a novel coaxial loop resonator for wireless power transfer

A Novel Coaxial Loop Resonator for Wireless PowerTransfer

Marco Dionigi, Mauro Mongiardo

D.I.E.I. Department of the University of Perugia, Perugia, Italy

Received 1 July 2011; accepted 4 October 2011

ABSTRACT: We present a novel resonator, based on a coaxial cable structure, suitable for

wireless power transfer (WPT) applications. Its advantages are compactness, ease of

manufacture, and low cost. WPT resonators generally operate in the HF, LF bands, using

magnetic field coupling and a lumped capacitance for achieving the sought resonance. In

our structure, the electric field is primarily confined inside the coaxial cable, whereas the

magnetic field is generated by the currents flowing on the external conductors. The article

describes the resonator structure, illustrates its design procedure, and discusses quality fac-

tor contributions. A few resonators with Q higher than 900, operating around 30 MHz,

have been built, simulated, and measured confirming the validity of the proposed

approach. VC 2012 Wiley Periodicals, Inc. Int J RF and Microwave CAE 22:345–352, 2012.

Keywords: resonator; wireless resonant power transfer; magnetic field coupling

INTRODUCTION

Wireless power transfer (WPT) is an emerging technology

that will allow one to power different appliances without

a direct connection with the electric power distribution.

Different applications have been presented so far, starting

with Nikola Tesla pioneering work in 1914 [1]. More

recent contributions [2–5] demonstrate mid range power

transmission by using resonant coils. These works have

stimulated several investigations: in [6] wireless electricity

(WiTricity) and its application to medical sensors and

implantable devices has been considered. It appears that

WiTricity is a suitable tool for providing wireless power

to a variety of medical sensors and implanted devices. A

different application has been proposed in [7], to realize a

sensor network that can operate with wireless resonant

energy transfer. Another type of application has been

investigated in [8], to apply wireless power transfer to

electric vehicles (EVs); in fact, the use of WiTricity pro-

vides a safe and convenient way to charge the EVs. Yet

another form of application has been proposed in [9],

where a procedure for inductive wireless powering of sin-

gle-chip systems has been presented.

In a more methodologically oriented approach [10,

11], we have found that dielectric structures, although

amenable of rigorous analytical study, possess resonant

fields which are mostly confined inside the dielectric,

especially when the Q is high. This has focused the atten-

tion on the case of inductive resonant coils for wireless

energy transfer [12, 13]. In particular, in [10, 11, 12, 14],

it has been noted that this type of subject is investigated

much better by using a network approach instead of

adopting the coupled mode theory originally proposed in

[2–4]. Moreover, in [13], it has also been observed that

additional, passive, resonators may help in extending the

range. Modelling of multi-resonators systems is quite

straightforward when using network theory, as it is well

known to microwave engineers.

Although it seems that WiTricity is an interesting and

viable way for several applications, not many resonator

structures have been proposed in literature. An ideal reso-

nator for WPT should possess a very high Q, no radiation,

only magnetic fields present outside the resonator. When

trying to realize such resonator, the problem seems to be

the following: if one uses distributed resonators (without

lumped capacitors) they tend to radiate energy, since coil

resonators have large radiation resistance. Moreover, the

coupling occurs also via electric fields thus giving an

unwanted sensitivity to different dielectrics. On the other

hand, resonators with lumped capacitance, although may

present high quality factors, require capacitances able to

withstand the high power present in the resonant circuit;

as a consequence they are quite bulky when a low reso-

nant frequency is required. A common solution adopted

Correspondence to: M. Dionigi; e-mail: [email protected]

VC 2012 Wiley Periodicals, Inc.

DOI 10.1002/mmce.20603Published online 23 February 2012 in Wiley Online Library

(wileyonlinelibrary.com).

345

so far is the one described in Figure 1, where a metallic

loop is used to generate the magnetic field and a lumped

capacitor is also employed. Unfortunately this structure is

limited in its quality factor Q to values of about 300–400.

We present in this article a novel resonator structure,

shown in Figure 2, based on a coaxial cable, with reduced

size, ease of manufacture and a higher quality factor. The

main interest of the proposed arrangement is that the elec-

tric field is almost entirely contained inside the coaxial

cable, whereas the currents present on the outside conduc-

tor of the coaxial cables generate the desired magnetic

fields outside of the resonator. This makes the resonator

particularly suitable for WPT applications.

The article is organized as following: in Section ‘‘The

resonator structure and its network model’’ the modelling

of the resonator is carried out and closed form expressions

are given for the computation of the relevant parameters;

in Section ‘‘Resonator Q factor optimization’’ the design

of high Q resonators is described and, finally, in section

‘‘Results,’’ an entire WPT system is designed and meas-

ured showing an excellent efficiency.

THE RESONATOR STRUCTURE AND ITSNETWORK MODEL

The resonator basic structure is depicted in Figure 2 with

its cross section geometry parameterization illustrated in

Figure 3. It is possible to compare the structure in Figure

2 with that in Figure 1 where we can identify the loop in-

ductor and the capacitor structure. The resonator of Figure

2 is composed of a coaxial cable section where the inner

conductor is connected with the outer conductor at the op-

posite end, and left unconnected at the other side. By

doing so we obtain a loop, where the external conductor

is used as loop inductor, and the coaxial section acts as a

capacitor. This structure is suitable to be used in WPT

systems when high quality coaxial cables are employed in

its manufacturing. If only a part of the circumference is

made of a coaxial cable extending for the angle d, as

shown in Figure 2, the rest of the circumference is

obtained from the external conductor connecting to the

Figure 1 Traditional loop resonator with lumped capacitor as-

sembly. This resonator is just a curved wire terminated at both

ends by a capacitance. The currents flow on the metallic (yellow)

conductor, which provides the inductive part. [Color figure can

be viewed in the online issue, which is available at

wileyonlinelibrary.com.]

Figure 2 Coaxial loop resonator structure composed of a

coaxial cable ring with internal conductor connected at one side

to the external conductor at the opposite end, and left open at the

other side. Metallization is sketched in yellow, dielectric is light

blue; d is the angular length of the coaxial K ¼ d2p : This type

of resonator is coupled to the source/load via magnetic

coupling with another loop, as shown in Figure 5. [Color

figure can be viewed in the online issue, which is avail-

able at wileyonlinelibrary.com.]

346 Dionigi and Mongiardo

International Journal of RF and Microwave Computer-Aided Engineering/Vol. 22, No. 3, May 2012

inner conductor at the other end. It is possible to define

the parameter coaxial length factor K ¼ d2p and, by adjust-

ing its length, it is possible to tune the capacitance value

and optimize the dimensions of the loop. The structure

can be modelled by a semi lumped model of the capacitor

and a rigorous analytical model of the loop inductance. In

the next subsection we will model the coaxial single loop

resonator to give an accurate prediction of resonant fre-

quency and quality factor.

Resonant Frequency ModellingReferring to Figure 4 we can compute the resonant fre-

quency by the classical formula:

fR ¼ 1

2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiLLoopCLoop

p (1)

By considering the external conductor as a single loop,

its inductance can be computed by the following formula

[15]:

LLoop ¼ lð2RLoop � RextÞ 1� k2

2

� �KðkÞ � EðkÞ

� �(2)

where K and E are complete elliptic integrals of the first

and second kinds and k is given by:

k ¼ 4RLoop RLoop � Rext

� �2RLoop � Rext

� �2 (3)

If the loop radius Rloop � Rext the following approxi-

mate formula can be used:

LLoop � RLoopl0lr ln8Rloop

Rext

� �� 2

� �(4)

From Figure 2 it is apparent that the coaxial cable is

left open at one end; by considering the cable cross

section sufficiently small, we can assume that it behaves

as an open circuited transmission line of length Len ¼2pRloop. It is noted that, apart for the open side fring-

ing capacitance, the coaxial cable adds the following

contribution:

CLoop � CLineLen K (5)

where Cline is the unit length capacitance of the coaxial

cable given by:

CLine ¼ 2pe0er

ln Rins

Rin

� (6)

and er is the dielectric constant of the insulator. When the

loop radius is smaller with respect to the external radius

the capacitance can be computed as illustrated in [16].

Quality Factor ModellingThe unloaded quality factor Q0 of the resonator depends

on three main factors: the conduction resistance of the

loop conductors RC, the radiation resistance Rrad, the

attenuation of the coaxial cable. Usually, these factors are

computed at the resonant frequency fR.The conduction resistance RC of the inductance loop

can be modelled by using the following equation [17]:

RC ¼ Len

2Rext

ffiffiffiffiffiffiffifRlpr

r(7)

Figure 3 Geometry of the coaxial capacitor section; we denote

by Rloop the loop radius, Rin the radius of the inner conductor,

Rins the inner radius of the external conductor, and Rext the exter-

nal conductor radius.

Figure 4 Loop equivalent circuit; we include in the equivalent circuit the loop radiation and conduction losses modelled as the resistan-

ces Rrad and Rc, respectively, and the capacitor losses modelled as Rcx; the loop inductance and coaxial capacitor are denoted, respectively,

by Lloop and Cloop.

A coaxial resonator for wpt 347

International Journal of RF and Microwave Computer-Aided Engineering DOI 10.1002/mmce

where l and r are, respectively, the permeability and the

conductivity of the external conductor of the coaxial

cable. The conduction quality factor QC can therefore be

computed from:

Qc ¼ 2p fRLLoopRc

(8)

A single circular loop shows a radiation resistance Rrad

given by the following equation [17]:

RRad ¼ 31170pR2

Loop

k2R

!2

(9)

where kR is the wavelength at the frequency fR. The radia-

tion quality factor Qrad can be computed by:

QRad ¼ 2p fRLLoopRRad

(10)

Let us now consider the losses introduced by the

coaxial section; by assuming a lossy coaxial, the input im-

pedance of a section of length KLen of low loss transmis-

sion line is given by the following equation:

Zin ¼ ZC coth cKLenð Þ (11)

where ZC is the characteristic impedance of the lossy

coaxial and c the complex propagation constant. We can

derive the expression for the equivalent series resistance

RESR of the capacitor as follows:

RESR ¼ Real ZC coth cKLenð Þ½ � (12)

From Eq. (12) the following expression holds for the

cable losses quality factor QESR:

QESR ¼ 2p fRLLoopRESR

(13)

In conclusion, we can recover the resonator unloaded

quality factor as follows:

Q0 ¼ 2p fRLLoopRC þ RRad þ RESRð Þ ¼

11

QESRþ 1

QRadþ 1

QC

(14)

Resonator Model Experimental TestTo test the resonator behaviour we have chosen a silver

plated SR-141 semi-rigid coaxial cable, filled with PTFE

insulator, with dimensions given in Table I. We have

made two resonators with radii Rloop ¼ 25.9 mm and

51.8 mm considering K ¼ 1. A simple test-fixture com-

posed of a resonator and a pair of coupling loops of diam-

eter 50 mm has been manufactured and it is shown in

Figure 5. The coupling loops and the resonator are coaxial

and the centre loops positions can be varied. We have

measured the insertion loss of the test fixture when the

distance between the coupling loops and the resonator is

70 mm. To find the peak frequency and the bandwidth of

the resonator a peak fitting function (Lorentzian) has been

adopted. The measured resonant frequency and quality

factor compared with the theoretical ones are given in

Table II showing a very good agreement. The test fixture

with resonator radii Rloop ¼ 25.9 mm at a distance

between the coupling loops of 50 mm has been simulated

in the band 100–200 MHz by a lumped elements (LE)

simulation [13]. The simple geometry considered allows

one to compute analytically loops inductance and mag-

netic couplings. The simulation and measurement results

are shown in Figure 6, from which is clear the very good

agreement between the computed and simulated values.

This confirms the accuracy of the resonator model.

The same structure has been simulated with an FDTD

simulator and surface currents and H fields are shown in

Figures 7a and 7b, respectively. It is worth to note that a

TABLE I Coaxial Cable Dimensions [18]

Cable Insulator 2Rext (mm) 2Rins (mm) 2Rin (mm)

SR-085 PTFE; er ¼ 2.1 tgd ¼ 0.0011 2.2 1.68 0.51

SR-141 PTFE; er ¼ 2.1 tgd ¼ 0.0011 3.58 3 0.92

SR-250 PTFE; er ¼ 2.1 tgd ¼ 0.0011 6.35 5.31 1.63

Custom Air; er ¼ 1 tgd ¼ 0 8 6 2

Figure 5 Test-fixture for the measurement of the coaxial reso-

nator. [Color figure can be viewed in the online issue, which is

available at wileyonlinelibrary.com.]

TABLE II Measured and Simulated SR-141 Resonators

R mm (MHz) (MHz) Q0sim Q0meas

25.9 132.92 132.88 388 388

51.8 57.14 57.35 347 382



resonant frequency of 133.66 MHz has been obtained

from the full-wave simulation, which is in good agree-

ment with the measured one.

RESONATOR Q FACTOR OPTIMIZATION

In this section we investigate the variation of the unloaded

Q factor and of the resonant frequency as function of the

loop radius and coaxial cable parameters to design a suita-

ble WPT resonator. We will show in the first subsection

only the results regarding full length (K ¼ 1) coaxial ca-

pacitor made of standard cables, whereas in the last sub-

section we will show the reduced length (K < 1) coaxial

airline capacitor characteristics.

Standard Cable ResonatorOnce the radius of the loop Rloop and the coaxial cable

size and losses are determined, the resonant frequency and

quality factor can be computed. We have chosen three

types of commercial semi-rigid cables as test cases, with

the cables identification codes and cross section dimen-

sions, given in Table I.

In Figure 8 the resonant frequency is plotted as func-

tion of the loop radius. It is clear that only minor differen-

ces in the resonant frequency occur for different cable

type. This is mainly due to the fact that all the cables

Figure 6 Measured and simulated (with lumped elements) val-

ues of the scattering parameters for a resonator made by SR141,

with R ¼ 25.9 mm radius. [Color figure can be viewed in the

online issue, which is available at wileyonlinelibrary.com.]

Figure 7 Full-wave simulation of a resonator realized withSR141 and R ¼ 25.9 mm radius; illustrated are the: (a) surface current and

(b) H field. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]



have the same dielectric insulator and the same conductors

radius ratio.

In Figure 9 are reported the unloaded quality factor for

resonators realized with different coaxial cables. It is

apparent that higher values of Q are obtained for small

radii. Nevertheless, these values of Q are not yet satisfy-

ing for WPT applications.

It is convenient to investigate the sources for the Qdegradation: in Figure 10 we have reported the various

types of contributions. From the latter figure it is apparent

that the main Q factor reduction is due to the cable losses,

thus it is clear that this parameter is critical for the reso-

nator performances.

From Figure 10 it is worth to note that the radiation fac-

tor Qrad decreases as the loop radius. This behaviour is pre-

dicted in Eq. (10). Combining Eqs. (9) and (10) we have:

QRad ¼ 2k5RLLoop31170 c pR4

Loop

; (15)

where c is the free space propagation speed. From Eqs.

(1) and (2) and Figure 8 it is clear that a reduction of the

radius Rloop diminishes the loop inductance Lloop and

increases the resonance frequency fR. As a consequence, a

reduction of the wavelength kR and of Lloop is obtained,

this causing the numerator of Eq. (15) to reduce faster

than the denominator, decreasing the Qrad factor.

Variable Length Airline ResonatorTo increase the resonator quality we can operate in two

ways: the first way is to reduce the cable insulator losses; the

second is to shorten the entire coaxial cable. The first action

will reduce the dielectric losses of the resonator, whereas the

second will reduce both the dielectric and conduction losses

inside the coaxial cable. A theoretical result is shown in Fig-

ure 11 where an airline with same cross section of a SR141

is considered. The Q factor of the cable is almost twice of

the standard cable and it shows a maximum for K ¼ 50%.

RESULTS

In this section we will show the same examples of resona-

tors and a custom airline resonator construction used to

realize a high efficiency WPT system.

Coaxial Airline ResonatorTo test the performances of a coaxial airline resonator we

have manufactured and measured a 200 mm diameter

loop with K ¼ 50% made of copper whose cross-section

Figure 8 Resonant frequency as function of loop radius for

different standard coaxial cables. [Color figure can be viewed in

the online issue, which is available at wileyonlinelibrary.com.]

Figure 9 Unloaded quality factor as function of loop radius for

different standard coaxial cables. [Color figure can be viewed in

the online issue, which is available at wileyonlinelibrary.com.]

Figure 10 SR141 resonator Q factors as function of loop ra-

dius. [Color figure can be viewed in the online issue, which is

available at wileyonlinelibrary.com.]

Figure 11 Unloaded quality factor of SR141 airline with

reduced length coaxial. [Color figure can be viewed in the online

issue, which is available at wileyonlinelibrary.com.]



dimensions are given in Table I. In this case the insulator

is air with er ¼ 1 and tgd ¼ 0. To insulate the internal

conductor from the external one a polyethylene sleeve of

0.07 mm thickness has been used. Figure 12 shows the

resonant frequency and quality factor as a function of the

loop radius and of the coaxial length factor K.From the simulated results of Figure 12 is clear the

considerable improvement in the quality factor obtained

with this resonator; it is also possible to observe that the

limiting components are now given by the radiation and

conduction losses.

The resonator has been measured obtaining a resonant

frequency of 30.8 MHz and a Q factor of about 910 instead

of the simulated one of 1450. This significant difference is

partially explained by considering that our in-house experi-

mental realization presents some deficiencies; in particular,

the distance between the central conductor and the shield

may vary consistently in our implementation. Nevertheless,

the high Q value achieved confirms the high quality of the

resonator structure. Moreover note that, due to the free

insertion of the inner conductor, it is also possible to adjust

the resonator radius, by sliding the internal conductor,

hence allowing accurate resonant frequency tuning.

WPT SystemOnce the validity of the model has been tested, we have

built and measured a simple WPT system. We have con-

sidered the source and load impedance of the value of 50

X. Under this assumption it is possible to define the effi-

ciency of the WPT as:

g ¼ S21j j2 � 100 (16)

We have realized the WPT system by using a couple of

resonators, as described in the previous section. The system

on the measure bench is shown in Figure 13, whereas the

WPT system measured efficiency is reported in Figure 14.

We have obtained an efficiency of 95% at a 0.5 diameter

distance between the resonators, 87% at a 1.5 diameter dis-

tance, and 40% at three diameter distances. Naturally, as

illustrated in [13], it is possible to improve these values by

adding additional resonators in between.

Figure 12 In the upper part (A) the calculated resonant fre-

quency is plotted as a function of the resonator radius; in the

lower figure (B) the computed quality factor of single loop reso-

nator, made by a custom coaxial cable with dimensions given in

Table I, is reported as a function of the resonant frequency (10 <

Rloop < 500 mm). Note that K is the coaxial length factor and K

¼ 100% corresponds to a full circumference length, whereas K ¼50% represents half circumference. [Color figure can be viewed

in the online issue, which is available at wileyonlinelibrary.com.]

Figure 13 WPT system on the measure bench. [Color figure

can be viewed in the online issue, which is available at

wileyonlinelibrary.com.]

Figure 14 Measured efficiency as function of frequency at dif-

ferent resonators distances D for coaxial resonator WPT system.

[Color figure can be viewed in the online issue, which is avail-

able at wileyonlinelibrary.com.]



CONCLUSIONS

We have presented a novel loop resonator for WPT sys-

tems. The resonator structure is extremely simple, neverthe-

less it satisfies the demands for compactness and high Q of

the WPT systems. In addition, it can be manufactured by

using any low loss coaxial cable. A simple modelling of

the single loop has been described providing good agree-

ment between simulated and measured results. An entire

WPT system has been built and measured confirming the

suitability of the structure for this type of applications.

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BIOGRAPHIES

Marco Dionigi has received the lau-

rea degree (110/110 cum laude) in

Electronic Engineering from the

University of Perugia. He achieved

at the same university the title of

Ph.D. In 1997 he became Assistant

Professor at the Faculty of Engi-

neering of the University of Peru-

gia. He took part in several research

project regarding the development of software tools for

waveguide and antenna fullwave simulation, the develop-

ment of permittivity and moisture microwave sensors,

the development of a SAR and ultrawideband antennas.

He was coauthor of a paper awarded of the ‘‘Young

Engineers Prize’’ at the European Microwave Conference

2005 in Paris. He is now involved in the study and

development of high efficient wireless electromagnetic

power transfer for industrial applications. He is author

of more than 50 papers on international journal and

conferences.

Mauro Mongiardo has received the

laurea degree (110/110 cum laude) in

1983, Ph.D. in 1991. He has been assist-

ant Professor of Electromagnetic Fields

from 1988, associate Professor from

1991 and full Professor from 2001. The

scientific interests of Mauro Mongiardo

have concerned primarily the numerical

modeling of electromagnetic wave propagation both in closed

and open structures. His research interests have involved also

the CAD of microwave components and antennas. His main

scientific contributions are in the modeling of microwave

propagation with modal techniques, integral equations and

hybrid numerical techniques. Other contributions have been

made in the analysis of complex electromagnetic problems via

a rigorous network approach. More recently he is involved in

wireless power transfer researches. He has served in the Tech-

nical Program Committee of the IEEE IMS from 1992; from

1994 he is member of the Editorial Board of the IEEE Trans-

actions on Microwave Theory and Techniques.



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a novel coaxial loop resonator for wireless power transfer

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