# Practical Model for Metamaterials in Wireless Power Transfer Systems
## 1 Introduction  
Metamaterials (MTMs) are novel artificial media consisting of a large number of microscopic unit cells and exhibiting negative permeability and/or negative permittivity on some frequencies [1; 2; 3; 4; 5; 6]. Since the first experimental demonstration of a negative index of refraction [7], numerous academic achievements on MTMs have been reported including transforming optics [8], invisibility cloaks [9], and novel MTM-based antennas [10]. These researches focus on high-frequency electromagnetics or optics. Recently, MTMs have been introduced to low-frequency near field applications such as wireless power transfer (WPT) and magnetic resonance imaging (MRI) [11; 12; 13; 14; 15; 16; 17; 18; 19; 20]. Unlike those in high-frequency electromagnetics, the electric and magnetic fields are almost decoupled in the near field, and thus the near field can be manipulated by single-negative MTMs [21]. The MTM-enhanced WPT system is one of the major applications of low-frequency MTMs in the near field realm. WPT attracts great attention due to its capability of feeding power easily to autonomous electronic devices, such as mobile phones, wearable products, implantable medical devices, and electric vehicles [22; 23; 24; 25]. However, the engineering applications of the WPT systems are usually restricted by the short transfer distance and the low transmission efficiency [26]. MTM slabs are introduced in the WPT systems to enhance the evanescent wave and thus improve the power transmission efficiency [11]. Theoretical analysis and simulations have shown that the power transferefficiency with MTMs can be improved by an order when the loss is limited [12]. Moreover, it has been experimentally confirmed that the MTM slabs enhance the WPT system efficiency by about 30%, and the location, as well as the quantity of the slabs, also has a great effect on the system performance [13; 17; 18; 19]. In low-frequency applications, researchers have made solid contributions to the analytical work of MTMs. In previous studies, MTMs have been considered as waveguides and used for the near field imaging at 21.3 MHz [27]. Furthermore, it has been shown that the properties of an MTM unit could be described by an equivalent circuit consisting of bulk and distributed elements [28], which provides suitable theoretical guidance for the MTMs in the low-frequency near-field. In [29], the near-field manipulating device was designed utilizing biatomic MTM structure, both simulations and experiments were carried out afterward to verify the analytical predictions at 45.8 MHz. Although some issues have been discussed and concluded in [27; 28; 29], the applications of MTMs in low-frequency near field systems are still faced up with many difficulties. Firstly, investigations on the characteristics of MTMs is restricted in the lower frequency (usually a few MHz) and near field range. Up to now, the equivalent circuit model has not been set up for the MTM-enhanced WPT systems. The theoretical models and parameters (for example, S-parameters) utilized in high-frequency scenarios were usually applied to low-frequency systems directly. Moreover, the measurement of the transmission efficiency is not intuitive, and the corresponding retrieval algorithm [30; 31] is complicated, making the calculation awkward and unpractical. Finally, in order to make the MTM slab working at negative permeability, the resonance frequency of the slab should be slightly lower than the working frequency of the WPT system, which holds out rather high requirements on the design of PCB MTMs [32; 33]. Due to the limitations of processing techniques, there is no flawless method to precisely decide the resonance frequency of the MTM slab. In a word, the research methodologies for MTM-enhanced WPT systems is still highly restricted. In order to solve the aforementioned problems, a practical model for MTMs used in WPT systems is proposed in this work. The relative permeability of MTMs is decided by the resonance frequency and the quality factor.