Analytical Method for Magnetic Field of Eccentric Magnetic Harmonic Gear

Eccentric magnetic harmonic gears (EMHG) can achieve high speed ratio transmission and large torque output by modulating the air gap length between permanent magnets (PM). The analytical method is based on the boundary perturbation method. The vector magnetic potential perturbation equation in the air gap region is established, and the general solution is obtained using the boundary conditions. According to the superposition principle, the air gap flux density of stator and rotor PMs acting alone is synthesized, and the electromagnetic torque and unbalanced magnetic pull are calculated. Last, we compare magnetic field distributions, electromagnetic torque and unbalanced magnetic pull computed by the analytical method with those obtained from finite-element method (FEM).


I. INTRODUCTION
Compared with mechanical gears, magnetic gears (MG) can offer several advantages, namely, low noise, less friction, no lubrication, high reliability etc. MG is a device that transfers torque and speed by magnetic coupling instead of mechanical gear meshing. The torque density of concentric magnetic gear (CMG) can be up to 100kN·m/m 3 [1], [2]. In recent years, eccentric magnetic harmonic gear (EMHG) has been proposed, their torque density and transmission ratio are much larger than CMGs [3], [4]. It is particularly suitable for gear ratios higher than about 20:1, the transmitted torque is ripple free and torque density can be up to 150kN·m/m 3 . Therefore, more and more scholars are paying attention to the EMHG.
Accurate calculation of air gap magnetic field is the key to optimum design of EMHG, whose structure is shown in Fig. 1. Because of the eccentric rotor revolution and rotation, the analysis of magnetic field will be more complicated. The calculation of air gap magnetic field includes finite element method (FEM) and analytical method. Although the FEM has high accuracy, it takes a long time and the mesh needs to be reconstructed when the rotor rotates. The analytical method has the advantages of fast calculation The associate editor coordinating the review of this manuscript and approving it for publication was Jenny Mahoney. and clear physical concepts [5]- [9]. The free rotation of the rotor can be achieved without the constraint of meshing. Therefore, it is also suitable for calculating the magnetic field of magnetic gear with eccentric rotor. In the calculation and analysis of electromagnetic field, because of the complexity of boundary shape or medium characteristics in the analytical region, it is difficult to obtain exact analytical solution for the analytical calculation of air gap magnetic field, so it is necessary to adopt approximate analytical method.
The boundary perturbation theory is used for the first time to analytically calculate the magnetic field of the rotor eccentricity of PM motor [10], [11]. In [12], Lubin proposed an analytical computation of the magnetic field distribution in a CMG, which is based on the resolution of Laplace's and Poisson's equations for each subdomain. The configuration and theoretical analysis of the CMG with Halbach PM Arrays has been discussed in [13], some parameters of CMG with Halbach PM Arrays have been optimized, and flux density and torque of CMG have been computed by an exact analytical method. Ref [14] analyzes and calculates the magnetic field and the electromagnetic torque of the CMG by adopting the exact analytical method, and optimizes the design of a CMG based on genetic algorithm toolbox of Matlab. In [15], Prof. Zhang proposed an analytical model of magnetic fields for magnetic harmonic gears, which is developed by the fractional linear transformation (FLT) method. In [16], the authors introduced new concept of complex relative air-gap permeance related to analytical field calculations in surface PM motors, which is developed from conformal transformation of the slot opening and used to accurately calculate the air-gap field for both radial and tangential components of the flux density in the slotted air gap. In [17], an analytical model is developed for the prediction of the air gap flux density in the eccentric permanent magnet inset machines at no-load and on-load conditions, which is fast and accurate.
In this paper, an analytical model of air gap magnetic field perturbation for EMHG is established. The eccentric perturbation ε in the analytical model is dimensionless. Firstly, when the PM on the low-speed rotor acts alone, according to the boundary conditions, the equation is solved in the rotor reference coordinate system, and the air gap flux density expression of the PM on the low-speed rotor can be obtained. Secondly, when the PM on the stator acts alone, the expression of the air gap flux density on the PM on the stator acts alone can be obtained in the stator reference coordinate system. Thirdly, according to the superposition principle, the air gap magnetic field of EMHG is synthesized and the electromagnetic torque and unbalanced magnetic pull are calculated. Finally, the analytical results are then verified with the finite-element method (FEM).

II. ANALYTICAL MODEL
As shown in Fig. 2, the EMHG is composed of 3 parts, a highspeed inner rotor, a low-speed outer rotor and a stator.
The low-speed rotor is mounted on the high-speed rotor through the bearing, the high-speed rotor and low-speed rotor rotate eccentrically relative to the stator, and the motion paths are the same. There is a sinusoidal time-varying air gap length between the low-speed rotor and the stator. The magnetic field produced by two groups of PMs will be modulated, the pole pairs of the asynchronous space harmonics formed by one group of PMs is equal to that of the other group of PMs, so that the torque transmission and speed can be realized. In order to maximize the torque transfer ability of EMHG and according to the principle of magnetic field modulation, the polar pairs of stator permanent magnet should be equal to that of space harmonic magnetic field, the relationship between the polar pairs of the two parts is as follows, where p s and p r are the number of pole pairs of stator and rotor, respectively. The gear ratio (G r ) is then given by, For simplicity of analysis, the following assumptions are adopted: 1) The analytical field is a 2-D plane, and the end effects are neglected.
2) The demagnetization curve of the PM is linear with relative permeability µ r = 1.
3) The permeability of stator and rotor iron is infinite. Fig. 1 shows the structure model of EMHG, which is divided into stator yoke subdomain I, stator PM subdomain II, air gap subdomain III, rotor PM subdomain IV and rotor yoke subdomain V. The meaning of each parameter is as follows: R rin indicates the inner radius of the iron yoke of high-speed rotor; R rout is the outer radius of the yoke of the low-speed rotor; R mr represents the outer radius of PM of low speed rotor; R ms is the inner radius of stator PM; R s indicates the inner radius of stator iron yoke; O r is the center of the rotor; O s is the center of the stator. Fig. 3 shows the analytical model of the EMHG. The analytical domain includes air gap subdomain III, stator PM subdomain II and rotor PM subdomain IV.
As shown in Fig. 3(a), when the low-speed rotor PMs act alone, the equivalent air gap domain includes air gap subdomain III and stator PM subdomain II. The X-Y coordinate system is established with the stator center O s as the coordinate origin, any point P in the domain of the equivalent eccentric air gap can be represented by the r-θ cylindrical coordinate system with the rotor center O r as the coordinate origin. The eccentricity distance between the rotor center and the stator center is a, and the eccentricity angle is ϕ. In Fig. 3(b), when the stator PMs act alone, the equivalent air gap domain includes the air gap subdomain III and the rotor PM subdomain IV. The X-Y coordinate system is established with the rotor center O r as the coordinate origin, any point P in the equivalent eccentric air gap domain can be represented by ξ -ψ cylindrical coordinate system with the stator center O s as the coordinate origin. The eccentricity distance between the rotor center and the stator center is a, and the eccentricity angle is ϕ.

B. PARTIAL DIFFERENTIAL EQUATIONS AND BOUNDARY CONDITIONS
When the low-speed rotor PMs act alone, from the geometric relationship [18], the equation of the inner circle track of the stator can be obtained as follows: Let r = R s + εδ(θ ), let the eccentric disturbance ε = a R s , and it is dimensionless, so, The stator boundary can be expressed as: The gradient solution of formula (5) can be obtained, (6) where e r and e θ are unit vectors in radial and tangential directions, respectively. In order to obtain the normal vector at the inner radius of stator yoke, the module of formula 6 can be calculated, (7) The boundary condition of inner radius of stator yoke is as follows: where r and θ represent radial and tangential components, respectively. i = 1 and i = 2 are the analytical domain of air gap and PMs, respectively. By substituting formula (7) with formula (8), the boundary conditions at the inner radius of stator yoke can be obtained, The radial flux density at the outer radius of the rotor is zero, and its boundary condition is, The radial and the tangential magnetic field strength at the interface of the PM surface are continuous, respectively. The boundary conditions at the interface are as follows: According to perturbation theory, vector magnetic potential A, magnetic flux density B and magnetic field intensity H can be expanded by power series of eccentric disturbance ε [19] where i = 1 indicates the air gap domain, i = 2 indicates the PM domain.
In r-θ polar coordinate system, vector magnetic potential A satisfies Laplace equation and Poisson equation: where v is the magnetoresistance, M is the magnetization of the PM, and its expression can be written, where M n = 2B r µ 0 α p sin(nπα p /2) nπα p /2 , B r is the remanence of PM, µ 0 is the vacuum permeability, α P is the polar arc coefficient, P is the number of poles of PM, θ 0 is the initial rotor deviation angle.
Equation (9) is expanded by Taylor series at r = R s , and the boundary conditions of the zero order and the first order equations at the inner radius of stator yoke can be obtained,

III. EXPRESSION OF AIR GAP FLUX DENSITY A. MAGNETIC FIELD ANALYSIS
The boundary condition of the zero order equation of the eccentric air gap magnetic field is, According to the boundary conditions, the equations (16) are solved. The analytical expressions of the zero order vector magnetic potential and the radial and tangential magnetic flux density in the air gap domain are as follows, where 1n are undetermined coefficients, see Appendix A for specific expressions.
The boundary condition of the first order equation of eccentric air gap magnetic field is, Similarly, the expression of the first order vector magnetic potential in the air gap domain is as follows, 1n r −(np r +2) ) sin(np r + 2)θ ] where M Therefore, the expressions of radial and tangential flux density in the first order air gap domain are as follows: 1n r −np r ) sin(np r − 1)θ + (np r + 1)(E (1) 1n r np r − F (1) 1n r −np r −2 ) cos(np r + 1)θ According to equation (14), we add the zero order air gap flux density and the first order air gap flux density, and eliminate the higher order infinitesimal of the eccentric disturbance ε. So the air gap flux density produced by the PM of the low speed rotor in the EMHG can be obtained,

B. AIR GAP FLUX DENSITY PRODUCED BY STATOR PM
When the stator PM acts alone, the eccentric air gap magnetic field can also be obtained similarly. Set the pole pairs of the PMs to p s . The eccentricity angle is ϕ, and the eccentricity disturbance ε = a/R rout . The radial flux density B ξ,1 and the tangential flux density B ψ,1 produced by the stator PM acting alone are as follows,

COMPOSITE AIR GAP FLUX DENSITY OF EMHG
By superposition of the air gap flux density when the stator PMs act alone and the air gap flux density when the rotor PMs acts alone, the superposition of flux density is shown in Fig. 4. The radial and tangential air gap flux density expressions B r and B θ of the EMHG are obtained as follows, where k = θ − ψ.

D. UNBALANCED MAGNETIC PULL AND ELECTROMAGNETIC TORQUE
According to the Maxwell stress tensor method, the unbalanced magnetic pull in the x-axis and y-axis direction can be obtained by integrating along the circular path through coordinate transformation.
The expression of electromagnetic torque is, where L ef is the axial length of EMHG.

IV. APPLICATION EXAMPLE
In order to verify the correctness of the analytical model, an EMHG with a transmission ratio of 8:1 is used for the finite element modeling in this paper. In this study, when the eccentricity is about 0.57, ϕ = 0, the magnetic density at the radius r = R mr + (g-a)/2 is studied with the center of the lowspeed rotor as the center. Table 1 lists the specific parameters of the EMHG.    6 shows the radial and tangential components of the magnetic field in the air gap of the EMHG. It can be seen that the analytical results of the magnetic density in the radial and tangential air gap are in good agreement with the results of the FEM. Fig. 7 shows the harmonic number of radial air gap flux density. It is obvious that the harmonic amplitude corresponding to the 8 th harmonic is the largest, which corresponds to the pole pairs of the rotor PM, and the harmonic amplitude corresponding to the 9 th harmonic is the second, which corresponds to the pole pairs of the stator PM. In order to achieve the maximum torque transmission of the EMHG, the pole pairs of the stator PM should meet P s = P r + 1.   Fig. 8 shows the static torque of the low speed rotor. In FEM, rotate the phase angle of the high-speed rotor from 0 • to 360 • and keep the position of the PM on the low-speed rotor unchanged. Calculate the electromagnetic torque every 10 • interval, and a total of 37 electromagnetic torque values are obtained. The analytical solution of static torque is obtained by formula (34). As can be seen from Fig. 8, the results of analytical method are in good agreement with those of FEM.

B. TORQUE AND UNBALANCED MAGNETIC PULL
Set the rotation angles of high-speed rotor and lowspeed rotor, respectively. When the eccentricity angle of the   high-speed rotor is changed from 0 • to 360 • , and the lowspeed rotor rotates (ϕ/8) • in the opposite direction to the high-speed rotor, the output constant electromagnetic torque of the low-speed rotor at various ϕ angles can be obtained. Fig. 9 shows the output constant torque of the low-speed rotor. The torque calculation result is 13.63N·m. It can be seen that the analytical results of the constant torque of the low-speed rotor are consistent with the wave trend of the FEM results. Fig. 10 shows the curve of unbalanced magnetic pull F x and F y , which are the unbalanced magnetic pull of low-speed rotor along the x-axis and y-axis, respectively. As shown in Fig. 10, the analytical results of unbalanced magnetic pull are in good agreement with the results of FEM.   11 shows the output torque of the low-speed rotor of the EMHG calculated by the analytical method under different eccentricity. As can be seen from that the torque value increases gradually with the increase of eccentricity. The torque ripple is very small in the transmitted torque. Table 2 shows the calculation time of analytical method and FEM. It can be seen from the Table that the time of analytical method is much less than that of FEM.

V. CONCLUSION
In this paper, a perturbation analytical method is proposed to compute the distribution of the air gap magnetic field and the electromagnetic torque for EMHG. The expressions of air gap magnetic field are determined by solving two-dimensional Laplace equation and Poisson equation. The eccentric circle trajectory equation can accurately describe the boundary conditions of inner radius of stator and outer radius of rotor, which takes the eccentric disturbing momentum as the perturbation variable. The correctness of the analytical method is verified by comparing the results of the analytical method with those of the FEM. The physical concept of the analytical model is clear and the calculation is convenient, which provides an effective method for the magnetic field calculation and optimization design of the EMHG.

APPENDIXES APPENDIX A
When the rotor PMs act alone, the expression of undetermined coefficient of the zero order equation, A 1n , B 1n , C 1n , D 1n are: The expression of undetermined coefficient of the first order equation, M 1n , N 1n , P 1n , Q 1n , A 1n , B 1n , C 1n , D 1n , E 1n , and Z (1) 1n : 2n are: The expression of undetermined coefficient of the first order equation, 2n are: Her research interest includes the stress and vibration analysis of motor cores considering magnetostriction and Maxwell electromagnetic stress. VOLUME 8, 2020