MODEL FOR A LOW CURRENT VACUUM ARC CATHODE REGION. EFFECT OF THE ELECTRONIC TEMPERATURE

Transcription

1 Acta Elctrotchnica t Informatica No. 3, Vol. 6, MODEL FOR A LOW CURRENT VACUUM ARC CATHODE REGION. EFFECT OF THE ELECTRONIC TEMPERATURE * Mohammd MESSAAD, * Ahmd Wahid BELARBI, ** André LEFORT, ** M hammd ABBAOUI * Dpartmnt of Elctrotchnics, Faculty of Elctrical Enginring, Univrsity of Scinc and Tchnology (USTO), P.O El-m naouar, Oran, Algria, bmssaad@yahoo.fr ** Elctric Arc and Thrmal Plasmas Laboratory, Blais Pascal Univrsity, Aubièr, Franc SUMMARY Exprimntal and thortical studis of arc cathod rgion hav bn mad during svral dcads and th task is not yt compltd, in spit of many fforts and progrss. In this work, a numrical modl dscribing th arc cathod rgion is dvlopd. Th modl is thn applid to a vacuum arc discharg intracting with a Cu cathod at low currnt (4-50A). Th modl yilds th tmpratur and lctric fild strngth at th cathod surfac, lctrons mittd and total currnt dnsity, cathod spot radius, diffrnt kinds of powr dnsitis hating and cooling th cathod, and th plasma lctron dnsity. Th comparison with xprimntal rsults shows good agrmnts. Kywords: vacuum arc, cathod spot, cathod rgion, currnt dnsity, powr dnsity, spot radius 1. INDRODUCTION Th arc discharg is a high-currnt, low-voltag discharg and a high plasma dnsity in th nar cathod rgion. Elctrons for th discharg ar supplid by th cathod spot. Th cathod spot, through which currnt ntrs th cathod is a highly contractd rgion (som micromtrs) oprating undr xtrm physical conditions (currnt dnsity A/m 2, powr dnsity W/m 2, lctric fild 10 9 V/m tc.). Th cathod voltag drop is only about (15-20 V for coppr cathod [1, 2]. Arc dischargs hav many important applications: arc wlding, high-prssur sodium lamps, fluorscnt lamp, tc. Arcs ar also involvd in powr switching. Whn a circuit carrying currnt is intrruptd, an arc always forms at th switch contacts. Th ffct of high tmpraturs producd at th last point on th contacts to opn, whn th currnt is confind to a small ara. So, a fundamntal undrstanding of th arc-cathod intractions is ruird. Th xprimntal study of cathod spot was aimd mainly at dtrmining th following basic spot charactristics: spot radius, th magnitud of th currnt dnsity at th spot, lctrical potntial gradint in front of th spot, th rat of th cathod rosion and th morphology of th rosion tracs, th lctron tmpratur, th lctron dnsity, and th distribution of ion charg stats in th vicinity of th spot, tc [1]. Thortically, th arc consists of thr zons: th cathod zon, th positiv column, and th anod zon. Th cathod rgion is considrd to b th most activ rgion in lctric arcs. Thus, th cathod phnomna hav bn studid for many yars. Unfortunatly, this rgion rprsnts a vry difficult subjct for xprimntal invstigation owing to its vry small dimnsion, high local prssur, high tmpratur gradint, intns radiation, tc. [1-3]. Thortical invstigation is hindrd by th multiplicity and divrsity of procss involvd. Th aim of this work is to prsnt a modl of th cathod rgion in vacuum arc discharg. 2. MODEL Th prsnt modl is inspird from th thortical study of numrous authors (as an xampl Hantzsch [4], Coulomb [5], Eckr [6], Zhou [7], Bolotov [8], Rthfld [9], Mittraur [10] and H[11] who hav studid th arc cathod rgion xtnsivly. Th modl assum th vaporation of cathod matrial from th spot is causd by its lvat surfac tmpratur. Th spot high tmpratur is attributd to Joul hating by th high currnt dnsity and th dposition of nrgy by ions and back diffusd lctrons straming to th cathod from th ionization rgion in front of it. Th modl assums apparanc of a positiv shath in front th spot that producs a high lctric fild (of th ordr of 10 9 V/m) which lowrs th work function and significantly amplifis lctrons mission from th cathod. Th postulatd mchanism is trmd fild assistd thrmionic (or TF) mission. According to th gnrally accptd dscription for vacuum arcs intracting with thir cathods, th cathod rgion can b dividd into thr sub-rgions (Fig. 1): cathod spot, spac charg zon and ionization zon. Th spac charg zon situatd in front of th cathod surfac is rsponsibl for th acclration of th mittd lctrons from th cathod surfac towards th ionization zon and th positiv ions formd in th ionization zon towards th cathod. A nt positiv spac charg of local dnsity forms in th spac charg zon as a rsult of th imbalanc btwn th ion and lctron dnsity. Th lctrons xtractd from th cathod travl mostly without collision in th spac charg zon. Whn ntring in th ionization zon, th mittd lctrons hav sufficint nrgy to ioniz th coppr atoms of vapor rlasd from th cathod matrial. Th rsultant

2 2 Modl for a Low Currnt Vacuum Arc Cathod Rgion Effct of th Elctronic Tmpratur positiv ions in th ionization zon ar also acclratd in th spac charg zon towards th cathod surfac. Th high lctric fild xisting at th cathod surfac which is ncssary for lctron mission from th cathod is cratd by ths positiv ions accumulatd in front of th cathod surfac. Th ionization zon is th rgion in which th ion flux to th cathod surfac is formd and th spac charg shath is th rgion in which ions going to th cathod and lctrons mittd from th cathod ar acclratd. In th nxt, w prsnt a modl for th vacuum arc cathod rgion in th cas of coppr cathod. This modl can b applid for othr mtals. Arc column LTE, n p Ionization zon n rs, n ms, n is, T i =T s, =1-4 V Spac charg zon ( m, c, vap, r ) ( i, r, J ) Cathod spot J i Jr Jm Cathod Shath dg Fig. 1 Modl for th vacuum arc cathod rgion Th lctric fild strngth at th cathod surfac is cratd by th positiv ions and it is valuatd using a simplifid uation of Mackown [10] Z i mi Ji U = c F 2. ε 0 (1) Whr m i is th positiv ion mass, is th cathod fall, ε is th dilctric constant, Z 0 i is th man ion charg, J i is th positiv ions currnt dnsity and is th lmntary charg. Th high tmpratur at th cathod surfac causs vaporation of Cu atoms and significant local rosion of th cathod matrial is obsrvd. During th procss of arc opration, th flux dnsity of coppr ( Γ v in Kg/m 2.s) atoms laving th cathod is dscribd by th Hrtz-Knudsn formula [5]: Pvap Γ v = 1 4 mcu kb Ts 3 (2) Hr, Ts is th local cathod surfac tmpratur, P vap is th uilibrium vapor of th cathod matrial that corrspond to local tmpratur, m cu is th atom mass of cathod matrial ( mcu = uma for Cu) and k B is th Boltzmann constant. Th cathod matrial vapor prssur ( P vap ) that corrspond to local cathod surfac tmpratur T s is valuatd with Langmuir formula [5]. P vap A ( + B C T ) ( T ) s = s 10 (3) Hr, A=17650, B=13.39, C= for Cu. Particls dnsitis at th shath dg n ms, n is, nrs ar rspctivly, th mittd lctrons, positiv ions and back-diffusd lctrons dnsitis. n s is th total lctron dnsity at th shath dg. n n n ms is Jm m 1 = ( ) 2 2 kb Ts +. Uc Ji Ji = = v. is kb( Ti + Zi. T) m i (4) (5) = Z. n (6) s i is nrs = ns nms = Zinis nms (7) Positiv ions ar tratd as mono-nrgtic particls and ntring th shath dg with th kb( Ti + Zi. T) Bohm[4] vlocity ( vis = ). It is mi assumd also that all ions rcombin at th cathod spot surfac [4, 5]. J m, J i, m, v i,, T i and Z i ar rspctivly, th mittd lctron currnt dnsity, th positiv ions currnt dnsity, th lctron mass, th vlocity of positiv ions, th tmpratur of lctrons and positiv ions at th shath dg and th man ion charg. Th lctron dnsity at th boundary ionization zon/arc column n p is [3]: 1/2 p = s ( ) (8) n n Currnt dnsitis Th lctrons ar mittd from th cathod surfac undr th combind action of th high surfac tmpraturs (T s ) and high surfac lctric fild strngths (F) maintaind by th ions prsnt in th shath. Such lctron mission mchanism is calld thrmo-fild (T-F) mission and was dscribd in dtail by Murphy and Good [12]. Th currnt dnsity of th mittd lctrons from th cathod dtrmind on th basis of th formalism of Murphy and Good rads:

3 Acta Elctrotchnica t Informatica No. 3, Vol. 6, jm = D( F, W) N( Ts, Φ, W) dw (9) W a Whr Wa = φ + W f. Hr, Ts is th cathod spot tmpratur, F is th lctric fild at th cathod spot surfac, Φ is th work function of cathod matrial ( Φ= 4.5 V ), f D F W is th lctron tunnling probability across th barrir and, N( Ts, F, W) is th numbr of Fmi-Dirac distributd lctrons having nrgy btwn W and W + dw and incidnt on th barrir pr unit tim and unit surfac ara. For mor dtails on ths xprssions and how to calculat th thrmo-fild W is th Frmi lvl, (, ) mittd lctrons s [12]. Th currnt dnsity ( J i ) of positiv ions which ar cratd in th ionization zon aftr th mtallic vapor hav bn ionizd by lctron impact is [4]: J = α β Z Γ (10) i i v Whr, α is th backflow cofficint of vapor towards th cathod and β is th ionization dgr of th plasma ( β 1 i.., all th mtal vapor is ionizd [4]). Th vlocity distribution of lctrons at th shath dg is assumd Maxwllian on, thus som 2U lctrons whos vlocitis ar highr than ( c m ) can ovrcom th cathod voltag drop and rach th cathod surfac. Thus, th currnt dnsity of back-diffusd lctrons from th plasma to th cathod surfac ( Jr ) is givn by [4, 11]: J r 1 8kBT Uc = nrs xp 4 π m kbt (11) 8 kbt Th uantity rprsnts th avrag π m thrmal vlocity of plasma lctrons at th shath dg. Th total currnt dnsity ( Jt ) is th sum of currnt dnsitis of mittd lctrons, back-diffusd lctrons and positiv ions. Jt = Jm + Ji Jr (12) Jt is calculatd also by assuming that th cathod spot has a circular shap with radius r s : All currnt dnsitis ar assumd to b uniformly distributd ovr th spot radius. I is th arc currnt 2 A = π. and A is th cathod spot surfac ( r s ) Powr dnsitis hating th cathod Th cathod is hatd by th impinging positiv ions and back-diffusd lctrons and by Joul hating. In th cathod fall potntial th ions tak th nrgy ( ). Raching th cathod, th ions giv thir kintic nrgy ( ) and th rcombination nrgy V ( i ZiΦ ff ) to th cathod surfac. Consuntly, th powr dnsity ( i ) of positiv ions is [5]: Ji 2.5kBTi ZikBT = V ZU Z W Z + Φ i i i c i ff v i (14) Hr, V i is th ionization potntial, F Φ ff =Φ is th ffctiv work function of 4πε 0 cathod matrial and Wv is th condnsation nrgy of coppr atom (3.5 V [5]). Th powr dnsity of back-diffusd lctrons r is: 5kT B = J Φ + 2 r r ff i and r (15) ar surfac powr phnomna hating th cathod. Additionally, within th cathod volum and spcially blow th cathod spot ara, powr is gnratd by Joul hating. In ordr to b comparabl with th othr componnts ( i and r ) this volum sourc is projctd onto th cathod spot surfac, rsulting in an uivalnt powr dnsity [4] of: ( ) 2 T s J k0 J t rs σ = (16) Whr σ is th lctric conductivity of coppr and can b approximatd by th Widmann-Franz law σ 0 as ( σ = ) with σ 0 bing th lctric conductivity T s at 300 K. k0 is a constant in th rang of ( ) [4]. Powr dnsitis cooling th cathod m, vap ar rspctivly, th powr dnsity of mittd lctron and powr dnsity of mtal vaporation: J I t = (13) A 5kT B s = J Φ + 2 m m ff (17)

4 4 Modl for a Low Currnt Vacuum Arc Cathod Rgion Effct of th Elctronic Tmpratur = W. Γ (18) vap vap v Powr loss by spot radiation ( r ) is calculatd assuming a blackbody radiation using th Stfan- Boltzmann law [10]. ( T ) 4 = σ. (19) r SB s σ SB is th Stfan-Boltzmann constant. Th Powr dnsity of hat conduction in th cathod body ( c ) can b stimatd as [8]: c Ts T0 = λ π (20) r s Whr, T s is th cathod spot tmpratur, T0 = 300K is th ambint tmpratur far from th cathod spot, r s is th cathod spot radius and λ is th thrmal conductivity of coppr. λ is takn tmpratur dpndnt [11, 12]. Th total powr dnsity hating th cathod is: = i J r (21) Th total powr dnsity cooling th cathod is: = m + vap + c + r (22) In th stationary cas, th powr balanc holds at th cathod surfac ( = ). + [6], Mittraur [10], and H [11]. Th valu of affcts apprciably th valus of th cathod spot tmpratur. As incrass th tmpratur of th cathod spot incrass. T s (K) =18Vv, Z i =2, α =1 =1 V =2 V =3 V 0 Fig. 2 Cathod spot tmpratur as function of currnt and Th lctric fild strngth in front of th cathod spot surfac F shows th sam tndncy as th cathod spot tmpratur. F prsntd in Fig. 3 9 varis in th rang (3 5) 10 V / m and incrass as arc currnt incrass. Th mor important is th lctronic tmpratur, th mor important is th lctric fild strngth. Th ordr of magnitud of th lctric fild agrs wll with th rsults of Hantzsch [4], Eckr [6], Zhou [7], Mittraur [10], and H [11]. 3. RESULTS AND DISCUSSIONS 5.0x10 9 =18 V, Z i Computation whr prformd for a coppr cathod with th following paramtrs: ( Uc = 15V, Ui = 15V, α = 1) For any valu of currnt, th cathod surfac is varid until th uantity + bcoms low + than In this work th ffct of th lctronic tmpratur on th vacuum arc paramtrs ar studid. This modl can show th importanc of th othr paramtrs: th man ion charg Z i and th uivalnt ionization potntial U i, th cathod voltag drop, th ionization dgr of th mtal vapor η, and th back-flow of th mtal vaporα. Figur 2 shows th cathod spot tmpratur for diffrnt arc currnt and for diffrnt valus of. Ths tmpraturs ar high compard with th vaporation tmpratur (2868 K) of coppr at ambint prssur (10 5 Pa). As arc currnt incrass, th cathod spot tmpratur incrass from 4400 K to 5000 K. This ordr of magnitud is in agrmnt with th thortical rsults of Hantzsch [4], Eckr F (V/m) 4.5x x x x10 9 =1 V =2 V =3 V Fig. 3 Elctric fild at th cathod spot as function of currnt and Figur 4 shows that th cathod spot radius dosn't incras many with th currnt as in Jüttnr s [14] and Daaldr s [15] rsults shown in th sam figur. At currnt of 10 A and =3V th spot radius is 5 m μ and bcoms 7 m μ at 50A. As T is incrasd, th spot radius is dcrasd and tnds to th xprimntally obtaind valus by Jüttnr [14] and Daaldr [15].

5 Acta Elctrotchnica t Informatica No. 3, Vol. 6, x x x x r s (m) 6.0x x x x x10-6 Juttnr rsult's[14] Daaldr rsult's[15] r s (modl) : =1v, =2v, =3v J r (A/m 2 ) T=1V T=2V T=3V Fig. 4 Spot radius as function of currnt and Th dpndnc of th currnt dnsity in th cathod spot on th arc currnt and th lctronic tmpratur is shown in Fig. 5. Th currnt dnsity incrass in th rang (5 10 ) to (4 10 A / m ) as arc currnt is incrasd from 4 to 50A. If is incrasd from 1V to 3V th currnt dnsity is multiplid by a factor two. Th most carful masurmnts of th cratr siz in function of currnt wr carrid out by Daaldr [14] and Jüttnr [15]. Ths authors found that at low currnt (4-50 A) th cratr siz is not apprciably affctd by th valu of th arc currnt. Th currnt dnsitis hav bn masurd by various authors [2-4, 16]. Th rsults wr obtaind undr diffrnt xprimntal conditions i.., natur and surfac uality of cathod, cathod anod distanc, natur and prssur of gas, mthod of masurmnt (spctroscopic or autographic), arc currnt rang, arc duration, tc. Owing to ths factors, on can s significant disparitis ( A/m 2 ) [1-4, 16] in th valus of currnt dnsity at th cathod obtaind xprimntally. Fig. 6 Currnt dnsity of back-diffusd lctrons as function of currnt for diffrnt valus of ( r Th powr dnsity of back-diffusd lctrons ) hating th cathod as function of currnt for diffrnt valus of is shown in Fig. 7. This powr dnsity which is proportional to th currnt dnsity of back-diffusd lctrons J r shows th sam bhavior as th latr. r is apprciably nhancd whn grows from 1V to 3V. Indd, r is nar 10 6 W/m 2 for =1V howvr for =3V it surpasss W/m 2 which is apprciabl. r (W/m 2 ) =1V =2V =3V 4.0x 3.5x =18 V, Z i J t (A/m 2 ) 3.0x 2.5x 2.0x 1.5x 1.0x 5.0x =1V =2V =3V Fig. 5 Currnt dnsity in th cathod spot as function of currnt for diffrnt valus of Th currnt dnsity of back-diffusd lctrons J r incrass slightly with currnt as shown in Fig 6. Howvr it is apprciably nhancd whn th lctronic tmpratur is incrasd. For =1V th currnt dnsity is low than 10 6 A/m 2 and surpasss A/m 2 for =3V. Fig. 7 Powr dnsity of back-diffusd lctrons hating th cathod as function of currnt for diffrnt valus of Th powr dnsitis of positiv ions, rtrodiffusd lctrons and Joul ffct hating th cathod ar shown in Fig. 8 for ( =1V) and Fig. 9 ( =3V). It can b sn that th cathod is hatd mainly by th positiv ions bombardmnt. Indd, th dnsity of powr providd by th bombardmnt of th positiv ions approachs (10 12 W/m 2 ). Th Joul ffct bcoms incrasingly significant in hating of cathod as th currnt incrass. Th contribution of th rtro-diffusd lctrons in hating th cathod is insignificant for =1V. Th situation chang for an lctronic tmpratur of =3V and th rtro-diffusd lctrons tak part in th contribution of hating th cathod. It xcds vn that of th Joul ffct as can b sn in Fig. 9.

6 6 Modl for a Low Currnt Vacuum Arc Cathod Rgion Effct of th Elctronic Tmpratur It is important to not that th ordrs of magnitud of powr dnsitis of ( à W/m 2 ) ar in good agrmnt with th rsults of Hantzsch [4, 16], Jüttnr [3] and Eckr[6] an xampl for a currnt of 25A and =3V, 6 Δ m is (1 10 kg / s). This rosion rat corrsponds to th valu of (40 μ g / C ) xprimntally obtaind and gnrally accptd by many authors (Daaldr [15], Jüttnr [2]). 10 U 12 c i, r, J (W/m 2 ) i (T=1V) J (T=1V) r (T=1V) Fig. 8 Powr dnsitis hating th cathod as function of currnt for =1V Q m, Q c, Q vap, Q ray (W/m 2 ) =3V: m, vap, cond, ray Fig. 10 Powr dnsitis cooling th cathod as function of currnt for =3V i, r, J (W/m 2 ) =3V i r Fig. 9 Powr dnsitis hating th cathod as function of currnt for =3V J =3V: m / -, c / - vap / -, r / - Th various dnsitis of powr cooling th cathod ar rprsntd on Fig. 10. It can b sn that th spot radiation can b nglctd with rspct to othr cooling phnomna of th cathod. Th cathod is coold fficintly by th mission of lctrons. Th cathod vaporization and hat conduction contribut also in th cooling th cathod. This conclusion is also sn in Fig. 11 showing th rlativ contribution of diffrnt powr dnsitis cooling th cathod as function of currnt for =3V. Figur 12 shows that th dnsity of th lctrons in th positiv column ( n p ) incrass with th currnt from to m but dcrass whn T incrass. Th ordr of magnitud of this dnsity is in good agrmnt with th rcnt xprimntal rsults of Popov [17] and thos of Jüttnr [3]. Th rosion rat ( Δ m) of th mtal from th cathod xprssd in kg pr scond incrass with th arc currnt as shown in Fig. 13. Th rosion rat dcrass whn T is nhancd from 1 to 3V. As 1E-5 Fig. 11 Rlativ contribution of diffrnt powr dnsitis cooling th cathod as function of currnt for =3V n p (m -3 ) 1.2x x x x x10 25 =1V =2V =3V 0 Fig. 12 Plasma lctron dnsity as function of currnt for diffrnt valus of

7 Acta Elctrotchnica t Informatica No. 3, Vol. 6, x x10-6 2x x10-6 Δm (Kg/s) x μg/(daaldr[15]) =1 V =2 V =3 V Fig. 13 Cathod rosion rat as function of currnt for diffrnt valus of 4. CONCLUSION In this work w stablishd a modl to calculat svral paramtrs of th vacuum arc cathod rgion at low currnt (4 to 50A). Th most important conclusions that can b drawn ar: Th tmpratur at th cathod spot is largr than th vaporation tmpratur of coppr and lis btwn 4300 and 4900 K. Th cathod tmpratur is nhancd with incrasing th arc currnt and th lctronic tmpratur. Th lctric fild at th cathod surfac lis in 9 th rang (3 5) 10 V / m and incrass with th arc currnt and th lctronic tmpratur. Th currnt dnsity at th cathod is vry high ( 5 10 to 4 10 A/ m ) and incrass with th arc currnt. Th currnt dnsity is apprciably nhancd whn th lctronic tmpratur grows from 1 to 3V. Th cathod spot radius grows slightly with arc currnt but dcrass whn th lctronic tmpratur is incrasd. Th spot radius rangs from 4 to 10 μ m. Ths valus ar in good agrmnts with th xprimntal rsults obtaind by Jüttnr [14] and Daaldr [15]. Th influnc of th lctronic tmpratur is bst sn on th currnt dnsity of th back-diffusd lctrons and consuntly th powr dnsity gaind by th cathod from th back-diffusd lctrons. As an xampl, for =1V th currnt dnsity is lowr than 10 6 A/m 2 but surpasss A/m 2 for =3V. Th currnt dnsity and th powr dnsity of th back-diffusd lctrons incras slightly with th arc currnt. Th cathod is hatd mainly by th positiv ions for low valus of th lctronic tmpratur. Hating by Joul ffct bcoms mor and mor fficint in th contribution of nrgy supply to th cathod as th arc currnt incrass. With an lctronic tmpratur of 1V, th contribution of backdiffusd lctrons rmains ngligibl. But, th situation is changd if th lctron tmpratur is raisd to 3V. Th contribution of th back-diffusd lctrons can surpass th on of th Joul ffct. Th spot radiation can b nglctd with rspct to th othr cooling phnomna of th cathod. Th mor fficint factor in cooling th cathod is th mission of lctrons spcially at high currnt. Th cathod vaporization and hat conduction contribut also in cooling th cathod. Th lctron dnsity in th cathod spot plasma is vry high ( 4 10 to 1 10 m ) which agrs with th rcnt masurmnts of Popov. Th rosion rat of th cathod matrial obtaind by th modl agrs wll with xprimntal and gnrally accptd valu of 40 μ g / C. Rsults of th modl ar in good agrmnt with rsults obtaind xprimntally by diffrnt authors. Th modl can giv many othr paramtrs (not rprsntd in this work) of th cathod rgion. Th valus of th paramtrs (cathod voltag drop, man ion charg Z i, uivalnt ionization potntial U i and th back-flow cofficint of th mtallic vapor towards th cathod α ) usd by many authors affct considrably th paramtrs of vacuum arc cathod rgion. So, xprimntal and thortical works must b undrtakn to dtrmin with mor prcision th paramtrs of th vacuum arc cathod rgion. ACKNOWLEDGMENTS This work was supportd by CNRS (Franc) and DEF (Algria) undr th projct N REFERENCES [1] Vacuié, S.: l Arc Elctriu, Enroylls, [2] Boxmann, R.L., D.M. Sandrs, J.P. Martin: Handbook of Vacuum Arc Scinc and Tchnology, Eds. Parc Ridg, N.J: Noys, [3] Jüttnr, B.: Cathod Spot of Elctric Arc, J.Phys. D: Appl. Phys, Vol. 34, 2001, pp [4] Hantzsch, E.: Cathod Spots, Thory of Cathod Spots, in Handbook of Vacuum Arc Scinc and Tchnology, Eds. Parc Ridg, N.J: Noys, [5] Coulomb, S., Munir, J.L: Importanc of High Local Cathod Spot Prssur on th Attachmnt of Thrmal Arcs on Cold Cathods, IEEE Trans on Plasma Scinc, Vol.25, N 5,1997, pp [6] Eckr, G.: Thortical Aspcts of th Vacuum Arc, Vacuum Arcs. Thory and Application, J. M. Laffrty, Ed. Nw York: Wily, 1980, pp [7] Zhou, X., Hbrlin, J.: Analysis of th Arc- Cathod Intraction of Fr Burning Arcs, Plasma Sourcs Sci. Tchnol, Vol.3, 1994, pp

8 8 Modl for a Low Currnt Vacuum Arc Cathod Rgion Effct of th Elctronic Tmpratur [8] Bolotov, A., Kozyrv, A., Korolv, Y.: A Physical modl of th Low-Currnt-Dnsity Vacuum Arc Coppr Cathods, IEEE Trans on Plasma scinc, Vol.23, N 6, 1995, pp [9] Rthfld, B., Wndlstorf, J., Klin, T., Simon, G.: A Slf-Consistnt Modl for th Cathod Fall Rgion of an Elctric Arc Calculation, J. Phys. D: Appl. Phys. 29, 1996, pp [10] Mittraur, J., Till, P.: Computr Simulation of th Dynamics of Plasma-Surfac Intraction in Vacuum Arc Cathod Spots, IEEE Trans on Plasma scinc, Vol. PS-15, N 5, 1987, pp [11] H, Z. J.: Contribution à l Étud Théoriu t Expérimntal d l Intraction Plasma Cathod dans un Arc Élctriu, Thès d Doctorat, Univ. Paris 6, [12] Murphy, E.L., Good, R.H.: Thrmionic Emission, Fild Emission and th Transitions Rgions, Phy.Rv., Vol. 102, 1956, pp [13] Andrs, A.: Ion Charg Stat Distribution of Vacuum Arc Plasmas: Th Origin of Spcis, Phys Rviw E, Vol.55, N 1,1997, pp [14] Jüttnr, B.: Invstigation of th Currnt Dnsity in th Cathod Spot of a Vacuum Arc, Contrib.Plasma Physics, Vol.25, N 5, 1985, pp [15] Daaldr, J.E.: Diamtr and Currnt Dnsity of Singl and Multipl Cathod Dischargs in Vacuum, IEEE Trans on Powr Appar. Syst PAS-93, 1974, pp [16] Hantzsch, E., Jüttnr, B.: Currnt Dnsity in Arc Spot, IEEE Trans on Plasma Scinc, Vol.13, N 5, 1985, pp [17] Popov, S.: Study of th Formation of Plasma Clustr in th Nar-Cathod Rgion of a Low- Currnt Vacuum Arc, IEEE Trans on Plasma Scinc, Vol.31, N 5, 2003, pp BIOGRAPHIES Mohammd Mssaad was born in Algria, in H rcivd th B.S in Elctrical Enginring from th Univrsity of Scinc and Tchnology of Oran (USTO), Algria, in 1992 and th M.S dgr from th sam Univrsity (USTO) in Now h is prusing th Doctorat dgr in lctrical nginring. Ahmd Wahid Blarbi was born in Algria, in H rcivd th B.S in Elctrical Enginring from th Univrsity of Scinc and Tchnology of Oran (USTO), Algria, in 1989 and th Doctorat dgr in Elctrical Enginring from th Univrsity Paul Sabatir (Toulous III) in Now h tachs at th Univrsity of Scinc and Tchnology of Oran (USTO), Algria. André Lfort was born in Franc in H rcivd th Doctorat dgr in scinc from th Univrsity Clrmont-Frrand, Franc, in H is working at th Arc Elctriu t Plasmas Thrmius Laboratory and h tachs as profssor at th Univrsity Institut of Tchnology of Clrmont-Frrand, Franc. M hammd Abbaoui was born in Morocco, in H rcivd th Doctorat dgr in scinc from th Univrsity Clrmont-Frrand, Franc, in H is working at th Arc Elctriu t Plasmas Thrmius Laboratory and h tachs as profssor at th Univrsity Institut of Tchnology of Clrmont-Frrand, Franc.