circuit coupling model containing equivalent eddy current

Circuit Coupling Model Containing Equivalent Eddy Current Loss

Impedance for Wireless Power Transfer in Seawater

Wangqiang Niu, Chen Ye, Wei GuKey Laboratory of Transport Industry of Marine Technology and Control Engineering,

Shanghai Maritime University,Shanghai, 201306

China

Abstract- Nowadays, as the whole world putmore emphasis on ocean resource exploration,the use of automatic underwater vehicles (UAVs)comes to be increasingly frequent. Inductivewireless power transfer (IWPT), as a powertransfer solution with high safety and flexibility,is quite promising applied in UAV power supply.However, when applied underwater, IWPT effi-ciency decreases due to eddy current loss (ECL)caused by high conductivity of water medium.In order to analyze IWPT output characteristicsin seawater, this paper proposes a coupling cir-cuit model involving equivalent eddy current lossimpedance (EECLI), which is derived via three-coil model. On the one hand, it is found thatsplitting frequency still exists in IWPT underseawater. On the other hand, EECLI is inde-pendent to coil distance, but proportional to op-eration frequency. The validity of the proposedmodel for IWPT system with coils in small size(coil outer diameter 12 cm, system resonant fre-quency 570 kHz) is verified by experiment, whichmeans it is available for IWPT system design andanalysis.

Keywords- inductive wireless power transfer,equivalent eddy current loss impedance, couplingcircuit model, frequency splitting, resonant cir-cuit in series.

I. Introduction

WITH the great development of electric and elec-tronic equipment age, people now have much

higher requirements on electric power transmission sce-narios, particularly for those mobile electrical applianceswhich need to work sustainably for a long time and arelimited by battery performance [1–3]. Then, inductivewireless power transfer technology seems to be an idealsolution. Especially when applied in underwater auto-matic vehicle (UAV), inductive wireless power transferhas much better safety, flexibility and reliability than

traditional wired power transfer since high conductivityand corrosiveness in water environment may cause dam-age to charging interface [4].

However, water environments with different conduc-tivity inevitably cause eddy current loss (ECL) and re-duce IWPT efficiency [5]. Thus, it is valuable to exploreeffects of high conductivity of seawater on IWPT. [6]investigates WPT characteristics in air, pipe water andseawater by experiment, finding that to coils with largeradius (decimeter level), both pipe water and seawaterhave significant attenuation effect on IWPT, even thatfrequency splitting phenomenon disappears compared toIWPT in air. Besides, when transmission distance keepsincreasing and exceeds a certain value, the output volt-age in liquid medium bounces back and trends to be astable value which is much higher than output in air.Similarly, [7] reports the results of experiments address-ing the effects of seawater conductivity on IWPT. Nev-ertheless, these researches lack theoretical details. Asfor frequency splitting in air, a common phenomenonto IWPT system with high quality factor, is discussedthoroughly in [8], but frequency splitting characteristicsunderwater is still in blank.

In order to further research IWPT underwater, the-oretical analysis based on electromagnetic field (EMF)calculation was proposed by researchers. In [9], ECLin underwater IWPT is analyzed by EMF theory andparameter optimization impedance of IWPT system inseawater is proposed as well. Specially, equivalent eddycurrent loss (EECLI) is introduced into circuit model tosimplify calculation procedure. While in [10], both ECLand detuning effect of seawater on IWPT are investi-gated. Moreover, the detuned system caused by seawaterconductivity is turned back to resonance through addinga compensation inductance. Furthermore, [11] proposesECL analysis of underwater WPT system with coil mis-alignments via geometry relation and Maxwell’ s equa-tion. On the other hand, ECL can be reduced throughfrequency optimization as well [12].

Nevertheless, plain EECLI model mentioned previ-ously is much too complex on calculation and lacks intu-itiveness on physical explanation. [13] applies a model re-

Received: March 30, 2021 Revised: April 14, 2021. Accepted: April 16, 2021. Published: April 23, 2021.

INTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNAL PROCESSING DOI: 10.46300/9106.2021.15.45 Volume 15, 2021

E-ISSN: 1998-4464 410

garding medium with high conductivity as a relay circuitin eddy current testing (ECT). It also reveals that themedium with high conductivity reduces self-inductanceon transmission side (Tx) and receiving side (Rx) in-creasing the resonance frequency.

In terms of underwater WPT designing, some workhas been done considering UAV application. E.g., a newcoil structure of 1×1×1 is proposed in [14] to reduce eddycurrent loss of WPT systems for UAV. In this research,the receiving coil is set between two parallel transmittingcoils and both simulation and experiment prove that thenew structure can effectively reduce ECL in the field. [15]proposes a voltage output stabilization realized by inte-gral circuit, which significantly reduces system volumewhile a good voltage stabilizing effect is guaranteed.

In this paper, a new WPT circuit model containingEECLI based on three-circuit model is proposed whichis suitable for WPT system with small coils where in-ductance variation can be ignored. In order to prove thevalidity of the new model, WPT experiment in air andseawater is carried out. In the experiment, the outputvoltage is observed under various frequency and coil dis-tance. It is found that frequency splitting exists both inair and seawater with little resonant frequency deviation.Meanwhile, the influence of EECLI on IWPT in seawatermainly occurs on the ridge of splitting. Furthermore, itis also found that EECLI has nothing to do with coil dis-tance, but is proportional to operation frequency. Thenew model is conducive to simplify analysis and optimizedesign of IWPT in seawater compared with EMF model.Our techniques can be combined with similar studies inElectrical Engineering like [16–18].

II. Circuit model analysis for IWPT inseawater

In this section, a calculation model for IWPT in sea-water based on three-coil circuit model is establishedfirst. Then, in order to evaluate EECLI caused by sea-water, the equivalent relay circuit is mapped into EECLIin primary and second side by formula derivation.

A. Three-coil modelDifferent from IWPT in air, it is found that various

extent of output attenuation usually occurs in IWPT un-derwater due to different conductivity of liquid mediumor coil dimensions. In this paper, we assume that bothTx and Rx coils have a coupling effect with seawatermedium, not only with each other. As a result, three-coil circuit model is introduced to analyze the outputcharacteristics of IWPT in seawater as shown in Fig. 1.In this model, the equivalent seawater relay circuit con-sists of equivalent seawater resistor rsea, inductor Lsea,mutual inductance Msea with Tx and Rx coils which isequal. In primary side, Us is AC voltage source, whileL1 and L2 represent the inductance of Tx and Rx coil.M means the mutual inductance between Tx and Rxcoils in air, which can be expressed as k

√L1L2 where k

is the coupling coefficient. To make the system resonantat a certain frequency, two capacitors are connected intoprimary and second side in series, whose capacitance is

Us

C C!

Uo

RB

RL

L1 L2

M

I

I!

rsea

Lsea

Msea

Isear1 r2

Fig. 1: The equivalent three-coil circuit model for IWPTin seawater

represented by C1 and C2, separately. In addition, RLis the load resistance, while RB is the resistance set forbalance, which means RL = RB here. r1 and r2 denotecoil internal resistance in primary and second side re-spectively. I1, I2 and Isea represent the current flowingthrough Tx, Rx and equivalent seawater loop, respec-tively.

According to Kirchhoff’ s law (KVL), three loops areable to list three equations which are expressed as fol-lowing matrix:Us0

0

=

Z1 −jωM −jωMsea

−jωM Z2 jωMsea

−jωMsea jωMsea Zsea

I1I2

Isea

(1)

where Z1, Z2 and Zsea represent the equivalentimpedance in three loops, respectively, of which expres-sions are given as: Z1 = RB + r1 + j(ωL1 − 1

ωC1), Z2 =

RL+r2+j(ωL2− 1ωC2

) and Zsea = rsea+jωLsea, while ωis the angular frequency. In order to obtain the theoreticEECLI, it is necessary to further transform (1). Solvingthe equations ,the eddy current in seawater is expressedas

Isea =jωMsea

ZseaI1 −

jωMsea

ZseaI2 (2)

Then, we haveUs = Z1I1 − jωMI2 − jωMsea( jωMsea

ZseaI1 − jωMsea

ZseaI2)

→ Us = (Z1 +(ωMsea)2

Zsea)I1 − jω(M −Msea

jωMsea

Zsea)I2

(3)In (3), the expression above can be divided into twoparts, which is similar to coupling circuit without re-lay circuit. Thus, the first part and second part can beregarded as the new impedance of the primary side andsecond side respectively, while the equivalent relay circuitis eliminated. Further, we can get the new impedance ofTx coil expression as

ZTxnew= r1new

+ jωL1new

= r1 + jωL1 +(ωMsea)2(rsea − jωLsea)

r2sea + (ωLsea)2(4)

→ r1 +(ωMsea)2rsear2sea + (ωLsea)2

+ jω(L1 −(ωMsea)2Lsear2sea + (ωLsea)2

)

(5)


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The same deriving procedure can be adopted to obtainthe new impedance expression in the second side as fol-lowing

ZRxnew= r2new

+ jωL2new

= r2 +(ωMsea)2rsear2sea + (ωLsea)2

+ jω(L2 −(ωMsea)2Lsear2sea + (ωLsea)2

)

(6)From (5) and (6), the new impedance and inductance ontwo sides can be summed up as following

rinew= ri +

(ωMsea)2rsear2sea + (ωLsea)2

(7)

and

Linew= Li −

(ωMsea)2Lsear2sea + (ωLsea)2

(8)

where i = 1, 2 representing original impedance andinductance on primary and second side respectively.Through analysis on three-coil model for WPT in sea-water, it can be seen that theoretically seawater has in-fluence on both impedance and inductance in WPT sys-tem. For impedance and inductance, they respectivelyincrease and reduce certain value which is relative withthe Msea, Lsea, rsea and ω, meanwhile the value of theformer three parameters are dependent on coil dimensionand seawater characteristics, e.g., conductivity, permit-tivity and permeability. In this case, the Lsea is quitetiny because the corresponding coil dimension is small.Thus, we ignore the variation of coil inductance.

B. Equivalent eddy current loss impedance modelIn order to analyze seawater attenuation effect on

WPT, the concept of equivalent eddy current lossimpedance Reddy is usually introduced. In [10], first theelectric field intensity E is obtained by EMF calcula-tion and then the eddy current loss can be derived asPeddy =

∫∫∫V

σE2dV , where V is the volume of ECL and

σ is the conductivity of seawater. Finally, the value ofEECLI is gained by the equation Reddy = Peddy/I whereI is the current in coil.

However, EMF methodology involves integration tothe square of Bessel function, which increases the com-plexity of calculation. In this paper, different from EMFmethodology, the proposed EECLI model is derived bypure coupling circuit model, which is much more intu-itive. From (7), the expression of EECLI can be ex-tracted as

Reddy =(ωMsea)2rsear2sea + (ωLsea)2

(9)

In this case, to simplify the EECLI model, we assumethat rsea is proportional to ω, namely rsea = αω, whereα is defined as resistance coefficient in seawater, thus theexpression of Reddy can be further transformed as

Reddy =M2seaα

α2 + L2sea

ω (10)

It can be seen that the value of Reddy is proportional

to ω and we can considerM2

seaαα2+L2

seaas a whole EECLI

coefficient β. Then Reddy can be expressed as

Reddy = βω (11)

Thus, a new circuit model containing EECLI can be ob-tained as shown in Fig. 2, which is transformed fromthree-coil model. According to KVL, two loops can listtwo equations as following matrix:

Us

C C

Uo

RB

RL

L1 L2

M

I!

I

Reddy1 Reddy2r1 r2

Fig. 2: The two-coil model containing equivalent eddycurrent loss impedance

(Us0

)=

(Z1 +Reddy1 −jωM−jωM Z2 +Reddy2

)(I1I2

)(12)

Thus, we can get complete circuit equations as below:Us − (r1 +Reddy1 +RB + jωL1 + 1

jωC1)I1 = jωMI2

I2(r2 +Reddy2 +RL + jωL2 + 1jωC2

) = −jωMI1(13)

where I1 =URB

RBand I2 = Uo

RL, where URB

is the volt-age on RB . Then the other expression of Reddy can bederived as

Reddy1 = |RB(RLUs−jωUo)URB

RL− Z1|

Reddy2 = | − jωM RLURB

R1Uo− Z2|

(14)

in order to verify theoretical value of Reddy since thephase and amplitude of URB

and Uo can be directlymeasured in experiment. To IWPT system in air, thereis no eddy current loss and Reddy = 0.

Through two-coil circuit model involving EECLI, the-oretical value of output voltage can be obtained usingnumerical software Wolfram Mathematica.

III. Experiment results and analysis

In order to further analyze the output characteristicsand EECLI variation of WPT in seawater, an experi-ment is carried out in air and seawater separately. Theexperiment and calculation results in different conditionsare compared and analyzed to verify the correctness oftheoretic model mentioned in the previous section.

A. Experimental Setup and ProcedureIn this case, the power source is set as AC voltage of

which peak-peak value (PPV) is 15.4 V with operationfrequency ranging from 300 kHz to 1 MHz. In the term ofTx and Rx coils, outer and inner diameter of the 15-turnplanar spiral coils wound by litze wire of 1.2 mm diame-ter are 8 cm and 12 cm respectively, as shown in Fig. 3.The self inductance of Tx and Rx coils are about 10.2 and10.4 µH measured by LCR meter (GW Instek:LCR821)


E-ISSN: 1998-4464 412

with 200 kHz frequency. Both two sides adopt 7.6 nFcapacitor to achieve resonant frequency around 570 kHz.In order to improve the quality factor of the circuit andobserve frequency splitting phenomenon, the external re-sistors in Tx and Rx are set as 5 Ω.

Fig. 3: Spiral coil

To imitate IWPT under seawater environment, thecoils are put into a 25*20*15 cm water tank made ofacrylic board and containing 13 cm high water level fullysubmerging two coils. Meanwhile, the conductivity ofsalt water is approximately 4.1 S/m measured by con-ductivity meter (LICHEN LC-DDB-1A), which is nearlysame as seawater. The complete experimental setup isdemonstrated in Fig. 4, which is composed of a signalgenerator (Tektronix: AFG3102) feeding sinusoidal sig-nals to a 10 W power amplifier (Rigol: PA1011), watertank, IWPT circuit prototype and oscilloscope (RIGOLDS4014) utilized to illustrate waveform of voltage on re-sistors in Tx and Rx. More specific parameters of IWPTcircuit protocol is illustrated in Table. 1, while the ex-perimental procedure is shown after.

Fig. 4: IWPT experimental setup

A..1 Self resonant frequency determination

First, without the receiving side, Tx coil is put intoair and the peak value of voltage on resistor is recordedwith the frequency to determine the actual value of theself resonant frequency.

Table 1: Circuit parametersComponents Parameters Values

Tx

RB 5.00 Ωr1 0.3 ΩC1 7.71 nFL1 10.2 µH

Rx

RL 5.02 Ωr2 0.28 ΩC2 7.66 nFL2 10.4 µH

A..2 Output voltage measurement on both sides

Next adding second side, the waveform of voltage onresistors in two sides is measured with various coil dis-tance and sweeping frequency. The frequency step is10 kHz and distance step is 0.5 cm. The experiment iscarried out in air and salt water to investigate seawa-ter ECL effects on WPT transmission characteristics. Inthis case, coil misalignment is not involved to simplifyanalysis procedure, therefore Tx coil is fixed while Rxcoil is moved coaxially with Tx coil.

A..3 Determination of EECLI coefficient

In this case, EECLI coefficient β is determined as6.5× 10−5 by fitting to one experiment value arbitrarily.

Fig. 5: Output voltage versus frequency and couplingcoefficient, where the black lines are measured data andthe continuous surface is theoretical values.

B. Experiment results

In the situation with merely first side, it is observedthat the peak value of voltage on resistor appears near570 kHz. When second side coupling with first side,Fig. 5 illustrates the relation in 3D among IWPT outputvoltage, frequency and coupling coefficient in seawater,which is obtained by model calculation and experiment,indicating that frequency splitting remains to exist [8].

The comparison between air and seawater is illus-trated in Fig. 6. It is clear that frequency splittingphenomenon appears in both air and seawater to coilswith diameter in centimeter level. Compared with IWPT


E-ISSN: 1998-4464 413

in air, seawater has a certain attenuation effect on out-put. In this case the splitting point emerges between 2 to2.5 cm (around 0.15 converting to coupling coefficient).Specifically, there is no significant resonant frequencyshifting observed, which corresponds to the theoreticalmodel where seawater equivalent inductance value is rel-atively tiny as the IWPT coils are in small scale. Onthe other hand, as shown in Fig. 7, extracting outputvoltage on second side under resonant frequency, as thecoil distance increases, both output voltage in air andseawater rises to the peak at splitting point, then de-cays to zero gradually. It is worth noting that EECLImainly affects on the splitting ridge of the output, oth-erwise the influence is comparatively weak, which meetsthe calculation model as well. Furthermore, in order

Fig. 6: Output voltage in air and seawater

Fig. 7: Output voltage in air and seawater when reso-nance

to obtain the experimental value of EECLI in two sides,voltage value on resistors of two sides is recorded andsubstituted into (14).

Therefore, the experimental value of EECLI versusdifferent coupling coefficient and frequency is obtained asshown in Fig. 8 (a) and (b), respectively, where the solidlines represent the calculation and the markers are theexperiment values in Tx and Rx. Fig. 8 (a) illustratesthe trends of EECLI value over coupling coefficient k

(a)

(b)

Fig. 8: EECLI variation: (a) EECLI value versus cou-pling coefficient k; (b) EECLI value versus frequency

from 0.1 to 0.5 at resonant and non-resonant frequency(370 kHz, 570 kHz and 870 kHz, respectively). At 370kHz, the mean EECLI values of Tx and Rx are 0.154 and0.148 Ω close to calculation value 0.152 Ω. At 570 kHz,the average EECLI of primary and second side are 0.242and 0.238 Ω which are consistent with the calculationvalue 0.234 Ω. Similarly at 870 kHz, the average EECLIof two sides are 0.356 and 0.346 Ω meeting the calculationvalue 0.357 Ω as well. It is noticeable that EECLI isindependent to k, regardless of whether the system is inresonance or not.

Fig. 8 (b) demonstrates the relationship betweenEECLI and operation frequency from 300 kHz to 1 MHz.Specifically, EECLI presents a linear growth from ap-proximately 0.1 to 0.4 Ω as frequency increases, wherethe experimental values conform to the calculation ones,which verifies the validity of (11). It is worth mentioningthat comparing the outcome in this paper with the oneby EMF method in [9], both shows positive correlationbetween EECLI and frequency, however quadratic rela-tion is found in [9] and proportional relation is found inthis study.


E-ISSN: 1998-4464 414

IV. Conclusions and future work

In this paper, output characteristics of IWPT withcoils in small size is analyzed via coupling circuit modelcontaining EECLI, which is based on three-coil couplingmodel, instead of EMF model. Frequency splitting canbe observed in IWPT under seawater as well as in air.The ECL mainly causes attenuation on the ridge of out-put. Most importantly, it is found that, for a symmetricsystem, EECL in two sides is equal and independent tocoil distance, but proportional to operation frequency,which is similar to the conclusion by EMF model [9]and implies the significance of frequency optimizationin the design of IWPT system in seawater. The pro-posed EECLI model omits the complex EMF calcula-tion process and substantially improves the convenienceof underwater IWPT analysis, while certain accuracy isensured.

However, the relation between seawater resistance co-efficient α and EECLI coefficient β is interesting to fur-ther discuss in the future. In addition, this paper ismerely aimed at IWPT with coils in small dimension.The characteristics of IWPT with large-radius coils maydiffer [6] and will be more worth researching next step.Then, the practical application on large underwater de-vices such as UAVs with high power can be considered.

Acknowledgment

The authors would like to acknowledge their col-leagues, for their assistant during the experiment.

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E-ISSN: 1998-4464 415

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W. Niu received the B.E. degree from Xi’an Aerotech-nical College, Xi’an, China, in 1998, the M.E. degreefrom Northwestern Polytechnical University, Xi’an, in2004, and the Ph.D. degree from Shanghai Jiao TongUniversity, Shanghai, China, in 2008.

From 2013 to 2014, he was a Visiting Lecturer withMcMaster University, Hamilton, ON, Canada. Since2008, he has been a Lecturer with Shanghai MaritimeUniversity, Shanghai, where he has been an AssociateProfessor, since 2017. His research interests includewireless power transfer, control of marine equipment,and prognostics and health management.

C. Ye received the B.E. degree from Shanghai Mar-itime University, Shanghai, China, in 2016. He is cur-rently pursuing the M.S. degree with Shanghai MaritimeUniversity, Shanghai, China.

His main research interest includes wireless powertransfer technology.

W. Gu received the B.E. and Ph.D. degrees fromShanghai Maritime University, Shanghai, China, in 1982and 2008, respectively.

Since 1982, he has been a Teacher with ShanghaiMaritime University, where he has been a Professor,since 1997, and is currently the Director of the Key Lab-oratory of Transport Industry of Marine Technology andControl Engineering. His research interest includes ma-rine information control technology.

Contribution of individual authors to the creationof a scientific article (ghostwriting policy)Wangqiang Niu did the study conceiving and methodol-ogy.Chen Ye carried out the experiment and original draftpreparation.Wei Gu is responsible for review and editing.In general, please, followhttp://naun.org/main/format/contributor-role.pdf

Creative Commons Attribution License 4.0 (Attribution 4.0 International, CC BY 4.0)

This article is published under the terms of the Creative Commons Attribution License 4.0 https://creativecommons.org/licenses/by/4.0/deed.en_US


E-ISSN: 1998-4464 416

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circuit coupling model containing equivalent eddy current

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