Progress In Electromagnetics Research M, Vol. 23, 263{277, 2012 THE PERMITTIVITY FOR ANISOTROPIC DIELECTRICS WITH PERMANENT POLARIZATION I. Bere \Politehnica" University, 300223, Bd. V. P^arvan, Nr. 2, Timi»soara, Romania Abstract|A new permittivity is de¯ned for anisotropic dielectrics with permanent polarization, which allows obtaining simple connec- tions between the quantities of electric ¯eld. As an application, using the de¯ned quantity, we will demonstrate advantageous forms of the refraction theorems of the two-dimensional electric ¯eld lines at the separation surface of two anisotropic dielectrics with permanent polar- ization, anisotropic by orthogonal directions. 1. INTRODUCTION We know that [1{3], for dielectrics with permanent polarization, the connection law, among electric flux density D, electric ¯eld strength E and polarization P , is given by D = "0E + P ¿ + P p; (1) where "0 is permittivity of the vacuum. The separation in temporary (P ¿ ) and permanent (P p) components is unique only if P p is independent of E, and P ¿ | which is depending on E | is null at the same time with E. From (1) follows that, for materials with permanent polarization (P p 6= 0), the relation between D and E (which for materials with P p = 0 represents the classic permittivity) is not univocally determined by material, because P p could have several values. In the case of the ferroelectric materials, for various electrical hysteresis cycles, the value of D for E = 0, i.e., remanent electric flux density ¯ ¯ Dr = D E=0 = P p; (2) may have multiple values, depending on the considered electric hysteresis cycle (Figure 1). Received 16 December 2011, Accepted 7 February 2012, Scheduled 28 February 2012 * Corresponding author: Ioan Bere ([email protected]). 264 Bere D Dr2 Dr1 E O E max 1 Emax 2 Figure 1. Remanent electric flux densities-speci¯cation. In this context it is useful to de¯ne another permittivity (for dielectrics with permanent polarization), with which the equations have advantageous forms, and it is possible to identify some useful analogies with simpler case of the materials without permanent polarization. 2. ANOTHER PERMITTIVITY FOR ANISOTROPIC DIELECTRICS WITH PERMANENT POLARIZATION The temporary polarization value of anisotropic materials depends on electric ¯eld, and the temporary polarization law is P ¿ = "0ÂeE; (3) where, for the nonlinear materials, the components of electric susceptivity Âe depend on electric ¯eld intensity. Consequently, in case of the dielectrics with permanent polarization, nonlinear and anisotropic, (1) becomes ³ ´ D = "0 1 + Âe E + Dr; (4) where the tensor's components are nonlinear functions depending on the components of E. If we introduce the calculation quantity Dp = D ¡ Dr = D ¡ P p; (5) (4) becomes ³ ´ Dp = "0 1 + Âe E: (6) Progress In Electromagnetics Research M, Vol. 23, 2012 265 From (4), (5), (6), the relative ("rp) and absolute ("p) calculation permittivity tensors of anisotropic dielectrics with permanent polarization are de¯ned with these equations: "rp = 1 + Âe; "p = "0"rp: (7) By de¯ning the vector Dp (in (5)) and new absolute permittivity "p (in (7)), for anisotropic dielectrics with permanent polarization we obtain Dp = "pE: (8) With classical quantities [1{3], for anisotropic dielectrics with permanent polarization, we have D = "E + P p, so (8) is a more concentrated expression and simpler. Formally, (8) is similar with the classical equation D = "E, but the latter written for the materials without permanent polarization. For isotropic materials, even if they are with permanent polarization, (8) becomes Dp = "pE which shows that the spectra lines of Dp and E are the same in this case. We know that for materials with permanent polarization (even if they are isotropic) the spectra lines of D and E are di®erent [1, 2, 4]. Following the polarization main directions, tensor Âe has only three components [1{3]. If we note these directions (that in many cases are rectangular) with x, y, z indices, from (4) we have Dυ = "0 (1 + Âeº) Eυ + Drυ; υ = x; y; or z; (9) and all the three components of tensor "rp are Dυ ¡ Drυ Dpυ "rpυ = = ; υ = x; y; z: (10) "0Eυ "0Eυ Because (10) also contains the components of permanent polarization P p (which means the components of Dr), with that new permittivity we have taken into account, in advantageously way, the nonlinearity of the depolarization curves of the dielectric with permanent polarization, any minor electric hysterezis cycles, i.e., for any P p = Dr. If the source of electric ¯eld is considered, a dielectric with permanent polarization for the operating point is obtained, D < Dr (for components: Dº < Drº) and E < 0 (for components: Eº < 0). The depolarization curve is the part from the second quadrant of the hysteresis cycle; the terminology is similar to that used in the magnetic ¯eld. Therefore, the components of tensor "rp are positive scalars. It is interesting to specify that we should determine the nonlinear functions "rpυ(E) following the procedure used by the author for the permeability of permanent magnets in [5], if we know all the three 266 Bere electric hysteresis cycles following the polarization main axes. For these three main directions x, y, z the nonlinear function plots have similar forms, but they will be quantitatively di®erent, as the depolarization curves following the three main directions of the anisotropic dielectric are di®erent. The numerical solution for the electric ¯eld problem in systems with permanent polarization is obtained with an iterative process, because the systems are, generally, nonlinear. The parameter used to control the convergence of the problem can be the relative permittivity de¯ned with (10). For anisotropic materials, it is clear that the convergence of the calculation is made with the components of tensor "rp following the polarization main axes. Trough this de¯ned calculation quantity we take, univocally and advantageously, into account the nonlinearity of the depolarization curves, no matter how the permanent polarization is (i.e., remanent electric flux density). Obviously, if lacking the permanent polarization (P p = 0), Dp is identical to D, and the calculation permittivity "p is identical to classical ", i.e., "rp ´ "r. If the temporary polarization is negligible, from (1), (6) and (8) follows that all the components of tensor "rp are approximated with 1. Figure 2. Refraction of Dp. Progress In Electromagnetics Research M, Vol. 23, 2012 267 3. APPLICATIONS FOR THE REFRACTION THEOREMS Consider two di®erent dielectric media 1 and 2 at rest, with permanent polarization, separated by smooth surface S12, without free electric charge. The demonstration refers the electric ¯eld lines of E and the calculation flux density Dp (de¯ned by (5)), for two-dimensional (2D) ¯eld, in dielectrics with permanent polarization, anisotropic by orthogonal directions. For medium 1, main directions of the polarization are noted with (x1, y1) and unit vectors (i1, j1) and for medium 2 are noted with (x2, y2) and unit vectors (i2, j2) (Figures 2 and 3). In order to express the normal and tangent components of the electrical ¯eld at the separation surface S12, in point 0 (where the refraction is analyzed) we attach the rectangular system ((n; t), with unit vectors n, t). The main axes of polarization in both media | therefore rectangular systems (x1, y1) and (x2, y2) | are di®erent from each other and di®erent from the system (n, t). In order to write the projections on axes of quantities Dp and E, Figure 3. Refraction of E. 268 Bere Figure 4. Components of Dp. we introduce the angles (see Figures 2 and 3): ¡ ¢ ®¸ = ^ Dp¸; n ; ¸ = 1 or 2; ¡ ¢ ¯¸ = ^ E¸; n ; ¸ = 1 or 2; (11) ¡ ¢ '¸ = ^ i¸; n ; ¸ = 1 or 2: Since the media in contact was anisotropic considered, the spectra lines of D, E and P p are di®erent, therefore also the spectra lines of Dp and E are di®erent. Consequently, generally ®¸ 6= ¯¸ (¸ = 1, 2). If we take into account the local form of electric flux law for a discontinuity surface without free electric charge, the normal components of electric flux density D at the separation surface S12 are preserved, which means D1n = D2n = Dn: (12) From the local form of electromagnetic law for the considered conditions result, the conservation of the components of E, E1t = E2t = Et: (13) For 2D ¯eld in anisotropic media with permanent polarization, if we write (8) for both media, we obtain the following relation: Dp¸ = "p¸E¸; ¸ = 1; 2; (14) Progress In Electromagnetics Research M, Vol. 23, 2012 269 where "p¸ = k"p¸x"p¸yk are the tensors of calculation absolute permittivity in both dielectrics with permanent polarization. If we emphasize the components following the main directions (see also (10), where "0"rpυ = "pυ), (14) becomes Dpλυ = "pλυEλυ; ¸ = 1; 2 and º = x; y (15) We remark that between Dp and E components, we can write relations similar to (15) only following the polarization main directions (º = x or y), but not following n and t directions. We must specify that Dp¸º and Ep¸º are the projections of vectors Dp¸ and E¸, following the polarization main directions, i.e., for the cases of 2D ¯eld showed in Figures 2 and 3 and the notices (11), we can write these equations (see also Figures 4 and 5): Dp1 = Dp1xi1 + Dp1yj1 ³ ¼ ´ = Dp1 cos (®1 ¡ '1) i1 + Dp1 cos ®1 ¡ '1 + j1 £ ¤ 2 = Dp1 cos (®1 ¡ '1) i1 ¡ sin (®1 ¡ '1) j1 ; Dp2 = Dp2xi2 + Dp2yj2 ³ ¼ ´ = Dp2 cos (®2 +2¼¡'2) i2 +Dp2 cos ®2 +2¼¡'2 + j2 £ ¤ 2 = Dp2 cos ('2 ¡ ®2) i2 + sin ('2 ¡ ®2) j ; 2 (16) E1 = E1xi1 + E1yj1 ³ ¼ ´ = E1 cos (¯1 ¡ '1) i1 + E1 cos ¯1 ¡ '1 + j1 £ ¤ 2 = E1 cos (¯1 ¡ '1) i1 ¡ sin (¯1 ¡ '1) j1 ; E2 = E2xi2 + E2yj2 ³ ¼ ´ = E2 cos (¯2 + 2¼ ¡ '2) i2 + E2 cos ¯2 + 2¼ ¡ '2 + j2 £ ¤ 2 = E2 cos ('2 ¡ ¯2) i2 + sin ('2 ¡ ¯2) j2 : It is obvious that Dp1, Dp2, E1 and E2 are the modules of vectors, so they are positive scalars.