Introduction To Modern Planar Transmission Lines. Anand K. Verma

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      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_s incurring an Ohmic loss in the conducting medium; and its imaginary gives the internal inductance L_i of a conducting medium:

      (4.5.40)

      For the unbounded medium, |R_s| = |ω L_i|, and the internal inductance L_i is 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_c for 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_z and E_y are 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)

      The field decreases by a factor e^−αx_, whereas the wave travels through a lossy medium. If the wave travels a distance x = δ = 1/α, known as the skin depth, the field is decreased by 1/e of its initial strength, i.e. approximately 37% of its initial field strength. However, the power decreases at a faster rate, i.e. by the factor e^−2αx. If the initial power density is S₀, the power density at distance x is

(4.5.44)

      The attenuation constant α in the above equation is used from equation (4.5.35b). The power loss of wave traveling a distance x is computed after computing the power loss at unit distance x = 1m:

      (4.5.45)

In the above equation, the value of e is 2.71828. The power loss is about 9 dB per skin‐depth. The attenuation constant α of a lossy medium is defined by equation (4.5.44a) as follows:

      (4.5.46)

4.6 Polarization of EM‐waves

      The uniform plane wave in the unbounded medium is the TEM‐type wave. The monochromatic EM‐wave is characterized by amplitude, phase, and polarization states. The microwave to optical wave devices can appropriately manipulate these characteristics to steer the EM‐waves in the desired direction with shaped wavefront. The modern metasurfaces, discussed in sections (22.5) and (22.6) of chapter 22, can achieve such controls on the reflected and transmitted waves.

      In general, both and fields have two orthogonal field components in a plane normal to the direction of propagation, the x‐direction, as shown in Fig. (4.9a and b). The field components are in the (y‐z)‐plane. For the TEM mode, it is possible to get either (E_y, H_z) or (E_z, H_y) pair of fields. Both pairs of field components can also exist. The orientation of the electric field component and the movement of the tip of the resultant E‐field determine the polarization of the EM‐wave. Thus, the (E_y, H_z) pair of the EM‐wave is called a y‐polarized wave, as only the E_y component of wave exists. The (E_z, H_y) pair of the EM‐wave is called the z‐polarized wave. The (y‐z)‐plane is known as the plane of polarization. It is normal to the direction of propagation, i.e. the x‐axis. Both these polarizations are linear polarization.

Figure (4.9a and c) show that for the EM‐wave propagating in the x‐direction, the tip of the E_y field component moves along the y‐axis from +E₀ to 0 to −E₀. The movement and rotation of the tip of the ‐vector could be seen using the instantaneous ‐vector of the wave propagating in the x‐direction. In Fig. (4.9c), the path of the tip of the E_y‐field vector, in the plane of polarization, traces a line with respect to time. The linear trace demonstrates the vertical linear polarization. However, if both E_y and E_z field components are present, any of the three kinds of wave polarizations can be obtained: (i) linear polarization, (ii) circular polarization, and (iii) elliptical polarization. These polarizations are shown in Fig. (4.10a–c). The type of wave polarization depends upon the magnitude and the relative phase of the orthogonal E_y and E_z field components. The polarization states are briefly discussed below.

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