Читать книгу Introduction To Modern Planar Transmission Lines - Anand K. Verma - Страница 162

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.