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γcos⋅H ( ) γcos⋅Δ+ HH

δγ coscos ⋅⋅H ( ) δγ coscos ⋅⋅Δ+ HH

H HH Δ+

The Analysis of Gradiometer Signal in Magnetic Field Measurement with Fluxgate Transducer

Georgiana Marin(Corresponding author)1,*, Serghei Radu2, Gheorghe Samoilescu3 and Octavian Baltag4 1Research Center for Navy, Department of ship physical fields, Constanta, Romania

2Barklav Maritime Agency, Constanta, Romania 3Naval Academy, Electrical engineering Department, Constanta, Romania

4University of Medicine and Pharmacy, Faculty of Medical Bioengineering, Iasi, Romania *Corresponding author (E-mail : georgianamarin01@gmail.com)

Abstract—Gradiometer techniques are generally employed to detect near sources of weak magnetic field, especially in biomagnetic measurements, but they are also effective in the detection of the magnetic signature of surface vessels and submarines. In this paper there is analyzed the recorded signal of a fluxgate gradiometer consisting of two coaxial transducers based on the rejection coefficient. There are also discussed error sources affecting the gradiometer signal.

The object of analysis is the magnetic signature gradient of a ship model. The measurement setup consists of a shielded room surrounded by a triaxial Helmholtz coil, with magnetic field automatic compensation system. The measuring device is a triaxial fluxgate magnetometer picking up the values of the magnetic signature components for the ship model. Magnetic field measurements were processed by the time derivative gradiometer technique. Thus there is determined the spatial variation of the magnetic signature components and of the total field signature, on the three axes of the rectangular coordinate system.

Keywords-fluxgate transducer; gradiometer; magnetic field; rejection factor; ship magnetic signature.

I. INTRODUCTION The issue of magnetic field gradient is imposed by

exploitation of this concept in detecting underwater ferromagnetic bodies by gradiometer equipment towed by surface vessels or aircraft, or in detecting magnetic field variations by magnetic mine sensors [1-5]. These underwater weapons have incorporated fluxgate sensors, with high sensitivity and sufficiently high signal-to-noise ratio [5]. Gradiometer techniques are generally employed to detect near sources of weak magnetic field [6, 9-11], especially in biomagnetic measurements [9, 10], but they are also effective in the detection of the magnetic signature of surface vessels and submarines [6, 7, 11].

Although the problem of magnetic signature determining and exploitation by underwater weapons is extensively dealt within the issue of ship protection against naval mines [1-5, 11-14], the signature gradient approach is found in a few references [5, 11]. Particularizing the ship magnetic field issue, the magnetic signature gradient for a scale ship model was analyzed. A triaxial fluxgate magnetometer was used for this purpose. Time derivative gradiometer signal technique [11]

described in the paper was applied. The need to employ this method in gradiometer measurements with fluxgate transducers is also discussed.

There are few references regarding the ship magnetic signature gradient [2-5]. The literature describing the magnetic mines detection of ship signature is usually classified [3]. For this purpose there was employed a ship model [12], instead of giving a detailed description of a particular ship magnetic signature and gradient. Based on triaxial magnetic measurements there were determined the magnetic field components gradients, as well as the total field gradient, on all axes of the orthogonal coordinate system.

II. MAGNETIC GRADIENT DETECTOR WITH FLUXGATETRANSDUCERS

For the analysis of the signal detected by a fluxgate gradiometer, there are considered two fluxgate transducers coaxially placed at distance l from each other, representing the gradiometer base. In fact, the transducers are not perfectly aligned, as shown in Fig. 1, nor identical, having different sensitivities [11]. Let us denote by S the sensitivity of the first transducer – on left side of Fig.1, and by S + ΔS, the sensitivity of the second one. Furthermore, the measured external field is not uniform, but there is a variation ΔH between the field values picked up by transducers. There is further analyzed the effect of misalignment errors on the gradiometer signal [7, 10]. The voltages measured at the two transducers pickup coils are proportional to the value of magnetic field component on the respective transducer axis [7].

Figure 1. The layout of fluxgate transducers composing the gradiometer (Baltag, 2009)

The measured voltage by the first fluxgate transducer is:

.coscos1 δγ ⋅⋅⋅= HSU (1)

Whereas the voltage measured by the second fluxgate transducer is:

( )( ) ( ) ( ).coscos2 βδαγ −−Δ+Δ+= HHSSU (2)

Here γ and δ denote the vertical and horizontal angles, respectively, between the magnetic field and the first transducer axis, whilst α and β denote the vertical and horizontal angles of misalignment, respectively, between the first and second transducer.

If the signal coils of the two transducers are connected in opposition, the resulting output voltage is the difference between voltages U2 and U1:

( )( ) ( ) ( )

.coscoscoscos

δγβδαγ

⋅⋅−−−−⋅Δ+Δ+=Δ

SHHHSSU

(3)

A feature of the fluxgate gradiometer is given by the ratio between the output voltage caused only by the field gradient ΔH, and the output voltage generated only by the field H applied simultaneously to the two transducers. This ratio is called rejection coefficient and can be written as:

( ) ( )

.1

coscoscoscos1

1

+⎥⎦

⎤⎢⎣

⎡−⋅−

⋅−Δ

⎟⎠⎞

⎜⎝⎛ +

ΔΔ

=

βδαγδγ

SS

SS

HH

F (4)

The rejection coefficient F is the gradiometer ability to measure only the gradient of field ΔH, in the presence of an external field H [10]. It is a function of the two transducers sensitivities, as well as their alignment. In order to determine the effect of transducers misalignment, there is analyzed the variation of the rejection ratio R for varying angles of misalignment α and β, where R is:

( ) ( )

.1

coscoscoscos1

1

+⎥⎦

⎤⎢⎣

⎡−⋅−

⋅−Δ

+Δ=

βδαγδγ

SS

SS

R (5)

Since we are interested in the effects of misalignment errors, we consider the magnetic field H oriented on first transducer axis - the reference axis, therefore γ = 0, δ = 0. The second fluxgate transducer axis is vertically rotated with angle α from the reference axis.

Figure 2. Variation of rejection factor depending on misalignment angles

The chart in Fig. 2 shows the variation of the R coefficient based on the vertical angle α, considering the non-identity of the two transducers signal windings being 0.1*ΔS = S. The rejection ratio variation for the horizontal deviation of the second transducer from the reference axis is analyzed in the following cases: β = 0°, β = 5°, β = 10° and β = 15°. It has a significant variation for angle α in the ranges 0° ÷ 45° and 315° ÷ 360 °, for all analyzed cases. The inset chart in Fig. 2 shows the detailed variation of R ratio in the range 0 ° ÷ 45 °. It is noticed the high magnitude of the coefficient R variation for β angle values of β = 5 ° and β = 15 ° and α angle values in the range 20 ° ÷ 25 °. In the case of transducers misalignment, low values of the angles α and β, of up to 5 ° or 10 °, are expected. In these variation intervals, the rejection ratio R is approximately equal to the ratio S/ΔS chosen for the analysis.

In order to overcome the obstacle of the fluxgate axes misalignment, the time derivative gradiometer technique can be used [11]. This employs a single magnetometer sensor and the distance-based derivation becomes equivalent to the time derivation. The gradiometer signal is given by the magnetic field variations recorded between successive positions, at distance Δx. Due to the non- simultaneity of recordings, total field variations include the Earth’s magnetic field fluctuations – the diurnal variations. To eliminate this disadvantage, it is preferable to perform measurements in a magnetically quiet period - usually in the afternoon [5, 15].

III. DESCRIPTION OF MEASUREMENT SET-UP The measurements occurred in the Laboratory of

Bioelectromagnetism inside the Faculty of Medical Bioengineering, Iasi. The system of measurement is illustrated in Fig. 3 and consists of a shielded room made entirely of non-ferromagnetic materials, surrounded by a triaxial Helmholtz coils system with magnetic field automatic compensation and control system [13, 14, 20-22]. For the magnetic signature analysis of a model ship, there was used a system allowing simultaneous measurement and recording of the three components of the magnetic field. A scale model of a specialized naval ship was constructed, reproducing the ship hull and superstructure [14], without the equipment onboard.

Figure 3. The measurement set-up inside the shielded chamber

The ratio between the original vessel and model main dimensions is 100. The measuring device, a triaxial fluxgate magnetometer is placed in the center of the shielded room and the Helmholtz system [9]. Through the means of the triaxial Helmholtz coil system and the compensation control system [14-16], the Earth’s magnetic field was compensated inside the shielded room. In order to eliminate the effect of geomagnetic field local variations, the magnetic field measurements were performed in the afternoon, in a magnetically quiet interval.

The ship model is fastened to an aluminum beam connected to a mobile platform. The magnetometer is mounted on a transversal wooden beam, independent to the mobile platform. The mobile system slides on the longitudinal x-axis, each measurement line on this axis representing a series of 103 points within 1 cm distance from each other. The system is then moved along the positive y-axis at fixed distance, following a further measurement line, parallel to the first one. Thus there are performed a set of runs on the longitudinal axis, generating a matrix of measurement points [14]. One end of the mobile system on the main axis (longitudinal) was fitted with an optical sensor generating a square wave for each measurement point. The square wave signal is used by the acquisition system to trigger measurement in the points of interest.

IV. MEASUREMENT RESULTS Further are illustrated the three components of the magnetic

signature – the magnetic flux density, at reference depth z = 7 cm [13]. The results are given in nT. In the longitudinal plane, there are 103 measurement points, upwards from bow to stern, representing 102 cm, thus covering nearly twice the ship model length of 60 cm.

Of the three components, the vertical one (Bz) is the dominant, reaching values in the range from 2.84 to 5.27 nT. This is due to the model ship magnetization in the Earth’s magnetic field in the geographical area the model ship was constructed, which also has as dominant component the vertical one [1, 2, 5]. The longitudinal component (Bx) ranges from 1.24 nT to 2.61 nT, whereas the transversal component (By) is low, compared to the other two. It is noted that the extreme values of Bz and Bx components are reached in the vicinity of bow - stern extremities of the model.

Figure 4. Longitudinal component Bx of the model ship magnetic signature at depth z = 7 cm

Figure 5. Vertical component Bz of the model ship magnetic signature at depth z = 7 cm

Figure 6. Transverse component By of the model ship magnetic signature at depth z = 7 cm

Figure 7. Total field B magnetic signature at depth z = 7 cm

The total value of the magnetic induction in each field point is computed with:

.222zyx BBBBB ++== (6)

From the measured components there can be obtained the total field value of the magnetic signature, illustrated in Fig. 7. It has similar shape to that of the vertical component, but higher values, in the range 3.3 to 5.7 nT. The dominant character of the vertical component Bz is thus proved.

V. THE MAGNETIC SIGNATURE GRADIENT Variation of magnetic signature components can be

emphasized by the magnetic field gradient calculation and representation. Modern magnetic mines sensors operate on the detection of magnetic field gradient, thus eliminating the influence of disturbing factors such as diurnal variation of the geomagnetic field and allowing rapid identification of near-field sources. There are further analyzed the signature components gradients and total field gradient, with measurement base on three axes of the Oxyz system. The gradient was determined by the time derivation technique.

A. The longitudinal gradient of the magnetic signature components Longitudinal gradient of the three components was

computed as the difference between field values in successive points on the longitudinal axis, with base distance Δx = 1 cm. Thus were obtained the longitudinal gradient of the longitudinal Gx(Bx), vertical Gx(Bz), and transversal component Gx(By). The computation formulas are given below:

( )

( )( )

.

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

ΔΔ=

∂∂=

ΔΔ

=∂

∂=

ΔΔ=

∂∂=

xB

xBBG

xB

xB

BG

xB

xBBG

zzzx

yyyx

xxxx

(7)

Figure 8. Longitudinal gradient of the longitudinal component Gx(Bx)

Figure 9. Longitudinal gradient of the vertical component Gx(Bz)

Figure 10. Longitudinal gradient of the transverse component Gx(By)

The negative extreme values of the longitudinal variation of component Bx are achieved below the ship keel, towards the bow. The positive extreme values are recorded at the aft extremity, near the ship sides. The gradient Gx(Bz) analysis in Fig. 9, shows that the maximum longitudinal variation of the vertical component takes place in the abscissa range of 10 to 30 cm, in the afore part of the ship. The negative extreme values are situated symmetrically with respect to the longitudinal plane, at transversal distance Δy = 6 cm approximately equal to half the vessel width. Similar variations within a smaller range occur in the aft area of the ship and behind it. The transverse component By illustrated in Fig. 6, varies from -0.74 to +0.26 nT. However, its variation on the longitudinal axis, graphically illustrated in Fig. 10, generates gradient values Gx(By) comparable to those of the longitudinal and vertical components. The extreme values are located approximately in the ship sides, near the fore – aft extremities.

Therefore, all three components of the magnetic signature of the ship model show longitudinal gradients in the range -20 to +10 nT / m and absolute values of positive or negative polarity registered near the ship afore and aft extremities.

B. The transverse gradient of the magnetic signature components The transverse gradient of the three components was

similarly computed, i.e. as the difference between successive points values on the transverse axis, with base distance Δy = 2 cm. Notations and computing relationships are similar to those of the longitudinal gradient.

Figure 11. Transverse gradient of the longitudinal component Gy(Bx)

Figure 12. Transverse gradient of the vertical component Gy(Bz)

Figure 13. Transverse gradient of the transverse component Gy(By)

Variation of the longitudinal component Bx on y-axis shows extreme values in the right afore-aft ends of the ship, one ship half-width from the longitudinal plane. The transverse gradient of the vertical component Gy(Bz) shows negative values in the forward half of the ship, near its sides. Maximum variation is recorded astern, at one ship width from the longitudinal plane. From the analysis of transverse gradients, there is noticed a variation interval narrower than of longitudinal gradients, but its order of magnitude is comparable to the latter.

C. The vertical gradient of the magnetic signature components The vertical gradient of the three components Bx, Bz, By was

computed as the difference between the field values in the horizontal planes z = 7 cm and z = 9 cm. Notations and formulas are similar to those of the previously computed gradients.

Figure 14. Vertical gradient of the longitudinal component Gz(Bx)

Figure 15. Vetical gradient of the vertical component Gz(Bz)

Figure 16. Vertical gradient of the transverse component Gz(By)

Variations in the z-axis of signature components are illustrated Figs. 14-16. The Gz(Bx) gradient has two half-waves, the positive polarity being located from amidships towards the stern, and beyond it to about 10 cm. Negative values lie afore, below the keel. The vertical component gradient in the z-axis, Gz(Bz) consists of two half-waves, similar to the longitudinal component variation, but with higher minimum value and covering a larger longitudinal distance x= 20 to 50 cm.

The transverse component variation in the z-axis exhibits four half-waves symmetrically situated from the model ship longitudinal plane. Positive half-waves are located at half bow, near the ship sides, whereas negative half-waves are arranged before the aft end, at 20 cm outside the edges of the vessel.

D. Total field gradient Based on the total field magnetic signature, the three axes

gradients can be computed, using the formulae similar to those for computing magnetic field components gradients. Spatial distribution of total field gradient is illustrated in Figs. 17 - 19.

Figure 17. Total field gradient on the x-axis Gx(B)

Figure 18. Total field gradient on the z-axis Gz(B)

Figure 19. Total field gradient on the y-axis Gy(B)

Comparison of total field gradients variation pattern to the components gradients reveals the major impact of the vertical component gradient in the appearance of the total field gradient on all three axes of the coordinate system. Thus, in all three cases, the shape of the vertical component gradient is reflected completely in the total field gradient.

VI. CONCLUSIONS There was analyzed the ability of a fluxgate gradiometer to

detect a signal in the presence of an external field, by means of the rejection factor. Since the perfect alignment of fluxgate transducers is difficult to achieve, it may cause errors in fluxgate gradiometers measurements. In order to overcome the obstacle of misaligning the fluxgate axes, a triaxial fluxgate magnetometer may be used along with the time derivative gradiometer technique.

There was applied the time derivative gradiometer technique to a set of magnetic signature measurements performed on a ship model, in order to determine the magnetic variation on the three axes of the orthogonal coordinate system Oxyz. Thus there were determined the gradients of the three signature components and the total field gradient. There is noticed the major impact of the vertical component gradient in the total field gradient, in all three axes of the coordinate system. The paper originality lies in the complete analysis of magnetic signature gradient of both the total field and its components. Such analysis is useful in interpreting the interaction between the magnetic field sensors fitted to underwater weapons and surface ships and submarines.

ACKNOWLEDGMENT G. Marin thanks the Bioelectromagnetism Laboratory

research team of Faculty of Medical Bioengineering Gr.T.Popa, Iasi, who took active part in magnetic field measurements.

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