Anand K. Verma - Introduction To Modern Planar Transmission Lines

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 xon the application of electric field intensity E x. The conduction current in the x‐direction is the rate of flow of total charge ΔQ econtained 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 mis 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 sis due to both electrons and holes leading to the following expression:

(4.2.26)

where N eand N hare numbers of electrons and holes per unit volume. The charges on electron and holes are equal q e= q h= e = 1.6 × 10 −19Coulombs. The electron and hole mobilities, in a semiconductor, are and , respectively. The signs of q eand are negative, whereas the signs of q hand are positive. However, the conductivity σ sof a semiconductor is always positive.

The circuit model helps to understand the electrical property of a medium. It is further useful for simulating the electrical responses of a medium. The electrical property of a dielectric medium is expressed through relative permittivity that shows the electric energy storage ability of the medium. The capacitor also stores electric energy. Therefore, a dielectric medium is modeled as a capacitor . The loss in a medium is due to the dissipation of energy that is modeled as a resistor. Similarly, the permeability of a medium, such as an inductor, shows its ability to store magnetic energy. Therefore, the permeability of a medium is modeled as an inductor [B.10–B.12].

Figure (4.6b)shows a parallel‐plate capacitor, containing a lossy dielectric medium with complex relative permittivity . It further shows its RC circuit model that is a parallel combination of the capacitor (C) and resistor (R). It is connected to a time‐harmonic voltage source v = v 0e jωtthat produces a time‐harmonic electric field, E = E 0e jωtin the dielectric medium. The displacement current density in the dielectric medium is

(4.3.1)

The displacement current density has two components with the quadrature phase:

(4.3.2)

The reactive current density (J cap) flows through , i.e. through the capacitor. It is shown by the presence of “j.” Thus, the real part of the relative permittivity is modeled as a capacitor C . The resistive current density (J r), causing a loss in the dielectric medium, flows through , i.e. through resistor R. The imaginary part of relative permittivity is modeled as a resistor R , parallel to the capacitor C. It is obvious from equation (4.3.1)that the loss caused by is a positive quantity only when the complex relative permittivity is defined as a subtractive combination of its real and imaginary parts mentioned in equation (4.2.18a). The current through the equivalent circuit is