Current Loop Off Axis Field Approximations With Excellent Accuracy and Low Computational Cost

The current loop is a fundamental building block of cylindrically symmetric magnetic calculations. However, the off-axis magnetic field involves the subtraction of elliptic integrals of the first and second kind, which is computationally expensive, hard to manipulate in equations, and difficult to visualize. By conducting a different binomial series expansion on the original loop integral, a series solution is created, which can be simplified to a set of approximation functions with useful characteristics: exactly correct along the axis and at distance, while in the current loop itself the relative error is limited, computationally simple, highly accurate, and easy to visualize for behavior or symmetry. For the radial field, the first and second orders fit where the parameters are optimized to minimize the relative peak error. Note the symmetry that occurs by expressing as function of the scaled expansion term$W$, which ranges from 0 to 1, allowing maximum relative errors of 0.025 for first order and 2.9E-4 for second order. For the axial field first order, due to the subtraction of terms, the accuracy is only modest but the second order has a 9E-4 maximum relative error, with zero error at the loop, on the$z =0$loop plane. The axial field relative error stays low within the loop, but the outside stays low for any h until it falls to 1% of the$z =0$plane value, then it loses accuracy as$z$is near where the field direction reverses sign due to slight differences in the zero crossing predicated coordinate, increasing again in accuracy as$z$moves further from the reveal. Higher order approximations for both radial and axial add more$W$terms with increasing accuracy—for radial, the third order is 1.8E-5, fourth order is 2.3E-6, and fifth order is 4.9E-7, while axial goes as third 4.6E-5, fourth 6E-6, and fifth 1.3E-6. Relative error plots in 3-D space are presented for all orders of approximations. The simplicity of these functions suggests new ways combining loops to optimize such things as field uniformity.

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