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IEEE TRANSACTIONS ON MAGNETICS, VOL. XX, NO. Y, MONTH 2005 101
Frequency-Domain Homogenization of Bundles of
Wires in 2D Magnetodynamic FE Calculations
Johan Gyselinck and Patrick Dular

Abstract— A general approach for the frequency-domain homogenization of multi-turn windings in 2D FE calcula- tions is presented. First a skin and proximity effect charac- terization of the individual conductors, of arbitrary cross- section and packing, is obtained using a representative 2D FE model. Herein three excitation modes are considered, viz current and flux density in two perpendicular directions. In practical cases, the three modes are independent and the obtained frequency-dependent impedance and complex re- luctivity tensor can be readily used in a FE model of the complete device. By way of example and validation, the method is applied to an inductor having an airgap and one of three different windings. The homogenized model produces global results (impedance versus frequency) that agree well with those obtained with a more precise FE model. In the latter each turn of the winding is explicitly modelled and finely discretized. Keywords— Skin effect, proximity effect, eddy currents, finite element methods. I. Introduction Multiturn windings in electromagnetic devices operated at sufficiently high frequency may display considerable eddy current losses. This constitues a major design aspect for, e.g., swiched mode power supply (SMPS) transform- ers. Unfortunately, the analytical approaches proposed in literature over the last decades do not garantee satisfac- tory results for all practical applications [1], [2], [3], [4], [5]. They may therefore be advantageously combined with numerical methods, in particular the finite element (FE) method [6], [7]. The brute-force FE approach, which consists in explic- itly modelling and finely discretizing each separate turn of a winding in a (2D of 3D) FE model, remains to date too computationally expensive for it to be a practical de- sign tool. The use of homogenization methods is impera- tive, see, e.g., [8] for a general and mathematical approach. Their application to windings is not yet common practice. In the frequency domain, proximity effect and associated losses can be effected by adopting a complex permeabil- ity (or reluctivity), the (frequency-dependent) expression of which is either obtained analytically [4] or using an el- ementary FE model [7], [9]. The latter method has been mainly used for a rectangular packing of round conductors. In this paper we propose a generalization of this ap- proach, allowing for an arbitrary packing (rectangular, hexagonal, . . . ) and conductors of an arbitrary cross- section (round, rectangular, . . . ) [10]. By way of exam- ple, this approach is used for characterising two different Manuscript received June 7, 2004. The research was carried out in the frame of the Inter-University Attraction Pole P5/34 for funda- mental research funded by the Belgian federal government. J. Gyselinck is with the Department of Electrical Engineering, Free University of Brussels, Belgium ([email protected]). P. Du- lar is with the Institut Montefiore, Department of Electrical Engineer- ing, University of Li`ege, Belgium; he is a Research Associate with the Belgian National Fund for Scientific Research (F.N.R.S.). conductors and packings, and will allow to homogenize the winding in a 2D axisymmetric model of an inductor. The results obtained are compared to those obtained with a fine FE model in which each turn is modelled and discretized. II. Skin and proximity effect A. Definitions and notations A sinusoidal regime at frequency f (or pulsation ω = 2πf ) is assumed throughout this paper. A sinusoidally varying quantity, e.g. a current I(t) = √2 Ie cos(ωt + φ), is represented by the complex number (in bold) I = Ie ei φ , where Ie is the r.m.s. or effective value of the current (Ie = √I I ∗ , * denotes conjugate value) and i the imaginary unit (i = √− 1). A winding of n periodically spaced identical conductors (wires, turns) is considered. The conductors have a cross- sectional surface area Ωc , length l, conductivity σ (or re- sistivity ρ = 1/σ) and permeability µ = µ0 (or reluctivity ν = ν0 = 1/µ0 ) . The insulating space between the conduc- tors is nonmagnetic (ν = ν0 ) ; the fill factor of the winding is denoted by λ. The n conductors, connected in series, carry the same (net) current I. For low frequencies, or for a direct cur- rent, the Joule losses P are given by RD C I 2 e = RD C |I|2 , where RD C is the DC resistance, i.e. ρl/Ωc for one conduc- tor and nρl/Ωc for the complete winding. Due to the time- variation of the current I(t) and the transverse magnetic in- duction B(t) “seen” by the conductors, the current density j(x, y, t) is non-uniform in their cross-section (in the xy- plane), which leads to higher Joule losses P = nl Ω c ρj 2 e dΩ. Commonly, two effects are distinguished and the a ssociated losses simply added: skin effect (I ̸= 0, B = 0) and prox- imity effect (I = 0, B ̸= 0). The analytical expressions for round conductors, given hereafter, will further serve as reference for the more gen- eral case. The equivalent radius of the conductors, of ar- bitrary cross-sectional shape, is given by r = Ω c / π. By dividing this radius by the skin depth δ = 2 / (σωµ0 ) , we obtain the unitless parameter X = r/δ = √ f · r√ π σµ0 , which will further be referred to as reduced frequency [3]. B. Isolated circular conductor: analytical solution We consider a single, isolated round conductor of (equiv- alent) radius r. The increase of Joule losses due to skin effect is a function of the reduced frequency X [9]: pI (X) = RA C RD C = X 2 Re (1 + i) J0 (1 + i)X J1 (1 + i)X , (1) with Jα the Bessel function of the first kind and o rder α. The conductor is now placed in a spatially uniform in-

102 IEEE TRANSACTIONS ON MAGNETICS, VOL. XX, NO. Y, MONTH 2005 duction B. The proximity losses are [9] PB = pB (X) π 4 σlr4 ω2 B2e , (2) where the dimensionless factor pB is given by pB (X) = 4 X 2 Im J2 (1 + i)X J0 (1 + i)X . (3) For sufficiently low frequency (X → 0), pB t ends to 1. age induction {B′x , B′ y } in the complete FE model can be imposed. In general, this latter induction value dif- fers (slightly) from {Bx , By }. In practice, by exploiting symmetry and/or linearity, a quickly converging iterative scheme can be used to find the excitation {I, B ′ x , B′y } pro- ducing the wanted values {I, Bx , By }. With {I, Bx , By } equal to {1, 0, 0}, {0, 1, 0} and {0, 0, 1} respectively, three elementary solutions (k = 1, 2, 3) and the associated current density jk (x, y) and induc-

duction B. The proximity losses are [9] PB = pB (X) π 4 σlr4 ω2 B2e , (2) where the dimensionless factor pB is given by pB (X) = 4 X 2 Im J2 (1 + i)X J0 (1 + i)X . (3) For sufficiently low frequency (X → 0), pB t ends to 1. C. Conductors of arbitrary cross-section: FE model In view of a complete eddy current effect characterization of conductors of arbitrary cross-section, a representative 2D FE model comprising a limited number of elementary cells is used. Each such elementary cell, of cross-sectional area Ω, contains one conductor cross-section and the surround- ing insulation (with λ = Ωc / Ω). The FE model consists of a central cell and nl surrounding layers of cells, with nl ≥ 0. Figs. 1 and 2 show the FE model with nl = 1 for the two different winding types that will be considered in the fol- lowing: round conductors with hexagonal packing and rect- angulars conductors with rectangular packing respectively. The conductor surface area Ωc = 1 mm2 (or equivalent ra- dius r = 0.5642 mm) and the fill factor λ = 0.8225 are the same in both cases. A conductivity σ = 6 · 107 S/m is further assumed. age induction {B′x , B′ y } in the complete FE model can be imposed. In general, this latter induction value dif- fers (slightly) from {Bx , By }. In practice, by exploiting symmetry and/or linearity, a quickly converging iterative scheme can be used to find the excitation {I, B ′ x , B′y } pro- ducing the wanted values {I, Bx , By }. With {I, Bx , By } equal to {1, 0, 0}, {0, 1, 0} and {0, 0, 1} respectively, three elementary solutions (k = 1, 2, 3) and the associated current density jk (x, y) and induc- tion {bx ( x, y), by (x, y)}k in the central cell, are readily ob- tained. The complex power S absorbed by the central cell, see e.g. [4], given by S = P + iQ = l Ω ρj 2 e + i ων0 b 2e dΩ , (5) is a quadratic form in {I, Bx , By }, the complex coefficients of which are obtained by integrating the product of the corresponding two elementary solutions over the central cell. With both the real and the imaginary part of S, active power P and reactive power Q, can be associated a 3 × 3 complex Hermitian matrix (transposed matrix equal to conjugate). These two matrices give a complete skin and proximity effect characterization of the winding. In the general case where the conductor does not have any sym- metry, the matrices have no other property besides the fact that they are Hermitian. Such a (rather academic) case, viz a round conductor with an eccentric hole, is treated in [10]. The coupling between skin and proximity effect is evidenced.

D. Conductors of symmetric cross-section

Fig. 1. Round conductors (radius r = 0.5642 mm) with hexagonal packing – elementary flux pattern for skin effect (left) and proximity effect (right), both for X = 2 Fig. 2. Rectangular conductors (1.4142 × 0.7071 mm2 ) with rectan- gular packing (1.4827 × 0.82 mm2 ) – elementary flux pattern for skin effect (left) and proximity effect (right), both for X = 2 Time-harmonic finite eddy current calculations using the well-known magnetic vector potential (MVP) formulation [11] are carried out. The three (scalar) quantities of interest for the homogenization are the net current I in the con- ductors and the two components of the average induction in the central cell , {Bx , By }. The current I is straightfor- wardly enforced in each conductor [11]. By means of the following condition on the outer boundary a(x, y) = yB′x − xB′y , (4) where a(x, y) is the z-component of the MVP, the aver- In more practical cases, as the ones treated in this paper (Figs. 1 and 2), the conductors are symmetric with respect to x and y axes. Thanks to the spatial orthogonality of the three elementary solutions, there is no coupling between the three excitation modes, and, in particular, skin and proximity effect are uncoupled. With the complex power absorbed in the central cell due to the current I flowing in the conductors can be associated a complex impedance Zs kin (X) : Zs kin = Ss kin I 2 e = pI (X) RD C + i qI (X) ω µ0 l 8π . (6) The expression for the imaginary part in (6), which is usu- ally negligible (see below), is based on the inductance of a round conductor (internal field only). The unitless factors pI (X) and qI (X) follow from the first elementary solution. Analogously, for an induction B in a certain direction, the complex power S can be written as Sp rox lΩ B2e = pB (X) 1 4 λσr2 ω2 + i qB (X) ων0 , (7) where the expression for the real part is based on the an- alytical expression (2) for low-frequency proximity losses in a round conductor (πR2 = λΩ). The unitless factors pB (X) and qB (X) for induction in x and y direction follow from the second and third elementary solution respectively. Figs. 3 and 4 give the frequency dependence (0.1 ≤ X ≤ 3) of the skin and proximity effect coefficients obtained with the FE models depicted in Figs. 1 and 2 respectively. (Note that for the dimensions chosen, X equal to 0.1, 1, 2 and 3 corresponds to f equal to 132.6 Hz, 13.26 kHz, 53.05 kHz and 119.4 kHz.) These results are obtained with one layer

GYSELINCK ET AL.: FREQUENCY-DOMAIN HOMOGENIZATION OF BUNDLES OF WIRES 103 of cells around the central cell (nl = 1). This is sufficiently accurate for the cases considered: e.g., for the round con- ductor and pB (X = 2), nl = 0 gives a 1.4% higher value, and nl = 2 a 0.015% lower value. The anisotropy of the round conductor with hexagonal packing has been found to be negligible, which is obviously not the case for the rectangular conductor. 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 0 0.5 1 1 .5 2 2.5 3 s k i n e f f e c t c o e f f i c i e n t Fig. 3. Skin effect coefficients pI and qI versus reduced frequency X for the two conductors 3.0 3.5 imity losses can be simply effected by means of a complex and frequency dependent reluctivity tensor νp rox : νp rox = qB (X) ν0 + i pB (X) 1 4 λσr2 ω (8) For round conductors, this tensor can be reduced to a scalar quantity. For rectangular conductors, with the sides paral- lel to x and y axes, the tensor is diagonal. Further, in the electrical circuit equation linking the ter- minal voltage of the winding, U , to the current I and the vector potential a [11], the DC resistance RD C has to be replaced by impedance Zs kin defined in (6). IV. Application example The homogenization method will now be applied to a 2D axisymmetric FE model of an inductor. Its 144-turn wind- ing is made of either of the two conductors characterized in section II (same cross-section and fill factor). The induc- tor model with round conductors and hexagonal packing is depicted in Fig. 5. In case of rectangular conductors and rectangular packing, the longest side of conductor cross- section is either in radial direction (along x-axis) or vertical (along y-axis), as shown in Fig. 6.

2.0

1.5   1.0  0.5 0.0

0 0.5         1.5    !  "# 2 1 2.5 3

Fig. 4. Proximity effect coefficients pB and qB versus reduced fre- quency X for the two conductors (for the rectangular conductors, flux along either x-axis or y-axis) Further calculations confirm some observations done in [9], namely that the skin effect losses (coefficient pI ) de- pend little on the fill factor λ, unlike the proximity losses (coefficients pB and qB ).

pend little on the fill factor λ, unlike the proximity losses (coefficients pB and qB ). III. Homogenization of winding 2D FE model In a 2D FE model, with either translational or axial sym- metry and either in the time or the frequency domain, the eddy currents in a multi-turn winding can be mod- elled directly and precisely by defining each conductor as a so-called massive conductor [11]. As a rule of thumb, the characteristic length of the discretization of the conductors, denoted by lc , has to be at least three times smaller than the skin depth δ. This quickly results in a prohibitively large number of unknowns. Furthermore, the additional electrical circuit equations for allowing a voltage supply of the winding or the coupling with an external circuit, have a detrimental effect on the conditioning of the system matrix and thus on the computation time. In the frequency domain, the eddy current effects can be easily taken into account by means of a homogenization technique [9], [4]. The complete winding is then modelled as a so-called stranded conductor, in which a uniform cur- rent density j = nI/Ωw , with n the number of conductors (wires, turns) and Ωw = nΩc / λ the overall surface area, is adopted [11]. A fine discretization of the winding domain Ωw , linked to the skin depth, is not required. On the basis of the above complex power considerations (5), the prox- Fig. 5. 2D axisymmetric model of inductor (round conductors and hexagonal packing, total inductor height 38 mm, outer radius 25 mm, height of winding window 27 mm, outer radius 20 mm, airgap thick- ness 3 mm and radius 10 mm) – flux lines (real part) with imposed unit current I = 1, at X = 0.1 (left) and X = 2 (right) This results in three different windings (having the same DC resistance of 0.22 Ω), for which time-harmonic calcu- lations with imposed unit current I = 1 are carried out, with the reduced frequency X ranging from 0.1 to 2. Both a fine model and a homogenized model are used. In the for- mer model, each of the 144 conductors is finely discretized (lc = r/8), leading to a total of 84000 to 92000 real un- knows and a computation time of 200 CPUs (on a Pen- tium 4, 2 GHz) for one time-harmonic calculation. In the homogenized model, the discretization of the winding cells is much coarser (lc = r),which results in ten times less un- knowns, viz 8000 to 9000, and a calculation time of only 1.5 s. Some flux patterns are shown in Figs. 5 and 6. In the lat- ter figure, one clearly observes the anisotropic eddy current effect. Figure 7 represents the vertical induction component along a radial line (10 mm ≤ x ≤ 20 mm, y = 0) situated in the horizontal symmetry plane and cutting the winding. The induction obtained with the fine model clearly shows

104 IEEE TRANSACTIONS ON MAGNETICS, VOL. XX, NO. Y, MONTH 2005

80 70 60

 50        40 30 20 10 0

0 0.5 1       1.5 2 0.5 1   !  "#$ %'& 1.5 2 3.95 3.90

Fig. 6. Winding with rectangular conductors (two orientations) – flux lines (real part, with imposed unit current I = 1, X = 2) in winding window, obtained with fine and homogenized model

inductance (mH)

3.85

the trace of the individual conductors located on the line
or adjacent to it.

3.80 3.75 3.70 3.65 3.60 0.05 3.55 0.04

3.50 Re induction (T)

0.03 0

Fig. 8. Resistance (up) and inductance (below) versus reduced fre- quency X, obtained with fine model (markers 3, +, ×) and homog- enized model (lines)

0.02 0.01 0.00

Im References

-0.02 -0.01 10 12 14 16 18 20 x coordinate (mm) Fig. 7. Vectical induction component (real and imaginary part) versus x-coordinate, with round conductors and X = 2, obtained with fine model (thin line) and homogenized model (thick line) The impedance of the inductor, Z = S/I 2 = R + iωL, is calculated via the complex power S. For the fine model, S is directly evaluated using (5); for the homogenized model, this is done using νp rox and Zs kin . Figure 8 shows the resis- tance R and the inductance L as a function of the reduced frequency X. For all three windings a very good agreement is observed between the impedance obtained with the fine model and with the homogenized model. Note that with increasing frequency, the contribution of the skin effect losses to the total Joule losses quickly be- comes negligible (compare pI (X)RD C with the total induc- tor resistance R for e.g. X = 1). One also verifies that the contribution of the qI -term in (6) to the total inductance is negligible: 1 µH (or less) compared to 3.5 mH (or more). V. Conclusions For windings with round and rectangular conductors, the homogenization method presented amounts to the usage of a frequency-dependent resistance and a complex frequency- dependent reluctivity tensor. The value of both is obtained using a computationally cheap 2D FE model. The real-life application example clearly demonstrates the ease of im- plementation, the accuracy and the favorable computation time of the homogenization method. After this numerical validation stage, future work will comprise experimental valdation and extension of the homogenization method to the time domain. References [1] P. Dowel, “Effects of eddy currents in transformer windings”, Proc. IEEE, vol. 113, pp. 1387–1394, August 1966. [2] F. Ferreira, “Improved analytical modeling of conductive losses in magnetic components”, IEEE Trans. Power Electron., vol. 9, no. 1, pp. 127–131, January 1994. [3] F. Robert, P. Mathys and J.-P. Schauwers, “Ohmic losses cal- culation in SMPS transformers: numerical study of Dowell’s approach accuracy”, IEEE Trans. on Magn., vol. 34, no. 4, pp. 1255–1257, July 1998. [4] O. Moreau, L. Popiel and J. L. Pages, “Proximity losses compu- tation with a 2D complex permeability modelling”, IEEE Trans. on Magn., vol. 34, no. 5, pp. 3616–3619, September 1998. [5] C. Larouci, J.-P. Keradec, J.-P. Ferrieux, L. Gerbaud and B. Lebedev, “Copper losses of flyback transformer: search for analytical expressions”, IEEE Trans. on Magn., vol. 39, no. 3, pp. 1745–1748, May 2003. [6] C. Sullivan, “Computationally efficient winding loss calcula- tion with multiple windings, arbitrary waveforms, and two- dimensional or three-dimensional field geometry”, IEEE Trans. on Power Electron., vol. 16, no. 1, pp. 142–150, January 2001. [7] X. Nan and C. Sullivan, “An improved calculation of proximity- effect loss in high-frequency windings of round conductors”, in Proc. IEEE 34th Annual Conference on Power Electronics Spe- cialist (PESC), Acapulco, Mexico, June 15–19, 2003, vol. 2, pp. 853–860. [8] M. Feddi, Z. Ren and A. Razek and A. Bossavit, “Homogeniza- tion technique for Maxwell equations in periodic structures”, IEEE Trans. on Magn., vol. 33, no. 2, pp. 1382–1385, March 1997. [9] A. Podoltsev and B. Lebedev, “Analysis of effective resistance and eddy-current losses in multiturn winding of high-frequency magnetic components”, IEEE Trans. on Magn., vol. 39, no. 1, pp. 539–548, January 2003. [10] J. Gyselinck and P. Dular, “Homogenisation of bundles of wires of arbitrary cross-section taking into account skin and proximity effect and their coupling”, in Proc. Fifth IEE International Con- ferenve on Computation in Electromagnetics (CEM), Stratford- upon-Avon, UK, April 19–22, 2004, pp. 137–138. [11] P. Lombard, G. Meunier, “A general purpose method for electric and magnetic combined problems for 2D, axisymmetric and tran- sient systems”, IEEE Trans. on Magn., vol. 29, no. 2, pp. 1737– 1740, March 1993.