We present the strong finite element method formulation for 2D closed electrostatic problems in anisotropic homogeneous dielectric. Hierarchical higher (arbitrary) order strong basis functions are applied. Results for effective permittivity and characteristic impedance of shielded striplines are given and compared with results obtained by a commercial software. © 2012 Wiley Periodicals, Inc. Microwave Opt Technol Lett 54:1001–1006, 2012; View this article online at wileyonlinelibrary.com. DOI 10.1002/mop.26676
We present strong and weak Finite Element Method formulations for 2D closed electrostatic problems. Special hierarchical higher (arbitrary) order basis functions are applied. Results for characteristic impedance of shielded planar transmission lines, obtained by the two formulations are compared with results from the literature, and their specific advantages briefly commented. © 2011 Wiley Periodicals, Inc. Microwave Opt Technol Lett 53:1114–1119, 2011; View this article online at wileyonlinelibrary.com. DOI 10.1002/mop.25917
In this paper the strong FEM formulation based on hierarchical basis functions of higher order and Galerkin method is proposed. This physically means that the boundary conditions -continuity of the electric potential V and of the normal component of vector D, on element boundaries are exactly satisfied. As a benchmark example, a coaxial transmission line of the square cross section is analyzed.
The bianisotropic media, magnetoelectric materials, having constitutive relations D = E E + x H and B = p H + x E (vector values are in bold) are theoretically predicted by Dzyaloscinskii (1959) and by Landau and Lifshitz (1957) and experimentally observed by Astrov in chromium oxide (1 960) [I] . The present paper shows a general procedure for low frequency (LF) electromagnetic (EM) field determination in bianisotropic media. Using Maxwell's equations: rot H = J , rot E = O , divD = p and divB =0, the following two independent excitation cases are observed: a) Electrostatic problems, where p # 0 and J = 0 ( p is the volume charge density and J is electric current density ) Introducing scalar potential functions, electric, 9, and magnetic, 9 , , scalar potential, the electric, E, and magnetic, H, field strength are defined as: E = -g r a d 9 and H=-grad qm ,whereAcp=-pl& ,AV,,,= X p / p E e a n d E e = & -X 2 / p . If Q is electric scalar potential in free space, A$ = -p / EO, then: cp = q Q I Ee and cp, = -x Q I p . Finally, E = E O E O / E~, Eo=-grad~$ , H = -x E / p , D = E~E and B=O. b) If p = 0, but LF current exists, J# 0, it is convenient to use vector potential functions, magnetic, A, and electric, F, vector potential.Then magnetic flux density, B, and electric displacement, D, are: B = r o t A and D = r o t F, where A A = -p J and A F = -x J. Vector potentials satisfy gauge conditions : div A = 0 and div F = 0 [2] IfA is free space solution, AA = -p~ J , then : In order to illustrate the present procedure several examples are used. 1) For isolated point charge, q, placed in coordinate origin: where r is radius vector of field point. 2) For isolated linear conductor having LF current I and infinite length ( L + m ) and placed on direction r = 0 : A 3) Bianisotropic spherical body with radius a in homogeneous magnetic field Ho = X -b z . Outside the body is free space. Using scalar potential functions and solving Laplace's equation by method of separation of variables with boundary conditions on the body surface that tangential components of electric and magnetic field strengths and normal components of magnetic flux density and electric displacement are unchanged, the potential scalar functions are C l r c o s e r s a CZrcosf3 r S a .={ C 3 c o s e / r 2 r > a and vm={-Ho r Wse + c4 case / r2 r 2 a , where C1= 3p,,x&/ 6 , C2 = -3p,,(~+2~0)Ho 1 6 , C3 = 3p,,xa3Ho/6, C4 = ~~H~[(E+~Eo)(~-I.Io)] 1 6 , 4) Infinite linear conductor with LF current I above bianisotropic half space (Fig.). 6 = (€+2Eo)(p+2I.IO) -x 2 . region 1 region 2 -C2 In r2 in 2 -C3ln rz in 1 -C3 In rl in 2 ' F ={ bianisotropic media Using integral transform method and satisfying all boundary conditions, the vector potential axial components are (all infinite components are ignored) :References:The obtained results give image theorem.
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