Current Loop Off Axis field Approximations with Excellent Accuracy and Low Computational Cost

Abstract

The current loop is a fundamental building block of cylindrically symmetric magnetic calculations. However the off axis field magnetic involves the subtraction of elliptical integrals and second kind which is computationally expensive, hard to manipulate in equations and difficult to visualize. By doing a different binomial series expansion on the original loop integral a series solution is created, which can be simplified to a set of approximations with useful characteristics: exactly correct at along the axis, and at distance, while at the current loop itself the relative error is limited, computational simple, easy to visualize the behavior and symmetry. Figure 1 shows for the radial field the first and second order fits where the parameters are optimized to give minimize the relative peak error. Note the symmetry that occurs by expressing as function of W, allowing maximum relative errors of 0.025 for 1st order and 2.9E-4 for second order (Figure 2 shows 2nd order relative error in all space). For axial field (Figure 1) the first order, due to the subtraction the accuracy is only modest but the 2nd order has a 9E-4 maximum relative error along, with zero error at the loop on the z=0 plane. The axial field relative error stays low within the loop but outside does, and stays low for any h until it falls to 1% of the z=0 plane value, then lose 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 further from the reveal. Higher order approximations add more W terms and see increasing accuracy – for radial 3rd order 1.8E-5, 4th 2,3E-6, 5th 4.9E-7, while axil goes as 3rd 4.6E-5, 4th 6E-6. Note that the simplicity of these functions suggests new ways combining loops to optimize such things as field uniformity. The final paper will show how these approximations are created, plus more details on the error distributions of these and higher orders.  Figure 1: Current loop off axis field approximations of order 1 and 2  Figure 2: Radial field 2nd order approximation relative error