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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.