In the above expression, fp is the plasma frequency that is a characteristic cut‐off frequency of the plasma medium [B.4, B.14]. The permeability of nonmagnetized plasma is μ = μ0. Other parameters are as follows‐ε0: permittivity of free space, N: electron density, e: electron charge, and me: electron mass. The propagation constant, phase velocity, and plasma wavelength λplasma of the EM‐wave wave in a plasma medium are given below:
In equation (3.3.8),
The phase velocity of the EM‐waves in a plasma medium is frequency‐dependent. Therefore, it is a dispersive medium that supports a fast‐wave. It is fast in the sense that the phase velocity is higher than the phase velocity of the EM‐wave in free space given by
The dispersion is a property of the wave‐supporting medium. The phase velocity of a wave in a dispersive medium can either decrease or increase with the increase in frequency. Thus, all dispersive media could be put into two groups – (i) normal dispersive medium and (ii) abnormal or anomalous dispersive medium.
Figure 3.21 Nature of normal (positive) dispersion.
Figure (3.21a and b) show the general behavior of a medium having normal dispersion. The relative permittivity of such a medium increases with frequency, i.e. dεr/df is positive, and the phase velocity decreases with frequency, i.e. dvp/df is negative. A microstrip line provides such a medium for the normal dispersion. The effective relative permittivity of a microstrip line increases with frequency leading to a decrease in the phase velocity with an increase in frequency. The microstrip is discussed in chapter 8.
Figure (3.22a and b) show the general behavior of an anomalous dispersive medium. The relative permittivity of such a medium decreases with an increase in frequency, i.e. dεr/df < 0 (negative). It leads to an increase in the phase velocity with an increase in frequency, i.e. dvp/df > 0 (positive). A microstrip line on a semiconductor substrate having the Metal, Insulator, Semiconductor (MIS) or the Schottky structure, in the transition region, is an anomalous dispersive medium [J.3, J.4].
It is emphasized that there is nothing abnormal with the anomalous dispersion. Both kinds of dispersions exist in reality. The normal dispersion is also called the positive dispersion as the gradient of εr with frequency is positive, i.e. dεr/df > 0. Similarly, the anomalous dispersion is called the negative dispersion with dεr/df < 0. The relative permittivity of material undergoes both kinds of dispersion depending upon the physical cause of dispersion. The dispersion is caused by several kinds of material polarizations – dipolar, ionic, electronic, and interfacial polarization. Once the frequency is varied from low‐frequency to the optical frequency, the material medium undergoes these polarization changes, and the propagating wave experiences both the normal and anomalous dispersion at different frequencies [B.17, B.18]. It is discussed in chapter 6.
Figure 3.22 Nature of anomalous (negative) dispersion.
The concept of phase velocity applies to a single frequency signal. Now the question is to apply it to a complex baseband signal and a modulated signal. It is possible to use the phase velocity concept to such waveforms through the Fourier series of a periodic signal and using the Fourier integral for a nonperiodic signal. Any signal, periodic or nonperiodic, is composed of a large number of sinusoidal signals. They have a definite amplitude and phase relationship with the fundamental frequency of the signal. A combination of all sinusoidal components gives a complex signal of definite wave‐shape. If the complex waveshape travels through a dispersive medium having frequency‐dependent attenuation constant α(f), the amplitude of each signal component changes differently. Similarly, in a dispersive medium having a frequency‐dependent propagation constant β(f), each signal component travels with a different velocity. It results in different phase‐change for each frequency component of the complex wave; so the shape of the wave changes while traveling on a line or through the medium. The numerical inverse Fourier transform provides the wave‐shape of a signal in the time‐domain at any location in the medium. Thus, the Fourier method helps to apply the concept of phase velocity to complex waveform propagation [J.5, J.6]. Such investigations are important to maintain the signal integrity on the IC and MMIC chips.
3.3.2 Group Velocity
A complex signal composed of two or more frequency components forms a wave‐packet. However, the frequency components should not be much different from each other like an amplitude modulated signal. Figure (3.23) shows the wave‐packet formed by a group of a narrowband complex signal. The wave‐packet has a central or a carrier frequency of higher value, superimposed with a low‐frequency envelope. The carrier wave travels with the phase velocity vp, whereas the envelope, i.e. the wave shape travels with the group velocity vg. In a nondispersive medium, the carrier wave and envelope both travel with the same velocity without a change in the waveshape. However, in the case of a dispersive medium, velocities of the carrier and envelope are different. Depending on the nature of dispersion, the wave could be the forward wave or the backward wave. If the medium has normal (positive) dispersion, the phase velocity, i.e. the velocity of the carrier wave, and the velocity of the envelope, i.e. the group velocity, are in the same direction, as shown in Fig (3.23). The wave is known as the forward wave. However, in the case of a highly anomalous (negative)
