## New Application for Skewed Motor

A new application will be available in Version 9 of Flux. This application allows taking into account skewed machine. Two different models have been developed: 2D N-slices model and extruded model. For each model, you can solve the machine for different working points, using the different magnetic applications:static, magnetoharmonic and transient. The rotor or the stator can be skeweed. In the first part, the methods are preseented. In the second part, the results of simulations of an induction motor are presented.

**Numerical methods: **The main idea is to Consider the case as a 2D problem extruded with the length L of the motor in the axial OZ direction.

The air regions including the end windings are not described by finite elements at the ends of the motor, where a tangential magnetic fieldcondition is imposed. The end windings of the rotor and the stator are taken into account by an electrical circuit coupling. In practice, the user of the 2Ds oftware describes the case at the z = 0 end of the motor as another 2D problem. Two models have been tested to take into account the skewed slots: the 2D N-slices model and the extruded model.

**A. N-slice model. **In the N-slice model, the base 2D surface mesh, which is perpendicular to the shaft, is propagated n times. The motor is considered to be composed of n slices. Thus each slice has a thickness of L/n and is represented by a disk with a surface mesh. The rotor mesh of two adjacent disks has been turned by an angle of Î±/n, where Î± corresponds to the total angle of the skew. The winding and bar currents are assumed to be continuous from one disk to another. The standard 2D Az one component magnetic vector potential formulation is used for this model.

**B. Extruded model. **In this approach, the base 2D surface mesh has been extruded to produce a 3D mesh. The 2D** **rotor mesh is extruded through a** **helical geometric transformation,** **and the 2D stator mesh through** **an OZ translation.** **

The bars of the rotor are solid conductors, coupled to the rotor circuit, and are described by a t-t0- Ï† formulation. The stator and rotor steel sheet regions are described by the total magnetic scalar potential formulation, and the air gap and the other air regions by the t0-Ï† reduced scalar potential formulation . The two skewed models described above have been implemented in FLUX software developed by CEDRAT and LEG (Laboratoire d’ Electrotechnique de Grenoble).

**Results**

A 5.5 kW induction motor has been modeled using the method described above. The motor has 2 pole pairs, 48 stator slots, and 28 rotor slots with a skewing of 0.8 of the stator slot pitch. Only one fourth of the mesh of the motor and one-fourth of the circuit have been described due to a periodicity. In this section only the N-slice model is presented, because of the time savings that it offers compared to the extruded model. The results obtained with the Nslice model have been compared on the one hand with experimental measurement and on the other hand with results obtained from a purely 2D model, i.e., one considering that the rotor slots are straight. (This model is referred to above as the straight slot 2D model, using FLUX, 2D application, version 8.)

**A. Results for rated load. **Our first interest concerns the** **results of the induction motor operating at constant speed. Let us consider the rated load test of the machine, which corresponds to a slip of 1.93%, i.e., a rotation speed of 1471.05 rpm. Results are focused on :

* Comparison of rms current and average torque,

* Torque ripple analysis.

A comparison of the results obtained by the N-slice model with measurements is provided in table I. It shows very good agreement at rated load condition. Indeed, the differences between measurements and calculations are relatively small: approximately 0.88% with regard to the current and 0.33% for the torque.

To show the effect of the skewing of rotor bars, in figure 3 the instantaneous evolution of the electromagnetic torque for the two cases of machine at the rated speed is given. The figure shows that the torque ripple in the skewed slots case model (1.3 N.m) is approximately five times less than that of the straight slots model (6 N.m).The Fourier decomposition of the two signals shows that the straight slots case is more prone to parasitic effects, which results in a wider harmonic spectrum compared to the skewed slots case.

**B. Results for a start up test**

In this section our interest is in the results obtained for a start-up operating mode that requires the electromechanical coupling.

The inertia of the rotor is J=0.4935 kg.m2, the resistant torque is Mr=3.5839 N.m and the friction coefficient is set to f=3.906.10-4 N.m.s.deg-1. Let us note that these coefficients also take into account the presence of the dumb-bell shaft and the load. Figures 5 and 6 show the variation of the torque and the angular speed from start-up until the synchronous speed (1500 rpm) is reached for the straight slots model and the skewed slots model, respectively.

To confirm the results obtained with these simulations, we carried out startup tests on the real machine(with skewed rotor slots).

Even if there is pulsation in both cases, we observed that they show very different mechanical characteristics. The starting torque ripple is much higher for the 2D straight slots model. In fact, the maximum torque ripple that occurs between t=0.6 s and t=0.8 s is equal to 66.63 N.m for the 2D straight slots model and 11.56 N.m for the N-slice skewed slots model.

It is clear that the curves presented in figure 7 are much closer to those obtained for the skewed slots model. In addition to the minimization of the torque ripple, the skewing of the rotor bars makes possible to minimize the harmonic distortions of the electric quantities. It is interesting to check the current versus time at no load. Theoretically, for the machine considered, the skewing of the rotor has as a consequence the diminution of the order harmonics (kN/pÂ±1)k integer, where N represents the number of stator or rotor slots. An attenuation of the harmonics order 13 and 15, then 23 and 25…is expected.

In table II we provide the harmonic decomposition of the signals that are calculated when the machine is at stationary operation.

It is clear that the harmonic contents of the signals extracted from the straight slot model are richer than those of the skewed slots model.

For the first model the maximum amplitude is reached at the 13^{th} order of the stator frequency. The amplitudes at the 15th, 23^{rd} and 25th harmonic orders are significant as well. The amplitudes for the skewed slots model are notably lower: for example, the amplitude of the 13^{th} harmonic order changes from 1.04% of the fundamental amplitude for the straight slots case to .21% for the skewed slots model.

**Conclusion**

In this paper, we have presented two methods, a 2D n-slice model and a t-t0-Ï† 3D skewed model. They take into account the coupling with the electric circuit, the nonlinear B(H) properties, the rotating motion and the inertia momentum conservation law. An induction motor has been simulated with these models. Two different cases have been analyzed: rated load and start-up. The current and torque of the rated load case with the N-slice model are in good agreement with measurements (less than 5% of differences). In the start-up case, the decreasing of the torque ripple and harmonic in the current has been observed between a straight model and a skewed model. With the N-slice model the results are closer to measurement, validating the implementation of this new model in FLUX software.

The next step will be the modeling of motors with complete 3D models, i.e. with end regions described by finite element regions.

**Acknowledgement**

The authors acknowledge Ludovic DOFFE of the LMBE laboratory of Amiens (France) for providing data and measurements.