Journal of Science & Technology 101 (2014) 047-053 Computation of Air-Gap Flux Density of Three-Phase Line Start Permanent Magnet Synchronous Motors Nguyen Vu Thanh", Bui Dinh Tieu, Pham Hung Phi, Nguyen Thanh Son Hanoi University of Science and Technology No Dai Co Viet Sir Ha Noi Viet Nam Received: January 10, 2014; accepted: April 22, 2014 Abstract When converting the design of three-phase squirrel cage rotor induction motors into the design of threephase line start permanent magnet synchronous motors, they have usually its stator fixed, mainly its rotor changed In this case, the pieces of permanent magnet have been inserted in the squirrel cage rotor Thus, the air-gap flux is mainly produced by the magnet Hence, it is very important to determine exactly the airgap flux density This paper presents a novel method for a detailed analysis of the air-gap flux density determination related to the steel lamination B-H characteristics Finally, the results are verified by the FEIvl tool m Ansoft Maxwell software Keywords: LSPMSMs, air-gap dux density, magnetomotive force, size of permanent magnet Introduction According to [1-5], the design of the Ime start permanent magnet synchronous motors (LSPMSMs) IS based on the induction motors (IMs) design procedures as the initial work In the design process of IMs, the magnitude of the air-gap flux density fundamental (Bg) is usually chosen in advance [15,16] However, in the case of LSPMSMs, choosing the magnitude is unreasonable It is caused by the presence of permanent magnets buried in the rotor In this case, the air-gap flux is mainly produced by the operatmg point flux of permanent magnet (PM) Thus, if the Bg is chosen in the LSPMSMs design process, usmg the FEM method to verify the Bg causes a significant enor and accuracy of design parameters in the next steps In references [6-12], the determinahon of the air-gap flux based on the analysis method often Ignores non-linear effects of steel lamination B-H charactenstics and only takes the PM and au^-gap into account Thus, analysis results become linear and inaccurate In this paper, the analysis method is introduced to determine the magnitude of the fundamental of the air-gap flux density with non-linear effect of steel lamination B-H curves in the steady mode In order to validate the proposed method, the FEM tool (Maxwell 2D) and CAD tool (RMxprt) of Ansoft Maxwell software is used due to the advantages of the software including the high accuracy and processing speed Computation of the peak value fundamental of the air-gap flux density * Conesponding Author Tel (+84)3869,2511, E-mail; thanh.nguyenvu@liust,edu 'f the 2.1 Analysis of permanent magnet fiux density (Bm) at the operating point The Bm plays a vital role in analyzing the Bg, The method of Bm determmation is shown in this section The model of the LSPMSM in this research is shown in Fig I Fig Mam flux path in LSPMSM ah IS the length of stator yoke (Isy), ah and be are the height of stator tooth (lu), hg and cd are the height of air-gap (g), gf and de are height of rotor tooth (l,,), fe is the length of rotor yoke (Iry), length of magnet (Lm), width of magnet (Wm), rip distance (Wf) Journal of Science & Technology 101 (2014) 047-053 To determine the operating point of permanent magnet conesponding to the demagnetization characteristics, the mam flux path is considered in the motor The flux goes from the north pole of the magnet to teeth and yoke of stator through an air-gap and through rotor teeth and yoke and back to the south pole, via an air gap In this process, the flux crosses the permanent magnet twice, the stator and rotor tooth length twice, the air gap twice and the stator and rotor yoke length once, as shown m Fig In Fig 1, the magnetic circuit is considered with the permanent magnet source Bg is determined by using Ampere's Law for the main flux path and taking the nonlinear B-H curve of the steel into account ^ Hdl = 2Hn,Ln, -I- 2F,r 4- 2Fts 4- 2Fg -IF.v+F,3, = After some manipulations, we obtain „ _ l-pHnit-n, lip A -^ Bn,Sn, = Kl^BgSg Sm: Area of permanent magnet Sg' Area of air-gap under pole pitch Substituting equation (2) into equation (4), the permanent magnet flux density of operating point is derived as lol-mKlmSe , , ^H^ (I) (2) (•*) where lioKimSgA 2Sn,ge (5) In addition, according to [6-12], the demagnetization line charactenstics can be written as follows: where B = Br + poHmH ge' Effective air gap [7,11] where E.-K,g fimi' Relative permeability of the magnet Kc = - ^ + ^ln H Ho' Permeability of air (po- 4Tt,10'' Tm/A) Tj! Tooth pitch (m) As the operating point of permanent magnet is the intersection of the air-gap Ime with the second quadrant demagnetization curve (demagnetization line), so we can obtain A = 2H„(B„)l,r + 2Hts(Bts)lts -IHsy(Bsy}lsy + Hry(Bry)lry where Hm Operatmg point permanent magnet field intensity, (A/m) Hn: Rotor tooth field intensity, (A/m) His' Stator tooth field mtensity, (A/m) Hsy; Stator yoke field intensity, (A/m) Hiy: Rotor yoke field intensity, (A/m) B^: Tooth flux density of stator, (T) BIT: Tooth flux density of rotor, (T) Bsy, Yoke flux density of stator, (T) B,y: Yoke flux density of rotor, (T) Fu' Mmf of rotor tooth, (A,T) F^; Mmf of stator tooth, (A.T) Fsy; Mmf of stator yoke, (A.T) ¥ry: Mmfof rotor yoke, (A.T) Fg Mmf of air-gap (A.T) Substituting equation (6) into equation (5) with some manipulations, we have airgap line Operating point (Bm, HM B, B DemagnetizaJiertiT i ne " ^H 1im Fig The operating point of permanent magnet (Bri Remanent flux density, HQ: Intrinsic coercive force) where On the other hand' (!>„, = Kim'Sg where Kin,: Leakage flux factor [17], (8) (9) Journal of Science & Technology 101 (2014) 047-053 Assummg that the half of air-gap flux under the pole pitch (fl>g) passes through the stator yoke (*I*sy) and rotor yoke (*ry) [12, 15, 16], as shown in Fig where T: pole pitch (m) In the case of stator: "l^g = Bry = k,y„B^ (22) where K: Stator tooth width (m) (23) Substituting equation (II) into equation (12) 2.3 Determining the A parameter •Jts = ^"l'\ •.'Kin, I B„ -^ B,s= k„Bm (13)where (14) In the case of rotor ' HrkpeLs btrl A2H„(B„)ltr + 2Ht5(Bt5)i,5 -i- H,y{Bsy)l,y -1Hry(Bry)0o'-2L^) (24) From Section B, it is easily found that the flux density of tooth and yoke is a function of the permanent magnet fiux density Journal of Science & Technology 101 (2014) 047-053 2.4 Calculating the operating point flux density of permanent magnet (B^ From the Sections A, B and C, the algorithm of determinmg Bj^ is shown in Fig 12 in which Bn, is the chosen flux density of permanent magnet, B ^ 'S the calculated flux density of permanent magnet If the relative enor between Bn, and B^^ is withm the range + 1%, the B ^ value is accepted If the relative enor is ± I %, B™ value is without the range re-selected so that Bm value is near the previous value Table The t lalytical results for the three c 2.2kWmotor Parameters Kta K, gc k,n km k, k,,n, a b c B.(T) B,.(T) ofBS Results and conclusions We investigate three cases for 2.2kW motor In these cases, the width (Wm) and the length (Lm) of magnet are varied > Case 1: L„, - mm , W„, = 54mm Case 2: L„^3mm,W„ B,(T) B.,(T) H„ (A/m) H„ (A/m) H,(A/m) H„(A/m) = 51mm Case 3: L„ = mm; W^ - 40mm The analytical results for the three cases are shown in table I The results companng the analytical method (AM) to the FEM method (FM) are shown in Table cti B„(T) FEM verification ofBm and B\ of2.2kW Bg is the magnitude of the first fundamental of the airgap flux density Bg value is found by Fast Fourier Transform (FFT) method "r Case 1: L^^ mm, Wm - 54mm, BS(T) B,(T) Table AM 0,89 >• Case L„ = mm; W„ = 51mm FM shown in Fig Fig S and Fig e(% 0,92 3.47 shown w Fig 4, Fig and Fig Case 3: L„, = mm fr„ - 40mm Casel 1,529 1,347 0,0006 1,132 0,940 1,617 0,962 6,609 6,395 0.0018 0,836 1,008 0,856 1,439 450 660 490 2450 0,75 Case 1,389 1,347 0.0006 1,083 0,871 1,547 0,921 9,794 Cases 1,161 1,347 0,0006 10,026 0,0019 0,827 1,029 0,875 1,470 550 840 590 4390 0,815 0,95 0,89 0,952 0,891 0,485 0,496 Comparative results for 2kW Bm(T) 0,95 0,98 3,46 shown in Fig 10, Fig 11 Fig 12 Fig Flux lines and air-gap flux density for case 1,08 1,06 1,47 0,48 0,48 0,61 Bs(T) 0,49 0,49 0,40 0,875 0,726 1,250 0,74t 30,019 33,082 0,0024 0,785 0,945 0,804 1,350 460 700 500 2850 0,94 1,01 1,084 0,4SS 0,45 0,45 0,65 Journal of Science & Technology 101 (2014) 047-053 Fig FFT for ak-gap flux density for case 1, with Bg = 0,488 (T) Fig 10 FFT for air-gap flux density for case 3, with Bg - 0,452 (T) 51 Journal of Science & Technology 101 (2014) 047-053 -F Fig, 11 Magnet flux density at the operating pomt for case 3, with Bn, = 1,068 (T) Finding L™ and Wm of the magnet, accordmg to a certain Vm Calculating the parameters b^, b,r, 1«, I„, g Calculatmg the parameters ksm, k™, kysn,, kyn., Kta Calculatmg d,e and c ii expression (8) Preset the magnet flux density, B™ i Determine the flux density B« B„, Bsy, B^ X Pick up H from B-H curve of steel, we get H^, Hlr, Hys, Hy, Calculating A parameter from above information X Calculating B^ from expression (8) Fig 12 Computmg algorithm ofBj Journal of Science & Technology 101 (2014) 047-053 From the obtained results in the table 2, it is easy to recognize that the analysis enor of air-gap flux density between the AM and the FM is less than 1% when taking steel lamination B-H curves into account The obtamed results are quite accurate when applying the analysis method The contributions of this paper include' 1) Fmding the some factors (k™, k™, ksym, kiym) which are used determine the relationship between B^, BE, B.^, B^y and Bn, 2) Proposing a computing algorithm to determine the operating pomt flux density of permanent magnet [5] [6] [7] [8] [9] {10] Fig 13 Rotor slot (ho = 0.4 mm, hi = 1.4 mm, dl -3.89 mm, d - 9 mm, hT= 12 mm) [11] [12] [13] Fig 14 Stator slot (hos = mm, hl2=l2.4mm, bos = 2.9 mm, d l - mm, d2 - 5.3 mm, hs= 13.6 mm) [14] [15] [16] [17] References [1] F Libert, J Soulard, and J Engstrom, "Design of a 4pole line start permanent magnet synchronous motor," Proc ICEMS 2002, Brugge, Belgium, Aug, 2002 [2] A,J Sorgdrager, A,J Grobler and R.J Wang, "Design procedure of a line start permanenl magnet synchronous machine" Proceedings of the 22nd South African UniversUies Power Engineering Conference, 2014 [3] W Hung, S H Mao, and M C Tsai, "Investigation of line start permanent magnet synchronous motors with Intenor-magnet rotors and surface-magnet rotors," Electncal Machines and Systems, ICEMS 2008, [4] Nedelcu, S , Tudorache, T., Ghita, C, "Influence of design parameters on a line start permanent magnet machine charactenstics", Optimization of Electrical and Electronic Equipment (OPTIM), IEEE, 2012 Guang Yang, Jun Ma, Jlan-Xin Shen, Yu Wang, "OpUmal Design and Expenmental Venficabon of a Line-Start Permanent Magnet Synchronous Motor", Electncal Machines and Systems, ICEMS 2008, R Knshnan, "Permanent Magnet Synchronous and BLOC dnve", CRC Taylor & Francis 2010, J R Hendershot va TJE Miller, "Design of bmshless permanent magnet motors", Magna physics pulishing, 2010 Lee Seong Taek, "Development and Analysis of Interior Permanenl Magnet Synchronous Motor with Field Excitation Structure", Doctoral Dissertations, University of Tennessee, 2009 Edward P Furlani, "Permanent Magnet and Electromechanical Devices, Materials, Analysis, and Applications", Academic Press, 2001, Duane Hanselman, "Bmshless Permanent Magnet Motor Design-Vers ion 2", Magna Physics Publishing, 2006 Jacek f Gieras, Mitchell Wing, "Permanent magne motor technology Design and Applications, Second Edition, Revised and Expanded", Marcel Dekker, Inc, 2002 Juha Pyrhonen, Tapani Jokinen, Valeria Hrabovcova, "Design of rotating electncal machines", John Wiley & Sons, 2008, Dan STOIA, Ovidm CHIRILA, Mihai CERNAT, Kay HAMEYER, Drago BAN, "The behaviour of the Power LSPMSM in asynchronous operation", Electronics and Motion Control Conference (EPE/PEMC) I4th International, 2010 Dan STOIA Mihai CERNAT, Kay HAMEYER, Drago BAN, "Analytical Design and Analysis of LineStart Permanent Magnet Synchronous Motors", AFRICON, IEEE, 2009 Ion Boldea, Syed a Nasar, "The Induction Machines Design Handbook, Second Edition", Taylor and Francis Group, 2010 PCS Trin Khanh Ha, TS Nguyin H6ng Thanh, 'ThiSl ke may dien", Nha xuat ban khoa hoc va kT Ihu?il, Ronghai Qu and Thomas A Lipo, "Analysis and Modeling of Augap Sc Zigzag Leakage Fluxes in a Surface Mouuted-PM Machine", IEEE, 2002,