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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Based on the core hysteresis features, the RTD-fluxgate core, while working, is repeatedly saturated with excitation field. When the fluxgate simulates, the accurate characteristic model of the core may provide a precise simulation result. As the shape of the ideal hysteresis loop model is fixed, it cannot accurately reflect the actual dynamic changing rules of the hysteresis loop. In order to improve the fluxgate simulation accuracy, a dynamic hysteresis loop model containing the parameters which have actual physical meanings is proposed based on the changing rule of the permeability parameter when the fluxgate is working. Compared with the ideal hysteresis loop model, this model has considered the dynamic features of the hysteresis loop, which makes the simulation results closer to the actual output. In addition, other hysteresis loops of different magnetic materials can be explained utilizing the described model for an example of amorphous magnetic material in this manuscript. The model has been validated by the output response comparison between experiment results and fitting results using the model.

The fluxgate sensor has been widely used in magnetic field measurements due to its high sensitivity, small size and low power consumption [

The shape of the hysteresis loop which relates to the features of magnetic core determines the final output characteristics of the RTD-fluxgate. Therefore, the precise fitting of the hysteresis loop determines the fluxgate simulation quality [

In order to obtain the accurate fitting of the dynamic changing rule of the hysteresis loop when the RTD-fluxgate is working and facilitate the numerical simulation of this kind of fluxgate, this paper proposes a new arc tangent model containing a dynamic permeability parameter via analysis of the working principle of RTD-fluxgate, ideal hysteresis loop model, and arc tangent model. Compared with the output response results of the RTD-fluxgate based on an ideal hysteresis loop model, the novel arc tangent model which contains a dynamic permeability parameter and fits the actual dynamic hysteresis loop improves the accuracy of the hysteresis loop simulation on soft magnetic materials and reduces the deviation of the output response simulation of the RTD-fluxgate sensor.

When the RTD-fluxgate is working, the core of the sensor is magnetized by a periodically alternating magnetic field to the states of two-way over-saturation. The target magnetic field can influence the residence time of the magnetic core in positive and negative saturation states. In practice, we may obtain the values of target magnetic fields by detecting the time difference of the output pulse signals which relate to the states. If the target magnetic field is zero, since the exciting magnetic field only exists in the axial direction of the sensor, the residence times of the magnetic core in positive and negative saturation states are the same, and the time difference between them is zero, ΔT = T^{+} − T^{−} = 0, and their sum is the excitation signal cycle, T^{+} + T^{−} = T, as is shown in _{x} exists along the axis of the sensor, this field is superimposed on the excitation magnetic field, so that the residence times of the magnetic core in positive and negative saturation states are different, then the time difference between them is not zero, ΔT = T^{+} − T^{−} ≠ 0, as is shown in

In _{c} is the coercive field, H_{x} is the target magnetic field, T^{+} is the time interval between the positive pulse and negative pulse of the output signal, and T^{−} is the time interval between the negative pulse and positive pulse of the output signal.

As described in the working principle of the RTD-fluxgate, the output response of the sensor is related to the two-way over-saturation and the target magnetic field is detected based on the difference between the residence times in two states. The dynamic hysteresis loop reflects the dynamic working process of the core and the states of output signal, but the magnetization process transition between the two stable states is not instantaneous, therefore, an accurate description of the hysteresis loop can affect the RTD-fluxgate research.

The features of the flux density

The expression of the output signal is shown in

According to the Faraday law of electromagnetic induction, under the condition that the sensing component parameters of the sensor and the amplitude of the excitation field are constant, the maximum dynamic permeability of the core determines the maximum amplitude of the output signal.

The ideal hysteresis loop is shown in _{c} of core in this direction but also needs to overcome the external magnetic field. That means the coercive field H_{c} is increased in this direction, and

In _{1max} and μ_{2max} represent the position of the maximum dynamic permeability when the core is reversely and forwardly magnetized, respectively. In order to verify the relationship between the hysteresis loop and the output signal, the experiments are implemented under two different conditions (there is the external magnetic field and there is no external magnetic field), as shown in

As shown in the comparison between _{1max}| ≠ |μ_{2max}|. _{1max} and μ_{2max} are moved and the greater the measured magnetic field is, the more distance the two positions will move toward the same orientation, then the time differences of the output signal will become greater, as shown in

In summary, the caculation of the level of variation of the maximum dynamic permeability between the forward magnetization curve and the reverse magnetization curve in a hysteresis loop can caculate the time difference of the sensor output signal. Because the forward magnetization time and the reverse magnetization time compose a magnetization cycle, the same as the cycle of the magnetic excitation field. The variation of the time difference of the output signal is reflected by the changing of the maximum permeability positions. Therefore, the fitting accuracy of the hysteresis loop, especially at the positions of the maximum permeability, directly affects the output response simulation quality of the RTD-fluxgate established by the fitting equation.

To improve the fluxgate simulation accuracy, Trujillo

In _{sat} is the saturation flux density, μ_{0} is the permeability in vacuum state, _{d} is the value of relative permeability of core when

Based on _{sat}_{0}. In order to improve the fitting accuracy of the hysteresis loop model, taking into account that the coercive field in the hysteresis loop is always hindering the relative change of the excitation magnetic field, the correction term ±H_{c} is included in the arc tangent model, so

In the model described by

According to Faraday's law of electromagnetic induction, the core flux density _{c} are known, the values of the permeability parameter

In a magnetically shielded room, a precision current source (KEITHLEY 6221) is used to drive a Helmholtz coil to generate a fixed external magnetic field. A high precision data acquisition module (NI PXI-4495) is used to collect the sensor output signals under the condition of a 100 mA, 5 Hz sine excitation field. The measured induction voltage is integrated to get the changing rules of the core flux density

_{D}

In _{n}_{D}_{d} in

By using the Matlab software, when the excitation magnetic field

In

In

As seen from _{sat} and coercive field H_{c} can be reflected intuitively by the BH curve. This is beneficial for the selection of core materials. The fitting data of the dynamic permeability parameter _{D}

Because the output signal peaks of RTD-fluxgate sensor correspond to the positions of maximum core dynamic permeability, and the fitting of dynamic permeability parameter affects the output response simulation of the RTD-fluxgate. As seen in

In the ideal condition that the maximum permeability of hysteresis loop at the position of coercive field is infinite, Andò

In _{e}_{c} is the coercive field. As the output response of the RTD fluxgate is established based on the ideal hysteresis loop model and the ideal excitation signal, the relative permeability of the actual hysteresis loop is not an infinite value which is changed while the vary of excitation magnetic field. Therefore, as the positions of the maximum permeability cannot be confirmed accurately from the time difference output response of the RTD-fluxgate which is based on the ideal hysteresis loop model. There is a time difference deviation owing to the fact the hysteresis state transition time of the core status can be neglected.

In order to verify the accuracy of the proposed model and analyze the output response of the RTD-fluxgate conveniently, another type of analysis of the output response of the RTD-fluxgate is obtained by using

The output response of the RTD-fluxgate is proportional to the magnetic field of the measured target, so the time difference can be obtained through the time point related to the output signal peaks by derivating

There exists only a numerical result, not an analytical result because

As shown in _{c} are known, the value of the actual dynamic permeability parameter _{n} can be calculated through _{D}

The experiments are validated under the conditions of excitation magnetic fields 100 mA sine at 5 Hz, 80 mA sine at 5 Hz and 100 mA sine at 10 Hz and a range of external magnetic fields from 0.08 A/m to 10 A/m with a 2.0 A/m interval in the magnetically shielded room. A couple of output signals are selected for fitting. The output time D-values of the RTD-fluxgate which are actually measured, and those calculated by

As seen from

The corresponding relative deviation contrast values are 2.9%–4.4% and 1.0%–3.4%, respectively. The sensitivity of the proposed RTD-fluxgate output response is closer to the practical application compared with the math model, so the proposed model can minimize the deviation between the magnetic features of the core material in simulation and the practical application materials.

A special arc tangent model is proposed to make the relative fitting deviation of the hysteresis loop less than ±3%. The described model changes the original permeability parameter to the dynamic permeability parameter and involves the coercive field in the excitation magnetic field. The physical parameters α, _{D}_{c} can describe the hysteresis loop more accurately. The model is useful for research on the output response of RTD-fluxgate sensors and selecting the core materials. The absolute values of relative deviation between the output time D-values of the RTD fluxgate and the actual values are less than 3.4%. In addition, the illustrated model makes up for the drawback that the fixing shape of the ideal hysteresis loop model could not accurately reflect the actual dynamic variation of the hysteresis loop, fits the core hysteresis loop more accurately, and minimizes the deviation between the magnetic features of the core material in simulation and practical application materials. The model provides a theoretical basis for research on the simulation of the sensors' output responses.

Thank the National Natural Science Foundation of China (No.41274183), the National Natural Science Foundation (No.40904053) and the Fundamental Research business of Central Universities (No.450060445213) for their support of the research described in this paper. We also extend our gratitude to the Key Laboratory of Geo-Exploration Instrumentation (Jilin University) Ministry of Education for their help.

The authors declare no conflict of interest.

The schematic diagram of the output pulse signals: (

(

The output signal of the sensor: (

The actual changing curve of the permeability parameter

The comparison between actual measuring curve and two fitting curves of the hysteresis loop.

The contrast curve of dynamic permeability parameters.

The relative deviation between the fitted dynamic permeability parameter and the actual dynamic permeability parameter.

The flow chart of the output response of the RTD-fluxgate by the fitting model.

The output time D-values contrast of the RTD-fluxgate. Data1 is the output time D-value in the (100 mA, 5 Hz) sine excitation magnetic field. Data2 is the output time D-value in the (80 mA, 5 Hz) sine excitation magnetic field. Data3 is the output time D-value in the (100 mA, 10 Hz) sine excitation magnetic field.

The relative deviation contrast curves between the output time D-values of the RTD-fluxgate under two different conditions s and the actual value.