Guillaume Parent

Stéphane Duchesne

Patrick Dular

Determination of Flux Tube Portions by Adjunction of Electric or Magnetic Multivalued Equipotential Lines

Keywords: Finite element analysis, field lines, flux tubes

In 2-D electrostatic and magnetostatic finite element problems, portions of flux tubes are defined by field lines together with their orthogonal lines. These two sets of lines can be obtained as the isovalue lines of scalar and vector potentials. One set is related to a multivalued potential that, when made single-valued via the definition of cuts, undergoes discontinuities, resulting in lines interruptions at the cuts. A method is proposed for obtaining continuous isovalue lines for the multivalued potentials. This then helps in locating selected portions of flux tubes and quantifying their equivalent properties, i.e., capacitances or reluctances, to be used as is or in equivalent circuits.

I. INTRODUCTION

W HEN modeling an electromagnetic device, the choice of one method or the other depends on several parameters such as the level of desired accuracy, computational time or complexity of the physical phenomena. The equivalent circuit modeling approach, whether electric or magnetic, usually offers the best balance between computation time and accuracy of the results, which is why it is intensively used for fast modeling [START_REF] Peng | Magnetic-equivalent-circuit modelling of switched reluctance machines with mutual coupling effects[END_REF], failure analyzes [START_REF] Sizov | Analysis and diagnostics of adjacent and nonadjacent broken-rotor-bar faults in squirrel-cage induction machines[END_REF] and design-optimization [START_REF] Bakr | Review of the space mapping approach to engineering optimization and modeling[END_REF]. Nevertheless, both accuracy and ruggedness of these studies greatly depend on the determination of the lumped parameters constituting the equivalent circuit. They can be determined by analytical relations, tubes and slices method [START_REF] Hammond | Tubes and slices: a new way of teaching the principles of electric and magnetic fields[END_REF] or from Finite Element (FE) modeling [START_REF] Gyselinck | Airgap reluctance identification for magnetic equivalent circuit modelling of induction machines[END_REF].

The method presented in this paper uses the latter approach but, differently from [START_REF] Gyselinck | Airgap reluctance identification for magnetic equivalent circuit modelling of induction machines[END_REF], it uses both static dual 2-D formulations to accurately determine flux tube portions, delimited by both flux walls and gates, that are related to lumped parameters. Indeed, an electromagnetic problem is usually solved through either a scalar or a vector potential formulation. In 2-D and in case of a total potential, the associated field is either parallel or perpendicular to its isovalue lines, which makes the determination of such lines a key point for the characterization of flux tubes portions. For a given problem, obtaining both sets of lines requires the use of dual formulations. Whereas the commonly used formulations (with electric scalar potential in electrostatics and magnetic vector potential in magnetostatics) involve single-valued potentials, their dual formulations (with electric vector potential and magnetic scalar potential, respectively) involve multivalued potentials, i.e., presenting discontinuities through some cut lines making their support domain simply connected [START_REF] Bossavit | Magnetostatic problems in multiply connected regions: some properties of the curl operator[END_REF]. The aim of this work is the development of a method allowing to adjunct the parts of the isovalue lines of the multivalued potentials at the cuts in order to be able to exploit them as boundaries of flux tubes portions.

Corresponding author: G. Parent (email: guillaume.parent@univ-artois.fr).

The complementary portions are directly obtained from the complementary classical single-valued potentials.

Electrostatic and magnetostatic test problems are solved to illustrate and validate the developed methodology.

II. STATIC DUAL FORMULATIONS AND FLUX TUBES A. Electrostatic and magnetostatic problems

Electrostatic and magnetostatic problems in a bounded domain Ω ⊂ R 2 of boundary Γ are governed by :

curl e = 0 , div d = ρ , d = e (1a) curl h = j s , div b = 0 , b = µh (1b)
where e is the electric field, d is the electric flux density, h is the magnetic field, b is the magnetic flux density, j s is a source current density, ρ is a source volume charge density, is the electric permittivity and µ is the magnetic permeability. Their weak formulations are obtained from Green's identities (2):

(f , grad g) Ω + (div f , g) Ω = n • f , g Γ (2a) (curl f , g) Ω + (f , curl g) Ω = n × f , g Γ (2b)
where f , g ∈ H 1 (Ω), g ∈ H 1 (Ω), (•, •) Ω and •, • Γ denote a volume integral in Ω and a surface integral on Γ of the product of their vector field arguments respectively and n is the unit normal vector exterior to Ω. The domain Ω is composed of conducting and nonconducting parts Ω c and Ω C c as Ω = Ω c ∪ Ω C c . Moreover, Ω s denotes the source parts for ρ or j s , with Ω s ⊂ Ω c in electrostatics and Ω s ⊂ Ω C c in magnetostatics.

From each of the systems (1a) or (1b), two dual formulations can be established -one involving a single-valued potential, the other one involving a multivalued potential -depending on the selected strong and weak formulations, and thus whether (2a) or (2b) is used [START_REF] Ren | A 3D vector potential formulation using edge element for electrostatic field computation[END_REF][START_REF] Durand | Magnétostatique[END_REF].

B. Single-valued potentials formulations

The single-valued potentials formulations, either in electrostatics or in magnetostatics, are the most commonly used, essentially for their ease of implementation and use.

In electrostatics, the e-conform formulation is obtained by introducing the electric scalar potential v such that e = -grad v. Then, in Ω C c , i.e., where ρ = 0, the weak form of div d = 0 can be obtained from (2a):

(-grad v , grad v ) Ω C c -n•d , v Γd = 0 , ∀v ∈ F 0 e (Ω C c ) (3) where F 0 e (Ω C c )
is the space of admissible potential v and v is its associated test function [START_REF] Ren | A 3D vector potential formulation using edge element for electrostatic field computation[END_REF].

In magnetostatics, the b-conform formulation is obtained by introducing the magnetic vector potential a such that b = curl a, with a gauge condition on a to ensure its uniqueness. The weak form of curl h = j s can be obtained from (2b):

(µ -1 curl a , curl a ) Ω -(j s , a ) Ωs + n × h , a Γh = 0 ∀a ∈ F 1 b (Ω) (4) where F 1 b (Ω)
is the space of admissible potential a and a is its associated test function [START_REF] Durand | Magnétostatique[END_REF].

C. Multivalued potentials formulations

The multivalued potentials formulations, in duality with the formulations presented in section II-B, usually require the introduction of source fields related to their associated sources [START_REF] Bossavit | Magnetostatic problems in multiply connected regions: some properties of the curl operator[END_REF][START_REF] Ren | A 3D vector potential formulation using edge element for electrostatic field computation[END_REF][START_REF] Dular | Dual magnetodynamic formulations and their source fields associated with massive and stranded inductors[END_REF]. These are non-zero in the source regions Ω s and can be extended to be non-zero in Ω C s = Ω \ Ω s , allowing the initially multivalued total potentials to become single-valued reaction potentials.

In electrostatics, the source field d s allows ensuring div d s = ρ in Ω c and div d s = 0 in Ω C c . Electric flux density d can then be expressed as the association of such a source field and a divergence-free reaction field, expressed as the curl of an electric vector potential u (with a gauge condition), i.e., d = d s + curl u [START_REF] Ren | A 3D vector potential formulation using edge element for electrostatic field computation[END_REF], leading to the d-conform formulation:

( -1 d s , curl u ) Ω C c + ( -1 curl u , curl u ) Ω C c + n × e , u Γe = 0 , ∀u ∈ F 1 d (Ω C c ) (5) where F 1 d (Ω C c )
is the space of admissible potential u and u is its associated test function. The same procedure is followed in magnetostatics [START_REF] Durand | Magnétostatique[END_REF]. In presence of a nonzero current source j s in Ω s , a source field h s is defined such that curl h s = j s . The reaction field can thus be expressed as the gradient of a magnetic scalar potential φ, leading to h = h sgrad φ. The resulting h-conform formulation is:

(µ grad φ , grad φ ) Ω -(µ h s , grad φ ) Ω + n • b , φ Γb = 0 ∀φ ∈ F 0 h (Ω) (6) where F 0 h (Ω)
is the space of admissible potential φ and φ is its associated test function.

D. Using potentials isovalue lines as flux tubes boundaries

For simplicity of presentation, one denotes F either the electric or magnetic flux density, f either the electric or magnetic vector potential, i.e., u or a, G either the electric or magnetic field, and g either the electric or magnetic scalar potential, i.e., v or φ. In 2-D, with electric and magnetic fields

(f 1 ) (f 2 ) (g 1 ) (g 2 )
iso-f lines iso-g lines ∆g ∆f Fig. 1. Flux tube boundaries definition

(f 1 ) (f 2 ) (f 3 ) (f 4 ) (g 1 ) (g 2 ) Cut (a) Cut on iso-f lines (electrostatics) (f 1 ) (f 2 ) (g 1 ) (g 2 ) (g 3 ) (g 4 )
Cut (b) Cut on iso-g lines (magnetostatics) Fig. 2. Discontinuities of f and g isovalue lines on cuts in the xy-plane, f can be written f = f z z = f z where z is the unit vector perpendicular to the studied plane. This implies that gauge condition div f = 0 is implicitly satisfied.

In regions where potentials f and g are total (with a zero source field), one can write:

F = curl f = grad f × z (7a) G = -grad g (7b)
From (7a), it is straightforward to deduce that iso-f lines and F are colinear, i.e., field lines of F are iso-f lines. In other words, iso-f lines represent flux walls for F . Similarly, it is straightforward to deduce from (7b) that iso-g lines are normal to G, i.e., iso-g lines represent flux gates for G. Hence, in the case of continuous iso-lines, the area bounded by iso-f and iso-g lines defines a flux tube (Fig. 1) and can then be related to its equivalent property, i.e., a capacitance or a reluctance in case of an electrostatic or a magnetostatic problem, respectively. Because (7a) only makes sense in 2-D, the method is not applicable as such to 3-D cases.

Among the defined potentials, those that can be multivalued are f ≡ u = u•z in electrostatics and g ≡ φ in magnetostatics. In general, multivalued potentials are not total in Ω C c because, as mentioned in section II-C, d-conform and h-conform formulations imply the introduction of source fields, that are non-zero in Ω C s . These can however be substituted with discontinuities of the associated potential f ≡ u or g ≡ φ through cuts making Ω C s simply connected [START_REF] Bossavit | Magnetostatic problems in multiply connected regions: some properties of the curl operator[END_REF][START_REF] Dular | Dual magnetodynamic formulations and their source fields associated with massive and stranded inductors[END_REF], which thus makes that potential total in the whole Ω C s . In electrostatics, the discontinuity ∆u cutc through a cut c is the total electric charge carried by the source conducting region connected only to that cut, with the plus or minus sign depending on the cut orientation. If more that one cut is connected to a source region, each of the related cut discontinuities is a portion of the carried total charge, their sum being the total charge. Similarly, in magnetostatics, the discontinuity ∆φ cutc through a cut c is the signed total electric current carried by the conducting region connected only to that cut or a portion of that current in case of multiple cuts.

The discontinuities of the potentials do not allow the direct definition of flux tubes in the vicinity of the cuts regions (Fig. 2). The adjunction of associated portions of isovalue lines on both sides of the cuts has to be done, which asks for a particular treatment as developed hereafter.

III. ADJUNCTION OF MULTIVALUED POTENTIAL LINES

From a calculated FE-wise distribution of a multivalued potential, e.g., f ≡ u, in Ω C c , pairs of additional FE-wise distributions are created by shifting the initial potential to plus or minus cut discontinuity ∆f cutc , for each cut c. The initial distribution with all the shifted distributions for all cuts fill a database of distributions D = {f, all cuts {f + ∆f cutc , f -∆f cutc }}.

Then, a line γ is selected from which iso-f lines are starting for a certain set S = {f 0 , ..., f n } of n + 1 discrete values f i , i = 0, ..., n, regularly incremented with a constant ∆f , i.e., f i+1 = f i + ∆f, i = 0, ..., n -1. Point p i denotes the point on γ at which value f i is found. One has f n -f 0 = ∆f cutc , and thus ∆f = ∆f cutc /n, when the starting surface is the one of a particular source region connected only to cut c. This selected set can possibly be reduced to values of f encountered along some particular surfaces in the studied problem, e.g., the surfaces of the stator and rotor teeth of an electrical machine for g ≡ φ in place of f , to focus on the flux tubes in a particular region, between these particular surfaces. For each value f i in the considered set S, the successive unique FE-wise portions P i,j (j = 0, ... is the portion index) of the related iso-f i line are searched for in the database D, starting with the portion that originates from point p i on the selected surface γ. Each added portion is the one in D starting from the end of the last added portion. Its end becomes the starting point for the next portion up to the end of the iso-f i line, usually located on the boundary of another source region or a symmetry plane. A summary of the whole procedure is given in Table I.

TABLE I POST-PROCESSING ADJUNCTION PROCEDURE

Step Action 1)

Build distribution set D = {f, all cuts {f + ∆fcut c , f -∆fcut c }} 2)
Select a line region γ in Ω C c with its set of discrete values S = {f 0 , ..., fn} of f , at respective positions {p 0 , ..., pn} 3)

For each f i in S index j = 0, point pstart = p i Do Search the FE-wise portion of iso-f i line in D starting at pstart If portion found P i,j = found portion j = j + 1, pstart = end point of found portion Else next f i While portion found

IV. APPLICATION EXAMPLES

A. Two-wire cable capacitance determination

As an electrostatic illustration example, a two-wire cable is considered. The linear capacitance of such a cable is:

C analytical = π 0 r ln( D r ) (8)
where D is the distance between the wires centers and r is their radius. The result of the problem solved with the d-conform formulation is presented in Fig. 3a. As stated in section II-D, the iso-u z lines (with u z = u) undergo a discontinuity through the cut linking the two conductors and cannot be used as flux walls as is (Fig. 3a). The lines portions are then adjuncted following the method described in section III to obtain continuous lines (Fig. 3b). At last, the flux tubes boundaries are properly defined by solving the problem with the e-conform formulation and combining both sets of isovalue lines, i.e., u z and v (Fig. 4a). For each flux tube portion, an elementary capacitance C ele = |∆uz| |∆v| can be defined, where ∆u z and ∆v are the fixed potential differences between its two flux walls and its two flux gates, respectively (Fig. 4b). The two-wire cable capacitance can be computed by properly associating the elementary ones, i.e., in parallel along flux gates lines at first and then in series along flux walls lines. As a numerical application, the capacitance obtained for D = 6 mm and r = 1 mm is 1.5525 × 10 -11 F m -1 and 1.5407 × 10 -11 F m -1 by using [START_REF] Durand | Magnétostatique[END_REF] and the flux tube method, respectively. It is worth noting that since the adjunction procedure is performed at the post-processing step, its accuracy only depends on the accuracy of the original finite element computations. Moreover, the distribution in the studied domain of potential values used to plot the iso-lines are obtained by classic interpolation from the basis functions. In other words, if the e-conform and d-conform computations preceding the adjunction procedure give valid results then the adjunction procedure also will, and it is totally robust no matter the geometry of the studied case.

Moreover, it can also be noted that due to its capability to discretize Ω C c into elementary portions, this method allows to compute the capacitance of a specific part of the studied domain instead of the global one, such as some insulated regions or regions with fringing field. This capability is highlighted in the following section.

B. Airgap reluctance identification

As a magnetostatic illustration example, the developed airgap of a switched reluctance machine is considered. The proper identification of the airgap reluctances is a key point when using Magnetic Equivalent Circuit (MEC) method as a modeling approach. Following the same methodology than the one presented in section IV-A, the problem is solved at first with the h-conform formulation to obtain the iso-φ lines portions (Fig. 5a) which are then adjuncted following the method described in section III (Fig. 5b). Magnetic flux tubes portions are then determined by combining iso-a z lines to reconstituted iso-φ lines (Fig. 6a). Each flux tube defines an elementary reluctance R ele = |∆φ| |∆az| . In the case of salient-pole machines, such as the considered one, computing the global reluctance of the airgap would not make any sense. Actually, in the case of a switched reluctance machine, different airgap reluctances can be defined [START_REF] Kokernak | Magnetic circuit model for the mutually coupled switched-reluctance machine[END_REF], such as overlap, fringing, slot leakage and pole-to-pole leakage reluctances. Moreover, each of them depends on the position of rotor teeth with respect to stator teeth. For this reason, the capability of the flux tubes method to focus on a specific region is an interesting property for the identification of such a machine reluctance equivalent network. Indeed, the considered regions can be selected by properly defining the tube walls, i.e., the min and max values of the iso-φ lines. As an example, Figs. 6b and 6c present the flux tubes to consider in order to determine the overlap and fringing reluctances respectively.

V. CONCLUSION

The contribution of the work presented in this paper is twofold. Firstly, a method allowing to eliminate the discontinuities undergone through cuts by multivalued potentials isovalue lines from their calculated FE-wise distribution has been presented. In particular, it has been shown that the developed adjunction method is totally generic since it is suited for any 2-D multivalued potential, i.e., electric or magnetic, for any 2-D geometry and for any number, shape and distribution of source regions. Secondly, it has been established that, once the adjuncted isovalue lines of the multivalued potential are obtained, they can be combined with the dual isovalue lines of the single-valued potential calculated from the dual formulation in order to define the boundaries of flux tube portions. Those flux tube portions can then be related to their equivalent physical property to constitute either a capacitance or a reluctance network. This network can be employed in many applications, such as the determination of the capacitance or the reluctance of a particular region of the studied domain, or the establishment of equivalent lumped parameters circuits.

Fig. 3 .Fig. 4 .

 34 Fig.3. Iso-uz lines for a two-conductor electrostatic problem (a cut (poly)line joins the conductors)

Fig. 5 .

 5 Fig. 5. Iso-φ lines inside an airgap (a cut (poly)line joins the conductors)

Fig. 6 .

 6 Fig.6. Magnetic flux tubes boundaries (iso-φ and iso-az lines)


Characterization and modeling of a virtual air gap by means of a reluctance network

Jean-Franc ¸ois Brudny 1 , Guillaume Parent 1 , and Inès Naceur 1 1 Univ. Artois, Laboratoire Systèmes Electrotechniques et Environnement (LSEE), F-62400, Béthune, France

This paper deals with an analytical model of the virtual air gap function in order to define design rules when this technique is used inside complex devices. The establishment of those design rules constitutes the main originality of the developments presented in the paper, which are about a single phase magnetic core. It is based on the definition of a nonlinear reluctance network that takes simultaneously into account the effects of DC and AC magnetic flux components in the disturbed area that contains the virtual air gap. The results are validated using a finite element analysis and experimentally.

Index Terms-Magnetic core, saturation, virtual air gap, variable inductance, magnetic equivalent circuit.

I. INTRODUCTION

T HE Virtual Air Gap (VAG) concept has been introduced in order to control the transformer inrush currents [START_REF] Molcrette | Reduction of inrush current in single-phase transformer using virtual air gap technique[END_REF]. Since then, this technique has been taken up in North America [START_REF] Dolan | Harmonic mitigation in a virtual air gap variable reactor via control current modulation[END_REF]- [START_REF] Avila-Montes | A novel compensation scheme based on a virtual air gap variable reactor for ac voltage control[END_REF]. The principle consists in arranging auxiliary windings inside a magnetic core excited by a DC current in order to generate variable local saturations. These saturations modify the R MC core reluctance seen from a main winding wounded on the magnetic core and supplied by an AC voltage source. That way, this magnetic core can be assimilated to a regular one including a mechanical air gap with a variable thickness g. Seen from the AC voltage source, the operation of the VAG can also be seen as a variable inductance L p . Due to the complexity of the magnetic phenomena occurring around the auxiliary windings the VAG behavior is usually studied by finite element analysis, although investigations were conducted to provide analytical characterizations [START_REF] Konrad | An improved method for virtual air gap length computation[END_REF]. Recently, two patents involving the VAG, in partnership with EDF R&D, were published. The first one [START_REF] Brudny | Series current limiter using a magnetic circuit comprising holes and windows[END_REF] concerns a series current limiter allowing to monitor the inrush current of induction machines directly connected to the electrical grid. More generally, it allows to limit the short-circuit currents in power grids. This aspect can be particularly advantageous when the power demanded by certain installations connected to this network increases. Indeed, in this case, this limiter makes it possible to minimize the costs generated by the necessary structural modifications, such as changing the transformer, without being obliged to change the upstream protective device. The second pattern [START_REF] Guuinic | Series voltage regulator with electronics protected against short-circuits by magnetic circuit-based decoupling using holes and windows[END_REF] is about the Dynamic Voltage Restorers, more especially the protection of their static power converter parts [START_REF] Majchrzak | Coupling transformer with a virtual air gap for the protection of dynamic voltage restorers[END_REF][START_REF]Design of a coupling transformer with a virtual air gap for dynamic voltage restorers[END_REF].

Note that the local aspect characterizing the nonlinear effects of the VAG leads to an interesting property: it is possible to use several VAG on one magnetic core in order to obtain a global effect equal to the sum of those produced by each VAG. This property can be very usefull when designing a device in which a significant VAG effect is required.

In order to define preliminary design rules, this paper establishes a model to analytically characterize the function Corresponding author: G. Parent (email: guillaume.parent@univ-artois.fr) provided by the VAG under stady-state operating conditions with sinusoidal supply for the main winding. First, the studied device is introduced. Then, an analytical modeling based on a nonlinear reluctance network [START_REF] Moallem | An improved magnetic equivalent circuit method for predicting the characteristics of highly saturated electromagnetic devices[END_REF] is reported. The third part presents several waveforms, deduced from a simple spreadsheet, characterizing the device operation. That makes it possible to formulate relationships representing the VAG behavior leading to a very easy characterization of auxiliary winding design, which constitute the main contribution of this paper. Finally, relevant specifications, useful for the design of such a device are presented and validated experimentally.

In the following, given a time-dependant electric or magnetic variable z, its peak value and RMS value are noted ẑ and Z respectively. II. VIRTUAL AIR GAP CHARACTERIZATION Fig. 1 presents a magnetic core on which a n p turn main winding supplied by an AC voltage source, noted v p , of frequency f , period T and angular frequency ω, is wound. All the results presented in the following correspond to a sinusoidal -50 Hz v p source. The magnetic core, whose dimensions (Fig. 1) are L = 164 mm, h = 284 mm, D = 66 mm and w = 66 mm, is made of an isotropic steel characterized by its first magnetization curve b(h). From this curve, the variations of the relative magnetic permeability µ r (b) are obtained and presented in Fig. 2. The magnetic core is charaterized by its cross-section S MC = Dw and its mean length

h D L w w φ p l VAG i p v p i A v A Magnetic
0 0.
L MC = 2(L + h -2w).
The local saturation of the core is achieved by two auxiliary windings of n A turns each located in four orifices crossed by a i A DC current. The axis of each auxiliary winding coincides with the y axis of the core. Moreover, the x axis equidistant from each auxiliary winding can be considered as magnetically neutral.

This specific geometry, which considers a small value for D with regard to w brings two advantages:

1) during conventional operation mode, i.e. i A = 0 A, it allows to limit the perturbations on the AC magnetic flux line distribution inside the magnetic core, 2) during the VAG operating mode, i.e. i A = 0 A, it allows to minimize the AC Ampere-turn losses imposed by i A along the x direction. n p and n A are 252 and 20 respectively. The two auxiliary windings are connected so that they produce opposite magnetic fields when fed by i A . That way, those fields cancel each other outside the area which covers a l VAG length, qualified as "disturbed", around the auxiliary windings when the effects generated by v p and i A interact. Fig. 3b shows the magnetic field lines distribution for i p = 0 A and Fig. 3c shows this distribution for i A = 20 A and V p = 240 V at a time step t such that b p = bp = 0.984 T, leading to i p = îp = 2.18 A. These field lines maps show that, on l VAG , it is possible to distinguish two different states, noted High (H) and Low (L) levels. On the H level, at a given time step t such that i p > 0 A, the very large majority of the flux establishes between the orifices, whereas on the L level it establishes on the orifice external sides. This flux distribution is inverted at t + T /2. In order to distinguish the parameters depending on whether they are related to the High or Low level, they will be noted with a H or L index respectively. Similarly, in order to distinguish the parameters depending on whether they are related to interior or exterior areas, they will be noted with an "int" or "ext" index. Moreover, in the following, a lower index "0" is used to denote a variable when i A = 0 A.

III. ANALYTICAL MODELING

The modeling is based on the reluctance network presented in Fig. 4. p = n p i p and A = n A i A denote the magnetic excitations produced by the main winding and the auxiliary windings respectively. The plus (+) and minus (-) signs associated to A refer to the previously stated magnetic opposition. The construction of the proposed reluctance network assumes magnetic flux circulating along the y axis inside the disturbed area. Then, three courses of action are possible to account for the distortions of field lines around the orifices' extremities along the x and x axes:

• The first way to proceed assumes, considering the effects produced by φ p generated by v p , that the increasing of the field lines length generated by these distortions is negligible compared to the ones we would have established in the unmodified magnetic circuit. Hence, those distortions do not significantly affect the magnetic effects produced by this flux component. Then, only an adjustment of the DC Ampère-turns (At) is necessary by considering a useful magneto-motive force λ A A with λ A < 1. Obtaining λ A is done based on Fig. 3b by considering the mean flux tube and calculating the magnetic potential difference appearing on the tube's elements oriented along the y axis. This leads to a first approximation of λ A as follows:

λ A = γ(γ + c) γ(γ + c) + a(a + d) (1)
• The second way to proceed consists in characterizing the reluctances R MCUA , R int and R ext which refer to the undisturbed magnetic core area, to the areas located inside and on the external sides of the auxiliary windings respectively, by considering the fictive heights lVAG 2 = 2γ + c + d superior to the ones of the orifices. Given the symmetries, R int and R ext are each defined with a section equal to 2Da.

• The third possibility consists in combining the two aforementioned methods and adjusting λ A and γ in order, if necessary, to match the results of the analytic expressions with the measurements, for a given pair (i A , V p ) Actually, whichever chosen procedure, determining γ remains a problem. In order to numerically compute it, a finite element analysis is done to determine the axis x position closest to the orifices for which the magnetic potential difference, along x , can be considered negligible.

The reluctance network presented in Fig. 4 is obtained using the third procedure. With n A = 20 and a current density of 10 A it leads to a = 14.5 mm, γ = 20 mm, c = 20 mm and d = 4 mm. The rated i A current is equal to 20 A.

Finally, to use the reluctance network shown in Fig. 4, the curve µ r (b) has been linearized by portions as shown in Fig. 2. For the k th straight line segment (SLS k ), µ r is defined by:

φ extH R extH λ A A+ R intH φ intH φ extL R extL λ A A- R intL φ intL p R MCUA
µ r = α k + β k b for b min k < b < b max k (2)
Table I provides the values of α k and β k with respect to k.

Let us neglect the main winding voltage drop. At a given voltage source v p = vp cos(ωt) it comes v p = dψ p dt where ψ p is the main winding linked flux. Let us then assume that the generated AC magnetic flux φ p is evenly distributed over the width w of the magnetic circuit in the undisturbed zone, it comes ψ p = n p φ p . Since φ p = b p S MC , it can be deduced:

b p = vp n p ωS MC sin(ωt) (3)
Considering the rule of flux conservation and neglecting leakage fluxes, the magnetic flux density b p in the disturbed area can be evaluated:

b p = b p w 4a (4)
Even if the values of b p and b p are known at any time, their distribution in the disturbed area remains unknown. In this area, considering for example the flux between and on the sides of the orifices related to the H level, the following relations has to be satisfied:

     φ p = φ extH + φ intH R extH φ extH = H R intH φ intH + λ A A+ = H ( 5
)
where H is the magnetic potential difference appearing on the H level terminals (Fig. 4). It can then be deduced:

b extH + b intH = 2b p (6)
By assuming that the operating points located between and on the sides of the orifices belong to a k and a k portion respectively, it comes:

       R intH = 1 µ 0 (α k + β k b intH ) w l VAG 4aS MC R extH = 1 µ 0 (α k + β k b extH ) w l VAG 4aS MC (7)
where µ 0 is the vacuum magnetic permeability. Combining ( 5) and ( 7) leads to:

b extH (α k + β k b extH ) - b intH (α k + β k b intH ) = 2µ 0 L VAG λ A A (8)
Introducing ( 6) in ( 8) and defining the following parameters:

                             K A = 2µ 0 λ A n A l V AG N 1 = 2a(β k -β k ) N 2 = -2a(α k + α k ) + b p w(β k -β k ) N 3 = b p wα k D 1 = -2aβ k β k D 2 = 2a(α k β k -α k β k ) + β k β k b p w D 3 = α k (2aα k + β k b p w) (9)
the following relation can be deduced:

(N 1 -K A i A D 1 )b intH 2 + (N 2 -K A i A D 2 )b intH + (N 3 -K A i A D 3 ) = 0 (10)
Setting:

∆ = (N 2 -K A i A D 2 ) 2 -4(N 1 -K A i A D 1 )(N 3 -K A i A D 3 ) (11)
leads to:

b intH = -(N 2 -K A i A D 2 ) ± √ ∆ 2(N 1 -K A i A D 1 ) (12)
Then, one has to check if the values of b extH and b intH belong to the intervals delimiting the k and k' portions according to Table I. If not, another couple [k,k'] is chosen and the procedure if performed again until relevant values are found. Then, the same procedure is performed for the L level.

IV. WAVEFORMS

The numerical exploitation of relations established in section III leads to the following equalities:

R intH = R extL R extH = R intL (13) b intH = b extL b extH = b intL (14)
Fig. 5a shows the variations, on a period T , of δv p = v p vp , b p and i p for i A = 20 A, V p = 240 V and λ A = 1. For comparison, this figure also shows the variation of i p 0 . Figs. 5b and5c give the variations of the magnetic potential differences H , L and VAG = L + H appearing in the disturbed area as well as the variations of b p , b intH and b extH respectively, during the same period. The variations of i p , i p 0 , H , L and VAG obtained from finite element computations are presented in Fig. 6. It can be noted that the reluctance network and the finite element computations lead to very similar results. It implicitly means that the second one of procedures stated in section III can be efficient, as a first approximation. Note that these waveforms can be qualified as "theoretical" because the effects of the main winding voltage drop are not taken into account and i A is an absolutely perfect DC current. Curves shown in Figs. 5a and 6a confirm the VAG efficiency considering the amplitude variations of i p compared to i p 0 . Figs. 5b and 6b highlight that VAG is almost equal to n A i A and -n A i A when b p > 0 and b p < 0 respectively. Then, it is possible as a first approximation to characterize the VAG operation by the relation:

n p i p = R MCUA φ p + n A i A sgn(φ p ) (15)
Since l VAG is small (128 mm) compared to L MC (1160 mm) it is possible to express relation (15) as follows:

n p i p = n p i p 0 + n A i A sgn(φ p ) (16)
Since R MC denotes the reluctance of the magnetic core seen from the primary winding, it comes: φ p = n p i p RMC . Moreover, since ψ p = n p i p it is possible to evaluate the main winding inductance value L p = n p 2 RMC . The reluctance R g of the mechanical equivalent air gap associated to the disturbed area is related to g as follows:

g = µ 0 S MC R g (17)
Then, the value of R g can be calculated by comparing the values of φ p and φ p 0 . Assuming that g << L MC leads to the equality R MC = R MC0 +R g allowing to compute the value of g. Fig. 7 show the variations of L p , L p 0 and g with time during a period T . b p is also shown in Fig. 7 in order to better find the position of the extrema of the deduced quantities which present quite disturbed variations that do not allow an easy pre-designing phase.

V. CHARACTERISTICS AND EXPERIMENTAL CHECKS

The following experimental checks concern the magnetic core designed in section III. Moreover, since the evolution laws of certain parameters are relatively complex, the following developments consider their fundamental component only. The latters are noted with a "(1)" index.

In order to propose a more suitable designing tool let us note i p (rms) the equivalent sinusoidal current associated to i p . By definition, those i p (rms) and i p have the same RMS value. In the cases considered in Figs. 5 and7 the fundamental peak value of i p is îp

(1) = 2.455 A. Moreover, I p = 1.767 A leads to îp (rms) = 2.498 A. About i p 0 , the computed values are îp 0(1) = 0.606 A and îp 0(rms) = 0.611 A. The difference between the peak values is less than 2 % for i p and around 1 % for i p 0 . Then the errors made when using i p (rms) instead of i p are compatible with the assumptions necessarely made to perform this analytical study.

Since only sinusoidal quantities are considered, the relations previously stated for the determination of L p and g instantaneous values lead to characterize those quantities by their mean values < L p > and < g > which present a constant value during a period T :

< L p >= vp îp (rms) ω (18)
< g >= µ 0 S MC n p2 ω vp ( îp (rms) -îp 0(rms) ) (19)
L p (H) V p = 70 V V p = 170 V V p = 240 V V p = 330 V V p = 430 V (a)
V p = 70 V V p = 170 V V p = 240 V V p = 330 V V p = 430 V (b) Variations of g with i A
Fig. 8. Variations with i A of external characteristics for V p = 70 V, 170 V, 240 V, 330 V and 430 V Fig. 8 shows the theoretical variations of < L p > (Fig. 8a) and < g > (Fig. 8b) deduced from (18) and ( 19), for several values of V p chosen so that for i A = 0 the operating point is:

• around the middle of SLS 1 : V p = 70 , bp = 0.3 T,

• around the middle of SLS 2 : V p = 170 V , bp = 0.7 T,

• on the upper level of SLS 3 : V p = 240 V , bp = 1 T,

• on the lower level of SLS 3 : V p = 330 V , bp = 1.35 T,

• around the middle of SLS 4 : V p = 430 V , bp = 1.8 T. Fig. 8b shows the efficiency of the VAG by highlighting that the performances in terms of equivalent air gap thickness increase with i A and decrease with V p . This result was expected since for high values of V p the saturation affects the whole magnetic core.

The experimental measurements about < g > come from several papers already published. Fig. 9a, taken from [START_REF] Konrad | Virtual air gap length computation with the finite-element method[END_REF], shows the variations of < g > with i A for several values of V p (30 V, 55 V, 110 V, 220 V), as well as the results obtained from a finite element analysis. Let us point out that the magnetic core used in [START_REF] Konrad | Virtual air gap length computation with the finite-element method[END_REF] is the same as the one described in this paper. Only the geometry of the auxiliary windings orifices differs. Moreover, the considered magnetic core includes two VAG instead of one in order to validate that the principle of summation of the VAG magnetic effects. Consequently, since the values of n p and n A are identical, the scale of the y-axis of the curves presented in [START_REF] Konrad | Virtual air gap length computation with the finite-element method[END_REF] was adapted (division by 2) to obtain Fig. 9a. By comparing Fig. 8b with Fig. 9a it can be noted that despite the differences between the two structures, the results given by the analytical model presented in this paper are in good accordance with those obtained experimentally using the concept of equivalent sinusoidal current. For example, < g > is of the order of 5 mm for i A = 15 A when V p is close to 220 V. This is indeed remarkable since, from an experimental point of view, the assumptions made about the main winding voltage drop and the ideal i A current are no longer satisfied. In order to make a rigorous comparison, the proposed model has been used at V p = 220 V. Fig. 9b shows the variations of < g > with i A obtained experimentally -deduced from [START_REF] Konrad | Virtual air gap length computation with the finite-element method[END_REF] -by the reluctance network proposed in this paper, by finite element analysis and by (20), the latter allowing to determinate an approached value of < g > deduced from ( 19) and (16) by substituting n A i A with n p ( îp (rms) -îp 0(rms) ):

< g >= µ 0 S MC n p n A ω vp i A (20)
It appears that all the methods underestimate the value of < g >. Moreover, the maximum difference between experiments and (20) is up to 30 %, for i A = 20 A, whereas the results given by the reluctance network are in good accordance with experiments. Then, if the aim of a study is to quickly obtain a first sizing, then one can use relation (20). If the study needs more accuracy, then the reluctance network should be used. Of course, it is still possible to model the whole structure using the finite element method, but this method is too time consuming for a design purpose.

VI. CONCLUSION

The purpose of this study is to propose a pre-designing tool of a device including a virtual air gap. This tool has been realized following several reflexions that came to us following the analysis of one-phase as well as three-phase real cases. The methodology consists in designing, in a first step, a magnetic core assuming that actual mechanical air gaps, with variable thickness, are inserted in it. In a second step, knowing the boundaries of this thickness allow to characterize the auxiliary windings and orifices, keeping in mind that it has to be technically buildable. In order to optimize the structure, this tool is able to quickly answer to questions such as: What is the impact of the orifice shapes on the results? What is the impact of the electrical steel grade on the virtual air gap performances? In the case of the use of grain oriented electrical steel, is it better to use the rolling direction or the transverse direction? Of course, final fine tuning of the design can still be performed using finite element simulations but the flexibility of the presented tool makes it an interesting alternative.

aFig. 3 .

 3 Fig. 3. Characterization of the disturbed area by the virtual air gap

Fig. 4 .

 4 Fig. 4. Reluctance network

Fig. 6 .Fig. 7 .

 67 Fig.[START_REF] Avila-Montes | A novel compensation scheme based on a virtual air gap variable reactor for ac voltage control[END_REF]. Variations with time of the main electric and magnetic variables for V p = 240 V and i A = 20 A

  Variations of L p with i A

Fig. 9 .

 9 Fig. 9. Variations of g with i A for various conditions

TABLE I CHARACTERIZATION

 I OF µR(b) BY STRAIGHT LINE SEGMENTS

	k	α k	β k	b min (T)	bmax (T)
	1	6050	100	0	0.5
	2	7627.75	-3055.55	0.5	0.86
	3	10830.5	-6779.66	0.86	1.517
	4	1372.55	-545.02	1.517	2.1


Mustapha Toudji

Guillaume Parent

Stéphane Duchesne

Patrick Dular

Determination of Winding Lumped Parameter Equivalent Circuit by Means of Finite Element Method

Keywords: Electrical parameters, equivalent circuit, machine windings, finite element analysis

In this paper, the finite element method is used in order to determine electrical parameters of a machine winding lumped element model. Thanks to this model, it will be possible to determine the electrical behavior of electrical machine winding fed by a power converter and thus to study the distribution of the turn-to-turn maximum voltage stress as early as the design process. The suitability of the method is validated by comparison with measurements.

I. INTRODUCTION

E LECTRICAL machines are more and more driven by power converters. Those converters are based on the use of semiconductor devices which have an increasingly high switching frequency (up to several MHz today). This high frequency can generate sudden voltage and current variations. The latter, when applied to the electrical machine windings, generate high electric field levels between the insulators, exceeding sometimes the Partial Discharge Inception Voltage (PDIV). Exceeding this threshold leads to the appearance of partial discharges which are particularly harmful to the turn-to-turn insulation [START_REF] Kaufhold | Failure mechanism of the interturn insulation of low voltage electric machines fed by pulse-controlled inverters[END_REF]. For this reason, it is critical to study the PDIV of the machine windings as early as the design process. Therefore, an accurate and not time consuming method which allows to develop an equivalent model of the winding representing all phenomena occurring in the electrical machine winding is required.

Numerous works focus on electrical machines windings modeling methods. The most used ones are based on the transmission line theory [START_REF] Islam | An equivalent circuit for calculation of interturn voltage distribution of stator windings in the presence of slot discharges[END_REF]- [START_REF] Gersem | Transmission-line modelling of wave propagation effects in machine windings[END_REF], the finite element (FE) method [START_REF] Vaseghi | Modeling of son-salient pm synchronous machines under stator winding inter-turn fault condition: dynamic model-fem model[END_REF]- [START_REF] Suresh | Predicting the transient effects of pwm-voltage waveform on the stator windings of random wound induction motors[END_REF] and lumped parameter equivalent circuits [START_REF] Magdun | Prediction of common mode ground current in motors of inverter-based drive systems[END_REF]- [START_REF] Mihaila | A simulation method to predict the turn-to-turn voltage spikes in a pwm-fed motor winding[END_REF]. The use of the transmission line theory is related to the wavelength at the frequency of the modeled phenomena. It is preferable to use it to model high power electrical machines windings. In addition, this model is no longer valid if a rearrangement of the turns occurs. The FE method has the advantage to be the most accurate one. However, a very fine mesh is needed in order to take into account all phenomena occurring in the winding which makes this method prohibitively time consuming for a design process.

The lumped parameter equivalent circuit method offers the best compromise between accuracy and computation time. There are two approaches: the systemic approach and the phenomenological approach. The systemic approach uses the black box model [START_REF] Liu | Modeling of converter transformers using frequency domain terminal impedance measurements[END_REF]. The internal structure of the system is thus unknown for the user and moreover its components are not necessarily linked to physical phenomena of machine windings. This approach is suitable for a design process but Corresponding author: G. Parent (email:guillaume.parent@univ-artois.fr).

has not been chosen in this work. The chosen approach is the phenomenological one which is the most appropriate method for a design process as well as the the most used one. It involves the development of an RLC-lumped parameter equivalent circuit that allows to describe all phenomena occurring in the winding of the electrical machine by representing each specific physical phenomenon by one of its elements. Nevertheless, the determination of the parameter values of the lumped elements model remains problematic since it is generally performed experimentaly or with empirical data. This is obviously a step that an electrical designer cannot afford. The aim of this paper is to show that it is possible to compute those values by means of an FE analysis considering both the geometry of the turn and its location in the coil, i.e., lumped parameter values that are specific to each turn, in order to provide a highly scalable tool that allows to predict partial discharges advent during the design process of the machine. Moreover, the results are compared to previous works [START_REF] Mihaila | A simulation method to predict the turn-to-turn voltage spikes in a pwm-fed motor winding[END_REF] in order to show the added value for parameters determined by means of an FE analysis.

II. LUMPED PARAMETER EQUIVALENT CIRCUIT

Physical phenomena occurring in an n-turn winding are: generation of Joule losses, iron losses as well as turn-to-turn and turn-to-ground dielectric losses. There are also electrostatic phenomena that create an electric field by turn-to-turn and turn-to-ground charge displacements. Moreover, an electromotive force appears accross each turn due to the current flowing through it. In the presented model, each turn of the n-turn winding is modeled using the elementary RLC lumped parameter equivalent cell represented in Fig. 2. It includes the turn resistance R s representing the Joule losses, the turn-to-turn and the turn-to-ground dielectric losses resistances, respectively noted R t and R m , and in presence of an iron core, the resistance R p representing the core losses. The equivalent cell also includes the self-inductance L p characterizing the generated electromotive force accross the turn and the mutual inductance M ij describing the induced voltage accross the j-th turn caused by the current flowing through the i-th turn, and thus the generation of an electromotive force accross the j-th turn. The model also contains capacitances: the turn-to-turn capacitance C t and the turn-to-ground capacitance C m corresponding to the turn-to-turn and turn-to-ground charge displacements, respectively. This model is used to simulate the transient behavior of windings in order to estimate all turn-to-turn voltages. The main purpose is to determine the peak value of the first voltage overshoot which corresponds to the turn-to-turn maximum stress.

Rs 1 Lp 1 Rp 1 Cm 1 Rm 1 Ct 1-2 Rt 1-2 Ct 1-3 Rt 1-3 Rs 2 Lp 2 Rp 2 Cm 2 Rm 2 Ct 2-3
As an illustration, the equivalent circuit of the 4-turn winding represented in Fig. 3 is shown in Fig. 1.

Rs

III. DETERMINATION OF PARAMETERS A. Resistances determination

Among the resistances of the equivalent circuit, the self resistance R si represents the Joule losses P i in the i-th turn. They are all determined using a magnetodynamic FE analysis in order to account for both the skin and proximity effects occurring in the winding. The Joule losses in each turn are

P i = 1 2 σ S J i .J * i dS ( 1
)
where σ is the electric conductivity [S m -1 ] and J i is the current density inside a cross section of the i-th turn [A m -2 ]. R si can then be evaluated as

R si = P i I 2 i ( 2
)
where I i is the current in the considered turn.

For now, the resistance values representing the turn-to-turn and the turn-to-ground dielectric losses (R t and R m ) are analytically estimated thanks to previous works [START_REF] Mihaila | A simulation method to predict the turn-to-turn voltage spikes in a pwm-fed motor winding[END_REF]. The calculated value is the same in each turn.

The iron losses resistance can be determined by estimating the total iron losses using the approach explained in [START_REF] Bottauscio | Iron losses in electrical machines: influence of different material models[END_REF]. In that approach, a specific curve describing iron losses variation according to the magnetic flux density is taken as a reference. After computing the magnetic flux density value in each FE by means of a magnetodynamic FE formulation, the iron losses in that FE can be easily determined using the reference curve. The total iron losses are then the sum of all FE losses.

B. Inductances determination

To calculate the self inductance L p i of each turn i, a magnetodynamic FE analysis is used. The computation is performed as follows. Each conductor is defined alternately as a current source in which a current of 1 A is imposed while the current flowing through the other conductors is set to 0 A. In a linear case the magnetic energy W m is

W m = 1 2 µ S B 2 dS ( 3
)
where µ is the magnetic permeability and B is the magnetic flux density. L p i can then be evaluated as

L p i = 2 W m I 2 i ( 4
)
The mutual inductance between the i-th and j-th turns is determined following the same methodology, the only difference being that both the considered turns are defined as current sources at the same time, carrying the same current of 1 A. As for the self inductance determination, the current in the other conductors is set to 0 A. M ij is then defined as

M ij = 1 I j I j (W m - 1 2 L p i I 2 i - 1 2 L p j I 2 j ) (5)

C. Capacitances determination

The capacitance matrix is computed using an electrostatic FE analysis. Each conductor is defined as a voltage source, set to 1 V, while all other conductors are set to 0 V. The computation allows the determination of the charge q carried by each conductor as well as the global electric energy W e . The turn-to-turn and turn-to-ground capacitances, C ti-j and C mi respectively, can then be evaluated as

C ti-j = q j V i ; C mi = q m V i (6)

IV. SIMULATION RESULTS AND VALIDATION

All the computations involved in this work are performed with the academic finite element solver GetDP [START_REF] Dular | A general environment for the treatment of discrete problems and its application to the finite element method[END_REF]. It allows to solve usual electromagnetic problems, i.e., electrostatics and magnetodynamics in our case, as well as circuit networks problems. The determination of every parameters presented in section III as well as the simulation of the equivalent circuit are thus solved with the same tool. The whole procedure is applied to the 56-turn winding presented in Fig. 4. These parameter values are reported in Figs. 5, 6 and7. The coil is tested without its magnetic core in a first approach. Parameters C m and R p are therefore neglected. In the simulation, all the conductors of the winding have the same circle shape with a diameter of 1.5 mm. The coil is supplied with a voltage step of 200 V, lasting 500 ns and with a rising time of 22 ns (Fig. 8). In order to simulate under the same conditions as measurements, an elementary model of a cable has been used. Its different parameters were determined by means of measurements.

Fig. 8 shows comparison between the measured and the simulated step response of the winding. In order to show the impact of the FE parameters determination, the simulation results of the model developed by [START_REF] Mihaila | A simulation method to predict the turn-to-turn voltage spikes in a pwm-fed motor winding[END_REF] are added. The parameters of this model were determined using analytical, experimental and numerical methods. The impedance variation of the 56-turn winding with respect to the frequency has been measured using the KEYSIGHT Technologies 4294A impedance analyzer and compared to simulation results in a frequency range from 50 Hz to 5 MHz. The comparison also includes simulation results of the model developed by [START_REF] Mihaila | A simulation method to predict the turn-to-turn voltage spikes in a pwm-fed motor winding[END_REF] (Fig. 9). Moreover, Table I shows the maximum impedance value with its respective frequency value evaluated with each method. FE determination of the parameters leads to a very good accordance with measurements, either in time or frequency domain. In comparison with the previous works [START_REF] Mihaila | A simulation method to predict the turn-to-turn voltage spikes in a pwm-fed motor winding[END_REF], this study highlights two points. Firstly, it is possible to accurately determine the lumped parameter values a priori, i.e., without having to build the winding. As previously stated, this is a key point for the winding designer. Secondly, the results clearly show the importance of considering lumped parameter values specific to each turn instead of using the same elementary cell (Fig. 2) for the whole winding.

V. DISCUSSION

The obtained results on the 56-turn winding validate the different studied assumptions on the physical phenomena occurring in the winding. Of course, the validation has to be confirmed for more complex windings (motors, transformers, etc.) in order to confirm the accuracy of the method. First of all, measurements made with coils placed on a magnetic circuit are needed to fully validate the method with all its parameters. In addition, further works have to be done to make the developed method operational for machines design process.

• The theoretical way used for the dielectric losses determination is probably responsible for the differences observed in the maximum impedance value in the frequency domain. In the time domain, this parameter seems to have a less visible influence.

• It can also be interesting to extend the method bandwidth in order to deal with the other resonance frequencies. Nevertheless, the use of this kind of model is made difficult by the necessity to take into account the wire feeding the coil. Establishing an efficient model for such wires in a large bandwidth is not an easy task. In this way, it would be useful to adapt the bandwidth of our model to the length of the supply wire, which is a critical point.

VI. CONCLUSION

A method has been presented for the a priori determination of the elements of an equivalent diagram, which allows the prediction of coils behaviour during a design process. The results show good accuracy in comparison with the measurements and a significant improvement of a previous method using globally defined parameters.

Fig. 1 .

 1 Fig. 1. Lumped parameter equivalent circuit of a 4-turn winding.

Fig. 2 .

 2 Fig. 2. Lumped parameter equivalent circuit of one turn.

Fig. 3 .

 3 Fig. 3. An example of a 4-turn winding.

Fig. 4 .

 4 Fig. 4. 56-turn winding.

Fig. 5 .Fig. 7 .

 57 Fig. 5. Values of Joule losses resistances.

Fig. 9 .

 9 Fig. 9. Measured and simulated impedances of the 56-turn winding.

TABLE I MAXIMUM

 I IMPEDANCE VALUES VERSUS THE FREQUENCY.

		350
		300
	Step response (V)	100 150 200 250
		50	developed model Measurement	Model from [10] Voltage supply
		0
		0	100	200	300	400	500
				Time (ns)
	Fig. 8. Measured and simulated voltages at the input of the 56-turn winding.
							Maximum impedance (kΩ) Frequency (MHz)
							Developed model	83.3	2.45
							Measurement	48.9	2.42
							Model from [10]	69.4	3.01
							As regards Figs. 8 and 9, the proposed method based on