# Magnetic Performance of Eddy Current Suppressing Structures in Additive Manufacturing

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Eddy Current Losses in Sheets

## 3. Space-Filling Curves

#### Comparison with Sheets

## 4. Fill Factor Comparison

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

AM | Additive Manufacturing |

CAD | Computer-Aided Design |

FE | Finite-Element |

SFC | Space-Filling Curve |

## References

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**Figure 1.**SFCs with their starting curve and two iterations of refinement each.

**Top**(

**a**–

**c**): Hilbert curve.

**Bottom**(

**d**–

**f**): Peano curve.

**Figure 2.**Schema of eddy currents ${I}_{eddy}$ (red) within the cross section of a beam due to the magnetic field B (cyan).

**Figure 3.**Model used for FE simulation: A cube made of soft-magnetic material (cyan), surrounded by a coil (brown) driven to achieve the peak flux density ${B}_{p}$, as well as air (gray) outside the coil and between the coil and the soft-magnetic material. A symmetry condition is applied to the red surface.

**Figure 4.**Flux density over the cross section of the soft-magnetic cube consisting of sheets with varying thickness d. (

**a**): $d=1\phantom{\rule{0.277778em}{0ex}}\mathrm{mm}$. (

**b**): $d=0.5\phantom{\rule{0.277778em}{0ex}}\mathrm{mm}$. (

**c**): $d=0.25\phantom{\rule{0.277778em}{0ex}}\mathrm{mm}$. (

**d**): $d=0.1\phantom{\rule{0.277778em}{0ex}}\mathrm{mm}$.

**Figure 5.**Current density over the cross section of the soft-magnetic cube consisting of sheets with varying thickness d. The size of the arrows is proportional to the in-plane eddy current magnitude. (

**a**): $d=1\phantom{\rule{0.277778em}{0ex}}\mathrm{mm}$. (

**b**): $d=0.5\phantom{\rule{0.277778em}{0ex}}\mathrm{mm}$. (

**c**): $d=0.25\phantom{\rule{0.277778em}{0ex}}\mathrm{mm}$. (

**d**): $d=0.1\phantom{\rule{0.277778em}{0ex}}\mathrm{mm}$.

**Figure 6.**Thickened SFCs and their resulting cross sections. The nominal structural width w, gap width ${w}_{g}$ and maximum structure width ${w}_{max}$ of the Hilbert-based structure are shown exemplarily. (

**a**,

**d**): Hilbert curve. (

**b**,

**e**): Peano curve. (

**c**,

**f**): Sierpiński curve.

**Figure 7.**Cross sections of the first, second and third order o of the Hilbert curve using the gap width ${w}_{g}=50\phantom{\rule{0.277778em}{0ex}}\mathsf{\mu}\mathrm{m}$ and nominal structural width $w\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.5\phantom{\rule{0.277778em}{0ex}}\mathrm{mm}$.

**Figure 8.**Flux density over the cross section of the soft-magnetic cube with structures based on SFCs. (

**a**): Third order Hilbert. (

**b**): Second order Peano.

**Figure 9.**Current density over the cross section of the soft-magnetic cube with structures based on SFCs. (

**a**): Third order Hilbert. (

**b**): Second order Peano.

**Figure 10.**Third order Hilbert curve (black) with length ${l}_{SFC}$ and proposed curve including arcs (red) used to estimate the effective length ${l}_{SFC}^{*}$ for calculating eddy currents.

**Figure 11.**Ratios of loss densities obtained by means of FE analysis and (3) depending on the ratio of the mean structure width $\overline{w}$ to the length of the SFC ${l}_{SFC}^{*}$. The lines are fitted functions excluding the data points exhibiting a significant skin effect.

**Figure 12.**Illustration of the modelling approach of the second order Hilbert SFC (

**a**) and the respective packaged sheets (

**b**) using squares (green) and rectangular connections (red) between them.

**Figure 14.**Fill factor ${F}_{sheets}$ of laminated sheets depending on sheet thickness d for a number of gap widths ${w}_{g}$. The currently available AM-based manufacturing methods are represented by the green line, ${w}_{g}=50\phantom{\rule{0.277778em}{0ex}}\mathsf{\mu}\mathrm{m}$, while conventional manufacturing methods are characterized by the blue line, ${w}_{g}=10\phantom{\rule{0.277778em}{0ex}}\mathsf{\mu}\mathrm{m}$.

**Figure 15.**Fill factors of SFCs for different orders o and AM-built sheets of thickness $d=w=0.5\phantom{\rule{0.277778em}{0ex}}\mathrm{mm}$ with a gap of ${w}_{g}=50\phantom{\rule{0.277778em}{0ex}}\mathsf{\mu}\mathrm{m}$. (

**Left**) Hilbert curve, (28), and AM-built sheets, (29). (

**Right**) Peano curve, (31), and AM-built sheets, (32).

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

Frequency | f | $10\phantom{\rule{0.277778em}{0ex}}\mathrm{Hz}$ |

Peak Flux Density | ${B}_{p}$ | $1\phantom{\rule{0.277778em}{0ex}}\mathrm{T}$ |

Electrical Resistivity | ${\rho}_{el}$ | $89.29\phantom{\rule{0.277778em}{0ex}}\mathrm{n}\mathsf{\Omega}\mathrm{m}$ |

Relative Magnetic Permeability | ${\mu}_{r}$ | 4000 |

Volumetric Mass Density | ${\rho}_{m}$ | $7870\phantom{\rule{0.277778em}{0ex}}\frac{\mathrm{kg}}{{\mathrm{m}}^{3}}$ |

Cube Side Length | l | $1\phantom{\rule{0.277778em}{0ex}}\mathrm{mm}$ |

d in mm | Specific Power Losses P in $\frac{\mathbf{mW}}{\mathbf{kg}}$ | ||
---|---|---|---|

Analytical | 2D Analytical | FE Simulation | |

1 | 234.09 | 117.05 | 97.84 |

0.5 | 58.52 | 46.82 | 40.01 |

0.25 | 14.63 | 13.77 | 12.33 |

0.1 | 2.34 | 2.32 | 2.20 |

SFC & Order | Specific Losses P in $\frac{\mathbf{mW}}{\mathbf{kg}}$ | ||
---|---|---|---|

FE Simulation | Analytical $\mathit{w}$ | Analytical $\overline{\mathit{w}}$ | |

Hilbert 1 | 56.22 | 55.52 | 66.73 |

Hilbert 2 | 66.16 | 58.33 | 79.97 |

Hilbert 3 | 71.64 | 58.51 | 81.47 |

Peano 1 | 60.44 | 57.92 | 69.26 |

Peano 2 | 64.41 | 58.52 | 71.67 |

Peano 3 | 64.97 | 58.52 | 71.84 |

**Table 4.**Specific eddy current losses obtained analytically and via FE simulation for a number of different SFCs and sheets of thickness $d=w$.

SFC & Order | Specific Losses P in $\frac{\mathbf{mW}}{\mathbf{kg}}$ | |||
---|---|---|---|---|

Analytical | FE Simulation | |||

SFC | Sheets | SFC | Sheets | |

Hilbert 1 | 66.73 | 47.71 | 56.21 | 40.97 |

Hilbert 2 | 79.97 | 55.52 | 66.16 | 49.89 |

Hilbert 3 | 81.47 | 57.76 | 71.64 | 54.22 |

Peano 1 | 69.26 | 53.32 | 60.44 | 46.96 |

Peano 2 | 71.67 | 57.92 | 64.41 | 54.70 |

Peano 3 | 71.84 | 58.46 | 64.97 | 57.32 |

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**MDPI and ACS Style**

Klein, C.; May, C.; Nienhaus, M.
Magnetic Performance of Eddy Current Suppressing Structures in Additive Manufacturing. *Actuators* **2024**, *13*, 94.
https://doi.org/10.3390/act13030094

**AMA Style**

Klein C, May C, Nienhaus M.
Magnetic Performance of Eddy Current Suppressing Structures in Additive Manufacturing. *Actuators*. 2024; 13(3):94.
https://doi.org/10.3390/act13030094

**Chicago/Turabian Style**

Klein, Carsten, Christopher May, and Matthias Nienhaus.
2024. "Magnetic Performance of Eddy Current Suppressing Structures in Additive Manufacturing" *Actuators* 13, no. 3: 94.
https://doi.org/10.3390/act13030094