Introduction To Modern Planar Transmission Lines. Anand K. Verma

Figure 4.1 The unit vector

is in the direction normal to the surface.

      Again, by using Gauss’s vector integral identity, the above expression is written below in the differential form:

      (4.1.5)

      However, the magnetic charges are not found in nature, i.e. ρ_m = 0. Therefore,

      (4.1.6)

      The amount of charge, or current, is an absolute quantity. It does not dependent on the material medium. Thus, the corresponding flux or the flux density is also not dependent on the surrounding medium. In brief, the charge and current create the electric and magnetic flux field, i.e. the flux densities ; and these are not influenced by a material medium.

      Experiments demonstrate that the electrically charged body, or a current‐carrying conductor, interacts with other charged body, or another current‐carrying conductor. Such interaction, i.e. the mutual force, is influenced by the medium surrounding these bodies. Therefore, medium‐independent flux densities cannot explain the interaction between two charged bodies or current‐carrying conductors. The interactions between the charges and current‐carrying conductors take place through the force fields, expressed by the electric field intensity and magnetic field intensity . The field intensities are also responsible for the electromagnetic (EM)‐power transportation through a medium. However, the field intensities are influenced by the electrical and magnetic properties of a medium.

      4.1.2 Constitutive Relations

      It is noted above that the charge and current create the electric and magnetic flux fields, described by the flux densities They also create the force field described by the electric and magnetic field intensities . The flux density parameters are related to the force field intensity parameters by the following constitutive relations:

(4.1.7)

where ε₀ and μ₀ are the permittivity and permeability of the free space. The permittivity of any medium is its ability to store electric energy, such as a capacitor. Therefore, it is identified as the capacitance of the free space. Any dielectric material medium can store more electric energy through the mechanism of electric polarization. The electric dipole is created during the process of polarization and the total induced charge is shown as electric flux density . Chapter 6 presents a detailed discussion of material polarization. The electric polarization of material under the influence of an external electric field gives a higher value of permittivity as compared to the permittivity of the free space. This is known as the relative permittivity ε_r of a medium. It is also known as the dielectric constant of a medium. The permittivity of a medium is ε = ε₀ε_r. For an isotropic dielectric medium, ε_r is a scalar quantity. However, for an anisotropic medium, it is a tensor quantity.

      Likewise, the free space has also an ability to store magnetic energy. It is expressed as its permeability. Magnetic material is magnetized by the process of magnetization under the influence of an external magnetic field. Thus, magnetic material stores more magnetic energy as compared to the free space. The ability of a magnetic material to store magnetic energy is expressed through its relative permeability. The permeability of the medium is μ = μ₀μ_r. The permittivity ε₀ and permeability μ₀ of the free space are the primary physical constants ε₀ = 8.854 × 10⁻¹² F/m, μ₀ = 4π × 10⁻⁷ H/m. Again, for the isotropic magnetic medium, the relative permeability μ_r is a scalar quantity, and for an anisotropic medium, it is a tensor quantity.

      It is interesting to note that the velocity of EM‐wave and the characteristic (intrinsic) impedance of the free space are given in terms of these primary constants,

      (4.1.8)

      A material medium with a finite conductivity dissipates energy in the form of heat. The finite conductivity of a medium is due to the presence of free charge carriers. The free space is considered as a lossless medium because it has no free charge carrier. The conduction current flows through a medium under the influence of an external electric field. The conduction current density () in a medium is related to the electric field intensity () by Ohm’s law:

(4.1.9)

      where σ is the conductivity of a conducting medium. The conductivity (σ) of the isotropic conducting medium is a scalar and for an anisotropic conducting medium it is a tensor. As free space has no conductivity, there is nothing like the relative conductivity of a medium. However, sometimes the conductivity of a medium is expressed in terms of the conductivity of copper.

      In summary, the electrical properties of a material are described by the relative permittivity (ε_r), relative permeability (μ_r), and conductivity (σ). A material can have all three properties at a time, or it can have one predominant property at a time. Assuming the case of one predominant property at a time, all materials are classified into three basic categories.

      4.1.3 Category of Materials

      Dielectric Materials

      The dielectric materials support electric polarization of bound charges and electric displacement current through it. At the micro‐level, the electric polarization creates dipole moments that appear as the permittivity of the dielectric material at the macro‐level. The permittivity is frequency‐dependent and lossy. So, permittivity is a complex quantity showing the lossy nature of dielectric materials. This is discussed in chapter 6. The relative permittivity of natural dielectric material is always positive and more than unity. However, engineered artificial dielectrics can have relative permittivity 0 < ε_r