Abstract:
Embodiments of the present invention relate to magnetic gears comprising a pair of rotors magnetically coupled in a geared manner via a magnetic space harmonic generated as a consequence of varying an air gap between sets of permanent magnets.

Description:
FIELD OF THE INVENTION 
       [0001]    Embodiments of the present invention relate to magnetic gears. 
       BACKGROUND TO THE INVENTION 
       [0002]    Mechanical gearboxes are extensively used to match the operating speed of prime-movers to the requirements of their loads for both increasing rotational speed such as, for example, in a wind-powered generator or reducing rotational speed such as, for example, in an electric-ship propulsion arrangement. It is usually more cost and weight effective to employ a high-speed electrical machine in conjunction with a mechanical gearbox to achieve requisite speed and torque characteristics. However, white such a high-speed electrical machine in conjunction with a mechanical gearbox allows high system torque densities to be realised, such mechanical gearboxes usually require lubrication and cooling. Furthermore, reliability can also be a significant issue. Consequently, direct drive electrical machines are employed in applications where a mechanical gearbox cannot be used. 
         [0003]    Several techniques of achieving magnetic gearing, using permanent magnets, are known within the art. For example,  FIG. 1  shows the most commonly used topology for magnetic gears. It can be appreciated that  FIG. 1  shows a magnetic gear  100  comprising a first, high-speed, rotor  102  bearing a plurality of permanent magnets  104  that is magnetically coupled, in a geared manner, to a second, low speed, rotor  106  comprising a number of permanent magnets  108 . A significant disadvantage of the magnetic gear  100  shown in  FIG. 1  is that the topology suffers from a very poor utilisation of the permanent magnets since very few of the permanent magnets simultaneously contribute to torque transmission at any given time. The very poor torque transmission capability has limited the use of magnetic gearing. 
         [0004]    The problem associated with the magnetic gear  100  of  FIG. 1  is solved by the magnetic gear  200  shown in  FIG. 2 .  FIG. 2  shows a rotary magnetic gear  200  comprising a first or inner rotor  202 , a second or outer rotor  204  and a number of pole pieces  206 , otherwise known as an interference or an interference means. The first rotor  202  comprises a support  208  bearing a respective first number of permanent magnets  210 . In the illustrated magnetic gear, the first rotor  202  comprises 8 permanent magnets or 4 pole-pairs arranged to produce a spatially varying magnetic field. The second rotor  204  comprises a support  212  bearing a respective second number of permanent magnets  214 . The second rotor  204  comprises 46 permanent magnets or 23 pole-pairs arranged to produce a spatially varying field. The first and second numbers of permanent magnets are different. Accordingly, there will be little or no useful direct magnetic coupling or interaction between the permanents magnets  210  and  214  such that rotation of one rotor will not cause rotation of the other rotor. 
         [0005]    The pole pieces  206  are used to allow the fields of the permanent magnets  210  and  214  to interact in a geared manner. The pole pieces  206  modulate the magnetic fields of the permanent magnets  210  and  214  so they interact to the extent that rotation of one rotor will induce rotation of the other rotor in a geared manner. Rotation of the first rotor  202  at a speed ω 1  will induce rotation of the second rotor  204  at a speed ω 2  where ω 1 &gt;ω 2  and visa versa. 
         [0006]    However, the magnetic gear topology shown in  FIG. 2  has the disadvantages that it is unsuitable for high gear ratios, it is relatively complex and has an unfavourable torque density especially when higher gear ratios are required. 
         [0007]    It is an object of embodiments of the present invention to at least mitigate one or more of the above problems of the prior art. 
       SUMMARY OF EMBODIMENTS OF THE INVENTION 
       [0008]    Accordingly, a first aspect of embodiment of the present invention provides a magnetic gear comprising first and second moveable members having associated first and second pluralities of permanent magnets respectively arranged such that the first and second pluralities of permanent magnets are separated by a varying distance that, in response to relative movement of the first and second moveable members, magnetically couples the first and second pluralities of permanent magnets in a geared manner via a common magnetic harmonic generated as a consequence of the relative movement. 
         [0009]    Advantageously, the magnetic gears according to embodiments of the present invention exhibit significant advantages, in terms of simplicity and torque density, especially when higher gear ratios are required as compared to the prior art. 
         [0010]    Other embodiments are described below and claimed in the claims. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0011]    Embodiments of the present invention will now be described, by way of example only, with reference to the accompanying drawings in which: 
           [0012]      FIG. 1  shows a conventional magnetic gear; 
           [0013]      FIG. 2  shows a further conventional magnetic gear; 
           [0014]      FIG. 3  shows a magnetic gear according to a first embodiment; 
           [0015]      FIG. 4  shows a graph of variation in normal flux density with circumference position of the embodiment shown in  FIG. 3 ; 
           [0016]      FIG. 5  illustrates a harmonic spectrum of the waveform shown in  FIG. 4 ; 
           [0017]      FIG. 6  depicts a magnetic gear according to a further embodiment; 
           [0018]      FIG. 7  shows another embodiment of a magnetic gear; 
           [0019]      FIG. 8  illustrates yet another embodiment of a magnetic gear; 
           [0020]      FIG. 9  depicts still another embodiment of a magnetic gear; 
           [0021]      FIG. 10  shows a graph of variation of pull-out torque with maximum air gap per metre of axial length for embodiments of the present invention; 
           [0022]      FIG. 11  shows a preferred embodiment of a magnetic gear; 
           [0023]      FIGS. 12(   a ) to ( d ) illustrate the operation of an embodiment of a magnetic gear; 
           [0024]      FIG. 13  depicts circumferential variation of normal flux density clue to movement of an intermediary rotor for a given point at the centre of a stator magnet according to an embodiment; 
           [0025]      FIG. 14  shows a magnetic harmonic spectrum of the waveform of  FIG. 13 ; and 
           [0026]      FIGS. 15(   a ) to ( d ) illustrate the gearing of a magnetic gear according to an embodiment. 
       
    
    
     DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS 
       [0027]      FIG. 3  shows a magnetic gear  300  according to a still further embodiment. The magnetic gear  300  comprises an inner rotor  302 , an outer rotor  304  and a stator  306 . The inner rotor  302  comprises a non-cylindrical shaft  308  arrange to rotate about an axis (not shown). The outer rotor  304  comprises a number of permanent magnets  310  mounted on a flexible substrate  312 . A plurality of bearings  314  are disposed between the inner rotor  302  and the outer rotor  304  to support relative rotation between the inner  302  and outer  304  rotors. The stator  306  comprises a plurality of permanent magnets  316 , mounted on a substrate  318 , that are magnetically coupled to the permanent magnets  310  of the outer rotor  304  to produce a geared rotation between the inner  302  and outer  304  rotors using the above described principles, that is, the circumference of the inner rotor is at least one of shaped and rotated at a predetermined speed to produce harmonics that couple the permanent magnets  310  of the rotor  304  to the permanent magnets  316  of the stator  306 , that is, selected pole-pairs of the outer rotor permanent magnets are coupled to corresponding pole-pairs of the stator permanent magnets. 
         [0028]    Preferably, the inner rotor  302  is a high-speed rotor. The high-speed rotor  302  is non-circular. The high-speed rotor  302 , also known as a waveform generator, is, as indicated above, shaped so as to have a predetermined profile. In the embodiment illustrated, the waveform generator  302  has a sinusoidal profile having a radius, r, measured relative to an axis of the shaft, given by 
         [0000]        r=r   ov   +r   b  cos(2θ)  (1) 
         [0000]    where r av  is the average radius and r b  is the maximum deviation from the average. It is worth noting that although for the embodiment shown in  FIG. 3  is profile given by equation (1) is adopted any profile which could be approximated by r=r av +r b  cos(nnθ), where nn is an integer would work. Therefore, the flux-density due to the low-speed rotor magnets  310  can be written as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         B 
                         = 
                           
                          
                         
                           
                             B 
                             m 
                           
                            
                           
                             
                               cos 
                                
                               
                                 ( 
                                 
                                   pp 
                                    
                                   
                                       
                                   
                                    
                                   θ 
                                 
                                 ) 
                               
                             
                              
                             
                               [ 
                               
                                 
                                   λ 
                                   0 
                                 
                                 + 
                                 
                                   
                                     λ 
                                     1 
                                   
                                    
                                   
                                     cos 
                                      
                                     
                                       ( 
                                       
                                         2 
                                          
                                         
                                             
                                         
                                          
                                         θ 
                                       
                                       ) 
                                     
                                   
                                 
                               
                               ] 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               B 
                               m 
                             
                              
                             
                               cos 
                                
                               
                                 ( 
                                 
                                   pp 
                                    
                                   
                                       
                                   
                                    
                                   θ 
                                 
                                 ) 
                               
                             
                              
                             
                               λ 
                               0 
                             
                           
                           + 
                           
                             
                               1 
                               2 
                             
                              
                             
                               B 
                               m 
                             
                              
                             
                               λ 
                               1 
                             
                              
                             
                               cos 
                                
                               
                                 ( 
                                 
                                   
                                     ( 
                                     
                                       pp 
                                       + 
                                       2 
                                     
                                     ) 
                                   
                                    
                                   θ 
                                 
                                 ) 
                               
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             1 
                             2 
                           
                            
                           
                             B 
                             m 
                           
                            
                           
                             λ 
                             1 
                           
                            
                           
                             cos 
                              
                             
                               ( 
                               
                                 
                                   ( 
                                   
                                     pp 
                                     - 
                                     2 
                                   
                                   ) 
                                 
                                  
                                 θ 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
         [0029]    Therefore, harmonics with pole-pairs of (pp+2) and (pp−2) are created that can interact with the stator magnets  316 . 
       Gear Ratio 
       [0030]    Writing equation (2) as a function of time gives 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         B 
                         = 
                           
                          
                         
                           
                             B 
                             m 
                           
                            
                           
                             
                               cos 
                                
                               
                                 ( 
                                 
                                   pp 
                                    
                                   
                                     ( 
                                     
                                       θ 
                                       - 
                                       
                                         
                                           ω 
                                           ls 
                                         
                                          
                                         t 
                                       
                                     
                                     ) 
                                   
                                 
                                 ) 
                               
                             
                              
                             
                               [ 
                               
                                 
                                   λ 
                                   0 
                                 
                                 + 
                                 
                                   
                                     λ 
                                     1 
                                   
                                    
                                   
                                     cos 
                                      
                                     
                                       ( 
                                       
                                         2 
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                                           ( 
                                           
                                             θ 
                                             - 
                                             
                                               
                                                 ω 
                                                 w 
                                               
                                                
                                               t 
                                             
                                           
                                           ) 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                               ] 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
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                           … 
                           + 
                           
                             
                               B 
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                                     ) 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           … 
                           + 
                           
                             
                               
                                 
                                   B 
                                   m 
                                 
                                  
                                 
                                   λ 
                                   1 
                                 
                               
                               2 
                             
                              
                             
                               cos 
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                                 [ 
                                 
                                   
                                     
                                       ( 
                                       
                                         pp 
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                                         2 
                                       
                                       ) 
                                     
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                                   + 
                                   
                                     
                                       ( 
                                       
                                         
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                           + 
                         
                       
                     
                   
                   
                     
                       
                           
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                               [ 
                               
                                 
                                   
                                     ( 
                                     
                                       pp 
                                       + 
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                                     ) 
                                   
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                                 - 
                                 
                                   
                                     ( 
                                     
                                       
                                         2 
                                          
                                         
                                             
                                         
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                                           ω 
                                           w 
                                         
                                       
                                       + 
                                       
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                               ] 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where ω is , is the speed of the low speed rotor  304  and
       ω w  is the speed of the high speed rotor  302  (wave-form generator)       
 
         [0032]    Therefore, in order for the harmonic of order (pp+2) to couple with the static field of the stator magnets  316 , the following relationship between the rotor speeds must hold: 
         [0000]    
       
         
           
             
               
                 
                   
                     ω 
                     ls 
                   
                   = 
                   
                     - 
                     
                       
                         2 
                          
                         
                             
                         
                          
                         
                           ω 
                           w 
                         
                       
                       pp 
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
         [0033]    If the gear is designed with q=(pp−2) pole-pairs on the stator  306 , the relationship, expressed in terms of such pole-pairs, between the rotor speeds becomes: 
         [0000]    
       
         
           
             
               
                 
                   
                     ω 
                     ls 
                   
                   = 
                   
                     + 
                     
                       
                         2 
                          
                         
                           ω 
                           w 
                         
                       
                       pp 
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
         [0034]    It should further be noted that the magnets  310  on the low-speed rotor  304  rotate with respective, different, speeds at each moment in time due to their different positions on the sinusoidal circumference or profile of the high-speed rotor  302 . Therefore, ω is  represents the average rotational speed of all magnets  310  of the low speed rotor  304 . 
         [0035]    There are a number of parameters associated with the magnetic gear  300  shown in  FIG. 3 . It should be noted that the flexible or low speed rotor  304  comprises a number of pole-pairs, pp. Secondly, the stator  306  comprises a number of pole-pairs, qq. Thirdly, the stator  306  has a predetermined outer radius, Ro. Fourthly, the magnets  316  on the stator  306  have a predetermined radial thickness, Lpm_stat. The magnets  310  of the low speed rotor  304  have a predetermined radial thickness, Lpm_low. Due to the noncircular shape of the high-speed rotor  302 , the radial gap between the permanent magnets  310  of the low speed rotor  304  and the permanent magnets  316  of the stator  306  varies. In the embodiment illustrated, the radial air gap varies from a minimum, Gap_min, to a maximum, Gap_max. Fifthly, the back-iron  314  of the stator  306  has a predetermined radial length or thickness, Liron_stat. The dimensions of the above parameters for an embodiment of the harmonic gear  300  may be as given in table 1 below. 
         [0000]    
       
         
               
             
               
               
             
           
               
                 TABLE 1 
               
               
                   
               
               
                 Dimensions of harmonic gear in FE predictions 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 Outer radius, Ro 
                 85 mm  
               
               
                 Minimal length of air-gap, gap_min 
                 1 mm 
               
               
                 Length of back-iron on stator, Liron_stat 
                 5 mm 
               
               
                 Minimal Length of back-iron on low-speed rotor, Liron_low 
                 5 mm 
               
               
                 Magnet thickness on stator, Lpm_stat 
                 5 mm 
               
               
                 Magnet thickness on low-speed rotor, Lpm_low 
                 5 mm 
               
               
                   
               
             
          
         
       
     
         [0036]    The number of pole-pairs, qq, on the stator  306  must be equal to (pp+2) or (pp−2), as has been deduced from equation (2), to produce torque between the stator and low-speed rotor magnets. To demonstrate this further,  FIG. 4  shows a graph  400  of the variation of normal flux density, which is due to the low-speed rotor magnets  312 , through or at the centre of the stator magnets  316  with circumferential position for an embodiment of a magnetic gear  300  with pp=20 and gap_max=9.5 mm. 
         [0037]      FIG. 5  shows a harmonic spectrum  500  of the waveform  402  shown in  FIG. 4 . It can be seen from the harmonic spectrum  500  that the (pp+2) harmonic  502  has the largest flux density amplitude. Therefore, an embodiment of a magnetic gear with qq=(pp+2) stator pole-pairs will produce the maximum torque for a low-speed rotor having pp pole-pairs. Table 2 compares predicted torques for embodiments of the gear when the stator has qq=(pp−2)=18 and qq=(pp+2)=22 pole-pairs. 
         [0000]        qq =( pp− 2)=18  qq =( pp+ 2)=22       Torque per meter 1170 Nm 3020 Nm         
         [0039]    Table 2, Comparison between predicted torque when stator has (pp+2) and (pp−2) pole-pairs 
         [0040]      FIGS. 6 to 9  show various magnetic gears according to embodiments of the present invention. The embodiments have the parameters as described above with reference to table 1 but with different respective values of gap_max. 
         [0041]    Referring to  FIG. 6 , there is shown an embodiment of a magnetic gear  600  comprising a first rotor  602 , a second rotor  604  and a stator  606 . The second rotor  604  comprises 40 pole pairs. The stator  606  comprises 42 pole-pairs and the maximum air gap is 5.5 mm. 
         [0042]    Referring to  FIG. 7 , there is shown an embodiment of a magnetic gear  700  comprising a first rotor  702 , a second rotor  704  and a stator  706 . The second rotor  704  comprises 30 permanent magnets. The stator  706  comprises 32 pole-pairs and the maximum air gap is 7 mm. 
         [0043]    Referring to  FIG. 8 , there is shown an embodiment of a magnetic gear  800  comprising a first rotor  802 , a second rotor  804  and a stator  806 . The second rotor  804  comprises 40 pole pair. The stator  806  comprises 22 pole-pairs and the maximum air gap is 9.5 mm. This arrangement is the same as that described with reference to  FIG. 3 . 
         [0044]    Referring to  FIG. 9 , there is shown an embodiment of a magnetic gear  900  comprising a first rotor  902 , a second rotor  904  and a stator  906 . The second rotor  904  comprises 10 pole pairs. The stator  906  comprises 12 pole-pairs and the maximum air gap is 16 mm. 
         [0045]      FIG. 10  illustrates a graph  1000  showing the variation of pull-out torque with maximum air gap per metre of axial length for the embodiments described with reference to  FIGS. 6 to 9 . A first curve  1002  illustrates the torque versus maximum air gap performance for the embodiment described with reference to  FIG. 6 . A second curve  1004  illustrates the torque versus maximum air gap performance for the embodiment described with reference to  FIG. 7 . A third curve  1006  illustrates the torque versus maximum air gap performance for the embodiment described with reference to  FIG. 3  or  8 . A fourth curve  1008  illustrates the torque versus maximum air gap performance for the embodiment described with reference to  FIG. 9 . 
         [0046]    Referring to  FIG. 11 , there is a shown a magnetic gear  1100  according to an embodiment having non-coaxial or eccentric rotors that rotate about respective axes, such that one axis orbits another axis. 
         [0047]    The magnetic gear  1100  shown in  FIG. 11  comprises first  1102  and second  1104  stages. The magnetic gear  1100  is illustrated using two end views  1106  and  1108  and a cross-sectional-axial view  1110 . 
         [0048]    The first stage  1102  comprises an input rotor  1112  having mounted thereon, via bearings  1114 , an inner or first rotor  1116 , also known as an intermediary rotor. The first stage  1102  also comprises a stator  1118 . It can be appreciated that the input rotor  1112  is coupled, in an eccentric manner, to a central shaft  1120 . The intermediary rotor  1116  comprises a plurality of permanent magnets  1126 . The stator  1118  comprises a soft magnetic material  1128  bearing a plurality of permanent magnets  1130 . Rotation of the input rotor  1112  around its axis  1124 , causes the intermediary rotor  1116  to orbit the axis  1124 . This, in turn, causes the intermediary rotor  1116  to rotate about its central axis  1122 , as a result of the magnetic coupling between the pluralities of permanent magnets  1126  and  1130 , caused by the varying radial airgap between them. It can be appreciated that the intermediary rotor  1116  bears, at an output end, that is, in the second stage  1104  of the magnetic gear  1100 , a second set of permanent magnets  1132  comprising a predetermined number of permanent magnets. The second stage  1104  of the magnetic gear  1100  comprises an output rotor  1134  bearing a plurality of permanent magnets  1136 . The rotation of the intermediary rotor  1116  around the axis  1122  causes the rotation of output rotor  1134  around axis  1124  as a result of the magnetic coupling between the pluralities of permanent magnets  1132  and  1136  caused by the varying radial airgap between them. It can be appreciated that the outer rotor  1134  is mounted to a respective output portion  1138  of the shaft  1120  via a bearing  1140 . The output portion  1138  is coaxial with the input rotor  1112  and, therefore, shares the common axis  1124 . 
         [0049]    The contour of the intermediary rotor  1116 , formulated from the centre of the stator  1118  or output rotor  1134 , can be approximated as a sinusoidal profile: 
         [0000]        r=r   av   +r   b  cos(θ)  (6) 
         [0050]    Therefore, the flux-density in the outer bore of the gear in stage 1 or stage 2, due to the intermediary rotor magnets  1126  or  1132 , can be written as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           B 
                           
                             1 
                             , 
                             2 
                           
                         
                         = 
                           
                          
                         
                           
                             B 
                             m 
                           
                            
                           
                             
                               cos 
                                
                               
                                 ( 
                                 
                                   
                                     pp 
                                     
                                       1 
                                       , 
                                       2 
                                     
                                   
                                    
                                   θ 
                                 
                                 ) 
                               
                             
                              
                             
                               [ 
                               
                                 
                                   λ 
                                   0 
                                 
                                 + 
                                 
                                   
                                     λ 
                                     1 
                                   
                                    
                                   
                                     cos 
                                      
                                     
                                       ( 
                                       θ 
                                       ) 
                                     
                                   
                                 
                               
                               ] 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               B 
                               m 
                             
                              
                             
                               cos 
                                
                               
                                 ( 
                                 
                                   
                                     pp 
                                     
                                       1 
                                       , 
                                       2 
                                     
                                   
                                    
                                   θ 
                                 
                                 ) 
                               
                             
                              
                             
                               λ 
                               0 
                             
                           
                           + 
                           
                             
                               1 
                               2 
                             
                              
                             
                               B 
                               m 
                             
                              
                             
                               λ 
                               1 
                             
                              
                             
                               cos 
                                
                               
                                 ( 
                                 
                                   
                                     ( 
                                     
                                       
                                         pp 
                                         
                                           1 
                                           , 
                                           2 
                                         
                                       
                                       + 
                                       1 
                                     
                                     ) 
                                   
                                    
                                   θ 
                                 
                                 ) 
                               
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             1 
                             2 
                           
                            
                           
                             B 
                             m 
                           
                            
                           
                             λ 
                             1 
                           
                            
                           
                             cos 
                              
                             
                               ( 
                               
                                 
                                   ( 
                                   
                                     
                                       pp 
                                       
                                         1 
                                         , 
                                         2 
                                       
                                     
                                     - 
                                     1 
                                   
                                   ) 
                                 
                                  
                                 θ 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where the subscripts 1 and 2 denote the 1 st  and 2 nd  stages of the permanent magnets respectively. 
         [0051]    Therefore, harmonics with pole-pairs of (pp 1,2 +1) and (pp 1,2 −1) are created at the outer magnets  1130  and  1136  of each stage of the magnetic gear  1100 . The former harmonic is generally larger than the latter. Hence, qq 1,2 =pp 1,2 +1 are been selected for realising such a magnetic gear  1100 . 
       Gear Ratio 
       [0052]    Equation (7) can be written as a function of time to give: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           B 
                           
                             1 
                             , 
                             2 
                           
                         
                         = 
                           
                          
                         
                           
                             B 
                             m 
                           
                            
                           
                             
                               cos 
                                
                               
                                 ( 
                                 
                                   
                                     pp 
                                     
                                       1 
                                       , 
                                       2 
                                     
                                   
                                    
                                   
                                     ( 
                                     
                                       θ 
                                       - 
                                       
                                         
                                           ω 
                                           m 
                                         
                                          
                                         t 
                                       
                                     
                                     ) 
                                   
                                 
                                 ) 
                               
                             
                              
                             
                               [ 
                               
                                 
                                   λ 
                                   0 
                                 
                                 + 
                                 
                                   
                                     λ 
                                     1 
                                   
                                    
                                   
                                     cos 
                                      
                                     
                                       ( 
                                       
                                         θ 
                                         - 
                                         
                                           
                                             ω 
                                             
                                               i 
                                                
                                               
                                                   
                                               
                                                
                                               n 
                                             
                                           
                                            
                                           t 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                               ] 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           … 
                           + 
                           
                             
                               B 
                               m 
                             
                              
                             
                               λ 
                               1 
                             
                              
                             
                               cos 
                                
                               
                                 ( 
                                 
                                   
                                     pp 
                                     
                                       1 
                                       , 
                                       2 
                                     
                                   
                                    
                                   
                                     ( 
                                     
                                       θ 
                                       - 
                                       
                                         
                                           ω 
                                           m 
                                         
                                          
                                         t 
                                       
                                     
                                     ) 
                                   
                                 
                                 ) 
                               
                             
                              
                             
                               cos 
                                
                               
                                 ( 
                                 
                                   θ 
                                   - 
                                   
                                     
                                       ω 
                                       
                                         i 
                                          
                                         
                                             
                                         
                                          
                                         n 
                                       
                                     
                                      
                                     t 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           … 
                           + 
                           
                             
                               
                                 
                                   B 
                                   m 
                                 
                                  
                                 
                                   λ 
                                   1 
                                 
                               
                               2 
                             
                              
                             
                               cos 
                                
                               
                                 [ 
                                 
                                   
                                     
                                       ( 
                                       
                                         
                                           pp 
                                           
                                             1 
                                             , 
                                             2 
                                           
                                         
                                         - 
                                         1 
                                       
                                       ) 
                                     
                                      
                                     θ 
                                   
                                   + 
                                   
                                     
                                       ( 
                                       
                                         
                                           ω 
                                           
                                             i 
                                              
                                             
                                                 
                                             
                                              
                                             n 
                                           
                                         
                                         - 
                                         
                                           
                                             pp 
                                             
                                               1 
                                               , 
                                               2 
                                             
                                           
                                            
                                           
                                               
                                           
                                            
                                           
                                             ω 
                                             m 
                                           
                                         
                                       
                                       ) 
                                     
                                      
                                     t 
                                   
                                 
                                 ] 
                               
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             
                               
                                 B 
                                 m 
                               
                                
                               
                                 λ 
                                 1 
                               
                             
                             2 
                           
                            
                           
                             cos 
                              
                             
                               [ 
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         pp 
                                         
                                           1 
                                           , 
                                           2 
                                         
                                       
                                       + 
                                       1 
                                     
                                     ) 
                                   
                                    
                                   θ 
                                 
                                 - 
                                 
                                   
                                     ( 
                                     
                                         
                                     
                                      
                                     
                                       
                                         ω 
                                         
                                           i 
                                            
                                           
                                               
                                           
                                            
                                           n 
                                         
                                       
                                       + 
                                       
                                         
                                           
                                             pp 
                                              
                                             
                                                 
                                             
                                           
                                           
                                             1 
                                             , 
                                             2 
                                           
                                         
                                          
                                         
                                           ω 
                                           m 
                                         
                                       
                                     
                                     ) 
                                   
                                    
                                   t 
                                 
                               
                               ] 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where ω m  is the speed of the intermediary rotor  1116  and
       ω in  is the speed of the input shaft  1112  (high-speed rotor).
 
Stage 1: in order for the harmonic of order (pp 1 +1) to couple with the static field of the stator magnets  1130 , the relationship between the rotor speeds can be derived as follows:
       
 
         [0000]    
       
         
           
             
               
                 
                   
                     ω 
                     
                       i 
                        
                       
                           
                       
                        
                       n 
                     
                   
                   = 
                   
                     - 
                     
                       
                         ω 
                         
                           i 
                            
                           
                               
                           
                            
                           n 
                         
                       
                       
                         pp 
                         1 
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
         [0054]    Stage2: In order for the harmonic of order (pp 2 +1) to couple with the field of the magnets  1136  on the output rotor  1134 , which rotates with a speed of ω out , the following equation must hold: 
         [0000]      (ω in   +pp   2 ω m )=( pp   2 +1)ω out   (10) 
         [0000]    which results in 
         [0000]    
       
         
           
             
               
                 
                   
                     ω 
                     out 
                   
                   = 
                   
                     
                       
                         1 
                         
                           ( 
                           
                             
                               pp 
                               2 
                             
                             + 
                             1 
                           
                           ) 
                         
                       
                        
                       
                         ω 
                         
                           i 
                            
                           
                               
                           
                            
                           n 
                         
                       
                     
                     + 
                     
                       
                         
                           pp 
                           2 
                         
                         
                           ( 
                           
                             
                               pp 
                               2 
                             
                             + 
                             1 
                           
                           ) 
                         
                       
                        
                       
                         ω 
                         m 
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
       Overall Gear Ratio: 
       [0055]    Combining equations (9) and (11) results in the overall gear ratio of the 2-stage harmonic gear given by equation (12) 
         [0000]    
       
         
           
             
               
                 
                   
                     ω 
                     out 
                   
                   = 
                   
                     
                       - 
                       
                         
                           
                             
                               pp 
                               2 
                             
                             
                               pp 
                               1 
                             
                           
                           - 
                           1 
                         
                         
                           ( 
                           
                             
                               pp 
                               2 
                             
                             + 
                             1 
                           
                           ) 
                         
                       
                     
                      
                     
                       ω 
                       
                         i 
                          
                         
                             
                         
                          
                         n 
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
         [0056]    An embodiment of a magnetic gear as depicted in  FIG. 11  was realised using the parameters of table 3 below. It can be appreciated from the eccentricity value and the minimum air gap value that the maximum air gap value is 6 mm. 
         [0000]    
       
         
               
             
               
               
               
             
               
               
               
               
             
           
               
                 TABLE 3 
               
             
             
               
                   
               
               
                 Values of parameters for the harmonic gear of FIG. 11. 
               
             
          
           
               
                 Parameter 
                 Description 
                 Value 
               
               
                   
               
             
          
           
               
                 e 
                 eccentricity (distance between centres of 
                 5 
                 mm 
               
               
                   
                 high-speed rotor and stator) 
               
               
                 pp 1   
                 Number of pole-pairs on intermediary 
                 20 
               
               
                   
                 rotor in stage 1 
               
               
                 qq 1   
                 Number of pole-pairs on stator 
                 21 
               
               
                 pp 2   
                 Number of pole-pairs on intermediary 
                 21 
               
               
                   
                 rotor in stage 2 
               
               
                 qq 2   
                 Number of pole-pairs on output rotor 
                 22 
               
               
                 Ro 
                 Outer radius 
                 85 
               
               
                 Gap_min 
                 Minimal length of air-gap 
                 1 
                 mm 
               
               
                 Lpm 
                 Magnet thickness 
                 5 
                 mm 
               
               
                 Liron 
                 Thickness of back-iron 
                 5 
                 mm 
               
               
                   
               
             
          
         
       
     
         [0057]    The gear ratios related to the harmonic gear with the parameters given in table 3 are shown in table 4. 
         [0000]    
       
         
               
             
               
               
               
               
             
           
               
                 TABLE 4 
               
               
                   
               
               
                 Gear ratios of harmonic gear 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 G 1   
                 Gear ratio of 1 st  stage of gear, ω in /ω in   
                 20 
               
               
                   
                 G 2   
                 Overall gear ratio of harmonic gear, ω in /ω out   
                 440 
               
               
                   
                   
               
             
          
         
       
     
         [0058]    It can be appreciated that relatively high gear ratios can be realised. 
         [0059]    Referring to the first stage  1102  of the magnetic gear  1100  of  FIG. 11 , the shaft  1120  and the intermediary rotor  1116  rotate in opposite directions. Therefore, an anticlockwise rotation of the shaft  1120  results in a clockwise rotation of the intermediary rotor  1116  and visa versa. This rotation is demonstrated schematically by  FIGS. 12(   a ) to  12 ( d ). Referring to  FIG. 12(   a ), two permanent magnets  1202  and  1204  are identified as reference points. They are associated with the intermediary rotor  1116  and the stator  1118  respectively. These permanent magnets are arbitrarily selected as being aligned at 0° prior to rotation of the shaft  1120 . The mutual positions of the two permanent magnets  1202  and  1204  can be seen to have changed slightly when the shaft has been rotated 90° anticlockwise such that the permanent magnet  1202  of the intermediary rotor  1116  has moved slightly in the clockwise direction relative to the permanent magnet  1204  of the stator  1118  as can be appreciated from  FIG. 12(   b ). Referring to  FIG. 12(   c ), the shaft  1120  has rotated through 180° and the permanent magnet  1202  of the intermediary rotor  1116  has moved even further away in a clockwise direction from the permanent magnet  1204  of the stator  1118 . Referring to  FIG. 12(   d ), the shaft  1120  has rotated through 261° and the permanent magnet  1202  of the intermediary rotor  1116  has moved still further away in a clockwise direction from the permanent magnet  1204  of the stator  1118 . 
         [0060]    A mathematical model of the first stage  1102  of the magnetic gear  1100  of  FIG. 11  was produced and simulations of the variations in the magnet fields were investigated.  FIG. 13  illustrates circumferential variation of the normal flux density due to movement of the intermediary rotor  1116  for a given point at the centre of a stator magnet such as the above described stator magnet  1204 . It can be appreciated that the intermediary rotor  1116  that is eccentrically positioned relative to the stator axis  1124  results in a varying air gap, which, in turn, results in a complex spatially distributed magnetic field  1302  that enables magnetic coupling/torque transmission between the permanent magnets  1126  and  1130 . 
         [0061]      FIG. 14  illustrates the magnetic harmonic spectrum  1400  of the waveform  1302  depicted in  FIG. 13 . It can be appreciated that the 21st harmonic  1402  is dominant. Referring to the second stage  1104  of the magnetic gear  1100  of  FIG. 11 , the shaft  1120  and the intermediary rotor  1116  rotate in opposite directions and the intermediary rotor  1116  and the output rotor  1134  rotate in the same direction but at different rates of rotation, that is, in a geared manner. Therefore, an anticlockwise rotation of the shaft  1120  results in a clockwise rotation of the intermediary rotor  1116  and the output rotor  1134  and visa versa. This rotation is demonstrated schematically by  FIGS. 15(   a ) to  15 ( d ). 
         [0062]    Referring to  FIG. 15(   a ), two permanent magnets  1502  and  1504  are identified as reference points. They are associated with the intermediary rotor  1116  and the output rotor  1134  respectively. These permanent magnets are arbitrarily selected as being aligned at 0′ prior to rotation of the shaft  1120 . The mutual positions of the two permanent magnets  1502  and  1504  can be seen to have changed slightly when the shaft has been rotated 261° anticlockwise such that the permanent magnet  1502  of the intermediary rotor  1116  has moved clockwise to a greater extent than the permanent magnet  1504  of the output rotor  1134  has moved clockwise as can be appreciated from  FIG. 15(   b ). Referring to  FIG. 15(   c ), the shaft  1120  has rotated through  2781 ′ and the permanent magnet  1502  of the intermediary rotor  1116  has moved even further in a clockwise direction as compared to the permanent magnet  1504  of the output rotor  1134 . Referring to  FIG. 15(   d ), the shaft  1120  has rotated through  5661 ′ and the permanent magnet  1502  of the intermediary rotor  1116  has moved still further in a clockwise direction as compared to the permanent magnet  1504  of the output rotor  1134 . It can be appreciated that the gearing between the rotation of the input shaft  1120  and the rotation of the output rotor  1134  is extremely large to the extent that the output rotor  1134  has rotated about 5° as compared to the 5661° of rotation of the input shaft  1120 . Therefore, extremely high and precise gearing can be realised using embodiments of the present invention. 
         [0063]    Also, although the above embodiments have been described with reference to radial field rotors and rotation, embodiments can equally well be realised using axial field rotors and rotation as well as translators and translation, that is, the principles of embodiments of the present invention can be realised in the context of linear gears. 
         [0064]    The above embodiments have been described with reference to the inner rotor driving the outer rotors. However, it will be appreciated that embodiments can be realised in which an outer rotor drives an inner rotor thereby reversing the gear ratio.