# Analytical Model to Calculate Radial Forces in Permanent-Magnet Synchronous Machines
## 5 Conclusions  
In this article, an analytical model based on spatial Fourier series for the calculation of radial magnetic forces and pressures in PMSMs is developed. First, magnetic flux densities created in both open-circuit and load operation conditions are represented by Fourier series. The radial magnetic forces and pressures originating from these magnetic flux densities on both the base of the stator tooth and on the rotor surface are then calculated by applying [PERSON]'s tensor law. The proposed analytical expressions for the calculus of flux densities, magnetic pressures, and magnetic forces are defined as a function of several design parameters, such as dimensions, magnetic properties of the materials, and so on. Hence, the model developed gives insight into the functioning principle of the electrical machines and the expressions developed enable design rules to be defined. Furthermore, the proposed analytical model makes it possible to face iterative calculus, which is very common in the designing process of electrical machines, within a relatively short time while ensuring accurate enough results. Figure 26: Analytical order identification of the experimental vibration measurements in the Qs36p15 machine. The proposed model was validated by both FEM simulations and experimental tests. Two types of machine were considered: a fractional machine comprising 36 slots in the stator and 30 poles in the rotor, and an integer machine comprising 48 slots in the stator and 16 poles in the rotor. Results given by the model showed good agreement with the FEM results. The results for flux densities, pressures, and forces were compared with those obtained by FEM and it was demonstrated that the magnitudes and frequencies of the main harmonics were computed with rather good accuracy. However, some differences were observed in the calculated forces and pressures in the stator. This was probably due to the nonlinearity of the problem, which was not considered in the proposed model. This nonlinearity is greater in slotted stators because the flux is concentrated at the edges of the slots saturating these regions. For this reason, significant differences in the pressures calculated at the edge of the slots were obtained, while the pressures computed in the middle of the slots were very accurate. In addition, two prototypes were built and vibrations were measured under load conditions. According to the results, it can be stated that the proposed model is able to identify the frequencies of the main force harmonics. Nevertheless, with these experimental results, the magnitudes of the harmonics could not be validated. This issue is challenging and as a future research prospect, it would be interesting to conduct additional research in order to experimentally validate the magnitudes of these magnetic forces. Furthermore, it is important to notice that the developed model estimates magnetic forces not only on the stator surface but also on the rotor surface. Interesting results were obtained when comparing the performances of both machines. In the integer machine, all estimated harmonics were even. In the case of the fractional machine, the harmonics of the force on the stator were even, but those on the rotor were fractional. These results were corroborated by the experimental tests. Vibrations were measured in both machines by placing an accelerometer on the surface of the stator. In the case of the fractional machine, fractional harmonics were clearly identified. The main advantages of the model compared to the FEM method are the rapidity of the calculi and the ease of relating components of the flux density and magnetic forces to design variables. Therefore, by employing the presented model, an iterative optimization of machine design may be accomplished in a relatively easy and quick way. Finally, as future prospect, it would be interesting to include in the model the amplitude of the forces created by the eccentricities. In this way, the model would be able to compute all magnetic forces that appear in PMSMs.