Melati Rabia

Hamid Azzedine

Thierry Lebey
email: thierry.lebey@laplace.univ-tlse.fr

Derkaoui Mokhtaria

DESIGN OF A NEW ELECTRICAL MODEL OF A FERROMAGNETIC PLANAR INDUCTOR FOR ITS INTEGRATION IN A MICRO CONVERTER

Keywords: DC-DC Power converter, Integration, Ferromagnetic inductor, Geometrical parameters, Technological parameters, New electrical model

This paper presents a new electrical model of a ferromagnetic planar inductor with opposite entry and exit. Our aim is the monolithic integration of this type of inductor in a buck micro-converter. Initially we choose the geometry of the planar spiral inductor which gives the highest inductance value; and by using the software FEMLAB3.1 we also simulate the electromagnetic effects in two types of inductors, one with a ferromagnetic core, and the other with-out, in order to choose the type of inductor which has better electromagnetic compatibility with the components of vicinity. A dimensioning of the geometrical parameters is carried out following a predimensioning. By inspiring on the model of Yue and Yong, we design the new electrical model which highlights all parasitic effects generated by the core and the substrate, and allows calculation of the technological parameters. This model is much simpler than the model of Yamaguchi and takes into account some parasitic effects which were neglected by Yue and Yamaguchi. To validate the results of dimensioning as well as the operation of the dimensioned inductor, we use PSIM6.0 software.

Introduction

The power converters cover more and more scopes of application, This imply different environments of operation, starting from the domestic uses (general public), to the high applications to requirement for reliability (aeronautical); According to the application for which the converter is dedicated, one or more objectives of dimensioning are often specified and must be optimized. The integration of various components of a static converter is actually the principal stake in the field of Power Electronics. Indeed, considering the development of distributed architectures or -Systems-On-Chip‖, new technics must be developed to reduce size and reach better efficiencies. Even if much progress have been done in this field, some technological obstacles still prevent reaching powerful supply in a smaller place. Planar spiral inductors are widely used in monolithic integrated circuits (MIC) for commercial wireless communications. However, these inductors often take up a large portion of the chip area, far more space than the active devices. In a typical amplifier MMIC (Monolithic Microwave Integrated Circuits) up to 80% of chip area is occupied by inductors. This is why the compact miniature inductors are highly desirable for low-cost, highly integrated MIC. In integration, the choice of the topology and configuration of the inductor has a great importance, because it must make it possible to reach strongest possible values of inductance, with negligible losses on a minimal section, and must have a good electromagnetic compatibility with the other components of vicinity. Its realization is also a parameter which must be taken into account.

The study of the literature on this subject, shows that spiral planar inductors are the most studied [START_REF] Estibals | Design and realisation of integrated inductor with low DC-resistance value for integrated power applications[END_REF][START_REF] Chunlai | Effect of the silicon substrate structure on chip spiral inductor[END_REF][START_REF] Jean-Marc | Contribution a l'integration d'une inductance sur silicium et etude de son convertisseur synchrone associe[END_REF][START_REF] Estibals | Conception, réalisation et caractérisation de micro-miroirs à déflexion localisée appliqués aux télécommunications optiques[END_REF][START_REF] Viallon | Optimisation de structures différentielles en technologie sige pour applications en bande millimétrique. Application à la conception d'un mélangeur doublement équilibré en bande k[END_REF][START_REF] Troussier | Integration de bobines sur silicium pour la conversion d'énergie[END_REF][START_REF] Salles | Conception d'éléments passifs magnétiques pour convertisseurs de faible puissance[END_REF][START_REF] Soltani | Integration of converter losses in the modelling of hybrid photovoltaic-wind generating system[END_REF][START_REF] Dangthai | Reseaux de micro-convertisseurs[END_REF][START_REF] Burghartz | On the design of spiral inductors on silicon[END_REF]; they are realized either on a magnetic core and a semiconductor substrate or only on a semiconductor substrate or between two layers of magnetic materials. In the design of this type of passive components, for applications of power, the objective is to obtain an inductor having the following performances: -High operating frequency (beyond the MHz), -Low resistances DC and AC, -Low Compactness.

-A good electromagnetic compatibility with the close components. The realization of a good compromise proves to be difficult, because the integration of such a component introduces additional disturbing elements compared to a regular coil, the substrate and the magnetic core; by their properties respectively semi-conductive and conductive, they cause additional losses of energies to high frequencies which will degrade not only the factor of quality, but also the operating frequency of inductor. The passive components concerned with this study are spiral planar ferromagnetic inductors that we will name -Inductor MOFS‖ (Metal_ Oxide _ Ferrite _ Semi-conductor), and our aim is the monolithic integration of this type of component in a micro-converter. This study does not relate to only the dimensioning of the geometrical parameters of the inductor, but takes into account the electromagnetic effects and the parasitic effects generated by stacking of various materials. To dimension the geometrical parameters with a minimum of energy losses, we use a new approach [START_REF] Melati | Modeling and dimensioning of a planar inductor for a monolithic integration[END_REF], and to calculate the technological parameters, we conceive a new electrical model of an inductor MOFS, this model highlight all the parasitic effects generated by the core, the substrate and the conductor. By using software PSIM6.0, we visualize the waveforms of the currents and voltages in the micro-converter and at the boundaries of the integrated inductor, the form of these signals gives us a clear idea on the operation of the inductor and the micro-converter.

Possibilities of integration

Due to the increase in request, the research efforts in integration multiplied, and gave rise to several technologies. There exist today two integrations types of power: Monolithic integration and hybrid integration. Monolithic integration consists in realizing on the same substrate of the specific functions of power and of circuits or systems traditional which fulfill the functions of filtering (Figure 1), protection and even of order, and their interconnections. In the hybrid approach, the component will result from the association of several functions in only one block, either by stacking, or by regrouping of functions (Figure 2), among the hybrid components, there exist the filters, the transformers, converters … etc. Whose advantages are: the reduction of the costs and dimensions, a simpler assembly…etc.

Methodology of optimized design of integrated inductor

The dimensioning and the realization of an integrated inductor is a big step in the realization of a power converter. The optimization of the inductance value for a given surface will thus depend on several big factors. The dimensioning of an inductor, whatever the type of application for which it is intended, includes three stages: _ Magnetic dimensioning, _ Electric dimensioning, _ Mechanical and thermal dimensioning. Magnetic dimensioning depends on the good choice of the magnetic material which makes it possible to store the quantity of wanted energy, of dimensions of the electrical circuit, and depends also of relative permeability and induction of saturation of the selected material. A good magnetic dimensioning guarantees the obtaining value of the wished inductance, the limitation of the losses energy, and especially avoids saturation of the component. Electrical dimensioning strongly depends on the geometrical parameters of inductor; its purpose is to minimize the losses by Joule effects in the conductor, and mitigate the capacitive, resistive and inductive effects which disturb the good performance of inductor. The choice of the semi-conductor constituting the substrate is also a crucial factor in the optimization of a planar inductor, because it must not to be only compatible with the realization of the active circuits which compose the micro-converter, but it must also generate a minimum of energy losses.

Presentation of the micro converter

We choose a Buck micro-converter continuous-continuous step-down (Figure 3). The inductor that we want to integrate will thus be dimensioned for this type of application. We will choose the schedule of conditions according to: Input Voltage: in The voltages and concerned currents are relatively weak. To increase the output of the micro converter, it is imperative to reduce to the maximum the energy losses inside this converter.

Needs and applications

The inductor that we wish to integrate falls under a general orientation tending towards the miniaturization and the total integration of the systems of small power. The applications concerned are the portable electronics, whose tendency always goes towards the reduction of the size and the number of components. Thus we will register this study in the field of the small power and more particularly that of the energy conversion. Because of ease of its realization compared to other planar inductors, and the presence of experimental results cited in the literature, and thanks to its good electromagnetic compatibility, we focused our studies on the modelling of planar spiral inductor.

Different geometries of planar spiral inductors

The planar spiral inductors are presented in different geometrical shapes, circular, square, octagonal, and hexagonal; they are all characterized by the same geometrical parameters: the number of turns n, the internal diameter d in , the external diameter out d of inductor, the width w and the thickness t of the rectangular coil, and the spacing inter-whorls s (Figure 4). Before making our choice, we will study the influence of the geometry on the value of the inductance.

Choice of the geometry

We will choose now the geometry of the planar inductor that we wish to integrate. In the literature, we find several analytical expressions such as the formula of Bryan [START_REF] Bryan | Printed inductors and capacitors[END_REF][START_REF] Basset | Conception, réalisation et test d'un micro système de transport alimenté à distance[END_REF], the formula of wheeler [START_REF] Wheeler | Simple inductance formulas for radio coils[END_REF] the formula of Mohan and al [START_REF] Mohan | The design, modelling and optimization of on-chip inductor and transformer circuit[END_REF]… etc. These expressions enable us to make a comparison of the value of inductance for various spiral geometries: square, circular, hexagonal or octagonal.

Wheeler's Formula

Wheeler's expression allows an evaluation of the inductance of an hexagonal, octagonal or square reel carried out in a discrete way.

A simplification can be operated in the integrated planar case [START_REF] Wheeler | Simple inductance formulas for radio coils[END_REF]. Inductance mw L given by the method of Wheeler has as an expression (1):

2 avg mw 1 0 r 2 m nd L k 1 k A     (1)
Where " m A " is the factor of form, defined by expression (2): We can consider in the Wheeler's formula, that the value of inductance is proportional to ratio 1 2 kk (Table 2). The greatest value of ratio 1 2 kk is allotted to the square geometry

out in m out in dd A dd    . ( 2

Monomial's Formula

The expression of Monomial [START_REF] Mohan | Simple accurate expressions for planar spiral inductances[END_REF] used to calculate the inductance is based on the relation (4): The values of these coefficients are given in Table3. We also notice that the greatest value of ratio

    3 4 1 2 5
     is allotted to the square geometry.

Mohan's Formula

The expression given by Mohan for the calculation of inductance is expressed by relation (5) [START_REF] Mohan | The design, modelling and optimization of on-chip inductor and transformer circuit[END_REF]:

2 0 avg 1 2 2 34 n d c c L ln c c 2              (5)
Where ρ is the factor of form expressed by formula [START_REF] Troussier | Integration de bobines sur silicium pour la conversion d'énergie[END_REF]:

out in out in dd dd    (6) 1 2 3 4
c , c , c , c are coefficients given according to the geometrical form (Table 5). c ln(c ) c c  .

The square geometry takes the greatest value of the product   c ln(c ) c c  ( Table 6). The four tables show that the square spiral geometry gives the greatest value of inductance, so will opt of a square spiral planar inductor.

Influence of the magnetic core in a planar inductor

A magnetic core in a planar inductor, not only allows to increase the value of inductance, to channel the magnetic flux, to store energy or to transmit it, but it also enables to decrease strongly the number of whorls, which mitigates the inter-whorls capacitive effect, as well as the capacitive effects due to the substrate and magnetic core; consequently it facilitates the realization of the inductor.

The choice of the used magnetic material determines the size of the component and confines the magnetic flux in the inductor, moreover containment of the lines of magnetic field is advantageous, because it attenuates the electromagnetic disturbances (EMI) of which the effects are accentuates with the increase of frequency; this causes an increase flow through the cross-section of the component (Figure 5) and notably an increase of the inductance value (by 10% to 100%) (Figure 6) for the same number of turns and the same occupied surface. 9. Simulation 3D of the different electromagnetic effects in a planar spiral inductor.

By using FEMLAB 3.1 software, we studied the electromagnetic compatibility of a planar inductor with the other micro converter's components. Firstly we visualized the electromagnetic phenomenon in a planar inductor without (Figures 8 and9) and with a magnetic core (Figures 11 and12). We observe in Figures 8 and9 the propagation of the magnetic field lines in all directions, that disturb the operating close components, while in Figures 11 and12 , the majority of the magnetic field lines are concentrated at the walls of the magnetic block, there are only some lines which overflow of the other part of the spiral. Thus the electromagnetic disturbance with respect to the close components is very attenuated, but another problem emerges; we note the presence of a current which flows in the mgnetic core. this current is generated by capacitive effect and by the eddy currents (Figure13). To circumvent this problem we proceed as follows :

To attenuate the eddy currents; we cut out the magnetic core in plates according to our choice, vertically (Figure 14) or horizontally (Figure 15); and the currents due to the capacitive effects will be attenuated thanks to a good dimensioning of the electrical circuit and magnetic circuit of the inductor (it is the objective of our work). It should be noted that the two directions of flaking do not present a difference from the magnetism point of view. On the other hand, we must choose the solution which facilitates the realization, and most adapted to planar technology. We conclude finally, that to have a good electromagnetic compatibility of the inductor with the close components, it is preferable to insert a magnetic or ferromagnetic layer as an inductor's core.

Predimensioning of a planar inductor

Different geometrical parameters of inductor directly influence the inductance and resistance values, in the case of integrated inductor using planar technology: conductor's width thickness, spacing, external and internal diameters, and number of turns.

Predimensioning in power electronics makes it possible to trim the design problem of the power converters. It especially makes it possible to ensure the functionality wished, as well as the respect of the principal specifications described in the schedule of conditions. So before carrying out a geometrical dimensioning, it is preferable to make a predimensioning.

Influence of the section of the conductor

For power applications, the conductor section (w.t) is an important factor, because it is this parameter which determines the mean levels of maxima current in front of being supported by the component. In addition, the study of the literature shows that if the conductor thickness has little influence on the value of inductance, it on the other hand will be determining on the value of resistance series. The reduction of the conductor's width increases the value of inductance but at the price of an increase of the current density and consequently heating of the inductor. For this reason, the section of the conductor must be considered, firstly for the execution of a schedule of conditions; it is then fixed according to the maximum density of current tolerated by the structure.

Once the section will be fixed, it remains to played only on the inter-whorls spacing -s‖, and the number of turns -n‖. To illustrate this study we choose a planar spiral inductor with an external diameter fixed at 900 µm and a frequency of operation equal to 1, 5 MHZ. The skin effect is a phenomenon of electromagnetic origin, which exists for all conductors crossed by an alternative current and depends strongly on the frequency (Figure16): it causes the decrease of the current density while going far from periphery of the conductor to its center. It results an increase from the conductor's resistance (Figure17); what leads to more important losses by Joule effect. The skin thickness determines the width of the zone where the current concentrates in a conductor (Figure 18).

The skin thickness is represented by relation [START_REF] Salles | Conception d'éléments passifs magnétiques pour convertisseurs de faible puissance[END_REF]:

f    (7) With 0r μ μ μ  ( 7 0 μ 4π10  
H/m), µ r is the relative permeability of material used as conductor.

So that the current flows in the entire conductor, it is necessary that one of the following conditions be filled:

w2  Or t2  .

If the two conditions can be filled at the same time, it is the ideal, but in the case where only one condition w2  or t2  can be realized; it is preferable to have the thickness "t" more important, or its width " w "? To slice, we will study the influence of the width -w ‖ (Figure 19) as well as the influence of the thickness -t‖ (Figure 20) on the ratio R/L, with constant external and internal diameters. The analysis of Figures 19 and20 shows that the widening of the conductor as well as the increase of the conductor thickness, for a constant internal and external diameters, involve the reduction of the ratio R/L. but they cause a stressing of the capacitive effects.

Influence of the number of turns "n" and spacing "s"

For an occupied surface, fixed by the authorized maximum external diameter, the number of turns -n‖ is finished. It remains to be seen if it is preferable to have an inductor with a strong rate of filling, or on the contrary to privilege a winding with few turns. Figures 19 and 20 show that when the number of turns increases, for tending towards a maximum filling, the ratio R/L decreases, but, an important number of turns in a planar inductor, accentuates capacitive effects, as the inter-turns capacitive effect whose influence appears when the operating frequency increases. It should be noted that figures 19 and 20 are traced for fixed internal and external diameters, therefore if the conductor width "w" increases, implies automatically decreasing of spacing inter-whorls -s‖, and also decreasing of ratio R/L. But a narrow spacing between the turns is also in favour of the accentuation of parasitic effects. . For this geometrical study, it emerges that: The maximum surface that must occupy a planar spiral inductor is determined by the volume of the substrate (or of the magnetic core in the case of a ferromagnetic inductor), the determination of a set of geometrical parameters suitable ensuring the lowest ratio R/L, must answer several rules. By studying the variations of the ratio R /L, according to the various geometrical parameters, several conclusions arise in order to establish a methodology of predimensioning. Thus, it is first of all necessary to translate the constraints imposed by a given current fixing in a final way the section of the conductor -w.t ‖. In the same way, minimum ratio R/L can be reached by decreasing spacing -s ‖ as much as possible, while making sure that the rate of filling of the spiral is maximum; but a narrow spacing accentuates the capacitive effect inter-turns which degrades the performances of the inductor. To circumvent this problem, we choose the condition t2  instead of w2  , because by decreasing the thickness -t ‖ we mitigate the inter-turns capacitive effect.

From this geometrical study, we will dimension a square planar spiral inductor for its integration in a microconverter. This inductor is made up of a copper winding, a magnetic core which is an alloy of 20% of iron and 80% of nickel ( ie Nf ) and a semi-conductor substrate of silicon.

Dimensioning of the inductor

The steps for the design of planar spiral inductor are represented by the following algorithm (Figure21).

Dimensioning of the magnetic circuit

From schedule of conditions, we define the specifications of the micro converter which constitutes the starting point for the dimensioning of the inductor. By taking into account of selected electric and magnetic characteristics, we evaluate the volume of the magnetic core, which enables us to define the section on which we will put the electrical circuit of the inductor; then we will evaluate the dimensions of this circuit allowing to answer the specifications of the converter in terms of magnetic storage of energy and losses in materials. The following equations enable us to calculate the value of the necessary inductance for the realization of the micro converter; the voltage across the inductor depends on the variation of current flowing in this latter (Equation 8).

     (11)
L min I 0,16A  , on I 0, 44A  and L max I 0,6A  Thus the micro converter functions in mode of continuous conduction, i.e. that the current in the reel is always positive and never cancels itself (Figure 22). Duration of the period T during which the switch M1 is closed (Figure 3), leads to an increase in energy stored in the inductor, this duration represents the cyclic ratio α; for a step-down we have:

out in V 0,5 V   (12) e on V (1 ) I L f     (13)
The undulation of current on I  is maximum for

       (14)
Thus we must carry out a micro reel whose value of inductance will be 1,89µH, this value makes it possible to determine the stored energy and consequently the volume of magnetic circuit, which enables us to evaluate the external diameter of inductor; The diameter being fixed, we can calculate the other geometrical parameters (Figure4) which are: the number of turns -n‖, the internal diameter -din‖ of inductor , the width -w‖ and the thickness -t ‖ of the rectangular coil, and the spacing inter-turns -s ‖. Evaluation of these parameters is the subject of our study. An insulator is placed between the core and the conductor on the same level in order, to isolate the electrical circuit from the magnetic circuit of the inductor.

Calculation of the volume of the core

The value of inductance being given; it is possible to calculate the maximum energy to be stored in this component by relation ( 15)

2 max LI 2 1 W  (15)
For an inductance of 1,89µH, crossed by a maximum current of 0,6 A, the inductor can store 0,34 µj.

To determine the volume of ferrite necessary to this stocking, we must know the voluminous density of energy characterizing this material. This density is given by equation ( 16)

2 max v max 0r B W 2   (16)
The necessary volume to store energy will be fixed by the maximum magnetic induction Bmax which can support the material, and its relative permeability µr. 0  is the magnetic permeability of the free space ). In the case of an alloy Ni-Fe (80% of nickel and of iron 20%) and without air-gap whose characteristics are: r 800  and max B 0,6 T  , we obtain:

3 vmax W 179Jm  
. The volume of the ferromagnetic core is given by relation 17.

1, 9 mm 3 of Ni-Fe is necessary to store 0, 34 µj. Let us note here that more the magnetic permeability will be raised, more the volume of the magnetic circuit will be important, for a given maximum induction.

Dimensioning of the electrical circuit of inductor

The volume of the ferromagnetic core being evaluated to 1,9 mm 3 , We will make a game of dimensions, we consider the core as a block having a thickness of 1,6 mm, what gives a square section equal to 1,1875 mm 2 that we name mag A it is the section on which we will put the spiral coil, each of its sides is equal to 1,089 mm; if we leave on each side a distance of 95 µm, the external diameter will of the inductor be equal to 900 µm.

The calculation of the values of other geometrical parameters of the electrical circuit is based on quite precise conditions.

Calculation of the number of turns

We find in the literature several formulas which allow us to calculate the number of turns according to the value of inductance, such as the formula of Bryan, the formula of wheeler, the formula of Mohan and formula of Monomial. If we calculate the number of turns with the three formulas, the results will be almost identical. So we are forced to choose one method. We chose the method of Wheeler; it allows an evaluation of the inductance of an hexagonal, octagonal, square reel, carried out in a discrete way. A simplification can be operated in the integrated planar case [START_REF] Wheeler | Simple inductance formulas for radio coils[END_REF]. Inductance given mw L by the method of Wheeler has as an expression (18):

2 avg mw 1 0 r 2 m nd L k 1 k A     (18)
The

  2 2 0 r 1 ou t 2L (1 c) k (1 c) n k d (1 c)        (19)
We choose as report c 0,25



, so with the external diameter out

Calculation of the thickness "t" and the width "w" of the conductor

The calculation of these two parameters depends on the skin thickness. To circumvent this problem, let us calculate the width -w‖ and the thickness -t‖ of the conductor according to the skin thickness, and to the surface density of current which flows in this last. For a rectangular conductor, the formulas are as follows [START_REF] Gautier | Modèles électriques pour la conception des circuits intégrés sur silicium[END_REF]: , and frequency f 1,5 MHz  we will obtain after calculation 53,57µm  So that the current flows in the entire conductor, it is necessary that one of the following conditions is carried out: w2  or t2  . We impose one of the two values -t ‖ or -w ‖; from the predimensioning it is preferable to give the smallest value to the thickness t ‖. In putting t 50 m  we can calculate the width -w ‖. In order that a maximum current I 0,6A  can flow in the coil conductor, it is necessary that the section S of this conductor meets the conditions [START_REF] Patrick Yue | Physical modeling of spiral inductors on silicon[END_REF]:

avg I S j  with S w.t  (22)
It should be noted that the acceptable current density in a planar inductor is higher than that in the large reels, because the losses by Joule effect which overheat the conductor are proportional to its volume. In the majority of cases, micro-conductors are in contact with substrates having good properties of temperature conduction.

According to dimensions and the form, the densities of current from 3 10 to 4 .

After calculation, the average current density and the section of the rectangular conductor are estimated at:  . Since the condition t2  is respected. We can increase the value of w. We thus put a width w= 130 m  .

12.2.1. Simulation of the section of the conductor.

By using software FEMM 4.2, we simulate the dimensioned conductor of the inductors, in order to ensure us that the skin effect is completely circumventing; the result is given by figure 23. The dimensioning of the conductor's section gives an encouraging result, because Figure 23 shows that at a frequency of 1.5MHZ, this section is completely crossed by the current, and there is not any deserted part, therefore the problem of increase in resistance series and heating of the ribbon conductor is eliminated.

Calculation of inter-turns spacing

From the geometry of the square inductor, we establish formula [START_REF] Yamaguchi | Two port type ferromagnetic RF integrated inductor[END_REF]:

out in out in d d 2wn d d 2wn 2s(n 1) s 2(n 1)          (23)

Calculation of the average conductor length.

The average length of the coil conductor in a square spiral inductor is deduced from the expression [START_REF] Patrick Yue | Physical modeling of spiral inductors on silicon[END_REF]:

  avg out l 4n d (n 1)s nw s      (24)
For s= 78 µm , Where avg l 4, 418 mm 

Results of dimensioning of electrical circuit

The essential parameters of planar inductor that we want to integrate are gathered in the table 7.

Table 7 shows that the computed values of the geometrical parameters are in the standards of integration, thus the result of dimensioning is encouraging.

Modeling of a spiral planar inductor 13.1. Planar inductor without magnetic core

After Nguyen and Meyer in 1990 [START_REF] Nguyen | Si IC-compatible inductors and LC passive filters[END_REF][START_REF] Couderc | Etude de matériaux doux à forte aimantation et à résistivité élevée pour les radio-fréquences. Applications aux inductances spirales planaires sur silicium pour réduire la surface occupée[END_REF] which were the first with proposed a simple model in -π‖ to describe the behaviour of a planar inductor integrated on silicon. An improved model was developed later by Ashby and Al [START_REF] Couderc | Etude de matériaux doux à forte aimantation et à résistivité élevée pour les radio-fréquences. Applications aux inductances spirales planaires sur silicium pour réduire la surface occupée[END_REF][START_REF] Ashby | High Q inductors for wireless applications in a complementary silicon bipolar process[END_REF]. However the parameters of the model need to be adjusted starting from the experimental curves, rather than to have a physical significance. More recently Yue and Yong [START_REF] Patrick Yue | Physical modeling of spiral inductors on silicon[END_REF] brought back a similar model (Figure 25) but with parameters more appropriate to the geometry of inductor. The electric diagram of a spiral planar inductor is deduced starting from its transverse section (Figure24).

Planar inductor with magnetic core

When a magnetic metal layer is built above (or below) of the inductor, the model becomes considerably complicated because of the interactions between layers stacking (Coil -Magnetic core -Substrate) constituting the inductor. The first equivalent diagram (Figure 26) was proposed by Yamaguchi and al [START_REF] Yamaguchi | Two port type ferromagnetic RF integrated inductor[END_REF]. This model takes into account the parasitic capacities C Sm1 and C Sm2 between the magnetic layer and the inductor which are due to the potential differences between the coil and the magnetic layer. Such capacities C pm1 and C pm2 also exist between the magnetic plan (potential floating) and the substrate (potential with the mass). These capacities are associated with resistances R pm1 and R pm2 which correspond to the losses associated by Joule effect in silicon. Another C mag capacity is also added in series with the main inductance (L mag ) if the magnetic material is split into separate components, this corresponds then to the capacity of these magnetic elements. This L mag inductance corresponds to the fraction of inductance (L S ) increased by the presence of the magnetic layer with strong permeability. It is also accompanied by a series resistance type R mag which takes into account the Joule effects in the magnetic layer and the losses in L m inductance due to the presence of this conducting magnetic layer.

New electrical model of a ferromagnetic inductor

In a physical model each circuit element is related directly to the physical layout. This property is very important when designing new inductors or transformers where measurements data are not available. The key to accurate physical modeling in this way is the ability to describe the behaviour of the inductance and the parasitic effects. Each lumped element of the model should be consistent with the physical phenomena occurring in the part of structure it represents. In a pure physical model the value of electrical lumped elements is only determined by the geometry and material constants of the structure.

During modeling, we have to be prudent about the accuracy and limitations of the model. Inspired by the model of Yu and Yong, we design a new electrical model of a spiral planar inductor with a ferromagnetic core and a silicon substrate. The aim of this new model is to minimise the number of components in the equivalent circuit. The basic idea is to model the whole winding as one lumped physical element with two ports. Accordingly, the inductance and all parasitics just relate to one physical element. The Figure 28 shows the equivalent circuit resulting from the transverse section (Figure 27). It represents an almost symmetrical pi-circuit with ports on both sides. We gave to this type of inductance the name MOFS -Metal-Oxide-Ferrite-Semi-conductor‖.

Discussion and interpretation

If we manage to integrate a planar inductor in an electronic device, this inductor must have two connections, one at its entry and an other at its exit. In literature, the exit of the spiral is carried out through a conductor buried in the insulator which is placed at the lower part of winding. Neither the model of Yue, nor the model of Yamaguchi took into account the parasitic effects which generates the buried conductor. In this work, we present a new electrical model of planar spiral ferromagnetic inductor in which entry and exit are opposites (Figure 32). The electrical model corresponding at a ferromagnetic spiral planar inductror where the entry and the exit are on the same side is in reference [START_REF] Melati | Modeling and dimensioning of a planar inductor for a monolithic integration[END_REF].

New electrical model of ferromagnetic planar inductor with opposite entry and exit

Starting from the transverse section of the ferromagnetic spiral planar inductor (Figure 31), we posed the corresponding electrical model with opposite entry and exit (Figure 32).

Calculation of Technological parameters

Before passing to the realization of an inductor following a dimensioning such as that described above, new elements must be taken into consideration in order to determine if the beforehand fixed geometrical parameters will induce a correct frequential behavior of the component. For that, we must be interested in other parasitic elements, which are mentioned in Figure 32. ε 0 is the permittivity of free space. The series resistance of the winding conductor is also a crucial problem in the design of inductors. Moreover, when inductor works at dynamic mode, the metal line suffers from skin effects and proximity effects, and series resistance s R becomes function of the frequency.

At first approximation the resistances s R is estimated by using the classical relation as in equation ( 26).

   .
The inductor which we wish to integrate being of type MOFS, an oxide is essential to ensure an electrical insulation between the electrical circuit and the ferrite, which is in its turn put on a semi-conductor substrate. This piled structure generates various parasitic elements which imply increasingly important losses of energy when the frequency increases. The study of the literature provides several expressions allowing the calculation of some of these parasitic elements. In this regard, the most cited reference work is those of Yue [START_REF] Patrick Yue | Physical modeling of spiral inductors on silicon[END_REF]. The substrate and ferrite resistances sub R and mag R represent respectively the ohmic losses in the substrate and the core (ferrite). They are caused by the current flowing between the winding conductor and the ground contact. Although the winding is put on a non-conducting dielectric, a current flows through the capacitive coupling between winding and substrate, and between the ferrite and ground contact. An appropriate expression for the substrate resistance is derived from reference [START_REF] Kehrer | Design of monolithic integrated lumped transformers in silicon-based technologies up to 20 GHz[END_REF].

The substrate resistance calculation is based on the area where the capacitive coupling acts on the substrate.

This area depends on the width -w ‖ and the average length -avg l ‖of the winding. These parasitic effects degrade the performances of inductor. The succession Metal-Oxide-Semi-conductor creates an oxide capacity ox C , in the same way the substrate material block between two ideal conductor plates, is represented by an electrical equivalent circuit of a physical configuration which consists of the resistance sub R and the shunt-capacitor sub C in parallel .

If the internal port of inductor is brought back towards outside by a buried contact, the potential difference between the whorls and the buried conductor can induce stray capacities. We also consider that the capacity v1 C is equivalent to the sum of the capacities of covering between the buried conductor and the different whorls, capacity v2 C is between the buried conductor ant the core (Figure 31).

The circuit in "π" being symmetrical (without capacities v1 C and v2 C ), we have:

ox ox1 ox 2 C CC 2   (28) sub sub1 sub2 C CC 2  ( 29
) sub1 sub2 sub R R 2R  (30) mag1 mag2 mag R R 2R  (31)
We model usually the capacities in an integrated inductor starting from the concept of capacity with parallel plates:

ox1 0 ox ox 1 A C 2 t    (32) sub1 0 si sub 1A C 2h    (33) avg s0 t l C s  (34)   2 v1 0 ox 12 n 1 w C t      (35) v2 0 ox [nw (n 1)s d]w C t        (36)
It should be noted that the width of the buried conductor is equal to the width -w‖.

The suitable expressions for the substrate and ferrite resistances are given by relations (37) and (38):

sub sub1 Si h R2 A  (37) mag mag1 NiFe h R2 A  (38)
The resistance of the buried portion of the conductor ribbon is expressed as follows:

b [nw (n 1)s d] Rb w h     (39)
Where sub h , mag h , ox t and b h represent respectively thickness of substrate Si ,ferrite N i F e , the oxide of silicon S i O 2 ,and the buried conductor of copper;



) are the respective relative permittivities of free space, of the silicon and of the oxide,avg l ‖ represents the average length of the coil , -w‖ its width, -n‖ is the number of turns. mag h =1,6 mm. In puting sub h = 50 µm, t 1-2 = t 2-3 =20µm ox t = 60 µm , b h =20µm, and d=95µm The results are grouped in Table 8.

Table.8:

Results of technological parameters of a -MOFS‖ inductor with opposite entry and exit.

Technological parameters Calculated values

C s (pF) 0,025 C ox1 (pF) 0,17 C sub1 (pF) 0,6 C v1 (pF) 0,03 pF C v2 (pF) 0,097 PF R s (Ω) 0,017 R b (Ω)
113.10 -5 R mg1 (MΩ) 56.10-5 R sub1 (KΩ) 2,05.10 -3

The following Table 9 gives the results of technological parameters of the -MOFS‖ inductor where the entry and the exit are on the same direction [START_REF] Couderc | Etude de matériaux doux à forte aimantation et à résistivité élevée pour les radio-fréquences. Applications aux inductances spirales planaires sur silicium pour réduire la surface occupée[END_REF].

It is noted that the values of these parameters are calculated with the same schedule of conditions as the precedent Table 8.

Table .9. Results of technological parameters of an inductor -MOFS‖ with entry and exit at the same direction.

Technological parameters

Calculated values C s (pF) 0,025 C ox1 (pF) 0,17 C sub1 (pF) 0,6 C v1 (pF) 0,056 C v2 (pF) 0,17 R s (Ω) 0,017 R b (Ω)
205.10 -5 R mg1 (MΩ) 56.10-5 R sub1 (KΩ) 2,05.10 -3

Interpretation of results

The aim of this work is to minimize the losses of energy in the inductor, i.e. to mitigate the capacitive and resistive effects due to the induced currents and the eddy currents; for that, the values of the capacitances ox C , sub C , v1 C and v2 C must be as weak as possible, to avoid all infiltrations of currents in the substrate and in the magnetic core; on the other hand the values of resistances mag R and sub R must be as high as possible to avoid the passage of the currents induced by capacitive effect in the substrate and the core; but resistances b R and s R must be very low to facilitate the flow of current in the conducting ribbon.

The computed values of the technological parameters which strongly depend on the geometrical parameters are in agreement with the wished objectives.

The values of resistances s R and b R are negligible, the values of the capacities ox C , sub C , v1 C and v2 C are very weak; on the other hand the resistance mag R is too low, therefore it facilitates the infiltration of the induced current towards the substrate, but the resistance sub R is sufficiently strong to make barrier with this current.

Simulation of the electrical circuit of the integrated inductor

To test the results of the dimensioning of the planar spiral ferromagnetic inductor, we use the PSIM6.0 software.

To validate the results of our work, we simulate with software PSIM6.0 the equivalent electrical circuit of the micro-converter containing the electrical circuit of the dimensioned square spiral planar inductance. This simulation is carried out using the results of Tables 8 and9, and values of micro-converter capacitance and a load resistance. This simulation enables us to make sure that the values allotted to the geometrical and technological parameters after dimensioning do not disturb the operation of the inductor. And also allows us to see that the integration of this inductance in the micro-converter ensures a good performance of this last. At the first time, we simulate the electrical circuit of a micro-converter containing an ideal inductor without any parasitic effects, to be sure of the good operating micro-converter (Figures 34 and35).

At the second time, we simulate the electrical circuit of a micro-converter containing a dimensioned -MOFS‖ inductor with all its parasitic effects,once, the -MOFS‖ inductor with entry and exit on the same direction (Figures 36 and 37); and another once the -MOFS‖ inductor in with opposite entry and exit (Figures 38 and39). After visualization of the waveforms of currents and voltages in the micro-converter, we record at two diffrent moments , the minimal and maximum values of these latters.

The comparison of the waveforms of voltages and currents, resulting from the three simulations, as well as their measered values , will be carried out thereafter.

To simulate the electrical circuit of the micro-converter, we must calculate the value of the capacitance as well as the value of the load resistance.

calculation of the value capacitance of the micro-converter

We want a constant voltage at the exit of the micro-converter , but in practice this is not the case.

The output current Is being almost constant, the great variation of the inductor current I L can be absorbed only by thecondenser. The current flowing in the condenser (I c = I L -I savg) is a triangular current, The voltage Vc on its terminal is thus made up of parabola arcs (figure33) . At the moment 1 ( Figure 33) the voltage is minimal, it goes up then and reaches its maximum when the total sufrace marked in red is added (Moment 3). The undulation amplitude is the surface of this triangle that is to say (0,5x base x Height).

out out 2 in V 1 1 T I 1 ( ) V 1 C 2 2 2 V 8Lf C      (41) Where out out in 2 out V V1 V C 8Lf V       (42)
The capacitance value is inversely proportional to the undulation of the output voltage; thus we can exploit the value of capacitance C to attenuate this undulation and to have a more or less constant output signal. The waveforms of Figure 35 reflect those of a Buck converter , the current at the boundaries of the reel is corrugated and never itself cancels, therefore the continuous conduction mode is respected; the terminals voltage of the capacitor and the output voltage are similar and continuous, with a light undulation. We also notice a continuous output current, and a transitory mode during the first micro seconds. 17.3. Simulation of a micro-converter circuit with a MOFS inductor with entry and exit at the same direction.

I S 3.56979e-1 3.77367e-1
The waveforms of figure 37 are also similar to those of a Buck converter; we must now check if the dimensioning of the inductor is perfect or not, for that, we will draw up a summary table which contains results of three simulations, then we compare the results of the two last with those of the first.

Simulation of a micro-converter circuit with a MOFS inductor with opposite entry and exit

I S 3.77491e-1 3.55993e-1
We notice in figure 39 that the waveforms are respected and reflect those of a Buck converter, we thus conclude that if we change the exit of the inductor in order to have entry and the exit in opposed directions, the operation of the latter will not be disturbed. We conclude finaly that the proposed new electrical circuit of a ferromagnetic inductor with opposite entry and exit functions perfectly.

Interpretation of results.

The table 13 gathers the results of the three simulations. To interpret the results of these simulations, we calculate the relative errors of the voltages and the output currents (Table 14).

. The average value of the output current in the inductor in which entry and the exit are at the same direction is very close to the desired value: 0.36717 A instead of 0.38 A, with a relative error of 5.55℅. The inductor in which entry and exit are opposed, comes in the second place with the same relative error and an average value of the output current from 0.36674 A instead of 0.38 A. while the ideal inductor presents a relative error of 7,76℅, and an average value of the output current slightly higher than the desired value.

If we take into account the voltages, the ideal inductor has the best results with an output voltage (we can say) equal to the wished output voltage , and an undulation relative error of 3,88 ℅ better than the desired relative error.

The two integrated inductors also present an output voltage, close to the desired value, 2.2005 V for the inductor with opposite entry and exit and 2.20303 V for the inductance in which entry and the exit are the samedirection, instead of 2.5V which is the desired value.

The undulation relative error is slightly larger than the desired error 5,5℅ instead of 5℅.

In conclution we can say that the new electrical circuit of an integrated ferromagnetic inductor with opposite entry and exit, that we conceived, respects the waves form currents and voltages, exactly like that of an ideal reel . It is a encouraging point.

Concerning the dimensioning of the ferromagnetic inductor, the results are very encouraging, because the average values of the tensions and currents at the output of the micro-converter, are very close to the desired values. but we can have better results.

To adjust the average value of the output current I avg , we can exploit the value of the load resistance , while respecting the continuous conduction mode.

To attenuate the undulations of the output voltage Vout, we can exploit the value of the capacity of the microconverter, while respecting the integration values for a condenser. And finally, to have more exact value of the output voltage, we can play on dimensions of the conductor width and of the spacing inter-whorls, so to mitigate to the maximum the capacitive and resistive effects in the ferromagnetic core, and substrate.

Comparison and Interpretation

The study of the literature about the losses of energies in a planar inductor enabled us to define the causes of these losses and to establish relations with the component geometry;

To circumvent this problem, we proposed a new approach as well as a new electrical model of a spiral planar inductor with a ferromagnetic core and a semiconductor substrate which enable us to dimension the inductor and mitigate all parasitic effects. This dimensioning is carried out by taking account of the acceptable current by inductor and of the electromagnetic characteristics of used materials. Several papers in literature treat the problem of dimensioning of planar inductors, but the majority of these papers are based on a set of dimensions. For example, the dimensioning suggested by Alain Salles rests on a successive research of the value of inductance and resistance by modification of a set of geometrical parameters. He regards as set of dimensions, the external diameter of inductor (3000µm), the conductor width (15µm), the spacing inter whorls (1µm) and the conductor thickness (15µm). By studying the variations of the inductance and the series resistance of the inductor according to its geometrical parameters (n, w, t and s); Alain presents his methodology of dimensioning. By using those parameters, he determines the optimal performances of an inductor; but various iterative calculations are then necessary .

The dimensioning of the technological parameters which are due to the parasitic effects generated by the core and the substrate was neglected. We can also say that the model presented by Allain Salles is limited because it provides the values of inductance and resistance only in low frequencies Troussier also treats the problem of dimensioning of a planar inductor, but it carries out only the dimensioning of the core and neglects completely the dimensioning of the spiral, although the geometrical parameters of the conductor play a crucial role in attenuation of the parasitic effects due to stacking of different materials of inductor.

Troussier studies only the parasitic effects related to the conductor (effect of skin and effect of proximity) by estimating the conductor dimensions to a section of 100µm X 100µm and a length of 15cm, but a length of 15cm is far from being integrable. The work presented in this paper is not based on a set of dimensions, but on the dimensioning of each geometrical parameter of inductor. This dimensioning is carried out according to the operating frequency of the micro-converter and of the maximum acceptable current by the inductor; the obtained results are in agreement with integration. The technological parameters are extracted thanks to the design from a new electrical model of a spiral planar inductor with a ferromagnetic core and a semiconductor substrate.

Inspired by the model of Yue and Yong, we design this new electrical model which highlights the parasitic effects generated by the core and the substrate, and allows us to calculate the technological parameters which are strongly related to the geometrical parameters. The calculation of the technological parameters gives us a clear idea on the validity of the dimensioning of geometrical parameters. The innovation in this electrical model is summarized at the following points: 1_ This model is much simpler than the model of Yamaguchi.

2_This model takes into account some parasitic effects which were neglected by Yamaguchi and Yue, these effects represent the capacitive effects between the spiral and the buried conductor, and between this last and the magnetic core, as well as the resistive effect of the buried conductor. 3_ this model is designed to function in high and very high frequencies.

The simulation of the currents and voltages in the micro-converter and integrated inductor allowed us to validate the good dimensioning of the geometrical and technological parameters, as well as the good performance of this electrical model.

The new approach that we present in this work associated with the new electrical model of MOFS inductor, opens the way with integration and consequently the realization of new magnetic components necessary for the next generations of structures of power conversion; Because in a micro converter, the value of inductance depends on the specifications of this last, it means that each specifications require a value of quite precise inductance.

Starting from this approach, we can work out a computer program which calculates all geometrical and technological parameters for each value of inductance, it means, the dimensioning a -MOFS‖ inductor for each category of converter.

If we manage to conceive a technique who facilitates the integration of the other components of the microconverter (Transistor, Diode, Condenser and Resistance) in the semiconductor substrate of the integrated inductance, we will obtain a new component of energy conversion -the Integrated Micro Converter‖ which can be marketed in the form of integrated circuit..

Conclusion

The aim of our work is the dimensioning of a ferromagnetic square spiral planar inductor for its monolithic integration in a micro-converter. We first determine the volume of the magnetic core necessary for the storage of energy, and following a predimensioning; we calculate the geometrical parameters while taking account of the maximum area that must occupy the planar inductor and the maximum allowable current in the coil. The good dimensioning allows reducing strongly the losses energy in the inductor.

To extract the technological parameters, we conceive a new electrical model of a ferromagnetic spiral planar inductor in which entry and exit are opposites. This model is much simpler than the model of Yamaguchi, and takes into account the capacitive and resistive effects generated by the buried conductor, these effects are not mentioned in the electrical model of Yamaguchi.

Using the software PSIM6.0, we validate the results of the dimensioning of the inductor. We compare the waveforms of currents and voltages of a micro-converter in three different electrical circuits. The first circuit is a micro-converter containing an ideal coil, without any parasitic effects; the second and the third are an equivalent circuits of the micro-converter with a dimensioned and integrated ferromagnetic spiral planar inductor, in the second inductor, the entry is opposite at the exit , in the third, the entry and the exit are at the same direction.

After this comparison, we conclude that the results are very encouraging, because the waveforms of currents and voltages are respected, what enables us to say that the new electrical circuit of the ferromagnetic inductor that we propose functions perfectly.

The measured values of currents and voltages at the output micro-converter are very close to the desired values, this reflects the good dimensioning of the inductor.

Fig. 1 :

 1 Fig.1: Microphotography of an integrated inductor and an integrated capacitor.

Fig. 2 :

 2 Fig.2: Principle and example of a converter developed by J.A. Ferreira.

Fig. 3 :

 3 Fig.3: Schematic diagram of the Buck converter.

Fig. 4 :

 4 Fig.4: Geometrical parameters of a spiral planar inductor.

  ) And avg d is the average diameter of coil winding defined by the internal diameter d in , and the external diameter d out by relation (coefficients which depend on the used geometrical form . The values of these two coefficients are given in table1.

     : are Coefficients which depends on the geometrical form of the planar spiral inductor

Fig. 5 :

 5 Fig.5: Distribution of the magnetic flux of same intensity in a plan of transverse section of a planar spiral with and without magnetic core.

Fig. 6 :

 6 Fig.6: Variation of inductance according to the frequency with and without magnetic core. (a) Result of simulation. (b) Result from literature [17].

Fig. 7 :

 7 Fig.7: Planar inductor to be simulated without magnetic core.

Fig. 8 :

 8 Fig.8: Propagation of magnetic field lines in a planar inductor without magnetic core (front view).

Fig. 9 :

 9 Fig.9: Propagation of magnetic field lines in a planar inductor without magnetic core (view of profile).

Fig. 10 :

 10 Fig.10: Planar inductor to be simulated with a magnetic core: front view.

Fig. 11 :

 11 Fig.11: Propagation of magnetic field lines in a planar inductor with a magnetic core (front view).

Fig. 12 :

 12 Fig.12: Propagation of magnetic field lines in a planar inductor with a magnetic core (front view).

Fig. 13 :

 13 Fig.13: Eddy currents and displacement current in the core and substrate induced by the flow of current in inductor.

Fig. 14 :Fig. 15 :

 1415 Fig.14: Magnetic core divided vertically into sheets. Fig.15: Magnetic core divided horizontally into sheets.

Fig. 16 :

 16 Fig.16: Variation of skin thickness according to the frequency.

Fig. 17 :

 17 Fig. 17: Variation of resistance according to the frequency.

Fig. 18 :

 18 Fig.18: Useful volume in a conductor delimited by the skin effect.

Fig. 19 :

 19 Fig.19: Variation of ratio R/L for different values of the width w .

Fig. 20 :

 20 Fig.20: Variation of ratio R/L for different values of the thickness t.

Fig. 21 :

 21 Fig.21: Procedure of dimensioning.

  During the state on, the current which flows through the reel increases according to relation[START_REF] Dangthai | Reseaux de micro-convertisseurs[END_REF]:At the end of the state on, the current L I increased: operating process is imposed by the average output current savg I the average current in the condenser is null in permanent mode, thus: savg Lavg ii 

Fig. 22 :

 22 Fig.22: Evolution of the voltages and currents with time in an ideal buck converter operating in continuous mode.



  and we obtain a number of turns: n 1,95 2   .

10 2(

 10 A / mm ) are then possible.What enables us to put as conditions to boundaries:

  conductor's section which makes flow a maximum current of 0.6 A with an average current density of 0.313.10 9 Am -2 , which gives a minimal width w=38,29 m

Fig. 23 :

 23 Fig. 23: Representation of the conductor after circumventing of skin effect.

Fig. 24 :

 24 Fig.24: Transverse section of a spiral planar inductor.

Fig. 25 :

 25 Fig.25: Model in -π‖ of a spiral planar inductor developed by Yue and Yong [22].

Fig. 26 :

 26 Fig.26: Model in -π‖ of a ferromagnetic planar inductor [23].

Fig. 27 :

 27 Fig.27: Transverse section of a ferromagnetic planar inductor.

Fig. 28 :

 28 Fig.28: New Model of a ferromagnetic inductor -MOFS‖.

Fig. 29 :

 29 Fig.29: Planar inductor of which entry and the exit are in the same direction (a) Front view, (b) View of profile.

Fig. 30 :

 30 Fig.30: Planar inductor with opposite entry and exit (a) Front view, (b) View of profile.

Fig. 31 :

 31 Fig.31: Transverse section of the ferromagnetic spiral planar inductor -MOFS‖ with opposite entry and exit.

Fig. 32 :

 32 Fig.32: New Model of an inductor -MOFS‖ with opposite entry and exit

First

  of all, by construction a planar inductor has an inter-turns capacity s C expressed by relation (25); whose influence appears when the use frequency of the component increases.

12 t

 12  and 23 t  are respectively distances between the spiral and the buried conductor, and between the buried conductor and the ferrite, A represent the conductor section which is in contact with insulator (

Fig. 33 : 2 .

 332 Fig.33: Residual undulations of the output voltage

Fig. 34 :

 34 Fig.34: Micro-converter circuit with an ideal reel.

Fig. 35 :

 35 Fig.35: Waveforms of the currents and voltages of a micro-converter containing an ideal reel.

Fig. 36 :

 36 Fig.36: Micro-converter circuit with a dimensioned MOFS inductor with entry and exit at the same side.

Fig. 37 :

 37 Fig.37: Waveforms of the currents and voltages of a micro-converter containing MOFS inductor with entry and exit at the same side.

Fig. 38 :

 38 Fig.38: Micro-converter circuit cwith a dimensioned MOFS inductor with opposites entry and exit.

Table . 1

 . : Values of the coefficients used in formula of Wheeler.

	Form	k	1	k	2
	Square	2,34	2,75
	Hexagonal	2,33	3,82
	Octagonal	2,25	3,55

Table . 2

 . : Values of ratio 1 2 kk according to the geometry.

	Geometry	square hexagonal	octagonal
	Rapport 1 2 kk	0.85	0.82	0.63

Table . 3

 . : Values of the coefficients used in formula of Monomial.

	Geometry Square	Hexagonal Octagonal
			1,62 10 -3 1,28 10 -3	1,33 10 -3
		1	-1,21	-1,24	-1,21
		2	-0,147	-0,174	-0,163
	Square	1296,9472
	Hexagonal	529,313
	Octagonal	586,23

Table . 5

 . : Values of the coefficients used in formula of Mohan.

						Geometry	1 c	c	2	c	3	c	4
						Square	1.27	2.07	0.18	0.13
						Hexagonal 1.09	2.23	0	0.17
						Octagonal 1.07	2.29	0	0.19
						Circular	1	2.46	0	0.20
	In Mohan's formula, we consider that the value of inductance is proportional to the product
	1		2	3	4	

Table . 7

 . : Results of the dimensioning of geometrical parameters

	Geometrical parameters	Results of dimensioning
	Number of turns: n	2
	Average length: l avg	4,418 mm
	Width: w	130 µm
	Thickness: t	50 µm
	Spacing inter-turns: s	78 µm
	External diameter : d out	900 µm
	Internal diameter : d in	225 µm

Table . 10:

 . Maximum and minimum values of voltages and currents

	Time (s)	1.97793e-5	2.01404e-5
	V L	-9.97817e-1	1.83677e+0
	I L	4.85022e-1	3.86323e-1
	V C	2.59393e+0	2.40003e+0
	I C	5.27001e-2	-1.36812e-2
	V S	2.59393e+0	2.40003e+0
	I S	4.32322e-1	4.00004e-1

  Table.11. Maximum and minimum measured values of voltages and currents

	Time (s)	2.01003e-5	1.98195e-5
	V L	6.55253e-1	-2.42465e+0
	I L	2.70421e-1	3.83511e-1
	V C	2.14187e+0	2.26420e+0
	I C	-8.65580e-2	6.14400e-3
	V S	2.14187e+0	2.26420e+0

  Table.12. Maximum and minimum measured values of voltages and currents.

	Time (s)	1.98195e-5	2.01404e-5
	V L	-2.42540e+0	2.00926e+0
	I L	3.83657e-1	3.31348e-1
	V C	2.26495e+0	2.13595e+0
	I C	6.16592e-3	-2.46442e-2
	V S	2.26495e+0	2.13595e+0

Table . 13

 . : Table of results comparison of a three simulations.

		T.I
	C.V	I.I	I.OEE	I. EESD SV
	I Lmax	0,48502	0.38365	0.38351	0.6
	∆I L	0.09869	0.05231	0.11309	0.16
	I savg	0,41616	0.36674	0.36717	0.38
	∆I s	0.03231	0.02149	0.02039	/
	V s.avg	2.49698	2.2005	2.20303	2.5
	∆V s	0.09695	0.129	0.12235	0.125
	T.I: Type of Inductor
	C.V: Current and Voltage
	I.I: Ideal Inductor
	I.OEE: Inductor with Opposite Entry and Exit
	I. EESD: Inductor with Entry and Exit at the Same Direction.

  Table.14: Relative errors of currents and voltages.

				T.I
	R.E	T.I	I.I	I.OEE	I.SEE	S.V
		Is		7.76℅	5.85℅	5.55℅	5℅
	Is
	Vs 	3.88℅	5.86℅	5.55℅	5℅
	Vs
	R.E: Relative Error.

Acknowledgment

The authors would like to thank N. Bentayeb for its collaboration in the translation of this paper.


Laurence Questienne

Anne Atas

Boris Burle

Wim Gevers

Objectifying the Subjective: Building Blocks of Metacognitive Experiences in Conflict Tasks

Keywords: Congruency, Introspection, Response competition, Electromyography

Metacognitive appraisals are essential for optimizing our information processing. In conflict tasks, metacognitive appraisals can result from different inter-related features (e.g. motor activity, visual awareness, response speed, etc.). Thanks to an original approach combining behavioral and electromyographic measures, the current study objectified the contribution of three features (reaction time, motor hesitation with and without response competition, and visual congruency) to the subjective experience of urge-to-err in a priming conflict task. Both reaction time and motor hesitation with response competition were major determinants of metacognitive appraisals. Importantly, motor hesitation in absence of response competition and visual congruency had limited effect. Because science aims to rely on objectivity, subjective experiences are often discarded from scientific inquiry. The current study shows that subjectivity can be objectified.

Imagine waiting at a multi-rows central traffic light. When the side-line traffic light turns green, you may have the feeling that you almost reacted to this irrelevant stimulation. In the lab, control processes are often studied through conflict tasks like the Stroop [START_REF] Stroop | Studies of interference in serial verbal reactions[END_REF], the Flanker (B. A. Eriksen & [START_REF] Eriksen | Effects of noise letters upon the identification of a target letter in a nonsearch task[END_REF] or the arrow priming task [START_REF] Vorberg | Different time courses for visual perception and action priming[END_REF]. These tasks have in common that participants have to respond to a task-relevant stimulus dimension while ignoring distracting information. In the arrow priming task, for instance, participants indicate the direction of a target arrow while ignoring a prime arrow presented just before. It is objectively more difficult (slower response and more errors) when arrows are incongruent (i.e. point in opposite directions) even when the prime is subliminal [START_REF] Vorberg | Different time courses for visual perception and action priming[END_REF].

Importantly, congruency effects are not limited to objective measures. Participants report stronger subjective experiences of "urge-to-err" or higher "difficulty" on incongruent trials [START_REF] Desender | Feeling the conflict: the crucial role of conflict experience in adaptation[END_REF][START_REF] Morsella | Subjective aspects of cognitive control at different stages of processing[END_REF][START_REF] Questienne | Introspection of Subjective Feelings Is Sensitive and Specific[END_REF], even when unaware of the congruency [START_REF] Desender | The temporal dynamics of metacognition: Dissociating task-related activity from later metacognitive processes[END_REF]. Which (combination of) features, simultaneously experienced (motor activity, visual awareness, speed, etc.), contribute to the reported experience remains to be deciphered.

Based on the literature, we identified a shortlist of features that could contribute to the experience of urge-to-err, within the context of an arrow priming task. This list is not meant to be exhaustive and can be generalized to any other conflict task. First, participants could use the evaluation of their reaction time (RT) (e.g. [START_REF] Corallo | Limits on introspection: distorted subjective time during the dual-task bottleneck[END_REF] to infer their urge-to-err (i.e. 'I was slow on this trial so it was difficult'). Second, motor hesitation (for a related concept, see 'response fluency' in e.g. Chambon and Haggard, 2012 could influence the urge-to-err. On incongruent trials, prime and target activate competing responses [START_REF] Botvinick | Conflict monitoring and cognitive control[END_REF] installing a motor hesitation that could trigger the urge-toerr (e.g. [START_REF] Morsella | Subjective aspects of cognitive control at different stages of processing[END_REF]. Motor hesitation can also occur in absence of response competition (see below for a definition). Such hesitations could also trigger the urge-toerr. Third, visual aspects of the congruency can relate to the urge-to-err, even with subliminal primes, because they still modulate visual processing [START_REF] Naccache | The priming method: imaging unconscious repetition priming re-veals an abstract representation of number in the parietal lobes[END_REF][START_REF] Ortells | The semantic origin of unconscious priming: Behavioral and event-related potential evidence during category congruency priming from strongly and weakly related masked words[END_REF]. Dissociating the contribution of these features to the reported urge-toerr is challenging because they are partly confounded (e.g. incongruent trials have slower RT, higher motor hesitation). Avoiding confounds requires separate objective measure of each feature. Visual congruency is determined by trial type. RT distribution analysis allows assessing the impact of RT on urge-to-err. But motor hesitation (with or without response competition) cannot be objectified solely on behavioral measures. Recording electromyographic (EMG) activity of the muscles involved in response execution objectively reveals motor hesitation (e.g. C. W. [START_REF] Eriksen | An electromyographic examination of response competition[END_REF]. First, on 15-20% of correct trials, subthreshold EMG activation occurs on the incorrect hand before the correct response (Figure 1b). Such 'partial-errors' occur more frequently on incongruent trials signaling the occurrence of response competition (e.g. [START_REF] Burle | Error negativity does not reflect conflict: a reappraisal of conflict monitoring and anterior cingulate cortex activity[END_REF]. Studying the urge-to-err as a function of these partial-errors allows relating the urge-toerr to motor hesitation implying response competition. Second, some correct trials contain several bursts of only correct EMG activity before button press (Figure 1c). Such multiple activations signal a motor hesitation, without necessarily involving response competition. These trials were not analyzed in detail yet (e.g. [START_REF] Servant | Using Covert Response Activation to Test Latent Assumptions of Formal Decision-Making Models in Humans[END_REF] but enable to study whether urge-to-err is specifically sensitive to response competition or to all motor hesitations.

Method Participants

Based on a previous study on subjective experience [START_REF] Questienne | Metacognition and cognitive control: behavioural adaptation requires conflict experience[END_REF], Experiment 2), twenty-seven healthy naive participants (6 men; 19.26 years, SD = 1.58) participated against e15/hr. This experiment was approved by the committee "CPP Sud-Méditerranée", agreement n • 1041 (RCB n • : 2010-A00745-34). Participants signed informed consent.

Material and procedure

Using the Psychopy toolbox [START_REF] Peirce | PsychoPy-Psychophysics software in Python[END_REF], stimuli were presented on a white screen on a 15-inch, 70 Hz CRT monitor. Viewing distance was 120 cm. Responses were given by thumb presses on buttons mounted on top of two cylinders grasped in each hand. EMG activity of flexor pollicis brevis was recorded (1024 Hz, Biosemi Active2, [START_REF] Servant | Using Covert Response Activation to Test Latent Assumptions of Formal Decision-Making Models in Humans[END_REF] for procedures)1 . Participants pressed as fast as possible the button corresponding to the direction of a target arrow, preceded by a prime arrow (Figure 2a). After each trial, participants orally judged to what extent they were about to make an error (i.e. urge-to-err) on an 8-points scale (1 = "no urge-to-err", 8 = "strong urge-to-err") or mentioned real errors (recorded as "9"). They performed 20 blocks of 64 randomized trials, preceded by 16 training trials. Congruency, arrow direction and spatial location (above or below Then, it becomes green and the button corresponding to the prime can be pressed , followed by the oral response to the Perceptual Awareness Scale (PAS).

fixation) were counterbalanced within each block. Participants were not informed about the presence of the primes. After this part, participants were asked whether they noticed their presence before performing an awareness test (Figure 2b). The stimulus presentation was identical, but participants indicated the prime direction. The response was given after 900 ms to decrease motor priming. This delay fitted with the minimum interval between the target and the subjective report in the first part. Then, participants orally evaluated the quality of their visual experience of the prime on the Perceptual Awareness Scale (PAS) [START_REF] Ramsøy | Introspection and subliminal perception[END_REF], a 4-point scale from (1 = "No visual experience" , 4 = "Absolutely clear image"). There were 2 blocks of 64 trials, preceded by 16 training trials.

Data processing

All Data are available at osf.io/wz9jp. Because of technical problems, 2.27% and 3.59% of trials were lost for 2 participants. RTs slower/shorter than Median ± 3 Median Absolut Deviation computed by participant were removed (6.46 %) [START_REF] Leys | Detecting outliers: Do not use standard deviation around the mean, use absolute deviation around the median[END_REF]. EMG signal was high-pass filtered (10hz). Onsets of the EMG activity were manually marked based on visual inspection (see e.g. [START_REF] Burle | Executive control in the Simon effect: an electromyographic and distributional analysis[END_REF]. Importantly, the experimenter was blind to the congruency and reported experience that she/he was looking at. Three categories of trials were selected: Pure-Correct (62.23%, SD = 12.94), Partial-Error (16.38%; SD = 7.15) and Correct-Correct (4.80%; SD = 2.66) (Figure 1). Other trials (16.58%, SD=7.91) were discarded because of errors, multiple EMG bursts and/or tonic activity. The size of EMG activity was estimated as the 90th percentile of the Figure 3. Distributional analysis of the urge-to-err. For different types of trials, the reaction time (RT) distributions were sorted in ascending order and binned in three classes of equal length [START_REF] Ratcliff | Group reaction time distributions and an analysis of distribution statistics[END_REF][START_REF] Vincent | The function of the vibrissae in the behavior of the white rat[END_REF]. The mean score of "urgeto-err" is computed for each bin and plotted against the mean of RT of its bin, separately for Pure-Correct, Partial-Error and Correct-Correct trials and for congruent (a) and incongruent trials (b). The same distributional graph is plotted for congruent and incongruent Pure-Correct trials for unaware (c) and aware participants (d). Distributional graphs are plotted for three sizes of Partial-Error (e). To define small, medium and large Partial-Error, incorrect EMG sizes were sorted in ascending order and divided in three equal bins. Error bars represent standard error. cumulative sum of the EMG signal on the 500 ms following the EMG onset (for more details, see [START_REF] Servant | Using Covert Response Activation to Test Latent Assumptions of Formal Decision-Making Models in Humans[END_REF].

Results

Overall performances

Typical congruency effects were observed on RTs, error rates, proportion of Pure-Correct and Partial-Error trials (Table 1). The congruency effect was also observed in the urge-to-err. There were also marginally more Correct-Correct trials on congruent than on incongruent trials.

Dynamics analysis of urge-to-err

Urge-to-err was first plotted as a function of congruency, EMG categories and RTs distribution (Figures 3a &3b). All slopes were positive: urge-to-err generally increased with RT. Importantly, they differed depending on the congruency and the EMG category (e.g. the Partial-Errors slope was steeper than other EMG categories).

To statistically assess these observations, we conducted a linear mixed model (LMM) with SAS software. Urge-to-err was the dependent variable. A log-transformation was applied to meet the normality assumption. Congruency, EMG category (Pure-Correct vs Partial-Error vs Correct-Correct), RT (of each individual trial) and all interactions were fixed effects. RTs were centered on the mean of all trials (as such, the effect of the other factors were evaluated for the RT mean). Participants and by-participant RT slope were selected as random effects based on restricted maximum likelihood method and Akaike's Information Criterion [START_REF] Akaike | A New Look at the Statistical Model Identification[END_REF].

Main effects were significant (RT: F(1,26) = 36.33, p < .001; Congruency: F(1,26835) = 30.65, p < .001; EMG categories, F(1,26835) = 307.35, p < .001) but modulated by higher order interactions, analyzed with contrasts (planned comparisons) and Bonferroni corrections (see Table 2). Six contrasts were conducted to analyze the interaction between EMG category and RT, F(1,26835) = 44.91, p < .001. Urge-to-err increased with RT for all EMG categories (slope Partial-Error = 2.30, slope Pure-Correct =1.28; slope Correct-Correct =1.12). Larger slopes for Partial-Errors indicated that they were associated with stronger urge-to-err, and this difference increased with slower RTs. Five contrasts were conducted to analyze the interaction between Congruency and RT, F(1,26835) = 4.77, p = .029, and the threeway interaction, F(1,26835) = 4.97, p = .007. Urge-to-err increased with RT for both congruent (slope = 1.44) and incongruent trials (slope = 1.69). While those slopes differ on Pure-Correct trials (larger for incongruent than for congruent trials, difference = -0.55), this was not the case on Partial-Errors (difference = 0.13), nor on Correct-Correct trials (dif-ference = -0.3484). Stimulus congruency influenced urge-toerr on Pure-Correct trials but not on Correct-Correct trials or Partial-Errors.

Prime awareness

Fifteen participants did not notice the prime (unaware group); twelve participants claimed they did (aware group). Detecting the primes was subjectively as difficult for both groups (PAS: 1.71, 95% CI [1.306,2.116] vs 1.79, 95% CI [1.319, 2.266]), t(25) = -0.285 , p = .778. However, the aware group detected the prime slightly above chance level (d' = 0.32, 95% CI [0.065, 0.582], t(11) = 2.76 , p = .019), while the unaware group did not (d' = 0.01, 95% CI [-0.151, 0.168], t(14) = 0.115 , p = .910). This difference was significant, t(25) = 2.353, p = .027.

Supplementary analyses

To assess whether aware participants drove the congruency effect on Pure-Correct trials, a specific LMM was conducted. The log-transformed level of urge-to-err on Pure-Correct trials was the dependent variable. Congruency, RT and group (aware vs unaware) and all interaction were fixed effects. Participants and by-participant RT slope were random effects. The three-way interaction was significant, F(1,19987) = 10.31, p = .001. Three contrasts (alpha = .05/3) showed that the slope difference between congruent and incongruent trials was larger in the aware group (difference = -0.61, 98.33% CI [-1.06, -0.15]), t(19987) = 3.21, p = .001, but significant in both groups (aware : difference = -0.95, 98.33% CI [-1.31, -0.59]), t(19987) = -6.34, p < .001, unaware: difference= -0.35, 98.33% CI [-0.62, -0.07]), t(19987) = -3.01, p = .003 (Figures 3c/3d). In sum, both aware and unaware participants contributed to the congruency effect observed on urge-to-err on Pure-Correct trials. Urge-to-err as a function of the incorrect EMG size of Partial-Errors was also analyzed. A specific LMM was conducted on Partial-Errors. The log-transformed urge-toerr was the dependent variable. Only RTs, the centered incorrect EMG size and their interaction were fixed effects. Congruency did not improve model fitting (LRT(4) = 3.19; p = .561). Participant and by-participant RT slope were random effects. Effect of RT, F(1,26) = 49.81, p < .001, and EMG size, F(1,5243) = 925.37, p < .001, were significant and interacted, F(1,5243) = 4.73, p = .030. The slower the RT, the more urge-to-err increased with the incorrect EMG size (Figure 3e). Finally, we checked whether the size of the correct EMG activity corresponding to the behavioral response could explain the modulation of urge-to-err with the EMG category. Wilcoxon tests (because of non-normal distributions) with adjusted alpha (.05/3) showed that the size of correct EMG burst of Partial-Errors trials differed from size of EMG burst of Pure-Correct trials, Z = -3.748, p < .001. However, size of correct EMG burst on Partial-Errors and size of the second correct EMG burst of Correct-Correct trials were similar, Z = -1.057; p = .290, while they were experienced with different urge-to-err (see above). Conversely, EMG activity of Pure-Correct and second correct EMG burst of Correct-Correct trials were different, Z = -2.643; p = .008, while they were experienced similarly (see above). Thus, correct EMG associated with the behavioral response cannot explain the relation between EMG categories and urge-to-err.

Discussion

The current study aimed to disentangle the contribution of RT, motor hesitation (with or without response competition) and visual conflict to the subjective experience of urge-to-err in an arrow priming task. Participants experienced congruent and incongruent trials differently [START_REF] Desender | Feeling the conflict: the crucial role of conflict experience in adaptation[END_REF][START_REF] Morsella | Subjective aspects of cognitive control at different stages of processing[END_REF][START_REF] Questienne | Introspection of Subjective Feelings Is Sensitive and Specific[END_REF]. However, the origin of these experiences has been conjectured rather than established. By combining EMG measures and RT distributions analyses, we objectified the specific role of several features and how they related to each other to form subjective experiences. So far, studies interested in the experience of urge-to-err considered RT as a confounding variable that has to be eliminated (e.g. [START_REF] Morsella | Subjective aspects of cognitive control at different stages of processing[END_REF]. Here, using RT distribution analyses we quantified the relation between RT and urge-to-err, which appears to be linear for all trial types. Whether this relationship results from an explicit strategy (i.e. 'I was slow, therefore I report stronger urgeto-err') can be debated. Interestingly, the strength of this relationship varies as a function of the EMG pattern. The presence of a Partial-Error overall increases the urge-to-err and steepens the relationship with RT. Crucially, this was true on congruent and incongruent trials. Partial-Errors on congruent trials do not result from the prime. They probably result from implicit/explicit response expectancies (e.g. [START_REF] Perruchet | Dissociating the effects of automatic activation and explicit expectancy on reaction times in a simple associative learning task[END_REF] or stimulus repetitions/alternations (e.g. [START_REF] Hommel | A feature-integration account of sequential effects in the Simon task[END_REF] etc. . . This indicates that urge-to-err is sensitive to response competition, irrespective of its origin. The present results likely generalize to other conflict tasks inducing response competition (e.g. Flanker task, Simon task, etc.). Whether they extend to other types of conflict, such as task-conflict [START_REF] Goldfarb | Evidence for task conflict in the Stroop effect[END_REF], remains an open issue. Importantly, Partial-Errors could have induced an offset of the RT slope (shifting the curve up) without affecting the slopes, suggesting that response competition and RTs acted independently on urge-to-err. Instead, Partial-Errors steepen the urge-to-err -RT curve, showing that both factors interact to generate metacognitive experiences. The analysis of Partial-Error size completes this pattern: the slope relating urgeto-err -RT increases with the size of the incorrect EMG of Partial-Errors, suggesting that the urge-to-err is not only sensitive to the presence, but also to the amount of response competition. In contrast, urge-to-err -RT curves for Correct-Correct trials (Figures 3a/3b) seem a continuation of the Pure Correct ones. Because Correct-Correct trials and Partial-Errors lead to similarly slow RTs (see abscissa on Figure 3a/3b), this further establishes that the increased urge-to-err on Partial-Errors is not a by-product of their longer RTs. Thus, while motor hesitation implying response competition influences the urge-to-err, motor hesitation without response competition does not. Finally, congruency contributes to urge-to-err in a limited way. A congruency effect remains on urge-to-err only on Pure-Correct trials, even for participants unaware of the primes; it hence cannot be attributed to intentional strategies (e.g. 'this was an incongruent trial, so it was difficult'). Rather, it could reflect a subjective sensitivity to some early visual processing triggered by subliminal primes [START_REF] Naccache | The priming method: imaging unconscious repetition priming re-veals an abstract representation of number in the parietal lobes[END_REF][START_REF] Ortells | The semantic origin of unconscious priming: Behavioral and event-related potential evidence during category congruency priming from strongly and weakly related masked words[END_REF]. However, partial muscular activation overrules this sensitivity: no effect of congruency is observed on Partial-Errors nor Correct-Correct trials. This last observation on Correct-Correct trials indicates that, although motor hesitation without response competition does not increase urge-to-err, it could still have an influence. Alternatively, the congruency effect on Pure-Correct trials could be due to residual response competition. On Pure-Correct incongruent trials competitive responses could still be activated but not strong enough to achieve the muscular level. This would mean that subjectivity is sensitive to response selection steps occurring before muscular peripheral activations [START_REF] Chambon | Sense of control depends on fluency of action selection, not motor performance[END_REF]. Note that, despite the easiness of the task (e.g. few errors and overall low urge-to-err), urge-to-err can still be related to different task-related features. However, while behavioral and EMG results are similar to experiments using similar tasks without introspection [START_REF] Burle | Error negativity does not reflect conflict: a reappraisal of conflict monitoring and anterior cingulate cortex activity[END_REF]Vor-berg et al., 2003), we cannot exclude that the introspective instruction somehow changed the task [START_REF] Grützmann | Response accuracy rating modulates ERN and Pe amplitudes[END_REF]. Especially, the urge-to-err could have become salient, leading to a general overestimation of this experience. While possible, for our purposes, knowing how different features modulate this experience was more important than its absolute value.

To conclude, by combining behavioral and EMG measures, we show that experiences of urge-to-err stem from several features that can be disentangled. Previous work suggested that subjective experiences could trigger cognitive adaptation [START_REF] Desender | Feeling the conflict: the crucial role of conflict experience in adaptation[END_REF][START_REF] Questienne | Metacognition and cognitive control: behavioural adaptation requires conflict experience[END_REF]. The current study discloses the building blocks of such metacognitive experiences, a crucial step to understand its relation to adaptation. More broadly, the current study opens perspectives to study metacognition and introspection in general. First, this work provides tools to describe subjective experiences in a fine-grained way. Second, subjective experiences are at the heart of human mental life, the object of psychology, but can only be accessed through introspective reports, often considered as unreliable [START_REF] Nisbett | Telling more than we can know: Verbal reports on mental processes[END_REF]. But introspective reports seem quite reliable when they are restricted to memories of cognitive processes, without implying complex inferences [START_REF] Ericsson | Verbal reports as data[END_REF]. The current study underlines this reliability by relating introspective reports to objective measures. Whereas science aims to rely on objectivity, the current study shows that subjectivity can be objectified.

Figure 1 .

 1 Figure 1. Example of each EMG category. a. Pure-Correct trials are trials with only one EMG activity on the correct hand. b. Partial-Error correspond to correct trials with EMG activity on the incorrect hand preceding the correct one. c. Correct-Correct trials are trials with a small activation on the correct hand occurring before the main EMG burst leading to the response.

  Note. RT=reaction time; CI= Confidence Interval. Repeated ANOVAs were performed, excepted on urge-to-err where a Wilcoxon non-parametric test was used because of deviation from normal distribution unresolved by data transformation. ANOVAs on error rate, Pure-Correct, Partial-Error and Correct-Correct rates were computed after arcsine square-root transformation for proportions.

Figure 2 .

 2 Figure 2. Example of trial in the first part (a):A trial begins with a fixation cross (700 ms), followed by a prime (1.8 • times 0.9 • ) (14 ms) and a white screen (28 ms). Next, the target (2.7 • × 1.1 • ) appears (142 ms). To mask the prime, the target has a central cutout corresponding to the shape of the superimposed left and right-pointing primes. The central fixation cross becomes green until the response. After the response, a black fixation cross lasts during at least 400 ms or longer until 900 ms passed from the onset of the target. Then, a question marked appears until the subjective response is given and encoded by the experimenter. Example of trial in the awareness test (b): The trial presentation is the same except that after the target, a black fix appears during 758 ms. Then, it becomes green and the button corresponding to the prime can be pressed , followed by the oral response to the Perceptual Awareness Scale (PAS).

Table 1

 1 Summary of the congruency effects on the different variables.

		Congruent trials	Incongruent trials
	Variables	Mean	95%CI Mean	95% CI Statistical test
	Behavioral Measures					F(1,26)	p	η 2 p
	RT (ms)	359	[341, 377]	412	[397, 427] 526.50 < .001	.95
	Error rate (%)	1.49	[0.97, 2.01]	8.84	[5.73, 11.96]	56.87 < .001	.69
	EMG categories					F(1,26)	p	η 2
						001	.86
	Partial-Error (%)	8.50	[6.29, 10.71] 24.50 [20.45, 28.55] 171.74 < .001	.87
	Correct-Correct (%)	5.16	[3.99, 6.34]	4.43	[3.39, 5.48]	3.97 < .057	.13
	Subjective experience					Z	p	r
	Urge-to-err	1.82	[1.50, 2.14]	2.43	[2.08, 2.78]	-4.54 < .001 -.61

p Pure-Correct (%) 76.15 [72.29, 80.01] 47.89 [40.84, 54.94] 161.02 < .

Table 2

 2 Results of the contrasts decomposing the interactions of the Linear Mixed Model used for the dynamic analyze of the urge-to-err

	Contrasts	Estimate (1-α)×100 % CI t(26835)	p Adjusted α
	RT slope PC	1.28	[0.60, 1.96]	4.97 < .001	.008
	RT slope CC	1.12	[0.34, 1.88]	3.88 < .001	.008
	RT slope PE	2.30	[1.58, 3.02]	8.45 < .001	.008
	RT slope CC -RT slope PE	-1.18	[-1.63, -0.74]	-7.00 < .001	.008
	RT slope PC-RT slope PE	-1.01	[-1.31, -0.72]	-9.08 < .001	.008
	RT slope PC -RT slope CC	-0.17	[-0.55 0.22]	-1.13	.259	.008
	RT slope Cong.	1.44	[0.57, 2.31]	5.42 < .001	.01
	RT slope Incong.	1.69	[0.81, 2.57]	6.33 < .001	.01
	RT slope Cong -Incong PC	-0.55	[-0.80, -0.29]	5.55 < .001	.01
	RT slope Cong -Incong CC	-0.35	[-1.05, 0.35]	1.28	.200	.01
	RT slope Cong -Incong PE	0.12	[-0.37, 0.63]	0.67	.505	.01

Note. RT=reaction time; PC = Pure-Correct trials; PE=Partial-Error trials; CC=Correct-Correct trials; Cong. = congruent trials; Incong. = incongruent trials; CI= Confidence Interval.

EEG activity was also recorded for other purposes and beyond the scope of the current study

L. Questienne is supported by the Fund for Scientific Research (F.R.S. -FNRS, Belgium, Grant Number 1.A.875.14F). This work was partially supported by the European Research Council under the European Community's Seventh Framework Program (FP7/2007-2013 Grant Agreement no. 241077), and by Wallonia-Brussels International and the Fund for Scientific Research(F.R.S.), the French Ministries for Foreign Affairs Higher Education and Research through a Hubert Curien partnership (PHC no. 35947SC). This work was not funded by a grant from NIH, Wellcome Trust or Research Councils UK. We thank Mathieu Servant for his valuable advices. Results discussed in this article were presented at two conferences (6th Motivation and Cognitive Control Symposium, 2016, St Andrew, Scotland and the European Workshop on Cognitive Neuropsychology, 2016, Bressanone, Italy