Improved error of electromagnetic shielding problems by a two-Process coupling subproblem technique

Science & Technology Development Journal, 23(2):524-527 Open Access Full Text Article Research Article Training Center of Electrical Engineering, School of Electrical Engineering, Hanoi, University of Science and Technology Correspondence Dang Quoc Vuong, Training Center of Electrical Engineering, School of Electrical Engineering, Hanoi, University of Science and Technology Email: vuong.dangquoc@hust.edu.vn History  Received: 2020-04-03  Accepted: 2020-05-11  Published: 2020-05-18 DOI : 10.32508/stdj.v23i2.2054 Copyright © VNU-HCM Press. This is an open- access article distributed under the terms of the Creative Commons Attribution 4.0 International license. Improved error of electromagnetic shielding problems by a two-process coupling subproblem technique Dang Quoc Vuong* Use your smartphone to scan this QR code and download this article ABSTRACT Introduction: The direct application of the classical finite element method for dealing with mag- neto dynamic problems consisting of thin regions is extremely difficult or even not possible. Many authors have been recently developed a thin shellmodel to overcome this drawback. However, this development generally neglects inaccuracies around edges and corners of the thin shell, which leads to inaccuracies of the magnetic fields, eddy currents, and joule power losses, especially in- creasing with the thickness. Methods: In this article, we propose a two-process coupling sub- problem technique for improving the errors that overcome thin shell assumptions. This tech- nique is based on the subproblem method to couple SPs in two-processes. The first scenario is an initial problem solved with coils/stranded inductors together with thin region models. The obtained solutions are then considered as volume sources for the second scenario, including ac- tual volume improvements that scope with the thin shell assumptions. The final solution is, to sum up, the subproblem solutions achieved from both scenarios. The extended method is ap- proached for the h-conformal magnetic formulation. Results: The obtained results of the method are checked/compared to be close to the reference solutions computed from the classical finite element method and the measured results. This can be pointed out in a very good agreement. Conclusion: The extended method has also been successfully applied to the practical problem (TEAM workshop problem 21, model B). Key words: Magnetic flux density, eddy current losses, Joule power losses, thin shells, finite element method, subproblem method (SPM) INTRODUCTION The direct application of the finite element method (FEM)1 for dealing with magneto dynamic problems consisting of thin regions is extremely difficult or even not possible. Many authors2 have been recently de- veloped a thin shell (TS) model in order to overcome this drawback. However, this development generally neglects inaccuracies around edges and corners of TS, which leads to inaccuracies of the local fields (mag- netic fields, eddy currents, and Joule power losses...). The aim of this study is to propose a two-process cou- pling subproblem (SP) technique for improving the errors appearing from the TS models that were de- veloped in2. The technique is herein based on the subproblem method (SPM) presented by many au- thors3–7. The technique allows to couple SPs in two- processes. The first scenario is an initial problem solved with coils/stranded inductors and thin region models; the obtained solutions are then considered as volume sources (VSs) (express as of permeabil- ity and conductivity material in conducting regions) for the second scenario including actual volume im- provements that scopewith the TS assumptions2. The final solution is, to sum up, the SP solutions achieved from both the scenarios. The extended method is im- plemented for the magnetic field density formulation and applied to a practical problem (TEAM workshop Problem 21, model B)8. COUPLING SUBPROBLEM TECHNIQUE In the strategy SP, a canonical magneto dynamic problem i, to be solved at procedure i, is solved in a domain Wi, with boundary ¶Wi = Gi = Gh;i[Gb;i: The eddy current belongs to the conducting part Wc;i (Wc;i  Wi), whereas the stranded inductors are the non-conducting WCC , with Wc;i = Wc;i [WCc;i. The Maxwell’s equations together with the following con- stitutive relations3–7. curl hi = ji; divbi = 0; curle ei =¶tbi (1a-b-c) bi = mihi+bs;i; ei = s1i ji+ es;i (2a-b) n eijGe = j f ;i (3) Cite this article : Quoc Vuong D. Improved error of electromagnetic shielding problems by a two- process coupling subproblem technique. Sci. Tech. Dev. J.; 23(2):524-527. 524 Science & Technology Development Journal, 23(2):524-527 where bi is the magnetic flux density, hi is the mag- netic field, ei is the electric field, ji is the electric cur- rent density, m i is themagnetic permeability, s i is the electric conductivity and n is the unit normal exterior toWi. The surface field j f ;i in (2 c) is a surface source (SS) expressed as changes of interface conditions (ICs) and is generally defined as a zero for classical homoge- neous boundary conditions (BCs). If nonzero, it can consider as SS that account for particular phenomena presenting at the idealized thin regions between the positive and negative sides of Gi (G+i and G i ). The source fields bs;i and es;i in (2 a-b) are VSs. In the SPM, the changes of materials from the TS region (i = 1; m1 and s1) to the volume improvement (i = 2; m2 and s2) can be defined via VSs4–7,9. bs;2 = (m2m1)h1; es;2 = (s12 s11 ) j1 (4 a-b) The total fields can be defined via a superposition method1, i.e. b= b1+b2 = m2(h1+h2) (5 a-b) e= e1+ e2 = s12 ( j1+ j2) (6 a-b) FINITE ELEMENTWEAK FORMULATION Magnetic field intensity formulation By starting from the Ampere’s law (1c), the weak con- form magnetic field formulation of SP i (i 1, 2) can be written as3–7: ¶t(mihi; h 0 i)Wi +(s 1 i curl hi; curl h 0 i)Wc;i ¶t(bs;i; h0i)Wi+(es;i; curl h 0 i)Wi +<[nei]gi ; h 0 i>Gi +Gigi=0; 8h 0 i2H1e;i(curl;Wi): (7) The magnetic field hi in (7) is decomposed into parts, hi = hs;i + hr;i, where h s;i is the source magnetic field defined via an imposed electric current density in the stranded inductorsWs;i, that is (curl hs;i; curl h 0 s;i)Ws;i = ( js;i; curl h 0 s;i)Ws;i ; 8h0s;i 2 H1e;i(curl; Ws;i) (8) and hr;i is the associated reaction magnetic field, which we have to define, i.e.( curl hs;i = js;i in Ws;i curl hr;i = 0 inWCc;iWs;i (9) In the non-conducting regionsWCc;i, the reaction field hr;iis thus defined via a scalar potential7. The function space H1e;i (...) in (7) and (8) is a curl - conform containing the basis functions for hi and hs;i as well as for the test function h 0 i and h 0 s;i (at the discrete level, this space is defined by finite edge ele- ments); notations (
,
) and are respectively a volume integral in and a surface integral of the prod- uct of their vector field arguments. The integral sur- face term Gigi on Gh in (7) is defined as a homoge- neous NeumannBC, e.g., imposing a symmetry condition of “zero magnetic flux”, i.e. neijGe = 0) nbijGe;i = 0: (10) The trace discontinuity<[nei]gi ; h 0 i>Gi appearing in (7) is considered as a TS model and given as1: <[nei]gi ; h 0 i>Gi=<mibi¶t(2hc;i+hd;i); h 0 c;i>Gi +< 1 2 [mibi¶t(2hc;i+hd;i)+ 1 sibi hd;i]; h 0 c;i>G+i (11) where hc;i and hd;i are continuous and discontinuous components of hi, and b i is a factor defined as bi = g1i tanh  digi 2  ; gi = 1+ j 2 ; di = s 2 wsimi (12) for di and d i being the local thickness of the TS and skin depth, respectively. Projected solutions between thin shell and volume improvement The obtained solution h1 (i=1) in sub-domain of the TS model W1 is now considered as a VS in a sub- domain of the volume improvement (current prob- lem) W2 (i=2). This means that at the discrete level, the source h 1 solved in the mesh of the W1 has to be projected in meshW2 via a projection method9. This can be done via its curl limited toW2, i.e. (curl h12; curl h 0 2)W2 = (curl h1; curl h 0 2)W2 ; 8 h02 2 H12(Curl; W2) (13) Where H12(Curl; W2) is a gauged curl-conform func- tion space for the projected source ￿12 and the test function h02. Magnetic field intensity formulation with volume improvement The solution in (7) with the TS model (solved from the first scenario) is forced as a VS for solving the second problem (that contains an actual vol- ume/volume improvement) through the volume in- tegrals ¶t(bs;i; h 0 i)Wi and (es;i;curla 0 i )Wi , where bs;i; e 525 Science & Technology Development Journal, 23(2):524-527 s;i are given in (4a-b). For that, the weak formulation for a volume improvement (for example, i =2) is then written as ¶t(m2h2; h 0 2)W2 +(s 1 2 curl h2; curl h2)Wc;i +¶t (m2m1)h1; h02)W2 +((s12 s11 ) j1; curl h 0 2)W2) +hn e2;h02iG2 = 0;8 h 0 2 2 H1e;2(curl; W2): (14) At the discrete level, the source fields h1 and j 1 de- fined in the mesh of the TS model (i = 1) via (7) are now projected in the mesh of the current SP/volume improvement SP (i = 2) via (13) shown in Section Pro- jected solutions between thin shell and volume im- provement. APPLICATION TEST The test problem is a TEAM workshop problem 21 (model B) 8, with two excitation coils and a magnetic steel plate (Figure 1). The thickness of the plate is 10 mm, and the electric conductivity is s = 6.484MS/m, the relative magnetic permeability m = 200, the fre- quency f = 50Hz, and the exciting current of 25A.The test problem is solved in a 3-D case. The distribution of the magnetic flux density b in a cut-plane due to the exciting/imposed current in the coils with a simplified mesh of the TS SP is shown in Figure 2. Figure 1: The Geometry of the 2-D and 3-D 8. The inaccuracies on the eddy current density and Joule power loss density between the TS and volume improvement along the vertical edge (z-direction), with effects of m , s and f (d = 10 mm), are improved by important volume improvements shown in Fig- ure 3. The significant errors near the edges and cor- ners reach 40% with d (skin-depth) = 2.1 mm and thickness d = 10 mm in Figure 3 (top), and 50% in Figure 3(bottom) as well. The volume improvements are then checked to be close to the reference solu- tions (computed from FEM) for different parameters in both Figure 3. The relative improvement of the power loss density along the thin plate is presented in Figure 4. It can obtain up to 65% near the edges and corners of the TS. Figure 2: Distribution of the magnetic flux den- sityb in (a cutplane)due to theexciting/imposed current in the coils, with m = 200,s = 6.484MS/m and f = 50 Hz. The results obtained on the magnetic flux density from the volume improvement are also compared with the measured results8 pointed out in Figure 5. The maximum and minimum errors between two method are proximately 10.9% and 1.5%, respectively. This is said that there is a very suitable validation of the extended method. DISCUSSION AND CONCLUSION A two-process coupling subproblem technique with the magnetic field formulation has been successfully extended for improving errors on the local fields of magnetic flux density, eddy current density and Joule power loss density around the edges and corners of the TS approximations proposed in 2. The obtained results of the method are checked to be close to the reference solution in computation of the classical FEM1 and are also compared to be similar the measured results from a TEAM workshop prob- lem 21 (model, B) proposed by many authors8. This 526 Science & Technology Development Journal, 23(2):524-527 Figure 3: Eddy current density (top) and Joule power loss density (bottom) between the TS and volume solution along the vertical edge (z- direction), with effects of m ,s and f (d = 10mm). Figure 4: Relative improvement of the Joule power loss density along the plate, with the ef- fects m , s and f (d = 10mm). Figure 5: The comparison of the volume im- provement (computed results) andmeasured re- sults 3 along z-direction {x =0.576m, y = 0m}. is also demonstrated that there is a very good agree- ment between the studied technique and experiment methods. The developed technique has been successfully car- ried out with the linear case in the frequency domain. The extension of the method could be also imple- mented in the time domain and the nonlinear case (proposed in10) in next study. All the steps of the technique have been validated and applied to international test problem (TEAM workshop problem 21, model B) 8. In particular, the achieved results is a good condition to analyze the in- fluence of the fields to around electrical/electronic de- vices when taking a shielding plate into account. COMPLETING INTERESTS The author declares that there is no conflict of interest regarding the publication of this paper. AUTHOR’S CONTRIBUTIONS All the main contents, source-codes and the com- puted results of this article have developed by the au- thor. REFERENCES 1. Koruglu S, Sergeant P, Sabarieqo RV, Dang VQ, Wulf MD. Influ- ence of contact resistance on shielding efficiency of shielding gutters for high-voltage cables. IET Electric Power Applica- tions. 2011;5(9):715–720. Available from: https://doi.org/10. 1049/iet-epa.2011.0081. 2. 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