Practical Model for Metamaterials in Wireless Power Transfer Systems

applied sciences

Article Practical Model for Metamaterials in Wireless Power Transfer Systems

Jingying Liu 1, Zhi Gong 1 , Shiyou Yang 1, Hui Sun 1 and Jing Zhou 1,2,* 1 College of Electrical Engineering, Zhejiang University, Hangzhou 310000, China; [email protected] (J.L.); [email protected] (Z.G.); [email protected] (S.Y.); [email protected] (H.S.) 2 Polytechnic Institute, Zhejiang University, Hangzhou 310000, China * Correspondence: [email protected]

Received: 29 October 2020; Accepted: 27 November 2020; Published: 28 November 2020

Abstract: Metamaterials (MTMs) with extraordinary electromagnetic properties are recently applied to wireless power transfer (WPT) systems to improve power transmission efficiency. Although theoretical progress has been made on MTMs in low frequency near field, in the operation frequency of most WPT systems (usually MHz), the design of MTMs still utilizes the model used in high-frequency applications. Therefore, a practical model of MTMs in low MHz band is proposed in this work. The resonance frequency and quality factor are used to describe the characteristics of an MTM slab. The near field WPT systems with MTMs are then modeled as electric circuits, the system efficiency is explicitly deduced, and optimization algorithms are employed to optimize the MTM resonance frequency and maximize the system efficiency. The proposed practical model is validated via a prototype wireless power transfer system operating at 6.78 MHz. Experiments show that the proposed MTM model has good accuracy for low MHz WPT systems compared with the high-frequency model. The proposed practical model of MTMs provides an accurate way to analyze the performance of MTM at low MHz frequencies and greatly benefits the future exploitation of MTM-based low-frequency near field applications.

Keywords: metamaterial; near ﬁeld; power engineering; wireless power transfer; low frequency

1. Introduction Metamaterials (MTMs) are novel artificial media consisting of a large number of microscopic unit cells and exhibiting negative permeability and/or negative permittivity on some frequencies [1–6]. Since the first experimental demonstration of a negative index of refraction [7], numerous academic achievements on MTMs have been reported including transforming optics [8], invisibility cloaks [9], and novel MTM-based antennas [10]. These researches focus on high-frequency electromagnetics or optics. Recently, MTMs have been introduced to low-frequency near field applications such as wireless power transfer (WPT) and magnetic resonance imaging (MRI) [11–20]. Unlike those in high-frequency electromagnetics, the electric and magnetic fields are almost decoupled in the near field, and thus the near field can be manipulated by single-negative MTMs [21]. The MTM-enhanced WPT system is one of the major applications of low-frequency MTMs in the near field realm. WPT attracts great attention due to its capability of feeding power easily to autonomous electronic devices, such as mobile phones, wearable products, implantable medical devices, and electric vehicles [22–25]. However, the engineering applications of the WPT systems are usually restricted by the short transfer distance and the low transmission efficiency [26]. MTM slabs are introduced in the WPT systems to enhance the evanescent wave and thus improve the power transmission efficiency [11]. Theoretical analysis and simulations have shown that the power transfer

Appl. Sci. 2020, 10, 8506; doi:10.3390/app10238506 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, 8506 2 of 13 efficiency with MTMs can be improved by an order when the loss is limited [12]. Moreover, it has been experimentally confirmed that the MTM slabs enhance the WPT system efficiency by about 30%, and the location, as well as the quantity of the slabs, also has a great effect on the system performance [13,17–19]. In low-frequency applications, researchers have made solid contributions to the analytical work of MTMs. In previous studies, MTMs have been considered as waveguides and used for the near field imaging at 21.3 MHz [27]. Furthermore, it has been shown that the properties of an MTM unit could be described by an equivalent circuit consisting of bulk and distributed elements [28], which provides suitable theoretical guidance for the MTMs in the low-frequency near-field. In [29], the near-field manipulating device was designed utilizing biatomic MTM structure, both simulations and experiments were carried out afterward to verify the analytical predictions at 45.8 MHz. Although some issues have been discussed and concluded in [27–29], the applications of MTMs in low-frequency near field systems are still faced up with many difficulties. Firstly, investigations on the characteristics of MTMs is restricted in the lower frequency (usually a few MHz) and near field range. Up to now, the equivalent circuit model has not been set up for the MTM-enhanced WPT systems. The theoretical models and parameters (for example, S-parameters) utilized in high-frequency scenarios were usually applied to low-frequency systems directly. Moreover, the measurement of the transmission efficiency is not intuitive, and the corresponding retrieval algorithm [30,31] is complicated, making the calculation awkward and unpractical. Finally, in order to make the MTM slab working at negative permeability, the resonance frequency of the slab should be slightly lower than the working frequency of the WPT system, which holds out rather high requirements on the design of PCB MTMs [32,33]. Due to the limitations of processing techniques, there is no flawless method to precisely decide the resonance frequency of the MTM slab. In a word, the research methodologies for MTM-enhanced WPT systems is still highly restricted. In order to solve the aforementioned problems, a practical model for MTMs used in WPT systems is proposed in this work. The relative permeability of MTMs is decided by the resonance frequency and the quality factor. A numerical model is proposed in order to obtain the quality factor of the MTM slab. A practical circuit model of WPT systems with MTMs is established. The power efficiency is calculated by the resonance frequency and the quality factor of the MTM slab. An optimization algorithm is used to optimize transmission power efficiency. The proposed MTM-enhanced WPT system is validated via physical experimental studies. Unlike the calculation methods in high-frequency electromagnetics, the proposed model is more suitable for MTM-based low-frequency near field systems. The circuit model provides a more accurate and suitable way to predict the performance of WPT systems with MTM slabs.

2. Practical Model for Low-Frequency Metamaterials

2.1. Lumped Impedance Parameters of the MTM Unit The Swiss roll, as shown in Figure1, is the most widely-used structure for low-frequency MTM units, which can be simply fabricated using thePrinted Circuit Board (PCB) technology. Two reverse spirals are printed on both sides of the FR4 (a kind of glass ﬁber composite material) substrate and connected by vias in order to reduce the MTM resonance frequency [34–36]. To facilitate easy adjustment of the MTM resonance frequencies, lumped capacitors are connected in series with the reverse Swiss rolls. Moreover, it has been experimentally conﬁrmed that the lumped capacitance dominates the resonance on low frequencies [37]. Therefore, one can use lumped parameters to characterize the MTM units. Appl. Sci. 2020, 10, 8506 3 of 13 Appl. Sci. 2020, 10, x FOR PEER REVIEW 3 of 14

FigureFigure 1.1. TheThe SwissSwiss rollroll structurestructure of of the the metamaterial metamaterial (MTM) (MTM) unit. unit. Two Two reverse reverse spirals spirals on on each each side side of Fr4 substrate. of Fr4 substrate. In order to design MTMs in a proper way in the near field, the lumped parameters are used to In order to design MTMs in a proper way in the near field, the lumped parameters are used to model the MTM unit. According to [37], the relative permeability µ of the MTM modeled by the model the MTM unit. According to [37], the relative permeability μr of the MTM modeled by the lumped parameters is given in (1): r lumped parameters is given in (1): 2 ω2 µ = 1 + F ω , (1) r μ =+1,F ⋅ 2 2 r · −+ωωωω22+ jω(()ωM/QM) ++ ωω (1) − jQMM/ MM whereωωM is the resonance frequency of MTMs (related to the effective inductance and capacitance), where M is the resonance frequency of MTMs (related to the effective inductance and capacitance), and Q is the quality factor of the MTMs. F is a coefficient related to the shape and the structure of and Q Mis the quality factor of the MTMs. F is a coefficient related to the shape and the structure of the unit:M the unit: P 2 µ n S 0 · i=1 i F = n 2 , (2) μ L⋅e() f f VunitS 0 i=1 i F = ， (2) L V S i where e f f is the effective inductance, unit is the volumeLVeff unit of an MTM unit, and i is the area of th turn in a unit with n turns in total. The coefficient F typically falls in the range of 0.16–0.50, according to [38]. whereSince theLeff lumpedis the effective capacitance inductance, is much V largerunit is the than volume the parasitic of an MTM capacitance unit, and of theSi is unit, the thearea latter of ith can turn be inneglected a unit with in this n turns case. in In total. other The words, coefficient if an MTM F typically unit is in falls a fixed in the shape range and of structured 0.16–0.50, (fixedaccordingF) with to [38].a lumped Since capacitor, the lumped the capacitance property of is its much relative larger permeability than the parasitic can be decided capacitance by the of resonance the unit, frequencythe latter can(calculated be neglected by Le fin f andthis capacitance)case. In other and words, the qualityif an MTM factor. unit is in a fixed shape and structured (fixed F) with a lumped capacitor, the property of its relative permeability can be decided by the resonance 2.2. Q-Based Design Theory in the MTM Slab frequency (calculated by Leff and capacitance) and the quality factor. MTMs used in WPT systems usually appear in the form of slabs composing of several MTM units 2.2.arranged Q-Based periodically, Design Theory which in the can MTM be regarded Slab as effective media [39]. In order to obtain the relative permeabilityMTMs used of the in MTMWPT systems slab, the usually common appear numerical in the model form utilizingof slabs composing electromagnetic of several waves MTM and unitsthe corresponding arranged periodically, periodic boundary which can conditions be regarded is usually as effective used asmedia in [37 [39].]. In In this order numerical to obtain model, the waverelative ports permeability are used asof the MTM excitation, slab, the and common perfect electronicnumerical conductormodel utilizing (PEC) electromagnetic and perfect magnetic waves andconductor the corresponding (PMC) are used periodic as symmetric boundary boundary conditions conditions is usually to obtain used S-parameters. as in [37]. In Thenthis numerical a retrieval model,algorithm wave is employed ports are to used calculate as the the excitation, relative permeability and perfect of electronic the MTM conductor slab. However, (PEC) this and technique perfect magneticis less rigorous conductor when (PMC) applied are toused systems as symmetric on low MHzboun frequencies.dary conditions The to distance obtain S-parameters. between the waveThen aports retrieval and thealgorithm MTM unit is employed is too large to tocalculate be modeled the relative in the lowpermeability frequencies. of the Moreover, MTM slab. modeling However, the thisentire technique MTM slab is at less the rigorous unit cell levelwhen is alsoapplied impractical to systems due toon the low limitation MHz frequencies. of computation Thecapability. distance betweenNevertheless, the wave accordingports and tothe the MTMQ-based unit design is too theorylarge [to38 ]be and modeled (1), the in relative the low permeability frequencies. of Moreover,the entire slabmodeling can be the easily entire and MTM accurately slab at the derived unit ce fromll level the is numericalalso impractical model due shown to the in limitation Figure2. ofThe computationQ factor of the capability. effective medium response is essentially equal to the Q factor of an individual resonant particle.Nevertheless, This implies according that by measuringto the Q-based a single design fabricated theory MTM [38] and unit, (1), one the can relative simply permeability and accurately of theestimate entirethe slab relative can be permeabilityeasily and accurately or permittivity derived offrom an MTMthe numerical slab without model complicated shown in Figure simulation, 2. The Q factor of the effective medium response is essentially equal to the Q factor of an individual resonant particle. This implies that by measuring a single fabricated MTM unit, one can simply and accurately Appl. Sci. 2020, 10, x FOR PEER REVIEW 4 of 14

Appl.estimate Sci. 2020 the, 10 relative, x FOR PEER permeability REVIEW or permittivity of an MTM slab without complicated simulation,4 of 14

Appl.fabrication, Sci. 2020, or10, 8506measurements. The lumped port is used as the excitation, and four PEC boundaries4 of 13 estimateare used thefor relative surrounding permeability outer boundaries. or permittivi Thety of solid an MTM model slab represents without complicateda two-dimensional simulation, (2D) fabrication,extension of or MTM measurements. units and is The a proper lumped way port to deis scribeused asthe the structure excitation, of an and MTM four slab. PEC Unlike boundaries those arefabrication,researches used for inheriting or surrounding measurements. high-frequency outer The boundaries. lumped analysis port The methods, is used solid as model thethis excitation,is representsa single-port and a four two-dimensionalsimulation PEC boundaries without (2D) are S- extensionusedparameters for surrounding of orMTM inversion units outer andcalculations. boundaries. is a proper After Theway solidadding to de modelscribe the representslumpedthe structure capacitor, a two-dimensional of an theMTM resonance slab. (2D) Unlike frequency extension those researchesofand MTM quality units inheriting factor and isof a high-frequencythe proper MTM way slab to describecananalysis be calculated themethods, structure by this ofthe is an admia MTMsingle-portttance slab. (Y-parameters) Unlike simulation those without researches from the S- parametersinheritingsimulation high-frequency resultsor inversion shown calculations. analysisin Figure methods, 3. AfterThe relative adding this is apermeabilitythe single-port lumped simulationcapacitor,of the MTM the without slab resonance is S-parametersthen concludedfrequency or andinversionby (1). quality calculations. factor of the After MTM adding slab the can lumped be calculated capacitor, by the the resonance admittance frequency (Y-parameters) and quality from factor the simulationof the MTM results slab can shown be calculated in Figure by 3. the The admittance relative permeability (Y-parameters) of fromthe MTM the simulation slab is then results concluded shown byin Figure(1). 3. The relative permeability of the MTM slab is then concluded by (1).

Figure 2. The setting of the designed MTM slab with four perfect electronic conductor (PEC) boundaries. FigureFigure 2. The 2. The setting setting of the of designed the designed MTM MTM slab with slab four with perfect four perfect electronic electronic conductor conductor (PEC) boundaries. (PEC) boundaries.

Figure 3. The admittance parameter (Y-parameter) of an MTM slab. The quality factor Q can be Figure 3. The admittance parameter (Y-parameter) of an MTM slab. The quality factor Q can be calculated from the graph. Take the resonance frequency f , and get the high limit frequency f and low calculated from the graph. Take the resonance frequencym fm, and get the high limit frequencyh fh and Figurelimit frequency 3. The admittancef l when the parameter amplitude (Y-parameter) of Y-parameter of drops an MTM to 0.707 slab. of itself,The quality then Q factor= Qffm=−/ Q( f h()can ff lbe) f. low limit frequency fl when the amplitude of Y-parameter drops to 0.707 of itself, then mhl− calculated from the graph. Take the resonance frequency fm, and get the high limit frequency fh and . It should be pointed out that after adding the PEC, in order to tune the MTM=− slab,() the value low limit frequency fl when the amplitude of Y-parameter drops to 0.707 of itself, then Qfmhl f f of the series capacitance is different from the previous results of the MTM unit. Under the periodic .It should be pointed out that after adding the PEC, in order to tune the MTM slab, the value of boundary, the value of capacitance cannot be simply calculated by ω = √LC because the influence the series capacitance is different from the previous results of the MTM unit. Under the periodic of theIt surroundingshould be pointed MTM out unit that needs after to adding be considered. the PEC, The in order proposed to tune simulation the MTM method slab, the can value exactly of boundary, the value of capacitance cannot be simply calculated byω= LC because the influence of thereflect series the capacitance difference due is different to the periodicity from the ofprev MTMious units,results which of the makes MTM theunit. design Under and the frequency periodic the surrounding MTM unit needs to be considered. The proposed simulation method can exactly modulation of MTMs more accurate and reliable. ω boundary,reflect the thedifference value of due capacitance to the periodicity cannot be of simply MTM calculatedunits, which by makes= LC thebecause design the and influence frequency of the surrounding MTM unit needs to be considered. The proposed simulation method can exactly 2.3.modulation Analysis of of MTM-EnhancedMTMs more accurate WPT Systemand reliable. reflect the difference due to the periodicity of MTM units, which makes the design and frequency modulation2.3. AnalysisA practical ofof MTMsMTM-Enhanced model more for WPT accurate WPT systems Systemand with reliable. MTM slabs is proposed and shown in Figure4. The model of the MTM slab is established using lumped circuit parameters, in which RL is the resistance of the 2.3.load Analysis andA practicalω is of the MTM-Enhanced model operational for WPT angular WPT systems System frequency. with MTM slabs is proposed and shown in Figure 4. The model of the MTM slab is established using lumped circuit parameters, in which RL is the resistance of theA load practical andω modelis the operationalfor WPT systems angular with frequency. MTM slab s is proposed and shown in Figure 4. The model of the MTM slab is established using lumped circuit parameters, in which RL is the resistance of the load andω is the operational angular frequency. Appl. Sci. 2020, 10, 8506 5 of 13 Appl. Sci. 2020, 10, x FOR PEER REVIEW 5 of 14

Figure 4.4. The proposed modelmodel forfor thethe MTM-enhancedMTM-enhanced wirelesswireless powerpower transfertransfer (WPT)(WPT) system.system. MMijij isis the mutualmutual inductanceinductance betweenbetween coilscoils andand MTMs.MTMs.

According to Kirchhoff’s Law, the circuit equation is given by (3), where ω0 is the operation According to Kirchhoff’s Law, the circuit equation is given by (3), where 0 is the operation frequency of the WPT system, ω1ωis the resonance frequency of both transmission and receiving coils, frequency of the WPT system, 1 is the resonance frequency of both transmission and receiving and ωM is the resonance frequency of MTMs. Assuming ω1 = ω0, the system efficiency η can be ω ωω= η calculatedcoils, and byM using is the (4).resonance Substituting frequency (3) into of MTMs. (4), one Assuming finds that10 the system, the system efficiency efficiency is related can to ωbeM calculatedand QM. Inby otherusing words, (4). SubstitutingωM and Q (3)M are into the (4), key one parameters finds that influencingthe system efficiency the efficiency is related of such to ω ω MTM-enhancedM and QM . In WPTother systems.words, M and QM are the key parameters influencing the efficiency of such MTM-enhanced WPT systems.         ω ωω0 ω1 R 0 0 I Lω0 1 ω0M ω0M I U  1s  1  −−ωωω1 1 ω1 ω0 1M 13  1   s   −  LMM11−− 0 1M 0 13         ωω ω0 ωM      0 R 0  I  + j ω0M L ω ω0M  I  =  0  (3)  M  M  101M M M ωM ω0 M3  M     −RI−00   − − ω ω  IU   011s 0 R  I  ω M ω M L ω 0 1  I1s  0  3L 3 0 13 ω0 M3ω 3 1 ω1 ω0 3 −−−− ωω0 M − ω−  00+RIjMLMM 01MMM 0M3M MI =0  ωω  (3) 00−RI M 2 0  I 0 3L  3 Pout I RL 3 η(ω , Q ) = = 3 .ω ω (4) ωωωMMLM M P I U −−0 1 013in 0M31 s 31ωω 10 In order to obtain the maximum power efficiency of MTM-enhanced WPT systems, we designed 2 an optimization algorithm by turning ωM. QM variesPIR only ωM because LM and RM are fixed when the ηω(),=Q out = 3L shape of the MTM unit is determined. AsMM for the relative permeability. of the MTM slab, it is exactly(4) PIUin1 s related to ωM and QM when the parameters of the WPT system is settled. Therefore, the optimization goal andIn order restrictions to obtain are the as maximum follows: power efficiency of MTM-enhanced WPT systems, we designed ω ω an optimization algorithm by turning M Max. QM variesη only M because LM and RM are fixed when the shape of the MTM unit is determined. Ass.t. for the Re re(µlativer) a permeability, of the MTM slab, it is exactly(5) ω ≤ related to and Q when the parameters of theIm WPT(µr) systemb is settled. Therefore, the optimization M M ≤ wheregoal anda and restrictionsb are the controllableare as follows: limitations. The optimal solution can be acquired using optimization algorithms such as genetic algorithm (GA)Max or particleη swarm optimization (PSO) to solve this problem. After calculation, we obtain the optimal ωM which()μ can≤ be tuned， by adjusting the additional capacitor s.t. Re r a (5) in the MTM slab, and then the efficiency of the WPT system with the MTM slab reaches its maximum Im ()μ ≤ b value. It should be pointed out that, the optimizationr result is dependent on the setting of limitations. Inwhere particular, a and the b parameterare the controllable b that represents limitations. the loss The of MTMs optimal needs solution to be practical, can be soacquired the results using can beoptimization verified by algorithms experiments. such as genetic algorithm (GA) or particle swarm optimization (PSO) to solve this problem. After calculation, we obtain the optimalω which can be tuned by adjusting the 3. Experimental Verification M additional capacitor in the MTM slab, and then the efficiency of the WPT system with the MTM slab reachesAprototype its maximum MTM-enhanced value. It should WPT be system pointed is ou developedt that, theto optimization verify the proposed result is dependent model. First, on the setting MTM unitsof limitations. are designed In particular, and fabricated, the parameter as shown b that in represents Figure5. The the sizeloss of of MTMs the MTM needs unit to isbe 20practical, mm 20so mmthe results1.6 mm can. be Two verified reverse by spirals, experiments. each with seven turns, are printed on both sides of the × × FR4 substrate and connected by a via and a lumped capacitor. The MTM units are then arranged as a Appl. Sci. 2020, 10, x FOR PEER REVIEW 6 of 14 Appl. Sci. 2020, 10, x FOR PEER REVIEW 6 of 14 3. Experimental Verification 3. Experimental Verification A prototype MTM-enhanced WPT system is developed to verify the proposed model. First, the MTM Aunits prototype are designed MTM-enhanced and fabricated WPT, assystem shown is developin Figureed 5. to The verify size the of theproposed MTM unitmodel. is 20 First, mm the × Appl.20MTM Sci.mm2020 units× ,1.610, 8506aremm. designed Two reverse and fabricatedspirals, each, as shownwith seven in Figure turns, 5. are The printed size of onthe both MTM sides unit of is the20 mmFR46 of × 13 substrate20 mm × and 1.6 connectedmm. Two byreverse a via andspirals, a lumped each with capacitor. seven Theturns, MTM are unitsprinted are onthen both arranged sides of as thea two- FR4

dimensionalsubstrate and slab, connected as is shown by a viain Figure and a lumped5b. The lumpedcapacitor. impedance The MTM parameter units are thenLeff of arranged an MTM as unit a two- is two-dimensional slab, as is shown in Figure5b. The lumped impedance parameter Le f f of an MTM aboutdimensional 4.89 μH slab, and Fas = is 0.37, shown correspondingly in Figure 5b. Theaccord lumpeding to impedancesimulation results.parameter Leff of an MTM unit is unit is about 4.89 µH and F = 0.37, correspondingly according to simulation results. about 4.89 μH and F = 0.37, correspondingly according to simulation results.

(a) (b) (a) (b) Figure 5. The fabricated MTMs, (a) the MTM unit. (b) The MTM slab. FigureFigure 5. The5. The fabricated fabricated MTMs, MTMs, (a ()a the) the MTM MTM unit. unit. ((bb)) TheThe MTM slab. slab. A WPT system resonating at 6.78 MHz is then established, shown in Figure 6. An A4WP spiral powerA WPTA transmit WPT system system resonator resonating resonating is used at at 6.78as 6.78 the MHz MHztransmission is is then then established, established, coil (Tx). In shown shownorder to inin eliminate Figure 66.. the An influenceA4WP spiral spiral of powerparasiticpower transmit transmit parameters, resonator resonator a metal is usedis usedspiral as as theis the designed transmission transmission as a receiving coil coil (Tx). (Tx). coil InIn orderorder(Rx), as to showneliminate in theFigure the influence inﬂuence 7. Both of of parasiticcoilsparasitic are parameters, printed parameters, on a PCB metal a metaland spiral adjusted spiral is designed is to designed resonate as a as receivingat a6.78 receiving MHz. coil Acoil (Rx), signal (Rx), as showngenerator as shown in Figureand in Figure an7 .amplifier Both 7. Both coils are used as the excitation. An impedance matching circuit is connected between the source and Tx. are printedcoils areon printed PCB andon PCB adjusted and adjusted to resonate to resonate at 6.78 MHz. at 6.78 A MHz. signal A generator signal generator and an ampliﬁerand an amplifier are used Aare 50 usedΩ resistor as the is excitation. loaded on Anthe impedancereceiving coil. matching circuit is connected between the source and Tx. as the excitation. An impedance matching circuit is connected between the source and Tx. A 50 Ω A 50 Ω resistor is loaded on the receiving coil. resistor is loaded on the receiving coil.

Appl. Sci. 2020, 10, x FOR PEER REVIEW 7 of 14 FigureFigure 6. 6.The The proposedproposed experimental system. system. Figure 6. The proposed experimental system.

(a) (b)

FigureFigure 7.7. ((a) The transmitting coil, coil, (b (b) )the the receiving receiving coil. coil.

the system are shown in Table 1. The optimized result is fM = 6.5 MHz. By tuning the capacitor in ω MTMs, the resonance frequency of the MTM slab ( M ) is changing as well. It is important to decide the lumped capacitance of each unit in the MTM slab according to the optimal resonance frequency. ω ω However, since M is influenced by the surrounding units, it is complicated to obtain M by ω calculation, and M is also difficult to be tested in experiments. Therefore, an experiment is designed to imitate the periodic boundaries around an MTM unit. A lumped port is located at the interface of the metal spiral, and 4 PEC boundaries are added around an MTM unit shown in Figure 8a, which corresponds to the simulation setting as presented in chapter 2. According to [40], a piece of metal foil is used as the PEC boundary which surrounds an MTM unit to form a cubic test box. As shown in Figure 8b, port 1 of a Vector Network Analyzer is connected to the lumped port of the MTM unit for excitation and testing.

Table 1. Parameters of the WPT system.

Parameter Value Parameter Value Parameter Value

R1 2.5 Ω L1 4.7 μH C1 132 pF

R3 5 Ω L3 26.7 μH C3 20 pF

Rs 5 Ω RLeq 50 Ω Q2 41.66 Appl. Sci. 2020, 10, 8506 7 of 13

3.1. Permeability Measurement In order to get the characteristics of the MTM slab, the relative permeability of the fabricated MTM is measured. Firstly, the genetic algorithm with the optimization goal and the restrictions presented in chapter 2 is used to find the optimal efficiency of the WPT system. The parameters of the system are shown in Table1. The optimized result is fM = 6.5 MHz. By tuning the capacitor in MTMs, the resonance frequency of the MTM slab (ωM) is changing as well. It is important to decide the lumped capacitance of each unit in the MTM slab according to the optimal resonance frequency. However, since ωM is influenced by the surrounding units, it is complicated to obtain ωM by calculation, and ωM is also difficult to be tested in experiments. Therefore, an experiment is designed to imitate the periodic boundaries around an MTM unit. A lumped port is located at the interface of the metal spiral, and 4 PEC boundaries are added around an MTM unit shown in Figure8a, which corresponds to the simulation setting as presented in chapter 2. According to [40], a piece of metal foil is used as the PEC boundary which surrounds an MTM unit to form a cubic test box. As shown in Figure8b, port 1 of a Vector Network Analyzer is connected to the lumped port of the MTM unit for excitation and testing.

Table 1. Parameters of the WPT system.

Parameter Value Parameter Value Parameter Value

R1 2.5 Ω L1 4.7 µH C1 132 pF R3 5 Ω L3 26.7 µH C3 20 pF Appl. Sci. 2020, 10, x FOR PEERRs REVIEW5 Ω RLeq 50 Ω Q2 41.66 8 of 14

Figure 8. The proposed measurement setup ( a) The PEC boundaries are made of metal foil with an MTM unit. ( b) The The MTM MTM unit is added added inside the boundaries. ( c) Port Port 1 1 connects connects to to the the vector vector network network analyzer to get the impedance parameters. The Y-parameters is obtained after adding a 135 pF lumped capacitor in the proposed MTM unit, The Y-parameters is obtained after adding a 135 pF lumped capacitor in the proposed MTM unit, and is showed in Figure9. It is obvious that the MTM slab resonates at 6.49 MHz with the lumped and is showed in Figure 9. It is obvious that the MTM slab resonates at 6.49 MHz with the lumped capacitor, whereas the MTM unit without boundaries resonates at 6.16 MHz. It can be seen that the capacitor, whereas the MTM unit without boundaries resonates at 6.16 MHz. It can be seen that the resonance frequency predicted by the experimental method is in good agreement with the simulation resonance frequency predicted by the experimental method is in good agreement with the simulation result. Since the utilization of copper foil introduces power loss, the quality factor obtained by the result. Since the utilization of copper foil introduces power loss, the quality factor obtained by the experiment has a deviation from the simulation result within tolerance. experiment has a deviation from the simulation result within tolerance.

Figure 9. The impedance results of MTMs.

ω Once the M and QM has been obtained, the relative permeability of the MTM slab is calculated and shown in Figure 10. The MTM slab resonates at 6.49 MHz, and the real part of relative permeability is negative at 6.78 MHz. Moreover, because of the power loss from copper foil in the experiment, the imaginary part (represents the loss of MTMs) of the experimental results is slightly bigger than the simulation results. This result is also consistent with the formula ω= LC . According to the experiment and simulation results, the effective inductance of an MTM unit is 4.9 μH. Because of the interaction between MTM units, the resonance frequency of the MTM slab is difficult to decide, so experimental verification is essential. Appl. Sci. 2020, 10, x FOR PEER REVIEW 8 of 14

Figure 8. The proposed measurement setup (a) The PEC boundaries are made of metal foil with an MTM unit. (b) The MTM unit is added inside the boundaries. (c) Port 1 connects to the vector network analyzer to get the impedance parameters.

The Y-parameters is obtained after adding a 135 pF lumped capacitor in the proposed MTM unit, and is showed in Figure 9. It is obvious that the MTM slab resonates at 6.49 MHz with the lumped capacitor, whereas the MTM unit without boundaries resonates at 6.16 MHz. It can be seen that the resonance frequency predicted by the experimental method is in good agreement with the simulation Appl.result. Sci. Since2020, 10 the, 8506 utilization of copper foil introduces power loss, the quality factor obtained by8 ofthe 13 experiment has a deviation from the simulation result within tolerance.

Figure 9. The impedance results of MTMs. Figure 9. The impedance results of MTMs.

Once the ωM and QM has been obtained, the relative permeability of the MTM slab is calculated Once the ω and Q has been obtained, the relative permeability of the MTM slab is and shown in FigureM 10. TheM MTM slab resonates at 6.49 MHz, and the real part of relative permeability iscalculated negative and at 6.78 shown MHz. in Figure Moreover, 10. The because MTM slab of the resonates power lossat 6.49 from MHz, copper and foilthe real in the part experiment, of relative thepermeability imaginary is part negative (represents at 6.78 the MHz. loss Moreover, of MTMs) because of the experimental of the power results loss from is slightly copper bigger foil in than the theexperiment, simulation the results. imaginary This part result (represents is also consistent the loss of with MTMs) the formulaof the experimentalω = √LC. Accordingresults is slightly to the ω experimentbigger than the and simulation simulation results. results, This the re esultffective is also inductance consistent of with an MTMthe formula unit is 4.9= µLCH.. BecauseAccording of to the experiment and simulation results, the effective inductance of an MTM unit is 4.9 μH. Because Appl. theSci. 2020 interaction, 10, x FOR between PEER REVIEW MTM units, the resonance frequency of the MTM slab is difficult to9 of decide, 14 soof the experimental interaction verification between MTM is essential. units, the resonance frequency of the MTM slab is difficult to decide, so experimental verification is essential.

(a) (b)

Figure 10. The relative permeability of the experimental MTM slab. (a) The experimental results. Figure(b )10. The The simulation relative permeability results. of the experimental MTM slab. (a) The experimental results. (b) The simulation results. 3.2. Experimental Verification of the MTM-Enhanced WPT System 3.2. Experimental Verification of the MTM-Enhanced WPT System After obtaining the characteristics of the MTM slab, the MTM-enhanced WPT system is established toAfter verify obtaining the proposed the model.characteristics According of tothe the MTM optimization slab, the results, MTM-enhanced when the fM isWPT 6.5 MHz, system the poweris establishedtransfer to effi verifyciency the of proposed the system model. is 26.4%. Acco Therding efficiency to the ofoptimization the system withoutresults, when MTMs the is 8.6%fM accordingis 6.5 MHz,to the (4). power The effi transferciency efficiency of the WPT of systemthe system has is increased 26.4%. The by 17.8%efficiency in theoretical of the system analysis. without Nevertheless, MTMs is 8.6%the couplingaccording coe toffi (4).cients The in efficiency this calculation of the are WPT obtained system by has magnetic increased field simulation,by 17.8% in which theoretical is diffi cult analysis.to measure Nevertheless, and sensitive the coupling to external coefficients disturbances, in this and calculation this inevitably are obtained results inbythe magnetic deviation field in the simulation,experimental which measurements. is difficult to measure and sensitive to external disturbances, and this inevitably results inTo the verify deviation the calculation in the experimental results, the measurements. experimental transmission efficiency of WPT is calculated byTo (4).verify The the input calculation power is results, measured the withexperimental a power meter, transmission and the efficiency output power of WPT is calculated is calculated through by (4). The input power is measured with a power meter, and the output power is calculated through the measurement of the output voltage. The designed MTM slab is added to the system afterwards and the parameters are the same as the simulation settings. First, when the operation frequency of the system changes, the variation of power transfer efficiency is shown in Figure 11. The distance between transceiver coils is 15 cm, and the MTM slab is placed 4 cm away from the Tx. The experimental results show that when the MTM slab is tuned to 6.5 MHz, the optimal working frequency of the system is 6.78 MHz. This is consistent with the previous simulation and numerical optimization results. Experimental results also demonstrate that the efficiency of the system is 18.2% when the distance between coils is 10 cm. Afterwards, the position of the slab is fixed and the distance between the transceiver coils is changed in the range of 5–20 cm. The variation of system efficiency with distance is shown in Figure 12. After adding the MTM slab, the transmission efficiency of the experimental WPT system has been greatly improved by about 10% at the transmission distance of 10 cm. Because of the loss in the input signal amplifier, the experimental results are slightly smaller than the results from optimization and theoretical calculations. Appl. Sci. 2020, 10, 8506 9 of 13 the measurement of the output voltage. The designed MTM slab is added to the system afterwards and the parameters are the same as the simulation settings. First, when the operation frequency of the system changes, the variation of power transfer efficiency is shown in Figure 11. The distance between transceiver coils is 15 cm, and the MTM slab is placed 4 cm away from the Tx. The experimental results show that when the MTM slab is tuned to 6.5 MHz, the optimal working frequency of the system is 6.78 MHz. This is consistent with the previous simulation and numerical optimization results. Experimental results also demonstrate that the efficiency of the system is 18.2% when the distance between coils is 10 cm. Afterwards, the position of the slab is fixed and the distance between the transceiver coils is changed in the range of 5–20 cm. The variation of system efficiency with distance is shown in Figure 12. After adding the MTM slab, the transmission efficiency of the experimental WPT system has been greatly improved by about 10% Appl.at the Sci. transmission 2020, 10, x FOR distance PEER REVIEW of 10 cm. Because of the loss in the input signal amplifier, the experimental10 of 14 resultsAppl. Sci. are2020 slightly, 10, x FOR smaller PEER REVIEW than the results from optimization and theoretical calculations. 10 of 14

Figure 11. Variation of power and efficiency with different frequencies. Figure 11. Variation of power and eﬃciency with diﬀerent frequencies. Figure 11. Variation of power and efficiency with different frequencies.

Figure 12. Variation of power and eﬃciency with diﬀerent distances between Tx and Rx. Figure 12. Variation of power and efficiency with different distances between Tx and Rx. Figure 12. Variation of power and efficiency with different distances between Tx and Rx. In order to verify the accuracy of the proposed practical model, we compared the power efficiencyIn order results to verifywith experimentalthe accuracy resultsof the andproposed the high-frequency practical model, calculation we compared results the(in powerwhich efficiency isresults obtained with by experimental measuring S-parametersresults and the) and high-frequency the comparison calculation is made inresults Figure (in 13. which The powerefficiency efficiency is obtained of the by WPT measuring system S-parameterswith the MTM) and slab the is comparisoncalculated by is |madeS21|2 (inS21 Figure is the positive13. The 2 transferpower efficiency factor. At ofradio the frequency,WPT system |S 21with| is theusually MTM used slab to is represent calculated the by positive |S21|2 (energyS21 is the transfer.), positive andtransfer the results factor. fromAt radio the proposed frequency, model |S21| are2 is calculatedusually used by to(4). represent It is clearly the shown positive that energy the results transfer.), from theand proposedthe results practicalfrom the proposedmodel have model much-impro are calculatedved byaccuracy (4). It is thanclearly those shown obtained that the from results the from S- parameterthe proposed method. practical Furthermore, model have the S-parameter-basedmuch-improved accuracy simulation than has thoserather obtainedhigh requirements from the onS- theparameter computer method. performance Furthermore, and is thextremelye S-parameter-based time-consuming. simu lation has rather high requirements on the computer performance and is extremely time-consuming. Appl. Sci. 2020, 10, 8506 10 of 13

In order to verify the accuracy of the proposed practical model, we compared the power efficiency results with experimental results and the high-frequency calculation results (in which efficiency is obtained by measuring S-parameters) and the comparison is made in Figure 13. The power efficiency 2 of the WPT system with the MTM slab is calculated by |S21| (S21 is the positive transfer factor. At radio 2 frequency, |S21| is usually used to represent the positive energy transfer.), and the results from the proposed model are calculated by (4). It is clearly shown that the results from the proposed practical model have much-improved accuracy than those obtained from the S-parameter method. Furthermore, the S-parameter-based simulation has rather high requirements on the computer performance and is

Appl.extremely Sci. 2020 time-consuming., 10, x FOR PEER REVIEW 11 of 14

FigureFigure 13. 13. ComparisonComparison of of different diﬀerent efficiency eﬃciency results after adding MTM slab in a WPT system.

4. Discussion Discussion The approximations and uncertainties of the proposedproposed modelmodel areare clarifiedclarified here.here. Studies on magneto-inductive (MI)(MI) waves waves have have pointed pointed out thatout the that existence the existence of MI waves of MI may waves add considerably may add considerablyto the losses of to the the MTMs losses [ 41of]. the Therefore, MTMs [41]. in MTM-enhanced Therefore, in MTM-enhanced WPT systems, the WPT losses systems, will cause the alosses drop willin system cause transmissiona drop in system efficiency. transmission This is also efficiency. the main This reason is also for the the disparity main reason between for experimentalthe disparity betweenresults and experimental theoretical results results. and However, theoretical incorporating results. However, MI theory incorporating into the MTM-enhanced MI theory into WPT the MTM-enhancedsystem would incur WPT complicated system would expressions incur complicate and complexd expressions results. According and complex to [ 41results.], the expressionAccording toof two-dimensional[41], the expression (or of even two-dimensional three-dimensional) (or ev MIen three-dimensional) waves involves various MI waves factors involves under di variousfferent factorsconditions. under In different order to solveconditions. this problem, In order we to simplified solve this the problem, MI theory we andsimplified kept the the “practical MI theory model” and keptconcise the and “practical the error model” was controlled concise underand the a permissibleerror was controlled level, which under makes a permissible it much more level, convenient which makesto be applied it much in more practice. convenient to be applied in practice. As the analytical method in high-frequency applications usually utilized the S parameter to represent the system efficiency, efficiency, the eefficiencyfficiency obtainedobtained using lumped parameters in this paper cannot be compared directly with previous results. The The higher higher efficiency efficiency shown shown in in Figure 13 is owing to differentdifferent reference standardsstandards ofof eefficiency.fficiency. As for the future development of the research researcheses on MTM-enhanced WPT systems, the proposed model cancan help help researchers researchers to clarifyto clarify the resonance the resonance frequency frequency of MTMs of and MTMs to estimate and theto estimate transmission the transmissionefficiency of the efficiency system. of In the addition, system. as In the addition, lumped as parameters the lumped are parameters adopted in are the adopted model, thein the effi model,ciency theof the efficiency system canof the be obtainedsystem can simply be obtained using circuit simply parameters using circuit (voltage parameters and current), (voltage without and resortingcurrent), withoutto the expensive resorting device, to the expensive which works device, in a demandingwhich works environment. in a demanding The environment. proposal of the The model proposal will ofbring the greatmodel convenience will bring togreat the convenience subsequent developmentto the subsequent of the development MTMs-enhanced of the WPT MTMs-enhanced systems. WPT systems.

5. Conclusions A practical model for MTMs in low-frequency near field WPT systems is proposed and validated in physical experiments in this paper. According to theoretical analysis, once the structure of an MTM unit is fixed, the characteristics of MTMs is merely determined by its resonance frequency and the quality factor. The resonance frequency can be tuned by the added compensation capacitor of the MTM unit. A circuit model for the MTM-enhanced WPT system is proposed and the power efficiency of the system is concluded in an equation that concerns the resonance frequency and the quality factor of MTMs. An optimization algorithm is utilized to find the optimal efficiency of the WPT system by tuning the resonance frequency of the MTM slab. The experimental results of the proposed model is in good agreement with the theoretical analysis. In conclusion, compared with the former works, the practical model and corresponding optimization procedure provide an accurate and simple way to analyze the MTM-enhanced WPT system in the low-frequency near field. Appl. Sci. 2020, 10, 8506 11 of 13

5. Conclusions A practical model for MTMs in low-frequency near field WPT systems is proposed and validated in physical experiments in this paper. According to theoretical analysis, once the structure of an MTM unit is fixed, the characteristics of MTMs is merely determined by its resonance frequency and the quality factor. The resonance frequency can be tuned by the added compensation capacitor of the MTM unit. A circuit model for the MTM-enhanced WPT system is proposed and the power efficiency of the system is concluded in an equation that concerns the resonance frequency and the quality factor of MTMs. An optimization algorithm is utilized to find the optimal efficiency of the WPT system by tuning the resonance frequency of the MTM slab. The experimental results of the proposed model is in good agreement with the theoretical analysis. In conclusion, compared with the former works, the practical model and corresponding optimization procedure provide an accurate and simple way to analyze the MTM-enhanced WPT system in the low-frequency near field.

Author Contributions: J.L. and S.Y. conceived and designed the study; J.L. and Z.G. gave the theoretical and data analysis; J.L. gave the design of metamaterials and wrote the paper; J.Z. and J.L. revised the whole manuscript; J.Z. and H.S. gave the financial support during the research. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the Zhejiang Key R&D Program under Grant No. 2019C01044. Acknowledgments: The authors would like to thank B. Wei at the Training Platform of Information and Microelectronic Engineering at the Polytechnic Institute of Zhejiang University. Conflicts of Interest: The authors declare no conflict of interest.

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