Abstract:
A method of commutating a motor includes operatively interfacing a stator and actuated component of the motor, arranging at least two winding sets relative to the actuated component, and independently controlling the at least two winding sets so that with the at least two winding sets the actuated component is both driven and centered.

Description:
[0001]    This application is related to U.S. patent application Ser. No. Attorney Docket: 390-012197-US (PAR), entitled: Commutation of An Electromagnetic Propulsion and Guidance System, filed Jun. 27, 2007. 
     
    
     BACKGROUND 
       [0002]    The disclosed embodiments relate to electromagnetic bearings and motors, in particular to a self bearing motor with a reduced number of winding sets. 
       Brief Description of Related Developments 
       [0003]    In certain applications utilizing motors, materials must be processed in a controlled, clean atmosphere where microscopic contaminates may represent a severe problem. In those applications, cleanliness may be directly related to yield, which may in turn affect cost. Other motor applications may include processing steps utilizing hostile atmospheres of highly corrosive gases and high temperatures. Motors with contact bearings may wear, produce particulate contamination, and eventually fail due to the hostile environment. Bearings may also exhibit an unacceptable amount of vibration and play before failing. 
         [0004]    U.S. Pat. No. 5,818,137, issued on Oct. 6, 1998 describes an integrated magnetic motoring and suspension system with a stator that has permanent magnets that lift the rotor, eight control windings that stabilize the radial position and tilt orientation of the rotor, and drive windings that actuate the rotor. U.S. Pat. No. 6,707,200, issued on Mar. 16, 2004 describes a rotational magnetic gimbal with an integral magnetic bearing that utilizes four equal length coil segments, each having at least two phases. Torque is applied to each of the coil segments independently (or to groups of coil segments, e.g., as in three-phase motors) to achieve both a desired rotational speed and a desired radial position. U.S. Pat. No. 7,078,839, issued on Jul. 18, 2006 describes a self bearing asymmetrical stepper motor with five coil segments having three phases each. Magnitudes of the currents supplied to the respective coils vary according to a geometric eccentricity of the center of the rotor, and accordingly different magnetic forces are generated from the respective electromagnets. Due to the difference in the magnetic forces, the rotor can be supported and compensation is provided for the geometric deviation. 
       SUMMARY 
       [0005]    The disclosed embodiments are directed to a method of commutating a motor including operatively interfacing a stator and actuated component of the motor, arranging at least two winding sets relative to the actuated component, and independently controlling the at least two winding sets so that with the at least two winding sets the actuator is both driven and centered. 
         [0006]    In another embodiment, an apparatus for commutating a motor includes at least two winding sets arranged relative to an actuated component of the motor, and commutation circuitry operable to independently control the at least two winding sets so that with the at least two winding sets the actuated component is both driven and centered. 
         [0007]    In still another embodiment a motor includes a stator having at least two independently controlled winding sets, an actuated component operatively interfacing the stator, and a controller communicatively connected to the at least two winding sets for controlling the at least two winding sets to independently drive and center the actuated component, where the at least two winding sets are arranged relative to the actuated component and the controller is programmed to control the at least two winding sets so that with the at least two winding sets the actuated component is both driven and centered. 
         [0008]    In other embodiments, a substrate processing apparatus has a motor with a stator having at least two independently controlled winding sets, an actuated component operatively interfacing the stator, and a controller communicatively connected to the at least two winding sets for controlling the at least two winding sets to independently drive and center the actuated component, wherein the at least two winding sets are arranged relative to the actuated component and the controller is programmed to control the at least two winding sets so that with the at least two winding sets the actuated component is both driven and centered. 
         [0009]    In other embodiments, a method of commutating a motor includes calculating an adjustment electrical angle, and utilizing the adjustment electrical angle in a common set of commutation equations to independently torque and actively center a rotor of the motor so that the rotor is actively centered with at least two of the motors winding sets. 
         [0010]    In another embodiment, a method of commutating a motor includes calculating an adjustment electrical angle and entering the adjustment electrical angle into commutation equations for commutating a motor to independently torque and actively center a rotor of the motor so that the rotor is actively centered with at least two of the motors winding sets. 
         [0011]    In another embodiment, an apparatus for commutating a motor includes circuitry for calculating an adjustment electrical angle, and a current amplifier operable to utilize the adjustment electrical angle in a common set of commutation equations so that the common set of commutation equations is capable of producing both torque and active centering forces in motors with at least two winding sets and in motors with at least three winding sets. 
         [0012]    In still another embodiment, a motor includes a rotor, and windings driven by a current amplifier, the current amplifier having circuitry for calculating an adjustment electrical angle, and an amplifier operable to utilize the adjustment electrical angle in a common set of commutation equations so that the common set of commutation equations is capable of producing both torque and active centering forces that are different from each other in the motor, where the motor is at least one of a two winding set motor or a three winding set motor. 
         [0013]    In yet another embodiment, a substrate processing apparatus includes a controller for commutating a motor including circuitry for calculating an adjustment electrical angle, and a current amplifier operable to utilize the adjustment electrical angle in a common set of commutation equations so that the common set of commutation equations is capable of producing both torque and active centering forces that are different from each other in the motor, wherein the motor is at least one of a two winding set motor or a three winding set motor. 
         [0014]    In yet a further embodiment, a substrate processing apparatus includes a motor having a rotor and windings driven by a current amplifier, the current amplifier having circuitry for calculating an adjustment electrical angle, and an amplifier operable to utilize the adjustment electrical angle in a common set of commutation equations so that the common set of commutation equations is capable of producing both torque and active centering forces that are different from each other in the motor, where the motor is at least one of a two winding set motor or a three winding set motor. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0015]    The foregoing aspects and other features of the presently disclosed embodiments are explained in the following description, taken in connection with the accompanying drawings, wherein: 
           [0016]      FIG. 1  shows a schematic diagram of an exemplary motor suitable for practicing the disclosed embodiments; 
           [0017]      FIG. 1A  shows an exemplary flow diagram of the operation of the embodiment of  FIG. 1 . 
           [0018]      FIG. 2  shows an exemplary motor embodiment having a linear configuration; 
           [0019]      FIG. 3  shows an exemplary motor embodiment with two winding sets offset by about 180 degrees; 
           [0020]      FIG. 4  shows an exemplary embodiment with winding sets distributed on the circumference of the motor; 
           [0021]      FIG. 5  shows an exemplary embodiment where the winding sets are divided into a number of winding subset sets; 
           [0022]      FIG. 6  shows an exemplary embodiment utilizing three winding sets; 
           [0023]      FIG. 7  shows an exemplary embodiment utilizing four winding sets; 
           [0024]      FIG. 8  shows an exemplary embodiment of a sensor system suitable for providing position feedback for the disclosed embodiments; and 
           [0025]      FIG. 9  shows a top view of an exemplary substrate processing apparatus  1000  incorporating features of the disclosed embodiments. 
       
    
    
     DETAILED DESCRIPTION 
       [0026]      FIG. 1  shows a schematic diagram of an exemplary motor  100  suitable for practicing the embodiments disclosed herein. Although the presently disclosed embodiments will be described with reference to the drawings, it should be understood that they may be embodied in many alternate forms. It should also be understood that any suitable size, shape or type of elements or materials could be used. 
         [0027]    Motor  100  includes an actuated component  110 , in this embodiment in the form of a rotor, and winding sets  115 ,  120 . For purposes of the disclosed embodiments, it should be understood that the term actuated component includes a device that performs a motion or applies a force in response to forces generated by the winding sets described herein. The rotors and platens of the disclosed embodiments are examples of actuated components. 
         [0028]    The embodiment of exemplary motor  100  depicted in  FIG. 1  is shown as having a rotary configuration, although other embodiments may include linear configurations as will be described below. Actuated component  110  also referred to as a rotor, may have any suitable construction. Winding sets  115 ,  120  may include one or more windings and may be driven by current amplifier  125  which may include software, hardware, or a combination of software and hardware suitable for driving the winding sets. 
         [0029]      FIG. 1A  shows an exemplary flow diagram of the operation of the embodiment of  FIG. 1 . Referring to  FIGS. 1 and 1A , in block  10 , controller  112  may operate to receive actuated component position information from one or more sensors or sensor systems  105  of motor  100  (block  60 ). The controller may then may determine force and torque commands from the position information and system control laws. In block  20 , the controller  112  may then apply one or more force transformation functions to generate a set of commanded winding forces. In block  30 , the commanded winding forces may then be used to determine commutation parameters, for example, I and Δ as will be explained below. The controller may then provide the commutation parameters to the current amplifier  125 . 
         [0030]    The current amplifier  125  may also include a processor  127 , a commutation function  130  and a current loop function  135  for driving the winding sets. The processor may generally control and coordinate the operations of commutation and current loop functions. The processor may provide the commutation parameters to the commutation function  130 , which as shown in block  40 , may determine commanded currents for one or more windings of each winding set according to a set of commutation equations. In block  50 , the current loop function  135  may provide actual currents to the motor windings  115 ,  120 . The current loop function may also provide a feedback and driving capability for maintaining the current through the windings as determined. Each current amplifier disclosed herein includes circuitry, hardware or software in any combination as required to perform the functions and computations for the disclosed embodiments. 
         [0031]      FIG. 2  shows another exemplary embodiment having a linear configuration. Motor  200  includes an actuated component  210 , in this embodiment having the form of a platen and winding sets  215 ,  220 . Similar to the embodiment of  FIG. 1 , actuated component  210  may be constructed in any suitable manner and winding sets  215 ,  220  may include one or more windings. For purposes of the disclosed embodiments, it should be understood that the rotor may be constrained along the axial direction. 
         [0032]    It should be understood that individual winding sets  215 ,  220  may have any suitable orientation. Current amplifier  225  may drive winding sets  215 ,  225 , and may include software, hardware, or any combination of software and hardware suitable for driving the winding sets. A processor  230 , a commutation function  235  and a current loop function  240  may be utilized by current amplifier  225  for driving winding sets  215 ,  225  similar to the embodiment of  FIG. 1 . The processor  230 , commutation function  235 , and current loop function  240  may also include circuitry for receiving feedback from one or more sensors or sensor systems that provide position information 
         [0033]    The disclosed embodiments are directed to a self-bearing motor that advantageously utilizes a minimal number of winding sets to affect the self bearing motor function. For example, one or more embodiments may utilize as few as two of a motor&#39;s winding sets to produce the forces required for self bearing. The windings may include any type of winding suitable for use in the disclosed embodiments. 
         [0034]    The disclosed embodiments may utilize the same windings to provide substantially decoupled torque forces and radial forces in order to actively control the center of rotation of the actuated component or rotor in the rotary embodiments. In the linear embodiments the same windings may be utilized to provide substantially decoupled linear forces in the direction of motion and guidance forces for actively controlling the platen around a distance across the gap between the stator and the platen. For purposes of the disclosed embodiments linear forces in the direction of motion and torque are referred to collectively as driving forces. For purposes of the disclosed embodiments, the radial forces for actively controlling the center of rotation of the rotor in the rotary embodiments, and the guidance forces across the gap between the stator and the platen are referred to collectively as centering forces. 
         [0035]    The disclosed motor embodiments may include segmented windings, for example, winding sets divided into one or more winding subset sets distributed along desired portions of the rotor or platen. Each winding subset set may include one or more windings and may be driven to produce substantially decoupled radial and tangential forces simultaneously for rotary embodiments and to provide substantially decoupled propulsion and guidance forces for linear embodiments. In at least one embodiment, an adjustment electrical angle may be calculated and utilized in a common set of commutation equations to energize the windings. Other commutation implementations may also be utilized. As mentioned above, the actuated components of the embodiments, for example, rotor  110  and platen  210  may be actively centered with as few as two of the motors winding sets. Additional embodiments of the self bearing motor may utilize a novel arrangement of windings and magnetic poles. 
         [0036]    Some embodiments of the self-bearing motor may utilize as few as two winding sets and as few as two control channels. One or more commutation functions may be utilized which provide forces along both tangential and radial directions to the actuated component in the form of a rotor, or along motion and guidance directions to the actuated component in the form of a platen. Lorentz and Maxwell type forces may be employed. Optimization techniques may be employed to improve efficiencies in generating driving and centering forces. 
         [0037]    A sensor system may be included that provides both position and gap feedback. For example, the sensor system may be configured to provide simultaneous position measurement along a first and second axis. The sensor system may also be capable of providing measurement along a third axis, either simultaneously with the first and second axis measurements or individually. In one or more embodiments, the sensor system may provide all measurements individually, simultaneously or in any combination. In at least one embodiment one axis may extend tangentially to the motor rotor or platen while another axis may be orthogonal to the surface of the rotor or platen. 
         [0038]    In further embodiments, position and gap feedback for the motor embodiments are obtained utilizing a limited number of sensors. In at least one embodiment, as few as two pairs of sensors may be used, eliminating the need for multiple numbers of sensor devices. Within each pair, the sensors may be positioned some number of mechanical or electrical degrees apart. Counterpart sensors in the first pair may be positioned a certain number of mechanical or electrical degrees apart from counterpart sensors in additional pairs. In at least one embodiment, two pairs of sensors are used to provide feedback for the disclosed motors. 
         [0039]    Applications of the disclosed embodiments may include any suitable motor application, for example, robotics in semiconductor processing. Self-bearing motors as disclosed may also be utilized as replacements for regular bearings. For example, self bearing motors may be advantageously used by drive shafts that control robotic arms. The elimination of contact bearings advantageously reduces particle generation and reducing the number of amplifier channels provides cost savings in hardware implementation. 
         [0040]      FIGS. 3 ,  4 ,  5 ,  6 , and  7  show additional embodiments of a self-bearing brushless motor as disclosed herein. For simplicity only one phase per motor is shown. The direction of current in the various conductors may be indicated by a dot indicating a direction out of the paper plane and an “x” indicating a direction into the paper plane. Each motor may include a permanent magnet actuated component, for example, rotor  310  having a sequence of alternating poles designated “N” and “S,” however, any suitable rotor structure may be utilized. 
         [0041]    In the exemplary embodiments, the stators may have an “iron” core (e.g. stator backing made of any suitable material, such as ferromagnetic material), or the motors may have coreless stators. The relationships for desired driving forces torque (T) and centering forces (F x ) along the x-axis and (F y ) along the y-axis for the embodiments shown in  FIGS. 3 ,  4 ,  5 ,  6 , and  7  are described herein for motor arrangements utilizing Lorentz forces and for arrangements using a combination of Lorentz and Maxwell forces. The relationships may be utilized to determine appropriate commutation operations for each winding set that will concurrently produce the desired torque T, force F x  and force F y . 
         [0042]    In at least one embodiment, the relationships may be utilized to determine appropriate phase current amplitudes (I j , j=A, B, C, . . . ) and electrical angle offsets (Δ j , j=A, B, C, . . . ) for each winding set that will concurrently produce decoupled torque T and radial forces F x , F y  in the motor. Once determined, the phase current amplitudes (I j , j=A, B, C, . . . ) and electrical angle offsets (Δ j , j=A, B, C, . . . ) may be used with a common set of commutation functions to produce a desired torque (T) and also desired centering forces (F x ) along the x-axis and (F y ) along the y-axis. 
         [0043]    In some embodiments, the motor may be driven by a 3 phase sinusoidal commutation. In other embodiments, as an alternative to sinusoidal commutation, space vector modulation may be used for wye-configured windings to produce equivalent sine-wave currents through the windings. 
         [0044]    Referring to  FIG. 3 , exemplary relationships for the desired torque (T) and centering forces F x  (along x-axis) and F y  (along y-axis) as a function of the radial and tangential forces F rj  and F tj  (j=A,B) utilizing Lorentz forces for 2 winding sets, winding set A  315 , and winding set B  320  are shown below. 
         [0045]    It should be understood that the relationships, calculations and resulting commutation schemes are exemplary and that any suitable commutation solutions may be utilized in the disclosed embodiments to determine decoupled driving and centering forces. While two winding sets are shown it should be understood that the motor may include additional windings or winding sets in addition to those used according to the disclosed embodiments. It should also be understood that any number of winding sets may be utilized to provide the desired decoupled driving and centering forces of the embodiments described herein. While winding sets A  315  and B  320  are shown offset by about 180 degrees it should be understood that other offsets may also be utilized. 
         [0046]    As noted before the stator on which the winding sets are disposed may have iron backing (e.g. a cored stator) or the stator may be coreless. Exemplary force relations, for this embodiment of  FIG. 3  utilizing Lorentz forces, may be expressed as follows, where R is the radius of the rotor. 
         [0000]        T=R ( F   tA   +F   tB )   (1) 
         [0000]        F   x   =F   tA   −F   tB    (2) 
         [0000]        F   y   =F   rA   −F   rB    (3) 
         [0047]    It can be seen that the force relations are expressed as 3 equations and 4 unknowns. An additional equation may be obtained, for example, by finding the minimum of the following cost function. 
         [0000]        J=F   rA   2   +F   rB   2    (4) 
         [0048]    It should be noted that the cost function used above is an example which conveniently distributes the y-direction force equally between the two winding sets, and prevents the different radial forces from interfering with each other. Other cost functions may also be used to solve the problem. For example, another cost function may be utilized that is based on currents drawn by the windings. 
         [0049]    The minimization of the cost function above will provide the minimum radial forces subjected to the constraints of (1) to (3): 
         [0000]    
       
         
           
             
               
                 
                   
                     F 
                     rB 
                   
                   = 
                   
                     - 
                     
                       
                         F 
                         y 
                       
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
             
               
                 
                   
                     F 
                     rA 
                   
                   = 
                   
                     
                       F 
                       y 
                     
                     2 
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
         [0050]    Solving (1) and (2) for F tA  and F tB : 
         [0000]    
       
         
           
             
               
                 
                   
                     F 
                     tA 
                   
                   = 
                   
                     
                       1 
                       2 
                     
                      
                     
                       ( 
                       
                         
                           T 
                           R 
                         
                         + 
                         
                           F 
                           x 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
             
               
                 
                   
                     F 
                     tB 
                   
                   = 
                   
                     
                       1 
                       2 
                     
                      
                     
                       ( 
                       
                         
                           T 
                           R 
                         
                         - 
                         
                           F 
                           x 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
         [0051]    The following force/current relations may be utilized for the embodiment utilizing Lorentz forces of  FIG. 3  (assuming 3-phase winding sets): 
         [0000]      F tA =1.5K t I A  cos Δ A    (9) 
         [0000]      F rA =1.5K r I A  sin Δ A    (10) 
         [0000]      F tB =1.5K t I B  cos Δ B    (11) 
         [0000]      F rB =1.5K r I B  sin Δ B    (12) 
         [0000]    where:
   I A =Amplitude of phase current for winding A (A)   I B =Amplitude of phase current for winding B (A)   Δ A =Electrical angle offset for winding set A (rad)   Δ B =Electrical angle offset for winding set B (rad)   
 
         [0056]    Using (7) to (12): 
         [0000]    
       
         
           
             
               
                 
                   
                     Δ 
                     A 
                   
                   = 
                   
                     a 
                      
                     
                         
                     
                      
                     
                       tan 
                       ( 
                       
                         
                           
                             F 
                             y 
                           
                           
                             
                               T 
                               R 
                             
                             + 
                             
                               F 
                               s 
                             
                           
                         
                          
                         
                           
                             K 
                             t 
                           
                           
                             K 
                             r 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
             
               
                 
                   
                     Δ 
                     B 
                   
                   = 
                   
                     a 
                      
                     
                         
                     
                      
                     
                       tan 
                       ( 
                       
                         
                           
                             F 
                             y 
                           
                           
                             
                               F 
                               x 
                             
                             - 
                             
                               T 
                               R 
                             
                           
                         
                          
                         
                           
                             K 
                             t 
                           
                           
                             K 
                             r 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
         [0057]    For purposes of the disclosed embodiments, all arc tangent functions (atan) described herein may also be interpreted as a four quadrant inverse tangent functions (atan2) and the corresponding arguments therefore. 
         [0058]    The following exemplary motor commutation equations may be utilized: 
         [0000]        i   Aj   =I   A  sin [θ A ( x,z )−Δ A +(2π/3) j], j= 0, 1, 2   (17) 
         [0000]        i   Bj   =I   B  sin [θ B ( x,z )−Δ B +(2π/3) j], j= 0, 1, 2   (18) 
         [0000]    where I A , Δ A , I B , Δ B  control magnitudes and orientations of force vectors produced by winding sets A and B, and where:
   j=the individual phases of each winding set.   
 
         [0060]    Thus, in the example, by adjusting the electrical angles θ A  and θ B  with the electrical angle offsets Δ A  and Δ B , a common or standard set of motor commutation equations may be used for producing desired torque (T) and active centering forces (F x ) along the x-axis and (F y ) along the y-axis utilizing, in this embodiment, only 2 winding sets producing Lorentz forces. Other commutation schemes using any suitable commutation implementation may also be employed. 
         [0061]    A suitable exemplary technique utilizing a standard set of commutation equations in this manner may be found in U.S. patent application Attorney Docket: 390-012197-US (PAR), entitled: Commutation of An Electromagnetic Propulsion and Guidance System, filed Jun. 27, 2007 which is incorporated by reference herein in its entirety. 
         [0062]    Again referring to  FIG. 3 , the motor arrangement in accordance with another exemplary embodiment may utilize a combination of Lorentz and Maxwell forces. The relationships for the desired torque (T) and centering forces (Fx) along the x-axis and (Fy) along the y-axis as a function of the radial and tangential forces (Frj, Ftj, j=A, B) are the same as shown in Equations (1) to (3). Similar to the embodiments above, it should be understood that the relationships, calculations and resulting commutation schemes are exemplary and that any suitable commutation solutions may be utilized in the disclosed embodiments. 
         [0063]    The following force/current relations may be utilized for the embodiment of  FIG. 3  utilizing Lorentz and Maxwell forces. 
         [0000]      F tA =1.5 K   t I A  cos Δ A    (19) 
         [0000]        F   rA =1.5 K   r   I   A  sin Δ A +1.5K M   I   A   2    (20) 
         [0000]      F tB =1.5K t I B  cos Δ B    (21) 
         [0000]        F   rB =1.5 K   r   I   B  sin Δ B +1.5K M   I   B   2    (22) 
         [0064]    The derivation may be separated into two cases, where F y &gt;0 and F y &lt;0 (with respect to the sign convention shown in  FIG. 3 ). These two cases are exemplary. It should be understood that other techniques and force distributions may also be used to determine solutions for I A , Δ A , I B , Δ B  while still utilizing a single winding set to produce both tangential and radial forces according to the disclosed embodiments. 
         [0065]    For the case where F y &gt;0, make 
         [0000]      Δ B =0   (23) 
         [0000]    in order, for example, to minimize the Maxwell force associated with winding set B  320  which would otherwise interfere with F y . 
         [0066]    For the case where F y &lt;0, make 
         [0000]      Δ A =0   (24) 
         [0000]    in order, for example, to minimize the Maxwell force associated with winding set A  315  which would otherwise interfere with F y . 
         [0067]    Therefore, for the exemplary case where F y &gt;0: 
         [0000]    
       
         
           
             
               
                 
                   
                     I 
                     A 
                   
                   = 
                   
                     
                       
                         
                           - 
                           
                             b 
                             A 
                           
                         
                         + 
                         
                           
                             
                               b 
                               A 
                               2 
                             
                             - 
                             
                               4 
                                
                               
                                 a 
                                 A 
                               
                                
                               
                                 c 
                                 A 
                               
                             
                           
                         
                       
                       
                         2 
                          
                         
                           a 
                           A 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
             
               
                 
                   
                     I 
                     B 
                   
                   = 
                   
                     
                       1 
                       
                         3 
                          
                         
                           K 
                           t 
                         
                       
                     
                      
                     
                       ( 
                       
                         
                           T 
                           R 
                         
                         - 
                         
                           F 
                           x 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
             
               
                 
                   
                     Δ 
                     A 
                   
                   = 
                   
                     a 
                      
                     
                         
                     
                      
                     
                       tan 
                       ( 
                       
                         
                           
                             K 
                             t 
                           
                           
                             K 
                             r 
                           
                         
                          
                         
                           
                             
                               F 
                               rA 
                             
                             - 
                             
                               1.5 
                                
                               
                                 K 
                                 M 
                               
                                
                               
                                 I 
                                 A 
                                 2 
                               
                             
                           
                           
                             F 
                             tA 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   27 
                   ) 
                 
               
             
             
               
                 
                   
                     Δ 
                     B 
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   28 
                   ) 
                 
               
             
           
         
       
     
         [0000]    and for the exemplary case where F y &lt;0: 
         [0000]    
       
         
           
             
               
                 
                   
                     I 
                     A 
                   
                   = 
                   
                     
                       1 
                       
                         3 
                          
                         
                           K 
                           t 
                         
                       
                     
                      
                     
                       ( 
                       
                         
                           T 
                           R 
                         
                         + 
                         
                           F 
                           x 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   29 
                   ) 
                 
               
             
             
               
                 
                   
                     I 
                     B 
                   
                   = 
                   
                     
                       
                         
                           - 
                           
                             b 
                             B 
                           
                         
                         + 
                         
                           
                             
                               b 
                               B 
                               2 
                             
                             - 
                             
                               4 
                                
                               
                                 a 
                                 B 
                               
                                
                               
                                 c 
                                 B 
                               
                             
                           
                         
                       
                       
                         2 
                          
                         
                           a 
                           B 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   30 
                   ) 
                 
               
             
             
               
                 
                   
                     Δ 
                     A 
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   31 
                   ) 
                 
               
             
             
               
                 
                   
                     Δ 
                     B 
                   
                   = 
                   
                     a 
                      
                     
                         
                     
                      
                     
                       tan 
                       ( 
                       
                         
                           
                             K 
                             t 
                           
                           
                             K 
                             r 
                           
                         
                          
                         
                           
                             
                               F 
                               rB 
                             
                             - 
                             
                               1.5 
                                
                               
                                 K 
                                 M 
                               
                                
                               
                                 I 
                                 B 
                                 2 
                               
                             
                           
                           
                             F 
                             tB 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   32 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where a j , b j  and c j , j=A, B, as a function of T, Fx and Fy are defined in Table 1 below: 
         [0000]    
       
         
               
               
               
             
           
               
                 TABLE 1 
               
               
                   
               
               
                 PA- 
                   
                   
               
               
                 RAMETERS 
                 F y  &gt; 0 
                 F y  &lt; 0 
               
               
                   
               
             
             
               
                 a j   
                 
                   
                     
                       
                         
                           a 
                           A 
                         
                         = 
                         
                           
                             ( 
                             
                               
                                 K 
                                 M 
                               
                               
                                 K 
                                 r 
                               
                             
                             ) 
                           
                           2 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           a 
                           B 
                         
                         = 
                         
                           
                             ( 
                             
                               
                                 K 
                                 M 
                               
                               
                                 K 
                                 r 
                               
                             
                             ) 
                           
                           2 
                         
                       
                     
                   
                 
               
               
                   
               
               
                 b j   
                 
                   
                     
                       
                         
                           b 
                           A 
                         
                         = 
                         
                           - 
                           
                             
                               ( 
                               
                                 
                                   
                                     4 
                                     3 
                                   
                                    
                                   
                                     F 
                                     rA 
                                   
                                    
                                   
                                     
                                       K 
                                       M 
                                     
                                     
                                       K 
                                       r 
                                       2 
                                     
                                   
                                 
                                 + 
                                 1 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           b 
                           B 
                         
                         = 
                         
                           - 
                           
                             
                               ( 
                               
                                 
                                   
                                     4 
                                     3 
                                   
                                    
                                   
                                     F 
                                     rB 
                                   
                                    
                                   
                                     
                                       K 
                                       M 
                                     
                                     
                                       K 
                                       r 
                                       2 
                                     
                                   
                                 
                                 + 
                                 1 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 c j   
                 
                   
                     
                       
                         
                           c 
                           A 
                         
                         = 
                         
                           
                             
                               ( 
                               
                                 
                                   F 
                                   tA 
                                 
                                 
                                   1.5 
                                    
                                   
                                     K 
                                     t 
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   F 
                                   rA 
                                 
                                 
                                   1.5 
                                    
                                   
                                     K 
                                     r 
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           c 
                           B 
                         
                         = 
                         
                           
                             
                               ( 
                               
                                 
                                   F 
                                   tB 
                                 
                                 
                                   1.5 
                                    
                                   
                                     K 
                                     t 
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   F 
                                   rB 
                                 
                                 
                                   1.5 
                                    
                                   
                                     K 
                                     r 
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 F tA   
                 
                   
                     
                       
                         
                           1 
                           2 
                         
                          
                         
                           ( 
                           
                             
                               T 
                               R 
                             
                             + 
                             
                               F 
                               x 
                             
                           
                           ) 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           1 
                           2 
                         
                          
                         
                           ( 
                           
                             
                               T 
                               R 
                             
                             + 
                             
                               F 
                               x 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                   
               
               
                 F tB   
                 
                   
                     
                       
                         
                           1 
                           2 
                         
                          
                         
                           ( 
                           
                             
                               T 
                               R 
                             
                             - 
                             
                               F 
                               x 
                             
                           
                           ) 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           1 
                           2 
                         
                          
                         
                           ( 
                           
                             
                               T 
                               R 
                             
                             - 
                             
                               F 
                               x 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                   
               
               
                 F rA   
                 
                   
                     
                       
                         
                           F 
                           y 
                         
                         + 
                         
                           
                             
                               K 
                               M 
                             
                             
                               6 
                                
                               
                                 K 
                                 t 
                                 2 
                               
                             
                           
                            
                           
                             
                               ( 
                               
                                 
                                   T 
                                   R 
                                 
                                 - 
                                 
                                   F 
                                   x 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             K 
                             M 
                           
                           
                             6 
                              
                             
                               K 
                               t 
                               2 
                             
                           
                         
                          
                         
                           
                             ( 
                             
                               
                                 T 
                                 R 
                               
                               + 
                               
                                 F 
                                 x 
                               
                             
                             ) 
                           
                           2 
                         
                       
                     
                   
                 
               
               
                   
               
               
                 F rB   
                 
                   
                     
                       
                         
                           
                             K 
                             M 
                           
                           
                             6 
                              
                             
                               K 
                               t 
                               2 
                             
                           
                         
                          
                         
                           
                             ( 
                             
                               
                                 T 
                                 R 
                               
                               - 
                               
                                 F 
                                 x 
                               
                             
                             ) 
                           
                           2 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             
                               K 
                               M 
                             
                             
                               6 
                                
                               
                                 K 
                                 t 
                                 2 
                               
                             
                           
                            
                           
                             
                               ( 
                               
                                 
                                   T 
                                   R 
                                 
                                 + 
                                 
                                   F 
                                   x 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                         - 
                         
                           F 
                           y 
                         
                       
                     
                   
                 
               
               
                   
               
             
          
         
       
     
         [0068]    By applying I A , Δ A , I B , Δ B  as derived in the commutation functions  130  to drive winding sets A  315  and B  320 , using for example (17) and (18) above, a common set of motor commutation equations may be used to produce desired torque (T) and active centering forces (F x ) along the x-axis and (F y ) along the y-axis utilizing, in this embodiment, only 2 winding sets producing both Lorentz and Maxwell forces. 
         [0069]    Thus, similar to the embodiment above, by adjusting the electrical angles θ A  and θ B  with the electrical angle offsets Δ A  and Δ B , a common or standard set of motor commutation equations may be used for producing desired torque (T) and active centering forces (F x ) along the x-axis and (F y ) along the y-axis utilizing, in this embodiment, only 2 winding sets producing Lorentz and Maxwell forces. Other commutation schemes using any suitable commutation implementation may also be utilized. 
         [0070]      FIG. 4  shows an exemplary embodiment where the winding sets  415 ,  420  are distributed at desired locations on the circumference of the rotor  410 . Each winding set  415 ,  420  may be segmented into any number of winding subsets, which may be distributed at any number of desired locations. The winding subsets may be grouped or distributed with any desired electrical or mechanical offset with respect to each other or with respect to the rotor  410 . Any number of winding sets and winding subsets and any suitable distribution may be utilized.  FIG. 4  shows an exemplary embodiment utilizing two winding sets  415 ,  420 , each arranged as two winding subsets,  425 ,  430  and  435 ,  440 , respectively. The two winding subsets in each winding set are coupled electrically and shifted with respect to each other by any suitable electrical and mechanical offset. 
         [0071]      FIG. 5  shows an exemplary embodiment utilizing two winding sets A  515 , and B  520 , each arranged as two winding subsets,  525 ,  530  and  535 ,  540 , respectively. The two winding subsets in each winding set are coupled electrically and shifted with respect to each other by about 90 electrical degrees. As a result, when one of the two winding subsets in the pair produces pure tangential force the other winding subset in the pair generates pure radial force, and vice versa. In the exemplary embodiment shown, the segments of each of the respective winding sets may be geometrically arranged at an angle of about 90°. In alternate embodiments the geometric angular offset and the electrical angle offset between winding segments of a respective winding set may be different from each other. In this embodiment, winding set A  515  has two winding subsets A 0    530  and A 90    525 , and winding set B  520  has two winding subsets B 0    540  and B 90    535 . Each of the winding sets A and B may be driven by a current amplifier similar to current amplifier  125  of  FIG. 1 . 
         [0072]    Exemplary relationships for the desired torque (T) and centering forces (F x ) along the x-axis and (F y ) along the y-axis for the segmented winding sets  515 ,  520  of the embodiment of  FIG. 5  utilizing Lorentz forces are as shown below. It should be understood that the relationships, calculations and resulting commutation schemes are exemplary and that any suitable commutation solutions may be utilized in the disclosed embodiments. While two winding sets are shown it should be understood that any number of winding sets may be utilized. Similarly, while four winding subsets are shown it should be understood that any number of winding subsets may be utilized. In addition, while winding subsets  525 ,  530 ,  535 ,  540  are shown offset by 90 degrees it should be understood that other offsets may also be utilized. 
         [0073]    The force relations for this embodiment of  FIG. 5  may be expressed as follows: 
         [0000]        T=R ( F   tA   0   +F   tB   90   +F   tB   0   +F   tA   90 )   (33) 
         [0000]        F   x   =F   tA   0   +F   rA   90   −F   tB   0   −F   rB   90    (34) 
         [0000]        F   y   =F   rA   0   +F   tA   90   −F   rB   0   −F   tB   90    (35) 
         [0074]    Note that the following equations hold because within each winding set the winding subsets are offset by 90 electrical degrees. 
         [0000]      F rA   0 =βF tA   90    (36) 
         [0000]      F rA   90 =βF tA   0    (37) 
         [0000]      F rB   0 =βF tB   90    (38) 
         [0000]      F rB   90 =βF tB   0    (39) 
         [0000]        β=K   r   /K   t    (40) 
         [0075]    The following force/current relations may be utilized for the embodiment of  FIG. 5  utilizing Lorentz forces: 
         [0000]      F tA   0 =1.5K t I A  cos Δ A    (41) 
         [0000]      F rA   0 =1.5K r I A  sin Δ A    (42) 
         [0000]      F tB   0 =1.5K t I B  cos Δ B    (43) 
         [0000]      F rB   0 =1.5K r I B  sin Δ B    (44) 
         [0076]    In order to find a solution for I A , Δ A , I B , Δ B  in terms of the radial and tangential forces, it is necessary to add one more equation since there are more variables than equations. There are many ways to solve this problem and different solutions can be obtained depending on the design criteria utilized. As an example, the additional equation is proposed to be obtained by minimizing the cost function below: 
         [0000]        J= ( F   rA   0 ) 2 +( F   rA   90 ) 2 +( F   rB   0 ) 2 +( F   rB   90 ) 2    (45) 
         [0077]    From equations (33) to (45) it is possible to arrive at the following solutions: 
         [0000]    
       
         
           
             
               
                 
                   
                     F 
                     rA 
                     0 
                   
                   = 
                   
                     
                       
                         β 
                         
                           4 
                            
                           R 
                         
                       
                        
                       
                         { 
                         
                           T 
                           - 
                           
                             
                               R 
                               
                                 β 
                                 + 
                                 1 
                               
                             
                              
                             
                               [ 
                               
                                 
                                   F 
                                   x 
                                 
                                 - 
                                 
                                   F 
                                   y 
                                 
                               
                               ] 
                             
                           
                         
                         } 
                       
                     
                     + 
                     
                       
                         β 
                         
                           4 
                            
                           
                             ( 
                             
                               β 
                               + 
                               1 
                             
                             ) 
                           
                         
                       
                        
                       
                         [ 
                         
                           
                             F 
                             x 
                           
                           + 
                           
                             F 
                             y 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   46 
                   ) 
                 
               
             
             
               
                 
                   
                     F 
                     rB 
                     0 
                   
                   = 
                   
                     
                       
                         β 
                         
                           2 
                            
                           R 
                         
                       
                        
                       
                         { 
                         
                           T 
                           - 
                           
                             
                               R 
                               
                                 β 
                                 + 
                                 1 
                               
                             
                              
                             
                               [ 
                               
                                 
                                   F 
                                   x 
                                 
                                 - 
                                 
                                   F 
                                   y 
                                 
                               
                               ] 
                             
                           
                         
                         } 
                       
                     
                     - 
                     
                       F 
                       rA 
                       0 
                     
                   
                 
               
               
                 
                   ( 
                   47 
                   ) 
                 
               
             
             
               
                 
                   
                     F 
                     tA 
                     0 
                   
                   = 
                   
                     
                       
                         F 
                         x 
                       
                       
                         β 
                         + 
                         1 
                       
                     
                     + 
                     
                       
                         F 
                         rB 
                         0 
                       
                       β 
                     
                   
                 
               
               
                 
                   ( 
                   48 
                   ) 
                 
               
             
             
               
                 
                   
                     F 
                     tB 
                     0 
                   
                   = 
                   
                     
                       
                         F 
                         rA 
                         0 
                       
                       β 
                     
                     - 
                     
                       
                         F 
                         y 
                       
                       
                         β 
                         + 
                         1 
                       
                     
                   
                 
               
               
                 
                   ( 
                   49 
                   ) 
                 
               
             
             
               
                 
                   
                     I 
                     A 
                   
                   = 
                   
                     
                       
                         
                           ( 
                           
                             
                               F 
                               rA 
                               0 
                             
                             
                               1.5 
                                
                               
                                 K 
                                 r 
                               
                             
                           
                           ) 
                         
                         2 
                       
                       + 
                       
                         
                           ( 
                           
                             
                               F 
                               tA 
                               0 
                             
                             
                               1.5 
                                
                               
                                 K 
                                 t 
                               
                             
                           
                           ) 
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   50 
                   ) 
                 
               
             
             
               
                 
                   
                     I 
                     B 
                   
                   = 
                   
                     
                       
                         
                           ( 
                           
                             
                               F 
                               rB 
                               0 
                             
                             
                               1.5 
                                
                               
                                 K 
                                 r 
                               
                             
                           
                           ) 
                         
                         2 
                       
                       + 
                       
                         
                           ( 
                           
                             
                               F 
                               tA 
                               0 
                             
                             
                               1.5 
                                
                               
                                 K 
                                 t 
                               
                             
                           
                           ) 
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   51 
                   ) 
                 
               
             
             
               
                 
                   
                     Δ 
                     A 
                   
                   = 
                   
                     A 
                      
                     
                         
                     
                      
                     
                       TAN 
                        
                       
                         ( 
                         
                           
                             
                               F 
                               rA 
                               0 
                             
                             
                               F 
                               tA 
                               0 
                             
                           
                            
                           
                             
                               K 
                               t 
                             
                             
                               K 
                               r 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   52 
                   ) 
                 
               
             
             
               
                 
                   
                     Δ 
                     B 
                   
                   = 
                   
                     A 
                      
                     
                         
                     
                      
                     
                       TAN 
                        
                       
                         ( 
                         
                           
                             
                               F 
                               rB 
                               0 
                             
                             
                               F 
                               tB 
                               0 
                             
                           
                            
                           
                             
                               K 
                               t 
                             
                             
                               K 
                               r 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   53 
                   ) 
                 
               
             
           
         
       
     
         [0078]    By applying the derived I A , Δ A , I B , Δ B  in a commutation function, for example  130  ( FIG. 1 ) to drive winding subsets  525 ,  530  and winding subsets  535 ,  540 , using for example (17) and (18) above, a common set of motor commutation equations may be used for producing desired torque (T) and active centering forces (F x ) along the x-axis and (F y ) along the y-axis with, in this embodiment, only 2 winding sets producing Lorentz forces. 
         [0079]    Thus, similar to the embodiments above, by adjusting the electrical angles θ A  and θ B  with the electrical angle offsets Δ A  and Δ B , a common or standard set of motor commutation equations may be used for producing desired torque (T) and active centering forces (F x ) along the x-axis and (F y ) along the y-axis utilizing, in this embodiment, only 2 winding sets producing Lorentz forces. Other commutation schemes using any suitable commutation implementation may also be employed. 
         [0080]      FIG. 6  shows an exemplary embodiment utilizing three winding sets A  615 , B  620 , and C  625  where the winding sets extend over 3 sectors of the rotor, designated A, B, and C as shown. It should be understood that any number of winding sets extending over any number of sectors may be included. Each of the winding sets A  615 , B  620 , and C  625  may be driven by a current amplifier  630 . Current amplifier  630  may include software, hardware, or a combination of software and hardware suitable for driving the winding sets. Current amplifier  630  may also include a processor  635 , a commutation function  640 , and a current loop function  645  suitable for driving the desired number of winding sets. Similar to current amplifier  125  in  FIG. 1 , the commutation function  640  may determine current for windings of each winding set according to a set of specified functions, while the current loop function  645  may provide a feedback and driving capability for maintaining the current through the windings as determined. The processor  635 , commutation function  640 , and current loop function  645  may also include circuitry for receiving feedback from one or more sensors or sensor systems that provide position information. 
         [0081]    As has been previously noted, the motor in the exemplary embodiment shown in  FIG. 6  may be configured to operate on the basis of Lorentz forces. Referring again to  FIG. 6 , exemplary relationships for the desired torque (T) and centering forces (Fx) along the x-axis and (Fy) along the y-axis utilizing Lorentz forces may be expressed as shown below. It should be understood that the relationships, calculations and resulting commutation schemes are exemplary and that any suitable commutation solutions may be utilized in the disclosed embodiments. While three winding sets are shown it should be understood that any number of winding sets may be utilized. While winding sets A  615 , B  620 , and C  625  are shown substantially equally distributed about the stator offset by about 120 degrees it should be understood that other offsets may also be utilized. In other alternate embodiments, the windings may be arranged in a configuration that is generally symmetric about a desired axis but unequally distributed around the stator perimeter. 
         [0082]    The force relations for this embodiment of  FIG. 6  utilizing Lorentz forces may be expressed for example as follows: 
         [0000]    
       
         
           
             
               
                 
                   T 
                   = 
                   
                     R 
                      
                     
                       [ 
                       
                         
                           F 
                           tA 
                         
                         + 
                         
                           F 
                           tB 
                         
                         + 
                         
                           F 
                           tC 
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   54 
                   ) 
                 
               
             
             
               
                 
                   
                     F 
                     x 
                   
                   = 
                   
                     
                       F 
                       tA 
                     
                     - 
                     
                       
                         1 
                         2 
                       
                        
                       
                         ( 
                         
                           
                             F 
                             tB 
                           
                           + 
                           
                             F 
                             tC 
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         
                           3 
                         
                         2 
                       
                        
                       
                         ( 
                         
                           
                             F 
                             rB 
                           
                           - 
                           
                             F 
                             rC 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   55 
                   ) 
                 
               
             
             
               
                 
                   
                     F 
                     y 
                   
                   = 
                   
                     
                       F 
                       rA 
                     
                     - 
                     
                       
                         
                           3 
                         
                         2 
                       
                        
                       
                         ( 
                         
                           
                             F 
                             tB 
                           
                           - 
                           
                             F 
                             tC 
                           
                         
                         ) 
                       
                     
                     - 
                     
                       
                         1 
                         2 
                       
                        
                       
                         ( 
                         
                           
                             F 
                             rB 
                           
                           + 
                           
                             F 
                             rC 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   56 
                   ) 
                 
               
             
           
         
       
     
         [0083]    Assume: 
         [0000]      F tA =F tB =F tC =F t    (57) 
         [0000]    for the purposes of equally distributing the tangential forces that produce torque among the three winding sets. It should be understood that the equal distribution is an exemplary solution and that other techniques and force distributions may also be used to determine solutions for I A , Δ A , I B , Δ B . Other force distributions may be utilized, for example, by introducing a cost function as shown in the embodiments above. For example, a cost function may be utilized based on currents through the windings. 
         [0084]    Using (54), (55), (56), and (57): 
         [0000]    
       
         
           
             
               
                 
                   
                     F 
                     t 
                   
                   = 
                   
                     T 
                     
                       3 
                        
                       R 
                     
                   
                 
               
               
                 
                   ( 
                   58 
                   ) 
                 
               
             
             
               
                 
                   
                     F 
                     x 
                   
                   = 
                   
                     
                       
                         3 
                       
                       2 
                     
                      
                     
                       ( 
                       
                         
                           F 
                           rB 
                         
                         - 
                         
                           F 
                           rC 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   56 
                   ) 
                 
               
             
             
               
                 
                   
                     F 
                     y 
                   
                   = 
                   
                     
                       F 
                       rA 
                     
                     - 
                     
                       
                         1 
                         2 
                       
                        
                       
                         ( 
                         
                           
                             F 
                             rB 
                           
                           + 
                           
                             F 
                             rC 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   60 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Similar to the embodiment of  FIG. 1 , for this embodiment of  FIG. 6  there are 2 equations and 3 unknowns. Therefore, one more equation is needed. The additional equation may be obtained by minimizing the following exemplary cost function. As mentioned previously, other cost functions and force distributions may be utilized. 
         [0000]        J=F   rA   2   +F   rB   2   +F   rC   2    (61) 
         [0000]    This gives the optimized solution as: 
         [0000]    
       
         
           
             
               
                 
                   
                     F 
                     rB 
                   
                   = 
                   
                     
                       1 
                       3 
                     
                      
                     
                       ( 
                       
                         
                           
                             3 
                           
                            
                           
                             F 
                             x 
                           
                         
                         - 
                         
                           F 
                           y 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   62 
                   ) 
                 
               
             
             
               
                 
                   
                     F 
                     rA 
                   
                   = 
                   
                     
                       2 
                       3 
                     
                      
                     
                       F 
                       y 
                     
                   
                 
               
               
                 
                   ( 
                   63 
                   ) 
                 
               
             
             
               
                 
                   
                     F 
                     rC 
                   
                   = 
                   
                     
                       - 
                       
                         1 
                         3 
                       
                     
                      
                     
                       ( 
                       
                         
                           
                             3 
                           
                            
                           
                             F 
                             x 
                           
                         
                         + 
                         
                           F 
                           y 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   64 
                   ) 
                 
               
             
           
         
       
     
         [0085]    The following force/current relations may be utilized for the embodiment utilizing Lorentz forces of  FIG. 6 : 
         [0000]      F tj =1.5I j K t  cos Δ j , j=A, B, C   (65) 
         [0000]      F rj =1.5I j K r  sin Δ j , j=A, B, C   (66) 
         [0086]    As a result: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       I 
                       j 
                     
                     = 
                     
                       
                         
                           
                             ( 
                             
                               
                                 F 
                                 rj 
                               
                               
                                 1.5 
                                  
                                 
                                   K 
                                   r 
                                 
                               
                             
                             ) 
                           
                           2 
                         
                         + 
                         
                           
                             ( 
                             
                               
                                 F 
                                 t 
                               
                               
                                 1.5 
                                  
                                 
                                   K 
                                   t 
                                 
                               
                             
                             ) 
                           
                           2 
                         
                       
                     
                   
                   , 
                   
                     j 
                     = 
                     A 
                   
                   , 
                   B 
                   , 
                   C 
                 
               
               
                 
                   ( 
                   67 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       Δ 
                       j 
                     
                     = 
                     
                       a 
                        
                       
                           
                       
                        
                       
                         tan 
                          
                         
                           [ 
                           
                             
                               
                                 F 
                                 rj 
                               
                               
                                 F 
                                 t 
                               
                             
                              
                             
                               
                                 K 
                                 t 
                               
                               
                                 K 
                                 r 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                   , 
                   
                     j 
                     = 
                     A 
                   
                   , 
                   B 
                   , 
                   C 
                 
               
               
                 
                   ( 
                   68 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where j represents each winding set and: 
         [0000]    
       
         
           
             
               
                 
                   
                     F 
                     t 
                   
                   = 
                   
                     T 
                     
                       3 
                        
                       R 
                     
                   
                 
               
               
                 
                   ( 
                   69 
                   ) 
                 
               
             
             
               
                 
                   
                     F 
                     rA 
                   
                   = 
                   
                     
                       2 
                       3 
                     
                      
                     
                       F 
                       y 
                     
                   
                 
               
               
                 
                   ( 
                   70 
                   ) 
                 
               
             
             
               
                 
                   
                     F 
                     rB 
                   
                   = 
                   
                     
                       1 
                       3 
                     
                      
                     
                       ( 
                       
                         
                           
                             3 
                           
                            
                           
                             F 
                             x 
                           
                         
                         - 
                         
                           F 
                           y 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   71 
                   ) 
                 
               
             
             
               
                 
                   
                     F 
                     rC 
                   
                   = 
                   
                     
                       - 
                       
                         1 
                         3 
                       
                     
                      
                     
                       ( 
                       
                         
                           
                             3 
                           
                            
                           
                             F 
                             x 
                           
                         
                         + 
                         
                           F 
                           y 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   72 
                   ) 
                 
               
             
           
         
       
     
         [0087]    Similar to the embodiments above, I j  and Δ j  where j=A, B, C may be applied in commutation function  640 , using for example commutation equations in the form of (17) as applicable for winding sets A, B and C, for providing the desired torque T and centering forces Fx, Fy using the three winding sets  615 ,  620 ,  625  producing Lorentz forces. 
         [0088]    As with the embodiments above, the electrical angle may be adjusted using the electrical angle offset Δ j  where j=A, B, C in order to provide the desired torque and centering forces using a common set of motor commutation equations. Other commutation schemes using any suitable commutation implementation may also be employed. 
         [0089]    Still referring to  FIG. 6 , the motor in another embodiment may be configured to operate with a combination of Lorentz and Maxwell forces. Exemplary relationships for the desired torque (T) and centering forces (F x ) along the x-axis and (F y ) along the y-axis as a function of the radial and tangential forces (Frj and Ftj, j=A, B, C) may be similar to the embodiment of  FIG. 3  utilizing a combination of Lorentz and Maxwell forces disclosed above. 
         [0090]    [In this embodiment, a solution may be obtained based on the location of the desired centering force vector (F x , F y ) relative to the three sectors A, B, and C defined for the motor of  FIG. 6 . For example, each sector may be 120 degrees, and three solutions may be found, one for each sector. 
         [0091]    The following force/current relations may be utilized for the embodiment of  FIG. 6  utilizing Lorentz and Maxwell forces. 
         [0000]      F tj =1.5I j K t  cos Δ j , j=A, B, C   (73) 
         [0000]        F   rj =1.5 I   j   K   r  sin Δ j +1.5 K   M   I   j   2   , j=A, B, C    (74) 
         [0092]    An example solution for the embodiment of  FIG. 6  utilizing Lorentz and Maxwell forces is presented in Table 2 below, where 
         [0000]    
       
         
           
             
               
                 a 
                 j 
               
               = 
               
                 
                   ( 
                   
                     
                       K 
                       M 
                     
                     
                       K 
                       r 
                     
                   
                   ) 
                 
                 2 
               
             
             , 
             
               
                 b 
                 j 
               
               = 
               
                 - 
                 
                   
                     ( 
                     
                       
                         
                           4 
                           3 
                         
                          
                         
                           F 
                           rj 
                         
                          
                         
                           
                             K 
                             M 
                           
                           
                             K 
                             r 
                             2 
                           
                         
                       
                       + 
                       1 
                     
                     ) 
                   
                   2 
                 
               
             
           
         
       
     
         [0000]    and 
         [0000]    
       
         
           
             
               
                 c 
                 j 
               
               = 
               
                 
                   
                     ( 
                     
                       
                         F 
                         tj 
                       
                       
                         1.5 
                          
                         
                           K 
                           t 
                         
                       
                     
                     ) 
                   
                   2 
                 
                 + 
                 
                   
                     ( 
                     
                       
                         F 
                         rj 
                       
                       
                         1.5 
                          
                         
                           K 
                           r 
                         
                       
                     
                     ) 
                   
                   2 
                 
               
             
             , 
             
               j 
               = 
               A 
             
             , 
             B 
           
         
       
     
         [0000]    
       
         
               
               
             
               
               
               
               
             
           
               
                   
                 TABLE 2 
               
             
             
               
                   
                   
               
               
                   
                 SECTORS 
               
             
          
           
               
                 PARAMETERS 
                 AB 
                 BC 
                 CA 
               
               
                   
               
               
                 I A   
                 
                   
                     
                       
                         
                           
                             
                               - 
                               
                                 b 
                                 A 
                               
                             
                             + 
                             
                               
                                 
                                   b 
                                   A 
                                   2 
                                 
                                 - 
                                 
                                   4 
                                    
                                   
                                     a 
                                     A 
                                   
                                    
                                   
                                     c 
                                     A 
                                   
                                 
                               
                             
                           
                           
                             2 
                              
                             
                               a 
                               A 
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         T 
                         
                           4.5 
                            
                           
                             K 
                             t 
                           
                            
                           R 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             
                               - 
                               
                                 b 
                                 B 
                               
                             
                             + 
                             
                               
                                 
                                   b 
                                   B 
                                   2 
                                 
                                 - 
                                 
                                   4 
                                    
                                   
                                     a 
                                     B 
                                   
                                    
                                   
                                     c 
                                     B 
                                   
                                 
                               
                             
                           
                           
                             2 
                              
                             
                               a 
                               B 
                             
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 I B   
                 
                   
                     
                       
                         
                           
                             
                               - 
                               
                                 b 
                                 B 
                               
                             
                             + 
                             
                               
                                 
                                   b 
                                   B 
                                   2 
                                 
                                 - 
                                 
                                   4 
                                    
                                   
                                     a 
                                     B 
                                   
                                    
                                   
                                     c 
                                     B 
                                   
                                 
                               
                             
                           
                           
                             2 
                              
                             
                               a 
                               B 
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             
                               - 
                               
                                 b 
                                 A 
                               
                             
                             + 
                             
                               
                                 
                                   b 
                                   A 
                                   2 
                                 
                                 - 
                                 
                                   4 
                                    
                                   
                                     a 
                                     A 
                                   
                                    
                                   
                                     c 
                                     A 
                                   
                                 
                               
                             
                           
                           
                             2 
                              
                             
                               a 
                               A 
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         T 
                         
                           4.5 
                            
                           
                             K 
                             t 
                           
                            
                           R 
                         
                       
                     
                   
                 
               
               
                   
               
               
                 I C   
                 
                   
                     
                       
                         T 
                         
                           4.5 
                            
                           
                             K 
                             t 
                           
                            
                           R 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             
                               - 
                               
                                 b 
                                 B 
                               
                             
                             + 
                             
                               
                                 
                                   b 
                                   B 
                                   2 
                                 
                                 - 
                                 
                                   4 
                                    
                                   
                                     a 
                                     B 
                                   
                                    
                                   
                                     c 
                                     B 
                                   
                                 
                               
                             
                           
                           
                             2 
                              
                             
                               a 
                               B 
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             
                               - 
                               
                                 b 
                                 A 
                               
                             
                             + 
                             
                               
                                 
                                   b 
                                   A 
                                   2 
                                 
                                 - 
                                 
                                   4 
                                    
                                   
                                     a 
                                     A 
                                   
                                    
                                   
                                     c 
                                     A 
                                   
                                 
                               
                             
                           
                           
                             2 
                              
                             
                               a 
                               A 
                             
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 Δ A   
                 atan(└K t └F rA  − 1.5K M I A   2 ┘┘/[K r F t ]) 
                 0 
                 atan(K t └F rB  − 1.5K M I B   2 ┘/[K r F t ]) 
               
               
                 Δ B   
                 atan(K t └F rB  − 1.5K M I B   2 ┘/[K r F t ]) 
                 atan(└K t └F rA  − 1.5K M I A   2 ┘┘/[K r F t ]) 
                 0 
               
               
                 Δ C   
                 0 
                 atan(K t └F rB  − 1.5K M I B   2 ┘/[K r F tB ]) 
                 atan(└K t └F rA  − 1.5K M I A   2 ┘┘/[K r F tA ]) 
               
               
                   
               
               
                 F rA   
                 
                   
                     
                       
                         
                           F 
                           y 
                         
                         + 
                         
                           
                             1 
                             3 
                           
                            
                           
                             [ 
                             
                               
                                 
                                   3 
                                 
                                  
                                 
                                   F 
                                   x 
                                 
                               
                               + 
                               
                                 
                                   
                                     2 
                                      
                                     
                                       K 
                                       M 
                                     
                                   
                                   9 
                                 
                                  
                                 
                                   
                                     ( 
                                     
                                       T 
                                       
                                         
                                           K 
                                           t 
                                         
                                          
                                         R 
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           2 
                           27 
                         
                          
                         
                           
                             
                               K 
                               M 
                             
                              
                             
                               ( 
                               
                                 T 
                                 
                                   
                                     K 
                                     t 
                                   
                                    
                                   R 
                                 
                               
                               ) 
                             
                           
                           2 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           2 
                           3 
                         
                          
                         
                           [ 
                           
                             
                               
                                 3 
                               
                                
                               
                                 F 
                                 x 
                               
                             
                             + 
                             
                               
                                 
                                   K 
                                   M 
                                 
                                 9 
                               
                                
                               
                                 
                                   ( 
                                   
                                     T 
                                     
                                       
                                         K 
                                         t 
                                       
                                        
                                       R 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                   
               
               
                 F rB   
                 
                   
                     
                       
                         
                           2 
                           3 
                         
                          
                         
                           [ 
                           
                             
                               
                                 3 
                               
                                
                               
                                 F 
                                 x 
                               
                             
                             + 
                             
                               
                                 
                                   K 
                                   M 
                                 
                                 9 
                               
                                
                               
                                 
                                   ( 
                                   
                                     T 
                                     
                                       
                                         K 
                                         t 
                                       
                                        
                                       R 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           F 
                           y 
                         
                         + 
                         
                           
                             1 
                             3 
                           
                            
                           
                             [ 
                             
                               
                                 
                                   3 
                                 
                                  
                                 
                                   F 
                                   x 
                                 
                               
                               + 
                               
                                 
                                   
                                     2 
                                      
                                     
                                       K 
                                       M 
                                     
                                   
                                   9 
                                 
                                  
                                 
                                   
                                     ( 
                                     
                                       T 
                                       
                                         
                                           K 
                                           t 
                                         
                                          
                                         R 
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           2 
                           27 
                         
                          
                         
                           
                             
                               K 
                               M 
                             
                              
                             
                               ( 
                               
                                 T 
                                 
                                   
                                     K 
                                     t 
                                   
                                    
                                   R 
                                 
                               
                               ) 
                             
                           
                           2 
                         
                       
                     
                   
                 
               
               
                   
               
               
                 F rC   
                 
                   
                     
                       
                         
                           2 
                           27 
                         
                          
                         
                           
                             
                               K 
                               M 
                             
                              
                             
                               ( 
                               
                                 T 
                                 
                                   
                                     K 
                                     t 
                                   
                                    
                                   R 
                                 
                               
                               ) 
                             
                           
                           2 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           2 
                           3 
                         
                          
                         
                           [ 
                           
                             
                               
                                 3 
                               
                                
                               
                                 F 
                                 x 
                               
                             
                             + 
                             
                               
                                 
                                   K 
                                   M 
                                 
                                 9 
                               
                                
                               
                                 
                                   ( 
                                   
                                     T 
                                     
                                       
                                         K 
                                         t 
                                       
                                        
                                       R 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           F 
                           y 
                         
                         + 
                         
                           
                             1 
                             3 
                           
                            
                           
                             [ 
                             
                               
                                 
                                   3 
                                 
                                  
                                 
                                   F 
                                   x 
                                 
                               
                               + 
                               
                                 
                                   
                                     2 
                                      
                                     
                                       K 
                                       M 
                                     
                                   
                                   9 
                                 
                                  
                                 
                                   
                                     ( 
                                     
                                       T 
                                       
                                         
                                           K 
                                           t 
                                         
                                          
                                         R 
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 F t   
                 
                   
                     
                       
                         T 
                         
                           3 
                            
                           R 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         T 
                         
                           3 
                            
                           R 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         T 
                         
                           3 
                            
                           R 
                         
                       
                     
                   
                 
               
               
                   
               
             
          
         
       
     
         [0093]    Commutation functions for this embodiment may be derived in a manner similar to the embodiments described above. For example, as with the embodiments above, an electrical angle may be adjusted using an electrical angle offset Δ j , j=A, B, C in order to provide the desired forces using a common set of motor commutation equations, for example, commutation equations in the form of (17) as applicable for winding sets A, B and C. Other commutation schemes using any suitable commutation implementation may also be employed. 
         [0094]      FIG. 7  shows an embodiment utilizing four winding sets A  715 , B  720 , C  725 , and D  730  where the winding sets extend over 4 sectors of the rotor, designated A, B, C, and D as shown. Each of the winding sets A  715 , B  720 , C  725 , and D  730  may be driven by a current amplifier  735  capable of driving four winding sets (though shown as evenly distributed around the stator perimeter, in alternate embodiments the winding sets may be arranged in any other desired manner). Current amplifier  735  may include software, hardware, or a combination of software and hardware suitable for driving the winding sets. Current amplifier  735  may also include a processor  740 , commutation function  745  and a current loop function  750  suitable for driving four winding sets. Similar to current amplifier  125 , the commutation function  745  may determine current for windings of each winding set according to a set of specified functions, while the current loop function  750  may provide a feedback and driving capability for maintaining the current through the windings as determined. The processor  740 , commutation function  745 , and current loop function  750  may also include circuitry for receiving feedback from one or more sensors or sensor systems that provide position information. 
         [0095]    In the exemplary embodiment shown in  FIG. 7 , the motor may be configured to operate on the basis of Lorentz forces. Exemplary relationships for the desired torque (T) and centering forces (F x ) along the x-axis and (F y ) along the y-axis utilizing Lorentz forces as a function of radial and tangential forces (Frj and Ftj, j=A, B, C, D) in the embodiment of  FIG. 7  are as shown below. Any other relationships, calculations and resulting commutation schemes may also be used. 
         [0000]        T=R ( F   tA   +F   tB   +F   tC   +F   tD )   (75) 
         [0000]        F   x   =F   tA   +F   rB   −F   tC   −F   rD    (76) 
         [0000]        F   y   =F   rA   +F   tB   −F   rC   +F   tD    (77) 
         [0096]    Assume: 
         [0000]      F tA =F tB =F tC =F tD =F t    (78) 
         [0000]    for the purposes of equally distributing the tangential forces that produce torque among the four winding sets. It should be understood that the equal distribution is an exemplary solution and that other techniques and force distributions may also be used to determine solutions for I j  and Δ j , j=A, B, C, D. Other force distributions may be utilized, for example, by introducing a cost function as shown in the embodiments above or another exemplary cost function. Another exemplary cost function may be based on currents through the windings. 
         [0097]    Therefore: 
         [0000]    
       
         
           
             
               
                 
                   
                     F 
                     t 
                   
                   = 
                   
                     T 
                     
                       4 
                        
                       R 
                     
                   
                 
               
               
                 
                   ( 
                   79 
                   ) 
                 
               
             
             
               
                 
                   
                     F 
                     x 
                   
                   = 
                   
                     
                       F 
                       rB 
                     
                     - 
                     
                       F 
                       rD 
                     
                   
                 
               
               
                 
                   ( 
                   80 
                   ) 
                 
               
             
             
               
                 
                   
                     F 
                     y 
                   
                   = 
                   
                     
                       F 
                       rA 
                     
                     - 
                     
                       F 
                       rC 
                     
                   
                 
               
               
                 
                   ( 
                   81 
                   ) 
                 
               
             
           
         
       
     
         [0098]    Assume: 
         [0000]    
       
         
           
             
               
                 
                   
                     F 
                     rA 
                   
                   = 
                   
                     
                       F 
                       y 
                     
                     2 
                   
                 
               
               
                 
                   ( 
                   82 
                   ) 
                 
               
             
             
               
                 
                   
                     F 
                     rC 
                   
                   = 
                   
                     - 
                     
                       
                         F 
                         y 
                       
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   83 
                   ) 
                 
               
             
             
               
                 
                   
                     F 
                     rB 
                   
                   = 
                   
                     
                       F 
                       x 
                     
                     2 
                   
                 
               
               
                 
                   ( 
                   84 
                   ) 
                 
               
             
             
               
                 
                   
                     F 
                     rD 
                   
                   = 
                   
                     - 
                     
                       
                         F 
                         x 
                       
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   85 
                   ) 
                 
               
             
           
         
       
     
         [0000]    for the purposes of equally distributing the centering forces between the windings on opposite sides. As mentioned above, it should be understood that the equal distribution is an exemplary solution and that other techniques and force distributions may also be used. 
         [0099]    The following force/current relations may be utilized for this embodiment of  FIG. 7  utilizing Lorentz forces. 
         [0000]      F tj =1.5I j K t  cos Δ j , j=A, B, C, D   (86) 
         [0000]      F rj =1.5I j K r  sin Δ j , j=A, B, C, D   (87) 
         [0100]    As a result: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       I 
                       j 
                     
                     = 
                     
                       
                         
                           
                             ( 
                             
                               
                                 F 
                                 rj 
                               
                               
                                 1.5 
                                  
                                 
                                   K 
                                   r 
                                 
                               
                             
                             ) 
                           
                           2 
                         
                         + 
                         
                           
                             ( 
                             
                               
                                 F 
                                 t 
                               
                               
                                 1.5 
                                  
                                 
                                   K 
                                   t 
                                 
                               
                             
                             ) 
                           
                           2 
                         
                       
                     
                   
                   , 
                   
                     j 
                     = 
                     A 
                   
                   , 
                   B 
                   , 
                   C 
                   , 
                   D 
                 
               
               
                 
                   ( 
                   88 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       Δ 
                       j 
                     
                     = 
                     
                       a 
                        
                       
                           
                       
                        
                       
                         tan 
                          
                         
                           [ 
                           
                             
                               
                                 F 
                                 rj 
                               
                               
                                 F 
                                 t 
                               
                             
                              
                             
                               
                                 K 
                                 t 
                               
                               
                                 K 
                                 r 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                   , 
                   
                     j 
                     = 
                     A 
                   
                   , 
                   B 
                   , 
                   C 
                   , 
                   D 
                 
               
               
                 
                   ( 
                   89 
                   ) 
                 
               
             
           
         
       
     
         [0000]    for each winding set j, j=A  715 , B  720 , C  725 , and D  730 , where 
         [0000]    
       
         
           
             
               
                 F 
                 r 
               
               = 
               
                 T 
                 
                   4 
                    
                   R 
                 
               
             
             , 
             
               
 
             
              
             
               
                 F 
                 rA 
               
               = 
               
                 
                   F 
                   y 
                 
                 2 
               
             
             , 
             
               
 
             
              
             
               
                 F 
                 rB 
               
               = 
               
                 
                   F 
                   x 
                 
                 2 
               
             
             , 
             
               
 
             
              
             
               
                 F 
                 rC 
               
               = 
               
                 
                   
                     - 
                     
                       
                         F 
                         y 
                       
                       2 
                     
                   
                    
                   
                       
                   
                    
                   and 
                    
                   
                       
                   
                    
                   
                     F 
                     rD 
                   
                 
                 = 
                 
                   - 
                   
                     
                       
                         F 
                         x 
                       
                       2 
                     
                     . 
                   
                 
               
             
           
         
       
     
         [0101]    I j  and Δ j , j=A, B, C, D may be applied in commutation function  745  for providing the desired torque T and centering forces Fx, Fy using the 4 winding sets  715 ,  720 ,  725 , and  430  producing Lorentz forces. 
         [0102]    As with the embodiments above, the electrical angle may be adjusted using the electrical angle offset Δ j , j=A, B, C, D in order to provide the desired torque and centering forces using a common set of motor commutation equations. Thus, the segmented windings A, B, C, D may be driven to produce decoupled torque and centering forces simultaneously to drive and center a rotor of the motor using the adjusted electrical angle in the common set of commutation equations. In addition, any other suitable commutation operations may also be utilized. 
         [0103]    Again referring to  FIG. 7 , the motor in another embodiment may utilize a combination of Lorentz and Maxwell forces. The relationships for the desired torque (T) and centering forces (F x ) along the x-axis and (F y ) along the y-axis as a function of the radial and tangential forces (Frj and Ftj, j=A, B, C, D) are identical to those of Equations (75) to (81). Similarly to above embodiments utilizing Lorentz and Maxwell forces, a solution may be obtained based on the location of the desired centering force vector (Fx, Fy) relative to four sectors defined for the embodiment of  FIG. 7 . In this embodiment, each sector may be 90 degrees, and four sets of solutions may be found, one for each sector. 
         [0104]    The following force/current relations may be utilized for the embodiment of  FIG. 7  utilizing Lorentz and Maxwell forces. 
         [0000]        F   rj =1.5 I   j   K   r  sin Δ j +1.5 K   M   I   j   2   , j=A, B, C    (90) 
         [0105]    An example solution for the embodiment of  FIG. 7  utilizing Lorentz and Maxwell forces is presented in Table 3 below, where, 
         [0000]    
       
         
           
             
               
                 a 
                 j 
               
               = 
               
                 
                   ( 
                   
                     
                       K 
                       M 
                     
                     
                       K 
                       r 
                     
                   
                   ) 
                 
                 2 
               
             
             , 
             
               
 
             
              
             
               
                 b 
                 i 
               
               = 
               
                 
                   - 
                   
                     
                       ( 
                       
                         
                           
                             4 
                             3 
                           
                            
                           
                             F 
                             rj 
                           
                            
                           
                             
                               K 
                               M 
                             
                             
                               K 
                               r 
                               2 
                             
                           
                         
                         + 
                         1 
                       
                       ) 
                     
                     2 
                   
                 
                  
                 
                     
                 
                  
                 and 
               
             
           
         
       
       
         
           
             
               
                 c 
                 j 
               
               = 
               
                 
                   
                     ( 
                     
                       
                         F 
                         tj 
                       
                       
                         1.5 
                          
                         
                           K 
                           t 
                         
                       
                     
                     ) 
                   
                   2 
                 
                 + 
                 
                   
                     ( 
                     
                       
                         F 
                         rj 
                       
                       
                         1.5 
                          
                         
                           K 
                           r 
                         
                       
                     
                     ) 
                   
                   2 
                 
               
             
             , 
             
               j 
               = 
               A 
             
             , 
             B 
           
         
       
     
       in Table 3. 
       [0106]      
         [0000]    
       
         
               
               
             
               
               
               
               
               
             
           
               
                 TABLE 3 
               
             
             
               
                   
               
               
                 PA- 
                 SECTORS 
               
             
          
           
               
                 RAMETERS 
                 AB 
                 BC 
                 CD 
                 DA 
               
               
                   
               
               
                 I A   
                 
                   
                     
                       
                         
                           
                             
                               - 
                               
                                 b 
                                 A 
                               
                             
                             + 
                             
                               
                                 
                                   b 
                                   A 
                                   2 
                                 
                                 - 
                                 
                                   4 
                                    
                                   
                                     a 
                                     A 
                                   
                                    
                                   
                                     c 
                                     A 
                                   
                                 
                               
                             
                           
                           
                             2 
                              
                             
                               a 
                               A 
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         T 
                         
                           6 
                            
                           
                             RK 
                             t 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         T 
                         
                           6 
                            
                           
                             RK 
                             t 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             
                               - 
                               
                                 b 
                                 B 
                               
                             
                             + 
                             
                               
                                 
                                   b 
                                   B 
                                   2 
                                 
                                 - 
                                 
                                   4 
                                    
                                   
                                     a 
                                     B 
                                   
                                    
                                   
                                     c 
                                     B 
                                   
                                 
                               
                             
                           
                           
                             2 
                              
                             
                               a 
                               B 
                             
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 I B   
                 
                   
                     
                       
                         
                           
                             
                               - 
                               
                                 b 
                                 B 
                               
                             
                             + 
                             
                               
                                 
                                   b 
                                   B 
                                   2 
                                 
                                 - 
                                 
                                   4 
                                    
                                   
                                     a 
                                     B 
                                   
                                    
                                   
                                     c 
                                     B 
                                   
                                 
                               
                             
                           
                           
                             2 
                              
                             
                               a 
                               B 
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             
                               - 
                               
                                 b 
                                 A 
                               
                             
                             + 
                             
                               
                                 
                                   b 
                                   A 
                                   2 
                                 
                                 - 
                                 
                                   4 
                                    
                                   
                                     a 
                                     A 
                                   
                                    
                                   
                                     c 
                                     A 
                                   
                                 
                               
                             
                           
                           
                             2 
                              
                             
                               a 
                               A 
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         T 
                         
                           6 
                            
                           
                             RK 
                             t 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         T 
                         
                           6 
                            
                           
                             RK 
                             t 
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 I C   
                 
                   
                     
                       
                         T 
                         
                           6 
                            
                           
                             RK 
                             t 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             
                               - 
                               
                                 b 
                                 B 
                               
                             
                             + 
                             
                               
                                 
                                   b 
                                   B 
                                   2 
                                 
                                 - 
                                 
                                   4 
                                    
                                   
                                     a 
                                     B 
                                   
                                    
                                   
                                     c 
                                     B 
                                   
                                 
                               
                             
                           
                           
                             2 
                              
                             
                               a 
                               B 
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             
                               - 
                               
                                 b 
                                 A 
                               
                             
                             + 
                             
                               
                                 
                                   b 
                                   A 
                                   2 
                                 
                                 - 
                                 
                                   4 
                                    
                                   
                                     a 
                                     A 
                                   
                                    
                                   
                                     c 
                                     A 
                                   
                                 
                               
                             
                           
                           
                             2 
                              
                             
                               a 
                               A 
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         T 
                         
                           6 
                            
                           
                             RK 
                             t 
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 I D   
                 
                   
                     
                       
                         T 
                         
                           6 
                            
                           
                             RK 
                             t 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         T 
                         
                           6 
                            
                           
                             RK 
                             t 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             
                               - 
                               
                                 b 
                                 B 
                               
                             
                             + 
                             
                               
                                 
                                   b 
                                   B 
                                   2 
                                 
                                 - 
                                 
                                   4 
                                    
                                   
                                     a 
                                     B 
                                   
                                    
                                   
                                     c 
                                     B 
                                   
                                 
                               
                             
                           
                           
                             2 
                              
                             
                               a 
                               B 
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             
                               - 
                               
                                 b 
                                 A 
                               
                             
                             + 
                             
                               
                                 
                                   b 
                                   A 
                                   2 
                                 
                                 - 
                                 
                                   4 
                                    
                                   
                                     a 
                                     A 
                                   
                                    
                                   
                                     c 
                                     A 
                                   
                                 
                               
                             
                           
                           
                             2 
                              
                             
                               a 
                               A 
                             
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 Δ A   
                 atan(K t └F rA  − 1.5K M I A   2 ┘/[K r F t ]) 
                 0 
                 0 
                 atan(K t └F rB  − 1.5K M I B   2 ┘/[K r F t ]) 
               
               
                 Δ B   
                 atan(K t └F rB  − 1.5K M I B   2 ┘/[K r F t ]) 
                 atan(K t └F rA  − 1.5K M I A   2 ┘/[K r F t ]) 
                 0 
                 0 
               
               
                 Δ C   
                 0 
                 atan(K t └F rB  − 1.5K M I B   2 ┘/[K r F t ]) 
                 atan(K t └F rA  − 1.5K M I A   2 ┘/[K r F t ]) 
                 0 
               
               
                 Δ D   
                 0 
                 0 
                 atan(K t └F rB  − 1.5K M I B   2 ┘/[K r F t ]) 
                 atan(K t └F rA  − 1.5K M I A   2 ┘/[K r F t ]) 
               
               
                   
               
               
                 F rA   
                 
                   
                     
                       
                         
                           F 
                           y 
                         
                         + 
                         
                           
                             
                               K 
                               M 
                             
                             24 
                           
                            
                           
                             
                               ( 
                               
                                 T 
                                 
                                   RK 
                                   t 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             K 
                             M 
                           
                           24 
                         
                          
                         
                           
                             ( 
                             
                               T 
                               
                                 RK 
                                 t 
                               
                             
                             ) 
                           
                           2 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             K 
                             M 
                           
                           24 
                         
                          
                         
                           
                             ( 
                             
                               T 
                               
                                 RK 
                                 t 
                               
                             
                             ) 
                           
                           2 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           F 
                           x 
                         
                         + 
                         
                           
                             
                               K 
                               M 
                             
                             24 
                           
                            
                           
                             
                               ( 
                               
                                 T 
                                 
                                   RK 
                                   t 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 F rB   
                 
                   
                     
                       
                         
                           F 
                           x 
                         
                         + 
                         
                           
                             
                               K 
                               M 
                             
                             24 
                           
                            
                           
                             
                               ( 
                               
                                 T 
                                 
                                   RK 
                                   t 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           F 
                           y 
                         
                         + 
                         
                           
                             
                               K 
                               M 
                             
                             24 
                           
                            
                           
                             
                               ( 
                               
                                 T 
                                 
                                   RK 
                                   t 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             K 
                             M 
                           
                           24 
                         
                          
                         
                           
                             ( 
                             
                               T 
                               
                                 RK 
                                 t 
                               
                             
                             ) 
                           
                           2 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             K 
                             M 
                           
                           24 
                         
                          
                         
                           
                             ( 
                             
                               T 
                               
                                 RK 
                                 t 
                               
                             
                             ) 
                           
                           2 
                         
                       
                     
                   
                 
               
               
                   
               
               
                 F rC   
                 
                   
                     
                       
                         
                           
                             K 
                             M 
                           
                           24 
                         
                          
                         
                           
                             ( 
                             
                               T 
                               
                                 RK 
                                 t 
                               
                             
                             ) 
                           
                           2 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           F 
                           x 
                         
                         + 
                         
                           
                             
                               K 
                               M 
                             
                             24 
                           
                            
                           
                             
                               ( 
                               
                                 T 
                                 
                                   RK 
                                   t 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           F 
                           y 
                         
                         + 
                         
                           
                             
                               K 
                               M 
                             
                             24 
                           
                            
                           
                             
                               ( 
                               
                                 T 
                                 
                                   RK 
                                   t 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             K 
                             M 
                           
                           24 
                         
                          
                         
                           
                             ( 
                             
                               T 
                               
                                 RK 
                                 t 
                               
                             
                             ) 
                           
                           2 
                         
                       
                     
                   
                 
               
               
                   
               
               
                 F rD   
                 
                   
                     
                       
                         
                           
                             K 
                             M 
                           
                           24 
                         
                          
                         
                           
                             ( 
                             
                               T 
                               
                                 RK 
                                 t 
                               
                             
                             ) 
                           
                           2 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             K 
                             M 
                           
                           24 
                         
                          
                         
                           
                             ( 
                             
                               T 
                               
                                 RK 
                                 t 
                               
                             
                             ) 
                           
                           2 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           F 
                           x 
                         
                         + 
                         
                           
                             
                               K 
                               M 
                             
                             24 
                           
                            
                           
                             
                               ( 
                               
                                 T 
                                 
                                   RK 
                                   t 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           F 
                           y 
                         
                         + 
                         
                           
                             
                               K 
                               M 
                             
                             24 
                           
                            
                           
                             
                               ( 
                               
                                 T 
                                 
                                   RK 
                                   t 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 F t   
                 
                   
                     
                       
                         T 
                         
                           4 
                            
                           R 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         T 
                         
                           4 
                            
                           R 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         T 
                         
                           4 
                            
                           R 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         T 
                         
                           4 
                            
                           R 
                         
                       
                     
                   
                 
               
               
                   
               
             
          
         
       
     
         [0107]    Commutation functions for this embodiment may be derived in a manner similar to the embodiments described above. For example, as with the embodiments above, an electrical angle may be adjusted using an electrical angle offset Δ j , j=A, B, C, D in order to provide the desired forces using a common set of motor commutation equations, such as commutation equations in the form of (17) as applicable for winding sets A, B, C and D. Other commutation schemes using any suitable commutation implementation may also be employed. 
         [0108]    As with the embodiments above, an electrical angle may be adjusted using the electrical angle offset Δ j , j=A, B, C, D in order to provide the desired forces using a common set of motor commutation equations. Thus, the winding sets A, B, C, D may be driven to produce decoupled torque and centering forces simultaneously to torque and actively center a rotor of the motor using the adjusted electrical angle in the common set of commutation equations. 
         [0109]      FIG. 8  shows an exemplary embodiment of a sensor system suitable for providing position feedback for the disclosed embodiments. The sensor system may include a plurality of sensors arranged in pairs. Each sensor pair may be positioned some number of mechanical or electrical degrees apart from another pair. Counterpart sensors in the first pair may be positioned a certain number of mechanical or electrical degrees apart from counterpart sensors in additional pairs. 
         [0110]    In the exemplary embodiment of  FIG. 8 , two pairs of Hall Effect sensors  810 ,  815  may provide feedback for determining rotor positions for the embodiments above. First pair of Hall Effect sensors  810  may include sensors Hall A 1  and Hall A 2 , while second pair of Hall Effect sensors  815  may include sensors Hall B 1  and Hall B 2 . In this embodiment, the first sensor pair  810  is positioned 90 mechanical degrees offset from second sensor pair  815 . Each sensor may also be positioned 90 electrical degrees offset from its pair counterpart. For example, sensor Hall A 1  may have a 90 degree electrical offset from sensor Hall B 1  and sensor Hall A 2  may have a 90 electrical degree offset from sensor Hall B 2 . Each sensor, Hall A 1 , Hall A 2 , Hall B 1 , and Hall B 2  may be connected to the appropriate current amplifier  125 ,  225 ,  630 ,  735  for providing position feedback. 
         [0111]    In the exemplary embodiment of  FIG. 8  sensors Hall A 1  and Hall A 2  are also positioned 90 electrical degrees apart from one another, producing phase-shifted position-dependent sinusoidal signals as the rotor rotates. The electrical position of rotor  110  may be computed as: 
         [0000]      θ= a  tan( A   1   /A   2 )   (91) 
         [0000]    where A 1  and A 2  are the signals from sensors Hall A 1  and Hall A 2 , respectively. 
         [0112]    The rotor displacement along the X and Y direction may be computed using both sets of sensors: 
         [0000]    
       
         
           
             
               
                 
                   
                     gap 
                     x 
                   
                   = 
                   
                     
                       K 
                       gX 
                     
                     
                       
                         
                           B 
                           1 
                           2 
                         
                         + 
                         
                           B 
                           2 
                           2 
                         
                       
                       4 
                     
                   
                 
               
               
                 
                   ( 
                   92 
                   ) 
                 
               
             
             
               
                 
                   
                     gap 
                     Y 
                   
                   = 
                   
                     
                       K 
                       gY 
                     
                     
                       
                         
                           A 
                           1 
                           2 
                         
                         + 
                         
                           A 
                           2 
                           2 
                         
                       
                       4 
                     
                   
                 
               
               
                 
                   ( 
                   93 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where A 1  and A 2  are the signals from sensors Hall A 1  and Hall A 2 , respectively, and B 1  and B 2  are the signals from sensors Hall B 1  and Hall B 2 , respectively, and where the constants K gX  and K gY  may be, for instance, experimentally determined against known gap references. 
         [0113]    Therefore, in this embodiment only two pairs of Hall Effect sensors provide the positional feedback required for the disclosed embodiments. The alternating poles of the rotor may produce a sinusoidal output from each sensor as the rotor rotates. Thus, each pair of sensors  810 ,  815  produces a phase shifted sine wave output from which the rotor position may be determined. As shown above, an angle determined by the arctangent of the ratio of the two signals of, for example, sensors Hall A 1  and Hall A 2  may be calculated. In addition, the gap along the x and y axes may be determined by, for example, dividing corresponding constants by the 4 th  root of the sum of the squares of the two sensor signals output by each sensor pair. 
         [0114]      FIG. 9  shows a top view of an exemplary substrate processing apparatus  1000  incorporating features of the disclosed embodiments. Substrate processing apparatus  1001  generally has an atmospheric section  1050 , which is open to the atmosphere, and an adjoining vacuum section  1100 , which is equipped to function as a vacuum chamber. Atmospheric section  1050  may have one or more substrate holding cassettes  1150 , and an atmospheric substrate transport apparatus  1200 . Vacuum section  1100  may have one or more processing modules  1250 , and a vacuum substrate transport apparatus  1300 . The embodiment shown in  FIG. 9  may have load locks  1350 ,  1400  for allowing substrates to pass between atmospheric section  1050  and vacuum section  1100  without violating the integrity of any vacuum that may be present in vacuum section  1100 . 
         [0115]    Substrate processing apparatus  1000  also includes a controller  1700  that controls the operation of substrate processing apparatus  1000 . Controller  1700  may include a processor  1730  and a memory  1780 . Controller  1700  may be connected to substrate processing system  1000  through link  1830 . For purposes of the disclosed embodiments a substrate may be for example, a semiconductor wafer (e.g. a 200 mm or 300 mm wafer), a flat panel display substrate, any other type of substrate suitable for processing by substrate processing apparatus  1000 , a blank substrate, or an article having characteristics similar to a substrate, such as certain dimensions or a particular mass. 
         [0116]    Atmospheric substrate transport apparatus  1200  may include one or more self-bearing motors, for example,  1600  according to the disclosed embodiments. Self-bearing motor  1600  may advantageously utilize a minimal number of winding sets to affect the self bearing motor function and may use the same windings to provide substantially decoupled driving and centering forces in accordance with the disclosed embodiments. Self-bearing motor  1600  may be driven by a current amplifier  1250 , which may be similar to the current amplifiers described above. Vacuum substrate transport apparatus  1300  may also include one or more self-bearing motors  1900  in accordance with the disclosed embodiments. Self-bearing motor  1900  may be driven by one or more current amplifiers  1500 , which are similar to the current amplifiers described above. Substrate processing apparatus  1000  may include other self bearing motors as disclosed herein wherever suitable. 
         [0117]    In summary, the disclosed embodiments are directed to a self-bearing motor that advantageously utilizes a minimal number of winding sets to affect the self bearing motor function. Reducing the number of winding sets is advantageous because, for example, associated complex and costly control systems and electronics for controlling independent windings are also reduced. In at least one embodiment as few as two winding sets of a motor may be used to effect the self bearing motor function. In the disclosed exemplary rotary embodiments the same windings may be used to provide substantially decoupled driving and centering forces in order to actively control the center of rotation of the rotor. The same windings may be utilized to provide substantially decoupled propulsion and guidance forces to control a platen in the exemplary linear embodiments. The disclosed motor embodiments may include segmented windings, for example, windings distributed along desired portions of the rotor or platen and divided to form winding sets having one or more winding subsets. In addition, an exemplary sensor system may be included that provides both position and gap feedback. For example, the sensor system may be configured to provide simultaneous position measurement along a first and second axis. The sensor system may also be capable of providing measurement along a third axis, either simultaneously with the first and second axis measurements or individually. 
         [0118]    It should be understood that the foregoing description is only illustrative of the present embodiments. Various alternatives and modifications can be devised by those skilled in the art without departing from the embodiments disclosed herein. Accordingly, the embodiments are intended to embrace all such alternatives, modifications and variances which fall within the scope of the appended claims.