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High-Q Self-Resonant Structure for Wireless Power
Transfer
Aaron L.F. Stein Phyo Aung Kyaw Charles R. Sullivan
Thayer School of Engineering
Dartmouth College
Hanover, NH 03755 USA
Email: {Aaron.L.Stein, Phyo.A.Kyaw.TH, Charles.R.Sullivan}@dartmouth.edu
Abstract—The range and efficiency of wireless power transfer
systems are limited by the quality factor of the transmit and
receive coils used. Multi-layer self-resonant structures have been
proposed as a low-cost method for creating high-Q coils for high-
frequency wireless power transfer. In these structures thin foil
layers are separated by a dielectric material in order to form a
capacitance that resonates with the inductance of the structure,
while also forcing equal current sharing between conductors. In
order to reduce winding loss, these structures are made with foil
layers much thinner than a skin depth, which makes the layers
of the structure extremely difficult to handle. In this paper, we
present a modified self-resonant structure in which the layered
conductors are made from standard PCB substrates with no
vias. The PCB substrates provide an inexpensive way to handle
thin conductive layers, and the modified self-resonant structure
ensures that the poor dielectric properties of the PCB substrates
do not impact the quality factor of the structure. The modified
self-resonant structure makes it feasible to achieve advantages
similar to litz wire, but at multi-MHz frequencies where effective
litz wire is not commercially available. Experimental results
show that the structure has a quality factor of 1177 at 7.08
MHz, despite only being 6.6 cm in diameter. The quality factor
normalized by the diameter is more than 6.5x larger than other
coils presented in the literature.
I. INT ROD UC TI ON
Wireless power transfer is of great interest for many appli-
cations including biomedical, automotive, and consumer hand-
held electronics [1]–[4]. In many of these applications a high-
frequency magnetically-coupled resonant system is the most
effective method of transmitting wireless power. The efficiency
and range of such a system is limited by the quality factor
and coupling coefficient of the resonant coils that generates
the electromagnetic coupling [3], [5], [6].
Traditional coils consist of a spiral loop of wire connected
to a ceramic or film capacitor. The quality factor of such a coil
increases linearly with the diameter of the coil [7], [8]. So we
propose a figure of merit Qd, which is the ratio of the quality
factor Qto the diameter dof the coil: Qd=Q
d. Experimental
data in the literature for high frequency coils around 6.78 MHz
have a Qdthat ranges from 3 to 28 cm−1[3], [9]–[13].
The Qdof conventional coils is limited by two main factors.
First, below 1 MHz coils are typically made from litz wire in
order to minimize losses due to skin and proximity effects.
However, the benefit of using litz wire is limited in the MHz
frequency range due to the need to have strand diameters
much smaller than the skin depth. Such small strand diameters
are not commercially available because they are difficult and
therefore expensive to manufacture. Second, in many designs,
eddy currents are induced in the capacitors due to their
proximity and orientation to the coil.
To mitigate these issues a multi-layer self-resonant structure
using thin sheets of conductors, capacitive ballasting, and low-
loss dielectrics was proposed in [14]. This structure consists of
alternating layers of C-shape conductors and dielectric rings
placed in a ferrite core, creating inductively coupled capacitors
in parallel with an inductor. The integration of capacitance in
the structure is similar to the integrated LC and LCT (inductor,
capacitor, transformer) passive power components discussed
in, for example, [15]–[17]. However, unlike the previous work,
multi-layer self-resonant structures use the capacitance not
only to implement the necessary capacitance, but also to make
the conductors more efficient by equalizing current sharing
between them. As a result, they not only provide a parts
count savings through integration, but also provide a dramatic
performance benefit. Furthermore, the self-resonant structure
reduces eddy currents by keeping thin foil layers parallel to the
magnetic field, and does not require inter-layer connections.
In this work, we present a modification to the self-resonant
structure that achieves similar performance, but simplifies
the construction process by allowing the thin conductive
layers to be constructed from standard PCB substrates. This
modification to the self-resonant structure led to the first
experimental implementation of a multi-layer self-resonant
structure at resonant frequency practical for wireless power
transfer.
A related type of self-resonant structure is the split-ring
resonator (SRR) [18]. A SRR is a pair of C-shaped conductors
that forms a simple resonator which can be arrayed to create
metamaterials. Metamaterials can be designed with unusual
and controllable electromagnetic properties, and have even
been proposed as a way to influence the coupling of resonant
inductive wireless power systems. However, although it can
be shown that an ideal negative-permeability material could
be used to enhance performance, the losses of practical SRRs
limit the usefulness of this approach [19]. Whereas an individ-
ual SRR comprises just two concentric C-shaped conductors,
the self-resonant structure configures them in a stack rather
than concentrically, and uses many layers to achieve low
losses, in conjunction with soft magnetic material shaping the
field for lowest losses.
The many layers of the self-resonant structure are made
from foil, which makes using conductors thinner than the skin
depth feasible even at high-frequencies; however, very thin
foil layers are difficult to handle. The practical construction
challenges associated with using such thin layers prevented us
from from experimentally validating the self-resonant structure
in [14] at the desired resonant frequency. One way to overcome
the challenges associated with using such thin copper layers is
to pattern the C-shapes by etching thin copper layers laminated
on substrates. In the resonant structure in [14], the capacitance
between adjacent conductors, such as those on opposite sides
of a substrate, provides the capacitance for resonance. Thus,
for high-Q resonance, the substrate dielectric must have a very
low dissipation factor. Unfortunately, common substrate mate-
rials such as FR4 and polyimide have high dissipation factors
(0.015 and 0.002) even at low frequencies and the performance
gets worse at higher frequencies. Copper laminates with low-
loss substrates such as PTFE are much more expensive.
In this paper, we present a variation of the self-resonant
structure described in [14] to allow fabrication with more
conventional methods and materials. Our new resonant struc-
ture uses high-loss but low-cost substrates such as FR4 and
polyimide to support thin conductor layers for easy handling
without adversely affecting the quality factor of the resonance.
The improved manufacturability of the modified structure
presented here allowed us to successfully implement a high-
Q 7 MHz resonant structure for wireless power transfer.
The modified structure is described in Section II, its loss
mechanisms are analyzed in Section III, and experimental
characterization results are presented in Section IV.
In addition to wireless power transfer applications, high-Q
resonant structures are of interest as passive components for
power converter applications. Our design work and test results
for similar structures used in resonant power conversion are
discussed in [20].
II. MO DI FIE D SEL F-RE SO NAN T STRUCTURE
The modified self-resonant coil is illustrated in Fig. 1. The
structure creates a parallel LC resonance. The inductance L
is equivalent to a single turn around the magnetic core, while
the total capacitance Cequiv is created by inductively coupling
each section of the structure as shown in the circuit model in
Fig. 2. A section of the structure is two C-shaped conductors
with opposing orientation that are separated by a low-loss
dielectric. For example, in Fig. 1, Layers 2, 3, and 4 form
one section. Each section forms two capacitors. One capacitor
is formed in each area the conductors overlap as illustrated in
Fig. 3. The capacitance between half of one C shape and the
facing half of the other C shape in the same section Csh, is
a function of the angle of overlap of the layers in radians θ
(shown in Fig. 3), the outer radius of the coil r2, the inner
radius r1, and the dielectric thickness td
Csh =ǫθ(r2
2−r2
1)
td2.(1)
Fig. 1. The layers of a 2 section modified self-resonant structure.
Csh
Csh
Csh
Csh
Csub
Fig. 2. Equivalent circuit model of a 2 section modified self-resonant
structure.
The modified self-resonant structure has msections, where
each section of the structure has two series connected Csh.
The equivalent capacitance Cequiv of the structure is
Cequiv =mCsh
2.(2)
This is the only capacitance which can be excited, and
in conjunction with the inductance determines the resonant
frequency of the structure. The resonant frequency of the
structure ωois given by
ωo=1
pLCequiv
.(3)
The proposed modified self-resonant structure allows the
θ
Fig. 3. In this figure two overlapping C-shaped conductors forming one
section is shown. Each section forms two capacitors Csh which are connected
in series. The angle of overlap of one capacitor θis marked .
200
200
600
600
600
600
1000
1000
1000
1000
1000
1000
1400
1400
1400
1400
1400
1800
20 40 60 80 100 120 140 160 180 200
Number of Sections
2
4
6
8
10
12
14
16
18
20
Conductor thickness (microns)
Fig. 4. The theoretical quality factor of the modified self-resonant structure at
7 MHz is plotted as a function of the conductor thickness and the number of
sections for a 6.6 cm pot described in Section II. The field weakening factor
is unity and the current crowding factor zero in this figure.
use of low-performance substrates such as FR4 or polyimide
without significantly affecting the Q of the structure. To
achieve this, any two conductor layers that are separated
by a high-loss substrate are oriented such that their gaps
are aligned. For example in Fig. 1, the top layer of copper
(layer 1) is separated from the second layer of copper (layer
2) by a high-loss substrate, and are both oriented such that
the gap is coming out of the page. A capacitance Csub is
formed between these two layers; however, the orientation
ensures that no strong electric field is generated between the
layers. The voltage induced in Csub is only due to the leakage
magnetic flux, which is a small fraction of the overall magnetic
flux. This allows the high-loss substrate to be integrated into
the self-resonant structure, without significantly affecting the
quality factor of the structure. Furthermore, the thickness of
the substrate does not affect the equivalent capacitance Cequiv,
so it can be selected based on considerations such as ease
of handling, and the overall compactness of the complete
structure.
III. LOS S MEC HA NI SM S
The performance of the modified self-resonant structure is
measured by the quality factor of the device at resonance. The
quality factor Qis
Q=ωoL/Rtotal,(4)
where Rtotal is sum of 3 equivalent series resistances (ESR)
that model winding resistance, core loss, and dielectric loss.
The ESR for each of these loss mechanisms is derived in this
section.
1) Winding Loss: The power lost in the winding is due
to both the low frequency resistance Rlf of the winding,
and eddy currents due to the high-frequency magnetic field
(proximity effect). The increased losses due to these eddy
currents can be modeled by a resistance Re. The power lost
in the winding Pwind can be expressed as
Pwind =I2
rmsRw ind =I2
rmsRlf +I2
rmsRe,(5)
where Rwind is the AC resistance of the structure. Simpli-
fication of (5) shows that the AC resistance factor Rac
Rlf is
1 + Re
Rlf . Therefore, the winding resistance of the structure
is the product of a low frequency resistance and an AC
resistance factor, and is given by
Rwind =Rlf 1 + Re
Rlf .(6)
Both Rlf and the AC resistance factor are derived in [14] for
the self-resonant structure. The winding resistance is given by
Rwind =2πρ
ln(r2
r1)tcm"k1+m2
9tc
δ4
k2#,(7)
where k1is (1−θ
3π), k2is (1 + θ
π), tcis the thickness of
the foil layers, δis the skin depth, and ρis the resistivity of
the conductor material. This analysis assumes that a magnetic
core with infinite permeability is placed directly adjacent to
the windings. In practice, there is a gap between the magnetic
core and the winding, and furthermore the permeability of the
high frequency magnetic material is not large enough to be
accurately modeled as infinite. Compared to the idealized case,
these practical consideration weaken the magnetic field and
prevent the magnetic field lines from being perfectly parallel
to the foil layers.
The weakening of the magnetic field reduces the power lost
due to the proximity effect. The impact of field weakening
on the winding resistance is modeled by a field weakening
factor Ffw . This factor is derived from a finite element analysis
described in Appendix A-A. In the idealized case, Ff w is 1,
and it decreases in practical scenarios. When the magnetic
field lines are not parallel to the foil layers horizontal current
crowding occurs in the winding. This is modeled with a current
crowding factor Fcc. This factor is also derived from a finite
element analysis, and its extraction process is described in
Appendix A-B. In the idealized case Fcc is 1, and it increases
in practical scenarios. In total the winding resistance is given
by
Rwind =2πρ
ln( r2
r1)tM "k1Fcc +Ffw M2
9tc
δ4
k2#.(8)
The winding loss expression illuminates important design
parameters: the thickness of the conductor and the number
of sections. If the conductor is too thick the proximity effect
losses will be high, whereas if the conductor is too thin the
DC resistance of the conductor will be large. Similarly, an
optimal number of sections exists. If too many sections are
used, the proximity effect losses will once again be high, and
if too few sections are used the conductor resistance will be
high. An optimal number of sections and conductor thickness
can be derived; however, in our prototyping work, we instead
constrained the design based on material thicknesses that are
readily available from stock. A contour plot in Fig. 4 illustrates
the impact of varying these parameters on the quality factor
of the structure.
2) Magnetic Core Loss: In this application both the real
part µ′and the imaginary part µ′′ of the magnetic core
permeability affect the quality factor of the structure. The loss
in the magnetic core is modeled by an ESR, which can be
derived from a reluctance model of the magnetic core. Using
this model, the single-turn inductance of the structure L∗is a
complex number given by
L∗=1
ℓeh
Aeµ0(µ′−jµ′′ )+Ra
,(9)
where ℓeh is the effective length of the core half (half the
effective length of a full pot core), Aeis the effective area of
the core, and Rais the reluctance of the air gap. The ESR
that models core loss is a function of the angular frequency ω
and is given by
Rcore =ℜ[jωL∗]
=
ωℓeh
µ0Aeµ′′
ℓeh
µ0Ae+Raµ′2+ (Raµ′′)2.(10)
The denominator of Rcore is dominated by (Raµ′)2. There-
fore, the ESR of the core is approximately proportional to
Rcore ∝
∼1
µ′Qmaterial
,(11)
where the quality factor of the material Qmaterial is µ′
µ′′ . In
order to reduce core loss, it is important to select a material
that both has a large Qmaterial, and a large real component
of magnetic permeability at the resonant frequency of the
structure.
3) Dielectric Loss: The modified self-resonant structure
does not use external capacitors, so the loss mechanisms as-
sociated with conductors of external capacitors do not impact
it. Instead, the losses created by the capacitance Cequiv of the
structure are due to the dielectric material, and can be modeled
with an ESR that is given by
Rdieletrcic =Dd
Cequiv ω,(12)
where Ddis the dissipation factor of the material. To reduce
the dielectric loss a material with a small dissipation factor
such as PTFE or polypropylene should be used. There are no
significant losses created by Csub, despite the use of the high-
loss substrate, because it is not involved in the resonance of
the structure.
IV. RES ULT S - IM PL EM ENTATION O F TH E MOD IFI ED
SEL F-RE SO NAN T STRUCTURE
The performance of the modified self-resonant structure was
experimentally validated. The device comprises three main
components. First a pot core was made from Fair-Rite’s 67
material. This material was chosen for its low loss at 7 MHz.
The pot core has a diameter of 6.6 cm, and a height of 1.62
cm. Next, the conductive layers of the structure were created
using 6 µm copper that, for ease of handling, was laminated on
both sides of a 25 µmpolyimide substrate and patterned into
C-shapes using standard PCB fabrication processes. Finally,
50.8 µmthick PTFE film was cut with a die cutter to form
the low-loss dielectric layers. A picture of the modified self-
resonant structure is shown in Fig. 5, and system parameter
values are shown in Table I.
For this implementation of the modified self-resonant struc-
ture, the analysis in Section III estimates the ESR of the
structure to be 4.9 mΩ. A finite element analysis of the
structure is used to derive the field weakening factor Ffw of
0.80, and the current crowding factor Fcc of 1.74, which results
in a predicted winding resistance of 1.6 mΩ. Experimentation
using FairRite’s 67 material found the imaginary component
of the relative permeability to be 0.07, which results in an core
loss ESR of 1.9 mΩ. Finally, using the dissipation factor from
a PTFE data-sheet, the ESR that model the dielectric loss is
1.4 mΩ. Given that the inductance of the structure is 155 nH,
(4) estimates that this structure will have a quality factor of
1407.
TABLE I
SEL F-RE SO NANT C OI L VARIA BL ES,T HE IR DE SC RIP TIO NS ,AND VAL UE S IN
TH E EXP ER IME NTAL S ETU P. THE W IND IN G,CO RE ,AND D IE LEC TR IC ES RS
AR E DER IV ED FRO M TH E MOD EL S IN SEC TI ON III.
Parameter Description Value
dStructure diameter 6.6 cm
mNumber of sections 48
Core window height 9.2 mm
Height of structure 16 mm
r2Coil outer radius 26.25 mm
r1Coil inner radius 14.85 mm
tConductor thickness 6 µm
Substrate thickness 25.4 µm
Stacked layers height 5.5 mm
θOverlap angle 2.97rad
δSkin depth 25 µm
ρConductor resistivity 16.8 nΩ-m
Ffw Field Weakening factor 0.80
Fcc Current crowding factor 1.74
Rwind Winding ESR 1.4 mΩ
LStructure inductance 155 nH
µ′Core relative permeability 40
µ′′ Imaginary relative permeability 0.07
ℓeh Effective length of core half 37.5 mm
AeEffective core area 717 mm2
RaReluctance of air path 5.4 MA
Wb
Rcore Core ESR 1.9 mΩ
Cequiv Structure capacitance 3.28 nF
tdDielectric thickness 25.4 µm
ǫDielectric permittivity 2.2ǫo
DdDielectric dissipation factor 2×10−4
Rdieletric Dielectric ESR 1.4 mΩ
Fig. 5. Picture of the modified self-resonant structure that was used for
experimental results.
A. Experimental Performance of the Modified Self-Resonant
Structure
The quality factor of the resonant structure was experimen-
tally found to be 1177. The quality factor was derived from
the magnitude of the impedance that is shown in Fig. 6, using
the ratio of the resonant frequency to the 3dB bandwidth. To
verify this measurement the quality factor was also derived by
measuring inductance, resonant frequency and magnitude of
the maximum impedance Zpk to compute Q=Zpk
ωoL= 1136.
The error between the theoretical and experimental quality
factor is 16.1%, which suggests good agreement with the
analysis presented in Section III. The Qdof the modified
resonant structure is 178 cm−1, which represents a factor of
6.35 improvement over the current state-of-the-art [3], [9]–
[13]. The experimental results are summarized in Table II.
TABLE II
SUM MARY O F EX PER IME NTAL R ES ULTS
Parameter Description Value
foResonant frequency 7.08 MHz
QQuality factor 1177
QdFigure of merit (FOM) 178
FOM percent improvement 635%
B. Impact of the Modified Self-Resonant Structure on Wireless
Power Transfer
The maximum achievable efficiency ηmax between two
coils of a wireless power transfer system is dependent on the
quality factor Qof the coils, and the coupling coefficient k.
The maximum efficiency is derived in [5], [6] and is given by
ηmax =(Qk)2
1 + p1 + (Qk)22.(13)
Therefore, to maximize the efficiency of a wireless power
transfer system the quality factor and coupling factor should
be maximized.
The modified self-resonant structure has been experimen-
tally demonstrated to have a quality factor that is 6.3 times
larger than conventional coils; however, to understand the
7.076 7.078 7.08 7.082 7.084 7.086
Frequency (MHz)
3500
4000
4500
5000
5500
6000
6500
7000
7500
8000
8500
Imepedance Magnitude
Modified Self Resonant Coil
Resonant Frequency
3dB Marker
7 7.05 7.1 7.15
Frequency (MHz) ×106
0
1000
2000
3000
4000
5000
6000
7000
8000
Imepedance Magnitude
Fig. 6. An Agilent 4294A impedance analyzer was used to measure the
impedance magnitude of the modified self-resonant structure around its
resonant frequency. The impedance magnitude is shown for two different
frequency ranges to illustrate the high-q nature of the resonance. The ex-
perimental measured quality factor of the modified self-resonant structure is
1177.
0 50 100 150
Coil Separation Distance (mm)
0.001
0.01
0.1
1
Coupling Factor (k)
Fig. 7. A finite element analysis is used to derive the coupling coefficient
(k) as a function of the distance between the magnetic coils.
impact of this on the efficiency of wireless power transfer the
coupling factor must also be considered. The coupling factor
is determined by the shape, orientation, and properties of the
magnetic cores. A finite element analysis of the magnetic core
used in this work shows that the coupling factor ranges from
0.875 to 0.0014 as the transmission distance increases from
1 mm to 150 mm. The coupling factor is plotted over this
entire range in Fig. 7, and pictures that demonstrate the relative
transmission distances compared to the core size are shown in
Fig. 8.
The maximum achievable efficiency, given by (13), using
the modified self resonant structure is compared to the current
state-of-the-art coil designs in Fig 9. For this comparison we
assume that each resonator is implemented with the same
magnetic core used for our modified self-resonant structure.
The maximum achievable efficiency using the modified self-
Fig. 8. A picture of the magnetic cores at a separation distances of a) 50
mm, b) 100 mm, and c) 150 mm illustrates the relative size of the coil to
the range of wireless power transfer discussed in this work.
resonant structure is derived using the experimental quality
factor of 1177 and the simulated coupling factor. The state-
of-the-art design uses the same coupling factor as the self-
resonant structure, but with a quality factor of 185, which is
derived by multiplying the state-of-the-art Qdby the diameter
of the core.
The modified self-resonant structure improves wireless
power transfer efficiency for any distance between the coils.
For example, if the coils are 20 mm apart the modified self-
resonant structure can achieve 98.7% efficiency, while the
current state-of-the-art coil coil technology can achieve 91.9%.
At longer distances, the difference is even more dramatic. At
a distance of 90 mm, the state-of-the-art coil design can only
achieve an efficiency of 22%, while the modified self resonant
structure achieves an efficiency of 77%. Furthermore the mod-
ified self-resonant structure can maintain high efficiency out
to longer distances. For example, the modified self resonant
structure can maintain at least 90%efficiency at a distance up
to 65 mm, while conventional coil designs can only achieve
90% efficiency at distances up to 30 mm.
V. CO NC LU SI ON
The efficiency and range of resonant wireless power transfer
is highly dependent on the quality factor of the resonant tank.
This work introduces a new high-Q self-resonant structure that
is both easy to manufacture and cost effective. To achieve this
goal the thin copper layers of the structure are created using
inexpensive substrates such as FR4 or polyimide laminated
with copper. Although inexpensive, these substrates are not
efficient dielectrics. By orienting the layers on the two sides
of the high-loss substrate differently than proposed in [14],
we avoid exciting the substrate capacitance and thus avoid
losses in it. Experimental results confirm the advantages of
the modified self-resonant structure, as the quality factor
normalized by the diameter of the structure is shown to be
more than 6.35 times higher than the current state-of-the-art
without using high-cost materials or manufacturing processes.
The improved quality factor of the modified self-resonant
structure improves the range over which wireless power can
be transferred. For example, compared to the current state-of-
the-art, the modified self resonant structure more than double
0 50 100 150
Coil Separation Distance (mm)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Efficiency
Q=1177 Self-Resonant Structure
Q=185 Current State-of-the-Art
Q=100
Fig. 9. The theoretical maximum wireless power transfer efficiency as a
function of transmission distance is shown for the modified self-resonant
structure, and the current state-of-the-art coil design. The drastically improved
quality factor of the modified self-resonant structure causes a significant
improvement in wireless power transfer efficiency, and improves the viable
range of wireless power transfer.
the range at which energy can be transferred at an efficiency
of at least 90%.
APP EN DI X A
FIN IT E ELE ME NT WI ND IN G LOSS EXTRAC TI ON
The field weakening and current crowding factors are ex-
tracted from a two-dimensional axisymmetric finite element
analysis (FEA). Accurate modeling of the magnetic core
properties, and the physical dimensions are important to the
result of the simulation. Each section of the modified self
resonant structure consists of two copper layers, so the FEA
model is an inductor with 2mturns of foil. The foil layers
of the model are connected in series in order to force equal
current sharing between layers, and are driven with an RMS
current Irms. The thickness of the foil windings is tc, and
therefore it is important to ensure that the FEA mesh size
within the winding is small enough to accurately model the
effects within the winding. Finally, a low frequency resistance
of the coil Rlf is needed for this analysis, and it can be derived
from a FEA simulation; however, to reduce computation time
an analytical expression is used and is given by
Rlf =4mπρ
ln ( r2
r1)tc
.(14)
A. Field Weakening Factor
The field weakening factor accounts for decreased proximity
effect loss due to a reduction in the magnetic field. The
proximity effect power loss is produced by the magnetic
field inside the winding area; however, the impact of field
weakening is derived by only considering the spatial average
of the square of the peak value of the magnetic field parallel to
the foil layers Dˆ
B2
rE. The power loss due to the parallel-flux
proximity effect Pprox is given by
Pprox =Dˆ
Br
2Eω2t2
c
24ρVf(15)
where Vfis the total volume of the foil. Ffw is derived by
equating the AC resistance calculated with the simulated field
strength (1 + Pprox
I2
rmsRlf )to the theoretical AC resistance factor
(1 + Ffw (2m)2
9tc
δ4). The field weakening factor is derived
from this relationship and is given by
Ffw =9Pprox
I2
rmsRlf (2m)2tc
δ4.(16)
B. Current Crowding Factor
The current crowding factor accounts for increased losses in
the conductors due to horizontal current crowding. This factor
is derived from the resistance Rfea of the FEA model at the
resonant frequency. The ratio of Rf ea to Rlf is
Rfea
Rlf
=Ffw(2m)2
9tc
δ4
+Fcc,(17)
so the current crowding factor Fcc is given by
Fcc =Rfea
Rlf
−Ffw(2m)2
9tc
δ4
.(18)
REF ER EN CE S
[1] J. S. Ho, A. J. Yeh, E. Neofytou, S. Kim, Y. Tanabe, B. Patlolla,
R. E. Beygui, and A. S. Poon, “Wireless power transfer to deep-tissue
microimplants,” Proceedings of the National Academy of Sciences, vol.
111, no. 22, pp. 7974–7979, 2014.
[2] M. Adeeb, A. Islam, M. Haider, F. Tulip, M. Ericson, and S. Islam,
“An inductive link-based wireless power transfer system for biomedical
applications,” Active and Passive Electronic Components, 2012.
[3] A. P. Sample, D. A. Meyer, and J. R. Smith, “Analysis, experimental
results, and range adaptation of magnetically coupled resonators for
wireless power transfer,” IEEE Transactions on Industrial Electronics,
vol. 58, no. 2, pp. 544–554, 2011.
[4] T. Imura, H. Okabe, and Y. Hori, “Basic experimental study on helical
antennas of wireless power transfer for electric vehicles by using mag-
netic resonant couplings,” in Vehicle Power and Propulsion Conference.
IEEE, 2009, pp. 936–940.
[5] E. Waffenschmidt and T. Staring, “Limitation of inductive power transfer
for consumer applications,” in 13th European Conference on Power
Electronics and Applications. IEEE, 2009, pp. 1–10.
[6] M. Kesler, “Highly resonant wireless power transfer: Safe efficient, and
over distance,” Witricity Corporation, pp. 1–32, 2013.
[7] C. R. Sullivan, B. A. Reese, A. L. Stein, and P. A. Kyaw, “On size and
magnetics: Why small efficient power inductors are rare,” in 3D Power
Electronics Integration and Manufacturing (3D-PEIM), International
Symposium on. IEEE, 2016, pp. 1–23.
[8] D. J. Perreault, J. Hu, J. M. Rivas, Y. Han, O. Leitermann, R. C. Pilawa-
Podgurski, A. Sagneri, and C. R. Sullivan, “Opportunities and challenges
in very high frequency power conversion,” in Applied Power Electronics
Conference and Exposition. IEEE, 2009, pp. 1–14.
[9] K. Fotopoulou and B. W. Flynn, “Wireless power transfer in loosely cou-
pled links: Coil misalignment model,” IEEE Transactions on Magnetics,
vol. 47, no. 2, pp. 416–430, 2011.
[10] C. Florian, F. Mastri, R. P. Paganelli, D. Masotti, and A. Costanzo,
“Theoretical and numerical design of a wireless power transmission link
with GaN-based transmitter and adaptive receiver,” IEEE Transactions
on Microwave Theory and Techniques, vol. 62, no. 4, pp. 931–946, 2014.
[11] A. Khripkov, W. Hong, and K. Pavlov, “Design of an integrated resonant
structure for wireless power transfer and data telemetry,” in Microwave
Workshop Series on RF and Wireless Technologies for Biomedical and
Healthcare Applications (IMWS-BIO). IEEE, 2013, pp. 1–3.
[12] A. Kurs, A. Karalis, R. Moffatt, J. D. Joannopoulos, P. Fisher, and
M. Soljaˇ
ci´
c, “Wireless power transfer via strongly coupled magnetic
resonances,” Science, vol. 317, no. 5834, pp. 83–86, 2007.
[13] S.-H. Lee and R. D. Lorenz, “Development and validation of model
for 95%-efficiency 220 watt wireless power transfer over a 30-cm air
gap,” IEEE Transactions on Industry Applications, vol. 47, no. 6, pp.
2495–2504, 2011.
[14] C. R. Sullivan and L. Beghou, “Design methodology for a high-Q
self-resonant coil for medical and wireless-power applications,” in 14th
Workshop on Control and Modeling for Power Electronics (COMPEL).
IEEE, 2013, pp. 1–8.
[15] J. A. Ferreira and J. D. Van Wyk, “Electromagnetic energy propagation
in power electronic converters: toward future electromagnetic integra-
tion,” Proceedings of the IEEE, vol. 89, no. 6, pp. 876–889, 2001.
[16] J. T. Strydom and J. D. Van Wyk, “Volumetric limits of planar integrated
resonant transformers: a 1 MHz case study,” IEEE Transactions on
Power Electronics, vol. 18, no. 1, pp. 236–247, 2003.
[17] E. Waffenschmidt and J. Ferreira, “Embedded passives integrated circuits
for power converters,” vol. 1, 2002, pp. 12–17.
[18] R. Marques, J. Martel, F. Mesa, and F. Medina, “Left-handed-media
simulation and transmission of EM waves in subwavelength split-ring-
resonator-loaded metallic waveguides,” Physical Review Letters, vol. 89,
no. 18, p. 183901, 2002.
[19] T. Oh and B. Lee, “Analysis of wireless power transfer using meta-
material slabs made of ring resonators at 13.56 mhz,” Journal of
electromagnetic engineering and science, vol. 13, no. 4, pp. 259–262,
2013.
[20] P. Kyaw, A. Stein, and C. R. Sullivan, “High-Q resonator with inte-
grated capacitance for resonant power conversion,” in Applied Power
Electronics Conference and Exposition. IEEE, 2016.