Laurent Taylor

Xavier Margueron
email: xavier.margueron@ec-lille.fr

Yvonnick Le Menach

Philippe Le Moigne

Numerical Modeling of PCB Planar Inductors: Impact of 3D Modeling on High Frequency Copper Loss Evaluation

Loss values are key parameters for designing high performance high frequency (HF) magnetic components for power electronics (PE) converters. With the increase of PE switching frequencies, copper losses have to be precisely quantified, ideally until some megahertz. In the literature, many 2D numerical simulations based on finite element analysis (FEA) are performed for such computations. 3D FEA studies of planar components are still limited because of modeling problems, computational resources and computing time. In this paper, quantitative comparisons between 2D and 3D simulation results for planar inductors are achieved focusing on copper loss computation. Results are compared in terms of simulation performances and accuracy. The aim of the paper is to highlight benefits of 2D and 3D FEA simulations in order to choose the appropriate model according to the studied problem.

Introduction

High Frequency (HF) planar magnetic components (PMC) are interesting solution for the integration of power electronic converters. Indeed, in comparison with other classical HF wounded components, planar inductors and transformers are low-profile and lead to more power density in smaller and lighter components [1,2]. They are particularly dedicated to embedded devices and transportation systems [1,[START_REF] Teu Magambo | Planar Magnetic Components in More Electric Aircraft: Review of Technology and Key Parameters for DC-DC Power Electronic Converter[END_REF], where size and weight are important criterias. The other main advantage, dealing with this technology, is the low dispersion characteristic linked to their fabrication process. The latter also leads to more controllable and predictable parasitic parameters [START_REF] Hurley | Transformers and Inductors for Power Electronics: Theory, Design and Applications[END_REF]. PMC now can be found in a lot of applications: Resonant converters [START_REF] Water | Improved High-Frequency Planar Transformer for Line Level Control (LLC) Resonant Converter[END_REF][START_REF] Ibanez | Novel technique for bidirectional series-resonant DC/DC converter in discontinuous mode[END_REF], gate drivers [START_REF] Am | Design methodology for optimising a high insulation voltage insulated gate bipolar transistor gate driver signal transmission function[END_REF], Electromagnetic compatibility (EMC) filters [START_REF] Chen | Improving the Characteristics of integrated EMI filters by embedded conductive Layers[END_REF][START_REF] Tan | Leakage Inductance Analytical Calculation for Planar Components with Leakage Layers[END_REF], high voltage or high current applications [START_REF] Feng | EMI Modeling and Simulation of High Voltage Planar Transformer[END_REF][START_REF] Wang | Prototype design for a high-voltage high-frequency rectifier transformer for high power use[END_REF][START_REF] Lefranc | Optimisation and characterisation of a planar transformer with a high voltage ratio and high output voltage for plasma reactors[END_REF][START_REF] Margueron | Current Sharing Between Parallel Turns of a Planar Transformer: Prediction and Improvement Using a Circuit Simulation Software[END_REF] for example.

From a practical point of view, PMC's windings can be made with multilayer printed circuit board (PCB) or copper foils. Due to these technologies, there are significant differences between its geometrical sizes: height, width and length of copper layers and magnetic core. This singularity enables to obtain components with a really good efficiency but leads to problem when dealing with numerical modeling. Indeed, to simulate resistive and inductive variations in HF, meshing has to be very thin (typically 20µm@30M Hz) in order to take into account skin and proximity effects inside conductors. Such problem requires a performing computing code and also important computational resources.

This paper investigates the problem of HF planar component numerical modeling. In the literature, numerous papers present 1D planar modeling [START_REF] Margueron | Current Sharing Between Parallel Turns of a Planar Transformer: Prediction and Improvement Using a Circuit Simulation Software[END_REF][START_REF] Prieto | Magnetic Component Model for Planar Structures Based on Transmission Lines[END_REF][START_REF] Chen | A Systematic Approach to Modeling Impedances and Current Distribution in Planar Magnetics[END_REF][START_REF] Tria | Implementation of a Nonlinear Planar Magnetics Model[END_REF] as well as comparison between ana-lytical calculations to 2D numerical computations for copper losses evaluation [START_REF] Dimitrakakis | High-Frequency Copper Losses in Magnetic Components With Layered Windings[END_REF][START_REF] Reatti | Comparison of various methods for calculating the AC resistance of inductors[END_REF][START_REF] Yan | Model and Design of PCB Parallel Winding for Planar Transformer[END_REF][START_REF] Han | A Practical Copper Loss Measurement Method for the Planar Transformer in High-Frequency Switching Converters[END_REF][START_REF] Podoltsev | Analysis of effective resistance and eddy-current losses in multiturn winding of high-frequency magnetic components[END_REF][START_REF] Wang | An Improved Calculation of Copper Losses in Integrated Power Inductors on Silicon[END_REF]. Although analytical models are precise in low frequency (LF), the error on copper loss calculations strongly increases with frequency and can reach more than 20% in HF [START_REF] Dimitrakakis | High-Frequency Copper Losses in Magnetic Components With Layered Windings[END_REF]. One of the reasons is linked to the space between conductors that cancel the 1-D hypothesis on the magnetic field, commonly known as the Dowell hypothesis [START_REF] Dowell | Effects of eddy currents in transformer windings[END_REF]. 2D numerical modeling can then be used to overcome this problem. However, differences still exist while comparing simulation results with measurements on real devices [START_REF] Dimitrakakis | High-Frequency Copper Losses in Magnetic Components With Layered Windings[END_REF][START_REF] Han | A Practical Copper Loss Measurement Method for the Planar Transformer in High-Frequency Switching Converters[END_REF]. The reason is that the 2D models are too simplistic and do not take into account the effect of the third dimension. For some years, 3D modeling of planar components has appeared in the literature [START_REF] Water | Improved High-Frequency Planar Transformer for Line Level Control (LLC) Resonant Converter[END_REF][START_REF] Aime | Numerical Methods for Eddy Currents Modeling of Planar Transformers[END_REF] but such simulations are still limited, especially when dealing with HF current density calculation and copper losses. The implementation of 3-D Finite Element Analysis (FEA) simulations is time-consuming and needs high performance computing resources. Even if computer performances still increase, it can be interesting to answer some questions: With regard to planar component HF copper loss computation, is a 3D modeling really necessary? Is a 2D approach sufficient for designers? What is really lost with such simplification?

In this paper, 2D and 3D FEA are performed on HF planar components to compute HF copper losses. Simulations results are analysed and compared to answer the previous questions. The paper is organized as follows: Section 2 starts with an introduction to HF copper loss numerical computation and problems dealing with the modeling of planar magnetic components. Then, in section 3, 2D and 3D simulations are performed on two types of devices: 2-layer PCB without magnetic core and complete planar inductors, including PCB's windings and ferrite cores. 2-D and 3-D FEA simulation results are precisely analysed to highlight differences between both kinds of FEA simulations. In section 4, impedance measurements are realized on prototypes to be compared to FEA results. Finally, all these comparisons lead to conclude on the benefits, drawbacks and usefulness of 3D modeling for HF planar components.

Problematic of HF Power Electronic Magnetic Component Modeling

Review of 3D Modeling Methods

In the literature, 2D FEA is widely performed to compute copper losses in magnetic components [START_REF] Dimitrakakis | High-Frequency Copper Losses in Magnetic Components With Layered Windings[END_REF][START_REF] Reatti | Comparison of various methods for calculating the AC resistance of inductors[END_REF][START_REF] Yan | Model and Design of PCB Parallel Winding for Planar Transformer[END_REF][START_REF] Han | A Practical Copper Loss Measurement Method for the Planar Transformer in High-Frequency Switching Converters[END_REF][START_REF] Podoltsev | Analysis of effective resistance and eddy-current losses in multiturn winding of high-frequency magnetic components[END_REF][START_REF] Wang | An Improved Calculation of Copper Losses in Integrated Power Inductors on Silicon[END_REF]. Such simulations enable to validate hypothesis used in analytical modeling. The study of HF magnetic component in 3D is still limited because of problems linked to modeling complexity, computing resources and calculation times. To avoid these difficulties and simplify the numerical problem, some methods have been developped based on the use of multiple 2D simulations [START_REF] Aime | Numerical Methods for Eddy Currents Modeling of Planar Transformers[END_REF][START_REF] Prieto | Study of 3-D magnetic components by means of "double 2-D" methodology[END_REF].

Regarding other numerical modeling method, the Partial Element Equivalent Circuit (PEEC) method [START_REF] Ruehli | Equivalent Circuit Models for Three-Dimensional Multiconductor Systems[END_REF] enables the computation of PCB without magnetic cores. Each conductor is divided into elementar ones, where the current is considered as uniform. Some equivalent circuits are then obtained for each elementary conductor and connected together. However, this method can not be directly applied for PMC because magnetic material can not be considered. Some extensions have been developped such as the the µPEEC method [START_REF] Ngoc | New PEEC formulation for modeling 2D core, transformer. Principles, academic and industrial applications[END_REF] which takes into account magnetic material but only for 2D modeling as well as the PEEC-BIM [START_REF] Kovacevic | Full PEEC Modeling of EMI Filter Inductors in the Frequency Domain, Magnetics[END_REF] that can be used for 3D computation but limited to toroidal cores. Recently, FEA and PEEC benefits were combined for the computation of Litz Wire Systems [START_REF] Roßkopf | Calculation of Power Losses in Litz Wire Systems by Coupling FEM and PEEC Method[END_REF]. Examples of 3D and 2D numerical models of planar components are presented in The Fig. 1(c).

The typical lenghts of Ferroxcube [START_REF]Soft Ferrites and Accessories[END_REF] planar cores (Fig. 1(d)) are listed in Table 1. Copper track thicknesses h t depend on the technology of PCB's manufacturers. Standard values of h t are 35µm, 70µm, 105µm or 210µm while w t is a design parameter that can vary up to the length of core's window w w .

A close look to all these values shows that there is a large difference between all these dimensions. This specificity leads to meshing problems while modeling such component. Indeed, meshes have to be very thin inside the copper conductors to take into account HF effects, while the mesh's size can be relaxed for the magnetic core. Between PCB and core, meshing has to be extended gradually, particularly in corners of PCB rectangular conductors. As a consequence, with such 3-D meshing, the number of unknows will increase exponentially and will lead to huge numerical problems to be solved.

Impact of Meshing

Meshing, i.e. the numerical modeling spatial discretization, is a key point to solve a finite element (FE) problem. In order to obtain valid FE simulations in wide frequency range, meshing has to be densified in areas where fields' variations (E, B) are important.

Let's take the example of an infinite plate (Fig. 2). The current density distribution is linked to equation (1) and the field (H) is tangential to the plate. Then, the volumic current density is one dimensional J(x, y, z, t) = J(x, t) (2). The evolution of the envelope of J at t = 0 is presented on Fig. 2 to highlight the effect of spatial discretization.

H 0 J(x) J0 π 2 δ πδ 3π 2 δ 2πδ
∆J = µσ ∂J ∂t (1)
where µ and σ are the permeability and the conductivity of the material, respectively.

J(x, t) = J 0 • e -x δ cos(ωt - x δ )e y (2)
where δ (3) is the skin depth.

δ = 1 πσf µ ( 3
)
where f is the frequency. In order to facilitate the interpretation of the skin effect, also known as the Kelvin effect, the concept of "fictive shell" (Fig. 3(a)) is proposed in [START_REF] Boucherot | Effet de peau[END_REF]. At the energetic level, it is as if J flows through a thickness δ (3) where ∀x ∈ [0, δ] , |J(x)| = J 0 / √ 2. Indeed, the power density P s is equivalent with both distributions (4) [START_REF] Water | Improved High-Frequency Planar Transformer for Line Level Control (LLC) Resonant Converter[END_REF]. However, the current density distribution is continuous and significant on thicknesses of many δ. The Fig. 3(a) highlights the impact of a coarse mesh on the computation of J. On this figure, s is the space between two nodes of the mesh. J is supposed to be well calculated on these nodes and intermediate points are determined based on linear interpolation. Error, r (P ref Cu , P

P s = ∞ 0 ρJ 2 (x)dx = ρ ∞ 0 J 2 (x)dx (4) ∞ 0 (J 0 ) 2 e -2x δ = δ 0 (J 0 ) 2 2 dx = J 2 0 δ 2 (5)
s δ
Cu ) , between the exact solution of copper loss calculation (P ref Cu ) and the one obtained with different mesh's size (P s δ Cu ) is plotted on Fig. 3(b). To obtain a computation's error below 1%, it appears that 3 meshes have to be included in the skin depth (3), at least. The table 2 lists some skin depth values and their associated maximum mesh value.

Finite Element Problem to be Solved

PMC are made of three different materials: electrical conductor (copper), magnetic conductor (ferrite) and insulating material (FR4, or Kapton for example) for PCB (Fig. 1(d)).

To calculate HF impedances, full Maxwell's equations [START_REF] Bossavit | Computational Electromagnetism[END_REF] have to be solved taking into account the permittivity of the PCB. For the frequency below 10 MHz, the finite element problem can be reduced to a time harmonic eddy current problem (6) (7) [START_REF] Chen | Improving the Characteristics of integrated EMI filters by embedded conductive Layers[END_REF]. In this case, the insulating

rotH = J s + J ed (7) divB = 0 (8)
where E, H and B are the electrical field, the magnetic field and the magnetic flux density, respectively. The current density vector (7) can be divided in two terms : one source (J s ) and one induced (J ed ). The boundary conditions are given by :

E ∧ n = 0| Γe (9) B.n = 0| Γ b (10)
In this paper, the eddy current problem is solved with potential formulation (T, Ω) [START_REF] Ren | Comparison of some 3D eddy current formulations in dual systems[END_REF]. To take into account the current I, the vector potential K and N are introduced such as

rotK = N (11) Γe Nds = 1 (12) divN = 0 in D c (13)
Consequently, one can write

J = J s + J ed = rot(KI + T) and H = KI + T + gradΩ (14)
Then the (T, Ω) formulation takes the following form

rot 1 σ rot(KI + T) + jωµ(KI + T + gradΩ) = 0 in D c divµ(KI + T + gradΩ) = 0 in D nc
with σ the electrical conductivity and µ the magnetic permeability. In our case I is fixed. The FE formulation is solved using code Carmel3D [START_REF]code Carmel3D[END_REF]. This FE code was first developed for low frequency electrotechnical machines. For HF magnetic components, code Carmel3D presents some advantages:

1. Both FE formulations (T, Ω) and (A, Φ) can be used even if the (A, Φ) one is unusable with small mesh (< 0.1mm).

2. code Carmel3D presents high performances in terms of number of unknows (some Millions).

3. The FE code is open and can be used for reduced order model [START_REF] Taylor | Model-Order Reduction of Magneto-harmonic Problems Based on POD. Application to Planar Magnetic Components[END_REF], useful for faster computations.

As a major drawback, the meshing has to be very thin in HF, as well as in other commercial codes [START_REF] Roßkopf | Calculation of Power Losses in Litz Wire Systems by Coupling FEM and PEEC Method[END_REF].

Modeling of Planar inductors

In this section, FEA is performed on 2D and 3D numerical models of planar components. Results are compared in terms of copper losses and current densities. Two types of devices are modeled:

• "PCB": For these devices, only windings of planar magnetics are modeled. PCB windings with 2-turns, 4-turns and 8-turns, splitted on 2-layers, are simulated in 2D and 3D. As an example, the 3D model of the 4-turn PCB is presented in Fig. 6(a).

• "Inductor": Planar inductors are modeled including PCB winding combined with lossless magnetic core. Examples are given in Fig. 5 for the 2D model and in Fig. 6(c) for the 3D one.

Components' Description

As mentioned before, PCB are made of 2, 4 and 8 turns on 2 layers of thickness h t = 35µm. For each case, conductor's width (w t ) is 7mm, 3mm and 1.5mm respectively. The three inductors are based on the use of these PCB for windings and a double E32/6/20 3F 3 ferrite magnetic core [START_REF]Soft Ferrites and Accessories[END_REF], whose dimensions are given in Table 1. All these devices are used to study the impact of geometry on copper loss distribution. The lenghts were set to present large discrepancy between minimal and maximal dimensions as well as element's size for the mesh.

For the numerical computation of HF copper losses, some simplifications are done:

1. Electromagnetic behavior of the epoxy FR4 (Fig. 1(d) from PCB is not taken into account (µ r = 1). Therefore, Epoxy is not modeled in FEA simulations.

2. Capacitive effects between conductors are not considered. FEA is only performed to focus on copper losses in a frequency range where the device will be used, including main harmonics. These parasitic capacitive effects between copper tracks and between copper and ferrite core appear for higher frequency. This topic is not in the scope of this study.

3. 3F 3 Ferrite magnetic material is supposed lossless and its permeability is constant (µ r = 2000). The ferrite core modify magnetic field around copper conductors and then copper losses. If magnetic core losses were considered in the FEA, it will be difficult to separate both types of losses in particular during the experimental validation stage. The inductance value an its variation with frequency is not either in the scope of this work.

2D Modeling

Only winding windows are modeled for these 2D simulations. For PCB device, only the copper tracks are modeled while for inductors, the PCB windings are included in a ferrite core. Conductor Copper loss densities [START_REF] Chen | A Systematic Approach to Modeling Impedances and Current Distribution in Planar Magnetics[END_REF] in the three inductors are presented in Fig. 5. Note that the evolution of losses with the frequency is different for the three models. Indeed, because of skin and proximity effects, the current is concentrated at the ends of the conductors. This effect varies according to the ratio of the width of the conductor w t (Fig. 1(d)) on the skin thickness (3). Thus, the planar inductor of 8 turns with w t = 1.5mm has a better HF behavior than the 2 turns one with w t = 7mm. The use of 2D simulations presents real benefits in terms of computations but needs to have homogenous losses (i.e. same current density on the tracks's lenghts) inside the component to be relevant.

ρ cu = J • E (15)

3D Modeling

For 3D FEA, the PCB and inductors under simulation are enclosed in an air cubic box of length 10cm. The magnetic field H is tangential to the outer surfaces of the box. A current of 1A rms value flows through the copper tracks and a very fine mesh of conductors is used to simulate the resistance variations in HF (until 30M Hz). As an example, the number of FE unknowns reaches 1.9 Million (with the use of 40 Go RAM) for 14µm meshes inside the conductors of the 2-turn inductor.

First, only copper layers on PCB are modeled. Such application, without magnetic core, can be interesting for contactless energy transfer [START_REF] Cove | Improving Wireless Power Transfer Efficiency Using Hollow Windings With Track-Width-Ratio[END_REF], HF drivers [START_REF] Koudra | Design of a driver for Thyristors Using a New Extra Flat Planar Transformer Without Magnetic Circuit[END_REF], bendable transformer [START_REF] Ho | Modeling and Analysis of the Bendable Transformer[END_REF] or very high frequency (VHF) transformer for example [START_REF] Sepahvand | 100 MHz isolated DC-DC resonant converter using spiral planar PCB transformer[END_REF]. Fig. 6(a) presents copper loss density distribution for the 4-turn PCB without core. Results are presented for 10kHz, 100kHz and 1M Hz, respectively. The 2D simulation loss homogenization hypothesis is verified for low frequency (LF) 10kHz and 100kHz (Fig. 6(a)). Indeed, the copper loss density plotted on a cross section perpendicular to the current flow direction is identical all along the copper tracks. The Fig. 6(b) presents these areas where this hypothesis is valid. It can be noted that in the copper track corners, the geometry is different and hence the copper loss density.

For 1M Hz, a current concentration is observed on the extremity of copper layers (Fig. 6(a)). That concentration in the PCB ends leads to an increase of the copper losses. Copper loss density is also stronger in the PCB corners. The area where the 2D simulation loss homogenization hypothesis is valid is limited to the PCB central part (Fig. 6(b)). 2D simulations will only give correct losses on approximately 25% of the 4-layer PCB.

Then, the ideal planar magnetic core is added to the PCB. Copper loss density repartition, obtained at 1M Hz, is shown in Fig. 6(c) for the 3 inductors with 2, 4 and 8 turns, respectively. A current concentration on the conductors' edge can be observed, as in the previous case without magnetic core. However, the losses distribution along the inductor is more homogeneous. Considering the copper tracks under the magnetic core, 2D simulations can give correct losses estimation on 50% of the whole component (Fig. 6(d)).

A local study enables to understand differences between 2D and 3D FEM simulations. Indeed, a close look at the current density vector field (Fig. 7) allows highlighting the origin the HF losses in windings' ends. For the 4-turn PCB case (Fig. 7(a)), the proximity effect between turns involves the emergence of current loops. A current flowing in the opposite direction is created on a large part of the inner conductors. This effect appears only when the conductors width is large enough compared to the skin depth (3). This phenomenon changes loss distribution and thus increase the copper losses. As a comparison, the current density vector field for the 4-turn inductor is shown in Fig. 7(b). Current loops are less important than for the case of the PCB alone. A current is still flowing in the opposite direction but its proportion is much lower. This effect is greater on the copper tracks outside the magnetic core. Under the core, the magnetic flux generated by a turn is concentrated in the magnetic material and have less impact on other close conductors. The 2D loss homogenization hypothesis is consistent inside the magnetic core's winding. Without core, the copper loss density varies with the third dimension.

Comparison and Analysis

In the previous section, 2D and 3D FEM simulations were performed for PCB and inductors with 2, 4 and 8 turns. Results are now compared together to deduce rules of 2D and 3D FEA usage.

Overall Analysis: Resistive Factor Comparison

The global comparison of 2D and 3D modeling results is made based on the relative resistance factor F r [START_REF] Tria | Implementation of a Nonlinear Planar Magnetics Model[END_REF]. This factor, applied to copper windings, allows quantifying the increase of resistance with frequency. The relative error r is also calculated to quantify the difference between simulations results and measurements.

F r (f ) = R AC (f ) R DC (16)
4.1.1. PCB: In order to compare FEA simulation results, impedance measurements are performed on a 4-turn 2-layer PCB. These measurements are achieved using impedance analyzer HP4294 [START_REF]Agilent 4294A Precision Impedance Analyser-Operation Manual[END_REF]. Resistance values are plotted on a large frequency range, from 1kHz to 30M Hz. The Fig. 8(a) presents the 4-turn PCB relative resistance factors for 2D, 3D FEA results and prototype's measurements. The relative errors ε r are plotted in Fig. 8(b). 2D simulations underestimate the resistance between 100kHz and 30M Hz. The resistance value and thus the copper looses are calculated with 10% error. The difference between simulation results and measurements for frequency higher than 5MHz can be explained by two main reasons:

1. The parasitic capacitance C p that exists between the two PCB layers leads to a resonance with 2. Meshing is also a source of errors. Indeed, for 3D simulations, meshes are set to 60µm while for 2D ones, meshes are 6 times thinner (about 10µm). As a consequence, the 3-D simulation is also underestimated from 8MHz.

4.1.2. Planar Inductors: Regarding inductors, measurements with ferrite magnetic core, can not be compared to FEA simulations results. Indeed, simulations were performed using ideal lossless magnetic core to highlight the modeling impact on copper losses. Resistance measurements on a planar inductor will be hidden by the influence of the ferrite losses in HF. The copper winding relative resistances obtained for 2D and 3D simulations are compared in Fig. 8(c). The difference between both modeling approaches is included in the range [5% -10%] until 4MHz (Fig. 8(d)). A significant error appears for higher frequency. The 3D inductor modeling presents same limitations than for the PCB alone.

Variation of copper losses along PCB conductors

The main hypothesis, that enables the use of 2D simulations to obtain precisely losses, is the homogeneous losses inside the component. That means that the current density and the copper losses are quite similar and do not vary all along the component's track. In order to validate this hypothesis, 3D simulation are performed to locally analysed variations of current density along PCB conductors. The evolution of ρ cu is calculated according to y values with a fixed z. The obtained values create a curve network that is plotted on Fig. 9(a) for the 4-turn PCB and on Fig. 9(b) for the 4turn inductor. The variation's band and the average value are also plotted in these figures. These curves enable to compare 2-D and 3-D modeling. Indeed, the curve network area depicts effects of 3-D modeling. Moreover, the average value ρ cu (x, y, z 0 ) y [START_REF] Dimitrakakis | High-Frequency Copper Losses in Magnetic Components With Layered Windings[END_REF] enables to directly compare results from 2D and 3D modeling. The range of variation for y is shown on Fig. 9(a) and Fig. 9(b). The average distribution along the x-axis is also presented, as well as the 2D simulation results. It should be noted that the 2D meshing is finer than in the case of 3D simulations because of computing memory limitations. This difference between both meshes can explain the obtained curves.

For the 4-turn PCB (Fig. 9(a)), one can note that losses seem fairly accurate on the inner edge (x = 5.1) and the outer one (x = 12.1). Difference between average value ρ cu and 2D modeling results is more significant for x = 8.1 and x = 9.1.

For the 4-turn inductor with magnetic core (Fig. 9(b)), the 3-D average value is very close to the 2D one. The magnetic material enables to minimize variation along the copper conductor.

ρ cu (x, y, z 0 ) y = ymax y min ρ cu (x, y, z 0 ) dy (17)
with z 0 located at the upper layer center.

Conductor's length: y-axis:

In order to quantify the differences between 2D and 3D modeling, copper loss average density ρ cu (x, y, z 0 ) x ( 18) is calculated along conductor length in the y direction. The ratio ρ cu3D /ρ cu2D is then obtained and shown in Fig. 9(c) for 2, 4 and 8 turns PCB and in Fig. 9(d) for the inductor at 1M Hz.

For the PCB without core (Fig. 9(c)), 2D modeling provides a good loss estimation on the range y ∈ [-5mm : 5mm]. Outside this interval, edge effects increase differences between both modeling until an error of 50%. Fluctuation of ρ cu x is due to the mesh.

With the ferrite magnetic core (Fig. 9(d)), 2D simulations overestimate losses about 4.6% on the interval y ∈ [-10mm : 13mm]. Outside this range, edge effects are increased and the obtained error with the 2D simulation is close to 10%. In such case, the homogeneous loss hypothesis appears to be more credible with magnetic core than the case without.

ρ cu (x, y, z 0 ) x = x=12.1 x=5.1 ρ cu (x, y, z 0 ) dx (18)

Discussion

In order to answer the three main questions, some results analysis can be discussed. Some conclusions are summarized below:

1. The increase of power electronics switching frequency leads to the use of numerical tools to correctly model planar components. Indeed, the hypothesis, traditionally made for analytical calculations, cannot be applied for such geometry and operating frequency.

2. 3D modeling of PCB enables to compute copper losses precisely ( r < 5% for f < 6M Hz). Same computation with 2D models can reach more than 21% at 5M Hz. 3. When a magnetic material is used, the differences between 2D and 3D simulations stay below 5%.

As a consequence, it seems necessary to model in 3D when the component does not contain magnetic material. Indeed, regarding components with magnetic core, the homogeneous loss hypothesis appears to be interesting and 2D modeling enables to obtain quite good approximation of copper losses. This conclusion is valid for traditional PCB and copper foil planar transformer. For less classical design [START_REF] Djuric | Design, Modeling, and Analysis of a Compact Planar Transformer[END_REF], 3D approach will always be necessary as well as for local studies of losses inside windings [START_REF] Teu Magambo | Impact of PCB Track Shape on HF Copper Losses of Planar Magnetic Components[END_REF]. 3D modeling seems also useful when studying impact of harmonics. Indeed with 2D, losses are underestimated for high frequency. Moreover, when frequency is too high, magnetic core might lose its properties. Then, 3D seems essential.

In this study, conditions applied to numerical simulations and impedance measurements are quite restrictive. Indeed, simulations were performed with pure sinus voltage and current. The same goes for impedance measurements, in low signal. Obviously, in a power converter, magnetic components are not supplied with pure sinus low voltage but with large signals containing HF harmonics. Furthermore, PE magnetic components are source of magnetic losses and other HF parasistic effect like capacitances for example. It could be interesting to perform FEA and measurements under real operating conditions to test validity of numerical models, including all these effects. Such simulation, in time-domain, is very complicated, time-consuming and necessits accurate models of material to be compared to measurements. From the practical point of view, even these measurements are difficult and needs specific appliance and tests to separate copper losses and magnetic material ones [START_REF] Keradec | Validating the power loss model of a transformer by measurement -Validation is key[END_REF]. Consequently, for these reasons, a frequency-domain analysis at harmonic values is usually prefered when designing magnetic components. Conclusions of this study are therefore directly applicable by designers.

Conclusion

Numerical modeling appears as the only alternative to model HF magnetic components, taking into account geometry and material characteristics. In this paper, planar magnetic component copper losses are numerically computed on a large frequency range based on 2D and 3D FEA. Simulations results and performances are compared on PCB and inductor devices. 3D FEA Modeling gives better results with errors lower than 5% in comparison with measurements on PCB prototype until few Megahertz. The 2D modeling leads to significant errors, close to 10% from 1MHz. For the inductor, i.e PCB and ferrite magnetic core, the winding outside the magnetic core is the main cause of differences between simulation results. The magnetic field is concentrated in magnetic material and thus limits differences between 2D and 3D simulation. In these cases, 2D models can finally be sufficient difficulties induced by 3D modeling.

  Fig. 1. Planar component a Planar inductor b PCB c 3D and 2D models d Typical lengths

Fig. 2 .

 2 Fig. 2. Volumic current density diffusion inside an infinite plate

Fig. 3 .

 3 Fig. 3. Impact of meshing a Computation of J with different mesh size b Error r due to mesh's size ( s δ )

Fig. 4 .

 4 Fig. 4. FE problem

Fig. 5 .

 5 Fig. 5. Copper loss density computed with 2D FEA as a function of frequency for the 3 planar magnetic inductors.

Fig. 6 .Fig. 7 .

 67 Fig. 6. Copper loss density repartition a 4-turn PCB without ferrite core. Impact of frequency (10kHz, 100kHz and 1M Hz) b Areas for the 2D simulation loss hypothesis -PCB w/o magnetic core c Inductors at 1M Hz. Impact of turn number (2 turns, 4 turns and 8 turns) d Area for the 2D simulation loss hypothesis -8-turn inductor w/ magnetic core

4. 2 Fig. 8 .

 28 Fig. 8. Copper winding relative resistance versus frequency a Comparison between 2D, 3D and measurement for the 4-turn PCB b Relative error compared to measurement for the 4-turn PCB c Comparison between 2D and 3D for the 4-turn inductor d Relative error for the 4-turn inductor

Fig. 9 .

 9 Fig. 9. Distribution of copper losses a Copper losses density as a function of x for the 4-turn 2-layer PCB w/o magnetic core at 1MHz b Copper losses density as a function of x for the 4-turn 2-layer inductor w/ magnetic core at 1MHz c Relative copper losses as a function of y for a 2, 4 and 8-turn PCB w/o magnetic core at 1MHz d Relative copper losses as a function of y for a 2, 4 and 8-turn inductor w/ magnetic core at 1MHz

[ 1 ]

 1 Ouyang, Z., Andersen, M. A. E.: 'Overview of Planar Magnetic Technology Fundamental Properties', IEEE Trans. Power Electron., 2014, 29, (9), pp. 4888-4900 [2] Quinn, C. , Rinne, K., O'Donnell, T., Duffy, M., Mathuna, C.O.: 'A review of planar magnetic techniques and technologies', Proc. Applied Power Electronics Conference and Exposition, APEC, 2001, vol.2, pp. 1175-1183

Table 1

 1 Planar Ferrite Typical Lengths (Data from[START_REF]Soft Ferrites and Accessories[END_REF])

	Core	w c	h c	d c	w w	h w
		(mm) (mm) (mm) (mm) (mm)
	EE14/3.5/5	14	7	5	4	4
	EE18/4/10	18	8	10	5	4
	EE22/6/16	22	11.4	15.8	5.9	6.4
	EE32/6/20 31.75 12.7	20.3	9.27	6.36
	EE38/8/25	38.1 16.52 25.4 11.31	8.9
	EE43/10/28 43.2	19	27.9	13.3	10.8
	EE58/11/38 58.4	21	38.1 20.95	13
	EE64/10/50	64	20.4	50.8	21.8	10.2

Table 2

 2 Skin depths and minimal mesh size for copper conductors at 300 • K

	Frequency (kHz)	1	10 100 10 3 10 4
	δ (µm)	2075 656 207 65.6 20.7
	Maximum mesh size for a 1% error (µm) 691 218 69	21	6.9


Wenhua Tan
email: wenhua.tan@yahoo.cn

Member, IEEE Xavier Margueron
email: xavier.margueron@ec-lille.fr

Nadir Idir, Member, IEEE Laurent Taylor
email: laurent.taylor@ec-lille.fr.

N Idir

Leakage Inductance Analytical Calculation for Planar Components with Leakage Layers

Keywords: Leakage inductance, planar component, magnetic shunt, Green's function

Planar magnetic components are promising solutions for the integration of power electronic systems. The leakage inductance of such components plays an essential role in power converters. In this paper, an analytical modeling method for leakage inductance computation is developped for planar components with plasto-ferrite leakage layers. This method is based on the solution of Poisson's equations for magneto-static using multilayered Green's functions. The obtained formulations are general and precise and have been validated by numerical tests. Experimental characterizations have been performed on two magnetic components: A planar LLC and planar common mode choke. The obtained results show that with the described method, the static leakage inductance of planar components can be accurately estimated.

I. INTRODUCTION

T HE trends toward integration of power electronic systems make planar magnetic components prevalent solutions to realize more compact power converters [START_REF] Aime | Numerical Methods for Eddy Currents Modeling of Planar Transformers[END_REF], [START_REF] Ouyang | Overview of Planar Magnetic Technology Fundamental Properties[END_REF]. These components exhibit low profile, high power density and high reproducibility compared to other types of magnetic components. The leakage inductance of such component is a topic of huge interest. Indeed, due to their low profile, their leakage inductance value is usually supposed to be low [START_REF] Carsten | The low leakage inductance of planar transformers; fact or myth?[END_REF]. For many applications, this low value can be an advantage because transformer's leakage inductance may cause extra losses, stress on components and ElectroMagnetic Interference (EMI) problems due to high frequency voltage oscillations [START_REF] Pahlevaninezhad | An Improved Layout Strategy for Common-Mode EMI Suppression Applicable to High-Frequency Planar Transformers in High-Power DC/DC Converters Used for Electric Vehicles[END_REF]- [START_REF] Ouyang | Optimal Design and Tradeoff Analysis of Planar Transformer in High-Power DCDC Converters[END_REF].

On the opposite, two main applications can find the usefulness of a high leakage inductance. For resonant converter LLC (Fig. 1(a)), the leakage inductance L lk can be used for Zero Voltage Switching (ZVS), without adding an extra inductor [START_REF] De Groot | Design of a 1-MHz LLC Resonant Converter Based on a DSP-Driven SOI Half-Bridge Power MOS Module[END_REF], even if this leakage inductance can also causes voltage imbalance [START_REF] Jung | Bifilar Winding of a Center-Tapped Transformer Including Integrated Resonant Inductance for LLC Resonant Converters[END_REF] or high voltage oscillations [START_REF] Park | Analysis on Center-Tap Rectifier Voltage Oscillation of LLC Resonant Converter[END_REF]. This leakage inductance must have then a specific value for typical softswitching converters [START_REF] Ferrell | The role of parasitic inductance in high-power planar transformer design and converter integration[END_REF] or Dual Active Bridge (DAB) for example [START_REF] Wang | Design of low-profile nanocrystalline transformer in high-current phase-shifted DC-DC converter[END_REF]. If its value is not sufficient, some supplementary resonant inductor has to be added [START_REF] Vuksic | The Multiresonant Converter Steady-State Analysis Based on Dominant Resonant Process[END_REF], [START_REF] Ouyang | Planar-Integrated Magnetics (PIM) Module in Hybrid Bidirectional DCDC Converter for Fuel Cell Application[END_REF].

For EMI filters, the leakage inductance of a common-mode (CM) choke can also be used for filtering differential mode (DM) perturbations [START_REF] Wang | Design Theory and Implementation of a Planar EMI Filter Based on Annular Integrated InductorCapacitor Unit[END_REF]- [START_REF] Luo | An integrated common mode and differential mode choke for EMI suppression using magnetic epoxy mixture[END_REF]. For example, in [START_REF] Wang | Design Theory and Implementation of a Planar EMI Filter Based on Annular Integrated InductorCapacitor Unit[END_REF], a leakage layer has been integrated between Printed Circuit Board (PCB) inside a pot core while in [START_REF] Tan | A Common-Mode Choke Using Toroid-EQ Mixed Structure[END_REF], the leakage layer has been inserted inside a mixed toroidal/EQ planar core. In both case, this leakage layer enable to increase the leakage inductance of the CM inductor. Other integrated EMI filters using CM leakage inductance to filter DM perturbations have also been reported in the past years, including flexible multilayers [START_REF] Wu | Design, modeling, and improvement of integrated EMI filter with flexiblemultilayer foils[END_REF], combined toroidal cores for DM and CM chokes [START_REF] Lai | An integrated EMI choke for differential-mode and common-mode noise suppression[END_REF] or using magnetic epoxy mixture [START_REF] Luo | An integrated common mode and differential mode choke for EMI suppression using magnetic epoxy mixture[END_REF].

As discussed earlier, regarding planar components (transformers or CM chokes), their leakage inductances can be then increased by adding some supplementary leakage layers (also called magnetic shunt) such as Ferrite Polymer Composite (FPC) between the component's windings [START_REF] Ouyang | Optimal Design and Tradeoff Analysis of Planar Transformer in High-Power DCDC Converters[END_REF], [START_REF] Tan | A Common-Mode Choke Using Toroid-EQ Mixed Structure[END_REF], [START_REF] Chen | Improving the Characteristics of integrated EMI filters by embedded conductive Layers[END_REF], [START_REF] Pavlovsky | Winding Losses in High-Current, High-Frequency Transformer Foil Windings with Leakage Layer[END_REF]. Resonant converters or CM inductors require a specific amount of leakage inductance so an accurate modeling method for calculating such L lk is necessary.

In the literature, many methods have been reported to assess the leakage inductance of a transformer, based on Finite Element Analysis (FEA) or analytical calculation [START_REF] Prieto | Taking into account all the parasitic effects in the design of magnetic components[END_REF]- [START_REF] Ouyang | Calculation of Leakage Inductance for High-Frequency Transformers[END_REF]. For FEA, computation can be made in 2D, 3D, or mixed 2D/3D for HF losses and inductance values [START_REF] Aime | Numerical Methods for Eddy Currents Modeling of Planar Transformers[END_REF], [START_REF] Prieto | Taking into account all the parasitic effects in the design of magnetic components[END_REF]- [START_REF] Doebbelin | Calculation of leakage inductance of core-type transformers for power electronic circuits[END_REF]. FEA is a powerful tool to study electromagnetic components but when dealing with optimization, using such tool can be very cumbersome and time consuming. Even if the numerical resolution is becoming faster with last generation computers, time for geometry's description and mesh can also be prohibitive. For magnetic component designers, analytical tool will be prefered, in particular for first design steps, when transformer parameters have to be tuned as desired. Most of the analytical methods for the L lk calculation of a transformer are based on 1D Dowell's assumption [START_REF] Dowell | Effects of eddy currents in transformer windings[END_REF] such that the magnetic flux is tangential to the surface of conductors [START_REF] Hurley | Transformers and Inductors for Power Electronics: Theory, design and applications[END_REF]- [START_REF] Chen | Integration of EMI filter for distributed power system (DPS) front-end converter[END_REF]. This assumption is true when the magnetic window is well filled with long conductors. However, in some cases, the winding arrangement may be very irregular such as low filling factor and non-alignment of conductors. Applying the Dowell's assumption will lead to errors in the results. To be more general, the method of Roth can be applied for 2D transformer's winding cross section. This method consists in using double Fourier series to solve the Poisson's equations in a rectangular region [START_REF] Hammond | Roth's method for the solution of boundary-value problems in electrical engineering[END_REF]. However, double Fourier series exhibit a low convergence rate so that they are computationally expensive. Besides, this method only works when the matter in the rectangular region is uniform. For planar components whose cross-sections are not magnetically homogeneous, like in the case of leakage layers, Roth's method can not be applied.

In such cases, only FEA are performed to determine the value of the leakage inductance [START_REF] Ouyang | Overview of Planar Magnetic Technology Fundamental Properties[END_REF]. No analytical formulation is available in the literature, except in [START_REF] Zhang | Leakage Inductance Calculation for Planar Transformers with a Magnetic Shunt[END_REF] where some reluctance model is used. This method is still limited to magnetic window well filled with long conductors.

In [START_REF] Margueron | Analytical Calculation of Static Leakage Inductances of H.F. Transformers Using PEEC Formulas[END_REF] and [START_REF] Margueron | Complete Analytical Calculation of Static Leakage Parameters: A step toward HF Transformer Optimization[END_REF], authors propose a general 2D analytical method for calculating the leakage inductance of planar transformers. This method is based on a PEEC-like formula for rectangular conductors. To account for the influence of magnetic core, the method of magnetic images is employed on the cross-section of the winding part inside the core [START_REF] Stratton | Electromagnetic Theory[END_REF], [START_REF] Roshen | Analysis of planar sandwich inductors by current images[END_REF]. This general method, completely independent of the conductor's arrangement, can accurately evaluate the magnetic field in the cross-section of the component for conventional power electronic transformers, planar ones or more recently for power transformers [START_REF] Lambert | Analytical Calculation of Leakage Inductance for Low-Frequency Transformer Modeling[END_REF]. However, similarly to Roth's method, the PEEC-like formula also requires a uniform environment of matter as prior condition.

If the cross-sections are magnetically multi-layered, the application of such method can be very cumbersome. For the above reasons, an improved method based on multilayered Green's function has been developed in this paper for calculating the leakage inductance of planar components having magnetically multi-layered structure.

This paper is organized as follows: in Section II, a review of the leakage inductance calculation and the PEEClike formulations are given. Section III introduces the multilayered Green's function with detailed analysis and numerical validations. In Section IV, two application examples, LLC and CM choke are presented to validate the proposed method. The paper will end with conclusion.

II. LEAKAGE CALCULATION PROBLEM ANALYSIS

A. Modeling of Leakage Inductance

The existing modeling methods for calculating the static leakage inductance rely on the evaluation of the magnetic energy stored in the component when the total ampere-turns are compensated [START_REF] Binns | The Analytical and Numrical Solution of Electric and Magnetic Fields[END_REF]. The relation between the leakage inductance and the magnetic energy W mag is expressed by:

W mag = 1 2 L lk I 2 .
(1) where L lk is the static leakage inductance and I the current in the winding. This magnetic energy W mag (1) can be determined by 2D or 3D finite element simulations (FEM), which are precise but very time-consuming or with analytical approaches. This computation is generally based on 2D approximation that the current direction is perpendicular to the cross-section of the studied component. Consequently, the magnetic energy W mag can be calculated by:

W mag = W s mag L mean (2)
where W s mag is the energy per unit length stored in the cross-section of the studied component whereas L mean is the mean length of the winding fixed from the location of the peak energy density.

In High Frequency (HF), it is well known that the leakage inductance value will decrease due to eddy current effects [START_REF] Bahmani | Accurate Evaluation of Leakage Inductance in High-Frequency Transformers Using an Improved Frequency-Dependent Expression[END_REF], [START_REF] Ouyang | Calculation of Leakage Inductance for High-Frequency Transformers[END_REF]. In this work, impact of HF magnetic fields on leakage inductances will not be studied. Indeed, the goal of the developped Green analytic tool is to give designers a fast estimation of their leakage value, as well as in [START_REF] Margueron | Complete Analytical Calculation of Static Leakage Parameters: A step toward HF Transformer Optimization[END_REF]. If this static value does not seem sufficient, the design has to be started again because in HF, leakage will be further reduced.

B. Review of 2D PEEC-like Modeling Method

In 2D magnetostatic analysis, the magnetic field over the cross-section of a planar component is governed by the following Poisson's equation:

∇ 2 A z = -µJ z . (3)
In order to solve this equation, authors proposed in [START_REF] Margueron | Analytical Calculation of Static Leakage Inductances of H.F. Transformers Using PEEC Formulas[END_REF], [START_REF] Margueron | Complete Analytical Calculation of Static Leakage Parameters: A step toward HF Transformer Optimization[END_REF] the 2D PEEC-like formulations, where the leakage inductance of planar transformers can be calculated analytically with no hypothesis on the conductors' topology. In free space, the potential vector A z in the space due to a rectangular conductor carrying current I (Fig. 2) can be expressed by [START_REF] Pahlevaninezhad | An Improved Layout Strategy for Common-Mode EMI Suppression Applicable to High-Frequency Planar Transformers in High-Power DC/DC Converters Used for Electric Vehicles[END_REF].

With this analytical formula, the distribution of the potential vector in the space due to the rectangular conductor can be determined. In case of multiple conductors, the superposition principle can be applied for the calculation. As long as the information of A z is obtained, the integration (5) can be performed to calculate the magnetic energy per unit length stored in the cross-section of the component.

W s mag = 1 2 Ω A z J z dxdy (5)
Thus, the leakage inductance of the component can be estimated by:

L lk ≈ 2W s mag L mean I 2 (6)
This method is general and gives accurate estimation of the leakage inductance value. However, just as stated in the introduction, this method can not be applied for multi-layered structure where the permeability of the window is not uniform. Although the magnetic image method [START_REF] Stratton | Electromagnetic Theory[END_REF], [START_REF] Roshen | Analysis of planar sandwich inductors by current images[END_REF] can applied for taking into account the influence of magnetic material, it can not be applied on a multi-layered structure where a ferrite polymer layer is present, for the following reasons:

1) With this method, four first-order images are employed due to direct reflections and four second-order images are added due to secondary reflections [START_REF] Margueron | Complete Analytical Calculation of Static Leakage Parameters: A step toward HF Transformer Optimization[END_REF]. Higher order images are neglected. However, when FPC layer is implemented in the region, the reflection phenomenon becomes much more complicated.

2) The aforementioned method assumes that the thickness of the ferrite is infinite. As long as the permeability of the ferrite material is large enough, this assumption is very close to the reality. However, since the FPC layer has a low permeability, the impact of its thickness has to be considered, which further complicates the calculation.

In view of these reasons, an analytical method for leakage inductance calculation based on Green's function for multilayered structure is introduced.

III. MULTI-LAYERED GREEN'S FUNCTION

A. Description of the Problem

The structure of planar components with PCB technology is shown in Fig. 1(b). The leakage layer in ferrite polymer composite is optional but can be used to increase the leakage inductance for passive component integration in the application of soft-switching converters or integrated CM and DM inductors for EMI filters. It should be noted that the FR-4 epoxy and the isolating layers are considered as magnetically transparent. As a consequence, the component cross-section is composed of three layers of matter, i.e. transparent-FPCtransparent. Similarly to [START_REF] Margueron | Analytical Calculation of Static Leakage Inductances of H.F. Transformers Using PEEC Formulas[END_REF], the current is supposed to be perpendicular to this cross-section and the system is described by the magnetostatic Poisson's equation [START_REF] Carsten | The low leakage inductance of planar transformers; fact or myth?[END_REF]. As the relative permeability of MnZn ferrite material is very high (in order of 10 3 -10 4 ), it can be considered as magnetic material with infinite permeability as an approximation. It can be shown that the boundary conditions (8) holds by using the continuity conditions on the tangential component of the magnetic field H (Fig. 3): The boundary conditions for the problem are derived as follows:

H T f errite -H T window = 0 (7)
    
∂A z ∂y = 0 over the horizontal surface

∂A z ∂x = 0 over the vertical surface (8)
As seen, homogeneous Neumann boundary conditions are obtained. However, this problem admits solutions if and only if the compatibility condition is satisfied:

∂Ω ∂A z ∂n dl + µ 0 Ω J z dS = 0 (9)
This can be proved by applying Gauss's Law on the Poisson's equation [START_REF] Carsten | The low leakage inductance of planar transformers; fact or myth?[END_REF]. With the homogeneous boundary conditions [START_REF] Jung | Bifilar Winding of a Center-Tapped Transformer Including Integrated Resonant Inductance for LLC Resonant Converters[END_REF], the first term of equation ( 9) is null, requiring the second term to be null too. In fact, this is the prior condition for the calculation of leakage inductance. Therefore, the Poisson's equation ( 3) with these boundary conditions admit solutions.

After solving the potential vector A z from (9), the equation ( 5) and ( 6) should be used to determine the leakage inductance of such component.

B. Generality on Green's Function

Green's function is a powerful tool for analyzing electromagnetic problems. It is a specially constructed function to solve partial differential equations [START_REF] Olver | Introduction to Partial Differential Equations[END_REF]. It can be regarded as the impulse response of an electromagnetic system to Dirac type excitation. For the magnetostatic Poisson's equation given in [START_REF] Carsten | The low leakage inductance of planar transformers; fact or myth?[END_REF], its associated Green's function is:

∇ 2 G = -µδ ( 10
)
where δ is a Dirac current source. With this Green's function, the solution of the original problem can be expressed by:

A z (x, y) = Ω G (x, y) J z dS - 1 µ ∂Ω A z ∂G ∂n dl (11)
A z (x, y; a, b) = -µ 0 I 16πab XY ln X 2 + Y 2 + X 2 arctan Y X + Y 2 arctan X Y X=x-a X=x+a Y =y-b Y =y+b (4)
From this equation, it is desirable to also provide homogeneous Neumann boundary conditions for the Green's function so that the second term of ( 11) is null and the potential vector A z can be represented by a convolution. However, this is not allowed since such boundary condition will violate the compatibility condition. Indeed, the following compatibility condition should be satisfied during the construction of the Green's function:

∂Ω ∂G ∂n dl = -µ 0 (12)
C. Multi-layered Green's Function

In [START_REF] Tan | Analytical modeling of parasitic capacitances for a planar common mode inductor in EMI filters[END_REF], [START_REF] Tan | Modeling and Design of Passive Planar Components for EMI Filters[END_REF], authors present a multi-layered Green's function for solving electrostatic Poisson's equation with homogeneous Dirichlet boundary conditions in a bounded multilayered region. As stated previously, the studied planar components have a magnetically multi-layered cross-section. In this paper, the method of [START_REF] Tan | Analytical modeling of parasitic capacitances for a planar common mode inductor in EMI filters[END_REF] will be extended for magnetostatic analysis. It has been shown that the boundary conditions should verify (8) to admit solutions. Therefore, the following boundary conditions are proposed:

       ∂G ∂y y=H = - µ 0 L ∂G ∂x x=0 = ∂G ∂x x=L = ∂G ∂y y=0 = 0 (13)
As seen, only the top edge of the rectangular region is not zero. With these boundary conditions, the compatibility conditions given in (12) can be satisfied.

According to the previous analysis, the cross-section of the planar component is a 3-layer structure. If a Dirac current source is introduced, the structure is further split into four layers. Therefore, only a 4-layer structure is studied in this work. As shown in Fig. 4(a), each layer has a height H k and a permeability µ k . Here, the local coordinates are applied for each layer and the origin is chosen at the lower left vertex of each rectangular region. The Dirac current source locates at the interface between layer j and j + 1 (j < 4). As a consequence, the Green's function in layer i satisfies:

∇ 2 G i = 0 (14)
The following boundary conditions can be written out:

1) Left and right boundaries of layer i: x = 0 and x = L:

∂G i ∂x x=0 = ∂G i ∂x x=L = 0 (15)
2) Top and bottom boundaries (in layer 4 and layer 1):

       ∂G 1 ∂y y=0 = 0 Layer 1 ∂G 4 ∂y y=H4 = - µ 0 L Layer 4 (16)
3) Continuity conditions on the interface between layer i and i + 1 (i < 4):

       G i y=Hi -G i+1 y=0 = 0 1 µ i ∂G i ∂y y=Hi - 1 µ i+1 ∂G i+1 ∂y y=0 = δ ij δ (x -x s ) (17)
where δ ij is the Chronecker Delta function whereas δ(xx s ) presents the Dirac current source.

From these boundary conditions [START_REF] Tan | A Common-Mode Choke Using Toroid-EQ Mixed Structure[END_REF], the general solution in layer i can be derived as follows:

G i = α i 0 y + β i 0 + ∞ n=1 α i n ch (k n y) + β i n sh (k n y) cos (k n x) (18
) where ch() and sh() are hyperbolic cosine and sinus functions respectively. Note that the term α i 0 y + β i 0 appears since the eigenvalue of ( 14) can be zero [START_REF] Tan | Modeling and Design of Passive Planar Components for EMI Filters[END_REF].

In order to determine the parameters α i 0 , β i 0 , α i n and β i n , the remaining boundary conditions are used. From ( 16), the following relations hold:

α 1 0 = 0 and β 1 n = 0 α 4 0 = - µ 0 L and β 4 n = -α 4 n tanh (k n H 4 ) (19)
The next step consists in using the continuity condition [START_REF] Lai | An integrated EMI choke for differential-mode and common-mode noise suppression[END_REF] to establish the link between the parameters of different layers. The detailed derivation is given in Appendix, only the final results are given here. The parameters α i 0 , β i 0 , α i n and β i n can be expressed by [START_REF] Dowell | Effects of eddy currents in transformer windings[END_REF]. Note that F l,l+1 n and F l,l+1 n -1 are the up-going and down-going transformation matrix, respectively:

F l,l+1 0 = µ l+1 µ l 0 H l 1 (20)
F l,l+1 0 -1 = µ l µ l+1 0 -µ l µ l+1 H l 1 (21)
F l,l+1 n = ch (k n H l ) sh (k n H l ) µ l+1 µ l sh (k n H l ) µ l+1 µ l ch (k n H l ) (22) F l,l+1 n -1 = ch (k n H l ) -µ l µ l+1 sh (k n H l ) -sh (k n H l ) µ l µ l+1 ch (k n H l ) (23)
The parameter α 1 n and α 4 n can be derived from the continuity condition when i = j.

D. Magnetic Field Calculation

The first integral can be performed on the conductor area where the current is non-zero. The second term corresponds to the average value of the potential vector on the boundary of the region, which is a constant. Neglecting this constant, the first integral of ( 11) can be evaluated analytically [due to the term cos(k n x s ) in ( 25)] along x-axis and numerically along y-axis by discretizing the conductor into K thin elements (Fig. 5). For element k, the y coordinate of the center point y sm = (y k + y k+1 )/2 is chosen for the integration along y-axis.

The magnetic induction B can then be calculated by deriving A z , as given by [START_REF] Ferrieux | Alimentations a Decoupage, Convertisseurs a Resonance[END_REF]. Note that the parameters α i 0 , β i 0 , α i n and β i n are only dependent to the coordinate of the source point. This property allows the separation of the mathematical treatment on the source conductor and the observation one. Therefore, the partial derivation on G (on the observation point) given in ( 26) can be performed directly on the expression whereas the integration on the source conductor is still evaluated numerically as done before.

         B x = ∂A z ∂y = -µ Ω ∂G (x, y) ∂y Jds B y = - ∂A z ∂x = µ Ω ∂G (x, y) ∂x Jds (26)
The final purpose involves estimating the total energy in the cross-section via the integral [START_REF] Colonel | Transformer and Inductor Design Handbook[END_REF]. The Greens function has to be integrated both on the source conductor and the observation conductor. The integration on the source conductor is performed numerically while the integration on the observation conductor can be performed analytically according to the expression [START_REF] Luo | An integrated common mode and differential mode choke for EMI suppression using magnetic epoxy mixture[END_REF]. However, three cases should be distinguished (Fig. 6):

1) The position of the source conductor element is higher than the observation conductor:

In this case, the observation conductor is completely in layer i+1, therefore the integration is performed with the Greens function G i+1 for layer i+1.

2) The position of the source conductor element is lower than the observation conductor:

In this case, the observation conductor is completely in layer i, the integration is thereby calculated with the Greens function G i for layer i.

3) The position of the source conductor element is included in the observation conductor:

In this case, as the observation conductor crosses the two layers, the integration is performed by two separate parts. Both the Greens function G i and G i+1 should be integrated on the corresponding parts. Note that during the calculation of A z , a constant appears on the final obtained results due to the Neumann type boundary conditions [START_REF] Landau | Electrodynamique des milieux continus[END_REF]. However, this constant will not affect the results of ( 5) since the integral of this constant over all the conductors will be null as the total ampere-turns is compensated in the component's window.

E. Numerical Validations

To validate the proposed method, the multi-layered Green's function is applied on two structures. The first case (Fig. 7) does not correspond to real component but is computed to test the developped approach. The second one, is closer to a real planar component (Fig. 10(a)). For both components, numerical simulations, performed with ANSYS Maxwell 2D [START_REF]User's Guide -Maxwell 2D Version 15[END_REF], are done to be compared to Green's calculation. For each FEA simulation, total ampere-turns are compensated.

1) Configuration test: As shown in Fig. 7, an FPC layer is implemented so that the structure is divided into three layers. The total ampere-turns of the component are compensated. To analyse the results, two test cuts are performed on this crosssection at y = 0.75 mm and y = 3.75 mm respectively. The obtained vector potential on these two test cuts are compared with Finite Elements simulations results, as shown in Fig. 8(a). As seen, a constant difference is observed between the calculated curves and the simulated ones. These results are foreseen due to the Neumann type boundary conditions. Removing this constant (Fig. 8(b)), a good agreement between these two curves can be found. The magnetic induction B x and B y can also be deduced, as illustrated in Fig. 9. One can note a good coherence between the calculated curves and simulated ones. After the numerical integration, the magnetic energy stored in the window is 2.29E -5J/m while the calculated one is similar. In the previous analysis, it is supposed that the permeability is infinite. Indeed, even though the practical permeability of ferrite material is in the order of 10 3 µ 0 , the infinite-permeability assumption still gives good approximation to the reality. FEA simulations have been performed to assess the influence of the magnetic core. Results are presented in Table I where the magnetic energy stored in the windows is computed with various permeability values. Comparing FEA results and the Green's calculations, it can be seen that the proposed method can accurately determine the magnetic field in the structures of interest.

2) Planar CM choke: This second test case is based on the design of a planar CM choke (Fig. 10(a)) made of 3F3 E32/I planar core [START_REF]Ferroxcube[END_REF]. Some FPC layer (height 0.8mm) is added between two windings. The total ampere-turns of the component are also compensated. The current in each conductor higher than the FPC layer is -1A whereas the current is 1A for the conductors lower than the FPC layer. The Fig 10(b) shows the magnetic field inside the component's window. From this simulation, it can be concluded that the field is not tangential to the conductors. Therefore, the 1D Dowell assumption is not directly applicable in such case. This hypothesis is confirmed when analysing Bx and By (Fig. 11) along the horizontal test cut at H = 1.5mm. Simulated results are also compared to the computed ones in Fig. 11, where both simulated and calculated fields show good agreement. After numerical integration, the calculated magnetic energy stored in the window is 2.43E -5J/m, while the simulated value is 2.57E -5J/m. The difference is equal only to 5%.

IV. APPLICATIONS

The previous method has been tested on two real components: A LLC transformer and a CM choke. Both devices are based on the use of ferrite planar cores [START_REF]Ferroxcube[END_REF], associated with FPC C350 [START_REF]EPCOS[END_REF] for the leakage layers.

A. Components' description

For the LLC transformer (Fig. 12(a)), two 3F3 E38-cores are used while for the CM inductor (Fig. 13(a)), two 3C90 E43-cores are preferred.

The transformer is made of 4 copper layers of 35µm with 8 turns for the primary and 2 for the secondary. The CM inductor is a symmetrical component, made of 8 identical 70µm copper layers with 4 turns on each layer. For both components a leakage layer made of FPC is sandwiched between the two windings to increase the leakage inductance. This FPC layer is set to 0.2mm and 0.96mm, for the transformer and the CM inductor, respectively. The winding arrangements are described in Fig. 12(b) and Fig. 13(c).

α i 0 β i 0 =        0 0 i ≤ j N -1 l=i F l,l+1 0 -1 -µ0 L 0 i > j and α i n β i n =          1 l=i-1 F l,l+1 n α 1 n 0 i ≤ j N -1 l=i F l,l+1 n -1 α 4 n -α 4 n tanh (k n H 4 ) i > j (24)
       j l=3 F l,l+1 0 -1 -µ0 L β 4 0 = j l=1 F l,l+1 0 0 β 1 0 + - µj+1 L 0 j l=3 F l,l+1 n -1 α 4 n -α 4 n tanh (k n H 4 ) = j l=1 F l,l+1 n α 1 n 0 + 0 - 2µj+1 nπ cos (k n x s ) (25)

B. Leakage inductance calculation

For the part inside the core, the previous Green's computation is applied. For the winding part outside the core, the FPC layer is not implemented. Therefore, the PEEC-like formulations can be employed for calculating the magnetic field as well as the magnetic energy per unit length. In fact, it can be shown that the PEEC-like formula (4) is the integral form of the free space Green's function (Green's theorem) over the rectangular region

[-a/2, a/2] × [-b/2, b/2] [51]: G (x, y; x s , y s ) = - µ 0 2π ln (x -x s ) 2 + (y -y s ) 2 (27)
The Table II presents energy computed inside the core using Green's functions W ic and the one calculated outside the core based on PEEC formulas W oc . The total leakage inductance is then calculated with (28) taking into account lengths of windings inside and outside the core (Fig. 14). For the LLC transformer, the leakage inductance is evaluated to 162nH while for the CM inductor, it is equal to 13.4µH.

L lk = 2 I 2 [2W ic l ic + W oc (l oc1 + l oc2 )] (28)
Regarding computation time, duration depends on the number of conductors. For example, the LLC computation lasts 3.45s while the CM inductor 5.14s. As a comparison, FEA performed on the example of Fig. 10(a analytical approach can then be interesting for component's optimization and its leakage inductance tuning.

C. Leakage inductance measurement

Both components are then characterized with an impedance analyzer [START_REF]Agilent 4294A Precision Impedance Analyser-Operation Manual[END_REF] and their leakage inductances are measured based on method proposed in [START_REF] Margueron | Identifying the Magnetic Part of the Equivalent Circuit of n-Winding Transformer[END_REF]: The leakage impedance Z s (Fig. 15) is calculated using open-circuit and short-circuit impedance measurements [START_REF] Li | Analysis of the leakage inductance of planar transformer[END_REF]. This method enables to cancel the effect on magnetizing inductance while calculating the leakage one [START_REF] Ouyang | The analysis and comparison of leakage inductance in different winding arrangements for planar transformer[END_REF]. with Z 0 the impedance measured from primary winding with secondary in open-circuit, Z 0 the impedance measured from secondary winding with primary in open-circuit and Z cc the impedance measured from primary winding with secondary in short-circuit. r p (Fig. 15) is deduced from Z 0 in low frequency while r s is calculated with Z 0 also in low frequency [START_REF] Margueron | Identifying the Magnetic Part of the Equivalent Circuit of n-Winding Transformer[END_REF].

Z s = Z 0 (Z cc -r p ) (Z 0 -r p ) -r s (29)
L lkmeas = Im( Z s 2πf ) (30)
The measured values are plotted in Fig. 16, where they are compared to the computed ones. In this figure, for low frequencies, the leakage inductance is masked by resistances while in HF, parasitic capacitances cancel the magnetic effect. The leakage value is quite constant on the frequency range [1kHz; 1M Hz].

In both components coupling coefficients k between windings are close to 1. Magnetizing inductances are evaluated to 431µH and 851µH for the LLC and the CM inductor, respectively. These values are widely stronger than the leakage ones. With such a difference, the total ampere-turn compensation hypothesis stays valid because magnetizing current can be neglected in case of short-circuit test.

D. Discussion

Regarding Fig. 16, it can be noted that computation for both cases gives good result with error lower than 15%. These differences can be justified for following reasons:

1) Errors related with dimensions: The dimensions reported in Fig. 12(b) and Fig. 13(c) have been measured directly on the prototypes and are not precises. 2) Simplification of the geometry: The window's description is not identical inside and outside the core. The part inside the magnetic core is well described but the one outside is simplified and the angles/corners are not really taken into account. 3) Sensitivity to the average length: The results are directly proportional to the energy density. They have to be multiplied by a lenght which is easy to determine inside the core (depth of the core) but which is more difficult to evaluate outside. A mean path is fixed, based on window's peak energy but this assumption is not precise. 4) Magnetic material disparities: As an example, the C350 material is given with a permeability of 9 ± 20% [START_REF]EPCOS[END_REF].

The same computation has been done again, taking into account this disparity. Calculation has been done with µ r = 10.8(9 + 20%) and µ r = 7.2(9 -20%). The Table III summarizes the results and shows that this parameter can lead to significant differences. Without the leakage layers, leakage inductance can be evaluated, based on [START_REF] Margueron | Complete Analytical Calculation of Static Leakage Parameters: A step toward HF Transformer Optimization[END_REF], to 128nH (W ic = 6.68E -7J/m) and 11.8µH (W ic = 5.287E-5J/m) for the LLC transformer and the CM inductor, respectively. The leakage layers enable then to increase the leakage inductance and can be analytically calculated with a good precision with the developped method. However, this increase can be considered as quite low while regarding relative differences between leakage inductances with and without leakage layers. As a percentage, this rise is equal to 26% for the LLC and 13% for the CM inductor. These values can still be increased by acting on two main factors:

1) Increase of FPC thickness: In both examples, FPC layers are limited to 0.2mm and 0.96mm. Adding some layers, can enable to increase the leakage inductance value. 2) Addition of FPC outside the core: In this work, only FPC inside the core was considered. Adding leakage layers outside the core will also increase the leakage value. But the present method will not be able to calculate its value. Only 3D FEA could be performed to do this assessment. These both solutions can also be discussed. Indeed, in most of power electronic converters, while power density is required, HF power transformers are made of interleaved layers to limit HF copper losses [START_REF] Prieto | Using parallel windings in planar magnetic components[END_REF], [START_REF] Margueron | Current Sharing Between Parallel Turns of a Planar Transformer: Prediction and Improvement Using a Circuit Simulation Software[END_REF]. Such interleaving reduces the leakage inductance. FPC can be added between each primary and secondary layers [START_REF] Pavlovsky | Partial Interleaving: A Method to Reduce High Frequency Losses and to Tune the Leakage Inductance in High Current, High Frequency Transformer Foil Windings[END_REF] but here also, the present method will be unusable. Some developpements have to be made on massively multi-layered windings as well as HF effects.

In connection with HF power transformers, problem of parallel windings should also be adressed. In planar transformers, parallel layers are a common practice to increase currents inside windings. The Green method can be applied with such conductor configurations. Indeed, only current values have to be settled, as in the example presented in Fig. 7. In low frequency, the current repartition will be homogeneous and current will be uniformly distributed inside conductors. It is not true in HF, but the developped Green method is also limited and can not be applied in HF.

V. CONCLUSION

In this paper, the leakage inductance of planar magnetic components including leakage layer is studied via an analytical method based on multilayered Green's function. Using this Green's function, the Poisson equation for a rectangular region with Neumanns type boundary conditions can be solved to determine the magnetic field in the cross-section of the studied component. Simulation and measurement results show that this method is very effective and accurate for planar component leakage inductance calculation when a leakage layer is inserted inside the component's windings.

APPENDIX A DERIVATION OF EQUATION

Recall the general solution of Greens function in layer i (18) and the continuity condition [START_REF] Lai | An integrated EMI choke for differential-mode and common-mode noise suppression[END_REF]. If i = j, δ ij = 0. In this case, the Dirac current source does not locate on the interface to analyze. The condition can be expressed by [START_REF] Chen | Integration of EMI filter for distributed power system (DPS) front-end converter[END_REF]. Based on the equality of the coefficients term by term, the following matrix systems can be derived:

α i+1 n β i+1 n = F i,i+1 n α i n β i n (n = 0, 1, 2...∞) (33)
Here the matrix F i,i+1 n is the upward transformation matrix that relates the parameters of layer i and layer i+1. Its expression can be found in [START_REF] Pavlovsky | Winding Losses in High-Current, High-Frequency Transformer Foil Windings with Leakage Layer[END_REF] and [START_REF] Prieto | Study of 3-D magnetic components by means of "double 2-D" methodology[END_REF]. The downward transformation matrix F i,i+1 n -1 can be defined similarly as:

α i n β i n = F i,i+1 n -1 α i+1 n β i+1 n
(n = 0, 1, 2...∞) [START_REF] Bahmani | Accurate Evaluation of Leakage Inductance in High-Frequency Transformers Using an Improved Frequency-Dependent Expression[END_REF] where F i,i+1 n -1 is given in [START_REF] Prieto | Taking into account all the parasitic effects in the design of magnetic components[END_REF] and [START_REF] Doebbelin | Calculation of leakage inductance of core-type transformers for power electronic circuits[END_REF].

If i = j, δ ij = 1. In this case, the Dirac current source locates on the studied interface. The condition [START_REF] Lai | An integrated EMI choke for differential-mode and common-mode noise suppression[END_REF] becomes [START_REF] De Leon | Leakage Inductance Design of Toroidal Transformers by Sector Winding[END_REF]. Integrate the second term of this equation ( 32) on [0, L]:

L µ j α j 0 - L µ j+1 α j+1 0 = 1 (35)
Multiply the two sides of the second equation of (32) by cos(knx), and integrate it on [0, L]: He is currently a Full Professor with IUT A of the University of Lille 1, where he teaches power electronics and electromagnetic compatibility. Since 1993, he has been with the Laboratory of Electrical Engineering and Power Electronics, University of Lille 1. His research interests include design methodologies for HF switching converters, power devices (SiC and GaN), electromagnetic compatibility of the static converters, HF modeling of the passive components, and EMI filter design methodologies.

Fig. 1 .

 1 Fig. 1. Integrated planar LLC. (a) Equivalent circuit. (b) Cross-section of the component.

Fig. 2 .

 2 Fig. 2. Rectangular conductor cross-section.

Fig. 3 .

 3 Fig. 3. Continuity conditions on the boundary in the core window.

Fig. 4 . 4 -

 44 Fig. 4. 4-layer structure for Green's function deduction. (a) Source point higher than FPC layer. (b) Source point lower than FPC Layer.

Fig. 5 .

 5 Fig. 5. Discretization along y-axis of the source conductor.

Fig. 6 .

 6 Fig. 6. Analytical integration on the observation conductor: Three cases.

Fig. 7 .

 7 Fig. 7. Test configuration.

Fig. 8 .

 8 Fig. 8. Comparison between calculated results and simulated results. (a) Before removing the constant. (b) After removing the constant.

Fig. 9 .

 9 Fig. 9. Comparison between calculated results and simulated results for the test case. (a) Test cut 1: Bx. (b) Test cut 1: By. (c) Test cut 2: Bx. (d) Test cut 2: By.

Fig. 10 .Fig. 11 .

 1011 Fig. 10. Planar CM choke cross section under simulation. (a) Window cross section. (b) Field inside the window.

Fig. 12 .Fig. 13 .

 1213 Fig. 12. LLC planar transformer (a) Component top view. (b) Cross section and dimensions.

Fig. 14 .

 14 Fig. 14. Typical winding length.

Fig. 15 .

 15 Fig. 15. Equivalent circuit for leakage measurement.

Fig. 16 .

 16 Fig. 16. Comparison between measurement and computation results.

  Xavier Margueron (M'09) was born in Chambery, France, in 1980. He received the diploma of Electrical Engineering from the Ecole Nationale Superieure d'Ingenieurs Electriciens de Grenoble, Saint Martin d'Heres, France, in 2003 and the PhD Degree in Electrical Engineering from the Universite Joseph Fourier, Grenoble, France, in 2006. Its principal fields of research related to passive components and HF modelling. Since September 2007, he has been an associate professor at the Ecole Centrale de Lille, France, where he carries out research in the L2EP laboratory. His research interests include modeling and optimization of HF magnetic passive components for power electronic converters and planar magnetic technology. Laurent Taylor was born in Dunkerque, France, in 1987. He received the M.S. degree from Ecole Normale Superieure de Cachan, France in 2011, and from Ecole Centrale de Lille, France in 2012. He's currently working toward Ph.D. degree in Power Electronics and Modeling at the Laboratoire d'Electrotechnique et d'Electronique de Puissance, Lille, France. His research interests include HF power electronics, passive component modeling and model order reduction methods. Nadir Idir (M'93) received the Ph.D. degree in electrical engineering from the University of Lille 1, Villeneuve-dAscq, France, in 1993.

TABLE I VARIATION

 I OF MAGNETIC ENERGY WITH PERMEABILITY

) Rearrange the equation ( 35), [START_REF] Hammond | Roth's method for the solution of boundary-value problems in electrical engineering[END_REF] and the first equation of [START_REF] De Leon | Leakage Inductance Design of Toroidal Transformers by Sector Winding[END_REF], the following matrix system are obtained:

To solve the value of α i 0 , β i 0 , α i n and β i n , the equations ( 37) and ( 38) are solved. On the interface between layers j and j+1 (j < 4), the equations ( 25) hold. It can be shown that β 1 0 = β 2 0 = β 3 0 = β 4 0 = constant , which agree with feature of a homogenous Neumann B.C. problem that an arbitrary constant will appear in the final solutions. In this study, they are set to be zero for simplicity. It is explained in the text that their value will not influence the final leakage inductance. Once the values of α 1 n and α 4 n are obtained, the relation equations ( 24) are applied to find all the parameters α i 0 , β i 0 , α i n and β i n . More details related to this development are given in [START_REF] Tan | Modeling and Design of Passive Planar Components for EMI Filters[END_REF].