STRANDED CORE TRANSFORMER LOSS ANALYSIS

Transcription

1 Univesity of Kentucky UKnowledge Univesity of Kentucky Maste's Theses Gaduate School 2008 STRANDED CORE TRANSFORMER LOSS ANALYSIS Xingxing Zhang Univesity of Kentucky, Click hee to let us know how access to this document benefits you. Recommended Citation Zhang, Xingxing, "STRANDED CORE TRANSFORMER LOSS ANALYSIS" (2008). Univesity of Kentucky Maste's Theses This Thesis is bought to you fo fee and open access by the Gaduate School at UKnowledge. It has been accepted fo inclusion in Univesity of Kentucky Maste's Theses by an authoized administato of UKnowledge. Fo moe infomation, please contact

2 ABSTRACT OF THESIS STRANDED CORE TRANSFORMER LOSS ANALYSIS We will pesent the appoaches used to investigating the powe loss fo the standed coe tansfomes. One advantage of using standed coe is to educe powe loss o enhance tansfome efficiency. One difficulty in the modeling of this type of tansfome is that the coe is not solid (thee ae small gaps between coe wies due to cicula coss section). A two dimensional finite element method with nodal basis function fo magnetostatic field was developed to study the effects of the small gaps between coe wies. The magnetic flux densities ae compaed fo the unifom (solid) coes and the standed coes fo vaious pemeability values. The effects of diffeent ai gap dimensions in standed coe to the magnitude of magnetic flux density wee also discussed. The esults of the two dimensional study wee applied to modify the B-H cuves in a 3D simulation with an equivalent simplified unifomed coe tansfome model via Ansoft Maxwell 3D. This is achieved by output the magnitude of magnetic flux density at fixed points of mesh cente. The total coe loss of a tansfome was pedicted by integation of the losses of all elements. KEYWORDS: Standed Coe Tansfome, FEM, B-H Cuve, Ion Loss, Ai Gap Xingxing Zhang Ap. 0 th. 2008

3 STRANDED CORE TRANSFORMER LOSS ANALYSIS By Xingxing Zhang Cai-Cheng Lu Diecto of Thesis Yu Ming Zhang Diecto of Gaduate Studies Ap. 0 th. 2008

4 RULES FOR THE USE OF THESES Unpublished theses submitted fo the Maste s degee and deposited in the Univesity of Kentucky Libay ae as a ule open fo inspection, but ae to be used only with due egad to the ights of the authos. Bibliogaphical efeences may be noted, but quotations o summaies of pats may be published only with the pemission of the autho, and with the usual scholaly acknowledgments. Extensive copying o publication of the thesis in whole o in pat also equies the consent of the Dean of the Gaduate School of the Univesity of Kentucky. A libay that boows this thesis fo use by its patons is expected to secue the signatue of each use. Name Date

5 THESIS Xingxing Zhang The Gaduate School Univesity of Kentucky 2008

6 STRANDED CORE TRANSFORMER LOSS ANALYSIS THESIS A thesis submitted in patial fulfillment of the equiements fo the degee of Maste of Science in Electical Engineeing in the College of Engineeing at the Univesity of Kentucky By Xingxing Zhang Lexington, Kentucky Diecto: D. Cai-Cheng Lu, Pofesso of Electical Engineeing Lexington, Kentucky 2008 Copyight Xingxing Zhang 2008

7 ACKNOWLEDGMENTS The following thesis, while an individual wok, benefited fom the insights and diection of seveal people. Fist, my Thesis Chai, D. Cai-Cheng Lu, exemplifies the high quality scholaship to which I aspie. In addition, I wish to thank the complete Thesis Committee: D. Yuan Liao and D. Stephen Gedney. Each individual povided insights that guided and challenged my thinking, substantially impoving the finished poduct. In addition to the technical and instumental assistance above, I eceived equally impotant assistance fom family and fiends. My boyfiend, Junwen Wang, povided on-going suppot thoughout the thesis pocess, as well as technical assistance citical fo completing the poject in a timely manne. My paents, Qingyan Zhang and Mei Wu, instilled in me, fom an ealy age, the desie and skills to obtain the Maste s. Finally, I wish to thank the espondents of my study (who emain anonymous fo confidentiality puposes). Thei comments and insights ceated an infomative and inteesting poject with oppotunities fo futue wok. iii

8 Table of Contents Acknowledgments... iii List of Tables...vi List of Figues...vii List of Files...ix Chapte. Intoduction.... Backgound....2 Thesis stuctue... 3 Chapte 2. B-H cuve Modification via Finite Element Method Theoy and fomulations Goven equation and weak fom Tiangula elements Linea intepolation function of magnetic potential Local element calculation Bounday Condition Validation of finite element method Ampee s cicuital law Test case Test case Test case B-H cuve Modification B field distibution in both standed coe and unifom coe models B-H cuve modification Chapte 3. Ansoft Maxwell 3D Modeling Intoduction to Ansoft Maxwell 3D... 4 iv

9 3.2 Simplified thee winding tansfome model Geneal fomulation of tansfome loss Maxwell 3D simulation and coe loss calculation Chapte 4. Conclusion...53 Refeences...54 Vita...56 v

10 List of Tables Table 2.. Oiginal B-H data fo standed coe Table 2.2. B eo between standed coe with μ =50 and unifom coe Table 2.3. B eo between standed coe with μ =300 and unifom coe Table 2.4. B eo between standed coe with μ =000 and unifom coe Table 2.5. Tested B- μ data fo standed coe Table 2.6. Tested B- μ data fo unifom coe Table 3.. Coe loss pe pound povided by manufactue vi

11 List of Figues Figue 2.. Paametes of a typical tiangle... 7 Figue 2.2. Example to illustate the integation on the inteio bounday... 3 Figue 2.3. Case geometic model... 7 Figue 2.4. Compaison between H value calculated via FEM and exact esult fo Case... 7 Figue 2.5. Mesh plot fo Case... 8 Figue 2.6. Detail mesh plot of Case... 8 Figue 2.7. Geometic model of Case Figue 2.8. Compaison between H value calculated via FEM and exact H value in Case Figue 2.9. Mesh plot fo Case Figue 2.0. Detailed mesh plot fo Case Figue 2.. Geometic model of Case Figue 2.2. Compaison between H value calculated via FEM and exact H value in Case Figue 2.3. Mesh plot fo Case Figue 2.4. Detailed mesh plot fo Case Figue 2.5. Mesh plot and detailed mesh plot fo the standed coe model Figue 2.6. Mesh plot and detailed mesh plot fo the unifom coe model Figue 2.7. B field distibution fo standed coe model Figue 2.8. B field distibution fo unifom coe model Figue 2.9. B eo between standed coe with μ =50 and unifom coe Figue B eo between standed coe with μ =300 and unifom coe vii

12 Figue 2.2. B eo between standed coe with μ =000 and unifom coe Figue Tested B- μ data fo standed coe and unifom coe Figue Tested B- μ data fo standed coe and unifom coe with a smalle gap Figue Oiginal B-H Cuve fo standed coe tansfome and Modified B-H Cuve fo unifom coe tansfome Figue 3.. Geomety and dimensions of eight-piece coe tansfome Figue 3.2. Simplified tansfome model Figue 3.3. Extenal cicuit of tansfome Figue 3.4. Mesh geneated by Maxwell 3D Figue 3.5. B magnitude distibution via Maxwell 3D Figue 3.6. Outputted B magnitude data at fixed points Figue 3.7. Powe loss data... 5 viii

13 List of Files File Name: MS_Thesis_by_XingxingZhang Type: Size: pdf.25mbytes ix

14 Chapte. Intoduction. Backgound The calculation of coe loss has been consideed as an impotant step in the designing of tansfome. Powe tansfome coe is geneally made of mateial which has a vey low ion loss. Howeve, as the numbes of unites applied as so lage, even a small loss can add up to significant amount. The ion loss is mainly caused by distotion, unequal distibution and otating of magnetic fluxes in a coe []. Many appoaches have been studied to educe ion loss. One of the widely used techniques is to build the coe using sheet ion. Using thin sheets can significantly educe the eddy cuent loss. Anothe new appoach, which has been poposed by Busswell Enegy LLC, is to use stand ion to build the coes. The fundamental benefit of using a magnetic ion coe made of standed ion in a utility tansfome is the eduction in the tansfome s ion loss [2]. The application of the finite element method (FEM) has bought a geat advance in analytical techniques fo powe tansfome fo powe loss analysis. The 3D finite element analysis has been well developed in the ecent thee decades to compute unknowns such as magnetic field, magnetostatic field and magnetic vecto potential [2-6]. Lately, finite element method has been impoved fo both accuacy and efficiency. In the application of solving thee-dimensional magnetostatics, the nodal scala potential function is mixed with a face-edge fomulation to obtain a moe accuate esult [7]. A tansient edge-based vecto fomulation is utilized to compute the induced eddy-cuent losses in the oto of a claw-pole altenato and the use of adaptive mesh optimization leads to a coect esult [8]. Mesh quality can diectly affect the accuacy of finite element analysis. A new mesh impovement system elated to potential benefits and costs ae investigated using a suite of electomagnetic benchmaks and mesh quality measues

15 theoetically linked to FEM accuacy [9]. The hybid finite element method and bounday integal method is widely used fo scatteing and adiation poblems while this method has a vey slow convegence ate since the finite element matix is ill-conditioned. The impovement of this method including the adoption of multi-fontal method makes the hybid method convege vey fast and still keeps accuacy [0]. In this thesis, ou goal is to pedict the ion loss of a thee winding standed coe tansfome with complex geomety. A commecial softwae Ansoft Maxwell 3D is used to compute magnetostatic field in thee dimension models. Howeve, the 3D modeling though Maxwell 3D can only model coes of tansfome with unifomed mateials. In this case, the B-H cuve of standed coe mateial, which is povided by the manufactue, need to be modified fo the unifom coe via two-dimensional finite element method. In this two-dimensional finite element appoach, the fist ode tiangle elements wee used with a nodal basis function. The basic field equation is vecto Poisson s equation [, 2]. We need to solve the magnetic vecto potential in Poisson s equation by dividing the field egion into small elements and appoximate the unknown by linea equation in evey element. The Diichlet bounday condition is imposed on the mesh teminal. By the definition of magnetic vecto potential, the magnetic flux density can be solved to modify the B-H cuve fo unifom coe. Maxwell 3D is an electomagnetic field simulation softwae used fo the design and analysis of 3D stuctues. In this thesis, we used Maxwell 3D to analysis and display field distibution of 3D tansfome model and output esult data at selected points. The pe-pocessing wok included coe volume discetization and cente points of each elements output. The post pocessing wok is to integate the coe loss pe unit volume which detemined by the magnitude of B field. 2

16 .2 Thesis stuctue In this thesis, we pesented a method to calculate the coe loss of a thee phase tansfome. The main pocedue of this method is as following: ) A finite element method using nodal basis function is deived and validated fo 2D magnetostatic field. 2) B-H cuve fo standed coe tansfome modified fo unifom coe. 3) Using Ansoft Maxwell3D to simulate the 3D simplified tansfome model. Find the magnetic flux density distibution in ode to calculate the coe loss. 3

17 Chapte 2. B-H cuve Modification via Finite Element Method In the pocess of 3D modeling using Maxwell 3D, unifom coe is used to substitute standed coe. Theefoe, the B-H cuve which epesents the mateial chaacteistics needs to be modified. In this chapte, a two-dimensional finite element appoach is used to modify the B-H cuve fo unifom coe. Equation Section 2 2. Theoy and fomulations 2.. Goven equation and weak fom The 2D magnetostatic poblem has been fomed in tems of the magnetic vecto potential A, which defined as: A = B A = 0 (2.) (2.2) It is assumed that the excitation J = J z which is independent of the vaiable z. Fo z this excitation, the vecto potential A has ẑ component only. The goven vecto Poisson equation fo A can be witten as: = μ Az μ0 J z Theefoe, the magnetic flux density can be calculated fom B= A z (2.3) [, 2]. In the above, μ is the elative pemeability which is a function of position. Intoducing a test function A a z, we can deive the weak fom of vecto Poisson equation as, 4

18 a a F( Az, Az ) = Az Az μ 0 Jz dv μ V (2.4) Using vecto identity A B = ( A B) + B A, Equation (2.4) can be simplified in a 2D case [], ( ), a a a a F A 0 ( ) ˆ z Az = Az AzdV μ Az JzdV Az Az ndl μ μ Γ V V (2.5) whee ˆn denotes the unit vecto nomal to Γ. Equation (2.5) is the weak fom of 2D vecto Poisson equation Tiangula elements Befoe the deivation of finite element analysis fo a 2D magnetostatic poblem, a useful aea coodinates( L, L, L is pesented below. 2 3) A convenient set of coodinates ( L, L, L ) fo a tiangle ( ) 2 3 defined by the following linea equations in Catesian system: ( 2 3 x = Lx + Lx 2 2+ Lx 3 3 y = L y + L y + L y = L + L2 + L , 2, 3 in Figue 2. is Evey set of L, L, L ) coesponds to a unique set of Catesian coodinates [3]. Solving Equation (2.6) fo x and y, we have (2.6) a+ bx + cy L = 2Δ a2 + b2x+ c2y L2 = 2Δ a3+ b3x+ c3y L3 = 2Δ (2.7) whee, 5

19 x y Δ= det x2 y2 = aea of tangle23 2 x y 3 3 and a = x2y3 x3y2 b = y2 y3 c = x3 x2 (2.8) a2 = x3y y3x b2 = y3 y c2 = x x3 (2.9) a3 = xy2 yx2 b3 = y y2 c3 = x2 x (2.0) Based on the above definition, we obseve that when point P on edge 23, L = 0 ; if it is on vetex, then L =. When point P on edge 3, L 2 = 0 ; if it is on vetex 2, then L =. 2 When point P on edge 2, L 3 = 0; if it is on vetex 3, then L =. 3 The majo advantage of tiangula elements is that they can be used in poblems with iegula geometies. In finite element pocedue, tiangula mesh is widely adopted and aea coodinates ae used to epesent both linea and nonlinea local functions. 6

20 3 Edge 3 ( x3, y 3) (,, ) P L L L 2 3 Edge 2 y ( x, y ) Edge 2 x ( x2, y 2) Figue 2.. Paametes of a typical tiangle 7

21 2..3 Linea intepolation function of magnetic potential In a tiangula element, the magnetic potential component Az ( x, y) at any point can be appoximated by the linea intepolation function defined at evey vetex. In the tiangle of Figue 2., the magnetic potential Az ( x, y) can be appoximated as: A( xy, ) = a+ bx+ cy (2.) If the potential has values of, and A at the vetices, 2 and 3 espectively, A A2 3 then we apply Equation (2.) to the thee vetices to obtain, A = a+ bx+ cy A2 = a+ bx2 + cy A3 = a+ bx3+ cy 2 3 (2.2) This will allow us to solve fo the expansion coefficients ( abc,, ). The esults ae listed as follows, a = a A + a A + a 2Δ b = ba + b2a2 + b3a3 2Δ c = ca + c2a2 + c3a3 2Δ ( A ) ( ) ( ) (2.3) Pluging Equation (2.3) in Equation (2.), we get, 3 A x y a bx c y A 2 (, ) = ( i + i + i ) i Δ (2.4) i= In tems of aea coodinates, Equation (2.4) becomes Axy (, ) = LA i i 3 i= (2.5) 8

22 Thus, we obtained the linea epesentation of the unknown potential using its values at the vetices of tiangles Local element calculation Fom Equation (2.5), nodal basis expansion in each element can be expected in such a fom: A CL x y z A L x y ẑ 3 3 e z = i i z = j i= j= ae (, ) ˆ and (, ) (2.6) whee C i is the unknown coefficient which needs to be detemined. Now, we discetize Equation (2.5) on each tiangle. The tiangle index e is ignoed fo some quantities fo simplicity. Define S ( (, ) ˆ) ( (, ) ˆ ij = Lj xyz Li xyzdv ) (2.7) μ V B = L x y z J dv (, ) ˆ j j z V (2.8) Γ (, ) ˆ ( (, ) ˆ ij = Lj x y z Li x y zˆ ) ndl (2.9) μ Γ whee ˆn denotes the unit vecto nomal to Γ. e ae Then, function F( Az, A z ) can be expanded in local element as F A A = CS μ B CΓ ( e 3 3 ae, ) ( z z i ij 0 j i ij i= j= ) (2.20) In the following pat, we will discuss each matix espectively. ) S matix calculation Using some vecto identities, S ij can be simplified as 9

23 S ( (, ) ˆ) ( (, ) ˆ ij = Lj x y z Li x y z) dv μ V = Lxˆ ˆ ˆ ˆ j Ly j Lx i Ly i dv μ V y x y x = Lj Li + Li Lj dv μ V y y x x (2.2) By the definition of aea coodinates, the diffeential opeation can be calculated as ai + bx i + ciy Li = = ci y y 2Δ 2Δ In the same way, we have, and ai + bx i + ciy Li = = bi x x 2Δ 2Δ y = c 2Δ L j j and x L j = b 2Δ j Imposing the esults in equation (2.2), Sij = ci cj + bi bj dv 2Δ 2Δ 2Δ 2Δ μ V L2 = c i c L L j bi bj gdldl = = + 2Δ 2Δ 2Δ 2Δ μ whee g = 2Δ, and Δ is the aea of the tiangle Then, 2) B matix calculation S 2 = cc + bb μ 4 Δ 4 Δ (2.22) ij i j i j Similaly, the esult of B matix calculation can be witten as B = L x, y zˆ J dv = J L dv ( ) j j z j V V L2 = J L gdldl L2= 0 L= 0 2 Δ = J g = J 6 3 In the above, J is the excitation cuent at the cente of the tiangles. j (2.23) 0

24 3) Γ matix calculation By definition, C (, ) ˆ ( (, ) ˆ) ˆ iγ ij = Lj x y z CiLi x y z ndl μ Γ Using vecto identities, ( ab) = a b b a and a ( b c) = ( a c) b ( a b) c we have, Thus, ( ) ( ) CLz = CL z z CL i i i i i i = C L z C ˆ ( ˆ) ˆ iγ ij = Ljz Ci Li z ndl μ Γ i i = ( L ˆ ˆ) ( ˆ ) ˆ jz z Ci Li Liz Ci L i ndl μ Γ = L ˆ j CiLi ndl μ Γ (2.24) (2.25) In the above, we have applied the fact the L is in x-y plane whee zˆ L= 0. Fo evey tiangula mesh, 3 e Az CΓ = L dl (2.26) n i ij j i= μ Γ An example is given in Figue 2.2. Two adjacent tiangula elements wee used fo illustation. In tiangle, In tiangle 2, C L dl L dl L dl 3 e node2 node3 node Az iγ ij = e j + j + j i= μ n node node2 node3 (2.27) C L dl L dl L dl 3 e2 node4 node2 node3 Az iγ ij = e2 j + j + i= μ n node2 node3 node4 j (2.28)

25 Fo two adjacent tiangle mesh which shae a same edge, at the inteface, the magnetic potential satisfies [2] e Az e2 = Az and A μ A e e2 z z = e e2 n μ n. Thus, the middle tems of the ight-hand side in Equation (2.27) and Equation (2.28) cancelled. Theefoe, Γ matix can be cancelled at all inteio edges. We will discuss the oute bounday condition late. 4) Magnetic flux density calculation Fom Equation (2.), B can be expanded in local element as, 3 i= i= ( ) e B = CL zˆ i i 3 = C ˆ i Lx i Liyˆ i= y x (2.29) 3 = C ( ˆ ˆ i cix biy) 2Δ The weak fom fo local element Equation (2.20) can be simplified as: F A A = CS B = C S μ B ( e 3 3 ae, ) ( μ ) z z i ij 0 j lem lem 0 lem i= j= To minimize the equation, we set F = 0. This leads to C = μ B S. lem 0 lem lem 2

26 3 ( x 3, y 3 ) Edge 3 Edge 2 Tiangle Tiangle 2 4 ( x 4, y 4 ) ( x, y) Edge 2 ( x2, y 2) Edge Figue 2.2. Example to illustate the integation on the inteio bounday 3

27 2..5 Bounday Condition Thee ae seveal absobing bounday conditions to be applied fo mesh tuncation. When the bounday is fa enough to the tansfome model, the Diichlet Bounday Condition, on the oute bounday can yield an accuate solution. A( x, y ) = 0 (2.30) 4

28 2.2 Validation of finite element method 2.2. Ampee s cicuital law Ampee s cicuital law states that the line integal of H about any closed path is exactly equal to the diect cuent enclosed by that path [4], The magnetic flux density is elated to H by H dl= I (2.3) B = μ μ Η (2.32) 0 In the goven vecto Poisson Equation Az = μ0 Jz, J z denotes the electic μ 2 cuent density with the unit A/m. If can be calculated fom J Δ, whee z J z is unifomly distibuted, the total cuent I Δ is the aea containing J z. We will test the FEM pogam with thee test cases and compae the esult with the exact H value computed by Ampee s cicuital law Test case The fist case is a conducto of cicula coss section with a adius a = 0.m which has a 2 elative pemeability μ =. A cuent density J = A/m is imposed on it. The backgound mesh is teminated at a cicle of adius g =.2m. The geomety is shown in Figue 2.3. The exact H can be calculated as H 2 Jπ Jπ = =, < a, (2.33) 2π 2 5

29 H 2 Jπ a Jπa = =, a, 2π 2 (2.34) Using the finite element pogam, H is calculated at fixed angles of 0,45 and 90, espectively. Figue 2.4 is the compaison of esults of FEM and the exact H value. Figue 2.5 and Figue 2.6 show the mesh plot [5]. The figues show that the esults calculated by FEM agee well with the exact esults. 6

30 Souce J Theta Rho Figue 2.3. Case geometic model compaison between H value calculated via FEM and exact H value 0 degee 45 degee 90 degee theoy H Magnetic field intensity H Rho paamete in pola coodinate Figue 2.4. Compaison between H value calculated via FEM and exact esult fo Case 7

31 Figue 2.5. Mesh plot fo Case Figue 2.6. Detail mesh plot of Case 8

32 2.2.3 Test case 2 The second case is a cicle coss section of a conducto with a adius a= 0.mwhich has a elative pemeability μ =. A cuent density 2 J =A/m is imposed on it. A mateial which has a elative pemeability μ = 900 suounds the souce with a adius b = 0.2m.The backgound is filled of ai with a adius g = m which has a elative pemeability μ =. The geomety is shown in Figue 2.7. The exact H can be calculated in the same way as in case. The finite element pogam calculated H at fixed angles of 0,45 and 90 espectively. Figue 2.8 is the compaison of esults of FEM and the exact H value. Figue 2.9 and Figue 2.0 show the mesh plot. The figues show that the esults calculated by FEM agee well with the exact esults 9

33 Souce J Theta Rho μ Figue 2.7. Geometic model of Case compaison between H value calculated via FEM and exact H value 0 degee 45 degee 90 degee theoy H Magnetic field intensity H Rho paamete in pola coodinate Figue 2.8. Compaison between H value calculated via FEM and exact H value in Case 2 20

34 Figue 2.9. Mesh plot fo Case Figue 2.0. Detailed mesh plot fo Case 2 2

35 2.2.4 Test case 3 The thid case is a ing coss section of a conducto with an inne adius a = 0.m and oute adius b = 0.2m which has a elative pemeability μ =. A cuent density 2 J = A/m is imposed on it. A mateial which has a elative pemeability μ = 900 suounds the souce with a adius c = 0.24m.The backgound is filled of ai with a adius g =.2m which has a elative pemeability μ =. The geomety is shown in Figue 2.. The exact H can be calculated fom H = 0, < a, (2.35) ( ) ( ) π a J a J H = =, a < b, (2.36) 2π ( ) ( ) π b a J b a J H = =, b, 2π 2 (2.37) The finite element pogam calculated H at fixed angles of 0,45 and 90 espectively. Figue 2.2 is the compaison of esults of FEM and the exact H value. Figue 2.3 and Figue 2.4 show the mesh plot. Fom Figue 2.2, we can see that the esults calculated via FEM match well with the exact esults. Though testing thee cases with egula geomety which have exact esults, the algoithm and pogam of this 2D finite element method ae demonstated accuate to solve magnetostatic field. In the next section, this method will be applied on two models with diffeent tansfome coe stuctues. 22

36 Souce J Theta Rho μ Figue 2.. Geometic model of Case compaison between H value calculated via FEM and exact H value 0 degee 45 degee 90 degee theoy H Magnetic field intensity H Rho paamete in pola coodinate Figue 2.2. Compaison between H value calculated via FEM and exact H value in Case 3 23

37 Figue 2.3. Mesh plot fo Case Figue 2.4. Detailed mesh plot fo Case 3 24

38 2.3 B-H cuve Modification In ode to pedict accuate coe loss fo a tansfome, a simple but accuate tansfome model including non-linea B-H chaacteistics, extenal exciting cicuit, and coe loss pe pound, must be available. In ou case, the standed coe model is had to model and simulate by softwae such as Maxwell 3D. Then, an equivalent but simple model is necessay to build. In the following pat, tansfome coe with unifom distibuted mateial will eplace the standed coe in the simulation of coe loss analysis. Fo an accuate esult, the B-H cuve needs to be modified fo a unifom coe tansfome B field distibution in both standed coe and unifom coe models The standed coe mesh plot is shown in Figue 2.5. The souce aea in the uppe pat of the model is imposed with a cuent density of 2 J = A/m while the othe souce aea in the lowe pat is imposed with a cuent density of. The mateial between souce aeas denotes the standed coe. The small gaps between coe sections and whole backgound ae filled with ai. 2 J 2 = A/m Figue 2.6 is the mesh plot fo the simplified unifom coe model. The souce and backgound ae the same as the standed coe model. The mateial of coe between souce aeas is unifom distibuted as shown in detailed mesh plot. We stat with a μ = 00 imposed on both the standed coe and unifom coe elements. Figue 2.7 and Figue 2.8 show the esult of B field distibution in a patch plot view in the cente coe aea. 25

39 (a). Mesh plot fo the standed coe model (b). Detailed mesh plot fo standed coe model Figue 2.5. Mesh plot and detailed mesh plot fo the standed coe model 26

40 (a). Mesh plot fo the unifom coe model (b). Detailed mesh plot fo the unifom coe model Figue 2.6. Mesh plot and detailed mesh plot fo the unifom coe model 27

41 Figue 2.7. B field distibution fo standed coe model Figue 2.8. B field distibution fo unifom coe model 28

42 2.3.2 B-H cuve modification Table 2. shows the B-H cuve supplied by manufactues which obtained fom measuing the standed coe tansfome. Since the model has been simplified as a unifom distibuted coe tansfome, the oiginal B-H cuve is no longe applicable to the new model. We need to modify the cuve based on the esults of B field calculation via finite element method. Same mesh was applied to the model of standed coe and the model of unifom coe. We calculate the eo of B when elative pemeability is fixed fo the mesh elements in the coe aea of standed coe model using B eo = N i= B ui N B si (2.38) whee N denotes the total numbe of elements in the coe aea of standed coe. B ui and B si epesent the magnetic flux density of unifom coe and the magnetic flux density of standed coe in the i th element espectively. Based on Equation (2.38), we calculate the eo of B between standed coe and unifom coe when elative pemeability changes. Figue 2.9, Figue 2.20 and Figue 2.2 espectively show that when the elative pemeability of standed coe is 50, 300 and 000, the eo of B between standed coe and unifom coe. The test esults ae in Table 2.2, Table 2.3 and Table 2.4 espectively. Then, we calculate B aveage fo both standed coe and unifom coe model using N i i= B avg = N B Δ i= Δ i i (2.39) 29

43 whee N denotes the total numbe of elements in the coe aea, Bi and Δ i epesent the magnetic flux density and aea of the i th element espectively. We change the value of μ fo standed coe model and ecalculate B avg. Table 2.5 lists the test values of μ and the coesponding B avg fo the standed coe. In the same way, Table 2.5 is the test values unifom coe. μ and the coesponding B avg fo the The Bavg - μ cuves fo standed coe model and unifomed coe model ae shown in Figue Fo the pupose of tansfome design, we also test μ and the coesponding B avg fo standed coe with a smalle gap and unifom coe in the same mesh discetization.. We can see that B avg is a fixed value when μ going to infinite. We will explain this phenomenon by intoducing the geneation of static magnetic fields [6, 7]. The basic laws of magnetostatics ae B = 0 (2.40) H = J (2.4) Consideing a suface S which enclosed by a path C, we can opeate suface integal of Equation (2.4) ove S. Hds = Jds (2.42) By applying Stokes theoem to Equation (2.42), we have Equation (2.43) often states as Ampee s cicuit law. s c s s Hdl = Jds = I (2.43) Apply Equation (2.43) to ou 2D coe model, we can get, Bl μ Bl + = J* aea (2.44) μ μ

44 whee B0 denotes the magnetic flux density in the ai and coil. B denotes the magnetic flux density in coe. is the path length in the ai and coil. l is the path length in the coe. l0 If the elative pemeability is infinite, Equation (2.44) becomes Bl J 0 0 *aea μ = (2.45) 0 Thus, B avg is a fixed value. Povided with the test data, the B-H cuve can be calculated though algoithm involving Fouie seies and finite element analysis iteation [8]. The initial value of the iteation is fixed on the B value when elative pemeability goes infinitely. The diffeence of 0 B μ = between standed coe and unifom coe is When the gap size educes to 5 6 of the oiginal gap size, the diffeence of B μ = between standed coe and unifom coe educes to as we expected. In ou modeling, based on the oiginal B-H cuve fo standed coe tansfome, we can adjust the oiginal B-H cuve popotional to the values of Bavg when μ is infinite. Figue 2.24 shows the oiginal B-H cuve fo standed coe tansfome and the modified B-H cuve fo unifom coe model. The next chapte shows the Maxwell 3D simulation using the modified B-H cuve and the esults of ion loss pediction. 3

45 .36 x Eo (T) Value of elative pemeability Figue 2.9. B eo between standed coe with μ =50 and unifom coe.34 x Eo (T) Value of elative pemeability Figue B eo between standed coe with μ =300 and unifom coe 32

46 .36 x Eo (T) Value of elative pemeability Figue 2.2. B eo between standed coe with μ =000 and unifom coe 33

47 3.2 x 0-9 B aveage value compaison fo standed coe and unofom coe unifom coe 3.5 standed coe 3. B aveage in coe aea Relative pemeability Figue Tested B- μ data fo standed coe and unifom coe 0 x 0-7 B aveage value compaison fo coe with smalle gap 9.5 unifom coe smalle gap standed coe smalle gap B aveage in coe aea Relative pemeability Figue Tested B- μ data fo standed coe and unifom coe with a smalle gap 34

48 2.4 B-H cuve B value (T) Oiginal B-H cuve modified B-H cuve H value (A/m) Figue Oiginal B-H Cuve fo standed coe tansfome and Modified B-H Cuve fo unifom coe tansfome 35

49 B ( T ) H( A/ m ) μ B μ = = μ μ H Table 2.. Oiginal B-H data fo standed coe 36

50 Relative pemeability of unifom coe B eo ( 0 9 ) Table 2.2. B eo between standed coe with μ =50 and unifom coe Relative pemeability of unifom coe B eo ( 0 9 ) Table 2.3. B eo between standed coe with μ =300 and unifom coe μ B eo ( 0 9 ) Table 2.4. B eo between standed coe with μ =000 and unifom coe 37

51 Bavg ( T ) 0 9 μ Table 2.5. Tested B- μ data fo standed coe 38

52 Bavg ( T ) 0 9 μ Table 2.6. Tested B- μ data fo unifom coe 39

53 Chapte 3. Ansoft Maxwell 3D Modeling In this chapte, we will calculate the ion loss of a thee winding tansfome in a 3D model. The manufactue povides measued data which include the dimensions of tansfome, density of coe mateials, ion loss pe pound in diffeent magnetic flux density at fequency of 60 Hz, and B-H cuve fo coe mateial. In the simulation of Ansoft Maxwell 3D, it is had to model the standed coe and the softwae will consume lots of time to compute fields in complex geomety. Theefoe, a unifom distibuted coe is a good choice of substitute fo standed coe. We have modified B-H cuve which is the main popety fo a coe mateial. The oiginal geomety is vey complex which will be descibed in section 3.2. Fo computational convenience, the model can be simplified as an equivalent model unde those two conditions: the equal of coe volume and the equal of mean length pe tun. The next step is to simulate the simplified model in Maxwell 3D and output data of B field. Although the softwae can geneate 3D mesh automatically, the mesh data can not be obtained. In that way, we need to discetize the coe space and compute the cente of evey mesh element by pogam. Maxwell 3D can output field value on fixed points. Then, the given B-powe loss pe pound data is obtained by intepolation. Accoding to B value at evey mesh cente, a coesponding powe loss pe pound value can be found. By assembling the volume and powe loss pe pound at evey point, the ion loss of tansfome can be achieved. 40

54 3. Intoduction to Ansoft Maxwell 3D Maxwell 3D is a softwae fo the simulation of electomagnetic fields which can be used to pedict the pefomance of electomagnetic and electomechanical component designs in a vitual envionment. It includes 4 solve modules: Tansient, AC Magnetic, DC Magnetic and electic field. They ae designed to solve poblems in both time and fequency domains. Each model uses 3D Finite Elements and automatic adaptive meshing techniques to compute the electical/electomagnetic behavio of low-fequency components. Maxwell 3D can solve fo electomagnetic-field paametes such as foce, toque, capacitance, inductance, esistance and impedance as well as geneate state-space model, visualize 3D electomagnetic fields, and optimize design pefomance [9]. In this chapte, we need to use this softwae to solve field fo the 3D simplified tansfome model and output the magnetic flux density at selected points in coe mateial. Equation Section 3 4

55 3.2 Simplified thee winding tansfome model The geomety and dimensions of oiginal tansfome model is shown in Figue 3.. The inne winding and oute winding ae both low voltage windings. The middle winding is high voltage winding. Outside the thee concentic windings ae eight identical pieces of coe. Ou goal is to output the magnitude of B in the cente of evey mesh element and the coesponding volume of the mesh element. The geomety of oiginal eight-piece tansfome is had to opeate and will involves many additional woks. Theefoe, an equivalent simplified tansfome model is constucted and analyzed fist. The conditions of equivalent fo two tansfomes ae stated as: the equal of coe volume and the equal of mean length pe tun. Based on these two conditions, we can build a simplified tansfome model shown in Figue

56 (a). Geomety of oiginal eight-piece coe tansfome 43

57 (b). Dimensions of eight-piece coe tansfome (top view) 44

58 (c). Dimensions of eight-piece coe tansfome (side view) Figue 3.. Geomety and dimensions of eight-piece coe tansfome 45

59 Figue 3.2. Simplified tansfome model 46

60 3.3 Geneal fomulation of tansfome loss The total loss of tansfome is mainly made of two components: coppe loss (winding) and coe loss. The coe loss is detemined by the volume integation of the coe loss pe unit volume, N P = p ( B) dv = p ( B) Δ V (3.) coe vt, vt, i i V i= whee N is the total numbe of mesh elements of the coe, B i is the flux density at the cente of the i-th element, ΔV i is the volume of the i-th element. Fo a unifom flux density distibution (the same value ove the entie coe egion), the total coe loss is the loss pe unit volume multiply the volume of the coe. If the coe s mass density is ρ, the estimated loss pe pound (fo coe) will be P H K B B K B 2 2 ω ω 2 lb = ( 2 c + π hyst p ) p + eddy p (3.2) ρ The coefficient K eddy may be detemined quite accuately fo laminate ion sheet and stands. Let σ Fe be the conductivity of the ion, then Fo laminate ion sheet with sheet thickness a, Keddy = 2 (3.3) 2 ( σ Fea ) Fo stands of cicula coss section of diamete d, Keddy = (3.4) 2 ( σ Fed ) 32 Fom Equation (3.3) and Equation (3.4), we can see the coefficient K eddy is educed which makes a notable eduction in the ion loss when standed coe mateial is used. 47

61 3.4 Maxwell 3D simulation and coe loss calculation In Maxwell 3D poject inteface, we daw the simplified model in the egion and input data of B-H cuve obtained fom Chapte fo the mateial of coe. The extenal cicuit is shown in Figue 3.3. Mesh geneated by Maxwell 3D is shown in Figue.3.4. The magnitude of B field distibution calculated by Maxwell 3D is posted in Figue 3.5. Figue 3.6 shows the mesh geneated by pogam and the magnitude of B at evey mesh cente in the coe outputted by Maxwell 3D. Table 3. lists the data of coe loss pe pound fo the tansfome model when B value changed. Then, the data of coe loss pe pound at evey point in Figue 3.6 can be calculated by intepolation. Figue 3.7 is the plot of powe loss pe pound function. The density of coe mateial is kg/dm. The 3 volume of coe is 0.03 m. Steps to calculate coe loss ae as follows:. Divide the coe space into moe than 2000 small elements and calculate the coodinate fo the cente of evey element. 2. Use Maxwell 3D to model the 3D simplified tansfome model and analyze B field in coe pace. 3. Input points calculated in step to Maxwell 3D and output the B field magnitude in selected points. 4. Accoding to the B field magnitude in selected points, find the coesponding powe loss pe pound values at cente points of elements fom Figue Plug all the data into Equation (3.) and we can obtain the final esult of ion loss. Pcoe = 8.3W (3.5) 48

62 Figue 3.3. Extenal cicuit of tansfome Figue 3.4. Mesh geneated by Maxwell 3D 49

63 Figue 3.5. B magnitude distibution via Maxwell 3D Figue 3.6. Outputted B magnitude data at fixed points 50

64 7 Ion loss data 6 5 Powe loss pe pound Magnitude of B field Figue 3.7. Powe loss data 5

65 B (T) Loss (W/Kg) Table 3.. Coe loss pe pound povided by manufactue 52

66 Chapte 4. Conclusion In this thesis, A 2D finite element method fo magnetostatic field and Ansoft Maxwell 3D ae adopted to pedict the coe loss of a thee winding tansfome. Model simplification and data appoximation ae impotant appoaches in engineeing. The standed coe which has a complex geometic dimension is simplified as a unifom distibuted coe. In ode to keep the accuacy of powe loss pediction, the B-H cuve of the standed coe which the manufactue povides needs to be coected fo unifom coe. The calculation of modified B-H cuve is implemented by finite element method which has been validated by thee magnetostatic cases. The oiginal 3D model of the thee winding tansfome has an inconvenient geomety which has eight pieces of coe. Theefoe, an equivalent model unde the ules of equal coe volume and equal mean length pe tun is poduced in a ectangula shape fo easy discetization of mesh. Maxwell 3D is used to ceate the equivalent model and solve fo the field. The output B field data on evey mesh cente and the coesponding powe loss pe pound need to be assembled with the density of coe mateial and volume of evey mesh element to acquie the pediction of coe loss in thee-winding standed coe tansfome. The whole pocedue still needs impovement. Fo the 2D nodal basis finite element method, the matix solving is vey time consuming. An impoved algoithm is necessay to enhance the efficiency. The accuacy of this pediction can be impoved by adopting numeical method to calculate the modified B-H cuve fo unifom coe. 53

67 Refeences [] M. Okabe, M. Okada and H. Tsuchiya, Effects of Magnetic Chaacteistics of Mateials on the Ion Loss in the Thee Phase Tansfome Coe, IEEE Tans. Magn, vol. MAG-9, no.5, pp , Septembe 983. [2] M. Chiampi, A. L. Nego and M. Tataglia, A Finite Element Method to Compute Thee Dimensional Magnetic Field Distibution in Tansfome Coes, IEEE Tans. Magn, vol.mag-6, no.6, pp , Novembe 980. [3] M. V. K. Chai, A. Konad, Z. J. Csendes, P. Silveste and M. A. Palmo, Thee-Dimensional Magnetostatic Field Analysis of Electical Machiney by the Finite-Element Method, IEEE Tansactions on Powe Appaatus and Systems, vol.pas-00, no.8, pp , August 98. [4] N. A. Demedash, T. W. Nehl, O. A. Mohammed and F. A. Fouad, Expeimental Veification and Application of the Thee Dimensional Finite Element Magnetic Vecto Potential Method in Electical Appaatus, IEEE Tansactions on Powe Appaatus and Systems, vol.pas-00, no.8, pp , August 98. [5] J. Coulomb, Finite Element Thee Dimensional Magnetic Field Computation, IEEE Tans. Magn, vol.mag-7, no.6, pp , Novembe 98. [6] N. A. Demedash, T. W. Nehl, F. A. Fouad and O. A. Mohammed, Thee Dimensional Finite Element Vecto Potential Fomulation of Magnetic Fields in Electical Appaatus, IEEE Tansactions on Powe Appaatus and Systems, vol.pas-00, no.8, pp , August 98. [7] S. Bobbio, P. Alotto, F. Delfino, P. Gidinio and P. Molfino, Equivalent Souce Methods fo 3-D Foce Calculation With Nodal and Mixed FEM in Magnetostatic Poblems, IEEE Tans. Magn, vol.37, no.5, pp , Septembe

68 [8] C. Kaehle and G.. Hennebege, Tansient 3-D FEM Computation of Eddy-Cuent Losses in the Roto of a Claw-Pole Altenato, IEEE Tans. Magn, vol.40, no.2, pp , Mach 2004/ [9] M. Doica and D. D. Giannacopoulos, Impact of Mesh Quality Impovement Systems of the Accuacy of Adaptive Finite-Element Electomagneitcs With Tetaheda, IEEE Tans. Magn, vol.4, no.5, pp , May [0] X. C. Wei, E. P. Li and Y. J. Zhang, Efficient Solution to the Lage Scatteing and Radiation Poblem Using the Impoved Finite-Element Fast Multipole Method, IEEE Tans. Magn, vol.4, no.5, pp , May [] J. Jin, The finite element method in electomagnetics (2 nd edition). New Yok: Wiley, [2] S. J. Salon and J. M. Schneide, Fomulation of Poisson s Equation, IEEE Tans. Magn, vol.mag-6, no.6, pp , Novembe 98. [3] O. C. Zienkiewicz, The finite element method (fouth edition). McGaw-Hill Book Company(UK), 989. [4] W. H. Hayt, J and John A. Buck, Engineeing Electomagnetics (6 th edition). New Yok: McGaw-Hill, 200. [5] Caicheng Lu, Mesh27 use manual (intenal efeence), Univesity of Kentucky, [6] M. Kihaa, An intoduction to Electomagnet Design, June [7] J. Kan and R. Wines, Finite Element Modeling of Pemeability Diffeences in CEBAF Dipoles, J-Lab, TN, 999. [8] G. Liu, X. Xu, Impoved Modeling of the Nonlinea B-H Cuve and Its Application in Powe Cable Analysis, IEEE Tans. Magn, vol.38, no.4, pp , July [9] Ansoft, Maxwell 3D, poduct oveview, Maxwell 3D use manual,

69 Vita The autho of this thesis was bon in the city of Xi an, P. R. China, June 2 th, 984. She finished he undegaduate study in Xidian Univesity at Xi an, China and got he Bachelo s degee in July Duing he undegaduate study, she woked in the Digital Signal Pocessing Lab in Xidian Univesity instucted by Pofesso Wanyou Guo (Dec Jun. 2006, Xi an, China). She was honoed with the Univesity Scholaships fo 3 consecutive yeas in Xidian Univesity ( ). 56