Abstract:
The object of invention is the method of sizing, selection and comparison of linear and rotary electrical machines. According to the invention, the machines can be sized, selected and compared by new specific parameters: electromagnetic specific motor constant k EMS , specific motor constant k S , electromagnetic normal motor constant k EMN , normal motor constant k N , electromagnetic specific volume motor constant k EMSV , specific volume motor constant k SV , electromagnetic specific mass motor constant k EMSM  specific mass motor constant k SM  and relative continuous force F RC . These parameters slightly depend on machine overall dimensions but mostly depend on machine design.

Description:
[0001]    I, Alexei Stadnik, claim priority of provisional application No. 61/281,175 
     
    
     BACKGROUND OF THE INVENTION 
       [0002]    For sizing the electrical machines the parameter called “motor constant” is widely used (see, for example, “A Practical Use Of The Motor Constant c” by George A. Beauchemin—Motion Control, Jul. 25, 2009; “How to speed up dc motor selection”—Machine Design, Oct. 5, 2000; “Snake-oil specs spell trouble for motor sizing” by William A. Flesher—Machine Design, Jun. 4, 1998). The methods of sizing base on motor constant which highly depends on electrical machine overall dimensions. Therefore, the choice of electrical machines depends on electrical machine envelope. For example, if overall dimensions of one electrical machine are less than another electrical machine, it will have smaller motor constant. However, small electrical machine may be much better design than larger one. 
       SUMMARY OF THE INVENTION 
       [0003]    The invention provides a method of sizing, selection and comparison of electrical machines. The invented method use the new parameters called electromagnetic specific motor constant k EMS , specific motor constant k S , electromagnetic normal motor constant k EMN , normal motor constant k N , electromagnetic specific volume motor constant k EMSV , specific volume motor constant k SV , electromagnetic specific mass motor constant k EMSM , specific mass motor constant k SM  and relative continuous force F RC . These parameters slightly depend on electrical machine overall dimensions but mostly depend on machine design. Therefore, comparing the electrical machines with different specific parameters shows the difference in machine design. The method used new specific parameters has next main advantages: 
         [0004]    1. Comparison of electrical machines. For two or more electrical machines with different overall dimensions new specific parameters show the difference in electrical machine design. If new specific parameters of one electrical machine more than other it is mean that electrical machine have better design. It is very useful for comparison of different electrical machines from various sources. 
         [0005]    2. Selection of electrical machines. Selection of the source for electrical machine very often is not easy because each source provides data with different overall dimensions. It is very useful for engineers to solve this problem using new specific parameters that show the goodness of machine design for different electrical machines. To select source of electrical machine with better design the engineers can select source with better new specific parameters. 
         [0006]    3. Electrical machines sizing. Very often the required motor constant does not meet any existing electrical machine from various sources or electrical machine overall dimensions do not fit the required envelope. The estimation of new motor constant or overall dimensions can be done using new specific parameters. 
     
    
     
       DESCRIPTION OF THE FIGURES 
         [0007]    FIG.  1 —is the partial case of slotless, brushless flat linear machine with three phase winding. 
           [0008]    FIG.  2 —is flat linear machine, forcer length less than magnet track length 
           [0009]    FIG.  3 —is flat linear machine, magnet track length less than forcer length 
           [0010]    FIG.  4 —is balanced linear machine 
           [0011]    FIG.  5 —is U-shape linear machine, forcer length less than magnet track length 
           [0012]    FIG.  6 —is U-shape linear machine, magnet track length less than forcer length 
           [0013]    FIG.  7 —is tube linear machine, forcer length less than magnet track length 
           [0014]    FIG.  8 —is tube linear machine, magnet track length less than forcer length 
           [0015]    FIG.  9 —is frameless rotary machine 
           [0016]    FIG.  10 —is housed rotary machine 
       
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENT 
       [0017]    The motor constant is defined as 
         [0000]    
       
         
           
             
               
                 
                   
                     k 
                     M 
                   
                   = 
                   
                     
                       F 
                       C 
                     
                     
                       P 
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
         [0018]    Where F C  is continuous force produced by linear machine, P is continuous heat dissipation. 
         [0019]    Consider the partial case of linear machine ( FIG. 1 ). The machine is slotless, brushless and flat with three phase winding. The following assumptions have been made:
       Number of slots per pole and phase is 1 (q=1).   Magnetic field has only components on X and Z axis B X  and B Z : B Y =0   There is no magnetic field outside of interval from −w mag /2 to w mag /2 along Y axis   The B Z  is sinusoidal along X axis   The B Z  along Z axis inside of coil is not changed   The commutation is sinusoidal   Forcer length is less than magnet track length       
 
         [0027]    Taking into account the assumptions above, one can get the analytical equation for motor constant at 25° C.: 
         [0000]    
       
         
           
             
               
                 
                   
                     k 
                     M 
                   
                   = 
                   
                     
                       3 
                       π 
                     
                      
                     
                       
                         1 
                         
                           2 
                         
                       
                       · 
                       
                         
                           
                             B 
                             MAX 
                           
                           · 
                           
                             
                               k 
                               fil 
                             
                           
                           · 
                           
                             
                               k 
                               Width 
                             
                           
                           · 
                           
                             
                               k 
                               Height 
                             
                           
                         
                         
                           
                             
                               ρ 
                               25 
                             
                           
                           · 
                           
                             
                               1 
                               + 
                               
                                 k 
                                 epw 
                               
                             
                           
                         
                       
                       · 
                       
                         
                           W 
                           · 
                           H 
                           · 
                           τ 
                           · 
                           
                             N 
                             FPoles 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where B MAX —maximum value of magnetic field inside coil, 
         [0000]    
       
         
           
             
               
                 k 
                 Width 
               
               = 
               
                 
                   w 
                   mag 
                 
                 W 
               
             
             , 
             
               
 
             
              
             
               w 
               mag 
             
           
         
       
     
         [0000]    —magnet width (see  FIG. 1 ), 
         [0000]    
       
         
           
             
               
                 k 
                 Height 
               
               = 
               
                 
                   h 
                   c 
                 
                 H 
               
             
             , 
           
         
       
     
         [0000]    h c —coil height (see  FIG. 1 ), ρ 25 —conductors specific resistivity at 25° C., N FPoles —number of forcer poles, H and W—linear machine overall dimensions, τ—motor pole pitch (see  FIG. 1 ). The parameter k fil  in (2) is coefficient of filling factor. By definition, 
         [0000]    
       
         
           
             
               
                 
                   
                     k 
                     fil 
                   
                   = 
                   
                     
                       3 
                        
                       
                         
                           N 
                           0 
                         
                         · 
                         
                           S 
                           c 
                         
                       
                     
                     
                       
                         h 
                         c 
                       
                       · 
                       τ 
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where N 0  is number of coil turns per pole and phase, S C  is area of cross-section of conductor without insulation. 
         [0028]    Another coefficient k epw  is called the coefficient of end parts and defined as 
         [0000]    
       
         
           
             
               
                 
                   
                     k 
                     epw 
                   
                   = 
                   
                     
                       
                         l 
                         turn 
                       
                       - 
                       
                         2 
                         · 
                         
                           w 
                           mag 
                         
                       
                     
                     
                       2 
                       · 
                       
                         w 
                         mag 
                       
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
         [0029]    Here l turn  is length of one turn. 
         [0030]    So, for slotless brushless flat linear electrical machine the following relation between motor dimensions and motor constant: 
         [0000]      k M ˜√{square root over (N FPoles ·τ·W·H)}  (5)
 
         [0000]      k M ˜√{square root over (N FPoles ·V Pole )}  (6)
 
         [0000]    where V Pole  is the volume of machine per pole pitch length. 
       Linear Motors, Electromagnetic Specific Motor Constant 
       [0031]    The specific parameter k EMS  is called “electromagnetic specific motor constant”. In contrast to motor constant, it does not depend on motor length, slightly depends on electrical machine dimension and reflects only the design of electrical machine. For electrical machines with forcer length less than magnet track length, electromagnetic specific motor constant is defined as 
         [0000]    
       
         
           
             
               
                 
                   
                     k 
                     EMS 
                   
                   = 
                   
                     
                       k 
                       M 
                     
                     
                       
                         
                           N 
                           FPoles 
                         
                         · 
                         τ 
                         · 
                         W 
                         · 
                         H 
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where k M  is motor constant, N FPoles  is number of forcer poles, τ is motor pole pitch, H and W are linear machine overall dimensions. 
         [0032]    For electrical machines with magnet track length less than forcer length, 
         [0000]    
       
         
           
             
               
                 k 
                 EMS 
               
               = 
               
                 
                   k 
                   M 
                 
                 
                   
                     k 
                     
                       MT 
                        
                       
                           
                       
                        
                       F 
                        
                       
                           
                       
                        
                       poles 
                     
                   
                   · 
                   
                     
                       
                         N 
                         FPoles 
                       
                       · 
                       τ 
                       · 
                       H 
                       · 
                       W 
                     
                   
                 
               
             
             , 
           
         
       
     
         [0000]    where 
         [0000]    
       
         
           
             
               
                 k 
                 
                   
                     MT 
                      
                     _ 
                      
                     F 
                   
                    
                   
                     _ 
                      
                     poles 
                   
                 
               
               = 
               
                 
                   N 
                   
                     MT 
                      
                     Poles 
                   
                 
                 
                   N 
                   FPoles 
                 
               
             
             , 
           
         
       
     
         [0000]    N MTPoles  is number of magnet track poles. 
         [0033]    Some examples of linear electrical machines are shown on  FIG. 2  (flat linear machine, forcer length less than magnet track length);  FIG. 3  (flat linear machine, magnet track length less than forcer length);  FIG. 4  (balanced linear machine);  FIG. 5  (U-shape linear machine, forcer length less than magnet track length);  FIG. 6  (U-shape linear machine, magnet track length less than forcer length);  FIG. 7  (tube linear machine, forcer length less than magnet track length) and  FIG. 8  (tube linear machine, magnet track length less than forcer length). 
       Linear Motors, Specific Motor Constant 
       [0034]    The specific parameter k S  is called “specific motor constant”. In contrast to motor constant, it slightly depends on machine dimension and reflects only the design of electrical machine. For electrical machines with forcer length less than magnet track length, specific motor constant is defined as 
         [0000]    
       
         
           
             
               
                 
                   
                     k 
                     S 
                   
                   = 
                   
                     
                       k 
                       M 
                     
                     
                       
                         
                           L 
                           F 
                         
                         · 
                         W 
                         · 
                         H 
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
         [0035]    Here k M  is motor constant, L F  is forcer length, H and W are linear machine overall dimensions. For machines with magnet track length less than forcer length, 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       k 
                       S 
                     
                     = 
                     
                       
                         k 
                         M 
                       
                       
                         
                           k 
                           
                             
                               MT 
                                
                               _ 
                                
                               F 
                             
                              
                             
                               _ 
                                
                               length 
                             
                           
                         
                         · 
                         
                           
                             
                               L 
                               F 
                             
                             · 
                             W 
                             · 
                             H 
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                     
                 
               
             
           
         
       
     
         [0000]    where 
         [0000]    
       
         
           
             
               
                 k 
                 
                   
                     MT 
                      
                     _ 
                      
                     F 
                   
                    
                   
                     _ 
                      
                     length 
                   
                 
               
               = 
               
                 
                   L 
                   MT 
                 
                 
                   L 
                   F 
                 
               
             
             , 
           
         
       
     
         [0000]    L MT  is magnet track length. 
         [0036]    Some examples of linear electrical machines are shown on  FIG. 2  (flat linear machine, forcer length less than magnet track length);  FIG. 3  (flat linear machine, magnet track length less than forcer length);  FIG. 4  (balanced linear machine);  FIG. 5  (U-shape linear machine, forcer length less than magnet track length);  FIG. 6  (U-shape linear machine, magnet track length less than forcer length);  FIG. 7  (tube linear machine, forcer length less than magnet track length) and  FIG. 8  (tube linear machine, magnet track length less than forcer length). 
       Linear Motors, Electromagnetic Normal Motor Constant 
       [0037]    The specific parameter k EMN  is called “electromagnetic normal motor constant”. In contrast to motor constant, it does not depend on motor length. For electrical machines with forcer length less than magnet track length, electromagnetic normal motor constant is defined as 
         [0000]    
       
         
           
             
               
                 
                   
                     k 
                     EMN 
                   
                   = 
                   
                     
                       k 
                       M 
                     
                     
                       
                         
                           N 
                           FPoles 
                         
                         · 
                         τ 
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where k M  is motor constant, N FPoles  is number of forcer poles, τ is motor pole pitch. 
         [0038]    For electrical machines with magnet track length less than forcer length, 
         [0000]    
       
         
           
             
               
                 k 
                 EMN 
               
               = 
               
                 
                   k 
                   M 
                 
                 
                   
                     k 
                     
                       
                         MT 
                          
                         _ 
                          
                         F 
                       
                        
                       
                         _ 
                          
                         poles 
                       
                     
                   
                   · 
                   
                     
                       
                         N 
                         FPoles 
                       
                       · 
                       τ 
                     
                   
                 
               
             
             , 
           
         
       
     
         [0000]    where 
         [0000]    
       
         
           
             
               
                 k 
                 
                   
                     MT 
                      
                     _ 
                      
                     F 
                   
                    
                   
                     _ 
                      
                     poles 
                   
                 
               
               = 
               
                 
                   N 
                   
                     MT 
                      
                     Poles 
                   
                 
                 
                   N 
                   FPoles 
                 
               
             
             , 
           
         
       
     
         [0000]    N MTPoles  is number of magnet track poles. 
         [0039]    Some examples of linear electrical machines are shown on  FIG. 2  (flat linear machine, forcer length less than magnet track length);  FIG. 3  (flat linear machine, magnet track length less than forcer length);  FIG. 4  (balanced linear machine);  FIG. 5  (U-shape linear machine, forcer length less than magnet track length);  FIG. 6  (U-shape linear machine, magnet track length less than forcer length);  FIG. 7  (tube linear machine, forcer length less than magnet track length) and  FIG. 8  (tube linear machine, magnet track length less than forcer length). 
       Linear Motors, Normal Motor Constant 
       [0040]    The specific parameter k N  is called “normal motor constant”. In contrast to motor constant, it slightly depends on forcer length. For electrical machines with forcer length less than magnet track length, normal motor constant is defined as 
         [0000]    
       
         
           
             
               
                 
                   
                     k 
                     N 
                   
                   = 
                   
                     
                       
                         k 
                         M 
                       
                       
                         
                           L 
                           F 
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
         [0041]    Here k M  is motor constant, L F  is forcer length. For machines with magnet track length less than forcer length, 
         [0000]    
       
         
           
             
               
                 k 
                 N 
               
               = 
               
                 
                   k 
                   M 
                 
                 
                   
                     k 
                     
                       
                         MT 
                          
                         _ 
                          
                         F 
                       
                        
                       
                         _ 
                          
                         length 
                       
                     
                   
                   · 
                   
                     
                       L 
                       F 
                     
                   
                 
               
             
             , 
           
         
       
     
         [0000]    where 
         [0000]    
       
         
           
             
               
                 k 
                 
                   
                     MT 
                      
                     _ 
                      
                     F 
                   
                    
                   
                     _ 
                      
                     length 
                   
                 
               
               = 
               
                 
                   L 
                   MT 
                 
                 
                   L 
                   F 
                 
               
             
             , 
           
         
       
     
         [0000]    L MT  is magnet track length. 
         [0042]    Some examples of linear electrical machines are shown on  FIG. 2  (flat linear machine, forcer length less than magnet track length);  FIG. 3  (flat linear machine, magnet track length less than forcer length);  FIG. 4  (balanced linear machine);  FIG. 5  (U-shape linear machine, forcer length less than magnet track length);  FIG. 6  (U-shape linear machine, magnet track length less than forcer length);  FIG. 7  (tube linear machine, forcer length less than magnet track length) and  FIG. 8  (tube linear machine, magnet track length less than forcer length). 
         [0000]    Linear Motors, Electromagnetic specific volume motor constant 
         [0043]    The specific parameter k EMSV  is called “electromagnetic specific volume motor constant”. For electrical machines with forcer length less than magnet track length, electromagnetic specific volume motor constant is defined as 
         [0000]    
       
         
           
             
               
                 
                   
                     k 
                     EMSV 
                   
                   = 
                   
                     
                       k 
                       M 
                     
                     
                       
                         
                           N 
                           FPoles 
                         
                         · 
                         
                           V 
                           Pole 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where k M  is motor constant, N FPoles  is number of forcer poles, V Pole  is volume of machine per pole pitch length. For machines with magnet track length less than forcer length, 
         [0000]    
       
         
           
             
               
                 k 
                 EMSV 
               
               = 
               
                 
                   k 
                   M 
                 
                 
                   
                     k 
                     
                       
                         MT 
                          
                         _ 
                          
                         F 
                       
                        
                       
                         _ 
                          
                         poles 
                       
                     
                   
                   · 
                   
                     
                       
                         N 
                         FPoles 
                       
                       · 
                       
                         V 
                         Pole 
                       
                     
                   
                 
               
             
             , 
           
         
       
     
         [0000]    where 
         [0000]    
       
         
           
             
               
                 k 
                 
                   
                     MT 
                      
                     _ 
                      
                     F 
                   
                    
                   
                     _ 
                      
                     poles 
                   
                 
               
               = 
               
                 
                   N 
                   
                     MT 
                      
                     Poles 
                   
                 
                 
                   N 
                   FPoles 
                 
               
             
             , 
           
         
       
     
         [0000]    N MTPoles  is number of magnet track poles. 
         [0044]    Some examples of linear electrical machines are shown on  FIG. 2  (flat linear machine, forcer length less than magnet track length);  FIG. 3  (flat linear machine, magnet track length less than forcer length);  FIG. 4  (balanced linear machine);  FIG. 5  (U-shape linear machine, forcer length less than magnet track length);  FIG. 6  (U-shape linear machine, magnet track length less than forcer length);  FIG. 7  (tube linear machine, forcer length less than magnet track length) and  FIG. 8  (tube linear machine, magnet track length less than forcer length). 
       Linear Motors, Specific Volume Motor Constant 
       [0045]    The specific parameter k SV  is called “specific volume motor constant”. For electrical machines with forcer length less than magnet track length, specific volume motor constant is defined as 
         [0000]    
       
         
           
             
               
                 
                   
                     k 
                     SV 
                   
                   = 
                   
                     
                       k 
                       M 
                     
                     
                       
                         V 
                         SF 
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where k M  is motor constant, V SF  is volume of machine reduced to forcer length. For machines with magnet track length less than forcer length, 
         [0000]    
       
         
           
             
               
                 k 
                 SV 
               
               = 
               
                 
                   k 
                   M 
                 
                 
                   
                     
                       k 
                       
                         
                           MT 
                            
                           _ 
                            
                           F 
                         
                          
                         
                           _ 
                            
                           length 
                         
                       
                     
                     · 
                     
                       V 
                       SMT 
                     
                   
                 
               
             
             , 
           
         
       
     
         [0000]    where 
         [0000]    
       
         
           
             
               
                 k 
                 
                   
                     MT 
                      
                     _ 
                      
                     F 
                   
                    
                   
                     _ 
                      
                     length 
                   
                 
               
               = 
               
                 
                   L 
                   MT 
                 
                 
                   L 
                   F 
                 
               
             
             , 
           
         
       
     
         [0000]    L MT  is magnet track length, L F  is forcer length, V SMT  is volume of machine reduced to magnet track length. 
         [0046]    Some examples of linear electrical machines are shown on  FIG. 2  (flat linear machine, forcer length less than magnet track length);  FIG. 3  (flat linear machine, magnet track length less than forcer length);  FIG. 4  (balanced linear machine);  FIG. 5  (U-shape linear machine, forcer length less than magnet track length);  FIG. 6  (U-shape linear machine, magnet track length less than forcer length);  FIG. 7  (tube linear machine, forcer length less than magnet track length) and  FIG. 8  (tube linear machine, magnet track length less than forcer length). 
       Linear Motors, Electromagnetic Specific Mass Motor Constant 
       [0047]    The specific parameter k EMSV  called “electromagnetic specific mass motor constant”. For electrical machines with forcer length less than magnet track length, electromagnetic specific mass motor constant is defined as 
         [0000]    
       
         
           
             
               
                 
                   
                     k 
                     EMSM 
                   
                   = 
                   
                     
                       k 
                       M 
                     
                     
                       
                         
                           N 
                           FPoles 
                         
                         · 
                         
                           M 
                           Pole 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where k M  is motor constant, N FPoles  is number of forcer poles, M Pole  is machine mass per pole pitch length. For machines with magnet track length less than forcer length, 
         [0000]    
       
         
           
             
               
                 k 
                 EMSM 
               
               = 
               
                 
                   k 
                   M 
                 
                 
                   
                     k 
                     
                       MT_F 
                        
                       _poles 
                     
                   
                   · 
                   
                     
                       
                         N 
                         FPoles 
                       
                       · 
                       
                         M 
                         Pole 
                       
                     
                   
                 
               
             
             , 
           
         
       
     
         [0000]    where 
         [0000]    
       
         
           
             
               
                 k 
                 
                   MT_F 
                    
                   _poles 
                 
               
               = 
               
                 
                   N 
                   MTPoles 
                 
                 
                   N 
                   FPoles 
                 
               
             
             , 
           
         
       
     
         [0000]    N MTPole  is number of magnet track poles. 
         [0048]    Some examples of linear electrical machines are shown on  FIG. 2  (flat linear machine, forcer length less than magnet track length);  FIG. 3  (flat linear machine, magnet track length less than forcer length);  FIG. 4  (balanced linear machine);  FIG. 5  (U-shape linear machine, forcer length less than magnet track length);  FIG. 6  (U-shape linear machine, magnet track length less than forcer length);  FIG. 7  (tube linear machine, forcer length less than magnet track length) and  FIG. 8  (tube linear machine, magnet track length less than forcer length). 
       Linear Motors, Specific Mass Motor Constant 
       [0049]    The specific parameter k SM  is called “specific mass motor constant”. For electrical machines with forcer length less than magnet track length, specific mass motor constant is defined as 
         [0000]    
       
         
           
             
               
                 
                   
                     k 
                     SM 
                   
                   = 
                   
                     
                       k 
                       M 
                     
                     
                       
                         M 
                         SF 
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where k M  is motor constant, M SF  is machine mass reduced to forcer length. For machines with magnet track length less than forcer length, 
         [0000]    
       
         
           
             
               k 
               SM 
             
             = 
             
               
                 k 
                 M 
               
               
                 
                   
                     k 
                     
                       MT_F 
                        
                       _length 
                     
                   
                   · 
                   
                     M 
                     SMT 
                   
                 
               
             
           
         
       
     
         [0000]    where 
         [0000]    
       
         
           
             
               
                 k 
                 
                   MT_F 
                    
                   _length 
                 
               
               = 
               
                 
                   L 
                   MT 
                 
                 
                   L 
                   F 
                 
               
             
             , 
           
         
       
     
         [0000]    L MT  is magnet track length, L F  is forcer length, M SMT  is machine mass reduced to magnet track length. 
         [0050]    Some examples of linear electrical machines are shown on  FIG. 2  (flat linear machine, forcer length less than magnet track length);  FIG. 3  (flat linear machine, magnet track length less than forcer length);  FIG. 4  (balanced linear machine);  FIG. 5  (U-shape linear machine, forcer length less than magnet track length);  FIG. 6  (U-shape linear machine, magnet track length less than forcer length);  FIG. 7  (tube linear machine, forcer length less than magnet track length) and  FIG. 8  (tube linear machine, magnet track length less than forcer length). 
       Linear Motors, Relative Continuous Force 
       [0051]    For comparing the force characteristics of linear machines with different overall dimensions, the parameter F RC  called “relative continuous force” is introduced. For electrical machines with forcer length less than magnet track length, relative continuous force is defined as 
         [0000]    
       
         
           
             
               
                 
                   
                     F 
                     RC 
                   
                   = 
                   
                     
                       F 
                       C 
                     
                     
                       
                         L 
                         F 
                       
                       · 
                       W 
                       · 
                       
                         H 
                       
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where F C  is continuous force produced by linear machine, L F  is forcer length, H and W are linear machine overall dimensions. For machines with magnet track length less than forcer length, 
         [0000]    
       
         
           
             
               
                 F 
                 RC 
               
               = 
               
                 
                   F 
                   C 
                 
                 
                   
                     L 
                     MT 
                   
                   · 
                   W 
                   · 
                   
                     H 
                   
                 
               
             
             , 
           
         
       
     
         [0000]    where L MT  is magnet track length. 
         [0052]    Some examples of linear electrical machines are shown on  FIG. 2  (flat linear machine, forcer length less than magnet track length);  FIG. 3  (flat linear machine, magnet track length less than forcer length);  FIG. 4  (balanced linear machine);  FIG. 5  (U-shape linear machine, forcer length less than magnet track length);  FIG. 6  (U-shape linear machine, magnet track length less than forcer length);  FIG. 7  (tube linear machine, forcer length less than magnet track length) and  FIG. 8  (tube linear machine, magnet track length less than forcer length). 
       Rotary Motors, Specific Motor Constant 
       [0053]    For rotary machines, the specific parameter called “specific motor constant” is introduced. It is defined as 
         [0000]    
       
         
           
             
               
                 
                   
                     k 
                     S 
                   
                   = 
                   
                     
                       k 
                       M 
                     
                     
                       D 
                       · 
                       
                         
                           D 
                           · 
                           L 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where k M  is motor constant, L is length of rotary machine or length of winding of frameless rotary machine, D is outside diameter or dimension of square side of rotary machine. Some examples of rotary electrical machines are shown on  FIG. 9  (frameless rotary machines) and  FIG. 10  (housed rotary machines). 
       Examples of Use 
       [0054]    1. Linear motor, forcer is shorter than magnet track. The existing motor series is defined by height H, width W, different forcer lengths, poles numbers, and motor constants. We are going to keep existing cross-section and estimate k M     —     new  for required poles number N FPoles     —     req  or forcer length L F     —     req  other than existed; or estimate poles number N FPoles     —     new  or forcer length L F     —     new  for required k M     —     req  other than existed. 
         [0055]    1.1. Estimation of motor constant k M     —     new  for required poles number: N FPoles     —     req    
         [0056]    Step 1—find electromagnetic specific motor constant k EMS    
         [0057]    Step 2—find 
         [0000]    
       
         
           
             
               k 
               M_new 
             
             = 
             
               
                 k 
                 EMS 
               
               · 
               
                 
                   
                     N 
                     FPoles_req 
                   
                   · 
                   τ 
                   · 
                   W 
                   · 
                   H 
                 
               
             
           
         
       
     
         [0058]    1.2. Estimation of poles number N FPoles     —     new  for required motor constant: k M     —     req    
         [0059]    Step 1—find electromagnetic specific motor constant k EMS    
         [0060]    Step 2—find 
         [0000]    
       
         
           
             
               N 
               FPoles_new 
             
             = 
             
               
                 Integer 
                  
                 
                     
                 
                 [ 
                 
                   
                     
                       ( 
                       
                         
                           k 
                           M_req 
                         
                         
                           k 
                           EMS 
                         
                       
                       ) 
                     
                     2 
                   
                   · 
                   
                     1 
                     
                       τ 
                       · 
                       W 
                       · 
                       H 
                     
                   
                 
                 ] 
               
               + 
               1 
             
           
         
       
     
         [0061]    1.3. Estimation of motor constant k M new  for required forcer length: L F req    
         [0062]    Step 1—find specific motor constant k S    
         [0063]    Step 2—find 
         [0000]    
       
         
           
             
               k 
               M_new 
             
             = 
             
               
                 k 
                 S 
               
               · 
               
                 
                   
                     L 
                     F_req 
                   
                   · 
                   W 
                   · 
                   H 
                 
               
             
           
         
       
     
         [0064]    1.4. Estimation of forcer length L F     —     new  for required motor constant: k M     —     req    
         [0065]    Step 1—find specific motor constant k S    
         [0066]    Step 2—find 
         [0000]    
       
         
           
             
               L 
               F_new 
             
             = 
             
               
                 
                   ( 
                   
                     
                       k 
                       M_req 
                     
                     
                       k 
                       S 
                     
                   
                   ) 
                 
                 2 
               
               · 
               
                 1 
                 
                   W 
                   · 
                   H 
                 
               
             
           
         
       
     
         [0067]    2. Linear motor, forcer is shorter than magnet track. The existing motors have different heights, widths, forcer lengths and motor constants. We are going to estimate k M     —     new  for required overall dimensions L F     —     req , W req , H req  other than existed; or estimate overall dimensions L F     —     new , W new , H new  for required k M     —     req  other than existed. 
         [0068]    2.1. Estimation of motor constant k M     —     new  for required overall dimensions L F     —     req , W req , H req . 
         [0069]    Step 1—find specific motor constant k S    
         [0070]    Step 2—find 
         [0000]    
       
         
           
             
               k 
               M_new 
             
             = 
             
               
                 k 
                 S 
               
               · 
               
                 
                   
                     L 
                     F_req 
                   
                   · 
                   
                     H 
                     req 
                   
                   · 
                   
                     W 
                     req 
                   
                 
               
             
           
         
       
     
         [0071]    2.2. Estimation of overall dimensions L F     —     new , W new , H new  for required motor constant k M     —     req . 
         [0072]    Step 1—find specific motor constant k S    
         [0073]    Step 2—find 
         [0000]    
       
         
           
             
               
                 L 
                 F_new 
               
               · 
               
                 W 
                 new 
               
               · 
               
                 H 
                 new 
               
             
             = 
             
               
                 ( 
                 
                   
                     k 
                     M_req 
                   
                   
                     k 
                     S 
                   
                 
                 ) 
               
               2 
             
           
         
       
     
         [0074]    2. Linear motor, forcer is shorter than magnet track. The existing motors have different heights, widths, forcer lengths, continuous forces. We are going to estimate F C     —     new  for required overall dimensions L F     —     req , W req , H req  other than existed; or estimate overall dimensions L F     —     new , W new , H new  for required F C     —     req  other than existed. 
         [0075]    2.1. Estimation of continuous force F C     —     new  for required overall dimensions L F     —     req , W req , H req . 
         [0076]    Step 1—find relative continuous force F RC    
         [0077]    Step 2—find 
         [0000]    
       
         
           
             
               F 
               
                 C 
                  
                 
                     
                 
                  
                 _ 
                  
                 
                     
                 
                  
                 new 
               
             
             = 
             
               
                 F 
                 RC 
               
               · 
               
                 L 
                 
                   F 
                    
                   
                       
                   
                    
                   _ 
                    
                   
                       
                   
                    
                   req 
                 
               
               · 
               
                 W 
                 req 
               
               · 
               
                 
                   H 
                   req 
                 
               
             
           
         
       
     
         [0078]    2.2. Estimation of overall dimensions L F     —     new , W new , H new  for required continuous force F C     —     req    
         [0079]    Step 1—find relative continuous force F RC    
         [0080]    Step 2—find 
         [0000]    
       
         
           
             
               
                 L 
                 
                   F 
                    
                   
                       
                   
                    
                   _ 
                    
                   
                       
                   
                    
                   New 
                 
               
               · 
               
                 W 
                 new 
               
               · 
               
                 
                   H 
                   new 
                 
               
             
             = 
             
               
                 F 
                 
                   C 
                    
                   
                       
                   
                    
                   _ 
                    
                   
                       
                   
                    
                   req 
                 
               
               
                 F 
                 RC 
               
             
           
         
       
     
         [0081]    3. Frameless radial rotary motors. The existing motors have different diameters, lengths and motor constants. We are going to estimate k M     —     new  for required overall dimensions D req ,L req , other than existed; or estimate overall dimensions D new ,L new  for required k M     —     req  other than existed. 
         [0082]    3.1. Estimation of motor constant k M     —     new  for required overall dimensions D req ,L req . 
         [0083]    Step 1—find specific motor constant k S    
         [0084]    Step 2—find 
         [0000]    
       
         
           
             
               k 
               
                 M 
                  
                 
                     
                 
                  
                 _ 
                  
                 
                     
                 
                  
                 new 
               
             
             = 
             
               
                 k 
                 s 
               
               · 
               
                 D 
                 req 
               
               · 
               
                 
                   
                     L 
                     req 
                   
                   · 
                   
                     D 
                     req 
                   
                 
               
             
           
         
       
     
         [0085]    3.2. Estimation of overall dimensions D new ,L new  for required motor constant k M     —     req    
         [0086]    Step 1—find specific motor constant k S    
         [0087]    Step 2—find 
         [0000]    
       
         
           
             
               
                 D 
                 new 
               
               · 
               
                 
                   
                     L 
                     new 
                   
                   · 
                   
                     D 
                     new 
                   
                 
               
             
             = 
             
               
                 k 
                 
                   M 
                    
                   
                       
                   
                    
                   _ 
                    
                   
                       
                   
                    
                   req 
                 
               
               
                 k 
                 S