International Journal of Applied Electromagnetics and Mechanics 1 (2018) 1–8
DOI 10.3233/JAE-170075
IOS Press
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Hysteresis measurements of soft magnetic
materials and estimation of residual stresses
Nasir Mehbooba,b,∗ , Muhammad Farooq Nasirb , Roland Grössingera , Peter Osera ,
Martin Kriegischa and Paul Fulmekc
a
Institute of Solid State Physics, TU Vienna, Wiedner Hauptstr. 8-10; A-1040, Austria
Department of Physics, Riphah International University, Islamabad, Pakistan
c
Institute of Sensor and Actuator Systems, TU Vienna, Gusshaustr. 25-29; A-1040, Austria
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Abstract. Frequency and temperature dependence of the hysteresis loops (both major and minor hysteresis loops) of wellknown materials such as pure Fe and pure Ni in the as-cast state and after a heat treatment, were measured and analyzed.
Furthermore temperature dependent magnetostriction of the as-cast and annealed Ni sample were measured and residual stresses
were estimated by assuming the magnetoelastic energy as the origin of coercivity H c . The hysteresis measurements were made
on fully automized hysteresigraph which is handled by a labview program. The system works with ring- or frame-shaped samples
in the frequency range from 0.1 Hz to 2 kHz and in the temperature range from −77 K to 493 K.
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1. Introduction
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Keywords: Magnetic hysteresis, coercivity, residual stresses, magnetostriction
An accurate measurement of the hysteresis loop of the soft magnetic materials is an old and in
principle well known topic, see e.g. [1]. Commercial magnetometers can be used for magnetization
measurements. Nevertheless, measurements of samples in a magnetically open system (e.g. vibrating
sample magnetometer) are fundamentally different from measurements in magnetically closed system
(ring or frame-shaped samples), because in closed system no stray field is produced. Standard commercial
magnetometers offer a sufficiently high magnetic field (electromagnet, superconducting magnet), which
allows the determination of the saturation magnetization (major loop); however the measurement of minor
loops is practically not possible with these systems. Minor loops can be traced by applying decaying field
to study the microscopic mechanism of magnetic reversal such as first order reversal curve (FORC). A
standardized method for soft magnetic materials is the well-known Epstein frame (IEC 60404-2), which
is generally used to measure losses as well as the hysteresis loops of Fe-Si transformer sheets [2]. The
disadvantage of this method is that it needs large size samples (25 cm to 50 cm long) of the sheets and
can only be used at room temperature. For scientific investigations smaller ring (toroid) or framed-shaped
sample are more convenient. Finite element calculations show that even at high exciting currents all flux
* Corresponding author: Nasir Mehboob, Riphah International University, I-14 Islamabad, Pakistan. Tel.: +92 51 8446000 7;
Fax: 92 51 5125170; E-mail: nasir.mehboob@riphah.edu.pk.
1383-5416/18/$35.00 © 2018 – IOS Press and the authors. All rights reserved
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N. Mehboob et al. / Hysteresis measurements of soft magnetic materials and estimation of residual stresses
remain inside the ring-shaped or magnetically closed sample. In such case one can gets a good and reliable
low field behavior (minor loops, recoil curves etc.) of the material. Nevertheless approaching saturation
magnetic field needs special care in winding of sample, otherwise the saturation magnetization value is
often less accurate.
Apart from measurement techniques and sample shape, magnetic properties of the materials such as
coercivity, remanence, permeability as well as shape of hysteresis loop are greatly influenced by residual
stresses, which develop during manufacturing, punching, drilling or assembling process etc [3,4]. The
movement of domain and domain wall (origin of coercivity H c ) are hindered by residual stresses through
magnetostrictive coupling, hence contributing additional magnetoelastic energy which causes an increase
in coercivity. The relation between coercivity and magnetostriction can be simply described by the
following equation [5–7].
𝜎𝜆 + ⟨𝐾⟩
μ𝑜 𝑀𝑠
(1)
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𝐻𝑐 =
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Where 𝜆 is magnetostriction, 𝜎 is the stress and product of the two (𝜎𝜆) is magnetoelastic energy, μo is
the vacuum permeability and ⟨K⟩ is effective magnetocrystalline anisotropy.
Number of destructive, semi destructive or nondestructive methods have been developed and discussed
in detail to measure residual stresses [8–11]. The relation (1) allows an analysis and estimation of the
residual stress by using the temperature dependence of the coercivity data at 0 Hz and by considering the
magnetoelastic energy (𝜎𝜆) as main contributor.
In present work we presented a temperature and frequency dependent hysteresis of as cast pure iron,
nickel and annealed iron and nickel at 400 °C for 4 hrs. Hysteresis measurements were performed on closed
ring-shaped samples in order to avoid effects of demagnetizing field. Magnetostriction were measured on
the same pure as-cast and annealed nickel samples as a function of temperature. Furthermore, on the basis
of temperature dependent hysteresis loop and magnetostriction, the level of residual stress of nickel sample
was estimated by applying a simple approach based on the correlation of magnetoelastic energy (𝜎𝜆) and
coercivity.
2. Experimental setup
Frequency and temperature dependent hysteresis loops of pure Fe, pure Ni samples in the as-cast and
annealed state were measured on fully automated hysteresigraph based on a similar system as describe
in [12–14]. The main parts of the hysteresigraph include a computer, a national instrument measurement
card (NI 6120) and power amplifier (APEX or KEPCO). KEPCO amplifier is used for generating the
magnetizing current. The whole setup is handled by Labview program. Block diagram of the automated
hysteresigraph is shown in Fig. 1(a). Such type of system allows to obtain the desired waveform of
current I (t) e.g. triangular, sinusoidal, saw tooth etc., frequency dependent measurements and automatic
demagnetization of the sample which is important for measuring accurate permeability. Moreover, taking
average over the whole loops improves signal–noise ratio. For comparison, commercial ferrite sample was
also measured.
The hysteresis loops were measured using ring shaped samples equipped with primary and secondary
windings (N 2 ). The magnetic field H produced by the current I is given by the formula: H = N 1 I∕lm
(where lm is magnetic path length and N 1 number of primary winding) and Induction field (B-field) was
𝑉𝑑𝑡
computed according to 𝐵 = ∫𝐴𝑁
where v is the induced voltage in secondary winding N 2 and A is area of
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N. Mehboob et al. / Hysteresis measurements of soft magnetic materials and estimation of residual stresses
Fig. 1. (a) Block diagram of the LabView controlled experimental setup. (b) Dependence of magnetization at external field as
function of number of primary winding of the ring-shaped pure Fe sample.
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cross-section of ring sample). The parameters N 1 , N 2 , lm , and A are given in result and discussion section
for each measured samples. The samples were demagnetized before each measurement.
The temperature dependence of the hysteresis loops in the range from room temperature (RT) to 220
°C were measured by placing the sample in a temperature controlled furnace. Temperature class 220
°C copper wire was used for winding. While hysteresis measurements at fixed low temperature were
performed by inserting sample in a Vacuum (thermos) flask containing freezing water (0 °C, 273 K), dry
ice (−78 °C, 195 K) or liquid nitrogen (−176 °C, 77K) can also be used depending on the requirement.
The sample temperature was determined by using a PT 100 sensor.
The magnetostriction was measured using a strain gauge method with a fast ac-bridge (Hottinger
KWS 3085A) in a pulse field system (pulse duration 50 ms, maximum field 5 T). For high temperature
measurements a vacuum isolated furnace was constructed. The strain gauges (HBM 1-LC11-3/120) can
be used up to 240 °C.
3. Fundamental materials
Measurements of hysteresis loop as a function of the frequency and temperature were performed on the
samples with following specifications: pure Fe (as cast), pure Fe (after annealing; 400 °C, 4 h) (A = 2. 1
× 10−5 m2 , lm = 0. 0785 m), pressed isotropic Ni (as cast), pressed isotropic Ni (after annealing; 400 °C,
4 h) (A = 1. 056 × 10−5 m2 , lm = 0. 04803 m) and commercial ferrite sample (WURTH ELEKTRONIK
product number 74270104) (A = 5. 16 × 10−5 m2 , lm = 0. 04542 m). The samples were measured from
RT up to 200 °C (473 K). The triangular form of H (t) was selected in order to obtain a constant dH∕dt,
which also permits to calculate the relative permeability of a sample as described in [5].
4. Results and discussion
The basic requirements while measuring hysteresis loop on above mentioned hysteresigraph are (i) to
find the most exact way to minimize the losses of induction field and (ii) to obtain the high field behavior.
N. Mehboob et al. / Hysteresis measurements of soft magnetic materials and estimation of residual stresses
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Fig. 2. Frequency dependent hysteresis loop measured on (a) pure Fe (as cast) and (b) commercial ferrite at room temperature.
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We measured the induction field in pure as cast Fe at 0.1 Hz, as function of number of primary windings
(N 1 ) while keeping number of secondary windings (N 2 ) constant. The magnetic field H produced by
the current I with increasing primary winding density, a systematic increase of the magnetization was
achieved as shown in Fig. 1(b). However this is not sufficient to obtain high field behavior as visible
by comparing the shape of the hysteresis loops – the hysteresis curves measured with different numbers
of primary windings are not identical (inset Fig. 1(b) flux density against number of windings). This
demonstrates that the use of a careful winding scheme is very important for measuring correct and accurate
magnetization curve. Comparative investigations using a single sheet tester gave strong deviations in the
shape and magnitude of the loop values [13]. The disadvantage in present system is that the unavoidable
air gap between adjacent windings and also between winding and sample which influences the shape of
the loops. In such case the “true” magnetic field inside of the sample can-not be estimated accurately.
Figure 2(a) shows the frequency dependence of the hysteresis loops measured on as cast Fe sample
at room temperature. The effect of eddy currents is obvious from widening of loops with increasing
frequency, contrast to the hysteresis loops of commercial ferrite (WURTH ELEKTRONIK) measured
with frequencies range 100 Hz to 2000 Hz as shown in Fig. 2(b). As can be expected the hysteresis loops
of ferrite are identical at all frequencies without significant eddy currents effect.
Figures 3 and 4 show a linear behavior of coercivity H c (half width of symmetric saturated hysteresis
loop) of pure as cast Fe and Ni versus square root of frequency (f 1∕2 ), which is characteristics of eddy
currents effect [5]. From the linear behavior “true” coercivity (H c at 0 Hz) was obtained by extrapolation
to zero frequency (see-inset Figs 3 & 4). Similar behavior was shown by heat treated pure Fe and Ni,
however the coercivity of pure Fe and heat treated Fe is about 10 kA/m and 4.5 kA/m respectively at a 50
Hz frequency. Similarly for pure Ni and heat treated Ni the value of coercivity is 6.7 kA/m and 1.47 kA/m
at 50 Hz respectively. The low coercivity in heat treated samples is due to decrease in stress anisotropy
and magnetoelastic energy contribution.
Figure 5(a and b) shows the temperature (well below the Curie temperature) dependence of the
coercivity obtained from hysteresis loops measured at different frequencies on pure as-cast Fe and ascast Ni samples respectively. It decreases linearly within measured temperature range. The decreased in
coercivity is due to release of deformation stresses and hence lessening of anisotropy and magnetoelastic
energy with increasing measuring temperature. Pure Ni sample shows a steep slope of coercivity with
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Fig. 3. The coercivity versus square root of the frequency measured on as cast pure Fe at different temperatures.
Fig. 4. The coercivity versus square root of the frequency measured on as cast pure Ni at different temperatures.
temperature as compared to pure Fe sample which might be due to relieve of some casting stresses at
higher rate in nickel sample.
In order to investigate average residual stress additionally the temperature dependence of magnetostriction was measured with the same polycrystalline pure Ni sample. Figure 6(a) shows magnetostriction data
obtained for polycrystalline pure Ni sample in comparison with literature values reported for Ni single
crystal [15–18]. The systematic deviation of magnetostriction data of present work (Fig. 6(a)) at higher
temperature may be due to problems with the absolute temperature measurement at elevated temperatures.
In order to estimate the stress (𝜎), the product of temperature dependent coercivity and saturation
magnetization (μo Ms . H c ) is plotted as a function of the temperature dependent magnetostriction as shown
in Fig 6(b). By using the relation between coercivity and magnetostriction described by the Eq. (1), one
can estimate the average residual stress in the sample. The slope of this plot by linear fitting gives the
average residual stress 𝜎 of the sample. For the as cast Ni sample residual stress value of about 25 MPa was
N. Mehboob et al. / Hysteresis measurements of soft magnetic materials and estimation of residual stresses
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Fig. 5. (a) Temperature dependence of the coactivity measured at different frequencies on as cast pure Fe and (b) as cast pure Ni.
Fig. 6. (a) Temperature dependence of magnetostriction of pure Ni in comparison with present measurement (black). (b) The
product µo Ms ∗ H c as a function of magnetostriction plotted for as cast pure Ni, and annealed Ni.
estimated whereas for the annealed Ni sample residual stress of about 13 MPa was obtained. The reduction
of residual stress in annealed sample is due to release of residual stresses and decrease fraction defects.
Hence, annealing eliminates the undesirable residual stresses, decreases the coercive force and increases
the permeability. Internal stresses of about 200 MPa are reported for rapidly quenched amorphous Ni
ribbons [19,20].
5. Conclusion
A Labview controlled hysteresigraph allows good and accurate measurements of the frequency and
temperature dependence hysteresis of soft magnetic materials. This system provide facility to avoid
demagnetizing field effect which influence the shape of hysteresis loop by using magnetically closed
sample. The reliability of the developed hystersigraph was checked by measuring the frequency and
N. Mehboob et al. / Hysteresis measurements of soft magnetic materials and estimation of residual stresses
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temperature dependent hysteresis loop of pure as-cast Fe, Ni, and annealed Fe, Ni samples and compare
with the hysteresis loop of commercial ferrite. No eddy current effect found in ferrite sample in contrast to
pure Fe and Ni sample. The coercivity at 0 Hz (“true” coercivity) for pure Fe was obtained by extrapolating
the linear behavior of coercivity versus square root of frequency.
Magnetic hysteresis and magnetostriction are stress sensitive properties of materials. Therefore temperature dependence of the coercivity and the magnetostriction were measured on pure as-cast and annealed
Ni sample and residual stresses were calculated from the relation of magnetoelastic energy and coercivity.
The Ni sample annealed at 400 K for 4 hrs exhibit almost 50% stress reduction as compared to as-cast
nickel.
Acknowledgements
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This work was supported in part by Austrian FWF under Grant Proj.Nr: S10406-NI6 and by the project
No. 101/09/1323 of the Grant Agency of the Czech Republic.
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