Modeling, Simulation and Verification of Contactless PowerTransfer Systems

J. Serrano(1,*), M. Perez-Tarragona(1), C. Carretero(2), J. Acero(1).

(1)Department of Electronic Engineering and Communications. Universidad de Zaragoza.Maria de Luna, 1. 50018 Zaragoza. Spain.

(2)Department of Applied Physics. Universidad de Zaragoza.Pedro Cerbuna, 12. 50009 Zaragoza, Spain.

(*)E-mail: jserrano@unizar.es

Abstract—This work presents the analysis of awireless power transfer system consisting of twocoupled coils and ferrite slabs acting as flux con-centrators. The study makes use of Finite ElementMethod (FEM) simulations to predict the key per-formance indicators of the system such as coupling,quality factor and winding resistance. The simula-tions results were compared against experimentalmeasurements on a prototype showing consistence.

Keywords—Wireless power transfer, Electro-magnetic modeling, FEM simulation, InductiveCharging.

I. INTRODUCTION

Wireless power transfer (WPT) applicationsare taking major importance among market trendsthanks to their versatility and user convenience.This solution allows the producers to removethe cables and connectors which are one of themain causes of breakdown. As this technologyconsolidates, WPT systems are becoming morecommon and available for a large number ofdevices. The modeling of these systems is,therefore, of great interest.

Important research efforts were made in thepast on the development of this WPT systems[1–4]. The main areas of interest include inductivecoupling between the coils [5], power lossesin the windings [6–10], and power electronics[11–15]. In terms of the intended application,these systems can be roughly classified in lowpower magnetic links and high power magneticlinks. Low power links are highly limited by sizerestrictions, i.e. small-sized devices and largeseparation between coils. These leads to looselycoupled coils and low efficiency in the inductivepower transference. On the other hand, highpower links are characterized the use of largercoils with higher coupling leading to higherinductive power transference efficiencies [16].

In this work, a simulation model for low powertransfer will be presented in order to predict thekey performance indicators of the system. These

Fig. 1. Wireless power transfer system.

will be later compared against measurements ona prototype.

In order to address this problem, in SectionII, key performance indicators will be defined.In Section III, the electromagnetic model will bepresented. In Section IV, the simulation procedurewith COMSOL will be detailed. In Section V,the model will be experimentally validated bycharacterizing a prototype. Finally, Section VIgathers the main conclusions of the study.

II. KEY PERFORMANCE INDICATORS OF LOWPOWER INDUCTIVE LINKS

The operating principle of WPT systems isbased on Lenz-Faraday’s Law, according towhich the induced electromotive force in anyclosed circuit is equal to the negative of the timerate of change of the magnetic flux enclosedby the circuit [17]. An alternating current isdriven through the primary coil generating andalternating magnetic field. This field causes avariable flux through the secondary coil, wherean electromotive force is induced. The systemcan be interpreted as a loosy coupled corelesstransformer.

The coupling between two coils, k, is definedas usual,

Excerpt from the Proceedings of the 2016 COMSOL Conference in Munich

k =M√L1L2

, (1)

where L1 is the self-inductance of the primary,L2 is the self-inductance of the secondary, M isthe mutual inductance.

Moreover, the quality factor of a coil is definedas,

Qi =ω0Li

Ri, (2)

where ω0 is the angular frequency of resonanceand i = 1, 2 is an index indicating primary orsecundary coil.

The efficiency of a WPT for a certain geometrywas proven [16] to be a function of the aforemen-tioned variables, being limited by the maximumefficiency [5]:

ηmax =1

1 + 2(kQ)2

(1 +

√1 + (kQ)

2

) (3)

where Q =√Q1Q2.

As can be inferred from Equation 3, the effi-ciency is maximized when the parameter kQ ismaximized. Therefore, the parameter kQ will bechosen as a key performance parameter, whichintroducing Equations 1 and 2, can be expressedas,

kQ =M√L1L2

√(ω0L1) (ω0L2)

R1R2. (4)

The electromagnetic model in the next sectionwill have as main objective to model and correctlypredict the variables involved in Equation 4: M ,L1, L2, R1 and R2 which are characteristic of thecoils.

III. ELECTROMAGNETIC MODELING

A. Self and mutual inductance calculation

High frequency devices usually employmulti-stranded litz wire due to its good reducedac-losses [10]. Litz wire ensures that all thestrands in the cable are equivalent amongthem [8], and therefore the current is equallyshared between all of the strands. Under thisassumption, the primary inductor can be modeledas a ring-shaped ideal domain with rectangularcross-section through which a constant currentdensity flows. This simplification is graphicallyrepresented in Fig. 2, where tf represents theferrite’s thickness, tw represents the winding’sthickness, rext and rint are the external andinternal radii respectively and indexes 1 and 2represent primary coil (transmitter) and secondarycoil (receiver). ∆z is the separation between coils

(a)

z∆

r∆

,2ft

,2wt

,1ft,1wt

int ,2r

ext ,2r

int ,1r

ext ,1r

(b)

Fig. 2. Cross-section schematic of (a) wireless powertransfer system with two coils wound with litz wire andtwo slabs of ferrite as flux concentrator. (b) Simulationmodel.

and ∆r represents misalignment.

Once the geometry is defined, the current den-sity can be defined as,

−→J =

n · I0tw,1 (rext,1 − rint,1)

ϕ (5)

being n the number of turns, I0 the currentand ϕ represents the azimuthal direction of acylindric coordinate system.

This current generates a magnetic field H andan electric field E, from which the induced elec-tromotive force can be calculated for both coils:

Vi = −∮ −→

E ·−→dl = − 1

Scoil,i

∫Vcoil,i

EϕdV (6)

From were the coil’s impedance can be ob-tained,

Zi =ViI0. (7)

The calculation of L1, L2 and M is immediate.When coil 1 is excited, L1 and M are obtainedas:

L1 ==m(Z1)

ω(8)

M ==m(Z2)

ω(9)

When coil 2 is excited, L2 can be calculatedand M can be verified:

L2 ==m(Z2)

ω(10)

M ==m(Z1)

ω(11)

Excerpt from the Proceedings of the 2016 COMSOL Conference in Munich

Note that as the inductors are ideal domains,no power losses take place on them. Therefore,the resulting resistance from Ri = <e(Zi) wouldbe null. The resistance of the losses are calculatedby a separate loss model.

B. Winding resistance calculation

The power losses in the winding can be dividedinto two groups, depending on their cause. Onone hand, dc and skin effect losses produce anassociated power loss referred as conduction loss,Pcond. On the other hand, induced currents due toproximity effects produce proximity losses, Pprox.Both can be modeled by resistances, Rcond andRprox, in such a way that Ri = Rcond,i +Rprox,i.

From previous studies [18, 19] the resistance ofa circular cross-sectional conductor per unit length(e.g. PCB tracks) is already known:

Rdc,skin,u.l. =1

πr20σΦdc,skin(r0/δ) (12)

Rprox,t,u.l. =4π

δΦprox(r0/δ)|Ht|2 (13)

where r0 is the strand radius, σ is the conductorelectrical conductivity, δ is the skin depth, Ht

is the transversal magnetic field and Φdc,skin

and Φprox are functions including the geometricand frequency dependences. Their analyticalexpressions where inferred in [20].

By integrating equations (12) and (13) in in-ductor domain, and expressing the resistances interms of cylindrical coordinates (where the indexk refers to the coordinates r and z ) the conduc-tion resistance Rcond,11 and proximity resistanceRprox,11 of one turn made of one strand can beobtained:

Rcond,11 =lavπr20σ

Φdc,skin(r0/δ) (14)

Rprox k,11 =4π

σΦprox (r0/δ)

⟨2πr

∣∣H0 i,1

∣∣2⟩(15)

where lav = π (rext + rint) represents theaverage length of the turns, H0 i,1 corresponds tothe magnetic field generated by the single-wireturn and

⟨2πr

∣∣H0 i,1

∣∣2⟩ is the averaged squaremagnetic field in the ideal ring-type inductorvolume.

Taking into account that an inductor consistof n turns connected in series, and ns strandsconnected in parallel, the relation for conductionlosses is straightforward: Rcond= n

nsRcond,11. On

the other hand, proximity losses are additiveboth with turns and strands and proportional to

Fig. 3. Model geometry.

TABLE I. System parameters.

Description Symbol ValueNr. Turns n 20Nr. Strands ns 105Strand radius rs 40 (µm)Ferrite thickness tf 2.5 (mm)Ferrite side length l 50 (mm)Coil external radius rext 22 mmCoil internal radius rint 10.12 mmCoil thickness tw 2.5 mmCoil separation ∆z 4 mmMisalignment ∆r 0-44 mm

the square of the magnetic field incident on theconductors, |H0|2 = n2|H0,1|2. Consequently,Rprox = n3 · ns ·Rprox,11.

The resistance of each coil is finally given by

Ri =n

nsRcond,11,i + n3 · ns ·Rprox,11,i. (16)

IV. SIMULATION IN COMSOL

The model was simulated in COMSOL makinguse of the AC/DC Module. The geometry isshown in Fig. 3 and the geometric parameters areshown in Table I.

The domains representing the coils are definedas quasi-ideal domains (σ ≈ 0 S/m, µr = 1),while the ferrites are characterized by theirphysical properties (σ ≈ 0 S/m, µr = 2000).Firstly, an external current density is set in theprimary coil. The current density is calculatedaccording to Equation (5), with a current ofI0 = 1 A and the geometric parameters in TableI. The simulations are run in frequency domain,sweeping the frequency from 1 kHz up to 300kHz. From this simulation the L1 and M can becalculated. In the second place, the simulationis repeated exciting the secondary coil (receiver)in order to obtain L2. Moreover, the magneticfield average is obtained in order to calculate thelosses according to Equation (15).

Excerpt from the Proceedings of the 2016 COMSOL Conference in Munich

(a) (b)

(c) (d)

Fig. 4. Simulation results for the magnetic field B. (a) Complete system. Primary coil is excited, secondary coil isin open circuit. (a) Complete system. Secondary coil is excited, primary coil is in open circuit. (c) Top view of theopen-circuited coil. (d) Top view of the excited coil.

(a) (b)

Fig. 5. Measured prototype. (a) Primary coil (b) Com-plete system.

Fig. 5b shows the obtained magnetic fieldwhen exciting the primary coil or transmitter andthe secondary coil or receiver.

V. EXPERIMENTAL VERIFICATON

In order to verify the model, a prototype wasmeasured with a Keysight E4980A LCR Meter.The system is composed of two coils whosegeometric parameters are shown in Table I. Bothcoils are equal, and therefore their resistance andinductance will be denoted as R = R1 = R2

and L = L1 = L2 respectively. The measuredprototype is shown in Fig. 5 and the measuringsetup is shown in Fig. 6.

The LCR Meter was set to measure seriesresistance and inductance from 10 kHz to 300kHz. The results of the self inductances an

Fig. 6. Measuring setup.

resistances are compared against the calculationsin Fig. 7 for different misalignments. As canbe observed, the resistances and and self-inductances slightly decrease with an increasingmisalignment. This has low impact in the kQfactor, and therefore in the maximal efficiency.However, the mutual inductance drastically dropswhen misaligning the coils. Fig. 8 shows acomparison between the calculated value of Mand the measured value, which was calculated asa quarter of the difference of the inductance ofthe series connection of both coils, Ls, and thecounter-series connection, Lcs [21]:

M =Ls − Lcs

4. (17)

Excerpt from the Proceedings of the 2016 COMSOL Conference in Munich

0 50 100 150 200 250 3000

100

200

300

400

500

f (kHz)

R (

mΩ

)

∆r = 0 mm∆r = 11 mm∆r = 22 mm∆r = 33 mm∆r = 44 mm

(a)

0 50 100 150 200 250 3000

10

20

30

40

f (kHz)

L (

µH)

∆r = 0 mm∆r = 11 mm∆r = 22 mm∆r = 33 mm∆r = 44 mm

(b)

Fig. 7. Impedance components of the coils as a function of frequency for different misalignments. Simulated impedancecomponents in solid line. Measured impedance components in circles. (a) Resistance. (b) Inductance.

0 10 20 30 40 50−5

0

5

10

15

20

25

30

∆r (mm)

M (

µH)

Sim.Exp.

Fig. 8. Mutual inductance as a function of misalignmentat 100 kHz. Simulated in solid line. Measured in circles.

In all cases, the simulations are consistent withthe experimental characterization.

VI. CONCLUSIONS

In this paper, a method for simulating WirelessPower Transfer systems in COMSOL is presented.The method combines analytical developmentswith Finite Element Method simulations andits main objective is to predict the value of thekey performance indicators of Wireless PowerTransfer system. In these systems, the maximumefficiency is highly dependent of the productbetween the coupling factor and the geometricaverage of the quality factors of both coils. Thesecan be expressed in terms of self-inductances,mutual inductances and resistances. Therefore, themethod proposed aims to accurately predict thesevariables. The simulations were compared againstempirical results showing good consistence.

VII. ACKNOWLEDGEMENTS

This work was partly supported by the SpanishMINECO under Projects TEC2013-42937-R andRTC-2014-1847-6, by the DGA-FSE, and by theBSH Home Appliances Group.

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Excerpt from the Proceedings of the 2016 COMSOL Conference in Munich