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 [1]. 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. [2] 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 [3]. 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 S-parameters and the efficiencies based on them.
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.
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 [4] has been derived and is given by
where θ_{1} is the incident angle and θ_{2} is the refraction angle, a, a', b, b' and are distances as defined in Fig. 1, d is the distance between the two loops (d =a + a' + b + b'), h is the height of the slab determined to capture as much magnetic fields from the loop as possible, and μ_{r} is the relative effective permeability usually realized with RRs or SRRs. The sum of a' and b' is the width (s) of the slab. The focusing of the fields from the loop mainly depends on the permeability of the slab, and not on the permittivity since magnetic fields are dominant in the near region of the loop.
Negative permeability can also be determined by using Eq. (1) for a given thickness of the slab (s) and a distance between two loops (d). Note also that the same focusing effect takes place as long as it is located between the two loops and the slab height is sufficiently long, independent of its position.
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 (r_{1}) of the loop is 10 cm and the radius (r_{ring}) of the ring is 1 cm. The inductance of the loop is 47.63 nH, the capacitance of the loaded chip capacitor is 383 pF, the ohmic resistance of the loop is 0.025 Ω, and the Q-factor is 1,838 at the resonant frequency of 13.56MHz. Table 1 summarizes the EM-simulated WPT efficiencies for different a, s, and load resistance (R_{L}) with a fixed d =48 cm. The optimum load (R_{L,opt}) for a maximum efficiency when the two loops are in free space is given by
where R_{2} is the resistance of the second loop, k is the coupling coefficient, Q_{1} and Q_{2} are the quality factors of the first and second resonator, respectively. The expression (2) is derived using circuit theory and has been checked to be equivalent to the expression in [2] written in terms of coupled mode theory. The optimum load (R_{L,opt}) for a maximum efficiency when using the slab has been determined from EM simulations. The WPT efficiencies have been evaluated with |S_{21}|^{2}/(1 ？ |S_{11}|^{2}), which remains the same for any reference impedance of port 1.
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 μ_{r} = ？2 with a = 16 cm. For the case with the slab, the resonant frequency has shifted somewhat downward. The efficiencies in Table 1 have been evaluated at about 12.3MHz. This can be tuned back to the required resonant frequency of 13.56MHz by adjusting the capacitance of the chip capacitor placed on the loop with almost the same efficiency. Fig. 2(b) shows the EM-simulated S-parameters for different distances (a and b) and different values of permeability. For the cases where μ = ？2 and ？3, s is determined to be 16 and 12 cm, respectively, using (1). This result also indicates that the S-parameters do not depend on the position a of the lossless slab.
Now, we deal with the case of μ_{r} = ？1 slabs considering loss effects. Fig. 3(a) shows the unit cell structure designed to have an isotropic effective permeability behavior. The copper ring resonator, used in the EM simulation has been designed to have the effective relative permeability of ？1 at 13.56MHz. Its inductance is 10.8 nH and the capacitance of the inserted chip capacitor is 1.26 pF. The side length of the unit-cell is 12 cm. The radii and thickness of the ring resonator are 4 and 1 cm, respectively. Fig. 3(b) shows the extracted Re[μ_{r}] and Im[μ_{r}], which are ？1 and 0.23 at the design frequency of 13.56MHz, respectively. In Fig. 3(c), we compare the EM-simulated S-parameters for an ideal isotropic slab (μ_{r} = ？1 ？ j0.23 irrespective of frequencies) with the EM-simulated and measured S-parameters for the RR bulk structure based on the configuration given by Fig. 1. For these simulations, r_{1} = r_{2} = 5 cm, d = 24 cm, a = b = 6 cm, s =12 cm, and the Q-factor of each loop is 473. Although the S-parameters for the cases of highly dispersive RR bulk structures are far more dependent on frequencies than are those of the ideal isotropic slab where permeability is independent of frequencies, they are still in good agreement at 13.36MHz. The slight difference between the EM-simulated and the measured S-parameters for RR bulk structures is judged to arise from fabrication errors. In Table 2, we summarize the WPT efficiencies for the various cases including those in Fig. 3(c) when R_{L} =50 Ω and R_{L} = R_{L,opt}. For the case of the ideal lossless slab (μ_{r} = ？1), a significant improvement in efficiencies is achieved up to 96%, due to the effect of magnetic field focusing. For the lossy slabs with the imaginary part of 0.23, however, the efficiencies decrease to about 20% due to absorption of power in the slabs. The critical imaginary part, where the maximum efficiency becomes that of free space (32%), is 0.07. The RRs used for this work is a planar type. If a ring type RR is employed, the loss effect decreases and a higher WPT efficiency may be obtained.
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 μ_{r} = ？1, ？2, and ？3, the WPT efficiencies are greatly improved to close to 100% due to the effect of focusing. For the lossy slabs consisting of planar type RRs, where effective permeability has been evaluated as μ_{r} = ？1 ？ j0.23, the WPT efficiencies are significantly decreased down to about 20% due to the power absorption by the slab. If a ring-type RR is used, the effective imaginary part of the slab permeability can be designed to be smaller and the WPT efficiency can be enhanced to that degree.