Analytical prediction and optimization of torque characteristic for flux-concentration cage-type eddy-current couplings with slotted conductor rotor topology

Abstract

In this paper, an analytical model is proposed to predict the magnetic field and torque characteristics of flux-concentration cage-type eddy-current couplings with slotted conductor rotor topology. Due to the presence of discontinuous boundary conditions in permanent magnet rotor domain and conductor rotor domain, the accurate sub-domain method is employed in polar coordinates. The permanent magnet domain and the conductor domain are divided into different sub-domain according to the number of poles and slots. The magnetic vector potential in each domain is obtained by solving the field equation based on the separated variable method. Then the magnetic field distributions and output torque are computed. By using the nonlinear 2D and 3D finite element analysis, the validity of such model is proved, and the cause of deviation is discussed. In the end, the proposed formula is used for a design optimization using particle swarm optimization algorithm.

Keywords

Eddy-current couplings sub-domain method finite-element method design optimization

1. Introduction

Permanent magnet (PM) eddy-current couplings can be used to transmit torque without any mechanical contact between two rotating parts. Because of their unique performances, they are frequently applied in harsh environments, such as pharmaceutical and chemical industries. In addition, such devices are also employed in speed adjustment system [1] and braking system [2].

Although PM eddy-current couplings have been studied for many years, surface-mounted types have always been the focus of attention for researchers. By comparison, few studies have been reported for the flux-concentration types [3]. As a matter of fact, compared with surface-mounted types, flux-concentration PM eddy-current couplings have some advantages such as no centrifugal force on the PMs, high irreversible demagnetization withstand, and robust rotor construction. On the other hand, to improve the torque density of such devices, the conductor rotor can be slotted and filled with iron yoke [4]. Inspired by these studies, this paper proposes the flux-concentration cage-type eddy-current couplings with slotted conductor rotor topology.

To evaluate the performance of such devices, numerical or analytical methods can be used. Although mature and powerful, the former is computationally intense and a lack of flexibility, especially in the optimal design of machines [5]. In most cases, numerical methods, for example, finite element analysis (FEA) is employed to verify and analyze the performance of an available or candidate design. The latter trades a better calculation speed for the accuracy. In recent years, all kinds of analytical model are built for surface-mounted PM eddy-current couplings without slotted conductor topology, for example, 2D analytical model [6,7], 3D model [8–10], and equivalent magnetic circuit model [11,12]. However, due to the special construction of PM rotor and conductor rotor, these models are not suitable for the proposed topology in this paper. Fortunately, the sub-domain model is an effective way to solve these issues, which has been applied in PM machines [13], and magnetic gears [14]. The simplified analytical model in Cartesian coordinates is established for surface-mounted PM eddy-current couplings, which neglects the curvature effects of 3D geometry, thus such model is not accurate [15]. We had proposed a hybrid analytical method for the flux-concentration PM eddy-current couplings without slotted conductor disk [16], and it can solve the PM with circumferential magnetization problem, but cannot solve the eddy-current problem in the conductor spoke.

A novel PM eddy-current couplings is concerned in this paper. As the accurate analytical model can be used to analyze performance and optimize design parameters, the analytical models of electromagnetic field and torque for flux-concentration cage-type eddy-current couplings with slotted conductor rotor topology is developed in polar coordinates. Based on the accurate sub-domain method, the magnetic vector potential in each domain is obtained using the separation of variables method. Then the explicit expressions for the flux density distribution and output toque are given. The nonlinear FEA and prototype test are applied to validate the torque model. In the end, the improved particle swarm optimization is introduced to optimize the torque performance.

2. Magnetic field model

2.1. Configuration

The real geometry of the considered couplings is shown in Fig. 1. As shown in Fig. 1, the PM rotor is inside the conductor rotor; each magnet is circumferentially magnetized and inserted into iron cores; the conductor rotor is slotted and filled with protrusions of back iron. The geometrical parameters of prototype to be investigated are as follows:

Fig. 1.

Geometry of the studied eddy-current couplings: (a) Exploded view (b) 2-Dgeometry.

(1) The inner radius of the PM rotor is R ₁;

(2) The outer radius of the PM rotor is R ₂;

(3) The inner radius of the conductor rotor is R ₃;

(4) The outer radius of the conductor rotor is R ₄;

(5) The numbers of conductor spokes and pole-pairs are respectively Q and p.

In order to simplify the calculation, the initial positions of the ith PM and qth conductor spoke are respectively given by $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\theta}_{i}=-\frac{{\beta}}{2}+\frac{i{\pi}}{p}\quad \text{for }1\leq i\leq 2p\end{eqnarray}$ (1) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\alpha}_{p}=\frac{2{\pi}(q-1)}{Q}\quad \text{for }1\leq q\leq Q.\end{eqnarray}$ (2)

2.2. Assumptions

To simplify the analytical modelling process, some reasonable assumptions, commonly employed in the modeling of such devices, are adopted as follows:

(1) Iron core and back iron have infinite permeability;

(2) The conductor bar, the PMs, and the air-gap have the same permeability.

(3) The eddy current density is distributed uniformly in the conductor bar, and only has the axial-direction component.

(4) All the interfaces are defined asthe constant.

Moreover, due to the presence of eddy current in the conductor bars, the field equation has to be converted into a Helmholtz equation. According to the previous study [15,17,18], the PM region can be treated as the source of a time-varying magnetic field. Then the magnetic vector potential in each sub-domain is associated with time variable, which can be expressed as $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \mathbf{A}_{1}=\Re[\tilde{A}_{1}(r,{\theta})e^{j{\lambda}_{0}t}]\overrightarrow{e_{z}}\quad \text{for region 1}\end{eqnarray}$ (3) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \mathbf{A}_{i}^{2}=\Re[\tilde{A}_{i}^{2}(r,{\theta})e^{j{\lambda}_{0}t}]\overrightarrow{e_{z}}\quad \text{for}∼i\text{th sub-domain in region 2}\end{eqnarray}$ (4) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \mathbf{A}_{3}=\Re[\tilde{A}_{3}(r,{\theta})e^{j{\lambda}_{0}t}]\overrightarrow{e_{z}}\quad \text{for region 3}\end{eqnarray}$ (5) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \mathbf{A}_{q}^{4}=\Re[\tilde{A}_{q}^{4}(r,{\theta})e^{j{\lambda}_{0}t}]\overrightarrow{e_{z}}\quad \text{for}∼q\text{th sub-domain in region 4},\end{eqnarray}$ (6) where, a tilde over a variable denotes its complex form and $j=\sqrt{-1}$ , $\Re$ denotes the real part, $\overrightarrow{e_{z}}$ is the unit vector in the z-direction, 𝜆₀ is the angular frequency and can be expressed as $\begin{eqnarray}\displaystyle {\lambda}_{0}=\displaystyle \frac{2{\pi}n_{1}mps}{60}, & & \displaystyle\end{eqnarray}$ (7) where, m is the mth time harmonic, n ₁ is the input speed, and s is the slip.

2.3. Analytical calculation of electromagnetic field

According to the assumption, in the shaft region (region 1), the field problem is to solve Laplace’s equation as follows $\begin{eqnarray}\displaystyle \displaystyle \frac{\partial ^{2}\tilde{A}_{1}}{\partial r^{2}}+\frac{1}{r}\frac{\partial \tilde{A}_{1}}{\partial r}+\frac{1}{r^{2}}\frac{\partial ^{2}\tilde{A}_{1}}{\partial {\theta}^{2}}=0,\quad \text{for}∼\left\{\begin{array}{@{}l@{}}0\leq r\leq R_{1}\\ 0\leq {\theta}\leq 2{\pi}.\end{array}\right. & & \displaystyle\end{eqnarray}$ (8)

In the PM region, due to the existence of the iron core, the boundary condition is not simple. Considering the boundary condition that the tangential component of the magnetic field at the interface between the shaft and the PM is continuous, and can be expressed as $\begin{eqnarray}\displaystyle \left.\frac{\partial \tilde{A}_{1}}{\partial r}\right|_{r=R_{1}}=f_{1}({\theta}), & & \displaystyle\end{eqnarray}$ (9) where, f ₁(θ) is a piecewise function, and can be expressed as $\begin{eqnarray}\displaystyle f_{1}({\theta})=\left\{\begin{array}{@{}l@{}}\left.\displaystyle \frac{\partial \tilde{A}_{i}^{2}}{\partial r}\right|_{r=R_{1}}+(-1)^{i}B_{r},\quad \forall {\theta}\in [{\theta}_{i},{\theta}_{i}+{\beta}]\\ 0,\quad \text{else}.\end{array}\right. & & \displaystyle\end{eqnarray}$ (10)

By taking into account (9), the general solution of (8) can be written as $\begin{eqnarray}\displaystyle \tilde{A}_{1}(r,{\theta})=\displaystyle \mathop{\sum }_{n=1}^{\infty }{\displaystyle \frac{R_{1}}{n}}\left({\displaystyle \frac{r}{R_{1}}}\right)^{n}[a_{n}^{1}\cos (n{\theta})+b_{n}^{1}\sin (n{\theta})], & & \displaystyle\end{eqnarray}$ (11) where, n is the number of spatial harmonic in the shaft region.

The integral constants in (11) are determined using a Fourier series expansion at the interface between the shaft and the PM over the interval [0, 2π]. Thus $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle a_{n}^{1}=\frac{1}{{\pi}}\int _{0}^{2{\pi}}f_{1}({\theta})\cos (n{\theta})d{\theta}\end{eqnarray}$ (12) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle b_{n}^{1}=\frac{1}{{\pi}}\int _{0}^{2{\pi}}f_{1}({\theta})\sin (n{\theta})d{\theta}.\end{eqnarray}$ (13)

In the ith PM sub-domain in region 2 (region 2i), the field problem is to solve Posson’s equation as follows $\begin{eqnarray}\displaystyle {\displaystyle \frac{\partial ^{2}\tilde{A}_{i}^{2}}{\partial r^{2}}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial \tilde{A}_{i}^{2}}{\partial r}}+\frac{1}{r^{2}}{\displaystyle \frac{\partial ^{2}\tilde{A}_{i}^{2}}{\partial {\theta}^{2}}}=-(-1)^{i}\frac{B_{r}}{r},\quad \text{for}∼\left\{\begin{array}{@{}l@{}}R_{1}\leq r\leq R_{2}\\ {\theta}_{i}\leq {\theta}\leq {\theta}_{i}+{\beta}.\end{array}\right. & & \displaystyle\end{eqnarray}$ (14)

According to the assumption, the boundary conditions at the interface between iron cores and PM are $\begin{eqnarray}\displaystyle \left.\displaystyle \frac{\partial \tilde{A}_{i}^{2}}{\partial {\theta}}\right|_{{\theta}={\theta}_{i}}=0\quad \text{and}\quad \left.\displaystyle \frac{\partial \tilde{A}_{i}^{2}}{\partial {\theta}}\right|_{{\theta}={\theta}_{i}+{\beta}}=0. & & \displaystyle\end{eqnarray}$ (15)

In addition, the magnetic vector potential at the interface between the ith PM and regions 1 and 3 are continuous. That is $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \left.\tilde{A}_{i}^{2}(r,{\theta})=\tilde{A}_{1}(r,{\theta})\right|_{r=R_{1}}\end{eqnarray}$ (16) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \left.\tilde{A}_{i}^{2}(r,{\theta})=\tilde{A}_{3}(r,{\theta})\right|_{r=R_{2}}.\end{eqnarray}$ (17)

Thus, the general solution of ((14)) can be written as $\begin{eqnarray}\displaystyle \tilde{A}_{i}^{2}(r,{\theta}) & = & \displaystyle a_{i0}^{2}+b_{i0}^{2}\ln r-r(-1)^{i}B_{r}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼\mathop{\sum }_{k=1}^{\infty }\left\langle a_{ik}^{2}\frac{\left(\frac{r}{R_{2}}\right)^{\frac{k{\pi}}{{\beta}}}-\left(\frac{R_{2}}{r}\right)^{\frac{k{\pi}}{{\beta}}}}{\left(\frac{R_{1}}{R_{2}}\right)^{\frac{k{\pi}}{{\beta}}}-\left(\frac{R_{2}}{R_{1}}\right)^{\frac{k{\pi}}{{\beta}}}}-b_{ik}^{2}\frac{\left(\frac{r}{R_{1}}\right)^{\frac{k{\pi}}{{\beta}}}-\left(\frac{R_{1}}{r}\right)^{\frac{k{\pi}}{{\beta}}}}{\left(\frac{R_{1}}{R_{2}}\right)^{\frac{k{\pi}}{{\beta}}}-\left(\frac{R_{2}}{R_{1}}\right)^{\frac{k{\pi}}{{\beta}}}}\right\rangle \cos \left[\frac{k{\pi}}{{\beta}}({\theta}-{\theta}_{i})\right],\end{eqnarray}$ (18) where, k is the number of spatial harmonic in the ith PM sub-domain. To simplify the analytical expressions, the following notationsare introduced $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle U(a,b,z)=(a/b)^{z}+(b/a)^{z}\end{eqnarray}$ (19) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle V(a,b,z)=(a/b)^{z}-(b/a)^{z}\end{eqnarray}$ (20) Therefore, ((18)) can be further expressed as $\begin{eqnarray}\displaystyle \tilde{A}_{i}^{2}(r,{\theta}) & = & \displaystyle a_{i0}^{2}+b_{i0}^{2}\ln r-r(-1)^{i}B_{r}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼\mathop{\sum }_{k=1}^{\infty }\left\langle a_{ik}^{2}\frac{V(r,R_{2},k{\pi}/{\beta})}{V(R_{1},R_{2},k{\pi}/{\beta})}-b_{ik}^{2}\frac{V(r,R_{1},k{\pi}/{\beta})}{V(R_{1},R_{2},k{\pi}/{\beta})}\right\rangle \cos \left[\frac{k{\pi}}{{\beta}}({\theta}-{\theta}_{i})\right].\end{eqnarray}$ (21)

The integral constants in (21) are determined using a Fourier series expansion at the interface between the ith PM and regions 1 and 3 over the interval [θ_i, θ_i + 𝛽]. They can be given by $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \left.a_{i0}^{2}+b_{i0}^{2}\ln r\right|_{r=R_{1}}=\frac{1}{{\beta}}\int _{{\theta}_{i}}^{{\theta}_{i}+{\beta}}[R_{1}(-1)^{i}B_{r}+\tilde{A}_{1}(R_{1},{\theta})]d{\theta}\end{eqnarray}$ (22) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \left.a_{i0}^{2}+b_{i0}^{2}\ln r\right|_{r=R_{2}}=\frac{1}{{\beta}}\int _{{\theta}_{i}}^{{\theta}_{i}+{\beta}}[R_{2}(-1)^{i}B_{r}+\tilde{A}_{3}(R_{2},{\theta})]d{\theta}\end{eqnarray}$ (23) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle a_{ik}^{2}=\frac{2}{{\beta}}\int _{{\theta}_{i}}^{{\theta}_{i}+{\beta}}[R_{1}(-1)^{i}B_{r}+\tilde{A}_{1}(R_{1},{\theta})]\cos \left[\frac{k{\pi}}{{\beta}}({\theta}-{\theta}_{i})\right]d{\theta}\end{eqnarray}$ (24) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle b_{ik}^{2}=\frac{2}{{\beta}}\int _{{\theta}_{i}}^{{\theta}_{i}+{\beta}}\left[R_{2}(-1)^{i}B_{r}+\tilde{A}_{3}(R_{2},{\theta})\right]\cos [\frac{k{\pi}}{{\beta}}({\theta}-{\theta}_{i})]d{\theta}.\end{eqnarray}$ (25)

In the air-gap region (region 3), the field problem is to solve Laplace’s equation as follows $\begin{eqnarray}\displaystyle \frac{\partial ^{2}A_{3}}{\partial r^{2}}+\frac{1}{r}\frac{\partial A_{3}}{\partial r}+\frac{1}{r^{2}}\frac{\partial ^{2}A_{3}}{\partial {\theta}^{2}}=0,\quad \text{for}∼\left\{\begin{array}{@{}l@{}}R_{2}\leq r\leq R_{3}\\ 0\leq {\theta}\leq 2{\pi}.\end{array}\right. & & \displaystyle\end{eqnarray}$ (26)

Considering the boundary condition that the tangential component of the magnetic field at the interface between the air-gap and the PM are continuous, which can be expressed as $\begin{eqnarray}\displaystyle \left.\frac{\partial \tilde{A}_{3}}{\partial r}\right|_{r=R_{2}}=f_{2}({\theta}), & & \displaystyle\end{eqnarray}$ (27) where, f ₂(θ) is a piecewise function, similarly to f ₁(θ), and can be expressed as $\begin{eqnarray}\displaystyle f_{2}({\theta})=\left\{\begin{array}{@{}l@{}}\left.\displaystyle \frac{\partial \tilde{A}_{i}^{2}}{\partial r}\right|_{r=R_{2}}+(-1)^{i}B_{r},\quad \forall {\theta}\in [{\theta}_{i},{\theta}_{i}+{\beta}]\\ 0,\quad \text{else}.\end{array}\right. & & \displaystyle\end{eqnarray}$ (28)

In addition, due to the iron-core protrusion, the physical quantities at the interface between the air-gap and conductor spoke are discontinuous, which is expressed as $\begin{eqnarray}\displaystyle \left.\displaystyle \frac{\partial \tilde{A}_{3}}{\partial r}\right|_{r=R_{3}}=f_{3}({\theta}), & & \displaystyle\end{eqnarray}$ (29) where, f ₃(θ) is also a piecewise function, which can be expressed as $\begin{eqnarray}\displaystyle f_{3}({\theta})=\left\{\begin{array}{@{}l@{}}\left.\displaystyle \frac{\partial \tilde{A}_{p}^{4}}{\partial r}\right|_{r=R_{3}},\quad \forall {\theta}\in [{\alpha}_{q},{\alpha}_{q}+{\beta}_{1}]\\ 0,\quad \text{else}.\end{array}\right. & & \displaystyle\end{eqnarray}$ (30)

Thus, the general solution of (26) can be written as $\begin{eqnarray}\displaystyle \displaystyle \tilde{A}_{3}(r,{\theta}) & = & \displaystyle \mathop{\sum }_{n=1}^{\infty }\left[a_{n}^{3}\frac{R_{2}}{n}\frac{V(r,R_{3},n)}{U(R_{2},R_{3},n)}+b_{n}^{3}\frac{R_{3}}{n}\frac{V(r,R_{2},n)}{U(R_{2},R_{3},n)}\right]\cos (n{\theta})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼\displaystyle \mathop{\sum }_{n=1}^{\infty }\left[c_{n}^{3}\frac{R_{2}}{n}\frac{V(r,R_{3},n)}{U(R_{2},R_{3},n)}+d_{n}^{3}\frac{R_{3}}{n}\frac{V(r,R_{2},n)}{U(R_{2},R_{3},n)}\right]\sin (n{\theta}),\end{eqnarray}$ (31) where, n is the number of spatial harmonic in this region.

As before, the integral constants are determined using a Fourier series expansion at the interface conditions. Thus $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \displaystyle a_{n}^{3}=\frac{1}{{\pi}}\int _{0}^{2{\pi}}f_{2}({\theta})\cos (n{\theta})d{\theta}\end{eqnarray}$ (32) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \displaystyle b_{n}^{3}=\frac{1}{{\pi}}\int _{0}^{2{\pi}}f_{3}({\theta})\cos (n{\theta})d{\theta}\end{eqnarray}$ (33) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \displaystyle c_{n}^{3}=\frac{1}{{\pi}}\int _{0}^{2{\pi}}f_{2}({\theta})\sin (n{\theta})d{\theta}\end{eqnarray}$ (34) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \displaystyle d_{n}^{3}=\frac{1}{{\pi}}\int _{0}^{2{\pi}}f_{3}({\theta})\sin (n{\theta})d{\theta}.\end{eqnarray}$ (35)

In the qth conductor bar sub-domain (region 4q), due to the time-varying magnetic field, the induced current will be generated in the conductor bars. Thus, the governing equation in the qth conductor bar sub-domain is expressed as follows $\begin{eqnarray}\displaystyle \displaystyle \frac{\partial ^{2}\tilde{A}_{q}^{4}}{\partial r^{2}}+\frac{1}{r}\frac{\partial \tilde{A}_{q}^{4}}{\partial r}+\frac{1}{r^{2}}\frac{\partial ^{2}\tilde{A}_{q}^{4}}{\partial {\theta}^{2}}=-{\mu}_{0}\tilde{J}_{q},\quad \text{for }\left\{\begin{array}{@{}l@{}}R_{3}\leq r\leq R_{4}\\ {\alpha}_{q}\leq {\theta}\leq {\alpha}_{q}+{\beta}_{1},\end{array}\right. & & \displaystyle\end{eqnarray}$ (36) and the induced current can be derived as $\begin{eqnarray}\displaystyle \displaystyle \tilde{J}_{q}=-{\sigma}\frac{\partial \tilde{A}_{q}^{4}}{\partial t}=-\frac{j{\pi}{\sigma}n_{1}smp}{30}\tilde{A}_{q}^{4}. & & \displaystyle\end{eqnarray}$ (37)

Therefore, (36) is further expressed as $\begin{eqnarray}\displaystyle \displaystyle \frac{\partial ^{2}\tilde{A}_{q}^{4}}{\partial r^{2}}+\frac{1}{r}\frac{\partial \tilde{A}_{q}^{4}}{\partial r}+\frac{1}{r^{2}}\frac{\partial ^{2}\tilde{A}_{q}^{4}}{\partial {\theta}^{2}}=\frac{j{\pi}{\mu}_{0}{\sigma}n_{1}smp}{30}\tilde{A}_{q}^{4}. & & \displaystyle\end{eqnarray}$ (38)

According to the assumption, the boundary conditions at the interface between the qth conductor bar and iron protrusions are obtained as $\begin{eqnarray}\displaystyle \left.\displaystyle \frac{\partial \tilde{A}_{q}^{4}}{\partial {\theta}}\right|_{{\theta}={\alpha}_{q}}=0\quad \text{and}\quad \left.\frac{\partial \tilde{A}_{q}^{4}}{\partial {\theta}}\right|_{{\theta}={\alpha}_{q}+{\beta}_{1}}=0. & & \displaystyle\end{eqnarray}$ (39)

Moreover, another boundary condition at the interface r = R ₄ is $\begin{eqnarray}\displaystyle \left.\displaystyle \frac{\partial \tilde{A}_{q}^{4}}{\partial {\theta}}\right|_{r=R_{4}}=0. & & \displaystyle\end{eqnarray}$ (40)

In addition, the magnetic vector potential at the interface between the qth conductor bar sub-domain and region3 is continuous. That is $\begin{eqnarray}\displaystyle \left.\tilde{A}_{q}^{4}(r,{\theta})=\tilde{A}_{3}(r,{\theta})\right|_{r=R_{3}}. & & \displaystyle\end{eqnarray}$ (41)

((38)) is an Helmholtz equation, using the separation variable method, whose general solution can be derived as $\begin{eqnarray}\displaystyle \displaystyle \tilde{A}_{q}^{4}(r,{\theta}) & = & \displaystyle a_{q0}^{4}\frac{J_{0}({\Lambda}_{1}r)-\dfrac{J_{1}({\Lambda}_{1}R_{4})}{N_{1}({\Lambda}_{1}R_{4})}N_{0}({\Lambda}_{1}r)}{J_{0}({\Lambda}_{1}R_{3})-\dfrac{J_{1}({\Lambda}_{1}R_{4})}{N_{1}({\Lambda}_{1}R_{4})}N_{0}({\Lambda}_{1}R_{3})}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼\mathop{\sum }_{g=1}^{\infty }a_{qg}^{4}\frac{J_{{\lambda}(g)}({\Lambda}_{1}r)-{\Lambda}_{2}N_{{\lambda}(g)}({\Lambda}_{1}r)}{J_{{\lambda}(g)}({\Lambda}_{1}R_{3})-{\Lambda}_{2}N_{{\lambda}(g)}({\Lambda}_{1}R_{3})}\cos [{\lambda}(g)({\theta}-{\alpha}_{q})],\end{eqnarray}$ (42) where, g is the number of spatial harmonic in qth conductor bar sub-domain; $a_{q0}^{4}$ and $c_{qg}^{4}$ are the integral constants to be determined; J _∗(. ) and N _∗(. ) are separately the first and second category Bessel function with the order ∗, and $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \displaystyle {\lambda}(g)=\frac{g{\pi}}{{\beta}_{1}}\end{eqnarray}$ (43) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \displaystyle {\Lambda}_{1}=\sqrt{\frac{-j{\pi}{\mu}_{0}{\sigma}n_{1}smp}{30}}\end{eqnarray}$ (44) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \displaystyle {\Lambda}_{2}=\frac{{\lambda}(g)J_{{\lambda}(g)}({\Lambda}_{1}R_{4})-{\Lambda}_{1}R_{4}J_{{\lambda}(g)+1}({\Lambda}_{1}R_{4})}{{\lambda}(g)N_{{\lambda}(g)}({\Lambda}_{1}R_{4})-{\Lambda}_{1}R_{4}N_{{\lambda}(g)+1}({\Lambda}_{1}R_{4})}.\end{eqnarray}$ (45)

The integral constants $a_{0}^{4q}$ and $a_{g}^{4q}$ are determined using a Fourier series expansion at the interface r = R ₃ over the interval [𝛼_q, 𝛼_q + 𝛽₁]. Thus $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \displaystyle a_{q0}^{4}=\frac{1}{{\beta}_{1}}\int _{{\alpha}_{q}}^{{\alpha}_{q}+{\beta}_{1}}A_{3}(R_{3},{\theta})d{\theta}\end{eqnarray}$ (46) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \displaystyle a_{qg}^{4}=\frac{2}{{\beta}_{1}}\int _{{\alpha}_{q}}^{{\alpha}_{q}+{\beta}_{1}}A_{3}(R_{3},{\theta})\cos [{\lambda}(g)({\theta}-{\alpha}_{q})]d{\theta}.\end{eqnarray}$ (47)

2.4. Flux density distribution

Although the numbers of spatial harmonics and time harmonic can be infinite, they will be compromises between computation time and accuracy. Assume that the numbers of spatial harmonics respectively are N in domain 1 and 3, K in sub-domain 3i, and G in sub-domain 4p, and the number of the time harmonic is M, the integration constants can be easily obtained by using the MATLAB software. According to the relationship between the flux density and the magnetic vector potential, the circumferential and radial of the flux density distribution in each domain can be calculated by $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \displaystyle B_{{\theta}}=-\mathop{\sum }_{m=1}^{M}\frac{\partial \tilde{A}}{\partial r}\end{eqnarray}$ (48) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \displaystyle B_{r}=\mathop{\sum }_{m=1}^{M}\frac{1}{r}\frac{\partial \tilde{A}}{\partial {\theta}}.\end{eqnarray}$ (49)

Therefore, in the region 1, the circumferential and radial of the fluxdensity distribution are derived by $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle B_{{\theta}}^{1}=-\mathop{\sum }_{m=1}^{M}\mathop{\sum }_{n=1}^{N}\left(\frac{r}{R_{1}}\right)^{n-1}[a_{n}^{1}\cos (n{\theta})+b_{n}^{1}\sin (n{\theta})]\end{eqnarray}$ (50) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle B_{r}^{1}=\mathop{\sum }_{m=1}^{M}\mathop{\sum }_{n=1}^{N}\left(\frac{r}{R_{1}}\right)^{n-1}[-a_{n}^{1}\sin (n{\theta})+b_{n}^{1}\cos (n{\theta})].\end{eqnarray}$ (51)

In the sub-domain 2i, the circumferential and radial of the flux density distribution are derived by $\begin{eqnarray}\displaystyle \displaystyle B_{i{\theta}}^{2}=\mathop{\sum }_{m=1}^{M}\left[(-1)^{i}B_{r}-\frac{b_{i0}^{2}}{r}\right]-\mathop{\sum }_{m=1}^{M}\mathop{\sum }_{k=1}^{K}\frac{k{\pi}}{r{\beta}}\left\langle \begin{array}{@{}c@{}}\displaystyle a_{ik}^{2}\frac{U(r,R_{2},k{\pi}/{\beta})}{V(R_{1},R_{2},k{\pi}/{\beta})}\\[10.79993pt] ∼-b_{ik}^{2}\displaystyle \frac{U(r,R_{1},k{\pi}/{\beta})}{V(R_{1},R_{2},k{\pi}/{\beta})}\end{array}\right\rangle \cos \left[\frac{k{\pi}}{{\beta}}({\theta}-{\theta}_{i})\right] & & \displaystyle\end{eqnarray}$ (52) $\begin{eqnarray}\displaystyle B_{ir}^{2}=-\mathop{\sum }_{m=1}^{M}\mathop{\sum }_{k=1}^{K}\frac{k{\pi}}{r{\beta}}\left\langle \begin{array}{@{}c@{}}\displaystyle a_{ik}^{2}\frac{V(r,R_{2},k{\pi}/{\beta})}{V(R_{1},R_{2},k{\pi}/{\beta})}\\[10.79993pt] ∼\displaystyle- b_{ik}^{2}\frac{V(r,R_{1},k{\pi}/{\beta})}{V(R_{1},R_{2},k{\pi}/{\beta})}\end{array}\right\rangle \sin \left[\frac{k{\pi}}{{\beta}}({\theta}-{\theta}_{i})\right]. & & \displaystyle\end{eqnarray}$ (53) In the region 3, the circumferential and radial of the flux density distribution are derived by $\begin{eqnarray}\displaystyle \displaystyle B_{{\theta}}^{3}=-\mathop{\sum }_{m=1}^{M}\mathop{\sum }_{n=1}^{N}\frac{1}{r}\left\langle \begin{array}{@{}c@{}}\displaystyle \left[a_{n}^{3}R_{2}\frac{U(r,R_{3},n)}{U(R_{2},R_{3},n)}+b_{n}^{3}R_{3}\frac{U(r,R_{2},n)}{U(R_{2},R_{3},n)}\right]\cos (n{\theta})\\[10.79993pt] +∼\displaystyle \left[c_{n}^{3}R_{2}\frac{U(r,R_{3},n)}{U(R_{2},R_{3},n)}+d_{n}^{3}R_{3}\frac{U(r,R_{2},n)}{U(R_{2},R_{3},n)}\right]\sin (n{\theta})\end{array}\right\rangle & & \displaystyle\end{eqnarray}$ (54) $\begin{eqnarray}\displaystyle B_{r}^{3}=-\mathop{\sum }_{m=1}^{M}\mathop{\sum }_{n=1}^{N}\frac{1}{r}\left\langle \begin{array}{@{}c@{}}\displaystyle \left[a_{n}^{3}R_{2}\frac{V(r,R_{3},n)}{U(R_{2},R_{3},n)}+b_{n}^{3}R_{3}\frac{V(r,R_{2},n)}{U(R_{2},R_{3},n)}\right]\sin (n{\theta})\\[10.79993pt] \displaystyle -\left[c_{n}^{3}R_{2}\frac{V(r,R_{3},n)}{U(R_{2},R_{3},n)}+d_{n}^{3}\displaystyle R_{3}\frac{V(r,R_{2},n)}{U(R_{2},R_{3},n)}\right]\cos (n{\theta})\\ \end{array}\right\rangle . & & \displaystyle\end{eqnarray}$ (55) In the sub-domain 4q, the circumferential and radial of the flux density distribution are derived by $\begin{eqnarray}\displaystyle \displaystyle B_{q{\theta}}^{4} & = & \displaystyle -\mathop{\sum }_{m=1}^{M}\mathop{\sum }_{g=1}^{G}a_{q0}^{4}\frac{-{\Lambda}_{1}J_{1}({\Lambda}_{1}r)+\frac{J_{1}({\Lambda}_{1}R_{4})}{N_{1}({\Lambda}_{1}R_{4})}{\Lambda}_{1}N_{1}({\Lambda}_{1}r)}{J_{0}({\Lambda}_{1}R_{3})-\frac{J_{1}({\Lambda}_{1}R_{4})}{N_{1}({\Lambda}_{1}R_{4})}N_{0}({\Lambda}_{1}R_{3})}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle -∼\mathop{\sum }_{m=1}^{M}\mathop{\sum }_{g=1}^{G}\frac{a_{qg}^{4}}{r}\left\langle \begin{array}{@{}c@{}}\displaystyle \frac{{\lambda}(g)J_{{\lambda}(g)}({\Lambda}_{1}r)-{\Lambda}_{1}rJ_{{\lambda}(g)+1}({\Lambda}_{1}r)}{J_{{\lambda}(g)}({\Lambda}_{1}R_{3})-{\Lambda}_{2}N_{{\lambda}(g)}({\Lambda}_{1}R_{3})}\\[10.79993pt] -\displaystyle {\Lambda}_{2}\frac{{\lambda}(g)N_{{\lambda}(g)}({\Lambda}_{1}r)-{\Lambda}_{1}rN_{{\lambda}(g)+1}({\Lambda}_{1}r)}{J_{{\lambda}(g)}({\Lambda}_{1}R_{3})-{\Lambda}_{2}N_{{\lambda}(g)}({\Lambda}_{1}R_{3})}\end{array}\right\rangle \cos [{\lambda}(g)({\theta}-{\alpha}_{q})]\end{eqnarray}$ (56) $\begin{eqnarray}\displaystyle B_{qr}^{4} & = & \displaystyle \mathop{\sum }_{m=1}^{M}\mathop{\sum }_{g=1}^{G}\left\langle \begin{array}{@{}c@{}}\displaystyle \frac{a_{q0}^{4}}{r}\frac{J_{0}({\Lambda}_{1}r)-\frac{J_{1}({\Lambda}_{1}R_{4})}{N_{1}({\Lambda}_{1}R_{4})}N_{0}({\Lambda}_{1}r)}{J_{0}({\Lambda}_{1}R_{3})-\frac{J_{1}({\Lambda}_{1}R_{4})}{N_{1}({\Lambda}_{1}R_{4})}N_{0}({\Lambda}_{1}R_{3})}\\[10.79993pt] \displaystyle -a_{qg}^{4}\frac{{\lambda}(g)}{r}\frac{J_{{\lambda}(g)}({\Lambda}_{1}r)-{\Lambda}_{2}N_{{\lambda}(g)}({\Lambda}_{1}r)}{J_{{\lambda}(g)}({\Lambda}_{1}R_{3})-{\Lambda}_{2}N_{{\lambda}(g)}({\Lambda}_{1}R_{3})}\sin [{\lambda}(g)({\theta}-{\alpha}_{q})]\end{array}\right\rangle .\end{eqnarray}$ (57)

3. Model validation

In order to evaluate the reliability and shortcomings of the proposed analytical model, nonlinear FE methods, including 2D and 3D, are used as the comparison. The geometric parameters listed in Table 1 are considered as a case study; and the ANSOFT Maxwell software is employed. In the numerical model, the nonlinear properties of iron material (steel_1010) are considered, and the PM and conductor have the same properties as those in analytical model. In addition, the parameters have not been optimized, and the aim of this paper is to validate the proposed model.

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

Symbols Parameters Values

R ₁ Inner radius of PM 15 mm

R ₂ Outer radius of PM 30 mm

𝛽 Opening angle of PM π∕12

𝛽₁ Width of conductor spoke π∕18

p Pole-pairs number 4

R ₃–R ₂ Length of air-gap 1 mm

R ₄–R ₃ Thickness of conductor bars 2 mm

L Active axial length 50 mm

L _o Axial length of overlapping 30 mm

B _r Remanence of PM (NdFeB) 1.27 T

σ_cs Conductivity of copper 58 MS/m

Q Number of conductor bars 18

M (m) Number of time harmonic 25

N (n) Number of spatial harmonic in region 1 and 3 30

K (k) Number of spatial harmonic in region 2i 25

G (g) Number of spatial harmonic in region 4i 20

Symbols	Parameters	Values
R ₁	Inner radius of PM	15 mm
R ₂	Outer radius of PM	30 mm
𝛽	Opening angle of PM	π∕12
𝛽₁	Width of conductor spoke	π∕18
p	Pole-pairs number	4
R ₃–R ₂	Length of air-gap	1 mm
R ₄–R ₃	Thickness of conductor bars	2 mm
L	Active axial length	50 mm
L _o	Axial length of overlapping	30 mm
B _r	Remanence of PM (NdFeB)	1.27 T
σ_cs	Conductivity of copper	58 MS/m
Q	Number of conductor bars	18
M (m)	Number of time harmonic	25
N (n)	Number of spatial harmonic in region 1 and 3	30
K (k)	Number of spatial harmonic in region 2i	25
G (g)	Number of spatial harmonic in region 4i	20

3.1. Flux density distribution

According to the analytical model, the radial and tangential components of flux density distributions in each sub-domain can be obtained. Figures 2 and 3 respectively show the radial and tangential components in the air-gap sub-domain (r = 30.5 mm) at the low slip speed case (n ₁ s = 30 r/min). Figures 4 and 5 show the high slip speed case (n ₁ s = 300 r/min). In order to ensure the credibility of the verification results, these state values are arbitrarily selected.

Fig. 2.

Radial component of flux density distributions at r = 30.5 mm for n ₁ s = 30 r/min.

As indicated in Figs 3 and 4, there are good agreements between the analytical model and those obtained by FEM. However, at the high slip speed case, as indicated in Figs 5 and 6, there will be slight deviations between the 3D-FEM and the proposed analytical model. The maximum error is less than 11%. The results show that the proposed model has sufficient accuracy to predict flux density distribution, especially at the low slip speed case. Further investigations show that, the ignorance of eddy current path in the end of conductor spokes is the primary cause of deviation. As the slip speed goes up, the eddy current density will increase, so does the deviation between the analytical model and 3D-FEM. However, it’s worth remembering that, in the practical application, the slip is relatively small, and generally ranges from 2% to 5% [12,14].

Fig. 3.

Tangential component of flux density distributions at r = 30.5 mm for n ₁ s = 30 r/min.

Fig. 4.

Radial component of flux density distributions at r = 30.5 mm for n ₁ s = 300 r/min.

Fig. 5.

Tangential component of flux density distributions at r = 30.5 mm for n ₁ s = 300 r/min.

3.2. Output torque

Considering the slot effects, the output torque of such devices consists of electromagnetic torque and cogging torque. The electromagnetic torque can be evaluated by the eddy-current loss in the conductor spokes. Thus, such torque can be derived by $\begin{eqnarray}\displaystyle T_{em}=\frac{Q_{loss}}{{\omega}_{s}}, & & \displaystyle\end{eqnarray}$ (58) with $\begin{eqnarray}\displaystyle Q_{loss}=\mathop{\sum }_{m=1}^{M}\mathop{\sum }_{q=1}^{Q}\frac{l_{o}m^{2}p^{2}}{{\sigma}}\int _{R_{1}}^{R_{2}}\int _{{\alpha}_{q}}^{{\alpha}_{q}+{\beta}_{1}}\tilde{A}_{q}^{4}(\tilde{A}_{q}^{4})^{\ast }d{\theta}dr, & & \displaystyle\end{eqnarray}$ (59) where (. )^∗ denotes the conjugate, ω_s is the relative angular velocity of conductor rotor.

According to the assumption, the 3Dgeometrical structure is simplified as 2D model, whereby the actual current paths in the overhang regions are not accounted for. Therefore, the torque with ((58)) is not accurate. To overcome this problem, the Russell–Norsworthy (R–N) correction factor has been widely used and achieved good results [19], which can be expressed as $\begin{eqnarray}\displaystyle k_{3D}=1-\frac{\text{tanh}\left(\frac{{\pi}L_{o}}{{\tau}}\right)}{\frac{{\pi}L_{o}}{{\tau}}\left(1+\text{tanh}\left(\frac{{\pi}L_{o}}{{\tau}}\right)\cdot \text{tanh}\left[\frac{{\pi}(L-L_{o})}{2{\tau}}\right]\right)}, & & \displaystyle\end{eqnarray}$ (60) where, τ is the pole pitch.

Therefore, the corrected electromagnetic torque is further expressed as $\begin{eqnarray}\displaystyle T_{em-3D}=k_{3D}T_{em}. & & \displaystyle\end{eqnarray}$ (61)

By using the Maxwell stress tensor [12], the cogging torque can be computed by $\begin{eqnarray}\displaystyle T_{cogg}=\mathop{\sum }_{q=1}^{Q}\frac{(R_{3}+R_{4})L_{o}}{4{\mu}_{0}}\int _{0}^{2{\pi}}B_{r}^{3}\left(\frac{R_{3}+R_{4}}{2},{\theta}\right)B_{{\theta}}^{3}\left(\frac{R_{3}+R_{4}}{2},{\theta}\right)d{\theta}. & & \displaystyle\end{eqnarray}$ (62)

The total output torque of such devices can be expressed as $\begin{eqnarray}\displaystyle T_{o}=T_{em-3D}+T_{cogg}. & & \displaystyle\end{eqnarray}$ (63)

In order to show the effectiveness of (63), the analytical results are compared with those obtained with 3D-FEM and 2D-FEM. Figure 6 shows the comparison results of transmitted torque versus slip speed. As illustrated in Fig. 6, because of the 3D correction factor (60), the analytical values are consistent with the 3D-FEM results for the low slip speed. It can be deduced that, in this case, the induced current produced in the iron protrusions has a little influence on torque values; and the R–N correction factor has a satisfactory compensation effect. However, for the high slip speed, there are still deviations between the proposed model and 3D-FEM. One of the main reasons is that the R–N correction factor has poor effect for the high slip speed, and some studies have illustrated the cause [12,14]. In any case, the analytical results are better than the 2D-FEM results.

Fig. 6.

Transmitted torque versus slip speed.

Fig. 7.

Comparison results of error rate.

The results obtained with the 3D-FEM is considered the reference, Fig. 7 shows the comparison results of error rate with 2D-FEM, analytical model with R–N correction, and analytical model without R–N correction. Figure 8 clearly shows the analysis results above. For the slip speed case, without R–N correction, the analytical results are close to those obtained with 2D-FEM, where the nonlinear characteristic of the materials is considered. For the high speed case, the deviations between (63) and 3D-FEM results will increase, but within the scope of the acceptance.

3.3. Curvature effects

As shown in Fig. 1, the device is essentially a 3D cylindrical topology. Thus the curvature effects have to be taken into consideration in analytical model. However, it is neglected in most of the literature for simplifying model. To consider a dimensionless number 𝜆, which is defined as the ratio of the radial excursion of the magnets around the mean radius to the pole pitch [9]: $\begin{eqnarray}\displaystyle {\lambda}=\frac{R_{2}-R_{1}}{{\tau}}\quad \text{with}∼{\tau}=\frac{{\pi}(R_{1}+R_{2})}{2p}. & & \displaystyle\end{eqnarray}$ (64)

The value of 𝜆 reflects the curvature situation of such devices, the larger 𝜆 is, the more significantly the curvature is. By changing the number of the pole-pairs, the torque-slip characteristic is discussed. The following settings are used:

(1) The results obtained with 3D-FEM are accurate and used as the benchmark;

(2) The low slip speed is considered;

(3) p ranges from 1 to 15, thus 𝜆 ranges from 0.21 to 3.18.

Fig. 8.

Error on the torque prediction between 3D-FEM and analytical model.

Figure 8 shows the prediction errors between the analytical model and the 3D-FEM results for different slip speed values, respectively, 50 r/min, 100 r/min, and 200 r/min. It can be observed in Fig. 8 that, with the increase of curvature coefficient 𝜆, the errors remain almost constant. Thus, it can be deduced that the precision of the proposed model is not affected by the curvature effects. It is no exaggeration to say that it’s one of the main advantages of analytical model in polar coordinates.

Fig. 9.

Overall structure of test platform system.

3.4. Experimental validation

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. 9, and the specific details are described in our previous studies [11,16]. It’s worth noting that the torque-speed characteristic is uniquely concerned to evaluate the accuracy of the analytical model in this paper.

Fig. 10.

Torque-speed characteristics by using the test platform system.

Figure 10 shows the torque-speed characteristics by using the experiment platform. Moreover, the deviation rates between the analytical model and the test results are given. From Fig. 10, it can be observed that the proposed model has good agreement with the actual values, especially at the low slip speed case. As previously described, such case is the normal working range. When the slip speed goes up, there may be imaginable error. Apart from the aforementioned reasons, temperature is one of the answers, which has been discussed in our previous studies [11]. Anyway, the proposed model can be used in the design optimization of such devices for its reducing computation time. Compared with 3D-FEM, the computing time of the proposed analytical model is nearly all less than 2s.

4. Design optimization

In fact, the parameters listed in Table 1 are rough, which can be further optimized based on the projects and customers requirement. It could be argued that it is the value of the analytical model. A hybrid Particle Swarm Optimization-Simplex Method (PSO-SM) algorithm [20] is employed in this paper, and the following multi-objective optimization problem is solved.

(1) The objective functions are maximize (T), and minimize (M _PM), which can be converted into a single objective: maximize (T∕M _PM). T is the torque for n ₁ s = 600 r/min and M _PM corresponds to the total mass of the PM.

(2) The design variables are pole-pairs (p), slot number (Q), opening angle of PM (𝛽), and angle of conductor spoke (𝛽₁), inner diameter of PM (R ₁).

(3) The constraints and bounds are p ∈ [1,20], Q ∈ [4,50], 𝛽 ∈ [π∕20, π], 𝛽₁ ∈ [π∕25, π∕2], R ₁ ∈ [5,30), p𝛽 < 2π, and Q𝛽₁ < 2π. In addition, the mass density of PM is equal to 7600 kg/m³.

The initialization parameters of PSO are given in Table 2.

Table 2
Parameter of PSO

Parameters Values

Dimensions of search space 5

Number of particles 25

Maximum number of iterations 1000

Acceleration constant c ₁ 2

Acceleration constant c ₂ 2

Decreasing linear inertia weight range 0.9:0.4

Parameters	Values
Dimensions of search space	5
Number of particles	25
Maximum number of iterations	1000
Acceleration constant c ₁	2
Acceleration constant c ₂	2
Decreasing linear inertia weight range	0.9:0.4

Table 3

Optimization results

Parameters	Values
Pole-pairs (p)	5
Slot number (Q)	24
Opening angle of PM (𝛽)	π∕14
Angle of conductor spoke (𝛽₁)	π∕20
Inner diameter of PM (R ₁)	18
Torque (T)	+17.5%
Total mass of PM (M _PM)	−9%

Fig. 11.

Fitness values versus generations.

The optimization procedure uses 25 particles with 1000 iterations, but 250 iterations are enough to reach a stable solution, as shown in Fig. 11. The computation time for achieving convergence condition is about 1 min. Table 3 shows the optimization results. Compared with the original performances with the parameters in Table 1, After optimization, about 17.5% improvement in the torque and a considerable reduction of about 9% in the total mass of PM have been achieved.

5. Conclusion

In this paper, a flux-concentration cage-type eddy-current coupling with slotted conductor rotor topology is studied. Based on the accurate sub-domain method, the analytical models of magnetic field in each domain and total output torque are established in polar coordinates. The analytical results are compared with nonlinear FEA. The result shows that the proposed model can save much computation time with ideal accuracy, thus it can be used to conveniently evaluate the performances of PM eddy-current couplings in their initial design stages. A special optimization case shows the analytical model plays a more important role in design.

Footnotes

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 61673404, 61876169), the Central Plains Thousand People Plan of Henan Province (Grant No. ZYQR201810162), the Key Scientific Research Projects in Colleges and Universities of Henan Province (Grant No. 19A120014), the Research Award Fund for Outstanding Young Teachers in Henan Provincial Institutions of Higher Education (2016GGJS-094), the Science and Technique Project of China National Textile and Apparel Council (Grant Nos. 2017054, 2018104).

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Analytical prediction and optimization of torque characteristic for flux-concentration cage-type eddy-current couplings with slotted conductor rotor topology

Abstract

Keywords

1. Introduction

2. Magnetic field model

2.1. Configuration

Table 2 Parameter of PSO Parameters Values Dimensions of search space 5 Number of particles 25 Maximum number of iterations 1000 Acceleration constant c 1 2 Acceleration constant c 2 2 Decreasing linear inertia weight range 0.9:0.4

Footnotes

Acknowledgements

References

Table 2
Parameter of PSO

Parameters Values

Dimensions of search space 5

Number of particles 25

Maximum number of iterations 1000

Acceleration constant c ₁ 2

Acceleration constant c ₂ 2

Decreasing linear inertia weight range 0.9:0.4