Analytical predictions and analysis of electromagnetic field and torque for eddy-current couplers with Quasi-Halbach magnet array

Abstract

Accurate analytical model is critical for the design and optimization of the electromagnetic devices. In this paper, a fully 3D analytical model for eddy-current couplers with a Quasi-Halbach magnet array is presented. The originality of this paper lies in the Halbach magnet array with arbitrary magnetization direction and the eddy currents generated in conductor rotor are considered in the 3D analytical computation. To avoid the complex computation with Bessel function, the 3D cylindrical geometry is linearized and discussed in 3D Cartesian coordinates. The magnetic scalar potential in the non-conductor region (PM, air-gap, and back iron) and magnetic field strength in eddy-current region (conductor disk and its back iron) are calculated based on the separation variable method. The electromagnetic field, axial force and torque are obtained. In the end, by using the 3D finite element analysis, the feasibility and validity of the general 3D analytical model are verified.

Keywords

Analytical model eddy-current couplers Halbach array

1. Introduction

By using the rare-earth permanent magnet (PM), such as Nd-Fe-B, the electromagnetic devices can simplify own structures, and obtain more favorable torque-to-weight ratio [1]. Compared with the standard vertical or radial magnetizations, the alternative Halbach arrays introduce some attractive benefits, for example, more self-shielding and a sinusoidal variation of the air-gap magnetic flux density. Therefore, they have be widely adopted in motors [2, 3], gears [4, 5], and couplers [6, 7].

Figure 1 shows the PM eddy-current couplers with three-segmented Halbach magnet. As the ideal Halbach magnet array is difficult to fabricate, a discrete segmented quasi-Halbach array is introduced in this paper. In order to evaluate their performances, there are two main approaches: numerical and analytical methods. The former, such as finite element method (FEM) [8, 9] has accurate analysis results, but depend on the refined mesh. Therefore, the accuracy of this method is affected by many indefinite factors, and requires a lot of computing time and resources, especially 3D-FEM in the optimal design of machines.

The accurate analytical model is the focus of attention of many scholars, because it can be easily employed in the performance analysis and parameter optimization in the initial design stage of electromagnetic devices [10]. In recent years, all kinds of analytical model are established for the PM eddy-current couplers with axial magnetization or radial magnetization, for example, 2D model [11, 12], 3D model [13, 14], and the lumped parameter equivalent magnetic circuit model (EMCM) [15, 16]. Among these methods, 2D analytical model is obtained by solving the simplified 2D Maxwell equations, but the results have to be corrected by a 3D coefficient, such as the Russel and Norsworthy’s correction factor [17]; the EMCM is more simple and intuitive, however, it only provides the approximate solution. 3D analytical model can give accurate results without additional correction, in particular in the 3D cylindrical coordinate system. However it will produce complex Bessel function operation. In addition, the preceding 3D analytical model has own limitations, and cannot be applied to the complex topology. At present, there is no 3D analytical model available for the PM eddy-current couplers with Halbach magnet array. Therefore, it is very meaningful to construct a general 3D analytical model of such topology.

In this paper, a PM eddy-current coupler with Halbach magnet array is concerned. To order to conveniently predict and analyse its performances, an accurate 3D analytical model is set up. The 3D cylinder structure is simplified into the linear 3D layer structure at the average radius of PM. In the 3D Cartesian coordinates, the magnetic scalar potential in the regions without induced current and the magnetic field strength in the eddy-current regions are obtained using the separation of variables method. In particular, the eddy current effect of back iron in the conductor side is taken into account. Then the explicit expressions for the flux density distribution and output toque are given. In the end, the 3D analytical results are compared with those obtained by the nonlinear 3D-FEM. It is shown that the proposed analytical model has the high accuracy. In addition, the prototype test further indicates that the maximum error of torque model is about 5%.

Figure 1.

Schematic of PM eddy-current couplers with three-segmented Halbach magnet (4 pole-pairs as the case).

2. 3-D analytical model of magnetic field

2.1 Description of geometry

The real geometry of the considered coupler is shown in Fig. 1. As shown in Fig. 1, as the magnetic source, the three-segment Halbach magnet array is posted in the conductor disk (copper). Differently from the traditional axial and radial magnetization, the magnets in this paper are magnetized with the arbitrary direction $\theta$ . The magnet array is fixed by the aluminum frame. The geometrical parameters of the case to be investigated are as follows:

The outer and inner radius of the PM array are $R_{1}$ and $R_{2}$ , thus the effective length of PM is $L=R_{1}-R_{2}$ .

The outer and inner radius of the conductor disk are $R_{3}$ and $R_{4}$ , thus the effective length of conductor disk is $H=R_{3}-R_{4}$ , and $H>L.$

The number of pole-pairs is $p$ , the middle section of the magnet array is magnetized axially, and the both sides of the magnet array are magnetized with the orientation $\theta$ , which is shown in Fig. 2. The pole pitch is $\tau$ ; the length of the middle section of the magnet array is $\tau_{2}$ ; the length of the both sides of the magnet array is $\tau_{1}$ , and $\tau_{2}+2\tau_{1}=\alpha\tau$ .

The thickness of back iron of the PM rotor is $a$ ; the thickness of PM is $b$ ; the length of air-gap is $c$ ; the thickness of copper disk is $d$ , and the thickness of back iron of the conductor rotor is $e$ .

Figure 2.

3D linear geometry of eddy-current couplers with three-segmented Halbach magnet.

2.2 Calculation assumptions

Most obviously the PM eddy-current couplers are in nature 3-D cylindrical geometry, the analytical expressions using the cylindrical coordinate system will be more accurate, but this will brings a high level of computation complexity, such as the Bessel functions [11]. To simplify the analytical modelling, the mean radius of the magnets $R_{av}$ is considered, and the topology is simplified into a 3D linear geometry shown in Fig. 2. As shown in Fig. 2, the whole model is divided into five layers, respectively back iron region (region I) , PM region (region II), air-gap region (region III), conductor region (region IV), and its back iron region (region V). Some reasonable assumptions are adopted as follows.

All the regions are composed of linear media.

The magnet, air-gap, and conductor have the same permeability $\mu_{0}$ ; the permeability of back iron in the PM rotor side is $\mu_{1}$ , and the permeability of back iron in the conductor disk side is $\mu_{2}$ .

The PM rotor and corresponding back iron are “stationary,” and the conductor disk and its back iron are “moving” with the velocity $\textbf{v}=R_{av}\omega\textbf{e}_{x}$ , where $\omega$ is the relative angular velocity.

The induced currents generated in the conductor disk and corresponding back iron are considered in the model. The conductivity of region IV is $\sigma_{0}$ , and the conductivity of region V is $\sigma_{1}$ .

All the interfaces are defined as the constant.

Figure 3.

Sectional view of the linear system of eddy-current coupler in: (a) $x$ - $z$ plane (b) $y$ - $z$ plane.

2.3 Boundary conditions

In order to solve the 3D electromagnetic field problems, the boundary conditions, including the $x$ -direction, $y$ -direction, and $z$ -direction, have to be defined. Figure 3 shows the sectional view of 3D linear geometry of eddy-current couplers in the $x$ - $z$ plane and $y$ - $z$ plane. Therefore, the corresponding boundary conditions are adopted in this paper as follows:

In the $x$ -direction, considering the periodicity of the magnetic source, the field variables also have the same periodic with source term, thus the boundaries are given in the $x=-\tau/2$ and $x=\tau/2$ . Moreover, they should satisfy the following relation,

$\displaystyle\left.{{\bm{\Omega}}_{1}:{\bf H}_{i}(x,y,x)}\right|_{x=\frac{\tau% }{2}}=\left.{-{\bf H}_{i}(x,y,x)}\right|_{x=\frac{-\tau}{2}}\ \text{for}\ i=% \text{I, II, III, IV, and V}$ (1)

In the $y$ -direction, due to $H>L$ , the magnetic field will weaken to zero at the $y=-H/2$ and $y=H/2$ , thus the boundaries are given in the $y=-H/2$ and $y=H/2$ . Moreover, they should satisfy the following relation

$\displaystyle\left\{{\begin{array}[]{l}\left.{{\bm{\Omega}}_{2}:{\bf H}_{i}(x,% y,x)}\right|_{y=\frac{-H}{2}}=0\\ \left.{{\bm{\Omega}}_{3}:{\bf H}f{}_{i}(x,y,x)}\right|_{y=\frac{H}{2}}=0\\ \end{array}}\right.\text{for}\ i=\text{I, II, III, IV, and V}$ (2)

In the $z$ -direction, on the basis of continuity of the tangential magnetic field strength and the normal magnetic flux density at the interfaces between two adjacent regions, the boundary conditions will be used to determine the unknown coefficients of magnetic scalar potential in region I, II and III and the magnetic field intensity in region IV and V. In addition, we assume that the flux line cannot leak out the back iron, which can be expressed by

$\displaystyle\left\{{\begin{array}[]{l}\left.{\Gamma_{1}:{\bf B}_{I}(x,y,x)}% \right|_{z=-a}=0\\ \left.{\Gamma_{5}:{\bf B}_{V}(x,y,x)}\right|_{z=b+c+d+e}=0\\ \end{array}}\right.$ (3)

Figure 4.

Magnetization distribution along the coordinate $x$ (a) $z$ -direction componen (b) $x$ -direction component.

2.4 Source terms

The magnetization M can be decomposed to the $z$ -direction component and the $x$ -direction component, as shown in Fig. 4. In the 3D Cartesian coordinate system, the magnetization M can be written as

$\displaystyle{\bf M}=M_{x}(x,y){\bf e}_{x}+M_{z}(x,y){\bf e}_{z}$ (4)

where, $M_{z}(x,y)$ and $M_{x}(x,y)$ respectively denote the $z$ -component and $x$ -component of M in the ( $x$ , $y$ ) area. According to the Fig. 4, the analytical expression of $M_{x}(x,y)$ and $M_{z}(x,y)$ in the space area ( $x$ , $y$ ) $\in{\{}-{\tau}/2<x<\tau/2$ , $-H/2<y<H/2$ } are given by

$\displaystyle M_{x}(x,y)=\left\{{\begin{array}[]{l}-\frac{B_{r}}{\mu_{0}}\sin% \theta,(x,y)\in\left\{-\frac{\tau_{2}}{2}-\tau_{1}\leqslant x\leqslant-\frac{% \tau_{2}}{2},-\frac{L}{2}\leqslant y\leqslant\frac{L}{2}\right\}\\ \frac{B_{r}}{\mu_{0}}\sin\theta,(x,y)\in\left\{\frac{\tau_{2}}{2}\leqslant x% \leqslant\frac{\tau_{2}}{2}+\tau_{1},-\frac{L}{2}\leqslant y\leqslant\frac{L}{% 2}\right\}\\ 0,\textit{else}.\\ \end{array}}\right.$ (5)

$\displaystyle M_{z}(x,y)=\left\{{\begin{array}[]{l}\frac{B_{r}}{\mu_{0}}\cos% \theta,(x,y)\in\{-\frac{\tau_{2}}{2}-\tau_{1}\leqslant x\leqslant-\frac{\tau_{% 2}}{2},-\frac{L}{2}\leqslant y\leqslant\frac{L}{2}\}\\ \frac{B_{r}}{\mu_{0}},(x,y)\in\{-\frac{\tau_{2}}{2}\leqslant x\leqslant\frac{% \tau_{2}}{2},-\frac{L}{2}\leqslant y\leqslant\frac{L}{2}\}\\ \frac{B_{r}}{\mu_{0}}\cos\theta,(x,y)\in\{\frac{\tau_{2}}{2}\leqslant x% \leqslant\frac{\tau_{2}}{2}+\tau_{1},-\frac{L}{2}\leqslant y\leqslant\frac{L}{% 2}\}\\ 0,\textit{else}.\\ \end{array}}\right.$ (6)

where, $B_{r}$ is the remanent flux density.

The magnetization distributions $M_{z}(x$ , $y)$ and $M_{x}(x$ , $y)$ can be expressed using a double Fourier series expansion in a complex form as follows:

$\displaystyle M_{z}(x,y)=\Re\left\{{\sum\limits_{n=1}^{\infty}{\sum\limits_{k=% 1}^{\infty}{M_{znk}\cos(\lambda_{Y_{n}}y)\exp(j\lambda_{X_{k}}x)}}}\right\}$ (7) $\displaystyle M_{x}(x,y)=\Re\left\{{\sum\limits_{n=1}^{\infty}{\sum\limits_{k=% 1}^{\infty}{M_{xnk}\cos(\lambda_{Y_{n}}y)\exp(j\lambda_{X_{k}}x)}}}\right\}$ (8)

where,

$\displaystyle M_{znk}=\frac{16B_{r}}{kn\mu_{0}\pi^{2}}\sin\left(\frac{\lambda_% {Y_{n}}L}{2}\right)\left\{{(1-\cos\theta)\sin\left(\frac{\lambda_{X_{k}}\tau_{% 2}}{2}\right)+\cos\theta\sin\left[\lambda_{X_{k}}\left(\frac{\tau_{2}}{2}+\tau% _{1}\right)\right]}\right\}$ (9) $\displaystyle M_{xnk}=\frac{16jB_{r}\sin\theta}{kn\mu_{0}\pi^{2}}\sin\left(% \frac{\lambda_{Y_{n}}L}{2}\right)\left\{{\cos\left(\frac{\lambda_{X_{k}}\tau_{% 2}}{2}\right)-\cos\left[\lambda_{X_{k}}\left(\frac{\tau_{2}}{2}+\tau_{1}\right% )\right]}\right\}$ (10)

with

$\displaystyle\lambda_{X_{k}}=\frac{k\pi}{\tau}$ (11) $\displaystyle\lambda_{Y_{n}}=\frac{n\pi}{H}$ (12)

where, $n$ and $k$ respectively indicate the harmonic numbers in the $x$ -direction and $y$ -direction; $j$ is an imaginary number, and $j=\sqrt{-1}$ ; $\Re$ denotes real component.

2.5 3D analytical solution

In the region I and III, there will be no induced currents and source terms, thus the magnetic scalar potential $\varphi$ is introduced to solve the field problem. To calculate the gradient of $\varphi$ , we can get

$\displaystyle{\bf H}_{I,III}=-\nabla\varphi_{I,III}$ (13)

In addition, we have

$\displaystyle\nabla\cdot{\bf B}=0\ \text{and}\ {\bf B}=\mu{\bf H}$ (14)

Therefore,

$\displaystyle\nabla\cdot(-\mu\nabla\varphi)=-\nabla\varphi\cdot\nabla\mu-\mu% \nabla\cdot\nabla\varphi$ (15)

According to the assumption Eq. (1), each region is the homogeneous media, Eq. (15) can be reduced to a Laplace equation, that is

$\displaystyle\frac{\partial^{2}\varphi_{I,III}}{\partial x^{2}}+\frac{\partial% ^{2}\varphi{}_{I,III}}{\partial y^{2}}+\frac{\partial^{2}\varphi_{I,III}}{% \partial z^{2}}=0$ (16)

According to the boundary conditions in the $x$ -direction and $y$ -direction, the general solution of Eq. (16) by using the separation of variables principle is written as

$\displaystyle\varphi_{I,III}(x,y,z)=\Re\left\{{\sum\limits_{n=1}^{\infty}{\sum% \limits_{k=1}^{\infty}{(a_{nk}^{I,III}e^{\lambda_{nk}z}+b_{nk}^{I,III}e^{-% \lambda_{nk}z})\cos(\lambda_{Y_{n}}y)\exp(j\lambda_{X_{k}}x)}}}\right\}$ (17)

where,

$\displaystyle\lambda_{nk}=\sqrt{(\lambda_{Y_{n}})^{2}+(\lambda_{X_{k}})^{2}}$ (18)

In the region II, due to the existence of the magnetic source, the magnetic flux density can be expressed by

$\displaystyle{\bf B}=\mu_{0}({\bf H}+{\bf M})$ (19)

Thus, Eq. (15) can be changed into the Poisson equation as follows

$\displaystyle\nabla^{2}\varphi_{II}=\nabla\cdot{\bf M}$ (20)

According to the boundary conditions in the $x$ -direction and $y$ -direction, and the magnetization distributions in the $x$ -direction and $z$ -direction, the general solution of Eq. (20) by using the separation of variables principle is written as

$\displaystyle\varphi_{II}(x,y,z)=\Re\left\langle{\sum\limits_{n=1}^{\infty}{% \sum\limits_{k=1}^{\infty}{\left\{{\begin{array}[]{l}a_{nk}^{II}e^{\lambda_{nk% }z}+b_{nk}^{II}e^{-\lambda_{nk}z}+\frac{16B_{r}\sin\theta}{n\mu_{0}\pi\tau% \lambda_{{}^{nk}}^{2}}\sin\left(\frac{\lambda_{Y_{n}}L}{2}\right)\times\\ \left({\cos\left(\frac{\lambda_{X_{k}}\tau_{2}}{2}\right)-\cos\left[\lambda_{X% _{k}}\left(\frac{\tau_{2}}{2}+\tau_{1}\right)\right]}\right)\\ \end{array}}\right\}}}}\right.\left.\cos(\lambda_{Y_{n}}y)\exp(j\lambda_{X_{k}% }x)\!\!\!\!\!\phantom{\frac{1}{2}}\right\rangle$ (21)

In the region IV and V, due to the existence of eddy currents, the magnetic field intensity H is introduced to solve the moving induced current problem, which can be expressed by

$\displaystyle{\bf H}_{IV,V}=H_{IV,Vx}{\bf e}_{x}+H_{IV,Vy}{\bf e}_{y}++H_{IV,% Vz}{\bf e}_{z}$ (22)

As the steady-state field is studied, we have

$\displaystyle\frac{\partial{\bf B}}{\partial t}=0\ \text{and}\ \frac{\partial{% \bf D}}{\partial t}=0$ (23)

The Maxwell equations can be written as

$\displaystyle\nabla\times{\bf H}={\bf J},\nabla\times{\bf E}=0,\nabla\cdot{\bf B% }=0$ (24)

and the additional equations are

$\displaystyle{\bf J}=\sigma({\bf E}+{\bf VB})\ {\bf B}=\mu{\bf H}$ (25)

Through a series of calculations, the field problem is to solve the diffusion equations as follows

$\displaystyle\nabla^{2}{\bf H}_{IV}=-\sigma_{0}\mu_{0}\nabla\times({\bf v}% \times{\bf H}_{IV})$ (26) $\displaystyle\nabla^{2}{\bf H}_{V}=-\sigma_{1}\mu_{2}\nabla\times({\bf v}% \times{\bf H}_{V})$ (27)

Equations (26) and (27) can be further written into differential equations in the Cartesian coordinates as

$\displaystyle\left\{{\begin{array}[]{l}\frac{\partial^{2}H_{IVx}}{\partial x^{% 2}}+\frac{\partial^{2}H_{IVx}}{\partial y^{2}}+\frac{\partial^{2}H_{IVx}}{% \partial z^{2}}=\sigma_{0}\mu_{0}R_{av}\omega\frac{\partial H_{IVx}}{\partial x% }\\ \frac{\partial^{2}H_{IVy}}{\partial x^{2}}+\frac{\partial^{2}H_{IVy}}{\partial y% ^{2}}+\frac{\partial^{2}H_{IVy}}{\partial z^{2}}=\sigma_{0}\mu_{0}R_{av}\omega% \frac{\partial H_{IVy}}{\partial x}\\ \frac{\partial^{2}H_{IVz}}{\partial x^{2}}+\frac{\partial^{2}H_{IVz}}{\partial y% ^{2}}+\frac{\partial^{2}H_{IVz}}{\partial z^{2}}=\sigma_{0}\mu_{0}R_{av}\omega% \frac{\partial H_{IVz}}{\partial x}\\ \end{array}}\right.$ (28)

$\displaystyle\left\{{\begin{array}[]{l}\frac{\partial^{2}H_{Vx}}{\partial x^{2% }}+\frac{\partial^{2}H_{Vx}}{\partial y^{2}}+\frac{\partial^{2}H_{Vx}}{% \partial z^{2}}=\sigma_{1}\mu_{2}R_{av}\omega\frac{\partial H_{Vx}}{\partial x% }\\ \frac{\partial^{2}H_{Vy}}{\partial x^{2}}+\frac{\partial^{2}H_{Vy}}{\partial y% ^{2}}+\frac{\partial^{2}H_{Vy}}{\partial z^{2}}=\sigma_{1}\mu_{2}R_{av}\omega% \frac{\partial H_{Vy}}{\partial x}\\ \frac{\partial^{2}H_{Vz}}{\partial x^{2}}+\frac{\partial^{2}H_{Vz}}{\partial y% ^{2}}+\frac{\partial^{2}H_{Vz}}{\partial z^{2}}=\sigma_{1}\mu_{2}R_{av}\omega% \frac{\partial H_{Vz}}{\partial x}\\ \end{array}}\right.$ (29)

Therefore, the general solutions of Eqs (28) and (29) by using the separation of variables principle are written as [13]

$\displaystyle\left\{\begin{array}[]{l}H_{IVx}(x,y,z)=\Re\left\{\sum\limits_{n=% 1}^{\infty}\sum\limits_{k=1}^{\infty}(a_{nk}^{IV}e^{\zeta_{nk}z}+b_{nk}^{IV}e^% {-\zeta_{nk}z})\cos(\lambda_{Y_{n}}y)\exp(j\lambda_{X_{k}}x)\right\}\\ H_{IVy}(x,y,z)=\Re\left\{\sum\limits_{n=1}^{\infty}\sum\limits_{k=1}^{\infty}-% \frac{1}{\lambda_{Y_{n}}}\left[j\lambda_{X_{k}}(a_{nk}^{IV}e^{\zeta_{nk}z}+b_{% nk}^{IV}e^{-\zeta_{nk}z})\right.\right.\\ \quad\left.\left.+\zeta_{nk}(c_{nk}^{IV}e^{\zeta_{nk}z}-d_{nk}^{IV}e^{-\zeta_{% nk}z})\right]\sin(\lambda_{Y_{n}}y)\exp(j\lambda_{X_{k}}x)\!\!\!\!\!\!\!% \phantom{\sum\limits^{n}_{i=1}}\right\}\\ \par H_{IVz}(x,y,z)=\Re\left\{\sum\limits_{n=1}^{\infty}\sum\limits_{k=1}^{% \infty}(c_{nk}^{IV}e^{\zeta_{nk}z}+d_{nk}^{IV}e^{-\zeta_{nk}z})\cos(\lambda_{Y% _{n}}y)\exp(j\lambda_{X_{k}}x)\right\}\\ \end{array}\right.$ (30)

$\displaystyle\left\{{\begin{array}[]{l}H_{Vx}(x,y,z)=\Re\left\{{\sum\limits_{n% =1}^{\infty}{\sum\limits_{k=1}^{\infty}{(a_{nk}^{V}e^{\zeta_{nk}^{\prime}z}+b_% {nk}^{V}e^{-\zeta_{nk}^{\prime}z})\cos(\lambda_{Y_{n}}y)\exp(j\lambda_{X_{k}}x% )}}}\right\}\\ \par H_{Vy}(x,y,z)=\Re\left\{\sum\limits_{n=1}^{\infty}\sum\limits_{k=1}^{% \infty}-\frac{1}{\lambda_{Y_{n}}}[j\lambda_{X_{k}}(a_{nk}^{V}e^{\zeta_{nk}^{% \prime}z}+b_{nk}^{V}e^{-\zeta_{nk}^{\prime}z})\right.\\ \quad\left.+\zeta_{nk}(c_{nk}^{V}e^{\zeta_{nk}^{\prime}z}-d_{nk}^{V}e^{-\zeta_% {nk}^{\prime}z})]\sin(\lambda_{Y_{n}}y)\exp(j\lambda_{X_{k}}x)\!\!\!\!\!\!\!% \phantom{\sum\limits^{n}_{i=1}}\right\}\\ H_{Vz}(x,y,z)=\Re\left\{\sum\limits_{n=1}^{\infty}\sum\limits_{k=1}^{\infty}(c% _{nk}^{V}e^{\zeta_{nk}^{\prime}z}+d_{nk}^{V}e^{-\zeta_{nk}^{\prime}z})\cos(% \lambda_{Y_{n}}y)\exp(j\lambda_{X_{k}}x)\right\}\\ \end{array}}\right.$ (31)

where,

$\displaystyle\zeta_{nk}=\sqrt{(\lambda_{nk})^{2}+j\sigma_{0}\mu_{0}R_{av}% \omega\lambda_{X_{k}}}$ (32)

and

$\displaystyle\zeta_{nk}^{\prime}=\sqrt{(\lambda_{nk})^{2}+j\sigma_{1}\mu_{2}R_% {av}\omega\lambda_{X_{k}}}$ (33)

2.6 Determination of unknown coefficients

Table 1
Dimensions and materials properties of the case-study coupling

Symbols	Parameters	Values
$a$	Thickness of back iron in PMs side	20 mm
$b$	Thickness of PMs	50 mm
$c$	Length of air-gap	10 mm
$d$	Thickness of copper disk	10 mm
$e$	Thickness of back iron in copper side	20 mm
$R_{1}$	Outer radius of PMs	270 mm
$R_{2}$	Inner radius of PMs	190 mm
$R_{3}$	Outer radius of copper disk	320 mm
$R_{4}$	Inner radius of copper disk	150 mm
$p$	Pole-pairs number	9
$\theta$	Angle of oblique magnetization	45o
$\sigma_{1}$	Conductivity of copper	58 MS/m
$\sigma_{2}$	Back iron in copper side	2 MS/m
$B_{r}$	Remanence of PM (NdFeB)	1.18 T
$\alpha_{1}$	Pole-arc ( $\tau_{1})$ to pole-pitch ration	0.25
$\alpha_{2}$	Pole-arc ( $\tau_{2})$ to pole-pitch ration	0.3
$N(n)$	Number of harmonic in $x$ -direction	9
$K(k)$	Number of harmonic in $y$ -direction	9

As shown in the analytical Eqs (17), (21), (30), and (31), $a_{nk}^{I}$ , $b_{nk}^{I}$ , $a_{nk}^{II}$ , $b_{nk}^{II}$ , $a_{nk}^{III}$ , $b_{nk}^{III}$ , $a_{nk}^{IV}$ , $b_{nk}^{IV}$ , $c_{nk}^{IV}$ , $d_{nk}^{IV}$ , $a_{nk}^{V}$ , $b_{nk}^{V}$ , $c_{nk}^{V}$ , and $d_{nk}^{V}$ are the unknown coefficients to determine. Therefore, fourteen independent linear equations have to be constructed. These equations can be obtained by the interface conditions in the $z$ -direction, which can be expressed as follows:

$\displaystyle\frac{\partial\varphi_{I}(x,y,-a)}{\partial z}=0\ \text{at}\ % \Gamma_{1}$ (34) $\displaystyle\left\{{\begin{array}[]{l}\frac{\varphi_{I}(x,y,0)}{\partial x}=% \frac{\varphi_{II}(x,y,0)}{\partial x}\\ \mu_{1}\frac{\partial\varphi_{I}(x,y,0)}{\partial z}=\mu_{0}\frac{\partial% \varphi_{II}(x,y,0)}{\partial z}-\mu_{0}M_{z}\\ \end{array}}\right.\ \text{at}\ \Gamma_{12}$ (35) $\displaystyle\left\{{\begin{array}[]{l}\frac{\varphi_{II}(x,y,b)}{\partial x}=% \frac{\varphi_{III}(x,y,b)}{\partial x}\\ \frac{\partial\varphi_{II}(x,y,b)}{\partial z}=\frac{\partial\varphi_{III}(x,y% ,b)}{\partial z}+M_{z}\\ \end{array}}\right.\ \text{at}\ \Gamma_{23}$ (36) $\displaystyle\left\{{\begin{array}[]{l}\frac{\partial\varphi_{III}(x,y,b+c)}{% \partial x}=-H_{x}^{IV}(x,y,b+c)\\ \frac{\partial\varphi_{III}(x,y,b+c)}{\partial y}=-H_{y}^{IV}(x,y,b+c)\\ \frac{\partial\varphi_{III}(x,y,b+c)}{\partial z}=-H_{z}^{IV}(x,y,b+c)\\ \end{array}}\right.\ \text{at}\ \Gamma_{34}$ (37) $\displaystyle\left\{{\begin{array}[]{l}H_{x}^{IV}(x,y,b+c+d)=H_{x}^{V}(x,y,b+c% +d)\\ H_{y}^{IV}(x,y,b+c+d)=H_{y}^{V}(x,y,b+c+d)\\ \mu_{0}H_{z}^{IV}(x,y,b+c+d)=\mu_{2}H_{z}^{V}(x,y,b+c+d)\\ \end{array}}\right.\ \text{at}\ \Gamma_{45}$ (38) $\displaystyle H_{z}^{V}(x,y,b+c+d+e)=0\ \text{at}\ \Gamma_{5}$ (39)

It’s worth noting that we assume the flux lines vertically pass through the interface between region IV and region V. To substitute the general solution for the magnetic scalar potential in region I, II, and III, and the magnetic field strength in region IV and V into Eqs (34)–(39), the linear system equation can be formed and described in the matrix form as

$\displaystyle{\bf MX}={\bf Y}$ (40)

where, M, X, and Y are the constant coefficient matrix, and the undetermined variable vector, and a constant source vector, respectively. Their expressions are given in Eqs (40)–(42). And the detailed expressions and descriptions of the elements in M and Y are given in the appendix.

$\displaystyle{\bf X}=[a_{nk}^{I},b_{nk}^{I},a_{nk}^{II},b_{nk}^{II},a_{nk}^{% III},b_{nk}^{III},a_{nk}^{IV},b_{nk}^{IV},c_{nk}^{IV},d_{nk}^{IV},a_{nk}^{V},b% _{nk}^{V},c_{nk}^{V},d_{nk}^{V}]^{T}$ (41) $\displaystyle{\bf Y}=[0,\Gamma^{(1)},\Gamma^{(2)},\Gamma^{(3)},\Gamma^{(4)},0,% 0,0,0,0,0,0,0,0]^{T}$ (42)

$\displaystyle{\bf M}=\left[{{\begin{array}[]{cccccccccccccc}1&{M^{(1)}}&0&0&0&% 0&0&0&0&0&0&0&0&0\\ 1&1&{-1}&{-1}&0&0&0&0&0&0&0&0&0&0\\ {M^{(2)}}&{-M^{(2)}}&1&{-1}&0&0&0&0&0&0&0&0&0&0\\ 0&0&{M^{(3)}}&1&{-M^{(2)}}&{-1}&0&0&0&0&0&0&0&0\\ 0&0&{-1}&{M^{(4)}}&1&{-M^{(4)}}&0&0&0&0&0&0&0&0\\ 0&0&0&0&{M^{(5)}}&1&{M^{(6)}}&{M^{(7)}}&0&0&0&0&0&0\\ 0&0&0&0&{M^{(8)}}&{M^{(9)}}&1&{M^{(10)}}&{M^{(11)}}&{M^{(12)}}&0&0&0&0\\ 0&0&0&0&0&0&{M^{(13)}}&1&0&0&0&0&0&0\\ 0&0&0&0&{M^{(14)}}&{M^{(15)}}&0&0&1&{M^{(10)}}&0&0&0&0\\ 0&0&0&0&0&0&{M^{(16)}}&{M^{(17)}}&{-M^{(13)}}&1&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0&1&{M^{(18)}}&0&0\\ 0&0&0&0&0&0&0&0&0&0&{M^{(19)}}&1&{M^{(20)}}&{M^{(21)}}\\ 0&0&0&0&0&0&0&0&{M^{(22)}}&{M^{(23)}}&0&0&1&{M^{(18)}}\\ 0&0&0&0&0&0&0&0&0&0&0&0&{M^{(24)}}&1\\ \end{array}}}\right]$ (43)

3. Model validation and analysis

In order to evaluate the proposed analytical model, nonlinear 3D-FE method is used as the comparison. In practice, the ANSOFT Maxwell software is employed. The geometric parameters listed in Table 1 are considered as a case study. In the numerical model, the nonlinear properties of iron material (steel_1010) shown in Fig. 5, and the curvature effects are considered, which are ignored in analytical model. In addition, the thickness of back iron is designed to avoid the magnetic saturation, thus the permeability of back iron is large enough.

Figure 5.

B-H curve for the ferromagnetic material.

Figure 6.

Flux density distribution along the $x$ -direction at $y=$ 0 and $z=$ 0.055 with slip speed 0 r/min.

3.1 Flux density distribution

According to the relationship between the flux density and the magnetic scalar potential, the three components of flux density distributions in air-gap region can be obtained as follows:

$\displaystyle B_{IIIx}=-\mu_{0}\frac{\partial\varphi_{III}(x,y,z)}{\partial x}% =\Re\left\{{\sum\limits_{n=1}^{\infty}{\sum\limits_{k=1}^{\infty}{-j\lambda_{X% _{k}}\mu_{0}(a_{nk}^{III}e^{\lambda_{nk}z}+b_{nk}^{III}e^{-\lambda_{nk}z})}}}% \right.\left.\cos(\lambda_{Y_{n}}y)\exp(j\lambda_{X_{k}}x)\!\!\!\!\!\phantom{% \frac{1}{2}}\right\}$ (44) $\displaystyle B_{IIIy}=-\mu_{0}\frac{\partial\varphi_{III}(x,y,z)}{\partial y}% =\Re\left\{{\sum\limits_{n=1}^{\infty}{\sum\limits_{k=1}^{\infty}{\mu_{0}% \lambda_{Y_{n}}(a_{nk}^{III}e^{\lambda_{nk}z}+b_{nk}^{III}e^{-\lambda_{nk}z})}% }}\right.\left.\sin(\lambda_{Y_{n}}y)\exp(j\lambda_{X_{k}}x)\!\!\!\!\phantom{% \frac{1}{2}}\right\}$ (45) $\displaystyle B_{IIIz}=-\mu_{0}\frac{\partial\varphi_{III}(x,y,z)}{\partial z}% =\Re\left\{{\sum\limits_{n=1}^{\infty}{\sum\limits_{k=1}^{\infty}{-\mu_{0}% \lambda_{nk}(a_{nk}^{III}e^{\lambda_{nk}z}-b_{nk}^{III}e^{-\lambda_{nk}z})}}}% \right.\left.\cos(\lambda_{Y_{n}}y)\exp(j\lambda_{X_{k}}x)\!\!\!\!\phantom{% \frac{1}{2}}\right\}$ (46)

Figures 6 and 7 respectively show the flux density distributions along the $x$ -direction at $y=$ 0 and $z=b+c$ /2 with the slip speed 0 r/min and 80 r/min while keeping the other parameters constant. As indicated in Figs 6 and 7, there are good agreements for the $x$ -component and $z$ -component between 3D analytical results and those obtained by 3D-FEM. However, for the $y$ -component, there will be slight deviations between the 3D-FEM and the proposed analytical model. One of the main reasons is that the curvature effects have been ignored in the simplified 3D mathematical model. In addition, the results also show that the induced current has distorted the flux density distribution at the high slip speed case.

Figure 7.

Flux density distribution along the $x$ -direction at $y=$ 0 and $z=$ 0.055 with slip speed 80 r/min.

Figures 8 and 9 respectively show the flux density distributions along the $y$ -direction at $x=$ 0 and $z=b+c$ /2 with the slip speed 0 r/min and 80 r/min while keeping the other parameters constant. As shown in Figs 8 and 9, there are good agreement between the analytical results and 3D-FEM results. There may appear deviations at the edge of conductor disk for $B_{y}$ , which is caused by the boundary condition (2). However, in any case, the proposed model has sufficient accuracy to predict flux density distributions.

Figure 8.

Flux density distribution along the $y$ -direction at $x=$ 0 and $z=$ 0.055 with slip speed 0 r/min.

Figure 9.

Flux density distribution along the $y$ -direction at $x=$ 0 and $z=$ 0.055 with slip speed 80 r/min.

3.2 Force and electromagnetic torque

Figure 10.

Variation laws of axial force with slip speed.

According to the Maxwell stress tensor method, the axial force is calculated using the integration surface $z=b$ as follows

$\displaystyle F_{axial}=p\mu_{0}\int_{-\tau}^{\tau}{\int_{-H/2}^{H/2}{\left% \langle{\frac{H_{IIIz}^{2}(x,y,b)-H_{IIIx}^{2}(x,y,b)-H_{IIIy}^{2}(x,y,b)}{2}}% \right\rangle}}dxdy$ (47)

where,

$\displaystyle\left\{{\begin{array}[]{l}H_{IIIx}(x,y,b)=\Re\left\{{\sum\limits_% {n=1}^{\infty}{\sum\limits_{k=1}^{\infty}{-j\lambda_{X_{k}}(a_{nk}^{III}e^{% \lambda_{nk}b}+b_{nk}^{III}e^{-\lambda_{nk}b})\cos(\lambda_{Y_{n}}y)\exp(j% \lambda_{X_{k}}x)}}}\right\}\\ H_{IIIy}(x,y,b)=\Re\left\{{\sum\limits_{n=1}^{\infty}{\sum\limits_{k=1}^{% \infty}{\lambda_{Y_{n}}(a_{nk}^{III}e^{\lambda_{nk}b}+b_{nk}^{III}e^{-\lambda_% {nk}b})\sin(\lambda_{Y_{n}}y)\exp(j\lambda_{X_{k}}x)}}}\right\}\\ H_{IIIz}(x,y,b)=\Re\left\{{\sum\limits_{n=1}^{\infty}{\sum\limits_{k=1}^{% \infty}{-\lambda_{nk}(a_{nk}^{III}e^{\lambda_{nk}b}-b_{nk}^{III}e^{-\lambda_{% nk}b})\cos(\lambda_{Y_{n}}y)\exp(j\lambda_{X_{k}}x)}}}\right\}\\ \end{array}}\right.$ (48)

Because the 3D geometrical structure of eddy-current couplers is considered in the analytic calculation, the toque can be derived by using the Maxwell stress tensor method without any correction factor, which can be expressed by

$\displaystyle T=pR_{av}\mu_{0}\int_{-\tau}^{\tau}{\int_{-H/2}^{H/2}{H_{IIx}(x,% y,b)H_{IIz}(x,y,b)}}dxdy$ (49)

For the axial-flux devices, the axial force affects the bearing losses of the drive. Figure 10 shows the axial force versus slip speed. It can be found that the axial force has the maximum value when the coupler is working under the “stationary” state; with the slip speed increasing, the axial force decreases rapidly, and will switch to being repulsive. Moreover, the analytical results agree with the 3D-FEM results, which show the effectiveness of the analytic expression Eq. (48).

Figure 11 shows the torque versus slip speed. As shown in Fig. 11, the torque is well predicted by the analytical Eq. (49). Be different from the 2D analytical methods [6, 7, 15, 16], the proposed 3D analytical model doesn’t need a three-dimensional correction, such as the R-N correction factor. In addition, it can be discovered that the eddy-current coupler with the parameters listed in Table 1 has the optimal torque at the slip speed 80 r/min. With the slip speed increasing, the torque will increase, and then decrease. The contribution of back iron on the torque is also studied. As shown in Fig. 11, with the increase of the slip speed, the percentage data clearly show a sharp decline trend. It is worth stressing that the contribution of back iron on torque cannot be ignored at the smaller slip speed, because the percentage may be up to 8%.

Figure 11.

Variation laws of torque with slip speed.

Figure 12.

Toque along with $\alpha_{1}$ and $\theta$ for $\omega=$ 80 r/min.

3.3 Influence of magnetic array on torque

In this section, the influence of geometric parameters of magnetic array on the torque will be discussed. According to the Fig. 3, $\tau_{2}+2\tau_{1}=\alpha\tau$ , where $\tau_{1}=\alpha_{1}\tau$ , and $\tau_{2}=\alpha_{2}\tau$ . What is clear is that the torque is the most optimal when the value of $\alpha$ is 1. Thus, we will discuss the effects of $\alpha_{1}$ and magnetization angle $\theta$ with $\alpha=$ 1.

Figure 12 shows the torque along with the parameters $\alpha_{1}$ and $\theta$ for the slip speed 80 r/min. As previously indicated, they can greatly affect the torque property of such devices. Based on the analytical results shown in Fig. 12, there will be the optimum combination of $\alpha_{1}$ and $\theta$ , where $\alpha_{1}=$ 0.3, and $\theta\in$ [0.5 0.6]. Therefore, the analytical model will provide us an effective way to select the proper parameters.

Figure 13.

Error rate between analytical torque and 3D-FEM with the change of $\delta$ .

3.4 Influence of curvature effects on the predicted results

As shown in Fig. 1, the device is essentially a 3-D cylindrical topology. When it is simplified into a linear 3D structure, the curvature effects will be ignored in the 3D analytical model in the Cartesian coordinate system. In order to discuss the influence of curvature effects on the model precision, a dimensionless number $\delta$ is introduced as follows [18]:

$\displaystyle\delta=\frac{R_{2}-R_{1}}{\tau}\ \text{with}\ \tau=\frac{\pi(R_{1% }+R_{2})}{2p}$ (50)

$\delta$ is the ratio of the radial length of PMs to the pole pitch at the average radius of PM, which implies the degree of curvature, the larger $\delta$ is, the more pronounced curvature is. From Eq. (50), we can see that $\delta$ is affected by three parameters: outer radius of PMs ( $R_{1}$ ), inner radius of PMs ( $R_{2}$ ), and the number of pole pairs ( $p$ ). Thus, we have to discuss their influences on the predicted results.

Figure 13 shows the error rate between the 3D analytical torque and 3D-FEM with the change of curvature coefficient $\delta$ . The values of $\delta$ are obtained by arbitrarily combination of $R_{1}$ , $R_{2}$ , and $p$ . As indicated in Fig. 13, when $\delta$ is less than 1.5, the error rate will be very small and no more than 3%. However, when $\delta$ is larger than 1.5, the error rate will be increased with $\delta$ . For the case in Table 1, $\delta=$ 0.997, thus the linearized 3D analytical model is relatively accurate.

3.5 Experimental testing results

To test and validate the real performance of the prototype based on the parameters in Table 1, an experiment platform has been designed and established, overall structure of which is shown in Fig. 14, and the specific details are described in our previous studies [15, 16]. It’s worth noting that the torque-speed characteristic is uniquely concerned to evaluate the accuracy of the analytical model in this paper.

Figure 14.

Test platform system and its composition.

Figure 15.

Test results with different air-gap length: $c=$ 10 mm, $c=$ 15 mm, and $c=$ 20 mm.

The measurement results of torque for eddy-current coupler are shown in Fig. 15. As can be observed, three values of air-gap length are considered, namely, 10 mm, 15 mm, and 20 mm. In addition, the average error rate is also given in Fig. 15. As shown in Fig. 15, the analytical results have good agreement with the test results, and the maximum discrepancy is about 5%. The discrepancy may come from the curvature effects and temperature factor, which is discussed in our previous study [19]. However, the proposed analytical model has enough precision to evaluate the performances of eddy-current couplers. It is important to note that the computation time for 3D-FEM, employing a desktop PC (32 G (RAM) with 8 core), is about 300 min (15 points), while the computing time of the proposed analytical model is nearly all less than 3s with $n=$ 9 and $k=$ 9.

The analytical model is also used to estimate the performances of eddy-current couplers with axial magnetization PM, at this time, the value of $\alpha_{1}$ is null. Therefore, the proposed model has wide applicability.

4. Conclusion

In this paper, an accurate 3D analytical model for the PM eddy-current couplers with quasi-Halbach magnet array is developed in the 3D Cartesian coordinates. The radial edge effects of 3D geometry and eddy-current effects of back iron are considered in the analytical model. The analytical expressions of magnetic field and output torque are formulated. Compared with the 3D-FEM, the 3D analytical model can save so much time and resources with the satisfactory accuracy, thus it can be used to conveniently evaluate and optimize the PM eddy-current couplers in their initial design stages.

Footnotes

Acknowledgments

The work was supported in part by National Key R&D program of China under Grant 2017YBF1300 900, and in part by the Doctoral Scientific Research Foundation of Liaoning Province No. 201601030.

Appendices

The detailed expressions and descriptions of the elements in M and Y will be given in this section.

(A.1) $\displaystyle M^{(1)}=-e^{2\lambda_{nk}a}$

(A.2) $\displaystyle M^{(2)}=\frac{-\mu_{1}}{\mu_{0}}$

(A.3) $\displaystyle M^{(3)}=e^{2\lambda_{nk}b}$

(A.4) $\displaystyle M^{(4)}=e^{-2\lambda_{nk}b}$ (A.5) $\displaystyle M^{(5)}=e^{2\lambda_{nk}(b+c)}$ (A.6) $\displaystyle M^{(6)}=\frac{e^{(\lambda_{nk}+\zeta_{nk})(b+c)}}{j\lambda_{X_{k% }}}$ (A.7) $\displaystyle M^{(7)}=\frac{e^{(\lambda_{nk}-\zeta_{nk})(b+c)}}{j\lambda_{X_{k% }}}$ (A.8) $\displaystyle M^{(8)}=\frac{\lambda_{Y_{n}}^{2}e^{(\lambda_{nk}-\zeta_{nk})(b+% c)}}{j\lambda_{X_{k}}}$ (A.9) $\displaystyle M^{(9)}=\frac{\lambda_{Y_{n}}^{2}e^{(-\lambda_{nk}-\zeta_{nk})(b% +c)}}{j\lambda_{X_{k}}}$ (A.10) $\displaystyle M^{(10)}=e^{-2\zeta_{nk}(b+c)}$ (A.11) $\displaystyle M^{(11)}=\frac{\zeta_{nk}}{j\lambda_{X_{k}}}$ (A.12) $\displaystyle M^{(12)}=\frac{-\zeta_{nk}e^{-2\zeta_{nk}(b+c)}}{j\lambda_{X_{k}}}$ (A.13) $\displaystyle M^{(13)}=e^{2\zeta_{nk}(b+c+d)}$ (A.14) $\displaystyle M^{(14)}=\lambda_{nk}e^{(\lambda_{nk}-\zeta_{nk})(b+c)}$ (A.15) $\displaystyle M^{(15)}=-\lambda_{nk}e^{-(\lambda_{nk}+\zeta_{nk})(b+c)}$ (A.16) $\displaystyle M^{(16)}=\frac{-j\lambda_{X_{k}}e^{2\zeta_{nk}(b+c+d)}}{\zeta_{% nk}}$ (A.17) $\displaystyle M^{(17)}=\frac{-j\lambda_{X_{k}}}{\zeta_{nk}}$ (A.18) $\displaystyle M^{(18)}=e^{-2\zeta_{nk}^{\prime}(b+c+d)}$ (A.19) $\displaystyle M^{(19)}=e^{2\zeta_{nk}^{\prime}(b+c+d)}$ (A.20) $\displaystyle M^{(20)}=\frac{\zeta_{nk}^{\prime}e^{2\zeta_{nk}^{\prime}(b+c+d)% }}{j\lambda_{X_{k}}}$ (A.21) $\displaystyle M^{(21)}=\frac{-\zeta_{nk}^{\prime}}{j\lambda_{X_{k}}}$ (A.22) $\displaystyle M^{(22)}=-\frac{\mu_{0}e^{(\zeta_{nk}-\zeta_{nk}^{\prime})(b+c+d% )}}{\mu_{2}}$ (A.23) $\displaystyle M^{(23)}=-\frac{\mu_{0}e^{-(\zeta_{nk}+\zeta_{nk}^{\prime})(b+c+% d)}}{\mu_{2}}$ (A.24) $\displaystyle M^{(24)}=e^{2\zeta_{nk}^{\prime}(b+c+d+e)}$ (A.25) $\displaystyle\Gamma^{(1)}=\frac{16B_{r}\sin\theta}{n\mu_{0}\pi\tau\lambda_{{}^% {nk}}^{2}}\sin\left(\frac{\lambda_{Y_{n}}L}{2}\right)\left({\cos(\frac{\lambda% _{X_{k}}\tau_{2}}{2})-\cos\left[\lambda_{X_{k}}\left(\frac{\tau_{2}}{2}+\tau_{% 1}\right)\right]}\right)$ (A.26) $\displaystyle\Gamma^{(2)}=\frac{-16B_{r}}{kn\mu_{0}\pi^{2}\lambda_{nk}}\sin% \left(\frac{\lambda_{Y_{n}}L}{2}\right)\left\{{(1-\cos\theta)\sin\left(\frac{% \lambda_{X_{k}}\tau_{2}}{2}\right)}\right.\quad\left.+\cos\theta\sin\left[% \lambda_{X_{k}}\left(\frac{\tau_{2}}{2}+\tau_{1}\right)\right]\right\}$ (A.27) $\displaystyle\Gamma^{(3)}=\frac{-16B_{r}\sin\theta e^{\lambda_{nk}b}}{n\mu_{0}% \pi\tau\lambda_{{}^{nk}}^{2}}\sin\left(\frac{\lambda_{Y_{n}}L}{2}\right)\left(% {\cos(\frac{\lambda_{X_{k}}\tau_{2}}{2})-\cos\left[\lambda_{X_{k}}\left(\frac{% \tau_{2}}{2}+\tau_{1}\right)\right]}\right)$ (A.28) $\displaystyle\Gamma^{(4)}=\frac{-16B_{r}e^{-\lambda_{nk}b}}{kn\mu_{0}\pi^{2}% \lambda_{nk}}\sin\left(\frac{\lambda_{Y_{n}}L}{2}\right)\left\{{(1-\cos\theta)% \sin\left(\frac{\lambda_{X_{k}}\tau_{2}}{2}\right)}\right.\quad\left.+\cos% \theta\sin\left[\lambda_{X_{k}}\left(\frac{\tau_{2}}{2}+\tau_{1}\right)\right]\right\}$

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Analytical predictions and analysis of electromagnetic field and torque for eddy-current couplers with Quasi-Halbach magnet array

Abstract

Keywords

1. Introduction

2.1 Description of geometry

Table 1 Dimensions and materials properties of the case-study coupling

Footnotes

Acknowledgments

Appendices

References

Table 1
Dimensions and materials properties of the case-study coupling