# Racetrack fluxgate sensor core demagnetization factor

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Sensors and Actuators A xxx (2007) xxx–xxx
Racetrack fluxgate sensor core demagnetization factor
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J. Kubı́k a,∗ , P. Ripka b
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a
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b
Q1
Tyndall National Institute, Lee Maltings, Prospect Row, Cork, Ireland
Faculty of Electrical Engineering, Czech Technical University in Prague, Technická 2, 16627 Praha 6, Czech Republic
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Received 23 August 2007; received in revised form 19 October 2007; accepted 19 October 2007
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Abstract
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Keywords: Fluxgate; Racetrack; Demagnetization factor
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DP
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The demagnetization factor of a fluxgate core plays an important role in the resulting sensor sensitivity and noise. The global (magnetometric)
demagnetization factor of fluxgate sensor ring cores was evaluated using finite element modelling. This new method was verified using measured
effective demagnetization factors for ring cores and subsequently used in the modelling of the global demagnetization factor of racetrack fluxgate
cores. The results of modelling and measurements were compared and show a very close match. Based on modelling, an empirical formula is
presented to quickly evaluate the fluxgate racetrack core global demagnetization factor based on core geometry and material permeability. Finally,
the modelling, measurement and empirical estimation were compared showing a good agreement on five measured cores. The derived empirical
formula can be easily utilized in designing the sensor racetrack core shape.
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1. Introduction
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The demagnetization factors of ferromagnetic objects have
been studied both analytically and experimentally in many
publications. When the ferromagnetic object is exposed to a
homogeneous magnetic field Hext , the magnetic field in the ferromagnetic object HC is smaller than the external field Hext . (In
general, also the direction of the HC vector may be different from
Hext , but we will not consider this case here.) The difference is
characterized by the demagnetization field DMC , acting against
the external field:
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HC = Hext − DMC .
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(1)
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HC =
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∗
Hext
Hext
=
1 + Dκ
1 + D(μ − 1)
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where D is a dimensionless demagnetization factor (in the general case it is a tensor and in our simplified case it is a number
between 0 and 1), and MC = κHC is the magnetization. From that
we can write:
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Corresponding author.
E-mail addresses: [email protected] (J. Kubı́k),
[email protected] (P. Ripka).
1
2
(2)
An important fact is that the demagnetizing field is homogeneous
only in the case of ellipsoidal objects. In that case, D is a constant
for the whole object and it is independent of the permeability and
depends only on the object geometry (supposing that the object
is made of isotropic and linear magnetically soft material. The
tables and formulas for the calculation of demagnetization factors for ellipsoidal shapes were proposed by Osborn [1]. In all
other cases than the ellipsoid, the local demagnetization field is
a function of position in the volume of the object, but the local
demagnetization factor is still independent of μr by definition.
Therefore, it is impossible to generalize the demagnetization
factor to such objects without some sort of averaging. The literature distinguishes between two types of averaging [2]: ballistic
(or local, fluxmetric) and magnetometric (or global) demagnetization factors. The magnetometric (global) demagnetization
factor concerns the whole average magnetization of the object
(corresponding to a “magnetometric” measurement with a large
search coil compared to the object’s dimensions) and the ballistic
demagnetization factor corresponds to the average magnetization in the object’s midplane (corresponding to measurement
with a very short search coil at the object’s midplane). Notice
the meaning of the “large search coil” for the traditional magnetometric insertion measurement: the magnetization of the object
is measured by inserting the object inside such a coil or turning
it by 180◦ . The voltage induced in the coil is then integrated.
doi:10.1016/j.sna.2007.10.066
Please cite this article in press as: J. Kubı́k, P. Ripka, Racetrack fluxgate sensor core demagnetization factor, Sens. Actuators A: Phys. (2007),
doi:10.1016/j.sna.2007.10.066
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Vi = −N2 μ0 HC S
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dμr (t)
dt
(3)
The longitudinal in-axis magnetic field HC in such a core is
equal to the longitudinal external magnetic field Hext . In case
of a different core shape, the demagnetization factor has to be
taken into account. This can be rewritten as the decrease of the
material relative permeability μr to the apparent permeability
μA [5]:
μr
μA =
1 + D(μr − 1)
(4)
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Vi = −N2 μ0 HS
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Eq. (5) clearly indicates that the demagnetization factor D of the
sensor core is one of the key factors in the fluxgate sensor sensitivity. In general, the highest sensor output would be obtained
for a demagnetization factor D equal to zero. Demagnetization
factor of 1 would yield a zero output.
Furthermore, the demagnetization factor plays an important
role in the sensor noise performance. The proportionality of the
external noise field to the demagnetization factor was shown in
Ref. [6].
One of the first attempts to estimate the demagnetization factor of the common fluxgate sensor core shape – the ring core –
was made in 1972 by Burger [10]. The effective demagnetization factor (the demagnetization factor with respect to a given
core and measurement coil geometry) was calculated from the
measured pick-up coil inductance with (Lcore ) and without (Lair )
inserted core [6]. The apparent core permeability μA and the
effective demagnetization factor D for a given sensor core with
cross-section area Acore and given coil with cross-section area
Acoil can be calculated using equations (from Ref. [6]):
CO
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(5)
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dμr (t)
1−D
[1 + D(μr (t) − 1)]2 dt
EC
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Substituting the apparent permeability for the relative permeability, and differentiating, the fluxgate output voltage becomes:
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Notice that both μA and D are in this case not exclusive properties of the core, but slightly depend also on the coil geometry.
For coils wound around the core and having a similar length as
the core, the effective demagnetization factor is not very different from the global demagnetization factor. This may not be true
for coils much shorter or longer. For completeness, notice that
the pick-up coil can also be located inside the core [8].
Experimentally measured data from other articles were collected and an empirical equation for the local demagnetization
factor Dlocal of ring core with cross-sectional area A and diameter d was derived (7) based on the experimental data covering
a large range of fluxgate ring-core dimensions [5].
Dlocal = 1.83
Lcore −Lair
Acoil
×
+1
μA =
Lair
Acore
(μr /μA )−1
and D=
μr −1
(6)
A
d2
(7)
The relationship between local and global demagnetization factors Dlocal and Dglobal was determined for a ring core in Ref.
[5]:
Dglobal = 2Dlocal +
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Analytical solutions of demagnetization factors were presented for several special cases of objects such as cylinders [2]
or rectangular prisms [3,4]. When a square cross-section prism
with the same length to width ratio is compared to an elongated
spheroid of the same length/width ratio, the demagnetization
factor of a rectangular prism is much higher.
Demagnetization factor plays an important role in fluxgate
sensor core design. The most interesting property for a fluxgate
sensor designer is the demagnetization factor in the direction of
the sensor sensitivity axis. Thus, from now on, the demagnetization factor in the direction of measured field will be discussed
(unless otherwise noted). The fluxgate sensor output formula (3)
derived from Faraday’s law of electric induction applies only
to sensors with infinitely long cores (with zero demagnetization factors). Vi denotes voltage induced to pick-up coil with
N2 turns, μr is the relative permeability of the sensor core with
cross-section area S.
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1
1
= 2Dlocal +
κ
μr − 1
(8)
Finally, the analytical approach to ring core geometry demagnetization has been recently published [11] showing a good
agreement with measured values.
The racetrack geometry is another common fluxgate sensor
core shape, but to the best of authors’ knowledge neither analytical calculations nor empirical estimations of demagnetization
factor have been published so far.
2. Modelling and veriﬁcation on ring cores
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FEM (finite element method) models of the demagnetization
field can be used for the calculation of global demagnetization
factors. We will first verify this method on a ring-core geometry
and later use it for racetrack sensor cores.
The ring cores data collected in Ref. [5] will be used for
verification of the models. FEM modelling can provide data on
magnetic field vector HC and magnetic flux density vector BC at
each point inside the ferromagnetic core. This data can be used
to determine the magnetometric demagnetization factor of any
core shape. The FEM model consists of a core with defined permeability exposed to an external homogeneous magnetic field
Hext in the direction of intended demagnetization factor determination. When considering the demagnetization factor in one
direction (z-axis in our case) then only collinear components of
HC (x,y,z) and BC (x,y,z) vectors are used for the calculation of
the magnetometric demagnetization factor Dm .
The relation (9) between external field Hext , the average of
core internal field collinear component HC (z) and core average
magnetization collinear component MC (z) over the whole core
volume is the basis of Dm calculation.
HC (z) = Hext − Dm MC (z)
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(9)
Please cite this article in press as: J. Kubı́k, P. Ripka, Racetrack fluxgate sensor core demagnetization factor, Sens. Actuators A: Phys. (2007),
doi:10.1016/j.sna.2007.10.066
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The substitution of the average magnetization (10) into (9) yields
(11).
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MC (z) =
BC (z)
− HC (z)
μ0
(10)
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This approach is not limited to linear materials and thus the
modelled material may have a permeability defined by a BH
curve.
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Dm =
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Fig. 2. Global demagnetization factors of selected ring cores with cross-section
area A and diameter d: comparison of measured values, empirical equation and
FEM model.
core used by Clarke [9]. The RMS errors of FEM modelled and
empirically estimated ring core magnetometric demagnetization
factors compared in Table 1 are 0.53 × 10−3 and 0.67 × 10−3 ,
respectively.
The FEM models provide magnetometric demagnetization
factor values for ring cores which show a strong correlation
with the measured values and also with the values obtained from
empirical formulas (7) and (8) by Primdahl et al. [5]. The FEM
models will be used when modelling the global demagnetization
factor of racetrack cores to provide data on a wide range of cores
to create an empirical estimation formula for this type of core
without actually physically creating cores. Real cores then will
be used for verification of this formula.
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The principle of the calculation of the global demagnetization factor from FEM models was verified using alreadyTable 1
measured fluxgate sensor cores demagnetization factors. Seven
of twenty-six ring cores presented in Ref. [5] with known permeability values covering a wide spectrum of dimensions were
chosen and FEM modelled. A linear ferromagnetic material was
used for modelling (constant relative permeability). The cores
were exposed to an external homogeneous field Bext = 50 ␮T
(corresponding to Hext ≈ 40 A/m). An example of the magnetic
field flux density (collinear to the applied field in z-axis direction) distribution for the thickest ring core measured by Primdahl
et al. [5] is shown in Fig. 1. An external magnetic flux density
Bext = 50 ␮T applied in the z-axis direction creates the maximum
core flux density Bmax = 15 mT. The magnetometric demagnetization factor was calculated from the results of the FEM analysis
using (11). The results of the measurements (from Ref. [5] and
references therein) recalculated to the global demagnetization
factors, FEM models and an empirical estimation using (7) and
(8) are compared in and Fig. 2 with the dimension notation of
Fig. 3. The maximum deviation of the FEM model from the
measured value was +18% for a ring core used by Ripka and
Primdahl [7] formed out of a stack of 8 rings etched from an
amorphous magnetic material which was a different construction than other thin-tape wound cores. The deviations from other
measured values are below 10%. Interestingly, a large deviation
of the empirical estimation from the measured demagnetization factor (+16%) also occurs for the ring core used by Ripka
and Primdahl [7] and a deviation of +18% occurs for a ring
EC
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(11)
RR
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HC (z) − Hext
HC (z) − (BC (z)/μ0 )
3. Racetrack core models
The modelled racetrack cores will be described using the
dimensions notation shown in Fig. 4. Two basic geometries are
modelled: the “regular” racetrack core with constant track width
and the asymmetrical core with a slight asymmetry in track width
(originally used for sensor feedthrough adjusting [12]). The
modelled geometries cover a wide spectrum of racetrack core
shapes with lengths l from 18 to 140 mm, core widths d ranging
from 6.2 to 12 mm, track widths T ranging from 1.2 to 2.4 mm
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Fig. 1. 17 mm diameter, 0.5 mm thick ring core surface magnetic flux density
in z-direction in [T] (Hext = 50 ␮T applied in the z-axis).
Fig. 3. Ring-core dimensions.
Please cite this article in press as: J. Kubı́k, P. Ripka, Racetrack fluxgate sensor core demagnetization factor, Sens. Actuators A: Phys. (2007),
doi:10.1016/j.sna.2007.10.066
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Table 1
Global demagnetization factor of ring cores selected from Ref. [5]
35,800
33,000
31,000
28,700
40,000
1,871
2,500
d [mm]
T [mm]
h [mm]
17.0
17.0
17.0
17.0
20.0
16.4
12.7
0.125
0.250
0.375
0.500
2.000
0.264
0.254
1
1
1
1
0.4
1
1.6
Q4 d, diameter; T, thickness, h, height.
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1.87
3.56
5.16
6.92
7.47
4.29
8.58
1.61
3.20
4.78
6.37
7.35
4.14
9.62
track core with a track width, T, equal to the asymmetrical core
average track width.
4. Racetrack cores measurement
A method for the measurement of an effective demagnetization factor of fluxgate sensor cores was proposed in Ref. [6]
and was used for this measurement. The plastic coil bobbin was
made with an inner hollow aperture slightly wider than the tested
racetrack cores (Fig. 6). Using 0.2 mm diameter copper wire, 131
turns were wound in a single layer around the bobbin to form
a 30 mm long coil equal to the length of the tested racetrack
cores. Thus it can be considered as “a large pick-up coil” and
the measured demagnetization factor as the global demagnetization factor of the sensor core. The core longitudinal axis (or
the sensitivity axes of the whole sensor) was placed horizontally
and oriented in the East-West direction during all demagnetization factor measurements to avoid longitudinal magnetization
of the core by the horizontal component of the geomagnetic
field.
A Hewlett-Packard 4284A precision LCR meter with Kelvin
clips was used to measure the air coil inductance at frequencies
ranging from 90 to 100 kHz. The self-inductance of the air
coil was (within 5%) constant in the whole frequency range
indicating only a minor influence of the parasitic capacitance.
Fig. 4. Racetrack dimensions: regular (left) and asymmetrical core (right).
Fig. 5. 30 mm long, 8 mm wide, 1.55 mm track wide, 25 ␮m thick racetrack core
surface magnetic flux density in z-direction in [T] (Hext = 50 ␮T applied in the
z-axis direction).
DP
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TE
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1.70
3.28
4.88
6.38
6.34
4.01
8.18
Empirical
EC
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RR
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0.0004
0.0009
0.0013
0.0017
0.0020
0.0010
0.0025
FEM
CO
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and core thicknesses t ranging from 25 to 125 ␮m. 23 geometries were modelled, in total. The 30 mm long asymmetrical
cores with Tmax = 1.7 mm and Tmin = 1.4 mm were also modelled and compared to a core with a track thickness T = 1.55 mm
equal to the average track thickness of the asymmetrical core.
At first, the linear magnetic material with relative permeability
μr = 25,000 was used, corresponding to a measured permeability μ4 (measured at Hmax = 0.4 A/m) of the Vitrovac 6025X
amorphous alloy [13] (the maximum measured amplitude permeability was μr = 63,000). An example of the distribution of
magnetic flux density in the racetrack core is shown in Fig. 5
for 30 mm long racetrack core. With a relative permeability of
25,000, a maximum flux density of BC = 0.24 T occurs in the core
when an external magnetic flux density Bext = 50 ␮T is applied
in the longitudinal direction (the model’s z-axis). Table 2 shows
the modelled global demagnetization factors of fluxgate racetrack cores either for linear magnetic material with μr = 25,000
and for nonlinear magnetic material characterized by a BH
curve based on amplitude permeability measurement of Vitrovac
6025X [13]. It may be seen that the differences are negligible and
thus the modelling with nonlinear material, which takes much
more computer time, is not necessary in this case. The largest
difference in the modelled magnetometric demagnetization factor occurs for very long and thin cores (such as 140 mm long,
25 ␮m thick: 0.03 × 10−3 vs. 0.05 × 10−3 ) which could saturate
even for applied field of Bext = 50 ␮T when a BH curve is used
When comparing a core with a thickness t = 25 ␮m to the
thicker core with a thickness t = 50 ␮m, the global demagnetization factor is approximately two times higher for the thicker core,
which corresponds to the behaviour of global demagnetization
factors in ring cores. Note that the magnetometric demagnetization factor Dglobal = 0.00027 for an asymmetrical core is equal
to the magnetometric demagnetization factor for a regular race-
Measured
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Dglobal × 103
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Primdahl et al. [6]
Primdahl et al. [6]
Primdahl et al. [6]
Primdahl et al. [6]
Ripka and Primdahl [7]
Nielsen et al. [8]
Clarke [9]
A/d2
Dimensions
μr
RO
Sample
Please cite this article in press as: J. Kubı́k, P. Ripka, Racetrack fluxgate sensor core demagnetization factor, Sens. Actuators A: Phys. (2007),
doi:10.1016/j.sna.2007.10.066
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Table 2
Magnetometric demagnetization factor of racetrack cores modelled with constant relative permeability and nonlinear BH curve
Dimensions
20 mm
20
18.2
6.4
6.2
1.2
1.8
25
25
75
136
0.45
0.70
0.47
0.69
30 mm
30
29.8
30
7.8
7.8
9
1.2
1.8
2.4
25
25
25
33
51
67
0.23
0.33
0.38
0.22
0.32
0.37
2
2
25
25
10
3
0.09
0.03
0.10
0.05
70
140
12
12
t [␮m]
20
18.2
6.4
6.2
1.2
1.8
50
50
150
272
30 mm
30
30
29.8
30
7.8
8
7.8
9
1.2
1.55
1.8
2.4
50
25
50
50
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43
101
133
2
2
50
50
70
140
12
12
29.8
30
30
7.8
9
9
1.8
2.4
2.4
75
75
100
As. 30 mm
As. 30 mm
As. 30 mm
As. 30 mm
As. 30 mm
30
30
30
30
30
8
8
8
8
8
1.55
1.55
1.55
1.55
1.55
25
50
75
100
125
TE
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256
257
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264
Q2
CO
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0.45
0.28
0.64
0.75
0.46
0.27
0.63
0.74
20
5
0.17
0.06
0.16
0.07
152
200
267
0.94
1.11
1.48
–
–
–
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172
215
0.28
0.55
0.81
1.07
1.33
0.27
0.55
0.80
1.06
1.32
5. Empirical formula for racetrack core
demagnetization factor
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5.1. Influence of shape
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The empirical estimation of a ring core local demagnetization factor and the relationship between the local and the global
demagnetization factor including the influence of relative permeability has been published in Ref. [5]. A racetrack core is a
more complicated shape and thus estimating the global demagnetization factor will require more input parameters representing
the geometrical dimensions. The FEM racetrack models with a
relative permeability μr = 25,000 will be used for the estimation.
Some similarity of racetrack core physical dimensions influence
on the core global demagnetization factor will probably be similar to a ring core (thickness, core width), others will be new. In
RR
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The measuring current in a constant current mode was 5 mA
corresponding to a maximum axial field of approximately
20.6 A/m in the centre and 10.4 A/m at the end of the coil. The
field values ratio corresponds to a long thin solenoid having
at the end 50% of the field in the centre [14]. The coil selfinductance at 10 kHz was then measured for five thicknesses
of the as-cast Vitrovac 6025X racetrack core (for dimensions
see the last five rows of Table 2), i.e. one to five etched cores
placed on each other yielding 25–125 ␮m thickness range. The
core permeability used for the global demagnetization factor
calculation (6) was μr = 63,000. This value was measured as the
maximum amplitude permeability at 10 kHz. Measurements
with the external wire-wound coil produced repeatable and
consistent results on the five cores. The global demagnetization
factor of a 30 mm long, 8 mm wide and 25 ␮m thick racetrack
core is 0.00030—see Table 3, Col. 5 for the complete results.
EC
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0.86
1.23
DP
30 mm
0.88
1.38
RO
20 mm
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T [mm]
Q5 “As.” denotes the asymmetrical core.
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Dglobal × 103 BH curve
d [mm]
70 mm
140 mm
250
Dglobal × 103 μr = 25,000
l [mm]
70 mm
140 mm
251
A/l2 × 106
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Table 3
The global demagnetization factor of five racetrack cores, l = 30 mm, d = 8 mm, T = 1.55 mm
Thickness
Dglobal × 103
t [␮m]
FEM
μr = 25,000
FEM
BH curve (nonlinear)
Empirically estimated
μr = 25,000
Empirically estimated
μr = 63,000
Measured
Vitrovac 6025X
25
50
75
100
125
0.28
0.55
0.81
1.07
1.33
0.27
0.55
0.80
1.06
1.32
0.28
0.54
0.80
1.06
1.32
0.28
0.54
0.79
1.05
1.31
0.30
0.55
0.82
1.03
1.30
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Fig. 7. Global demagnetization factor as a function of physical dimensions of
racetrack core. Comparison of FEM models and estimation.
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• thicker cores mean higher demagnetization factor (t parameter),
• wider tracks mean higher demagnetization factor (T parameter) and
• longer cores mean lower demagnetization factor (l parameter).
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Dglobal (l, T, t) ≈ k1G1
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EC
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Starting with these assumptions, an analogy to ring-core formulas (7) and (8) can be utilized with arbitrary parameters k1G and
k2G :
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2Tt
+ k2G1
l2
(12)
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Dglobal (l, d, T, t) ≈ k1G2
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2Tt
+ k2G2
(l + k3G2 d)2
Dglobal (μr ) ≈
0.2
+ Dmin
μr
(14)
The RMS errors are 2.1 × 10−6 for 30 mm core and 6.7 × 10−6
for 70 mm core.
5.3. Combining geometry and permeability influence
Comparing the empirical Eqs. (13) and (14) we get the empirical estimation of the global demagnetization factor for fluxgate
racetrack cores as a function of core geometry and permeability:
Dglobal (l, d, T, t, μr ) ≈ k1
2Tt
0.2
+
+ k2
2
μr
(l + k3 d)
(15)
(13)
Minimizing the RMS error yielded the k1G , k2G and k3G constants k1G2 = 6.58, k2G2 = 23 × 10−6 and k3G2 = 1.8 with the
RMS error of eG2 = 1.82 × 10−5 . Note that the RMS error here is
almost three times lower than for the previous estimation. Fig. 7
shows the comparison of the global demagnetization factor calculated by FEM to the global demagnetization factor estimated
using (13).
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Based on the global demagnetization factors modelled on 23
various racetracks, k1G1 and k2G1 were acquired by the least
squares method: k1G1 = 2.70 and k2G1 = 55 × 10−6 . The RMS
error of this empirical formula is eG1 = 5.2 × 10−5 . The increasing racetrack width d for the racetrack constant length l increases
the distance between the two tracks thus decreasing their interaction. Therefore, we can expect that d decreases the global
demagnetization factor in a similar fashion to l. This is reflected
by an arbitrary constant k3G2 in the proposed formula (13).
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The dependence of the global demagnetization factor on the
relative permeability of two selected cores’ FEM models (30 mm
long, track 1.55 mm wide and 25 ␮m thick and 70 mm long, track
2 mm wide and 50 ␮m thick) is displayed in Fig. 8 for a given
core. This relation can be expressed as (14), where Dmin is the
global demagnetization factor for a given core shape with very
high relative permeability.
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general, these rules will apply:
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5.2. Influence of permeability
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Fig. 6. Coil dimensions (in mm) of wire-wound coil for global demagnetization
factor measurement of 30 mm long racetrack fluxgate sensor cores.
Fig. 8. Global demagnetization factor as a function of relative permeability
for 30 mm (Dmin = 0.271 × 10−3 ) and 70 mm (Dmin = 0.161 × 10−3 ) long core.
Comparison of discrete points of FEM models and solid lines of estimation (14).
Please cite this article in press as: J. Kubı́k, P. Ripka, Racetrack fluxgate sensor core demagnetization factor, Sens. Actuators A: Phys. (2007),
doi:10.1016/j.sna.2007.10.066
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lower relative permeability as indicated by the models (Fig. 8).
Fig. 9 further illustrates the close match of FEM models, empirical estimations and measurement of the global demagnetization
factors of racetrack cores.
The comparison with ring core fluxgate cores is very interesting: the demagnetizing factor of racetrack fluxgate cores is lower
compared to all the ring cores in Table 1. This fact indicates that
long, thin racetrack fluxgate cores may be an advisable option
for highly sensitive sensors and magnetometers.
where
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k1 = 6.58,
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k2 = 23 × 10
−6
0.2
−
= 15 × 10−6 ,
25, 000
k3 = 1.8.
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(16)
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Eq. (15) with corresponding parameters (16) provides a quick
estimation of the fluxgate racetrack core global demagnetization
factor without the need for FEM modelling. The input parameters are only the basic geometrical dimensions defined in Fig. 4
and the relative permeability of the core material.
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6. Comparison of models, estimation and measurement
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Q3
The comparison of the global demagnetization factors of five
racetrack cores is shown in Table 3. The table contains the global
demagnetization factors acquired by linear and nonlinear FEM
models (Cols. 2 and 3), the empirical estimation using Eq. (15)
with two relative permeability values corresponding to μ4 and
μmax (Cols. 4 and 5) and the global demagnetization factors
measured on real sensor cores using the method with external pick-up coil described above (Col. 6). The Col. 2 global
demagnetization factors were modelled with μr = 25,000 corresponding to the measured μ4 of the used material. The Col.
3 global demagnetization factors were modelled with nonlinear
hysteresis-free BH curve (hysteresis-free BH curve was required
by the FEM software). The BH curve was constructed by connecting the endpoints of measured BH loops with gradually
increasing H field. The RMS errors of the FEM models compared with measured values are 2.6 × 10−5 and 2.1 × 10−5 using
linear and nonlinear magnetic material model, respectively. The
RMS errors of empirically estimated values compared with
measured values are 2.0 × 10−5 and 2.1 × 10−5 using relative
permeability values of μr = 25,000 and μr = 63,000 for the estimation, respectively. The modelled and estimated values of the
global demagnetization factor show a strong correlation with
the measured values with a maximum error of 9%. The nonlinear FEM model shows only slightly better results than the linear
model with μr = 25,000. The influence of the relative permeability on the empirically estimated global demagnetization factors
is negligible due to the use of high-permeability material. Much
more caution is necessary when using a core material with a
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The fluxgate sensor racetrack core magnetometric (global)
demagnetization factor was modelled by FEM and an empirical estimation relation has been fitted (Eq. (15) with constants
(16)). This empirical estimation is based on the geometrical
dimensions of the racetrack core and the core relative permeability. The results of the measurements on the racetrack core
samples are in agreement with the models and the empirical estimations. The empirical formula provides a ready-to-use solution
for a quick estimation of the global demagnetization factor of
a given fluxgate sensor racetrack core prior to FEM modelling,
manufacturing and measurement of the actual core.
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Fig. 9. The global demagnetization factor for 30 mm long racetrack fluxgate
core. Comparison of FEM model, empirical estimation and measured values.
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7. Conclusions
References
[1] J.A. Osborn, Demagnetizing factors of the general ellipsoid, Phys. Rev. 67
(11–12) (1945) 351–357.
[2] D.-X. Chen, J.A. Brug, R.B. Goldfarb, Demagnetizing factors for cylinders,
IEEE Trans. Magn. 27 (4) (1991) 3601–3619.
[3] A. Aharoni, Demagnetizing factors for rectangular ferromagnetic prisms,
J. Appl. Phys. 83 (6) (1998) 3432–3434.
[4] A. Aharoni, “Local” demagnetization in a rectangular ferromagnetic prism,
Phys. Status Solidi (b) 229 (3) (2002) 1413–1416.
[5] F. Primdahl, P. Brauer, J.M.G. Merayo, O.V. Nielsen, The fluxgate ring-core
internal field, Meas. Sci. Technol. (13) (2002) 1248–1258.
[6] F. Primdahl, B. Hernando, O.V. Nielsen, J.R. Petersen, Demagnetising factor and noise in the fluxgate ring-core sensor, J. Phys. E Sci. Instrum. (22)
(1989) 1004–1008.
[7] P. Ripka, F. Primdahl, Tuned current-output fluxgate, Sens. Act A 82 (2000)
161–166.
[8] O.V. Nielsen, J.R. Petersen, F. Primdahl, P. Brauer, B. Hernando, A.
Fernandez, J.M.G. Merayo, P. Ripka, Development, construction and analysis of the ‘Ørsted’ fluxgate magnetometer, Meas. Sci. Technol. 6 (1995)
1099–1115.
[9] D.B. Clarke, Demagnetization factors of ringcores, IEEE Trans. Magn. 35
(1999) 4440–4444.
[10] J.R. Burger, The theoretical output of a ring core fluxgate sensor, IEEE
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[11] M. De Graef, M. Beleggia, The fluxgate ring-core demagnetization field,
J. Magn. Magn. Mater. 305 (2006) 403–409.
[12] P. Ripka, Race-track fluxgate with adjustable feedthrough, Sens. Act A 85
(2000) 227–231.
[13] J. Kubı́k, PCB fluxgate sensors, Dissertation thesis, CTU in Prague, 2006,
pp. 56–58.
[14] D. Jiles, Introduction to magnetism and magnetic materials, Chapman &
Hall, London, 1996, pp. 19, ISBN 0 412 38640 2.
Biographies
Jan Kubı́k received his Ing. degree (M. Eng. equivalent) in the field of computercontrolled measurement systems in March 2003 at the Faculty of Electrical
Please cite this article in press as: J. Kubı́k, P. Ripka, Racetrack fluxgate sensor core demagnetization factor, Sens. Actuators A: Phys. (2007),
doi:10.1016/j.sna.2007.10.066
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Pavel Ripka received an Ing. degree in 1984, a CSc (equivalent to PhD) in
1989 and Prof. degree in 2001 at the Czech Technical University, Prague, Czech
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Republic. He works at the Department of Measurement, Faculty of Electrical
Engineering, Czech Technical University as a full professor, teaching courses
in electrical measurements and instrumentation, engineering magnetism and
sensors. He also worked as visiting scientist at Danish Technical University
(1990–1993), National University of Ireland (2001) and in the Institute for the
Protection and the Security of the Citizen, European Commission Joint Research
Centre in Italy (2005/6). His main research interests are magnetic measurements
and magnetic sensors, especially fluxgate. He is an author of >50 SCI journal papers, 2 books and 5 patents. He is a member of IEEE, Elektra society,
Czech Metrological Society, Czech National IMEKO Committee and Eurosensors Steering Committee. He served as an associate editor of the IEEE Sensors
Journal. He was a General Chairman of Eurosensors 2002 conference.
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Engineering of Czech Technical University in Prague. He successfully defended
his PhD thesis “PCB Fluxgate Sensors” under supervision of Prof. Pavel Ripka in
December 2006 at the same university. He also worked at the National University
of Ireland in Galway (2001 and 2002) and at the Lappeenranta University of
Technology in Lappeenranta, Finland (2003). He has joined the Tyndall National
Institute in Cork in September 2006 as a researcher. His research interests range
from magnetic field sensors design and characterization to computer-controlled
measurement systems development, FEM modelling of magnetic systems and
microcontroller applications. He is the main author or co-author of more than
35 journal and conference papers.
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Please cite this article in press as: J. Kubı́k, P. Ripka, Racetrack fluxgate sensor core demagnetization factor, Sens. Actuators A: Phys. (2007),
doi:10.1016/j.sna.2007.10.066
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