Axial Cylindrical Three-Coil Systems for Producing a Uniform Magnetic Field

This paper investigates the magnetic field generated by symmetric three-coil systems. The coils constituting such systems have parallel planes, and their centers are located on the same line (axis). The general solution is derived under the assumption that the coil cross-sections are infinitely small. It is shown that solutions obtained by expressing coil fields in terms of elliptic integrals are more advantageous than those obtained using the formulation based on spherical harmonic series. The advantage of the former formulation is that the coil radii, positions, and ampere-turn ratios are treated as system parameters, which simplifies the coil system analysis and design. Additionally, it is shown that for finding the parameters of a sixth-order system, it suffices to solve a third-order polynomial equation, from the solution of which one or two (depending on the coil radii ratio) physically meaningful solutions (parameter sets) can be readily deduced. The proposed approach considerably facilitates the evaluation of magnetic field uniformity and several field uniformity estimation criteria have been successfully applied. The results show that the field uniformity requirements depend on the dimensions of real-life coil systems and currents, which is crucial for biomedical applications.