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JOURNAL OF OPTOELECTRONICS AND ADVANCED MATERIALS, Vol. 11, No. 1, January 2008, p. 45 - 55
Magnetostrictive behaviour of ribbons and wires:
analytical modelling and experimental validation
A. MAMALIS, E. HRISTOFOROU*
National Technical University of Athens, Zografou Campus, Athens 15780, Greece
In this paper the magnetostrictive behaviour of ribbons and wires is revisited and studied, proposing an analytical modelling
followed by experimental validation using the magnetostrictive delay line (MDL) method, due to the needs for engineering
applications of magnetostriction, like sensors and transducers. The obtained results may be used for the determination of
the M(H) and λ(H) functions as well as their uniformity distribution along the length of magnetostrictive ribbons and wires,
which is a key-factor for the characteristics of sensors based on the magnetostriction effect.
(Received November 24, 2008; accepted January 21, 2009)
Keywords: Magnetostriction, Analytical modeling, Ribbons, Wires, Sensors
1. Introduction
Sensors and transducers have an increasing interest
because of their importance in many technological
applications [1]. All modern vehicles and transportation
means use a vast variety of sensors and transducers, thus
allowing a safer and more comfortable way of driving and
commuting. The operation of all medical instruments is
based on sensors. Industry is also employing more and
more transducers for the monitoring and control of
production lines. In the literature, sensors have been
categorized in several ways [2]. In the present work, they
are categorized according to the following three principles:
the first one is the subject of measurement, the most
significant divisions being physical and chemical sensors;
the second principle concerns the physical phenomenon
and material on which the operation of the sensor is based,
the main categories being conducting, semiconducting,
dielectric, magnetic and superconducting sensors; and the
third one concerns their applications, the main categories
being industrial, transportation, automotive, medical,
military, domestic and environmental sensors.
Magnetic sensors play a significant role in physical
measurements used in all kinds of applications [3, 4]. The
most often used magnetic phenomena in today’s magnetic
sensor technology are the magneto-resistance [5], the
magneto-impedance [6], the magnetostriction [7], the
electromagnetic induction [8] and the Hall effect [9]. The
dynamics of magnetic domains is the main mechanism
responsible for magnetic effects used in sensing
applications [10]. Any possible use of the dynamic
response of this mechanism can result in a sensing
element. There are two distinct cases of domain dynamics,
one of which is the domain wall dynamics and the other
one the domain rotation dynamics. There also exist
dependent effects derived from these dynamics, both
macroscopic and microscopic.
Magnetostriction, a particular effect in magnetic
materials, has been thoroughly investigated in terms of
theory and modeling as well as in terms of experimental
details and applications [11-15]. The theory of
magnetostriction is mainly based on the principles of
micromagnetics [16]. The applications concern sensors
and actuators, requiring materials of engineering
magnetostriction constant in the order of 10 ppm and 1000
ppm, respectively.
A technique that utilizes the magnetostriction effect in
the design and development of sensors measuring
displacement, stress and field is the magnetostrictive delay
line (MDL) technique [17]. In this work, the
magnetostrictive behavior of ribbons and wires able to
operate as MDLs is analytically modeled, followed by
experimental validation. The presented model may be used
for the determination of the M(H) and λ(H) loops as well
as their non non-uniformity distribution along their length.
2. Magnetization effects contributing to
magnetostriction
The main magnetization effects contributing to
magnetostriction are based on domain wall and domain
rotation dynamics as well as the macroscopic and
microscopic mechanisms dependent on domain dynamics.
These effects are briefly presented bearing in mind that a
key parameter in the magnetostrictive behavior is the
hysteresis in their response. Hysteresis should be
negligible in applications like mechanical and field sensors
in order to improve the uncertainty level of the sensors,
but it should be heavily present in applications like
security sensors to improve the stability of the stored
information.
46 A. Mamalis, E. Hristoforou
2.1.Domainwalldynamics.
The dynamics of domain walls and their
corresponding use in sensor applications concern their
nucleation and mobility or propagation in the magnetic
substance [18]. There are two mechanisms of domain wall
propagation, namely the bowing process and the parallel
motion of the domain walls. The mode of propagation
depends on the energy stored in these walls. Low energy
walls propagate through the bowing process as shown in
Fig. 1a while high energy walls propagate more rigidly as
shown in Figure 1b. The bowing process is more likely to
occur in soft magnetic materials, which are low pinning
materials, while the more rigid motion occurs in the harder
ones. The reversibility of the domain wall propagation
determines the presence or not of hysteresis in the
phenomenon used in the sensing element and depends
mainly on the defects in the magnetic substance and the
pinning effect of magnetic dipoles. Domain wall dynamics
are used for small field measurements as well as for
mechanical sensors based on small field measurements
[19].
Fig. 1. Modes of propagation of domain walls. (a) Low
energy walls (b) high energy walls
Therefore, the sensor designer using domain wall
dynamics should tailor the magnetic material with respect
to the application in request. If the case is a sensor based
on domain wall propagation with hysteresis in the
minimum possible amplitude, the material should include
as less defects as possible and be as soft as possible. This
may be controlled through the composition of the material,
as well as through annealing of the material in order to
minimize the internal stresses generated by the above-
mentioned defects, targeting coercive fields of the order of
1 A/m [20]. In this case, the material should have low
magnetostriction and correspondingly low magneto-elastic
response to avoid cross-talk with possibly uncontrollable
stray magneto-elastic waves. One can fulfil both
requirements by using FeCoSiB wires of magnetostriction
in levels of 0.1 ppm, after thermal annealing and
sometimes magnetic field annealing [21], while using a
magnetostrictive substance may result in high levels of
Barkhausen noise. Typical annealing conditions are of the
order of 30-60 oC/min for the rising temperature, steady
state conditions of 300oC – 750 oC for 10 – 60 minutes and
finally slow cooling in Ar atmosphere for about 12-24
hours. Typical field conditions during annealing are 800 –
8000 A/m. Another technique also used in the material
tailoring is the stress – current annealing, with typical
values of tensile stress and current of 100 – 500 MPa and
100 – 300 mA respectively [22]. On the contrary, in
security sensors the pinning defects or the controllable
introduction of defects on the surface of the material can
result in a significant improvement of the sensor stability.
2.2.Domainrotationdynamics.
Domain rotation dynamics have two distinct areas of
operation, the irreversible and the reversible area [23].
Irreversible rotation occurs when the magnetic domains,
oriented along a given easy axis A, re-orient along another
easy axis B, closer to the axis of the external field H,
because of the presence of this field, as shown in Figure2a.
Reversible domain rotation occurs after the irreversible
rotation process has taken place. Since the new easy axis B
is, in general, not the same as the axis of the external field
H, the magnetic dipoles rotate reversibly towards the axis
of the external field H, as shown in Fig. 2b. After the
removal of the external field, the magnetic domains rotate
back to the easy axis direction B, along which they had
been initially and irreversibly re-orientated. In general,
magnetic domains do not return back to their initial easy
axis A. Both reversible and irreversible processes are
associated with the presence of magnetostriction. The
irreversible process is additionally responsible for the
small or large Barkhausen jumps, introducing magnetic
noise in the sensing element. Employing the irreversible
process results in hysteresis in magnetic rotation, as well
as in a relatively higher level of noise with respect to the
reversible process. Both hysteresis and noise affect the
uncertainty of any possible magnetic device used for
sensing. Therefore, if the aim is the development of a
sensor, where hysteresis and noise should be minimized,
only the reversible area of domain rotation should be used.
On the contrary if the aim is high hysteresis, the
irreversible area of the domain rotation should be used.
The domain rotation effect has found applications mainly
in the field of mechanical sensors [24]. The dynamic
behaviour of these processes can result in elastic waves,
propagating along the magnetic substance. This is
precisely the basis of the MDL technique [25-28]. The
MDL technique has been extensively studied in order to
understand its operation and optimize its performance [29-
32] using a variety of methods [33-37].
Magnetostrictive behaviour of ribbons and wires: analytical modelling and experimental validation 47
Fig. 2. Irreversible and reversible rotation in magnetic
domains. (a) Irreversible orientation along easy axis A or
B, (b) Reversible small magnetization angle rotation
(SMAR)
A vast variety of magnetostrictive materials have been
developed up to now. Today, the materials exhibiting the
largest possible magnetostriction are the recently
developed magnetic shape memory (MSM) alloys [38],
exhibiting dimensional changes in the order of 1% to 10%.
Their operation is based on the martensitic – austenitic
transformation even at room temperature due to the change
of the biasing field.
Before the development of MSM alloys, the materials
exhibiting the largest magnetostriction were the rare earth
– transition metal alloys, with saturation magnetostriction
in the order of 800 ppm to 2000 ppm [39-41].
Combination of rare earth elements and magnetic
substances, like iron, nickel and cobalt, resulted in the
development of Terfenol and other similar alloys, which
have been extensively used in engineering applications.
Soft magnetostrictive alloys based on iron, nickel and
cobalt exhibit a relatively low magnetostriction in the
order of 30-100 ppm and are generally used as MDLs. The
needs in modern sensor development would not allow for
the use of the classical polycrystalline materials and led to
the development of the amorphous magnetostrictive
alloys, like ribbons, wires and glass-covered wires,
prepared by rapid quenching techniques [42-46].
The amorphicity of the used magnetostrictive material
helps in the minimization of the irreversible rotation
process, because of the minimization of the coercive field
and the field range responsible for the irreversible domain
rotation. The need for miniaturization has led research
groups to develop MDL arrangements in thin film
structure, thus enhancing the possibility of developing
integrated systems [47-49]. Furthermore, the need for
better sensor characteristics has led to the development of
nano-crystalline magnetostrictive ribbons and wires [50,
51] with even lower hysteresis, which can be used as
MDLs, provided that they can exhibit magnetostriction.
Recently, an interesting composite material including
magnetostrictive substance in a non-magnetic matrix has
been proposed for magnetoelastic applications [52].
When designing a sensor using the effect of domain
rotation, one should tailor the magnetic material in order to
minimize the amplitude of the external field responsible
for the irreversible rotation process and correspondingly
maximize the external field range for reversible rotation.
Annealing techniques have been employed, targeting the
proper tailoring of the magnetostrictive elements [53-55].
These techniques mainly include heat annealing, field
annealing and stress-current annealing, not only
eliminating the defects of the material, but also re-
orienting the magnetic anisotropy in order to eliminate the
irreversible swift process of the domain rotation. The
elimination of the irreversible process occurs simply
because of the absence of an easy axis direction near the
direction of the external field H. In this case, the magnetic
field in the field annealing process should be perpendicular
to the easy axis of the material. The magnitudes of
temperature and field are similar to those in the case of
domain wall dynamics.
The λ(H) function is the most important characteristic
regarding the MDL operation, since it can model the
operation of an MDL set-up and consequently a sensor
arrangement based on the MDL technique. Proper tailoring
should take into consideration the dynamic response of
λ(H) with respect to frequency and not only the saturation
magnetostriction constant λs or the static magnetostriction
function. A number of instruments have been developed in
order to measure the dynamic characteristics of this
function as well as the engineering magnetostriction
constant λs [56-58]. An excellent review of such
measurement techniques is given in [59].
2.3. Dependent mechanisms.
Apart from the domain wall and domain rotation
dynamics, there are other dependent magnetic effects,
which can be measured and used as macroscopic electrical
and magnetic properties of the material.
The most well-known and used effect is the magneto-
resistance effect [60], observed mainly in magnetic thin
films. According to this effect, the dc electrical resistance
of a magnetic film changes about 2-3%, with respect to the
externally applied magnetic field, due to the magnetic
48 A. Mamalis, E. Hristoforou
domain rotation and in some cases due to domain wall
nucleation. The most significant magneto-resistive effect,
the “giant” magneto-resistive effect, appears in magnetic
thin film multi-layers where the change in resistance
hovers in the range of 50-80% at room temperature. This
“giant” effect is due to the perpendicular anisotropy of the
magnetic layers causing a large magnetic moment rotation.
Recently, the “colossal” magneto-resistive effect has been
observed in magnetic oxides, offering even larger changes
in resistance, but in cryogenic environments. The
magneto-resistive effect is mainly used in field sensors and
recording media applications. Another effect, which has
also found applications, is the ac magneto-resistance effect
or magneto-impedance effect [61-63]. According to this
effect, the ac resistance or impedance of a magnetic
substance varies with the applied field. This effect also
exists in non-ferromagnetic materials due to the skin
effect, although its amplitude is much smaller than in
ferromagnetic materials.
In some zero-magnetostrictive wires, with
circumferential magnetic anisotropy, the magneto-
impedance changes more than 100% with respect to the
applied external field. Although this effect has only
recently been studied, it has already been used in industrial
and automotive applications due to its great sensitivity in
magnetic field.
Recently, another effect attracts the interest of the
field sensor market and mainly the recording media
market. This is the spin valve effect [64-66], according to
which an especially designed magnetic arrangement
exhibits non-symmetrical B-H response. This property
allows very well localized field measurements with an
acceptable accuracy. Apart from that, the spin tunneling
effect has also been used in recording media and accurate
field measurements [67].
Apart from these, the rather classical inductive effects
[68] have been implemented in the form of the
fluxgate set-up [69] for accurate field detection and linear
variable differential transformer (LVDT) for displacement
sensing [70]. Other related electromagnetic effects such as
the Hall effect, the Quantum Hall effect and the SQUID
are also able to detect field. With the exception of
magneto-elasticity, which is used for direct detection of
mechanical sizes, the main sensing application of magnetic
effects and materials is the detection of magnetic field.
Once the field or field change has been measured, one can
map the measurement to another physical size, like
displacement, stress, flow etc.
3. Analytical modeling of the basic MDL
arrangement
The most classical MDL arrangement, the coil-coil
MDL set-up, is shown in Fig. 3. A short excitation coil and
a short search (named also detecting or receiving) coil are
placed around each one of the two ends of the MDL. The
delay line is terminated using latex adhesive to eliminate
acoustic reflections. Details of the various versions of such
arrangements can be found in [71, 72] and are presented
hereinafter.
Fig. 3. The basic MDL arrangement. (1) Excitation coil,
(2) Magnetostrictive delay line, (3) Search coil.
Magnetostrictive materials subjected to either low or
high frequency fields, tend to undergo either domain wall
motion or magnetic domain rotation respectively, always
towards the direction of the externally applied field. Thus,
applying external bias or pulsed field along the MDL axis
results initially in Barkhausen jumps, which contribute to
the hysteretic and irreversible part of the λ(H) function
and consequently in small angle rotation which is the
anhysteretic and reversible part of the said λ(H) function.
Fig. 4. A typical
λ
(H) function.
Therefore, polarizing the MDL with a dc bias field
Hdcx, results in an elongation of the material δλo, illustrated
as a point (δλo, Hdcx) on the λ(H) function, shown on Fig.
4. When a pulsed field He(t) is additionally applied at the
region where the bias field has been applied, a similar but
dynamic elongation δλ(t) occurs, as shown in Fig. 5,
resulting in an elastic wave propagating along the MDL,
following the classical wave equation, as shown in Fig. 6.
Fig. 5. Microstrain with respect to space.
Magnetostrictive behaviour of ribbons and wires: analytical modelling and experimental validation 49
In classical magnetostrictive materials, the optimum
pulsed field width is in the order of μs. Thus, the
wavelength of the propagating elastic wave is in the order
of several mm. Therefore, in the most common MDL
elements, where the MDL cross section is a tenth of mm2,
a Lamb wave is propagating. Using materials with higher
frequency response or larger cross section can result in
surface acoustic wave propagation. Skin effect plays an
important role in modeling and tailoring the behaviour of
the microstrain generation and propagation.
Fig. 6. Propagating elastic pulse along the length of the
MDL.
The pulsed field along the MDL, responsible for the
elastic wave generation follows a decaying profile
extending from the fully magnetized central region to a
limit which is practically of the order of the excitation coil
diameter, indicating the active region of the
magnetostrictive material involved in the microstrain
generation.
This elastic wave propagates along the length of the
MDL, mainly as a longitudinal elastic wave, because of
the shape of the acoustic wave guide: the short cross
section with respect to the wavelength and the dimensions
of the MDL eliminate any transverse and quasi-transverse
waves.
Fig. 7. Voltage output with respect to time.
The propagating elastic wave, in its course, changes
the local magnetization component along the MDL axis,
provided that the MDL is locally magnetized. The total,
macroscopic change of the magnetic flux along the axis of
the wire is the result of the statistical sum of local
infinitesimal changes in the orientation of magnetic
dipoles, in the course of the propagating elastic wave.
Thus, the magnitude of the biasing field determines the
change of the local magnetization component along the
MDL axis. This is actually the inverse magnetostriction
effect. In some materials, the earth’s field can be enough
to polarize and consequently cause the presence of such
effect.
Thus, if an inductive means, like a search coil, is set
around the MDL, a pulsed voltage proportional to the first
derivative of the flux is induced across its ends, as shown
in Fig. 7. The search coil should be set at a distance x from
the elastic wave point of origin (PO), which ought to be
small enough to cause negligible attenuation and large
enough to avoid electromagnetic coupling between
excitation and detection means. Such pulsed voltage
output is received with a delay time proportional to the
distance x and inversely proportional to the longitudinal
sound velocity of the magnetostrictive element. A real
pulsed voltage output waveform, with the corresponding
delay time from the excitation pulse observed as impulse
response, is shown in Fig. 8. In this case, the relatively
small waveforms following the main pulse are due to
reflections of the propagating elastic pulse.
Fig. 8. The detected MDL pulsed voltage output. The
first impulse response is due to the pulsed excitation
field. The main pulsed voltage output follows, with a
characteristic amplitude Vo. The small waveforms
following the main pulsed output are reflections of the
propagating elastic pulse at the ends of the
magnetostrictive medium (Time units in seconds and
voltage amplitude in Volts).
3.1. Magnetostriction modeling
At the atomic level, magnetostriction is the aggregate
result of the deformations of the crystal lattices inside the
domains that tend to align with the domain magnetization.
The deformation of a crystal lattice is due to the
interactions between the atomic moments occupying its
sites that result in altering the bond lengths. When the
bonds lie at an angle
φ
to the domain magnetization, the
magnetoelastic energy tends to align the bonds with the
domain magnetization, but is counterbalanced by the
elastic bond energy. At the macroscopic level, one can
think of the energy added to the system because of an
50 A. Mamalis, E. Hristoforou
externally applied field, m
EΔ, as being counterbalanced
by the change in elastic bond energy, el
EΔ along the
MDL axis:
2
2
2
1
2
1⎟
⎠
⎞
⎜
⎝
⎛
Δ
Δ
=
Δ
Δ
⇔Δ=Δ⋅Δ⇔Δ=Δ H
k
H
M
kHMEE elm
λ
λ
(1)
where k is the macroscopic elastic constant of the material,
related to Young's modulus EY, and Δλ is the elongation
caused by the change in magnetization ΔΜ.
For dH
dM
dH
d
H∝→Δ
λ
,0
The derivative dM
dH corresponds to the differential
susceptibility diff
χ
, of the magnetic material, which can
be described by a function of the form:
c
H
diff e
c
H
dH
dM −
⎟
⎠
⎞
⎜
⎝
⎛+∝= 0
χχ
(2)
where c is a fitting constant with field dimensions related
to 1
K and
s
M
[73] and 0
χ
the initial susceptibility. The
above mentioned equation is the solution to a second order
linear differential equation whose characteristic equation
is: 20xcxc++=
, as in the critical damping case in
resonance. Thus
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛+−
−+=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛+∝ 0
02
1
0
2
0
χ
χ
χλχλ
λ
c
H
s
c
H
se
c
H
ee
c
H
dH
d
(3)
where λs the saturation magnetostriction constant. Hence:
⎟
⎟
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎜
⎜
⎝
⎛
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛+⋅−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛+⋅
=
=⋅+∝=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛+−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛+−
∫∫
0
2
1
0
2
2
1
0
2
0
0
0
0
2
24
2
1
)(
χ
χ
λ
χλ
λ
λ
χ
χ
χ
χ
c
H
e
c
H
erfu
e
dHe
c
H
e
dH
d
H
c
H
s
c
H
s
(4)
where ()erf x is the error function and:
()
∑
∫∞
=
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛+
+
−
==
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛+
0
12
0
0)12(!
122
2
10
n
n
nx
c
H
x
nn
dxe
c
H
erf
ππ
χ
χ
(5)
Thus:
(
)
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛+=
⎟
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎜
⎝
⎛
+⋅−
+
−
⋅
∝⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛+−
∞
=
+
∑
00
2
1
0
12
2
2
1
,2
22
)12(!
1
)( 0
0
χχ
π
λλ
χ
χ
c
H
x
c
H
e
x
nn
u
eH c
H
n
n
n
s
(6)
Experimental data, have illustrated that in the case of
anhysteretic behaviour, the ()
H
λ
function can be fitted
by
(
)
0,1)( 2>−= −ceH cH
s
λλ
, where the positive
number c is an adaptive parameter. In the case of
hysteretic evidence this model could become
(
)
0,1)( 2
)( >−= ±− ceH c
HHc
s
λλ
.
The Energetic Model (EM) described in [74] relates
the fitting constant c to microscopic parameters of the
material. At weak fields,
2
0
1
2
0
2
0⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅
⋅
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅
⋅
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
s
q
s
s
M
Kc
M
kqM
c
μμχ
(7)
and at strong fields,
2
0
1
'
2
2
2
42
2⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅
⋅
=≈
+−+
⋅
=
s
g
s
s
M
Kc
H
ggg
Hg
c
μ
(8)
where g, h, k, q are the parameters of the EM:
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛⋅
⋅=
1
2
K
M
cg s
g
μ
, )2lnexp( ⋅
=g
H
hs,
2
sYk Eck
λ
⋅⋅= , 2
1
sY
k
q
E
K
c
c
q
λ
⋅
⋅= (9)
with
g
c, h
c, k
c and q
c being the model's dimensionless
microscopic constants. With the anisotropy
field
s
kM
K
H⋅
=
0
1
2
μ
, the saturation field 2
'
kg
s
Hc
H⋅
=
('
g
c is a proportionality constant) and q
c and '
g
c being of
the same order of magnitude, an average value for c at
both weak and strong fields is defined by
2
'
2
0
1
'
'
2⎟
⎠
⎞
⎜
⎝
⎛
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅
⋅
+
⋅
=c
H
M
K
cc
cc
cs
sgq
gq
μ
In order to prove the principle of the described
formalism, experimental data were obtained using a
Fe78Si7B15 amorphous ribbon MDL, exhibiting λs~30-32
ppm. The sample was previously stress-current annealed
under 350 MPa and 100 mA, to remove internal stresses
and improve its magnetostrictive behavior and uniformity
of magnetic domains. The MDL set-up was operating by
Magnetostrictive behaviour of ribbons and wires: analytical modelling and experimental validation 51
incrementing the peak value of the pulsed current Ie from 1
to 13.6A and back. The bias field at the arrangement was
varied from 0 to 130 A/m at each Ie amplitude. The output
voltage is related to the dynamic response of the
anhysteretic λ(H) function and is a function of the pulsed
current waveform.
The peak voltage is maximum at a bias field of 16 A/m for
all Ie. Considering the induced voltage,
()
00
max
ee
dc dc dc
eee
dH dH
dB d d d d
Vt H H V H A
dt dt dH dt dt dH dH
λ
λλλ
⎧⎫
∝∝ = ⇒= =
⎨⎬
⎩⎭ (9)
where A0 is a constant related to Ie and λs.
Fig. 9. Comparison of experimental and theoretical data
for biasing field using equation 1.1 to 1.5.
Fig. 9 shows the experimental and theoretical data,
concerning the dependence of the MDL peak amplitude Vo
on the DC bias field at the region of the receiving coil, at
various values of Ie, producing the field He, for several
amplitudes of Ie. As expected, the value of c turns out to
be the same for all theoretical curves: c=15 A/m. This
suggests that c is indeed related to material constants. A0
exhibits a linear dependence on Ie with signs of saturation
for higher currents.
3.2. An alternative way of modeling the coil-coil MDL
arrangement
Following the basic MDL set-up as illustrated in Fig.
3, the MDL is activated by transmitting pulsed current
)()( tfHtH ee ⋅= , through the excitation coil or the
pulsed current conductor. Pulsed current generates a
pulsed magnetic field along the magnetostrictive element.
This field generates a pulsed microstrain at the region of
excitation of the magnetostrictive element,
()
)(t
eoe Η+Η
λ
due to the magnetostriction effect. Since
the magnetostrictive material is in the shape of cylinder or
ribbon, it can operate as acoustic waveguide. Therefore,
the pulsed microstrain propagates along the length of the
magnetostrictive element as longitudinal acoustic pulse.
As soon as it arrives at the region of the search coil, it is
detected as pulsed voltage output, proportional to the first
derivative of the propagating pulse, due to the inverse
magnetostriction effect. The generation and detection of
the pulsed microstrain is possible and repeatable due to the
presence of biasing fields at the acoustic stress point of
origin and the search area, oe
H and or
H respectively,
which orient the magnetic dipoles in a given direction.
The propagating pulsed microstrain induces stresses
)(
λ
σ
in the MDL. These stresses act as effective field
)(
σ
σ
fH
=
in the MDL, added in the already existing
biasing field along its length. Provided that the microstrain
propagates without dispersion and after effects, which is
applicable for the front acoustic wave, it arrives at the
region of the search coil, inducing such an effective field
σ
H along the length of the MDL. Thus, the flux within
the magnetic region inside the search coil is:
)()()(
σ
μ
HHHSt oror +⋅
⋅
=
Φ
(10)
where S is the cross section of the magnetostrictive
element. Thus, the voltage output Vo(t) at the search coil
is:
dt
dH
HA
dt
d
tV oro
σ
μ
⋅⋅−=
Φ
−= )()( (11)
Where
A
includes S and search coil parameters. Provided
that excitation pulsed field is relatively small, the effective
field and stress are assumed to be proportionally related:
(
)
)()( tHHafH eoe +
⋅
=
=
λ
σ
σ
(12)
Thus )(tVo becomes:
(
)
(
)
dt
tdf
H
dH
d
HaA
dt
tHHd
dH
d
HaA
dt
tHHd
HaAtV
eor
eoe
or
eoe
oro
)(
)(
)(
)(
)(
)()(
⋅⋅⋅⋅⋅−=
+
⋅⋅⋅−=
+
⋅⋅⋅−=
λ
μ
λ
μ
λ
μ
(13)
Thus the peak to peak magnitude of )(tVo, o
V is given
by:
dH
d
HHcaAV oreo
λ
μ
⋅⋅⋅⋅⋅−= )( (14)
where c is the maximum of
dt
tdf )( .
In the case that oee HH , are not changing and or
H
changes, o
V becomes:
)()( 11 ororeo HCHHccaAV
μ
μ
⋅=
⋅
⋅
⋅
⋅
⋅
−
=
(15)
52 A. Mamalis, E. Hristoforou
where constant max1 )/( dHdc
λ
=. Coefficient 1
C is a
constant, mainly dependent on the material and the fields
at the excitation regions. Under these conditions )( or
H
μ
is proportional to o
V.
In case that ore HH , are constant and oe
H changes, o
V
becomes:
dH
d
C
dH
d
HHcaAV oreo
λ
λ
μ
⋅=⋅⋅⋅⋅⋅−= 2
)( (16)
where 2
C is a constant, mainly dependent on the material
and the excitation field and biasing field at the excitation
and receiving regions respectively. Under these conditions
d
H
d
λ
is proportional to o
V. When oroe HH , are constant
and e
H changes, o
V becomes:
dH
d
HC
dH
d
HHcaAV eoreo
λ
λ
μ
⋅⋅=⋅⋅⋅⋅⋅−= 3
)( (17)
Thus
dH
d
C
H
V
e
o
λ
⋅= 3, where 3
C is a constant, mainly
dependent on the material and the biasing fields at the
excitation and receiving regions.
Apart from being useful in MDL behaviour
description, this approach can also lead towards the use of
this MDL arrangement for the experimental determination
of the M-H and λ-H loops of magnetostrictive ribbons and
wires as well as their corresponding uniformity functions.
All the above mentioned equations used for the coil-coil
MDL arrangement can also be used for the case of the next
presented arrangement concerning conductors
perpendicular to MDLs.
4. The MDL modeling used for the
determination of magnetic properties
Following the theory developed in the previous
chapter, the voltage output ()
o
Vt at the search coil is
given by equation 11:
()
()
dt
tHHd
d
H
d
HaA
dt
tHHd
HaAtV
eoe
or
eoe
oro
)(
)(
)(
)()(
+
⋅⋅⋅−=
=
+
⋅⋅⋅−=
λ
μ
λ
μ
(18)
It will be hereinafter shown how this procedure can
result in the experimental determination of the M-H and λ-
H loops of magnetostrictive ribbons and wires as well as
their corresponding uniformity functions [75].
M-H loop
Keeping the excitation and biasing fields at the
excitation region He and Hoe respectively constant, while
the biasing field Hor at the receiving region changes, the
peak amplitude of the MDL pulsed voltage output Vo is
given by equation 18 corresponding to the first derivative
of M(Hor). Normalizing Vo as well as its integral function
and calibrating the MDL set-up using a standard Ni
magnetostrictive wire of known M-H loop, the μ-H and
M-H loops at the region of the receiving coil of the MDL
can be determined. Since the applied biasing field is dc,
the method determines the dc μ-H and M-H loops. As the
sample is vibrated by the propagating elastic pulse, the
method is an alternative vibrating sample magnetometer
(VSM) technique, so it can be named MDL-VSM
technique. A number of magnetostrictive ribbons and
wires have been tested according to this method. In this
report, indicative data of amorphous positive
magnetostrictive ribbons and wires of the rather typical
Fe78Si7B15 composition are presented. Fig. 10a and 10b
illustrate the dependence of the normalized MDL voltage
output, which is equal to the magnetic permeability μ, as
well as the magnetization M loops on the biasing field H,
concerning an amorphous Fe78Si7B15 magnetostrictive
ribbon, after stress-current annealing under 400 MPa and
0.5 Å for 10 minutes. Fig. 11a and 11b illustrate the same
response for the case of amorphous Fe78Si7B15 ribbon after
thermal annealing in 450 oC and Ar atmosphere for 1 h
and consequent slow rate cooling. The observed hysteresis
may be attributed to the partial crystallization of the
ribbon. Such a technique can be used for studying various
hysteretic properties of magnetostrictive materials.
-1,2
-0,7
-0,2
0,3
0,8
-2000 -1500 -1000 -500 0 50 0 1000 1500 2000
Biasing field (A/m)
Normalized MDL voltage outpu
t
(a)
-1,2
-0,7
-0,2
0,3
0,8
-2000 -1500 -1000 -500 0 500 1000 1500 2000
Biasing fie ld (A/m)
Norm alized M-H loo
p
(b)
Fig. 10. Permeability (a) and magnetization loops
(b) concerning Fe78Si7B15 amorphous wire after stress-
current annealing.
Magnetostrictive behaviour of ribbons and wires: analytical modelling and experimental validation 53
As an example the sharp and bistable behaviour of as-
cast amorphous magnetostrictive Fe78Si7B15 wires,
corresponding to the Large Barkhausen jump, can be
observed with this experiment, allowing the ability of
observing the uniformity of the bistable behaviour along
the length of the wire.
0
5
10
15
20
25
30
35
40
45
50
0 200 400 600 800 1000 1200 1400 1600
Biasing field (A/m)
MDL voltage output (mV
)
(a)
0
5
10
15
20
25
30
0 200 400 600 800 1000 1200 1400 1600
Biasing field (A/m)
Magnetization (a.u.
)
(b)
Fig. 11. Permeability (a) and magnetization loops (b)
concerning Fe78Si7B15 amorphous ribbon after thermal
annealing.
λ
-H loop
Keeping He and Hor constant, while the biasing field
Hoe changes, the peak amplitude of the MDL pulsed
voltage output Vo is given by equation 16, being
proportional to d
λ
(Hoe)/dH. Normalization process and
calibration against a standard Ni magnetostrictive wire of
known
λ
-H loop results in the dc
λ
-H function
determination. Among various tested magnetostrictive
materials, indicative results are presented concerning
amorphous as-cast positive Fe78Si7B15 magnetostrictive
wires. Fig. 12a and 12b illustrate the normalized MDL
response and the
λ
dependence on Hoe respectively.
Maintaining the basing fields Hoe and Hor steady and
changing the excitation field He, the peak amplitude of
Vo/He is given by equation 17. Thus, the integral of Vo/He
on He is proportional to the magnetostriction
λ
.
Normalization and calibration against a standard Ni
magnetostrictive wire of known
λ
-H loop results in the
λ
-
H loop determination. Fig. 13a and 13b demonstrate
indicatively the normalized MDL response and its integral
corresponding to the λ-H function for the case of as-cast
amorphous Fe78Si7B15 wires.
0
0,2
0,4
0,6
0,8
1
1,2
-2000 -1500 -1000 -500 0 500 1000 1500 2000
Biasing field (A/m)
Normalized MDL voltage output
(a)
0
0,2
0,4
0,6
0,8
1
1,2
-2000 -1500 -1000 -500 0 500 1000 1500 2000
Biasing field (A/m)
Normalized lamda-H loop
(b)
Fig. 12. Normalized MDL response on biasing field at
the excitation point (a) and integration of the MDL
voltage output corresponding to the dc
λ
-H loop (b).
0
0,2
0,4
0,6
0,8
1
1,2
-2000 -1500 -1000 -500 0 500 100 0 1500 2000
Pulsed field (A/m)
Normalized MDL voltage outpu
t
(a)
0
0,2
0,4
0,6
0,8
1
1,2
-2000 -1500 -1000 -500 0 500 1000 1500 2000
Pulsed field (A/m)
Normalized lamda-H functio
n
(b)
Fig. 13. Normalized Vo/He MDL response on the pulsed
field (a) and integration of Vo/He corresponding to the ac
λ
-H loop (b).
54 A. Mamalis, E. Hristoforou
On the M(H) and
λ
(H) results
The main advantage of the MDL-VSM technique with
respect to the VSM technique is the by-design ability of
non-destructive magnetic testing. Normally, in a classic
VSM, the sample has to be cut in small pieces in order to
be accommodated inside the VSM holder. Another
significant advantage is the ability of measuring
permeability, magnetization and flux density uniformity of
the under test specimen, by moving the position of the
receiving coil and the surrounding biasing coil.
Using this technique it is also possible to measure the
M-H loop of magnetostrictive elements, not having the
shape of acoustic waveguide by gluing them on a glass
substrate. Thus the elastic pulse generated either by
magnetostrictive or piezoelectric means, is coupled to the
under test magnetostrictive specimen via the glass
substrate and therefore, the dependence of μ and M may
also be determined. This method can be also applied for
the stress dependence determination of the M-H and λ-H
loop.
Controlling the temperature of the set-up, one can
determine the dependence of the μ-H, M-H and λ-H loops
on temperature. Accordingly, changing the biasing field
with a given frequency, always less than the frequency
corresponding to the pulsed current excitation period,
which is of the order of 1 ms thus corresponding to 1 kHz
maximum limit of biasing field frequency, the dependence
of μ-H, M-H and λ-H loops on frequency may also be
determined. Temporal dependence tests of μ-H, M-H and
λ-H loops may also be performed. All above mentioned
applications may improve the implementation of
magnetism and magnetic effects in engineering
applications [76-79].
5. Conclusions
The generation, propagation and detection of an
elastic pulse in the simplest MDL arrangements have been
investigated. Such a study has resulted in understanding
the mechanisms and parameters of elastic pulse generation
and detection, namely being the excitation and biasing
fields as well as the mechanical action on the MDL.
Understanding these mechanisms and parameters allowed
for the conceiving and development of new methods for
determining the magnetic properties of magnetostrictive
ribbons and wires.
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____________________________
*Corresponding author: eh@metal.ntua.gr
