## 4 Validation
### No-Load Flux Densities
#### 4.2.1 Stator-Segmented Case  
For the estimation of the magnetic flux density created by magnets, it is necessary to adapt the permeance function. Figure 16 presents both permeance function used for the SS-machine with \(p_{gap}\) = 50% and \(n_{gap}\) = 7 and 14 gaps, respectively.  
Figure 17 shows the comparison between the magnetic flux density due to PM in the air gap for a fixed rotor position as a function of the angular position, given by analytical and numerical (2D FEM) models for stator-segmented structures with different \(p_{gap}\) and \(n_{gap}\) (waveforms and spectrum).  
Figure 15: Studied machines. (**a**) Non-segmented machine with \(S_{pp}\) = 2/5. (**b**) Stator-segmented machine with 25% of magnetic gap. (**c**) Non-segmented machine with \(S_{pp}\) = 1/2. (**d**) Rotor-segmented machine with around 20% of magnetic gap.  
The use of extended permeance functions for flux density calculation give similar results than numerical simulations. Indeed waveforms are similar, and harmonics contents and values calculated by the two methods are comparable. The permeance function allows replicating well the effects due to slotting, as well as the effect of the magnetic gaps. The spectral analysis done on the obtained waveforms shows that all the phenomena related to permeance variation are taken into account using the analytical method.  
To complete the study, Figures 18 and 19 present the comparison of the peak values for the magnetic flux density created by magnets in the air gap calculated by 2D FEM and analytical model with variations on \(p_{gap}\) and \(n_{gap}\) from the initial design.  
There is a small difference between peak values of \(B_{/R}\) calculated by analytical and numerical methods which can also be seen in the curves presented in Figure 17. However, the relative difference between the peak values obtained by 2D FEM and analytical model is less than 8%.  
Figures 18b and 19b show the correlation between the waveforms of the magnetic flux density due to PM in the air gap given by analytical and numerical models. Results show that analytical model gives a great estimation of \(B_{/R}\) in the air gap (stator internal radius) (correlation value greater than 0.99 for all the studied \(p_{gap}\) and \(n_{gap}\)). However, on one hand, the increase of \(p_{gap}\) leads to a decrease of the correlation value in the case of stator segmentation with a magnetic material. This is due to approximation on the permeance function related to magnetic gaps. On the other hand, the increase of the number of gaps leads to an increase of the correlation between analytical and numerical results.  
Figure 16: Permeance function for the SS-machine with \(p_{gap}\) = 0.50. (**a**) \(n_{gap}\) = 7 gaps. (**b**) \(n_{gap}\) = 14 gaps.  
Figure 17: Waveforms and spectral analysis of magnetic flux density for the SS-machine with different \(p_{gap}\) and \(n_{gap}\). (**a1,a2**) Reference machine. (**b1,b2**) SS-machine with \(p_{gap}=0.25\) and \(n_{gap}=7\) gaps. (**c1,c2**) SS-machine with \(p_{gap}=0.50\) and \(n_{gap}=7\) gaps. (**d1,d2**) SS-machine with \(p_{gap}=0.50\) and \(n_{gap}=14\) gaps.  
Figure 18: Comparison between numerical and analytical calculation of magnetic flux densities due to PM for the SS-machine with \(n_{gap}=7\) gaps and with different \(p_{gap}\). (**a**) Peak values. (**b**) Correlation.