Abstract:
An electric motor having reduced cogging torque. Motors have stator teeth separated by a slot, or space, which is filled with air and/or copper. The magnetic field present in the space is reduced, compared to that present in the iron cores. Thus, when the rotor rotates, it sees a changing magnetic field as it passes a tooth face, then a slot, and then a neighboring tooth face. This change can produce a force or cogging torque in the motor. The invention changes the geometry of the tooth face and slot to reduce variation of reluctance and increase overall flux of the stator.

Description:
BACKGROUND OF THE INVENTION  
       [0001]     In many types of electric motor, the rotor does not spin freely when electric power is absent. When one manually rotates such a rotor, one feels a succession of detents, as it were. Each detent represents a preferred position at which the rotor seeks to rest.  
         [0002]     Perhaps the simplest illustration of this phenomenon is found in the permanent-magnet stepper motor, a simplified version of which is schematically illustrated in  FIGS. 1-3 .  FIG. 1  illustrates four pairs of coils. A wire  6  connects coil-pair members  3 A and  3 B. The corresponding wires for the other pairs are not shown, to avoid clutter. Each coil is wound around a core  9 , which is constructed of iron, or other high-permeability material.  
         [0003]     When a current I passes through coil-pair  3 A and  3 B, a magnetic field B 1  is generated. In one mode of operation, the coil-pairs are energized in sequence, as shown in  FIG. 2 . That is, field B 1  is first created. Then field B 1  terminates, and field B 2  is generated. Then field B 2  terminates, and field B 3  is generated, and so on. A rotating magnetic field is created.  
         [0004]      FIG. 3  illustrates schematically the rotor of the motor, in the form of a single bar magnet  12 . The bar magnet  12  tends to align itself with the rotating magnetic field, not shown in  FIG. 3 , and assumes the successive positions  13 ,  14 ,  15 , and  15  indicated in  FIG. 4 .  
         [0005]     The net effect is that the rotating magnetic field of  FIG. 2  induces rotation in the bar magnet  12  of  FIG. 4 .  
         [0006]     However, when no current passes through any coil, the bar magnet  12  does not assume one of an infinite number of possible rest positions. Instead, the magnet  12  preferentially aligns itself with a pair of coils, as in  FIG. 3 . If one manually displaces the bar magnet  12  slightly from this rest position, and then releases the bar magnet  12 , the bar magnet  12  will return to its previous rest position.  
         [0007]     A simple explanation is that the bar magnet is attracted to the iron in the cores, and pulls itself toward the nearest iron available. The Detailed Description of the Invention, below, offers a more complex explanation, applicable to more general cases. This pull of the bar magnet into a preferred rest position is variously given the term cogging torque, detent torque, salient pole torque, reluctance torque, and possibly other terms.  
         [0008]     This torque also is present when the motor is operating. This torque is superimposed on the torque induced by the rotating magnetic field. Thus, each time the bar magnet  12  passes a pair of iron cores, such as pair  3 A and  3 B, a small cogging torque accelerates the bar magnet  12  slightly as the bar magnet  12  approaches the pair, and later another small cogging torque decelerates the bar magnet  12  slightly as the bar magnet  12  departs from that same pair.  
         [0009]     This repeated acceleration and deceleration creates vibration in the motor, which is not desirable in some situations. This vibration is particularly undesirable in electric power assistance steering (EPAS). One reason is that, since the motor drives an EPAS, and since the vibration actually takes the form of periodic accelerations and decelerations of the motor, the vibration can cause small pressure pulses in the EPAS. These pulses may be detected in the driver&#39;s hands on the steering wheel, and may cause annoyance. Also, depending on the particular hydraulic linkage existing between the EPAS and the forward wheels, the periodic pressure pulses can cause slight periodic changes direction of the vehicle, causing tire wear and wheel vibration.  
       OBJECTS OF THE INVENTION  
       [0010]     An object of the invention is to reduce cogging torque in electric motors.  
       SUMMARY OF THE INVENTION  
       [0011]     In one form of the invention, radial slot openings between adjacent stator teeth in an electric motor are changed to a non-radial configuration, to increase magnetic flux exiting the slots.  
         [0012]     These and other objects and advantages of the invention will be apparent from the following description, the accompanying drawings and the appended claims.  
     
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0013]      FIGS. 1-4  are highly simplified representations of a prior-art permanent magnet motor, such as a stepper motor;  
         [0014]      FIG. 5  illustrates one form of the invention;  
         [0015]      FIG. 6  illustrates how a magnetic flux is computed;  
         [0016]      FIG. 7  illustrates a discontinuous ring  21 F, and is used to show how gap G forces the magnetic flux to be smaller, compared with  FIG. 6 ;  
         [0017]      FIG. 8  illustrates an electrical model used to compute the magnetic flux of  FIG. 7 ;  
         [0018]      FIGS. 9 and 10  illustrate an electromagnet in two configurations;  
         [0019]      FIG. 11  is a schematic representation of stator teeth  70  and a rotor pole  78  in a prior-art electric motor;  
         [0020]      FIG. 12  is a representation of magnetic flux lines in the apparatus of  FIG. 11 ;  
         [0021]      FIG. 13  illustrates one form of the invention;  
         [0022]      FIG. 14  is a representation of magnetic flux lines in the apparatus of  FIG. 13 ;  
         [0023]      FIGS. 15-19  illustrate how magnetic flux density decreases as the two iron bodies  100  and  105  are rotated about point  125 ;  
         [0024]      FIG. 20  illustrates the magnetic flux of  FIG. 19 , but placed in the slot opening  74  in  FIG. 11 ;  
         [0025]      FIG. 21  illustrates the slot opening of the invention of  FIG. 13 ;  
         [0026]      FIG. 22  illustrates the magnetic flux of the type shown in  FIGS. 15-18 , placed in the slot of  FIG. 21 ;  
         [0027]      FIGS. 23-25  illustrate one conception of how one form of the invention can be constructed; and  
         [0028]      FIG. 26  illustrates reference lines and directions for one form of the invention.  
     
    
     DETAILED DESCRIPTION OF THE INVENTION  
       [0029]      FIG. 5  illustrates one form of the invention. An analysis will be given which explains, from one perspective, why this form of the invention reduces cogging torque. At the basis of this analysis are the following three concepts.  
         [0030]     One concept is that when a magnetic field is present in a system, forces arise which tend to move the components of the system into a configuration in which magnetic reluctance is reduced, and minimal if possible. For example, when an ordinary horseshoe magnet attracts an iron nail to its legs, the magnetic reluctance of the system is reduced when the nail is in contact with the legs of the magnet, compared to the reluctance when the nail is one foot away.  
         [0031]     Another concept is that when a magnetic field is created by an electric current in a system, forces are created which tend to move the components of the system into a configuration which increases, or maximizes, inductance of the system.  
         [0032]     A third concept is that in applying either of the above two concepts, if a change in configuration causes a relatively large change in inductance or reluctance, then relatively large forces are involved. Conversely, if a change in configuration causes a relatively small change in inductance or reluctance, then relatively small forces are involved. Thus, if two mechanically identical systems A and B are compared, and if movement in system A causes a small change in reluctance or inductance, compared with system B, then the forces in system A will be less than those in system B.  
         [0033]     These concepts will explain how reluctance and inductance of a generalized system change as internal parts of the system move, thereby creating forces. This analysis will then be applied to the device of  FIG. 5  to explain how movement of internal parts causes the forces responsible for cogging torque.  
         [0034]      FIG. 6  shows five copies  21 A- 21 E of a square iron ring having sides of length L. A coil  23  is wound around ring  21 A, and contains 10 turns.  
         [0035]     When a current I is generated in the coil  23 , a magnetic field H is generated, indicated in ring  21 B. Field H is called the magnetic field intensity, and can be calculated using the following equation. 
 
NI=4LH 
 
         [0036]     N is the number of turns,  10 ; I is the current; and H is the magnetic field intensity.  
         [0037]     In  FIG. 5 , L is indicated as the outer dimension of the square ring  21 A. For simplicity, length 4L is taken as the average path length traversed by a path running through the center of each leg. This path, of course, is slightly less than 4L.  
         [0038]     The expression NI represents the current I multiplied by the number of turns in the coil  23 . It should be observed that the current I passing ten times around ring  21 B is, for present purposes, identical in effect to a sheet current ten times as large, passing around ring  21 C once, as indicated by the single arrow  24  wrapped around ring  21 C.  
         [0039]     In this example, if I equals one amp, then H=10/4L. If L is one meter, then H=2.5. The units of H are amperes, or ampere-turns, per meter. Thus, H equals 2.5 ampere-turns per meter. H is also called a magneto-motive force, MMF.  
         [0040]     The H-field is accompanied by another field, the B-field indicated in ring  21 D. The B-field represents magnetic field density, as opposed to magnetic field intensity, represented by H.  
         [0041]     The B-field in ring  21 D in  FIG. 5  can be computed using the following equation: 
 
B=μ r μ 0 H 
 
 wherein 
        μ r  is the relative permeability of the iron of ring  21 D,     μ 0  is the permeability of vacuum, and     H is the magnetic field intensity, computed above.        
 
         [0046]     The constant μ r  for iron lies in the range of 4,000, and for modern high-permeability materials can be as high as one million, or more. For air, the relative permeability μ r  is very close to unity.  
         [0047]     A significant fact, which will be applied in greater detail later, can be observed here. If ring  21 D were constructed of air, the magnetic field density, B, would have a certain value, determined from the equation B=μ 0 H, since, as just stated, μ r  for air is unity for practical purposes.  
         [0048]     However, if ring  21 D were constructed of iron, or other high permeability material, the magnetic field density B can be 4,000 to one million times larger, because the following equation applies, and μ r  is far greater than unity. 
 
B=μ r μ 0 H 
 
         [0049]     Restated, placing iron, or other high-permeability material, into a region occupied by an H-field will increase the magnetic field density, namely, the B-field, and will increase the B-field by a factor of 100 to one 5000.  
         [0050]     Thus, if a system can rearrange itself so that more iron, or other high-permeability material, becomes positioned in a path occupied by an H-field, then a larger B-field will be created. Consequently, according to the two concepts described above, forces will be generated which promote this rearrangement, since the rearrangement (1) decreases reluctance, (2) increases inductance, or both.  
         [0051]     Stated more simply, if a system can rearrange itself to place high permeability material into a path occupied by an H-field, forces will arise which promote that rearrangement.  
         [0052]     Once B is computed, which gives the field density in terms of Webers per square meter, one computes the total magnetic flux.  
         [0053]     The computation, in mathematical form, is analogous to computing total force applied by a given pressure. For example, a pressure of 100 pounds per square inch may be present. If that pressure is applied to 9 square inches, then the total force applied is 100×9, or 900 pounds.  
         [0054]     Similarly, a given B-field may be X Webers per square meter. If that B-field is applied to 9 square meters, then the total flux (termed flux) is 9× Webers. In both cases, a parameter per unit area is present: pounds per square inch in the case of pressure, and Webers per square meter, in the case of the magnetic field. One finds the total value of the parameter (force or flux) by multiplying by the area over which the parameter is applied.  
         [0055]     In  FIG. 5 , ring  21 E is shown cut away. The area in question is labeled A. If B, shown in ring  21 D, is multiplied by A, the result is the total flux φ, in units of Webers (assuming B to be uniform across area A).  
         [0056]     A specific example will be given. This example will be contrasted with a slightly different example, given later for a discontinuous ring. For the first example, the following values are assumed: 
        μ r =4,000     μ 0 =4×PI× 10   −7  Henries per meter     A=0.01 square meter     Variable H was computed above, and is 2.5 ampere-turns per meter. B thus equals (4,000)(4×PI× 10   −7 )(2.5), or 0.0126 Webers/meter 2  Flux, φ, equals BA, or (0.0126)(0.01), or 0.000126 Weber.     Inductance equals φ/I, the ratio of the flux to the current producing it. Since the current is one amp, the inductance is 0.000126 Henry.        
 
         [0062]     It will now be shown how this value of inductance decreases when an air gap G is inserted into ring  21 F, as in  FIG. 7 . However, the computation will be done differently, in order to explain the concept of reluctance.  
         [0063]     The flux φ in the ring  21 F is analogous to electric current, and the system can be modeled as shown in  FIG. 8 . The parenthetical symbols refer to the electrical model. In the electrical model, where two resistors R G  and R L  are connected in series, the same current passes through both resistors.  
         [0064]     Similarly, in the magnetic case, (1) the iron of the ring and (2) the air in the gap G are connected in series, and the same flux φ passes through both.  
         [0065]     Mathematically, the two cases are identical. In the electrical case, Ohm&#39;s Law is obeyed: 
 
 V=I ( R   G   +R   Fe ) 
 
 wherein 
        V is the voltage,     I is the current, and     R G  and R Fe  are resistors.        
 
         [0070]     In the magnetic case, an equation of the same form is used: 
 
 MMF =φ( REL   Fe   +REL   G ) 
 
 wherein 
        MMF is the Magneto Motive Force,     φ is the flux, and     REL Fe  and REL G  are the magnetic reluctances of the iron and air gap G.        
 
         [0075]     For the magnetic case, each reluctance REL is computed using the following equation: 
 
REL=L/Aμ r μ 0  
 
 wherein 
        L is the length of the material (and not to be confused with inductance L),     A is the cross-sectional area, and     μ r  and μ 0  are defined above.        
 
         [0080]     Assume that the length of gap G in  FIG. 6  is 0.01 meter, or one centimeter. The reluctance of the gap G is thus 
 
0.01/(0.01)(1)(4×PI×10 −7 ) or 796,178. 
 
         [0081]     Reluctance has the units of inverse-Henry, H −1 .  
         [0082]     The reluctance of the discontinuous iron ring will be computed. Since the length of G is very small compared to 4L (ie, one centimeter compared to 400 centimeters, or 0.25 percent), the length of the iron will be treated as 4L for simplicity. The reluctance of the iron is thus 
 
4/(0.01)(4,000)(4×PI×10 −7 ) or 79,617H −1 . 
 
 The reluctance of the iron is about ten percent of that of the air gap. 
 
         [0084]     The total reluctance in  FIG. 7  is thus the sum, or 875,795H −1 . If the MMF is 10 ampere-turns, as above, then the flux is determined by the magnetic equivalent of Ohm&#39;s law: 
 
fluxΦ=10 amp-turn/(reluctance)=10/875,795=1.14×10 −5  Webers 
 
         [0085]     The inductance of the structure in  FIG. 7  is, as above, φ/I. Substituting numerical values gives an inductance of 1.14×10 −5  Henries. In contrast, the inductance of the solid ring of  FIG. 6  was computed as 0.000126 Henry, roughly ten times larger.  
         [0086]     Therefore, the Inventor points out that the inductance of the structure of  FIG. 7  is significantly less than that of  FIG. 6 , even though the amount of iron in the magnetic circuit is practically the same in both cases.  
         [0087]     The air gap G in  FIG. 7  is responsible for reducing the inductance. If one returns to the electrical model, it is clear that, if resistance R G  in  FIG. 8  is extremely large, compared with R Fe , the former dominates the total series resistance. The current is drastically reduced, compared to the case where R G  is absent, or small.  
         [0088]     Similarly, the reluctance of the air gap G is very large, because of the low permeability of the air, namely, μ 0 . The high reluctance of the air gap G dominates the total series reluctance of the air-plus-iron in  FIGS. 6 and 7 . The high reluctance of the air gap G drastically reduces the flux φ. The reduction in flux reduces the inductance.  
         [0089]     These facts can be used to explain how a force can be created in a magnetic circuit, according to the first two concepts outlined at the beginning of this discussion.  
         [0090]     As a simple example, if the discontinuous ring  21 F in  FIG. 7  were split along dashed line  43 , so that the two parts could pivot about point  44 , application of a current would cause the gap G to decrease. One reason, according to the concepts outlined at the beginning of this discussion, is that reducing the gap G will decrease the reluctance of the gap G, REL G . The system prefers to assume a configuration of reduced reluctance.  
         [0091]     Another reason is that the reduced reluctance increases the flux φ in the ring, for a given current. That increases inductance. The system prefers to assume a configuration of increased inductance.  
         [0092]     Another example will be given with reference to  FIG. 9 , which illustrates an electromagnet  45 , having a stationary iron section  50  and a freely movable iron section  55 , movable in the direction of arrows  60 . When a current I is applied, the movable iron section moves into the position shown in  FIG. 10 , closing the gap G of  FIG. 9 .  
         [0093]     This movement occurs because the configuration of  FIG. 9  has a relatively low inductance, similar to that of  FIG. 7 . From another point of view, the reluctance seen by the H-field (not shown) in traversing the path around the iron-plus-air-gap-G in  FIG. 9  is relatively high, because of the presence of the air gap G, in the same manner of  FIG. 7 .  
         [0094]     In contrast, the system of  FIG. 10  has a relatively low reluctance, because of the absence of the air gap, analogous to  FIG. 6 . The low reluctance of  FIG. 10  creates a higher flux φ, not shown, with a corresponding higher inductance, because inductance is defined as φ/I.  
         [0095]     Forces are created which rearrange the system into the configuration of  FIG. 10 , compared with  FIG. 9 , based on the first two concepts described at the beginning of this Detailed Description. The system seeks a configuration of high inductance, low reluctance, or both.  
         [0096]     This discussion will now apply the preceding principles to the present invention.  
         [0097]      FIG. 11  illustrates a partial cross-section of a motor in the prior art. Iron stator teeth,  70 , carrying coils  73 , are separated by slot opening  74 . In some embodiments, the slots  74  may contain copper current-carrying bars (not shown). A pole  78  of the rotor is shown.  
         [0098]      FIG. 12  shows flux lines superimposed on the structure of  FIG. 11 . The flux lines were drawn by the Inventor using flux mapping techniques.  
         [0099]      FIG. 11  shows the rotor pole  78  in a particular position, wherein the pole  78  is positioned directly across from slot opening  74 , or in a mid-slot position, between two teeth  70 . If the rotor pole  78  is rotated so that point P 1  becomes adjacent point P 2 , then pole  78  is no longer directly across from slot opening  74 . In this position, a larger flux (not shown) enters the tooth  70  from the pole  78 , compared with the situation of  FIGS. 11 and 12 .  
         [0100]     One explanation for this larger flux is that more iron is present in the path which the flux follows from pole  78  to the tooth  70 , because slot opening 74  is at the fringes of that flux when point P 1  is aligned with P 2 .  
         [0101]     With the larger flux present, a lower reluctance, a higher inductance, or both, are found, based on the principles described above.  
         [0102]     Thus, one or more forces exist tending to move the system from the configuration of  FIGS. 11 and 12  to the configuration where point P 1  is adjacent P 2 . These types of forces are responsible for the cogging torque.  
         [0103]      FIG. 13  illustrates one form of the invention, and  FIG. 14  shows superimposed flux lines. A comparison of  FIG. 14  with  FIG. 12  shows that an extra flux line exists in  FIG. 14 , and is labeled Path A. Thus, the reluctance of the system of  FIGS. 13 and 14 , in the position shown, is lower than that of  FIGS. 11 and 12 , in the position shown.  
         [0104]     Forces still exist in  FIGS. 13 and 14  tending to rotate the pole  83  so that P 3  is adjacent P 4 . However, these forces are reduced, compared with the corresponding forces in  FIGS. 11 and 12 . One reason is that the change in inductance (from the situation of  FIG. 13  to that in which P 3  is aligned with P 4  in the same Fig.) which accompanies this rotation of pole  83  is less under the invention, compared with the corresponding change in reluctance (from the situation of  FIG. 11  to that in which P 1  is aligned with P 2 ) in  FIGS. 11 and 12 .  
         [0105]     Stated another way, when P 3  becomes aligned with P 4  in  FIG. 13 , a change in reluctance occurs, compared with the configuration actually shown in that same Fig. However, that change is less than the corresponding change in  FIG. 11 . One reason is that the flux entering tooth  80  in  FIG. 14  is larger, compared with the corresponding flux in  FIG. 12 , because of the added flux indicated by Path A.  
         [0106]     Consequently, when the rotor pole  83  rotates so that points P 3  and P 4  become aligned in  FIG. 13 , the change in flux is not so great as the corresponding change in  FIG. 11 . Thus, the change in reluctance is not so great either. A simple numerical example will illustrate.  
         [0107]     Assume the values shown in Table 1:  
                       TABLE 1                               Flux Value       Case   Rotor Position   (arbitrary units)                   1       F       2       F + A       3   P1 and P2 aligned ( FIG. 11 )   X       4   P3 and P4 aligned ( FIG. 13 )   X                  
 
 The term “Flux Value” refers to the flux entering the rotor pole  78  or  83 . “A” represents the extra flux in Path A in  FIG. 13 . 
 
         [0109]     It is assumed that the position where P 1  and P 2  are aligned (case 3) experiences the same flux as where P 3  and P 4  (case 4) are aligned. This is considered reasonable, because the invention provides no significant structural change at those aligned points.  
         [0110]     The change in flux in the prior-art system is found by subtracting Case 3 from Case 1, and equals F−X. Under the invention, the corresponding change is Case 4-Case 2, and equals F−X−A.  
         [0111]     Under the invention, the change in flux is less by quantity A. Consequently, the cogging torque is correspondingly smaller under the invention, based on the third concept discussed at the beginning of the Detailed Description of the Invention.  
         [0112]     This reduction can be explained from another perspective. The pole  78  in  FIG. 11 , and the corresponding pole  83  in  FIG. 13 , create a given amount of flux. The quantity of flux can be computed from the equation 
 
Flux=MMF/Reluctance 
 
         [0113]     The magnet pole strength (MMF) in  FIGS. 13 and 14  are the same, compared with the corresponding magnet pole strength in  FIGS. 11 and 12 . As explained above, MMF in the equation immediately above is equal to the current, multiplied by some constants. For permanent magnet poles this may be considered to be an equivalent current. However, in  FIG. 14 , flux has increased, as indicated by the extra flux line following Path A.  
         [0114]     Thus, according to the equation immediately above, reluctance in  FIG. 13  has decreased, compared with  FIG. 11 . Thus, a given change in reluctance occurs when the rotor pole  83  in  FIG. 13  moves into the position of that Fig., from the position where points P 3  and P 4  were aligned. Another change in reluctance occurs when the rotor pole  78  in  FIG. 11  moves into the position of that Fig., from the position where points P 1  and P 2  were aligned.  
         [0115]     The former change in reluctance is less than the latter. Thus, by the third concept discussed at the beginning of the Detailed Description of the Invention, the forces involved in the former are less than in the latter. The invention reduces cogging torque.  
         [0116]     The invention can be explained from yet another perspective.  FIG. 15  illustrates two blocks of iron  100  and  105 . Block  105  corresponds to the inner surface  110  of a stator tooth in  FIG. 13 . Block  100  corresponds to the outer surface  115  of the rotor pole  83 . The solid lines in  FIG. 15  represent the B-field. The dashed lines represent equipotential surfaces of U, magnetic potential. Dashed box  120  represents a flux tube.  
         [0117]     Flux tubes are described, for example, in the text  Electromagnetics , by John D. Kraus (McGraw-Hill, 1992, ISBN 0-07-035621-1). This text is hereby incorporated by reference.  
         [0118]     If block  105  is progressively rotated about point  125 , as in  FIGS. 16-18 , the width W of the flux tubes near the separating ends  130  and  135  increases.  
         [0119]     One reason is that, under the rules for drawing flux tubes as outlined in the text just identified, the tubes are constructed of stacks of approximately square blocks, termed curvilinear squares. However, the number of squares in each stack remains approximately constant. Thus, if the length of a stack increases, as occurs in tube  8  during the change from  FIG. 15  to  FIG. 16 , the width of the stack must also increase.  
         [0120]     Thus, the sequence of  FIGS. 15-18  shows that the flux density in the tubes is decreasing, because flux density is inversely proportional to the width W of the tube.  
         [0121]     When block  105  reaches the position shown in  FIG. 19 , it corresponds roughly to a side face  140  of a prior-art slot opening  75  in  FIG. 11 . The width W in  FIG. 19  of the flux tube  2  is much greater, compared with that in  FIGS. 16-18 . The flux density is significantly reduced in  FIG. 19 .  
         [0122]      FIG. 19  is not drawn to scale, but is used to show the general principle that the flux density in a flux tube such as tube  2  is greatly reduced, compared with a tube in the corresponding position on block  100  in, say,  FIG. 17 .  
         [0123]      FIG. 20  is a representation of the flux tubes of  FIG. 19 , but positioned within a slot opening  74  of  FIG. 11 . Block  100  corresponds to the rotor pole  78  of  FIG. 11 . In contrast,  FIG. 21  shows a slot opening  150  according to the invention, adjacent the rotor pole  83  (corresponding to block  100  in  FIG. 18 .  FIG. 22  shows flux tubes of the type in  FIG. 16, 17 , or  18  superimposed on the slot opening of  FIG. 21 .  
         [0124]     Plainly, in  FIG. 22 , a larger number of flux tubes reach block/rotor-pole  83 , compared with  FIG. 20 . The larger flux means that reluctance of the system of  FIG. 22  is reduced, compared with that of  FIG. 20 .  
         [0125]     In  FIG. 22 , the flux lines reaching the rotor pole  83  originate from a high-permeability body, or bulge, 80 which is radially outward of the pole  83 . One definition of “high permeability” is that the relative permeability is 3,000 or greater. Another definition is that a “high permeability” material is suitable for use as transformer iron.  
         [0126]     In contrast, no such radially outward body of high permeability material exists in  FIG. 20 . Pure air is radially outward in  FIG. 20 .  
         [0127]     The invention can be characterized in yet another manner.  FIG. 23  slot opening  75  illustrates the slot opening  74  of  FIG. 11 . Slot opening  75  runs along a radial axis  180 , and has a radially inner mouth, exit, or opening, indicated by dashed box  175 .  
         [0128]     Under one form of the invention, a high-permeability body  190  is positioned within the slot opening  75 , as in  FIG. 24 . Body  190  is magnetically continuous with the adjacent material  192  of the stator phase. In addition, material  195  in  FIG. 25  may be removed, forming a slot opening  210  having generally parallel walls  215  and  220 .  
         [0129]     In one form of the invention, body  190  is also physically continuous with the adjacent material  192  of the stator tooth. That is, if two high-permeability bodies are physically adjacent, but separated by a very thin low-permeability sheet, it could be stated that they are nevertheless magnetically continuous. For example, if the air gap G of  FIG. 7  were extremely thin, so that it did not reduce the flux to any significant extent, it could be said that the ring  21 F is magnetically continuous.  
         [0130]     In contrast, ring  21 D, for example, is not only magnetically continuous, but physically continuous. No foreign material splits the ring, as in ring  21 F.  
         [0131]     Several definitions of magnetically continuous are the following. Preferably, the ring  21 F in  FIG. 7  is considered magnetically continuous if air gap G does not reduce the flux by more than 5 percent, compared with  FIG. 6 , given comparable dimensions and currents. In other embodiments, the percentage of five just stated can be increased to any values between 6 and 20.  
       Additional Considerations  
       [0132]     1. One view of the invention is that, when the rotor pole is aligned as in  FIG. 13 , that is, when the center of the rotor pole is aligned with the inner opening of the slot opening, the rotor is located at a mid-slot position. It is midway between adjacent stator teeth.  
         [0133]     2. A central axis  225  can be defined in  FIG. 26 , which is located mid-way in the slot opening  210 . If the slot opening is tapered, the axis  225  can follow the midline between the slot opening walls  215  and  220  in  FIG. 25 .  
         [0134]     The central axis  225  is non-radial, as is the slot opening itself. Further, a radially inner part  235  may cross a radius  230 . A radially outer part  240  may be spaced from the radius  230  by distance D.  
         [0135]     3.  FIG. 11  indicated that coil  73  surrounds a stator tooth  70 . In one form of the invention the stator coil may reside in a slot opening of  FIG. 13  instead of, or in addition to, a corresponding coil in  FIG. 13 .  
         [0136]     4. From one perspective, the invention provides a stator core  80  in  FIG. 13 , which has a compound inner face which includes two surfaces F 1  and F 2 . Surface F 1  faces radially inward, and follows a constant radius R 1 . Surface, or facet, F 2 , is located radially outward of the exit  175  in  FIG. 24 , and follows an increasing radius R 2  in  FIG. 13 .  
         [0137]     From another perspective, as one moves circumferentially, in the direction of arrow A 1  in  FIG. 13 , one encounters an inner face F 1  of constant radius R 1 , and then a facet F 2  of progressively increasing radius R 2 . Facet F 2  borders the slot opening  210  in  FIGS. 25 and 26 , which separates the cores  80  in  FIG. 13 .  
         [0138]     Numerous substitutions and modifications can be undertaken without departing from the true spirit and scope of the invention. What is desired to be secured by Letters Patent is the invention as defined in the following claims.