Coupling Coefficient Measurement Method with Simple Procedures

energies

Article Coupling Coeﬃcient Measurement Method with Simple Procedures Using a Two-Port Network Analyzer for a Multi-Coil WPT System

Seon-Jae Jeon and Dong-Wook Seo *

Department of Radio Communication Engineering, Korea Maritime and Ocean University (KMOU), 727 Taejong-ro, Yeongdo-gu, Busan 49112, Korea; [email protected] * Correspondence: [email protected]; Tel.: +82-51-410-4427

Received: 10 September 2019; Accepted: 16 October 2019; Published: 17 October 2019

Abstract: In this paper, we propose a measurement method with a simple procedure based on the deﬁnition of the impedance parameter using a two-port network analyzer. The main advantage of the proposed measurement method is that there is no limit on the number of measuring coils, and the method has a simple measurement procedure. To verify the proposed method, we measured the coupling coeﬃcient among three coils with respect to the distance between the two farthest coils at 6.78 and 13.56 MHz, which are frequencies most common for a wireless power transfer (WPT) system in high-frequency band. As a result, the proposed method showed good agreement with results of the conventional S-parameter measurement methods.

Keywords: coupling coefficient; impedance matrix; multiple coils; mutual inductance; scattering matrix; transfer impedance; wireless power transfer

1. Introduction The conventional wireless power transfer (WPT) system with two coils has obvious limits; the system is very sensitive to the transmission distance and alignment between the coils [1,2]. To resolve these problems, adaptive or tunable matching networks have been adopted [3,4], and a transmitting module not with a single coil, but with multiple coils has been also introduced into the WPT system [5,6]. Especially, the magnetic beamforming, which focuses magnetic fields from the multiple transmitting coils to the receiving coil, has recently attracted lots of attention. For an ideal multi-coil WPT system, the coupling coefficients among transmitting coils should be zero, but in reality, they are not. The non-zero coupling coefficient among transmitting coils is one of the major causes of deterioration of the power transfer efficiency (PTE) in the magnetic beamforming. To minimize the effect of non-zero coupling coefficients, the phase and magnitude of signals input to transmitting coils should be adjusted to the optimum values that are estimated from the measured coupling coefficients among coils. The adjustment makes the all current of transmitting coils in phase, and results in a maximum receiver current and maximum PTE [5,6]. Therefore, the accurate and simple procedures for measurement of the coupling coefficient among multiple coils is essential for implementing the magnetic beamforming of a multi-coil WPT system. For a WPT system with an operating frequency of a low-frequency (LF) band such as 110 to 205 kHz, the coupling coefficient (or mutual inductance) is usually measured by an LCR meter [7] or an impedance analyzer [8]. On the other hand, in a high-frequency (HF) band such as 6.78 or 13.56 MHz, most coils have a frequency-dependent characteristic due to the effect of ac resistance and parasitic elements. Therefore, a vector network analyzer is commonly used to measure the coupling coefficient in HF bands [9]. In the case of multi-coil WPT systems, the measurement issue arises primarily from

Energies 2019, 12, 3950; doi:10.3390/en12203950 www.mdpi.com/journal/energies Energies 2019, 12, 3950 2 of 10

Energies 2019, 12, x FOR PEER REVIEW 2 of 10 the common vector network analyzer (VNA) having only two ports. Although a multiport VNA can measure thethe couplingcoupling coe coefficientfficient among amon ag greatera greater number number of coils,of coils, it is it still is still only only available available for coils for withcoils equalwith equal or less or than less thethan number the number of ports. of ports. In thisthis paper,paper, wewe simulatesimulate the the e effectffect of of inaccurately inaccurately measured measured coupling coupling coe coefficientsfficients on on the the PTE PTE of aof multi-coil a multi-coil WPT WPT system, system, and and propose propose a method a method to readily to readily measure measure the coupling the coupling coefficients coefficients among multipleamong multiple coils using coils a two-portusing a two-port VNA based VNA on based the definition on the definition of the impedance of the impedance matrix. In matrix. addition, In thisaddition, measurement this measurement method can method overcome can overcome the limitation the limitation of the performance of the performance of a multi-port of a multi-port VNA for moreVNA coilsfor more than coils the number than the of number ports. We of ports. verify We the validityverify the of validity the proposed of the methodproposed from method the results from thatthe results the proposed that the method proposed has the method same performance has the same as conventionalperformance measurement as conventional methods measurement according tomethods distance. according to distance.

2. Effect Effect of Inaccurate Coupling CoeCoefficientfficient onon PTEPTE ofof aa Multi-CoilMulti-Coil WPTWPT SystemSystem In this section, section, we we simulate simulate the the PTE PTE of of a amult multi-coili-coil WPT WPT system system with with magnetic magnetic beamforming beamforming by byapplying applying inaccurate inaccurate coupling coupling coefficient coefficient informat informationion to to examine examine the the importance importance of of an accurateaccurate coupling coecoefficientfficient measurement.measurement. Consider the multiple-input single-outputsingle-output (MISO) WPT system configuredconfigured as four transmitting coils and a single receiving coil, where all transmittingtransmitting coils are modeled as an inductance of 1.1 µμH and a parasiticparasitic resistance of 0.40.4 ohm.ohm. Additionally, we assume that all couplingcoupling coecoefficientsfficients amongamong transmitting coilscoils are fixedfixed to 0.05, and coupling co coeefficientsfficients between transmitting and receiving coils are also fixedfixed toto 0.05.0.05. The assumption that all coupling coecoefficientsfficients among transmitting coils are the same for all transmitting coils is didifficultfficult to realizerealize in practice,practice, but this assumption is eeffectiveffective in simulating thethe PTE PTE of of a multi-coil a multi-coil WPT WPT system, system, because because the input the signals input ofsignals all transmitting of all transmitting resonators haveresonators the same have value the same when value the magnetic when the beamforming magnetic beamforming algorithm isalgorithm applied. is Figure applied.1 shows Figure the 1 schematicshows the diagram schematic of thediagram MISO WPTof the system MISO forWPT circuit syst simulationem for circuit using simulation the Keysight’s using ADS the (AdvancedKeysight’s DesignADS (Advanced System). Design System).

MUTIND P_AC C PORT1 C_Tx1 L Num=1 C=c L_Tx1 Mutua l Z=50 Ohm L=L M_Tx1_Rx Pac=polar(dbmtow(0),0) R=Rp K=k_Tx_Rx Freq=freq M=k_Tx_Rx*L Inductor1="L_Tx1" Inductor2="L_Rx"

MUTIND

P_AC C PORT2 C_Tx2 L Mutua l Num=2 C=c L_Tx2 M_Tx2_Rx Z=50 Ohm L=L K=k_Tx_Rx R=Rp Pac=polar(dbmtow(0),0) M=k_Tx_Rx*L Vout Freq=freq Inductor1="L_Tx2" C R L Inductor2="L_Rx" C_Rx RL L_Rx C=c R=3 Ohm L=L MUTIND R=Rp

C P_AC Mutua l PORT3 C_Tx3 L C=c L_Tx3 M_Tx3_Rx Num=3 K=k_Tx_Rx Z=50 Ohm L=L R=Rp M=k_Tx_Rx*L Pac=polar(dbmtow(0),0) Inductor1="L_Tx3" Freq=freq Inductor2="L_Rx"

MUTIND

P_AC C M_Tx3_Tx4 C_Tx4 L Mutua l PORT4 M_Tx4_Rx K=k_Tx_Tx Num=4 C=c L_Tx4 M=k_Tx_Tx*L L=L K=k_Tx_Rx Z=50 Ohm M=k_Tx_Rx*L Inductor1="L_Tx3" Pac=polar(dbmtow(0),0) R=Rp Inductor2="L_Tx4" Inductor1="L_Tx4" Freq=freq Inductor2="L_Rx"

Figure 1. Schematic diagram of a a multiple-input multiple-input single-output single-output (MISO) wireless power transfer (WPT) system for the advanced designdesign systemsystem (ADS)(ADS) circuitcircuit simulation.simulation.

To apply magnetic beamforming to the MISO WPT system, the voltage of transmitting coils must be given by [6]:

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Energies 2019, 12, x FOR PEER REVIEW 3 of 10 To apply magnetic beamforming to the MISO WPT system, the voltage of transmitting coils must be given by [6]:   ω2 T ! bbf f =+ω2MMMTM bfb, f viTTZT → (1) →v T = ZT + R i T , (1) RRR wherewhereM is M the is mutualthe mutual inductance inductance matrix matrix between between the transmittingthe transmitting and and receiving receiving coils; coils;RR is R theR is the total total b f bf → i T resistanceresistance of theof the receiving receiving resonator; resonator;i T Tisthe is the transmit transmit current current and and is proportionalis proportional to toM M; and; andZ TZ is is thethe inter-coupling inter-coupling matrix matrix among among transmitting transmitting coils coils and and isis constructedconstructed asas thethe couplingcoupling coecoefficientsfficients (or (or mutual inductances)inductances) amongamong transmittingtransmitting coils. coils. That That is, is, the the amplitude amplitude and and phase phase of theof the transmit transmit voltagevoltage is dominantly is dominantly determined determined by couplingby coupling coe fficoefficientcient information. information. Therefore, Therefore, the the more more accurate accurate thethe coupling coupling coe coefficientfficient is, theis, the more more accurately accurately the the transmit transmit voltage voltage for for the the magnetic magnetic beamforming beamforming is is obtained.obtained. By applyingBy applying the the coupling coupling coe fficoefficientcient with with error error term term to the to the magnetic magnetic beamforming, beamforming, we simulatedwe simulated the PTEthe PTE degradation degradation of the of MISO the WPTMISO system WPT atsystem 13.56 MHz.at 13.56 The MHz. simulated The simulated results are shownresults inare Figure shown2. in WhenFigure the exact2. When coupling the exact coeffi couplingcient of 0.05coefficient is used of for 0. the05 is beamforming, used for the beamforming, the PTE achieves the about PTE achieves 77%. As theabout error 77%. rate As of the couplingerror rate coe offfi thecient coupling increases, coefficient the PTE decreases.increases, the Coupling PTE decreases. coefficient Coupling error ratescoefficient of 10% and error 20% rates result of in 10% PTE degradationsand 20% result of 7%in andPTE 15%,degradations respectively, of 7% compared and 15%, with respectively, no error. 2 −2 Sincecompared the coupling with coeno ffierror.cient Since among the coilscoupling usually coef hasficient a low among level coils with usually 10− order, has errora low ratelevel of with 10% 10 3 −3 meansorder, 10− errororder rate that of 10% is quite means challenging 10 orderto that measure. is quite Additionally,challenging tothe measure. larger Additionally, the number of the coils larger usedthe in number the HF band,of coils the used more in ditheffi cultHF band, and cumbersome the more difficult it is to and measure cumbersome the coupling it is to coe measurefficients. the Therefore,coupling a highlycoefficients. accurate Therefore, measurement a highly method accurate of couplingmeasurement coeffi methodcients with of coupling simple procedurescoefficients iswith verysimple important procedures to achieve is very the idealimportant PTE ofto aachieve multi-coil the WPTideal system.PTE of a multi-coil WPT system.

Figure 2. Simulated power transfer efficiency (PTE) with respect to error rate of coupling coefficients Figure 2. Simulated power transfer efficiency (PTE) with respect to error rate of coupling coefficients between coils in the MISO WPT system. between coils in the MISO WPT system. 3. Conventional and Proposed Methods 3. Conventional and Proposed Methods This section presents conventional and proposed measurement methods of coupling coefficients among multipleThis section coils. presents For the conventional measurement and of coupling proposed coe measurementfficients, we methods use some of assumptions. coupling coefficients First, theamong measured multiple coupling coils. coe Forfficient the meas is noturement by electrical of coupling coupling coefficients, but by magnetic we use some coupling. assumptions. In the case First, of thethe radio-frequency measured coupling identification coefficient (RFID) is not by with electrical a few millimeters coupling but distance by magnetic between coupling. coils, electrical In the case couplingof the shouldradio-frequency be considered identification [10]. However, (RFID) with a WPT a few system millimeters in HF distance band with between the transmission coils, electrical distancecoupling of several should tens be of considered centimeters [10]. usually However, considers a onlyWPT magneticsystem in coupling. HF band Second, with the measuredtransmission mutualdistance inductance of several is not tens the idealof centimeters inductance, usually but the considers mutual inductance only magnetic affected coupling. by the frequency Second, the andmeasured surrounding mutual objects inductance by parasitic is not elements. the ideal inductance, This is because but the the mutual values inductance used for the affected magnetic by the beamformingfrequency ofand the surrounding multi-coil WPT objects system by areparasitic the mutual elements. inductance This is abecauseffected by the parasitic values elements.used for the magnetic beamforming of the multi-coil WPT system are the mutual inductance affected by parasitic elements.

3.1. Conventional Measurement Methods

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Energies3.1. Conventional 2019, 12, x FOR Measurement PEER REVIEW Methods 4 of 10 In HF bands, the conventional methods to measure coupling coefficients among three or more In HF bands, the conventional methods to measure coupling coefficients among three or more coils can be categorized into two types, namely, those that use multi-port and those that use two-port coils can be categorized into two types, namely, those that use multi-port and those that use two-port VNAs. The former method simply measures the n n [S] matrix for n coils that are directly connected VNAs. The former method simply measures the n ×× n [S] matrix for n coils that are directly connected to ports of a multi-port VNA without an additional resonant capacitor. Then, the measured [S] matrix to ports of a multi-port VNA without an additional resonant capacitor. Then, the measured [S] matrix is converted into the [Z] matrix on an external PC. Consequently, the coupling coefficient between the is converted into the [Z] matrix on an external PC. Consequently, the coupling coefficient between ith and jth coils is obtained by: the ith and jth coils is obtained by: n o M Mij Im{}ZijIm Zij k ==k = ij = , , (2) ij ij p q n o (2) LL⋅ Li Lj {}⋅ {} ij ImZZiiIm ImZ jj Im Z · { ii} · jj where Mij is the mutual inductance between the ith and jth coils, and Li and Lj are the self-inductances where Mij is the mutual inductance between the ith and jth coils, and Li and Lj are the self-inductances of the coils, respectively. Here, Zij is the ij entry of the [Z] matrix, and it is often referred to as the of the coils, respectively. Here, Zij is the ij entry of the [Z] matrix, and it is often referred to as the transfertransfer impedance impedance for for i × j. The The otherother conventionalconventional method method repeatedly repeatedly measures measures 2 2 2× [S]2 [S] matrices matrices for × × forall all possible possible sets sets of twoof two coils, coils, where where the the other other coils coils must must be be terminated terminated with with a 50 a 50 ohm ohm load load from from the thedefinition definition ofthe of the S parameter, S paramete andr, thenand then synthesize synthesize the n then [S]n × matrixn [S] matrix for the for total then totalcoils. n This coils. method This × methodis theoretically is theoretically equivalent equivalent to the former to the methodformer method [11,12]. [11,12].

3.2.3.2. Proposed Proposed Measurement Measurement Method Method FromFrom Equation Equation (2), (2), it it is is confirmed confirmed that that the the coupling coupling coefficient coefficient is is calculated calculated using using only only elements elements ofof the the [Z] [Z] matrix matrix without without th thee need need for for the the [S] [S] matrix. matrix. Furthermore, Furthermore, the the calculation calculation of of kkij ijrequiresrequires only only impedanceimpedance parameters parameters related related to to the the ithith and and jjthth coils. coils. The The Z Z parameters, parameters, often often called called open-circuit open-circuit impedanceimpedance parameters, parameters, are are measured oror calculatedcalculated by by applying applying current current to to one one port port and and measuring measuring the theresulting resulting voltages voltages at all at otherall other ports ports opened. opened. That Th is,at to is, obtain to obtain the impedance the impedance parameters parameters related related the ith theand ithjth and coils, jth thecoils, others the others port must port bemust open-circuited. be open-circu Inited. addition, In addition, the required the required Z parameters Z parameters can be canobtained be obtained by converting by converting the measured the measured 2 2 [S] 2 × matrix 2 [S] matrix into the into 2 the2 [Z]2 × matrix.2 [Z] matrix. × × ConsiderConsider a a transmitter transmitter array array configured configured as as four four coils, coils, as as shown shown in in Figure Figure 33.. To To measure the couplingcoupling coefficient coefficient (k (k1414) )between between coils coils 1 1and and 4, 4, both both the the coils coils 1 1and and 4 4 are are connected connected to to a a two-port two-port networknetwork without without any any resonant resonant capacitor, capacitor, and and all all ot otherher coils coils are are open-termina open-terminated.ted. Because Because most most VNAs VNAs provideprovide a a function function to to mathematically mathematically manipulate manipulate me measuredasured data, data, such such as as Keys Keysight’sight’s equation equation editor, editor, wewe can can immediately immediately check thethe couplingcoupling coecoefficientfficient between between coils coils 1 and1 and 4 on4 on the the display display of theof the two-port two- portVNA VNA by converting by converting a 2 a 22 [S]× 2 matrix[S] matrix to a to 2 a 2 [Z]× 2 [Z] matrix matrix and and applying applying Equation Equation (2). (2). × ×

Figure 3. Circuit diagram for measuring coupling coeﬃcient between coils 1 and 4 for four coils using Figure 3. Circuit diagram for measuring coupling coefficient between coils 1 and 4 for four coils using a two-port vector network analyzer (VNA). a two-port vector network analyzer (VNA). 3.3. Simplicity of the Proposed Method 3.3. Simplicity of the Proposed Method Figure4 shows the measurement procedures of the proposed method and two conventional methodsFigure for 4 couplingshows the coe measurementﬃcients among procedures multiple of coils. the proposed Method I usingmethod a multi-portand two conventional VNA is very methodsconvenient for becausecoupling all coefficients relationships among among multiple coils are co measuredils. Method at once. I using However, a multi-port if the number VNA is of very coils convenient because all relationships among coils are measured at once. However, if the number of coils is larger than the number of ports of the network analyzer, this method cannot be applied. Additionally, the measured high-order [S] matrix is not instantaneously converted to the corresponding [Z] matrix on the common VNA, so the [S] matrix should be saved in an SnP

Energies 2019, 12, 3950 5 of 10 is larger than the number of ports of the network analyzer, this method cannot be applied. Additionally, theEnergies measured 2019, 12,high-order x FOR PEER [S]REVIEW matrix is not instantaneously converted to the corresponding [Z] matrix5 of on10 the common VNA, so the [S] matrix should be saved in an SnP (touchstone) ﬁle format and converted on an(touchstone) external PC. file Method format IIand using converted a two-port on an VNA exte requiresrnal PC. repetitive Method two-portII using a measurements two-port VNAn Crequires2 times forrepetitive all possible two-port sets ofmeasurements two coils and nC all2 times unconnected for all possible coils terminated sets of two with coils a and 50 ohm all unconnected load. In addition, coils theterminated total n withn [S] a matrix50 ohm should load. In be addition, synthesized the total using n × 2 n [S]2 [S]matrix matrices. should On be the synthesized other hand, using in the2 × × × proposed2 [S] matrices. method, On the only other two hand, coils arein the connected proposed with method, a two-port only two VNA, coils and are the connected coupling with coeﬃ a cienttwo- obtainedport VNA, from and Equation the coupling (2) can coefficient be immediately obtained checked from Equation on the display (2) can ofbe the immediately VNA according checked to theon distancethe display between of the coils.VNA Additionally,according to thisthe distance method isbetween not limited coils. to Additionally, the number ofthis coils. method Therefore, is not thelimited proposed to the methodnumber hasof coils. very Therefore, simple procedures. the proposed method has very simple procedures.

Figure 4. Measurement procedures for the proposedproposed and conventional methods.methods.

4. Experimental Setup and Results This section presentspresents experimentalexperimental setupsetup andand resultsresults toto validatevalidate thethe proposedproposed method.method. 4.1. Experimental Setup for Measurement 4.1. Experimental Setup for Measurement To simplify the problem, consider that there are three coils mutually coupled with each other. To simplify the problem, consider that there are three coils mutually coupled with each other. Figure5 shows the experiment setup to verify the performance of the proposed method. Three coils Figure 5 shows the experiment setup to verify the performance of the proposed method. Three coils were aligned co-axially. The coupling coeﬃcients between the ﬁrst and third coils were measured by were aligned co-axially. The coupling coefficients between the first and third coils were measured by changing the distance between the two coils in the HF band, in which the second coil was always changing the distance between the two coils in the HF band, in which the second coil was always located in the middle of the two coils. The parameters of the coils are summarized in Table1. The coil located in the middle of the two coils. The parameters of the coils are summarized in Table 1. The coil with the inductance of 1.1 µH was designed to have an high quality factor of about 250 at 6.78 MHz, with the inductance of 1.1 μH was designed to have an high quality factor of about 250 at 6.78 MHz, and it has three turns and a single layer printed on the FR-4 substrate, of which the dimension is and it has three turns and a single layer printed on the FR-4 substrate, of which the dimension is 95.7 95.7 mm 105.7 mm. The coil of 3.8 µH was fabricated to verify the proposed method for coils mm × 105.7× mm. The coil of 3.8 μH was fabricated to verify the proposed method for coils with high with high inductance, and it has six turns and two layers printed on the FR-4 substrate of the same inductance, and it has six turns and two layers printed on the FR-4 substrate of the same dimension dimension as the coil of 1.1 µH. as the coil of 1.1 μH.

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Figure 5. Experimental setup for measuring the coupling coefficient of three coils.

FigureFigure 5 5.. ExperimentalExperimentalTable setup setup 1. Parameters for for measuring measuring of coils the the used coupling coupling in measurement. coefficient coeﬃcient of of three three coils. coils.

Table 1. Parameters Coil 1 of coils used in Coil measurement. 2 Coil 3 Table 1. Parameters of coils used in measurement. L = 1.1 μH L = 1.1 μH L = 1.1 μH Case #1 Coil 1 Coil 2 Coil 3 CoilRdc = 10.08 Ω CoilRdc = 20.08 Ω CoilRdc = 30.08 Ω L = 1.1 µH L = 1.1 µH L = 1.1 µH Case #1 L = L1.1 = 1.1μH μ H L = L1.1 = 3.8μH μ H L = L1.1 = 1.1μH μ H CaseCase #1 #2 Rdc = 0.08 Ω Rdc = 0.08 Ω Rdc = 0.08 Ω Rdc = 0.08 Ω Rdc = 0.08 Ω Rdc = 0.08 Ω L =R1.1dc =µ H0.08 Ω L = 3.8RµdcH = 0.19 ΩL = 1.1 µRHdc = 0.08 Ω Case #2 L = 1.1 μH L = 3.8 μH L = 1.1 μH Case #2 Rdc = 0.08 Ω Rdc = 0.19 Ω Rdc = 0.08 Ω 4.2. Changing Coupling CoefficientR betweendc = 0.08 Coils Ω Rdc = 0.19 Ω Rdc = 0.08 Ω 4.2. ChangingFigure Coupling6 shows Coethe ffiresultscient between measured Coils by Method I using Keysight’s four-port VNA, E5071C, for 4.2. Changing Coupling Coefficient between Coils CasesFigure #1,6 #2, shows and only the resultstwo coils measured without bya middle Method coil. I usingIdeally, Keysight’s the coupling four-port coefficient VNA, between E5071C, two Figure 6 shows the results measured by Method I using Keysight’s four-port VNA, E5071C, for forcoils Cases must #1, #2, maintain and only a twoconstant coils without value regardless a middle coil. of Ideally,the frequency the coupling and coeexistencefficient betweenof other twocoils. Cases #1, #2, and only two coils without a middle coil. Ideally, the coupling coefficient between two coilsHowever, must maintain as the adistance constant between value regardless the farthest of the coils frequency decreases, and th existencee measured of other coupling coils. However,coefficients coils must maintain a constant value regardless of the frequency and existence of other coils. asdeviate the distance from between that of the farthesttwo coils coils without decreases, a mi theddle measured coil. The coupling deviation coe increasesfficients deviatefurther fromas the However, as the distance between the farthest coils decreases, the measured coupling coefficients thatfrequency of the two increases coils without and the a middleinductan coil.ce of The the deviation middle coil increases increases. further as the frequency increases deviate from that of the two coils without a middle coil. The deviation increases further as the and the inductance of the middle coil increases. frequency increases and the inductance of the middle coil increases.

Figure 6. Measured coupling coefficient using Method I for three coils of Cases #1, #2, and two coils. Figure 6. Measured coupling coefficient using Method I for three coils of Cases #1, #2, and two coils. 4.3. Comparison Results and Discussion 4.3.Figure Comparison 6. Measured Results coupling and Discussion coefficient using Method I for three coils of Cases #1, #2, and two coils. The coupling coefficients measured by the conventional and proposed methods with respect to The coupling coefficients measured by the conventional and proposed methods with respect to 4.3.the Comparison distance between Results the and farthest Discussion coils are shown in Figure7. We can notice that the proposed method producesthe distance almost between the same the farthest results ascoils the are conventional shown in Figure methods. 7. We can For notice a more that accurate the proposed comparison, method weproduces calculatedThe coupling almost the meancoefficients the same and standardresults measured as deviation the by convention the conv of dientionalfferenceal methods. betweenand proposed For thea more results methods accurate of Method with comparison, respect I and theto we theproposedcalculated distance method, between the mean and the and the farthest statisticalstandard coils resultsaredeviation shown are ofin summarized Figuredifference 7. We inbetween can Table notice2 .the It isthatresults confirmed the proposedof Method that bothmethod I and the the produces almost the same results as the conventional methods. For a more accurate comparison, we calculated the mean and standard deviation of difference between the results of Method I and the

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Energiesproposed 2019 , method,12, x FOR PEERand theREVIEW statistical results are summarized in Table 2. It is confirmed that both7 of the10 meanmean and and standard standard deviation deviation are are less less than than 0.001. 0.001. This This means means that that the the proposed proposed method method provides provides preciseproposedprecise and and method, repeatable repeatable and coupling thecoupling statistical coe coefficientsﬃ cientsresults among are among summarized multiple multiple coils. incoils. Table 2. It is confirmed that both the mean and standard deviation are less than 0.001. This means that the proposed method provides precise and repeatable coupling coefficients among multiple coils.

(a) (b)

FigureFigure 7. 7Measured. Measured coupling coupling(a) coe coefficientsﬃcients of of three three coils coils as as a a function function of of distance distance using( busing) the the conventional conventional andand proposed proposed methods methods at at 6.78 6.78 and and 13.56 13.56 MHz: MHz: (a ()a Case) Case #1 #1 and and (b ()b Case) Case #2. #2. Figure 7. Measured coupling coefficients of three coils as a function of distance using the conventional TableandTable proposed 2. 2. Mean methods and and standard standard at 6.78 deviation and deviation 13.56 of MHz: difference of difference (a) Case betw #1 betweeneen and the (b results) theCase results #2.of Method of Method I and the I andproposed the proposed method. method. Table 2. Mean and standard deviation of difference between the results of Method I and the proposed atat 6.78 6.78 MHz MHz at 13.56 MHz at 13.56 MHz method. meanMean St Dev.st dev. Mean mean St Dev. st dev. at−4 46.78 MHz 4 −4 4 at −13.564 MHz4 −4 Case #1Case #1 4.14.1 × 1010− 5.21 5.2110 −× 10 5.11 10 5.11− × 103.75 10− 3.75 × 10 mean× 4 ×st dev.4 × mean4 × 4 st dev. Case #2Case #2 5.215.21 × 1010−4 5.94 5.9410 −× 10−4 12.26 10 12.26− ×11.73 10−4 10− 11.73 × 10−4 Case #1 4.1 × ×10−4 5.21× × 10−4 × 5.11 × 10−4 × 3.75 × 10−4 Figure 8 shows the reactance of transfer impedances measured by the conventional and Case #2 5.21 × 10−4 5.94 × 10−4 12.26 × 10−4 11.73 × 10−4 proposedFigure 8methods shows the with reactance respect of to transfer the frequency impedances at the measured distance by of the 8 conventionalcm and 16 cm and between proposed the Figure 8 shows the reactance of transfer impedances measured by the conventional and methodsfarthest withcoils. respectIn spite toof thethe frequencysimple measuremen at the distancet procedure, of 8 cm the and proposed 16 cm between method theprovides farthest accurate coils. proposed methods with respect to the frequency at the distance of 8 cm and 16 cm between the Inresults spite of that the are simple similar measurement to those of procedure, the conventional the proposed methods. method That provides is, the proposed accurate resultsmethod that can are be farthest coils. In spite of the simple measurement procedure, the proposed method provides accurate similarused to to measure those of thethe conventionaltransfer impedance, methods. as Thatwell is,as thethe proposedcoupling methodcoefficient. can be used to measure the results that are similar to those of the conventional methods. That is, the proposed method can be transfer impedance, as well as the coupling coeﬃcient. used to measure the transfer impedance, as well as the coupling coefficient.

(a) (b)

Figure 8. Measured transfer(a) impedance Z13 as a function of frequency for the( bfarthest) distance of 8 cm and 16 cm using the conventional and proposed methods: (a) Case #1 and (b) Case #2. FigureFigure 8.8. MeasuredMeasured transfertransfer impedanceimpedanceZ Z1313 asas a function ofof frequencyfrequency forfor thethe farthestfarthest distancedistance ofof 88 cmcm and 16 cm using the conventional and proposed methods: (a) Case #1 and (b) Case #2. andBecause 16 cm usingthe transfer the convention impedanceal and isproposed ideally methods:represented (a) Case by #1Zij and = j ω(bM) Caseij, the #2. transfer impedance should be proportional to the frequency. However, in reality, the transfer impedance is not linear for Because the transfer impedance is ideally represented by Z = jωM , the transfer impedance the Becausefrequency the because transfer of impedance the effect ofis theideally parasi representedtic components by Zij ij of= jtheωM ijijcoil., the These transfer effects impedance become should be proportional to the frequency. However, in reality, the transfer impedance is not linear shouldstronger be in proportional Case #2 than to Case the frequency. #1 and at higherHowever, frequency in reality, than the at transfer lower frequency. impedance is not linear for the frequency because of the effect of the parasitic components of the coil. These effects become stronger in Case #2 than Case #1 and at higher frequency than at lower frequency.

Energies 2019, 12, 3950 8 of 10

Energiesfor the 2019 frequency, 12, x FOR PEER because REVIEW of the effect of the parasitic components of the coil. These effects8 become of 10 stronger in Case #2 than Case #1 and at higher frequency than at lower frequency. ToTo validate validate the the accuracy accuracy of ofthe the proposed proposed proc procedure,edure, the the measured measured Z parameters Z parameters were were converted converted intointo the the S parameters S parameters and and they they were were compared compared with with the the results results obtained obtained from from Methods Methods I and I and II. II.First, First, in inorder order to toobtain obtain a 3 a × 3 3 [S]3 [S] matrix matrix from from the the pr proposedoposed measurement measurement method, method, a set a set of oftwo two coils coils was was × connectedconnected to toa two-port a two-port VNA, VNA, and and a 2 a × 2 2 [Z]2 [Z]matrix matrix was was measured measured with with the the other other coils coils open-circuited. open-circuited. × ForFor the the other other set set of ofcoils, coils, 2 × 2 2 [Z]2 [Z] matrices matrices were were measured, measured, and and a 3 a × 3 3 [Z]3 [Z] matrix matrix was was constructed constructed by by × × combiningcombining three three 2 × 2 2 [Z]2 [Z] matrices. matrices. Finally, Finally, a 3 a × 3 3 [S3] [S] matrix matrix was was obtained obtained from from a 3 a × 3 3 [Z]3 [Z] matrix matrix by by × × × applyingapplying conversion conversion equations. equations. Figure Figure 9a,b9a,b shows shows the the S13 S 13fromfrom Methods Methods I and I and II, II,and and converted converted from from thethe proposed proposed method method inin aa range range from from 1 to 1 15to MHz 15 MHz on the on Smith the chartSmith and chart rectangular and rectangular plot, respectively. plot, respectively.S13 is the transmission S13 is the transmission coefficient coefficient from port 3from to port port 1 3 when to port all 1 other when coils all other are terminated coils are terminated in matched in loadmatched of 50 load ohms. of 50 However, ohms. However, in reality, in the reality, coils the of a coils WPT of systems a WPT systems are not terminatedare not terminated with 50 with ohms, 50and ohms, then and S13 thenhas S no13 has special no special physical physical meaning. meaning. Neverthless, Neverthless, the [S] the matrix [S] matrix converted converted from thefrom [Z] thematrix [Z] matrix obtained obtained by the by proposed the proposed method method is in goodis in good agreement agreement with with the [S] the matrix [S] matrix obtained obtained by the byconventional the conventional methods methods in the in frequency the frequency range range from the from LF the to HF LF band.to HF This band. means This thatmeans the that proposed the proposedmethod method provides provides accurate accu measurementrate measurement results. results.

(a)

(b)

FigureFigure 9. S 9. parameterS parameter results results for three for three coils coils from from the pr theoposed proposed and conventional and conventional methods methods at 8 cm: at 8(a cm:) Smith(a) Smith chart chartform formand (b and) real (b )and real imaginary and imaginary values. values.

InIn the the proposed proposed method, thethe critical critical assumption assumption is thatis that unconnected unconnected coils coils with awith two-port a two-port network networkanalyzer analyzer must be must perfectly be perfectly open-terminated. open-termina Inted. the measurement,In the measurement, we realized we realized the open the condition open condition by un-connecting anything from the coil’s SMA connector. This method does not fully satisfy the assumption for wide frequency band or high frequency due to the parasitic capacitance of the SMA connector, whereas the un-connecting method can be still powerful at a low frequency, including LF and HF bands used in most WPT systems.

Energies 2019, 12, 3950 9 of 10 by un-connecting anything from the coil’s SMA connector. This method does not fully satisfy the assumption for wide frequency band or high frequency due to the parasitic capacitance of the SMA connector, whereas the un-connecting method can be still powerful at a low frequency, including LF andEnergies HF 2019 bands, 12,used x FOR in PEER most REVIEW WPT systems. 9 of 10 In order to investigate the effect of non-perfect open, we applied the proposed method to the measurementIn order to investigate of the transfer the effect impedance of non-perfec of threet coilsopen, with we applied open-termination the proposed and method without to any the termination.measurement The of used the opentransfer termination impedance is the of OPEN three partcoils of with the Keysight’s open-termination 85052C, ofand which without the error any istermination. less than 0.65 The forused DC open to 3 termination GHz. Figure is 10 the shows OPEN the part normalized of the Keysight’s coupling 85052C, coefficients of which with the respect error ± ◦ tois frequenciesless than ±0.65° that for were DC obtainedto 3 GHz. fromFigure the 10 measured shows the transfernormalized impedance coupling using coefficients Equation with (2) respect and normalizedto frequencies to the that maximum were obtained coupling from coe ffithecient meas forured comparison transfer of impedance the effect ofusing the open Equation termination. (2) and Thenormalized difference to between the maximum the coupling coupling coe fficoefficientcients of thefor twocomparison cases is within of the effect approximately of the open 5%, termination. except for aroundThe difference 20 MHz between and 50 the MHz, coupling which coefficients are the self-resonant of the two cases frequencies is within of approximately the coils of 3.8 5%,µH except and 1.1forµ H,around respectively. 20 MHz Inand other 50 MHz, words, which this meansare the that self-resonant the proposed frequencies method of provides the coils reliable of 3.8 resultsμH and without1.1 μH, idealrespectively. open termination, In other words, but must this notmeans be usedthat the near proposed the SRF ofmethod the coils. provides Since reliable it is common results towithout operate ideal WPT open systems termination, at a lower but frequency must not be than used the near SRFs the of SRF coils, of the coils. proposed Since method it is common can be to effoperateectively WPT applied systems to measure at a lower the coupling frequency coe ffithancient th ore mutualSRFs of inductance coils, the betweenproposed the method coils used can inbe WPTeffectively systems. applied to measure the coupling coefficient or mutual inductance between the coils used in WPT systems.

(a) (b)

FigureFigure 10. 10.Measured Measured normalized normalized coupling coupling coeﬃ coefficientcient at distance at distance of 8 cm of for 8 open-terminated cm for open-terminated methods: (amethods:) Case #1 ( anda) Case (b) Case#1 and #2. (b) Case #2.

5.5. Conclusions Conclusions InIn this this paper, paper, a coupling a coupling coeﬃ coefficientcient measurement measurement method method with a simplewith a procedure simple procedure was proposed was toproposed overcome to theovercome inconvenience the inconvenience and limitation and oflimi conventionaltation of conventional measurement measurement methods of methods coupling of coecouplingﬃcients coefficients among coils among in multi-coil coils in WPT multi-coil systems. WPT To verify systems. the potentialTo verify of the the potential proposed of measurement the proposed method,measurement a comparative method, analysisa comparative was performed analysis bywas graph performed and mean by andgraph standard and mean deviation and standard values withdeviation the conventional values with measurementthe conventional methods measurement with respect methods to distance. with respect As ato result, distance. the As proposed a result, methodthe proposed was demonstrated method was todemonstrated achieve competitive to achieve performance competitive with performance good accuracy. with good If the accuracy. proposed If methodthe proposed is applied method to the is magnetic applied to beamforming the magnetic of be a multi-coilamforming WPT of a systems, multi-coil it canWPT be systems, a very powerful it can be alternativea very powerful to obtain alternative prompt results.to obtain prompt results.

Author Contributions: Conceptualization, D.-W.S.; methodology, D.-W.S.; software, S.-J.J.; validation, S.-J.J.; Author Contributions: Conceptualization, D.-W.S.; methodology, D.-W.S.; software, S.-J.J.; validation, S.-J.J.; formal analysis, S.-J.J.; investigation, S.-J.J.; resources, S.-J.J.; data curation, D.-W.S.; writing—original draftformal preparation, analysis, S.-J.J.; D.-W.S.; investigation, writing—review S.-J.J.; andresources, editing, S.-J.J.; D.-W.S.; data visualization, curation, D.-W.S.; S.-J.J.; writing—original supervision, D.-W.S.; draft projectpreparation, administration, D.-W.S.; D.-W.S.;writing—review funding acquisition,and editing, D.-W.S. D.-W.S.; visualization, S.-J.J.; supervision, D.-W.S.; project administration, D.-W.S.; funding acquisition, D.-W.S. Funding: This work was supported by a National Research Foundation of Korea (NRF) grant funded by the KoreaFunding: Government This work (MSIT) was (No.supported 2018R1C1B6003854). by a National Research Foundation of Korea (NRF) grant funded by the ConﬂictsKorea Government of Interest: (MSIT)The authors (No. 2018R1C1B6003854) declare no conﬂicts of interest.

Conflicts of Interest: The authors declare no conflicts of interest.

References

1. Kalwar, K.A.; Aamir, M.; Mekhilef, S. A design method for developing a high misalignment tolerant wireless charging system for electric vehicles. Measurement 2018, 118, 237–245.

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References

1. Kalwar, K.A.; Aamir, M.; Mekhilef, S. A design method for developing a high misalignment tolerant wireless charging system for electric vehicles. Measurement 2018, 118, 237–245. [CrossRef] 2. Mao, S.; Wang, H.; Zhu, C.; Mao, Z.-H.; Sun, M. Simultaneous wireless power transfer and data communication using synchronous pulse-controlled load modulation. Measurement 2017, 109, 316–325. [CrossRef][PubMed] 3. Anowar, T.I.; Barman, S.D.; Reza, A.W.; Kumar, N. High-Eﬃciency Resonant Coupled Wireless Power Transfer via Tunable Impedance Matching. Int. J. Electron. 2017, 104, 1607–1625. 4. Lee, J.H.; Seo, D.W. Adaptive Wireless Power Transfer System without Feedback Information using Single Matching Network. IEICE Trans. Commun. 2019, E102-B, 257–265. [CrossRef] 5. Jadidian, J.; Katabi, D. Magnetic MIMO: How To Charge Your Phone in Your Pocket. In Proceedings of the 20th Annual International Conference on Mobile Computing and Networking (MOBICOM), Maui, HI, USA, 7–11 September 2014; pp. 495–506. 6. Shi, L.; Kabelac, Z.; Katabi, D.; Perreault, D. Wireless Power Hotspot that Charges All of Your Devices. In Proceedings of the 21th Annual International Conference on Mobile Computing and Networking (MOBICOM), Paris, France, 7–11 September 2015; pp. 2–13. 7. Acero, J.; Carretero, C.; Lope, I.; Alonso, R.; Lucia, O.; Burdio, J.M. Analysis of the mutual inductance of planar-lumped inductive power transfer systems. IEEE Trans. Ind. Electron. 2013, 60, 410–420. 8. Theodoulidis, T.; Ditchburn, R.J. Mutual impedance of cylindrical coils at an arbitrary position and orientation above a planar conductor. IEEE Trans. Magn. 2007, 43, 3368–3370. [CrossRef] 9. Fu, M.; Zhang, T.; Ma, C.; Zhu, X. Eﬃciency and optimal loads analysis for multiple-receiver wireless power transfer systems. IEEE Trans. Microw. Theory Tech. 2015, 63, 801–812. 10. Hackl, S.; Lanschutzer, C.; Raggam, P.; Randeu, W.L. A Novel Method for Determining the Mutual Inductance for 13.56 MHz RFID Systems. In Proceedings of the 2008 6th International Symposium on Communication Systems, Networks and Digital Signal Processing, Graz, Austria, 25 July 2008; pp. 297–300. 11. Tippet, J.C.; Speciale, R.A. A rigorous technique for measuring the scattering matrix of a multiport device with a 2-port network analyzer. IEEE Trans. Microw. Theory Tech. 1982, 30, 661–667. [CrossRef] 12. Rolfes, I.; Schiek, B. Multiport method for the measurement of the scattering parameters of n-ports. IEEE Trans. Microw. Theory Tech. 2005, 53, 1990–1996. [CrossRef]

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