Design and analysis of metamaterials for wireless power transfer with same resonance frequency but different L/C combinations

Abstract

Metamaterial is an artificial material with unique material properties that cannot be found in a naturally existing one, and hence it is able to perform functions that a conventional material cannot do. Consequently, it has shown great potential in various engineering applications such as resonance coupled mid-range wireless power transmission (R-WPT) systems. However, the analysis and design of an application-oriented metamaterial include both the unit cells and their combinations, and thus are very complex. Moreover, the performance of a metamaterial is very sensitive to uncertainties in both manufacturing and operating conditions. In this regard, a two-phase methodology for robust design optimizations of metamaterial slabs in R-WPT applications is proposed and applied to the design optimization of a prototype R-WPT system with promising results.

Keywords

Metamaterial robust optimization response surface model

1. Introduction

Metamaterial (MM) is an artificial periodic material with unique material properties that cannot be found in a natural one when working at its resonance frequencies. MM structure often consists of unit cells that evenly repeat themselves in 3D space. Because the footprint of the unit cell in MM structure is much smaller than the wavelength of the resonance frequency, the structure therefore can be regarded as a block of a homogenous, isotropic material with designed electromagnetic properties, such as a negative permeability and or permittivity, at the resonance frequency [1]. The unit cell of a MM consists of two parts, a dielectric substrate that offers a mechanical support for the entire macroscopic structure, and a metallic pattern deposit on the substrate, where the inductance and capacitance of the unit cell come from. Hence, the metallic pattern of the unit cell determines the resonance nature of the MM and plays the most important role in the MM design and optimization [2]. Because of the possibility of adjustments from positive to negative effective properties, MM has shown great potential in various engineering applications; for instance, in miniaturized antenna systems and resonance coupled mid-range wireless power transmission (R-WPT) systems; and has been studied extensively in the past decade [3,4]. A R-WPT system utilizes the near magnetic field as the medium for energy transmissions and works base on the resonance between the receiving and transmitting coils. The resonance of the receiving and transmitting coils significantly increases the transmission range of a R-WPT system from 40 to 200 mm [5]. The purpose of adding MMs in between the receiving and transmitting coils in R-WPT systems is to enhance the power transfer efficiency (PTE) of the systems via the resonance coupling between transmission coils and the MM slabs or blocks [6,7]. By tuning the resonance frequency of the MM blocks, the transmission can be enhanced, reduced or even blocked accordingly.

The metallic pattern in the MM unit cell can be viewed as a series RLC circuit, and the resonance frequency of a series RLC circuit is inversely proportional to the product of the inductance (L) and the capacitance (C) of the circuit, and hence to have a relatively low working frequency (below 30 MHz), a large inductance and capacitance are necessary, leading to a large device footprint. On the other hand, in order to maintain the homogeneity and isotropy of the electromagnetic properties throughout the MMs, the dimensions of the metallic pattern in MM unit cell have to be strictly constrained. One way to overcome this contradiction is the series addition of on-chip circuit components, such as lumped capacitors, to increase the capacitance. However, for an RLC circuit having a high capacitance, its Q-factor, describing its frequency selectivity, would be relatively low. Due to the aforementioned reasons, the design and optimization processes of a MM unit cell, especially the metallic structure to obtain an appropriate RLC combination, are crucial for the performance and efficiency enhancements. In [8], the effects of different Q-factors to the performances of MM slabs in high frequency (GHz range) applications have been analysed. However, there is a lack of a similar analysis and optimization methodology for MM structures in the low frequency applications, where both the footprint of the MM unit cell and the energy loss in the lumped element are non-negligible. Moreover, since the band of the working frequency of the MMs is relatively narrow, the robustness of the MM performance against uncertainties in both the manufacturing and operating tolerances should be carefully considered. In this regard, this paper investigates the influence of the L/C ratios and Q-factor of metamaterials on the efficiency and robustness of the R-WPT systems, and a new methodology to design and optimize a MM unit cell and the R-WPT system is proposed based on a surrogate model based robust optimization methodology.

2. Metamaterials in resonant-wireless power transmission

The MM unit cell can be regarded as a miniaturized RLC circuit. The circulating current conduction path, which is formed by the metallic pattern of the unit cell, creates the inductance in the circuit. Moreover, when the both sides of the substrate are deposited with the metallic strips, this sandwich structure creates a capacitance in the circuit. It should be pointed out that the parasitic capacitances between each layers of the spiral metallic strips are often very small as compared to the capacitance created by the sandwich structure. In the other occasions, where the metallic pattern is only deposited on one side of the substrate, the only capacitor available is the parasitic capacitor and is relatively small, and hence a lumped capacitance is added to the MM unit cell to reduce the resonance frequency. However, such approach will incur an additional device footprint and an energy cost. Generally, two types of MM unit cells are commonly used in electromagnetic applications, namely the split ring resonator and the spiral resonator. The spiral resonator contains a plain helix metallic pattern that is similar to that of a planer helix coil, and has a larger inductance and resistance with a compact unit cell size, which makes it more appropriate for low frequency applications. In this paper, the spiral resonator is chosen as the topology for the MM unit cell. Moreover, one of the main advantages of WPT system is the removal of the transmission medium of energy transmissions and hence the freedom of movements of the transmission devices. However, if too large MM structures are added in between the coils, this advantage will be forfeited. Hence, in this paper, the MM unit cells repeat themselves in x and y directions and form a MM slab with only one layer of MM unit cells in the energy transmission direction of the R-WPT system.

To control the inductance and the capacitance of the MM unit cell, the number of the turns for the spiral pattern is selected as the design parameter. The equivalent inductance of the unit cell is calculated from [8]: $\begin{eqnarray}\displaystyle L=\frac{{\mu}_{0}}{2{\pi}}\left\{4l-\left[2(N+1)-\frac{3}{N}\right](g+w)\right\}\left[\ln \left(\frac{\left\{4l-\left[2(N+1)-{\displaystyle \frac{3}{N}}\right](g+w)\right\}}{2w}\right)+\frac{1}{2}\right]. & & \displaystyle\end{eqnarray}$ (1) where l, g, and w, are, respectively, the maximum side length, the air gap width and the strip width of the spiral pattern; N is the number of turns of the spiral, and μ₀ is the free space permeability. Since the metallic pattern is only deposited on one side of the substrate, the only capacitance of the metallic pattern is the parasitic capacitance between the adjacent metal strips. Although relatively small, this capacitance still plays a significant role in the resonance circuit. It can be calculated from [8]: $\begin{eqnarray}\displaystyle C=\frac{l}{4(g+w)}\frac{N^{2}}{N^{2}+1}\left[l(N-1)-\frac{N^{2}-1}{2}(g+w)\right]{\varepsilon}_{0}{\varepsilon}_{r}\frac{K\left(\sqrt{1-k^{2}}\right)}{K(k)} & & \displaystyle\end{eqnarray}$ (2) where ϵ₀, ϵ_r are, respectively, the permittivity of the air and the relative permittivity of the substrate; k is a constant, K is the complete elliptical integral of the first kind: $\begin{eqnarray}\displaystyle \begin{array}{@{}l@{}}k=g/(2w+g).\\[5.0pt] K(x)=\displaystyle \int _{0}^{2{\pi}}d{\theta}/\sqrt{1-x^{2}\sin ^{2}{\theta}}.\end{array} & & \displaystyle\end{eqnarray}$ (3)

The resistance of the metallic pattern is estimated using [8]: $\begin{eqnarray}\displaystyle R=\frac{{\rho}}{wt}l. & & \displaystyle\end{eqnarray}$ (4)

In a R-WPT system, the working frequency is often relatively low, i.e. in tens of MHz range, and hence to reach a resonance frequency as low as this value limit, a large inductance and a large capacitance are required. However, the intrinsic inductance and capacitance of the MM unit cell are not able to reach to such high values, and an external lumped capacitance is thus connected to the two ends of the metallic pattern, and hence in parallel with the parasitic capacitance of the MM unit cell. The equivalent circuit model for the MM unit cell is thus given in Fig. 1.

Fig. 1.

The equivalent circuit of a MM unit cell.

3. Robust design optimization of the metamaterial

The Q-factor of an RLC circuit is defined as [9]: $\begin{eqnarray}\displaystyle Q=\frac{{\omega}L}{R}=\frac{1}{{\omega}RC}. & & \displaystyle\end{eqnarray}$ (5)

The Q-factor of a resonance circuit describes its selectivity performance with respect to the working frequency. A circuit of a high Q-factor would have only prominent circuit features, such as negative permeability in the case of the MM structure, at frequency very close to the resonance frequency; while a circuit of a low Q-factor however will have a relatively large working frequency band. Conventionally, a high Q-factor resonance circuit is preferable, so sharp variation performances can be achieved only at the designed and targeted frequency. However, as explained previously, because there is a high probability of having manufacturing errors in the fabrication process of the miniaturized metallic pattern, the dimensions of the MM unit cell and its resonance frequency are no longer deterministic, and hence the robustness of the MM structures is also important in WPT applications. Generally, the design and optimization of MM structures in WPTs can be formulated as: $\begin{eqnarray}\displaystyle \begin{array}{@{}l@{}}\max \mathit{PTE}(L,C)=|S_{21}|^{2}\times 100\%.\\ \text{s.t.}∼f=f_{working}\end{array} & & \displaystyle\end{eqnarray}$ (6) where, S₂₁ is the transmission coefficient of the WPT system, f_working is the working frequency. As the capacitance value is controlled by the lumped capacitor and is fixed, the main sources for the uncertainty is the inductance of the MM unit cell. Moreover, the robustness counterpart of the original optimization problem (6), using the weighted mean value of the objective function over a design region, is used in the present work and formulated as: $\begin{eqnarray}\displaystyle \begin{array}{@{}l@{}}\max \mathit{Mean}(\mathit{PTE}(L,C,f))={\displaystyle \frac{1}{n}}\displaystyle \mathop{\sum }_{n}w_{i}\times \mathit{PTE}(L+{\rm\Delta}L_{i},C,f).\\[8.0pt] \text{s.t.}∼f=f_{working}\end{array} & & \displaystyle\end{eqnarray}$ (7) where w_i is the weighting factor for the ith candidate. The total n number of candidates are evenly sampled from the neighbour of a design for the robustness performance calculation. To calculate the objective values, the transmission coefficient, S₂₁, is evaluated using a 3D FE analysis. Obviously, the robustness performance calculation requires an excessive number of FEM evaluations, leading to a slow optimization speed. Response surface model, being one of the most effective surrogate models, has been widely employed in electromagnetic inverse problems. In this regard, an improved response surface model (RSM) is proposed to construct the surrogate model of the system, and to replace the FEM simulation and hence increase the optimization speed.

In the R-WPT application, the MM slab is generally consisted of a number of unit cells (to see Fig. 2b). Moreover, our primary study has revealed that the MM unit cells structure with the same resonant frequency but different inductance-capacitance combinations outperforms the one with an identical inductance-capacitance combination in view of performance enhancements. In this point of view, a MM slab consisting of unit cells of the same resonance frequency but different L/C combinations is proposed. However, this will lead to a very complex design optimization process of the MM slab. To address this issue, the whole optimal procedure of the MM slab design is divided into two phases.

The objectives of the first design phase are the computation of the performance and robustness parameters of different unit cells, and the classification of the unit cell. In this stage, using the FEM simulation results of a homogeneous MM slab, where all the unit cells have identical structures, a third order interpolation is constructed using: $\begin{eqnarray}\displaystyle y_{j}=C_{0}+\mathop{\sum }_{i=1}^{3}C_{i}x_{j}^{i}+e. & & \displaystyle\end{eqnarray}$ (8) where y_j is the objective value of the jth sample, C₀, C_i are coefficients in the interpolation, and e is an error compensation term. The robustness of each unit cell design is thus calculated using this interpolation model. According to the objective values and the robustness performances, the unit cell designs are classified into “objective dominated design with high objective value and low robustness”, and “robustness dominated design with high robustness and low objective values”. Moreover, to simplify the optimal design, the unit cells of the MM slab are divided into two groups: one centre cell and eight side cells (Fig. 2b).

Fig. 2.

(a) MM unit cell and (b) the MM slab structure.

In the second phase of the optimization process, the inductances of the centre cell and side cells are set to be different to allow a hybrid MM slabs design. Because the unit cells no longer share the same design, the interpolation model constructed in the first stage is no longer valid, and hence a multi-start RSM is proposed and employed. The proposed method has two control parameters: the inductance of the centre cell and the inductance of the side cells, and the initial design space is divided into four different sub spaces based on the objective values and robustness of the centre cell and side cell obtained at the first stage, namely the objective dominated-objective dominated, the objective dominated-robustness dominated, the robustness dominated-objective dominated, and the robustness dominated-robustness dominated. In each sub-space, 3², total nine samples are simulated using 3D FEM and the corresponding local second order RSM is constructed using: $\begin{eqnarray}\displaystyle y={\beta}_{0}+\mathop{\sum }_{i=1}^{k}{\beta}_{i}x_{i}+\mathop{\sum }_{j=1}^{k}\mathop{\sum }_{i=1,i<=j}^{k}{\beta}_{i,j}x_{i}x_{j}+e & & \displaystyle\end{eqnarray}$ (9) where 𝛽₀, 𝛽_i and 𝛽_{i, j} are coefficients, k is the number of the total coefficients. Based on the RSM of each sub-space, the performances of all possible design in each subspace is evaluated and the global optimal solution can be eventually found by using an exhaustive method. Since the RSM reduces significantly the computational time, an exhaustive exploration of the entire design space in details will not incur too heavy computation cost to the optimization problem and hence is feasible in the proposed methodology.

4. Numerical results

The MM unit cell used in this paper consists a piece of substrate and a piece of metal strip deposited on one side of the substrate. The substrate, which is made by FR4, has a constant thickness of 4 mm, a constant side width of 37.2 mm, and a dielectric constant of 4.4. The metallic pattern is made by a copper strip of 0.0035 mm thickness, a common standard for PCB fabrication, and a width, w, of 0.5 mm and a spacing, g, of 0.5 mm. The largest square of the metallic pattern has a side width of 36.2 mm, 1 mm smaller than that of the substrate, and the number of turns of the spiral varies from 2 to 15, and the incremental is set to be integer. The overall MM slab consist of 3 × 3 (9) unit cells. The MM structure with all the annotations are depicted in Fig. 2.

The MM slabs are placed right at the middle of a conventional four-coil R-WPT systems. In the four-coil system, a transmission coil and a receiving coil are added in between the source and the load coils to focus the magnetic field and hence increase the transmission distance [11]. The source and load coils are copper rings with a diameter of 60 mm, placed 200 mm apart, and the transmission and receiver coils are identical copper helix coils with a diameter of 120 mm, an axial length of 30 mm and a cross-sectional area of 1 mm², placed 140 mm apart. A 50 Ohms load is used. Figure 3 depicts schematic diagram of the studied 13.56 MHz four coil WPT system and the MM slabs with 9-unit cell of metamaterials for PTE enhancements.

Fig. 3.

Four coil WPT system with a metamaterial slab in this paper.

The proposed optimal methodology is applied to find the optimal inductance-capacitance combinations of the 9-unit cells of the MM slab to obtain the best performance and robustness of the prototype R-WPT system.

In the first phase of the optimization process, to model the uncertainty, the upper and lower limits of the inductance for each unit cell are set to be 10% of its nominal value; The RSM is generated using 14 potential designs, and the numerical results of typical runs indicate that the error between the predicted results of the proposed RSM and the computed PTEs of the WPT system using FEM is less than 3%. Under such implement conditions, the PTE and the corresponding robust performance of the proposed prototype WPT system under an identical turn unit cell MM slab are computed and tabulated in Table 1. It should be pointed out that the CPU time used by the proposed RSM to compute once robust performance of a design is 10 seconds while that for a FEM is 30–40 minutes.

From the numerical results of Table 1, the unit cells with turn numbers of 2–8 are classified into objective-dominated designs, while those with turn numbers of 9–15 are grouped as robustness-dominated designs, and four RSMs are constructed. The optimization procedure continues to phase two.

Table 1

PTE and its degradation under different turn unit cell MM slabs

Number of turns	Inductance (H)	PTE	PTE (robust)	Performance degradation (%)
2	0.000143589	0.247742206	0.205285481	17.13746119
3	0.000140607	0.241794507	0.198114255	18.06503095
4	0.000137931	0.170512353	0.18572433	−8.921333975
5	0.000135381	0.145181665	0.170976641	−17.76737808
6	0.000132897	0.130704472	0.155421737	−18.91080289
7	0.000130452	0.127094254	0.140047108	−10.19153369
8	0.000128036	0.124926212	0.12921911	−3.436347462
9	0.000125639	0.122928538	0.1243486	−1.15519327
10	0.000123259	0.120821324	0.121739	−0.75953099
11	0.000120892	0.118708268	0.120177852	−1.237979686
12	0.000118538	0.11815806	0.118858428	−0.592738474
13	0.000116194	0.117339153	0.117653319	−0.267742259
14	0.000113859	0.116723016	0.116563745	0.136451741
15	0.000111535	0.115917788	0.11560561	0.269309749

In the second stage of the optimization process, a multi-start RSM is constructed and employed; and an exhaustive method is used to find the global optimal solution in the whole design space. More specially, it is found that when the side cells have two turns and the centre cell has six turns, the PTE of the system reaches a maximum value of over 25%, while the robustness of this solution remains strong, i.e., the PTE degradation is relatively low (<10%). Obviously, a significant higher quality design is obtained in terms of both performance and robustness parameters, which are not achievable by a homogeneous slab. Moreover, compared with the case where these is no MM slab presented in the WPT system, the PTE of the system increases significantly from 3.16% to over 25%, indicating the necessity of the addition in the WPT systems.

5. Conclusion

This paper strives to understand the influence of different LC ratios and Q-factors of MM unit cell on the performance, especially the robustness of the PTE, of a MM assisted low frequency R-WPT system; and proposes a two phase methodology to optimize a R-WPT application oriented MM slab based on RSM surrogate model. The numerical results of a 13.56 MHz four-coil WPT system indicate that: 1. The LC combinations do play a crucial role in affecting the robustness of the overall WPT system. Although the working frequencies are the same, different LC combinations can offer different working bandwidth, and hence different ability to withstand uncertainties and errors in the devices, and hence is a good way for designing and optimizing the MM structure; 2. The proposed methodology is able to search for the robust solutions within a short period of time and a relatively low number of FEM simulations. The results show that by optimizing the hybrid MM slabs, where the unit cell structures are not identical, in WPT system, the system power transfer efficiency and its robustness can be significantly enhanced.

Footnotes

Acknowledgements

This work was supported by the Research Grant Council of the Hong Kong SAR Government under projects PolyU 152254/16E, G-YBPM and G-YBY7.

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