Optimization Method Based on Zonal Harmonics for Axially Symmetric Magnets

An optimization method based on the zonal harmonic description of the magnetic field, suitable for the design of axially symmetric magnetic sources, is presented. Expressions of both axial $B_{z}$ and radial $B_{\rho }$ components of the magnetic field are obtained as the infinite sum of the zonal harmonics. For arrangements having as the basic building block an axially magnetized cylinder, analytical formulas for all zonal momenta $M_{l}$ have also been derived up to the 11th order ( $l=11$ ). Through the relationships established by these formulas, a procedure of optimization is initiated to find arrangements of permanent magnet rings that fulfill wished goals of design. Two examples of application have been presented: a pair of rings arrangement (PRA) for a maximally homogeneous gradient field and the four-ring arrangement (FRA) for a maximally homogeneous magnetic field. Several optimized configurations have been found, both for the PRA and FRA cases. For the PRA of thin thickness, geometrical conditions found reproduce those of the known Maxwell pair of current loops for the generation of gradients. The uniformity in gradients so achieved reaches 0.5% within a region with radius 28% of the gap between the lower and upper rings. For the FRA configurations with a gap of 300 mm, seven configurations are reported, providing a field strength between 98.1 and 82.2 mT. The inhomogeneity is 100 ppm at 40% of the gap in the best case. A match of the field lines calculated using finite-element analysis (FEA) with those calculated through the serial expressions of the radial and axial components confirms the compliance of the found FRA configurations with the goals of the design imposed. Furthermore, the homogeneity obtained compares well with the other designs reported in other axially symmetric configurations for magnetic resonance imaging (MRI) presented in the literature. The found FRA configurations, however, distinguish themself by their more simple structure. Discussions about the application of the method in the design of magnetic lenses for electron optics or magnetic actuators have also been included.

I. Introduction

In the design of magnetic systems with purposes of varied nature, axial symmetry is a feature common to be found. It offers a simplicity that is advantageous both from the constructive and the theoretical perspectives, as it has been shown for the construction of magnetic lenses [1], construction of an electromechanical actuator [2], containment of low-density plasma [3], calculation of the self-energy in a sphere with no volume charge in the magnetization [4], and design of thick solenoids as sources of constant magnetic field [5]. Particularly, magnetic resonance imaging (MRI) has driven the demand for spatially homogeneous magnetic fields, leading to the designs of magnet sources with this kind of symmetry: starting with four thick solenoids [6] for the very first whole-body MRI scanner at 0.15 T, through a combination of coils, permanent magnets, and iron yokes [7] until to a well-known Helmholtz pair of coils [8]. Still within the realm of MRI, the Maxwell pair of coils is, on the other hand, another example of designs with this symmetry, but with the goal of producing spatially uniform gradients of magnetic fields [9]. Both pairs of coils are also the result of an optimization. In general terms, a procedure of optimization is the search of a minimum (global or local) in a set of variables over an objective function, which complies with several constraints imposed upon it. Within the realm of the type of magnetic sources hereby concerned, such a procedure has been executed over solutions for the magnetic field obtained through finite-element analysis (FEA) [2], [10] or through a series of zonal harmonics for the magnetic field [5]. From a more theoretical perspective, an optimization method was proposed by Jensen [12] for the same type of source based on a generalized family of orthonormal functions, which allows him to derive both the Halbach structure [11] and a tightly assembled arrangement of magnet rings of counteracting remanence out of the same formalism. The variety of methods, aiming either at the optimization itself or at the calculation of the field, is under the current state of the research noticeable.