## 2 Proposed WPT System with the MMs
### Analysis and Design of the MMs  
It is generally known that the MMs with negative permeability have the supernatural ability that changes the direction of the magnetic propagation and contributes to distance enhancement and efficiency improvement. Reference [20; 21; 22] used the numerical models and analytical theory to explain the principle of MM's focusing property. According to Maxwell equations:  
\[\left\{ \begin{array}{l}k\cdot E=0\\ k\cdot H=0\\ k\times E=\mu\omega H\\ k\times H=-\varepsilon\omega E\end{array} \right. \tag{8}\]  
where \(w\) is the plane electromagnetic wave frequency, \(k\) is wave vector, \(\mu\) is the dielectric permeability, \(\varepsilon\) is the dielectric constant, \(E\) is the electric field intensity, and \(H\) is the magnetic field intensity. According to the formula, \(n^{2}=\mu_{r}\varepsilon_{r}\), when \(k\), \(E\), and \(H\) form the left hand system, and the refractive index, \(n\), is negative. When the electromagnetic wave enters the surface of the medium, according to [PERSON]'s law:  
\[\frac{\tan\theta_{1}}{\tan\theta_{2}}=\frac{\mu_{r1}}{\mu_{r2}} \tag{9}\]  
\(\theta_{1}\) and \(\theta_{2}\) are the angle of incidence and refraction. As shown in Figure 6, the refraction occurs when electromagnetic waves pass through two dielectric interfaces with opposite chirality. Figure 6c manifests that the reflected wave and the incident wave are in the same direction when the relative permeability is negative. If the magnetic field can be restrained effectively and controlled artificially to solve the problem of rapid divergence and attenuation of the magnetic field, the transmission distance and efficiency of the magnetic resonance wireless transfer system can be improved effectively [18]. To achieve the desired MMs slab, the HFSS (High Frequency Simulator Structure) software was adopted to simulate the MM structure. The 5-parameter can be obtained and finally converted into the relationship between the relative permeability and frequency [25; 26; 27]. To make the MM unit cell elastic and compact, the proposed MM unit cell was designed to be square with a length of 120 mm and a series capacitance of 2.27 nF was welded on both ends of the copper strip to tune the resonant frequency of the MM unit cell. The structure diagram of the MM unit cell is shown in Figure 7a. The substrate was made of FR-4 whose relative dielectric constant was \(\varepsilon\tau\) = 4.4 and the loss tangent was tan\(\delta\) = 0.025 and its thickness was 2 mm. The extracted relative permeability curve depending on the frequency of the MM unit cell is illustrated in Figure 7b. The structure parameters of the MM unit cell are listed in Table 1.  
Figure 5: The efficiency as a function of the \(k_{12}\) and the \(k_{23}\). To optimize the MM unit cell design, a variety of parameter structures were compared for resonance frequency control, as shown in Figure 8. When the width, the interval and the turn of the copper strip, and the thickness of the substrate varied, the resonance frequency of the MM unit cell changed correspondingly. It can be seen that the width and interval had no obvious influence on the resonance frequency, whereas the change of the turn made the resonance frequency deviate largely. The main reason was that the increase of the turn number made the inductance increase more distinct. In addition, the thickness of the substrate had less impact for the resonance frequency. In this paper, the MM unit cell adopting the above structure parameters was a comprehensive consideration of size, flexibility, and firmness in fabrication. To accommodate the coil size, the MMs slab consisted of a \(5\times 5\) unit cell, and the diameter of the \(Tx\) (\(Rx\)) coil was 500 mm. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Parameter** & **Value** \\ \hline S & Length of side (mm) & 120 \\ g & Interval (mm) & 3 \\ w & Width (mm) & 4 \\ N & Turn & 3 \\ T & Thickness of substrate & 2 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The structure parameters of the MM unit cell.