Patent Publication Number: US-10782116-B2

Title: Discrete magnetic angle sensor device, a magnetic angle sensor arrangement, a method for generating an angle signal and a method for providing a sensor signal

Description:
CROSS REFERENCE TO RELATED APPLICATION 
     This application is a continuation of U.S. patent application Ser. No. 15/223,586, filed Jul. 29, 2016, which is a continuation of U.S. patent application Ser. No. 13/944,410, filed Jul. 17, 2013, now U.S. Pat. No. 9,671,214. These application are each incorporated herein by reference in their entirety. 
    
    
     TECHNICAL FIELD 
     Embodiments relate to a discrete magnetic angle sensor device, a magnetic angle sensor arrangement, a method for generating an angle signal and a method for providing a sensor signal, which may, for instance, be used to determine a rotation angle of a first component with respect to a second component as, for instance, in the field of automotive applications. 
     BACKGROUND 
     In many applications a first component rotates or pivots with respect to a second component. Examples come, for instance, from the field of automotive applications such as determination of a steering angle. For instance, it may be advisable for a control unit of a car to determine the steering angle by measuring a rotation angle of the steering column. Conventionally, a magnet may be mounted on an end of a corresponding axle or shaft with a magnetization such that a magnetic field sensor arranged along the rotation axis of the axle or shaft is capable of detecting a change of the magnetic field in response to the rotation of the steering column. Such an arrangement is also referred to as an on-axis angle sensor since the magnetic field sensor is mounted along the rotation axis of the shaft or axle to the end of which the magnet is mounted. 
     However, for instance, due to installation space restrictions or rather circumstances such an on-axis angle sensor might not always be implementable. In such a case, an off-axis angle sensor with a through-shaft magnet may be used. This arrangement may be referred to as an off-axis sensor since the magnetic field sensor is not placed along the rotation axis of the respective shaft or axle. 
     Due to the arrangement of the magnet with respect to the magnet field sensor, the sensor might suffer from non-optimal magnetic field conditions, which may lead to a higher error of the angle of the magnetic field created by the magnet compared to a more conventional design. 
     Therefore, a demand exists to improve an accuracy of a determination of a rotation angle, for instance, of a magnetic field with respect to a reference direction. Similar technical challenges and a similar demand may also exist, when, for instance, the magnet is in an off-axis angle sensor arrangement attached to an end of a shaft or an axle or in an on-axis arrangement. 
     SUMMARY 
     A discrete magnetic angle sensor device according to an embodiment comprises a first magnetic field gradiometer and a second magnetic field gradiometer. The first and second gradiometers are of different types of a group of gradiometer types. Being comprised in a discrete magnetic angle sensor device, the first and second magnetic field gradiometers may be accommodated in a sensor package. 
     A discrete magnetic angle sensor device according to an embodiment is based on the finding that using at least two gradiometers of different types in a discrete magnetic angle sensor device may improve an accuracy and, therefore, decrease an error of an angle determination with respect to the angle of a magnetic field. Optionally, the discrete magnetic angle sensor device according to an embodiment may further comprise a sensor circuit configured to generate a sensor signal indicative of a rotation angle, for instance an angle of the magnetic field, based on a first gradiometer signal of a first magnetic field gradiometer and based on a second gradiometer signal of a second magnetic field gradiometer. The sensor signal may, thus, be generated to be robust against a background magnetic interference. Or, to put it in other words, the sensor circuit may optionally be configured such that the sensor signal is robust against background magnetic interference. Such an optional embodiment may be based on the finding that an implementation of an angle sensor device into a system or an arrangement may be simplified by integrating such a sensor circuit. This may enable a determination of the angle of the magnetic field without having to rely on gradiometer signals from external sources. Hence, the determination may be more accurate. Additionally or alternatively, by using two gradiometers of different types it may be possible to base the determination of the angle of the magnetic field on two different angle dependencies of the gradiometer signals. Moreover, it may also be possible to determine the angle more robustly against external interfering magnetic fields. 
     A magnetic angle sensor arrangement according to an embodiment comprises at least one discrete magnetic angle sensor device, the discrete magnetic angle sensor device comprising a first magnetic field gradiometer, a second magnetic field gradiometer, which may be accommodated in a package, and a sensor circuit configured to generate a sensor signal indicative of a rotation angle, which may be given by an angle of the magnetic field with respect to a reference direction or an angular position of a magnetic field source (e.g. a magnet or a coil), based on a first gradiometer signal of a first magnetic field gradiometer and on a second gradiometer signal of the second magnetic field gradiometer. The first magnetic field gradiometer and the second magnetic field gradiometer are of different types of a group of gradiometer types. The at least one discrete magnetic angle sensor device is arranged fixedly with respect to rotation axis around which a magnet is rotatably mountable such that the at least one discrete magnetic angle sensor is arranged off axis with respect to the rotation axis. For instance, the rotation axis and its mathematical extension may be oriented such that it does not penetrate or go through at least one of the discrete angle sensor(s), one or more of its sensor elements, effective positions of the gradiometers or similar center points of the sensor elements used. 
     A magnetic angle sensor arrangement according to an embodiment is based on the finding that by using at least one discrete magnetic angle sensor device as described before, an accuracy of the angle may be improved in an off-axis arrangement of the at least one discrete magnetic angle sensor device with respect to the rotation axis. Optionally a plurality of discrete magnetic angle sensor devices may be used, which may eventually increase an accuracy further, for instance, by making the arrangement more robust against external influences outside the plurality of discrete magnetic angle sensor devices. 
     A method for generating an angle signal according to an embodiment comprises providing a magnetic field. It further comprises generating a plurality of sensor signals, each sensor signal generated by an individual discrete magnetic angle sensor device, the sensor signals being indicative of a rotation angle. The sensor signals of the individual discrete magnetic angle sensor devices may be optionally essentially independent of a homogeneous external magnetic disturbance field. They may be optionally measured at the respective positions of the discrete magnetic angle sensor devices. The method further comprises generating the angle signal indicative of the angle of the magnetic field based on the plurality of sensor signals. 
     The method for generating an angle signal according to an embodiment is based on the finding that by employing a plurality of sensor signals generated by individual discrete magnetic angle sensor devices, each of the sensor signals being indicative of a rotation angle at a position of the discrete magnetic angle sensor device, an accuracy of the rotation angle, for instance of a magnetic field, which may be generated by a magnet or a coil, may be improved. In other words, by using a plurality of sensor signals an accuracy caused, for instance, by a misalignment or accuracy deteriorating effects may be reduced. 
     A method for providing a sensor signal according to an embodiment comprises generating a first gradiometer signal using a first gradiometer. The method further comprises generating a second gradiometer signal using a second gradiometer. It further comprises generating the sensor signal indicative of a rotation angle based on the first and second gradiometer signals, wherein the first magnetic field gradiometer and the second magnetic field gradiometer are of different types of a group of gradiometer types and the first and second gradiometers are comprised in a discrete magnetic angle sensor device. 
     The method for providing the sensor signal according to an embodiment is based on the finding that an accuracy of the rotation angle may be improved by using two gradiometers of different types, since it may be possible to base the determination of the angle of the magnetic field on two different angle dependencies of the gradiometer signals. Moreover, it may also be possible to determine the angle more robustly against external interfering magnetic fields. It may also be possible to increase an over-all sensitivity and, hence, an accuracy of the angle of the magnetic field to be determined. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Several embodiments of the present invention will be described in the enclosed Figures. 
         FIG. 1  shows a schematic view of a discrete magnetic angle sensor device according to an embodiment. 
         FIG. 2  shows a flowchart of a method for providing a sensor signal according to an embodiment. 
         FIG. 3  shows a schematic diagram of a magnetic angle sensor arrangement according to an embodiment in an off-axis configuration. 
         FIG. 4  shows a flowchart of a method for generating an angle signal according to an embodiment. 
         FIG. 5  shows a schematic representation of a magnetic angle sensor arrangement in an off-axis configuration. 
         FIG. 6  shows a schematic representation of a further magnetic angle sensor arrangement according to an embodiment in an off-axis configuration. 
         FIG. 7  shows a schematic representation of a magnetic angle sensor arrangement according to an embodiment in an off-axis configuration. 
         FIG. 8  shows a schematic representation of a magnetic angle sensor arrangement comprising a discrete magnetic angle sensor device according to an embodiment in an on-axis configuration. 
         FIG. 9  shows a schematic plan view of a discrete magnetic angle sensor device according to an embodiment. 
         FIG. 10  shows a schematic plan view of a discrete magnetic angle sensor device according to an embodiment. 
         FIG. 11  shows a schematic plan view of a discrete magnetic angle sensor device  100  according to a further embodiment. 
         FIG. 12  shows a diagram of two magnetic field components along a rotation angle for a magnet. 
         FIG. 13  shows a diagram of the two magnetic field components along the rotation angle for the magnet of  FIG. 12  at a different location along a rotation axis. 
         FIG. 14  illustrates amplitudes of different magnetic field components as a function of a distance from a mid-plane of the magnet. 
         FIG. 15  shows a diagram of amplitudes of field differences or gradients taken at a distance of 1.5 mm for three different magnetic field components as a function of a distance from a mid-plane. 
         FIG. 16  shows a schematic diagram of a discrete magnetic angle sensor device according to an embodiment. 
         FIG. 17  shows a semi-transparent perspective view of a magnetic angle sensor arrangement according to an embodiment in an off-axis configuration. 
         FIG. 18  shows a perspective view of a magnetic angle sensor arrangement shown in  FIG. 17 . 
         FIG. 19  shows a close-up of the perspective view of  FIG. 18  with a mold compound removed from two discrete magnetic angle sensor devices according to an embodiment removed. 
         FIG. 20  shows a side view of the arrangement shown in  FIG. 19 . 
         FIG. 21  shows a diagram of amplitudes of magnetic field components of a magnet as a function of a distance from a mid-plane. 
         FIG. 22  shows a schematic diagram of a further discrete magnetic angle sensor device according to an embodiment. 
         FIG. 23  shows a diagram of different gradients as a function of a distance from a center plane for a magnet M 1 . 
         FIG. 24  shows a diagram of different gradients as a function of a distance from a center plane for a magnet M 2 . 
         FIG. 25  shows a diagram of different gradients as a function of a distance from a center plane for a magnet M 3 . 
         FIG. 26  shows a semitransparent perspective view of a magnetic angle sensor arrangement according to an embodiment in an off-axis configuration. 
         FIG. 27  shows an enlarged part of  FIG. 26 . 
         FIG. 28  shows a perspective view of the portion shown in  FIG. 27 . 
         FIG. 29  shows the arrangement of the substrates of the discrete magnetic angle sensor devices of the magnetic angle sensor arrangement of  FIGS. 27 and 28 . 
         FIG. 30  shows an enlarged perspective view of a substrate of a discrete magnetic angle sensor device of the magnetic angle sensor arrangement according to an embodiment shown in  FIGS. 26 to 29 . 
         FIG. 31  shows a diagram of different gradients as a function of a distance from a center plane for a magnet M 1 . 
         FIG. 32  shows a diagram of different gradients as a function of a distance from a center plane for a magnet M 2 . 
         FIG. 33  shows a diagram of different gradients as a function of a distance from a center plane for a magnet M 3 . 
         FIG. 34  shows a perspective view of a magnetic angle sensor arrangement according to an embodiment comprising eight discrete magnetic angle sensor devices in an off-axis configuration. 
         FIG. 35  shows an enlarged perspective view of the magnetic angle sensor arrangement of  FIG. 34  with the mold compound removed from two discrete magnetic angle sensor devices. 
         FIG. 36  shows a semi-transparent perspective view of a magnetic angle sensor arrangement according to an embodiment comprising two discrete magnetic angle sensor devices in an off-axis configuration. 
         FIG. 37  shows a solid perspective view of the magnetic angle sensor arrangement of  FIG. 36 . 
         FIG. 38  shows an enlarged view of the magnetic angle sensor arrangement as shown in  FIGS. 36 and 37  with the mold compounds removed from the discrete magnetic angle sensor devices. 
         FIG. 39  shows a diagram of different gradients as a function of a distance from a center plane for a magnet M 1 . 
         FIG. 40  shows a diagram of different gradients as a function of a distance from a center plane for a magnet M 2 . 
         FIG. 41  shows a diagram of different gradients as a function of a distance from a center plane for a magnet M 3 . 
         FIG. 42  shows a diagram illustrating an amplitude as a function of a distance along the z-direction at a radial distance of 4.5 mm. 
         FIG. 43  shows a diagram illustrating an amplitude as a function of a distance along the z-direction at a radial distance of 8 mm. 
         FIG. 44  shows a diagram of amplitudes of a gradiometer output as a function of the z-direction for a radial distance of 4.5 mm. 
         FIG. 45  shows a diagram of amplitudes of a gradiometer output as a function of the z-direction for a radial distance of 8 mm. 
         FIG. 46  shows a diagram of amplitudes of gradiometer outputs along the radial direction taken as differences at the radial distances of 6 mm and 4.5 mm as a function of a distance along the z-axis. 
         FIG. 47  shows a diagram of amplitudes of gradiometer outputs along the radial direction taken as differences at the radial distances of 8 mm and 6.5 mm as a function of a distance along the z-axis. 
         FIG. 48  shows a transparent perspective view of a further magnetic angle sensor arrangement comprising three SMD-type discrete magnetic angle sensor devices according to an embodiment in an off-axis configuration. 
         FIG. 49  shows a solid perspective view of the magnetic angle sensor arrangement of  FIG. 48 . 
         FIG. 50  shows a perspective view of a magnetic angle sensor arrangement of  FIGS. 48 and 49  without the magnet. 
         FIG. 51 a    shows an enlarged perspective view of the magnetic angle sensor arrangement shown in  FIGS. 48 to 50  with the mold compounds removed from the discrete magnetic angle sensor devices. 
         FIG. 51 b    shows a side view of the magnetic angle sensor arrangement shown in  FIGS. 48 to 50  and  FIG. 51   a.    
         FIG. 52  shows a diagram of gradients of the radial, the tangential and the axial magnetic field components with respect to the radial direction for magnet M 3 . 
         FIG. 53  shows a diagram of gradients of the radial, the tangential and the axial magnetic field components with respect to the tangential direction for magnet M 3 . 
         FIG. 54  shows a diagram of gradients of the radial, the tangential and the axial magnetic field components with respect to the axial direction for magnet M 3 . 
         FIG. 55  shows a diagram of gradients of the radial, the tangential and the axial magnetic field components with respect to the radial direction for magnet M 3  in the bore of the magnet. 
         FIG. 56  shows a diagram of gradients of the radial, the tangential and the axial magnetic field components with respect to the tangential direction for magnet M 3  in the bore of the magnet. 
         FIG. 57  shows a diagram of gradients of the radial, the tangential and the axial magnetic field components with respect to the axial direction for magnet M 3  for the bore of the magnet. 
         FIG. 58  shows the radial, the tangential and the axial component of the magnetic field as a function of a distance from a mid-plane of a magnet based on a FEM-simulation. 
         FIG. 59  shows a further result of a FEM-simulation. 
         FIG. 60  shows a plan view of a substrate comprising two gradiometers being part of a discrete magnetic angle sensor device according to an embodiment in an off-axis configuration. 
         FIG. 61  shows a plan view of a substrate comprising two gradiometers being part of a discrete magnetic angle sensor device according to an embodiment in an off-axis configuration. 
         FIG. 62 a    shows a plan view of a substrate comprising two gradiometers being part of a discrete magnetic angle sensor device according to an embodiment in an off-axis configuration. 
         FIG. 62 b    shows a plan view of a substrate comprising two gradiometers being part of a discrete magnetic angle sensor device according to an embodiment in an off-axis configuration. 
         FIG. 62 c    shows a plan view of a substrate comprising two gradiometers being part of a discrete magnetic angle sensor device according to an embodiment in an off-axis configuration. 
         FIG. 63  shows a perspective partial sectional view of a shaft, a diametrically magnetized magnet and a disc. 
         FIG. 64  shows a perspective view of a magnetic angle sensor arrangement according to an embodiment in an off-axis configuration comprising a ferrous disc. 
         FIG. 65  shows a further perspective view of a magnetic angle sensor arrangement of  FIG. 64 . 
         FIG. 66  shows a perspective partial sectional view of a magnetic angle sensor arrangement shown in  FIGS. 72 and 73 . 
         FIG. 67  shows a sideways partial sectional view of the magnetic angle sensor arrangement shown in  FIGS. 72 to 75 . 
         FIG. 68  shows a perspective partial sectional view of a magnet for a magnetic angle sensor arrangement according to an embodiment. 
         FIG. 69  shows a perspective view of a multi-part magnet. 
         FIG. 70  shows a diagram of magnetic field components along an x-direction and a y-direction in a mid-plane of the magnet shown in  FIG. 69  as a function of a rotation angle of the magnet. 
     
    
    
     DETAILED DESCRIPTION 
     In the following, embodiments according to the present invention will be described in more detail. In this context, summarizing reference signs will be used to describe several objects simultaneously or to describe common features, dimensions, characteristics, or the like of these objects. The summarizing reference signs are based on their individual reference signs. Moreover, objects appearing in several embodiments or several figures, but which are identical or at least similar in terms of at least some of their functions or structural features, will be denoted with the same or similar reference signs. To avoid unnecessary repetitions, parts of the description referring to such objects also relate to the corresponding objects of the different embodiments or the different figures, unless explicitly or—taking the context of the description and the figures into account—implicitly stated otherwise. Therefore, similar or related objects may be implemented with at least some identical or similar features, dimensions, and characteristics, but may be also implemented with differing properties. 
       FIG. 1  shows a schematic view of a discrete magnetic angle sensor device  100  according to an embodiment. The discrete magnetic angle sensor device  100  comprises a first magnetic field gradiometer  110 - 1 , which is sensitive to a gradient of a first magnetic field component  120 - 1  of a magnetic field. It further comprises a second magnetic field gradiometer  110 - 2 , which is sensitive to a gradient of a second magnetic field component  120 - 2 , which is different from the first magnetic field component  120 - 1 . The first and second magnetic field components  120 - 1 ,  120 - 2  are illustrated in  FIG. 1  by arrows indicating first and second directions, respectively, along which the magnetic field components  120  are taken. Since the magnetic field components  120  are different, also the corresponding directions and, hence, the corresponding errors are not parallel aligned or—to put it in other words—non-collinear. The discrete magnetic angle sensor  100  may, therefore, be referred to as a two-dimensional gradient sensor or sensor device  100 . 
     In other words, the two gradiometers  110  are of different types of a group of gradiometer types. The group of gradiometer types comprises a gradiometer of a first order, a gradiometer of a second order and a gradiometer of n-th order, wherein n is an integer larger than 2, each gradiometer being sensitive at least either to a gradient of a first magnetic field component of a magnetic field or to a gradient of a second magnetic field component of the magnetic field different from the first magnetic field component the gradiometers further comprising at least one sensor element of a group of sensor types, wherein the group of sensor types comprises a vertical Hall-sensor, a horizontal Hall-sensor, a giant magneto-impedance device and a magnetoresistive sensor element. In other words, a gradiometer of a certain type is at least defined by an order of the gradiometer, at least one magnetic field component, to which it is sensitive and a type of sensor element  180  it comprises. For example, two gradiometers  110  may be of different types, when the gradiometers  110  are responsive to two different magnetic field components  120 , are of different order or comprise different types of sensor elements (e.g. Hall sensor elements and anisotropic magnetoresistive sensor elements (AMR)). 
     As will be laid out in more detail below, a component of a vector is given by a direct product of the corresponding vector with a unity vector or direction vector having a length of 1 along the corresponding direction of the component. A gradiometer  110  of a first order may detect a difference of a (physical) quantity at two different locations. When these two different locations are located sufficiently close enough to one another, a gradiometer signal of a gradiometer  110  of the first order corresponds to a slope of the quantity along the direction of the two different locations between the two locations. It may be equal to the value of the slope at a midpoint (e.g. center) between the two locations multiplied by the distance between the two locations. The midpoint may be considered the place, where the gradiometer  110  essentially measures, for instance, a slope or a higher derivative of the quantity based on the order of the gradiometer. 
     Accordingly, a gradiometer signal of a gradiometer  110  of the second order is indicative of a difference of two gradiometer signals of gradiometers  110  of a first order, which are once again taken at a location between the midpoint of the two gradiometers  110 . Hence, a gradiometer signal of a gradiometer of a second order may be indicative of a second derivative of the physical quantity along the corresponding gradiometer direction  150 . This system continues for gradiometers  110  of higher orders, for instance, of a n-th order with n being an integer larger than 2. 
     Naturally, the physical quantity may be a component of a magnetic field, an angle of the magnetic field with reference to a reference direction or an angle of a projection of the magnetic field onto a plane in view of the reference direction. In other words, the angle may be indicative of the angular position of a magnet, a coil or—in more general terms—of magnetic source comprising, for instance, a coil and/or a magnet. 
     The discrete magnetic angle sensor  100  may optionally comprise a substrate  130  comprising a main surface  140  on which the first and second field gradiometers  110 - 1 ,  110 - 2  may be formed. However, alternatively or additionally, the first and second field gradiometers  110 - 1 ,  110 - 2  may also be formed within the substrate  130 . As a consequence, it may be possible to fabricate the discrete magnetic angle sensor device  100  according to an embodiment using thin film processing steps such as depositing layers of metallic, semiconducting or insulating materials, patterning the same, milling the pattern structures, doping or other process steps. Additionally or alternatively, the gradients of the first and second magnetic field components  120 - 1 ,  120 - 2  may be taken along a first and second gradiometer direction  150 - 1 ,  150 - 2 , respectively, by the first and second magnetic field gradiometer  110 - 1 ,  110 - 2 , which are arranged parallel to the main surface  140  of the substrate  130 . The gradiometer directions  150 - 1 ,  150 - 2  of the first and second magnetic field gradiometer  110 - 1 ,  110 - 2  may, for instance, coincide with an extension of the magnetic field gradiometers  110 - 1 ,  110 - 2 , respectively. 
     The substrate  130  may comprises a plate-shape comprising a thickness perpendicular to the main surface  140  smaller than extensions on the substrate  130  parallel to the main surface  140 . The substrate  130  may be, for instance, a semiconductor substrate. For instance, the substrate may be a silicon substrate (Si) or a silicon-on-insulator substrate (SOI). However, it may also be possible to use an insulating substrate depending on the application in mind. 
     As mentioned before, embodiments relate to a discrete magnetic angle sensor device  100 . Being a discrete device, the discrete magnetic angle sensor device  100  may be implemented as an electrical or electronic system, which is arranged in a package. For instance, the substrate  130 , which may be referred to as die or chip in embodiments, may be PEL plastic encapsulated to name just one example of an encapsulating material. In other words, the discrete magnetic angle sensor device  100  forms a single unit which can easily be handled and integrated into a more complex electronic circuit or mounted onto a printed circuit board or another carrier. Typically, dimensions of a discrete magnetic angle sensor device  100  according to an embodiment are at the most 50 mm, at the most 30 mm or at the most 20 mm, for instance, 4 mm. Naturally, a discrete magnetic angle sensor device  100  may comprise more than one substrate  130 . In other words, it may be implemented as a multichip module comprising two or more chip substrates  130  housed in a same package. The two or more substrates may be arranged parallel or orthogonally with respect to one another. For instance, one or more substrate  130  may be arranged in a vertical position while another substrate  130  may be arranged horizontally. Such an arrangement may, for instance, be used when the discrete magnetic angle sensor device  100  according to an embodiment may be used as a compass sensor in a cellular phone. It is to be understood that in embodiments, discrete devices such as the discrete magnetic angle sensor device described herein may have implemented logic circuit elements for example to provide compensation, self-calibration, signal processing for the sensor device. These functionalities may be on a same chip substrate as the sensing element or on a different substrate housed within a same module package. 
     A discrete sensor device may be, for instance, contained within or formed on a single substrate. It may also be distributed over several substrates with the substrates being arranged or contained in a single package. For instance, all parts of the sensor may be manufactured in a single process sequence, such as a semiconductor wafer process to fabricate the discrete sensor device  100 . Sometimes, parts of the sensors may be manufactured after a typical microelectronic wafer manufacturing process. For instance, magnetic flux concentrators may be glued to a top of a wafer or magneto-resistors may be sputtered on top of a wafer after the last interconnect layer has been manufactured. In order not to pollute the wafer fab, these parts may be done immediately after the ordinary wafer process, yet these processing steps may still be closely linked to the wafer fabrication, particularly, if a final passivation layer protecting circuit and sensor elements are applied afterwards. Another possible feature of a discrete sensor  100  may be that it has undergone a magnetic test, before it is assembled together with a rotatable magnet in the final angle sensor system. If such a test has been carried out, the individual parts that went through this test may be regarded as discrete sensors. For instance, the test may comprise a simplified test procedure allowing to verify if the discrete sensor devices work and if their performance is in their expected limits. In other words, the test may be used to see if an additional calibration may be unnecessary, advisable or perhaps even necessary. However, it may be interesting to try to avoid an additional calibration to avoid implementing an additional memory or other storage cells to store the calibration data. This may, for instance, be avoided by using a set of discrete sensor devices having similar properties and/or characteristics within a specified, application-specific margin. For instance, the discrete sensor devices may be coupled to the same power source and/or being fabricated from during the same process steps. 
     In other words, a discrete magnetic angle sensor device  100  according to an embodiment may be implemented as an angle sensor contained within a sensor package comprising two magnetic field gradiometers sensitive to two different magnetic field directions. An accuracy of the angle determination may be improved by more perfectly aligning the first and second gradiometers  110 - 1 ,  110 - 2 . This may be achievable cost-efficiently and accurately by implementing the gradiometers  110 - 1 ,  110 - 2  on the same substrate  130 , for instance, when they are manufactured simultaneously. For instance, the sensor elements may be placed within a predefined accuracy of, for instance, 10 μm or better. They may, additionally or alternatively, be placed in a single package. 
     The gradients or, to be more precise, the first gradiometer direction  150 - 1  and the second gradiometer direction  150 - 2  may be optionally arranged parallel to the main surface  140  on the substrate  130  or the chip surface. The same may also apply to the magnetic field components  120 - 1 ,  120 - 2  or their corresponding first and second directions. However, in the embodiment shown in  FIG. 1 , the first magnetic field component  120 - 1  and its first direction is perpendicular to the main surface  140  of the substrate  130  while the second magnetic field component  120 - 2  and its second direction is parallel oriented to the main surface  140 . As a consequence, in the embodiment shown in  FIG. 1 , the first magnetic field component  120 - 1  is perpendicular to the second magnetic field component  120 - 2 . This may also be implemented, when, for instance, one of the magnetic field components  120  is not oriented perpendicularly to the main surface  140  or even oriented parallel to the main surface  140 . By configuring the magnetic field gradients  110  accordingly, it may be possible to determine an angle of the magnetic field more accurately than in different configurations. 
     A magnetic field may be split in three orthogonal components, for example, in orthogonal components Bx, By and Bz along a x-direction, a y-direction and a z-direction of an orthogonal Cartesian coordinate system, or, for example, along a radial component BR, a tangential component Bpsi and a z-component Bz along orthogonal directions of a cylindrical coordinate system. Mathematically, the components may be derived by determining the direct product of the vector indicative of the magnetic field and a unity vector along the corresponding direction. 
     Moreover, the gradient, for instance, along the x-direction may be obtained by differentiating the dependency with respect to x (e.g. dBx/dx and dBpsi/dx). To put it in more general terms, the gradient of a (physical) quantity T along the directions {nx, ny, nz} are given by the inner product or direct product of the gradient of T (i.e. grad(T)=)={dT/dx,dT/dy,dT/dz}) and the directional vector {nx,ny,nz} (i.e. nx*dT/dx+ny*dT/dy+nz*dT/dz), where the directional vector {nx,ny,nz} is normalized to have a length l. The gradient may be approximated by a difference. For instance, grad(T) along the x-direction may be approximated by dT/dx=(T(x+dx)−T(x))/dx. As a consequence, it may be possible to determine or approximate a gradient along a direction by detecting or measuring to values of the corresponding quantity at two locations separated from one another by a certain distance, and by dividing the difference of the two measured values by the distance. 
     In terms of magnetic field components, a gradient may also be determined based on the Maxwell equations. For instance, based on the Gauss&#39;s law for magnetism (absence of magnetic monopoles) dBx/dx+dBy/dy+dBz/dz=0, by measuring magnetic field gradients dBx/dx and dBy/dy along a substrate  130  with the main surface  140  arranged parallel to the x-y-plane, the gradient dBz/dz may be calculated. Therefore, it may be possible to measure the gradient perpendicular to the main surface  140  of the substrate  130 . 
     Based on the further Maxwell equations, for instance, the Ampere&#39;s circuital law (magnetic B-field being curl-free) may allow to determine the gradients Bx and By perpendicular to the substrate  130  (along the z-direction) based on the equations dBx/dz=dBz/dx and dBy/dz=dBz/dy. As a consequence, a device or system may be capable of determining a gradient along a certain direction based on a more complex arrangement of two or more sensor elements  180 . 
     Optionally, the discrete magnetic angle sensor device  100  according to an embodiment may further comprise a sensor circuit which may be, for instance, formed in or on the die of the magnetic angle sensor device  100  and configured to generate a sensor signal SS indicative of the angle of the magnetic field based on a first gradiometer signal GS 1  of the first magnetic field gradiometer  110 - 1  and on a second gradiometer signal GS 2  of the second magnetic field gradiometer  110 - 2 . To facilitate this, the first gradiometer  110 - 1  may be configured to determine the gradient of a first magnetic field component of a magnetic field along the first direction and generate the first gradiometer signal GS 1  indicative of the gradient accordingly. Similarly, the second gradiometer  110 - 2  may also be configured to determine the gradient of the second magnetic field component  120 - 2  of the magnetic field along the second direction and to generate the second gradiometer signal GS 2  indicative of this gradient. To be able to receive the first and second gradiometer signals GS 1 , GS 2 , the sensor circuit  170  may be electrically coupled to the first and second magnetic field gradiometers  110 - 1 ,  110 - 2 , respectively. An electrical coupling may be facilitated by directly connecting the respective components with one another or by indirectly coupling the respective components with one another, for instance, by connecting in between one or more further components, electrical elements or circuits. 
     As will be laid out in more detail below, by implementing the sensor circuit  170 , the discrete magnetic angle sensor device  100  may be able to provide the sensor signal SS indicative of the rotation angle, of the angle of the magnetic field and/or of the angular position of the magnetic field source (e.g. comprising a magnet and/or a coil) more accurately, since by integrating the first and second magnetic field gradiometers  110 - 1 ,  110 - 2  into the discrete magnetic angle sensor device  100 , an evaluation of the angle may be performed by the sensor circuit  170 . In other words, a processing of the respective gradiometer signals GS 1 , GS 2  may be performed inside the discrete magnetic angle sensor device  100  making an arrangement comprising such a discrete magnetic angle sensor device  100  more robust against mismatch of magnetic sensitivities of the gradiometers  110 , misalignments and other mounting tolerances to name just a few examples. Due to using magnetic field gradiometers  110 , the discrete magnetic angle sensor device  100  provides an output signal (sensor signal SS) which is robust against or independent of homogeneous interfering magnetic fields. 
     The sensor circuit  170  may be in such a case optionally configured to generate the sensor signal SS based on a combination of the first and second gradiometer signals GS 1 , GS 2  such that a common dependency of the first and second gradiometer signals GS 1 , GS 2  is cancelled out, such as a common gain factor, to name just one example. Such a common factor or—in more general terms—a common dependency, may be for instance the result of different magnetic sensitivities of the magnetic field gradiometers  110 , which may create angle errors if these different magnetic sensitivities are not considered. However, by implementing the sensor circuit  170  such that such a common dependency is considered, different magnetic sensitivities in different discrete magnetic angle sensor devices  100  might not lead to an additional angle error in a system or arrangement comprising more than one discrete magnetic angle sensor device  100 . 
     In other words, by implementing the sensor circuit  170 , the gradiometer signals GS 1 , GS 2  which may be based on different magnetic sensitivities, are processed inside the discrete magnetic angle sensor device  100 . In other words, these sensitivities may be taken care of during a calibration process of the discrete magnetic angle sensor device  100 , which can be carried out on a device-specific basis rather than a system-wide or arrangement-wide calibration when compared to an implementation in which a processing of the signals is not carried out inside the discrete devices  100 . 
     The previously-mentioned combination of the first and second gradiometer signals GS 1 , GS 2  may be based on an arctan-determination of a quotient of information comprised in the first and second gradiometer signals GS 1 , GS 2 . Such an arctan-determination may, for instance, be implemented based on look-up tables, a CORDIC algorithm (CORDIC=Coordinate Rotation Digital Computer) or a similar implementation. 
     The first and second gradiometer signals GS 1 , GS 2  may, for instance, comprise a sine dependency or a sine-like dependency and a cosine dependency or a cosine-like dependency, respectively, versus a rotational position of the magnet or versus an azimuthal coordinate in a cylindrical reference frame concentric with the rotation axis or versus the angle of a magnetic field. Naturally, the same may also hold true vice-versa. A sine-like or cosine-like dependency on a physical quantity is a dependency with similar properties of a sine or a cosine dependency, respectively. However, deviations may, for instance, occur from an ideal sine or cosine dependency in terms of maximal values (amplitude), zero-crossings or the like with respect to each other and/or with respect to an ideal sine or cosine dependency. According to Fourier-series expansions, a sine-like or cosine-like dependence may be regarded as a sum of several sinusoidal terms, where the fundamental period is the basic sine or cosine, respectively, and all higher harmonics are deviations from this sine. These higher harmonics may have smaller amplitudes (typically ten times smaller) than the fundamental term and may be phase shifted against the fundamental term. 
     In other words, the discrete magnetic angle sensor device  100  is configured to generate the sensor signal SS based on the gradient of a first magnetic field component  120 - 1  with respect to the first gradiometer direction  150 - 1 . Moreover, the discrete magnetic angle sensor device  100  may also be configured to generate the sensor signal SS based on the gradient of a second magnetic field component  120 - 2  with respect to the second gradiometer direction  150 - 2 . However, in different embodiments, the second gradiometer direction  150 - 2  may be parallel or coincide with the first gradient direction  150 - 1 . In such a case, the discrete magnetic angle sensor device  100  may be configured to generate the sensor signal SS based on the gradient of the second magnetic field component  120 - 2  with respect to the first gradiometer direction  150 - 1 . In other words, the gradiometer directions  150 - 1 ,  150 - 2  may be identical, parallel or non-collinear depending on the concrete implementation of a discrete magnetic angle sensor device  100 . 
     The gradiometer directions  150  may be parallel or non-collinear with respect to the direction of the corresponding magnetic field component  120  to which the respective magnetic field gradiometer  110  is sensitive. In other words, as shown in  FIG. 1 , the respective gradiometer directions  150  may be non-collinear with the magnetic field components  120  of the magnetic field gradients  110 . To be even more precise, optionally, the magnetic field components  120  and the corresponding gradiometer direction  150  of the respective magnetic field gradiometer  110  may be perpendicular. However, in different embodiments of a discrete magnetic angle sensor device  100  the gradiometer direction  150  may be parallel or identical to a direction of a magnetic field component  120  of one or more magnetic field gradiometers  110 . 
     Basically, the magnetic field gradiometers  110 - 1 ,  110 - 2  may be independently of one another implemented as gradiometers of arbitrary orders. For instance, a first order gradiometer is capable of providing information concerning a slope of a physical quantity along the respective gradiometer direction  150 . A first order gradiometer therefore typically comprises at least two sensor elements capable of detecting the respective physical quantity at two different locations. In comparison, a second order gradiometer may also be able to provide information concerning a curvature along the respective gradiometer direction. Such a gradiometer typically comprises at least three corresponding sensor elements at three different locations. Higher order gradiometers may comprise additional sensor elements and, as a consequence, may be able to provide additional information on the spatial dependency of the respective physical quantity. For instance, a N-order gradiometer may comprise typically at least (N+1) sensor elements. 
     Although embodiments of a discrete magnetic angle sensor device  100  may utilize gradiometers independently of one another of any order, by implementing at least one of the first magnetic field gradiometer  110 - 1  and the second magnetic field gradiometer  110 - 2  as a first order gradiometer, implementational complexity and evaluation of the respective gradiometer signals GS 1 , GS 2 , respectively, may be simplified without sacrificing the achievable accuracy in determining the angle significantly. For instance, both, the first and second magnetic field gradiometers  110 - 1 ,  110 - 2  may be implemented as first order gradiometers. 
     Naturally, a discrete magnetic angle sensor device  100  may comprise further magnetic field sensor elements comprised in or associated with the first and/or the second gradiometer  110 - 1 ,  110 - 2 . It may also comprise one or more additional gradiometers  110 . Of course, the roles of the first and second gradiometers  110 - 1 ,  110 - 2  may be exchanged in other embodiments. 
     However, other discrete magnetic angle sensor devices  100  according to an embodiment may comprise exactly two gradiometers  110  or, to be more precise, the first magnetic field gradiometer  110 - 1  and the second magnetic field gradiometer  110 - 2 . These gradiometers may, however, be implemented as first order gradiometers or fully or partially gradiometers of a higher order. 
     Optionally, in a discrete magnetic angle sensor device  100  according to an embodiment, the first magnetic field gradiometer  110 - 1  may comprise a first magnetic sensor element  180 - 1  and a second magnetic sensor element  180 - 2 , which are of the same sensor type of a group of sensor types. Accordingly, the second magnetic field gradiometer  110 - 2  may comprise a third magnetic sensor element  180 - 3  and a fourth magnetic sensor element  180 - 4 , which are of the same sensor type of the group of sensor types. The group of sensor types comprises a vertical Hall-sensor, a horizontal Hall-sensor, a giant magneto-impedance device (GMI), a magnetic field effect transistor (MAG-FET) and a magneto-resistive sensor element (xMR), such as an anisotropic magneto-resistance sensor element (AMR), a giant magneto-resistance sensor element (GMR), an extraordinary magneto-resistance sensor element (EMR) and a tunneling magneto-resistive sensor element (TMR). In other words, each of the gradiometers  110 - 1 ,  110 - 2  may comprise at least two sensor elements of the same sensor type. However, the sensor elements  180  of different gradiometers  110  may differ from one another. For instance, the first and second sensor elements  180 - 1 ,  180 - 2  may be implemented as lateral or horizontal Hall-sensor elements, while the third and fourth sensor element  180 - 3 ,  180 - 4  may be implemented as vertical Hall-sensor elements. A lateral Hall sensor or lateral Hall sensor element  180  is typically responsive to a magnetic field component perpendicular to a main surface  140  of the die or substrate  130 . By choosing the appropriate sensor elements, it may be possible to configure the discrete magnetic angle sensor device  100  such that it may be designed to be sensitive to specific magnetic field components  120 . In some cases it may also be favorable if several gradiometers share common elements, for instance, if two sensor elements are arranged along a first direction to constitute the first gradiometer  110 - 1  and two sensor elements of the same type are arranged in a second perpendicular direction to constitute the second gradiometer  110 - 2  one may arrange the sensor elements  180  in the ends and a corner of an L-shape, whereby in the corner two sensor elements would result, and they could be replaced by a single one that is part of both gradiometers  110 . Optionally, the first or second gradiometers  110  may comprise at least one sensor element  180  responsive or sensitive to an in-plane magnetic field component  120  with respect to the main surface  140  of the substrate  130 . 
     The first and second magnetic sensor elements  180 - 1 ,  180 - 2  may be optionally arranged along the first gradiometer direction  150 - 1 . Accordingly, the third and fourth magnetic sensor elements  180 - 3 ,  180 - 4  may be arranged along the first gradiometer direction  150 - 1  or along the second gradiometer direction depending on whether the first and second gradiometer directions  150 - 1 ,  150 - 2  are parallel or identical or non-collinear. However, to simplify the description of embodiments, sometimes the second gradiometer direction  150 - 2  also may be parallel or identical to the first gradiometer direction  150 - 1 . 
     In the embodiment shown in  FIG. 1 , the third and fourth magnetic sensor elements  180 - 3 ,  180 - 4  are arranged along the second gradiometer direction  150 - 2  perpendicular to the first gradiometer direction  150 - 1 . 
     It should be noted that the first and second gradiometers  110  may share one or more sensor elements  180 . 
       FIG. 2  shows a flowchart of a method for providing a sensor signal SS according to an embodiment. After optionally providing a magnetic field in a sub-process S 100 , the method for providing the sensor signal SS comprises in a sub-process S 110  generating the first gradiometer signal GS 1  indicative of the gradient of the first magnetic field component  120 - 1  of the magnetic field with respect to the first gradiometer direction  150 - 1  using the first gradiometer  110 - 1 . The method further comprises in a sub-process  120  generating the second gradiometer signal GS 2  indicative of the gradient of the second magnetic field component  120 - 2  of the magnetic field different from the first magnetic field component  120 - 1  with respect to the first gradiometer direction  150 - 1  or with respect to the second gradiometer direction  150 - 2  different from the first gradiometer direction  150 - 1  using the second gradiometer  110 - 2  depending on the implementation of the gradiometers  110 - 1 ,  110 - 2 . The first and second gradiometers  110 - 1 ,  110 - 2  are comprised in a discrete magnetic angle sensor device  100 . 
     The method further comprises in a sub-process S 130  generating the sensor signal SS indicative of the angle of the magnetic field based on the first and second gradiometer signals GS 1 , GS 2 . 
     However, although the flowchart of  FIG. 2  shows a sequence of the sub-processes S 100 , S 110 , S 120  and S 130 , the individual sub-processes may be carried out simultaneously, timely overlapping or in a different order than depicted in  FIG. 2 . For instance, generating the first and second gradiometer signals GS 1 , GS 2  in the sub-processes S 110  and S 120 , respectively, may be carried out simultaneously or quasi-simultaneously. However, since generating the sensor signal SS in sub-process S 130 , is based on the first and second gradiometer signals GS 1 , GS 2 , it may be advisable, however, not under all circumstances necessary, to carry out sub-process S 130  after sub-processes S 110  and S 120 . 
       FIG. 3  shows a schematic diagram of a magnetic angle sensor arrangement  200  according to an embodiment. The magnetic angle sensor arrangement  200  comprises at least one discrete magnetic angle sensor device  100  according to an embodiment as outlined before. In other words, the discrete magnetic angle sensor device  100  comprises a first magnetic field gradiometer  110 - 1 , sensitive to a gradient of a first magnetic field component  120 - 1  of a magnetic field, a second magnetic field gradiometer  110 - 2  sensitive to a gradient of a second magnetic field component  120 - 2  of the magnetic field different from the first magnetic field component  120 - 1 , and a sensor circuit  170  configured to generate a sensor signal indicative of an angle of the magnetic field based on a first gradiometer signal GS 1  of a first magnetic field gradiometer  110 - 1  and on a second gradiometer signal GS 2  of the second magnetic field gradiometer  110 - 2 . 
     The at least one discrete magnetic angle sensor device  100  is fixedly arranged with respect to a rotation axis  210  around which a magnet  220  is rotatably mountable such that the at least one discrete magnetic angle sensor device  100  is arranged off-axis with respect to the rotation axis  210 . In other words, the at least one discrete magnetic angle sensor device  100  is radially displaced with respect to the magnet  200  when mounted. 
     As outlined before, a discrete magnetic angle sensor device  100  may, for instance, be used in an off-axis configuration of the discrete magnetic angle sensor device  100  with respect to the magnet  200 . Due to the implementation of the first and second gradiometers  110 - 1 ,  110 - 2 , different magnetic field components  120 - 1 ,  120 - 2  may improve an accuracy of an angle determination in such an off-axis configuration. However, as outlined before, discrete magnetic angle sensor devices  100  may also be used in different applications, such as an on-axis configuration, in which the discrete magnetic angle sensor device according to an embodiment is aligned along the rotation axis  210 . 
     Optionally, the magnetic angle sensor arrangement according to an embodiment may comprise more than one discrete magnetic angle sensor devices  100 . In other words, it may comprise a plurality of discrete magnetic angle sensor devices  100  arranged according to a predefined pattern around the rotation axis  210 . In the schematic diagram of  FIG. 3 , the magnetic angle sensor arrangement  200  comprises a first discrete magnetic angle sensor device  100 - 1  and a second discrete magnetic angle sensor device  100 - 2 . 
     To be even more precise, in the embodiment shown in  FIG. 3 , the magnetic angle sensor arrangement  200  according to an embodiment comprises two discrete magnetic sensor devices  100  which are diametrically arranged at an angle of 180° with respect to each other around the rotation axis  210  such that—under ideal conditions—a connecting line  230  connecting a common central point of the first and second gradiometers  110 - 1 ,  110 - 2  of the discrete magnetic angle sensor devices  100  with one another intersects the rotation axis  210 . Using this predefined pattern may be able to compensate at least partially an eccentricity of the magnet  220  with respect to its rotation axis  210 . In other words, this configuration may further improve an accuracy of the angle determination. 
     To put it in more general terms, whenever a plurality of discrete magnetic angle sensor devices  100  is implemented, optionally, the discrete magnetic angle sensor devices  100  may be arranged equidistantly around the rotation axis  210  in terms of an angle around or perpendicular to the rotation axis  210 .  FIG. 3  shows this arrangement in the case of two discrete magnetic angle sensor devices  100 . By arranging the discrete magnetic angle sensor devices  100  equidistantly around the rotation axis  210 , it may be possible to improve an accuracy of an angle determined by the magnetic angle sensor arrangement  200  to compensate an eccentric mounting of the magnet  220  with respect to the rotation axis  210 . In other words, when the magnetic angle sensor arrangement  200  comprises n discrete magnetic angle sensor devices  110 - 1 , . . . ,  110 - n , it may be advisable to arrange the discrete magnetic angle sensor devices  100  such that two neighboring discrete magnetic angle sensor devices  100  comprise an angle between them of 360°/n. In yet other words, an equidistant or regular azimuthal arrangement may be advisable. Or to put it in yet other words, optionally, the arrangement  200  may comprise a n-fold rotational symmetry with respect to the rotation axis  210  with n being an integer. 
     Optionally, the discrete magnetic angle sensor devices  100  may further be arranged at an essentially constant radial distance from and around the rotation axis  210 , around which the magnet is rotatable when mounted. The radial arrangement of the discrete magnetic angle sensor devices  100  with respect to the rotation axis  210  as shown, for instance, in  FIG. 3 , may also beneficially influence an accuracy of the angle determination. Due to the same radial distance from the rotation axis  210  in the case of an ideally-mounted magnet  220  all the discrete magnetic angle sensor devices  100  will be subjected to the same gradient of the magnetic field with respect to the first and second magnetic field components  120 - 1 ,  120 - 2  when the magnetic  220  rotates around the rotation axis  210 . As a consequence, the discrete magnetic angle sensor devices  100  may sense the same or a similar amplitude of the gradient of the magnetic field components  120 . As a consequence, similar signal strengths may be achievable by all the discrete magnetic angle sensor devices  100 . A ratio of the amplitudes of the sinusoidal gradiometer signals GS in a sensor package may depend on a radial position. It may be advisable to arrange the sensor devices  100  such that the ratios of all satellites (devices  100 ) may be identical or similar. This may result in a more accurate average or similar determination of an over-all angle, as will be laid out in more detail below, since it may be possible to more evenly cancel out error when calculating an angle signal AS indicative of the angle. 
     The magnetic angle sensor arrangement according to an embodiment may additionally or alternatively comprise a control circuit  240 , which is configured to receive the sensor signals SS 1 , SS 2  from the plurality of discrete magnetic angle sensor devices  100 . It is further configured to generate an angle signal AS indicative of the angle of the magnetic field provided or generated by the magnet  220  based on the sensor signals SS 1 , SS 2 , received from the plurality of discrete magnetic angle sensor devices  100 . 
     To enable this, the control circuit  240  may be coupled to the discrete magnetic angle sensor devices  100  in such a way that the control circuit  240  is capable of receiving an information-comprising signal from the discrete magnetic angle sensor devices  100 . This may be fully or partially comprised transmitting the respective signals, such as the sensor signals SS 1 , SS 2 , electrically, optically, inductively or by a radio-based transmission. Once again, coupling the discrete magnetic angle sensor devices  100  with the control circuit  240  directly or indirectly via one or more additional components or circuits can be effected. Such additional components or circuits may, for instance, comprise a receiver, a transmitter, an amplifier or a signal-influencing circuit. Naturally, although  FIG. 3  shows the control circuit  240  using a point-to-point contact, also other communication networks such as a data bus or a daisy-chain configuration may be used to send, receive or interchange signal, to name just a few examples. 
     To illustrate this, the gradiometer signals GS 1  and GS 2  may be considered to be two independent coordinates. A pair of values (GS 1 , GS 2 ) may be associated to a point in a GS 1 -GS 2 -plane. A position of this point, its distance from the origin of the GS 1 /GS 2 -coordinate axes, as well as an angle with respect to a reference direction (e.g. GS 1 -axis) may be determined. The distance may be of no or lesser importance compared to the angle of the point, which may be transmitted to the controller circuit  240 . However, this may be more complex, since it may comprise determining the arctan of the ratio. Transmitting the ratio of the two values GS 1  and GS 2  may be more feasible. However, this may be more problematic, when the denominator becomes (approximately) zero. Moreover, information might be lost, since, for instance, by mirroring the point (GS 1 , GS 2 ) with respect to both coordinate axes leading to point (−GS 1 , −GS 2 ) still leads to the same ratio, since the sign cancels out. However, it may be possible to transmit a quotient of the minimum of the absolute values of GS 1  and GS 2  divided by the maximum of the absolute values of GS 1  and GS 2  (min (abs(GS 1 ),abs(GS 2 ))/max (abs(GS 1 ),abs(GS 2 ))). In a two-dimensional plane only 8 different points per radius value from the origin exist with the same value indicated above. Hence, it may be sufficient to additionally transmit a piece of information concerning the 45°-spanning sub-quadrant comprising the respective (GS 1 ,GS 2 )-point. For that, only three additional bits may be necessary. 
     The sensor signals SS 1 , SS 2  of the discrete magnetic angle sensor devices  100  may comprise information on a ratio of values comprised in the first and second gradiometer signals GS 1 , GS 2  and on a quadrant of the angle of the magnetic field and on a quadrant or sub-quadrant of these gradiometer signals GS. However, in other embodiments it may be sufficient to use information on a sign of the values comprised in both gradiometer signals GS 1  and GS 2  of a single or some sensor devices  100 . However, when, for instance, the sign-related information may be distorted by an offset error or other influences, for instance, during or in the vicinity of a zero-crossing, it may be advisable to obtain and use information on the sign of at least a further sensor device  100 . 
     If the system is to be robust against a random failure of any sensor device  100 , it may be advisable to implement more than one satellite providing sign-related information. However, when, for instance, an angle range to be detected is sufficiently small, it may be possible to implement the arrangement without using sign-related information. 
     By implementing the discrete magnetic angle sensor devices  100 , accordingly, it may be eventually possible not to implement the angle determination, for instance, not to implement the look-up table for the arctan-computation or the previously-mentioned CORDIC algorithm. In such a case, the discrete magnetic angle sensor devices  100 , which are also referred to as satellites will only provide the ratios of a gradiometer signal along with the respective information on the quadrant or sub-quadrant of the angle of the magnetic field while the master or control circuit  240  combines these signals, for instance, using a linear combination, wherein the coefficients of this linear combination may be functions of the information regarding the quadrant or sub-quadrant of the angle. Afterwards, the control circuit  240  may implement the arctan-function or an equivalent look-up table as outlined before, to generate the angle signal AS. 
     Instead or additionally implementing the control circuit  240 , it may also be possible to implement at least one of the discrete magnetic angle sensor devices  100  in such a way that this at least one discrete magnetic angle sensor device  100  is configured to receive the sensor signals SS from the other discrete magnetic angle sensor devices of the plurality of discrete magnetic angle sensor devices  100 . This at least one discrete magnetic angle sensor device may then also be configured to generate the angle signal AS indicative of the angle of the magnetic field of the magnet  220  based on the sensor signals SS received from the other discrete magnetic angle sensor devices  100  and based on the sensor signal and based on its own determined gradients of the first and second magnetic field components  120 - 1 ,  120 - 2 . This may, for instance, be based on the first and second gradiometer signals GS 1 , GS 2 , or the sensor signal SS of said discrete magnetic angle sensor device  100 . 
     While implementing a separate control circuit  240  may eventually provide the opportunity of simplifying the individual discrete magnetic angle sensor devices  100  according to an embodiment, using at least one discrete magnetic angle sensor device  100  capable of performing the previously-described generation of the angle signal AS based on the sensor signals of the other discrete magnetic angle sensor devices  100  may reduce a number of electrical connections and simplify wiring of the magnetic angle sensor arrangement  200 . Hence, depending on the circumstances and boundary conditions, it may or may not be advisable to implement a separate control circuit  240 . Or to put it in other words, the control circuit  240  may be comprised in one or more of the discrete magnetic angle sensor devices  100 . 
     Optionally, the control circuit  240  or at least one of the discrete magnetic angle sensor devices  100  depending on the implementation chosen, may further be configured to generate a triggering signal TS. The discrete magnetic angle sensor devices of a plurality of discrete magnetic angle sensor devices may further be configured to generate their respective sensor signals SS indicative of synchronous measurements by the discrete magnetic angle sensor devices  100  based on the trigger signal TS. However, the individual sensor signals SS may be transmitted sequentially and asynchronously. The triggering signal TS causes the devices  100  (satellites) to start, carry out or to provide synchronous measurements of the devices  100 . This may further improve an accuracy of the angle determination since by providing sensor signals indicative of synchronous measurements errors or inaccuracies due to deviations in terms of points of time, when the measurements are taken, may be reduced. 
     In other words, in this case, one of the discrete magnetic angle sensor devices  100  will be the master controlling the other discrete magnetic angle sensor devices  100 , which are then considered the slaves or satellites. 
     The discrete magnetic angle sensor devices  100  may be configured to generate their sensor signals SS periodically, intermittently or, for instance, when a predefined condition is fulfilled. Any parameter or the condition may eventually be programmable or changeable. 
     In the embodiment shown in  FIG. 3 , the control circuit  240  is furthermore coupled to the discrete magnetic angle sensor devices  100  to provide the triggering signal TS to the respective discrete magnetic angle sensor devices  100 . Upon receiving the triggering signals TS, the discrete magnetic angle sensor devices  100  may perform the respective determination of the gradients of the first and second magnetic field components  120 - 1 ,  120 - 2  by, for instance, sampling the respective gradiometer signals GS 1 , GS 2  or by another similar technical implementation. After receiving the triggering signal TS, the sensor signals SS corresponding to the previously-mentioned synchronous measurements may then be transmitted to the control circuit  240  or the discrete magnetic angle sensor device  100  operating as the master. However, the measurements may be made in the past, for instance, 10 μs or 50 μs before. In yet other words, the triggering signal TS may provide the opportunity of making the satellites (the discrete magnetic angle sensor devices  100 ) to carry out the measurements preferably at the same moment of time. Under ideal circumstances, the triggering signal TS starts the data acquisition in all or at least some of the devices  100 . However, this may be difficult to achieve, since the devices  100  may operate asynchronously, unless they may be provided with a common clock signal. Each data acquisition may take a period of time. Depending on the moment with respect to the period for the data acquisition, the sensor  100  may provide the previously determined value, the present value or any combination of these and optionally of earlier data. Such a combination may comprise an interpolation or an extrapolation to a point of time being identical to the time the trigger signal TS is received or shifted by a defined timespan against this time. 
     Depending on the implementation, the control circuit  240  or the discrete magnetic angle sensor device  100  acting as the master may be configured to generate the angle signal AS taking the predefined pattern of the discrete magnetic angle sensor devices  100  into account. This may be, for instance, done based on averaging the angles comprised in the sensor signals of the discrete magnetic angle sensor devices  100  after subtracting the positions of the discrete magnetic angle sensor devices from the angles comprised in the sensor signals according to the predefined patter. In other words, the angle signal AS may be indicative of an angle which is given by an average of a difference between an angle ψi, of which the respective sensor signal of the i-th discrete magnetic angle sensor device  100 - i  is indicative of, and an angle position ψi 0 , under which the respective discrete magnetic angle sensor device  100 - i  is arranged with respect, for instance, to a reference direction. The averaging may be done, for instance, based on an arithmetic averaging, a weighted arithmetic averaging or another plain or weighted averaging algorithm. 
     However, in this context, it is important to note that the previously-described averaging and subtracting do not have to be implemented as indicated above. It is merely sufficient that the control circuit  240  or the discrete magnetic angle sensor device  100  acting as the master is capable of performing a calculation or determination based on this or a similar equation this optional feature is implemented. For instance, the averaging may comprise manipulating analog voltages or currents manipulating digital values simply indicative of the respective angles. For instance, these calculations may be carried out using a linear combination or another more easily implemented function based on the respective sensor signals themselves, without calculating the explicit angles of which the sensor signals are indicative. However, in other embodiments, the previously-described averaging may, for instance, be carried out in terms of the actual angle values. 
     Optionally, the discrete magnetic angle sensor devices  100  may be configured to generate the sensor signal based on the gradient of the first magnetic field component  120 - 1  with respect to a first gradiometer direction  150 - 1  of a discrete magnetic angle sensor device  100 . The discrete magnetic angle sensor devices  100  may further be configured to generate the sensor signal based on the gradient of the second magnetic field component  120 - 2  with respect to the first gradiometer direction  150 - 1  of the discrete magnetic angle sensor device  100  or with respect to a second gradiometer direction  150 - 2  of the discrete magnetic angle sensor device different from the first gradiometer direction  150 - 1 . The discrete magnetic angle sensor devices  100  may in such a case be arranged such that all first magnetic field components  120 - 1  are of the same type of magnetic field components and all second magnetic field components  120 - 2  are also of the same type of magnetic field component. The group of magnetic field components comprises an axial component with respect to the rotation axis  210  and a position of a respective discrete magnetic angle sensor device  100 , a radial component with respect to the rotation axis  210  and the position of the respective discrete magnetic angle sensor device  100 , and a tangential component with respect to the rotation axis  210  and the position of the respective discrete magnetic angle sensor device  100 . Moreover, all first gradiometer directions  150 - 1  may be of a same type of direction and all second gradiometer directions  150 - 2  may also be of a same type of direction. Similar to the group of magnetic field components, the group of types of directions comprises an axial direction with respect to the rotation axis  210  and the position of the respective discrete magnetic angle sensor device  100 , a radial direction with respect to the rotation axis  210  and the position of the respective discrete magnetic angle sensor device  100 , and a tangential direction with respect to the rotation axis  210  and the position of the respective discrete magnetic angle sensor device  100 . Hence, all first types of gradiometers  110  may be sensitive to the same cylindrical components of the magnetic field (either radial, azimuthal, or axial) and the gradients of all gradiometers  110  may be along the same cylindrical directions (either radial, azimuthal, or axial), whereby the circular cylindrical reference frame is concentric to the rotation axis. 
     The magnetic angle sensor arrangement  200  may optionally comprise the rotatably mounted magnet  220 . The magnet  220  may comprise an essentially diametrical magnetization  250  oriented such that the magnetization  250  is essentially oriented perpendicularly to the rotation axis  210 . Optionally, it may further be advisable for the magnet  220  to comprise along with optional ferromagnetic parts in its vicinity an essentially rotational symmetrical shape, as will be outlined below in more detail. 
     In other words, the magnet  220  as shown in  FIG. 3  does not comprise a radial magnetization, but a magnetization which is essentially oriented perpendicular to the rotation axis  210  being essentially constant in terms of its direction throughout the whole magnet  220 . Such a magnet  220  may be especially beneficial for using in an off-axis configuration as shown in  FIG. 3 . 
     The first gradiometer  110 - 1  of the discrete magnetic angle sensor devices  100  may comprise a first lateral Hall sensor element and a second lateral Hall sensor element as the first and second magnetic sensor elements  180 - 1 ,  180 - 2 , respectively. Accordingly, the second gradiometer  110 - 2  may comprise a first vertical Hall sensor element and a second vertical Hall sensor element as the third and fourth magnetic sensor elements  130 - 3 ,  130 - 4 , respectively. In this case, the at least one discrete magnetic angle sensor device  100  may be arranged such that an amplitude of the gradient of the first magnetic field component  120 - 1  is smaller than an amplitude of the gradient of the second magnetic field component  120 - 2  with respect to a maximum specified rotation angle range of the magnet  220 . The maximum specified rotation angle range may, for instance, comprise a full rotation (i.e. 360°), for instance for an application in which the magnet  220  can rotate freely around the rotation axis  210 , or it may be an angle range smaller than 360°, for instance for applications in which the angle is to be determined only with respect to a smaller maximum specified rotation angle range. Examples may, for instance, come from the field of windscreen wipers in the field of automotive applications. 
     In other words, the discrete magnetic angle sensor devices  100  may be arranged with respect to the magnet  220  such that the magnetic field gradient having the larger amplitude will be monitored or detected by the vertical Hall sensor elements, while the lateral Hall sensor elements are arranged such that these sensor elements detect the weaker gradient. Only in very rare cases, both orthogonal gradients along the first and second directions (first magnetic field component  120 - 1  and second magnetic field component  120 - 2 ) are essentially equally strong. In other words, in most applications, always one gradient is significantly stronger than the other one. In such a case, it may be advisable to use the Hall sensor elements having a lower off-set error—in other words, the lateral Hall sensor elements—for the weaker gradient and the Hall elements with the larger off-set error—in other words, the vertical Hall sensor elements—for the stronger gradients to improve the overall accuracy compared to the opposite arrangement. Naturally, this may also be true for other sensor elements  180 . Generally speaking, it may be advisable to use sensor elements  180  with a higher accuracy for a gradiometer  110  having the lower amplitude of the two gradiometers  110 . The accuracy of a sensor element may be predominantly influenced by a hysteresis, an off-set error, a gain error, noise or crosstalk. The amplitudes of the gradiometer signals GS 1 , GS 2  may be the amplitude values of the sine- or cosine-like gradients of the magnetic field component(s) in question. 
     An embodiment of an arrangement  200  may comprise more than one device  100 . It may, for instance, comprise N discrete magnetic sensor devices  100  with N being an integer larger than 1. The devices  100  may be arranged at a constant radial distance with respect to the rotation axis  210  having angle differences of 360°/N between neighboring devices  100 . The devices  100  may be attached or arranged on a stator, where a diametrically magnetized magnet  220  may be mounted rotatably against the stator. The magnet  220  may comprise a geometry with a M-fold rotational symmetry with respect to the rotation axis  210 , where M is an integer, which may be larger than or equal to N. The rotor may be pivotable with respect to the stator. The arrangement may be configured to generate the angle signal AS and/or to determine an angle of the magnet  220  relative to the stator by combining the sensor signals SS of the discrete magnetic sensor devices  100 . An object may comprise an M-fold rotational symmetry, when an axis exists such that the object may be mapped in a mathematical sense by rotating same around the axis by 360°/M. The axis may coincide with the rotation axis  210 . Naturally, the magnet may comprise exactly one part, but also more than one part. 
       FIG. 4  shows a flowchart of a method for generating an angle signal AS. The method comprises in a sub-process S 200  providing a magnetic field. In a sub-process S 210 , the method comprises generating a plurality of sensor signals SS, each sensor signal SS being generated by an individual discrete magnetic angle sensor device according to an embodiment. Accordingly, the sensor signals SS are indicative of an angle of the magnetic field at a position of the discrete magnetic angle sensor device  100  based on the gradient of the first magnetic field component  120 - 1  of the magnetic field with respect to the first gradiometer direction  150 - 1 . It is further based on the gradient of the second magnetic field component  120 - 2  of the magnetic field different from the first magnetic field component  120 - 1  with respect to the first gradiometer direction  150 - 1  or the second gradiometer direction  150 - 2  different from the first gradiometer  150 - 1  depending on the design and implementation of the respective discrete magnetic angle sensor devices  100 . In a further sub-process S 220 , the method further comprises generating the angle signal AS indicative of the angle of the magnetic field based on the plurality of sensor signals SS. 
     As indicated above and outlined in more detail below, by employing a plurality of discrete magnetic angle sensor devices  100  it may be possible to increase an accuracy of the angle determination. This may, for instance, be beneficial in the case of an off-axis configuration. For instance, providing the magnetic field may optionally comprise providing the magnetic field by a rotatably mounted magnet  220 , wherein the magnet  220  comprises an essentially diametrical magnetization  250  oriented such that the magnetization is essentially oriented perpendicular to a rotation axis  210 , around which the magnet  220  is rotatable. In such a case, it may be advisable to implement the magnet  220  along with ferromagnetic parts in its closer vicinity to be essentially rotationally symmetric or having at least M-fold symmetry with M being an integer larger than 1. 
     As outlined before, the method may optionally also or alternatively comprise in a sub-process S 230  generating a triggering signal TS. In such a case it may be advisable that generating the plurality of sensor signals SS in sub-process S 210  comprising generating each sensor signal SS indicative of a synchronous measurement based on the triggering signal. As a consequence, it may once again be possible to increase an accuracy of the angle determination. 
     However, generating and using the triggering signal TS is optional. The sensor devices  100  may periodically generate the sensor signals SS indicative of their most recent measurements or at least based on the most recent measurements. The sensor signals SS may comprise a time index, and the controller circuit  240  or the device  100  acting as the master may use the data based on the time indices, for instance, to extrapolate or interpolate the data. 
     Naturally, also the sub-processes described in the context of  FIG. 4  are not required to be executed in the order as indicated by  FIG. 4 . The individual sub-processes may be executed simultaneously, timely overlapping, in a different order or any combination thereof. 
     In the following, further embodiments will be described in more detail focusing on both the theoretical background but also on technical details. For instance, in many of the following embodiments, vertical Hall sensor elements, which are also referred to as vertical Hall elements or vertical Hall sensors, and lateral Hall sensor elements, which are also referred to as horizontal or lateral Hall sensors or Hall plates, will be described. However, these magnetic sensor elements may, for instance, be exchanged or supplemented by other types of magnetic sensor elements  180  such as the previously mentioned magneto-resistive sensor elements (xMR) or giant magneto-impedance sensor elements (GMI). Therefore, whenever a Hall sensor element is described, it may be possible to replace this Hall sensor element by another type of magnetic sensor element  180  as described and mentioned before. The same may also apply to other components such as the sensor circuit  170 , which may, for instance, be implemented by a processor or another circuit. Therefore, it is to be noted that the specific details of the following embodiments may equally well be implemented differently. 
       FIG. 5  shows a schematic representation of a magnetic angle sensor arrangement  200  according to an embodiment. As outlined earlier, embodiments relate to magnetic angle sensors  100 , which may also be referred to as a two-dimensional gradient sensor, and arrangements as, for instance, shown in  FIG. 5 . A magnet  220  is mounted, for instance, on a through-shaft  260  and sensor elements  180  arranged as gradiometers  110  are placed aside or off a rotation axis  210 . The gradiometers  110  are arranged on or inside a substrate  130  of the magnetic angle sensors  100 , which may be implemented as a silicon die. For the sake of simplicity, the rotation axis  210  is identical to a z-axis of a corresponding coordinate system. 
     Such systems may be used when the end of the shaft  210  is not available for attachment of the magnet  220  or when there is no space available to place the sensor  100  on the axis  210  of rotation.  FIG. 5  shows the sensor die  130  arranged perpendicular to the rotation axis  210  or, in other words, with an orthogonal orientation of sensor die  130 . In other words,  FIG. 5  shows an off-axis angle sensor  100  with through-shaft magnet  220 . The magnet  220  comprises a diametrical magnetization  250 , which is oriented in a plane perpendicular to the z-axis. In the situation shown in  FIG. 5 , the magnetization  250  is oriented along a y-axis, which along with a x-axis and the z-axis forms a orthogonal coordinate system. 
       FIG. 6  shows a schematic representation of a further magnetic angle sensor arrangement  200  according to an embodiment, in which the sensor die  130  is aligned parallel to the rotation axis  210 . A main surface  140  of the silicon die  130  is, hence, arranged parallel to the rotation axis  210  (axial orientation of sensor die  130 ), while in the arrangement  200  shown in  FIG. 5 , the main surface  140  of the silicon die  130  was arranged orthogonally to the rotation axis  210 . The arrangement  20  of  FIG. 6  comprises an off-axis angle sensor  100  according to an embodiment with diametrically magnetized through-shaft magnet  220 . 
     In principle, there are infinitely many such axial orientations of the sensor die  130 . One edge of the sensor die  130  may be kept parallel to the rotation axis  210  and the die  130  may be turned around this edge. In particular, two such directions comprise the cases of the main surface  140  of the die  130  being arranged tangential to a cylindrical surface, where the axis of the cylinder coincides with the rotation axis  210 , and the main surface  140  of the die  130  being arranged perpendicular to this cylindrical surface. 
     Naturally, the magnet  220  may also be mounted on the end of the shaft  260  as shown in  FIG. 7 . The orientation and the number of the sensors  100  may be chosen independently.  FIG. 7 , hence, shows an embodiment of an off-axis angle sensor  100  with a diametrically magnetized end-of-shaft magnet  220 . The sensor die  130  is axial oriented with respect to the rotation axis  210 . 
     Embodiments may offer the possibility of increasing a robustness against external magnetic disturbances. The sensor  100  may be configured to estimate the rotation angle of the magnet  220  independent of superimposed homogeneous magnetic fields in any direction up to a limiting magnetic field strength. Depending on the magnetic sensor elements  180  used, the limiting magnetic field strength may be about 100 mT. In other cases, it may be higher or lower. Conventional sensor may be more sensitive to external magnetic disturbances. 
       FIG. 8  shows a schematic representation of a magnetic angle sensor arrangement comprising a discrete magnetic angle sensor device  100  or sensor  100  according to an embodiment in an on-axis configuration. Here, the main surface  140  of the sensor die  130  is essentially perpendicular arranged to the rotation axis  210 . The arrangement comprises an on-axis angle sensor  100  with a diametrically magnetized end-of-shaft magnet  220 . 
     To outline the operation of embodiments more closely, in the following a magnet  220  will be assumed, which is mirror symmetric to the plane z=0. Therefore, the z-axis coincides with the rotation axis  210  as shown before. Of course in many real-world implementations this is true only in an approximate sense, because many magnets  220  may have mounting tolerances and comprise slight asymmetries of the geometry. 
     The magnet  220  is diametrically magnetized. Without loss of generality, we assume that it is magnetized in the y-direction as shown above. The magnetization  250  may be homogeneous, although in reality there may be often some small inhomogeneity present. Therefore, it will be assumed that the magnet  220  comprises at least a significant dipole moment in the y-direction and all relevant higher multi-pole moments are small compared to the dipole moment. The number of embodiments, magnet arrangements and configurations is, nevertheless, vast. 
     The dipole moment is typically equivalent to the volume average of the magnetization  250 . It is often the most important magnetic contribution to the magnetic field. However, if the magnet has a shape that differs from a homogeneously magnetized sphere, the magnetic field can be decomposed into dipole and higher multi-pole contributions. Typically, the magnets for angle sensors have the shape of tablets (short cylinders), flat parallel epipeds or general blocks. Consequently, these magnets  220  typically comprise a dipole moment and higher multi-pole moments. 
     It might be advisable that the dipole moment is present and dominates all other moments. Higher multi-pole moments may have partly beneficial and partly deteriorating effects on the angle measurement. It is explained in the references “Inaccuracies of Anisotropic Magneto-Resistance Angle Sensors due to Assembly Tolerances” (U. Ausserlechner, Progr. Electromagn. Research B, vol. 40, pp. 79-99 (2012)), “Inaccuracies of Giant Magneto-Resistive Angle Sensors Due to Assembly Tolerances” (U. Ausserlechner, IEEE Trans. Magn., May 2009, vol. 45, no. 5, pp. 2165-2174), and “The Optimum Layout for Giant Magneto-Resistive Angle Sensors” (U. Ausserlechner, IEEE Sens. J., vol. 10, no. 10, pp. 1571-1582 (2010)), which are incorporated by reference, how the higher multi-pole moments affect the accuracy of the angle sensor. The origin of higher multi-pole moments may be due to the magnet geometry and an inhomogeneous magnetization  250 . The origin of inhomogeneous magnetization may be mainly attributed to the finite coercivity of the magnet  220  and/or an insufficiently large magnetization pulses. 
     By “relevant higher multi-pole moments” those are meant, which lead to magnetic field components that affect the magnetic field sensor elements in the angle sensor. For instance, if the sensor elements  180  are Hall plates then they may be affected only by magnetic field components perpendicular to the sensor die  130 , as long as the other components are below a critical value of, for instance, about 100 mT. On the other hand magneto-resistive (MR) devices are typically not affected by out-of-plane magnetic field components  120  as long as they are below another critical value of, for instance, about 500 mT. These may be considered irrelevant. 
     In an ideal case the magnets  220  should have a shape of rotational symmetry. Yet it will become clear that deviations from this ideal symmetry may be allowable at the costs of accuracy in the angle estimation by the sensor  100 . 
     An angle sensor system  100  may comprise at least four angle sensor elements  180 , which may be preferably located on a single sensor die  130  at at least two positions thereon. The sensor die  130  may have an axial orientation such that a line perpendicular to the die  130  and through a center point of the sensor  100 , which may be considered “the gravity center” of these sensor elements  180 , crosses the rotation axis  210 . A first set of two magnetic field sensor elements  180 - 1 ,  180 - 2  of a first type may be shifted along a first gradiometer direction  150 - 1 . They may form a first gradiometer  110 - 1 . A second set two magnetic field sensor elements  180 - 3 ,  180 - 4  of a second type may be shifted along the first gradiometer direction  150 - 1  or a second gradiometer direction  150 - 2 . Assuming that the first and second gradiometer directions  150 - 1 ,  150 - 2  may also be parallel or essentially identical, these directions  150  may be parallel or orthogonal. 
     The first type of magnetic field sensor elements  180  may, for instance, mainly respond to an out-of-plane magnetic field component  120 , while the second type of magnetic field sensor elements  180  may mainly respond to an in-plane magnetic field component  120 . The second type of magnetic field sensor elements  180  may mainly respond to that in-plane magnetic field component  120 , which is parallel to the second gradiometer direction  150 - 2 . 
     A sensor circuit  170  may compute a first difference of magnetic field components  120 - 1  detected by the first two magnetic field sensor elements  180 - 1 ,  180 - 2  and it may also compute a second difference of magnetic field components  120 - 2  detected by the second two magnetic field sensor elements  180 - 3 ,  180 - 4 . The sensor circuit  170  may, for instance, compute an arc-tangent of a ratio of these first and second differences. The first type of magnetic field sensor elements  180  may be Hall plates, while the second type of magnetic field sensor elements  180  may be vertical Hall effect devices or magneto-resistors. The shaft  260  may be ferrous or non-ferrous. In other words, it may have a relative permeability between μr=1 and, as an example, μr=10000. 
     Embodiments facilitate a concept, which will be explained in more detail now. Assuming the simplest case of a non-magnetic shaft  260  along the z-direction, around which a magnet  220  of rotational geometry is mounted with a homogeneous magnetization  250  in the y-direction, the magnetic field can be expressed by the following equations. In the general case (like e.g. a cone) the radius R′ of the magnet is a function of the z′-coordinate. In cylindrical coordinates one arrives at the following formulas: 
                           ⁢         B   x     ⁡     (     R   ,   ψ   ,   z     )       =       B   rem     ⁢     sin   ⁡     (     2   ⁢   ψ     )       ⁢     b   1                 (   1   )                       ⁢         B   y     ⁡     (     R   ,   ψ   ,   z     )       =       B   rem     ⁢     {       b   0     -       cos   ⁡     (     2   ⁢   ψ     )       ⁢     b   1         }                 (   2   )                       ⁢         B   z     ⁡     (     R   ,   ψ   ,   z     )       =       B   rem     ⁢   sin   ⁢           ⁢   ψ   ⁢           ⁢     b   2                 (   3   )                       ⁢   with                               b   0     ⁡     (     R   ,   z   ,   H   ,       R   ′     ⁡     (     z   ′     )         )       =       1   16     ⁢       ∫       z   ′     =       -   H     /   2         H   /   2       ⁢       R   ′     ⁢         3   ⁢     Rk   2     ⁢       F   1     ⁡     (       5   /   4     ,     7   /   4     ,   2   ,     k   2       )         -     4   ⁢           ⁢     R   2   ′     ⁢       F   1     ⁡     (       3   /   4     ,     5   /   4     ,   1   ,     k   2       )               (       R   2     +     R   ′2     +       (     z   -     z   ′       )     2       )       3   /   2         ⁢     dz   ′                   (   4   )                   b   1     ⁡     (     R   ,   z   ,   H   ,       R   ′     ⁡     (     z   ′     )         )       =       R   8     ⁢       ∫       z   ′     =       -   H     /   2         H   /   2       ⁢       R   ′2     ⁢               3   ⁢     R   2     ⁢       F   1     ⁡     (       5   /   4     ,     7   /   4     ,   2   ,     k   2       )         -                 (     15   /   8     )     ⁢           ⁢     R   ′     ⁢     k   2     ⁢       F   1     ⁡     (       7   /   4     ,     9   /   4     ,   3   ,     k   2       )                   (       R   2     +     R   ′2     +       (     z   -     z   ′       )     2       )       5   /   2         ⁢     dz   ′                   (   5   )                   b   2     ⁡     (     R   ,   z   ,   H   ,       R   ′     ⁡     (     z   ′     )         )       =         3   ⁢   R     4     ⁢       ∫       z   ′     =       -   H     /   2         H   /   2       ⁢           (     z   -     z   ′       )     ⁢           ⁢     R   2   ′2     ⁢       F   1     ⁡     (       5   /   4     ,     7   /   4     ,   2   ,     k   2       )             (       R   2     +     R   ′2     +       (     z   -     z   ′       )     2       )       5   /   2         ⁢     dz   ′                   (   6   )               
and with k=2RR′/(R 2 +R′ 2 +(z−z′) 2 ), H being the thickness of the magnet, and B rem  the remanence.  2 F 1 (a,b,c,x) denotes the hypergeometric function. The primed coordinates (R′, z′) denote the source points whereas the unprimed coordinates (R, ψ, z) denote the test point. Equations (1)-(6) are based on the assumption that the magnet  220  has no bore. A magnet  220  with a bore can be treated as a superposition of the magnet  220  without the bore and another magnet in the shape of the bore, but with a negative remanence. The radial and azimuthal field components are given by:
 
 B   R ( R,ψ,z )= B   rem  sin ψ( b   0   +b   1 )   (7)
 
 B   ψ ( R,ψ,z )= B   rem  cos ψ( b   0   −b   1 )   (8)
 
     For magnets  220  with mirror symmetry to the mid-plane z=0 it holds b 2 (R,0)=0 and ∂b 0 (R,0)/∂z=0 and ∂b 1 (R,0)/∂z=0. This follows from equations (7) and (8) when we insert z=0 and assume mirror symmetry R′(z′)=R′(−z′). For instance, for ∂b 0 (R,0)/∂z=0 it follows: 
     
       
         
           
             
               
                 
                   
                     
                       ∫ 
                       
                         
                           z 
                           ′ 
                         
                         = 
                         
                           
                             - 
                             H 
                           
                           / 
                           2 
                         
                       
                       
                         H 
                         / 
                         2 
                       
                     
                     ⁢ 
                     
                       
                         ∫ 
                         
                           
                             ψ 
                             ′ 
                           
                           = 
                           0 
                         
                         
                           2 
                           ⁢ 
                           π 
                         
                       
                       ⁢ 
                       
                         
                           
                             
                               ( 
                               
                                 
                                   R 
                                   / 
                                   
                                     2 
                                   
                                 
                                 - 
                                 
                                   
                                     R 
                                     ′ 
                                   
                                   ⁢ 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ψ 
                                     ′ 
                                   
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               R 
                               ′ 
                             
                             ⁢ 
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ψ 
                               ′ 
                             
                             ⁢ 
                             
                               z 
                               ′ 
                             
                           
                           
                             
                               ( 
                               
                                 
                                   R 
                                   2 
                                 
                                 + 
                                 
                                   R 
                                   ′2 
                                 
                                 + 
                                 
                                   
                                     2 
                                   
                                   ⁢ 
                                   
                                     
                                       RR 
                                       ′ 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           cos 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           
                                             ψ 
                                             ′ 
                                           
                                         
                                         + 
                                         
                                           sin 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           
                                             ψ 
                                             ′ 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 + 
                                 
                                   z 
                                   ′2 
                                 
                               
                               ) 
                             
                             
                               5 
                               / 
                               2 
                             
                           
                         
                         ⁢ 
                         d 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ψ 
                           ′ 
                         
                         ⁢ 
                         
                           dz 
                           ′ 
                         
                       
                     
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     This condition may be fulfilled if the magnet  220  is mirror-symmetric with respect to its mid-plane (z=0), because then the integrand in the lower half has opposite sign than the integrand in the upper half. Therefore, it might be advisable to look for a magnet  220 , which is mirror-symmetric to its mid-plane (z=0). 
     With equations (3), (7) and (8), it follows: 
     
       
         
           
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         
                           B 
                           z 
                         
                         ⁡ 
                         
                           ( 
                           
                             R 
                             , 
                             ψ 
                             , 
                             
                               
                                 Δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 z 
                               
                               2 
                             
                           
                           ) 
                         
                       
                       - 
                       
                         
                           B 
                           z 
                         
                         ⁡ 
                         
                           ( 
                           
                             R 
                             , 
                             ψ 
                             , 
                             
                               
                                 
                                   - 
                                   Δ 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 z 
                               
                               2 
                             
                           
                           ) 
                         
                       
                     
                     ≅ 
                     
                       
                         B 
                         rem 
                       
                       ⁢ 
                       Δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       z 
                       ⁢ 
                       
                         
                           ∂ 
                           
                             
                               b 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               
                                 R 
                                 , 
                                 
                                   z 
                                   = 
                                   0 
                                 
                               
                               ) 
                             
                           
                         
                         
                           ∂ 
                           z 
                         
                       
                       ⁢ 
                       sin 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       ψ 
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         B 
                         R 
                       
                       ⁡ 
                       
                         ( 
                         
                           R 
                           , 
                           
                             ψ 
                             + 
                             
                               
                                 Δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 ψ 
                               
                               2 
                             
                           
                           , 
                           
                             z 
                             = 
                             0 
                           
                         
                         ) 
                       
                     
                     - 
                     
                       
                         B 
                         R 
                       
                       ⁡ 
                       
                         ( 
                         
                           R 
                           , 
                           
                             ψ 
                             - 
                             
                               
                                 Δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 ψ 
                               
                               2 
                             
                           
                           , 
                           
                             z 
                             = 
                             0 
                           
                         
                         ) 
                       
                     
                   
                   ≅ 
                   
                     
                       B 
                       rem 
                     
                     ⁢ 
                     Δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ψ 
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               b 
                               0 
                             
                             ⁡ 
                             
                               ( 
                               
                                 R 
                                 , 
                                 
                                   z 
                                   = 
                                   0 
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               b 
                               1 
                             
                             ⁡ 
                             
                               ( 
                               
                                 R 
                                 , 
                                 
                                   z 
                                   = 
                                   0 
                                 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                     
                     ⁢ 
                     cos 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     ψ 
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         B 
                         ψ 
                       
                       ⁡ 
                       
                         ( 
                         
                           R 
                           , 
                           
                             ψ 
                             + 
                             
                               
                                 Δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 ψ 
                               
                               2 
                             
                           
                           , 
                           
                             z 
                             = 
                             0 
                           
                         
                         ) 
                       
                     
                     - 
                     
                       
                         B 
                         ψ 
                       
                       ⁡ 
                       
                         ( 
                         
                           R 
                           , 
                           
                             ψ 
                             - 
                             
                               
                                 Δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 ψ 
                               
                               2 
                             
                           
                           , 
                           
                             z 
                             = 
                             0 
                           
                         
                         ) 
                       
                     
                   
                   ≅ 
                   
                     
                       - 
                       
                         B 
                         rem 
                       
                     
                     ⁢ 
                     Δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ψ 
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               b 
                               0 
                             
                             ⁡ 
                             
                               ( 
                               
                                 R 
                                 , 
                                 
                                   z 
                                   = 
                                   0 
                                 
                               
                               ) 
                             
                           
                           - 
                           
                             
                               b 
                               1 
                             
                             ⁡ 
                             
                               ( 
                               
                                 R 
                                 , 
                                 
                                   z 
                                   = 
                                   0 
                                 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                     
                     ⁢ 
                     sin 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     ψ 
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     For angle sensing, determination or estimation, it might be advisable to use a pair of equations comprising sine and cosine dependency, respectively. Thus, several possibilities of varying implementation principles exist, which will be outlined below. 
       FIG. 9  shows a schematic plan view of a discrete magnetic angle sensor device according to an embodiment.  FIG. 9  is similar to  FIG. 1 , but differs with respect to some aspects. On the substrate or sensor die  130 , once again two gradiometers  110 - 1 ,  110 - 2  are arranged comprising two magnetic sensor elements  180  each. Similar to the embodiment shown in  FIG. 1 , the first gradiometer  110 - 1  is responsive or sensitive to the magnetic field component  120 - 1  perpendicular to the sensor die  130 . In other words, the first magnetic field component  120 - 1  may be an out-of-plane magnetic field component. The magnetic sensor elements  180 - 1 ,  180 - 2  may, for instance, be Hall plates or rather out-of-plane magnetic field sensor elements  180 . 
     The second gradiometer  110 - 2  comprising the magnetic sensor elements  180 - 3 ,  180 - 4  is responsive to magnetic field components  120 - 2  in the z-direction. Accordingly, the magnetic sensor elements  180 - 3 ,  180 - 4  may, for instance be implemented as vertical Hall effect devices or vertical Hall sensor elements. However, also other in-plane magnetic field sensor elements  180  may be used since the corresponding second magnetic field component  120 - 2  is oriented along the main surface  140  of the die  130 , because of which the second magnetic field component  120 - 2  is also referred to as an in-plane magnetic field component. 
     The first and second magnetic sensor elements  180 - 1 ,  180 - 2  are oriented along the first gradiometer direction  150 - 1  which is, in the embodiment shown in  FIG. 9 , oriented along the circumferential direction of the magnet. In other words, the horizontal direction corresponds to the direction ψ. The two Hall plates  180 - 1 ,  180 - 2 , therefore, are spaced from one another at a distance of R×Δψ, whereby R is the radial distance of point  270  from the rotation axis. Correspondingly, the third and fourth magnetic sensor element  180 - 3 ,  180 - 4  or, in other words, in the embodiment shown in  FIG. 9 , the two vertical Hall element  180 - 3 ,  180 - 4 , are separated by a distance Δz along the z-axis. 
     A center point  270  of the magnetic sensor elements  180 - 1 ,  180 - 2 ,  180 - 3 ,  180 - 4  of the two gradiometers  110 - 1 ,  110 - 2 , which may, for instance, correspond to an intersecting point of the connecting lines between the sensor elements  180  of a first gradiometer  110 - 1  and the corresponding connecting line of the sensor elements  180  of the second gradiometer  110 - 2 . As outlined before, the center point  270  may, for instance, be oriented such that it lies on a plane being in the center of the magnet  220  in the arrangement as depicted, for instance, in  FIG. 6 or 7 . 
       FIG. 10  shows a similar schematic plan view of a further discrete magnetic angle sensor device  100 , which differs from the one shown in  FIG. 9  by several aspects. For instance, the second gradiometer  110 - 2  is now oriented along the first gradiometer direction  150 - 1 . To be more precise, a distance between the third and fourth magnetic sensor elements  180 - 3 ,  180 - 4  is now oriented along the first gradiometer direction  150 - 1 . However, a distance between the third and fourth magnetic sensor elements  180 - 3 ,  180 - 4  differs from the corresponding distance of the first and second magnetic sensor elements  180 - 1 ,  180 - 2  as illustrated by the arrow indicating the first gradiometer direction  150 ′- 1  corresponding to the second gradiometer  110 - 2 . 
     Irrespective of this aspect, the discrete magnetic angle sensor device  100  once again comprises a common center point  270  for both, the first and second gradiometers  110 - 1 ,  110 - 2 . The common center point  270  is situated along half of a connecting line between the first and second magnetic sensor elements  180 - 1 ,  180 - 2  and along the connecting line connecting the third and fourth magnetic sensor elements  180 - 3 ,  180 - 4 . 
     A common center point  270  is situated at a center point of a connecting line connecting the first and second magnetic sensor elements  180 - 1 ,  180 - 2  along the first gradiometer direction  150 - 1  and at a midpoint of a connecting line connecting the third and fourth magnetic sensor elements  180 - 3 ,  180 - 4  of the second gradiometer  110 - 2  along the second gradiometer direction  150 - 2 . In other words, the center point  270  corresponds to both midpoints of the two gradiometers  110 - 1 ,  110 - 2  and, simultaneously, to all of the four magnetic center elements  180 - 1 , . . . ,  180 - 4 . It, therefore, can be considered a center of gravity of the sensor device  100 . 
     Moreover, the second magnetic field component  120 - 2  is tilted by 90° compared to the sensor device  100  shown in  FIG. 9  in a plane parallel to the main surface  140 . Therefore, also the third and fourth magnetic sensor elements  180 - 3 ,  180 - 4  are sensitive to an in-plane magnetic field component  120  parallel to the main surface  140  of the silicon die  130 . 
     However, it should be noted that the gradiometers  110 - 1 ,  110 - 2  may be arranged in such a way that the midpoints of the connecting lines and, hence, their center points  270  do not coincide. This may, however, be under some circumstances suboptimal. 
       FIG. 11  shows a corresponding schematic plan view of a discrete magnetic angle sensor device  100  according to an embodiment, which differs from the one shown in  FIG. 10  such that the gradiometers  110 - 1 ,  110 - 2  and their magnetic sensor elements  180 - 1 , . . . ,  180 - 4  are displaced in terms of their midpoints of the connecting lines such that their center points are displaced along the horizontal axis corresponding to the R·ψ direction. Hence, the previously-described common center point  270  is replaced by two center points  270 - 1 ,  270 - 2  of the first and second gradiometers  110 - 1 ,  110 - 2 , respectively. In the embodiment shown in  FIG. 11 , the gradiometers  110 - 1 ,  110 - 2  are arranged such that their center points  270 - 1 ,  270 - 2  are symmetrically arranged with respect to the vertical axis corresponding to the z-axis in the case of an implementation in a magnetic angle sensor arrangement  200  as shown, for instance, in  FIGS. 6 and 7 . 
     Embodiments of a magnetic angle sensor device  100  may, therefore, comprise two in-plane magnetic field sensor elements  180  spaced apart by a first distance along a first gradiometer direction  150 - 1  and, for instance, two out-of-plane magnetic field sensor elements  180  spaced apart by a second distance along the first gradiometer direction  150 - 1  or a second gradiometer direction  150 - 2 , which may be different from the first gradiometer direction  150 - 1 . It may further comprise sensor circuit  170  configured to compute or determine a difference of magnetic fields or—in other words—a gradient of the magnetic field based on the two in-plane magnetic field sensor elements  180 . Similarly, the sensor circuit  170  may be configured to compute the difference of fields or the corresponding gradient based on the two out-of-plane magnetic field sensor elements  180 . It may further be configured to estimate the angular position the magnet  220 , whose magnetic field the sensor elements  180  detect, on the basis of these two differences or gradients based on an algorithm. This may be done based on an arc-tangent computation or based on a lookup table. However, there are numerous ways to perform this computation or estimation. 
     Furthermore, a magnetic angle sensor arrangement  200  according to an embodiment may comprise a magnet  220 . A position of the at least one sensor device  100  with respect to the magnet  220  may be chosen based on a large number of parameters, some of which will be outlined below in more detail. 
     A corresponding magnetic angle sensor arrangement  200  may comprise at least one magnetic angle sensor device  100  according to an embodiment as outlined before along with a diametrically magnetized magnet  220  mounted or mountable on a rotor. The sensor die  130  of the magnetic angle sensor device  100  may be mounted on a stator. Naturally, also a configuration is possible with the at least one sensor device  100  being mounted or mountable to the rotator, while the magnet  220  is mounted or mountable on to the stator. 
     The die  130  of the at least one magnetic angle sensor device  100  may, for instance, comprise two in-plane magnetic field sensors elements  180  spaced apart by a first distance, and two out-of-plane magnetic field sensor elements  180  spaced apart by a second distance, whereby “in-plane” and “out-of-plane” refers to the main surface of the substrate, onto or in which they are located. As outlined before, the angle sensor device  100  may further comprise a circuit  170 , which may be configured to subtract the magnetic field components  120  detected by the two in-plane magnetic field sensor elements  180  to obtain a first difference or gradient. It may further be configured to subtract the magnetic field components  120  detected by the two out-of-plane magnetic field sensor elements  180  to obtain a second difference or gradient and estimate the angular position of the rotor against the stator using an algorithm operating on the basis of the first and second differences or gradients. 
     Optionally, the die  130  may be placed off-axis with respect to the rotation axis  210 . The rotation axis  210  may be parallel to the main surface  140  of the die  130 . In other words, the discrete magnetic sensor device  100  may be used as a differential out-of-axis angle sensor. 
     Embodiments of a magnetic angle sensor device  100  may, for instance, use four magnetic field sensor elements  180 , two of which constitute a first group or the first gradiometer  110 - 1 . They may be sensitive to a first component  120 - 1  of two orthogonal components  120 - 1 ,  120 - 2  of the magnetic field. The other two magnetic field sensor elements  180  constitute a second group or the second gradiometer  110 - 2 . They may be sensitive to the second component  120 - 2  of the magnetic field. 
     In embodiments, the spacing of the sensor elements  180  of one discrete magnetic angle sensor device may be often less than 10 mm or less than 5 mm, since the magnetic sensor elements  180  are typically arranged on or in a single die  130 . This arrangement of the sensor elements  180  may be beneficial, since the sensor elements  180  may be more easily matched, for instance, in terms of their magnetic sensitivities. This may lead to a more efficient suppression of homogeneous background magnetic fields and other disturbances. Moreover, the spacing of the sensor elements  180  may be kept smaller, which may also improve a suppression of background magnetic fields. 
     Embodiments of a magnetic angle sensor device  100  and a corresponding arrangement  200  may use sensors sensitive to tangential magnetic field components  120 . This may be beneficial, since a differences of radial field components  120  may be smaller for magnets of a large diameter. Embodiments may utilize for radial components horizontal Hall sensor elements  180  for these components, which may be better suited for weaker magnetic field components than, for instance, vertical Hall sensor elements. 
     In the embodiments shown in  FIGS. 10 and 11 , the gradiometers  110 - 1 ,  110 - 2  are both oriented such that the sensor elements  180  of the two gradiometers  110 - 1 ,  110 - 2  are offset along the movement direction of the magnet  220 . In the embodiment shown in  FIG. 9 , only the two sensor elements  180 - 1 ,  180 - 2  of the first gradiometer  110 - 1 , which may be implemented as horizontal or lateral Hall plates for the gradient along the radially oriented first gradiometer direction  150 - 1 , may be offset along the movement direction (ψ-direction), whereas the other elements  180 - 3 ,  180 - 4  may be implemented as vertical Hall elements for the magnetic component along the z-direction, which are offset in the z-direction. 
     As the  FIGS. 10 and 11  have shown, the sensor elements may be arranged such that the center points  270  of the two gradiometers  110  coincide or may be offset from one another. In other words, the center spots  270  are not necessarily at the same spot, although this may also be implemented. 
     In embodiments, the signals in the nominator and denominator of a possible arctan-implementation (cf. the signals in equations (10)-(12)) may also be sinusoidal. Yet for general magnets, these signals may have different amplitudes. This may have to be addressed in an implementation. The question of the different amplitudes may have to be separately considered as will be outlined below. 
     However, embodiments are based on measuring slopes or gradients of magnetic field components  120 . In other words, the arctan-determination is not directly carried out on the magnetic field components  120 , but the differences or gradients accessible by using the gradiometers  110  are used instead. Therefore, although an angle sensor device  100  according to an embodiment uses four sensor elements  180 , two slopes or gradients at two spots (the center points  270 ), which are close nearby are used for the angle determination. The points are located on the same die  130  so that they are less than the size of the die  130  separated from one another. 
     Naturally, using a gradiometer  110  to determine the gradient results in an approximation of the gradient by sensing the respective magnetic field component  120  in two spots given by the locations of the magnetic sensor elements  180 , using their difference and dividing same by the distance between the relevant sensor devices  180 . Hence, taking two sensor elements  180 , which are sensitive to the same magnetic field component  120  at two locations, then the subtraction of both divided by their spacing gives an approximation for the slope or gradient of the respective field component  120  at the midpoint or center point  270  between both elements  180 . 
     According to an embodiment, the rotational position of the magnet  270  may, for instance, be estimated based on pairs of (dB R /dψ,dB ψ /dψ) or with (dB R /dψ,dB z /dz), to name just some examples. The first pair of numbers are slopes along the same direction d/dψ, whereas the second pair of numbers are slopes along orthogonal directions d/dψ and d/dz. 
     On the other hand the amplitude of the curve dB z /dz may depend mainly on an axial dimension or thickness of the magnet  220 . So the thickness of the magnet  220  may offer a further degree of freedom in the design of the sensor system or arrangement  200 . It may be possible to trim it in order to adjust the amplitude of the dB z /dz-curve to values close to the amplitudes of the dB R /dψ-curve. 
     These considerations open up an insight into the general operational principles of differential off-axis angle sensors  100  according to an embodiment. Gradients in other directions (e.g. radial direction) may also be used. With regard to equations (3), (7) and (8) the following table is derivable, that gives the psi-dependencies (ψ-dependencies) of the slopes of the magnetic field components  120 : 
                                                 BR   Bv   B,                                                            d/dR   sin ψ   cos ψ   sin ψ           d/dψ   cos ψ   sin ψ   cos ψ           d/dz   sin ψ   cos ψ   sin ψ                        
From the three components B R , B ψ , B z  and three derivatives d/dR, d/dψ, d/dz corresponding to three gradiometer directions  150 , when integrated into a magnetic angle sensor arrangement  200  or a similar arrangement, a total of nine combination arises. As the table shows, this gives five signals with sine-of-psi-dependency (sine signals) and four signals with cosine-of-psi-dependency (cosine signals). For an angle sensor, in principle arbitrary pairs including one sine- and one cosine-function may be chosen.
 
     However, not all of the nine functions in this table are independent. According to Maxwell&#39;s equations the curl of the B-field vanishes in free space. This leads to the equations dB z /dψ=RdB ψ /dz and dB z /dR=dB R /dz. Therefore, the two fields in the table with identical or similar accentuated borders are identical. 
     From a sensor point of view, it may be interesting to use horizontal Hall sensor elements  180 , because their errors (e.g. offset and noise) are smaller than those of vertical Hall sensor elements. This may mean that systems detecting slopes of the first magnetic field component  120 - 1  (along a first direction, which is the out-of-plane direction) in second and third directions (gradiometer directions  150 ) might be interesting, whereby the three directions are mutually perpendicular. For instance, if R is the first direction, using the pair (dB R /dψ, dB R /dz) may be interesting. If ψ (psi) is the first direction, using the pair (dB ψ /dR,dB ψ /dz) may be interesting, yet both signals have a cosine dependency—so this pair might not work for an angle sensor  100 . If z is the first direction, using the pair (dB z /dR, dB z /dψ) may be interesting. 
       FIGS. 12 and 13  show results of a three dimensional simulation of a magnet  220 . The magnet  220  has an outer diameter of about 30 mm with a 6 mm diameter bore and an iron shaft  260  through the bore along the z-axis. The magnet  220  is 8 mm thick along the z-axis and comprises a remanence Brem of 220 mT pointing in the y-direction.  FIG. 12  shows diagrams for BR and Bpsi as a function of psi for the plane z=0, which is the mid-plane of the magnet  220 .  FIG. 13  shows the curves for the same magnetic field components  120  at a shifted plane shifted by a distance of 3.5 mm to 4 mm. 
       FIG. 14  shows at radial distance of 17 mm from the rotation axis  210  amplitudes of the three magnetic field components  120  versus their axial position (z-coordinate along the z-axis), where the position z=0 corresponds to the mid-plane of the magnet  220 . Starting from the mid-plane (z=0), the amplitude of the Bz-component increases linearly until the test point reaches roughly the position of the top surface of the magnet  220 , where it falls again. At about z=t/2 with t being the thickness of the magnet  220 , the Bz-field has the largest amplitude, which is nearly twice as large as the Bpsi-field component there, and about as large as the BR-field component. 
       FIG. 15  shows a field difference of the three magnetic field components at a 1.5 mm radial distance with one point being located at a distance of R=17 mm and one point being located at R=18.5 mm. So this is the relevant magnetic input for gradiometers with radial gradiometer direction. Here the dBR-signal is strongest in a range of about z=0 mm to about z=2 mm, whereas the dBz-signal is strongest near z=4.0 mm to 4.5 mm. Near z=3.5 mm both signals are equally or comparably strong according to this simulation. 
       FIG. 16  shows a schematic diagram of a discrete magnetic angle sensor device  100  according to an embodiment, which may be used in the framework of a differential out-of-axis sensor arrangement  200  according to an embodiment. The magnetic angle sensor device  100  comprises once again a sensor die  130  with a main surface  140 , on which a first gradiometer  110 - 1  and a second gradiometer  110 - 2  are implemented. The gradiometers  110 - 1 ,  110 - 2  are both arranged along a first gradiometer direction  150 - 1 ,  150 ′- 1 , respectively, which is parallel to a radial direction with the rotation axis being the symmetry center. 
     The first gradiometer  110 - 1  comprises a first and a second magnetic sensor element  180 - 1 ,  180 - 2 , which is sensitive or responsive to a first magnetic field component  120 - 1  along the z-direction. These magnetic sensor elements  180 - 1 ,  180 - 2  may, for instance, be implemented as vertical Hall effect devices. The two magnetic sensor elements  180 - 1 ,  180 - 2  are arranged along the first gradiometer direction  150 - 1  at a distance along the radial direction of ΔR z . The index Z indicates that the two magnetic sensor elements  180 - 1 ,  180 - 2  are arranged to be responsive to the first magnetic field component  120 - 1  along the z-direction. Center points of the first and second gradiometers  110 - 1 ,  110 - 2  are shifted along the z-direction on purpose to allow determining the gradients dBR/dR at z=0 and dBz/dR at z=4.5 mm as shown in  FIG. 15 . In other words, a position and/or an orientation of the devices  100  may be adapted according to the geometry of the arrangement  200 . 
     The second gradiometer  110 - 2  comprises a third and a fourth magnetic sensor elements  180 - 3 ,  180 - 4  which are also arranged along the first gradiometer direction  150 ′- 1 , which are responsive to a second magnetic field component  120 - 2 , which is oriented in the arrangement depicted in  FIG. 16  along the radial direction (R). In other words, the magnetic sensor elements  180 - 3 ,  180 - 4  are responsive to the radial magnetic field component, which may, for instance, be detected by appropriately oriented vertical Hall effect devices. Therefore, the magnetic sensor elements  180 - 3 ,  180 - 4  may be implemented as vertical Hall effect sensor elements. They are separated along the gradiometer direction  150 ′- 1  by a distance ΔR R , where the index R indicates the sensitivity of the magnetic sensor elements  180  to the radial component. 
     The first and second gradiometers  110 - 1 ,  110 - 2  are furthermore separated along the z-axis corresponding to the actual direction by a distance Δz. A center point  270  of the four magnetic sensor elements  180 - 1 , . . . ,  180 - 4  is furthermore shown in  FIG. 16 , corresponding to a “gravity center” of these sensor elements  180 . 
     The sensor die  130  may be placed near the perimeter of the magnet  220  with the z-axis parallel to the rotation axis  210  and the R-axis aligned such that its projection goes through the rotation axis  210 . The axial position of the sensor die  130  may be adjusted so that the BR-sensor-pair (gradiometer  110 - 2 ) may be as close as possible to the mid-plane (z=0) of the magnet  220  and the Bz-sensor-pair (gradiometer  110 - 1 ) may be close to the maximum of the amp(dBz)-curve shown in  FIG. 15  (z=4 mm). 
     Theoretically, this might call for Δz=4 mm, which may often be too large for usual silicon dies. A cheap silicon die often offers a size of about 1.6 mm·1.6 mm so that the distances may be in such a case Δz=ΔR z =ΔR, =1.5 mm. Under these boundary conditions it may be advisable if the gravity center or center spot  270  has a z-position close to where the amp(dBR/dR) and amp(dBz/dR)-curves cross each other. In other words, it may be located near z=3.5 mm. Then both signals may have an amplitude of about 13 mT in the case outlined above. If the vertical Hall sensor elements used as the sensor elements  180  have, for instance, 75 μT worst case offset error, this may correspond to 75μ/13 m·180°/π=0.33° worst case angle error with π=3.1415 . . . . If four such devices are placed at psi-angles of 0°, 90°, 180° and 270° the offset error may effectively halve, which may give quite a good angle accuracy. 
       FIGS. 17, 18, 19 and 20  show a semi-transparent perspective view, a perspective view, a close-up and a side view of a magnetic angle sensor arrangement  200  according to an embodiment comprising five of the previously described discrete magnetic angle sensor devices  100 . The magnetic angle sensor devices  100  are arranged equally spaced around a shaft  260  to which a magnet  220  is mounted. As outlined before, the magnet  220  is diametrically magnetized and fixed to the shaft  260  in a through-hole configuration. The magnet  220  is furthermore in a ring-like shape as outlined before. 
     The discrete magnetic angle sensor devices  100  are arranged on a carrier plate  280 , which is oriented essentially perpendicular to the shaft  260  and, hence, to the rotation axis  210  of the shaft  260 . As, for instance,  FIGS. 19 and 20  show, the discrete magnetic angle sensor devices  100  are mechanically and electrically coupled to the carrier plate  280  by three pins  290 . A middle pin  290 ′ is mechanically and electrically coupled to a leadframe  300  onto which the substrate or die  130  of the discrete magnetic angle sensor device  100  is mounted.  FIGS. 19 and 20  furthermore show the positions and orientations of the four magnetic sensor elements  180 - 1 , . . . ,  180 - 4  as previously outlined in the context of  FIG. 16 . 
     To show the inner structure of the discrete magnetic angle sensor devices  100 , in  FIGS. 19 and 20  two of these devices  100  are shown without a surrounding mold component  310  encapsulating the leadframe  300  and the die  130 . Naturally, instead of a mold compound  310  also other encapsulating techniques may be used. 
       FIG. 21  shows a diagram of amplitudes of psi-gradients of magnetic field components of the same magnet as shown in  FIG. 14, 15  as a function of a distance from a mid-plane. As  FIG. 21  shows, the slopes of the magnetic field components  120  versus psi-direction may be smaller than in  FIG. 15 . In the case shown here, the psi-gradient of the radial field component  120  has the largest slope which lies at the mid-plane (z=0) at a radial distance of about 17 mm, but its value is still only a third of the slope in the R-direction. The signals (dB/dpsi*1.5 mm/R) are essentially identical to the signals B multiplied by a factor (1.5 mm/R=1.5/17=1/11.3). Thus, the difference fields are only 1/11 th  of the absolute field. In other words, if the mismatch between both sensors  180  of the difference pair (gradiometer  110 ) is about 0.1% it may add a contribution of about 1.13% to the difference signal, which is in quadrature to it and which might give an additional angle error. 
     In the z=0 plane or—in other words—near the mid-plane of the magnet  220 , the signal amplitudes acquire the following absolute values (sign of the signals is not accounted for): 
                                         in {R = 17.75                   mm, z = 0 mm}:   B R     B ψ     B z                    d/dR × 1.5 mm   18.9 mT × sinψ   6.2 mT × cosψ   0 × sinψ       d/dψ × 1.5 mm/R   5.8 mT × cosψ   2.1 mT × sinψ   0 × cosψ       d/dz × 1.5 mm   0 × sinψ   0 × cosψ   12.0 mT × sinψ       Absolute field   65.3 mT ×sinψ     24.3 mT ×cosψ     0 ×sinψ         amplitudes                    
The largest gradient signals are 1.5 mm×dB R /dR and 1.5 mm×dB z /dz, yet they are both in phase and therefore not appropriate for an angle sensor. Either of these two signals may be combined with 1.5 mm×dR or (1.5 mm/R)×dB R /dψ to make up an angle sensor  100  with sine- and cosine-signals. Although both signals are nearly equally strong it might be better to use 1.5 mm×dB ψ /dR, because Bpsi may be much smaller than BR and therefore the Bpsi-versus-R-gradiometer might be less affected by mismatches of the sensor pair then the BR-versus-psi-gradiometer.
 
     Similarly, for the sine-signal it might be advisable to use 1.5 mm×dB z /dz over 1.5 mm×dB R /dR. Although the latter signal may be stronger, it might be more affected by a mismatch of the sensors in the BR-pair due to the large BR-field. This may give a sensor that is similar to the sensor  100  described above on context of  FIG. 9 . 
       FIG. 22  shows a schematic diagram of a magnetic angle sensor device  100  according to an embodiment, which is similar to the device  100  shown in  FIG. 9 . However, compared to the device  100  in  FIG. 9 , the second gradiometer  110 - 2  (the Bz-sensor-pair) is shifted to the left side of the die  130 . In other words, the second gradiometer  110 - 2  is shifted along the first gradiometer direction  150 - 1  or the radial direction (R). Here, the first magnetic sensor element  180  of the first gradiometer  110 - 1  and the magnetic sensor elements  180 - 3 ,  180 - 4  of the second gradiometer  110 - 2  are arranged on a straight line perpendicular to the first gradiometer direction  150 - 1 . Hence, the die  130  may be oriented in such a way that the horizontal axis points in the radial and not in the azimuthal direction. 
     As a consequence, in the plane at the z=4 mm coordinate or—in other words—near the top surface of the magnet  220 , the signal amplitudes may assume the following absolute values (sign of the signals is not accounted for). 
     
       
         
           
               
               
               
               
             
               
                   
               
               
                 in {R = 17.75 
                   
                   
                   
               
               
                 mm, z = 4 mm}: 
                 B R   
                 B ψ   
                 B z   
               
               
                   
               
             
            
               
                 d/dR × 1.5 mm 
                 9.3 mT × sinψ 
                 4.3 mT × cosψ 
                 14.2 × sinψ 
               
               
                 d/dψ × 1.5 mm/R 
                 3.6 mT × cosψ 
                 1.7 mT × sinψ 
                 3.2 × cosψ 
               
               
                 d/dz × 1.5 mm 
                 15.7 × sinψ 
                 3.0 × cosψ 
                 4.9 mT × sinψ 
               
               
                 Absolute field 
                 41.3 mT × sinψ 
                 19.7 mT × cosψ 
                 36.0 × sinψ 
               
               
                 amplitudes 
               
               
                   
               
            
           
         
       
     
     The largest gradient signals are 1.5 mm×dB R /dz and 1.5 mm×dB z /dR. Both are in phase, so it might be advisable to use an additional cosine(psi)-signal. Since the Bz-field is 15% smaller than the BR-field (radial field component), it may be slightly better to use 1.5 mm×dB z /dR, because it might be less affected by a mismatch of the sensor elements  180  in the Bz-pair (gradiometer  110 - 1 ). Unfortunately, all cosine-signals may be weak (4.3 mT or less) so that it is not clear, which combination of signals might give the best available results. Generally, the gradiometers  110  may be chosen so that one has sine-like and the other cosine-like dependence and from the various possibilities of sine-like dependence (or cosine-like dependence) one may choose a gradiometer with large gradiometer amplitudes and not too large amplitudes of the field component, to which the gradiometer is sensitive. So if the gradient, for instance, detects dBx/dy (x and y denoting any of the R-, psi- and the z-direction) then of course the amplitude of dBx/dy should be sufficiently large (amp(dBx/dy)), but also the ratio of amplitudes of Bx and dBx/dy should be small (amp(Bx)/amp(dBx/dy)), for instance, smaller than 10. 
     As shown above, there are numerous sensor layouts. Embodiments may utilize one or more basic ideas. Embodiments may, for instance, be based on sensor units  110  or nodes, which measure the gradients of two magnetic field components  120 , for instance, orthogonal ones like BR, Bpsi, Bz. However, it is not necessary to use orthogonal components. It is sufficient when they are not collinear. This may be done, for instance, in two locations, that are close together. In other words, the two locations are on the same sensor die  130  and typically less than 3 mm distant from one another. Often, a distance between them is less than the radius of the magnet  220 . Therefore, their spacing may be defined very accurately, for instance, with micrometer precision due to the accuracy achievable with which modern integrated circuit technology defining the geometry and positions of the sensor elements  180 . The gradiometers  110  may cancel homogeneous background magnetic disturbances so that all gradiometer outputs may be combined arbitrarily without getting errors due to background fields. 
     Each sensor unit  100  may estimate the rotation angle by an algebraic combination, which cancels out common multiplicative factors on both gradiometer outputs (e.g. ratio or arctangent of ratio). This class of algebraic combinations may cancel out magnetic field strength. Therefore, the sensor  100  might not be affected by lifetime or temperature drifts of magnetization of the magnet  220 . Furthermore, this class of algebraic combinations may allow to combine angles obtained by different sensors  100  or sensor dies  130 , even if the magnetic sensitivity of magnetic field sensors  100  may vary between the dies  130 . 
     For example, when a first sensor unit  100  with a first gradiometer output GN 1  and a second gradiometer output GT 1 , both outputs may be proportional to the magnetization M of the magnet  220  and to the sensitivity S 1  defined by a common bias circuit, and to a function of the rotation angle N 1 (psi), T 1 (psi). Thus one may obtain
 
 GN 1= M*S 1* N 1(psi)  (13)
 
 GT 1= M*S 1* T 1(psi)  (14)
 
     Since the algebraic combination cancels common multiplicative factors, the first sensor unit  100  may estimate the rotation angle according to
 
psi1= f 1( GN 1/ GT 1)= f 1( N 1(psi)/ T 1(psi)).  (15)
 
     Thereby f 1  is some function, such as the arctangent or a more complicated algorithm. The strength of the magnet  220  and the sensitivity of the magnetic field sensors  100  on sensor unit  1  cancel out. 
     Having a second sensor unit  100 , it may have again a first and second gradiometer outputs GN 2  and GT 2 . With the equations
 
 GN 2= M* 52* N 2(psi)  (16)
 
 GT 2= M* 52* T 2(psi)  (17)
 
the circuit may compute the angle
 
psi2= f 2( GN 2/ GT 2)= f 2( N 2(psi)/ T 2(psi)).  (18)
 
Consequently, the sensitivity of the field sensors  180  on sensor unit  2 —namely S 2 —cancels out. This may be used to achieve a higher accuracy, because it is typically almost not possible to have perfectly identical sensitivities (S 1 =S 2 ) on different discrete sensors (e.g. different sensor units  100  and/or different sensor dies  130 ). Embodiments may cope with this fact simply by eliminating the sensitivities in the estimation of the angles (psi 1 , psi 2 ).
 
     Imperfect magnetizations and assembly tolerances may also lead to large errors in the estimated angle of a single sensor unit  100 . However, when several sensor units are arranged around the magnet  220  and their outputs is combined, for instance by averaging their outputs taking the positions of the sensor units  100  into account, this combining of outputs may lead to drastically smaller angle errors. For instance, an eccentricity of the magnet  220  with respect to the rotation axis  210  when being mounted to the shaft  260  may be eventually counteracted by placing, for instance, at least two sensor elements  180  diametrically at an angle of 180° around the rotation axis  210 . Thereby it may be possible to increase an accuracy by averaging the angle estimations of the two sensors  100 . Similarly, material inhomogeneities within the magnet  220 , such as voids, may eventually be counteracted by placing several sensors  100  on a circle around the axis  210  and average their angle estimations. By these measures it may be possible to at least partially counteract deviations from a pure sine wave-like magnetic field dependency when the magnet  220  rotates around the rotation axis  210 . 
     The previously described averaging merely represents one way of combining the outputs of the sensor devices or units  100 . Other schemes of combing the information comprised in the outputs may be used, which are indicative of different positions spread over a larger area (e.g. around the circumference of the magnets or around the rotation axis). 
     The main surface  140  of the sensor die  130  may be tangential to the magnet  220 —thus the surface normal may be identical with a radial ray that is sent from the rotation axis  210  outwards through the sensor  100 . It may be apt for leaded packages. Thus the gradiometers  110  may be able to detect, for instance, two sorts of slopes directly (d/dpsi and d/dz) and the third one (d/dR) indirectly by exploiting Maxwell relations between the field components—for each one of the components BR, Bpsi, Bz. This may give 2·3=6 direct gradiometer outputs. 
     As outlined before, various orientations of the sensor die  130  may be used. For instance, horizontal Hall sensor elements  180  only, vertical Hall sensor elements  180  only or mixed Hall sensor elements  180  may be used. Depending on the carrier for the sensor devices  100 , the orientation of the sensor dies  130 , the available gradients and possible combinations of sensor elements  180 , different package types may eventually be used. For instance, for an (R,z)-plane oriented sensor die  130 , magnetic field gradients dB{R,psi,z}/d{R,z} may be directly available. The gradients dB{R,psi,z}/dpsi may be computed out of the gradients dB{R,psi,z}/d{R,z} by use of Maxwell&#39;s equations. These may be detected by horizontal and vertical Hall sensor elements  180 . A leaded package type may be used. For a (R,psi)-plane oriented sensor die  130 , magnetic field gradients dB{R,psi,z}/d{R,psi} may be directly available. These may be detected by horizontal and/or vertical Hall sensor elements  180 . An SMD-package type may be used (SMD=surface mountable device). For a (psi,z)-plane oriented sensor die  130 , magnetic field gradients dB{R,psi,z}/d{psi,z} may be directly available. These may be detected by horizontal and/or vertical Hall sensor elements  180 . A leaded package type may be used. 
     In order to figure out, which magnetic field gradients are strongest, simulations concerning three magnets will be outlined in the sequel. Magnet M 1  has a 4 mm inner diameter, a 12 mm outer diameter, and 3 mm thickness. The remanence Brem is 220 mT. Magnet M 2  has an 8 mm inner diameter, a 28 mm outer diameter, a 7 mm thickness, and a remanence Brem of 220 mT. Magnet M 3  has a 30 mm inner diameter, a 60 mm outer diameter, a 10 mm thickness with a remanence Brem of 220 mT. Magnet M 3  has a relative permeability μ r , of 1.5. For some simulations, it will be assumed to be mounted on an iron shaft  260  with a 24 mm diameter and an iron disk 1 mm thick. 
     A radial position of the sensor die  130  is 7 mm for M 1  (1 mm air gap), 15.5 mm for M 2  (1.5 mm air gap), and 32 mm for M 3  (2 mm air gap). The gradiometers  110  are assumed to be made up of two point-like sensor elements at 1.5 mm spacing. 
       FIGS. 23, 24 and 25  show diagrams of several gradients as a function of a distance from a mid-plane (z=0) for the magnets M 1 , M 2  and M 3 , respectively. As the simulations show, two signals are large for all magnets: 
             1.5   ⁢           ⁢   mm   ×       dB   R     dz     ⁢           ⁢   and   ⁢           ⁢   1.5   ⁢           ⁢   mm   ×         dB   z     dz     .           
Unfortunately, both have a sine(psi)-dependence. The 3 rd  largest gradient signal is
 
                   1.5   ⁢           ⁢   mm     R     ×       dB   R       d   ⁢           ⁢   ψ         ,         
which has a cosine(psi)-dependence. In order to boost this signal, it may be possible to increase the spacing along the psi-direction from 1.5 mm to, for instance, 2.5 mm (increases the signal by a factor of approximately 1.67) and try to reduce the thickness of the magnet  220 . Therefore, one embodiment, which may be advisable to use under some circumstances, uses the signals dBR/dpsi and dBz/dz at z=0 (i.e. in the mid-plane of the magnet  220 ). Then even for a large magnet diameter of 60 mm amplitudes up to 8 mT may be possible.
 
     A positive aspect of this embodiment may be that a small signal amplitude dBR/dpsi may be detectable by horizontal Hall sensor elements  180 , whereas the larger signals 
             1.5   ⁢           ⁢   mm   ×       dB   z     dz           
may be detectable by, for instance, vertical Hall sensor elements  180 . A ratio of signals may be 40/14=2.9 for M 1 , 22/8=2.8 for M 2 , and 16/4=4 for M 3 . These ratios agree with the offset errors of the horizontal to vertical Hall sensor elements  180 . The signal three-times larger may be detected by vertical Hall sensor elements  180  with three-times larger offset error. Thus, it gives the same angle error, which may optimize the overall angle error of the sensor  100 . In any case, it may be advisable or interesting to center the sensor  100  around the mid-plane of the magnet (z=0). Then small shifts in the z-direction change only little in the amplitude of the signals. If it is assumed that a dBR/dpsi-signal amplitude of 6 mT and a worst case offset of 15 μT for this gradiometer  110  of horizontal Hall sensor elements  180  is present, it may give a worst case angle error of 15/6000*180/pi=0.14° for a single sensor unit  100 . The layout of such a sensor  100  may be identical to the embodiment shown in  FIG. 9 . The arrangement of a complete sensor composed of four sensor units at 0°, 90°, 180°, 270° is shown in  FIG. 26-30 . Of course, additional sensor units  100 , for instance, midway between the ones shown may also be added, for instance, at additional offset angles of 45° to the previously mentioned angles to further improve the accuracy.
 
     Here,  FIG. 26  shows a transparent perspective view of a magnetic angle sensor arrangement  200  as the whole or total system comprising four discrete magnetic angle sensor devices  100 . The sensor devices  100  are equidistantly arranged around the rotation axis  210  around which the shaft  260  with the magnet  220  attached to it, may rotate. The magnet  220  is once again arranged as a through-hole magnet  220  with a diametrical magnetization, which leads, essentially to a sinusoidal dependency of the magnetic field components when rotating the magnet  220  around its rotation axis  210 . The discrete magnetic angle sensor devices  100  are once again arranged on a carrier plate  280  with their silicon dies  130 —as the enlarged portion of  FIG. 26  as shown in  FIG. 27  shows—facing the rotation axis  210 . Although it may be interesting under some conditions that the distance of the sensor elements from the magnet may be minimized, it may also be chosen to place the sensor packages in such a way that the rear side of the die  150  or the die-paddle faces the magnet  220 . This may reduce the risk that the sensor die  150  may be damaged by the magnet  220  during rotation if magnet or sensor are displaced laterally by some fault condition during operation. In other words, a line beginning at the rotational axis  210  and radially outward pointing may intersect the main surfaces  140  of the silicon dies  130  perpendicularly. 
       FIG. 28  shows a solid perspective view of total system of  FIG. 27 , while  FIG. 29  shows the view of  FIG. 28  with the magnet  220  and the mold compounds  310  of the discrete magnetic angle sensor devices  100  removed.  FIG. 29  illustrates once again that the dies  130  are mounted on a leadframe  300  and coupled via pins  290  to the carrier plate  280 . Moreover,  FIG. 29  illustrates the positions of the magnetic sensor elements  180 . 
     The magnet  220  is a small magnet, for instance magnet M 1  as outlined above. In other words, it may be made from a ferrite with remanence Brem of 0.22 T, an inner diameter of 4 mm, an outer diameter of 12 mm and a thickness of 3 mm. 
       FIG. 30  shows an enlarged portion of  FIG. 29  showing one of the discrete magnetic angle sensor devices  100  along with its die  130 . The die  130  is mounted to the leadframe  300  which is connected to the carrier plate  280  by pins  290 . For the sake of simplicity, the bond wires are not shown in  FIGS. 26 to 30 . 
     As outlined in the context of  FIG. 9 , the discrete magnetic angle sensor device  100  comprises once again two gradiometers  110  comprising the previously-mentioned four magnetic sensor elements  180 - 1 , . . . ,  180 - 4 . 
     During the assembly of the sensor system or arrangement  200 , a gauge may be used, which holds the sensor packages  100  in place during a solder or assembly procedure. Thereby, the sensor packages may eventually be placed with highest possible accuracy with respect to the magnet  220  or at least with respect to the hole in the carrier plate  280  configured to accommodate the magnet  220  afterwards. The carrier plate  280  may be, for instance, a circuit board. The gauge may define the R-, psi-, and z-positions of the sensor dies  130 . 
     The two sensor elements  180 - 1 ,  180 - 2  constituting the dBR/dpsi-gradiometer  110 - 1  may be implemented as horizontal Hall sensor devices. The two sensor elements  180 - 3 ,  180 - 4  constituting the dBz/dz-gradiometer  110 - 2  may, for instance, be implemented as vertical Hall sensor elements. Ideally, the horizontal Hall sensor elements  180 - 1 ,  180 - 2  may be in the mid-plane of the magnet  220  (i.e. at z=0). Alternatively, the sensor elements  180 - 3  and  180 - 4  may be implemented by AMR-resistor strips with Barber-poles and optional flipping coils and/or field compensation coils across the AMR-resistor strips. 
     The sensor elements  180  of the gradiometers  110  may be separated from one another by 1.5 mm. With an offset error of, for example, 14 μT for the horizontal Hall sensor elements  180 - 1 ,  180 - 2 , in the dBR/dpsi-gradiometer it may be possible to realize an angle error of 0.1°. Similarly, with an offset of, for example, 40 μT for the vertical Hall sensor elements  180 - 3 ,  180 - 4 , in the dBz/dz-gradiometer it may also be possible to realize an angle error of 0.1°. In other words, it is a mixed vertical/horizontal Hall sensor system, that might realize a system error of 0.1° if four satellites (discrete sensor devices  100 ) are used. 
     Alternatively, one could center the sensor  100  near the z-positions at the maxima of 
               1.5   ⁢           ⁢   mm   ×       dB   R     dz     ⁢           ⁢   and   ⁢           ⁢   1.5   ⁢           ⁢   mm   ×       dB   ψ     dz       ,         
which are also in quadrature with respect to the psi-direction. Also the pair
 
             1.5   ⁢           ⁢   mm   ×       dB   R     dz     ⁢           ⁢   and   ⁢           ⁢       1.5   ⁢           ⁢   mm     R     ×       dB   z       d   ⁢           ⁢   ψ             
may have comparable amplitudes, and they are in quadrature. However, they might suffer from weak signals, which are even weaker than above. Moreover, they might have to be detected by vertical Hall sensor elements  180 , whereas the stronger signal
 
             1.5   ⁢           ⁢   mm   ×       dB   R     dz           
may be detected by horizontal Hall sensor elements  180 . So these systems might need to measure the weak fields with the less appropriate sensor type and this might increases angle errors.
 
     A further embodiment of a sensor arrangement  200  and/or a device  100  according to an embodiment may use a first order gradiometer  110  for dBpsi/dpsi and a second order gradiometer  110  for d 2 Bpsi/dpsi 2  at z=0. The gradiometer amplitudes may be small, but AMR-based sensor elements  180  comprising Barber poles, flipping coils and/or compensation coils may be used very well. The AMR sensor elements  180  may be arranged as depicted in  FIG. 16 , arranged at position z=0. The pairs of sensor elements  180  may be overlapping or their center points may be arranged slightly offset. 
     Similarly, gradiometers  110  of first and second orders may be used for the psi-direction, the AMR sensor elements  180  of which are tilted by 90° to detect the BR-component. This may allow to use very large magnets  220 , leading to very small gradiometer signals. As sensor devices  100  leaded-packages as well as SMD-packages may be used (SMD=surface mountable devices). 
     Assuming a sensor die  130 , the main surface  140  of which is parallel to the (R,z)-plane, which is, for instance, apt for leaded packages. Thus the gradiometers  110  can detect two sorts of slopes directly: d/dR and d/dz for each one of the components BR, Bpsi, Bz. The magnets  220  may be identical, yet the radial positions may be different, because the sensors  100  might not be placed so close to the magnet  220  any more due to package constraints. For instance, it might not be possible to position a Hall device closer than approximately 0.55 mm to the edge of a leaded package, like e.g. PG-SSO-3. Thus, it might be necessary to add 0.25 mm to the radial positions for d/dz-gradiometers  110  and 1.0 mm for d/dR-gradiometers  110  compared to the above given values for tangential die orientation. 
     The radial position of the sensor may be 7.25 mm and 8.0 mm for M 1 , 15.75 mm and 16.5 mm for M 2 , and 32.25 mm and 33 mm for M 3 . Assuming that the sensors  100  are placed the closest possible way to the magnet  220  (i.e. minimum R-positions), for the magnets M 1 , M 2 , M 3  the gradients as a function of a distance from a mid-plane as shown in  FIGS. 31, 32 and 33  may result. 
       FIGS. 31, 32 and 33  show that there are more strong signals than in the tangential die orientation. However, these curves all have sin(psi)-dependencies. The strongest cos(psi)-signal is 
               1.5   ⁢           ⁢   mm   ×       dB   ψ     dR     ⁢   cos   ⁢           ⁢   ψ     ,         
which calls for z=0 position (mid-plane of magnet) for the sensor devices  100 . This component may, however, be detected by horizontal Hall sensors  180 . Together with the vertical Hall sensor elements  180 , that detect
 
             1.5   ⁢           ⁢   mm   ×       dB   z     dz     ⁢   sin   ⁢           ⁢   ψ   ⁢           ⁢   or   ⁢           ⁢   1.5   ⁢           ⁢   mm   ×       dB   R     dR     ⁢   sin   ⁢           ⁢   ψ         
one can implement an angle sensor  100 . The magnetic field components detected by the vertical Hall sensor elements  180  may be three- to four-times larger than the components to be detected by the horizontal Hall effect sensors  180 , which may be favorable to achieve a good angle accuracy.
 
     However, the difference field for the horizontal Hall sensor element may only be 4 mT for magnet M 3 . Like in the case of tangential die orientation, one may increase the distance of the gradiometer (in R-direction) in order to improve the smaller amplitude by a factor of 2. Then the signals may become similar like in the case of tangential die orientation. 
     Layout and arrangement versus magnet  220  of such an arrangement  200  according to an embodiment is shown in  FIGS. 34 and 35 . Each of the equidistantly distributed eight magnetic angle sensor devices  100  comprise {dBz/dz, dBpsi/dR}-gradiometers  110  as shown in  FIG. 22 . Like outlined above, the sensor elements  180  may comprise horizontal Hall sensor elements  180  aligned in the mid-plane of the magnet (i.e. at z=0) and vertical Hall sensor elements  180  sensitive to the z-component of the magnetic field Bz. The layout is similar to the one shown in  FIG. 22  with the dBz/dz-gradiometer  110  being shifted towards the magnet  220 . 
     This layout could also be used for the tangential die orientation discussed above—so it might be possible to implement this layout more generally than the original layout of the embodiment shown in  FIG. 9 . The  FIGS. 34 and 35  show eight discrete magnetic angle sensors  100  positioned at integer multiples of 45°, yet more or less discrete sensors  100  may also be possible, which might lead to a higher or lower accuracy. The magnet  220  may be magnetized in a diametrical direction. 
     The size of the complete angle sensor arrangement  200  may be slightly larger, because the sensor packages  100  are aligned in radial direction (6 mm in diameter). Perhaps this is not relevant in practice, because the circuit board  280  itself might need a certain width for structural stability. 
     An alternative version of a magnetic angle sensor arrangement  200  is shown in  FIGS. 36, 37 and 38  comprising two discrete magnetic angle sensor devices  100  according to an embodiment. The arrangement  200  uses {dBR/dR, dBpsi/dR}-gradiometers  110 , which results in a layout similar to those shown in  FIGS. 10 and 11 , but with gradiometers  110  separated from each other along the z-direction. This layout might enable both types of sensor elements  180  being spaced apart along the same direction, here the radial direction, so that for a given die size, the longer edge of the die  130  may be radially arranged. This increased R-spacing might give larger signals and smaller errors. The magnet  220  is again diametrically magnetized. The sensor system or arrangement  200  may comprise only two sensor units  100  at opposite sides of the magnet. Using two satellites or discrete sensor devices  100  diametrically arranged at positions offset by 180°, may already strongly reduce angle errors caused by an eccentricity of the magnet  220  with respect to the rotation axis  210 . Using an even number of discrete sensor devices  100  may allow placing them such that to each discrete sensor device  100  with respect to the rotation axis, a diametrically placed discrete sensor device  100  exists. In other words, with an integer L indicative of the number of discrete sensor devices  100 , it might be favorable under at least some circumstances to use an even number. This may allow a more efficient way of canceling out eccentricity errors caused, for instance, by the magnet or a carrier of the discrete sensor devices  100 . However, also an odd number of discrete sensor devices  100  (L) may be advisable to implement under some circumstances. 
     In  FIG. 38  the mold compound  310  is removed. The first gradiometer  110 - 1  comprises the two magnetic sensor elements  180 - 1 ,  180 - 2 , which may be implemented as horizontal Hall sensor elements, which detect the magnetic field components  120  perpendicular to the die surface  140  (i.e. Bpsi). The second gradiometer  110 - 1  comprises the magnetic sensor elements  180 - 3 ,  180 - 4 , which may be implemented as vertical Hall sensor elements, which detect the radial magnetic flux component BR. They may be shifted in axial direction (i.e. z-direction) slightly in order not to overlap. However, it is also possible to place them on the same z-position and shift them slightly along the R-direction.  FIG. 38  shows that both sensor elements  180  may be oriented in the same azimuthal direction (e.g. in clockwise direction), yet one of them may also be rotated by 180° around its symmetry axis going through the center pin so that eventually one sensor is oriented clockwise and the other one counterclockwise. This may lead to simpler layout on the circuit board. 
     Based on the assumption that the magnetic sensor elements  180  of the gradiometers  110  separated from one another by a distance of 1.5 mm along the gradiometer directions, with an offset error of 9.5 μT for the horizontal Hall sensor elements for the dBR/dpsi-gradient, an angle error of approximately 0.1° may be achievable. With an offset error of 15 μT for the vertical Hall sensor elements for the dBz/dz-gradient, an angle error of 0.3° may be achievable, yielding a total error of approximately 0.2° for this 2 satellite mixed vertical/horizontal Hall element system. 
     For a system comprising 8 or 12 mixed vertical/horizontal Hall element satellites with a distance of 1.5 mm of the magnetic sensor elements  180  of the gradiometers  110  along the gradiometer directions, with a small magnet (e.g. M 1 ) and an offset error of 9.5 μT a for the horizontal Hall sensor elements for dBpsi/dR, an angle error of 0.1° may be achievable. Similarly, for the vertical Hall sensor elements with an offset error of 32 μT for the dBz/dz-gradient or 30 μT for the dBR/dR-gradient, an angle error of 0.1° may be achievable, yielding a system error of 0.1°. 
     Finally, an embodiment assumes a sensor die  130 , the main surface  140  of which is parallel to the (R,psi)-plane and which is apt for SMD-packages,  FIGS. 39, 40 and 41  show different gradients as function of a distance from a mid-plane for different magnets  220 . Thus, the gradiometers  110  may be able to detect two sorts of slopes directly: d/dR and d/dpsi for each one of the components BR, Bpsi, Bz. The magnets  220  are assumed to be identical to the above discussed magnets M 1 , M 2  and M 3 . Moreover, the radial positions are identical to the last case for the d/dR-gradiometers  110 , whereas the d/dpsi-gradiometers may need radial positions like the d/dz-gradiometers previously discussed. 
     The radial positions of the sensor elements  180  are 7.25 mm and 8.0 mm for M 1 , 15.75 mm and 16.5 mm for M 2  and 32.25 mm and 33 mm for M 3 . Assuming that the discrete sensors  100  are placed the closest possible way to the magnet  220  (i.e. minimum R-position), for the magnets M 1 , M 2 , M 3  the gradients as a function of a distance from the mid-plane are shown in  FIGS. 39, 40 and 41 . 
     There are again two large signals, which are in phase: 
             1.5   ⁢           ⁢   mm   ×       dB   R     dR     ⁢   sin   ⁢           ⁢   ψ   ⁢           ⁢   and   ⁢           ⁢   1.5   ⁢           ⁢   mm   ×       dB   z     dR     ⁢   sin   ⁢           ⁢     ψ   .           
At z=0, embodiments may utilize
 
             1.5   ⁢           ⁢   mm   ×       dB   R     dR     ⁢   sin   ⁢           ⁢   ψ         
and, as cosine-signal, either
 
                 1.5   ⁢           ⁢   mm     R     ×       dB   R       d   ⁢           ⁢   ψ       ⁢   cos   ⁢           ⁢   ψ   ⁢           ⁢   or   ⁢           ⁢   1.5   ⁢           ⁢   mm   ×       dB   ψ     dR     ⁢   cos   ⁢           ⁢     ψ   .           
However, both options may not yield the best results, because they might favor vertical Hall sensor elements  180 . This might give larger angle errors than the arrangements discussed above.
 
     However, it may also be possible to place the sensors  100  next to the shaft  260 , but below or above the magnet  220 . This may avoid large radial distances, and thus increase dpsi=1.5 mm/R for the azimuthal gradiometer  110 . 
     For instance, for a magnet  220  having an inner diameter of 6 mm, an outer diameter of 30 mm, a thickness of 8 mm and an iron shaft  260  through it with a relative permeability of μ r  of 1000 or more, it may be possible to increase the signal strength. The magnet is characterized by a remanence Brem=0.22 T, μ r (recoil)=1.5. 
     The BR-field may be strong near the shaft  260 , but also at larger radial distances. Yet the Bpsi- and Bz-fields may be strong only when the position is not too close to the shaft  260 , since the shaft might short the field lines, as illustrated in  FIGS. 42 and 43  for radial positions of 4.5 mm and 8 mm, respectively. The gradiometer outputs along the psi-direction are comparably weak, except for dBR/dpsi close to the shaft  260 , as illustrated in  FIGS. 44 and 45  for the two radial distances mentioned before.  FIGS. 46 and 47  finally show the gradiometer outputs along the R-direction for 1.5 mm-gradiometers located at R=4.5 mm and 6.5 mm, respectively, with the second sensor element  180  of the gradient  110  being placed 1.5 mm further outwards. Their output is also comparably weak. 
     Both Bpsi/dR and dBR/dpsi dependencies are comparably strong for small radial distances. However, they are both cos(psi)-functions. 
     However, it might be a better solution to use the pair of gradients {dBR/dpsi, dBz/dR} of two gradiometers  110 . As outlined in context of  FIG. 39 , they might have cosine and sine dependencies versus angle psi and their amplitudes are approximately 13 mT and 4 mT, respectively, whereby the weaker signal might be detected by horizontal Hall sensor elements, if the package is an SMD-type as shown in  FIGS. 48-51 . 
       FIGS. 48 and 49  show perspective view of a magnetic angle sensor arrangement  200  according to an embodiment comprising three equidistantly arranged discrete magnetic angle sensor devices  100  as surface-mounted devices (SMD) arranged on a carrier plate  280  which may be, for instance, implemented as a printed circuit board. The magnet  220  is once again implemented as a through-hole magnet  220  mechanically fixed to a shaft  260 . However, the discrete magnetic angle sensor devices  100  are, arranged along the rotation axis  210  between the magnet  220  and the carrier plate  280 , as also shown in  FIG. 51   b.    
       FIG. 50  shows a perspective view of the magnetic angle sensor arrangement  200  with the magnet  200  removed.  FIG. 50  illustrates that the discrete magnetic angle sensor devices  100  comprise a mold compound  310 , which has been removed to offer an insight into the inner structure of the discrete magnetic angle sensor devices  100  in the enlarged perspective view of  FIG. 51 a   .  FIG. 51 a    illustrates once again that the discrete magnetic angle sensor devices  100  comprise a sensor die  130  with a main surface  140  onto which the magnetic sensor elements  180  are provided. The die  130  is mounted onto a leadframe  300  coupled to three pins  290 , where—for the sake of simplicity—the bond wires are once again not shown in  FIG. 51 a   . A central pin  290  is coupled to the leadframe  300  providing, for instance, a ground potential for the discrete magnetic angle sensor device  100 . Naturally, the number of pins  290  may be chosen differently from three in this embodiment as well as the previously-described embodiments and the embodiments described below. Moreover it is also possible to reverse the orientation of the sensor packages, so that the die is placed face-down and the die-paddle of the leadframe faces the moving magnet. This might be more reliable if the air-gap between moving magnet and sensor package is small, because then the sensor die is protected not only by the mold compound but also by the die paddle from the moving magnet. 
     The pins  290  of the sensor packages  100  may be bent towards a surface of the circuit board  280 , where they may be soldered to copper traces. However, this bending is not shown in  FIGS. 48-51 . Instead of a package  100  with all pins  290  on a single side, one may also use conventional SMD-packages with pins  290  on opposite sides. It is also possible to use VQFN types of packages, which have no leads standing off the package body, but only lands or pads or gull-wings. Hence, instead of pins  290  also other coupling structures may be implemented to mechanically and/or electrically couple the discrete sensor devices  100  to a system. In other words, the term “coupling structure” serves as a summarizing term for all these structures including the pins  290 , which may be fully or partially replaced in embodiments by appropriate coupling structures. 
     The shaft may be non-magnetic (μ r =1) or ferrous (μ r &gt;1000). The magnet  220  may be magnetized in diametrical direction. The system comprises three sensor units  100  placed at integer multiples of 120° around the shaft  260 . A gap between the sensor packages  100  and the bottom surface of the magnet  220  may be chosen to be about 1 mm. The sensor elements  180 - 1 ,  180 - 2  on the die  130  may be implemented as horizontal Hall sensor elements detecting the Bz-component, whereas the sensor elements  180 - 3 ,  180 - 4  on the die  130  may be implemented as vertical Hall sensor elements, that detect Bpsi-components. 
     In order to find locations, where the gradiometer outputs are large, magnetic field derivatives in the (R,z)-plane for magnet M 3  are determined.  FIGS. 52, 53 and 54  show the gradients of the three magnetic field components with respect to the radial, the tangential and the axial direction, respectively, as a function of a parameter s. In  FIG. 52 , the wide curve illustrates the dBz/dR*1.5 mm-dependency, the thin light curve the dBpsi/dR*1.5 mm-dependency, and the thin dark line the dBR/dR*1.5 mm-dependency. Similarly, in  FIG. 53 , the wide curve illustrates the dBz/dpsi*1.5 mm/R-dependency, the thin light curve the dBpsi/dpsi*1.5 mm/R-dependency, and the thin dark line the dBR/dpsi*1.5 mm/R-dependency. Finally, in  FIG. 53 , the wide curve illustrates the dBz/dz*1.5 mm-dependency, the thin light curve the dBpsi/dz*1.5 mm-dependency, and the thin dark line the dBR/dz*1.5 mm-dependency. 
     The gradients decay with larger distance to the surface of the magnet  220 . Therefore, it might be possible to compare the gradients along an envelope in 1 mm distance to the surface. In the following, the parameter s is the length along the following path: Beginning at s=0, which corresponds to (R,z)=(0, 6 mm). Then s increases up to s=31 for (R,z)=(31 mm, 6 mm). For larger s, the location is (R,z)=(31 mm, 37 mm-s). Thus, for example, s=31 corresponds to (R,z)=(31 mm, 6 mm) and s=37 corresponds to (R,z)=(31 mm, 0 mm). 
     From these plots, the derivatives at a specific location can be determined and they can be ranked according to their strength. Finding strong sin-cos-pairs may offer a possibility of implementing a corresponding angle sensor device  100  or a corresponding arrangement  200 . Mainly locations are interesting, which have zero slope (d/ds=0, i.e. the extremes), because they represent points, which are robust against misplacement in s-direction. However, a problem of misplacement orthogonal to the s-direction may still exist. 
     Summary for s=37 mm which is (R,z)=(31 mm, 0 mm): 
                                                    1.5 mm * dBR/dR   22 mT   Sin (psi)           1.5 mm * dBz/dz   18 mT   Sin (psi)           1.5 mm * dBpsi/dR    5 mT   Cos (psi)           1.5 mm/R * dBR/dpsi   4.5 mT    Cos (psi)                        
Summary for s=33 mm to 34 mm which is (R,z)=(31 mm, 3 mm to 4 mm):
 
                                                    1.5 mm * dBz/dz   28 mT   Sin (psi)           1.5 mm * dBR/dR   24 mT   Sin (psi)           1.5 mm/R * dBR/dpsi   4.4 mT    Cos (psi)                        
Summary for s=32 mm to 33 mm which is (R,z)=(31 mm, 4 mm to 5 mm):
 
                                                    1.5 mm * dBR/dz    45 mT   Sin (psi)           1.5 mm * dBz/dR    32 mT   Sin (psi)           1.5 mm/R * dBz/dpsi   3.7 mT   Cos (psi)           1.5 mm * dBpsi/dz   3.7 mT   Cos (psi)                        
Summary for s=30 mm which is (R,z)=(30 mm, 6 mm):
 
                                                    1.5 mm * dBR/dR   40 mT   Sin (psi)           1.5 mm * dBz/dz   28 mT   Sin (psi)           1.5 mm/R * dBz/dpsi    4 mT   Cos (psi)                        
Summary for s=16 mm which is (R,z)=(16 mm, 6 mm):
 
                                                    1.5 mm * dBz/dR    27 mT   Sin (psi)           1.5 mm * dBR/dz    17 mT   Sin (psi)           1.5 mm * dBpsi/dR   4.7 mT   Cos (psi)           1.5 mm/R * dBR/dpsi   4.7 mT   Cos (psi)                        
Summary for s=15 mm which is (R,z)=(15 mm, 6 mm):
 
     
       
         
           
               
               
               
               
             
               
                   
                   
               
             
            
               
                   
                 1.5 mm * dBR/dR 
                 39 mT 
                 Sin (psi) 
               
               
                   
                 1.5 mm * dBz/dz 
                 30 mT 
                 Sin (psi) 
               
               
                   
                 1.5 mm/R * dBz/dpsi 
                 7.5 mT  
                 Cos (psi) 
               
               
                   
                 1.5 mm * dBpsi/dz 
                  5 mT 
                 Cos (psi) 
               
               
                   
                   
               
            
           
         
       
     
     Thus, the strongest cosine-signal occurs at R=15 mm. The small radial distance increases the factor “1.5 mm/R” in front of dBz/dpsi. Accordingly, an even stronger gradient may be found in the bore of the magnet, as illustrated in  FIGS. 55, 56 and 57  corresponding to FIGS.  52 ,  53  and  54 , respectively. However, instead of the diagrams being a function of the parameter s, in these figures the axial distance from the mid-plane of the magnet  220  is used. 
     Summary for (R,z)=(14 mm, 0 mm): 
     
       
         
           
               
               
               
               
             
               
                   
                   
               
             
            
               
                   
                 1.5 mm * dBz/dz 
                   17 mT 
                 Sin (psi) 
               
               
                   
                 1.5 mm * dBR/dR 
                 12.6 mT 
                 Sin (psi) 
               
               
                   
                 1.5 mm/R * dBR/dpsi 
                  6.7 mT 
                 Cos (psi) 
               
               
                   
                   
               
            
           
         
       
     
     Thus, the pair {dBz/dz, dBR/dpsi} may be a good alternative for a tangential die orientation, which may fit well into the small gap between the magnet  220  and the shaft  260 . If the sensor packages  100  are, for instance, arranged on a circle with a radius of 14 mm (R=14 mm) and each package  100  has about 6 mm width, a maximum number of 14 sensor units  100  can be accommodated. Thus, an angular spacing of 30° with a total of 12 discrete sensor units  100  may still be possible and it should be enough to achieve very good angular accuracy, if the angle estimations of these 12 sensor units are combined (e.g. averaged) by a control unit. 
     In the following, results of a finite element method simulation (FEM-simulation) will be discussed. These simulations are based on an iron shaft  260  with μ r  of 1000 or more and a diameter of 24 mm. The magnet  220  has an inner diameter of 30 mm, an outer diameter of 60 mm, a thickness of 10 mm and it is magnetized with remanence Brem=0.22 T (μ(recoil)=1.5). 
       FIG. 58  shows a first result of the FEM-simulation. A curve  320  shows the gradiometer signal 1.5 mm/R*dBR/dpsi as a function of a distance from the mid-plane of the magnet  220  and at a radial distance R=14 mm, given in units of Tesla. A curve  322  shows the gradiometer signal proportional to dBz/dpsi and a curve  324  the gradiometer signal proportional to dBx/dpsi. The curve  320  shows the value 0.007783 T at a position z=0 mm. 
     This is approximately 16% larger than in the case of a shaft with μ r =1. The other two components (1.5 mm/R*dBz/dpsi, 1.5 mm/R*dBpsi/dpsi) are smaller due to the iron shaft having values of approximately 57% and 38% of the values for the non-magnetic shaft  260 , respectively. 
     However, also the sine-signal 1.5 mm*dBz/dz (curve  330  in  FIG. 59 ) is much weaker due to the ferrous shaft: only 6.3 mT at R=14 mm and z=0. 
     Hence, a ferrous shaft  260  may reduce one of the two gradiometer signals. Therefore, it might be advisable not to place the sensor units  100  near the inner diameter of a ring-shaped magnet  220 , if it is mounted on a ferrous shaft  260 . If the shaft  260  is non-magnetic, it might be advisable to place the sensor units  100  in the mid-plane of the magnet ring  220  near the inner diameter. 
       FIG. 60  shows a schematic plan view of a discrete magnetic angle sensor device  100  comprising three lateral Hall sensor devices  180 - 1  (H 1 ),  180 - 2  (H 2 ) and  180 - 3  (H 3 ). The three Hall sensor devices  180 , which may be replaced by other magnetic sensor devices  180  such as AMR sensor devices  180  in other embodiments, are arranged on a die  130  and form a first gradiometer  110 - 1  comprising the first and third Hall sensor element  180 - 1 ,  180 - 3 , and a second gradiometer  110 - 2  comprising all three Hall sensor devices  180  used to estimate an angle of a magnetic field. The first gradiometer  110 - 1  is a first order gradiometer  110 , the second gradiometer  110 - 2  a second order gradiometer  110 . 
     The Hall sensor elements may be arranged on common line  340  equidistantly separated from one another along the line  340 . Optionally, as shown in  FIG. 60 , the second Hall sensor element  180 - 2  may be displaced perpendicular to the line  340 . For instance, the three sensor elements  180  may be arranged on a circular arc, the center of which may coincide with the rotation axis  210 . 
     A first gradiometer signal GS 1  may be based on the signals H 1 , H 3  of the first and third sensor elements  180 - 1 ,  180 - 3 , respectively. The first gradiometer signal GS 1  may be equal to GS 1 =H 1 -H 3 . A second gradiometer signal GS 2  may be based on the signals H 1 , H 2 , H 3  of all sensor elements  180 - 1 ,  180 - 2 ,  180 - 3 , respectively, and equal to GS 2 =H 1 +H 3 −2*H 2 . As a consequence, the first gradiometer signal GS 1  may be proportional to sin(psi), the second gradiometer signal GS 2  to cos(psi), when, for instance, the line  340  is tangentially oriented to a circle being concentrically arranged around the rotation axis  210 . Accordingly, both gradiometers  110  are arranged along a common gradiometer direction  150  being collinear to the line  340 . 
     As outlined before, the first gradiometer signal GS 1  may be proportional to sin(psi). As a consequence, the second gradiometer signal GS 2  is then proportional to cos(psi), since it is (approximately) proportional to the derivative with respect to psi of the first gradiometer signal GS 1 . Hence, based on both signals GS 1  and GS 2  the angle may be derivable based on a arctan-computation (psi=arctan (GS 1 , k*GS 2 ) with a factor k) or a similar calculation as outlined before. 
     The device  100  may be arranged with respect to the magnet  220  as shown, for instance, in  FIG. 8 or 60 . Such a package may, for instance, be arranged under the magnet  220  (not shown in  FIG. 60 ). Alternatively, the device  100  may be moved radially outwards such that the rotation axis  210  does not intersect the die  130  as shown in  FIG. 60 . Hence, the device  100  may eventually be used as an on-axis sensor or as an off-axis sensor. In the latter case, two discrete devices  100  may be used, when the shaft  260  is arranged in between the two devices  100 . 
     It may be interesting to use such a device  100 , due to the small number of sensor devices  180  that may offer a limited space and a small energy consumption compared to other systems and devices. Moreover, only sensor elements  180  of the same type may be used, which may be fabricated having similar technology-related and thermal properties as well as similar expected life times. 
       FIG. 61  shows a plan view of a further device  100 , which may be based on vertical Hall sensor elements  180 , AMR-sensor elements  180  or sensor elements  180  of another type. The device  100  comprises four sensor elements  180 - 1 ,  180 - 2 ,  180 ′- 2 ,  180 - 3 , which may be single sensor elements  180  or comprise series connections of, for instance, xMR-sensor elements such as AMR-sensor elements. Accordingly, the sensor elements  180 - 1 ,  180 - 2  may form a first bridge  345 - 1 , while the sensor elements  180 ′- 2  and  180 - 3  form a second bridge  345 - 2 . Depending on whether the sensor elements  180  shown in  FIG. 61  are single elements  180  or series connections of sensor elements  180  forming half bridges, the bridges  345  are half bridges or full bridges. The sensor elements  180  may be arranged on a die  130 . 
     Assuming the sensor elements  180 - 1 ,  180 - 2 ,  180 ′- 2  and  180 - 3  provide the signals A 1 , A 2 , A 2 ′ and A 3 , respectively, two gradiometer signals GS 1 , GS 2  may be derivable based on the bridges  345 - 1 ,  345 - 2  providing the signals B 1 =A 1 -A 2 , B 2 =A 2 ′-A 3 , respectively. GS 1  may be given by GS 1 =B 1 +B 2 =(A 1 -A 2 )+(A 2 ′-A 3 ) and GS 2  by GS 2 =B 1 -B 2 =(A 1 -A 2 )−(A 2 ′-A 3 ). In other words, the signals B 1  and B 2  are first order gradiometer signals. The signal GS 1  is also a first order gradiometer signal and it is identical to the sum of the first order gradiometer signals B 1  and B 2 . The signal GS 2  is also a second order gradiometer signal and it is identical to the difference of the first order gradiometer signals B 1  and B 2 . 
     A further embodiment of a device  100  may comprise four sensor elements  180  arranged on a circle, two of which are, for instance, diametrically arranged lateral Hall sensor elements  180  and two of which are diametrically arranged magnetic field effect transistors (MAG-FET). Naturally, the magnetic field effect transistors may be replaced by triangular Hall sensor elements, a signal is derivable at a node in between the two triangular Hall sensor elements. 
       FIGS. 62 a , 62 b  and 62 c    show plan views of further devices  100  according to an embodiment, which will be described below. These devices  100  use the pair {dB R /dz, dB R /dψ}. If the die  130  is parallel to the (psi,z)-plane, it may be possible to use a single type of magnetic field sensor  180  sensitive to, for instance, BR. This type of sensor element  180  may be a horizontal Hall plate, which may have a smaller error than vertical Hall sensor elements. In this case it might be interesting to use a rectangular die  130 , where the spacing of the pair aligned in the z-direction is smaller than the spacing of the pair aligned in psi-direction. In the example discussed above, when, for instance, the first one has a spacing of only 0.9 mm and the second one has a spacing of 2.45 mm, this may give an amplitude of 15.7*0.9/1.5=9.4 mT for the gradiometer in z-direction and 3.6*2.45/1.5=5.9 mT for the gradiometer in psi-direction. Thus, the difference of signal amplitudes for both gradiometers  110  may be smaller than given above. If each gradiometer  110  has a worst case offset error of 15 this may give a worst case angle error of 15μ/5.9 m*180°/pi=0.15°. 
     As  FIG. 62 b    shows, it may even be possible to use only three sensor elements  180  in order to reduce the power consumption and the area consumption of the circuit. In this case the sensor  100  may compute GS 1 =BR(right)−BR(left) to obtain a first sinusoidal signal as the gradiometer signal GS 1  of the first order gradiometer  110  and GS 2 =BR(center)−(BR(right)+BR(left))/2 to obtain a second sinusoidal signal that is in quadrature to the first sinusoidal signal (cosine signal) as the gradiometer signal GS 2  of the second order gradiometer  110 . 
     Alternatively, as shown in  FIG. 62 c   , the upper sensor element  180  may be placed asymmetrically forming an “L”-shaped design. 
     Then the sensor may compute GS 1 =BR(lower right)−BR(lower left) to obtain a first sinusoidal signal and GS 2 =BR(upper left)−(BR(lower right)+BR(lower left))/2 to obtain a second sinusoidal signal. Alternatively, the second sinusoidal signal by be obtained by GS 2 ′=BR(upper left)−BR(lower left). GS 2 ′ is a gradiometer signal of a first order gradiometer, whereas GS 2  is an approximate second order gradiometer signal. These two sinusoidal signals may not be exactly 90° phase shifted (with respect to psi) any more but for a given magnet this orthogonality error can be accounted for. Naturally, the gradiometer signals GS 1 , GS 2  may be used as outlined above in context with  FIG. 62   b.    
       FIG. 63  shows a perspective view of a shaft  260 , a disc-like magnet  220  with a central bore, through which the shaft  260  runs. The magnet  220  is once again diametrically magnetized. However, the arrangement shown further comprises a disc  350  having the same outer diameter as the ring-shaped magnet  220 . The disc  350  furthermore comprises a sensor bore through which the shaft  260  also extends. The disc  350  comprises a thickness along the rotation axis  210 , which is smaller than the outer diameter of the disc  350 . 
     The disc  350  may be ferromagnetic or ferrous, for instance, made from a ferrous material and having a thickness of 1 mm along the rotation axis  210  (z-direction). It is fixed on the shaft  260 . Being produced from a ferrous material, it may comprise a relative permeability μ r  of 1000 or more. Based on the structure shown in  FIG. 63 , a FEM-simulation has been carried out. 
     The presence of the ferrous disk  350  alone—using a non-magnetic shaft  260 —reduces dBR/dpsi and increases dBz/dz. This may eventually degrade an achievable accuracy. To sum up, the strongest signals may be obtained, if the shaft  260  and the disk  350  are ferrous. Then the amplitudes of the {dBz/dz, dBR/dpsi}-gradiometer pair  110  with a spacing of 1.5 mm may be 15 mT and 9.5 mT, respectively. If, for instance, a tangential die orientation is used, this may fit perfectly into the narrow gap. However, a limitation may be present for the dBz/dz-gradiometer, because it might only be detectable by vertical Hall sensor elements  180 , whereas a dBR/dpsi-gradient may be detectable by horizontal Hall sensor elements  180 . 
     Assuming a worst case vertical Hall sensor element offset error of 75 μT, an angle error of 75/15000·180°/π=0.3° may result, while in the case of a horizontal Hall sensor element offset of 15 μT may give an angle error of 15/9500*180°/π=0.1°. In this case one may increase the z-spacing of the vertical Hall sensor elements of the corresponding gradiometer and reduce the psi-spacing of the horizontal Hall sensor elements. Additionally or alternatively, one may reduce the thickness of the magnet  220  to increase the gradient dBz/dz. 
       FIG. 64  shows a perspective view of a mechanical setup of a magnetic angle sensor arrangement  200 . The arrangement  200  comprises a shaft  260 , which may optionally comprise a ferrous material, such as iron. The material may comprise a relative permeability μ r  of 1000 or more (μ r &gt;1000). 
     The arrangement  200  further comprises a disk  350 , which may optionally and additionally or alternatively comprise a ferrous material as well. The arrangement  200  further comprises an optionally diametrically magnetized magnet  220 . The disk  350  may have the same outer diameter as the magnet  220 , which may be ring- or cylinder-shaped. The arrangement  200  may further comprise a carrier plate  280 , which may be implemented as a single or more than one circuit board. In the embodiment shown in  FIG. 64-67 , the carrier plate  280  comprises two separate printed circuit boards. The carrier plate  280  or—in other words—the printed circuit board(s) may hold the sensor units or discrete magnetic angle sensor devices  100  according to an embodiment. The arrangement  200  may comprise one or more sensor devices  100 . The carrier plate  280  may further be configured to make electrical connections to and possibly also between the sensor devices  100 . 
       FIG. 65  shows a bottom view with parts of the leads or pins  290  of the sensor units  100  going through the circuit board. The sensor devices  100  may be positioned on a circle around the shaft  260 . Consequently, the sensor devices  100  are in this embodiment not outside the magnet  220 , but inside the bore of the magnet  220 , which has the shape of a ring. The circuit boards  280  are split in two halves, because this may simplify an assembly procedure of the arrangement  200 . The two halves may be held together by some mechanical fixture or they are mounted on a single frame, which holds them together, to name just two alternative approaches. 
     Of course, it may be advisable, if not vital that the position tolerances of the sensor units  100  versus the magnet  220  and the shaft  260  are not unduly impaired by the splitting of the circuit board. Naturally, the circuit board (carrier plate  280 ) may be split asymmetrically such that one part (the larger one) may hold at least two opposite sensor units  100 . Then it may be possible to compare the readings of the opposite two units  100  in a test after assembly with the readings of other opposite pairs. If they are not similar, this might be an indication that the two boards are not positioned accurately enough or symmetrically enough around the rotation axis  210 . 
     In the  FIGS. 66 and 67  a quarter of the magnet  220  is cut away and one of the two circuit boards is hidden in order to enable a better view to the sensor units  100 . The sensor dies  130  are oriented in a tangential way so that their main surfaces  140  are parallel to the (psi,z)-plane. The two gradiometers are shown. The gradiometers comprise two sensor elements  180 , which may, for instance, be implemented as horizontal Hall sensor elements making up the dBR/dpsi-gradiometer  110 . The discrete sensor device  100  further comprises the two sensor elements  180 ′, which may be implemented as vertical Hall sensor elements making up the dBz/dz-gradiometer  110 . 
     The active or main surface  140  of the dies  130  is here oriented towards the inner diameter of the magnet  220  and not towards the shaft  260 . In fact, from a magnetic point of view, it may be beneficial, if the gap between the sensor elements  180  and the magnet  220  is small, whereas the gap between sensor elements  180  and the shaft  260  may be larger. 
       FIG. 67  shows the axial position of the elements  180 . The centers of both gradiometers  110  may be in the mid-plane of the magnet  220 . 
     Additionally, a further disk may be added below the circuit board or—in other words—on a side facing away from the disk  350 . The further disk may optionally comprise or be fabricated from a ferrous material as outlined before. For specific dimensions this may increase the signals and it may also increase a shielding against external fields. It may—additionally or alternatively—improve the mechanical robustness of the system, although it might make the assembly of the arrangement  200  more complicated. 
     Hysteresis effects of all ferrous parts fixed to the magnet  220  may be irrelevant, since they are fixed with respect to the magnetic field they are not likely to distort the functional relationship of signals-versus-rotation-angle. 
     In the  FIG. 64-67  the ferrous disk  350  has a larger diameter than the magnet  220 . Yet it may also have a smaller one or an identical one. The outer diameter may not be so important for magnetic reasons, since the fields may be evaluated at the inner diameter. However, it may be advisable to use specific profiling of the disk  350  near the inner diameter. However, such a profile might be independent of the rotation angle psi (no psi-dependence). It may have an (R,z)-dependence, for instance, the disk  350  may have a conical shape at diameters between the magnet  220  and the shaft  260 . Alternatively or additionally, it may approach the discrete sensor units  100  in axial direction by, for instance, about 1 mm there. 
     Implementing such a disk  350 , which is also referred to as a swashplate, may be freely combined with embodiments described before, for instance, having several sensor units  100 , each one outputting an angle estimation and a control unit  170 , which combines all these estimations to get an improved angle estimation. 
     Since more than one sensor unit  100  may be used, they should provide means to obtain synchronous readings. For instance, as outlined before, an external control unit like a microprocessor (μC) may be able to send them a synchronization pulse or triggering signal that initiates a synchronized measurement or sampling of magnetic fields or estimated rotation angles at all sensor units  100 . Since the discrete sensors  100  usually run asynchronously according to their own clock generated in the individual sensor units  100 , they may have algorithms on-board, which interpolate or extrapolate synchronized angle values from former readings. If it is assumed that, for instance, a discrete sensor  100  computes updates of values of rotation angle every 30 μs, it may store a predefined number of preceding values, for instance the preceding five values, in an on-chip register. If a synchronization pulse is applied to the sensor unit  100  22 μs after the last update was computed, the sensor unit  100  may fit, for instance, a 2 nd -order polynomial through the five values in the register and compute the extrapolated value at the time of the synchronization pulse. This value is then communicated to the microprocessor. Thus the microprocessor may obtain synchronized values of all sensor units  100  and it may use these readings to get an improved estimate of rotation angle by various algorithms, for instance, simply by taking the average of all values with respect to their position in the arrangement  200 . 
       FIG. 68  shows a perspective view of a magnet  220 , having a generic form, which may offer a diametrical and/or homogeneous magnetization along with a rotational shape. However, it may not be a magnet practical to implement. However, it may allow purely sinusoidal fields according to the following equations:
 
 B   R ( R,ψ,z )= b   R ( R,z )sin ψ  (19)
 
 B   ψ ,( R,ψ,z )= b   ψ ( R,z )cos ψ  (20)
 
 B   z ( R,ψ,z )= b   z ( R,z )sin ψ  (21)
 
       FIG. 69  shows a further magnet  220  comprising three toroidal parts  360 - 1 ,  360 - 2 ,  360 - 3  along with results of a numerical simulation of the magnetization based on a real demagnetization curve of a typical ferrite material. FIG. illustrates that the magnets  220  shown before, may, for instance, also be implemented based on more than one part  360 . Moreover,  FIG. 70  illustrates the sinusoidal dependencies of the x- and y-components of the magnetic field Bx, By. 
     The description and drawings merely illustrate the principles of the invention. It will thus be appreciated that those skilled in the art will be able to devise various arrangements that, although not explicitly described or shown herein, embody the principles of the invention and are included within its spirit and scope. Furthermore, all examples recited herein are principally intended expressly to be only for pedagogical purposes to aid the reader in understanding the principles of the invention and the concepts contributed by the inventor(s) to furthering the art, and are to be construed as being without limitation to such specifically recited examples and conditions. Moreover, all statements herein reciting principles, aspects, and embodiments of the invention, as well as specific examples thereof, are intended to encompass equivalents thereof. 
     Functional blocks denoted as “means for . . . ” (performing a certain function) shall be understood as functional blocks comprising circuitry that is adapted for performing or to perform a certain function, respectively. Hence, a “means for s.th.” may as well be understood as a “means being adapted or suited for s.th.”. A means being adapted for performing a certain function does, hence, not imply that such means necessarily is performing said function (at a given time instant). 
     The methods described herein may be implemented as software, for instance, as a computer program. The sub-processes may be performed by such a program by, for instance, writing into a memory location. Similarly, reading or receiving data may b e performed by reading from the same or another memory location. A memory location may be a register or another memory of an appropriate hardware. The functions of the various elements shown in the Figures, including any functional blocks labeled as “means”, “means for forming”, “means for determining” etc., may be provided through the use of dedicated hardware, such as “a former”, “a determiner”, etc. as well as hardware capable of executing software in association with appropriate software. When provided by a processor, the functions may be provided by a single dedicated processor, by a single shared processor, or by a plurality of individual processors, some of which may be shared. Moreover, explicit use of the term “processor” or “controller” should not be construed to refer exclusively to hardware capable of executing software, and may implicitly include, without limitation, digital signal processor (DSP) hardware, network processor, application specific integrated circuit (ASIC), field programmable gate array (FPGA), read only memory (ROM) for storing software, random access memory (RAM), and non-volatile storage. Other hardware, conventional and/or custom, may also be included. Similarly, any switches shown in the Figures are conceptual only. Their function may be carried out through the operation of program logic, through dedicated logic, through the interaction of program control and dedicated logic, the particular technique being selectable by the implementer as more specifically understood from the context. 
     It should be appreciated by those skilled in the art that any block diagrams herein represent conceptual views of illustrative circuitry embodying the principles of the invention. Similarly, it will be appreciated that any flow charts, flow diagrams, state transition diagrams, pseudo code, and the like represent various processes, which may be substantially represented in computer readable medium and so executed by a computer or processor, whether or not such computer or processor is explicitly shown. 
     Furthermore, the following claims are hereby incorporated into the Detailed Description, where each claim may stand on its own as a separate embodiment. While each claim may stand on its own as a separate embodiment, it is to be noted that—although a dependent claim may refer in the claims to a specific combination with one or more other claims—other embodiments may also include a combination of the dependent claim with the subject matter of each other dependent claim. Such combinations are proposed herein unless it is stated that a specific combination is not intended. Furthermore, it is intended to include also features of a claim to any other independent claim even if this claim is not directly made dependent to the independent claim. 
     It is further to be noted that methods disclosed in the specification or in the claims may be implemented by a device having means for performing each of the respective steps of these methods. 
     Further, it is to be understood that the disclosure of multiple steps or functions disclosed in the specification or claims may not be construed as to be within the specific order. Therefore, the disclosure of multiple steps or functions will not limit these to a particular order unless such steps or functions are not interchangeable for technical reasons. 
     Furthermore, in some embodiments a single step may include or may be broken into multiple substeps. Such substeps may be included and part of the disclosure of this single step unless explicitly excluded.