Читать книгу Introduction To Modern Planar Transmission Lines - Anand K. Verma - Страница 136

The backward wave supporting CL‐line, discussed above, has no cut‐off frequency. Figure (3.29) shows the modified CL‐ line by adding a shunt inductor L_sh, inside the gray box, across the series‐connected capacitor C. It is an HPF type CL‐line that supports the backward wave with a cut‐off frequency. The propagation characteristics of this line are obtained from the series impedance and shunt admittance p.u.l.:

(3.4.19)

The propagation constant of the line is

(3.4.20)

The cut‐off frequency is . The phase velocity of the usual LC‐line is . Therefore, the propagation constant of the shunt inductor L_sh in the series arm loaded CL‐line is

(3.4.21)

The inductor loaded CL‐line behaves like a high‐pass filter. The wave propagates for ω > ω_c. For the frequency below cut‐off, i.e. for ω < ω_c, the wave is in the evanescent mode. The (ω − β) diagram of the inductor loaded CL‐line is similar to the (ω − β) diagram of Fig (3.28c). However, the cut‐off frequency is not shown in Fig (3.28c). A reader can easily add the cut‐off frequency ω_c in the dispersion diagram of Fig (3.28c). Unlike the unloaded CL, the present loaded CL line shows the cut‐off frequency behavior. The present HPF type loaded CL line also supports the dispersive backward wave with phase velocity and group velocity opposite to each other. The propagation constant β decreases with frequency, whereas the phase velocity increases with frequency. It shows that the loaded CL‐line has anomalous dispersion. The phase and group velocities of the backward wave are

(3.4.22)

In summary, the inductor loaded CL‐line shown in Fig (3.29) supports the backward wave. Above the cut‐off frequency, i.e. for ω > ω_c, and for the limiting case ω → ∞, equations (3.4.22a) and (3.4.22b) reduce to equations (3.4.17) and (3.4.18), respectively. The phase velocity of the unloaded CL‐line is . For ω_C = 0 also, the above equations are reduced to equations (3.4.17) and (3.1.18). Therefore, above the cut‐off frequency, i.e. for ω > ω_c, the inductive loading of the transmission line shown in Fig (3.29) supports the backward slow‐wave outside the light cone and the backward fast‐wave within the light cone. Its dual structure, shown in Fig (3.27a), always supports the forward fast‐wave.