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Mauricio Esguerra
EPCOS AG, Ferrites Division, Product Design Department
P. O. Box 801709, D-81617 Munich, GERMANY
Abstract – An extension to Hodgdon´s hysteresis equation and its solution for minor
loops in terms of measured major loops is presented. Various application relevant
parameters for soft ferrite materials are derived and the results compared to well-
known models for the small signal regime.
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Many attempts have been done in the
past to quantitatively describe
hysteresis loops. One of the most
promising approaches is the
description of the memory-effect
inherent to hysteresis by means of
differential equations [1,2]. The one
formulated by Hodgdon with the B-
field as dependent variable,
(
)
(
)
(
)
G+
G%
G%
GW
I % + J %= ⋅
− +
α
sgn
(1)
has two major advantages:
• direct integrability
• applicable to most practical
problems, where voltage (and
consequently flux density) rather
than current is the known signal
Hodgdon´s own solution starts from a
parametrization of the curve, whereas
the author’s approach is based on
regarding the major loop as the
particular solution of the equation [3].
From there all other solutions, e.g.
minor loops, can be obtained without
explicitly considering the material
functions I% and J%. The upper
+¶ % and lower +¶ % minor loop
branches can be therefore expressed
in terms of the corresponding
measured major loop branches + %
and + %.
The aim of the present work is to
improve two major aspects. On the
one hand calculated minor loops close
to saturation deviate systematically
from measured ones. On the other
hand the quality and/or number of
points of the measured major loops is
often not enough for an accurate minor
loop calculation.
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In order to remedy the first deficiency,
the differential equation has been
modified by regarding the parameter
α
as a function of % rather than of the
amplitude
%
ˆ
[3], the equation being still
integrable. From the solution for sym-
metric minor loops given in [3], eq. (3)
follows for the integral over
α
%:
)
ˆ
()
ˆ
(
ˆ
)
ˆ
(
ˆ
)
ˆ
(
ˆ
ˆ
)(
%+%+
%+%+
H
−
−
=
∫
−
′′
α
(2)
According to the symmetry of eq. (1)
the function
α
%is even, such that the
general integral is given by:
)(
ˆ
)(
)()(
ˆ
)()(
ˆ
)(
ˆ
)(
),(
11
11
22
22
)(
21
2
1
−
−
⋅
−
−
=
∫
≡
′′
α
(3)
The commutation curve needs also to
be modified with respect to [3], eq. (5)
in order to account for the above
mentioned improvement of the
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221
saturation behavior. The following
DQVDW] takes this into consideration:
(
)
(
)
(
)
ξ
+%+%+
ˆ
1
ˆˆˆ
−−=
(4)
The exponent ξ can be determined
from the slope of the commutation
curve around the origin:
)
11
(
1
;
00
+
%
µµµ
ααξ
−== (5)
The solution for a (not necessarily
symmetric) minor loop reads finally:
( ) ( )
(
)
),( %%&
%++
%+%+
−
+=
′
(6a)
( ) ( )
(
)
),( %%&
%++
%+%+
−
+=
′
(6b)
The extreme field strength points +
and + are given by
),(
1
),(
)()(
),(
)(
),()(
),(
−
−+−⋅
=
(7a)
),(),( −−−=
(7b)
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To improve accuracy, empirical fit
functions for the lower and the upper
branches of the major loop have been
found as an extension of the “linear”
approximation given by eq. 8 in [3]:
(8b)
(8a)
+
%
%
%
%+
+
%
%
%
%+
−
−
⋅=
+
−
⋅=
1
1
)(
1
1
)(
µµ
µµ
The parameters + (coercive field),
slope at + ) and % (saturation flux
density) have an obvious physical
meaning, whereas D and E describe
the squareness of the loop. Fig. 1
shows an example for Material N87 at
100°C with following parameters: µ
i
=
3980, % =0.398 T, D=4.91, E=4.32,
+ =11.9 A/m, =5148. The fit curves
(lines) and the measurement (points)
show an excellent agreement.
20 0 20 40 60 80 100 120 140 160 180 200
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
H [A/m]
B [T]
Fig. 1: Major hysteresis loop´s fit functions
200 150 100 50 0 50 100 150 200
0.4
0.3
0.2
0.1
0
0.1
0.2
0.3
0.4
H [A/m]
B [mT]
Fig. 2: Symmetric Minor Loop with = =0.25 T (solid line)
with major loop (dotted line)
200 150 100 50 0 50 100 150 200
0.4
0.3
0.2
0.1
0
0.1
0.2
0.3
0.4
H [A/m]
B [mT]
Fig. 3: Non-Symmetric Minor Loop with =-0.11 T and
=0.39 T (solid line) with major loop (dotted line)
Figs. 2 and 3 show a symmetric and a
non-symmetric minor loops with
∆
%=0.5 T for N87 at 100°C. With the
present model it is possible to yield
good agreement not only in the
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222
symmetric case, but especially in the
way the minor upper branch follows
the major branch close to saturation.
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The main purpose of hysteresis
modeling is the evaluation of core
losses. However, the calculation of the
enclosed area of a minor loop cannot
be evaluated by a closed integral. In
the approximation of symmetric, small
excitations the losses are given by:
( ) ( )
[ ]
3
2
0
ˆ
0
ˆ
3
4
2
α
≈
′
−
′
=
∫
(9)
This result coincides with the previous
derivation [3] and is in accordance with
Rayleigh´s law [4].
The hysteresis material constant (in
the limit
0
ˆ
→%
) is related to the
harmonic distortion and reads:
2
0
11
3
4
−=
+
µµπµ
η
(10)
Both the core losses and η
B
increase
with the difference between the inverse
values of and , i. e. the “real” and
the “ideal” permeabilities [3].
The reversible permeability under dc-
bias can be given in the approximation
D
≈
E and
0
ˆ
→%
by:
µ
µ
µ µ
α
( )
( )
=
+ − ⋅
−
⋅ +
−
⋅ − −
⋅ −
⋅
−
1 1
1
1
1
1 2 1
1 1
2
1
(11)
This equation shows that the reversible
permeability depends from two terms
representing both loss-independent
(squareness parameter D) and loss-
dependent (1/ – 1/ ) contributions.
Thus, improving the loss quality of a
material does not necessarily yield an
improvement of its dc-bias behavior.
Fig. 4 shows the normalized reversible
permeability as a function of dc-flux
density referred to saturation for
material T55 at 25°C. For reference
the universal curve after Gans [4] is
also shown with an excellent
agreement.
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[1] D.C. Jiles and D.L. Atherton, J.
Magnetism Magn. Mater. 61 (1986) 48.
[2] M. L. Hodgdon, IEEE Trans. Magn.
24 (1988) 3120.
[3] M. Esguerra, J. Magnetism Mater.
157/158 (1996) 366-368.
[4] R.M. Bozorth: Ferromagnetism,
D.van Norstrand Co.Inc., New York
1951
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
B/Bs
µrev/µi
Fig. 4: Calculated normalized reversible
permeability vs. flux density (solid line) in
comparison to the Gans curve (dotted line)