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 2@ 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