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http://ieeexplore.ieee.org/Xplore Analysis of the Torque Production Mechanism for Flux-Switching Permanent Magnet Machines James D. McFarland jdmcfarland@wisc.edu T. M. Jahns jahns@engr.wisc.edu Wisconsin Electric Machines & Power Electronics Consortium Dept. of Electrical and Computer Engineering University of Wisconsin – Madison Madison, WI USA Abstract—This paper investigates the principles underlying torque production in a flux-switching permanent magnet (FSPM) machine. Because the phase windings and permanent magnets in FSPM machines are both located on the stator, the torque production mechanism is not the same as for a conventional PM synchronous machine. Spatial harmonic analysis is applied to examine the frequency components present in the electric loading and magnetic loading of the machine. Since torque is proportional to the product of the electric and magnetic loading, understanding the source of the principal harmonics in these waveforms yields powerful insights into the components that result in torque production. The analysis is first presented for a specific FSPM machine (12-slot, 10-pole) and then extended to a general FSPM machine. The primary torque-producing harmonics in the airgap flux density waveform are found to be the heterodyned harmonics of the MMF produced by the stator magnets and the airgap permeance seen by the stator looking into the rotor. Analytical results are compared to results from finite element (FE) analysis and exhibit good agreement. I. INTRODUCTION Flux-switching permanent magnet (FSPM) synchronous machines have received considerable attention in the literature in recent years as an alternative to other types of PM machines. This type of machine was introduced in 1955 [1] and has since been examined by many researchers [2-9]. FSPM machines present a number of potential benefits compared to conventional PM synchronous machines with magnets located on the rotor. The rotor structure of a FSPM machine is relatively simple, consisting of a stack of steel laminations with salient poles. As a result, the rotor is mechanically robust and capable of high-speed operation without the magnet retention issues associated with surface PM (SPM) and interior PM (IPM) synchronous machines. In addition, the location of the magnets on the stator periphery makes them easier to cool using a stator cooling jacket or similar techniques. FSPM machines have demonstrated advantages in performance compared to an SPM machine for a high-speed generator application, due in significant part to mitigation of This work was supported by the Wisconsin Electric Machines and Power Electronics Consortium (WEMPEC) and this material is based on work supported by the Department of Energy Under Award Number: DEE0005573. Ayman M. EL-Refaie elrefaie@research.ge.com GE Global Research Center Niskayuna, NY USA magnet retention problems during high-speed operation [10]. In addition, FSPM machines have been designed to achieve attractive torque density values that are higher than other types of doubly-salient PM machines [11]. A lumped parameter magnetic circuit model has been developed for the analysis of FSPM machines, and the model has been extended to three dimensions to account for end effects [12]. Sizing equations and a general design procedure for FSPM machines have also been developed, including use of the dq-axis theory [13]. However, despite the recent increased interest in this machine type, the exact mechanism responsible for torque production in FSPM machines is not well described in the literature. The purpose of this paper is to present the results of an investigation into the principles underlying torque production in FSPM machines. Concepts of electric and magnetic loading in the spatial frequency domain provide powerful tools for gaining insights into the machine’s torque production mechanism. It will be shown that torque production in FSPM machines exhibits similarities to that in Vernier-type PM machines as well as magnetic gears, as documented in [14]. The goal is to build on an improved understanding of this torque production mechanism to gain a deeper appreciation of the key design factors that, in turn, will lead to better FSPM machines. II. FSPM MACHINE ANALYTICAL PROCEDURE For a PM synchronous machine without significant magnetic saliency (i.e., capability to produce reluctance torque), the electromagnetic torque is proportional to the product of the magnetomotive force (MMF) produced by the stator windings and the airgap flux density produced by the rotor permanent magnets (PMs) [15]. Since FSPM machines have negligible saliency, this torque production mechanism is appropriate for this analysis [16]. The airgap MMF produced by the stator windings is calculated as the product of the winding function for each phase and the current flowing through each phase [17]. The total stator winding MMF for the machine is the sum of the contributions of the individual phases and is calculated as a function of position. Spatial Fourier analysis is performed on the waveform to calculate the magnitudes of each spatial harmonic component of the stator winding MMF waveform. 978-1-4799-5776-7/14/ 31.00 2014 IEEE Digital Object Identifier 10.1109/TIA.2015.2411655 0093-9994 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications standards/publications/rights/index.html for more information.

This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication. The final version of record is available at http://dx.doi.org/10.1109/TIA.2015.2411655 Figure 2 – Cross-section view of a 12-slot, 10-pole FSPM machine. Figure 1 – Sketch of permanent magnets in an FSPM machine stator and the corresponding plot of normalized PM MMF vs. spatial angular position The MMF produced by the PMs is calculated in a similar manner, illustrated in Fig. 1. The magnet MMF plotted in Fig. 1 is normalized to simplify the spatial harmonic analysis, shown in more detail later in the paper. At the circumferential centerline of each magnet, the magnet MMF (MMFPM) is incremented or decremented by 2 if the north side of the magnet is facing in the positive or negative circumferential direction, respectively. The normalized airgap permeance for the machine as seen from the stator looking into the rotor can be formulated as unity at the location of a rotor tooth and zero away from a rotor tooth. This simplified and normalized airgap permeance allows for the analysis method to be generalized for varying rotor pole numbers and for a number of geometry parameters. It is worth noting that the rotor permeance is a function of both space and time since the rotor is free to move with time. The airgap magnetic flux density due to the stator PMs can be calculated as the product of the magnet MMF and the airgap permeance function. Since the airgap permeance function is time-varying, the airgap magnetic flux density will be time-varying as well. Finally, the electromagnetic torque produced by the machine is proportional to the product of the stator winding MMF and the airgap flux density contributed by the stator-mounted PMs. III. ANALYSIS OF TORQUE PRODUCTION MECHANISM IN FSPM MACHINES A cross-section view of a 12-slot, 10-pole FSPM machine is shown in Fig. 2. For each coil side in the stator slots, the associated phase and winding direction is indicated by the letter and the polarity sign in the cross-sectional view. The magnetization direction for each magnet is indicated by a thick red arrow in the figure (i.e., the arrowhead is a north pole). Figure 3 – Stator winding MMF vs. angular position for a 12-slot, 10-pole FSPM machine, including the two lowest-order spatial harmonic components (upper) and the associated spatial harmonic content of the stator winding MMF waveform (lower). The MMF produced by the three-phase stator winding in this machine is shown in the upper half of Fig. 3 for one time instant as a function of spatial location. The MMF waveform is modeled as a sequence of perfect steps because the winding MMF contributed by each slot has been idealized to be entirely concentrated at an angle corresponding to the center of each slot opening. The two lowest-order spatial harmonics of the MMF waveform (4th and 8th) are superimposed on this same spatial MMF plot. It is worth noting that one rotor pole pitch corresponds to 360 elec. deg. for this type of machine, so the x-axis of the MMF plot corresponds to a full mechanical rotation (10 x 360 3600 elec. deg). The magnitudes of the first fifty spatial harmonic components of the stator MMF waveform are plotted in lower half of Fig. 3. The primary spatial harmonic component of the stator winding MMF waveform is the 4th harmonic for the 12-slot, 10-pole machine. In general, for an FSPM machine of this type with non-overlapping end-turns, the primary spatial harmonic component is equal to the ratio of the number of stator slots to the number of phases (12/3 4). Non-triplen integer multiples of the primary spatial harmonic are also present (8, 16, 20, 28, etc). Copyright (c) 2015 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing pubs-permissions@ieee.org.

This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication. The final version of record is available at http://dx.doi.org/10.1109/TIA.2015.2411655 Figure 4 – Normalized magnet MMF vs. angular position for a 12-slot, 10pole FSPM machine, including plots of the two lowest-order spatial harmonic components (upper) and the associated spatial harmonic content of the magnet MMF waveform (lower). The airgap MMF produced by the magnets in the 12slot, 10-pole FSPM machine is shown in the upper half of Fig. 4 as a function of spatial angular position in the airgap. The two lowest-order spatial harmonics of the MMF waveform (6th and 18th) are superimposed on this same plot, and the magnitudes of the first fifty spatial harmonic components are plotted in the lower half of Fig. 4. As shown in Fig. 4, the primary spatial harmonic component of the magnet MMF is the 6th harmonic for the 12-slot, 10-pole machine. In general, for an FSPM machine of this type with circumferentially magnetized magnets alternating in direction, the primary spatial harmonic component is equal to half the number of slots (12/2 6). The simplified and normalized airgap permeance looking into the rotor for the 12-slot, 10-pole FSPM machine is shown in the upper half of Fig. 5 for one instant in time as a function of spatial angular position in the airgap. The two lowest-order spatial harmonics of the permeance waveform (10th and 20th) are also superimposed in this figure, and the magnitudes of the first 50 spatial harmonic components are plotted in the lower half of Fig. 5. Figure 5 indicates that the primary spatial harmonic component of the airgap permeance waveform is the 10th harmonic for the 12-slot, 10-pole machine. In general, for an FSPM machine of this type, the primary spatial harmonic component is equal to the number of salient rotor poles (10). The normalized airgap flux density due to the stator PMs is equal to the product of the magnet MMF waveform (Fig. 4) and the airgap permeance waveform (Fig. 5). The resulting spatial airgap flux density waveform for the 12slot, 10-pole FSPM machine is shown in the upper half of Fig. 6 for one time instant as a function of spatial angular position in the airgap. The 2 lowest-order spatial harmonics of the permeance waveform (4th and 6th) are also super- Figure 5 – Normalized airgap permeance looking into the rotor for a 12slot, 10-pole FSPM machine, including plots of the two lowest-order spatial harmonic components (upper) and the associated spatial harmonic content of the airgap permeance waveform (lower). Figure 6 – Normalized airgap flux density vs. angular position for a 12-slot, 10-pole FSPM machine, including plots of the two lowest-order spatial harmonic components (upper) and the associated spatial harmonic content of the airgap flux density waveform (lower). imposed in this figure, and the magnitudes of the first 50 harmonic components are plotted in the lower half of Fig. 6. Each harmonic component in the normalized airgap flux density waveform can be identified to be a frequency order equal to either a sum or a difference of one harmonic component from the magnet MMF waveform and one from the airgap permeance waveform. For example, the 4th spatial harmonic component of the airgap flux density waveform is due to the difference between the 10th harmonic in the airgap permeance waveform and the 6th harmonic in the magnet MMF waveform, and the 16th harmonic is due to their sum. This sum and difference frequency modulation results from taking the product of two sinusoidal waveforms Copyright (c) 2015 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing pubs-permissions@ieee.org.

This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication. The final version of record is available at http://dx.doi.org/10.1109/TIA.2015.2411655 Figure 7 – FE-calculated airgap flux density waveform for a 12-slot, 10-pole FSPM machine (upper) and the associated spatial harmonic content of the FE-calculated airgap flux density waveform (lower). of differing frequencies, manifesting itself as a heterodyning phenomenon. Next, it should be noted that both the airgap flux density (Fig. 6) and the stator winding MMF (Fig. 3) waveforms contain spatial harmonics of 4th and 16th order. These flux density and winding MMF harmonics rotate at the same temporal frequency and in the same direction, thereby interacting to produce non-zero average torque. In contrast, none of the other spatial frequencies match in both frequency and direction, so none of them contribute any average torque. In general, for an FSPM machine of this type, the torque production is due to two heterodyned frequencies that are the sum and difference of the number of rotor poles and half the number of slots (10 - 12/2 4; 10 12/2 16). These harmonic orders are also equal to two integral, non-triplen multiples of the ratio of the number of stator slots to the number of phases (12/3 4; 4 * 12/3 16). Generalizing, an FSPM machine with a candidate combination of slot, pole, and phase numbers can produce non-zero average torque only if integers exist that meet both of these criteria. The use of these relationships to develop a general expression for identifying the combinations of number of stator phases, stator slots, and rotor poles that can produce non-zero average torque is discussed in more detail in Section V. The 12-slot, 10-pole machine shown in Fig. 2 has been simulated using FE software under no-load conditions. The airgap flux density waveform from this simulation is shown in Fig. 7, along with the magnitude of the first fifty spatial harmonic components. The waveform from this FE analysis and the associated harmonic component values show reasonable agreement with the waveform shown in Fig. 6. Differences in the two airgap flux density waveforms are primarily due to fringing effects in the airgap near the salient rotor teeth that are not accounted for in the simplified analytical model for the airgap permeance function. Figure 8 – Graphic visualization of the definition of airgap surface occupation factor for a flux-switching PM machine. IV. DESIGN METHODS TO INCREASE TORQUE PRODUCTION CAPABILITY IN FSPM MACHINES It is desirable to design electric machines with the highest possible torque density. This allows for minimum volume and/or mass for a given application, which can be of critical importance in a variety of applications including electric traction motors. Design guidelines for machines help simplify this process and allow for quick selection of attractive topologies and geometry parameters. One such design guideline is the torque- producing winding factor(s) for a machine winding configuration. These factors largely dictate the effective armature (stator) MMF, or electric loading, in the machine and higher factors generally lead to increases in torque density. A second design factor that has a significant impact on the torque production capabilities of ac machines with singly- and doubly-salient topologies is the airgap surface occupation factor (ASOF), which will be explored in this section. The ASOF is defined in Fig. 8 as the ratio of the airgap surface occupied by rotor teeth to the total airgap surface area. This factor is closely linked to the concept of magnetic loading, which reflects how much useful magnetic flux enters the machine’s airgap. Results from the previous sections show that the torque production capability of a flux-switching PM machine is heavily dependent on the MMF produced by current flowing in the stator windings (Fig. 3), the MMF at the airgap from the permanent magnets (Fig. 4), and the permeance waveform looking into the rotor from the stator (Fig. 5). The MMF from current in the stator windings is fixed primarily by the geometry of the machine stator and is ultimately limited by thermal constraints imposed by the conductive losses caused by those same currents. Similarly, the MMF Copyright (c) 2015 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing pubs-permissions@ieee.org.

This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication. The final version of record is available at http://dx.doi.org/10.1109/TIA.2015.2411655 30.3% increase 12s/10p 12s/14p 12% increase FE analysis uses 12s/10p with: Constant widths of stator teeth, stator slots, and stator magnets Variable rotor teeth width Figure 9 – Airgap magnetic flux density factor KB,gap plotted as a function of airgap surface occupation factor (ASOF). from the permanent magnets is fixed by the stator geometry and is limited by the material properties and geometry of the magnets. The possibilities for raising the torque production by increasing either of these quantities are inherently limited by material properties and thermal constraints. In contrast, the permeance waveform of the rotor is not necessarily set by the stator geometry and is not subject to the same material limitations, making it possible to consider using the rotor permeance characteristics to increase the machine’s torque production capability. The ASOF is closely related to the airgap permeance waveform, providing a figure of merit that can be related to the machine’s torque production capabilities. The airgap magnetic flux density factor (KB,gap) is defined to be the magnitude of the primary harmonic of the airgap permeance that results in torque production (the 10 th harmonic in a 12-slot, 10-pole machine, and the harmonic of order equal to the number of rotor poles in general). The value of KB,gap is heavily dependent on the airgap surface occupation factor, as evident in the following expression: ଶ ܭ ǡ ൌ ݊݅ݏ ሺߨ ή ܨܱܵܣ ሻ గ (1) The airgap magnetic flux density factor is important to torque production and the relationship in (1) is plotted in Fig. 9. Readers familiar with signal analysis will recognize this as the Fourier transform of the boxcar function. In terms of machine analysis, this means the primary harmonic component of the airgap permeance waveform will be larger as the ASOF nears 0.5. Thus, a larger rotor tooth span up to the limit of 0.5 is beneficial for the airgap permeance. A span larger than half the rotor pole pitch would be unlikely for a FSPM machine of this type. The permeance waveform and harmonic content was shown in Fig. 5 for the 12-slot/10-pole machine and the 10th harmonic component is the primary harmonic of interest. As discussed previously, this permeance waveform plays a key Figure 10 – Results of study of variable ASOF effect on torque production capability for FSPM machines, including analytical, FE, and experimental results. role in determining the airgap flux density waveform (Fig. 6) and, ultimately, the average torque. In fact, both the airgap permeance Pg and the average torque T are proportional to the airgap magnetic flux density factor KB,gap defined in (1), and these relationships are summarized as follows: ܲ Ƚ ܭ ǡ (2) ܤ ߙ ܲ (3) ଶ ܶ Ƚ ܭ ǡ ൌ ݊݅ݏ ሺߨ ή ܨܱܵܣ ሻ గ (4) The above analysis suggests that the torque production capability of a FSPM machine is closely linked to how much of the airgap surface is occupied by rotor teeth. In order to maximize torque production, it appears beneficial for the ASOF to be as close to 0.5 as possible. However, it is important to correlate these results with both experimental data and finite element analysis to validate the findings. As an initial step, the increase in torque capability provided by modifying the pole number of a FSPM machine can be compared with the predictions from the preceding analysis. For this simplified analysis the circumferential width values of the stator teeth, stator magnets, stator slots, and rotor teeth are held equal and constant. A 12-slot, 10pole FSPM machine has an ASOF value of 0.208, which leads to KB,gap 0.388 according to (1). For a 12-slot, 14pole FSPM topology, the ASOF value is equal to 0.292, which leads KB,gap 0.505. This analysis predicts a torque increase of 30.3% solely from increasing the rotor pole number from 10 to 14. However, results from [18] show an increase in torque production of only 12%, considerably lower that the predictions of this simplified ASOF-based analysis. These results are summarized in Fig. 10, where the torque produced by the 12-slot, 10-pole machine is used as the base value for normalization. Examination of the Copyright (c) 2015 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing pubs-permissions@ieee.org.

This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication. The final version of record is available at http://dx.doi.org/10.1109/TIA.2015.2411655 Rotor Tooth Width (pu) ASOF 0.5 1.0 1.5 0.104 0.208 0.313 Figure 11 – Detailed view of rotor teeth for study of variable ASOF effect on torque production capability of 3-ph 12-slot/10-pole FSPM machines. difference between the analytical predictions and experimental results has led to the conclusion that magnetic saturation plays a significant role as discussed in the following paragraphs. Another approach to studying the effect of ASOF on torque production is to vary the rotor tooth width while holding all of the other geometry parameters constant – including maintaining a stator slot count of 12 and a rotor pole number of 10. This exercise modifies the ASOF and allows the change in torque to be analyzed using finite element analysis. In the base case, the stator slot opening, stator tooth width, permanent magnet width, and rotor tooth width are equal. For this study, the rotor tooth width is varied from 50% of the base value to 50% larger than the base. A detailed view of the rotor teeth for the extreme values as well as the base value are shown in Fig. 11. The results of the finite element study of variable ASOF effects on torque production capability have been plotted in Fig. 10. The results are normalized so that they coincide for the 12-slot, 10-pole machine at ASOF 0.208 where the normalized torque is 1 pu. The red normalized torque line represents the predictions of the preceding ASOF analysis, the blue torque line represents the results of the variable ASOF FE study, and the black x’s represent experimental results for 12-slot, 10-pole and 12-slot, 14-pole FSPM machines [18]. There is significant difference between the results predicted by the ASOF analysis and those from the finite element analysis. However, the finite element study results match those from [18] very well despite the change in rotor pole number from 10 to 14. This agreement suggests that the ASOF value plays a major role in determining the torque production independent of the rotor pole number. Closer examination has confirmed that the deviation between the red normalized torque curve and the blue FE results curve is primarily attributable to increased magnetic saturation of the stator teeth caused by increases in the rotor tooth width. Substantial saturation is not accounted for in the ASOF analysis, but is included for the finite element results and, inevitably, for the experimental results. Rotor Tooth Width (pu) ASOF Max. Stator Tooth B [T] MMF Drop Along Stator Tooth [A-turns] 0.5 1.0 1.5 0.104 0.208 0.313 2.15 2.25 2.29 130 440 820 Figure 12 – Detailed view of FE-calculated magnetic saturation in stator and rotor teeth for study of variable ASOF effect on torque production capability of 3-ph 12-slot/10-pole FSPM machines. Flux density plots for the same cases in Fig. 11 are shown in Fig. 12. The FE results reveal that increasing the rotor tooth width significantly increases the magnetic saturation in the stator teeth. This is expected, since increasing the rotor tooth width increases the flux passing into the rotor tooth from the stator tooth. Since the stator tooth width is held constant, the increase in magnetic flux causes the stator tooth flux density to rise, resulting in increased magnetic saturation. However, this increase in magnetic saturation reduces the torque capability of the machine. In contrast, the simplified ASOF analysis assumes the stator is a stiff source of MMF, equivalent to an assumption that a majority of the MMF drop in the magnetic circuit occurs in the airgap and rotor steel. Increased rotor tooth width and the resulting increased magnetic saturation in the stator teeth makes this assumption less and less true as the rotor tooth width is increased. It should be noted that the maximum stator tooth flux density shown in Fig. 12 increases only modestly with the growth in rotor tooth width due to the saturated condition of the stator and rotor teeth. However, Fig. 12 confirms that the MMF drop along the length of the stator tooth grows significantly as the rotor tooth width is increased due to the resulting increase in stator tooth magnetic flux enabled by the widened rotor teeth. Further FE analysis has subsequently been performed to investigate the sensitivity of the ASOF analysis method to stator iron saturation. When the stator iron is replaced with ideal (non-saturating) iron, the normalized FE-predicted results match the analytical ASOF predictions much more closely. Copyright (c) 2015 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing pubs-permissions@ieee.org.

This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication. The final version of record is available at http://dx.doi.org/10.1109/TIA.2015.2411655 TABLE I: FSPM MACHINES WITH THREE PHASES CAPABLE OF NON-ZERO AVERAGE TORQUE Slots Poles SPP Winding Factor ASOF 6 5 0.4 6 7 0.286 12 2 2 12 10 0.4 12 14 0.286 18 3 2 18 15 0.4 24 4 2 24 20 0.4 30 5 2 6 5 0.4 6 7 0.286 0.866 0.208 0.866 0.292 0.866 0.042 0.866 0.208 0.866 0.292 0.866 0.042 0.866 0.208 0.866 0.042 0.866 0.208 0.866 0.042 0.866 0.208 0.866 0.292 TABLE II: FSPM MACHINES WITH FOUR PHASES CAPABLE OF NON-ZERO AVERAGE TORQUE Slots Poles SPP Winding Factor ASOF 4 3 0.333 8 2 1.000 8 6 0.333 12 3 1.000 12 9 0.333 16 4 1.000 16 12 0.333 20 5 1.000 20 15 0.333 24 6 1.000 4 3 0.333 8 2 1.000 0.707 0.188 0.707 0.063 0.707 0.188 0.707 0.063 0.707 0.188 0.707 0.063 0.707 0.188 0.707 0.063 0.707 0.188 0.707 0.063 0.707 0.188 0.707 0.063 TABLE III: FSPM MACHINES WITH FIVE PHASES CAPABLE OF NON-ZERO AVERAGE TORQUE Slots Poles SPP Winding Factor ASOF 10 3 0.667 10 7 0.286 10 9 0.222 20 2 2.000 20 6 0.667 20 14 0.286 20 18 0.222 30 3 2.000 30 9 0.667 40 4 2.000 10 3 0.667 10 7 0.286 0.588 0.075 0.588 0.175 0.951 0.225 0.951 0.025 0.588 0.075 0.588 0.175 0.951 0.225 0.951 0.025 0.588 0.075 0.951 0.025 0.588 0.075 0.588 0.175 TABLE IV: FSPM MACHINES WITH SIX PHASES CAPABLE OF NON-ZERO AVERAGE TORQUE Slots Poles SPP Winding Factor ASOF V. 6 2 0.5 6 4 0.25 6 5 0.2 12 2 1 12 4 0.5 12 8 0.25 12 10 0.2 18 3 1 18 6 0.5 18 12 0.25 6 2 0.5 6 4 0.25 0.500 0.083 0.500 0.167 0.866 0.208 0.866 0.042 0.500 0.083 0.500 0.167 0.866 0.208 0.866 0.042 0.500 0.083 0.500 0.167 0.500 0.083 0.500 0.167 SLOT, POLE, AND PHASE NUMBER COMBINATIONS Results from Section IV suggest that a flux-switching PM machine of this type can produce non-zero average torque if the numbers of stator slots (S1), rotor poles (p), and electrical phases (m) satisfy the following equation, with n defined as an integer not divisible by 3, with 1 or 2 being preferred. ݊ ௌభ ൌ േ ט ௌభ ଶ (5) The list of flux-switching machines that meet this requirement can be catalogued for a variety of stator slot, rotor pole, and phase count combinations. For each combination, an initial incomplete assessment of machine torque production capability can be inferred from a combination of two key quantities: 1) the winding factor for the stator windings that determines the effective MMF produced by current flowing in the stator; and 2) the airgap surface occupation factor (ASOF), discussed in Section IV. In order to result in an attractive machine design, a combination should ideally have a high value for both metrics. Several of the flux-switching PM machine configurations with three phases that sat

Fig. 6 for one time instant as a function of spatial angular position in the airgap. The 2 lowest-order spatial harmonics of the permeance waveform (4th and 6th) . The calculated performance characteristics resulting from the preliminary design of a five-phase, 20-slot, 18-pole flux-switching PM machine have been helpful .

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