> FOR CEFC 2012DIGESTS, 1(c), 6(a), 11(b)<1Optimization of the Eddy Current Testing
Hiroki KIKUCHIHARA1, Iliana MARINOVA2, Yoshifuru SAITO1,
Manabu OHUCH3, Hideo MOGI3 and Yoshiro OIKAWA3
1

Graduate School of Hosei University, Tokyo 184-8584, Japan
2
Technical University of Sofia, Sofia 1756, Bulgaria
3
Denshijiki Industry Co., Ltd, Tokyo 115-0051, Japan

Eddy current testing (ECT) is one of the most representative nondestructive testing methods for metallic materials, parts, structuresand so on. Operating principle of ECT is based on the two major properties of magnetic field. One is that alternating magnetic field
induces eddy current in all of the conducting materials. Thereby, an input impedance of the magnetic field source, i.e., electric source,
depends on the eddy current path. Second is that the magnetic field distribution depends only on the exciting but also the reactive
magnetic fields caused by the eddy currents in targets. Former and latter are the impedance sensing and magnetic flux sensing types,
respectively.
This paper concerns with an optimization of a new magnetic flux sensing type sensor named "∞ coil" whose exciting and sensing
coils are composed of the ∞ shape and finite length solenoid coil wound ferrite bar, respectively. Development of this ∞ coil fully
depends on the 2D and 3D finite elements methods. According to the simulation results, we have worked out two types of ∞ coils.
Practical experiments reflect on the validity of both simulation and design aims, quite well.
Index Terms— Eddy current, Nondestructive testing

I. INTRODUCTION

II. NEW ECT SENSOR COIL

odern engineering products such as air-plane,
automobile, smart building, high speed train and so on
are essentially composed of metallic materials for forming theshape of product, suspending the mechanical stress andconstructing the structural frames. In particular, the masstransportation vehicles, e.g. large air plane, high-speed train,
express highway bus and so on, carrying a large number ofpeople are required ultimately high safety as well as reliability.
To keep the safety, nondestructive testing to the metallic
materials is one of the most important technologies because
most of the structure materials are composed of the metallic
materials.
Various nondestructive testing methods, such as eddy current
testing (ECT), electric potential method, ultrasonic imaging
and x-ray tomography, are currently used. Among thesemethods, ECT needs not complex electronic circuits and directcontact to target. Furthermore, a target whose major frameparts are composed of conductive metallic materials can beselectively inspected by ECT [1,3].
In this paper, a new ECT sensor coil named "∞ coil" is
proposed. Development of this ∞ coil fully depends on the 2D
and 3D finite elements methods and optimize and the eddy
current testing.
The most important key idea of the ∞ coil is that setting thesensor coil in the lowest magnetic field space caused by the
exciting coils, because this sensor coil is capable of catch only
the magnetic fields caused by the detour eddy currents flowing
around a defect. This paramount important key idea has beenyields intensive 2D finite elements simulation. After that 3D
finite elements simulation has been carried out to check up thevalidity of this key idea.
Thus, we have succeeded in innovating and optimizing a newhigh sensibility eddy current sensor "∞ coil" by means of the2D and 3D finite elements simulations [2,3].

1) Operating principle of ECTOperating principle of the separately installed sensing coiltype is fundamentally based on that the sensing coil catches
the magnetic field intensity variation caused by the detour
eddy currents flowing around a defect in the target metallic
materials.
To realize this, three methodologies may be considered. The
first detects the variation of entire magnetic fields caused byboth exciting and eddy currents. In this case, the sensing coildetects the magnetic fields distribution caused by the detour
eddy currents around defects among the entire magnetic fields.
Second is that the sensing coil surface is installed in aperpendicular direction to an exciting coil surface. This meansthat the sensing coil never induce an electric voltage caused by
the exciting fields because the surface of sensing coil always
parallels to the exciting magnetic fields, so that this sensing
coil is capable of selectively catching the magnetic fields
caused by the detour eddy currents around a defect. This typeof sensor has high liftoff characteristics compared with thoseof the first one. The final third one is that the sensing coil is
set on the zero exciting magnetic fields space in addition to thesecond one coil layout. This final one is capable of catching
only the magnetic fields caused by the detour eddy currents
around a defect, and has been innovated in this paper by asequential intensive finite elements simulation.
2) 2D simulation
At the beginning of our new sensor project, to concentrate
the magnetic fields, as shown in Fig.1, the two exciting coils
whose magneto motive forces becomes in additive were
employed.

M

> FOR CEFC 2012DIGESTS, 11(b) <23) 3D simulation of the ∞ coil
To evaluate and validity check of the ∞ coil performance,
we employed a 3D simulation model shown in Fig.3.

Fig.1 Two exciting coils
To decide a location of sensing coil whose surface isorthogonal to the exciting magnetic fields, we have carried out
a lot of 2D finite elements simulations. As a result, it is foundthat there is nearly zero exciting fields space between the two
exciting coils. This leads to our ∞ coil. Since the zero exciting
magnetic fields condition in addition to the orthogonal coilsurface layout to the exciting magnetic fields enhances thesensibility of ∞ coil in an ultimately.
Fig 2(a) shows an exciting magnetic field intensity
distribution. In this figure, it is possible to find the zeromagnetic fields space between the two parallel exciting coils.
According to this simulation result, we put on a ferrite bar at
the bottom surface of the two exciting coils in order toenhance the sensibility of sensor coil. As shown in Fig.2(b),setting the ferrite bar hardly disturb the exciting magnetic
fields intensity distribution. Thus, winding around the sensor
coil around the ferrite core leads to an ultimately optimized
high sensibility ECT sensor, i.e., ∞ coil.

Fig.3 3D simulation model of the ∞ coil
Table 1 various constants used in the 3D simulation.
Exciting coil
Coil outer diameter
22.4mmCoil inner diameter
20mm
Coil length
10mm
Number of turn75
Input current(peak)
250mA
Frequency
256kHzSensing coil
Coil outer diameter
1.4mm×2.4mm
Coil inner diameter
1mm×2mm
Coil length
6mm
Number of turn100
Axis core
JFEferrite_MB1H _23℃

Table 1 lists various constants used in the 3D simulation.
The eddy currents in a plane target located under the two
exciting coil surfaces are shown in Fig. 4, where the two
adjacent exciting coils side to the no-defect, 0 degree, 90
degree and 45 degree line defects are shown in Figs.
4(a),4(b),4(c) and 4(d), respectively.

(a) Without ferrite bar(a) No defect(b) With ferrite bar
Fig. 2 Magnetic fields intensity

> FOR CEFC 2012DIGESTS, 11(b) <3(b) 0 degree defect to the two adjacent exciting coils(b) 0 degree defect to the two adjacent exciting coils(c) 90 degree defect to the two adjacent exciting coils(c) 90 degree defect to the two adjacent exciting coils(d) 45 degree defect to the two adjacent exciting coils
Fig.4 Eddy currents in a plane metallic target.

(d) 45 degree defect to the two adjacent exciting coils
Fig.5 Magnetic flux density vector distributions in the ferritebar

Fig.5 shows the magnetic flux density vector distribution
corresponding to that of eddy currents in Fig. 4. Observe themagnetic flux density vector distributions in Fig.5 reveals thatthe sensing coil wound around the ferrite bar could be induced
the electric voltages not the cases (a), (b), (c) and (d).

(a) No defect

The induced voltages of the sensor coil under the conditions(a)-(d) in Figs.4 and 5 are shown in Fig.6. By observing theinduced voltages in Fig.6 reveals that the case (d) yields thehighest voltage.

Fig.6 Induced voltages in the sensor coil

> FOR CEFC 2012DIGESTS, 11(b) <4III. EXPERIMENT
1) Tested target peace and ∞ coil
We employed two coppers plate having 1mm thickness astarget metal plates. One has no defect and the other has a line
defect having 2mm width. Also, we worked out a prototype of
the ∞ coil. Table 2 lists various constants of the prototype ∞coil. We used two exciting coils and one sensing coil having
ferrite core. Fig. 7 shows a picture of the prototype ∞ coil. The
physical dimensions of this prototype ∞ coil are corresponding
to the 3D simulation model shown in Fig.3.
Fig.8 Measured voltages of the practical sensing coil.
Table 2 Various constants of the prototype ∞ coil.
Exciting coil

Sensing coil

Conductor length
4.7m
Diameter of conductor
0.4mm
Coil outer diameter
23mm
Coil inner diameter
20mm
Coil length
10mm
Number of turn75
Number of coil layers
3
Number of coils
2
Conductor length
60cm
Diameter of conductor
0.1mm
Axis core
Ferrite bar (MnZn)
Coil outer diameter
1.4mm×2.4mm
Coil inner diameter
1mm×2mm
Coil length
6mm
Number of turn100
Number of coil layers
2

IV. CONCLUSIONAs shown above, we have succeeded in innovating andoptimizing a new high sensibility eddy current sensor " ∞coil" by means of the 2D and 3D finite elements simulations.
Thus, we have elucidated that the finite elements
simulation technology is one of the most effective andpowerful tools to research and develop the newelectromagnetic devices. All of the 2D and 3D finite elements
simulations were carried out by the finite element package"femtet" produced by Murata Software Co. Ltd.
REFERENCES
[1]
[2]

[3]

Fig.7 Picture of the prototype ∞ coil.
2) Experimental results
Fig.8 shows the measured voltages corresponding to that ofcases in Figs. 4 and 5. Even though the peak measured voltage
is somewhat smaller than those of Fig.6, sensing
characteristics of the prototype ∞ coil are well corresponding
to that of Fig. 6.

I.Marinova, S.Hayano and Y.Saito, Ployphase eddy current testing,
Journal of Applied Physics, Vol. 75, No.10, pp. 5904-5906, 1994.
N.Burais and A.Nicolas, Electromagnetic field analysis in remote fieldeddy current testing systems, IEEE Transactions on Magnetics, Vol.25,
No.4, pp.3010-3012, 1989.
S.McFee and J.P.Webb, Automatic mesh generation for h-p adaption,
IEEE Transactions on Magnetics, Vol.29, No.2, pp.1894-1897, 1993.

> FOR CEFC 2012DIGESTS, 11(b) <5

Fluctuation Frequency Analysis of the Barkhausen Signals
under Static and Dynamic Stresses
Jun KAWAZOE1, Iliana MARINOVA2, and Yoshifuru SAITO1
1

Graduate School of Hosei University, Tokyo 184-8584, Japan
2
Technical University of Sofia, Sofia 1756, Bulgaria

Ferromagnetic materials are widely used for various artificial products such as cars, trains, ships and so on. Because of its
mechanical property, iron steel is most popular in use for frame materials. Nondestructive testing of iron steel is an extremely
important way to maintain their mechanical reliability. As is well known fact, the Barkhausen signals are emitted from only theferromagnetic materials having magnetic domain structures. Also this signal varies depending upon their past mechanical as well asradioactive stress histories.
In the present paper, we have applied a generalized frequency fluctuation analysis to the Barkhausen signals to detect the various
mechanical stresses. Surprisingly, it has been succeeded in clarifying that application of our frequency fluctuation analysis to the
Barkhausen signals makes it possible to detect the two kinds of mechanical stress.
Index Terms—Barkhausen signal, Frequency fluctuation, Signal processing, Least squares

V. INTRODUCTION

B

arkhausen signal is popularly observed in the
ferromagnetic materials composed of the magnetic
domain structures, e.g. iron, nickel, cobalt and garnet, when
they are magnetizing. Also, it is well known that the
Barkhausen signals are very sensitive to the physical external
input, such as mechanical stress and radioactive damage, tothe ferromagnetic materials [1].
According to the past researches concerning to a
relationship between the Barkhausen signal and appliedmechanical stress, it has been revealed that Barkhausen signals
are very sensitive to the mechanical stress and radioactive
damage but any deterministic regularity has not been found.
Recently, only a macroscopic regularity has been reported
by means of a 1st order frequency fluctuation analysis
approach [2]. Principal purpose of this paper is thatconventional frequency fluctuation analysis employing only
1st order fluctuation is generalized to the frequency fluctuation
analysis employing the nth order fluctuations.
As a result, we succeed in extracting the distinct difference
between the stressed and not stressed Barkhausen signals.
Also, we succeed in visualizing in the most clear manner on
the four dimensional space, whose coordinates are composedof the terms representing the higher order frequency
fluctuation terms of Barkhausen signals.
One of the most famous frequency fluctuations is the 1/f
frequency fluctuation, which can be observed in most of thenatural phenomena such as natural wind, sea water waves,river flow sound and so on gives a healing effect to thementalities via human sensibilities [3].
Conventional 1/f frequency analysis is that application of
the 1st order least squares to the both Fourier power spectrum
and frequency extracts the 1st order frequency fluctuation, i.e.,
Log of Fourier power spectrum is approximated by Log of a0+ a1f, yields a 1st order frequency fluctuation characteristic,

where a0 and a1 are the 0th and a 1st order frequency
fluctuation terms, respectively. If the frequency fluctuation
term a1 takes a1=1, then we have the 1/f frequency fluctuation.
On the other side, we generalize this conventional 1st orderfrequency fluctuation to the nth order frequency fluctuation
characteristics, i.e., Log of Fourier power spectrum isapproximated by a Log of a0 + a1f+a2f2+ ...+ anfn, where a0, a1,
a2, ,..., an are the 0th, 1st 2nd, ... ,nth order frequency fluctuation
terms, respectively. Careful examination of the coefficients a0,
a1, a2, ,..., an leads to the precise frequency fluctuation
characteristic of the Barkhausen signal.
Second important key point of our approach is how tovisualize the dependency of the 0th, 1st, 2nd, ... ,nth orderfrequency fluctuation terms to the externally applied forces.
According to our experimental results on this research, mostof the frequency fluctuation characteristics are sufficiently
represented up to the 4th order terms. So that let thenormalized 1st, 2nd, 3rd, 4th order frequency fluctuation
coefficients be respectively the coordinate values on the x-, y-,z-axes, and point color, then up to the 4th order frequency
fluctuation characteristics locate the three dimensional spacecoordinate position and point color. Thus, comparison among
the different specimens of this diagram visualizes an each of
the distinct characteristics depending on their mechanicalstress conditions.

VI. FREQUENCY FLUCTUATION ANALYSIS
1) Basic equations
Let us consider an arbitrary signal g(t) and its Fourier power
spectrum G(f), and take the logarithm of both Fourier
spectrum G(f) and frequency f. Plot the log G(t) on the log fcoordinate represents a frequency characteristic of the signal
g(t) on the frequency domain, i.e., the frequency characteristicof the original signal is represented on a x-y plane coordinate
system whose horizontal x- and vertical y-axes are

> FOR CEFC 2012DIGESTS, 6(f)<6corresponding to the logarithms of Fourier power spectrum
G(f) and of frequency f, respectively.
In order to represent a global frequency characteristic of theoriginal signal g(t) in frequency domain, apply a following
power series function:h  f   a0  a1 f  a2 f 2  ...  an f n .

(1)This x-y plane makes it possible to evaluate the higher orderfrequency fluctuation analysis. The coefficients a0, a1 a2, ,..., anin (1) are determined by least squares as
1

A  CT C  CT Y .

(2)A superscript T denotes a matrix transpose; the vectors A,
Y and matrix C are respectively given byA   a0

a1 . an  ,T(3)Y  h  f0  h  f1  . h  fm  ,
T

1

1
C  1

.
1


f0
f1

f 02 .
f12 .

f2
.
fm

f 22 .
. .
f m2 .

f 0n 
f1n 
f 2n  ,. 
f mn 

(4)(5)

m  n.
In (3)-(5), a number of equations m, i.e. number of thesampled frequencies f0, f1 f2, ,..., fn, always greater than those of
unknowns n, i.e., number of the coefficients a0, a1 a2, ,..., an. In
most case, this condition is satisfied because of the number ofsampled frequency m is much greater than those of the order nof the function h(f) in (1).
2) Classical 1st order frequency fluctuation
Fig. 1 shows a Barkhausen signal measurement devices.
The tested specimens are the silicon steels with the 0.35mm
thickness, 30mm width and 100mm length. The tested
specimen is put on the upper two head surfaces of U shape
ferrite core wound the 300 turns exciting coil. The specimen in
Fig.1 is excited by flowing a 1A sinusoidal alternating current
through this exciting coil.
Fig. 2 shows a typical frequency characteristic of a
Barkhausen signal under no stress.

Fig. 1. Barkhausen signal measurement devices.

Fig. 2. An example of typical frequency characteristic of a Barkhausen
signal under no stress.

The frequency characteristic in Fig. 2 is obviously divided
into two frequency regions. One is lower frequency region
whose frequency fluctuation can be approximated by f-2.26
derived from 1st order approximation in (1), and the otherhigher one is a dispersing signal whose frequency fluctuation
can be approximated by f 0, i.e. white noise.
On the other side, Fig. 3 shows a typical frequency
characteristic of Barkhausen signal under 3kg stressed.
The frequency characteristic in Fig. 3 is also divided into
two frequency regions. One is lower frequency region whose
1st order frequency fluctuation can be approximated by f-1.69,
and the other higher one is a dispersing signal whose
frequency fluctuation can be approximated by f 0.

Fig. 3. An example of typical frequency characteristic of a Barkhausen
signal under 3kg stressed.

Consideration of the difference between the first orderfrequency fluctuations f-2.26 and f-1.69 suggests that an
application of the 3kg stress to the tested silicon steel changes
the steep 1st order gradient -2.26 to the smoothly gradient 1.68. This result had been verified by the 1st order frequency
fluctuation analysis of the 30 distinct sample specimens [2].
Thus, it is possible to detect the stress on the ferromagnetic
materials by the classical 1st order frequency fluctuation
analysis.
However, the difficulty is still remaining, i.e., how todecide the frequency range for which to be computed thefrequency fluctuation characteristic. This problem is solved inthis paper by employing one of the optimization methods:, i.e.,k-means method based on Euclid distance [4].

> FOR CEFC 2012DIGESTS, 6(f)<3) Stress visualization
We have carried out the random stresses visualization inmuch the same as described in Section 2.2. These stresses
were applied to the specimen by handling the random weights
by a string as shown in Fig. 4 while the specimen is fixed bywood blocks to keep the same flux path conditions.
Fig. 5 shows one of the frequency characteristic of the
Barkhausen signals under stress. The frequency characteristic
in Fig. 5 is also divided into the two frequency regions. One islower frequency region and the other is higher one. The higher
frequency region is a tendency same as white noise. Thereby itis essential to reduce influence of the higher one to judge
under stress otherwise no stress.
Fig. 6 shows that Fig. 5 is divided into two groups by the kmeans method based on Euclid distance [4].
In much the same way as the 1st order frequency fluctuation,
the left side group is a meaningful frequency region. Thereby
we have carried out the stresses visualization only applying
the higher order frequency fluctuation analysis to thismeaningful frequency region in Fig. 6. Because of too small
coefficient of over 4th order terms, a frequency fluctuation
characteristic under stress along with the meaningfulfrequency region is obtained by up to the 4th order power
series function in (1). To check the reproducibility of thecoefficients a1 , a2 , a3 , a4 , we have 10~12 times independently
measured and computed these coefficients a1 , a2 , a3 , a4 in (1)to the same specimen. After normalizing all of the coefficients
a1 , a2 , a3 , a4 to be the values between 1 and 0, let thenormalized coefficients a1 ' , a2 ' , a3 ' , a4 ' be respectively
corresponded to the x-, y-, z-axes values and point color, thenthis makes it possible to visualize a dispersion characteristic of
the coefficients a1 ' , a2 ' , a3 ' , a4 ' . Fig. 7 shows a dispersion
characteristics of coefficients a1 ' , a2 ' , a3 ' , a4 ' .
The magnitude of 4th order normalized frequency
fluctuation coefficients in Fig. 7 is rainbow colored.
Fig. 8 shows the normalized frequency fluctuation
coefficients in terms of the black and light gray colors. The
black and light gray colors are corresponding to the specimens
under no stress and stress, respectively.

7

Fig. 5. One of the frequency characteristic of
the Barkhausen signals under stress.

Fig. 6. Classified into two groups denoted lightand dark colors by means of the k-means method basedon Euclid distance.

(a) x-y plane

Fig. 4. The stresses are applied by hanging the weights.
(b) x-z plane
Fig. 7. The dispersion characteristics of coefficients.

> FOR CEFC 2012DIGESTS, 6(f)<8(a) x-y plane(b) x-z plane
Fig. 9. Classified into two groups of the normalized
frequency fluctuation coefficients difference between the stressed (light gray)and no stressed (dark gray) groups by means of the k-means
method based on Euclid distance. The centroid of each of the 2clusters is shown by a black point.

Observation of the dispersion characteristics of the stressed
specimens suggests that the normalized coefficientsa1 ' , a2 ' , a3 ' , a4 ' in (1) do not disperse but focus on a relatively
small area centered at (0.2,0.8,0.1). On the contrary, thenormalized coefficients a1 ' , a2 ' , a3 ' , a4 ' in (1) of the nostressed specimens disperse the vicinity of a centroid(0.6,0.2.0.6).
This means that the externally applied stress suppresses thedispersion and confines them into the small space in Fig. 9.
(b) x-z plane
Fig. 8. The normalized frequency fluctuation coefficients difference between
the stressed and no stressed groups.

The dispersion characteristics of coefficients divided in the
stressed group and no stressed group in Fig. 8 are clustered by
the k-means method based on Euclid distance as shown in Fig.
9.

VII. CONCLUSION
Previously, we have succeeded in detecting the appliedstress on the silicon steel by means of the 1st order frequency
fluctuation method [2].
This paper has generalized this 1st order analysis method tothe nth order frequency fluctuations analysis method.
Up to the 4th order frequency fluctuation characteristics
have been visualized in a three-dimensional space. Further it
has been revealed that each of the distinct differences among
the specimen not stressed and stressed is clarified by the kmeans method based on Euclid distance.
REFERENCES
[4]
[5]

[6]

[7]

(a) x-y plane

R M. Bozorth, Ferromagnetism, IEEE PRESS, p. 462, 1993.
S. Nojima and Y. Saito, Application of Frequency Fluctuation to
Barkhausen Signals and its Application, J. Magn. Soc. Jpn., 35, 380-385,
2011.
M. Teranishi, K. Maruyama, S. Hayano, and Y. Saito: Visualization of
1/f Frequency Component in Dynamic Image of Natural Phenomena,
The Visualization Society of Japan, B108, July , 2005.
J.B.MacQueen, Some Methods for classification and Analysis ofMultivariate Observations, Proceedings of 5th Berkeley Symposium on
Mathematical Statistics and Probability, University of California Press,
pp. 281–297, 1967.

> FOR CEFC 2012DIGESTS, 6(f)<9Optimization of the Secondary Resonant Contactless Transformer
Tatsuya OHASHI 1, Yoshifuru SAITO 1, and Iliana MARINOVA2
1

Graduate School of Hosei University, Tokyo 184-8584, Japan
2
Technical University of Sofia, Sofia 1756, Bulgaria

Contactless power supplier is composed of a transformer having the distinct primary and secondary coils separated by air gap.
Because of the electromagnetic compatibility problem, it is essential to keep the leakage magnetic fields around the contactless power
supplier as low as possible. Previously, we have reported that an application of the discrete wavelets transform to the magnetic field
vector distributions around the transformer having open magnetic flux paths clarifies that the soup plate like ferrite cores embedding
spirally wound coils gives a far excellent magnetic field vector distribution compared with those of the conventional U shape cores. As
a result, we have succeeded in obtaining one of the reasonable core shapes by observing the wavelets spectra of measured magnetic
field vector distributions. Furthermore, it has been revealed that a tested trial transformer gives nearly 80 percent power transmission
efficiency even though the primary and secondary coils are separated by 10mm air gap. This paper is one of the success researchsolutions to overcome the specific absorption rate (SAR) problem based on the finite elements and optimization methodologies. Namely,
a contactless flat shaped transformer whose primal and secondary ferrite cores are separated has been successfully optimized by
combining the finite elements with linear programming optimization approaches.
Index Terms— contactless transformer, magnetic field visualization, SAR optimization

VIII. INTRODUCTION

D

evelopment of modern semiconductor technology makes
it possible to realize the small and light weight electronic
devices equipped with a large variety of smart functions
such as smart cellar phone, ultra mobile computers, etc.
Although these electronics provide the highly efficient job
environment, entertainments and convenient electronic
consumer life, environments around human life are filling up
with the electromagnetic fields. Particularly, because of themany electric power suppliers to energize the electronic
products, it is essentially accompanied the electric power lines
jangle, which leads to SAR problems.
One of the solutions of this electromagnetic compatibility
problem in the human life environment is to work out thecordless contactless power suppliers having low SAR level.
This paper concerns with development of a transformer
composed of the separated primary and secondary cores by air
gap. Further, this paper is one of the success research solutions
to overcome the SAR problem based on the finite elements
and linear programming optimization methodologies. Namely,
a contactless flat shaped transformer whose primal andsecondary ferrite cores are separated has been successfully
developed by the modern numerical computation approaches.
To realize the contactless transformer exhibiting low SARlevel, we employ flat shape primal and secondary ferrite cores.
Also, to obtain a high efficiency, we employ a secondary
resonant circuit between the secondary leakage inductance andexternally attached capacitor [1].
This transformer having the open magnetic flux paths
between the primary and secondary core is of the paramountimportant part to realize the contact-less power suppliers.
Previously, we have reported that an application of thediscrete wavelets transform to the magnetic field vector
distributions around the transformer having open magnetic
flux paths clarifies that the soup plate like ferrite cores
embedding spirally wound coils gives a far excellent magnetic
field vector distribution compared with those of theconventional U shape cores. Further, it has been revealed thatthe flat transformer employing soup plate shape ferrite coresand spirally wound coils confines the magnetic fields at thecenter of the flat transformer, which minimizes the magnetic
fields around the transformer [2,3].
In this paper, we have carried out an intensive finite elements
simulation to evaluate the leakage magnetic fluxes, and to
optimize the core size, core shape, number of turns, diameterof conductor and the circuit parameters of the transformer. As
a result, it is revealed that numerical and experimental studies
employing a secondary resonant technique reveal that our flattransformer with 1cm air gap and 0.69 coupling factor iscapable of transmitting 79 percent of the input power to thesecondary circuits [4].

IX. FUNDAMENTAL CHARACTERISTICS
1) Core shape design
In the contactless power suppliers, the primal and secondary
distinct magnetic cores are always used to transmit electric
power through air gap. Further, high frequency drive of thetransformer is always employed in order to reduce the weightof the magnetic cores.

Fig. 1

A tested soup plate like ferrite core transformer.

One of the magnetic materials having good high frequency
characteristic is a ferrite so that our contactless transformer
has employed the two distinct flat shape ferrites as shown in

> FOR CEFC 2012DIGESTS, 6(f)<10

Fig.1. The primal and secondary magnetic cores are the same
core whose shape is like a soup plate filling up the spirally
wound coils instead of soup. To evaluate the optimum coresize, number of turns, and the circuit parameters of thetransformer, we have carried out an intensive finite elements
simulation.

When we connect each of the terminals in Fig.3 to additive
polarity as shown in Fig. 4(a), it is possible to obtain thefollowing relationship

2) The magnetic field vector distribution
Fig.2 shows a measured magnetic field vectors distribution
at some instance of the tested flat transformer under the 1Ω
pure resistive load and secondary resonant conditions.
Although a relatively large air gap of 1cm between the
primary and the secondary cores, Fig.2 reveals that major
magnetic fluxes are concentrated at the center of flat cores.
This suggests that our flat cores keep the leakage magnetic
fluxes in small, i.e., keep the low level of SAR.
Further, it has been possible to observe the significant
magnetic field vectors as shown in Fig.2, even thoughmagnetic fields caused by the secondary load current cancel
the primal magnetic fields.

where La, L1, L2 and M are the additive-, primary self-,
secondary self- and mutual inductances, respectively.

La  L1  L2  2M ,

Fig.4

(1)

Series connection of the primary and secondary coilsin Fig. 3.

Also, when we connects each of the terminals in Fig. 3 tosubtractive polarity as shown in Fig. 4(b), we have

Ls  L1  L2  2M ,Fig.2 Magnetic field vector distribution of the tested transformer under the1Ω pure resistive load and secondary resonant conditions.

3) Coupling factor
A coupling factor κ is one of the most importanttransformer characteristics, which indicates smallness of theleakage magnetic field vectors between the primary andsecondary coils. Namely, as possible as large coupling factor κmeans as possible as small leakage magnetic field vector
distribution around the transformer.
Let us consider a simplified circuit model of the transformer
shown in Fig. 3 to evaluate the coupling factor κ. The dots '∙'shown over the primary and secondary coils in Fig.3 indicate
the positive induced voltages at each of the coil terminals.
awhere Ls is a subtractive-inductance.
Further, the primary and secondary self-inductances can bemeasured independently. Thereby, combination of (1) and (2)leads to the mutual inductance M and coupling factor, which
are respectively given by the following equations

La  Ls
,
4
M
k
.
L1 L2
M 

(3)

Table 1 lists the measured inductances La, Ls, L1, L2 andcoupling factor κ changing the air-gap length between the
primary and secondary core surfaces.
TABLE I Coupling factor of the tested transformer employing flat shapecores measured at 30kHz.

c
●

Fig.3

(2)

Gap[mm]

0

1

3

5

7

10

L1[μH]

578.6

348.2

231.1

181.6

169.9

133.9

L2[μH]

572.7

348.1

229.4

181.0

168.3

133.3

Ls[μH]

2297.4 1358.2

881.8

669.4

617.6

450.8

Lo[μH]

16.9

26.1

41.4

56.0

61.1

84.3

Κ

0.99

0.96

0.91

0.84

0.82

0.69

●

b
d
Simplified circuit model of a transformer to evaluate the coupling
factor κ.

> FOR CEFC 2012DIGESTS, 6(f)<11

The results listed in Table 3 suggest that the tested
transformer is capable of keeping the good coupling factors κ≈ 0.7, even if the primary and secondary coils are separated by
air-gap of 1cm.
4) Power transmission rate
Power transmission rate is the other important characteristic,
which indicates the efficiency of the transformer.
To improve the transformer efficiency, a secondary
resonance between the capacitor and secondary leakage
inductance is widely used and well known technique [4].
Fig.5 shows a simplified circuit model attaching a resonantcapacitor C.
L1●

suggest that computed and experimented primal selfinductances exhibit the similar tendency to the change of airgap length. Thus, it has been shown that the simulation modelin Fig. 6 is well corresponding to the practical one. Fig.8
shows one of the magnetic flux density distributions aroundthe tested transformer.

L2
●

C

R
Fig.7 The computed primal self-inductances along withthe measured ones.

Fig. 5 Circuit diagram model of a secondary resonant type transformer.

Attaching 40μF resonant capacitor in parallel to thesecondary coil terminal of the tested transformer leads to anearly ε ≈ 80% input power transmission rate defined bySecondary output power
100 %
Pr imary input power.

(4)

X. THE FINITE ELEMENTS AND OPTIMIZATION
1) The finite elements simulation
We have carried out an intensive finite elements simulation
to evaluate the leakage magnetic fluxes, and to optimallydesign the core size, core shape, number of turns, diameter ofconductor and the circuit parameters of the tested transformer.
This numerical simulation has been carried out by a free
finite element software FEMM (Finite Element Method
Magnetics). Fig.6 shows a cross-section of the simulation
model.

Fig.6 Tested transformer employing the soup plate likeferrite cores.

Fig.7 shows the primal self-inductance changes inaccordance with the air-gap length changing between the
primary and secondary core surfaces. The results in Fig.7

Fig.8 Magnetic field vectors distributions around the tested transformer.
Color bar unit is tesla.

1) Linear programming optimization
The power transmission rate from the entire primary input
to the secondary output powers of the transformer is definedby (4). By utilizing the circuit parameters, e.g., inductances
obtained by the finite elements simulations, the linear
programming optimization of (4) with respect of frequency
and capacitance yields a result shown in Fig.9.
The results in Fig.9 suggest that our contactless transformer
having a relatively large 1cm air gap is capable of obtaining
over 80% efficiency ε. This result by numerical optimization
method was verified by the practical experiments as describedin 2.4 of this paper. Thus, we have succeeded to demonstrate
the usefulness and validities of modern computational
approach to the electrical device design.

> FOR CEFC 2012DIGESTS, 6(f)<12
Our proposed transformer suppresses the leakage magnetic
fields around the core, because it is a natural extension of theshell type core structure. Thus, we have succeeded indemonstrating not only the usefulness and validities of modern
computational approach to the electrical device design but also
working out the optimum discrete core transformer.
REFERENCES[8]

Fig.9 Power transmission efficiency evaluated by the linear programming
optimization.

XI. CONCLUSION
To realize the contactless transformer exhibiting low SAR,
this paper has worked out one of the reasonable transformers
having open magnetic flux path by means of the finite
elements approach.

S.Takada, Y.Saito and K.Horii、 Visualization of the magnetic field
vectors around the contact-less power suppliers, Japan Society of
Visualization, Proceedings of the Visualization Symposium in 2000,
Paper No. P01-001.
[9] S.Matsuyama, S.Matsuyama, Y.Saito, Data Handling Methodology for
Discrete Wavelets and Its Application to The Dynamic Vector Fields,
International Journal of Wavelets, Multiresolution and Information
Processing, Vol.4, No.2, June (2006) pp.263-271. World Scientific
Publishing Company.
[10] S.Matsuyama, Y.Oguchi, Y.Saito, T.L.Kunii, Handling Technique of
the Dynamic Color Computer Graphics by the Wavelets transform,
Japan Society of Visualization, Proceedings of the Visualization
Symposium in 1999, Paper No. 206.
[11] J.L. Harrison, A new resonance transformer, Electron Devices, IEEE
Transactions on Issue Date: Oct 1979 Vol.26 Issue: 10, pp. 1545 - 1