Improved 3-D Analytical Model for Axial-Flux Eddy-Current Couplings With Curvature Effects

TLDR: An improved 3D analytical model for axial-flux permanent-magnet eddy-current couplings is presented in this paper, which directly takes into account the radial edge effects and the curvature effects on the torque prediction without the need of any correction factor.

Abstract: An improved 3-D analytical model for axial-flux permanent-magnet eddy-current couplings is presented in this paper. As the problem is solved in a 3-D cylindrical coordinate system, the proposed model directly takes into account the radial edge effects and the curvature effects on the torque prediction without the need of any correction factor. It is shown that, the new analytical model is very accurate, even for the geometries where the curvature effects are very pronounced. Another advantage of the proposed model is the great reduction of computation time compared to 3-D finite-elements simulations and an easier adaptation for parametric studies and optimization.

Figures

Fig. 1. Cross section of the studied axial-flux permanent magnet eddy-current coupling.

TABLE I PARAMETERS OF THE STUDIED EDDY-CURRENT COUPLING

Fig. 4. PMs distribution (a), and Torque-slip speed characteristic (b) for p = 4 (λ = 1.14) and c = 5mm.

Fig. 11. Torque versus temperature for p = 4, c = 1mm and 300rpm.

Fig. 10. Conductivity of copper as a function of the temperature.

Fig. 2. Five-layer field model of the studied magnetic coupling in the r-z and r-θ planes for one pole-pitch.

Frequently asked questions

Q1. What is the boundary condition for the magnetic field in the nonconducting regions?

It is important to note that for regions 4 and 5, the boundary condition (5) corresponds to a zero value for the radial component of the induced current at r = R3. 

Q2. What is the magnetic field in the nonconducting regions?

B. Magnetic Field in the Nonconducting RegionsFor the nonconducting regions (i = 1, 2, 3), the magneticfield is based on the magnetostatic Maxwell’s equations0i⋅ =B∇ 0i× =H∇ (9)From (9), the magnetic field strength can be written in termsof a magnetic scalar potential iΦ , which is defined asi i= − ΦH ∇ (10)For the permanent magnets, the authors consider a linearcharacteristic with a relative permeability near unity (NdFeB magnets) such as0 0i i iµ µ= +B H M ( )0 for 2iM i≠ = (11)where Mi is the remanent magnetization vector defined in (1). 

Q3. how long does it take to obtain the torque-slip characteristic?

Using (49), the torque-slip characteristic given in Fig. 9 is obtained in a few tens of milliseconds whereas it needs more than one hour with the 3-D FE model with the same assumptions. 

Q4. How long does it take to compute the torque-speed characteristics?

The torque-speed characteristics given in Fig. 5(b) have been computed in 0.1s when using 3-D analytical model whereas it takes more than 3 hours with 3-D FE model. 

Q5. What is the formulation to solve the problem in the conducting regions?

The authors have also shown in [14] that the induced currents in the conducting regions are laminar and flow in the r-θ planes, therefore the current density presents only two components:( , , , ) ( , , , )i ri iJ r z t J r z tθθ θ= +rJ e eθ (24)Knowing that 0i⋅ =∇ J , it is therefore easier to address this problem by choosing a J-formulation. 

Q6. What is the torque expression given in this paper?

For these geometries and for an air-gap value c = 5mm, the authors have computed the torque-slip characteristic with three different models:- the 3-D FE model which is considered as the referencemodel,- the torque formula (39) given in [14] for which thecurvature effects was neglected,- the torque expression (44) given in this paper whichconsiders the curvature effects. 

Q7. What is the mathematical expression of Mn(r)?

The mathematical expression of Mn(r), which depends on the magnetization distribution as shown in Fig. 3, will be developed in the next section. 

Q8. What is the value of the back-iron thickness?

As the back-iron thickness have been chosen to avoid magnetic saturation, the authors consider a constant value for the relative permeability of the ferromagnetic parts µrb = 1000.