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Li Hua, Zhang Shu-Lin, Zhang Chao-Xiang, Kong Xiang-Yan, Xie Xiao-Ming. An efficient calibration method for SQUID measurement system using three orthogonal Helmholtz coils. Chinese Physics B, 2016, 25(6): 068501  
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An efficient calibration method for SQUID measurement system using three orthogonal Helmholtz coils
Li Hua1, 2, Zhang Shu-Lin1, †, , Zhang Chao-Xiang1, Kong Xiang-Yan1, Xie Xiao-Ming1
State Key Laboratory of Functional Materials for Informatics, Shanghai Institute of Microsystem and Information Technology (SIMIT), Chinese Academy of Sciences (CAS), Shanghai 200050, China
University of Chinese Academy of Sciences, Beijing 100049, China

 

† Corresponding author. E-mail: zhangsl@mail.sim.ac.cn

Project supported by the “Strategic Priority Research Program (B)” of the Chinese Academy of Sciences (Grant No. XDB04020200) and the Shanghai Municipal Science and Technology Commission Project, China (Grant No. 15DZ1940902).

Abstract
Abstract

For a practical superconducting quantum interference device (SQUID) based measurement system, the Tesla/volt coefficient must be accurately calibrated. In this paper, we propose a highly efficient method of calibrating a SQUID magnetometer system using three orthogonal Helmholtz coils. The Tesla/volt coefficient is regarded as the magnitude of a vector pointing to the normal direction of the pickup coil. By applying magnetic fields through a three-dimensional Helmholtz coil, the Tesla/volt coefficient can be directly calculated from magnetometer responses to the three orthogonally applied magnetic fields. Calibration with alternating current (AC) field is normally used for better signal-to-noise ratio in noisy urban environments and the results are compared with the direct current (DC) calibration to avoid possible effects due to eddy current. In our experiment, a calibration relative error of about 6.89 × 10−4 is obtained, and the error is mainly caused by the non-orthogonality of three axes of the Helmholtz coils. The method does not need precise alignment of the magnetometer inside the Helmholtz coil. It can be used for the multichannel magnetometer system calibration effectively and accurately.

PACS: 85.25.Dq;07.55.Ge;06.20.fb;07.55.Db
Keyword:SQUID;magnetometer;Tesla/volt coefficient calibration;three-dimensional Helmholtz coil
1. Introduction

As an extremely sensitive magnetic sensor, superconducting quantum interference devices (SQUIDs) are widely used for the very weak magnetic field measurements, such as magnetocardiography (MCG), fetal MCG, magnetoencephalography (MEG), etc.[1,2] SQUID is usually driven in flux-locked loop (FLL) read-out electronics, so that the output voltage varies linearly with the magnetic field which is scaled by the Tesla/volt coefficient. As a magnetic sensor, the sensing field magnitude should be precisely known from the voltage output of the SQUID channel. Therefore, the Tesla/volt coefficient of the SQUID system is very important and must be accurately calibrated, especially for a multichannel system.

In order to calibrate a multichannel magnetometer system, two kinds of methods were previously studied. A most traditional method used the single Helmholtz coils. A uniform field was imposed right on the magnetometer and the corresponding voltage output was measured.[3–5] Although effective, the direction of the uniform field must be precisely along the normal direction of the SQUID pickup coil. The other method was based on the measurements of signals generated by many small calibration coils. The unknown sensor parameters, including the Tesla/volt coefficient, were numerically estimated by using complex algorithms.[6–10] Additionally, all of the calibration was performed under a certain AC frequency.

In this paper, we propose a simple and yet efficient calibration method for the SQUID magnetometer system by using three orthogonal Helmholtz coils. The Tesla/volt coefficient can be treated as the magnitude of a vector pointing to the normal direction of the magnetometer. Based on the orthogonal uniform field offered by the three-dimensional Helmholtz coil, the Tesla/volt coefficient can be directly calculated. In our experiment, calibration under different frequencies and the traditional single Helmholtz coils, is measured and compared.

2. Calibration method
2.1. Theory

The magnetometer relies on the integrated pickup coil that picks up the measuring field and couples it to the SQUID. The voltage response of the magnetometer is proportional to the flux passing through the pickup coil area, which can be expressed as:

where B, S, and kϕ are the magnetic field, the pickup coil area, and the flux-to-voltage ratio, respectively. By introducing the normal direction of the magnetometer pickup coil, the Tesla/volt coefficient (KTesla/volt) can be treated as the magnitude of the following vector:

Figure 1 shows the calibration schematic for a SQUID magnetometer. The SQUID magnetometer can be arbitrarily placed in the orthogonal uniform magnetic field. By respectively applying the uniform field, the KTesla/volt is decomposed into three orthogonal components of Kx, Ky, and Kz.

Fig. 1. Calibration schematic for a SQUID magnetometer.

By the orthogonal synthesis, the KTesla/volt can be expressed by

where Bx, By, and Bz are the magnitudes of the applied three orthogonal uniform fields; Vx, Vy, and Vz are the corresponding voltage responses of the magnetometer under the applied field Bx, By, and Bz respectively. By recording the applied magnetic fields and voltage responses, the Tesla/volt coefficient can be directly calculated.

2.2. Numerical simulation

In this method, the calibration accuracy is directly influenced by two key parameters of the field uniformity and orthogonality of the three-dimensional Helmholtz coil. In order to evaluate the influences, digital simulation on these two parameters is first performed.

2.2.1. Field uniformity calculation

The field uniformity of one pair of Helmholtz coils is defined by the uniformity error Δε(x,y,z), which is expressed as

where BAxi(0,0,0) is the magnetic field at the center of the Helmholtz coil and BAxi(x,y,z) is an analytical expression of the axial magnetic field for the Helmholtz coil in all space, which is given by[11,12]

with N being the coil turns, I0 the current value, a the coil side length, and d the coil distance. In our simulation, a volume of 10 cm × 10 cm × 10 cm in the central area is chosen to estimate the field uniformity. Table 1 gives the Δε(x,y,z) under different coil side lengths. For a higher calibration accuracy, the side length of the Helmholtz coil should be larger than 1.5 m.

Table 1.

Values of uniformity error Δε of Helmholtz coil under different coil sizes.

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2.2.2. Field orthogonality calculation

For the field orthogonality of three-dimensional Helmholtz coil, a parameter of the orthogonality error Δε(α, β, γ) is used. Figure 2 shows the relationship between orthogonal field B and non-orthogonal field B′. The θ(α,β,γ) is defined as the non-orthogonal angle of the three-dimensional Helmholtz coil. The field B can be calculated by the field B′ according to Eq. (6).[13]

Fig. 2. Relationship between orthogonal field B and non-orthogonal field B′. α is the angle between O′X′ and XOY; β is the angle between O′Y′ and OY; γ is the angle between O′X′ and YOZ.

The Δε(α,β,γ) corresponding to θ(α,β,γ) is defined as follows:

which can directly reflect the effect of the non-orthogonal field on the calibration accuracy. Table 2 gives the values of orthogonality error Δε (α,β,γ) under several different values of θ(α,β,γ). It shows that the orthogonality error rises fast with the non-orthogonal angle.

Table 2.

Calculated values of orthogonality error Δε(x,y,z).

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2.3. Experiment setup

Figure 3 shows the calibration experiment setup. Consider the independence of sensor orientation in our method, then three vector magnetometers will be selected to be calibrated. The magnetometers are fixed on a ceramic cube holder with a coverage volume of about 2 cm × 2 cm × 2 cm, which is placed in the center of the three-dimensional Helmholtz coil. In our experiment, the coil side lengths of the three pairs of Helmholtz coil are selected to be 1.9 m, 2.0 m, and 2.1 m respectively. The uniformity error is calculated to be less than 2.5 × 10−5. The non-orthogonal angle of the three-dimensional Helmholtz coil is within 0.5° with which the orthogonality error is calculated to be less than 7.2 × 10−4.

Fig. 3. The calibration setup.

Based on the experiment setup, the calibration under different applied frequencies from DC to several tens Hz are performed and compared with each other. Moreover, the independence of sensor orientation is also evaluated by rotating the cryostat along the XY-plane.

3. Calibration results and discussions

Calibration with AC field is normally used for better signal-to-noise ratio in noisy urban environments. In order to avoid possible effects due to eddy current, the AC calibration results are compared with DC calibration results. The calibration results under different frequencies from DC to 30 Hz are shown in Table 3. The calibration relative error is defined as dividing the maximum value minus the minimum value by the average value. During the DC calibration, although field fluctuation at low frequency is inevitable under an unshielded environment, the coefficient can be obtained by linear fitting of the voltage response with the field under different DC currents. The coefficient KTesla/volt of this magnetometer is measured to be 0.2266 V/μT with a relative error of 3.098 × 10−4. Moreover, the calibration results are shown to be independent of frequency.

Table 3.

Several calibration results under different frequencies from DC to 30 Hz.

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In order to illustrate the particularity with respect to the traditional method, comparison measurements are carried out. Figure 4 shows the comparison results between the traditional method and our method. The magnetometers are rotated along the XY plane 10 times. The rotation angle between magnetometer 1 and the X axis is controlled within ±5° which is the same as the case between magnetometer 2 and the Y axis. Magnetometer 3 is always along the Z axis. Figure 4(a) shows the calibration results of the three magnetometers using the traditional method. The calibration relative errors of magnetometer 1 and 2 are 1.26 × 10−2 and 1.30 × 10−2 respectively. However, the calibration relative error is 1.45 × 10−4 for magnetometer 3, which is right along the field direction. It is obvious that the calibration relative error of the traditional method is greatly dependent on the relative angle between the magnetometer and the field direction. Figure 4(b) shows the corresponding calibration results of our method. On the contrary, the calibration results are not influenced by the changing of magnetometer angle. The maximum calibration relative error of the three magnetometers is 6.89 × 10−4. Based on previous digital simulation, the error was mainly caused by the non-orthogonal direction of the three-dimensional Helmholtz coil. In the calibration of the multichannel system, the SQUID magnetometers are placed in the uniform field area of the three-dimensional Helmholtz coil and three orthogonal uniform fields are applied respectively. There is no need to consider the different placements and orientations of SQUID magnetometer in Dewar, the Tesla/volt coefficient of each channel can be simultaneously calibrated.

Fig. 4. Comparison of measurement result between the traditional method (a) and our method (b): 1, 2, and 3 are three magnetometers; measurement points correspond to different rotation angles.
4. Conclusions

We propose an efficient calibration method for the SQUID measurement systems using three orthogonal Helmholtz Coils. A calibration relative error of about 6.89 × 10−4 is achieved, which is mainly due to the non-orthogonal applied field. The results are shown to be independent of the applied frequency and sensor orientation. Without requiring the precise alignment of the magnetometer and complex algorithm, this method will be very convenient and effective for multichannel SQUID magnetometer system calibration.

Reference
1Andrä WNowak H2007Magnetism in Medicine: A Handbook2nd ednBerlinWILEY-VCH Verlag Berlin GmbH101152101–5210.1002/9783527610174
2Qiu YLi HZhang S LWang Y LKong X YZhang C XZhang Y SXu X FYang KXie X M 2015 Chin. Phys. 24 078501
3Melo C F DAraújo R LArdjomand L MQuoirin N S RIkeda MCosta A A 2009 Measurement 42 1330
4Tao WGao JWei Y KLiu Z HZhang CZhang YWang HMa P201210th International Symposium on Antennas, Propagation and EM Theory (ISAPE), IEEEOctober 22–26, 2012Xian, China27710.1109/ISAPE.2012.6408763
5Zhang Y SQiu YZhang C XLi HZhang S LWang Y LXu X FDing H SKong X Y 2014 Acta Phys. Sin. 63 228501 (in Chinese)
6Adachi YHiguchi MOyama DHaruta YKawabata SUehara G 2014 IEEE Trans. Mag. 50 5001304
7Vivaldi VSorrentino ASommariva SPiana MRombetto SRusso M2014Biomag2014, August 24–28, 2014 Halifax, Canada
8Pasquarelli AMelis MMarzetti LMuller H PErné S N2004Neurol. Clin. Neurophysiol2004:94, PMID: 16012682
9Bruno A CCosta Ribeiro P 1991 Rev. Sci. Instrum. 62 1005
10Chella FZappasodi FMarzetti LDella Penna SPizzella V 2012 Phys. Med. Biol. 57 4855
11Alldred J CScollar I 1967 J. Sci. Instrum. 44 755
12Lee S GKang C SChang J W 2007 IEEE Trans. Appl. Supercond. 17 3769
13Russell C TAnderson BBaumjohann Wet al. 2014 Space Sci. Rev.