Calculation of the influence of slot geometry on the magnetic flux density of the air gap of electrical machines: three-dimensional study

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1 Calulation of the influene of geometry on the magneti flux denity of the air gap of eletrial mahine: three-dimenional tudy Rodrigo A. Lima, A. C. Paulo Coimbra, Tony Almeida, Viviane Margarida Gome, Thiago M. Pereira, Aylton J. Alve and Weley Paheo Calixto Abtrat The objetive of thi work i to invetigate the influene of ted air gap ontrutive parameter on magneti flux denity of ating mahine. For thi purpoe, different approahe were ued to olve the air gap field diagram uing finite element method and the magneti field ditribution uniformity wa evaluated by Carter' fator alulation on twodimenional and three-dimenional model. itivity analyi of ontrutive parameter wa performed and reult how that geometry modifie the magneti flux on air gap and hift the air gap magneti equipotential midline of double ted mahine. Finally, minimization of Carter fator on twodimenional model preent an optimized geometry with a near uniform magneti flux denity ditribution. Index Term Carter' fator, finite element method, ating mahine. M I. INTRODUCTION AGNETIC iruit of ating mahine are briefly ompoed by it ferromagneti part (or and tator) and air gap [1]. Thu, analyti equation and numerial method imulation are both ued on magneti iruit deign proe to determine parameter uh a torque and mahine exitation urrent [], [3]. Magneti flux ditribution on air gap ha great influene on mahine performane beaue the mot part of magneti energy ditribution i ontained in air gap domain [1], [4]. One there are magneti flux denity flutuation in air gap domain due the preene of in ferromagneti part, air gap relutane i dependent of the or and tator relative poition [5], [6]. Thi work wa upported by the Coordination for the Improvement of Higher Eduation Peronnel-CAPES PDSE Proe number: /014-00, National Counil of Sientifi and Tehnologi Development of Brazil CNPq and Reearh Support Foundation of Goiá State-FAPEG. R. A. Lima (orreponding author), T. M. Pereira and W. P. Calixto are with Eletrial and Computer Engineering Shool, Federal Univerity of Goia, UFG, Av. Univeritaria, 1488 Qd. 86 Bl. A Zip , Goiania, Goia, Brazil and with Experimental & Tehnologial Reearh and Study Group, NExT of Federal Initute of Goia, IFG, Rua 75, 46, Centro, Zip: , Goiania, Goia, Brazil ( rodrigo.lima@ifg.edu.br, thiago.pereira@ifg.edu.br, wpalixto@gmail.om ). A. C. P. Coimbra and T. Almeida are with Intitute of Sytem and Roboti, ISR of Coimbra Univerity, UC, Rua Silvio Lima, Zip , Coimbra, Portugal ( aoimbra@dee.u.pt, tony@dee.u.pt ). V.M. Gome and A. J. Alve are with Experimental & Tehnologial Reearh and Study Group, NEXT IFG ( vivianemargarida@gmail.om, aylton.alve@ifg.edu.br ). Fig. 1. Simplified air gap of a mahine with on tator' urfae, where, t i tator tooth pith, w / w tooth/ width, h tator t height and R / R tator/or yoke height. Blue line repreent uniform magneti flux,. N r Although analyti deription for magneti flux denity in air gap domain i not an eay tak, F. W. Carter preented an analyti equation to quantify the magneti flux redution in ted air gap by introduing the onept of equivalent air gap [7]- [9]. Carter onidered that magneti flux redution i equivalent to replae the ted air gap with length g for an equivalent mooth air gap with length geq k g ( k 1). The term k i alled Carter' fator and it value i influened by air gap magneti flux ditribution, where k 1 i equivalent to a uniform magneti flux ditribution. The purpoe of thi paper i to preent a tudy of geometry influene on magneti flux denity ditribution in air gap dominion uing Carter' fator to evaluate the flux ditribution uniformity. The finite element method (FEM) wa ued to olve magneti flux ditribution in air gap domain in two and three dimenional model and different approahe to evaluate Carter' fator were ompared. Additionally, it wa performed a enitivity analyi to tudy the influene of air gap ontrutive parameter influene on Carter fator TRANSACTIONS ON ENVIRONMENT AND ELECTRICAL ENGINEERING ISSN Vol, No 1 (017) Rodrigo A. Lima, A. C. Paulo Coimbra, Tony Almeida, Viviane Margarida Gome, Thiago M. Pereira, Aylton J. Alve and Weley Paheo Calixto

2 onidering two different pattern. Finally, double-ted air gap geometry effet wa onidered on a ae tudy where different pattern were ombined. w / g (3) 5 w / g Fig.. Diretized domain repreentation of air gap uing FEM. Red boundarie and green boundarie repreent Dirihlet' known potential and blue boundarie repreent Neumann boundary ondition. Fig. 3. Magneti flux diagram determined by FEM in air gap domain with length L. Green line repreent intermediate magneti equipotential line with length II. CARTER S FACTOR Carter' original tudy onider a implified air gap geometry to quantify the magneti flux redution by the preene. Fig.1 preent the ontrutive relation onidered in air gap. To deribe the magneti flux denity in air gap domain, Carter onidered that region loe to teeth have uniform magneti flux (blue arrow in Fig. 1) while region near of opening have null magneti flux. Null flux region will redue the mean magneti flux in air gap domain and the Carter' fator i defined by [1], [4], [7]. t k. t w (1) The value of on (1) depend on air gap ontrutive parameter and i related to the magneti flux ditribution near the teeth. Carter found () uing Shwartz-Chritoffel' onformal mapping on a implified geometry [7]. Other methodologie may be found in literature to determine. However, thi work will onider the empirial expreion (3) propoed by Langdorff for omparion of analyti reult [10]. w artan g w ln 1 l i numerially determined. g w g () Fig. 4. Domain repreentation and urrent denity onidered. Flux diagram determined by FEM. On double ted air gap, the Carter fator i defined by the produt of the or Carter fator, k, and the tator Carter fator, k. In thi ae, k and r k are independently alulated uing (1) and i determined by equation () or (3) [1], [5], [6]. k k k (4) r Although many author onider uffiient in pratie, the reult obtained by expreion (1), (), (3) and (4) are not aurate [11]- [15]. More aurate method involve air gap magneti field diagram olution by numerial method. In thi work the finite element method i applied on different approahe for magneti field diagram olution in air gap domain. The firt approah, uing the field diagram olution, onider that neither eletri harge nor urrent exit within the air gap domain. In thi ae, the problem i totally haraterized by etting magneti alar potential V. Magneti indution line are mapped by olving two dimenional Laplae' equation with repetive Neumann' and Dirihlet' boundarie ondition on analogy to eletati behavior [1], [6], [13]. Fig. illutrate Neumann boundary ondition (blue and green line), and red region repreent the known potential of Dirihlet ondition applied to the or and the tator urfae. Fig. 3 illutrate the magneti equipotential line mapping (blue line) and the magneti flux denity B (red line). The intermediate equipotential line obtained in diagram ha greater length than intermediate equipotential line of a mooth air gap. r m

3 The intermediate equipotential line length i given by li and i determined numerially [13]. The ratio of the atual intermediate equipotential line length l by theoretial mooth air gap length L alulate Carter fator [16], [17]. li k (5) L Another Carter fator definition uing FEM i the ratio between magneti flux denity peak value in air gap, B, and the magneti flux denity' average along the tator pith [1], [14]. B k (6) B where av av i B i the average value of magneti flux denity along the tooth pith axi x, given by [1], [14], 1 t Bav B( x) dx (7) t 0 The eond approah for the air gap field diagram alulation onider the eletri urrent in the winding and map the indution line onidering both air gap and ferromagneti part of the mahine. Fig. 4a indiate the diretion of urrent denity ( F ) of eah winding. The reult i hown in Fig. 4b and ine thi approah do not onider equipotential line, Carter' fator i determined only by (6) and (7). Although reult obtained by two-dimenional magneti field mapping provide more aurate reult than analytial method they don t deribe the dynamial behavior of ating mahine. On double ted air gap the magneti flux depend on relative poition of or and tator. The quirrel age or ha an angular deviation of one tep to redue the magnetomotive fore lo in mahine. However, the angular deviation of or break the axial ymmetry and two-dimenional approximation of Laplae equation are not poible beaue magneti alar potential and magneti flux denity ha axial dependeny. Thi paper propoe a three-dimenional analyi of magneti field diagram in airgap domain and the generalization of equation (5), (6) and (7) for Carter fator alulation. For magneti alar potential approah, the equation (6) an be extended to three-dimenional domain taking into onideration magneti equipotential urfae in plae of magneti alar equipotential line of twodimenional approah. Amef k (8) A where, g Amef i the intermediate magneti equipotential urfae numerially determined in airgap domain and A i the g theoretial intermediate magneti potential of the mooth air gap. The three-dimenional evaluation of Carter fator by (8) i equivalent to the mean of the reult of (5) taken into infiniteimal angular diplaement in the one tep interval. In addition, three-dimenional generalization of exitation urrent approah lead to the modifiation of the alulu of the mean magneti field denity in axial oordinate 1 Bav B( rmean,, z) ddz (9) A where, g rmean () (d) Fig. 5. Generi repreentation and main parameter to be varied S 10. Reulting geometry for w w w and the axial oordinate equivalent to the radiu oordinate of minimum axial ditane between or and tator teeth. The imum value of magneti flux denity B i determined on axial oordinate r and the Carter fator i mean then alulated by (6). III. METHODOLOGY A generi wa reated to tudy the geometry effet in air gap magneti flux. Changing it main ontrutive parameter, it i poible to oberve hange in pattern. In Fig. 5 are preented ome poible hange in the geometry. The variation of parameter h, w, R, w, w and med S reate new air gap pattern. Thu, the field diagram i olved for magneti potential approah and magnetizing urrent approah. Carter fator med R. () R w and S 70 (Pattern I). (d) S 60 and R w (Pattern II). h (mm) TABLE I SLOT CONSTRUCTIVE INITIAL VALUES w (mm) w (mm) t (mm)

4 then quantifie flux uniformity [1], [13], [14]. A mall indution mahine tator inpired the initial value of the. The ontrutive parameter are hown in Table I. Value of h and w are fixed a indiated in Table I to enure that the winding will be aommodated. The variation of the parameter R and S i divided in two pattern. Firt pattern ha R and S relation given by (10a) and (10b) and i illutrated in Fig. 5. R w (10a) w S 180,arin (10b) R Seond pattern i illutrated in Fig. 5d and the parameter R and S are given by (11a) and (11b), repetively. R w, h (11a) w S arin R Both pattern relate opening opening wmed by (1a) and Eq.(1b). (11b) w and intermediate w [ w, w ] (1a) min 7w wmin 3w 10wmin wmed w (1b) 3( w wmin ) 3( w wmin ) where (1b) ame from geometrial analyi for wmed w when w w. In (1a) wa adopted wmin 1 mm. The viable pae i repreented by equation (10b), (10a) and (1a). Five ae tudie analye the effet of air gap variation uing both magneti field mapping approahe. The parameter variation and domain diretization are defined by (11) and (1) and reulted in the enitivity tudy of the ontrution parameter [18]. The enitivity analyi tudy i baed on the fatorial experiment deign. For textual omprehenion, fator in the fatorial experiment deign i related to the deiion variable onidered in enitivity tudy. On thi paper, Carter fator i onidered the ytem repone or experimental reult of the fatorial experimental deign. The fatorial experiment deign take on all poible level ombination of deiion variable and hematially organize their ombination in a planning diagram. The planning diagram relate the deiion variable ombined level with their repetive ytem repone. Conidering two generi variable P and Q, with two direte level, the experiment i alled fatorial deign and ha four ytem repone Y. In thi ae, the planning diagram j an be piture by Fig. 6. The prinipal effet of deiion variable i given by, Y Y4 Y1 Y3 P (13a) Y3 Y4 Y1 Y Q (13b) The repetive enitivity analyi of P and Q onider the mean of different fatorial deign prinipal effet alulated by (13) and i expreed by it relative value. The repetive mean relative perentage value i denoted by and, repetively for P and Q. P A. Cae I: w Fig. 6. Sytem repone diagram for a Q The firt ae tudy i a enitivity analyi on R, S and. In thi ae, air gap magneti equipotential line are mapped applying the alar magneti potential formalim V m and Carter' fator i determined by (5) [1], [13], [14]. The enitivity analyi i then performed by taking a diretized domain from equation (10) and (11). For pattern I, the prinipal effet of S and w are evaluated uing (13) and the enitivity of eah parameter i alulated a the average of everal fatorial deign in ditint two level domain. Analogouly, the enitivity analyi of R and w i performed to pattern II. Additionally, thi ae tudy, magneti flux denity line are mapped on the air gap domain. The imum magneti flux denity B and mean magneti flux denity B by av FEM and the Carter fator i then evaluated uing (6). The enitivity analyi tudy i then performed. B. Cae II The enitivity analyi on R, S and w for air gap magneti flux etting the winding urrent. Carter' fator i determined by (6) [1], [13], [14]. Althought the winding urrent approah i alo a FEM method to determine magneti flux diagram i not poible determine alar magneti equipotential urve and Carter fator determination from (5). The enitivity analyi i then performed only to the exitation urrent approah by the alulation of the repetive prinipal effet of deiion variable in the diretized domain in analogou form of Cae I. C. Cae III: fatorial experiment deign. Thi ae tudy how that air-gap equipotential midline,

5 defined by the magneti equipotential line with minor influene by the air gap geometry, i not alway loated in the middle of the airgap. Different geometrie of double ted air Fig. 7 Three-dimenional model of double ted airgap. Detailed view of finite element meh on airgap domain. gap were onidered and the reult how that magneti equipotential middle line poition approah to the or urfae or tator urfae [16], [17]. D. Cae IV On thi ae tudy an air gap geometry i propoed by uing ontrained optimization of Carter fator by real-oded geneti algorithm [19]. Thi ae tudy onider the minimization problem min k (14) o.. x where x ( R, S, w, w ) i the deiion variable vetor, med and i the viable pae defined by equation (10), (11) and (1). On an optimization proe, eah deiion vetor x i an individual belonging to a finite population that evolve by roover and mutation agent on a natural eletion heme inpired by Darwin evolution theory. The population evolve by ueive iteration alled generation until an individual atifie a determined top riteria. On thi ae tudy a population of 50 individual evolve until the error defined by 1 ha a value minor to 10 3 k or the number of iteration ahieve 100 generation. The Carter fator i alulated uing magneti field diagram by FEM and (6). On the firt geometry propoed, the ontrutive parameter on Table I are fixed and the optimization proe take into onideration only the parameter R and S. The objetive of thi tudy i to ompare the Carter fator of an optimized geometry with the original geometry from the real indution mahine. On the eond geometry propoal, (14) i olved to x. On both geometrie, the Carter fator obtained by FEM i ompared to analytial alulation by (1), () and (3). E. Cae V The final ae tudy ue a generi three-dimenional model of a double ted air-gap reated in FEM. The air gap ontrutive parameter were parametrized with initial value given in Table I. Uing magneti alar equipotential approah the magneti field i mapped the intermediate equipotential urfae are numerially determined. The Carter fator for the theoretial double ted air gap i then evaluated by (8) and ompared with the reult evaluated by the analytial approah. Additionally, the effet of the or twit angle,, analyzed varying the value of from 5º to 40º. The Fig. 7a repreent the generi air gap model and the detailed view of finite element meh i depited on Fig. 7b. The air gap region wa divided on four layer to inreae the number of finite element on the air gap domain. A. Cae I IV. RESULTS ibility analyi of Carter fator due to geometrial parameter variation wa performed alulating the relative deviation of k by an aoiated perturbation on parameter viable pae. Carter' fator wa alulated by (5) uing magneti alar potential mapping by FEM. Reult for pattern I are preented in the firt and eond olumn of Table II, where and are the repetive mean S w enibility of S and w parameter taken by the n fatorial ombination of perturbation in viable pae. Additionally, a relative deviation in S with repet the initial value of Table I and their repetive relative deviation on k i depited in Fig. 8. TABLE II CASE STUDY I MAGNETIC POTENTIAL APPROACH S Pattern I w R Pattern II w.013% % 0.075% 99.95%

6 On the other hand, enitivity analyi of R and w parameter to pattern II are preented in third and fourth olumn of Table II. It i poible to oberve that in both the reulting relative deviation in k. itivity of S and w for pattern I are preented in the firt and eond Fig. 8. itivity analyi of S parameter (blue axi) and enitivity analyi of w parameter (blak axi) for pattern I by magneti alar potential alulation of k by (5). pattern the major influene for k provided by w parameter, although in pattern I S ha ignifiant ontribution when ompared with R parameter in pattern II. After the magneti alar potential analyi, the magneti field denity wa evaluated in the air gap tooth axi and Carter' fator i determined by (6). itivity analyi of Carter' fator uing the magneti field denity approah wa arried out analogouly to analyi performed for the magneti alar potential approah. Table III preent the repetive enitivity of S and w for pattern I in the firt and eond olumn. Additionally, the enitivity of R and w parameter for pattern II are preented in third and fourth olumn of Table III. TABLE III CASE STUDY I MAGNETIC FIELD DENSITY APPROACH Pattern I Pattern II S B. Cae II w R w 0.3% 99.7% 0.008% 99.99% Magneti field denity determination ue FEM and exitation urrent approah in the tooth axi of air gap. Beaue the magneti field i diretly determined on thi method, Carter' fator i determined only by (6). Following the ame methodology deribed in Cae I, enitivity analyi wa performed for both pattern onidering relative perturbation in their repetive viable pae and alulating () Fig. 9. Double ted air-gap pattern and their repetive equipotential midline (red line). Slot A and B, A and C, () A and D. olumn of Table IV. The enitivity value of R and w are preented in the third and fourth olumn of Table IV.

7 S C. Cae III Thi etion preent the analyi of double air gap geometry influene in air gap magneti equipotential midline poition. Equipotential midline i defined by the equipotential line that have minor influene by preene in air gap ferromagneti urfae. Due to pae limitation, the four pattern illutrated in Fig.9 were ombined to form only three different type of air gap geometrie. Table V ummarize the ontrutive parameter. TABLE V SLOT PATTERN The red line depited in Fig. 9 repreent the equipotential midline mapped by FEM in air gap domain. Table VI preent the repetive axial poition of midline and the relative deviation from theoretial poition. Axial poition y i line meaured from the or urfae to the equipotential midline unperturbed region and i preented in eond olumn of Table VI. Third olumn how the deviation from the line theoretial poition by relative to the total ize of the air gap. D. Cae IV Pattern I TABLE IV CASE STUDY II w R Pattern II w 0.6% 99.74% 0.0% 99.98% Pattern w (mm) R (mm) S (deg) A B C 3 70 D Pattern TABLE VI MIDLINE POSITION AND DISPLACEMENT y (mm) line (mm) line A & B A & C A & D Optimized geometrie are preented in Fig. 10. The optimization of ontrutive parameter reult in an error 6 of 1 10 and i equivalent to a uniform magneti field flux on air gap. Table VII ummarize ontrutive parameter and ompare their repetive Carter Fator by FEM and analytial method. The reult of Carter fator alulation in different methodologie how that the propoed geometry have improved the imulated magneti field uniformity. However, only the FEM methodology an determine aurately the ontribution of R and S parameter. E. Cae V On thi ae tudy the magneti field mapping of the geometry depited in Fig.7a with parameter indiated by Table I reult in an intermediate equipotential urfae. The total area of the equipotential wa numerially determined by FEM. Table VIII preent the omparion of Carter fator evaluated to the three-dimenional air gap methodology (3D- FEM) and the analytial equation preented by Carter and Langdorff. Fig. 10. Optimized geometry by geneti algorithm and Carter fator alulation by FEM. TABLE VII SLOT CONSTRUCTIVE PARAMETERS AND CARTER S FACTOR COMPARISON Parameter Original Optimized R 3.00 mm mm w.00 mm 1.00 mm S º 9.8º w 3.89 mm 3.33 mm med k (6) k (1) and () k (1) and (3) The reult evaluated by the 3D-FEM methodology how little divergene to the analytial methodology. The reaon for the divergene on the value of Carter fator an be explained by the magneti field mapping in the air gap domain. The twit angle of one tep allow to evaluate different relative poition between the tator and or. The reulting intermediate magneti alar equipotential urfae alo repreent the dynamial harater of the magneti flux uniformity on thi ae, one the equipotential urfae area i onerved with the angular diplaement of the or. The effet of twit angle i depited on the Fig. 11, where an 18 or i repreented with twit angle of 5º, 0º and 40º, repetively. Uually the value of i equivalent to one tep in the mahine. However, uing a a ontrutive parameter of the three-dimenional imulation the effet of on magneti flux uniformity an be analyzed. It i poible to oberve from Fig.1 that Carter fator derement aymptotially with twit angle. A poible explanation i that with greater twit angle more tator and

8 or enounter in an alignment poition and the ditortion of the magneti flux i ompenated. TABLE VIII CARTER S FACTOR COMPARISON EVALUATED ON THREE-DIMENSIONAL AIRGAP Approah Equation k Carter (1), () and (4) Langdorff (1), (3) and (4) D-FEM (8) of the magneti flux ditribution by alulating Carter' fator. Magnetotati approah how that ontrutive parameter type I an ontribute in % for the enitivity of the value of the Carter fator. Thu, even though w, h and w are the mot reponible for the magneti flux uniformity on the air gap it i poible that earh for optimized geometrie ha to take into onideration parameter intrinially linked to the geometry. Furthermore, geometry ha great influene on magneti equipotential midline in doubled ted air gap. Deviation of 3% and 0% wa oberved, even on imilar pattern of Fig.1 Graphi repreentation of the effet of twit angle on Carter fator. () Fig.11 Different twit angle in a or with 18. 5º, 0º and () 40º. V. CONCLUSIONS Thi paper preented a tudy of air gap geometry influene on the magneti field flux denity on ating mahine. Finite element method wa uefully ued to meaure uniformity on or and tator urfae howing that the equipotential midline i not neearily loated in the middle of the gap length. Additionally, geneti algorithm optimization proe and Carter fator alulation preented a geometry with an approximately uniform magneti flux on air gap. Although real deign mahine proe demand more retrition ondition on air gap ontrutive parameter the propoed optimization method i eaily adapted inluding penalization fator to optimization problem and offer an effiient method to air gap optimization proe. Finally, the three-dimenional magneti field mapping on the double air gap domain wa performed to meaure the Carter fator. The propoed methodology alulate the Carter fator uing the intermediate magneti alar potential urfae in air gap domain. The effet of the or twit angle wa analyzed and how that Carter fator dereae aymptotially with twit angle. Although the teady-tate tudy performed in the imulation, the reult repreent the dynami harater of the magneti flux uniformity in the mahine, one the area of alar magneti equipotential urfae remain ontant over angular diplaement of the or.

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