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Under such combined dispersion, and nonlinearity, a pulse can propagate without changing its shape. Such robust EM‐waves are called the solitons. The solitons are useful in optical communication. However, the solitons have also been generated in the microwaves frequency range

      4.2.5 Non‐lossy and Lossy Medium

      All physical dielectric and magnetic material media have some amount of loss. Normally, the substrates used in the microwave planar technology do not have a high loss, except the doped semiconducting substrates. The lossy dielectrics are known as the imperfect dielectrics. The losses in the dielectric and magnetic materials change the relative permittivity and permeability into complex quantities:

(4.2.18)

      The imaginary parts of the relative permittivity and permeability are taken as negative quantities due to our choice of time‐harmonic function e^jωt. However, and are always positive quantities in a passive lossy medium. The lossy capacitor and the lossy inductor can model the complex relative permittivity and permeability of a medium. This is called the circuit modeling of a material medium. The modeling of the relative permittivity of a medium by the capacitor shows that both the medium and capacitor can store electric energy. Likewise, the relative permeability of medium and the corresponding inductor model both are the magnetic energy storage components.

      4.2.6 Static Conductivity of Materials

Figure (4.6a) shows a cylindrical section of the lossy material of conductivity (σ), i.e. resistivity ρ = 1/σ. Its cross‐sectional area and length are A and h, respectively. The lossy material can be modeled as a resistor R, also shown in Fig. (4.6a). The conduction current density J_c flowing through the conductor, due to the free mobile charges, is J_c = I_c/A_, where conduction current is I_c = V/R. The electric field intensity in the material is . On combining these expressions, the following relation is obtained:

(4.2.19)

The above expression (4.2.19b) is Ohm's law for a lossy medium. The following expression of the equivalent resistance of a lossy material, in terms of its resistivity, follows from the above equation (4.2.19c):

      (4.2.20)

      In general, the Ohm’s law for the anisotropic medium is written in the vector form as . Further, the expression is also obtained below for the conductivity σ of the material in terms of the basic parameters of mobile charges in a lossy material, i.e. in terms of electron charge and its mobility. The expression is also applicable to semiconductors. In the case of a semiconductor, the conduction current is due to both the electrons (negative charges) and holes (positive charges).

Figure (4.6a) considers h = Δx length of a cylindrical section of conducting material with a cross‐sectional area ΔA. The free charges move in the direction x with a velocity v_x on the application of electric field intensity E_x. The conduction current in the x‐direction is the rate of flow of total charge ΔQ_e contained in a volume, ΔV = ΔA × Δx.

Figure 4.6 Circuit model, parameters of a dielectric medium.

      (4.2.21)

      In the limiting case, . Thus, and the conduction current density is

      (4.2.22)

      In a material, the charge movement is random due to the scattering and so forth. However, an average motion is assumed, giving the drift of charges in the x‐direction with a drift velocity . The drift velocity is proportional to the electric field intensity that provides another expression for J_c:

(4.2.23)

where constant μ_m is called the mobility of a charge. It is noted that μ is also used as a symbol for permeability. On comparing equations (4.2.19b) and (4.2.23b), the following expression for conductivity is obtained:

      (4.2.24)

      If N is the number of free charges per unit volume, with charge q on each carrier, the charge density is ρ_e = Nq. The equation (4.2.24) of the conductivity is changed to

      (4.2.25)

      In the case of a conductor, the charge carrier is electron, i.e. q = q_e (the electron charge) and μ = μ^e_m (electron mobility). However, for a semiconductor, its conductivity σ_s is due to both electrons and holes leading to the following expression:

      (4.2.26)

      where N_e and N_h are numbers of electrons and holes per unit volume. The charges on electron and holes are equal q_e = q_h = e = 1.6 × 10⁻¹⁹ Coulombs. The electron and hole mobilities, in a semiconductor, are and , respectively. The signs of q_e and are negative, whereas the signs of q_h and are positive. However, the conductivity σ_s of a semiconductor is always positive.

      

4.3 Circuit Model of Medium

The

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