ANALYTICAL SOLUTION FOR EDDY CURRENT PROBLEM, USING SPACE EIGENFUNCTIONS EXPANSION
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1 ANALYTICAL SOLUTION FOR EDDY CURRENT PROBLEM, USING SPACE EIGENFUNCTIONS EXPANSION MARILENA STĂNCULESCU, MIHAI MARICARU, VALERIU ŞTEFAN-MINCULETE, STELIAN MARINESCU, IOAN FLOREA HĂNŢILĂ Key wods: Analyical mehod, Space eienfuncions expansion, Eddy cuen poblem. Time peiodic EM fields in linea conducin media ae usually deemined by usin ime Fouie seies of non-sinusoidal peiodic wavefoms. We pesen a mehod leadin o a ime peiodic soluion involvin space eienfuncion expansion. Fo ansien poblems, he mehod leads o a vey elean fom of he analyical soluion. Fo ime-peiodic poblems, if he space eienfuncions can be easily deemined, he mehod may be an ineesin alenaive o Fouie seies decomposiion, when he peiodic exciaion has a wide hamonic specum. The soluion may be obained as a poduc beween a maix dependin only by he domain popeies, and a maix descibin he coil cuen. These wo appoaches ae complemenay, he lae one povin an acceleaed conveence. An illusaive example concenin an aluminium cylindical case fo boh sinusoidal and ianula wavefom is included.. INTRODUCTION Eddy cuen poblems compuaion is pefomed in ime domain, by solvin a numbe of Helmholz poblems, a each ime sep. Fo he peiodic seady-sae case, one can use he hamonic analysis mehod fo which he elecomaneic field quaniies may be decomposed in ime Fouie seies, o he bue foce mehod fo which he asympoic soluion is bein seached in ime domain. The fis mehod has wo main disadvanaes: () i equies a hue amoun of compuaion when a lae numbe of hamonics should be eained; () he hamonics can no lone be sepaaed fo nonlinea media case. The second mehod equies a lae compuin ime if he analyzed sucue pesens a lae ime consan. Moeove, i may be insable unless one uses a sufficienly small division, fac ha leads o an inceased compuaion ime. The mahemaical modellin of he seady-sae poblems may be done by usin diffeenial equaions, o by usin ineal equaions. Finie Elemen Mehod o Finie Diffeence Mehod in space and ime ae he mos used pocedues fo space disceizaion. Poliehnica Univesiy of Buchaes, 33 Splaiul Independenţei, 64 Buchaes, Romania, mailena.sanculescu@upb.o ICPE SA, 33 Spl. Uniii, 338 Buchaes, Romania Rev. Roum. Sci. Techn. Élecoechn. e Éne., 58,, p. 3 34, Bucaes, 3
2 4 Mailena Sănculescu e al. The analyical mehods have he advanae of obainin some symbolical soluions, bu hey may be efficienly applied only fo a few foms of he compuaion domains, bounded by ohoonal coodinaes iven by quadics o ooidal coodinaes. The bounday condiions mus be associaed o hese coodinaes. The peiodic seady-sae soluion is descibed by Fouie seies of he soluions of he hamonics. The numeical compuaion of he soluion implies summin up he Fouie seies, which, fo a wide hamonics specum, may lead o many compuaions and o insabiliy in he aea of he exciaion jumps. Mihai Vasiliu povided [] he ime peiodic soluion wo-dimensional EM fields fo a ecanula ba unde a peiodic sequence of ecanula pulses. As fo he Fouie seies case, eienfuncion expansions ae ohoonal and, as a consequence, he diffeenial equaion of he elecomaneic field poblem can be sepaaed on seies componens. A few impoan advanaes of his mehod ae: () he eienfuncion expansion is fas conveen, equiin only a few ems in he seies; () i leads o a vey elean analyical soluion fo he peiodic seady-sae soluion o fo ansien soluion. The main disadvanaes of his mehod ae: () i equies he deeminaion of he eienfuncions and eienvalues of he ellipic opeao defined in he compuaion domain, wih he bounday condiions bein imposed by he poblem; () he mehod canno be applied fo compuaion domains ha conain boh conducin and insulain media; (3) he numbe of indexes fo he seies summin is iven by he space dimension of he compuaion domain, while fo mos of he analyical mehods, he numbe of indexes fo seies summin is smalle wih one uni. This pape analyses he linea poblem of induced cuens in an aluminium case by a peiodic cuen exciaion. Efficien pocedues fo eienvalues equaion compuaion ae bein poposed.. THE PRINCIPLE OF THE METHOD The elecomaneic field quasi-saionay equaions lead o he followin equaion: E + σ E =, µ () wih he bounday condiions imposed upon he anenial componen of E o E and wih he iniial condiion fo E, wih aue condiion σe =, whee σ and µ ae he conduciviy and he maneic pemeabiliy of he medium fom he compuaion domain, assumed all o be posiive and consan in ime. Equaion () can be also wien as ( ( σe ) σe) + =. One can show σ µ σ
3 3 Eddy cuen poblem soluion, usin space eienfuncions 5 [] ha, fo null bounday condiions, he mappin ( ) is σ µ σ posiive and symmeical. One admis ha is even posiively definie. Then, i exis he eienfuncions Ψ and he eienvalues λ, ha veify he equaion: Ψ = λ σ µ σ Ψ and he aue condiion σψ =. The eienfuncions have zeo bounday condiions, ae ohoonal and hey can be nomalized, he inne poduc bein: Ψ, Ψ > = Ψ Ψ i d Ω. The soluion decomposed in eienfucion expansion is: < i Ω σ E(, ) = E (, ) + C ( ) Ψ ( ), () f whee E f veifies he bounday condiions. By inoducin () in () and ain he inne poduc wih Ψ, esul he odinay diffeenial equaions of C () componens: whee d d C + C F = ( ) f Ψ E f + σ µ σ λ = F, (3) E,. Equaion (3) has he soluion λ ( τ) λ dτ + C ()e C ( ) = F ( τ)e. The iniial values of C componens esul fom elaion (): () Ψ, σ E(,) E f (,). If one seaches fo he peiodic soluion, hen he condiion C (T ) = C () is imposed, whee T is he peiod and i esuls: C = ( ) T λ C () = ( F ( )e (T τ) T τ dτ) /( e λ ). The above pesened ae also valid even if one uses he modified veco poenial: A = Ed, wih A ( ) = B(). The soluion obained by decomposin in eienfuncions seies has an elean fom, bu he eienfuncions and eienvalues poblem compuaion is a vey complicaed one. In addiion, he index m passes houh all he naual coodinaes fom N, whee m is he numbe of dimensions of he space ha belons o he compuaion domain. This numbe is ipled when he eienfuncions ae vecos. Fo he case in which he compuaion domain conains also insulain sub-domains, is pefeable ha one uses he
4 6 Mailena Sănculescu e al. 4 maneic veco poenial, his havin diffeen equaions fo conducin and isolain media. Even if he opeao defined on sub-domains is posiive and symmeical, he deeminaion of he eienfuncions and eienvalues is a vey complex poblem, which can be compued only numeically. The eienfuncion expansion decomposiion mehod canno be applied. Howeve, fo simple cases, he eienvalues and eienfuncions may be easily deemined. I is he case of some D poblems. Below we analyse such an example. 3. EDDY CURRENTS FROM THE CYLINDRICAL CASE A coil wih inne adius c, havin he cuen i, suounds concenically he case ha will be heaed usin eddy cuens (see Fi. ). Because he impoan lenh has axial diecion, one adops a D model. The elecomaneic field quaniies depend only on he adial coodinae ρ. The case hicness is, he inne diamee is a, he conduciviy is σ and he maneic pemeabiliy µ is consan. The case oue diamee is b = a +. Coil conducin media a ϕ ρ i c Fi. Compuaion domain. The maneic field inensiy H is axially oiened, on oz axis diecion, H = H z = H ( ρ, ), whee is he oz axis uni veco, and he elecic field inensiy E and he cuen densiy J ae oiened on he diecion of ϕ coodinae, E u ϕ E = u E( ρ, ). Usin pola coodinaes and he maneic veco poenial = ϕ ϕ ( B = A ), we have:
5 5 Eddy cuen poblem soluion, usin space eienfuncions 7 ( ρa) = µ H. ρ ρ (4) The maneic field senh is consan in space wih ai. So, on he oue suface of he case we have he followin bounday condiion: H b = in / l, whee N is he numbe of uns of he coil and l is he lenh of he coil. Tain ino accoun elaionship (4), i esuls he bounday condiion fo ρ = b : ( ρa) ρ b ρ= b ρ= N = µ i. (5) l The line ineal of he veco poenial on he inne bounday is equal o he maneic flux in he ineio of he case: πaa = πa µ H and, ain ino accoun he elaionship (4), i esuls he bounday condiion fo ρ = a : whee ( ρa) ρ ρ= a = µ µ is he case elaive maneic pemeabiliy. A ρ= a, (6) 4. THE USE OF THE CARTESIAN COORDINATES If he hicness of he case is much smalle han he inne adius, hen one may simplify he poblem by adopin Caesian coodinaes. The maneic field senh H is axially oiened, on oz axis diecion, H = H z = H ( x, ) and he maneic veco poenial A and he cuen densiy J ae oiened on he diecion of y coodinae, A = ja y = ja( x, ). We can admi ha he oiin moves alon he case inne bounday, so fo ρ = a, we have x =, and fo ρ = b, we have x =. The equaion of he maneic veco poenial is: A A = µσ, (7) x and he bounday condiions (5) and (6) becomes: and A x x= µn = i l = x = µ x x= (8) a A A. (9)
6 8 Mailena Sănculescu e al. 6 d ( ) I can be poved ha he mappin is posiively defined and symmeical on he dx Hilbe space L (, ) of he zeo bounday condiions funcions, whee he inne poduc is < U, V > = Equaion () has he soluions: D" = α D', whee So (main D =D), UVd x. The eienfuncions d Φ dx Φ = λ = D' sin λ α aλ = µ Φ and eienvalues Φ λ equaion is:. () x + D" cosλ x. Fom (9) we have. () Φ = D(sinλ x + α cosλ x). () Fom he homoenous Neumann condiion () we have: c λ = α. (3) Tain ino accoun (8), he nom of he funcion Φ is: Φ = D a = D (sin λ x + α cosλ x) dx = ( α + ) + =, fom whee i esuls µ D and he expession of he eienfuncions: whee Z = + α a + µ Φ. sin λ x + α = Z cosλ x, (4) 5. SOLVING OF EIGENVALUES EQUATION Equaion (3) may be eplaced by soluion fo each aumen λ π/ * N λ π aλ =, ha has only one µ ( ) π π/,( ) π, belonin ineval ( ). Denoin by z = λ π/ ( ) π, equaion
7 7 Eddy cuen poblem soluion, usin space eienfuncions 9 [( π/ + ( ) π + z )/ ] ( a / ) z = µ * has only one soluion z ( π/,) fo each N. Then, insead his, i is moe useful o use he followin equaion: π + ( ) π + z ( ) ac a f z = z + =. (5) µ df The funcion f has he popey: aµ (, + ), =. dz ( ) ( ) µ + a( ) π We can apply he Picad-Banach aloihm (fo ex. [3, 4]) fo solvin equaion (5), seachin he fixed poin of he funcion ( z ) = z γ f ( z), whee γ is chosen such ha o be a conacion. The opimum value fo γ is γ = /( + ), obainin he conacion faco: θ = /( + ). If we sop he ieaions of he Picad-Banach seies a ieaion n, he eo wih espec o he exac soluion may be evaluaed usin he elaionship: z ( n ) z < θ θ ( n) ( n ) z z. We can obseve ha he conacion faco deceases wih he ode of he eienvalues. Fo > 3 hee ae sufficien a mos 4 ieaions in ode o obain an almos zeo eo ( < ). 6. ALUMINIUM CASE ELECTROMAGNETIC FIELD EQUATION COMPUTATION A. Buildin he funcion ha conains he bounday condiions. One wies: A ( x, ) A f ( x, ) + C ( ) Φ ( x). (6) = A simple bounday funcion A f may be: A f = qx + p. Tain ino accoun he bounday condiions (8) and (9), we have: A f ( x, ) = q ( ) ( x + a / µ ), (7) whee q () = ( µ N / l) i( ). (8) B. Deeminaion of C coefficiens. One eplaces (6) in equaion (7) and, ain ino accoun () and (6), i esuls
8 3 Mailena Sănculescu e al. 8 dq a dc λc ( ) Φ ( x) = µσ x + + Φ ( x). d µ d Tain he inne poduc wih Φ and ain ino accoun ha he eienfuncions ae oho-nomalized, esuls: dc β C () + dq a dq = x Φdx d d + = W, (9) µ d a + sin λ λ a λ whee β = and W = x Φ dx µσ + = µ. Equaion µ Z (9) has he soluion: β dq β ( τ) C ( ) = C () e W e dτ. () dτ If one wans o deemine he ansien soluion, hen he iniial value of he veco poenial mus be iven: A (x,). We wie he elaionship (6) fo = and, ain he inne poduc wih Φ, i esuls: a C ( ) = A( x,) p() x + Φ dx = A q ()W, () µ whee A = A( x,) Φ dx. If we wan o obain he peiodic soluion of he equaion (9), hen, imposin he peiodiciy condiion C (T ) = C () in (), we T have C () = dq β ( T τ) W τ βt e d. By eplacin in () i esuls C () e dτ T µ N = β di β ( T τ) di β ( τ) τ + τ W U e l e d τ e d, whee U =. β T d dτ e 7. EXAMPLES We choose: a =.565m, =.4m, l =.m, ρ =.6-8 Ω m, µ =, N = 4, fo an aluminium case. Fo each, we deemine he fixed poin of funcion. Fo > 3, he eienvalues aleady pesen he asympoic
9 9 Eddy cuen poblem soluion, usin space eienfuncions 3 behaviou λ π. Then one immediaely obains he coespondin values fo α, β, Z, W. All hese values depend only on he eomey and no on he sinal ype (exciaion cuen). Havin a able of values ( n, i n), n =,, M+, (, i ) = (, i ( )), ( M +, im + ) = ( T, i()) = ( T, i ), we conside a piecewise linea funcion o descibe he in+ in ime behaviou of he exciaion cuen.: i = in + ( n). We have: n + n d β ( τ ) i e dτ Γ (T ) = T / dτ β e i β β n+ in n+ n = ( ) β di β ( T ) e d d choose a consan ime sep β whee ϕ ( ) e e, n+ n+ n n+ M in+ in β ( T n+ ) β ( T n ) = ( e e ) n+ n, n =,, M fom whee. Fo simpliciy, we β n=, so n Γ (T ) = Γ ( M ) = ϕ n = ( n ) and we have: M n= β + ( M n) n in) e ( i, () = e / β. If is in he middle of he ineval, ], we have: Γ = Γ ( m / ) ) () β ( τ) ( = i ( τ)e dτ = m + β + ( m n) n in) e [ m m+ = ψ ( im+ im) ϕ ( i, m =,,, M, n= β whee he ih membe sum is zeo fo m = and ψ = e / β. To educe he compuaions, he sum m Θ, m = ϕ n= β + ( m n) n in) e (3) ( i can be ecusively β β compued: Θ, =, Θ, m = ϕ ( i m im ) e + e Θ, m, m =,, M. The value of C () coefficien in he middle of he ineval, ] is heefoe: µ N l Fo an abiay coodinae x, we have: C A [ m m+ ( ) β ( m / ) (( m / ) ) = W U e Γ ( M ) + Γ (( m / ) ) ( x ( m / ) ), = N i + i a x + µ µ m+ m + ( m / ) ) l. (4) C ( Φ ( x). (5)
10 3 Mailena Sănculescu e al. A. Sinusoidal exciaion cuen. In ode o veify he esuls, we supposed an sinusoidal exciaion cuen i( ) = sin( ω), wih he fequency f = Hz. In his case, he soluion obained fom complex imae of equaion (7) may be immediaely obained and coincides wih he soluion obained by he mehod descibed in his pape. Only a few odes of he eienvalues and eienfuncions ae enouh in ode o obain a hih accuacy soluion (Fi., fo x = 3.99 mm)..e-5 A (T.m) complex.e-5 Kmax=3 Kmax=6.E E-5 i (A ) E-5 (ms) Fi. Influence of he maximum consideed eienvalues numbe K max on he maneic veco poenial alon fo sinusoidal exciaion and compaison wih he complex imae equaion soluion. - (ms) Fi. 3 Tianula peiodic cuen. B. The ianula cuen. Fo ianula peiodic cuen descibed in Fi. 3, he compuaion by complex imaes implies Fouie analysis. The mehod poposed in his pape, eeps he values fo λ, α, β, Z, W, U, ϕ, ψ, ecompuin only Γ, he only expession dependin on he exciaion cuen. The maneic veco poenial is iven in Fi. 4. A (T.m).E-5.E-5.E E-5 -.E-5 (ms) x= mm x= mm x= mm x=3 mm x=4 mm Fi. 4 Maneic veco poenial alon fo he cuen iven in Fi.. A (T.m).E-5.E-5 8.E-6 4.E-6.E+ x= x=mm x= mm x=3 mm x=4 mm (ms) Fi. 5 Vaiaion wih ime of he maneic veco poenial alon. C. Tansien sae. We suppose a ianula cuen impulse: i( ) =, fo 5 7 [, ) ; i( ) =, fo [ 5, 5 5 ) s ; i ( ) =, fo s. Suppose ha he iniial condiion is zeo: A (x,) =. Fom (8) and () i esuls C () and fom (6 8) i esuls: 7
11 Eddy cuen poblem soluion, usin space eienfuncions 33 µ N a A ( x, ) = di β ( τ) i( ) x + Φ τ µ W l e d. (6) dτ The value of he veco poenial fo any coodinae x, in he middle of he ineval [ m, m+ ] is iven also by he elaionship (5) bu, fo he ansien seady-sae soluion, we have: µ N C (( m / ) ) = WΓ (( m / ) ). (7) l In Fi. 5 he ime behaviou of he maneic veco poenial in diffeen poins of he case is pesened. I is emaable he simpliciy of he soluion (6) in compaison wih ha obained fom soluion of Laplace imae of equaion (7). 8. CONCLUSIONS The advanaes of his mehod, compaed o he nown analyical mehods, ae: only a few ems in he eienfuncion expansion (a mos 6) ae needed; he eienvalues ae vey quicly deemined (fo ode bie han 4, only 3 ieaions ae enouh o obain a soluion wih almos zeo eo); he efficiency of he mehods does no depend on he exciaion shape (fo Fouie analysis, a lae numbe of hamonics ae needed if he exciaion has jumps). The elaionships (), (3), (4) and (5) undeline a vey ineesin advanae of he mehod: fo iven eomey and fequency, one can choose a ime sep and one can compue a few vecos which do no depend on he exciaion and which can be used o analyze any shape of cuen. The ansien soluion is specaculaly efficien deemined in compaison wih he case when one uses Laplace ansfom and i easily acceps any exciaion shape. Unfounaely, he mehod can be applied fo simple domains, whee he eienvalues and he eienfuncions ae easy o be deemined. Fo he aluminium case, a vey efficien Picad-Banach echnique, fo solvin eienvalues poblem has been poposed. ACKNOWLEDGMENTS This wo was suppoed in pa by he Secoal Opeaional Poamme Human Resouces Developmen 7 3 of he Romanian Minisy of Labou, Family and Social Poecion houh he Financial Aeemen POSDRU/89/.5/S/6557. Received on Novembe,
12 34 Mailena Sănculescu e al. REFERENCES. M. Vasiliu, Nonsinusoidal wo-dimensional eddy cuens in a ecanula ba, Rev. Roum. Sci. Techn. Élecoechn. e Éne., 36, 3, pp , 99.. F.I. Hănţilă, N. Vasile, B. Cânanu, I. Gheoma, T. Leuca, M. Silahi, Elemene de cicui cu efec de câmp, Ediua ICPE, F.I. Hănţilă, Livia Bandici, T. Leuca, Tehnici infomaice în inineia elecică, Edi. Univesiăţii din Oadea,. 4. F.I. Hănţilă, F. Consaninescu, A.G. Gheohe, Miuna Niţescu, M. Maicau, A new aloihm fo fequency domain analysis of nonlinea cicui, Rev. Roum. Sci. Techn. Élecoechn. e Éne., 54,, pp , 9.
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