In a communication network, several kinds of electrical signals propagate on a transmission line. The signal could be a modulated or unmodulated carrier wave, the baseband analog signal, or the digital pulses. The TEM mode transmission lines, and also various kinds of non‐TEM waveguide structures support wave propagation. The parameters defining these transmitting media could be either frequency‐independent or frequency‐dependent. The property of the medium has a significant impact on the nature of wave propagation through a medium. The wave velocity has no simple or unique meaning, like the meaning of the velocity of a particle. There are several kinds of wave velocities – phase velocity, group velocity, energy velocity, signal velocity, etc., applied to wave propagation. The significance of several types of wave velocities is inherent both in the complexity of a signal and also in the complexity of the wave supporting medium. This section focuses attention on the meaning of the phase and group velocities only. Section (3.4) demonstrates these two wave velocities as applied to several kinds of the artificial linear dispersive transmission lines.
3.3.1 Phase Velocity
The concept of phase velocity is applicable to a single frequency wave, i.e. to a monochromatic wave discussed in Section (2.1) of chapter 2. The phase velocity is just the movement of the wavefront. The wavefront is a surface of constant phase, like maximum, minimum, or zero‐level points shown in Fig (2.3). It is given by equation (2.1.8) of chapter 2 and reproduced below:
(3.3.1)
The propagation constant β is influenced by the wave‐supporting medium. For a lossless TEM transmission line and lossless unbounded space, β is given by
(3.3.2)
where ε and μ are permittivity and permeability of a medium. Thus, pairs (L, C) and (ε, μ) are the parameters that characterize the electrical property of the wave supporting‐media. The unbounded medium supports the plane wave propagation. If these parameters are not frequency‐dependent, the medium is known as nondispersive. In such a medium, the phase velocity remains constant at every frequency. However, if any of these parameters are frequency‐dependent, the propagation constant β is frequency‐dependent and consequently, the phase velocity is frequency‐dependent. The medium that supports the frequency‐dependent phase velocity is known as the dispersive medium. Normally, the characteristic impedance or intrinsic impedance of a dispersive medium is also frequency‐dependent. The parameters (L, C) and (ε, μ) are usually independent of signal strength. Such a medium is called a linear medium, whereas the signal strength dependent medium is a nonlinear medium. The characteristics of the medium are discussed in Section (4.2) of chapter 4. The present discussion is only about the linear and dispersive transmission lines.
Why a medium becomes dispersive? One reason for dispersion is the loss associated with a medium. The geometry of a wave supporting inhomogeneous structures, commonly encountered in the planar technology, is another source of the dispersion. In the case of a transmission line, the parameters R and G are associated with losses and they make propagation constant β frequency‐dependent. Likewise, losses make permittivity ε and permeability μ of material medium frequency‐dependent complex quantities. However, a low‐loss dielectric medium can be nondispersive in the useful frequency band. For such cases, the attenuation and propagation constants are given by
In equation (3.3.3), c is the velocity of EM‐wave in free space. The above expressions are obtained from equations (4.5.16) and (4.5.19) of chapter 4. The complex permittivity is given by ε = ε′ − jε″. The imaginary part, showing a loss in a medium, is related to the conductivity of a medium through relation σ = ωε″. If ε′ and ε″ are not frequency‐dependent, the propagation constant β is not frequency‐dependent and the phase velocity is also not frequency‐dependent. However, if they are frequency dependent, the phase velocity is also frequency‐dependent. The phase velocity in the low‐loss dielectric medium is
(3.3.4)
Therefore, the presence of loss decreases the phase velocity of EM‐wave. This kind of wave is known as the slow‐wave. The slow‐wave can be dispersive or nondispersive. However, it is associated with a loss. This aspect is further illustrated through the EM‐wave propagation in a high conductivity medium. The conducting medium is discussed in subsection (4.5.5) of chapter 4. The attenuation (α), phase constant (β), and phase velocity (vp,con) of a highly conducting medium are given by equation (4.5.35b) of chapter 4 [B.3]:
(3.3.6)
The above expressions are obtained for a highly conducting medium, σ/ωε >> 1. It does not apply to the lossless medium with σ = 0. The wave propagation is associated with significant loss (α) given by equation (3.3.5). Moreover, both the attenuation constant and phase velocity are frequency dependent, so a conducting medium is highly dispersive. However, the periodic structures and other mechanisms give slow‐wave structures with a small loss. Such structures are useful for the development of compact microwave components and devices [B.1, ]. The slow‐wave periodic transmission line structures are discussed in chapter 19.
Some EM‐wave supporting media have cut‐off property. They support the wave propagation only above the certain characteristic frequency of a medium or a structure. These media and structures are also dispersive. For instance, the nonmagnetic plasma medium has such cut‐off property [B.4, B.14]. The plasma medium is discussed in the subsection (6.5.2) of chapter 6. The permittivity of a plasma medium is given by equation (6.5.16 ):
(3.3.7)
