TWO DIMENSIONAL ANALYSIS OF AN E-CORE INDUCTOR R. Mertens, K. Hameyer and R. Belmans Katholieke Universiteit Leuven, Dep. EE (ESAT), Div. ELEN, Kardinaal Mercierlaan 94, B-3001 Heverlee, Belium ABSTRACT: One of the aims of a CAD laboratory in electromanetics is to make students familiar with commercial CAD packaes often based on the finite element technique. To avoid an unquestionable believe in computer results, it is useful to compare the classical approaches with the finite element approach. An E-core inductor with its rather simple construction is chosen as a subject of the CAD laboratory and the inductance of a two dimensional approximation is calculated in different ways. INTRODUCTION In spite of its rather simple construction, a core of laminated iron and a coil of copper (fi. 1), an E-core inductor has a lot to offer as a subject of a CAD laboratory. In the classical approach, the inductor is seen as a manetic circuit (flux tube model), or a superposition of flux-mmf plots of the different tubes of which the inductor is made. Different approximations either include or exclude the manetomotive force in the iron parts, the non-linear behaviour of the material, the leakae flux or the frinin effect in the air ap. No effects in the third dimension are taken into account. Because a two dimensional finite element calculation deals inherently with those effects, it is used as reference. The students have to compare and discuss the results and the restrictions of the various methods. Fiure 1: Three dimensional view of the E-core inductor. CLASSICAL APPROACH Manetic circuit a/ a/ a a/ A manetic circuit based on Hopkinson's law is a first approach to calculate the inductance L. The manetic circuit is described by a flux tube model. Reluctances represent the different flux tubes of the model. The inductance is calculated from the total reluctance R m. c N L R m where N is the number of turns of the coil. A first approximation is obtained by only taken the air ap reluctances into account (fi. 3). It can be noted that the manetic flux throuh the riht or the left le is half of the flux throuh the centre le. This is a parallel connection in terms of a manetic circuit. (1) thickness: b a/ Fiure : Cross section of the E-core inductor. Table 1: Rated values and dimensions of the E-core inductor. L 150 mh I 10 A N 88 3.3 mm a 6 cm b 9 cm c 9 cm
The total reluctance is iven by R R + R m m m1. () is the reluctance of the air ap beneath the centre le and R m is the reluctance of the air ap beneath the riht or left le. The cross-section of the air ap is taken different from the core cross-section to take the frinin effect in the air ap into account. The dimension of the core cross-section is increased with a fraction of the air ap lenth at each side. The crosssection S 1 of the air ap beneath the centre le becomes (nelectin frinin in the third dimension) ( ) S1 a+ 1K b. (3) 1 R m R m Fiure 3: Simplified manetic circuit of the inductor. µ 0 ( a+ 1K ) b R m a µ 0 + 1K b (4) (5) 1 Table 1 and fi. show the dimensions of the inductor used in the CAD laboratory. Table shows the result of the inductance calculation in which the dimension of the core cross-section is increased with the air ap lenth at each side. R m R m Leakae flux Fiure 4: Improved manetic circuit of the inductor. The reluctance of a rectanular slot with current carryin conductors is calculated to deal with the leakae flux in the E-core inductor, resultin in an improved manetic circuit of the inductor (fi. 4). The reluctance is obtained startin from Ampère's law [4]. This exercise is very useful because it stresses that the eneral formulae for the reluctance are only valid in case of an uniform field distribution. a 3 µ bc 0 (6) The total reluctance is iven by 1 1 1 R R 1 + R + m m R m m3. (7) Table 3 shows the result of the inductance calculation based on this improved manetic circuit. Frinin effect based on the Carter factor Two different factors can be used to obtain an improved approximation of the frinin effect in the air ap (fi. 5). The Carter factor, the first one, is defined as the ratio of two air ap lenths, the real lenth of the air ap and the equivalent lenth δ of a smooth air ap (without slots) in which the same manetic enery is stored. a) b) c) Fiure 5: Flux plot of a zone in the air ap of the inductor (a), calculation of the Carter factor (b) and a Carter based factor (c). Table : Inductance calculation based on a simplified manetic circuit of the inductor. 4.381 10 5 H -1 R m 8.367 10 5 H -1 R m 7.97 10 5 H -1 L 0.099 H Table 3: Inductance calculation based on an improved manetic circuit of the inductor. 4.381 10 5 H -1 R m 7.97 10 5 H -1 8.84 10 6 H -1 R m 7.036 10 5 H -1 L 0.118 H
k c δ (8) The Carter factor is defined as [5, 6]: λ kc λ α, (9) ' R' m3 1 with 4 α βarctan( β) ln 1+ β (10) π β w (11) slot width w a (1) R' m1 Fiure 6: Approximation of the frinin effect usin the Carter based factor. slot pitch λ 3 a. (13) 1 The Carter factor can also be calculated by the finite element method (fi. 5b) and is as model very suitable to make the students familiar with the CAD software. Dirichlet boundary conditions are applied at the left and riht hand side of the model. The manetic flux Φ is forced by applyin the correct values for the manetic vectorpotential at the left and riht side of the finite element model. An equivalent manetic reluctance is calculated out the stored manetic enery W. ( ) Φ Aleft Ariht b (14) W 1 RmΦ (15) The same air ap is seen in two different ways, but with the same reluctance, to determine the correction factor k f of the cross-section of the air ap. kc µ λ b µ λ w k b ( + f ) 0 0 (16) k f β α. (17) The cross-section S 1 of the air ap beneath the centre le of the E-core becomes (nelectin frinin in the third dimension) ( f ) S a+ k b. (18) 1 Table 4 shows the result based on the Carter factor. Frinin effect based on a Carter-like factor The second factor is based on the Carter factor and can only be calculated usin the finite element method. Instead of applyin the Dirichlet boundary condition at the left hand side of the model, it is applied at the top side (fi. 5c). R m8 R m7 R m R m6 R m4 R m5 R m6 R m8 R m7 R m Fiure 7: Manetic circuit of the inductor with inclusion of the MMF drop in the iron. Table 4: Inductance calculation with a better approximation of the frinin effect based on the Carter factor. k c 1.7 k f 1.61 4.133 10 5 H -1 R m 7.187 10 5 H -1 8.84 10 6 H -1 R m 6.579 10 5 H -1 L 0.16 H k c (FEM) 1.8 with W 3.357 10 3 J Φ 0.09 Wb Table 5: Inductance calculation with a better approximation of the frinin effect based on a Carter-like factor. W.39 10 3 J Φ 0.09 Wb k f 0.91 4.43 10 5 H -1 R m 8.111 10 5 H -1 8.84 10 6 H -1 R m 7.114 10 5 H -1 L 0.117 H Table 6: Inductance calculation with inclusion of the MMF drop in the iron parts. k f 0.91 µ r 970 4.43 10 5 H -1 R m 8.111 10 5 H -1 8.84 10 6 H -1 R m4 9.115 10 3 H -1 R m5 6.837 10 3 H -1 R m6.735 10 4 H -1 R m7 1.367 10 4 H -1 R m8 3.646 10 4 H -1 R m 7.579 10 5 H -1 L 0.109 H
The equivalent manetic reluctance can be seen as a parallel connection (fi. 7) of the reluctance R' m1 of the air ap, frinin included, and the reluctance R' m3 of a rectanular slot with uniform field distribution. The value of R' m1 is used to obtain the reluctance of the air ap with a better approximation of the frinin effect. Table 5 shows the result based on the Carter-like factor. Inclusion of the manetomotive force drop in the iron parts Fiure 8 shows a possible manetic circuit includin the manetomotive force drop of the iron parts. The iron is assumed to be linear. Further refinement of this model is not necessary because a rouh finite element model takes over. The averae lenth of a flux line in each iron part is determined, to calculate the reluctance of that part. Table 6 ives the results of this calculation. The knowlede of the Y- transformation is needed to simplify the manetic circuit. Inclusion of the non-linear behaviour of the iron parts results in flux dependent reluctances. The inductance can be obtained by solvin a non-linear system of equations. Superposition of flux-mmf plots Another approach includin the non-linear behaviour of the iron parts but not the leakae flux, is the superposition of the flux-mmf plot of the air aps with the flux-mmf plots of the iron parts. Table 7 shows the workin scheme. Due to the difference in manetic flux throuh the left or riht le and the middle le, both paths are seperately handled. The relation between the fluxes defines how the flux-mmf plots have to be combined. The flux-mmf plot of an air ap is defined by its reluctance. The flux-mmf plot of an iron part is obtained startin from the manetization curve (fi. 8). Fiure 9 shows the flux-mmf plots. The last step results in the inductance versus the current (fi. 10). Table 8 ives the value of the inductance for the rated value of the current. FINITE ELEMENT APPROACH The calculation of the inductance is based on the identification of the manetic enery W stored in the inductor expressed in manetic quantities W H B d d V, (18) and the same enery expressed in electric quantities W 1 LI. (19) Fiure 11 shows the flux plot of the finite element solution of the inductor, while table 9 shows the result of the finite element calculation of the inductance. B - manetic flux density [T] 1.8 1.6 1.4 1. 1.0 0.8 0.6 0.4 0. 0.0 0 5000 10000 15000 0000 5000 H - manetic field [A/m] Fiure 8: Non-linear manetization curve of the iron parts. 0.010 0.008 0.006 0.004 0.00 Φ - manetic flux [Wb] iron paths air aps total 0.000 0 4000 8000 1000 16000 0000 MMF - manetomotive force [A] Fiure 9: Total and partial flux-mmf plots. 0.100 0.080 0.060 0.040 0.00 0.000 L - inductance [H] 0 10 0 30 40 50 60 I - current [A] Fiure 10: Calculation of the inductance by superposin the flux-mmf plots. Table 7: Workin scheme for the superposin of flux-mmf plots. Step 1 H B Step MMF H l ΦBS Step 3 MMF Φ R MMF Step 4 MMF Φ MMF N Φ Step 5 I L N I Table 8: Calculation of the inductance by superposin the flux-mmf plots. L 0.097 H m
CONCLUSION The understandin of a simple manetic system such as an inductor forms the basis of more difficult manetic systems as electrical motors and ives an idea on what can be nelected durin a finite element calculation. Comparin the different results shows the importance of frinin, leakae, MMF drop and non-linear behaviour. When a device can not be considered as infinity lon as the E-core inductor, and a three dimensional finite element calculation is required, becomes also clear. ACKNOWLEDGEMENT The authors are rateful to the Belian Nationaal Fonds voor Wetenschappelijk Onderzoek for its financial support of this work and the Belian Ministry of Scientific Research for rantin the IUAP No. 51 on Manetic Fields. Fiure 11: Flux plot of the inductor. Table 9: Two dimensional finite element calculation of the inductance. W 5.94 J L 0.119 H REFERENCES 1. Binns, K. J., Lawrenson, P. J. and Trowbride, C. W., The Analytical and Numerical Solutions of Electric and Manetic Fields, John Wiley & Sons (199).. Lowther, D. A. and Silvester, P. P., Computer-Aided Desin in Manetics, Spriner-Verla (1986). 3. Silvester, P. P. and Ferrari, R. L., Finite elements for Electrical Enineers, Cambride University Press (1990). 4. de Jon, H. C. J., Lipo, T. A., Novtny, D. and Richter, E., AC Machine Desin, University of Wisconsin-Madison, Collee of Enineerin, Auust 3-5 (1993). 5. Carter, F. W., A Note on Airap and Interpolar Induction, JIEE 9, 95 (1900). 6. Carter, F. W., The Manetic Field of the Dynamo-Electric Machine, JIEE 64, 100 (196).