Power Transfer Analysis of an Asymmetric Wireless Transmission System Using the Scattering Parameters

Article Power Transfer Analysis of an Asymmetric Wireless Transmission System Using the Scattering Parameters

Živadin Despotović , Dejan Reljić* , Veran Vasić and Djura Oros

Department of Power, Electronic and Telecommunication Engineering, Faculty of Technical Sciences, University of Novi Sad, Novi Sad 21000, Serbia; [email protected] (Ž.D.); [email protected] (V.V.); [email protected] (D.O.) * Correspondence: [email protected]; Tel.: +381-21-485-4545

Abstract: The most widely adopted category of the mid-range wireless power transmission (WPT) systems is based on the magnetic resonance coupling (MRC), which is appropriate for a very wide range of applications. The primary concerns of the WPT/MRC system design are the power transfer capabilities. Using the scattering parameters based on power waves, the power transfer of an asymmetric WPT/MRC system with the series-series compensation structure is studied in this paper. This approach is very convenient since the scattering parameters can provide all the relevant characteristics of the WPT/MRC system related to power propagation. To maintain the power

transfer capability of the WPT/MRC system at a high level, the scattering parameter S21 is used to determine the operating frequency of the power source. Nevertheless, this condition does not coincide with the maximum possible power transfer efficiency of the system. In this regard, the WPT/MRC system is thereafter configured with a matching circuit, whereas the scattering parameter 0 S21 S21’is used to calculate and then adjust the matching frequency of the system. This results in the maximum available power transfer efficiency and thereby increases the overall performance

Citation: Despotović,Ž.; Reljić,D.; of the system. Theoretical investigations are followed by numerical simulation and experimental Vasić,V.; Oros, D. Power Transfer validation. Analysis of an Asymmetric Wireless Transmission System Using the Keywords: wireless power transfer; asymmetric system; magnetic resonance; two-port networks; Scattering Parameters. Electronics S-parameters 2021, 10, 906. https://doi.org/ 10.3390/electronics10080906

Academic Editor: Massimo Donelli 1. Introduction Wireless power transmission (WPT) is an emerging technology that has been studied Received: 17 March 2021 since the work of Tesla and his ideas for wireless transmission [1]. Up to now, WPT has Accepted: 5 April 2021 Published: 10 April 2021 drawn a lot of attention, while much research has been devoted to improving power transfer capability (PTC) and power transfer efﬁciency (PTE), including transmission range.

Publisher’s Note: MDPI stays neutral It is expected that the WPT will be used on a bigger scale in the times to come. with regard to jurisdictional claims in The most popular WPT method is based on the concept of inductive coupling. Since published maps and institutional afﬁl- the WPT via magnetic resonance coupling (MRC) was reported in [2], it has attracted iations. researches more than ever before. WPT/MRC is a system that operates at resonance to make the circuit behave as the purely resistive. In many publications, one can ﬁnd WPT/MRC as a category of inductive power transfer (IPT) [3] called resonant inductive power transfer [4]. The main advantage of the WPT/MRC system over the well-known IPT is a higher PTE. This advantage has provided the implementation of the WPT/MRC systems not only Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. in low-power devices [5–7], but also in high-power applications, such as electric vehicle This article is an open access article charging solutions [8,9]. In order to increase the PTE at the medium distance, resonant distributed under the terms and coils (resonant relays) were introduced. Therefore, WPT/MRC system may consist of one conditions of the Creative Commons or more resonant coils. With regard to the compensation topology, two-coil WPT/MRC Attribution (CC BY) license (https:// systems are categorized as series-series (SS), series-parallel (SP), parallel-series (PS), or creativecommons.org/licenses/by/ parallel-parallel (PP) compensated systems as shown in Figure1. In addition, few hybrid 4.0/). compensation topologies have been introduced [3]. Based on the transmitter and receiver

Electronics 2021, 10, 906. https://doi.org/10.3390/electronics10080906 https://www.mdpi.com/journal/electronics Electronics 2021, 10, 906 2 of 16 Electronics 2021, 10, x FOR PEER REVIEW 2 of 17

parameters,parameters, the the WPT/MRC systems systems are are catego categorizedrized as as symmetric, symmetric, quasi-symmetric, quasi-symmetric, and asymmetricasymmetric (Figure (Figure 11).). The symmetric system isis composedcomposed ofof identicalidentical coilscoils (inductances(inductances L and L ) and compensation capacitors (capacitances C and C ), which results in the same 𝐿1 and 𝐿2) and compensation capacitors (capacitances1 𝐶 and2 𝐶), which results in the resonant frequencies ( f and f02) on transmitter and receiver sides. In a quasi-symmetric same resonant frequencies01 (𝑓 and 𝑓) on transmitter and receiver sides. In a quasi-sym- metricsystem, system, the transmitter the transmitter and receiver and receiver have alsohave thealso same the same resonant resonant frequency, frequency, but theybut theyconsist consist of non-identical of non-identical coils coils and and capacitors. capacitors. The The difference difference between between quasi-symmetric quasi-symmetric and andasymmetric asymmetric systems systems is that is that the transmitterthe transmitte andr and receiver receiver of the of asymmetricthe asymmetric system system do notdo nothave have the the same same resonant resonant frequencies. frequencies.

WPT/MRC

SS SP PS PP

Symmetric Quasi-symmetric Asymmetric

L1=L2 L1≠L2 L1≠L2 C1=C2 C1≠C2 C1≠C2 f01=f02 f01=f02 f01≠f02

FigureFigure 1. 1. TheThe magnetically magnetically coupled coupled resonant resonant wireless wireless power power transmission transmission (WPT/MRC) (WPT/MRC) system system classification.classiﬁcation. Series-series Series-series (SS), (SS), series-parallel series-parallel (SP), (SP), parallel-series parallel-series (PS), (PS), and and parallel-parallel parallel-parallel (PP) (PP) compensated structures. compensated structures.

NumerousNumerous studies studies of of symmetric symmetric and and quasi-sy quasi-symmetricmmetric systems systems have have been been reported reported in thein theliterature literature over over the theyears. years. The Theoperating operating frequency frequency of such of such systems systems is equal is equal to the to res- the onantresonant frequency frequency in inthe the under-coupled under-coupled and and critical-coupled critical-coupled regime. regime. In In the the over-coupled over-coupled regime,regime, however, thethe transmitting transmitting power power at at the the resonant resonant frequency frequency is significantly is significantly reduced; reduced;thus, the thus, operating the operating frequency frequency needs needs to be adoptedto be adopted to provide to provide the maximum the maximum PTC PTC [10]. [10].The frequencyThe frequency splitting splitting phenomena, phenomena, caused caus byed the by over-coupled the over-coupled regime, regime, has a significanthas a sig- nificantinfluence influence on the PTC on the and PTC has and been has widely been studiedwidely studied [11–13]. [11–13]. ManyMany techniques techniques for for tracking tracking and and eliminatin eliminatingg issues issues of of the the freq frequencyuency splitting have beenbeen proposed. proposed. In In [14,15], [14,15], the the analysis analysis of of frequency frequency splitting splitting and and the the PTE PTE of of two-coil two-coil and and four-coilfour-coil systems systems have have been been reported. reported. In In [14] [14],, a a symmetric symmetric system system has has been been analysed in in termsterms of of voltage voltage gains gains and and output output power. power. With With regard regard to to the the PTE, PTE, defined defined as as the the ratio of loadload power power and and available available source source power, power, an an optimal optimal ratio ratio of of load and source resistances hashas been investigated, keepingkeeping thethe originaloriginal resonant resonant frequency. frequency. However, However, matching matching circuit cir- cuitbetween between the sourcethe source and the and system, the system, which provideswhich provides the flow the of availableflow of available source power source to powerthe system, to the has system, not been has considered. not been considered. The similar The analysis similar has analysis been reported has been in reported [15], but forin [15],a four-coil but for system. a four-coil system. ToTo overcome overcome issues issues of of the the over-coupled over-coupled regime, regime, a anew new WPT/MRC WPT/MRC method method based based on splittingon splitting frequencies frequencies is proposed is proposed in [16]. in [16 Th].e The power power transfer transfer performances performances have have been been in- vestigatedinvestigated theoretically theoretically and and experimentally. experimentally. Yet, Yet, only only the thesymmetric symmetric system system has been has been discussed.discussed. In the In case the caseof different of different quality quality factors factors of the of transmitter the transmitter and the and receiver, the receiver, the exact the calculationexact calculation of splitting of splitting frequencies frequencies is rather is rathercomplex complex [16]. [16]. WithWith the the aim aim of of increasing increasing the the PTE PTE of of th thee quasi-symmetric quasi-symmetric system, system, a a mixed-resonant mixed-resonant structurestructure (with thethe shuntshunt capacitor capacitor added added in in the the SS SS circuit circuit arrangement) arrangement) has beenhas been proposed pro- in [17]. The scattering parameter S has been used for the analysis of the PTE and critical posed in [17]. The scattering parameter21 𝑆 has been used for the analysis of the PTE and criticalcoupling coupling factor derivationfactor derivation of the quasi-symmetricof the quasi-symmetric system system (also valid(also forvalid the for symmetric the sym- metricsystem). system). In an effortIn an toeffort eliminate to eliminate the frequency the frequency splitting, splitting, researchers researchers have have also proposedalso pro- posedother topologiesother topologies of the quasi-symmetricof the quasi-symmetric system, system, such as such thecircuit as the structurecircuit structure based on based non- onidentical non-identical transmitting transmitting and receiving and receiving coils [18,19 coils]. The [18,19]. authors The in authors [18] have in used [18] appropriatehave used appropriatecoil configuration coil configuration in a mixed-resonant in a mixed-resonant structure of the structure WPT/MRC of the to WPT/MRC eliminate frequency to elimi- splitting, while the magnitude of the scattering parameter S21 has been used to calculate nate frequency splitting, while the magnitude of the scattering parameter 𝑆 has been

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and analyse the PTE. However, without the matching circuit between the source and the system, the PTE cannot be directly calculated from the S21 scattering parameter. Likewise, in [19], a pair of non-identical transmitting and receiving coils has been proposed as an approach to avoid the over-coupled regime. Nowadays, the diversity of wirelessly charged electrical consumer devices [20–27] requires greater focus on asymmetric WPT/MRC systems. Unlike the symmetric and quasi-symmetric systems that are well studied and reported in the literature, the analysis of the asymmetric circuits is more complex and more challenging. Much less research has been carried out on the asymmetric structure. The analysis resulting from symmetric cases cannot be used straightforwardly for the system with asymmetric structure as they will not be accurate enough [28]. Therefore, the calculation and selection of the asymmetric system’s operating frequency depends on system parameters, which need to be optimized [29]. Hav- ing this in mind, this paper aims to establish further theoretical research on the asymmetric WPT/MRC system and provide solutions for the maximum PTC and PTE of the proposed system. Compared with the recent researches discussed above, which are mainly dedicated to the symmetric and quasi-symmetric systems, this paper focuses on the power transfer analysis of the two-coil WPT/MRC asymmetric system with the SS compensation topology. Herein, the traditional impedance (Z) parameters approach is not quite appropriate for the comprehensive characterization of WPT/MRC systems since it is difficult to involve frequency-dependent parasitic effects associated with the circuit components. Therefore, the concept of scattering (S) parameters is introduced as it can provide all the relevant characteristics of the WPR/MRC system related to power transfer. These parameters describe the electrical behaviour of the entire system (parasitic effects are included), thus greatly simplifying the power transfer analysis of the system under test. What is more, the S-parameters can be easily and accurately obtained by a vector network analyser (VNA), which is an additional benefit of the proposed approach. Using the electric circuit theory and the concept of the S-parameters, the model of the asymmetric WPT/MRC system is derived. The model is used to determine the operating frequency of the system to provide higher power transferred to the load, thus increasing the PTC of the system. It is shown that, for the characterisation of the PTC, the scattering parameter S21 is of paramount importance. By using the impedance matching circuit, the power transfer performance of the asymmetric WPT/MRC system can be greatly improved. Having determined the matching frequency and assuming that the system is matched to the power source, input power to the asymmetric WPT/MRC system will be equal to the available source power, and it will be transmitted to the load with the maximum possible PTE the system can provide. Using the concept of the S-parameters based on the power waves, the PTE of the system can be calculated effectively, by squaring the magnitude of the generalised 0 scattering parameter S21. Lastly, the theoretical investigations are verified by numerical simulations and experimental results. The remainder of this paper is organized as follows. In Section2, the theoretical analysis of the asymmetric system is introduced, followed by the PTC and PTE analyses using the generalized S-parameters concept. In Section3, the theoretical investigations are validated by numerical simulations, followed by experimental tests on the prototype of the two-coil asymmetric WPT/MRC system with the SS compensation structure. Finally, the conclusion is drawn in Section4. AppendixA provides appropriate mathematical explanations related to Section2, while the methodology for the recalculation of the S- parameters for an arbitrary normalisation impedance is presented in AppendixB.

2. Theoretical Analysis of the Asymmetric WPT/MRC System The two-coil WPT/MRC asymmetric system consists of two electromagnetic subsys- tems with different resonant frequencies (Figure1). In order to improve PTC and PTE of the WPT system, the compensation circuit is required in both transmitter and receiver sides [30]. The selection of adequate compensation topology depends on a given range of applications. ElectronicsElectronics 2021 2021, 10, 10, x, xFOR FOR PEER PEER REVIEW REVIEW 4 4of of 17 17

Electronics 2021, 10, 906 4 of 16

applications.applications. ThisThis paperpaper focusesfocuses onon the the SSSS compensationcompensation structure structure ofof thethe two-coiltwo-coil WPT/MRC asymmetric system, although the analysis concept is also valid for any com- WPT/MRCThis paper focusesasymmetric on the system, SS compensation although the structure analysis of theconcept two-coil is also WPT/MRC valid for asymmetric any compensation topologies. pensationsystem, although topologies. the analysis concept is also valid for any compensation topologies. TheTheThe equivalent equivalentequivalent circuit circuitcircuit of ofof the thethe two-coil two-coiltwo-coil SS SSSS compensatedcompensated compensated WPT/MRC WPT/MRC WPT/MRC asymmetric asymmetricasymmetric system systemsystem isis shown shown in in Figure Figure 2. 2. The The system system consis consiststs of of the the transmitting transmitting coil coil inductor inductor 𝐿 𝐿 and and its its se- se- is shown in Figure2. The system consists of the transmitting coil inductor L1 and its series- ries-connectedries-connected compensation compensation capacitor capacitor 𝐶 𝐶. On. On the the receiver receiver side, side, the the compensation compensation capac- capac- connected compensation capacitor C1. On the receiver side, the compensation capacitor itoritor 𝐶 𝐶 is is series-connected series-connected with with the the receiving receiving coil coil inductor inductor 𝐿 𝐿. .Mutual Mutual inductance inductance be- be- C2 is series-connected with the receiving coil inductor L2. Mutual inductance between tweentween these these two two coils coils is is denoted denoted as as 𝑀 𝑀. Here,. Here, 𝑅 𝑅, ,𝑅 𝑅, and, and 𝑅 𝑅, ,𝑅 𝑅 represent represent equiva- equiva- these two coils is denoted as M12. Here, RL1, RL2, and RC1, RC2 represent equivalent series lentresistanceslent series series resistances resistances associated associated withassociated transmitting with with tran tran andsmittingsmitting receiving and and coilsreceiving receiving and compensationcoils coils and and compensation compensation capacitors, capacitors,respectively,capacitors, respectively, respectively, while the distributed while while the the distributed parasiticdistributed capacitance parasitic parasitic capacitance ofcapacitance the coils canof of the the be coils includedcoils can can be inbe in- the included in the compensation capacitance. AC voltage source 𝑉 , with the inner impedance cludedcompensation in the compensation capacitance. capacitance. AC voltage sourceAC voltageVS, with source the inner𝑉, with impedance the innerR impedanceS, supplies 𝑅 𝑅 𝑅the, supplies, primarysupplies the sidethe primary primary (transmitter side side (transmitter resonator), (transmitter while resonator), resonator), the voltage while while of the the voltage voltage load R Lof ofon the the the load load secondary 𝑅 on on 𝑉 thesidethe secondary secondary is denoted side side as Vis Lis denoted. denoted For the as sake as 𝑉 of. For. For simplicity, the the sake sake andof of simplicity, withoutsimplicity, much and and without loss without of generality, much much loss loss it of isof generality,adoptedgenerality, that it it is theis adopted adopted source that andthat the thethe source loadsource impedances and and the the load load are impedances impedances purely resistive. are are purely purely resistive. resistive.

RRC1C 1 CC1 1 RRL1L 1 MM1212 RRL2L 2 CC2 2 RRC2C 2

I1I 1 I2I 2 RRS S L1L 1 L2L 2 VVL L RRL L + + VVS S

FigureFigureFigure 2. 2.2. The TheThe equivalent equivalentequivalent circuit circuitcircuit model modelmodel of of the the magnetically magnetically coupled coupled resonant resonant wireless wireless power power trans- trans- transmission asymmetric system with the series-series compensation structure. missionmission asymmetric asymmetric system system with with the the series-series series-series compensation compensation structure. structure.

2.1.2.1.2.1. PTC PTCPTC Analysis AnalysisAnalysis of ofof the thethe WPT/MRC WPT/MRCWPT/MRC System SystemSystem TheTheThe analytical analyticalanalytical model model of of of the the the asymmetric asymmetric asymmetric WPT/MRC WPT/MRC WPT/MRC system system system is is of isof key ofkey keyimportance importance importance for for theforthe power thepower power transfer transfer transfer performance performance performance analyses. analyses. analyses. Many Many Manytheories theories theories have have been havebeen adopted beenadopted adopted in in the the litera- inlitera- the ture.literature.ture. The The equivalent equivalent The equivalent lumped-element lumped-element lumped-element circuit circuit circuitmodel model modeldepicted depicted depicted in in Figure Figure in Figure2 2is is described described2 is described using using theusingthe electric electric the electric circuit circuit circuittheory theory theorywith with the withthe Z-parameters. Z-parameters. the Z-parameters. This This is Thisis a acommon common is a common approach approach approach at at low low at fre-low frequencies.frequencies.quencies. At At higher At higher higher operating operating operating frequencies, frequencies, frequencies, ho ho however,wever,wever, these these these parameters parameters parameters are are not not well wellwell suited suitedsuited tototo characterize characterizecharacterize WPT/MRC WPT/MRCWPT/MRC system system because because it it is is difficult difﬁcultdifficult to toto perform performperform their theirtheir measurements. measurements.measurements. ParasiticParasiticParasitic effects effectseffects of ofof the thethe circuit circuitcircuit components componentscomponents limit limitlimit their theirtheir efficient efﬁcientefficient use.use. use. Therefore, Therefore,Therefore, the thethe concept conceptconcept ofofof S-parameters S-parametersS-parameters is isis much muchmuch more moremore convenient. convenient.convenient. The TheThe S-parameters S-parametersS-parameters describe describedescribe correlations correlationscorrelations of ofof a a a newnewnew set setset of ofof variables variablesvariables in inin terms termsterms of ofof forward forwardforward and andand backward backwardbackward waves, waves,waves, rather ratherrather than thanthan their theirtheir terminal terminalterminal variablevariablevariable values. values.values. In In In the the literature, literature, thethe the S-parametersS-para S-parametersmeters are are are usually usually usually based based based on on voltageon voltage voltage and and and current cur- cur- renttravellingrent travelling travelling waves waves waves [31– 34[31–34]. [31–34].]. This This descriptionThis description description is practical, is is practical, practical, since since thesince S-parameters the the S-parameters S-parameters can be can can easily be be easilyandeasily accurately and and accurately accurately measured measured measured by the by by VNA the the VNA [17VNA]. In[1 [17]. this7]. In In paper, this this paper, paper, however, however, however, the concept the the concept concept of the of S-of theparametersthe S-parameters S-parameters based based onbased the on poweron the the power wavespower [waves 35waves–37] [3 is[35–37] adopted.5–37] is is adopted. adopted. This approach This This approach approach is more preferredis is more more preferredherein,preferred due herein, herein, to its due due usefulness to to its its usefulness usefulness for the powerfor for th the e propagationpower power propagation propagation analysis analysis analysis [31,32], [31,32], [31,32], as well as as as well well the asobservationas the the observation observation of the of frequencyof the the frequency frequency splitting splitting splitting phenomena phenomena phenomena [19]. [19]. [19]. TheTheThe equivalent equivalentequivalent lumped-element lumped-elementlumped-element circuit circuitcircuit model modelmodel from fromfrom Figure FigureFigure 2 2 cancan can be be be represented represented represented as as a a passivepassivepassive linear linearlinear two-port two-porttwo-port network networknetwork with withwith the thethe S-parameters, S-parameters,S-parameters, as asas shown shownshown in in in Figure FigureFigure 33. .3.

PortPort 1 1 Port Port 2 2

I I 1 I I 2 R 1 2 RS S Two-portTwo-port VV 1 VV 2 RRL L + + 1 networknetwork 2 VVS S

FigureFigureFigure 3. 3.3. The TheThe magnetically magneticallymagnetically coupled coupledcoupled resonant resonantresonant wireless wirelesswireless power powerpower transmission transmissiontransmission system system represented represented as as a a two-porttwo-porttwo-port passive passivepassive linear linearlinear network networknetwork model modelmodel with withwith the thethe S-parameters. S-parameters. S-parameters.

Electronics 2021, 10, 906 5 of 16

The conversion of the Z-parameter matrix into the generalized S-parameter matrix can be done by transformation [35,38]:

∗ −1 −1 S = F(Z − Z0)(Z + Z0) F , (1)

where Z0 represents the reference impedance matrix used for the normalisation process (symbol * denotes complex conjugate). Based on Equation (1), it is also possible to convert S-parameters into Z-parameters representation. The appropriate transformation can be obtained by rearranging Equation (1) and is provided in [35]. For the two-port network from Figure3, matrices in Equation (1) have the following general forms: S S S = 11 12 , (2) S21 S22   1 Z Z RL1 + RC1 + j ωL1 − ωC jωM12 Z = 11 12 =  1 , Z21 Z22 jωM R + R + j ωL − 1 12 L2 C2 2 ωC2 (3) Z01 0 Z0 = , (4) 0 Z02   √ 1 0 2 R {Z } F =  e 01 , (5) 0 √ 1 2 Re{Z02}

where Z01 and Z02 are reference impedances for Port 1 and Port 2, respectively. In the following analysis it is assumed to be Z01 = RS (for Port 1) and Z02 = RL (for Port 2), i.e., real-valued reference impedances are selected for the normalisation process. The S-parameters can provide valuable performance information on energy transmission through the WPT/MRC network. Among all S-parameters (S11, S22, S12, S21), the S21 parameter is of particular importance. This parameter is the ﬁgure of merit in regards to the PTC since it is commonly used to describe the frequency splitting phenomena and the power delivered from Port 1 to Port 2 of the WPT/MRC system (Figure3). Generally, the 2 PTC is a function of the squared magnitude of the S21 parameter, i.e., |S21| represents the normalised load power (load power scaled by the available power of the source). Hence, a 2 higher value of |S21| leads to higher power transferred to the load. By rearranging Equations (1)–(5), the S21 parameter can be expressed in terms of the Z-parameters as: √ 2Z12 Z01Z02 S21 = . (6) (Z11 + Z01)(Z22 + Z02) − Z12Z21 2 In order to ﬁnd the frequency at which |S21| reaches the maximum value, the following equation has to be solved: ∂|S |2 21 = 0, (7) ∂ω where ω is the operating angular frequency (S-parameters are frequency-domain quantities). The Equation (7) has a bi-quartic form and can be solved using the Kulkarni method [39,40], with the previous reduction to the quartic equation form by introducing the auxiliary variable x, where x = ω2:

4 3 2 x + a3x + a2x + a1x + a0 = 0. (8)

Coefﬁcients in Equation (8) are as follows:

a3 = 0, (9) Electronics 2021, 10, 906 6 of 16

" # 4 2 2 ω01ξ 2 2 1 2 1 2 a2 = + − − 2 1 − k − ξ − 1 , (10) 2 2 2 2 2 2 2 (1 − k ) Q1 Q2 Q1Q2 ξ

6 2 " 2 # ω01ξ 1 ξ 2 a1 = 2 − + 2 ξ + 1 , (11) 2 2 2 2 (1 − k ) Q2 Q1 8 4 ω01ξ a0 = −3 , (12) (1 − k2)2 where: ω ξ = 02 , (13) ω01

ω01L1 Q1 = , (14) Rs + RL1 + RC1

ω02L2 Q2 = , (15) RL + RL2 + RC2 M k = √ 12 , (16) L1L2

and ω01 and ω02 are resonant angular frequencies of the transmitter and the receiver sides, respectively. Coefficients in Equations (13)–(16) are expressed in terms of the frequency asymmetry factor (ξ), loaded quality factors of the transmitter and the receiver (Q1 and Q2, respectively), and the coupling coefficient (k) between the two coils. These factors provide more information about the transmitter and receiver sides including their asymmetry, in contrast to lumped-elements. As far as solutions of Equation (8) are concerned, only real- and positive-valued solutions can be selected. Therefore, there are two scenarios: either there is only one real- and positive-valued solution (x1), or there are three different real- and positive-valued solutions√ (x1, x2, and x3) of Equation (8). In the first case, there is only one angular frequency (ω1 = x1), which is actually the operating angular frequency of the power source. It is obvious that the WPT/MRC system is not in the over-coupled regime. However, in the second case, the frequency splitting phenomena occurs, i.e., the WPT/MRC system is in the over-coupled regime. One has to√ calculate√ the magnitude√ of the S21 parameter for all three angular frequencies (ω1 = x1, ω2 = x2, ω3 = x3) and to select the one at which the magnitude of the S21 parameter reaches the maximum value. Thereafter, the operating angular frequency of the power source is tuned to the previously selected one. Remaining angular frequencies correspond to a local minimum and a local maximum of the S21 magnitude, and are not of interest. Detailed expressions of the solutions of Equation (8), including coefficients in Equations (10)–(12), are listed in AppendixA. One important 2 note to keep in mind, however, is that in the previous analysis, |S21| does not represent the PTE of the WPT/MRC system. In other words, without the matching circuit, the maximum value of the |S21| will not provide the maximum PTE of the system.

2.2. PTE Analysis of the WPT/MRC System In order to achieve the maximum power transfer with the maximum available PTE of the WPT/MRC asymmetric system, it is, therefore, necessary to match the system to the power source. The common approach is to introduce an impedance matching network (IMN) between the power source and the primary side of the WPT/MRC system, as illustrated by the schematic representation in Figure4. The matching conditions can be derived using the maximum power transfer theorem, that is, the output impedance of the matching network (with the source impedance included) has to be a complex conjugate ∗ matched to the input impedance of the WPT/MRC system, i.e., ZSIMN = Zin. Electronics 2021, 10, x FOR PEER REVIEW 7 of 17

trated by the schematic representation in Figure 4. The matching conditions can be derived using the maximum power transfer theorem, that is, the output impedance of the Electronics 2021 10 , , 906 matching network (with the source impedance included) has to be a complex conjugate7 of 16 ∗ matched to the input impedance of the WPT/MRC system, i.e., 𝑍 =𝑍.

Zin Port 1 Port 2

I1 I2 RS Two-port IMN V1 V2 RL + network VS

ZSIMN FigureFigure 4. 4.The The magnetically magnetically coupled coupled resonant resonant wireless wireless power power transmission transmission system system with with the impedancethe imped- matchingance matching network network (IMN), (IMN), represented represented as a two-port as a two-port passive passive linear linear network network model model with with the gener- the alisedgeneralised S-parameters. S-parameters.

InIn general, general, the the impedance impedanceZSIMN 𝑍(the (the source source impedance impedance seen seen by the by two-port the two-port network) net- canwork) be can represented be represented by a complexby a complex number. number. In order In order to studyto study the the PTE PTE of of the the matched matched WPT/MRCWPT/MRC system,system, it it is is now now more more appropriate appropriate to useto use the the complex complex normalisation normalisation impedance imped- associatedance associated with Port with 1 inPort Equation 1 in Equation (4), i.e., Z 01(4),= Zi.e.,SIMN 𝑍. Thus,=𝑍 the. PTEThus, of thethe WPT/MRCPTE of the systemWPT/MRC can besystem calculated can be in calculated a more convenient in a more way.convenient Anticipating way. Anticipating that the IMN that in Figurethe IMN4 willin Figure provide 4 thewill matching provide conditionthe matching of the condition WPT/MRC of the system WPT/MRC to the power system source, to the the power PTE (deﬁnedsource, the as the PTE ratio (defined of the loadas the power ratio and of availablethe load sourcepower power)and available in terms source of S-parameters power) in becomes:terms of S-parameters becomes: 0 2 η = S , (17) T ′ 21 𝜂 =𝑆 , (17) 2 where S0 represents the transducer power gain [37]. Since the current analysis neces- where |𝑆21 | represents the transducer power gain [37]. Since the current analysis neces- sitates the use of complex reference impedance, the S matrix in Equation (2) is denoted sitates the use of complex reference impedance, the 𝑺matrix in Equation (2) is denoted as as S0 to distinguish it from the previous S matrix with real-valued reference impedances. 𝑺 to distinguish it from the previous 𝑺 matrix with real-valued reference impedances. Both matrices are, however, related to the power waves. Parameter S0 is the element of 𝑆21 theBoth generalized matrices are,S0 matrix. however, The rela conversionted to the ofpower the Z waves.matrix Parameter of the WPT/MRC is the asymmetric element of 𝑺 0 𝒁 the generalized matrix. SThe conversion of the matrix of the WPT/MRCS asymmetric system into the generalized matrix is done according to Equation (1), where is replaced systemS0 into the generalized 𝑺 matrix is done according to Equation (1), where 𝑺 is re- with . placedThe withS0 parameter𝑺 . can be expressed in terms of the Z-parameters as [41]: The 𝑆21′ parameter can be expressed in terms of the Z-parameters as [41]: √ Z R R 0 2𝑍2 21 𝑅 SIMN𝑅 L S21 = , (18) 𝑆 = (Z11 + ZSIMN)(Z22 + RL) − Z12Z, 21 (18) 𝑍 +𝑍𝑍 +𝑅 −𝑍𝑍 R = R {Z } wherewhere 𝑅SIMN =𝑅e𝑍SIMN, while, while other other Z-parameters Z-parameters in Equation in Equation (18) (18) are arepreviously previously de- deﬁnedfined inin Equation Equation (3). (3). UsingUsing thethe circuitcircuit theory,theory, the the input input impedance impedance of of the the WPT/MRC WPT/MRC asymmetricasymmetric systemsystem (Figures(Figure 22 andand4 Figure) can be 4) expressed can be expressed as: as: 1 𝜔𝑀2 2 1 ωM12 Z𝑍in == (R𝑅L1 +𝑅+ RC1)++𝑗j𝜔𝐿ωL 1−− + + , , (19)(19) 𝜔𝐶ωC1 𝑍 Z2 where: where: 1 1 Z2 = (RL2 + RC2 + RL) + j ωL2 − . (20) 𝑍 = 𝑅 +𝑅 +𝑅 + 𝑗 𝜔𝐿 − ω.C 2 (20) 𝜔𝐶 Assuming that Z = Z∗ (the maximum power transfer theorem is adopted), and Assuming that 𝑍SIMN =𝑍∗in (the maximum power transfer theorem is adopted), and using Equations (19)–(20), the PTE of the WPT/MRC system displayed in Figure4 can be using Equations (19)–(20), the PTE of the WPT/MRC system displayed in Figure 4 can be written as: written as: RL RL(RL1 + RC1) ηT = − , (21) RL2 + RC2 + RL a(RL2 + RC2 + RL) where: ω2 M2 (R + R + R ) = { } = + + 12 L2 C2 L a Re Zin RL1 RC1 2 . (22) |Z2| Electronics 2021, 10, 906 8 of 16

The angular frequency at which the system should be matched can be found by solving the equation: ∂η T = 0. (23) ∂ω The following analytical solution of Equation (23) is obtained: v u 1 = u ωm t 2 , (24) (C2(RL2+RC2+RL)) L2C2 − 2

where ωm represents the angular frequency at which the matching condition of the WPT/MRC system with the IMN occurs. This frequency corresponds to the power source operating angular frequency. Equation (24) can be rewritten in terms of the loaded quality factor (Q2) and resonant angular frequency (ω02) of the receiver, as follows: s 1 1 = − ωm 1 2 , (25) ω02 2Q2

where Q2 is defined in Equation (15). The next step is to design the IMN at the angular frequency of ωm. The IMN is usually accomplished using a simple L-section circuit, consisting of lumped-elements, such as capacitors and inductors. Since the approach of the L-section matching circuit design is well studied in the literature, it will only be briefly discussed in terms of the L-section topology. The choice of the appropriate L-section configuration depends on the values of RS and the parameter a (see Equation (22)). If RS < a, the matching inductance Lm is placed in a series with the source, while the matching capacitor Cm is placed in parallel with the primary side of the WPT/MRC system (configuration 1 in Figure5). On the other hand, if RS > a, then the L-section topology needs to be designed by adding the matching capacitor Cm in parallel to the source and the matching inductance Lm is series with the Electronics 2021, 10, x FOR PEER REVIEWprimary side of the WPT/MRC system (configuration 2 in Figure5). Calculation of9 theof 17 L-section matching network elements can be performed either analytically or graphically (using the Smith chart).

IMN IMN

Lm Lm

RS RS C C + m + m VS VS

ZSIMN ZSIMN Configuration 1 Configuration 2 FigureFigure 5.5. L-section matching network conﬁguration.configuration.

AfterThe IMN the systemprovides is no being reflected matched power to the waves. power Thus, source, the power the power tran ofsferred the WPT/MRC to the load systemis maximised. at Port The 1 will PTE be can equal now to thebe calculated available sourceexactly power according (Pavs to), givenEquation in terms (17), ofwhich the sourcerepresents voltage the (maximumVS) by: available PTE of the WPT/MRC system (maximum available 2 transducer power gain). |VS| Pavs = . (26) 8RS 3. NumericalThe IMN Simulation provides no and reﬂected Experimental power waves. Results Thus, the power transferred to the load is maximised.To validate The the PTE theoretical can now research, be calculated the power exactly transfer according analysis to Equation of the asymmetric (17), which representsWPT/MRC thesystem maximum is investigated available using PTE numeri of the WPT/MRCcal simulations, system followed (maximum by experimental available transducervalidation. powerThe S-parameters gain). simulations are conducted with the help of the Advanced Design System (ADS) software tool, while the analytical calculations are obtained based on the presented theoretical analysis using Matlab software. The experimental prototype of the asymmetric WPT/MRC system with the SS compensation structure is designed to verify analytical calculations.

3.1. Numerical Results The asymmetric WPT/MRC system is designed according to the circuit model in Fig- ure 2. The parameters of the simulated system are provided in Table 1 and are considered to be frequency-independent. It is worthwhile to mention that these parameters correspond to those of the experimental setup. The resonant frequencies of the transmitter and receiver (𝑓 and 𝑓) are 281.7 kHz and 256.9 kHz, respectively. The sweep range of the power source operating frequency (𝑓) is 100–500 kHz, while the coupling coefficient varies from 0.01–0.7. The inner resistance of the power source (𝑅) and the load resistance (𝑅) are arbitrarily chosen to be 2.2 Ω. The equivalence of source and load resistances is irrelevant for the power transfer analysis.

Table 1. Parameters of the magnetically coupled resonant wireless power transmission system.

Component Parameter Value

Transmitter coil inductance 𝐿 31.3 µH Receiver coil inductance 𝐿 18.9 µH Transmitter coil resistance 𝑅 0.316 Ω Receiver coil resistance 𝑅 0.237 Ω Transmitter compensation capacitor 𝐶 10.2 nF Receiver compensation capacitor 𝐶 20.3 nF Transmitter compensation capacitor resistance 𝑅 0.044 Ω Receiver compensation capacitor resistance 𝑅 0.022 Ω

Figure 6 presents the simulation result of the magnitude of the 𝑆 parameter (marked as 1) with respect to the operating frequency and the coupling coefficient of the WPT/MRC system depicted in Figure 2. Results of the analytical calculation based on

Electronics 2021, 10, 906 9 of 16

3. Numerical Simulation and Experimental Results To validate the theoretical research, the power transfer analysis of the asymmetric WPT/MRC system is investigated using numerical simulations, followed by experimental validation. The S-parameters simulations are conducted with the help of the Advanced Design System (ADS) software tool, while the analytical calculations are obtained based on the presented theoretical analysis using Matlab software. The experimental prototype of the asymmetric WPT/MRC system with the SS compensation structure is designed to verify analytical calculations.

3.1. Numerical Results The asymmetric WPT/MRC system is designed according to the circuit model in Figure2. The parameters of the simulated system are provided in Table1 and are considered to be frequency-independent. It is worthwhile to mention that these parameters correspond to those of the experimental setup. The resonant frequencies of the transmitter and receiver ( f01 and f02) are 281.7 kHz and 256.9 kHz, respectively. The sweep range of the power source operating frequency ( f ) is 100–500 kHz, while the coupling coefﬁcient varies from 0.01–0.7. The inner resistance of the power source (RS) and the load resistance (RL) are arbitrarily chosen to be 2.2 Ω. The equivalence of source and load resistances is irrelevant for the power transfer analysis.

Table 1. Parameters of the magnetically coupled resonant wireless power transmission system.

Component Parameter Value

Transmitter coil inductance L1 31.3 µH Receiver coil inductance L2 18.9 µH Transmitter coil resistance RL1 0.316 Ω Receiver coil resistance RL2 0.237 Ω Transmitter compensation capacitor C1 10.2 nF Receiver compensation capacitor C2 20.3 nF Transmitter compensation capacitor resistance RC1 0.044 Ω Receiver compensation capacitor resistance RC2 0.022 Ω

Electronics 2021, 10, x FOR PEER REVIEW 10 of 17 Figure6 presents the simulation result of the magnitude of the S21 parameter (marked as 1) with respect to the operating frequency and the coupling coefﬁcient of the WPT/MRC system depicted in Figure2. Results of the analytical calculation based on Equation (6), Equationincluding (6), Equations including (8)–(16), Equations are presented(8)–(16), are on presented the same Figureon the6 samewith theFigure solid 6 blackwith the line solid(marked black asline 2). (marked as 2).

2 1 Simulation results 1.0

0.8 2 Analytical results

| 0.6

[p.u.] 1

2 0.4

S |

0.2 0.0 1 0.5 k 0.3 500 k [p.u.] 400 0.1 300 200 f [kHz] 100 kHz] S21 FigureFigure 6. 6.TheThe magnitude magnitude of of the the scattering scattering parameter parameter S21 (1)(1) and and the the trajectory trajectory of of its its maximum maximum val- values ues(2) (2) expressed expressed in in terms terms of of the the operating operating frequency frequency (f ) ( andf) and coupling coupling coefﬁcient coefficient (k ).(k).

As discussed above, the PTC and the frequency splitting phenomena can be described with the 𝑆 parameter. This is clearly demonstrated by the 3D surface plot in Figure 6 (marked as 1). As the result marked as 1 in Figure 6 shows, for the same values of the coupling coefficient, the magnitude of the 𝑆 parameter reaches not only the global maximum but also local extrema. These results fully correspond to the presented theoretical analysis. On the same 3D plot in Figure 6, the trajectory of the absolute maximum values of the magnitude of the 𝑆 parameter is outlined at the top of the surface (marked as 2). As can be seen, this trajectory coincides with the points in the simulation results. Moreover, the trajectory denotes the system’s operating points, which should be selected to provide higher power transferred to the load. The square magnitude of the 𝑆 parameter represents the normalised load power. The system depicted in Figure 2 is not designed to operate at the maximum PTE, but with the maximum PTC. To achieve the maximum PTE as well, it has been suggested in the previous section to incorporate the IMN to the WPT/MRC system (schematic representation in Figure 4). Using Equation (24) and the parameters of the simulated asymmetric WPT/MRC system (Table 1), the matching frequency (𝑓) of 257.4 kHz is calculated. Thereafter, the L-section matching circuit elements (𝐿 and 𝐶 in Figure 5) are determined for this value of the frequency and each value of the coupling coefficient in the range of 0.01–0.7. The operating frequency of the power source is tuned at the value of the previously calculated matching frequency. The PTE of the matched asymmetric WPT/MRC system versus the coupling coefficient is presented graphically in Figure 7, where red cross markers denote simulation results, while the solid black line corresponds to the calculation results of the PTE obtained by the theoretical analysis with the help of Equations (17)–(18).

Electronics 2021, 10, 906 10 of 16

As discussed above, the PTC and the frequency splitting phenomena can be described with the S21 parameter. This is clearly demonstrated by the 3D surface plot in Figure6 (marked as 1). As the result marked as 1 in Figure6 shows, for the same values of the coupling coefficient, the magnitude of the S21 parameter reaches not only the global maximum but also local extrema. These results fully correspond to the presented theoretical analysis. On the same 3D plot in Figure6, the trajectory of the absolute maximum values of the magnitude of the S21 parameter is outlined at the top of the surface (marked as 2). As can be seen, this trajectory coincides with the points in the simulation results. Moreover, the trajectory denotes the system’s operating points, which should be selected to provide higher power transferred to the load. The square magnitude of the S21 parameter represents the normalised load power. The system depicted in Figure2 is not designed to operate at the maximum PTE, but with the maximum PTC. To achieve the maximum PTE as well, it has been suggested in the previous section to incorporate the IMN to the WPT/MRC system (schematic representation in Figure4). Using Equation (24) and the parameters of the simulated asymmetric WPT/MRC system (Table1), the matching frequency ( fm) of 257.4 kHz is calculated. Thereafter, the L-section matching circuit elements (Lm and Cm in Figure5) are determined for this value of the frequency and each value of the coupling coefficient in the range of 0.01–0.7. The operating frequency of the power source is tuned at the value of the previously calculated matching frequency. The PTE of the matched asymmetric WPT/MRC system versus the coupling coefficient is presented graphically in Figure7, Electronics 2021, 10, x FOR PEER REVIEWwhere red cross markers denote simulation results, while the solid black line corresponds11 of 17 to the calculation results of the PTE obtained by the theoretical analysis with the help of Equations (17) and (18).

f=fm=257.4 kHz 1.0

0.8 2 p.u.] | '

21 0.6 S = | 0.4 Τ [ η Simulation results 0.2 Analytical results 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 k [p.u.] k FigureFigure 7. 7.PowerPower transfer transfer efficiency efﬁciency of ofthe the matched matched asymmetric asymmetric wireless wireless power power transmission transmission sys- system temversus versus the the coupling coupling coefﬁcient coefficient (k). (k).

It Itcan can be be observed observed from from Figure Figure 77 thatthat numerical numerical and and analytical analytical results results correspond correspond withwith each each other, other, thus thus validating validating the the theore theoreticaltical analysis. analysis. The The PTE PTE can can now now be be expressed expressed 0 asas the the square square magnitude magnitude of ofthe the 𝑆S21 parameter,parameter, whereas whereas the the power power at at Port Port 1 1is is equal equal to to the the availableavailable source source power power and and it it is transmitted toto PortPort 2 2 with with the the maximum maximum possible possible PTE PTE that thatthe the system system can can provide. provide. This This is consistentis consistent with with the the theoretical theoretical investigation. investigation. If Ifthe the frequency frequency of of the the power power source source is isvari varied,ed, the the IMN IMN elements elements have have to to be be designed designed forfor each each operating operating frequency frequency and and the couplingcoupling coefﬁcientcoefficient values, values, so so as as to to maintain maintain the the PTE PTEof theof the WPT/MRC WPT/MRC system system at a at high a high level. level. Figure Figure8 depicts 8 depicts the simulated the simulated PTE of thePTE matched of the matchedsystem system (marked (marked as 1) with as 1) respect with respect to the wideto the range wideof range the operatingof the operating frequency frequency and the andcoupling the coupling coefﬁcient. coefficient. Under the Under assumption the assumption that the system that the is matchedsystem is to matched the power to source, the powerthe PTE source, can alsothe bePTE calculated can also be according calculated to Equations according (17) to Equations and (18). Therefore,(17)–(18). Therefore, on the same onplot the insame Figure plot8, in the Figure solid black8, the line solid (marked black li asne 2)(marked represents as 2) the represents trajectory the of thetrajectory maximum of thepossible maximum PTE possible of the system, PTE of whichthe system, corresponds which corresponds to the matching to the frequency matching provided frequency by providedEquation by (24). Equation As can (24). be seen,As can this be line seen, coincides this line with coincides the points with inthe the points simulation in the simu- results. lationFollowing results. this Following line, the this system line, reachesthe system the reaches maximum the maximum possible transmission possible transmission power and power and efficiency. This is in accordance with the results shown in Figure 7. On the other hand, the PTE of the matched asymmetric WPT/MRC system decreases as the operating frequency gets away from the value of the matching frequency determined by Equa- tion (24). This is clearly observable from simulation results in Figure 8.

1 Simulation results 1.0 2 Analytical results

0.8 2

2 |

p.u.] '

1 0.6 2

S |

0.4 =

[ 0.2 η 0.0 1 0.5 k 0.3 500 k [p.u.] 400 0.1 300 200 f [kHz] 100 Hz] Figure 8. Power transfer efficiency of the matched asymmetric wireless power transmission system (1) and the trajectory of its maximum values (2) expressed in terms of the operating frequency (f) and coupling coefficient (k).

3.2. Experimental Results

Electronics 2021, 10, x FOR PEER REVIEW 11 of 17

f=fm=257.4 kHz 1.0

0.8 2 p.u.] | '

21 0.6 S = | 0.4 Τ [ η Simulation results 0.2 Analytical results 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 k [p.u.] k Figure 7. Power transfer efficiency of the matched asymmetric wireless power transmission system versus the coupling coefficient (k).

It can be observed from Figure 7 that numerical and analytical results correspond with each other, thus validating the theoretical analysis. The PTE can now be expressed as the square magnitude of the 𝑆 parameter, whereas the power at Port 1 is equal to the available source power and it is transmitted to Port 2 with the maximum possible PTE that the system can provide. This is consistent with the theoretical investigation. If the frequency of the power source is varied, the IMN elements have to be designed for each operating frequency and the coupling coefficient values, so as to maintain the PTE of the WPT/MRC system at a high level. Figure 8 depicts the simulated PTE of the matched system (marked as 1) with respect to the wide range of the operating frequency and the coupling coefficient. Under the assumption that the system is matched to the power source, the PTE can also be calculated according to Equations (17)–(18). Therefore, on the same plot in Figure 8, the solid black line (marked as 2) represents the trajectory of Electronics 2021, 10, 906 the maximum possible PTE of the system, which corresponds to the matching frequency11 of 16 provided by Equation (24). As can be seen, this line coincides with the points in the simulation results. Following this line, the system reaches the maximum possible transmission powerefﬁciency. and efficiency. This is in accordance This is in withaccordance the results with shown the results in Figure shown7. On in the Figure other 7. hand, On the the otherPTE hand, of the the matched PTE of asymmetric the matched WPT/MRC asymmetric system WPT/MRC decreases system as thedecreases operating as the frequency oper- atinggets frequency away from gets the away value from of the the matching value of frequencythe matching determined frequency by determined Equation (24). by Equa- This is tionclearly (24). observableThis is clearly from observable simulation from results simulation in Figure results8. in Figure 8.

1 Simulation results 1.0 2 Analytical results

0.8 2

2 |

p.u.] '

1 0.6 2

S |

0.4 =

[ 0.2 η 0.0 1 0.5 k 0.3 500 k [p.u.] 400 0.1 300 200 f [kHz] 100 Hz] FigureFigure 8. 8.PowerPower transfer transfer efficiency efﬁciency of of the the matched matched asymmetric asymmetric wireless wireless power power transmission transmission sys- system tem(1) (1) and and the the trajectory trajectory of its of maximumits maximum values values (2) expressed(2) expressed in terms in terms of the of operating the operating frequency frequency (f ) and Electronics 2021, 10, x FOR PEER REVIEW(f) and coupling coefficient (k). 12 of 17 coupling coefﬁcient (k). 3.2.3.2. Experimental Experimental Results Results ToTo validatevalidate thethe conducted power power transfer transfer analysis analysis of of the the asymmetric asymmetric WPT/MRC WPT/MRC sys- system,tem, an an experimental experimental prototype prototype is is constructe constructed,d, while while the the power transfer performanceperformance isis evaluatedevaluated by by experimental experimental tests. tests. The The prototype prototyp ise is displayed displayed in Figurein Figure9. The 9. The parameters parameters of

theof the prototype prototype are are consistent consistent with wi thoseth those presented presented in Table in Table1. 1.

Coils

5 Port 1 Port 2

6 2 3 4

FigureFigure 9.9.Experimental Experimental setup setup of of the the two-coil two-coil asymmetric asymmetric wireless wireless power power transmission transmission system system (1–DC (1 – powerDC power supply, supply, 2–high-frequency 2 – high-frequency inverter, inverter, 3–source 3 – source impedance, impedance, 4–impedance 4 – impedance matching matching network, network, 5 – transmitting and receiving coils, 6 – load impedance). 5–transmitting and receiving coils, 6–load impedance).

TheThe experimental experimental setup, setup, depicted depicted in in Figure Figure9, consists9, consists of DCof DC power power supply supply (marked (marked as 1),as high-frequency1), high-frequency inverter inverter (marked (marked as 2), sourceas 2), impedancesource impedance (marked (marked as 3), IMN as (marked3), IMN as(marked 4), transmitting as 4), transmitting and receiving and resonators receiving (coilsresonators with their (coils compensation with their compensation circuits, marked cir- ascuits, 5), and marked load as impedance 5), and load (marked impedance as 6). Transmitter(marked as and6). Transmitter receiver air-core and receiver coils are air-core made withcoils aare planar made spiral with structure a planar withspiral the structure same 100 with mm the inner same diameter. 100 mm The inner outer diameter. diameters The ofouter the transmitterdiameters of and the receiver transmitter coils and are receiver 175 mm coils and 155are mm,175 mm respectively. and 155 mm, Transmitter respectively. coil hasTransmitter 13 turns andcoil has the 13 receiver turns and coil hasthe receiver 10 turns. co Bothil has coils 10 turns. are arranged Both coils in are the arranged same axis in andthe aresame aligned. axis and For are the aligned. entireduration For the entire of experimental duration of tests, experimental the distance tests, between the distance coils wasbetween ﬁxed coils to 35 was mm, fixed which to provides 35 mm, which the mutual prov inductanceides the mutual between inductance them of between 8.66 µH. Thisthem givesof 8.66 the µH. constant This gives coupling the constant coefﬁcient coupling of 0.356. coefficient The IMN consistsof 0.356. of The capacitor IMN consists and inductor of capacitor and inductor arrays, which are manually configured for each operating frequency. A regulated DC power supply provides the output voltage of 10 V. The inner resistance of the power source is set to be 2.2 Ω via an external resistor, while the load impedance is also purely resistive with the resistance of 2.2 Ω. As previously mentioned, the equivalence of source and load resistances is irrelevant for the power transfer analysis. The high- frequency inverter is used to drive the WPT/MRC system and is controlled via a square wave signal generated by the function generator (marked as 7 in Figure 9). All of the above-mentioned electrical parameters of the system were measured by the impedance analyser at the frequency of 250 kHz. During the experiment, input and output voltage and current of the resonators are measured with an oscilloscope. Due to the high bandwidth, the Rogowski probe is used for the current measurement, while the oscilloscope voltage probe is used for the voltage measurement. Thus, the input and output instantaneous power of the WPT/MRC system are obtained as the product of the corresponding instantaneous values of voltage and current. The active power is determined as the average value of the calculated instantaneous power, so the PTE of the WPT/MRC system can be readily calculated. The S-parameters measurement of the WPT/MRC system is performed in a 50 Ω impedance system with the Bode 100 Omicron Lab VNA. However, in this paper, the reference impedances used for the normalisation process are not equal to the normalisation impedance of 50 Ω. Moreover, in the previous section it has been explained that the generalized 𝑺 matrix is defined using the complex reference impedance. Therefore, the measured S-parameters in a 50 Ω impedance system have to be renormalized. An exact renormalization procedure is provided in Appendix B.

Electronics 2021, 10, 906 12 of 16

arrays, which are manually configured for each operating frequency. A regulated DC power supply provides the output voltage of 10 V. The inner resistance of the power source is set to be 2.2 Ω via an external resistor, while the load impedance is also purely resistive with the resistance of 2.2 Ω. As previously mentioned, the equivalence of source and load resistances is irrelevant for the power transfer analysis. The high-frequency inverter is used to drive the WPT/MRC system and is controlled via a square wave signal generated by the function generator (marked as 7 in Figure9). All of the above-mentioned electrical parameters of the system were measured by the impedance analyser at the frequency of 250 kHz. During the experiment, input and output voltage and current of the resonators are measured with an oscilloscope. Due to the high bandwidth, the Rogowski probe is used for the current measurement, while the oscilloscope voltage probe is used for the voltage measurement. Thus, the input and output instantaneous power of the WPT/MRC system are obtained as the product of the corresponding instantaneous values of voltage and current. The active power is determined as the average value of the calculated instantaneous power, so the PTE of the WPT/MRC system can be readily calculated. The S-parameters measurement of the WPT/MRC system is performed in a 50 Ω impedance system with the Bode 100 Omicron Lab VNA. However, in this paper, the reference impedances used for the normalisation process are not equal to the normalisation impedance of 50 Ω. Moreover, in the previous section it has been explained that the Electronics 2021, 10, x FOR PEER REVIEW 0 13 of 17 generalized S matrix is defined using the complex reference impedance. Therefore, the measured S-parameters in a 50 Ω impedance system have to be renormalized. An exact renormalization procedure is provided in AppendixB. InIn the the first first experiment, experiment, the the asymmetric WPT/MRCWPT/MRC systemsystem fromfrom Figure Figure9 was9 was operated oper- atedwithout without the the IMN. IMN. To exploreTo explore the the PTC PTC of the of the system, system, the the load load power power was was measured measured over overthe the operating operating frequency frequency range range of of 210–390 210–390 kHz kHz andand thenthen was scaled scaled by by the the available available sourcesource power, power, thus thus leading leading to to the the normalised normalised load load power. power. The The results results of of the the experiment experiment areare depicted depicted with with red red cross cross markers markers in in Figure Figure 10 10 (Method (Method 1). 1). Likewise, Likewise, the the PTC PTC of of the the WPT/MRCWPT/MRC system system was was investigated investigated through through the the measurement measurement of ofthe the 𝑆S 21parameter.parameter. The The resultsresults of of the the square square magn magnitudeitude ofof the the renormalized renormalized 𝑆S21 parameterparameter are are also also presented presented in in FigureFigure 10 10 (Method (Method 2), 2), denoted denoted with with blue blue diamond diamond markers. markers.

k = 0.356 [p.u.] 0.8 Analytical results 0.6 p.u.] ] Experiment-Method 1 [p.u.]

p.u. 0.4 Experiment-Method 2 [

2 | 0.2 21

[ |S 0.0

100 200 300 400 500 f [kHz] f [kH ] FigureFigure 10. 10. NormalisedNormalised load load power power of ofthe the asymmetric asymmetric wireless wireless power power transmission transmission system system versus versus operatingoperating frequency frequency (f). (f ).

AsAs is isto to be be expected, expected, the the results results of of the the previous previous two two experimental experimental tests tests are are almost almost identical.identical. This This indicates indicates that that the the PTC PTC anal analysisysis of of the the WPT/MRC WPT/MRC system system can can completely completely relyrely on on the the S-parameters S-parameters concept, concept, which which is isof of great great practical practical importance. importance. This This fact fact is isbased based notnot only only on on the the experimental experimental results, results, but but also also on on the the presented presented theoretical theoretical investigation. investigation. OnOn the the same same plot plot in in Figure Figure 10,10, the analytical calculationcalculation ofof the the square square magnitude magnitude of of the theS21 𝑆parameter, parameter, with with respect respect to to the the operating operating frequency, frequency, is depictedis depicted with with a solida solid black black line. line. As Ascan can be be seen, seen, the the theoretical theoretical results results are inare excellent in excellent agreement agreement with with the experimental the experimental results. results. In the next experiment, the PTE of the matched asymmetric WPT/MRC system was evaluated to compare theoretical and experimental results. The WPT/MRC system was connected to the power source via the tuneable IMN, so as to ensure the maximum PTE. The input power of the WPT/MRC system (power at Port 1 in Figure 9) and the load power were measured over the operating frequency range of 220–290 kHz, in a step of 10 kHz. The results of the PTE are shown in Figure 11 with red cross markers (Method 1). It can be observed that, for the constant value of the coupling coefficient, the PTE varies with the operating frequency, resulting in the maximum of around 0.87 at the frequency of about 250 kHz. This is in accordance with the previous numerical calculations (Figure 7). Since the WPT/MRC system was matched to the power source, the PTE can be observed with the generalized S-parameters. After measuring the 𝑆 parameter and its renormalization to the desired reference impedance, the PTE of the system can be expressed as the square magnitude of the 𝑆 parameter. The results of the PTE are depicted in Figure 11 with blue diamond markers (Method 2). It can be noticed that both experimentally obtained results of the PTE (Method 1 and Method 2) are in a very good agreement. This proves the effectiveness of the S-parameters concept analysis of the PTE. For comparison purposes, the results of the analytical calculation are also presented in Figure 11 (solid black line). The analytical results are in quite good correlation with the experimental ones (Figure 11). The slight difference between them exists at higher operating frequencies. This is caused by variations of the overall WPT/MRC system parameters due to frequency effect. Notwithstanding, both experimental and theoretical results are basically consistent with each other. Finally, this confirms that the power transfer analysis of the matched

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In the next experiment, the PTE of the matched asymmetric WPT/MRC system was evaluated to compare theoretical and experimental results. The WPT/MRC system was connected to the power source via the tuneable IMN, so as to ensure the maximum PTE. The input power of the WPT/MRC system (power at Port 1 in Figure9) and the load power were measured over the operating frequency range of 220–290 kHz, in a step of 10 kHz. The results of the PTE are shown in Figure 11 with red cross markers (Method 1). It can be observed that, for the constant value of the coupling coefﬁcient, the PTE varies with the operating frequency, resulting in the maximum of around 0.87 at the frequency of about 250 kHz. This is in accordance with the previous numerical calculations (Figure7). Since the WPT/MRC system was matched to the power source, the PTE can be observed with the generalized S-parameters. After measuring the S21 parameter and its renormalization to the desired reference impedance, the PTE of the system can be expressed 0 as the square magnitude of the S21 parameter. The results of the PTE are depicted in Figure 11 with blue diamond markers (Method 2). It can be noticed that both experimentally obtained results of the PTE (Method 1 and Method 2) are in a very good agreement. This proves the effectiveness of the S-parameters concept analysis of the PTE. For comparison purposes, the results of the analytical calculation are also presented in Figure 11 (solid black line). The analytical results are in quite good correlation with the experimental ones Electronics 2021, 10, x FOR PEER REVIEW 14 of 17 (Figure 11). The slight difference between them exists at higher operating frequencies. This is caused by variations of the overall WPT/MRC system parameters due to frequency effect. Notwithstanding, both experimental and theoretical results are basically consistent with asymmetriceach other. WPT/MRC Finally, this system conﬁrms can that be theproperly power performed transfer analysis using the of the generalized matched asymmetric S-parameters.WPT/MRC system can be properly performed using the generalized S-parameters.

k = 0.356 [p.u.] 1.0

] 0.9 p.u.] p.u.] p.u. [

| 0.8 ' 21

S 0.7 Analytical results [ [ = |

Experiment-Method 1 Τ 0.6 η Experiment-Method 2 0.5 200 220 240 260 280 300 f [kHz] f [kH ] FigureFigure 11. 11. PowerPower transfer transfer efficiency efﬁciency of of the the matched matched asymmetric asymmetric wireless wireless power power transmission transmission sys- system temversus versus operating operating frequency frequency (f ). (f).

4.4. Conclusions Conclusions WPTWPT technique technique based based on onthe theMRC MRC is identified is identified as a key as atechnology key technology for various for various com- mercialcommercial applications. applications. However, However, this concept this concept faces many faces issues. many issues.The main The concern main concernabout WPT/MRCabout WPT/MRC systems systemsis power isefficiency. power efficiency. It lays at Itthe lays root at of the most root of of the most WPT/MRC of the WPT/MRC system designsystem challenges. design challenges. Therefore, Therefore, a deep an aalysis deep of analysis PTC and of PTCPTE andis essential. PTE is essential. InIn this this paper, paper, the the power power transfer transfer of of the the two-coil two-coil asymmetric asymmetric wireless wireless transmission transmission systemsystem was was considered considered in indetails details based based on the on theS-parameters. S-parameters. It was It shown was shown that the that pro- the posedproposed methodology methodology is quite is quite adequate adequate for forthe the performance performance characterisation characterisation of ofthe the WPT/MRCWPT/MRC system, system, since since it entirely it entirely provides provides information information on on the the power power transfer transfer capacity capacity viavia the the S-parameters, S-parameters, including including frequency frequency splitting splitting phenomena. phenomena. Unlike Unlike other other methods methods usedused to to characterise characterise power power transfer transfer properti propertieses of of the the transmission transmission system, system, the the proposed proposed methodmethod is ishighly highly favourable favourable as as the the S-paramete S-parametersrs are are simply simply obtained obtained with with the the VNA. VNA. With With thethe use use of of the the S-parameters S-parameters based based on on the the po powerwer waves, waves, the the analytical analytical model model of of the the SS SS compensatedcompensated WPT/MRC WPT/MRC asymmetric asymmetric system system was was used used to todefine define the the operating operating frequency frequency ofof the the system system to to provide provide higher higher power power transferred transferred to to the the load, load, therefore therefore increasing increasing the the PTC.PTC. The The analysis analysis was was further further extended extended on on th thee system system with with the the incorporated incorporated IMN. IMN. These These yield zero reflected power at the network input, making the WPT/MRC system operate at the maximum possible PTE, with the maximal power transferred to the load at the same time. Although the efficiency of the matched WPT/MRC asymmetric system varies with the operating frequency, the paper demonstrated that the PTE can still be observed with the generalized S-parameters. The validity of all presented theoretical analyses was successfully proven by numerical simulations, as well as experimentally. The proposed theoretical investigation concept on the power transfer performance is applicable not only for the asymmetric WPT/MRC system with the SS compensation circuit but also for other WPT/MRC structures with different compensation topologies.

Author Contributions: Conceptualization, Ž.D.; methodology, Ž.D., D.R. and V.V.; software, Ž.D.; validation, Ž.D. and D.R.; formal analysis, Ž.D.; investigation, Ž.D. and D.R.; resources, D.O.; writing—original draft preparation, Ž.D. and D.R; writing—review and editing, V.V. and D.O.; visualization, D.R.; supervision, V.V. All authors have read and agreed to the published version of the manuscript. Funding: This research has been supported by the Ministry of Education, Science and Technological Development of the Republic of Serbia through the project No. 451-03-9/2021-14/ 200156: “Innova- tive scientific and artistic research from the FTS activity domain”. Conflicts of Interest: The authors declare no conflict of interest.

Appendix A

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yield zero reﬂected power at the network input, making the WPT/MRC system operate at the maximum possible PTE, with the maximal power transferred to the load at the same time. Although the efﬁciency of the matched WPT/MRC asymmetric system varies with the operating frequency, the paper demonstrated that the PTE can still be observed with the generalized S-parameters. The validity of all presented theoretical analyses was successfully proven by numerical simulations, as well as experimentally. The proposed theoretical investigation concept on the power transfer performance is applicable not only for the asymmetric WPT/MRC system with the SS compensation circuit but also for other WPT/MRC structures with different compensation topologies.

Author Contributions: Conceptualization, Ž.D.; methodology, Ž.D., D.R. and V.V.; software, Ž.D.; validation, Ž.D. and D.R.; formal analysis, Ž.D.; investigation, Ž.D. and D.R.; resources, D.O.; writing— original draft preparation, Ž.D. and D.R; writing—review and editing, V.V. and D.O.; visualization, D.R.; supervision, V.V. All authors have read and agreed to the published version of the manuscript. Funding: This research has been supported by the Ministry of Education, Science and Technological Development of the Republic of Serbia through the project No. 451-03-9/2021-14/200156: “Innovative scientific and artistic research from the FTS activity domain”. Conflicts of Interest: The authors declare no conflict of interest.

Appendix A Coefﬁcients in Equation (8) can be expressed in terms of inductances, capacitances, and resistances of the WPT/MRC system components, including the inner resistance of the power source and the load resistance, and have the following forms:

a3 = 0, (A1)

L L L L 2 M2 −2R2 L2 − 2R2 L1 + R2R2 2 1 2 + 1 + 2 − 2 12 1 C2 2 C1 1 2 C1C2 C2 C1 C1C2 a2 = + , (A2) 2 2 2 2 − L1L2 − M12 − L1L2 − M12 2 2 R2 R1 4 L1 L2 2 2 + 2 2 − C C C + C C1 C2 1 2 2 1 a1 = , (A3) 2 2 − L1L2 − M12 3 a0 = , (A4) 2 2 2 2 − L1L2 − M12 C1C2 where: R1 = RL1 + RC1 + RS, (A5)

R2 = RL2 + RC2 + RL. (A6) The general solutions of Equation (8) are: q √ √ 2 √ − b1 − p + b1 − p − 4 b0 − pc0 x = , (A7) 1 2 q √ √ 2 √ − b1 − p − b1 − p − 4 b0 − pc0 x = , (A8) 2 2 q √ √ 2 √ − b1 + p + b1 + p − 4 b0 + pc0 x = , (A9) 3 2 q √ √ 2 √ − b1 + p − b1 + p − 4 b0 + pc0 x = , (A10) 4 2 Electronics 2021, 10, 906 15 of 16

where: b1 = 0, (A11) r n √ r n √ 2a p = 3 − + D + 3 − − D − 2 , (A12) 2 2 3 a2 m = −4a − 2 , (A13) 0 3 2a3 8a a n = − 2 + 2 0 − a2, (A14) 27 3 1 n2 m3 D = + , (A15) 4 27 1 b = (a + p), (A16) 0 2 2 a c = − 1 . (A17) 0 2p

Appendix B The S-parameters are measured in a 50 Ω impedance system. Their renormalization to S-parameters for an arbitrary reference impedance can be performed by using:

−1 h −1 ∗ −1i h −1 −1i −1 Sn = Fn (I − S) (I + S) − Z0nZ0 · (I − S) (I + S) + Z0nZ0 Fn . (A18)

where: Sn represents S matrix in the new reference impedance system, S0 represents S matrix in the 50 Ω impedance system, I is the identity matrix, and Z0 and Z0n are the original (50 Ω) and new reference impedance, respectively. Symbol * denotes a complex conjugate. The matrix Fn is given by:   √ 1 0 2 Re{Z0n} Fn =  . (A19) 0 √ 1 2 Re{Z0n}

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