Introduction To Modern Planar Transmission Lines. Anand K. Verma
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Figure 3.29 Inductor loaded CL‐line.
Series Connected Parallel Lsh‐C Type Line
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 Lsh, 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
(3.4.21)
The inductor loaded CL‐line behaves like a high‐pass filter. The wave propagates for ω > ωc. For the frequency below cut‐off, i.e. for ω < ωc, 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 ωc 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
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 > ωc, 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
Series Capacitor Loaded LC‐Line
The normal LC‐line can also be loaded with a series capacitor Cs in the series arm. Figure (3.30) shows the series capacitor loaded LC line. The propagation parameters of the loaded line are computed using the circuit analysis. The series arm impedance and the shunt arm admittance p.u.l. are given below:
Figure 3.30 Series capacitor loaded LC‐line.
(3.4.23)
The propagation constant of the capacitor loaded LC‐line is
where the phase velocity of the unloaded LC‐line is
Equations (3.4.24) and (3.4.25) for β, vp, and vg are identical to equations (3.4.10a) and