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Tailoring the magnetization linearity

Journal of Magnetism and Magnetic Materials 500 (2020) 166327
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Journal of Magnetism and Magnetic Materials
journal homepage: www.elsevier.com/locate/jmmm
Research articles
Tailoring the magnetization linearity of Finemet type nanocrystalline cores
by stress induced anisotropies
Lajos K. Varga
Wigner Research Center for Physics, PO Box 49, 1525 Budapest, Hungary
Continuous stress annealing
Stress induced anisotropy
Linearity of magnetization curve
Distribution of anisotropy field
In this paper a theoretical and experimental investigation will be presented on the linearity of the flat hysteresis
loop obtained by continuous stress annealing of Finemet type nanocrystalline ribbon. The stability of the effective permeability versus the magnetizing field depend on the residual random distribution of local magnetization and can be characterized by i) the coefficient a in the expression of magnetization approaching the
saturation: M/Ms = 1 − a/H, valid in a restricted region, for Hlin < H < Hsat, where Hlin is the limit of
linearity and by ii) the distribution of the anisotropy field ΔHK. The theoretical upper limit of linearity is given by
the anisotropy field HK = Bs/µo.µeff. The linearity limit, Hlin, can be expressed as a difference between the value
HK and the half width of the anisotropy field distribution, ΔHK. The linearity will be measured by the ratio
R = Hlin/HK. The parameter R is almost constant (around 0.72) for large applied stresses and carefully selected
annealing parameters (furnace geometry, temperature distribution along the furnace and pulling velocity). For
small applied stresses (i.e. for effective permeability’s above 4000) however this linearity parameter is reduced
to zero and a potbellied loop appears. A possible explanation will be given for the stress dependence of linearity
parameter based on the back stress model of stress annealing.
1. Introduction
L·I p2
The emergence of new active components based on GaN opens the
way for increasing the switching frequency (several MHz) which leads
to the decrease of the passive components size like inductors and
transformers in power electronics. In general, at high frequency, spinel
ferrites are considered the best low losses magnetic materials. For example, Ni0.35Zn0.55Cu0.10Co0.014Fe2- δ O4-δ can be mentioned
which works at several MHz having a permeability around 250 [1].
However the ferrites work at relative low peak induction and the
maximal working temperature is also low because of low Curie temperature. These ferrite planar cores with Copper winding integrated in
PCB are suitable for low power distributed supplies only. For high
power applications cut and powder cores from metallic magnetic alloys
and nanocrystalline alloys with transversal induced anisotropy are
more promising for size reduction of power converter devices.
For size reduction of the energy converter inductive element in
switch mode power supplies (SMPS) reduction of the effective permeability is necessary as can be seen from the stored energy expression in
two different forms:
2·μo μeff
where L is the inductivity, Ip, is the peak current, Bp is the peak inductivity and V is the volume of the core. It turns out that the ratio of
the effective permeability versus the volume of the core is constant for a
given design of the SMPS:
μo ·L·I p2
The core volume can only be reduced by reducing the effective
permeability, i.e. by increasing the flatness of the hysteresis loop. This
is why in modern SMPS devices cores with effective permeability below
or around 100 are used. Such cores are available as cut cores, powder
cores and stress annealed nanocrytalline Finemet cores. The linearity of
these µ vs HDC curves can be characterized by the expanse of HDC = Hlin
, where the permeability is almost constant and decreases a certain Δµ
value (5% for example) only. The good linearity is a pre-requisite in
many power electronic applications like audio, solar and wind power
converters, filters, etc.
In this study, we propose to study the stress annealed Finemet
ribbon with low losses, relatively large saturation (Bs = 1.23 T)
E-mail address: varga.lajos@wigner.mta.hu.
Received 3 September 2019; Received in revised form 18 December 2019; Accepted 19 December 2019
Available online 20 December 2019
0304-8853/ © 2019 The Author. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license
Journal of Magnetism and Magnetic Materials 500 (2020) 166327
L.K. Varga
compared to ferrite materials and with variable permeability as a
function of applied stress maintaining the low coercive field (below
10 A/m). Applying an optimal nanocrystallization heat treatment, the
starting amorphous ribbon becomes a two - phase structure with about
70 vol% Fe-Si nano-sized crystallites embedded in a residual amorphous matrix. The stress annealed ribbon is brittle but still manageable
to wound up in toroidal core. The influence of processing parameters on
the magnetic properties of continuously stress annealed (CSA) nanocrystalline Finemet type alloy has been the subject of detailed study
since the first paper appeared in this topic [2]. Continuous or flash
annealing is an industrial applicable process where the ribbon passes
between two reels through an open tubular furnace. It was established
that the induced anisotropy, K, and the resulting effective permeability
(µeff·µo = Js2/2K) depend on the applied tensile stress only and does not
depend on the pulling velocity, whereas the elongation of the ribbon
during the tensile stress annealing does depend on the pulling velocity.
It should be mentioned that in the literature a general view is that the
stress induced anisotropy is due to the creep [3]. This is contradicted by
the fact that the induced anisotropy does not depend on the stress induced elongation, it depends on the magnitude of the applied stress
only [4].
Fig. 1. Schematic diagram of continuous stress annealing equipment.
2. Theoretical part
The magnetization curve for stress annealed samples is composed
from two regions only: the linear part
J= μeff ·μo ·H if H< Hlim
and the approach to saturation part
J/Js = 1 − a/H
if Hlim < H< Hsat ,
where “a” is the Neel constant and represent the heterogeneities of free
volume distributions whose internal stress fields produces a distribution
of magnetostrictive anisotropies preventing alignment to saturation.
The larger the parameter “a” the more difficult obtaining the saturation
and more restricted is the linear part of magnetization.
It is worth mentioning that equation (2) can be taken as the first
term of the expansion of the exponential:
J= Js ∗ exp( −a/H),
Fig. 2. Invers permeability as a function of applied stress.
d 2J
dH 2
μo ·μ
Our home built continuous (or flash) annealing equipment [7] is
similar to that applied by others [6] with some practical modifications.
The equipment presented schematically in Fig. 1 includes an open
tubular furnace cut along the length of the cylinder in order to place the
running amorphous precursor ribbon into the middle of the furnace. No
protective gas is applied during annealing. The ribbon is pulled reel-to
reel at constant velocities under a constant tensile stress. The sample
coils are rolled up subsequently from the bobbin containing the nanocrystalline ribbon. The pulling force between 1 and 40 N is measured
with a dynamometer F and is kept constant through a feed-back to the
motor with controlled torque. Both servo-motors have incremental
encoder which make possible to read the position values of the motors.
From these positions the positive or negative strain of the ribbon can be
determined. The negative strain appears at small stresses where the
contraction due to the amorphous –crystalline transformation is larger
than the elongation due to the creep.
(Fe73.5Si15.5B7Nb3Cu1) have been used for stress annealing experiments.
The ribbon width was 20 mm and the thickness of the ribbon was
constant 20 µm. The pulling velocity depends on the length of the
which is inversely proportional to the steepness of the magnetization
curve, µ.
The experimental determined linearity, Hli n = HKexp , however extend
up to a smaller value determined by the width of the p(HK) distribution,
ΔHK, where:
HKexp = HKtheo −
3. Experimental
Its width at half amplitude will be ΔHK.
The upper limit for the linearity is given by the anisotropy field HK:
HKtheo =
In the following we will study experimental these parameters of the
linearity as a function of induced anisotropy, i.e. as a function of stress
applied during annealing.
A single or two exponentials (corresponding to domain rotation, DR,
and domain wall movement, DWR) can be used to simulate the magnetization region between Hlin and Hsat.
The approach to saturation can also be characterized by the width of
the distribution of anisotropy field, ΔHK. The distribution of HK will be
determined by the Barandiaran method [5] as
p (HK ) = −H
For the characterization of the linearity we have adopted the ratio of
the experimental and theoretical determined anisotropy fields:
Journal of Magnetism and Magnetic Materials 500 (2020) 166327
L.K. Varga
Fig. 3. Quasi-DC hysteresis loop (a) and the differential permeability (b) at small pulling stress of 2.5 MPa (F = 1 N).
Fig. 4. Quasi-DC hysteresis loop (a) and the differential permeability (b) at medium tensile stress of 100 MPa (F = 40 N).
F = 40 N
B*H = 1.57*H - 926
a = 590
Fig. 5. The distributions of the anisotropy fields as a function of the applied
pulling forces.
Fig. 6. Determination of the curvature parameter, a, for the sample annealed
under stress σ = 100 MPa (F = 40 N).
tubular furnace and is set in order to obtain 4–6 s for the running time
through the furnace. The annealing is in air and the temperature is set
between 873 and 973 K. The pulling velocity of ribbon under tensile
stress annealing can be increased up to 1 m/s which makes the productivity of this method competitive with the magnetic field annealing.
Ku = − λsFeSi ·σi·νcr =
2μo ·μ
where λsFeSi is the saturation magnetostriction of the FeSi phase, σi is
the internal residual stress and νcr, is the crystalline fraction, Js is the
saturation polarization. The theoretical value for the product (µ·σ) can
be calculated from equation (8) as:
4. Results and discussion
First we have checked the back stress theory constant (µ·σ) for our
sample. The stress induced anisotropy, Ku, is of magnetoelastic origin
μ·σ =
3μo ·|λs|·υcr
Journal of Magnetism and Magnetic Materials 500 (2020) 166327
L.K. Varga
Fig. 7. a) Representation of curvature parameters ΔHK and “a” as a function of applied stress and b) relationship between these two parameters.
µ·σ has been experimentally determined to be 53850, as it can be
obtained from the steepness of the linear dependence of the invers
permeability versus applied stress presented in Fig. 2. Theoretically, a
similar value, µ·σ = 53535, can be obtained considering: Bs = 1,23 T,
λs = −10 ppm and νcr = 0,7. These values of the saturation magnetostriction and crystallized fraction are in accordance with the literature values [3].
In the following we will present some characteristic quasi – DC
magnetization curves using a sinusoidal excitation at f = 0,01 Hz frequency on a toroidal sample with 30 mm outer and 20 mm inner diameters.
For small applied pulling stress (2.5 MPa) see Fig. 3 the induced
anisotropy was not developing in the whole section of the sample,
therefore a potbelly hysteresis curve is obtained without a linear initial
part. The differential permeability shows no plateau as a function of DC
field, its value for small exciting field reaches a high value around
10 000.
For intermediate and large applied stresses the hysteresis loop
shows a beautiful linearity. The extent of linearity increases with the
applied stress, but interestingly the linearity ratio, R, remains almost
constant, for properly prepared samples, around 0,70–0,75.
Applying the Barandiaran method (Eq. (4)) the distributions of
Fig. 8. Stress annealed magnetization curves in normalized representation:
relative induction versus relative field. HK is the theoretical anisotropy field
given by Eq. (5).
Fig. 9. Comparison of the normalized magnetization curve a), and of normalized anisotropy field distribution b), for three commercially available cores with flat
loops: Cobalt – based zero lambda amorphous, stress annealed nanocrystalline Finemet samples from Magnetec and Imphy companies.
Journal of Magnetism and Magnetic Materials 500 (2020) 166327
L.K. Varga
anisotropy field, p(Hk), was obtained as a function of applied pulling
force. The results are presented in Fig. 5 showing that the higher is the
applied force the larger is the width of the p(HK) distribution.
Unfortunately, the second description of the approach to saturation
using equation (2) is not unequivocal. The parameter a, characterizing
the extent of the curvature between the linear and saturated parts of the
magnetizing curve can be determined by linearizing the B-H curve,
within a range of fields, Hlin < H < Hsat, taken rather arbitrarily, and
in addition the parameter Js resulting from the linear fit is higher than
the experimental determined Js = 1.23 T. Eq. (2) can be written after
linearization as:
cores with similar linearity to the Imphy’s cores.
5. Conclusions
- The minimal stress region should be further studied to understand
the pot belly loop and the lack of plateau in the field dependence of
differential permeability.
- The linearity ratio, R, for the stress annealed nanocrystalline cores is
almost the same for medium and high applied stresses. For our stress
annealing technology R is around 0.72–0.74.
- On this base, it is possible to represent the DC hysteresis loops in
normalized form J/Js versus H/HK, where the loops are almost coincident for a given stress annealing technology.
J ·H = Js ·H − Js ·a
Taking the data of the magnetization curve from the Fig. 4, the
value for the parameter “a” can be obtained from the intercept of the fit
and the apparent saturation from the steepness of the line (see Fig. 6.).
The curvature parameters determined as a function of the applied
stress are presented in Fig. 7a, together with the width, ΔHK, of the
anisotropy field distribution. Within the error of determinations both
parameter increases linearly as a function of the applied stress indicating a close relationship between these parameters (see Fig. 7b):
ΔHK ~ 1.6*a.
The linearity ratio can be calculated as:
HK −
2 HK
Declaration of Competing Interest
The authors declare that they have no known competing financial
interests or personal relationships that could have appeared to influence the work reported in this paper.
This study was partially supported by the Hungarian National
Research, Development and Innovation Office, Grant Nr. KFI_16-12016-0079 and by the mobility grant of Hung. Acad. Sciences Nr. NKM91/2019 and by Hungarian OTKA grant No. K 128229.
Inserting the stress dependence of ΔHK from Fig. 7a:
ΔHK = 53+ 8.71*σ and HK from equation (9): HK = (3λsνcr /Js)*
σ = 17* σ in equation (11) we get
R = 0.74 − σ , which is almost independent from the applied stress
for σ > 10 MPa.
This allows us to present all the magnetization curves in normalized
form as B/Bs versus H/HK (see Fig. 8). In this depiction the magnetization curves almost coincide, having an effective permeability,
µeff = 1.
For comparison 3 other commercially available cores with transversal induces anisotropies have been measured also with similar effective permeability’s: magnetic field annealed Co-based (zero lambda)
amorphous (µ = 1500) core and two stress annealed Finemet samples
from Magnetec Hungary (µ = 1600) and from Aperam Alloys Imphy
(France) (µ = 1400). In Fig. 9a the normalized magnetization curves
are presented and in the Fig. 9b the normalized anisotropy field distributions are shown determined from the double derivative of Fig. 9a
using equation (4). The linearity parameter R can be obtained in a
straightforward way as the place of the half width at half amplitude of
the normalized p(HK) distribution. The linearity is the best for the
amorphous Co based core (R = 0.90). Comparing the two stress annealed sample, the linearity of the Magnetec core (R = 0.80) is far
better than that of Imphy’s core (R = 0.72). Our technology provides
Appendix A. Supplementary data
Supplementary data to this article can be found online at https://
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