Abstract:
A method and apparatus for reducing or eliminating the effects of torque ripple and cogging torque and otherwise improving performance in an electromechanical machine such as a motor or generator. The rotor and/or stator is conceptually sectionalized and the sections spaced apart by amount sufficient to alleviate deleterious aspects of cogging torque and torque ripple. Positioning of the stator teeth or rotor magnets is determined based on the calculated spacing. Conceptual sections may be formed as physically individual segments. Unwound teeth may be disposed in end spaces between sections occupying less than the entire area of the end space.

Description:
RELATED APPLICATION DATA 
     This application is a divisional of U.S. Nonprovisional patent application Ser. No. 13/240,731, filed Sep. 22, 2011, entitled “Sectionalized Electromechanical Machines Having Low Torque Ripple and Log Cogging Torque Characteristics,” (which will issue as U.S. Pat. No. 8,912,704 on Dec. 16, 2014); and which application claims the benefit of priority of U.S. Provisional Patent Application Ser. No. 61/385,660, filed on Sep. 23, 2010, and titled “Sectionalized Electromechanical Machines Having Low Torque Ripple and Low Cogging Torque Characteristics.” Each of these applications is incorporated by reference herein in its entirety. 
    
    
     FIELD OF THE INVENTION 
     The present invention generally relates to the field of electromechanical machines. In particular, the present invention is directed to a sectionalized electromechanical machine having low torque ripple and low cogging torque characteristics. 
     BACKGROUND 
     Electromechanical machines, i.e., generators or motors, may be designed and configured in a wide variety of sizes—from, for example, relatively small diameter servomotors and automobile alternators to relatively large diameter hydroelectric turbines and powerful electric motors for ship propulsion or other uses. Given the size of some of these machines, there has been interest in creating machines from a compellation of smaller components, such as segmented rotors or stators, to allow for ease in manufacturing, installation, and maintenance. However, segmented designs to date have had a variety of drawbacks such as increased expense due to more difficult manufacture and maintenance, greater possibility of damage to coils, lost torque, and/or increase in torque ripple and cogging torque. One known way to simplify manufacture and maintenance, and to reduce cost, is to employ a less complex design. 
     One example of a less complex design for electromechanical machines is a single tooth, concentric wound design. In this design each tooth of the stator or rotor is wound with a single coil of wire concentrically around the tooth. While this design can be less expensive than more complex winding patterns, it can suffer from even more significant negative effects due to cogging torque and torque ripple. To address these problems, various tooth skewing schemes have been developed but these again raise cost of the design and can be incompatible with segmented designs due to the difficulty of assembly. 
     SUMMARY OF THE DISCLOSURE 
     The present disclosure is directed to an electromechanical machine having rotor poles and stator poles that cooperate to generate electricity from rotation or convert electricity to rotation, comprising inner and outer structures, at least one of said structures rotating relative to the other of said structures with the rotor poles disposed on the rotating structure and the stator poles disposed on the other structure, at least one of said structures including a plurality of sections, wherein the poles of said structure including the plurality of sections are arranged in sets of wound poles formed on teeth having a depth; a set of said poles is disposed in each one of said plurality of sections and each set has two end poles, one at each section end; the end poles are spaced from the section ends by an end space at each section end, the end space having a width of approximately the distance from the end pole winding to the section end, and a depth of approximately the depth of the wound pole teeth to define an end space area; and an unwound tooth is disposed in each said end space, the unwound teeth occupying not more than 90% of said end space area. 
     In another implementation, the present disclosure is directed to a stator for an electromechanical machine. The stator includes wound stator poles formed on teeth having a depth, wherein the wound poles are arranged a plurality of sections each comprising a set of wound poles with two wound end poles, one at each section end; the wound end poles are spaced from the section ends by an end space at each section end, the end space having a width of approximately the distance from the end pole winding to the section end, and a depth of approximately the depth of the wound pole teeth to define an end space area; and at least one unwound tooth is disposed in the end spaces defined by adjacent section end, the unwound teeth occupying not more than 90% of the total end space area of the adjacent sections. 
     In yet another implementation, the present disclosure is directed to a method for arranging wound stator poles in an electromechanical machine to reduce torque ripple, the wound poles being formed on teeth having a depth. The method includes conceptually dividing the stator poles into a plurality of sections to provide a sectionalized stator with each section having ends adjacent a next section; arranging the poles of the sectionalized stator in a plurality of sets of poles, each set disposed within one said section and having two end poles, one at each section end; and 
     positioning the poles within each set at a section pole pitch, the section pole pitch being a function of the frequency of the torque ripple harmonic to be at least reduced in operation of the machine; positioning the end poles of adjacent sections at a boundary pole pitch, the boundary pole pitch being a function of the section pole pitch; spacing the end poles from the section ends by an end space at each section end, the end space having a width of approximately the distance from the end pole winding to the section end, and a depth of approximately the depth of the wound pole teeth to define an end space area; and disposing an unwound tooth in each said end space, the unwound teeth occupying not more than 90% of said end space area. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       For the purpose of illustrating the invention, the drawings show aspects of one or more embodiments of the invention. However, it should be understood that the present invention is not limited to the precise arrangements and instrumentalities shown in the drawings, wherein: 
         FIG. 1  is a flow diagram illustrating a method of constructing a sectionalized electromechanical machine having low torque ripple and low cogging torque characteristics according to an embodiment of the present invention; 
         FIG. 1A  is another flow diagram illustrating a method of stator design in an alternative embodiment of the present invention. 
         FIG. 2  is a plan view of an electromechanical machine having a segmented stator according to an embodiment of the present invention; 
         FIG. 3  is a plan view of a portion of an electromechanical machine having a segmented stator according to an alternative embodiment of the present invention; 
         FIG. 4  is a plan view of an electromechanical machine having a sectionalized stator according to another embodiment of the present invention; 
         FIG. 5  is a plan view of an electromechanical machine having a segmented rotor design according to an embodiment of the present invention; and 
         FIGS. 6A-C  are partial plan views of alternative embodiments of the present invention employing varying end tooth configurations. 
     
    
    
     DETAILED DESCRIPTION 
     An electromechanical machine as referred to herein is a machine for the generation of electrical energy from mechanical work, e.g., producing electricity from the rotation of turbine blades, or the conversion of electrical energy to mechanical work, e.g., the use of electricity to rotate an object. Generally, the present disclosure is directed to an apparatus and method including sectionalized stator and rotor designs, and for reducing cogging torque and torque ripple in an electromechanical machine. Embodiments of the invention have particular application to single tooth concentric wound devices, although it may be applicable to other designs as well. 
     The terms sectionalizing and segmenting are used in many ways in the art with respect to electromechanical machines. For example, they can refer to constructing a rotor or stator of a machine in a plurality of arched or curved sections that can be installed into a structural frame to create a stator or rotor. Sectionalizing, as used herein, means that the rotor or stator are provided in conceptual sections, or, in other words, that the stator or rotor have repeated arched design portions corresponding to each section, but may be generally unitary structures, i.e., the stator or rotor sections are not made from separate pieces. In some embodiments, as described herein below, the sections can be formed as individual segments that are assembled to form a complete rotor or stator. 
     Sectionalizing a machine in accordance with embodiments of the present invention involves an analysis of the physical layout, e.g., position and numbers of stator and rotor poles, etc., and an electrical layout, e.g., the number of phases, number of circuits, and the coil arrangements, etc. Before describing possible exemplary embodiments of physical and electronic layouts for an electromechanical machine according to embodiments of the invention, terminology is defined and embodiments of exemplary design processes for such a machine are described with respect to  FIGS. 1 and 1A . 
     Those trained in the art of electric machine design will understand the concept of poles and phases. Poles on the stator and rotor are referred to, respectively, as stator poles and rotor poles. For simplicity and clarity of description, the illustrative embodiments discussed herein have three phases on the stator and two phases on the rotor. Other combinations may be used in practice and still fall within the invention. In the exemplary embodiments described herein, the rotor phases are north and south magnetic poles and the stator phases are single tooth, concentrically wound poles. As will be appreciated by persons of ordinary skill in the art, when using magnets for the poles it may be practical to have just two phases, but in a rotor having wound poles there could be any number of phases. If the stator has wound poles, it is possible to have any number of phases as well. 
     Again, for simplicity and clarity of the concepts and exemplary embodiments described herein, exemplary stator designs use single coil, single tooth concentric wound poles. This also may be a desirable structure for many large machines. However, as will be appreciated by persons of ordinary skill in the art, it is possible to have a single pole formed from a number of teeth and a number of coils. In such a case, the concepts discussed for the present invention are the same, but the description and details are more complex. Thus, for simplicity we use the terms rotor pole and stator pole to refer to either a permanent magnet pole or a wound coil on a tooth on the rotor or stator, respectively. Persons of ordinary skill in the art will be able to apply the teaching of the present invention to even more complex designs. 
     Certain aspects of the present invention relate to the spacing of the stator or rotor poles. For uniform application, spacing is measured between the pole center lines. For the purposes of the descriptions herein, the center of wound poles are at the center of the teeth around which the wire is wound. The center of the magnetic poles is at the center of the magnets. Previously, the centers of the poles of a machine were generally equally spaced around the machine. However, aspects of the present invention include a structure and method for using unequal spacing to lower torque ripple and cogging torque while minimizing the decrease in torque or power density resulting from the new arrangement. 
     With reference to  FIG. 1 , in embodiments of the present invention, as is typical of many design processes, designing an electromechanical machine begins with a determination of the desired electrical characteristics of the machine, which dictate the physical layout. For instance, the design for a generator may include, at step  14 , an initial determination of the basic configuration, such as the winding type, e.g., single tooth concentric wound, a number of phases, a desired voltage, and/or desired current for the power delivered. As another example, for a motor, step  14  may include an initial determination of the number of phases, a desired torque, or desired speed of the motor shaft. Other initial considerations may be taken into account as determined applicable by persons of ordinary skill in the art. 
     From the initial desired design of step  14 , physical and electrical characteristics may be determined. In one embodiment, a next step  16  would be the selection of the number of poles on each the rotor and stator. Generally, from an electrical layout perspective, and as would be readily understood by those skilled in the art, a next step, step  18 , would be a determination of the sectional structure, such as the number of sections and arrangement of poles, including the number of poles per series string and number of sections and circuits. Typically each pole is composed of coil wrapped around a core, frequently called a tooth, a plurality of the coils can be connected together in series to create a string. Each string is then connected in parallel to obtain the desired design result. In an exemplary embodiment, each string of poles should have the same number of poles and an equal number of end poles (poles at the end of a section) so that the strings share current properly. The width of the poles (see, e.g.,  FIG. 2 , reference numeral  144 ) is another physical parameter that may also be set at this time. 
     As one parameter, a determination of the number of conceptual sections may be included in step  18 . In an exemplary embodiment of design process  10 , the number of sections of either the rotor or the stator is a whole number multiple of the number of phases. The choice for the number of sections also may be influenced, in part, by the needs of manufacturing and assembly. Note that in embodiments wherein the winding configuration includes reversed coils, the number of stator phases may be twice the number of electrical phases. In general, persons of ordinary skill in the art may determine the number of sections based on their own applications and preferred parameters taking into account the teachings of the present invention. 
     In one exemplary embodiment, generally applicable to generators or motors that have tooth wound configurations that are sectionalized, step  18  may be broken down into a series of sub-steps for stator design as illustrated in  FIG. 1A . In order to provide balanced currents in each parallel circuit of a machine it may be desirable to have an equal number of end poles in each parallel circuit and in each electrical phase of the machine. Although three phase designs generally predominate, it is possible to design and use machines with any integer number of phases. But regardless of the number of phases to be applied in a particular embodiment, in one exemplary embodiment of the invention, the number of poles per section and the arrangement of the poles within the sections is selected with the following design requirements in mind: First, the number of poles per section should be an integer that is not a whole number multiple of the number of phases. Second, each series string of poles should have the same number of poles. Third, each series string of poles should include equal numbers of end poles. 
     These design requirements can be met by the following sub-steps as illustrated for step  18  in  FIG. 1A . In step  20 , the number of sections is set to an integer multiple of the number of phases. In step  22 , the maximum number of circuits is set as follows: if the number of poles per section is an even number, set the maximum number of circuits as equal to twice the number of sections divided by the number of phases; alternatively, if the number of poles per section is an odd number, set the maximum number of circuits as half the above number. Then, in step  24 , the number of circuits is selected from the possible number of circuits, which are the factors of the maximum number of circuits. It will be appreciated by persons of ordinary skill in the art that the steps illustrated in  FIG. 1A  are just one exemplary method for arriving at a workable arrangement of coils, sections and circuits, in the case that the characteristics of the end pole are different enough to cause possible imbalances in the phase or the circuit&#39;s currents, which in turn would cause complex torque ripple patterns. If the end pole characteristics are of no significance, then many other techniques may be employed for arriving at a workable arrangement without departing from the scope of the present invention as set forth in the appended claims. 
     One result of the above process is illustrated by the following non-limiting example. In an exemplary device having nine sections, with seven coils per section and three phases, the poles may be arranged as 
     |ABCABCA|BCABCAB|CABCABC|ABCABCA|BCABCAB|CABCABC|ABCABCA|BCABCAB|CABCABC| 
     where each letter A, B, or C represents a phase that is connected to each pole and each “|” represents a section boundary. Note that for each three sections there are seven of each letter ABC and two of each phase are next to a section boundary. Since in this design the number of coils per section is odd, the maximum number of circuits would be three. Following the design parameters set forth above, there are only two options: one or three circuits. 
     With reference again to  FIG. 1 , the overall design process according to an embodiment of the present invention may continue with step  26 . The ratio of the number of coils to the number of magnets (pole ratio) generates a characteristic torque ripple at one or more harmonic integer multiples of the fundamental frequency and a resultant cogging torque harmonic. The harmonic integer of the frequency that is to be eliminated or reduced is identified at step  26 . In general, and as would be apparent to one skilled in the art, an electromechanical machine generates numerous harmonics of the characteristic torque ripple frequency, each of which has a varying amount of influence on machine operation. Thus, the chosen harmonic torque ripple frequency to be canceled will generally be the one with the greatest influence on the operation of the machine, e.g., the dominant harmonic torque ripple frequency. 
     Torque ripple and cogging torque may then be reduced or eliminated by appropriate arrangement of the spacing of poles within sections and across the boundaries between adjacent sections in either the rotor or stator. In one embodiment of the invention, this can be accomplished at steps  30  and  34  based on only a small number of selected parameters: the number of poles, the number of phases, the number of sections and a balancing factor. 
     Using the above parameters, the pole pitch within each section is calculated in step  30  using Equation 1 as follows: 
     
       
         
           
             
               
                 
                   PPs 
                   = 
                   
                     
                       360 
                       P 
                     
                     × 
                     
                       ( 
                       
                         1 
                         - 
                         
                           
                             N 
                             × 
                             Ph 
                             × 
                             X 
                           
                           
                             H 
                             × 
                             P 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   [ 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     1 
                   
                   ] 
                 
               
             
           
         
       
     
     where:
         PPs is the section pole pitch;   P is the number of poles;   Ph is the number of phases;   H is the harmonic number to be eliminated;   N is the number of sections; and   X is a factor in the range of greater than zero to one.       

     When cogging torque is reduced by the sizing of the boundary pole pitch as described above, it can also be accompanied by a reduction in the torque density produced by the electromechanical machine. Such a reduction in torque density may or may not be desirable in particular, case-specific applications. Balancing factor X in Equation 1 allows such considerations to be taken into account to balance desired cogging torque reduction with the undesired loss of torque density when determining the section pole pitch in particular designs. In applications where torque ripple is a primary concern, X may be set as 1. However, in applications where reduction in torque density is a factor to be considered, the section pole pitch may be increased by selecting X at a value less than one. As can be seen from Equation 2 below, increasing the section pole pitch also reduces the boundary pole pitch to provide a more even spacing of the poles around the machine, which will provide increased torque density while lowering the reduction in cogging torque to a degree. A person skilled in the art will be able to select balancing factor X to achieve the best balance between parameters for a particular application based on the teachings contained herein. In one exemplary embodiment, balancing factor X is selected to be in the range of about 0.45 to about 0.95. In another embodiment, balancing factor X is in the range of about 0.60 to about 0.90. 
     Next, at step  34 , the boundary pole pitch, the center to center distance between adjacent end poles across section divisions, is calculated according to Equation 2 based on the calculated section pole pitch as follows: 
     
       
         
           
             
               
                 
                   PPb 
                   = 
                   
                     
                       ( 
                       
                         
                           360 
                           - 
                           
                             ( 
                             
                               P 
                               × 
                               PPs 
                             
                             ) 
                           
                         
                         N 
                       
                       ) 
                     
                     + 
                     PPs 
                   
                 
               
               
                 
                   [ 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     2 
                   
                   ] 
                 
               
             
           
         
       
     
     where:
         PPb is the boundary pole pitch;   PPs is the section pole pitch;   P is the number of poles; and   N is the number sections.       

     Using Equations 1 and 2 above, the sections of either a rotor or stator, wound or permanent magnet, may be arranged to reduce or eliminate a selected torque ripple harmonic. (In the case of a permanent magnet rotor the number of phases will be two—north and south). To construct the rotor/stator section(s) at step  38 , the poles are arranged within each section with their centers at the section pole pitch and the last pole at each end of the section spaced from the section edge by one-half the boundary pole pitch. Tooth/coil or magnet sizes as previously selected still apply with the only caveat being that they should not be so large as to overlap when spaced at the calculated section pole pitch. Embodiments and examples of the construction of rotor and stator sections, whether segmented or not, are described in more detail below. If the stator or rotor are individual segments, the segments may then be assembled at step  42 . It will be appreciated by persons of ordinary skill in the art that the design process steps illustrated in  FIGS. 1 and 1A  need not be followed linearly as depicted. Steps may be looped or reordered as appropriate to a particular design or process. 
       FIG. 2  shows an exemplary embodiment in which electromechanical machine  100  has a sectionalized and segmented stator design according to the present invention. Electromechanical machine  100  has, among other things, a rotor assembly  104  and a stator assembly  108 . In this embodiment, the wound stator poles of the stator assembly are arranged in sections  109 , wherein each section is an individual segment  124 A-C as described below. Typically, rotor assembly  104  and stator assembly  108  have a common central axis and reside within a housing (not shown). In general, rotor assembly  104  is rotated by a shaft  106 , or other mechanism known in the art, that is acted upon by an external force, e.g., the wind, or the rotor assembly may rotate the shaft by being acted upon by stator assembly  108 , e.g., by periodic excitement of the coils  136 . In the above described exemplary configuration, stator assembly  108  remains stationary. Stator assembly  108  includes a stator frame  120 , carrying plural stator segments  124 A-C that each include a stator back iron  128  and a plurality of inwardly directed teeth  132  (also sometimes called cores). Coils  136  are wound on teeth  132  to provide wound poles  144 . End poles  146  are located at the end of each section. In the case of wound poles  144 , an end space  148  is presented between the end coils on end poles  146  that represent a space where a skipped tooth or pole would have been disposed in a conventional design. 
     Stator back iron  128  may be fabricated out of materials known in the art, such as various metals, composites, or other materials with sufficient rigidity and electrical characteristics, with laminated or solid construction, to meet the desired design specifications. The teeth (cores) may be disposed on the back iron or fabricated as a unitary body with the back iron, and are also made of materials known in the art. 
     Rotor assembly  104  includes plural magnets  116  disposed at least about equally around the surface of rotor member  112 . Magnets  116  are typically, although not necessarily, permanent magnets composed of materials known in the art, such as composite metals or rare earth magnets. Rotor member  112  is generally annular and may be fabricated out of materials readily apparent to those skilled in the art such as various metals, composites, or other materials with sufficient rigidity to maintain the annular shape of rotor member  112  under rotational stress. The attachment of magnets  116  to rotor member  112  may be accomplished by means well known in the art. 
     With reference again to the design process as described above, as an example, assume machine  100  in  FIG. 2  is a three-phase machine with a pole ratio of stator poles  144  to magnets  116  of three to two; for example, twenty-four stator poles and sixteen rotor poles. A machine with this configuration might be expected to have a dominant torque ripple frequency at the sixth harmonic integer of the electrical frequency. To at least substantially cancel this frequency, the stator poles may be arranged according to Equation 1 above. Based on already determined design parameters, the number of poles (P), in this case stator poles, is 24, the number of phases (Ph) is 3, the harmonic to be eliminated (H) is 6 and the number of sections (N) is 3. Thus, for maximum cogging torque reduction with balancing factor X set as 1, the section pole pitch (PPs) within each section is calculated to be 14.0625 mechanical degrees. 
     The end poles  146  are then spaced from the section ends  126  by a distance equal to one-half the boundary pole pitch (PPb), calculated according to Equation 2 above. Based on the above parameters, the boundary pole pitch is 21.5625 mechanical degrees, one-half of which is 10.78125 mechanical degrees. If the width of the teeth  132  is hypothetically selected at 7.0 degrees, the slot between teeth in each section would be 7.0625 degrees, and the space at the end of each section, to the outside of the end teeth would be 7.28125 mechanical degrees. The unoccupied end space  148  between windings at the ends of segments or sections may or may not be occupied by unwound end teeth as discussed in more detail below. 
     Alternatively, as described above, if it were determined desirable to increase the torque density while lowering the reduction in cogging torque, balance factor X may be set at a value of less than one, for example at 0.85. With a balancing factor X of 0.85 and all other parameter being as established above, the section pole pitch is 14.2031 mechanical degrees. Using this modified section pole pitch, the boundary pole pitch is again calculated as above using Equation 2. (Note: setting the value of X to 0 yields a result of PPs=PPb=15 degrees, which is equally spaced poles as in a traditional machine.) 
     Although the configuration of electromechanical machine  100 , as shown in  FIG. 2 , indicates that stator assembly  108  with the previously specified components encircles rotor assembly  104  (described as an “inside rotating configuration”), other configurations of electromechanical machine  100  would be readily apparent to those skilled in the art. For instance, the rotor assembly may encircle the stator assembly (described as an “outside rotating configuration”), or the rotor assembly may include a plurality of cores and a plurality of coils as poles, while the stator assembly may include a plurality of magnets as poles. Described below are further exemplary embodiments with inside rotating configurations having stator assemblies including cores and coils, and rotor assemblies including magnets, although the invention is not limited by these exemplary configurations. In addition, more than one rotor assembly/stator assembly may be used in an electromechanical machine according to the invention, such as where rotor assemblies and stator assemblies exist in a stacked configuration (not shown). Those skilled in the art would readily appreciate that the embodiments and methodology taught herein are applicable to electromechanical machines utilizing multiple rotor assemblies and stator assemblies in various configurations. In another alternative, although only three segments are shown in  FIG. 2 , as would be apparent to those skilled in the art, more or fewer segments are possible. 
     Another exemplary embodiment is illustrated in  FIG. 3 , wherein the configuration of the rotor assembly is arranged according to the exemplary methodology to eliminate or reduce undesired torque ripple frequencies. In this example, machine  200  includes a rotor assembly  204  with a rotor member  224  comprising plural sections  224 A and  224 B and a plurality of poles  216  disposed there around. In this example there are sixteen poles and two rotor sections. Although machine  200  shows two distinct rotor segments forming sections  224 A and  224 B, in other embodiments the rotor sections may be conceptual sections that form an undivided rotor member. Stator assembly  208  may include a stator frame  220 , a stator back iron  228 , a plurality of teeth  232  and a plurality of coils  236  that surround teeth  232  to form stator poles  244 . Teeth  232 , generally, although not necessarily, extend radially toward the central axis of stator assembly  208  an amount sufficient to provide for coils  236 , but not so far as to interfere with the movement of rotor assembly  204 . 
     By way of non-limiting example, the design and configuration of a machine consisting of sixteen rotor poles  216  and twenty-four stator poles  244  may, if constructed by typical procedures used in the art, generate, as a predominant frequency, a sixth harmonic torque ripple frequency. In this example, in order to at least substantially cancel the sixth harmonic torque ripple frequency, the configuration of the rotor assembly is modified in accordance with the exemplary methodology described above as follows. 
     Applying Equation 1 to the rotor assembly and permanent magnets (rotor poles) (P=16, N=2, Ph=2, H=6), the pole pitch in each section (PPs) is calculated to be 21.5625 mechanical degrees if the balancing factor X is set at 1. Next, using Equation 2, the boundary pole pitch (PPb) is calculated to be 29.0625 mechanical degrees. Thus the centers of each pole at the end of each section are spaced at one-half the boundary pole pitch, or 14.53125 mechanical degrees from each section edge  226 . 
     In this case the magnet width corresponds to the width of the rotor poles. Thus, the distance between magnets follows from the section pole pitch. For example if the magnet width were selected as being 20 mechanical degrees, when spaced at a section pole pitch of 21.5625 mechanical degrees, the distance between magnets would be 1.5625 degrees. Using these parameters, the exemplary machine may be constructed and assembled as described at steps  38  and  42  of  FIG. 1 . 
     Electromechanical machine  300  as shown in  FIG. 4  illustrates a further alternative exemplary embodiment of the present invention in which stator assembly  308  is formed with sections  309  provided as individual segments. The design and configuration of stator assembly  308  and machine  300  is similar to that of machine  100 , as machine  300  includes a rotor assembly  304  that may include a rotor member  312  and a plurality of magnets  316 . Stator assembly  308  may include a stator frame  320 , and a plurality of stator segments  324  with a stator back iron  328 , a plurality of inwardly extending teeth  332  and a plurality of coils  336 . With the segmented design, section ends  326  also correspond to the physical ends of the segments. In this example, the basic structure is selected as a three-phase machine comprising eighteen separate stator segments  324 A-R (all segments not shown), each including ten poles for a total of 180 poles. Teeth  332  are selected with a width of 1.0 degrees. Teeth  332  and coils  336  are paired to form stator poles  344 . In this embodiment, the space between end poles  346  is occupied by unwound end teeth  348 , which are formed by two halves, one on each section/segment end  326 . 
     Again, using Equation 1 with balancing factor X set as 1, and assuming the sixth harmonic integer is to be eliminated or reduced, the section pole pitch (PPs) is calculated to be 1.90 mechanical degrees (P=180, Pu=3, N=18, H=6). Based on this determination, the center of the end pole  346  in each section is spaced 0.45 mechanical degrees from section end  336 , i.e. one-half the boundary pole pitch (PPb), as calculated with Equation 2. From these dimensions a person of ordinary skill in the art can construct an electromechanical machine according to this example that at least substantially eliminates torque ripple at the sixth harmonic frequency. As seen in  FIG. 4 , each segment  324  has end teeth  348 . When machine  300  is assembled, end teeth  348  from neighboring segments may combine to form a divided unwound, skip tooth. 
     Persons of ordinary skill in the art will be able to derive any number of working formulas based on the basic approach as taught above in order to provide methodology that may be more convenient to specific design considerations without departing from the spirit and scope of the present invention. For example, in one alternative derivation applicable in particular to designs wherein the arrangement of wound stator poles is to be adjusted, the spacing of end poles may be initially determined dividing 360 electrical degrees by the identified harmonic integer corresponding to the frequency to be reduced, which results in the spacing being approximately equal to the torque ripple frequency wave length. Alternatively, the spacing is determined in mechanical degrees according to Equation 3:
 
 Ss=X* 720/( H*Pr )  [Equation 3]
 
     where:
         Ss is the sector spacing in mechanical degrees for the stator (this parameter corresponds to the boundary pole pitch minus the sector pole pitch (PP b −PPs) as calculated above);   H is the harmonic integer associated with the harmonic torque ripple frequency to be reduced or eliminated;   Pr is the number of rotor poles; and   X is a factor in the range of greater than zero to one.
 
As described above, balancing factor X may be applied to adjust the balance between cogging torque reduction and loss of torque density.
       

     Using this derivation, the pole pitch on the stator may then be determined by subtracting the spacing, Ss, from the electrical degree span of the segment, i.e., angular measurement around the circumference of the machine in electrical degrees divided by the number of segments, and then dividing that result by the number of poles in the segment. In mechanical degrees, the pole pitch within each segment can alternatively be determined using Equation 4: 
     
       
         
           
             
               
                 
                   PPs 
                   = 
                   
                     
                       
                         ( 
                         
                           360 
                           / 
                           N 
                         
                         ) 
                       
                       - 
                       Ss 
                     
                     Ps 
                   
                 
               
               
                 
                   [ 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     4 
                   
                   ] 
                 
               
             
           
         
       
     
     where:
         PPs is the stator pole pitch within a sector in mechanical degrees;   N is the number of sections;   Ss is the sector spacing in mechanical degrees; and   Ps is the number of poles per stator segment.       

     Given this information, the stator segments may then be constructed. To do so, the distance, D, between the teeth of wound poles may be calculated using Equation 5:
 
 D=PP−Wt   [Equation 5]
 
     where:
         PP is the stator pole pitch in mechanical degrees;   Wt is the angular width of the stator tooth; and   D is the angular distance between stator teeth, e.g. the slots of a stator core.
 
With the distance, D, determined, a person of ordinary skill in the art may construct the physical or conceptual sections with the determined pole pitch and distance between poles to at least substantially eliminate the selected torque ripple harmonic.
       

       FIG. 5  shows a further alternative embodiment of machine  400  having a three stator pole to two rotor pole pole ratio, i.e., twenty-four coils to sixteen magnets. Machine  400  includes a rotor assembly  404  with a rotor member  412  that supports magnets  416 . Stator assembly  408  may include a stator frame  420 , and a plurality of stator sections  424  that include a stator back iron  428 , a plurality of teeth  432  and a plurality of coils  436  as previously described. However, in this case sections  424  are not formed as individual segments as the stator is a single piece construction. Teeth  432  and coils  436  are once again paired to form stator poles  442 . 
     As would be understood by those skilled in the art, in a typical prior art rotor, with this basic design, rotor poles  416  normally would be spaced about equally around the circumference at, in this example, about 22.5 mechanical degrees (180 electrical degree) intervals. The twenty-four stator poles  442  also would be typically spaced equally around the stator assembly  408 , at about 15.0 mechanical degree (120 electrical degree) intervals. However, in this exemplary embodiment of the invention, in order to at least substantially cancel a torque ripple frequency, skip teeth  440  are added and sized using Equations 3-5 as described above. For example, if one desires to cancel the sixth harmonic frequency to the greatest extent and thus sets the balancing factor X as 1, spacing Ss will be 7.5 mechanical degrees. Pole pitch, PP, will thus be 14.0625 mechanical degrees and the slot width, D, between teeth will be 7.0625 degrees, assuming a tooth width, Wt, of 7.0 degrees. Skip teeth  440  are disposed within the space between end poles  446 , but since the stator is of unitary construction, skip teeth  440  are also unitary. 
     The configuration of skip or unwound teeth, such as end teeth  348  in  FIG. 4  or skip teeth  440  in  FIG. 5 , can be adjusted to further optimize machine performance in further alternative embodiments. Exemplary designs of skip teeth in this regard are shown in  FIGS. 6A-C , illustrating three alternative embodiments each with magnets  516 , teeth  532 , coils  536  and back irons  528 . In the case of a full skip tooth, e.g., as in  FIG. 5 , the leakage inductance of the end poles will be somewhat higher than the other poles since the end slots are narrower than the others. Conversely in the case of a very short or no end tooth, as in  FIG. 2 , the leakage inductance is lower. Thus, an intermediate shape can be chosen that will provide similar leakage inductance closer to the internal sector poles. 
     A wide variety of shapes may be selected for skip or unwound end teeth to influence leakage inductance. For example,  FIG. 6A  shows a short skip tooth  540 A,  FIG. 6B  shows a tapered skip tooth  540 B, and  FIG. 6C  shows a thin skip tooth  540 C. These are just a limited sampling of the many shapes that may be selected by a person of ordinary skill in the art based on the teachings herein. Any shape can be tuned to provide a desired leakage inductance for the stator. Leakage inductance is just one of the parameters that affect the performance of this end coil. Another parameter is rotor magnet flux coupling. The rotor flux coupling can generally be controlled by adjusting the amount of the skip tooth that is near the rotor and how far it is from the rotor. The exact shape and dimensions of the skip tooth are functions of many generator design details and must be adjusted for each design. It will depend on choices of parameters such as the air gap between the rotor and stator, the tooth width, the slot width, and the desired performance at full or partial power as may be determined by a person of ordinary skill in the art (for a particular application) based on the teachings continued herein. 
     In one exemplary embodiment, skip teeth may be configured to employ features of each of the embodiments of  FIGS. 6A-C . In other words, the skip teeth may be short skip teeth with a trapezoidal shape wherein the base is narrower than the available end space. In one such exemplary embodiment the skipped teeth may have a depth from stator back iron  528  of about 125 mm as compared to the full depth of about 133 mm of wound teeth  532  (i.e. about 8 mm shorter), such that the depth of the skip tooth is about 94% of the depth of the wound teeth, a base that is 50% less than the available width of the end space, and, also taking into account the trapezoidal shape, the end space area between coils  536  of the end poles may be only approximately 68% filled by a skip tooth. In general the end space area may be not more than 90% occupied by an unwound tooth. In other embodiments the end space area may be between about 55% and about 80% filled by a skip tooth. When a short tooth type design is employed as illustrated in  FIG. 6A , the short skip tooth may have a depth of about 90% to about 98% of the wound teeth. Alternatively, the skipped teeth may be about 5-15 mm shorter than adjacent wound teeth. It will be appreciated by persons of ordinary skill in the art that the solid skip teeth as shown in  FIGS. 6A-C  may also be formed in parts when segmented designs are employed as illustrated, for example, by end teeth  348  in  FIG. 4 . For example, the skipped tooth may be added as a separate component. 
     Exemplary embodiments have been disclosed above and illustrated in the accompanying drawings. It will be understood by those skilled in the art that various changes, omissions and additions may be made to that which is specifically described herein without departing from the spirit and scope of the present invention. For example, as will be readily appreciated by persons of ordinary skill in the art, the decision to cancel the 6th harmonic in the foregoing examples is a design parameter based on specific applications utilized only for illustration purposes herein. Other individual harmonics may be targeted as determined for a particular design, without departing from the scope of the invention.