# Practical Model for Metamaterials in Wireless Power Transfer Systems
## 2 Practical Model for Low-Frequency Metamaterials
### Analysis of MTM-Enhanced WPT System  
A practical model for WPT systems with MTM slabs is proposed and shown in Figure 4. The model of the MTM slab is established using lumped circuit parameters, in which \(R_{\mathrm{L}}\) is the resistance of the load and \(\omega\) is the operational angular frequency.  
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 \(f_{\mathrm{m}}\), and get the high limit frequency \(f_{\mathrm{h}}\) and low limit frequency \(f_{\mathrm{i}}\) when the amplitude of Y-parameter drops to 0.707 of itself, then \(Q=f_{\mathrm{m}}/(f_{\mathrm{h}}-f_{\mathrm{i}})\).  
Figure 2: The setting of the designed MTM slab with four perfect electronic conductor (PEC) boundaries.  
According to Kirchhoff's Law, the circuit equation is given by (3), where \(\omega_{0}\) is the operation frequency of the WPT system, \(\omega_{1}\) is the resonance frequency of both transmission and receiving coils, and \(\omega_{M}\) is the resonance frequency of MTMs. Assuming \(\omega_{1}=\omega_{0}\), the system efficiency \(\eta\) can be calculated by using (4). Substituting (3) into (4), one finds that the system efficiency is related to \(\omega_{M}\) and \(Q_{M}\). In other words, \(\omega_{M}\) and \(Q_{M}\) are the key parameters influencing the efficiency of such MTM-enhanced WPT systems.  
\[\left[\begin{array}{ccc}-R_{1s}&0&0\\ 0&-R_{M}&0\\ 0&0&-R_{3L}\\ \end{array}\right]+\left[\begin{array}{ccc}I_{1}\\ \omega_{0}I_{1}\\ I_{M}\\ I_{3}\\ \end{array}\right]+\left[\begin{array}{ccc}-L_{1}\omega_{1}\big{(}\frac{\omega _{0}}{\omega_{1}}-\frac{\omega_{1}}{\omega_{0}}\big{)}&\omega_{0}M_{1M}&\omega _{0}M_{13}\\ \omega_{0}M_{1M}&-L_{M}\omega\big{(}\frac{\omega_{0}}{\omega_{M}}-\frac{ \omega_{M}}{\omega_{0}}\big{)}&\omega_{0}M_{M3}\\ \omega_{0}M_{13}&\omega_{0}M_{M3}&-L_{3}\omega_{1}\big{(}\frac{\omega_{0}}{ \omega_{1}}-\frac{\omega_{1}}{\omega_{0}}\big{)}\\ \end{array}\right]\left[\begin{array}{c}I_{1}\\ I_{M}\\ I_{3}\\ \end{array}\right]=\left[\begin{array}{c}U_{s}\\ 0\\ 0\\ \end{array}\right] \tag{3}\]  
\[\eta(\omega_{M},Q_{M})=\frac{P_{out}}{P_{in}}=\frac{I_{3}^{2}R_{\rm L}}{I_{1} U_{s}}. \tag{4}\]  
In order to obtain the maximum power efficiency of MTM-enhanced WPT systems, we designed an optimization algorithm by turning \(\omega_{M}\). \(Q_{M}\) varies only \(\omega_{M}\) because \(L_{\rm M}\) and \(R_{\rm M}\) 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:  
\[\begin{array}{ccc}\text{Max}&\eta\\ \text{s.t.}&\text{Re}(\mu_{\rm r})\leq a\\ &\text{Im}(\mu_{\rm r})\leq b\\ \end{array}, \tag{5}\]  
where \(a\) and \(b\) are the controllable limitations. The optimal solution can be acquired using optimization algorithms such as genetic algorithm (GA) or particle swarm optimization (PSO) to solve this problem. After calculation, we obtain the optimal \(\omega_{M}\) which can be tuned by adjusting the additional capacitor in the MTM slab, and then the efficiency of the WPT system with the MTM slab reaches its maximum value. It should be pointed out that, the optimization result is dependent on the setting of limitations. In particular, the parameter b that represents the loss of MTMs needs to be practical, so the results can be verified by experiments.