IEEE TRANSACTIONS ON MAGNETICS, VOL. 33, NO. 5, SEPTEMBER 1997

Demagnetizing Field of a Ringcore Fluxgate Magnetometer

H. How ElectroMagnetic Applications, Inc., Boston, MA 02109

L. Sun and C. Vittoria ECE Department, Northeastern University, Boston, MA 021 IS

Absfruct-The demagnetizing field inside a ringcore sensor has been solved numerically and the performance of a ringcore magnetometer has been analyzed under triangular- current excitations. The gated signals appear at the knees of the magnetization curves where maximum nonlinearity occurs. The gated signals show constant amplitude whose width decreases linearly with the applied DC signal.

I. INTRODUCTION

The demagnetizing field within a ferromagnetic body can only be analytically calculated for simple sample geometries [I]. The demagnetizing field within a ringcore sensor can hardly be addressed and the past analyses on sensor performance always assumed a simple geomeiq of the core consisting of a pair of long fmmagnetic rods in the parallel-gate configuration [Z]. There is a need to rigorously solve the ringcore geometry in which the demagnetizing field can be accurately accounted for. We present in this paper such a theory where the ringcore fluxgate sensor has been analytically modeled to provide calculations on the gated signals, resulting in an estimate on the sensor resolution.

11. FORMLILATIONS

The Conventional demagnetizing factor, DM, was defined for a magnetic body bounded by a second-degree surface, for example, an ellipsoid, where the external magnetizing field, 9. was applied along one of its principal axes [I]. As such, the demagnetizing field, HM, is uniform within the body arising from a magnetization distribution 4nM, parallel to K, given by

HM = DM. 4nM. (1) I f q is not along the principal axis, (1) still holds, provided that the demagnetizing factor D, in (1) becomes a tensor of rank 2, i.e., HM will be no longer parallel to 4nM or )la. For a magnetic sample of arbitrary shape, for example, a toroidally shaped magnetic core as we are mainly concerned in the present calcula- tions, the concept of demagnetizing factor is of very little use, This is because neither 4nM nor HM is miform within the sample core, and hence it is meaningless to defme a global constant relating their ratio as depicted in (1).

In this section we proceed to calculate the induced magneti- zation 4nM and the demagnetizing field HM in a ring core sample in the presence of a uniform external magnetic field 9. Instead of calculating a global ratio between HM and 4nM, we calculate the local distribution of HM inside the core material such that the

Manuscript received January 29,1997.

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total magnetic field H = HM + is obtainable. Knowledge of H is required to quantitatively estimate the fluxgate operation where the ring core is employed as the sensor element.

For a normal ringcore fluxgate magnetometer the core is made of high-permeability magnetic metals or metallic glasses in the form of thin ribbon layers. As a model calculation, we consider the core is made of permalloy metal of width w = 1 mm and thickness d = 0.3 mm. The radius of the ring core is R = 2 cm. Since R is much larger than w and d, we may assume that the variation of the fields along core width and thickness can be ignored comparing to that along the circumference, the 4 - direction, of the ring. That is, the magnetization of the core can be written as

(2)

where e,, and e+ represent the unit vectors along the p and @ directions, respectively. The above one dimensional assumption of (2) can largely simplify the following calculations. However, even if this one-dimensional assumption does not rigorously hold, the associated demagnetizing field will be modified in the transverse direction, and hence it will not affect much the calcu- lated results. The other parameters that are needed are the saturation magnetization of the permalloy core, 4nM, = 10.8 kG, and the magnetic susceptibility x = 100,000.

M(r) = M,(N ep + qb(4) e+.

From Maxwell equation

V B = V * (H + 4nM) = O , (3) one can, analogous to electric charges, define the magnetic charge density

such that p M = -v * M, (4)

V H = 4xpM. ( 5 ) Therefore, when ( 2 ) is used as the expression for magnetization, Eq.(4) dictates the following magnetic sources:

where pM and uM denote the volume magnetic monopole charge density and volume magnetic dipole charge density, respectively. Therefore, the resultant polarization (demagnetizing) field is

where r denotes the distance between the source at r' and the

0018-9464/97$10.00 0 1997 IEEE

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1 0 !2 LL p 0.9998

E, 2

= 0.999

- N +

0.9996 m

n 0.g994

- E 2 0.9992 5

Posilion in Ring (degree)

Fig1 Calculated demagnetizing field as a function of the core position.

observer at r, and e, is the unit vector pointing from r' to r. Using (6) and (7 ) the demagnetizing field inside the ring can be now written as

i aM, a w

4, -[1 - cos($ -$')I ,

Let the external magnetic field W, be applied along the x-direc- tion, We have, therefore,

(11)

(12)

M p = x [Kl cos4 + ~ ~ , ) , l , M+ = x [ - H a sinCS + (Hd+I

The induced core magnetization, 4nM, and the demagnetizing , can be then completely solved from the above integral

equations, (9) to (12). For a sensor core made 01 permalloy (magnetic susceptibility x = 100000) most of the applied field will be screened out from the cole mateiial, only about 0 1 % of the external field can penetrate into the core rmg near the vertical positions (4 = go", 270"). This is shown 111 Fig 1 where the normalized demagnetizing field is plotted as a function of the azimuthal angle Cp. Ths screening effect is found to increase lincarly with x for x z 1000, as shown in Fig.2.

Hence, although the use of large x matcrial can increase the sensing ability of a fluxgate magnetometer, it can also result in small penetration field to be effectively utilized by the sensor.

. \ \

Slope 4 = -1

I_

10' i o z i o 3 IO' 105

Core Magnetic Susceptibility

Fig2 Calculated demagnetizing field as a function of the core susceptibility.

These two tendencies counter balance each other and for large x values, say, above 1000, the sensitivity of the magnetometer is independent of x, to be limited by other factors, for example, the noise figure of the core materials. The induced electromotive force in the detector coil can be calculated from Faraday law,

-1 d o e.m$ = -- c dt

In (1 3 ) N denotes the number of wmdmgs in the detector coil, R is the radius of the sensor core whose cross sectional area is denoted as A, c is the speed of light m vacuum, and the subscripts

1 'I and "2" denote the volume elements of the core, at azimuthal angle j=@, intercepted by the sensing circuit of the detector coils spreadmg over a distance dz In (1 3 ) B,(t) and B,(t) are magnetic induction fields given by

B , ( r ) - 4 ~ ~ ( t ) + ~ ~ ~ ( t ) + H = [ 1 ~L)(#;xj] sm@, (14)

B,(O =4nM,(O + f L X & ) +K[1 -Q47X)l s w k (15) where H,, denotes the excitation field of the sensor coil, Ha the applied DC field tu be detected, and D( &x) the demagnetizing

for example The second terms in (1 4) and (1 5) cancel out and the last terms have no explicit time dependence The mduced electro- motive force can be finally wntten as

factm oft& s a w r cure whose functional form IS plotted in Fig 1,

Here d w has been used for the core-cross sectional area A and the magnetic field H is comprised of the following two components

N ( f ) = H,,(t) +Ha D( 4;;~) sin 4. (17)

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H, = 0.1 Oe 0.003 I

0.002

0.00 1

0

-0.00 1

-0.002

A

L E

-0.003 ' ' ' ' ' ' ' -15 0 0.2 0.4 0.6 0.8 1

t/T

Fig.3 Calculated fluxgate signal in a permalloy ringcore for a DC field of 0.1 Oe.

Therefore, when the excitation current in the sensor coil is known, the induced em$ can be calculated using (1 6) and (1 7).

We assume N = 200 turns/cm and the sensor core is to be excited by triangular currents. That is, we assume Hac is of triangular fonn whose amplitude is 5H, with a period denoted as T. For permalloy, H,, the coercive force, is 0.1 Oe. Fig.3 shows the calculated e.m$ for Ho = 0.1 Oe (left coordinate axis) together with the change of magnetization of the sensor core as a function of time (right coordinate axis). The waveform is similar to that calculated by Gordon et a1 assuming a parallel-gated core configuration [Z]. A fluxgate sensor is a nonlinear device, since the excitation fi-equency has been doubled by the gated signals, as clearly indicated in Fig.3.

In Fig.4 we show waveforms of the gated signals (the corresponding first upward one in Fig.3) for three applied DC fields: H, = 0.1,O.O 1, and 0.001 Oe. It is seen in Fig.4 that the signals have sharp falling edges at right sides which are fixed at acommon time instant t = 0.358 T. From Fig.3 we notice that at t = 0.358 T the magnetization of the fluxgate core starts to saturate, and hence its magnetic susceptibility drops from a high

0.003

I

(Unsaturated Core) . 0

0.3579 0.358 t /T

ig.4 Gated signals viewed in fine scale for the appliedDC field Ha = 0.1, 0.01, and 0.001 Oe.

I i o 8 i o 6 i o 4 i o 2 loo

Applied DC Field (Oe)

Fig5 Calculated fluxgate signal width as a function of the applied DC field.

value, x = 100,000 to zero. As such, the applied field penetrates into the core abruptly beyond this point and the total magnetic field of the core depicted by (1 7) is too large to effectively excite the core near its active region possessing maximum nonlinearity. Actually, the right-half portions of the signal waveforms in Fig.4 do not vanish abruptly; they are reduced in amplitude by a factor of [ 1 - D(@ = 90"; x = lOO,OOO)] (2 0.001) with width being correspondingly enlarged to the right by the same factor. This point has never been addressed in the literature in the past, since most of the authors assumed the demagnetizing field is a constant during the whole magnetization process of the core material.

Fig.4 also indicates that the amplitude of the gated signals is independent of the applied DC field. This amplitude is determined by the core material as well as other structural parameters, including the fluxgate dimensions and wiring configurations. Fig.5 plots the signal width as a function of the applied DC field, H,. It is seen in Fig.5 that the signal width decreases linearly with H,. Sensitive sensors in the order of IO-' Oe are possible if the detecting circuit can collect signals near the gated signal peaks [3].

111. DISCUSSIONS

We have included the demagnetizing field in the calculation of a ring-core fluxgate magnetometer. The external field can penetrate into the core only in the regions whose circumference is not perpendicular to the external field. The gated signals only show half the waveform as compared to the previous calculations. Our calculations indicate that sensitive sensors can be constructed if the signals are sampled near the knees of the magnetization nonlinearity.

REFERENCES

[ 11 B. Lax and K. J. Button, "Microwave Ferrites and Ferrirnagnetica"

[2] D. I. Gordon and R.H. Lundsten, IEEE Trans. Magnetics, Map-l,

[3] 0. V. Nielsen, J.R. Petersen, A. Femandez, B. Hernando, P. Spisak,

Mcciraw M11, New York, 1962.

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F. Primdhl, and N. Moser, Meas. Sci. Technol., 2,435,1991.