In recent years, wireless power transmission (WPT) technology has become one of the fastest and highest impact technologies. The concept of WPT was initiated by N. Tesla in 1914, but it had not resulted in any practical applications until recently, due to its low efficiency. A magnetically coupled WPT based on the coupled mode theory was first investigated in 2007 by Prof. Soljacic and his research group at MIT . However, a given WPT system typically consisting of two resonators with a certain separation in free space has a limit of WPT efficiencies. Much research has been performed in an attempt to overcome this problem. For example, metamaterial has been considered as one solution that can address the low efficiency problem.
The realization of media that have negative permittivity and permeability (also called left-handed metamaterial) has become feasible since Pendry et al.  proposed a new method in 1999 that used an array of thin wires and split ring resonators (SRRs). A focusing problem for a magnetostatic case with an effective permeability of ？1 was treated in . The present paper derives a simple design formula for the case with general negative permeability based on [4,5]. The provided formula facilitates the design of two-loop WPT configurations that employ metamaterial slabs with arbitrary negative effective permeability (for instance, ？1, ？2, and ？3). The WPT with lossless and lossy slabs are examined in terms of their
The metamaterial slabs are assumed to be ideally isotropic or realized by periodically arrayed ring resonators (RRs). The effects of losses when using RRs are also estimated.
Ⅱ. WPT SYSTEM USING METAMATERIAL SLABS FOR MAGNETIC FLUX FOCUSING
At the very low frequencies (severalMHz down to kHz) used in WPT systems, the system size is very small compared with the wavelength. For this case, the quasistatics may be the governing rule. The loop resonator (loop antenna with a chip capacitor loaded onto it) used in this work may be best modeled in the framework of quasi-magnetostatics. It may also be understood as a magnetic dipole. Fig. 1 shows the magnetically coupled WPT system using the metamaterial slabs with negative permeability. A generalized formula based on the magnetostatic field refraction  has been derived and is given by
Negative permeability can also be determined by using Eq. (1) for a given thickness of the slab (
Ⅲ. SIMULATION RESULTS AND DISCUSSIONS
The WPT system used in electromagnetic (EM) simulations is shown in Fig. 1. The two loop-resonators are made of copper and are designed at 13.56MHz. The radius (
[Fig. 1.] Magnetically coupled wireless power transmission with metamaterial slab of negative permeability.
Table 1 shows that the WPT efficiencies are greatly enhanced compared to the free space case due to the effect of magnetic field focusing when the lossless slabs are inserted between the two loops.
In addition, the results show that the use of the optimum load (2) is crucial for further enhancement of the WPT efficiencies. Fig. 2 shows the EM-simulated S-parameters for the cases in Table 1. Fig. 2(a) shows the S-parameters without the slab and with a slab having
[Table 1.] Comparison of WPT efficiencies when using lossless slabs (f0 =13.56MHz, r1 =10 cm, d=48 cm, Q=1,838)
Comparison of WPT efficiencies when using lossless slabs (f0 =13.56MHz, r1 =10 cm, d=48 cm, Q=1,838)
[Fig. 2.] Comparison of |S11|’s and |S21|’s as a function of frequency for the cases in Table 1. (a) Cases without and with slab (μr= ？2) when d = 48 cm, s = 16 cm, and a = 16 cm. (b) Cases with slabs (μr = ？2, ？3) for different a’s.
Now, we deal with the case of
The effects of magnetic flux focusing when using a slab with negative permeability in WPT systems have been analyzed based on a formulation assuming magnetostatics. For the lossless slabs with
[Fig. 3.] Characteristics of RR and comparison of S-parameters (d = 24 cm, a=b=6 cm, s=12 cm). (a) Isotropic unit cell of RR. (b) Extracted μr for RR bulk. (c) Comparison of S-parameters for the cases of using ideal slab and RR bulks when RL =50 Ω.
[Table 2.] Summary of the WPT efficiencies (d=24 cm, a=b=6 cm, s=12 cm)
Summary of the WPT efficiencies (d=24 cm, a=b=6 cm, s=12 cm)