3-D Magnetic Field Mathematical Model Considering the Eccentricity and Inclination of Magnetic Gears

Eccentricity and inclination are common fault types in magnetic gears, serving as prerequisites for the stable transmission of cycloidal magnetic gears (CyMG) and nutation magnetic gears (NMGs). Eccentricity and inclination alter the permeance of the magnetic gear air gap, leading to the generation of complex harmonic magnetic fields within it. To clarify the modulation effects of eccentricity and inclination on the air gap magnetic field, and address the limitation of current 2-D magnetic field models, which fail to calculate the 3-D magnetic field distribution caused by magnetic gear inclination. This article presents a 3-D mathematical model for magnetic gears’ eccentricity and inclination, termed the permeance coefficient-based improved subdomain method (PC-ISM). First, this method computes the 3-D magnetic field of the coaxially facing magnetic gear. Then, by mapping it onto the 3-D magnetic field of the nonuniform air-gap magnetic gear pair using the magnetic permeance coefficient, it mitigates the challenge of calculating the 3-D magnetic field arising from eccentricity and inclination while preserving accuracy. The magnetic permeance coefficient serves not only as a 3-D magnetic field mapping for various magnetic gear setups but also as a descriptor of the modulation effect of nonuniform air gaps on the magnetic field. This article examines the magnetic field characteristics, including distribution, intensity, and order, along with the mechanical characteristics such as torque, torque ripple, and axial force, and verifies these through finite element simulation. The study found that both CyMG and NMG can convert the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p_{\text {pm}}$ </tex-math></inline-formula>th order fundamental magnetic field into the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p_{\text {pm}}\pm 1$ </tex-math></inline-formula>th order harmonic magnetic field. Thus, to ensure stable transmission, the number difference of magnetic pole pairs must be 1. When the minimum air gap is constant, the average harmonic magnetic field intensity in the nutation air gap exceeds that in the cycloid air gap, indicating that the nutation angle enhances the harmonic magnetic field intensity. In the mechanical characteristics analysis, the maximum transmission torque of the NMG reaches 13.82 N<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\cdot $ </tex-math></inline-formula>m, and the calculated volume torque density equals 189.90 kN<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\cdot $ </tex-math></inline-formula>m/m3. At different input speeds, the output speed of magnetic gear 1 (MG1) quickly stabilizes, with a transmission ratio of −8.71 after stabilization.