Symmetry of Magnetostatic Fields Generated by Toroidal Helicoidal Magnets

Abstract

In this paper the axial symmetry of the magnetic field generated by a permanent magnet of helicoidal toroidal kind is shown. In the first part of the paper we illustrate the shape of the magnet and the number of areas where the field is calculated to demonstrate the symmetry. We define quantitatively the size of the toroidal helical magnet and the regions where the magnetostatic field is evaluated. The field is carried out for each angular sector that represents the regions where the magnetic flux density is computed. This calculation is performed with reference to a matrix of points belonging to each sector. Two sets of evaluations are performed. The first one is referred to a less dense matrix of points relative to all the regions. The aim of this computation is to demonstrate the axial symmetry of the field. The second set of calculations concerns the field evaluation by using a much higher dense matrix of points. By using this data we are able to interpolate the same field with a high precision. This second evaluation of the field is carried out with reference only to the flat region facing the first coil of the helical toroidal magnet. The use of an interpolation surface through the final points of the magnetic induction vectors previously computed allows a very fast evaluation of the field virtually in all the infinite points of the angular sector. The symmetry enables us to drastically reduce the time computation of the magnetostatic field in the points of interest

Evaluation of Forces and Torques Generated by Toroidal Helicoidal Magnetic Fields

“…Figure 5 shows the field B Sa generated by A and evaluated on S a with the magnetic charge method. The evaluation of B Sa has previously been performed on all the 70 polarized sectors illustrated in the figure.The detailed procedure of the B Sa computation and the results only relative to the surface S a are illustrated in [6]. In Appendix A a short explanation of the B Sa computation is reported.…”

“…Finally, we observe that edges b of E are not parallel to the tangent s-s to the circumference Γ R at the point U m . All the symbols used in Figure 6 are tuned to those considered in [6].…”

“…The choice of the points was performed following the procedure described in [6]. Table 2 in [6]). All the dimensions of the system are those considered in [6].…”

“…These phenomena also depend on the geometric configuration of the device. Consequently, as discussed in [6], the attempt for evaluating if there are other possible architectures for magnetic systems that minimize the previously mentioned negative effects of the factors i), ii), and iii) can be beneficial. Therefore, with reference to the development of new configurations of electrical machines, a toroidal helicoidal arrangement of permanent magnets has been considered.…”

“…Before we illustrate the mathematical physical model developed to evaluate forces and torques, a short description of the magnetic system is given. The system has already been partially described in [6], but for the convenience of the reader, its working principle is summarized in the following. We can also see that the knowledge of forces and moments caused by this particular configuration of the magnetic field can be advantageous not only when searching for new configurations of electrical machines with high efficiency.…”