2728 IEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 6, JUNE 2007 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. I. INTRODUCTION AR MAGNETS have been applied to control circuits or processes galvanically separated , . 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 . Thereby high precision, resolution, and stability from a statistical point of view are expected . 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 preferred. B Fig. 1. Hall sensor array measurement setup. II. INDIRECT POSITION MEASUREMENT 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 . The sensors have a sensitivity of and sampling rate Hz. Perpendicular 100 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 with was estimated from The magnet position using the polytop or Nelder–Mead simplex algorithm region from a constant initialization point within the 1e . terminated at a maximum parameter tolerance III. FEM SIMULATIONS AND PARAMETER ESTIMATION 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 with bar magnet can be modeled as a solenoid of length turns and an azimuthal current . 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 (1) MARSCHNER AND FISCHER: INDIRECT MEASUREMENT OF A BAR MAGNET POSITION 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. 2729 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 at in a distance , or magnetic dipole if . It can be formed to 0 (2) 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 . The approx. dipole model and current , e.g., used by , , or , 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 ulated (3) is obtained using the magnetic scalar potential and  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  implemented by Barrowes (Matlab function mcomelp.m) was used. 0 0 4 3 with 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 model (4) 0 match could be corrected to . For the distance mismm 2730 IEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 6, JUNE 2007 IV. CONCLUSION 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 Parameter 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 . ACKNOWLEDGMENT 0 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 I = 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 ( 111 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 . to 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. REFERENCES  T. Kaulberg and G. Bogason, “A silicon potentiometer for hearing aids,” in Analog Integrated Circuits and Signal Processing. Berlin, Germany: Springer Science+Business Media B.V., Jan. 1996, vol. 9, pp. 31–38, no. 1.  H. Grüger, U. Vogel, S. Ulbricht, and H. Niemann, “Hall array based linear movement detektion,” in Proc. 6th Eur. Magnetic Sensors & Actuators Conf. (EMSA’06), Bilbao, 2006.  U. Marschner, W.-J. Fischer, R. Gottfried-Gottfried, and W. Pufe, “Model-based signal processing in a cmos hall sensor row microsystem,” in 6th Int. Conf. MICRO SYSTEM Technologies’98, Potsdam, Germany, 1998.  A. Van den Bos, “Parametric statistical model-based measurement,” Measurement, vol. , no. 14, pp. 55–61, 1994.  J. Vanderlinde, Classical Electromagnetic Theory. New York: Wiley, 1993.  S. Zhang and J. Jin, Computation of Special Functions. New York: Wiley, 1996.  G. Wunsch and H.-G. Schulz, Elektromagnetische Felder. Berlin, Germany: Verlag Technik, 1989.  H. Raab, E. Blood, T. Steiner, and J. H. R. , “Magnetic position and orientation tracking system,” IEEE Trans. Aerosp. Electron. Syst., vol. AES-15, no. 5, Sep. 1979.  C. Cordier, L. Mechin, C. Gunther, M. Sing, D. Bloyet, and V. Mosser, “Hall sensor response to an inhomogeneous magnetic field,” IEEE Sensors J., vol. 5, no. 5, pp. 934–941, 2005.  V. Schlageter, P.-A. Besse, R. S. Popovic, and P. Kucera, “Tracking system with five degrees of freedom using a 2d-array of hall sensors and a permanent magnet,” Sens. Act. A, vol. 2951, pp. 1–6, 2001. Manuscript received October 31, 2006; revised February 19, 2007 (e-mail: [email protected]; [email protected]).