# A Comparison of Methods to Measure the Coupling Coefficient of Electromagnetic Vibration Energy Harvesters

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## Abstract

**:**

## 1. Introduction

## 2. Electromagnetic Coupling

#### 2.1. Theory

#### 2.2. Four Methods of Measuring the Electromagnetic Coupling Coefficient

## 3. Experimental and Simulative Validation

#### 3.1. Energy Harvester Implementation

#### 3.2. Finite Element Simulation

#### 3.3. Measurement Details

#### 3.4. Propagation of Uncertainty

## 4. Results and Discussion

#### 4.1. Damping Influence and Optimum Load

#### 4.2. Measuring ${K}_{\mathrm{i}}$

#### 4.3. Comparison and Discussion

## 5. Summary

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

**Figure A1.**Cross-section of an air coil. The grey area is filled with windings (some wire cross-sections are shown as examples).

## References

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**Figure 2.**(

**a**) Model of a magnetic circuit with coil (copper), magnets (grey), and iron (dark). (

**b**) FE simulation result of the magnetic flux with Ansys Electromagnetics.

**Figure 3.**Measured resonance curve of the induced voltage in an electromagnetic energy harvester and best fit by a curve of the type given in Equation (6) for setup 1L from Section 3.3 below.

**Figure 4.**Photograph of the cantilever energy harvester used for experimental tests. The reader sees the clamped copper beam, two magnets with golden coating, the iron legs of the magnetic circuit, and the copper coil in the gap between the magnets (connected to the base).

**Figure 5.**Measured load-resistance influence in harvester setup 1L. (

**a**) Voltage amplitude, electrical damping factor, and maximum load power as functions of the load resistance. All quantities have been normalized for unity. The voltage uncertainties are negligible, and therefore have not been indicated. (

**b**) Frequency dependence of the power delivered to the load for different load resistances. The optimum load resistance for harvester excitement at its resonance frequency of about 51 Hz is 8 kΩ.

**Figure 6.**Identification of the coupling coefficient by way of a magnetic-force measurement: (

**a**) Magnetic force for setup 2 as a function of the coil current and identified coupling coefficient K

_{i}; (

**b**) results for K

_{i}in five subsequent measurements for every setup. The sample uncertainty (red error bar) is much bigger for small clamping lengths, which indicates effects not included in the single-measurement uncertainty Equation (10).

**Figure 7.**Comparison of the simulated and measured coupling coefficients of five harvester implementations and the measured and predicted optimum load resistance.

Setup No. | Description | ℓ/mm | f_{r}/Hz | N | D_{w}/µm | R_{C}/Ω | â/m/s^{2} |
---|---|---|---|---|---|---|---|

1L | Coil 1, long | 27 | 51.2 | 1300 | 50 | 226 | 0.75 |

1M | Coil 1, medium | 22 | 65.3 | 1300 | 50 | 226 | 1 |

1S | Coil 1, short | 18 | 81.2 | 1300 | 50 | 226 | 1 |

2 | Coil 2 | 27 | 51.2 | 2115 | 40 | 880 | 0.75 |

3 | Coil 3 | 27 | 51.2 | 3620 | 30 | 1707 | 0.75 |

**Table 2.**Measured and calculated optimum load resistances and measured maximum load power at an acceleration amplitude of â = 0.75 m/s

^{2}. Setups 1M and 1S were characterized at higher amplitudes (â = 1 m/s²), but the results were scaled to â = 0.75 m/s

^{2}.

Setup No. | R_{C}/kΩ | R_{L,opt}/kΩ | R_{L,opt,c}/kΩ | P_{max}/µW |
---|---|---|---|---|

1L | 0.23 | 8 | 7.1 | 257 |

1M | 0.23 | 6 | 5.9 | 179 |

1S | 0.23 | 5 | 5.2 | 94 |

2 | 0.88 | 18 | 19.6 | 223 |

3 | 1.7 | 60 | 60.6 | 253 |

**Table 3.**Simulated and measured coupling coefficients. The data are visualized in Figure 7.

Setup No. | Coupling Coefficient in Wb/m | ||||
---|---|---|---|---|---|

K_{sim} | K_{oc} | K_{sc} | K_{R} | K_{i} | |

1L | 7.1 | 7.1 | 7.3 | 7.6 | 7.2 |

1M | 7.1 | 6.8 | 7.1 | 7.2 | 7.4 |

1S | 7.1 | 6.9 | 6.6 | 7.0 | 7.4 |

2 | 11.5 | 11.0 | 11.3 | 11.0 | 12.0 |

3 | 19.7 | 19.8 | 20.2 | 19.6 | 20.2 |

© 2019 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/).

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**MDPI and ACS Style**

Mösch, M.; Fischerauer, G.
A Comparison of Methods to Measure the Coupling Coefficient of Electromagnetic Vibration Energy Harvesters. *Micromachines* **2019**, *10*, 826.
https://doi.org/10.3390/mi10120826

**AMA Style**

Mösch M, Fischerauer G.
A Comparison of Methods to Measure the Coupling Coefficient of Electromagnetic Vibration Energy Harvesters. *Micromachines*. 2019; 10(12):826.
https://doi.org/10.3390/mi10120826

**Chicago/Turabian Style**

Mösch, Mario, and Gerhard Fischerauer.
2019. "A Comparison of Methods to Measure the Coupling Coefficient of Electromagnetic Vibration Energy Harvesters" *Micromachines* 10, no. 12: 826.
https://doi.org/10.3390/mi10120826