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Indirect Measurement of a Bar Magnet Position Using a Hall Sensor Array
Uwe Marschner and Wolf-Joachim Fischer
Semiconductor and Microsystems Technology Laboratory, Dresden University of Technology, Dresden, 01062 Germany
The measured magnetic induction field of a bar magnet can be used to compute or measure indirectly the position of the magnet. A
model-based parameter estimation approach requires a precise induction field model and knowledge of the model error. In this paper,
different analytic models describing the axial component of the induction field in front of a small cylindrical bar magnet are compared
with Hall sensor array measurements and finite-element simulations. The used solenoid model matched the finite-element method simulations but requires a high numerical effort. The approximate dipole model is suitable when a proposed correction function for the
estimated distance is applied and current and position are estimated simulaneously.
Index Terms—Electromagnetic induction, parameter estimation, permanent magnet, position tracking.
AR MAGNETS have been applied to control circuits or
processes galvanically separated [1], [2]. Based on induction field measurements gained by an integrated 1-D Hall sensor
array or row the position of a bar magnet in two dimensions can
be estimated [3]. Thereby high precision, resolution, and stability from a statistical point of view are expected [4]. The precision depends on random and systematic measurement errors,
the used numerical data formats, the estimation algorithm, and
the mathematical model. This paper focuses on distance estimation errors caused by known models which describe the axial
component of the bar magnet. The induction field was simulated with finite elements to determine the model errors without
measurement errors. The investigation is a prerequisite for a
new intelligent position sensor which measures the magnet position indirectly via nonlinear parameter estimation. Due to limited computational capabilities a simple mathematical model is
Fig. 1. Hall sensor array measurement setup.
was measured with 32
The induction field
single spinning current Hall sensors in a row
mm of 7.75 mm total length
as shown in Fig. 1 [2]. The sensors have a sensitivity of
and sampling rate
Hz. Perpendicular
to the sensor array a cylindrical bar magnet was positioned
and fixed during the measurement. Two Nd-Fe-B magnets
mm mm for magnet
with the geometries
1 (NE152) and (2 mm, 10 mm) for magnet 2 (NE210),
kA/m and relative
were investigated.
remanent permeability
was estimated from
The magnet position
using the polytop or Nelder–Mead simplex algorithm
from a constant initialization point within the
terminated at a maximum parameter tolerance
Figs. 3 and 4 depict the FEM simulated flux lines of the
NE152 and a comparable 1.68 kA ring current positioned at half
Fig. 2. FEM-simulated (o) und LS-estimated distributions of the z component
of the NE210 magnetic induction B (x ) at a distance z = 0:1 mm with the
solenoid model (—, z^ = 5:093 mm), exact dipole model (1 4 1; z^ = 0:496
mm) and approximate dipole model (-x-, z^ = 1:666 mm).
magnet length. The components of of the magnet simulamm and
tion at positions
served as references for the evaluation of
sensor locations
for 0.1 mm disthe analytical models. Fig. 2 shows
tance, Figs. 5 and 7 the estimated currents and distances.
It has been shown that the discontinuity in magnetization ,
assuming a smooth distribution of molecular magnetic dipoles,
contributes an effective surface current. Thus a cylindrical
bar magnet can be modeled as a solenoid of length
turns and an azimuthal current
[5]. Applying
Biot-Savarts law and using the magnetic vector potential the
component of the axisymmetric described magnetic induction
is given by
at the point
Digital Object Identifier 10.1109/TMAG.2007.893632
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
0018-9464/$25.00 © 2007 IEEE
Fig. 3. Axisymmetric FEM simulated magnetic flux lines of the NE152 bar
magnet and Hall sensor locations; solenoid flux lines are nearly equal. The rotational axis is between sensor 16 (second quadrant) and 17.
Fig. 4. Ring current flux lines and Hall sensor locations. Close to the related
NE152 magnet front the flux lines differ strongly.
with a winding distance , radius and permeability . The disand a -shift to
. 177 A (NE152)
tance relates to
and 172 A (NE210) were estimated for .1
Setting to one turn and
, equation (1) describes
of a single ring current
in a distance , or magnetic
dipole if
. It can be formed to
Fig. 5. Deviation z^ z of the estimated NE152 magnet distance and estimated
currents I for the solenoid model (x), exact dipole model ( ), approximate
dipole model with current estimation (+), and fixed current of 1.689 kA ( ,
chosen to give z^
z = L=2 at z = 18 mm) and corrected dipole model
(FEM: , measurements: ).
with complete elliptic integrals and of the first and second
kind in Legendre normal form2 [7]. The approx. dipole model
and current , e.g., used by [8], [9], or [10],
with distance
around the magnet, although open boundary (inf) elements bordered it to decrease the simulation error. Up to 40 mm the simat the middle axis confirmed
is obtained using the magnetic scalar potential and
[7] and is preferred.
As it can be observed in Figs. 2, 5, and 7 the solenoid model
matched the FEM simulation; 1 m air
was simulated
1Solenoid integration with MATLAB quadl.m: adaptive Gauss/Lobatto
qudrature rule, applied absolute error tolerance = 1:0 e-4. w = 10 (NE152)
and w = 50 (NE210); the integration limits were changed to w + =2 to
w =2 for a slightly improved fit defining w~ = 9:5 and w~ = 49:5.
2The series expansion given by Zhang [6] implemented by Barrowes (Matlab
function mcomelp.m) was used.
T for the NE210.
Using the dipole models in the close-up range of the mag, or locanets the estimated distance deviations
tion of the current ring related to the magnet front varied. By
in a corrected dipole
estimating simultaneously current and
mm mm with
match could be corrected to
. For
the distance mismm
Fig. 6. NE152 parameter estimation residuals. The marker assignment is the
same as in Fig. 5.
It can be concluded that the position of a small bar magnet
with low relative remanent permeability can be estimated from
low width induction field measurements with a high precision
using the approximate dipole model when the ring current is
estimated, too, and the estimated distance corrected. The simultaneous estimation of distance and current with the dipole
model—instead of fixing the current—lead to smaller residuals
as plotted in Figs. 6 and 8. The lower estimated distance is balanced by a lower current.
gives the ring location for far distances. For
these FEM simulations that was not the half magnet length.
The influence of the measurement noise to was much less
for the NE210 than for the NE152. The longer and little wider
magnet improved the signal to noise ratio at larger distances
where the Hall sensor noise dominated the measurements
leading to a higher imprecision of the position estimation. Improved sensors and statistical signal processing should reduce
the variance of and .
Fig. 7. Deviation z^ z of the estimated NE210 magnet distance and estimated
currents. The marker assignment is the same as in Fig. 5. The fixed dipole current
= 8:51 kA relates to the solenoid current.
Fig. 8. NE210 parameter estimation residuals. The marker assignment is the
same as in Fig. 5. The fixed dipole current I was set to 8.51 kA ( —) and
) matching I^ at z = 18 mm.
7.64 kA (
a precision of 5 m could be achieved for the NE210 and 2 m
for the NE152 based on the FEM simulations.
The measurements were averaged 100 times within
for each sensor and the individual offsets subtracted. They
showed an inclined sensor row in direction with an angle
rad and offset
mT and
mT. Caused by hole tolerances x-shifts up
mm were estimated simultaneously with
and .
The work was supported in part by the German Bundesministerium für Wirtschaft und Technologie (BMWi), InnoNet
program Grant No. 16IN0287. The authors would like to thank
Prof. Albrecht Reibiger, Dr. Barbara Adolphi, Dr. Matthias
Plötner and Richard Zahn (all TU Dresden), Dr. Rudolf Schäfer,
Dr. Dietrich Hinz (both Leibnitz Institute IFW Dresden), and
Dr. Birger Jettkant (Universitätsklinik Bergmannsheil Bochum)
for valuable hints, Michael Loeper (TU Dresden) for manufacturing the measurement setup, and Dr. Uwe Vogel (Fraunhofer
IPMS Dresden) for providing several Hall sensor arrays.
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Manuscript received October 31, 2006; revised February 19, 2007 (e-mail:
[email protected]; [email protected]).
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