Optimized Design and Calibration of the Triaxis Induction Magnetometer with Crosstalk and Nonorthogonality Compensation

An optimized triaxis induction magnetometer has been designed and calibrated to minimize the influences from the nonorthogonality and the magnetic flux crosstalk. Utilizing the nonlinear least square method, contributions due to the nonorthogonal assembly of three transducers are cancelled. The magnetic flux crosstalk is a frequency-dependent error component in the calibration of the triaxis inductionmagnetometer. Influences from the assembly density, the frequency, and the feedback amount are analyzed theoretically, and an optimized sensor configuration which has a smaller crosstalk is achieved. Moreover, a mathematical compensation algorithm has also been utilized to suppress the residues crosstalk ulteriorly. To validate the theoretical analysis, a triaxis induction magnetometer was manufactured and the experiment setup has also been built. The experiment results show that the cross-outputs of the transverse induction magnetometers have been significantly decreased about two orders, indicating that the proposed method is applicable for the triaxis induction magnetometer.


Introduction
It is the objective of this paper to achieve minimum contributions from the magnetic flux crosstalk among three transducers and nonorthogonality of the assembly accuracy.Nonorthogonality error occurs because of the mechanical precision of the individual magnetometer axes [1][2][3][4][5][6][7][8][9][10].Magnetic flux crosstalk is a frequency-dependent component and is due to an unexpected magnetic flux distribution among three transducers [11,12].
Various calibration algorithms have been proposed for the nonorthogonality in the past years.Transformation matrix between the orthogonality coordinate and the triaxis magnetometer coordinate is derived theoretically.Then, the calibration is considered as nonlinear estimation and calibration parameters can be achieved with different algorithms.Pylvänäinen introduced a calibration method based on recurving fitting of an ellipsoid to collected samples from the sensor [1].Crassidis et al. developed alternative realtime algorithms based on the extended Kalman filter (EKF) and unscented Kalman filter (UKF).Both algorithms provide accurate integer resolution in real time, but the UKF is more robust to large initial condition errors than the EKF [2].Pang compared the calibration performance of UKF, twostep algorithm, and nonlinear least square methods, and the experimental results show that the nonlinear least square method has the smallest error average and standard deviation [3][4][5].
The magnetic crosstalk is another limitation for the measurement accuracy of the triaxis induction magnetometer.Regarding Grosz et al. 's research, a maximum crosstalk 27.7% is reached just above the resonant frequency of the primary coil where the secondary flux crosstalk dominates.Conversely, at a frequency just below the resonant frequency, the minimum crosstalk is achieved because of the compensation of the primary and secondary crosstalk flux [11].Numerous investigations about the crosstalk have been published in recent years [11][12][13][14][15][16][17][18][19].A strong enough negative-feedback flux is an effective way to decrease the crosstalk near the resonant frequency.And the secondary flux generated by the primary

R
The equivalent electronic circuit of the induction magnetometer.
coil above the resonant frequency has been suppressed.However, an unwanted increasing crosstalk below the resonant frequency emerges concomitantly.Paperno et al. proposed a mathematical compensation method for the crosstalk in 2011 [12].The compensation is based on deriving crosstalkfree magnetometer outputs from a system of equations describing the magnetometer total outputs as a function of the applied field and the parameters of the triaxis induction magnetometer.But the dynamic range of the transducer output is too large and the resolution of the magnetometer is limited.The purpose of this paper is to develop an optimized design and calibration method of the triaxis induction magnetometer.The magnetic flux-feedback principle has been utilized to suppress the crosstalk near the resonant frequency.Moreover, contributions from the assembling density, frequency, and feedback amount have been analyzed and an optimized sensor configuration with smaller crosstalk has been achieved.Moreover, a calibration method, in which, besides the mathematical compensation algorithms for the crosstalk, the nonorthogonality compensation has also been considered synthetically, is proposed to improve the calibration performance of the triaxis induction magnetometer.
To validate the proposed method, a triaxis induction magnetometer was manufactured.And the experiment results show that, after calibration, the cross-outputs of the transverse induction magnetometers have been significantly decreased by two orders, indicating that the proposed method is applicable to the triaxis induction magnetometer.

Principle of the Triaxis Induction Magnetometer
The triaxis induction magnetometer and its equivalent electronic circuit are as shown in Figures 1 and 2 [20][21][22][23][24]. Three induction magnetometers are assembled orthogonally to sense the varied magnetic field in the space., , and  are the resistance, inductance, and capacitance of the induction coil separately,  f and  f are the inductance of the feedback coil and the feedback resistance,  is the mutual inductance between the induction and feedback coils,  is the gain of the preamplifier, and   is the output of the preamplifier.  is calculated as follows: ,, is the magnetic flux density in -axis, -axis, and axis transducers,  app is the apparent permeability of the core, and Φ p is the primary magnetic flux in every transducer.

Optimized Design and Calibration of the Triaxis Induction Magnetometer
3.1.Calibration of the Nonorthogonality.As shown in Figure 3, - is the orthogonal coordinate and -      denotes the triaxis induction magnetometer coordinate.The vertical axis   is defined to be completely aligned with the -axis.The   -axis lies in the plane defined by -axis and -axis.And  is the angle between   -axis and the forward direction of -axis. is the angle between   -axis and the plane, which is defined by -axis and -axis.
And  is defined as the angle between the projection of   -axis in the plane and the forward direction of -axis.
The transformation between two coordinates can be expressed as in (2).  is the coordinate transformation matrix.  ,   , and   are the measured magnetic flux density in -axis, -axis, and -axis, and    ,    , and    are the real magnetic flux density in -axis, -axis, and -axis: ] . (2)

Optimized Design and Calibration of the Crosstalk.
The crosstalk, which is due to an unexpected magnetic flux distribution in three transducers, is a specific error component of the triaxis induction magnetometer.Its strength mainly depends on the assembly density ( c in Figure 1), the feedback amount ( f ), and the frequency ().

Crosstalk Evaluation of the Triaxis Induction Magnetometer.
The crosstalk due to the primary and the secondary magnetic flux is simulated by the FEM software Maxwell 15.0 and shown as in Figure 4.
As for the crosstalk due to the applied flux (Figure 4(a)), we can see that the applied vertical flux also flows in the transverse transducers and provides an additional contribution for the transverse output.Moreover, there is also secondary flux, generated by the current in the vertical induction coil, flowing in the transverse transducers (Figure 4(b)).
According to Grosz et al. 's research, the relative crosstalk  c is defined by the ratio of the crosstalk flux in the transverse transducer, to the primary flux in the vertical transducer, shown as follows [11,12]: cp is the crosstalk coefficient due to unit primary flux,  cs is the crosstalk coefficient due to unit secondary flux, Φ p is the primary flux in the vertical transducer, Φ s is the secondary flux generated by the induction coil in the vertical transducer,  p is the relative primary crosstalk compared to unit Φ p and its value is equal to  cp , and  s is the relative secondary crosstalk compared to unit Φ p .Φ s is estimated as follows: Subjecting ( 1) and ( 4) to (3),  c can be expressed as The magnetic field in every axis can be estimated as in (6).Φ  , Φ  , and Φ  are the measured magnetic flux in -axis, axis, and -axis, and Φ   , Φ   , and Φ   are the crosstalk-free magnetic flux: The values of the apparent permeability  app and the core cross section  are equal compared with each other.Based on equation Φ =  app ⋅  ⋅ , (6) can be simplified as

Optimized Design of the Assembly Density 𝐷 c . 𝜙 cp and
cs depend on a set of transducer parameters, such as the core's material, the aspect ratio, and the geometry structure of the sensor configuration.In this paper, the core material is permalloy and the cross diameter is 8.5 mm.To achieve an optimized sensor configuration with a smaller crosstalk, the crosstalk with different  c is simulated with the help of the FEM simulation software Maxwell 15.0, as shown in Figures 5 and 6.
As shown in Figures 5 and 6, the magnetic flux density in -axis has no significant differences with different  c .The crosstalk magnetic flux density in -axis and -axis varies from positive values to negative values (or from negative values to positive values), and the zero crosstalk point moves in concert with  c .The configuration with  c = 98.5 mm selected for the crosstalk in the transverses transducers can be mostly cancelled, and the residues crosstalk is  cp = 0.64% and  cs = 0.93%.

The Influence of the Feedback Amount and Frequency.
Based on (3), the total relative crosstalk  c depends on the relative primary crosstalk  p and secondary crosstalk  s .For a given triaxis induction magnetometer,  p has a constant amplitude and phase in the whole frequency band, while  s is varying with different feedback amount and frequencies, as shown in Figure 7.
From Figure 7 we can see that, compared with  p ,  s has large phase differences below the resonant frequency, the same phase at high frequency band.Moreover, with the feedback resistance increasing,  s is decreasing at low frequency band and increasing at resonant frequency.At frequency band higher than the resonant frequency, the feedback has no significant influence and  s approaches  cs .
Moreover,  c with different feedback amount and frequencies has also been calculated, as shown in Figure 8.
At frequency band below the resonant frequency,  c is equal to  p approximately for  s is so small that it can be neglected (see Figure 7).The minimum value is approached at frequency just below the resonant frequency, where  p is mostly cancelled by the reversed  s .At the frequency just above the resonant frequency,  c reaches its maximum value due to the maximum  s .At frequency band higher than the resonant frequency,  c reaches ( p +  s ) for  p is in phase with  s .
Therefore, if a low crosstalk below the resonant frequency is needed, or in some specific situation we want to get a high sensitivity near the resonant frequency, a light feedback is a better choice.But if you want to achieve a smaller crosstalk near the resonant frequency, a deep feedback is preferred.

Calibration of the Triaxis Induction
Magnetometer.Considering the calibration equations for the nonorthogonality and the crosstalk, as evaluated in (2) and ( 7), the calibration model of the triaxis induction magnetometer can be derived as follows: ,   , and   are the outputs of the triaxis induction magnetometer, and   ,   , and   are the sensitivities of the -axis, -axis, and -axis induction magnetometers.
Based on (8), the output voltages of the triaxis induction magnetometer are calibrated and compensated.

Experiment
4.1.Experimental Setup.To validate the method proposed above, a triaxis induction magnetometer was manufactured as shown in Figure 1, and its parameters are shown in "Notations" section.The preamplifier circuit (-axis as an example) of the sensor is as shown in Figure 9. Dual-JFET is utilized in the first-stage for it has a high input impendence.
The experiment setup is also established as shown in Figure 10.
The applied magnetic field  is generated by the Helmholtz coils (green part, driven by current source Keithley 6221).The yellow square indicates the area where the magnetic field is uniform.The red circuit in the uniform field area is the platform to fix the triaxis induction magnetometer (blue cube in the red circle).The outputs of the sensor are measured by the Agilent 35670A and processed utilizing the nonlinear least square method (programmed by the MATLAB software) [3].

Experiment Results.
The sensitivities of different axes of the triaxis induction magnetometer were calculated, as shown in Figure 11.We can see that it has a sensitivity of 40 mV/nT @ 100 Hz and 100 mV/nT @ float part.
The calibration of the triaxis induction magnetometer, including the nonorthogonality and the crosstalk compensation, has also been done.In detail, the magnetic flux is   applied in -axis direction, and the normalized cross-outputs of the transverse induction magnetometers (-axis and axis) have been measured and calibrated based on (8), as shown in Figure 12.
We can see that, compared with utilizing the magnetic flux-feedback, the crosstalk of the -axis and -axis induction magnetometers is decreased about two decades approximately in the whole frequency band.Moreover, compared with utilizing the compensation algorithm proposed by Paperno et al. [12], a lower crosstalk is achieved near the resonant frequency (3 kHz approximately in this paper).It is because the high crosstalk near the resonant frequency has been suppressed by the negative-feedback flux.Then, the residues crosstalk is compensated using the compensation algorithm.

Conclusions
To improve the performance of the triaxis induction magnetometer, an optimized design and calibration method are proposed and validated in this paper.The nonorthogonality is cancelled based on the coordinate transformation.With the help of the FEM software Maxwell, an optimized sensor configuration which has a smaller crosstalk is achieved.Finally, the calibration model including the nonorthogonality and the crosstalk compensation of the triaxis induction magnetometer is built.The experiment results show that the cross-outputs have been decreased about two decades in the whole frequency band, which is much more efficient than using either the negative-feedback flux or the mathematical compensation algorithm.
The crosstalk factor due to the applied and the secondary flux is evaluated with the help of the FEM simulation software in this paper.As for further investigation, it will be meaningful to build a numerical model used for the crosstalk coefficients evaluation.

Figure 5 :Figure 6 :
Figure 5: The crosstalk due to the applied flux (the blue/purple/red solid lines, double crossed lines, and dotted lines indicate the primary flux in -axis/-axis/-axis with  c = 28.5 mm, 68.5 mm, and 98.5 mm separately).

Figure 7 :Fr
Figure 7:  s simulation with different feedback amount and frequencies.

Figure 8 :
Figure 8: The crosstalk with different feedback amount.

Figure 12 :
Figure 12: The calibration result of the triaxis induction magnetometer.