Journal of Soft Computng and Informaton Technology Vol. 3, No. 4, Wnter 15 GA Based Pole Shape Optmzaton for Sound Nose Reducton n Swtched Reluctance Motors Ahmad Dadpour 1, and Kourosh Ansar 1 Ferdows Unversty of Mashhad, a.dadpour@gu.ac.r Ferdows Unversty of Mashhad, k.ansar@um.ac.r Abstract: In ths paper an optmzed pole shape s presented to reduce acoustc nose of swtched reluctance motor (SRM). The optmzaton s based on the genetc algorthm and by consderng both radal force and torque rpple reducton. A two-dmensonal (-D) fnte element (FE) analyss s carred out to smulate the 6/4 SRM for each soluton of the populaton generated by GA. To decrease the acoustc nose n SRM, arcs on the rotor and stator teeth are desgned n three steps ncludng: rotor wth arcuate teeth, stator wth arcuate teeth, and both stator and rotor wth arcuate teeth. In the case of the stator wth arcuate teeth, torque rpple and radal force decrease n comparson wth the base motor whle the average torque for ths model s the same as the base motor. In the case of the rotor wth arcuate teeth, torque rpple ncreases and radal force decreases. However, the radal force and torque rpple mght be vared n the same or opposte drecton. The best soluton produced by the GA has been mplemented on a real motor. Expermental results on a real motor demonstrate the valdty of our proposed GA based optmzed pole shape. Keywords: Swtched reluctance motor, Genetc algorthm, Radal force, Torque rpple, Acoustc nose. 1. Introducton Swtched reluctance motors (SRMs) develop torque through an nteracton between the electromagnetc exctaton from the stator poles and the rotor teeth. Once a partcular combnaton of phase currents s establshed and mantaned n the stator, the rotor teeth wll be attracted nto algnment wth the stator poles n a partcular poston. Ths attracton force can be dvded nto tangental and radal force components relatve to the rotor. The tangental force s converted nto the rotatonal torque. It contans a sgnfcant radal force component n addton to the requred tangental force [1]. The domnant source of the acoustc nose n the SRM has been shown to be the dstorton of the stator by radal magnetc force. The other problem for SRMs s torque rpple whch causes ncreased undesrable acoustc nose. It s also caused by the salency of the stator and rotor []. Revewng of lteratures durng recent years about acoustc nose reducton n SRM drve shows that some researchers worked on operatng parameters of a swtched reluctance drve. They changed some parameters such as the magntude of phase currents and the tme whch these currents turned on or off. As a result of these changes, torque rpple was mnmzed [3, 4]. Some researchers mnmzed the radal force by changng of these parameters [5]. In several papers, the shapes of rotor and stator poles were studed to decrease the torque rpple [6-1] or magnetc radal force [11]. All of the mentoned studes have done separately on ether the torque rpple or radal magnetc force reducton. On the other hand, n these works torque rpple was decreased wthout any research on the radal force or counter. However, when torque rpple s decreased the radal force may be ncreased or decreased. In ths paper, the geometry of low magnetc radal force together wth torque rpple s studed and a motor havng optmzed arcuate teeth s proposed. Its characterstcs are smulated by fnte element method (FEM) analyss and compared wth SRMs havng the conventonal shape. Extensve two-dmensonal (FE) analyss for a motor are requred to determne the best stator and rotor pole arc values. Therefore, the optmum arcs of the stator and rotor pole of a SRM are obtaned by usng the genetc algorthm (GA). A genetc algorthm was employed n [1] for optmzng the shape of magnets to mnmze the coggng torque. Applcatons of genetc optmzaton algorthm n estmaton of the parameters of servo electrcal drves and dynamc state of DC motor were proposed n [13] and [14], respectvely. GA was used for optmzaton because of ts power n searchng whole soluton space wth more probablty to fndng the global optmum [13]. In [15] a speed controller desgn for a swtched reluctance (SR) motor n order to acheve mnmum torque rpple and hgh control performance was presented. Genetc algorthm optmzed the turn-on and turn-off degrees of each phase, the parameters of PID controller n transent state, and parameters of PID controller that consdered for reducng the torque rpple n steady state. Also, GA obtaned the optmum parameters of three nonlnear gans consdered for fuzzy swtchng between the two PID controllers. In ths paper the genetc algorthm s used to optmze the pole shape to reduce acoustc nose of swtched reluctance motor (SRM). The rest of the paper s organzed as follows: Domnant sources of the acoustc nose n SR motors are ntroduced n secton. Smulaton of the SR motor by usng FE method s descrbed n secton 3. Three new SR motor desgns are presented n 1
GA Based Pole Shape Optmzaton for Sound Nose Reducton n......ahmad Dadpour et al. secton 4. In secton 5, the optmzaton of stator and rotor arcs by usng the GA s descrbed. Expermental system and concluson are gven n sectons 6 and 7, respectvely.. Magnetc Sources of Acoustc Nose Radal force and torque rpple are the domnant sources of the acoustc nose n SRM. Each of these sources s carred out n ths secton. a) Radal force The magnetc flux n the SRM passes across the ar gap n an approxmate radal drecton producng radal, tangental, and lateral forces on the stator and rotor. To determne the relatonshp between these forces and the SRM dmensons, the SRM s assumed to be operatng n the lnear regon. Fgure 1 llustrates the varous dmensons nvolved n the dervaton of these forces. It can be seen that the co-energy, W (, ) s a state functon of the four ndependent varable,, l g, and L. Thus, ts dfferental (radal force, lateral force and electromagnetc torque) can be expressed as: W (,,l g ) L r T ph F r (3) l g l g W r (,,L ) T ph F y (4) L l g T W (, ) rl T ph (5) l g The tangental force s obtaned by dvdng the tangental torque by the radus of the rotor pole, yeldng: L T T ph Ft (6) r l g Fg. 1: Varous dmensons of SRM Consder that ron s nfntely permeable and has zero reluctance whch leaves only the ar gap to provde reluctance n the crcut. The ar gap flux densty at a gven stator and rotor pole overlap angle (θ), ar gap ( l g ), and current () s gven as Bg (,,l g ). Let r be the outer radus of the rotor, L be the stack length or ron length n the z drecton, T ph be the number of turns n one phase of the machne, H g be the magnetc feld strength, be the flux, and be the permeablty of ar. Then the flux lnkage s derved as: (, ) T ph T ph Bg A T ph H g L r T ph T ph L r L T ph r l g l (1) g The co-energy s gven by: T ph L W (, ) (, )d r.d l g T ph L r () l g The equatons (3), (4), and (6) show that radal force s usually multple tmes that of the tangental and lateral forces n the SRM. Such a large force causes stator vbratons. Moreover, the equatons (3) and (5) show that radal force and torque are dependent on desgn parameters and square of phase current. Some of these parameters are common n two equatons, and some lke and l g s able dfferent effect on radal force and torque. In addton to these parameters, the leakage flux, ron crcut reluctance, and saturaton effect on radal force and torque. It s dffcult to get the relaton between these all phenomena and radal force and torque. So we should smulated the machne to see the effect of all these on the radal force and/or torque. The nductance s obtaned by dvdng the flux lnkage by the phase current, yeldng: (, ) T ph L L r (7) l g Then, substtutng Eq. (7) n Eq. (5), the electromagnetc torque s obtaned as: 1 dl T (8) d b) Acoustc nose ntensty Sound power radated by an electrc machne can be expressed as [16]:
Journal of Soft Computng and Informaton Technology.... Vol. 3, No. 4, Wnter 15 P 4..c.. f.l rel exc.dcrcum.rout (9) Where c s the travelng speed of sound (m/s) n the medum, ρ s the densty of ar, Rout s the outer radus of the stator (m), L s the stack length or ron length n the z drecton, f exc s the exctaton frequences (Hz) and rel s the relatve sound ntensty and equal to: k rel 1 k Where k s the wave number and equal to:.rout. f k exc c Ampltude of dynamc deflecton equal to: 1 F ( f exc ) Rc 3 r per ( Rc ) m 4 E hs Dccum (1) f f exc ( exc ( 1 ( ) ) ) f f m m where F r s the Ampltude of radal force wave per (N/m), s the logarthmc decrement and equal to s the dampng rato and equal to: c 4KM where K and M are the equvalent stffness and mass. n m N rp f exc ( n ) n f p (11) 6 where N rp, m of the machne (r/mn) and are the number of rotor poles and speed f p s the fundamental frequency of phase current (Hz). The equatons (9) and (1) show that sound power radated s proportonal to the square ampltude of radal force. Therefore, to decrease the sound power radated we must fnd a way to decrease the radal force. c) Torque rpple When motor s runnng and torque s not constant, we have torque rpple and nose. The expresson for the torque rpple s [7]: T max T Torque Rpple mn (1) T av where T max, T mn are the maxmal value and mnmal value of total torque, T av s the average value of total torque. To nose reducton, we must decrease the torque rpple. The average torque s found out over that porton of the torque profle, whch s actually utlzed n the motor. In a 6/4 SRM, one phase wll be excted for 3 or may be a lttle bt more. The authors would lke to select that 3 where the torque s nearly constant. Ths choce wll ensure maxmum average torque wth mnmum rpple. 3. SRM Smulaton by usng FE Method Because the magnetc feld of SRM vares wth rotor poston. In the analyss of electromagnetc feld followng assumptons are presented. (a) The end-wndng magnetc feld effects of the SRM are neglected. The magnetc feld dstrbutes nvarable along the longtudnal axs. The magnetc vector potental A and the current densty J only have the axal components Az and J z. The magnetc nducton ntensty only has the component B y. Bx and component (b) The materal of the stator core s sotropc, and has sngle-valued B-H cure. (c) Magnetc feld outsde the motor s neglected. The outer dameter crcle of the stator and the nternal dameter crcle of the rotor are zero-vector magnetc lne. On the bass of above assumpton, boundary value problem about calculaton of two-dmensonal electrostatc feld can be expressed as the followng [17] 1 1 ( ( Az ) ( ( Az ) J z x x y y (13) Az 1 where μ s the permeablty of the materal. 1 s the outer dameter crcle of the stator of the SRM. s the nternal dameter crcle of the rotor of the SRM. Then the calculaton model for two dmenson feld numercal analyss s establshed. Table I shows dmensons of a three phase 6/4 SRM that has been used. Both the permeablty of the ar and the wndng are 1. The materal of stator and rotor lamnatng s.5mm defned by usng BH magnetzng curve have been showed n Fg.. A fnte element analyss software, ANSYS, has been used to smulate a nonlnear magnetc -D model of ths SRM for each soluton produced by the GA. 3
GA Based Pole Shape Optmzaton for Sound Nose Reducton n......ahmad Dadpour et al. Table I. Dmensons and Materal Data of the Orgnal Motor Number of stator poles 6 Number of rotor poles 4 Stator outer dameter 1.4 cm Ar gap length.4 cm Stator nner dameter 7.16 cm Number of turns/phase 18 Stator pole heght 1.6 cm Stator pole arc 4 degree Stator back ron thckness 1 cm Rotor pole arc 4 degree Rotor outer dameter 7.1 cm Rotor pole heght.1 cm Stator and rotor core materal DBII steel Stack depth 3.74 cm Flux densty / T 1.6 1.4 1. 1.8.6 Fg. 3: Desgn of rotor havng arcuate teeth. r = cm.4. 5 1 15 5 3 35 4 45 Fg. 4: Flux densty profles for dfferent radus of arcuate rotor teeth. 18 Fg. : BH magnetzng curve of stator and rotor lamnatng. 4. New Desgn of SRM To decrease the acoustc nose n SRM, arcs on the rotor and stator teeth are desgned n three steps. a) Rotor wth arcuate teeth Arcs are desgned only on the rotor teeth (Fg. 3). The radus of arcs can vary from (straght lne) to half of the rotor pole heght (half crcle), Rr 1.5cm. To have more percepton about the effect of curvng rotor and stator on the SRM characterstcs, three sets of ANSYS based smulatons are performed. In the frst smulaton set, the stator s not arcuate and the rotor has arc values equal to, 3,, and 1.5 cm. The correspondng smulaton results are llustrated n fgures 4-7. Fgures 4, 5, 6, and 7 show the flux densty, nductance, torque, and radal force of the SRM, respectvely where phase current s equal to A. Fgure 4 shows decreasng the radus of arcuate rotor ncreases flux densty. As shown n fgure 5, the ncremental nductance decreases by reducton of the radus of arcuate rotor. Fgure 6 shows that the torque ncreases n unalgned postons and decreases n algned postons. Fnally, fgure 7 llustrates that decreasng the radus of arcuate rotor reduces radal force and as a result, the sound power decreases. Inductance / mh Torque / N.m 16 14 1 1 8 6 4 r = cm 5 1 15 5 3 35 4 45 Fg. 5: Inductance profles for dfferent radus of arcuate rotor teeth..7.6.5.4.3..1 r = cm -.1 5 1 15 5 3 35 4 45 Fg. 6: Statc torque profles for dfferent radus of arcuate rotor teeth. 4
Journal of Soft Computng and Informaton Technology.... Vol. 3, No. 4, Wnter 15 45 4 Radal force / N 35 3 5 15 1 r = cm Inductance / mh 18 16 14 1 1 8 6 r = cm 5 4 5 1 15 5 3 35 4 45 Fg. 7: Radal Force profles for dfferent radus of arcuate rotor teeth. b) Stator wth arcuate teeth In the second step, arcs on the stator teeth are desgned (Fg. 8). The radus of arcs can vary from (straght lne) to half of the stator pole heght (half crcle), Rs. 81cm. In the second smulaton set, the rotor s not arcuate and the stator has arc values equal to, 3,, and 1.5 cm. The correspondng smulaton results are gven n fgures 9-1. Fgures 9, 1, 11, and, 1 show the flux densty, nductance, torque, and the radal force of the SRM, respectvely. Fgure 9 shows decreasng the radus of arcuate stator ncreases the flux densty n algned postons. As shown n fgure 1, the nductance s not vared by reducton of the radus of the arcuate stator. Fgure 11 shows that the average torque s nearly constant. Fnally, fgure 1 llustrates that decreasng the radus of arcuate stator reduces the ampltude of radal force and as a result, the sound power decreases. Flux densty / T 1.6 1.4 1. 1.8.6.4 Fg. 8: Desgn of stator havng arcuate teeth. r = cm. 5 1 15 5 3 35 4 45 Fg. 9: Flux densty profles for dfferent radus of arcuate stator teeth. Torque / N.m 5 1 15 5 3 35 4 45 Fg. 1: Inductance profles for dfferent radus of arcuate stator teeth..7.6.5.4.3..1 5 1 15 5 3 35 4 45 Fg. 11: Statc Torque profles for dfferent radus of arcuate stator teeth. Radal force / N 45 4 35 3 5 15 1 5 r = cm r = cm 5 1 15 5 3 35 4 45 Fg. 1: Radal Force profles for dfferent radus of arcuate stator teeth. c) Stator and Rotor wth arcuate teeth In the thrd step, arcs on the stator and rotor teeth are desgned (Fg. 13). The radus of rotor and stator arcs can vary from (straght lne) to half of the rotor pole heght and half of the stator pole heght (half crcle), respectvely ( Rr 1.5cm and Rs. 81cm ). In the thrd smulaton set, both rotor and stator are arcuate and have the same arc values equal to, 3,, and 1.5 cm. The correspondng smulaton results are gven n fgures 14-17. 5
GA Based Pole Shape Optmzaton for Sound Nose Reducton n......ahmad Dadpour et al. 45 4 35 Fg. 13: Desgn of Rotor and stator havng arcuate teeth. Radal force / N 3 5 15 r = cm Flux densty / T Inductance / mh Torque / N.m 1.6 1.4 1. 1.8.6.4. 5 1 15 5 3 35 4 45 Fg 14: Flux densty profles for dfferent radus of arcuate stator and rotor teeth. 18 16 14 1 1 8 6 4 r = cm 5 1 15 5 3 35 4 45 Fg. 15: Inductance profles for dfferent radus of arcuate stator and rotor teeth..7.6.5.4.3..1 r = cm r = cm 5 1 15 5 3 35 4 45 Fg. 16: Statc Torque profles for dfferent radus of arcuate stator and rotor teeth. 1 5 5 1 15 5 3 35 4 45 Fg. 17: Radal Force profles for dfferent radus of arcuate stator and rotor teeth. Fgures 14, 15, 16, and 17 show the flux densty, nductance, torque, and radal force of the SRM, respectvely. Fgure 14 shows decreasng the radus of arcuate rotor ncreases flux densty. As shown n fgure 15, the nductance s nearly constant n unalgned postons and decreased n algned postons by reducton of the radus of arcuate rotor. Fgure 16 shows that the average torque decreases. Fnally, fgure 17 llustrates that decreasng the radus of arcuate rotor reduces the ampltude of radal force and as a result, the sound power decreases. 5. The Optmal Desgn of Stator and Rotor Pole In these desgns when the shape of poles vares, the mass of motors remans constant. Sound power radated s lowest when the radal force and torque rpple are mnmzed. It s mportant to note that when the shape of poles vares durng nose emsson decrease, torque should not be decreased. Snce, the sound power radated s proportonal wth square of ampltude of radal force, the objectve functon for SRM sound power radated s defned as: F obj F r Rpple T av T (18) The mnmum nose and maxmum torque wll occur when the objectve functon s mnmzed. As dscussed before, one phase of a 6/4 SRM s selected equal to 3 n whch the torque s nearly constant. The whole rotor angle s 45 degrees. Hence, the nterval of a phase would be [º-3º] to [15º-45º]. Therefore, we face to multvarable problem whch must be optmzed. To solve ths problem, the genetc algorthm s used. The GA have to found the optmum arc values for both stator and rotor and also the best phase nterval so that the average torque does not reduce and the objectve functon, F obj, would be mnmzed. The optmtool of MATLAB was used to mplement our GA 6
Journal of Soft Computng and Informaton Technology.... Vol. 3, No. 4, Wnter 15 optmzaton. Because of the type of our optmzaton problem varables, the populaton type of the GA was selected as double vector. The reason s that the rotor and stator arcs may have very large values. The populaton sze was selected equal to 1 from whch two best chldren are remaned for the next generaton. 6% of the rest populatons are generated from crossover and the rest 4% from mutatons. To have the lowest rsk of holdng n local mnma, the mutaton should be sgnfcant. Hence, a unform mutaton functon wth the rate of.3 was selected. Also, the scattered crossover functon was chosen to generate chldren from parents. By usng ths settng of GA parameters, the algorthm reached to the best soluton after only 38 epochs. Fgure 18 shows the best and mean ftness functon value (F obj ) for 1 epochs. The best arc values for stator and rotor were obtaned equal to 1.41cm and, respectvely. In other words, the best result occurs f only the stator s arcuate. In the GA optmzaton procedure, the arc values greater than cm are consdered as. Also, the optmum nterval was obtaned between 4.85º and 34.85º. A comparson between the conventonal and the optmzed motors s gven n table II. As shown n ths table, the average torque values for both motors are approxmately equal. The optmzed motor causes only about 6% and 87.5% of the torque rpple and radal force of the conventonal motor, respectvely. Consequently, the ftness value of the optmzed motor s about 45% of the ftness of the conventonal motor. It demonstrates that the sound nose of the motor wth the arcuate stator s much lower than the conventonal motor, whle ther average torques are equal. 6. Expermental System The specfcaton of expermental model of SR motor was A, -V, 1-hp, 6/4. The rotor was manufactured by steel layer havng arcuate poles so that rotor dmensons are equal to the base rotor. Ths rotor has four poles, and ts pole arcs are degree greater than stator pole arcs. The arcuate rotor was assembled n the stator nstead of the base rotor. Fgure 19 shows the SR motor wth a rotor havng arcuate teeth. Fgure shows the base rotor of the SRM prototype. Fgure 1 shows the drve of SR motor. After runnng the motor, the sound nose was measured by sound level meter. Comparson between the noses of the base motor and the motor wth the arcuate rotor showed that the sound nose decreased when the rotor has arcuate poles. 7. Concluson In ths paper, an optmum desgn of SRM for sound nose reducton s presented based on the genetc algorthm. The best GA soluton determned that a motor the straght rotor and the arcuate stator wth the arc value equal to 1.41cm. Also, the optmum nterval was obtaned between 4.85 and 34.85 degrees. By usng the GA optmzaton procedure, the mnmum ftness functon obtaned about 45% of the conventonal motor after 38 generatons. Experments on a real arcuate stator motor valdate the obtaned optmum results. The real arcuate stator motor has much lower sound nose wth about equal average torque than the conventonal motor. Fg. 19: A SR motor contanng a rotor havng arcuate teeth. Fg. 18: The best and mean ftness functon values (F obj) obtaned n each generaton of the genetc algorthm Table II: Comparson between the conventonal and the optmzed motors T av Rpple T F r F obj R r = R s =.5318.1484 43 5186 R r =, R s = 1.41cm.531.884 378 3537.5 Fg. : Manufactured base rotor. 7
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