Introduction To Modern Planar Transmission Lines. Anand K. Verma

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‖ is parallel (||) to the normal (y‐axis) of an anisotropic substrate surface located in the (x‐z) plane. In the (x‐z) plane, the remaining two relative permittivity components have identical values ε_r⊥. The relative permittivity components ε_r⊥ are in the plane normal (⊥) to the y‐axis, and also normal to the external electric field E_y. The y‐axis is the main axis. It is also known as the C‐axis, or the optic‐axis, or the extraordinary axis. In the direction of the optic axis, the permittivity is different. The other two x‐ and z‐axes are known as the ordinary axes. The x‐ and z‐polarized EM‐waves, known as the ordinary waves, in the (x‐z)‐plane travel with the same velocity, whereas y‐polarized EM‐wave, known as the extraordinary wave, travels with another velocity. The nonhomogeneous medium as shown in Fig. (4.3) is also a uniaxial medium with the x‐axis as the optic axis, supporting the extraordinary wave propagation. The y and z‐axes are the ordinary axes, supporting the ordinary wave propagation.

      Figure (4.4) shows a microstrip line of width w on an anisotropic substrate of thickness h. It forms a parallel‐plate capacitor placed in the (x‐z)‐plane. It views ε_r‖ component of the uniaxial relative permittivity. Whereas, if the parallel plates of the capacitor are placed either in the (x‐y)‐plane or the (y‐z)‐plane, it will view the ε_r⊥ component of a uniaxial substrate medium. Thus, a uniaxial dielectric medium offers two different values of capacitance, depending on the placement of the parallel plates, connected to a voltage source. Normally, manufacturers provide data for ε_r‖ and ε_r⊥ of the uniaxial substrates. The constitutive relation for the uniaxial medium aligned to the physical axes is

      (4.2.9)

      In the case of the crystal axes ξ, η of an anisotropic substrate, shown in Fig. (4.4), have an angle θ = θ₁ = θ₂ with respect to the physical axes x, y; the relative permittivity components could be computed by the following expressions [B.24]:

      (4.2.10)

      Type III: Biaxial materials. For such an anisotropic medium, all three principal relative permittivity components are different, i.e. ε_rξ ≠ ε_rη ≠ ε_rζ. Such a medium is known as the biaxial medium [B.3].

      The above nomenclatures are also applicable to the permeability of a magnetic material.

      Anti‐symmetric Anisotropic Materials

      The plasma medium, i.e. the electron gas model of metal or ionosphere, is treated as an isotropic medium. However, in the presence of a static biasing magnetic field, it becomes a uniaxial dielectric material with off‐diagonal elements for the permittivity matrix. This permittivity matrix is anti‐symmetric. It does not support linearly polarized characteristic waves as the normal modes. It forms a gyroelectric medium, i.e. electrically gyrotropic medium. Similarly, in the presence of a static biased magnetic field in the z‐direction, ferrite medium becomes a gyromagnetic medium, i.e. magnetically gyrotropic medium. Both plasma and ferrite media, under the magnetic biasing, support propagation of circularly polarized characteristic waves as the normal modes [B.2–B.4, B.21, B.22]. The permittivity and permeability tensors in matrix form for these media are summarized below.

      Gyroelectric Medium

The relative permittivity matrix for gyroelectric medium, under the applied magnetic field in the z‐direction, is expressed as follows:

      (4.22)

      In the above expressions, ω_p and ω_c are the plasma frequency and cyclotron frequency. The cyclotron frequency is also called the gyrofrequency. The jκ is a cross‐coupling factor responsible for the gyration property of the relative permittivity of a medium. Other parameters are N: electron density, e: electron charge, m: electron mass, and B₀: DC magnetic field. In the case of the extremely high magnetic field, ω_c → ∞ and ε_r = 1, κ = 0. Also, in the absence of magnetic field, B₀ = 0, κ = 0_, and the gyrotropic medium is reduced to a uniaxial dielectric medium. In some cases, there is no plasma medium in the z‐direction and ε_{r, zz} = 1.

      Gyromagnetic Medium

      The relative permeability matrix of a gyromagnetic ferrite medium is given below:

      (4.2.12)

      In the above equations, H₀ is the biasing magnetic field in the z‐direction, M_s is the saturation magnetization, and the parameter γ is the gyromagnetic ratio. Again the cross‐coupling factor jκ is responsible for the gyration property of the relative permeability of a medium. The permeability function shows a singularity at the frequency ω = ω₀. It is suppressed in the presence of losses. In the case of unbiased and demagnetized ferrite, H₀ = 0 and M_s = 0 leading to k = 0 and μ_r = 1. The ferrite, in this case, acts as a dielectric medium with permeability μ = μ₀.

      Magneto‐dielectric Composite Materials

      The magneto‐dielectric (MD) materials have both electric and magnetic properties. These materials are characterized by the simultaneous presence of permittivity and permeability tensors. These tensors for uniaxial MD materials are expressed as follows:

      (4.2.13)

      The above‐given expressions for the uncoupled independent relative permittivity and permeability tensors show that there is no coupling between the electric and magnetic fields.