A Tool for Load Modeling in Induction Hardening Equipment Driven by Power Semiconductor Systems

Transcription

1 1 A Tool for Load Modling in Induction Hardning Equipmnt Drivn by Powr Smiconductor Systms Robrto Suárz-Antola and Digo Suárz-Bagnasco, Mmbr, IEEE Abstract-- Klvin ffct (Skin ffct) is usd in surfac hardning producd by induction hating of gars, camforms, camshafts and othr work pics of fairly complx gomtris. Th induction hating quipmnt for surfac hardning of mtals and alloys (using LF or mdium frquncis in th jargon of induction hating) is composd by a coil or coil assmbly and a powr smiconductor driving systm up to 50kHz. Th load sn by th driving systm is quivalnt to a transformr. Th primary corrspon to th xcitation coil or coil assmbly, and th work pic corrspond to a short-circuitd scondary. To asss th lctrical load it is ncssary to dtrmin th variations in skin dpth from plac to plac du to local curvatur ffcts in th work pic, and its variations in spac and tim du to variations in conductivity and magntic proprtis coupld with thrmal ffcts. In ths and othrs tchnical applications of Klvin ffct it is oftn ncssary to b abl to rlat local skin dpths with local curvaturs of th surfac of lctrically conductiv bodis. Th purpos of this papr is twofold. First, to driv a closd form analytical formula that rlats th local skin dpth with th local man curvatur and th wll known skin dpth for a flat conductiv body. Th limits of applicability of this formula ar discussd. Th prdictd skin dpths ar compard with availabl xprimntal rsults obtaind in th framwork of surfac hardning procsss. Scond, to apply th abov mntiond formula to dscrib th lctrical load of th induction hating quipmnt in th conditions usd for surfac hardning. In th choic or dsign of an induction hating systm th paramtrs of th intndd procss (dpth of Klvin ffct, tmpraturs to b rachd and duration of th hating procss, amongst othrs) put rstrictions ovr th coils and th powr driving systm. To dtrmin th bst shap and siz of induction coils or coil assmblis, th complx thrmal and lctromagntic procsss producd in th work pics must b simulatd using a computr cod. Th possibl applications of th formula givn in this papr, during th planning of ths digital simulation is discussd. In th dsign of th lctronic powr systm, driving th coils with th work pics insid, th load impdanc must b spcifid. Th formula givn hr is applid to asss som charactristics of th load that may b of intrst in th choic or dsign of th driving systm. Indx Trms-- Induction hating, Load modling, Skin ffct. I. INTRODUCTION Induction hating is a non-contact hating mthod. An lctrically conductiv work pic (a mtal or a smiconductor) is locatd in an altrnating magntic fild of a coil or coil assmbly. By lctromagntic induction th fild producs ddy currnts in th matrial, which is hatd as a rsult of Joul ffct. In frrous mtals, hystrsis loss hating occurs, but this ffct is in gnral fairly small compard with th hating ffct of th ddy- currnts. Th inducd currnt dnsity and its hating ffcts, ar non uniform. Th currnt dnsity is maximum at th surfac of th work pic narst to th coil conductors. In th cas of a simply connctd body, th currnt dnsity is zro at its cntr. As th local dnsity of thrmal powr is proportional to th squar of th local lctric currnt dnsity, th hating ffct is also non - uniform. If th frquncy of th altrnating magntic fild is high nough, th inducd lctric currnts ar dvlopd in a thing layr adjacnt to th abov mntiond surfac of th work pic. This is th skin or Klvin ffct, which is usd in nginring manufacturing industry for th hardning of baring surfacs, for wlding, for brazing, and similar hat tratmnt procsss [1]. Th dpth of pntration of ddy currnts can b charactrizd by th so calld skin dpth. It is a function of th frquncy of th fild, as wll as, th physical proprtis and gomtry of th work pic [1]. For surfac hardning, th matrial is hatd during a fw scon. In th arlir stags of th procss, th hat is confind in a layr with th skin dpth. Th hatd ara is thn coold vry fast, so that a surfac hardning ffct is obtaind whil th bulk of th matrial rmains ductil. Th distribution of hating in th body can b controlld adjusting th frquncy of th altrnating magntic fild. Frquncis in th rang 50 Hz to 1 MHz ar usd for svral purposs, from through hating at low frquncis to surfac hating at high frquncis. In this papr, surfac hardning using powr sourcs up to 50 khz will b considrd. Ths ar th so calld mdium frquncy hating quipmnts. Thy ar basd on high powr thyristors. In som of ths quipmnts, th rctifid 50 Hz supply is choppd by th thyristors and fd into a rsonant circuit formd by th ractants of th loadd work coil (rsistiv and inductiv) togthr with tuning capacitors. Th

2 load circuit sn by th driving systm is quivalnt to a short circuitd transformr. Th primary corrspon to th xcitation coil or coil assmbly. Th work pic can b considrd as a short circuitd scondary. Th impdanc of this short circuitd transformr dpn on th gomtry and physical proprtis of th work pic, th frquncy of opration, and th gomtry and numbr of turns of th coils. In th choic or dsign of an induction hating systm to b applid in surfac hardning procsss, th paramtrs of th intndd procss (dpth of Klvin ffct, tmpraturs to b rachd and duration of th hating procss, amongst othrs) put rstrictions ovr th coils and th driving systm. In surfac hardning and othr tchnical applications of Klvin ffct, it is oftn ncssary to b abl to rlat th variations of th skin dpth from plac to plac, du to local curvatur ffcts in th hatd body, and its variations in spac and tim du to variations in conductivity and magntic proprtis coupld with thrmal ffcts. A complt dscription of th problms posd by th mthod can b found in rfrnc []. Th practical importanc of finding a rlation btwn th local skin dpth and th local curvatur of th boundary of th work pic was strssd in rfrnc [3], whr an intrsting qualitativ dscription is givn. Th purpos of this papr is two fold. First, to driv a closd form analytical formula that rlats th local skin dpth with th local man curvatur and with th wll known skin dpth for a flat conductiv body. Scond, apply th abov mntiond formula to dscrib th load impdanc that must b drivn by th induction hating quipmnt in th usual conditions for surfac hardning of gars and othr work pics of fairly complx gomtris. II. AN ANALYTICAL FORMULA THAT RELATES SKIN DEPTH WITH LOCAL MEAN CURVATURE In ordr to study th Klvin ffct in low frquncy fil, as in th cas of induction hating, displacmnt currnt may r S = E H r B S = E J H b nglctd [4]. In this cas if vctor, from Maxwll quations w obtain: is th Poynting Lt us considr th work pic locatd insid th working coil. If an altrnating magntic fild is producd in th coil, th lctromagntic fild in th air outsid th work pic but insid th coil may b dscribd by quasi - static quations. In th surfac of th work pic th lctromagntic nrgy ntrs to th matrial along th normal to th considrd point, du to th big diffrnc in phas vlocity btwn th air and th conductiv body (rfraction with absorption in th boundary) [5]. As a consqunc, th Poynting vctor in th boundary of th work pic is prpndicular to this surfac and points towar th intrior. Lt us considr now th lins of th fild S r insid th (1) work pic, and th congrunc of surfacs orthogonal to ths lins (on of ths surfacs is th boundary of th hatd S = S bing r S a unit vctor and S th norm body). But S of th Poynting vctor. On surfac blonging to th abov mntiond congrunc pass through th point in which S r is r is th normal vctor to th surfac at this locatd, and S point. According to a rsult of diffrntial gomtry and classical fild thory applicabl to a congrunc of rgular surfacs, th divrgnc of th normal vctor is twic th man curvatur of th surfac [6] [7]. Thn, = S r ds S S + S S = + S ds Hr r dirction of S, and s rprsnts th arc lngth along th corrsponding fild lin. From (1) and (), substituting J = σe and taking tim avrags it follows: = B E σ (4) = (5) ωµσ () is th dirctional drivativ of S in th ds B = S σ E H (3) whr σ is th conductivity of th matrial. If th magntic hystrsis is ngligibl, H may b put qual to zro. Thn it follows that: ( ) d ln S S Th spatial scals that charactriz th variation of th fil in a dirction tangnt to th boundary of th work pic ar of th sam ordr of magnitud of a charactristic dimnsion of th hatd body. Th spatial scal of variation of th fil along th dirction of th Poynting vctors is of th sam ordr of magnitud of th skin dpth for a flat boundary [4]: Hr ω is th angular frquncy of th altrnating magntic fild producd by th quipmnt and µ is th magntic prmability of th matrial. In th cas of surfac hardning, th skin dpth is svral ordrs of magnitud lss than a charactristic dimnsion of th hatd body. As a consqunc, in a first approximation, th lctric and magntic fil can b dscribd by th formula corrsponding to a flat boundary [4] [8]. Thn

3 3 1 E = E and S E H Hr th factor =. rprsnts th RMS valu of th magnitud, and 1 appars bcaus th lctric fild has a phas lag of π/4 rlativ to th magntic fild (nglcting magntic hystrsis). Btwn th lctric and magntic fil thr is th rlation E = ζ H (6) whr [4] [8] ωµ ζ = (7) σ From (4) to (7) it follows that: d ( ln S ) = = If th boundary is flat, 0 and from (8): s S () s = S ( 0) (9) givs th vanishing of th avrag magnitud of th Poynting vctor going towar th intrior of th body from its surfac. S is th magnitud of th avrag Poynting vctor in a ( 0) point of th surfac of th work pic and ( s) (8) S is th corrsponding magnitud at a point of th sam fild lin at a distanc s form th surfac. This rsult suggsts th introduction of an quivalnt skin dpth that corrspon to th cas of a curvd surfac: + 1 = 1 or = (10) 1+ This formula givs th rlation btwn th local skin dpth, th local man curvatur of th boundary and th skin dpth corrsponding to a flat boundary btwn th work pic and th surrounding air. As far as w know, this rlation is a nw on. III. A DESCRIPTION OF THE LOAD IMPEDANCE SEEN BY THE INDUCTION HEATING POWER SOURCE Now, lt us considr th work pic as a short-circuitd scondary of a transformr quivalnt to th loadd coil. In th conditions prvailing during surfac hardning th skin dpth is much smallr than th gomtric dimnsions of th hatd body. In this cas w can considr a bar of non-circular cross sction. Th boundary of ach cross sction is givn by th sam curv that may b dscribd in th x-z plan of an orthogonal systm of coordinats xyz. Th y axis is paralll to th axis of th bar. Lt h b th lngth of th bar. It is possibl to introduc a surfac dnsity of lctric currnt intgrating th volumtric dnsity in dpth following th lins of th Poynting vctor fild. If Ĵ s is th phasor that rprsnts th surfac dnsity, and Î is th global currnt that is inducd in th work pic, thn Iˆ = hjˆ s (11) Th lctric currnts flows paralll to th x-z plan, in ach cross sction of th bar. If Ê s is th phasor that rprsnts th lctric fild tangnt to th boundary of th hatd body (also containd in th x-z plan), th inducd lctromotiv forc is givn by: I = Eˆ s dl (1) In th cas of a half infinit mdium with a flat boundary, it is possibl to show that [9] [10]: Eˆ 1+ j = ˆ (13) σ s J s If th Klvin ffct is strong nough, and th curvatur is lss than 1/, w can suppos that (13) may b applid to th bar substituting by th local skin dpth givn by (10). Thn multiplying and dividing by h and taking into account (11) it follows: j dl Eˆ 1+ I = sdl Iˆ = h (14) σ So w obtain th following formula for th impdanc of th work pic: + j dl Zˆ 1 = (15) hσ l is th lngth of th curv, from (10) and (15) w If ( ) obtain l( ) Zˆ = ( 1+ j ) h σ (16) Hr, by dfinition = 1+ (17) This is an avrag skin dpth that corrspon to an avrag man curvatur: 1 = dl l ( ) (18)

4 4 IV. DISCUSSION AND CONCLUSIONS In ordr to apply th rsults obtaind abov for th skin dpth and for th impdanc of a work pic, w will considr first th simplst and mor common typ of gar: spur gars [11]. Spur gars ar usd to transmit rotary motions btwn paralll shafts, with gars tth that ar locatd on th circumfrnc of a circular plat. At th tth lvl th boundary is convx. But if th boundary is convx, th man curvatur is ngativ. Thn, th local skin dpth, according to formula (10) will b gratr than th skin dpth of a flat boundary. If th boundary is concav, th man curvatur will b positiv. Thn th local skin dpth will b smallr than th skin dpth of a flat boundary. Now, lt us suppos that this toothd whl is locatd insid a coil in ordr to b hatd by induction. Th inducd lctric currnt will circulat in a layr adjacnt to th surfac of th tth and th notchs btwn th tth, vanishing towar th intrior of th whl. As th local skin dpth is highr at th tth lvl than at th notchs lvl, th ral part of th local valu of th impdanc (th so calld activ rsistanc) will b lowr at th tth lvl than at th notchs lvl. Howvr, th sam lctric currnt circulats vrywhr (so that th lctric currnt dnsity at th tth lvl will b smallr than th lctric currnt dnsity at th notchs lvl). As a consqunc, th matrial adjacnt to th notchs will b hatd mor than th matrial of th tth. This fact is important from th standpoint of induction hardning. Morovr, from th rsults drivd in this papr it is possibl to calculat th distribution of hat powr in a layr adjacnt to th surfac of th work pic. This distribution may b usd as an input in th quation of hat transfr, and solving this quation th corrsponding tmpratur fild may b calculatd. If th frquncy of opration is high nough, th skin dpth for a flat boundary will b so small that will b ngligibl in comparison with 1. In that cas, th local skin dpth will b qual to and th hat powr producd by th inducd lctric currnt will b th sam in all locations of th surfac of th gar. Ths consquncs of th analytical formula ar confirmd both by xprimntal rsults and by digital simulations of th procss of surfac hardning of gars by induction hating. [] [3]. Th assumptions mad in ordr to driv (10) suggst that th formula for th local skin dpth can b applid only if th product is small nough, prhaps lss than 0.3. This guss may b substantiatd in a comparison btwn th approximat analytical formula obtaind in this papr for bars of any cross sctions and th xact rsults obtaind for th Klvin ffct in infinit circular cylindrs, in two diffrnt situations. In on of thm, an infinit cylindrical conductor carris an altrnating currnt paralll to its axis. Th currnts produc an altrnating magntic fild prpndicular to th axis of th cylindr [3] [8]. In th othr situation, an infinit cylindrical conductor is locatd in an altrnating magntic fild, uniform and paralll to its axis. Th inducd currnts ar containd in th circular cross sctions of th cylindr. This cas, studid by Förstr and othrs [1], is a simplifid mathmatical modl that may b applid in th framwork of crtain lctromagntic mtho for non dstructiv tsting of mtals. Howvr, it has a closr connction with th induction hating cas. A dtaild analysis of th rlation btwn th approximat formula obtaind hr and th consquncs of th xact solutions for fil and currnts in ths cass will b don lswhr. Hr w only not that for a cylindr th man curvatur is constant and qual to twic th rciprocal of r 0 th radius of th cylindr, so that from (10) it follows: = (19) 1+ ro Both xact solutions for th fil in th infinit cylindrs, th classical on for th Klvin ffct in an altrnating currnt conductor and th solution obtaind by Förstr for th conductor insid an infinit coil, ar constructd using modifid Bssl functions [8] [9] [1]. From an analysis of th asymptotic xprssions of this Bssl functions, formula (19) is obtaind, as will b shown in othr work. Formula (16) givs an analytical stimat of th work pic impdanc. If th lin intgral of th local man curvatur, takn ovr th curv, is ngativ (as is th cas of a prdominantly convx body) both th rsistiv and th inductiv parts of th impdanc will b lss than th corrsponding impdanc componnts for a flat body. If th lin intgral is positiv (as is th cas of a prdominantly concav body) both th rsistiv and inductiv parts will b gratr than th corrsponding impdanc componnts of a flat body. If th frquncy is high nough, ths diffrncs vanish. In ordr to calculat th impdanc sing by th lctronic driving systm, th work pic impdanc must b rflctd from th scondary to th primary circuit (th coil) of th quivalnt transformr, using a wll known procdur. End ffcts ar nglctd in this analytical approach, as wll as th variations in conductivity and magntic prmability producd by local tmpratur incras and magntic hystrsis. Hystrsis may b dscribd using a complx magntic prmability in th formula for th skin dpth [9]. On consqunc is that th activ rsistanc is now highr than th inductiv ractanc in th impdanc givn by (16).Magntic saturation may b takn into account using a suitably smallr valu for th magntic prmability in th formula for th skin dpth and for th work pic impdanc. Anyhow, to dtrmin th bst shap and siz of induction coils or coil assmblis, in ordr to hat by induction th work pics of complx gomtry as found in practic, th analytical approach is clarly insufficint. Th complx thrmal and lctromagntic procsss producd in th work pics and th intraction btwn th work pics and th coils must b simulatd using computr cods. Th usd of ths cods is not straight forward, and wrong rsults can b

5 5 obtaind from th numrical calculations. Nvrthlss, th analytical approach affor gnral rsults that may b usd as guid lins about what to xpct and about what to sarch in rlation with th rsults of th digital simulations. Also th analytical approach givs us usful lowr and uppr boun for th valus of paramtrs such as th local skin dpths or th impdancs that may b usd to slct induction hat quipmnt and to stablish som main charactristics of th hating procss. All this sms to dsrv furthr study. V. REFERENCES [1] M.Laughton and R.Warn (E) Elctrical Enginr s Rfrnc Handbook, Oxford: Nwns, 003, Chaptr 9. [] V. Rudnv. D. Lovlss, R. Cook and M. Black, Handbook of induction hating, Nw York: Marcl Dkkr, 003. [3] A. Ntushil and K. Polivanov, Principios d lctrotcnia, volumn 3 (Toría dl campo lctromagnético), Bunos Airs: Cartago, 1959, pp [4] M. Brdov, V. Rumiantsv, and I. Toptigin, Elctrodinámica clásica, Moscú: Mir, 1986, pp [5] V.V. Nikolski, Elctrodinámica y propagación d ondas d radio, Moscú: Mir, 1985, pp [6] J.Eriksn, Tnsor fil, an Appndix to C.Trusdall and R.Toupin, Classical fild thoris, Encyclopdia of Physics, Volum 3, Brlin: Springr, [7] R.Suárz Ántola, Osmosis, filtration and fractur of porous mdia, in Procdings of th XIV Congrss of Mchanical Enginring,1997 [8] L.Landau and E.Lifchitz, Elctrodinamiqu ds miliux continus,tours d Physiqu Thoriqu, Volum 8, Moscú : Mir,1969,Chaptr [9] A. Ntushil and K. Polivanov, Principios d lctrotcnia, volumn 3 (Toría dl campo lctromagnético), Bunos Airs: Cartago, 1959, pp [10] V.V. Nikolski, Elctrodinámica y propagación d ondas d radio, Moscú: Mir, 1985, pp [11] W.Stadlr, Analytical Robotics and Mchatronics, Nw York :Mc Graw-Hill,1995, pp [1]R.Hochschild, Elctromagntic mtho of tsting mtals, in Progrss in Non Dstructiv Tsting, Volum 1, London: Hywood, 1959, pp VI. BIOGRAPHIES Dr. Robrto Suárz-Antola was born in Montvido, Uruguay, in H works as a physicist and nuclar nginr and is th had of th Dpartmnt of Elctrical Enginring of th Catholic Univrsity of Uruguay whr h works as Profssor of Enginring and Applid Mathmatics. H is consultant physicist in th Dircción Nacional d Enrgía y Tcnología Nuclar of th Uruguayan Ministry of Industry, Enrgy and Mins. H shars with Dr. O. Fiandra a patnt rlativ to concav lctrods to stimulat and sns th lctrical activity of biological tissus. H lcturd in svral countris of Latin Amrica, Europ and Asia. Digo Suárz-Bagnasco was born in Montvido, Uruguay, in H is an lctronic nginr from Catholic Univrsity of Uruguay whr h works as Assistant Profssor. H works as a frlanc nginr and as tchnical advisor of Arnaldo C. Castro S.A. amongst othrs firms. H shard with J. Alvarz- Alonso and Dr. R. Suárz-Antola an honorary priz of th Uruguayan Acadmy of Enginring. H is a mmbr of IEEE sinc 004 and is a founding mmbr of IEEE-Gold Uruguay.