Computationally-Efficient, Generalized Expressions for the Proximity-Effect in Multi-Layer,...

5404 IEEE TRANSACTIONS ON MAGNETICS, VOL. 49, NO. 11, NOVEMBER 2013

Computationally-Efficient, Generalized Expressions forthe Proximity-Effect in Multi-Layer, Multi-Turn Tubular

Coils for Wireless Power Transfer SystemsZeljko Pantic and Srdjan Lukic

Electrical and Computer Engineering Department, North Carolina State University, Raleigh, NC 27604 USA

Wireless power transfer (WPT) based on magnetic induction is used in numerous applications where physical contact between thepower source and the load is not desired. For efficient power transfer, the resonant coils should have a low equivalent series resistance atthe resonant frequency and have a high packing factor while being simple to manufacture. Coils made from hollow copper tubes might bean acceptable alternative to Litz wire designs due to low skin-effect resistance, easy manufacturing, and simplicity in implementing activecooling; however, the lack of an analytical model for complex coil designs poses a difficulty in systematically assessing its benefits andlimitations. This paper presents a new method, based on the Fourier series, for evaluating proximity-effect losses in a multi-turn, multi-layer tubular coil. The model evaluates the proximity factor as a function of coil and tube parameters, which can be incorporatedinto the design and optimization procedures. The derivations are supported by simulations that compare analytic and finite elementmodels (FEM) of current density distribution in the coil. The model is further validated via experimental measurements of the resultingequivalent series resistance for two prototype coils.

Index Terms—Model of a tubular coil, skin and proximity effect modeling, tubular conductor, wireless power transfer.

I. INTRODUCTION

R ECENTLY, the notion of powering devices wirelessly hasroused a great deal of interest in academia and industry

for its ability to make charging easier for a range of devices,from cell phones to plug-in electric vehicles. A key componentallowing efficient wireless power transfer (WPT) is the designof high quality-factor coils that compensate for the low couplingcoefficient between the source and the receiver. High qualityfactors can be achieved by minimizing the skin and proximityeffect in the coil that cause a rise in the coil equivalent seriesresistance. Both skin and proximity effects lead to uneven cur-rent distribution in the conductors, with the bulk of the currentflowing far from the center and far from the other turns of thecoil, resulting in a poor utilization of the cross-section of theconductor.An effective way to reduce skin-effect resistance is to

design the resonant coils using Litz wires [1]–[3]. Currentis distributed equally among the strands of the Litz wire bytwisting the bundles and by keeping the strand diameter belowthe skin depth of the conductive material [4]. However, asthe frequency increases, the strand diameter must be reduced,which adds cost, complicates the manufacturing process, andlowers the packing factor as the insulation occupies a largerportion of the bundle cross-section area. In addition there is astrong strand-to-strand proximity effect that further displacesthe current from a uniform distribution. In [5], [6] authorshave proposed to plate the strands with magnetic material tofurther minimize the strand-to-strand proximity effect. Thougheffective, this method further complicates the coil design andfurther reduces the packing factor. An interesting alternative

Manuscript received December 12, 2012; revised March 05, 2013; acceptedMay 06, 2013. Date of publication May 21, 2013; date of current version Oc-tober 21, 2013. Corresponding author: S. Lukic (e-mail: [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TMAG.2013.2264486

to Litz wire is a coil made of hollow copper tube. Tubularconductors have lower skin-effect compared to other com-monly used conductor shapes with the same cross sectionalarea [7], [8]. If the proximity effect losses are minimized, atubular conductor might prove to be superior to Litz wires insome WPT applications. Since optimization procedures formulti-turn, multi-layer Litz-wires exist [1], [9], a key issue isto offer a similar design procedure for hollow copper tubes thatmaximizes the coil quality factor, given the operating frequencyand spatial constrains of the application.Modeling of a WPT system includes the derivation of the

expressions for total ac resistance of the coils, self-inductance,self-capacitance, and mutual coupling as a function of coil ge-ometry and operating frequency. Expressions for coil self-in-ductance have been presented in [10], for self-capacitance in [2],and for mutual inductance in [11], [12]. In the case of Litz-wire amodel for ac resistance is given in [2]. For hollow copper tubes,while all other parameters are easily adopted from literature,finding the coil ac resistance due to skin and proximity effectis a major challenge. Finite element modeling (FEM) softwarepackages provide a potential solution; unfortunately, it becomesunfeasible to incorporate high-fidelity FEM simulations in anoptimization procedure.In the past, researchers have developed methods to evaluate

the ac resistance of a single or multiple tubular conductors (typ-ically two or three) arranged in some particular formation. Solidconductors were thoroughly investigated in the pioneering workby Manneback [13], Carlson [14] and Dwight [15]. Mannebackapplied a network theory concept to solve for the current dis-tribution caused by proximity and skin effect in two parallelcylindrical solid wires. Carson applied Poyinting’s theorem ofenergy flow to work out a closed analytical solution for resis-tance factor of a two wire system with opposing currents (asystem known as “go-and-return”). Dwight considered varioussolid and tubular conductive configurations and solved the re-sulting integral equations in closed form by using a method ofsuccessive approximations. The problem is addressed in a sim-ilar way by Arnold in [16]–[18]. The main characteristic of the

0018-9464 © 2013 IEEE

PANTIC AND LUKIC: PROXIMITY-EFFECT IN MULTI-LAYER, MULTI-TURN TUBULAR COILS FOR WIRELESS POWER TRANSFER SYSTEMS 5405

contributions in [13]–[19] is that they deal with a specific geom-etry and a limited number of solid or tubular conductors, rarelymore than three. In [20] the authors discuss the eddy current phe-nomenon in many different structures (solid conductor, tubularconductors, sphere, and plane sheets). The radiation resistanceis discussed in detail, but is of secondary importance for anWPTsystem as shown in [9]. A very interesting analysis of coil de-signs that results in frequency independent coil resistances forapplication in ac instrumentation is given in [21]. However, dueto very low frequencies (less than 3000 Hz), small coil induc-tance, and large resistance (1–10 000 ), the results are not ap-plicable in WPT systems. In [22] authors use different coils fortransmitters and receivers in the radio frequency range. Theyemphasize that tubular conductors are ideally suited for watercooling in high power transmitters. They also underscore lackof an analytic solution for resistance calculation in multi-turntubular coils.Smith [19] looked at a single layer of multiple solid conduc-

tors and was the first to use the Fourier series expansion of thecurrent density distribution. He found an exact solution that isvalid for a two-conductor system and proposed an approxima-tion for the multi-turn case. The accuracy of the applied approx-imation depends on the number of harmonic terms included inthe Fourier series. Smith suggested the point-matching or theleast squares procedures to calculate the Fourier series coef-ficients. Although numerically burdensome, this approach re-veals that the current density distribution for a limited numberof round conductors is always a periodic function of anglewith a period of , which allows the evaluation of Fourier se-ries coefficients to match the actual current distribution. Un-fortunately, the exact solution of the integral equation can beobtained only for several geometrically simple topologies [23],[24], while a numerical approach is used in [25], [26]. In otherwords, existing proximity effect models are well-suited for eval-uating single, two or three turn coil designs. However, if used inthe general form to evaluate a more complex coil arrangement,they become numerically inefficient to integrate within an opti-mization procedure.In this work, we derive an expression for the current den-

sity distribution in multi-turn multi-layer coil. We generalizethe Fourier series expansion derived in [27] to model the cur-rent distribution in a thin tube surrounded by a network of fila-ments representing the other turns of the coil. Replacing othercoil turns with filaments is justified by recognizing that the mag-netic field outside the conductor can be represented by a currentfilament positioned in the center of the conductor that carriesthe same current . However, due to the strong proximity ef-fect in the multi-turn coil, the current is no longer distributeduniformly. Therefore, we develop a method to appropriatelyplace the filament to best approximate the effect of the cur-rent distribution in the tube on the rest of the system. Finallywe derive an expression for the resistance factor and accountfor the proximity effect of the curvature of the tubular coil. Wevalidate the analytic model through finite element simulationsand though experiment for two-coil designs that make use oftubular conductors with wall thicknesses differing by two or-ders of magnitude.The main contribution of the proposed method over using the

finite element approach is the substantial reduction in the com-

Fig. 1. Layout of the coil.

putational burden. Modeling tubular conductors is especiallychallenging using FEM software due the requirement for a veryfine mesh to accurately model the skin effect. At very high fre-quencies the skin depth requires custom meshes to ensure suffi-cient mesh density in the current conducting portion of the tube.Therefore, it becomes difficult to determine the frequency de-pendence of the coil resistance in a frequency sweep. The useof very fine meshes requires substantial processing power. Forexample a forty-point frequency sweep of a six-turn coil takesmore than one hour to complete on an Intel i7 2.93 GHz, 8GB processor, compared to five seconds execution time for theMATLAB® code implementation of the proposed algorithm.As a result the analytic approach can be incorporated in anautomated optimization procedure, which would be very chal-lenging with FEM software.The remainder of the paper is outlined as follows. Section II

defines the system geometry as well as the conditions requiredfor the decoupling of the skin and proximity effects, whichis necessary for the analytic derivation. Section III presents anew algorithm for modeling the proximity effect, which usesFourier series expansion to solve the integral equation and de-rive a closed-form solution for the ac resistance of a multi-turn,multi-layer tubular coil. By replacing the surrounding tubeswith correctly positioned current filaments, the equivalentstructure can be significantly simplified. In Section IV theanalytic model is validated through a set of COMSOL Mul-tiphysics® simulations for three different coil structures.Experimental verification of the suggested resistance model ispresented in Section V. Finally, Section VI briefly summarizesthe contributions of the presented work.

II. PROBLEM STATEMENT

We define our coil in general terms as a uniform multi-turn,multi-layer coil of inner radius , and the longitudinal andradial length of winding area and . The specification of thecoil turns assumes hollow tube inner radius , outer radius ,wall thickness , and distances between turns in bothdirections and . These parameters are visualized in Fig. 1.The numeration of the turns inside a coil is depicted in Fig. 2.A method to decouple the skin and proximity effect for

tubular coils was thoroughly discussed in [28]. Following the


Fig. 2. Turns numeration in a winding.

decoupling principle proposed in [29], [9] and [30] the authorsin [28] split the resulting current density as follows:

(1)

where the local conductor field modulates the current in ra-dial direction and the external field defines the current inazimuthal direction. Consequently, the approximate formula fortotal ac resistance is in the form:

(2)

where is the proximity factor [9]. Analyzing the skin effectfirst, we evaluated multiple methods to calculate the skin effectin a tubular conductor in [28] and compared the results to FEMsim ulations. We concluded that the modified Dwight’s formulabased on the Bessel function [31] is most accurate for the fre-quency range of interest (up to 4 MHz). The main focus of thispaper is the derivation of an expression for as a function ofthe coil and tube parameters.

III. DERIVATION OF THE EXPRESSION FOR THE PROXIMITYFACTOR AND CURRENT DENSITY DISTRIBUTION

We build on the approach in [27] where the authors derivedthe current distribution for a system containing two parallelinfinitely long elements: a thin tube and a current filament(Fig. 3). The Tube–current filament problem was solved earlierby Dwight [15] but his solution requires successive approxima-tions that cannot be generalized for the multi-layer, multi-turncase.

A. Derivation of Formula for Current Density DistributionAffected by Multiple Filaments

Following the approach outlined in [27], the current throughthe cross section of the tubular conductor in Fig. 3 can be decom-posed into two parts by applying the superposition principle:

(3)

Fig. 3. Conductive tube and current filament [27].

where represents skin effect current distribution andis current in the tubular conductor caused by the filament.

If the thickness of the tube wall is small enough, radial changein current density can be neglected, which results inconstant

over the conductor cross section. Furthermore, it can beshown that average value of over the same tube crosssection is zero. In other words, azimuthally modulatescurrent through the conductor to form a final current densitylayout. The expression for is a solution of the integralequation (4) that describes the current density at an arbitrarypoint of the tubular conductor:

(4)

The terms on the right side of the equation can be explainedas follows [27]: the first term exists even if the current throughfilament B is zero. Therefore, the first term represents the in-fluence that current in the tube has on itself. The second termdescribes redistribution of current in the tube caused by currentin the filament B. Constant cancels the constant componentin the first two terms that should not be present in the expressionfor .Equation (4) can be simplified by replacing:

(5)

(6)

Finally, the authors in [27] were able to simplify (4) into

(7)

The authors in [27] solved integral equation (7) in a very ef-fective way. They applied Fourier series expansion discussedabove and exploited the axial symmetry to reduce the whole


Fig. 4. A conductive tube and a filament in an arbitrary position.

series to cosine terms. Unfortunately, that approach is suitableonly for the simple case of one tube and one filament. Since ourgoal is to generalize the approach, we further develop (7) intoa form that is suitable for multi-turn, multi-layer coils. Let usassume that each arbitrary coil turn is replaced with afilament carrying a current . For an arbitrary coil at a position

let us use polar coordinates andto describe the position of an arbitrary filament , as shownin Fig. 4. Parameters , and belong to the ranges:

and . Now, we modify expression (7) to aform that describes the current density redistribution caused bythe current in the equivalent filament.

(8)

All other tubular turns (except the one at position ) willbe replaced with the filaments in the same way and their influ-ence on the tube at position will be modeled as in (8). Byusing superposition, the proximity current distribution in tube

can be rewritten in the form:

(9)

or more explicitly as:

(10)

The first term on the right side of the formula can be simplifiedby exchanging the order of the summation and the integral:

(11)

Now, let us modify the second term on the right side of (10)and transform it into an infinite series of cosine and sine terms.The first step is to modify (10) into a form appropriate for seriesexpansion:

(12)

Now, by applying complex Taylor’s series [27]:

(13)

to the function in (12), where, and equating the real

parts of the result in (13), we get:

(14)

Using trigonometric relations (12) can be further simplified to:

(15)


By exchanging the order of summation, one can identify theterms that represent coefficients for cosine and sine expansion:

(16)

where:

(17)

(18)

(19)

It is evident that the solution of contains the entireFourier series expansion (both sine and cosine terms are incor-porated). As explained earlier, the average value ofover cross section of conductor is zero, so the term isnullified:

(20)

The result is an integral equation in the following form:

(21)

Constant should be selected to cancel out the constant termon the right side of the (21) resulting in:

(22)

Two integrals on the right side of (21) will be solved by ap-plying partial integration and using the solution of the resultingsingular integral from [27], [32]:

(23)

(24)

Combining (17)–(24) we identify coefficients of the Fourierseries:

(25)

(26)

Finally, total current through a turn at position is the sumof the skin effect current density (equal for allturns) and a proximity current : Formula (27), shownat the bottom of the page, shows that proximity effect modifiesboth phase and magnitude of the current density phasor.

B. Correction of the Filament Position

From the previous discussion it is not clear how to positioneach filament to best represent the tubular conductor. When thecurrent distribution in a tubular conductor is uniform, an ade-quate replacement is a filament in the center of the tube. For anon-uniform current distribution , the filament positionmust be changed. Let us assume that the tube is positioned in the

(27)


Fig. 5. A tubular conductor replaced by an equivalent current filament.

origin (0, 0) of an coordinate system and that its currentdistribution is described by (27).The goal is to derive the position of the equivalent filament

expressed in terms of a position vector in the system(Fig. 5). The current through the equivalent filament should bethe same as the tube current. Equation (27) shows that at any an-gular position the current density is a complex numberwith frequency dependent magnitude and phase. It is easy to de-rive the ratio between real and imaginary components of the thharmonic:

(28)

This ratio is the smallest for the first harmonic andincreases for the higher order harmonics. In the case ofWPT res-onant coils and are commensurable and much smaller than. We conclude that for , (28) is typically much greaterthan zero and that the imaginary term is much smaller than thereal part. Therefore, we neglect the imaginary part of the firstharmonic when finding the location of the current filament. Al-though the imaginary component in the higher order harmonicsincreases, their overall contribution to declines rapidlymaking the imaginary part of the higher order harmonics neg-ligible. In conclusion, for the derivation of , we consideronly the real part of (see (29) at the bottom of the page).

can be further simplified by expressing it in a complexform of Fourier series:

(30)

where:

(31)

In order to derive an expression for , we exploit the con-cept of the center of mass of a mechanical object. More specifi-cally, we determine the currentfilament position by calculating avector as an “average” position of the current filament thatcan replace continuously distributed current through a tubularconductor. Expressed mathematically, the well-known formulafor the center of mass for a two-dimensional object:

(32)

is modified into a formula for “center of distributed current”:

(33)

Combining (30), (31), (33) and switching the order of summa-tion and integration results in:

(34)

One can easily see that the previous integral gives a non-zerovalue only for :

(35)

Therefore, the final expression for the correction vector will be:

(36)

(29)


The derivation introduced so far merits further discussion.First, we note that (36) determines the shift of filamentdue to the proximity effect of all other turns of the copper tuberesonant coil. The real and imaginary components of vector

shift the position of the filament from the physicalcenter of the tubular conductor in the longitudinal and radial di-rections, respectively. In deriving the correction factor for fila-ment from (36) we locate all other filaments making up thehollow copper tube at the centers of the respective tube they rep-resent. This simplification introduces an acceptably small errorin the current distribution and resistance calculation, as we showlater. This simplification results in a procedure that does not con-tain iterations, significantly simplifying the proximity factor andresistance calculations. Finally, with the corrected position of allfilaments determined from (36), the current distribution withinthe hollow copper tube can be calculated from (29).

C. Derivation of an Expression for Proximity-Effect Resistance

Resistance per length that models proximity losses in the con-ductors is:

(37)

where is an infinitesimally small area of the conductor. Com-plex form of the complete Fourier expressions is ap-plied again:

(38)

(39)

resulting in:

(40)

The previous expression can be simplified by recognizing thatonly the product of the same harmonics from two sums gener-ates a nonzero result :

(41)

Substituting (39) into (41) allows us to express as a func-tion of current distribution coefficients and :

(42)

Finally, by recognizing that , the ratio be-tween the proximity and dc resistance will be (43), shown at thebottom of the page. It is easy to see that results presented in [27]can be derived from (43) as a special case for a two-turn coil.The total ac resistance that combines skin and proximity effectscan be obtained now by using (43) for proximity effect factor,and (2) to combine these two phenomena into one expressionfor ac resistance.

D. Effect of the Coil Curvature to Proximity Factor

The previous analysis is based on the assumption of straight,parallel conductors. However, a coil contains curvatures andsome sort of “return” set of conductors at the distance of approx-imately that must be accounted for. In a typical mid-rangeWPT resonant coil, radius is much bigger than the dimen-sions of the winding area and . Therefore, the effect of the“return” set of conductors is insignificant compared to the fieldof adjacent ones. However, for coils of a smaller radii, the influ-ence of the “return” increases, and should be accounted for. Inorder to develop a sense of the impact of the “return” conduc-tors on the resistance of the coil, let us assume that a coil hasa radius cm, and that it contains two turns of radius

cm at a distance of cm (center-to-center distance).From (27) one can easily see that contribution of the adjacentand “return” turns to current density distribution are quantifiedwith factors, respectively:

(44)

(45)

(43)


Therefore, with the exception of the first harmonic, the influ-ence of the “return” turns on current distribution is negligible.Specifically in this example, the second harmonic contributionof the “return” turns is 150 times lower than the adjacent con-ductors. Therefore, our analysis will be focused on the modifi-cation of the first harmonic in the expression for current densitydistribution (27) and, consequently, in the expression for prox-imity-effect resistance (43).Although the accurate distance and angle

of each particular turn can be included into theexpression, it would unreasonably increase the calculationburden without a significant contribution to accuracy. Instead,

“return” current carrying conductors will be replacedwith the one turn in the middle of the “return” winding area,which carries , as shown in Fig. 6. Further on,angle for all the turns is very close to 180 , whichsuggests that they mostly contribute to cosine term in (27) and(43). Therefore, the angles will be replaced with 180 , whichfurther simplifies the final expression. Finally, the modifiedexpressions for current density distribution and proximityresistance are given in (46) and (47), while the correction factoris defined in (48) (see equations at the bottom of the page).

IV. SIMULATION RESULTS

To verify the derived formulas, current distribution and re-sistance per length for three different configurations of parallelconductors are analyzed by using FEM simulations and analyt-ical expressions (29) and (43). Importance of the modificationsgiven by (46) and (47) will also be highlighted. The configu-rations contain two, three, or four conductors. The specifica-tions of configurations are given in Table I. All tubular con-ductors have the same dimensions, and distances between themare kept constant and equal to the diameter of the tube. Firstten harmonics are used in (29) and (43), although

Fig. 6. Modification of the “return” set of conductors.

we show that incorporating only the first two harmonics is typ-ically sufficient. Figs. 7–9 show the FEM—2D cross sectionplot of the current distribution for each of the configurations.These FEM models are used for an angular current density ex-traction at a radius . In order to avoid condition-ally stable solutions, very low convergence errors were allowedin the FEM simulations and they have been tested for differentmesh types. Additionally, infinite elements modeling has beenapplied to capture all the magnetic energy of the system and theboundary layer property has been used for tube meshing sinceit allows variable maximum mesh dimensions and a very finemesh at the outer surface of the conductor [33].Here we emphasize and prove the importance of filament po-

sition correction . The easiest way to achieve this isto calculate current distribution and ac resistance error for the

(46)

(47)

(48)


Fig. 7. FEM model—current density magnitude—Configuration 1.



case when the correction is not applied. Current distribution forConfiguration 2 is given in Fig. 10. As one can see, outside con-ductors are almost unaffected by omission of correction. The

TABLE ISPECIFICATIONS OF TEST TUBULAR COIL CONFIGURATIONS

Fig. 10. A comparison of the analytical and FEM results of current distributionfor Configuration 2 when the correction of the filament position is not applied.

reason is very simple: the middle conductor that is the most in-fluential to their current distribution has, due to configurationsymmetry, zero correction vector . However, cur-rent distribution in the middle conductor deviates significantlyfrom FEM results, due to the incorrect location of the adjacentouter current filaments. Macroscopically, neglecting the correc-tion vector always results in a higher total ac re-sistance, and the effect is more obvious for tightly packed coils.All in all, much higher relative errors presented in Table II sug-gest that the proposed filament location correction is necessaryfor accurate results.Comparing the analytic expression with the filament position

correction to the FEM results in Figs. 11–13 shows excellentagreement with an almost complete overlap between FEM andanalytical results. Maximum relative errors between FEM andanalytic results are summarized in Table II ( is used as a basevalue). A small error is the result of neglecting the imaginarycomponent in the derivation of the filament position. Althoughsmall, it changes the current phase significantly in the part of thetube where real component approaches its minimum as demon-strated in Fig. 14, which shows the magnitude and phase of thecurrent distribution in Configuration 1. Similar results are seenfor Configurations 2 and 3.In order to validate the expression for proximity factor ,

the ac resistance was computed analytically and using FEM forthree configurations in the frequency range up to 4 MHz. Theresults are presented in Figs. 15–17. Maximum relative errors ofanalytical methods are summarized in Table II. Since very smallerror for skin effect resistance has been already demonstrated in[28], results from Table II can be accepted as an indirect proofthat (43) is sufficiently accurate.In order to validate the derivation in (46)–(48), accounting for

the coil curvature, a new FEM model was constructed that usescircular instead of straight tubular conductors. The test coil hassix turns wound in a plane, resulting in the coil inner diameter of20 cm and outer diameter of approximately 30 cm. Inner radius

is comparable to winding dimension , which guarantees a


Fig. 11. A comparison of the analytical and FEM results of current distributionfor Configuration 1.



Fig. 14. The magnitude and phase of the current distribution for Configuration1—analytical results.

significant impact of the coil curvature. Tubular conductor hasan inner diameter of 4 mm, and outer diameter of 6 mm. Thesnapshot of the current distribution among the turns is given inFig. 10 (only a half of the coil is shown) for 50 kHz signal fre-quency. From the image one can easily see that the effect of the

TABLE IICOMPARISONS BETWEEN FEM AND ANALYTICAL SOLUTION

(L: LEFT, R: RIGHT, MID: MIDDLE, B: BOTTOM)

Fig. 15. Total ac resistance for Configuration 1.



“return” coil impairs the symmetry in current distribution be-tween three left and three right conductors. The comparison be-tween the FEM total ac resistance and analytically calculated acresistance with and without modification is given in Fig. 19 forfrequency range of up to 1 MHz. While modified results closelyfollow FEM results, the analytical values without modificationhave an error of approximately 8%.We finally look at the minimum number of harmonics that

should to be incorporated into (43) to get an acceptably small


Fig. 18. Current distribution of planar coil (only half of the coil is presented).

Fig. 19. AC resistance of a coil with and without modification for the coil cur-vature—comparison to FEM results.

error. The current distribution diagrams given in Figs. 15, 16 and17 demonstrate that for two- and four-conductor topologies, thefirst harmonic is obviously dominant, while middle conductor ofthe three-conductor topology shows almost no evidence of thefirst harmonic. Therefore, one can conclude that the number ofrelevant harmonics depends on the coil topology, and distancebetween turns. For a coil with more close turns, more harmonicsshould probably be included. A minimum number of harmonicsthat reduces the error to less than 1% of the value when 100harmonics are included is given in Table II. One would probablychoose 10 as an empiric number of harmonics, which is enoughfor almost any configuration.

V. EXPERIMENTAL VERIFICATION

In order to experimentally verify the model for proximity ef-fect resistance derived in Section III, two experimental proto-types are built. The wall thickness selection allows the proposedanalytical approach to be tested for two extreme cases, since oneprototype uses a very thin tubular conductor (

m), while the other is built from a thick tube ( mm).The thin tubular coil is realized by using a thin copper tape thatis glued around a rubber tube of a specified radius. On the otherhand, the thicker tube has was purchased off-the-shelf and thenshaped into a planar coil. Specification of the prototypes is sum-marized in Table III, while their photos are shown in Figs. 20and 21.

Fig. 20. Prototype coil I.

Fig. 21. Prototype coil II.

TABLE IIIDESIGN SPECIFICATIONS FOR PROTOTYPES

The prototypes are tested by using Agilent 4294A PrecisionImpedance Analyzer [34]. At themoment of testing, only fixtureextension 16047E was available. Unfortunately, it is intendedfor testing small objects that can be fixed directly to the fixture.Therefore, additional connectors had to be used to connect thecoil to the 16047E fixture, and consequently to the impedanceanalyzer. An additional problem is that large dimensions of thecoil prototypes make them extremely susceptible to the parasiticfields. This problem can be alleviated, to some extent, by usinglonger connecting cables and adjusting the frequency range toless than 1 MHz for Prototype I and 0.5 MHz for Prototype II.However, the additional connectors increase dc and skin effectresistance, which was compensated for by offsetting the model


Fig. 22. Measured values: A snapshot for Prototype I.

Fig. 23. Measured values: A snapshot for Prototype II.

results to match the dc resistance and by rescaling the results toaccount for the skin-effect resistance of the connector.Two snapshots captured by the frequency analyzer are shown

in Figs. 22 and 23, while a comparative review of measured andcalculated values for Prototype I and II are given in Figs. 24 and25, respectively. Analytical results predict a significant changeof ac resistance for the thick tubular conductor, and almost con-stant values for the thin one throughout the most of frequencyrange. The increase of for the thick tube is a consequence ofsimultaneous growth of both skin and proximity resistance fac-tors. As the frequency approaches 1 MHz, the growth of isslowed down and the slope of the curve is reduced. For the thintube and 0–0.5 MHz frequency range, the skin effect is almostnon-existent since the skin depth exceeds the wall thickness bymore than twice. Further on, for frequency greater than 100 kHz,from (5) is much greater than one, making the proximity factorfrom (43) almost insensitive to frequency. Consequently, a verysmall change in can be observed for Prototype II.

Fig. 24. Measured and analytical values—a comparative preview for PrototypeI.

Fig. 25. Measured and analytical values—a comparative preview for PrototypeII.

Based on the results presented, we conclude that the proposedanalytical model matches almost perfectly with the FEM results,but differs from the measured results for the prototypes. We at-tribute the discrepancy, in order of importance, to:• Manufacturing tolerances of the coils. The coils used weremade by hand, and using copper tubes with unknown wallthickness tolerance.

• Limitations of the Agilent 4294A Precision ImpedanceAnalyzer in measuring the parameters of large antennas.Agilent recommends E5061B ENA Series Network Ana-lyzer, which was not available to the authors.

• Although an effort has been made to place the coil so that itdoes not interact with external fields, there is a possibilitythat the field induced inside the coil has not been eliminatedcompletely.

• The estimation of from the ESR measured by the in-strument requires the calculation of the parasitic capaci-tance which also involves some error.

• Practical coils contain additional connectors that were sol-dered at the ends of the coils, and used to link the coil andthe impedance analyzer. Obviously, the algorithm is notable to model them. Additional Litz wire conductors wereused to separate instrument and tested coil and the resis-tance of these conductors is measured and subtracted fromthe values obtained for the entire system.

VI. CONCLUSION

In this work we presented an analytic model for a multi-layermulti-turn hollow tubular resonant coil for WPT applications.More specifically, we present a new derivation of a closed-formformula for the coil ac resistance. The main contribution of theresearch is presented in Section III where we derive a new way


to model the proximity effect resistance. The modeling processstarts from current density expansion into a Fourier series withrespect to azimuthal angle , which finally results in a closedform for ac resistance per length and proximity factor as afunction of the coil parameters. The effect of the coil curvaturehas been taken into account by using a simple correction factorfor first harmonic only. The accuracy of the derived expressionhas been tested through a set of comparisons to FEM and ex-perimental results for the same coil designs. The FEM resultsmatch very well with the analytic model over a wide frequencyrange. The experimental results show a very good match withthe analytical and FEMmodels at lower frequencies, and exhibitthe same tendencies in resistance increase at higher frequencies.The possible reasons for the mismatch at higher frequencies isoutlined in the previous section, and are mostly a function ofcoil manufacturing method, and measuring equipment limita-tions. The main advantage of the proposed analytic approach isits computational efficiency, allowing for its incorporation intooptimization algorithms, thus providing a much better solutionthan the FEM-based iteration procedure.

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