Introduction To Modern Planar Transmission Lines. Anand K. Verma

(ENZ) materials. Under certain conditions, it can also acquire negative permittivity, creating the epsilon negative (ENG) medium [B.16]. It is discussed in section (5.5) of chapter 5.

      The electric displacement current flowing through a dielectric medium does not involve the flow of current through free charges. It is associated with time‐dependent electric fields. The electric displacement current density is a vector quantity. It expressed as follows:

      (4.1.10)

      Magnetic Materials

      The magnetic materials support magnetization, i.e. magnetic polarization by the magnetic field. It also supports magnetic displacement current due to the time‐dependent magnetic field:

      (4.1.11)

A diamagnetic material has permeability μ ≤ μ₀ and a paramagnetic material has permeability μ > μ₀. The diamagnetic behavior is caused by induced magnetic dipole moments, creating a magnetic field that opposes the externally applied magnetic field. However, the paramagnetic property comes into existence due to the alignment of already randomly existing magnetic dipole moments in the material. The conducting ferromagnetic materials have much larger permeability due to domain formation. However, the engineered artificial magnetic materials can have relative permeability less than unity, 0 < μ_r < 1. These are known as the mu near‐zero (MNZ) materials [B.16]. Again, the engineered materials can also have mu negative, giving the mu negative (MNG) materials. It is discussed in section (5.5) of chapter 5.

      Conductors

It supports the flow of the free charge carrier, i.e. electrons. The free charge carriers are responsible for the conduction current density. It is given by equation (4.1.9). The free electrons confined within a neutral conductor form the plasma medium. The artificial dielectrics, including the wire‐medium (i.e. rodded medium), are also modeled as the plasma medium. The ionosphere also supports the plasma medium [B.2]. The plasma medium is modeled using the Drude model, discussed in subsection (6.5.2) of chapter 6. The plasma medium provides a means to engineer materials with negative permittivity. These materials are called the epsilon negative materials, i.e. the ENG materials.

4.2 Electrical Property of Medium

      Any material can be electrically characterized by the relative permittivity (ε_r), relative permeability (μ_r) and conductivity (σ), or resistivity (ρ). However, these electrical parameters are not constants for any given material. For instance, these are both temperature and frequency‐dependent. Also, these may not be uniform throughout the volume of a material. Further, the characterizing parameters may depend on the field intensity, the direction of the applied field, operating frequency, working temperature, and pressure. The parameters can also depend on the history of a medium. However, the static value of these parameters, at room temperature, is treated as constant. A special kind of material, called chiral material, requires another parameter called chirality, i.e. the handedness of materials for its characterization [B.13, B.17]. Several properties of the medium are described briefly in this section.

      4.2.1 Linear and Nonlinear Medium

The relative permittivity (ε_r) and the relative permeability (μ_r) of a linear material do not depend on the magnitude of the electric or magnetic field intensity, respectively. However, for nonlinear materials, the relative permittivity and relative permeability are functions of the electric and magnetic field intensity, respectively, and expressed as ε_r(E) and μ_r(H). Thus, for an isotropic nonlinear medium, equation (4.1.7a) is written as

(4.2.1)

where ε_r(E) = ε_r1 + ε_r2E + ε_r3E² + ⋯ and so forth. The coefficients ε_r2, ε_r3, … indicate the order of nonlinearity in the nonlinear relative permittivity. In the case of a time‐harmonic electric field, i.e. E = E₀e^jωt, equation (4.2.1) is written as,

      (4.2.2)

It is obvious that while the input signal has only one frequency ω, shown in Fig. (4.2), the output of a nonlinear medium has several harmonically related frequency components, ω, 2ω, 3ω, … and so forth. Thus, a sinusoidal input signal gets distorted, once it passes through a nonlinear medium. Such distortion also occurs in an amplifier in the nonlinear region.

      Similarly, the relative permeability of a nonlinear magnetic medium is a function of the amplitude of the magnetic field. The constitutive relation, given by equation (4.1.7b), is written as

(4.2.3)

Figure 4.2 Response of nonlinear medium showing generation of harmonics.

Figure 4.3 Inhomogeneous medium showing a step variation of relative permittivity with substrate height.

where μ_r(H) = μ_r1 + μ_r2H + μ_r3H² + ⋯ and so forth. The coefficients μ_r2, μ_r3, … indicate the order of nonlinearity in the nonlinear relative permeability of a magnetic medium.

      4.2.2 Homogeneous and Nonhomogeneous Medium

The relative permittivity (ε_r) and the permeability (μ_r) are not necessarily uniform throughout the volume of a medium. These parameters could also be position‐dependent. The variation in ε_r and μ_r could be in discrete steps, or they could be continuous functions of the position. Likewise, the conductivity of a medium can also be a function of position. If the parameters ε_r, μ_r, and σ are uniform throughout the medium, the medium is called homogeneous; otherwise, it is a nonhomogeneous medium or an inhomogeneous medium. A multilayer dielectric medium, forming a parallel capacitor, as shown