[2007] IEEE. Reprinted, with permission, from [Jiaxin Chen, Jianguo Zhu, Youguang Guo, A 2-D nonlinear FEA tool embedded in Matlab/Simulink

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1 [2007] IEEE. Repinted, with pemission, fom [Jiaxin Chen, Jianguo Zhu, Youguang Guo, A 2-D nonlinea FEA tool embedded in Matlab/Simulin suounding fo application of electomagnetic field analysis in powe convetes, Electical Machines and Systems, ICEMS. Intenational Confeence on, 8-11 Oct. 2007]. This mateial is posted hee with pemission of the IEEE. Such pemission of the IEEE does not in any way imply IEEE endosement of any of the Univesity of Technology, Sydney's poducts o sevices. Intenal o pesonal use of this mateial is pemitted. Howeve, pemission to epint/epublish this mateial fo advetising o pomotional puposes o fo ceating new collective wos fo esale o edistibution must be obtained fom the IEEE by witing to pubs-pemissions@ieee.og. By choosing to view this document, you agee to all povisions of the copyight laws potecting it

2 A 2-D Nonlinea FEA Tool Embedded in Matlab/Simulin Suounding fo Application of Electomagnetic Field Analysis in Powe Convetes Jiaxin Chen 1,2, Jianguo Zhu 2, Youguang Guo 2 1 College of Electomechanical Engineeing, Donghua Univesity, Shanghai , China 2 Faculty of Engineeing, Univesity of Technology, Sydney, NSW 2007, Austalia Abstact-This pape pesents a 2-D nonlinea finite element analysis (FEA) tool embedded in Matlab/Simulin suounding fo the application of electomagnetic field analysis in powe convetes. Compaing with othe FEA tools such as ANSYS and ANSOFT, as diffeent contol aithmetic has been ealized in Matlab/Simulin suounding, the significant advance by using the FEA tool embedded in Matlab/Simulin suounding is that the field analysis can be moe easily intefaced with the extenal contol aithmetic. Consideing that the chaacteistics of most field analyses in powe convetes which ae static electomagnetic equipments, not only the geneal application pocedue of the 2-D FEA tool is intoduced, but also some impovements fo stengthening its function is poposed. As an example, the poposed model is implemented fo the pefomance analysis of a flybac switching AC-DC convete. By unning the poposed model in Simulin suounding, seveal pefomances can be obtained efficiently. I. INTRODUCTION With the development of moden electonic equipments, moe and moe powe convetes ae poduced. In most off-line powe convetes, thee is geneally a tansfome used fo the isolation between the input and output. Unde the influence of the satuation and nonlineaity of magnetic mateials, the nonlinea finite element analysis (FEA) has often been used to obtain the accuate pefomance of electomagnetic equipment [1], and many FEA tools have been pogammed. As a contol system, a typical convete is constituted by both the tansfome and its feeding contol cicuit. Howeve, because most FEA tools ae pogammed in geneal languages such as C and FORTRAN, o ae suppoted by commecial softwaes such as ANSOFT and ANSYS, the field analysis and the analysis of contol system often belong to two diffeent softwae systems. This bings some difficulties in the analysis. Consideing that the contol aithmetic can be ealized in the Matlab/Simulin suounding easily, and an FEA tool can also be embedded in the Simulin, the analysis esults obtained fom the FEA tool embedded in Matlab/Simulin suounding would be vey convenient to be tansfeed to the contol system bloc, which can also be ealized in Simulin [2]. Although the function of FEA tool is wea that only static 2-D nonlinea FEA analysis can be pocessed, to most powe convetes as a static electomagnetic equipment, its function may be sufficient. Fo that, in this pape, based on the intoduction of the geneal application pocedue of the 2-D FEA tool, some impovements fo stengthening its function is pesented. As an example, the poposed model is implemented fo the pefomance analysis of a flybac switching AC-DC convete, and seveal pefomances ae obtained efficiently. II. FINITE ELEMENT FORMULATION A. Basic Field Equations The geneal equations fo a soft magnetic mateial can be expessed as B= µ ( H+ M) = µ (1 + M/ HH ) = µµ H (1) µ = 1+ M H whee B is the magnetic flux density, M the magnetization with espect to the magnetic field stength, H, µ 0 the pemeability of ai, and µ the elative pemeability of the magnetic mateial, which is the function of the magnetization and magnetic field stength and can be decided by the Peisach model diectly. Accoding to the Maxwell equations, one has (2) H = J 0 (3) whee J 0 is the cuent density. The magnetic flux density B can be expessed as the ciculation of a magnetic potential vecto A: B= A (4) Based on (1)-(4), the govening equation is given by A ( ) = µµ Fo the 2-D case, this is educed to o 0 A ( ) = µµ 0 J (5) J (6) 0

3 1 Az 1 Az ( ) + ( ) = µ x µ x y µ y J (7) 0 z whee A z is the axial component of the magnetic vecto potential, and J z is the cuent density in the axial diection. By applying Galein s weighted esidual appoach to (6), the system matix can be obtained as whee K ij Fi = [ K] {A} = {F} = Ω (8) Ω 1 ( N u ( N iµ 0 N J ) dω z i j ) dω, (9) and N i is a shape function. Consideing that the enegy method has high accuacy, by using (8) and (9), the flux densities in two consecutive time steps can be obtained. Then the diffeential inductance of the pimay winding in the tansfome can be calculated by dψ ψ Lp L i ) = = di i ψ = 2W / i ψ i = ( 1 1 (10) whee ψ is the flux linage of the pimay winding of the tansfome, i the cuent flowing though the pimay winding, W the enegy stoed in the tansfome, and the uppe scipts () and (-1) efe to the th and (-1)th steps, espectively. B. FEA Tool in Matlab/Simulin Suounding [3] In Matlab envionment, thee is a patial diffeential equation (PDE) toolbox. The coe of the PDE toolbox is a PDE solve that uses the finite element method (FEM) fo poblems defined on bounded domains in the plane. The toolbox can solve fou inds of PDE equations: elliptic equation, paabolic equation, hypebolic system, and eigenvalue equation. The magnetostatic field equation of (6) belongs to the elliptic equation. The toolbox povides a use-fiendly inteface between the computing envionment of MATLAB and the technical pocedues of the FEM. The technical pocedues of the FEM shown in Fig.1 ae divided into seven steps including: (1) Geomety desciption; (2) Bounday conditions; (3) Mesh geneation; (4) PDE coefficients; (5) Solve paametes; (6) Plot flags and use data stings; and (7) Solve PDE. In Fig. 1, the ectangles stand fo functions and ellipses give data epesented by matices o M-files. Aows indicate data necessay fo the functions. As thee is a definite diection in this diagam, one can cut it by pesenting the needed data sets, and then continue downwad. Fig. 1. Technical pocedues of the FEM

4 C. Impovement of FEA Tool in Matlab/Simulin Suounding The PDE Toolbox is designed fo both beginnes and advanced uses. It has the gaphical use inteface (GUI), which is a self-contained gaphical envionment fo PDE solving. Fo common applications one can use the specific physical tems athe than abstact coefficients, and solving a PDE guides you though an example step by step. Fo the application in which some paametes such as geomety ae often changed, this method by which all the modification must be done in its gaphical envionment appeas to be discommodious. Accoding to the help document of the PDE tool, advanced applications ae also possible by downloading the domain geomety, bounday conditions, and mesh desciption to the MATLAB wospace. Fom the command line (o M-files) one can call functions fom the toolbox to do the had wo, e.g., geneating meshes, discetizing the poblem, pefoming intepolation, and plotting data on unstuctued gids, etc., while etaining full contol ove the global numeical algoithm. In this pape, by using expeiment pogam, the PDE tool can be done with a sub-function which can be called by its main function. As some paametes ae pocessed, the PDE tool can be set as global vaiables and tansfeed fom the main function. This will incease the efficiency of the PDE tool calculation. III. NONLINEAR INDUCTANCE OF TRANSFORMER CONSIDERING THE MAGNETIC SATURATION In ode to descibe the poposed method easily, a flybac switching AC-DC convete with input voltages of VAC/50Hz, ated output of 15 VDC/1.2 A, and switching fequency of 100 Hz is used as an example in this pape. The design of the tansfome is divided into two steps. Fistly, by using the accustomed analytical method, the initial paametes of the tansfome ae detemined as shown in Table I. Then by using the FEA tool embedded in Matlab suounding, the accuate paametes of the tansfome ae obtained. The FEA meshing is plotted in Fig. 2(a), and Fig. 2(b) shows the 2D magnetic foce lines of an E25-tansfome with ai gap. Both the diffeential inductance and appaent inductance of tansfome ae shown in Fig. 2(c). In ode to obtain the same inductance shown in Table I, the length of ai gap has been changed to 0.15 mm. (a) (b) TABLE I. MAJOR PARAMETERS OF THE TRANSFORMER IN A FLYBACK CONVERTER No. Paamete Value 1 Magnetic Coe type EE-25 2 Magnetic mateial type TDK PC40 3 Bobbin Type EE-25 (8 pin) 4 Inductance of the pimay winding, L mh 5 Numbe of tuns of windings, n1:n2:n3 58:12:9 6 Fequency of PWM, f 100 KHz 7 Length of ai gap: l g mm 8 Maximal magnetic flux density, B max mt 9 Maximal PWM duty atio, D max (c) Fig. 2. Calculation of nonlinea inductance: (a) meshing plot, (b) magnetic foce lines, and (c) appaent and diffeential inductances vs. pimay cuent. IV. MODEL IMPLEMENTATION Fig. 3 illustates the typical topology of a flybac AC-DC convete with pea cuent-contolled mode, whee the dashed line connects contol bloc to the main cicuit, C 0 is the input filte capacitance, and R 0 is the paasitical esistance in the input cicuit of the convete.

5 Fig. 3. Typical cicuit of flybac AC to DC convete A. Modelling of the Flybac Switching AC-DC Convete Pape [2] has intoduced a simulation model of flybac DC-DC convete in Matlab/Simulin suounding. As the cicuit of flybac AC-DC convete (Fig. 3) may be divided into two pats as shown in two dashed fames sepaately: one is the ectifie, and the othe is the flybac DC-DC convete, the complete simulation model of flybac AC-DC convete can be obtained only by adding the model of ectifie cicuit to that in [2], and it is shown in Fig. 4. The simulation model poposed in [2] is shown in the subsystem of flybac DC-DC convete and all the left pats constitute the model of the ectifie cicuit. The two pats in Fig. 4 coespond to those in Fig. 3, and the same ae the physical vaiables and electonic pats such as the esisto, R 0 and the output voltage of ectifie, V 1. The diffeential inductance of the tansfome obtained fom the PDE tool is put into a table which can be sampled by the simulation model though the method of looup table. Fig. 4. Simulation model of flybac AC-DC convete B. Pefomance analysis Othe ey paametes of the convete include: R1=160 KΩ/1W, C1=10 uf/25 V, C2=63 µf/35 V, R S =1.3 Ω, R3=1.2 KΩ, C3=100 pf, MOSFET: SSS6N60A, R3=0 Ω, D1: MUR1620, D2: UF4006, D3: 1N4148, R2=100 KΩ/1 W, C2=3.3 nf/1000 V, and C o =1000 µf/25 V. All the left paametes of electonic pats in the flybac DC-DC convetes ae the same as those in [2]. By eplacing the above paametes into the model and unning this model in MATLAB/Simulin suounding, seveal pefomances ae obtained as shown in Fig. 5. Unde the input voltage of 220 V, Fig. 5(a) illustates the output voltage of the ectifying cicuit and the voltage acoss the input filte capacito, C2, Figs. 5(b) and 5(c) show the output voltage and output cuent, Fig. 5(d) shows the cuent wavefoms flowing though the pimay winding of convete with the ated load, and Fig. 5(e) shows the input powe of the convete with the ated load. (a)

6 (b) (c) (e) Fig. 5. Simulation esults unde the input voltage V ac=220 V: (a).output voltage of ectifie, V 1 and the voltage acoss the capacito, V c0; (b) Output voltage, V o; (c) Output cuent of flybac DC-DC convete, I o; (d) Cuent wavefoms flowing though the pimay winding of convete with ated load; and (e) Input powe of the convete with the ated load. V. CONCLUSION This pape pesents the 2-D nonlinea finite element analysis (FEA) tool embedded in Matlab/Simulin suounding fo the application of electomagnetic field analysis in powe convetes. The significant advance of using the FEA tool embedded in Matlab/Simulin suounding is that the field analysis can be easily intefaced with the extenal contol aithmetic. The geneal application pocedue of the 2-D FEA tool is intoduced and some impovements fo stengthening its function ae pesented. As an example, the pefomance of a flybac AC-DC convete is analyzed by a simulation model in Simulin suounding, and seveal pefomances ae obtained efficiently. (d) REFERENCES [1] P. Kuo-Peng, N. Sadowsi, and N. J. Batistela, A geneal method fo coupling static convetes with electomagnetic stuctues, IEEE Tans. Magn., vol. 33, no. 2, pp , Ma [2] J. X. Chen, Y. G. Guo, and J. G. Zhu, A genealized dynamic model fo flybac switching convete based on nonlinea finite element analysis, in Poc. Int. Technology and Innovation Conf., Advanced Manufactuing Technology, Hangzhou, China, Nov. 6-7, 2006, Pape No [3] Patial Diffeential Equation Toolbox Use's Guide, the MathWos, Inc.,