Modeling Magnetic Fields with Helical Solutions to Laplace's Equation

Abstract

The series solution to Laplace's equation in a helical coordinate system is derived and refined using symmetry and chirality arguments. These functions and their more commonplace counterparts are used to model solenoidal magnetic fields via linear, multidimensional curve-fitting. A judicious choice of functional forms, a small number of free parameters and sparse input data can lead to highly accurate, fine-grained modeling of solenoidal magnetic fields, including helical features arising from the winding of the solenoid, with overall field accuracy at better than one part per million.

A Parametric Integral Formulation to Approximate the Magneto Quasi-Static Fields of 3-D Cylindrical Solenoids With Helical Winding

“…The work by [14] proposes a model to represent the solenoidal magnetic fields via a series solution to Laplace's equation in a helical coordinate system using the separation of variables technique. As stated in section IV of [14], the method requires determination of unknown separation constants of the series-expanded scalar potential through leastsquare fitting of the magnetic field ⃗ B on a subset of 3D spatial grid points inside the solenoid, where ⃗ B is calculated through numerical evaluation of the Biot-Savart integral in discretized 3D space. As noted in section V of [14], the computational cost (in memory, processor, and compute time)…”

“…As stated in section IV of [14], the method requires determination of unknown separation constants of the series-expanded scalar potential through leastsquare fitting of the magnetic field ⃗ B on a subset of 3D spatial grid points inside the solenoid, where ⃗ B is calculated through numerical evaluation of the Biot-Savart integral in discretized 3D space. As noted in section V of [14], the computational cost (in memory, processor, and compute time)…”

“…The work by [15] proposes a model for solving the solenoidal magnetic fields via numerical evaluation of the Biot-Savart integral in the rectangular coordinate system, where the helical contour is discretized as straight line segments in 3D space; this technique appears to be similar to the field-generation step described in section IV of [14]. The formulation targets modeling of the magnetic field of two-wire and three-wire helix structures (e.g., twisted-wire pairs) and includes examination of a helical structure with core diameter of 200 mm.…”

“…The work by [16] provides expressions for multi-pole expansion of a single helical current conductor to describe the magnetic field of a helical dipole with an infinite length. Similar to [14], this work also relies on the separation of variables technique, where the unknown separation constants appear to be determined by application of periodic boundary conditions for a coil of infinite length. The formulation assumes an infinitesimally thin helical current filament and includes examination of the fields due to a half-length helical dipole with a core radius of 177.8 mm.…”

A Parametric Integral Formulation to Approximate the Magneto Quasi-Static Fields of 3D Cylindrical Solenoids with Helical Winding

“…The work by [14] proposes a model to represent the solenoidal magnetic fields via a series solution to Laplace's equation in a helical coordinate system using the separation of variables technique. As stated in section IV of [14], the method requires determination of unknown separation constants of the series-expanded scalar potential through leastsquare fitting of the magnetic field ⃗ B on a subset of 3D spatial grid points inside the solenoid, where ⃗ B is calculated through numerical evaluation of the Biot-Savart integral in discretized 3D space.…”