Design and measurement of three-dimensional uniform-field coils based on swarm intelligence algorithms under ferromagnetic boundaries.

• The uniform-field coils design parameters under ferromagnetic boundaries are optimized by swarm algorithms based on the target-field method. • The equipotential division equation of the flow function is given considering the actual needs of the magnetic shield with a through-hole. • The effect of the shield permeability and the endcap opening size on the magnetic field distribution at the center of the coils is analyzed. • The maximum relative errors between the measured field generated by the WOA-TFMUFY coil and the simulation results were no higher than 2.2 %. Combination systems of high permeability magnetic shields and coils are widely used in fields such as the magnetic field standard metrology and the zero magnetic room construction. To solve the problem that the magnetic field of uniform-field coils under the ferromagnetic boundary of magnetic shields is difficult to design precisely and the uniformity is difficult to improve, three-dimensional uniform field coils are designed based on the magnetic field model of coils under the ferromagnetic boundary using the target-field method (TFM), and the weight coefficient vectors determining the coils' structure are optimized using intelligent optimization algorithms, which improves the uniformity of the magnetic field in the target region by 2 orders of magnitude through whale optimization algorithm (WOA). Then, the radial WOA-TFMUFY coil and the axial WOA-TFMUFZ coil are designed considering sidewall optical holes, and their magnetic field distribution is analyzed and verified under different permeabilities and axial opening sizes of the shield. Compared to the traditional saddle coil and Lee-whiting coil, the maximum magnetic field non-uniformity in the target uniform field region is reduced by 82.2 % and 92.8 % for the WOA-TFMUFY coil and WOA-TFMUFZ coil. The maximum relative errors between the measured magnetic field generated by the designed coils and the finite element method (FEM) simulation results are no higher than 2.2%. This novel method can provide more precise and uniform magnetic field metrology standard elements. [ABSTRACT FROM AUTHOR]