Analysis of Double and Single Sided Induction Heating Systems by Layer Theory Approach

Transcription

1 J Electomagetic Aalysis & Applicatios, 00,, doi:0436/jemaa00705 Published Olie July 00 ( 403 Aalysis of Double ad Sigle Sided Iductio eatig Systems by Laye Theoy Appoach Layth Jameel Bui Qasee Geeal & Theoetical Electical Egieeig Depatmet, Uivesity of Duisbug-Esse, Duisbug, Gemay Received Febuay th, 00; evised Apil 8 th, 00; accepted Apil 5 th, 00 ABSTRACT The iteative laye theoy appoach is applied to the aalysis of double sided ad sigle sided iductio heatig systems fo cotiuous heatig of thi metal stips The excitatio is tasvese to the diectio of stip motio ad ca be thee phase o sigle phase Nomagetic as well as feomagetic stips ae employed The impotat system paametes, amely, stip esistace, eactace, iduced powe ad electomagetic foce ae itoduced Accuacy of the method is veified with measuemet of pactical iductio heatig system togethe with compaiso to umeical ad aalytical methods Keywods: Eddy Cuets, Electomagetic Aalysis, Eegy Covesio, Iductio eatig Itoductio TREE phase iductio heatig systems such as tasvese flux iductio heatig (TFI) systems ad tavelig wave iductio heatig (TWI) systems have bee extesively studied i ecet yeas While umeical techiques ae moe popula ad paticulaly useful fo ivestigatig the iduced cuet ad powe distibutios takig ito accout logitudial ad tasvese edge effects, aalytical methods ae moe coveiet fo the itegal paametes detemiatio ad aalysis 3-D fiite elemet method (FEM) has bee employed i the aalysis of TFI systems [-8] while -D ad 3-D FEM have bee employed i the aalysis of TWI systems [9-3] Few papes elatig to aalytical methods fo the aalysis of sigle phase ad tavelig wave but cylidical iductio heatig systems have bee published [4-7] Oly a few eseaches pay attetio to this aea i the wold The TWI is ot fully appeciated with espect to thei mai advatages ad possible idustial applicatios [8] A Ali, V Bukai fom St Petesbug Electotechical Uivesity i Russia ad F Dughieo, M Foze, S Lupi, P Siega, V Nemkov fom Uivesity of Padua i Italy have obtaied sigificat achievemets i this aea I the vey ecet yeas, Takamitsu Sekie, ideo Tomita, Shuji Obata ad Yokio Saito fom Tokyo De- ki Uivesity i Japa have desiged a excellet tavelig wave iductio heatig system ad caied out expeimet [9] A aalytical method based o the decompositio of the mai magetic flux imposed by meas of a excitatio coil ito patial magetic fluxes alog diffeet egios that compise the assembly The basic cicuit paametes that featue the electic pefomace i iductio heatig devices havig a excitatio axial widig as foud i iductio motos fo geeatig otay magetic fields ae mathematically modeled [0] Mode aalytical appoaches usig tasmissio lie temiology [-3] ae cofied to lossless o low coductivity (dielectic) media whee displacemet cuets ae pomiet at micowave fequecies i the ode of hudeds of gigahetz, which is ot the case as i this appoach whee iduced powe is the majo objective of iductio heatig, moeove these methods ae pimaily applied to isotopic media while the laye theoy is applied to both isotopic ad aisotopic media, also it is ot metioed i these efeeces whethe these appoaches may be used i the case of thee phase (tavelig wave) excitatio The laye theoy appoach has bee maily used fo the aalysis of liea, tubula liea ad helical motio iductio motos as give i [4-6] TWI systems widely commo i liteatue ae of the double sided iductio heatig (DSI) system type as it employs uppe ad lowe excitatio iductos with es- Copyight 00 SciRes

2 404 Aalysis of Double ad Sigle Sided Iductio eatig Systems by Laye Theoy Appoach pect to the log ad thi cotiuously movig stip Sigle sided iductio heatig (SSI) systems ae TWI systems that employ oe iducto fo excitig the metal stip while sigle phase iductio heatig systems ae commoly kow as logitudial flux iductio heatig (LFI) systems The pimay object of this pape is to popose a geeal mathematical model fo the iductio heatig system usig the actual topology fo sigle phase ad thee phase excitatios with ay umbe of poles fo SSI ad DSI systems As a secod object, the pape employs the multi-laye appoach with the appopiate cuet sheet to calculate the flux desity compoets, iduced powe i the stip, temial impedace ad the magetic foce actig o the stip i the diectio of field tavel i the case of TWI systems Mathematical Model Thee Phase Excitatio A geeal multi-egio poblem is aalyzed Figue shows a coss-sectio of the N-egio model used i the theoy The model is take to be a set of plaa egios The cuet sheet lies betwee egios ad + x z z z z g z g z gn gn g z g z g N y σ N Nμ N N σ N- N μ N- N 3 σ 3 μ σ μ σ μ BN BN N N B B B B B J Figue Geeal model with cuet sheet at bouday z=g The cuet sheet vaies siusoidally i the y-diectio ad with time It is of ifiite extet i the x-diectio ad ifiitesimally thi i the z-diectio Regios -N ae layes of mateials whee the geeal egio has a coductivity σ ad aisotopic elative pemeability µ The aisotopy is a appoximatio made i ode to deal with slotted egios The egios ae tavelig at velocity ( s ) f elative to a statioay efeece fame whee λ is the wavelegth of applied field, f is the fequecy ad s is the slip i egio defied as f s f whee f is fequecy of the field expeieced by egio I this fame the tavelig field has a velocity f It is assumed that displacemet cuet is egligible ad magetic satuatio is eglected Maxwell s equatios fo ay egio i the model ae J () B E () t B 0 (3) E 0 (4) J E (5) B (6) The bouday coditios may be summaized as follows ) The omal compoet of the magetic flux desity B z is cotiuous acoss a bouday ) All field compoets vaish at z 3) The tagetial compoet of magetic field stegth y is cotiuous acoss a bouday, but allowace must be made fo the cuet sheet i the mae explaied i Sectio 3 Excitatio Cuet Desity It is assumed that the widig poduces pefect siusoidal tavelig wave The lie cuet desity may be epeseted as J Re Jexp[ j( t ky)] (7) whee J, ad k ae the lie cuet desity, agula fequecy ad wave legth facto espectively The lie cuet desity is give by J 6 N eff I p whee I, p, τ ad N eff ae the ms value of the phase cuet, umbe of poles, pole pitch ad effective umbe Copyight 00 SciRes

3 Aalysis of Double ad Sigle Sided Iductio eatig Systems by Laye Theoy Appoach 405 of seies tus pe phase espectively The wave legth facto is defied as k (8) Field Equatio of a Geeal Regio As a fist step i the aalysis the field compoets of a geeal egio ae deived, assumig that all fields vay as exp[ j(ωt ky)], ad omittig this facto fo simplicity easos fom all the field expessios that follow Takig oly the x- compoet fom both sides of () yields Fom (3) we have which leads to Bz z B 0 x jkb B z B y jk (9) z z Takig the z- compoet fom both sides of () yields Ex Bz (0) k Takig oly the x-compoet fom both sides of () yields B B z y Ex y z Theefoe, usig (9) ad (0) ito () yields whee B z z y α BZ α k jωμ0 μσ () () The solutio is give by Bz Acosh( αz) Csih( αz) (3) whee A ad C ae abitay costats to be detemied fom the bouday coditios Fom (3) we get α y Asih( αz) Ccosh( αz) (4) jkμ μ 0 3 Field Calculatios at the Regio Boudaies Figue shows a geeal egio of thickess S the omal compoet of magetic flux desity o the lowe bouday is B - ad the tagetial compoet of magetic field stegth is - The coespodig values o the uppe bouday ae B ad Fom (3) ad (4) Bz, Acosh( αz) Csih( αz) (5) z g z g egio f S Figue Geeal egio y, jkμ0 μ B B α Asih( α z ) Ccosh( α z ) (6) Equivalet expessios fo B z,- ad z,- ca be foud by eplacig z by z - Now fo the egios whee o N o whee B cosh( α S ) sih( α S ) B z, z, β (7) y, βsih( αs)cosh( αs) y, Bz, Bz, T y, y, β α (8) (9) jμ0 μk ece give the values of B z ad y at the lowe bouday of a egio, the values of B z ad y at the uppe bouday ae immediately obtaiable fom this simple tasfe matix elatio At the boudaies whee o excitatio cuet sheet exists, B z ad y ae cotiuous; thus fo example, if two egios ae cosideed with o cuet sheet at the commo bouday, kowig B z ad y at the lowe bouday of the fist egio, B z ad y at the uppe bouday of the secod egio ca be calculated by successive use of the udelyig two tasfe matices Cosideig the cuet sheet to be at z g, the, (0) ad y, y, J, () y, y, whee is the tagetial magetic field stegth i y, close lowe poximity to the bouday ad z, is the tagetial magetic field stegth i close uppe poximity to the bouday Give the cuet sheet excitatio at z g the oveall stuctue divides ito a uppe pat, which is modeled accodig to Copyight 00 SciRes

4 406 Aalysis of Double ad Sigle Sided Iductio eatig Systems by Laye Theoy Appoach Bz, Bz, T T T y, y, J () ad a ie pat which suppots the followig elatio Bz, Bz, T T T y, (3) y, If the top egio is ow cosideed, the as z, tah( αz) ad all field quatities ted to zeo, hece o the bouday gn the field quatities ae elated by β B (4) N N N Theefoe at ay z withi egio N the field quatities become B B exp α g z z N N N y N N N N β B exp{ α ( g z)} Cosideig the bottom egio whee =, the field quatities ae elated by βb (5) ad at ay z withi egio B B exp α z g z β B exp{ α ( z g )} y 4 Suface Impedace Calculatios The suface impedace lookig outwads at a bouday of z g is defied as Ex, Bz, Z (6) y, k y, ad the suface impedace lookig iwads is defied as Z Ex, Bz, (7) k y, y, Usig the method obtaied i [7] with the values of B z,n-, y,n-, B z,, y, ad [T ] as deived i the pevious sectio the Z Z Zi (8) Z Z whee Z i is the iput suface impedace at the cuet sheet ad Z + ad Z ae the suface impedaces lookig outwads ad iwads at the cuet sheet Substitutig fo Z ad Z + usig (6) ad (7) espectively, ad eaagig the tems yields Ex, Zi (9) y, y, Substitutig () ito (9) yields Ex, Zi (30) J Thus the iput suface impedace at the cuet sheet has bee detemied This meas that all field compoets ca be foud by makig use of this ad (7), (), (3) 5 Temial Impedace, Powe ad Tagetial Foce The temial impedace pe phase pe mete of axial legth ca be deived [7] i tems of Z i as 4N eff Zt Zi Ω/m (3) λp avig foud E x, B z ad y at all boudaies, it is the a simple matte to calculate the powe eteig a egio though the cocept of Poytig vecto The time aveage powe desity passig though a suface is give by P Re E * W/m ece the time aveage powe desity flowig upwads fom the cuet sheet at z g is give by P * i, Re Ex, y, ad the time aveage powe desity flowig dowwads fom the cuet sheet at z g is give by P * i, Re Ex, y, The et powe desity i a egio is the diffeece betwee the powe i ad powe out ω * * Pi ReBz, y, Bz, y, (3) k It follows that the tagetial foce desity F y actig o the stip is the et powe desity iduced divided by tavelig wave velocity λf F P i y N/m (33) λf 5 Sigle Phase Excitatio This is a simple poblem tha the thee phase oe ad oly egios below o above the cuet sheet (depedig whee the stip is located) eed be cosideed Othe egios do ot i ay way affect the field distibutio The excitatio is povided by a sigle coil of N tus pe mete of axial legth cayig alteatig cuet i the tasvese diectio The lie cuet desity takes the fom Copyight 00 SciRes

5 Aalysis of Double ad Sigle Sided Iductio eatig Systems by Laye Theoy Appoach 407 with J Re{ Jexp( jωt)} (34) J NI I this case k 0, all field elatios, Maxwell s equatios ad bouday coditios hold The solutio is give by Bz Acosh( αz) Csih( αz) (35) whee α jωμ0 μσ (36) If the plae z 0 passes though the cetal axis of the stip the Bb/ Acosh( αb/ ) Csih( αb / ) whee b is the thickess of the stip ad B b/ is the axial (tagetial) compoet of magetic flux desity at the uppe suface of the stip The axial compoet of magetic flux desity at the lowe suface of the stip is give by Bb/ Acosh( αb/ ) Csih( αb / ) ece we ca wite Bb/ Bb/ B0 cosh( αb / ) (37) whee B 0 is the axial compoet of magetic flux desity at the cete of the stip The iput suface impedace at the cuet sheet becomes Zi Z (38) whee Z is obtaied usig (7) The et powe desity iduced i the stip is obtaied usig the cocept of Poytig vecto ad theefoe the et powe desity i the stip is * * Pi Re{ Ex, y, Ex, y, } (39) J Re{ αtah( αb/ )} w m σ The temial impedace is give by the elatioship 3 Numeical Results α Zt tah( αb / ) (40) σ The solutio pocedue that has bee descibed i the pevious sectios is used to aalyze two examples to check validity ad accuacy Oe example is a sigle phase pactical iductio heatig system with feomagetic stip [7] The impotace of this example is that measuemet is available i additio to calculatio The othe example is based o FEM solutio fo DSI system [9] Fo compaiso easos, FEM computatio is adopted i ou aalysis which is widely used as a umeical techique fo this kid of applicatios I ou implemetatio, the field domai is divided ito a umbe of egios, each beig defied by its coodiates, pemeability ad coductivity Each egio is descitized usig fist ode tiagula elemets [8] The iduced powe i the stip is obtaied though the solutio of goveig diffeetial equatio fo each odal magetic vecto potetial Thee values of powe ae computed: the powe itegated ove the coil, the ai gap powe ad the powe itegated ove the stip The solutio is assumed to be coveget whe these thee values do ot diffe by moe tha % which is temed as the powe mismatch o powe imbalace 3 Pactical Sigle Phase Iductio eatig System Poblem data ae give i Table Results obtaied usig the laye theoy appoach ad FEM ae i good ageemet with measuemet as show i Table This ageemet is attibuted to the fact that stip thickess is vey small compaed to stip legth ad width which coicides with the assumptios made i the mathematical model Table Poblem data fo pactical iductio heatig system (based o [7], Ex 3) Stip thickess, (mm) 56 Stip width, (mm) 0 Stip legth, (mm) 70 Relative pemeability of stip 50 Mea stip coductivity, (S/m) Poductio ate, (to/h) 907 eat cycle, (sec) 75 Speed of stip, (m/s) 069 Fequecy, (z) 9600 Coil axial legth, (mm) 70 Coil width, (mm) 70 Ai gap legth, (mm) 733 Amplitude of lie cuet desity, (ka/m) 383 Table Computed paametes of the pactical sigle phase iductio heatig system Paamete Stip powe (kw) Stip esistace (Ω) Reactace (Ω) Measued value [8] Value calculated by empiical fomula [8] FEM Laye theoy value Copyight 00 SciRes

6 408 Aalysis of Double ad Sigle Sided Iductio eatig Systems by Laye Theoy Appoach Figue 3 shows the vaiatio of the axial compoet of magetic flux desity alog stip depth at mid coil axial legth fo sigle phase model usig FEM aalysis ad the laye theoy appoach The ageemet betwee the esults of both methods may be cosideed good with a maximum elative deviatio of 49% It is show i this figue that the axial compoet of magetic flux desity deceases apidly (expoetially) fom the suface of the stip fo both sides due to ski effect Obviously thee is o omal compoet fo sigle phase iductio heatig system ad this ca be deived diectly fom Maxwell s equatios 3 Sigle Sided ad Double Sided Tavelig Wave Iductio eatig Systems Refeece [9] employed a double sided iductio heatig system whose data ae give i Table 3 Axial Flux Desity (T) Cetal axial plae of stip laye theoy fiite elemet Stip Depth (mm) Figue 3 Vaiatio of axial flux desity compoet with stip depth fo sigle phase iductio heatig model Fo the sake of compaiso, the same model is adopted as a sigle sided iductio heatig system usig the same lie cuet desity by emovig oe of the iductos alog with its backig io Table 4 shows the computed paametes fo both systems usig FEM ad the laye theoy appoach Agai the esults coelate well as discussed i Subsectio 3 Figue 4 ad Figue 5 show espectively the vaiatio of omal ad tagetial (axial) flux desity compoets alog magetic gap legth The maximum deviatio betwee the esults of both methods is foud to be 5% Figue 6 ad Figue 7 show the vaiatio of omal ad tagetial (axial) flux desity compoets alog distace omal to the stip Agai both methods coelate well withi 4% I both systems the axial flux desity compoet i the ai gap is geate tha the omal compoet, this may be attibuted to the fact that the pole pitch is much geate tha the ai gap legth i both systems ad i this case these systems ae cosideed as axial flux machies It is clea fom these figues that the axial compoet of magetic flux desity is deceased withi the stip due to ski effect which is ot effectively poouced i the omal compoet to the stip Table 4 Computed paametes fo tavelig wave iductio heatig systems Paamete FEM Value Laye Theoy Value Pe phase DSI stip esistace (Ω) Pe phase DSI eactace (Ω) DSI stip powe (kw) DSI axial foce (N) Pe phase SSI stip esistace (Ω) Pe phase SSI eactace (Ω) SSI stip powe (kw) SSI axial foce (N) Table 3 Poblem data fo tavelig wave DSI ad SSI systems (based o Refeece [9]) Stip thickess, (mm) Stip width, (mm) 000 Stip legth, (mm) 960 Relative pemeability of stip Mea stip coductivity, (S/m) Axial pole pitch, (mm) 480 Slot pitch, (mm) 60 Slot width, (mm) 80 Slot depth, (mm) 40 Slots pe pole pe phase Numbe of axial poles Numbe of coductos pe slot 8 Fequecy, (z) 50 Iducto axial legth, (mm) 960 Iducto width, (mm) 000 Magetic yoke depth, (mm) 80 Ai gap legth betwee yoke & stip, (mm) 5 Amplitude of lie cuet desity, (ka/m) 00 Iput phase voltage, (V) 0 Nomal Flux Desity (mt) magetic gap laye theoy fiite elemet Nomal Distace (mm) Figue 4 Vaiatio of omal flux desity compoet alog omal distace to stip fo double sided iductio heatig system Copyight 00 SciRes

7 Aalysis of Double ad Sigle Sided Iductio eatig Systems by Laye Theoy Appoach 409 Axial Flux Desity (mt) stip thickess magetic gap laye theoy fiite elemet Nomal Distace (mm) Figue 5 Vaiatio of axial flux desity compoet alog omal distace to stip fo double sided iductio heatig system Nomal Flux Desity (mt) fiite fiite elemet elemet laye laye theoy theoy stip thickess ie yoke suface Nomal Distace (mm) Figue 6 Vaiatio of omal flux desity compoet alog omal distace to stip fo sigle sided iductio heatig system Axial Flux Desity (mt) fiite elemet laye theoy stip thickess 50 ie yoke suface Nomal Distace (mm) Figue 7 Vaiatio of axial flux desity compoet alog omal distace to stip fo sigle sided iductio heatig system 4 Coclusios The laye theoy appoach has bee used fo the aalysis of sigle sided, double sided tavelig wave ad sigle phase iductio heatig systems This method has bee applied to compute electical paametes of vaious iductio heatig systems with feomagetic ad omagetic thi stips The esults show clealy that the theoetical esults coelate well with fiite elemet method esults i additio to expeimetal oe This may be cosideed as fai justificatio to the aalysis method poposed i this pape 5 Ackowledgemets The autho is deeply idebted to Pofesso Daiel Ei, his host ad co-woke at the Uivesity of Duisbug-Esse fo advice ad ecouagemet REFERENCES [] F Dughieo, M Foza ad S Lupi, 3-D Solutio of Electomagetic ad Themal Coupled Field Poblems i the Cotiuous Tasvese Flux eatig of Metal Stips, IEEE Tasactios o Magetics, Vol 33, No, 997, pp [] V Bukai, F Dughieo, S Lupi, V Nemkov ad P Siega, 3D-FEM Simulatio of Tasvese-Flux Iductio eates, IEEE Tasactios o Magetics, Vol 3, No 3, 995, pp [3] F Dughieo, M Foza, S Lupi ad M Tasca, Numeical ad Expeimetal Aalysis of a Electo-Themal Coupled Poblem fo Tasvese Flux Iductio eatig Equipmet, IEEE Tasactios o Magetics, Vol 34, No 5, 998, pp [4] N Biachi ad F Dughieo, Optimal Desig Techiques Applied to Tasvese-Flux Iductio eatig Systems, IEEE Tasactios o Magetics, Vol 3, No 3, 995, pp [5] Z Wag, X Yag, Y Wag, ad W Ya, Eddy Cuet ad Tempeatue Field Computatio i Tasvese Flux Iductio eatig Equipmet fo Galvaizig Lie, IEEE Tasactios o Magetics, Vol 37, No 5, 00, pp [6] Z Wag, W uag, W Jia, Q Zhao, Y Wag ad W Ya, 3-D Multifields FEM Computatios of Tasvese Flux Iductio eatig fo Movig Stips, IEEE Tasactios o Magetics, Vol 35, No 3, 999, pp [7] D Schulze ad Z Wag, Developig a Uivesal TFI Equipmet Usig 3D Eddy Cuet Field Computatio, IEEE Tasactios o Magetics, Vol 3, No 3, 996, pp [8] S Galui, M Zlobia ad K Bliov, Numeical Model Appoaches fo I-Lie Stip Iductio eatig, Poceedigs of 009 IEEE EUROCON Cofeece, Sait- Petesbug, 8-3 May 009, pp Copyight 00 SciRes

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