Expansion Of The Magnetic Flux Density Field In Toroidal Harmonics

CERN (Conseil Européen pour la Recherche Nucléaire) is recognized worldwide as the main research laboratory in the field of particle physics. Inevitably, all this requires the use of the most advanced technologies, both from the point of view of the instruments and the analytical descriptive methods. One of the numerous potentials of the work carried out at CERN concerns the possibility of exploiting the aforementioned technologies even in contexts distant from the physics of particles, with the result of influencing the technological advancement of many areas. For example, one of the most widely employed theories at CERN, regarding the analytical description of the magnetic flux density inside solenoidal magnets (or approximable as such under suitable assumptions) for the acceleration of particles, is the so-called multipole expansion. This is a two-dimensional or three-dimensional analysis of the distribution of the magnetic flux density generated by the windings of a magnet. The magnet in question can be either resistive or superconductive; clearly, the context in which it is employed does not influence the applicability of the aforementioned theory, provided that are satisfied requirements which mainly concern the geometry of the system. The theory of multipole expansion can also be used in the case of toroidal magnetic configurations. However, in this case it is necessary to develop the analysis in a coordinate system much more complex and not at all intuitive. In this context the work of the present thesis is inserted, which aims to develop a methodology for the description of the magnetic flux density generated by a toroidal magnet, using the theory of multipole expansion. The study begins with a general description of the toroidal geometry and some of its relevant applications. The possible coordinate systems that can be used in the case of toroidal magnets are also described, in order to identify the most suitable coordinate set for the multipole expansion. Following an overview of the magnetic flux density generated by solenoidal magnets, the study focuses on the existing characterizations for what concerns toroidal magnets. After an accurate analysis, it is identified a characterization which allows to describe in the most general form the magnetic flux density inside the system, in order to exploit this study for most of the toroidal configurations, existing and in the design phase. The main advantage of an analytical description based on the expansion in multiples concerns the possibility of identifying the components of the dipole, quadrupole, sestupole, etc., by evaluating the multipole coefficients (also known as multipolar moments or coefficients) that appear expansion. Because of the analytical complexities introduced by this type of description and the geometry considered, numerous verifications are necessary to validate the expressions of the magnetic flux density components. In the central part of the work, a method for the calculation of multipolar coefficients is proposed; it is based on fitting procedures. The only requirement for the application of this technique is the knowledge of the magnetic flux density values (or of the scalar potential) in defined points of a toroidal shape surface. Furthermore, a possible numerical strategy is introduced for the acquisition of the magnetic flux density values on the points of the mentioned grid. Finally, the results obtained from the application of the proposed fitting procedure are presented and discussed, in the case of toroidal magnets formed by 8 and 50 coils.