# Practical Model for Metamaterials in Wireless Power Transfer Systems

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Practical Model for Low-Frequency Metamaterials

#### 2.1. Lumped Impedance Parameters of the MTM Unit

#### 2.2. Q-Based Design Theory in the MTM Slab

#### 2.3. Analysis of MTM-Enhanced WPT System

_{L}is the resistance of the load and $\omega $ is the operational angular frequency.

_{M}and R

_{M}are fixed when the shape of the MTM unit is determined. As for the relative permeability of the MTM slab, it is exactly related to ${\omega}_{M}$ and ${Q}_{M}$ when the parameters of the WPT system is settled. Therefore, the optimization goal and restrictions are as follows:

## 3. Experimental Verification

#### 3.1. Permeability Measurement

#### 3.2. Experimental Verification of the MTM-Enhanced WPT System

_{21}|

^{2}(S

_{21}is the positive transfer factor. At radio frequency, |S

_{21}|

^{2}is usually used to represent the positive energy transfer.), and the results from the proposed model are calculated by (4). It is clearly shown that the results from the proposed practical model have much-improved accuracy than those obtained from the S-parameter method. Furthermore, the S-parameter-based simulation has rather high requirements on the computer performance and is extremely time-consuming.

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The Swiss roll structure of the metamaterial (MTM) unit. Two reverse spirals on each side of Fr4 substrate.

**Figure 2.**The setting of the designed MTM slab with four perfect electronic conductor (PEC) boundaries.

**Figure 3.**The admittance parameter (Y-parameter) of an MTM slab. The quality factor Q can be calculated from the graph. Take the resonance frequency f

_{m}, and get the high limit frequency f

_{h}and low limit frequency f

_{l}when the amplitude of Y-parameter drops to 0.707 of itself, then $Q={f}_{\mathrm{m}}/\left({f}_{\mathrm{h}}-{f}_{\mathrm{l}}\right)$.

**Figure 4.**The proposed model for the MTM-enhanced wireless power transfer (WPT) system. M

_{ij}is the mutual inductance between coils and MTMs.

**Figure 8.**The proposed measurement setup (

**a**) The PEC boundaries are made of metal foil with an MTM unit. (

**b**) The MTM unit is added inside the boundaries. (

**c**) Port 1 connects to the vector network analyzer to get the impedance parameters.

**Figure 10.**The relative permeability of the experimental MTM slab. (

**a**) The experimental results. (

**b**) The simulation results.

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

R_{1} | 2.5 $\mathsf{\Omega}$ | L_{1} | 4.7 $\mathsf{\mu}\mathrm{H}$ | C_{1} | 132 pF |

R_{3} | 5 $\mathsf{\Omega}$ | L_{3} | 26.7 $\mathsf{\mu}\mathrm{H}$ | C_{3} | 20 pF |

R_{s} | 5 $\mathsf{\Omega}$ | R_{Leq} | 50 $\mathsf{\Omega}$ | Q_{2} | 41.66 |

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**MDPI and ACS Style**

Liu, J.; Gong, Z.; Yang, S.; Sun, H.; Zhou, J.
Practical Model for Metamaterials in Wireless Power Transfer Systems. *Appl. Sci.* **2020**, *10*, 8506.
https://doi.org/10.3390/app10238506

**AMA Style**

Liu J, Gong Z, Yang S, Sun H, Zhou J.
Practical Model for Metamaterials in Wireless Power Transfer Systems. *Applied Sciences*. 2020; 10(23):8506.
https://doi.org/10.3390/app10238506

**Chicago/Turabian Style**

Liu, Jingying, Zhi Gong, Shiyou Yang, Hui Sun, and Jing Zhou.
2020. "Practical Model for Metamaterials in Wireless Power Transfer Systems" *Applied Sciences* 10, no. 23: 8506.
https://doi.org/10.3390/app10238506