Introduction To Modern Planar Transmission Lines. Anand K. Verma

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and so forth are also anisotropic because and vectors are not in the same direction. Several authors have treated the properties of anisotropic medium and EM‐wave propagation through such media in detail [B.1–B.4, B.9, B.11, B.13–B.15, B.17–B.23]. This subsection reviews basic concepts related to anisotropic media.

      The relative permittivity and relative permeability of these anisotropic media are not scalar quantities. They are tensor quantity, described by 3 × 3 matrices. The constitutive relations of such electric and magnetic media are written as follows:

(4.2.4)

In general, elements of permittivity and permeability matrices could be complex quantities and also frequency‐dependent, accounting for the losses and dispersion in the material medium. Equation (4.2.4a) shows that for the anisotropic dielectric medium, the electric flux density is not parallel to electric field intensity . Likewise, equation (4.2.4b) shows that the vector is not parallel to the vector . For instance, if the incident field on an anisotropic dielectric medium has only E_x component, i.e. x‐polarized incident E‐field, it generates all three components of electric flux density – D_x, D_y, and D_z. The same applies to the anisotropic magnetic medium.

      The above equations can be written in a more compact form as

      (4.2.5)

      The above permittivity and permeability matrices could be either symmetric or anti‐symmetric. Thus, the anisotropic materials could be divided into two broad groups: (i) symmetric anisotropic materials and (ii) anti‐symmetric anisotropic materials. The symmetric anisotropic materials support linearly polarized EM‐waves propagating as the normal modes of the homogeneous unbounded medium. However, circularly polarized EM‐waves are the normal modes of the anti‐symmetric anisotropic medium. The normal modes of media travel without any change in polarization.

      Symmetric Anisotropic Materials

      The complex permittivity matrix , showing the permittivity tensor, of symmetric anisotropic dielectric material is a Hermitian symmetric matrix, i.e. the following relation holds:

      (4.2.6)

      In the above equation, the matrix elements are the complex conjugate of the matrix elements . The superscript T shows the transpose of the permittivity matrix. The above relation also holds for a real permittivity tensor of symmetric anisotropic dielectric material. In a dielectric material case, the off‐diagonal elements of the matrix are symmetrical, i.e. ε_{r, xy} = ε_{r, yx}, and so forth. Similar expression can also be obtained for the symmetric anisotropic magnetic material. However, some media do not follow this symmetry rule.

To describe an anisotropic medium, two sets of the coordinate systems are used: one set is for the crystal structure axes of the anisotropic medium; and another set is used for the physical axes of the line structure. Figure (4.4) shows the crystal axes (ξ, η, ς), i.e. (xi, eta, zeta), and the physical axes (x, y, z) of a microstrip line on an anisotropic substrate.

      The crystal axes (ξ, η, ς) are rotated with respect to the physical axes (x, y, z) by the angles θ₁, θ₂, and θ₃. The off‐diagonal elements of [ε_r] in equation (4.2.4) are present due to the nonalignment of two coordinate systems. However, if they are aligned, i.e. θ₁ = θ₂ = θ₃ = 0, then the off‐diagonal elements of [ε_r] are zero; and the constitutive relation (4.2.4a) reduces to

      (4.2.7)

The crystal axes (ξ, η, ς) are also known as the principal coordinate system of a material medium; and the diagonal relative permittivity components ε_rξ, ε_rη and ε_rζ are known as the principal relative permittivity components [B.1, B.3, B.10, B.12–B.14, B.24]. Normally, all relative permittivity components are positive quantities. However, it is possible to get one component as a negative quantity in the engineered composites that provide unique EM‐wave characteristics [J.1, J.2, B.16]. Along the principal axes, the components of the vector are parallel to vector .

Figure 4.4 The crystal axes (ξ, η, ς) and the physical axes (x, y, z) of a planar anisotropic sheet.

      The dielectric materials are further classified into three categories:

      Type I: Isotropic materials. The relative permittivity components of these materials are identical, i.e. ε_rζ = ε_rη = ε_rζ. Thus, the relative permittivity of isotropic material is a scalar quantity.

      Type II: Uniaxial materials. These are anisotropic materials with relative permittivity components ε_rζ = ε_rξ = ε_r⊥, and ε_rη = ε_r‖. It is shown within a box in Fig. (4.4). In the case of alignment of crystal axes along the physical axes, permittivity components are expressed as ε_xx = ε_zz = ε_r⊥, ε_yy = ε_r‖. Thus, the permittivity tensor of the uniaxial anisotropic substrate is expressed as follows

      (4.2.8)

      For a uniaxial substrate, shown in Fig. (4.4), the applied external electric field E_y faces the relative permittivity component ε_r‖. The permittivity component ε_r‖

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