Anand K. Verma - Introduction To Modern Planar Transmission Lines

Equations (4.5.31a)and (4.5.31b)are solved to get the vector algebraic form of wave equation as follows:

(4.5.32)

Likewise, the wave equation for could be written that is useful for the EM‐wave propagation in an anisotropic medium. Equation (4.5.32b)is an eigenvalue equation, and the nontrivial solution for E ≠ 0 provides the eigenvalue of k belonging to the propagating waves in the unbounded isotropic medium. The medium supports two numbers of linearly y‐polarized waves known as normal modes propagating in ±x‐directions with same phase velocity (v p) given below:

(4.5.33)

In the above equation, is the wavenumber in free space. Using the intrinsic impedance with equation (4.5.31a), the magnetic field vector and Poynting vector are obtained below:

(4.5.34)

In the case of the propagation of waves in an isotropic medium, the wavevector and Poynting vector both are in the same direction . It provides the phase and group velocities in the same direction.

The loss‐tangent (tan δ), given in equation (4.5.10), is much greater than unity, i.e. tan δ ≫ 1 for a highly conducting medium. It means a contribution of the conduction current is much more than that of the displacement current in a conducting medium, i.e. . However, the approximation for a low‐loss medium is taken differently. The propagation constants of the EM‐wave in a highly conducting and also in a low‐loss medium are obtained from equation (4.5.4)as follows:

(4.5.35)

In a lossy medium, the plane wave propagates in the x‐direction with the uniform field components in the (y‐z)‐plane as shown in Fig. (4.9a). The field components are given by equation (4.5.24), incorporating the conductivity σ of a medium. They are modified as,

(4.5.36)

Using the field solutions from equations (4.5.20), the above equations are reduced to the following forms:

(4.5.37)

In the above equations, the complex propagation constant γ is given by equation (4.5.4).

The conducting medium is highly dispersive, whereas the low‐loss medium is nondispersive. Using equations ( 4.5.35a,b)with equation ( 4.5.12a), the wave equation and the phase velocity in a conducting medium are given below:

(4.5.38)

It shows that the conducting medium is dispersive, and the phase velocity increases with an increase in frequency.

The characteristic impedance ( intrinsic impedance ) of the low‐loss and highly conducting media are obtained from equations (4.5.37)and (4.5.35a)as follows:

(4.5.39)

The characteristic impedance, i.e. the intrinsic impedance η c, of a high‐loss conducting medium is a complex quantity, with an equal magnitude of real and inductive imaginary parts. The real part of is known as the surface resistance , R sincurring an Ohmic loss in the conducting medium; and its imaginary gives the internal inductance L iof a conducting medium:

(4.5.40)

For the unbounded medium, |R s| = |ω L i|, and the internal inductance L iis due to the penetration of the magnetic field in the medium. It is further discussed in subsection (8.4.2) of chapter 8. The expressions for the magnetic and electric fields and Poynting vector in a conducting medium are given below:

(4.5.41)

The α, β ,and η cfor a highly conducting medium are given by equations (4.5.35b)and (4.5.39b). The uniform plane wave in an unbounded conducting medium is still TEM‐type. However, field components H zand E yare not in‐phase. These are in‐phase in a dielectric medium shown in Fig. (4.9a). The power transported per unit area, in a conducting medium, in the x‐direction is also given by the following expression:

(4.5.42)

The power transmitted through the conducting lossy medium is a complex quantity. Its real part gives the power that comes out from the medium of length x, whereas the imaginary part gives stored energy due to the field penetration in the conductor. The input power density available at x = 0 is . The power density after traveling distance x in a highly conducting medium is

(4.5.43)