GENERALIZED MATHEMATICAL MODELS OF THE ELECTROMAGNETIC FIELD IN NONLINEAR MEDIA FOR THE STATIONAR, QUASISTATIONARY AND TIME VARIABLE REGIMES

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1 U.P.B. Sci. Bull., Series C, Vol. 73, Iss. 3, 011 ISSN x GENERALIZED MATHEMATICAL MODELS OF THE ELECTROMAGNETIC FIELD IN NONLINEAR MEDIA FOR THE STATIONAR, QUASISTATIONARY AND TIME VARIABLE REGIMES Gabriel BARBU 1 Î lucrare sut prezetate şi aalizate modele matematice geeralizate ale câmpului electromagetic î diverse regimuri. Petru aceste modele matematice sut formulate teoreme de existeta si uicitate. Ecuaţiile de câmp sut scrise ca distribuţie. Cosiderăm că relaţia costitutivă poate fi eliiară. This paper presets ad aalyzes geeralized mathematical models of the electromagetic field i differet regimes. Existece ad uiqueess theorems are formulated for these mathematical models. Field equatios are writte as a distributio. We cosider that the costitutive relatioship may be oliear. Keywords: existece ad uiqueess theorems, statioary electromagetic field, periodic quasistatioary regime, variable regime 1. Itroductio The correct formulatio of problems of the electromagetic field aalysis ivolves, i additio to defiig the basic pheomeology of the problem, establishig a pheomeological model which relates to the field system, idetifyig sources of field ad models of the material, obtaiig a mathematical model of electromagetic field, which plays a importat role i this issue [1]. The mathematical model cosists of a operatioal equatio; the ukow (primary size is a state of the field size or other associated scalar or vector. I additio to the above, ad depedig o the type of operatioal equatio satisfied by the primary size, supplemetary coditios ecessary for the uiqueess of the solutio equatio may be eumerated. For the correctess of the mathematical model, there must be demostrated existece ad uiqueess theorems ad stability of the solutio, which should esure the arrowig of the solutio deviatios i case of data corruptio. 1 PhD studet, Teacher, Idustrial High School Timpuri Noi, Bucharest, Romaia, gaby.barbu@gmail.com

2 4 Gabriel Barbu For simplicity of presetatio, we cosider the boudary coditios zero ad i additio ito the steady-state coditios, we cosider the curret desity ad charge zero. Uiqueess theorems have bee proposed ad demostrated log before, by Răduleţ R., Al. Timoti, ad A. Ţugulea [], [3]. However these theorems imposed the restrictio accordig to which the dimesios of the electromagetic field belog to C 1 class o subdomais, while at their borders coditios of trasitio are established. Obviously, imposig coditios of trasitio requires, i tur, restrictios o surfaces. I case of oliear media, must be imposed restrictios that all costitutive relatios must be C 1 class. From mathematical poit of view, the legitimate questio arises, whether it is possible for the aforemetioed restrictios ot to allow ay solutios at all. So it ca be expected that expadig the class of the fuctios i which vector fields are defied, may prove useful. We propose for the geeralized magetic field to be a ordered pair (B, H S, where S = L ( Ω L ( Ω. Obviously, at this poit the risk of the uiqueess of solutio gettig lost might occur. I this case, the derivatives must be treated i a geeral way: ( divz,ϕ ( rotz,ψ =, gradϕ = Z rotψ Z, ( K,, ( K ϕ (1 Ψ ( where rot ad div take the meaigs of operators. K ad K are spaces of the scalar fuctios (vectors idefiitely derivable ad havig compact support i Ω. Equatios of statioary regime of the magetic field (assumig we also have a magetic charge ρ are: div B = ρ, rot H = J, B = F (H L is a oliear relatio of material. Relatios (1, ( are local forms ( differetials of the equatios of the magetic field. where ρ ad J are distributios, ad F : ( Ω L ( Ω 1.1. Notatios ad defiitios Let be the domai Ω R 3, bouded by the closed area Ω ad let be the iterval [0,T] R. 0,T 0,t 0,T. We ote Π = Ω ] [ ad Π t = Ω ] [, with t ] [

3 Geeralized mathematical models of the electromagetic field i oliear media 5 x : L the Hilbert space of the fuctio 3 Ω E, with ier product x, y x y dv L = ; we cosider C k (Ω the ( Ω Let E 3 be the Euclidea space, ad ( Ω space of the cotiuous fuctio x : Ω E, havig k-cotiuous Frechet differetiable o Ω, 0 k (so they have cotiuous partial derivatives up to ordial k; K(Ω, the space of the fuctio belogig to C (Ω ad havig compact support Ω. The spaces C k (Ω ad K(Ω are dese i L ( Ω. We defie similarly the spaces L ( Π, C k (Π, K(Π. We use the otatios: Ω 3 ( Π k C ( Π x t = x t, x dτ = x L is the space of the fuctios etc. We deote u ( Π Obviously, u L ( Π ( Π 0 x, where x L ( Π ad similarly u L (, L the completed space C 1 (Π, i the average uiform topology. L. Theorem 1: The operator ( dτ : L ( Π u ( Π t 0 L is cotiuous. If E 3 is replaced by R, we defie similarly the space L ( Ω, K(Ω, L ( Π, etc. Where the fuctio defied o Ω (or Π are values i R. Let H a Hilbert space. We ote a material relatioship as a cotiuous fuctio if: ad ˆF : H H. The relatioship Fˆ is lipschitzia ad uiformly mootoe Fˆ ( x Fˆ ( y sup x,y H = Θ < (3 x y Fˆ ( x Fˆ ( y, x y if x, y H = θ >0 (4 x y We say that Fˆ U(H.

4 6 Gabriel Barbu Basically, the applicatio Fˆ U (L (Ω or ( Fˆ U (L (Π, ca be obtaied from a local material relatioship F: Ω E 3 E 3 ( sau F: Ω 0, T E 3 E 3.. The geeralized mathematical model of the statioary field [4] [5] We call a geeralized statioary magetic field the ordered pair [s] = (B, H, where B,H L (Ω. We deote S F the vector space of pairs [s]. We say that a series ([ s ] 1 is coverget if ( B 1 ad ( H 1 are coverget i L (Ω. Let be S h Ω cosistig of a fiite umber (k of disjoit surfaces (S h = k S hi ad Shi Shj = 0 for i j. i=1 We deote S F F the set of pairs [s] with B,H C 1 (Ω, where [3]: ( H(P = 0, for P S h (5 B(P = 0, for P S b = Ω\S h (6 S hi B(P da = 0, for i = 1,,, k-1 (7 div B = 0, rot H = 0 (8 Let be H,B C ( Ω with properties rot H=0, div B=0 ad ull coditios o the border. The fields B,H may be ozero because the costitutive relatio may be oliear ad F(H 0. We say (B,H S F F. Let be SF F the completed space i S F F topology. SF F is a subspace of S F. Equatios (1, ( are local forms of the statioary regime equatios. It should also be iterestig to fid a itegral formulatio for the equatios of the quasistatioary system. Affiliatio to the L (Ω spaces implies the existece of Lebesque sets of ull measure o which the vector fields may ot be defied. These sets ca be surfaces or curves. For this reaso, the formulatio of the itegral forms must resort to Fubii theorem. ( 0 R Let there be,u [, 1] S u a family of surfaces, who ca be represeted by the family of sheets r = r ( u,ξ,η, where r is the positio vector ad r : [0,1] 3 E 3 is C 1 class. The family of surfaces has the property (PS: c > 0

5 Geeralized mathematical models of the electromagetic field i oliear media 7 r so that u [0,1] ad P S u,, c, where is the ormal to surface u S u. So the family of surfaces fills a ω domai. Let be B L (Ω. For almost ay u [0, 1] there are the Φ = B da fluxes (i the Lebesque s sese. Let be a series ( B the average fluxes Φ ~ 1 = du B da, will coverge for Φ ~. 0 S u 1 who coverges to B, the The itegral form of the magetic flux law: Let there be (,u [ 0, 1] σ u a family of closed surfaces with (PS property ad [s] SF F. The Φ ~ = 0 ad for almost ay u [0,1], Φ σ u = 0. The itegral form of the div B = 0 relatio for [s] SF F field: Let be,m 0, 1 R a family of curves which ca be represeted by a family of ( C M [ ] roads r = r ( u,ξ,η, ξ ad η beig the coordiates of poit M ad r : [0,1] 3 E 3 is C 1 class. The family of curves has the (PC property: c > 0 so that M ad u r r r r c (9 ξ η u u Let be H L (Ω [4]; for almost ay M [0,1] there is the circulatio (i the Lebesque s sese = Hdr H 1 L (Ω who U CM. Let be a series ( CM coverges to H, the average circulatios ~ U = [ 0, 1] da M C M ~ H dr coverge for U. ~ ~ U U ν( ω H H L ( ω (10 c where v(ω is the volume of the domai ω. The itegral form of the rot H = 0 relatio for [s] SF F field: Let be ( γ,m [ 0, 1] M a family of closed curves with (PC property ad [s] SF F. The V γ M = 0 ad for almost ay M [0,1], U γ M = 0. C M 1 Theorem (uiqueess ad existece: Let be U U(L (Ω; there is oe ad oly oe field [s] SF F where H = U (B. S u CM

6 8 Gabriel Barbu 3. Geeralized mathematical model for quasistatioary electromagetic field [6] Let be QF the vectorial space of the ordered quartets [q]=(e, b, h, j C QF where: rot e = - b, rot h = j, div b = 0 (11 j 1 i L (, ad We say that ([ q ] 1 is a Cauchy series, if ( e 1, ( ( b 1, ( h 1 are Cauchy ito ul (. Let be QF the completed QF space of this topology. Let be [q] QF We ame periodic quasistatioary electromagetic field ay [q] QF elemet. The local equatios of the quasistatioary electromagetic field [3] Let there be [q] QF ; φ, ψ K( ad χ K( ; the followig relatios are valid: e,rotϕ L ( = b,ϕ L ( - the iductio law (1 h,rotψ L ( = j,ψ L ( - the magetic circuit law (13 b,gradχ L ( = 0 - the magetic flux law (14 Let be S e Ω ad S h Ω / S e.we deote QF F the set of fields [q] = QF wherefore: ( e( P = 0 for P e = S e [ 0,T ] (15 ( h( P = 0 for P h = S h [ 0,T ] (16 b t=0 = 0 (17 Let QF be the completed space of QF F F i QF topology QF. QF is a QF F subspace. Theorem 3 : Let be A, B two positive ad liiar operators ad A is hermitia. If [q] QF F fulfils the relatio e = A j ad b = B h, the [q] = 0.

7 Geeralized mathematical models of the electromagetic field i oliear media 9 Corollary: Let be s L ( ad, i ul ( A, B two positive operators ad A hermitia; there is at most [q] ad b = B h + i. QF field where e = A j + s F Theorem 4 (of existece i QF F We presume fulfilled the followig coditio: ϕ K(, the equatio M x = rotrot x+ x = ϕ (18 has a solutio x C so that ( x(p = 0, for P e (19 ( rot x ( P = 0, for P h (0 Equatio (18 has uique solutio because: x,m 1 x L = rot x( t (Π t therefore defied positive. L ( Ω + x ad the operator M is L ( Π t Theorem 5: Let be s K (Π ad i L ( Π QF F, with e, j u L ( Π ad b, h ( Π. There is a oly oe field [q] L, where e = ρ j + s, b = μ h + i ad ρ, μ > 0. The fuctio : L ( Π L ( Π Ψˆ ( i = ( b is a cotractio. = h+ i b μ (ad Ψˆ defied by 4. Geeralized mathematical model of the electromagetic field i variable regime [7] Let GF be the vectorial space of the ordered quitets [g] = (e, j, d, b, h, with e, j, d, b, h C (Π, where rot e = - b, rot h = j + d, div b = 0 (1 We say that ([ g ] 1 is a Cauchy series if ( e 1, ( d 1, ( ( h 1, are Cauchy i ul (Π ad ( j 1 is Cauchy i L (Π. Let be GF the completed space of the GF i this topology. b 1,

8 30 Gabriel Barbu We call geeralized electromagetic field ay elemet [g] GF. Geeralized local forms of the electromagetic field equatios Let be [g] = (e, j, d, b, h GF ; ϕ, ψ K(Π ad χ K(Π the followig relatioships are valid: e,rotϕ L (Π = b,ϕ L (Π - the iductio law ( h,rotψ L (Π = j,ψ L (Π - d,ψ L (Π the magetic circuit law (3 Let be S e for which: b,gradχ L (Π = 0 - the magetic flux law (4 Ω ad S h Ω e( P = 0, for P e = S e [ 0,T ] h( P = 0, for P h = S h [ 0,T ] / S e. We deote GF F the set of fields [g] = GF (5 (6 b t = 0 = 0 (7 d t = 0 = 0 (8 We deote GF F the completed space of the GF F i this topology GF. GF F is a GF subspace. with Let be B, C T(L (Ω. We presume the followig coditio is met: ϕ K(Π ad σ > 0, ( x C (Π the solutio of equatio: Nˆ x = σ x + ( = 0 rot B rot x + C x = φ (9 x P, for P e (30 ( = 0 rotx P, for P h (31 x t = 0 = 0 (3

9 Geeralized mathematical models of the electromagetic field i oliear media 31 I the previous coditio we cosidered that x C (Π, which implies restrictios o operators B, C (these restrictios could be partially avoided, cosiderig that x is a elemet of a Sobolev-type space of orm: 1 + rot x + x L ( Π L ( Π L ( Π C B x. (33 The N operator is defied positive ad the equatio solutio (9 is uique. Let W (Π be the Sobolev space of orm [7] 1 ( x x (34 + rot L ( Π L ( Π Π The whatever s L (Π, p L (, m W (Π, there is a field [g] GF F, with e = ρ j + s, h = B b m, d = C e + p. Π Theorem 6.(of existece: Let there be p L (, m W (Π, C, B T(L (Ω ad U, which for ay t applies Lipschitzia L (Π t i L (Π t ; additioally U is causal, amely, for ay t, y belogs i [0, t], it is determied oly by x i the [0, t] iterval (through y = U (x. There is [g] GF F with j = U (e, h = B b m, d = C e + p. 6. Coclusios For the fairess of the mathematical model, there must be demostrated theorems of the existece, uiqueess ad stability of the solutio, which should esure the arrowig of the solutio deviatios i case of data corruptio. Takig ito cosideratio the local forms of the equatios of the magetic field, is iterestig to fid itegral formulatios for them. Affiliatio to the L (Ω spaces implies the existece of Lebesque sets of ull measure o which the vector fields may ot be defied. These sets ca be surfaces or curves. For this reaso, the formulatio of the itegral forms must resort to Fubii theorem. The coditios for the solutio of the electromagetic field i order to esure its uiqueess may be too restrictive ad the existece of the solutio

10 3 Gabriel Barbu might be usafe, especially whe the costitutive relatio is oliear. Because of these restrictios, the compoets of the electromagetic field are described by fuctios with cotiuous derivatives o subdomais so that local forms may be writte for the field equatios. I order there should be a solutio which preserves, at the same time, its uiqueess, lighter restrictios will therefore be adopted for the solutio of the field L (Ω. R E F E R E N C E S [1] C. Mocau, Teoria câmpului electromagetic, Editura Didactică şi Pedagogică, Bucureşti, 1991 («The theory of the electromagetic field» i Romaia [] A. Tugulea, Al. Timoti, Codiţii de uicitate î determiarea câmpului electrostatic şi magetic cvasistaţioar î materiale eliiare cu polarizare reversibilă şi magetizare reversibilă, St. Cerc. Eerg. Elth., r. 3, 1965, p («Coditios of uiqueess i determiig the electrostatic ad quasistatioary magetic field i oliear media with reversible polarizatio ad reversible magetizatio» - i Romaia [3] R. Răduleţ., A. Ţugulea, Al.Timoti, Teoreme de uicitate petru regimuri variabile ale câmpului electromagetic, St. Cerc. Eerg. Elth., r. 1, 1971, p («Theorems of uiqueess for variable regimes of the electromagetic field» i Romaia [4] F. Hăţilă, Mathematical models o the relatio betwee B ad H for o/liear media, Revue Roumaie des Scieces Techiques - Serie Électrotechique et Éergétique, r. 3, 1974, p [5] F. Hăţilă, O the uiqueess theorems of the statioary ad quasistatioary electromagetic fields i oliear media, Revue Roumaie cieces Techiques - Serie Électrotechique et Éergétique, o., 1975, p [6] F. Hăţilă, Cotribuţii asupra teoriei maşiilor de curet cotiuu cu mageţi permaeţi, Teză de doctorat, Bucureşti, 1976 («Cotributios regardig the CC electrical machies with permaet magets theory», PhD Thesis i Romaia [7] S.L. Sobolev, Ecuaţiile fizicii matematice, Editura Tehică, Bucureşti, 1955 («Mathematical physics equatios» i Romaia.

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