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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$B_{z}$ </tex-math></inline-formula> and radial <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$B_{\rho }$ </tex-math></inline-formula> 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$M_{l}$ </tex-math></inline-formula> have also been derived up to the 11th order (<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$l=11$ </tex-math></inline-formula>). 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.