Torque Density Analysis and Verification of Coaxial Magnetic Gear Based on Analytical Electromagnetic Model

Abstract

The conventional types of electric machine systems for low speed and high torque applications are the direct-drive machines and the electric machines with the mechanical reduction gears. The direct-drive machines have the advantages in uncomplicated structure and simple controllability, but the disadvantages in large volume and required high output capacity of inverter. The mechanical reduction gear enables us to decrease the system volume, but have the issues in low torque density in high gear ratio, relatively low maximum operating speed, fatigue failure, regular maintenace, oil leakage, and increased backlash after a long-term operation. As an alternative fascinating solution, the magnetic gear receives attention. Other than the mechanical reduction gears, the magnetic gear does not require for the physical contact between the input shaft and the output shaft. Thus, it is possible for the magnetic gear to operate in high speed region. Also, it exhibits the high torque density in high gear ratio. Furthermore, it shows the advantages in high system reliability, low noise and vibration, inherent overload protection, and semi- permanent operation without regular maintenance. The coaxial magnetic gear exhibits the highest torque density out of all types of magnetic gears. However, the ferromagnetic pole piece or the flux modulator modulates the magnetic flux density distribution in air-gap region, and thus makes the energy conversion principle complicated. Due to its complex torque generation mechanism, the majority of prior academic papers focus on extensive, application specific multi-objective optimization methods for coaxial magnetic gear design. It has scarcely been discussed that which magnetic flux density harmonics contribute to torque about how much. As a result, we raise question that we should consider all the harmonics and all the phase shifts for every cases or applications. To solve the problems, this paper presents an analytical electromagnetic model for torque density computation of coaxial magnetic gear. For analytical approach, the pole pair and pole piece combination is limited to pin + pout = np. The static magnetic field or magnetostatic field and linear permeability of magnetic cores are assumed. The eddy currents and the corresponding eddy current loss are neglected. The tangential magnetic flux density distribution was obtained indirectly from radial magnetic flux density distribution. Using the aforementioned assumptions, the radial and the tangential magnetic flux density distributions are analytically expressed using Fourier Series. Based on the analytical expressions, the harmonic order, the angular velocity, the amplitude, and the phase angle of 6 major components are determined. Using the equivalent frequency characteristics, each component can be illustrated on the complex plane in phasor form. Maxwell Stress Tensor is employed for the analytical torque expression. Using the orthogonality of trigonometric functions and inner products, it is noted that the kpin th order and kpout th order of magnetic flux density harmonics contribute to the input torque and the output torque, respectively (k : odd number). It also can be approximately represented using pin th order and pout th order. Furthermore, the phase shift between the inner and the outer rotors for maximum torque generation is defined as pinθ0(in) - poutθ0(out) = ±mπ (m : interger). Consequently, the electromagnetic torque density can be expressed using the split ratio, the amplitudes of the pout th radial and tangential magnetic flux density harmonics induced by the inner and outer PMs. The corresponding phasor diagram is also presented. In addition, through partial differentiation of the torque density expression for each component, the coherence functions are also developed. Both finite element analysis and experiment are accomplished for verification of proposed analytical electromagnetic model for torque density. The prototype have the gear ratio of 9.33:1. The maximum error between analytical, numerical, and experimental results is about 5%. On the other hand, the proposed model has the limitations that it cannot consider and reflect any kinds of losses generated in coaxial magnetic gear. However, it is expected that the proposed model is useful for the coaxial magnetic gear design with high torque density capacity hereafter.