Current Loop Off-Axis Field Approximations

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The idealized thin current loop is the foundation for cylindrically symmetric applied coil magnetics. The field equations along the coil axis are easy to solve both numerically and analytically, however, the equations for the fields away from the axis are significantly more complex. The off-axis field equation can be derived from Biot-Savart law and the exact solution involves the difference of elliptical integrals of the first and second kind which make the field difficult to visualize and numerically expensive to calculate.

We have derived a new set of approximations to the current loop equations using a binomial expansion of the elliptical integrals. The resulting terms of the expansion are fitted to minimize the maximum relative error of the approximation over all space. The resulting approximation is exactly correct along the loop axis, at infinity, and it is highly accurate with a low computational cost everywhere else. This approximation is broken down into 5 orders of increasing accuracy and computat onal cost for the radial and axial components of the magnetic field. For the radial field, the first and second order maximum relative errors are approximately 2.5×10-2 and 2.9×10-4 respectively. The higher ordered radial approximations add additional corrective terms with the third, fourth, and fifth ordered approximations having a maximum relative error of 1.8×10-5, 4.3×10-6, and 4.9×10-7 respectively.

Similarly, the axial approximations are highly accurate between the loop and coil axis (h<a) as well as on the loop plane. However, off the plane it diverges as we approach the location where the axial field changes polarity. This is due to the slight difference in location where the axial field changes polarity given by the exact equations and approximations. Therefore, when calculating the relative error of the axial field at the exact polarity changing location, the approximation’s axial is slightly non-zero. This leads to a significantly large relative near these locations although the absolute error is negligible. Outside of the polarity reversing locations, the axial relative error decays to zero. Both the axial and radial approximations are computationally simple algebraic functions. Additionally, 3D relative error plots for the magnetic field inside and outside of the loop are presented and discussed for all orders of approximation to visualize their behaviour.

We have applied these approximations to a common engineering system, the Helmholtz coil, which is a pair of identical coils separated by a distance equal to their radii. As our approximation is exactly correct along the axis, deriving the Helmholtz coil spacing from the approximation is computationally simple. Furthermore, we have shown that the third order approximation offers the highest accuracy for a nearly negligible computational cost when compared to second order. Using the first order approximation, we have derived a simple algorithm allows engineers to quickly calculate what coil radius is required to produce a sufficiently uniform field for a known volume. This algorithm has also been expanded to the second and third ordered approximations if an extremely high accuracy is required at the expense of a small increase in computational cost.
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