Introduction To Modern Planar Transmission Lines. Anand K. Verma

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4 Waves in Material Medium – I: (Waves in Isotropic and Anisotropic Media, Polarization of Waves)

      Introduction

      The characteristics of EM‐wave propagating on a planar line are strongly dependent on the nature of the materials used in planar technology. The familiarity with the characteristics of the medium and EM‐wave propagation in the unbounded medium is important to understand the working of the planar transmission lines. These topics are extensively covered in several books [B.1–B.15].

      Broadly speaking, the present chapter covers basic electrical characteristics of the material media and the EM‐waves propagation in the unbounded dielectric media – both isotropic and anisotropic. In the first part of the present chapter, and also in chapter 6, attention is paid to the physical processes and the circuit models to understand the electrical properties of the material medium. The electrical and magnetic properties of the materials appear as the responses to the electric and magnetic excitations. Such excitations could be in the form of the circuit sources, such as the voltage and a current source. It could also be in the form of the field sources, such as the electric field intensity (E) and magnetic field intensity (H). The excitation could be any of three forms, namely (i) time‐independent, i.e. the static or DC type; (ii) frequency‐dependent, i.e. the time‐harmonic dependent, or AC (phasor) type; and (iii) arbitrary time‐dependent, i.e. the transient type. The discussion is limited to the static and time‐harmonic type of responses of the materials, i.e. the material response and behavior in the frequency‐domain.

      Objectives

       To review the EM‐field quantities and medium parameters.

       To review the basic electrical properties of media.

       To obtain elementary circuit models of media.

       To review Maxwell’s equations.

       To present the wave equation in the unbounded lossless and lossy isotropic dielectric medium.

       To review wave polarizations.

       Jones matrix description of polarization states.

       To present the wave equation in the unbounded lossless anisotropic dielectric medium.

4.1 Basic Electrical Quantities and Parameters

      The electrical charge and the electric current are the primary electrical sources for the creation of the electric field and the magnetic field, respectively. The charge, also current (displacement current), is described by the flux field, i.e. the flux density (). Two electrically charged bodies or two current‐carrying conductors interact through the force fields, i.e. the field intensity (). The static charge creates the static electric field around itself, whereas the electric current creates the magnetic field around itself. The magnetic charge does not exist in nature. Sometimes, we talk about the magnetic charge, only as a mathematical source for the magnetic field. It is a hypothetical creation to maintain the symmetry of the field equations. The charge and current, i.e. flux fields are not determined by a medium, whereas the electric and magnetic interactions, i.e. force fields, between two separate bodies in a medium, are influenced by the electromagnetic parameters of the medium.

      4.1.1 Flux Field and Force Field

The electric and magnetic fields are visualized through the line of flux. Thus, the electric charge (Q_e) is described by the electric flux (Ψ_e). The total charge (Q_T) on a physical body and the corresponding electric flux are related by Gauss’s law:

      (4.1)

If the charge is distributed throughout the volume of a body, in the form of volume charge density ρ_e, the elemental charge in the volume element dv is ρ_edv. The flux is further expressed as the electric flux density (D), i.e. the flux per unit area. The flux through the elemental surface area is It is shown in Fig. (4.1). The flux density is a vector quantity. It is also called the electric displacement vector.

      Gauss’s Law for Electric Flux

      Total electric flux coming out of a closed surface = Total charge enclosed inside the volume of a closed surface, i.e.

      (4.1.2)

      The above expression is the integral form of Gauss’s law. It can be converted to the differential form by using Gauss’s vector integral identity,

      (4.1.3)

      Gauss’s Law for Magnetic Flux

      Similar to the electric charge distribution, the magnetic charge distribution can be assumed in a volume of the body. The magnetic charge density is expressed as ρ_m. The magnetic charge creates a magnetic flux Ψ_m. Similar to the case of the electric charge, the elemental magnetic charge in the volume dv is ρ_mdv, and the elemental magnetic flux coming out of the surface is where is the magnetic flux density as ds is the elemental surface of the enclosed volume v. It is also called the magnetic displacement vector. The Gauss’s law for the magnetic charge and magnetic flux can be written in the integral form as follows:

      (4.1.4)