The fluxgate ring-core demagnetization field

ARTICLE IN PRESS

Journal of Magnetism and Magnetic Materials 305 (2006) 403–409 www.elsevier.com/locate/jmmm

The fluxgate ring-core demagnetization field M. De Graefa,, M. Beleggiab a

Department of Materials Science and Engineering, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 15213-3890, USA b Center for Functional Nanomaterials, Brookhaven National Laboratory, Upton, NY 11973, USA Received 11 July 2005; received in revised form 24 January 2006 Available online 21 February 2006

Abstract The local demagnetization factor for the ring-core flux gate is derived analytically, based on a tangential magnetization model. The results are in good agreement with experimental data for a wide range of ring shape parameters. Approximate expressions in the limit of a narrow, thin ring are obtained, and indicate that the local demagnetization factor scales with the ratio of the cross-sectional area to the total area of the ring. Analytical modeling of the demagnetization factors for a uniform magnetization state results in an underestimate of the local cross-section averaged demagnetization factors by 50% or more. r 2006 Elsevier B.V. All rights reserved. Keywords: Demagnetization factors; Fluxgate; Internal field; Shape amplitude

1. Introduction Fluxgate magnetometers are currently used to measure small DC or low-frequency AC magnetic fields with highfield resolution or sensitivity of about 0:1 nT [1]. One class of magnetometers employs the ring-core geometry, the properties of which are described in Refs. [1–3]. In most theoretical models for the ring-core geometry, it is assumed that the internal field in the core is equal to the applied field corrected by a suitable local demagnetization factor and a geometrical factor. For the geometry shown in Fig. 1(a), the relation between the internal (H i ) and applied (H a ) field is assumed to be (in the notation of Ref. [1]): Hi ¼

H a cos y , 1 þ wD

(1)

where w is the susceptibility and D the local demagnetization factor at the location arrowed in Fig. 1(a). This relation, which we will refer to as the cos y-model, was experimentally verified by measuring the internal field as a function of position around the ring [1]. Experimental values for D were reported by several authors and are typically around 10
3 for representative ring-core dimensions. Primdahl et al. [3] tabulated results from various Corresponding author. Tel.: +1 412 268 8527; fax: +1 412 268 7596.

E-mail address: [email protected] (M. De Graef). 0304-8853/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2006.01.111

groups, and a portion of their Table 1 is repeated in this paper as Table 1. A least-squares analysis of the demag factors revealed that the factors scale as the ratio of the ring cross-section A to the square of the mean diameter d (the average of the inner and outer diameters) [3], A . (2) d2 The thickness-to-width ratio of the ring is reported to have a small effect, if any, on the demagnetization factor D. As stated in Ref. [3], satisfactory theoretical calculations should corroborate this result, which is the main goal of the present paper. We will employ a Fourier space approach to the computation of the local demagnetization factor for the cos y-model, and compare the resulting demag factors with both experimental data and relation (2). Our main finding is that the dependence of the demag factors on the ring geometry is more correctly described by a proportionality to the ratio between the cross-sectional area and the square of the outer diameter instead of the average diameter. D ¼ 1:826

2. Theoretical model To facilitate comparison between the ring dimensional parameters used in this paper, and the parameters used by

ARTICLE IN PRESS M. De Graef, M. Beleggia / Journal of Magnetism and Magnetic Materials 305 (2006) 403–409

404

Primdahl et al. [3], we have listed the conversions in Table 2. In particular, we will use the ring radius ratio s and the aspect ratio Z as the main dimensional parameters. Fig. 1(b) shows the main ring parameters graphically. In this paper, we are mostly interested in the cross-section indicated in gray. In cylindrical coordinates, the crosssection is described by the coordinate ranges ðR1 oroR2 ; y ¼ 0;
hozohÞ. In this section, we will first compute the effective demag factor for the rectangular ring using the cos y magnetization model. Then, we compute an approximate solution for s ! 1. If we assume that the tangential magnetization in the ring is described by M y ðyÞ ¼ M 0 cos y, which in Cartesian coordinates reads secondary pickup coil

θ

z

θ

2h

r R1 R2

Ha (a)

(b)

Fig. 1. Schematic diagram of the ring with rectangular cross-section. (a) Shows the relation between the external field and the location of the secondary pickup coil. In (b), the shaded area corresponds to the crosssection for which the computations in this paper are carried out. Other dimensional parameters are listed in Table 2.

MðrÞ ¼ M 0 ð
sin y cos y; cos2 y; 0Þ,

(3)

for jzjoh and R1 oroR2 , then the magnetization Fourier transform (FT) can be written as " # k2x
k2y 2kx ky sinðhkz Þ g2 ; g0
g2 ; 0 , (4) MðkÞ ¼ 2pM 0 kz k2 k2 where Z gn ¼

R2

r drJ n ðkrÞ,

(5)

R1

where J n ðxÞ is a Bessel function of the first kind. Note that, in this model, the magnitude of the magnetization varies with y. It has been argued in Ref. [1], based on experimental data, that this should be a reasonable approximation to the real magnetization. The axial and radial components of the magnetization are assumed to have negligible importance, since the pick-up coil is only sensitive to the tangential component. Strictly speaking, for non-uniform magnetizations, the demagnetization tensor is not defined. In fact, only when a body is uniformly magnetized, is the demagnetization field a function of shape alone. And only in this case, the general relationship B ¼ m0 ðM þ HÞ becomes simply Bi ¼ m0 ðM i
N ij M j Þ, where N ij is the demagnetization tensor field. For a non-uniform magnetization, the demagnetization field is a function of shape and magnetization topography. In this case, no simple relationship, valid everywhere, can be established between magnetization and demagnetization

Table 1 Ring dimensions and experimental demagnetization factors from Ref. [3]; the approximate and exact theoretical values as well as the relative difference between experiment and exact theory are also shown  103

#

d

t

w

A d2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

15.90 12.70 25.40 17.00 17.00 17.00 17.00 17.00 17.00 17.00 17.00 17.00 16.60 26.00 26.00 26.00 25.50 16.74 17.00 17.00 17.00 17.00 20.00

0.1270 0.2540 0.1270 0.0500 0.1000 0.1750 0.2500 0.3750 0.5000 0.6000 0.2750 0.2750 0.1016 0.3250 0.4250 0.5250 0.1016 0.2640 0.1250 0.2500 0.3750 0.5000 2.0000

1.600 1.600 3.174 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.580 1.000 1.000 1.000 1.580 1.000 1.000 1.000 1.000 1.000 0.400

0.804 2.520 0.625 0.173 0.346 0.606 0.865 1.298 1.730 2.076 0.952 0.952 0.583 0.481 0.629 0.777 0.247 0.943 0.433 0.865 1.298 1.730 2.000

s

Z

3 N exp ff  10

N approx:  103 ff

N theor:  103 ff

0.984 0.961 0.990 0.994 0.988 0.980 0.971 0.957 0.943 0.932 0.968 0.968 0.988 0.975 0.968 0.960 0.992 0.969 0.985 0.971 0.957 0.943 0.818

0.0998 0.1235 0.1243 0.0587 0.0585 0.0582 0.0580 0.0576 0.0571 0.0568 0.0579 0.0579 0.0946 0.0380 0.0378 0.0377 0.0617 0.0588 0.0584 0.0580 0.0576 0.0571 0.0182

1.500 3.890 2.000 0.440 0.720 1.200 1.720 2.570 3.190 3.990 1.810 1.840 1.130 1.250 1.630 1.980 0.520 1.740 0.836 1.625 2.422 3.171 3.184

1.607 4.591 1.170 0.408 0.811 1.409 1.997 2.959 3.896 4.629 2.191 2.191 1.189 1.241 1.612 1.978 0.572 2.163 1.011 1.997 2.958 3.896 5.149

1.591 4.518 1.166 0.404 0.798 1.371 1.925 2.810 3.652 4.299 2.106 2.106 1.179 1.182 1.517 1.840 0.567 2.082 0.991 1.925 2.810 3.652 3.853

Data were taken from the following sources: rings 1–2 [9], 3 [10], 4–17 [11], 18 [1], 19–22 [2], and 23 [12].

% dev. 6.07 16.1
41.7
8.10 10.9 14.3 11.9 9.35 14.5 7.74 16.4 14.5 4.33
5.41
6.95
7.03 8.98 19.6 18.6 18.5 16.0 15.2 21.0

ARTICLE IN PRESS M. De Graef, M. Beleggia / Journal of Magnetism and Magnetic Materials 305 (2006) 403–409

405

Table 2 Ring geometry parameters as used in Ref. [3] and in this paper Quantity

Primdahl et al

This paper

Reduced parameters

Inner radius Outer radius Radius ratio

d
t 2 dþt 2 d
t dþt

R1 R2

R2 s R2 s

Average diameter Ring height Aspect ratio

d w

R1 þ R2 2h

w dþt

h R2

Ring width Cross-sectional area A Ring area Ar

t wt

R2
R1 2hðR2
R1 Þ

p 4 ðd

R1 R2

þ tÞ2

pR22

R2 ð1 þ sÞ 2R2 Z Z R2 ð1
sÞ 2R22 Zð1
sÞ –

The last column uses the dimensionless parameters s and Z and the outer radius R2 .

fields. The analysis of non-uniform topographies is, obviously, much more complex. There are, however, particular shapes and topographies, where the symmetry involved allows us to define an effective demagnetization factor. Under these circumstances, and for particular selected points of high symmetry of the body, we can write B ¼ m0 ð1
NÞM, where N describes how strong the demagnetization effect is at this particular point. This effective factor N only makes sense when the magnetization and the demagnetization fields are parallel to each other, pointing in opposite direction. In the general case, M and H are never parallel, except, again, in particular symmetry points of the body. In a ring with a cos y magnetization, this happens only along the x-axis, where the magnetization is along y and the demagnetizing field is along
y. For any point in the ðx; zÞ cross-section, therefore, we can define an effective demagnetization factor Nðx; zÞ simply as the normalized demagnetizing field: Nðx; zÞ ¼


1 H y ðx; 0; zÞ M0

(6)

in such a way that, along the x-axis of the ring, we can write By ðx; 0; zÞ ¼ m0 ½1
Nðx; zÞM 0 . Although very restrictive, this definition allows for a comparison with experiments. From the magnetization FT, we can directly evaluate the demagnetization field from [4] ZZZ 1 k dk HðrÞ ¼
3 ½MðkÞ  keirk . (7) 2 8p k þ k2z Since we have M  k ¼ 2pM 0 ky ðg0 þ g2 Þ sinðhkz Þ=kz , we obtain
 ZZZ M0 k dk sinðhkz Þ irk HðrÞ ¼
2 ky ðg0 þ g2 Þ (8) e , kz 4p k2 þ k2z which we evaluate in the y ¼ 0 plane: ZZ dkx dky M0 Hðx; 0; zÞ ¼
2 ðkx ; ky ; 0Þ½ky ðg0 þ g2 Þ 4p k2 ½1
e
kh coshðkzÞeixkx .

For symmetry reasons, the H x and H z components vanish, so that we are only left with H y : ZZ M0 k dk dy½sin2 yðg0 þ g2 Þ H y ðx; 0; zÞ ¼
4p ½1
e
kh coshðkzÞeixk cos y , ð10Þ hence, the effective demagnetization factor is finally Z 1 dk J 1 ðkxÞðg0 þ g2 Þ½1
e
kh coshðkzÞ, Nðx; zÞ ¼ 2x

or, in dimensionless units w  x=R2 , z  z=R2 , K  kR2 : Z 1 dK J 1 ðKwÞ½J 0 ðsKÞ
J 0 ðKÞ Nðs; Z; w; zÞ ¼ w K2 ½1
e
ZK coshðKzÞ. ð12Þ Since the rings which have been studied experimentally are, in general, of small thickness, rather than taking the effective demagnetization factor in the cross-section ðsowo1;
ZozoZÞ, we choose to consider the averaged factor over the ring thickness. The final expression turns out to be Z 1 dK N av ðs; Z; wÞ ¼ J 1 ðKwÞ½J 0 ðsKÞ
J 0 ðKÞ 2Zw K 3 ð13Þ ð2ZK
1 þ e
2ZK Þ. Although this integral has no full analytical solution, we can proceed with the evaluation of the effective averaged demagnetization factor in a parameter range similar to where the experiments were performed, namely s close to 1. We begin by writing the integral as the sum of three terms: Z 1 dK I1 ¼ J 1 ðKwÞ½J 0 ðsKÞ
J 0 ðKÞ, w K2 I2 ¼


ð9Þ

(11)

I3 ¼

1 2Zw

1 2Zw

Z

Z

dK J 1 ðKwÞ½J 0 ðsKÞ
J 0 ðKÞ, K3

dK J 1 ðKwÞ½J 0 ðsKÞ
J 0 ðKÞe
2ZK . K3

(14)

ARTICLE IN PRESS M. De Graef, M. Beleggia / Journal of Magnetism and Magnetic Materials 305 (2006) 403–409

The first integral is done easily   1 s2 1
2
log w2 I1 ¼ 4 w the second one can be found in [5, Eq. ð6:574:18 Þ]
  1 3 1 s2 2w2 F 1
;
; 1; 2 I2 ¼ 12Z 2 2 w   2 1 1 s
32 F 1
;
; 2; 2 , 2 2 w

(15)

the effective demagnetization factor, to first order in 1
s and Z is
  
   1 16 1 16 A ln 2
1 ð1
sÞZ ¼ ln 2
1 . N¼ 2p Z 4 Z Ar (20)

ð16Þ

Note that this relation contains the factor A=Ar , where Ar is the area of the ring, as defined in Table 2. In the notation of Ref. [3] we have
   1 16 A ln 2
1 N¼ . (21) p Z ðd þ tÞ2

where 2 F 1 ½. . . is a hypergeometric function. To compute the average over the radial direction of the ring, these expressions can be integrated readily over w from s to 1. Before taking care of the remaining integral, we may note that when s ! 1, I 1 and I 2 exhibit a weak and smooth dependence on the coordinate w. For instance, I 1 decreases monotonically from 0.0527 to 0.0475 as w varies from s ¼ 0:9 to 1. To estimate a reliable value to compare with experiments, we can then choose either to take the mean value of I 1 and I 2 averaged along w between s and one, or take the exact value of I 1 and I 2 at the midpoint w ¼ ð1 þ sÞ=2: to first order in
 ð1
sÞ they coincide. Finally, we expand I 1 and I 2 for
! 0, in order to have the correct limit:
I 1 , 2 I 2


2
. 3pZ

(17)

The third and final integral I 3 requires special care. The exponential factor expð
2ZKÞ represents a damping factor, so that for numerical integration, the range can be taken to be 0oKo5=Z. Note that this exponential factor is necessary to insure the convergence of the integral, and cannot be expanded for Z ! 0. If this is attempted, the logarithmic behavior in Z of the demagnetization field (and of all other quantities associated with it, such as the magnetostatic energy), which is to be expected in the thinfilm limit of magnetostatics, is lost. The correct procedure is as follows: first, we choose to evaluate the integral at the midpoint w ¼ ð1 þ sÞ=2 ¼ 1

=2 (again, to first order in
, this coincides with taking the average along w), then we expand to first order in
, integrate along K, and, finally, we expand in Z Z 1 dK J 1 ½Kð1

=2Þ I3 ¼ fJ 0 ½ð1

ÞK
J 0 ½Kge
2ZK 2Z K 3 ð1

=2Þ Z
dK 2  J ðKÞe
2ZK ð18Þ 2Z K 2 1  
1 1 1
¼ 2 F 1
; ; 2;
2
2 2 2 Z 2
   2

Z
16
þ ln 2
1 .  ð19Þ 3pZ 2 2p Z Not surprisingly, the I 1 and I 2 contributions compensate exactly the first two terms of I 3 , so that the final result for

Comparing this equation to Eq. (2), we note two differences. First, the demag factor is proportional to the ratio of the cross-section to the square of the total diameter d þ t instead of the mean diameter d. And, second, as suggested in Ref. [3], there is a weak dependence on the aspect ratio Z of the ring. To compare the predictions of the s ! 1 model with the experimental data in Table 1, we select two subsets of rings with nearly equal aspect ratio Z. The first set contains 14 rings (numbers 4–12 and 18–22), with average aspect ratio Z¯ ¼ 0:0579; the second set contains three rings (numbers 14–16) with an average aspect ratio of Z¯ ¼ 0:0378. The experimental demag factors for these rings are plotted with respect to A=Ar in Fig. 2. The slope of the lines fitted through the data sets are 1:993 for the first set, and 2:664 for the second set. The predicted slopes according to Eq. (21) are 2:378 and 2:649, respectively. The agreement is rather good for the subset with the smaller average aspect ratio. In Fig. 3, we compare the linear regression and cos y models with the experimental data set for all 23 entries of Table 1. The experimental demag factors are plotted along the horizontal axis, the fitted or modeled demag factors

4

10 9 22

3 8 N x 103

406

21

16

2

12 7 11 2018

15 14 6

1 4 0

5 2.5

19

5.0

7.5

10.0

12.5

15.0

17.5

20.0

104 x A/(d+t)2 Fig. 2. Linear fit of the experimental demag factors for two subsets of Table 1 to the approximate relation of Eq. (21). The solid lines are the results of linear regression, the dashed lines are computed from Eq. (21) using the average aspect ratio Z¯ for each subset. The large solid circles are the experimental data points; the smaller gray dots are calculated demag factors for the corresponding rings obtained by numerical integration of Eq. (13), averaged over w.

ARTICLE IN PRESS M. De Graef, M. Beleggia / Journal of Magnetism and Magnetic Materials 305 (2006) 403–409

5 x 10-3

5 x 10-3 2

Std. Dev. = 3.05 x 10-4

23

22

9

3

22

2 20

1 17 4 0

21

1811

8

12

1 7 16 6 15 3 13 19 14 5

10

23

10 Exact cos θ model

Primdahl linear regression

2

Std. Dev. = 3.10 x 10-4 4

4

0

407

3

9

21 8

2

19

1

13

11 12 18 20 16 7 1 6 15 14

3

5 17 4

1

2 3 Experimental demag factor

4

0 5 x 10-3

0

1

2 3 Experimental demag factor

4

5 x 10-3

Fig. 3. Comparison between the experimental demag factors (horizontal axis) and the Primdahl linear regression fit D ¼ 1:826A=d 2 (left side), and the exact results from the cos y model (right side). Each point is labeled with a number corresponding to the entries in Table 1. The closer a point is to the solid diagonal line, the better the agreement between the model and the experimental results. The standard deviations for both approaches are similar. The standard deviation of the approximate results (not shown) amounts to 4:95  10
4 .

along the vertical axis. The closer a point is located to the straight diagonal line, the better the agreement between model and experiment. For the linear regression model, N ¼ 1:826A=d 2 , most of the data points are located below the diagonal line, with a standard deviation of 3:05  10
4 . The cos y model has most data points above the diagonal line, with a similar standard deviation of 3:10  10
4 . For comparison, the standard deviation of the approximate model amounts to 4:95  10
4 . It is not too difficult to understand why several points are located rather far from the straight line. For instance, points 2 and 3 correspond to rings with aspect ratios Z that are more than twice as large as the average value. With increasing aspect ratio, it becomes less likely that the cos y model provides an accurate description of the magnetization state, since more complex magnetization configurations become possible. For ring 23, the thinnest ring with the smallest value of s, the agreement between theory and experiment is also rather poor, presumably for similar reasons. Overall, the agreement between the experiments and the cos y model is rather good, indicating that the dependence on the dimensional combination Zð1
sÞ (or A=ðd þ tÞ2 ) is consistent with experimental results. We have also compared the results of the cos y model with analytical computations of the demagnetization tensor field N ij ðrÞ for a uniformly magnetized ring. Using the shape amplitude method described in Ref. [4], the demag tensor components can be expressed in terms of elliptic integrals and, after averaging over the cross-section of the ring, compared with the results from the cos y model. The explicit expressions are omitted here because they are lengthy and not particularly illuminating. When the tangential demag factor for the uniform magnetization

state is expanded for the case of a thin disk with s close to 1, the resulting expression is (the subscript u stands for uniform)
   1 16 ln 2 þ 1 ð1
sÞZ N u ðs; ZÞ ¼ 4p Z
   1 16 A ln 2 þ 1 . ð22Þ ¼ 8 Z Ar It is noteworthy that this expression shows the same dependence on Zð1
sÞ as the cos y model, suggesting that the correct dependence on the ring dimensions is A=ðd þ tÞ2 instead of A=d 2 . Numerical analysis of this expression shows that the uniform magnetization model underestimates significantly the values of the experimental demag factors, with relative errors of 50% or more. A comparison of the Nðw; zÞ demag factors in the ring cross-section for the uniform and cos y models is shown in the contour plots of Fig. 4 for ring shape parameters s ¼ 0:9 and Z ¼ 0:1. For the uniform magnetization state, the demag factor in the cross-section plane has two extrema, a minimum for w slightly less than s, and a maximum for w slightly larger than 1. Negative values to the left of center compensate for the positive values on the right side, resulting in a small (but positive) average demag factor across the cross-section. The cos y magnetization model has a different configuration of constant demag factor contours, with a shallow maximum slightly off to the left of center. The cross-section averaged demag factors are 0:00541 for the uniform magnetization case, and 0:00967 for the cos y model. It is clear that the uniform magnetization model gives rise to a very different field configuration than the cos y model.

ARTICLE IN PRESS M. De Graef, M. Beleggia / Journal of Magnetism and Magnetic Materials 305 (2006) 403–409

408

Cos θ Magnetization Model

Uniform Magnetization Model 0.1

0.0

-0.1 0.90

0.95

1.00

0.90

0.95

1.00

Fig. 4. Contour plot of the demag tensor element N yy for a ring with shape parameters s ¼ 0:9 and Z ¼ 0:1. The demag factors are shown as a function of position inside the ring for sowo1 and
ZozoZ. The point ðw; zÞ ¼ ð0:95; 0:0Þ corresponds to the center of the ring. The figure on the left shows the demag factor for the uniform magnetization model and the figure on the right corresponds to the cos y model (Eq. (12)).

3. Discussion and conclusions The cos y model results in an overestimate of the experimental demag factors by up to 20%, indicating that this model might not contain all the necessary features to fully describe the true demag factors. One could argue that the deviations between the cos y model and the experimental results are due to experimental errors or uncertainties. If we assume that the relative error in the measurement of the ring dimensions is 0:1%, then this translates into relative errors on the values of s and Z. For the 23 rings of Table 1, none of the experimental demag factors fall within the uncertainty ellipse centered on the theoretical value for the cos y model, indicating that the relative errors on the ring dimensions are not the cause for the residual discrepancy between theory and experiment. The relatively tight clustering of all the experimental data points, acquired over a period of many years by different methods, indicates that the experimental data are of good quality. An exact complete computation of the total field at the cross-section plane due to the superposition of a homogeneous external field and the inhomogeneous field in the ring core has not yet been carried out. While it would be possible to perform this computation numerically, using a micromagnetics approach, the problem is highly non-linear and non-local, which means that an analytical solution is not easy to obtain. The cos y computation presented in this paper shows a better agreement with the experimental data than several approximate models discussed in detail in Ref. [3], and shows a standard deviation similar to that of the

least-squares fit of Ref. [3]. Our attempts to formulate more complex models, based on an onion-like magnetization state, have been unsuccessful so far. In summary, we have presented a complete analytical computation of the local demagnetization factor for the ring-core flux gate based on a cos y magnetization model. The results are in good agreement with experimental data for a wide range of ring shape parameters. Approximate expressions in the limit of a narrow, thin ring have been obtained, and indicate that the demag factor scales with the ratio of the cross-sectional area to the total area of the ring. We have also shown that the tangential demagnetization tensor field component of the ring core for a uniform magnetization state results in an underestimate of the local cross-section averaged demagnetization factors by 50% or more. Acknowledgments The authors would like to acknowledge stimulating interactions with F. Primdahl, S. Tandon and Y. Zhu. Financial support was provided by the US Department of Energy, Basic Energy Sciences, under contract numbers DE-FG02-01ER45893 and DE-AC02-98CH1-886. References [1] O.V. Nielsen, J.R. Petersen, F. Primdahl, P. Brauer, B. Hernando, A. Fernandez, J.M.G. Merayo, P. Ripka, Meas. Sci. Technol. 6 (1995) 1099.

ARTICLE IN PRESS M. De Graef, M. Beleggia / Journal of Magnetism and Magnetic Materials 305 (2006) 403–409 [2] F. Primdahl, B. Hernando, O.V. Nielsen, J.R. Petersen, J. Phys. E 22 (1989) 1004. [3] F. Primdahl, P. Brauer, J.M.G. Merayo, O.V. Nielsen, Meas. Sci. Technol. 13 (2002) 1248. [4] M. Beleggia, M. De Graef, J. Magn. Magn. Mater. 263 (2003) L1. [5] I.S. Gradshteyn, I.M. Ryzhik, Tables of Integrals, Series, and Products, sixth ed., Academic Press, San Diego, 2000.

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[9] D.B. Clarke, IEEE Trans. Magn. 35 (1999) 4440. [10] S.V. Marshall, in: Proceedings of the National Electronics Conference, Chicago, IL, 1969, p. 236. [11] P. Brauer, The ringcore fluxgate sensor, Ph.D. Thesis, Department of Automation, The Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark, 1997. [12] P. Ripka, F. Primdahl, Sensors Actuators A 82 (2000) 161.