# Practical Model for Metamaterials in Wireless Power Transfer Systems
## 3 Experimental Verification
### Permeability Measurement  
In order to get the characteristics of the MTM slab, the relative permeability of the fabricated MTM is measured. Firstly, the genetic algorithm with the optimization goal and the restrictions presented in chapter 2 is used to find the optimal efficiency of the WPT system. The parameters of the system are shown in Table 1. The optimized result is \(f_{M}=6.5\) MHz. By tuning the capacitor in MTMs, the resonance frequency of the MTM slab (\(\omega_{M}\)) is changing as well. It is important to decide the lumped capacitance of each unit in the MTM slab according to the optimal resonance frequency. However, since \(\omega_{M}\) is influenced by the surrounding units, it is complicated to obtain \(\omega_{M}\) by calculation, and \(\omega_{M}\) is also difficult to be tested in experiments. Therefore, an experiment is designed to imitate the periodic boundaries around an MTM unit. A lumped port is located at the interface of the metal spiral, and 4 PEC boundaries are added around an MTM unit shown in Figure 8a, which corresponds to the simulation setting as presented in chapter 2. According to [40], a piece of metal foil is used as the PEC boundary which surrounds an MTM unit to form a cubic test box. As shown in Figure 8b, port 1 of a Vector Network Analyzer is connected to the lumped port of the MTM unit for excitation and testing.  
The Y-parameters is obtained after adding a 135 pF lumped capacitor in the proposed MTM unit, and is showed in Figure 9. It is obvious that the MTM slab resonates at 6.49 MHz with the lumped capacitor, whereas the MTM unit without boundaries resonates at 6.16 MHz. It can be seen that the resonance frequency predicted by the experimental method is in good agreement with the simulation result. Since the utilization of copper foil introduces power loss, the quality factor obtained by the experiment has a deviation from the simulation result within tolerance.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** & **Parameter** & **Value** \\ \hline \(R_{1}\) & \(2.5\,\Omega\) & \(L_{1}\) & \(4.7\,\mathrm{\SIUnitSymbolMicro H}\) & \(C_{1}\) & 132 pF \\ \(R_{3}\) & \(5\,\Omega\) & \(L_{3}\) & \(26.7\,\mathrm{\SIUnitSymbolMicro H}\) & \(C_{3}\) & 20 pF \\ \(R_{8}\) & \(5\,\Omega\) & \(R_{\mathrm{Leq}}\) & \(50\,\Omega\) & \(Q_{2}\) & 41.66 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of the WPT system.  
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.  
Once the \(\omega_{M}\) and \(Q_{M}\) has been obtained, the relative permeability of the MTM slab is calculated and shown in Figure 10. The MTM slab resonates at 6.49 MHz, and the real part of relative permeability is negative at 6.78 MHz. Moreover, because of the power loss from copper foil in the experiment, the imaginary part (represents the loss of MTMs) of the experimental results is slightly bigger than the simulation results. This result is also consistent with the formula \(\omega=\sqrt{LC}\). According to the experiment and simulation results, the effective inductance of an MTM unit is 4.9 \(\mu\)H. Because of the interaction between MTM units, the resonance frequency of the MTM slab is difficult to decide, so experimental verification is essential.