Experimental study of wireless power transfer with metamaterials and resonators

Abstract

Inserting metamaterials into the path of wireless power transfer (WPT) is an effective way to improve the power transfer efficiency (PTE) or extend the power transfer distance. The resonant frequency f ₀ of the metamaterial is a critical parameter to distinguish the positive or negative permeability regions, which should be determined accurately. A new method to measure f ₀ is developed and validated in theory and simulation. The paramagnetic region (where the effective permeability is positive) is a frequency band below f ₀, and is used as the operating region of WPT with insertion of the metamaterial. According to experiments, the PTE is improved by comparison to the case without metamaterial. More interestingly, the PTE improvement effects at different operating regions are compared. It is shown that the region, where the forced-resonant frequency f _RS is close to f ₀, has better effect than the other regions where f _RS is lower or higher than f ₀.

Keywords

Wireless power transfer resonator metamaterial

1 Introduction

Wireless power transfer (WPT) has become a promising technology in many potential applications such as consumer electronics [1], implantable devices [2], and electronic vehicles [3], for its advantage of “wireless”. The power transfer efficiency (PTE) and the power transfer distance (PTD), which are the two most concerned parameters, are mutually exclusive conditioned since the PTE decreases with the PTD. Under the condition of keeping enough PTE and power delivered to the load, many efforts have been made to extend the PTD, including optimization of the coil or ferrite core geometries [4–7], impedance matching circuits [8], and optimal selection of the operating frequency [9]. In addition to these approaches, inserting intermediate coils on the power transfer path is another scheme to improve the PTE or extend the PTD. Since the magnetic field in the near-field region (which is the work region of the inductively coupled WPT) attenuates fast at a cubic rate with the distance, the intermediate coils have to segment the distance linearly to many parts, although some extra loss exists in the intermediate coils themselves.

There are two commonly used intermediate coils, resonators [10,11] or metamaterials [12]. The resonators have the same resonant frequencies with the resonant frequency of the power transmitting (Tx) and receiving (Rx) coils as well as the operating frequency of WPT. The metamaterial, consisting of an array of resonators, has the property of negative permeability and permittivity [13] within a narrow band at a frequency that is slightly higher than the resonant frequency of the metamaterial. Consequently the metamaterial has negative refractive index, resulting in focusing of the electromagnetic wave when passing through the metamaterial, and amplifying the evanescent wave in the near-field case [14,15]. The latter is used in inductive WPT to improve PTE. Because of the decoupling property of electric and magnetic fields in the near-field region, only the negative permeability is required [15]. Furthermore, bulky three-dimensional (3D) metamaterials [15] for WPT is also not necessarily needed for evanescent field amplification [16], as the loss increases with the thickness of the structure. One slab metamaterial has similar yet even better effect than 3D one on improving the PTE of WPT [17].

The main difference between the resonator and metamaterial schemes is whether the operating frequency is equal to or larger than the resonant frequency of the inserted coils respectively. Different from the negative permeability metamaterial for WPT, the positive permeability phenomenon for WPT was studied in [18], called the paramagnetic response, in which an additional resonant coil with a resonant frequency substantially higher than the operating frequency of WPT is added close to the Tx coils. The experimental results in [18] showed that the paramagnetic response has also the potential to improve the PTE like the negative permeability metamaterial case.

The resonant frequency of metamaterial is a critical frequency to distinguish the polarity of permeability (i.e., positive or negative) [12]. At the same time, the operating frequency of WPT is designed around this critical frequency to obtain a prescribed effective permeability 𝜇_eff (e.g., 𝜇_eff = −1) [12]. However, the permeability varies fast at the frequencies around the critical frequency, which makes the frequency band of negative permeability be narrow [19]. As a consequence, the design of the resonant frequency of the unit cell or the metamaterial by virtue of calculations or simulations may readily lead to a wrong operating region (i.e., whether positive or negative permeability regions) since the calculations and simulations may differ from the reality or the experiments.

In this work, an experimental method to measure the resonant frequency of the unit cell or the metamaterial is developed firstly in Section 2. As a consequence, the operating frequency at the positive or negative permeability region can be identified confidently. Different from the negative permeability region which has been studied a lot, the positive permeability region is studied experimentally in Section 3, where the metamaterial is assumed to be placed at various positions along the power transfer path of the WPT. The study in [18] is a special case of this present work since only one cell was inserted and its position was fixed (e.g., at a location very close to the Tx coil) in [18]. More interestingly, the WPT that operates at different frequency regions are compared in Section 4, followed by the conclusions in Section 5.

2 Design and test of metamaterial

A typical diagram of inductively coupled WPT system with insertion of a metamaterial slab is shown in Fig. 1(a), where the power is transferred from the Tx coil to the Rx coil. The metamaterial is an array comprised of many cells. Split-ring resonators [13] or spiral coils [12] are two common structures of the cells. The more compact structure, i.e., the spiral coil structure shown in Fig. 1(b) is herein chosen as the unit cell because of its lower self-resonant frequency.

Some theoretical models to predict the effective permeability, 𝜇_eff, at a limited accuracy were reported [19,20]. Instead, 𝜇_eff can also be extracted from the S-parameter characteristics of a unit cell of the metamaterial [21], computed by electromagnetic (EM) simulation tool (e.g., HFSS), because of the periodic structure property of metamaterials. Specifically, 𝜇_eff can be expressed in terms of refractive index n and impedance z via the S-parameters as follows: $\begin{eqnarray}\begin{array}{@{}c@{}}\displaystyle {\mu}_{\text{eff}}=nz\\[12.0pt] \displaystyle n=\frac{1}{kd}\arccos \left[\frac{1}{2S_{21}}\left(1-S_{11}^{2}+S_{21}^{2}\right)\right]\\[12.0pt] \displaystyle z=\sqrt{\frac{\left(1+S_{11}\right)^{2}-S_{21}^{2}}{\left(1-S_{11}\right)^{2}-S_{21}^{2}}},\end{array}\end{eqnarray}$ (1) where k and d are the wavenumber and the maximum length of the unit cell, respectively.

In the simulation setup, perfect magnetic conductor boundaries should be set to simulate a periodic environment [23], which, however, cannot be realized in reality. An alternative approximate method in experiments is placing the unit cell between two loop coils [17]. The magnetic fields produced by the two loop coils are perpendicular to the unit cell. Under this setup, the S-parameters are measured and used to retrieve the effective permeability. However, this measured result depends on the setup, e.g., different loss of the two loops leads to different S ₁₁ and consequently different effective permeability. That is, it is difficult to make the frequency be at a certain required 𝜇_eff, like 𝜇_eff = −1 [12], especially when 𝜇_eff changes drastically with frequency. Different with determining an accurate magnitude of 𝜇_eff, the self-resonant frequency (i.e., critical frequency) of the unit cell is concerned firstly herein for determining the positive or negative 𝜇_eff.

Some efforts [24–26] have been made in modelling the spiral coils to avoid time-consuming EM simulation. However, the errors between the simulation or modelling and the reality result in wrong operating frequency region setup. A new method is herein developed to accurately measure the self-resonant frequency of a unit cell of a metamaterial as shown in Fig. 2(a) where a one-turn test coil is closely placed near the unit cell. The impedance of the test coil, Z _T, is measured by using a vector network analyzer (VNA, Agilent E5071C). The equivalent circuit model is shown in Fig. 2(b) where R _T (R _U), L _T (L _U) and C _T (C _U) are the parasitic resistance, the self-inductance, and the parasitic capacitance of the test coil (the unit cell), respectively, and M _TU is the mutual inductance between the test coil and the unit cell. The analytical modelling can be found in the Appendix. When neglecting the small value of the parasitic capacitance C _T, the test coil impedance Z _T can then be expressed as $\begin{eqnarray}\displaystyle Z_{\text{T}} & = & \displaystyle R_{\text{T}}+j{\omega}L_{\text{T}}\left(1+\frac{k_{\text{TU}}^{2}}{{\omega}_{0}^{2}/{\omega}^{2}-1+j/Q_{\text{U}}}\right)\nonumber\\ \displaystyle & = & \displaystyle R_{\text{T}}+\frac{\left({\omega}L_{\text{T}}/Q_{\text{U}}\right)\cdot k_{\text{TU}}^{2}}{\left({\omega}_{0}^{2}/{\omega}^{2}-1\right)^{2}+1/Q_{\text{U}}^{2}}\text{+}j{\omega}L_{\text{T}}\left(1+\frac{\left({\omega}_{0}^{2}/{\omega}^{2}-1\right)k_{\text{TU}}^{2}}{\left({\omega}_{0}^{2}/{\omega}^{2}-1\right)^{2}+1/Q_{\text{U}}^{2}}\right)\end{eqnarray}$ (2) with $\begin{eqnarray}\begin{array}{@{}c@{}}\displaystyle Q_{\text{U}}={\omega}L_{\text{U}}/R_{\text{U}}\\[12.0pt] \displaystyle Z_{\text{U}}=R_{\text{U}}+j{\omega}L_{\text{U}}+1/j{\omega}C_{\text{U}}\\[12.0pt] \displaystyle k_{\text{TU}}=M_{\text{TU}}/\sqrt{L_{\text{T}}L_{\text{U}}},\end{array}\end{eqnarray}$ (3) where ω =2πf and ω₀ =2πf ₀ are the frequency and self-resonant frequency of the unit cell respectively, Q _U is the quality factor of the unit cell, and k _TU is the coupling coefficient between the test coil and the unit cell. It can be observed from (2) and (3), when the frequency crosses ω₀, the imaginary part of Z _U crosses zero at ω₀. Consequently, peaks of the real and imaginary parts of Z _T, i.e., Re(Z _T) and Im(Z _T), appear at a frequency very close to ω₀ because the denominators in (2) reach the minimum values.

As an example, a unit cell of metamaterial is fabricated with 11-turns, 4-mm pitch between adjacent turns, 3-mm trace width, and 35-μm thickness copper, on 1.6-mm thickness FR4 substrate with relative permittivity ϵ_r of 4.4. The outer diameter of the spiral coil and the size of the unit cell are 100 mm and 120 mm, respectively, as shown in Fig. 1(b). The calculated, simulated, and measured values of Re(Z _T) and Im(Z _T) are shown in Figs. 3(a) and 3(b) respectively. The simulated results are obtained from the EM simulation tool HFSS. Both the calculated and simulated self-resonant frequencies, as observed from Fig. 3, are 49.0 MHz, which is the same as the result from (A.6) in the Appendix. The measured self-resonant frequency is 51 MHz, which is slightly higher than the calculated and simulated ones. Note that the calculated result in (A.6) does not always have very high accuracy. When the substrate is thin, the error increases because the effective permittivity of the environment surrounding the printed coil is no longer (1 +ϵ _r)∕2. Nevertheless, the (rough) calculation can be still used to approximate the self-resonant frequency of the spiral coil to accelerate the unit cell design at the initial trial and can be improved by simulation further.

The expression of impedance Z _T in (2) can also be expressed as $Z_{\text{T}}=R_{\text{T}}+j{\omega}L_{\text{T}}\cdot (1+{\mu}_{\text{eff}}^{\prime })$ , where ${\mu}_{\text{eff}}^{\prime }$ can be considered as the effective permeability in the space surrounding the test coil when the unit cell is close to the test coil. Therefore, when the frequency is higher than ω₀, the effective permeability is negative, and vice versa. The critical frequency to distinguish the positive or negative permeability, i.e., ω₀ of the unit cell, is also checked for a coil array (e.g., metamaterial) by using the same method. The setup of measuring the critical frequency of the metamaterial is shown in Fig. 4(a), where a test coil is close to the metamaterial that is comprised of a 2 × 2 array. The equivalent circuit model is shown in Fig. 4(b), in which R _n, L _n, C _n are the parasitic resistance, self-inductance, and parasitic capacitance of the n-th unit cell, respectively. The impedance Z _T is given by $\begin{eqnarray}\displaystyle Z_{\text{T}} & = & \displaystyle R_{\text{T}}+j{\omega}L_{\text{T}}\left(1+\frac{\mathop{\sum }_{1}^{N_{\text{M}}}k_{\text{T}n}^{2}}{{\omega}_{0}^{2}/{\omega}^{2}-1+j/Q_{\text{U}}}\right)\nonumber\\ \displaystyle & = & \displaystyle R_{\text{T}}+\frac{\left({\omega}L_{\text{T}}/Q_{n}\right)\cdot \mathop{\sum }_{1}^{N_{\text{M}}}k_{\text{T}n}^{2}}{\left({\omega}_{0}^{2}/{\omega}^{2}-1\right)^{2}+1/Q_{n}^{2}}\text{+}j{\omega}L_{\text{T}}\left(1+\frac{\left({\omega}_{0}^{2}/{\omega}^{2}-1\right)\mathop{\sum }_{1}^{N_{\text{M}}}k_{\text{T}n}^{2}}{\left({\omega}_{0}^{2}/{\omega}^{2}-1\right)^{2}+1/Q_{n}^{2}}\right)\end{eqnarray}$ (4) where k _Tn is the coupling coefficient between the test coil and the n-th unit cell of totally N _M cells, and Q _n = Q _U is the quality factor of the n-th unit cell. The Z _T expression of (4) is the same as (2) if the summation of k _Tn in (4) is replaced with k _TU. That is, the critical frequency of the metamaterial, at which Re(Z _T) reaches a peak and Im(Z _T) reaches zero, is the same as the case of the unit cell. The simulated and measured results of Z _T are shown in Fig. 5, under the measurement setup shown in Fig. 4(a). The simulated and measured self-resonant frequencies are 48.0 MHz and 50.0 MHz, respectively, slightly lower than the results of the unit cell (i.e., 49.0 MHz and 51 MHz, respectively). The difference of the self-resonant frequencies between the metamaterial and the unit cell is induced by the mutual inductance among the unit cells in the metamaterial.

The effective permeability of the metamaterial can be obtained by specifying $Z_{\text{T}}=R_{\text{T}}+j{\omega}L_{\text{T}}\cdot (1+{\mu}_{\text{eff}}^{\prime \prime })$ . Actually, this effective permeability is related to the test coil itself, because the coupling coefficient is related to both the test coil and the cells. That is, ${\mu}_{\text{eff}}^{\prime \prime }$ is not the effective permeability of the metamaterial. Nevertheless, the consistency of the critical frequency between the unit cell and the metamaterial is proved.

The effective permeability, 𝜇_eff, of the metamaterial can be obtained through the HFSS simulation [21] by extracting the S-parameters of the unit cell according to (1) as shown in Fig. 6, where the boundary between the regions of positive and negative 𝜇_eff is 48 MHz, which is the same as the simulated self-resonant frequency shown in Fig. 5. Thus, one can infer that the measured 𝜇_eff changes the polarity at about 50 MHz frequency. One can also observe from Fig. 6 that the real part of 𝜇_eff, i.e., Re(𝜇_eff), reaches peak at 47 MHz in simulation (while about 49 MHz in measurement), and the imaginary part of 𝜇_eff, i.e., Im(𝜇_eff), is close to zero at 46 MHz in simulation (while about 48 MHz in measurement). Moreover, Re(𝜇_eff) is negative from 48 to 49 MHz (while about from 50 to 51 MHz in measurement). The region of negative Re(𝜇_eff) is used for WPT in the previous studies. Herein, another region, where Re(𝜇_eff) is positive and larger than 1 when the frequency is lower than ω₀ (called as the paramagnetic metamaterial herein), is studied.

3 Insertion of paramagnetic metamaterial

Experimental setup of an inductively coupled WPT is shown in Fig. 7(a), where two coils are placed coaxially at a 20-cm distance. The two single-turn coils with the same geometrical parameters are printed on PCB by using 35-μm thickness and 2-mm trace width copper. The radius of the coil is 10 cm. The calculated self-inductance and self-resonance frequency of the coil are 690 nH and 239 MHz, respectively, according to [22]. In order to make the working frequency of the WPT system be in the paramagnetic response region of the pre-designed metamaterial, two 18-pF capacitors are connected to the Tx and Rx coils in series, which leads to measured and calculated forced-resonant frequencies of 43.75 MHz and 45 MHz, respectively. At 43.75-MHz operating frequency of the WPT, according to the analysis in section 2 and Fig. 6, Re(𝜇_eff) is positive and larger than 1, and Im(𝜇_eff) is close to 0, i.e., the metamaterial is in the paramagnetic response region.

The metamaterial designed in Section 2 is inserted into the path of the WPT. The experimental setup is shown in Fig. 7(b). The PTE (i.e., 𝜂) can be measured by using a VNA according to the relationship between the PTE and the S-parameters [27]: $\begin{eqnarray}{\eta}=|S_{21}|^{2}/(1-|S_{11}|^{2}).\end{eqnarray}$ (5)

The results of the measured PTE with metamaterials comprised of 2 × 2 and 3 × 3 cells are shown in Figs. 8(a) and 8(b), respectively. The cases with various distances (i.e., 5 cm and 10 cm) between the metamaterial and the Rx coil as well as the case without metamaterial are compared in these figures. At the operating frequency of the WPT, i.e., 43.75 MHz, the PTE with metamaterial can be improved about 10% by comparison to the results from the case without metamaterial.

Besides the peak at 43.75 MHz, the other 2 peaks are located at 48 MHz and 50 MHz, respectively, as shown in Fig. 8(a). At 48 MHz where Re(𝜇_eff) is close to the maximum value and Im(𝜇_eff) still is low according to Fig. 6 and the analysis in Section 2, the PTE is also improved although the operating frequency deviates the resonant frequency of the Tx and Rx coils (i.e., 43.75 MHz). At 50 MHz, which is actually the self-resonant frequency of the metamaterial as shown in Fig. 5, the PTE with metamaterial can be improved more than 5 times by comparing to the results without metamaterial. In Fig. 8(b), there exist two obvious peaks at 43.75 MHz and near 50 MHz. At 48 MHz, the PTE with metamaterial is improved too although no obvious peak can be observed.

The measured PTEs with metamaterial comprised of 2 × 2 cells at different power transfer distance are shown in Fig. 9, where the metamaterial is placed at the middle of the path between the Tx and Rx coils. From Fig. 9, one can observe that the insertion of the metamaterial can considerably improve the PTEs at various power transfer distances.

From Fig. 8, we can observe that the peaks appear near the resonant frequency of the coils (Tx/Rx coils or unit cell of metamaterials). If the resonant frequency of Tx/Rx coils close to the resonant frequency of metamaterials, it looks that the improvement of power transfer efficiency will be larger which will be studied further in Section 3.

4 PTE improvement effect at different operating regions of metamaterial

According to the experiments in Section 3, the PTE can be improved when the operating frequency is at the positive permeability region of the metamaterial. Another interesting thing is that the PTE is improved drastically near the resonant frequency of the metamaterial. Previous studies also show that the PTE is improved by insertion of negative-permeability metamaterials [17] or resonators [11]. However, until now no comparison is checked for the improvement effects of all the three operating regions of metamaterial (i.e., positive, zero, and negative permeability), which will be done in this section.

The forced-resonant frequency f _RS is set to 50.75 MHz and 53 MHz by connecting 13-pF and 12-pF capacitors to the Tx and Rx coils in series, respectively. The forced-resonant frequency f _RS is selected as the operating frequency of the WPT to realize the resonant coupled WPT system. The PTE without metamaterial is plotted in Fig. 10(a), whereas Figs. 10(b) and 10(c) depict the PTEs when the distance between the metamaterial and the Rx coil is set to be 10 cm and 5 cm, respectively.

Some conclusions can be drawn from Fig. 10 as follows. Firstly, the PTE can be maximized when the operating frequency is close to the self-resonant frequency of the metamaterial. Here, the case with the forced-resonant frequency of 50.75 MHz has the best PTE. This is similar with the case that some resonators with the same resonant frequency make up a wireless power resonator-domino system [11]. Actually, the metamaterial is comprised of resonators. When the metamaterial is used in the near-field WPT, it behaves like a relay. Compared to the cases the metamaterial with positive or negative permeability, the zero-permeability case is the most effective type. Secondly, when the forced-resonant frequency is lower than the self-resonant frequency of the metamaterial (e.g., f _RS = 43. 75 MHz), the PTE can be improved at this frequency by comparison to the case without metamaterial. However, when the forced-resonant frequency is higher than the self-resonant of the metamaterial (e.g., f _RS = 53 MHz), the PTE is attenuated by comparison to the case the operating frequency is close to the self-resonant frequency of the metamaterial. Finally, regardless the position of the inserted metamaterial, the previous two conclusions are still held by comparing Fig. 10(b) and Fig. 10(c).

5 Conclusion

The resonant frequency of the metamaterial, f ₀, is a critical parameter to distinguish the positive or negative permeability regions. The errors from the calculated and simulated resonant frequencies may possibly result in wrong operating region of metamaterial in the WPT applications. A simple and efficient measurement method has been herein developed and verified in both theory and simulation. Then, the WPT with insertion of metamaterial into the power transfer path has been studied. In particular, the WPT that was operated at the frequency lower than the resonant frequency of the metamaterial (i.e., in the paramagnetic region of the metamaterial) was experimentally studied. It has been found that the power transfer efficiency is improved about 10% by comparison to the case without metamaterial. Furthermore, different operating regions of the metamaterial in WPT are compared. It was shown that the case that f _RS is close to f ₀ has better PTE improvement than the cases that f _RS is lower or higher than f ₀.

Footnotes

Acknowledgements

This work was supported in part by the National Natural Science Foundation of China No. 61771175, in part by the Zhejiang Provincial Natural Science Foundation of China No. LY17F010018.

Appendix

The measurement setup shown in Fig. 2(a) is analytically modeled by the equivalent circuit shown Fig. 2(b). The resistance of a single-turn circular coil with circular cross-section at frequency ω is modeled under the impact of skin-effect [28]: (A.1) $\begin{eqnarray}\displaystyle \begin{array}{@{}l@{}}\displaystyle R_{\text{T}}=\frac{m}{2}\left(\frac{ber\left(m\right)bei^{\prime }\left(m\right)-bei\left(m\right)ber^{\prime }\left(m\right)}{\left(ber^{\prime }\left(m\right)\right)^{2}+\left(bei^{\prime }\left(m\right)\right)^{2}}\right)\cdot \frac{l_{\text{wT}}}{{\sigma}{\pi}r_{\text{wT}}^{2}}\\[24.0pt] \displaystyle m=\sqrt{{\omega}{\mu}_{\text{r}}{\mu}_{0}{\sigma}}r_{\text{wT}},\end{array} & & \displaystyle\end{eqnarray}$ where r _wT and σ are the radius and conductivity of the wire, respectively, l _wT is the length of the test coil, 𝜇₀ and 𝜇_r are the free space permeability and relative permeability of the conductor, respectively, and ber (m), bei (m), ber ^′(m), and bei ^′(m) are Kelvin functions [29]. The self-inductance of the test coil, L _T, is given as [22] (A.2) $\begin{eqnarray}L_{\text{T}}=\frac{{\mu}_{0}a}{2}\left(\ln \left(\frac{8a}{r_{\text{w}}}\right)-1.75\right),\end{eqnarray}$ where a is the radius of the test coil. The lumped parasitic capacitance, C _T, is modeled from the self-resonant frequency, f _SRFT, i.e., C _T = 1∕L _T∕(2πf _SRFT)². At the self-resonant frequency, a half of the wavelength of signal travelling along the loop equals the loop perimeter [30]: (A.3) $\begin{eqnarray}f_{\text{SRFT}}=\frac{c}{4{\pi}a},\end{eqnarray}$ where c is the velocity of electromagnetic wave.

The parasitic resistance of the unit cell (i.e., the spiral coil) is the resistance of a conductor with rectangular cross-section, (A.4) $\begin{eqnarray}\displaystyle \begin{array}{@{}l@{}}\displaystyle R_{\text{U}}=\frac{l_{\text{wU}}}{{\sigma}tw}\cdot \frac{t}{{\delta}\left(1-e^{-t/{\delta}}\right)}\\[12.0pt] \displaystyle {\delta}=\sqrt{0.5{\sigma}{\mu}_{\text{r}}{\mu}_{0}{\omega}},\end{array} & & \displaystyle\end{eqnarray}$ where δ is the skin-depth, w and t are the trace width and thickness of the wire, and s is the space between the turns as shown in Fig. 1(b). The length of the spiral coil, l _wU, is approximated as πr ² _out∕2∕p [31], where r _out and p are the outer radius of the unit cell and the pitch as shown in Fig. 1(b), respectively. The self-inductance [32] of a multi-turn spiral coil is (A.5) $\begin{eqnarray}\displaystyle \begin{array}{@{}l@{}}\displaystyle L_{\text{U}}=0.5{\mu}_{0}N^{2}\left(r_{\text{out}}+r_{\text{in}}+w\right)\left(\ln \left(\frac{2.46}{{\rho}}\right)+0.2{\rho}^{2}\right)\\[12.0pt] \displaystyle {\rho}=\frac{r_{\text{out}}-r_{\text{in}}}{r_{\text{out}}+r_{\text{in}}+w},\end{array} & & \displaystyle\end{eqnarray}$ where r _in is the inner radius of the unit cell as shown in Fig. 1(b). When modeling the parasitic capacitance, similar to the case in the test coil, the self-resonant frequency of the unit cell, f _SRFU, can be obtained firstly [30], (A.6) $\begin{eqnarray}f_{\text{SRFU}}=\frac{c}{2l_{\text{wU}}\sqrt{\displaystyle \frac{1+{\varepsilon}_{\text{r}}}{2}}}=\frac{pc}{2{\pi}r_{\text{out}}^{2}}\sqrt{\frac{2}{1+{\varepsilon}_{\text{r}}}},\end{eqnarray}$ where ϵ_r is the relative permittivity of the substrate on which the spiral coil is printed. Then the lumped parasitic capacitance, C _U is C _U = 1∕L _U∕(2πf _SRFU)².

The mutual inductance between two coaxial loops can be found in [33,34]. By considering the unit cell as a coil with effective radius b _eff = (r _out + r _in)∕2 and N turns, the mutual inductance can be approximated as, (A.7) $\begin{eqnarray}M_{\text{TU}}=\frac{{\mu}_{0}{\pi}a^{2}b_{\text{eff}}^{2}}{2\left(a^{2}+b^{2}+z^{2}\right)^{1.5}}\left(1+\frac{15}{32}{\gamma}_{1}^{2}+\frac{315}{1024}{\gamma}_{1}^{4}\right),\end{eqnarray}$ where 𝛾₁ = 2ab _eff∕(a ² + b _eff ² + z ²), and z is the axial distance between the two coils.

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