# A New Scale Factor Adjustment Method for Magnetic Force Feedback Accelerometer

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## Abstract

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## 1. Introduction

^{2}), which is too small for a full tensor gradient measurement. When considering the full tensor measurement application, using magnetic force to rebalance the accelerometer with a large measuring range might be a better choice. The modified Model VII accelerometer used in Bell’s GGI is one specific application [4,5]. To adjust the scale factor dynamically, the accelerometer is disassembled, and an additional trimming coil is wound around the permanent magnet [11,12]. An appropriate current is calculated, which flows through the trimming coil to change the strength of the magnetic field acting on the torque coil. Thus, the scale factor can be adjusted. However, imperfect assembly will cause more nonlinear and cross-coupling effects, and the current in the trimming coil may cause extra unwanted heat effects, which will affect the relative permeability of the magnet. Hence, the stability of the magnetic field in the feedback loop will deteriorate.

## 2. The Principle of the Scale Factor Adjustment Method by Trimming the Feedback Current

_{f}related to the movement of the PM. The current I

_{f}flows through the torque coil and generates an equal and opposite force to compensate the movement. The related signal I

_{f}is therefore taken as the output of the accelerometer.

_{0}is the strength of the permanent magnet, and L is the circumference of the torque coil. The physical parameters of the accelerometer appears in Table 1.

_{t}flowing through the trimming coil generates an additional magnetic strength ∆B. Hence, the magnetic field in which the torque coil acts is strengthened or weakened. Therefore, the force rebalance equation varies with the strength of the magnetic field, and the feedback force is given by

_{i}

_{0}into the Equation (3), the trimmed scale factor K

_{i}has the following relationship

_{t}.

_{f}is acquired with an Analog-Digital-Converter (ADC), multiplied by a scale proportionality coefficient p provided by the Field Programmable Gate Array (FPGA) according to the actual requirement, and subsequently transferred to analog current ${I}_{t}^{\prime}$ with a Digital–Analog Converter (DAC). The adjustment current ${I}_{t}^{\prime}$ and the feedback current I

_{f}are summed and injected into the torque coil to rebalance the PM. The adjusting current ${I}_{t}^{\prime}$ and the feedback current I

_{f}satisfy the relationship

_{tot}injected into the torque coil consists of two parts, I

_{f}and ${I}_{t}^{\prime}$, and they satisfy the relationship given by

_{i}’ is the scale factor after trimming the feedback current. It can be seen from the expressions that the acceleration-to-current transfer function varies with the coefficient p. Comparing Equation (8) with Equation (4), we find that these two methods have similar formulas for the scale factor and are adjustable in common.

_{c}, H

_{pid}, H

_{a}and H

_{t}, respectively. The gain of H

_{t}is adjustable by FPGA, and must satisfy the relationship H

_{t}= p × H

_{a}to ensure that Equation (5) is satisfied. After that, the adjusting current ${I}_{t}^{\prime}$ and the output current I

_{f}are summed as I

_{tot}and injected into the torque coil. To trim the scale factor of the accelerometer, all that needs to be done is to change the proportional coefficient p in FPGA.

## 3. Experimental Verification

_{i}. The adjustment unit (blue, Figure 1) consists of a digital circuit with a high-precision, 20-bit ADC AD7703, FPGA, and 20-bit DAC AD5791.

_{in}= V

_{out}/K, where V

_{out}and K are the output voltage and the scale factor of the reference accelerometer respectively. The scale factor of the reference accelerometer used was 100 V/g. The tilt table was inclined an additional angle of about 9 mrad each time, and was kept still for about 90 s at each position to record the data. The calibration results are shown in Figure 4. The output currents of the trimmed accelerometer with different proportional coefficient p vary with time, as shown in Figure 4a. When taking the input acceleration derived from the output of the reference accelerometer as the x-axis and the output current of the trimmed accelerometer as the y-axis, accordingly, the slope of the line is the scale factor of the trimmed accelerometer, as shown in Figure 4b.

_{i}

_{0}of the accelerometer. By a least-squares linear fitting, the result is ${K}_{i0}/{K}_{i}^{\prime}=1.0001(2)+1.0001(6)p$, which results in good agreement with the expected value. The fitting results verify the proposed scale factor adjustment method. The changes of the scale factor are 33% smaller at p = 0.5 and 100% larger at p = −0.5 than the original one, which is in perfectly experimental agreement with Equation (8). In general cases of GGI, the scale factors of the accelerometers are able to achieve a consistency of ±10%, which corresponds to the range of p from +0.11 to −0.09. The scale factor balance loop would be able to calculate an appropriate coefficient p for each accelerometer in the GGI automatically [10]. Thus, this method would meet the requirement to expediently adjust the scale factor of the accelerometers in the development of the GGI.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**The diagram of a magnetic force rebalance accelerometer with the scale factor trimming. The red signal flow is the Bell’s scale factor adjusting method with trimming coil, while the blue signal flow is the new method with the trimming feedback current presented in this paper.

**Figure 2.**The block of the transfer function for the scale factor trimming. The value of the coefficient p can be provided numerically by FPGA according to the actual needs.

**Figure 3.**The calibration system. (

**a**) Principle of the calibration with the tilting method. (

**b**) The setup of the calibration system.

**Figure 4.**Calibration results of the scale factor of the trimmed accelerometer. (

**a**) The output of the trimmed accelerometer with different proportional coefficient p. (

**b**) The output of the trimmed accelerometer varies with changes in acceleration, and the slope of each line is the scale factor of the trimmed accelerometer corresponding to the proportional coefficient p. The standard deviation of each current step is at the order of 0.01 mA.

**Figure 5.**Linear fitting the ratio of ${K}_{i0}/{K}_{i}^{\prime}$ varying with the coefficient p according to Equation (8). The standard deviation of each point is at the order of 10

^{−4}.

Parameter | Design |
---|---|

Proof mass m (mg) | 10 |

Effective area of the differential capacitive (mm^{2}) | 350 |

Gap between the plates (ìm) | 86 |

Strength of the magnet (T) | 0.25 |

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## Share and Cite

**MDPI and ACS Style**

Huang, X.; Deng, Z.; Xie, Y.; Li, Z.; Fan, J.; Tu, L. A New Scale Factor Adjustment Method for Magnetic Force Feedback Accelerometer. *Sensors* **2017**, *17*, 2471.
https://doi.org/10.3390/s17112471

**AMA Style**

Huang X, Deng Z, Xie Y, Li Z, Fan J, Tu L. A New Scale Factor Adjustment Method for Magnetic Force Feedback Accelerometer. *Sensors*. 2017; 17(11):2471.
https://doi.org/10.3390/s17112471

**Chicago/Turabian Style**

Huang, Xiangqing, Zhongguang Deng, Yafei Xie, Zhu Li, Ji Fan, and Liangcheng Tu. 2017. "A New Scale Factor Adjustment Method for Magnetic Force Feedback Accelerometer" *Sensors* 17, no. 11: 2471.
https://doi.org/10.3390/s17112471