ТОП просматриваемых книг сайта:
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
(3.4.9)
The phase and group velocities of the wave on the loaded dispersive line are
Figure (3.27b) shows the dispersion behavior, i.e. the function ω = f(β), on the (ω − β) diagram for the shunt inductor loaded line. It shows the frequency‐dependent behavior of the propagation constant β and attenuation constant α. The dashed line is the light line. Its slope ψ at the origin is the phase velocity of the EM‐wave in free space, i.e. vp = v0 = c. The propagation constant β exists above the light line only for ω > ωc and wave attenuation occurs for ω < ωc. At location P, the local slope ϕ provides the group velocity vg, whereas the slope ψ of the point P at the origin O provides the phase velocity vp of the propagating EM‐wave on the loaded transmission line.
Figure 3.27 Shunt inductor loaded line and its characteristics.
Figure (3.27c) shows the variations in the phase velocity and the group velocity with frequency. For ω → ωc, vp → ∞ , vg → 0, and for ω → ∞, both wave velocities move toward the light line, i.e. toward the phase velocity (v0) of the unloaded host transmission line. From equation (3.4.10a) dvp/dω is negative, i.e. the phase velocity decreases with frequency, whereas from the equation (3.4.10b), dvg/dω is positive, i.e. the group velocity increases with frequency. However, both velocities tend toward v0, i.e. phase velocity in the unloaded host medium. Figure (3.27b) also shows that the propagation constant β increases with frequency. Thus, the shunt inductor loaded line has normal dispersion. The wave considered is a forward‐moving wave with both the phase and group velocities in the same direction, as both anti‐clockwise gradients ψ and ϕ defining the phase and group velocities respectively are positive. Above the cut‐off frequency, i.e. above the light line in the fast‐wave region, the phase velocity is higher than the phase velocity of the unloaded L‐C type transmission line. The shunt inductor loaded line supports the fast‐wave. The wave is fast as compared to the wave velocity, shown as the slope of the light line, on an unloaded transmission line.
The group velocity vg is also frequency‐dependent, causing dispersion in the envelope of a wave‐packet. Therefore, if a voltage wave‐packet, say a Gaussian wave‐packet, propagates on a shunt inductor loaded line, it disperses while moving forward. Figure (3.27d) shows that its amplitude decreases and the pulse width increases. This is known as the group dispersion. The time delay of the envelope over the distance d is
(3.4.11)
In the present subsection, the propagation constant of the loaded line is obtained by solving the wave equation. However, the dispersion relation and also the characteristics impedance of the loaded line could be obtained from the circuit analysis. This simple method is applicable to several interesting cases of loaded lines forming the basis for the modern Electromagnetic bandgap (EBG) materials and metamaterials.
3.4.2 Circuit Models of Dispersive Transmission Lines
The above discussion demonstrates that the reactive loading of a line modifies the electrical characteristics of an unloaded host line. This section considers a few such modifications.
Shunt Inductor Loaded Line
The total series impedance and total shunt admittance of the shunt inductor loaded dispersive line, shown in Fig (3.27a), are given by
(3.4.12)
The series impedance and shunt admittance per unit length (p.u.l.) are
(3.4.13)
The complex propagation constant of the wave on the shunt inductor loaded line is
(3.4.14)
The propagation constant β for ω > ωc is identical to equation (3.4.8). The attenuation constant α is obtained for ω < ωc. The expressions for the phase velocity and group velocity follow from the expression of β as discussed previously. The characteristic impedance of the loaded dispersive transmission line Zod(ω) is given by
(3.4.15)
For the case, ω < ωc, the characteristic impedance Zod is an imaginary quantity that stops the signal transmission through the line. The line behaves like a high‐pass filter (HPF). For the case ω > ωc, the characteristic impedance Zod is a real quantity that allows the signal propagation on the line. However, the characteristic impedance Zod in the pass‐band is frequency‐dependent.
Backward Wave Supporting Line
Figure (3.28a) shows the circuit model of the standard low‐pass filter (LPF) type LC transmission line that can also be realized by cascading several lumped elements LC unit cells. However, the distance between two units cells should be a fraction of wavelength, i.e. number of LC unit cells should 10 or more per wavelength. Figure (3.28b) shows the dual of the LC line realized again by cascading several lumped elements CL unit cells. It is called the CL transmission line. It is a high‐pass filter (HPF) type transmission line. Its propagation characteristic is obtained using the circuit analysis:
Equation (3.4.16) gives the rectangular hyperbolic relationship for ω and β in the first and second quadrants of the (ω − β) diagram shown in Fig (3.28c). The phase velocity is