# Topology Choice and Optimization of a Bearingless Flux-Switching Motor with a Combined Winding Set

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## Abstract

**:**

## 1. Introduction

#### 1.1. History and Description of the Bearingless Motor

#### 1.2. Bearingless Permanent Magnetic Flux-Switching Slice Motor

#### 1.3. New Topology with Combined Windings

## 2. Setup

#### 2.1. Stator Teeth Geometry

#### 2.2. Rotor Geometry

#### 2.3. Motor Height

#### 2.4. Air Gap

#### 2.5. Magnetic Saturation

- use lower coil currents;
- change the back yoke of the stator iron (parameter ${w}_{fr}$) to increase the cross-sections of the back yoke.

## 3. Simulations

#### 3.1. Simulation Parameters

#### 3.2. Motor Optimization Criteria

#### 3.2.1. Passive Stiffness Values

- Axial stiffness (${k}_{z}$ in N/mm) is defined as stabilizing force per axial deflection of the rotor in the z-direction. In the FEM simulations, the rotor is displaced 1 mm from its center position. The value of the axial stiffness is negative and thus stabilizing. Hence, it is desirable to achieve high absolute values;
- Tilting stiffness (${k}_{\phi x}$ and ${k}_{\phi y}$ in Nm/deg) is defined as the stabilizing motor torque per degree due to PM reluctance forces when the rotor is tilted around the x- and y-axis. In the FEM simulations, the rotor is tilted 1 deg around the x- and y-axis, respectively, and the value of the tilting stiffness is negative, indicating stabilizing behavior. Again, higher absolute values are preferable;
- Radial stiffness (${k}_{x}$ and ${k}_{y}$ in N/mm) is defined as destabilizing radial force per deflection on the rotor pulling the rotor and the stator together. Active bearing forces have to be generated in order to overcome these forces. The radial reluctance forces are different from the other described passive forces as they destabilize the system; thus, it is desired that the radial stiffness features smaller absolute values. In simulations, the rotor is radially displaced 1 mm in the x- and y-directions to compute the radial stiffness values.

#### 3.2.2. Active Suspension Forces and Drive Torque

## 4. Topology Choice

#### 4.1. Results

- The cogging forces are the forces exerted on the rotor only by the PM field (${F}_{x,cogg}$ (0 At) and ${F}_{y,cogg}$ (0 At) in Table 3) and should be small and exceeded by the active motor active forces with 1000 At in one coil.

#### 4.2. Favorable Topologies

## 5. Optimization

- Motor height ${m}_{height}$, optimization of the parameters that define the motor height (in the axial z-direction), for both the stator and the rotor: ${s}_{height}$ and ${r}_{height}$;
- Stator geometry, optimization of the stator parameters: ${w}_{lrratio},{w}_{ironratio}$ and ${w}_{pmratio}$;
- Rotor geometry, optimization of the rotor parameters: ${M}_{r},{M}_{rheight}$ and ${\beta}_{Ri}$.

#### 5.1. FEM Simulations and Optimization Results

- Step 1:
- Motor height.

- Step 2:
- Stator and rotor height.

- Step 3:
- Stator geometry.

- Step 4:
- Rotor geometry.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

FEM | Finite element method |

PM | Permanent magnet |

FSPM | Flux-switching slice motor |

EMF | Electromotive force |

MMF | Magneto-motive force |

## References

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**Figure 2.**PM flux path depending on the rotor angle [9].

**Figure 3.**Cross-section of a bearingless flux-switching slice motor with twelve stator teeth and ten rotor poles [16].

**Figure 8.**Simulation results for increasing motor height: (

**a**) axial stiffness, (

**b**) rotor torque, (

**c**) radial stiffness in the x-direction, (

**d**) radial stiffness in the y-direction, (

**e**) tilting stiffness in the x-direction, (

**f**) tilting stiffness in the y-direction, (

**g**) radial bearing force in the x-direction and (

**h**) radial bearing force in the y-direction.

**Figure 9.**(

**a**) Tilting stiffness in the x-direction with decreasing rotor height; (

**b**) tilting stiffness in the y-direction with increasing stator height.

Parameter | Description |
---|---|

z_move | axial deflection of the rotor (z-axis), mm |

rotate_z | rotor angle, deg |

x_move | radial deflection of the rotor (x-axis) |

y_move | radial deflection of the rotor (y-axis) |

tilting_x | rotor tilting (x-axis) |

tilting_y | rotor tilting (y-axis) |

Parameter | Value |
---|---|

$\delta $ | 3 mm |

${m}_{height}$ | 10 mm |

${s}_{height}$ | 0 mm |

${r}_{height}$ | 0 mm |

${d}_{Ro}$ | 150 mm |

${w}_{lrratio}$ | 1 |

${w}_{ironratio}$ | 1 |

${w}_{pmratio}$ | 1 |

${M}_{r}$ | 1 |

${M}_{height}$ | 0.5 |

${\beta}_{Ri}$ | 0.3 |

6 Stator Segments | |||||

4 rotor poles | 5 rotor poles | 6 rotor poles | 7 rotor poles | 8 rotor poles | |

${F}_{x,cogg}$ (0 At) (N) | 0.309 | 4.593 | 0.987 | 2.219 | 0.688 |

${F}_{y,cogg}$ (0 At) (N) | 0.311 | 3.952 | 0.465 | 0.301 | 0.447 |

${F}_{x,i}$ (1000 At) (N) | 11.55 | 18.85 | 22.38 | 17.75 | 9.066 |

${F}_{y,i}$ (1000 At) (N) | 8.420 | 7.253 | 5.696 | 1.197 | 4.419 |

${T}_{z,cogg}$ (0 At) (mNm) | 345.322 | 10.429 | 440.84 | 17.617 | 91.61 |

${T}_{z,i}$ (1000 At) (mNm) | 264.46 | 287.109 | 410.527 | 378.566 | 292.489 |

8 Stator Segments | |||||

6 rotor poles | 7 rotor poles | 8 rotor poles | 9 rotor poles | 10 rotor poles | |

${F}_{x,cogg}$ (0 At) (N) | 0.223 | 6.190 | 0.337 | 0.630 | 0.347 |

${F}_{y,cogg}$ (0 At) (N) | 0.168 | 5.965 | 0.336 | 0.638 | 0.363 |

${F}_{x,i}$ (1000 At) (N) | 12.62 | 15.66 | 15.962 | 14.841 | 11.375 |

${F}_{y,i}$ (1000 At) (N) | 5.757 | 5.707 | 5.205 | 4.179 | 1.483 |

${T}_{z,cogg}$ (0 At) (mNm) | 7.712 | 5.544 | 534.333 | 15.222 | 19.967 |

${T}_{z,i}$ (1000 At) (mNm) | 209.39 | 272.37 | 338.366 | 331.124 | 296.037 |

10 Stator Segments | |||||

8 rotor poles | 9 rotor poles | 10 rotor poles | 11 rotor poles | 12 rotor poles | |

${F}_{x,cogg}$ (0 At) (N) | 0.269 | 8.716 | 0.346 | 0.512 | 0.228 |

${F}_{y,cogg}$ (0 At) (N) | 0.272 | 9.037 | 0.291 | 0.493 | 0.196 |

${F}_{x,i}$ (1000 At) (N) | 10.99 | 12.515 | 11.884 | 11.854 | 9.964 |

${F}_{y,i}$ (1000 At) (N) | 4.638 | 4.983 | 4.536 | 4.099 | 2.838 |

${T}_{z,cogg}$ (0 At) (mNm) | 44.22 | 13.731 | 623.352 | 6.857 | 23.391 |

${T}_{z,i}$ (1000 At) (mNm) | 210.273 | 238.515 | 271.287 | 278.891 | 261.102 |

Simulation error: ±1 |

**Table 4.**Topologies separated based on the results of Table 3.

Feasible Topologies | Unfeasible Topologies |
---|---|

S6R7, S8R6, S8R9, S8R10, S10R11, S10R12 | S6R4, S6R5, S6R6, S6R8, S8R7, S8R8, S10R8, S10R9, S10R10 |

${\mathit{m}}_{\mathit{height}}$ | |
---|---|

Simulation range | 10–20 mm |

Step size | 1 mm |

Simulation error: ±1 |

${\mathit{s}}_{\mathit{height}}=0$ mm, ${\mathit{r}}_{\mathit{height}}=-4$ mm | ${\mathit{s}}_{\mathit{height}}=4$ mm, ${\mathit{r}}_{\mathit{height}}=0$ mm | Unit | Increase | |
---|---|---|---|---|

rotor height | 10 | 14 | mm | |

stator height | 14 | 18 | mm | |

${k}_{z}$ | $-7.7$ | $-9.5$ | N/mm | 23% |

${k}_{x}$ | $-38$ | $-62$ | N/mm | 61% |

${k}_{y}$ | $-50$ | $-82$ | N/mm | 64% |

${k}_{\phi x}/{k}_{\phi y}$ | $-180$ | $-190$ | mNm/deg | 5.5% |

${F}_{x,i}$ (1000 At) | 24 | 35 | N | 45% |

${F}_{y,i}$ (1000 At) | 2 | 3 | N | 50% |

${T}_{z,i}$ (1000 At) | 550 | 800 | mNm | 45% |

${m}_{height}=14$ mm | ||||

Simulation error: ±1 |

Parameter | Range | Step |
---|---|---|

${w}_{lrratio}$ | 0.5–1.5 | 0.2 |

${w}_{ironratio}$ | Equation (9) | |

${w}_{pmratio}$ | 0.7–1.2 | 0.1 |

${m}_{height}=14$ mm, ${s}_{height}=4$ mm, ${r}_{height}=0$ mm | ||

Simulation error: ±1 |

${\mathit{w}}_{\mathit{pmratio}}$ | 0.7 | 0.8 | 0.9 | 1 | 1.1 | 1.2 | Unit |
---|---|---|---|---|---|---|---|

${k}_{z}$ | $-10.2$ | $-10.8$ | $-11.8$ | $-12$ | $-13$ | $-12.8$ | N/mm |

${k}_{x}$ | 86.787 | 91.04 | 92.669 | 92.133 | 91.07 | 87.433 | N/mm |

${k}_{y}$ | 88.65 | 91.7 | 95.5 | 100 | 105.8 | 108.1 | N/mm |

${k}_{\phi x}$ | $-170$ | $-182$ | $-182$ | $-208.8$ | $-215$ | $-235$ | mNm/deg |

${k}_{\phi y}$ | $-112$ | $-150$ | $-180$ | $-205$ | $-270$ | $-315$ | mNm/deg |

${F}_{x,i}$ (1000 At) | 38.17 | 38.756 | 40.61 | 40.43 | 40 | 39.087 | N |

${F}_{y,i}$ (1000 At) | 1.98 | 2.48 | 3.331 | 2.566 | 3.157 | 3.41 | N |

${T}_{z,i}$ (1000 At) | 781.1 | 815.49 | 849.4 | 890.16 | 890.626 | 880.33 | mNm |

${w}_{lrratio}=1$ | |||||||

Simulation error: ±1 |

Parameter | Range | Step |
---|---|---|

${M}_{r}$ | 0.5–1.5 | 0.2 |

${M}_{rheight}$ | 0.2–1 | 0.2 |

${\beta}_{Ri}$ | 0.2–1.8 | 0.2 |

${m}_{height}=14$ mm, ${s}_{height}=4$ mm, ${r}_{height}=0$ mm, ${w}_{lrratio}=1,{w}_{pmratio}=0.9,{w}_{ironratio}=1.05$ | ||

Simulation error: ±1 |

${\mathit{M}}_{\mathit{r}}$ | 0.5 | 0.7 | 0.9 | 1.1 | 1.3 | 1.5 | Unit |
---|---|---|---|---|---|---|---|

${k}_{z}$ | $-9.75$ | $-9.88$ | $-9.6$ | $-10.25$ | $-11.25$ | $-11.75$ | N/mm |

${k}_{x}$ | 53.7 | 51.8 | 56 | 70 | 88.8 | 97.3 | N/mm |

${k}_{y}$ | 71 | 74.5 | 81.5 | 86.7 | 90.8 | 91 | N/mm |

${k}_{\phi x}$ | $-193$ | $-187.5$ | $-198$ | $-194.5$ | $-203.8$ | $-206$ | mNm/deg |

${k}_{\phi y}$ | $-163$ | $-187.8$ | $-194$ | $-197$ | $-187$ | $-195.5$ | mNm/deg |

${F}_{x,i}$ (1000 At) | 22.923 | 29.3685 | 33.503 | 36.5 | 37.20674 | 26.33018 | N |

${F}_{y,i}$ (1000 At) | 1.7562 | 2.92512 | 3.522144 | 4.0175 | 4.504442 | 4.253711 | N |

${T}_{z,i}$ (1000 At) | 464.8362 | 631.9887 | 759.4341 | 835.322 | 894.7061 | 865.8532 | mNm |

Simulation error: ±1 |

${\mathit{M}}_{\mathit{rheight}}$ | 0.2 | 0.4 | 0.6 | 0.8 | 1 | Unit |
---|---|---|---|---|---|---|

${k}_{z}$ | $-18.8$ | $-10.4$ | $-9.8$ | $-9.5$ | $-9.2$ | N/mm |

${k}_{x}$ | 74 | 64 | 62.5 | 61 | 61.8 | N/mm |

${k}_{y}$ | 93 | 86 | 82.5 | 77.5 | 71.8 | N/mm |

${k}_{\phi x}$ | $-230$ | $-203$ | $-188.5$ | $-190$ | $-188$ | mNm/deg |

${k}_{\phi y}$ | $-242$ | $-208$ | $-195$ | $-188$ | $-170$ | mNm/deg |

${F}_{x,i}$ (1000 At) | 27.25042 | 34.195 | 35.84779 | 34.97191 | 33.84293 | N |

${F}_{y,i}$ (1000 At) | 1.60687 | 3.62255 | 4.3294 | 4.634218 | 5.47 | N |

${T}_{z,i}$ (1000 At) | 536.4645 | 741.24 | 876.3362 | 889.4663 | 889.0948 | mNm |

${M}_{r}=1.1$ | ||||||

Simulation error: ±1 |

${\mathit{\beta}}_{\mathit{Ri}}$ | 0.2 | 0.4 | 0.6 | 0.8 | Unit |
---|---|---|---|---|---|

${F}_{x,i}$ (1000 At) | 29.46025 | 39.79425 | 40.84454 | 40.8778 | N |

${F}_{y,i}$ (1000 At) | 8.289963 | 4.013222 | 3.328296 | 2.582815 | N |

${T}_{z,i}$ (1000 At) | 860.901 | 884.3387 | 865.9317 | 857.9403 | mNm |

${M}_{r}=1.1,{M}_{rheight}=0.6$ | |||||

Simulation error: ±1 |

Bearingless Motor Preview before First Optimization Steps | Optimized Bearingless Motor Preview | Unit | Increase | |
---|---|---|---|---|

rotor height | 10 | 14 | mm | |

stator height | 14 | 18 | mm | |

${k}_{z}$ | $-7.7$ | $-9.5$ | N/mm | 23% |

${k}_{x}$ | $-38$ | $-62$ | N/mm | 61% |

${k}_{y}$ | $-50$ | $-82$ | N/mm | 64% |

${k}_{\phi x}/{k}_{\phi y}$ | $-180$ | $-190$ | mNm/deg | 5.5% |

${F}_{x,i}$ (1000 At) | 24 | 35 | N | 45% |

${F}_{y,i}$ (1000 At) | 2 | 3 | N | 50% |

${T}_{z,i}$ (1000 At) | 550 | 800 | mNm | 45% |

${m}_{height}=14$ mm | ||||

Simulation error: ±1 |

Parameter | Value before First Optimization Step | Value before First Optimization Step |
---|---|---|

${m}_{height}$ | 10 mm | 14 mm |

${s}_{height}$ | 0 mm | 4 mm |

${r}_{height}$ | 0 mm | 0 mm |

${w}_{lrratio}$ | 1 | 1 |

${w}_{ironratio}$ | 1 | 1.05 |

${w}_{pmratio}$ | 1 | 0.9 |

${M}_{r}$ | 1 | 1.1 |

${M}_{height}$ | 0.5 | 0.6 |

${\beta}_{Ri}$ | 0.3 | 0.4 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Jurdana, V.; Bulic, N.; Gruber, W. Topology Choice and Optimization of a Bearingless Flux-Switching Motor with a Combined Winding Set. *Machines* **2018**, *6*, 57.
https://doi.org/10.3390/machines6040057

**AMA Style**

Jurdana V, Bulic N, Gruber W. Topology Choice and Optimization of a Bearingless Flux-Switching Motor with a Combined Winding Set. *Machines*. 2018; 6(4):57.
https://doi.org/10.3390/machines6040057

**Chicago/Turabian Style**

Jurdana, Vedran, Neven Bulic, and Wolfgang Gruber. 2018. "Topology Choice and Optimization of a Bearingless Flux-Switching Motor with a Combined Winding Set" *Machines* 6, no. 4: 57.
https://doi.org/10.3390/machines6040057