Methods, systems, and apparatus for reducing cogging torque in an electric machine

1. An interior permanent magnet machine comprising:

a rotor rotatable about a central machine axis and comprising a plurality of permanent magnet openings and a plurality of permanent magnets disposed therein; and

a stator disposed coaxially with said rotor, said stator comprising a plurality of stator teeth that define a plurality of stator slots therebetween, each of said plurality of stator teeth comprising a stator tooth tip, said stator tooth tip comprising an interior surface configured to facilitate reducing cogging torque, said stator separated from said rotor by a radial air gap between said rotor and said interior surface, said stator tooth tip comprising a circumferential first end, a circumferential second end, and a center section therebetween, said center section concentric about said rotor such that said radial air gap is constant within said center section, said interior surface of said circumferential first end and said circumferential second end are planar and are angled relative to said interior surface of said center section such that said radial air gap between said rotor and said stator tooth tip increases as a circumferential distance from said center section increases, said circumferential first end and said circumferential second end each comprising a planar circumferential end tip having a radial thickness, a radial distance is a distance between a maximum air gap at said planar circumferential end tip and said radial air gap within said center section, a first angle θ that is an angle measured from a first line to a second line, the first line extends between an axial center of the stator and an intersection between said center section and at least one of said circumferential first end and said circumferential second end and the second line extends between the axial center of the stator and a point equidistant from said planar circumferential end tip of at least one of said circumferential first end and said circumferential second end of the stator tooth and said planar circumferential end tip of at least one of said circumferential first end and said circumferential second end of an adjacent stator tooth, wherein an angle α of said interior surface of said circumferential first end relative to said interior surface of said center section is determined based at least partially on a frequency content of a cogging torque generated by the machine, said first angle θ, and said radial distance, wherein said radial thickness of said planar circumferential end tips are determined based at least partially on a frequency content of a torque wave generated by the electric machine.

2. A machine in accordance with claim 1 , wherein said circumferential first end located a first distance in a first circumferential direction from said center section of said stator tooth tip and said circumferential second end located a second distance in a second circumferential direction from said center section of said stator tooth tip, the first circumferential direction opposite to the second circumferential direction.

3. A machine in accordance with claim 2 , wherein said radial air gap is larger between said rotor and said circumferential first end of said stator tooth tip than said radial air gap is between said rotor and said center section of said stator tooth tip.

4. A machine in accordance with claim 2 , wherein said radial air gap is larger between said rotor and said circumferential second end of said stator tooth tip than said radial air gap is between said rotor and said center section of said stator tooth tip.

5. A machine in accordance with claim 1 , wherein the angle of said circumferential first end relative to said interior surface of said center section is determined to facilitate reducing an amplitude of the cogging torque at least one of a native frequency, a pole pass frequency, a slot order frequency, and a stator winding back EMF harmonic frequency.

6. A machine in accordance with claim 1 , wherein said stator tooth tip is configured to at least one of reduce and maintain an amplitude of the cogging torque at multiple harmonic frequencies.

7. A machine in accordance with claim 1 , wherein said permanent magnet openings are separated by rotor webs configured to facilitate reducing leakage flux through said rotor webs.

8. A machine in accordance with claim 7 , wherein a profile of said rotor webs is substantially trapezoidal.

9. A machine in accordance with claim 1 , wherein said plurality of permanent magnets are centered within said plurality of permanent magnet openings, said plurality of permanent magnet openings configured to restrict movement of said plurality of permanent magnets disposed therein.

10. A machine in accordance with claim 1 , wherein said radial distance is greater than said radial thickness of said planar circumferential end tips.