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Introduction To Modern Planar Transmission Lines. Anand K. Verma
Читать онлайн.Название Introduction To Modern Planar Transmission Lines
Год выпуска 0
isbn 9781119632474
Автор произведения Anand K. Verma
Издательство John Wiley & Sons Limited
Figure 3.25 ω − β diagram to get phase and group velocities.
(3.3.12)
At the cut‐off frequency ωp, β → 0, results in an infinite extent of the phase velocity, while the group velocity is zero, vg = 0. However, away from the cut‐off frequency, the phase velocity decreases to a limiting value
As the propagation constant increases from β = 0 at ωp to
The group velocity is less than the velocity of the EM‐wave in the free space, vg ≤ 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:
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 (vp) is a function of frequency; therefore,
Figure (3.26a and b) show the dispersive behaviors of the phase velocity (vp) and propagation constant (β). The expression for the group velocity in a dispersive medium, i.e. a medium with frequency‐dependent refractive index
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)