Preferable PMSM rotor geometry for reduced axial flux components in the stator core

The four design modifications employed in the axial end region are investigated using the rotor of a permanent magnet synchronous machine. The reference machine is an 8‑pole synchronous machine with interior permanent magnets in a V‑arrangement. Figure 4 shows the cross section of the PMSM concerned. A reference machine with a simple rotor geometry was selected to better allow a focus on the possibilities and design modifications in the axial end region. The rotor, including the shaft, was constructed from one material and designed as a solid model without lamination, as in additive manufacturing. The main data of the machine are listed in Table 1.

In general, the lamination of the stator will prevent the flux from passing through the stator in the axial direction. This is implemented in the FEA by a packing factor \(k_{\mathrm{p}}\). This results in anisotropy of permeability in the tangential and normal directions. The modification of the effective permeability in the tangential direction \(\mu_{\mathrm{t}}\) and of the effective permeability in the normal direction \(\mu_{\mathrm{n}}\) due to the packing factor \(k_{\mathrm{p}}\) is determined by and where \(\mu_{\mathrm{Fe}}\) is the relative permeability of the stator core and \(\mu_{0}\) is the magnetic constant.

The mathematical model is based on [1] and has been extended to include the step and curve design modifications. The parameters were determined for no-load operation so as to exclude the effect of the stator currents. The axial magnetic stator flux coefficient \(\sigma_{\mathrm{axi}}\) is determined by where \(\phi_{\mathrm{1,axi}}\) represents the axial magnetic stator flux component and \(\phi_{\updelta}\) the total flux passing from the rotor into the stator. The magnetic flux coefficient \(\lambda_{\mathrm{ref}}\) is determined by where \(\phi_{\updelta,\mathrm{A}}\) represents the total flux passing from the rotor into the stator which has been reduced by the design modifications and \(\phi_{\updelta,\mathrm{ref}}\) the total flux without design modifications. The area ratio \(\alpha_{\mathrm{mat}}\) is determined by where \(A\) represents the reduced area resulting from the design modifications. The parameter \(A_{\mathrm{max}}\) defines the maximum reducible area on the reference machine when the number of steps is equal to one. The maximum step height is limited by the height of the edge bridges so as to avoid modifying the permanent magnets. \(\alpha_{\mathrm{mat}}\) = 0% describes the unchanged reference machine and \(\alpha_{\mathrm{mat}}\) = 100 % results when \(A\) equals \(A_{\mathrm{max}}\). The reduced area \(A_{\mathrm{s}}\) resulting from a variable number of steps is determined by where \(N_{\mathrm{s}}\) is the number of steps and \(h_{\mathrm{s}}\) is the step height, as shown in Fig. 3a. The radial step height is set equal to the axial step length so as to create square steps. The reduced area \(A_{\mathrm{c}}\) resulting from a curve is determined by where \(r_{\mathrm{axi,in}}\) represents the variable inner radius of the reduced area, as shown in Fig. 3b. The reduced area \(A_{\mathrm{r}}\) resulting from a variable outer radius is determined by which reflects the subtraction of the area of a quarter circle from a square with side length \(r_{\mathrm{axi,out}}\), as shown in Fig. 3c. The reduced area \(A_{\mathrm{ch}}\) resulting from a variable chamfer is determined by where \(h_{\mathrm{ch,rad}}\) is the radial chamfer length and \(h_{\mathrm{ch,axi}}\) is the axial chamfer length, as shown in Fig. 3d. In case of a 45-degree chamfer, \(h_{\mathrm{ch,rad}}\) equals \(h_{\mathrm{ch,axi}}\). In [1], it was shown that the 45-degree chamfer is preferable to a different axial and radial chamfer lengths.

In Figs. 5, 6 and 7, the results for \(\sigma_{\mathrm{axi}}\) are plotted against \(\alpha_{\mathrm{mat}}\) while varying the number of steps. In comparison with the maximum \(\alpha_{\mathrm{mat}}\), a higher number of steps achieves lower area ratios according to (6). The crosses represent the calculation results from the 3D-FEA. The trend line shows a similar asymptotic behaviour for all numbers of steps. In Figs. 8, 9 and 10, the results for the reduced \(\lambda_{\mathrm{ref}}\) are plotted against \(\alpha_{\mathrm{mat}}\). Especially at small area ratios, results of the FEA for five steps show an almost constant \(\sigma_{\mathrm{axi}}\), but a greatly reduced \(\lambda_{\mathrm{ref}}\).

Figures 11 and 12 compare the results for the variation of the number of steps. The trend lines for three and five steps were extended to the theoretical maximum area ratio. The trend line for \(\sigma_{\mathrm{axi}}\) shows similar results. The results for one step and for three steps are almost identical. Up to an area ratio of about 50% all trend lines are similar. The trend line for five steps shows an asymptotic course up to \(\sigma_{\mathrm{axi}}\) = 0.52%. However, up to an area ratio of 10% the axial magnetic stator flux coefficient is almost constant. The comparison of the results for \(\lambda_{\mathrm{ref}}\) shows significant differences. It can be seen that the higher the number of steps, the higher the reduction in the magnetic flux coefficient. Due to the proportionality between the reference flux and the torque, it can be seen from Figs. 11 and 12 that a single step would be preferable.

In Fig. 13, the results for \(\sigma_{\mathrm{axi}}\) are plotted against \(\alpha_{\mathrm{mat}}\). In comparison with the maximum \(\alpha_{\mathrm{mat}}\) and with regard to the height of the edge bridges, the varied radius of the curve achieves a maximum area ratio of about 80% according to (7). The crosses represent the calculation results from the 3D-FEA simulation. The trend line shows an asymptotic course up to \(\sigma_{\mathrm{axi}}\) = 0.51%. In Fig. 14, the results for the reduced \(\lambda_{\mathrm{ref}}\) are plotted against \(\alpha_{\mathrm{mat}}\). The magnetic flux coefficient decreases by 0.11%.

In Fig. 15, the results for \(\sigma_{\mathrm{axi}}\) are plotted against \(\alpha_{\mathrm{mat}}\) for the reference PMSM. In comparison to the maximum \(\alpha_{\mathrm{mat}}\) and with regard to the height of the edge bridges, the varied radius achieves a maximum area ratio of about 22% according to (8). The crosses represent the calculation results from the 3D-FEA simulation. The trend line shows an asymptotic course up to \(\sigma_{\mathrm{axi}}\) = 0.56%. In Fig. 16, the results for the reduced \(\lambda_{\mathrm{ref}}\) are plotted against \(\alpha_{\mathrm{mat}}\). As the area ratio increases, the reference flux decreases. However, the influence on the reference flux is minimal. The magnetic flux coefficient decreases by 0.11% only.

In Fig. 17, the results for the design modification using a chamfer for \(\sigma_{\mathrm{axi}}\) are plotted against \(\alpha_{\mathrm{mat}}\). In comparison with the maximum \(\alpha_{\mathrm{mat}}\) and with regard to the height of the edge bridges, the varied chamfer achieves an area ratio of about 50% according to (9). The crosses represent the calculation results from the 3D-FEA simulation. The trend line shows an asymptotic course up to \(\sigma_{\mathrm{axi}}\) = 0.53%. In Fig. 18, the results for the reduced \(\lambda_{\mathrm{ref}}\) are plotted against \(\alpha_{\mathrm{mat}}\). The magnetic flux coefficient decreases by 0.12%.

Figures 19 and 20 compare the results for the variation of the design modifications without varying the step counts. At small area ratios, the results for the axial magnetic stator flux coefficient obtained using the curve, the chamfer and the radius are similar to the variation of the number of steps. Up to an area ratio of 15%, the radius generates a smaller \(\sigma_{\mathrm{axi}}\). Between an area ratio of 15% and 35%, the design modification using a chamfer has a smaller axial magnetic stator flux coefficient. At higher area ratios, the design modification using a curve is the better variant to reduce the axial magnetic stator flux. The comparison of \(\lambda_{\mathrm{ref}}\) shows that there are significant differences between the design modifications. Under \(\alpha_{\mathrm{mat}}\) = 5% the trend lines are almost similar. However, the reduction of \(\lambda_{\mathrm{ref}}\) caused by the curve is lowest for every area ratio, followed by the design modification using a chamfer. The results for the radius show the greatest reduction in \(\lambda_{\mathrm{ref}}\). From this, it can be deduced that, with the design modification using the radius, the torque would decrease the most. However, the reduction in \(\lambda_{\mathrm{ref}}\) is very small compared to the reduction in the axial magnetic stator flux components, and so still fulfills the intention of the design modifications. To make best use of the three design modifications in the reduction of the axial magnetic stator flux components while ensuring a small reduction of the magnetic flux, combinations of these would be possible.

In general, when compared to [1], it can be seen that both, \(\sigma_{\mathrm{axi}}\) and the reduction of \(\lambda_{\mathrm{ref}}\), are lower for a PMSM than for a cylindrical rotor synchronous machine. This may be due to the different sizes of the reference machines or the end winding geometry of the field winding.