This document proposes and validates an equivalent circuit model for a wireless power transfer system capable of transferring 220W of power over a 30cm air gap with 95% efficiency. The model represents the transmitter and receiver coils as inductors with low mutual coupling. Analytical expressions for the model are derived and validated using finite element analysis and experimental results. Loss analysis is also performed to investigate skin effect and proximity effect losses at high operating frequencies. A new coil spatial design is proposed to reduce such losses compared to conventional coil designs.

1. Development and Validation of Model for 95% Efficiency, 220 W Wireless Power
Transfer over a 30cm Air-gap
Seung-Hwan Lee Robert D. Lorenz
Student Member, IEEE Fellow, IEEE
University of Wisconsin-Madison, WEMPEC
1513 University Avenue, Madison, WI, 53706
lorenz@engr.wisc.edu
Abstract – Although 60W wireless power transfer was
demonstrated in 2007, still there is no equivalent circuit model
for a sub-meter air-gap, hundreds of Watts, high efficiency
wireless system. A design-oriented circuit model is needed for
this technology to evolve. This paper proposes an equivalent
circuit model for the wireless system and analyzes the system
based on the proposed model. The proposed model and its
analysis are validated by means of FEA and experimental
results. Furthermore, as a viable solution for high power (over
10kW) applications, losses in the wireless power transfer system
are investigated in the following section. Because of the high
operating frequency (MHz), skin- and proximity effect were
shown to be dominant. New spatial layout of a coil is proposed
that significantly reduces losses caused by skin- and proximity
effect. Proposed coil design is evaluated by means of FEA.
Index Terms— equivalent circuit, large air-gap, proximity
effect, skin effect, spatial layout, wireless power transfer.
I. INTRODUCTION
Removing the cable between a power source and a load is
very important in many applications; e.g. for a plug-in hybrid
vehicle or for a plug-in electric vehicle battery charger, the
power cable is the “bottleneck” for reliability and
maintenance performance, since the exposed power cable is
subjected to mechanical deterioration caused by outdoor
environments and users. Thus, wireless power transfer with
air-gaps greater than ten centimeters, kWatt rating, and high
efficiency is a very important technology.
One of the earliest research papers about sub-meter, high
power, and efficient wireless power transfer was published by
Karalis et al. in 2007 [1]. In that paper, authors presented a
non-radiative scheme based on magnetic resonance coupling.
They analyzed the strong coupling regime of two resonating
dielectric disks, capacitively-loaded conducting-wire loops
with coupled mode theory (CMT). Since they used CMT,
the characteristic equations of the two resonating objects
were decoupled, the analysis of the system was very simple
and quality factor and power transfer efficiency of coupled
objects could be easily calculated. However, the paper did
not provide a design-oriented model for such wireless power
transfer systems, suitable for designing or validating a design.
In 2009, Waffenschmidt and Staring [2] published a paper
about the limitations of the inductive power link system in
consumer electronics. They derived optimal system
parameters for efficient power transfer and investigated the
effects of distance and size of coil on power transfer
efficiency. However, their analysis only used a resonant
circuit on the receiver coil, not on the transmitter, and as a
result, they concluded that inductive power transfer is only
effective over short distances. They did not explore how
power transfer capacity, distance, and efficiency of a wireless
power transfer system can be improved significantly if both
the transmitter and the receiver are tuned LC resonant circuits
as demonstrated in [1]. Yang et al. published a paper about
inductor modeling in wireless links for implantable devices
design [3]. However, in general, the rated power of these
applications was a few milliWatts and the distance between
two coils was just a few centimeters [4]. As a result, the
models in these papers are not applicable to larger distances
and higher power systems.
In the first part of this paper, a model for a 30cm air-gap,
220W, high efficiency wireless power transfer system is
proposed. Based on electric and magnetic field analysis, the
two coils of the system are modeled as two inductors having
low coupling coefficients. With the equivalent circuit
model, the efficiency of wireless power transfer system using
two tuned LC resonant tanks is analyzed. Using closed
form expressions for inductance and resistance of the coils,
theoretical power transfer efficiency is presented. By means
of finite element analysis (FEA) and experimental results, the
proposed model is verified.
In the second part of this paper, losses in the wireless
power transfer are analyzed. In a low power application,
losses are not the critical limitation in implementing the
system because the system voltage and current are usually
small. There are no significant difficulties in building a
power converter, a transmitter, and receiver coil. However,
in high power applications, loss is the key limiting factor in
system implementation. The required voltage and current
specifications of switches in a power converter increases
978-1-4244-5287-3/10/$26.00 ©2010 IEEE 885

2. significantly as the loss increases. Also, the overall size of
the system increases because of the high rated voltage,
current, and losses. Although loss analysis is very important
in implementing the wireless power transfer system for high
power applications (over 10kW), it has not been covered in
the literature [1-4]. This paper investigates the origins of
losses in the non-radiative magnetic resonant wireless power
transfer system as its second major focus. Based on detailed
loss analysis, a new spatial configuration of a coil is proposed
and its suitability is evaluated by comparing its equivalent
series resistance to that of a conventional spiral shaped coil.
The paper begins by developing a lumped parameter
equivalent circuit model for non-radiative coupled magnetic
resonance wireless power transfer systems based on electric
and magnetic field equations. In the following section, the
developed theoretical model is evaluated by FEA and
experimental results. In the last section, the origins of losses
in the system are analyzed. A new spatial design of a coil is
proposed, having low loss when compared to conventional
helical or spiral shape coils. That design approach is
evaluated with FEA results.
II. WIRELESS POWER TRANSFER SYSTEM MODELING
Because the non-radiative wireless power transfer (WPT)
scheme presented in [1] utilizes the near-field of a current
carrying coil as a power transfer medium, the characteristics
of an electric and magnetic field in the near-field of a coil can
be formulated to build an equivalent circuit model.
According to Balanis [5], phasor expressions of the electric
and magnetic fields of a current carrying circular coil in
spherical coordinates follows (1) and (2),
Hr = j
ka2I0cosθ
2r2 ⎣
⎡
⎦
⎤1+
1
jkr e-jkr, Hφ = 0
Hθ = -j
(ka)2I0sinθ
4r ⎣
⎡
⎦
⎤1+
1
jkr -
1
(kr)2 e-jkr, (1)
Er = Eθ = 0,
Eφ = η0
(ka)4I0sinθ
4r ⎣
⎡
⎦
⎤1+
1
jkr e-jkr. (2)
where: a is radius of a circular coil,
r is the distance from the center of the coil to an
observing point,
k is the wave number,
η0 is the intrinsic impedance of air, and
I0 is the current in the coil.
From (1) and (2), the total radiated power over a closed
spherical surface s follows (3),
P=
⌡
⌠1
2(E×H*)⋅ds = η0
π
12(ka)4| |I0 2
⎣
⎡
⎦
⎤1+j
1
(kr)3 r^. (3)
1L
M
2L
2R1R
(a) Large air-gap WPT system (b) Equivalent circuit model
Fig. 1. Schematics of 30cm air-gap wireless power transfer system
1
1
sC
1sL
sM
2sL
2R1R
2
1
sC
LRsV
1LV
1LISI 2LI LRIRV
1
msC
msL
inY
2LV
Fig. 2. Steady state circuit model of 30cm air-gap wireless power transfer
system
As can be seen in (3), the second, purely imaginary term is
dominant in the near-field zone (k⋅r<<1), and the real term is
dominant in the far-field zone (k⋅r>>1), so the radiated power
in the near-field is reactive and inductive.
Furthermore, Inan showed in [6] that a circuit can be
approximated by an equivalent lumped parameter circuit if its
characteristic length is less than one-tenth of the relevant
wavelength. This is because spatial distribution of current
and voltage do not change along the circuit elements if the
characteristic length is very small. As a result, the
transmitter and receiver coil used in this research can be
modeled as lumped elements because their diameter is
usually smaller than one-tenth of a wavelength.
Consequently, two coils of the non-radiative power transfer
scheme of [1] can be represented as two lumped inductors as
shown in Fig. 1(a) and (b). The non-radiative wireless
power transfer system of this paper, two coaxial coils with
30cm air-gap, can be represented as two inductors and
parasitic resistors with weak mutual-inductance between the
coils. Parasitic resistors R1 and R2 are caused by skin-
and proximity effect losses and radiation losses because the
operating frequency is as high as a few MHz.
To maximize power transfer capacity and efficiency, two
tuned LC resonant tanks are used as the transmitter and
receiver as discussed in [1]. The steady state circuit
diagram is depicted in Fig. 2. An impedance matching L-
network, Lm and Cm, was added in the transmitter circuit of
Fig. 2, to maximize input power at the resonant frequency.
Based on the circuit in Fig. 2, steady state expressions for
the voltage and current of every node and branch can be
calculated. The power transfer efficiency follows (4),
η =
PRL
Pin
=
| |IRL
2RL
| |IL1
2R1+| |IL2
2R2+| |IRL
2RL
. (4)
If the resonant frequency of two tanks ω01 and ω02 are
equal, i.e. ω01 = ω02 = ω0 and the load resistance RL is
greater than ω0L2, the expression for efficiency of the system
886

3. can be simplified to (5),
η ≈
1
1+
RL
ω02M2(1+ω02C22RL2)
R1 +
RL
ω02L22R2
(5)
As can be seen in (5), resonant frequency ω0, mutual-
inductance M, and self-inductance L2 must be increased for
high efficiency, and the parasitic resistance R1 and R2
should be decreased. However, it should be noted that there
are trade-off relationships between the efficiency and the
system parameters; e.g. the Ohmic losses caused by skin- and
proximity effects as the resonant frequency increases. In
addition, the radiation losses increase significantly as the
frequency increases, therefore, the efficiency begins
decreasing as the frequency exceeds a certain limit. Another
important point is that there is an optimal load resistance that
maximizes power transfer efficiency in a given system. The
optimal load resistance expression was derived by
differentiating (4),
RL max
efficiency
=
L2 - C2M2ω02 - LmC12M2ω04
C1C22M2ω04+C12C22LmM2ω06
(6)
III. DEVELOPED MODEL EVALUATION
The developed model was evaluated using a lab testbed
wireless power transfer system. Theoretical efficiency was
calculated based on the equivalent model. The theoretical
efficiency was compared with finite element analysis (FEA)
results and experimental results. The radius of the coil of
the testbed system was 17cm, the diameter of the wire was
2mm, turn spacing was 2cm, and the number of turns was
three. The distance between two coils was 30cm and the
resonant frequency of two resonant tanks was 3.7MHz.
A. Analytical Model of the Testbed Coil
To validate the model, electric parameters were estimated
based on existing models [7-13]. According to Miller [7],
self- and mutual inductance of coils follow (7) and (8),
L =
μ0N2a2
3c ⎩
⎨
⎧
⎭
⎬
⎫dc
a2[F(k)-E(k)] +
4d
c E(k) –
8a
c [H] (7)
M =
2μ0
γ
(ab)1/2[(1 -
γ2
2 )F(γ) – E(γ)] [H] (8)
where a, b are the radii of the two coils, c is axial length of
the coils, d = (4a2 + c2)1/2, N is number of turns, μ0 is
permeability of air, k = 2a/d, γ2 = 4ab/[(a+b)2+z2], z is the
distance between coils and F(k) and E(k) are the complete
elliptic integrals of the first and second kind, respectively.
Radiation loss is calculated per Balanis [5],
Rrad = 20π2N2(C/λ)4 (9)
where C is the perimeter of coil, λ is the wavelength, and N is
the number of turns of the coil.
(a) Isolated coil (b) Two coupled coils
Fig. 3. Drawings of an isolated coil and mutually coupled coils
TABLE I
ANALYTICALLY CALCULATED, CALCULATED FROM FEA, AND
EXPERIMENTALLY MEASURED PARAMETERS
Analytic
Calculation
FEA
Results
Experimental
Results
Self-inductance (μH) 5.06 5.3 5.1
deviation (%) - 4.7 0.8
Mutual-inductance (μH) 0.28 0.285 0.31
deviation (%) - 1.8 11
Parasitic Resistance (Ω) 0.126 0.12 0.15
deviation (%) - 4.7 19
Equation (10) for the skin effect loss of solid circular
conductors was that presented by Kelvin [11].
Rac= Rdc
mr
2
bei′(mr)ber(mr)- ber′(mr)bei(mr)
bei′2(mr) + ber′2(mr)
(10)
where r is the radius of the conductor, m = ωμσ, ω = 2πf, f
is the excitation frequency, μ is the permeability and σ is the
conductivity of the conductor, ber and bei are the real and
imaginary parts of Bessel functions and prime in the above
equation is the first derivative with respect to mr.
The proximity effect loss of a single-layer solenoid coil is
calculated per Smith [12]. An additional factor was
included in the total ac resistance expression as ROhmic =
Rac(1+Gp), where Gp is the proximity effect factor [12].
Calculated self- and mutual inductance of two coaxial
coils were 5.06μH and 0.28μH, respectively. The resistance
calculated from the Ohmic loss terms was 0.09Ω, and the
radiation resistance was 3.4μΩ at 3.7MHz. When RL is
2kΩ, the peak theoretical efficiency at the resonant frequency
was 97%. To validate the proposed lumped circuit model
and theoretical value of the parameters, FEA results and
experimentally measured parameters values are compared in
Table I.
B. Finite Element Analysis of the Test Coil
To check and validate the analytical model, the testbed
system was modeled using the JMAG-designer 3-D FEA
package. At first, a 17cm radius, 4mm wire-diameter, 3 turn,
solid circular cross-sectional single copper coil was
implemented in a three dimensional space as shown in Fig.
3(a). Self-inductance of the coil was obtained by static
analysis. A current of 1Ap-p was applied to the wire and
total flux linkage was measured.
887

4. [μT]
(a) Sectional
view
(b) 3.34MHz
(c) 3.68MHz
(resonant frequency)
Fig. 4. Magnetic flux density dependence on operating frequency
The resistance of the coil was determined by frequency
analysis. At 3.7MHz, 1Vp-p was applied to the coil and
total power loss of the single coil was measured and used to
calculate resistance. Then, a 30cm distance coupled coil
model was tested to measure mutual inductance of the coils
as shown in Fig. 3(b). Resultant parameters are shown in
Table I.
By FEA, a virtual wireless power transfer experiment via
resonant magnetic field was performed as well as parameter
extraction. The magnetic flux density of the coupled
resonating coils in the cross-sectional plane A-A is presented
in Fig. 4. It should be noted that the resonating magnetic
field is the key component in the power transfer. If the
input frequency is not same as the resonant frequency of the
two coils, the magnitude of induced magnetic flux density in
secondary coil (upper coil in Fig. 4(b)) is small whereas it is
very large when the operating frequency is the same as the
resonant frequency of the transmitter and receiver resonant
tanks in Fig. 4(c).
C. Experimental Results
Fig. 5 shows the configuration of the experimental setup.
A HF transceiver, ICOM-718, is used as the power source.
It generates radio frequency power and its maximum power
rating is 100W. A linear amplifier, AL-811, is used to
amplify input power up to 600W. MFJ-993b automatic
antenna tuner is used for the impedance matching L-network.
Inductance of the L-network was 4μH and the capacitance
was 470pF.
Self-inductance and resistance of the sample system was
measured by an HP4263A LCR meter. L1 and L2 was
5.1μH and R1 and R2 was 0.15Ω at 3.7MHz. Mutual-
inductance of the system was measured by a short circuit test
whereby the secondary coil was shorted out, and then the
voltage of the primary side and the current of the primary and
secondary side were measured. From this, the mutual-
inductance calculated as 0.31μH. Measured parameters are
shown in Table I. Based on (6), a load resistance RL of
3kΩ was selected. Fig. 6 shows the photos of the HF
transceiver, power amplifier, antenna tuner, and two resonant
tanks.
In Fig. 7, measured input and output voltage and current
are shown. Simulated input and output voltage and current,
based on the proposed model, are shown in Fig. 8.
Fig. 5. The block diagram of experimental setup
(a) HF transceiver
(b) Antenna tuner
(c) Linear Amplifier (d) Two resonant tanks
Fig.6. Photos of the experimental setup
By comparing Fig. 7 and 8, it is shown that the proposed
model is quite accurate in high power, large air-gap wireless
power transfer system. In Fig. 7, measured average input
power was 220W, and the measured average output power
was 215W which shows 96% power transfer efficiency.
In Fig. 9, analytical, FEA, and experimental power
transfer efficiencies were compared. This result
experimentally verified that 30cm air-gap, 220W, wireless
system has 96% power transfer efficiency.
IV. COIL DESIGN FOR HIGH POWER APPLICATIONS
There are two components causing losses in a wireless
power transfer system: radiation loss and Ohmic loss. These
two loss components are very critical in MHz operating
frequency since radiation loss is proportional to the fourth
power of the operating frequency, and Ohmic loss, especially
skin effect loss, is nearly proportional to the square root of
the frequency. However, radiation loss is negligible in the
non-radiative wireless power transfer system because the
ratio of the perimeter of the transmitter and receiver coil to
the wavelength of electromagnetic field is usually less than
one-tenth. Then the radiation resistance of the coil is
approximately equal to 10-4Ω. Thus, Ohmic losses caused
by skin- and proximity effect are the dominant parts of losses
while the radiation to the far-field is negligible in this system.
Ohmic losses depend on the cross-sectional shape and the
spatial configuration of each turn of a coil. In this section,
the effect of cross-sectional shape on the skin- and proximity
effect loss variation is investigated by means of FEA.
888

5. Inputvoltage[V]
Inputcurrent[A]
Time [μs]
(a) Measured input voltage and current
Outputvoltage[V]
Outputcurrent[A]
Time [μs]
(b) Measured output voltage and current
Power[W]
Legend:
Blue - Pin
Red – Pout
Pin : 220W
Pout: 215W
Time [μs]
(c) Measured instantaneous input and output power
Fig. 7. Measured input and output voltage and current waveforms
Inputvoltage[V]
Inputcurrent[A]
Time [μs]
(a) Simulated input voltage and current
Outputvoltage[V]
Outputcurrent[A]
Time [μs]
(b) Simulated output voltage and current
Power[W]
Legend:
Blue - Pin
Red – Pout
Time [μs]
(c) Simulated input and output power
Fig.8. Simulated input and output voltage and current waveforms
Efficiency[%]
Legend:
Solid line –
Theoretical
efficiency
-o- – FEA efficiency
-Δ- – Measured
efficiency
Solid circular cross-
section coil
Pin : 220W
Frequency [MHz]
Fig. 9. Theoretical, FEA and measured efficiency comparison
Fig. 10. Schematic of a single turn coil with 5 different cross-sections
A. Skin Effect Loss Depending on Cross-sectional Shape
To investigate the impact of skin effect on equivalent
series resistances (ESR) of a single turn circular loop coil,
five different cross-sectional shapes are modeled and their
ESR are compared using FEA. Five different cross-
sectional shapes are shown in Fig. 10. For fast simulation
and comparison, a solid conductor having very small radius
(200μm) is used as a baseline in the FEA. All the cross-
sectional shapes are hollow tube type except the baseline. The
wall thicknesses of the hollow tubes are equal to the skin
depth: 34μm at the test frequency, 3.6MHz. The areas of all
the different shapes are identical to the area of the solid
circular shape in order to have same DC resistance. To
maintain the same cross-sectional area, the outer radius of the
hollow circular cross section and the length of the sides in
each polygon were adjusted. JMAG-Studio, 2-D axis-
symmetric simulation was used for finite element analysis.
In Fig. 11, current density contours of the single turn coil
for each cross-sectional shape are shown. It should be noted
that in Fig. 11(a), a negative current is flowing in the center
of the solid circular (SC) coil while the current densities of
the other four coils are positive in the entire cross-section.
The negative current in SC is caused by an internal magnetic
field which is generated by the current flowing in the skin
depth layer [6]. It is known as skin effect and the negative
current in SC causes heavily concentrated current distribution
on the outer circumference of the coil to compensate the
negative current. The effect of the concentrated current
density on equivalent series resistance is shown in Fig. 12.
It is showing that using a solid conductor in high frequency
889

6. (a) Solid circular
(SC)
(b) Hollow circular
(HC)
(c) Hollow octagonal
(HO)
[A/mm2]
(d) Hollow hexagonal
(HH)
(e) Hollow square
(HS)
Fig. 11. Current density distribution of a single turn coil for 5 different
cross-sections
ACresistance[Ω]
Legend:
SC: Solid circular
HC: Hollow circular
HO: Hollow octagon
HH: Hollow hexagon
HS: Hollow square
Fig. 12. Single turn coil resistances depending on the cross-sectional shape
and high power applications is inefficient. In Fig. 12, it
should be noted that the hollow circular (HC) cross-sectional
shape has the lowest AC resistance compared to solid circular
conductor and any other polygonal shape conductors. The
HC’s ESR is three times smaller than the SC’s ESR. A
primary reason for this is because there is no current density
concentration caused by skin effect in HC conductor as
shown in Fig. 11(b). Since the wall-thickness of the hollow
tube is equal to the skin depth, current is distributed
uniformly. Secondly, since the current density distributions
of polygonal conductors are concentrated on the sharp
corners as shown in Fig. 11 (c-e), their ESR is higher than
hollow circular type coil.
Because of its low skin effect loss, a hollow circular tube
having skin depth equal to the wall thickness will be used as
the basic geometry for the following discussion of spatial
configuration.
B. Proximity Effect Loss Depending on the Spatial
Configuration of Each Coil Turn
Another Ohmic loss mechanism of the coil in wireless
power transfer system, proximity effect loss, is caused by
interaction of magnetic fields between adjacent current
carrying conductors. Proximity effect induces additional
non-uniformity on the conductor current distribution [12].
To investigate proximity effects’ role on ESR, a spiral coil
having a hollow circular cross-section is analyzed in this
paper. An isometric view of the spiral coil and its A-A
Fig. 13. Conventional spiral wound coil with 5 different numbers of turns.
[A/mm2]
(a) Two turn coil (HC)
(b) Three turn coil (HC)
Fig. 14. Current density distribution of conventional spiral wound coils
(a) Schematic of magnetic field interaction between adjacent current
carrying conductors
[μT]
(b) Magnetic flux intensity contour for three turn conventional spiral coil
Fig. 15. Magnetic flux re-distribution due to an adjacent conductor
cross-sectional drawing is depicted in Fig. 13. In this
analysis, five different cases were tested, from two-turn to
six-turn coils.
Turn-to-turn spacing was 40μm, which is almost the same
as the skin depth. The resulting current distribution for the
two-turn and three-turn coil cases are shown in Fig. 14. The
non-uniformity of the current density distribution for the two-
turn coil is clearly shown by comparing Fig. 11(b) and Fig.
14(a). In Fig. 14 (a), current density is very high at the outer
circumference of each coil while it is low in the center
890

7. ACresistance[Ω]
Frequency: 3.6MHz
Cross-section shape:
Hollow circular
Coil shape: spiral
Number of turns [turns]
Fig. 16. ESR depending on the number of turns
ACResistance[Ω]
Frequency: 3.6MHz
Cross-section shape:
Hollow circular
Coil shape: spiral
Number of turn: 2
2×Rskin Rproximity
(a) Two-turn
ACResistance[Ω]
Frequency: 3.6MHz
Cross-section shape:
Hollow circular
Coil shape: spiral
Number of turn: 6
6×Rskin Rproximity
(b) Six-turn
Fig. 17 ESR change due to proximity effect
region and it is even negative on the right side of the left turn
coil. The reason for non-uniformity is shown in Fig. 15.
The magnetic fields generated by turn 1 (solid line) and turn
2 (dotted line) are interacting destructively in this center
region (region 2) while they interacts constructively in region
1 and 3. As a result, the magnetic field intensity is very
high in region 1 and 3 while it is very low in region 2, which
causes high current density in blue area and low current
density in green area of Fig 15 (a). The situation degrades
as the number of turns increases as can be seen in Fig. 14 (b)
and Fig. 15 (b). As depicted in Fig. 16, the ESR of a spiral
wound coil increases non-linearly as the number of turns of
the coil increases.
Fig. 17 shows the impact of proximity effect on ESR by
comparing the resistance of a coil with the resistance
assuming only skin effect exists. If proximity effect does
not exist, the ESR of a two-turn coil should be equal to two
times of the skin effect resistance (R skin ). However,
because of the proximity effect, total resistance is 1.3 times
larger than 2*Rskin for the two-turn coil and it is 3.3 times
larger than 6*Rskin for the six-turn coil.
C. Proposed Conductor Design
According to Fig. 14 - 17, ESR of a current-carrying coil
Fig. 18 Schematic of the proposed conductor design
[μT]
Fig. 19. Magnetic field intensity contour of the three-turn plated coil
increases rapidly if the magnetic field of a turn interacts with
an adjacent turn’s magnetic field (proximity effect loss). It
should be noted that both constructive and destructive
interaction causes an increase in the ESR because of
concentration of current density in a small part of the
conductor. Also, it is shown in Fig. 11 and 12, that skin
effect loss is significantly increased if a solid circular
conductor is used instead of a hollow tube conductor.
Based on analysis of skin- and proximity effect losses, a
new spatial layout of three-turn plated coil is proposed in Fig.
18. In this design, an individual turn constitutes an arc
component of total circular hollow cross-section. For
example, each turn of a three-turn coil occupies 120 degrees
as shown in Fig. 18. By this spatial layout, magnetic flux
interactions between the turns are significantly decreased
because the inside of the circular coil has nearly zero
magnetic field by symmetry and the magnetic fields of the
outside of the circular coil are connected smoothly without
adding or subtracting the fields. As shown by FEA in Fig.
19, the magnetic flux contour does not change abruptly while
the magnetic flux of the three-turn conventional spiral coil
changed significantly in the neighboring area of the each turn
as shown in Fig. 15 (b). Fig. 20 shows the current density
distribution of the proposed design coil. As expected,
current distribution is almost uniform and there is no reverse
direction current flow. Resultant ESR of the proposed
design is shown in Fig. 21 with ESR of other conventional
spiral coils. By this comparison, it is demonstrated that the
proposed spatial layout of the plated multi-turn coil decreases
the losses caused by skin- and proximity effect. Although
only a three-turn example is shown here, this spatial layout
principle can be extended to any multi-turn coil layout.
Implementation of the proposed conductor geometry can
be achieved by plating copper on a dielectric tube since the
wall-thickness is equal to the skin depth. Because of the
891

8. [MA/m2]
Fig. 20. Current density distribution of the proposed three-turn plated coil
ACResistance[Ω]
Legend:
Conventional:
HC: Hollow circular
HO: Hollow octagon
HH: Hollow hexagon
HS: Hollow square
Proposed:
PC: Plated circular
Frequency: 3.6MHz
Fig. 21. Resistance comparison of three-turn coils
Isometric view Bottom view
Fig. 22. 3-D drawing of the proposed conductor layout
high operating frequency, (several MHz), skin depth is about
30μm that is suitable for plating. By transposing each turn
by 360°/N degrees, where N is the number of turns, an N turn
inductor can be built easily.
An example spatial layout for three-turn inductor is shown
in Fig. 22. The first turn, (yellow face), is transposed by
120 degrees to meet the second turn, (red face). The second
turn is transposed by 120 degrees to meet the third turn, (blue
face).
CONCLUSIONS
The first focus of this paper is on development of an
equivalent circuit model for 30 cm distance, 220W, high
efficiency, wireless power transfer system. By using
electric and magnetic field analysis of the current carrying
coils in the system, it was shown that the coils can be
represented as two lumped inductors and resistors with very
weak mutual inductance. All the inductances and
resistances were also estimated by closed form expressions.
Based on the proposed model and the estimated circuit
parameters, the equation for power transfer efficiency was
derived. It was shown by finite element field analysis and
experiments that for high efficiency power transfer over large
air gaps, matched tuning of the resonant transmitter and
resonant receiver is critical.
As a second focus, the origins of losses in a non-radiative
mid-range wireless power transfer system were investigated.
Skin- and proximity effect losses were shown to be dominant.
A new spatial layout for multi-turn coils was proposed that
has very low losses compared to conventional coil geometries.
The proposed spatial layout of the new multi-turn coil was
evaluated by comparing its ESR to the ESR of conventional
designs since that is a primary metric for the power
conversion efficiency.
ACKNOWLEDGMENT
The authors wish to acknowledge the motivation provided
by the Wisconsin Electric Machines and Power Electronics
Consortium (WEMPEC) of the University of Wisconsin-
Madison.
REFERENCES
[1] A. Karalis, J. D. Joannopoulos, and M. Soljačić, “Efficient wireless
non-radiative mid-range energy transfer,” Ann. Phys., vol. 323, pp34-
48, Jan. 2008.
[2] E. Waffenschmidt and T. Staring, “Limitation of Inductive Power
Transfer for Consumer Applications,” in Proc. 13th European Conf. on
Power Electronics and Applications, EPE 2009, Barcelona, Spain, Sept.
2009.
[3] Z. Yang, W. Liu, and E. Basham, “Inductor Modeling in Wireless
Links for Implantable Electronics,” IEEE Trans. Magnetics, vol. 43,
pp.3851-3860, Oct. 2007.
[4] R. R. Harrison, “Designing Efficient Inductive Power Links for
Implantable Devices,” in Proc. 2007 IEEE Intl. Symposium on Circuits
and Systems (ISCAS 2007), New Orleans, LA, pp. 2080-2083, 2007.
[5] C. A. Balanis, Antenna Theory: Analysis and Design, 3rd ed., New
York: Wiley, 2005, pp. 231-246.
[6] U. S. Inan and A. S. Inan, Engineering Electromagnetics and Waves,
Boston: Addison Wesley, 1999.
[7] H. C. Miller, “Inductance Formula for a Single-Layer Coil,” in Proc.
IEEE, vol. 75, no.2, pp.256-257, Feb. 1987
[8] R. Lundin, “A Handbook Formula for the Inductance of a Single-Layer
Circular Coil,” in Proc. IEEE, vol. 73, no. 9, Sep. 1985.
[9] F. W. Grover, “Tables for the Calculation of the Mutual Inductive of
Any Two Coaxial Single-Layer Coils,” in Proc. Inst. Radio Engineers,
vol. 21, no. 7, Jul. 1933.
[10] F. W. Grover, “The Calculation of the Inductance of Single-Layer Coils
and Spirals Would With Wire of Large Cross Sections,” in Proc. Inst.
Radio Engineers, vol. 17, no. 11, Nov. 1929.
[11] L. F. Woodruff, Principles of Electric Power Transmission, 2nd ed.
New York: John Wiley & Sons In., 1938.
[12] G. S. Smith, “Radiation Efficiency of Electrically Small Multiturn
Loop Antennas,” IEEE Trans on Ant. and Prop., vol. 20, pp.656-657,
Sep. 1972.
[13] G. S. Smith, “Proximity Effect in Systems of Parallel Conductors,” J.
Appl. Phys., vol. 43, no. 5, pp. 2196-2203, 1972.
[14] D. Kurschner and C. Rathge, “Contactless Energy Transmission
Systems with Improved Coil Positioning Flexibility for High Power
Applications,” in Proc. IEEE PESC, pp.4326-4332, Rhodes, Greece,
Jun. 2008.
[15] H. Abe, H. Sakamoto, and K. Harada, “A Noncontact Charger Using a
Resonant Converter with Parallel Capacitor of the Secondary Coil,”
IEEE Trans. on Ind. App., vol. 36, no.2, pp. 444-451, 2000.
[16] C. M. Zierhofer and E. S. Hochmair, “Geometric Approach for
Coupling Enhancement of Magnetically Coupled Coils”, IEEE Trans.
on Biomedical Eng., vol. 43, no. 7, pp. 708-714, 1996.
[17] B. L. Cannon, J. F. Hoburg, D. D. Stancil, and S. C. Goldstein,
“Magnetic Resonant Coupling As a Potential Means for Wireless
Power Transfer to Multiple Small Receivers,” IEEE Trans on Power
Electronics, vol.24, no.7, Jul. 2009.
[18] A. Kurs, R. Moffatt, and M. Soljačić, “Simultaneous mid-range power
transfer to multiple devices,” in Appl. Phys. Lett. 96, vol. 96, 044102,
2010.
892