Investigation on Slot Pole Combinations of a PM Vernier Motor with Fractional-Slot Concentrated Winding Configurations

Transcription

1 energies Article Investigation on Slot Pole Combinations a PM Vernier Motor Fractional-Slot Concentrated Winding Configurations Byungtaek Kim Department Electrical Engineering, Kunsan National University, Gunsan-si 5450, Korea; btkim@kunsan.ac.kr; Tel.: Received: 0 August 07; Accepted: 30 August 07; Published: September 07 Abstract: This paper presents a new method to find available slot pole combination a permanent magnet (PM) vernier motor fractional-slot concentrated winding (FSCW) configurations instead conventional design rule. To this aim, for a common structure PM vernier machines FSCW, air gap flux densities including modulation flux due to vernier effects are investigated from magnetic view points n a general condition to effectively use modulation flux is derived. Under obtained condition, specific design condition for slot pole combinations are established for most popular FSCW configurations coil spans ±π/3. Using established condition, all available vernier structures including those which could not be found by previous rule are obtained, back-electromotive force (EMF) each structure is analytically estimated to check vernier effects. During se procedures, it is also revealed that some general FSCW-PM motors possess vernier effects thus can be classified into vernier motors. To verify proposed ideas, characteristics back EMF are analyzed through finite element (FE)-simulations for various models, compared ir analytical calculation results. Finally, characteristics torque regarding to slot pole combinations vernier motors are discussed. Keywords: fractional-slot concentrated winding; modulation flux; permanent magnet; vernier motor. Introduction The vernier PM motor is being recently considered a strong cidate able to meet needs for higher power density, especially in low-speed electro-mechanical systems such as electric vehicles wind power generators. The motor uniquely makes use a modulation flux through reaction between slot harmonic air gap permeance magneto-motive force (MMF) rotor PM, so called vernier effect, as well as a common flux due to PM, consequently resulting in higher back EMF thus higher power. In order to bring about modulation flux, well-known rule p m Q s = p w is commonly used where p m, Q s, p w are numbers magnet pole pairs, stator slots (or called flux modulation pole), winding pole pairs, respectively [ 4]. This rule can be readily applied to distributed-winding machines, however, se windings generally employ lengthy end-turns higher copper losses, thus fractional-slot concentrated windings (FSCW) are preferred [5]. On or h, re can be an uncertainty when one applies previous rule to machine FSCWs, because winding pole-pairs p w FSCW configurations can be ambiguous as follows. In a PM machine traditional distributed windings, p w is very apparent equal to magnet pole-pairs p m, which implies that power conversion takes place interaction between both magnetic fields from stator windings PM same pole pairs. Meanwhile, as simplest examples, two popular types FSCW-PM machines can be considered, as follows. It is assumed that both machines have identical stator three slots, but one has two PM poles Energies 07, 0, 30; doi:0.3390/en

2 Energies 07, 0, 30 Energies 07, 0, 30 assumed that both machines have identical stator three slots, but one has two PM poles or has four PM poles. According to field-interaction principle, it is reasonable to think or has four PM poles. According to field-interaction principle, it is reasonable to think that that one two PM pole has two winding poles, or four PM poles has four one two PM pole has two winding poles, or four PM poles has four winding winding poles, demonstrating identical stators can have different winding poles. However, in poles, demonstrating identical stators can have different winding poles. However, in quite many quite many studies, FSCW-PM vernier machines built upon previous rule have been proposed studies, FSCW-PM vernier machines built upon previous rule have been proposed under under assumption that three slots a FSCW machine make up one winding pole pair [6 0]. assumption that three slots a FSCW machine make up one winding pole pair [6 0]. This This assumption is obviously useful as well as convenient to design vernier machine, but re assumption is obviously useful as well as convenient to design vernier machine, but re is still is still uncertainty wher conventional rule under assumption provides every available uncertainty wher conventional rule under assumption provides every available slot pole slot pole combinations for vernier machine. combinations for vernier machine. In this study, to avoid this ambiguity associated winding pole pairs to find every In this study, to avoid this ambiguity associated winding pole pairs to find every available vernier structure FSCW configurations, firstly, both useful air gap fluxes are available vernier structure FSCW configurations, firstly, both useful air gap fluxes are expressed for a generalized PM vernier structure, n induced voltage a single conductor expressed for a generalized PM vernier structure, n induced voltage a single conductor due to both fluxes are investigated from an electro-magnetic view point. The single conductor due to both fluxes are investigated from an electro-magnetic view point. The single conductor voltage equation leads to necessary condition to utilize modulation flux common PM voltage equation leads to necessary condition to utilize modulation flux common PM flux as well. By coupling condition a given slot angle for three phase FSCWs, criterion flux as well. By coupling condition a given slot angle for three phase FSCWs, criterion for slot pole combination vernier machines can be specified. As a useful example, very for slot pole combination vernier machines can be specified. As a useful example, very popular slot angle ±π/3 FSCWs, criterion for combination is presented. Using criterion, popular slot angle ±π/3 FSCWs, criterion for combination is presented. Using criterion, available combinations for vernier machines are obtained results are compared those available combinations for vernier machines are obtained results are compared those obtained from previous rules, which reveals that proposed method helps find many more obtained from previous rules, which reveals that proposed method helps find many more available slot pole combinations for FSCW vernier PM machine. In addition, from results, available slot pole combinations for FSCW vernier PM machine. In addition, from results, it it is also revealed that some conventional FSCW-PM machines can be classified into vernier is also revealed that some conventional FSCW-PM machines can be classified into vernier machines. To investigate individual effects modulation flux conventional PM flux on machines. To investigate individual effects modulation flux conventional PM flux on back EMF, well-organized back EMF equations are provided back EMF obtained models back EMF, well-organized back EMF equations are provided back EMF obtained models are analytically calculated. Finally, FE simulations are carried out for various models to check are analytically calculated. Finally, FE simulations are carried out for various models to check proposed, results are compared analytical calculation results. proposed, results are compared analytical calculation results.. Air Gap Flux a FSCW-PM Vernier Machine. Air Gap Flux a FSCW-PM Vernier Machine.. General Structure a SPM Vernier Machine FSCWs.. General Structure a SPM Vernier Machine FSCWs Figure shows a surface PM (SPM) vernier machine p m magnet pole pairs in which three adjacent Figure stator shows slots a provide surface PM area (SPM) for vernier three-phase machine slot-concentrated pm magnet pole windings. pairs in Towhich generalize three adjacent structure stator slots vernier provide machine, area it is for assumed three-phase that eachslot-concentrated stator tooth haswindings. n aux auxiliary To generalize poles (teeth), structure thus vernier total flux machine, modulation it is assumed poles Qthat fmp each is equal stator to ntooth aux Qhas s where naux auxiliary Q s is again poles (teeth), number thus stator slots. total flux modulation poles Qfmp is equal to nauxqs where Qs is again number stator slots. Stator B-phase A-phase C-phase cond. 'a' n-auxiliary poles when p m < Q fmp B ver Slot opening, o when p m > Q fmp θ=0 θ B conv θ m :moving Rotor p m pole pairs.. Air Gap Flux Density Equation Figure.. Conceptual structure a FSCW-PM vernier motor.

3 Energies 07, 0, Air Gap Flux Density Equation In SPM structure Figure, magneto-motive force(mmf) rotor PMs can be approximately represented by () fundamental component MMF wave F.m, where θ θ m st for angular position in air gap angular movement rotor, respectively. F mag F.m cos{p m (θ θ m )} () In (), F.m is expressed as () where g a is air gap length, g m is magnet thickness, B r is remanence flux density PM, µ rec µ 0 are recoil permeability PM permeability vacuum, respectively. F m. = 4 B r g m () π µ rec µ 0 Next, air gap is uneven due to stator teeth auxiliary poles. If reference position θ = 0 is at center slot all slot-open widths are same as shown in Figure, specific permeance air gap is provided as: ( ) Λ g P 0 P cos Q fmp θ, (3) where coefficient P 0 P reflect average first harmonic values permeance, respectively, y can be expressed in classical form using conformal mapping method []. The resulting values are given as (4) (5) in terms geometries machine where g m+a is effective gap length considering magnet thickness, given by g m /µ rec + g a. The factor c 0 is ratio slot open o to slot pitch πr g /Q fmp. Finally, β in (4) (5) is a non-linear coefficient, given as (6). P 0 = µ 0 (.6βc 0 ) g m+a (4) P = µ ( ) 0 β 0.39 sin(.6πc g m+a π 0.39 c 0 ) 0 (5) β = ( ) (6) 4 + o g m+a By multiplying PM s MMF () specific air gap permeance (3), air gap flux wave is obtained as (7), where anor wave p m + Q fmp pole pairs whose speed is p m /(p m + Q fmp ) ω m has been ignored since its speed is quite low thus develops negligible back EMF []. { ( )} B g F.m P 0 cos{p m (θ θ m )} F.mP cos (P m Q FmP ) θ P m P m Q θ fmp m (7) = B conv + B ver Equation (7) clearly shows both fluxes, B conv p m pole pairs B ver p ver (= p m Q fmp ) pole pairs, ir mechanical angular speeds are ω m (=dθ m /dt) p m /(p m Q fmp ) ω m, respectively. In addition, it should be noted that rotational direction B ver depends on sign p m Q fmp. In or words, traveling directions both flux are same when p m Q fmp > 0 opposite when p m Q fmp < 0 as depicted in Figure. Despite speed difference, time derivatives both fluxes at any position have same frequency. For example, B g at θ = 0 is given by (8), showing both same frequency p m ω m. In particular, both waves are passing ir (+) ( ) maximum values at instance θ m = 0. B g (θ = 0) F.m P 0 cos(p m θ m ) F.mP cos(p m θ m ) = ˆB conv cos(p m θ m ) ˆB ver cos(p m θ m ) (8)

4 Energies 07, 0, Slot Pole Combinations a PM Vernier Machine Concentrated Windings 3. Slot Pole Combinations a PM Vernier Machine Concentrated Windings 3.. Necessary Condition for Vernier Effects Utilization 3.. Necessary Condition for Vernier Effects Utilization As mentioned (7), re are both flux waves in air gap. Thus, so as to obtain higher back As mentioned (7), re are both flux waves in air gap. Thus, so as to obtain higher back EMF, both back EMFs from Bconv Bver need to be additive. To find condition for this operation, it EMF, both back EMFs from B conv B ver need to be additive. To find condition for this operation, is convenient to check induced voltage conductor a at position θ = 0 at instance it is convenient to check induced voltage conductor a at position θ = 0 at instance θm = 0, since both fluxes have ir (+) ( ) peak values (B θ m = 0, since both fluxes have ir (+) ( ) peak values B ( ˆB conv ) as explained already ˆB ver ) as explained (8). By using principle that induced voltage a conductor e = vb, where v is relative already (8). By using principle that induced voltage a conductor e = vb, where v is speed B to conductor. Since relative speeds waves are given as vconv = rgωm vver = relative speed B to conductor. Since relative speeds waves are given as v conv = r g ω m rg pm/(pm Qfmp) ωm, instantaneous voltage ea in conductor unit length is given by (9). v ver = r g p m /(p m Q fmp ) ω m, instantaneous voltage e a in conductor unit length is given by (9). ea = vconvbˆ conv vverbˆ verp e a = v conv ˆB conv v ver ˆB, ( verp (9) = rgω m ), (9) = r g ω m ( Bˆ conv + G rb ˆ ver ˆB conv + G r ˆB vre ) where rg r g is air air gap gap radius radius Gr G r = = pm/(pm p m m Qfmp) Q which fmp ) whichis is so-called so-calledgear gearratio ratio [9] [9] which contributes higher back EMF vernier PM machines than conventional PM machines. From (9), (9), it itis isclear clearthat thatboth bothemfs EMFsare areadditive additive magnitude magnitude ea eis maximized a is maximized when Gr when > G0 r > alternatively, Qfmp 0 alternatively, Q pm fmp p> 0. m > Therefore, 0. Therefore, relation Qfmp relation Q pm fmp p> 0 m > is 0 is necessary necessarycondition conditionto to effectively utilize back EMF due to Bver, B ver, which whichalso alsoindicates both bothwaves need needto tomove in inopposite direction each each or. Meanwhile, this thiscondition is isobtained from fromonly onlyone oneconductor a a in infigure Figure,, so so it it is is also also necessary necessary to check to check wher wher EMFs EMFs or conductors or conductors still additive still additive under under condition. condition. It can be proved It can be byproved investigating by investigating slot angle slot which angle arewhich angles are between angles between both adjacent both adjacent slots for slots B for Bconv conv B ver. Figure Bver. Figure demonstrates demonstrates slot angles slot αangles αconv conv α αver ver for for both flux both waves. flux waves. The angle The angle αconv α conv for Bfor Bconv conv pm p m poles poles can can be easily be easily represented represented as (0). as (0). On On or or h, h, angle angle αver α ver for for Bver B ver pver p ver = = pm p m Qfmp Q fmp should shouldbe beobtained obtained consideration consideration its itsopposite oppositetraveling travelingdirection to to Bconv, B conv, can canfinally finallybe beobtained obtainedas as () () by byusing usingrelation Qfmp Q = fmp = nauxqs. n aux Q s. α conv π π α conv = p m P (0) Q m (0) s s α ver = π π P ver π π α (P ( Q) Q s s ) = π ver = pver = pm nauxqs = pmp m () Qs Q s QsQ s cond. 'a' A-phase Stator Bver αver B conv α conv Figure Figure.. Slot Slotangles anglesfor forboth bothflux waves. waves. From (0) (), it is clear both angles αconv αver are always same under necessary From (0) (), it is clear both angles α conv α ver are always same under necessary condition Qfmp pm > 0 thus back EMFs are additive for every conductor. It reveals anor condition Q fmp p m > 0 thus back EMFs are additive for every conductor. It reveals anor interesting fact that most FSCW-PM machines considered as general PM motors can be classified interesting fact that most FSCW-PM machines considered as general PM motors can be classified as as vernier machines. For example, machines very popular ratios three slots/two vernier machines. For example, machines very popular ratios three slots/two magnet magnet poles three slots/four magnet poles are commonly considered as general FSCW-PM poles three slots/four magnet poles are commonly considered as general FSCW-PM machines. machines. Therefore, in most previous references or textbooks, characteristics se machines are explained only conventional PM flux (which is Bconv in this study) out considering

5 Energies 07, 0, 30 5 Therefore, in most previous references or textbooks, characteristics se machines are explained only conventional PM flux (which is B conv in this study) out considering modulation flux, consequently re was no difference in fundamental magnitudes back EMFs both machines [ 4]. However, slot pole combinations machines satisfy necessary condition Q fmp p m > 0 where Q fmp are both 3 because n aux =, p m are, respectively, which means y possess modulation flux as well as conventional PM flux. Thus, y can be classified as vernier machines in a special case n =, it is necessary to consider modulation flux effects on back EMF machines which will be investigated in chapter IV this paper. 3.. Derivation Design Criteria for Various Auxiliary Poles Based on obtained necessary condition, complete criteria for slot pole combinations FSCW-PM vernier machines are proposed. To this end, slot angle α conv (=α ver ) for flux B conv is assumed to be π/3 or 4π/3(= π/3) which are most popular winding configurations since it has one slot coil pitch. With se angles, minimum number slots for a complete machine is three. If criteria are developed for three slots (Q s = 3), y are easily exped to machines whose slot number is q multiple three. For structure n aux auxiliary poles in Figure, angle α ver (0) can be given as () by using Q s = Q fmp /n. α ver (= α conv ) = nπ P m Q fmp = ± π 3 + kπ, () where n =,, 3,..., k = 0, ±, ±, ±3,... By solving () for p m, complete criteria for combinations FSCW vernier machines slot angle ±π/3 are obtained as (3). It should be mentioned that obtained criteria can be used out considering winding pole pair p w in conventional rule, p m Q fmp = p w. p m = ± + 3k 3n aux Q fmp = ±( + 3k) for Q fmp p m = 3n aux p m > 0 (3) For various auxiliary poles n aux = ~3, every available number p m for a FSCW-PM vernier machine three slots (Q s = 3) are acquired by using (3), resulting in combinations in total. They are listed in Table additional parameters such as p ver G r. These combinations can be compared combinations from previous rules. If previous rule p m = Q fmp p w = n aux Q s p w is used assumption p w = for three slots (Q s = 3), only three combinations can be obtained for same n aux = ~3, which are p m =, 5, 8, respectively. It means proposed method widens range choice slot pole combinations, helps to check existence vernier effects in any PM machine. Table. Proposed p m (PM pairs) for a three-slot FSCW vernier motor various n aux. Aux. Pole Numbers n aux k PM Pairs p m p ver = Q fmp p m G r = p m /p ver 3 0 / 0 5 /5 4 / /8 7 / / /4 7 7/ 3 8 8

6 Energies 07, 0, 30 6 Besides, as expected in last section, models p m = are included for n aux = in Table as vernier structures. In fact, re is a reason why y have not been considered as vernier machines so far. It is that vernier effects in both models are not significant, which can be explained as follows. Since P 0 is certainly larger than P in (3), it is apparent that ˆB conv is quite greater than ˆB ver from (7), thus significance vernier effects in total back EMF depends on gear ratio G r in (8). In this respect, values G r both models in Table are relatively small at / respectively, thus ˆB conv is still dominant rar than G r ˆB ver in (8). With same reason, models p m = 5 for n aux = p m = 8 for n aux = 3 whose G r values are 5 8 respectively are expected to produce quite higher back EMF comparing to model p m = for n =. 4. Analytical Calculation Vernier Effects Verifications through FE-Simulations To check vernier effects on back EMF obtained models, back EMF each models in Table are analytically calculated analytical method, n analytical calculation results are compared FE-simulation results. To this end, firstly, geometries analysis models are assumed to have common specifications in Table. Every model has same air gap radius r g, air gap length g a, same PM thickness g m same stack length l stk, which keeps magnetic loading same for each models. The ferrite magnet is assumed for PM whose B r is 0.4T. The slot open ratio c 0, in particular, is set to 0.5 because vernier effects tend to be maximized at value according to previous studies [,3]. To keep same electric loading, every model has same number series-turns/phase N ph, same current is applied. With se conditions, back EMF each model is finally calculated at 500rpm rotational speed. Table. Common specifications analysis models. Quantity Value Quantity Value Air gap radius r g 5 mm PM residual flux density B r 0.4T (ferrite) Air gap length g a 0.5 mm Magnet thickness g m mm Stack length l stk 00 mm Total series number turns/phase N ph 0 Rotational speed 500 rpm Armature phase current A 4.. Analytical Calculations Back EMF using a Single Conductor Voltage In order to calculate back EMF, coefficients P 0, P, β for given geometries are firstly investigated by using (4) (5) number auxiliary poles, n aux as shown in Figure 3, where it should be noted that P 0, P, β are independent number magnet pole pairs, p m. It illustrates as n aux increases, coefficient β decreases because slot opening o decreases n aux while gap is kept constant as represented in (6). From (4) (5), it is natural that tendencies specific permeance, P 0 P, in Figure 3b are opposite each or. With obtained P 0 P, both peak flux densities ˆB conv ˆB ver are calculated by using (7) for combinations in Table. Using (8), induced voltages ê a.conv ê a.ver due to both flux densities ê conv ˆB ver, ir sum, that is, peak induced voltage ê a conductor a is calculated at 500 rpm in which factor G r in Table is used.

7 naux while gap is kept constant as represented in (6). From (4) (5), it is natural that tendencies specific permeance, P0 P, in Figure 3b are opposite each or. With obtained P0 P, both peak flux densities B B are calculated by using (7) for combinations in Table. Using (8), induced voltages e. e. due to both flux densities B Energies B 07, 0,, ir sum, that is, peak induced voltage e 30 conductor a is calculated 7 at 500 rpm in which factor Gr in Table is used. Coefficient β Number auxiliar poles (a) Specific permeance [H/m ] P Number auxiliar poles (b) P Figure 3. Variation specific permeance auxiliary poles. (a) β; (b) Specific permeance Figure 3. Variation β specific permeance auxiliary poles. (a) β; (b) Specific permeance P0 P P. 0 P. The calculated results through procedure explained above are listed in Table 3. In case n aux =, two models p m = have same voltage from general PM flux. On or h, ir voltages due to vernier effects are quite different, thus model p m = has higher back EMF than one p m =. Consequently, it is necessary to consider vernier effects in calculation back EMF model ratio three slots/four magnet poles even though voltage from common PM flux is still dominant. Among models in Table, two models p m = 5 n aux = p m = 8 n aux = 3 have higher voltage from modulation flux than that from conventional PM flux, resulting in substantially higher net back EMF than or ones. In particular, model p m = 8 for n aux = 3 is almost.5 times back EMF model p m = at n aux =. Table 3. Flux densities induced voltages a conductor. Aux. Pole Numbers n PM Pairs p m ˆB conv (T) ˆB ver (T) ê a.conv (V/m) ê a.ver (V/m) ê a (V/m) From results Table 3, it can be summarized that EMFs from modulation flux as well as conventional PM flux increases increase n aux. The model G r less than has negligible vernier effects which are those p m = n aux =, p m = n aux =, p m =,, 4, 5 n aux = 3 while ors show meaningful vernier effects in back EMF. For convenient comparison, back EMF results in Table 3 are plotted in Figure 4.

8 From results Table 3, it can be summarized that EMFs from modulation flux as well as conventional PM flux increases increase naux. The model Gr less than has negligible vernier effects which are those pm = naux =, pm = naux =, pm =,, 4, 5 naux = 3 while ors show meaningful vernier effects in back EMF. For convenient comparison, back EMF results in Table 3 are plotted in Figure 4. Energies 07, 0, p m =8 Induced voltage [V] p m =5 p m = p m =4 p m = p m = p m = p m =7 p m =5 p m =4 p m = m = Number auxiliar poles Figure Figure Induced Induced voltages voltages aa conductor conductor vernier vernier machines. machines. Using back EMF e a single conductor in Table 3 Figure 4, back EMF in phase Using back EMF ê winding each model can a a single conductor in Table 3 Figure 4, back EMF in phase be easily calculated (4) considering winding factor due to winding each model can be easily calculated (4) considering winding factor due to slot slot angle ±/3π which will be compared FE simulation results in next section. angle ±/3π which will be compared FE simulation results in next section. π E ˆ b = sin NPhealstk (4) E b = sin π 3 3 N Phê a l stk (4) Models Models for for Finite Finite Element Element Analysis Analysis Simulation Results Results Energies 07, 0, 30 8 Among models in Table 3, 3, five five models in in total total are are chosen for for FE-analysis, ir detailed specifications are same as as provided in in Table.. Two Two m m are are machines machines p m pm = n aux which are chosen to one p m naux = which are chosen to confirm vernier effects one pm = since y have been commonly considered as conventional machines. In In or words, ir back EMFs should be same according to to classical classicalmachine machineory, but but will will be be different different according according to to results results in Table in Table 3. The 3. The or or three three models models p m = 5 n aux =, p m = 7 8 n aux = 3 are chosen to verify pm = 5 naux =, pm = 7 8 naux = 3 are chosen to verify substantial substantial back EMF boost back EMF effects. boost The effects. stators The analysis stators models analysis models n aux =,, 3 are shown in naux =,, 3 are shown in Figure 5, Figure respectively 5, respectively where where slot open slot ratio open each ratio model each is 0.5. model In addition, is 0.5. In addition, thickness thickness back yoke back is determined yoke is determined considering considering flux per pole, flux so per it is pole, thinner so it as is thinner n aux increases, which partially naux increases, which partially compensates compensates reduced slot area reduced slot model area high model naux. high n aux. 5mm 50mm 5mm 50mm 5mm 50mm (a) (b) (c) Figure 5. Stators different auxiliary poles naux for FE simulations. (a) naux = ; (b) naux = ; (c) naux = 3. Figure 5. Stators different auxiliary poles n aux for FE simulations. (a) n aux = ; (b) n aux = ; (c) n aux = 3. For analysis models stators in Figure 5 rotors having different pm, flux density equipotential lines are solved FEM presented in Figure 6. It describes that For analysis models stators in Figure 5 rotors having different p flux density core teeth is around 0.4~0.7 T, so re is no concern about saturation effects. m, flux density equipotential lines are solved FEM presented in Figure 6. It describes that flux density core teeth is around 0.4~0.7 T, so re is no concern about saturation effects.

9 (a) (b) (c) Figure 5. Stators different auxiliary poles naux for FE simulations. (a) naux = ; (b) naux = ; (c) naux = 3. For analysis models stators in Figure 5 rotors having different pm, flux Energies density 07, 0, 30 equipotential lines are solved FEM presented in Figure 6. It describes that 9 flux density core teeth is around 0.4~0.7 T, so re is no concern about saturation effects. (a) (b) (c) (d) (e) Figure 6. Flux density equipotential lines vernier motors for FE simulations. (a) pm = (naux = Figure 6. Flux density equipotential lines vernier motors for FE simulations. (a) p ); (b) pm = (naux = ); (c) pm = 5 (naux = ); (d) pm = 7 (naux = 3); (e) pm = 8 (naux = 3). m = (n aux = ); (b) p m = (n aux = ); (c) p m = 5 (n aux = ); (d) p m = 7 (n aux = 3); (e) p m = 8 (n aux = 3). Under established simulations conditions, no-load back EMF analysis models at 500rpm Under are analyzed established simulations back EMF conditions, waveforms are plotted no-load in Figure back EMF 7a. It shows analysis that models at pm = 8 develops highest back EMF, next is model pm 500rpm are analyzed back EMF waveforms are plotted in Figure = 5, 7a. It shows last is that model model pm =. Especially, model pm = has apparently less back EMF than model pm p m = 8 develops highest back EMF, next is model p m = 5, last is model =, clearly demonstrating existence vernier effects on back EMF. The magnitudes p m =. Especially, model p m = has apparently less back EMF than model first harmonics back EMF waveforms in Figure 7a are obtained by discrete Fourier expansion Energies p m =, 07, clearly 0, 30 demonstrating existence vernier effects on back EMF. The magnitudes 9 first harmonics back EMF waveforms in Figure 7a are obtained by discrete Fourier expansion compared back EMFs analytically calculated from (4) in Figure 7b. It Itshows that both results are in good accordance whose difference is less is less than than 5%. 5%. Consequently, it canit be can said be that said every that every model model found found by using by using proposed proposed method method includes includes vernier vernier effects, effects, even though even though effects effects first one first (p m one = ) (pm = ) G r = / Gr = are / very are very negligible. negligible. Back EMF[V] p m =8 p m =5 p m =7 p m = p m = back EMF [V] FEM Analytical period [t/t] (a) 0 Pm= Pm= Pm=5 Pm=7 Pm=8 (b) Figure 7. Comparison Back EMF results. (a) Waveforms (FEM); (b) Magnitudes (FEM Figure 7. Comparison Back EMF results. (a) Waveforms (FEM); (b) Magnitudes (FEM analytical). analytical). In In addition, addition, characteristics characteristics torque torque are are calculated calculated for for analysis analysis models. models. FE-simulations FE-simulations are are carried carried out out phase phase current current A (RMS) A (RMS) in phase in phase back EMF, back EMF, simulation simulation results results are depicted are depicted in Figure in Figure 8. It shows 8. It shows that that model model p m = 8 pm = 8 n aux = 3 naux = is 3 about is about.5.5 times times torque torque model model p m = pm = n naux aux =,, overall overall characteristics characteristics coincide coincide well well back back EMF EMF results, results, which which supports supports validity validity proposed proposed assertion assertion effectively effectively describes describes advantage advantage vernier vernier machines higher power density. It is also can be seen that torque ripple characteristics are getting better increasing number PMs. 3 p m =8

10 analytical). In addition, characteristics torque are calculated for analysis models. FE-simulations are carried out phase current A (RMS) in phase back EMF, simulation Energies results 07, are 0, depicted 30 in Figure 8. It shows that model pm = 8 naux = 3 is about.5 times torque 0 model pm = naux =, overall characteristics coincide well back EMF results, which supports validity proposed assertion effectively describes advantage vernier machines higher power density. It It is is also can be seen that torque ripple characteristics are are getting getting better better increasing number PMs. 3.5 p m =8 p m =5 Torque [Nm].5 p m =7 p m = p m = period [t/t] Figure 8. Torque characteristics vernier motors ( A phase current). Figure 8. Torque characteristics vernier motors ( A phase current). 5. Conclusions 5. Conclusions In this study, to improve oretical uncertainty restrictions in case that previous design In this rule study, is applied to improve to machines oretical slot-concentrated uncertainty windings, restrictions in necessary case that condition previous for design effective ruleutilization is applied tovernier machines effects, Qfmp > slot-concentrated pm, is firstly obtained, n under necessary condition, condition for effective new criteria utilization for slot pole vernier combinations effects, Qwere fmp > proposed p m, is firstly for obtained, most popular case n under slot angle condition, ±/3π new criteria for slot pole combinations were proposed for most popular case slot angle ±/3π in expression well-organized form. It should be noted that winding pole pairs, p w FSPM machine is not necessary in proposed criteria. Using proposed criteria, every structure having vernier effects was found for a base FSCW PM machine three stator slots various auxiliary poles n aux. The investigation obtained FSCW-PM vernier machines are summarized as follows. First, in obtained structures, many ors are included which couldn t be found previous rules, which shows versatility proposed criteria. Notably, it is shown that PM machines slot/pole ratios 3/ 3/4 are also included which have been considered as general ones out vernier effects. In addition, model p m = 7 n aux = 3 is newly found has quite big vernier effects which is hardly found by previous design rule. Using obtained slot pole combinations, practical FSCW-PM machines are designed same electric magnetic loadings. The analytical calculations FE analysis for back EMF models are performed, both results are in good accordance, indicating usefulness air gap specific permeance functions expressed in terms machine geometries. From calculation results, it was proven that obtained models from proposed method have vernier effect as expected, effects are dependent on slot pole combinations. As expected, model three slots/four PM poles has higher back EMF than that three slots/two PM poles due to higher gear ratio G r. Therefore, it is necessary to consider vernier effects in accurate expression performance characteristics such as back EMF those machines. The torque characteristics obtained through FE simulations presents good agreement back EMF results, describing that vernier motors have much higher power density than conventional PM machines. Finally, it can be said that proposed method can be used in search for more cidates PM vernier machines, helps to make decisions about best slot pole combination in various conditions.

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