Olivier Dezellus

Raymundo Arroyave

Suzana G Fries

Thermodynamic modelling of the Ag-Cu-Ti ternary system

Keywords: Calphad, braze alloys, phase diagram, Ag-Cu-Ti

Ag-Cu-Ti system is important for brazing applications and mainly for ceramic joining.

This system is characterized by numerous intermetallics in the Cu-Ti binary and the existence of a miscibility gap in the liquid phase. For applications, the knowledge of phase equilibria, invariant reactions in the temperature range of interest and thermodynamic activity values (mainly of Ti) are important. Thermodynamic model parameters for all the stable phases in the Ag-Cu, Cu-Ti and Ag-Ti systems, previously obtained using the Calphad method and available in the literature are used. New thermodynamic description for the ternary interaction parameter of the liquid is obtained from experimental informations. Ti 2 Cu and Ti 2 Ag which have the same crystallographic structure were modelled as a single phase. The same was done for TiCu and TiAg. Finally, solid solubility of Ag in the Ti-Cu intermetallics is taken into account. The parameters obtained in this assessment are later used for the calculation of selected sections that can be useful for research and applications in the field of joining with Ti activated Ag-Cu braze.

Introduction

The Ag-Cu binary consists in a simple eutectic ( [START_REF] Hayes | [END_REF] modified by [2] and hereby noted as [3] in the following). Ag-Ti is characterized by the existence of two intermetallics with peritectic decomposition (see figure 1 from [START_REF] Arroyave | Thermodynamics and Kinetics of Ceramic/Metal Interfacial Interactions[END_REF]) while in the Cu-Ti system 6 intermetallic phases are stable, where only one melts congruently (see figure 2 from [START_REF] Kumar | [END_REF]). is the continuous solid solution between the Ti 2 Cu and Ti 2 Ag compounds, the second is the existence of a miscibility gap in the liquid state between a Ag-rich and a Ti-rich liquids.

Since this miscibility gap is not observed in the binary sub-systems, it is assumed to have a closed topology.

Alloys of the Ag-Cu-Ti ternary system are often used for brazing ceramics to metals in the temperature range 800-900 • C [START_REF] Eustathopoulos | Wettability at High Temperatures[END_REF]. The low liquidus temperature of the Ag-Cu eutectic allows brazing at reasonably low temperatures, while the presence of Ag increases the activity of Ti very noticeably [7], promoting interfacial reactions with most of ceramic materials [START_REF] Eustathopoulos | Wettability at High Temperatures[END_REF]8]. Depending on its activity, Ti can form compounds with a partially metallic character on various ceramic solids thus leading to an improvement of wettability [9,10]. A thorough knowledge of the Ag-Cu-Ti phase diagram from room temperature up to 900 • C is then required when trying to understand the mechanisms of reactive wetting or to develop high performance metal/ceramic brazed joints, more especially when the metal is a titanium base alloy [11,12,13]. Although the Ag-Cu-Ti system has been modelled before [14], no thermodynamic description has been made available. In this paper, a model for the ternary Ag-Cu-Ti system is proposed and phase diagram and thermochemical calculations are compared with the experimental evidence available.

Experimental data

The only systematic experimental study of this system is due to Eremenko et al. [15,16,17].

The system was later critically reevaluated by Chang et al. [18] and more recently by Kubachewski et al. [19]. Only some formal changes have been made and no new experimental information have been given. One of the main experimental feature reported by Eremenko et al. is the existence of a liquid miscibility gap that divides the liquid into Ag rich and Ti rich solutions. This miscibility gap has been confirmed experimentally by Paulasto et al. [14].

No ternary compounds has been found in the Ag-Cu-Ti system. Crystallographic data of the phases of the Ag-Ti and Cu-Ti systems are listed in Table 1 (data from [19]). According to Eremenko et al. a continuous solid solution exists between iso-structural Ti 2 Ag and Ti 2 Cu and the dependence of lattice parameters in this solid solution obeys Vegards law [15,17,16]. The two other iso-structural solids TiCu and TiAg do not form a continuous solid solution and are not even in equilibrium with each other at any temperature. However, both phases exhibit noticeable penetration in the ternary composition triangle: Eremenko et al. reported solubility limits at 700 • C which are respectively 5 at.% Ag in TiCu and 2 at.% Cu in TiAg [17]. The solubility of Ag in TiCu is highly sensitive to the temperature and it has been measured as about 13 at.% at 950 Recent experimental investigations on the Ag-Cu-Ti system by isothermal diffusion experiments have reported an Ag content in the Cu-rich Cu-Ti compounds (Ti 3 Cu 4 , Ti 2 Cu 3 and TiCu 4 ) up to 1.5 at.% at 790 • C [20,21] that confirm previous results. The decomposition of Ti 3 Cu 4 , Ti 2 Cu 3 and TiCu 4 compounds in the ternary system occurs by ternary transition reaction with the liquid phase and Ag solid solution at temperatures varying from 783 • C to 860 • C according to [16,19] (see table 2).

3 Thermodynamic modelling

Literature survey

In the Ag-Cu-Ti system, reliable descriptions for the Cu-Ti [START_REF] Kumar | [END_REF] and Ag-Cu [3] binaries are available in the literature and will be used as a starting description in this work. The third Ag-Ti binary had been thermodynamically assessed both by Arroyave [START_REF] Arroyave | Thermodynamics and Kinetics of Ceramic/Metal Interfacial Interactions[END_REF] and Li et al. [22] and the parameters of the first one are used in the following. The first attempt of assessment [19]).

of the Ag-Cu-Ti ternary systems has been perfomed by Paulasto et al. in 1995 [14]. They proposed some isothermal sections at high temperature and optimised the value of an excess ternary parameter in the liquid phase in order to describe satisfactorily the miscibility gap.

Unfortunately, the whole set of parameters used was not reported. More recently, Arroyave started a new assessment of this ternary system that has been partly reported in its PhD [START_REF] Arroyave | Thermodynamics and Kinetics of Ceramic/Metal Interfacial Interactions[END_REF].

Since this period, the work has been further pursued and the whole assessment is detailed in the present paper.

Unary phases

For the thermodynamic functions of the pure elements in their stable and metastable states, the phase stability equations compiled by Dinsdale [23] were used.

The solution phases

The liquid phase and the solid solution phases (fcc,hcp and bcc) were described by the Redlich-Kister substitutional solution model.. The Gibbs energy function of the solution phase Φ (Φ = liquid, bcc, hcp, and fcc) for 1 mole of atoms is described by the following expression:

G Φ m = i x Φ i • G Φ i + RT i x Φ i ln(x Φ i ) + ex G Φ i (1) ex G Φ i = n-1 i n j=i-1 x Φ i x Φ j L Φ i,j + x Φ i x j =i =k x Φ k =i =j L i,j,k (2)
where elements Ag, Cu and Ti are identified as 1,2,3; n is equal to 3, x Φ i is the molar fraction of element; • G Φ i corresponds ot the Gibbs energy of the pure element in the state Φ; ex G Φ i is the excess Gibbs energy which is expressed in the Redlich-Kister polynomial; and L Φ i,j the binary interaction parameter between elements i and j that can be further expanded as:

L Φ i,j = k k L Φ i,j (x Φ i -x Φ j ) k (3) k L Φ i,j = k a + k bT (4)
In order to describe the existence of a liquid miscibility gap in the central region, a ternary interaction parameter ( Φ L i,j,k in equation 2) was incorporated into the description of the excess Gibbs energy of the liquid phase. For solid solution phases (fcc, hcp, and bcc), the existing descriptions of the binaries are used [3,[START_REF] Kumar | [END_REF][START_REF] Arroyave | Thermodynamics and Kinetics of Ceramic/Metal Interfacial Interactions[END_REF] and no ternary parameter added.

3.4 Binary phases extending into the ternary system

Extension of stoichiometric binary compounds into the ternary

In the Cu-Ti binary system, Ti 3 Cu 4 and Ti 2 Cu 3 are essentially stoichiometric and therefore they are modelled as line compounds. In this work, in order to take into account the small, but important Ag solubility detected in those phases [20,17,14,21], the models are modified by considering the presence of Ag in the Cu rich sublattice, leading to (Ti) p (Cu,Ag) q where p and q are stoichiometric numbers, respectively 3-4 and 2-3 for Ti 3 Cu 4 and Ti 2 Cu 3 . As a consequence the expression of the Gibbs energy function may be written as:

G T ipCuq = y ′ T i y ′′ Cu • G T ipCuq T i:Cu + y ′ T i y ′′ Ag • G T ipAgq T i:Ag + RT (y ′ T i ln y ′ T i + y ′′ Cu ln y ′′ Cu ) + RT (y ′ T i ln y ′ T i + y ′′ Ag ln y ′′ Ag ) + ex G T ipCuq (5)
where y s i is the site fraction of component i in sublattice s, and

• G T ipCuq T i:Cu , • G T ipAgq
T i:Ag are the Gibbs energies of the stoichiometric compounds Ti p Cu q and Ti p Ag q formed when each of the sublattices is occupied by only one component:

• G T ipAgq T i:Ag = p • G hcp T i + q • G f cc Ag + A (6)
where A is an optimised parameter. As the Gibbs energies of Ti 2 Cu 3 and Ti 3 Cu 4 in the binary are approximately the same, and because their compositions and decomposition temperatures are also very close in the ternary, the same value of parameter A was assumed for both of them. ex G T ipCuq is the excess Gibbs energy and its composition dependence is assumed to conform with a Redlich-Kister polynomial. Due to the restricted number of experimental data the temperature dependence of the excess Gibbs energy is neglected and only the subregular solution term of the Redlich-Kister serie was used:

ex G T ipCuq = y ′′ Cu y ′′ Ag (y ′ T i L T ipCuq T i:Cu,Ag ) (7)
L T ipCuq T i:Cu,Ag = a 0 pq (y ′′ Cu -y ′′ Ag ) (8)
Finally, for Ti 3 Cu 4 and Ti 2 Cu 3 the parameters to be optimised are A (eq. 6) and the two Redlich-Kister coefficient a n (eq. 8).

Eremenko et al. [17] concluded that the Ti 2 Cu and Ti 2 Ag phases are iso-structural with a complete solid solubility between the two phases. Therefore, these two phases are modeled as a single phase by using the sublattice formalism (Ti) 2 (Cu,Ag) 1 [24], allowing random mixing of Cu and Ag in the second sublattice. The derived expressions of the Gibbs energy functions are similar to those obtained for Ti 3 Cu 4 and Ti 2 Cu 3 and detailed above (see eqns.

5 and 6).

Extension of non-stoichiometric binary compounds in the ternary

In the Cu-Ti binary system, TiCu 4 and TiCu are non stoichiometric compounds with an homogeneity range of, respectively, about 3 at.% Ti and 4 at.% Ti around their ideal compositions of 20 at.% Ti and 50 at.%Ti. As Ti 2 Cu and Ti 2 Ag, the TiCu and TiAg phases are iso-structural [17] and TiAg is also non stoichiometric with an homogeneity range of about 2 at.% Ti. Therefore, they are described as a single phase using the model proposed by Hari Kumar et al. [START_REF] Kumar | [END_REF] allowing for mixing of all three atoms in the two sublattices:

(Cu,Ag,Ti) 1 (Cu,Ag,Ti) 1 . Therefore the Gibbs energies for this phase can be represented by:

G T iM = y ′ Cu y ′′ Cu • G T iM Cu:Cu + y ′ T i y ′′ Cu • G T iM T i:Cu + y ′ Cu y ′′ T i • G T iM Cu:T i + y ′ Ag y ′′ Ag • G T iM Ag:Ag + y ′ T i y ′′ Ag • G T iM T i:Ag + y ′ Ag y ′′ T i • G T iM Ag:T i + y ′ Ag y ′′ Cu • G T iM Ag:Cu + y ′ Cu y ′′ Ag • G T iM Cu:Ag + RT ( i=T i,Cu,Ag (y ′ i ln y ′ i + i=T i,Cu,Ag (y ′′ i ln y ′′ i ) + i=Ag,Cu,T i y ′ i y ′′ Cu y ′′ T i L T iM i:Cu,T i + i=Ag,Cu,T i y ′ Cu y ′ T i y ′ i L T iM Cu,T i:i + i=Ag,Cu,T i y ′ i y ′′ Ag y ′′ T i L T iM i:Ag,T i + i=Ag,Cu,T i y ′ Ag y ′ T i y ′ i L T iM Ag,T i:i + y ′ Cu y ′ Ag y ′′ Ag L T iM Cu,Ag:Ag + y ′ Cu y ′ Ag y ′′ Cu L T iM Cu,Ag:Cu + y ′ Cu y ′ Ag y ′′ T i L T iM Cu,Ag:T i + y ′ Cu y ′′ Cu y ′′ Ag L T iM Cu:Cu,Ag + y ′ Ag y ′′ Cu y ′′ Ag L T iM Ag:Cu,Ag + y ′ T i y ′′ Cu y ′′ Ag L T iM T i:Cu,Ag (9)
where the parameters • G T iM i:j with i, j = Cu or Ag, L T iM i:Cu,Ag and L T iM Cu,Ag:i are the parameters to be optimised, the other ones coming either from [START_REF] Kumar | [END_REF] for the TiCu phase or from [START_REF] Arroyave | Thermodynamics and Kinetics of Ceramic/Metal Interfacial Interactions[END_REF] for the TiAg phase.

Concerning the last TiCu 4 compound, the existing description assumed the existence of defects on both sublattices that are anti-structure atoms represented by the sublattice notation (Ti%,Cu) 1 :(Cu%,Ti) p [START_REF] Kumar | [END_REF]. Introduction of Ag in this compound can lead to a dramatic increase of the number of interaction parameters between the elements in each sublattice. However, the only available experimental data that can be used during the optimisation to refine these parameters is the value of Ag solubility. It is thus important to restrict the number of adjustable parameters. For that purpose, in the present work it was firstly assumed Ag is present only in the second sublattice, where Cu is the major component leading to (Ti%,Cu) 1 :(Cu%,Ag,Ti) 4 in sublattice notation. Assuming that the interaction between constituents in different sublattice are independent, the Gibbs energy of this phase is expressed as in Eq. 10:

G T iCu 4 = y ′ T i y ′′ Cu • G T iCu 4 T i:Cu + y ′ T i y ′′ Ag • G T iCu 4 T i:Ag + y ′ T i y ′′ T i • G T iCu 4 T i:T i + y ′ Cu y ′′ Cu • G T iCu 4 Cu:Cu + y ′ Cu y ′′ Ag • G T iCu 4 Cu:Ag + y ′ Cu y ′′ T i • G T iCu 4 Cu:T i + RT [(y ′ T i ln y ′ T i + y ′ Cu ln y ′ Cu ) + 4(y ′′ T i ln y ′′ T i + y ′′ Cu ln y ′′ Cu + y ′′ Ag ln y ′′ Ag ) + i=T i,Cu y ′ i y ′′ T i y ′′ Cu L T iCu 4 i:T i,Cu + i=T i,Cu y ′ i y ′′ T i y ′′ Ag L T iCu 4 i:T i,Ag + i=T i,Cu y ′ i y ′′ Cu y ′′ Ag L T iCu 4 i:Cu,Ag + i=T i,Cu,Ag y ′ T i y ′ Cu y ′′ i L T iCu 4 T i,Cu:i (10)
where the parameters • G T iCu 4 i:j with i, j = Cu or Ti and L T iCu 4 i:T i,Cu are directly taken from the Cu-Ti binary system obtained by Hari Kumar et al. [START_REF] Kumar | [END_REF]. The two other lattice stabilities are given by:

• G T iCu 4 T i:Ag = • G hcp T i + 4 • G f cc Ag + 5B (11)
• G T iCu 4 Cu:Ag = • G f cc Cu + 4 • G f cc Ag + 5B (12)
B, L T iCu 4 i:T i,Ag , L T iCu 4 i:Cu,Ag and L T iCu 4 T i,Cu:i are the parameters to be optimised. Note that only the regular-solution terms in the Redlich-Kister serie were considered.

Optimisation procedure

Optimisation was performed using the PARROT module of the Thermo-Calc software [25], that uses weighted experimental data as input. For this optimisation, the different sets of phases were optimised in different stages. For the Ti 2 M phase, a positive zeroth-order interaction parameter was used to allow the phase to become less stable at the central regions of the Ti 2 Ag-Ti 2 Cu iso-compositional line in order to avoid making the phase undesirably [19] and calculated in this work.

Reaction Type Reference T ( • C) Phase Composition ( at.%) Ag Cu Ti L 1 + (βTi) --⇀ ↽ --L 2 +
more stable in the central region than at the end members because of the ideal entropic contribution. As noted above, the liquid phase presents a liquid miscibility gap in the central region of the system because of Cu-Ti interactions that are negative whereas the Ag-Ti are positive. Due to this fact, it was decided to introduce a ternary interaction parameter as shown in equation 2. For the TiM phase, in order to limit the number of adjustable parameters, all the interactions parameters L T iM i:Cu,Ag and L T iM Cu,Ag:i involving only Ag and Cu (i = Ag or Cu) are considered as equal. Finally, the L T iM T i:Cu,Ag parameter is described by the zeroth-and first order interaction coefficients. All these parameters were optimised during a first step by using the phase equilibria, invariant reaction [15,16,14] and activity values [7] at high temperature (above 900 • C).

Next, the invariant reactions at lower temperature (see table 2) and Ag solubilities in Ti-Cu compounds were used to optimised the remaining parameters of the Ti 3 Cu 4 , Ti 2 Cu 3 and TiCu 4 phases (about 1.5 at.% at 790 • C -see section 2). To perform this optimisation step, it was necessary to reduce the number of independent model parameters by introducing certain assumptions. For the lattice stabilities corresponding to hypothetical compounds like Ti p Ag q and TiAg 4 or CuAg 4 , the parameters A and B in Eqns. 6 and 12 were arbitrarily assumed to be +5000 J.mol -1 . For the same reasons, the following simplifications were introduced in the description of TiCu 4 :

• L T iCu 4 Cu:T i,Ag = 0 L T iCu 4 Cu:Cu,Ag = • L T iCu 4 T i:T i,Ag = • L T iCu 4 T i:Cu,Ag = • L T iCu 4 i:j,Ag (13)
Finally, three regular-solution interaction parameters were optimised in this second step:

• L T i 3 Cu 4 T i:Cu,Ag , • L T i 2 Cu 3 T i:Cu,Ag and • L T iCu 4 i:j,Ag .
At the end of optimisation, rounding off of the optimised coefficients was done according to the procedure described in [START_REF] Lukas | Computational Thermodynamics: The Calphad Method[END_REF].

5 Results and discussion

Comparison with experimental results

Optimised values of the whole set of model parameters are given in appendix A. tal data points obtained from the literature (See [17,14]). In general, the agreement is quite good. Figure 3 shows the calculated and experimental [17] the composition of TiM and Ti 2 M phases involved in the three phase equilibrium with the Ag-rich liquid are well-reproduced. Composition of the liquid is given by Paulasto et al. [14] as being the composition of the Ag solid solution after solidification. However, the fine precipitation of Ti-Cu intermetallics during cooling leads to an underestimation of the Ti content in the liquid that could explain the discrepancy observed in figure 4. The direction of the tie-lines indicating regions across the miscibility gap at thermochemical equilibrium also correspond to reported experimental results [17].

The consistency between experimental and calculated invariant equilibria constitutes a good indicator of the validity of the thermodynamic descriptions used. Table 2 presents a comparison between the calculated and experimental invariant points (from [19]), including reaction temperatures (see also figure 5 for partial vertical section focused on the temperature range interesting for brazing). The agreement is fairly good on the whole temperature range and mainly at high temperature, although the discrepancy increases at lower temperature. The biggest discrepancy between experiments and calculations is the nature of the invariant reaction involving Ti 2 Cu, TiCu and the liquid phase. Eremenko proposed a eutectic reaction whereas this reaction has a transition type character (L 3 +L 4 --⇀ ↽ --Ti 2 Cu+TiCu) in our assessment. This is a direct consequence of the stabilization of TiCu by addition of Ag.

In brazing of ceramics to metals, the chemical activity of Ti in the liquid braze is perhaps the most important factor when trying to predict the reaction products formed at the ceramic/metal interface. Thus, a reasonable calculation of the chemical activity of Ti in liquid Ag-Cu-Ti melts is desired. Figure 6 shows the comparison between experimental values (EMF measurement from [7]) and calculated Ti chemical activities in eutectic Ag-Cu melts.

The agreement can be considered as being satisfactory. It is important to note, however, that the calculated values are higher than the experimental ones. This implies a higher calculated positive deviation from ideality than experimentally observed. It should be noted that Rongti et al. [START_REF] Rongti | [END_REF] reported activity values that are 1 or 2 order of magnitude lower.

This major discrepancy between two experimental studies, that highlights the difficulty of The projection on the Gibbs triangle of the monovariant lines is presented in figure 8. Some discrepancies exist with the projection reported by Eremenko et al. [15]: the most important is the extension of the liquid miscibility gap at low temperature. The calculated value is about 838 • C against 850 • C. The difference is small but has important consequences as the calculated miscibility gap intercepts more monovariant lines descending from the invariant points in the Cu-Ti binary to the Ag-Cu eutectic, leading to the appearance of two new invariant reactions that are not reported in the reaction scheme suggested by Eremenko et al. [15]. However, if the experimental results of Eremenko et al. allow the determination of the primary phase cristallization, the extension of liquid miscibility gap at low temperature can hardly be deduced. Moreover, the existence of successive invariant peritectic transfor-mations at relatively low temperature and in narrow temperature and composition ranges can also hardly be detected by thermal analysis method or micrographs obtained on as-cast samples. For these reasons, the calculated projection can be considered as acceptable even if some additional experimental results would be necessary to clarify the extension of the liquid miscibility gap.

Calculations with the assessed thermodynamic model

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Some consequences on brazing processes

Thermodynamic approach can be useful to understand the mechanism of a classical brazing process with Ti activated braze at about 850 • C. First of all, it has been recently shown by Differential Thermal Analysis (performed on Mettler-Toledo TMA/SDTA 840) in the range 750 to 800 • C and by isothermal diffusion at 790 • C that TiCu 4 is the first phase to form when a transient Ag-Cu eutectic liquid spreads onto solid Ti between 780 and 790 • C [20]. Next, during the temperature rise toward 850 • C, dissolution of Ti in the AgCu melt proceeds across the Ti x Cu y intermetallic compounds that formed by solid state diffusion between TiCu 4 and Ti [21]. The kinetics of enrichment in Ti of the liquid phase is limited by the dissolution rate of Ti x Cu y and in particular TiCu 4 . As a consequence, the Ti amount available in the liquid state is strongly dependent on the way Ti is brought to the brazing filler (thickness and size are not the same for colaminated sheets, powder embedded in a paste, screen printing of a Ti paste, etc.) and on the temperature used for the isothermal step of the brazing process. For example, the results of Kozlova et al. at 840 • C, [29] show that the dissolution of 1-20 µm Ti particles is not complete after 15 min of contact (presence of Ti 3 Cu 4 phase embedded in big round TiCu 4 particles formed during heating and isothermal treatment). At the opposite, Shiue et al. [30] observed complete dissolution after only 180 s of a few micrometers thick Ti foil at 900 • C, a temperature higher than the decomposition temperature of TiCu 4 .

Figure 9 and 10 presents the isothermal sections with the respective plots of the Ti activity (reference state is hcp-Ti) at 800 and 850 • C respectively which are data valuable for direct use during active brazing process (u Ag =

x Ag

x Ag +x Cu ). It appears that the Ti activity of a liquid in equilibrium with TiCu 4 is about 0.18 in the 800-850 • C temperature range. This activity is particularly important for reactive brazing of ceramic materials because the nature of the reaction product that is formed at the liquid/solid interface is determined by this value [9,29,31]. Fortunately, the Ti activity varies hardly with the nominal Ti amount in Ag-Cu braze whatever the nature of the Ti-Cu intermetallic phase involved in the liquid/solid equilibrium. This is revealed by the upper limit of liquid domain on figure 10 that is nearly flat. However, in order to avoid a dramatic decrease of Ti activity because of its consumption by chemical reaction with the substrates, it is recommended to use high Ti amount and to bring Ti to the brazing filler in the form of small particles. The temperature can hardly be increased above 850 • C because of the appearance of a liquid miscibility gap between a Ag rich and a Cu-Ti rich liquid [14]. Finally, it should be emphasized that in the case of heterogeneous brazing of a ceramic with a metal, the interaction between the reactive element and the elements contained in the metallic solid have to be taken into account.

0 L Liquid Ag,Cu = +17323.4 -4.46819 T [3] 1 L Liquid Ag,Cu = +1654.38 -2.35285 T [3] 0 L Liquid Ag,Cu,Ti = 32400 [4] 0 L Liquid Ag,Ti = 25632 [4] 1 L Liquid Ag,Ti = -327 [4] 0 L Liquid Cu,Ti = -19330 + 7.651 T [5] 2 L Liquid Cu,Ti = +9382 -5.448 T [5] FCC -A1 (Ag,Cu,Ti) 1 (Va) 1 • G FCC-A1 Ag:VA -• H FCC-A1 Ag = +GHSERAG [23] • G FCC-A1 Cu:VA -• H FCC-A1 Cu = +GHSERCU [23] • G FCC-A1 Ti:VA -• H HCP-A3 Ti = +6000 -. T + GHSERTI [23] 0 L FCC-A1 Ag,Cu:VA = +36061.88 -10.44288 T [3] 1 L FCC-A1 Ag,Cu:VA = -4310.12 [3] 0 L FCC-A1 Ag,Ti:VA = 23405 [4] 0 L FCC-A1 Cu,Ti:VA = -9882 [5] 1 L FCC-A1
Cu,Ti:VA = 15777 [START_REF] Kumar | [END_REF] HCP -A3 (Ag,Cu,Ti) 1 (Va) .5

• G HCP-A3 Ag:VA -• H FCC-A1 Ag = +300 + .3 T + GHSERAG [23] • G HCP-A3 Cu:VA -• H FCC-A1 Cu = +600 + .2 T + GHSERCU [23] • G HCP-A3 Ti:VA -• H HCP-A3 Ti = +GHSERTI [23] 0 L HCP-A3 Ag,Cu:VA = +35000 -8 T [3] 0 L HCP-A3 Ag,Ti:VA = 50050 [4] 1 L HCP-A3 Ag,Ti:VA = 27000 [4] 0 L HCP-A3
Cu,Ti:VA = 16334 [START_REF] Kumar | [END_REF] BCC -A2 (Ag,Cu,Ti) 1 (Va) 3

• G BCC-A2 Ag:VA -• H FCC-A1 Ag = +3400 -1.05 T + GHSERAG [23] • G BCC-A2 Cu:VA -• H FCC-A1 Cu = +4017 -1.255 T + GHSERCU [23] • G BCC-A2 Ti:VA -• H HCP-A3 Ti = [ 23
T 3 + 1400501 T -1 0 L BCC-A2 Ag,Cu:VA = +35000 -8 T [3] 0 L BCC-A2 Ag,Ti:VA = +22900 + .5 T [4] 1 L BCC-A2 Ag,Ti:VA = 1500 [4] 0 L BCC-A2 Cu,Ti:VA = 3389 [5] Cu 2 Ti (Cu) 2 (Ti) 1 • G Cu 2 Ti Cu:Ti -2 • H FCC-A1 Cu -• H HCP-A3 Ti = -17628 +2 GHSERCU + GHSERTI [5] Cu 3 Ti 2 (Ag,Cu) 3 (Ti) 2 • G Cu 3 Ti 2 Ag:Ti -3 • H FCC-A1 Ag -2 • H HCP-A3 Ti = +25000 +3 GHSERAG + 2 GHSERTI This work • G Cu 3 Ti 2 Cu:Ti -3 • H FCC-A1 Cu - 2

Figure 1 : 2 →Figure 2 :

 122 Figure 1: Calculated Ag-Ti phase diagram, using thermodynamic assessment by Arroyave [4].

Figure 3 :

 3 Figure 3: Calculated and experimental phase diagram of the Ag-Cu-Ti system at 700 • CThe experimental data was obtained from[17].

Figure 3

 3 Figure 3 and 4 presents the calculated Ag-Cu-Ti phase diagram, along with experimen-

Figure 4 :

 4 Figure 4: Calculated and experimental phase diagram for the Ag-Cu-Ti system at 950 • C. Experimental data after [14].

4 →Figure 5 :Figure 6 :

 456 Figure 5: Calculated partial vertical section for 60 at.% of Ag.

Figure 7 Figure 7 :

 77 Figure7presents the calculated isothermal section of the ternary system at 850 • C that is typically the recommended temperature for brazing with a Ti activated Ag-Cu braze. Since the formation of the first liquid droplet, at 780 • C in the Ag-Cu binary, the domain of liquid has extended widely with the appearance of a horn reflecting the tendency to demixion. The miscibility gap is already formed and all the Ti-Cu intermetallics are still in equilibrium with the liquid phase. This figure, associated with the list of invariant points (see 2) highlights the complexity of processes that occur for this system in a very narrow range of temperature

Figure 8 :

 8 Figure 8: Calculated projection of the monovariant lines of the Ag-Cu-Ti system according to the present description. The labelling of the invariant reactions corresponds to table 2. Temperatures are given in Celsius.

4 →Figure 9 : 4 →Figure 10 : 14 ≤

 4941014 Figure 9: Activity of Ti as a function of u Ag =

1 • G Cu 4 • G Cu 4 - 4 •Cu 4 3 • G Cu 4 - 3 •CuTi 2 ( 2 • G CuTi 2 - 2 •- 2 •

 1444434322222 • H HCP-A3 Ti = -46245 +10.86 T + 3 GHSERCU + 2 GHSERTI [5] 0 L Cu 3 Ti 2 Ag,Cu:Ti = 27800 This work Cu 4 Ti (Ag,Cu,Ti) 4 (Cu,Ti) Ti Ag:Cu-4 • H FCC-A1 Ag -• H FCC-A1 Cu = +25000 +4 GHSERAG + GHSERCU This work • G Cu 4 Ti Cu:Cu -5 • H FCC-Ti Ti:Cu -• H FCC-A1 Cu H HCP-A3 Ti = +80055 -11.693 T + GHSERCU + 4 GHSERTI [5] • G Cu 4 Ti Ag:Ti -4 • H FCC-A1 Ag -• H HCP-A3 Ti = +25000 +4 GHSERAG + GHSERTI This work • G Cu 4 Ti Cu:Ti -4 • H FCC-A1 Cu -• H HCP-A3 Ti = -30055 +11.693 T + 4 GHSERCU + GHSERTI [5] • G Cu 4 Ti Ti:Ti -5 • H HCP-A3 Ti Ti 3 (Ag,Cu) 4 (Ti) Ti 3 Ag:Ti -4 • H FCC-A1 Ag -3 • H HCP-A3 Ti = +25000 +4 GHSERAG + 3 GHSERTI This work • G Cu 4 Ti 3 Cu:Ti -4 • H FCC-A1 Cu H HCP-A3 Ti = -68236 +15.946 T + 4 GHSERCU + 3 GHSERTI [5] 0 L Cu 4 Ti 3Ag,Cu:Ti = 51300 This work CuTi (Ag,Cu,Ti) 1 (Ag,Cu,Ti) 1• G CuTi Ag:Ag -2 • H FCC-Ti:Ag -• H FCC-A1 Ag -• H HCP-A3 Ti = +24080 -1.32 T +GHSERAG + GHSERTI [4] • G CuTi Ag:Cu -• H FCC-A1 Ag -Ag,Cu) 1 (Ti) Ag:Ti -• H FCC-A1 Ag H HCP-A3 Ti = -8325 + 3.3 T +2 GHSERTI + GHSERAG [4]• G CuTi 2 Cu:Ti -• H FCC-A1 Cu H HCP-A3 Ti = -36393 +14.064 T + GHSERCU + 2 GHSERTI [5] 0 L CuTi 2 Ag,Cu:Ti = 14500 This work Corresponding author Dr. O. Dezellus LMI UMR CNRS 5615, University Lyon 1 43 Bd du 11 novembre 1918 69622 Villeurbanne, France Tel: (+33) 4 72 44 83 86 Fax: (+33) 4 72 44 06 18 e-mail: olivier.dezellus@univ-lyon1.fr

  • C [14]. Concerning the other solid phases (Cu rich Cu x Ti y compounds) the Ag solubility seems to decrease continuously as Cu increases: at 700 • C Eremenko et al. gave a generic value that is lower than 2 at.% while Paulasto et al. measured an Ag content up to 2.8 at.% in Ti 3 Cu 4 at 950 • C.

Table 1 :

 1

	Strukturbericht Diagram	Symbol used in	Pearson symbol/	Lattice
		symbol Thermo-Calc data file	Space group/	parameter (pm)
				Prototype
	A1	(Ag)	FCC A1	cF4	a=408.57
		(Cu)	Fm3m		a=361.46
				Cu
	A3	α-Ti	HCP A3	hP2	a=330.65
				P6 3 /mmc
				Mg
	A2	β-Ti	BCC A2	cI2	a=295.06
				Im3m	c=468.35
				W
		TiCu 4	TiCu 4	oP20	a=452.5
				Pnma	b=434.1
				ZrAu 4	c=1295.3
		TiCu 2	TiCu 2	oC12	a=436.3
				Amm2	b=797.7
				VAu 2	c=447.3
		Ti 2 Cu 3	Ti 2 Cu 3	tP10
				P4/nmm	c=1395
				Ti 2 Cu 3
		Ti 3 Cu 4	Ti 3 Cu 4	tI14	a=313
				I4/mmm	c=1994
				Ti 3 Cu 4
	C11 b	Ti 2 Cu	Ti 2 M	tI6	a=295.3
				I4/mmm	c=1073.4
		Ti 2 Ag		MoSi 2	a=295.2
					c=1185
	B11	TiAg	TiM	tP4	a=290.3
				P4/nmm	c=574
		TiCu		TiCu	a=310.8 to 311.8
					c=588.7 to 592.1

Symbols and crystal structures of the stables phases in the ternary Ag-Cu-Ti system (lattice parameters from

Table 2 :

 2 Invariant equilibria in the Ag-Cu-Ti ternary system. Comparison of the values accepted by

  900.00 ≤ T < 1155.00 : +4382.601 + 126.0791 T -23.9887 T ln( T ) -.0042033 T 2 -9.0876 • 10 -8 T 3 + 42680 T -1 1155.00 ≤ T < 1300.00 : +13103.253+60.0676 T -14.9466 T ln( T ) -.0081465 T 2 + 2.02715 • 10 -7 T 3 -1477660 T -1 1300.00 ≤ T < 1941.00 : +369519.198-2553.9505 T +342.059267 T ln( T ) -.163409355 T 2 + 1.2457117 • 10 -5 T 3 -67034516 T -1 1941.00 ≤ T < 4000.00 : -19887.066 + 298.8087 T -46.29 T ln( T )

  ] 298.15 ≤ T < 1155.00 : -1272.064+134.78618 T -25.5768 T ln( T ) -6.63845 • 10 -4 T 2 -2.78803 • 10 -7 T 3 + 7208 T -1 1155.00 ≤ T < 1941.00 : +6667.385+105.438379 T -22.3771 T ln( T ) +.00121707 T 2 -8.4534 • 10 -7 T 3 -2002750 T -1 1941.00 ≤ T < 4000.00 : +26483.26-182.354471 T +19.0900905 T ln( T ) -.02200832 T 2 + 1.228863 • 10 -6

Acknowledgements

The authors acknowledge Bo Sundman for his valuable technical assistance and for fruitful discussions on refinement strategy. One of the authors O.D. acknowledges Pr. Ingo Steinbach from the Interdisciplinary Centre for Advanced Materials Simulation (ICAMS) at Ruhr-Universitat (Bochum) for financial support.

List of Figures


Slim Hrigua
email: slim.hrigua@satie.ens-cachan.fr

Cyrille Gautier

Bertrand Revol

François Costa

MODELING OF POWER SEMICONDUCTORS IN VHDL-AMS FOR EMI SIMULATIONS

Keywords: Power components, VHDL-AMS modeling, EMI simulation

published or not. The documents may come

NOMENCLATURE

INTRODUCTION

The simulation of power converter functioning can be done according to increasing levels of complexity depending on the represented phenomena. In the EMC oriented simulation case, it requires precise electric models for all elements that contribute in the converter functioning. Furthermore, the necessity to represent accurately the commutation phenomena over several switching periods leads to significant simulation time. All these elements require a compromise between model complexity and accuracy in order to keep a reasonable simulation time.

Since the semiconductor components switching are the main EMI sources, their models must be able to represent them with a high precision.

In this work, we are going to propose a dynamic IGBT model in VHDL-AMS. The IGBT referenced IRGPC40UD is chosen to establish the model. It integrates an n-channel IGBT chip as well as a fast recovery anti-parallel diode. Some simplifications will be added to the original IGBT model in order to reduce the simulation time. Afterward, both of these models will be implemented in a buck converter in order to observe the evolution of the conducted EMI.

VHDL-AMS MODELING

As an on-set of VHDL, VHDL-AMS adds simultaneous instructions permitting to affect, in continuous time, the values transported by the objects 'QUANTITY'. It also provides the object 'TERMINAL' that is considered as a node and associated to a 'NATURE' that defines its physical domain.

The VHDL-AMS offers the possibility to insert equations descended from physical studies as simultaneous instructions while giving the possibility to introduce some imperfections on the parameters to evaluate their influences on the model performances. Thanks to this multiabstractions approach, the VHDL-AMS puts at stake several physical domains in low and highlevel models and allows regrouping them in a global model having a structural approach. These properties permit to reduce the computing time while detailing some part more finely, whereas others can be modeled in a more abstract manner. All the modeling works in this paper are done using the VHDL-AMS programming language.

LUMPED-CHARGE DIODE MODEL

Based on a physical approach, the lumped charge modeling technique leads to a considerable reduction of model complexity while maintaining the internal carrier transport process of the power semiconductor device. Several applications of this technique show good results in literature [START_REF] Yuan | An Improved Lumped-Charge Model and Parameter Extraction Approach of PIN Diodes[END_REF]- [START_REF] Iannuzzo | Physical CAD Model for High-Voltage IGBTs Based on Lumped-Charge Approach[END_REF]. We referred for this work to the Ma-Laurizen diode model [START_REF] Ma | Modeling of Power Diodes with the Lumped-Charge Modeling Technique[END_REF]. This model was developed to have a better representation of the forward and reverse recovery. The Inclusion of these phenomena is critical in the power converters design and for the simulation of the conducted EMI. diode structure [START_REF] Ma | Modeling of Power Diodes with the Lumped-Charge Modeling Technique[END_REF].

Thanks to this technique, the basic semiconductor equations are simplified while using a nodal arrangement (Fig. 1). These relations are transformed in lumped-charge equations for every region. In fact, since the weakly doped zone N - stores the carriers in excess that determines the diode transient features, three nodes (2, 3 and 4) are chosen for this region. The nodes 2 and 4 are connection nodes and the node 3 is the charge storage node. For the highly doped regions, only one node is sufficient to act at the same time as connection and storage charge node. This is due to the weak spatial variations of the carriers concentration. The nodes 1 and 5 are considered sufficient for the N + and the P + regions.

With regard to the VHDL-AMS diode model, we leaned on the described models in [START_REF] Lauritzen | Compact Models for Power Semiconductor Devices[END_REF] used on SABER simulator. A test circuit, described in the literature [START_REF] Hrigua | Using of VHDL-AMS to Model Power Semiconductors for EMI Simulation[END_REF], was used to evaluate the inverse transient characteristic of the diode model from a null forward current. In the coming works, we will use this diode model in a DC-DC converter as a freewheeling diode.

IGBT DYNAMIC MODEL

Up to now, an interval still exists between the industrial development and the valid models on the simulators. As for the IGBT modeling, there is a compromise between the accurateness and the model simplicity. Different approaches exist at the IGBT modeling time, offering for each its own performances to retranscribe at best the real behavior of the device. Departing from this idea, we are trying to pursue these recent efforts by understanding the internal mechanism in order to identify the relevant parameter of the device for the EMI analysis.

For these IGBT modeling works, we lean up on the Hefner physical model [START_REF] Hefner | An Experimentally Verified IGBT Model Implemented in the Saber Circuit Simulator[END_REF] which is classified among the first complete model with charge control [START_REF] Pittet | Modélisation Physique d'un Transistor de Puissance IGBT -Traînée en Tension à l'Enclenchement[END_REF] that shows excellent results in literature [START_REF] Hefner | An Investigation of the Drive Circuit Requirements for the Power Insulated Gate Bipolar Transistor (IGBT)[END_REF] and on the VHDL-AMS modeling works [START_REF] Ibrahim | Contribution au Développement de Modèles pour l'Électronique de Puissance en VHDL-AMS[END_REF].

This model describes the n-channel IGBT as an nchannel MOSFET that supply the base current of the PNP bipolar transistor. Nevertheless, these internal components behave differently of their microelectronic counter-parts.

VDMOSFET CHARACTERISTICS

The expression of the current Imos crossing the VDMOSFET channel is described in literature [START_REF] Hefner | An Experimentally Verified IGBT Model Implemented in the Saber Circuit Simulator[END_REF]. Fig. 2 and3 show the VDMOSFET capacitive elements. The Cgd capacitance is equal to the oxide capacitance Coxd for Vds ≤ Vgs െ V ୢ , for except that for Vds > Vgs െ V ୢ , the Cgd capacitance will be constituted by a serial recombination of the Coxd capacitance and the depletion capacitance Cgdj at the level of the grid-drain overlap. The expression of this capacitance is described like follows:

• ܸ ௗ௦ ܸ ௦ െ V ୢ ݀݃ܥ ൌ Coxd • ܸ ௗ௦ ܸ ௦ െ V ୢ ݀݃ܥ ൌ C ୭୶ୢ ܥ ௗ C ୭୶ୢ ܥ ௗ (1) ݆݀݃ܥ ൌ A ୢ ε ୱ୧ ܹ݆݃݀ (2)
The sum of the oxide capacitance Coxs at the level of the grid-source overlap and the source metallization capacitance Cm forms the grid-source capacitance Cgs expressed by equation [START_REF] Iannuzzo | Physical CAD Model for High-Voltage IGBTs Based on Lumped-Charge Approach[END_REF].

ݏ݃ܥ ൌ Cm Coxs (3)
The depletion capacitance of the drain-source junction Cds is given by:

ݏ݀ܥ ൌ A ୢୱ ε ୱ୧ ܹ݀s݆ (4)
The Cgd and Cds capacitances are also proportional to Agd and Ads surfaces with Wdsj and Wgdj are respectively the widths of the depletion zone for the drain-source and the grid-drain junctions.

BIPOLAR TRANSISTOR CHARACTERISTICS

Contrary to the conventional bipolar transistor, the base length is not anymore too small in relation to the diffusion length of its minority carriers. The transistor is thus separated to an anode-base junction, a base-cathode junction and a resistive base Rb. The equivalent diagram used for the IGBT modeling in VHDL-AMS is detailed on Fig. 2.

Contrary to the Iccer current that flows through the redistribution capacitance Ccer ( 6), the representation of the steady-state BJT collector current Icss doesn't depend on the variations of the anode-cathode potential. These two currents that flow through the transistor base are given by the following expressions:

ݏݏܿܫ ൌ ൬ 1 1 b ൰ ܫ ் ൬ b 1 b ൰ 4D ୮ ܹ ଶ ܳ ݎ݁ܿܿܫ ൌ ݎ݁ܿܥ ܸ݀ ݐ݀ (5) (6)
On the other hand, the anode-base junction is represented as a steady-state current Ibss set in parallel with the Iceb current. These currents are given by the following expressions:

ݏݏܾܫ ൌ ܳ τ ୌ ܳ ଶ ܳ ଶ 4ܰ ௦ ଶ n ୧ ଶ ݁݊ݏܫ ܾ݁ܿܫ ൌ ݀ܳ ݐ݀ (7)

CONDUCTED EMI SIMULATION

As known, the main goal of the virtual prototyping is to reproduce a device functioning with adjustable levels of accuracy. Nonetheless, every semiconductor is considered as permanent or temporary carriers reservoir and its responses facing the external excitations depend of several physical constraints (non-linear behavior, temperature sensitivity…). These constraints become increasingly perplex in complex switches case (complex geometric structure, several doping level of the layers, elevated number of layers…) what makes the approximations taken at the behavioral modeling time a fatal source of errors on the electromagnetic perturbations analysis especially for the transient behavior. Those constraints justify our tendency to physical models for EMI simulation.

As a matter of fact, a physical model whose responses are highly accurate is essential to conduct an EMC study. The two main problems that appear when using physical models for semiconductor devices is the knowledge of the exact parameters needed to reproduce the real functioning of the device and the reduction of simulation time.

SIMULATION CIRCUIT TOPOLOGY

The different simulations presented in this survey have been done by using a buck converter circuit presented in Fig. 4. The switching cell is constituted of an IGBT transistor, a freewheeling diode and an inductive load.

To define the models parameters, we used the manufacturer's technical files [START_REF]IRG4PC40UD: Insulated Gate Bipolar Transistor with Ultrafast Soft Recovery Diode[END_REF]- [START_REF]HFA15TB60/HFA15TB60-1: Ultrafast Soft Recovery Diode[END_REF] as a reference. The IGBT and the diode modeling can be achieved in an independent way since they represent two detached components in the same package [START_REF]IRG4PC40UD: Insulated Gate Bipolar Transistor with Ultrafast Soft Recovery Diode[END_REF] and it allows us to use the IGBT body diode as a freewheeling diode. At this phase of work, we referred to the literature [START_REF] Bryant | Two-Step Parameter Extraction Procedure With Formal Optimization for Physics-Based Circuit Simulator IGBT and p-in Diode Models[END_REF]- [START_REF] Chibante | A Simple and Efficient Parameter Extraction Procedure for Physics Based IGBT Models[END_REF] in order to approximate some IGBT parameters from the datasheet. The SGP30N60 IGBT already studied in the literature [START_REF] Ibrahim | Contribution au Développement de Modèles pour l'Électronique de Puissance en VHDL-AMS[END_REF] was taken as a reference for our IGBT parameters since they have the same manufacturing technology and approximately the same characteristics. In further works, we will focus on the extraction and the adjustment of the parameters by an optimization approach in order to minimize the differences between the experimentations and simulations.

A LISN circuit has been inserted between the supply and the converter in order to calculate the conducted EMI generated by the circuit in both differential and common modes. The circuit was simulated with considering of the non-ideal electrical interconnections. In older works, we detailed the functioning of the switching cell by making a temporal analysis of the models responses [START_REF] Hrigua | Using of VHDL-AMS to Model Power Semiconductors for EMI Simulation[END_REF].

The different simulations of these models are achieved on the Portunus® software. Conducted EMI versus time are extracted from the LISN. A simple Fast Fourier Transformation (FFT) calculation of these voltages allows us to deduce the EMI spectrum. These calculations were carried out on the Matlab® software.

EFFECT OF THE IGBT MODEL ON THE CONDUCTED EMI

During this observation we are interested in the amplitude variations that appear on the EMI spectrum which frequency band reaches 100 MHz. These variations are mainly due to the switching at the level of the non-linear components. Therefore, the frequencies and the amplitude of noise peaks could depend on some parameters of their models.

The duty cycle of the IGBT switching frequency is 0.5 generated by the control supply E1for a grid voltage included between 0V and 15V. The inductance of the load Lch is set to 200µH. The maximum step chosen for temporal sampling was 100ps.

In order to test the reliability of the IGBT model for the EMI simulation, we have varied the value of the grid resistance Rg (10Ω/60Ω), the load resistance Rch (30Ω/100Ω) for a fixed values of voltage source E2 (500V) and switching frequency (40kHz), then we observed the evolution of the conducted EMI at the LISN for each variation time.

As shown in Fig. 5, the blue spectrum is gotten by using the already described IGBT model.

The variation of the grid resistance Rg proves that the switching time appears as an important factor for the spread spectrum. Also, the variation of the load resistance Rch proves the high current switching effect on high frequency current spike.

The major problem that we found during the conducted EMI extraction was the simulation time.

The simulation time for 30µs of temporal representation at 40kHz of switching frequency was approximately 45 minutes to 55 minutes and for 60µs of temporal representation at 20kHz of switching frequency it largely exceeds 1 hour.

In order to reduce the simulation time, we focused on the dynamic behavior of the IGBT current during transient conditions. The BJT emitter-base current Iceb expressed in equation ( 8) represent the time rate-of-change of the base charge. The voltage that characterizes this junction is expressed by nonlinear expressions and it describes implicitly its capacitance [START_REF] Hefner | An Experimentally Verified IGBT Model Implemented in the Saber Circuit Simulator[END_REF]. Contrary to the Iceb current, Iccer (6) depends on the non-linear redistribution capacitance Ccer times the time rate-of-change of the BJT emitter-collector voltage. As regard to the VDMOSFET, the currents that flow through the non-linear Cgd (1) and Cds (4) capacitances depend on the depletion widths and on the time rate-ofchange of the junction's voltages. On the other hand, the current that flows through the static Cgs capacitance depends only on the time rate-ofchange of the grid-source voltage.

With the purpose of minimizing the simulation time, we separated the static and the transient functioning of the IGBT model. In fact, since the Iccer, Icgd, Icds and the Icgs current are voltage time rate-of-change dependent, their major effect takes place during the switching phases, elsewhere their influences are minimal and can be neglected during the steady state behavior of the device.

In order to define the transition conditions from each state we are based on the value of the instantaneous excess carrier base charge Q and of the grid-source junction potential [START_REF] Yue | An Analytical Insulated-Gate Bipolar Transistor (IGBT) Model for Steady-State and Transient Applications Under All Free-Carrier Injection Conditions[END_REF]. During the turn-on transition, the applied Vgs voltage exceeds the threshold voltage; the free carriers are injected from the BJT emitter to base region. Ones reaching the background mobile carrier base charge Qb, these free carriers will pass through the BJT basecollector junction or recombine with the electrons supplied from the MOS channel. Furthermore, since the doping concentration in the base is relatively low, the free carriers density is higher than the doping concentration which characterizes the state of high-level free carriers injection. So over this phase we have Q>Qb.

During the turn-off transition, no electrons current pass through the MOS channel and the IGBT behave like an open-base BJT. Subsequently, the anode current is only composed of the holes current due to the electrons-holes recombination and its decay time depends on free carriers life time in the base. Over this phase we can define Q included in [αQb>Qb, Qb] with α is an empirical value for the ratio Q/Qb. Beyond this, Q<Qb and thus the base doping concentration is much higher than the hole concentration until the next Vgs turning-on condition. As a conclusion, we can use these different statements to have an autonomous model able to detect its own transient instances.

The simplified model was implemented in the same circuit and tested at the same conditions as the original model. The red EMI spectrum is gotten after simplifying the model (Fig. 5). As we can see, the agreement between the two spectra validates the approximations imposed to simplify the initial model.

In low grid resistance case Rg (10Ω) we got the same results for the two models and for higher value of Rg (60Ω) the spectra are quite similar. The simplified model was also tested under other value of voltage source E2 (200V) and switching frequency (20kHz) and it also shows a very good agreement with the initial model. Furthermore, the simulation time for 30µs of temporal representation at 40kHz passed approximately to 7 minutes and to 8 minutes for 60µs of temporal representation at 20kHz.

At this level, the model does not consider the component thermal variations, but it does not prevent that the silicon temperature-dependent properties and the IGBT temperature-dependent parameters [START_REF] Hefner | A Dynamic Electro-Thermal Model for the IGBT[END_REF] could be taken into account even with the simplifications that we made. These results allowed us to test the robustness of the IGBT model facing the external excitations, to show the considerable effect of semiconductor switching on EMI spectrums and in another step to identify the elements that mainly contribute to the resonances.

•

  Ads: Body region area [m 2 ]. • Agd: Gate-drain overlap area [m 2 ]. • b=µ n /µ p : Ambipolar mobility ratio [-]. • Ccer: Collector-emitter redistribution capacitance [F]. • Cds: Drain-source depletion capacitance [F]. • Cgd: Gate-drain capacitance [F]. • Cgdj: Gate-drain overlap depletion capacitance [F]. • Cgs: Gate-source capacitance [F]. • Coxd: Gate-drain overlap oxide capacitance [F]. • Coxs: Exceeding oxide capacity [F]. • Dp: Hole diffusivity [m 2 .s -1 ]. • Ibss: Steady-state base current [A]. • Iceb: Emitter-base capacitor current [A]. • Iccer: Collector-emitter redistribution current [A]. • Icss: Steady-state collector current [A]. • I sne : Emitter electron saturation current [A]. • I T : Anode current [A]. • n i : Intrinsic carrier concentration [m -3 ]. • Nscl: Collector-base space charge concentration [m -3 ]. • q: Electronic charge [C]. • Q: Instantaneous excess carrier base charge [C]. • Q B : Background mobile carrier base charge [C]. • Vds: Drain-source voltage [V]. • Vec: Emitter-collector voltage [V]. • Vgs: Gate-source voltage [V]. • V Td : Gate-drain overlap depletion threshold voltage [V]. • W: Quasi-neutral base width [m]. • Wdsj: Drain-source depletion width [m]. • Wgdj: Gate-drain overlap depletion width [m]. • ε si : Silicon permittivity [F/m]. • τ HL : Base high-level lifetime [s].

Fig. 1 .

 1 Fig. 1. Lumped-charge distribution in a 1-D P + N -N + diode structure [4].

( 8 )Fig. 2 .Fig. 3 .

 823 Fig. 2. Equivalent circuit model of the IGBT.

Fig. 4 .

 4 Fig. 4. Switching circuit.

Fig. 5 .

 5 Fig. 5. EMI spectrum for F=40kHz and E2=500V. Rg=10Ω (a)Rch=30Ω, (b)Rch=100Ω Rg=60Ω (c)Rch=30Ω, (d)Rch=100Ω

CONCLUSION AND PERSPECTIVES

The IGBT structure is similar to that of an nchannel VDMOSFET with the exception that the ntype drain contact is replaced by the p-type anode region for the IGBT. In contrast with microelectronic BJT, the internal IGBT epitaxial layer has a lightly doped wide base that add, compared to VDMOSFET, more complexity to the internal consideration of the bipolar phenomena which makes the modeling works, physical considerations and simplification efforts more prevalent and necessary in the case of bipolar components.

The EMI analysis at the conception stage of static converters circuits requires the use of accurate models for the semiconductor components. The previously developed physics-based IGBT model provides the basic criteria to lead off an EMC survey on a real prototype.

The model has been used to describe the different EMI spectra for various external circuit conditions. These preliminary results perform well for the range of static and dynamic conditions in which the device is intended to be operated. The validations of the proposed model in relation to experimental measurements are necessary at this stage.