Abstract:
A rotation angle detecting device includes a signal generator, a magnetic rotor and a rotation angle calculating unit that calculates a rotation angle θ of the rotor based on the output signals of the signal generator. The signal generator includes a magnetic rotor that has a permanent magnet and a shaft connectable with the rotating object and at least three magnetic sensor elements disposed in the magnetic field to generate a plurality of output signals when the rotor rotates. The rotation angle calculating unit calculates a first candidate of the rotation angle θ from a first pair of the three output signals, examines whether the first candidate of the rotation angle θ is normal or not and calculates a next candidate of the rotation angle θ from another pair of the three output signals if the first candidate of the rotation angle θ is not normal, examines whether the next candidate of the rotation angle θ is normal or not, repeats the above two steps until a next candidate of the rotation angle θ is judged normal, and outputs one of the first and next candidates of the rotation angle θ as a formal rotation angle if it is judged normal.

Description:
CROSS REFERENCE TO RELATED APPLICATION 
     The present application is based on and claims priority from Japanese Patent Applications 2007-133380, filed May 18, 2007, the contents of which are incorporated herein by reference. 
     BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates to a rotation angle detecting device that detects the rotation angle of a rotating object. 
     2. Description of the Related Art 
     JP-A-2003-75108 discloses a prior art rotation angle detecting device that is constructed of a disk-shaped permanent magnet and two or three magnetic sensors. The magnetic sensors are respectively disposed at portions to confront the permanent magnet so as to generate output signals according to the magnetic field strength. A rotation angle of a rotating object is calculated by a certain arithmetic expression based on the output signals of the magnetic sensors. However, it is impossible for such a prior art rotation angle detecting device to detect a rotation angle if one of the magnetic sensors fails. 
     SUMMARY OF THE INVENTION 
     Therefore, an object of the invention is to provide an improved rotation angle detecting device that can detect a rotation angle even if one of the magnetic sensors fails, without increasing the number of the magnetic sensors. 
     According to a feature of the invention, a rotation angle detecting device includes a housing, a signal generator and a rotation angle calculating unit that calculates a rotation angle θ of the rotor based on the output signals. The signal generator includes a rotor having a permanent magnet, a shaft connectable with a rotating object and three magnetic sensor elements. The rotation angle calculating unit is configured to carry out the following steps: calculating a first candidate of the rotation angle θ from a first pair of the three output signals; examining whether the first candidate of the rotation angle θ is normal or not; calculating a next candidate of the rotation angle θ from another pair of the three output signals if the first candidate of the rotation angle θ is not normal; examining whether the next candidate of the rotation angle θ is normal or not; repeating the above two steps until a next candidate of the rotation angle θ is judged normal; and outputting one of the first and next candidates of the rotation angle θ as a formal rotation angle if it is judged normal. 
     In the above rotation angle detecting device, the signal generator is configured to generate the output signals in sinusoidal shapes; and the rotation angle calculating unit is configured to provide sin θ and cos θ from one of the first and next candidates of the rotation angle θ to calculate the rotation angle θ from arctan (tan θ). In addition, the rotation angle calculating unit examines the normality of the rotation angle by comparing sin θ, cos θ and the rotation angle θ with preset values. The permanent magnet may provide the magnet field extending in a direction perpendicular to the rotation axis of the rotor, and the magnetic sensor elements respectively have sensing directions that are perpendicular to the rotation axis of the rotor and different from each other. In this arrangement, the permanent magnet is preferably disposed inside the rotor around the rotation axis. 
     The rotation angle detecting device may further include means for changing amplitude of the output signals as the rotor rotates, and the rotation angle calculating unit is arranged to calculate the number of turns of the rotor based on the amplitude of the output signals. The means for changing amplitude may include a mechanism of shifting the permanent magnet relative to the magnetic sensor elements to change strength of the magnetic field as the rotor rotates. For this purpose, the permanent magnet may have a conical inner surface the diameter of which linearly increases or decreases as the inner surface shifts in parallel to the rotation axis. The mechanism of shifting the permanent magnet may include a sleeve and a pair of male and female screws formed on portions of the sleeve and the rotor that are in contact with each other. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Other objects, features and characteristics of the present invention as well as the functions of related parts of the present invention will become clear from a study of the following detailed description, the appended claims and the drawings. In the drawings: 
         FIG. 1  is a longitudinal cross-section of a rotation angle detecting device according to the first embodiment of the invention; 
         FIG. 2  is a schematic cross-sectional plan view of a signal generator of the rotation angle detecting device illustrated in  FIG. 1  cut along line II-II; 
         FIG. 3  is a graph showing wave shapes of the output signals of three magnetic sensor elements; 
         FIG. 4  is a graph showing a relation between a rotation angle θ, an output signal Va in the vertical axis and a signal Vab in horizontal axis; 
         FIG. 5  illustrates a longitudinal cross-section of a rotation angle detecting device according to the second embodiment of the invention; 
         FIG. 6  is a schematic cross-sectional plan view of a signal generator and a gear of the rotation angle detecting device shown in  FIG. 5  cut along line VI-VI; 
         FIG. 7  is an enlarged view of a longitudinal cross-section of the signal generator of the rotation angle detecting device illustrated in  FIG. 5 ; 
         FIG. 8  is an enlarged plan view of the signal generator illustrated in  FIG. 6 ; 
         FIG. 9  is a longitudinal cross-section of the rotation angle detecting device according to the second embodiment of the invention in operation; 
         FIG. 10  is a graph showing wave shapes of the output signals of three magnetic sensor elements of the rotation angle detecting device according to the second embodiment of the invention; and 
         FIG. 11  is a graph showing a relation between a rotation angle θ, an output signals Va in the vertical axis and a signal Vab in horizontal axis. 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     A couple of preferred embodiments of the invention will be described with reference to the appended drawings. 
     A rotation angle detecting device  1  according to the first embodiment of the invention will be described with reference to  FIG. 1-FIG .  4 . 
     As shown in  FIGS. 1 and 2 , the rotation angle detecting device  1  includes a signal generator  11 , a rotation angle calculating unit  12 , a housing  13  and a support member  14 . 
     The signal generator  11  is comprised of a rotor  10  and three magnetic sensors  113 - 115 . In other words, the signal generator  11  is comprised of the permanent magnet  110 , the yoke  112  and the magnetic sensor elements  113 - 115 . 
     The rotor  10  is rotatably supported by the housing  13  and includes a permanent magnet  110 , a yoke  112  and a rotary shaft  101  connected with a rotating object (not shown). The permanent magnet  110  is a cylindrical member made of ferrite that is magnetized in a direction perpendicular to the rotation axis M of the rotor  10 . Accordingly, a magnetic pole N is formed on one portion of the inside surface of the cylindrical permanent magnet  110  and a magnetic pole S is formed on another portion of the same inside surface that confronts the magnetic pole N with the rotation axis being between the two poles. That is, a magnetic field φ is formed to extend along a radial direction as shown in an arrow in  FIGS. 1 and 2 . 
     The yoke  112  has a cylindrical magnetic member that surrounds the permanent magnet  110  at an axial end of the rotor  10  and a disk-like bottom member from which the rotary shaft  101  projects to the other axial end so as to rotate about the rotation axis M. The magnetic sensor elements  113 - 115  are Hall IC elements, which are fixed to the support member  14  to be disposed at a space surrounded by the permanent magnet  110  around the rotation axis M. Each of the sensor elements  113 - 115  has a sensing direction that is perpendicular to the rotation axis M and 120 degrees in angle different from those of other sensor elements  113 - 115 . 
     The signal generator  11  generates three output signals, each of which has a phase that is different from the others and not anti-phase of others, according to the rotation angle θ that changes between 0 degree and 360 degrees. 
     When the rotor  10  rotates, the sensor elements  113 - 115  respectively output signals Va, Vb, Vc, which are expressed by the following expression E1.
 
Va=A sine θ
 
 Vb=A  sine(θ+120°)
 
 Vc=A  sine(θ−120°),  [E1]
 
where:
 
A is an amplitude of the output signals; and θ is a rotation angle of the rotor  10  between 0° and 360°.
 
     The rotation angle calculation unit  12  is comprised of a microcomputer to calculate the rotation angle θ from two of the output signals Va, Vb, Vc that are selected in turn. The rotation angle calculation unit  12  examines if the calculated rotation angle θ is correct or not before finally providing the rotation angle θ. The rotation angle calculation unit  12  converts the output signals Ma, Vb, for example, to digital data and calculates Vab by the following expression E2
 
 Vab=Va /tan 60 °+Vb /sin 60 °=A  cos θ[E2]
 
     Further, the rotation angle θ or the rotor  10  is calculated from Vab and Va by the following expression E3. 
     
       
         
           
             
               
                 
                   
                     
                       
                         θ 
                         = 
                           
                         ⁢ 
                         
                           arctan 
                           ⁡ 
                           
                             ( 
                             
                               tan 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               θ 
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           arctan 
                           ⁡ 
                           
                             ( 
                             
                               A 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 θ 
                                 / 
                                 A 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               θ 
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           arctan 
                           ⁡ 
                           
                             ( 
                             
                               Vab 
                               / 
                               Va 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     E 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     3 
                   
                   ] 
                 
               
             
           
         
       
     
     When the output signals Va, Vb are normal, a point (Vab, Va) that is defined by Vab and Va can be depicted on a circle having whose radius is the amplitude A in rectangular coordinates, as shown in  FIG. 4 . The angle between the straight line that joins the point (Vab, Va) and the origin 0 of the rectangular coordinates and the horizontal Vab axis becomes the rotation angle θ of the rotor  10 . That is, there is a correspondence relationship among Vab, Va and the rotation angle θ. Therefore, by determining whether a proper correspondence relationship exists among Vab, Va and the rotation angle θ, the rotation angle calculation unit  12  examines whether the rotation angle θ is normal or not based on the relationship between Vab, Va and the rotation angle θ. The rotation angle calculation unit  12  compares Vab. Va and the rotation angle θ with preset expected values that indicate the proper correspondence relationship among Vab, Va and the rotation angle θ. Thereby, the rotation angle calculation unit  12  determines whether the rotation angle is normal or not. In the above, the expected values are set to values that are indicative of the correspondence relationship among Vab, Va and the rotation angle θ and that are represented by circle locus, as shown in  FIG. 4 . Therefore, based on sine value (A sin θ) and cosine value (A cos θ) each corresponding to the rotation angle θ, and based on the rotation angle θ, it is determined whether the rotation angle θ is normal. 
     Likewise, the rotation angle calculation unit  12  also converts the output signals Vb, Vc to digital data and calculates Vbc. Then the rotation angle θ is calculated and examined in the same manner as above to examine whether it is normal or not. Likewise, the rotation angle calculation unit  12  also calculate Vca and the rotation angle θ from the output signals Vc, Va and examines the rotation angle θ in the same manner as above. 
     If the magnetic sensor element  113  fails, the output signal Va is judged not normal, while the output signals Vb and Vc are judged normal. In this case, the rotation angle calculation unit  12  provides the rotation angle θ based on the output signals Vb and Vc. 
     A rotation angle detecting device  2  according to the second embodiment of the invention will be described with reference to  FIG. 5-FIG .  11 . 
     As shown in  FIG. 5 , the rotation angle detecting device  2  includes a shaft  20 , a signal generator  23 , a rotation angle calculating unit  24 , a housing  25  and a support member  26 . The signal generator  23  is comprised of a gear  21 , a rotor  22  and three magnetic sensors  234 - 236 . The shaft  20  carries the gear  21  at the middle thereof and is connected with a rotating object at one end so as to rotate the gear  21  when the rotating object rotates. The shaft  20  is rotatably supported by the housing  25 . The rotor  22  is also rotatably supported by the housing via a sleeve  2331 . The rotor  22  includes a cylindrical permanent magnet  230 , a cylindrical yoke  232  and a yoke shifting mechanism  233 . The cylindrical yoke  232  has gear teeth on the outer periphery thereof in engagement with the gear  21 . 
     The signal generator  23  generates three output signals, each of which has a phase that is different from the others and not anti-phase of others, according to the rotation angle θ of the rotor  22  between 0 degree and an angle much larger than 360 degrees, such as 720 degrees (two turns) or 1080 degrees (three turns). 
     The permanent magnet  230  is a cylindrical member made of ferrite that is magnetized in a direction perpendicular to the rotation axis M of the rotor  22 . Accordingly, a magnetic pole N is formed on one portion of the inside surface of the cylindrical permanent magnet  110  and a magnetic pole S is formed on another portion of the same inside surface that confronts the magnetic pole N across the rotation axis, which is disposed between the two poles. As a result, a magnetic field φ is formed in a radial direction as shown in an arrow in  FIGS. 7 and 8 . 
     The yoke  232  is a cylindrical magnetic member that surrounds the permanent magnet  230 . The permanent magnet  230  is fixed to the inner surface of the yoke  232 . The permanent magnet  230  has a conical inner surface the diameter of which linearly increases as the inner surface shifts upward in parallel to the rotation axis M. 
     The yoke shifting mechanism  233  shifts the yoke  232  up or down as the rotor  22  rotates in one or the other direction. The yoke shifting mechanism  233  is comprised of a sleeve  2331  that has a female screw  2332  and a male screw  2330  that is formed on a portion of the gear teeth of the yoke  232  in engagement with the female screw  2332  of the sleeve  2331 . The sleeve  2331  is an arc-shaped member fixed to the cylindrical inner surface of the housing  25  that confronts the yoke  232  so as to rotatably support the rotor  22 . 
     The magnetic sensor elements  234 - 236  are Hall IC elements, which are fixed to the support member  26  to be disposed at a space surrounded by the permanent magnet  230  around the rotation axis M. Each of the sensor elements  234 - 236  has a sensing direction that is perpendicular to the rotation axis M and 120 degrees in angle different from those of other sensor elements  234 - 236 . When the rotor  22  rotates, the rotor  22  shifts downward to decrease the strength of the magnetic field and the sensor elements  234 - 236  respectively output signals Va, Vb, Vc, which are shown in  FIG. 10  and expressed by the following expression E4.
 
 Va=f (θ)sine θ
 
 Vb=f (θ)sine(θ+120°)
 
 Vc=f (θ)sine(θ−120°),  [E4]
 
where:
 
θ is a rotation angle of the rotor  22 ; and f(θ) is an amplitude of the output signals that changes as the rotation angle θ changes.
 
     The rotation angle calculation unit  24  is comprised of a microcomputer to calculate the rotation angle θ from two of the output signals Va, Vb, Vc that are selected in turn. The rotation angle calculation unit  12  examines if the calculated rotation angle θ is correct or not before finally providing the rotation angle θ. The rotation angle calculation unit  24  converts the output signals Va, Vb, for example, to digital data and calculates Vab by the following expression E5
 
 Vab=Va /tan 60 °+Vb /sin 60 °=f (θ)cos θ  [E5]
 
     Further, the rotation angle θ 1  of the rotor  22 , which is a rotation angle θ less than 360 degrees, is calculated from Vab and Va by the following expression E6. 
     
       
         
           
             
               
                 
                   
                     
                       
                         θ 
                         = 
                           
                         ⁢ 
                         
                           arctan 
                           ⁡ 
                           
                             ( 
                             
                               tan 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               θ 
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           arctan 
                           ⁡ 
                           
                             ( 
                             
                               
                                 f 
                                 ⁡ 
                                 
                                   ( 
                                   θ 
                                   ) 
                                 
                               
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 θ 
                                 / 
                                 
                                   f 
                                   ⁡ 
                                   
                                     ( 
                                     θ 
                                     ) 
                                   
                                 
                               
                               ⁢ 
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               θ 
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           arctan 
                           ⁡ 
                           
                             ( 
                             
                               Vab 
                               / 
                               Va 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     E 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     6 
                   
                   ] 
                 
               
             
           
         
       
     
     The rotation angle calculation unit  24  calculates the number of turns from the amplitude f(θ) of one of the output signals Va and Vb. The rotation angle θ of the rotor  22  is calculated from the one-turn rotation angle θ 1  and the number of turns. 
     When the output signals Va, Vb are normal, a point (Vab, Va) that is defined by Vab and Va can be depicted on a whirl having a variable distance f(θ) from the origin in rectangular coordinates, as shown in  FIG. 11 . The angle between the straight line that joins the point and the origin of the rectangular coordinates and the horizontal Vab axis becomes the rotation angle θ of the rotor  22 . The rotation angle calculation unit  24  judges whether the rotation angle θ is normal or not based on the relationship between Vab, Va and the rotation angle θ. 
     The rotation angle calculation unit  24  also converts the output signals Vb, Vc to digital data and calculates Vab. Then the rotation angle θ is calculated and examined in the same manner as above to examine whether it is normal or not. The rotation angle calculation unit  24  also calculate Vca and the rotation angle θ from the output signals Vc, Va and examines the rotation angle θ in the same manner as above. 
     If the magnetic sensor element  234  fails, the rotation angle calculation unit  24  judges that the output signal Va is not normal, while it judges that the output signals Vb and Vc are normal. In this case, the rotation angle calculation unit  12  provides the rotation angle θ based on the output signals Vb and Vc. 
     A modification of the rotation angle detecting device  2  according to the first embodiment of the invention will be described below. 
     In this modified embodiment, the three magnetic sensor elements  113 - 115  are disposed so that each of the sensing direction is in an angle other than 120 degrees different from those of other sensor elements  113 - 115 . For example, if the phase difference between the output signal Va′ of the sensor element  113  and the output signal Vb′ of the sensor element  114  is α°+90°, the following expressions E7 and E8 may be given, in which A′ is an amplitude of the output signals, and θ′ is a rotation angle of the rotor. 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                             
                         
                         ⁢ 
                         
                           
                             X 
                             ′ 
                           
                           = 
                             
                           ⁢ 
                           
                             
                               [ 
                               
                                 
                                   
                                     
                                       Vab 
                                       ′ 
                                     
                                   
                                 
                                 
                                   
                                     
                                       Va 
                                       ′ 
                                     
                                   
                                 
                               
                               ] 
                             
                             = 
                             
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         
                                           A 
                                           ′ 
                                         
                                         ⁢ 
                                         cos 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           θ 
                                           ′ 
                                         
                                       
                                     
                                   
                                   
                                     
                                       
                                         
                                           A 
                                           ′ 
                                         
                                         ⁢ 
                                         sin 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           θ 
                                           ′ 
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                               = 
                               
                                 R 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   2 
                                   · 
                                   G 
                                   · 
                                   R 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   1 
                                   · 
                                   X 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         X 
                         = 
                           
                         ⁢ 
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     Vb 
                                     ′ 
                                   
                                 
                               
                               
                                 
                                   
                                     Va 
                                     ′ 
                                   
                                 
                               
                             
                             ] 
                           
                           = 
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       A 
                                       ′ 
                                     
                                     ⁢ 
                                     sin 
                                     ⁢ 
                                     
                                       { 
                                       
                                         
                                           θ 
                                           ′ 
                                         
                                         + 
                                         
                                           ( 
                                           
                                             
                                               90 
                                               ⁢ 
                                               ° 
                                             
                                             + 
                                             α 
                                           
                                           ) 
                                         
                                       
                                       } 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       A 
                                       ′ 
                                     
                                     ⁢ 
                                     sin 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       θ 
                                       ′ 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           R 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           1 
                         
                         = 
                           
                         ⁢ 
                         
                           [ 
                           
                             
                               
                                 
                                   cos 
                                   ⁡ 
                                   
                                     [ 
                                     
                                       
                                         - 
                                         π 
                                       
                                       / 
                                       4 
                                     
                                     ] 
                                   
                                 
                               
                               
                                 
                                   - 
                                   
                                     sin 
                                     ⁡ 
                                     
                                       [ 
                                       
                                         
                                           - 
                                           π 
                                         
                                         / 
                                         4 
                                       
                                       ] 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   sin 
                                   ⁡ 
                                   
                                     [ 
                                     
                                       
                                         - 
                                         π 
                                       
                                       / 
                                       4 
                                     
                                     ] 
                                   
                                 
                               
                               
                                 
                                   cos 
                                   ⁡ 
                                   
                                     [ 
                                     
                                       
                                         - 
                                         π 
                                       
                                       / 
                                       4 
                                     
                                     ] 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         G 
                         = 
                           
                         ⁢ 
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     
                                       sin 
                                       ⁡ 
                                       
                                         [ 
                                         
                                           π 
                                           / 
                                           4 
                                         
                                         ] 
                                       
                                     
                                     ⁢ 
                                     
                                       cos 
                                       ⁡ 
                                       
                                         [ 
                                         
                                           
                                             π 
                                             / 
                                             4 
                                           
                                           - 
                                           
                                             α 
                                             / 
                                             2 
                                           
                                         
                                         ] 
                                       
                                     
                                   
                                   
                                     
                                       sin 
                                       ⁡ 
                                       
                                         [ 
                                         
                                           π 
                                           / 
                                           6 
                                         
                                         ] 
                                       
                                     
                                     ⁢ 
                                     cos 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     α 
                                   
                                 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 
                                   
                                     
                                       sin 
                                       ⁡ 
                                       
                                         [ 
                                         
                                           π 
                                           / 
                                           4 
                                         
                                         ] 
                                       
                                     
                                     ⁢ 
                                     
                                       sin 
                                       ⁡ 
                                       
                                         [ 
                                         
                                           
                                             π 
                                             / 
                                             4 
                                           
                                           - 
                                           
                                             α 
                                             / 
                                             2 
                                           
                                         
                                         ] 
                                       
                                     
                                   
                                   
                                     
                                       sin 
                                       ⁡ 
                                       
                                         [ 
                                         
                                           π 
                                           / 
                                           6 
                                         
                                         ] 
                                       
                                     
                                     ⁢ 
                                     cos 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     α 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         
                           R 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           2 
                         
                         = 
                           
                         ⁢ 
                         
                           [ 
                           
                             
                               
                                 
                                   cos 
                                   ⁡ 
                                   
                                     [ 
                                     
                                       
                                         π 
                                         / 
                                         4 
                                       
                                       - 
                                       
                                         α 
                                         / 
                                         2 
                                       
                                     
                                     ] 
                                   
                                 
                               
                               
                                 
                                   - 
                                   
                                     sin 
                                     ⁡ 
                                     
                                       [ 
                                       
                                         
                                           π 
                                           / 
                                           4 
                                         
                                         - 
                                         
                                           α 
                                           / 
                                           2 
                                         
                                       
                                       ] 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   sin 
                                   ⁡ 
                                   
                                     [ 
                                     
                                       
                                         π 
                                         / 
                                         4 
                                       
                                       - 
                                       
                                         α 
                                         / 
                                         2 
                                       
                                     
                                     ] 
                                   
                                 
                               
                               
                                 
                                   cos 
                                   ⁡ 
                                   
                                     [ 
                                     
                                       
                                         π 
                                         / 
                                         4 
                                       
                                       - 
                                       
                                         α 
                                         / 
                                         2 
                                       
                                     
                                     ] 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   [ 
                   E7 
                   ] 
                 
               
             
             
               
                 
                   
                     θ 
                     ′ 
                   
                   = 
                   
                     
                       arctan 
                       ⁡ 
                       
                         ( 
                         
                           tan 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             θ 
                             ′ 
                           
                         
                         ) 
                       
                     
                     = 
                     
                       
                         arctan 
                         ⁡ 
                         
                           ( 
                           
                             
                               A 
                               ′ 
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               θ 
                               / 
                               
                                 A 
                                 ′ 
                               
                             
                             ⁢ 
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               θ 
                               ′ 
                             
                           
                           ) 
                         
                       
                       = 
                       
                         arctan 
                         ⁡ 
                         
                           ( 
                           
                             
                               Vab 
                               ′ 
                             
                             / 
                             
                               Va 
                               ′ 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   [ 
                   E8 
                   ] 
                 
               
             
           
         
       
     
     If four or more magnetic sensor elements are available, an accurate rotation angle can be provided even if two or more sensor elements fail. 
     The magnetic sensor elements can be replaced by other sensor elements that provide sinusoidal output signals, such as optical sensor elements. 
     The shape of permanent magnet can be replaced by other than cylindrical shape, such as a rectangular shape if it provides a magnetic field that is perpendicular to the rotation axis of the rotor. 
     In the foregoing description of the present invention, the invention has been disclosed with reference to specific embodiments thereof. It will, however, be evident that various modifications and changes may be made to the specific embodiments of the present invention without departing from the scope of the invention as set forth in the appended claims. Accordingly, the description of the present invention is to be regarded in an illustrative, rather than a restrictive, sense.