# A Statistical Investigation into Assembly Tolerances of Gradient Field Magnetic Angle Sensors with Hall Plates

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Magnetic Field Solution

**T**$(\overrightarrow{n},\alpha )$ is the transformation matrix for rotations.

**T**$(\overrightarrow{n},\alpha )$ can be found in [18,19]. Equation (3) is applicable for any rotation by an angle $\alpha $ around an axis given by its vector $\overrightarrow{n}={[{n}_{1},{n}_{2},{n}_{3}]}^{T}$ with unit length.

**T**$(\overrightarrow{n},\alpha )$ becomes

**Remark**

**1.**

#### 2.2. Performing Monte Carlo Simulations

`size`defines the dimension

`[rows,columns]`of the array created.

## 3. Model Input

- Diameter $D=6$ mm
- Height $H=2.5$ mm

**Remark**

**2.**

## 4. Results

## 5. Discussion

- The distribution of $\Delta \phi $ is clearly non-Gaussian: The angle errors of rare outliers are much bigger than one would predict with a Gaussian. Non-Gaussian error distributions have also been reported for MEMS based inclinometers by [23].
- A variation of the reading radius has less influence on $\Delta \phi $ than assembly tolerances.
- Typical angle errors (= standard deviation) and rare outliers ($=99.9\%$ percentiles) are similarly affected by $RR$, whereby small $RR$ give slightly smaller angle errors.

^{nd}order terms, such as the product of eccentricity of the magnet times tilt of magnet or the eccentricity of magnet times the eccentricity of the sensor. These mixed error terms have an important consequence: the total error is more than the sum of individual errors.

**Example**

**1.**

^{st}system is perfectly accurate except for an eccentricity of the magnet, the angle error is

^{nd}system is perfectly accurate except for an eccentricity of the sensor, the angle error is

^{rd}system is perfectly accurate except for the eccentricities of the magnet and sensor, the angle error is

^{st}system comprises one term and the error of the 2

^{nd}system also comprises one term; however, the error of the 3

^{rd}system comprises not just two terms, but three terms. The 3

^{rd}term is proportional to the product of eccentricities of sensor and magnet. Therefore, the distribution of $\Delta {\phi}_{3}$ is wider than the statistical sum of $\Delta {\phi}_{1}$ and $\Delta {\phi}_{2}$. For ${x}_{M}={y}_{M}=0.3$ mm, ${x}_{S}=-0.2$ mm, ${y}_{S}=0.1$ mm, and ${\tilde{E}}^{axial}=35000$ m${}^{-2}$, it follows that $\Delta {\phi}_{1}={0.045}^{\circ}$, $\Delta {\phi}_{2}={0.016}^{\circ}$, and $\Delta {\phi}_{3}={0.109}^{\circ}$.

^{st}order error terms means that the typical angle error is small. If a typical tilt occurs, it gives an angle error, which is proportional to the square of the small tilt.

${e}_{M}$ | worst case angle error $\Delta \phi $ if only the magnet is placed eccentrically |

${e}_{S}$ | worst case angle error $\Delta \phi $ if only the sensor is placed eccentrically |

${t}_{M}$ | worst case angle error $\Delta \phi $ if only the magnet is tilted |

${t}_{S}$ | worst case angle error $\Delta \phi $ if only the sensor is tilted |

^{st}line; however, all four assembly parameters give 10 error terms, sic of which are mixed errors. It is obvious that in the rare cases, when all four assembly parameters are large, we will end up with a much larger total error due to the six mixed error terms. This is also verified numerically in Figure 5.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

AE | Average Angle Error: $\left[max\right(\Delta \phi )$ - $min(\Delta \phi )]$/2 |

AG | Air gap |

${B}_{rem}$ | Remnant magnetization |

D | Magnet diameter |

DoF | Degree(s) of Freedom |

EMI | Electromagnetic Interference |

H | Magnet height |

${H}_{cb}$ | Magnetic coercivity |

GMR | Giant Magnetoresistance |

MDPI | Multidisciplinary Digital Publishing Institute |

ME | Maximum Angle Error: $max\left(\right|max(\Delta \phi )|,|min(\Delta \phi )\left|\right)$ |

MEMS | Microelectromechanical Systems |

TMR | Tunnel Magnetoresistance |

RR | Reading Radius |

w.r.t. | with respect to |

## Appendix A

- Diameter $D=10$ mm
- Height $H=2$ mm

**Figure A1.**Complementary Cumulative Probability Density Function (CCDF) versus ME-angle error for the approaches presented in this publication and the one presented in [13] for $RR=0.1\phantom{\rule{0.277778em}{0ex}}\mathrm{mm}$ at $AG=1\phantom{\rule{0.277778em}{0ex}}\mathrm{mm}$. Magnet parameters: $D=10\phantom{\rule{0.277778em}{0ex}}\mathrm{mm}$ and $H=2\phantom{\rule{0.277778em}{0ex}}\mathrm{mm}$. Assembly tolerances according to Table 2; 10,000 Monte Carlo samples taken.

**Table A1.**Comparison of ME-angle errors extracted from the data in Figure A1.

CCDF Value | This Publication | In [13] |
---|---|---|

0.1 | ${0.31}^{\circ}$ | ${0.32}^{\circ}$ |

0.01 | ${0.54}^{\circ}$ | ${0.56}^{\circ}$ |

0.001 | ${0.78}^{\circ}$ | ${0.81}^{\circ}$ |

## Appendix B

**Table A2.**Comparison of assembly parameters used in [13] with corresponding variables in this publication.

This Publication | In [13] |
---|---|

${\alpha}_{M}$ | $\beta \phantom{\rule{0.277778em}{0ex}}cos\left(\alpha \right)$ |

${\beta}_{M}$ | $\beta \phantom{\rule{0.277778em}{0ex}}sin\left(\alpha \right)$ |

${\eta}_{M}=arcta{n}_{2}({\alpha}_{M},{\beta}_{M})$ | $\alpha $ |

${\delta}_{M}$ | $\beta $ |

${\alpha}_{S}$ | $\lambda \phantom{\rule{0.277778em}{0ex}}cos\left(\gamma \right)$ |

${\beta}_{S}$ | $\lambda \phantom{\rule{0.277778em}{0ex}}sin\left(\gamma \right)$ |

${\eta}_{S}=arcta{n}_{2}({\alpha}_{S},{\beta}_{S})$ | $\gamma $ |

${\delta}_{S}$ | $\lambda $ |

${\gamma}_{S}$ | $\vartheta $ |

${x}_{S}$ | ${\u03f5}_{x}={\u03f5}_{r}cos\left(\chi \right)$ |

${y}_{S}$ | ${\u03f5}_{y}={\u03f5}_{r}sin\left(\chi \right)$ |

$arcta{n}_{2}({x}_{S},{y}_{S})$ | $\chi $ |

${z}_{S}$ | ${\u03f5}_{z}$ |

${x}_{M}$ | ${\delta}_{x}={\delta}_{r}\phantom{\rule{0.277778em}{0ex}}cos\left(\eta \right)$ |

${y}_{M}$ | ${\delta}_{y}={\delta}_{r}\phantom{\rule{0.277778em}{0ex}}sin\left(\eta \right)$ |

$arcta{n}_{2}({x}_{M},{y}_{M})$ | $\eta $ |

${z}_{M}$ | ${\delta}_{z}$ |

**Remark**

**A1.**

## Appendix C

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**Figure 3.**Cumulative probability distribution of the angle error $\Delta \phi $ at the rotation angle $\phi ={33.75}^{\circ}$ for various reading radii RR at an air gap $AG$ of (

**a**) $1.5\phantom{\rule{0.277778em}{0ex}}\mathrm{mm}$ and (

**b**) $2.5\phantom{\rule{0.277778em}{0ex}}\mathrm{mm}$. Magnet parameters: $D=6$ mm and $H=2.5$ mm. Assembly tolerances are according to Table 2. The dashed straight line denotes a Gaussian distribution with equal average and standard deviation. Note to legend: “+”: Monte Carlo simulation; “- - -”: Gaussian distribution with the same standard deviation as Monte Carlo simulation for $RR=1$ mm; 100,000 Monte Carlo runs per configuration.

**Figure 4.**(

**a**) Standard deviation and (

**b**) $99.9\%$ percentile of $\Delta \phi $ (at $\phi ={33.75}^{\circ}$) over reading radius RR for two air gaps $AG$. Data are from Figure 3. The ratio of percentile versus standard deviation is $\approx 5.03$ and it varies about $\pm 3.5\%$ versus both $AG$.

**Figure 5.**Cumulative probability distribution of angle errors for assembly tolerances of Table 2. $\Delta {\phi}_{ME}$ = angle error for magnet eccentricity only, $\Delta {\phi}_{SE}$ for sensor eccentricity only, $\Delta {\phi}_{ST}$ for sensor tilt only, $\Delta {\phi}_{MT}$ for magnet tilt only, $sum$ = $|\Delta {\phi}_{ME}|+|\Delta {\phi}_{SE}|+|\Delta {\phi}_{ST}|+|\Delta {\phi}_{MT}|$, and $total$ = real angle error if all assembly tolerances of Table 2 are simultaneously present; 1000 Monte Carlo runs conducted.

**Figure 7.**Measured AE angle error vs. AG for a single piece of test chip with RR = 1 mm. Each curve was measured for a specific sensor eccentricity ${x}_{S}$, ${y}_{S}$ in the range of $[-0.3,0.3]$ mm. All other assembly tolerances were too small to be measured, but it is very likely that they did not vanish altogether (magnet eccentricity < $0.1$ mm and tilts up to a few degrees). During the experiment, the sensor was mounted only once on the translation stage, and therefore all assembly tolerances were constant, except for ${x}_{S}$ and ${y}_{S}$. Note that the combined action of inevitable assembly tolerances and intentionally changed sensor eccentricity produces a wide variety of curves that fill up the entire area below the curve with maximum sensor eccentricity. The curve for $({x}_{S},{y}_{S})=(0,0)$ mm is flat for medium and large air gap, but it increases at small air gap, which is probably due to inevitable tilt errors. The curve for $({x}_{S},{y}_{S})=(-0.3,0.3)$ mm has identical behavior at small air gap; yet, at large air gap, we clearly see the strong effect of eccentricity proportional to the rise in ${\tilde{E}}^{axial}$ versus air gap.

**Figure 8.**$\Delta \phi $ versus rotation angle for (

**a**) $({x}_{S},{y}_{S})=(-0.3,0.3)$ mm and (

**b**) $({x}_{S},{y}_{S})=(0,0)$ mm from Figure 7. Two different patterns are detectable: ${\tilde{T}}^{axial}$ dominates for small $AG$ and ${\tilde{E}}^{axial}$ dominates for large $AG$. Tilts and magnet eccentricity are not quantified but small and equal for all curves.

Sensor DoF | Magnet DoF |
---|---|

Position ($\overrightarrow{{d}_{S}}$) | Position ($\overrightarrow{{d}_{M}}$) |

${\mathit{x}}_{\mathit{S}}$ | ${\mathit{x}}_{\mathit{M}}$ |

${\mathit{y}}_{\mathit{S}}$ | ${\mathit{y}}_{\mathit{M}}$ |

${z}_{S}$ | ${z}_{M}$ |

Orientation (w.r.t. shaft CS) | Orientation (w.r.t. shaft CS) |

${\mathit{\alpha}}_{\mathit{S}}$ | ${\mathit{\alpha}}_{\mathit{M}}$ |

${\mathit{\beta}}_{\mathit{S}}$ | ${\mathit{\beta}}_{\mathit{M}}$ |

${\gamma}_{S}$ | ${\gamma}_{M}$ |

**Table 2.**Overview of assembly tolerances (eccentricities and tilts) for the sensor and magnet. Note that $\mathcal{N}(0,0.01)$ mm means an average of 0 mm and a variance of $0.01$ mm${}^{2}$, which is equivalent to a standard deviation of $0.1$ mm.

Sensor Tolerances | Magnet Tolerances |
---|---|

${x}_{S}\sim \mathcal{N}(0,0.01)\phantom{\rule{0.277778em}{0ex}}\mathrm{mm}$ | ${x}_{M}\sim \mathcal{N}(0,0.01)\phantom{\rule{0.277778em}{0ex}}\mathrm{mm}$ |

${y}_{S}\sim \mathcal{N}(0,0.01)\phantom{\rule{0.277778em}{0ex}}\mathrm{mm}$ | ${y}_{M}\sim \mathcal{N}(0,0.01)\phantom{\rule{0.277778em}{0ex}}\mathrm{mm}$ |

${z}_{S}\sim \mathcal{N}(0,0)\phantom{\rule{0.277778em}{0ex}}\mathrm{mm}$ | ${z}_{M}\sim \mathcal{N}(0,0)\phantom{\rule{0.277778em}{0ex}}\mathrm{mm}$ |

${\delta}_{S}\sim \mathcal{N}{(0,1)}^{\circ}$ | ${\delta}_{M}\sim \mathcal{N}{(0,1)}^{\circ}$ |

${\eta}_{S}\sim \mathcal{U}{(0,360)}^{\circ}$ | ${\eta}_{M}\sim \mathcal{U}{(0,360)}^{\circ}$ |

${\alpha}_{S}\sim {\delta}_{S}\phantom{\rule{0.277778em}{0ex}}cos\left({\eta}_{S}\right)$ | ${\alpha}_{M}\sim {\delta}_{M}\phantom{\rule{0.277778em}{0ex}}cos\left({\eta}_{M}\right)$ |

${\beta}_{S}\sim {\delta}_{S}\phantom{\rule{0.277778em}{0ex}}sin\left({\eta}_{S}\right)$ | ${\beta}_{M}\sim {\delta}_{M}\phantom{\rule{0.277778em}{0ex}}sin\left({\eta}_{M}\right)$ |

${\gamma}_{S}\sim \mathcal{N}{(0,0)}^{\circ}$ | ${\gamma}_{M}\sim \mathcal{N}{(0,0)}^{\circ}$ |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Ergun, S.; Ausserlechner, U.; Holliber, M.; Granig, W.; Zangl, H.
A Statistical Investigation into Assembly Tolerances of Gradient Field Magnetic Angle Sensors with Hall Plates. *Mathematics* **2019**, *7*, 968.
https://doi.org/10.3390/math7100968

**AMA Style**

Ergun S, Ausserlechner U, Holliber M, Granig W, Zangl H.
A Statistical Investigation into Assembly Tolerances of Gradient Field Magnetic Angle Sensors with Hall Plates. *Mathematics*. 2019; 7(10):968.
https://doi.org/10.3390/math7100968

**Chicago/Turabian Style**

Ergun, Serkan, Udo Ausserlechner, Michael Holliber, Wolfgang Granig, and Hubert Zangl.
2019. "A Statistical Investigation into Assembly Tolerances of Gradient Field Magnetic Angle Sensors with Hall Plates" *Mathematics* 7, no. 10: 968.
https://doi.org/10.3390/math7100968