# Design Optimization and Comparison of Cylindrical Electromagnetic Vibration Energy Harvesters

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. Electromagnetic Energy Harvester Model

#### 2.2. Coil–Magnet Structure

## 3. Design Optimization

#### 3.1. Optimization Method

#### 3.2. Design Parameters

#### 3.3. Implementation Procedure

## 4. Results and Discussions

#### 4.1. Validation

#### 4.2. Power Optimization

#### 4.3. Electromagnetic Coupling Coefficient Maximization

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Equivalent circuit model of an electromagnetic energy harvester, including representation of the mechanical and electrical domain.

**Figure 2.**Basic arrangement of a magnet in-line coil structure with the cylindrical magnet and hollowed, cylindrical coil.

**Figure 3.**Flow chart of an expensive black-box optimization algorithm with surrogate model. Adopted and generalized from [38].

**Figure 4.**Magnetic structure of the investigated cylindrical electromagnetic energy harvesters: (

**a**) Configuration DM1S1C. (

**b**) Configuration H3M1C. (

**c**) Configuration H5M1C. (

**d**) Configuration H5M3C. Right-hand side depicts a 2D representation with relevant dimensions.

**Figure 5.**Exemplified model implementation in ANSYS Maxwell. The models are implemented in 2D and utilize the axisymmetric property of the configurations to result in time efficient simulations.

**Figure 6.**Implementation of the differential equations of the electromagnetic energy harvester for output power calculation.

**Figure 7.**Interaction diagram between the different tools of the multi-tool optimization process. ANSYS Maxwell and MATLAB Simulink are called directly from the Python environment, and thus the user only interacts with Python.

**Figure 8.**The relationship between the flux linkage and the displacement for the H3M1C configuration with parameters listed in Table 4.

**Figure 9.**Best objective function value vs. number of function evaluation for different configurations: (

**a**) Configuration DM1S1C. (

**b**) Configuration H3M1C. (

**c**) Configuration H5M1C. (

**d**) Configuration H5M3C. The objective function is the inverse of the output power.

**Figure 10.**Resulting structures of the four configurations after output power optimization: (

**a**) Configuration DM1S1C. (

**b**) Configuration H31C. (

**c**) Configuration H5M1C. (

**d**) Configuration H5M3C.

**Figure 11.**Resulting structures of the four configurations after electromagnetic coupling maximization: (

**a**) Configuration DM1S1C. (

**b**) Configuration H3M1C. (

**c**) Configuration H5M1C. (

**d**) Configuration H5M3C.

**Table 1.**Constant parameters of the electromagnetic energy harvester based on dimensions and materials according to [33].

Parameters | Values |
---|---|

Constrained volume | 30.15 cm${}^{3}$ |

Constrained height | 24 mm |

Constrained base radius | 20 mm |

Gap between coil and magnet | 2 mm |

Gap between coil and back shield | 2 mm |

Magnet inner radius | 2 mm |

Back shield thickness | 1.5 mm |

Back shield height | 24 mm |

Density of magnet | 7500 kg m${}^{-3}$ |

Density of steel 1010 | 7872 kg m${}^{-3}$ |

Coil wire diameter | 0.1 mm |

Coil fill factor | 0.65 |

Parameters | Values |
---|---|

Excitation amplitude (RMS) | 0.2 g |

Frequency of vibration | 55 Hz |

Mechanical damping ratio | 0.00457 |

**Table 3.**Relevant design parameters of the harvester configurations, as well as their upper and lower bounds.

Design Parameter | Configurations | Lower Boundary | Upper Boundary |
---|---|---|---|

R_{i}/R_{o} | DM1S1C, H3M1C, H5M1C, H5M3C | 0.2 | 0.9 |

h_{spacer}/h | DM1S1C | 0.1 | 0.9 |

h_{coil}/h | DM1S1C, H3M1C, H5M1C | ||

h_{mag_rad}/h | H3M1C | ||

h_{mag_rad_total}/h | H5M1C, H5M3C | ||

h_{coil_total}/h | H5M3C | ||

h_{mag_rad_out}/h_{mag_rad} | H5M1C, H5M3C | 0.1 | 1 |

h_{coil_out}/h_{coil} | H5M3C | ||

g_{coil}/g_{coil_max} | H5M3C |

**Table 4.**Parameters of Halbach array configuration with three magnets and one coil (H3M1C) for verification. Parameters are selected in accordance with [33] to evaluate model implementation.

Parameters | |
---|---|

Magnet outer radius | 12 mm |

Magnet height | 8 mm |

Moving mass | 79.2 g |

Coil inner radius | 14 mm |

Coil outer radius | 16.5 mm |

Coil height | 12 mm |

Coil number of turns | 2483 |

Coil resistance | 510 $\Omega $ |

Remnant flux density | 1.45 T |

Coercive field strength | 976 kA/m |

**Table 5.**Verification results of the non-optimized H3M1C configuration with previously reported results according to [33].

Our Work | Ordoñez et al. [33] | |
---|---|---|

Optimal load resistance ($\mathrm{k}\Omega $) | 34.3 | 37.55 |

Coupling coefficient (Wb m^{−1}) | 91.94 | 95.83 |

Output power (mW) | 23.95 | 23.8 |

Configuration | Configuration | Configuration | Configuration | |
---|---|---|---|---|

DM1S1C | H3M1C | H5M1C | H5M3C | |

R_{i}/R_{o} | 0.899 | 0.899 | 0.899 | 0.899 |

h_{coil}/h | 0.541 | 0.46 | 0.437 | - |

h_{coil_total}/h | - | - | - | 0.614 |

h_{coil_out}/h_{coil} | - | - | - | 0.16 |

h_{spacer}/h | 0.364 | - | - | - |

h_{mag_rad}/h | - | 0.245 | - | - |

h_{mag_rad_total}/h | - | - | 0.261 | 0.311 |

h_{mag_rad_out}/h_{mag_rad} | - | - | 0.1 | 0.14 |

g_{coil}/g_{coil_max} | - | - | - | 0.99 |

w_{mag} (mm) | 11.0355 | 11.0355 | 11.0355 | 11.0355 |

h_{spacer} (mm) | 8.736 | - | - | - |

h_{mag_rad} (mm) | - | 5.88 | 5.22 | 5.83 |

h_{mag_rad_out} (mm) | - | - | 0.522 | 0.816 |

h_{mag_axial} (mm) | 7.632 | 9.06 | 8.868 | 8.269 |

w_{coil} (mm) | 1.46 | 1.46 | 1.46 | 1.46 |

h_{coil} (mm) | 12.984 | 11.04 | 10.488 | 11.16 |

h_{coil_out} (mm) | - | - | - | 1.78 |

g_{coil} (mm) | - | - | - | 4.58 |

N_{coil_turn} | 1573 | 1338 | 1271 | 1353 |

R_{coil} ($\Omega $) | 333 | 284 | 269 | 296 |

N_{coil_out_turn} | - | - | - | 216 |

R_{coil_out} ($\Omega $) | - | - | - | 41 |

R_{load_opt} ($\mathrm{k}\Omega $) | 13.2 | 14.9 | 13.7 | 17.6 |

Configuration | Configuration | Configuration | Configuration | |
---|---|---|---|---|

DM1S1C | H3M1C | H5M1C | H5M3C | |

Output power (mW) | 28.62 | 28.3 | 28.28 | 28.278 |

Output voltage (V) | 19.4 | 20.55 | 19.7 | 22.26 |

Volume power density (mW cm^{−3}) | 0.949 | 0.939 | 0.938 | 0.938 |

Mass power density (mW g^{−1}) | 0.299 | 0.3018 | 0.3015 | 0.3015 |

**Table 8.**Design and performance parameters of the four configurations after optimization for maximum electromagnetic coupling.

Configuration | Configuration | Configuration | Configuration | |
---|---|---|---|---|

DM1S1C | H3M1C | H5M1C | H5M3C | |

R_{i}/R_{o} | 0.7 | 0.69 | 0.68 | 0.734 |

h_{coil}/h | 0.69 | 0.7 | 0.64 | - |

h_{coil_total}/h | - | - | - | 0.887 |

h_{coil_out}/h_{coil} | - | - | - | 0.285 |

h_{spacer}/h | 0.106 | - | - | - |

h_{mag_rad}/h | - | 0.26 | - | - |

h_{mag_rad_total}/h | - | - | 0.187 | 0.254 |

h_{mag_rad_out}/h_{mag_rad} | - | - | 0.751 | 0.79 |

g_{coil}/g_{coil_max} | - | - | - | 0.956 |

w_{mag} (mm) | 8.15 | 8 | 7.86 | 8.64 |

h_{spacer} (mm) | 2.54 | - | - | - |

h_{mag_rad} (mm) | - | 6.24 | 1.79 | 2.36 |

h_{mag_rad_out} (mm) | - | - | 1.35 | 1.86 |

h_{mag_axial} (mm) | 10.73 | 8.88 | 9.75 | 8.96 |

w_{coil} (mm) | 4.35 | 4.5 | 4.64 | 3.86 |

h_{coil} (mm) | 16.56 | 16.8 | 15.36 | 13.56 |

h_{coil_out} (mm) | - | - | - | 3.86 |

g_{coil} (mm) | - | - | - | 1.3 |

N_{coil_turn} | 5962 | 6250 | 5898 | 4328 |

R_{coil} ($\Omega $) | 1148 | 1197 | 1124 | 847.6 |

N_{coil_out_turn} | - | - | - | 1233 |

R_{coil_out} ($\Omega $) | - | - | - | 241.6 |

R_{load_opt} ($\mathrm{k}\Omega $) | 74.2 | 81.5 | 63.6 | 81.68 |

Coupling coefficient (Wb m^{−1}) | 115.07 | 117.65 | 100.15 | 125 |

Output voltage (V) | 35.5 | 36.6 | 31.78 | 39 |

Output power (mW) | 16.82 | 16.74 | 15.89 | 18.65 |

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**MDPI and ACS Style**

Phan, T.N.; Aranda, J.J.; Oelmann, B.; Bader, S.
Design Optimization and Comparison of Cylindrical Electromagnetic Vibration Energy Harvesters. *Sensors* **2021**, *21*, 7985.
https://doi.org/10.3390/s21237985

**AMA Style**

Phan TN, Aranda JJ, Oelmann B, Bader S.
Design Optimization and Comparison of Cylindrical Electromagnetic Vibration Energy Harvesters. *Sensors*. 2021; 21(23):7985.
https://doi.org/10.3390/s21237985

**Chicago/Turabian Style**

Phan, Tra Nguyen, Jesus Javier Aranda, Bengt Oelmann, and Sebastian Bader.
2021. "Design Optimization and Comparison of Cylindrical Electromagnetic Vibration Energy Harvesters" *Sensors* 21, no. 23: 7985.
https://doi.org/10.3390/s21237985