2908 zyxwvutsrqponm IEEE zyxwvut TRANSACTIONS ON MAGNETICS, VOL. 29, zyx NO. zyx 6, NOVEMBER 1993 Field Distortion Caused by Magnetization Tolerances of Permanent Magnets Manlio G. Abele and Henry Rusinek New York University, New York NY 10016, USA Franc0 Bertora Esaote Biomedica, 16153 Genova, Italy zyxwvuts Abstract zyxwvutsrqp - Factors contributing to distortion of the field generated by permanent magnets, such as fabrica- tion and material tolerances, are analyzed using the har- monic series expansion of the magnetostatic potential. The power spectrum is influenced by the number of discrete blocks and the number of layers of the magnetic structure. Ferromagnetic plates, acting as spatial fllters inserted in the magnetic structure, reduce the distortions due to magnetization tolerances by approximately 60%. I. INTRODUCTION Permanent magnets based on materials with quasi-linear demagnetization characteristics, such as rare earth alloys, can be designed to generate strong magnetic fields of the order of the remanence of the material [l]. Several factors contribute to distort the ideal field expected inside the mag- net cavity, including geometrical factors and demagnetiza- tion characteristics that can be corrected in the design phase. Fabrication and material tolerances, on the other hand, must be corrected by shimming the assembled mag- net. Methods for correcting field distortions are of particular importance in applications where the field must be gen- erated with high precision, such as for MR imaging in medicine, where the requirements of field uniformity are of the order of 50 parts per million or better. Magnetization tolerances result from imperfection of fabrication and magnetization processes of the magnetic materials. A powerful magnet is typically assembled from a large number uniformly magnetized blocks. The designer must expect differences in the remanences of the order of a few percent in magnitude and of the order of several degrees in orientation from block to block. The remanence zyxwvu 7 in each block can be written as zyxwvutsrq f = yo + @, where 70 is the nominal uniform remanence and e is the random fluctuation. Manuscript received Fkbruary 15, 1993. This work was sponsored by Esaote Biomedica, Italy. The distribution of zyxwv 5'0 is dictated by the design of the magnetic structure. Its magnitude is usually uniform throughout the structure. This paper models the magnetiza- tion tolerances by assuming that the magnitude of ss' is the same in each block and that its orientation obeys a uniform probability distribution over a solid angle 4%. If the material has a linear demagnetization characteristic with a small susceptibility, the effect of ss' can be analyzed independently of the field generated by Yo. The field dis- tortion within the magnet cavity caused by ss' is character- ized by the expansion in spherical harmonics of the distor- tion &D of the scalar potential: where p, 8 and w are the spherical polar coordinates of a point of the magnet cavity, The coefficients q?? and q# are given by where the sum is extended to the total number N of indivi- dual blocks, sk is the surface of the block where ss' = Gk, and 7t is perpendicular to sk and oriented outwards with respect to S,. Pi is the Legendre associated function of the first kind. Goefficients qiI[r2 determine the spatial spec- trum of the field distortion generated by 0. II. YOKELESS CYLINDRICAL MAGNET A yokeless magnet is a structure designed to generate a uniform field within its cavity and to confine the field within its external boundary without the need of a fer- romagnetic yoke. An example is the cylindrical structure where the magnetic material is confined between two coax- ial cylinders of radii rl and r2. The magnet is composed of uniformly magnetized blocks arranged in concentric layers. The field within the cavity is uniform, for example, if the remanence J: is chosen as rl J, = Jo zyxwvu (7) cosy , 2 r 1 r 2 J, = Jo (-) siny , (3) where r, y are polar coordinates and Jo is a constant [21. 0018-9464/93$03.00 8 1993 IEEE