Читать книгу Introduction To Modern Planar Transmission Lines - Anand K. Verma - Страница 130

A wave‐packet is formed by a linear combination of two signals of equal magnitude with a small difference in angular frequency and phase constant. It is shown in Fig (3.24). The composite voltage wave is given by

Figure 3.24 Formation of a wave‐packet.

(3.3.9)

The carrier wave has frequency ω₀ and propagation constant β₀. The above expression applies to a narrow‐band signal, Δω << ω₀. The envelope frequency is Δω and its propagation constant is Δβ. The carrier wave inside the envelope, shown in Fig (3.23), moves with phase velocity:

(3.3.10)

The velocity of the envelope, i.e. the group velocity, is obtained from the constant phase point on the envelope

(3.3.11)

Both the phase velocity and group velocity are represented over the (ω − β) dispersion diagram of the wave‐supporting medium. The (ω − β) diagram plays a very significant role in the wave propagation in periodic and metamaterial media. These are discussed in the chapters 18–22. Figure (3.25) shows the dispersion relation given by equation (3.3.8a). The point P on the dispersion diagram locates ( β₀, ω₀) . The slope ψ of a point P on the dispersion diagram, with respect to the origin, gives the phase velocity of a carrier wave, whereas the local slope ϕ at ω = ω₀, i.e. a local tangent at point P, provides the group velocity:

Figure 3.25 ω − β diagram to get phase and group velocities.

(3.3.12)

At the cut‐off frequency ω_p, β → 0, results in an infinite extent of the phase velocity, while the group velocity is zero, v_g = 0. However, away from the cut‐off frequency, the phase velocity decreases to a limiting value . It is the slope of the light–line as shown in Fig (3.25). The light‐line divides the (ω − β) diagram into two regions – within the light‐cone it is the fast‐wave region, and outside the light‐cone, it is the slow‐wave region. At much higher frequency, both velocities become equal. For the plasma medium, equations (3.3.8a) and (3.3.8b) provide the propagation constant and phase velocity.

As the propagation constant increases from β = 0 at ω_p to , and the phase velocity decreases from v_p ≈ ∞ at ω_p to for ω → ∞, the plasma forms a normal, i.e. the positive dispersive medium. However, the phase velocity above the cut‐off frequency in a plasma medium is always greater than the velocity of EM‐wave in the free space, v_p ≥ c. This wave is a fast‐wave. The group velocity is computed as follows:

(3.3.13)

The group velocity is less than the velocity of the EM‐wave in the free space, v_g ≤ c. In summary, the plasma forms a fast‐wave normal dispersive medium that supports the forward wave having the same direction for both the phase and group velocities. Equations (3.3.8d) and (3.3.13c) for the wave propagation in a nonmagnetized plasma with μ = μ₀, ε₀ and v = c show that the phase and group velocities are related through the following expression:

(3.3.14)

In equation (3.3.14), the general isotropic medium supports EM‐wave propagation with velocity .

A more general relation between the phase and group velocities could be obtained. In the dispersive medium, phase velocity (v_p) is a function of frequency; therefore,

(3.3.15)

Figure (3.26a and b) show the dispersive behaviors of the phase velocity (v_p) and propagation constant (β). The expression for the group velocity in a dispersive medium, i.e. a medium with frequency‐dependent refractive index , can be rewritten as follows:

(3.3.16)

In equation (3.3.16), c is the velocity of EM‐wave in free space. The cases of dispersion are considered below:

 Case‐I: . It is the no dispersion case. In this case, the above equation provides vg = vp. Figure (3.26a) shows that the phase velocity in a nondispersive medium remains unchanged with angular frequency. Figure (3.26b) shows that the propagation constant (β) is a linear function of angular frequency (ω). The free space is one such medium.

 Case‐II: . It is the case of normal, i.e. the positive dispersion. Figure (3.26b) indicates that the slope of the propagation constant β in the normal dispersive medium increases nonlinearly with angular frequency ω. Therefore, the phase velocity decreases with angular frequency, i.e. dvp/dω is negative. It is shown in Fig (3.26a). For this case, equations (3.3.15) and (3.3.16) show that vg < vp. It is the case applicable to a dispersive microstrip line.Figure 3.26 Nature of dispersion on (ω − β) the diagram.

 Case‐III: . It is the case of the anomalous (abnormal), i.e. the negative dispersion. The propagation constant β of such an anomalous dispersive medium increases nonlinearly with angular frequency ω. However, on the (ω − β) diagram, shown in Fig (3.26b), its value is below the dispersionless medium of the case‐I. The slope of the propagation constant β, in the nonlinear region, decreases with an increase in frequency ω, i.e. dn/dω < 0. Figure (3.26a) shows that the phase velocity increases with angular frequency, i.e. dvp/dω is positive. Equations (3.3.15) and (3.3.16) show that vg > vp. However, the group velocity in an anomalous dispersive medium is not the velocity of energy transportation. This case is applicable to the dispersive MIS or Schottky microstrip lines in the transition region [J.3, J.4]. The equations (3.3.15) and (3.3.16) further indicate a possibility of backward wave propagation (vg negative) in an anomalous dispersive medium, if dn/dω is significantly negative, i.e. the medium has a very large negative dispersion. This is the special condition for the existence of backward wave in the anomalous dispersive medium. However, such a medium is very lossy. Lorentz model, discussed in subsection (6.5.1) of chapter 6, explains the phenomenon.

The relations between two velocities are obtained below by using the propagation constant (β) and wavelength (λ), instead of the angular frequency (ω):

(3.3.17)

The condition for the normal dispersion is dv_p/dβ < 0, dv_p/dλ > 0 and for the anomalous dispersion, it is dv_p/dβ > 0, dv_p/dλ < 0.