Euler–Lagrange analysis of metamaterial slabs in near field systems

Abstract

The Euler–Lagrange method is proposed for the rigorous modeling of low-frequency metamaterial (MTM) slabs in near field systems. The mechanisms of the extraordinary performance-enhancing capabilities of MTM slabs in near field systems are analyzed in numerical studies. The proposed method is verified by experimental studies of a prototype MTM-coil system. This work focuses on low-frequency MTM slabs and enables rigorous, flexible, and feasible analysis of MTM-based near field systems in electrical engineering.

Keywords

Metamaterial magnetic field numerical modelling

1. Introduction

Metamaterials (MTMs) have become a research focus in the past decades. MTM slabs (metasurfaces) exhibit advantages of small spatial volume and less power loss compared with the three-dimensional (3D) MTMs, and thus can be used in place of the 3D counterparts in various applications [1]. Recently, MTM-based low-frequency near field systems, such as wireless power transfer (WPT) [2] and magnetic field shielding [3] systems, have been proposed. MTM slabs exhibit extraordinary properties and enhance the performances of these systems, showing great potential in low-frequency near field devices and systems.

However, the available modeling methods for low-frequency MTM slabs still face challenges. First, in the commonly-used continuum media modeling method, the equivalent thickness of the MTM slabs has theoretical vagueness since slabs only have a few layers and can hardly be treated as homogeneous in the direction of thickness. The flexibility of continuum media modeling is also highly restricted because the unit cell simulations assume all unit cells to be identical [4], thus the hybrid slabs [5] can hardly be rigorously studied using the continuum media modeling method. In addition, circuit-level parameters of near field systems are hard to calculate if the MTM slabs are modeled as continuum media. Second, although MTM slabs can be alternatively modeled as infinite thin surfaces with boundary conditions [1], the deductions are still based on the wave theory and can hardly be directly applied to the non-propagating near field. Third, the full-system unit-cell level electromagnetic field simulations, like those in high-frequency cases [6], are inefficient and even impractical for low-frequency MTMs because the spiral unit cells usually require extremely fine mesh.

In this regard, the Euler–Lagrange method is introduced in this work to provide a new perspective to model low-frequency MTM slabs. The Euler–Lagrange method is well-developed in mechanics and has been applied to optical MTMs in analyzing the inter-unit coupling and energy levels in THz range [7,8]. In this work, the Euler–Lagrange is applied to low-frequency MTM slabs working in low MHz near field systems. The performance-enhancing capabilities of MTM slabs are revealed and interpreted by studying the induced current in all unit cells. The accuracy of the Euler–Lagrange method is verified by the subsequent experiments. The proposed method can be easily extended to hybrid MTMs with different unit cells. The main contribution of this work is the rigorous and flexible modeling method for low-frequency MTM slabs in near field engineering applications.

2. Euler–Lagrange modeling of low-frequency metamaterials

For low-frequency MTM slabs, spiral coils connected by capacitors C, are used as the unit cells. Each unit cell is approximated as an RLC circuit, as depicted in Fig. 1(a)–(b). L, R, and C are the spiral self-inductance, resistance, and the lumped capacitance, respectively. e is the electromotive force generated by the external magnetic field, and e’ represents the inter-unit inductive coupling. It is worth emphasizing that this approximation is valid only in the low-frequency near field realm since the stray parameters of the spirals have to be considered in the equivalent circuit on high frequencies and the simple one in Fig. 1(b) becomes inaccurate.

Fig. 1.

The low-frequency MTM unit cell. (a) The MTM unit cell. (b) The equivalent RLC series circuit of an MTM unit cell. (c) An MTM slabs consisting of multiple unit cells.

One considers an MTM slab consisting of n unit cells, as shown in Fig. 1(c). All unit cells are coupled with each other by mutual-inductances. The Lagrangian L of the slab is given by $\begin{eqnarray}\boldsymbol{L}=\frac{1}{2}\mathop{\sum }_{j=1}^{n}L_{j}\dot{q}_{j}^{2}-\mathop{\sum }_{j=1}^{n}\frac{q_{j}^{2}}{2C_{j}}+\frac{1}{2}\mathop{\sum }_{j\neq k}^{n}M_{jk}\dot{q}_{j}\dot{q}_{k}+\mathop{\sum }_{j=1}^{n}e_{j}(t)q_{j}\end{eqnarray}$ (1) where, L_j and C_j are the self-inductance and the lumped capacitance of the jth unit, M_jk is the mutual inductance between the ith and the jth units, e_j(t) is the electromotive force in the jth unit, q_j is the accumulated electric charge on the lumped capacitor in the jth unit, and $\dot{q}_{j}$ is the first order time derivative of q_j [9]. The dissipation function F is: $\begin{eqnarray}\boldsymbol{F}=\frac{1}{2}\mathop{\sum }_{j=1}^{n}R_{j}\dot{q}_{j}^{2}.\end{eqnarray}$ (2)

The Euler–Lagrange equations of the MTM slab are $\begin{eqnarray}\frac{d}{dt}\frac{\partial \boldsymbol{L}}{\partial \dot{q}_{j}}-\frac{\partial \boldsymbol{L}}{\partial q_{j}}+\frac{\partial \boldsymbol{F}}{\partial \dot{q}_{j}}=0\quad (j=1,2,\ldots ,n).\end{eqnarray}$ (3)

Substituting (1) and (2) into (3), one has the motion equation of the MTM slab as $\begin{eqnarray}\displaystyle \begin{array}{@{}l@{}}\left[\begin{array}{@{}ccc@{}}L_{1} & \cdots ∼ & M_{1,n}\\ \vdots & \ddots & \vdots \\ M_{n,1} & \cdots ∼ & L_{n}\end{array}\right]∼\left[\begin{array}{@{}c@{}}\ddot{q}_{1}\\ \vdots \\ \ddot{q}_{n}\end{array}\right]+\left[\begin{array}{@{}c@{}}R_{1}\\ \ddots \\ R_{n}\end{array}\right]∼\left[\begin{array}{@{}c@{}}\dot{q}_{1}\\ \vdots \\ \dot{q}_{n}\end{array}\right]+\left[\begin{array}{@{}c@{}}1/C_{1}\\ \ddots \\ 1/C_{n}\end{array}\right]\left[\begin{array}{@{}c@{}}q_{1}\\ \vdots \\ q_{n}\end{array}\right]=\left[\begin{array}{@{}c@{}}e_{1}(t)\\ \vdots \\ e_{n}(t)\end{array}\right].\\[18.0pt] \Rightarrow [\boldsymbol{M}]\{\ddot{\boldsymbol{X}}\}+[\boldsymbol{R}]\{\dot{\boldsymbol{X}}\}+[\boldsymbol{K}]\{\boldsymbol{X}\}=\{\boldsymbol{E}(t)\}\quad \boldsymbol{X}\in R^{n}\end{array} & & \displaystyle\end{eqnarray}$ (4) where $\ddot{q}_{j}$ is the second order time derivative of q_j. In the implementation of the method, one first assembles all matrices in (4). The capacitance in [ K ] is equal to the lumped capacitance in each MTM unit cell. The inductance and the resistance can be solved via partial element equivalent circuit programs. Since the algorithms for resistance and inductance usually have much smaller computation cost than full-system electromagnetic field simulations, the numerical analysis of low-frequency MTM slabs using the Euler–Lagrange method is simpler and faster than the unit-cell level electromagnetic field simulations.

The resonance frequencies of the MTM slab are obtained by solving the equation: $\begin{eqnarray}|\boldsymbol{M}{\lambda}^{2}+\boldsymbol{K}|=0\end{eqnarray}$ (5) where 𝜆 is the eigenvalue. The resonance frequencies are obtained from the imaginary part of 𝜆. For an MTM slab with n unit cells, a total of n resonance frequencies are obtained from (5). Each resonance frequency corresponds to a resonance mode which further affects the performance of the MTM slab. Moreover, one can easily calculate the induced current in each unit cell and analyze the performance of the MTM slab by using (4) and invoking the relation $\begin{eqnarray}\left\{\begin{array}{@{}l@{}}I_{i}(t)=\dot{q}_{i}(t)\\[3.0pt] I_{i}({\omega})=j{\omega}q_{i}({\omega})\end{array}\right.\end{eqnarray}$ (6)

3. Low-frequency near field systems with metamaterial slabs

The state-of-art applications of low-frequency MTM slabs in near field systems can be generalized as MTM-coil systems. A numerical prototype consisting of an MTM slab and an excitation coil is analyzed using the proposed method to reveal the mechanisms of such systems. The MTM unit cell consists of two reverse copper spirals, each with 8 turns, as shown in Fig. 2(a). The two reverse spirals are connected by a via on the outer ring and by a 225 pF lumped capacitor on the inner ring. The substrate is in the size of 20 mm × 20 mm × 1.6 mm. The unit cell has a 3 pF stray capacitance which is much smaller than the lumped capacitance, thus the approximations in Fig. 1(b) are valid for the unit cell. The unit cells are then arranged as a 3 × 3 array and numbered as shown in Fig. 2(b). A 5-turn coil with the radius of 50.00 mm is coaxially placed above the MTM as an excitation, as illustrated in Fig. 2(c). The distance between the coil and the MTM slab is 50.00 mm.

Fig. 2.

The solid models of the numerical analysis. (a) The solid model of a MTM unit cell. (b) An MTM slab consisting of 9 unit cells. All units are numbered. (c) The solid model of the MTM-coil system.

Fig. 3.

The calculation results of a 9-unit MTM slab. (a) The resonance frequencies. The x-coordinate of each vertical line represents the resonance frequencies. (b) The resonance modes.The y-axis represents the normalized resonance strength of each unit cell.

First, the resonance frequencies and the resonance modes of the MTM slab are calculated using ((5)). As is shown in Fig. 3(a), the number of resonance frequencies equals to the number of unit cells, which agrees with the previous analysis in Section 2. The resonance modes in Fig. 3(b) suggest that the unit cells do not behave exactly in the same way, and they even have very complicated behaviors on some frequencies. The highest mode (Mode 9 at 5.785 MHz) is of the uppermost interest in practical MTM systems since all induced current are almost in the same phase and thus the MTM slab have the strongest magnetic field manipulation capabilities. Other resonance may affect the system performance, but are weak in strength.

To compare the proposed method with other related analysis one, the resonance frequency of the same MTM unit is also computed by using the widely-used continuum media method. Since an infinite periodic arrangement of unit cells is assumed in the continuum media method, only one resonance (7.08 MHz, denoted as $f_0'$ ) is obtained through the continuum media theory. Moreover, $f_0'$ does not agree with the measured resonance frequency f₀ = 6.3 MHz of the 8 × 8 MTM slab, whereas Euler–Lagrange method captures the resonance frequency accurately, as given in Section 4. Consequently, Euler–Lagrange modeling demonstrates a clearer physical view of the MTM resonance. The frequency-dependent induced current in unit cells are then calculated by reiterating ((4)) as: $\begin{eqnarray}-{\omega}^{2}[\boldsymbol{M}]\{\boldsymbol{X}\}+j{\omega}[\boldsymbol{R}]\{\boldsymbol{X}\}+[\boldsymbol{K}]\{\boldsymbol{X}\}=\{\boldsymbol{E}({\omega})\}\quad \boldsymbol{X}\in R^{n}\end{eqnarray}$ (7) where ω is the angular frequency. The elements in $\{\boldsymbol{E}({\omega})\}$ denote the external excitation on each MTM unit cell and can be calculated by $\begin{eqnarray}E_{i}({\omega})=-j{\omega}M_{ex,i}I_{ex}\quad i=1,2,\ldots ,9,\end{eqnarray}$ (8) where M_{ex, i} is the mutual inductance between the ith unit cell and the excitation coil, and I_ex is the current in the excitation coil. The calculated induced current are plotted in Fig. 4. A strong resonance is observed on f₀ = 5.785 MHz which agrees with the highest resonance frequency shown in Fig. 3(a). The induced current in each MTM unit cell reaches maximum and has a −90^° phase shift on the highest resonance frequency. Some weak resonances are also observed between 4.5 MHz and 5.5 MHz and have trivial effect on the entire system.

Fig. 4.

The induced current in all MTM units. (a) The magnitude, (b) the phase, and (c) the real part of the induced current. Modeling and calculations were performed using ANSYS and MATLAB.

The mechanisms of MTM-based systems can be studied by the in-phase component of the induced current with respect to the excitation I_ex, as shown in Fig. 4(c). When the frequency rises to just below f₀, the in-phase component increases, and the reaction field of the MTM slab enhances the excitation magnetic field. When the frequency rises above f₀, the in-phase component decreases and the counter-phase component increases, thus the MTM generates a reaction field inverse to the excitation field. In this case, the MTM slab acts as a shield.

The calculated induced current also benefits the circuit-level design of MTM-based systems. For example, one can readily obtain the frequency-dependent input impedance: $\begin{eqnarray}Z_{in}=Z_{coil}+j{\omega}([M_{ex,1}\cdots M_{ex,n}][\boldsymbol{I}_{1}\cdots \boldsymbol{I}_{n}]^{T})/I_{ex}\end{eqnarray}$ (9) where Z_coil is the impedance of the excitation coil, and I _i is the calculated complex induced current in the ith unit cell. By contrast, if the MTM slab is homogenized as continuous media, the analytical computation of Z_coil becomes very complicated and one often has to conduct full-system electromagnetic field computations to obtain Z_in.

4. Experimental verifications

To verify the proposed method, a prototype MTM-coil system is developed in Fig. 5. MTM unit cells identical to those in Fig. 2 are fabricated and arranged as an 8 × 8 array slab. A standard WPT coil, loaded with a sinusoidal voltage, is used as the excitation coil (Tx). A capacitor is connected in series with Tx to improve the input performance for easier measurement. An open-circuited 5-turn coil (Rx) identical to that in Fig. 2(c) is used as a magnetic field sensor which generates higher voltage with stronger magnetic field. The Tx, MTM slab, and the Rx are placed coaxially. Both the distance between the Tx and the MTM and the distance between the MTM slab and the Rx are 50 mm. One conducts a frequency sweep of the system from 4 MHz to 7 MHz. On each frequency, the magnitude of input voltage on Tx (denoted as V_in), the open-circuit induced voltage on Rx (denoted as V_out), the magnitude of input current on Tx (denoted as I_in), and the phase shift (denoted as 𝛼) from I_in to V_out are measured using oscilloscope, high-frequency current probes, and differential voltage probes. The magnitude of Z₂₁ and the system input impedance are calculated on each frequency by using the measured voltages and currents.

In the implementation of the proposed Euler–Lagrange method, one conducts a frequency sweep on (7) and calculates the transmission impedance by: $\begin{eqnarray}Z_{21}=(j{\omega}[M_{Rx,1}∼\cdots ∼M_{Rx,n}∼∼M_{Rx,Tx}][\boldsymbol{I}_{1}∼\cdots ∼\boldsymbol{I}_{n}∼\boldsymbol{I}_{ex}]^{T})/\boldsymbol{I}_{ex}\end{eqnarray}$ (10) where M_{Rx, i} is the mutual inductance between the Rx and the ith MTM unit cell, M_{Rx, Tx} is the mutual inductance between the Rx and the Tx, and I _ex is the current in Tx.

Fig. 5.

The MTM units and the MTM-coil system. (a) The fabricated MTM units. Multiple lumped capacitors can be soldered in parallel. (b) The MTM slab with 8 × 8 unit cells. (c) The prototype MTM-coil system.

Fig. 6.

The measured and calculated results. (a) The magnitude of Z₂₁. (b) The input resistance. (c) The input reactance. (d) The calculated resonance frequencies of the MTM slab.

The experimental and calculated results are compared in Fig. 6. A strong resonance near 6.3 MHz is observed in the experiments and also accurately calculated by the Euler–Lagrange method. Figure 6(a) shows the coupling between Tx and Rx can be enhanced or reduced depending on the system operation frequency. The magnetic field is enhanced when Z₂₁ with MTM is greater than that without MTM, and is weakened when Z₂₁ with MTM is smaller than that without MTM. Consequently, one can change the lumped capacitors in the unit cells to tune the MTM slab to behave either field-enhancing or shielding at a fixed frequency. The input impedance near the MTM resonance is also accurately calculated by the Euler–Lagrange method, as shown in Fig. 6(b)–(c). Moreover, the highest resonance frequency in Fig. 6(d) agrees to the resonance frequency in Fig. 6(a).

5. Conclusion

The Euler–Lagrange method is introduced to provide a new modeling approach for low-frequency MTM slabs in near field systems. The resonance frequencies are readily calculated through the characteristic equation, and the MTM slab has the strongest resonance on the highest resonance frequency. The response of each MTM unit under external excitations are calculated using the Euler–Lagrange equations. The mechanisms of MTM-based near field systems are interpreted through the study on an MTM-coil system. This work enables the accurate design of MTM-based WPT and magnetic field shield systems, benefiting the future applications of MTM slabs in low frequency near field systems.

Footnotes

Acknowledgements

The authors acknowledge support from the National Natural Science Foundation of China (Grant No. 52077192), the National Key Research and Development Project (Grant No. 2020AAA0109001), and Zhejiang Province Key R & D programs (No. 2021C05004).

References

Holloway

C.L.

, A discussion on the interpretation and characterization of metafilms/metasurfaces: The two-dimensional equivalent of metamaterials, Metamaterials3 (2009), 100–112.

Guan

Liu

Yang

and Chen

, Efficiency enhancement of wireless power transfer system by using resonant coil and negative permeability metamaterial, Int. J. Appl. Elctrom.59 (2019), 647–655.

, Investigation of negative and near-zero permeability metamaterials for increased efficiency and reduced electromagnetic field leakage in a wireless power transfer system, IEEE Trans. Electromag. Compat.61 (2019), 1438–1446.

Numan

A.B.

and Sharawi

M.S.

, Extraction of material parameters for metamaterials using a full-wave simulator, IEEE Antenn. Propag. M.55 (2013), 202–211.

Cho

, Thin hybrid metamaterial slab with negative and zero permeability for high efficiency and low electromagnetic field in wireless power transfer systems, IEEE Trans. Electromag. Compat.60 (2018), 1001–1009.

Almoneef

T.S.

and Ramahi

O.M.

, A 3-dimensional stacked metamaterial arrays for electromagnetic energy harvesting, Prog. Electromagn. Res.146 (2014), 109–115.

Giessen

and Liu

, Coupling effects in optical metamaterials, Angew. Chem. Int. Ed.49 (2010), 9838–9852.

Sersic

, Electrical and magnetic dipole coupling in near-infrared split-ring metamaterial arrays, Phys. Rev. Lett.103 (2009), 213902.

Goldstein

Poole

and Safko

, Variational principles and Lagrange’s equations, in: Classical Mechanics, Addison Wesley, 2000, pp. 34–64.