Patent Publication Number: US-10309803-B2

Title: Detecting sensor error

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     This application claims the benefit under 35 U.S.C. § 119(e) of U.S. Provisional Application No. 62/118,963, filed Feb. 20, 2015, and titled “DETECTING SENSOR ERROR,” the entirety of which is hereby incorporated by reference. 
    
    
     BACKGROUND 
     Technical Field 
     The described technology generated relates to sensors and, more specifically, to apparatus and methods for detecting sensor errors through various combinations of sensing elements. 
     Description of the Related Art 
     Magnetic sensors can be implemented to obtain linear or circular position or angle information of a mechanical component, such as a shaft, in various applications, including an automotive steering system. Magnetic sensing elements used in magnetic angle sensors often suffer from changing sensitivity levels and non-linearity errors due to, for example, temperature change, and it is desirable to implement a sensor error detection mechanism for magnetic sensors. 
     SUMMARY OF CERTAIN INVENTIVE ASPECTS 
     The methods and devices of the described technology each have several aspects, no single one of which is solely responsible for its desirable attributes. 
     One embodiment is an apparatus for detecting sensor error, the apparatus including a first channel including a first magnetic sensing element, the first channel configured to provide first channel data, a second channel including a second magnetic sensing element, the second channel configured to provide second channel data, a third channel configured to receive a first signal from the first sensing element and a second signal from the second sensing element, the third channel configured to provide third channel data, and a processor. The processor is configured to receive first channel data from the first channel, second channel data from the second channel, and third channel data, compute expected third channel data from the first channel data and the second channel data, and detect sensor error based at least partly on a comparison of the expected third sensor data with the third channel data. 
     Each of the first and second sensing elements can be a magnetoresistive sensor. For instance, the first and second sensing elements can be anisotropic magnetoresistance (AMR) sensors, giant magnetoresistive (GMR) sensors, or tunneling magnetoresistive (TMR) sensors. The processor can be further configured to determine an angle and a radius based at least partly on the first sensor data and the second sensor data, compute expected third sensor data based at least partly on the angle and the radius, and detect the sensor error by comparing the expected third sensor data with the third sensor data. The first sensing element can include a first full bridge and the second sensing element can include a second full bridge, and the first signal is an output from the first full bridge and the second signal is an output from the second full bridge. The third signal is an output from a half-bridge of the first full bridge and a half-bridge of the second full bridge. The first sensing channel can be configured to process two half bridge outputs of the first full bridge and the second sensing channel is configured to process two half bridge outputs of the second full bridge. 
     Each of the first, second, and third sensing channels can further include an amplifier and a sampling circuit. The first and second sensing channels can be embodied in a single die. The first and second sensing elements can be collocated. Each of the first and second elements can be configured to provide a differential output. The apparatus can be configured to provide an error flag responsive to the detected error satisfying a threshold. Another embodiment can be an electronic power assisted steering system including the abovementioned apparatus for detecting sensor error. 
     Another embodiment is a method for detecting sensor error, the method including receiving first sensor data from a first channel and second sensor data from a second channel, determining an angle and a radius based at least partly on the first sensor data and the second sensor data, computing the expected third sensor data based at least partly on the angle and the radius, and detecting sensor error based at least partly on a comparison of the expected third sensor data with the third sensor data. The first channel includes a first magnetic sensing element. The second channel includes a second magnetic sensing element, the second magnetic sensing element oriented at a first angle relative to the first sensing element. The third channel is configured to receive a first signal from the first magnetic sensing element and a second signal from the second magnetic sensing element. 
     The method for detecting sensor error can further include determining the angle, which can include computing the angle using an arctan function. The method can further include providing an error flag responsive to the detected error satisfying a threshold. Each of the first and second sensing elements can be an anisotropic magnetoresistance (AMR) sensor. The first angle can be approximately 45 degrees. 
     Another embodiment is an apparatus for detecting sensor error, the apparatus including a first magnetic sensing element comprising a first full bridge having two half bridge outputs, a second magnetic sensing element comprising a second full bridge having two half bridge outputs, three differential amplifiers, and a processor in communication with the three differential amplifiers. Each of the three differential amplifiers are configured to receive two signals from the half bridges of the first magnetic sensing element and the second magnetic sensing element, and wherein each of the differential amplifiers are configured to receive different combinations of the half bridge signals. The processor is configured to receive data associated with outputs of two of the three differential amplifiers, compute expected data associated with an output of the third of the three differential amplifiers based on the received data, detect sensor error based at least partly on a comparison of the expected data associated with the output of the third differential amplifier and data associated with an output of the third differential amplifier. The first magnetic sensing element and the second magnetic sensing element can be oriented at approximately 45 degrees to each other when they are AMR sensors. The first magnetic sensing element and the second magnetic sensing element can be oriented at approximately 90 degrees to each other when they are GMR sensors or TMR sensors. 
     Another embodiment is an apparatus for detecting sensor error, the apparatus includes a first channel comprising a first magnetic sensing element comprising a first full bridge having two half bridge outputs, a second channel comprising a second magnetic sensing element comprising a second full bridge having two half bridge outputs, a third channel configured to receive a first signal from one of the half bridge outputs of the first full bridge and a second signal from one of the half bridge outputs of the second full bridge, and a processor. The processor is configured to receive first channel data from the first channel, second channel data from the second channel, and third channel data from the third channel. The processor is further configured to compute a first value from two of the first channel data, second channel data, and third channel data and compute a second value from another two of the first channel data, second channel data, and third channel data, and detect sensor error based at least partly on a comparison of the first value with the second value. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       These drawings and the associated description herein are provided to illustrate specific embodiments of the invention and are not intended to be limiting. 
         FIG. 1A  is a diagram of an example implementation of the technology disclosed herein according to one embodiment. 
         FIG. 1B  is a diagram of an example implementation of the sensing circuit  102  of  FIG. 1A  according to one embodiment. 
         FIG. 1C  is a diagram of an example implementation of the sensing circuit  102  of  FIG. 1A  in part according to another embodiment. 
         FIG. 2A  is a block diagram of an example implementation that includes three channels and a processor according to one embodiment. 
         FIG. 2B  is a block diagram of another example implementation that includes three channels and a processor according to one embodiment. 
         FIG. 2C  is a block diagram of another example implementation that includes three channels and a processor according to one embodiment. 
         FIG. 2D  is a diagram of an example sensing circuit configuration according to one embodiment. 
         FIG. 2E  is a diagram of another example sensing circuit configuration according to one embodiment. 
         FIG. 3A  is a flowchart of an example implementation of the error detection method described herein according to one embodiment. 
         FIG. 3B  is a flowchart of an example implementation of the error detection method described herein according to another embodiment. 
     
    
    
     DETAILED DESCRIPTION OF CERTAIN EMBODIMENTS 
     Various aspects of the novel systems, apparatuses, and methods are described more fully hereinafter with reference to the accompanying drawings. Aspects of this disclosure may, however, be embodied in many different forms and should not be construed as limited to any specific structure or function presented throughout this disclosure. Rather, these aspects are provided so that this disclosure will be thorough and complete, and will fully convey the scope of the disclosure to those skilled in the art. Based on the teachings herein, one skilled in the art should appreciate that the scope of the disclosure is intended to cover any aspect of the novel systems, apparatuses, and methods disclosed herein, whether implemented independently of or combined with any other aspect. For example, an apparatus may be implemented or a method may be practiced using any number of the aspects set forth herein. In addition, the scope is intended to encompass such an apparatus or method which is practiced using other structure, functionality, or structure and functionality in addition to or other than the various aspects set forth herein. It should be understood that any aspect disclosed herein may be embodied by one or more elements of a claim. 
     Although particular aspects are described herein, many variations and permutations of these aspects fall within the scope of the disclosure. Although some benefits and advantages of the preferred aspects are mentioned, the scope of the disclosure is not intended to be limited to particular benefits, uses, or objectives. Rather, aspects of the disclosure are intended to be broadly applicable to different wired and wireless technologies, system configurations, networks, including optical networks, hard disks, and transmission protocols, some of which are illustrated by way of example in the figures and in the following description of the preferred aspects. The detailed description and drawings are merely illustrative of the disclosure rather than limiting, the scope of the disclosure being defined by the appended claims and equivalents thereof. 
     In this description, reference is made to the drawings where like reference numerals can indicate identical or functionally similar elements. It will be understood that elements illustrated in the figures are not necessarily drawn to scale. Moreover, it will be understood that certain embodiments can include more elements than illustrated in a drawing and/or a subset of the elements illustrated in a drawing. Further, some embodiments can incorporate any suitable combination of features from two or more drawings. 
     In some embodiments, the disclosure herein may be applied to an apparatus having a rotating shaft, which may include one or more magnetic elements such as dipole or ring magnets. A measurement of the magnetic field by the sensing elements can be used to determine the angle of rotation of the shaft and a radius representative of synchronous multichannel sensor outputs, for example. In some embodiments, the magnetic angle of rotation ANGLE MAG  can be calculated as: 
                     ANGLE   MAG     =       arctan   ⁡     (       V   SIN       V   COS       )       2             (     Equation   ⁢           ⁢   1     )               
and the sensor output radius V RAD  can be determined as:
 
                     V   RAD     =           (       V   SIN     -       V   REF     2       )     2     +       (       V   COS     -       V   REF     2       )     2                 (     Equation   ⁢           ⁢   2     )               
where V SIN  and V COS  are respective output voltages of two channels having respective sensing elements, for example, with their sensing elements oriented at a physical angle of about 45° with each other, the sensor output radius V RAD  is a voltage representative of synchronized outputs of the two channels having respective sensing elements, and where V REF  is the reference voltage applied to the sensing elements. Each of the bridge outputs can have a sinusoidal response to magnetic angle, and in embodiments having the first and second sensing elements oriented at about 45° relative to each other, the two bridge outputs can be represented by the following equations:
 
 V   SIN   =V   RAD ×sin(2×ANGLE MAG ×π/180)  (Equation 3)
 
 V   COS   =V   RAD ×sin(2×(ANGLE MAG +45°)×π/180)  (Equation 4)
 
where V SIN  and V COS  are full bridge outputs of the two channels.
 
     The sensor output radius V RAD  can be substantially constant at a given temperature when the two channel data (V SIN  and V COS ) are synchronized, and a significant deviation from the calculated radius voltage at a given temperature may indicate an error or a defect in the system. In other embodiments, the error detection disclosed herein can be applied to an apparatus measuring parameters other than a rotation angle, such as length or current. 
       FIG. 1A  shows a diagram of an example implementation of the technology disclosed herein according to one embodiment. The illustrated implementation  100  includes a shaft  106 , a magnet  104 , and a sensing circuit  102 . A rotation angle  108  is also illustrated in  FIG. 1A . Embodiments of the sensing circuit  102  are further described in connection with  FIGS. 1B to 2B  below. The magnet  104  can be attached to the rotating shaft  106 . The rotating shaft  106  can be associated with a steering wheel of a motor vehicle in certain implementations. The sensing circuit  102  can sense changes in position of the magnet  104  and provide an indication of rotation of the rotating shaft  106 . 
     In embodiments including magnetic elements on a rotating shaft (e.g., the shaft  106 ), magnetic sensors such as anisotropic magnetoresistive (AMR) sensors, giant magnetoresistive (GMR), or tunneling magnetoresistive (TMR) sensors, or any other suitable magnetoresistive (XMR) sensors can be used as sensing elements to implement the disclosed apparatus and/or methods. In some embodiments, the magnetic sensors can measure the magnetic field based on the change in resistivity that is proportional to the perpendicular magnetic field. By using magnetic sensors, such as AMR sensors, and various combinations of sensing elements of the magnetic sensor, an error can be detected according to aspects of this disclosure. For instance, embodiments described herein can overcome challenges related to sensor errors due to shorting of resistive sensing elements of AMR sensors, electrostatic discharge (ESD) leakage of output circuit(s), damages to one or more filters (e.g., electromagnetic compatibility (EMC) filter), gain errors, nonlinearity errors, the like, or any combination thereof. 
     In other embodiments, other types of sensing elements that obtain positional information from a differential Sin and/or Cos output may be used to implement the error detection apparatus and methods described herein. For example, elements such as a resolver, Hall Effect sensing elements, optical sensing elements, the like or any combination thereof may be used. Furthermore, the disclosure herein can provide self-checking of sensing error regardless of sensor variations due to temperature, for example. 
     Aspects of this disclosure can be implemented in a magnetic angle sensor including, for example, the sensing circuit  102 . Such a magnetic sensor can include AMR sensing elements, signal conditioning electronics and an interface. In one application, the interface can be a peripheral acceleration sensor (PAS) interface, such as a PAS4 interface for remote accelerometers used for airbag deployment in vehicles, for communication with a host electronic control unit (ECU). The magnetic angle sensor can be implemented in an automotive Electric Power Assisted Steering (ePAS) system. Such a system has a functional safety specification defined in accordance with ISO-26262, which is a functional safety standard for road vehicles. The principles and advantages discussed herein can be implemented to verify angle and/or radius measurements (e.g., rotation of the shaft  106  associated with a steering wheel) in an ePAS system to satisfy a functional safety specification. In other instances, the illustrated implementation  100  can be used in other systems complying with other specifications or standards, or systems requiring relatively low error in and/or robust cross-checking of angle sensing. In some embodiments, the sensing ranges of magnetic, mechanical, and/or electrical angles can be different from each other. For example, in an embodiment using AMR sensors, such as illustrated in the sensing circuit  102  of  FIG. 1A , a 90° rotation in electrical angle may correspond to a 45° rotation in magnetic angle. 
     As described herein “electrical angle” can be understood as the angle calculated from arctan(V SIN /V COS ), and as described above “magnetic angle” can be understood as ANGLE MAG  calculated according to Equation 1 above. The “magnetic angle” can be the “electrical angle” divided by 2. Also as described herein “mechanical angle” can be understood as the angle determined based on the magnet design, which can be reflected in a mechanical-to-magnetic angle transfer function. In embodiments using a single pole-pair magnet, the “mechanical angle” and “magnetic angle” are the same. In embodiments using a multi-pole magnet, such a pole-ring, the “mechanical angle” can be expressed as (ANGLE MAG /N), where N is the number of pole pairs. 
       FIG. 1B  shows a diagram of example implementation of the sensing circuit  102  of  FIG. 1A  according to one embodiment. A sensing circuit may have a plurality of half bridges, and the example sensing circuit  102  illustrated in  FIG. 1B  includes a system of half bridges having sensing elements of two full bridges (or four half bridges) and four half bridge outputs V 1 , V 2 , V 3 , and V 4 . A plurality of half bridges (or full bridges which are paired half bridges) of the half bridge system can be oriented at an angle relative to one another and configured to be sensitive to a magnetic field in different directions. In  FIG. 1B , the full bridges are oriented at a 45° angle relative to each other. One sensing element being oriented at an angle relative to another sensing element encompasses being oriented at any suitable non-zero angle relative to another sensing element. As further described below, a plurality of combinations of the half bridge outputs may be used to determine the angle of rotation and/or sensor radius and detect sensor errors. In various implementations of half bridge systems, one or more of the combinations of the half bridge outputs may be interdependent, and a plurality of non-interdependent half bridge combinations may be selected. In other embodiments, the system of half bridges may include more or fewer half bridges than the embodiment illustrated in  FIG. 1B . 
     In the example illustrated in  FIG. 1B , it is possible to treat each of the full bridge elements as two half bridges. In  FIG. 1B , each of the resistors in the bridges have been labeled. In the following description, the full bridge composed of resistors R 1 -R 4  will be called Bridge 1 and the full bridge composed of resistors R 5 -R 8  will be called Bridge 2. Each of the resistors in the bridge responds to magnetic angle as shown below:
 
 R   x   =R±ΔR ·sin(2α)  Bridge 1:
 
 R   x   =R±ΔR ·sin(2(α+45))  Bridge 2:
 
where α is the magnetic stimulus angle, which, for example, may correlate to the rotation angle  108  of  FIG. 1A , and R can represent the nominal resistance value of each resistor in the bridge. The system of half-bridges can be named as indicated in the table below:
 
     
       
         
           
               
             
               
                 TABLE 1 
               
             
            
               
                   
               
               
                 Half bridge naming convention 
               
            
           
           
               
               
               
            
               
                   
                 Half Bridge Name 
                 Resistors 
               
               
                   
                   
               
               
                   
                 HB1 
                 R1, R3 
               
               
                   
                 HB2 
                 R2, R4 
               
               
                   
                 HB3 
                 R5, R7 
               
               
                   
                 HB4 
                 R6, R8 
               
               
                   
                   
               
            
           
         
       
     
     If the sensor is treated as two full bridges, at least two pieces of useful information are available, namely (V 2 −V 1 ) and (V 4 −V 3 ). 
     If the sensor is treated as a set of half-bridges, we have six pieces of available information, as summarized in Table 2 below. 
     
       
         
           
               
             
               
                 TABLE 2 
               
             
            
               
                   
               
               
                 Half-bridge combination voltage naming convention 
               
            
           
           
               
               
               
            
               
                   
                 Combination Voltage Name 
                 Output combination 
               
               
                   
                   
               
               
                   
                 V B1   
                 V2-V1 
               
               
                   
                 V B2   
                 V4-V3 
               
               
                   
                 V B3   
                 V3-V1 
               
               
                   
                 V B4   
                 V4-V2 
               
               
                   
                 V B5   
                 V3-V2 
               
               
                   
                 V B6   
                 V4-V1 
               
               
                   
                   
               
            
           
         
       
     
     The voltages at nodes providing outputs V 1  to V 4  of the half-bridges can be as shown below: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                             
                         
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                                           ( 
                                           
                                             α 
                                             + 
                                             45 
                                           
                                           ) 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             V 
                             REF 
                           
                           · 
                           
                             [ 
                             
                               0.5 
                               + 
                               
                                 
                                   
                                     Δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     R 
                                   
                                   
                                     2 
                                     ⁢ 
                                     R 
                                   
                                 
                                 · 
                                 
                                   sin 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       2 
                                       ⁢ 
                                       
                                         ( 
                                         
                                           α 
                                           + 
                                           45 
                                         
                                         ) 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     8 
                   
                   ) 
                 
               
             
           
         
       
     
     In Equations 5 to 8, V REF  can represent a reference voltage applied to the sensing elements, and R can represent the nominal resistance value of each resistor in the bridge. 
     In the circuit illustrated in  FIG. 1B , V B1  and V B2  should be the same as when the sensor is treated as two full bridges. 
     V B1  can be computed as shown in Equations 9 and 10: 
     
       
         
           
             
               
                 
                   
                     V 
                     
                       B 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       1 
                     
                   
                   = 
                   
                     
                       
                         V 
                         2 
                       
                       - 
                       
                         V 
                         1 
                       
                     
                     = 
                     
                       0.5 
                       + 
                       
                         
                           
                             Δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             R 
                           
                           
                             2 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             R 
                           
                         
                         · 
                         
                           sin 
                           ⁡ 
                           
                             ( 
                             
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               α 
                             
                             ) 
                           
                         
                       
                       - 
                       
                         [ 
                         
                           0.5 
                           - 
                           
                             
                               
                                 Δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 R 
                               
                               
                                 2 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 R 
                               
                             
                             · 
                             
                               sin 
                               ⁡ 
                               
                                 ( 
                                 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   α 
                                 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     9 
                   
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       V 
                       
                         B 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         1 
                       
                     
                     = 
                     
                       
                         
                           V 
                           REF 
                         
                         · 
                         
                           
                             Δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             R 
                           
                           
                             2 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             R 
                           
                         
                         · 
                         sin 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ( 
                         
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           α 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     10 
                   
                   ) 
                 
               
             
           
         
       
     
     V B2  can be computed as shown in Equations 11 and 12: 
     
       
         
           
             
               
                 
                   
                     V 
                     
                       B 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       2 
                     
                   
                   = 
                   
                     
                       
                         V 
                         4 
                       
                       - 
                       
                         V 
                         3 
                       
                     
                     = 
                     
                       0.5 
                       + 
                       
                         
                           
                             
                               Δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               R 
                             
                             
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               R 
                             
                           
                           · 
                           sin 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           
                             2 
                             ⁢ 
                             
                               ( 
                               
                                 α 
                                 + 
                                 45 
                               
                               ) 
                             
                           
                           ) 
                         
                       
                       - 
                       
                         [ 
                         
                           0.5 
                           - 
                           
                             
                               
                                 
                                   Δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   R 
                                 
                                 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   R 
                                 
                               
                               · 
                               sin 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ( 
                               
                                 2 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     α 
                                     + 
                                     45 
                                   
                                   ) 
                                 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     11 
                   
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       V 
                       
                         B 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         2 
                       
                     
                     = 
                     
                       
                         
                           V 
                           REF 
                         
                         · 
                         
                           
                             Δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             R 
                           
                           
                             2 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             R 
                           
                         
                         · 
                         sin 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ( 
                         
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ( 
                             
                               α 
                               + 
                               45 
                             
                             ) 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     12 
                   
                   ) 
                 
               
             
           
         
       
     
     The extra information available from the half-bridges can be as shown below, and V B3  can be computed as shown in Equations 13 and 14: 
                     V     B   ⁢           ⁢   3       =         V   3     -     V   1       =       V   REF     ⁢     {     0.5   -           Δ   ⁢           ⁢   R       2   ⁢           ⁢   R       ·   sin     ⁢           ⁢     (     2   ⁢           ⁢     (     α   +   45     )       )       -     [     0.5   -         Δ   ⁢           ⁢   R       2   ⁢           ⁢   R       ·     sin   ⁡     (     2   ⁢           ⁢   α     )           ]       }                 (     Equation   ⁢           ⁢   13     )                       ⁢       V     B   ⁢           ⁢   3       =       -     V   REF       ·       Δ   ⁢           ⁢   R       2   ⁢           ⁢   R       ·     [       sin   ⁢           ⁢     (     2   ⁢           ⁢     (     α   +   45     )       )       -     sin   ⁢           ⁢     (     2   ⁢           ⁢   α     )         ]                 (     Equation   ⁢           ⁢   14     )               
To simplify the expression of V B3 , the following trigonometric identity can be used:
 
                       sin   ⁢           ⁢     z   1       -     sin   ⁢           ⁢     z   2         =     2   ⁢           ⁢   cos   ⁢           ⁢     (         z   1     +     z   2       2     )     ⁢           ⁢   sin   ⁢           ⁢     (         z   1     -     z   2       2     )               (     Equation   ⁢           ⁢   15     )               
Thus, V B3  can be further computed as shown in Equations 16, 17, and 18:
 
     
       
         
           
             
               
                 
                   
                     V 
                     
                       B 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       3 
                     
                   
                   = 
                   
                     
                       - 
                       
                         
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           R 
                         
                         
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           R 
                         
                       
                     
                     · 
                     
                       V 
                       REF 
                     
                     · 
                     
                       [ 
                       
                         
                           2 
                           · 
                           
                             cos 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   
                                     2 
                                     ⁢ 
                                     
                                       ( 
                                       
                                         α 
                                         + 
                                         45 
                                       
                                       ) 
                                     
                                   
                                   + 
                                   
                                     2 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     α 
                                   
                                 
                                 2 
                               
                               ) 
                             
                           
                         
                         ⁢ 
                         
                           sin 
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   2 
                                   ⁢ 
                                   
                                     ( 
                                     
                                       α 
                                       + 
                                       45 
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   α 
                                 
                               
                               2 
                             
                             ) 
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     16 
                   
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       V 
                       
                         B 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         3 
                       
                     
                     = 
                     
                       
                         - 
                         
                           
                             Δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             R 
                           
                           
                             2 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             R 
                           
                         
                       
                       · 
                       
                         V 
                         REF 
                       
                       · 
                       
                         [ 
                         
                           
                             cos 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   α 
                                 
                                 + 
                                 45 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             sin 
                             ⁡ 
                             
                               ( 
                               45 
                               ) 
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     17 
                   
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       V 
                       
                         B 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         3 
                       
                     
                     = 
                     
                       
                         - 
                         
                           
                             Δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             R 
                           
                           
                             
                               2 
                             
                             ⁢ 
                             R 
                           
                         
                       
                       · 
                       
                         V 
                         REF 
                       
                       · 
                       
                         [ 
                         
                           cos 
                           ⁡ 
                           
                             ( 
                             
                               
                                 2 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 α 
                               
                               + 
                               45 
                             
                             ) 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     18 
                   
                   ) 
                 
               
             
           
         
       
     
     V B4  can be computed as shown in Equations 19 and 20: 
                     V     B   ⁢           ⁢   4       =         V   4     -     V   2       =       V   REF     ⁢     {     0.5   +         Δ   ⁢           ⁢   R       2   ⁢           ⁢   R       ·     sin   ⁡     (     2   ⁢     (     α   +   45     )       )         -     [     0.5   +         Δ   ⁢           ⁢   R       2   ⁢           ⁢   R       ·     sin   ⁡     (     2   ⁢   α     )           ]       }                 (     Equation   ⁢           ⁢   19     )                       ⁢       V     B   ⁢           ⁢   4       =         Δ   ⁢           ⁢   R       2   ⁢           ⁢   R       ·     V   REF     ·     [       sin   ⁡     (     2   ⁢     (     α   +   45     )       )       -     sin   ⁡     (     2   ⁢           ⁢   α     )         ]                 (     Equation   ⁢           ⁢   20     )               
To simplify the expression of V B4 , the following trigonometric identity can be used:
 
                       sin   ⁢           ⁢     z   1       -     sin   ⁢           ⁢     z   2         =     2   ⁢           ⁢   cos   ⁢           ⁢     (         z   1     +     z   2       2     )     ⁢     sin   ⁡     (         z   1     -     z   2       2     )                 (     Equation   ⁢           ⁢   21     )               
Thus, V B4  can be further computed as shown in Equations 22, 23, and 24:
 
     
       
         
           
             
               
                 
                   
                     V 
                     
                       B 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       4 
                     
                   
                   = 
                   
                     
                       
                         Δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         R 
                       
                       
                         2 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         R 
                       
                     
                     · 
                     
                       V 
                       REF 
                     
                     · 
                     
                       [ 
                       
                         
                           2 
                           · 
                           
                             cos 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   
                                     2 
                                     ⁢ 
                                     
                                       ( 
                                       
                                         α 
                                         + 
                                         45 
                                       
                                       ) 
                                     
                                   
                                   + 
                                   
                                     2 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     α 
                                   
                                 
                                 2 
                               
                               ) 
                             
                           
                         
                         ⁢ 
                         
                           sin 
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   2 
                                   ⁢ 
                                   
                                     ( 
                                     
                                       α 
                                       + 
                                       45 
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   α 
                                 
                               
                               2 
                             
                             ) 
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     22 
                   
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       V 
                       
                         B 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         4 
                       
                     
                     = 
                     
                       
                         
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           R 
                         
                         
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           R 
                         
                       
                       · 
                       
                         V 
                         REF 
                       
                       · 
                       
                         [ 
                         
                           
                             cos 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   α 
                                 
                                 + 
                                 45 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             sin 
                             ⁡ 
                             
                               ( 
                               45 
                               ) 
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     23 
                   
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       V 
                       
                         B 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         4 
                       
                     
                     = 
                     
                       
                         
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           R 
                         
                         
                           
                             2 
                           
                           ⁢ 
                           R 
                         
                       
                       · 
                       
                         V 
                         REF 
                       
                       · 
                       
                         [ 
                         
                           cos 
                           ⁡ 
                           
                             ( 
                             
                               
                                 2 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 α 
                               
                               + 
                               45 
                             
                             ) 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     24 
                   
                   ) 
                 
               
             
           
         
       
     
     V B5  can be computed as shown in Equations 25 and 26: 
                     V     B   ⁢           ⁢   5       =         V   3     -     V   2       =       V   REF     ⁢     {     0.5   -         Δ   ⁢           ⁢   R       2   ⁢           ⁢   R       ·     sin   ⁡     (     2   ⁢     (     α   +   45     )       )         -     [     0.5   +         Δ   ⁢           ⁢   R       2   ⁢           ⁢   R       ·     sin   ⁡     (     2   ⁢           ⁢   α     )           ]       }                 (     Equation   ⁢           ⁢   25     )                       ⁢       V     B   ⁢           ⁢   5       =       -       Δ   ⁢           ⁢   R       2   ⁢           ⁢   R         ·     V   REF     ·     [       sin   ⁡     (     2   ⁢     (     α   +   45     )       )       +     sin   ⁡     (     2   ⁢           ⁢   α     )         ]                 (     Equation   ⁢           ⁢   26     )               
To simplify the expression of V B5 , the following trigonometric identity can be used:
 
                       sin   ⁢           ⁢     z   1       +     sin   ⁢           ⁢     z   2         =     2   ⁢           ⁢     sin   ⁡     (         z   1     +     z   2       2     )       ⁢     cos   ⁡     (         z   1     -     z   2       2     )                 (     Equation   ⁢           ⁢   27     )               
Thus, V B5  can be further computed as shown in Equations 28, 29, and 30:
 
     
       
         
           
             
               
                 
                   
                     V 
                     
                       B 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       5 
                     
                   
                   = 
                   
                     
                       - 
                       
                         
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           R 
                         
                         
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           R 
                         
                       
                     
                     · 
                     
                       V 
                       REF 
                     
                     · 
                     
                       [ 
                       
                           
                       
                       ⁢ 
                       
                         
                           2 
                           · 
                           
                             cos 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   
                                     2 
                                     ⁢ 
                                     
                                       ( 
                                       
                                         α 
                                         + 
                                         45 
                                       
                                       ) 
                                     
                                   
                                   + 
                                   
                                     2 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     α 
                                   
                                 
                                 2 
                               
                               ) 
                             
                           
                         
                         ⁢ 
                         
                           sin 
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       α 
                                       + 
                                       45 
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   α 
                                 
                               
                               2 
                             
                             ) 
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     28 
                   
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       V 
                       
                         B 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         5 
                       
                     
                     = 
                     
                       
                         - 
                         
                           
                             Δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             R 
                           
                           
                             2 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             R 
                           
                         
                       
                       · 
                       
                         V 
                         REF 
                       
                       · 
                       
                         [ 
                         
                           
                             cos 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   α 
                                 
                                 + 
                                 45 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             sin 
                             ⁡ 
                             
                               ( 
                               45 
                               ) 
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     29 
                   
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       V 
                       
                         B 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         5 
                       
                     
                     = 
                     
                       
                         - 
                         
                           
                             Δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             R 
                           
                           
                             
                               2 
                             
                             ⁢ 
                             R 
                           
                         
                       
                       · 
                       
                         V 
                         REF 
                       
                       · 
                       
                         [ 
                         
                           cos 
                           ⁡ 
                           
                             ( 
                             
                               
                                 2 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 α 
                               
                               + 
                               45 
                             
                             ) 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     30 
                   
                   ) 
                 
               
             
           
         
       
     
     V B6  can be computed as shown in Equations 31 and 32: 
                     V     B   ⁢           ⁢   6       =         V   4     -     V   1       =       V   REF     ⁢     {     0.5   +         Δ   ⁢           ⁢   R       2   ⁢           ⁢   R       ·     sin   ⁡     (     2   ⁢     (     α   +   45     )       )         -     [     0.5   -         Δ   ⁢           ⁢   R       2   ⁢           ⁢   R       ·     sin   ⁡     (     2   ⁢           ⁢   α     )           ]       }                 (     Equation   ⁢           ⁢   31     )                       ⁢       V     B   ⁢           ⁢   6       =         Δ   ⁢           ⁢   R       2   ⁢           ⁢   R       ·     V   REF     ·     [       sin   ⁡     (     2   ⁢     (     α   +   45     )       )       +     sin   ⁡     (     2   ⁢           ⁢   α     )         ]                 (     Equation   ⁢           ⁢   32     )               
To simplify the expression of V B6 , the following trigonometric identity can be used:
 
                       sin   ⁢           ⁢     z   1       +     sin   ⁢           ⁢     z   2         =     2   ⁢           ⁢     sin   ⁡     (         z   1     +     z   2       2     )       ⁢     cos   ⁡     (         z   1     -     z   2       2     )                 (     Equation   ⁢           ⁢   33     )               
Thus, V B6  can be further computed as shown in Equations 34, 35, and 36:
 
     
       
         
           
             
               
                 
                   
                     V 
                     
                       B 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       6 
                     
                   
                   = 
                   
                     
                       
                         Δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         R 
                       
                       
                         2 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         R 
                       
                     
                     · 
                     
                       V 
                       REF 
                     
                     · 
                     
                       [ 
                       
                         
                           2 
                           · 
                           
                             sin 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   
                                     2 
                                     ⁢ 
                                     
                                       ( 
                                       
                                         α 
                                         + 
                                         45 
                                       
                                       ) 
                                     
                                   
                                   + 
                                   
                                     2 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     α 
                                   
                                 
                                 2 
                               
                               ) 
                             
                           
                         
                         ⁢ 
                         
                           cos 
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   2 
                                   ⁢ 
                                   
                                     ( 
                                     
                                       α 
                                       + 
                                       45 
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   α 
                                 
                               
                               2 
                             
                             ) 
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     34 
                   
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       V 
                       
                         B 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         6 
                       
                     
                     = 
                     
                       
                         
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           R 
                         
                         R 
                       
                       · 
                       
                         V 
                         REF 
                       
                       · 
                       
                         [ 
                         
                           
                             sin 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   α 
                                 
                                 + 
                                 45 
                               
                               ) 
                             
                           
                           ⁢ 
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ( 
                             45 
                             ) 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     35 
                   
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       V 
                       
                         B 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         6 
                       
                     
                     = 
                     
                       
                         
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           R 
                         
                         
                           
                             2 
                           
                           ⁢ 
                           R 
                         
                       
                       · 
                       
                         V 
                         REF 
                       
                       · 
                       
                         [ 
                         
                           sin 
                           ⁡ 
                           
                             ( 
                             
                               
                                 2 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 α 
                               
                               + 
                               45 
                             
                             ) 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     36 
                   
                   ) 
                 
               
             
           
         
       
     
     In all of the above calculations, the radius can be computed as follows: 
     
       
         
           
             
               
                 
                   
                     
                       
                         Δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         R 
                       
                       R 
                     
                     · 
                     
                       V 
                       REF 
                     
                   
                   = 
                   radius 
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     37 
                   
                   ) 
                 
               
             
           
         
       
     
     Still referring to  FIG. 1B , there are a number of methods to use the extra pieces of information from the half-bridges. One method is to use the extra information to verify the primary channel data. When using this method, the angle and radius can be calculated from V B1  and V B2 . The expected values for one or more of V B3  to V B6  could be calculated based on the angle and radius determined from V B1  and V B2 . An auxiliary analog channel, which could include multiple paths, a single multiplexed path, or a single path that receives outputs from half bridges of different full bridges, could then be used to determine V B3  to V B6 , and the outputs of each measurement could be compared to the expected value. It can be observed from the equations for V B3  to V B6  that these values can be determined from computing two of these values from sensor data and then deriving the other two values from the two values computed from sensor data, as V B4  is the additive inverse of V B3 , and V B6  is the additive inverse of V B5 . 
       FIG. 1C  shows in part a diagram of another example implementation of the sensing circuit  102  of  FIG. 1A . A half bridge system having more than five half bridges as its sensing elements may be configured to determine which half bridge is defective or producing an error, for example. In  FIG. 1C , the sensing circuit having XMR sensing elements has three full bridges (six half bridges), which are each oriented at an angle to one another. In this example, the three full bridges are oriented at physical angles of 120° to one another. In other embodiments may include different number of sensing elements (i.e., full/half bridges) at different relative orientation angles (e.g., 22.5°, 45°, 90°). In some embodiments, resistor matching with the resistors close together and/or interlaced can be beneficial. 
       FIG. 2A  is a block diagram of an example implementation according to one embodiment. The illustrated circuit  200   a  includes sensing elements  102   a  and  102   b , which can be part of the sensing circuit  102  described above in connection with  FIGS. 1A-1C , a multiplexer  202 , and a processor  210  performing various functions described herein. The illustrated circuit  200   a  further includes differential amplifiers  204   a ,  204   b , and  204   c , sampling circuits  206   a ,  206   b , and  206   c , filters  220 , and an interface  222 . A channel can include an amplifier, a sampling circuit and/or any other circuitry to process signals from the sensing circuit  102  in a signal path to the processor  210 . In some embodiments, a first channel can include the sensing element  102   a , the amplifier  204   a , the sampling circuit  206   a , at least part of the filters  220 , and at least part of the interface  222 . A second channel can include the sensing element  102   b , the amplifier  204   b , the sampling circuit  206   b , at least part of the filters  220 , and at least part of the interface  222 . A third channel can include the multiplexer  202 , the amplifier  204   c , the sampling circuit  206   c , at least part of the filters  220 , and at least part of the interface  222  as shown in  FIG. 2A . The third channel can alternatively be implemented without the multiplexer  202  as shown in  FIG. 2C . Although not illustrated in  FIGS. 2A-2C , signals from the sensing elements  102   a  and  102   b  as described herein can be signals directly from the sensing elements  102   a  and  102   b  or by way of one or more intervening circuit elements following the sensing elements  102   a  and  102   b , such as the one or more pre-processing or intervening circuits illustrated in  FIG. 2E  and described below. 
     The sampling circuits  206   a ,  206   b , and  206   c  can each include, for example, an analog-to-digital converter (ADC), such as a sigma-delta ADC. It should be noted that  FIGS. 2A-2C  do not illustrate physical orientation of the sensing elements  102   a  and  102   b  as the sensing elements (e.g.,  102   a  and  102   b ; three or more sensing elements in other embodiments) are oriented at an angle with respect to one another, for example, as described in connection with  FIGS. 1B-1C  above. The multiplexer  202  receives four half bridge outputs from the sensing elements  102   a ,  102   b  (i.e., first and second full bridges). The multiplexer  202  may select one combination of the half bridge outputs that is different from the combination sent to the amplifier  204   a  and also different from the combination sent to the amplifier  204   b . The multiplexer  202  can receive at least three of the four half bridge outputs and select a combination of those outputs that is different from the combination sent to the amplifier  204   a  and also different from the combination sent to the amplifier  204   b  in another embodiment. According to yet another embodiment, one of the differential inputs of amplifier  204   c  can receive a half bridge output and the other differential input of the amplifier  204   c  can be electrically connected to a multiplexer arranged to select one of two or more half bridge outputs to provide to the other differential input of the amplifier  204   c . The multiplexer  202  can be optional as  FIG. 2C  described below illustrates an embodiment without the multiplexer  202 .  FIGS. 2A-2C  may further include analog circuit elements to reduce computation by the processor  210 , for example, as further illustrated in  FIG. 2E  below. 
       FIG. 2B  is a block diagram of another example implementation according to one embodiment. The circuit  200   b  of  FIG. 2B  can implement any of the principles and advantages of the circuit  200   a  of  FIG. 2A . As illustrated in  FIG. 2B , the processor  210  includes a measurement calculation module  212  and an error detection module  214  to perform the various functions described herein. Each of these modules can be implemented with physical hardware. The measurement calculation module  212  can compute angle and radius, for example, in accordance with Equations 1 and 2 above. The measurement error detection module  214  can compare a value associated with data from the third channel that includes the amplifier  204   c  and the sampling circuit  206   c  with data associated with the first and second channels that include the amplifier  204   a  and the sampling circuit  206   a  and the amplifier  204   b  and the sampling circuit  206   b , respectively, to detect whether there is a sensor error. The error detection module  214  can also perform any suitable computations to support the comparison. For instance, the error detection module  214  can compute an expected value of the third channel data from one of the combinations of expected half bridge values (e.g., V B3  to V B6 ) different from the ones used to calculate the angle and radius. The expected third channel data based, for example, on two of V B3  to V B6 , can be computed in accordance with Equations 18, 24, 30, 36, and/or 37 using the angle and radius values determined by the measurement calculation module  212 . The third channel data can be data received from the third channel or a value computed based on data received from the third channel. The processor  210  in any of the illustrated embodiments can be implemented by a host electronic control unit (ECU) of an Electric Power Assisted Steering (ePAS) system, for example. 
       FIG. 2C  is a block diagram of another example implementation according to one embodiment. The illustrated circuit  200   c  is implemented without the multiplexer  202  ( FIGS. 2A-2B ). The differential amplifier  204   c  in  FIG. 2C  receives its first input from the sensing element  102   a  (i.e., the first full bridge) and its second input from the sensing element  102   b  (i.e., the second full bridge). In other embodiments, the differential amplifier  204   c  may receive its first input from the sensing element  102   b  (i.e., second full bridge) and its second input from the sensing element  102   a  (i.e., first full bridge). In this embodiment, the angle and radius can be calculated from the outputs of the amplifiers  204   a  and  204   b , and based on the angle and radius values, the expected data associated with the amplifier  204   c  can be determined, which is compared with the actual data associated with the amplifier  204   c . As this embodiment does not include a multiplexer, it may advantageously conserve power, for example. 
       FIG. 2D  is a block diagram illustrating double bridge calculation. Angle can be determined as angle=arctan 2(S−C′·(√{square root over (2)}−1), C′−S·(√{square root over (2)}−1)) where C′=C*·√{square root over (2)}. Two full bridges  102   a ,  102   b  of  FIG. 2D  each includes two half bridges. S is a signal provided by a differential amplifier  252  configured to receive two half bridge outputs and C* is a signal provided by another differential amplifier  250  configured to receive a two half bridge outputs, of which at least one is different than the half bridge outputs received by the other amplifier. The angle determined from the circuit shown in  FIG. 2D  can be compared to an angle computed from a different combination of the half bridge outputs. For instance, the other combination of half bridge outputs can correspond to the signals provided to amplifiers  204   a  and  204   b  of  FIGS. 2A-2C . When the difference between the two computed angles satisfies a threshold, an error flag can be set to indicate a sensor error. As shown in  FIGS. 2D and 2E , in some embodiments, the determination of the full bridge values (e.g., V SIN  and V COS ) of the first and the second channels, and in turn, the determination the angle and radius values can be implemented with one or more intermediate analog circuits. For example, the differential amplifiers  250  and  252  in  FIG. 2D  may output intermediate values S and C*, which then can be received by following circuit elements, such as the ones shown in  FIG. 2E , to output Sin (V SIN ) and Cos (V COS ) signals. 
     In another embodiment, the angle and radius computed using the differential amplifiers  250  and  252  shown in  FIG. 2D  can be used to compute an expected value associated with a third sensing channel that receives a different combination of two half bridge outputs from the sensing elements  102   a  and  102   b . The expected value can be one of the values associated with a processed version of one of the voltages of Table 2, for example. The expected value can then be compared to a value generated by the third sensing channel. 
       FIG. 2E  shows a circuit configured to process half bridge outputs. This circuit can be implemented in connection with any of the principles and advantages associated with half bridges discussed herein for purposes of computing angle and/or radius.  FIG. 2E  illustrates an analog circuit for correcting the phase shift that can reduce time consuming calculation. In some embodiments, at least some of the computations that can be performed by the processor  210  can be performed by the analog circuit illustrated in  FIG. 2E  to output Sin (V SIN ) and Cos (V COS ) signals. It can be advantageous to implement at least some of the computations to be performed by an analog circuit similar to the one disclosed in  FIG. 2E , which can, for example, save computation time. 
       FIG. 3A  shows an example flowchart implementing an error detection method described herein according to one embodiment. The method can be performed with the circuit elements and one or more modules of the processor illustrated in  FIGS. 2A-2C  and/or in connection with the circuits illustrated in  FIGS. 2D-2E . 
     In operation  302 , two output signals from two combinations of two of four half-bridge outputs of magnetic sensor elements are computed. For example, the two output signals computed in operation  302  may correspond to the outputs from the first and the second channels. In operation  304 , angle and radius can be determined based on the two output signals from the operation  302 . In operation  306 , the expected third output signal can be determined based on the angle and radius determined in operation  304 . In operation  305 , the third output signal or data associated with the third channel can be determined or obtained. In operation  308 , the expected outputs and the determined or obtained output of the third channel can be compared to, for example, compute the difference between the expected and determined values. In operation  310 , it can be determined whether the difference between the expected and determined outputs of the third channel is greater than a threshold. If the difference is greater than the threshold, an error can be indicated in operation  314 . If the difference is not greater than the threshold, the process  300  can proceed to operation  312  to use the calculated angle. 
     In some embodiments, operations in  FIG. 3A  can be performed by the various elements of the circuits shown in  FIGS. 2A-2E . For example, operation  304  can be performed by the measurement calculation module  212  in  FIGS. 2B and 2C , and operations  306 ,  308 , and  310  can be performed, for example, by the error detection module  214  in  FIGS. 2B and 2C . 
       FIG. 3B  shows another example flowchart implementing the error detection method described herein according to one embodiment. The method can be performed with the analog circuit elements and one or more modules of the processor illustrated in  FIGS. 2A-2C  and/or in connection with the circuits illustrated in  FIGS. 2D-2E . 
     In operation  352 , two different output signals from different combinations of two of four half-bridge magnetic sensor element outputs can be determined. In operation  354 , two magnetic field angles can be determined separately based on the two output signals from operation  352 . In operation  356 , the two separately determined magnetic field angles can be compared to, for example, compute the difference between the two magnetic field angles. In operation  358 , it can be determined whether the difference between the two magnetic field angles is above a threshold. If the difference is greater than the threshold, an error can be indicated in operation  362 . If the difference is not greater than the threshold, the process  350  can proceed to operation  360  to use the one of the calculated angles. 
     In some embodiments, operations in  FIG. 3B  can be performed by the various elements of the circuits shown in  FIGS. 2A-2E . For example, operations  352  and  354  can be performed by the measurement calculation module  212  in  FIGS. 2B and 2C , and operations  356  and  358  can be performed, for example, by the error detection module  214  in  FIGS. 2B and 2C . 
     Aspects of this disclosure can be implemented in various electronic devices. Examples of the electronic devices can include, but are not limited to, consumer electronic products, parts of the consumer electronic products, electronic test equipment, vehicular electronics systems, etc. Examples of the electronic devices can include, but are not limited to, computing devices, communications devices, electronic household appliances, automotive electronics systems, etc. Further, the electronic devices can include unfinished products. 
     Unless the context clearly requires otherwise, throughout the description and the claims, the words “comprise,” “comprising,” “include,” “including,” and the like are to be construed in an inclusive sense, as opposed to an exclusive or exhaustive sense; that is to say, in the sense of “including, but not limited to.” Additionally, the words “herein,” “above,” “below,” and words of similar import, when used in this application, shall refer to this application as a whole and not to any particular portions of this application. Where the context permits, words in the above Detailed Description of Certain Embodiments using the singular or plural number may also include the plural or singular number respectively. Where the context permits, the word “or” in reference to a list of two or more items is intended to cover all of the following interpretations of the word: any of the items in the list, all of the items in the list, and any combination of the items in the list. 
     Moreover, conditional language used herein, such as, among others, “can,” “could,” “might,” “may,” “e.g.,” “for example,” “such as” and the like, unless specifically stated otherwise, or otherwise understood within the context as used, is generally intended to convey that certain embodiments include, while other embodiments do not include, certain features, elements and/or states. Thus, such conditional language is not generally intended to imply that features, elements and/or states are in any way required for one or more embodiments or that one or more embodiments necessarily include logic for deciding, with or without author input or prompting, whether these features, elements and/or states are included or are to be performed in any particular embodiment. 
     The foregoing description and claims may refer to elements or features as being “connected” or “coupled” together. As used herein, unless expressly stated otherwise, “connected” means that one element/feature is directly or indirectly connected to another element/feature, and not necessarily mechanically. Likewise, unless expressly stated otherwise, “coupled” means that one element/feature is directly or indirectly coupled to another element/feature, and not necessarily mechanically. Thus, although the various schematics shown in the Figures depict example arrangements of elements and components, additional intervening elements, devices, features, or components may be present in an actual embodiment (assuming that the functionality of the depicted circuits is not adversely affected). 
     As used herein, the term “determining” encompasses a wide variety of actions. For example, “determining” may include calculating, computing, processing, deriving, investigating, looking up (e.g., looking up in a table, a database or another data structure), ascertaining and the like. Also, “determining” may include receiving (e.g., receiving information), accessing (e.g., accessing data in a memory) and the like. Also, “determining” may include resolving, selecting, choosing, establishing and the like. Further, a “channel width” as used herein may encompass or may also be referred to as a bandwidth in certain aspects. 
     The various operations of methods described above may be performed by any suitable means capable of performing the operations, such as various hardware and/or software component(s), circuits, and/or module(s). Generally, any operations illustrated in the Figures may be performed by corresponding functional means capable of performing the operations. 
     The various illustrative logical blocks, modules, and circuits described in connection with the present disclosure may be implemented or performed with a general purpose processor, a digital signal processor (DSP), an application specific integrated circuit (ASIC), a field programmable gate array signal (FPGA) or other programmable logic device (PLD), discrete gate or transistor logic, discrete hardware components or any combination thereof designed to perform the functions described herein. A general purpose processor may be a microprocessor, but in the alternative, the processor may be any commercially available processor, controller, microcontroller or state machine. A processor may also be implemented as a combination of computing devices, e.g., a combination of a DSP and a microprocessor, a plurality of microprocessors, one or more microprocessors in conjunction with a DSP core, or any other such configuration. 
     The methods disclosed herein comprise one or more steps or actions for achieving the described method. The method steps and/or actions may be interchanged with one another without departing from the scope of the claims. In other words, unless a specific order of steps or actions is specified, the order and/or use of specific steps and/or actions may be modified without departing from the scope of the claims. 
     It is to be understood that the implementations are not limited to the precise configuration and components illustrated above. Various modifications, changes and variations may be made in the arrangement, operation and details of the methods and apparatus described above without departing from the scope of the implementations. 
     Although this invention has been described in terms of certain embodiments, other embodiments that are apparent to those of ordinary skill in the art, including embodiments that do not provide all of the features and advantages set forth herein, are also within the scope of this invention. Moreover, the various embodiments described above can be combined to provide further embodiments. In addition, certain features shown in the context of one embodiment can be incorporated into other embodiments as well.