Cylindrical harmonic analysis of the magnetic field in the aperture of the superconducting winding of an electromagnet

Abstract

Получены аналитические выражения для средне-интегральных по длине коэффициентов цилиндрических гармоник магнитной индукции, создаваемой сверхпроводящей токовой обмоткой внутри апертуры дипольного или квадрупольного прямого электромагнита на основе геометрических параметров обмотки. Показана пропорциональность средне-интегральных вкладов в магнитную индукцию, создаваемых неосновными гармониками, средне-интегральному значению основного поля.It is of practical interest to create such models of the electromagnetic field of an electromagnet that help correct the mean integral coefficients of the harmonics of magnetic induction by the geometric parameters of the magnet. The aim of the study was to obtain analytical expressions for the coefficients of the mean integral lengths of the cylindrical harmonics of magnetic induction created by the current in the superconducting winding inside the aperture of a dipole or quadrupole direct electromagnet on the basis of the geometric parameters of the winding. The analytical solution found is based on the sectorial spherical harmonics of the internal solution of the Laplace equation for the scalar potential of the magnetic field and allows associating them with a series of polar harmonics. The study shows the proportionality between the mean integral contributions to the magnetic induction, produced by uncharacteristic harmonics, and the mean integral value of the ground field. It has been determined that the contribution to the mean integral values of the harmonics coefficients from the end elements of the winding appears only in the presence of the iron framework as a result of its saturation. The received analytical representations make it possible to calculate necessary correction of geometrical parameters of a current winding when optimizing the magnetic field inside the aperture of dipole and quadrupole electromagnets with given mean integral factors.