Passive Shimming by Eliminating Spherical Harmonics Coefficients of all Magnetic Field Components Generated by Correction Iron Pieces

When a magnet generating highly homogeneous magnetic field is designed, a shimming is required. The shimming is usually performed so that the magnetic field only in the axis direction is compensated by iron pieces and/or coils. The compensation is commonly achieved by eliminating the coefficients of the spherical harmonics expansion of the magnetic field generated by the main magnet. Some papers showed the coefficients of the spherical harmonics expansion in only the z-direction for passive shimming. However, recently some magnets generate a magnetic field tilted from the z-axis, such as a magic-angle-spinning nuclear magnetic resonance/MRI. It is, therefore, necessary to eliminate the coefficients of the spherical harmonics expansion in the x- and y-direction.