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IEEE TRANSACTIONS ON MAGNETICS, VOL. 26, NO. 5, SEPTEMBER 1990 2041 ANNULAR MAGNET POSITION SENSOR G. LEMARQUANDand V. LEMARQUAND L.S.M. Universite de Savoie BP 240 74942 ANNECY LE VIEUX CEDEX FRANCE Abstract. This paper presents an absolute permanent magnet angular position sensor. I t s original geometry was optimized to obtain a perfectly linear variation law of the normal induction in the air gap. This sensor i s characterized by a simple electronic treatment, a low realization cost, and a resolution on the order of a tenth of a degree. INTRODUCTION This paper dealswith an absolute angular postion sensor made with a permanent magnet. This type of sensor has many industrial applications. Absolute angle position sennsors are needed for automobile steering shafts, for tool machines and robotics, and for the control of rotatjng systemsl. They.are part of the development of automation today. Linear sensor The variation of the magnetic induction versusthe angle 0 i s linear. To have information over a whole turn one possibility is to have two signals in quadrature and with triangular variation. A comparison of the respectivesignsof V1 and V2 determines the quadrant in which the angle is located. In each quadrant, a comparison of the absolute values of V1 and V2 determines the octant in which the angle is located. So, the variations of B have t o be treated only over a eighth of the period. In fact, the determination of 8 is always obtained by treating the signal that has the smallest modulus. So the important'point is to have a variation as linear as ossible in a lT/2 wide interval around the zero of the signal fiigure 2). IT\ Induction B SENSOR DESCRIPTION The sensor consists in one permanent magnetic ring fixed t o the rotor shaft. It is uniformly magnetized and,the magrretization axis i s perpendicular t o the ring axis. Two linear Hall effect voltage generators are fixed on the stator and positioned 90" apart. The sensor geometry is shown in figure 1. Figure 2 : Work intervals Sinusoidalsensor The variation of the induction versusangle is sinusoidal. The two signals from the Hall generators, V1 and V2, are propohional to the cosine and sine of the an le, respectively. The octant, in which the angle is locatedj i s determined as described in previous paragraph. The work interval is then (O,Tr/4). Over this interval the ratio of the two signals is used t o determine the tangent of the angle. This new signal is independant of the possible drifts of the multiplicative coefficient K of the two signals, V1 and V2 (cf. equation 1). Once the tan ent is known, the angle is then di itized using Cordic's aygorithm or using Tchebycheff's posynomials algrorithm. MODELLING Because there are two classes of sensors, those which have a linear response and those which have a sinusoidal response, each type of sensor must be optimized separately. Figure 1 : Sensor geometry C :core radius ; R : rotor radius ;5 :inner stator radius. The Hall voltage generators deliver two voltagesthat depend on the normal magnetic field in the air gap, i .e. on the rotor position : where 0 is the relative an le between rotor and stator. Both rotor and stator areiuilt with magnetic material (iron) t o guide the magnetic flux. ELECTRONIC TREATMENT Absolute sensors generally need the variation of an analog signal. There are two large families of sensors : for the first, the variation i s linear, for the second, it i s sinusoidal. Sinusoidalsensor The modelling of the geometry shown in figure 1 by the finite element program FLUXZD shows that the normal magnetic field in the air gap is perfectly sinusoidal. In fact, the program gives an easy method by which a stud of the normal component of the magnetic induction as a Lnction of sensor geometry can be performed. In this study.the following parameters are varied : C, the radius of the iron core ; R, the radius of the rotor ; and 5, the inner radius of the stator. The results from this study show that the variations of the magnetic induction are always sinusoidal. The error between the calculated curve and a sinusoidal curve is 0.5 %. This error is insi nificant and can be attributed t o the error introduced %y the finite element program. Figure 3 shows the variations of the normal induction in the middle of the air gap for a normalized magnet with remanent induction of 1 T. OO18-9464/90/09OO-2041$01.OO 0 1990 IEEE -_ -- _ _ ~ 2042 Induction B f x E 0.98 o = 0 $ = 0.92 Es 4- ES = 0.25 A ES E 0.5 = 0.75 Figure 4 : '0 0.5 ' C R Figure 3 : Normal induction in the middle of the air gap versus ratio CI R with normalized magnet The ratio, OR, gives the proportion of iron in the rotor. When C /R is equal t o zero, the rotor is only made with permanent magnet and the induction is maximum. When C / R is equal to unity, the rotor i s made exclusively with iron and the induction is zero. Three centers design C :core radius ; R :rotor radius ; 5 : inner stator radius ;X eccentricity In order t o obtain good accuracy, it is necessary tor the slope, p, of the linear part of the B versus €3curve to be as large as possible. For a given ratio R / S the maximum slope is reached when the rotor is only made with permanent magnet. The value of the slope decreases when the radius of the iron core increases. The slope also decreases when the ratio R / S decreases while C/R i s held constant. As can be expected, a thinner air gap and larger magnet volume gives a larger slope (figure 8). The amplitude of the signals delivered by the Hall generators is, for this variation law, multiplicated by a term sin 9 / 0, where [email protected] i s the angular width of the active element of the Hall generator. Linearsensor To obtain a triangular variation, it i s necessary the permanent magnet not to have a constant width. This can be achieved with a three centers design as shown in figure 4. This geometry i s easy to produce by turning. The representative curve, induction, B, versus 0,calculated by the program i s compared to the straight line L which is tangential to the curve when the induction i s equal to zero. The maximum of the relative error, E, between calculated curve and line L always occurs for 0 = T T I 4 When E i s negative, the curve is below the line L ; when E is positive, the curve i s above the line L. For X equal t o zero, the stator geometry i s circular and the variation of B i s sinusoidal. The error, E, has a maximum value which is negative. When X is equal to unity, the core is tangential t o the rotor (see figure 5) and the permanent magnet has a thickness equal tdzero for two points. For any given Rand 5, a value of the eccentricity, which causes the error, E, to go to zero, always exists. There exists a value of the eccentricit for which whatever the ratio C/R is, the error E i s the same [point A on figure 6). If the ratio, WS,is held constant while the ratio, U R , i s decreased, then the value of the eccentricity for which the error, E, is equal to zero also decreases. Thus, as the magnet becomes a larger proportion of the rotor, the eccentricity decreases. If the ratio, U R , i s held constant while the ratio, R/S, is decreased, then the value of the eccentricity for which the error, E, is equal to zero increases. Thus, as the size of the rotor decreases, the eccentricity increases. Figure 5 : Three centers design for X = 1 2043 THE PROTOTYPES Two prototypes were built. The first one i s fully circular and has a sinusoidal variation of B versus 8. I t s dimensions are as follows : C=5mm R=llmm -E -- 0.9 S=18mm The permanent magnet is a plastic ferrite whose remanent induction is 0.4T. The performance of the sensor i s as follow: resolution precision 0.1 degree 0.5 degree Thissensor is very stable when the temperature varies. The second prototype has a three centersdesign and the following characteristics : S = 18mm R = 16.2mm 'C = 4mm X = 3mm The variation of the normal induction in the air gap is perfectly linear, in ood agreement with the modelling. The precision ofthissensor is 0.1 degree, its resolution is 0.02 degree. So it is more precise than the sinusoidal one, with a simpler electronic treatment, but it is more subject t o drifts when the temperature varies. The study of an original geometry for a permanent magnet angle position sensor has determined the optimum dimensions for linear and sinusoidal variation o f the ma netic induction in the airgap. The linear sensor has a higier precision and simpler accompanying electronics than the sinusoidal sensor, but tends t o drift more during temperature variations. The resolution and precision of the sensorsare0.l degree and 0.5 degreefor the sinusoidal one and 0.02-degree and 0.1 degree for the linear one. 1 R = 0.9 Figure 7 : Error, E, between calculated curve and line versus X I R-C fortwovaluesof R I S 0 g = 0.9 /$=0.5 E t = 0.9 5.10-~ '0 Figure8: R - C (-0.1 Figure 6 : X R-C 0.5 -0.1 CONCLUSION ror / / Error, E, between calculated curve and line versus XIR-Cforreveralvaluesof C I R ; R I S = 0.9 Curve (a) corresponds to a rotor built exclusively with magnet. 0.5 -CR Slope, p, of the line for the value of the eccentricity that makes the error E equal to zero versus the ratio C I S REFERENCES (1) R.S. Davidson, R.D. Gourlay "Applying the Hall effectto angular transducers" Solid state electronics, Pergamon Press vol9., pp 471-484, Printed in Great Britain 1966 (2) A. Petersen "Magnetic field sensors in cars", record of procceding Capteurs 89, APIST, Paris, France, june 6-9,1989. (3) "Sensors" Philips composants, RTC, Philips