# Investigation of High-Efficiency Wireless Power Transfer Criteria of Resonantly-Coupled Loops and Dipoles through Analysis of the Figure of Merit

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## Abstract

**:**

## 1. Introduction

## 2. Theory

#### 2.1. Loops

#### 2.2. Dipoles

## 3. Results

#### 3.1. Loops

**Table 1.**Maximum figure of merit values, ${U}_{m,max}$, for the parameter sweeps of the loop system. The loop radii, ${a}_{max}$, and resonant frequencies, ${f}_{0,max}$, corresponding to each maximum are also given.

d (m) | 0.01 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | |
---|---|---|---|---|---|---|---|

Coaxial | ${U}_{m,max}$ | 510 | 155 | 77.0 | 43.7 | 26 | 16.3 |

${a}_{max}$ (m) | 0.09 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |

${f}_{0,max}$ (MHz) | 71 | 30 | 30 | 30 | 30 | 30 | |

Coaxial (extended range) | ${U}_{m,max}$ | - | 190 | 141 | 119 | 104 | 95 |

${a}_{max}$ (m) | - | 0.9 | 1.98 | 2.62 | 2.73 | 3.76 | |

${f}_{0,max}$ (MHz) | - | 10 | 5 | 4 | 4 | 3 | |

Parallel | ${U}_{m,max}$ | 47.1 | 14.4 | 7.42 | 4.48 | 2.95 | 2.06 |

${a}_{max}$ (m) | 0.09 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | |

${f}_{0,max}$ (MHz) | 71 | 30 | 30 | 30 | 30 | 30 | |

Parallel (extended range) | ${U}_{m,max}$ | - | 17.6 | 13.0 | 10.96 | 9.68 | 8.80 |

${a}_{max}$ (m) | - | 0.8 | 1.62 | 2.57 | 3.57 | 3.71 | |

${f}_{0,max}$ (MHz) | - | 11 | 6 | 4 | 3 | 3 |

**Figure 3.**Variation of (a) the maximum efficiency resonant frequency ${f}_{0,max}$ with respect to the loop radius a and (b) the variation in the electrical size of the loop ${S}_{0,max}$ when operating at ${f}_{0,max}$ with respect to loop radius a.

**Figure 4.**Variation of the figure of merit ${U}_{m}$ for the loop system with respect to resonant frequency ${f}_{0}$ and loop radius a for (a) $d=0.01$ m, (b) $d=0.1$ m and (c) $d=0.5$ m for the coaxial configuration.

**Figure 5.**Variation of the figure of merit ${U}_{m}$ for the loop system with respect to resonant frequency ${f}_{0}$ and loop radius a for (a) $d=0.01$ m, (b) $d=0.1$ m and (c) $d=0.5$ m for the parallel configuration.

#### 3.2. Dipoles

**Figure 6.**Variation of the figure of merit ${U}_{m}$ for the dipole system with respect to resonant frequency ${f}_{0}$ and dipole length l for (a) $d=0.01$ m, (b) $d=0.1$ m and (c) $d=0.5$ m for the coaxial configuration.

**Figure 7.**Variation of the figure of merit ${U}_{e}$ for the dipole system with respect to resonant frequency ${f}_{0}$ and dipole length l for (a) $d=0.01$ m, (b) $d=0.1$ m and (c) $d=0.5$ m for the parallel configuration.

**Table 2.**Maximum figure of merit values, ${U}_{m,max}$, for the parameter sweeps of the dipole system at $d=0.1-0.5$ m for both coaxial and parallel configurations. The dipole lengths, ${l}_{max}$, and resonant frequencies, ${f}_{0,max}$, at which each maximum occurs are also provided.

d (m) | 0.01 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | |
---|---|---|---|---|---|---|---|

Coaxial | ${U}_{e,max}$ | 6.48 ×10${}^{6}$ | 2.76 × 10${}^{5}$ | 6.87 × 10${}^{4}$ | 3.05 × 10${}^{4}$ | 1.17 ×10${}^{4}$ | 1.00 ×10${}^{4}$ |

${l}_{max}$ (m) | 0.05 | 0.06 | 0.11 | 0.17 | 0.22 | 0.28 | |

${f}_{0,max}$ (MHz) | 1 | 1 | 1 | 1 | 1 | 1 | |

Parallel | ${U}_{e,max}$ | 5.01 × 10${}^{7}$ | 7.56 × 10${}^{5}$ | 1.87 × 10${}^{5}$ | 8.22 × 10${}^{4}$ | 4.34 × 10${}^{4}$ | 2.53 × 10${}^{4}$ |

${l}_{max}$ (m) | 0.05 | 0.18 | 0.36 | 0.5 | 0.5 | 0.5 | |

${f}_{0,max}$ (MHz) | 1 | 1 | 1 | 1 | 1 | 1 | |

Parallel (extended range) | ${U}_{e,max}$ | - | - | - | 8.25 × 10${}^{4}$ | 4.60 × 10${}^{4}$ | 2.91 × 10${}^{4}$ |

${l}_{max}$ (m) | - | - | - | 0.54 | 0.72 | 0.89 | |

${f}_{0,max}$ (MHz) | - | - | - | 1 | 1 | 1 |

#### 3.3. Coupling Coefficients and Q-Factors of Loops and Dipoles

**Figure 8.**Relative difference ${k}_{diff}$ between the coupling coefficients of the loops and dipoles for (a) the parallel dipole and coaxial loop configurations and (b) the coaxial dipole and parallel loop configurations.

#### 3.4. Efficiency

**Figure 10.**Efficiencies for each configuration of the loop system when operating at ${U}_{m,max}$. The efficiencies were computed using the data in Table 1.

**Figure 11.**Matched-load efficiencies of the coaxial and parallel configurations of the loop and dipole systems operating at ${f}_{0}=46.1$ MHz and $l=2a=0.3$ m.

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Moorey, C.; Holderbaum, W.; Potter, B.
Investigation of High-Efficiency Wireless Power Transfer Criteria of Resonantly-Coupled Loops and Dipoles through Analysis of the Figure of Merit. *Energies* **2015**, *8*, 11342-11362.
https://doi.org/10.3390/en81011342

**AMA Style**

Moorey C, Holderbaum W, Potter B.
Investigation of High-Efficiency Wireless Power Transfer Criteria of Resonantly-Coupled Loops and Dipoles through Analysis of the Figure of Merit. *Energies*. 2015; 8(10):11342-11362.
https://doi.org/10.3390/en81011342

**Chicago/Turabian Style**

Moorey, Charles, William Holderbaum, and Ben Potter.
2015. "Investigation of High-Efficiency Wireless Power Transfer Criteria of Resonantly-Coupled Loops and Dipoles through Analysis of the Figure of Merit" *Energies* 8, no. 10: 11342-11362.
https://doi.org/10.3390/en81011342