Polygonal Coil Systems for Magnetic Fields with Homogeneity of the Fourth to the Eighth Order

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Abstract
Systems of polygonal coils have some practical advantages for the production of relatively weak magnetic fields of very large volume, with high homogeneity. They are easier to construct than large circular systems and they have slightly greater axial spacing and correspondingly greater error limits than similar systems using the inscribed circular elements. But they require greater wire length and power input, for the same current, field strength, and error limits. Bloom et al. showed that a polygonal system with N sides can be designed with field homogeneity e, not greater than N, by calculations that use only the axial derivatives of the field at the origin, and they computed two eighth order systems with octagonal filaments. No other polygonal systems with homogeneity above the fourth order have been known, since all proposed sixth order solutions with four square filaments are invalidated by an algebraic error. We have extended the spherical harmonic theory of polygonal systems and of the contours of total vector error in their magnetic fields. For the regular polygon of N sides, we have listed formulas for the axial field derivatives through the tenth order, which suffice to define the contours, for small errors, in circular approximation. For N=4,6,8 we formulate midplane derivatives through the eighth order, to define the N‐fold azimuthal distortion. For larger errors, we describe simple procedures to construct the complete contours from field nets, directly computed, in the midplane and in two planes through the axis. We have computed about 1000 systems with N=4,6,8, or ∞, with two, three, or four elements. They belong to more than a score of classes, with homogeneities from the fourth to the eighth order. The elements may be filaments and/or solenoids. The two‐element systems include Maxwell pairs to generate field gradients with fourth order uniformity. Nearly 300 systems are tabulated, including true sixth order systems with four square filaments. The tables list system parameters with second differences for interpolation, field strength, axial error limits, and efficiency. For fourth order systems, correction factors are added to permit the use of small finite cross sections.