Magnetic Model of the CERN Proton Synchrotron

In this thesis, we present development of the first magnetic model of the main Proton Synchrotron magnets. The operation of the machine and conducting research on increasing its performance require a significant amount of the magnetic field data for the optics modelling. For magnets of the Proton Synchrotron, such amount of data cannot be provided by the magnetic measurements, therefore, we have to rely on magnetic models. We have developed the model based on the numerical analysis of the magnetic field in the magnet and performed the validation with the use of the available magnetic measurements data. The results of the 2D quasi-static analysis were used to decompose the multipolar contributions of different magnet circuits on different levels fo the iron core magnetization within the whole range of the PS operation energies. Established formulas of every circuit contributions take into account the global magnetization of the yoke and its saturation as well as a local magnetization changes caused by contributions of other circuits. The 3D numerical analysis was setup to study the influence of the magnet stray field and gaps between the magnet blocks on the integrated multipolar field. Its results were used to derive appropriate model corrections of the integrated multipolar field with respect to the field in the reference cross-section and to approximate the effective magnetic lengths of the auxiliary circuits of the PS magnet. In order to perform a non-linear chromaticity analysis and to take the full advantage of the developed model, we have modified the existing optics model of the PS lattice. We have integrated magnetic model into the optics calculation with two simulation adjustment methods, which optimize the main coil current in similar way as it is done for the real magnets and adjust four free parameters to match the measured parameters of the initial working point. We have successfully validated the combined magnetic and optics models for various beam momentum levels and magnet powering conditions. With the use of the new magnetic model, for the first time in the history of the PS we are able to predict a higher-order chromaticity function for any energy. Base on the optics modelling results, we have derived working point transfer matrices for the non-linear chromaticity, which allow to predict a set of powering current offsets required to apply foreseen working point adjustment. We were also able to model an offset of the measured transfer matrices due to a change of the radial position chosen for the beam production.