# Optimization of Axial Magnetic Bearing Actuators for Dynamic Performance

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## Abstract

**:**

## 1. Introduction

## 2. Thrust Magnetic Bearing Model and Performance

#### 2.1. Axial Actuator Model

#### 2.2. The Importance of Magnetic Actuator Bandwidth

## 3. Optimization

#### 3.1. Design Constraints

- Outer radius constraint—the outer radius (${r}_{3}$) must not exceed a bound determined either by the available space within the housing or the allowable hoop stress in the thrust disk at maximum speed;
- Inner radius constraint—the inner radius (${r}_{0}$) must be sufficient to accommodate the shaft and allow for assembly;
- Axial length constraint—the total length of the actuator must not exceed a bound determined from the machine design and its rotordynamics;
- Peak force constraint—adequate pole face area must be provided so as to generate the specified maximum force. Furthermore, the cross-sectional area of all segments of the flux path must be greater than or equal to that of the pole face, thus, ensuring that the material’s saturation flux density can be achieved at the pole face;
- Continuous force constraint—the cross-sectional area of the coil must be sufficiently large so as to provide the number of ampere-turns required to generate the specified continuous force without causing coil overheating.

#### 3.2. Optimization Method

#### 3.3. Pareto-Optimal Curves

## 4. Results

#### 4.1. Design Example

#### 4.2. Pareto-Optimal Results

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

## Appendix B

^{2}.

## References

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**Figure 1.**Axisymmetric geometry of thrust disk and stator electromagnet of an axial magnetic bearing. Only one electromagnet of the opposing pair shown (flux-carrying path in iron cross hatched).

**Figure 2.**Loop gain and sensitivity constraints (Equations (9) and (10)) for performance limitations analysis.

**Figure 3.**Pareto-optimal curves showing maximum achievable actuator bandwidth as a function of the maximum allowable disk outer radius for various values of maximum force.

**Figure 4.**Pareto-optimal curves showing maximum achievable actuator bandwidth as a function of maximum force for various values of axial gap.

**Figure 5.**Pareto-optimal curves showing maximum achievable actuator bandwidth as a function of maximum force for various values of the maximum allowable disk outer radius.

**Figure 7.**Pareto-optimal curve showing maximum achievable bandwidth as a function of relative magnetic permeability ${\mu}_{r}$ of the actuator’s ferromagnetic material.

Constraint | Mathematical Expression |
---|---|

Outer radius | ${r}_{3}\le {\rho}_{1}$ |

Inner radius | ${r}_{0}\ge {\rho}_{2}$ |

Axial length | ${d}_{1}+2{d}_{2}+2{d}_{3}\le {\rho}_{3}$ |

Peak force | ${A}_{flux}<\pi \left({r}_{1}^{2}-{r}_{0}^{2}\right)$ ${A}_{flux}<\pi \left({r}_{3}^{2}-{r}_{2}^{2}\right)$ ${A}_{flux}<2\pi {r}_{1}{d}_{1}$ ${A}_{flux}<2\pi {r}_{1}{d}_{3}$ where ${A}_{flux}=\frac{{f}_{peak}{\mu}_{0}}{{B}_{sat}^{2}}$ |

Continuous force | ${A}_{coil}<\left({r}_{2}-{r}_{1}\right){d}_{2}$ where ${A}_{coil}=\left(\frac{1}{\mathsf{\Lambda}}\right)\left\{\left({B}_{bias}{A}_{min}{R}^{0}\right)+\frac{{\mu}_{0}{R}^{0}{f}_{cont}}{2\left(\frac{{A}_{min}}{{A}_{i}}+\frac{{A}_{min}}{{A}_{o}}\right){B}_{bias}}\right\}$ ${A}_{min}=\mathrm{min}\left({A}_{0},{A}_{i}\right)$ ${A}_{i}=\pi \left({r}_{1}^{2}-{r}_{0}^{2}\right)$ ${A}_{o}=\pi \left({r}_{3}^{2}-{r}_{2}^{2}\right)$ |

Parameters | |

Gap (mm) | 1.158 |

Saturation flux density (Tesla) | 1.2 |

Relative permeability | 1000 |

Iron conductivity (MS/m) | 2 |

Maximum coil current density (A/cm^{2}) | 300 |

Constraints | |

Outer radius ${\rho}_{1}$ (mm) | 150 |

Inner radius ${\rho}_{2}$ (mm) | 50 |

Axial length ${\rho}_{3}$ (mm) | not active |

Peak force ${f}_{peak}$ (N) | 6600 |

Continuous force ${f}_{cont}$ (N) | $0.6\text{}{f}_{peak}$ |

**Table 3.**Dimensions and performance characterization variables for axial magnetic actuator designs. Nominal (starting) values and values after optimization.

Dimension | Nominal Value | Optimized Value |

${r}_{0}$ (mm) | 51.7 | 114.7 |

${r}_{1}$ (mm) | 68.3 | 122.5 |

${r}_{2}$ (mm) | 90.9 | 143.8 |

${r}_{3}$ (mm) | 101.3 | 150.0 |

${d}_{1}$ (mm) | 16.5 | 7.5 |

${d}_{2}$ (mm) | 26.3 | 28.0 |

${d}_{3}$ (mm) | 15.2 | 7.5 |

Performance | Nominal Value | Optimized Value |

${\omega}_{-3dB}$ (Hz) | 51.8 | 206.5 |

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**MDPI and ACS Style**

Spece, H.; Fittro, R.; Knospe, C. Optimization of Axial Magnetic Bearing Actuators for Dynamic Performance. *Actuators* **2018**, *7*, 66.
https://doi.org/10.3390/act7040066

**AMA Style**

Spece H, Fittro R, Knospe C. Optimization of Axial Magnetic Bearing Actuators for Dynamic Performance. *Actuators*. 2018; 7(4):66.
https://doi.org/10.3390/act7040066

**Chicago/Turabian Style**

Spece, Henry, Roger Fittro, and Carl Knospe. 2018. "Optimization of Axial Magnetic Bearing Actuators for Dynamic Performance" *Actuators* 7, no. 4: 66.
https://doi.org/10.3390/act7040066