Optimal overhang structure considering overhang effects in the axial flux permanent magnet motor

Abstract

In this paper, the overhang structure, which has been used in the radial flux permanent magnet motor, was applied to the axial flux permanent magnet (AFPM) motor. Moreover, the overhang effect was investigated for different overhang structures. Because the AFPM motor has an enlarged form along radial direction, the leakage flux and the saturation effect are considerably different at inner and outer radius of the stator. Therefore, inner and outer overhang part could have different overhang effects. This can lead to the optimization problem with optimal overhang length. Because the AFPM motor requires 3D finite element analysis from its 3D structure, an efficient optimization algorithm is required for optimal overhang structure. In order to increase the convergence speed of the optimization algorithm, this paper proposed the gradient assisted contour method.

Keywords

Axial flux permanent magnet motor finite element method overhang effect overhang structure

1 Introduction

From the existence of the end winding, a useless space would be generated from the end of stack to the housing of the motor. This space becomes larger when the motor adopts the distributed winding instead of the concentrated winding [1]. However, this inefficient space can be converted into an efficient one by applying a structure called the overhang. The overhang means a structure, which has a rotor longer than a stator [2,3].

In the RFPM motor, when the permanent magnet (PM) is mounted on the surface of the rotor, the length of effective air-gap will be increased because the PM has a permeability similar to air [4]. In addition, due to the leakage flux at the end of the stack, the air-gap magnetic flux density would be decreased on average [5].

The overhang structure generates additional magnetic flux over the end of stack due to its configuration. Thus, the loss of leakage flux can be offset and the air-gap magnetic flux density can be increased on average [6]. From this overhang effect, characteristics of the RFPM motor would be increased without addition of inverter capacity.

In this paper, the overhang structure, which has been applied in the RFPM motor, is employed to the axial flux permanent magnet (AFPM) motor. In the RFPM motor, the overhang structure has a rotor longer than a stator in the axial direction. However, in the AFPM motor, the overhang structure has a rotor longer than a stator in the radial direction.

2 Overhang effect in the AFPM motor

2.1 Magnetic saturation effect

In the RFPM motor, when the overhang structure is applied, it can be provided with the same amount of permanent magnet (PM) in the axial direction. However, because the AFPM motor has an enlarged shape along radial direction, the overhang structure has different volume of the PM although the same overhang length is applied to inner and outer side of the rotor. Therefore, this could generate different saturation effects.

Figure 1 presents the AFPM motor with the overhang structure. Table 1 shows the basic specification of the AFPM motor. In Fig. 1, l _in and l _out are the length of inner and outer overhang part. In general, the cost of ferrite PMs is much cheaper than rare-earth PMs. The remanence flux density B _r of ferrite PMs is much smaller than those of rare-earth PM. If rare-earth PMs have to be used to increase the motor performance with the overhang structure, the fabrication cost will be significantly increased. Moreover, it will be sufficient to saturate the air-gap. Therefore, the rare-earth PMs are not suitable for overhang structure. From these effects, this paper employed ferrite-PMs in order to reduce the fabrication cost and increase the motor performance.

The overhang effect will give different contributions at inner and outer radius of the stator. Moreover, because it will generate different influence from no-load and load condition, the overhang effect was investigated for different overhang structures and load conditions.

When the overhang structure was applied, the overhang length could be limited by space between the shaft and the end winding. In this paper, the overhang effect was investigated for different overhang length and load condition, when the volume of the PM was increased by 5% and 10%, respectively.

Table 2 presents a specification of analysis models. The volume of PMs is based on it of Case 1. In Case 2 and Case 4, the overhang structure is applied along the inner direction. However, In Case 3 and Case 5, it is only employed along the outer direction. Figure 2 and Table 3 present the comparison of results.

From results, it showed that the overhang effect increased the performance compared to one without overhang structure. However, it presented that no load and load characteristic was not linearly enhanced as the volume of the PM was increased. Moreover, it generated better performance when overhang structure was applied at the outer direction than at the inner direction, while using the same amount of PM. Even, with a smaller amount of the PM, similar overhang effects were obtained. Although Case 4 used a larger volume of the PM than it of Case 3, Case 4 does not show better characteristics compared to Case 3. These were due to leakage and saturation effect of the AFPM motor.

The AFPM motor extends its form along radial direction. However, because the structure of slot is parallel, the ratio of the teeth to the pole pitch is not constant along radial direction. Even, it has the smallest value at the inner radius of the stator. From these effects, the leakage flux and the saturation effect are considerably different toward inner and outer radius of the AFPM motor.

Figures 3 and 4 show distribution of magnetic flux density on no load and load condition. When the overhang length was more increased toward inner than outer radius of the AFPM motor, it showed that the magnetic saturation was increased. From these results, the overhang structure should be employed considering the saturation effect at inner one and outer radius of the AFPM motor. Even, these effects could lead to the optimization problem with optimal overhang length.

2.2 Demagnetization effect

In the PM, a permeance coefficient was calculated as follows: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle B_{n}=B_{x,n}\times \cos ({\alpha})+B_{y,n}\times \cos ({\alpha})\end{eqnarray}$ (1) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle H_{n}=H_{x,n}\times \cos ({\alpha})+H_{y,n}\times \cos ({\alpha})\end{eqnarray}$ (2) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle P_{n}=\displaystyle \frac{B_{n}}{H_{n}}\end{eqnarray}$ (3) where B _n and H _n are the magnetic flux density and the magnetic field intensity in the nth element of the PM, B _x,n and B _y,n are the x- and y-axis components of the magnetic flux density, H _x,n and H _y,n are the x- and y-axis components of the magnetic field intensity, 𝛼 is the magnetizing angle considering the position of rotor, and P _n is a permeance coefficient in the nth element, respectively.

In this paper, the analysis model employs the ferrite PM. The maximum energy product of a ferrite PM is reported to be about 10% of a rare-earth PM [7]. The coercive force H _c of the ferrite PM are much smaller than one of a rare-earth PM [7]. Therefore, from the viewpoint of an irreversible demagnetization, the ferrite PM is less advantageous than the rare-earth PM.

The overhang part which was exposed to the end of stack has high magnetic reluctance. Especially, the PM suffers low permeance coefficient. From low permeance coefficient, the operating point of the PM would be decreased. Moreover, the PM can be demagnetized by demagnetizing field when the field weakening control is applied.

Figure 5 shows a distribution of the permeance coefficient when inner and outer radius of the rotor are 145 mm and 290 mm, respectively. Figure 5 presents that the permeance coefficient was more decreased in overhang parts than non-overhang parts and it could contribute to the irreversible demagnetization of the PM although the overhang part was small. From these results, a consideration of the irreversible demagnetization in overhang structure using the ferrite PM is essential. Therefore, this effect could also lead to the optimization problem with optimal overhang length against the irreversible demagnetization of the PM.

3 Gradient assisted climb method

A two-dimensional (2D) finite element (FE) analysis is much faster than a three-dimensional (3D) approach in terms of the modeling and computation costs. However, the 2D FE analysis guarantee accuracy because overhang structure cannot be taken into consideration in a 2D. The 3D FE analysis needs much computational cost. Even, the optimal design using 3D FE analysis will be challengeable one.

In the optimization problem, the main issue may be on finding the optimal solution. In the AFPM motor, the optimal solution can involve the optimum with the best quality and optima with sufficient qualification. In this paper, the optimization problem refers to finding both optimum and optima.

The AFPM motor requires 3D finite element (FE) analysis because it has basically 3D structure. Moreover, it needs an iterative FE analysis in order to converge toward optima. Therefore, an efficient optimization algorithm is required for the optimal overhang length of the AFPM motor.

In this paper, the proposed algorithm is based on the climb method which is inspired from climbing a mountain [8]. The climb method can find optima quickly and efficiently. However, there is a defect caused by using the annealing method as the main operator. The concept of annealing method was based on mutation operators in the evolution strategy [9]. According to the evaluation of a local solution in the evolution range, the limit of the exploration range is decided. However, the direction from the previous optimum to the candidate optimum is not determined.

In order to increase the convergence speed into optima, the gradient method can be combined with a deterministic or a stochastic optimization algorithm. In order to calculate the gradient using the numerical technique, additional evaluations are required. Each evaluation means 3D FE analysis in optimization problem of the AFPM motor. It generates computationally expensive evaluations.

To address these problems, the gradient assisted climb method, which does not require additional evaluations to calculate the gradient, is proposed.

In a 2D problem, gradients can be expressed by (4)–(8) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle u_{1}=u_{11}e_{1}+u_{12}e_{2}\end{eqnarray}$ (4) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle u_{2}=u_{21}e_{1}+u_{22}e_{2}\end{eqnarray}$ (5) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle D_{u_{1}}f(x_{1},x_{2})=u_{11}\displaystyle \frac{\partial f(x_{1},x_{2})}{\partial x_{1}}+u_{12}\displaystyle \frac{\partial f(x_{1},x_{2})}{\partial x_{2}}\end{eqnarray}$ (6) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle D_{u_{2}}f(x_{1},x_{2})=u_{21}\displaystyle \frac{\partial f(x_{1},x_{2})}{\partial x_{1}}+u_{22}\displaystyle \frac{\partial f(x_{1},x_{2})}{\partial x_{2}}\end{eqnarray}$ (7) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \left(\begin{array}{@{}c@{}}D_{e_{1}}f(x_{1},x_{2})\\ D_{e_{2}}f(x_{1},x_{2})\end{array}\right)=\left(\begin{array}{@{}cc@{}}u_{11} & u_{12}\\ u_{21} & u_{22}\end{array}\right)^{-1}\left(\begin{array}{@{}c@{}}D_{u_{1}}f(x_{1},x_{2})\\ D_{u_{2}}f(x_{1},x_{2})\end{array}\right)\end{eqnarray}$ (8) where u _i is the vector toward local peak in the contour region of the climb method, e _i is unit vector, and u _ij is the amplitude. D _ui f (x ₁, x ₂) is the directional derivative of f (x ₁, x ₂).

In the n-dimensional problem, the gradients for the n vectors can be expressed by (9) $\begin{eqnarray}\displaystyle \left(\begin{array}{@{}c@{}}D_{e_{1}}f(x_{1},\cdots ∼,x_{n})\\ \vdots \\ D_{e_{n}}f(x_{1},\cdots ∼,x_{n})\\ \end{array}\right)=\left(\begin{array}{@{}ccc@{}}u_{11} & \cdots ∼ & u_{1n}\\ \vdots & \ddots & \vdots \\ u_{n1} & \cdots ∼ & u_{nn}\\ \end{array}\right)^{-1}\left(\begin{array}{@{}c@{}}D_{u_{1}}f(x_{1},\cdots ∼,x_{n})\\ \vdots \\ D_{u_{n}}f(x_{1},\cdots ∼,x_{n})\\ \end{array}\right). & & \displaystyle\end{eqnarray}$ (9)

In Fig. 6, the directional derivative is calculated as follows: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle D_{u_{1}}f(x_{1},x_{2})=\displaystyle \frac{f_{best}(x_{1},x_{2})-f_{worst}(x_{1},x_{2})}{\left\Vert u_{1}\right\Vert }\end{eqnarray}$ (10) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle D_{u_{2}}f(x_{1},x_{2})=\displaystyle \frac{f_{best}(x_{1},x_{2})-f(x_{1},x_{2})}{\left\Vert u_{2}\right\Vert }\end{eqnarray}$ (11) where f _best(x ₁, x ₂) is the value at the local optimum and the f _worst(x ₁, x ₂) is the value of contour level. The f (x ₁, x ₂) is a random one in the contour region. Because the f _best(x ₁, x ₂), f _worst(x ₁, x ₂) and f (x ₁, x ₂) are already evaluated one to construct the metamodel in the climb method, it does not need additional evaluations. $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle D_{1}=D_{u_{1}}f(x_{1},x_{2})\cdot \displaystyle \frac{u_{1}}{\left\Vert u_{1}\right\Vert },\quad D_{2}=D_{u_{2}}f(x_{1},x_{2})\cdot \displaystyle \frac{u_{2}}{\left\Vert u_{2}\right\Vert },\ldots\end{eqnarray}$ (12) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle D=\mathop{\sum }_{i=1}^{n}D_{i}\end{eqnarray}$ (13) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\alpha}_{i}^{\prime }=D\cdot {\alpha}_{i}.\end{eqnarray}$ (14) Therefore, the proposed algorithm provides a new alternative as an operator in the climb method. Especially, about the ith design variable, the directional information can be provided with (14).

4 Results

The proposed algorithm is verified by an application to the mathematical function and an optimization problem of the AFPM motor with the overhang structure.

4.1 Mathematical function

The proposed algorithm is compared with results in [9]. The formula for the test function is $\begin{eqnarray}\displaystyle f=50-(x_{1}-5)^{2}+(x_{2}-5)^{2}+\mathop{\sum }_{i=1}^{2}5\cos [2{\pi}(x_{i}-5)] & & \displaystyle\end{eqnarray}$ (15) where 2.5 ≤ x, y ≤ 7.5. Figure 7 shows the shape of the mathematical function.

Table 4 presents the comparison of results. Although the climb method generates next candidates in the random direction, it makes a good contribution to the quality of the metamodel. Moreover, the climb method can converge to the optimal solution because the quality of the metamodel generally improves according to function calls.

However, the proposed method can directly converge to the optimal solution from the next candidate with directional information. Therefore, the proposed method can converge to the optimal solution with a small number of function calls.

From Table 4, about 14% of function calls can be reduced when it is compared to that in the climb method. Therefore, in an optimization problem of the AFPM motor, the proposed method is expected to save a significant amount of 3D FE analysis.

4.2 Optimization problem of AFPM motor

As a practical optimization problem, the AFPM motor was selected. The optimal design can be ideally achieved only when all of design variables are considered. Moreover, the object can be more than one. When the AFPM motor which was already constructed was limited from electric loading, the only way that the AFPM motor can improve its characteristic is to replace the rotor part as one with the overhang structure. It is the best way to increse the motor performance without the significant increase of the cost from a new AFPM motor. Therefore, we selected design variables regarding the overhang structure. Figure 8 shows design variables. l _in and l _out are the length of inner and outer overhang part. l _m was fixed as the length of the stack. h _m is the height of PMs. Because the analysis model adopted the concentrated winding, the overhang length was limited to it of the end winding.

Table 5 shows the optimization results. We select Type 3 as the best compromise one because this candidate has high peak value and low THD of EMF in the limited volume and non-irreversible demagnetization of PMs. Table 5 showed that it had generally better performance when overhang structure was applied toward the outer direction, while considering the saturation effect of the AFPM motor.

5 Conclusion

In this paper, the overhang effect was investigated for different overhang structures. In order to analyze the overhang effect in the AFPM, it takes a lot of time. Therefore, the overhang effect has not been studied for AFPM motors in detail. For the optimal overhang structure in the AFPM motor, the algorithm combined with gradient method was proposed. The proposed algorithm can directly converge to the optimal solution through the directional information in design variables. It could reduce the computational cost of the optimization process. This reduction made the proposed algorithm suitable for the optimization problem of the AFPM motor with overhang structure. From results, we presented new guidelines for the overhang structure in the AFPM motor.

Footnotes

Acknowledgements

This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No. 2017R1C1B5015672) and Dongil Culture and Scholarship Foundation.

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