.

The choice of optimization technique depends on several factors such as the number of variables, parameter space size, how far away the starting point is from the optimum etc. The Nelder Mead Simplex method was chosen for this optimization. An added advantage using this method is its ability to continue optimization even if for some parameter settings the model cannot be evaluated i.e. infeasible results.

The average torque (Goal > 400 Nm) and torque ripple (Goal < 40 Nm) results are passed on to the optimizer in the form of a goal function which is minimized. The goal values may be arbitrarily selected to instruct the optimizer to find the maximum average torque even if this value is not obtainable with the parameter constraints.

Figure 4 shows the sum of all goals where a convergence can be clearly seen.

Figure 5: Initial and optimized steady-state torque versus time

Figure 5 shows the steady state torque for both the initial and optimized geometries.

Figure 6: Initial and optimized rotor geometries

In the initial model, the permanent magnets were buried intentionally deep within the rotor which clearly leads to poor torque production in the motor. In figure 6, the biggest improvement can be seen in the average torque. The improvement in the torque ripple is marginal. This is to be expected since the magnets are much closer to the surface of the rotor in the optimized configuration and the interaction between the magnets and stator teeth becomes significant.

Figure 7: Time variation of the absolute magnetic flux density

Monitors are available to allow the magnetic fields and other quantities to be extracted. Figure 7 shows the absolute value of the magnetic flux density versus time. Further post-processing of this data may be performed such as the extraction of field values at points or on faces.

Figure 8: Back-EMF per phase versus time

In contrast to a magnetostatic parametric analysis, a transient motion solution enables the automatic calculation of the time-varying inductance and the Back-EMF in each phase as shown in figure 8.

Figure 9: Peak and RMS Back-EMF (V) versus Speed (RPM) at No-Load

The usefulness of the Back-EMF is also exhibited in figure 9 in which its variation is plotted as a function of speed.

This articles serves to demonstrate some of the concepts and functionality required to perform the electromagnetic field simulation and optimization of a typical permanent magnet synchronous machine. Through the optimization of the flux barriers geometry, the stray flux was reduced and so the average torque increased. To further reduce the torque ripple, other approaches may be applied such as modifying the rotor surface profile.

In conjunction with modern optimization tools, finite element simulation allows the designer to investigate and improve current designs or develop new motor topologies and concepts.

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