Abstract:
An apparatus for coupling mechanical power between the rotor of a Brushless DC Motor and an external mechanical load comprises: a) two concentric rings; b) an equal number of magnets connected to the inner ring and to the outer ring; and c) an opposite orientation of the poles of each couple of facing magnets, wherein one magnet is placed on the inner ring, and its facing magnet is placed on the outer ring; wherein the first of said two concentric rings is rotatable around an axis by the application of a force not applied by the second ring, and wherein when said first concentric ring rotates, the second ring rotates as well by the action of magnetic forces.

Description:
FIELD OF THE INVENTION 
       [0001]    The present invention relates to a magnetic clutch architecture designed to couple mechanical power between the rotor of Brushless DC Motors (BLDC) and an external mechanical load, without using direct or indirect mechanical connection such as gears, wheels, strips or other similar arrangements. 
       BACKGROUND OF THE INVENTION 
       [0002]    In many common systems, the connection between different parts of the system is performed by mechanical components. A significant disadvantage of using such connecting parts is the energy loss, caused by friction. Another disadvantage caused by friction is the wear of the connecting surfaces of the parts. As the speed and force between the parts increase, so does the friction and therefore the damage to their surfaces, until they often can no longer function properly. 
         [0003]    In systems operating at high speeds, like motors that usually operate in extremely high speeds, the friction and its outcomes are substantial, resulting in the need for many maintenance services and frequent change of parts, which require a great investment of both time and money. 
         [0004]    The present invention relates to a device used in BLDC motors, such as the motor described in PCT patent application No. PCT/IL2013/050253 
         [0005]    It is an object of the present invention to provide a device and method that overcome the drawbacks of the prior art. 
         [0006]    Other objects and advantages of the invention will become apparent as the description proceeds. 
       SUMMARY OF THE INVENTION 
       [0007]    An apparatus for coupling mechanical power between the rotor of a Brushless DC Motor and an external mechanical load, comprises:
       a) two concentric rings;   b) an equal number of magnets connected to the inner ring and to the outer ring; and   c) an opposite orientation of the poles of each couple of facing magnets, wherein one magnet is placed on the inner ring, and its facing magnet is placed on the outer ring;   wherein the first of said two concentric rings is rotatable around an axis by the application of a force not applied by the second ring, and wherein when said first concentric ring rotates, the second ring rotates as well by the action of magnetic forces.       
 
         [0012]    In one embodiment of the invention the rings are flat ring-shaped plates. In another embodiment of the invention each couple of facing magnets are of the same size. 
         [0013]    In some embodiments of the invention the magnetic strengths of two facing magnets are essentially the same. In another embodiment of the invention each of the magnets in the inner ring has a facing magnet in the outer ring. 
         [0014]    The connecting means, in some embodiments of the invention, connect one of the rings to an external system. In other embodiments of the invention a ring which is not connected to the external system is driven by the rotation of the ring that is connected to the external system and the driven ring is forced to move because of the magnetic force between two coupled magnets. 
         [0015]    Typically, the distances between the components of the apparatus are consistent with the desired forces and in some embodiments of the invention the distance between two adjacent magnets on the ring is not the same as the distance between two other adjacent magnets on the same ring. 
         [0016]    The invention also encompasses a brushless motor coupled with a clutch comprising two concentric rings, an equal number of magnets connected to the inner ring and to the outer ring, and an opposite orientation of the poles of each couple of facing magnets, wherein one magnet is placed on the inner ring, and its facing magnet is placed on the outer ring, and wherein the first of said two concentric rings is rotatable around an axis by the application of a force not applied by the second ring, and wherein when said first concentric ring rotates, the second ring rotates as well by the action of magnetic forces. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0017]    In the drawings: 
           [0018]      FIG. 1  shows two concentric rings, provided with magnets, according to one embodiment of the invention, in a static state; 
           [0019]      FIG. 2  shows the two rings of  FIG. 1  in a dynamic state; 
           [0020]      FIG. 3  shows the measurements of the force on a single couple of magnets mounted at distance d from each other and shifted linearly; 
           [0021]      FIG. 4  shows the measurements of the force in a demo system, according to another embodiment of the invention; 
           [0022]      FIG. 5  shows exemplary physical measures of the components in a BLDC demo system, according to another embodiment of the invention; 
           [0023]      FIG. 6  shows a schematic setup of two magnets, according to another embodiment of the invention; 
           [0024]      FIG. 7  shows solenoids illustrated as consisting of a collection of infinitesimal current loops, stacked one on top of the other; and 
           [0025]      FIG. 8  shows two loops of infinitesimal thickness, each one belonging to a magnet. 
       
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
       [0026]      FIG. 1  shows two concentric rotating rings  101  and  102  at rest. The inner ring  101  consists of the rotor of a BLDC motor (which can be, for example, the motor of PCT/IL2013/050253-WO/2013/140400), and the outer ring  102  is connected to a mechanical load and provides the power for it. A number of permanent magnets, equal to the number of the magnets in the rotor of the BLDC motor, are mechanically fixed on the outer ring  102  with their S-N axes oriented tangentially to the circumference. 
         [0027]    At rest, each one of the magnets  104  located on the outer ring  102 , is facing the corresponding magnet  103  located on the rotor  101 . The S-N axis orientation of each magnet  104  on the outer ring  102  is opposite to the S-N axis orientation of the corresponding (facing) magnet  103  on the rotor  101 . As a result, the magnets  104  on the outer ring  102  are positioned with alternating polarity. It should be emphasized that there is no physical connection between the rotor  101  and the outer ring  102 . For reasons that will be thoroughly explained later on in this description, based on the laws of magnetostatics, the relative position of the rotor  101  with respect to the outer ring  102 , depends on the state of the system—if the system is in a static state or a dynamic state, as will be further described. 
         [0028]    In a static state—when the BLCD rotor is at rest, each magnet  104  on the outer ring  102  is exactly aligned in front of the corresponding magnet  103  on the rotor  101 , as shown in  FIG. 1 . In a dynamic state—when the BLCD rotor  101  turns, while the outer ring  102  is connected to a load (not completely free to move), the relative position of each magnet  103  on the rotor ring  101  with respect to the corresponding magnet  104  on the load ring  102 , will change and will stabilize to a new state. 
         [0029]    The corresponding magnets  103  and  104  will no longer be perfectly aligned. The relative position of the magnets will shift in a quasi-linear fashion tangentially to the circumference of the rings  101  and  102 . The magnets  103  and  104  will reach an offset h as shown in  FIG. 2 , and will stabilize there. The offset h will depend on the opposing force exercised by the load. It will be seen that under proper conditions h will increase directly proportionally to the force needed to make the load ring  102  rotate along with the rotor ring  101 . 
         [0030]    It will be presented that in the range of interest, the offset h is roughly directly proportional to the force transfer, and as long as h is not too large, the rotor ring  101  will be able to “pull along” the load ring  102 , without the occurrence of any physical contact between the two ring  101  and  102 . When the size of h approaches the width of the gap between the magnets  103  and  104 , the force transferred drops. The maximal force that the rotor ring  101  will be able to apply to the load ring  102  will depend on the strength and on the geometry of the permanent magnets, on the number of magnets, as well as on the gap between the two rings  101  and  102 . 
         [0031]      FIG. 3  shows the measurements of the force on a single couple of magnets mounted at distance d from each other and shifted linearly. The shaded area  301  shows the range for which the pulling force between the magnets  103  and  104  is roughly proportional to the offset h. 
         [0032]    To illustrate the order of magnitude of the forces involved, two magnets with front-to-front separation of  29 mm, can provide roughly a maximal force transfer of 140N (about 14 Kg) in direction tangential to the ring. 
         [0033]    In the BLDC motor demo system built according to the invention, there are  8  magnets were provided with face-to-face separation of about 30 mm. The demo system is capable to apply a force of 140×8=1120N (about 112 Kg). Since the outer ring  102  in the demo system has a radius of about 420 mm, the magnetic clutch should be able to transfer a torque of about 470 N-m. 
         [0034]    In a measurement carried out on the BLDC demo system, and as shown in  FIG. 4 , the inventors did not try to achieve and measure the maximal power transfer, however, they showed force transfer measurements of the order of 600N, which is in good agreement with the order of magnitude of the maximal possible force (1120N) predicted by the measurements on one couple of magnets. Also it shows that the total force is proportional to the relative offset. 
         [0035]    The physical measures of the components in the BLDC demo system, as provided by the inventors, are shown in  FIG. 5 . From the figure one can see that the system includes 8 magnets, and the separation between the rotor ring  101  and the load ring  102  is 30 mm. 
         [0036]    Magnetostatic computations are among the most difficult and complex tasks to be carried out analytically, and even when a closed-form analytical expression can be found, the resulting formulas are often too complex to provide a clear understanding of the phenomena. Moreover, most often, one can only perform computerized simulations obtained by numerically solving the field equations. Numerical solutions, however, although precise for a specific setup, do not provide an insight to the general behavior of the system. 
         [0037]    Fortunately, in the specific case under consideration, general conclusions can be drawn by means of a relatively simple mathematical analysis. This is made possible because, in the system under consideration, the magnets are free to move only along a direction tangential to their S-N axis, and they are fixed in all other directions. Therefore, it is only needed to compute the component of the force in a direction parallel to the S-N axes of the magnets, which results in major mathematical simplifications that allow us to draw conclusion regarding general system features, without the need of actually solving the complex three-dimensional integrals involved. 
         [0038]    What was analyzed is the setup shown in  FIG. 6 . {circumflex over (x)}, ŷ and {circumflex over (z)} are mutually perpendicular unit vectors. Two cubic magnets  601  and  602  are positioned so that their S-N axes are parallel to direction {circumflex over (z)}. Their S-N orientation is opposite, and they are displaced with an offset h in direction {circumflex over (z)}. The magnets  601  and  602  are assumed cubic, for the purpose of this exemplary analysis, however the general conclusions hold true for other shapes as well. The measurements shown in  FIG. 3  have been carried out on a similar setup. 
         [0039]    Under this setup, as long as the offset h is small relatively to the physical dimension of the gap between the magnets  601  and  602 , the component of the force acting on either magnet  601  and  602  in the direction {circumflex over (z)}, is directly proportional to the offset h. The size of h is relatively small, roughly when the offset h is less than  1 / 3  of the separation d between the magnets  601  and  602 . As the offset becomes larger than that, the force reaches a maximal value, and then decreases with increasing h. 
         [0040]    As a first step, by using the Amperian model, a permanent magnet with magnetization M in direction {circumflex over (z)}, may be modeled in the form of a uniform surface current density J s  flowing on the surface of the magnet in direction perpendicular to {circumflex over (z)}. M is the net magnetic dipole moment per unit volume, and J s  is the equivalent surface current per unit length. Therefore we may replace each magnet  601  and  601  in  FIG. 6  by the equivalent “solenoids” shown in  FIG. 7 , with equal currents in opposite directions. 
         [0041]    Each solenoid  701  in  FIG. 7  can be represented as consisting of a collection of infinitesimal current loops, stacked one on top of the other, carrying currents of amplitudes dl=J s dz and dl′=J s dz′, flowing in the {circumflex over (x)}ŷ plane in opposite directions. Let us consider now, two loops of infinitesimal thickness, each one belonging to one of the magnets as shown in  FIG. 8 . 
         [0042]    The force caused on the left-side loop L located at vertical position z by the right-side loop L′ located at vertical position z′, is directly derived from Ampere&#39;s law of force, and is given by the expression 
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         [0000]    and           and          ′ are infinitesimal lengths in the direction of the current flow in the corresponding loops, and therefore they lie in the {circumflex over (x)}ŷ plane. 
         [0043]    Now, referring to  FIG. 8 , it points out several preliminary remarks: 
         [0044]    1. We know that |y−y′|≧d and we denote R {circumflex over (x)}ŷ ≡√{square root over ((x−x′) 2 +(y−y′) 2 )}{square root over ((x−x′) 2 +(y−y′) 2 )}. It follows that R {circumflex over (x)}ŷ ≧d. R {circumflex over (x)}ŷ (x, x′, y, y′) is independent from z and z′ , and we may write 
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         [0045]    2. In the present setting, d is comparable to the size of the magnet, and we assume offsets small enough so that h 2 &lt;&lt;d 2  (for instance 
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         [0046]    3. Since we are interested only in the force in the {circumflex over (z)} direction, the only relevant component of {circumflex over (r)}−{circumflex over (r)}′ in the numerator of the integrand, is the one in direction {circumflex over (z)}. All other forces are of no interest, since the magnets cannot move in other directions. Thus, in order to compute the force acting on the magnets in {circumflex over (z)} direction, we may replace {circumflex over (r)}−{circumflex over (r)} in the numerator of the integrand by (z−z′){circumflex over (z)}. 
         [0047]    4.           and          ′ are incremental vectors in the {circumflex over (x)}ŷ plane. More precisely, in the present setting of square magnets, the scalar product (         ·         ′) is either ±dxdx′ or ±dydy′. Therefore z and z′ are constant with respect to the integration variables when integrating over the path of the loops. Moreover, if dx,dx′ have opposite signs, their direction of integration is opposite too, and therefore, the limit of the corresponding integrals are reversed, and similarly for dy ,dy′. The outcome is that the sign of the integral for all the various sub-integration ranges defined by (         ·         ′) remains unchanged. Therefore the sign value of the double integral over the loop paths, is the same as the sign of the integrand. 
         [0048]    With the above understanding, the force ΔF z  in direction {circumflex over (z)} acting on the current loop L because of the current loop L′, is the result of the following integral: 
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         [0049]    The cumulative force ΔF {circumflex over (z)},L  applied by all the current loops on the right side on one single current loop L on the left side (see  FIG. 8 ) is given by 
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         [0050]    The total force F {circumflex over (z)} (h) acting on the magnet located at the origin is the sum of all the forces on its loops 
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                                       ( 
                                       
                                         z 
                                         - 
                                         
                                           z 
                                           ′ 
                                         
                                       
                                       ) 
                                     
                                   
                                   
                                     
                                       [ 
                                       
                                         
                                           R 
                                           
                                             
                                               x 
                                               ^ 
                                             
                                              
                                             
                                               y 
                                               ^ 
                                             
                                           
                                           2 
                                         
                                         + 
                                         
                                           
                                             ( 
                                             
                                               z 
                                               - 
                                               
                                                 z 
                                                 ′ 
                                               
                                             
                                             ) 
                                           
                                           2 
                                         
                                       
                                       ] 
                                     
                                     
                                       3 
                                       / 
                                       2 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                            
                           
                              
                             
                               z 
                               ′ 
                             
                           
                         
                       
                       ] 
                     
                      
                     
                        
                       z 
                     
                   
                 
               
             
           
         
       
     
         [0051]    Changing the order of integration we obtain 
         [0000]    
       
         
           
             
               
                 F 
                 
                   z 
                   ^ 
                 
               
                
               
                 ( 
                 h 
                 ) 
               
             
             = 
             
               
                 - 
                 
                   
                     μ 
                      
                     
                         
                     
                      
                     
                       J 
                       s 
                       2 
                     
                   
                   
                     4 
                      
                     π 
                   
                 
               
                
               
                   
               
                
               
                 
                   ∫ 
                   L 
                 
                  
                 
                   
                     ∫ 
                     
                       L 
                       ′ 
                     
                   
                    
                   
                       
                   
                    
                   
                     
                       [ 
                       
                         
                           ∫ 
                           0 
                           a 
                         
                          
                         
                           
                             ( 
                             
                               
                                 ∫ 
                                 h 
                                 
                                   h 
                                   + 
                                   a 
                                 
                               
                                
                               
                                 
                                   
                                     ( 
                                     
                                       z 
                                       - 
                                       
                                         z 
                                         ′ 
                                       
                                     
                                     ) 
                                   
                                   
                                     
                                       [ 
                                       
                                         
                                           R 
                                           
                                             
                                               x 
                                               ^ 
                                             
                                              
                                             
                                               y 
                                               ^ 
                                             
                                           
                                           2 
                                         
                                         + 
                                         
                                           
                                             ( 
                                             
                                               z 
                                               - 
                                               
                                                 z 
                                                 ′ 
                                               
                                             
                                             ) 
                                           
                                           2 
                                         
                                       
                                       ] 
                                     
                                     
                                       3 
                                       / 
                                       2 
                                     
                                   
                                 
                                  
                                 
                                    
                                   
                                     z 
                                     ′ 
                                   
                                 
                               
                             
                             ) 
                           
                            
                           
                              
                             z 
                           
                         
                       
                       ] 
                     
                      
                     
                         
                     
                      
                     
                       ( 
                       
                         
                            
                           
                              
                             ^ 
                           
                         
                         · 
                         
                             
                         
                          
                         
                            
                           
                             
                                
                               ^ 
                             
                             ′ 
                           
                         
                       
                       ) 
                     
                   
                 
               
             
           
         
       
     
         [0052]    Noting that R {circumflex over (x)}ŷ   2  is independent from z and z′, and therefore is constant when integrating with respect to dz and dz′, the inner integrals can be computed analytically, and yield 
         [0000]    
       
         
           
             
               
                 ∫ 
                 0 
                 a 
               
                
               
                 
                   ( 
                   
                     
                       ∫ 
                       h 
                       
                         h 
                         + 
                         a 
                       
                     
                      
                     
                       
                         
                           ( 
                           
                             z 
                             - 
                             
                               z 
                               ′ 
                             
                           
                           ) 
                         
                         
                           
                             [ 
                             
                               
                                 R 
                                 
                                   
                                     x 
                                     ^ 
                                   
                                    
                                   
                                     y 
                                     ^ 
                                   
                                 
                                 2 
                               
                               + 
                               
                                 
                                   ( 
                                   
                                     z 
                                     - 
                                     
                                       z 
                                       ′ 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                             
                             ] 
                           
                           
                             3 
                             / 
                             2 
                           
                         
                       
                        
                       
                          
                         
                           z 
                           ′ 
                         
                       
                     
                   
                   ) 
                 
                  
                 
                    
                   z 
                 
               
             
             = 
             
               ln 
                
               
                 { 
                 
                   
                     
                       [ 
                       
                         
                           ( 
                           
                             ɛ 
                             - 
                             A 
                           
                           ) 
                         
                         + 
                         
                           
                             1 
                             + 
                             
                               
                                 ( 
                                 
                                   ɛ 
                                   - 
                                   A 
                                 
                                 ) 
                               
                               2 
                             
                           
                         
                       
                       ] 
                     
                      
                     
                       [ 
                       
                         
                           ( 
                           
                             ɛ 
                             + 
                             A 
                           
                           ) 
                         
                         + 
                         
                           
                             1 
                             + 
                             
                               
                                 ( 
                                 
                                   ɛ 
                                   + 
                                   A 
                                 
                                 ) 
                               
                               2 
                             
                           
                         
                       
                       ] 
                     
                   
                   
                     
                       ( 
                       
                         ɛ 
                         + 
                         
                           
                             1 
                             + 
                             ɛ 
                           
                         
                       
                       ) 
                     
                     2 
                   
                 
                 } 
               
             
           
         
       
     
         [0000]    where we used 
         [0000]    
       
         
           
             
               A 
               = 
               
                 a 
                 
                   R 
                   
                     
                       x 
                       ^ 
                     
                      
                     
                       y 
                       ^ 
                     
                   
                 
               
             
             , 
             
               
                 and 
                  
                 
                     
                 
                  
                 ɛ 
               
               = 
               
                 
                   h 
                   
                     R 
                     
                       
                         x 
                         ^ 
                       
                        
                       
                         y 
                         ^ 
                       
                     
                   
                 
                 . 
               
             
           
         
       
     
         [0000]    Since R {circumflex over (x)}ŷ ≧d, then if h 2 &lt;&lt;d 2 ≦R {circumflex over (x)}ŷ   2  (for instance 
         [0000]    
       
         
           
             
               h 
               &lt; 
               
                 d 
                 3 
               
             
             ) 
           
         
       
     
         [0000]    then 
         [0000]    
       
         
           
             
               
                 
                   h 
                   2 
                 
                 
                   R 
                   
                     
                       x 
                       ^ 
                     
                      
                     
                       y 
                       ^ 
                     
                   
                   2 
                 
               
               = 
               
                 
                   ɛ 
                   2 
                 
                  
                 1 
               
             
             , 
           
         
       
     
         [0000]    and we may expand the last expression in a first-order Taylor series as follows 
         [0000]    
       
         
           
             
               
                 ∫ 
                 0 
                 a 
               
                
               
                 
                   ( 
                   
                     
                       ∫ 
                       h 
                       
                         h 
                         + 
                         a 
                       
                     
                      
                     
                       
                         
                           ( 
                           
                             z 
                             - 
                             
                               z 
                               ′ 
                             
                           
                           ) 
                         
                         
                           
                             [ 
                             
                               
                                 R 
                                 
                                   
                                     x 
                                     ^ 
                                   
                                    
                                   
                                     y 
                                     ^ 
                                   
                                 
                                 2 
                               
                               + 
                               
                                 
                                   ( 
                                   
                                     z 
                                     - 
                                     
                                       z 
                                       ′ 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                             
                             ] 
                           
                           
                             3 
                             / 
                             2 
                           
                         
                       
                        
                       
                          
                         
                           z 
                           ′ 
                         
                       
                     
                   
                   ) 
                 
                  
                 
                    
                   z 
                 
               
             
             = 
             
               
                 
                   
                     2 
                      
                     
                       
                         1 
                         - 
                         
                           
                             1 
                             + 
                             
                               A 
                               2 
                             
                           
                         
                       
                       
                         
                           1 
                           + 
                           
                             A 
                             2 
                           
                         
                       
                     
                      
                     ɛ 
                   
                   + 
                   
                     O 
                      
                     
                       ( 
                       
                         ɛ 
                         3 
                       
                       ) 
                     
                   
                 
                 ≈ 
                 
                   2 
                    
                   
                     
                       
                         1 
                         - 
                         
                           
                             1 
                             + 
                             
                               
                                 a 
                                 2 
                               
                               / 
                               
                                 R 
                                 
                                   
                                     x 
                                     ^ 
                                   
                                    
                                   
                                     y 
                                     ^ 
                                   
                                 
                                 2 
                               
                             
                           
                         
                       
                       
                         
                           1 
                           + 
                           
                             
                               a 
                               2 
                             
                             / 
                             
                               R 
                               
                                 
                                   x 
                                   ^ 
                                 
                                  
                                 
                                   y 
                                   ^ 
                                 
                               
                               2 
                             
                           
                         
                       
                     
                     · 
                     
                       h 
                       
                         R 
                         
                           
                             x 
                             ^ 
                           
                            
                           
                             y 
                             ^ 
                           
                         
                       
                     
                   
                 
               
               = 
               
                 
                   g 
                    
                   
                     ( 
                     
                       x 
                       , 
                       
                         x 
                         ′ 
                       
                       , 
                       y 
                       , 
                       
                         y 
                         ′ 
                       
                     
                     ) 
                   
                 
                 · 
                 h 
               
             
           
         
       
     
         [0053]    Since 
         [0000]    
       
         
           
             
               
                 
                   1 
                   + 
                   
                     
                       a 
                       2 
                     
                     / 
                     
                       R 
                       
                         
                           x 
                           ^ 
                         
                          
                         
                           y 
                           ^ 
                         
                       
                       2 
                     
                   
                 
               
               &gt; 
               1 
             
             , 
           
         
       
     
         [0000]    it follows that the function g(x,x′,y,y′) is some negative function of x,x′, y, y′, namely g(x,x′, y, y′)=|g(x,x′,y,y′)|. Therefore, recalling that the sign of the double integral over x,x′,y,y′ is the same as the sign of the integrand, and setting) ∫ L ∫ L′ |g(x,x′,y,y′)|(         ·         ′)=K 2 , the total force f {circumflex over (z)} (h), acting on the magnet at the origin, due to the offset of the other magnet, has the form 
         [0000]    
       
         
           
             
               
                 
                   
                     F 
                     
                       z 
                       ^ 
                     
                   
                    
                   
                     ( 
                     h 
                     ) 
                   
                 
                 ≈ 
                 
                   
                     
                       μ 
                        
                       
                           
                       
                        
                       
                         J 
                         s 
                         2 
                       
                        
                       h 
                     
                     
                       4 
                        
                       π 
                     
                   
                    
                   
                       
                   
                    
                   
                     
                       ∫ 
                       L 
                     
                      
                     
                       
                         ∫ 
                         
                           L 
                           ′ 
                         
                       
                        
                       
                           
                       
                        
                       
                         
                            
                           
                             g 
                              
                             
                               ( 
                               
                                 x 
                                 , 
                                 
                                   x 
                                   ′ 
                                 
                                 , 
                                 y 
                                 , 
                                 
                                   y 
                                   ′ 
                                 
                               
                               ) 
                             
                           
                            
                         
                          
                         
                           ( 
                           
                             
                                
                               
                                  
                                 ^ 
                               
                             
                             · 
                             
                                 
                             
                              
                             
                                
                               
                                 
                                    
                                   ^ 
                                 
                                 ′ 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               = 
               
                 
                   
                     
                       K 
                       2 
                     
                      
                     μ 
                      
                     
                         
                     
                      
                     
                       J 
                       s 
                       2 
                     
                   
                   
                     4 
                      
                     π 
                   
                 
                  
                 h 
               
             
             , 
             
               
 
             
              
             
               
                 h 
                 2 
               
                
               
                 d 
                 2 
               
             
           
         
       
     
         [0000]    where K is some proportionality constant. Finally, recalling that M=J s  is the net magnetization per unit volume in the {circumflex over (z)} direction, and referring to  FIG. 6 , the force acting on the left magnet is 
         [0000]    
       
         
           
             
               
                 
                   F 
                   
                     z 
                     ^ 
                   
                 
                  
                 
                   ( 
                   h 
                   ) 
                 
               
               = 
               
                 
                   
                     
                       K 
                       2 
                     
                      
                     μ 
                      
                     
                         
                     
                      
                     
                       M 
                       2 
                     
                   
                   
                     4 
                      
                     π 
                   
                 
                  
                 h 
               
             
             , 
             
                 
             
              
             
               
                 h 
                 2 
               
                
               
                 d 
                 2 
               
             
           
         
       
     
         [0054]    Thus, for any offset h&lt;d/3, the force transferred by the clutch is directly proportional to the offset h and to the square magnetization per unit volume. Moreover, the force is in direction of the offset itself. 
         [0055]    All the above description has been provided for the purpose of illustration and is not meant to limit the invention in any way. The computations shown above are provided as an aid in understanding the invention, and should not be construed as intending to limit the invention in any way.