Active Shimming for the Entire Components of Magnetic Field Using Tilted Ellipsoidal Correction Coils

Magnets creating highly homogeneous and high magnetic field have been used in MRI, NMR, accelerator, and so on. Coils have various shapes, such as solenoid, dipole, D-shape, toroid. When employing a high-field superconducting magnet, it is hard to use ferromagnetic material for shimming because of its magnetic saturation. Commonly, a superconducting magnet is shimmed with currents, called “active shimming.” Usually, for shimming, a magnetic field is expressed with spherical harmonics function. To achieve a highly homogeneous field, the higher order spherical harmonic coefficients than the 0th are vanished by suitably arranging shim coils. So far, a formulation of spherical harmonic expression of a circular coil is presented in many papers. However, the circular coil is restricted to eliminate inhomogeneity coefficients. Moreover, only the axial component of the spherical harmonics is represented in these papers. Recently, a few magnets are under development to create a magnetic field tilted from the axis. It is, therefore, necessary to vanish the inhomogeneous field of the entire components. This paper presents the spherical harmonic expression of the entire field components for an ellipsoidal coil, and the translation and rotation of spherical harmonics. Finally, an example of active shimming is shown.