Fig. 4: Formulation of rotor field isoline functions.
From: Magnetic localization and manipulation of locking synchronous motors
a Magnetic field isolines and arrows in the rotor’s magnetization plane (magnetization and rotation axes). b Field isolines are 2D cross-sections of tri-axial ellipsoidal field isosurfaces, which rotate about a motor rotation axis \(({\hat{{{\boldsymbol{\omega }}}}}_{{{\rm{m}}}})\) with an angle (ϕ) during activation. c In situ rotation of field ellipsoidal isosurfaces generates rotating fields (b(p, ϕ)) at any position (p), which itself forms an ellipse with principal axes defined by the maximum and minimum field strengths \(({b}_{\max },{b}_{\min })\). d Schematic representation of the magnetization plane, with an isoline ellipse traced by a position (pe(biso, β)) defined by the isoline field strength (biso) and position angle (β) from the magnetization axis \((\hat{{{\boldsymbol{\mu }}}})\). Any isoline position (pe) is additionally associated with vectors of maximum field \(({{{\boldsymbol{b}}}}_{\max })\), minimum field \(({{{\boldsymbol{b}}}}_{\min })\), field rotation axis \(({\hat{{{\boldsymbol{\omega }}}}}_{{{\rm{b}}}})\), and scalars of field ratio \(({b}_{{{\rm{ratio}}}}=\parallel {{{\boldsymbol{b}}}}_{\max }\parallel /\parallel {{{\boldsymbol{b}}}}_{\min }\parallel )\) and field angle \((\alpha := \angle {\hat{{{\boldsymbol{\omega }}}}}_{{{\rm{b}}}},{{{\boldsymbol{p}}}}_{{{\rm{e}}}})\). A 3D representation is provided in Supplementary Fig. 5. e Analytical function fit to isoline position vectors \(({{{\boldsymbol{p}}}}_{{{\rm{e}}}}({b}_{{{\rm{iso}}}},\beta ):{{\mathbb{R}}}^{2}\to {{\mathbb{R}}}^{2})\), showing distances from the rotor magnet to isoline positions. f Analytical function fit to field ratio \(({b}_{{{\rm{ratio}}}}(\beta ):{\mathbb{R}}\to {\mathbb{R}})\) and field angle \((\alpha (\beta ):{\mathbb{R}}\to {\mathbb{R}})\), considering that α(β) = α(biso, β) ≡ α(pe).
