Switched Reluctance Electrical Motors

Abstract

A switched reluctance electrical motor comprises a rotor having a substantially circular cylindrical envelope with a diameter in a range of 50 mm to 300 mm and a length in the range of 20 mm to 250 mm. The rotor comprises a plurality of radially extending rotor teeth. A switched reluctance electrical motor further comprises a stator surrounding the rotor and comprising a plurality of stator poles. The rotor teeth are circumferentially-spaced apart from each other to define slots between adjacent teeth that, expressed in normalised angular and radial coordinates, have a cross-sectional profile transverse to an axis of rotation of the rotor, lying within a well-defined polygonal region.

Description

[0001] Switched Reluctance Electrical Motors Technical Field The present disclosure relates to rotors for switched reluctance electrical motors and switched reluctance electrical motors comprising such rotors. Background A switched reluctance motor (SRM) is a type of electrical motor. The stator of a SRM has windings to which switching currents are delivered to drive the motor, while the rotor has no windings or permanent magnets. It is known that conventional SRM designs suffer from torque ripple, i.e. a periodic increase or decrease in output torque as the rotor and motor shaft rotate. Torque ripple can be particularly strong and problematic in high-torque SRMs. Thus, there is a need to provide improved SRMs that deliver high torque and exhibit low torque ripple. Summary According to a first aspect of the invention, there is provided a switched reluctance electrical motor. The switched reluctance electrical motor comprises a rotor and a stator. The rotor has a substantially circular cylindrical envelope with a diameter in a range of 50 mm to 300 mm and a length in the range of 20 mm to 250 mm, and comprises a plurality of radially extending rotor teeth. The stator surrounds the rotor and comprises a plurality of stator poles. The rotor teeth are circumferentially-spaced apart from each other to define slots between adjacent teeth that, expressed in normalised angular and radial coordinates, have a cross-sectional profile transverse to an axis of rotation of the rotor, lying within a polygonal region having a corresponding Well-known text, WKT, representation of: POLYGON ((0.42242 -0.00244, 0.413150.00244, 0.377380.01113, 0.312210.06593, 0.27282 0.09056, 0.21648 0.12963, 0.11751 0.20699, 0.11737 0.24054, 0.11385 0.28571, 0.092770.34885, 0.079810.4127, 0.029340.55067, -0.028080.69306, - 0.04690.75422, -0.057490.80562, -0.059860.84371, -0.056290.87459, -0.04921 0.90035, -0.038590.92121, -0.002350.97436, -0.003381.00302, 0.004611.00413, 0.00934 0.974, -0.00366 0.94692, -0.02113 0.92063, -0.03404 0.89499, -0.03991 0.87057, -0.039910.83394, -0.032860.79365, -0.02230.75824, -0.005870.71551, 0.00822 0.68376, 0.06916 0.56091, 0.09615 0.49741, 0.11259 0.45224, 0.12671 0.40451, 0.13498 0.3602, 0.14082 0.33968, 0.14904 0.29816, 0.15258 0.26618, 0.16080.24664, 0.173710.22222, 0.210090.18559, 0.41080.08059, 0.480050.0464, 0.60211 0.12088, 0.65747 0.20825, 0.67606 0.23077, 0.70775 0.28083, 0.7208 0.31071, 0.73963 0.34012, 0.77844 0.39027, 0.80047 0.41514, 0.83099 0.47009, 0.86385 0.53724, 0.89319 0.61661, 0.91432 0.70208, 0.94603 0.86837, 0.96484 0.94541, 0.97672 0.97992, 0.99448 1.00208, 0.9972 1.00479, 1.00055 1.00474, 1.0045 1.0019, 1.00464 0.99842, 1.00117 0.99512, 0.98592 0.96825, 0.98005 0.94139, 0.9730.88034, 0.96830.78621, 0.963610.73126, 0.95420.67131, 0.94131 0.61905, 0.94012 0.60538, 0.93073 0.5602, 0.91667 0.50916, 0.91075 0.48072, 0.89076 0.41467, 0.86963 0.35973, 0.85197 0.32177, 0.83451 0.2906, 0.82746 0.26984, 0.81455 0.07204, 0.81559 0.0617, 0.81103 0.05861, 0.80013 0.04309, 0.796120.04157, 0.672760.01345, 0.612790.00367, 0.57054 -0.00122, 0.55634 - 0.00122, 0.54941 -0.00366, 0.48122 -0.00366, 0.42242 -0.00244)). Embodiments of the present invention may enable switched reluctance motors that deliver high torque and exhibit low torque ripple. Embodiments may achieve this by providing rotor teeth having a transverse cross-sectional profile that is irregular but which is selected from a well-defined group of profiles. Each of the rotor teeth may have a smooth, curved, radially outer surface and asymmetric side walls. Each rotor tooth may have one side wall of the asymmetric side walls that has an overhanging portion that projects from the rotor tooth in one rotational direction, for example a principal rotational direction. In some implementations, the number of rotor teeth may be 12 and the number of stator poles may be 18. In other implementations, the number of rotor teeth may be 4 and the number of stator poles may 6. In other implementations, the number of rotor teeth may be 31 and the number of stator poles may be 48. In other implementations, the number of rotor teeth may be 48 and the number of stator poles may be 72. In some implementations, the substantially circular cylindrical envelope of the rotor may have a diameter in a range of 100 mm to 200 mm. In some implementations, the rotor may have a length in the range of 80 mm to 150 mm (e.g. a length of 90 mm). In some implementations, the number of rotor teeth may be 12, the number of stator poles may be 18, and the substantially circular cylindrical envelope of the rotor may have a diameter of 90 mm. In some implementations, said polygonal region has a corresponding WKT representation of: POLYGON ((0.422420.00000, 0.413150.00244, 0.377380.01113, 0.312210.06593, 0.27282 0.09056, 0.21648 0.12963, 0.11751 0.20699, 0.11737 0.24054, 0.11385 0.28571, 0.092770.34885, 0.079810.4127, 0.029340.55067, -0.028080.69306, - 0.04690.75422, -0.057490.80562, -0.059860.84371, -0.056290.87459, -0.04921 0.90035, -0.038590.92121, -0.002350.97436, -0.003381.00000, 0.004611.00000, 0.00934 0.974, -0.00366 0.94692, -0.02113 0.92063, -0.03404 0.89499, -0.03991 0.87057, -0.039910.83394, -0.032860.79365, -0.02230.75824, -0.005870.71551, 0.00822 0.68376, 0.06916 0.56091, 0.09615 0.49741, 0.11259 0.45224, 0.12671 0.40451, 0.13498 0.3602, 0.14082 0.33968, 0.14904 0.29816, 0.15258 0.26618, 0.16080.24664, 0.173710.22222, 0.210090.18559, 0.41080.08059, 0.480050.0464, 0.60211 0.12088, 0.65747 0.20825, 0.67606 0.23077, 0.70775 0.28083, 0.7208 0.31071, 0.73963 0.34012, 0.77844 0.39027, 0.80047 0.41514, 0.83099 0.47009, 0.86385 0.53724, 0.89319 0.61661, 0.91432 0.70208, 0.94603 0.86837, 0.96484 0.94541, 0.97672 0.97992, 0.99448 1.00000, 0.9972 1.00000, 1.00055 1.00000, 1.0045 1.00000, 1.00464 0.99842, 1.00117 0.99512, 0.98592 0.96825, 0.98005 0.94139, 0.9730.88034, 0.96830.78621, 0.963610.73126, 0.95420.67131, 0.94131 0.61905, 0.94012 0.60538, 0.93073 0.5602, 0.91667 0.50916, 0.91075 0.48072, 0.89076 0.41467, 0.86963 0.35973, 0.85197 0.32177, 0.83451 0.2906, 0.82746 0.26984, 0.81455 0.07204, 0.81559 0.0617, 0.81103 0.05861, 0.80013 0.04309, 0.79612 0.04157, 0.67276 0.01345, 0.61279 0.00367, 0.57054 0.00000, 0.55634 0.00000, 0.549410.00000, 0.481220.00000, 0.422420.00000)). According to a second aspect of the invention, there is provided a system comprising the switched reluctance motor according to the first aspect and an inverter configured to provide electrical drive currents to the switched reluctance motor. The drive currents are periodic with a waveform that, expressed as normalised current values over one period, lies between a lower bound waveform and an upper bound waveform, the lower bound waveform and the upper bound waveform being defined by respective piecewise cubic functions ^^(t;{^^^^},{^^ji}) having respective breakpoints {^^^^} and respective coefficients {^^ji}, and the piecewise cubic functions satisfying: where the respective breakpoints {^^^^} and the respective coefficients {^^ji} for the upper bound waveform are: and the respective breakpoints {^^^^} and the respective coefficients {^^ji} for the lower bound waveform are: Brief Description of the Drawings Figure 1 is a cross-sectional view of a stator and a rotor according to an embodiment; Figure 2 illustrates a detail of the stator and rotor of Figure 1; Figure 3 illustrates a surface profile of a slot between two teeth of the rotor of Figures 1 and 2; Figure 4 illustrates upper and lower bounds of a drive current waveform for driving stator poles of the stator of Figure 1; Figures 5 illustrates simulated performance parameters; Figure 6 illustrates a conventional design; Figure 7 illustrates simulated performance parameters; Figure 8 shows a polygon defining a range of rotor slot profiles; and Figures 9A-11B illustrate further example rotors. Detailed Description A rotor of a conventional switched reluctance motor (SRM) is typically provided with regularly shaped teeth, i.e. with each tooth having a flat or arcuate tip opposed to the stator and having generally straight or arcuate sidewalls. To enable electrical motors that deliver high torque and exhibit low torque ripple, embodiments described herein propose a rotor with rotor teeth of an irregular shape. In other words, the teeth, and therefore the slots between adjacent teeth, have a transverse cross-sectional profile that is irregular but which is selected from a well-defined group of profiles. A particular example embodiment is described below with reference to Figures 1 to 4. The identified group of rotor slot profiles is then described in detail with reference to Figures 5, 7, and 8, and further example embodiments are described with reference to Figures 9A to 11B. With reference to Figure 1, a switched reluctance motor comprising an inner rotor 1 and an outer stator 2 will now be described. The rotor 1 is configured to be rotatable around a rotational axis of the rotor 1 (which is perpendicular to the plane of the cross-sectional view of Figure 1) and to cooperate with the stator 2, during use of the electrical motor, to provide a torque to a shaft 3. In particular, during use of the electrical motor, the rotor 1 may rotate counterclockwise (in the view of Figure 1). It is to be understood that the electrical motor may also be operated to rotate the rotor 1 clockwise (however, in general, the “irregular” rotor teeth described in this specification are predominantly beneficial for one rotational direction, i.e. to the “counterclockwise” operation of the electrical motor in the view of Figure 1). The stator 2 has a generally tubular shape and comprises an inner surface 4 which, in the cross-sectional view of Figure 1, has a generally circular shape. The stator 2 defines a plurality of circumferentially-spaced slots which extend from the inner surface 3 radially outwards to form a corresponding plurality of stator teeth 5. Respective wound field coils 6 are provided around the stator teeth 5. These coils 6 (e.g. copper coils) are conductive, and are configured to provide an electromagnetic field within the motor when driven by a current. By selectively driving the coils 6, the magnetic field within the motor may be rotated, causing the rotor 1 to be driven to rotate (Figure 1 includes crosses “x” and dots “.” within the coils 6 to respectively represent electrical currents coming out or going into the respective plane at any given moment in time, when the motor is in use). It is to be understood that the stator 2 may be a conventional SRM stator and the stator teeth may have a conventional, regular shape. In the embodiment of Figure 1, the stator 2 has 18 stator teeth, an outer radius of 132.0 mm and an inner radius of 91.0 mm (i.e. the inner surface 4 of the stator 2 has a radius of 91.0 mm in the view of Figure 1). The rotor 1 and the stator 2 have a length (also known as “stack length”) of 90 mm. The rotor 1 has a substantially circular cylindrical envelope and comprises a plurality of radially extending rotor teeth 7 circumferentially-spaced apart from each other to define slots 8 between adjacent teeth. The slots 8 extend co-axially along an otherwise circular cylindrical rotor 1. This means that the teeth 7 (and correspondingly the slots 8) have the same cross- sectional profile along the entire length of the rotor 1 (rotor skew can be implemented as required). Figure 2 is a close-up of a portion 9 of the SRM of Figure 1, showing one of the rotor teeth 7 in more detail (each rotor tooth 7 has generally the same shape). In the cross- sectional view of Figure 2, the teeth 7 have a generally smooth, curved radially outer surface 10 and “irregularly” shaped sidewalls 11,12. In the illustrated example, the sidewalls 11,12 of any given tooth are asymmetric, i.e. the “sidewalls” 11, 12 are not mirror images of each other (e.g. unlike the rightmost sidewall 12, the leftmost sidewall 11 comprises an overhanging portion 13). The rotor 1 and the stator 2 are dimensioned so that a small air gap is provided between the rotor teeth 7 and the stator 2. In the embodiment of Figure 1, the air gap is 1 mm. As the cross-section of the rotor 1 generally exhibits a nr-fold rotation symmetry (where nris the number of rotor teeth 7 or slots 8, i.e. nr=12 in the embodiment of Figure 1) around the rotational axis of the rotor 1, the surface profile 14 of the slots 8 may be expressed conveniently using a polar coordinate system centred on the rotational axis of the rotor 1, that is using polar coordinates comprising a radial coordinate defining a distance from the centre of the rotational axis and an angular coordinate defining an angle from a reference direction. Further, the rotor slot surface profile 14 may be expressed in normalised radial and angular coordinates. In particular, the “normalised radial coordinate rnorm” may be defined by where r0 is the distance from the rotational axis, rmin is minimum radius of the rotor 1 (i.e. the shortest distance from the rotational axis to the surface profile 14 of the rotor slot 8), and rmax is the rotor’s outer diameter, as illustrated in Figure 1. For the particular embodiment ofFigures 1 and 2, the values for minimum and maximum radius are 90.0^mm (the diameter of the shaft 3 of Figure 1 is 25 mm). Similarly, the “normalised angular coordinate φnorm” may be defined by where φ0 is the angle from a reference direction (by convention may increase in counter- clockwise direction), are angles indicating the end points of one example slot 8 in clockwise direction (as illustrated in Figure 1). It is to be understood that ^^^^^^^^^^is independent of the choice of the reference direction. The value of the normalised angular coordinate φnorm is not restricted to the interval between 0 and 1. For the particularembodiment of Figures 1 and 2, the values for 24.0 deg. The line 15 in the graph of Figure 3 represent the surface profile 14 of opposed sidewalls of a pair of adjacent rotor teeth 7 and which together define a slot 8, expressed in the above described normalised radial and angular coordinates. Other embodiments may comprise rotor slots having the profile 15 but having different values for the parameters rmin, rmax, φmin, and φmax(corresponding example embodiments are described below with reference to Figures 9A to 11B). The profile 15 can be expressed as a piecewise-linear function defined by a sequence of coordinates (i.e. each coordinate is an ordered pair comprising a normalised angular value and a normalised radial value). The profile 15 can readily be reconstructed from the sequence by linear interpolation between successive coordinates. The “Well-known text (WKT) representation” is a commonly employed representation that uses such sequences of coordinates to represent vector geometry objects (e.g. lines, polygons and other geometrical objects). A definition of the WKT format is included in the ISO-IEC 13249-3:2016 standard. The WKT representation of the profile 15, in ( coordinates, reads: LINESTRING (01, 0.003730.97444, -0.010240.94839, -0.02650.92328, -0.039430.89703, -0.04526 0.86906, -0.04594 0.84067, -0.04295 0.81238, -0.03731 0.78436, -0.02967 0.75667, -0.020480.72929, -0.010040.70223, 0.001410.67544, 0.01370.64889, 0.02612 0.6224, 0.03835 0.59585, 0.050340.56922, 0.06206 0.54252, 0.073450.51572, 0.08445 0.4888, 0.09494 0.46175, 0.104760.43453, 0.113640.40709, 0.121060.37937, 0.12719 0.35143, 0.132930.32342, 0.138280.29537, 0.143240.26726, 0.147810.23911, 0.15199 0.21092, 0.181150.18429, 0.210940.15813, 0.241460.13256, 0.272850.10771, 0.30526 0.08379, 0.338960.06109, 0.374220.0401, 0.411510.02164, 0.451330.00728, 0.49386 0.00013, 0.536990.00382, 0.576130.01912, 0.6066 0.04458, 0.627930.07579, 0.64228 0.10965, 0.6517 0.14468, 0.681360.16329, 0.70986 0.18309, 0.735620.20527, 0.74613 0.2338, 0.75517 0.26362, 0.76738 0.29267, 0.78068 0.32139, 0.7946 0.34991, 0.80891 0.37829, 0.82352 0.40657, 0.83777 0.43497, 0.85102 0.4637, 0.8633 0.49273, 0.87467 0.522, 0.88515 0.5515, 0.8948 0.58119, 0.90366 0.61105, 0.91176 0.64105, 0.91914 0.67118, 0.92583 0.70142, 0.93188 0.73176, 0.9373 0.76217, 0.94216 0.79266, 0.9471 0.82313, 0.952370.85357, 0.95810.88394, 0.96448 0.91424, 0.971870.94438, 0.98125 0.97415, 11). To generate torque, the coils 6 may be driven by temporally varying electrical currents (synchronised with the rotational position of the rotor 1). To this end, the SRM motor may comprise an inverter unit (not shown in Figure 1) configured to provide the corresponding drive currents. Any known and suitable inverter unit may be used. In general, the drive currents are periodic and may be defined by specifying a waveform (i.e. a temporal profile of current values over one cycle). More specifically, the SRM of Figure 1 can be operated as a 3-phase SRM. In this case, the coils 6 may be driven with respectively phase-shifted current waveforms (coils with the same reference numeral 6K, where { } in Figure 1 aredriven with the same phase-shifted waveform), i.e. coils 6A, 6B and 6C are driven by a corresponding current ^^^^(^^) (where ^^ is a rotation angle of the rotor 1) which satisfies The current waveform is continuous and periodic, for all angles ^^.Further, the nr=12-fold symmetry of the rotor 1 means that the current waveform repeats itself 12-fold over one revolution of the rotor 1, i.e. the period of the current waveform is given by (where f is the frequency of rotation of the rotor 1), and the rotational angle of rotor 1at time t is thereforeThus, the drive currents for the SRM of Figure 1 may be defined by a single-valued function specifying current values over one period ^ A suitable waveform for driving the motor of Figure 1 may be readily identified by performing an appropriate computational simulation of the motor during operation (conventional and commercially available simulation software may be used to perform such simulations). For example, such simulations may be used to iteratively improve / optimise an initial waveform “guess” (the simulation may take limitations of the inverter into account, for example, the inverter output may be limited to positive voltages / currents). In this way, example current waveforms for driving the motor of Figure 1 have been identified by the inventors and are now described with reference to Figure 4. The waveforms 41 and 43 shown in the graph of Figure 4 represent respectively an upper waveform bound and a lower waveform bound which specify normalised current values over one period T (i.e. the current values are normalised with respect to a nominal drive current value which may represent a drive current at full drive power, e.g.200 A). The SRM of Figure 1 may be operated with a drive current waveform which lies in between the upper and lower bounds 41, 43, i.e. The upper and lower bounds 41, 43 may be defined using a sequence of piecewise polynomial functions. More specifically, the upper and lower bounds 41, 43 may be expressed as piecewise cubic polynomials on an evenly spaced grid. Mathematically, piecewise cubic functions may be defined by breakpoints {^^^^}, coefficients {^^ji}, and the following construction: In this notation, the time “t” is the query point at which the piecewise cubic functions is evaluated, ^^^^ is i-th breakpoint in the piecewise cubic polynomial, and coefficients ^^jiare the four coefficients of the cubic polynomial for the i-th interval. In broad terms, to evaluate the piecewise cubic functions for time “t”, the function k(t) is used to find the index of the first breakpoint to the left of “t”. Then, the so-found index (say index “i”) is used to retrieve the numerical values for the relevant breakpoint ^^^^ and the four coefficients from a respective table. The respective breakpoints {^^^^} and coefficients {^^ji} for the upper bound and lower bounds 41, 43 of Figure 4 are given respectively in Table 1 and Table 2. Table 1:

[0002] Table 2:

[0003] Further, by simulating the operation of the switched reluctance motors, the inventors have found that the output of such motors can be improved over designs employing conventional regularly shaped rotor teeth, with the use of irregularly shaped teeth such as those illustrated in Figures 1 to 3. Figure 5 illustrates the results for a set of simulated tooth shapes, each point in the plot showing a result for a given tooth shape. In particular, Figure 5 illustrates the results of approximately one million different tooth shapes (the outer diameter rmax of the rotor and the number of rotor teeth nr is same for all shown data points). The large number of simulations performed and illustrated means that the data points in the Figure are not necessarily individually identifiable and the data set is shown more as shading where darker areas indicate a greater density of data points and lighter areas indicate a lower density of data points. More specifically, the simulation results shown in Figures 5 and 7 correspond to a predicted performance of the motor at an operating point of f = 2768 rotations per minute and a nominal drive current of Inom =200 A. The circle 51 in Figure 5 indicates a conventional design (illustrated in Figure 6) with regularly shaped rotor teeth which results in a predicted output torque of less than 205 N / m with a torque ripple of more than 14%. Figure 7 shows a group of data points 53 of extracted from Figure 5 which are identified as a group of optimal tooth shapes, specifically shapes resulting in a predicted output torque exceeding 227 N / m and torque ripple of less than 2% (when driven with a drive current waveform that lies within the upper and lower bounds 41, 43 described above). In particular, the circle 55 indicates the predicted performance for the tooth shape shown in Figures 1 to 3 which results in a predicted torque exceeding 228 N / m and torque ripple of less than 0.5%. The identified group of tooth shapes correspond to a set of slot profiles that, when expressed in normalised radial and angular coordinates, are contained within a region 57 enclosed by a polygon 59 as shown in Figure 8. Thus, the polygon 59 defines boundaries of the enclosed region 57 in the (normalised) polar coordinate system and embodiments lie within the region 57. In other words, all slot profiles that lie within the polygon 59 of Figure 8 exhibit desired torque and torque ripple characteristics, e.g. a predicted output torque exceeding 227 N / m and torque ripple of less than 2% (for f = 2768 rotations per minute and Inom=200 A). The polygon 59 can be defined by a sequence of normalised coordinates which represent the polygon’s vertices (each coordinate is an ordered pair comprising a normalised angular value and a normalised radial value. The polygon 59 can then be readily reconstructed from the sequence by linear interpolation between vertices (i.e. by connecting successive coordinates with a straight line). The WKT representation of the polygon 59, in (^^^^^^^^^^, ^^^^^^^^^^) coordinates, reads: POLYGON ((0.42242 -0.00244, 0.41315 0.00244, 0.37738 0.01113, 0.31221 0.06593, 0.272820.09056, 0.216480.12963, 0.117510.20699, 0.117370.24054, 0.113850.28571, 0.092770.34885, 0.079810.4127, 0.029340.55067, -0.028080.69306, -0.04690.75422, - 0.057490.80562, -0.059860.84371, -0.056290.87459, -0.049210.90035, -0.038590.92121, -0.002350.97436, -0.003381.00302, 0.004611.00413, 0.009340.974, -0.003660.94692, - 0.021130.92063, -0.034040.89499, -0.039910.87057, -0.039910.83394, -0.032860.79365, -0.02230.75824, -0.005870.71551, 0.008220.68376, 0.069160.56091, 0.096150.49741, 0.112590.45224, 0.12671 0.40451, 0.134980.3602, 0.140820.33968, 0.149040.29816, 0.15258 0.26618, 0.1608 0.24664, 0.17371 0.22222, 0.21009 0.18559, 0.4108 0.08059, 0.480050.0464, 0.602110.12088, 0.657470.20825, 0.676060.23077, 0.707750.28083, 0.7208 0.31071, 0.739630.34012, 0.778440.39027, 0.800470.41514, 0.830990.47009, 0.863850.53724, 0.893190.61661, 0.914320.70208, 0.946030.86837, 0.964840.94541, 0.97672 0.97992, 0.99448 1.00208, 0.9972 1.00479, 1.00055 1.00474, 1.0045 1.0019, 1.00464 0.99842, 1.00117 0.99512, 0.98592 0.96825, 0.98005 0.94139, 0.973 0.88034, 0.9683 0.78621, 0.96361 0.73126, 0.9542 0.67131, 0.94131 0.61905, 0.94012 0.60538, 0.930730.5602, 0.916670.50916, 0.910750.48072, 0.890760.41467, 0.869630.35973, 0.85197 0.32177, 0.83451 0.2906, 0.82746 0.26984, 0.81455 0.07204, 0.81559 0.0617, 0.811030.05861, 0.800130.04309, 0.796120.04157, 0.672760.01345, 0.612790.00367, 0.57054 -0.00122, 0.55634 -0.00122, 0.54941 -0.00366, 0.48122 -0.00366, 0.42242 - 0.00244)). Conventional and readily available software tools may be used to plot the polygon 59 or the profile 15 (e.g. online WKT plotting tools such as on https: / / www.geometrymapper.com) or to test whether a particular surface profile (expressed in (^^^^^^^^^^, ^^^^^^^^^^) coordinates) lies within the area enclosed by the polygon 59. For example, the C / C++ library “GEOS” may be used to read in i) a WKT representation of the surface profile in question and ii) the above WKT representation of the polygon 59, and to implement the test “GEOSContains” to determine whether the profile in question lies (completely) within the polygonal area 57. Referring back to the graph of Figure 7, the points 53 show predicted torque and toque ripple values for more than 10,000 different rotor slot profiles selected from the group of profiles defined by polygon 59. It is evident that each of these rotor slot profiles has improved predicted torque and torque ripple values (specifically output torque exceeding 227 N / m and torque ripple of less than 2% for the above described operating conditions) compared to the aforementioned conventional design. Various modifications will be apparent to those skilled in the art. As one example, in some embodiments, the slots between adjacent teeth may have a cross-sectional profile transverse to an axis of rotation of the rotor that, expressed in normalised angular and radial coordinates, lies within a polygonal region having a corresponding WKT representation of the polygon 59 described above, except that the values of the radial coordinate are clipped to be within the range 0 to 1 (e.g. a WKT representation of the corresponding polygonal region may be obtained by replacing, in the above-mentioned WKT representation of the polygon 59, negative values of the radial coordinate with zero and values of the radial coordinate greater than 1, with 1). As another example, in some embodiments, the number of rotor teeth and stator poles may be different than the corresponding numbers in the embodiment of Figure 1 (i.e. the rotor may have more than or less than 12 rotor teeth and the stator may have more than or less than 18 stator poles). By way of example only, the following provides indicative numbers of slots and poles for different SRM designs: three-phase SRMs: 6 / 2; 6 / 4; 6 / 8; 6 / 14; 12 / 8; 18 / 12; 24 / 16; four-phase SRMs: 8 / 6; 8 / 10; 16 / 12; 24 / 18; 32 / 24; five-phase SRMs: 10 / 4; 10 / 6; 10 / 8; 10 / 12; six-phase SRMs: 12 / 10; 12 / 14; 24 / 20; seven-phase SRMs: 14 / 10; 14 / 12; 14 / 16. It is noted in particular that it may be advantageous to have a number of rotor teeth (poles) Nr that is greater than the number of stator poles Ns. Such designs may have relatively high efficiency, output power and lower torque ripple. For example, Nr=2*Ns-2, so 10 / 6, 14 / 8, and 18 / 10 etc. Figures 9A to 11B illustrate variations of the above described rotor 1. Figure 9A is cross- sectional view of rotor 61 comprising four rotor teeth. The slots between adjacent teeth of the rotor 61 have the transverse cross-sectional profile 15 of Figure 4 (i.e. a profile that lies within the polygon 59). Figure 9B is a table containing numerical values of parameters specifying the rotor 61 such as the minimum radius rmin, the outer diameter rmax, the angles φminand φmaxindicating the end points of a slot, and the shaft diameter. Similarly, Figures 10A and 11A are cross-sectional views of rotors 63, 65 respectively comprising 31 and 48 rotor teeth (the slots between adjacent teeth of the rotors 63, 65 have the transverse cross-sectional profile 15 of Figure 4). The tables of Figures 10B and 11B respectively contain numerical values of parameters specifying the rotors 63, 65. It is to be understood that the rotors 61, 63, 65 of Figures 9A, 10A and 11A are intended to be used with appropriately configured stators (e.g. conventional stators that are appropriately dimensioned and that have a suitable number of stator poles). For example, a corresponding stator for use with rotor 61, 63 or 65 may have respectively 6, 48 or 72 stator poles. In embodiments, the substantially circular cylindrical envelope of the above described rotors may have a diameter in a range of 20 mm to 250 mm. In some embodiments, the substantially circular cylindrical envelope of the above described rotors may have a diameter in a range 100 mm to 200 mm. In embodiments, the length of the above described rotors may be in the range of 20 mm to 250 mm (i.e. the rotors may extend along their rotational axis for a length in the range of 20 mm to 250 mm). In some embodiments, the rotor length may be in the range of 80 mm to 150 mm. The above described embodiments may be manufactured with known manufacturing methods (e.g. die casting, machining, additive manufacturing, and the like). It is to be understood that, as in conventional SRMs, in the above described embodiments the rotors, e.g. rotor 1, may not comprise permanent magnets or windings but may comprise a ferroelectric material (e.g. iron).

Claims

Claims 1. A switched reluctance electrical motor comprising: a rotor having a substantially circular cylindrical envelope with a diameter in a range of 50 mm to 300 mm and a length in the range of 20 mm to 250 mm, and comprising a plurality of radially extending rotor teeth, and a stator surrounding the rotor and comprising a plurality of stator poles, and the rotor teeth are circumferentially-spaced apart from each other to define slots between adjacent teeth that, expressed in normalised angular and radial coordinates, have a cross- sectional profile transverse to an axis of rotation of the rotor, lying within a polygonal region having a corresponding Well-known text, WKT, representation of: POLYGON ((0.42242 -0.00244, 0.413150.00244, 0.377380.01113, 0.312210.06593, 0.27282 0.09056, 0.21648 0.12963, 0.11751 0.20699, 0.11737 0.24054, 0.11385 0.28571, 0.092770.34885, 0.079810.4127, 0.029340.55067, -0.028080.69306, - 0.04690.75422, -0.057490.80562, -0.059860.84371, -0.056290.87459, -0.04921 0.90035, -0.038590.92121, -0.002350.97436, -0.003381.00302, 0.004611.00413, 0.00934 0.974, -0.00366 0.94692, -0.02113 0.92063, -0.03404 0.89499, -0.03991 0.87057, -0.039910.83394, -0.032860.79365, -0.02230.75824, -0.005870.71551, 0.00822 0.68376, 0.06916 0.56091, 0.09615 0.49741, 0.11259 0.45224, 0.12671 0.40451, 0.13498 0.3602, 0.14082 0.33968, 0.14904 0.29816, 0.15258 0.26618, 0.16080.24664, 0.173710.22222, 0.210090.18559, 0.41080.08059, 0.480050.0464, 0.60211 0.12088, 0.65747 0.20825, 0.67606 0.23077, 0.70775 0.28083, 0.7208 0.31071, 0.73963 0.34012, 0.77844 0.39027, 0.80047 0.41514, 0.83099 0.47009, 0.86385 0.53724, 0.89319 0.61661, 0.91432 0.70208, 0.94603 0.86837, 0.96484 0.94541, 0.97672 0.97992, 0.99448 1.00208, 0.9972 1.00479, 1.00055 1.00474, 1.0045 1.0019, 1.00464 0.99842, 1.00117 0.99512, 0.98592 0.96825, 0.98005 0.94139, 0.9730.88034, 0.96830.78621, 0.963610.73126, 0.95420.67131, 0.94131 0.61905, 0.94012 0.60538, 0.93073 0.5602, 0.91667 0.50916, 0.91075 0.48072, 0.89076 0.41467, 0.86963 0.35973, 0.85197 0.32177, 0.83451 0.2906, 0.82746 0.26984, 0.81455 0.07204, 0.81559 0.0617, 0.81103 0.05861, 0.80013 0.04309, 0.796120.04157, 0.672760.01345, 0.612790.00367, 0.57054 -0.00122, 0.55634 - 0.00122, 0.54941 -0.00366, 0.48122 -0.00366, 0.42242 -0.00244)).

2. The switched reluctance motor according to claim 1, wherein the number of rotor teeth is 12 and the number of stator poles is 18.

3. The switched reluctance motor according to claim 1, wherein the number of rotor teeth is 4 and the number of stator poles is 6.

4. The switched reluctance motor according to claim 1, wherein the number of rotor teeth is 31 and the number of stator poles is 48.

5. The switched reluctance motor according to claim 1, wherein the number of rotor teeth is 48 and the number of stator poles is 72.

6. The switched reluctance motor according to any preceding claim, wherein the substantially circular cylindrical envelope of the rotor has a diameter in a range of 100 mm to 200 mm.

7. The switched reluctance motor according to any preceding claim, wherein the rotor has a length in the range of 80 mm to 150 mm.

8. A system comprising: the switched reluctance motor according to any preceding claim; and an inverter configured to provide electrical drive currents to the switched reluctance motor, wherein the drive currents are periodic with a waveform that, expressed as normalised current values over one period, lies between a lower bound waveform and an upper bound waveform, the lower bound waveform and the upper bound waveform being defined by respective piecewise cubic functions ^^(t;{^^^^},{^^ji}) having respective breakpoints {^^^^} and respective coefficients {^^ji}, and the piecewise cubic functions satisfying:where the respective breakpoints {^^^^} and the respective coefficients {^^ji} for the upper bound waveform areand the respective breakpoints {^^^^} and the respective coefficients {^^ji} for the lower bound waveform are.

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