Equivalent Complex Permeability and Conductivity of Litz Wire in Wireless Power Transfer Systems

Mohammad Etemadrezaei and Srdjan M. Lukic FREEDM Center, ECE department

North Carolina State University Raleigh, NC, USA

metemad, smlukic@ncsu.edu

Abstract— In this paper the skin and proximity effect losses for Litz wire winding used in wireless power transfer system is calculated using exact 2-D method. Based on those losses the equivalent complex permeability and conductivity are calculated for each strand in the bundle. Due to specific properties of Litz wire, the whole bundle is homogenized using isotropic complex permeability and conductivity and is simulated numerically in Finite Element (FE) verifying the analytical and experimental results.

I. INTRODUCTION Efficiency is considered to be a major criterion in

designing wireless power transfer (WPT) systems, which have shown promising industry potential in many applications such as charging of electric vehicles [1]-[2]. Due to high frequency of the WPT system, the AC losses (skin and proximity effects) exist in the resonant coils which reduce the quality factor and efficiency of the WPT system. Thickening the wire does not necessarily decrease its resistance; it does so until the conductor diameter reaches the skin depth and after that the resistance increases. Further reduction of resistance in high frequency systems can be done by using Litz wire. Fig. 1 shows a typical WPT resonant coil and the Litz wire wound inside it. The Litz wire is made of many insulated strands twisted together in sub-bundles in a way that each strand occupies all the positions inside the bundle in a full twist, due to radial and azimuthal movements.

Great effort has been made to analytically calculate the skin and proximity losses in high frequency systems windings [3]-[13]; primarily, by introducing the orthogonality between skin and proximity effects, [3], which states that skin and proximity losses can be separated if the conductor cross section is symmetric and the external magnetic field is uniform and perpendicular to its axis of symmetry. There are two well-known methods to analytically calculate AC losses, the exact 2-D [3], and approximate 1-D methods, [6]. The exact 2-D method is

based on Bessel functions and is precise when the electromagnetic field in the problem domain has two directions [3]-[5]. Another method uses approximate Bessel functions where the electromagnetic field is parallel to the surface of the conductors by representing the conductors parallel to the magnetic field with an equal-area 1-D foil [6]-[13]. In applications such as transformers, [8]-[10], and induction heating devices, [12]-[13], where the field has basically one direction, the approximate 1-D solution is more practical and simpler. However, the application of Litz wire in this paper is WPT system, in which the field has two directions, and the exact 2-D method is more precise though complex. Skin and proximity losses in the Litz wire are divided into bundle and strand level effects. Twisting the strands can mitigate the bundle level proximity loss and using sub-bundles reduces the bundle level skin effect. Therefore only the strand level AC losses are of concern in this paper. Since each strand of the Litz wire occupies all the positions in the bundle due to its radial and azimuthal displacements, the current is uniformly distributed among the strands; therefore, the skin effect loss is the same in each strand [7]. For the same reason the internal (due to the rest of strands in the bundle) and external (due to neighboring conductors) proximity effects can be calculated separately and are the same for each strand [13].

Figure 1. The WPT resonant coil and the Litz wire wound inside it.

This work made use of ERC shared facilities supported by the National. Science Foundation under Award Number EEC-08212121

978-1-4673-0803-8/12/$31.00 ©2012 IEEE 3833

At low frequencies the ratio of strand diameter to skin depth is small, and the AC loss calculation is similar to a magnetostatic problem. But at higher frequencies where the ratio of strand diameter to skin depth is close to one, the problem is no longer a magnetostatic one and the solution depends on the calculation of induced local eddy and displacement currents, which is quite extensive in numerical and Finite Element (FE) tools. An imperative method to overcome this adversity is to represent each strand with a material with complex permeability and permittivity that has the same active and reactive power as the original strand [14]-[20]. This way, the field calculation is simplified to a problem similar to the magnetostatic one, with all the values in phasors, and is no longer dependent on solving for induced local eddy and displacement currents. Any material with a B-H loop can be represented by a complex permeability; the real part represents the magnetic energy storage and the imaginary part deals with magnetic losses (proximity loss in a conductor). Similarly, any material with a D-E loop can be represented by complex permittivity; its real part accounts for electrical energy storage and its imaginary part corresponds to conduction losses (skin loss in the conductor).

Yet still there is another difficulty in simulating the Litz wire in FE tools. Diameter of strand in the Litz wire in high frequency applications can be as small as few micrometers, and to take the skin depth into account requires huge meshes and is computationally very time consuming and hardly feasible. A domineering solution is to homogenize the whole Litz bundle with a solid material, with the same inner diameter as of the Litz wire, and an equivalent complex permeability and permittivity to show the same active and reactive power as the Litz bundle. Several authors have tried to homogenize and simulate the Litz winding in FE tools for different applications such as transformers by finding their equivalent complex quantities [17]-[20]. However, in all these works, the homogenization is performed for a 1-D field application using approximate solution. But the application in this paper is wireless power transfer which has a totally 2-D field orientation and 1-D assumption is not valid in this case. Therefore the exact 2-D field solution is utilized to extract the homogenized complex quantities of the Litz bundle and to calculate the AC losses.

A detailed analytical derivation of skin and proximity losses using exact 2-D method is explained in section II. Moving to frequency domain, section III describes the concept of complex permeability and permittivity for magnetic materials focusing on the skin and proximity effects contributions in the Litz wire strand. Using the results in sections II and III, the homogenization of the Litz wire is brought in section IV followed by analytical, FE and experimental results and model verification for a designed and built WPT system in section V.

II. AC LOSSES IN LITZ WIRE In high frequency applications, such as wireless power

transfer system, the skin and proximity effect losses

dominate the DC loss and contribute to a reduction in efficiency. However, using Litz wire can attenuate such AC losses to a level comparable to DC loss. There are two analytical methods to calculate the AC losses in the Litz wire windings: the exact and approximate methods [3]. In the exact solution the changing of magnetic field across the conductor surface area due to local eddy current is taken into account while in approximate one, this change is neglected. At low frequencies where the conductor diameter is less than the skin depth, both solutions have quite the same accuracy; however, at high frequencies that the skin depth is comparable to the conductor diameter, the exact solution is preferred to the approximate one due to its higher precision.

In [7]-[10] the AC losses of Litz bundle are calculated for transformer winding, in which the approximate 1-D solution is precise enough. However, the application of Litz wire in this paper is WPT system winding with complete 2-D electromagnetic field orientation; therefore, the exact 2-D solution is more precise. The skin effect loss of a round conductor (here Litz wire strand), with a current magnitude of I (A) flowing through it, is defined in [4] as

1 [ ( ) ( ) ( ) ( )] 2, (1)_ 2 22 2 [ ( ) ( )]

R ber bei bei berdcP Iskin st ber bei

γ γ γ γ γ

γ γ

′ ′−=

′ ′+

in which Rdc is the DC resistance of the strand in ohms; the functions bern and bein are the real and imaginary parts of Bessel function of first kind and order n; and γ is related to the ratio of conductor diameter and skin depth as

. (2)2

dsγδ

=

where ds is the strand diameter and δ is the skin depth. If the conductor is cylindrical and the magnetic field is perpendicular to its axis of symmetry and uniform over its cross section, the proximity and skin effect losses are orthogonal [3]. Satisfying these conditions, the proximity loss per unit of length of a round strand (with conductivity of σ in S/m) exposed to a sinusoidal varying magnetic field, with amplitude H (A/m) at the surface of strand, is shown in [4] to be

[ ( ) ( ) ( ) ( )]2 22 2 . (3)_ 2 2[ ( ) ( )]

ber ber bei beiP Hprox st ber bei

γ γ γ γπγσ γ γ

′ ′+−=

+

The skin and proximity effect losses are divided in to bundle and strand level effects. The Litz wires are woven in such a way that each strand has both radial and azimuthal movements and occupies all the positions in the bundle. This property leads to uniform distribution of current in all strands and negligence of bundle level proximity effect. The bundle level skin effect can be controlled and diminished using more complex methods such as sub-bundles. Therefore, only the strand level skin and proximity effects are of concern for AC losses in Litz wire. The strand level skin effect resistance

3834

is the same for all strands and since they are connected in parallel, the total skin effect resistance of Litz bundle is

_2

21 . (4)skin stskin

s

PR

n I=

in which ns is the total number of strands in the bundle. The total skin loss of the Litz bundle is the number of strands times the skin loss in each strand, due to equal resistance and current of each strand.

A strand in the Litz wire is subjected to two sources of proximity fields, one from the adjacent strands in the bundle (internal field) and the other one from nearby conductors (external field), intH H Hext= + . Each strand occupying all the positions in the bundle leads to several useful observations in the proximity effect losses in the bundle:

• Whether or not the external field is constant across the bundle cross section area, the external proximity effect loss is the same for all the strands.

• The internal field is symmetric inside the bundle and the spatial average of the squared field is exposed to each strand,

( )22 , (5)int 2 22

n IsHdb

βπ

=

in which β is a factor representing strand, bundle and filler packing factors for a Litz wire with inner diameter db according to

2. (6)2

dsns db

β =

• The internal and external proximity losses are orthogonal and can be calculated separately,

2 2 2 . (7)intH H Hext= +

• The proximity loss in each strand is the same and the total proximity loss is

_ . (8)prox s prox stP n P=

III. MAGNETIC MATERIALS CHARACTERISTICS Any material has various loss components, losses relating

to the B-H curve of the material are due to magnetic properties of the material and are classified into magnetic losses and losses related to the D-E curve are in the electrical nature and are classified into electrical losses. Any magnetic material with an elliptical B-H loop, which is also called the hysteresis loop, can be represented using complex permeability, iμ μ μ′ ′′= − , that represents both magnetic energy storage and magnetic losses of the material. The use

of complex permeability assumes that the flux in the material is sinusoidal and the material is linear, which cannot accommodate the saturation effects. The elliptical representation of B-H curve means that the material can be represented by a magnetic energy storage element in combination with a series resistive part (lossy inductor). The real part of the complex permeability deals with the energy storage and the imaginary part represents the magnetic loss in the material (proximity effect loss).

In a completely analogous approach, any electric material with an elliptical D-E loop can be represented with a complex permittivity, iε ε ε′ ′′= − , that represents both the electrical energy storage and electrical losses of the component. The elliptical representation of the D-E loop means that the material can be represented with an electrical energy storage element in combination with a parallel resistive part (lossy capacitor). The real part of complex permittivity corresponds to the electrical energy storage and the imaginary part represents the conduction loss (skin effect loss).

The general differential Maxwell equations in the stationary matter can be used to compute and relate different electrical and magnetic losses and stored energies in the Litz wire. However, these equations are too complex and time consuming and are practical only in the powerful numerical field computation tools. Another approach is to use the conservation of energy by applying Poynting Vector theorem, which in point form is written in [21] at time domain as

( ) , (9)t tE H H B J E E D− ∇ ⋅ × = ⋅ ∂ + ⋅ + ⋅ ∂

in which H (A/m) is the magnetic and E (V/m) is the electric field strengths, respectively. Due to sinusoidal field excitation, the field vectors can be represented in frequency domain. The integral form of the Poynting vector theorem in a linear, time-invariant and isotropic material, re-written by the phasor magnitudes of complex field quantities is

2 221 1 1 1ˆ ˆ ˆ ˆ ˆ( ) 2 , (10)

2 2 4 4S V V

E H ds E dv i H E dvσ ω μ ε∗ ∗⎡ ⎤− × ⋅ = + −⎢ ⎥⎣ ⎦∫ ∫ ∫

By using the real and imaginary parts of complex permeability and permittivity, (10) is simplified to

2 2 22

2 2

1 1ˆ ˆ ˆ ˆ ˆ( )2 2

1 1ˆ ˆ2 , (11)4 4

S V

V

E H ds E H E dv

i H E dv

σ ωμ ωε

ω μ ε

∗ ⎡ ⎤′′ ′′− × ⋅ = + +⎢ ⎥⎣ ⎦

⎡ ⎤′ ′+ −⎢ ⎥⎣ ⎦

∫ ∫

∫

And by denoting the total average complex power of the volume V bounded by surface S as <Psource>, the average dissipated power as <Pd>, the average stored magnetic energy as <Wm>, and the average electrical stored energy as <We>, the Poynting vector can be re-written as

( )2 , (12)source d m eP P i W Wω= + −

3835

in which

2 2 22

2

2

1 ˆ ˆ( ) ,2

1 ˆ ˆ ˆ ,2

1 ˆ ,4

1 ˆ . (13)4

sourceS

dV

mV

eV

P E H d s

P E H E dv

W H dv

W E dv

σ ωμ ωε

μ

ε

∗= − × ⋅

⎡ ⎤′′ ′′= + +⎢ ⎥⎣ ⎦

⎡ ⎤′= ⎢ ⎥⎣ ⎦

⎡ ⎤′= ⎢ ⎥⎣ ⎦

∫

∫

∫

∫

The average dissipated power loss, <Pd>, can be expressed with two terms that describe the magnetic and electric losses inside the conductor. The magnetic loss which accounts for proximity loss in the strand can be expressed as

21 ˆ , (14)2dm

V

P H dvωμ⎡ ⎤′′= ⎢ ⎥⎣ ⎦∫

that considers the imaginary part of complex permeability. The electrical loss that includes the skin effect loss in the strand can be written as

2 21 1ˆ ˆ( ) , (15)2 2de

V V

P E dv E dvσ ωε σ⎡ ⎤ ⎡ ⎤′′ ′= + =⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦∫ ∫

which uses the imaginary part of complex permittivity. The skin effect loss can also be modified using complex conductivity according to (15) in which σ’ is the real part of complex conductivity as

( ) . (16)i iσ σ σ σ ω ε ω ε′ ′′ ′′ ′= − = + +

The real part of complex conductivity accounts for electrical conduction current (static and frequency dependent conduction) which in electrical conductors represents DC and skin effect conductivities; and the imaginary part considers the displacement currents in the matter.

By knowing the complex quantities of the material, the different stored energies and losses can be computed. However, to get a simpler view of the material, it is useful to represent the material by quantities of interest such as resistance and inductance. The total resistance of a material can be computed from the dissipated power in the material along with a current flowing through.

IV. CONTINUUM REPRESENTATION OF LITZ WIRE A Litz wire can contain thousands of strands inside it

with diameters too small to be feasible to construct in the Finite Element (FE) tools. It’s theoretically possible but practically impossible to directly simulate and analyze the Litz wire in the FE, because it requires intensive meshes and is computationally too expensive. At low frequencies, where the diameter of strand is much smaller than the skin depth, the windings can be simulated in magnetostatic 2-D FE with windings having the same complex permeability and permittivity as those of the air. However, at high frequencies, that the strand diameter is comparable to skin depth, the effect of conductor eddy and displacement currents on the main field makes the calculation of main field at the surface of conductor impossible and dependent on the computation of local eddy and displacement currents. To take the skin depth properly into account requires huge meshes; however, this adversity can be beaten by using the equivalent complex permeability and permittivity for the strand. These complex quantities are used to find different components of loss and stored energy besides the field change due to local eddy and displacement currents without solving for them. Thus, the field computation is simplified to a problem close to magnetostatic one with all values in phasors. Still, the excessive number of strands makes the calculation hardly feasible. An imperative method is to homogenize the whole Litz bundle by finding its equivalent complex quantities. Then the bundle can be represented by a block of round solid material with equivalent complex permeability and conductivity. Fig. 2 shows the homogenization process. In the first step each strand is replaced by an equivalent material represented by complex permeability and conductivity and then the whole bundle is homogenized with its equivalent complex quantities.

Great effort has been put to homogenize and simulate the Litz windings in FE tools for different applications such as transformers by finding their equivalent complex quantities. Reference [14] used the complex permeability method to substitute the rectangular conductor with a lossy ferromagnetic material. Then [17] used the analytical proximity loss solution from modified Dowell’s method [6], to find the imaginary part of complex permeability for a layered transformer winding. The homogenization concept

TABLE I. LITZ PROPERTIES

Parameter VALUE

Number of strands 2625

Strand AWG 44

Bundle inner diameter 3.556 mm

Bundle serving 0.968 mm

Figure 2. Homogenization of Litz wire using equivalent complex quantities.

3836

was further extended in [20] by finding equivalent complex permeability of Litz bundle and the total winding. In [19] the homogenization for a hexagonally packed winding is initiated from a foil which is exposed to 1-D field direction. In all these works, the homogenization was performed for a 1-D field application using approximate solution. But the application in this paper is wireless power transfer which has a totally 2-D field direction and 1-D assumption is not valid in this case. Therefore the exact 2-D field solution is utilized to extract the homogenized complex quantities of the Litz bundle and to calculate the AC losses.

In section II the skin and proximity effect losses for each strand and the Litz bundle were derived using exact 2-D method. In section III the same losses were represented in frequency domain for each strand. Based on those losses, the equivalent complex permeability and conductivity can be calculated for each strand. By equating (1) and (15) (the skin effect loss per unit of length), the real part of complex conductivity for each strand is

2 22 [ ( ) ( )] , (17)[ ( ) ( ) ( ) ( )]st

ber beiber bei bei ber

σ γ γσγ γ γ γ γ

′ ′+′ =′ ′−

The complex conductivity of each strand can be homogenized to the whole bundle by having the DC resistance of the Litz bundle equal to that of a homogenized one

. (18)h stσ β σ′ ′=

The homogenization is based on the fact that the skin loss is the same for each strand in the Litz wire. Table I shows the properties of the Litz wire used to extract the complex quantities. This Litz bundle has 5 sub-bundles, each having 5 more sub-bundles, each having additional 3 sub-bundles with each sub-bundle having 35 strands of 44 AWG wire (5×5×3/44).

Fig. 3 shows the real part of the homogenized complex conductivity versus gamma (γ) for the Litz wire mentioned in table I obtained by two methods: 1-D approximate [17], and 2-D exact solutions. The two methods lead to almost same results for low values of γ until it reaches 0.7 (f = 1.66 MHz), and after that the conductivity obtained by 1-D solution drops more rapidly compared to 2-D exact solution. The reason is due to the initial assumption in dimension of the problem domain in 1-D method. The 1-D assumption is for applications in which the field is in one direction, such as transformers [17]-[20], and is not valid for application with complete two directional field orientations, such as wireless power transfer systems. As can be seen from the figure, the conductivity is constant until the skin depth reaches the strand diameter and then decreases as frequency increases; this is called the skin effect loss.

According to (16) the real part of complex conductivity contains the imaginary part of complex permittivity. The real and imaginary parts of complex permittivity form the Kramers-Kronig relation (due to the casual, linear and time-

invariant connection between polarization and electric field), which states that the real and imaginary parts are Hilbert pair. Due to the small effect of parasitic currents on the main field, real part of complex permittivity is neglected in this paper.

For the proximity loss, the imaginary part of complex permeability takes the role. By equating (3) and (14) (the proximity effect loss per unit of length), the imaginary part of complex permeability for each strand is found to be

2 22 2 2

[ ( ) ( ) ( ) ( )]16 , (19)[ ( ) ( )]st

s

ber ber bei beid ber bei

γ γ γ γγμσ ω γ γ

′ ′+−′′ =+

The complex permeability depends on the excitation frequency and the geometry of Litz bundle, but not on the coil configuration and dimension. The proximity loss in all the strands are equal to each other and the total proximity loss of the Litz wire is ns times the loss of each strand. The homogenization can be done using a solid material of the same diameter as the Litz bundle assuming the same field amplitude is applied to the Litz bundle before and after homogenization

_

2 22 21 1ˆ ˆ , (20)

2 4 2 4

prox s prox strand

s st s h Litz

P n P

n d H d Hπ πω μ ω μ

=

′′ ′′= =

with the homogenized permeability (imaginary part) to be

. (21)h stμ β μ′′ ′′=

Fig. 4 shows the imaginary part of homogenized relative complex permeability as a function of γ for the Litz wire, with properties mentioned in table I, using two methods of 1-D approximate [6], and 2-D exact [3]. The 1-D approximate solution is close to 2-D exact one for low values of γ up to 1.4 (f = 6.63 MHz), after which it falls below the 2-D solution. The imaginary part of complex permeability is approximately proportional to frequency f, at low strand diameter to skin depth ratio and is proportional to f-0.5 at higher values of strand diameter to skin depth ratio. The imaginary part of complex permeability deals with proximity loss and in order to keep this effect low, the Litz wire should be chosen such that the ratio of strand diameter and skin depth is below one (γ ≈ 0.7).

Due to the casual, linear and time-invariant relation between magnetization and magnetic field, the real and imaginary parts of complex permeability form Kramers-Kronig relation and the real part can be computed using Hilbert transform

2 20

( )2( ) . (22)hh cte

x x dxxμμ ω μ

π ω

∞ ′′′ ′= +−∫

in which ω is the angular frequency and µ’cte is a physical

constant that makes sure the real part of relative complex permeability go to 1 at very low frequencies. Fig. 5 shows

3837

the real part of relative complex permeability versus γ for the two methods.

According to Fig. 5, the real part of relative complex permeability goes to one at low frequencies where the strand diameter is much smaller than the skin depth. In other words, the Litz bundle inductance is almost constant for low values of γ up to 0.7 (f = 1.66 MHz). At higher frequencies the real part reduces to a constant that depends on the Litz bundle geometry and frequency.

V. MODEL VERIFICATION To verify the analytical model, the Litz wire (with

properties mentioned in table I) is wound in a one-layer coil of 9 turns, Fig. 1. The coil frame is made of Polyvinyl Chloride (PVC) which has a considerable loss tangent; however, in this paper it is neglected in the analytical and FE models. The spacing between the slots of winding is 7.7 mm, filled with PVC. Homogenization of the 2625-strand Litz bundle with complex permeability and conductivity simplifies the modeling of the winding in the FE analysis significantly. Fig. 6 shows the FE simulation of the homogenized Litz winding (2-D axisymmetric vertical view) in COMSOL® 4.2 at 2 MHz frequency.

In the light of homogenization constitution, the total AC resistance (skin and proximity effects) of the winding is calculated in the FE simulation, and is compared with the analytical exact 2-D, and approximate 1-D, [9], methods. Fig. 7 shows the calculated AC resistance by the analytical and FE methods for a frequency range of 40 Hz up to 2 MHz. 2 MHz frequency is far beyond most of the WPT system operational frequencies and it corresponds to gamma of 0.77 which is larger than 0.7 to dignify the difference between the 1-D and 2-D analytical methods. As is obvious from Fig. 7 the AC resistance in FE method is closer to 2-D exact method than to the 1-D approximate one. The reason is that in the 1-D method the field is assumed to be in one direction which is not applicable in WPT systems. Using (4) and (8) the skin and proximity effect resistances are calculated separately and shown in Fig. 8, compared with FE results.

Figure 3. Real part of complex conductivity for 1-D approximate and 2-D exact methods.

Figure 4. Imaginary part of relative complex permeability for 1-D approximate and 2-D exact solutions.

Figure 5. Real part of relative complex permeability for 1-D approximate and 2-D exact solutions.

Figure 6. 2-D Axisymmetric FE simulation of winding; surface: magnetic flux density (T), contour: magnetic field strength (A/m).

10-3

10-2

10-1

100

1010

0.5

1

1.5

2

2.5

3

3.5

4x 107

Gamma, γ

Rea

l co

nd

uct

ivit

y, σ

' h (

S/m

)

2-D Exact1-D Approximate

10-3

10-2

10-1

100

1010

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Gamma, γ

Imag

inar

y p

erm

eab

ility

, μ" h

/ μ 0

2-D Exact1-D Approximate

10-3

10-2

10-1

100

1010

0.2

0.4

0.6

0.8

1

Gamma, γ

Rea

l per

mea

bili

ty, μ

' h /

μ 0

2-D Exact1-D Approximate

3838

Figure 7. Total AC resistance of the winding vs. frequency for the 1-D approximate, 2-D exact analytical and FE methods.

Figure 8. Winding skin, proximity and total AC resistances using 2-D exact method compared with FE simulation.

Figure 9. The measured equivalent series impedance of the winding; dark blue: Rs (Ω) and light blue: Xx (Ω).

Figure 10. The actual electric circuit model of the Litz winding.

Figure 11. The calculated equivalent series resistance of the winding using analytical method, verified by FE experimental tests.

According to Fig. 8, the proximity effect resistance

contains the major part of AC resistance and is perfectly close to FE result. The skin effect part does not increase that much with frequency and is very close to DC resistance. The built Litz wire coil is tested with Agiltron® 4294A frequency response analyzer, and the frequency dependent series impedance (Zs =Rs+iXs) of the system is measured, Fig. 9. During the measurement, the coil is set about 30 cm above the ground to mitigate the interfering induced electromagnetic field of surrounding objects on the coil impedance.

The actual AC resistance of the Litz winding is different than what is shown in Fig. 9. Fig. 9 is the equivalent series impedance of the winding; however, there is capacitance C, in the winding between all the turns and this capacitance has a resistive part (PVC material, RC) which is neglected in the analytical and FE models. Therefore the actual electric circuit model of the Litz winding includes all these elements and is shown in Fig. 10.

Knowing the loss tangent of the PVC at room temperature 0.028, and assuming the inductance of the winding L, and capacitance C, constant for the frequency range of 40 Hz to 1 MHz (they are obtained by the frequency analyzer), the series equivalent resistance of Litz winding is calculated using analytical and FE results for RAC and is shown in Fig. 11, compared with experimental result.

0 0.5 1 1.5 2

x 106

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency ( Hz )

AC

Res

ista

nce

( O

hm

)

2-D Exact1-D ApproximateFE Method

0 0.5 1 1.5 2

x 106

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency ( Hz )

Res

ista

nce

( O

hm

)

Total AC resistance

Skin effectresistance

Proximity effectresistance

2-D Exact- - - FE Method

0 2 4 6 8 10

x 105

0

0.5

1

1.5

2

2.5

3

Frequency ( Hz )S

erie

s R

esis

tan

ce (

Oh

m )

2-D ExactFE MethodExperiment

3839

As is illustrated in Fig. 11, the experimental result is very close to FE and analytical results and the small difference is due to the connectors’ conduction resistance and any other objects nearby the test setup that interfere the electromagnetic field on the Litz wire and change the resistance.

VI. CONCLUSION

In this paper 2-D field simulation for a wireless power transfer coil with Litz wire winding was performed both analytically and numerically. It was concluded that the 2-D exact method is more precise and applicable than the 1-D approximate method. Due to the difficulty of numerically simulating Litz wire in FE, the method of complex permeability and conductivity for each strand was used. And because of special properties of Litz wire, the Litz bundle was homogenized with isotropic complex quantities. The system was then numerically simulated without solving for local eddy and displacement currents. The series equivalent impedance of the winding was measured using frequency analyzer and by getting the inductance, capacitance and loss tangent of the winding and its frame, the AC resistance of the winding was obtained. There was a good match between experimental, FE and analytical results.

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