Harmonic Balance for Magnetization Characteristics
Exhibiting Hysteretic Property
Hideaki Nemori and Yoshifuru Saito

Iliana Marinova

Graduate School of Electrical and Electronics Engineering
Hosei University
Tokyo 184-8584,Japan
Email address:hideaki.nemori.4p@stu.hosei.ac.jp

Department of Electrical Apparatus
Technical University of Sofia
Sofia, Bulgaria
E-mail address: iliana@tu-sofia.bg

Abstract—The present paper proposes one of the Domain
based hysteretic constitutive equations utilizing the orthogonal
property of the sinusoidal and cosinusoidal periodic function. At
first, we assume a magnetic domain based constitutive equation.
Second, to determine the coefficients of the assumed constitutive
equation, we applied orthogonal properties of the sinusoidal andcosinusoidal functions. To check the validity of our constitutive
equation, we carried out numerical as well as experimental
verifications.
Keywords—harmonic balance; hysteretic property; orthogonal
property

I. INTRODUCTION
As is well known, a representative ferromagnetic material
is the iron which is commonly used as the frame structural
materials of many artificial products. Ferromagnetic materials
exhibit a lot of complex physical properties, such as
magnetization, magnetostriction and magneto-thermodynamic
properties. Ferromagnetic materials are composed ofmagnetic domains and exhibit nonlinear hysteretic
magnetization characteristics. Particularly, their nonlinear
magnetization characteristics are function of externallyimpressed stresses. This is because some external physicalenergy is added to their domain structures so that theirstructures are essentially stimulated to change. This means thatparameterswhichcharacterizethemagnetization
characteristics of ferromagnetic materials may be considered
as one of the possible soundness figures to represent their
physical situation, e.g. under stressed or not.
We introduce a magnetic domain based constitutive
equation to model the magnetization properties offerromagnetic materials. And also, we describe this domain
based model in conjunction with harmonic balance method.
According to our formulation, a harmonic balance
representation of the domain based model could be derived bymeans of the orthogonal property between the odd and evenfunctions. Numerical as well as experimental verification of
our method has been carried out. Further, we have revealed thedifferences of its parameters under stressed or not.

II. DOMAIN BASED MODEL
A. Domain Based Model and Its Parameters
Previously, we have proposed two types of constitution
models for representing the ferromagnetic properties [1,2].
This paper employs the later proposed domain based model[2], and this domain based model is represented in terms of themagnetic field H[A/m] and flux density B[T]. The domain
based magnetization model is given by

H

1



B

1 dB r dH

,
s dt
s dt

(1)

where μ, μr and s are the permeability measured in the ideal
magnetization curve, reversible permeability measured alongwith the ideal magnetization curve, and hysteresis coefficient,
respectively.
B. Harmonic Balance ModelingFig. 1. Simple input and output system

Let us consider an input and output system shown in Fig.1.
When the input and output of this system are respectively
given by
n

n

i 0

i 0

n

n

i 0

i 0

f  t    ai sin  it    bi cos  it  ,g  t    ci sin  it    d i cos  it  ,

(2)

(3)

we represent this system by a following constitutive
equation

978-1-4799-5817-7/14/$31.00 ©2014 IEEEf (t ) 1g (t ) 

1 dg(t )  r df (t )

.
s dt
s dt

(4)

Let us introduce a phaser notation, i.e., a symbol ^ refers tocomplex quantities, then (10) could be reduced into^

An alternative form of (4) isf (t )  g (t )  (5)j  1 .

^g i t  

T



T

0

Similarly multiply the time derivative of output function
dg(t)/dt to both sides of (3) and integrate from 0 to T yields

0

dt





T dg (t )dt0g i t  f (t )dt
g (t )dt  

T dg (t ) dg (t )



dt

0

dt

dt  

T dg (t ) df (t )



0

dt

dt

dt

(7)
Furether, multiply the input function f(t) to both sides ofEquation.(3) and integrate from 0 to T yieldsT f (t ) f (t )dt
dg(t )
df (t )   f (t ) g (t )dt    f (t )dt    f (t )
dt
dt
dt

j tan 1

fi ,Imf i ,r2

fi ,r 2  fi ,Im 2

 j tan 1  i  j tan 1

(13)

ii

i

,

i.e.,
^

T dg (t )2 i   ii 
2

0

(6)

1  ji i  f^ t i  jii  i
1   i i 

0

0

(12)

Thus, the output is the real part of g (t ) f (t )dt.
dg(t )
df (t )   g (t ) g (t )dt    g (t )dt    g (t )
dt
dt
dt
T

(11)

where

dg(t )
df (t )
dt
dt

To determine the parameter α, β, γ, in (3), multiply theoutput function g(t) to both sides of (3) and integrate from 0 to
T yields.

T^1  ji i  f   i  jii  g ,1   i i 2

 i 2   ii 

2

fi ,r 2  fi ,Im 2fii 
cos  it  tan 1 i ,Im  tan 1 i ri  tan 1
,

fi ,r
 i 


(14)
Entire sum of the output (14) for i-th harmonics gives theoutput g of (3).

0

T

T

T

0

0

0

(8)
Substituting the input (2) and output (3) functions into theequations (6), (7) and (8), it is possible to set up a system of
equations for i-th harmonics as ai ci  bi di   ci  di
b c  a d   
0

i i
 i i
2
2

 ai  bi   a c  b d
 i i i i
2

 i   i   i 20  ci  di 22  ai di  bi ci   ai di  bi ci    ai ci  bi di  

0


.

(9)
Input and output relation for i-th harmonics is given byf i (t )   i g i (t )   i

dgi (t )df (t )
i i .
dt
dt

(10)

(a) Original(b) Recovered

Fig. 2. Validity check of the model (4) or (5).

Figure 2 shows one of the exmples of the recoverbility ofour domain based harmonic balance solution. According to theresult shown in Fig.(1), it is obvious that the nonlinear
magnetization characteristics exhibiting the hysteretic
properties of ferromagnetic fields could be solved by means ofphaser transform method.
C. Experiment
Figure 3 shows an experimental device and Table 1 lists its
various constants. The tested specimens are the silicon steelswith the 0.35mm thickness, 30mm width and 100mm length.
The tested specimen was 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 0.45A sinusoidal
alternating current through this exciting coil.
Figure 4 shows a comparison between the experimented
and recovered hysteresis loops.

Fig. 3. Specimen and U shape ferrite core

TABLE I.

SPECIFICATION OF THE MESUREMENT DEVICES

Specimen
Material: silicon steels
Length: 100cm
Width: 30mm
Thickness: 0.35mm
Number of coil turns: 300 turns
Diameter of conductor: 0.2mm

U shape ferrite core
Material: ferrite
Number of coil turns: 300 turns
Diameter of conductor: 0.6mm

The stresses were applied to the specimen by putting on thewood weights to the specimen as shown in Figure 5. The usedweights are 450 and 900 gram.
Figure 6 shows the parameters αi,βi,γi in (10) to each of
the harmonics. According to the variation of βi shown in Fig.
6(b), it is found that the parameterβi is highly sensitive to theapplied stress.
The parameter β i corresponds to the 1st time derivative
term as (5) so that this represents the hysteretic property of theferromagnetic materials.
Thus, the external stress gives a large effect to the magnetic
hysteresis, i.e., iron loss may be greatly increased by theexternally applied stress. In the other words, the iron losses inthe electrical machines distribute depending on the stress
distribution and take the large in value at highly stressed
positions.

(a) Parameter α(a) Experimented(b) Recovered by (14)Fig. 4. Comparison between the experimental hysteresis loop andrecovered hysteresis loop

Recovered hysteresis loop by (14) is well corresponding toexperimental one. This means that the hysteretic nonlinear
magnetization problems in ferromagnetic fields could be
solved by means of the harmonic balance methodology.
D. Stress Visualization
Figure 5 shows an experimental scheme illustration tovisualize the applied stress. A U shape ferrite core wound theexciting coil just same as shown in Fig.3. The specimen whichwound the search coil is put on the upper two head surfaces of
U shape ferrite core.

(b) Parameter β(c) Parameter γ
Fig. 5. The applied stress visualize scheme.

Fig. 6. The relationships between the parametersαi,βi,γi in (10) and theorder of harmonics when applied the different external stress.

III. CONCLUSIONSThe hysteretic nonlinear problems in ferromagnetic fields
could be solved by means of the harmonic balance approach.
The parameters μ, μr and s in the domain based model
could be determined in each of the harmonics by the Fourier
approach.
According to our approach, it has been revealed that theparameter related to the time derivative term of the domain
based model is highly sensitive to the applied stress. Namely,
the iron losses in the electrical machines distribute dependingon the stress distribution and take the large in value at thehighly stressed positions.
IV. REFERENCES[1]

Y. Saito, Three dimensional analysis of magnetodynamic fields inelectromagnetic devices taking into account the dynamic hysteresis
loops, IEEE Transaction on Magnetics Vol.MAG-18 No.2 March 1982,
pp.546-551.

[2]

S. Hayano. M.Namiki and Y.Saito, A magnetization model forcomputational magnetodynamics, Journal of Applied Physics Vo. 69,
No.8, April 1991, pp.4614-4616.

Backside Defect Searching by Means of the Low
Frequency ∞ Coil Excitation
Shunichi Hamanaka, Yoshifuru Saito
Graduate School of Electrical and Electronics Engineering
Hosei University,
Tokyo, Japan,

andIliana Marinova

Manabu Ohuch, Hideo Mogi
andYoshiro Oikawa
Denshijiki Industry Co, Ltd
Tokyo, Japan,

Technical University of Sofia 1756,
Bulgaria
Abstract— This paper describes the backside defect searchingby means of the low frequency ∞ coil excitation. The low
frequency ∞ coil excitation confronts to a noise processing
problem in the practical tests. To overcome this difficulty, thispaper employs two methodologies. One is an averaged sum andthe other is the Fourier transform signal processing methods toreduce the higher frequency components compared to theexcitation one. Thus, we have succeeded in enhancing the S/Nratio and detecting the signals caused by the backside defects ofthe targets. As a result, we have elucidated that the backside
defect searching is possible by employing the low frequency
excitation to our ∞ coil. Experimental as well as numericalverification along with intensive three-dimensional finite element
method are carried out to confirm our results.
Keywords—Backside defectNondestructive testing, ∞ coilsearching,Eddycurrent,

V. INTRODUCTION
Modern 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 mass transportation vehicles, e.g. large
air plane, high-speed train, express highway bus, carrying alarge number of people are required the ultimate high safety as
well as reliability.
To keep the high safety and reliability, nondestructive
testing to the metallic materials is one of the most important
key maintenance technologies because most of the structurematerials are composed of the metallic materials.
Various nondestructive testing methods, such as eddy
current testing (ECT), electric potential method, ultrasonicimaging and x-ray tomography are currently used to themodern airplane, high-speed -train and express high bus
maintenance. Among these methods, ECT needs not complex
electronic circuits and direct contact to the targets. And also,

978-1-4799-5817-7/14/$31.00 ©2014 IEEE

most of the targets whose major frame parts are composed of
conductive metallic materials can be selectively inspected by
ECT [1-3].
Operation principle of ECT is fundamentally based on themagnetic field distribution change detection due to the defectin the targets. To realize this principle, we have two
methodologies. One detects the defect in the target as a change
of input impedance of the exciting coil. This is because themagnetic field distribution is changed by the detour eddy
currents flowing around the defect in the target whichcorresponds to the secondary circuit of a single phasetransformer [2-3]. The other type ETC sensor equips a sensing
coil to detect the magnetic field change caused by the detour
eddy currents flowing around the defect. The former and latterare called the impedance sensing and sensing coil types,
respectively.
The sensing coil type is further classified into twovariations. Most popular sensing coil type employs adifferential coil, and also the other type sets the sensing coilsurface perpendicularly to those of the exciting coil. As is wellknown the differential coil detects the uniformity of themagnetic field distribution. Similarly the perpendicularly
installed sensing coil surface to those of exciting coil detects
only the magnetic fields caused by the detour eddy currents
due to the defect in the target.
Our developed ∞ coil one of the latter types, i.e., detects
only the magnetic fields caused by the detour eddy currents
due to the defect in the target. A key idea of our ∞ coil is thatthe sensing coil wound around a ferrite bar is installed at thelowest magnetic field intensity region between the north andsouth poles of exciting coils [1].
In the present paper, to search for the backside defect of atarget, we have employed a low frequency excited ∞ coil. As aresult, it is revealed that the ∞ coil has versatile capability, i.e.,
low frequency excitation of the ∞ coil makes it possible todetect the backside defects along with the signal processing
methodologies.

VI. SIGNAL PROCESSING METHODS

defectno-defect

A. Averaged Sum Method
In any signal measurement devices, it is essential toconfront to the mixing of noise problems. Particularly, the low
frequency excitation of ∞ coil measures a serious noise signal
quantity compared with those of the desired one.
One of the common natures of any noise never repeat thesame magnitude in value at the same periodic signal points.
This means that an average of the same periodic signals
reduces the noise signals. This signal processing methodology
is the averaged sum approach, which is carried out to a M-thtimes measured signal x(k) byx(k ) 

M

M

M

1
1
1
x i (k ) s i (k )   n i (k ) ,
M i 1
M i 1
M i 1(1)

Fig1 Schematic diagram of the 3D FEM simulation
Table 1 Various constants of the tested ∞coil
Exciting coil
Coil outer diameter
Coil inner diameter
Coil length
Number of turnInput voltage (peak)
Frequency

where s and n are the correct noise components, respectively.
B. Fourier Transform Method
The other methodology to remove the noise components
from the measured signal is the Fourier transform method.
Even though Fourier approach requires some mathematicaloperation or electronic circuits, it is one of the basetechnologies to synthesis the low-, high- and band pass filters.

22.4mm
20mm
10mm
75
3V
2kHz
Sensing coil

Coil outer diameter

0.9mm×2.4mm

Coil inner diameter

0.5mm×2.0mm

Coil length

6.0mm

Number of turn
Axis core100
Mn-Zn ferrite 300

To our ECT signal processing, we utilize the Fouriertransform approach to pick up a particular frequency
component from the measure signals, where the particularfrequency is the same as those of the exciting frequency.
VII. EXPERIMENT
A theoretical background of the low exciting ECT is theskin effects, i.e., low frequency magnetic fields induce theeddy currents in the deep location of the metallic materials.
Namely, the low frequency excitation of ECT makes it
possible to detect the defects located in the deep or backside of
the target metallic materials.
In our experiments, we have employed the two copper
plates, one is the normal copper plane having 2mm thickness,
10cm length, 2mm width, and the other is the target having a
1mm defect at the backside.
Consideration of the skin depth to these targets has led to
employ the 2kHz excitation frequency.
A

Averaged Sum Processing
Figure 1 shows a schematic diagram and Table 1 lists thevarious constants for the 3D finite element computations.
Figure 2 shows one of the computed results, i.e., a large
amplitude 5.1mV sensor signal has been obtained without any
noise effects where the sensor is on the backside defect and is
not on any defect.
Thus, theoretically it is possible to detect the backside
defect by the low frequency excitation of the ∞ coil.

Fig 2 Induced voltage in the exciting coil

Table 2 lists the various constants of the prototype ∞coil.
Figure 3 shows the target piece and a prototype of ∞ coil. The
normal target piece has been emulated by overlapping the twocopper plates having 2mm width, 1mm depth. Also, the defectpiece has been emulated by overlapping the one normal sheet
to the other sheet having 1mm depth defect. The various
constant of the prototype ∞ coil are the same used in the 3D
finite element simulations.

Table 2 Various constants of the prototype ∞coil
Exciting coil

Conductor length

4.7mm

Diameter of conductor

0.4mm

Coil outer diameter

23mm

Coil inner diameter

20mm

Coil length

10mm

Number of turnSensing coil75

Number of col layers

3

Number of coils

2

Conductor length

60cm

Diameter of conductor

0.1mm

Axis core

Ferrite bar (MnZn)

To remove the noise components from the experimentallyobtained signals in Fig. 4, we employed the averaged sum
method (1).
Figure 5 shows the refined signals in Fig. 4 by theaveraged sum signal processing. In Fig. 5, 200 times averaged
was carried out for the defect signal, and 400 times averaged
was carried out for the no defect signals. Difference of thenumber of averaging between them is that a large signal needs
not a large number of averaging because of good SN ration but
a small signal requires a large number of averaging because of
the bad SN ratio.
Even though the peak signal voltage of experimental results
is somewhat larger that those of computed ones, theexperimental signals exhibit the similar tendency to that ofcomputed ones.

Coil outer diameter 2.4mm×2.4mm
Coil inner diameter 1.4mm×1.4mm
Coil length

6.0mm

Number of turn
Number of coil layers

100
2

2mm
Fig 5 Induced voltage in sensing coil (averaged sum processed)
No-defect copper plate(a) Target pieceB(b) Prototype of the ∞ coilFig. 3 The target tested piece and ∞ coilFigure 4 show the results of experiments. It looks like theresults of the simulation but includes a lot of high frequency
noise, where the sensor is on the backside defect and is not on
any defect.

Fouier Transfom Processing
To carry out the experiment in more practical manner, thesensor coil was moved to point to point locations along a
straight line as shown in Fig. 6. The sensor induced signal ateach of the points was measured, where the sensor was locatedat the 5 points along a 20mm line.
Figure 7 shows the induced voltages at the 5 point
locations. Obviously, the maximum induced voltage at theposition 3 suggests the backside defect position of the target.
Figure 8 shows the signal measurement positions on theplane copper sheet. These positions are as that of the Fig. 6 toverify our approach.

Even though the experimented signals mixed with noise in
Fig.4, it is obvious that the backside defect could be detected.

Fig 6 1D flaw detection model (simulation)

Fig 4 Induced voltage in sensing coil (experiment)

Fig 7 Sensor output voltage (simulation)

depth. Also, the defect piece has been emulated byoverlapping the one normal sheet to the other sheet having
1mm depth defect.
In any practical measurements, it is essentially measured
the signals containing noise. To remove the noise signals nothaving the exciting current frequency, we employed theFourier signal processing approach.
Figure 9 shows the experimentally obtained sensor output
signals whose noise components were removed by the Fourier
signal processing.
Comparison the results in Figs. 7 and 9 shows that theirmaximum values are somewhat different but their position is
the same No. 3 point.
Thus, our backside defect searching by the low frequency
excitation is succeeded in a first step of the project.
VIII. CONCLUTION
In the present paper, we have employed the low frequency ∞coil excitation to search for the backside defect of the target.

Fig 8 1D flaw detection model (experiment)

As a result, it has been clarified that the low frequency ∞ coil
excitation makes it possible to search for the backside defects alongwith the signal processing procedures enhancing the S/N ratio.
IX. REFERENCES
[1]

[2]

[3]
Fig 9 sensor output voltage (experiment)

The normal target piece has been emulated byoverlapping the two copper plates having 2mm width, 1mm

Hiroki Kikuchihara, Iliana Marinova, and Yoshifuru Saito, Manabu
OHUCH, Hideo MOGI and Yoshiro OIKAWA, Optimization of the
Eddy Current Testing, Digest of The 15th Biennial IEEE Conference on
Electromagnetic Field Computation WC4-4 pp.495, Oita Japan
November 11-14 2012,
Hiroki Kikuchihara and Yoshifuru Saito, Enhance the Sensibility of the
ECT Sensor, Journal of the Japan Society of Applied Electromagnetics
and Mechanics, Vol.21, No.3 , 2013.
Hiroki Kikuchihara, Iliana Marinova, and Yoshifuru Saito, Enhance theSensibility of the Resonance type ECT Sensor, JAPMED'8 pp. 130-131,
2013.