Patent Publication Number: US-10309802-B2

Title: Angle sensors, systems, and methods

Description:
REFERENCE TO RELATED APPLICATION 
     This application is a continuation of U.S. application Ser. No. 13/944,391 filed Jul. 17, 2013, the contents of which are incorporated by reference in their entirety. 
    
    
     TECHNICAL FIELD 
     The invention relates generally to magnetic field sensors and more particularly to magnetic field angle sensors for sensing an angle of rotation of a shaft or other object. 
     BACKGROUND 
     Magnetic field sensors can be used to sense an angle of rotation of a shaft. For example, a magnet can be mounted on the shaft such that it rotates with the shaft, and a magnetic field sensor can be arranged proximate the magnet in order to sense a magnetic field induced by the magnet as it rotates with the shaft. When the magnetic field sensor is mounted next to or adjacent the shaft, i.e., off of the axis of rotation of the shaft, the sensor is often referred to as an “off-axis” magnetic field angle sensor. Off-axis magnetic field angle sensors often are implemented when the end of the shaft is unavailable as a location for the sensor or there simply is not space available on the shaft. 
     In many applications there is a general preference for magnetic field angle sensors, including off-axis magnetic field angle sensors, to be inexpensive and non-complex while also being robust with respect to external magnetic fields and other disturbances. A drawback of some conventional approaches, then, is a requirement of at least two sensor substrates with sensor elements having the same magnetic sensitivity. The required matched magnetic sensitivity is difficult to obtain and in combination with the need for multiple sensor substrates is more expensive to produce. 
     SUMMARY 
     Embodiments relate to magnetic field angle sensors, systems and methods. In an embodiment, a magnetic field angle sensor comprises a magnet rotatable about an axis of rotation; a least one sensor unit arranged proximate the magnet, the at least one sensor unit comprising two magnetic field sensor elements arranged on a first continuous surface of a substrate and spaced apart from one another along a direction of movement of the magnet, the two magnetic field sensor elements responsive to the same magnet field component; and circuitry coupled to the two magnetic field sensor elements to determine an absolute angle of rotation of the magnet from a sum and a difference of output signals of the two magnet field sensor elements. 
     In an embodiment, a magnetic field angle sensor comprises a magnet rotatable about an axis of rotation; a plurality of sensor units arranged proximate the magnet and spaced apart from one another about the axis of rotation by (360 degrees/n), where n is the number of sensor units and is greater than or equal to 2, wherein each of the at least two sensor units comprises at least two magnetic field sensor elements each responsive to the same magnetic field component induced by the magnet; and circuitry coupled to the plurality of sensor units to determine, for at least one sensor unit, a first coarse estimation of an angle of rotation of the magnet from a sum and a difference of output signals of the at least two magnetic field sensor elements, and to determine a second refined estimation of the angle of rotation of the magnet from at least one first coarse estimation. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Embodiments may be more completely understood in consideration of the following detailed description of various embodiments of the invention in connection with the accompanying drawings, in which: 
         FIG. 1A  is a perspective diagram of an off-axis angle sensor according to an embodiment. 
         FIG. 1B  is a perspective diagram of an off-axis angle sensor according to an embodiment. 
         FIG. 2A  is a top view block diagram of a sensor unit according to an embodiment. 
         FIG. 2B  is a top view block diagram of a sensor unit according to an embodiment. 
         FIG. 2C  is a top view block diagram of a sensor unit according to an embodiment. 
         FIG. 3A  is a perspective and partially transparent view of an angle sensor according to an embodiment. 
         FIG. 3B  is an enlarged view of a portion of  FIG. 3A  with a magnet omitted according to an embodiment. 
         FIG. 3C  is an enlarged view of a portion of  FIG. 3B . 
         FIG. 3D  is a flowchart according to an embodiment. 
         FIG. 4  is an angle diagram related to a axial magnetic field according to an embodiment. 
         FIG. 5  is a block diagram of a sensor according to an embodiment. 
         FIG. 6A  is a perspective and partially transparent view of an angle sensor according to an embodiment. 
         FIG. 6B  is a view of a portion of  FIG. 6A  with a magnet omitted according to an embodiment. 
         FIG. 6C  is an enlarged view of a portion of  FIG. 6B   
         FIG. 6D  is a side view of the sensor of  FIG. 6A . 
         FIG. 7A  is a partial perspective view of an angle sensor according to an embodiment. 
         FIG. 7B  is a side view of the angle sensor of  FIG. 7A . 
         FIG. 8  is a perspective block diagram of an on-axis angle sensor according to an embodiment. 
         FIG. 9  is a top view block diagram of a sensor unit according to an embodiment. 
         FIG. 10A  is a top view block diagram of a vertical Hall sensor unit according to an embodiment. 
         FIG. 10B  is a top view block diagram of another vertical Hall sensor unit according to an embodiment. 
         FIG. 10C  is a top view block diagram of another vertical Hall sensor unit according to an embodiment. 
         FIG. 10D  is a top view block diagram of another vertical Hall sensor unit according to an embodiment. 
     
    
    
     While the invention is amenable to various modifications and alternative forms, specifics thereof have been shown by way of example in the drawings and will be described in detail. It should be understood, however, that the intention is not to limit the invention to the particular embodiments described. On the contrary, the intention is to cover all modifications, equivalents, and alternatives falling within the spirit and scope of the invention as defined by the appended claims. 
     DETAILED DESCRIPTION 
     Embodiments relate to magnetic field angle sensors, including off-axis and on-axis sensors. In an embodiment, a magnetic field angle sensor comprises two sensor units, and each sensor unit comprises two sensor elements. The sensor units are arranged proximate a magnet and spaced apart from one another in a direction of movement of the magnet, wherein the two sensor elements of each sensor unit are responsive to the same magnetic field component induced by the magnet coupled to a shaft as the shaft rotates. In each sensor unit, a sum and a difference of the output signals of the two sensor elements can be calculated to determine a coarse estimation of a rotation angle, and a more refined estimation can be obtained by combining the coarse estimations of each sensor unit. In embodiment, the magnetic field angle sensor comprises a control unit or other circuitry to carry out this combining. 
     Referring to  FIGS. 1A and 1B , an off-axis magnetic field angle sensor  100  is depicted. Sensor  100  comprises a magnet  102  mounted or otherwise affixed to a shaft  104  such that magnet  102  rotates with shaft  104 . Magnet  102  has a diametric magnetization in the direction indicated in  FIGS. 1A and 1B , i.e., the y-direction. A sensor unit  105  comprises a substrate  106  and is arranged proximate magnet  102  and shaft  104 . As depicted in both  FIGS. 1A and 1B , a major plane of substrate  106 , i.e., the surface on which magnetic field sensor elements  108  are mounted, is arranged perpendicularly to the rotation axis of shaft  104  as illustrated. In embodiments, substrate  106  can comprise a substrate, die, circuit board or other suitable structure, though for convenience the term substrate generally will be used herein but it not be considered limiting. In one embodiment, two magnetic field sensor elements  108   a  and  108   b  are arranged on substrate  106 . 
     The relative position of substrate  106  and magnet  102  can vary in embodiments, as can the arrangement and orientation of sensor elements  108   a  and  108   b . In  FIG. 1A , substrate  106  is adjacent to shaft  104  and magnet  102  and can be coplanar with magnet  102 . In  FIG. 1B , substrate  106  is again adjacent shaft  104  and is arranged below magnet  102 .  FIGS. 1A and 1B  are merely exemplary of some embodiments, and the relative positions of magnet  102 , shaft  104 , substrate  106  and/or sensor elements  108  can vary in still other embodiments. For example, in other embodiments a plurality of sensor units  105  having sensor elements  108   a  and  108   b  can be used, arranged on other sides of magnet  102  and shaft  104 . Moreover,  FIG. 1  is not to scale and a simplified conceptual depiction to illustrate basic components and features of an embodiment of sensor  100 . Other embodiments, including on-axis embodiments, will be discussed elsewhere herein. 
       FIG. 2A  depicts one embodiment of a sensor unit  105 . Sensor unit  105  comprises substrate  106  on which two magnetic field sensor elements  108   a  and  108   b , which can comprise Hall plates, magnetoresistive elements, MAG-FETs or other suitable out-of-plane magnetic field sensor elements, as sensor elements  108   a  and  108   b  are responsive to a magnetic field component which is perpendicular to the main plane of substrate  106  as depicted in  FIG. 2 . In other embodiments, in-plane sensor elements can be used, such as vertical Hall elements, magnetoresistive elements such as AMR, GMR, TMR or others, or still other suitable sensor elements. For example,  FIGS. 2B and 2C  depict sensor units  105  comprising vertical Hall sensor elements  108   a  and  108   b , with the white arrows in each figure indicating the sensitivity direction of each sensor element  108   a  and  108   b . Generally in  FIGS. 2B and 2C , as well as in  FIG. 2A , sensor elements  108   a  and  108   b  are spaced apart on substrate  106  in a direction of relative movement of magnet  102  when it rotates. In still other embodiments, sensor elements  108   a  and  108   b  can be configured to be operated as dynamic offset compensation circuits (e.g., in spinning current, spinning voltage or other suitable operational schemes). 
     In  FIG. 3 , another relative configuration of sensor units  105 , magnet  102  and shaft  104  is depicted, with  FIG. 3A  being an overall perspective view,  FIG. 3B  a detailed perspective view with magnet  102  not depicted, and  FIG. 3C  a detailed perspective view of a sensor unit  105   a . In  FIG. 3A , a plurality of sensor units  105   a ,  105   b ,  105   c  and  105   d  are used. The number of relative configuration of sensor units  105  can vary in other embodiments, such that more or fewer sensor units  105  can be used and their relative spacing and arrangement on a substrate  110  can vary. In general, sensor units  105  are evenly spaced at (360 degrees/n), where n is the number of sensor units  105  and is greater than or equal to 2. For example, in another embodiment two sensor units  105   a  and  105   c  are used, arranged opposite one another across magnet  102  and shaft  104  and spaced apart by (360/2) or 180 degrees. As depicted, sensor units  105   a - d  are mounted to substrate  110  which comprises an aperture  112  (see  FIGS. 3B and 3C ) in which shaft  104  and magnet  102  are arranged such that sensor units  105   a - d  are proximate magnet  102  as magnet  102  rotates with shaft  104 . In embodiments, care can be taken with respect to the accuracy of positioning of sensor units  105   a - d  with respect to magnet  102  and/or aperture  112  during assembly. 
     The depiction in  FIG. 3 , as well as the rest of the drawings included here, is not necessarily to scale, nor are all components, features and details depicted. For example, traces on substrate  110 , including those coupling sensor units  105   a - 105   d , is not specifically depicted but nevertheless understood by those skilled in the art. 
       FIG. 3C  is an enlarged view of one sensor unit  105   a . Sensor unit  105   a  comprises substrate  106  on which two magnetic field sensor elements  108   a  and  108   b  are arranged. More or fewer sensor elements  108  can be used in other embodiments, and sensor elements  108  can comprise magnetic field sensor elements such as Hall-effect sensor elements, including horizontal Hall plates or vertical Hall elements, magnetoresistive sensor elements or other suitable sensor elements in various embodiments. As appreciated by those skilled in the art, the arrangement and positioning of sensor elements  108  (and/or sensor unit  105   a  and substrate  106 ) relative to magnet  102  and shaft  104  can vary according to the type of sensor element used in any particular embodiment. In examples discussed herein, an ordinary horizontal Hall plate embodiment generally will be used, though any examples and discussion are not limiting with respect to other embodiments. In  FIG. 3C , sensor elements  108   a  and  108   b  are arranged in the mid-plane of magnet  102 , i.e., at z=0 in an embodiment. 
     Sensor elements  108   a  and  108   b  can form a gradiometer in some embodiments for sensing a gradient of a magnetic field induced by magnet  102 , which is diametrically magnetized. In the embodiment of  FIG. 3C , a circular cylindrical coordinate system can be used, having a radial component B R , azimuthal component B ψ  (i.e., B ψ  with reference to  FIG. 2 ) and axial component B z  of a magnetic field B. Magnetic field components of diametrically magnetized magnet  102 , including B R , B ψ  and B z , vary sinusoidally with respect to the azimuthal position. Therefore, the gradients of B R , B ψ  and B z , which are dB R /dψ, dB ψ /dψ and dB z /dψ, also vary sinusoidally and are in quadrature with the original curves. Thus, {B R , dB R /dψ} is a pair of signals in which one has a sin(ψ)-dependence and the other a cos(ψ)-dependence. The same is true for {B ψ , dB ψ /dψ} and {B z , dB z /dψ}. 
     In embodiments, one or more of these pairs of signals can be used to obtain cosine and sine signals, from which a rotation angle can be derived. For example, ψ=arctan [sine-signal, k*cosine-signal] according to the CORDIC (COordinate Rotation Digital Computer) algorithm, with k being a scaling factor used to equalize the amplitudes of the sine and cosine signals. The following definition of the arctangent is used in embodiments: 
     
       
         
           
             
                 
             
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     Referring again to  FIG. 3C , sensor elements  108   a ,  108   b  of sensor unit  105   a  (as well as of sensor units  105   b - d ) are the same in embodiments, such that both are responsive to the same magnetic field component. Referring also to  FIG. 2 , sensor element  108   a  is spaced apart from sensor element  108   b  in the psi-direction, where d=R×Δψ, with R being the radial distance of sensor elements  108   a  and  108   b  from the axis of rotation of shaft  104 . Sensor  100  therefore can determine a difference between output signals of sensor elements  108   a  and  108   b  and use that difference as an estimate or approximation of dB R /dψ. Sensor  100  and/or circuitry coupled thereto, such as a control unit comprising, e.g., a microprocessor, also can average the output signals of sensor elements  108   a  and  108   b  in order to estimate or approximate the magnetic field component B R . 
     In other words, and referring to  FIG. 3D , at  10  sensor  100  can obtain a coarse angle estimation or measurement from each sensor unit  105 , i.e., sensor elements  108   a  and  108   b  of sensor unit  105   a , such as by adding and subtracting the output signals of each sensor element  108   a - 108   b  as discussed in more detail below. Then, as also discussed in more detail herein below, sensor  100  can determine at 20 a refined or more precise angle estimation or measurement by combining, such as by averaging, the coarse angle estimations of each sensor unit  105   a - 105   d.    
     Herein throughout, the calculations and computations mentioned as being determined or carried out by sensor  100  can be carried out wholly within sensor  100  or a component thereof, by circuitry coupled thereto, or by some combination thereof, which may vary according to a particular embodiment, implementation or application. For example, a control unit, such as a microprocessor or other suitable circuitry, can be incorporated within sensor  100  or coupled thereto, as is discussed elsewhere herein with reference to the example embodiment of  FIG. 5 . Therefore, any particular example or mention of sensor  100  is not limiting. 
     Thus, sensor  100  can compute 
               ψ   ′     =     arctan   ⁡     [         B   R     ⁡     (     ψ   0     )       ,     k   ⁢         dB   R     ⁡     (     ψ   0     )         d   ⁢           ⁢   ψ           ]             
in which the position of sensor unit  105 , i.e., the intersection of the z-axis and the ψ-axis which is the midpoint between sensor elements  108   a  and  108   b  as depicted in  FIG. 2 , is assumed to be at the azimuthal angle ψ 0 . Then it follows that
 
               ψ   ′     =     arctan   ⁡     [           OUT   1     +     OUT   2         2   ×   S       ,     k   ⁢         OUT   1     -     OUT   2         S   ×       d   12     /   R             ]             
where OUT 1  and OUT 2  are the output signals of the sensor elements  108   a  or  108   b  (which are sensitive to the magnetic field component perpendicular to substrate  106 , e.g, Hall plates in one embodiment), S is the magnetic sensitivity of each sensor element  108   a  and  108   b  defined by OUT=S×B R , indices 1 and 2 denote sensor element  108   a  (e.g., 1) or sensor element  108   b  (e.g., 2), d 12  is the spacing between sensor elements  108   a  and  108   b , and R as above is the radial distance of sensor elements  108   a  and  108   b  from the axis of rotation of shaft  104 . Because the same sensor output signals OUT 1  and OUT 2  are used in both calculations above, though one is added and the other subtracted, the absolute calibration of sensor elements  108   a  and  108   b  cancels out, assuming that both sensor elements  108   a  and  108   b  match. Thus, it holds that
 
               ψ   ′     =     arctan   ⁡     [           OUT   1     +     OUT   2       2     ,     k   ⁢         OUT   1     -     OUT   2           d   12     /   R           ]             
which is independent of the magnetic sensitivity of sensor elements  108   a  and  108   b  on substrate  106 .
 
     From the signals B R  and dB R /dψ, an estimated rotation angle can be obtained. The above calculations, however, have not taken into consideration any effects of background or external magnetic field disturbances. In embodiments, such as the one depicted in  FIGS. 3A-3C , a plurality of sensor units  105  can be used in sensor  100  in order to increase immunity of sensor  100  with respect to external magnetic field disturbances. 
     Consider, for example, sensor units  105   a  and  105   c  in  FIGS. 3B and 3C  and a sensor layout such as that depicted in  FIG. 2B , where one, e.g., sensor unit  105   a  is positioned at ψ 0  and the other, e.g., sensor unit  105   b , is positioned at ψ 0 +180 degrees. In other words, the sensor units  105   a  and  105   b  are opposite one another with respect to the axis of rotation of shaft  104 , and their radial positions are identical. If a homogenous background magnetic field is applied, it has no effect on sensor units  105   a  and  105   b  if the direction of the field is orthogonal to B Z . The magnetic field component parallel to a line on which both sensor units are located affects the B Z  signal, but not the dB Z /dψ signal, because the latter is a difference of fields on sensor elements  108   a  and  108   b  in any one sensor unit  105   a  or  105   c  such that a homogeneous background disturbance cancels out. A ratio of homogeneous applied magnetic field over B Z  amplitude can be denoted as ε, and if ε=0.1 the disturbance magnetic field is 10% of the amplitude of B Z  induced by magnet  102 . Thus, the radial magnetic field on sensor unit  105   a  (at ψ 0 ) is B Z *(1+ε), while the radial magnetic field on sensor unit  105   c  (at ψ 0 +180 degrees) is B Z *(1−ε). This is illustrated in  FIG. 4 . In k*dB Z /dψ, k is the aforementioned scaling factor intended to equalize the amplitudes of B Z  and dB Z /dψ. 
     Thus, an error can be introduced into the angles sensed by sensor units  105   a  and  105   c , where ψ′ 1  is the angle measured by sensor unit  105   a , and ψ′ 2  is the angle measured by sensor unit  105   b:  
 
ψ′ 1 =arctan [sin ψ+ε, cos ψ]
 
ψ′ 2 =arctan [sin ψ−ε, cos ψ]
 
     Assuming still that ε=0.1, both ψ′ 1  and ψ′ 2  can have a relatively significant angle error of arctan(0.1)=5.7 degrees near the worst-case positions of psi=0 degrees and 180 degrees, with the angle at sensor unit  105   a  being 5.7 degrees too large and at sensor unit  105   c  the opposite, 5.7 degrees too small. If sensor  100  calculates the average of both signals, the result is accurate, though in other rotational positions this will not necessarily be the case. For arbitrary rotational positions, the error is given by: 
             Δψ   =         -   1     2     ⁢   arctan   ⁢         ɛ   2     ⁢   sin   ⁢           ⁢     (     2   ⁢   ψ     )         1   +       ɛ   2     ⁢           ⁢   cos   ⁢           ⁢     (     2   ⁢   ψ     )                   
where the error is largest at odd multiples of ψ=45 degrees. For ε=0.1, the maximum angle error is 0.29 degrees, which is 20 times smaller than the maximum angle error of sensor unit  105   a  alone.
 
     Therefore, in a system such as sensor  100  in  FIGS. 3A-3C , comprising four sensor units  105   a - 105   d  arranged at integer multiples of 90 degrees around shaft  104 , then sensor unit  105   a  is at ψ 0 , sensor unit  105   b  is at ψ 0 +90 degrees, sensor unit  105   c  is at ψ 0 +180 degrees and sensor unit  105   d  is at ψ 0 +270 degrees, and the latter two are orthogonal to the assume background magnetic field and therefore not influenced by this disturbance. If sensor  100  computes the total angle again, but now as an average over all four angles of sensor units  105   a - 105   d , the angle error of 0.29 degrees is reduced significantly to only 0.15 degrees. In other embodiments, more than four sensor units  105   a - 105   d  can be used, which can further reduce the angle error, though not at the same rate as illustrated here in increasing from two sensor units  105   a  and  105   c  to four sensor units  105   a - 105   d . In addition to an even number of sensor units  105 , in embodiments an odd number of sensor units  105  (e.g., 3, 5, 7, etc.) also can significantly suppress background magnetic interference in the estimated rotation angle. 
     Additionally or alternatively, in embodiments different combinations of ψ 1  and ψ 2  can be used to reduce the effects of external disturbance magnetic fields. For example, in the presence of some external disturbance and for k=1, sensor  100  can determine
 
ψ′ 1 =arctan [sin ψ+ε, cos ψ]
 
by
 
               tan   ⁢           ⁢     ψ   1   ′       =         sin   ⁢           ⁢   ψ     +   ɛ       cos   ⁢           ⁢   ψ             
which is the inverse of the arctan [x, y] operation. The same can be applied to ψ′ 2  by
 
               tan   ⁢           ⁢     ψ   2   ′       =         sin   ⁢           ⁢   ψ     -   ɛ       cos   ⁢           ⁢   ψ             
The average of both tangents is independent of the background field ε:
 
     
       
         
           
             
               
                 
                   tan 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ψ 
                     1 
                     ′ 
                   
                 
                 + 
                 
                   tan 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ψ 
                     2 
                     ′ 
                   
                 
               
               2 
             
             = 
             
               
                 
                   1 
                   2 
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       
                         
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ψ 
                         
                         + 
                         ɛ 
                       
                       
                         cos 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         ψ 
                       
                     
                     + 
                     
                       
                         
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ψ 
                         
                         - 
                         ɛ 
                       
                       
                         cos 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         ψ 
                       
                     
                   
                   ) 
                 
               
               = 
               
                 tan 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 ψ 
               
             
           
         
       
     
     Thus, in embodiments in which sensor  100  comprises at least two sensor units  105   a  and  105   c  arranged at ψ 0  and ψ 0 +180 degrees, respectively (or, e.g.,  105   b  and  105   d  arranged at ψ 0 +90 degrees and ψ 0 +270 degrees, respectively, or other opposing psi-positions 180 degrees apart), sensor  100  can compute the following as a better estimation of the rotation angle, in which homogenous background magnetic fields can cancel perfectly or nearly so in embodiments: 
     
       
         
           
             arctan 
             ⁢ 
             
                 
             
             ⁢ 
             
               ( 
               
                 
                   
                     tan 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ψ 
                       1 
                       ′ 
                     
                   
                   + 
                   
                     tan 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ψ 
                       2 
                       ′ 
                     
                   
                 
                 2 
               
               ) 
             
           
         
       
     
     Another factor that can influence the accuracy of sensor  100  in embodiments is imperfect (e.g., non-homogenous) magnetization of magnet  102 . This non-homogeneity can result in magnetic field components induced thereby to have higher odd harmonics according to the following:
 
 BR =sin(psi)+ b 3*sin(3*psi+kappa3)+ b 5*sin(5*psi+kappa5)+ . . . .
 
in which b3, b5, etc., are small but not negligible. If magnet  102  is mounted eccentrically with respect to the rotation axis of shaft  104 , additional even-order harmonics can result. Thus, in embodiments a plurality of sensor units  105 , such as discussed above, can be used in sensor  100  and evenly distributed. For example, in, e.g.,  FIGS. 3A-3C  four sensor units  105   a - 105   d  are spaced apart by 90 degrees. In an embodiment comprising five sensor units  105 , the spacing can be every 72 degrees or, generally, for N units at integer multiples of 360 degrees/N with output signals of psi1, psi2 . . . psiN, which are coarse estimates of the rotation angle of shaft  104 .
 
     A control unit of or coupled to sensor  100  can receive the psi values output by each sensor unit  105  and combine them to obtain a more accurate estimate of the rotation angle of shaft  104 . In an embodiment, the psi values can be averaged, similar to as discussed above with respect to an embodiment comprising two sensor units  105   a  and  105   c . In another embodiment, the N sensor units can be grouped into opposing pairs with respect to their orientation around shaft  104 , similar to sensor units  105   a  and  105   c , and  105   b  and  105   d . Then, the arctangent for each pair can be determined according to the following: 
             arctan   ⁢           ⁢     (         tan   ⁢           ⁢     ψ   1   ′       +     tan   ⁢           ⁢     ψ   2   ′         2     )           
and the arctangents of all of the groups can be averaged. In yet another embodiment, the average of all of the tangents for each sensor unit  105  can be determined, with the arctangent of the average then calculated.
 
     As previously mentioned, a scaling factor k can be used in embodiments to equalize the amplitudes of B R  and dB R /dψ. This feature also can help to reduce the cost of sensor units  105 . For example, k depends on the diameter of magnet  102  and the placement of sensor units  105 , more particularly sensor elements  108 , with respect to magnet  102 . Thus, it is difficult to pre-program a single value for k given that these factors can vary, but the alternative can be more complicated and expensive (i.e., to determine k for each individual sensor  100  or even sensor unit  105  and to incorporate memory circuitry, such as EEPROM or other suitable memory, into each sensor unit  105 ). 
     Therefore, in embodiments k can be considered to comprise two factors, ka and kb, such that k=ka*kb. In one embodiment, and referring to  FIG. 5 , ka is an approximation of an appropriate ratio of amplitudes of B R  and dB R /dψ, and kb is a corrective factor (e.g., about 1 in embodiments) stored not in sensor unit  105  but in memory circuitry  122  of a control unit  120  which is part of sensor  100 . In another embodiment, control unit  120  and/or memory circuitry  122  can be external but coupled to sensor  100 , and/or memory circuitry  122  can be external to control unit  120 , and/or external or internal to any one or more sensor unit  105 , with the depiction in  FIG. 5  being exemplary of but one embodiment. Moreover, sensor  100  can comprise more or fewer components than are depicted in the simplified block diagram of  FIG. 5 . Control unit  120  then combines the output signals of all of sensor units  105   a - 105   n  to obtain a more precise estimation of the rotation angle of shaft  104 . In practice, kb may be different for each sensor unit  105   a - 105   n , or it may be the same for some or all. 
     Thus, in an embodiment sensor unit  105   a  provides an output signal which is: 
     
       
         
           
             
               ψ 
               1 
             
             = 
             
               arctan 
               ⁡ 
               
                 [ 
                 
                   
                     
                       B 
                       R 
                     
                     ⁡ 
                     
                       ( 
                       
                         ψ 
                         0 
                       
                       ) 
                     
                   
                   , 
                   
                     
                       k 
                       a 
                     
                     ⁢ 
                     
                       
                         
                           dB 
                           R 
                         
                         ⁡ 
                         
                           ( 
                           
                             ψ 
                             0 
                           
                           ) 
                         
                       
                       
                         d 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         ψ 
                       
                     
                   
                 
                 ] 
               
             
           
         
       
     
     Sensor unit  105   c , arranged opposite sensor unit  105   a  with respect to the axis of rotation of shaft  104  (as in, e.g.,  FIGS. 3A-3C ), provides an output signal which is: 
     
       
         
           
             
               ψ 
               2 
             
             = 
             
               arctan 
               ⁡ 
               
                 [ 
                 
                   
                     
                       B 
                       R 
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           ψ 
                           0 
                         
                         + 
                         
                           180 
                           ⁢ 
                           ° 
                         
                       
                       ) 
                     
                   
                   , 
                   
                     
                       k 
                       a 
                     
                     ⁢ 
                     
                       
                         
                           dB 
                           R 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               ψ 
                               0 
                             
                             + 
                             
                               180 
                               ⁢ 
                               ° 
                             
                           
                           ) 
                         
                       
                       
                         d 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         ψ 
                       
                     
                   
                 
                 ] 
               
             
           
         
       
     
     Control unit  120  then determines the following: 
                           ⁢       [         B   R     ⁡     (     ψ   0     )       ,       k   a     ⁢         dB   R     ⁡     (     ψ   0     )         d   ⁢           ⁢   ψ           ]     ⁢     /     ⁢           (       B   R     ⁡     (     ψ   0     )       )     2     +       (       k   a     ⁢         dB   R     ⁡     (     ψ   0     )         d   ⁢           ⁢   ψ         )     2                                           ⁢   and                             [         B   R     ⁡     (       ψ   0     +     180   ⁢   °       )       ,       k   a     ⁢         dB   R     ⁡     (       ψ   0     +     180   ⁢   °       )         d   ⁢           ⁢   ψ           ]     ⁢     /     ⁢           (       B   R     ⁡     (       ψ   0     +     180   ⁢   °       )       )     2     +       (       k   a     ⁢         dB   R     ⁡     (       ψ   0     +     180   ⁢   °       )         d   ⁢           ⁢   ψ         )     2                                 
whereby the square roots normalize the length of the signals to one. In another embodiment, each sensor unit  105  can directly provide the sine and cosine values to control unit  120 , which can reduce computational complexity in sensor units  105  but impose higher data protocol demands between the sensor units  105  and control unit  120 .
 
     Next, control unit  120  adjusts the scaling factor k by determining 
     
       
         
           
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       [ 
                       
                         
                           
                             B 
                             R 
                           
                           ⁡ 
                           
                             ( 
                             
                               ψ 
                               0 
                             
                             ) 
                           
                         
                         , 
                         
                           
                             k 
                             a 
                           
                           ⁢ 
                           
                             k 
                             b 
                           
                           ⁢ 
                           
                             
                               
                                 dB 
                                 R 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   ψ 
                                   0 
                                 
                                 ) 
                               
                             
                             
                               d 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               ψ 
                             
                           
                         
                       
                       ] 
                     
                     ⁢ 
                     
                       / 
                     
                     ⁢ 
                     
                       
                         
                           
                             ( 
                             
                               
                                 B 
                                 R 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   ψ 
                                   0 
                                 
                                 ) 
                               
                             
                             ) 
                           
                           2 
                         
                         + 
                         
                           
                             ( 
                             
                               
                                 k 
                                 a 
                               
                               ⁢ 
                               
                                 
                                   
                                     dB 
                                     R 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       ψ 
                                       0 
                                     
                                     ) 
                                   
                                 
                                 
                                   d 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   ψ 
                                 
                               
                             
                             ) 
                           
                           2 
                         
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   and 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     [ 
                     
                       
                         
                           B 
                           R 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               ψ 
                               0 
                             
                             + 
                             
                               180 
                               ⁢ 
                               ° 
                             
                           
                           ) 
                         
                       
                       , 
                       
                         
                           k 
                           a 
                         
                         ⁢ 
                         
                           k 
                           b 
                         
                         ⁢ 
                         
                           
                             
                               dB 
                               R 
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   ψ 
                                   0 
                                 
                                 + 
                                 
                                   180 
                                   ⁢ 
                                   ° 
                                 
                               
                               ) 
                             
                           
                           
                             d 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             ψ 
                           
                         
                       
                     
                     ] 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     / 
                   
                   ⁢ 
                   
                     
                       
                         
                           ( 
                           
                             
                               B 
                               R 
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   ψ 
                                   0 
                                 
                                 + 
                                 
                                   180 
                                   ⁢ 
                                   ° 
                                 
                               
                               ) 
                             
                           
                           ) 
                         
                         2 
                       
                       + 
                       
                         
                           ( 
                           
                             
                               k 
                               a 
                             
                             ⁢ 
                             
                               
                                 
                                   dB 
                                   R 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       ψ 
                                       0 
                                     
                                     + 
                                     
                                       180 
                                       ⁢ 
                                       ° 
                                     
                                   
                                   ) 
                                 
                               
                               
                                 d 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 ψ 
                               
                             
                           
                           ) 
                         
                         2 
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
           
         
       
     
     In embodiments, the scaling factor portion kb can be determined from a calibration operation, such as during manufacturing and/or end-of-line testing of sensor  100 , in which control unit  120  determines the amplitudes of the following, and therefrom calculates kb as a ratio between the two: 
     
       
         
           
             
               
                 B 
                 R 
               
               ⁡ 
               
                 ( 
                 
                   ψ 
                   0 
                 
                 ) 
               
             
             ⁢ 
             
               / 
             
             ⁢ 
             
               
                 
                   
                     ( 
                     
                       
                         B 
                         R 
                       
                       ⁡ 
                       
                         ( 
                         
                           ψ 
                           0 
                         
                         ) 
                       
                     
                     ) 
                   
                   2 
                 
                 + 
                 
                   
                     ( 
                     
                       
                         k 
                         a 
                       
                       ⁢ 
                       
                         
                           
                             dB 
                             R 
                           
                           ⁡ 
                           
                             ( 
                             
                               ψ 
                               0 
                             
                             ) 
                           
                         
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ψ 
                         
                       
                     
                     ) 
                   
                   2 
                 
               
             
           
         
       
       
         
           
             
               k 
               a 
             
             ⁢ 
             
               
                 
                   dB 
                   R 
                 
                 ⁡ 
                 
                   ( 
                   
                     ψ 
                     0 
                   
                   ) 
                 
               
               
                 d 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 ψ 
               
             
             ⁢ 
             
               / 
             
             ⁢ 
             
               
                 
                   
                     ( 
                     
                       
                         B 
                         R 
                       
                       ⁡ 
                       
                         ( 
                         
                           ψ 
                           0 
                         
                         ) 
                       
                     
                     ) 
                   
                   2 
                 
                 + 
                 
                   
                     ( 
                     
                       
                         k 
                         a 
                       
                       ⁢ 
                       
                         
                           
                             dB 
                             R 
                           
                           ⁡ 
                           
                             ( 
                             
                               ψ 
                               0 
                             
                             ) 
                           
                         
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ψ 
                         
                       
                     
                     ) 
                   
                   2 
                 
               
             
           
         
       
     
     Then, in one embodiment, control unit  120  determines new angles for sensor units  105   a  and  105   c  using the adjusted scaling factor k=ka*kb, according to 
                     ψ   1   ′     =     arctan   ⁡     [           B   R     ⁡     (     ψ   0     )       ⁢     /     ⁢           (       B   R     ⁡     (     ψ   0     )       )     2     +       (       k   a     ⁢         dB   R     ⁡     (     ψ   0     )         d   ⁢           ⁢   ψ         )     2           ,           ⁢       k   a     ⁢           ⁢     k   b     ⁢           ⁢         dB   R     ⁡     (     ψ   0     )         d   ⁢           ⁢   ψ       ⁢           ⁢     /     ⁢           (       B   R     ⁡     (     ψ   0     )       )     2     +       (       k   a     ⁢         dB   R     ⁡     (     ψ   0     )         d   ⁢           ⁢   ψ         )     2             ]                                       ⁢   and                             ψ   2   ′     =                 arctan   ⁢           [           ⁢                 B   R     ⁡     (       ψ   0     +     180   ⁢   °       )       ⁢     /     ⁢           (       B   R     ⁡     (       ψ   0     +     180   ⁢   °       )       )     2     +       (       k   a     ⁢         dB   R     ⁡     (       ψ   0     +     180   ⁢   °       )         d   ⁢           ⁢   ψ         )     2           ,                 k   a     ⁢     k   b     ⁢         dB   R     ⁡     (       ψ   0     +     180   ⁢   °       )         d   ⁢           ⁢   ψ       ⁢     /     ⁢           (       B   R     ⁡     (       ψ   0     +     180   ⁢   °       )       )     2     +       (       k   a     ⁢         dB   R     ⁡     (       ψ   0     +     180   ⁢   °       )         d   ⁢           ⁢   ψ         )     2                 ⁢           ]                                 
and from there determines an improved angle estimate by
 
     
       
         
           
             
               ψ 
               ′ 
             
             = 
             
               
                 
                   ψ 
                   1 
                   ′ 
                 
                 + 
                 
                   ( 
                   
                     
                       ψ 
                       2 
                       ′ 
                     
                     - 
                     
                       180 
                       ⁢ 
                       ° 
                     
                   
                   ) 
                 
               
               2 
             
           
         
       
     
     This can reduce the effects of homogeneous background fields, though it may not entirely cancel them. For example, if a background field is 10% of the amplitude of the BR field, the resulting angle error in ψ′ will be less than 0.3 degrees, which may be acceptable in some or most situations. 
     Nevertheless, in another embodiment a second approach can more completely cancel out homogeneous background fields, in which control unit  120  calculates the sum of in-phase signals {tilde over (B)} R  and the sum of quadrature signals {tilde over (Q)} R  by: 
     
       
         
           
             
               
                 B 
                 ~ 
               
               R 
             
             = 
             
               
                 
                   
                     B 
                     R 
                   
                   ⁡ 
                   
                     ( 
                     
                       ψ 
                       0 
                     
                     ) 
                   
                 
                 ⁢ 
                 
                   / 
                 
                 ⁢ 
                 
                   
                     
                       
                         ( 
                         
                           
                             B 
                             R 
                           
                           ⁡ 
                           
                             ( 
                             
                               ψ 
                               0 
                             
                             ) 
                           
                         
                         ) 
                       
                       2 
                     
                     + 
                     
                       
                         ( 
                         
                           
                             k 
                             a 
                           
                           ⁢ 
                           
                             
                               
                                 dB 
                                 R 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   ψ 
                                   0 
                                 
                                 ) 
                               
                             
                             
                               d 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               ψ 
                             
                           
                         
                         ) 
                       
                       2 
                     
                   
                 
               
               + 
               
                 
                   
                     B 
                     R 
                   
                   ⁡ 
                   
                     ( 
                     
                       
                         ψ 
                         0 
                       
                       + 
                       
                         180 
                         ⁢ 
                         ° 
                       
                     
                     ) 
                   
                 
                 ⁢ 
                 
                   / 
                 
                 ⁢ 
                 
                   
                     
                       
                         ( 
                         
                           
                             B 
                             R 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 ψ 
                                 0 
                               
                               + 
                               
                                 180 
                                 ⁢ 
                                 ° 
                               
                             
                             ) 
                           
                         
                         ) 
                       
                       2 
                     
                     + 
                     
                       
                         ( 
                         
                           
                             k 
                             a 
                           
                           ⁢ 
                           
                             
                               
                                 dB 
                                 R 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     ψ 
                                     0 
                                   
                                   + 
                                   
                                     180 
                                     ⁢ 
                                     ° 
                                   
                                 
                                 ) 
                               
                             
                             
                               d 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               ψ 
                             
                           
                         
                         ) 
                       
                       2 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 Q 
                 ~ 
               
               R 
             
             = 
             
               
                 
                   k 
                   a 
                 
                 ⁢ 
                 
                   k 
                   b 
                 
                 ⁢ 
                 
                   
                     
                       dB 
                       R 
                     
                     ⁡ 
                     
                       ( 
                       
                         ψ 
                         0 
                       
                       ) 
                     
                   
                   
                     d 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     ψ 
                   
                 
                 ⁢ 
                 
                   / 
                 
                 ⁢ 
                 
                   
                     
                       
                         ( 
                         
                           
                             B 
                             R 
                           
                           ⁡ 
                           
                             ( 
                             
                               ψ 
                               0 
                             
                             ) 
                           
                         
                         ) 
                       
                       2 
                     
                     + 
                     
                       
                         ( 
                         
                           
                             k 
                             a 
                           
                           ⁢ 
                           
                             
                               
                                 dB 
                                 R 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   ψ 
                                   0 
                                 
                                 ) 
                               
                             
                             
                               d 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               ψ 
                             
                           
                         
                         ) 
                       
                       2 
                     
                   
                 
               
               + 
               
                 
                   k 
                   a 
                 
                 ⁢ 
                 
                   k 
                   b 
                 
                 ⁢ 
                 
                   
                     
                       dB 
                       R 
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           ψ 
                           0 
                         
                         + 
                         
                           180 
                           ⁢ 
                           ° 
                         
                       
                       ) 
                     
                   
                   
                     d 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     ψ 
                   
                 
                 ⁢ 
                 
                   / 
                 
                 ⁢ 
                 
                   
                     
                       
                         ( 
                         
                           
                             B 
                             R 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 ψ 
                                 0 
                               
                               + 
                               
                                 180 
                                 ⁢ 
                                 ° 
                               
                             
                             ) 
                           
                         
                         ) 
                       
                       2 
                     
                     + 
                     
                       
                         ( 
                         
                           
                             k 
                             a 
                           
                           ⁢ 
                           
                             
                               
                                 dB 
                                 R 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     ψ 
                                     0 
                                   
                                   + 
                                   
                                     180 
                                     ⁢ 
                                     ° 
                                   
                                 
                                 ) 
                               
                             
                             
                               d 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               ψ 
                             
                           
                         
                         ) 
                       
                       2 
                     
                   
                 
               
             
           
         
       
     
     Only the in-phase signals are affected by homogeneous background magnetic fields, as the quadrature signals are gradiometric signals in which homogeneous background magnetic fields cancel out. Thus, control unit  120  can also use one of the following instead: 
     
       
         
           
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         Q 
                         ~ 
                       
                       R 
                     
                     = 
                     
                       2 
                       ⁢ 
                       
                         k 
                         a 
                       
                       ⁢ 
                       
                         k 
                         b 
                       
                       ⁢ 
                       
                         
                           
                             dB 
                             R 
                           
                           ⁡ 
                           
                             ( 
                             
                               ψ 
                               0 
                             
                             ) 
                           
                         
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ψ 
                         
                       
                       ⁢ 
                       
                         / 
                       
                       ⁢ 
                       
                         
                           
                             
                               ( 
                               
                                 
                                   B 
                                   R 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     ψ 
                                     0 
                                   
                                   ) 
                                 
                               
                               ) 
                             
                             2 
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   k 
                                   a 
                                 
                                 ⁢ 
                                 
                                   
                                     
                                       dB 
                                       R 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         ψ 
                                         0 
                                       
                                       ) 
                                     
                                   
                                   
                                     d 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     ψ 
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   or 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       Q 
                       ~ 
                     
                     R 
                   
                   = 
                   
                       
                   
                   ⁢ 
                   
                     2 
                     ⁢ 
                     
                       k 
                       a 
                     
                     ⁢ 
                     
                       k 
                       b 
                     
                     ⁢ 
                     
                       
                         
                           dB 
                           R 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               ψ 
                               0 
                             
                             + 
                             
                               180 
                               ⁢ 
                               ° 
                             
                           
                           ) 
                         
                       
                       
                         d 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         ψ 
                       
                     
                     ⁢ 
                     
                       / 
                     
                     ⁢ 
                     
                       
                         
                           
                             ( 
                             
                               
                                 B 
                                 R 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     ψ 
                                     0 
                                   
                                   + 
                                   
                                     180 
                                     ⁢ 
                                     ° 
                                   
                                 
                                 ) 
                               
                             
                             ) 
                           
                           2 
                         
                         + 
                         
                           
                             ( 
                             
                               
                                 k 
                                 a 
                               
                               ⁢ 
                               
                                 
                                   
                                     dB 
                                     R 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         ψ 
                                         0 
                                       
                                       + 
                                       
                                         180 
                                         ⁢ 
                                         ° 
                                       
                                     
                                     ) 
                                   
                                 
                                 
                                   d 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   ψ 
                                 
                               
                             
                             ) 
                           
                           2 
                         
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
           
         
       
     
     The homogeneous background magnetic field also cancel out in the sum B R  of in-phase signals, such that both {tilde over (B)} R  and {tilde over (Q)} R  are free of background interference. Thus, control unit  120  can calculate the angle according to
 
ψ′=arctan [ {tilde over (B)}   R   ,{tilde over (Q)}   R ]
 
     Referring to  FIG. 6 , an alternative to the embodiment of  FIGS. 3A-3C  comprises surface-mounted devices (SMDs), with the sensor units  105   a - 105   c  arranged below (as depicted) or above magnet  102 . The views of  FIGS. 6A, 6B and 6C  are similar to the views of  FIGS. 3A, 3B and 3C , respectively. The configuration of  FIG. 6  can be advantageous in some applications, for example when magnet  102  has a larger diameter, such as greater than about 15 mm, or when SMDs are desired or available for some other reason. Additionally, the particular SMD depicted in  FIG. 6  can vary in other embodiments, as can the package or other configuration in other, non-SMD embodiments. For example, SMDs with pins on both rather than a single side can be used in embodiments. In still other embodiments, Very Thin Quad Flat Non-Leaded (VQFN) packages can be used, or other suitable packages and/or configurations known or developed and as appreciated by those skilled in the art. 
     In the embodiment of  FIG. 6 , three sensor units  105   a - 105   c  are depicted, spaced apart from one another at integer multiples of 120 degrees around shaft  104 , which can be nonmagnetic or ferrous in embodiments. The number of sensor units  105  and their relative arrangements can vary in other embodiments. For example, and similar to  FIG. 3 , four sensor units  105   a - 105   d  can be used in the configuration of  FIG. 6  in another embodiment. 
     In embodiments, a distance d between the package of sensor units  105   a - 105   c  and magnet  102  is in a range of about 0.5 mm to about 3 mm, such as about 1 mm in an embodiment. As in other embodiments, magnet  102  is magnetized diametrically, and sensor elements  108   a  and  108   b  of each sensor unit  105   a - 105   c  are configured to detect the B z , or axial, component of the magnetic field induced by magnet  102 . Operation of sensor  100  can be similar to as is discussed herein with respect to other embodiments. 
     In yet another embodiment depicted in  FIGS. 7A and 7B , sensor units  105  are arranged within the bore of magnet  102 , which can comprise an annular cylinder having a first, inner radius (i.e., of the bore) and a second, outer radius. The leads of sensor units  105  are coupled to a printed circuit board (PCB)  111  such that the sensor units  105  extend into the bore of magnet  102 , which is spaced apart from PCB  111 . Magnet  102  is coupled to a disk portion  103  of shaft  104  such that magnet  102  rotates with shaft  104  in operation. Magnet can be coupled to shaft  104  in some other way in other embodiments, and/or the particular configuration of disk portion  103  can vary, e.g., a diameter of disk portion  103  with respect to a diameter of magnet  102 , with the depiction in  FIGS. 7A and 7B  being merely one example. For example, a profile, shape, dimension or other configuration of the inner bore portion of magnet  102 , proximate sensor units  105 , can vary in embodiments. In one embodiment, the inner bore profile of magnet  102  has a conical-type shape opposite shaft  104 . In another embodiment, the inner bore profile of magnet  102  is extended or angled in an axial direction toward sensor units  105 , such as by less than about 3 mm, for example extending about 1 mm in such a direction in one embodiment. 
     In still other embodiments, with respect to  FIG. 7  and others discussed herein, the material of disk  103  and/or shaft  104  can vary. For example, one or both can comprise a ferrous or otherwise magnetic material, such as iron, in embodiments. In other embodiments, however, one or both of disk  103  and shaft  104  can comprise a non-magnetic material. Though this can change the strength of the magnetic field and effects of external fields, it can have little effect on angle accuracy and can, in fact, reduce production costs if less expensive materials are used. 
     PCB  111  can comprise a unitary piece, or can comprise a plurality of pieces, e.g., two halves, which can be easier to manufacture, though care should be taken in assembly of sensor  100  with respect to position tolerances, such as for sensor units  105  relative to magnet  102 . In  FIG. 7A , only half of PCB  111  is shown, and a portion of magnet  102  is cut-away such that sensor units  105  within the bore can be seen. Though not depicted, a second disk can be included below PCB  111  in embodiments, which can improve shielding of external magnetic fields. In one embodiment, PCB  111  can comprise a unitary structure having an aperture comprising a first portion, such as near the center of PCB  11 , in which shaft  104  is arranged and a second portion which extends from an edge or perimeter of PCB  111  to the first portion, enabling shaft  104  to be arranged in the first portion of the aperture without having to pass PCB  111  over an end of shaft  104 , which can simply manufacturing and assembly. A width or other dimension of the second portion can be slightly larger than a diameter of shaft  104 , e.g., at least about 1 mm larger in embodiments. The size and placement of the second portion of the aperture also should be considered with respect to the arrangement of sensor units  105  on PCB  111 . 
     As depicted, substrates  106  of sensor units  105  are arranged such that the surface of substrate  106  on which sensor elements  108  are mounted is parallel with the [ψ, z]-plane. Sensor elements  108   a  and  108   b , as in other embodiments, can comprise Hall plates or other suitable magnetic field sensor elements, for example ordinary Hall plates as depicted. The active surface of each of substrates  106  of sensor units  105  is arranged opposite (e.g., facing) magnet  102 , rather than shaft  104 , in one embodiment as depicted. In embodiments, sensor units  105  are arranged such that a center line of sensor elements  108   a  and  108   b  is generally in line with a midplane of magnet  102 . 
     The embodiments discussed thus far have generally related to off-axis angle sensors, though it has been discovered that these principles also can apply to embodiments of on-axis angle sensors as well. An example on-angle sensor  200  is depicted in  FIG. 8 , with the same or similar reference numerals used to refer to the same or similar components, parts and/or features. In sensor  200 , sensor unit  205  is arranged coaxially with shaft  104 , with substrate  206  oriented perpendicularly thereto. 
     Analogous to  FIG. 2 ,  FIG. 9  depicts an example layout of sensor unit  205  comprising at least two sensor elements, here four: sensor elements  208   a ,  208   b ,  208   c  and  208   d . The four sensor elements  208 - 208   d  can be considered to comprise two sensor units (i.e., similar to sensor units  105   a  and  105   b  discussed above), with sensor elements  208   a  and  208   b  comprising a first sensor unit, and sensor elements  208   c  and  208   d  comprising a second sensor unit. In one embodiment, sensor elements  208 - 208   d  can be selected such that they are sensitive to the z-component of the magnetic field inducted by magnet  102 , or sensor elements  208   a - 208   d  can be selected in other embodiments to be sensitive to the R-component or to the ψ-component. As in off-axis angle sensor embodiments, sensor elements  208   a - 208   d  can comprise Hall plates, vertical Hall elements, magnetoresistive elements, or other suitable magnetic field sensor elements. 
     Sensor elements  208   a - 208   d  can be considered to be arranged on a circle with the rotation angle ψ as shown, such that sensor element  208   a  is shifted in the ψ-direction with respect to each of sensor elements  208   b - d , sensor element  208   b  is shifted in the ψ-direction with respect to sensor elements  208   a ,  208   c  and  208   d , etc. Sensor elements  208   a - 208   d  are arranged around the rotation axis of shaft  102 , which is aligned with the intersection of the x-axis and the y-axis in  FIG. 9 , which is also the center point of all of sensor elements  208   a - 208   d . The mid-points between sensor elements of each sensor unit, i.e., the mid-point between sensor elements  208   a  and  208   b  and the midpoint between sensor elements  208   c  and  208   d , are aligned with one another but at opposing positive and negative y-axis positions. In other words, the azimuthal position of the midpoint of sensor elements  208   a  and  208   b  and the azimuthal position of the midpoint of sensor elements  208   c  and  208   d  differ by 180 degrees. Though the radial positions may vary in some embodiments or implementations, in other embodiments the sensor elements are arranged to be mirror-symmetric with respect to the rotation axis of shaft  102  as follows: sensor elements  208   a  and  208   c  are mirror-symmetric with one another, and sensor elements  208   b  and  208   d  are mirror-symmetric with one another. 
     In operation, sensor  200  is similar to sensor  100  discussed herein above. Sensor  100 , such as a control unit similar to control unit  120  or other circuitry of or coupled to sensor  100 , determines a difference between and a sum of the output signals of each sensor unit, here for sensor elements  208   a  and  208   b  (e.g., Ba+Bb, and Ba−Bb, where Ba is the output signal of sensor element  208   a  and Bb is the output signal of sensor element  208   b , etc.), and again for sensor elements  208   c  and  208   d  (e.g., Bc+Bd, and Bc−Bd). Then, an angle ψab and an angle ψcd are determined for each sensor unit according to
 
arctan [( Ba+Bb ), k *( Ba−Bb )]
 
This is repeated for sensor elements  208   c  and  208   d:  
 
arctan [( Bc+Bd ), k *( Bc−Bd )]
 
Both angles then can be combined as discussed above according to:
 
ψ′=(ψ ab+ψcd− 180 degrees)/2
 
or
 
ψ′=arctan((tan(ψ ab )+tan(ψ cd )/2)
 
     In another embodiment, sensor  100  can simply determine the following:
 
ψ′=arctan [( Ba+Bb )−( Bc+Bd ), k *(( Ba−Bb )−( Bc−Bd ))]
 
     In other on-axis embodiments, vertical Hall sensor elements can be used. In general, vertical Hall sensor elements are responsive to magnetic field components parallel to the plane of the sensor element, as opposed to perpendicular for ordinary Hall plate devices discussed elsewhere herein. Referring to  FIG. 10A , one example configuration of a sensor element  305  utilizing vertical Hall devices is depicted. Here as in other embodiment, the same or similar reference numerals are used to refer to the same or similar elements or features. Sensor element  305  comprises a substrate  306  on which are arranged four vertical Hall sensor elements  308   a  (also referred to herein as H_1),  308   b  (H_2),  308   c  (H_3) and  308   d  (H_4). In the embodiment of  FIG. 10A , sensor elements  308   a - 308   d  comprise BR sensor elements, given their orientations as depicted. 
     Similar to  FIG. 9 , sensor elements  308   a - 308   d  of  FIG. 10A  can be considered to be arranged on a circle with the rotation angle ψ as shown, such that sensor element  308   a  is shifted in the ψ-direction with respect to each of sensor elements  308   b - d , sensor element  308   b  is shifted in the ψ-direction with respect to sensor elements  308   a ,  308   c  and  308   d , etc. Sensor elements  308   a - 308   d  are arranged around the rotation axis of shaft  102 , which is aligned with the intersection of the x-axis and the y-axis in  FIG. 10A , which is also the center point of all of sensor elements  308   a - 308   d . Though the radial positions may vary in some embodiments or implementations, in other embodiments the sensor elements are arranged to be mirror-symmetric with respect to the rotation axis of shaft  102 . 
     Another embodiment of sensor unit  305  comprising vertical Hall sensor devices is depicted in  FIG. 10B , in which the orientations of sensor elements  308   a - 308   d  are rotated counter-clockwise 90 degrees, such that sensor elements  308   a - 308   d  are Bψ, rather than BR, sensor elements. Other features, including the relative arrangement of sensor unit  305  with respect to magnet  102  and shaft  14 , are similar to or the same as those of  FIG. 10A  unless otherwise specified herein. 
     The magnetic field of magnet  102  of rotational symmetry and with homogeneous magnetization in the y-direction is given by
 
 B   R ( R,ψ,z )= B   rem  sin ψ( b   0   +b   1 )
 
 B   ψ ( R,ψ,z )= B   rem  cos ψ( b   0   −b   1 )
 
 B   z ( R,φ,z )= B   rem  sin ψ b   2  
 
where b0, b1, and b2 are functions of R and ψ only. Referring to  FIG. 10A , if the field on sensor element  308   a  (H_1) is denoted as B R1 ,  308   b  (H_2) as B R2 ,  308   c  (H_3) as B R3  and  308   d  (H_4) as B R4 , the following results:
 
                   B     R   ⁢           ⁢   1       +     B     R   ⁢           ⁢   2       -     (       B     R   ⁢           ⁢   3       +     B     R   ⁢           ⁢   4         )           B     R   ⁢           ⁢   1       -     B     R   ⁢           ⁢   2       -     (       B     R   ⁢           ⁢   3       -     B     R   ⁢           ⁢   4         )         =           sin   ⁢           ⁢   ψ     +     sin   ⁡     (     ψ   +     90   ⁢   °       )       -     (       sin   ⁡     (     ψ   +     180   ⁢   °       )       +     sin   ⁡     (     ψ   -     90   ⁢   °       )         )           sin   ⁢           ⁢   ψ     -     sin   ⁡     (     ψ   +     90   ⁢   °       )       -     (       sin   ⁢           ⁢     (     ψ   +     180   ⁢   °       )       -     sin   ⁡     (     ψ   -     90   ⁢   °       )         )         =           sin   ⁢           ⁢   ψ     +     cos   ⁢           ⁢   ψ           sin   ⁢           ⁢   ψ     -     cos   ⁢           ⁢   ψ         =     -     tan   ⁡     (     ψ   +     45   ⁢   °       )                   
whereby sensor element  308   a  (H_1) is located at position ψ and ψ=0 denotes the x-axis. This means that sensor  100  comprising sensor unit  305  can estimate the angular position of magnet  102  by:
 
ψ′=arctan [ B   R3   +B   R4 −( B   R1   +B   R2 ), B   R1   −B   R2 −( B   R3   −B   R4 )]−45°.
 
     Conversely, and referring to  FIG. 10B , if the field on sensor element  308   a  (H_1) is denoted as B ψ1 , sensor element  308   b  (H_2) as B ψ2 , sensor element  308   c  (H_3) as B ψ3  and  308   d  (H_4) as B ψ4 , the following results: 
     
       
         
           
             
               
                 
                   B 
                   
                     ψ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     1 
                   
                 
                 + 
                 
                   B 
                   
                     ψ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     2 
                   
                 
                 - 
                 
                   ( 
                   
                     
                       B 
                       
                         ψ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         3 
                       
                     
                     + 
                     
                       B 
                       
                         ψ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         4 
                       
                     
                   
                   ) 
                 
               
               
                 
                   B 
                   
                     ψ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     1 
                   
                 
                 - 
                 
                   B 
                   
                     ψ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     2 
                   
                 
                 - 
                 
                   ( 
                   
                     
                       B 
                       
                         
                             
                         
                         ⁢ 
                         ψ3 
                       
                     
                     - 
                     
                       B 
                       
                         ψ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         4 
                       
                     
                   
                   ) 
                 
               
             
             = 
             
               
                 
                   
                     cos 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     ψ 
                   
                   + 
                   
                     cos 
                     ⁡ 
                     
                       ( 
                       
                         ψ 
                         + 
                         
                           90 
                           ⁢ 
                           ° 
                         
                       
                       ) 
                     
                   
                   - 
                   
                     ( 
                     
                       
                         cos 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           
                             ψ 
                             + 
                             
                               180 
                               ⁢ 
                               ° 
                             
                           
                           ) 
                         
                       
                       + 
                       
                         cos 
                         ⁡ 
                         
                           ( 
                           
                             ψ 
                             - 
                             
                               90 
                               ⁢ 
                               ° 
                             
                           
                           ) 
                         
                       
                     
                     ) 
                   
                 
                 
                   
                     cos 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     ψ 
                   
                   - 
                   
                     cos 
                     ⁡ 
                     
                       ( 
                       
                         ψ 
                         + 
                         
                           90 
                           ⁢ 
                           ° 
                         
                       
                       ) 
                     
                   
                   - 
                   
                     ( 
                     
                       
                         cos 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           
                             ψ 
                             + 
                             
                               180 
                               ⁢ 
                               ° 
                             
                           
                           ) 
                         
                       
                       - 
                       
                         cos 
                         ⁡ 
                         
                           ( 
                           
                             ψ 
                             - 
                             
                               90 
                               ⁢ 
                               ° 
                             
                           
                           ) 
                         
                       
                     
                     ) 
                   
                 
               
               = 
               
                 
                   
                     
                       cos 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       ψ 
                     
                     - 
                     
                       sin 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       ψ 
                     
                   
                   
                     
                       cos 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       ψ 
                     
                     + 
                     
                       sin 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       ψ 
                     
                   
                 
                 = 
                 
                   tan 
                   ⁡ 
                   
                     ( 
                     
                       
                         - 
                         ψ 
                       
                       + 
                       
                         45 
                         ⁢ 
                         ° 
                       
                     
                     ) 
                   
                 
               
             
           
         
       
     
     whereby sensor element  308   a  (H_1) is located at position ψ and ψ=0 denotes the x-axis. Thus, sensor  100  comprising sensor unit  305  can estimate the angular position of magnet  102  by:
 
ψ′=arctan └ B   ψ3   +B   ψ4 −( B   ψ1   +B   ψ2 ), B   ψ1   −B   ψ2 −( B   ψ3   −B   ψ4 )┘−45°.
 
     Finally, if sensor elements  308   a - 308   d  are arranged to be responsive instead to the z-component of the magnetic field, such as similarly to as is depicted in  FIG. 9 , the field on sensor element  308   a  (H_1) is denoted as B z1 , sensor element  308   b  (H_2) as B z2 ,  308   c  (H_3) as B z3  and  308   d  (H_4) as B z4 , the following results: 
     
       
         
           
             
               
                 
                   B 
                   
                     z 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     1 
                   
                 
                 + 
                 
                   B 
                   
                     z 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     2 
                   
                 
                 - 
                 
                   ( 
                   
                     
                       B 
                       
                         z 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         3 
                       
                     
                     + 
                     
                       B 
                       
                         z 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         4 
                       
                     
                   
                   ) 
                 
               
               
                 
                   B 
                   
                     z 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     1 
                   
                 
                 - 
                 
                   B 
                   
                     z 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     2 
                   
                 
                 - 
                 
                   ( 
                   
                     
                       B 
                       z3 
                     
                     - 
                     
                       B 
                       
                         z 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         4 
                       
                     
                   
                   ) 
                 
               
             
             = 
             
               
                 
                   
                     sin 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     ψ 
                   
                   + 
                   
                     sin 
                     ⁡ 
                     
                       ( 
                       
                         ψ 
                         + 
                         
                           90 
                           ⁢ 
                           ° 
                         
                       
                       ) 
                     
                   
                   - 
                   
                     ( 
                     
                       
                         sin 
                         ⁡ 
                         
                           ( 
                           
                             ψ 
                             + 
                             
                               180 
                               ⁢ 
                               ° 
                             
                           
                           ) 
                         
                       
                       + 
                       
                         sin 
                         ⁡ 
                         
                           ( 
                           
                             ψ 
                             - 
                             
                               90 
                               ⁢ 
                               ° 
                             
                           
                           ) 
                         
                       
                     
                     ) 
                   
                 
                 
                   
                     sin 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     ψ 
                   
                   - 
                   
                     sin 
                     ⁡ 
                     
                       ( 
                       
                         ψ 
                         + 
                         
                           90 
                           ⁢ 
                           ° 
                         
                       
                       ) 
                     
                   
                   - 
                   
                     ( 
                     
                       
                         sin 
                         ⁡ 
                         
                           ( 
                           
                             ψ 
                             + 
                             
                               180 
                               ⁢ 
                               ° 
                             
                           
                           ) 
                         
                       
                       - 
                       
                         sin 
                         ⁡ 
                         
                           ( 
                           
                             ψ 
                             - 
                             
                               90 
                               ⁢ 
                               ° 
                             
                           
                           ) 
                         
                       
                     
                     ) 
                   
                 
               
               = 
               
                 
                   
                     
                       sin 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       ψ 
                     
                     + 
                     
                       cos 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       ψ 
                     
                   
                   
                     
                       sin 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       ψ 
                     
                     - 
                     
                       cos 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       ψ 
                     
                   
                 
                 = 
                 
                   - 
                   
                     tan 
                     ⁡ 
                     
                       ( 
                       
                         ψ 
                         + 
                         
                           45 
                           ⁢ 
                           ° 
                         
                       
                       ) 
                     
                   
                 
               
             
           
         
       
     
     whereby sensor element  308   a  (H_1) is located at position ψ and ψ=0 denotes the x-axis. Thus, sensor  100  comprising sensor unit  305  can estimate the angular position of magnet  102  by:
 
ψ′=arctan [ B   z3   +B   z4 −( B   z1   +B   z2 ), B   z1   −B   z2 −( B   z3   −B   z4 )]−45°.
 
     In an embodiment, these three systems (BR for radial, Bψ for azimuthal and Bz for axial) can be combined. For example, instead of having a single sensor element  308   a  (H_1), sensor unit  305  comprises three sensor elements each sensitive to one of a radial, azimuthal, and axial magnetic field component. The same applies to the sensor elements  308   b  (H_2),  308   c  (H_3) and  308   d  (H_4). Then sensor  100  can estimate the rotation angle according to the respective formula given above. Finally, sensor  100  can compare the three angle values and combine them in a way to be more precise in spite of assembly tolerances, background fields (particularly inhomogeneous background fields), and errors associated with the sensor elements (e.g., offset, noise, thermal drift, etc.). In one embodiment, this combination can comprise averaging, though other methods can be used in other embodiments. 
     The configurations of  FIGS. 10A and 10B  can vary in other embodiments. For example,  FIGS. 10C and 10D  depict two additional vertical Hall arrangements, though still other arrangements and configurations can be used in other embodiments. The particular examples given herein are merely examples and are not be considered limiting. In  FIG. 10C , sensor elements  308   a - 308   d  are arranged proximate the corners of substrate  306 . In  FIG. 10D , sensor elements  308   a - 308   d  are instead arranged proximate a center or midpoint of substrate  306 , which can be aligned with a rotation axis of shaft  104  in on-axis embodiments, as discussed elsewhere herein. 
     In general, a sensor system according to embodiments comprises n sensor units located at azimuthal positions of 0 degrees, 360 degrees/n, 2*360°/n, . . . (n−1)*360°/n, relative to a concentric axis of rotation of a magnet. Each sensor unit can provide at least one output signal that encodes a measured angle in a range of 0 degrees to 360 degrees. In the absence of angle errors and for small rotation angles of ψ 0 &lt;360 degrees/n, a first sensor unit measures ψ 0 , a second sensor unit measures ψ 0 +360 degrees/n, a third sensor unit measures ψ 0 +2*360 degrees/n, and sensor unit n measures ψ 0 +(n−1)*360 degrees/n, which is still less than 360 degrees. 
     If the rotation angle increases, the angles measured by all sensor units will not be monotonous: Suppose n=3 and ψ 0 =150 degrees. The first sensor unit measures 150 degrees, the second sensor unit measures 150+120=270 degrees, the third sensor unit measures mod(150+240, 360)=30 degrees. Thus, the measured angles are 150 degrees, 270 degrees, and 30 degrees. They are not monotonous, because 30 degrees is less than 270 degrees. In such a case, the sensor system needs to precondition the measured angles: It adds integer multiples of 360 degrees to each measured angle until all angles are monotonous, i.e. angle(#1)&lt;angle(#2)&lt;angle(#3)&lt; . . . &lt;angle(#n). After this manipulation, the sensor system subtracts the azimuthal position from each angle. Thus, angle(#1) remains unchanged; from angle(#2) the system subtracts 360 degrees/n, from angle(#3) the system subtracts 2*360 degrees/n, and from angle(#n) the system subtracts (n−1)*360 degrees/n. Finally, all of these values can be averaged and the remainder after division by 360 degrees is taken as the result. Hence, this sequence of operations can be essentially an averaging in embodiments, but it can consider some pre- and post-conditioning to account for the finite interval [0 degrees, 360 degrees] of acceptable angle ranges. Thus, it can be referred to as a conditioned averaging. In embodiments, the subtraction of azimuthal positions can be omitted because it only shifts the zero point of the measured angle. 
     Various embodiments of systems, devices and methods have been described herein. These embodiments are given only by way of example and are not intended to limit the scope of the invention. It should be appreciated, moreover, that the various features of the embodiments that have been described may be combined in various ways to produce numerous additional embodiments. Moreover, while various materials, dimensions, shapes, configurations and locations, etc. have been described for use with disclosed embodiments, others besides those disclosed may be utilized without exceeding the scope of the invention. 
     Persons of ordinary skill in the relevant arts will recognize that the invention may comprise fewer features than illustrated in any individual embodiment described above. The embodiments described herein are not meant to be an exhaustive presentation of the ways in which the various features of the invention may be combined. Accordingly, the embodiments are not mutually exclusive combinations of features; rather, the invention can comprise a combination of different individual features selected from different individual embodiments, as understood by persons of ordinary skill in the art. Moreover, elements described with respect to one embodiment can be implemented in other embodiments even when not described in such embodiments unless otherwise noted. Although a dependent claim may refer in the claims to a specific combination with one or more other claims, other embodiments can also include a combination of the dependent claim with the subject matter of each other dependent claim or a combination of one or more features with other dependent or independent claims. Such combinations are proposed herein unless it is stated that a specific combination is not intended. Furthermore, it is intended also to include features of a claim in any other independent claim even if this claim is not directly made dependent to the independent claim. 
     Any incorporation by reference of documents above is limited such that no subject matter is incorporated that is contrary to the explicit disclosure herein. Any incorporation by reference of documents above is further limited such that no claims included in the documents are incorporated by reference herein. Any incorporation by reference of documents above is yet further limited such that any definitions provided in the documents are not incorporated by reference herein unless expressly included herein. 
     For purposes of interpreting the claims for the present invention, it is expressly intended that the provisions of Section 112, sixth paragraph of 35 U.S.C. are not to be invoked unless the specific terms “means for” or “step for” are recited in a claim.